In this paper a new form of the Hosszu-Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszu-Gluskin chain formula is not unique and can be generalized ("deformed") using a parameter q which takes special integer values. A version of the "q-deformed" analog of the Hosszu-Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The "q-deformed" homomorphism theorem is also given.
A new form of the Hosszu-Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszu-Gluskin chain formula is not unique and can be generalized (“deformed”) using a parameter q which takes special integer values. A version of the
“q-deformed” analog of the Hosszu-Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The “q-deformed” homomorphism theorem is also given.
A new form of the Hosszu-Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszu-Gluskin chain formula is not unique and can be generalized (“deformed”) using a parameter q which takes special integer values. A version of the
“q-deformed” analog of the Hosszu-Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The “q-deformed” homomorphism theorem is also given.
2013 IEEE International Symposium on Information TheoryJoe Suzuki
The document discusses generalizing Bayesian measures to situations where random variables may be neither discrete nor continuous. It defines generalized density functions that allow estimating probabilities without assuming an underlying distribution. A main theorem states that universal Bayesian measures gn can estimate densities fn such that the log ratio of fn/gn approaches 0 as n increases, for any stationary ergodic distribution. This generalizes the Bayesian solution to testing independence of random variables without requiring a known distribution.
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but "quantized". The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
Plan: 1. Classification of general polyadic systems and special elements. 2. Definition of n-ary semigroups and groups. 3. Homomorphisms of polyadic systems. 4. The Hosszú-Gluskin theorem and its “q-deformed” generalization. 5. Multiplace generalization of homorphisms - heteromorpisms. 6. Associativity quivers. 7. Multiplace representations and multiactions. 8. Examples of matrix multiplace representations for ternary groups. 9. Polyadic rings and fields. 10. Polyadic analogs of the integer number ring Z and the Galois field GF(p). 11. Equal sums of like powers Diophantine equation over polyadic integer numbers.
This document discusses multilinear twisted paraproducts, which are generalizations of classical paraproduct operators to higher dimensions. It begins by reviewing classical paraproducts on the real line and their generalization to higher dimensions using dyadic squares. It then discusses complications that arise, such as twisted paraproducts. The document presents a unified framework for studying such operators using bipartite graphs and selections of vertices. It proves a main boundedness result and discusses special cases like classical dyadic paraproducts and dyadic twisted paraproducts. It introduces tools like Bellman functions and calculus of finite differences to analyze estimates for paraproduct-like operators on finite trees of dyadic squares.
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsVjekoslavKovac1
This document summarizes a talk given by Vjekoslav Kovač at a joint mathematics conference. The talk concerned establishing T(1)-type theorems for entangled multilinear Calderón-Zygmund operators. Specifically, Kovač discussed studying multilinear singular integral forms where the functions partially share variables, known as an "entangled structure." He outlined establishing generalized modulation invariance and Lp estimates for such operators. The talk motivated further studying related problems involving bilinear ergodic averages and forms with more complex graph structures. Kovač specialized his techniques to bipartite graphs, multilinear Calderón-Zygmund kernels, and "perfect" dyadic models.
Entity Linking via Graph-Distance MinimizationRoi Blanco
Entity-linking is a natural-language--processing task that consists in identifying strings of text that refer to a particular
item in some reference knowledge base.
One instance of entity-linking can be formalized as an optimization problem on the underlying concept graph, where the quantity to be optimized is the average distance between chosen items.
Inspired by this application, we define a new graph problem which is a natural variant of the Maximum Capacity Representative Set. We prove that our problem is NP-hard for general graphs; nonetheless, it turns out to be solvable in linear time under some more restrictive assumptions. For the general case, we propose several heuristics: one of these tries to enforce the above assumptions while the others try to optimize similar easier objective functions; we show experimentally how these approaches perform with respect to some baselines on a real-world dataset.
Clustering in Hilbert geometry for machine learningFrank Nielsen
- The document discusses different geometric approaches for clustering multinomial distributions, including total variation distance, Fisher-Rao distance, Kullback-Leibler divergence, and Hilbert cross-ratio metric.
- It benchmarks k-means clustering using these four geometries on the probability simplex, finding that Hilbert geometry clustering yields good performance with theoretical guarantees.
- The Hilbert cross-ratio metric defines a non-Riemannian Hilbert geometry on the simplex with polytopal balls, and satisfies information monotonicity properties desirable for clustering distributions.
We combined: low-rank tensor techniques and FFT to compute kriging, estimate variance, compute conditional covariance. We are able to solve 3D problems with very high resolution
This note explicates some details of the discussion
given in Appendix B of E. Jang, S. Gu, and B. Poole.
Categorical representation with Gumbel-softmax.
ICLR, 2017.
Tales on two commuting transformations or flowsVjekoslavKovac1
1) The document summarizes recent work on ergodic averages and flows for commuting transformations. It discusses convergence results for single and double linear ergodic averages in L2 and almost everywhere, as well as providing norm estimates to quantify the rate of convergence.
2) It also considers double polynomial ergodic averages and provides proofs for almost everywhere convergence in the continuous-time setting. Open problems remain for the discrete-time case.
3) An ergodic-martingale paraproduct is introduced, motivated by an open question from 1950. Convergence in Lp norm is shown, while almost everywhere convergence remains open.
Tailored Bregman Ball Trees for Effective Nearest NeighborsFrank Nielsen
This document presents an improved Bregman ball tree (BB-tree++) for efficient nearest neighbor search using Bregman divergences. The BB-tree++ speeds up construction using Bregman 2-means++ initialization and adapts the branching factor. It also handles symmetrized Bregman divergences and prioritizes closer nodes. Experiments on image retrieval with SIFT descriptors show the BB-tree++ outperforms the original BB-tree and random sampling, providing faster approximate nearest neighbor search.
The Universal Measure for General Sources and its Application to MDL/Bayesian...Joe Suzuki
1) The document presents a new theory for universal coding and the MDL principle that is applicable to general sources without assuming discrete or continuous distributions.
2) It constructs a universal measure νn that satisfies certain conditions to allow generalization of universal coding and MDL.
3) This generalized framework is applied to problems that previously separated discrete and continuous cases, such as Markov order estimation using continuous data sequences and mixed discrete-continuous feature selection.
This document summarizes research on computing stochastic partial differential equations (SPDEs) using an adaptive multi-element polynomial chaos method (MEPCM) with discrete measures. Key points include:
1) MEPCM uses polynomial chaos expansions and numerical integration to compute SPDEs with parametric uncertainty.
2) Orthogonal polynomials are generated for discrete measures using various methods like Vandermonde, Stieltjes, and Lanczos.
3) Numerical integration is tested on discrete measures using Genz functions in 1D and sparse grids in higher dimensions.
4) The method is demonstrated on the KdV equation with random initial conditions. Future work includes applying these techniques to SPDEs driven
This document discusses different geometric structures and distances that can be used for clustering probability distributions that live on the probability simplex. It reviews four main geometries: Fisher-Rao Riemannian geometry based on the Fisher information metric, information geometry based on Kullback-Leibler divergence, total variation distance and l1-norm geometry, and Hilbert projective geometry based on the Hilbert metric. It compares how k-means clustering performs using distances derived from these different geometries on the probability simplex.
This document provides an overview of Internet technologies and Java programming. It discusses the evolution of the Internet from early protocols like FTP and email to the development of the World Wide Web. Key Internet tools are explained including browsers, web servers, and HTML. Java is introduced as a portable, object-oriented programming language that allows software to be delivered via the web. The document outlines Java's core features and compares it to other languages. It also differentiates between Java applications and applets, and provides examples of basic Java code.
R-Biopharm Rhône is an innovative company based in Glasgow that produces test kits for detecting contaminants, antibodies, vitamins, and mycotoxins. The company was founded in 1986 as a spin-off from Strathclyde University and has experienced significant growth. During the visit, the Managing Director and Product Manager discussed the company's specialized facilities, unique product offerings, and international markets served. They also addressed future challenges as the company continues to expand globally.
2013 IEEE International Symposium on Information TheoryJoe Suzuki
The document discusses generalizing Bayesian measures to situations where random variables may be neither discrete nor continuous. It defines generalized density functions that allow estimating probabilities without assuming an underlying distribution. A main theorem states that universal Bayesian measures gn can estimate densities fn such that the log ratio of fn/gn approaches 0 as n increases, for any stationary ergodic distribution. This generalizes the Bayesian solution to testing independence of random variables without requiring a known distribution.
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but "quantized". The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
Plan: 1. Classification of general polyadic systems and special elements. 2. Definition of n-ary semigroups and groups. 3. Homomorphisms of polyadic systems. 4. The Hosszú-Gluskin theorem and its “q-deformed” generalization. 5. Multiplace generalization of homorphisms - heteromorpisms. 6. Associativity quivers. 7. Multiplace representations and multiactions. 8. Examples of matrix multiplace representations for ternary groups. 9. Polyadic rings and fields. 10. Polyadic analogs of the integer number ring Z and the Galois field GF(p). 11. Equal sums of like powers Diophantine equation over polyadic integer numbers.
This document discusses multilinear twisted paraproducts, which are generalizations of classical paraproduct operators to higher dimensions. It begins by reviewing classical paraproducts on the real line and their generalization to higher dimensions using dyadic squares. It then discusses complications that arise, such as twisted paraproducts. The document presents a unified framework for studying such operators using bipartite graphs and selections of vertices. It proves a main boundedness result and discusses special cases like classical dyadic paraproducts and dyadic twisted paraproducts. It introduces tools like Bellman functions and calculus of finite differences to analyze estimates for paraproduct-like operators on finite trees of dyadic squares.
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsVjekoslavKovac1
This document summarizes a talk given by Vjekoslav Kovač at a joint mathematics conference. The talk concerned establishing T(1)-type theorems for entangled multilinear Calderón-Zygmund operators. Specifically, Kovač discussed studying multilinear singular integral forms where the functions partially share variables, known as an "entangled structure." He outlined establishing generalized modulation invariance and Lp estimates for such operators. The talk motivated further studying related problems involving bilinear ergodic averages and forms with more complex graph structures. Kovač specialized his techniques to bipartite graphs, multilinear Calderón-Zygmund kernels, and "perfect" dyadic models.
Entity Linking via Graph-Distance MinimizationRoi Blanco
Entity-linking is a natural-language--processing task that consists in identifying strings of text that refer to a particular
item in some reference knowledge base.
One instance of entity-linking can be formalized as an optimization problem on the underlying concept graph, where the quantity to be optimized is the average distance between chosen items.
Inspired by this application, we define a new graph problem which is a natural variant of the Maximum Capacity Representative Set. We prove that our problem is NP-hard for general graphs; nonetheless, it turns out to be solvable in linear time under some more restrictive assumptions. For the general case, we propose several heuristics: one of these tries to enforce the above assumptions while the others try to optimize similar easier objective functions; we show experimentally how these approaches perform with respect to some baselines on a real-world dataset.
Clustering in Hilbert geometry for machine learningFrank Nielsen
- The document discusses different geometric approaches for clustering multinomial distributions, including total variation distance, Fisher-Rao distance, Kullback-Leibler divergence, and Hilbert cross-ratio metric.
- It benchmarks k-means clustering using these four geometries on the probability simplex, finding that Hilbert geometry clustering yields good performance with theoretical guarantees.
- The Hilbert cross-ratio metric defines a non-Riemannian Hilbert geometry on the simplex with polytopal balls, and satisfies information monotonicity properties desirable for clustering distributions.
We combined: low-rank tensor techniques and FFT to compute kriging, estimate variance, compute conditional covariance. We are able to solve 3D problems with very high resolution
This note explicates some details of the discussion
given in Appendix B of E. Jang, S. Gu, and B. Poole.
Categorical representation with Gumbel-softmax.
ICLR, 2017.
Tales on two commuting transformations or flowsVjekoslavKovac1
1) The document summarizes recent work on ergodic averages and flows for commuting transformations. It discusses convergence results for single and double linear ergodic averages in L2 and almost everywhere, as well as providing norm estimates to quantify the rate of convergence.
2) It also considers double polynomial ergodic averages and provides proofs for almost everywhere convergence in the continuous-time setting. Open problems remain for the discrete-time case.
3) An ergodic-martingale paraproduct is introduced, motivated by an open question from 1950. Convergence in Lp norm is shown, while almost everywhere convergence remains open.
Tailored Bregman Ball Trees for Effective Nearest NeighborsFrank Nielsen
This document presents an improved Bregman ball tree (BB-tree++) for efficient nearest neighbor search using Bregman divergences. The BB-tree++ speeds up construction using Bregman 2-means++ initialization and adapts the branching factor. It also handles symmetrized Bregman divergences and prioritizes closer nodes. Experiments on image retrieval with SIFT descriptors show the BB-tree++ outperforms the original BB-tree and random sampling, providing faster approximate nearest neighbor search.
The Universal Measure for General Sources and its Application to MDL/Bayesian...Joe Suzuki
1) The document presents a new theory for universal coding and the MDL principle that is applicable to general sources without assuming discrete or continuous distributions.
2) It constructs a universal measure νn that satisfies certain conditions to allow generalization of universal coding and MDL.
3) This generalized framework is applied to problems that previously separated discrete and continuous cases, such as Markov order estimation using continuous data sequences and mixed discrete-continuous feature selection.
This document summarizes research on computing stochastic partial differential equations (SPDEs) using an adaptive multi-element polynomial chaos method (MEPCM) with discrete measures. Key points include:
1) MEPCM uses polynomial chaos expansions and numerical integration to compute SPDEs with parametric uncertainty.
2) Orthogonal polynomials are generated for discrete measures using various methods like Vandermonde, Stieltjes, and Lanczos.
3) Numerical integration is tested on discrete measures using Genz functions in 1D and sparse grids in higher dimensions.
4) The method is demonstrated on the KdV equation with random initial conditions. Future work includes applying these techniques to SPDEs driven
This document discusses different geometric structures and distances that can be used for clustering probability distributions that live on the probability simplex. It reviews four main geometries: Fisher-Rao Riemannian geometry based on the Fisher information metric, information geometry based on Kullback-Leibler divergence, total variation distance and l1-norm geometry, and Hilbert projective geometry based on the Hilbert metric. It compares how k-means clustering performs using distances derived from these different geometries on the probability simplex.
This document provides an overview of Internet technologies and Java programming. It discusses the evolution of the Internet from early protocols like FTP and email to the development of the World Wide Web. Key Internet tools are explained including browsers, web servers, and HTML. Java is introduced as a portable, object-oriented programming language that allows software to be delivered via the web. The document outlines Java's core features and compares it to other languages. It also differentiates between Java applications and applets, and provides examples of basic Java code.
R-Biopharm Rhône is an innovative company based in Glasgow that produces test kits for detecting contaminants, antibodies, vitamins, and mycotoxins. The company was founded in 1986 as a spin-off from Strathclyde University and has experienced significant growth. During the visit, the Managing Director and Product Manager discussed the company's specialized facilities, unique product offerings, and international markets served. They also addressed future challenges as the company continues to expand globally.
Manufacturer and Supplier of a wide assortment of Automation Sensors & Products. These products cater to several industries such as automobile, cement chemical, steel, textile and pharmaceuticals such as Slot Sensors, Diffuse Scan Sensors, Through Scan Sensor, Retroreflective Scan Sensor, Electronic Timers, Time Totaliser / Hour Meter, Sequential Timer, Both Side Display Counters.
This is about the new Apple programming language.
You won't learn everything about, you know that's not possible. But you'll learn the basics, which is good enough to know if you want to learn more about it!
Kickstarter - US vs UK (1st half of 2013)ICO Partners
The document analyzes data from video game projects on Kickstarter from January to July 2013, comparing US and UK trends. It finds that in the first half of 2013, the UK had fewer overall and successful video game projects than the US, but UK projects on average raised more money per backer. Additionally, one UK game, Elite: Dangerous, significantly influenced overall funding totals due to its success raising over $2.5 million.
This annual report from Rochambeau French International School summarizes the 2014-2015 school year. It discusses outstanding academic results including high university acceptance rates and exam scores. It also provides information on the diverse international student body, financial aid awarded, and fundraising results. The school saw increased enrollment, surplus finances, and continued investment in infrastructure and technology. Overall, 2014-2015 was described as a great year of growth and success for Rochambeau.
The document advertises the Great Wall Cup, an international youth football tournament held annually in Beijing, China. It provides information on the tournament dates, age categories, tournament format, pitch locations, and emphasizes the opportunity for cultural exchange and experiencing China beyond just football. Sponsorship from China Sports Tour is acknowledged for supporting the tournament. Teams can register on the official website to participate.
Conficasa is an international real estate and mortgage company that provides mortgages to Mexican workers in the US to buy property in Mexico. It was founded in 1998 to realize the American Dream in Mexico. Through its RAICES program developed with the Mexican government, Conficasa qualifies Mexican descendants living in the US for mortgages to purchase homes in Mexico if they have relatives living there. The company sees a large untapped market and opportunity to address Mexico's housing needs by facilitating financing for Mexican workers in the US. Conficasa is seeking capital to expand its infrastructure, processing capabilities, and marketing to take advantage of this sizeable opportunity.
NSAA Planet Speaker Webinar 1 Dec 2010Craig Rispin
The document discusses emerging technology trends impacting the speaking industry, including globalization, shifting decision makers, virtual delivery methods, increasing multimedia sophistication, hybrid meetings, improved information access, enhanced networking and interaction capabilities, and green meetings. It also provides information on and screenshots of various online tools and services, such as Google Apps, Setster, MailChimp, BatchBook, ShoeBoxed, Squarespace, Evernote, ReQall, and Aviary.
Since 1998 the company Behn + Bates has been the packing machine specialist for the food and petfood industries in the internationally operating Haver & Boecker Group.
Field trip in Chicago May 22 and 23, 2007 (en anglais)ve-finance
- Veolia Environnement North America provides environmental services across North America, including water, waste management, energy, and transportation.
- It has approximately $3.5 billion in annual revenue and 30,000 employees across the region.
- Veolia is a leader in various sectors such as water treatment and transportation, operating the largest public transportation contracts in cities like Las Vegas and Boston.
Improve Customer Experiences With Big Data #DataTalkExperian_US
Join our #DataTalk on Thursdays at 5 p.m. ET. This week, we learned from Matt Seeley, President of Experian Marketing Services, Mark Johnson, President and CEO of Loyalty 360, and Ginger Conlon, Editor-in-Chief of Direct Marketing News.
Baker v. Carr was a 1962 Supreme Court case that challenged the constitutionality of Tennessee's legislative apportionment. The plaintiffs argued that the state's failure to reapportion its legislature in over 60 years violated the Equal Protection Clause, as urban populations had greatly increased but rural areas retained control of the legislature. The Supreme Court ruled that issues of legislative apportionment presented a justiciable question within the Court's competence to decide, overturning prior decisions, and that Tennessee's apportionment scheme did violate equal protection. This landmark decision paved the way for the "one person, one vote" principle and altered the political landscape across the United States.
Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed.
MSC classes: 16T05, 16T25, 17A42, 20N15, 20F29, 20G05, 20G42, 57T05
Density theorems for Euclidean point configurationsVjekoslavKovac1
1. The document discusses density theorems for point configurations in Euclidean space. Density theorems study when a measurable set A contained in Euclidean space can be considered "large".
2. One classical result is that for any measurable set A contained in R2 with positive upper Banach density, there exist points in A whose distance is any sufficiently large real number. This has been generalized to higher dimensions and other point configurations.
3. Open questions remain about determining all point configurations P for which one can show that a sufficiently large measurable set A contained in high dimensional Euclidean space must contain a scaled copy of P.
We consider a general approach to describing the interaction in multigravity models in a D-dimensional space–time. We present various possibilities for generalizing the invariant volume. We derive the most general form of the interaction potential, which becomes a Pauli–Fierz-type model in the bigravity case. Analyzing this model in detail in the (3+1)-expansion formalism and also requiring the absence of ghosts leads to this bigravity model being completely equivalent to the Pauli–Fierz model. We thus in a concrete example show that introducing an interaction between metrics is equivalent to introducing the graviton mass.
We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be “entangled” such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique, which was provided previously. For polyadic semigroups and groups we introduce two external products: (1) the iterated direct product, which is componentwise but can have an arity that is different from the multipliers and (2) the hetero product (power), which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. We show in which cases the product of polyadic groups can itself be a polyadic group. In the same way, the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), in which all multipliers are zeroless fields. Many illustrative concrete examples are presented. Thу proposed construction can lead to a new category of polyadic fields.
The Probability that a Matrix of Integers Is DiagonalizableJay Liew
The Probability that a
Matrix of Integers Is Diagonalizable
Andrew J. Hetzel, Jay S. Liew, and Kent E. Morrison
1. INTRODUCTION. It is natural to use integer matrices for examples and exercises
when teaching a linear algebra course, or, for that matter, when writing a textbook in
the subject. After all, integer matrices offer a great deal of algebraic simplicity for particular
problems. This, in turn, lets students focus on the concepts. Of course, to insist
on integer matrices exclusively would certainly give the wrong idea about many important
concepts. For example, integer matrices with integer matrix inverses are quite
rare, although invertible integer matrices (over the rational numbers) are relatively
common. In this article, we focus on the property of diagonalizability for integer matrices
and pose the question of the likelihood that an integer matrix is diagonalizable.
Specifically, we ask: What is the probability that an n × n matrix with integer entries is
diagonalizable over the complex numbers, the real numbers, and the rational numbers,
respectively?
sublabel accurate convex relaxation of vectorial multilabel energiesFujimoto Keisuke
This document summarizes a presentation on the paper "Sublabel-Accurate Convex Relaxation of Vectorial Multilabel Energies". It discusses how the paper proposes a method to efficiently solve high-dimensional, nonlinear vectorial labeling problems by approximating them as convex problems. Specifically, it divides the problem domain into subregions and approximates each subregion with a convex function, yielding an overall approximation that is still non-convex but with higher accuracy. This lifting technique transforms the variables into a higher-dimensional space to formulate the data and regularization terms in a way that allows solving the problem as a convex optimization.
The Gaussian or normal distribution is one of the most widely used in statistics. Estimating its parameters using
Bayesian inference and conjugate priors is also widely used. The use of conjugate priors allows all the results to be
derived in closed form. Unfortunately, different books use different conventions on how to parameterize the various
distributions (e.g., put the prior on the precision or the variance, use an inverse gamma or inverse chi-squared, etc),
which can be very confusing for the student. In this report, we summarize all of the most commonly used forms. We
provide detailed derivations for some of these results; the rest can be obtained by simple reparameterization. See the
appendix for the definition the distributions that are used.
We examine the effectiveness of randomized quasi Monte Carlo (RQMC) to improve the convergence rate of the mean integrated square error, compared with crude Monte Carlo (MC), when estimating the density of a random variable X defined as a function over the s-dimensional unit cube (0,1)^s. We consider histograms and kernel density estimators. We show both theoretically and empirically that RQMC estimators can achieve faster convergence rates in
some situations.
This is joint work with Amal Ben Abdellah, Art B. Owen, and Florian Puchhammer.
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented as block diagonal matrices (resulting in the Wedderburn decomposition), general forms of polyadic structures are given by block-shift matrices. We combine these forms to get a general shape of semisimple nonderived polyadic structures (“double” decomposition of two kinds). We then introduce the polyadization concept (a “polyadic constructor”), according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The “deformation” by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.
We research behavior and sharp bounds for the zeros of infinite sequences of polynomials orthogonal with respect to a Geronimus perturbation of a positive Borel measure on the real line.
This document discusses various topics related to integration including indefinite and definite integrals, power rules, properties of integrals, integration by parts, u-substitution, and definitions. Some key points covered are:
- Integration was developed independently by Newton and Leibniz in the late 17th century.
- The indefinite integral finds an antiderivative, while the definite integral evaluates the area under a function between bounds.
- Common integration techniques include power rules, integration by parts, and u-substitution.
- Integration rules and properties allow integrals to be transformed and simplified.
Application of the Monte-Carlo Method to Nonlinear Stochastic Optimization wi...SSA KPI
This document describes a method for solving nonlinear stochastic optimization problems with linear constraints using Monte Carlo estimators. The key aspects are:
1) An ε-feasible solution approach is used to avoid "jamming" or "zigzagging" when dealing with linear constraints.
2) The optimality of solutions is tested statistically using the asymptotic normality of Monte Carlo estimators.
3) The Monte Carlo sample size is adjusted iteratively based on the gradient estimate to decrease computational trials while maintaining solution accuracy.
4) Under certain conditions, the method is proven to converge almost surely to a stationary point of the optimization problem.
5) As an example, the method is applied to portfolio optimization with
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMSTahia ZERIZER
In this article, we study boundary value problems of a large
class of non-linear discrete systems at two-time-scales. Algorithms are given to implement asymptotic solutions for any order of approximation.
This is a journal concise version (without diagrams and figures) of the preprint arXiv:1308.4060.
Abstract: Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary generalizations of quantum groups and the Yang-Baxter equation are presented.
The Multivariate Gaussian Probability DistributionPedro222284
The document discusses the multivariate Gaussian probability distribution. It defines the distribution and provides its probability density function. It then discusses various properties including: functions of Gaussian variables such as linear transformations and addition; the characteristic function and how to calculate moments; marginalization and conditional distributions. It also provides some tips and tricks for working with Gaussian distributions including how to calculate products.
This document studies module-algebra structures of the quantum universal enveloping algebra Uq(sl(m+1)) on the coordinate algebra of quantum n-dimensional vector spaces Aq(n). The main results are:
1) A complete classification is given of Uq(sl(m+1)) module-algebra structures on Aq(3) when m ≥ 2 and ki acts as an automorphism of Aq(3).
2) All module-algebra structures of Uq(sl(m+1)) on Aq(2) are characterized with the same method.
3) The module-algebra structures of Uq(sl(m+1)) on Aq(n)
Shape restrictions such as monotonicity often naturally arise. In this talk, we consider a Bayesian approach to monotone nonparametric regression with a normal error. We assign a prior through piecewise constant functions and impose a conjugate normal prior on the coefficient. Since the resulting functions need not be monotone, we project samples from the posterior on the allowed parameter space to construct a “projection posterior”. We obtain the limit posterior distribution of a suitably centered and scaled posterior distribution for the function value at a point. The limit distribution has some interesting similarity and difference with the corresponding limit distribution for the maximum likelihood estimator. By comparing the quantiles of these two distributions, we observe an interesting new phenomenon that coverage of a credible interval can be more than the credibility level, the exact opposite of a phenomenon observed by Cox for smooth regression. We describe a recalibration strategy to modify the credible interval to meet the correct level of coverage.
This talk is based on joint work with Moumita Chakraborty, a doctoral student at North Carolina State University.
This document presents a summary of a talk on building a harmonic analytic theory for the Gaussian measure and the Ornstein-Uhlenbeck operator. It discusses how the Gaussian measure is non-doubling but satisfies a local doubling property. It introduces Gaussian cones and shows how they allow proving maximal function estimates for the Ornstein-Uhlenbeck semigroup in a similar way as for the heat semigroup. The talk outlines estimates for the Mehler kernel of the Ornstein-Uhlenbeck semigroup and combines them to obtain boundedness of the maximal function.
Similar to S. Duplij, A q-deformed generalization of the Hosszu-Gluskin theorem (20)
We introduce a polyadic analog of supersymmetry by considering the polyadization procedure (proposed by the author) applied to the toy model of one-dimensional supersymmetric quantum mechanics. The supercharges are generalized to polyadic ones using the n-ary sigma matrices defined in earlier work. In this way, polyadic analogs of supercharges and Hamiltonians take the cyclic shift block matrix form, and they can describe multidegenerated quantum states in a way that is different from the N-extended and multigraded SQM. While constructing the corresponding supersymmetry as an n-ary Lie superalgebra (n is the arity of the initial associative multiplication), we have found new brackets with a reduced arity of 2<=m<n and a related series of m-ary superalgebras (which is impossible for binary superalgebras). In the case of even reduced arity m we obtain a tower of higher order (as differential operators) even Hamiltonians, while for m odd we get a tower of higher order odd supercharges, and the corresponding algebra consists of the odd sector only.
https://arxiv.org/abs/2406.02188
We generalize σ-matrices to higher arities using the polyadization procedure proposed by the author. We build the nonderived n-ary version of SU(2) using cyclic shift block matrices. We define a new function, the polyadic trace, which has an additivity property analogous to the ordinary trace for block diagonal matrices and which can be used to build the corresponding invariants. The elementary Σ-matrices introduced here play a role similar to ordinary matrix units, and their sums are full Σ-matrices which can be treated as a polyadic analog of σ-matrices. The presentation of n-ary SU(2) in terms of full Σ-matrices is done using the Hadamard product. We then generalize the Pauli group in two ways: for the binary case we introduce the extended phase shifted σ-matrices with multipliers in cyclic groups of order 4q (q>4), and for the polyadic case we construct the correspondent finite n-ary semigroup of phase-shifted elementary Σ-matrices of order 4q(n-1)+1, and the finite n-ary group of phase-shifted full Σ-matrices of order 4q. Finally, we introduce the finite n-ary group of heterogeneous full Σ^het-matrices of order (4q(n-1))^4. Some examples of the lowest arities are presented.
https://arxiv.org/abs/2403.19361. *) https://www.researchgate.net/publication/360882654_Polyadic_Algebraic_Structures, https://iopscience.iop.org/book/978-0-7503-2648-3.
CONTENTS 1. INTRODUCTION 2. PRELIMINARIES 3. POLYADIC SU p2q 4. POLYADIC ANALOG OF SIGMA MATRICES 4.1. Elementary Σ-matrices 4.2. Full Σ-matrices 5. TERNARY SUp2q AND Σ-MATRICES 6. n-ARY SEMIGROUPS AND GROUPS OF Σ-MATRICES 6.1. The Pauli group 6.2. Groups of phase-shifted sigma matrices 6.3. The n-ary semigroup of elementary Σ-matrices 6.4. The n-ary group of full Σ-matrices 7. HETEROGENEOUS FULL Σ-MATRICES REFERENCES
We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley-Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, that is proportional to the intermediate arities. Second, a new polyadic product of vectors in any vector space is defined. Endowed with this product the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process ("imaginary tower"), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call "half-quaternions" and "half-octonions". The latter are not subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced "half-quaternion" norm we obtain the ternary analog of the sum of two squares identity. We prove that the unitless ternary division algebra of imaginary "half-octonions" we have introduced is ternary alternative.
https://arxiv.org/abs/2312.01366
https://www.amazon.com/s?k=duplij
This document introduces hyperpolyadic structures, which are n-ary analogs of binary division algebras like the reals, complexes, quaternions, and octonions. It proposes two constructions:
1) A matrix polyadization procedure that increases the dimension of a binary algebra to obtain a corresponding n-ary algebra by using cyclic shift block matrices.
2) An "imaginary tower" construction on subsets of binary division algebras that gives nonderived ternary division algebras of half the original dimension, called "half-quaternions" and "half-octonions."
178 pages, 6 Chapters. DOI: 10.1088/978-0-7503-5281-9. This book presents new and prospective approaches to quantum computing. It introduces the many possibilities to further develop the mathematical methods of quantum computation and its applications to future functioning and operational quantum computers. In this book, various extensions of the qubit concept, starting from obscure qubits, superqubits and other fundamental generalizations, are considered. New gates, known as higher braiding gates, are introduced. These new gates are implemented as an additional stage of computation for topological quantum computations and unconventional computing when computational complexity is affected by its environment. Other generalizations are considered and explained in a widely accessible and easy-to-understand way. Presented in a book for the first time, these new mathematical methods will increase the efficiency and speed of quantum computing.Part of IOP Series in Coherent Sources, Quantum Fundamentals, and Applications. Key features • Provides new mathematical methods for quantum computing. • Presents material in a widely accessible way. • Contains methods for unconventional computing where there is computational complexity. • Provides methods to increase speed and efficiency. For the light paperback version use MyPrint service here: https://iopscience.iop.org/book/mono/978-0-7503-5281-9, also PDF, ePub and Kindle. For the libraries and direct ordering from IOP: https://store.ioppublishing.org/page/detail/Innovative-Quantum-Computing/?K=9780750352796. Amazon ordering: https://www.amazon.de/gp/product/0750352795
Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories.
https://www.mdpi.com/books/book/6455
We investigate finite field extensions of the unital 3-field, consisting of the unit element alone, and find considerable differences to classical field theory. Furthermore, the structure of their automorphism groups is clarified and the respective subfields are determined. In an attempt to better understand the structure of 3-fields that show up here we look at ways in which new unital 3-fields can be obtained from known ones in terms of product structures, one of them the Cartesian product which has no analogue for binary fields.
https://arxiv.org/abs/2212.08606
Abstract: Algebraic structures in which the property of commutativity is substituted by the me- diality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or e-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with (n-1) associators of the arity (2n-1) satisfying a (n^2-1)-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.
In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become (m, n)-rings. Second, we introduce a possible p-adic analog of the residue class modulo a p-adic integer. Then, we find the relations which determine, when the representatives form a (m, n)-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.
Книга «Поэфизика души» представляет собой полное, на момент издания 2022 г., собрание прозаических произведений автора. Как рассказы, так и миниатюры на полстраницы, пронизаны эмоциями и искренними чувствами на грани срыва, что заставляет возвращаться к ним вновь. Буквально каждое слово рисует уникальные картины нетривиальных внутренних миров автора, которые перетекают друг в друга поэтическими образами, воплощенными в прозе. Также включены юмористические путевые заметки о поездке в Китай.
Книга "Поэфизика души", Степан Дуплий – полное собрание прозы 2022, 230 стр. вышла в Ridero: https://ridero.ru/books/poefizika_dushi и Kindle Edition file на Амазоне: https://amazon.com/dp/B0B9Y4X4VJ . "Бумажную" книгу можно заказать на Озоне https://ozon.ru/product/poefizika-dushi-682515885/?sh=XPu-9Sb42Q и на ЛитРес: https://litres.ru/stepan-dupliy/poefizika-dushi-emocionalnaya-proza-kitayskiy-shtrih-punktir . Google books: https://books.google.com/books?id=9w2DEAAAQBAJ .
Книгу можно заказать из-за рубежа на AliExpress: https://aliexpress.com/item/1005004660613179.html .
Книга «Гравитация страсти» представляет собой полное собрание стихотворений автора на момент издания (август, 2022). Стихотворения пронизаны эмоциями и искренними чувствами на грани срыва, что заставляет возвращаться к ним вновь. Буквально каждое слово рисует уникальные картины нетривиальных внутренних миров автора, которые перетекают друг в друга необычными поэтическими образами.
Книга "Гравитация страсти", Степан Дуплий - полное собрание стихотворений 2022, 338 стр. вышла в Ridero: https://ridero.ru/books/gravitaciya_strasti
. Книга в мягкой обложке доступна для заказа на Ozon.ru: https://ozon.ru/product/gravitatsiya-strasti-707068219/?oos_search=false&sh=XPu-9TbW9Q
, на Litres.ru: https://www.litres.ru/stepan-dupliy/gravitaciya-strasti-stihotvoreniya , за рубежом на AliExpress: https://aliexpress.com/item/1005004722134442.html , и в электронном виде Kindle file на Amazon.com: https://amazon.com/dp/B0BDFTT33W .
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in the block-diagonal matrix form (Wedderburn decomposition), a general form of polyadic structures is given by block-shift matrices. We combine these forms in a special way to get a general shape of semisimple nonderived polyadic structures. We then introduce the polyadization concept (a "polyadic constructor") according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The "deformation" by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.
We generalize the Grothendieck construction of the completion group for a monoid (being the starting point of the algebraic $K$-theory) to the polyadic case, when an initial semigroup is $m$-ary and the corresponding final class group $K_{0}$ can be $n$-ary. As opposed to the binary case: 1) there can be different polyadic direct products which can be built from one polyadic semigroup; 2) the final arity $n$ of the class groups can be different from the arity $m$ of initial semigroup; 3) commutative initial $m$-ary semigroups can lead to noncommutative class $n$-ary groups; 4) the identity is not necessary for initial $m$-ary semigroup to obtain the class $n$-ary group, which in its turn can contain no identity at all. The presented numerical examples show that the properties of the polyadic completion groups are considerably nontrivial and have more complicated structure than in the binary case.
In book: S. Duplij, "Polyadic Algebraic Structures", 2022, IOP Publishing (Bristol), Section 1.5. See https://iopscience.iop.org/book/978-0-7503-2648-3
https://arxiv.org/abs/2206.14840
The book is devoted to the thorough study of polyadic (higher arity) algebraic structures, which has a long history, starting from 19th century. The main idea was to take a single set, closed under one binary operation, and to 'generalize' it by increasing the arity of the operation, called a polyadic operation. Until now, a general approach to polyadic concrete many-set algebraic structures was absent. We propose to investigate algebraic structures in the 'concrete way' and provide consequent 'polyadization' of each operation, starting from group-like structures and finishing with the Hopf algebra structures. Polyadic analogs of homomorphisms which change arity, heteromorphisms, are introduced and applied for constructing unusual representations, multiactions, matrix representations and polyadic analogs of direct product. We provide the polyadic generalization of the Yang–Baxter equation, find its constant solutions, and introduce polyadic tensor categories.
Suitable for university students of advanced level algebra courses and mathematical physics courses.
Key features
• Provides a general, unified approach
• Widens readers perspective of the possibilities to develop standard algebraic structures
• Provides the new kind of homomorphisms changing the arity, heteromorphisms, are introduced and applied for construction of new representations, multiactions and matrix representations
• Presents applications of 'polyadization' approach to concrete algebraic structures
https://arxiv.org/abs/2201.08479
We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be "entangled" such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique provided earlier. For polyadic semigroups and groups we introduce two external products: 1) the iterated direct product which is componentwise, but can have arity different from the multipliers; 2) the hetero product (power) which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. It is shown in which cases the product of polyadic groups can itself be a polyadic group. In the same way the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), when all multipliers are zeroless fields, which can lead to a new category of polyadic fields. Many illustrative concrete examples are presented.
This document proposes a new mechanism for "deforming" or breaking commutativity in algebras called "membership deformation". It involves taking the underlying set of an algebra to be an "obscure/fuzzy set" with elements having membership functions between 0 and 1 rather than a crisp set. The membership functions are incorporated into the commutation relations such that elements with equal membership functions commute, while others do not. This provides a continuous way to deform commutativity. The approach is then generalized to ε-commutative algebras and n-ary algebras. Projective representations of n-ary algebras are also studied in relation to this new type of deformation.
We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions. Finally, we introduce the sandwich higher polyadic regularity with generalized idempotents.
This document discusses generalizing the concept of regularity for semigroups in two ways: higher regularity and higher arity (polyadic semigroups).
For binary semigroups, higher n-regularity is defined such that each element has multiple inverse elements rather than a single inverse. However, for binary semigroups this reduces to ordinary regularity. For polyadic semigroups, several definitions of regularity and higher regularity are introduced to account for the higher arity operations. Idempotents and identities are also generalized for polyadic semigroups. It is shown that the definitions of regularity for polyadic semigroups cannot be reduced in the same
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.
A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
BREEDING METHODS FOR DISEASE RESISTANCE.pptxRASHMI M G
Plant breeding for disease resistance is a strategy to reduce crop losses caused by disease. Plants have an innate immune system that allows them to recognize pathogens and provide resistance. However, breeding for long-lasting resistance often involves combining multiple resistance genes
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...Travis Hills MN
Travis Hills of Minnesota developed a method to convert waste into high-value dry fertilizer, significantly enriching soil quality. By providing farmers with a valuable resource derived from waste, Travis Hills helps enhance farm profitability while promoting environmental stewardship. Travis Hills' sustainable practices lead to cost savings and increased revenue for farmers by improving resource efficiency and reducing waste.
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
Phenomics assisted breeding in crop improvementIshaGoswami9
As the population is increasing and will reach about 9 billion upto 2050. Also due to climate change, it is difficult to meet the food requirement of such a large population. Facing the challenges presented by resource shortages, climate
change, and increasing global population, crop yield and quality need to be improved in a sustainable way over the coming decades. Genetic improvement by breeding is the best way to increase crop productivity. With the rapid progression of functional
genomics, an increasing number of crop genomes have been sequenced and dozens of genes influencing key agronomic traits have been identified. However, current genome sequence information has not been adequately exploited for understanding
the complex characteristics of multiple gene, owing to a lack of crop phenotypic data. Efficient, automatic, and accurate technologies and platforms that can capture phenotypic data that can
be linked to genomics information for crop improvement at all growth stages have become as important as genotyping. Thus,
high-throughput phenotyping has become the major bottleneck restricting crop breeding. Plant phenomics has been defined as the high-throughput, accurate acquisition and analysis of multi-dimensional phenotypes
during crop growing stages at the organism level, including the cell, tissue, organ, individual plant, plot, and field levels. With the rapid development of novel sensors, imaging technology,
and analysis methods, numerous infrastructure platforms have been developed for phenotyping.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
The binding of cosmological structures by massless topological defects
S. Duplij, A q-deformed generalization of the Hosszu-Gluskin theorem
1. Filomat xx (yyyy), zzz–zzz
DOI (will be added later)
Published by Faculty of Sciences and Mathematics,
University of Niˇs, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
A “q-deformed” generalization of the Hossz´u-Gluskin theorem
Steven Duplija
aMathematisches Institut, Universit¨at M¨unster, Einsteinstr. 62, D-48149 M¨unster, Germany
Abstract. In this paper a new form of the Hossz´u-Gluskin theorem is presented in terms of polyadic powers
and using the language of diagrams. It is shown that the Hossz´u-Gluskin chain formula is not unique and
can be generalized (“deformed”) using a parameter q which takes special integer values. A version of the
“q-deformed” analog of the Hossz´u-Gluskin theorem in the form of an invariance is formulated, and some
examples are considered. The “q-deformed” homomorphism theorem is also given.
Contents
1 Introduction 1
2 Preliminaries 2
3 The Hossz´u-Gluskin theorem 9
4 “Deformation” of Hossz´u-Gluskin chain formula 14
5 Generalized “deformed” version of the homomorphism theorem 20
1. Introduction
Since the early days of “polyadic history” [1–3], the interconnection between polyadic systems and
binary ones has been one of the main areas of interest [4, 5]. Early constructions were confined to building
some special polyadic (mostly ternary [6, 7]) operations on elements of binary groups [8–10]. A very special
form of n-ary multiplication in terms of binary multiplication and a special mapping as a chain formula was
found in [11] and [12, 13]. The theorem that any n-ary multiplication can be presented in this form is called
the Hossz´u-Gluskin theorem (for review see [14, 15]). A concise and clear proof of the Hossz´u-Gluskin
chain formula was presented in [16].
In this paper we give a new form of the Hossz´u-Gluskin theorem in terms of polyadic powers. Then we
show that the Hossz´u-Gluskin chain formula is not unique and can be generalized (“deformed”) using a
parameter q which takes special integer values. We present the “q-deformed” analog of the Hossz´u-Gluskin
theorem in the form of an invariance and consider some examples. The “q-deformed” homomorphism
theorem is also given.
2010 Mathematics Subject Classification. 08B05, 17A42, 20N15
Keywords. (polyadic group, polyad, polyadic power, D¨ornte relations, q-deformed number, homomorphism theorem)
Received: dd Month yyyy; Accepted: dd Month yyyy
Communicated by (name of the Editor, mandatory)
Research supported by the project “Groups, Geometry and Actions” (SFB 878).
Email address: duplijs@math.uni-muenster.de, http://wwwmath.uni-muenster.de/u/duplij (Steven Duplij)
2. S. Duplij / Filomat xx (yyyy), zzz–zzz 2
2. Preliminaries
We will use the concise notations from our previous review paper [17], while here we repeat some
necessary definitions using the language of diagrams. For a non-empty set G, we denote its elements by
lower-case Latin letters i ∈ G and the n-tuple (or polyad) 1, . . . , n will be written by
(
1, . . . , n
)
or using
one bold letter with index (n)
, and an n-tuple with equal elements by n
. In case the number of elements
in the n-tuple is clear from the context or is not important, we denote it in one bold letter without indices.
We omit ∈ G, if it is obvious from the context.
The Cartesian product
n
G × . . . × G = G×n
consists of all n-tuples
(
1, . . . , n
)
, such that i ∈ G, i =
1, . . . , n. The i-projection of the Cartesian product Gn
on its i-th “axis” is the map Pr
(n)
i
: G×n
→ G such that(
1, . . . i, . . . , n
)
−→ i. The i-diagonal Diagn : G → G×n
sends one element to the equal element n-tuple
−→
( n
)
. The one-point set {•} is treated as a unit for the Cartesian product, since there are bijections
between G and G × {•}×n
, where G can be on any place. In diagrams, if the place is unimportant, we denote
such bijections by ϵ. On the Cartesian product G×n
one can define a polyadic (n-ary or n-adic, if it is necessary
to specify n, its arity or rank) operation µn : G×n
→ G. For operations we use small Greek letters and place
arguments in square brackets µn
[ ]
. The operations with n = 1, 2, 3 are called unary, binary and ternary. The
case n = 0 is special and corresponds to fixing a distinguished element of G, a “constant” c ∈ G, and it is
called a 0-ary operation µ
(c)
0
, which maps the one-point set {•} to G, such that µ
(c)
0
: {•} → G, and (formally) has
the value µ
(c)
0
[{•}] = c ∈ G. The composition of n-ary and m-ary operations µn ◦ µm gives a (n + m − 1)-ary
operation by the iteration µn+m−1
[
, h
]
= µn
[
, µm [h]
]
. If we compose µn with the 0-ary operation µ
(c)
0
, then
we obtain the arity “collapsing” µ
(c)
n−1
[ ]
= µn
[
, c
]
, because is a polyad of length (n − 1). A universal
algebra is a set which is closed under several polyadic operations [18]. If a concrete universal algebra has
one fundamental n-ary operation, called a polyadic multiplication (or n-ary multiplication) µn, we name it a
“polyadic system”1)
.
Definition 2.1. A polyadic system G =
⟨
set|one fundamental operation
⟩
is a set G which is closed under polyadic
multiplication.
More specifically, a n-ary system Gn =
⟨
G | µn
⟩
is a set G closed under one n-ary operation µn (without
any other additional structure).
For a given n-ary system
⟨
G | µn
⟩
one can construct another polyadic system
⟨
G | µ′
n′
⟩
over the same set
G, but with another multiplication µ′
n′ of different arity n′
. In general, there are three ways of changing the
arity:
1. Iterating. Composition of the operation µn with itself increases the arity from n to n′
= niter > n. We
denote the number of iterating multiplications by ℓµ and call the resulting composition an iterated
product2)
µ
ℓµ
n (using the bold Greek letters) as (or µ•
n if ℓµ is obvious or not important)
µ′
n′ = µ
ℓµ
n
def
=
ℓµ
µn ◦
(
µn ◦ . . .
(
µn × id×(n−1)
)
. . . × id×(n−1)
)
, (2.1)
where the final arity is
n′
= niter = ℓµ (n − 1) + 1. (2.2)
1)A set with one closed binary operation without any other relations was called a groupoid by Hausmann and Ore [19] (see, also
[20]). Nowadays the term “groupoid” is widely used in the category theory and homotopy theory for a different construction, the
so-called Brandt groupoid [21]. Bourbaki [22] introduced the term “magma”. To avoid misreading we will use the neutral notation
“polyadic system”.
2)Sometimes µ
ℓµ
n is named a long product [3].
3. S. Duplij / Filomat xx (yyyy), zzz–zzz 3
There are many variants of placing µn’s among id’s in the r.h.s. of (2.1), if no associativity is assumed.
An example of the iterated product can be given for a ternary operation µ3 (n = 3), where we can
construct a 7-ary operation (n′
= 7) by ℓµ = 3 compositions
µ′
7
[
1, . . . , 7
]
= µ3
3
[
1, . . . , 7
]
= µ3
[
µ3
[
µ3
[
1, 2, 3
]
, 4, 5
]
, 6, 7
]
, (2.3)
and the corresponding commutative diagram is
G×7 µ3×id×4
- G×5 µ3×id×2
- G×3
HHHHHHHHHHH
µ′
7
=µ3
3
j
G
µ3
?
(2.4)
In the general case, the horizontal part of the (iterating) diagram (2.4) consists of ℓµ terms.
2. Reducing (Collapsing). To decrease arity from n to n′
= nred < n one can use nc distinguished elements
(“constants”) as additional 0-ary operations µ
(ci)
0
, i = 1, . . . nc, such that3)
the reduced product is defined
by
µ′
n′ = µ
(c1...cnc )
n′
def
= µn ◦
nc
µ
(c1)
0
× . . . × µ
(cnc )
0
× id×(n−nc)
, (2.5)
where
n′
= nred = n − nc, (2.6)
and the 0-ary operations µ
(ci)
0
can be on any places in (2.5). For instance, if we compose µn with the
0-ary operation µ
(c)
0
, we obtain
µ
(c)
n−1
[ ]
= µn
[
, c
]
, (2.7)
and this reduced product is described by the commutative diagram
G×(n−1)
× {•}
id×(n−1)
×µ
(c)
0- G×n
G×(n−1)
ϵ
6
µ
(c)
n−1 - G
µn
?
(2.8)
which can be treated as a definition of a new (n − 1)-ary operation µ
(c)
n−1
= µn ◦ µ
(c)
0
.
3. Mixing. Changing (increasing or decreasing) arity by combining the iterating and reducing (collaps-
ing) methods.
Example 2.2. If the initial multiplication is binary µ2 = (·), and there is one 0-ary operation µ
(c)
0
, we can construct
the following mixing operation
µ
(c)
n
[
1, . . . , n
]
= 1 · 2 · . . . · n · c, (2.9)
which in our notation can be called a c-iterated multiplication4)
.
3)In [23] µ
(c1...cnc )
n is called a retract, which is already a busy and widely used term in category theory for another construction.
4)According to [24] the operation (2.9) can be called c-derived.
4. S. Duplij / Filomat xx (yyyy), zzz–zzz 4
Let us recall some special elements of polyadic systems. A positive power of an element (according to
Post [4]) coincides with the number of multiplications ℓµ in the iteration (2.1).
Definition 2.3. A (positive) polyadic power of an element is
⟨ℓµ⟩ = µ
ℓµ
n
[
ℓµ(n−1)+1
]
. (2.10)
Example 2.4. Let us consider a polyadic version of the binary q-addition which appears in study of nonextensive
statistics (see, e.g., [25, 26])
µn
[ ]
=
n∑
i=1
i +
n∏
i=1
i, (2.11)
where i ∈ C and = 1 − q0, q0 is a real constant (we put here q0 1 or 0). It is obvious that ⟨0⟩
= , and
⟨1⟩
= µn
[
n−1
, ⟨0⟩
]
= n + n
. (2.12)
So we have the following recurrence formula
⟨k⟩
= µn
[
n−1
, ⟨k−1⟩
]
= (n − 1) +
(
1 + n−1
)
⟨k−1⟩
. (2.13)
Solving this for an arbitrary polyadic power we get
⟨k⟩
=
(
1 +
n − 1 1−n
) (
1 + n−1
)k
−
n − 1 2−n
. (2.14)
Definition 2.5. A polyadic (n-ary) identity (or neutral element) of a polyadic system is a distinguished element ε
(and the corresponding 0-ary operation µ
(ε)
0
) such that for any element ∈ G we have [27]
µn
[
, εn−1
]
= , (2.15)
where can be on any place in the l.h.s. of (2.15).
In polyadic systems, for an element there can exist many neutral polyads n ∈ G×(n−1)
satisfying
µn
[
, n
]
= , (2.16)
where may be on any place. The neutral polyads are not determined uniquely. It follows from (2.15) and
(2.16) that εn−1
is a neutral polyad.
Definition 2.6. An element of a polyadic system is called ℓµ-idempotent, if there exist such ℓµ that
⟨ℓµ⟩ = . (2.17)
It is obvious that an identity is ℓµ-idempotent with arbitrary ℓµ. We define (total) associativity as invariance
of the composition of two n-ary multiplications
µ2
n
[
, h, u
]
= invariant (2.18)
under placement of the internal multiplication in the r.h.s. with a fixed order of elements in the whole
polyad of (2n − 1) elements t(2n−1)
=
(
, h, u
)
. Informally, “internal brackets/multiplication can be moved on
any place”, which gives
µn ◦
(
i=1
µn × id×(n−1)
)
= µn ◦
(
id ×
i=2
µn × id×(n−2)
)
= . . . = µn ◦
(
id×(n−1)
×
i=n
µn
)
, (2.19)
where the internal µn can be on any place i = 1, . . . , n. There are many other particular kinds of associativity
which were introduced in [4, 28] and studied in [29, 30] (see, also [31]). Here we will confine ourselves to
the most general, total associativity (2.18).
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Definition 2.7. A polyadic semigroup (n-ary semigroup) is a n-ary system whose operation is associative, or
G
semi rp
n =
⟨
G | µn | associativity (2.18)
⟩
.
In general, it is very important to find the associativity preserving conditions, when an associative initial
operation µn leads to an associative final operation µ′
n′ while changing the arity (by iterating (2.1) or reducing
(2.5)).
Example 2.8. An associativity preserving reduction can be given by the construction of a binary associative operation
using a (n − 2)-tuple c as
µ
(c)
2
[
, h
]
= µn
[
, c, h
]
. (2.20)
The associativity preserving mixing constructions with different arities and places were considered in
[23, 30, 32].
In polyadic systems, there are several analogs of binary commutativity. The most straightforward one
comes from commutation of the multiplication with permutations.
Definition 2.9. A polyadic system is σ-commutative, if µn = µn ◦σ, where σ is a fixed element of Sn, the permutation
group on n elements. If this holds for all σ ∈ Sn, then a polyadic system is commutative.
A special type of the σ-commutativity
µn
[
, t, h
]
= µn
[
h, t,
]
(2.21)
is called semicommutativity. So for a n-ary semicommutative system we have
µn
[
, hn−1
]
= µn
[
hn−1
,
]
. (2.22)
If a n-ary semigroup G
semi rp
n is iterated from a commutative binary semigroup with identity, then G
semi rp
n
is semicommutative. Another possibility is to generalize the binary mediality in semigroups
(
11 · 12
)
·
(
21 · 22
)
=
(
11 · 21
)
·
(
12 · 22
)
, (2.23)
which follows from the binary commutativity. For n-ary systems, it is seen that the mediality should contain
(n + 1) multiplications, that it is a relation between n × n elements, and therefore that it can be presented in
a matrix from.
Definition 2.10. A polyadic system is medial (or entropic), if [33, 34]
µn
µn
[
11, . . . , 1n
]
...
µn
[
n1, . . . , nn
]
= µn
µn
[
11, . . . , n1
]
...
µn
[
1n, . . . , nn
]
. (2.24)
In the case of polyadic semigroups we use the notation (2.1) and can present the mediality as follows
µn
n [G] = µn
n
[
GT
]
, (2.25)
where G = ij is the n × n matrix of elements and GT
is its transpose.
The semicommutative polyadic semigroups are medial, as in the binary case, but, in general (except
n = 3) not vice versa [35].
Definition 2.11. A polyadic system is cancellative, if
µn
[
, t
]
= µn [h, t] =⇒ = h, (2.26)
where , h can be on any place. This means that the mapping µn is one-to-one in each variable.
6. S. Duplij / Filomat xx (yyyy), zzz–zzz 6
If , h are on the same i-th place on both sides of (2.26), the polyadic system is called i-cancellative. The
left and right cancellativity are 1-cancellativity and n-cancellativity respectively. A right and left cancellative
n-ary semigroup is cancellative (with respect to the same subset).
Definition 2.12. A polyadic system is called (uniquely) i-solvable, if for all polyads t, u and element h, one can
(uniquely) resolve the equation (with respect to h) for the fundamental operation
µn [u, h, t] = (2.27)
where h can be on any i-th place.
Definition 2.13. A polyadic system which is uniquely i-solvable for all places i = 1, . . . , n in (2.27) is called a n-ary
(or polyadic) quasigroup.
It follows, that, if (2.27) uniquely i-solvable for all places, then
µ
ℓµ
n [u, h, t] = (2.28)
can be (uniquely) resolved with respect to h being on any place.
Definition 2.14. An associative polyadic quasigroup is called a n-ary (or polyadic) group.
In a polyadic group the only solution of (2.27) is called a querelement5)
of and is denoted by ¯ [3], such
that
µn
[
h, ¯
]
= , (2.29)
where ¯ can be on any place. Obviously, any idempotent coincides with its querelement ¯ = .
Example 2.15. For the q-addition (2.11) from Example 2.4, using (2.29) with h = n−1
we obtain
¯ = −
(n − 2)
1 + n−1
. (2.30)
It follows from (2.29) and (2.16), that the polyad
n(¯) =
(
n−2
, ¯
)
(2.31)
is neutral for any element , where ¯ can be on any place. If this i-th place is important, then we write n( ),i.
More generally, because any neutral polyad plays a role of identity (see (2.16)), for any element we define
its polyadic inverse (the sequence of length (n − 2) denoted by the same letter −1
in bold) as (see [4] and by
modified analogy with [15, 36])
n( ) =
(
−1
,
)
=
(
, −1
)
, (2.32)
which can be written in terms of the multiplication as
µn
[
, −1
, h
]
= µn
[
h, −1
,
]
= h (2.33)
for all h in G. It is obvious that the polyads
n( k
) =
((
−1
)k
, k
)
=
(
k
,
(
−1
)k
)
(2.34)
5)We use the original notation after [3] and do not use “skew element”, because it can be confused with the wide usage of “skew”
in other, different senses.
7. S. Duplij / Filomat xx (yyyy), zzz–zzz 7
are neutral as well for any k ≥ 1. It follows from (2.31) that the polyadic inverse of is
(
n−3
, ¯
)
, and one of
¯ is
(
n−2
)
, and in this case is called querable. In a polyadic group all elements are querable [37, 38].
The number of relations in (2.29) can be reduced from n (the number of possible places) to only 2 (when
is on the first and last places [3, 39]), such that in a polyadic group the D¨ornte relations
µn
[
, n( ),i
]
= µn
[
n( ),j,
]
= (2.35)
hold valid for any allowable i, j, and (2.35) are analogs of · h · h−1
= h · h−1
· = in binary groups. The
relation (2.29) can be treated as a definition of the (unary) queroperation ¯µ1 : G → G by
¯µ1
[ ]
= ¯, (2.36)
such that the diagram
G×n µn
- G
Prn
3
G×n
id×(n−1)
× ¯µ1
6
(2.37)
commutes. Then, using the queroperation (2.36) one can give a diagrammatic definition of a polyadic group
(cf. [40]).
Definition 2.16. A polyadic group is a universal algebra
G
rp
n =
⟨
G | µn, ¯µ1 | associativity, D¨ornte relations
⟩
, (2.38)
where µn is n-ary associative operation and ¯µ1 is the queroperation (2.36), such that the following diagram
G×(n) id×(n−1)
× ¯µ1
- G×n ¯µ1×id×(n−1)
G×n
G × G
id ×Diag(n−1)
6
Pr1
- G
µn
?
Pr2
G × G
Diag(n−1)×id
6
(2.39)
commutes, where ¯µ1 can be only on the first and second places from the right (resp. left) on the left (resp. right) part
of the diagram.
A straightforward generalization of the queroperation concept and corresponding definitions can be
made by substituting in the above formulas (2.29)–(2.36) the n-ary multiplication µn by the iterating multi-
plication µ
ℓµ
n (2.1) (cf. [41] for ℓµ = 2 and [42]).
Let us define the querpower k of recursively by [43, 44]
¯⟨⟨k⟩⟩
=
(
¯⟨⟨k−1⟩⟩
)
, (2.40)
where ¯⟨⟨0⟩⟩
= , ¯⟨⟨1⟩⟩
= ¯, ¯⟨⟨2⟩⟩
= ¯,... or as the k composition ¯µ◦k
1
=
k
¯µ1 ◦ ¯µ1 ◦ . . . ◦ ¯µ1 of the unary
queroperation (2.36). We can define the negative polyadic power of an element by the recursive relationship
µn
[
⟨ℓµ−1⟩, n−2
, ⟨−ℓµ⟩
]
= , (2.41)
or (after the use of the positive polyadic power (2.10)) as a solution of the equation
µ
ℓµ
n
[
ℓµ(n−1)
, ⟨−ℓµ⟩
]
= . (2.42)
8. S. Duplij / Filomat xx (yyyy), zzz–zzz 8
The querpower (2.40) and the polyadic power (2.42) are connected [45]. We reformulate this connection
using the so called Heine numbers [46] or q-deformed numbers [47]
[[k]]q =
qk
− 1
q − 1
, (2.43)
which have the “nondeformed” limit q → 1 as [[k]]q → k and [[0]]q = 0. If [[k]]q = 0, then q is a k-th root of
unity. From (2.40) and (2.42) we obtain
¯⟨⟨k⟩⟩
= ⟨−[[k]]2−n⟩
, (2.44)
which can be treated as the following “deformation” statement:
Assertion 2.17. The querpower coincides with the negative polyadic deformed power with the “deformation” param-
eter q which is equal to the “deviation” (2 − n) from the binary group.
Example 2.18. Let us consider a binary group G2 =
⟨
G | µ2
⟩
, we denote µ2 = (·), and construct (using (2.1) and
(2.5)) the reduced 4-ary product by µ′
4
[ ]
= 1 · 2 · 3 · 4 ·c, where i ∈ G and c is in the center of the group G2. In the
4-ary group G′
4 =
⟨
G, µ′
4
⟩
we derive the following positive and negative polyadic powers (obviously ⟨0⟩
= ¯⟨⟨0⟩⟩
= )
⟨1⟩
= 4
· c, ⟨2⟩
= 7
· c2
, . . . , ⟨k⟩
= 3k+1
· ck
, (2.45)
⟨−1⟩
= −2
· c−1
, ⟨−2⟩
= −5
· c−2
, . . . , ⟨−k⟩
= −3k+1
· c−k
, (2.46)
and the querpowers
¯⟨⟨1⟩⟩
= −2
· c−1
, ¯⟨⟨2⟩⟩
= −4
· c, . . . , ¯⟨⟨k⟩⟩
= (−2)k
· c[[k]]−2
. (2.47)
Let Gn =
⟨
G | µn
⟩
and G′
n′ =
⟨
G′
| µ′
n′
⟩
be two polyadic systems of any kind. If their multiplications are
of the same arity n = n′
, then one can define the following one-place mappings from Gn to G′
n (for many-place
mappings, which change arity n n′
and corresonding heteromorphisms, see [17]).
Suppose we have n + 1 mappings Φi : G → G′
, i = 1, . . . , n + 1. An ordered system of mappings {Φi} is
called a homotopy from Gn to G′
n, if (see, e.g., [34])
Φn+1
(
µn
[
1, . . . , n
])
= µ′
n
[
Φ1
(
1
)
, . . . , Φn
(
n
)]
, i ∈ G. (2.48)
A homomorphism from Gn to G′
n is given, if there exists a (one-place) mapping Φ : G → G′
satisfying
Φ
(
µn
[
1, . . . , n
])
= µ′
n
[
Φ
(
1
)
, . . . , Φ
(
n
)]
, i ∈ G, (2.49)
which means that the corresponding (equiary6)
) diagram is commutative
G
φ
- G′
G×n
µn
6
(φ)
×n
- (G′
)×n
µ′
n
6
(2.50)
It is obvious that, if a polyadic system contains distinguished elements (identities, querelements, etc.), they
are also mapped by φ correspondingly (for details and a review, see, e.g., [42, 48]). The most important
application of one-place mappings is in establishing a general structure for n-ary multiplication.
6)The map is equiary, if it does not change the arity of operations i.e. n = n′, for nonequiary maps see [17] and refs. therein.
9. S. Duplij / Filomat xx (yyyy), zzz–zzz 9
3. The Hossz´u-Gluskin theorem
Let us consider possible concrete forms of polyadic multiplication in terms of lesser arity operations.
Obviously, the simplest way of constructing a n-ary product µ′
n from the binary one µ2 = (∗) is ℓµ = n
iteration (2.1) [8, 49]
µ′
n
[ ]
= 1 ∗ 2 ∗ . . . ∗ n, i ∈ G. (3.1)
In [3] it was noted that not all n-ary groups have a product of this special form. The binary group
G∗
2 =
⟨
G | µ2 = ∗, e
⟩
was called a covering group of the n-ary group G′
n =
⟨
G | µ′
n
⟩
in [4] (see, also, [50]), where
a theorem establishing a more general (than (3.1)) structure of µ′
n
[ ]
in terms of subgroup structure of the
covering group was given. A manifest form of the n-ary group product µ′
n
[ ]
in terms of the binary one
and a special mapping was found in [11, 13] and is called the Hossz´u-Gluskin theorem, despite the same
formulas having appeared much earlier in [4, 51] (for the relationship between the formulations, see [52]).
A simple construction of µ′
n
[ ]
which is present in the Hossz´u-Gluskin theorem was given in [16]. Here
we follow this scheme in the opposite direction, by just deriving the final formula step by step (without
writing it immediately) with clear examples. Then we introduce a “deformation” to it in such a way that a
generalized “q-deformed” Hossz´u-Gluskin theorem can be formulated.
First, let us rewrite (3.1) in its equivalent form
µ′
n
[ ]
= 1 ∗ 2 ∗ . . . ∗ n ∗ e, i, e ∈ G, (3.2)
where e is a distinguished element of the binary group ⟨G | ∗, e⟩, that is the identity. Now we apply to (3.2)
an “extended” version of the homotopy relation (2.48) with Φi = ψi, i = 1, . . . n, and the l.h.s. mapping
Φn+1 = id, but add an action ψn+1 on the identity e of the binary group ⟨G | ∗, e⟩. Then we get (see (2.7) and
(2.9))
µn
[ ]
= µ
(e)
n
[ ]
= ψ1
(
1
)
∗ ψ2
(
2
)
∗ . . . ∗ ψn
(
n
)
∗ ψn+1 (e) =
∗
n∏
i=1
ψi
(
i
)
∗ ψn+1 (e) . (3.3)
In this way we have obtained the most general form of polyadic multiplication in terms of (n + 1)
“extended” homotopy maps ψi, i = 1, . . . n + 1, such that the diagram
G×(n)
× {•}
id×n
×µ
(e)
0- G×(n+1) ψ1×...×ψn+1
- G×(n+1)
G×(n)
ϵ
6
µ
(e)
n
- G
µ×n
2
?
(3.4)
commutes. A natural question arises, whether all associative polyadic systems have this form of multipli-
cation or do we have others? In general, we can correspondingly classify polyadic systems as:
1) Homotopic polyadic systems which can be presented in the form (3.3). (3.5)
2) Nonhomotopic polyadic systems with multiplication of other than (3.3) shapes. (3.6)
If the second class is nonempty, it would be interesting to find examples of nonhomotopic polyadic systems.
The Hossz´u-Gluskin theorem considers the homotopic polyadic systems and gives one of the possible
choices for the “extended” homotopy maps ψi in (3.3). We will show that this choice can be extended
(“deformed”) to the infinite “q-series”.
The main idea in constructing the “automatically” associative n-ary operation µn in (3.3) is to express
the binary multiplication (∗) and the “extended” homotopy maps ψi in terms of µn itself [16]. A simplest
binary multiplication which can be built from µn is (see (2.20))
∗t h = µn
[
, t, h
]
, (3.7)
10. S. Duplij / Filomat xx (yyyy), zzz–zzz 10
where t is any fixed polyad of length (n − 2). If we apply here the equations for the identity e in a binary
group
∗t e = , e ∗t h = h, (3.8)
then we obtain
µn
[
, t, e
]
= , µn [e, t, h] = h. (3.9)
We observe from (3.9) that (t, e) and (e, t) are neutral sequences of length (n − 1), and therefore using
(2.32) we can take t as a polyadic inverse of e (the identity of the binary group) considered as an element (but
not an identity) of the polyadic system
⟨
G | µn
⟩
, that is t = e−1
. Then, the binary multiplication constructed
from µn and which has the standard identity properties (3.8) can be chosen as
∗ h = ∗e h = µn
[
, e−1
, h
]
. (3.10)
Using this construction any element of the polyadic system
⟨
G | µn
⟩
can be distinguished and may serve as
the identity of the binary group, and is then denoted by e (for clarity and convenience).
We recognize in (3.10) a version of the Maltsev term (see, e.g., [18]), which can be called a polyadic Maltsev
term and is defined as
p
(
, e, h
) def
= µn
[
, e−1
, h
]
(3.11)
having the standard term properties [18]
p
(
, e, e
)
= , p (e, e, h) = h, (3.12)
which now follow from (3.9), i.e. the polyads
(
e, e−1
)
and
(
e−1
, e
)
are neutral, as they should be (2.32). Denote
by −1
the inverse element of in the binary group ( ∗ −1
= −1
∗ = e) and −1
its polyadic inverse in a
n-ary group (2.32), then it follows from (3.10) that µn
[
, e−1
, −1
]
= e. Thus, we get
−1
= µn
[
e, −1
, e
]
, (3.13)
which can be considered as a connection between the inverse −1
in the binary group and the polyadic
inverse in the polyadic system related to the same element . For n-ary group we can write −1
=
(
n−3
, ¯
)
and the binary group inverse −1
becomes
−1
= µn
[
e, n−3
, ¯, e
]
. (3.14)
If
⟨
G | µn
⟩
is a n-ary group, then the element e is querable (2.33), for the polyadic inverse e−1
one can choose(
en−3
, ¯e
)
with ¯e being on any place, and the polyadic Maltsev term becomes [53] p
(
, e, h
)
= µn
[
, en−3
, ¯e, h
]
(together with the multiplication (3.10)). For instance, if n = 3, we have
∗ h = µ3
[
, ¯e, h
]
, −1
= µ3
[
e, ¯, e
]
, (3.15)
and the neutral polyads are (e, ¯e) and (¯e, e).
Now let us turn to build the main construction, that of the “extended” homotopy maps ψi (3.3) in terms
of µn, which will lead to the Hossz´u-Gluskin theorem. We start with a simple example of a ternary system
(3.15), derive the Hossz´u-Gluskin “chain formula”, and then it will be clear how to proceed for generic n.
Instead of (3.3) we write
µ3
[
, h, u
]
= ψ1
( )
∗ ψ2 (h) ∗ ψ3 (u) ∗ ψ4 (e) (3.16)
11. S. Duplij / Filomat xx (yyyy), zzz–zzz 11
and try to construct ψi in terms of the ternary product µ3 and the binary identity e. We already know the
structure of the binary multiplication (3.15): it contains ¯e, and therefore we can insert between , h and u
in the l.h.s. of (3.16) a neutral ternary polyad (¯e, e) or its powers
(
¯ek
, ek
)
. Thus, taking for all insertions the
minimal number of neutral polyads, we get
µ3
[
, h, u
]
= µ2
3
,
∗
↓
¯e , e, h, u
= µ4
3
,
∗
↓
¯e , e, h, ¯e,
∗
↓
¯e , e, e, u
= µ7
3
,
∗
↓
¯e , e, h, ¯e,
∗
↓
¯e , e, e, u, ¯e, ¯e,
∗
↓
¯e , e, e, e
. (3.17)
We show by arrows the binary products in special places: there should be 1, 3, 5, . . . (2k − 1) elements in
between them to form inner ternary products. Then we rewrite (3.17) as
µ3
[
, h, u
]
= µ3
3
,
∗
↓
¯e , µ3 [e, h, ¯e] ,
∗
↓
¯e , µ2
3 [e, e, u, ¯e, ¯e] ,
∗
↓
¯e , µ3 [e, e, e]
. (3.18)
Comparing this with (3.16), we can exactly identify the “extended” homotopy maps ψi as
ψ1
( )
= , (3.19)
ψ2
( )
= φ
( )
, (3.20)
ψ3
( )
= φ
(
φ
( ))
= φ2 ( )
, (3.21)
ψ4 (e) = µ3 [e, e, e] , (3.22)
where
φ
( )
= µ3
[
e, , ¯e
]
, (3.23)
which can be described by the commutative diagram
{•} × G × {•}
µ
(e)
0
×id ×µ
(e)
0- G×3 id×2
× ¯µ1
- G×3
G
ϵ
6
φ
- G
µ3
?
(3.24)
The mapping ψ4 is the first polyadic power (2.10) of the binary identity e in the ternary system
ψ4 (e) = e⟨1⟩
. (3.25)
Thus, combining (3.18)–(3.25) we obtain the Hossz´u-Gluskin “chain formula” for n = 3
µ3
[
, h, u
]
= ∗ φ (h) ∗ φ2
(u) ∗ b, (3.26)
b = e⟨1⟩
, (3.27)
12. S. Duplij / Filomat xx (yyyy), zzz–zzz 12
which depends on one mapping φ (taken in the chain of powers) only, and the first polyadic power e⟨1⟩
of
the binary identity e. The corresponding Hossz´u-Gluskin diagram
G×3
× {•}3
id ×φ×φ2
×
(
µ
(e)
0
)×3
- G×6 id×3
×µ3
- G×4
G × G × G
ϵ
6
µ3
- G
µ×3
2
?
(3.28)
commutes.
The mapping φ is an automorphism of the binary group ⟨G | ∗, e⟩, because it follows from (3.15) and
(3.23) that
φ
( )
∗ φ (h) = µ3
[
µ3
[
e, , ¯e
]
, ¯e, µ3 [e, h, ¯e]
]
= µ3
3
[
e, , ¯e,
neutral
(¯e, e) , h, ¯e
]
= µ2
3
[
e, , ¯e, h, ¯e
]
= µ3
[
e, ∗ h, ¯e
]
= φ
(
∗ h
)
, (3.29)
φ (e) = µ3 [e, e, ¯e] = µ3
[
e,
neutral
(e, ¯e)
]
= e. (3.30)
It is important to note that not only the binary identity e, but also its first polyadic power e⟨1⟩
is a fixed point
of the automorphism φ, because
φ
(
e⟨1⟩
)
= µ3
[
e, e⟨1⟩
, ¯e
]
= µ2
3
[
e, e, e,
neutral
(e, ¯e)
]
= µ3 [e, e, e] = e⟨1⟩
. (3.31)
Moreover, taking into account that in the binary group (see (3.15))
(
e⟨1⟩
)−1
= µ3
[
e, e⟨1⟩, e
]
= µ2
3 [e, ¯e, ¯e, ¯e, e] = ¯e, (3.32)
we get
φ2 ( )
= µ2
3
[
e, e, , ¯e, ¯e
]
= µ2
3
[
e, e,
neutral
(e, ¯e) , ¯e, ¯e
]
= e⟨1⟩
∗ ∗
(
e⟨1⟩
)−1
. (3.33)
The higher polyadic powers e⟨k⟩
= µk
3
[
e2k+1
]
of the binary identity e are obviously also fixed points
φ
(
e⟨k⟩
)
= e⟨k⟩
. (3.34)
The elements e⟨k⟩
form a subgroup H of the binary group ⟨G | ∗, e⟩, because
e⟨k⟩
∗ e⟨l⟩
= e⟨k+l⟩
, (3.35)
e⟨k⟩
∗ e = e ∗ e⟨k⟩
= e⟨k⟩
. (3.36)
We can express the even powers of the automorphism φ through the polyadic powers e⟨k⟩
in the following
way
φ2k ( )
= e⟨k⟩
∗ ∗
(
e⟨k⟩
)−1
. (3.37)
This gives a manifest connection between the Hossz´u-Gluskin “chain formula” and the sequence of cosets
(see, [4]) for the particular case n = 3.
13. S. Duplij / Filomat xx (yyyy), zzz–zzz 13
Example 3.1. Let us consider the ternary copula associative multiplication [54, 55]
µ3
[
, h, u
]
=
(1 − h)u
(1 − h)u + (1 − )h(1 − u)
, (3.38)
where i ∈ G = [0, 1] and 0/0 = 0 is assumed7)
. It is associative and cannot be iterated from any binary group.
Obviously, µ3
[
3
]
= , and therefore this polyadic system is ℓµ-idempotent (2.17) ⟨ℓµ⟩ = . The querelement is
¯ = ¯µ1
[ ]
= . Because each element is querable, then
⟨
G | µ3, ¯µ1
⟩
is a ternary group. Take a fixed element e ∈ [0, 1].
We define the binary multiplication as ∗ h = µ3
[
, e, h
]
and the automorphism
φ
( )
= µ3
[
e, , e
]
= e2 1 −
e2 − 2 e +
(3.39)
which has the property φ2k
= id and φ2k+1
= φ, where k ∈ N. Obviously, in (3.39) can be on any place in the
product µ3
[
e, , e
]
= µ3
[
e, e,
]
= µ3
[
e, e,
]
. Now we can check the Hossz´u-Gluskin “chain formula” (3.26) for the
ternary copula
µ3
[
, h, u
]
=
(((
∗ φ (h)
)
∗ u
)
∗ e
)
= µ•
3
[
, e, e2 1 − h
e2 − 2he +
, e, (u, e, e)
]
= µ•
3
[
,
(
e, e2 1 − h
e2 − 2he +
, e
)
, u
]
= µ3
[
, φ2
(h) , u
]
= µ3
[
, h, u
]
. (3.40)
The language of polyadic inverses allows us to generalize the Hossz´u-Gluskin “chain formula” from
n = 3 (3.26) to arbitrary n in a clear way. The derivation coincides with (3.18) using the multiplication (3.10)
(with substitution ¯e → e−1
), neutral polyads
(
e−1
, e
)
or their powers
((
e−1
)k
, ek
)
, but contains n terms
µn
[
1, . . . , n
]
= µ•
n
1,
∗
↓
e−1
, e, 2, . . . , n
= µ•
n
1,
∗
↓
e−1
, e, 2, e−1
,
∗
↓
e−1
, e, e, 3, . . . , n
= . . .
= µ•
n
1,
∗
↓
e−1
, e, 2, e−1
,
∗
↓
e−1
, e, e, 3, . . . ,
∗
↓
e−1
,
n−1
e, . . . , e, n,
n−1
e−1
, . . . , e−1
,
∗
↓
e−1
,
n
e, . . . , e
. (3.41)
We observe from (3.41) that the mapping φ in the n-ary case is
φ
( )
= µn
[
e, , e−1
]
, (3.42)
and the last product of the binary identities µn [e, . . . , e] is also the first n-ary power e⟨1⟩
(2.10). It follows
from (3.42) and (3.10), that
φn−1 ( )
= e⟨1⟩
∗ ∗
(
e⟨1⟩
)−1
. (3.43)
In this way, we obtain the Hossz´u-Gluskin “chain formula” for arbitrary n
µn
[
1, . . . , n
]
= 1 ∗ φ
(
2
)
∗ φ2 (
3
)
∗ . . . ∗ φn−2 (
n−1
)
∗ φn−1 (
n
)
∗ e⟨1⟩
=
∗
n∏
i=1
φi−1 (
i
)
∗ e⟨1⟩
. (3.44)
7)In this example all denominators are supposed nonzero.
14. S. Duplij / Filomat xx (yyyy), zzz–zzz 14
Thus, we have found the “extended” homotopy maps ψi from (3.3) as
ψi
( )
= φi−1 ( )
, i = 1, . . . , n, (3.45)
ψn+1
( )
= ⟨1⟩
, (3.46)
where we put by definition φ0
( )
= . Using (3.31) and (3.44) we can formulate the Hossz´u-Gluskin
theorem in the language of polyadic powers.
Theorem 3.2. On a polyadic group Gn =
⟨
G | µn, ¯µ1
⟩
one can define a binary group G∗
2 =
⟨
G | µ2 = ∗, e
⟩
and its
automorphism φ such that the Hossz´u-Gluskin “chain formula” (3.44) is valid, where the polyadic powers of the
identity e are fixed points of φ (3.34), form a subgroup H of G∗
2, and the (n − 1) power of φ is a conjugation (3.43)
with respect to H.
The following reverse Hossz´u-Gluskin theorem holds.
Theorem 3.3. If in a binary group G∗
2 =
⟨
G | µ2 = ∗, e
⟩
one can define an automorphism φ such that
φn−1 ( )
= b ∗ ∗ b−1
, (3.47)
φ (b) = b, (3.48)
where b ∈ G is a distinguished element, then the “chain formula”
µn
[
1, . . . , n
]
=
∗
n∏
i=1
φi−1 (
i
)
∗ b (3.49)
determines a n-ary group, in which the distinguished element is the first polyadic power of the binary identity
b = e⟨1⟩
. (3.50)
4. “Deformation” of Hossz´u-Gluskin chain formula
Let us raise the question: can the choice (3.45)-(3.46) of the “extended” homotopy maps (3.3) be gener-
alized? Before answering this question positively we consider some preliminary statements.
First, we note that we keep the general idea of inserting neutral sequences into a polyadic product (see
(3.17) and (3.41)), because this is the only way to obtain “automatic” associativity. Second, the number of
the inserted neutral polyads can be chosen arbitrarily, not only minimally, as in (3.17) and (3.41) (as they are
neutral). Nevertheless, we can show that this arbitrariness is somewhat restricted.
Indeed, let us consider a polyadic group
⟨
G | µn, ¯µ1
⟩
in the particular case n = 3, where for any e0 ∈ G
and natural k the sequence
(
¯ek
0
, ek
0
)
is neutral, then we can write
µ3
[
, h, u
]
= µ•
3
[
, ¯ek
0, ek
0, h, ¯elk
0 , elk
0 , u, ¯emk
0 , emk
0
]
. (4.1)
If we make the change of variables ek
0
= e, then we obtain
µ3
[
, h, u
]
= µ•
3
[
, ¯e, e, h, ¯el
, el
, u, ¯em
, em
]
. (4.2)
Because this should reproduce the formula (3.16), we immediately conclude that ψ1
( )
= id, and the
multiplication is the same as in (3.15), and e is again the identity of the binary group G∗
= ⟨G, ∗, e⟩.
Moreover, if we put ψ2
( )
= φ
( )
, as in the standard case, then we have a first “half” of the mapping φ,
that is φ
( )
= µ3
[
e, h, something
]
. Now we are in a position to find this “something” and other “extended”
homotopy maps ψi from (3.16), but without the requirement of a minimal number of inserted neutral
polyads, as it was in (3.17). By analogy, we rewrite (4.2) as
µ3
[
, h, u
]
= µ•
3
[
, ¯e, (e, h, ¯eq
) , ¯e, eq+1
, u, ¯em
, em
]
, (4.3)
15. S. Duplij / Filomat xx (yyyy), zzz–zzz 15
where we put l = q + 1. So we have found the “something”, and the map φ is
φq
( )
= µ
ℓφ(q)
3
[
e, , ¯eq]
, (4.4)
where the number of multiplications
ℓφ
(
q
)
=
q + 1
2
(4.5)
is an integer ℓφ
(
q
)
= 1, 2, 3 . . ., while q = 1, 3, 5, 7 . . .. The diagram defined φq (e.g., for q = 3 and ℓφ
(
q
)
= 2)
{•} × G × {•}3
µ
(e)
0
×id ×
(
µ
(e)
0
)3
- G×5
id×2
×(¯µ1)
3
- G×5
G
ϵ
6
φq
- G
µ3×µ3
?
(4.6)
commutes (cf. (3.24)). Then, we can find power m in (4.3)
µ3
[
, h, u
]
= µ•
3
[
, ¯e, (e, h, ¯eq
) , ¯e, (e, u, ¯eq
)q+1
, ¯e, eq(q+1)+1
]
, (4.7)
and therefore m = q
(
q + 1
)
+ 1. Thus, we have obtained the “q-deformed” maps ψi (cf. (3.19)–(3.22))
ψ1
( )
= φ
[[0]]q
q
( )
= φ0
q
( )
= , (4.8)
ψ2
( )
= φq
( )
= φ
[[1]]q
q
( )
, (4.9)
ψ3
( )
= φ
q+1
q
( )
= φ
[[2]]q
q
( )
, (4.10)
ψ4
( )
= µ•
3
[
q(q+1)+1
]
= µ•
3
[
[[3]]q
]
, (4.11)
where φ is defined by (4.4) and [[k]]q is the q-deformed number (2.43), and we put φ0
q = id. The corresponding
“q-deformed” chain formula (for n = 3) can be written as (cf. (3.26)–(3.27) for “nondeformed” case)
µ3
[
, h, u
]
= ∗ φ
[[1]]q
q (h) ∗ φ
[[2]]q
q (u) ∗ bq, (4.12)
bq = e⟨ℓe(q)⟩, (4.13)
where the degree of the binary identity polyadic power
ℓe
(
q
)
= q
[[2]]q
2
= ℓφ
(
q
) (
2ℓφ
(
q
)
+ 1
)
(4.14)
is an integer. The corresponding “deformed” chain diagram (e.g., for q = 3)
G×3
× {•}13
id ×φq×φ4
q×
(
µ
(e)
0
)×13
- G×16
id×3
×µ6
3- G×4
G × G × G
ϵ
6
µ3
- G
µ×3
2
?
(4.15)
commutes (cf. the Hossz´u-Gluskin diagram (3.28)). In the “deformed” case the polyadic power e⟨ℓe(q)⟩ is
not a fixed point of φq and satisfies
φq
(
e⟨ℓe(q)⟩
)
= φq
(
µ•
3
[
eq2
+q+1
])
= µ•
3
[
eq2
+2
]
= e⟨ℓe(q)⟩ ∗ φq (e) (4.16)
16. S. Duplij / Filomat xx (yyyy), zzz–zzz 16
or
φq
(
bq
)
= bq ∗ φq (e) . (4.17)
Instead of (3.33) we have
φ
q+1
q
( )
∗ e⟨ℓe(q)⟩ = µ•
3
[
eq+1
,
]
= µ•
3
[
eq+2
]
∗ = e⟨ℓe(q)⟩ ∗ φ
q+1
q (e) ∗ (4.18)
or
φ
q+1
q
( )
∗ bq = bq ∗ φ
q+1
q (e) ∗ . (4.19)
The “nondeformed” limit q → 1 of (4.12) gives the Hossz´u-Gluskin chain formula (3.26) for n = 3.
Now let us turn to arbitrary n and write the n-ary multiplication using neutral polyads analogously to
(4.3). By the same arguments, as in (4.2), we insert only one neutral polyad
(
e−1
, e
)
between the first and
second elements in the multiplication, but in other places we insert powers
((
e−1
)k
, ek
)
(allowed by the chain
properties), and obtain
µn
[
1, . . . , n
]
= µ•
n
[
1, e−1
, e, 2, . . . , n
]
= µ•
n
[
1, e−1
,
(
e, 2,
(
e−1
)q)
, e−1
, eq+1
, 3, . . . , n
]
= . . .
= µ•
n
1, e−1
,
(
e, 2,
(
e−1
)q)
, e−1
,
eq+1
, 3,
q(q+1)
e−1
, . . . , e−1
e−1
, eq(q+1)+1
, 3, . . .
. . . ,
qn−2
+...+q+1
e, . . . , e , n−1,
q(qn−2
+...+q+1)
e−1
, . . . , e−1
, e−1
,
qn−1
+...+q+1
e, . . . , e , n,
q(qn−1
+...+q+1)
e−1
, . . . , e−1
, e−1
,
qn
+...+q+1
e, . . . , e
. (4.20)
So we observe that the binary product is now the same as in the “nondeformed” case (3.10), while the map
φ is
φq
( )
= µ
ℓφ(q)
n
[
e, ,
(
e−1
)q]
, (4.21)
where the number of multiplications
ℓφ
(
q
)
=
q (n − 2) + 1
n − 1
(4.22)
is an integer and ℓφ
(
q
)
→ q, as n → ∞, in the nondeformed case ℓφ (1) = 1, as in (3.42). Note that the
“deformed” map φq is the a-quasi-endomorphism [56] of the binary group G∗
2, because from (4.21) we get
φq
( )
∗ φq (h) = µ•
n
[
e, ,
(
e−1
)q
, e−1
, e, h,
(
e−1
)q]
= µ•
n
[
e, , e−1
,
(
e, e,
(
e−1
)q)
, e−1
, h,
(
e−1
)q]
= φq
(
∗ a ∗ h
)
, (4.23)
where
a = µ
ℓφ(q)
n
[
e, e,
(
e−1
)q]
= φq (e) . (4.24)
In general, a quasi-endomorphism can be defined by
φq
( )
∗ φq (h) = φq
(
∗ φq (e) ∗ h
)
. (4.25)
17. S. Duplij / Filomat xx (yyyy), zzz–zzz 17
The corresponding diagram
G × G
µ2
- G φq
G
G × G
φq×φq
6
ϵ
- G × {•} × G
id ×µ
(e)
0
×id
- G × G × G
µ2×µ2
6
(4.26)
commutes. If q = 1, then φq (e) = e, and the distinguished element a turns to the binary identity a = e, such
that the a-quasi-endomorphism φq becomes an automorphism of G∗
2.
Remark 4.1. The choice (4.21) of the a-quasi-endomorphism φq is different from [56], the latter (in our notation)
is φk
( )
= µn
[
ak−1
, , an−k
]
, k = 1, . . . , n − 1, it has only one multiplication and leads to the “nondeformed” chain
formula (3.44) (for semigroup case).
It follows from (4.20), that the “extended” homotopy maps ψi (3.3) are (cf. (4.8)–(4.11))
ψ1
( )
= φ
[[0]]q
q
( )
= φ0
q
( )
= , (4.27)
ψ2
( )
= φq
( )
= φ
[[1]]q
q
( )
, (4.28)
ψ3
( )
= φ
q+1
q
( )
= φ
[[2]]q
q
( )
, (4.29)
...
ψn−1
( )
= φ
qn−3
+...+q+1
q
( )
= φ
[[n−2]]q
q
( )
, (4.30)
ψn
( )
= φ
qn−2
+...+q+1
q
( )
= φ
[[n−1]]q
q
( )
, (4.31)
ψn+1
( )
= µ•
n
[
qn−1
+...+q+1
]
= µ•
n
[
[[n]]q
]
. (4.32)
In terms of the polyadic power (2.10), the last map is
ψn+1
( )
= ⟨ℓe⟩
, (4.33)
where (cf. (4.22))
ℓe
(
q
)
= q
[[n − 1]]q
n − 1
(4.34)
is an integer. Thus the “q-deformed” n-ary chain formula is (cf. (3.44))
µn
[
1, . . . , n
]
= 1 ∗ φ
[[1]]q
q
(
2
)
∗ φ
[[2]]q
q
(
3
)
∗ . . . ∗ φ
[[n−2]]q
q
(
n−1
)
∗ φ
[[n−1]]q
q
(
n
)
∗ e⟨ℓe(q)⟩. (4.35)
In the “nondeformed” limit q → 1 (4.35) reproduces the Hossz´u-Gluskin chain formula (3.44). Let us
obtain the “deformed” analogs of the distinguished element relations (3.47)–(3.48) for arbitrary n (the case
n = 3 is in (4.16)–(4.18)). Instead of the fixed point relation (3.48) we now have from (4.21), (4.34) and (4.32)
the quasi-fixed point
φq
(
bq
)
= bq ∗ φq (e) , (4.36)
where the “deformed” distinguished element bq is (cf. (3.50))
bq = µ•
n
[
e[[n]]q
]
= e⟨ℓe(q)⟩. (4.37)
The conjugation relation (3.47) in the “deformed” case becomes the quasi-conjugation
φ
[[n−1]]q
q
( )
∗ bq = bq ∗ φ
[[n−1]]q
q (e) ∗ . (4.38)
18. S. Duplij / Filomat xx (yyyy), zzz–zzz 18
This allows us to rewrite the “deformed” chain formula (4.35) as
µn
[
1, . . . , n
]
= 1 ∗ φ
[[1]]q
q
(
2
)
∗ φ
[[2]]q
q
(
3
)
∗ . . . ∗ φ
[[n−2]]q
q
(
n−1
)
∗ bq ∗ φ
[[n−1]]q
q (e) ∗ n. (4.39)
Using the above proof sketch, we formulate the following “q-deformed” analog of the Hossz´u-Gluskin
theorem:
Theorem 4.2. On a polyadic group Gn =
⟨
G | µn, ¯µ1
⟩
one can define a binary group G∗
2 =
⟨
G | µ2 = ∗, e
⟩
and (the
infinite “q-series” of) its automorphism φq such that the “deformed” chain formula (4.35) is valid
µn
[
1, . . . , n
]
=
∗
n∏
i=1
φ[[i−1]]q
(
i
)
∗ bq, (4.40)
where (the infinite “q-series” of) the “deformed” distinguished element bq (being a polyadic power of the binary identity
(4.37)) is the quasi-fixed point of φq (4.36) and satisfies the quasi-conjugation (4.38) in the form
φ
[[n−1]]q
q
( )
= bq ∗ φ
[[n−1]]q
q (e) ∗ ∗ b−1
q . (4.41)
In the “nondeformed” case q = 1 we obtain the Hossz´u-Gluskin chain formula (3.44) and the corre-
sponding Theorem 3.2.
Example 4.3. Let us have a binary group ⟨G | (·) , 1⟩ and a distinguished element e ∈ G, e 1, then we can define a
binary group G∗
2 = ⟨G | (∗) , e⟩ by the product
∗ h = · e−1
· h. (4.42)
The quasi-endomorphism
φq
( )
= e · · e−q
(4.43)
satisfies (4.25) with φq (e) = e2−q
, and we take
bq = e[[n]]q
. (4.44)
Then we can obtain the “q-deformed” chain formula (4.40) (for q = 1 see, e.g., [52]).
We observe that the chain formula is the “q-series” of equivalence relations (4.40), which can be for-
mulated as an invariance. Indeed, let us denote the r.h.s. of (4.40) by Mq
(
1, . . . , n
)
, and the l.h.s. as
M0
(
1, . . . , n
)
, then the chain formula can be written as some invariance (cf. associativity as an invariance
(2.18)).
Theorem 4.4. On a polyadic group Gn =
⟨
G | µn, ¯µ1
⟩
we can define a binary group G∗
=
⟨
G | µ2 = ∗, e
⟩
such that
the following invariance is valid
Mq
(
1, . . . , n
)
= invariant, q = 0, 1, . . . , (4.45)
where
Mq
(
1, . . . , n
)
=
µn
[
1, . . . , n
]
, q = 0,
∗
n∏
i=1
φ[[i−1]]q
(
i
)
∗ bq, q 0,
(4.46)
and the distinguished element bq ∈ G and the quasi-endomorphism φq of G∗
2 are defined in (4.37) and (4.21)
respectively.
19. S. Duplij / Filomat xx (yyyy), zzz–zzz 19
Example 4.5. Let us consider the ternary q-product used in the nonextensive statistics [26]
µ3
[
, t, u
]
=
(
+ t + u − 3
)1
, (4.47)
where = 1 − q0, and , t, u ∈ G = R+, 0 q0 1, and also + t + u − 3 0 (as for other terms inside brackets
with power 1
below). In case → 0 the q-product becomes an iterated product in R+ as µ3
[
, t, u
]
→ tu. The
quermap ¯µ1 is given by
¯ =
(
3 −
)1
. (4.48)
The polyadic system Gn =
⟨
G | µ3, ¯µ1
⟩
is a ternary group, because each element is querable. Take a distinguished
element e ∈ G and use (3.15), (4.47) and (4.48) to define the product
∗ t =
(
− e + t
)1
(4.49)
of the binary group G∗
2 =
⟨
G | µ2 = (∗) , e
⟩
.
1) The Hossz´u-Gluskin chain formula (q = 1). The automorphism (3.23) of G∗
is now the identity map φ = id.
The first polyadic power of the distinguished element e is
b = e⟨1⟩
= µ3
[
e3
]
=
(
3e − 3
)1
. (4.50)
The chain formula (3.26) can be checked as follows
µ3
[
, t, u
]
=
(((
∗ t
)
∗ u
)
∗ b
)
=
(((
− e + t
)
− e + u
)
− e + b
)1
=
(
− e + t − e + u − e + 3e − 3
)1
=
(
+ t + u − 3
)1
. (4.51)
2) The “q-deformed” chain formula (for conciseness we consider only the case q = 3). Now the quasi-endomorphism
φq (4.4) is not the identity, but is
φq=3
( )
=
(
− 2e + 3
)1
. (4.52)
In case q = 3 we need its 4th (= q + 1) power (4.12)
φ4
q=3
( )
=
(
− 8e + 12
)1
. (4.53)
The deformed polyadic power e⟨ℓe⟩
from (4.12) is (see, also, (4.11))
bq=3 = e⟨5⟩
= µ5
3
[
e13
]
=
(
13e − 18
)1
. (4.54)
Now we check the “q-deformed” chain formula (4.12) as
µ3
[
, t, u
]
= ∗ φq=3 (t) ∗ φ4
q=3 (u) ∗ bq=3 =
(((
∗ φq=3 (t)
)
∗ φ4
q=3 (u)
)
∗ bq=3
)
(4.55)
=
(
− e +
(
t − 2e + 3
)
− e +
(
u − 8e + 12
)
− e +
(
13e − 18
))1
(4.56)
=
(
+ t + u − 3
)1
. (4.57)
In a similar way, one can check the “q-deformed” chain formula for any allowed q (determined by (4.22) and (4.34)
to obtain an infinite q-series of the chain representation of the same n-ary multiplication.
20. S. Duplij / Filomat xx (yyyy), zzz–zzz 20
5. Generalized “deformed” version of the homomorphism theorem
Let us consider a homomorphism of the binary groups entering into the “deformed” chain formula
(4.40) as Φ∗
: G∗
2 → G∗′
2 , where G∗′
2 = ⟨G′
| ∗′
, e′
⟩. We observe that, because Φ∗
commutes with the binary
multiplication, we need its commutation also with the automorphisms φq in each term of (4.40) (which fixes
equality of the “deformation” parameters q = q′
) and its homomorphic action on bq. Indeed, if
Φ∗
(
φq
( ))
= φ′
q
(
Φ∗ ( ))
, (5.1)
Φ∗
(
bq
)
= b′
q, (5.2)
then we get from (4.40)
Φ∗ (
µn
[
1, . . . , n
])
= Φ∗ (
1
)
∗′
Φ∗
(
φ
[[1]]q
q
(
2
)
)
∗′
. . . ∗′
Φ∗
(
φ
[[n−1]]q
q
(
n
)
)
∗′
Φ∗
(
bq
)
= Φ∗ (
1
)
∗′
φ
′[[1]]q
q
(
Φ∗ (
2
))
∗′
. . . ∗′
φ
[[n−1]]q
q
(
Φ∗ (
n
))
∗′
b′
q
= µ′
n
[
Φ∗ (
1
)
, . . . , Φ∗ (
n
)]
, (5.3)
where ′
∗′
h′
= µ′
n
[
′
, e′−1
, h′
]
, φ′
q
( ′
)
= µ
′ ℓφ(q)
n
[
e′
, ′
,
(
e′−1
)q]
, b′
q = µ′•
n
[
e′ [[n]]q
]
. Comparison of (5.3) and
(2.49) leads to
Theorem 5.1. A homomorphism Φ∗
of the binary group G∗
2 gives rise to a homomorphism Φ of the corresponding
n-ary group Gn, if Φ∗
satisfies the “deformed” compatibility conditions (5.1)–(5.2).
The “nondeformed” version (q = 1) of this theorem and the case of Φ∗
being an isomorphism was
considered in [23].
Acknowledgments. The author is thankful to J. Cuntz for kind hospitality at the University of M¨unster,
where the work in its final stage was supported by the project “Groups, Geometry and Actions” (SFB 878).
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