ABSTRACT: This study researches a representation of set indexed Brownian motion { : } X X A A A via orthonormal basis, based on reproducing kernel Hilbert space (RKHS). The RKHS associated with the set indexed Brownian motion X is a Hilbert space of real-valued functions on T that is naturally isometric to 2 L ( ) A . The isometry between these Hilbert spaces leads to useful spectral representations of the set indexed Brownian motion, notably the Karhunen-Loève (KL) representation: [ ] X e E X e A n A n where { }n e is an orthonormal sequence of centered Gaussian variables. In addition, we present two special cases of a representation of a set indexed Brownian motion, when ([0,1] ) d A A and A = A( ) Ls .
On the k-Riemann-Liouville fractional integral and applications Premier Publishers
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary non-integer order. The subject is as old as differential calculus and goes back to times when G.W. Leibniz and I. Newton invented differential calculus. Fractional integrals and derivatives arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of a complex medium. Very recently, Mubeen and Habibullah have introduced the k-Riemann-Liouville fractional integral defined by using the -Gamma function, which is a generalization of the classical Gamma function. In this paper, we presents a new fractional integration is called k-Riemann-Liouville fractional integral, which generalizes the k-Riemann-Liouville fractional integral. Then, we prove the commutativity and the semi-group properties of the -Riemann-Liouville fractional integral and we give Chebyshev inequalities for k-Riemann-Liouville fractional integral. Later, using k-Riemann-Liouville fractional integral, we establish some new integral inequalities.
In this lecture, I will describe how to calculate optical response functions using real-time simulations. In particular, I will discuss td-hartree, td-dft and similar approximations.
The document describes a control framework called the "stack of tasks" which provides hierarchical task-based control for real-time redundant manipulators. It allows implementation of a data flow graph controlled by Python scripting. Tasks are defined as functions of the robot configuration, time, and other parameters that should converge to zero. The framework computes joint velocities to minimize higher priority tasks while satisfying lower priority tasks when possible. It has been tested on robots including HRP-2, Nao, and Romeo.
A Commutative Alternative to Fractional Calculus on k-Differentiable FunctionsMatt Parker
This document presents a method for creating a commutative operator that acts parallel to fractional calculus operators on continuous functions. It defines spaces Ck that contain images of continuous functions and combines these into a space Cdiff that contains a subset isomorphic to the space of continuous functions C(R). An operator Dk is defined on Cdiff that commutes with itself and acts equivalently to fractional derivatives on C(R) up to the differentiability of the function. This provides a commutative alternative to fractional calculus on continuous functions.
In this paper we have defined Dk
integral and proved the integration by parts formula.
Key Words and phrases: Absolutely Continuous function, Generalised absolutely continuous function,
Denjoy integration. 2000 Mathematics subject Classification: Primary 26A24 Secondary 26A21, 26A48,
44A10.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
The document presents a theorem that generalizes previous results on the L1-convergence of modified trigonometric cosine sums. Specifically, it obtains a necessary and sufficient condition for the L1-convergence of a modified cosine sum where the coefficients belong to the Fp class. This condition is that an log n = o(1) as n approaches infinity. The theorem also deduces Fomin's previous result as a corollary.
On complementarity in qec and quantum cryptographywtyru1989
The document discusses complementarity in quantum error correction and quantum cryptography. It begins with an introduction and outline. It then covers the Stinespring dilation theorem, purification of mixed states, conjugate/complementary channels, private quantum codes including for single qubits, and the complementarity of quantum codes. An example is also provided to illustrate complementarity between a swap channel and its conjugate.
On the k-Riemann-Liouville fractional integral and applications Premier Publishers
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary non-integer order. The subject is as old as differential calculus and goes back to times when G.W. Leibniz and I. Newton invented differential calculus. Fractional integrals and derivatives arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of a complex medium. Very recently, Mubeen and Habibullah have introduced the k-Riemann-Liouville fractional integral defined by using the -Gamma function, which is a generalization of the classical Gamma function. In this paper, we presents a new fractional integration is called k-Riemann-Liouville fractional integral, which generalizes the k-Riemann-Liouville fractional integral. Then, we prove the commutativity and the semi-group properties of the -Riemann-Liouville fractional integral and we give Chebyshev inequalities for k-Riemann-Liouville fractional integral. Later, using k-Riemann-Liouville fractional integral, we establish some new integral inequalities.
In this lecture, I will describe how to calculate optical response functions using real-time simulations. In particular, I will discuss td-hartree, td-dft and similar approximations.
The document describes a control framework called the "stack of tasks" which provides hierarchical task-based control for real-time redundant manipulators. It allows implementation of a data flow graph controlled by Python scripting. Tasks are defined as functions of the robot configuration, time, and other parameters that should converge to zero. The framework computes joint velocities to minimize higher priority tasks while satisfying lower priority tasks when possible. It has been tested on robots including HRP-2, Nao, and Romeo.
A Commutative Alternative to Fractional Calculus on k-Differentiable FunctionsMatt Parker
This document presents a method for creating a commutative operator that acts parallel to fractional calculus operators on continuous functions. It defines spaces Ck that contain images of continuous functions and combines these into a space Cdiff that contains a subset isomorphic to the space of continuous functions C(R). An operator Dk is defined on Cdiff that commutes with itself and acts equivalently to fractional derivatives on C(R) up to the differentiability of the function. This provides a commutative alternative to fractional calculus on continuous functions.
In this paper we have defined Dk
integral and proved the integration by parts formula.
Key Words and phrases: Absolutely Continuous function, Generalised absolutely continuous function,
Denjoy integration. 2000 Mathematics subject Classification: Primary 26A24 Secondary 26A21, 26A48,
44A10.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
The document presents a theorem that generalizes previous results on the L1-convergence of modified trigonometric cosine sums. Specifically, it obtains a necessary and sufficient condition for the L1-convergence of a modified cosine sum where the coefficients belong to the Fp class. This condition is that an log n = o(1) as n approaches infinity. The theorem also deduces Fomin's previous result as a corollary.
On complementarity in qec and quantum cryptographywtyru1989
The document discusses complementarity in quantum error correction and quantum cryptography. It begins with an introduction and outline. It then covers the Stinespring dilation theorem, purification of mixed states, conjugate/complementary channels, private quantum codes including for single qubits, and the complementarity of quantum codes. An example is also provided to illustrate complementarity between a swap channel and its conjugate.
Code of the multidimensional fractional pseudo-Newton method using recursive ...mathsjournal
The following paper presents one way to define and classify the fractional pseudo-Newton method through a group of fractional matrix operators, as well as a code written in recursive programming to implement this method, which through minor modifications, can be implemented in any fractional fixed-point method that allows solving nonlinear algebraic equation systems.
The mp-quantales were introduced in a previous paper as an abstraction of the lattices of ideals in mp-rings and the lattices of ideals in conormal lattices. Several properties of m-rings and conormal lattices were generalized to mp-quantales. In this paper we shall prove new characterization theorems for mp-quantales and for semiprime mp-quantales (these last structures coincide with the P F-quantales). Some proofs reflect the way in which the reticulation functor (from coherent quantales to bounded distributive lattices) allows us to export some properties from conormal lattices to mp-quantales.
Modular representation theory of finite groupsSpringer
This document discusses the Cartan-Brauer triangle in representation theory. It introduces some key concepts and objects of study, including:
- Group rings R[G], K[G], and k[G] over different coefficient rings R, K, and k.
- G-invariant lattices in a K[G]-module V, which are R[G]-submodules of V.
- A theorem stating that for any two G-invariant lattices L and L' in a K[G]-module V, their classes [L/πR L] and [L'/πR L'] are equal in Rk(G).
- This theorem is used to construct a
A crystallographic group is a group acting on R^n that contains a translation subgroup Z^n as a finite index subgroup. Here we consider which Coxeter groups are crystallographic groups. We also expose the enumeration in dimension 2 and 3. Then we shortly give the principle under which the enumeration of N dimensional crystallographic groups is done.
The Persistent Homology of Distance Functions under Random ProjectionDon Sheehy
Given n points P in a Euclidean space, the Johnson-Lindenstrauss lemma guarantees that the distances between pairs of points is preserved up to a small constant factor with high probability by random projection into O(log n) dimensions. In this paper, we show that the persistent homology of the distance function to P is also preserved up to a comparable constant factor. One could never hope to preserve the distance function to P pointwise, but we show that it is preserved sufficiently at the critical points of the distance function to guarantee similar persistent homology. We prove these results in the more general setting of weighted k-th nearest neighbor distances, for which k=1 and all weights equal to zero gives the usual distance to P.
Exact Quasi-Classical Asymptoticbeyond Maslov Canonical Operator and Quantum ...ijrap
This document discusses exact quasi-classical asymptotic solutions to the Schrodinger equation beyond the WKB and Maslov canonical operator approximations. It presents Colombeau solutions to the Schrodinger equation that can explain the nature of quantum jumps without additional postulates. The solutions are represented using path integral formulations involving Feynman propagators and Maslov canonical operators. Limiting quantum trajectories and averages are defined from the Colombeau solutions that correspond to measurement outcomes, providing an explanation for quantum jumps from the Schrodinger equation alone.
This document summarizes and compares different distances that can be used in generative adversarial networks (GANs). It introduces the Wasserstein distance, also known as the Earth Mover (EM) distance or Wasserstein-1 distance. The document shows that the Wasserstein distance is more meaningful than other distances like total variation, Kullback-Leibler divergence, and Jensen-Shannon divergence when the real and generated distributions start to differ but their support still overlap. It also demonstrates that training GANs with the Wasserstein distance provides improved stability during training compared to other distances. Several theorems and examples are provided to illustrate properties of the Wasserstein distance such as Lipschitz continuity.
Nevenko Bilić "Tachyon inflation on the holographic braneworld"SEENET-MTP
This document provides an overview of tachyon inflation in a holographic braneworld. It discusses how holographic cosmology arises from the AdS/CFT correspondence between gravity in a bulk AdS spacetime and a conformal field theory on the boundary. The cosmology is governed by Einstein's equations on the boundary, with the stress-energy of the boundary CFT serving as a perfect fluid. This results in a holographic Friedmann equation that avoids the Big Bang singularity. Tachyon inflation may occur in this braneworld scenario.
Variants of the Christ-Kiselev lemma and an application to the maximal Fourie...VjekoslavKovac1
1. The document discusses variants of the Christ-Kiselev lemma and its application to maximal Fourier restriction estimates.
2. The Christ-Kiselev lemma allows block-diagonal and block-triangular truncations of operators while controlling their operator norms.
3. These lemmas can be used to prove maximal and variational estimates for the restriction of the Fourier transform to surfaces, which has applications in harmonic analysis.
This document proposes a modular beamforming architecture for ultrasound imaging that uses FPGA DSP cells to overcome limitations of previous designs. It interleaves the interpolation and coherent summation processes, reducing hardware resources. This allows implementing a 128-channel beamformer in a single FPGA, achieving flexibility like FPGAs but with lower power consumption like ASICs. The design is scalable, allowing a tradeoff between number of channels, time resolution, and resource usage.
Quantitative norm convergence of some ergodic averagesVjekoslavKovac1
The document summarizes quantitative estimates for the convergence of multiple ergodic averages of commuting transformations. Specifically, it presents a theorem that provides an explicit bound on the number of jumps in the Lp norm for double averages over commuting Aω actions on a probability space. The proof transfers the structure of the Cantor group AZ to R+ and establishes norm estimates for bilinear averages of functions on R2+. This allows bounding the variation of the double averages and proving the theorem.
Reciprocity Law For Flat Conformal Metrics With Conical SingularitiesLukasz Obara
The document is a thesis that establishes an analogue of Weil's reciprocity law for flat conformal metrics with conical singularities on Riemann surfaces. It introduces the necessary mathematical background, including definitions of isothermal coordinates and metrics with conical singularities. The main result proves a relationship between three flat conformally equivalent metrics, each with different conical singularities.
Introduction to Fourier transform and signal analysis宗翰 謝
The document discusses Fourier analysis techniques. It introduces continuous and discrete Fourier transforms, and covers properties like orthogonality, completeness of basis functions (e.g. cosines and sines), and Fourier series representations of periodic functions like step functions. It also defines the Fourier transform and its properties like linearity, translation, modulation, scaling, and conjugation. Concepts like Dirac delta functions and convolution theory are explained in relation to Fourier analysis.
The document discusses scaling sets and MRA wavelet sets, which are measurable sets associated with multiresolution analyses and wavelets. It provides definitions and theorems characterizing scaling sets and MRA wavelet sets. Some simple examples of scaling sets and MRA wavelet sets are given as finite unions of intervals. The document then poses questions about the properties of general wavelet sets and provides counterexamples to ideas about possible restrictions on their structure. Finally, more complex examples of scaling sets and MRA wavelet sets are constructed using Rademacher functions.
Trilinear embedding for divergence-form operatorsVjekoslavKovac1
The document discusses a trilinear embedding theorem for divergence-form operators with complex coefficients. It proves that if matrices A, B, C are appropriately p,q,r-elliptic, then there is a bound on the integral of the product of the gradients of the semigroups associated with the operators. The proof uses a Bellman function technique and shows the relationship to the concept of p-ellipticity. It generalizes previous work on bilinear embeddings to the trilinear case.
This document discusses signal-space analysis and representation of bandpass signals. It can be summarized as follows:
1) A bandpass real signal x(t) can be represented using its complex envelope x(t) and carrier frequency fc. This results in an in-phase (I) and quadrature-phase (Q) representation of the signal.
2) Signals can be viewed as vectors in a vector space. Basic algebra concepts like groups, fields, and vector spaces are introduced.
3) Key concepts discussed include orthonormal bases, projection theorems, Gram-Schmidt orthonormalization, and representing signals in inner product spaces which allows defining notions of length and angle between signals.
The document provides an overview of concepts in functional analysis that will be covered in a math camp, including: function spaces, metric spaces, dense subsets, linear spaces, linear functionals, norms, Euclidean spaces, orthogonality, separable spaces, complete metric spaces, Hilbert spaces, and convex functions. Examples are given for each concept to illustrate the definitions.
Code of the multidimensional fractional pseudo-Newton method using recursive ...mathsjournal
The following paper presents one way to define and classify the fractional pseudo-Newton method through a group of fractional matrix operators, as well as a code written in recursive programming to implement this method, which through minor modifications, can be implemented in any fractional fixed-point method that allows solving nonlinear algebraic equation systems.
The mp-quantales were introduced in a previous paper as an abstraction of the lattices of ideals in mp-rings and the lattices of ideals in conormal lattices. Several properties of m-rings and conormal lattices were generalized to mp-quantales. In this paper we shall prove new characterization theorems for mp-quantales and for semiprime mp-quantales (these last structures coincide with the P F-quantales). Some proofs reflect the way in which the reticulation functor (from coherent quantales to bounded distributive lattices) allows us to export some properties from conormal lattices to mp-quantales.
Modular representation theory of finite groupsSpringer
This document discusses the Cartan-Brauer triangle in representation theory. It introduces some key concepts and objects of study, including:
- Group rings R[G], K[G], and k[G] over different coefficient rings R, K, and k.
- G-invariant lattices in a K[G]-module V, which are R[G]-submodules of V.
- A theorem stating that for any two G-invariant lattices L and L' in a K[G]-module V, their classes [L/πR L] and [L'/πR L'] are equal in Rk(G).
- This theorem is used to construct a
A crystallographic group is a group acting on R^n that contains a translation subgroup Z^n as a finite index subgroup. Here we consider which Coxeter groups are crystallographic groups. We also expose the enumeration in dimension 2 and 3. Then we shortly give the principle under which the enumeration of N dimensional crystallographic groups is done.
The Persistent Homology of Distance Functions under Random ProjectionDon Sheehy
Given n points P in a Euclidean space, the Johnson-Lindenstrauss lemma guarantees that the distances between pairs of points is preserved up to a small constant factor with high probability by random projection into O(log n) dimensions. In this paper, we show that the persistent homology of the distance function to P is also preserved up to a comparable constant factor. One could never hope to preserve the distance function to P pointwise, but we show that it is preserved sufficiently at the critical points of the distance function to guarantee similar persistent homology. We prove these results in the more general setting of weighted k-th nearest neighbor distances, for which k=1 and all weights equal to zero gives the usual distance to P.
Exact Quasi-Classical Asymptoticbeyond Maslov Canonical Operator and Quantum ...ijrap
This document discusses exact quasi-classical asymptotic solutions to the Schrodinger equation beyond the WKB and Maslov canonical operator approximations. It presents Colombeau solutions to the Schrodinger equation that can explain the nature of quantum jumps without additional postulates. The solutions are represented using path integral formulations involving Feynman propagators and Maslov canonical operators. Limiting quantum trajectories and averages are defined from the Colombeau solutions that correspond to measurement outcomes, providing an explanation for quantum jumps from the Schrodinger equation alone.
This document summarizes and compares different distances that can be used in generative adversarial networks (GANs). It introduces the Wasserstein distance, also known as the Earth Mover (EM) distance or Wasserstein-1 distance. The document shows that the Wasserstein distance is more meaningful than other distances like total variation, Kullback-Leibler divergence, and Jensen-Shannon divergence when the real and generated distributions start to differ but their support still overlap. It also demonstrates that training GANs with the Wasserstein distance provides improved stability during training compared to other distances. Several theorems and examples are provided to illustrate properties of the Wasserstein distance such as Lipschitz continuity.
Nevenko Bilić "Tachyon inflation on the holographic braneworld"SEENET-MTP
This document provides an overview of tachyon inflation in a holographic braneworld. It discusses how holographic cosmology arises from the AdS/CFT correspondence between gravity in a bulk AdS spacetime and a conformal field theory on the boundary. The cosmology is governed by Einstein's equations on the boundary, with the stress-energy of the boundary CFT serving as a perfect fluid. This results in a holographic Friedmann equation that avoids the Big Bang singularity. Tachyon inflation may occur in this braneworld scenario.
Variants of the Christ-Kiselev lemma and an application to the maximal Fourie...VjekoslavKovac1
1. The document discusses variants of the Christ-Kiselev lemma and its application to maximal Fourier restriction estimates.
2. The Christ-Kiselev lemma allows block-diagonal and block-triangular truncations of operators while controlling their operator norms.
3. These lemmas can be used to prove maximal and variational estimates for the restriction of the Fourier transform to surfaces, which has applications in harmonic analysis.
This document proposes a modular beamforming architecture for ultrasound imaging that uses FPGA DSP cells to overcome limitations of previous designs. It interleaves the interpolation and coherent summation processes, reducing hardware resources. This allows implementing a 128-channel beamformer in a single FPGA, achieving flexibility like FPGAs but with lower power consumption like ASICs. The design is scalable, allowing a tradeoff between number of channels, time resolution, and resource usage.
Quantitative norm convergence of some ergodic averagesVjekoslavKovac1
The document summarizes quantitative estimates for the convergence of multiple ergodic averages of commuting transformations. Specifically, it presents a theorem that provides an explicit bound on the number of jumps in the Lp norm for double averages over commuting Aω actions on a probability space. The proof transfers the structure of the Cantor group AZ to R+ and establishes norm estimates for bilinear averages of functions on R2+. This allows bounding the variation of the double averages and proving the theorem.
Reciprocity Law For Flat Conformal Metrics With Conical SingularitiesLukasz Obara
The document is a thesis that establishes an analogue of Weil's reciprocity law for flat conformal metrics with conical singularities on Riemann surfaces. It introduces the necessary mathematical background, including definitions of isothermal coordinates and metrics with conical singularities. The main result proves a relationship between three flat conformally equivalent metrics, each with different conical singularities.
Introduction to Fourier transform and signal analysis宗翰 謝
The document discusses Fourier analysis techniques. It introduces continuous and discrete Fourier transforms, and covers properties like orthogonality, completeness of basis functions (e.g. cosines and sines), and Fourier series representations of periodic functions like step functions. It also defines the Fourier transform and its properties like linearity, translation, modulation, scaling, and conjugation. Concepts like Dirac delta functions and convolution theory are explained in relation to Fourier analysis.
The document discusses scaling sets and MRA wavelet sets, which are measurable sets associated with multiresolution analyses and wavelets. It provides definitions and theorems characterizing scaling sets and MRA wavelet sets. Some simple examples of scaling sets and MRA wavelet sets are given as finite unions of intervals. The document then poses questions about the properties of general wavelet sets and provides counterexamples to ideas about possible restrictions on their structure. Finally, more complex examples of scaling sets and MRA wavelet sets are constructed using Rademacher functions.
Trilinear embedding for divergence-form operatorsVjekoslavKovac1
The document discusses a trilinear embedding theorem for divergence-form operators with complex coefficients. It proves that if matrices A, B, C are appropriately p,q,r-elliptic, then there is a bound on the integral of the product of the gradients of the semigroups associated with the operators. The proof uses a Bellman function technique and shows the relationship to the concept of p-ellipticity. It generalizes previous work on bilinear embeddings to the trilinear case.
This document discusses signal-space analysis and representation of bandpass signals. It can be summarized as follows:
1) A bandpass real signal x(t) can be represented using its complex envelope x(t) and carrier frequency fc. This results in an in-phase (I) and quadrature-phase (Q) representation of the signal.
2) Signals can be viewed as vectors in a vector space. Basic algebra concepts like groups, fields, and vector spaces are introduced.
3) Key concepts discussed include orthonormal bases, projection theorems, Gram-Schmidt orthonormalization, and representing signals in inner product spaces which allows defining notions of length and angle between signals.
The document provides an overview of concepts in functional analysis that will be covered in a math camp, including: function spaces, metric spaces, dense subsets, linear spaces, linear functionals, norms, Euclidean spaces, orthogonality, separable spaces, complete metric spaces, Hilbert spaces, and convex functions. Examples are given for each concept to illustrate the definitions.
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Code of the multidimensional fractional pseudo-Newton method using recursive ...mathsjournal
The following paper presents one way to define and classify the fractional pseudo-Newton method through a group of fractional matrix operators, as well as a code written in recursive programming to implement this
method, which through minor modifications, can be implemented in any fractional fixed-point method that allows
solving nonlinear algebraic equation systems.
Code of the Multidimensional Fractional Quasi-Newton Method using Recursive P...mathsjournal
The following paper presents one way to define and classify the fractional quasi-Newton method through a group of fractional matrix operators, as well as a code written in recursive programming to implement this method, which through minor modifications, can be implemented in any fractional fixed-point method that allows solving nonlinear algebraic equation systems.
The document discusses properties of lattices and proofs related to lattices. It contains 6 questions with solutions. Question 1 defines sets A, B, C and verifies their intersections. Question 2 proves an identity related to general progressions. Question 3 determines the number of possible two-digit numbers that can be formed with and without repetition of digits. Question 4 solves a recurrence relation. Question 5 checks if given sets of vectors are linearly dependent or independent. Question 6 proves that an algebraic lattice forms a partially ordered set.
This section define a level subring or level ideals obtain a set of necessary and sufficient condition for the
equality of two ideals and characterizes field in terms of its fuzzy ideals. It also presents a procedure to construct
a fuzzy subrings (fuzzy ideals) from any given ascending chain of subring ideal. We prove that the lattice of
fuzzy congruence of group G (respectively ring R) is isomorphic to the lattice of fuzzy normal subgroup of G
(respectively fuzzy ideals of R).In Yuan Boond Wu wangrning investigated the relationship between the fuzzy
ideals and the fuzzy congruences on a distributive lattice and obtained that the lattice of fuzzy ideals is
isomorphic to the lattice of fuzzy congruences on a generalized Boolean algebra. Fuzzy group theory can be
used to describe, symmetries and permutation in nature and mathematics. The fuzzy group is one of the oldest
branches of abstract algebra. For example group can be used is classify to all of the forms chemical crystal can
take. Group can be used to count the number of non-equivalent objects and permutation or symmetries. For
example, the number of different is switching functions of n, variable when permutation of the input are
allowed. Beside crystallography and combinatory group have application of quantum mechanics.
HARMONIC ANALYSIS ASSOCIATED WITH A GENERALIZED BESSEL-STRUVE OPERATOR ON THE...irjes
This document summarizes a research paper that considers a generalized Bessel-Struve operator on the real line. It defines generalized Bessel-Struve and Weyl integral transforms, which are shown to be transmutation operators relating the generalized Bessel-Struve operator to derivatives. These tools are then used to develop a new harmonic analysis associated with the generalized Bessel-Struve operator, including generalized Sonine integral transforms. Key results proven include Paley-Wiener theorems and properties of the various integral transforms.
This document summarizes a research paper that introduces new concepts related to characterizing kernel sets in fuzzy topological spaces. Specifically, it defines kernelled fuzzy points, boundary kernelled fuzzy points, and derived kernelled fuzzy points of a subset. Using these notions, the paper presents a new definition of the kernel set of a fuzzy topological space as the union of the set itself and the set of all boundary kernelled fuzzy points. It also introduces the concept of a fuzzy topological kr-space and uses these concepts to characterize weak separation axioms between TF0 and TF1 spaces.
On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This document summarizes a research article that defines extendable sets in the real numbers (R) and applies this concept to the Lyapunov stability comparison principle of ordinary differential equations. It begins with the author's own definition of extension on R and a basic result called the basic extension fact for R. It then reviews existing definitions and theorems on extension, including Urysohn's lemma and Tietze's extension theorem. The document concludes by extensively applying these results to prove some important results relating to the comparison principle of Lyapunov stability theory in ordinary differential equations.
This document presents research on extendable sets in the real numbers (R) and their application to the Lyapunov stability comparison principle of ordinary differential equations. It begins with definitions of the real numbers and extendable sets. It then reviews existing definitions of extension, including Urysohn's lemma and Tietze extension theorem. The main result proved is that every compact subset of R is extendable, while non-compact subsets are not. It concludes by extensively applying these results to prove important theorems regarding the comparison principle of Lyapunov stability theory in ordinary differential equations.
This document summarizes a research article that defines extendable sets in the real numbers (R) and applies this concept to proofs involving the Lyapunov stability comparison principle of ordinary differential equations. It begins with the author's own definition of an extension on R and a basic result called the basic extension fact for R. Existing definitions and theorems on extension, such as Urysohn's lemma and Tietze's extension theorem, are then reviewed. The document concludes by extensively applying these concepts to resolve proofs involving the comparison principle of Lyapunov stability theory.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Math 511 Problem Set 4, due September 21Note Problems 1 tAbramMartino96
Math 511 Problem Set 4, due September 21
Note: Problems 1 through 7 are the ones to be turned in. The remainder of the problems are
for extra functional analytic goodness.
1. Fix a,b ∈ R with a < b. Show that {1, t, t2, . . . , tn} is a linearly independent subset of
C[a,b]. From this conclude that {1, t, t2, t3, . . .} is a linearly independent set in C[a,b]. Give
an example of a function f ∈ C[a,b] so that f /∈ span{1, t, t2, . . .}.
2. Prove that if 1 ≤ p1 ≤ p2 ≤∞ then lp1 ⊆ lp2 .
3. Consider C[0, 2] with the function ‖ ·‖1 defined by
‖f‖1 =
∫ 2
0
|f(x)|dx, for f ∈ C[0, 2].
(a) Prove that ‖ ·‖1 is a norm.
(b) Prove that the normed linear space (C[0, 2],‖·‖1) is not complete (and thus not a Banach
space) by considering the sequence of functions
fn(x) =
1, x ≤ 1 − 1
n
n−nx, 1 − 1
n
< x < 1 + 1
n
−1, x ≥ 1 + 1
n
.
Show these are continuous functions, this sequence is a Cauchy sequence in the metric
derived from ‖ ·‖1, but that this sequence does not converge in C[0, 2] with this metric.
4. Let V be a vector space over R or C. A subset A ⊆ V is convex if for any v,w ∈ A and any
λ ∈ [0, 1] then λv + (1 −λ)w ∈ A, i.e. the segement connecting v and w is also in A.
(a) Let W be a vector subspace of V . Show that W is convex.
(b) Let X be a normed linear space. Show that the unit ball B1(0) is convex.
5. show that c ⊆ l∞ is a vector subspace of l∞ (see 1.5-3 for the definition of c) and so is c0, the
set of all sequences (xn) so that limn→∞ xn = 0.
6. Let 1 ≤ p < ∞ and en ∈ lp be the sequence with 1 in the nth place and 0 in all othe coordinates.
Show that {en : n ∈ N} is a Schauder basis for lp.
7. Now if X is a Banach space and (yn) a sequence in X, prove that
∑∞
n=1 ‖yn‖ < ∞ does imply
the convergence of
∑∞
n=1 yn. Thus in Banach spaces, absolute convergence implies convergence
of the series.
The following questions are for you to think about and not to be turned in.
1001. What is the completion of (0, 1) as a metric subspace of R with the euclidean metric?
Explain.
1002. Show that the discrete metric on a nontrivial vector space cannot be obtained from a norm.
1003. Show that if a normed vector space has a Schauder basis, then the space is separable. (You
can use a similar argument to your proof that lp is separable for 1 ≤ p < ∞.)
1004. Prove the general Hölder inequality: Suppose 1 ≤ r < p < ∞, and assume that
1
p
+
1
q
=
1
r
.
Show that for x = (x1,x2, . . .) and y = (y1,y2, . . .), and if we define the componentwise product
xy = (x1y1,x2y2, . . .), then
‖xy‖r ≤‖x‖p‖y‖q.
You may assume that x ∈ lp and y ∈ lq, although this is not necessary. (Hint: 1 = 1p
r
+ 1q
r
, and
use the regular Hölder inequality on particular sequences).
(Note: We can extend this to let p = r, and in this case q = ∞. The result will still hold.)
1005. Give an example of a subspace of l∞ which is not closed. Repeat for l2. (Hint: Look at
problem 3, p. 70)
1006. Let X be a normed vector space. Show that the convergenc ...
God bless... It is often useful to assert that something exists and .pdfxlynettalampleyxc
God bless... It is often useful to assert that something exists and is unique. For instance, There
exists a unique course at UW with the course number CSE 311. (a) 4 Points] Let the domain of
discourse be all vases. Let B be a constant representing \"the painted vase in Shayan\'s office.\"
Translate \"If there is a vase that is painted, then it is the painted vase in Shayan\'s office.\" into
first-order logic. Don\'t forget to define predicates Hint: Remember that a \"constant\', is a
particular object in the domain. For example, \"0\" and .1\" are constants in the domain of natural
numbers. (b) [4 Points] Let the domain of discourse be all university courses. Now, use an idea
similar to part (a) to translate the sentence about the CSE 311 course number into first-order
logic. Don\'t forget to define predicates.
Solution
Ans-
We prove only the results for the inverse image of a union and the image of an intersection; the
proof of the remaining two results is similar. If x f 1 S jJ Yj , then there exists y S jJ Yj such
that f(x) = y. Then y Yj for some j J and x f 1 (Yj ), so x S jJ f 1 (Yj ). It follows that f 1 [ jJ Yj
[ jJ f 1 (Yj ). 10 1. Sets and Functions Conversely, if x S jJ f 1 (Yj ), then x f 1 (Yj ) for some j
J, so f(x) Yj and f(x) S jJ Yj , meaning that x f 1 S jJ Yj . It follows that [ jJ f 1 (Yj ) f 1 [ jJ Yj
, which proves that the sets are equal. If y f T iI Xi , then there exists x T iI Xi such that f(x) =
y. Then x Xi and y f(Xi) for every i I, meaning that y T iI f (Xi). It follows that f \\ iI Xi ! \\ iI
f (Xi). The only case in which we don’t always have equality is for the image of an intersection,
and we may get strict inclusion here if f is not one-to-one. Example 1.25. Define f : R R by f(x)
= x 2 . Let A = (1, 0) and B = (0, 1). Then A B = and f(A B) = , but f(A) = f(B) = (0, 1), so f(A)
f(B) = (0, 1) 6= f(A B). Next, we generalize the Cartesian product of finitely many sets to the
product of possibly infinitely many sets. Definition 1.26. Let C = {Xi : i I} be an indexed
collection of sets Xi . The Cartesian product of C is the set of functions that assign to each index
i I an element xi Xi . That is, Y iI Xi = ( f : I [ iI Xi : f(i) Xi for every i I ) . For example, if I =
{1, 2, . . . , n}, then f defines an ordered n-tuple of elements (x1, x2, . . . , xn) with xi = f(i) Xi ,
so this definition is equivalent to our previous one. If Xi = X for every i I, then Q iI Xi is simply
the set of functions from I to X, and we also write it as XI = {f : I X} . We can think of this set
as the set of ordered I-tuples of elements of X. Example 1.27. A sequence of real numbers (x1,
x2, x3, . . . , xn, . . .) R N is a function f : N R. We study sequences and their convergence
properties in Chapter 3. Example 1.28. Let 2 = {0, 1} be a set with two elements. Then a subset
A I can be identified with its characteristic function A : I 2 by: i A if and only if A(i) = 1. Thus,
A 7 A is a one-to-one map from P(I) onto 2 I . Before giving.
Interval valued intuitionistic fuzzy homomorphism of bf algebrasAlexander Decker
This document discusses interval-valued intuitionistic fuzzy homomorphisms of BF-algebras. It begins with introducing BF-algebras and defining interval-valued intuitionistic fuzzy sets. It then defines interval-valued intuitionistic fuzzy ideals of BF-algebras and provides an example. Interval-valued intuitionistic fuzzy homomorphisms of BF-algebras are introduced and some properties are investigated, including showing that the image of an interval-valued intuitionistic fuzzy ideal under a homomorphism is also an interval-valued intuitionistic fuzzy ideal if it satisfies the "sup-inf" property.
This document provides the contents page for a book on integral equations. It lists 22 chapters covering topics such as Volterra integral equations, Fredholm integral equations, approximate solution methods, Green's functions, and singular integral equations. It was translated from the Russian original published in 1971 in Moscow by MIR Publishers.
In this paper, we introduce the concept of residual quotient of intuitionistic fuzzy subsets of ring and module and then define the notion of residual quotient intuitionistic fuzzy submodules , residual quotient intuitionistic fuzzy ideals. We study many properties of residual quotient relating to union, intersection, sum of intuitionistic fuzzy submodules (ideals). Using the concept of residual quotient, we investigate some important characterization of intuitionistic fuzzy annihilator of subsets of ring and module. We also study intuitionistic fuzzy prime submodules with the help of intuitionistic fuzzy annihilators. Many related properties are defined and discussed
This document provides an overview of preliminary topological concepts needed for applied mathematics. It defines topological spaces and metric spaces, and introduces key topological notions like open and closed sets, bases for topologies, convergence of sequences, accumulation points, interior and closure of sets, and dense sets. Metric spaces are shown to induce a natural topological structure, though not all topologies come from a metric. Examples are provided to illustrate various definitions and properties.
Abstract Quadripartitioned single valued neutrosophic (QSVN) set is a powerful structure where we have four components Truth-T, Falsity-F, Unknown-U and Contradiction-C. And also it generalizes the concept of fuzzy, initutionstic and single valued neutrosophic set. In this paper we have proposed the concept of K-algebras on QSVN, level subset of QSVN and studied some of the results. In addition to this we have also investigated the characteristics of QSVN Ksubalgebras under homomorphism.
Similar to Reproducing Kernel Hilbert Space of A Set Indexed Brownian Motion (20)
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Reproducing Kernel Hilbert Space of A Set Indexed Brownian Motion
1. International
OPEN ACCESS Journal
Of Modern Engineering Research (IJMER)
| IJMER | ISSN: 2249–6645 www.ijmer.com | Vol. 7 | Iss. 5 | May. 2017 | 22 |
Reproducing Kernel Hilbert Space of A Set Indexed
Brownian Motion
Arthur Yosef
Tel Aviv-Yaffo Academic College, 2Rabenu Yeruham St., Tel Aviv-Yaffo, Israel
I. INTRODUCTION
In this article, we present the representation of set indexed Brownian motion { : }AX X A A via
orthonormal basis, based on reproducing kernel Hilbert space (RKHS). Set indexed Brownian motion is a
natural generalization of planar Brownian motion where A is a collection of compact subsets of a fixed
topological space ( , )T . The frame of a set-indexed Brownian motion is not only a new stepto generalize
theclassical Brownian motion, but it was proven asa new look upon a Brownian motion (see [Yo09], [Yo15],
[MeYo], [He], [IvMe], [Kh], [MeNu]).
RKHS is a robust tooland can be used in a wide variety of areas such as curve fitting,signal analysis
and processing, function estimation and model description, differential equations, probability, statistics,
nonlinear Burgers equations, empirical risk minimization, fractals, machine learning and etc. (see [Par67],
[Par],[Ha],[Sc], [Va], [ScSm], [Dan], [Be], [Cu], [Ge], [Ad]).
Let’s assume we havea set indexed Brownian motion on topological and separable spaceT , with a
continuous covariance kernel :R A A . We can associate a Hilbert space, which is the reproducing
kernel Hilbert space of real-valued functions on T that is naturally isometric to
2
( )L A .The isometry between
these Hilbert spaces leads to useful spectral representations of the set indexed Brownian motion, notably the
Karhunen-Loève (KL) theorem.(The KL theorem is a representation of a stochastic process as an infinite linear
combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded
interval).In this work, the KL representation of a set indexed Brownian motion is:
[ ]A i A iX e E X e
Where 1{ }n ne
an orthonormal sequences of centered Gaussian variables. In addition, in this study we present
two special casesof a KL representationof a set indexed Brownian motion, when:
(a) [0,1]d
T and ([0,1] ) {[0, ]: [0,1] }d d
x x A A
(b) [0,1]d
T and ( )LsA = A
In the first case, the KL representation of a set indexed Brownian motion is:
ABSTRACT: This study researches a representation of set indexed Brownian motion
{ : }AX X A A via orthonormal basis, based on reproducing kernel Hilbert space (RKHS). The
RKHS associated with the set indexed Brownian motion X is a Hilbert space of real-valued functions
on T that is naturally isometric to
2
( )L A . The isometry between these Hilbert spaces leads to useful
spectral representations of the set indexed Brownian motion, notably the Karhunen-Loève (KL)
representation: [ ]A n A nX e E X e where { }ne is an orthonormal sequence of centered Gaussian
variables.
In addition, we present two special cases of a representation of a set indexed Brownian motion, when
([0,1] )d
A A and ( )LsA = A .
Keywords: Brownian motion, orthonormal basis, Hilbert Space, Karhunen-Loève.
2. Reproducing Kernel Hilbert Spaceof A Set Indexed Brownian Motion
| IJMER | ISSN: 2249–6645 www.ijmer.com | Vol. 7 | Iss. 5 | May. 2017 | 23 |
1 2
2 2 1 1
[0, ] [0, ] ... [0, ] 2(2 1)
1 1
( ) sin( (2 1) )d
d
d
d
A x x x n in
n i
X X e n x
In the second case:
1 2
2 2 2
2 2 1 1
( ) 2(2 1) 2([0, ] [0, ] ... [0, ])
1 1
lim( ) sin( (2 1) )i
kk k d mdm m m m m m m
d
kd
A g A nng m
n i
X X X e n
.
II. PRELIMINARIES
As in earlier set-indexed works (see [IvMe]), processes and filtrations will be indexed by a class A
whose elements are compact subsets of a fixed -compact metric space T . In addition, we assume A satisfies
several natural conditions. We use the definition and notation from [IvMe] and all this section comefrom there.
Definition 1.Let ( , )T be a non-void sigma-compact connected topologicalspace. A nonempty class A of
compact, connected subsets of T is called an indexed collection if it satisfies the following:
1. A . In addition, there is an increasing sequence ( )nB of sets in A s.t. 1n nT B
.
2. A is closed under arbitrary intersections and if ,A BA are nonempty, then A B is nonempty. If
( )iA is an increasing sequence in A and if there exists n such that i nA B for everyi , then i iA A .
3. ( ) A = B where B is the collection of Borel sets ofT .
4. Separability from above: There exist an increasing sequence of finite sub-classes 1{ ,..., }n
n n
n kA A A A
closed under intersection with ( )n n,B A u ( ( )nA u is the class of union of sets in nA ), and a sequence
of functions : ( )n ng T A A u such that:
(i) ng preserves arbitrary intersections and finite unions.
(ii) For each , ( )nA A g A A
and ( )n nA g A , ( ) ( )n mg A g A if n m .
(iii) ( ) 'ng A A A if , 'A A A and ( ) 'n ng A A A if A Aand ' nA A .
(iv) ( )ng for all n .
Examples.
a. The classical example is
d
T and ( ) {[0, ]: }d d
x x A = A (or [0,1]d
T and
([0,1] )d
A = A ). This example can be extended to
d
T and ( ) {[0, ]: }d d
x x A = A ,
which will give rise to a sort of 2d
-sides process.
b. The example (a) may be generalized as follows. Let
d
T or
d
T or [0,1]d
T and take A to be
the class of compact lower sets, i.e. the class of compact subsets A of T satisfying t A implies
[0, ]t A. (We denote the class of compact lower sets by ( )LsA ).
c. Additional examples have been given when T is a “continuous” rooted tree (see [Sl]) and T a subspace of
the Skorokhod space, [0,1]D (see [IvMe]).
We will require other classes of sets generated by A . The first is ( )A u , which is the class of finite
unions of sets in A . We note that ( )A u is itself a lattice with the partial order induced by set inclusion. Let C
consists of all the subsets of T of the form
, , ( )C A B A B A A u .
Any A -indexed function which has a (finitely) additive extension to C will be called additive (and is
easily seen to be additive on ( )C u as well). For stochastic processes, we do not necessarily require that each
sample path be additive, but additivity will be imposed in an almost suresense:
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A set-indexed stochastic process { : }AX X A A is additive if ithas an (almost sure) additive extension to
C : 0X and if 1 2, ,C C C C with 1 2C C C and 1 2C C then almost surely 1 2C C CX X X
. In particular, if C C and 1 n
i iC A A , 1, ,..., nA A A A then almost surely
11
... ( 1) n
i i j i i
n n
C A A A A A A A Ai i j
X X X X X
.
We shall always assume that our stochastic processes are additive. We note that a process with an (almost sure)
additive extension to C also has an (almost sure) additive extension to ( )C u .
Definition 2. A positive measure on ( , )T B is called strictly monotone on A if: ' 0 and A B
for all A B , ,A BA . The collection of these measures is denoted by ( )M A . ( ,' A A A A , note
that ' )
The classical examples for definition is Lebesgue measure or Radon measure when
d
T and
( ) {[0, ]: }d d
x x A = A
Definition 3. Let ( )M A . We say that the A -indexed process X is a Brownian motion with variance
if X can be extended to a finitely additive process on ( )C u and if for disjoint sets 1,..., nC C C,
1
,..., nC CX X are independent mean-zero Gaussian random variables with variances 1
,..., nC C , respectively.
(For any ( )M A , there exists a set-indexed Brownian motion with variance [IvMe]).
III. RKHS OF A SET INDEXED BROWNIAN MOTION
2
( )L A is a separable space,and thusit must have a countable orthonormal basis. Then every set
indexed stochastic process { : }AX X A A has the representation
1
,A i i i
i
X a a
where i are
random variables. Our purpose is to find the representation forthe set indexed Brownian motion via orthonormal
basis.
We restart with the RKHS of a set indexed Brownian motion. A RKHSis a Hilbert Space of functions. It can be
thought of as a space containing smoother function than the general Hilbert space.
Let A A and { : }AX X A A be a set indexed Brownian motion. We define the function
( , ) :R A A by ( , ) ( , ) [ ]A AR A Cov X X E X X .
Now, we define the set
1
{ : : ( ) ( , ), , , 1}
n
i i i i
i
f f a R A A a n
A A . Define an inner
product on by:
1 1
, ( , )
n m
i j i j
i j
f g a b R A B
when
1 1
( ) ( , ), ( ) ( , )
n m
i i j i
i j
f a R A g b R B
(1)
The fact that R is non-negative definite implies 2
1
, ( , ) 0
n
i i i
i
f f a R A A
for all f .From the inner
product, we get the following property:
1 1
( ) ( , ) ( , ), ( , ) , ( , )
n n
i i i i
i i
f B a R A B a R A R B f R B
(2)
Then for all f , B A,
2 2
| ( ) | | , ( , ) | , ( , ), ( , )f B f R B f f R B R B . The inequality
being merely the Schwartz inequality for semi-inner products, which holds as long as , 0f f . Thus, if
, 0f f then (2) implies that 0f for all B A. Consequently, (1) defines a proper inner product on
, and so we thus obtain a norm:
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,f f f
For 1{ }n nf
we have:
2 22 2
| ( ) ( ) | | , ( , ) | ( ) ( ) ( , )n m n m n mf B f B f f R B f B f B R B
2
( ) ( ) ( , )n mf B f B R B B
Thusitfollows that if 1{ }n nf
is Cauchy in then it is a pointwise Cauchy. The closure of denoted
by H . SinceT is separable and R continuous then H is also separable.Since H is a separable Hilbert space, it
must have a countable orthonormal basis.
Define ( )H X , the so-called “linear part” of the 2
( )L A space of the set indexed stochastic process X , as the
closure in 2
( )L A of
1
, , , 1i
n
i A i i
i
a X A a n
A ,
thinned out by identifying all elements indistinguishable in 2
( )L A (In other word, elements ,x y for which
2
[( ) ] 0E x y ). This contains all distinguishable random variables, with finite variance, obtainable as linear
combination of values of the process. There is a linear, one-one mapping
2
: ( )L T A defined by:
1 1
( ) ( , ) i
n n
i i i A
i i
f a R A a X
Τ T
Note that T is clearly norm preserving and so extends to all H with range equal to all of ( )H X . (In
other words, there is exists Tr such that | Tr T , ( )Dom HTr and Im( ) ( )H XTr ).
Since H is separable, we now know that ( )H X isalso separable. We can use this to build an orthonormal
basis for ( )H X . If 1{ }n n
is an orthonormal basis for H then 1{ }n ne
an orthonormal basis for ( )H X when
( )n ne T . X is a set indexed Brownian motion therefore [ ] 0nE e for all n and
1
[ ]A i A i
i
X e E X e
almost surely,
where the series converges in 2
( )L A . Easy to see that 1{ }n ne
an orthonormal sequences of centered Gaussian
variables. Since T was an isometry, it follows that
[ ] ( , ), ( )A i i iE X e R A A for all i
Then
1
( )A i i
i
X e A
almost surely.
RKHS and the KL representation of a set indexed Brownian motion on ([0,1] )d
A and on ( )LsA = A
Now, we willpresent two special cases of a RKHS and the KL representation of a set indexed Brownian motion,
when:
i. [0,1]d
T and ([0,1] ) {[0, ]: [0,1] }d d
x x A (see Examples after Definition 1).
ii. [0,1]d
T and ( )LsA = A (see Examples afterDefinition 1).
First case: [0,1]d
T and ([0,1] ) {[0, ]: [0,1] }d d
x x A = A
Let { : }AX X A A be a set indexed Brownian motion when [0,1]d
T and
([0,1] ) {[0, ]: [0,1] }d d
x x A = A .
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Let 1 2 1 2[0, ] [0, ] ... [0, ] , [0, ] [0, ] ... [0, ]d dA a a a B b b b A A then
1 1 2 2( , ) [ ] min{ , }min{ , } min{ , }A B d dR A B E X X a b a b a b
Values in set are:
1 1 2 2
1 1
( ) ( , ) min{ ( ), }min{ ( ), } min{ ( ), }
n n
i i i d d
i i
f A a R A A a i a a i a a i a
1 1 2 2
1 1
( ) ( , ) min{ ( ), }min{ ( ), } min{ ( ), }
n n
j j j d d
j i
g A R B A b j a b j a b j a
When
1 2[0, ( )] [0, ( )] ... [0, ( )]i dA a i a i a i A and 1 2[0, ( )] [0, ( )] ... [0, ( )]j dB b j b j b j A
The inner product on is:
1 1
, ( , )
n m
i j i j
i j
f g R A B
1 1 2 2
1 1
min{ ( ), ( )}min{ ( ), ( )} min{ ( ), ( )}
n m
i j d d
i j
a i b j a i b j a i b j
but
1
[0, ( )] [0, ( )]
0
min{ ( ), ( )} ( ) ( )k kk k a i b ja i b j I x I x dx for 1,2,...,k d
When DI is the indicator function of DA (
1 ,
( )
0 ,
D
x D
I x
x D
).
Therefore, we can rewrite the above as follows:
1 1
1 1
[0, ( )] 1 [0, ( )] 1 1 [0, ( )] [0, ( )]
1 1 0 0
, ( ) ( ) ( ) ( )d d
n m
i j a i b j a i d b j d d
i j
f g I x I x dx I x I x dx
1 1[0, ( )] 1 [0, ( )] 1 [0, ( )] [0, ( )] 1
1 1 [0,1]
( ) ( ) ( ) ( )d d
d
n m
i j a i b j a i d b j d d
i j
I x I x I x I x dx dx
1 1[0, ( )] 1 [0, ( )] [0, ( )] 1 [0, ( )] 1
1 1[0,1]
[ ( ) ( )] [ ( ) ( )]d d
d
n m
i a i a i d j b j b j d d
i j
I x I x I x I x dx dx
1 1 1
[0,1] [0,1]
( ,..., ) ( ,..., )
d d
I I I I
n n df x x g x x dx dx f g dS
When
11 2 [0, ( )] 1 [0, ( )]
1
( , ,..., ) [ ( ) ( )d
n
I
d i a i a i d
i
f x x x I x I x
11 2 [0, ( )] 1 [0, ( )]
1
( , ,..., ) [ ( ) ( )d
m
I
d j b j b j d
j
g x x x I x I x
Finally, ,f g
[0,1]d
I I
f g dS .
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We define 2
[0,1]
{ : : ( ) , ( ) }
d
I I I
f f f dS f dS
A then ( , ) I
R A and
11 [0, ] 1 [0, ] 1
[0,1]
( ) , ( , ) ( ,..., ) ( ) ( )d
d
I
d a a d df A f R A f x x I x I x dx dx then
I
. The mapping
2
: ( )L T A is
1
( ) i
n
I I
i A
i
f f g dS X
Τ T and from that we get in the same way
1
[ ]A i A i
i
X e E X e
almost surely.
Karhunen-Loève expansion of a set indexed Brownian motion:
Let 1 2, ,..., d and 1 2, ,..., d be a eigenvalues and normalized eigenfunctions of operator
2 2
: ( ) ( )L L A A
1 2( ( )) ( ( , ,..., ))dA x x x
[0,1]
( , ) ( )
d
R A S S dS
( , )
1 1 2 2 1 2 1 2
[0,1]
min{ , }min{ , } min{ , } ( , ,..., )
d
R A S
d d d ds x s x s x s s s ds ds ds
When 1 2[0, ] [0, ] ... [0, ]dA x x x A , 1 2[0, ] [0, ] ... [0, ] .dS s s s A
That is i and i solve the integral equation
1 2 1 1 2 2 1 2 1 2
[0,1]
( , ,..., ) min{ , }min{ , } min{ , } ( , ,..., )
d
d d d d dx x x s x s x s x s s s ds ds ds
And
1 2 1 2 1 2
[0,1]
1 ,
( , ,..., ) ( , ,..., )
0 ,d
i d j d d
i j
s s s s s s ds ds ds
i j
Suppose that, there exist a 1{ }d
i i such that 1 2 1 1 2 2( , ,..., ) ( ) ( ) ( )d d dx x x x x x for all
1 2( , ,..., ) [0,1]d
dx x x then
1 1
1 1 1 1 1 1 1
0 0
( ) ( ) ( )min{ , } ( )min{ , }d d d d d d dx x s s x ds s s x ds (3)
Denote
d
. If for all i ,
1
0
( ) ( )min{ , }i i i i i i ix s s x ds then we get the required on (3).It is clearthat,
1 1
0 0
( ) ( )min{ , } ( )
i
i
x
i i i i i i i i i i i i i
x
x s s x ds s ds x s ds for all i . Differentiating both sides with
respect to ix generates:
1
'( ) ( )
i
i i i i i
x
x s ds , ''( ) ( )i i i ix x for all i , togetherwith boundary
condition (0) 0i . The solutions of this pair of differential equations are given by:
1
, 2( ) 2sin( (2 1) )i n i ix n x ,
2
2
(2 1)n n
Then
1
1 2 1, 1 2, 2 , 2
1
( , ,..., ) ( ) ( ) ( ) 2 sin( (2 1) )
d
n d n n d n d i
i
x x x x x x n x
,
2
2
(2 1)
d
n n
The Karhunel-Loève expansion of X is obtained by setting i i i , we get:
7. Reproducing Kernel Hilbert Spaceof A Set Indexed Brownian Motion
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1
( )A n n n
n
X e A
In other words, 1 2
2 2 1 1
[0, ] [0, ] ... [0, ] 2(2 1)
1 1
( ) sin( (2 1) )d
d
d
d
x x x n in
n i
X e n x
(4)
Second case: [0,1]d
T and ( )LsA = A
Let 1 2( , ,..., ) [0,1]d
dx x x x , wedenote 1 2[0, ] [0, ] [0, ] ... [0, ]dx x x x . There existsan increasing
sequence of finite sub-classes 1([0,1] ) { ,..., }m
m d m m
kA A A Aclosed under intersection:
2
([0,1] ) {[0, ]: , 0 2 , 1,2,..., }i
m
km d m
i ix x k i d A .From the Definition 1 we derive that:
1 1 2
2 2 2 2 2
([0, ]) [0,( ,..., )] [0, ] [0, ] ... [0, ]d d
m m m m m
k kk k k
mg x , ( )mg and for all ( )A LsA ,
( )m mA g A .
Let ( )A LsA then from (4),
1 21 2
2 2 2
2 2 1 1
([0, ] [0, ] ... [0, ]) 2(2 1) 2[0, ] [0, ] ... [0, ]
1 1
( ) sin( (2 1) )i
kk k d mdm d
m m m
d
kd
g x x x nn
n i
X X e n
The process X is continuous and then from Definition 1 we derive:
1 2
2 2 1 1
([0, ] [0, ] ... [0, ]) 2(2 1) 2
1 1
lim lim( ) sin( (2 1) )i
d m
m d
d
kd
g x x x nnm m
n i
X e n
1 2
2 2 2
2 2 1 1
( ) 2(2 1) 2([0, ] [0, ] ... [0, ])
1 1
lim( ) sin( (2 1) )i
kk k d mdm m m m m m m
d
kd
A g A nng m
n i
X X X e n
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