This document summarizes the BTW sandpile model on a square lattice and defines relevant concepts. It describes how sandpiles are represented as height functions on the lattice, stable configurations, toppling rules, addition of sandpiles, and the group structure of recurrent sandpiles. Algorithms to find the identity of this group are developed in later sections.
1) The document discusses double integrals and methods for calculating them, including using iterated integrals and Riemann sums. Double integrals can represent volumes under surfaces.
2) Examples are provided to demonstrate calculating double integrals over rectangles and general regions using iterated integrals and partitioning the region.
3) There are two types of general regions: type I defined by ≤≤≤≤ and type II defined by ≤≤≤≤. The document provides methods for calculating double integrals over these region types.
The document summarizes the Frame-Stewart algorithm for solving the generalized Tower of Hanoi puzzle with n disks on k pegs. It begins by introducing the standard 3-peg Tower of Hanoi puzzle and recursive solution. It then describes Henry Dudeney's 4-peg variation and the Frame-Stewart algorithm from 1939 for solving the problem with n disks on any number of pegs k. The algorithm uses recursion and finding the optimal partition of disks to minimize the number of moves. The document proves properties of the number of additional moves between problems sizes.
The document summarizes research on magnetic monopoles in noncommutative spacetime. It begins by motivating noncommutative spacetime as a way to incorporate quantum gravitational effects. It then shows that attempting to quantize spacetime by imposing noncommutativity of coordinates leads to inconsistencies when trying to define a Wu-Yang magnetic monopole in this framework. Specifically, the potentials describing the monopole fail to simultaneously satisfy Maxwell's equations and transform correctly under gauge transformations when expanded to second order in the noncommutativity parameter. This suggests the Dirac quantization condition cannot be satisfied in noncommutative spacetime. Possible reasons for this failure and directions for future work are discussed.
Double integrals are used to calculate properties of planar laminas such as mass, center of mass, and moments of inertia by integrating a density function over a region. The inner integral is evaluated first, treating the other variable as a constant. Properties include:
1) Total mass by double integrating the density function over the region.
2) Center of mass coordinates by taking moments about axes and dividing by total mass.
3) Moments of inertia by double integrating the distance squared from an axis times the density.
Coincidence points for mappings under generalized contractionAlexander Decker
1. The document presents a theorem that establishes conditions for the existence of coincidence points between multi-valued and single-valued mappings.
2. It generalizes previous results by Feng and Liu (2004) and Liu et al. (2005) by relaxing the contraction conditions.
3. The main theorem proves that if mappings T and f satisfy generalized contraction conditions involving α and β functions, and the space is orbitally complete, then the mappings have a coincidence point.
The document introduces the concept of generalized quasi-nonexpansive (GQN) maps. Some key results are:
1) GQN maps generalize quasi-nonexpansive maps but the fixed point set may not always be closed or convex.
2) If a subset satisfies certain conditions, it is a GQN-retract of the space.
3) Under these conditions, the class of GQN-retracts is closed under intersection and the common fixed point set of an increasing sequence of GQN maps is a GQN-retract.
The document discusses applications of graphs with bounded treewidth. It covers the following key points:
1) Courcelle's theorem shows that many NP-complete graph problems can be solved in linear time for graphs of bounded treewidth using monadic second-order logic. This includes problems like independent set, coloring, and Hamiltonian cycle.
2) The treewidth of a graph is closely related to its largest grid minor - graphs with large treewidth contain large grid minors. There are polynomial relationships between treewidth and largest grid minor for planar graphs.
3) Planar graphs have bounded treewidth if and only if they exclude some grid configuration as a contraction. This helps characterize planar graphs of bounded treewidth
1) The document discusses double integrals and methods for calculating them, including using iterated integrals and Riemann sums. Double integrals can represent volumes under surfaces.
2) Examples are provided to demonstrate calculating double integrals over rectangles and general regions using iterated integrals and partitioning the region.
3) There are two types of general regions: type I defined by ≤≤≤≤ and type II defined by ≤≤≤≤. The document provides methods for calculating double integrals over these region types.
The document summarizes the Frame-Stewart algorithm for solving the generalized Tower of Hanoi puzzle with n disks on k pegs. It begins by introducing the standard 3-peg Tower of Hanoi puzzle and recursive solution. It then describes Henry Dudeney's 4-peg variation and the Frame-Stewart algorithm from 1939 for solving the problem with n disks on any number of pegs k. The algorithm uses recursion and finding the optimal partition of disks to minimize the number of moves. The document proves properties of the number of additional moves between problems sizes.
The document summarizes research on magnetic monopoles in noncommutative spacetime. It begins by motivating noncommutative spacetime as a way to incorporate quantum gravitational effects. It then shows that attempting to quantize spacetime by imposing noncommutativity of coordinates leads to inconsistencies when trying to define a Wu-Yang magnetic monopole in this framework. Specifically, the potentials describing the monopole fail to simultaneously satisfy Maxwell's equations and transform correctly under gauge transformations when expanded to second order in the noncommutativity parameter. This suggests the Dirac quantization condition cannot be satisfied in noncommutative spacetime. Possible reasons for this failure and directions for future work are discussed.
Double integrals are used to calculate properties of planar laminas such as mass, center of mass, and moments of inertia by integrating a density function over a region. The inner integral is evaluated first, treating the other variable as a constant. Properties include:
1) Total mass by double integrating the density function over the region.
2) Center of mass coordinates by taking moments about axes and dividing by total mass.
3) Moments of inertia by double integrating the distance squared from an axis times the density.
Coincidence points for mappings under generalized contractionAlexander Decker
1. The document presents a theorem that establishes conditions for the existence of coincidence points between multi-valued and single-valued mappings.
2. It generalizes previous results by Feng and Liu (2004) and Liu et al. (2005) by relaxing the contraction conditions.
3. The main theorem proves that if mappings T and f satisfy generalized contraction conditions involving α and β functions, and the space is orbitally complete, then the mappings have a coincidence point.
The document introduces the concept of generalized quasi-nonexpansive (GQN) maps. Some key results are:
1) GQN maps generalize quasi-nonexpansive maps but the fixed point set may not always be closed or convex.
2) If a subset satisfies certain conditions, it is a GQN-retract of the space.
3) Under these conditions, the class of GQN-retracts is closed under intersection and the common fixed point set of an increasing sequence of GQN maps is a GQN-retract.
The document discusses applications of graphs with bounded treewidth. It covers the following key points:
1) Courcelle's theorem shows that many NP-complete graph problems can be solved in linear time for graphs of bounded treewidth using monadic second-order logic. This includes problems like independent set, coloring, and Hamiltonian cycle.
2) The treewidth of a graph is closely related to its largest grid minor - graphs with large treewidth contain large grid minors. There are polynomial relationships between treewidth and largest grid minor for planar graphs.
3) Planar graphs have bounded treewidth if and only if they exclude some grid configuration as a contraction. This helps characterize planar graphs of bounded treewidth
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.
The document discusses uncertainty quantification (UQ) using quasi-Monte Carlo (QMC) integration methods. It introduces parametric operator equations for modeling input uncertainty in partial differential equations. Both forward and inverse UQ problems are considered. QMC methods like interlaced polynomial lattice rules are discussed for approximating high-dimensional integrals arising in UQ, with convergence rates superior to standard Monte Carlo. Algorithms for single-level and multilevel QMC are presented for solving forward and inverse UQ problems.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
This document provides examples and explanations of double integrals. It defines a double integral as integrating a function f(x,y) over a region R in the xy-plane. It then gives three key points:
1) To evaluate a double integral, integrate the inner integral first treating the other variable as a constant, then integrate the outer integral.
2) The easiest regions to integrate over are rectangles, as the limits of integration will all be constants.
3) For non-rectangular regions, the limits of integration may be variable, requiring more careful analysis to determine the limits for each integral.
Kittel c. introduction to solid state physics 8 th edition - solution manualamnahnura
1. The document discusses crystallographic planes and directions in a cube, the Miller indices of planes with respect to primitive axes, and the spacing between dots projected onto different planes of a crystal structure.
2. Key concepts from crystallography such as Miller indices, primitive lattice vectors, reciprocal lattice vectors, and the first Brillouin zone are defined. Calculations of interplanar spacing and lattice parameters are shown for simple cubic and face-centered cubic lattices.
3. Binding energies, cohesive energies, and equilibrium properties are calculated and compared for body-centered cubic and face-centered cubic crystal structures. Approximations made in describing crystal binding using Madelung energies and pair potentials are
This document summarizes the concept of bidimensionality and how it can be used to design subexponential algorithms for graph problems on planar and other graph classes. It discusses how bidimensionality can be defined for parameters that are closed under minors or contractions by relating their behavior on grid graphs. It presents examples like vertex cover and dominating set that are bidimensional. It also discusses how bidimensionality can be extended to bounded genus graphs and H-minor free graphs using grid-minor/contraction theorems.
1) The document discusses proximal algorithms for solving inverse problems in probability spaces, where the goal is to estimate an unknown variable x given noisy measurements y.
2) It describes using Bayesian methods like maximum a posteriori (MAP) estimation and Markov chain Monte Carlo (MCMC) to account for uncertainty, where the posterior distribution p(x|y) is assumed to be log-concave.
3) Proximal algorithms like the unadjusted Langevin algorithm (ULA) and proximal ULA (MYULA) are proposed for sampling from the posterior in high dimensions when p(x|y) is not differentiable.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
This document provides information about multiple integrals and examples of evaluating them. It begins by defining multiple integrals and iterated integrals. It then gives 4 examples of evaluating double and triple integrals over different regions. These regions include rectangles, triangles, and solids. The document also discusses Fubini's theorem, which allows reversing the order of integration in certain cases. It concludes by providing an example of converting an integral from Cartesian to polar coordinates.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
This document summarizes part of a paper by D.G. Northcott on the notion of a first neighbourhood ring, with an application to the Af+Bφ theorem. It introduces the concept of a first neighbourhood ring and superficial elements. The key points are:
1) It defines a superficial element as one where amv+s = mv+s for large v, generalizing the definition to allow zero divisors.
2) It proves three properties are equivalent for an element a to be superficial: the form ideal no:(φ) is the isolated component no, amv = mv+s for large v, and nv+s:(a) = mv for large v.
3) It shows
The document discusses computational aspects of stochastic phase-field models. It begins by motivating the inclusion of thermal noise in phase-field simulations through examples of dendrite formation. It then provides background on the deterministic phase-field and Allen-Cahn models before introducing the stochastic Allen-Cahn equation with additive white noise. The remainder of the document discusses the importance of studying this problem both theoretically and computationally, as well as outlining the topics to be covered in more depth.
This document summarizes the analysis of invariant mass distributions and decay angles of relativistically boosted psi and B meson decays using ROOT. It examines the invariant mass distributions of daughter particles from two-body ψ(3770) decays, three-body B+ decays, and four-body D0 decays. The decay angles of particles are also analyzed in ψ(3770) and B decays. Monte Carlo simulations were performed to generate the decays and ROOT libraries were used to calculate invariant masses and analyze the results.
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
- The document discusses the 4/3 problem as it relates to the gravitational field of a uniform massive ball moving at constant velocity.
- It derives expressions for the gravitational field potentials both inside and outside the moving ball using the superposition principle and Lorentz transformations.
- Calculations show that the effective mass of the gravitational field found from the field energy does not equal the effective mass found from the field momentum, with a ratio of approximately 4/3, demonstrating that the 4/3 problem exists for gravitational fields as it does for electromagnetic fields.
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.
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.
Non-informative reparametrisation for location-scale mixturesChristian Robert
1) The document proposes a new reparameterization of location-scale mixtures in terms of the global mean and variance of the mixture distribution. This constrains the component parameters to a specific region of parameter space.
2) A Metropolis-within-Gibbs algorithm is developed for the reparameterized mixture model and implemented in the Ultimixt R package. This allows accurate estimation of component parameters through MCMC.
3) The reparameterization is shown to work well for mixtures of normal distributions, allowing good mixing of chains and convergence to the true densities, even in overfitted cases.
- The document introduces Gaussian processes for regression and classification.
- Gaussian processes assume a probabilistic relationship between input and output variables, and place a probability distribution directly over functions.
- Key properties are that any finite number of function values have a joint Gaussian distribution, and the covariance between values is specified by a kernel function.
- Inference yields a Gaussian posterior distribution over functions, from which predictions at new points can be made analytically as Gaussian distributions.
The document provides a detailed history of psychology from ancient times through the 19th century. Some key points covered include:
- Ancient Greek philosophers like Plato and Hippocrates made early contributions to the field by introducing principles of scientific medicine and suggesting the brain is the seat of mental processes.
- During medieval times, Islamic physicians developed early concepts of clinical psychiatry and psychotherapy and built some of the first psychiatric hospitals.
- In the 16th-17th centuries, philosophers like Descartes introduced mind-body dualism while others like Spinoza argued the mind and body are one.
- The 18th century saw the coining of the term "psychology" and early empirical studies of the
The document discusses labor relations in the new economy and issues faced by independent contractors, such as misclassification by employers to avoid responsibilities. It explores debates around how independent contractors are distinguished from employees under labor laws. The emergence of organizations like the Freelancers Union is examined as an example of how workers in the new economy are self-organizing in nontraditional ways outside of traditional employee protections.
Data jumlah dan nama sekolah pengajuan rehab 2012 2013Tarara Salala
Dokumen tersebut berisi data sekolah dan rencana prioritas rehabilitasi, pembangunan perpustakaan, serta pengadaan alat peraga pendidikan beberapa sekolah di Kecamatan Kota Kudus termasuk MI NU Tahfidhul Qur'an TBS yang memiliki 237 siswa dan membutuhkan rehabilitasi gedung yang rusak sedang.
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.
The document discusses uncertainty quantification (UQ) using quasi-Monte Carlo (QMC) integration methods. It introduces parametric operator equations for modeling input uncertainty in partial differential equations. Both forward and inverse UQ problems are considered. QMC methods like interlaced polynomial lattice rules are discussed for approximating high-dimensional integrals arising in UQ, with convergence rates superior to standard Monte Carlo. Algorithms for single-level and multilevel QMC are presented for solving forward and inverse UQ problems.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
This document provides examples and explanations of double integrals. It defines a double integral as integrating a function f(x,y) over a region R in the xy-plane. It then gives three key points:
1) To evaluate a double integral, integrate the inner integral first treating the other variable as a constant, then integrate the outer integral.
2) The easiest regions to integrate over are rectangles, as the limits of integration will all be constants.
3) For non-rectangular regions, the limits of integration may be variable, requiring more careful analysis to determine the limits for each integral.
Kittel c. introduction to solid state physics 8 th edition - solution manualamnahnura
1. The document discusses crystallographic planes and directions in a cube, the Miller indices of planes with respect to primitive axes, and the spacing between dots projected onto different planes of a crystal structure.
2. Key concepts from crystallography such as Miller indices, primitive lattice vectors, reciprocal lattice vectors, and the first Brillouin zone are defined. Calculations of interplanar spacing and lattice parameters are shown for simple cubic and face-centered cubic lattices.
3. Binding energies, cohesive energies, and equilibrium properties are calculated and compared for body-centered cubic and face-centered cubic crystal structures. Approximations made in describing crystal binding using Madelung energies and pair potentials are
This document summarizes the concept of bidimensionality and how it can be used to design subexponential algorithms for graph problems on planar and other graph classes. It discusses how bidimensionality can be defined for parameters that are closed under minors or contractions by relating their behavior on grid graphs. It presents examples like vertex cover and dominating set that are bidimensional. It also discusses how bidimensionality can be extended to bounded genus graphs and H-minor free graphs using grid-minor/contraction theorems.
1) The document discusses proximal algorithms for solving inverse problems in probability spaces, where the goal is to estimate an unknown variable x given noisy measurements y.
2) It describes using Bayesian methods like maximum a posteriori (MAP) estimation and Markov chain Monte Carlo (MCMC) to account for uncertainty, where the posterior distribution p(x|y) is assumed to be log-concave.
3) Proximal algorithms like the unadjusted Langevin algorithm (ULA) and proximal ULA (MYULA) are proposed for sampling from the posterior in high dimensions when p(x|y) is not differentiable.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
This document provides information about multiple integrals and examples of evaluating them. It begins by defining multiple integrals and iterated integrals. It then gives 4 examples of evaluating double and triple integrals over different regions. These regions include rectangles, triangles, and solids. The document also discusses Fubini's theorem, which allows reversing the order of integration in certain cases. It concludes by providing an example of converting an integral from Cartesian to polar coordinates.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
This document summarizes part of a paper by D.G. Northcott on the notion of a first neighbourhood ring, with an application to the Af+Bφ theorem. It introduces the concept of a first neighbourhood ring and superficial elements. The key points are:
1) It defines a superficial element as one where amv+s = mv+s for large v, generalizing the definition to allow zero divisors.
2) It proves three properties are equivalent for an element a to be superficial: the form ideal no:(φ) is the isolated component no, amv = mv+s for large v, and nv+s:(a) = mv for large v.
3) It shows
The document discusses computational aspects of stochastic phase-field models. It begins by motivating the inclusion of thermal noise in phase-field simulations through examples of dendrite formation. It then provides background on the deterministic phase-field and Allen-Cahn models before introducing the stochastic Allen-Cahn equation with additive white noise. The remainder of the document discusses the importance of studying this problem both theoretically and computationally, as well as outlining the topics to be covered in more depth.
This document summarizes the analysis of invariant mass distributions and decay angles of relativistically boosted psi and B meson decays using ROOT. It examines the invariant mass distributions of daughter particles from two-body ψ(3770) decays, three-body B+ decays, and four-body D0 decays. The decay angles of particles are also analyzed in ψ(3770) and B decays. Monte Carlo simulations were performed to generate the decays and ROOT libraries were used to calculate invariant masses and analyze the results.
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
- The document discusses the 4/3 problem as it relates to the gravitational field of a uniform massive ball moving at constant velocity.
- It derives expressions for the gravitational field potentials both inside and outside the moving ball using the superposition principle and Lorentz transformations.
- Calculations show that the effective mass of the gravitational field found from the field energy does not equal the effective mass found from the field momentum, with a ratio of approximately 4/3, demonstrating that the 4/3 problem exists for gravitational fields as it does for electromagnetic fields.
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.
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.
Non-informative reparametrisation for location-scale mixturesChristian Robert
1) The document proposes a new reparameterization of location-scale mixtures in terms of the global mean and variance of the mixture distribution. This constrains the component parameters to a specific region of parameter space.
2) A Metropolis-within-Gibbs algorithm is developed for the reparameterized mixture model and implemented in the Ultimixt R package. This allows accurate estimation of component parameters through MCMC.
3) The reparameterization is shown to work well for mixtures of normal distributions, allowing good mixing of chains and convergence to the true densities, even in overfitted cases.
- The document introduces Gaussian processes for regression and classification.
- Gaussian processes assume a probabilistic relationship between input and output variables, and place a probability distribution directly over functions.
- Key properties are that any finite number of function values have a joint Gaussian distribution, and the covariance between values is specified by a kernel function.
- Inference yields a Gaussian posterior distribution over functions, from which predictions at new points can be made analytically as Gaussian distributions.
The document provides a detailed history of psychology from ancient times through the 19th century. Some key points covered include:
- Ancient Greek philosophers like Plato and Hippocrates made early contributions to the field by introducing principles of scientific medicine and suggesting the brain is the seat of mental processes.
- During medieval times, Islamic physicians developed early concepts of clinical psychiatry and psychotherapy and built some of the first psychiatric hospitals.
- In the 16th-17th centuries, philosophers like Descartes introduced mind-body dualism while others like Spinoza argued the mind and body are one.
- The 18th century saw the coining of the term "psychology" and early empirical studies of the
The document discusses labor relations in the new economy and issues faced by independent contractors, such as misclassification by employers to avoid responsibilities. It explores debates around how independent contractors are distinguished from employees under labor laws. The emergence of organizations like the Freelancers Union is examined as an example of how workers in the new economy are self-organizing in nontraditional ways outside of traditional employee protections.
Data jumlah dan nama sekolah pengajuan rehab 2012 2013Tarara Salala
Dokumen tersebut berisi data sekolah dan rencana prioritas rehabilitasi, pembangunan perpustakaan, serta pengadaan alat peraga pendidikan beberapa sekolah di Kecamatan Kota Kudus termasuk MI NU Tahfidhul Qur'an TBS yang memiliki 237 siswa dan membutuhkan rehabilitasi gedung yang rusak sedang.
Dokumen tersebut membahas tentang penggunaan konsep integral dalam menghitung luas daerah di bawah kurva dan volume benda putar. Termasuk menjelaskan rumus-rumus untuk menghitung luas daerah yang dibatasi oleh dua kurva, sumbu x, dan batas integral. Juga memberikan contoh soal dan pembahasan untuk menghitung luas daerah tertentu.
The document appears to be a slide from a Haiku Deck presentation that contains photo credits from various photographers and sources. Towards the bottom it encourages the viewer to create their own Haiku Deck presentation on SlideShare and provides a link to get started.
Process safety helps to avoid catastrophic accidents in process plants. Maxims and quotes carry important truths. These maxims emphasize fundamental points for facility employees.
CRE8 is a strategic creative solutions company established in 2001 in the UAE. It offers integrated branding, design, photography, and event management services to clients. Led by Managing Director Richard Shirazian, CRE8 has an experienced in-house design team. The company has expanded its photography studio and digital printing capabilities. CRE8's philosophy is to strive for innovation and creative solutions that meet clients' needs. It works on various projects and aims to deliver distinctive solutions on time and on budget.
This document provides a chronological overview of the history of psychology from ancient times through the 1890s. Some key points mentioned include:
- Ancient Greek philosophers like Plato and Hippocrates made early contributions to the study of the mind and mental processes.
- Beginning in the Middle Ages, Islamic scholars made advances in clinical psychology and established early psychiatric hospitals.
- In the 16th-17th centuries, philosophers like Descartes introduced mind-body dualism and theories of consciousness.
- The 18th-19th centuries saw the development of fields like psychophysics and physiological psychology. Pioneers included Wundt, Fechner, Helmholtz.
- The late 19th century saw the
Simone Fonseca is a design and communications specialist based in the UK. She has experience pitching new business opportunities, including presenting a Brazilian steakhouse concept to property developers and hosting an art exhibition in a landmark building. Her portfolio and contact details are provided.
This document appears to be a lesson plan that includes questions about clothing items and a transition to the first set of slides. It ends by saying goodbye to students and looking forward to the next class.
Iraz Öksüz has over 20 years of experience working with SAP modules such as FI, CO, AA, PC, SD, PP, MM, BW, and SM. He has extensive experience implementing SAP solutions, managing projects, and providing support. Currently, he works as an Application Support Manager at Unilever in Istanbul, where he is responsible for user access management, security reviews, incident management, and ensuring compliance.
This document provides estimates for several number theory functions without assuming the Riemann Hypothesis (RH), including bounds for ψ(x), θ(x), and the kth prime number pk. The following estimates are derived:
1) θ(x) - x < 1/36,260x for x > 0.
2) |θ(x) - x| < ηk x/lnkx for certain values of x, where ηk decreases as k increases.
3) Estimates are obtained for θ(pk), the value of θ at the kth prime number pk, showing θ(pk) is approximately k ln k + ln2 k - 1.
1) Show that the derivative of the function xy = ex-y is log(x)/(log(xe))^2
2) Calculate the rate of increase of the height of a cone formed by sand pouring from a pipe when the height is 4cm.
3) Find the equation of the tangent line to the curve y = √3x - 2 that is parallel to the line 4x - 2y + 5 = 0.
4) Prove the identity tan-1(1) + tan-1(2) + tan-1(3) = π.
This chapter introduces discrete and continuous dynamical systems through examples. Discrete examples include rotations and expanding maps of the circle, as well as endomorphisms and automorphisms of the torus. Continuous examples include flows generated by autonomous differential equations. Periodic points are also defined and analyzed for specific examples. Basic constructions for building new dynamical systems from existing ones are described.
(1) The document discusses various integration techniques including: review of integral formulas, integration by parts, trigonometric integrals involving products of sines and cosines, trigonometric substitutions, and integration of rational functions using partial fractions.
(2) Examples are provided to demonstrate each technique, such as using integration by parts to evaluate integrals of the form ∫udv, using trigonometric identities to reduce powers of trigonometric functions, and using partial fractions to break down rational functions into simpler fractions.
(3) The key techniques discussed are integration by parts, trigonometric substitutions to transform integrals involving quadratic expressions into simpler forms, and partial fractions to decompose rational functions for integration. Various examples illustrate the
Some properties of two-fuzzy Nor med spacesIOSR Journals
The study sheds light on the two-fuzzy normed space concentrating on some of their properties like convergence, continuity and the in order to study the relationship between these spaces
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?
The document discusses complex eigenvalues and eigenvectors for systems of linear differential equations. It shows that if the matrix A has complex conjugate eigenvalue pairs r1 and r2, then the corresponding eigenvectors and solutions will also be complex conjugates. This leads to real-valued fundamental solutions that can express the general solution. An example demonstrates these concepts, finding the complex eigenvalues and eigenvectors and expressing the general solution in terms of real-valued functions. Spiral points, centers, eigenvalues, and trajectory behaviors are also summarized.
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1. ALGORITHMICALLY FINDING THE IDENTITY OF THE BTW
SANDPILE GROUP
JACOB HAVEN AND ATTILA P´OR
Abstract. Algorithms to find the identity of the group of recurrent BTW
sandpiles are described. These algorithms are used to provide experimental
data, from which conjectures about the structure of the identity are tested.
1. Introduction
The BTW sandpile model was introduced by Bak, Tang and Wiesenfeld in
1988 [1] to study the 1/f noise. In their paper, they describe one, two and three
dimensional sandpiles as grids of heights (indexed by d-tuples, where d is the di-
mension), with each site toppling sand onto its 2d neighbors should it ever reach
a height of 2d. Since its introduction, the BTW sandpile model has been stud-
ied extensively by physicists for its demonstration of self-organized criticality, the
tendency to approach a critical state from many starting positions. Many mod-
ifications of the BTW model have been created, but the most important is the
generalization by Dhar in [5], [4], and [6] of the BTW model to the Abelian Sand-
pile Model (ASM). The ASM is defined for arbitrary directed graphs, represented
as toppling matrices, ∆, which describe the rules for when to topple at a site and
which sites (neighbors) to topple to. The algebra of the ASM can be studied to give
general mathematical results that may then be applied to special cases, such as the
BTW model. A good introduction to the general ASM, upon which the simplified
model here is based, is given in [8]. Most relevant to the topic is the finding that
the recurrent elements of the ASM have a group structure whose identity has been
studied, with some results, in [2] and [7].
Outline. The remainder of this article is organized as follows. Section 2 precisely
defines the specific BTW model studied and gives account of relevant previous
results. Section 3 describes the derivation of the algorithms used to find the identity
of the group of recurrent states. Section 4 analyses the identities produced to give
conjectures about their structure and fractal nature. Finally, Section 5 introduces
some further questions warranting future study.
2. BTW Sandpile Model on a Square Lattice
2.1. Sandpiles. We will be discussing the BTW sandpile model on the n × n grid,
so we shall define the set of indices of this grid. Let Nn be the set of all integers
from 1 to n. N2
n = Nn ×Nn is thus our set of indices. A sandpile on the n×n grid is
a height function from the indices of the grid to the natural numbers (starting with
0). Let Sn be the set of all n × n sandpiles. For an arbitrary sandpile, η ∈ Sn and
2000 Mathematics Subject Classification. 82B20.
1
2. 2 JACOB HAVEN AND ATTILA P ´OR
a position x = (i, j) ∈ N2
n, η(x) = η(i, j) = z, where z ∈ N is the height (number
of sand grains) of the sandpile at x. A stable sandpile is one which has maximum
height 3. Let Ωn be the set of all stable sandpiles, which is clearly a subset of Sn.
Let Qn and Zn be sandpile sets defined similarly to Sn, with the sandpiles returning
rational numbers and integers respectively, instead of natural numbers. We assign
the names αnand ωnto the minimum and maximum stable sandpiles that return 0
and 3 respectively.
Nn = {k ∈ Z+
| k ≤ n} = {1, 2, . . . , n}
N2
n = Nn × Nn = {(1, 1), . . . , (n, n)}
Sn = {η | η : N2
n → N}
Ωn = {η | η : N2
n → {0, 1, 2, 3}}
Qn = {η | η : N2
n → Q}
Zn = {η | η : N2
n → Z}
∀x ∈ N2
n :
αn(x) = 0
ωn(x) = 3
(1)
We will now define sandpile equality and partial ordering
Definition 2.1 (Sandpile Comparison). For all η, ζ ∈ Qn:
η = ζ if and only if η(x) = ζ(x) for all x ∈ N2
n
The comparisons ≤, ≥, <, and > are defined similarly.
max(η) = η(xmax), where η(xmax) ≥ η(x), for all x ∈ N2
n
min(η) is defined similarly.
For convenience, we will define the standard matrix representations of an arbi-
trary sandpile to be the matrix of all values it returns.
Definition 2.2 (Matrix Form). For η ∈ Qn and {x1, x2, . . . , xn2 } = N2
n:
(2) mat(η) =
η(1, 1) . . . η(1, n)
...
...
...
η(n, 1) . . . η(n, n)
2.2. Toppling. For sandpiles on the grid, we are interested in the von Neumann
neighborhood, N(x), of each point x ∈ N2
n. This neighborhood contains all the
points directly above, below, and to the left or right of x. The number of neighbors
of x, | N(x)|, is thus four in the center of the grid, three on the edges, and two in
the corners. Also note that y ∈ N(x) if and only if x ∈ N(y).
Definition 2.3 (von Neumann Neighborhood). For all For all x = (x1, x2) ∈ N2
n
and y = (y1, y2) ∈ N2
n, let dp = p
|x1 − y1|p + |x2 − y2|p. Thus, d1 = |x1 − y1| +
|x2 − y2|
N(x) = {y ∈ N2
n | d 1xy = 1}
The model will be represented using the n2
×n2
lattice laplacian toppling matrix,
∆n
, which completely describes the relationships between each lattice point. A
3. FINDING THE SANDPILE IDENTITY 3
point relates to itself with the number of its neighbors should the boundary be
removed (i.e.: all 4s), and to its neighbors with -1. Notice that as neighbor relations
are symmetric (the graph is undirected), so is ∆n
Definition 2.4 (Lattice Laplacian Toppling Matrix). For all indices x ∈ N2
n, and
y ∈ N2
n:
∆n
x,y =
4 if x = y
−1 if y ∈ N(x)
0 otherwise
With this toppling matrix, we may describe a types of operation on sandpiles
called firing and toppling rules. A firing rule acts on single site, x, by removing four
grains of sand from it and adding one to each of its neighbors, or in other words,
it fires x. A toppling rule fires only unstable sites.
Definition 2.5 (Toppling Rules). A firing rule is an operation Fx : Zn → Zn
For all η ∈ Zn and x, y ∈ N2
n:
Fx(η)(y) = η(y) − ∆n
x,y
A toppling rule is an operation Tx : Sn → Sn such that for all η ∈ Sn:
Tx(η) =
Fx(η) if η(x) ≥ 4
η otherwise
By composing a finite number of toppling rules, we may obtain a toppling func-
tion that relaxes an arbitrary sandpile, η ∈ Sn to a unique stable sandpile, called
the relaxation of η.
Definition 2.6 (Toppling Function). The toppling function, T∆n : Sn → Ωn, is
defined for some minimal sequence (x1, x2, . . . , xN ), where xi ∈ N2
n, to be:
T∆n =
N
i=1
T xi = T xi ◦ T x2 ◦ . . . ◦ T xN
(x1, x2, . . . , xN ) is minimal in the sense that for η ∈ (Sn ⊕ Ωn) (i.e.: η is unstable).
T∆n (η) =
N−1
i=1
T xi (η) ∈ Ωn
For convenience, we will denote the relaxation of η ∈ Sn as η = T∆n (η).
Also, η = ζ may be denoted η → ζ.
It is proven in section 2.3 of [8] that T∆n is well defined by Definition 2.6, using
the fact the toppling rules are commutative (i.e.: Tx ◦ Ty = Ty ◦ Tx). It is also
worthwhile to note that Tx ◦T∆n = T∆n , for all x ∈ N2
n, as toppling rules only act
when a site is unstable. Following from this, we can see that T k
∆n = T∆n for all
k ≥ 1.
4. 4 JACOB HAVEN AND ATTILA P ´OR
2.3. Addition. Let the operation of adding a single grain of sand at the site x ∈ N2
n
be denoted px : Qn → Qn. Let ax : Sn → Ωn, be the operation of adding a grain
of sand at x and allowing the sandpile to collapse. ax = T∆n ◦ px. Thus, for all
η ∈ Sn, px(η) → ax(η). Note that px and ax are both associative and commutative,
and that:
a
∆n
x,x
x =
y∈N(x)
a
−∆n
x,x
y
a4
x =
y∈N(x)
ay
(3)
The sum of two sandpiles is simply the sum of their heights at all grid points.
The sum with relaxation is the summation, followed by relaxation. Clearly, both
operations are associative and commutative.
Definition 2.7. Let us extend px, so that pk
x corresponds to adding k ∈ Q grains
of sand when k in nonnegative, and subtracting it is negative. For sandpiles η ∈ Sn
and ζ ∈ Sn, the summation of η and ζ is the result of summing the heights at all
points x ∈ N2
n.:
(η + ζ)(x) = η(x) + ζ(x) = ζ(x) + η(x)
η + ζ =
x∈N2
n
pζ(x)
x
(η) ∈ Sn.
Scalar multiplication for sandpiles is defined similarly to scalar multiplication
for vectors. For k ∈ Q:
(k · η)(x) = k · η(x)
k · η =
x∈N2
n
pk·ζ(x)
x
(η) ∈ Zn.
Subtraction of η ∈ Qn and ζinQn is defined as:
η − ζ = η + (−1) · ζ
Definition 2.8. For sandpiles η ∈ Sn and ζ ∈ Sn, the sum with relaxation is
η ⊕ ζ = η + ζ .
For convenience, let us also define the scalar multiplication by k ∈ Z+
k ⊗ η =
k η’s
η ⊕ . . . ⊕ η = k · η
Parentheses may be omitted because ⊕ is associative.
The following translations from operations to sandpile sums can prove useful:
∀η ∈ Qn px η = (px αn) + η(4)
∀η ∈ Sn ax η = (ax αn) ⊕ η(5)
5. FINDING THE SANDPILE IDENTITY 5
Theorem 2.9. For all η ∈ Sn, and ζ ∈ Sn:
η ⊕ ζ =
x∈N2
n
aζ(x)
x
(η)
Proof.
η ⊕ ζ = η + ζ =
T∆n
x∈N2
n
pζ(x)
x
(η)
We can now use the property that T∆n = T k
∆n for all k ≥ 1 to obtain:
η ⊕ ζ =
T n2
∆n
x∈N2
n
pζ(x)
x
(η) =
x∈N2
n
T∆n ◦ pζ(x)
x
(η) =
x∈N2
n
aζ(x)
x
(η).
Theorem 2.10. For η ∈ Sn and ζ ∈ Sn:
η ⊕ ζ = η ⊕ ζ = η ⊕ ζ
Proof. By Theorem 2.9:
η ⊕ ζ =
x∈N2
n
aζ(x)
x
(η)
Notice that in this form, Equation 3 is the same as applying Tx to ζ, where ζ(x) ≥ 4.
Thus, we may apply any number of Tx’s to ζ and still obtain the same result. Thus:
η ⊕ ζ =
x∈N2
n
aT∆n (ζ)(x)
x
(η) =
x∈N2
n
a ζ (x)
x
(η) = η ⊕ ζ
We may now reverse the order order of η and ζ and apply the same reduction to
obtain:
η ⊕ ζ = η ⊕ ζ = η ⊕ ζ
Corollary 2.11. From Theorem 2.10, we can see that in a relaxed sum of sandpiles,
only the outer relaxation must remain, with relaxation of the addends being optional.
Thus, η + ζ + θ = η + ζ + θ = η + ζ + θ = η + ζ + θ =. . .
2.4. Firing Sandpiles. The set Qn is an n2
dimensional vectorspace over the
rational numbers. Similarly Zn ⊂ Qn is an n2
dimensional vectorspace over the
integers.
Let δn
x ∈ Zn be the firing sandpile at x ∈ N2
n, corresponding to the xth row of
∆n
.
(6) ∀x, y ∈ N2
n δn
x (y) = ∆n
x,y
Let ϕ be the complete toppling operation on a sandpile that transforms it with
toppling matrix, ∆n
.
6. 6 JACOB HAVEN AND ATTILA P ´OR
Definition 2.12. ϕ : Qn → Qn. For all η ∈ Qn:
ϕ(η) = (∆n
)T
η =
x∈N2
n
η(x) · δn
x =
x∈N2
n
Fx
(η) ∈ Rn.
Since ∆n
is invertible, ϕ is a non-degenerate linear transformation of Qn and
∀η ∈ Qn ϕ-1
(η) = (∆n
)−1
η.
For any η ∈ Sn, ϕ-1
(η) gives the number of firings necessary at each point to
obtain the minimum sandpile.
(7)
x∈N2
n
Fϕ-1
(η)(x)
x
(η) = αn
Lemma 2.13. η ∈ Q+
n if and only if ϕ-1
(η)(x) ∈ Q+
n .
Proof. ϕ-1
(η)(x) ∈ Q+
n implies η ∈ Q+
n as:
η =
x∈N2
n
ϕ-1
(η)(x) · δn
x ≥
x∈N2
n
δn
x ≥ 0
Now, we will prove ϕ-1
(η)(x) ∈ Q+
n . Assume to the contrary that m = min(ϕ-1
(η)) <
0. Let ma = {x ∈ N2
n | ϕ-1
(η)(x) = m}. If x ∈ ma then
η(x) = ϕ(ϕ-1
(η))(x) = 4 ϕ-1
(η)(x) −
y∈N(x)
ϕ-1
(η)(y)
= 4m −
y∈N(x)
ϕ-1
(η)(y) ≤ (4 − | N(x)|) · m ≤ 0
Thus, η(x) ≤ 0, with equality if and only if | N(x)| = 4 and ϕ-1
(η)(y) = m for all
y ∈ N(x). Since η(x) ≥ 0 we have N(x) ⊂ ma whenever x ∈ ma. Thus, ma = N2
n,
meaning for all y ∈ N2
n, η(y) = m and | N(y)| = 4. Since there exist points on the
edges with fewer than 4 neighbors, this is a contradiction. Thus, m ≥ 0.
Now that we have the operator ϕ-1
, how we can be represent sandpiles as combi-
nations of the rows of ∆n
is of interest. Thus, we define Dn ⊂ Zn to be all integer
combinations of the rows of ∆n
, i.e.: the rowspace of ∆n
over Z. Let D+
n = Dn ∩Sn.
Lemma 2.14. η ∈ Dn if and only if ϕ-1
(η) ∈ Zn.
Proof. η ∈ Dn means there exists some ζ ∈ Zn such that (∆n
)T
ζ = η. Thus,
ϕ(ζ) = η and η = ϕ-1
(ζ).
Following directly from Lemmas 2.13 and 2.14:
Corollary 2.15. ϕ-1
(η) ∈ Sn if and only if η ∈ D+
n
Lemma 2.16. If η ∈ Ωn and ζ ∈ D+
n then ϕ-1
(η) ≤ ϕ-1
(η ⊕ ζ).
Proof. For some firing sequence (x1, x2, . . . , xN ), Let ζ0 = ζ and ζi = ζi−1 − δn
xi
and let η0 = η + ζ0 and ηi = η + ζi for all 0 ≤ i ≤ N:
η0 = η + ζ, η1 = η + ζ − δn
x1
, . . . , ηN = η + ζ −
N
i=1
δn
xi
= η ⊕ ζ
ζi ∈ Dn for all 0 ≤ i ≤ N by the following induction: ϕ-1
(ζ0) = ϕ-1
(ζ) ∈ D+
n .
Assuming ϕ-1
(ζk) ∈ Dn, then ϕ-1
(ζk+1) = p−1
xk
ϕ-1
(ζk) ∈ Dn.
7. FINDING THE SANDPILE IDENTITY 7
We show that ϕ-1
(ζi) ∈ Sn by induction.
From Corollary 2.15, ϕ-1
(ζ0) ∈ Sn.
Let us assume ϕ-1
(ζk) ∈ Sn for some 0 ≤ k < N.
As ϕ-1
(ζk+1) = p
(| N(xk+1)|−4)
xk+1 ϕ-1
(ζk), ϕ-1
(ζk+1) /∈ Sn if and only if ϕ-1
(ζk)(xk+1) <
4 − | N(xk+1)|) and ζk(xk+1) ≥ 4, and may thus be fired.
ζk(xk+1) =
y∈N2
n
ϕ-1
(ζk)(y)δn
y
(xk+1)
= 4 · ϕ-1
(ζk)(xk+1) −
z∈N(xk+1)
ϕ-1
(ζk)(z)δn
z
(xk+1)
≤ 4 · ϕ-1
(ζk)(xk+1) − ϕ-1
(ζk)| N(xk+1)|
≤ 4 · (4 − | N(xk+1|) − (4 − | N(xk+1|) · | N(xk+1)| − 1
= 16 − 8| N(xk+1)| + | N(xk+1)|2
− 1 ≤ 16 − (8) ∗ 2 + (2)2
− 1 = 3 < 4
Thus, if a firing at xk+1 would make ϕ-1
(ζk) /∈ Sn, that site is already stable,
and thus no firing will take place. Thus ϕ-1
ζk+1 ∈ Sn. And thus, by induction,
ϕ-1
ζN ∈ Sn.
As ϕ-1
(ζN ) ≥ αn,
ϕ-1
(η ⊕ ζ) = ϕ-1
(η + ζN ) ≥ ϕ-1
(η)
2.5. Group Properties. We will now define the reachability of a sandpile from
another sandpile. This coincides with the intuitive notion of a ”larger” sandpile,
up to relaxation.
Definition 2.17 (Reachability). A sandpile, η ∈ Ωn is reachable from ζ ∈ Sn if
and only if there there exists a sandpile θ ∈ Ωn (By Theorem 2.10, this is equivalent
to θ ∈ Sn), such that η = ζ ⊕ θ. This is denoted ζ → η.
ζ and η are said to communicate (ζ ∼ η) if and only if ζ → η and η → ζ.
Rechability may be used to define a class of stable sandpiles, known as reccurent
sandpiles, that are reachable from all stable sandpiles.
Definition 2.18 (Reccurent States). A recurrent sandpile, η ∈ Ωn, is one that is
reachable from all ζ ∈ Ωn. The set of all recurrent sandpiles is thus
Rn = {η ∈ Ωn | ∀ζ ∈ Ωn ζ → η}
Note that for all η ∈ Rn and ζ ∈ Rn, η ∼ ζ.
Two sandpiles are called equivalent if and only if there exists some sequence of
firings sandpile ϕ-1
(η − ζ) that transform between the η ∈ Qn and ζ ∈ Qn are
equivalent if and only if η − ζ ∈ Dn, denoted η ζ. Note that if ζ ∈ Ωn, η → ζ
implies η ζ, as some integer number of firings can performed on η to give ζ.
Corollary 2.19. If η ∈ Ωn, ζ ∈ Rn, and η ζ, then ϕ-1
(η) ≤ ϕ-1
(ζ). If η ∈ Rn,
then η = ζ.
Proof. For some θ ∈ D+
n , ζ = η ⊕ θ. Thus, ϕ-1
(ζ) = ϕ-1
(η ⊕ θ). By Lemma 2.16,
ϕ-1
(η) ≤ ϕ-1
(ζ)
If η ∈ Rn, then by the symmetry of , ϕ-1
(ζ) ≤ ϕ-1
(η). Thus, η = ζ.
8. 8 JACOB HAVEN AND ATTILA P ´OR
Corollary 2.20. If η, ζ ∈ Ωn, η ζ, and ϕ-1
(η) ≤ ϕ-1
(ζ), then η /∈ Rn.
Proof. Assume to the contrary that η ∈ Rn. Thus, by Corollary 2.19, ϕ-1
(ζ)(x) ≤
ϕ-1
(η), which is a contradiction.
Theorem 2.21. Rnwith the operation ⊕ forms an abelian group.
Proof. Let η ∈ Rn and ζ ∈ Rn
(1) ⊕ must be associative.
By Corollary 2.11:
a ⊕ (b ⊕ c) = a + b + c = a + b + c = (a ⊕ b) ⊕ c
Thus ⊕ is associative
(2) ⊕ must be commutative.
η ⊕ ζ = η + ζ = ζ + η = ζ ⊕ η, thus ⊕ is commutative,
(3) Rn must be closed under addition:
By the definition of a recurrent sandpile: For any β1, β2 ∈ Ωn, there
exists θ1, θ2 ∈ Ωn such that:
η = β1 ⊕ θ1
ζ = β2 ⊕ θ2
Thus, η ⊕ ζ = (β1 ⊕ β2) ⊕ (θ1 ⊕ θ2), and (β1 ⊕ β2) → (η ⊕ ζ).
As β2 is varying over all values in Ωn, we may take it to be αn, and thus
get: (β1 ⊕ αn) → (η ⊕ ζ).
Thus, for all β1 ∈ Ωn, β1 → (η ⊕ ζ). Thus (η ⊕ ζ) ∈ Rn.
(4) There exists a unique −η ∈ Rn such that η ⊕ −η ⊕ ζ = η η ⊕ ζ = ζ As
proven by Creutz in [3], −η = (|∆n
|−1)⊗η, where |∆n
| is the determinant
of the toppling matrix and the number of recursive states.
(5) There exists an identity en ∈ Rn, with the property that en ⊕ ζ = ζ.
From the previous, we see that en = η η.
We can now extend our notation of scalar multiplication for recurrent states.
Definition 2.22 (Scalar Multiplication for Rn). For all η ∈ Rn and k ∈ Z+
:
0 ⊗ η = en
(−k) ⊗ η = k ⊗ −η
(8)
3. Algorithms to Find the Identity
Definition 3.1. jn ∈ Dn is the sandpile such that jn(x) = 4 − | N(x)| for all
x ∈ N2
n. This definition implies that ϕ-1
(jn)(x) = 1 for all x ∈ N2
n.
mat(jn) =
2 1 · · · 1 2
1 0 0 1
...
...
...
1 0 0 1
2 1 · · · 1 2
Lemma 3.2. For all η ∈ Rn, jn ⊕ η = η.
9. FINDING THE SANDPILE IDENTITY 9
Proof. This follows from Corollary 2.19 as jn ⊕ η is a recurrent state and it is
equivalent to η.
Lemma 3.3. k ⊗ jn ∈ Rn for some k ∈ N.
Proof. Let ηi = i ⊗ jn for all i ∈ N. As ηi = ηi−1 ⊕ jn, the sequence ϕ-1
(ηi) is
monotonically increasing by Lemma 2.16. Since ηi ∈ Ωn and |Ωn| = 4n2
< ∞ the
sequence ϕ-1
(ηi) must eventually become constant
Theorem 3.4. k ⊗ jn = en for all k ≥ N. k, N ∈ N.
Proof. By the previous lemma for some N we have a = k ⊗ jn a recurrent state.
Since a ⊕ a = 2k ⊗ jn = a therefor a = en and the Theorem follows.
Theorem 3.4 leads directly to an algorithm (Algorithm 1) for finding the iden-
tity: simply start with jn and keep adding jn until no changes are made, which
indicates by Lemma 3.2 that a reccurent state has been reached. A similar algo-
rithm (Algorithm 2), starts with jn, and doubles (with toppling) until no changes
are made.
Let χn ∈ Sn be defined such that χn(x) = 4 for all x ∈ N2
n.
Theorem 3.5. en = 4 ⊗ (χn − χn )
Proof. Let a = χn − χn ∈ Tn. For any x ∈ N2
n the height a(x) = χn(x) −
χn (x) = 4 − χn (x) ≥ 4 − 3 = 1. Therefore 4 · a has height 4 at every position of
the grid and 4 · a is a recurrent state. Since it is also in Tn it is the unit en.
It is useful to analyse the identity as its toppling matrix, ϕ-1
(en). From ex-
perimental results, this has a convex paraboloid-like structure, especially in the
center, but it is much lower on the sides. Also of interest is the relationship
between ϕ-1
(en) and ϕ-1
(en+2). Let us define the maximum height on the side,
smax = max(ϕ-1
(en)(1, j)), j ∈ N. ϕ-1
(en+2)(i + 1, j + 1) ≈ ϕ-1
(en) + smax(i, j),
for 1 ≤ i, j ≤ n. Let us define τ1 to be this expansion along the sides of ϕ-1
(en).
τ1;n+2(i, j) =
ϕ-1
(en) + smax(i, j) if 2 ≤ i, j ≤ n
0 otherwise
(9)
en+2 = k · jn ⊕ ϕ(τ1)(10)
This provides a good estimate in the center of the sandpile, but toward the edges
it overcompensates and requires much toppling. Let us define a better estimate for
the sides of the sandpile, τ2, that expands along the center.
τ2;n+2(i, j) =
ϕ-1
(en)(i, j) if 1 ≤ i, j ≤ n/2 − 1
ϕ-1
(en)(i + 2, j + 2) if n/2 + 1n/2 ≤ i, j ≤ n ϕ-1
(en)(i ± 1, j)
if |i − ( n/2 + 1)| < 1
ϕ-1
(en)(i, j ± 1) if |j − ( n/2 + 1)| < 1
(11)
en+2 = k · jn ⊕ ϕ(τ2)
(12)
10. 10 JACOB HAVEN AND ATTILA P ´OR
4. Results From Computational Model
As can be seen in Appendix B, the identities approach a stable, fractal state,
with much symmetry. Of particular note is how much 2’s and 3’s (green and
red) dominate. In general, there is a large square of 2’s at the center, with four
triangular patterns of 3’s radiating outward to the edges. e2n+1 is related to e2n
in the following way: after seperating e2n into it’s four main symmetrical regions
(top left, top right, bottom left, bottom right), create a single empty column and
a single empty row, both in the center. Place 0 in the very center, with a single
column and row cross of 1’s inside of the square of 2’s, and 2’s filling up the rest.
Let us define Cn(i) = {x ∈ Sn | en(x) = i ∈ {0, 1, 2, 3}}. From the previous,
|C2n+1(i)| ≥ C2n(i)| for all i ∈ {0, 1, 2}, and |C2n+1(3)| ≥ C2n(3)|. Graphs of all
|Cn| for 3 < n < 125 Let Cn = |Cn|
n2 be the normalized |Cn|, such that
3
i=0 Cn = 1.
5. Further Questions
Let us define a function σ : [0, 1]2
→ {0, 2, 3} (where [0, 1]2
is the unit square)
as follows:
Definition 5.1. For i = 2, 3, σ(x) := i if and only if there exists an ε > 0 and an
N ∈ N such that for all n ≥ N and all y ∈ N2
n with the property d2(x, y
n ) < ε,
en(y) = i.
σ(x) = i ⇐⇒ ∃ε > 0, N ∈ Z+
∀n ≥ N∀y ∈ N2
n d2(x,
y
n
) < ε → en(y) = i
σ(x) = 0 otherwise.
(13)
Using σ, we may now define subsets of [0, 1]2
.
Definition 5.2. Let Ai = {x | σ(x) = i}, for i = 1, 2, 3 and x ∈ [0, 1]2
. Both sets
A2 and A3 are open subsets of the unit square by definition. Let B2 ⊂ A0 and
B3 ⊂ A0 be the boundaries of A2 and A3, respectively.
Let B = B2 ∩ B3 ⊂ A0 be the common boundary of A2 and A3.
We conjecture that the Hausdorff dimension of B is greater then one, but smaller
then two.
References
[1] P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality, Physical review A 38 (1988),
no. 1, 364–374.
[2] S. Caracciolo, G. Paoletti, and A. Sportiello, Explicit characterization of the identity config-
uration in an Abelian sandpile model, Journal of Physics A Mathematical General 41 (2008),
5003.
[3] M. Creutz, Abelian Sandpiles, Computers in Physics 5 (1991), no. 2, 198.
[4] D. Dhar, Self-organized critical state of sandpile automaton models, Phys. Rev. Lett. 64
(1990), no. 14, 1613–1616.
[5] D. Dhar and R. Ramaswamy, Exactly solved model of self-organized critical phenomena, Phys-
ical Review Letters 63 (1989), no. 16, 1659–1662.
[6] D. Dhar, P. Ruelle, S. Sen, and D.N. Verma, Algebraic aspects of abelian sandpile models,
Journal of Physics A: Mathematical and General 28 (1995), 805–831.
[7] Y. Le Borgne, On the identity of the sandpile group, Discrete Mathematics 256 (2002), no. 3,
775–790.
[8] R. Meester, F. Redig, and D. Znamenski, The abelian sandpile; a mathematical introduction,
Markov Processes and Related Fields 7 (2001), 509.
11. FINDING THE SANDPILE IDENTITY 11
Appendix A. Algorithms
Algorithm 1 Find Identity by adding jn.
en ← jn
repeat
olden ← en
en ← en ⊕ jn
until olden = en
return en
Algorithm 2 Find Identity by starting with jn and doubling.
en ← jn
repeat
olden ← en
en ← 2 · en
until olden = en
return en
Appendix B. Identities with their toppling vectors and other Data
e150
14. 14 JACOB HAVEN AND ATTILA P ´OR
ϕ-1
(e152) − τ1;152
ϕ-1
(e152) − τ2;152
15. FINDING THE SANDPILE IDENTITY 15
|Cn|(i)for3 ≤ n ≤ 125 Even n are fit with:
|Cn|(0) = 0.0638703x2
.10212
|Cn|(1) = 0.166812x1
.64857
|Cn|(2) = 0.642312x1
.86251
|Cn|(3) = 0.332719x2
.09833
Odd n are fit with:
|Cn|(0) = 0.0565012x2.12427
|Cn|(1) = 0.439039x1.48653
|Cn|(2) = 0.674312x1.855
|Cn|(3) = 0.289106x2.12412
16. 16 JACOB HAVEN AND ATTILA P ´OR
Cn(i) for 3 ≤ n ≤ 125 Even n are fit with:
Cn(0) = 0.0403779x0.208401
Cn(1) = 0.197368x−0.372473
Cn(2) = 0.610848x−0.128
Cn(3) = 0.306736x0.117251
Odd n are fit with:
Cn(0) = 0.0485335x0.158368
Cn(1) = 0.840877x−0.667482
Cn(2) = 0.563351x−0.104149
Cn(3) = 0.18834x0.22276
Gatton Academy of Mathematics and Science, Western Kentucky University
E-mail address: jacob.haven670@wku.edu
Department of Mathematics, Western Kentucky University
E-mail address: attila.por@wku.edu