Invariant Partitions Research Articles

Variational principles of invariance pressures on partitions

We investigate the relations between Bowen and packing invariance pressures and measure-theoretical lower and upper invariance pressures for invariant partitions of a controlled invariant set respectively. We mainly show that Bowen and packing invariance pressures can be determined via the local lower and upper invariance pressures of probability measures, which are analogues of Billingsley's Theorem for the Hausdorff dimension; and give variational principles between Bowen and packing invariance pressures and measure-theoretical lower and upper invariance pressures under some technical assumptions.

A Classification of Countable Lower 1-transitive Linear Orders

This paper contains a classification of countable lower 1-transitive linear orders. This is the first step in the classification of countable 1-transitive trees given in Chicot and Truss (2009): the notion of lower 1-transitivity generalises that of 1-transitivity for linear orders, and it is essential for the structure theory of 1-transitive trees. The classification is given in terms of coding trees, which describe how a linear order is fabricated from simpler pieces using concatenations, lexicographic products and other kinds of construction. We define coding trees and show that a coding tree can be constructed from a lower 1-transitive linear order (X, leqslant ) by examining all the invariant partitions on X. Then we show that a lower 1-transitive linear order can be recovered from a coding tree up to isomorphism.

Effective and Efficient Image Copy Detection with Resistance to Arbitrary Rotation

For detecting the image copies of a given original image generated by arbitrary rotation, the existing image copy detection methods can not simultaneously achieve desirable performances in the aspects of both accuracy and efficiency. To address this challenge, a novel effective and efficient image copy detection method is proposed based on two global features extracted from rotation invariant partitions. Firstly, candidate images are preprocessed by an averaging operation to suppress noise. Secondly, the rotation invariant partitions of the preprocessed images are constructed based on pixel intensity orders. Thirdly, two global features are extracted from these partitions by utilizing image gradient magnitudes and orientations, respectively. Finally, the extracted features of images are compared to implement copy detection. Promising experimental results demonstrate our proposed method can effectively and efficiently resist rotations with arbitrary degrees. Furthermore, the performances of the proposed method are also desirable for resisting other typical copy attacks, such as flipping, rescaling, illumination and contrast change, as well as Gaussian noising.

The homogeneous weight partition and its character-theoretic dual

The homogeneous weight on a finite Frobenius ring naturally induces a partition which is invariant under left and right multiplication by units. It is shown that the character-theoretic left-sided dual of this partition coincides with the right-sided dual, and even more, the left- and right-sided Krawtchouk coefficients coincide. An example is provided showing that this is not the case for general invariant partitions if the ring is not semisimple.

Reasoning with Computer Code: a new Mathematical Logic

Abstract A logic is a mathematical model of knowledge used to study how we reason, how we describe the world, and how we infer the conclusions that determine our behavior. The logic presented here is natural. It has been experimentally observed, not designed. It represents knowledge as a causal set, includes a new type of inference based on the minimization of an action functional, and generates its own semantics, making it unnecessary to prescribe one. This logic is suitable for high-level reasoning with computer code, including tasks such as self-programming, objectoriented analysis, refactoring, systems integration, code reuse, and automated programming from sensor-acquired data. A strong theoretical foundation exists for the new logic. The inference derives laws of conservation from the permutation symmetry of the causal set, and calculates the corresponding conserved quantities. The association between symmetries and conservation laws is a fundamental and well-known law of nature and a general principle in modern theoretical Physics. The conserved quantities take the form of a nested hierarchy of invariant partitions of the given set. The logic associates elements of the set and binds them together to form the levels of the hierarchy. It is conjectured that the hierarchy corresponds to the invariant representations that the brain is known to generate. The hierarchies also represent fully object-oriented, self-generated code, that can be directly compiled and executed (when a compiler becomes available), or translated to a suitable programming language. The approach is constructivist because all entities are constructed bottom-up, with the fundamental principles of nature being at the bottom, and their existence is proved by construction. The new logic is mathematically introduced and later discussed in the context of transformations of algorithms and computer programs. We discuss what a full self-programming capability would really mean. We argue that self-programming and the fundamental question about the origin of algorithms are inextricably linked. We discuss previously published, fully automated applications to self-programming, and present a virtual machine that supports the logic, an algorithm that allows for the virtual machine to be simulated on a digital computer, and a fully explained neural network implementation of the algorithm.

CRANK 0 PARTITIONS AND THE PARITY OF THE PARTITION FUNCTION

A well-known problem regarding the integer partition function p(n) is the parity problem, how often is p(n) even or odd? Motivated by this problem, we obtain the following results: (1) A generating function for the number of crank 0 partitions of n. (2) An involution on the crank 0 partitions whose fixed points are called invariant partitions. We then derive a generating function for the number of invariant partitions. (3) A generating function for the number of self-conjugate rank 0 partitions.

Bi-parametric convex quadratic optimization

In this paper, we consider the convex quadratic optimization problem with simultaneous perturbation in the right-hand side of the constraints and the linear term of the objective function with different parameters. The regions with invariant optimal partitions as well as the behaviour of the optimal value function on the regions are investigated. We show that identifying these regions can be done in polynomial time in the output size. An algorithm for identifying all invariancy regions is presented. Some implementation details as well as a numerical example are discussed.

Invariant energy partitions in chemical reactions and cluster dynamics simulations

A separation of intra and intermolecular degrees of freedom, based on an invariant phase-space hyperangular momentum approach, is proposed as useful alternative to the usual roto-vibrational mode analysis. Two partitions (hyperspherical and projective) of the kinetic energy are here illustrated for simulations of chemical reactions and shown interesting to analyze caloric curves of clusters. Case studies are Ar13 and Ar38.

On the entropy theory of finitely-generated nilpotent group actions

The entropy for actions of a finitely-generated nilpotent group G is investigated. The Pinsker algebra of such actions is described explicitly. The systems with completely positive entropy are shown to have a sort of ‘asymptotic independence property’, just as in the case of \mathbb{Z}^d-actions. The invariant partitions are used to prove that the property of completely positive entropy is equivalent to the property of the K-system (the property of the existence of special ‘good’ partitions). A complete spectral characterization of K-systems is given. A construction of examples of completely positive non-Bernoullian actions of general countable nilpotent groups and a class of solvable groups is presented.

1-Perfect Uniform and Distance Invariant Partitions

Let F n be the n-dimensional vector space over ℤ2. A (binary) 1-perfect partition of F n is a partition of F n into (binary) perfect single error-correcting codes or 1-perfect codes. We define two metric properties for 1-perfect partitions: uniformity and distance invariance. Then we prove the equivalence between these properties and algebraic properties of the code (the class containing the zero vector). In this way, we characterize 1-perfect partitions obtained using 1-perfect translation invariant and not translation invariant propelinear codes. The search for examples of 1-perfect uniform but not distance invariant partitions enabled us to deduce a non-Abelian propelinear group structure for any Hamming code of length greater than 7.

Opposite Littlewood-Richardson sequences and their matrix realizations

We introduce the concept of opposite Littlewood-Richardson sequences. Given the partitions a 0, a 1,…, a t and the nonnegative integers m 1 ⩽ … ⩽ m t , that concept gives a necessary and sufficient condition for the existence of a sequence of matrices A 0, B 1,…, B t with invariant partitions a 0, (1 m 1 ),…, (1 m t ) such that a i is the invariant partition of A 0 B 1 … B i for i = 1,…, t, and (1 m 1 ) + ··· + (1 m t ) is the invariant partition of B 1 B 2 … B t . We also present an explicit construction of a sequence of matrices which realizes a previously given opposite Littlewood-Richardson sequence. This result is a generalization of a well-known result of T. Klein and J. A. Green on extensions of p-modules.

Purely random time partitions and 1/ f-noise

A new statistical model of a random flow of events generating 1/ f-noise is constructed using scale invariant time partitions. The probability and spectral characteristics are considered. A peculiarity of the model is the coexistence of a non-smoothable time distribution of the events and the finiteness of all the statistical moments of the event density.

The spacetime approach to quantum mechanics

Feynman's sum-over-histories formulation of quantum mechanics is reviewed as an independent statement of quantum theory in spacetime form. It is different from the usual Schrödinger-Heisenberg formulation that utilizes states on spacelike surfaces because it assigns probabilities to different sets of alternatives. In a sum-over-histories formulation, alternatives at definite moments of time are more restricted than in usual quantum mechanics because they refer only to the coördinates in terms of which the histories are defined. However, in the context of the quantum mechanics of closed systems, sum-over-histories quantum mechanics can be generalized to deal with spacetime alternatives that are not “at definite moments of time”. An example in field theory is the set of alternative ranges of values of a field averaged over a spacetime region. An example in particle mechanics is the set of the alternatives defined by whether a particle never crosses a fixed spacetime region or crosses it at least once. The general notion of a set of spacetime alternatives is a partition (coarse-graining) of the histories into an exhaustive set of exclusive classes. With this generalization the sum-over-histories formulation can be said to be in fully spacetime form with dynamics represented by path integrals over spacetime histories and alternatives defined as spacetime partitions of these histories. When restricted to alternatives at definite moments of times this generalization is equivalent to Schrödinger-Heisenberg quantum mechanics. However, the quantum mechanics of more general spacetime alternatives does not have an equivalent Schrödinger-Heisenberg formulation. We suggest that, in the quantum theory of gravity, the general notion of “observable” is supplied by diffeomorphism invariant partitions of spacetime metrics and matter field configurations. By generalizing the usual alternatives so as to put quantum theory in fully spacetime form we may be led to a covariant generalized quantum mechanics of spacetime free from the problem of time.

Computing sylow subgroups of permutation groups using homomorphic images of centralizers

The ability to construct the Sylow subgroups of a large finite permutation group is fundamental to a number of important algorithms for analysing the structure of such a group. In this paper we describe an algorithm which constructs a Sylow subgroup by restricting the problem to images of centralizers under the natural homomorphisms given by actions on invariant sets and partitions of points. The algorithm is essentially as efficient as that used for computing centralizers in permutation groups.

Combinatorial Structures and Group Invariant Partitions

If a group acts on a set, an action of the group is induced on the partitions of the set. A formula is developed for the number of partitions invariant under this action. The formula is extended to count combinatorial objects such as labeled rooted trees or permutations defined on the invariant partitions.

Combinatorial structures and group invariant partitions

If a group acts on a set, an action of the group is induced on the partitions of the set. A formula is developed for the number of partitions invariant under this action. The formula is extended to count combinatorial objects such as labeled rooted trees or permutations defined on the invariant partitions.

Computational Algorithms for the Enumeration of Group Invariant Partitions

Let G be a finite group acting on a finite set S and hence on $\Pi (S)$, the lattice of partitions of S. Computational methods are developed for enumerating the invariants of this action.MSC codespartitions of a setgroup actions

On the relationship between zero entropy and quasi-discrete spectrum for affine transformations

Introduction. If T is a measure-preserving transformation of a Lebesgue space (M, (3, m) let 7r(T) be the maximum partition such that T.(T) has zero entropy. In ?2 we prove a result, which was stated without proof in [12], concerning the behaviour of the partitions 7(Tr.) associated with an increasing sequence of invariant measurable partitions {f ,n}. If T is an ergodic affine transformation of a compact connected metric abelian group G let (T) be the maximum partition such that T,,(T) has quasi-discrete spectrum. In ?3 we prove, using the result of ?2 and a method introduced by Rohlin in [8], that q(T) -7r(T). This result was first obtained by Parry [6] who also proved that the maximum partition y(T) such that T7(T) has the distal property is also 7r(T). My thanks are due to Dr. W. Parry for his guidance during this work.

On the coincidence of three invariant 𝜎-algebras associated with an affine transformation

Introduction. This paper is concerned with an ergodic affine transformation T of a compact metric connected abelian group X. T is therefore a Haar measure preserving homeomorphism of X onto itself and we can associate with T, three invariant u-algebras or partitions r, a, i: these are the maximum c-algebras such that the factors T,, T,, Ts have zero entropy, quasi-discrete spectrum, the distal property, respectively. It will be shown that the three u-algebras are identical. It is assumed that the reader is familiar with the theory of entropy of ergodic measure preserving transformations, For an account of this theory and its notations and for the related notions of measurable partition and factor transformation cf. [1]. Let (X, 8, m) be a measure space point isomorphic to the unit interval with Lebesgue sets and Lebesgue measure. A compact metric abelian group with normalized complete Haar measure is of this type. Let T be an invertible ergodic measure preserving transformation of X onto itself. Associated with each sub-a-algebra DC& there is an unique (mod 0) measurable partition 8 of X and conversely. (D = 8(8), = e(j)).) D is finite if and only if 8 is finite. The relations TODCOD, T= D correspond to the relations TS _ 8, TS =8 (mod 0), and if v, X, e are the trivial partition, trivial a-algebra and partition into one point sets, respectively, then v =E(01), Ot =8(v), E=E(8), 8=8(e).