609 research outputs found

    The MacWilliams Identity for Krawtchouk Association Schemes

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    The weight distribution of an error correcting code is a crucial statistic in determining its performance. One key tool for relating the weight of a code to that of its dual is the MacWilliams Identity, first developed for the Hamming association scheme. This identity has two forms: one is a functional transformation of the weight enumerators, while the other is a direct relation of the weight distributions via eigenvalues of the association scheme. The functional transformation form can, in particular, be used to derive important moment identities for the weight distribution of codes. In this thesis, we focus initially on extending the functional transformation to codes based on skew-symmetric and Hermitian matrices. A generalised b-algebra and new fundamental homogeneous polynomials are then identified and proven to generate the eigenvalues of a specific subclass of association schemes, Krawtchouk association schemes. Based on the new set of MacWilliams Identities as a functional transform, we derive several moments of the weight distribution for all of these codes

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Advances and Applications of DSmT for Information Fusion. Collected Works, Volume 5

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    This fifth volume on Advances and Applications of DSmT for Information Fusion collects theoretical and applied contributions of researchers working in different fields of applications and in mathematics, and is available in open-access. The collected contributions of this volume have either been published or presented after disseminating the fourth volume in 2015 in international conferences, seminars, workshops and journals, or they are new. The contributions of each part of this volume are chronologically ordered. First Part of this book presents some theoretical advances on DSmT, dealing mainly with modified Proportional Conflict Redistribution Rules (PCR) of combination with degree of intersection, coarsening techniques, interval calculus for PCR thanks to set inversion via interval analysis (SIVIA), rough set classifiers, canonical decomposition of dichotomous belief functions, fast PCR fusion, fast inter-criteria analysis with PCR, and improved PCR5 and PCR6 rules preserving the (quasi-)neutrality of (quasi-)vacuous belief assignment in the fusion of sources of evidence with their Matlab codes. Because more applications of DSmT have emerged in the past years since the apparition of the fourth book of DSmT in 2015, the second part of this volume is about selected applications of DSmT mainly in building change detection, object recognition, quality of data association in tracking, perception in robotics, risk assessment for torrent protection and multi-criteria decision-making, multi-modal image fusion, coarsening techniques, recommender system, levee characterization and assessment, human heading perception, trust assessment, robotics, biometrics, failure detection, GPS systems, inter-criteria analysis, group decision, human activity recognition, storm prediction, data association for autonomous vehicles, identification of maritime vessels, fusion of support vector machines (SVM), Silx-Furtif RUST code library for information fusion including PCR rules, and network for ship classification. Finally, the third part presents interesting contributions related to belief functions in general published or presented along the years since 2015. These contributions are related with decision-making under uncertainty, belief approximations, probability transformations, new distances between belief functions, non-classical multi-criteria decision-making problems with belief functions, generalization of Bayes theorem, image processing, data association, entropy and cross-entropy measures, fuzzy evidence numbers, negator of belief mass, human activity recognition, information fusion for breast cancer therapy, imbalanced data classification, and hybrid techniques mixing deep learning with belief functions as well

    Clones over Finite Sets and Minor Conditions

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    Achieving a classification of all clones of operations over a finite set is one of the goals at the heart of universal algebra. In 1921 Post provided a full description of the lattice of all clones over a two-element set. However, over the following years, it has been shown that a similar classification seems arduously reachable even if we only focus on clones over three-element sets: in 1959 Janov and Mučnik proved that there exists a continuum of clones over a k-element set for every k > 2. Subsequent research in universal algebra therefore focused on understanding particular aspects of clone lattices over finite domains. Remarkable results in this direction are the description of maximal and minimal clones. One might still hope to classify all operation clones on finite domains up to some equivalence relation so that equivalent clones share many of the properties that are of interest in universal algebra. In a recent turn of events, a weakening of the notion of clone homomorphism was introduced: a minor-preserving map from a clone C to D is a map which preserves arities and composition with projections. The minor-equivalence relation on clones over finite sets gained importance both in universal algebra and in computer science: minor-equivalent clones satisfy the same set identities of the form f(x_1,...,x_n) = g(y_1,...,y_m), also known as minor-identities. Moreover, it was proved that the complexity of the CSP of a finite structure A only depends on the set of minor-identities satisfied by the polymorphism clone of A. Throughout this dissertation we focus on the poset that arises by considering clones over a three-element set with the following order: we write C ≤_{m} D if there exist a minor-preserving map from C to D. It has been proved that ≤_{m} is a preorder; we call the poset arising from ≤_{m} the pp-constructability poset. We initiate a systematic study of the pp-constructability poset. To this end, we distinguish two cases that are qualitatively distinct: when considering clones over a finite set A, one can either set a boundary on the cardinality of A, or not. We denote by P_n the pp-constructability poset restricted to clones over a set A such that |A|=n and by P_{fin} we denote the whole pp-constructability poset, i.e., we only require A to be finite. First, we prove that P_{fin} is a semilattice and that it has no atoms. Moreover, we provide a complete description of P_2 and describe a significant part of P_3: we prove that P_3 has exactly three submaximal elements and present a full description of the ideal generated by one of these submaximal elements. As a byproduct, we prove that there are only countably many clones of self-dual operations over {0,1,2} up to minor-equivalence

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning

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    The paper introduces the application of information geometry to describe the ground states of Ising models by utilizing parity-check matrices of cyclic and quasi-cyclic codes on toric and spherical topologies. The approach establishes a connection between machine learning and error-correcting coding. This proposed approach has implications for the development of new embedding methods based on trapping sets. Statistical physics and number geometry applied for optimize error-correcting codes, leading to these embedding and sparse factorization methods. The paper establishes a direct connection between DNN architecture and error-correcting coding by demonstrating how state-of-the-art architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range arena can be equivalent to of block and convolutional LDPC codes (Cage-graph, Repeat Accumulate). QC codes correspond to certain types of chemical elements, with the carbon element being represented by the mixed automorphism Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix are elaborated upon in detail. The Quantum Approximate Optimization Algorithm (QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous to the back-propagation loss function landscape in training DNNs. This similarity creates a comparable problem with TS pseudo-codeword, resembling the belief propagation method. Additionally, the layer depth in QAOA correlates to the number of decoding belief propagation iterations in the Wiberg decoding tree. Overall, this work has the potential to advance multiple fields, from Information Theory, DNN architecture design (sparse and structured prior graph topology), efficient hardware design for Quantum and Classical DPU/TPU (graph, quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text overlap with arXiv:2109.08184 by other author

    Clase de pesos multilineales asociados a propiedades de continuidad de conmutadores de operadores fraccionarios generalizados

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    Estudiamos propiedades de continuidad para conmutadores de orden superior asociados a operadores fraccionarios generalizados que resultan ser una extensión del operador integral fraccionario IalphamI_alpha^m en el contexto multilineal. Las acotaciones son entre un producto de espacios de Lebesgue pesados y ciertos espacios de tipo Lipschitz pesados, extendiendo estimaciones previas de la literatura para el caso lineal. Este estudio incluye dos tipos diferentes de conmutadores y condiciones suficientes en los pesos involucrados para garantizar las acotaciones referidas anteriormente. También se incluye el rango óptimo de los parámetros involucrados, que se entiende en el sentido de describir una región fuera de la cual los pesos son triviales. El análisis incluye también ejemplos de pesos que abarcan esta región de optimalidad.Fil: Recchi, Diana Jorgelina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: Berra, Fabio Martín. Universidad Nacional del Litoral. Facultad de Ingeniería Química. Departamento de Matemáticas; ArgentinaFil: Pradolini, Gladis Guadalupe. Universidad Nacional del Litoral. Facultad de Ingeniería Química. Departamento de Matemáticas; ArgentinaXVII Congreso Antonio MonteiroBahía BlancaArgentinaUniversidad Nacional del SurInstituto de Matemática de Bahía Blanc

    A survey of congruences and quotients of partially ordered sets

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    The literature on congruences and quotients of partially ordered sets contains a large and profilerating array of approaches, but little in the way of systematic exposition and examination of the subject. We seek to rectify this by surveying the different approaches in the literature and providing philosophical discussion on requirements for notions of congruences of posets. We advocate a pluralist approach which recognises that different types of congruence arise naturally in different mathematical situations. There are some notions of congruence which are very general, whilst others capture specific structure which often appears in examples. Indeed, we finish by giving several examples where quotients of posets appear naturally in mathematics

    Complex-valued neural operator assisted soliton identification

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    The numerical determination of solitary states is an important topic for such research areas as Bose-Einstein condensates, nonlinear optics, plasma physics, etc. In this paper, we propose a data-driven approach for identifying solitons based on dynamical solutions of real-time differential equations. Our approach combines a machine-learning architecture called the complex-valued neural operator (CNO) with an energy-restricted gradient optimization. The former serves as a generalization of the traditional neural operator to the complex domain, and constructs a smooth mapping between the initial and final states; the latter facilitates the search for solitons by constraining the energy space. We concretely demonstrate this approach on the quasi-one-dimensional Bose-Einstein condensate with homogeneous and inhomogeneous nonlinearities. Our work offers a new idea for data-driven effective modeling and studies of solitary waves in nonlinear physical systems.Comment: 9 pages, 5 figure
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