Generating series and asymptotics of classical spin networks

Francesco Costantino; Julien Marché

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Global stability of Clifford-valued Takagi-Sugeno fuzzy neural networks with time-varying delays and impulses

Ramalingam Sriraman, Asha Nedunchezhiyan (2022)

Kybernetika

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In this study, we consider the Takagi-Sugeno (T-S) fuzzy model to examine the global asymptotic stability of Clifford-valued neural networks with time-varying delays and impulses. In order to achieve the global asymptotic stability criteria, we design a general network model that includes quaternion-, complex-, and real-valued networks as special cases. First, we decompose the $n$ -dimensional Clifford-valued neural network into $2^{m} n$ -dimensional real-valued counterparts in order to solve the...

Self-similarly expanding networks to curve shortening flow

Oliver C. Schnürer, Felix Schulze (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point, where they form $120$ degree angles, and expands homothetically under curve shortening flow. We also prove uniqueness of these networks.

Exponential expressivity of ${ReLU}^{k}$ neural networks on Gevrey classes with point singularities

Joost A. A. Opschoor, Christoph Schwab (2024)

Applications of Mathematics

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We analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains $D \subset ℝ^{d}$ , $d = 2, 3$ . We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in $D$ , comprising the countably-normed spaces of I. M. Babuška and B. Q. Guo. As intermediate result, we prove that continuous, piecewise polynomial...

On the number of ground states of the Edwards–Anderson spin glass model

Louis-Pierre Arguin, Michael Damron (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Ground states of the Edwards–Anderson (EA) spin glass model are studied on infinite graphs with finite degree. Ground states are spin configurations that locally minimize the EA Hamiltonian on each finite set of vertices. A problem with far-reaching consequences in mathematics and physics is to determine the number of ground states for the model on $ℤ^{d}$ for any $d$ . This problem can be seen as the spin glass version of determining the number of infinite geodesics in first-passage percolation...

New results on stability of periodic solution for CNNs with proportional delays and $D$ operator

Bo Du (2019)

Kybernetika

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The problems related to periodic solutions of cellular neural networks (CNNs) involving $D$ operator and proportional delays are considered. We shall present Topology degree theory and differential inequality technique for obtaining the existence of periodic solution to the considered neural networks. Furthermore, Laypunov functional method is used for studying global asymptotic stability of periodic solutions to the above system.

Approximation of mixing point configuration spaces over $Z^{d}$ by spaces of set configurations

K. Häsler (1989)

Banach Center Publications

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On the $L^{p}$ index of spin Dirac operators on conical manifolds

André Legrand, Sergiu Moroianu (2006)

Studia Mathematica

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We compute the index of the Dirac operator on a spin Riemannian manifold with conical singularities, acting from $L^{p} (Σ ⁺)$ to $L^{q} (Σ ¯)$ with p,q > 1. When 1 + n/p - n/q > 0 we obtain the usual Atiyah-Patodi-Singer formula, but with a spectral cut at (n+1)/2 - n/q instead of 0 in the definition of the eta invariant. In particular we reprove Chou’s formula for the L² index. For 1 + n/p - n/q ≤ 0 the index formula contains an extra term related to the Calderón projector.

Spin representations and binary numbers

Henrik Winther (2024)

Archivum Mathematicum

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We consider a construction of the fundamental spin representations of the simple Lie algebras $𝔰𝔬 (n)$ in terms of binary arithmetic of fixed width integers. This gives the spin matrices as a Lie subalgebra of a $ℤ$ -graded associative algebra (rather than the usual $ℕ$ -filtered Clifford algebra). Our description gives a quick way to write down the spin matrices, and gives a way to encode some extra structure, such as the real structure which is invariant under the compact real form, for some $n$ ....

The magnetization at high temperature for a p-spin interaction model with external field

David Márquez-Carreras (2007)

Applicationes Mathematicae

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This paper is devoted to a detailed and rigorous study of the magnetization at high temperature for a p-spin interaction model with external field, generalizing the Sherrington-Kirkpatrick model. In particular, we prove that $⟨ σ_{i} ⟩$ (the mean of a spin with respect to the Gibbs measure) converges to an explicitly given random variable, and that ⟨σ₁⟩,...,⟨σₙ⟩ are asymptotically independent.

Estimating composite functions by model selection

Yannick Baraud, Lucien Birgé (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the problem of estimating a function $s$ on ${[- 1, 1]}^{k}$ for large values of $k$ by looking for some best approximation of $s$ by composite functions of the form $g \circ u$ . Our solution is based on model selection and leads to a very general approach to solve this problem with respect to many different types of functions $g, u$ and statistical frameworks. In particular, we handle the problems of approximating $s$ by additive functions, single and multiple index models, artificial neural networks, mixtures...

Quantum Singularity Theory for $A_{(r - 1)}$ and $r$ -Spin Theory

Huijun Fan, Tyler Jarvis, Yongbin Ruan (2011)

Annales de l’institut Fourier

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We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the $r$ -spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity $W$ of type $A$ our construction of the stack of $W$ -curves is canonically isomorphic to the stack of $r$ -spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an $r$ -spin virtual class. Therefore, the Faber-Shadrin-Zvonkine...

Particles in the superworldline and BRST

Eugenia Boffo (2022)

Archivum Mathematicum

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In this short note we discuss $N$ -supersymmetric worldlines of relativistic massless particles and review the known result that physical spin- $N / 2$ fields are in the first BRST cohomology group. For $N = 1, 2, 4$ , emphasis is given to particular deformations of the BRST differential, that implement either a covariant derivative for a gauge theory or a metric connection in the target space seen by the particle. In the end, we comment about the possibility of incorporating Ramond-Ramond fluxes in the background. ...

The Collatz-Wielandt quotient for pairs of nonnegative operators

Shmuel Friedland (2020)

Applications of Mathematics

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In this paper we consider two versions of the Collatz-Wielandt quotient for a pair of nonnegative operators $A, B$ that map a given pointed generating cone in the first space into a given pointed generating cone in the second space. If the two spaces and two cones are identical, and $B$ is the identity operator, then one version of this quotient is the spectral radius of $A$ . In some applications, as commodity pricing, power control in wireless networks and quantum information theory, one needs...

On symmetric functions and the spin characters of $S_{n}$

John R. Stembridge (1990)

Banach Center Publications

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