Related papers: Finite Information Numbers through the Inductive C…
A set is introreducible if it can be computed by every infinite subset of itself. Such a set can be thought of as coding information very robustly. We investigate introreducible sets and related notions. Our two main results are that the…
We discuss the procedure of different partitions in the finite set of $N$ integer numbers and construct generic formulas for a bijective map of real numbers $s_y$, where $y=1,2,\ldots,N$, $N=\prod \limits_{k=1}^{n} X_k$, and $X_k$ are…
As an application of generalised statistical mechanics, it is studied a possible route toward a consistent generalised information theory in terms of a family of non-extensive, non-parametric entropies $H^\pm_D(P)$. Unlike other proposals…
The problem of ranking can be described as follows. We have a set of combinatorial objects $S$, such as, say, the k-subsets of n things, and we can imagine that they have been arranged in some list, say lexicographically, and we want to…
We propose a procedure for automated implicit inductive theorem proving for equational specifications made of rewrite rules with conditions and constraints. The constraints are interpreted over constructor terms (representing data values),…
Real-world physical signals are continuous and high-dimensional, yet the statistical-mechanics machinery of associative memory operates on discrete Ising spins. We bridge this divide through a multilayer Ising framework that couples a…
Understanding the limits imposed on information storage capacity of physical systems is a problem of fundamental and practical importance which bridges physics and information science. There is a well-known upper bound on the amount of…
Motility-induced phase separation (MIPS) is a purely non-equilibrium phenomenon in which self-propelled particles phase separate without any attractive interactions. One surprising feature of MIPS is the emergence of polar, nematic, and…
In this paper, we present an approach to integer factorization using distributed representations formed with Vector Symbolic Architectures. The approach formulates integer factorization in a manner such that it can be solved using neural…
We introduce a theory of integration with respect to the fixed point index, offering a substantial improvement over previous approaches based on the Lefschetz number. This framework eliminates several restrictive assumptions -- such as the…
We propose Weak and Entropy PINNs (WE-PINNs) for the approximation of entropy solutions to nonlinear hyperbolic conservation laws. Standard physics-informed neural networks enforce governing equations in strong differential form, an…
Current discrete randomness and information conservation inequalities are over total recursive functions, i.e. restricted to deterministic processing. This restriction implies that an algorithm can break algorithmic randomness conservation…
We introduce semi-parametric inducing point networks (SPIN), a general-purpose architecture that can query the training set at inference time in a compute-efficient manner. Semi-parametric architectures are typically more compact than…
The fundamental result of Li, Long, and Srinivasan on approximations of set systems has become a key tool across several communities such as learning theory, algorithms, computational geometry, combinatorics and data analysis. The goal of…
In this paper, we show that there is a large class of fermionic systems for which it is possible to find, for any dimension, a finite closed set of eigenoperators and eigenvalues of the Hamiltonian. Then, the hierarchy of the equations of…
Numerical modeling errors are unavoidable in finite element analysis. The presence of model errors inherently reflects both model accuracy and uncertainty. To date there have been few methods for explicitly quantifying errors at points of…
In this paper, we focus on approximating a natural class of functions that are compositions of smooth functions. Unlike the low-dimensional support assumption on the covariate, we demonstrate that composition functions have an intrinsic…
In this paper we provide a finite-sample and an infinite-sample representer theorem for the concatenation of (linear combinations of) kernel functions of reproducing kernel Hilbert spaces. These results serve as mathematical foundation for…
In this paper, we introduce a formulation of Physics-Informed Neural Networks (PINNs), based on learning the form of the Fourier decomposition, and a training methodology based on a spread of randomly chosen boundary conditions. By training…
In this paper we develop cyclic proof systems for the problem of inclusion between the least sets of models of mutually recursive predicates, when the ground constraints in the inductive definitions belong to the quantifier-free fragments…