36 Results Found

A dense $G_{\delta }$ subset of a Polish group is large in the sense of Baire category. We study a configuration consequence of this largeness. If G is a Polish abelian group, $R\subseteq G$ is dense $G_{\delta }$, and $A\subseteq G$ is countable, then the set of translat...

We introduce Lehmer Activation Units (LAUs), a class of aggregation-based neural activations derived from the Lehmer transform that unify feature weighting and nonlinearity within a single differentiable operator. Unlike conventional pointwise activa...

In this paper, we address the challenge of solving differential equations using physics-informed neural networks (PINNs), an innovative approach that integrates known physical laws into neural network training. The PINN approach involves three main s...

This paper presents an optimal $L_2$-induced control problem for systems with multiple sector-bounded nonlinearities. Sufficient boundedness conditions for the $L_2$-induced norm are derived in terms of a specific system of linear matrix inequalities (LMIs...

A new technique is proposed to increase the prediction accuracy of artificial neural networks (ANNs). This technique applies a stepwise regression (SR) procedure to the input data variables, which adds nonlinear terms into the input data in a way tha...

Let $\mathfrak{a}=\{a_m:m\in \mathbb{N}\}$ be a multiplicative periodic sequence of complex numbers. In this paper, we consider the approximation of analytic functions defined in the strip $\{s=\sigma +it:1/2<\sigma <1\}$ by shifts $\zeta (s+i\tau ;\mathfrak{a})$ of the zeta-function...

This paper is concerned with the sequence of positive linear operators obtained by certain generating functions of polynomials and with investigation of its approximation properties in detail. Initially, the convergence theorem is expressed for the s...

The famous Selberg class is defined axiomatically and consists of Dirichlet series satisfying four axioms (Ramanujan hypothesis, analytic continuation, functional equation, multiplicativity). The Selberg–Steuding class $\mathcal{S}$ is a complemented Selbe...

Deep neural networks have shown very successful performance in a wide range of tasks, but a theory of why they work so well is in the early stage. Recently, the expressive power of neural networks, important for understanding deep learning, has recei...

The distribution of word probabilities in the monkey model of Zipf’s law is associated with two universality properties: (1) the exponent in the approximate power law approaches −1...

In this study, the approximation of a pair of analytic functions defined on the strip $\{s=\sigma +it\in \mathbb{C}:1/2<\sigma <1\}$ by shifts $\left(\zeta \right(s+i\tau ),\zeta (s+i\tau ,\alpha \left)\right)$, $\tau \in \mathbb{R}$, of the Riemann and Hurwitz zeta-functions with transc...

Let F be the normalized Hecke-eigen cusp form for the full modular group and $\zeta (s,F)$ be the corresponding zeta-function. In the paper, the joint universality theorem on the approximation of a collection of analytic functions by shifts $(\zeta (s+i{h}_1\tau ,F),\dots ,\zeta$...

The relationship between Rough Set (RS) and algebraic systems has been long studied by mathematicians. RS is a growing research area that encourages studies into both real-world applications and the theory itself. In RS, a universe subset is characte...

The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function $\zeta \left(s\right)$ (zeros different from $s=-2m$, $m\in \mathbb{N}$) lie on the critical line $\sigma =1/2$. In this paper, combining the universality property of $\mathrm{\&zet}$...

Symmetry is presented in many works involving differential and integral equations. Whenever a human is involved in the design of an integral equation, they naturally tend to opt for symmetric features. The most common examples are the Green functions...

We study practical approximations of Kolmogorov prefix complexity (K) using IMP2, a high-level programming language. Our focus is on investigating the optimality of the interpreter for this language as the reference machine for the Coding Theorem Met...

Machine learning has become indispensable across various domains, yet understanding its theoretical underpinnings remains challenging for many practitioners and researchers. Despite the availability of numerous resources, there is a need for a cohesi...

In this work, we formulate a systematic expectation-value framework for dynamical systems whose probability densities evolve according to linear partial differential equations, such as the Fokker-Planck and Liouville equations. The approach is based...

A persistent concentration of scale-critical quantities is widely regarded as a necessary precursor to finite-time singularity formation in the three-dimensional incompressible Navier–Stokes equations. This paper develops a conditional structur...

In the paper, we consider the approximation of analytic functions by shifts from the wide class $\tilde{S}$ of L-functions. This class was introduced by A. Selberg, supplemented by J. Steuding, and is defined axiomatically. We prove the so-called joint...

Let $\mathfrak{a}=\left\{a_m\right\}$ and $\mathfrak{b}=\left\{b_m\right\}$ be two periodic sequences of complex numbers, and, additionally, $\mathfrak{a}$ is multiplicative. In this paper, the joint approximation of a pair of analytic functions by shifts $({\zeta }_{n_T}(s+i\tau ;\mathfrak{a}),{\zeta }_{n_T}(s+i\tau ,\alpha ;\mathfrak{b}))$ of absolu...

We prove that, within the class of pair potential Hamiltonians, the excess entropy is a universal, temperature-independent functional of the density and pair correlation function. This result extends Henderson’s theorem, which states that the free en...

A maximum entropy (ME) methodology was used to infer the Hubble constant from the temperature anisotropies in cosmic microwave background (CMB) measurements, as measured by the Planck satellite. A simple cosmological model provided physical insight a...

Medical image segmentation requires spatial transformations that remain effective across heterogeneous image statistics and boundary conditions. We introduce FunKAN, a KAN-inspired operator that acts directly on spatial feature maps: analytical Hermi...

We present a flexible mathematical method that extends the Rosenstark method and enables the analysis of any electrical network with feedback and all its key parameters such as gain, frequency response, input/output impedances, and noise. Unlike the...

Novel approaches to estimate information measures using neural networks are well-celebrated in recent years both in the information theory and machine learning communities. These neural-based estimators are shown to converge to the true values when e...

This paper investigates the structural properties of solutions of vector equilibrium systems and mixed variational inequalities in topological vector spaces. Based on Himmelberg-type fixed point theorem, combined with the analysis of set-valued mappi...

ANNs succeed in several tasks for real scenarios due to their high learning abilities. This paper focuses on theoretical aspects of ANNs to enhance the capacity of implementing those modifications that make ANNs absorb the defining features of each s...

Increasingly encompassing models have been suggested for our world. Theories range from generally accepted to increasingly speculative to apparently bogus. The progression of theories from ego- to geo- to helio-centric models to universe and multiver...

Learning in neural networks with locally-tuned neuron models such as radial Basis Function (RBF) networks is often seen as instable, in particular when multi-layered architectures are used. Furthermore, universal approximation theorems for single-lay...

Critical flow in irrotational motion is important in theoretical hydrodynamics and dam hydraulics. Therefore, the commented paper is relevant in theory and practice. It deals with an approximation for critical flow based on a set of simplified irrota...

A new dynamical paradigm merging quantum dynamics with cosmology is discussed. We distinguish between a universe and its background space-time. The universe here is the subset of space-time defined by ${\mathsf{\Psi }}_{\mathsf{\tau }}\left(x\right)\ne 0$, where ${\mathsf{\Psi }}_{\mathsf{\tau }}\left(x\right)$ is a solu...

Investigating causal interactions between entities is a crucial task across various scientific domains. The traditional causal discovery methods often assume a predetermined causal direction, which is problematic when prior knowledge is insufficient....

With the rapid development of offshore engineering, marine cranes are widely deployed in critical maritime operations, such as the precision installation of wind turbine blades. However, their highly coupled three-dimensional spatial dynamics, uncert...

Pressure sensors embodied in very tiny packages are deployed in a wide range of advanced applications. Examples of applications range from industrial to altitude location services. They are also becoming increasingly pervasive in many other applicati...

We propose an exploratory framework for a Bohmian model of quantum matter propagating on a classical curved spacetime background. The gravitational sector is governed by classical Einstein field equations throughout; no quantisation of spacetime is a...