Continuous Functions Research Articles

Deep Learning Evidence for Global Optimality of Gerver’s Sofa

The moving sofa problem, introduced by Leo Moser in 1966, seeks to determine the maximal area of a 2D shape that can navigate an L-shaped corridor of unit width. Joseph Gerver’s 1992 solution, providing a lower bound of approximately 2.2195, is the best known, though its global optimality remains unproven. This paper leverages neural networks’ approximation power and recent advances in invexity optimization to explore global optimality. We propose two approaches supporting Gerver’s conjecture that his sofa is the unique global maximum. The first approach uses continuous function learning, discarding assumptions about the monotonicity, symmetry, and differentiability of sofa movements. The sofa area is computed as a differentiable function using our “waterfall” algorithm, with the loss function incorporating both differential terms and initial conditions based on physics-informed machine learning. Extensive training with diverse network initialization consistently converges to Gerver’s solution. The second approach applies discrete optimization to the Kallus–Romik upper bound, improving it from 2.37 to 2.3337 for five rotation angles. As the number of angles increases, our model asymptotically converges to Gerver’s area from above, indicating that no larger sofa exists.

The curvature entropy inequalities of convex bodies

Abstract There are many entropy inequalities in geometry, some of them can be seen as the Minkowski inequalities in the form of entropy, which play important roles in convex geometry. In this article, let P P and Q Q be convex bodies, we introduce the mixed curvature entropy M f ( P , Q ) {M}_{f}\left(P,Q) and the curvature entropy E f ( P , Q ) {E}_{f}\left(P,Q) with respect to a continuous function f f . Moreover, we establish the log \log -Minkowski inequalities of the mixed curvature entropy and the curvature entropy for two classes of three-dimensional convex bodies.

On a variant of extremal disconnectedness in locales and spaces

We study a variant of extremal disconnectedness in frames, defined by requiring that any two disjoint open sublocales should be separable by disjoint open sublocales each of which is the interior of some zero-sublocale. In spaces, this notion was recently introduced in [1] in order to study some properties of frames of z-ideals of rings of continuous functions. The point-free approach adopted here brings to the fore some topological results not considered in [1], such as that a dense z-embedded subspace of a Tychonoff space has this property if and only if the space containing it has the property. In [1], this was proved only for the embedding X ⊆ βX. We give a ring-theoretic characterization by defining a property of f-rings in a transparent manner which shows why the property should be defined the way we do. The property is that if two annihilator ideals meet trivially, then the annihilator of each of the annihilator ideals contains a positive element such that the sum of the two elements is a non-zerodivisor.

Noise-resilient designs and analysis for optical neural networks

Abstract All analog signal processing is fundamentally subject to noise, and this is also the case in next generation implementations of Optical Neural Networks (ONNs). Therefore, we propose the first hardware-based approach to mitigate noise in ONNs. A tree-like and an accordion-like design are constructed from a given Neural Network (NN) that one wishes to implement. Both designs have the capability that the resulting ONNs give outputs close to the desired solution. To establish the latter, we analyze the designs mathematically. Specifically, we investigate a probabilistic framework for the tree-like design that establishes the correctness of the design, i.e., for any feed-forward NN with Lipschitz continuous activation functions, an ONN can be constructed that produces output arbitrarily close to the original. ONNs constructed with the tree-like design thus also inherit the universal approximation property of NNs. For the accordion-like design, we restrict the analysis to NNs with linear activation functions and characterize the ONNs’ output distribution using exact formulas. Finally, we report on numerical experiments with LeNet ONNs that give insight into the number of components required in these designs for certain accuracy gains. The results indicate that adding just a few components and/or adding them only in the first (few) layers in the manner of either design can already be expected to increase the accuracy of ONNs considerably. To illustrate the effect we point to a specific simulation of a LeNet implementation, in which adding one copy of the layers components in each layer reduces the Mean Squared Error (MSE) by 59.1% for the tree-like design and by 51.5% for the accordion-like design. In this scenario, the gap in accuracy of prediction between the noiseless NN and the ONNs reduces even more: 93.3% for the tree-like design and 80% for the accordion-like design.

Stochastic representation for solutions of a system of coupled HJB-Isaacs equations with integral–differential operators

In this paper, we focus on the stochastic representation of a system of coupled Hamilton–Jacobi–Bellman–Isaacs (HJB–Isaacs (HJBI), for short) equations which is in fact a system of coupled Isaacs’ type integral-partial differential equation. For this, we introduce an associated zero-sum stochastic differential game, where the state process is described by a classical stochastic differential equation (SDE, for short) with jumps, and the cost functional of recursive type is defined by a new type of backward stochastic differential equation (BSDE, for short) with two Poisson random measures, whose wellposedness and a prior estimate as well as the comparison theorem are investigated for the first time. One of the Poisson random measures μ appearing in the SDE and the BSDE stems from the integral term of the HJBI equations; the other random measure in BSDE is introduced to link the coupling factor of the HJBI equations. We show through an extension of the dynamic programming principle that the lower value function of this game problem is the viscosity solution of the system of our coupled HJBI equations. The uniqueness of the viscosity solution is also obtained in a space of continuous functions satisfying certain growth condition. In addition, also the upper value function of the game is shown to be the solution of the associated system of coupled Isaacs’ type of integral-partial differential equations. As a byproduct, we obtain the existence of the value for the game problem under the well-known Isaacs’ condition.

Spin relaxation dynamics with a continuous spin environment: The dissipaton equation of motion approach.

We investigate the quantum dynamics of a spin coupling to a bath of independent spins via the dissipaton equation of motion (DEOM) approach. The bath, characterized by a continuous spectral density function, is composed of spins that are independent level systems described by the su(2) Lie algebra, representing an environment with a large magnitude of anharmonicity. Based on the previous work by Suarez and Silbey [J. Chem. Phys. 95, 9115 (1991)] and by Makri [J. Chem. Phys. 111, 6164 (1999)] that the spin bath can be mapped to a Gaussian environment under its linear response limit, we use the time-domain Prony fitting decomposition scheme to the bare-bath time correlation function (TCF) given by the bosonic fluctuation-dissipation theorem to generate the exponential decay basis (or pseudo modes) for DEOM construction. The accuracy and efficiency of this strategy have been explored by a variety of numerical results. We envision that this work provides new insights into extending the hierarchical equations of motion and DEOM approach to certain types of anharmonic environments with arbitrary TCF or spectral density.

Algorithm XXXX: The Delaunay Density Diagnostic

Accurate approximation of a real-valued function depends on two aspects of the available data: the density of inputs within the domain of interest and the variation of the outputs over that domain. There are few methods for assessing whether the density of inputs is sufficient to identify the relevant variations in outputs—i.e., the “geometric scale” of the function—despite the fact that sampling density is closely tied to the success or failure of an approximation method. In this paper, we introduce a general purpose, computational approach to detecting the geometric scale of real-valued functions over a fixed domain using a deterministic interpolation technique from computational geometry. The algorithm is intended to work on scalar data in moderate dimensions (2-10). Our algorithm is based on the observation that a sequence of piecewise linear interpolants will converge to a continuous function at a quadratic rate (in \(L^{2}\) norm) if and only if the data are sampled densely enough to distinguish the feature from noise (assuming sufficiently regular sampling). We present numerical experiments demonstrating how our method can identify feature scale, estimate uncertainty in feature scale, and assess the sampling density for fixed (i.e. static) datasets of input–output pairs. We include analytical results in support of our numerical findings and have released lightweight code that can be adapted for use in a variety of data science settings.

Dini’s Theorem for Fuzzy Number-Valued Continuous Functions

This work aims to provide several versions of Dini’s theorem for fuzzy number-valued continuous functions defined on a compact set K. In this context, there is a wide variety of possibilities since, unlike the real line, we can consider different topologies and orders on the set of fuzzy numbers. For example, we will show that the fuzzy Dini’s theorem holds for the usual partial orders and the most commonly used topologies but does not hold for all orders in general.

Perturbations of norm-additive maps between continuous function spaces

Perturbations of norm-additive maps between continuous function spaces

On the representability of a continuous multivariate function by sums of ridge functions

In this paper, new conditions are found for the representability of a continuous multivariate function as a sum of ridge functions. Using these conditions, we give a new proof for the earlier theorem solving the problem, posed by A.Pinkus in his monograph “Ridge Functions”, up to a multivariate polynomial. That is, we show that if a continuous multivariate function has a representation as a sum of arbitrarily behaved ridge functions, then it can be represented as a sum of continuous ridge functions and some multivariate polynomial.

On multi-bump solutions for a class of (N,q)-Laplacian equation with critical exponential growth in [formula omitted]

This article deals with the following (N,q)-Laplacian equation with exponential critical growth in RN of the form:−ΔNu−Δqu+(μV(x)+B(x))(|u|N−2u+|u|q−2u)=h(u)inRN, where 2≤N<q, μ∈(0,∞), h is a continuous function with exponential critical growth and V:RN→R is a continuous function verifying some hypotheses. Assuming that the nonnegative function B has a potential well int(B−1({0})) composed of k disjoint components Ω1,Ω2,⋯,Ωk. With the help of the variational methods and Morse iteration technique, the existence and multiplicity of positive multi-bump solutions are obtained if the parameter μ>0 is large enough.

Nonlinear normal modes and response to random inputs of systems with bilinear stiffness

This work seeks to provide a comprehensive review of the effects that stiffness bilinearity can have on the nonlinear modes of a system, its response in a random vibration environment, and the connection between the two. Stiffness bilinearity here refers to a continuous piecewise linear force vs displacement function that is composed of two linear regions. This work focuses on a bilinear stiffness function that is regularized so there is a smooth transition at the point where the two linear regions meet. A single-degree-of-freedom system (SDOF) and a two-degree-of-freedom (2DOF) system are explored and several interesting behaviors are shown. The SDOF bilinear spring model is characterized by four parameters: the low amplitude frequency, the ratio of the linear stiffnesses on either side of the transition, the displacement at which the transition occurs, and the rate or sharpness of the transition. The effect of each parameter on the shape of the NNM is described. In a 2DOF system, these parameters have similar effects, but modal coupling is found to play a significant role. When a bilinear system is subjected to random excitation, many harmonics appear in the response for both the SDOF and 2DOF cases. The root-mean-square (RMS) response of the bilinear system can be larger or smaller than the corresponding linear case depending on the values of the parameters and the type of forcing (broadband, bandlimited, in the shape of a vibration mode, etc.). However, many cases were observed in which the RMS response of the bilinear system was almost the same as that of a linear system, and hence the response could be predicted well using linear analysis. It is hoped that the results presented herein can assist engineers in helping to determine when linear analysis would be adequate to predict the failure of a system, when a rigorous nonlinear analysis is required, and what phenomena are likely to be observed in the latter case.

Existence Results of Some Nonlinear Minimization Problems with Application to a System of PDE

A new class of non-self mappings, called proximal condensing operators, is introduced and used to investigate the existence of best proximity points for non-self mappings in reflexive Banach spaces by applying an appropriate measure of noncompactness. In this way, we present generalizations of well-known Schauder and Darbo fixed point problems. We then define the notion of best proximity points for multiplication of two non-self mappings in the setting of Banach algebras and present some sufficient conditions to ensure the existence of such points. Finally, we renorm a Banach space consists of all real valued and continuous functions with two variables to obtain the strict convexity condition and then we survey the existence of an optimal solution for a system of partial differential equations.

A Modified Bubnov-Galerkin Method for Solving Boundary Value Problems with Linear Ordinary Differential Equations

Introduction. The paper considers the solution of boundary value problems on an interval for linear ordinary differential equations, in which the coefficients and the right-hand side are continuous functions. The conditions for the orthogonality of the residual equation to the coordinate functions are supplemented by a system of linearly independent boundary conditions. The number of coordinate functions m must exceed the order n of the differential equation. Materials and Methods. To numerically solve the boundary value problem, a system of linearly independent coordinate functions is proposed on a symmetric interval [−1,1], where each function has a unit Chebyshev’s norm. A modified Petrov-Galerkin method is applied, incorporating linearly independent boundary conditions from the original problem into the system of linear algebraic equations. An integral quadrature formula with twelfth-order error is used to compute the scalar product of two functions. Results. A criterion for the existence and uniqueness of a solution to the boundary value problem is obtained, provided that n linearly independent solutions of the homogeneous differential equation are known. Formulas are derived for the matrix coefficients and the coefficients of the right-hand side in the system of linear algebraic equations for the vector expansion of the solution in terms of the coordinate function system. These formulas are obtained for second- and third-order linear differential equations. The modified Bubnov-Galerkin method is formulated for differential equations of arbitrary order. Discussion and Conclusions. The derived formulas for the generalized Bubnov-Galerkin method may be useful for solving boundary value problems involving linear ordinary differential equations. Three boundary value problems with second- and third-order differential equations are numerically solved, with the uniform norm of the residual not exceeding 10–11.

Fractional Sobolev type spaces of functions of two variables via Riemann-Liouville derivatives

AbstractWe introduce and study the spaces of fractionally absolutely continuous functions of two variables of any order and the fractional Sobolev type spaces of functions of two variables. Our approach is based on the Riemann-Liouville fractional integrals and derivatives. We investigate relations between these spaces as well as between the Riemann-Liouville and weak derivatives.

A Solution to Dynamic Weapon Assignment Problem Based on Game Theory for Naval Platforms

Weapon target assignment is a critical challenge in military contexts. Traditionally, commanding officers manually decide weapon assignments, but the problem’s complexity has grown over time. To address this, automated systems have been introduced. These systems fall into two categories, which are static (time-independent) and dynamic (considering changes over time). Static systems solve the problem for a single time step without considering temporal changes. Dynamic systems incorporate time as a variable, adapting to evolving scenarios. Two main approaches exist, which are asset-based and target-based. Asset-based approach maximizes the survival probability of assets, which is our focus in this study. We propose a solution using game theory that spans the entire area and all time frames. We employ game theory, treating continuous functions of time as utility functions for vessels. Continuous probability-to-kill values for weapons are defined across the area. Threat trajectories yield continuous kill probabilities for the weapons, translating to vessel utility. To avoid inefficiencies, we align individual vessel utility with global utility. The Nash Equilibrium provides the optimal weapon assignment strategy. Our study uses a naval environment for analysis. In summary, our research leverages game theory to dynamically assign weapons to naval vessels, aiming to maximize asset survival.

A generalization of prime ideals in MV-algebras

In this paper, we introduce a special type of prime ideals in [Formula: see text]-algebras. We define the concept of a [Formula: see text]-prime ideal, which is a proper ideal [Formula: see text] that preserves the property of being prime with respect to the ideal [Formula: see text]. If [Formula: see text] is not a subset of [Formula: see text], then we call the [Formula: see text]-prime ideal an [Formula: see text]-prime ideal. We investigate the relationships between these ideals and their connections to minimal prime ideals and maximal ideals. Additionally, we explore various properties such as extension, transitivity, meet, join, and annihilator. Finally, we study prime ideals in [Formula: see text]-algebras of continuous functions.

Bayesian Optimization Using Simulation-Based Multiple Information Sources over Combinatorial Structures

Bayesian optimization due to its flexibility and sample efficiency has become a standard approach for simulation optimization. To reduce this problem, one can resort to cheaper surrogates of the objective function. Examples are ubiquitous, from protein engineering or material science to tuning machine learning algorithms, where one could use a subset of the full training set or even a smaller related dataset. Cheap information sources in the optimization scheme have been studied in the literature as the multi-fidelity optimization problem. Of course, cheaper sources may hold some promise toward tractability, but cheaper models offer an incomplete model inducing unknown bias and epistemic uncertainty. In this manuscript, we are concerned with the discrete case, where fx,wi is the value of the performance measure associated with the environmental condition wi and p(wi) represents the relevance of the condition wi (i.e., the probability of occurrence or the fraction of time this condition occurs). The main contribution of this paper is the proposal of a Gaussian-based framework, called augmented Gaussian process (AGP), based on sparsification, originally proposed for continuous functions and its generalization in this paper to stochastic optimization using different risk profiles for combinatorial optimization. The AGP leverages sample and cost-efficient Bayesian optimization (BO) of multiple information sources and supports a new acquisition function to select the new source–location pair considering the cost of the source and the (location-dependent) model discrepancy. An extensive set of computational results supports risk-aware optimization based on CVaR (conditional value-at-risk). Computational experiments confirm the actual performance of the MISO-AGP method and the hyperparameter optimization on benchmark functions and real-world problems.

Order-boundary characterization of the linear lattice of Riemann 𝜇-integrable functions as a certain completion of the linear lattice of bounded continuous functions

The linear lattice of Riemann μ \mu -integrable functions on a completely regular space with a bounded positive Radon measure μ \mu is viewed as an extension of the linear lattice of bounded continuous functions. To characterize this Riemann extension, the new functional analysis category of c c -latlineals with refinements ( ≡ c r \equiv cr -latlineals) is introduced. On this basis, the procedure of c r cr -completion of certain order-boundary type is defined. The μ \mu -Riemann extension is shown to be the result of applying this procedure to the c r μ cr_{\mu } -latlineal of bounded continuous functions.

Bernstein-Riemann interpolation formula for arbitrary continuous functions on a segment

Bernstein-Riemann interpolation formula for arbitrary continuous functions on a segment