This paper concerns applications of advanced techniques of pseudo-Jacobians for studying a nonsmooth and nonconvex general optimization problem in Banach spaces. To this end, calculus rules for pseudo-Jacobians in infinite-dimensional spaces are presented; in particular, thwee chain rule for pseudo-Jacobians is developed. Then, using the idea of pseudo-Jacobian, the Robinson, Mangasarian- Fromovitz and Abadie constraint qualifications are generalized. The relations between these constraint qualifications and the local error bound property are investigated. Furthermore, a necessary optimality condition is derived under the Abadie constraint qualification. As an application, new constraint qualifications and necessary optimality conditions are obtained for nonsmooth and nonconvex infinite programming problems. In this way, a pseudo-Jacobian for the pointwise supremum of an arbitrary family of real-valued functions defined on a Banach space is provided. Some examples are given to clarify the results.

Abstract We introduce a quadratic form Q on the space of functions on the gap poset G of the numerical semigroup $$\langle a,b\rangle $$ ⟨ a , b ⟩ . We prove combinatorially that when evaluated on the indicator function of an upward closed subset D , this quadratic form precisely recovers the Gorsky–Mazin $$\texttt {dinv} $$ dinv statistic of D , viewed as a Young subdiagram of G . Furthermore, we prove Theorem 1.2 that when evaluated on a pair of subdiagrams of G , the symmetric bilinear form associated with Q is equal to a novel cross- $$\texttt {dinv} $$ dinv statistic, which is non-negative. Combining these, we prove the inequality $$\begin{aligned} Q(\mathbf {\textit{n}})\ge \dfrac{1}{|G|}\,\Vert \mathbf {\textit{n}}\Vert _\infty ^2 \end{aligned}$$ Q ( n ) ≥ 1 | G | ‖ n ‖ ∞ 2 if $$\mathbf {\textit{n}}$$ n is a real-valued decreasing function on G , showing an effective positive definiteness of Q on the corresponding cone. Theorem 1.2, the main engine of the paper, was autoformalized in Lean/Mathlib by AxiomProver.

Abstract This paper explores the reconstruction of a real-valued function f defined over a domain $$\varOmega \subset \mathbb {R}^2$$ Ω ⊂ R 2 using bivariate polynomials that satisfy triangular histopolation conditions. More precisely, we assume that only the averages of f over a given triangulation $$\mathcal {T}_N$$ T N of $$\varOmega $$ Ω are available and seek a bivariate polynomial that approximates f using a histopolation approach, potentially flanked by an additional regression technique. This methodology relies on the selection of a subset of triangles $$\mathcal {T}_M \subset \mathcal {T}_N$$ T M ⊂ T N for histopolation, ensuring both the solvability and the well-conditioning of the problem. The remaining triangles can potentially be used to enhance the accuracy of the polynomial approximation through a simultaneous regression. We will introduce histopolation and combined histopolation-regression methods using the Padua points, discrete Leja sequences, and approximate Fekete nodes. The proposed algorithms are implemented and evaluated through numerical experiments that demonstrate their effectiveness in function approximation.

Except a few, the majority of the literature on monitoring ordinal data consider independent and identically distributed processes, where samples of data are collected sequentially in time. However, stationary ordinal processes exhibiting serial dependence are also common in many real-world process monitoring applications. This study proposes three classes of novel control charts for monitoring serially dependent stationary ordinal processes. Instead of sample statistics, individual observations are utilized. Exponentially weighted moving-average smoothing of a sequence of estimates is used for estimating the probability mass (cumulative distribution) function of the process. Defined real-valued functions of the probability mass (cumulative distribution) estimates are then used as the statistic plotted on the control charts. The methods are designed to be sensitive to a shift in the marginal distribution. Average run length performance of the control charts are computed under a comprehensive set of data-generating process models, which are inspired by real-world examples and exhibit quite different serial dependence structures. The performances of the proposed control charts are evaluated and compared to provide recommendations for implementations. The results show that the class of demerit-type charts generally perform better than the others. To illustrate the application and interpretation of the proposed methods, a real-world data example on monitoring of heating, ventilation, and air conditioning systems in passenger rail coaches is discussed.

This work develops an analytical framework for the Hartley transform, which, unlike the Fourier transform, converts real-valued functions into real-valued functions. We derive Parseval–Goldstein-type identities for suitable classes of functions and establish Abelian theorems in the setting of compactly supported distributions and generalized functions. Moreover, a finite formulation of the Hartley transform is constructed and analyzed within the spaces of distributions of compact support and generalized functions.

In this paper, we investigate a (k,Υ) fractional quadratic integral equation in the Banach space of real-valued continuous functions on [0,1]. By using a measure of noncompactness associated with monotonicity and Darbo’s fixed point theorem, we provide sufficient conditions for the existence of at least one monotonic solution and analyze its stability. Finally, an illustrative example is presented to demonstrate the theoretical results, including several particular cases.

In this study, we introduce new versions of fuzzy fractional integral operators for both one- and two-variable cases. Using these operators, several Hermite–Hadamard-type (H-type) inclusions are established for fuzzy-number-valued convex functions (F·N⋅V-functions) and F·N⋅V-coordinated convex functions. These results are obtained by employing F·N⋅V-weighted functions within the framework of the newly defined Hadamard and generalized fractional integrals in one- and two-dimensional settings. The use of generalized fractional integral operators provides a unified approach that encompasses a wide class of classical and modern fractional integrals, including the fuzzy Riemann–Liouville and Hadamard types. This unified setting enables the derivation of more comprehensive and flexible inequality results in the fuzzy-number context. The inclusions obtained in this work significantly extend and generalize several known H⋅H-type inequalities previously established for real-valued and interval-valued functions (I⋅V-functions). Furthermore, the proposed results yield a variety of meaningful special cases by specifying suitable kernel functions and parameters of the generalized fractional integrals. In particular, we derive new weighted H⋅H-type inclusions involving logarithmic functions in the fuzzy-number framework. These findings underscore the effectiveness of generalized fractional integrals in capturing nonlocal behavior and uncertainty, and they provide new tools for further investigations in fuzzy analysis, fractional calculus, and generalized convexity.

An averaging operator on an associative algebra A is an algebraic abstraction of the time average operator on the space of real-valued functions defined in time-space. In this paper, we consider relative averaging operators on a bimodule M over an associative algebra A . A relative averaging operator induces a diassociative algebra structure on the space M . The full data consisting of an associative algebra, a bimodule and a relative averaging operator is called a relative averaging algebra. We define bimodules over a relative averaging algebra that fits with the representations of diassociative algebras. We construct a graded Lie algebra and an L ∞ -algebra that are respectively controlling algebraic structures for a given relative averaging operator and relative averaging algebra. We also define cohomologies of relative averaging operators and relative averaging algebras and find a long exact sequence connecting various cohomology groups. As applications, we study deformations and abelian extensions of relative averaging algebras. Finally, we define homotopy relative averaging algebras and show that they induce homotopy diassociative algebras.

Summary Let f be a real-valued function such that f ′ ( a ) exists. In this manuscript, we discuss whether or not it is the case that (7) lim n → ∞ f ( x n ) − f ( y n ) x n − y n = f ′ ( a ) when both sequences { x n } n = 1 ∞ and { y n } n = 1 ∞ converge to a. First, we obtain a theorem revealing that when x n and y n lie on different sides of a, the limit in [7] always holds true. However, in the case that both sequences { x n } n = 1 ∞ and { y n } n = 1 ∞ lie on the same side of a, we establish a second theorem and implement our findings to solve intricate calculus problems, illustrating the applications of our findings. Finally, we present a complex example which demonstrates that the limit in equation (7) may not hold if the conditions stated in our theorems are not satisfied.

We continue the investigation into the computational status of the existence of moduli of regularity (and their use for rates of convergence) in the sense of Kohlenbach, Lopez and Nicolae (2019), carried out w.r.t. classical reverse mathematics and Weihrauch degrees in a previous paper and determine the amount of LEM involved. We also show that the existence of a modulus of regularity always yields an algorithm for the computation of a zero in the case of continuous real-valued functions F on a compact metric space K (in F equipped with a modulus of uniform continuity and K given in standard representation) whenever such a zero exists. If K is a compact subset of a uniformly convex Banach space X and the zero set of F is convex one can compute even the zero of minimal norm. A modulus of regularity can also be used to compute the left-most infinite path of an infinite 0/1-tree. We also show that there is no proof-theoretically tame nonstandard uniformity principle which would make it possible to replace in the regularity assumption compactness by metric boundedness and still guarantee classically correct bounds.

Generalizing the notion of a multiplicative inequality among minors of a totally positive matrix, we describe, over full rank cluster algebras of finite type, the cone of Laurent monomials in cluster variables that are bounded as real-valued functions on the positive locus of the cluster variety. We prove that the extreme rays of this cone are the u -variables of the cluster algebra. Using this description, we prove that all bounded ratios are bounded by 1 and give a sufficient condition for all such ratios to be subtraction free. This allows us to show in Gr ( 2 , n ) , Gr ( 3 , 6 ) , Gr ( 3 , 7 ) , and Gr ( 3 , 8 ) that every bounded Laurent monomial in Plücker coordinates factors into a positive integer combination of so-called primitive ratios. In Gr ( 4 , 8 ) this factorization does not exists, but we provide the full list of extreme rays of the cone of bounded Laurent monomials in Plücker coordinates.

In this work, we establish the conditions for ensuring the existence and uniqueness of common best proximity points for non-self-mappings defined on the general metric spaces. A unified theoretical framework is formulated to cover a broad class of contraction mappings. We describe the required conditions on the real-valued functions (ℵ, Φ) : [0,∞) → R and verify that these secure the existence of common best proximity points for (ℵ, Φ)−interpolative contractions in complete metric spaces. The study further extends this concept by examining multiple forms of interpolative proximal-type contractions, such as proximal, Ćirić —Reich—Rus, Kannan, and Hardy-Rogers variants, through the use of the auxiliary functions (ℵ, Φ). Several illustrated examples are included to demonstrate the applicability of our findings. Finally, we conclude with an application involving a nonlinear fractional differential equation, showing that it fully satisfies the assumption of our main result.

In the context of functions between metric spaces, we introduce a new class of functions which we called Cauchy approachable function which happens to lie between the classes of continuous functions and Cauchy regular functions. A significant characteristic of these functions is that they preserve Cauchy connectedness, despite being a weaker form of Cauchy regularity. We say that a metric space (X, d) is a Cauchy approachable space if every real-valued continuous function on X is Cauchy approachable. Furthermore, by utilizing the concept of Cauchy approachable spaces, we are able to obtain a partial answer to an open question namely, [Question 14.4] posed in [3], in the summary of the last section.

Multi-composite activated neural network operators can be understood as positive linear operators, allowing them to be analyzed using standard, established theory. Formed by composing multiple general activation functions, these operators act upon continuous real-valued functions defined on a compact interval. This work presents a quantitative analysis of how quickly these operators converge to the unit operator. Utilizing general inequalities based on the modulus of continuity—applicable to either the function itself or its derivative—this study establishes both uniform and Lp approximation results. Furthermore, the analysis incorporates the convexity of functions to produce related, specific results.

Abstract We prove new results about comparing the efficiency of general state space Markov chain Monte Carlo algorithms that randomly select a possibly different reversible method at each step (previously known only for finite state spaces). We also provide new, simpler, more accessible proofs of key results, and analyse numerous examples. We provide a full proof of the formula for the asymptotic variance for real-valued functionals on $\varphi$ -irreducible reversible Markov chains, first introduced by Kipnis and Varadhan (1986, Commun. Math. Phys. 104 , 1–19). Given two Markov kernels P and Q with stationary measure $\pi$ , we say that the Markov kernel P efficiency-dominates the Markov kernel Q if the asymptotic variance with respect to P is at most the asymptotic variance with respect to Q for every real-valued functional $f\in L^2(\pi)$ . Assuming only a basic background in functional analysis, we prove that for two reversible Markov kernels P and Q , P efficiency-dominates Q if and only if the operator $\mathcal{Q}-\mathcal{P}$ , where $\mathcal{P}$ is the operator on $L^2(\pi)$ that maps $f\mapsto\int f(y)P(\cdot,\mathrm{d}y)$ and similarly for $\mathcal{Q}$ , is positive on $L^2(\pi)$ , i.e. $\langle f,\left(\mathcal{Q}-\mathcal{P}\right)f\rangle\geq0$ for every $f\in L^2(\pi)$ (previous proofs for general state spaces use technical results from monotone operator function theory). We use this result to show that under mild conditions, sandwich variants of data augmentation algorithms efficiency-dominate the original algorithm. We also provide other easy-to-check sufficient conditions for efficiency dominance, some of which are generalized from the finite state space case. We also provide a proof based on that of Tierney (1998, Ann. Appl. Prob. 8 , 1–9) that Peskun dominance is a sufficient condition for efficiency dominance for reversible kernels. Using these results, we show that Markov kernels formed by random selection of other ‘component’ Markov kernels will always efficiency-dominate another Markov kernel formed in this way, as long as the component kernels of the former efficiency-dominate those of the latter. These results on the efficiency dominance of combining component kernels generalizes the results on the efficiency dominance of combined chains introduced by Neal and Rosenthal (2024, J. Appl. Prob. 62 , 188–208) from finite state spaces to general state spaces.

The first aim of this study is to point out new aspects of approximation theory applied to a few classes of holomorphic functions via Vitali’s theorem. The approximation is made with the aid of the complex moments of the functions involved, which are defined similarly to the moments of a real-valued continuous function. By applying uniform approximation of continuous functions on compact intervals via Korovkin’s theorem, the hard part concerning uniform approximation on compact subsets of the complex plane follows according to Vitali’s theorem. The theorem on the set of zeros of a holomorphic function is also applied. In the end, the existence and uniqueness of the solution for a multidimensional moment problem are characterized in terms of limits of sums of quadratic expressions. This is the application appearing at the end of the title. Consequences resulting from the first part of the paper are pointed out with the aid of functional calculus for self-adjoint operators.

We characterize the upper semicontinuous representability of a semiorder ≺ as an interval order (namely, by a pair (u,v) of upper semicontinuous real-valued functions) on a topological space with a countable basis of open sets, where one of the representing functions is a one-way utility for the characteristic weak order ≺0 associated with the semiorder. Such a description generalizes the upper semicontinuous threshold representation. To this end, we introduce a suitable upper semicontinuity condition concerning a semiorder, namely strict upper semicontinuity. We further characterize the mere existence of an upper semicontinuous one-way utility for this characteristic weak order, with a view to the identification of maximal elements on compact metric spaces.

Integral inequalities, in general, serve as powerful tools for various applications. Specifically, when an integral operator is used as a predictive tool, an integral inequality can play a key role in defining, quantifying, and analyzing such processes. Real-valued functions over a fuzzy domain, also referred to as real-valued fuzzy functions, offer a valuable approach for incorporating uncertainty into prediction models. In this paper, using a straightforward proof method over a newly defined triangular LPL_PLP​ fuzzy space, we establish several new refinements for integral forms of the classical Hölder’s and newly defined triangular Hölder’s-like inequality. Numerous existing inequalities linked with the triangular Hölder’s-like inequality over a fuzzy domain can be improved through the newly obtained ones, as illustrated through applications such as the triangular Hölder’s power-mean-like integral inequality, triangular Cauchy–Schwarz-like inequality, triangular Minkowski’s-like inequality, and triangular Beckenbach’s-like inequality over a fuzzy domain. Additionally, our outcomes represent significant progressions in the field of mathematics.

Concerning the stability analysis for systems with time-varying delays, a negative-determination method for a class of generalized-convex (generalized-concave) matrix-valued polynomial function is proposed, which is an extension of real-valued quadratic function negative-determination lemma. Initially, the definition of generalized-convex matrix-valued function is presented, and its first-order and second-order determination conditions are established. Subsequently, negative definite sufficient conditions for generalized-convex (generalized-concave) matrix-valued polynomials with high degrees are developed. Further, stability criteria with less conservatism for linear time-delay systems are constructed by applying the proposed method. Finally, a numerical example and a load frequency control example are presented to verify the feasibility and the advantage of the designed method on achieving less conservative stability criterion.

Mappings from biological sequences (DNA, RNA, protein) to quantitative measures of sequence functionality play an important role in contemporary biology. We are interested in the related tasks of (i) inferring predictive sequence-to-function maps and (ii) decomposing sequence-function maps to elucidate the contributions of individual subsequences. Because each sequence-function map can be written as a weighted sum over subsequences in multiple ways, meaningfully interpreting these weights requires “gauge-fixing,” i.e., defining a unique representation for each map. Recent work has established that most existing gauge-fixed representations arise as the unique solutions to L_2-regularized regression in an overparameterized “weight space” where the choice of regularizer defines the gauge. Here, we establish the relationship between regularized regression in overparameterized weight space and Gaussian process approaches that operate in “function space,” i.e. the space of all real-valued functions on a finite set of sequences. We disentangle how weight space regularizers both impose an implicit prior on the learned function and restrict the optimal weights to a particular gauge. We show how to construct regularizers that correspond to arbitrary explicit Gaussian process priors combined with a wide variety of gauges and characterize the implicit function space priors associated with the most common weight space regularizers. Finally, we derive the posterior distribution of a broad class of sequence-to-function statistics, including gauge-fixed weights and multiple systems for expressing higher-order epistatic coefficients. We show that such distributions can be efficiently computed for product-kernel priors using a kernel trick.