This paper addresses sum-of-squares representations of nonnegative functions that are definable in o-minimal structures on ( R , + , ⋅ ) . Namely, let f , g 1 , … , g l , h 1 , … , h m : R n → R be definable C p functions ( p ≥ 2 ), and assume that f is nonnegative on the set S := { x ∈ R n | g 1 ( x ) ≥ 0 , … , g l ( x ) ≥ 0 , h 1 ( x ) = 0 , … , h m ( x ) = 0 } . Under some natural hypotheses on zeros of f in S, we show that f is expressible in the form f = ϕ 0 + ∑ i = 1 l ϕ i g i + ∑ j = 1 m ψ j h j , where ϕ i , ψ j : R n → R are definable C p − 2 -functions and each ϕ i is a sum of squares of definable C p − 2 -functions. As a consequence, we derive global optimality conditions which generalize the Karush–Kuhn–Tucker optimality conditions for nonlinear convex optimization.

Objective.Determining the intricate structure and function of neural circuits requires the ability to precisely manipulate circuit activity. Two-photon holographic optogenetics has emerged as a powerful tool for achieving this via flexible excitation of user-defined neural ensembles. However, the precision of two-photon optogenetics has been constrained by off-target stimulation (OTS), an effect where proximal non-target neurons can be unintentionally activated due to imperfect spatial confinement of light onto target neurons. New approaches are therefore needed to resolve the OTS problem.Approach.Here, we introduce a real-time computational method for mitigating OTS that first empirically samples each neuron's sensitivity to stimulation at proximal locations, and then optimizes stimulation sites using a fast, interpretable model based on adaptive non-negative basis function regression (NBFR).Main results.NBFR is highly scalable, completing model fitting for hundreds of neurons in just a few seconds and then optimizing stimulation sites in several hundred milliseconds per stimulus-fast enough for most closed-loop behavioral experiments. We characterize the performance of our approach in both simulations andin vivoexperiments in mouse hippocampus, showing its efficacy under realistic experimental conditions.Significance.Our results thus establish NBFR-based photostimulus optimization as an important addition to an emerging computational toolkit for precise yet scalable holographic optogenetics.

In this paper, we study a fractional Kirchhoff problem with a Hardy-type singular potential and general nonlinearities depending on the solution and its gradient: M∫∫RN×RN|u(x)−u(y)|q|x−y|N+qsdxdy(−Δ)su=λu|x|2s+f(x,u,∇u)inΩ, where Ω⊂RN is a bounded domain containing the origin, s∈(0,1), q∈(1,2] with N>2s, λ>0, and f is a measurable non-negative function satisfying suitable hypotheses. The main objective is to establish the existence of positive solutions for the largest possible class of nonlinearities f without imposing restrictions on λ. Two main cases areconsidered: (I)−f(x,u,∇u)=up+μ,and(II)−f(x,u,∇u)=|∇u|p+μg. Existence is proved under suitable hypotheses on q,p and the data g,μ. The results are new, including for the local case s=1.

We prove the multiplicity and concentration of normalized solutions of critical biharmonic equations with combined nonlinearities in \mathbb{R}^{N} , \begin{equation*}\Delta^{2}u+V(\varepsilon x)u=\lambda u+\mu |u|^{q-2}u+|u|^{2^{**}-2}u \text{ in }\mathbb{R}^{N}, \quad \int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\end{equation*} where \Delta^{2} is the biharmonic operator, N\geq5 , \mu,c>0 , \varepsilon>0 , \lambda\in\mathbb{R} , q\in(2,2+\frac{8}{N}) , and 2^{**}=\frac{2N}{N-4} is the Sobolev critical exponent. The potential V is a bounded and continuous nonnegative function, satisfying some suitable global conditions. Using minimization techniques and a truncation argument, we show that the number of normalized solutions is not less than the number of global minimum points of V when the parameter \varepsilon is sufficiently small. To overcome the loss of compactness of the energy functional due to the critical growth, we apply the concentration-compactness principle. To the best of our knowledge, this study is the first contribution regarding the concentration and multiplicity properties of normalized solutions of critical biharmonic equations with combined nonlinearities in \mathbb{R}^{N} . To some extent, the main results included in this paper complement several recent contributions to the study of biharmonic equations with combined nonlinearities.

We give a nonnegative step function with 575 equally spaced intervals such that ‖ f ∗ f ‖ L 2 ( R ) 2 ‖ f ∗ f ‖ L ∞ ( R ) ‖ f ∗ f ‖ L 1 ( R ) ≥ 0.901564. This improves upon a recent result of Deepmind’s AlphaEvolve from May 2025, which found a nonnegative step function with 50 equally space intervals for which the left hand side is ≥ 0.8962 . Our function was found using simulated annealing and gradient based methods.

In this work, we investigate the compatibility with the second law of thermodynamics of certain evolution equations for heat flux which frequently appear in the literature when treating problems of extended thermodynamics, namely in dual-phase-lag (DPL) and in three-phase-lag (TPL) theories for a rigid thermal conductor. For each one of these two cases, we propose a concrete form for the free energy function, in which heat flux enters as an independent variable side by side with temperature, and a corresponding non-negative quadratic dissipation function. An important aspect of the present work is that all the introduced material tensors in the formulation of the free energy are kept to the simplest form possible, and can be calculated based on experimental data. The present results demonstrate that both DPL and TPL approximate models for the evolution equation of the heat flux considered here can be rendered thermodynamically admissible through a class of free energy functions to satisfy the Second Law, providing a flexible and consistent modeling of non-Fourier heat conduction. In any case, it is shown that temperature and heat flux are determined simultaneously from a set of coupled, nonlinear partial differential equations, and that obtaining an equation for temperature independently from heat flux could be achieved only in special cases and under some simplifying assumptions. Within this context, the heat conductivity is assumed a linear function of temperature, an essential feature from an experimental point of view. The used methodology, consisting of going from evolution equations for the heat flux to the construction of dissipation functions and free energies which satisfy the requirements of the second law, is, in our belief, a useful trend to treat more difficult cases for higher approximation theories, or for couplings with the other fields, i.e. dynamics, electromagnetism and other. In spite of slight resemblance with published work, the suggested concrete forms of the free energies and dissipation functions, as well as the governing nonlinear set of partial differential equations are not mentioned in the available literature to the authors knowledge. It turns out that the only requirement of consistency with the second law of thermodynamics for the treated cases is that the thermal relaxation times be non-negative, and that the thermal relaxation time related to thermal displacement in TPL satisfies a certain inequality. Any further requirements could be linked only to the stability of solutions of the arising governing partial differential equations. A one-dimensional application in a half-space solves the newly suggested nonlinear system of governing equations, giving a physical insight to the presented model, and pointing out at its adequacy for describing the propagation of thermal waves with finite speed.

This paper explores the implications of guaranteeing privacy by imposing a lower bound on the information density between the private and the public data. We introduce a novel and operationally meaningful privacy measure called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pointwise maximal cost</i> (PMC) and demonstrate that imposing an upper bound on PMC is equivalent to enforcing a lower bound on the information density. PMC quantifies the information leakage about a secret to adversaries who aim to minimize non-negative cost functions after observing the outcome of a privacy mechanism. When restricted to finite alphabets, PMC can equivalently be defined as the information leakage to adversaries aiming to minimize the probability of incorrectly guessing randomized functions of the secret. We study the properties of PMC and apply it to standard privacy mechanisms to demonstrate its practical relevance. Through a detailed examination, we connect PMC with other privacy measures that impose upper or lower bounds on the information density. These are pointwise maximal leakage (PML), local differential privacy (LDP), and (asymmetric) local information privacy. In particular, we show that a mechanism satisfies LDP if and only if it has both bounded PMC and bounded PML. Overall, our work fills a conceptual and operational gap in the taxonomy of privacy measures, bridges existing disconnects between different frameworks, and offers insights for selecting a suitable notion of privacy in a given application.

The team formation problem is a very important problem in the labor market that has been proved to be NP-hard. This paper proposes an efficient bicriteria streaming algorithm aimed at striking a balance between gain and cost in team formation problems with cardinality constraints on the integer lattice. To address this, we utilized a model optimized for maximizing the difference between a nonnegative normalized monotone submodular function and a nonnegative linear function. We further consider the case where the first function of the object function is weakly submodular. Combining the lattice binary search with the threshold method, we present an online algorithm called bicriteria streaming algorithmsalgorithm. Concomitantly, we comprehensively analyze both models.

Let 0 ≤ α &lt; n , M α be the fractional maximal operator, M ♯ be the sharp maximal operator and b be the locally integrable function. Denote by [ b , M α ] and [ b , M ♯ ] be the commutators of the fractional maximal operator M α and the sharp maximal operator M ♯ . In this paper, we show some necessary and sufficient conditions for the boundedness of the commutators [ b , M α ] and [ b , M ♯ ] on slice spaces when the function b is the Lipschitz function, by which some new characterizations of the non-negative Lipschitz function are obtained.

Classical sequence spaces such as l∞ , c, and c0 have been extensively studied and recognized as fundamental in the advancement of functional analysis and related areas of mathematics. In this article, we investigate the algebraic properties together with para- norm structures of the sequence spaces W0 (∆, f )(C2), W (∆, f )(C2), and W∞ (∆, f )(C2) in a bi-complex setting, which are induced by a non-negative real-valued function ϕ.

A connected graph with the same order and size is known as a unicyclic graph.For a vertex u in a graph, we denote its degree by du.For a unicyclic graph G of a given order, we investigate an extremal problem concerning the degree-based graph invariants of the form BID f (G) = vwE(G) f (dv, dw), where E(G) represents the edge set of G and f is a symmetric non-negative function that depends on the degrees of adjacent vertices of G.These graph invariants are known as bond incident degree indices.One of our results provides a partial solution to an open problem proposed by Ergoti and Doli in [MATCH Commun.

This paper focuses on the following double critical Schrödinger–Poisson system: [Formula: see text] where [Formula: see text] and [Formula: see text] is a nonnegative weight function. Moreover, by the celebrated global bifurcation theorem due to Rabinowitz [Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971) 487–513], we prove existence of (weak) solutions for the above system. To the best of our knowledge, it seems to be the first time contribution to considering the bifurcation and existence of solutions for Schrödinger–Poisson systems with doubly critical growth by the global bifurcation theorem. To some extent, our main theorems complement some results established in [Y. Meng and X. He, Normalized solutions for the Schrödinger–Poisson system with doubly critical growth, Topol. Methods Nonlinear Anal. 62 (2023) 509–534; Y. Meng and X. He, Multiplicity of normalized solutions for the fractional Schrödinger–Poisson system with doubly critical growth, Acta Math. Sci. Ser. B Engl. Ed. 44 (2024) 997–1019; P. Pucci, L. Wang and B. Zhang, Bifurcation and existence for Schrödinger–Poisson systems with doubly critical nonlinearities, Z. Angew. Math. Phys. 75 (2024) 170; P. Pucci, L. Wang and B. Zhang, Bifurcation and regularity analysis of the Schrödinger–Poisson equation, Nonlinearity 37 (2024) 035011; L. Wang and Y. Xing, Bifurcation analysis of fractional Kirchhoff–Schrödinger–Poisson systems in [Formula: see text], Electron. J. Qual. Theory Differ. Equ. 3 (2024) 1–17].

Abstract We develop a unified strategy to obtain the geometric logarithmic Hardy inequality on any open set of a stratified group , provided the validity of the Hardy inequality in this setting, where the so‐called “weight” is regarded to be any measurable nonnegative function on . Provided the legitimacy of the latter for some , we also show an inequality that is an extension of the ‘generalized Poincaré inequality’ introduced by Beckner with the addition of the weight , and this is referred to as the “geometric Hardy‐Poincaré inequality.” The aforesaid inequalities become explicit in the case where , the half‐space of , when , and in the case where , when is the “horizontal norm” on the first stratum of . For the second case, the semi‐Gaussian analog of the derived inequalities is proved, when the Gaussian measure is regarded with respect to the first stratum of . Applying our results to the case where (abelian case), we generalize the classical probabilistic Poincaré inequality by adding weights.

In this paper, we consider the discrete Hardy and Copson operators on the cone of nonnegative monotone sequences. We prove that weighted inequalities of the form lp→ lq for discrete Hardy and Copson operators on the cone of monotone sequences, in the case 0 &lt; q &lt; ∞, 0&lt;p&lt;1, can be reduced to the corresponding inequalities on the cone of nonnegative sequences. The latter possess a broader basis for proof, which significantly extends the possibilities for their analysis. Weighted inequalities for the integral Hardy operator (in the continuous setting) on the cone of nonnegative nonincreasing functions have been studied previously by many authors. Reduction theorems for inequalities involving Hardy-type integral operators on the cone of nonincreasing functions to inequalities on the cone of nonnegative functions are also well known. We present various theorems concerning the equivalence of inequalities for discrete Hardy and Copson operators on the cone of nonnegative nonincreasing sequences and inequalities on the cone of nonnegative sequences. Our proofs differ substantially from those in the continuous case. Methods applicable in the continuous setting do not always work in the discrete setting. For the case p&gt;1, analogous results were obtained by the authors earlier.

We study a three-weight inequality for a superposition of the Copson, Hardy, and Tandori operators. The goal of this paper is to prove a complete characterization of the boundedness of the operator that is a combination of these three operators in weighted Lebesgue spaces from to . The main focus is on determining necessary and sufficient conditions under which this inequality holds for all non-negative measurable functions on the positive real axis. The notion of a fundamental function of a Borel measure with respect to an increasing function is used substantially. Since the Tandori operator is not a linear operator, we cannot use the duality methods used in earlier works. To solve this problem, we develop a new, simplified discretization method that avoids the complexities of previously known methods. An explicit form of the best constant in the inequality is obtained, demonstrating the accuracy and optimality of the results. By establishing necessary and sufficient conditions for the boundedness of these composite operators, we improve the inequalities previously established in the works of Gogatishvili A., Pick L., Opic B. [1]. The results obtained in the paper extend and complement existing research in the field of weighted inequalities and operator analysis in function spaces and offer potential applications in approximation theory, harmonic analysis and related areas.

Bohr’s Inequalities Associated with the Set of All Sequences of Nonnegative Continuous Functions

This paper presents a new general formulation of the Radon–Nikodym theorem in the setting of abstract measure theory. We introduce the notion of weak localizability for a measure and show that this property is both necessary and sufficient for the validity of a Radon–Nikodym-type representation under a natural compatibility relation between measures. The proof relies solely on elementary tools, such as Markov’s inequality and the monotone convergence theorem. In addition to establishing the main result, we provide a constructive approach to envelope functions for families of non-negative measurable functions supported on sets of finite measure.

The main objective of this paper is to introduce the novel concepts of double asymptotically deferred statistically equivalent and double strongly deferred asymptotically statistical equivalence by considering two non-negative real-valued Lebesgue measurable functions defined on (1,∞)×(1,∞). Furthermore, we investigate several inclusion theorems related to these new notions, which have not been previously explored in the literature.

UDC 517.5 We establish several sharp versions of Bohr's inequalities for the class of $K$-quasiconformal sense-preserving harmonic mappings on the unit disk $\mathbb{D} := \{z \in \mathbb{C}\colon |z| &lt; 1\}$ by using a sequence $\{\Psi_n(r)\}_{n=0}^\infty$ of nonnegative continuous functions defined on $[0,1)$ and such that the series $\sum_{n=0}^\infty \Psi_n(r)$ converges locally uniformly in $[0,1).$ As an application, we deduce several well-known results, as well as numerous improved and refined Bohr's inequalities for harmonic mappings in the unit disk $\mathbb{D}.$

We derive interior and boundary \(L^q\)-regularity estimates for double obstacle problems involving quasilinear operators of \(p\)-Laplacian type and lower-order terms with nonnegative potential functions satisfying a reverse Holder type condition. We prove that the \(L^q\) norms of the gradient of a solution to the obstacle problem, as well as the lower-order term, can be estimated by the \(L^q\) norms of the data and the gradients of the obstacles. Moreover, the relevant constants in the \(L^q\) estimates depend only on the constant in the reverse Holder type condition for the potential and are independent of the potential. For more information and the latex files, see https://ejde.math.txstate.edu/Volumes/2025/102/abstr.html