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By Foissy's work, the bidendriform structure of the Word Quasisymmetric Functions Hopf algebra (WQSym) implies that it is isomorphic to its dual. However, the only known explicit isomorphism due to Vargas does not respect the bidendriform structure. This structure is entirely determined by so-called totally primitive elements (elements such that the two half-coproducts vanish). In this paper, we construct two bases indexed by two new combinatorial families called red (dual side) and blue (primal side) biplane forests in bijection with packed words. In those bases, primitive elements are indexed by biplane trees and totally primitive elements by a certain subset of trees. We carefully combine red and blue forests to get bicolored forests. A simple recoloring of the edges allows us to obtain the first explicit bidendriform automorphism of WQSym.Mathematics Subject Classifications: 05A05, 05A19, 05E05, 05E18Keywords: Bidendriform Hopf algebras, Word Quasisymmetric Functions, packed words, permutation, primitive elements, duality, tree, forest, global descents

In this paper, we define and study a variant of multiple zeta values (MZVs) of level four, called alternating multiple mixed values or alternating multiple M -values (AMMVs), forming a Q [ i ] -subspace of the colored MZVs of level four. This variant includes the alternating version of Hoffman's multiple t -values, Kaneko–Tsumura's multiple T -values, and the multiple S -values studied by the authors previously as special cases. We exhibit nice properties similar to the ordinary MZVs such as the generalized duality, integral shuffle and series stuffle relations. After setting up the algebraic framework we derive the regularized double shuffle relations of the AMMVs by adopting the machinery from color MZVs of level four. As an important application, we prove a parity result for AMMVs previously conjectured by two of us. We also investigate several alternating multiple S - and T -values by establishing some explicit relations of integrals involving arctangent function. At the end, we compute the dimensions of a few interesting subspaces of AMMVs for weight less than 9. Supported by theoretical and numerical evidence aided by numerical and symbolic computation, we formulate a few conjectures concerning the dimensions of the above-mentioned subspaces of AMMVs. These conjectures hint at a few very rich but previously overlooked algebraic and geometric structures associated with these vector spaces. [ABSTRACT FROM AUTHOR]

This paper studies some results on solutions associated with interval-valued optimal control problems driven by generalized invariant convex (for short, invex) functionals. Necessary conditions of optimality are stated for the considered optimization problem. Also, the sufficiency of the necessary optimality conditions is investigated. Moreover, various duality relationships between the dual models and the primal optimization problem are deliberated. [ABSTRACT FROM AUTHOR]

This paper is devoted to the optimization of the Mayer problem with hyperbolic differential inclusions of the Darboux type and duality. We use the discrete approximation method to get sufficient conditions of optimality for the convex problem given by Darboux differential inclusions and the polyhedral problem for a hyperbolic differential inclusion with state constraint. We formulate the adjoint inclusions in the Euler-Lagrange inclusion and Hamiltonian forms. Then, we construct the dual problem to optimal control problem given by Darboux differential inclusions with state constraint and prove so-called duality results. Moreover, we show that each pair of primal and dual problem solutions satisfy duality relations and that the optimal values in the primal convex and dual concave problems are equal. Finally, we establish the dual problem to the polyhedral Darboux problem and provide an example to demonstrate the main constructions of our approach. [ABSTRACT FROM AUTHOR]

The theory of duality is of fundamental importance in the study of vector optimization problems and vector equilibrium problems. A Mond–Weir-type dual model for such problems is important in practice. Therefore, studying such problems with a dual approach is really useful and necessary in the literature. The goal of this article is to formulate Mond–Weir-type dual models for the minimization problem (P), the constrained vector optimization problem (CVOP) and the constrained vector equilibrium problem (CVEP) in terms of ϵ -upper convexificators. By applying the concept of ϵ -pseudoconvexity, some weak, strong and converse duality theorems for the primal problem (P) and its dual problem (DP), the primal vector optimization problem (CVOP) and its Mond–Weir-type dual problem (MWCVOP), the primal vector equilibrium problem (P) and its Mond–Weir-type dual problem (MWCVEP) are explored. [ABSTRACT FROM AUTHOR]

It was recently shown that integrating out M2 states on Calabi‐Yau manifolds captures non‐perturbative topological string physics in the free energy. In this note, It has been shown that the resulting expression manifests a certain duality symmetry: the free energy at strong string coupling is equal to the Calabi‐Yau period at weak string coupling. The duality yields the appropriate prescription for completing the integrating out in the ultraviolet. [ABSTRACT FROM AUTHOR]

For two Polish state spaces E X and E Y , and an operator G X , we obtain existence and uniqueness of a G X -martingale problem provided there is a bounded continuous duality function H on E X × E Y together with a dual process Y on E Y which is the unique solution of a G Y -martingale problem. For the corresponding solutions (X t) t ≥ 0 and (Y t) t ≥ 0 , duality with respect to a function H in its simplest form means that the relation E x [ H (X t , y) ] = E y [ H (x , Y t) ] holds for all (x , y) ∈ E X × E Y and t ≥ 0. While duality is well-known to imply uniqueness of the G X -martingale problem, we give here a set of conditions under which duality also implies existence without using approximating sequences of processes of a different kind (e.g. jump processes to approximate diffusions) which is a widespread strategy for proving existence of solutions of martingale problems. Given the process (Y t) t ≥ 0 and a duality function H , to prove existence of (X t) t ≥ 0 one has to show that the r.h.s. of the duality relation defines for each y a measure on E X , i.e. there are transition kernels (μ t) t ≥ 0 from E X to E X such that E y [ H (x , Y t) ] = ∫ μ t (x , d x ′) H (x ′ , y) for all (x , y) ∈ E X × E Y and all t ≥ 0. As examples, we treat resampling and branching models, such as the Fleming–Viot measure-valued diffusion and its spatial counterparts (with both, discrete and continuum space), as well as branching systems, such as Feller's branching diffusion. While our main result as well as all examples come with (locally) compact state spaces, we discuss the strategy to lift our results to genealogy-valued processes or historical processes, leading to non-compact (discrete and continuum) state spaces. Such applications will be tackled in forthcoming work based on the present article. [ABSTRACT FROM AUTHOR]

Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanisms, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure Λ for each of the two types. Λ -reproduction here means that a whole fraction of the population is replaced at a reproductive event. Our approach consists of constructing a Λ -asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Provided the measure are ordered stochastically, we can couple them. This allows us to construct the central object of this paper, the Λ − asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time Λ -asymmetric Moran model with its ancestral process. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinuous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types. [ABSTRACT FROM AUTHOR]

The epistemic thesis is the thesis that a 'might' counterfactual like (1) has the same meaning as (2). (1) If Matt had gone to the parade, David might have gone to the parade. (2) Maybe, if Matt had gone to the parade, David would have gone to the parade. In this paper, I give a compositional semantics for 'might' counterfactuals that predicts the epistemic thesis. I offer a new theory of the counterfactual interpretation of the modal 'might' — the interpretation it receives in (1) — and I show that, when coupled with a plausible semantics for 'if' clauses, my theory validates the epistemic thesis. [ABSTRACT FROM AUTHOR]

A technique for developing convex dual variational principles for the governing PDE of nonlinear elastostatics and elastodynamics is presented. This allows the definition of notions of a variational dual solution and a dual solution corresponding to the PDEs of nonlinear elasticity, even when the latter arise as formal Euler–Lagrange equations corresponding to non-quasiconvex elastic energy functionals whose energy minimizers do not exist. This is demonstrated rigorously in the case of elastostatics for the Saint-Venant Kirchhoff material (in all dimensions), where the existence of variational dual solutions is also proven. The existence of a variational dual solution for the incompressible neo-Hookean material in 2-d is also shown. Stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy are computed using the dual methodology. In particular, we show the stability of a dual elastodynamic equilibrium solution for which there are regions of non-vanishing length with negative elastic stiffness, i.e. non-hyperbolic regions, for which the corresponding primal problem is ill-posed and demonstrates an explosive 'Hadamard instability;' this appears to have implications for the modeling of physically observed softening behavior in macroscopic mechanical response. [ABSTRACT FROM AUTHOR]

Risk-averse multistage problems and their applications are gaining interest in various fields of applications. Under convexity assumptions, the resolution of these problems can be done with trajectory following dynamic programming algorithms like Stochastic Dual Dynamic Programming (SDDP) to access a deterministic lower bound, and dual SDDP for deterministic upper bounds. In this paper, we leverage the dual SDDP algorithm to compute a policy with guaranteed risk-adjusted performance for multistage stochastic linear problems. [ABSTRACT FROM AUTHOR]

AbstractMost existing classification methods aim to minimize the overall misclassification error rate. However, in applications such as loan default prediction, different types of errors can have varying consequences. To address this asymmetry issue, two popular paradigms have been developed: the Neyman-Pearson (NP) paradigm and the cost-sensitive (CS) paradigm. Previous studies on the NP paradigm have primarily focused on the binary case, while the multi-class NP problem poses a greater challenge due to its unknown feasibility. In this work, we tackle the multi-class NP problem by establishing a connection with the CS problem via strong duality and propose two algorithms. We extend the concept of NP oracle inequalities, crucial in binary classifications, to NP oracle properties in the multi-class context. Our algorithms satisfy these NP oracle properties under certain conditions. Furthermore, we develop practical algorithms to assess the feasibility and strong duality in multi-class NP problems, which can offer practitioners the landscape of a multi-class NP problem with various target error levels. Simulations and real data studies validate the effectiveness of our algorithms. To our knowledge, this is the first study to address the multi-class NP problem with theoretical guarantees. The proposed algorithms have been implemented in the R package npcs, which is available on CRAN. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work. [ABSTRACT FROM AUTHOR]

This work focuses on a non-smooth conic semi-infinite programming problem having vanishing constraints. Using the limiting constraint qualification, we establish a necessary optimality condition for the optimization model. Subsequently, the concept of generalized convexity over cones is introduced, followed by the development of sufficient optimality conditions. Wolfe's and Mond-Weir type dual models are also formulated for the considered semi-infinite optimization problem, and weak, strong and converse duality results are established under generalized Q-convexity/ pseudoconvexity/Q-quasiconvexity assumptions. The article incorporates numerical illustrations at appropriate places to validate the results. [ABSTRACT FROM AUTHOR]

We investigate the notion of aWasserstein median as an alternative to the Wasserstein barycenter, which has become popular but may be sensitive to outliers. In terms of robustness to corrupted data, we indeed show that Wasserstein medians have a breakdown point of approximately 1/2. We give explicit constructions of Wasserstein medians in dimension one which enable us to obtain Lp estimates (which do not hold in higher dimensions). We also address dual and multimarginal reformulations. In convex subsets of Rd, we connect Wasserstein medians to a minimal (multi)flow problem à la Beckmann and a system of PDEs of Monge--Kantorovich-type, for which we propose a p-Laplacian approximation. Our analysis eventually leads to a new numerical method to compute Wasserstein medians, which is based on a Douglas--Rachford scheme applied to the minimal flow formulation of the problem. [ABSTRACT FROM AUTHOR]

Rudolf Emil Kalman was a Hungarian-American electrical engineer and mathematician who revolutionized the field of control theory by introducing advanced mathematical techniques for studying control systems. In particular, he popularized the "Time-Domain Approach" for studying Linear Time-Invariant (LTI) systems over the traditional "Frequency-Domain Approach" that was typically used before his work. To contrast these approaches briefly, in the former approach, one works with the LTI system directly in the time-domain, whereas in the latter approach, one works with the transfer function, which relates the input of the LTI system to its output in the frequency domain. This was, in fact, one of his major contributions to control theory, since he showed that many fundamental properties of LTI systems can be deduced directly by taking his alternative approach, which otherwise would have been difficult (or maybe even impossible) to establish using the traditional approach. This expository article is an attempt to highlight only a few of his ingenious contributions to control theory. [ABSTRACT FROM AUTHOR]

This paper deals with a certain class of multiobjective semi-infinite programming problems with switching constraints (in short, MSIPSC) in the framework of Hadamard manifolds. We introduce Abadie constraint qualification (in short, ACQ) for MSIPSC in the Hadamard manifold setting. Necessary criteria of weak Pareto efficiency for MSIPSC are derived by employing ACQ. Further, sufficient criteria of weak Pareto efficiency for MSIPSC are deduced by using geodesic quasiconvexity and pseudoconvexity assumptions. Subsequently, Mond–Weir type and Wolfe type dual models are formulated related to the primal problem MSIPSC, and thereafter, several duality results are established that relate MSIPSC and the corresponding dual models. Several non-trivial examples are furnished in the framework of well-known Hadamard manifolds, such as the set consisting of all symmetric positive definite matrices and the Poincaré half plane, to illustrate the importance of the results derived in this article. To the best of our knowledge, this is the first time that optimality conditions and duality results for MSIPSC have been studied in the setting of Hadamard manifolds. [ABSTRACT FROM AUTHOR]

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The current paper examines how corporate social responsibility (CSR) through three dimensions (social, environmental and governance) could influence technological and non‐technological innovations. The study is conducted on a longitudinal dataset of firms listed on the SBF1201 index between 2005 and 2016. It shows significant associations between CSR performance, particularly social and governance performances, and technological innovations introducing new products. The implementation of new processes is positively and significantly associated with environmental performance. Findings highlight that the three CSR dimensions have significant effects on management innovations. However, they did not influence marketing innovations. Finally, undertaking socially responsible projects does not drive significant changes in R&D strategy, specifically R&D expenditures and the number of hired scientists and experts, but it increases significantly the number of patents. [ABSTRACT FROM AUTHOR]

Model training algorithms which observe a small portion of the training set in each computational step are ubiquitous in practical machine learning, and include both stochastic and online optimization methods. In the vast majority of cases, such algorithms typically observe the training samples via the gradients of the cost functions the samples incur. Thus, these methods exploit are the slope of the cost functions via their first-order approximations. To address limitations of gradient-based methods, such as sensitivity to step-size choice in the stochastic setting, or inability to exploit small function variability in the online setting, several streams of research attempt to exploit more information about the cost functions than just their gradients via the well-known proximal operators. However, implementing such methods in practice poses a challenge, since each iteration step boils down to computing the proximal operator, which may not be as easy as computing a gradient. In this work we devise a novel algorithmic framework, which exploits convex duality theory to achieve both algorithmic efficiency and software modularity of proximal operator implementations, in order to make experimentation with incremental proximal optimization algorithms accessible to a larger audience of researchers and practitioners, by reducing the gap between their theoretical description in research papers and their use in practice. We provide a reference Python implementation for the framework developed in this paper as an open source library at on GitHub (https://github.com/alexshtf/inc%5fprox%5fpt/releases/tag/prox%5fpt%5fpaper) Shtoff (Efficient implementation of incremental proximal point methods arXiv:2205.01457, 2024), along with examples which demonstrate our implementation on a variety of problems, and reproduce the numerical experiments in this paper. The pure Python reference implementation is not necessarily the most efficient, but is a basis for creating efficient implementations by combining Python with a native backend. [ABSTRACT FROM AUTHOR]

In this paper, we completely characterize L p , q -boundedness of (maximal) Fock projections on C for 1 ≤ p , q ≤ ∞ . As applications, we identify the dual space of mixed norm space F α p , q of entire functions and present an alternative proof of the Littlewood–Paley formula for F α p , q . [ABSTRACT FROM AUTHOR]

Coupled Wright–Fisher diffusions have been recently introduced to model the temporal evolution of finitely-many allele frequencies at several loci. These are vectors of multidimensional diffusions whose dynamics are weakly coupled among loci through interaction coefficients, which make the reproductive rates for each allele depend on its frequencies at several loci. Here we consider the problem of filtering a coupled Wright–Fisher diffusion with parent-independent mutation, when this is seen as an unobserved signal in a hidden Markov model. We assume individuals are sampled multinomially at discrete times from the underlying population, whose type configuration at the loci is described by the diffusion states, and adapt recently introduced duality methods to derive the filtering and smoothing distributions. These respectively provide the conditional distribution of the diffusion states given past data, and that conditional on the entire dataset, and are key to be able to perform parameter inference on models of this type. We show that for this model these distributions are countable mixtures of tilted products of Dirichlet kernels, and describe their mixing weights and how these can be updated sequentially. The evaluation of the weights involves the transition probabilities of the dual process, which are not available in closed form. We lay out pseudo codes for the implementation of the algorithms, discuss how to handle the unavailable quantities, and briefly illustrate the procedure with synthetic data. [ABSTRACT FROM AUTHOR]

In this note we explore duality in reverse convex optimization with reverse convex inequality constraints. While we are examining the special case of a finite index set of inequality constraints, we are primarily interested in the general case of an arbitrary infinite index set with neither extra topological structure nor extra measure structure. [ABSTRACT FROM AUTHOR]

We introduce truncated Besov and Triebel–Lizorkin function spaces and study their main properties: embeddings, interpolation, duality, lifting, traces. These new scales allow us to improve several known results in functional analysis and PDE's. In particular, we obtain a full solution to the trace/extension problem in the critical case as well as sharp Sobolev-type embeddings with critical smoothness parameters. [ABSTRACT FROM AUTHOR]

This article sheds new light on the debate between the Discovery and Creation views of entrepreneurial opportunity by drawing on quantum theory. We develop the Quantum view of opportunity, which explains how opportunity is both discovered and created. The Quantum view holds that the ontology and epistemology of opportunity are fundamentally inseparable, which explains why opportunity can never be fully specified. We argue, similar to the Discovery view, that opportunity exists as latent states irrespective of entrepreneurs and that, similar to the Creation view, opportunity is instantiated through entrepreneurial action, which changes opportunity. We use the Quantum view as a thought-provoking metaphor that facilitates the breaking out of the mold of ingrained thinking and moves beyond the Discovery-Creation dichotomy to further our understanding of entrepreneurship. We discuss how the Quantum view relates to established theoretical and empirical research in the entrepreneurship field. Plain English Summary: Rethinking Entrepreneurial Opportunity: A Quantum view! Discover how opportunities are both found and made. #QuantumOpportunity. Entrepreneurial opportunities have long been debated: are they discovered or created? Inspired by quantum mechanics, this article introduces a fresh perspective called the "Quantum view." Instead of seeing opportunities as either discovered or created, the Quantum view suggests that they are both created and discovered. It urges us to rethink conventional wisdom. Instead of just "finding" or "making" opportunities, the Quantum view suggests that opportunities exist in potential states and are realized through entrepreneurial enactment. Entrepreneurs can gather information about opportunities, but they can never fully specify them because every time they interact with an opportunity, they change it. Thus, the Quantum view emphasizes that we can learn about, but never fully specify, opportunities. This helps researchers think differently about opportunities and avoid extreme positions such as that entrepreneurs create something from nothing or that some entrepreneurs can accurately predict the future. [ABSTRACT FROM AUTHOR]

This paper studies the dual relation between risk-sensitive control and large deviation control of maximizing the probability for out-performing a target for Markov Decision Processes. To derive the desired duality, we apply a non-linear extension of the Krein-Rutman Theorem to characterize the optimal risk-sensitive value and prove that an optimal policy exists which is stationary and deterministic. The right-hand side derivative of this value function is used to characterize the specific targets which make the duality to hold. It is proved that the optimal policy for the "out-performing" probability can be approximated by the optimal one for the risk-sensitive control. The range of the (right-hand, left-hand side) derivative of the optimal risk-sensitive value function plays an important role. Some essential differences between these two types of optimal control problems are presented. [ABSTRACT FROM AUTHOR]

Bilevel optimization problems embed the optimality of a subproblem as a constraint of another optimization problem. We introduce the concept of near-optimality robustness for bilevel optimization, protecting the upper-level solution feasibility from limited deviations from the optimal solution at the lower level. General properties and necessary conditions for the existence of solutions are derived for near-optimal robust versions of general bilevel optimization problems. A duality-based solution method is defined when the lower level is convex, leveraging the methodology from the robust and bilevel literature. Numerical results assess the efficiency of exact and heuristic methods and the impact of valid inequalities on the solution time. [ABSTRACT FROM AUTHOR]

The financial challenges facing the Jordanian economy require careful attention and strategic responses. Addressing these challenges may necessitate increased investment. This study explores the relationship between intellectual capital efficiency and firms’ dividend policies and the potential impact of CEO characteristics on this relationship. An analysis was based on data from 90 Jordanian service and manufacturing companies from 2015 to 2019. The study employs the value-added intellectual capital coefficient (VAIC) to measure intellectual capital efficiency and uses the dividend payout ratio to represent dividend policy. The findings indicate a positive relationship between VAIC and dividend policy, suggesting that companies with higher intellectual capital efficiency tend to distribute higher dividends. However, CEO characteristics, such as age, tenure, and educational background, do not significantly affect this relationship. These results imply that strong corporate governance mechanisms are likely in place, ensuring effective decision-making processes and protecting stakeholders’ interests. By focusing on intellectual capital, firms can enhance their operational performance and attractiveness to investors, indirectly supporting economic stability.

This article proposes a novel robust invariance condition for uncertain linear discrete-time systems with state and control constraints, utilizing a method of semidefinite programming duality. The approach involves approximating the robust invariant set for these systems by tackling the dual problem associated with semidefinite programming. Central to this method is the formulation of a dual programming through the application of adjoint mapping. From the standpoint of semidefinite programming dual optimization, the paper presents a novel linear matrix inequality (LMI) conditions pertinent to robust positive invariance. Illustrative examples are incorporated to elucidate the findings. [ABSTRACT FROM AUTHOR]

In this paper, we study a robust optimization problem whose constraints include nonsmooth and nonconvex functions and the intersection of closed sets. Using advanced variational analysis tools, we first provide necessary conditions for the optimality of the robust optimization problem. We then establish sufficient conditions for the optimality of the considered problem under the assumption of generalized convexity. In addition, we present a dual problem to the primal robust optimization problem and examine duality relations. [ABSTRACT FROM AUTHOR]

We present a general convex relaxation approach to study a wide class of Unbalanced Optimal Transport problems for finite non-negative measures with possibly different masses. These are obtained as the lower semicontinuous and convex envelope of a cost for non-negative Dirac masses. New general primal-dual formulations, optimality conditions, and metric-topological properties are carefully studied and discussed. [ABSTRACT FROM AUTHOR]

In the 7th century BCE, the Kushite king Tanwetamani commissioned his "Dream Stela", which was to be erected in the Amun Temple of Jebel Barkal. The lunette of the stela features a dualistic artistic motif whose composition, meaning, and significance are understudied despite their potential to illuminate important aspects of royal Kushite ideology. On the lunette, there are two back-to-back offering scenes that appear at first glance to be nearly symmetrical, but that closer inspection reveals to differ in subtle but significant ways. Analysis of the iconographic and textual features of the motif reveals its rhetorical function in this royal context. The two strikingly similar but meaningfully different offering scenes represented the two halves of a Kushite "Double Kingdom" that considered Kush and Egypt together as a complementary geographic dual, with Tanwetamani presiding as king of both. This "Mirrored Motif" encapsulated the duality present in the Kushite ideology of kingship during the Twenty-Fifth Dynasty, which allowed Tanwetamani to reconcile the present imperial expansion of Kush with the history of Egyptian activity in Nubia. The lunette of the Dream Stela is therefore political art that serves to advance the Kushite imperial agenda. [ABSTRACT FROM AUTHOR]

Göring–Helmberg–Wappler introduced optimization problems regarding embeddings of a graph into a Euclidean space and the first nonzero eigenvalue of the Laplacian of a graph, which are dual to each other in the framework of semidefinite programming. In this paper, we introduce a new graph-embedding optimization problem, and discuss its relation to Göring–Helmberg–Wappler's problems. We also identify the dual problem to our embedding optimization problem. We solve the optimization problems for distance-regular graphs and the one-skeleton graphs of the C 60 fullerene and some other Archimedian solids. [ABSTRACT FROM AUTHOR]

A projective variety whose Gauss map has positive dimensional fibres corresponds to a special kind of scroll called Gaussian. A Gaussian scroll is a member of a canonical derived Gaussian flag. We introduce a duality in the class of Gaussian scrolls and flags and study its consequences. In particular, a Gaussian scroll is dual to the derived or tangent developable scroll of a Gaussian scroll in the dual projective space, and is the 'leading edge' or antiderived scroll of its derived stationary scroll. [ABSTRACT FROM AUTHOR]

We introduce a framework based on Rockafellar's perturbation theory to analyze and solve general nonsmooth convex minimization and monotone inclusion problems involving nonlinearly composed functions as well as linear compositions. Such problems have been investigated only from a primal perspective and only for nonlinear compositions of smooth functions in finite-dimensional spaces in the absence of linear compositions. In the context of Banach spaces, the proposed perturbation analysis serves as a foundation for the construction of a dual problem and of a maximally monotone Kuhn–Tucker operator, which is decomposable as the sum of simpler monotone operators. In the Hilbertian setting, this decomposition leads to a block-iterative primal-dual algorithm that fully splits all the components of the problem and appears to be the first proximal splitting algorithm for handling nonlinear composite problems. Various applications are discussed. Funding: The work of L. M. Briceño-Arias was supported by Agencia Nacional de Investigación y Desarrollo-Chile [Grant Fondo Nacional de Desarrollo Científico y Tecnológico 1190871, Grant Centro de Modelamiento Matemático ACE210010, Grant Centro de Modelamiento Matemático FB210005, and basal Funds for Centers of Excellence], and the work of P. L. Combettes was supported by the National Science Foundation [Grant DMS-1818946]. [ABSTRACT FROM AUTHOR]

This paper explores Percival Everett's James through themes of identity, duality, and self-perception. Focusing on James, a character torn between his racial and cultural heritage and personal aspirations, it analyses how Everett addresses the tension between societal expectations and individual autonomy. Using unreliable narration and shifting perspectives, the novel emphasizes the fluidity of identity and the challenge of self-coherence in a world shaped by stereotypes. The study also examines the novel's fragmented structure, reflecting the complexities of James's identity, while contributing to discussions on race, memory, and the human experience in contemporary literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a set-valued mathematical programming problem with parameters (PSVP), where set-valued functions are used for the constraints and objective functions. Set valued ρ-locally connected maps over cone (ρ-k-LC) are introduced and using these maps and contingent epiderivatives, the Karush-Kuhn-Tucker criteria of sufficiency is investigated for the above-mentioned problem. Lastly, using the same presumptions, the Mond-Weir dual is developed and duality theorems are demonstrated. Primary 26B25; Secondary 49N15. [ABSTRACT FROM AUTHOR]

This paper is concerned with a parabolic evolution equation of the form A(ut)+B(u)=f$A(u_t) + B(u) = f$, settled in a smooth bounded domain of Rd$\mathbb {R}^d$, d≥1$d\ge 1$, and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, −B$-B$ stands for a diffusion operator, possibly nonlinear, which may range in a very wide class, including the Laplacian, the m$m$‐Laplacian for suitable m∈(1,∞)$m\in (1,\infty)$, the "variable‐exponent" m(x)$m(x)$‐Laplacian, or even some fractional order operators. The operator A$A$ is assumed to be in the form [A(v)](x,t)=α(x,v(x,t))$[A(v)](x,t)=\alpha (x,v(x,t))$ with α$\alpha$ being measurable in x$x$ and maximal monotone in v$v$. The main results are devoted to proving existence of weak solutions for a wide class of functions α$\alpha$ that extends the setting considered in previous results related to the variable exponent case where α(x,v)=|v(x)|p(x)−2v(x)$\alpha (x,v)=|v(x)|^{p(x)-2}v(x)$. To this end, a theory of subdifferential operators will be established in Musielak–Orlicz spaces satisfying structure conditions of the so‐called Δ2$\Delta _2$‐type, and a framework for approximating maximal monotone operators acting in that class of spaces will also be developed. Such a theory is then applied to provide an existence result for a specific equation, but it may have an independent interest in itself. Finally, the existence result is illustrated by presenting a number of specific equations (and, correspondingly, of operators A$A$, B$B$) to which the result can be applied. [ABSTRACT FROM AUTHOR]

Bernal et al. provided a necessary and sufficient condition for a linear code to be realized as an ideal in a finite group algebra and De La Cruz and Willems proved a similar result for ideals in twisted group algebras. In this paper, we extend this characterization to crossed products. Furthermore, we determine conditions for some crossed product codes to be self-dual. [ABSTRACT FROM AUTHOR]

This manuscript demonstrates robust optimality conditions, Wolfe and Mond–Weir type robust dual models for a robust mathematical programming problem involving vanishing constraints (RMPVC). Further, the theorems of duality are examined based on the concept of generalized higher order invexity and strict invexity that establish relations between the primal and the Wolfe type robust dual problems. In addition, the duality results for a Mond–Weir type robust dual problem based on the concept of generalized higher order pseudoinvex, strict pseudoinvex and quasiinvex functions are also studied. Furthermore, numerical examples are provided to validate robust optimality conditions and duality theorems of Wolfe and Mond–Weir type dual problems. [ABSTRACT FROM AUTHOR]

The purpose of article's study is to discover Logos phenomenon which is present in the nature of thinking using an empirical method in the value system of F. Mathews & G. Berkeley from the point of view of modern ethics, historical experience, modern psychology and culture of ecology. "Logos" is considered as a segment of the field of thinking, a certain event in the field of meaning. The meaning of "Logos" as an event in the ontology of F. Mathews is considered from a metaphysical point of view and acquires a special semantic value in the context of G. Berkeley philosophy. In the space-time continuum, "Logos" exists in a moving field of meanings and is opposed to "Nonthought." In the course of history, the manifestation of "Logos" was absorbed by everyday life and mediocrity, but F. Mathews and G. Berkeley are trying to prevent the disappearance of "Logos" by creating an "Ecological Model of Thinking", revealing perception from internal semantic structures towards natural phenomena. An ecological model of thinking can prevent the formation of a "perceptual dead zone" where human genius is devalued by self-centered mediocrity. The phenomenon of the integrity of the "Logos" is an ethical expression of intuitive perception. The causes and consequences of "ontological inertia" are also revealed. [ABSTRACT FROM AUTHOR]

In this paper, we establish optimality conditions and sensitivity analysis of set-valued optimization problems in terms of higher-order radial derivatives. First, we obtain the optimality conditions with separated derivatives for a set-valued optimization problem, here separated derivatives means the derivatives of objective and constraint functions are different. Then, some duality theorems for a mixed type of primal-dual set-valued optimization problem are gained. Finally, several results concerning higher-order sensitivity analysis are presented. The main results of this paper are illustrated by some concrete examples. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a class of multiobjective semi-infinite programming problems on Hadamard manifolds (in short, (MOSIP-HM)). We derive some alternative theorems analogous to Tucker's theorem, Tucker's first and second existence theorem, and Motzkin's theorem of alternative in the framework of Hadamard manifolds. We employ Motzkin's theorem of alternative to establish necessary and sufficient conditions that characterize KKT pseudoconvex functions using strong KKT vector critical points and efficient solutions of (MOSIP-HM). Moreover, we formulate the Mond-Weir and Wolfe-type dual problems related to (MOSIP-HM) and derive the weak and converse duality theorems relating (MOSIP-HM) and the dual problems. Several non-trivial numerical examples are provided to illustrate the significance of the derived results. The results deduced in the paper extend and generalize several notable works existing in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we first consider the existence conditions, the construction and the enumeration of Hermitian self-dual θ -cyclic and θ -negacyclic codes over I F p 2 , where p is a prime number and θ is the Frobenius automorphism over I F p 2 . We then give necessary and sufficient conditions for the existence of Hermitian self-dual θ -cyclic and θ -negacyclic codes over I F p e where e is an even integer greater than 2. [ABSTRACT FROM AUTHOR]

In this work, optimality conditions and classical results from duality theory are derived for continuous-time linear optimization problems with inequality constraints. The optimality conditions are given in the Karush–Kuhn–Tucker form. Weak and strong duality properties as well as the complementary slackness theorem are established. A result concerning the existence of solutions is also stated. [ABSTRACT FROM AUTHOR]

Haagerup's proof of the non commutative little Grothendieck inequality raises some questions on the commutative little inequality, and it offers a new result on scalar matrices with non negative entries. The theory of completely bounded maps may be used to show that the commutative Grothendieck inequality follows from the little commutative inequality, and that this passage may be given a geometric form as a relation between a pair of compact convex sets of positive matrices, which, in turn, characterizes the little constant k G C . [ABSTRACT FROM AUTHOR]

A graph parameter is self-dual in some class of graphs embeddable in some surface if its value does not change in the dual graph by more than a constant factor. We prove that, in the class of connected hypergraphs without bridges and loops that are embeddable in some surface of Euler genus at most g , branchwidth is a (1 , g) -self-dual parameter, i.e., for every hypergraph G in the class, the branchwidth of its dual is at most the branchwidth of G plus g. This is the first proof that branchwidth is an additively self-dual width parameter. [ABSTRACT FROM AUTHOR]

This paper demonstrates that robust control based on only a priori information about the object's uncertainty can be significantly improved through the additional use of experimental data. Generalized H∞-optimal controllers are designed for an unknown linear time-varying system on a finite horizon. These controllers optimize the damping level of exogenous and/or initial disturbances as well as the maximum deviation of the terminal state of the system. The design method does not require the persistent excitation condition or the rank condition, which ensure the identifiability of the system. As a result, the amount of experimental data can be significantly reduced. [ABSTRACT FROM AUTHOR]

We present a generalized Farkas' Lemma for an inequality system involving distributions. This lemma establishes an equivalence between an infinite-dimensional system of moment inequalities and a semi-definite system, assuming that the support for the distributions is a spectrahedron. To the best of our knowledge, it is the first extension of Farkas' Lemma to the distributional paradigm. Applying the new Lemma, we then establish no-gap duality results between a class of moment optimization problems and numerically tractable semi-definite programs. [ABSTRACT FROM AUTHOR]

Given a probability space (Ω , F , P) and a rearrangement-invariant quasi-Banach function space X, the authors of this article first prove the α -atomic ( α ∈ [ 1 , ∞) ) characterization of weak martingale Hardy spaces W H X (Ω) associated with X via simple atoms. The authors then introduce the generalized weak martingale BMO spaces which proves to be the dual spaces of W H X (Ω) . Consequently, the authors derive a new John–Nirenberg theorem for these weak martingale BMO spaces. Finally, the authors apply these results to the generalized grand Lebesgue space and the weighted Lorentz space. Even in these special cases, the results obtained in this article are totally new. [ABSTRACT FROM AUTHOR]

We consider a sequence of block-separable convex programming problems describing the resource allocation in multiagent systems. We construct several iterative algorithms for setting the resource prices. Under various assumptions about the utility functions and resource constraints, we obtain estimates for the average deviation (regret) of the objective function from the optimal value and the constraint residuals. Here the average is understood as the expectation for independent identically distributed data and as the time average in the online optimization problem. The analysis of the problem is carried out by online optimization methods and duality theory. The algorithms considered use the information concerning the difference between the total demand and supply that is generated by agents' reactions to prices and corresponds to the constraint residuals. [ABSTRACT FROM AUTHOR]