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We prove that solutions to several shape optimization problems in the plane, with a convexity constraint on the admissible domains, are polygons. The main terms of the shape functionals we consider are either the Dirichlet energy E f ⁢ (Ω) of the Laplacian in the domain Ω or the first eigenvalue λ 1 ⁢ (Ω) of the Dirichlet-Laplacian. Usually, one considers minimization of such functionals (often with measure constraint), as for example for the famous Saint-Venant and Faber-Krahn inequalities. By adding the convexity constraint (and possibly other natural constraints), we instead consider the rather unusual and difficult question of maximizing these functionals. This paper follows a series of papers by the authors, where the leading idea is that a certain concavity property of the shape functional that is minimized leads optimal shapes to locally saturate their convexity constraint, which geometrically means that they are polygonal. In these previous papers, the leading term in the shape functional was usually the opposite of the perimeter, for which the aforementioned concavity property was rather easy to obtain through computations of its second order shape derivative. By carrying classical shape calculus, a similar concavity property can be observed for the opposite of E f ⁢ (Ω) or λ 1 ⁢ (Ω) when shapes are smooth and convex. The main novelty in the present paper is the proof of a weak convexity property of E f ⁢ (Ω) and λ 1 ⁢ (Ω) among planar convex shapes, namely rather nonsmooth shapes. This involves new computations and estimates of the second order shape derivatives of E f ⁢ (Ω) and λ 1 ⁢ (Ω) interesting for themselves. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the following degenerate elliptic Monge-Ampère equation: ... Under suitable structure conditions on f(t), we can show that and the solutions of linearized equation have the same symmetric property as the domain. Moreover, we also achieve some uniqueness results for homogenous nonlinearity f(t) = tp (p > n) in general convex domain. Compared to the results in [G. Huang, Calc. Var. Partial Differential Equations, 58 (2019), 73], we dispose the "uniformly convex" condition imposed on the domain. With a subtle approximation procedure and variant form of Hopf's lemma, we overcome the technical difficulties caused by the degeneracy of the above equation. What's more, we can also get the uniqueness property for domains whose geometry is close enough to the symmetric convex domain. [ABSTRACT FROM AUTHOR]

In this paper, we study the numerical method for the bi-Laplace problems with inhomogeneous coefficients; particularly, we propose finite element schemes on rectangular grids respectively for an inhomogeneous fourth-order elliptic singular perturbation problem and for the Helmholtz transmission eigenvalue problem. The new methods use the reduced rectangle Morley (RRM for short) element space with piecewise quadratic polynomials, which are of the lowest degree possible. For the finite element space, a discrete analogue of an equality by Grisvard is proved for the stability issue and a locally-averaged interpolation operator is constructed for the approximation issue. Optimal convergence rates of the schemes are proved, and numerical experiments are given to verify the theoretical analysis. • A discrete analogue of an equality (1.3) by Grisvard [1] on H 2 functions is proved for the reduced rectangular Morley (RRM for short in the sequel) element functions. This discrete equality makes the RRM space usable for bi-Laplacian problems with inhomogeneous coefficients. • Based on piecewise quadratic polynomials, the RRM scheme is the lowest-degree finite element scheme for the inhomogeneous bi-Laplace problems. Compared to other kinds of methods, it does not need tuning parameter or using indirect differential operators. • As revealed by [3] , the RRM element space does not admit a locally-defined projective interpolator. In this paper, however, a locally-defined stable interpolator (not projective) is carefully constructed for the RRM element space, and an optimal approximation is proved rigorously on both convex and nonconvex domains. [ABSTRACT FROM AUTHOR]

A convex fair partition of a convex polygonal region is defined as a partition on which all regions are convex and have equal area and equal perimeter. The existence of such a partition for any number of regions remains an open question. In this paper, we address this issue by developing an algorithm to find such a convex fair partition without restrictions on the number of regions. Our approach utilizes the normal flow algorithm (a generalization of Newton's method) to find a zero for the excess areas and perimeters of the convex hulls of the regions. The initial partition is generated by applying Lloyd's algorithm to a randomly selected set of points within the polygon, after appropriate scaling. We performed extensive experimentation, and our algorithm can find a convex fair partition for 100% of the tested problem. Our findings support the conjecture that a convex fair partition always exists. [ABSTRACT FROM AUTHOR]

F-concavity is a generalization of power concavity and, actually, the largest available generalization of the notion of concavity. We characterize the F-concavities preserved by the Dirichlet heat flow in convex domains on {\mathbb R}^n, and complete the study of preservation of concavity properties by the Dirichlet heat flow, started by Brascamp and Lieb in 1976 and developed in some recent papers. More precisely, we discover hot-concavity, which is the strongest F-concavity preserved by the Dirichlet heat flow; we show that log-concavity is the weakest F-concavity preserved by the Dirichlet heat flow; quasi-concavity is also preserved only for n=1; we prove that if F-concavity is strictly weaker than log-concavity and n\ge 2, then there exists an F-concave initial datum such that the corresponding solution to the Dirichlet heat flow is not even quasi-concave, hence losing any reminiscence of concavity. Furthermore, we find a sufficient and necessary condition for F-concavity to be preserved by the Dirichlet heat flow. We also study the preservation of concavity properties by solutions of the Cauchy–Dirichlet problem for linear parabolic equations with variable coefficients and for nonlinear parabolic equations such as semilinear heat equations, the porous medium equation, and the parabolic p-Laplace equation. [ABSTRACT FROM AUTHOR]

This paper delves into the numerical analysis of the extended Fisher-Kolmogorov (EFK) equation within open bounded convex domains R ⊂ R d , where d ≤ 3. Two distinct finite element schemes are introduced, namely the semi-discrete and fully-discrete finite element approximations. The existence and uniqueness of solutions are established for both the semi-discrete and fully-discrete finite element approximations. Error bounds are investigated across different scenarios, including comparisons between the semi-discrete and exact solutions, as well as between the semi-discrete and fully-discrete solutions, along with the fully-discrete and exact solutions. An effective algorithm has been proposed to solve the nonlinear system resulting from the fully-discrete finite element approximation at each time step. The paper also provides computed numerical error results and showcases a variety of numerical experiments to further illustrate and support the findings. [ABSTRACT FROM AUTHOR]

This paper investigates the space-time residual-based a posteriori error bounds of the mixed finite element method for the optimal control problem governed by the parabolic equation in a bounded convex domain. For the spatial discretization of the state and co-state variables, the lowest-order Raviart-Thomas spaces are utilized, although for the control variable, variational discretization technique is used. The backward-Euler implicit method is applied for temporal discretization. To provide a posteriori error estimates for the state and control variables in the L ∞ (L 2) -norm, an elliptic reconstruction approach paired with an energy strategy is utilized. The reliability and efficiency of the a posteriori error estimators are discussed. The effectiveness of the estimators is finally confirmed through the numerical tests. [ABSTRACT FROM AUTHOR]

In the presented paper, we investigate efficient solutions to optimization problems with multiple criteria and bounded trade-offs. A nonlinear optimization problem to find the relationships between the upper bound for trade-offs and objective functions is presented. Due to this problem, we determine some properly efficient points that are closer to the ideal point. To this end, we apply the extended form of the generalized Tchebycheff norm. Note that all the presented results work for general problems and no convexity assumption is needed. [ABSTRACT FROM AUTHOR]

In this paper we prove that among all convex domains of the plane with two axes of symmetry, the maximizer of the first non-trivial Neumann eigenvalue \mu _1 with perimeter constraint is achieved by the square and the equilateral triangle. Part of the result follows from a new general bound on \mu _1 involving the minimal width over the area. Our main result partially answers to a question addressed in 2009 by R. S. Laugesen, I. Polterovich, and B. A. Siudeja. [ABSTRACT FROM AUTHOR]

In this paper, over an arbitrary number field, we prove subconvexity bounds for selfdual GL3 L-functions in the t-aspect and for self-dual GL3 x GL2 L-functions in the GL2 Archimedean aspect. [ABSTRACT FROM AUTHOR]

Multi-term fractional diffusion equations can be regarded as a generalisation of fractional diffusion equations. In this paper, we develop an efficient meshless method for solving the multi-term time-space fractional diffusion equation. First, we use the Laplace transform method to deal with the multi-term time fractional operator, we transform the time into complex frequency domain by Laplace transform. The properties of the Laplace transform with respect to fractional-order operators are exploited to deal with multi-term time fractional-order operators, overcoming the dependence of fractional-order operators with respect to time and giving better results. Second, we proposed a meshless method to deal with space fractional operators on convex region based on quintic Hermite spline functions based on the theory of polynomial functions dense theorem. Meanwhile, the approximate solution of the equation is obtained through theory of the minimum residual approximate solution, and the error analysis are provided. Third, we obtain the numerical solution of the diffusion equation by inverse Laplace transform. Finally, we first experimented with a single space-time fractional-order diffusion equation to verify the validity of our method, and then experimented with a multi-term time equation with different parameters and regions and compared it with the previous method to illustrate the accuracy of our method. [ABSTRACT FROM AUTHOR]

Motivated by problems where jumps across lower dimensional subsets and sharp transitions across interfaces are of interest, this paper studies the properties of fractional bounded variation (BV)-type spaces. Two different natural fractional analogs of classical BV are considered: BVa, a space induced from the Riesz-fractional gradient that has been recently studied by Comi-Stefani; and bva, induced by the Gagliardo-type fractional gradient often used in Dirichlet forms and Peridynamics--this one is naturally related to the Caffarelli-Roquejoffre-Savin fractional perimeter. Our main theoretical result is that the latter bva actually corresponds to the Gagliardo-Slobodeckij space Wα>1. As an application, using the properties of these spaces, novel image denoising models are introduced and their corresponding Fenchel predual formulations are derived. The latter requires density of smooth functions with compact support. We establish this density property for convex domains. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fully parabolic nu'trient taxis system: ut = d1Δu − ∇ · (ϕ(u, v)∇v), vt = d2Δv − ξug(v) − μv + r(x, t), x ∈ Ω, t > 0 under homogeneous Neumann boundary conditions in a convex bounded domain with smooth boundary. We show that the system possesses a global bounded classical solution in domains of arbitrary dimension and at least one global generalized solution in high-dimensional domain. In addition, the asymptotic behavior of generalized solutions is discussed. Our results not only generalize and partly improve upon previously known findings but also introduce new insights. [ABSTRACT FROM AUTHOR]

In this paper, we give a local estimate for the Kobayashi distance on a bounded convex domain of finite type, which relates to a local pseudodistance near the boundary. The estimate is precise up to a bounded additive term. Also, we conclude that the domain equipped with the Kobayashi distance is Gromov hyperbolic that gives another proof of the result of Zimmer. [ABSTRACT FROM AUTHOR]

In this paper, we prove two separate lower bounds — one for nondegenerate convex domains and the other for nondegenerate ℂ -convex (but not necessarily convex) domains — for the squeezing function that hold true for all domains in ℂ n , for a fixed n ≥ 2 , of the stated class. We provide explicit expressions in terms of n for these estimates. [ABSTRACT FROM AUTHOR]

This paper deals with the following parabolic-parabolic-parabolic chemotaxis system with singular sensitivity and Lotka-Volterra competition kinetics (0.1) { u t = Δ u − χ 1 ∇ ⋅ (u w ∇ w) + μ 1 u (1 − u − a 1 v) , t > 0 , x ∈ Ω , v t = Δ v − χ 2 ∇ ⋅ (v w ∇ w) + μ 2 v (1 − v − a 2 u) , t > 0 , x ∈ Ω , w t = Δ w − w + u + v , t > 0 , x ∈ Ω , ∂ u ∂ ν = ∂ v ∂ ν = ∂ w ∂ ν = 0 , t > 0 , x ∈ ∂ Ω , u (0 , x) = u 0 (x) , v (0 , x) = v 0 (x) , w (0 , x) = w 0 (x) , x ∈ Ω , where Ω ⊂ R N (N ≥ 2) is a bounded smooth convex domain, and the parameters χ 1 , χ 2 , μ 1 , μ 2 , a 1 and a 2 are positive constants. It is shown that the system possesses globally bounded classical solutions under the following conditions χ 1 , χ 2 ∈ (0 , 1 2) for N = 2 , 3 or χ 1 , χ 2 ∈ (0 , N − 2 N − 1) for N ≥ 4. Moreover, if min ⁡ { μ 1 , μ 2 } > max ⁡ { χ 1 , χ 2 } 2 4 , we obtain the uniformly lower bound for w. Finally, when χ 1 , χ 2 are suitably small, it is proved that if 0 < a 1 , a 2 < 1 , then the solution (u , v , w) converges to (1 − a 1 1 − a 1 a 2 , 1 − a 2 1 − a 1 a 2 , 2 − a 1 − a 2 1 − a 1 a 2 ) in L ∞ norm as t → ∞ ; if 0 < a 2 < 1 ≤ a 1 , then the solution (u , v , w) converges to (0 , 1 , 1) in L ∞ norm as t → ∞. [ABSTRACT FROM AUTHOR]

This paper presents the concept of region stability and provides criteria for region stability of linear time delay systems, which can reveal the dynamic and steady-state performance of the systems more precisely. Corresponding design schemes for stabilization and tracking control that can accurately control various performance of time delay systems have also been explored. First, in the light of the connection between the poles and the dynamic properties of the system, the concept of region stability is given to describe the finer dynamic behavior of time delay systems. The criteria for the region stability are also presented. Second, the region stabilization methods are investigated, which can ensure that the system satisfies a certain dynamic performance by setting the eigenvalues in a certain convex region. Third, a precise tracking control of the linear time delay systems is addressed as an application of region stabilization. It can control the steady state performance and transient response of the tracking signal more precisely. Finally, three instances are provided to display the superiority of the new method for the performance indexes of the linear time delay systems. • The region stability criteria are proposed that reflect more performance of systems. • Precise controllers according to different requirements are provided. • Convergent rate of the linear time delay systems can be controlled effectively. • A precise tracking control approach is addressed as an application. [ABSTRACT FROM AUTHOR]

A walk u0, u1, . . ., uk−1, uk in a graph G is a weakly toll walk if u0ui ∈ E(G) implies ui = u1 and ujuk ∈ E(G) implies uj = uk−1. A set S of vertices of G is weakly toll convex if for any two nonadjacent vertices x, y ∈ S, any vertex in a weakly toll walk between x and y is also in S. The weakly toll convexity is the graph convexity defined over weakly toll convex sets. If S and S \ {x} are convex sets, then x is an extreme vertex of S. A graph convexity is said to be a convex geometry if it satisfies the Minkowski-Krein-Milman property, which states that every convex set is the convex hull of its extreme vertices. It is known that chordal, Ptolemaic, weakly polarizable, and interval graphs can be characterized as convex geometries with respect to the monophonic, geodesic, m3 , and toll convexities, respectively. Inspired by previous results in (Alcón, Brešar, Gologranc, Gutierrez, Šumenjak, Peterin, and Tepeh, 2015), in this paper we prove that a graph is a convex geometry with respect to the weakly toll convexity if and only if it is a proper interval graph. Furthermore, some well-known graph invariants are studied with respect to the weakly toll convexity, namely the weakly toll number and the weakly toll hull number. In particular, we determine these invariants for trees and we find bounds for interval graphs. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p -Laplace operator, namely: λ 1 (β , Ω) = min ψ ∈ W 1 , p (Ω) ∖ { 0 } ⁡ ∫ Ω F (∇ ψ) p d x + β ∫ ∂ Ω | ψ | p F (ν Ω) d H N − 1 ∫ Ω | ψ | p d x , where p ∈ ] 1 , + ∞ [ , Ω is a bounded, anisotropic mean convex domain in R N , ν Ω is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on R N. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on geometrical quantities associated to Ω. More precisely, we prove a lower bound of λ 1 in the case β > 0 , and a upper bound in the case β < 0. As a consequence, we prove, for β > 0 , a lower bound for λ 1 (β , Ω) in terms of the anisotropic inradius of Ω and, for β < 0 , an upper bound of λ 1 (β , Ω) in terms of β. [ABSTRACT FROM AUTHOR]

In this paper, we apply the so-called Alexandrov–Bakelman–Pucci (ABP) method to establish some geometric inequalities. We first prove a logarithmic Sobolev inequality for closed |$n$| -dimensional minimal submanifolds |$\Sigma $| of |$\mathbb S^{n+m}$|⁠. As a consequence, it recovers the classical result that |$|\mathbb S^{n}| \leq |\Sigma |$| for |$m = 1,2$|⁠. Next, we prove a Sobolev-type inequality for positive symmetric two-tensors on smooth domains in |$\mathbb R^{n}$|⁠ , which was established by D. Serre when the domain is convex. In the last application of the ABP method, we formulate and prove an inequality related to quermassintegrals of closed hypersurfaces of the Euclidean space. [ABSTRACT FROM AUTHOR]

In this paper, we can obtain curvature estimates for spacelike admissible graphic hypersurfaces in the (n+1)$(n+1)$‐dimensional Lorentz–Minkowski space R1n+1$\mathbb {R}^{n+1}_{1}$, and through which the existence of spacelike admissible graphic hypersurfaces, with prescribed 2‐nd Weingarten curvature and Dirichlet boundary data, defined over a strictly convex domain in the hyperbolic plane Hn(1)⊂R1n+1$\mathcal {H}^{n}(1)\subset \mathbb {R}^{n+1}_{1}$ of center at origin and radius 1, can be proven. [ABSTRACT FROM AUTHOR]

In this paper, Schwarz–Pick estimates are obtained for harmonic mappings from the unit ball of ℝn into origin-symmetric bounded convex domains Ω ⊂ ℝm, which extends the recent results of Liu [7], Chen and Hamada [2]. This approach is new and functional, which is based on some geometric properties of convex domains; and the domains discussed are rather general. Notice that when Ω is the unit p-ball in ℝm, our main theorem reduces to that of Chen and Hamada [2]. [ABSTRACT FROM AUTHOR]

The exponential stabilization of stochastic neural networks in mean square sense with saturated impulsive input is investigated in this paper. Firstly, the saturated term is handled by polyhedral representation method. When the impulsive sequence is determined by average impulsive interval, impulsive density and mode-dependent impulsive density, the sufficient conditions for stability are proposed, respectively. Then, the ellipsoid and the polyhedron are used to estimate the attractive domain, respectively. By transforming the estimation of the attractive domain into a convex optimization problem, a relatively optimum domain of attraction is obtained. Finally, a three-dimensional continuous time Hopfield neural network example is provided to illustrate the effectiveness and rationality of our proposed theoretical results. • The impulsive control sequence is defined by (mode-dependent) impulsive density. • Estimating attractive domain's size is converted into solving optimization problems. • Introduce a convex set to handle mode-dependent saturated impulsive input strategies. [ABSTRACT FROM AUTHOR]

We consider finite Morse index solutions to semilinear elliptic questions, and we investigate their smoothness. It is well‐known that: ‐For n=2$n=2$, there exist Morse index 1 solutions whose L∞$L^\infty$ norm goes to infinity.‐For n≥3$n \ge 3$, uniform boundedness holds in the subcritical case for power‐type nonlinearities, while for critical nonlinearities the boundedness of the Morse index does not prevent blow‐up in L∞$L^\infty$. In this paper, we investigate the case of general supercritical nonlinearities inside convex domains, and we prove an interior a priori L∞$L^\infty$ bound for finite Morse index solution in the sharp dimensional range 3≤n≤9$3\le n\le 9$. As a corollary, we obtain uniform bounds for finite Morse index solutions to the Gelfand problem constructed via the continuity method. [ABSTRACT FROM AUTHOR]

This paper investigates boundary blow-up problem of the k -Hessian equation σ k (μ (D 2 u)) = b (z) h (u) in a uniformly (k − 1) -convex domain D ⊂ R N , where b (z) grows like δ − α (z) as the distance function δ (z) → 0 and h (u) grows like u k (ln u) β as u → ∞. Boundary estimate of large solution u is obtained by the method of sub- and super- solutions. Compared to previous research (Zhang et al., 2022), this paper provides a more accurate estimate of the large solution when 0 < α < 1 , β > 2 k + 1. [ABSTRACT FROM AUTHOR]

In [23] , the first author and the third author introduced and studied the horospherical p -Minkowski problem for smooth horospherically convex domains in hyperbolic space. In this paper, we introduce and solve the discrete horospherical p -Minkowski problem in hyperbolic space for all p ∈ (− ∞ , + ∞) when the given measure is even on the unit sphere. [ABSTRACT FROM AUTHOR]

This paper presents a multiplayer Homicidal Chauffeur reach-avoid differential game, which involves Dubins-car pursuers and simple-motion evaders. The goal of the pursuers is to cooperatively protect a planar convex region from the evaders, who strive to reach the region. We propose a cooperative strategy for the pursuers based on subgames for multiple pursuers against one evader and optimal task allocation. We introduce pursuit enclosure functions (PEFs) and propose a new enclosure region pursuit (ERP) winning approach that supports the forward analysis for the strategy synthesis in the subgames. We show that if a pursuit coalition is able to defend the region against an evader under the ERP winning, then no more than two pursuers in the coalition are necessarily needed. We also propose a steer-to-ERP approach to certify the ERP winning and synthesize the ERP winning strategy. To implement the strategy, we introduce a positional PEF and provide the necessary parameters, states, and strategies that ensure the ERP winning for both one pursuer and two pursuers against one evader. Additionally, we formulate a binary integer program using the subgame outcomes to maximize the captured evaders in the ERP winning for the pursuit task allocation. Finally, we propose a multiplayer receding-horizon strategy where the ERP winnings are checked in each horizon, the task is allocated, and the strategies of the pursuers are determined. Numerical examples are provided to illustrate the results. [ABSTRACT FROM AUTHOR]

In a foundry, optimizing the charge mix is critical to achieving consistent quality, cost-efficiency, and desired qualities in the final metal or alloy product. This paper describes a data analytics-driven strategy for optimizing the charge mix by lowering the cost of the scrap used to prepare the molten metal while maintaining the required chemical composition, tensile strength, and hardness required by the foundry for manufacturing gray cast iron products (Grade FG 220). The linear programming approach is used for this purpose where all the constraints are strictly met. Three categories of constraints are used for this purpose, i.e., composition constraint, foundry constraint, and material grade constraint. In the linear programming approach, the feasible region is considered as an ellipsoidal region and the developed convex optimization problem is iteratively solved. The result showed potential cost savings could be obtained, accompanied by the needed alloy chemical composition and quality. [ABSTRACT FROM AUTHOR]

Stochastic (sub)gradient methods require step size schedule tuning to perform well in practice. Classical tuning strategies decay the step size polynomially and lead to optimal sublinear rates on (strongly) convex problems. An alternative schedule, popular in nonconvex optimization, is called geometric step decay and proceeds by halving the step size after every few epochs. In recent work, geometric step decay was shown to improve exponentially upon classical sublinear rates for the class of sharp convex functions. In this work, we ask whether geometric step decay similarly improves stochastic algorithms for the class of sharp weakly convex problems. Such losses feature in modern statistical recovery problems and lead to a new challenge not present in the convex setting: the region of convergence is local, so one must bound the probability of escape. Our main result shows that for a large class of stochastic, sharp, nonsmooth, and nonconvex problems a geometric step decay schedule endows well-known algorithms with a local linear (or nearly linear) rate of convergence to global minimizers. This guarantee applies to the stochastic projected subgradient, proximal point, and prox-linear algorithms. As an application of our main result, we analyze two statistical recovery tasks—phase retrieval and blind deconvolution—and match the best known guarantees under Gaussian measurement models and establish new guarantees under heavy-tailed distributions. [ABSTRACT FROM AUTHOR]

We prove that there exists a bounded convex domain \Omega \subset \mathbb {R}^3 of fixed volume that minimizes the first positive curl eigenvalue among all other bounded convex domains of the same volume. We show that this optimal domain cannot be analytic, and that it cannot be stably convex if it is sufficiently smooth (e.g., of class C^{1,1}). Existence results for uniformly Hölder optimal domains in a box (that is, contained in a fixed bounded domain D\subset \mathbb {R}^3) are also presented. [ABSTRACT FROM AUTHOR]

• Novel OPF model based on power injection-based equations for bipolar DC distribution network. • For the first time, SOCP is employed to relax the non-convex terms in the power injection-based equations of bipolar DC distribution networks. • An algorithm devised for adaptively updating initial points and boundaries in the solution process. • The relaxation error can be reduced by nearly 1000 times through the improved sequence bound tightening algorithm. Optimal power flow (OPF) is a fundamental tool for the operation analysis of bipolar DC distribution network (DCDN). However, existing OPF models based on current injection-based equations face challenges in reflecting the power distribution and exchange of bipolar DCDN directly since its decision variables are voltage and current. This paper addresses this issue by establishing a convex OPF model that can be used for the planning and operation of bipolar DCDN. As a novel OPF model, the power flow characteristics of bipolar DCDN are revealed through power injection rather than current injection, ensuring its applicability and convenience in power optimization problems. Furthermore, the original OPF model based on power injection-based equations is introduced, and for the first time, second-order cone programming (SOCP) is utilized to relax the non-convex terms within it. Notely, McCormick envelopes are used to restrict the feasible region of the convex model, thereby reducing the influence of SOCP relaxation. To enhance the tightness of the feasible region of the convex model, the refined sequence bound tightening algorithm (SBTA) is employed to adjust the boundaries within McCormick envelopes. This refinement aims to enhance the efficacy of the boundaries associated with the second-order cone constraints. The effectiveness of the proposed OPF model of the bipolar DCDN is verified through two typical power optimization problems related to operation and planning, i.e., capacity configuration of distributed generation (DG) and operation optimization of the bipolar DCDN. Compared to employing the SOCP relaxation method alone, the relaxation gap is reduced by nearly 1000 times. [ABSTRACT FROM AUTHOR]

We show that optimal polynomial meshes exist for every convex body in R d , confirming a conjecture by A. Kroó. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R 2 and let L ⊂ Ω be a one-dimensional set with finite length L = | L | . We are interested in minimizers of an energy functional that measures the size of a set projected onto itself in all directions: we are thus asking for sets that see themselves as little as possible (suitably interpreted). Obvious minimizers of the functional are subsets of a straight line but this is only possible for L ≤ diam (Ω) . The problem has an equivalent formulation: the expected number of intersections between a random line and L depends only on the length of L (Crofton's formula). We are interested in sets L that minimize the variance of the expected number of intersections. We solve the problem for convex Ω and slightly less than half of all values of L: there, a minimizing set is the union of copies of the boundary and a line segment. [ABSTRACT FROM AUTHOR]

This work is devoted to show the efficiency of a new numerical approach in solving geometrical shape optimization problems constrained to partial differential equations, on a family of convex domains. More precisely, we are interested to an improved numerical optimization process based on the new shape derivative formula, using the Minkowski deformation of convex domains, recently established in Boulkhemair and Chakib (J. Convex Anal. 21( n ∘ 1 ), 67–87 2014), Boulkhemair (SIAM J. Control Optim. 55( n ∘ 1 ), 156–171 2017). This last formula allows to express the shape derivative by means of the support function, in contrast to the classical one expressed in term of vector fields Henrot and Pierre 2005, Delfour and Zolésio 2011, Sokolowski and Zolesio 1992. This avoids some of the disadvantages related to the classical shape derivative approach, when one use the finite elements discretization for approximating the auxiliary boundary value problems in shape optimization processes Allaire 2007. So, we investigate here the performance of the proposed shape optimization approach through the numerical resolution of some shape optimization problems constrained to boundary value problems governed by Laplace or Stokes operator. Notably, we carry out a comparative numerical study between its resulting numerical optimization process and the classical one. Finally, we give some numerical results showing the efficiency of the proposed approach and its ability in producing good quality solutions and in providing better accuracy for the optimal solution in less CPU time compared to the classical approach. [ABSTRACT FROM AUTHOR]

We consider morphisms of smooth projective varieties over . We show that if π has at most one singular fibre, then X is uniruled and π admits sections. We reach the same conclusions, but with genus zero multisections instead of sections, if π has at most two singular fibres, and the first Chern class of X is supported in a single fibre of π. To achieve these result, we use action completed symplectic cohomology groups associated to compact subsets of convex symplectic domains. These groups are defined using Pardon's virtual fundamental chains package for Hamiltonian Floer cohomology. In the above setting, we show that the vanishing of these groups implies the existence of unirulings and (multi)sections. [ABSTRACT FROM AUTHOR]

We generalize the Yao–Yao partition theorem by showing that for any smooth measure in R d there exist equipartitions using (t + 1) 2 d - 1 convex regions such that every hyperplane misses the interior of at least t regions. In addition, we present tight bounds on the smallest number of hyperplanes whose union contains the boundary of an equipartition of a measure into n regions. We also present a simple proof of a Borsuk–Ulam type theorem for Stiefel manifolds that allows us to generalize the central transversal theorem and prove results bridging the Yao–Yao partition theorem and the central transversal theorem. [ABSTRACT FROM AUTHOR]

We investigate the convexity up to symplectomorphism (called symplectic convexity) of star-shaped toric domains in ℝ 4 . In particular, based on the criterion from Chaidez–Edtmair in [Invent. Math. 229(1) (2022) 243–301] via Ruelle invariant and systolic ratio of the boundary of star-shaped toric domains, we provide elementary operations on domains that can kill the symplectic convexity. These operations only result in small perturbations in terms of domains' volume. Moreover, one of the operations is a systematic way to produce examples of dynamically convex but not symplectically convex toric domains. Finally, we are able to provide concrete bounds for the constants that appear in Chaidez–Edtmair's criterion. [ABSTRACT FROM AUTHOR]

This paper provides a novel region stability criterion on the basis of the pole (eigenvalue) distribution of the linear stochastic system (LSS). Corresponding stabilization control and tracking control are also discussed, which are more suitable for practical engineering applications. First, in the light of the connection between the eigenvalues of the system and the dynamic properties of LSSs, the concept of region stability is given to describe the finer dynamic behavior of LSSs. The criterion for the region stability is also presented. Second, the region stabilization method is investigated according to region stability, which can ensure that the system satisfies a certain dynamic performance by setting the eigenvalues in a certain convex region. Third, a precise tracking control method of LSSs is addressed as an application of region stabilization. It can regulate the steady state properties and transient response of the tracking signal more precisely. Two examples are given to demonstrate the effectiveness of the new methods. • The region stability criterion can manifest the system performance more elaborately than the existing stability criteria. • The corresponding controller design method can control the dynamic performance of stochastic system more accurately. • The new tracking control can control the performance index of transition response during the tracking process. [ABSTRACT FROM AUTHOR]

In this paper, we propose a controller that stabilizes a holonomic robot with single-integrator dynamics to a target position in a bounded domain, while preventing collisions with convex obstacles. We assume that the robot can measure its own position and heading in a global coordinate frame, as well as its relative position vector to the closest point on each obstacle in its sensing range. The robot has no information about the locations and shapes of the obstacles. We define regions around the boundaries of the obstacles and the domain within which the robot can sense these boundaries, and we associate each region with a virtual potential field that we call a local navigation-like function (NLF), which is only a function of the robot's position and its distance from the corresponding boundary. We also define an NLF for the remaining free space of the domain, and we identify the critical points of the NLFs. Then, we propose a switching control law that drives the robot along the negative gradient of the NLF for the obstacle that is currently closest, or the NLF for the remaining free space if no obstacle is detected. We derive a conservative upper bound on the tunable parameter of the NLFs that guarantees the absence of locally stable equilibrium points, which can trap the robot, if the obstacles' boundaries satisfy a minimum curvature condition. We also analyze the convergence and collision avoidance properties of the switching control law and, using a Lyapunov argument, prove that the robot safely navigates around the obstacles and converges asymptotically to the target position. We validate our analytical results for domains with different obstacle configurations by implementing the controller in both numerical simulations and physical experiments with a nonholonomic mobile robot. [ABSTRACT FROM AUTHOR]

In this paper, we give a counter-example to show that coisotropic Ekeland–Hofer–Zehnder capacities do not satisfy superadditivity for hyperplane cuts in higher dimension. Next, we show that the coisotropic Ekeland–Hofer–Zehnder capacity relative to R n , k is not asymptotic equivalent to any normalized symplectic capacity in general case. But we show on the class of centrally symmetric convex domain in R 2 n , the coisotropic Ekeland–Hofer–Zehnder capacity is asymptotic equivalent to the Ekeland–Hofer–Zehnder capacity. Finally, we give some other results about the coisotropic Ekeland–Hofer–Zehnder capacity. [ABSTRACT FROM AUTHOR]

The C 1 , 1 estimate of the Dirichlet problem for degenerate k -Hessian equations with non-homogenous boundary conditions is an open problem, if the right hand side function f is only assumed to satisfy f 1 / (k − 1) ∈ C 1 , 1. In this paper, we solve this problem for convex solutions defined in the strictly convex bounded domain. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the problem of discrete-time stochastic control with multiplicative noise, where the control domains may not be convex. We propose a novel approach inspired by the classical spike variation in continuous-time cases, but adapted to the discrete-time setting by perturbing in a random scale instead of time scale. Our approach allows us to obtain the maximum principle recursively without the need for variation equations or adjoint equations, which are typically required in previous methods. Moreover, we provide two maximum principles, one for the general case and the other when feedback forms exist. Our approach covers the previous results when the control domains are convex. Finally, we demonstrate the effectiveness of our new results through an example. [ABSTRACT FROM AUTHOR]

We study the homogenization of the equation A(\cdot\varepsilon): D2u\varepsilon =f posed in a bounded convex domain Ω\subset Rn subject to a Dirichlet boundary condition and the numerical approximation of the corresponding homogenized problem, where the measurable, uniformly elliptic, periodic, and symmetric diffusion matrix A is merely assumed to be essentially bounded and (if n > 2) to satisfy the Cordes condition. In the first part, we show existence and uniqueness of an invariant measure by reducing to a Lax--Milgram-type problem, we obtain L2-bounds for periodic problems in doubledivergence-form, we prove homogenization under minimal regularity assumptions, and we generalize known corrector bounds and results on optimal convergence rates from the classical case of H\"older continuous coefficients to the present case. In the second part, we suggest and rigorously analyze an approximation scheme for the effective coefficient matrix and the solution to the homogenized problem based on a finite element method for the approximation of the invariant measure, and we demonstrate the performance of the scheme through numerical experiments. [ABSTRACT FROM AUTHOR]

We examine universal positivity constraints on 2 → 2 scattering in 4d planar N = 4 supersymmetric Yang-Mills theory with higher-derivative corrections. We present numerical evidence that the convex region of allowed Wilson coefficients (the "EFT-hedron") flattens completely along about one-third of its dimensions when an increasing number of constraints on the spectral density from crossing-symmetry are included. Our analysis relies on the formulation of the positivity constraints as a linear optimization problem, which we implement using two numerical solvers, SDPB and CPLEX. Motivated by the flattening, we propose a novel partially resummed low-energy expansion of the 2 → 2 amplitude. As part of the analysis, we provide additional evidence in favor of the conjecture [1] that the Veneziano amplitude is the only amplitude compatible with both S-matrix bootstrap constraints and string monodromy. [ABSTRACT FROM AUTHOR]

We study semigroups (\phi _t)_{t\geq 0} of holomorphic self-maps of the unit disk with Denjoy-Wolff point on the boundary. We show that the orthogonal speed of such semigroups is a strictly increasing function. This answers a question raised by F. Bracci, D. Cordella, and M. Kourou, and implies a domain monotonicity property for orthogonal speeds conjectured by Bracci. We give an example of a semigroup such that its total speed is not eventually increasing. We also provide another example of a semigroup having total speed of a certain asymptotic behavior, thus answering another question of Bracci. [ABSTRACT FROM AUTHOR]

The method of Kerzman and Trummer (J. Comp. Appl. Math. 14, 111–123, 1986) for computing the Riemann mapping function of a smooth domain is extended to include the case of simply connected convex regions with corners, in particular convex polygons. The connection between the Szegö kernel and the Riemann mapping function is classical. The integral equation in Kerzman and Trummer (J. Comp. Appl. Math. 14, 111–123, 1986) that determines the Szegö kernel is no longer defined in the presence of corners. We modify the equation by using new oblique projections. This approach is equivalent to employing preliminary mappings. [ABSTRACT FROM AUTHOR]

For a linear hyperbolic equation of fourth order, a Dirichlet type boundary problem in an orthogonally convex domain is investigated. Sharp sufficient conditions guaranteeing solvability and well-posedness of the problem under consideration are established. [ABSTRACT FROM AUTHOR]

We obtain new concavity results, up to a suitable transformation, for a class of quasi-linear equations in a convex domain involving the p-Laplace operator and a general nonlinearity satisfying concavity-type assumptions. This provides an extension of results previously known in the literature only for the torsion and the first eigenvalue equations. In the semilinear case p = 2 the results are already new since they include new admissible nonlinearities. [ABSTRACT FROM AUTHOR]