Search

Saddle points on high-dimensional potential energy surfaces (PES) play a determining role in the activated dynamics of molecules and materials. Building on approaches dating back more than 50 years, many open-ended transition-state search methods have been developed to follow the direction of negative curvature from a local minimum to an adjacent first-order saddle point. Despite the mathematical justification, these methods can display a high failure rate: using small deformation steps, up to 80% of the explorations can end up in a convex region of the PES, where all directions of negative curvature vanish, while if the deformation is aggressive, a similar fraction of attempts lead to saddle points that are not directly connected to the initial minimum. In high-dimension PES, these reproducible failures were thought to only increase the overall computational cost, without having any effect on the methods' capacity to find all saddle points surrounding a minimum. Using activation-relaxation technique nouveau (ARTn), we characterize the nature of the PES around minima, considerably expanding on previous knowledge. We show that convex regions can lie on activation pathways and that not exploring beyond them can introduce significant bias in the saddle-point search. We introduce an efficient approach for traversing the convex regions, almost eliminating exploration failures, while multiplying by almost 10 the number of identified unique and connected saddle points as compared to the standard ARTn, thus underlining the importance of correctly handling convex regions for completeness of saddle point explorations. [ABSTRACT FROM AUTHOR]

We study continuous, equality knapsack problems with uniform separable, non-convex objective functions that are continuous, antisymmetric about a point, and have concave and convex regions. For example, this model captures a simple allocation problem with the goal of optimizing an expected value where the objective is a sum of cumulative distribution functions of identically distributed normal distributions (i.e., a sum of inverse probit functions). We prove structural results of this model under general assumptions and provide two algorithms for efficient optimization: (1) running in linear time and (2) running in a constant number of operations given preprocessing of the objective function. [ABSTRACT FROM AUTHOR]

In this study, we investigate a hybrid-type anisotropic weakly over-penalised symmetric interior penalty method for the Poisson equation on convex domains. Compared with the well-known hybrid discontinuous Galerkin methods, our approach is simple and easy to implement. Our primary contributions are the proposal of a new scheme and the demonstration of a proof for the consistency term, which allows us to estimate the anisotropic consistency error. The key idea of the proof is to apply the relation between the Raviart–Thomas finite element space and a discontinuous space. In numerical experiments, we compare the calculation results for standard and anisotropic mesh partitions. [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]

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]

In this article, a finite‐time optimal containment control method is proposed for nonlinear multi‐agent systems with prescribed performance. First, a neural network‐based reinforcement learning algorithm is developed under the optimized backstepping framework. The algorithm employs an identifier‐critic‐actor architecture, where the identifiers, critics and actors are used to estimate the unknown dynamics, evaluate the system performance, and optimize the system, respectively. Subsequently, in order to guarantee the transient performance of the tracking error, the original system is converted into an equivalent unconstrained system. Then, the tracking errors are allowed to converge to a prescribed set of residuals in finite time by combining prescribed performance control and finite‐time optimal control techniques. Furthermore, by using the Lyapunov stability theorem, it is verified that all signals are semi‐globally practical finite‐time stable, and all followers can converge to a convex region formed by multiple leaders. Finally, the effectiveness of the proposed scheme is demonstrated by a practical example. [ABSTRACT FROM AUTHOR]

The doubly degenerate nutrient taxis model { u t = ∇ ⋅ (u v ∇ u) − ∇ ⋅ (S (u) v ∇ v) + ℓ u v , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , is considered under no-flux boundary conditions in a bounded convex planar domain Ω with smooth boundary. It is firstly shown that within a general framework involving a conveniently large class of nonlinearities S , under an a priori assumption on regularity of solutions to certain approximate problems one can construct a global weak solution whose first component is uniformly bounded. Applications of this to particular settings thereafter reveal global existence of such bounded weak solutions when either S ∈ C 1 ([ 0 , ∞)) satisfies lim sup ξ → ∞ S (ξ) ξ α < ∞ with some α < 2 and the initial data (u 0 , v 0) are reasonably regular but arbitrary large, or S (ξ) = ξ 2 for ξ ≥ 0 and the initial data are such that v 0 satisfies an appropriate smallness condition. [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]

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]

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]

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]

The traditional approach to the exponential synthesis problem for the space of solutions of a homogeneous convolution-type equation in a convex domain assumes that this space is invariant under some differential operator. This assumption makes it possible to reduce the problem of exponential synthesis to the problem of spectral synthesis. Is this assumption due to the method used to solve the problem, or is the invariance of the solution space necessary for a positive answer to the exponential synthesis problem? To answer this question, we consider special equations of the convolution type, the equations with q-sided convolution. For these equations, we show that the requirement of invariance for the solution space is necessary and cannot be omitted if we assume that the solution space admits exponential synthesis with a free choice of the convex region and the characteristic function of the equation. [ABSTRACT FROM AUTHOR]

It has been a widely-accepted belief that for a planar convex domain with two coordinate axes of symmetry, the location of maximal norm of gradient of torsion function is either linked to contact points of largest inscribed circle or connected to points on boundary of minimal curvature. However, we show that this is not quite true in general. Actually, we derive the precise formula for the location of maximal norm of gradient of torsion function on nearly ball domains in ℝn, which displays nonlocal nature and thus does not inherently establish a connection to the aforementioned two types of points. Consequently, explicit counterexamples can be straightforwardly constructed to illustrate this deviation from conventional understanding. We also prove that for a rectangular domain, the maximum of the norm of gradient of torsion function exactly occurs at the centers of the faces of largest (n - 1)-volume. [ABSTRACT FROM AUTHOR]

We prove a sharp global W2, p estimate for potentials of optimal transport maps that take a certain class of nonconvex planar domains to convex ones. [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]

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]

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]

We develop and analyse the HulC, an intuitive and general method for constructing confidence sets using the convex hull of estimates constructed from subsets of the data. Unlike classical methods which are based on estimating the (limiting) distribution of an estimator, the HulC is often simpler to use and effectively bypasses this step. In comparison to the bootstrap, the HulC requires fewer regularity conditions and succeeds in many examples where the bootstrap provably fails. Unlike sub-sampling, the HulC does not require knowledge of the rate of convergence of the estimators on which it is based. The validity of the HulC requires knowledge of the (asymptotic) median bias of the estimators. We further analyse a variant of our basic method, called the Adaptive HulC, which is fully data-driven and estimates the median bias using sub-sampling. We discuss these methods in the context of several challenging inferential problems which arise in parametric, semi-parametric, and non-parametric inference. Although our focus is on validity under weak regularity conditions, we also provide some general results on the width of the HulC confidence sets, showing that in many cases the HulC confidence sets have near-optimal width. [ABSTRACT FROM AUTHOR]

In a smoothly bounded convex domain Ω ⊂ R n with n ≥ 1, a no-flux initial-boundary value problem for u t = Δ u ϕ (v) , v t = Δ v − u v , is considered under the assumption that near the origin, the function ϕ suitably generalizes the prototype given by ϕ (ξ) = ξ α , ξ ∈ [ 0 , ξ 0 ]. By means of separate approaches, it is shown that in both cases α ∈ (0, 1) and α ∈ [1, 2] some global weak solutions exist which, inter alia, satisfy C (T) ≔ ess sup t ∈ (0 , T) ∫ Ω u (⋅ , t) ln ⁡ u (⋅ , t) < ∞ for all T > 0 , with supT>0C(T) < ∞ if α ∈ [1, 2]. [ABSTRACT FROM AUTHOR]

We prove that the cylindrical capacity of a dynamically convex domain in agrees with the least symplectic area of a disk-like global surface of section of the Reeb flow on the boundary of the domain. Moreover, we prove the strong Viterbo conjecture for all convex domains in which are sufficiently C3 close to the round ball. This generalizes a result of Abbondandolo-Bramham-Hryniewicz-Salomão establishing a systolic inequality for such domains. [ABSTRACT FROM AUTHOR]

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3): (a) The number of extreme points in an n-point order type, chosen uniformly at random from all such order types, is on average 4 + o(1). For labeled order types, this number has average 4 − 8/n²−n+2 and variance at most 3. (b) The (labeled) order types read off a set of n points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e., such sampling typically encounters only a vanishingly small fraction of all order types of the given size. Result (a) generalizes to arbitrary dimension d for labeled order types with the average number of extreme points 2d +o(1) and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods show the following relative of the Erdős- Szekeres theorem: for any fixed k, as n →∞, a proportion 1−O(1/n) of the n-point simple order types contain a triangle enclosing a convex k-chain over an edge. For the unlabeled case in (a), we prove that for any antipodal, finite subset of the two-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral, or one of A4, S4, or A5 (and each case is possible). These are the finite subgroups of SO(3) and our proof follows the lines of their characterization by Felix Klein. [ABSTRACT FROM AUTHOR]

We study search problems that can be solved by performing Gradient Descent on a bounded convex polytopal domain and show that this class is equal to the intersection of two well-known classes: PPAD and PLS. As our main underlying technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a continuously differentiable function over the domain [0, 1]² is PPAD∩ PLS-complete. This is the first non-artificial problem to be shown complete for this class. Our results also imply that the class CLS (Continuous Local Search) – which was defined by Daskalakis and Papadimitriou as a more “natural” counterpart to PPAD∩ PLS and contains many interesting problems – is itself equal to PPAD ∩ PLS. [ABSTRACT FROM AUTHOR]

In this note, we investigate the following incompressible chemotaxis-Navier-Stokes system with density-suppressed motility { n t + u ⋅ ∇ n = ∇ ⋅ (ϕ (c) ∇ n) − ∇ ⋅ (n χ (c) ∇ c) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n c , x ∈ Ω , t > 0 , u t + κ (u ⋅ ∇) u + ∇ P = Δ u + n ∇ φ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 in a bounded convex domain Ω ⊂ R 2 with smooth boundary under the no-flux boundary conditions for n , c and the Dirichlet boundary condition for u. We showed that for general (large) regular initial data, this system admits a global classical solution, which is uniformly bounded with respect to κ ∈ [ 0 , 1 ]. In particular, we removed the smallness assumption on the initial data ‖ c (⋅ , 0) ‖ L ∞ (Ω) in Liu-Zhao (J. Differ. Equ., 344 (2023) 608-657). [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]

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]

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

A representation of a finitely generated group into the projective general linear group is called convex co‐compact if it has finite kernel and its image acts convex co‐compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3‐manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean ×$\times$ Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples. [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]

This work is carried out to investigate the anisotropic hardening behavior of MS1180 steel through experiment and analytical modeling. Dogbone and hydraulic bulging specimens were machined to test the mechanical behavior of MS1180 steel plate under different loading directions and stress states. The loading process was recorded by the three‐dimensional digital image correction system. The result uncovers that the strength under equi‐biaxial tension is larger than that under uniaxial tension. Mechanical behavior under different loading directions presents the obvious difference related to the plastic strain. Hardening behavior under uniaxial tension and equi‐biaxial tension is characterized by the Swift‐Voce and polynomial equations, respectively. Three yield functions are adopted to describe the anisotropic behavior, including Yld2000‐2d, S‐Y2009, and CQN‐Chen. A convexity analysis method based on geometric definition is used to determine the convex region of the CQN‐Chen yield surface under large strain. By comparing the yield surfaces of the three yield functions under different plastic strain, the CQN‐Chen yield function possesses a more flexible and reasonable characterization ability than the Yld2000‐2d and S‐Y2009 yield functions. The anisotropic hardening behavior of MS1180 steel plate is modeled with an error of less than 0.03 by the CQN‐Chen yield function, showing high prediction accuracy. [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]

The first aim is to prove that the Roper-Suffridge extension operator preserves ε -starlike property on general domains given by convex functions. The second is to construct the generalized Roper-Suffridge extension operator on Reinhard domains Ω p 1 , p 2 , ⋯ , p n = { (z 1 , ... , z n) ∈ C n : ∑ j = 1 n | z j | p j < 1 } , p 1 , ⋯ , p n ≥ 1. By using a refined Schwarz-Pick lemma, we prove that the operator preserves important properties, e.g., subordination property and spirallikeness. This solves a problem of Gong and Liu. Our result improves many known results from p 1 = 2 to 1 ≤ p 1 < ∞ . As applications, we obtain growth and covering results associated with the extension operator on p-unit ball. Finally, by obtaining geometric and analytic properties of bounded symmetric domains, we generalize the Pfaltzgraff-Suffridge extension operator over bounded symmetric domains and prove Loewner chains and starlikeness are also preserved with a new idea. Further, we propose two conjectures for convexity property. [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]

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]

When dealing with the discretization of differential equations on non‐rectangular domains, a careful treatment of the boundary is mandatory and may result in implementation difficulties and in coefficient matrices without a prescribed structure. Here we examine the numerical solution of a two‐dimensional constant coefficient distributed‐order space‐fractional diffusion equation with a nonlinear term on a convex domain. To avoid the aforementioned inconvenience, we resort to the volume‐penalization method, which consists of embedding the domain into a rectangle and in adding a reaction penalization term to the original equation that dominates in the region outside the original domain and annihilates the solution correspondingly. Thanks to the volume‐penalization, methods designed for problems in rectangular domains are available for those in convex domains and by applying an implicit finite difference scheme we obtain coefficient matrices with a 2‐level Toeplitz structure plus a diagonal matrix which arises from the penalty term. As a consequence of the latter, we can describe the asymptotic eigenvalue distribution as the matrix size diverges as well as estimate the intrinsic asymptotic ill‐conditioning of the involved matrices. On these bases, we discuss the performances of the conjugate gradient with circulant and τ$$ \tau $$‐preconditioners and of the generalized minimal residual with split circulant and τ$$ \tau $$‐preconditioners and conduct related numerical experiments. [ABSTRACT FROM AUTHOR]

We consider an optimal distributed control problem in a strictly convex planar domain with a smooth boundary and a small parameter multiplying a highest derivative of an elliptic operator. A zero Dirichlet condition is set on the boundary of the domain, and control is additively involved in the inhomogeneity. The set of admissible controls is the unit ball in the corresponding space of square integrable functions. The solutions of the obtained boundary value problems are considered in the generalized sense as elements of a Hilbert space. The optimality criterion is the sum of the squared norm of the deviation of the state from a given state and the squared norm of the control with some coefficient. Due to this structure of the optimality criterion, the role of the first or second term of the criterion can be strengthen, if necessary. It is more important to achieve a given state in the first case and to minimize the resource cost in the second case. The asymptotics of the problem generated by the sum of a second-order differential operator with a small coefficient at a highest derivative and a zero-order differential operator is studied in detail. [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]

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]

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 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]

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]

In this article,we prove that nonlinear resolvents of infinitesimal generators on bounded and convex subdomains of C n are decreasing Loewner chains. Furthermore, we consider the problem of the existence of nonlinear resolvents on unbounded convex domains in C . In the case of the upper half-plane, we obtain a complete solution by using that nonlinear resolvents of certain generators correspond to semigroups of probability measures with respect to free convolution. [ABSTRACT FROM AUTHOR]

In this note we prove uniqueness of the critical point for positive solutions of elliptic problems in bounded planar domains: we first examine the Poisson problem $ -\Delta u = f(x, y) $ finding a geometric condition involving the curvature of the boundary and the normal derivative of $ f $ on the boundary to ensure uniqueness of the critical point. In the second part we consider stable solutions of the nonlinear problem $ -\Delta u = f(u) $ in perturbation of convex domains. [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 note, we study a nonlinear system involving the curl operator in a bounded convex domain in ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$, which is derived from the steady‐state approximation for Bean critical‐state model for type‐ II$$ \mathrm{II} $$ superconductors. We show the existence and establish the L∞$$ {L}^{\infty } $$ estimate of the curl of solutions to this system. [ABSTRACT FROM AUTHOR]

Our aim is to study a posteriori error estimates for the finite element method of the parabolic boundary control problems on a bounded convex polygonal domain. For discretization, piecewise linear and continuous finite elements are used to approximate the state and the adjoint-state variables, while piecewise constant functions are employed to approximate the control variable. The backward Euler implicit scheme is applied to discretize the time derivative. An adaptation of a novel elliptic reconstruction technique plays a key role in deriving the error estimates. The residual type a posteriori error estimates for the state and adjoint-state variables are derived in the L ∞ (0 , T ; H 1 (Ω)) -norm. Furthermore, an error bound for the control variable is established in the L ∞ (0 , T ; L 2 (Γ)) -norm. The numerical experiments illustrate the performance of the derived estimators. The adaptive mesh generated via the error indicators is very less in comparison to the uniform mesh, which shows the effectiveness of our derived estimators. [ABSTRACT FROM AUTHOR]

We describe spaces that are dual to some subspaces of the space C∞(D) relative to the inductive limits, where inductive limits, where D ⊂ R p is a bounded convex domain. For any logarithmically convex space of positive numbers M = M k , k ∈ Z + p we introduce the normed space C D , M of functions f ∈ C ∞ D and prove that the Fourier-Laplace transform establishes a topological isomorphism between the strongly dual space and the projective limit of the normed spaces. [ABSTRACT FROM AUTHOR]