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This paper is a corrigendum to the article "On the ideal theorem for number fields". The main result of this paper proves to be untrue and is replaced by an estimate of a weighted sum with an improved error term. [ABSTRACT FROM AUTHOR]

Convex in x domain, eigenfunction, level sets, local rearrangement Erratum to the paper "Convexity in x of the level sets of the first Dirichlet eigenfunction": (Published in Math. Nachr. Keywords: convex in x domain; eigenfunction; level sets; local rearrangement EN convex in x domain eigenfunction level sets local rearrangement 1025 1025 1 03/21/23 20230301 NES 230301 Lemma 3.2 is incorrect. [Extracted from the article]

In this tutorial note we consider the class of non-convex semi-infinite optimization problems which are defined by (one or) finitely many objective functions as well as infinitely many constraints in a finitedimensional space. We present an overview of recent results on the so-called p-power transformation which changes the original problem equivalently locally around a solution point. This transformation is a convexification procedure for the Lagrangian where the functions in the original problem are substituted by their p-th powers. As a consequence, the convexity of the so-transformed Lagrangian allows the application of local duality theory and corresponding solution methods locally around this solution point of the original problem. [ABSTRACT FROM AUTHOR]

The context of this work is to characterize the content and the structure of audiovisual documents by analysing the temporal relationships between basic events resulted from different segmentations of the same document. For this objective, we need to represent and reason about time. We propose a parametric representation of temporal relation between segments (points or intervals) in which the parameters are used to characterize the relationship between two non-convex intervals corresponding to two segmentations in the video analysis domain. The relationship is represented by a co-occurrences matrix noted as Temporal Relation Matrix (TRM). Each document is represented by a set of TRMs computed between each couple of segmentations of the same document using different features. The TRMs are analysed later to detect semantic events, highlight clues about the video content structure or to classify documents based on their types. For higher-level semantic events and documents' structure, we needed to apply some operations on the basic temporal relations and TRMs such as composition, disjunction, complement, intersection, etc. These operations brought to light more complex patterns; e.g. event 1 occurs at the same time of event 2 followed by event 3. In the work presented in this paper, we define a temporal relation algebra including its set of operations based on the parametric representation and TRM defined above. Several experimentations have been done on different audio and video documents to show the efficiency of the proposed representation and the defined operations for audiovisual content analysing. [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 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]

Prospect theory emerged as one of the leading descriptive decision theories that can rationalize a large body of behavioral regularities. The methods for eliciting prospect theory parameters, such as its value function and probability weighting, are invaluable tools in decision analysis. This paper presents a new simple method for eliciting prospect theory's value function without any auxiliary/simplifying parametric assumptions. The method is applicable both to choice under ambiguity (Knightian uncertainty) and risk (when events are characterized by objective probabilities). Our new elicitation method is implemented in a simple paper-and-pencil experiment (via an iterative multiple price list format). This is one of the first experiments that elicits non-parametric prospect theory's value function with salient rewards. The collected data generally confirm findings in the existing literature: the value function is S-shaped (concave in the gain domain and convex in the loss domain) though there is a weaker loss aversion on the aggregate level and a substantial heterogeneity in loss aversion on the individual level (41% loss averse, 6% loss neutral and 53% gain seeking). [ABSTRACT FROM AUTHOR]

This paper concerns the chemotaxis-Navier-Stokes system$\begin{equation} \begin{cases} n_t+u\cdot {\nabla } n = \Delta n - {{\nabla }\cdot}(n\chi(c) {\nabla } c),\\ c_t+u\cdot {\nabla } c = \Delta c -nf(c),\\ u_t+\kappa (u\cdot {\nabla }) u = \Delta u + {\nabla } P +n {\nabla }\Phi, \quad {{\nabla }\cdot} u = 0 \end{cases} \end{equation} \;\;\;\;\;\;\;\;\;\;\;\;(\star)$in a smoothly convex bounded domain under Neumann/Neumann/Dirichlet boundary conditions. The existing literature reveals that $ (\star) $ possesses the global solution under some mild assumptions on $ \chi $ and $ f $ [25,28]. It is shown in this paper that even though $ \chi $ admits some singularity (e.g. $ \chi(c) = c^{-\alpha} $ for some $ \alpha>0 $), appropriate use of some quantity involving $ \chi $ and $ f $ guarantees the existence of the global solution to $ (\star) $ for any choice of suitably regular (nonnegative) initial data. [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, the relationship between r -convexity and Schur-convexity is investigated first. Moreover, the r -convexity and Schur-convexity of a class of functions are studied. As applications, some new inequalities on Minkowski's inequality are established. [ABSTRACT FROM AUTHOR]

In this paper, we study the convexity of the trace function A → Tr B* f (A)B. And we get an extension of the Peierls-Bogolyubov inequality and the joint convexity of the trace geometric mean. [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 manuscript, we aim to establish a connection between the concept of inequalities and the novel Center-Radius order relation. The idea of a Center-Radius (CR)-order interval-valued Godunova-Levin (GL) function is introduced by referring to a total order relation between two intervals. Consequently, convexity and nonconvexity contribute to different kinds of inequalities. In spite of this, convex theory turns to Godunova-Levin functions because they are more efficient at determining inequality terms than other convexity classes. Our application of these new definitions has led to many classical and novel special cases that illustrate the key findings of the paper. Using total order relations between two intervals, this study introduces CR-(h1; h2)-Goduova-Levin functions. It is clear from their properties and widespread usage that the Center-Radius order relation is an ideal tool for studying inequalities. This paper discusses various inequalities based on the Center-Radius order relation. With the CR-order relation, we can first derive Hermite-Hadamard (H.H) inequalities and then develop Jensen-type inequality for interval-valued functions (IVF S) of type (h1; h2)-Godunova- Levin function. Furthermore, the study includes examples to support its conclusions. [ABSTRACT FROM AUTHOR]

In this paper, the variation of functions has been defined, whose domain is a convex and compact set in the plane. Furthermore, in addition to presenting properties that satisfy this variation, the vector space formed by functions with finite variation is studied, demonstrating that it is a Banach space and its elements can be expressed as the difference of non-decreasing functions. [ABSTRACT FROM AUTHOR]

The concept of convexity plays an important role in the study of economics and consumer theory. For the most part, such studies have been conducted on the assumption that consumer preferences are a binary relation that is complete, reflexive and transitive on the set X of consumption choices. However, each consumer is a biological being with multidimensional physiological needs so that possible consumptions from different dimensions cannot be compared by using preferences. By removing that unrealistic assumption, this paper examines how the various concepts of convex preferences and relevant properties can be re-established. We derive a series of 10 formal propositions and construct 6 examples to show that (a) a weighted combination of two possible consumptions is not necessarily comparable with any of the consumptions; (b) not every convergent sequence of a consumer's preferred consumptions asymptotically preserves that consumer's preference preordering; (c) not all preferences satisfy either positive multiplicativity or additive conservation; (d) three types of preference convexities – weak convexity, convexity and strong convexity – can all be introduced into general convex spaces. This paper concludes with some research topics of expected significance for future works. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some f-divergence measures that are related to the Jensen's divergence introduced by Burbea and Rao in 1982. We establish their joint convexity and provide some inequalities between these measures and a combination of Csisz'ar's f-divergence, f-midpoint divergence and f-integral divergence measures. [ABSTRACT FROM AUTHOR]

In this paper, we study the nearly critical Lane-Emden equations (⁎) { − Δ u = u p − ε in Ω , u > 0 in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N with N ≥ 3 , p = N + 2 N − 2 and ε > 0 is small. Our main result is that when Ω is a smooth bounded convex domain and the Robin function on Ω is a Morse function, then for small ε the equation (⁎) has a unique solution, which is also nondegenerate. As for non-convex domain, we also obtain exact number of solutions to (⁎) under some conditions. In general, the solutions of (⁎) may blow-up at multiple points a 1 , ⋯ , a k of Ω as ε → 0. In particular, when Ω is convex, there must be a unique blow-up point (i.e., k = 1). In this paper, by using the local Pohozaev identities and blow-up techniques, even having multiple blow-up points (non-convex domain), we can prove that such blow-up solution is unique and nondegenerate. Combining these conclusions, we finally obtain the uniqueness, multiplicity and nondegeneracy of solutions to (⁎). [ABSTRACT FROM AUTHOR]

This paper is concerned with finite element error estimates for second order elliptic PDEs with inhomogeneous Dirichlet boundary data in convex polygonal domains. The Dirichlet boundary data are assumed to be irregular such that the solution of the PDE does not belong to H 2 ⁢ (Ω) but only to H r ⁢ (Ω) for some r ∈ (1 , 2) . As a consequence, a discretization of the PDE with linear finite elements exhibits a reduced convergence rate in L 2 ⁢ (Ω) and H 1 ⁢ (Ω) . In order to restore the best possible convergence rate we propose and analyze in detail the usage of boundary concentrated meshes. These meshes are gradually refined towards the whole boundary. The corresponding grading parameter does not only depend on the regularity of the Dirichlet boundary data and their discrete implementation but also on the norm, which is used to measure the error. In numerical experiments we confirm our theoretical results. [ABSTRACT FROM AUTHOR]

We consider the following problem stated in 1993 by Buttazzo and Kawohl (Math Intell 15:7–12, 1993): minimize the functional ∫ ∫ Ω (1 + | ∇ u (x , y) | 2 ) - 1 d x d y in the class of concave functions u : Ω → [0,M], where Ω ⊂ R 2 is a convex domain and M > 0. It generalizes the classical minimization problem, which was initially stated by I. Newton in 1687 in the more restricted class of radial functions. The problem is not solved until now; there is even nothing known about the structure of singular points of a solution. In this paper we, first, solve a family of auxiliary 2D least resistance problems and, second, apply the obtained results to study singular points of a solution to our original problem. More precisely, we derive a necessary condition for a point being a ridge singular point of a solution and prove, in particular, that all ridge singular points with horizontal edge lie on the top level and zero level sets. [ABSTRACT FROM AUTHOR]

In the paper we consider minimisation ofU-statistics with the weighted Lasso penalty and investigate their asymptotic properties in model selection and estimation. We prove that the use of appropriate weights in the penalty leads to the procedure that behaves like the oracle that knows the true model in advance, i.e. it is model selection consistent and estimates nonzero parameters with the standard rate. For the unweighted Lasso penalty, we obtain sufficient and necessary conditions for model selection consistency of estimators. The obtained results strongly based on the convexity of the loss function that is the main assumption of the paper. Our theorems can be applied to the ranking problem as well as generalised regression models. Thus, usingU-statistics we can study more complex models (better describing real problems) than usually investigated linear or generalised linear models. [ABSTRACT FROM PUBLISHER]

In the present paper, an interesting subclass of meromorphic univalent functions defined on a punctured unit disk E={z:|z|1>1} has been considered and studied. A sufficient condition for these functions to be univalent and sense preserving in the class has been obtained. Certain geometric properties of the functions of the subclass of meromorphic functions has been discussed, such as coefficient inequality, starlike-ness, convexity, growth and distortion, convex linear combination and extreme points of the functions of the class by fixing some Taylor coefficients. [ABSTRACT FROM AUTHOR]

Clipping has great importance in computer graphics. the clip window is generally of rectangular shape but it may be in any shape. for rectangular clip window we have many popular algorithms like Cohan-Sutherland algorithm. Cyrus Beck algorithm. Liang Barsky algorithms, but it is not necessary that clip window will be always in shape of rectangle it may be of any type of convex region. One of the most common geometrical entities is circle if clip window is in circular shape, then how clipping can be done? How we can identify the position of line segment with respect to circular clip window? Thi paper presents a simple algorithm using parametric equation of the line to determine the position of the line with respect to the circular clip window and discuss the method to clip the clipping candidate line. The proposed algorithm is very simple and less computationally expensive. [ABSTRACT FROM AUTHOR]

This paper discusses an outer-approximation guided optimization method for constrained neural network inverse problems with rectified linear units. The constrained neural network inverse problems refer to an optimization problem to find the best set of input values of a given trained neural network in order to produce a predefined desired output in presence of constraints on input values. This paper analyzes the characteristics of optimal solutions of neural network inverse problems with rectified activation units and proposes an outer-approximation algorithm by exploiting their characteristics. The proposed outer-approximation guided optimization comprises primal and dual phases. The primal phase incorporates neighbor curvatures with neighbor outer-approximations to expedite the process. The dual phase identifies and utilizes the structure of local convex regions to improve the convergence to a local optimal solution. At last, computation experiments demonstrate the superiority of the proposed algorithm compared to a projected gradient method. [ABSTRACT FROM AUTHOR]

This paper is devoted to a comparison between the normalized first (non-trivial) Neumann eigenvalue |Ω|μ1(Ω) for a Lipschitz open set Ω in the plane, and the normalized first (non-trivial) Steklov eigenvalue P(Ω)σ1(Ω). More precisely, we study the ratio F(Ω):=|Ω|μ1(Ω)/P(Ω)σ1(Ω). We prove that this ratio can take arbitrarily small or large values if we do not put any restriction on the class of sets Ω. Then we restrict ourselves to the class of plane convex domains for which we get explicit bounds. We also study the case of thin convex domains for which we give more precise bounds. The paper finishes with the plot of the corresponding Blaschke-Santaló diagrams (x,y)=(|Ω|μ1(Ω),P(Ω)σ1(Ω)). [ABSTRACT FROM AUTHOR]

In this paper, an algorithm capable of modelling shadows from nearby obstructions onto photovoltaic arrays is proposed. The algorithm developed is based on the calculation of the solar position in the sky for any given instant in order to obtain the shadow projection for any object point. The computation is based on considering the shadows as convex regions and on a rasterization process to evaluate the shadowed area of the array. The idea is extended to provide the shading patterns for a desired range of time and to calculate the efficiency rate of the irradiation power incident on the array in comparison with the non-shadowed case. The algorithm has interesting applications, such as optimizing array positioning and orientation, evaluating the impact of new obstructions on pre-existing array installations, allowing precise and practical data for control strategies and MPPT techniques for partially shaded systems, calculating more realistically constrained payback scenarios and finding the optimal PV array interconnection. The results are illustrated by three numerical examples, in which the effects of a nearby building in the irradiation received by a photovoltaic array throughout the year, panel relocation and different interconnections are analysed. [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 investigates the coverage path planning problem for a fixed-wing UAV in convex polygon regions with several practical task requirements in photogrammetry considered. A typical camera model pointing forward-down for photogrammetric application is developed. In addition, the coordinates of the region vertices are converted from the WGS-84 coordinate system to the local ENU coordinate system for path planning convenience. The relationship between the minimum turning radius and the camera footprint is fully studied and the span coefficient of the fixed-wing UAV is first proposed. A novel flight pattern, named as the interlaced back-and-forth pattern in this paper, is presented accordingly. The proposed algorithm is compared with a traditional back-and-forth pattern in mathematics and several important results are given. Then, a practical low-computation algorithm for waypoints generation is developed. Finally, simulation results validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we improve and generalize some existing inequalities for unitarily invariant norms by using the convexity of the function f (v) = ||AvXB1--v+ A 1--vXBv|| on the interval [0,1] . [ABSTRACT FROM AUTHOR]

This paper introduces a new mapping that is weaker than nonexpansive mapping. The new mapping is different from the Suzuki's generalized nonexpansive mapping. This paper introduces a new iteration process for the fixed point, and gives fixed point theorems and Convergence theorems for a new generalized nonexpansive mapping in Banach space, insteading of uniformly convex Banach space. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the differential properties of the scalar flux ϕ (x) defined over a two-dimensional bounded convex domain, as a solution to the integral radiation transfer equation. Estimates for the derivatives of ϕ (x) near the boundary of the domain are given based on Vainikko's regularity theorem. The optimal pointwise error estimates in terms of the scalar flux are presented for the two classic finite difference methods: diamond difference (DD) and step difference (SD). Numerical results indicate the implication of the solution smoothness on the numerical convergence behavior. [ABSTRACT FROM AUTHOR]

In this paper, we study some properties such as the monotonicity, logarithmically complete monotonicity, logarithmic convexity, and geometric convexity, of the combinations of gamma function and power function. The obtained results generalize some related known results for parameters with specific values. [ABSTRACT FROM AUTHOR]

The performance of branch-and-bound algorithms for solving non-convex optimization problems greatly depends on convex relaxation techniques. They generate convex regions which are used for improving the bounds of variable domains. In particular, convex polyhedral regions can be represented by a linear system A. x = b . Then, bounds of variable domains can be improved by minimizing and maximizing variables in the linear system. Reducing or contracting optimally variable domains in linear systems, however, is an expensive task. It requires solving up to two linear programs for each variable (one for each variable bound). Suboptimal strategies, such as preconditioning, may offer satisfactory approximations of the optimal reduction at a lower cost. In non-square linear systems, a preconditioner P can be chosen such that P. A is close to a diagonal matrix. Thus, the projection of the equivalent system P. A. x = P. b over x, by using an iterative method such as Gauss–Seidel, can significantly improve the contraction. In this paper, we show how to generate an optimal preconditioner, i.e., a preconditioner that helps the Gauss–Seidel method to optimally reduce the variable domains. Despite the cost of generating the preconditioner, it can be re-used in sub-regions of the search space without losing too much effectiveness. Experimental results show that, when used for reducing domains in non-square linear systems, the approach is significantly more effective than Gauss-based elimination techniques. Finally, the approach also shows promising results when used as a component of a solver for non-convex optimization problems. [ABSTRACT FROM AUTHOR]

This paper presents a new methodology to obtain prediction regions of the output of a dynamical system. The proposed approach uses stored past outputs of the system and it is entirely data-based. Only two hyperparameters are necessary to apply the proposed methodology. These scalars are chosen so that the size of the obtained regions is minimized while fulfilling the desired empirical probability in a validation set. In this paper, methods to optimally estimate both hyperparameters are provided. The provided prediction regions are convex and checking if a given point belongs to a computed prediction region amounts to solving a convex optimization problem. Also, approximation methods to build ellipsoidal prediction regions are provided. These approximations are useful when explicit descriptions of the regions are necessary. Finally, some numerical examples and comparisons for the case of a non-linear uncertain kite system are provided to prove the effectiveness of the proposed methodology. • A method to compute prediction regions for multivariate systems is presented. • These regions contain future values of the outputs with a certain probability. • The proposed method does not need any knowledge of the system dynamics. • A procedure to obtain ellipsoidal approximations of the regions is also presented. • The results are illustrated by means of numerical examples. [ABSTRACT FROM AUTHOR]

We consider the shape derivative formula for a volume cost functional studied in previous papers where we used the Minkowski deformation and support functions in the convex setting. In this work, we extend it to some non-convex domains, namely the star-shaped ones. The formula happens to be also an extension of a well-known one in the geometric Brunn-Minkowski theory of convex bodies. At the end, we illustrate the formula by applying it to some model shape optimization problem. [ABSTRACT FROM AUTHOR]

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This paper proposes a regularization of the Monge–Ampère equation in planar convex domains through uniformly elliptic Hamilton–Jacobi–Bellman equations. The regularized problem possesses a unique strong solution u_\varepsilon and is accessible to the discretization with finite elements. This work establishes uniform convergence of u_\varepsilon to the convex Alexandrov solution u to the Monge–Ampère equation as the regularization parameter \varepsilon approaches 0. A mixed finite element method for the approximation of u_\varepsilon is proposed, and the regularized finite element scheme is shown to be uniformly convergent. The class of admissible right-hand sides are the functions that can be approximated from below by positive continuous functions in the L^1 norm. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions u. [ABSTRACT FROM AUTHOR]

This paper is first concerned with one kind of discrete-time stochastic optimal control problem with convex control domains, for which necessary condition in the form of Pontryagin's maximum principle and sufficient condition of optimality are derived. The results are then extended to two kinds of discrete-time stochastic games. Two illustrative examples are studied, for which the explicit optimal strategies are given. This paper establishes a rigorous version of discrete-time stochastic maximum principle in a clear and concise way and paves a road for further related topics. [ABSTRACT FROM AUTHOR]

This paper deals with the blow-up phenomena connected to the following porous-medium problem with gradient terms under Robin boundary conditions: { u t = Δ u m + k 1 u p − k 2 | ∇ u | q in Ω × (0 , t ∗) , ∂ u ∂ ν + γ u = 0 on ∂ Ω × (0 , t ∗) , u (x , 0) = u 0 (x) ≥ 0 in Ω ‾ , where Ω ⊂ R n (n ≥ 3 ) is a bounded and convex domain with smooth boundary ∂Ω. The constants p, q, m are positive, and p > q > m > 1 , q > 2 . By making use of the Sobolev inequality and the differential inequality technique, we obtain a lower bound for the blow-up time of the solution. In addition, an example is given as an application of the abstract results obtained in this paper. Our results can be regarded as an answer to the open question raised by Li et al. in (Z. Angew. Math. Phys. 70:1–18, 2019). [ABSTRACT FROM AUTHOR]