Narrow Results

In this paper, we define $X$-convex and quasi-$X$-convex functions. This class of real functions is the generalization of a family of convex functions. Additionally, we provide a thorough analysis of the core characteristics of these functions in this paper, along with numerous examples that help to illustrate the ideas. Furthermore, quasi-$X$-convex, semistrictly quasi-$X$-convex, and strictly quasi-$X$-convex functions are used to study and show its applications on optimization problems.

We consider functions on the unit disc whose images are contained in a strip of the complex plane. Under an additional condition, such functions are constants. We also consider appropriate operator valued versions. Applications are found to the theory of semigroups.

Let $M_0(n)$ (resp. $M_1(n)$) denote the number of partitions of $n$ with even (resp. odd) crank. Choi, Kang and Lovejoy established an asymptotic formula for $M_0(n)-M_1(n)$. By utilizing this formula with the explicit bound, we show that $M_k(n-1)+M_k(n+1)>2M_k(n)$ for $k=0$ or $1$ and $n\ge 39$. This result can be seen as the refinement of the classical result regarding the convexity of the partition function $p(n)$, which counts the number of partitions of $n$. We also show that $M_0(n)$ (resp. $M_1(n)$) is log-concave for $n\ge 94$ and satisfies the higher order Turán inequalities for $n\ge 207$ with the aid of the upper bound and the lower bound for $M_0(n)$ and $M_1(n)$.

Given a strictly concave rational PL function $\varphi$ on a complete $n$-dimensional fan $\mathrm{\Sigma }$, we construct an exact symplectic structure of finite volume on ${({ℂ}^{\times })}^n$ and a family of functions $H_{\varphi ,𝜖}$ called polyhedral Hamiltonians. We prove that for each $𝜖$ the one-periodic orbits of $H_{\varphi ,𝜖}$ come in families corresponding to finitely many primitive lattice points of $\mathrm{\Sigma }$ and determine their topology. When $\varphi$ is negative on the rays of $\mathrm{\Sigma }$, we show that the level sets of polyhedral Hamiltonians are hypersurfaces of contact type. As a byproduct, this construction provides a dynamical model for the singularities of toric varieties obtained as degenerations of Fano manifolds in any dimension via Okounkov bodies.

This is a review of the geometry of quantum states using elementary methods and pictures. Quantum states are represented by a convex body, often in high dimensions. In the case of $n$ qubits, the dimension is exponentially large in $n$. The space of states can be visualized, to some extent, by its simple cross sections: Regular simplexes, balls and hyper-octahedra.^a When the dimension gets large, there is a precise sense in which the space of states resembles, almost in every direction, a ball. The ball turns out to be a ball of rather low purity states. We also address some of the corresponding, but harder, geometric properties of separable and entangled states and entanglement witnesses.

“All convex bodies behave a bit like Euclidean balls.”

Keith Ball

Given a C*-algebra with trace τ, we compute the first and second variation formulas for the p-energy functional F_p of the unitary group of , for p = 2n an even integer, namely:

In this paper we address a relaxation theorem for a new integral functional of the calculus of variations defined on the space of functions in whose gradient is an L^p-vector field with distributional divergence given by a Radon measure. The result holds for integrand of type f(x, Δu) without any coerciveness condition, with respect to the second variable, and C¹-continuity assumptions with respect to the spatial variable x.

For continuous functions $A$ and $B$, which are allowed to change sign, we consider the nonlocal differential equation

We propose time-optimal algorithms for a number of convex polygon problems on meshes with multiple broadcasting. Specifically, we show that on a mesh with multiple broadcasting of size n × n, the task of deciding whether an n-gon is convex, deciding whether two convex n-gons edge-intersect, deciding whether one convex n-gon lies in the interior of another, as well as variants of the tasks of computing the intersection and union of two convex n-gons can be accomplished in Θ(log n) time. We also show that detecting whether two convex n-gons are separable takes O(1) time.

The mathematical program with switching constraints (MPSC), which has been introduced recently, is a difficult class of optimization problems. The reason for the difficulty is that the existence of the switching constraint makes the standard constraint qualifications invalid at local minimizers. This paper proposes the Wolfe-type dual (WD) model of this problem without switching constraints. Under the assumptions of convexity and strict convexity, we derive the weak, strong, converse, restricted converse, and strict converse duality results between MPSC and WD.

The work is devoted to the study and solution of one class of nonlinear integral equations with monotone and noncompact Hammerstein-type operators on whole line. Equations of this kind arise in many branches of natural science. In particular, such equations are applied in the dynamic theory of $p$-adic open–closed strings, in the kinetic theory of gases, in the mathematical theory of the spatial-temporal spread of pandemic. A constructive theorem for existence of non-negative nontrivial and essentially bounded solution is proved. The asymptotic behavior of constructed solution at $\pm \infty$ is studied. It is also proved that the difference between the limit in $+\infty$ and the solution represents summable function on $(-\infty ,+\infty ).$ Using results obtained a non-negative integrable and bounded solution for new class of integral equations on whole line with monotone and sign-alternating nonlinearity is constructed. Under additional restrictions on nonlinearity, a uniqueness theorem for a solution in a certain class of continuous and bounded functions is also proved. At the end of the work, the specific examples of above equations, satisfying all conditions of the formulated theorem are given.

Duck eggs are rich in good protein and deeply favored by consumers. Among them, double-yolked duck eggs have high commercial value but are not easy to incubate, so they are often picked out by inspection before processing, circulation or being incubated in egg products industry. This paper used the computer vision technology on single- and double-yolked duck eggs with similar shape and size that were hard to distinguish merely by their external characteristics, studied the effective image processing algorithm and built a cognition model. Then, we collected duck eggs’ transmission light color image, extracted B channel image, carried out iterative morphological open reconstruction and threshold segmentation to acquire yolk area, used convexity defects to identify double-yolked eggs and finally utilized watershed and ellipse fitting to separate yolk. Through inspection, 150 single- and double-yolked eggs were distinguished by using this method, with correct recognition rate reaching to 99% and 100%, respectively. The experimental results indicate that this method has a high accuracy in recognizing double-yolked duck eggs and can provide technical support in nondestructive identification of double-yolked duck egg.

Strong visibility is a generalization of standard visibility, defined with respect to a fixed set of line orientations. We investigate computational properties of this generalized visibility, as well as the related notion of strong convexity, and describe algorithms for the following tasks:

1. Testing the strong visibility of two points in a polygon.

2. Finding the strong convex hull of a point set or polygon.

3. Constructing the strong kernel of a polygon.

4. Identifying the points that are strongly visible from a given point.

A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most $\pi$. We can thus talk about the convexity of a set of points in terms of its min-max interior angle measure. The main result presented here is a nontrivial upper bound of the min-max value in terms of the number of points in the set. Motivated by a particular construction, we also pose a natural conjecture for the best upper bound.

We introduce the Extended Convex Differences Tree (ECDT) representation for n-dimensional polyhedra. The ECDT is simultaneously a representation and an efficiency scheme. A set is represented by a tree. Every node holds a convex bound to the set which it represents (not necessarily the convex hull). The union of the sets represented recursively by the children is the set difference between the parent's convex bound and the set the parent represents. The fact that a node holds a bound to its set is useful for avoidance of unnecessary computations. This bound is convex, permitting efficient algorithms. The ECDT uses convex differences and is therefore able lo look at concave areas as being convex. The ECDT can be viewed as an extension to the Convex Differences Tree scheme, without its drawbacks, or as a restricted form of GSG. We show how Boolean operations are performed directly on the ECDT and how a CSG tree is converted to ECDT form. Various geometric operations on the ECDT are detailed, including point membership classification, slicing by a hyper-plane, and boundary evaluation.

An algorithm for the recognition of a digital image of a planar curve which has undergone an affine transformation is presented. The algorithm is based on affine-invariant extremal geometric properties of curves, utilizes existing computational-geometric methods, and is relatively insensitive to noise. Its time complexity is linear in the number of image pixels on the curve. Extensions of our algorithm to deal with some cases of occlusion of the image curve and recognition under perspective transformations are also described. These algorithms are almost linear in the number of image pixels on the curve.

This paper investigates computational aspects of the well-known convexity theorem due to Helly, which states that the existence of a point in the common intersection of n convex sets is guaranteed by the existence of points in the common intersection of each combination of d+1 of these sets. Given an oracle which accepts d+1 convex sets and either returns a point in their common intersection, or reports its non-existence, we give two algorithms which compute a point in the common intersection of n such gets. The first algorithm runs in O(n^d+1T) time and O(n^d) space, where T is the time required for a single call to the oracle. The second algorithm is a multi-stage variant of the first by which the space complexity may be reduced to O(n) at the expense of an increase in the time complexity by a factor independent of n. We also show how these algorithms may be adapted to construct linear and spherical separators of a collection of sets, and to construct a translate of a given object which either contains, is contained by, or intersects a collection of convex sets.

We analyze the complexity of deciding the convexity of polynomials over simplexes. For polynomials of degree four or higher, the decision problem requires accomplishing an NP-hard task; when the degrees of the polynomials are less than four, the problem can be solved in NC time.

We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a linear-time algorithm which, given an n-vertex 3-connected plane G (with n ≥ 3), finds such a straight-line convex embedding of G into a (n - 2) × (n - 2) grid.

This paper first studies the admissible state set ${𝒢}$ of relativistic magnetohydrodynamics (RMHD). It paves a way for developing physical-constraints-preserving (PCP) schemes for the RMHD equations with the solutions in ${𝒢}$. To overcome the difficulties arising from the extremely strong nonlinearities and no explicit formulas of the primitive variables and the flux vectors with respect to the conservative vector, two equivalent forms of ${𝒢}$ with explicit constraints on the conservative vector are skillfully discovered. The first is derived by analyzing roots of several polynomials and transferring successively them, and further used to prove the convexity of ${𝒢}$ with the aid of semi-positive definiteness of the second fundamental form of a hypersurface. While the second is derived based on the convexity, and then used to show the orthogonal invariance of ${𝒢}$. The Lax–Friedrichs (LxF) splitting property does not hold generally for the nonzero magnetic field, but by a constructive inequality and pivotal techniques, we discover the generalized LxF splitting properties, combining the convex combination of some LxF splitting terms with a discrete divergence-free condition of the magnetic field. Based on the above analyses, several 1D and 2D PCP schemes are then studied. In the 1D case, a first-order accurate LxF-type scheme is first proved to be PCP under the Courant–Friedrichs–Lewy (CFL) condition, and then the high-order accurate PCP schemes are proposed via a PCP limiter. In the 2D case, the discrete divergence-free condition and PCP property are analyzed for a first-order accurate LxF-type scheme, and two sufficient conditions are derived for high-order accurate PCP schemes. Our analysis reveals in theory for the first time that the discrete divergence-free condition is closely connected with the PCP property. Several numerical examples demonstrate the theoretical findings and the performance of numerical schemes.