Convex Lattice Research Articles

Properties of Convex Lattice Sets under the Discrete Legendre Transform

The discrete Legendre transform is a powerful tool for analyzing the properties of convex lattice sets. In this paper, for t>0, we study a class of convex lattice sets and establish a relationship between vertices of the polar of convex lattice sets and vertices of the polar of its t−dilation. Subsequently, we show that there exists a class of convex lattice sets such that its polar is itself. In addition, we calculate upper and lower bounds for the discrete Mahler product of a class of convex lattice sets.

Reconstruction of Convex Sets from One or Two X-rays

We consider a class of problems of Discrete Tomography which has been deeply investigated in the past: the reconstruction of convex lattice sets from their horizontal and/or vertical X-rays, i.e. from the number of points in a sequence of consecutive horizontal and vertical lines. The reconstruction of the HV-convex polyominoes works usually in two steps, first the filling step consisting in filling operations, second the convex aggregation of the switching components. We prove three results about the convex aggregation step: (1) The convex aggregation step used for the reconstruction of HV-convex polyominoes does not always provide a solution. The example yielding to this result is called the bad guy and disproves a conjecture of the domain. (2) The reconstruction of a digital convex lattice set from only one X-ray can be performed in polynomial time. We prove it by encoding the convex aggregation problem in a Directed Acyclic Graph. (3) With the same strategy, we prove that the reconstruction of fat digital convex sets from their horizontal and vertical X-rays can be solved in polynomial time. Fatness is a property of the digital convex sets regarding the relative position of the left, right, top and bottom points of the set. The complexity of the reconstruction of the digital convex sets which are not fat remains an open question.

The Residue Theorem

Cauchy's Residue Theorem, commonly known as the residue Theorem, is a key theorem in complex variables because it enables people to determine the enclosed curve line of integrals for functionalities. Real integrals and infinite series may also be calculated using it. Our ability to handle complicated analysis is improved by the residue theorem. By connecting the internal Ehrhart polynomials to the closures, we illustrate these tetrahedron Ehrhart-Macdonald reciprocity laws. We calculate the Earhart coefficient of the codimensions to demonstrate our methodology. We conclude by demonstrating how to use our ideas to locate any paste applied with a convex lattice.

Convolution on distribution spaces characterized by regularization

AbstractLocally convex convolutor spaces are studied which consist of those distributions that define a continuous convolution operator mapping from the space of test functions into a given locally convex lattice of measures. The convolutor spaces are endowed with the topology of uniform convergence on bounded sets. Their locally convex structure is characterized via regularization and function‐valued seminorms under mild structural assumptions on the space of measures. Many recent generalizations of classical distribution spaces turn out to be special cases of the general convolutor spaces introduced here. Recent topological characterizations of convolutor spaces via regularization are extended and improved. A valuable property of the convolutor spaces in applications is that convolution of distributions inherits continuity properties from those of bilinear convolution mappings between the locally convex lattices of measures.

Inverse Limit Shape Problem for Multiplicative Ensembles of Convex Lattice Polygonal Lines

Convex polygonal lines with vertices in Z+2 and endpoints at 0=(0,0) and n=(n1,n2)→∞, such that n2/n1→c∈(0,∞), under the scaling n1−1, have limit shape γ* with respect to the uniform distribution, identified as the parabola arc c(1−x1)+x2=c. This limit shape is universal in a large class of so-called multiplicative ensembles of random polygonal lines. The present paper concerns the inverse problem of the limit shape. In contrast to the aforementioned universality of γ*, we demonstrate that, for any strictly convex C3-smooth arc γ⊂R+2 started at the origin and with the slope at each point not exceeding 90∘, there is a sequence of multiplicative probability measures Pnγ on the corresponding spaces of convex polygonal lines, under which the curve γ is the limit shape.

Congruences for Consecutive Coefficients of Gaussian Polynomials with Crank Statistics


 
 
 In this paper, we establish infinite families of congruences in consecutive arithmetic progressions modulo any odd prime $\ell$ for the function $p\big(n,m,N\big)$, which enumerates the partitions of $n$ into at most $m$ parts with no part larger than $N$. We also treat the function $p\big(n,m,(a,b]\big)$, which bounds the largest part above and below, and obtain similar infinite families of congruences.
 For $m \leq 4$ and $\ell = 3$, simple combinatorial statistics called "cranks" witness these congruences. We prove this analytically for $m=4$, and then both analytically and combinatorially for $m = 3$. Our combinatorial proof relies upon explicit dissections of convex lattice polygons.
 
 For $m \leq 4$ and $\ell = 3$, simple combinatorial statistics called ``cranks" witness these congruences. We prove this analytically for $m=4$, and then both analytically and combinatorially for $m = 3$. Our combinatorial proof relies upon explicit dissections of convex lattice polygons. 
 
 

Multisensory Flexible Braille Interactive Device Based on Liquid Crystal Elastomers

Traditional electronic braille devices’ rigid and complex structure limits their portability while satisfying braille output. Main-chain monodomain liquid crystal elastomers (m-LCEs) can achieve high contraction deformation under external stimulation conditions; it has broad application prospects in the human–machine interaction with the biological tissue safety and flexibility of LCEs. A multisensory flexible braille interactive device based on LCEs is proposed in this article. It utilizes the good reversible shape-memory performance of m-LCEs to solve the problems of complex structure design, poor portability, high cost, and low control stability in traditional rigid mechanical braille readers. The fixed-point stretchable convex lattice and the nonactuation performance-based film are accurately distinguished and fabricated by compression molding. Combined with the ultrathin flexible electric heating layer, the reversible contraction of the braille point is realized, and the braille point is given the dual perception of height difference and temperature difference. Moreover, matching circuit design can realize braille input by press and braille output by convex–concave conversion. Furthermore, we try to introduce thermochromic powders, referring to the principle of light-emitting diode (LED) imaging, to transform the pattern concept of visually impaired people in minds into color information that ordinary people can observe.

Convexity of Distinct Sum Sets

We study a combinatorial notion where given a set S of lattice points one takes the set of all sums of p distinct points in S, and we ask the question: ‘if S is the set of lattice points of a convex lattice polytope, is the resulting set also the set of lattice points of a convex lattice polytope?’ We obtain a positive result in dimension 2 and a negative result in higher dimensions. We apply this to the corner cut polyhedron.

Degenerations of Grassmannians via Lattice Configurations

AbstractWe study degenerations of Grassmannians constructed using convex lattice configurations in Bruhat–Tits buildings. Using techniques from quiver representations, we analyze their special fibers, which are explicitly described as quiver Grassmannians. For a class of lattice configurations, called the locally linearly independent configurations, we show that our construction coincide with Mustafin degenerations, thus generalizing a result of Faltings. In such cases, our analysis of special fibers also generalizes results of Cartwright et al. As an application, we prove a smoothing criterion for limit linear series on arbitrary reducible nodal curves.

Stability of $\varepsilon$-isometries on the positive cones of finite-dimensional Banach spaces

A weak stability bound for the $\varepsilon$-isometry $f$ form the positive cone of a reflexive, strictly convex and Gateaux smooth Banach lattice $X$ to a Banach space $Y$ is presented. This result is used to prove the stability theorem for the $\varepsilon$-isometry $f:(\mathbb{R}^n)^+\rightarrow Y$, where $\mathbb{R}^n$ is the $n$-dimensional space equipped with a $1$-unconditional norm and $Y$ is a n-dimensional, strictly convex and Gateaux smooth space.

Convex lattice polygons with all lattice points visible

Two lattice points are visible to one another if there exist no other lattice points on the line segment connecting them. In this paper we study convex lattice polygons that contain a lattice point such that all other lattice points in the polygon are visible from it. We completely classify such polygons, show that there are finitely many of lattice width greater than 2, and computationally enumerate them. As an application of this classification, we prove new obstructions to graphs arising as skeleta of tropical plane curves.

Sectional genus and the volume of a lattice polytope

For a convex lattice polytope having at least one interior lattice point, a lower bound for its volume is derived from Hibi’s lower bound theorem for the $$h^{*}$$ -vector. On the other hand, it is known that the sectional genus of a polarized variety has an upper bound, which is an extension of the Castelnuovo bound for the genus of a projective curve. In this paper, we prove the equivalence of these two bounds. Namely, a polarized toric variety has maximal sectional genus if and only if its associated polytope has minimal volume. This is a generalization of the known fact that polytopes corresponding to the anticanonical bundles of Gorenstein toric Fano varieties are reflexive polytopes (whose typical examples are minimal volume polytopes with only one interior lattice point).

Ambiguous reconstructions of hv-convex polyominoes

The core problem of discrete tomography, i.e., the faithful reconstruction of an unknown discrete object from projections along a given set of directions, is ill-posed in general. When further constraints are imposed on the object or on the employed directions, uniqueness of the reconstruction can be obtained. It is the case, for example, of convex lattice sets in Z2, for which a theorem by Gardner and Gritzmann assures the faithful reconstruction when suitable sets of directions are considered. It was conjectured that a similar result holds for the class of hv-convex polyominoes. In this paper we are concerned with this conjecture, providing new 4-tuples of discrete directions that do not lead to a unique reconstruction of hv-convex polyominoes, underlining the relevant structural difference with the class of convex sets. Our result is based on the recursive definition of new hv-convex switching components on discrete sets along four directions.

The polar of convex lattice sets

Let $K$ be a convex lattice set in $\mathbb{Z}^n$ containing the origin as the interior of its convex hull. In this paper, the definition of the polar of a convex lattice set $K$ is given both in $\mathbb{Q}^n$ and $\mathbb{Z}^n$. Some properties and inequalities about the convex lattice sets and their polar are established.

An extremal property of lattice polygons

We find the critical number of vertices of a convex lattice polygon that guarantees that the polygon contains at least one point of a given square sublattice. As a tool, we prove Diophantine inequalities relating the number of edges of a broken line and the coordinates of its endpoints.

A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture in Euclidean space

A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture in Euclidean space

Some problems about geometric lattice

In this paper, the survey about some results of the convex lattice set are given and the invariance of projection problem of convex lattice set is also obtained. And combining a famous result in the graph theory, several conjectures about the convex lattice set are presented.

Asymptotics of convex lattice polygonal lines with a constrained number of vertices

A detailed combinatorial analysis of planar convex lattice polygonal lines is presented. This makes it possible to answer an open question of Vershik regarding the existence of a limit shape when the number of vertices is constrained.

Order Bornological Locally Convex Lattice Cones

In this paper, we introduce the concepts of $us$-lattice cones and order bornological locally convex lattice cones. In the special case of locally convex solid Riesz spaces, these concepts reduce to the known concepts of seminormed Riesz spaces and order bornological Riesz spaces, respectively. We define solid sets in locally convex cones and present some characterizations for order bornological locally convex lattice cones.

An analogue of the Aleksandrov projection theorem for convex lattice polygons

An analogue of the Aleksandrov projection theorem for convex lattice polygons