Convex Geometry Research Papers - Academia.edu (original) (raw)

2025, SIAM International Conference on Data Mining

Inverse classification is the process of perturbing an instance in a meaningful way such that it is more likely to conform to a specific class. Historical methods that address such a problem are often framed to leverage only a single classifier, or specific set of classifiers. These works are often accompanied by naive assumptions. In this work we propose generalized inverse classification (GIC), which avoids restricting the classification model that can be used. We incorporate this formulation into a refined framework in which GIC takes place. Under this framework, GIC operates on features that are immediately actionable. Each change incurs an individual cost, either linear or non-linear. Such changes are subjected to occur within a specified level of cumulative change (budget). Furthermore, our framework incorporates the estimation of features that change as a consequence of direct actions taken (indirectly changeable features). To solve such a problem, we propose three real-valued heuristic-based methods and two sensitivity analysis-based comparison methods, each of which is evaluated on two freely available real-world datasets. Our results demonstrate the validity and benefits of our formulation, framework, and methods.

2025, Proceedings of the Indian Academy of Sciences. Section A, Physical Sciences

In this paper we establish Minkowski inequality and Brunn-Minkowski inequality for p-quermassintegral differences of convex bodies. Further, we give Minkowski inequality and Brunn-Minkowski inequality for quermassintegral differences of... more

2025, CiteSeer X (The Pennsylvania State University)

The main result of this paper is that in order to prove the local uniformization theorem for local rings it is enough to prove it for rank one valuations. Our proof does not depend on the nature of the class of local rings for which we want to prove local uniformization. We prove also the reductions for different versions of the local uniformization theorem.

2025, HAL (Le Centre pour la Communication Scientifique Directe)

✂✁ ☎✄ ✝✆ ✞✄ ✝✟ ✡✠ be a finite family of sets. We establish an improved inclusion-exclusion identity for each closure operator on the power set of ☛ having the unique base property. The result generalizes three improvements of the... more

2025, Discrete Mathematics & Theoretical Computer Science

Let (A_v)_v ∈ V be a finite family of sets. We establish an improved inclusion-exclusion identity for each closure operator on the power set of V having the unique base property. The result generalizes three improvements of the inclusion-exclusion principle as well as Whitney's broken circuit theorem on the chromatic polynomial of a graph.

2025

We present a new and elementary proof of some recent improvements of the classical inclusion-exclusion bounds. The key idea is to use an injective mapping, similar to the bijective mapping in Garsia and Milne's "bijective" proof of the... more

2025, Advances in Applied Mathematics

In this paper, we show that any convex geometry (dual antimatroid) gives rise to a lace map and deduce some recent variants of the inclusion-exclusion principle from Zeilberger's abstract lace expansion.  2002 Elsevier Science (USA)

2025, e-prints

This paper presents an exploration of convex polyhedra defined by the intersection of symmetric slabs derived from a specific initial set of 22 planes related to cubic, octahedral, and hexagonal symmetries. We define and analyze a primary parametric family of polyhedra, P (C), as the intersection of all 22 corresponding slabs, each with tunable half-width C. A computational study of P (C) reveals unexpected behavior: an empty set for 0 < C < 1, a single point for C = 1, and a consistent octahedron for C > 1. This surprising simplification, despite the complexity of the 44 defining inequalities, is discussed with a geometric interpretation. We document practical computational challenges encountered during the analysis, highlighting potential numerical sensitivities in vertex enumeration for complex hyperplane arrangements. Motivated by these findings, we investigate polyhedra defined by intersections of specific subsets of the initial slabs with independent parameters. This leads to the discovery and detailed characterization of a specific non-uniform orthogonal triangulated octagonal prismatoid with 16 vertices, 42 edges, and 28 faces, obtained from the intersection of cubic and hexagonal slabs with parameters C cube = 1.5 and C hex = 1.5. Its combinatorial properties are rigorously determined using computational geometry methods, including vertex enumeration and face analysis, confirming its structure as a V = 16, E = 42, F = 28 polyhedron with all 28 faces being triangular. Its geometric features, including its prismatoid nature with octagonal bases and observed symmetry, are described based on visual analysis. To the best of our knowledge, this specific geometric realization of a (16, 42, 28)-polyhedron, derived from this precise construction method and set of parameters, has not been previously documented in the literature, suggesting potential novelty. This work introduces a novel approach to defining and studying families of polyhedra from structured hyperplane arrangements, provides interesting case studies, highlights relevant computational challenges, and contributes a specific, well-characterized non-uniform polyhedron to the body of known examples. We discuss potential avenues for future research.

2025

I have a weakness you could call an intellectual disability, or, for short, my
mathematical idiocy. It is formulated as The Principle of Least Thought: I
crave understanding through the shortest route of steps, where the steps are no larger than a small amount. That small amount is much smaller for me than it is for the smart, more intuitive people, the people who make big leaps in thought without much exertion. By “shortest route” I mean absence of extraneous fluff, no extra symbols, no irrelevant, no distracting ideas. This is a pedagogical optimization problem.

2025, arXiv (Cornell University)

We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let A be the positive symmetric matrix taking the outer ellipsoid to the inner one. If tr A = 1, there exists a bijection between the orthogonal group O(n) and the set of such labeled simplices. Similarly, if tr A 2 = 1, there are families of parallelotopes and of cross polytopes, also indexed by O(n).

2025

Let V be a finite set and M a collection of subsets of V. Then M is an alignment of V if and only if M is closed under taking intersections and contains both V and the empty set. If M is an alignment of V, then the elements of M are called convex sets and the pair (V,M) is called an alignment or a convexity. If S ⊆ V, then the convex hull of S is the smallest convex set that contains S. Suppose X ∈ M. Then x ∈ X is an extreme point for X if X \ {x} ∈ M. A convex geometry on a finite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. Let G = (V,E) be a connected graph and U a set of vertices of G. A subgraph T of G containing U is a minimal U-tree if T is a tree and if every vertex of V (T) \U is a cut-vertex of the subgraph induced by V (T). The monophonic interval of U is the collection of all vertices of G that belong to some minimal U-tree. Several graph convexities are defined using minimal U-trees and structural ...

2025, arXiv (Cornell University)

We focus on the analysis of local minimizers of the Mahler volume, that is to say the local solutions to the problem min{M (K) := |K||K • | / K ⊂ R d open and convex, K = -K}, where K • := {ξ ∈ R d ; ∀x ∈ K, x • ξ < 1} is the polar body of K, and | • | denotes the volume in R d . According to a famous conjecture of Mahler the cube is expected to be a global minimizer for this problem. In this paper we express the Mahler volume in terms of the support functional of the convex body, which allows us to compute first and second derivatives of the obtained functional. We deduce from these computations a concavity property of the Mahler volume which seems to be new. As a consequence of this property, we retrieve a result which supports the conjecture, namely that any local minimizer has a Gauss curvature that vanishes at any point where it is defined (first proven by Reisner, Schütt and Werner in 2012, see ). Going more deeply into the analysis in the two-dimensional case, we generalize the concavity property of the Mahler volume and also deduce a new proof that any local minimizer must be a parallelogram (proven by Böröczky, Makai, Meyer, Reisner in 2013, see [3]).

2025, arXiv (Cornell University)

Motivated by Gentzen's disjunction elimination rule in his Natural Deduction calculus and reading inequalities with meet in a natural way, we conceive a notion of distributivity for join-semilattices. We prove that it is equivalent to a notion present in the literature. In the way, we prove that those notions are linearly ordered. We finally consider the notion of distributivity in join-semilattices with arrow, that is, the algebraic structure corresponding to the disjunction-conditional fragment of intuitionistic logic. the use of that notion. Note that Hickman used the term mild distributivity for Hdistributivity. The paper is structured as follows. After this introduction, in Section 2 we provide some notions and notations that will be used in the paper. In Section 3 we show how to arrive to our notion of ND-distributivity for join-semilattices. In Section 4 we compare the different notions of distributivity for join-semilattices that appear in the literature. We prove that one of those is equivalent to the notion of ND-distributivity found in Section 3. Finally, in Section 5 we consider what happens with the different notions of distributivity considered in Section 4 when join-semilattices are expanded with a natural version of the relative meet-complement. In this section we provide the basic notions and notations that will be used in the paper. Let J = (J; ≤) be a poset. For any S ⊆ J, we will use the notations S l and S u to denote the set of lower and upper bounds of S, respectively. That is, S l = {x ∈ J : x ≤ s, for all s ∈ S} and S u = {x ∈ J : s ≤ x, for all s ∈ S}. Lemma 1. Let J = (J; ≤) be a poset. For all a, b, c ∈ J the following statements are equivalent: (i) for all x ∈ J, if x ≤ a and x ≤ b, then x ≤ c, (ii) {a, b} l ⊆ {c} l , (iii) c ∈ {a, b} lu . A poset J = (J; ≤) is a join-semilattice (resp. meet-semilattice) if sup{a, b} (resp. inf{a, b}) exists for every a, b ∈ J. A poset J = (J; ≤) is a lattice if it is both a join-and a meet-semilattice. As usual, the notations a ∨ b (resp. a ∧ b) shall stand for sup{a, b} (resp. inf{a, b}). Given a join-semilattice J = (J; ≤), we will use the following notions: • J is downwards directed iff for any a, b ∈ J, there exists c ∈ J such that c ≤ a and c ≤ b. • A non empty subset I ⊆ J is said to be an ideal iff (1) if x, y ∈ I, then x ∨ y ∈ I and (2) If x ∈ I and y ≤ x, then y ∈ I. • The principal ideal generated by an element a ∈ A, noted (a], is defined by (a] = {x ∈ A : x ≤ a}. • Id(J) will denote the set of all ideals of J. • Id f p (J) will denote the subset of ideals that are intersection of a finite set of principal ideals, that is, Id f p (J) = {(a 1 ] ∩ • • • ∩ (a k ] : a 1 , ...a k ∈ J}.

2025, Proceedings of symposia in pure mathematics

Tube formulas (by which we mean an explicit formula for the volume of an ε-neighbourhood of a subset of a suitable metric space) have been used in many situations to study properties of the subset. For smooth submanifolds of Euclidean space, this includes Weyl's celebrated results on spectral asymptotics, and the subsequent relation between curvature and spectrum. Additionally, a tube formula contains information about the dimension and measurability of rough sets. In convex geometry, the tube formula of a convex subset of Euclidean space allows for the definition of certain curvature measures. These measures describe the curvature of sets which are not too irregular to support derivatives. In this survey paper, we describe some recent advances in the development of tube formulas for self-similar fractals, and their applications and connections to the other topics mentioned here.

2025, Nature Reviews Physics

Real networks are finite metric spaces. Yet the geometry induced by shortest path distances in a network is definitely not its only geometry. Other forms of network geometry are the geometry of latent spaces underlying many networks, and the effective geometry induced by dynamical processes in networks. These three approaches to network geometry are all intimately related, and all three of them have been found to be exceptionally efficient in discovering fractality, scale-invariance, self-similarity, and other forms of fundamental symmetries in networks. Network geometry is also of great utility in a variety of practical applications, ranging from the understanding how the brain works, to routing in the Internet. Here, we review the most important theoretical and practical developments dealing with these approaches to network geometry in the last two decades, and offer perspectives on future research directions and challenges in this novel frontier in the study of complexity.

2025

Focus is on two parties with Hilbert spaces of dimension d, i.e. “qudits”. In the state space of these two possibly entangled qudits an analogue to the well known tetrahedron with the four qubit Bell states at the vertices is presented. The simplex analogue to this magic tetrahedron includes mixed states. Each of these states appears to each of the two parties as the maximally mixed state. Some studies on these states are performed, and special elements of this set are identified. A large number of them is included in the chosen simplex which fits exactly into conditions needed for teleportation and other applications. Its rich symmetry – related to that of a classical phase space – helps to study entanglement, to construct witnesses and perform partial transpositions. This simplex has been explored in details for d = 3. In this paper the mathematical background and extensions to arbitrary dimensions are analysed. PACS numbers: 03.67Mn, 03.67.Hk

2025, Quantum Information Processing

The entanglement of a quantum system can be valuated using Mermin polynomials. This gives us a means to study entanglement evolution during the execution of quantum algorithms. We first consider Grover's quantum search algorithm, noticing that states during the algorithm are maximally entangled in the direction of a single constant state, which allows us to search for a single optimal Mermin operator and use it to evaluate entanglement through the whole execution of Grover's algorithm. Then the Quantum Fourier Transform is also studied with Mermin polynomials. A different optimal Mermin operator is searched at each execution step, since in this case there is no single direction of evolution. The results for the Quantum Fourier Transform are compared to results from a previous study of entanglement with Cayley hyperdeterminant. All our computations can be replayed thanks to a structured and documented open-source code that we provide.

2025, arXiv (Cornell University)

The intersection L of two different non-opposite hemispheres G and H of a d-dimensional sphere S d is called a lune. By the thickness of L we mean the distance of the centers of the (d -1)-dimensional hemispheres bounding L. For a hemisphere G supporting a convex body C ⊂ S d we define widthG(C) as the thickness of the narrowest lune or lunes of the form G ∩ H containing C. If widthG(C) = w for every hemisphere G supporting C, we say that C is a body of constant width w. We present properties of these bodies. In particular, we prove that the diameter of any spherical body C of constant width w on S d is w, and that if w < π 2 , then C is strictly convex. Moreover, we are checking when spherical bodies of constant width and constant diameter coincide.

2025, Discrete Mathematics

Let V be a finite set and C a collection of subsets of V . The ordered pair (V , C) is an alignment if C is closed under taking intersections and contains both ∅ and V . If (V , C) is an alignment, then C is a convexity for V , and the elements of C are referred to as the convex sets of the convexity C. A convex set A is a half-space if V -A is convex. The following separation properties have been defined for a given convexity C of V . (S 1 ) For every x ∈ V , the set {x} is convex. (S 2 ) For every pair a, b ∈ V , there exist complementary half-spaces A, B such that a ∈ A and b ∈ B. (S 3 ) For every convex set A and b ∈ V -A, there exist complementary half-spaces A ′ , B ′ in C such that A ⊆ A ′ and b ∈ B ′ . (S 4 ) For every pair A, B ∈ C of disjoint convex sets, there exist complementary half-spaces All well-known graph convexities satisfy property S 1 . Properties of graphs satisfying separation properties S 2 , S 3 , and S 4 with respect to the two most well-known graph convexities, namely, the geodesic and monophonic convexities, have been studied. In this paper we establish properties of graphs satisfying separation properties S 2 , S 3 , and S 4 relative to the 3-Steiner convexity and the 3-monophonic convexity of a graph.

2025, Mathematical Programming

This note settles an open problem about cut-generating functions, a concept that has its origin in the work of Gomory and Johnson from the 1970's and has received renewed attention in recent years.

2024, arXiv: Probability

We study the log-concave measures, their characterization via the Pr\'ekopa-Leindler property and also define a subset of it whose elements are called super log-concave measures which have the property of satisfying a logarithmic Sobolev inequality. We give some results about their stability. Certain relations with measure transportation of Monge-Kantorovitch and the Monge-Amp\'ere equation are also indicated with applications.

2024, MATHEMATICA SCANDINAVICA

If Delta\DeltaDelta stands for the region enclosed by the triangle in mathsfR2{\mathsf R}^2mathsfR2 of vertices (0,0)(0,0)(0,0), (0,1)(0,1)(0,1) and (1,0)(1,0)(1,0) (or simplex for short), we consider the space mathcalP(2Delta){\mathcal P}(^2\Delta)mathcalP(2Delta) of the 2-homogeneous polynomials on mathsfR2{\mathsf R}^2mathsfR2 endowed with the norm given by ∣ax2+bxy+cy2∣Delta:=sup∣ax2+bxy+cy2∣:(x,y)inDelta\|ax^2+bxy+cy^2\|_\Delta:=\sup\{|ax^2+bxy+cy^2|:(x,y)\in\Delta\}∣ax2+bxy+cy2∣Delta:=sup∣ax2+bxy+cy2∣:(x,y)inDelta for every a,b,cinmathsfRa,b,c\in{\mathsf R}a,b,cinmathsfR. We investigate some geometrical properties of this norm. We provide an explicit formula for ∣cdot∣Delta\|\cdot\|_\Delta∣cdot∣Delta, a full description of the extreme points of the corresponding unit ball and a parametrization and a plot of its unit sphere. Using this geometrical information we also find sharp Bernstein and Markov inequalities for mathcalP(2Delta){\mathcal P}(^2\Delta)mathcalP(2Delta) and show that a classical inequality of Martin does not remain true for homogeneous polynomials on non symmetric convex bodies.

2024, ArXiv

Given a planar point set XXX, we study the convex shells and the convex layers. We prove that when XXX consists of points independently and uniformly sampled inside a convex polygon with kkk vertices, the expected number of vertices on the first ttt convex shells is O(ktlogn)O(kt\log{n})O(ktlogn) for t=O(sqrtn)t=O(\sqrt{n})t=O(sqrtn), and the expected number of vertices on the first ttt convex layers is Oleft(kt3logfracnt2right)O\left(kt^{3}\log{\frac{n}{t^2}}\right)Oleft(kt3logfracnt2right). We also show a lower bound of Omega(tlogn)\Omega(t\log n)Omega(tlogn) for both quantities in the special cases where k=3,4k=3,4k=3,4. The implications of those results in the average-case analysis of two computational geometry algorithms are then discussed.

2024, arXiv (Cornell University)

At a first glance, the problem of illuminating the boundary of a convex body by external light sources and the problem of covering a convex body by its smaller positive homothetic copies appear to be quite different. They are in fact two sides of the same coin and give rise to one of the important longstanding open problems in discrete geometry, namely, the Illumination Conjecture. In this paper, we survey the activity in the areas of discrete geometry, computational geometry and geometric analysis motivated by this conjecture. Special care is taken to include the recent advances that are not covered by the existing surveys. We also include some of our recent results related to these problems and describe two new approaches-one conventional and the other computer-assisted-to make progress on the illumination problem. Some open problems and conjectures are also presented.

2024, arXiv (Cornell University)

2024, Information Geometry

In Riemannian geometry geodesics are integral curves of the Riemannian distance gradient. We extend this classical result to the framework of Information Geometry. In particular, we prove that the rays of level-sets defined by a pseudo-distance are generated by the sum of two tangent vectors. By relying on these vectors, we propose a novel definition of a canonical divergence and its dual function. We prove that the new divergence allows to recover a given dual structure (g, ∇, ∇ * ) of a dually convex set on a smooth manifold M. Additionally, we show that this divergence coincides with the canonical divergence proposed by Ay and Amari in the case of: (a) self-duality, (b) dual flatness, (c) statistical geometric analogue of the concept of symmetric spaces in Riemannian geometry. For a dually convex set, the case (c) leads to a further comparison of the new divergence with the one introduced by Henmi and Kobayashi.

2024, arXiv (Cornell University)

Spherical localisation is a technique whose history goes back to M.Gromov and V.Milman. It's counterpart, the Euclidean localisation is extensively studied and has been put to great use in various branches of mathematics. The purpose of this paper is to quickly introduce spherical localisation, as well as demonstrate some of its applications in convex and metric geometry. Spherical localisation is the spherical counterpart of what is known as Euclidean localisation. It was first used in . The canonical sphere S n is the Riemannian sphere with sectional curvature equal to 1 and its Riemannian metric structure. For simplicity, we view this space as a probability space upon which we have normalised the Riemannian volume. We shall denote the normalised measure by dµ. First we need to understand what the counterpart of a needle on the sphere is.

2024, Far East Journal of Mathematical Sciences (FJMS)

Spherical localisation is a technique whose history goes back to M. Gromov and V. Milman. Its counterpart, the Euclidean localization, is extensively studied and has been put to great use in various branches of mathematics. The purpose of this paper is to quickly introduce spherical localisation, as well as demonstrate some of its applications in convex and metric geometry. 0.1. Spherical Localisation. Spherical localisation is the spherical counterpart of what is known as Euclidean localisation. It was first used in [2]. The canonical sphere n S is the Riemannian sphere with sectional curvature equal to 1 and its Riemannian metric structure. For simplicity, we view this space as a probability space upon which we have normalised the Riemannian volume. We shall denote the normalised measure by dμ. First we need to understand what the counterpart of a needle on the sphere is. Definition 0.1 (Spherical needle). A k-dimensional spherical needle is a

2024, arXiv (Cornell University)

Toric manifolds with dual defect are classified. The associated polytopes, called defect polytopes, are proved to be the class of Delzant integral polytopes for which a combinatorial invariant vanishes.

2024, Arxiv preprint math/0305150

2024

Our aim is to give a simple view on the basics and applications of convex analysis. The essential feature of this account is the systematic use of the possibility to associate to each convex object---such as a convex set, a convex function or a convex extremal problem--- a cone, without loss of information. The core of convex analysis is the possibility of the dual description of convex objects, geometrical and algebraical, based on the duality of vectorspaces; for each type of convex objects, this property is encoded in an operator of duality, and the name of the game is how to calculate these operators. The core of this paper is a unified presentation, for each type of convex objects, of the duality theorem and the complete list of calculus rules. Now we enumerate the advantages of the `cone'-approach. It gives a unified and transparent view on the subject. The intricate rules of the convex calculus all flow naturally from one common source. We have included for each rule a pr...

2024, Contemporary Mathematics

The recently introduced and characterized scalable frames can be considered as those frames which allow for perfect preconditioning in the sense that the frame vectors can be rescaled to yield a tight frame. In this paper we define m-scalability, a refinement of scalability based on the number of nonzero weights used in the rescaling process, and study the connection between this notion and elements from convex geometry. Finally, we provide results on the topology of scalable frames. In particular, we prove that the set of scalable frames with "small" redundancy is nowhere dense in the set of frames.

2024, Contemporary Mathematics

The paper gives an account of the work of Vyacheslav Pavlovich Zakharyuta in the domain of complex analysis, in particular pluripotential theory, showing the influence of his research during several decades.

2024, arXiv (Cornell University)

2024, Advances in Applied Probability

In this paper we investigate the asymptotic behavior of sequences of successive Steiner and Minkowski symmetrizations. We state an equivalence result between the convergences of those sequences for Minkowski and Steiner symmetrizations. Moreover, in the case of independent (and not necessarily identically distributed) directions, we prove the almost-sure convergence of successive symmetrizations at exponential rate for Minkowski, and at rate with c > 0 for Steiner.

2024

A graph is said to be (k, 1) if its vertex set can be partitioned into k independent sets and 1 cliques. The class of (k, 1) graphs appears as a natural generalization of split graphs. In this paper, we describe a characterization for chordal (2,1) graphs. This characterization leads to a O(nm) recognition algorithm, where n and m are the numbers of vertices and edges of the input graph, respectively.

2024, Annales de l’institut Fourier

2024

In this paper, Using Buşneag's model ([1, 2, 3]), we introduce the notion of pseudo-valuation (valuation) on a subtraction algebra and a pseudo-metric is induced by a pseudo-valuation on subtraction algebras.

2024, arXiv (Cornell University)

We provide a new characterization of convex geometries via a multivariate version of an identity that was originally proved by Maneva, Mossel and Wainwright for certain combinatorial objects arising in the context of the k-SAT problem. We thus highlight the connection between various characterizations of convex geometries and a family of removal processes studied in the literature on random structures.

2024, arXiv: Combinatorics

Let V be a finite set and M a collection of subsets of V. Then M is an alignment of V if and only if M is closed under taking intersections and contains both V and the empty set. If M is an alignment of V , then the elements of M are called convex sets and the pair (V,M) is called an aligned space. If SV , then the convex hull of S is the smallest convex set that contains S. Suppose X 2 M. Then x 2 X is an extreme point for X if X\{x} 2 M. The collection of all extreme points of X is denoted by ex(X). A convex geometry on a finite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. Let G = (V,E) be a connected graph and U a set of vertices of G. A subgraph T of G containing U is a minimal U-tree if T is a tree and if every vertex of V (T) \ U is a cut-vertex of the subgraph induced by V (T). The monophonic interval of U is the collection of all vertices of G that belong to some minimal U-tree. A set S of vertices in a ...

2024, Remote Sensing of Environment

This paper reports on the use of linear spectral mixture analysis for the retrieval of canopy leaf area index (LAI) in three flux tower sites in the Boreal Ecosystem-Atmosphere Study (BOREAS) southern study area: Old Black Spruce, Old Jack Pine, and Young Jack Pine (SOBS, SOJP, and SYJP). The data used were obtained by the Compact Airborne Spectrographic Imager (CASI) with a spatial resolution of 2 m in the winter of 1994. The convex geometry method was used to select the endmembers: sunlit crown, sunlit snow, and shadow. Along transects for these flux tower sites, the fraction of sunlit snow was found to have a higher correlation with the field-measured canopy LAI than the fraction of sunlit crown or the fraction of shadow. An empirical equation was obtained to describe the relation between canopy LAI and the fraction of sunlit snow. There is a strong correlation between the estimated LAI and the field-measured LAI along transects (with R 2 values of 0.54, 0.71, and 0.60 obtained for the SOBS, SYJP, and SOJP sites, respectively). The estimated LAI for the whole tower site is consistent with that obtained by the inversion of a canopy model in our previous study where values of 0.94, 0.92, and 0.63 were obtained for R 2 for the SOBS, SYJP and SOJP sites, respectively. The CASI 2-m summer data over the SOBS site was also employed to investigate the possibility of deriving canopy LAI from the summer data using linear mixture analysis. At a spatial resolution of 10 m, the correlation between the field-measured LAI and the estimated LAI along transects is small at R 2 less than 0.3, while R 2 increases to 0.6 at a spatial resolution of 30 m. The difficulty in canopy LAI retrieval from the summer data at a spatial resolution of 10 m is likely due to the variation of the understory reflectance across the scene, although spatial misregistration of the CASI data used may also be a possible contributing factor.

2024

The Brunn-Minkowski theory is a core part of convex geometry. At its foundation lies the Minkowski addition of convex bodies which led to the definition of mixed volume of convex bodies and to various notions and inequalities in convex geometry. Various matrix analogs of these notions and inequalities have been well known for a century. We present a few new analogs. The major theorem presented here is the matrix analog of the Kneser-Süss inequality.

2024, arXiv (Cornell University)

Define an embedding of graph G = (V, E) with V a finite set of distinct points on the unit circle and E the set of line segments connecting the points. Let V 1 ,. .. , V k be a labeled partition of V into equal parts. A 2-factor is said to be cycling if for each u ∈ V , u ∈ V i implies u is adjacent to a vertex in V i+1 (mod k) and a vertex in V i−1 (mod k). In this paper, we will present some new results about cycling 2-factors including a tight upper bound on the minimum number of intersections of a cycling 2-factor for k = 3. 2 Structural Results Definition 2.1 Let u and v be distinct vertices in a geometric graph G. The edge uv ∈ E(G) induces a partition of V (G) by V (G) =

2024

Define an embedding of graph G=(V,E)G=(V,E)G=(V,E) with VVV a finite set of distinct points on the unit circle and EEE the set of line segments connecting the points. Let V1,ldots,VkV_1,\ldots,V_kV1,ldots,Vk be a labeled partition of VVV into equal parts. A 2-factor is said to be {\em cycling} if for each uinVu\in VuinV, uinViu\in V_iuinVi implies uuu is adjacent to a vertex in Vi+1:(mod:k)V_{i+1\: (mod \: k)}Vi+1:(mod:k) and a vertex in Vi−1:(mod:k)V_{i-1\: (mod\: k)}Vi−1:(mod:k). In this paper, we will present some new results about cycling 2-factors including a tight upper bound on the minimum number of intersections of a cycling 2-factor for k=3k=3k=3.

2024, MATHEMATICA SCANDINAVICA

2024, Advances in Applied Probability

2024

In this paper we present necessary and sufficient conditions to guarantee the existence of invariant cones, for semigroup actions, in the space of the k-fold exterior product. As consequence we establish a necessary and sufficient condition for controllability of a class of bilinear control systems. AMS 2020 subject classification: 22E46, 93B05, 20M20

2024, IEEE Transactions on Geoscience and Remote Sensing

Several of the more important endmember-finding algorithms for hyperspectral data are discussed and some of their shortcomings highlighted. A new algorithm-iterated constrained endmembers (ICE)-which attempts to address these shortcomings is introduced. An example of its use is given. There is also a discussion of the advantages and disadvantages of normalizing spectra before the application of ICE or other endmember-finding algorithms.