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In this work, Ramanujan type congruences modulo powers of primes p ≥ 5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t -core partitions known to satisfy Ramanujan type congruences for p = 5 , 7 , 11. The vectors of exponents corresponding to products that are modular forms for Γ 1 (p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ 1 (p) of weights 1 ≤ k ≤ 5 for primes 5 ≤ p ≤ 19 and whose Fourier coefficients satisfy Ramanujan type congruences for all powers of the primes. For each product satisfying a congruence, cyclic permutations of the exponents determine additional products satisfying congruences. Common forms among the exponent sets lead to products satisfying Ramanujan type congruences for a broad class of primes, including p > 19. Canonical bases for modular forms of level 5 ≤ p ≤ 19 are constructed by summing weight one Hecke Eisenstein series of levels 5 ≤ p ≤ 19 and expressing the result as a quotient of Klein forms. Generating sets for the graded algebras of modular forms for Γ 1 (p) and Γ (p) are formulated in terms of permutations of the exponent sets. A sieving process is described by decomposing the space of modular forms of weight 1 for Γ 1 (p) as a direct sum of subspaces of modular forms for Γ (p) of the form q r / p Z [ [ q ] ]. Since the relevant bases generate the graded algebra of modular forms for these groups, the weight one decompositions determine series dissections for modular forms of higher weight that lead to additional classes of congruences. [ABSTRACT FROM AUTHOR]

A new intrinsic volume metric is introduced for the class of convex bodies in ℝ n . As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a restricted number of vertices under this metric. This result improves the best known estimate, and shows that dropping the restriction that the polytope is contained in the ball or vice versa improves the estimate by at least a factor of dimension. The same phenomenon has already been observed in the special cases of volume, surface area and mean width approximation of the ball. [ABSTRACT FROM AUTHOR]

Integrated energy systems offer significant potential for synergistic utilisation of electricity, heat, and gas, leading to improved energy efficiency. However, effectively controlling numerous small-capacity devices within these systems presents operational challenges. This paper proposes a novel multi-time scale optimisation strategy that aggregates small-capacity controllable devices to address this issue. To enable aggregation, we utilise an approximate Minkowski sum method to unify devices with similar control characteristics. A two-stage scheduling strategy is developed, including day-ahead and intra-day stages. The day-ahead stage involves optimisation based on forecasted information, aiming for economic and low-carbon operation. In the intra-day stage, a rolling optimisation method utilising model predictive control is employed, incorporating a penalty term to achieve precise load response through power adjustment. Extensive simulation experiments validate the feasibility and reliability of the proposed strategy in effectively controlling small-capacity devices within integrated energy systems. [ABSTRACT FROM AUTHOR]

We study polytopes defined by inequalities of the form ∑ i ∈ I z i ≤ 1 for I ⊆ [ d ] and nonnegative z i where the inequalities can be reordered into a matrix inequality involving a column-convex { 0 , 1 } -matrix. These generalize polytopes studied by Stanley, and the consecutive coordinate polytopes of Ayyer, Josuat-Vergès, and Ramassamy. We prove an integral equivalence between these polytopes and flow polytopes of directed acyclic graphs G with a Hamiltonian path, which we call spinal graphs. We show that the volumes of these flow polytopes are given by the number of upper (or lower) G-cyclic orders defined by the graphs G. As a special case we recover results on volumes of consecutive coordinate polytopes. We study the combinatorics of k-Euler numbers, which are generalizations of the classical Euler numbers, and which arise as volumes of flow polytopes of a special family of spinal graphs. We show that their refinements, Ramassamy's k-Entringer numbers, can be realized as values of a Kostant partition function, satisfy a family of generalized boustrophedon recurrences, and are log concave along root directions. Finally, via our main integral equivalence and the known formula for the h ∗ -polynomial of consecutive coordinate polytopes, we give a combinatorial formula for the h ∗ -polynomial of flow polytopes of non-nested spinal graphs. For spinal graphs in general, we present a conjecture on upper and lower bounds for their h ∗ -polynomial. [ABSTRACT FROM AUTHOR]

Let N be a weakly unitarily invariant norm (i.e. invariant for the coadjoint action of the unitary group) in the space of skew-Hermitian matrices u n (C). In this paper we study the geometry of the unit sphere of such a norm, and we show how its geometric properties are encoded by the majorization properties of the eigenvalues of the matrices. We give a detailed characterization of norming functionals of elements for a given norm, and we then prove a sharp criterion for the commutator [ X , [ X , V ] ] to be in the hyperplane that supports V in the unit sphere. We show that the adjoint action V ↦ V + [ X , V ] of u n (C) on itself pushes vectors away from the unit sphere. As an application of the previous results, for a strictly convex norm, we prove that the norm is preserved by this last action if and only if X commutes with V. We give a more detailed description in the case of any weakly Ad -invariant norm. [ABSTRACT FROM AUTHOR]

We introduce the following cycle selection problem which is motivated by an application to kidney exchange problems. Given a directed graph G = (V , A) , a cycle selection is a subset of arcs B ⊆ A forming a union of directed cycles. A related optimization problem, the Maximum Weighted Cycle Selection problem can be defined as follows: given a weight w i , j ∈ R for all arcs (i , j) ∈ A , find a cycle selection B which maximizes w (B). We prove that this problem is strongly NP-hard. Next, we focus on cycle selections in complete directed graphs. We provide several ILP formulations of the problem: an arc formulation featuring an exponential number of constraints which can be separated in polynomial time, four extended compact formulations, and an extended non compact formulation. We investigate the relative strength of these formulations. We concentrate on the arc formulation and on the description of the associated cycle selection polytope. We prove that this polytope is full-dimensional, and that all the inequalities used in the arc formulation are facet-defining. Furthermore, we describe three new classes of facet-defining inequalities and a class of valid inequalities. We also consider the consequences of including additional constraints on the cardinality of a selection or on the length of the associated cycles. [ABSTRACT FROM AUTHOR]

A common tool in the theory of numerical semigroups is to interpret a desired class of semigroups as the lattice points in a rational polyhedron in order to leverage computational and enumerative techniques from polyhedral geometry. Most arguments of this type make use of a parametrization of numerical semigroups with fixed multiplicity m in terms of their m-Apéry sets, giving a representation called Kunz coordinates which obey a collection of inequalities defining the Kunz polyhedron. In this work, we introduce a new class of polyhedra describing numerical semigroups in terms of a truncated addition table of their positive sporadic elements. Applying a classical theorem of Ehrhart to slices of these polyhedra, we prove that the number of numerical semigroups with n sporadic elements and Frobenius number f is polynomial up to periodicity, or quasi-polynomial, as a function of f for fixed n. We also generalize this approach to higher dimensions to demonstrate quasi-polynomial growth of the number of affine semigroups with a fixed number of elements, and all gaps, contained in an integer dilation of a fixed polytope. [ABSTRACT FROM AUTHOR]

In this paper, we present an O ( n 2 ) -time algorithm for finding the exact shortest paths from a fixed source point to all other vertices on a triangulated polyhedral surface of a polytope in three-dimensional space, where n is the number of faces of the surface. Our algorithm builds a funnel tree by level, to compute such shortest paths. The funnel tree is built by recursively splitting funnels, which are generated as described by An [Optimization, 2018]. Because their left borders are straightest geodesics, funnels are determined explicitly by the law of cosines. Known approaches such as the planar unfolding technique, source images, or projections of ones are avoided in our algorithm. Although these approaches also run in O ( n 2 ) time in the worst case, our algorithm outperforms the others in practice. Some numerical examples are presented. [ABSTRACT FROM AUTHOR]

We analyze self-dual polyhedral cones and prove several properties about their slack matrices. In particular, we show that self-duality is equivalent to the existence of a positive semi- definite (PSD) slack. Beyond that, we show that if the underlying cone is irreducible, then the corresponding PSD slacks are not only doubly nonnegative matrices (DNN) but are extreme rays of the cone of DNN matrices, which correspond to a family of extreme rays not previously described. More surprisingly, we show that, unless the cone is simplicial, PSD slacks not only fail to be completely positive matrices but they also lie outside the cone of completely positive semidefinite matrices. Finally, we show how one can use semidefinite programming to probe the existence of self-dual cones with given combinatorics. Our results are given for polyhedral cones but we also discuss some consequences for negatively self-polar polytopes. [ABSTRACT FROM AUTHOR]

In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. First, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple ordering condition, which generalizes the notion of perfect elimination order of graphs. This unifies the Savage--Squire--West construction with a recent algorithm for generating elimination trees of chordal graphs. Second, we consider quotients of lattices of acyclic orientations of chordal graphs, and we provide a Gray code for them, addressing a question raised by Pilaud. This also generalizes a recent algorithm for generating lattice congruences of the weak order on the symmetric group. Our algorithms are derived from the Hartung--Hoang--M\"utze--Williams combinatorial generation framework, and they yield simple algorithms for computing Hamilton paths and cycles on large classes of polytopes, including chordal nestohedra and quotientopes. In particular, we derive an efficient implementation of the Savage--Squire--West construction. Along the way, we give an overview of old and recent results about the polyhedral and order-theoretic aspects of acyclic orientations of graphs and hypergraphs. [ABSTRACT FROM AUTHOR]

A generic rectangulation is a partition of a rectangle into finitely many interior-disjoint rectangles, such that no four rectangles meet in a point. In this work we present a versatile algorithmic framework for exhaustively generating a large variety of different classes of generic rectangulations. Our algorithms work under very mild assumptions, and apply to a large number of rectangulation classes known from the literature, such as generic rectangulations, diagonal rectangulations, 1-sided/area-universal, block-aligned rectangulations, and their guillotine variants, including aspect-ratio-universal rectangulations. They also apply to classes of rectangulations that are characterized by avoiding certain patterns, and in this work we initiate a systematic investigation of pattern avoidance in rectangulations. Our generation algorithms are efficient, in some cases even loopless or constant amortized time, i.e., each new rectangulation is generated in constant time in the worst case or on average, respectively. Moreover, the Gray codes we obtain are cyclic, and sometimes provably optimal, in the sense that they correspond to a Hamilton cycle on the skeleton of an underlying polytope. These results are obtained by encoding rectangulations as permutations, and by applying our recently developed permutation language framework. [ABSTRACT FROM AUTHOR]

Darwinian evolution is driven by random mutations, genetic recombination (gene shuffling) and selection that favors genotypes with high fitness. For systems where each genotype can be represented as a bitstring of length L, an overview of possible evolutionary trajectories is provided by the L-cube graph with nodes labeled by genotypes and edges directed toward the genotype with higher fitness. Peaks (sinks in the graphs) are important since a population can get stranded at a suboptimal peak. The fitness landscape is defined by the fitness values of all genotypes in the system. Some notion of curvature is necessary for a more complete analysis of the landscapes, including the effect of recombination. The shape approach uses triangulations (shapes) induced by fitness landscapes. The main topic for this work is the interplay between peak patterns and shapes. Because of constraints on the shapes for L = 3 imposed by peaks, there are in total 25 possible combinations of peak patterns and shapes. Similar constraints exist for higher L. Specifically, we show that the constraints induced by the staircase triangulation can be formulated as a condition of universal positive epistasis, an order relation on the fitness effects of arbitrary sets of mutations that respects the inclusion relation between the corresponding genetic backgrounds. We apply the concept to a large protein fitness landscape for an immunoglobulin-binding protein expressed in Streptococcal bacteria. [ABSTRACT FROM AUTHOR]

We study dilations of cyclic polytopes with the vertices defined by a generator of the simplest cubic fields. In particular, for a specific range of values, we give a precise number of the contained lattice points. [ABSTRACT FROM AUTHOR]

Finding a directed acyclic graph (DAG) that best encodes the conditional independence statements observable from data is a central question within causality. Algorithms that greedily transform one candidate DAG into another given a fixed set of moves have been particularly successful, for example, the greedy equivalence search, greedy interventional equivalence search, and max-min hill climbing algorithms. In 2010, Studeny\', Hemmecke, and Lindner introduced the characteristic imset (CIM) polytope, CIMp, whose vertices correspond to Markov equivalence classes, as a way of transforming causal discovery into a linear optimization problem. We show that the moves of the aforementioned algorithms are included within classes of edges of CIMp and that restrictions placed on the skeleton of the candidate DAGs correspond to faces of CIMp. Thus, we observe that greedy equivalence search, greedy interventional equivalence search, and max-min hill climbing all have geometric realizations as greedy edge-walks along CIMp. Furthermore, the identified edges of CIMp strictly generalize the moves of these algorithms. Exploiting this generalization, we introduce a greedy simplex-type algorithm called greedy CIM, and a hybrid variant, skeletal greedy CIM, that outperforms current competitors among hybrid and constraint-based algorithms. [ABSTRACT FROM AUTHOR]

A set S of points in R n is called a rationally parameterizable hypersurface if there is vector function σ : R n - 1 → R n having as components rational functions defined on some common domain D such that S = { σ (t) : t ∈ D } . A generalized n-dimensional polytope in R n is a union of a finite number of convex n-dimensional polytopes in R n . The Fourier–Laplace transform of such a generalized polytope P in R n is defined by F P (z) = ∫ P e z · x dx . Let γ be a fixed nonzero complex number. We prove that F P 1 (γ σ (t)) = F P 2 (γ σ (t)) for all t ∈ O implies P 1 = P 2 if O is an open subset of D satisfying some well-defined conditions and we present similar results for the null set of the Fourier–Laplace transform of P . Moreover we show that this theorem can be applied to quadric hypersurfaces that do not contain a line, but at least two points, i.e., in particular to spheres. [ABSTRACT FROM AUTHOR]

The problem of calculating exact lower bounds for the number of k-faces of dpolytopes with n vertices, for each value of k, and characterizing the minimizers has recently been solved for n not exceeding 2d. We establish the corresponding result for n = 2d+1; the nature of the lower bounds and the minimizing polytopes are quite different in this case. As a byproduct, we also characterize all d-polytopes with d + 3 vertices and only one or two edges more than the minimum. [ABSTRACT FROM AUTHOR]

We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We determine the inequality descriptions, number of facets, and face lattices of these polytopes. We also study partial permutohedra that we show arise naturally as projections of these polytopes. We enumerate facets and also characterize the face lattices of partial permutohedra in terms of chains in the Boolean lattice. Finally, we have a result and a conjecture on the volume of partial permutohedra when one parameter is fixed to be two. [ABSTRACT FROM AUTHOR]

In this paper we study some geometrical questions about the polytope of bi-capacities. For this, we introduce the concept of pointed order polytope, a natural generalization of order polytopes. Basically, a pointed order polytope is a polytope that takes advantage of the order relation of a partially ordered set and such that there is a relevant element in the structure. We study which are the set of vertices of pointed order polytopes and sort out a simple way to determine whether two vertices are adjacent. We also study the general form of its faces. Next, we show that the set of bi-capacities is a special case of pointed order polytope. Then, we apply the results obtained for general pointed order polytopes for bi-capacities, allowing to characterize vertices and adjacency, and obtaining a bound for the diameter of this important polytope arising in Multicriteria Decision Making. [ABSTRACT FROM AUTHOR]

Mixed-integer nonlinear optimization formulations of the disjunction between the origin and a polytope via a binary indicator variable is broadly used in nonlinear combinatorial optimization for modeling a fixed cost associated with carrying out a group of activities and a convex cost function associated with the levels of the activities. The perspective relaxation of such models is often used to solve to global optimality in a branch-and-bound context, but it typically requires suitable conic solvers and is not compatible with general-purpose NLP software in the presence of other classes of constraints. This motivates the investigation of when simpler but weaker relaxations may be adequate. Comparing the volume (i.e., Lebesgue measure) of the relaxations as a measure of tightness, we lift some of the results related to the simplex case to the box case. In order to compare the volumes of different relaxations in the box case, it is necessary to find an appropriate concave upper bound that preserves the convexity and is minimal, which is more difficult than in the simplex case. To address the challenge beyond the simplex case, the triangulation approach is used. [ABSTRACT FROM AUTHOR]

All SL (n) contravariant vector valuations on polytopes in R n are completely classified without any additional assumptions. The facet vector is defined. It turns out to be the unique class of such valuations for n ≥ 3 . In dimension two, the classification corresponds to the known case of S L (2) covariant valuations. [ABSTRACT FROM AUTHOR]

Given an undirected simple graph G = (V , E) and integer values f v , v ∈ V , a node subset D ⊆ V is called an f -tuple dominating set if, for each node v ∈ V , its closed neighborhood intersects D in at least f v nodes. We study the polytope that is defined as the convex hull of the incidence vectors in R V of the f -tuple dominating sets in G. New families of valid inequalities are introduced and a complete formulation is given for the case of stars. A corollary of our results is a proof that the conjecture reported in Argiroffo (2013) on a complete formulation of the 2-tuple dominating set polytope of trees does not hold. Preliminary computational results are also reported. [ABSTRACT FROM AUTHOR]

This paper investigates the passivity analysis problem of a class of fractional-order neural networks with interval parameter uncertainties (FONNs-IPUs). The previous IPU processing methods are mainly divided into three categories, i.e., directly using the maximum upper bound matrix or the minimum lower bound matrix, using the combination of these two matrices, and using the maximum norm bound matrix. These methods have some shortcomings, for example, the coupling of information between different intervals and the effect of the negative sign of connection weight are not considered. In order to better analyze the passivity of FONNs-IPUs, firstly, a novel treatment approach of IPUs called interval matrix polytope is proposed, which considers all possible boundary information matrices and the sign of IPUs. Secondly, based on the interval matrix polytope approach, the passivity criteria of FONNs-IPUs are established in linear matrix inequality forms. Finally, the effectiveness of the theoretical results is illustrated by simulation example. [ABSTRACT FROM AUTHOR]

Any integral convex polytope P in ℝ N provides an N -dimensional toric variety X P and an ample divisor D P on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on X P , obtained by evaluating global section of the line bundle corresponding to D P on every rational point of X P . This work presents an extension of toric codes analogous to the one of Reed–Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with nonzero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope P and an algorithmic technique to get a lower bound on the minimum distance is described. [ABSTRACT FROM AUTHOR]

The logarithmic capacitary Minkowski problem asks for necessary and sufficient conditions on a finite Borel measure on the unit sphere so that it is the logarithmic capacitary measure of a convex body. This article completely solves the case of discrete measures whose support sets are in general position. [ABSTRACT FROM AUTHOR]

Speakman and Lee (2017) gave a formula for the volume of the convex hull of the graph of a trilinear monomial, y = x 1 x 2 x 3 , over a box in the nonnegative orthant, in terms of the upper and lower bounds on the variables. This was done in the context of using volume as a measure for comparing alternative convexifications to guide the implementation of spatial branch-and-bound for mixed integer nonlinear optimization problems. Here, we introduce an alternative method for computing this volume, making use of the rich theory of mixed volumes. This new method may lead to a natural approach for considering extensions of the problem. [ABSTRACT FROM AUTHOR]

The Polytopes by Iannis Xenakis is the collective name of a series of spatial multimedia installations and musical land art pieces realised on the second half of the 20th century, which take part in the tradition that links Richard Wagner's conception of the total art work, the Gesamtkunstwerk. In the first decade of the same century, Alexander Scriabin envisioned his interpretation of a total art work, the utopian multimedia and multisensorial symphony Mysterium. By providing a brief description of the Concept of Polymediality (2003) and representative compositions associated with its two dimensions, this paper forms the hypothesis that all aforementioned works come under the contemporary notion of 'augmented reality'. This is understood within a hybrid and polyaesthetic context, in which the composer defines a spatial unity of all media elements interacting with the music. [ABSTRACT FROM AUTHOR]

This paper proposes the design of two coordinated wide-area damping controllers (CWADCs) for damping low frequency oscillations (LFOs), while accounting for the uncertainties present in the power system. The controllers based on Deep Neural Network (DNN) and Deep Reinforcement Learning (DRL), respectively, coordinate the operation of different local damping controls such as power system stabilizers (PSSs), static VAr compensators (SVCs), and supplementary damping controllers for DC lines (DC-SDCs). The DNN-CWADC learns to make control decisions using supervised learning; the training dataset consisting of polytopic controllers designed with the help of linear matrix inequality (LMI)-based mixed $H_2/H_\infty$ optimization. The DRL-CWADC learns to adapt to the system uncertainties based on its continuous interaction with the power system environment by employing an advanced version of the state-of-the-art deep deterministic policy gradient (DDPG) algorithm referred to as bounded exploratory control-based DDPG (BEC-DDPG). The studies performed on a 33 machine, 127 bus equivalent model of the Western Electricity Coordinating Council (WECC) system-embedded with different types of damping controls demonstrate the effectiveness of the proposed CWADCs. [ABSTRACT FROM AUTHOR]

We provide a necessary and suffcient condition for the uniqueness of penalized least-squares estimators whose penalty term is given by a norm with a polytope unit ball, covering a wide range of methods including SLOPE, PACS, fused, clustered and classical LASSO as well as the related method of basis pursuit. We consider a strong type of uniqueness that is relevant for statistical problems. The uniqueness condition is geometric and involves how the row span of the design matrix intersects the faces of the dual norm unit ball, which for SLOPE is given by the signed permutahedron. Further considerations based this condition also allow to derive results on sparsity and clustering features. In particular, we define the notion of a SLOPE pattern to describe both sparsity and clustering properties of this method and also provide a geometric characterization of accessible SLOPE patterns. [ABSTRACT FROM AUTHOR]

Group-based models appear in algebraic statistics as mathematical models coming from evolutionary biology, namely in the study of mutations of genomes. Motivated also by applications, we are interested in determining the algebraic degrees of the phylogenetic varieties coming from these models. These algebraic degrees are called phylogenetic degrees. In this paper, we compute the phylogenetic degree of the variety X G , n with G ∈ { Z 2 , Z 2 × Z 2 , Z 3 } and any n -claw tree. As these varieties are toric, computing their phylogenetic degree relies on computing the volume of their associated polytopes P G , n. We apply combinatorial methods and we give concrete formulas for these volumes. [ABSTRACT FROM AUTHOR]

We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint matrix, which in some cases improves previously known results. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a “giant component” of vertices, with measure 1-o(1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1-o(1)$$\end{document} and polynomial diameter. Both bounds rely on spectral gaps—of a certain Schrödinger operator in the first case, and a certain continuous time Markov chain in the second—which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables. [ABSTRACT FROM AUTHOR]

We explore the polytope structures for genuine entanglement, biseparability, full biseparability and Bell inequality of multi-qubit GHZ diagonal states. We first show that biseparable GHZ diagonal states make hypersimplices inside the simplices consisting of all GHZ diagonal states. Next, we consider full biseparability which is equivalent to positive partial transpose for GHZ diagonal states, and show that they make the convex hulls of simplices and cubes. We also visualize which part of the simplex violates multipartite Bell inequality. Finally, we compute precise volumes for genuine entanglement, biseparability, full biseparability and states violating Bell inequality among all GHZ diagonal states. [ABSTRACT FROM AUTHOR]

Pasteurella multocida is the causative agent of a range of animal, and occasionally human, diseases. Problems with antimicrobial treatment of P. multocida highlight the need to find other possible ways, such as prophylaxis, to manage infections. Current vaccines against P. multocida include inactivated bacteria, live attenuated and nonpathogenic bacteria; they have disadvantages such as lack of immunogenicity, reactogenicity, or reversion to virulence. Using bioinformatics approaches, potentially immunogenic and protective epitopes were identified and merged to design the most optimally immunogenic triple epitope PlpE fusion protein of P. multocida as a vaccine candidate. This triple epitope (PlpE1 + 2 + 3) was cloned into the pBAD/gIII A plasmid (pBR322-derived expression vectors designed for regulated, secreted recombinant protein expression and purification in Escherichia coli), expressed in Top 10 E. coli and purified in denatured form using Ni-NTA chromatography and 8 M urea. The immunogenicity of the purified proteins in BALB/c mice was assayed by measuring immunoglobulin G (IgG) responses. The protection potential was evaluated by challenging with 10 LD50 of serotype A:1, X-73 strain of P. multocida and compared with commercially available inactivated fowl cholera vaccine and PlpE protein. IgG levels elicited by the polytope fusion protein of P. multocida PlpE were higher than both commercially available inactivated fowl cholera vaccine and PlpE protein. Surprisingly, protection was independent of IgG level; commercially available inactivated fowl cholera vaccine (100% protection) was more protective than the polytope fusion protein (69% protection) and PlpE protein (69% protection). These results also confirm that IgG level is not a reliable indicator of protection. Further studies to evaluate the other antibody classes, such as immunoglobulin A or M, are required. The role of cell-mediated immunity should also be considered as a potential protection pathway. [ABSTRACT FROM AUTHOR]

A polytope P in some euclidean space is called quasi-regular if each facet F of P is regular and the symmetry group G (F) of F is a subgroup of the symmetry group G (P) of P . Further, P is of full rank if its rank and dimension are the same. In this paper, the quasi-regular polytopes of full rank that are not regular are classified. Similarly, an apeirotope of full rank sits in a space of one fewer dimension; the discrete quasi-regular apeirotopes that are not regular are also classified here. One curiosity of the classification is the difference between even and odd dimensions, in that certain families are present in E d if d is even, but are absent if d is odd. [ABSTRACT FROM AUTHOR]

In 5-axis machining, the initial orientation and position of the part in the fixture on the machine table are chosen to avoid collisions and to ensure that axes ranges are respected. However, kinematics of the machine is rarely considered for workpiece setup optimization although it affects the tool path execution and machining time. Indeed, for complex surfaces, actual feedrate is often lower than the programmed one and can present strong slowdowns, which are critical for the tool cutting conditions and therefore the part quality. This article investigates the use of kinematic manipulability criteria to determine the best orientation of the workpiece setup to maximize the actual feedrate and reduce machining time. The modelling of maximum velocity, acceleration, and jerk of each axis by means of polytopes makes it possible to take advantage of the whole kinematic space of the machine more intuitively. Simulations and experiments are carried out in 3 + 2-axis machining on test parts with a ball-end tool. For stretched surfaces, while the tool centre motion is given by the machining strategy, the tool axis orientation is optimized jointly with the workpiece setup. Experiments confirm that actual feedrate raises faster to better respect the programmed cutting conditions along each path. As feedrate is also higher, machining time is reduced significantly. [ABSTRACT FROM AUTHOR]

Špacapan recently showed that there exist 3‐polytopes with non‐Hamiltonian prisms, disproving a conjecture of Rosenfeld and Barnette. By adapting Špacapan's approach we strengthen his result in several directions. We prove that there exists an infinite family of counterexamples to the Rosenfeld–Barnette conjecture, each member of which has maximum degree 37, is of girth 4, and contains no odd‐length face with length less than k for a given odd integer k. We also show that for any given 3‐polytope H there is a counterexample containing H as an induced subgraph. This yields an infinite family of non‐Hamiltonian 4‐polytopes in which the proportion of quartic vertices tends to 1. However, Barnette's conjecture stating that every 4‐polytope in which all vertices are quartic is Hamiltonian still stands. Finally, we prove that the Grünbaum–Walther shortness coefficient of the family of all prisms of 3‐polytopes is at most 59/60. [ABSTRACT FROM AUTHOR]

The Kimura 3-parameter model is one of the most fundamental phylogenetic models in algebraic statistics. We prove that all algebraic varieties associated to this model are projectively normal, confirming a conjecture of Micha\lek. [ABSTRACT FROM AUTHOR]

We consider partial matchings, which are finite graphs consisting of edges and vertices of degree zero or one. We consider transformations between two states of partial matchings. We introduce a method of presenting a transformation between partial matchings. We introduce the notion of the lattice presentation of a partial matching, and the lattice polytope associated with a pair of lattice presentations, and we investigate transformations with minimal area. [ABSTRACT FROM AUTHOR]

We study sign-restricted matrices (SRMs), a class of rectangular (0 , ± 1) -matrices generalizing the alternating sign matrices (ASMs). In an SRM each partial column sum, starting from row 1, equals 0 or 1, and each partial row sum, starting from column 1, is nonnegative. We determine the maximum number of nonzeros in SRMs and characterize the possible row and column sum vectors. Moreover, a number of results on interchange operations are shown, both for SRMs and, more generally, for (0 , ± 1) -matrices. The Bruhat order on ASMs can be extended to SRMs with the result a distributive lattice. Also, we study polytopes associated with SRMs and some relates decompositions. [ABSTRACT FROM AUTHOR]

The principal aim of this work is to provide a proof of a theorem that computes the coefficients of the Ehrhart polynomial in general form. An application for this computation in any dimension is given along with the procedure to calculate the number of lattice points in n-dimensional space with the aid of the Ehrhart polynomial of the output PQ, for two polytopes P and Q whose dimensions are n and m, respectively. [ABSTRACT FROM AUTHOR]

The market shares of the inverter air conditioners (ACs) have increased in recent years due to their high energy efficiency ratio. The inverter ACs have great potentialities of participation in the demand response programs in power systems. This article proposes an evaluation method of the aggregated power and the response potential for aggregated inverter AC loads with the consideration of load heterogeneities. The clustering algorithm is firstly used to group the inverter ACs according to their daily load profiles. The individual geometrical polytope is adopted to model the response potential of each individual inverter AC. And then the response potentialities of other inverter ACs are replaced by the isomorphism of the given convex polytope. Finally, the aggregated response potential of the inverter ACs is obtained with the Minkowski summation algorithm. The simulation results prove that the proposed method has a better estimation accuracy of the aggregated response potential, and the maximum potential of each inverter AC is better utilized. [ABSTRACT FROM AUTHOR]

The geodesic complexity of a metric space X is the smallest k for which there is a partition of X × X into locally compact sets E0, . . ., Ek on each of which there is a continuous choice of minimal geo-desic σ(x0, x1) from x0 to x1. We prove that the geodesic complexity of an n-dimensional Klein bottle Kn equals 2n. The topological complexity of Kn remains unknown for n > 2. [ABSTRACT FROM AUTHOR]

Weighted cone-volume functionals are introduced for the convex polytopes in R n. For these functionals, geometric inequalities are proved and the equality conditions are characterized. A variety of corollaries are derived, including extremal properties of the regular polytopes involving the L p surface area. Some applications to crystallography and quantum theory are also presented. [ABSTRACT FROM AUTHOR]

In this paper we propose a way to triangulate a pointed order polytope. Pointed order polytopes are a generalization of order polytopes that include some important groups of polytopes appearing when bipolar scales arise in Decision Making or Game Theory, as the set of bi-capacities or the set of normalized bi-games, even for cases with restricted cooperation. Triangulating polytopes is an important and difficult problem that allows an elegant way to generate uniform random points in the polytope. For order polytopes, there exists a nice result that allows a way to triangulate this family of polytopes based on generating linear extensions. In this paper we prove a similar result for pointed order polytopes. The results in this paper allow to derive a procedure to generate random points inside a pointed order polytope that depends only on the structure of the subjacent poset, a problem that usually is simpler to tackle. In particular, this could be applied to generate bi-capacities or bi-capacities belonging to some subfamilies (e.g. k -symmetric, k -interactive,...) in a random way. [ABSTRACT FROM AUTHOR]

Objective(s): Since leishmaniasis is one of the health problems in many countries, the development of preventive vaccines against it is a top priority. Peptide vaccines may be a new way to fight the Leishmania infection. In this study, a silicon method was used to predict and analyze B and T cells to produce a vaccine against cutaneous leishmaniasis. Materials and Methods: Immunodominant epitope of Leishmania were selected from four TSA, LPG3, GP63, and Lmsti1 antigens and linked together using a flexible linker (SAPGTP). The antigenic and allergenic features, 2D and 3D structures, and physicochemical features of a chimeric protein were predicted. Finally, through bioinformatics methods, the mRNA structure was predicted and was produced chemically and cloned into the pLEXY-neo2 vector. Results: Results indicated, polytope had no allergenic properties, but its antigenicity was estimated to be 0.92%. The amino acids numbers, molecular weight as well as negative and positive charge residuals were estimated 390, ~41KDa, 41, and 30, respectively. The results showed that the designed polytope has 50 post-translationally modified sites. Also, the secondary structure of the protein is composed of 25.38% alpha-helix, 12.31% extended strand, and 62.31% random coil. The results of SDS-PAGE and Western blotting revealed the recombinant protein with 41 kDa. The results of Ramachandran plot showed that 96%, 2.7%, and 1.3% of amino acid residues were located in the preferred, permitted, and outlier areas, respectively. Conclusion: It is expected that the TLGL polytope will produce a cellular immune response. Therefore, the polytope could be a good candidate for an anti-leishmanial vaccine. [ABSTRACT FROM AUTHOR]

We define treetopes, a generalization of the three-dimensional roofless polyhedra (Halin graphs) to arbitrary dimensions. Like roofless polyhedra, treetopes have a designated base facet which intersects every face of dimension greater than one in more than one point. We prove an equivalent characterization of the 4-treetopes using the concept of clustered planarity from graph drawing, and we use this characterization to recognize the graphs of 4-treetopes in polynomial time. This result provides one of the first classes of 4-polytopes, other than pyramids and stacked polytopes, that can be recognized efficiently from their graphs. Additionally we show that every d-dimensional treetope (with d ≥ 3 ) has at most d + 1 base facets, and that despite not having any forbidden minors the 4-treetopes obey a separator theorem like the one for planar graphs. [ABSTRACT FROM AUTHOR]

We consider geometric and computational measures of complexity for sets of integer vectors, asking for a qualitative difference between f-vectors of simplicial and general d-polytopes, as well as flag f-vectors of d-polytopes and regular CW (d - 1) -spheres, for d ≥ 4 . [ABSTRACT FROM AUTHOR]

Grünbaum, Barnette, and Reay in 1974 completed the characterization of the pairs (f i , f j) of face numbers of 4-dimensional polytopes. Here we obtain a complete characterization of the pairs of flag numbers (f 0 , f 03) for 4-polytopes. Furthermore, we describe the pairs of face numbers (f 0 , f d - 1) for d-polytopes; this description is complete for even d ≥ 6 except for finitely many exceptional pairs that are "small" in a well-defined sense, while for odd d we show that there are also "large" exceptional pairs. Our proofs rely on the insight that "small" pairs need to be defined and to be treated separately; in the 4-dimensional case, these may be characterized with the help of the characterizations of the 4-polytopes with at most eight vertices by Altshuler and Steinberg (1984). [ABSTRACT FROM AUTHOR]

We describe the geometry of an arrangement of 24-cells inscribed in the 600-cell. In Section 7 we apply our results to the even unimodular lattice E8 and show how the 600-cell transforms E8/2E8, an 8-space over the field F2, into a 4-space over F4 whose points, lines and planes are labeled by the geometric objects of the 600-cell. [ABSTRACT FROM AUTHOR]

This paper is concerned with the extreme points of the polytopes of stochastic tensors. By a tensor we mean a multi-dimensional array over the real number field. A line-stochastic tensor is a nonnegative tensor in which the sum of all entries on each line (i.e. one free index) is equal to 1; a plane-stochastic tensor is a nonnegative tensor in which the sum of all entries on each plane (i.e. two free indices) is equal to 1. In enumerating extreme points of the polytopes of line- and plane-stochastic tensors of order 3 and dimension n, we consider the approach by linear optimization and present new lower and upper bounds. We also study the coefficient matrices that define the polytopes. [ABSTRACT FROM AUTHOR]

Matrices in R n × n with determinant equal to ∑ i ≤ n μ (i) , where μ represents the Möbius function, have been studied for decades. The motivation to study such matrices is the close connection that they have to the prime number theorem, Dirichlet convolution, and related concepts. We introduce two parameterized families of bordered Hermitian matrices that possess similar properties. Each family is comprised of matrices M s ∈ C (n − 1) × (n − 1) that satisfy det ⁡ M 0 = ∑ i ≤ n | μ (i) | , det ⁡ M 1 = (∑ i ≤ n μ (i)) 2 , and we show det ⁡ M s is a quadratic polynomial in s. We apply the Cauchy interlacing theorem to show that, for each matrix in one of the families, the product of all of the subdominant eigenvalues is bounded above by 6 / π 2 + O (n − 1 / 2). [ABSTRACT FROM AUTHOR]