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The volume decomposition functionals that were recently posed in Liu et al. (Adv Math 389:107902, 2021) to study the affine isoperimetric problem of polytopes, are shown to be uniquely represented as linear combinations of certain products of the LYZ functionals, which were originally introduced in Lutwak et al. (Trans Am Math Soc 353:1767–1779, 2001) to attack the longstanding Schneider projection problem in convex geometry. [ABSTRACT FROM AUTHOR]

In 1967, Grünbaum conjectured that any d-dimensional polytope with d + s ⩽ 2 d vertices has at least ϕ k (d + s , d) = d + 1 k + 1 + d k + 1 - d + 1 - s k + 1 k-faces. This conjecture along with the characterization of equality cases was recently proved by the author (A proof of Grünbaum's lower bound conjecture for general polytopes. Israel J. Math. 245(2), 991–1000 (2021)). In this paper, several extensions of this result are established. Specifically, it is proved that lattices with the diamond property (e.g., abstract polytopes) and d + s ⩽ 2 d atoms have at least ϕ k (d + s , d) elements of rank k + 1 . Furthermore, in the case of face lattices of strongly regular CW complexes representing normal pseudomanifolds with up to 2d vertices, a characterization of equality cases is given. Finally, sharp lower bounds on the number of k-faces of strongly regular CW complexes representing normal pseudomanifolds with 2 d + 1 vertices are obtained. These bounds are given by the face numbers of certain polytopes with 2 d + 1 vertices. [ABSTRACT FROM AUTHOR]

The circumcenter of mass of a simplicial polytope P is defined as follows: triangulate P, assign to each simplex its circumcenter taken with weight equal to the volume of the simplex, and then find the center of mass of the resulting system of point masses. The so obtained point is independent of the triangulation. The aim of the present note is to give a definition of the circumcenter of mass that does not rely on a triangulation. To do so we investigate how volumes of polytopes change under Möbius transformations. [ABSTRACT FROM AUTHOR]

We investigate the convex polytope Ω n (123 ‾) of doubly stochastic matrices spanned by the n × n permutation matrices that avoid an increasing pattern of length 3, the 123 ‾ -permutation matrices. We determine some of its facets and other faces, including faces of small dimension, and their connection to facets and faces of the polytope Ω n of all n × n doubly stochastic matrices. The paper concludes with some relations concerning the frequencies of the positions of the 1's in n × n 123 ‾ -permutation matrices, and the vertex-edge graph of Ω n (123 ‾). [ABSTRACT FROM AUTHOR]

MINLO (mixed-integer nonlinear optimization) formulations of the disjunction between the origin and a polytope via a binary indicator variable have broad applicability in nonlinear combinatorial optimization, for modeling a fixed cost c associated with carrying out a set of d activities and a convex variable cost function f associated with the levels of the activities. The perspective relaxation is often used to solve such models to optimality in a branch-and-bound context, especially in the context in which f is univariate (e.g., in Markowitz-style portfolio optimization). But such a relaxation typically requires conic solvers and are typically not compatible with general-purpose NLP software which can accommodate additional classes of constraints. This motivates the study of weaker relaxations to investigate when simpler relaxations may be adequate. Comparing the volume (i.e., Lebesgue measure) of the relaxations as means of comparing them, we lift some of the results related to univariate functions f to the multivariate case. Along the way, we survey, connect and extend relevant results on integration over a simplex, some of which we concretely employ, and others of which can be used for further exploration on our main subject. [ABSTRACT FROM AUTHOR]

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]

The paper considers the convex hull of a set of schedules for servicing identical requests by parallel devices. Precedence conditions are given on the set of requests. All requests enter the service queue simultaneously and have the same service duration. Interruptions in request servicing are prohibited. Time is discrete. The polyhedral properties of some previously constructed classes of valid inequalities are studied. The "depth" cuts are compared, and the strongest subclasses of cuts are found. The relative position of the schedule polytope and hyperplanes generated by inequalities is also studied. [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]

The harmonic polytope and the bipermutahedron are two related polytopes that arose in the Lagrangian geometry of matroids. We study the bipermutahedron. We show that it is a simple polytope whose faces are in bijection with the vertex-labeled and edge-labeled multigraphs with no isolated vertices; the generating function for its \(f\)-vector is a simple evaluation of the three variable Rogers-Ramanujan function. We introduce the biEulerian polynomial, which counts bipermutations according to their number of descents, and equals the \(h\)-polynomial of the bipermutahedral fan. We construct a unimodular triangulation of the product \(\Delta \times \cdots \times \Delta\) of triangles that is combinatorially equivalent to (the triple cone over) the bipermutahedral fan. Ehrhart theory then gives us a formula for the biEulerian polynomial, which we use to show that this polynomial is real-rooted and that the \(h\)-vector of the bipermutahedral fan is log-concave and unimodal. We describe all the deformations of the bipermutahedron; that is, the ample cone of the bipermutahedral toric variety. We prove that among all polytopes in this family, the bipermutahedron has the largest possible symmetry group. Finally, we show that the Minkowski quotient of the bipermutahedron and the harmonic polytope equals 2.Mathematics Subject Classifications: 52B20, 52B05, 05A15Keywords: Polytope, bipermutahedron, bipermutations, descents, \(f\)-vector, \(h\)-vector, unimodular triangulation, Ehrhart polynomial, real-rooted polynomial, deformation cone

Despite a full characterization of the face vectors of simple and simplicial polytopes, the face numbers of general polytopes are poorly understood. Around 1997, Bárány asked whether for all convex d-polytopes P and all 0 ≤ k ≤ d - 1 , f k (P) ≥ min { f 0 (P) , f d - 1 (P) } . We answer Bárány's question in the affirmative and prove a stronger statement: for all convex d-polytopes P and all 0 ≤ k ≤ d - 1 , f k (P) f 0 (P) ≥ 1 2 [ ⌈ d 2 ⌉ k + ⌊ d 2 ⌋ k ] , f k (P) f d - 1 (P) ≥ 1 2 [ ⌈ d 2 ⌉ d - k - 1 + ⌊ d 2 ⌋ d - k - 1 ]. In the former, equality holds precisely when k = 0 or when k = 1 and P is simple. In the latter, equality holds precisely when k = d - 1 or when k = d - 2 and P is simplicial. [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]

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]

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]

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]

Let the double hyperbolic space D H n , proposed in this paper as an extension of the hyperbolic space H n , contain a two-sheeted hyperboloid with the two sheets connected to each other along the boundary at infinity. We propose to extend the volume of convex polytopes in H n to polytopes in D H n , where the volume is invariant under isometry but can possibly be complex valued. We show that the total volume of D H n is equal to i n V n (S n) for both even and odd dimensions, and prove a Schläfli differential formula (SDF) for D H n . For n odd, the volume of a polytope in D H n is shown to be completely determined by its intersection with ∂ H n and induces a new intrinsic volume on ∂ H n that is invariant under Möbius transformations. [ABSTRACT FROM AUTHOR]

Continuity and discontinuity of two-dimensional domains are thoroughly investigated for accuracy and convergence rate using two prominent discretization methods, namely smoothed and scaled boundary finite element. Because of their capability and versatility when compared to primitive elements, N-sided polygonal elements discretized from modified DistMesh and PolyMesher schemes are used. In terms of accuracy and convergence rate, NSFEM and SBFEM are found to be superior to CSFEM and ESFEM regardless of meshing alternative. The best accuracy occurs at NSFEM and SBFEM, and the obtained convergence rates are optimal. Particularly, in the smoothing domain, it is believed that DistMesh has more promising potential than PolyMesher does; yet, in the discontinuity domain, PolyMesher has been discovered to be more powerful while maintaining its efficiency.

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]

In this paper, an exact algorithm based on the method of orienting curves is developed for solving the convex non-differentiable optimization problem on the closed unit cube in a finite dimensional space: finding the shortest path joining two points going through a sequence of adjacent triangles in 3D. As a result, the global solution of the problem is determined successively by some orienting curves and final curve, which can be exactly constructed with ruler and compass. A detailed numerical example is presented. [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]

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]

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]

Continuity and discontinuity of two-dimensional domains are thoroughly investigated for accuracy and convergence rate using two prominent discretization methods, namely smoothed and scaled boundary finite element. Because of their capability and versatility when compared to primitive elements, N-sided polygonal elements discretized from modified DistMesh and PolyMesher schemes are used. In terms of accuracy and convergence rate, NSFEM and SBFEM are found to be superior to CSFEM and ESFEM regardless of meshing alternative. The best accuracy occurs at NSFEM and SBFEM, and the obtained convergence rates are optimal. Particularly, in the smoothing domain, it is believed that DistMesh has more promising potential than PolyMesher does; yet, in the discontinuity domain, PolyMesher has been discovered to be more powerful while maintaining its efficiency. [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]

Let G = (V , E) be an undirected graph. The spanning trees polytope P(G) is the convex hull of the characteristic vectors of all spanning trees of G. In this paper, we describe all facets of P(G) as a consequence of the facets of the bases polytope P(M) of a matroid M, i.e., the convex hull of the characteristic vectors of all bases of M. [ABSTRACT FROM AUTHOR]

This paper contributes to the polyhedral aspect of the Maximum-Weight Bounded-Degree Rooted Tree Problem, where only edge-indexed variables are considered. An initial formulation is given, followed by an analysis of the dimension and a facial study for the polytope. Several families of new valid inequalities are proposed, which enables us to characterize the polytope on trees and cycles with a totally dual integral system. [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]

Aim: The envelope protein of novel coronavirus 2 (nCoV2) was reported to be highly conserved compared to its spike (S) protein which was shown to undergo several alterations in their amino acid sequences in the span of one year (2020–2021). Therefore, it is aimed to consider highly conserved structural protein of nCov2 namely envelope (E) protein to design the polytope for the formulation of the vaccine against coronavirus disease 2019 (Covid-19). Methods: Online in silico tools were employed to decipher the conservancy and antigenicity of E-protein of nCoV2. They are: to evaluate the molecular affinities among the chosen representatives of alpha and beta coronaviruses, the Molecular Evolutionary Genetics Analysis (MEGA) X 10.1.1 was used. Immune Epitope Database (IEDB)-NetMHCpan (ver. 4.1) tool was used to predict the epitopes of E protein binding to the frequently distributed major histocompatibility complex (MHC) I alleles. ProtParam, VaxJen, ToxinPred and AllerTop online tools were used to assess the physicochemical features, antigenicity, non-toxin and non-allergen aspects of constructed polytope. Secondary structure analysis and homology modelling validation of polytope were done using Phyre2 online tool. Discontinuous and linear epitopes of the designed polytope were predicted through IEDB Ellipro tool. Population coverage of epitopes of the polytope was performed using IEDB online tool with the frequent distribution of human leukocyte antigen (HLA) I alleles in the South Indian Asian population. Results: The phylogeny of envelope proteins of chosen representatives of Coronaviridae confirmed its conservancy and possible origin of nCoV2 from alpha coronaviruses through vampire CoV2. The designed polytope of E-protein was with 53 amino acid residues. The same was developed by linking with cysteine and serine (CS) residues in between epitopes. Conclusion: The antigenicity, non-allergen, non-toxin, homology modelling, discontinuous and linear epitopes of the designed polytope authenticate to explore the envelope protein for prophylactic measures. The epitopes of polytope were found to restrict to MHC I alleles occurring frequently among South Indian Asians.

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]