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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]

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]