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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]

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]

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]