On the complexity of equational decision problems for finite height complemented and orthocomplemented modular lattices

We study the computational complexity of the satisfiability problem and the complement of the equivalence problem for complemented (orthocomplemented) modular lattices L and classes thereof. Concerning a simple L of finite height, $$\mathcal {NP}$$ NP -hardness is shown for both problems. Moreover, both problems are shown to be polynomial-time equivalent to the same feasibility problem over the division ring D whenever L is the subspace lattice of a D-vector space of finite dimension at least 3. Considering the class of all finite dimensional Hilbert spaces, the equivalence problem for the class of subspace ortholattices is shown to be polynomial-time equivalent to that for the class of endomorphism $$*$$ ∗ -rings with pseudo-inversion; moreover, we derive completeness for the complement of the Boolean part of the nondeterministic Blum-Shub-Smale model of real computation without constants. This result extends to the additive category of finite dimensional Hilbert spaces, enriched by adjunction and pseudo-inversion.

Searching Monotone Arrays: A Survey

Given a monotone ordered multi-dimensional real array A and a real value k, an important question in computation is to establish if k is a member of A by sequentially searching A by comparing k with some of its entries. This search problem and its known results are surveyed, including the case when A has sizes not necessarily equal. Worst case search algorithms for various types of arrays of finite dimension and sizes are reported. Each algorithm has order strictly less than the product of the sizes of the array. Present challenges and open problems in the area are also presented.

A fast thermoelastic model based on the half-space theory applied to elastohydrodynamic lubrication of line contacts involving free boundaries

This study allows contact models based on semi-analytical methods including the impacts of thermoelastic deformations in contacts of finite dimension bodies. The proposed method controls heat flows crossing free boundaries. A comparison with FEA reveals that the proposed method can reduce the calculation times by more than 98%. The paper introduces the thermoelasticity effects into thermal-elastohydrodynamic lubrication (TEHL) modeling of line contact problems. The analysis reveals that including thermoelastic deformations changes the pressure profile and tends to localize the pressure close to the distribution center. Compared to TEHL simulations, the examined configurations caused an overall increase in the maximum pressure by about 9%, an overall film thickness reduction of about 7%, and an overall temperature increase of about 2 K.

Support and separation properties of convex sets in finite dimension

This is a survey on support and separation properties of convex sets in the n-dimensional Euclidean space. It contains a detailed account of existing results, given either chronologically or in related groups, and exhibits them in a uniform way, including terminology and notation. We first discuss classical Minkowski’s theorems on support and separation of convex bodies, and next describe various generalizations of these results to the case of arbitrary convex sets, which concern bounding and asymptotic hyperplanes, and various types of separation by hyperplanes, slabs, and complementary convex sets.

On a discrete game problem with non-convex control vectograms

In a normed space of finite dimension, a discrete game problem with fixed duration is considered. The terminal set is determined by the condition that the norm of the phase vector belongs to a segment with positive ends. In this paper, a set defined by this condition is called a ring. At each moment, the vectogram of the first player's controls is a certain ring. The controls of the second player at each moment are taken from balls with given radii. The goal of the first player is to lead a phase vector to the terminal set at a fixed time. The goal of the second player is the opposite. In this paper, necessary and sufficient termination conditions are found, and optimal controls of the players are constructed.

Quasiexcellence implies strong generation

We prove that the bounded derived category of coherent sheaves on a quasicompact separated quasiexcellent scheme of finite dimension has a strong generator in the sense of Bondal–Van den Bergh. This simultaneously extends two results of Iyengar–Takahashi and Neeman and is new even in the affine case. The main ingredient includes Gabber’s weak local uniformization theorem and the notions of boundedness and descendability of a morphism of schemes.

Topological and Dynamical Aspects of Jacobi Sigma Models

The geometric properties of sigma models with target space a Jacobi manifold are investigated. In their basic formulation, these are topological field theories—recently introduced by the authors—which share and generalise relevant features of Poisson sigma models, such as gauge invariance under diffeomorphisms and finite dimension of the reduced phase space. After reviewing the main novelties and peculiarities of these models, we perform a detailed analysis of constraints and ensuing gauge symmetries in the Hamiltonian approach. Contact manifolds as well as locally conformal symplectic manifolds are discussed, as main instances of Jacobi manifolds.