We prove strong completeness results for some modal logics with the universal modality, with respect to their topological semantics over 0-dimensional dense-in-themselves metric spaces. We also use failure of compactness to show that, for some languages and spaces, no standard modal deductive system is strongly complete.

Relational program verification is a variant of program verification where one can reason about two programs and as a special case about two executions of a single program on different inputs. Relational program verification can be used for reasoning about a broad range of properties, including equivalence and refinement, and specialized notions such as continuity, information flow security, or relative cost. In a higher-order setting, relational program verification can be achieved using relational refinement type systems, a form of refinement types where assertions have a relational interpretation. Relational refinement type systems excel at relating structurally equivalent terms but provide limited support for relating terms with very different structures. We present a logic, called relational higher-order logic (RHOL), for proving relational properties of a simply typed λ-calculus with inductive types and recursive definitions. RHOL retains the type-directed flavor of relational refinement type systems but achieves greater expressivity through rules which simultaneously reason about the two terms as well as rules which only contemplate one of the two terms. We show that RHOL has strong foundations, by proving an equivalence with higher-order logic, and leverage this equivalence to derive key meta-theoretical properties: subject reduction, admissibility of a transitivity rule, and set-theoretical soundness. Moreover, we define sound embeddings for several existing relational type systems such as relational refinement types and type systems for dependency analysis and relative cost, and we verify examples that were out of reach of prior work.

Akbari and Alipour [1] conjectured that any Latin array of order n with at least n2/2 symbols contains a transversal. For large n, we confirm this conjecture, and moreover, we show that n399/200 symbols suffice.

The Friedgut–Kalai–Naor (FKN) theorem states that if ƒ is a Boolean function on the Boolean cube which is close to degree one, then ƒ is close to a dictator, a function depending on a single coordinate. The author has extended the theorem to the slice, the subset of the Boolean cube consisting of all vectors with fixed Hamming weight. We extend the theorem further, to the multislice, a multicoloured version of the slice.

As an application, we prove a stability version of the edge-isoperimetric inequality for settings of parameters in which the optimal set is a dictator.

In this paper, we consider exponentiated location-scale model and obtain several ordering results between extreme order statistics in various senses. Under majorization type partial order-based conditions, the comparisons are established according to the usual stochastic order, hazard rate order and reversed hazard rate order. Multiple-outlier models are considered. When the number of components are equal, the results are obtained based on the ageing faster order in terms of the hazard rate and likelihood ratio orders. For unequal number of components, we develop comparisons according to the usual stochastic order, hazard rate order, and likelihood ratio order. Numerical examples are considered to illustrate the results.

The hedgehog Ht is a 3-uniform hypergraph on vertices $1, \ldots ,t + \left({\matrix{t \cr 2}}\right)$ such that, for any pair (i, j) with 1 ≤ i < j ≤ t, there exists a unique vertex k > t such that {i, j, k} is an edge. Conlon, Fox and Rödl proved that the two-colour Ramsey number of the hedgehog grows polynomially in the number of its vertices, while the four-colour Ramsey number grows exponentially in the square root of the number of vertices. They asked whether the two-colour Ramsey number of the hedgehog Ht is nearly linear in the number of its vertices. We answer this question affirmatively, proving that r(Ht) = O(t2 ln t).

In engineering design, surrogate models are often used instead of costly computer simulations. Typically, a single surrogate model is selected based on the previous experience. We observe, based on an analysis of the published literature, that fitting an ensemble of surrogates (EoS) based on cross-validation errors is more accurate but requires more computational time. In this paper, we propose a method to build an EoS that is both accurate and less computationally expensive. In the proposed method, the EoS is a weighted average surrogate of response surface models, kriging, and radial basis functions based on overall cross-validation error. We demonstrate that created EoS is accurate than individual surrogates even when fewer data points are used, so computationally efficient with relatively insensitive predictions. We demonstrate the use of an EoS using hot rod rolling as an example. Finally, we include a rule-based template which can be used for other problems with similar requirements, for example, the computational time, required accuracy, and the size of the data.

Suppose there are n players in an ongoing competition, with player i having value vi, and suppose that a game between i and j is won by i with probability vi/(vi + vj). Consider the winner plays competition where in each stage two players play a game, and the winner keeps playing in the next game. We consider two models for choosing its opponent, analyze both models as Markov chains, and determine their stationary probabilities as well as other quantities of interest.

Statistical inference on graphs often proceeds via spectral methods involving low-dimensional embeddings of matrix-valued graph representations such as the graph Laplacian or adjacency matrix. In this paper, we analyze the asymptotic information-theoretic relative performance of Laplacian spectral embedding and adjacency spectral embedding for block assignment recovery in stochastic blockmodel graphs by way of Chernoff information. We investigate the relationship between spectral embedding performance and underlying network structure (e.g., homogeneity, affinity, core-periphery, and (un)balancedness) via a comprehensive treatment of the two-block stochastic blockmodel and the class of K-blockmodels exhibiting homogeneous balanced affinity structure. Our findings support the claim that, for a particular notion of sparsity, loosely speaking, “Laplacian spectral embedding favors relatively sparse graphs, whereas adjacency spectral embedding favors not-too-sparse graphs.” We also provide evidence in support of the claim that “adjacency spectral embedding favors core-periphery network structure.”