Ascending chain condition for -pure thresholds on a fixed strongly -regular germ

In this paper, we prove that the set of all $F$-pure thresholds on a fixed germ of a strongly $F$-regular pair satisfies the ascending chain condition. As a corollary, we verify the ascending chain condition for the set of all $F$-pure thresholds on smooth varieties or, more generally, on varieties with tame quotient singularities, which is an affirmative answer to a conjecture given by Blickle, Mustaţǎ and Smith.

Commuting Regularity degree of finite semigroups

A pair (x, y) of elements x and y of a semigroup S is said to be a commuting regular pair, if there exists an element z ∈ S such that xy = (yx)z(yx). In a finite semigroup S, the probability that the pair (x, y) of elements of S is commuting regular will be denoted by dcr(S) and will be called the Commuting Regularity degree of S. Obviously if S is a group, then dcr(S) = 1. However for a semigroup S, getting an upper bound for dcr(S) will be of interest to study and to identify the different types of non-commutative semigroups. In this paper, we calculate this probability for certain classes of finite semigroups. In this study we managed to present an interesting class of semigroups where the probability is 1/2. This helps us to estimate a condition on non-commutative semigroups such that the commuting regularity of (x, y) yields the commuting regularity of (y, x).

THE VLASOV CONTINUUM LIMIT FOR THE CLASSICAL MICROCANONICAL ENSEMBLE

For classical Hamiltonian N-body systems with mildly regular pair interaction potential (in particular, [Formula: see text] integrability is required), it is shown that when N → ∞ in a fixed bounded domain Λ ⊂ ℝ3, with energy [Formula: see text] scaling as [Formula: see text], then Boltzmann's ergodic ensemble entropy [Formula: see text] has the asymptotic expansion SΛ(N,N2ε) = - N ln N + sΛ(ε) N + o(N). Here, the N ln N term is combinatorial in origin and independent of the rescaled Hamiltonian, while sΛ(ε) is the system-specific Boltzmann entropy per particle, i.e. –sΛ(ε) is the minimum of Boltzmann's H function for a perfect gas of energy ε subjected to a combination of externally and self-generated fields. It is also shown that any limit point of the n-point marginal ensemble measures is a linear convex superposition of n-fold products of the H-function-minimizing one-point functions. The proofs are direct, in the sense that (a) the map [Formula: see text] is studied rather than its inverse [Formula: see text]; (b) no regularization of the microcanonical measure [Formula: see text] is invoked, and (c) no detour via the canonical ensemble. The proofs hold irrespective of whether microcanonical and canonical ensembles are equivalent or not.

Matrix-Free Proof of a Regularity Characterization

The central concept in Szemerédi's powerful regularity lemma is the so-called $\epsilon$-regular pair. A useful statement of Alon et al. essentially equates the notion of an $\epsilon$-regular pair with degree uniformity of vertices and pairs of vertices. The known proof of this characterization uses a clever matrix argument. This paper gives a simple proof of the characterization without appealing to the matrix argument of Alon et al. We show the $\epsilon$-regular characterization follows from an application of Szemerédi's regularity lemma itself.

Holes in Graphs

The celebrated Regularity Lemma of Szemerédi asserts that every sufficiently large graph $G$ can be partitioned in such a way that most pairs of the partition sets span $\epsilon$-regular subgraphs. In applications, however, the graph $G$ has to be dense and the partition sets are typically very small. If only one $\epsilon$-regular pair is needed, a much bigger one can be found, even if the original graph is sparse. In this paper we show that every graph with density $d$ contains a large, relatively dense $\epsilon$-regular pair. We mainly focus on a related concept of an $(\epsilon,\sigma)$-dense pair, for which our bound is, up to a constant, best possible.

New neurosurgical operating glasses and fiberoptic headlight

✓ A new type of neurosurgical operating glasses is described, which can be mounted on a regular pair of glasses and give high magnification (× 4.4). Good illumination is provided by means of a newly designed fiberoptic headlight.