Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

In this paper, we show that if the Laplacian and gradient of the warping function of a compact warped product submanifold Ω p + q in the hyperbolic space ℍ m − 1 satisfy various extrinsic restrictions, then Ω p + q has no stable integral currents, and its homology groups are trivial. Also, we prove that the fundamental group π 1 Ω p + q is trivial. The restrictions are also extended to the eigenvalues of the warped function, the integral Ricci curvature, and the Hessian tensor. The results obtained in the present paper can be considered as generalizations of the Fu–Xu theorem in the framework of the compact warped product submanifold which has the minimal base manifold in the corresponding ambient manifolds.

Generalized bases of finite groups

Motivated by recent results on the minimal base of a permutation group, we introduce a new local invariant attached to arbitrary finite groups. More precisely, a subset $$\Delta $$ Δ of a finite group G is called a p-base (where p is a prime) if $$\langle \Delta \rangle $$ ⟨ Δ ⟩ is a p-group and $$\mathrm {C}_G(\Delta )$$ C G ( Δ ) is p-nilpotent. Building on results of Halasi–Maróti, we prove that p-solvable groups possess p-bases of size 3 for every prime p. For other prominent groups, we exhibit p-bases of size 2. In fact, we conjecture the existence of p-bases of size 2 for every finite group. Finally, the notion of p-bases is generalized to blocks and fusion systems.

Random bases for coprime linear groups

The minimal base size {b(G)} for a permutation group G is a widely studied topic in permutation group theory. Z. Halasi and K. Podoski [Every coprime linear group admits a base of size two, Trans. Amer. Math. Soc. 368 2016, 8, 5857–5887] proved that {b(G)\leq 2} for coprime linear groups. Motivated by this result and the probabilistic method used by T. Burness, M. W. Liebeck and A. Shalev, it was asked by L. Pyber [Personal communication, Bielefeld, 2017] whether or not, for coprime linear groups {G\leq GL(V)}, there exists a constant c such that the probability that a random c-tuple is a base for G tends to 1 as {\lvert V\rvert\to\infty}. While the answer to this question is negative in general, it is positive under the additional assumption that G is primitive as a linear group. In this paper, we show that almost all 11-tuples are bases for coprime primitive linear groups.

Minimal bases and minimal sub-bases for topological spaces

This paper investigates minimal bases and minimal sub-bases for topological spaces. First, a necessary and sufficient condition is derived for the existence of minimal base for a general topological space. Then the concept of minimal sub-base for a topological space is proposed and its properties are discussed. Finally, for Alexandroff spaces, some special results with respect to minimal bases and minimal sub-bases are illustrated.

Numerical and Experimental Studies on Extrusion of Fan-Shaped Shell of Magnesium Alloy

This paper presents a case study of optimizing the forming process for a fan-shaped shell component. Numerical simulation was used to study the backward extrusion process of a fan-shaped shell. The underfill defect produced at the opening of the extruded shell due to the billet shape was solved and the minimal base thickness required to avoid the presence of the underfill defect at the bottom corner of the component was defined through the numerical simulation. The extrusion drawing and forming process of the fan-shaped shell were designed on the basis of the results of the numerical simulation. Forming experiments had been performed on the fan-shaped shell at 380 °C and cracking was found on the outside wall in the center of the extruded shell. Choked groove on the inner wall of the die and reducing the lubrication had been used to avoid the presence of cracking. The fan-shaped shell of AZ31 magnesium alloy has been successfully formed by the three-stage forming process of hot upsetting, hot backward extrusion and cold sizing.

On the base size for the symmetric group acting on subsets

Let k, n be natural numbers with k ≦ n/2 and let Xn,k denote the set of k-element subsets of {1, 2, … n}. The symmetric group Sn acts in a natural way on the set Xn,k. Motivated by a question of Robert Guralnick, we investigate the size of a minimal base for this action. We give constructions providing a minimal base if n = 2k or if n ≧ k2. We also describe a general process providing a base of size at most c times bigger than the size of a minimal base for some universal constant c

GRANULAR OPEN FILTERS ON A HORIZONTAL BED UNDER WAVE AND CURRENT LOADING

Rubble mound coastal structures typically contain granular filters in one or more layers. These filters are normally geometrically tight (to prevent material washout), often difficult to realize in the field, and expensive. An alternative is a geometrically open filter (i.e. a large ratio of the size of toplayer material and underlayer material), designed in such a way that it fulfills its filter functions with only minimal base material loss or settlement. Potential applications of open filters include bed protections and toe & slope configurations of coastal structures. Proper guidelines on the design of open filters under wave and current loading could lead to significant cost and material savings, and to a more practical application of filters in the field. The physical model tests conducted in this study focus on granular open filters on a horizontal sand bed under wave and combined wave & current loading.