Logarithmic diameter bounds for some Cayley graphs

Let S ⊂ GL n ⁢ ( Z ) S\subset\mathrm{GL}_{n}(\mathbb{Z}) be a finite symmetric set. We show that if the Zariski closure of Γ = ⟨ S ⟩ \Gamma=\langle S\rangle is a product of special linear groups or a special affine linear group, then the diameter of the Cayley graph Cay ⁡ ( Γ / Γ ⁢ ( q ) , π q ⁢ ( S ) ) \operatorname{Cay}(\Gamma/\Gamma(q),\pi_{q}(S)) is O ⁢ ( log ⁡ q ) O(\log q) , where 𝑞 is an arbitrary positive integer, π q : Γ → Γ / Γ ⁢ ( q ) \pi_{q}\colon\Gamma\to\Gamma/\Gamma(q) is the canonical projection induced by the reduction modulo 𝑞, and the implied constant depends only on 𝑆.

Origin of the Turbulence Structure in Wall-Bounded Flows, and Implications toward Computability

Coordinate-transformed analysis of turbulence transport is developed, which leads to a symmetric set of gradient expressions for the Reynolds stress tensor components. In this perspective, the turbulence structure in wall-bounded flows is seen to arise from an interaction of a small number of intuitive dynamical terms: transport, pressure and viscous. Main features of the turbulent flow can be theoretically prescribed in this way and reconstructed for channel and boundary layer flows, with and without pressure gradients, as validated in comparison with available direct numerical simulation data. A succinct picture of turbulence structure and its origins emerges, reflective of the basic physics of momentum and energy balance if placed in a specific moving coordinate frame. An iterative algorithm produces an approximate solution for the mean velocity, and its implications toward computability of turbulent flows is discussed.

Classification of Vertex-Transitive Zonotopes

We give a full classification of vertex-transitive zonotopes. We prove that a vertex-transitive zonotope is a $$\Gamma $$ Γ -permutahedron for some finite reflection group $$\Gamma \subset {{\,\mathrm{O}\,}}(\mathbb {R}^d)$$ Γ ⊂ O ( R d ) . The same holds true for zonotopes in which all vertices are on a common sphere, and all edges are of the same length. The classification of these then follows from the classification of finite reflection groups. We prove that root systems can be characterized as those centrally symmetric sets of vectors, for which all intersections with half-spaces, that contain exactly half the vectors, are congruent. We provide a further sufficient condition for a centrally symmetric set to be a root system.

On Arithmetic Progressions in Symmetric Sets in Finite Field Model

We consider two problems regarding arithmetic progressions in symmetric sets in the finite field (product space) model. First, we show that a symmetric set $S \subseteq \mathbb{Z}_q^n$ containing $|S| = \mu \cdot q^n$ elements must contain at least $\delta(q, \mu) \cdot q^n \cdot 2^n$ arithmetic progressions $x, x+d, \ldots, x+(q-1)\cdot d$ such that the difference $d$ is restricted to lie in $\{0,1\}^n$. Second, we show that for prime $p$ a symmetric set $S\subseteq\mathbb{F}_p^n$ with $|S|=\mu\cdot p^n$ elements contains at least $\mu^{C(p)}\cdot p^{2n}$ arithmetic progressions of length $p$. This establishes that the qualitative behavior of longer arithmetic progressions in symmetric sets is the same as for progressions of length three.

Painlevé–Kuratowski Convergence of Solutions for Perturbed Symmetric Set-Valued Quasi-Equilibrium Problem via Improvement Sets

This paper is devoted to study the Painlevé–Kuratowski convergence of solution sets for perturbed symmetric set-valued quasi-equilibrium problems (SSQEP)[Formula: see text] via improvement sets. By virtue of the oriented distance function, the sufficient conditions of Painlevé–Kuratowski convergence of efficient solution sets for (SSQEP)[Formula: see text] are obtained through a new nonlinear scalarization technical. Then, under [Formula: see text]-convergence of set-valued mappings, the Painlevé–Kuratowski convergence of weak efficient solution sets for (SSQEP)[Formula: see text] is discussed. What’s more, with suitable convergence assumptions, we also establish the sufficient conditions of lower Painlevé–Kuratowski convergence of Borwein proper efficient solution sets for (SSQEP)[Formula: see text] under improvement sets. Some interesting examples are formulated to illustrate the significance of the main results.

Symmetric Fuzzy Stochastic Differential Equations with Generalized Global Lipschitz Condition

The paper contains a discussion on solutions to symmetric type of fuzzy stochastic differential equations. The symmetric equations under study have drift and diffusion terms symmetrically on both sides of equations. We claim that such symmetric equations have unique solutions in the case that equations’ coefficients satisfy a certain generalized Lipschitz condition. To show this, we prove that an approximation sequence converges to the solution. Then, a study on stability of solution is given. Some inferences for symmetric set-valued stochastic differential equations end the paper.

Approach to classify, separate, and enrich objects in groups using ensemble sorting

The sorting of objects into groups is a fundamental operation, critical in the preparation and purification of populations of cells, crystals, beads, or droplets, necessary for research and applications in biology, chemistry, and materials science. Most of the efforts exploring such purification have focused on two areas: the degree of separation and the measurement precision required for effective separation. Conventionally, achieving good separation ultimately requires that the objects are considered one by one (which can be both slow and expensive), and the ability to measure the sorted objects by increasing sensitivity as well as reducing sorting errors. Here we present an approach to sorting that addresses both critical limitations with a scheme that allows us to approach the theoretical limit for the accuracy of sorting decisions. Rather than sorting individual objects, we sort the objects in ensembles, via a set of registers which are then in turn sorted themselves into a second symmetric set of registers in a lossless manner. By repeating this process, we can arrive at high sorting purity with a low set of constraints. We demonstrate both the theory behind this idea and identify the critical parameters (ensemble population and sorting time), and show the utility and robustness of our method with simulations and experimental systems spanning several orders of scale, sorting populations of macroscopic beads and microfluidic droplets. Our method is general in nature and simplifies the sorting process, and thus stands to enhance many different areas of science, such as purification, enrichment of rare objects, and separation of dynamic populations.

An Identity in Commutative Rings with Unity with Applications to Various Sums of Powers

Let R=(R,+,·) be a commutative ring of characteristic m>0 (m may be equal to +∞) with unity e and zero 0. Given a positive integer n<m and the so-called n-symmetric set A=a1,a2,…,a2l-1,a2l such that al+i=ne-ai for each i=1,…,l, define the rth power sum Sr(A) as Sr(A)=∑i=12lair, for r=0,1,2,…. We prove that for each positive integer k there holds ∑i=02k-1(-1)i2k-1i22k-1-iniS2k-1-i(A)=0. As an application, we obtain two new Pascal-like identities for the sums of powers of the first n-1 positive integers.