A New Approach to the Creation and Propagation of Exponential Moments in the Boltzmann Equation

We study the creation and propagation of exponential moments of solutions to the spatially homogeneous d-dimensional Boltzmann equation. In particular, when the collision kernel is of the form |v ? v _*|^β b(cos (θ)) for β ∈ (0, 2] with cos (θ) = |v ? v _*|^?1(v ? v _*)·σ and σ ∈ 𝕊 ^d?1, and assuming the classical cut-off condition b(cos (θ)) integrable in 𝕊 ^d?1, we prove that there exists a > 0 such that moments with weight exp (amin {t, 1}|v|^β) are finite for t > 0, where a only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients.     

On the Support Problem for Drinfeld Modules

Let Φ be a Drinfeld A-module over an A-field K of generic characteristic. We will prove the following two results which are analogous to ones in number fields. Case 1. Φ is of rank one. Suppose that P and Q are two nontorsion points in Φ(K). If for any element a ? A and almost all prime ideals 𝒫 in  one has that Φ _a (P) ≡ 0 (mod 𝒫) ? Φ _a (Q) ≡ 0 (mod 𝒫), then Q = Φ _m (P) for some m ? A. Case 2. Φ is of general rank ≥ 1. Let x, y ? Φ(K) be two K-rational points. Denote  = End _K (Φ) which is commutative and Λ =  · y which is a cyclic -module. Let red _v :Φ(K) → Φ(k _v ) be the reduction map at a place v of K with residue field k _v . If red _v (x) ? red _v (Λ) for almost all places v of K. Then f(x) = g(y), for some nonzero elements f and g in .     

Orbits of the Actions of Odd-Order Solvable Groups

Suppose that V is a finite faithful irreducible G-module where G is a finite solvable group of odd order. We prove if the action is quasi-primitive, then either F(G) is abelian or G has at least 212 regular orbits on V. As an application, we prove that when V is a finite faithful completely reducible G-module for a solvable group G of odd order, then there exists v ∈ V such that C _G (v) ? F ₂(G) (where F ₂(G) is the 2nd ascending Fitting subgroup of G). We also generalize a result of Espuelas and Navarro. Let G be a group of odd order and let H be a Hall π-subgroup of G. Let V be a faithful G-module over a finite field of characteristic 2, then there exists v ∈ V such that C _H (v) ? O _π(G).     

Principal Functions for Bi-free Central Limit Distributions

We find the principal function of the completely non-normal operator l(v₁) + l(v₁)* + i(r(v₂) + r(v₂)*) on a subspace of the full Fock space \({\mathcal{F}}({\mathcal{H}})\) which arises from a bi-free central limit distribution. As an application, we find the essential spectrum of this operator.     

On Solvability of Finite Groups with Few Non-Normal Subgroups

For a finite group G, let v(G) denote the number of conjugacy classes of non-normal subgroups of G and vc(G) denote the number of conjugacy classes of non-normal noncyclic subgroups of G. In this paper, we show that every finite group G satisfying v(G) ≤2|π(G)| or vc(G) ≤ |π(G)| is solvable, and for a finite nonsolvable group G, v(G) = 2|π(G)| +1 if and only if G ? A ₅.     

Commuting Values of Generalized Derivations on Multilinear Polynomials

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, U the right Utumi quotient ring of R, f(x ₁,…, x _n ) a noncentral multilinear polynomial over K, and G a nonzero generalized derivation of R. Denote f(R) the set of all evaluations of the polynomial f(x ₁,…, x _n ) in R. If [G(u)u, G(v)v] = 0, for any u, v ∈ f(R), we prove that there exists c ∈ U such that G(x) = cx, for all x ∈ R and one of the following holds: 1. f(x ₁,…, x _n )² is central valued on R;

2. R satisfies s ₄, the standard identity of degree 4.

     

Multiple positive solutions of four point boundary value problems for finite difference equations†

In this paper, we consider the following second-order four-point boundary-value problems Δ² u(k ? 1)+f(k,u(k), Δu(k)) = 0,k ∈ {1,2,…,T}, u(0) = au(l ₁), u(T+1) = bu(l ₂). We give conditions on f to ensure the existence of at least three positive solutions of the given problem by applying a new fixed-point theorem of functional type in a cone. The emphasis is put on the nonlinear term involved with the first-order difference.     

Functorial Properties of Star Operations

We define a universal star operation to be an assignment *: A ? * _A of a star operation * _A on A to every integral domain A. Prime examples of universal star operations include the divisorial closure star operation v, the t-closure star operation t, and the star operation w = F _∞ of Hedstrom and Houston. For any universal star operation *, we say that an extension B ? A of integral domains is *-ideal class linked if there is a group homomorphism Cl_{* A} (A) → Cl_{* B} (B) of star class groups induced by the map I ? (IB)^{* _B} on the set of * _A -ideals I of A. We study several natural subclasses of the class of *-ideal class linked extensions.     

PrÜfer-Like Domains and the Nagata Ring of Integral Domains

A subring A of a Prüfer domain B is a globalized pseudo-valuation domain (GPVD) if (i) A?B is a unibranched extension and (ii) there exists a nonzero radical ideal I, common to A and B such that each prime ideal of A (resp., B) containing I is maximal in A (resp., B). Let D be an integral domain, X be an indeterminate over D, c(f) be the ideal of D generated by the coefficients of a polynomial f ∈ D[X], N = {f ∈ D[X] | c(f) = D}, and N _v  = {f ∈ D[X] | c(f)^?1 = D}. In this article, we study when the Nagata ring D[X] _N (more generally, D[X] _{N v} ) is a GPVD. To do this, we first use the so-called t-operation to introduce the notion of t-globalized pseudo-valuation domains (t-GPVDs). We then prove that D[X] _{N v} is a GPVD if and only if D is a t-GPVD and D[X] _{N v} has Prüfer integral closure, if and only if D[X] is a t-GPVD, if and only if each overring of D[X] _{N v} is a GPVD. As a corollary, we have that D[X] _N is a GPVD if and only if D is a GPVD and D has Prüfer integral closure. We also give several examples of integral domains D such that D[X] _{N v} is a GPVD.     

When the v-Operation Distributes over Intersections

Let D be an integral domain. We investigate when (∩ A_α) ?¹ = ∑ A_α ?¹ or (∩ A_α) ?¹ =(∑ A_α ?¹)_v (equivalently, (∩ A _α) _v  = ∩(A _α) _v ) for certain families {A _α} of nonzero fractional ideals of D.     

Further results on the existence of nested orthogonal arrays

Nested orthogonal arrays provide an option for designing an experimental setup consisting of two experiments, the expensive one of higher accuracy being nested in a larger and relatively less expensive one of lower accuracy. We denote by OA_(λ, μ)(t, k, (v, w)) (or OA(t, k, (v, w)) if λ = μ = 1) a (symmetric) orthogonal array OA _λ (t, k, v) with a nested OA _μ (t, k, w) (as a subarray). It is proved in this article that an OA(t, t + 1,(v, w)) exists if and only if v ≥ 2w for any positive integers v, w and any strength t ≥ 2. Some constructions of OA_(λ, μ)(t, k, (v, w))′s with λ ≠ μ and k ? t > 1 are also presented.     

Scaling Limits for Some P.D.E. Systems with Random Initial Conditions

Let X = {X(x, t), x ? R ⁿ , t ? R ₊} be the R ²-valued spatial-temporal random field X = (u, v) arising from a certain two-equation system of parabolic linear partial differential equations with a given random initial condition X ₀ = (u ₀, v ₀). We discuss the scaling limit of X under suitable conditions on X ₀. Since the component fields u, v are dependent, even when the initial data u ₀, v ₀ are independent, the scaling limit is not readily reduced to the known single equation case. The correlated structure of random vector (u(x, t), v(x′, t′)) and the Hermite expansion associated with (u ₀, v ₀) play the essential roles in our study. The work shows, in particular, the non-Gaussian scenario proposed by Anh and Leonenko [2 Anh , V.V. , and Leonenko , N.N. 1999 . Non-Gaussian scenarios for the heat equation with singular initial data . Stochastic Process. Appl. 84 : 91 – 114 .[Crossref], [Web of Science ®] , [Google Scholar]] for the single heat equation can be discussed for the two-equation system, in a significant way.     

Numerical Semigroup Rings and Almost Prüfer v-Multiplication Domains

Let D be an integral domain with quotient field K, X be an indeterminate over D, Γ be a numerical semigroup with Γ ? ?₀, D[Γ] be the semigroup ring of Γ over D (and hence D ? D[Γ] ? D[X]), and D + X ⁿ K[X] = {a + X ⁿ g∣a ∈ D and g ∈ K[X]}. We show that there exists an order-preserving bijection between Spec(D[X]) and Spec(D[Γ]), which also preserves t-ideals. We also prove that D[Γ] is an APvMD (resp., AGCD-domain) if and only if D[X] is an APvMD (resp., AGCD-domain) and char(D) ≠ 0. We show that if n ≥ 2, then D is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain) and char(D) ≠ 0 if and only if D + X ⁿ K[X] is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain). Finally, we give some examples of APvMDs which are not AGCD-domains by using the constructions D[Γ] and D + X ⁿ K[X].     

Vertex coloring complete multipartite graphs from random lists of size 2

Let K_s×m be the complete multipartite graph with s parts and m vertices in each part. Assign to each vertex v of K_s×m a list L(v) of colors, by choosing each list uniformly at random from all 2-subsets of a color set C of size σ(m). In this paper we determine, for all fixed s and growing m, the asymptotic probability of the existence of a proper coloring φ, such that φ(v)∈L(v) for all v∈V(K_s×m). We show that this property exhibits a sharp threshold at σ(m)=2(s−1)m.     

Fitting Height of a Finite Group with a Metabelian Group of Automorphisms

Let M = FH be a finite group that is a product of a normal abelian subgroup F and an abelian subgroup H. Assume that all elements in M?F have prime order p, and F has at most one subgroup of order p. Examples of such groups are dihedral groups for p = 2 and the semidirect product of a cyclic group F by a group H of prime order p such that C _F (H) = 1 or |C _F (H)| =p and C _{F/C F (H)}(H) = 1. Suppose that M acts on a finite group G in such a manner that C _G (F) = 1. We prove that the Fitting height h(G) of G is at most h(C _G (H))+ 1. Moreover, the Fitting series of C _G (H) coincides with the intersection of C _G (H) with the Fitting series of G.     

Lower Bounds on the Blow-Up Rate of the Axisymmetric Navier–Stokes Equations II

Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in ?³ with non-trivial swirl. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies, for some 0 ≤ ε ≤ 1, |v (x, t)| ≤ C _? r ^?1+ε |t|^?ε/2 for ? T ₀ ≤ t < 0 and 0 < C _? < ∞ allowed to be large. We prove that v is regular at time zero.     

On the Annihilator Graph of a Commutative Ring

Let R be a commutative ring with nonzero identity, Z(R) be its set of zero-divisors, and if a ∈ Z(R), then let ann _R (a) = {d ∈ R | da = 0}. The annihilator graph of R is the (undirected) graph AG(R) with vertices Z(R)* = Z(R)?{0}, and two distinct vertices x and y are adjacent if and only if ann _R (xy) ≠ ann _R (x) ∪ ann _R (y). It follows that each edge (path) of the zero-divisor graph Γ(R) is an edge (path) of AG(R). In this article, we study the graph AG(R). For a commutative ring R, we show that AG(R) is connected with diameter at most two and with girth at most four provided that AG(R) has a cycle. Among other things, for a reduced commutative ring R, we show that the annihilator graph AG(R) is identical to the zero-divisor graph Γ(R) if and only if R has exactly two minimal prime ideals.     

A lower bound on the independence number of arbitrary hypergraphs

We present a lower bound on the independence number of arbitrary hypergraphs in terms of the degree vectors. The degree vector of a vertex v is given by d(v) = (d₁(v), d₂(v), …) where d_m(v) is the number of edges of size m containing v. We define a function f with the property that any hypergraph H = (V, E) satisfies α(H) ≥ Σ_v∈V f(d(v)). This lower bound is sharp when H is a match, and it generalizes known bounds of Caro/Wei and Caro/Tuza for ordinary graphs and uniform hypergraphs. Furthermore, an algorithm for computing independent sets of size as guaranteed by the lower bound is given. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 213–221, 1999     

Asymptotics for Solutions of Elliptic Equations in Double Divergence Form

We consider weak solutions of an elliptic equation of the form ? _i ? _i (a _ij u) = 0 and their asymptotic properties at an interior point. We assume that the coefficients are bounded, measurable, complex-valued functions that stabilize as x → 0 in that the norm of the matrix (a _ij (x) ? δ _ij ) on the annulus B _2r \ B _r is bounded by a function Ω(r), where Ω²(r) satisfies the Dini condition at r = 0, as well as some technical monotonicity conditions; under these assumptions, solutions need not be continuous. Our main result is an explicit formula for the leading asymptotic term for solutions with at most a mild singularity at x = 0. As a consequence, we obtain upper and lower estimates for the L ^p -norm of solutions, as well as necessary and sufficient conditions for solutions to be bounded or tend to zero in L ^p -mean as r → 0.