Finitely presented algebras and groups defined by permutation relations

The class of finitely presented algebras over a field K with a set of generators a₁,…,a_n and defined by homogeneous relations of the form a₁a₂?a_n=a_σ(1)a_σ(2)?a_σ(n), where σ runs through a subset H of the symmetric group Sym_n of degree n, is introduced. The emphasis is on the case of a cyclic subgroup H of Sym_n of order n. A normal form of elements of the algebra is obtained. It is shown that the underlying monoid, defined by the same (monoid) presentation, has a group of fractions and this group is described. Properties of the algebra are derived. In particular, it follows that the algebra is a semiprimitive domain. Problems concerning the groups and algebras defined by arbitrary subgroups H of Sym_n are proposed.     

Nested matrices and the existence of least majorized elements

We characterize the matrices A for which X(b)={x∣x∈Rⁿ, x?0, Ax?b, σⁿ_i=1x_i=1} contains a least majorized element for all vectors b satisfying X(b)≠?.     

Partitions of natural numbers with the same representation functions

For a set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A₀ be the set of all nonnegative integers a with even D(a) and A₁ be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R₂(A,n)=R₂(N?A,n) for all n?2N−1, then R₂(A,n)=R₂(N?A,n)?1 for all n?12N²−10N−2 except for A=A₀ or A=A₁; (b) if R₃(A,n)=R₃(N?A,n) for all n?2N−1, then R₃(A,n)=R₃(N?A,n)?1 for all n?12N²+2N. Several problems are posed in this paper.     

The critical exponents for the quasi-linear parabolic equations with inhomogeneous terms

This paper deals with the critical exponents for the quasi-linear parabolic equations in Rn and with an inhomogeneous source, or in exterior domains and with inhomogeneous boundary conditions. For n?3, σ>−2/n and p>max{1,1+σ}, we obtain that pc=n(1+σ)/(n−2) is the critical exponent of these equations. Furthermore, we prove that if max{1,1+σ}<p?pc, then every positive solution of these equations blows up in finite time; whereas these equations admit the global positive solutions for some f(x) and some initial data u₀(x) if p>pc. Meantime, we also demonstrate that every positive solution of these equations blows up in finite time provided n=1,2, σ>−1 and p>max{1,1+σ}.     

Homothetic interval orders

We give a characterization of the non-empty binary relations ? on a N^*-set A such that there exist two morphisms of N^*-sets u₁,u₂:A→R₊ verifying u₁?u₂ and x?y⇔u₁(x)>u₂(y). They are called homothetic interval orders. If ? is a homothetic interval order, we also give a representation of ? in terms of one morphism of N^*-sets u:A→R₊ and a map such that x?y⇔σ(x,y)u(x)>u(y). The pairs (u₁,u₂) and (u,σ) are “uniquely” determined by ?, which allows us to recover one from each other. We prove that ? is a semiorder (resp. a weak order) if and only if σ is a constant map (resp. σ=1). If moreover A is endowed with a structure of commutative semigroup, we give a characterization of the homothetic interval orders ? represented by a pair (u,σ) so that u is a morphism of semigroups.     

Divisibility properties of power GCD matrices and power LCM matrices

Let a,b and n be positive integers and the set S={x₁,…,x_n} of n distinct positive integers be a divisor chain (i.e. there exists a permutation σ on {1,…,n} such that x_σ(1)|…|x_σ(n)). In this paper, we show that if a|b, then the ath power GCD matrix (S^a) having the ath power (x_i,x_j)^a of the greatest common divisor of x_i and x_j as its i,j-entry divides the bth power GCD matrix (S^b) in the ring M_n(Z) of n×n matrices over integers. We show also that if a?b and n?2, then the ath power GCD matrix (S^a) does not divide the bth power GCD matrix (S^b) in the ring M_n(Z). Similar results are also established for the power LCM matrices.     

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u_t-?u=|u|^p-1u in R^N. The initial data is of the form u₀=λ?, where ?∈C₀(R^N) is fixed and λ>0. We first take 1<p<p_f, where p_f is the Fujita critical exponent, and ?∈C₀(R^N)∩L¹(R^N) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<p_s, where p_s is the Sobolev critical exponent, and ?(x) decaying as |x|^-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial.     

A Tauberian theorem related to Weyl's criterion

Let {x_n} be a sequence of real numbers and let a(n) be a sequence of positive real numbers, with A(N) = Σ_n=1^Na(n). Tsuji has defined a notion of a(n)-uniform distribution mod 1 which is related to the problem of determining those real numbers t₀ for which A(N)^?1 Σ_n=1^Na(n)e^?it₀x_n → 0 as N → ∞. In case f(s) = Σ_n=1^∞a(n)e^?sx_n, s = σ + it, is analytic in the right half-plane 0 < σ, and satisfies a certain smoothness condition as σ → 0 +, we show that f(σ)^?1f(σ + it₀) → 0 as σ → 0 + if and only if A(N)^?1 Σ_n=1^Na(n)e^?it₀x_n → 0 as N → ∞.     

A new class of multiset Wilf equivalent pairs

We say the pair of patterns (σ,τ) is multiset Wilf equivalent if, for any multiset M, the number of permutations of M that avoid σ is equal to the number of permutations of M that avoid τ. In this paper, we find a large new class of multiset Wilf equivalent pairs, namely, the pair (σ_n-2(n-1)n, σ_n-2n(n-1)), for n?3 and σ_n-2 a permutation of {1^x₁,2^x₂,…,(n-2)^x_n-2}. It is the most general multiset Wilf equivalence result to date.     

Differential commutator identities

I.N. Herstein proved that if R is a prime ring satisfying a differential identity , with d a nonzero derivation of R, then R embeds isomorphically in M₂(F) for F a field. We consider a natural generalization of this result for the class of polynomials E_n(X)=[E_n-1(x₁,…,x_n-1),x_n]. Using matrix computations, we prove that if R satisfies a differential identity , or with some restrictions, then R must embed in M₂(F), but that differential identities using [[E_n,E_m],E_s] with m,n,s>1 need not force R to embed in M₂(F). These results hold if the expressions are identities for a noncommutative Lie ideal of R, rather than for R itself.     

Morphic rings and unit regular rings

A ring R is called left morphic if for every a∈R. A left and right morphic ring is called a morphic ring. If M_n(R) is morphic for all n≥1 then R is called a strongly morphic ring. A well-known result of Erlich says that a ring R is unit regular iff it is both (von Neumann) regular and left morphic. A new connection between morphic rings and unit regular rings is proved here: a ring R is unit regular iff R[x]/(xⁿ) is strongly morphic for all n≥1 iff R[x]/(x²) is morphic. Various new families of left morphic or strongly morphic rings are constructed as extensions of unit regular rings and of principal ideal domains. This places some known examples in a broader context and answers some existing questions.     

Generalized weakly symmetric rings

A ring R is defined to be GWS   if abc=0$abc=0$ implies bac⊆N(R)$bac\subseteq N(R)$ for a,b,c∈R$a,b,c\in R$, where N(R)$N(R)$ stands for the set of nilpotent elements of R. Since reduced rings and central symmetric rings are GWS, we study sufficient conditions for GWS rings to be reduced and central symmetric. We prove that a ring R is GWS   if and only if the n×n$n\times n$ upper triangular matrices ring _Un(R,R)$U_n(R,R)$ is GWS for any positive integer n. It is proven that GWS rings are directly finite and left min-abel. For a GWS ring R, R is a strongly regular ring if and only if R is a von Neumann regular ring if and only if R is a left SF   ring and J(R)=0$J(R)=0$; R is an exchange ring if and only if R is a clean ring. Finally, we show that GWS exchange rings have stable range 1 and a GWS semiperiodic ring R   with N(R)≠J(R)$N(R)\ne J(R)$ is commutative.     

Weighted pseudo almost periodic mild solutions of semilinear evolution equations with nonlocal conditions

In this paper, applying the theory of semigroups of operators to evolution families and Banach fixed point theorem, we prove the existence and uniqueness of the weighted pseudo almost periodic mild solution of the semilinear evolution equation x^′(t)=A(t)x(t)+f(t,x(t)) with nonlocal conditions x(0)=x₀+g(x) in Banach space X under some suitable hypotheses.     

Some stably tame polynomial automorphisms

We study the structure of length three polynomial automorphisms of R[X,Y] when R is a UFD. These results are used to prove that if SL_m(R[X₁,X₂,…,X_n])=E_m(R[X₁,X₂,…,X_n]) for all n≥0 and for all m≥3 then all length three polynomial automorphisms of R[X,Y] are stably tame.     

Gamma series associated to elements satisfying regularized double shuffle relations

Let K be a field of characteristic 0, and R be a commutative K-algebra. Let Φ(x₀,x₁) be an element in R《x₀,x₁》 with regularized double shuffle relations. We define a gamma series ΓΦ(s)∈1+s²R?s? associated to Φ. We prove that the associated beta series is just the image of ΦY(x₀,x₁) in the commutative formal power series ring R?x₀,x₁?, where if Φ=1+Φ₀x₀+Φ₁x₁, then ΦY=1+Φ₁x₁. We also give some equivalent conditions for the reflection formula of the gamma series ΓΦ(s).     

On the diameters of commuting graphs

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all non-central elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let D be a division ring and n ? 3. In this paper we investigate the diameters of Γ(M_n(D)) and determine the diameters of some induced subgraphs of Γ(M_n(D)), such as the induced subgraphs on the set of all non-scalar non-invertible, nilpotent, idempotent, and involution matrices in M_n(D). For every field F, it is shown that if Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) ? 6. We conjecture that if Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) ? 5. We show that if F is an algebraically closed field or n is a prime number and Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) = 4. Finally, we present some applications to the structure of pairs of idempotents which may prove of independent interest.     

Some families of strongly clean rings

A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute with each other. For a commutative local ring R and for an arbitrary integer n?2, the paper deals with the question whether the strongly clean property of M_n(R[[x]]), , and M_n(RC₂) follows from the strongly clean property of M_n(R). This is ‘Yes’ if n=2 by a known result.     

Lipschitz lower sigma-exponents of linear differential systems

For the lower sigma-exponent of the linear differential system ? = A(t)x, x ∈ R ⁿ , t ≥ 0, defined by the formula Δ_σ(A) ≡ inf_λ[Q]≤-σ λ ₁(A + Q), σ > 0, on the basis of the lower characteristic exponents λ ₁(A+Q) of perturbed linear systems with Lyapunov exponents λ[Q] ≤ ?σ < 0 of perturbations Q, we prove the following general form as a function of the parameter σ > 0. For any nondecreasing bounded function f(σ) of the parameter σ ∈ (0,+∞) that coincides with a constant on some infinite interval (σ ₀,+∞), σ ₀ ≥ 0, and satisfies the Lipschitz condition on the complementary interval (0, σ ₀], we prove the existence of a linear system with coefficient matrix A _f (t) bounded on the half-line [0,+∞) whose lower sigma-exponent Δ_σ(A _f ) coincides with the function f(σ) on the entire interval (0,+∞).     

Four applications of majorization to convexity in the calculus of variations

The resemblance between the Horn-Thompson theorem and a recent theorem by Dacorogna-Marcellini-Tanteri indicates that Schur-convexity and the majorization relation are relevant for applications in the calculus of variations and its related notions of convexity, such as rank one convexity or quasiconvexity. In Theorem 6.6, we give simple necessary and sufficient conditions for an isotropic objective function to be rank one convex on the set of matrices with positive determinant.Majorization is used in order to give a very short proof of a theorem of Thompson and Freede [R.C. Thompson, L.J. Freede, Eigenvalues of sums of Hermitian matrices III, J. Res. Nat. Bur. Standards B 75B (1971) 115-120], Ball [J.M. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics, in: R.J. Knops (Ed.), Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. 1, Res. Notes Math., 17, Pitman, 1977, pp. 187-241], or Le Dret [H. Le Dret, Sur les fonctions de matrices convexes et isotropes, CR Acad. Sci. Paris, Série I 310 (1990) 617-620], concerning the convexity of a class of isotropic functions which appear in nonlinear elasticity.Next we prove (Theorem 7.3) a lower semicontinuity result for functionals with the form _∫Ωw(D?(x))dx, with w(F)=h(lnV_F). Here F=R_FU_F=V_FR_F is the usual polar decomposition of F∈gl(n,R), and lnV_F is Hencky’s logarithmic strain.We close this paper with a compact proof of Dacorogna-Marcellini-Tanteri theorem, based only on classical results about majorization. The mentioned resemblance of this theorem with the Horn-Thompson theorem is thus explained.