Distortion Properties of Holomorphic Functions of Several Complex Variables

Let Q(D) be a class of functions q, q(0) = 0, |q(z)| < 1 holomorphic in the Reinhardt domain D ? C ⁿ, a and b — arbitrary fixed numbers satisfying the condition — 1 ≤ b < a ≤ 1. ??(a, b; D) — the class of functions p such that p ? ??(a, b; D) iff for some q ? Q(D) and every z ? D. S*(a, b; D) — the class of functions f such that f ? S*(a, g; D) iff S^c(a, b; D) — the class of functions q such that q ? S^c(a, b; D) iff , where p ε ??(a, b; D) and K is an operator of the form for z=z₁,z₂,…z_n. The author obtains sharp bounds on |p(z)|, f(z)| g(z)| as well as sharp coefficient inequalities for functions in ??(a, b; D), S*(a, b; D) and S^c(a, b; D).     

ON K 2 OF DIVISION ALGEBRAS

ABSTRACT

In this paper, it is proved that if F is a global field, then for any integer n > 3, there is an extension field E over F of degree n such that K ₂ E is not generated by the Steinberg symbols {a, b} with a ∈ F*, b, ∈ E*. If however, F is a number field and D is a finite-dimensional central division F-algebra with square free index, then K ₂ D is always generated by the Steinberg symbols {a, b} with a ∈ F*, b ∈ D*. Finally, the tame kernels of central division algebras over F are expressed explicitly.     

Extraction of smooth solutions of a class of regular equations in a rectangle

This work is devoted to the study of two-dimensional, regular, almost hypoelliptic operators P(D) = P(D ₂, D ₂) with regular Newton polyhedrons. It is proved that all generalized (weak) solutions of the equation P(D)u = f from a several weighted Sobolev space are infinitely differentiable functions in the rectangle {x ∈ E ²: −a < x ₁ < a, −b < x ₂ < b} in the variable x ₂, in which the function f is infinitely differentiable.     

Cauchy’s residue theorem for a class of real valued functions

Let [a, b] be an interval in ℝ Rand let F be a real valued function defined at the endpoints of [a, b] and with a certain number of discontinuities within [a, b]. Assuming F to be differentiable on a set [a, b] | E to the derivative f, where E is a subset of [a, b] at whose points F can take values ±∞ or not be defined at all, we adopt the convention that F and f are equal to 0 at all points of E and show that KH-vt ∝ _a ^b f = F(b) − F(a), where KH-vt denotes the total value of the Kurzweil-Henstock integral. The paper ends with a few examples that illustrate the theory.     

Invariant sets for a class of perturbed differential equations of retarded type

LetX be a Banach space and leta, b, q be real numbers such thata<b,q>0. Denote byD a locally closed subset ofX. A necessary and sufficient condition for the existence of a mild solutionu∈C([a−q, b ₁],X),a<b ₁<b, to the differential equationdu(t)/dt=Au(t)+f(t, u _t), such thatu:[a,b ₁]→D, u _a=ϕ is given. The linear operatorA is the generator of aC ₀ semigroupT(t), t≧0, withT(t) compact fort>0,f: [a, b)×C([−q,0],D _λ)→X is continuous and ϕ∈C([−q,0],D _λ) with ϕ(0)∈D. D _λ is a neighbourhood ofD. Applications to parabolic partial differential equations with retarded argument are given.     

A NOTE ON GRAPHS WHOSE DESTRUCTIONS YIELD COMPLETE SUBGRAPHS

Abstract

A connected graph G of order p =|V| and sise q =| E | is said to be (a_i, b_i)-destructible (with respect to E_i and V_i say) if a_i,b_i are integral factors of p and an a_i-set of edges E_i exists whose removal from G results in exactly b_i components isomorphic to K_i i.e. whose removal from G isolates the vertices in a b_i-set V_i. The operation of removing E_i and V_i from G results in either Ø or a subgraph H of G and is called an (a_i , b_i)-destruction of G. In this paper we show that the only graphs whose every (a_i,b_i)- destruction results in a complete subgraph are K (1,2) and K₄—e, where e ε K₄.     

Exceptional Sequences Determined by their Cartan Matrix

We investigate complete exceptional sequences E=(E ₁,¨,E _n ) in the derived category D ^b of finite-dimensional modules over a canonical algebra, equivalently in the derived category D ^b X of coherent sheaves on a weighted projective line, and the associated Cartan matrices C(E)=( [E _i ],[E _j ]). As a consequence of the transitivity of the braid group action on such sequences we show that a given Cartan matrix has at most finitely many realizations by an exceptional sequence E, up to an automorphism and a multi-translation (E ₁,¨,E _n )(E ₁[i ₁],¨,E _n [i _n ]) of D ^b . Moreover, we determine a bound on the number of such realizations. Our results imply that a derived canonical algebra A is determined by its Cartan matrix up to isomorphism if and only if the Hochschild cohomology of A vanishes in nonzero degree, a condition satisfied if A is representation-finite.     

Divisibility Graph for Symmetric and Alternating Groups

Let X be a nonempty set of positive integers and X* = X?{1}. The divisibility graph D(X) has X* as the vertex set, and there is an edge connecting a and b with a, b ∈ X* whenever a divides b or b divides a. Let X = cs(G) be the set of conjugacy class sizes of a group G. In this case, we denote D(cs(G)) by D(G). In this paper, we will find the number of connected components of D(G) where G is the symmetric group S _n or is the alternating group A _n .     

The dependence on parameters of the modulus problem for families of several classes of curves

Let, where A={a₁,..., a_n} and B={b₁,...,b_m} are systems of distinguished points, and let H be a family of homotopic classes H_i, i=1, ..., j + m, of closed Jordan curves in C, where the classes H_j+, =1, ..., m, consist of curves that are homotopic to a point curve in b. Let ={₁,...,_j+m} be a system of positive numbers. By P=P(,A,B) we denote the extremal-metric problem for the family H and the numbers : for the modulusU=U(,A,B) of this problem we have the equality , whereD ^*={D ₁ ^* ,...,D _j+m ^* } is a system of domains realizinga maximum for the indicated sum in the family of all systemsD={D ₁,...,D _j+m } of domains, associated with the family H (byU(D _i )) we denote the modulus of the domain D_i, associated with the class H_i). In the present paper we investigate the manner in whichU=U(,A,B) and the moduliU=(D ₁ ^* ) depend on the parameters _i, a_k, b; moreover, we consider the conditions under which some of the doubly connected domains D _i ^* ,i=1,...,j, from the system D^* turn out to be degenerate (Theorems 1–3). In particular, one obtains an expression for the gradient of the function M, as function of the parameter a=a_k (Theorem 4). One gives some applications of the obtained results (Theorem 5).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 136–148, 1985.     

PI (non)equivalence and Gelfand-Kirillov dimension in positive characteristic

The verbally prime algebras are well understood in characteristic 0 while over a field of positive characteristic p > 2 little is known about them. In previous papers we discussed some sharp differences between these two cases for the characteristic; we showed that the so-called Tensor Product Theorem cannot be extended for infinite fields of positive characteristic p > 2. Furthermore we studied the Gelfand-Kirillov dimension of the relatively free algebras of verbally prime and related algebras. In this paper we compute the GK dimensions of several algebras and thus obtain a new proof of the fact that the algebras M _a,a (E) ⊗ E and M _2a (E) are not PI equivalent in characteristic p > 2. Furthermore we show that the following algebras are not PI equivalent in positive characteristic: M _a,b (E) ⊗ M _c,d (E) and M _ac+bd,ad+cb (E); and M _a,b (E) ⊗ M _c,d (E) and M _{e, f} (E) ⊗ M _g,h (E) when a ≥ b, c ≥ d, e ≥ f, g ≥ h, ac + bd = eg+ f h, ad +bc = eh + fg and ac ≠ eg. Here E stands for the infinite dimensional Grassmann algebra with 1, and M _a,b (E) is the subalgebra of M _a+b (E) of the block matrices with blocks a × a and b × b on the main diagonal with entries from E ₀, and off-diagonal entries from E ₁; E = E ₀ ⊗ E ₁ is the natural grading on E. Partially supported by CNPq 620025/2006-9. This paper was written during the author’s PhD study at the UNICAMP, under the supervision of P.Koshlukov, to whom he expresses his sincere thanks.     

On the Matlis duals of local cohomology modules and modules of generalized fractions

Let (R, m) be a commutative Noetherian local ring with non-zero identity, a a proper ideal of R and M a finitely generated R-module with aM ≠ M. Let D(−) ≔ Hom _R (−, E) be the Matlis dual functor, where E ≔ E(R/m) is the injective hull of the residue field R/m. In this paper, by using a complex which involves modules of generalized fractions, we show that, if x ₁, …, x _n is a regular sequence on M contained in α, then H _{(x1, …,xnR} ⁿ D(H _a ⁿ (M))) is a homomorphic image of D(M), where H _b ⁱ (−) is the i-th local cohomology functor with respect to an ideal b of R. By applying this result, we study some conditions on a certain module of generalized fractions under which D(H _{(x1, …,xn)R} ⁿ (D(H _a ⁿ (M)))) ⋟ D(D(M)).     

Almost Schreier monoids

In an earlier paper, we introduced and studied the class of commutative integral domains D having the following property: if a, b₁, b₂ ? D{a, b_1, b_2 \in D} and a|b ₁ b ₂, there exist an integer k ≥ 1 and a₁, a₂ ? D{a_1, a_2 \in D} such that a ^k = a ₁ a ₂ and a_i|b_i^k , i = 1, 2{a_i|b{_i}^k , i = 1, 2} . In this paper, we show that many of our earlier results are purely multiplicative in the sense they can be extended to the setting of commutative cancellative monoids.     

On Extending Peano Derivatives

Let E be a closed set with inf E = a and sup E = b, and k be a positive integer. Let f : E Rbe such that the k-th Peano derivative of f relative to E, f _(k) (x, E), exists. It is proved under certain condition on the function f, that an extension F : [a, b] Rof f exists such that the ordinary derivative of F of order k, F ^<k> (x) exists on [a, b] and is continuous on [a, b], and f _<> (x, E) = F ^<i> (x) on E, for i = 1, 2, &, k.     

Bi-Parametric Potentials, Relevant Function Spaces and Wavelet-Like Transforms

We introduce new potential type operators J^a_b = (E+(-D)^b/2)^-a/bJ^{\alpha}_{\beta} = (E+(-\Delta)^{\beta/2})^{-\alpha/\beta}, (α > 0, β > 0), and bi-parametric scale of function spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta , p}({\mathbb{R}}^n) associated with J^α_β. These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization of the spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta, p}({\mathbb{R}}^n) is given with the aid of a special wavelet–like transform associated with a β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1).     

Connection between Fermat quotients and Euler numbers

In the paper, we obtain a congruence modulo p ³ among Euler numbers E _p−1, E _2p−2, and Fermat quotients Q ₂, Q _a where p = a ² + 4b ².      

The problem of product of conformal radii of nonoverlapping domains

Let a₁, a₂, a₃, b be distinct points in and let D be the family of all triples of nonoverlapping domains D₁, D₂, D₃ in \ {b} such that a_k∈ D_k, k=1,2,3. For this family we consider the problem on the maximum of the functional I=R₁R₂R₃, where R_k=R(D_k, a_k) is the conformal radius of D_k with respect to a_k. Geometrical properties of the extremal triple of domains are described. We prove that the maximum of I monotonically depends on the position of the point b and find the maximum in some special cases Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 114–128 Translated by N. Yu. Netsvetaev     

Note on the solution of a certain boundary-value problem

A simple algorithm is described for inverting the operatorD _x D _y (D _x andD _y here and subsequently denote partial differentiation with respect tox andy respectively) which occurs in the iterative solution of the equationD _x D _y f (x, y)=g (x, y, f, D _x f, D _x ² f,D _x D _y f, D _y ² f) when boundary values off(x,y) are given along the sides of the rectangle in thexy-plane whose corners are at the points (a,b); (a+(n+1)k,b); (a+(n+1)k,b+(n+1)h); (a,b+(n+1)h).Communication M. R. 43 of the Computation Department of the Mathematical Centre, Amsterdam.     

Cyclic antiautomorphisms of directed triple systems

A transitive triple, (a,b,c), is defined to be the set {(a,b), (b,c), (a,c)} of ordered pairs. A directed triple system of order v, DTS(v), is a pair (D,β), where D is a set of v points and β is a collection of transitive triples of pairwise distinct points of D such that any ordered pair of distinct points of D is contained in precisely one transitive triple of β. An antiautomorphism of a Directed triple system, (D,β), is a permutation of D that maps β to β⁻¹, where β ⁻¹ = {(c,b,a)|(a,b,c) E β}. In this article we give necessary and sufficient conditions for the existence of a Directed triple system of order v admitting an antiautomorphism consisting of a single cycle of length d and having v − d fixed points. Further, we give a more general result for partial Directed triple systems in which the missing ordered pairs are precisely those containing two fixed points. © 1996 John Wiley & Sons, Inc.