A Chebyshev Type Upper Estimate for Prime Elements in Additive Arithmetic Semigroups

 Let G(n) and Λ(n) be two sequences of nonnegative numbers which satisfy G(0)=1 and an additive convolution equation . A Chebyshev-type upper estimate for prime elements in an additive arithmetic semigroup is essentially a tauberian theorem on Λ(n) and G(n). Suppose with real constants . The theorem proved here states that and that holds with an explicit function R(n) of order <1 in n. This theorem is sharp. It has several applications. (Received 31 March 1999; in revised form 21 October 1999)     

The Category of Lattices over a Lattice-Finite Ring

It is well-known that for a lattice-finite order Λ over a complete discrete valuation domain, the radical of Λ-lat (the category of Λ-lattices) is nilpotent modulo projectives. Iyama has shown that this property merely depends on the combinatorial data given by the Auslander–Reiten quiver of Λ. Moreover, he established a criterion for a finite (symmetrizable) translation quiver Q to be the Auslander–Reiten quivers of an order Λ. We improve his characterization by showing that the remaining conditions on Q can be replaced by the existence of an additive function on the vertices of Q (Theorem 4). Our proof rests on a functorial theory of ladders, expressing the Auslander–Reiten structure of Λ-lat by means of an adjoint pair of functors L⊣L⁻ in the homotopy category of two-termed complexes over Λ-lat. Presented by I. Reiten Mathematics Subject Classifications (2000) Primary: 16G70, 16G30; secondary: 16G60.     

Completeness of the system {z^λn} in L^2[B,α0]

Let B be an unbounded domain located outside an angle domain with vertex at the origin, A ={λn}（n = 1,2,...） be a sequence of complex numbers satisfying sup ｜ arg（λn）｜ 〈 α 〈 π/2 and denote by M（∧） = {z^λ, λ ∈ ∧} the corresponding system of functions z^λ（λ∈∧）. Let α0（z） be a weight function defined on B. We obtain a completeness theorem for the system M（∧） in the Hilbert space L^2 [B, α0].     

One-Parameter Families of Modules for Tame Algebras and Bocses

Let Λ be a finite-dimensional algebra over an algebraically closed field k. We denote by mod Λ the category of finitely generated left Λ-modules. Consider the family ℱ(u) of the indecomposables M∈mod Λ such that , where is the subspace of morphisms which factorize through semisimple modules. If P,Q are projectives in mod Λ, ℱ(u)(P,Q) is the family of those modules M∈ℱ(u) such that a minimal projective presentation is of the formf_M: P→Q. We prove that if Λ is of tame representation type then each ℱ(P,Q) has only a finite number of isomorphism classes or is parametrized by μ(u,P,Q) one-parameter families. We give an upper bound for this number in terms of u,P and Q. Then we give some sufficient conditions for tame of polynomial growth type. For the proof we consider similar results for bocses. Presented by Y. Drozd Mathematics Subject Classifications (2000) 16G60, 16G70, 16G20.     

Gradings and Derived Categories

Let G{{\mathcal G}} be a group, Λ a G{{\mathcal G}}-graded Artin algebra and gr(Λ) denote the category of finitely generated G{{\mathcal G}}-graded Λ-modules. This paper provides a framework that allows an extension of tilting theory to D^b(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)) and to study connections between the tilting theories of D^b(L){{\mathcal D}}^b(\Lambda) and D^b(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)). In particular, using that if T is a gradable Λ-module, then a grading of T induces a G{{\mathcal G}}-grading on End_Λ(T), we obtain conditions under which a derived equivalence induced by a gradable Λ-tilting module T can be lifted to a derived equivalence between the derived categories D^b(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)) and D^b(gr(End_L(T))){{\mathcal D}}^b(\rm gr(\rm End_{\Lambda}(\textit T))).     

Grothendieck groups of unbounded complexes of finitely generated modules

Let Λ be a left Artinian ring, D⁺(mod Λ) (resp., D⁻(mod Λ), D(mod Λ)) the derived category of bounded below complexes (resp., bounded above complexes, unbounded complexes) of finitely generated left Λ-modules. We show that the Grothendieck groups K₀(D⁺(mod Λ)), K₀(D⁻(mod Λ)) and K₀(D(mod Λ)) are trivial. Received: 7 April 2005     

On two classes of permutations with number-theoretic conditions on the lengths of the cycles

Let Λ be an arbitrary set of positive integers andS _n (Λ) the set of all permutations of degreen for which the lengths of all cycles belong to the set Λ. In the paper the asymptotics of the ratio |S _n (Λ)|/n! asn→∞ is studied in the following cases: 1) Λ is the union of finitely many arithmetic progressions, 2) Λ consists of all positive integers that are not divisible by any number from a given finite set of pairwise coprime positive integers. Here |S _n (Λ)| stands for the number of elements in the finite setS _n (Λ). Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 881–891, December, 1997. Translated by A. I. Shtern     

Exponential sums over primes formed with coefficients of primitive cusp forms

Let Af（n） be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay（n） twisted with an additive character e（α√n）, α 〉0, i.e. Ef（x） = Σx〈n〈2x Af（n）e（α√n）.     

On local frequencies related to farey fractions

LetP ₁ andP ₂ be two sets of prime numbers and let ω(m,P_i)=#{p: p/m, pεP_i} (i=1,2) be two related additive functions ofm. For an irreducible positive fractionm/n, defineh(m/n)=ω(m, P ₁)+ω(n, P₂). In this paper the local frequenciesv _x{h(m/n)=s}=#{m/n ∈ F_x:h(m/n)=s}/#F_x are considered, whereF _x denotes the classical Farey series. Using the mean-value theorem for multiplicative functions of rational argument, a local limit theorem forv _x{h(m/n)=s} is proved. Research supported by the Lithuanian State Science and Studies Foundation. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 1, pp. 113–131, January–March, 2000. Translated by V. Stakènas     

On a question of K. Bezdek and T. Odor on Γ-parallelotops

K. Bezdek and T. Odor proved the following statement in [1]: If a covering ofE ³ is a lattice packing of the convex compact bodyK with packing lattice Λ (K is a Λ-parallelotopes) then there exists such a 2-dimensional sublattice Λ′ of Λ which is covered by the set ∪(K+z∣z ∈ Λ′). (K ∪L(Λ′) is a Λ′-parallelotopes). We prove that the statement is not true in the case of the dimensionsn=6, 7, 8. Supported by Hung. Nat. Found for Sci. Research (OTKA) grant no. 1615 (1991).     

Dilations and Completions for Gabor Systems

Let be a full rank time-frequency lattice in ℝ ^d ×ℝ ^d . In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L ²(ℝ ^d ) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)≤1, and to a dual Gabor Riesz basis pair for a Λ-shift invariant subspace containing M when v(Λ)>1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419–433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel–Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ)∪(∪ _j=1 ^N G(g _j ,Λ)) for L ²(ℝ ^d ). We show that this is true whenever v(Λ)≤N. In particular, when v(Λ)≤1, any Bessel–Gabor system is a subset of a tight Gabor frame G(g,Λ)∪G(h,Λ) for L ²(ℝ ^d ). Related results for affine systems are also discussed. Communicated by Chris Heil.     

Geometry of real grassmann manifolds. Parts I, II

The paper deals with the properties of the exterior algebra ℝ(Λⁿ) related to the Euclidean structure on ℝ(Λⁿ) induced by the scalar product in ℝ(Λⁿ). A geometric interpretation of inner multiplication for simple p-vectors is given. An invariant form of the Cartan criterion for the simplicity of a p-vector is given. The Plücker model realizing the real Grassmann manifold as a submanifold of the Euclidean space ℝ(Λⁿ), and an isometry of this submanifold onto the classical Grassmann manifold with SO(n)-invariant metric are described. A canonical decomposition of bivectors is given. Bibliography: 12 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 84–107. Translated by N. Yu: Netsvetaev.     

Imbedding of power series spaces and spaces of analytic functions

The diametral dimension of a nuclear Fréchet spaceE, which satisfies (DN) and (Ω), is related to power series spaces Λ₁(ε) and Λ_∞(ε) for some exponent sequence ε. It is proved thatE contains a complemented copy of Λ_∞(ε) provided the diametral dimensions ofE and Λ_∞(ε) are equal and ε is stable. Assuming Λ₁(ε) is nuclear, any subspace of Λ₁(ε) which satisfies (DN), can be imbedded intoE. Applications of these results to spaces of analytic functions are given. Support of Turkish Scientific and Technical Research Council is gratefully acknowledged.     

The inverse spectral problem for finite-rank Hankel operators

The following theorem is proved. Let Λ be a divisor of n points of the unit disk and let σ₁, σ₂,...σ _n be a finite sequence of nonzero complex numbers. Then there exists a Hankel operator Γ of rank n such that the divisor of the poles of its symbol is Λ and the eigenvalues of Γ (counted with the multiplicities) are σ₁, σ₂,...σ _n Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 5–15     

A splitting theorem for connected moravaK-theories

LetX be a connected, locally finite spectrum and letk(n) (n>-1) denote the (−1)-connected cover of then-th MoravaK-Theory associated to the primep.k(n) is aBP-module spectrum with π_*(k(n)) ≅ ℤ _p [υ _n ] where |v _n | = 2(p ⁿ -1). We prove the following splitting theorem: Thek(n) ^*-torsion ofk(n) ^* (X) is already annihilated byv _n ^e (e≥1) if and only ifk(n)ΛX is homotopy equivalent to a wedge of spectrak(n) and ^r k(n) (0≤r≤e-1) where ^r k(n) denotes ther-th Postnikov factor ofk(n). Moreover we investigate splitting conditions for ^r k(n)ΛX.     

The subgroups of the special linear group over a skew field that contain the group of diagonal matrices

For any (noncommutative) skew field T, the lattice of subgroups of the special linear group Λ=SL(n,T) that contain the subgroup Δ=SD(n,T) of diagonal matrices (with Dieudonné determinants equal to 1) is studied. It is established that for any subgroup H, Δ≤H≤Λ, there exists a uniquely determined unital net σ such that Λ(σ)≤H≤N(σ), where Λ(σ) is the net subgroup associated with the net σ and N(σ) is its normalizer in Λ. Bibliography: 11 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 91–103. Translated by Bui Xuan Hai.     

The ham sandwich theorem revisited

This paper continues the search, started in [10], for relatives of the ham sandwich theorem. We prove among other results, the following implications {fx21-1} whereK(n, k) is an important instance of the Knaster’s conjecture so thatK(n, n − 1) reduces to the Borsuk-Ulam theorem,B(n, k) is a R. Rado type statement about (k + 1) measures inR ⁿ whereB(n, n − 1) turns out to be the ham sandwich theorem andC(n, k) is a topological statement, established in this paper in the caseC(n, n − 2),n = 3 orn ≥ 5.     

On F-almost split sequences

Let A be an Artinian algebra and F an additive subbifunctor of Ext,（-, -） having enough projectives and injectives. We prove that the dualizing subvarieties of mod A closed under F-extensions have F-almost split sequences. Let T be an F-cotilting module in mod A and S a cotilting module over F = End（T）. Then Horn（-, T） induces a duality between F-almost split sequences in ⊥FT and almost sl31it sequences in ⊥S, where addrS = Hom∧（f（F）, T）. Let A be an F-Gorenstein algebra, T a strong F-cotilting module and 0→A→B→C→0 and F-almost split sequence in ⊥FT.If the injective dimension of S as a Г-module is equal to d, then C≌（ΩCM^-dΩ^dDTrA^＊）^＊,where（-）^＊=Hom（g,T）.In addition, if the F-injective dimension of A is equal to d, then A≌ΩMF^-dDΩFop^-d TrC≌ΩCMF^-d ≌F^d DTrC.     

The normalizer of the automorphism group of a module arising under extension of the base ring

Let Λ te an arbitrary associative ring with unity and let R be its unital subring contained in the center of Λ. Further, let M=λ M be a left free Λ-module of finite rank. In this paper, the normalizer of the subgroupAut(λM) of automorphisms of the module λM in the groupAut(_RM) of automorphisms of the module_RM is computed. If the ring Λ is additively generated by its invertible elements, then the above normalizer coincides with the semidirect product of the normal subgroupAut(λM) and a subgroup isomorphic to the groupAut(Λ/R) of all ring automorphisms of the ring Λ that are identical on R. Bibliography: 1 title. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 133‐135. Translated by A. I. Skopin.