A characterization of the finite simple group L<Subscript>16</Subscript>(2) by its prime graph

In this paper as the main result we prove that the projective special linear group L ₁₆(2) is uniquely determined by its prime graph. In fact we give a positive answer to an open problem arose in Zavarnitsin (Algebra Logic 43(4), 220–231, 2006) and we obtain a first example of a finite group with connected prime graph which is uniquely determined by its prime graph. This research was in part supported by a grant from IPM (No. 86200023).     

Homology of GL<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>: injectivity conjecture for GL<Subscript>4</Subscript>

The homology of GL _n (R) and SL _n (R) is studied, where R is a commutative ‘ring with many units’. Our main theorem states that the natural map H ₄(GL₃(R), k) → H ₄(GL₄(R), k) is injective, where k is a field with char(k) ≠ 2, 3. For an algebraically closed field F, we prove a better result, namely, is injective. We will prove a similar result replacing GL by SL. This is used to investigate the indecomposable part of the K-group K ₄(R).     

A Siegel modular threefold and Saito-Kurokawa type lift to <Emphasis Type="Italic">S</Emphasis><Subscript>3</Subscript>(Γ<Subscript>1,3</Subscript>(2))

Hulek and others conjectured that the unique differential three-form F (up to scalar) on the Siegel threefold associated to the group Γ_1,3(2) comes from the Saito-Kurokawa lift of the elliptic newform h of weight 4 for Γ₀(6). This F have been already constructed as a Borcherds product (cf. Gritsenko and Hulek in Int Math Res Notices 17:915–937, 1999). In this paper, we prove this conjecture by using the Yoshida lift and we settle a conjecture which relates our theorem. A remarkable fact is that the Yoshida lift using the usual test function cannot give the Saito-Kurokawa type lift of weight 3 associated to the group Γ_1,3(2). So important task is to find special test functions for the Yoshida lift at the bad primes 2 and 3. Dedicated to Professor Tomoyoshi Ibukiyama on his 60th birthday.     

Applications of discrete maximal <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> regularity for finite element operators

In this paper, we present applications of discrete maximal L _p regularity for finite element operators. More precisely, we show error estimates of order h ² for linear and certain semilinear problems in various L _p (Ω)-norms. Discrete maximal regularity allows us to prove error estimates in a very easy and efficient way. Moreover, we also develop interpolation theory for (fractional powers of) finite element operators and extend the results on discrete maximal L _p regularity formerly proved by the author. The author was supported by the DFG-Graduiertenkolleg 853.     

Modulus Semigroups and Perturbation Classes for Linear Delay Equations in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper we study C₀-semigroups on X × L_p( − h, 0; X) associated with linear differential equations with delay, where X is a Banach space. In the case that X is a Banach lattice with order continuous norm, we describe the associated modulus semigroup, under minimal assumptions on the delay operator. Moreover, we present a new class of delay operators for which the delay equation is well-posed for p in a subinterval of [1,∞). Dedicated to the memory of H. H. Schaefer     

<Emphasis Type="Italic">p</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-sequences of algebras with one fundamental operation

We show that if (p₀, p₁, ...) is the p_n-sequence of a nontrivial algebra with one fundamental operation, then p₁ ≥ p₀. Moreover, if , then p₁ > 2p₀. Received April 21, 2003; accepted in final form November 28, 2005.     

Approximation in the <Emphasis Type="BoldItalic">M</Emphasis><Subscript>2</Subscript>/<Emphasis Type="BoldItalic">G</Emphasis><Subscript>2</Subscript>/1 Queue with Preemptive Priority

The main purpose of this paper is to use the strong stability method to approximate the characteristics of the M ₂/G ₂/1 queue with preemptive priority by those of the classical M/G/1 queue. The latter is simpler and more exploitable in practice. After perturbing the arrival intensity of the priority requests, we derive the stability conditions and next obtain the stability inequalities with an exact computation of constants. From those theoretical results, we elaborate an algorithm allowing us to verify the approximation conditions and to provide the made numerical error. In order to have an idea about the efficiency of this approach, we consider a concrete example whose results are compared with those obtained by simulation.     

Automorphisms of Chevalley groups of types <Emphasis Type="Italic">B</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> over local rings

In the paper, we prove that every automorphism of any adjoint Chevalley group of type B ₂ or G ₂ is standard, i.e., it is a composition of an “inner” automorphism, a ring automorphism, and a central automorphism. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 3–29, 2007.     

On <Emphasis Type="Italic">c</Emphasis>-normal maximal and minimal subgroups of Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups of finite groups

A subgroup H of a finite group G is called c-normal in G if there exists a normal subgroup N of G such that G = HN and $H \cap N \leq H_{G} = {\rm core}_{G}(H)$. In this paper, we investigate the class of groups of which every maximal subgroup of its Sylow p-subgroup is c-normal and the class of groups of which some minimal subgroups of its Sylow p-subgroup is c-normal for some prime number p. Some interesting results are obtained and consequently, many known results related to p-nilpotent groups and p-supersolvable groups are generalized.     

The Exceptional Compact Symmetric Spaces <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">SO</Emphasis>(4) as Tubes

In the present paper we discuss in detail the cohomogeneity one isometric actions of the Lie groups SU(3) × SU(3) and SU(3) on the exceptional compact symmetric spaces G₂ and G₂/SO(4), respectively. We show that the principal orbits coincide with the tubular hypersurfaces around the totally geodesic singular orbits, and the symmetric spaces G₂ and G₂/SO(4) can be thought of as compact tubes around SU(3) and P², respectively. Moreover, we determine the radii of these tubes and describe the shape operators of the principal orbits. Finally, we apply these results to compute the volumes of the two symmetric spaces.The author was partially supported by the Hungarian National Science and Research Foundation OTKA T032478.     

Novikov superalgebras with <Emphasis Type="Italic">A</Emphasis><Subscript>0</Subscript> = <Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript><Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript>

Novikov superalgebras are related to quadratic conformal superalgebras which correspond to the Hamiltonian pairs and play a fundamental role in completely integrable systems. In this note we show that the Novikov superalgebras with A ₀ = A ₁ A ₁ and dim A ₁ = 2 are of type N and give a class of Novikov superalgebras of type S with A ₀ = A ₁ A ₁.     

<Emphasis Type="Italic">D</Emphasis><Subscript>∞</Subscript>-differential <Emphasis Type="Italic">E</Emphasis><Subscript>∞</Subscript>-algebras and Steenrod operations in spectral sequences

This paper is devoted to the introduction of a D _∞-differential analog of the notion of an E _∞-(co)algebra and to the construction of generalized Steenrod operations in terms of multiplicative spectral sequences. In this paper, we investigate basic homotopy properties of D _∞-differential E _∞-(co)algebras and construct a spectral sequence of a D _∞-differential E _∞-(co)algebra. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 43, Topology and Its Applications, 2006.     

Coxeter group actions on <Subscript>4</Subscript><Emphasis Type="Italic">F</Emphasis><Subscript>3</Subscript>(1) hypergeometric series

In this paper we investigate a certain linear combination K([(x)\vec])=K(a;b,c,d;e,f,g)K(\vec{x})=K(a;b,c,d;e,f,g) of two Saalschutzian hypergeometric series of type ₄ F ₃(1). We first show that K([(x)\vec])K(\vec{x}) is invariant under the action of a certain matrix group G _K , isomorphic to the symmetric group S ₆, acting on the affine hyperplane V={(a,b,c,d,e,f,g)∈ℂ⁷:e+f+g−a−b−c−d=1}. We further develop an algebra of three-term relations for K(a;b,c,d;e,f,g). We show that, for any three elements μ ₁,μ ₂,μ ₃ of a certain matrix group M _K , isomorphic to the Coxeter group W(D ₆) (of order 23040) and containing the above group G _K , there is a relation among K(m₁[(x)\vec])K(\mu_{1}\vec{x}), K(m₂[(x)\vec])K(\mu_{2}\vec{x}), and K(m₃[(x)\vec])K(\mu_{3}\vec{x}), provided that no two of the μ _j ’s are in the same right coset of G _K in M _K . The coefficients in these three-term relations are seen to be rational combinations of gamma and sine functions in a,b,c,d,e,f,g.     

On hamiltonicity of <Emphasis Type="Italic">P</Emphasis><Subscript>3</Subscript>-dominated graphs

We introduce a new class of graphs which we call P ₃-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P ₃-dominated graph. We prove that G is hamiltonian if α(G ²) ≤ κ(G), with two exceptions: K _2,3 and K _1,1,3. We also prove that G is hamiltonian, if G is 3-connected and |V(G)| ≤ 5δ(G) − 5. These results extend known results on (quasi-)claw-free graphs. This paper was completed when both authors visited the Center for Combinatorics, Nankai University, Tianjin. They gratefully acknowledge the hospitality and support of the Center for Combinatorics and Nankai University. The work of E.Vumar is sponsored by SRF for ROCS, REM.     

Localization of <Emphasis Type="Italic">LM</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-algebras

The aim of this paper is to define the localization LM _n -algebra of an LM _n —algebra L with respect to a topology F on L; in Section 5 we prove that the maximal LM _n -algebra of fractions (defined in [3]) and the LM _n -algebra of fractions relative to an Λ—closed system (defined in Section 2) are LM _n -algebras of localization.     

Inducing <Emphasis Type="Italic">W</Emphasis>-graphs

 Let ℋ be the Hecke algebra associated with a Coxeter group W. Many interesting ℋ-modules can be described using the concept of a W-graph, as introduced in the influential paper [4] of Kazhdan and Lusztig. In particular, Kazhdan and Lusztig showed that the regular representation of ℋ has an associated W-graph. The purpose of this note is to show that if W _J is a parabolic subgroup of W and V is a module for the corresponding Hecke algebra ℋ _J , then a W _J -graph structure for V gives rise to a W-graph structure for the induced module ℋ⊗ _ℋJ V. In the case that W _J is the identity subgroup and V has dimension 1, our construction coincides with that given by Kazhdan and Lusztig for the regular representation. For arbitrary J and V of dimension 1 we recover the constructions of Couillens [1] and Deodhar [3]. Received: 14 June 2002; in final form: 13 August 2002 / Published online: 1 April 2003 Mathematics Subject Classification (2000): 20C08     

Decompositions of the 3-uniform hypergraphs <Emphasis Type="Italic">K</Emphasis><Stack><Subscript>v</Subscript><Superscript>(3)</Superscript></Stack> into hypergraphs of a certain type

In this paper, we investigate a generalization of graph decomposition, called hypergraph decomposition. We show that a decomposition of a 3-uniform hypergraph K(3)v into a special kind of hypergraph K(3)4 - e exists if and only if v ≡ 0, 1, 2 (mod 9) and v ≥ 9.     

Regularity for weakly (K<Subscript>1</Subscript>, K<Subscript>2</Subscript>)-quasiregular mappings

In this paper, we first give the definition of weakly (K₁,K₂-quasiregular mappings, and then by using the Hodge decomposition and the weakly reverse Hölder inequality, we obtain their regularity property: For anyq ₁ that satisfies\(0< K_1 n^{(n + 4)/2} 2^{n + 1} \times 100^{n^2 } [2^{3n/2} (2^{5n} + 1)](n - q_1 )< 1\), there existsp ₁=p ₁(n,q ₁,K ₁,K ₂)>n, such that any (K₁, K₂)-quasiregular mapping\(f \in W_{loc}^{1,q_1 } (\Omega ,R^n )\) is in fact in\(W_{loc}^{1,p_1 } (\Omega , R^n )\). That is, f is (K₁,K₂)-quasiregular in the usual sense.     

Compression and denoising using <Emphasis Type="Italic">l</Emphasis><Subscript>0</Subscript>-norm

In this paper, we deal with l ₀-norm data fitting and total variation regularization for image compression and denoising. The l ₀-norm data fitting is used for measuring the number of non-zero wavelet coefficients to be employed to represent an image. The regularization term given by the total variation is to recover image edges. Due to intensive numerical computation of using l ₀-norm, it is usually approximated by other functions such as the l ₁-norm in many image processing applications. The main goal of this paper is to develop a fast and effective algorithm to solve the l ₀-norm data fitting and total variation minimization problem. Our idea is to apply an alternating minimization technique to solve this problem, and employ a graph-cuts algorithm to solve the subproblem related to the total variation minimization. Numerical examples in image compression and denoising are given to demonstrate the effectiveness of the proposed algorithm.