A construction for modified generalized Hadamard matrices using QGH matrices

Let G be a group of order mu and U a normal subgroup of G of order u. Let G/U = {U ₁,U ₂, . . . ,U _m } be the set of cosets of U in G. We say a matrix H = [h _ij ] of order k with entries from G is a quasi-generalized Hadamard matrix with respect to the cosets G/U if \({\sum_{1\le t \le k} h_{it}h_{jt}^{-1} = \lambda_{ij1}U_1+\cdots+\lambda_{ijm}U_m (\exists\lambda_{ij1},\ldots, \exists \lambda_{ijm} \in \mathbb{Z})}\) for any i ≠ j. On the other hand, in our previous article we defined a modified generalized Hadamard matrix GH(s, u, λ) over a group G, from which a TD _λ (uλ, u) admitting G as a semiregular automorphism group is obtained. In this article, we present a method for combining quasi-generalized Hadamard matrices and semiregular relative difference sets to produce modified generalized Hadamard matrices.     

MULTITRANSITIVITY OF CALOGERO-MOSER SPACES

Let G be the group of unimodular automorphisms of a free associative ?-algebra on two generators. A theorem of G. Wilson and the first author [BW] asserts that the natural action of G on the Calogero-Moser spaces C _n is transitive for all n ? ?. We extend this result in two ways: first, we prove that the action of G on C _n is doubly transitive, meaning that G acts transitively on the configuration space of ordered pairs of distinct points in C _n ; second, we prove that the diagonal action of G on \( {C}_{n_1}\times {C}_{n_2}\times \cdots \times {C}_{n_m} \) is transitive provided n ₁,?n ₂,?…,?n _m are pairwise distinct numbers.     

On $$\mathfrak {F}_{hq}$$-supplemented subgroups of a finite group

A subgroup H of a finite group G is quasinormal in G if it permutes with every subgroup of G. A subgroup H of a finite group G is \(\mathfrak {F}_{hq}\)-supplemented in G if G has a quasinormal subgroup N such that HN is a Hall subgroup of G and \((H\cap N)H_{G}/ H_{G} \le Z_{\mathfrak {F}}(G/H_{G})\), where \(H_{G}\) is the core of H in G and \({Z}_{\mathfrak {F}} (G/H_{G})\) is the \(\mathfrak {F}\)-hypercenter of \({G/H}_{G}\). This paper concerns the structure of a finite group G under the assumption that some subgroups of G are \(\mathfrak {F}_{hq}\)-supplemented in G.     

On dihedrants admitting arc-regular group actions

We consider Cayley graphs Γ over dihedral groups, dihedrants for short, which admit an automorphism group G acting regularly on the arc set of Γ. We prove that, if D _2n ≤G≤Aut(Γ) is a regular dihedral subgroup of G of order 2n such that any of its index 2 cyclic subgroups is core-free in G, then Γ belongs to the family of graphs of the form \((K_{n_{1}}\otimes\cdots\otimes K_{n_{\ell}})[K_{m}^{\mathrm{c}}]\), where 2n=n ₁???n _? m, and the numbers n _i are pairwise coprime. Applications to 1-regular dihedrants and Cayley maps on dihedral groups are also given.     

Generalizations of some zero sum theorems

Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weight A, denoted by D _A (G), is defined to be the least positive integer t such that, for every sequence (x ₁,..., x _t ) with x _i ?∈?G, there exists a non-empty subsequence \((x_{j_1},\ldots, x_{j_l})\) and a _i ?∈?A such that \(\sum_{i=1}^{l}a_ix_{j_i} = 0\). Similarly, for an abelian group G of order n, E _A (G) is defined to be the least positive integer t such that every sequence over G of length t contains a subsequence \((x_{j_1} ,\ldots, x_{j_n})\) such that \(\sum_{i=1}^{n}a_ix_{j_i} = 0\), for some a _i ?∈?A. When G is of order n, one considers A to be a non-empty subset of {1,..., n???1 }. If G is the cyclic group \({\Bbb Z}/n{\Bbb Z}\), we denote E _A (G) and D _A (G) by E _A (n) and D _A (n) respectively.In this note, we extend some results of Adhikari et al (Integers 8 (2008) Article A52) and determine bounds for \(D_{R_n}(n)\) and \(E_{R_n}(n)\), where \(R_n = \{x^2 : x \in (\mathbb{Z}/n\mathbb{ Z})^*\}\). We follow some lines of argument from Adhikari et al (Integers 8 (2008) Article A52) and use a recent result of Yuan and Zeng (European J. Combinatorics 31 (2010) 677–680), a theorem due to Chowla (Proc. Indian Acad. Sci. (Math. Sci.) 2 (1935) 242–243) and Kneser’s theorem (Math. Z. 58 (1953) 459–484; 66 (1956) 88–110; 61 (1955) 429–434).     

Cartan coherent configurations

The Cartan scheme \(\mathcal{X}\) of a finite group G with a (B, N)-pair is defined to be the coherent configuration associated with the action of G on the right cosets of the Cartan subgroup \(B\cap N\) by right multiplication. It is proved that if G is a simple group of Lie type, then asymptotically the coherent configuration \(\mathcal{X}\) is 2-separable, i.e., the array of 2-dimensional intersection numbers determines \(\mathcal{X}\) up to isomorphism. It is also proved that in this case, the base number of \(\mathcal{X}\) equals 2. This enables us to construct a polynomial-time algorithm for recognizing Cartan schemes when the rank of G and the order of the underlying field are sufficiently large. One of the key points in the proof is a new sufficient condition for an arbitrary homogeneous coherent configuration to be 2-separable.     

Korovkin-Type Approximation Theory in Riesz Spaces

In this paper, we prove the following Riesz spaces’ version of the Korovkin theorem. Let E and F be two Archimedean Riesz spaces with F uniformly complete, let W be a nonempty subset of \(E^{+}\), and let \((T_{n})\) be a given sequence of (r-u)-continuous elements of \(\mathcal {L(}E,F)\), such that \(\left| T_{n}-T_{m}\right| x=\left| (T_{n}-T_{m})x\right| \mathcal {\ }\)for all \(x\in E^{+},\) \(m,n\ge n_{0}\) (for a given \(n_{0}\in \mathbb {N} )\). If the sequence \((T_{n}x)_{n}\) \((r-u)\)-converges for every \(x\in W\), then \((T_{n})\) \((r-u)\)-converges also pointwise on the ideal \(E_{W}\), generated by W, to a linear operator \(S_{0}:E_{W}\rightarrow F\). We also prove a similar Korovkin-type theorem for nets of operators. Some applications for f-algebras and orthomorphisms are presented.     

Hrmander Type Theorem for Fourier Multipliers with Optimal Smoothness on Hardy Spaces of Arbitrary Number of Parameters

The main purpose of this paper is to establish the Hormander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy spaces for k≥ 3 using the multiparameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k = 3, and the method works for all the cases k≥ 3:■where x =(x_1,x_2,x_3)∈R~(n_1)×R~(n_2)×R~(n_3) and ξ =(ξ_1,ξ_2,ξ_3)∈R~(n_1)×R~(n_2)×R~(n_3). One of our main results is the following:Assume that m(ξ) is a function on R~(n_1+n_2+n_3) satisfying ■ with s_i n_i(1/p-1/2) for 1≤i≤3. Then T_m is bounded from H~p(R~(n_1)×R~(n_2)×R~(n_3) to H~p(R~(n_1)×R~(n_2)×R~(n_3)for all 0 p≤1 and ■ Moreover, the smoothness assumption on s_i for 1≤i≤3 is optimal. Here we have used the notations m_(j,k,l)(ξ)=m(2~jξ_1,2~kξ_2,2~lξ_3)Ψ(ξ_1)Ψ(ξ_2)Ψ(ξ_3) and Ψ(ξ_i) is a suitable cut-off function on R~(n_i) for1≤i≤3, and W~(s_1,s_2,s_3) is a three-parameter Sobolev space on R~(n_1)×R~(n_2)× R~(n_3).Because the Fefferman criterion breaks down in three parameters or more, we consider the L~p boundedness of the Littlewood-Paley square function of T_mf to establish its boundedness on the multi-parameter Hardy spaces.     

The Automorphism Group of a Class of Nilpotent Groups with Infinite Cyclic Derived Subgroups

The automorphism group of a class of nilpotent groups with infinite cyclic derived subgroups is determined. Let G be the direct product of a generalized extraspecial Z-group E and a free abelian group A with rank m, where E ={(1 kα_1 kα_2 ··· kα_nα_(n+1) 0 1 0 ··· 0 α_(n+2)...............000...1 α_(2n+1)000...01|αi∈ Z, i = 1, 2,..., 2 n + 1},where k is a positive integer. Let AutG G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G of G, and AutG/ζ G,ζ GG be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then(i) The extension 1→ Aut_(G') G→ AutG→ Aut(G')→ 1 is split.(ii) Aut_(G') G/Aut_(G/ζ G,ζ G)G≌Sp(2 n, Z) ×(GL(m, Z)■(Z~)m).(iii) Aut_(G/ζ G,ζ GG/Inn G)≌(Z_k)~(2n)⊕(Z)~(2nm).     

Multiplicity and the pull-back problem

We discuss a formula of S. Spodzieja and generalize it for the isolated improper Achilles-Tworzewski-Winiarski intersection index. As an application we give a simple proof of a result of P. Ebenfelt and L. Rothschild: if \({F\colon (\mathbb{C}^m,0)\to (\mathbb{C}^m,0)}\) is a finite holomorphic map, W a germ of a complex variety at zero such that F ^?1(W) is a smooth germ and the Jacobian of F does not vanish identically on it, then W is smooth too.     

Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups

Let G = N ? A, where N is a stratified group and A = ? acts on N via automorphic dilations. Homogeneous sub-Laplacians on N and A can be lifted to left-invariant operators on G, and their sum is a sub-Laplacian Δ on G. We prove a theorem of Mihlin–Hörmander type for spectral multipliers of Δ. The proof of the theorem hinges on a Calderón–Zygmund theory adapted to a sub-Riemannian structure of G and on L¹-estimates of the gradient of the heat kernel associated to the sub-Laplacian Δ.     

Note: A conjecture on partitions of groups

We conjecture that every infinite group G can be partitioned into countably many cells \(G = \bigcup\limits_{n \in \omega } {A_n }\) such that cov(A _n A _n ^?1 ) = |G| for each n ∈ ω Here cov(A) = min{|X|: X} ? G, G = X A}. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.     

Products of F*(<Emphasis Type="Italic">G</Emphasis>)-subnormal subgroups of finite groups

A subgroup H of a finite group G is called F*(G)-subnormal if H is subnormal in HF*(G). We show that if a group Gis a product of two F*(G)-subnormal quasinilpotent subgroups, then G is quasinilpotent. We also study groups G = AB, where A is a nilpotent F*(G)-subnormal subgroup and B is a F*(G)-subnormal supersoluble subgroup. Particularly, we show that such groups G are soluble.     

The Calderón problem for Hilbert couples

We prove that intermediate Banach spaces\(\mathcal{A}\) and\(\mathcal{B}\) with respect to arbitrary Hilbert couples\(\bar {H}\) and\(\bar {K}\) are exact interpolation if and onlyif they are exactK-monotonic, i.e. the condition\(f^0 \in \mathcal{A}\) and the inequality\(K(t,g^0 ;\bar {K}) \leqslant K(t,f^0 ;\bar {H}),t > 0\), implyg⁰∈B and ‖g⁰‖B≤‖f⁰‖ _A (K is Peetre’sK-functional). It is well known that this property is implied by the following: for each ?>1 there exists an operator\(T:\bar {H} \to \bar {K}\) such thatTf⁰=g⁰, and\(K(t,Tf;\bar {K}) \leqslant \rho K(t,f;\bar {H}),f \in \mathcal{H}_0 + \mathcal{H}_1 ,t > 0\). Verifying the latter property, it suffices to consider the “diagonal case” where\(\bar {H} = \bar {K}\) is finite-dimensional, in which case we construct the relevant operators by a method which allows us to explicitly calculate them. In the strongest form of the theorem it is shown that the statement remains valid when substituting ?=1. The result leads to a short proof of Donoghue’s theorem on interpolation functions, as well as Löwner’s theorem on monotone matrix functions.     

Ideal counting function in cubic fields

For a cubic algebraic extension K of ?, the behavior of the ideal counting function is considered in this paper. More precisely, let a _K (n) be the number of integral ideals of the field K with norm n, we prove an asymptotic formula for the sum \(\sum\nolimits_{n_1^2 + n_2^2 \leqslant x} {a_K \left( {n_1^2 + n_2^2 } \right)} \).     

On $$\mathcal {}$$-subgroups of finite groups

Let G be a finite group. A subgroup H of G is s-permutable in G if H permutes with every Sylow subgroup of G. A subgroup H of G is called an \(\mathcal {SSH}\)-subgroup in G if G has an s-permutable subgroup K such that \(H^{sG} = HK\) and \(H^g \cap N_K (H) \leqslant H\), for all \(g \in G\), where \(H^{sG}\) is the intersection of all s-permutable subgroups of G containing H. We study the structure of finite groups under the assumption that the maximal or the minimal subgroups of Sylow subgroups of some normal subgroups of G are \(\mathcal {SSH}\)-subgroups in G. Several recent results from the literature are improved and generalized.     

An Engel condition with skew derivations for one-sided ideals

We apply the theory of generalized polynomial identities with automorphisms and skew derivations to prove the following theorem: Let A be a prime ring with the extended centroid C and with two-sided Martindale quotient ring Q, R a nonzero right ideal of A and \(\delta \) a nonzero \(\sigma \)-derivation of A, where \(\sigma \) is an epimorphism of A. For \(x,y\in A\), we set \([x,y] = xy - yx\). If \([[\ldots [[\delta (x^{n_0}),x^{n_1}],x^{n_{2}}],\ldots ],x^{n_k}]=0\) for all \(x\in R\), where \(n_{0},n_{1},\ldots ,n_{k}\) are fixed positive integers, then one of the following conditions holds: (1) A is commutative; (2) \(C\cong GF(2)\), the Galois field of two elements; (3) there exist \(b\in Q\) and \(\lambda \in C\) such that \(\delta (x)=\sigma (x)b-bx\) for all \(x\in A\), \((b-\lambda )R=0\) and \(\sigma (R)=0\). The analogous result for left ideals is also obtained. Our theorems are natural generalizations of the well-known results for derivations obtained by Lanski (Proc Am Math Soc 125:339–345, 1997) and Lee (Can Math Bull 38:445–449, 1995).     

Boundary value problems for complete quasi-parabolic differential equations of odd order with variable domains of operators

We prove the well-posed solvability in the strong sense of the boundary value Problems

$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$

where the unbounded operators A _s (t), s > 0, in a Hilbert space H have domains D(A _s (t)) depending on t, are subordinate to the powers A ^1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A ₀(t) and the symmetric operators A _s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that

$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$

, where the smoothing operators are defined by

$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$

.

     

Solution of the Ulam Stability Problem for an Euler Type Quadratic Functional Equation

In 1968 S.M. Ulam proposed the problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?’’. In 1978 according to P.M. Gruber this kind of problems is of particular interest in probability theory and in the case of functional equations of different types. In 1997 W. Schuster established a new vector quadratic identity on the basis of the well-known Euler type theorem on quadrilaterals: If ABCD is a quadrilateral and M, N are the mid-points of the diagonals AC, BD as well as A′, B′, C′, D′ are the mid-points of the sides AB, BC, CD, DA, then |AB|² + |BC|² + |CD|² + |DA|² = 2|A′C′|² + 2|B′D′|² + 4|MN|². Employing in this paper the above geometric identity we introduce the new Euler type quadratic functional equation

$\begin{array}{l}2{[}Q(x_{0} - x_{1}+Q(x_{1}-x_{2})+Q(x_{2}- x_{3})+Q(x_{3}-x_{0}){]}\\\qquad = Q(x_{0}-x_{1}-x_{2}+x_{3})+Q(x_{0} + x_{1}-x_{2}-x_{3})+2Q(x_{0}-x_{1}+ x_{2}-x_{3})\end{array}$

for all vectors (x₀, x₁, x₂, x₃) X⁴, with X and Y linear spaces. For every x ∈ R set Q(x) = x². Then the mapping Q : X → Y is quadratic. Note also that if Q : R → R is quadratic, then we have Q(x) = Q(1)x². Besides note that the geometric interpretation of the special example

$\begin{array}{l}2{[}(x_{0} - x_{1})^{2}+ (x_{1}-x_{2})^{2}+ (x_{2}-x_{3})^{2}+(x_{3}-x_{0})^{2}{]}\\\qquad = (x_{0}-x_{1}-x_{2} + x_{3})^{2}+(x_{0} + x_{1}-x_{2}-x_{3})^{2} + 2(x_{0}-x_{1}+ x_{2}-x_{3})^{2}\end{array}$

leads to the above-mentioned Euler type theorem on quadrilaterals ABCD with position vectors x₀, x₁, x₂, x₃ of vertices A, B, C, D, respectively. Then we solve the Ulam stability problem for the afore-mentioned equation, with non-linear Euler type quadratic mappings Q : X → Y.