On quasi-similarity of subnormal operators

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.     

Valuations on algebras with involution

Let A be a central simple algebra with involution σ of the first or second kind. Let v be a valuation on the σ-fixed part F of Z(A). A σ-special v-gauge g on A is a kind of value function on A extending v on F, such that g(σ(x)x) = 2g(x) for all x in A. It is shown (under certain restrictions if the residue characteristic is 2) that if v is Henselian, then there is a σ-special v-gauge g if and only if σ is anisotropic, and g is unique. If v is not Henselian, it is shown that there is a σ-special v-gauge g if and only if σ remains anisotropic after scalar extension from F to the Henselization of F with respect to v; when this occurs, g is the unique σ-invariant v-gauge on A.     

Suppression of instability in rotatory hydromagnetic convection

Recently discovered hydrodynamic instability [1], in a simple Bénard configuration in the parameter regime under the action of a nonadverse temperature gradient, is shown to be suppressed by the simultaneous action of a uniform rotation and a uniform magnetic field both acting parallel to gravity for oscillatory perturbations whenever (Qσ ₁/π² + J/π⁴) > 1 and the effective Rayleigh numberR(1 -T ₀α₂) is dominated by either 27π⁴(1 + l/σ₁/4 or 27π⁴/2 according as σ₁ ≥1 or σ₁ ≤ 1 respectively. HereT ₀is the temperature of the lower boundary while α₂ is the coefficient of specific heat at constant volume due to temperature variation and σ₁,R,Q andJ respectively denote the magnetic Prandtl number, the Rayleigh number, the Chandrasekhar number and the Taylor number.     

On Point-Cyclic Resolutions of the 2-(63,7,15) Design Associated with PG(5,2)

 A t(v,k,λ) design is a set of v points together with a collection of its k-subsets called blocks so that t points are contained in exactly λ blocks. PG(n,q), the n-dimensional projective geometry over GF(q) is a 2(q ⁿ +q ⁿ⁻¹ +⋯+q+1,q ²+q+1, q ⁿ⁻² + q ⁿ⁻³ +⋯+q+1) design when we take its points as the points of the design and its planes as the blocks of the design. A 2(v,k,λ) design is said to be resolvable if the blocks can be partitioned as ℱ={R ₁,R ₂,…,R _s }, where s=λ(v−1)/(k−1) and each R _i consists of v/k disjoint blocks. If a resolvable design has an automorphism σ which acts as a cycle of length v on the points and ℱ^σ=ℱ, then the design is said to be point-cyclically resolvable. The design consisting of points and planes of PG(5,2) is shown to be point-cyclically resolvable by enumerating all inequivalent resolutions which are invariant under a cyclic automorphism group G=〈σ〉 where σ is a cycle of length v. These resolutions are shown to be the only resolutions which admit point-transitive automorphism group. Received: November 10, 1999 Final version received: September 18, 2000 Acknowledgments. The author would like to thank A. Munemasa for his assistance in writing computer programs on constructing projective spaces and searching for partial spreads. Moreover, she's thankful to T. Hishida and M.␣Jimbo for helpful discussions and for verifying the results of this paper. Present address: Mathematics Department, Ateneo de Manila University, Loyola Heights, Quezon City 1108, Philippines. e-mail: jumela@mathsci.math.admu.edu.ph     

Microscopic theory of superconductivity in MgB<Subscript>2</Subscript>-type systems in a magnetic field: A neighborhood of <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis>2</Subscript>

We calculate the upper critical magnetic field H _c2 in the framework of a microscopic superconductivity theory with two energy bands of different dimensions on the Fermi surface with the cavity topology typical of the compound MgB₂ taken into account (an anisotropic system). We assume an external magnetic field parallel to the crystallographic z axis. We obtain analytic formulas in the low-temperature range (T/T_c ≪ 1) and also near the critical temperature ((T-T_c)/T_c ≪ 1). We compare the temperature dependence of H_c2 for a two-band anisotropic system with that of H _c2 ⁰ corresponding to a two-band isotropic system (with Fermi-surface cavities of the same topology). We determine the role of the band-structure anisotropy, the positive curvature of the upper critical field near the critical temperature, and the important role of the ratio v₁/v₂ of the velocities on the Fermi surface in determining H_c2. We also obtain the values of the parameters Δ₁ and Δ₂ along the line of the critical magnetic field. This paper is dedicated to the 90th birthday of Professor D. N. Zubarev __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 113–128, January, 2008.     

A 2-factor with Short Cycles Passing Through Specified Independent Vertices in Graph

For a graph G, we define σ₂(G) := min{d(u) + d(v)|u, v ≠ ∈ E(G), u ≠ v}. Let k ≥ 1 be an integer and G be a graph of order n ≥ 3k. We prove if σ₂(G) ≥ n + k − 1, then for any set of k independent vertices v ₁,...,v _k , G has k vertex-disjoint cycles C ₁,..., C _k of length at most four such that v _i ∈ V(C _i ) for all 1 ≤ i ≤ k. And show if σ₂(G) ≥ n + k − 1, then for any set of k independent vertices v ₁,...,v _k , G has k vertex-disjoint cycles C ₁,..., C _k such that v _i ∈ V(C _i ) for all 1 ≤ i ≤ k, V(C ₁) ∪...∪ V(C _k ) = V(G), and |C _i | ≤ 4 for all 1 ≤ i ≤ k − 1. The condition of degree sum σ₂(G) ≥ n + k − 1 is sharp. Received: December 20, 2006. Final version received: December 12, 2007.     

Motivic torsors

The torsorP _δ=Hom^⊗ (H _DR,H _σ) under the motivic Galois groupG _σ=Aut^⊗ H _δ of the Tannakian category generated by one-motives related by absolute Hodge cycles over a field κ with an embedding σ:k↪ℂ is shown to be determined by its projectionP _σ→P _σ/G _σ ⁰ to a Gal( )-torsor, and by its localizationsP _σ⊕_k k _ξ at a dense subset of orderings ϕ of the field κ, provided κ has virtual cohomological dimension (vcd) one. This result is an application of a recent local-global principle for connected linear algebraic groups over a field κ of vcd ≤1.     

On subgroups of the general linear group that contain a maximal nonsplit torus

The paper deals with the structure of intermediate subgroups of the general linear group GL(n, k) of degree n over a field k of odd characteristic that contain a nonsplit maximal torus related to a radical extension of degree n of the ground field k. The structure of ideal nets over a ring that determine the structure of intermediate subgroups containinga transvection is given. Let K = k( ⁿ?{d} ) K = k\left( {\sqrt[n]{d}} \right) be a radical degree-n extension of a field k of odd characteristic, and let T =(d) be a nonsplit maximal torus, which is the image of the multiplicative group of the field K under the regular embedding in G =GL(n, k). In the paper, the structure of intermediate subgroups H, T ≤ H ≤ G, that contain a transvection is studied. The elements of the matrices in the torus T = T (d) generate a subring R(d) in the field k.Let R be an intermediate subring, R(d) ⊆ R ⊆ k, d ∈ R. Let σ_R denote the net in which the ideal dR stands on the principal diagonal and above it and all entries of which beneath the principal diagonal are equal to R. Let σ^R denote the net in which all positions on the principal diagonal and beneath it are occupied by R and all entries above the principal diagonal are equal to dR. Let E(σ_R) be the subgroup generated by all transvections from the net group G(σ_R). In the paper it is proved that the product TE(σ_R) is a group (and thus an intermediate subgroup). If the net σ associated with an intermediate subgroup H coincides with σ_R,then TE(σ_R) ≤ H ≤ N(σ_R),where N(σ_R) is the normalizer of the elementary net group E(σ_R) in G. For the normalizer N(σ_R),the formula N(σ_R)= TG(σ^R) holds. In particular, this result enables one to describe the maximal intermediate subgroups. Bibliography: 13 titles.     

Normal Quadratics in Ore Extensions,Quantum Planes,and Quantized Weyl Algebras

Let R be a Noetherian domain and let (σ,δ) be a quasi-derivation of R such that σ is an automorphism. There is an induced quasi-derivation on the classical quotient ring Q of R. Suppose F=t ²−v is normal in the Ore extension R[t;σ,δ] where v∈R. We show F is prime in R[t;σ,δ] if and only if F is irreducible in Q[t;σ,δ] if and only if there does not exist w∈Q such that v=σ(w)w−δ(w). We apply this result to classify prime quadratic forms in quantum planes and quantized Weyl algebras.     

Removability of singularities for a class of fully non-linear elliptic equations

In this paper we address the problem of understanding the singularities of the fully non-linear elliptic equation σ _k (v) = 1. These σ _k curvature are defined as the symmetric functions of the eigenvalues of the Schouten tensor of a Riemannian metric and appear naturally in conformal geometry, in fact, σ₁ is just the scalar curvature.Here we deal with the local behavior of isolated singularities. We give a sufficient condition for the solution to be bounded near the singularity. The same result follows for a more general singular set Λ as soon as we impose some capacity conditions. The main ingredient is an estimate of the L ^∞ norm in terms of a suitable L ^p norm. Mathematics Subject Classification (2000) 35J60, 53A30     

The Sigma Chromatic Number of a Graph

For a nontrivial connected graph G, let ${c: V(G)\to {{\mathbb N}}}For a nontrivial connected graph G, let c: V(G)? \mathbb N{c: V(G)\to {{\mathbb N}}} be a vertex coloring of G, where adjacent vertices may be colored the same. For a vertex v of G, let N(v) denote the set of vertices adjacent to v. The color sum σ(v) of v is the sum of the colors of the vertices in N(v). If σ(u) ≠ σ(v) for every two adjacent vertices u and v of G, then c is called a sigma coloring of G. The minimum number of colors required in a sigma coloring of a graph G is called its sigma chromatic number σ(G). The sigma chromatic number of a graph G never exceeds its chromatic number χ(G) and for every pair a, b of positive integers with a ≤ b, there exists a connected graph G with σ(G) = a and χ(G) = b. There is a connected graph G of order n with σ(G) = k for every pair k, n of positive integers with k ≤n if and only if k ≠ n − 1. Several other results concerning sigma chromatic numbers are presented.     

Schreier type theorems for bicrossed products

We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups (H;G; α; β) is deformed using a combinatorial datum (σ; v; r) consisting of an automorphism σ of H, a permutation v of the set G and a transition map r: G → H in order to obtain a new matched pair (H; (G; *); α′, β′) such that there exists a σ-invariant isomorphism of groups H _α⋈β G ≅H _{α′⋈β′} (G, *). Moreover, if we fix the group H and the automorphism σ ∈ Aut H then any σ-invariant isomorphism H _α⋈β G ≅ H _{α′⋈β′} G′ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for bicrossed products of groups are given.     

Pentadiagonal Oscillatory Matrices with Two Spectrum in Common

We denote the spectrum of an square matrix A by σ(A), and that of the matrix obtained by deleting the first i rows and columns of A by σ_i(A). It is known that a symmetric pentadiagonal oscillatory (SPO) matrix may be constructed from σ, σ₁ and σ₂. The pairs σ, σ₁ and σ₁, σ₂ must interlace; the construction is not unique; and the conditions on the data which ensure that A is oscillatory are extremely complicated. Given one SPO matrix A, the paper shows that operations may be applied to A to construct a family of such matrices with σ and σ₁ in common. Moreover, given one totally positive (TP) matrix A, we construct a family of TP matrices with σ, σ₁ and σ₂ in common.     

On the ideal (<Emphasis Type="Italic">v</Emphasis><Superscript>0</Superscript>)

The σ-ideal (v ⁰) is associated with the Silver forcing, see [5]. Also, it constitutes the family of all completely doughnut null sets, see [9]. We introduce segment topologies to state some resemblances of (v ⁰) to the family of Ramsey null sets. To describe add(v ⁰) we adopt a proof of Base Matrix Lemma. Consistent results are stated, too. Halbeisen’s conjecture cov(v ⁰) = add(v ⁰) is confirmed under the hypothesis t = min{cf(c), r}. The hypothesis cov(v ⁰) = ω ₁ implies that (v ⁰) has the ideal type (c, ω ₁, c).      

A Springer Theorem for Hermitian forms

Throughout this paper, we let (D,σ) be a central F -division algebra with involution σ such that F^σ={d ∈ F|σ(d)=d} is a Henselian valued field. By [11], the valuation on F^σ extends uniquely to a valuation on D. We denote this valuation by v. Moreover, we assume that the characteristic of the residue field, , is not 2. If the valuation on F is discrete, then any quadratic form q can be written as q= q₁⊥πq₂, where π is a uniformizer and q_i are unit forms. Springer's Theorem states that q is isotropic if and only if at least one of the residue forms and is isotropic. In this paper we generalize this result to ɛ -Hermitian forms. In Section 4, we use the connection between involutions on algebras and ɛ-Hermitian forms to prove an analog of the Springer Theorem for involutions. This paper was part of the author's doctoral dissertation at New Mexico State University. The author wishes to thank his advisor Pat Morandi for his tireless help.     

On tests of hypotheses about treatment effects and treatmentX places interactions,in two heteroscedastic experiments

Summary Suppose we have two independent experiments conducted with a set of ‘t’ treatments each, at different places. This paper deals with two interesting problems of testing of hypotheses associated with these experiments. The first problem deals with the test of the equality of the respective treatment effects in the two experiments. The second problem is concerned with the testing of the equality of treatment into places interactions. Though we assume normality, the variance σ ₁ ² in one experiment is assumed different from the variance σ ₂ ² in the other experiment. When no information is available aboutR=σ ₁ ² /(σ ₁ ² +σ ₂ ² ) except that 0≦R≦1, tests known as ‘bilateral tests’ are proposed in the literature, to test the hypotheses mentioned above. This paper studies some important small sample properties of these bilateral tests. More specifically we study the probability of the first and second kind of error of these bilateral tests as a function ofR. When the two experiments have the same number of observations on each treatment, the bilateral test is shown to control the first kind of error. Fort=1,2, the level of these tests is a strictly convex function ofR and hence these tests can be very conservative. Some power properties of these tests are also obtained. Two tests which are equivalent to the bilateral tests for large sample sizes, and which are superior to the bilateral tests for small sample sizes, are obtained.     

Intégration analytique d'un système d'équations différentielles non linéaires dans le voisinage d'un point singulier,II

Résumé Nous avons formé dans le premier mémoire une solution formelle de(28.1) ordonnée suivant les puissances entière des solutions de(E). En annulant les Z_j qui correspondent aux polynomes de degrés différents de σ_v*(<σ_i*), nous obtenons une solution formelle (F _v). Dans le chapitre III, nous cherchons la condition suffisante pour la convergence uniforme de la série formelle (F _v). Dans le chapi tre IV, ce résultat est étendu au cas où la solution formelle dépend essentiellement des polynomes λ_j(x) de divers degrés. Le dernier chapitre est consacré à l'extension du théorème d'existence deM. M. Hukuhara qui joue un r?le important dans notre Mémoire.     

A bifurcation problem governed by the boundary condition I

We deal with positive solutions of Δu = a(x)u ^p in a bounded smooth domain subject to the boundary condition ∂u/∂v = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ₁ where, roughly speaking, σ₁ is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile of the solution as λ → σ₁. Supported by DGES and FEDER under grant BFM2001-3894 (J. García-Melián and J. Sabina) and ANPCyT PICT No. 03-05009 (J. D. Rossi). J.D. Rossi is a member of CONICET.     

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     

Additive maps between standard operator algebras compressing certain spectral functions

We characterize the surjective additive maps compressing the spectral function Δ（·） between standard operator algebras acting on complex Banach spaces, where Δ（·） stands for any one of nine spectral functions σ（·）, σl（·）, σr（·）,σl（·） ∩ σr（·）, δσ（·）, ησ（·）, σap（·）, σs（·）, and σap（·） ∩ σs（·）.