On a new family of complete G 2-holonomy Riemannian metrics on S 3 × ℝ4

Studying a system of first-order nonlinear ordinary differential equations for the functions determining a deformation of the standard conic metric over S ³ × S ³, we prove the existence of a one-parameter family of complete G ₂-holonomy Riemannian metrics on S ³ × ?⁴.     

The Semifields of Order q 4 Where q is a Power of 2

The aim of this article is to investigate all semifields A of order q ⁴ where q is a power of 2, having the properties that A contains GF(q ²) in its left nucleus and admits a four-group E ? Z ₂ × Z ₂ of automorphisms of A acting freely on A.     

Timelike surfaces of constant mean curvature ±1 in anti-de Sitter 3-space ℍ3 1(−1)

It is shown that timelike surfaces of constant mean curvature ± in anti-de Sitter 3-space ?³ ₁(?1) can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in ?SL₂? via Bryant type representation formulae. These Bryant type representation formulae are used to investigate an explicit one-to-one correspondence, the so-called Lawson–Guichard correspondence, between timelike surfaces of constant mean curvature ± 1 and timelike minimal surfaces in Minkowski 3-space E ³ ₁. The hyperbolic Gauß map of timelike surfaces in ?³ ₁(?1), which is a close analogue of the classical Gauß map is considered. It is discussed that the hyperbolic Gauß map plays an important role in the study of timelike surfaces of constant mean curvature ± 1 in ?³ ₁(?1). In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gauß map and timelike surface of constant mean curvature ± 1 in ?³ ₁(?1) is studied.     

The periodicity in stable equivariant surgery

The classical surgery theory (see [5] and [23]) computes the structure set S_m (M, rel ?) of manifolds homotopy equivalent to M relative to the boundary. Siebenmann showed that in topological category, the structure set is 4-periodic: S_m(M, rel ?) ? S_m+4(M × D⁴, rel ?) up to a copy of ?; see [12]. Cappell and Weinberger gave a geometric interpretation of this periodicity in [8]. By using Weinberger's stratified surgery theory (see [24]), we extend this to an equivariant periodicity result for topological manifolds with homotopically stratified actions by compact Lie groups, with D⁴ replaced by the unit ball of certain group representations. In particular, if G is an odd order group acting on a topological manifold M, then the equivariant stable structure sets satisfy S (M, rel ?) ? S(M × D(?⁴ ? ?G), rel ?) up to copies of ?. © 1993 John Wiley & Sons, Inc.     

On Generators of the Tame Automorphism Group of Free Metabelian Lie Algebras

We prove that the tame automorphism group TAut(M _n ) of a free metabelian Lie algebra M _n in n variables over a field k is generated by a single nonlinear automorphism modulo all linear automorphisms if n ≥ 4 except the case when n = 4 and char(k) ≠ 3. If char(k) = 3, then TAut(M ₄) is generated by two automorphisms modulo all linear automorphisms. We also prove that the tame automorphism group TAut(M ₃) cannot be generated by any finite number of automorphisms modulo all linear automorphisms.     

Thermal expansion of ferroelectrics

Using the interference method the thermal expansion of lithium hydrazinium sulphate [Li(N₂H₅)SO₄] has been investigated in the temperature range ?170° C. to +220° C. The principal expansion coefficients along the crystallographic axes have been determined and at room temperature (25° C.) the values ofa _a,a _b anda _c are respectively 17·7, 13·5, and 43·0, ×10^?6 (°C.)^?1. The expansion coefficients are found to vary nonlinearly, and, in particular, along thec-axis the thermal expansion shows an anomalous behaviour between ?160°C. and ?60°C. and also in the neighbourhood of +130°C. These anomalies are explained as due to homomorphous transitions in which there is a reorientation of the ?NH ₊ ³ group about the N-N axis or a re-alignment of the ?NH₂ group.     

On the second cohomology of an algebraic group and of its lie algebra in a positive characteristic

Necessary and sufficient isomorphism conditions for the second cohomology group of an algebraic group with an irreducible root system over an algebraically closed field of characteristic p ≥ 3h ? 3, where h stands for the Coxeter number, and the corresponding second cohomology group of its Lie algebra with coefficients in simple modules are obtained, and also some nontrivial examples of isomorphisms of the second cohomology groups of simple modules are found. In particular, it follows from the results obtained here that, among the simple algebraic groups SL₂(k), SL₃(k), SL₄(k), Sp₄(k), and G ₂, nontrivial isomorphisms of this kind exist for SL₄(k) and G ₂ only. For SL₄(k), there are two simple modules with nontrivial second cohomology and, for G ₂, there is one module of this kind. All nontrivial examples of second cohomology obtained here are one-dimensional.     

The Autotopism Group of A Semifield of Order p 4, p is an Odd Prime

The aim of this article is to investigate the autotopism group of a semifield of order p ⁴, p is an odd prime, admitting a four-group of automorphisms E? Z ₂ × Z ₂ acting freely on A.     

Global continuation for first order systems over the half‐line involving parameters

Let X be one of the functional spaces W^1,p ((0, ∞), ?^N ) or C₀¹ ([0, ∞), ?^N ), we study the global continuation in λ for solutions (λ, u, ξ) ∈ ? × X × ?^k of the following system of ordinary differential equations: where ?^N = X₁ ⊕ X₂ is a given decomposition, with associated projection P: ?^N → X₁. Under appropriate conditions upon the given functions F and φ, this problem gives rise to a nonlinear Fredholm operator which is proper on the closed bounded subsets of ? × X × ?^k and whose zeros correspond to the solutions of the original problem. Using a new abstract continuation result, based on a recent degree theory for proper Fredholm mappings of index zero, we reduce the continuation problem to that of finding a priori estimates for the possible solutions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Schurity of Finite Abelian Groups

A finite group G is called a Schur group, if any Schur ring over G is associated in a natural way with a subgroup of Sym(G) that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this article, it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any noncyclic abelian Schur group of odd order is isomorphic to ?₃ × ?_{3 ^k} or ?₃ × ?₃ × ? _p where k ≥ 1 and p is a prime. In addition, we prove that ?₂ × ?₂ × ? _p is a Schur group for every prime p.     

A polarographic study of methionine complexes of cadmium and indium

The stability constants of methionine complexes of cadmium were determined polarographically by ths method of DeFord and Hume as β₁ =6·5 × 10³, β₂ = 1·7 × 10⁶ and β₃ = 2·1 · 10⁸. The indium complexes were studied by the modified method of Momoki and Ogawa and two complexes, with β₁ = 1·7 × 10⁸ and β₂ = 8·4 × 10¹³, were identified in the concentration range studied. The haf-wave potential of uncomplexed indium ion which cannot be measured directly owing to the irreversible nature of the reduction was calculated as — 0·503 Vvs. SCE.     

Commutative Rings Whose Cozero-Divisor Graphs are Unicyclic or of Bounded Degree

Let R be a commutative ring with unity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex set W*(R), where W*(R) is the set of all nonzero and nonunit elements of R, and two distinct vertices a and b are adjacent if and only if a ? Rb and b ? Ra, where Rc is the ideal generated by the element c in R. Recently, it has been proved that for every nonlocal finite ring R, Γ′(R) is a unicyclic graph if and only if R ? ?₂ × ?₄, ?₃ × ?₃, ?₂ × ?₂[x]/(x ²). We generalize the aforementioned result by showing that for every commutative ring R, Γ′(R) is a unicyclic graph if and only if R ? ?₂ × ?₄, ?₃ × ?₃, ?₂ × ?₂[x]/(x ²), ?₂[x, y]/(x, y)², ?₄[x]/(2x, x ²). We prove that for every positive integer Δ, the set of all commutative nonlocal rings with maximum degree at most Δ is finite. Also, we classify all rings whose cozero-divisor graph has maximum degree 3. Among other results, it is shown that for every commutative ring R, gr(Γ′(R)) ∈ {3, 4, ∞}.     

The n �� n KdV hierarchy

We introduce two new soliton hierarchies that are generalizations of the KdV hierarchy. Our hierarchies are restrictions of the AKNS n × n hierarchy coming from two unusual splittings of the loop algebra. These splittings come from automorphisms of the loop algebra instead of automorphisms of sl (n, \mathbbC){sl (n, \mathbb{C})} . The flows in the hierarchy include systems of coupled nonlinear Schr?dinger equations. Since they are constructed from a Lie algebra splitting, the general method gives formal inverse scattering, bi-Hamiltonian structures, commuting flows, and B?cklund transformations for these hierarchies.     

Well‐posedness for the 2‐D damped wave equations with exponential source terms

In this paper we study well‐posedness of the damped nonlinear wave equation in Ω × (0, ∞) with initial and Dirichlet boundary condition, where Ω is a bounded domain in ?²; ω?0, ωλ₁+µ>0 with λ₁ being the first eigenvalue of ?Δ under zero boundary condition. Under the assumptions that g(·) is a function with exponential growth at the infinity and the initial data lie in some suitable sets we establish several results concerning local existence, global existence, uniqueness and finite time blow‐up property and uniform decay estimates of the energy. Copyright © 2010 John Wiley & Sons, Ltd.     

Taylor–Couette problem and related topics

The O(2)×S¹-equivariant degree is applied to study the Hopf bifurcation for the Taylor–Couette problem. Isotypical crossing numbers and equivariant bifurcation invariants are evaluated and are applied to classify the equivariant Hopf bifurcation for the Taylor–Couette flow.     

On the finiteness of the number of orbits on quasihomogeneous (ℂ*)k × SL 2(ℂ)-manifolds

We obtain an effective criterion for the finiteness of the number of orbits contained in the closure of a given G-orbit for the case of a rational linear action of the group G: = (?*)^k × SL ₂(?) on a finite-dimensional linear space as well as on the projectivization of such a space.     

Equivariant Pieri Rule for the homology of the affine Grassmannian

An explicit rule is given for the product of the degree two class with an arbitrary Schubert class in the torus-equivariant homology of the affine Grassmannian. In addition a Pieri rule (the Schubert expansion of the product of a special Schubert class with an arbitrary one) is established for the equivariant homology of the affine Grassmannians of SL _n and a similar formula is conjectured for Sp _2n and SO _2n+1. For SL _n the formula is explicit and positive. By a theorem of Peterson these compute certain products of Schubert classes in the torus-equivariant quantum cohomology of flag varieties. The SL _n Pieri rule is used in our recent definition of k-double Schur functions and affine double Schur functions.     

The Second Cohomology of Simple SL 3-Modules

Let G be the simple, simply connected algebraic group SL ₃ defined over an algebraically closed field K of characteristic p > 0. In this article, we find H ²(G, V) for any irreducible G-module V. When p > 7, we also find H ²(G(q), V) for any irreducible G(q)-module V for the finite Chevalley groups G(q) = SL(3, q) where q is a power of p.     

A polarographic study of methionine complexes of cadmium in mixed solvents

Methionine complexes of cadmium in 25 and 50 per cent aqueous mixtures of ethyl and methyl alcohol and dioxan have been studied. The half-wave potentials measured in both the alcohols were the same and the reduction was reversible. Three complex species withβ ₁=1·0×10⁴,β ₂=1·1×10⁷ andβ ₃=1·2×10⁹ were found in 25 per cent alcohol while four complexes withβ ₁=3·0×10⁴,β ₂=4·3×10⁷,β ₃=4·0×10⁹ andβ ₄=1·6×10¹¹ were observed in 50 per cent solutions. In the case of dioxan, the reduction was quasi-reversible (k _s=1·0×10^?3 cm sec^?1) in 25 per cent and irreversible (k _s=2·0×10^?4 cm sec^?1) in 50 per cent solutions. The stability constants, evaluated using the formal potentials, wereβ ₁=7·0×10³,β ₂-3·9×10⁵;β ₂=3·9×10⁸ andβ ₄=3·4×10¹⁰ in 25 per cent dioxan andβ ₁=1·5×10⁴,β ₂=3·4×10⁷.β ₃=7·5×10⁹ andβ ₄=9·0×10¹¹ in 50 per cent solutions.     

On the several differences between primes

Enumeration of the primes with difference 4 between consecutive primes, is counted up to 5×10¹⁰, yielding the counting function π_2,4(5 × 10¹⁰) = 118905303. The sum of reciprocals of primes with gap 4 between consecutive primes is computedB ₄(5×10¹⁰)=1.197054473029 andB ₄=1.197054±7×10^?6. And Enumeration of the primes with difference 6 between consecutive primes, is counted up to 5×10¹⁰, yielding the counting function π_2,6(5 × 10¹⁰) = 215868063. The sum of reciprocals of primes with gap 6 between consecutive primes is computedB ₆(5×10¹⁰)=0.93087506039231 andB ₆=1.135835±1.2×10^?6.