Random process in a homogeneous Gaussian field

We consider a random process in a spatial-temporal homogeneous Gaussian field V (q , t) with the mean E V = 0 and the correlation function W(|q ? q′|, |t ? t′|) ≡ E[V (q, t)V (q′, t′)], where \( \bold{q} \in {\mathbb{R}^d} \), \( t \in {\mathbb{R}^{+} } \), and d is the dimension of the Euclidean space \( {\mathbb{R}^d} \). For a “density” G(r, t) of the familiar model of a physical system averaged over all realizations of the random field V, we establish an integral equation that has the form of the Dyson equation. The invariance of the equation under the continuous renormalization group allows using the renormalization group method to find an asymptotic expression for G(r, t) as r → ∞ and t → ∞.     

Propagation of sound pulses in a semiinfinite stratified medium

We consider the propagation of sound pulses due to a line source in the inhomogeneous, semi-infinite mediumy ≥ 0 with the boundary conditionφ=0 or?φ/(?y)=0 aty=0, whereφ is the acoustic velocity potential. We suppose that the velocity of wave propagation,c, is given byc ^?2=p ?qe ^?ay , wherep, q, α are real and positive andp>q. The method of dual integral transformation is used. The solution in terms of pulse propagation modes yields the diffracted pulse and the method of steepest descents gives the geometrical acoustic field.     

Study of anomalous kinetics of the annihilation reaction <Emphasis Type="Italic">A</Emphasis> + <Emphasis Type="Italic">A</Emphasis> → Ø

Using the perturbative renormalization group, we study the influence of a random velocity field on the kinetics of the single-species annihilation reaction A + A → Ø at and below its critical dimension d_c = 2. We use the second-quantization formalism of Doi to bring the stochastic problem to a field theory form. We investigate the reaction in spaces of dimension d ～ 2 using a two-parameter expansion in ε and Δ, where ε is the deviation from the Kolmogorov scaling parameter and Δ is the deviation from the space dimension d = 2. We evaluate all the necessary quantities, including fixed points with their regions of stability, up to the second order of the perturbation theory.     

On the derived length of units in group algebra

Let G be a finite group G, K a field of characteristic p ≥ 17 and let U be the group of units in KG. We show that if the derived length of U does not exceed 4, then G must be abelian.     

Descent properties of hermitian Witt groups in inseparable extensions

Let k be a field of characteristic ≠ 2, A be a central simple algebra with involution σ over k and W(A, σ) be the associated Witt group of hermitian forms. We prove that for all purely inseparable extensions L of k, the canonical map \({r_{L/k}: W(A, \sigma) \longrightarrow W(A_L, \sigma_L)}\) is an isomorphism.     

A remark on the capitulation problem

We determine explicitly an infinite family of imaginary cyclic number fields k, such that the 2-class group of k is elementary with arbitrary large 2-rank and capitulates in an unramified quadratic extension K. The infinitely many number fields k and K have the same Hilbert 2-class field and an infinite Hilbert 2-class field tower.     

The Uncountable Lattice of All Subfunctors of the Functor of Rational Representations

In set theory the cardinality of the continuum \(|{\mathbb R}|\) is the cardinal number of some interesting sets, like the Cantor set or the transcendental numbers. We will prove that the cardinal number of all subfunctors of the functor of rational representations \(k \otimes_{{\mathbb Z}} R_{{\mathbb Q}}\), taking values on odd order groups over the field k of characteristic 2, is equal to \(|{\mathbb R}|\). When the characteristic q?>?0 of the field k is not necessarily even, we will present a formula giving the dimension of the evaluations S _C,k(G), of the simple functor S _C,k, at any group G of order prime to q and being associated to a suitable cyclic group C.     

Periodic Groups Saturated with the Groups <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Given an indexing set I and a finite field K_α for each α ∈ I, let ? = {L₂(K_α) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L₂(P) (respectively SL₂(P)) for a suitable locally finite field P.     

On spectra of almost simple groups with symplectic or orthogonal socle

Finite groups are said to be isospectral if they have the same sets of the orders of elements. We investigate almost simple groups H with socle S, where S is a finite simple symplectic or orthogonal group over a field of odd characteristic. We prove that if H is isospectral to S, then H/S presents a 2-group. Also we give a criterion for isospectrality of H and S in the case when S is either symplectic or orthogonal of odd dimension.     

QUANDLE VARIETIES,GENERALIZED SYMMETRIC SPACES,AND <Emphasis Type="Italic">φ</Emphasis>-SPACES

We define a quandle variety as an irreducible algebraic variety Q endowed with an algebraically defined quandle operation ?. It can also be seen as an analogue of a generalized affine symmetric space or a regular s-manifold in algebraic geometry.Assume that Q is normal as an algebraic variety and that the action of its inner automorphism group Inn(Q) has a dense orbit. Then we show that there is an algebraic group G acting on Q with the same orbits as Inn(Q) such that each G-orbit is isomorphic to the quandle (G/H, ?φ) associated to the group G, an automorphism φ of G and a subgroup H of Gφ.     

On the Győry-Sárközy-Stewart conjecture in function fields

We consider function field analogues of the conjecture of Gy?ry, Sárközy and Stewart (1996) on the greatest prime divisor of the product (ab+1)(ac+1)(bc+1) for distinct positive integers a, b and c. In particular, we show that, under some natural conditions on rational functions F,G,H ∈ ?(X), the number of distinct zeros and poles of the shifted products FH +1 and GH +1 grows linearly with degH if degH > max{deg F, degG}. We also obtain a version of this result for rational functions over a finite field.     

Algebraic conditions for additive functions over the reals and over finite fields

Let C be an affine plane curve. We consider additive functions \(f{:}\; K\rightarrow K\) for which \(f(x)f(y)=0\), whenever \((x,y)\in C\). We show that if \(K=\mathbb {R}\) and C is the hyperbola with defining equation \(xy=1\), then there exist nonzero additive functions with this property. Moreover, we show that such a nonzero f exists for a field K if and only if K is transcendental over \(\mathbb Q\) or over \(\mathbb {F}_p\), the finite field with p elements. We also consider the general question when K is a finite field. We show that if the degree of the curve C is large enough compared to the characteristic of K, then f must be identically zero.     

Stability of Discontinuous Groups Acting on Homogeneous Spaces

Suppose given a nilpotent connected simply connected Lie group G, a connected Lie subgroup H of G, and a discontinuous group Γ for the homogeneous space M = G/H. In this work we study the topological stability of the parameter space R(Γ,G,H) in the case where G is three-step. We prove a stability theorem for certain particular pairs (Γ,H). We also introduce the notion of strong stability on layers making use of an explicit layering of Hom(Γ,G) and study the case of Heisenberg groups.     

The unramified computation of Rankin–Selberg integrals expressed in terms of Bessel models for split orthogonal groups: Part I

In this paper, we compute the local integrals, with normalized unramified data, over a p-adic field F, arising from general Rankin–Selberg integrals for SO _m × GL_r+k+1, where the orthogonal group is split over F, \(k \leqslant \left[ {\frac{{m - 1}}{2}} \right]\), and the irreducible representation of SO _m (F) has a Bessel model with respect to an irreducible representation of the split orthogonal group SO_m?2k?1(F). Our proof is by “analytic continuation from the unramified computation in the generic case”. We let the unramified parameters of the representations involved vary, and express the local integrals in terms of the Whittaker models of the representations, which exist at points in general position. Then we apply analytic continuation and the known unramified computation in the generic case. We discuss some applications to poles of partial L-functions and functorial lifting.     

On the cohomology of tori over local fields with perfect residue field

If T is an algebraic torus defined over a discretely valued field K with perfect residue field k, we relate the K-cohomology of T to the k-cohomology of certain objects associated to T. When k has cohomological dimension ≤ 1, our results have a particularly simple form and yield, more generally, isomorphisms between Borovoi’s abelian K-cohomology of a reductive group G over K and the k-cohomology of a certain quotient of the algebraic fundamental group of G.     

<Emphasis Type="Italic">k</Emphasis>-invariant nets over an algebraic extension of a field <Emphasis Type="Italic">k</Emphasis>

Let K be an algebraic extension of a field k, let σ = (σ _ij ) be an irreducible full (elementary) net of order n ≥ 2 (respectively, n ≥ 3) over K, while the additive subgroups σ _ij are k-subspaces of K. We prove that all σij coincide with an intermediate subfield P, k ? P ? K, up to conjugation by a diagonal matrix.     

Central extensions of generalized orthosymplectic Lie superalgebras

We completely determine the universal central extension of the generalized orthosymplectic Lie superalgebra osp _m|2n (R,-) that is coordinatized by an arbitrary unital associative superalgebra (R,-) with superinvolution. As a result, an identification between the second homology group of the Lie superalgebra osp _m|2n (R,-) and the first skew-dihedral homology group of the associative superalgebra (R,-) with superinvolution is created for positive integers m and n with (m,n) ≠ (1,1) and (m, n) ≠ (2,1). The second homology groups of the Lie superalgebras osp_1|2(R,-) and osp_2|2(R,-) are also characterized explicitly.     

On hereditary superradical formations

A formation F is superradical provided that: (1) F is a normally hereditary formation; (2) each group G = AB, where A and B are F-subnormal F-subgroups in G, belongs to F. We give an example of a hereditary superradical formation that is not soluble saturated. This gives a negative answer to Problem 14.99(b) in The Kourovka Notebook.     

On collineation groups of finite projective spaces containing a Singer cycle

By a result of Kantor, any subgroup of GL(n, q) containing a Singer cycle normalizes a field extension subgroup. This result has as a consequence a projective analogue, and this paper gives the details of this deduction, showing that any subgroup of PΓL(n, q) containing a projective Singer cycle normalizes the image of a field extension subgroup GL(n/s, q^s) under the canonical homomorphism GL(n, q) → PGL(n, q), for some divisor s of n, and so is contained in the image of ΓL(n/s, q^s) under the canonical homomorphism ΓL(n, q) → PΓL(n, q). The actions of field extension subgroups on V (n, q) are also investigated. In particular, we prove that any field extension subgroup GL(n/s, q^s) of GL(n, q) has a unique orbit on s-dimensional subspaces of V (n, q) of length coprime to q. This orbit is a Desarguesian s-partition of V (n, q).