Shmel’kin embeddings for abstract and profinite groups

The Magnus embedding is well known: given a group A=F/R, where F is a free group, the group F/[R, R] can be represented as a subgroup of a semidirect product AT, where T is an additive group of a free Z A-module. Shmel’kin genralized this construction and found an embedding for F/V(R), where V(R) is the verbal subgroup of R corresponding to a variety V. Later, he treated F as a free product of arbitrary groups, and on condition that R is contained in a Cartesian subgroup of the product, pointed out an embedding for F/V(R). Here, we combine both these Shmel’kin embeddings and weaken the condition on R, by assuming that F is a free product of groups A_i (iεI) and a free group X, and that its normal subgroup R has trivial intersection with each factor A_i. Subject to these conditions, an embedding for F/V(R) is found; we cell it the generalized Shmel’kin embedding. For the case where V is an Abelian variety of groups, a criterion is specified determining whether elements of AT belong to an embedded group F/V(R). Similar results are proved also for profinite groups. Supported by RFFR grant No. 99-01-00567. Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 598–612, September–October, 1999.     

Cauchy problem for equations with multiple characteristics

This paper contains a proof of γ^n(χ) correctness of the noncharacteristic Cauchy problem for nonstrictly hyperbolic equations with analytic coefficients under the condition that its characteristic roots are smooth and under some additional assumptions on the lower-order terms. There are two extreme cases: (1) $\text{χ <}\text{r}\text{r}\text{? 1}$. In this case condition (0.6) is “void,” and we do not require conditions on P_s for s < m. For this case, see [3, 8]. (2) Case of constant multiplicity of characteristic roots and χ = +∞. In this case condition (0.6) implies conditions on P_s, where s = m, m ? 1,…, m ? r + 1, i.e., up to the same order as the necessary condition for C^∞-correctness [2]. Recall that in the case of equations with characteristics of constant multiplicity condition (0.6) (Levi's condition in this case) for χ = ∞ is necessary [2, 4] and sufficient [1] for C^∞-correctness.     

On the compact-open and admissible topologies

It is known (see, for example, [H. Render, Nonstandard topology on function spaces with applications to hyperspaces, Trans. Amer. Math. Soc. 336 (1) (1993) 101-119; M. Escardo, J. Lawson, A. Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143 (2004) 105-145; D.N. Georgiou, S.D. Iliadis, F. Mynard, Function space topologies, in: Open Problems in Topology 2, Elsevier, 2007, pp. 15-23]) that the intersection of all admissible topologies on the set C(Y,Z) of all continuous maps of an arbitrary space Y into an arbitrary space Z, is always the greatest splitting topology (which in general is not admissible). The following, interesting in our opinion, problem is arised: when a given splitting topology (for example, the compact-open topology, the Isbell topology, and the greatest splitting topology) is the intersection of k admissible topologies, where k is a finite number. Of course, in this case this splitting topology will be the greatest splitting.In the case, where a given splitting topology is admissible the above number k is equal to one. For example, if Y is a locally compact Hausdorff space, then k=1 for the compact-open topology (see [R.H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945) 429-432; R. Arens, A topology for spaces of transformations, Ann. of Math. 47 (1946) 480-495; R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31]). Also, if Y is a corecompact space, then k=1 for the Isbell topology (see [P. Lambrinos, B.K. Papadopoulos, The (strong) Isbell topology and (weakly) continuous lattices, in: Continuous Lattices and Applications, in: Lect. Notes Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1984, pp. 191-211; F. Schwarz, S. Weck, Scott topology, Isbell topology, and continuous convergence, in: Lect. Notes Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1984, pp. 251-271]).In [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31] a non-locally compact completely regular space Y is constructed such that the compact-open topology on C(Y,S), where S is the Sierpinski space, coincides with the greatest splitting topology (which is not admissible). This fact is proved by the construction of two admissible topologies on C(Y,S) whose intersection is the compact-open topology, that is k=2.In the present paper improving the method of [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31] we construct some other non-locally compact spaces Y such that the compact-open topology on C(Y,S) is the intersection of two admissible topologies. Also, we give some concrete problems concerning the above arised general problem.     

Über Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1)u′(t)=F(t,u) for )≤t≤∞ together with a generalized initial condition (2)u(t)=?(t) forr≤t≤0 or a generalized Nicoletti condition (3)N u=η. Here,N is a linear operator; in the case of a system ofn equations the classical Nicoletti operator is given byN u=(u ₁(t ₁),...,u _n(t _n)), with givent _i. The functionsu, F ? are Banach space valued, the functionF(t, z) is defined fort≥0 andz∈C ⁰[r,∞). The main point is that the value ofF(t, z) may depend on the values ofz(s) forr≤s≤t+σ(t), where σ(t)>0. Simple examples show that without a restriction on the magnitude of the advancement σ(t) there is neither existence nor uniqueness. Our results show that when σ(t) is properly bounded and when the solution is to satisfy a certain growth condition which depends on σ(t), then there exists exactly one solution, and it depends continuously on the given data. In the case of the Nicoletti problem (1), (3) there is convergence to the solution satisfyingu(0)=η if 0≤t _i≤T andT→0 (this holds in infinite-dimensional spaces, too). These results are true ifF satisfies a Lipschitz condition of the form $$\left| {F(t,z) - F(t,y)} \right| \leqslant h(t)\max \left\{ {\left| {z(s) - y(s)} \right|:r \leqslant s \leqslant t + \delta (t)} \right\}.$$ . In the case where (1) is a finite system andF is only continuous, an existence theory is developped based onSchauder's fixed point theorem. Again, growth conditions play an essential role here.     

The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates

For a second-order symmetric strongly elliptic operator A on a smooth bounded open set in Rn, the mixed problem is defined by a Neumann-type condition on a part Σ₊ of the boundary and a Dirichlet condition on the other part Σ₋. We show a Kre?n resolvent formula, where the difference between its resolvent and the Dirichlet resolvent is expressed in terms of operators acting on Sobolev spaces over Σ₊. This is used to obtain a new Weyl-type spectral asymptotics formula for the resolvent difference (where upper estimates were known before), namely , where C0,+ is proportional to the area of Σ₊, in the case where A is principally equal to the Laplacian.     

On the greatest splitting topology

It is well known that (see, for example, [H. Render, Nonstandard topology on function spaces with applications to hyperspaces, Trans. Amer. Math. Soc. 336 (1) (1993) 101-119; M. Escardo, J. Lawson, A. Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143 (2004) 105-145; D.N. Georgiou, S.D. Iliadis, F. Mynard, in: Elliott Pearl (Ed.), Function Space Topologies, Open Problems in Topology, vol. 2, Elsevier, 2007, pp. 15-22]) the intersection of all admissible topologies on the set C(Y,Z) of all continuous maps of an arbitrary space Y into an arbitrary space Z, is always the greatest splitting topology. However, this intersection maybe not admissible. In the case, where Y is a locally compact Hausdorff space the compact-open topology on the set C(Y,Z) is splitting and admissible (see [R.H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945) 429-432; R. Arens, A topology for spaces of transformations, Ann. of Math. 47 (1946) 480-495; R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31]), which means that the intersection of all admissible topologies on C(Y,Z) is admissible. In [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31] an example of a non-locally compact Hausdorff space Y is given having the same property for the case, where Z=[0,1], that is on the set C(Y,[0,1]) the compact-open topology is splitting and admissible. This space Y is the set [0,1] with a topology τ, whose semi-regular reduction coincides with the usual topology on [0,1]. Also, in [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31, Theorem 5.3] another example of a non-locally compact space Y is given such that the compact-open topology on the set C(Y,[0,1]) is distinct from the greatest splitting topology.In this paper first we construct non-locally compact Hausdorff spaces Y such that the intersection of all admissible topologies on the set C(Y,Z), where Z is an arbitrary regular space, is admissible. Furthermore, for a Hausdorff splitting topology t on C(Y,Z) we find sufficient conditions in order that t to be distinct from the greatest splitting topology. Using this result, we construct some concrete non-locally compact spaces Y such that the compact-open topology on C(Y,Z), where Z is a Hausdorff space, is distinct from the greatest splitting topology. Finally, we give some open problems.     

On the conjugacy problem in the group F/N 1 ∩ N 2

Let N ₁ (N ₂) be the normal closure of a finite symmetrized set R ₁ (R ₂, respectively) in a finitely generated free group F = F(A). As is known, if R _i satisfies condition C(6), then the conjugacy problem is decidable in F/N _i . In the paper, it is proved that, if one adds to condition C(6) on the set R ₁ ∪ R ₂ the atoricity condition for the presentation 〈A | R ₁, R ₂〉, then the conjugacy problem is decidable in the group F/N ₁ ∩ N ₂ as well. In particular, for the decidability of the conjugacy problem in F/N ₁ ∩ N ₂, it is sufficient to assume that the set R ₁ ∪ R ₂ satisfies condition C(7).     

Gauss–Newton method for convex composite optimizations on Riemannian manifolds

A notion of quasi-regularity is extended for the inclusion problem ${F(p)\in C}$ , where F is a differentiable mapping from a Riemannian manifold M to ${\mathbb R^n}$ . When C is the set of minimum points of a convex real-valued function h on ${\mathbb R^n}$ and DF satisfies the L-average Lipschitz condition, we use the majorizing function technique to establish the semi-local convergence of sequences generated by the Gauss-Newton method (with quasi-regular initial points) for the convex composite function h ? F on Riemannian manifold. Two applications are provided: one is for the case of regularities on Riemannian manifolds and the other is for the case when C is a cone and DF(p ₀)(·) ? C is surjective. In particular, the results obtained in this paper extend the corresponding one in Wang et?al. (Taiwanese J Math 13:633?C656, 2009).     

Some observations on maximum weight stable sets in certain P5-free graphs

The maximum weight stable set problem (MWS) is the weighted version of the maximum stable set problem (MS), which is NP-hard. The class of P₅-free graphs – i.e., graphs with no induced path of five vertices – is the unique minimal class, defined by forbidding a single connected subgraph, for which the computational complexity of MS is an open question. At the same time, it is known that MS can be efficiently solved for (_P5,F)$(P_5\text{,}F)$-free graphs, where F is any graph of five vertices different to a C₅. In this paper we introduce some observations on P₅-free graphs, and apply them to introduce certain subclasses of such graphs for which one can efficiently solve MWS. That extends or improves some known results, and implies – together with other known results – that MWS can be efficiently solved for (_P5,F)$(P_5\text{,}F)$-free graphs where F is any graph of five vertices different to a C₅.     

Local boundedness of monotone bifunctions

We consider bifunctions ${F : C\times C\rightarrow \mathbb{R}}$ where C is an arbitrary subset of a Banach space. We show that under weak assumptions, monotone bifunctions are locally bounded in the interior of their domain. As an immediate corollary, we obtain the corresponding property for monotone operators. Also, we show that in contrast to maximal monotone operators, monotone bifunctions (maximal or not maximal) can also be locally bounded at the boundary of their domain; in fact, this is always the case whenever C is a locally polyhedral subset of ${\mathbb{R}^{n}}$ and F(x, ·) is quasiconvex and lower semicontinuous.     

Sobolev spaces on an arbitrary metric measure space: Compactness of embeddings

We formulate a new definition of Sobolev function spaces on a domain of a metric space in which the doubling condition need not hold. The definition is equivalent to the classical definition in the case that the domain lies in a Euclidean space with the standard Lebesgue measure. The boundedness and compactness are examined of the embeddings of these Sobolev classes into L _q and C _α . We state and prove a compactness criterion for the family of functions L _p (U), where U is a subset of a metric space possibly not satisfying the doubling condition.     

Problème de Cauchy sur un conoïde caractéristique pour des équations semi-linéaires hyperboliques

The aim of this work is to find a semi-global solution to the Cauchy problem (P) on a characteristic conoid C₀, that is which is defined not only neighbourhood of the sumitt O, but all around the conoid parts C₀ which shows the Cauchy data. In this respect, we will use the method of Kirchhoff's formulae constructed by Y. Choquet-Bruhat [4] and the results obtained from the linear case by F. Cagnac [2], Here, it is shown that all solution of (P), five times derivable, verify as such, as its derivatives up to the third order, a system of integral equations. Next this system is solved by the method of successive approximations. In this work, we do not shown that (P) has a solution. However, in a particular semilinear case (where f do not depend on the first partial derivatives), we can show that the solution of the integral system is a generalized solution of the Cauchy problem.     

Unique representation in convex sets by extraction of marked components

After introducing the basic concepts of extraction and marking for convex sets, the following marked representation theorem is established: Let C be a lineally closed convex set without lines, the face lattice of which satisfies some descending chain condition, and let μ be some marking on C. Then every point of C can be represented in unique way as a convex (nonnegative) linear combination of points (directions) of C which are μ-independent, and this representation can be determined by an algorithm of successive extractions. In particular, if C is a finite dimensional closed convex set without lines and μ marks extreme points (directions) only, then the marked representation theorem contains some well-known results of convex analysis as special cases, and it yields in the case where C is a polyhedral triangulation which extends available results on polytopes to the unbounded case. The triangulation of unbounded polyhedra then is applied to a certain class of parametric linear programs.     

Approximation of functions of finite variation by superpositions of a Sigmoidal function

The aim of this note is to generalize a result of Barron [1] concerning the approximation of functions, which can be expressed in terms of the Fourier transform, by superpositions of a fixed sigmoidal function. In particular, we consider functions of the type h(x) = ∫ℝ^d ƒ (〈t, x〉)dμ(t), where μ is a finite Radon measure on ℝ^d and ƒ : ℝ → ℂ is a continuous function with bounded variation in ℝ We show (Theorem 2.6) that these functions can be approximated in L₂-norm by elements of the set G_n = {Σ_i=0^staggeredn c_ig(〈a_i, x〉 + b_i) : a_iℝ^d, b_i, c_iℝ}, where g is a fixed sigmoidal function, with the error estimated by C/n^1/2, where C is a positive constant depending only on f. The same result holds true (Theorem 2.9) for f : ℝ → ℂ satisfying the Lipschitz condition under an additional assumption that ∫ℝ^d6t6e^d|u(t)| > ∞     

TheC ℓ Bailey transform and Bailey lemma

TheC _? nonterminating C_? summation theorem is derived by appropriately specializing Gustafson's₆ψ₆ summation theorem for bilateral basic hypergeometric series very well-poised on symplecticC _? groups. From this, the terminating₆?₅ and, hence, terminating₄?₃ summation theorem is obtained. A suitably modified₄?₃ is then used to derive theC _? generalization of the Bailey transform. The transform is then interpreted as a matrix inversion result for two infinite, lower-triangular matrices. This result is used to motivate the definition of theC _? Bailey pair. TheC _? generalization of Bailey's lemma is then proved. This result is inverted, and the concept of the bilateral Bailey chain is discussed. TheC _? Bailey lemma is then used to obtain a connection coefficient result for generalC _? littleq-Jacobi polynomials. All of this work is a natural extension of the unitaryA _?, or equivalentlyU(?+1), case. The classical case, corresponding toA ₁ or equivalentlyU(2), contains an immense amount of the theory and application of one-variable basic hypergeometric series, including elegant proofs of the Rogers-Ramanujan-Schur identities. TheC _? nonterminating₆?₅ summation theorem is also used to recover C. Krattenthaler's multivariable summation which he utilized in deriving his refinement of the Bender-Knuth and MacMahon generating functions for certain sets of plane partitions.     

C-GV triviality of function germs and Newton polyhedra

We provide estimates on the degree of C l GV determinacy ( G is one of Mather’s groups R or K ) of function germs which are defined on analytic variety V and satisfies a non-degeneracy condition with respect to some Newton polyhedron. The result gives an explicit order such that the C l geometrical structure of a function germ is preserved after higher order perturbations, which generalizes the result on C l G triviality of function germs given by M.A.S.Ruas.     

On a conjecture of Erdős and Simonovits: Even cycles

Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let C _k denote a cycle of length k, and let C _k denote the set of cycles C _?, where 3≤?≤k and ? and k have the same parity. Erd?s and Simonovits conjectured that for any family F consisting of bipartite graphs there exists an odd integer k such that ex(n,F ∪ C _k ) ～ z(n,F) — here we write f(n) ～ g(n) for functions f,g: ? → ? if lim _n→∞ f(n)/g(n)=1. They proved this when F ={C ₄} by showing that ex(n,{C ₄;C ₅})～z(n,C ₄). In this paper, we extend this result by showing that if ?∈{2,3,5} and k>2? is odd, then ex(n,C _2? ∪{C _k }) ～ z(n,C _2? ). Furthermore, if k>2?+2 is odd, then for infinitely many n we show that the extremal C _2? ∪{C _k }-free graphs are bipartite incidence graphs of generalized polygons. We observe that this exact result does not hold for any odd k<2?, and furthermore the asymptotic result does not hold when (?,k) is (3, 3), (5, 3) or (5, 5). Our proofs make use of pseudorandomness properties of nearly extremal graphs that are of independent interest.     

On the existence of singular solutions of the stationary Navier–Stokes problem

We consider the steady Navier–Stokes equations in the punctured regions (?) Ω?=?Ω ₀ \ {o} (with {o} ∈ Ω ₀) and (??) $ \varOmega ={{\mathbb{R}}^2}\backslash \left( {{{\overline{\varOmega}}_0}\cup \left\{ o \right\}} \right) $ (with $ \left\{ o \right\}\notin {{\overline{\varOmega}}_0} $ ), where Ω ₀ is a simple connected Lipschitz bounded domain of $ {{\mathbb{R}}^2} $ . We regard o as a sink or a source in the fluid. Accordingly, we assign the flux $ \mathcal{F} $ through a small circumference surrounding o and a boundary datum a on Γ?=??Ω ₀ such that the total flux $ \mathcal{F}+\int\nolimits_{\varGamma } {\boldsymbol{a}\cdot \boldsymbol{n}} $ is zero in case (?). We prove that if $ \left| \mathcal{F} \right|<2\pi \nu $ and $ \left| \mathcal{F} \right|+\left| {\int\nolimits_{\varGamma } {\boldsymbol{a}\cdot \boldsymbol{n}} } \right|<2\pi \nu $ in (?) and (??), respectively, where ν is the kinematical viscosity, then the problem has a C ^∞ solution in Ω, which behaves at o like the gradient of the fundamental solution of the Laplace equation.     

On the closedness of the algebraic difference of closed convex sets

We characterize in a reflexive Banach space all the closed convex sets C₁ containing no lines for which the condition C₁^∞∩C₂^∞={0} ensures the closedness of the algebraic difference C₁−C₂ for all closed convex sets C₂. We also answer a closely related problem: determine all the pairs C₁, C₂ of closed convex sets containing no lines such that the algebraic difference of any sufficiently small uniform perturbations of C₁ and C₂ remains closed. As an application, we state the broadest setting for the strict separation theorem in a reflexive Banach space.