1,031,016 research outputs found
Arhangel'ski\u{\i} sheaf amalgamations in topological groups
We consider amalgamation properties of convergent sequences in topological
groups and topological vector spaces. The main result of this paper is that,
for arbitrary topological groups, Nyikos's property is
equivalent to Arhangel'ski\u{\i}'s formally stronger property . This
result solves a problem of Shakhmatov (2002), and its proof uses a new
perturbation argument. We also prove that there is a topological space such
that the space of continuous real-valued functions on , with the
topology of pointwise convergence, has Arhangel'ski\u{\i}'s property
but is not countably tight. This result follows from results of
Arhangel'ski\u{\i}--Pytkeev, Moore and Todor\v{c}evi\'c, and provides a new
solution, with remarkable properties, to a problem of Averbukh and Smolyanov
(1968) concerning topological vector spaces. The Averbukh--Smolyanov problem
was first solved by Plichko (2009), using Banach spaces with weaker locally
convex topologies.Comment: Final version (minor changes
Non-Point Invertible Transformations and Integrability of Partial Difference Equations
This article is devoted to the partial difference quad-graph equations that
can be represented in the form , where the map
is injective. The transformation relates any
of such equations to a quad-graph equation. It is proved that this
transformation maps Darboux integrable equations of the above form into Darboux
integrable equations again and decreases the orders of the transformed
integrals by one in the -direction. As an application of this fact, the
Darboux integrable equations possessing integrals of the second order in the
-direction are described under an additional assumption. The transformation
also maps symmetries of the original equations into symmetries of the
transformed equations (i.e. preserves the integrability in the sense of the
symmetry approach) and acts as a difference substitution for symmetries of a
special form. The latter fact allows us to derive necessary conditions of
Darboux integrability for the equations defined in the first sentence of the
abstract
An existence result for a nonlinear transmission problems
Let and be open bounded subsets of of
class such that the closure of is contained in
. Let be a function in and let
and be continuous functions from to
. By exploiting an argument based on potential theory and on the
Leray-Schauder principle we show that under suitable and completely explicit
conditions on and there exists at least one pair of continuous
functions such that where the last equality is attained in certain weak sense. In a simple
example we show that such a pair of functions is in general
neither unique nor local unique. If instead the fourth condition of the problem
is obtained by a small nonlinear perturbation of a homogeneous linear
condition, then we can prove the existence of at least one classical solution
which is in addition locally unique
Convexity of Quotients of Theta Functions
For fixed and such that , the monotonicity of the
quotients of Jacobi theta functions, namely, , , on has been established
in the previous works of A.Yu. Solynin, K. Schiefermayr, and Solynin and the
first author. In the present paper, we show that the quotients
and are convex on .Comment: 17 pages, 6 figure
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