1,011 research outputs found
Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets
A new class of plurisubharmonic functions on the octonionic plane O^2= R^{16}
is introduced. An octonionic version of theorems of A.D. Aleksandrov and Chern-
Levine-Nirenberg, and Blocki are proved. These results are used to construct
new examples of continuous translation invariant valuations on convex subsets
of O^2=R^{16}. In particular a new example of Spin(9)-invariant valuation on
R^{16} is given.Comment: 35 pages. The definition of octonionic Hessian is replaced with the
transposed matrix as it should be. Other minor correction
Index transforms with the squares of Kelvin functions
New index transforms, involving squares of Kelvin functions, are
investigated. Mapping properties and inversion formulas are established for
these transforms in Lebesgue spaces. The results are applied to solve a
boundary value problem on the wedge for a fourth order partial differential
equation.Comment: arXiv admin note: substantial text overlap with arXiv:1711.0193
Certain identities, connection and explicit formulas for the Bernoulli, Euler numbers and Riemann zeta -values
Various new identities, recurrence relations, integral representations,
connection and explicit formulas are established for the Bernoulli, Euler
numbers and the values of Riemann's zeta function. To do this, we explore
properties of some Sheffer's sequences of polynomials related to the
Kontorovich-Lebedev transform
Index transforms with the square of Bessel functions
New index transforms, involving the square of Bessel functions of the first
kind as the kernel are considered. Mapping properties such as the boundedness
and invertibility are investigated for these operators in the Lebesgue spaces.
Inversion theorems are proved. As an interesting application, a solution to the
initial value problem for the third order partial differential equation,
involving the Laplacian, is obtained.Comment: arXiv admin note: text overlap with arXiv:1509.0276
Towards the Casas- Alvero conjecture
We investigate necessary and sufficient conditions for an arbitrary
polynomial of degree to be trivial, i.e. to have the form . These
results are related to an open problem, conjectured in 2001 by E. Casas-
Alvero. It says, that any complex univariate polynomial, having a common root
with each of its non-constant derivative must be a power of a linear
polynomial. In particular, we establish determinantal representation of the
Abel-Goncharov interpolation polynomials, related to the problem and having its
own interest. Among other results are new Sz.-Nagy type identities for complex
roots and a generalization of the Schoenberg conjectured analog of Rolle's
theorem for polynomials with real and complex coefficients
Theory of valuations on manifolds, IV. New properties of the multiplicative structure
This is the fourth part in the series of articles math.MG/0503397,
math.MG/0503399, math.MG/0509512 where the theory of valuations on manifolds is
developed. In this part it is shown that the filtration on valuations
introduced in math.MG/0503399 is compatible with the product. Then it is proved
that the Euler-Verdier involution on smooth valuations introduced in
math.MG/0503399 is an automorphism of the algebra of valuations. Then an
integration functional on valuations with compact support is introduced, and a
property of selfduality of valuations is proved. Next a space of generalized
valuations is defined, and some basic properties of it are proved. Finally a
canonical imbedding of the space of constructible functions on a real analytic
manifold into the space of generalized valuations is constructed, and various
structures on valuations are compared with known structures on constructible
functions.Comment: 40 pages; errors corrected in the proof of Theorems 7.4.1 and 8.4.
Theory of valuations on manifolds: a survey
This is a non-technical survey of a recent theory of valuations on manifolds
constructed in math.MG/0503397, math.MG/0503399, math.MG/0509512,
math.MG/0511171 and actually a guide to this series of articles. We review also
some recent related results obtained by a number of people. We formulate some
open questions.Comment: 19 pages; revised version: order of sections has been changed; typo
Quantization of fields over de Sitter space by the method of generalized coherent states. I. Scalar field
A system of generalized coherent states for the de Sitter group obeying
Klein-Gordon equation and corresponding to the massive spin zero particles over
the de Sitter space is considered. This allows us to construct the quantized
scalar field by the resolution over these coherent states; the corresponding
propagator can be computed by the method of analytic continuation to the
complexified de Sitter space and coincides with expressions obtained previously
by other methods. We show that this propagator possess the de Sitter-invariance
and causality properties.Comment: 11 pages, LATEX, using ioplppt.sty and iopfts.sty. v.2: some
misprints correcte
New structures on valuations and applications
An overview of some of the recent developments in the theory of valuations on
convex sets and its generalizations to manifolds is given. The exposition is
focused towards applications to integral geometry; several of such applications
are discussed.Comment: Lecture notes of the Advanced Course on Integral Geometry and
Valuation Theory at CRM, Barcelona. 39 pages. A new Section 2.8 added; other
minor correction
New inversion, convolution and Titchmarsh's theorems for the half-Hilbert transform
While exploiting the generalized Parseval equality for the Mellin transform,
we derive the reciprocal inverse operator in the weighted L_2-space related to
the Hilbert transform on the nonnegative half-axis. Moreover, employing the
convolution method, which is based on the Mellin-Barnes integrals, we prove the
corresponding convolution and Titchmarsh's theorems for the half-Hilbert
transform. Some applications to the solvability of a new class of singular
integral equations are demonstrated. Our technique does not require the use of
methods of the Riemann-Hilbert boundary value problems for analytic functions.
The same approach will be applied in the forthcoming research to invert the
half-Hartley transform and to establish its convolution theorem
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