Search results for "Formula"
showing 10 items of 755 documents
L∞-variational problem associated to dirichlet forms
2012
On the numerical evaluation of algebro-geometric solutions to integrable equations
2011
Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical evaluation of these solutions are studied. In the case of hyperelliptic surfaces efficient algorithms exist even for almost degenerate surfaces. This allows the numerical study of solitonic limits. For general real Riemann surfaces, the choice of a homology basis adapted to the anti-holomorphic involution is important for a convenient formulation of the solutions and smoothness conditions. Since existing algorithms for algebraic curves produce a homology basis no…
Pappus type theorems for hypersurfaces in a space form
2002
In order to get further insight on the Weyl’s formula for the volume of a tubular hypersurface, we consider the following situation. Letc(t) be a curve in a space formM λ n of sectional curvature λ. LetP 0 be a totally geodesic hypersurface ofM λ n throughc(0) and orthogonal toc(t). LetC 0 be a hypersurface ofP 0. LetC be the hypersurface ofM λ n obtained by a motion ofC 0 alongc(t). We shall denote it byC PorC Fif it is obtained by a parallel or Frenet motion, respectively. We get a formula for volume(C). Among other consequences of this formula we get that, ifc(0) is the centre of mass ofC 0, then volume(C) ≥ volume(C),P),and the equality holds whenC 0 is contained in a geodesic sphere or…
A Path-Integral Approach to the Cameron-Martin-Maruyama-Girsanov Formula Associated to a Bilaplacian
2012
We define the Wiener product on a bosonic Connes space associated to a Bilaplacian and we introduce formal Wiener chaos on the path space. We consider the vacuum distribution on the bosonic Connes space and show that it is related to the heat semigroup associated to the Bilaplacian. We deduce a Cameron-Martin quasi-invariance formula for the heat semigroup associated to the Bilaplacian by using some convenient coherent vector. This paper enters under the Hida-Streit approach of path integral.
A geometrical constructive approach to infinitesimal analysis: epistemological potential and boundaries of tractional motion
2014
Recent foundational approaches to Infinitesimal Analysis are essentially algebraic or computational, whereas the first approaches to such problems were geometrical. From this perspective, we may recall the seventeenth-century investigations of the “inverse tangent problem.” Suggested solutions to this problem involved certain machines, intended as both theoretical and actual instruments, which could construct transcendental curves through so-called tractional motion. The main idea of this work is to further develop tractional motion to investigate if and how, at a very first analysis, these ideal machines (like the ancient straightedge and compass) can constitute the basis of a purely geome…
Infinite Dimensional Holomorphy
2019
We give an introduction to vector-valued holomorphic functions in Banach spaces, defined through Frechet differentiability. Every function defined on a Reinhardt domain of a finite-dimensional Banach space is analytic, i.e. can be represented by a monomial series expansion, where the family of coefficients is given through a Cauchy integral formula. Every separate holomorphic (holomorphic on each variable) function is holomorphic. This is Hartogs’ theorem, which is proved using Leja’s polynomial lemma. For infinite-dimensional spaces, homogeneous polynomials are defined as the diagonal of multilinear mappings. A function is holomorphic if and only if it is Gâteaux holomorphic and continuous…
Holomorphic Functions on Polydiscs
2019
This is a short introduction to the theory of holomorphic functions in finitely and infinitely many variables. We begin with functions in finitely many variables, giving the definition of holomorphic function. Every such function has a monomial series expansion, where the coefficients are given by a Cauchy integral formula. Then we move to infinitely many variables, considering functions defined on B_{c0}, the open unit ball of the space of null sequences. Holomorphic functions are defined by means of Frechet differentiability. We have versions of Weierstrass and Montel theorems in this setting. Every holomorphic function on B_{c0} defines a family of coefficients through a Cauchy integral …
Algebraic aspects and coherence conditions for conjoined and disjoined conditionals
2019
We deepen the study of conjoined and disjoined conditional events in the setting of coherence. These objects, differently from other approaches, are defined in the framework of conditional random quantities. We show that some well known properties, valid in the case of unconditional events, still hold in our approach to logical operations among conditional events. In particular we prove a decomposition formula and a related additive property. Then, we introduce the set of conditional constituents generated by $n$ conditional events and we show that they satisfy the basic properties valid in the case of unconditional events. We obtain a generalized inclusion-exclusion formula and we prove a …
Toeplitz band matrices with small random perturbations
2021
We study the spectra of $N\times N$ Toeplitz band matrices perturbed by small complex Gaussian random matrices, in the regime $N\gg 1$. We prove a probabilistic Weyl law, which provides an precise asymptotic formula for the number of eigenvalues in certain domains, which may depend on $N$, with probability sub-exponentially (in $N$) close to $1$. We show that most eigenvalues of the perturbed Toeplitz matrix are at a distance of at most $\mathcal{O}(N^{-1+\varepsilon})$, for all $\varepsilon >0$, to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix.
Unitary Representations of the Modular and Two-Particle Q-Deformed Toda Chains
2001
The paper deals with the analytic theory of the quantum two-particle q-deformed Toda chains. This is the simplest nontrivial example clarifying the role of the modular duality concept (first discovered by L.Faddeev) in the representation theory of noncompact semisimple quantum groups. Explicit formulae for the Whittaker vectors and Whittaker functions are presented in terms of the double sine functions.