297,818 research outputs found
Functional it{\^o} versus banach space stochastic calculus and strict solutions of semilinear path-dependent equations
Functional It\^o calculus was introduced in order to expand a functional
depending on time , past and present values of
the process . Another possibility to expand
consists in considering the path as an
element of the Banach space of continuous functions on and to use
Banach space stochastic calculus. The aim of this paper is threefold. 1) To
reformulate functional It\^o calculus, separating time and past, making use of
the regularization procedures which matches more naturally the notion of
horizontal derivative which is one of the tools of that calculus. 2) To exploit
this reformulation in order to discuss the (not obvious) relation between the
functional and the Banach space approaches. 3) To study existence and
uniqueness of smooth solutions to path-dependent partial differential equations
which naturally arise in the study of functional It\^o calculus. More
precisely, we study a path-dependent equation of Kolmogorov type which is
related to the window process of the solution to an It\^o stochastic
differential equation with path-dependent coefficients. We also study a
semilinear version of that equation.Comment: This paper is a substantial improvement with additional research
material of the first part of the unpublished paper arXiv:1401.503
Qutrit Dichromatic Calculus and Its Universality
We introduce a dichromatic calculus (RG) for qutrit systems. We show that the
decomposition of the qutrit Hadamard gate is non-unique and not derivable from
the dichromatic calculus. As an application of the dichromatic calculus, we
depict a quantum algorithm with a single qutrit. Since it is not easy to
decompose an arbitrary d by d unitary matrix into Z and X phase gates when d >
2, the proof of the universality of qudit ZX calculus for quantum mechanics is
far from trivial. We construct a counterexample to Ranchin's universality
proof, and give another proof by Lie theory that the qudit ZX calculus contains
all single qudit unitary transformations, which implies that qudit ZX calculus,
with qutrit dichromatic calculus as a special case, is universal for quantum
mechanics.Comment: In Proceedings QPL 2014, arXiv:1412.810
Pseudo-differential operators with nonlinear quantizing functions
In this paper we develop the calculus of pseudo-differential operators
corresponding to the quantizations of the form where is a general function. In
particular, for the linear choices , , and
this covers the well-known Kohn-Nirenberg,
anti-Kohn-Nirenberg, and Weyl quantizations, respectively. Quantizations of
such type appear naturally in the analysis on nilpotent Lie groups for
polynomial functions and here we investigate the corresponding calculus
in the model case of . We also give examples of nonlinear
appearing on the polarised and non-polarised Heisenberg groups, inspired by the
recent joint work with Marius Mantoiu.Comment: 26 page
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