306 research outputs found
Elementary solution to the time-independent quantum navigation problem
A quantum navigation problem concerns the identification of a time-optimal Hamiltonian that realizes a required quantum process or task, under the influence of a prevailing ‘background’ Hamiltonian that cannot be manipulated. When the task is to transform one quantum state into another, finding the solution in closed form to the problem is nontrivial even in the case of timeindependent Hamiltonians. An elementary solution, based on trigonometric analysis, is found here when the Hilbert space dimension is two. Difficulties arising from generalizations to higher-dimensional systems are discussed
Nilpotent Classical Mechanics
The formalism of nilpotent mechanics is introduced in the Lagrangian and
Hamiltonian form. Systems are described using nilpotent, commuting coordinates
. Necessary geometrical notions and elements of generalized differential
-calculus are introduced. The so called geometry, in a special case
when it is orthogonally related to a traceless symmetric form, shows some
resemblances to the symplectic geometry. As an example of an -system the
nilpotent oscillator is introduced and its supersymmetrization considered. It
is shown that the -symmetry known for the Graded Superfield Oscillator (GSO)
is present also here for the supersymmetric -system. The generalized
Poisson bracket for -variables satisfies modified Leibniz rule and
has nontrivial Jacobiator.Comment: 23 pages, no figures. Corrected version. 2 references adde
Verifiable conditions of -recovery of sparse signals with sign restrictions
We propose necessary and sufficient conditions for a sensing matrix to be
"s-semigood" -- to allow for exact -recovery of sparse signals with at
most nonzero entries under sign restrictions on part of the entries. We
express the error bounds for imperfect -recovery in terms of the
characteristics underlying these conditions. Furthermore, we demonstrate that
these characteristics, although difficult to evaluate, lead to verifiable
sufficient conditions for exact sparse -recovery and to efficiently
computable upper bounds on those for which a given sensing matrix is
-semigood. We concentrate on the properties of proposed verifiable
sufficient conditions of -semigoodness and describe their limits of
performance
Hamilton-Jacobi formalism for Linearized Gravity
In this work we study the theory of linearized gravity via the
Hamilton-Jacobi formalism. We make a brief review of this theory and its
Lagrangian description, as well as a review of the Hamilton-Jacobi approach for
singular systems. Then we apply this formalism to analyze the constraint
structure of the linearized gravity in instant and front-form dynamics.Comment: To be published in Classical and Quantum Gravit
Direct approach to the problem of strong local minima in Calculus of Variations
The paper introduces a general strategy for identifying strong local
minimizers of variational functionals. It is based on the idea that any
variation of the integral functional can be evaluated directly in terms of the
appropriate parameterized measures. We demonstrate our approach on a problem of
W^{1,infinity} weak-* local minima--a slight weakening of the classical notion
of strong local minima. We obtain the first quasiconvexity-based set of
sufficient conditions for W^{1,infinity} weak-* local minima.Comment: 26 pages, no figure
First Order Actions: a New View
We analyse systems described by first order actions using the Hamilton-Jacobi
(HJ) formalism for singular systems. In this study we verify that generalized
brackets appear in a natural way in HJ approach, showing us the existence of a
symplectic structure in the phase spaces of this formalism
Finite reduction and Morse index estimates for mechanical systems
A simple version of exact finite dimensional reduction for the variational
setting of mechanical systems is presented. It is worked out by means of a
thorough global version of the implicit function theorem for monotone
operators. Moreover, the Hessian of the reduced function preserves all the
relevant information of the original one, by Schur's complement, which
spontaneously appears in this context. Finally, the results are
straightforwardly extended to the case of a Dirichlet problem on a bounded
domain.Comment: 13 pages; v2: minor changes, to appear in Nonlinear Differential
Equations and Application
Thermodynamics of an ideal generalized gas:II Means of order
The property that power means are monotonically increasing functions of their
order is shown to be the basis of the second laws not only for processes
involving heat conduction but also for processes involving deformations. In an
-potentail equilibration the final state will be one of maximum entropy,
while in an entropy equilibrium the final state will be one of minimum . A
metric space is connected with the power means, and the distance between means
of different order is related to the Carnot efficiency. In the ideal classical
gas limit, the average change in the entropy is shown to be proportional to the
difference between the Shannon and R\'enyi entropies for nonextensive systems
that are multifractal in nature. The -potential, like the internal energy,
is a Schur convex function of the empirical temperature, which satisfies
Jensen's inequality, and serves as a measure of the tendency to uniformity in
processes involving pure thermal conduction.Comment: 8 page
Six topics on inscribable polytopes
Inscribability of polytopes is a classic subject but also a lively research
area nowadays. We illustrate this with a selection of well-known results and
recent developments on six particular topics related to inscribable polytopes.
Along the way we collect a list of (new and old) open questions.Comment: 11 page
Notes about the Caratheodory number
In this paper we give sufficient conditions for a compactum in
to have Carath\'{e}odory number less than , generalizing an old result of
Fenchel. Then we prove the corresponding versions of the colorful
Carath\'{e}odory theorem and give a Tverberg type theorem for families of
convex compacta
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