1,897 research outputs found
A non-associative quantum mechanics
A non-associative quantum mechanics is proposed in which the product of three
and more operators can be non-associative one. The multiplication rules of the
octonions define the multiplication rules of the corresponding operators with
quantum corrections. The self-consistency of the operator algebra is proved for
the product of three operators. Some properties of the non-associative quantum
mechanics are considered. It is proposed that some generalization of the
non-associative algebra of quantum operators can be helpful for understanding
of the algebra of field operators with a strong interaction.Comment: one typo in Eq. (23) is correcte
Mirror Descent and Convex Optimization Problems With Non-Smooth Inequality Constraints
We consider the problem of minimization of a convex function on a simple set
with convex non-smooth inequality constraint and describe first-order methods
to solve such problems in different situations: smooth or non-smooth objective
function; convex or strongly convex objective and constraint; deterministic or
randomized information about the objective and constraint. We hope that it is
convenient for a reader to have all the methods for different settings in one
place. Described methods are based on Mirror Descent algorithm and switching
subgradient scheme. One of our focus is to propose, for the listed different
settings, a Mirror Descent with adaptive stepsizes and adaptive stopping rule.
This means that neither stepsize nor stopping rule require to know the
Lipschitz constant of the objective or constraint. We also construct Mirror
Descent for problems with objective function, which is not Lipschitz
continuous, e.g. is a quadratic function. Besides that, we address the problem
of recovering the solution of the dual problem
Quasigroups, Asymptotic Symmetries and Conservation Laws in General Relativity
A new quasigroup approach to conservation laws in general relativity is
applied to study asymptotically flat at future null infinity spacetime. The
infinite-parametric Newman-Unti group of asymptotic symmetries is reduced to
the Poincar\'e quasigroup and the Noether charge associated with any element of
the Poincar\'e quasialgebra is defined. The integral conserved quantities of
energy-momentum and angular momentum are linear on generators of Poincar\'e
quasigroup, free of the supertranslation ambiguity, posess the flux and
identically equal to zero in Minkowski spacetime.Comment: RevTeX4, 5 page
Quantum Effective Action in Spacetimes with Branes and Boundaries
We construct quantum effective action in spacetime with branes/boundaries.
This construction is based on the reduction of the underlying Neumann type
boundary value problem for the propagator of the theory to that of the much
more manageable Dirichlet problem. In its turn, this reduction follows from the
recently suggested Neumann-Dirichlet duality which we extend beyond the tree
level approximation. In the one-loop approximation this duality suggests that
the functional determinant of the differential operator subject to Neumann
boundary conditions in the bulk factorizes into the product of its Dirichlet
counterpart and the functional determinant of a special operator on the brane
-- the inverse of the brane-to-brane propagator. As a byproduct of this
relation we suggest a new method for surface terms of the heat kernel
expansion. This method allows one to circumvent well-known difficulties in heat
kernel theory on manifolds with boundaries for a wide class of generalized
Neumann boundary conditions. In particular, we easily recover several lowest
order surface terms in the case of Robin and oblique boundary conditions. We
briefly discuss multi-loop applications of the suggested Dirichlet reduction
and the prospects of constructing the universal background field method for
systems with branes/boundaries, analogous to the Schwinger-DeWitt technique.Comment: LaTeX, 25 pages, final version, to appear in Phys. Rev.
On representations of the feasible set in convex optimization
We consider the convex optimization problem where is convex, the feasible set K is convex and Slater's
condition holds, but the functions are not necessarily convex. We show
that for any representation of K that satisfies a mild nondegeneracy
assumption, every minimizer is a Karush-Kuhn-Tucker (KKT) point and conversely
every KKT point is a minimizer. That is, the KKT optimality conditions are
necessary and sufficient as in convex programming where one assumes that the
are convex. So in convex optimization, and as far as one is concerned
with KKT points, what really matters is the geometry of K and not so much its
representation.Comment: to appear in Optimization Letter
Improved algorithm for quantum separability and entanglement detection
Determining whether a quantum state is separable or entangled is a problem of
fundamental importance in quantum information science. It has recently been
shown that this problem is NP-hard. There is a highly inefficient `basic
algorithm' for solving the quantum separability problem which follows from the
definition of a separable state. By exploiting specific properties of the set
of separable states, we introduce a new classical algorithm that solves the
problem significantly faster than the `basic algorithm', allowing a feasible
separability test where none previously existed e.g. in 3-by-3-dimensional
systems. Our algorithm also provides a novel tool in the experimental detection
of entanglement.Comment: 4 pages, revtex4, no figure
Advances in low-memory subgradient optimization
One of the main goals in the development of non-smooth optimization is to cope with high dimensional problems by decomposition, duality or Lagrangian relaxation which greatly reduces the number of variables at the cost of worsening differentiability of objective or constraints. Small or medium dimensionality of resulting non-smooth problems allows to use bundle-type algorithms to achieve higher rates of convergence and obtain higher accuracy, which of course came at the cost of additional memory requirements, typically of the order of n2, where n is the number of variables of non-smooth problem. However with the rapid development of more and more sophisticated models in industry, economy, finance, et all such memory requirements are becoming too hard to satisfy. It raised the interest in subgradient-based low-memory algorithms and later developments in this area significantly improved over their early variants still preserving O(n) memory requirements. To review these developments this chapter is devoted to the black-box subgradient algorithms with the minimal requirements for the storage of auxiliary results, which are necessary to execute these algorithms. To provide historical perspective this survey starts with the original result of N.Z. Shor which opened this field with the application to the classical transportation problem. The theoretical complexity bounds for smooth and non-smooth convex and quasi-convex optimization problems are briefly exposed in what follows to introduce to the relevant fundamentals of non-smooth optimization. Special attention in this section is given to the adaptive step-size policy which aims to attain lowest complexity bounds. Unfortunately the non-differentiability of objective function in convex optimization essentially slows down the theoretical low bounds for the rate of convergence in subgradient optimization compared to the smooth case but there are different modern techniques that allow to solve non-smooth convex optimization problems faster then dictate lower complexity bounds. In this work the particular attention is given to Nesterov smoothing technique, Nesterov Universal approach, and Legendre (saddle point) representation approach. The new results on Universal Mirror Prox algorithms represent the original parts of the survey. To demonstrate application of non-smooth convex optimization algorithms for solution of huge-scale extremal problems we consider convex optimization problems with non-smooth functional constraints and propose two adaptive Mirror Descent methods. The first method is of primal-dual variety and proved to be optimal in terms of lower oracle bounds for the class of Lipschitz-continuous convex objective and constraints. The advantages of application of this method to sparse Truss Topology Design problem are discussed in certain details. The second method can be applied for solution of convex and quasi-convex optimization problems and is optimal in a sense of complexity bounds. The conclusion part of the survey contains the important references that characterize recent developments of non-smooth convex optimization
Effective action and heat kernel in a toy model of brane-induced gravity
We apply a recently suggested technique of the Neumann-Dirichlet reduction to
a toy model of brane-induced gravity for the calculation of its quantum
one-loop effective action. This model is represented by a massive scalar field
in the -dimensional flat bulk supplied with the -dimensional kinetic
term localized on a flat brane and mimicking the brane Einstein term of the
Dvali-Gabadadze-Porrati (DGP) model. We obtain the inverse mass expansion of
the effective action and its ultraviolet divergences which turn out to be
non-vanishing for both even and odd spacetime dimensionality . For the
massless case, which corresponds to a limit of the toy DGP model, we obtain the
Coleman-Weinberg type effective potential of the system. We also obtain the
proper time expansion of the heat kernel in this model associated with the
generalized Neumann boundary conditions containing second order tangential
derivatives. We show that in addition to the usual integer and half-integer
powers of the proper time this expansion exhibits, depending on the dimension
, either logarithmic terms or powers multiple of one quarter. This property
is considered in the context of strong ellipticity of the boundary value
problem, which can be violated when the Euclidean action of the theory is not
positive definite.Comment: LaTeX, 20 pages, new references added, typos correcte
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