Squeezed states and their applications to quantum evolution

In this paper, we consider quantum multidimensional problems solvable by using the second quantization method. A multidimensional generalization of the Bogolyubov factorization formula, which is an important particular case of the Campbell-Baker-Hausdorff formula, is established. The inner product of multidimensional squeezed states is calculated explicitly; this relationship justifies a general construction of orthonormal systems generated by linear combinations of squeezed states. A correctly defined path integral representation is derived for solutions of the Cauchy problem for the Schrödinger equation describing the dynamics of a charged particle in the superposition of orthogonal constant (E,H)-fields and a periodic electric field. We show that the evolution of squeezed states runs over compact one-dimensional matrix-valued orbits of squeezed components of the solution, and the evolution of coherent shifts is a random Markov jump process which depends on the periodic component of the potential.     

Generalized squeezed states and multimode factorization formula

We consider the multimode generalization of the normally ordered factorization formula of squeezings. This formula allows us to establish relationships between various representations of squeezed states, to calculate partial traces, mean values, and variations. The main results are expressed in terms of the matrix representation of canonical transformations, which is a convenient and numerically stable mathematical tool. Explicit representations are given for the inner product and the composition of generalized multimode squeezings. Explicitly solvable evolution problems are considered.     

Calculation of Berry's phase in squeezed states

A simple method based on generalized coherent states is proposed for calculation of Berry's phase. In this paper we calculate Berry's phase for a translated oscillator in standard coherent states as well as Berry's phase in squeezed states and spin coherent states, i.e., coherent states for the SU(1, 1) and SU(2) groups, respectively.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 51–59, 1988.     

A multilevel block incomplete cholesky preconditioner for solving normal equations in linear least squares problems

An incomplete factorization method for preconditioning symmetric positive definite matrices is introduced to solve normal equations. The normal equations are form to solve linear least squares problems. The procedure is based on a block incomplete Cholesky factorization and a multilevel recursive strategy with an approximate Schur complement matrix formed implicitly. A diagonal perturbation strategy is implemented to enhance factorization robustness. The factors obtained are used as a preconditioner for the conjugate gradient method. Numerical experiments are used to show the robustness and efficiency of this preconditioning technique, and to compare it with two other preconditioners.     

Entropy and entanglement of an effective two-level atom interacting with two quantized field modes in squeezed displaced fock states

We have studied the influences of ac-Stark shifts on the field quantum entropy, with “squeezed displaced Fock states” (SDFSs) basis. By a unitary transformation we derive a Raman-coupled Hamiltonian perturbatively in coupling constants. The exact results are employed to perform a careful investigation of the temporal evolution of entropy. A factorization of the initial density operator is assumed, with the privileged field mode being in the SDFS. We invoke the mathematical notion of maximum variation of a function to construct a measure for entropy fluctuations. The results show that the effect of the SDFS changes the quasiperiod of the field entropy evolution and entanglement between the atom and the field. The Rabi oscillation frequency, the collapse and revival times of the atomic coherence are found to have strikingly different photon-intensity dependent than those found previously. The general conclusions reached are illustrated by numerical results.     

Hierarchical Interpolative Factorization for Elliptic Operators: Integral Equations

This paper introduces the hierarchical interpolative factorization for integral equations (HIF‐IE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU decomposition that permits the efficient application of the discretized operator and its inverse. HIF‐IE is based on the recursive skeletonization algorithm but incorporates a novel combination of two key features: (1) a matrix factorization framework for sparsifying structured dense matrices and (2) a recursive dimensional reduction strategy to decrease the cost. Thus, higher‐dimensional problems are effectively mapped to one dimension, and we conjecture that constructing, applying, and inverting the factorization all have linear or quasilinear complexity. Numerical experiments support this claim and further demonstrate the performance of our algorithm as a generalized fast multipole method, direct solver, and preconditioner. HIF‐IE is compatible with geometric adaptivity and can handle both boundary and volume problems.© 2016 Wiley Periodicals, Inc.     

Finite state bilaterally deterministic strongly mixing processes

Triviality of the two-sided tail field of a stationary process is not an invariant property under factorization ([4]). In this paper we give an example of a bilaterally deterministic process with finitely many states which is strongly mixing. This extends and complements a result of Bradley ([1]).     

ILUT: A dual threshold incomplete LU factorization

In this paper we describe an Incomplete LU factorization technique based on a strategy which combines two heuristics. This ILUT factorization extends the usual ILU(O) factorization without using the concept of level of fill-in. There are two traditional ways of developing incomplete factorization preconditioners. The first uses a symbolic factorization approach in which a level of fill is attributed to each fill-in element using only the graph of the matrix. Then each fill-in that is introduced is dropped whenever its level of fill exceeds a certain threshold. The second class of methods consists of techniques derived from modifications of a given direct solver by including a dropoff rule, based on the numerical size of the fill-ins introduced, traditionally referred to as threshold preconditioners. The first type of approach may not be reliable for indefinite problems, since it does not consider numerical values. The second is often far more expensive than the standard ILU(O). The strategy we propose is a compromise between these two extremes.     

Time optimal factorizations on compact Lie groups

In this paper we study the relationship between factorization problems on SU(2ⁿ) or more generally on compact Lie groups G and time optimal control problems. Both types of problems naturally arise in physics, such as in quantum computing and in controlling coupled spin systems (NMR‐spectroscopy). In the first part we show that certain factorization problems can be reformulated as time optimal control problems on G. In the second part a necessary condition for the existence of finite optimal factorizations is discussed. At the end we illustrate our results by an example on Euler angle factorizations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rings of invariants for representations of quivers

In this Note we compute the generators of the ring of invariants for quiver factorization problems, generalizing results of Le Bruyn and Procesi. In particular, we find a necessary and sufficient combinatorial criterion for the projectivity of the associated invariant quotients. Further, we show that the non-projective quotients admit open immersions into projective varieties, which still arise from suitable quiver factorization problems. To cite this article: M. Halic, M.S. Stupariu, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Sparse Approximations of the Schur Complement for Parallel Algebraic Hybrid Solvers in 3D

<正>In this paper we study the computational performance of variants of an algebraic additive Schwarz preconditioner for the Schur complement for the solution of large sparse linear systems.In earlier works,the local Schur complements were computed exactly using a sparse direct solver.The robustness of the preconditioner comes at the price of this memory and time intensive computation that is the main bottleneck of the approach for tackling huge problems.In this work we investigate the use of sparse approximation of the dense local Schur complements.These approximations are computed using a partial incomplete LU factorization.Such a numerical calculation is the core of the multi-level incomplete factorization such as the one implemented in pARMS. The numerical and computing performance of the new numerical scheme is illustrated on a set of large 3D convection-diffusion problems;preliminary experiments on linear systems arising from structural mechanics are also reported.     

The importance of structure in incomplete factorization preconditioners

In this paper, we consider level-based preconditioning, which is one of the basic approaches to incomplete factorization preconditioning of iterative methods. It is well-known that while structure-based preconditioners can be very useful, excessive memory demands can limit their usefulness. Here we present an improved strategy that considers the individual entries of the system matrix and restricts small entries to contributing to fewer levels of fill than the largest entries. Using symmetric positive-definite problems arising from a wide range of practical applications, we show that the use of variable levels of fill can yield incomplete Cholesky factorization preconditioners that are more efficient than those resulting from the standard level-based approach. The concept of level-based preconditioning, which is based on the structural properties of the system matrix, is then transferred to the numerical incomplete decomposition. In particular, the structure of the incomplete factorization determined in the symbolic factorization phase is explicitly used in the numerical factorization phase. Further numerical results demonstrate that our level-based approach can lead to much sparser but efficient incomplete factorization preconditioners.     

Error estimate for the modified Newton method with applications to the solution of nonlinear,two-point boundary-value problems

The solution of nonlinear, two-point boundary value problems by Newton's method requires the formation and factorization of a Jacobian matrix at every iteration. For problems in which the cost of performing these operations is a significant part of the cost of the total calculation, it is natural to consider using the modified Newton method. In this paper, we derive an error estimate which enables us to determine an upper bound for the size of the sequence of modified Newton iterates, assuming that the Kantorovich hypotheses are satisfied. As a result, we can efficiently determine when to form a new Jacobian and when to continue the modified Newton algorithm. We apply the result to the solution of several nonlinear, two-point boundary value problems occurring in combustion.     

Approximate Solution of a Single-Base Multi-indentured Repairable-item Inventory System

In this paper we model a single-base multi-indentured repairable item inventory system. The base has a maximum number of identical on-line machines, and each machine consists of several module types. Our objective is to calculate the steady-state operating characteristics of the system. The usual Markovian approach leads to multidimensional state spaces that are large even for relatively small problems. Solving such a multidimensional system is very difficult because of the huge number of states. Consequently, we propose an approximation technique that allows us to solve large problems relatively quickly. Although the resulting solution is only approximate, a variety of test problems indicates that the algorithm is quite accurate.     

Toward an arithmetic of polynomials

Summary ForR a commutative ring, which may have divisors of zero but which has no idempotents other than zero and one, we consider the problem of unique factorization of a polynomial with coefficients inR. We prove that, if the polynomial is separable, then such a unique factorization exists. We also define a Legendre symbol for a separable polynomial and a prime of commutative ring with exactly two idempotents in such a way that the symbols of classical number theory are subsumed. We calculate this symbol forR = Q in two cases where it has classically been of interest, namely quadratic extensions and cyclotomic extensions. We then calculate it in a situation which is new, namely the so called generalized cyclotomic extensions from a paper by S. Beale and D. K. Harrison. We study the Galois theory in the general ring situation and in particular define a category of separable polynomials (this is an extension of a paper by D. K. Harrison and M. Vitulli) and a cohomology theory of separable polynomials.     

工件的释放时间和加工时间具有一致性的单机在线排序问题研究

工件的释放时间和加工时间具有一致性, 是指释放时间大的工件其加工时间不小于释放时间小的工件的加工时间, 即若$r_{i}\geq r_{j}$, 则$p_{i}\geq p_{j}$。本文在该一致性约束下, 研究最小化最大加权完工时间单机在线排序问题, 和最小化总加权完工时间单机在线排序问题, 并分别设计出$\frac{\sqrt{5}+1}{2}$-竞争的最好可能在线算法。     

Toeplitz operators associated with analytic crossed products II (invariant subspaces and factorization)

In [16], we introduced the notion of Toeplitz operators associated with analytic crossed products. In this paper, we study the structure of invariant subspaces with respect to the analytic crossed products. We also investigate the inner-outer factorization problems for analytic Toeplitz operators, the factorization problem for non-negative Toeplitz operators and Szegö's infimum problem.This work was supported in part by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education.     

Factorization in Integral Domains,IV

Let D be an integral domain such that every nonzero nonunit of D is a finite product of irreducible elements. In this article, we introduce and study several unifying concepts for the theory of nonunique factorization in D. They give a new way to measure, in some sense, how far an half-factorial domain (resp., bounded factorization domain, atomic domain) D is from being a UFD (resp., finite factorization domain, Cohen–Kaplansky domain) based on equivalence relations on the set of irreducible elements of D.     

Optimization of unconstrained functions with sparse hessian matrices-newton-type methods

Newton-type methods for unconstrained optimization problems have been very successful when coupled with a modified Cholesky factorization to take into account the possible lack of positive-definiteness in the Hessian matrix. In this paper we discuss the application of these method to large problems that have a sparse Hessian matrix whose sparsity is known a priori. Quite often it is difficult, if not impossible, to obtain an analytic representation of the Hessian matrix. Determining the Hessian matrix by the standard method of finite-differences is costly in terms of gradient evaluations for large problems. Automatic procedures that reduce the number of gradient evaluations by exploiting sparsity are examined and a new procedure is suggested. Once a sparse approximation to the Hessian matrix has been obtained, there still remains the problem of solving a sparse linear system of equations at each iteration. A modified Cholesky factorization can be used. However, many additional nonzeros (fill-in) may be created in the factors, and storage problems may arise. One way of approaching this problem is to ignore fill-in in a systematic manner. Such technique are calledpartial factorization schemes. Various existing partial factorization are analyzed and three new ones are developed. The above algorithms were tested on a set of problems. The overall conclusions were that these methods perfom well in practice.     

Factorizations,Localizations, and the Orthogonal Subcategory Problem

In this paper factorization structures of an abstract category are considered, depending on a class ?? of morphisms which is not necessarily closed under composition; as soon as it is one obtains the usual factorization systems defined by the diagonal-fill-in property. General existence criteria for those factorization structures are proved, in particular for monotone-light factorizations which are defined for abstract categories and which are considered in more detail. Finally, sufficient conditions for a positive solution of the Orthogonal Subcategory Problem are derived from the existence of certain factorization structures.