Operator (computer programming)

About: Operator (computer programming) is a research topic. Over the lifetime, 40896 publications have been published within this topic receiving 671452 citations. The topic is also known as: operator symbol & operator name.

All point correlation functions in SYK

Abstract: Large N melonic theories are characterized by two-point function Feynman diagrams built exclusively out of melons. This leads to conformal invariance at strong coupling, four-point function diagrams that are exclusively ladders, and higher-point functions that are built out of four-point functions joined together. We uncover an incredibly useful property of these theories: the six-point function, or equivalently, the three-point function of the primary O(N ) invariant bilinears, regarded as an analytic function of the operator dimensions, fully determines all correlation functions, to leading nontrivial order in 1/N , through simple Feynman-like rules. The result is applicable to any theory, not necessarily melonic, in which higher-point correlators are built out of four-point functions. We explicitly calculate the bilinear three-point function for q-body SYK, at any q. This leads to the bilinear four-point function, as well as all higher-point functions, expressed in terms of higher-point conformal blocks, which we discuss. We find universality of correlators of operators of large dimension, which we simplify through a saddle point analysis. We comment on the implications for the AdS dual of SYK.

Approximate inverse for linear and some nonlinear problems

Abstract: In this paper we present a method for solving problems such as Af = g by constructing an approximate inverse which maps the data g to a regularized solution of this equation of the first kind. No discretization for f is needed. The solution operator can be precomputed independently of the data. This works for linear problems and for nonlinear problems with a special structure. The regularization is achieved by computing mollified versions of the (minimum-norm) solution. It is shown that this class of regularization operators contains, as special cases, the classical methods such as Tikhonov - Phillips, iteration methods and also discretization methods. In the case where the operator has some invariance properties the storage needs are dramatically reduced.

Asymptotic Behavior of Solutions of Evolution Equations

Abstract: Publisher Summary This chapter presents the methods of investigation of the asymptotic behavior of solutions of evolution equations, endowed with a dissipative mechanism, based on the study of the structure of the ω-limit set of trajectories of the evolution operator generated by the equation. The dissipative mechanism usually manifests itself by the presence of a Liapunov functional, which is constant on ω -limit sets; the central idea of the approach presented in the chapter is to use this information in conjunction with properties of ω -limit sets such as invariance and minimality. The chapter discusses two examples of wave equations with weak damping for which the scheme set up by Hale applies. In particular, this requires that the Liapunov functional be continuous on phase space. The chapter explores the case of a hyperbolic conservation law that generates a semigroup on space. It also presents a survey of various applications and extensions of these ideas that may serve as a guide to those interested in learning more about the method.

Designing optimal quantum detectors via semidefinite programming

Abstract: We consider the problem of designing an optimal quantum detector to minimize the probability of a detection error when distinguishing among a collection of quantum states, represented by a set of density operators. We show that the design of the optimal detector can be formulated as a semidefinite programming problem. Based on this formulation, we derive a set of necessary and sufficient conditions for an optimal quantum measurement. We then show that the optimal measurement can be found by solving a standard (convex) semidefinite program. By exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum, the optimal measurement can be computed very efficiently in polynomial time within any desired accuracy. Using the semidefinite programming formulation, we also show that the rank of each optimal measurement operator is no larger than the rank of the corresponding density operator. In particular, if the quantum state ensemble is a pure-state ensemble consisting of (not necessarily independent) rank-one density operators, then we show that the optimal measurement is a pure-state measurement consisting of rank-one measurement operators.

Some new classes of complex symmetric operators

Abstract: We say that an operator T E B(H) is complex symmetric if there exists a conjugate-linear, isometric involution C: ℌ→ℌ so that T = CT * C. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data (dim ker T, dim ker T*).