We present applications of matrix methods to the analytic theory of polynomials. We first show how matrix analysis can be used to give new proofs of a number of classical results on roots of polynomials. Then we use m...
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We present applications of matrix methods to the analytic theory of polynomials. We first show how matrix analysis can be used to give new proofs of a number of classical results on roots of polynomials. Then we use matrix methods to establish a new log-majorization result on roots of polynomials. The theory of multiplier sequences gives the common link between the applications. (C) 2011 Elsevier Inc. All rights reserved.
Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that...
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Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefits can be expected for polynomial eigenvalue problems, for which the concept of an invariant subspace needs to be replaced by the concept of an invariant pair. Little has been known so far about numerical aspects of such invariant pairs. The aim of this paper is to fill this gap. The behavior of invariant pairs under perturbations of the matrix polynomial is studied and a first-order perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe efficient refinement procedures directly based on the polynomial formulation. Numerical experiments with matrix polynomials from a number of applications demonstrate the effectiveness of our extraction and refinement procedures. (C) 2010 Elsevier Inc. All rights reserved.
The work in this paper is to initiate a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by certain economically compact structure has a multilin...
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ISBN:
(纸本)9783642226151
The work in this paper is to initiate a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by certain economically compact structure has a multilinear monomial in its sum-product expansion. The complexity aspects of this problem and its variants are investigated with two objectives. One is to understand how this problem relates to critical problems in complexity, and if so to what extent. The other is to exploit possibilities of applying algebraic properties of polynomials to the study of those problems. A series of results about Pi Sigma Pi and Pi Sigma polynomials are obtained in this paper, laying a basis for further study along this line.
This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of mo...
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ISBN:
(纸本)9783642226151
This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its sum-product expansion. The complexity aspects of this problem and its variants have been investigated in our first paper by Chen and Fu (2010), laying a foundation for further study. In this paper, we present two pairs of algorithms. First, we prove that there is a randomized O*(p(k)) time algorithm for testing p-monomials in an n-variate polynomial of degree k represented by an arithmetic circuit, while a deterministic O*((6.4p)(k)) time algorithm is devised when the circuit is a formula, here p is a given prime number. Second, we present a deterministic O*(2(k)) time algorithm for testing multilinear monomials in Pi(m)Sigma(2)Pi(t) x Pi(k)Sigma(3) polynomials, while a randomized O*(1.5(k)) algorithm is given for these polynomials. Finally, we prove that testing some special types of multilinear monomial is W[1]-hard, giving evidence that testing for specific monomials is not fixed-parameter tractable.
There are a number of graphical languages for process algebras to specify distributed mobile real-time systems: Ambient calculus, KELL for MA, Bigraph for pi-calculus, etc. However there are some limitations in repres...
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ISBN:
(纸本)9780769544397
There are a number of graphical languages for process algebras to specify distributed mobile real-time systems: Ambient calculus, KELL for MA, Bigraph for pi-calculus, etc. However there are some limitations in representation of large and complex such systems in these languages, since only the most immediately available actions of processes are graphically represented. Further the temporal properties are represented numerically, not graphically. In order to overcome the limitations, this paper proposes a new graphical language, Onion, to represent visually all the actions of each process in the process. In Onion, the timed actions of each process are represented graphically by circularly layered leaves, like those of real onion, and their interactions and movements as edges among the leaves. Further the temporal properties of the actions are visually and quantitatively represented in a separate graph. Onion makes the understanding of the systems visually quantitative between the in-the-large and the in-the-small perspectives of the systems. Onion can be considered one of the most integrated graphical languages for process algebras.
We present a robust secure methodology for computing functions that are represented as multivariate polynomials where parties hold different variables as private inputs. Our generic efficient protocols are fully black...
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ISBN:
(纸本)9783642215537
We present a robust secure methodology for computing functions that are represented as multivariate polynomials where parties hold different variables as private inputs. Our generic efficient protocols are fully black-box and employ threshold additive homomorphic encryption;they do not assume honest majority, yet are robust in detecting any misbehavior. We achieve solutions that take advantage of the algebraic structure of the polynomials, and are polynomial-time in all parameters (security parameter, polynomial size, polynomial degree, number of parties). We further exploit a "round table" communication paradigm to reduce the complexity in the number of parties. A large collection of problems are naturally and efficiently represented as multivariate polynomials over a field or a ring: problems from linear algebra, statistics, logic, as well as operations on sets represented as polynomials. In particular, we present a new efficient solution to the multi-party set intersection problem, and a solution to a multi-party variant of the polynomial reconstruction problem.
An instance of the (Generalized) Post Correspondence Problem is during the decision process typically reduced to one or more other instances, called its successors. In this paper we study the reduction tree of GPCP in...
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An instance of the (Generalized) Post Correspondence Problem is during the decision process typically reduced to one or more other instances, called its successors. In this paper we study the reduction tree of GPCP in the binary case. This entails in particular a detailed analysis of the structure of end blocks. We give an upper bound for the number of end blocks, and show that even if an instance has more than one successor, it can nevertheless be reduced to a single instance. This, in particular, implies that binary GPCP can be decided in polynomial time.
polynomials are perhaps the most important family of functions in mathematics. They feature in celebrated results from both antiquity and modern times, like the insolvability by radicals of polynomials of degree ? 5 o...
ISBN:
(数字)9781601984814
ISBN:
(纸本)9781601984807
polynomials are perhaps the most important family of functions in mathematics. They feature in celebrated results from both antiquity and modern times, like the insolvability by radicals of polynomials of degree ? 5 of Abel and Galois, and Wiles" proof of Fermat"s "last theorem". In computer science they feature in, e.g., error-correcting codes and probabilistic proofs, among many applications. The manipulation of polynomials is essential in numerous applications of linear algebra and symbolic computation. Partial Derivatives in Arithmetic Complexity and Beyond is devoted mainly to the study of polynomials from a computational perspective. It illustrates that one can learn a great deal about the structure and complexity of polynomials by studying (some of) their partial derivatives. It also shows that partial derivatives provide essential ingredients in proving both upper and lower bounds for computing polynomials by a variety of natural arithmetic models. It goes on to look at applications which go beyond computational complexity, where partial derivatives provide a wealth of structural information about polynomials (including their number of roots, reducibility and internal symmetries), and help us solve various number theoretic, geometric, and combinatorial problems. Partial Derivatives in Arithmetic Complexity and Beyond is an invaluable reference for anyone with an interest in polynomials. Many of the chapters in these three parts can be read independently. For the few which need background from previous chapters, this is specified in the chapter abstract.
Ten years ago, Ko et al. described a Diffie-Hellman like protocol based on decomposition with respect to a non-commutative semigroup law. Instantiation with braid groups had first been considered, however intense rese...
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