搜索结果：Computation

Outsourcing computation has gained significant popularity in recent years due to the prevalence of cloud computing. There are two main security concerns in outsourcing computation: how to guarantee the cloud server performs the computation correctly and how to keep the client's data secret. The {\em single-server verifiable computation} (SSVC) of Gennaro, Gentry and Parno (Crypto'10) enables a client to delegate the computation of a function $f$ on any input $x$ with both concerns highly relieved, but only results in {\em computationally secure} schemes that {\em lack practical efficiency}. While the SSVC schemes use a single server, in this paper we develop a {\em multi-server verifiable computation} (MSVC) model where the client shares both $f$ and $x$ among multiple servers, each server performs a set of computations on its shares, and finally the client reconstructs $f(x)$ from all servers' results. In this MSVC model we propose a generic construction for outsourcing computations of the form $F{\bf x}$, where $F$ is a matrix and $\bf x$ is a vector. Our generic construction achieves {\em information-theoretic security, input privacy} and {\em function privacy}. By optimizing th

The classical theory of Kosambi-Cartan-Chern (KCC) developed in differential geometry provides a powerful method for analyzing the behaviors of dynamical systems. In the KCC theory, the properties of a dynamical system are described in terms of five geometrical invariants, of which the second corresponds to the so-called Jacobi stability of the system. Different from that of the Lyapunov stability that has been studied extensively in the literature, the analysis of the Jacobi stability has been investigated more recently using geometrical concepts and tools. It turns out that the existing work on the Jacobi stability analysis remains theoretical and the problem of algorithmic and symbolic treatment of Jacobi stability analysis has yet to be addressed. In this paper, we initiate our study on the problem for a class of ODE systems of arbitrary dimension and propose two algorithmic schemes using symbolic computation to check whether a nonlinear dynamical system may exhibit Jacobi stability. The first scheme, based on the construction of the complex root structure of a characteristic polynomial and on the method of quantifier elimination, is capable of detecting the existence of the Ja

Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of invariants and designing methods and algorithms to compute them remains an active area of ongoing research with an abundance of applications. In this incredibly vast topic, we focus on two particular themes displaying a fruitful interplay between the differential and algebraic invariant theories. First, we show how an algebraic adaptation of the moving frame method from differential geometry leads to a practical algorithm for computing a generating set of rational invariants. Then we discuss the notion of differential invariant signature, its role in solving equivalence problems in geometry and algebra, and some successes and challenges in designing algorithms based on this notion.

This paper is devoted to the study of infinitesimal limit cycles that can bifurcate from zero-Hopf equilibria of differential systems based on the averaging method. We develop an efficient symbolic program using Maple for computing the averaged functions of any order for continuous differential systems in arbitrary dimension. The program allows us to systematically analyze zero-Hopf bifurcations of polynomial differential systems using symbolic computation methods. We show that for the first-order averaging, $\ell\in\{0,1,\ldots,2^{n-3}\}$ limit cycles can bifurcate from the zero-Hopf equilibrium for the general class of perturbed differential systems and up to the second-order averaging, the maximum number of limit cycles can be determined by computing the mixed volume of a polynomial system obtained from the averaged functions. A number of examples are presented to demonstrate the effectiveness of the proposed algorithmic approach.

A computationally simple approach to inference in state space models is proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation of an intractable likelihood by matching summary statistics for the observed data with statistics computed from data simulated from the true process, based on parameter draws from the prior. Draws that produce a 'match' between observed and simulated summaries are retained, and used to estimate the inaccessible posterior. With no reduction to a low-dimensional set of sufficient statistics being possible in the state space setting, we define the summaries as the maximum of an auxiliary likelihood function, and thereby exploit the asymptotic sufficiency of this estimator for the auxiliary parameter vector. We derive conditions under which this approach - including a computationally efficient version based on the auxiliary score - achieves Bayesian consistency. To reduce the well-documented inaccuracy of ABC in multi-parameter settings, we propose the separate treatment of each parameter dimension using an integrated likelihood technique. Three stochastic volatility models for which exact Bayesian inference is either computationally chal

Graphs are common mathematical structures that are visual and intuitive. They constitute a natural and seamless way for system modelling in science, engineering and beyond, including computer science, biology, business process modelling, etc. Graph computation models constitute a class of very high-level models where graphs are first-class citizens. The aim of the International GCM Workshop series is to bring together researchers interested in all aspects of computation models based on graphs and graph transformation. It promotes the cross-fertilizing exchange of ideas and experiences among senior and young researchers from the different communities interested in the foundations, applications, and implementations of graph computation models and related areas.

In this paper, we examine the structure of systems that are weighted homogeneous for several systems of weights, and how it impacts the computation of Gröbner bases. We present several linear algebra algorithms for computing Gröbner bases for systems with this structure, either directly or by reducing to existing structures. We also present suitable optimization techniques. As an opening towards complexity studies, we discuss potential definitions of regularity and prove that they are generic if non-empty. Finally, we present experimental data from a prototype implementation of the algorithms in SageMath.

In recent years dynamical modelling has been provided with a range of breakthrough methods to perform exact Bayesian inference. However it is often computationally unfeasible to apply exact statistical methodologies in the context of large datasets and complex models. This paper considers a nonlinear stochastic differential equation model observed with correlated measurement errors and an application to protein folding modelling. An Approximate Bayesian Computation (ABC) MCMC algorithm is suggested to allow inference for model parameters within reasonable time constraints. The ABC algorithm uses simulations of "subsamples" from the assumed data generating model as well as a so-called "early rejection" strategy to speed up computations in the ABC-MCMC sampler. Using a considerate amount of subsamples does not seem to degrade the quality of the inferential results for the considered applications. A simulation study is conducted to compare our strategy with exact Bayesian inference, the latter resulting two orders of magnitude slower than ABC-MCMC for the considered setup. Finally the ABC algorithm is applied to a large size protein data. The suggested methodology is fairly general an

This paper is a sequel to "Computing diagonal form and Jacobson normal form of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We present a new fraction-free algorithm for the computation of a diagonal form of a matrix over a certain non-commutative Euclidean domain over a computable field with the help of Gröbner bases. This algorithm is formulated in a general constructive framework of non-commutative Ore localizations of $G$-algebras (OLGAs). We split the computation of a normal form of a matrix into the diagonalization and the normalization processes. Both of them can be made fraction-free. For a matrix $M$ over an OLGA we provide a diagonalization algorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$ holds and $D$ is diagonal. The fraction-free approach gives us more information on the system of linear functional equations and its solutions, than the classical setup of an operator algebra with rational functions coefficients. In particular, one can handle distributional solutions together with, say, meromorphic ones. We investigate Ore localizations of common operator algebras over $K[x]$ and use them in the unimodularity analys

This paper reviews the state-of-the-art of semantic change computation, one emerging research field in computational linguistics, proposing a framework that summarizes the literature by identifying and expounding five essential components in the field: diachronic corpus, diachronic word sense characterization, change modelling, evaluation data and data visualization. Despite the potential of the field, the review shows that current studies are mainly focused on testifying hypotheses proposed in theoretical linguistics and that several core issues remain to be solved: the need for diachronic corpora of languages other than English, the need for comprehensive evaluation data for evaluation, the comparison and construction of approaches to diachronic word sense characterization and change modelling, and further exploration of data visualization techniques for hypothesis justification.

The problem of solving a system of polynomial equations is one of the most fundamental problems in applied mathematics. Among them, the problem of solving a system of binomial equations form a important subclass for which specialized techniques exist. For both theoretic and applied purposes, the degree of the solution set of a system of binomial equations often plays an important role in understanding the geometric structure of the solution set. Its computation, however, is computationally intensive. This paper proposes a specialized parallel algorithm for computing the degree on GPUs that takes advantage of the massively parallel nature of GPU devices. The preliminary implementation shows remarkable efficiency and scalability when compared to the closest CPU-based counterpart. Applied to the "master space problem of $\mathcal{N}=1$ gauge theories" the GPU-based implementation achieves nearly 30 fold speedup over its CPU-only counterpart enabling the discovery of previously unknown results. Equally important to note is the far superior scalability: with merely 3 GPU devices on a single workstation, the GPU-based implementation shows better performance, on certain problems, than a s

In recent years, several successful applications of the Artificial Neural Networks (ANNs) have emerged in nuclear physics and high-energy physics, as well as in biology, chemistry, meteorology, and other fields of science. A major goal of nuclear theory is to predict nuclear structure and nuclear reactions from the underlying theory of the strong interactions, Quantum Chromodynamics (QCD). With access to powerful High Performance Computing (HPC) systems, several ab initio approaches, such as the No-Core Shell Model (NCSM), have been developed to calculate the properties of atomic nuclei. However, to accurately solve for the properties of atomic nuclei, one faces immense theoretical and computational challenges. The present study proposes a feed-forward ANN method for predicting the properties of atomic nuclei like ground state energy and ground state point proton root-mean-square (rms) radius based on NCSM results in computationally accessible basis spaces. The designed ANNs are sufficient to produce results for these two very different observables in 6Li from the ab initio NCSM results in small basis spaces that satisfy the theoretical physics condition: independence of basis spac

We report on our experiences exploring state of the art Groebner basis computation. We investigate signature based algorithms in detail. We also introduce new practical data structures and computational techniques for use in both signature based Groebner basis algorithms and more traditional variations of the classic Buchberger algorithm. Our conclusions are based on experiments using our new freely available open source standalone C++ library.

This paper introduces a simple and computationally efficient algorithm for conversion formulae between moments and cumulants. The algorithm provides just one formula for classical, boolean and free cumulants. This is realized by using a suitable polynomial representation of Abel polynomials. The algorithm relies on the classical umbral calculus, a symbolic language introduced by Rota and Taylor in 1994, that is particularly suited to be implemented by using software for symbolic computations. Here we give a MAPLE procedure. Comparisons with existing procedures, especially for conversions between moments and free cumulants, as well as examples of applications to some well-known distributions (classical and free) end the paper.

While evolution has inspired algorithmic methods of heuristic optimisation, little has been done in the way of using concepts of computation to advance our understanding of salient aspects of biological phenomena. We argue that under reasonable assumptions, interesting conclusions can be drawn that are of relevance to behavioural evolution. We will focus on two important features of life--robustness and fitness--which, we will argue, are related to algorithmic probability and to the thermodynamics of computation, disciplines that may be capable of modelling key features of living organisms, and which can be used in formulating new algorithms of evolutionary computation.

We present two new results on the computational limitations of affine automata. First, we show that the computation of bounded-error rational-values affine automata is simulated in logarithmic space. Second, we give an impossibility result for algebraic-valued affine automata. As a result, we identify some unary languages (in logarithmic space) that are not recognized by algebraic-valued affine automata with cutpoints.

We show that in quantum computation almost every gate that operates on two or more bits is a universal gate. We discuss various physical considerations bearing on the proper definition of universality for computational components such as logic gates.

Quantum conference is a process of securely exchanging messages between three or more parties, using quantum resources. A Measurement Device Independent Quantum Dialogue (MDI-QD) protocol, which is secure against information leakage, has been proposed (Quantum Information Processing 16.12 (2017): 305) in 2017, is proven to be insecure against intercept-and-resend attack strategy. We first modify this protocol and generalize this MDI-QD to a three-party quantum conference and then to a multi-party quantum conference. We also propose a protocol for quantum multi-party XOR computation. None of these three protocols proposed here use entanglement as a resource and we prove the correctness and security of our proposed protocols.

Let $f_1,...,f_s \in \mathbb{K}[x_1,...,x_m]$ be a system of polynomials generating a zero-dimensional ideal $\I$, where $\mathbb{K}$ is an arbitrary algebraically closed field. We study the computation of "matrices of traces" for the factor algebra $\A := \CC[x_1, ..., x_m]/ \I$, i.e. matrices with entries which are trace functions of the roots of $\I$. Such matrices of traces in turn allow us to compute a system of multiplication matrices $\{M_{x_i}|i=1,...,m\}$ of the radical $\sqrt{\I}$. We first propose a method using Macaulay type resultant matrices of $f_1,...,f_s$ and a polynomial $J$ to compute moment matrices, and in particular matrices of traces for $\A$. Here $J$ is a polynomial generalizing the Jacobian. We prove bounds on the degrees needed for the Macaulay matrix in the case when $\I$ has finitely many projective roots in $\mathbb{P}^m_\CC$. We also extend previous results which work only for the case where $\A$ is Gorenstein to the non-Gorenstein case. The second proposed method uses Bezoutian matrices to compute matrices of traces of $\A$. Here we need the assumption that $s=m$ and $f_1,...,f_m$ define an affine complete intersection. This second method also works