2,637 research outputs found
Detecting Simultaneous Integer Relations for Several Real Vectors
An algorithm which either finds an nonzero integer vector for
given real -dimensional vectors such
that or proves that no such integer vector with
norm less than a given bound exists is presented in this paper. The cost of the
algorithm is at most exact arithmetic
operations in dimension and the least Euclidean norm of such
integer vectors. It matches the best complexity upper bound known for this
problem. Experimental data show that the algorithm is better than an already
existing algorithm in the literature. In application, the algorithm is used to
get a complete method for finding the minimal polynomial of an unknown complex
algebraic number from its approximation, which runs even faster than the
corresponding \emph{Maple} built-in function.Comment: 10 page
Gradual sub-lattice reduction and a new complexity for factoring polynomials
We present a lattice algorithm specifically designed for some classical
applications of lattice reduction. The applications are for lattice bases with
a generalized knapsack-type structure, where the target vectors are boundably
short. For such applications, the complexity of the algorithm improves
traditional lattice reduction by replacing some dependence on the bit-length of
the input vectors by some dependence on the bound for the output vectors. If
the bit-length of the target vectors is unrelated to the bit-length of the
input, then our algorithm is only linear in the bit-length of the input
entries, which is an improvement over the quadratic complexity floating-point
LLL algorithms. To illustrate the usefulness of this algorithm we show that a
direct application to factoring univariate polynomials over the integers leads
to the first complexity bound improvement since 1984. A second application is
algebraic number reconstruction, where a new complexity bound is obtained as
well
Complete hierarchies of efficient approximations to problems in entanglement theory
We investigate several problems in entanglement theory from the perspective
of convex optimization. This list of problems comprises (A) the decision
whether a state is multi-party entangled, (B) the minimization of expectation
values of entanglement witnesses with respect to pure product states, (C) the
closely related evaluation of the geometric measure of entanglement to quantify
pure multi-party entanglement, (D) the test whether states are multi-party
entangled on the basis of witnesses based on second moments and on the basis of
linear entropic criteria, and (E) the evaluation of instances of maximal output
purities of quantum channels. We show that these problems can be formulated as
certain optimization problems: as polynomially constrained problems employing
polynomials of degree three or less. We then apply very recently established
known methods from the theory of semi-definite relaxations to the formulated
optimization problems. By this construction we arrive at a hierarchy of
efficiently solvable approximations to the solution, approximating the exact
solution as closely as desired, in a way that is asymptotically complete. For
example, this results in a hierarchy of novel, efficiently decidable sufficient
criteria for multi-particle entanglement, such that every entangled state will
necessarily be detected in some step of the hierarchy. Finally, we present
numerical examples to demonstrate the practical accessibility of this approach.Comment: 14 pages, 3 figures, tiny modifications, version to be published in
Physical Review
Identifying parameter regions for multistationarity
Mathematical modelling has become an established tool for studying the
dynamics of biological systems. Current applications range from building models
that reproduce quantitative data to identifying systems with predefined
qualitative features, such as switching behaviour, bistability or oscillations.
Mathematically, the latter question amounts to identifying parameter values
associated with a given qualitative feature.
We introduce a procedure to partition the parameter space of a parameterized
system of ordinary differential equations into regions for which the system has
a unique or multiple equilibria. The procedure is based on the computation of
the Brouwer degree, and it creates a multivariate polynomial with parameter
depending coefficients. The signs of the coefficients determine parameter
regions with and without multistationarity. A particular strength of the
procedure is the avoidance of numerical analysis and parameter sampling.
The procedure consists of a number of steps. Each of these steps might be
addressed algorithmically using various computer programs and available
software, or manually. We demonstrate our procedure on several models of gene
transcription and cell signalling, and show that in many cases we obtain a
complete partitioning of the parameter space with respect to multistationarity.Comment: In this version the paper has been substantially rewritten and
reorganised. Theorem 1 has been reformulated and Corollary 1 adde
A paradox in bosonic energy computations via semidefinite programming relaxations
We show that the recent hierarchy of semidefinite programming relaxations
based on non-commutative polynomial optimization and reduced density matrix
variational methods exhibits an interesting paradox when applied to the bosonic
case: even though it can be rigorously proven that the hierarchy collapses
after the first step, numerical implementations of higher order steps generate
a sequence of improving lower bounds that converges to the optimal solution. We
analyze this effect and compare it with similar behavior observed in
implementations of semidefinite programming relaxations for commutative
polynomial minimization. We conclude that the method converges due to the
rounding errors occurring during the execution of the numerical program, and
show that convergence is lost as soon as computer precision is incremented. We
support this conclusion by proving that for any element p of a Weyl algebra
which is non-negative in the Schrodinger representation there exists another
element p' arbitrarily close to p that admits a sum of squares decomposition.Comment: 22 pages, 4 figure
Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tort with quadratic and cubic frequencies
We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector omega = (1, Omega), where Omega is a quadratic irrational number, or a 3-dimensional torus with a frequency vector w = (1, Omega, Omega(2)), where Omega is a cubic irrational number. Applying the Poincare-Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which Q is the so-called cubic golden number (the real root of x(3) x - 1= 0), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases.Postprint (published version
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