Kth Step Research Articles

The local persistence length of semi-flexible self-avoiding walks on the square lattice

By applying the pruned-enriched Rosenbluth Monte Carlo simulation method, we have studied the local persistence length of semi-flexible linear polymers presented by self-avoiding walks (SAWs) on the square lattice, where the stiffness property is characterised by the weight s assigned to each bend of the walk. In this model, the local persistence length λN(k) of N-step SAWs is formulated as an ensemble average of the projection of the end-to-end position vector onto the oriented kth step of the SAW path. We have found that the persistence length λN(k)(s) decreases with s and increases with k and N. Regarding the chain position parameter k, we have scrutinised two particular cases: the first case is when k has a constant (fixed) value, independent of the chain length, and the second one is the situation when k is comparable with the SAW length k∼N . For fixed k, we have learnt that in the limit N→∞ , the persistence length λN(k)(s) tends towards a constant value λp(k)(s) behaving as k1/2 . In the other analysed case k∼N , in the same limit, λN(k)(s) diverges as N ϕ , with φ=1/2 . We have also examined the dependence of the studied quantity on s and discussed our results in the context of previous findings and predictions related to the persistence length behaviour of SAWs in Euclidean spaces.

Replica symmetry breaking in neural networks: a few steps toward rigorous results

In this paper we adapt the broken replica interpolation technique (developed by Francesco Guerra to deal with the Sherrington-Kirkpatrick model, namely a pairwise mean-field spin-glass whose couplings are i.i.d. standard Gaussian variables) in order to work also with the Hopfield model (i.e. a pairwise mean-field neural-network whose couplings are drawn according to Hebb’s learning rule): this is accomplished by grafting Guerra’s telescopic averages on the transport equation technique, recently developed by some of the authors. As an overture, we apply the technique to solve the Sherrington-Kirkpatrick model with i.i.d. Gaussian couplings centered at J0 and with finite variance J; the mean J0 provides a ferromagnetic contribution to be detected in a noisy environment tuned by J, hence making this model a natural test-case to be investigated before addressing the Hopfield model. For both the models, an explicit expression of their quenched free energy in terms of their natural order parameters is obtained at the Kth step (K arbitrary, but finite) of replica-symmetry-breaking. In particular, for the Hopfield model, by assuming that the overlaps respect Parisi’s decomposition (following the ziqqurat ansatz) and that the Mattis magnetization is self-averaging, we recover previous results obtained via replica-trick by Amit, Crisanti and Gutfreund (1RSB) and by Steffan and Kühn (2RSB).

Stabilization of cycles for difference equations with a noisy PF control

Difference equations, such as a Ricker map, for an increased value of the parameter, experience instability of the positive equilibrium and transition to deterministic chaos. To achieve stabilization, various methods can be applied. Proportional Feedback control suggests a proportional reduction of the state variable at every kth step. First, if k≠1, a cycle is stabilized rather than an equilibrium. Second, the equation can incorporate an additive noise term, describing the variability of the environment, as well as multiplicative noise corresponding to possible deviations in the control intensity. The present paper deals with both issues, it justifies a possibility of getting a stable blurred k-cycle. Presented examples include the Ricker model, as well as equations with unbounded f, such as the bobwhite quail population models. Though the theoretical results justify stabilization for either multiplicative or additive noise only, numerical simulations illustrate that a blurred cycle can be stabilized when both multiplicative and additive noises are involved.

A compressed cyclic reduction for QBDs with low rank upper and lower transitions

Consider a quasi-birth-and-death (QBD) Markov chain [6], having probability transition matrix where Bi, Ai, i = ?1, 0, 1, are m x m matrices. In the numerical solution of QBD Markov chains a crucial step is the efficient computation of the minimal nonnegative solution R of the quadratic matrix equation X = X 2 A?1 + XA 0 + A 1 . (1) To this purpose, many numerical methods, with different properties, have been designed in the last years (see for instance [1, 2, 3, 4]). However, many of these numericalmethods are defined for general block coefficients A?1, A0 and A1, and do not exploit the possible structure of these blocks. Recently, some attention has been addressed to the case where A ?1 has only few non-null columns, or A1 has only few non-null rows. These properties are satisfied when the QBD has restricted transitions to higher (or lower) levels. In particular, in [7] the authors exploit these properties of the matrix A ?1 , or A 1 , to formulate the QBD in terms of an M/G/1 type Markov chain, where the block matrices have size smaller than m; in particular, when both A?1 and A1 have the desired property, the latter M/G/1 type Markov chain reduces to a QBD. In [5] the structure of A ?1 is used in order to reduce the computational cost of some algorithms for computing R. Here we assume that both the matrices A ?1 and A 1 have small rank with respect to their size m. In particular, if A?1 and A1 have only few non-null columns and rows, respectively, they have small rank. We show that, under this assumption, the matrix R can be computed by using the cyclic reduction algorithm, where the matrices A(k) i , i = ?1, 0, 1, generated at the kth step of the algorithm, can be represented by small rank matrices. In particular, if r ?1 is the rank of A ?1 , and if r 1 is the rank of A 1 , then each step of cyclic reduction can be performed by means of O((r ?1+r1 ) 3 ) arithmetic operations. This cost estimate must be compared with the cost of O(m3) arithmetic operations, needed without exploiting the structure of A ?1 and A 1 . Therefore, if r 1 and r 1 /are much smaller than m, the advantage is evident. It remains an open issue to understand how the structure can be exploited in the case where only one between A ?1 and A 1 has low rank.

Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones

We prove that if \({U\subset \mathbb {R}^n}\) is an open domain whose closure \({\overline U}\) is compact in the path metric, and F is a Lipschitz function on ∂U, then for each \({\beta \in \mathbb {R}}\) there exists a unique viscosity solution to the β-biased infinity Laplacian equation $$\beta |\nabla u| + \Delta_\infty u=0$$ on U that extends F, where \({\Delta_\infty u= |\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}}\). In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the β-biased \({\epsilon}\)-game as follows. The starting position is \({x_0 \in U}\). At the kth step the two players toss a suitably biased coin (in our key example, player I wins with odds of \({\exp(\beta\epsilon)}\) to 1), and the winner chooses xk with \({d(x_k,x_{k-1}) < \epsilon}\). The game ends when \({x_k \in \partial U}\), and player II pays the amount F(xk) to player I. We prove that the value \({u^{\epsilon}(x_0)}\) of this game exists, and that \({\|u^\epsilon - u\|_\infty \to 0}\) as \({\epsilon \to 0}\), where u is the unique extension of F to \({\overline{U}}\) that satisfies comparison with β-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with β-exponential cones if and only if it is a viscosity solution to the β-biased infinity Laplacian equation.

A fast and efficient algorithm for Slater determinant updates in quantum Monte Carlo simulations

We present an efficient low-rank updating algorithm for updating the trial wave functions used in quantum Monte Carlo (QMC) simulations. The algorithm is based on low-rank updating of the Slater determinants. In particular, the computational complexity of the algorithm is O(kN) during the kth step compared to traditional algorithms that require O(N(2)) computations, where N is the system size. For single determinant trial wave functions the new algorithm is faster than the traditional O(N(2)) Sherman-Morrison algorithm for up to O(N) updates. For multideterminant configuration-interaction-type trial wave functions of M+1 determinants, the new algorithm is significantly more efficient, saving both O(MN(2)) work and O(MN(2)) storage. The algorithm enables more accurate and significantly more efficient QMC calculations using configuration-interaction-type wave functions.

Sequential unconstrained minimization algorithms for constrained optimization

The problem of minimizing a function f(x):RJ → R, subject to constraints on the vector variable x, occurs frequently in inverse problems. Even without constraints, finding a minimizer of f(x) may require iterative methods. We consider here a general class of iterative algorithms that find a solution to the constrained minimization problem as the limit of a sequence of vectors, each solving an unconstrained minimization problem. Our sequential unconstrained minimization algorithm (SUMMA) is an iterative procedure for constrained minimization. At the kth step we minimize the function to obtain xk. The auxiliary functions gk(x):D ⊆ RJ → R+ are nonnegative on the set D, each xk is assumed to lie within D, and the objective is to minimize the continuous function f:RJ → R over x in the set , the closure of D. We assume that such minimizers exist, and denote one such by . We assume that the functions gk(x) satisfy the inequalities for k = 2, 3, …. Using this assumption, we show that the sequence {f(xk)} is decreasing and converges to . If the restriction of f(x) to D has bounded level sets, which happens if is unique and f(x) is closed, proper and convex, then the sequence {xk} is bounded, and , for any cluster point x*. Therefore, if is unique, and . When is not unique, convergence can still be obtained, in particular cases. The SUMMA includes, as particular cases, the well-known barrier- and penalty-function methods, the simultaneous multiplicative algebraic reconstruction technique (SMART), the proximal minimization algorithm of Censor and Zenios, the entropic proximal methods of Teboulle, as well as certain cases of gradient descent and the Newton–Raphson method. The proof techniques used for SUMMA can be extended to obtain related results for the induced proximal distance method of Auslander and Teboulle.

Transferable step potentials for perfluorinated hydrocarbons

The step potential equilibria and discontinuous molecular dynamics (SPEADMD) model is adapted for characterizing the interaction potentials of perfluorocarbons and their mixtures with n-alkanes. We seek to explain the peculiar behavior of these systems, especially with regard to the unfavorable mixing behavior. The methodology is based on discontinuous molecular dynamics (DMD) and second order thermodynamic perturbation theory (TPT). DMD simulation is applied to the repulsive part of the potential. The effects of disperse attractions and hydrogen bonding are treated by TPT, discretizing the attractive potential into four distinct wells of variable depth. This approach accelerates the molecular simulations in general and the parameterization of the transferable potentials in particular. C 3–C 8 straight chain perfluorocarbons are characterized along with perfluorobenzene, perfluorocyclobutane, and heptafluoropropane applying explicit atom models for all fluorine atoms. Each compound is simulated at 21 densities. Interpolation with density combines with TPT to give a complete equation of state. The depths of the attractive wells are optimized by iterating on their values until the vapor pressures computed by the resulting equation of state provide the minimum deviation from experimental data. Step potentials offer the prospect of setting the step depths without preconceived notions regarding the shape of the potential. In the case of perfluorocarbons, it is found that the minimum of the potential is reached at roughly 1.8 σ, with the range from σ to 1.8 σ acting as a soft “shoulder” region which is weakly attractive, where σ designates the site diameter. This shoulder region can explain the unusually low heat of vaporization of perfluorocarbons and high vapor pressures relative to compounds with similar molecular volume. Application of this approach results in average absolute deviations of 8% in vapor pressure and 3% in liquid density. Coincidentally, the soft shoulder also explains the weak mixture interactions between perfluorocarbons and n-alkanes. The Lorentz–Berthelot combining rule assumed to describe site–site interactions between corresponding steps leads to ɛ ijk ∼ ( ɛ iik ɛ jjk ) 1/2, where ɛ ijk is the depth of the potential well between sites i and j in the kth step. For steps at short range, the alkane steps are at their deepest, but the perfluorocarbon interactions are weak, making the mixed interaction weak. For steps at long range, the perfluorocarbon steps are deeper, but the alkane interactions are weak, again making the mixed interaction weak.

Tame functions are semismooth

Superlinear convergence of the Newton method for nonsmooth equations requires a “semismoothness” assumption. In this work we prove that locally Lipschitz functions definable in an o-minimal structure (in particular semialgebraic or globally subanalytic functions) are semismooth. Semialgebraic, or more generally, globally subanalytic mappings present the special interest of being γ-order semismooth, where γ is a positive parameter. As an application of this new estimate, we prove that the error at the kth step of the Newton method behaves like $$O(2^{-{(1+\gamma)}^k})$$.

Twisted homology

D-branes are classified by twisted K-theory. Yet twisted K-theory is often hard to calculate. We argue that, in the case of a compactification on a simply-connected six manifold, twisted K-theory is isomorphic to a much simpler object, twisted homology. Unlike K-theory, homology can be twisted by a class of any degree and so it classifies not only D-branes but also M-branes. Twisted homology classes correspond to cycles in a certain bundle over spacetime, and branes may decay via Kachru-Pearson-Verlinde transitions only if this cycle is trivial. We provide a spectral sequence which calculates twisted homology, the kth step treats D(p-2k)-branes ending on Dp-branes.

A Computation of the Static-Feedback of the kth Step of RDEA

A Computation of the Static-Feedback of the kth Step of RDEA

Crossover Scaling in Piling Block Games

In a previous paper, Iwasaki and one of the authors (K.H.) have clarified that a game with piling blocks plays a role as the simplest model showing the self-organized criticality and has properties of scaling and universality for nonequilibrium phenomena. The model of pilng block game is constructed as follows: We use blocks with fixed length L. When putting a block on the topmost block of a pile, its center of mass is shifted by the unit length. The probability of shifting to the right is p and one to the left is q 1⁄4 1 p. Of course, p and q are exchangeable each other because of the right-left symmetry. Let yk be the one-dimensional coordinate of the center of mass of the block in the kth step. The random variable k takes +1 with the probability p and 1 with q. Then

A note on the generalized Josephus problem

1. This is a continuation, with supplementary notes, of our previous paper [3] in which some solutions have been provided for the generalized Josephus problem. If we are given two integers mミ2and n ~ 1, supposing that n objects, numbered from 1 to n, are arranged in a circle, starting then with object number 1 and counting each object in turn around the circle, clockwise or anticlockwise in the same direction fix.ed once for al, we eliminate every mth object until al the objects are removed.τhe kth Josephus number am(k,n) (1豆k歪n)is defined to be equal to 1 (1壬l壬n),if 1 isthe number attached to the object to be removed at the kth step of reduction. We have plainly

Random walks on the torus with several generators

AbstractGiven n vectors {i} ∈ [0, 1)d, consider a random walk on the d‐dimensional torus 𝕋d = ℝd/ℤd generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q*k) between the kth step distribution of the walk and Haar measure is bounded below by D(Q*k) ≥ C1k−n/2, where C1 = C(n, d) is a constant. If the vectors are badly approximated by rationals (in a sense we will define), then D(Q*k) ≤ C2k−n/2d for C2 = C(n, d, j) a constant. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004

On updating the least singular value: a lower bound

Let {A k } k=1 p , , be a sequence of m × k matrices such that A k+1 is obtained from A k by appending a column. In this paper a lower bound δ k+1 for the least singular value of each matrix A k+1 is derived from the value δ k . The kth step of the algorithm is based on the solution of the least squares problem A k x k =b k , using the QR updating scheme, so that the algorithm requires O(mp 2 + p 3) operations. Lastly, an error analysis is performed in order to detect when the computed lower bound is a good approximation of the exact least singular value.

Computations with Gohberg-Semencul-type formulas for Toeplitz matrices

The inverse of a Toeplitz matrix T n can be represented in different ways by Gohberg-Semencul-type formulas as the sum of products of upper and lower triangular Toeplitz matrices. If we have given such a formula, we can solve every equation T nξ = b in O( n log n) steps. There are three main questions arising with such representations of T n −1: (1) which special linear equations can be solved in order to get generating vectors for Gohberg-Semencul-type formulas, (2) which algorithm we want to apply to solve these questions, and (3) which special Gohberg-Semencul-type formula we want to use for evaluating T n −1 b. In this paper we present an elementary approach to derive all Gohberg-Semencul-type formulas. Then we introduce representations of T n −1 with special properties. In particular we prove that there exists a Gohberg-Semencul-type formula such that the generating vectors are pairwise orthogonal. Finally, based on the previous results, we give new fast and stable algorithms for solving linear Toeplitz systems that can be used to compute generating vectors for Gohberg-Semencul-type formulas. In the case of a breakdown or near-breakdown in the kth step, the new method introduces a perturbation in the kth entry t k of T n such that the perturbed submatrix T̂ k is well conditioned. By using the Sherman-Morrison-Woodbury formula it is possible to recover the original problem as soon as the corresponding submatrix T k + r is again nonsingular.

Computations with Gohberg-Semencul-type formulas for Toeplitz matrices

The inverse of a Toeplitz matrix Tn can be represented in different ways by Gohberg-Semencul-type formulas as the sum of products of upper and lower triangular Toeplitz matrices. If we have given such a formula, we can solve every equation Tnξ = b in O(n log n) steps. There are three main questions arising with such representations of Tn−1: (1) which special linear equations can be solved in order to get generating vectors for Gohberg-Semencul-type formulas, (2) which algorithm we want to apply to solve these questions, and (3) which special Gohberg-Semencul-type formula we want to use for evaluating Tn−1b. In this paper we present an elementary approach to derive all Gohberg-Semencul-type formulas. Then we introduce representations of Tn−1 with special properties. In particular we prove that there exists a Gohberg-Semencul-type formula such that the generating vectors are pairwise orthogonal. Finally, based on the previous results, we give new fast and stable algorithms for solving linear Toeplitz systems that can be used to compute generating vectors for Gohberg-Semencul-type formulas. In the case of a breakdown or near-breakdown in the kth step, the new method introduces a perturbation in the kth entry tk of Tn such that the perturbed submatrix T̂k is well conditioned. By using the Sherman-Morrison-Woodbury formula it is possible to recover the original problem as soon as the corresponding submatrix Tk + r is again nonsingular.

Pulse-by-pulse reoptimization of the synthesis filter in pulse-based coders

Two iterative algorithms for pulse-based linear predictive (PB-LP) analysis are developed. At the kth step, one of the algorithms assumes that the pulse locations of k pulses are known and calculates the LP coefficients jointly with k pulse amplitudes. In contrast, the other algorithm assumes that the previous k-1 pulse amplitudes are known and jointly estimates the LP filter coefficients and the kth pulse amplitude. These algorithms are proposed for synthesis filter reoptimization in pulse-based LP coders, and their effectiveness in multipulse coding is demonstrated through several simulations.

Estimating an Eigenvector by the Power Method with a Random Start

This paper addresses the problem of approximating an eigenvector belonging to the largest eigenvalue of a symmetric positive definite matrix by the power method. We assume that the starting vector is randomly chosen with uniform distribution over the unit sphere. This paper provides lower and upper as well as asymptotic bounds on the randomized error in the ${\cal L}_p$ sense, $p\in[1,+\infty]$. We prove that it is impossible to achieve sharp bounds that are independent of the ratio between the two largest eigenvalues. This should be contrasted to the problem of approximating the largest eigenvalue, for which Kuczynski and Wozniakowski [ SIAM J. Matrix Anal. Appl., 13 (1992), pp. 1094--1122] proved that it is possible to bound the randomized error at the kth step with a quantity that depends only on k and on the size of the matrix. We prove that the rate of convergence depends on the ratio of the two largest eigenvalues, on their multiplicities, and on the particular norm. The rate of convergence is at most linear in the ratio of the two largest eigenvalues.

Probabilistic Bounds on the Extremal Eigenvalues and Condition Number by the Lanczos Algorithm

The authors analyze the Lanczos algorithm with a random start for approximating the extremal eigenvalues of a symmetric positive definite matrix. They present some bounds on the Lebesgue measure (probability) of the sets of these starting vectors for which the Lanczos algorithm gives at the kth step satisfactory approximations to the largest and smallest eigenvalues. Combining these bounds gets similar estimates for the condition number of a matrix.