Relating time complexity of protein folding simulation to approximations of folding time

Finkelstein and Badretdinov [A.V. Finkelstein, A.Y. Badretdinov, Rate of protein folding near the point of thermodynamic equilibrium between the coil and the most stable chain fold, Fold. Des. 2 (1997) 115-121] approximated the folding time of protein sequences of length n by exp(λ⋅n2/3±χ⋅n1/2/2)ns, where λ and χ are constants close to unity. Recently, Fu and Wang [B. Fu, W. Wang, A ²O(n1−1/d⋅logn) time algorithm for d-dimensional protein folding in the HP-model, in: J. Daz, J. Karhumäki, A. Lepistö, D. Sannella (Eds.), Proceedings of the 31st International Colloquium on Automata, Languages and Programming, in: Lecture Notes in Comput. Sci., vol. 3142, Springer-Verlag, Heidelberg, 2004, pp. 630-644] published an exp(O(n1−1/d)⋅lnn) algorithm for d-dimensional protein folding simulation in the HP-model, which is close to the folding time approximation by Finkelstein and Badretdinov and can be seen as a justification of the HP-model for investigating general complexity issues of protein folding. We propose a stochastic local search procedure that is based on logarithmic simulated annealing. We obtain that after (m/δ)^a⋅D Markov chain transitions the probability to be in a minimum energy conformation is at least 1−δ, where m?b(d)⋅n is the maximum neighbourhood size (b(d) small integer), a is a small constant, and D is the maximum value of the minimum escape height from local minima of the underlying energy landscape. We note that the time bound is instance-specific, and we conjecture D<n1−1/d as a worst case upper bound. We analyse experimentally on selected benchmark problems for the d=2 case.     

Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations

If a partial differential equation is reduced to an ordinary differential equation in the form u^′(ξ)=G(u,θ₁,…,θm) under the traveling wave transformation, where θ₁,…,θm are parameters, its solutions can be written as an integral form . Therefore, the key steps are to determine the parameters' scopes and to solve the corresponding integral. When G is related to a polynomial, a mathematical tool named complete discrimination system for polynomial is applied to this problem so that the parameter's scopes can be determined easily. The complete discrimination system for polynomial is a natural generalization of the discrimination △=b²−4ac of the second degree polynomial ax²+bx+c. For example, the complete discrimination system for the third degree polynomial F(w)=w³+d₂w²+d₁w+d₀ is given by and . In the paper, we give some new applications of the complete discrimination system for polynomial, that is, we give the classifications of traveling wave solutions to some nonlinear differential equations through solving the corresponding integrals. In finally, as a result, we give a partial answer to a problem on Fan's expansion method.     

On certain properties of Hurwitz determinants for interval polynomials

Consider the stable interval polynomialsF _n (z)=z ⁿ +a ₁ z ^n?1 +...+a _n?1 z+a _n wherea _i are real numbers, satisfying the inequalities α _i ≤a _i ≤β _i ,i=1,2, ...,n. In this paper we prove that mind _n (a) is the same foraεD andaεD ₁, whereD=[α₁, β₁]×[α₂, β₂]×...×[α _n , β _n ],D={(γ₁, γ₂,...γ _n )∈D:γ₁=α₁∨γ₁=β₁,... γ _n =α _n ∨γ _n =β _n }d _n (a)=detH, aεD, H—Hurwitz matrix for the polynomialF _n (z).     

Parabolic equations with double variable nonlinearities

The paper is devoted to the study of the homogeneous Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions:

ut=div(a(x,t,u)|u^|α(x,t)|∇u^|p(x,t)−2∇u)+f(x,t)     

On the cubicity of bipartite graphs

A unit cube in k-dimension (or a k-cube) is defined as the Cartesian product R₁×R₂×?×Rk, where each Ri is a closed interval on the real line of the form [ai,ai+1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. Many NP-complete graph problems can be solved efficiently or have good approximation ratios in graphs of low cubicity. In most of these cases the first step is to get a low dimensional cube representation of the given graph.It is known that for a graph G, . Recently it has been shown that for a graph G, cub(G)?4(Δ+1)lnn, where n and Δ are the number of vertices and maximum degree of G, respectively. In this paper, we show that for a bipartite graph G=(A∪B,E) with |A|=n₁, |B|=n₂, n₁?n₂, and Δ^′=min{ΔA,ΔB}, where ΔA=maxa∈Ad(a) and ΔB=maxb∈Bd(b), d(a) and d(b) being the degree of a and b in G, respectively, cub(G)?2(Δ^′+2)⌈lnn₂⌉. We also give an efficient randomized algorithm to construct the cube representation of G in 3(Δ^′+2)⌈lnn₂⌉ dimensions. The reader may note that in general Δ^′ can be much smaller than Δ.     

Binary search with delayed and missing answers

How many questions are necessary and sufficient to guess an unknown number x in the set S={1,2,…,n}, by using only comparison questions, that is questions of the type “Is x?a?”, a∈S, when answers to questions are received with a delay of d time units and up to c of the answers can be lost, i.e., can not be received at all? We exactly solve this problem for all integers d?0 and c=1.     

Validation of SeaWiFS chlorophyll a concentrations in the Southern Ocean: A revisit

Surface chlorophyll a concentrations (C_a, mg m^− 3) in the Southern Ocean estimated from SeaWiFS satellite data have been reported in the literature to be significantly lower than those measured from in situ water samples using fluorometric methods. However, we found that high-resolution (∼ 1 km²/pixel) daily SeaWiFS C_a (C_a^SWF) data (SeaDAS4.8, OC4v4 algorithm) was an accurate measure of in situ C_a during January-February of 1998-2002 if concurrent in situ data measured by HPLC (C_a^HPLC) instead of fluorometric (C_a^Fluor) measurements were used as ground truth. Our analyses indicate that C_a^Fluor is 2.48 ± 2.23 (n = 647) times greater than C_a^HPLC between 0.05 and 1.5 mg m^− 3 and that the percentage overestimation of in situ C_a by fluorometric measurements increases with decreasing concentrations. The ratio of C_a^SWF/C_a^HPLC is 1.12 ± 0.91 (n = 96), whereas the ratio of C_a^SWF/C_a^Fluor is 0.55 ± 0.63 (n = 307). Furthermore, there is no significant bias in C_a^SWF (12% and − 0.07 in linear and log-transformed C_a, respectively) when C_a^HPLC is used as ground truth instead of C_a^Fluor. The high C_a^Fluor/C_a^HPLC ratio may be attributed to the relatively low concentrations of chlorophyll b (C_b/C_a = 0.023 ± 0.034, n = 482) and relatively high concentrations of chlorophyll c (C_c/C_a = 0.25 ± 0.59, n = 482) in the phytoplankton pigment composition when compared to values from other regions. Because more than 90% of the waters in the study area, as well as in the entire Southern Ocean (south of 60° S), have C_a^SWF between 0.05 and 1.5 mg m^− 3, we consider that the SeaWiFS performance of C_a retrieval is satisfactory and for this C_a range there is no need to further develop a “regional” bio-optical algorithm to account for the previous SeaWiFS “underestimation”.     

Nearly tight bounds on the number of Hamiltonian circuits of the hypercube and generalizations

It has been shown that for every perfect matching M of the d-dimensional n-vertex hypercube, d?2, n=d², there exists a second perfect matching M^′ such that the union of M and M^′ forms a Hamiltonian circuit of the d-dimensional hypercube. We prove a generalization of a special case of this result when there are two dimensions that do not get used by M. It is known that the number Md of perfect matchings of the d-dimensional hypercube satisfies and, in particular, (2d/n)^n/2(n/2)!?Md?(d!)^n/(2d). It has also been shown that the number Hd of Hamiltonian circuits of the hypercube satisfies 1?limd→∞(logHd)/(logMd)?2. We finally strengthen this result to a nearly tight bound ((dlog2/(eloglogd))(1−on(1)))?Hd?(d!)^n/(2d)((d−1)!)^n/(2(d−1))/2 proving that limd→∞(logHd)/(logMd)=2. This means that the bound Hd?Md is improved to a nearly tight , so the number of Hamiltonian circuits in the hypercube is nearly quadratic in the number of perfect matchings. The proofs are based on a result for graphs that are the Cartesian product of squares and arbitrary bipartite regular graphs that have a Hamiltonian cycle. We also study a labeling scheme related to matchings.     

Asymmetric quantum Reed-Solomon and generalized Reed-Solomon codes

Two new families of asymmetric quantum codes are constructed in this paper. The first one is derived from the Calderbank-Shor-Steane (CSS) construction applied to classical Reed-Solomon (RS) codes, providing quantum codes with parameters [[N = l(q ^l −1), K = l(q ^l −2d + c + 1), d _z ≥ d/d _x ≥ (d−c)]] _q , where q is a prime power and d > c + 1, c ≥ 1, l ≥ 1 are integers. The second family is derived from the CSS construction applied to classical generalized RS codes, generating quantum codes with parameters [[N = mn, K = m(2k−n + c), d _z ≥ d/d _x ≥ (d−c)]] _q , where q is a prime power, 1 < k < n < 2k + c ≤ q ^m , k = n − d + 1, and n, d > c + 1, c ≥ 1, m ≥ 1 are integers. Although the second proposed construction generalizes the first one, the techniques developed in both constructions are slightly different. These new codes have parameters better than or comparable to the ones available in the literature. Additionally, the proposed codes can be utilized in quantum channels having great asymmetry, that is, quantum channels in which the probability of occurrence of phase-shift errors is large when compared to the probability of occurrence of qudit-flip errors.     

Bipanconnectivity and edge-fault-tolerant bipancyclicity of hypercubes

A bipartite graph is bipancyclic if it contains a cycle of every even length from 4 to |V(G)| inclusive. It has been shown that Q_n is bipancyclic if and only if n?2. In this paper, we improve this result by showing that every edge of Q_n−E′ lies on a cycle of every even length from 4 to |V(G)| inclusive where E′ is a subset of E(Q_n) with |E′|?n−2. The result is proved to be optimal. To get this result, we also prove that there exists a path of length l joining any two different vertices x and y of Q_n when h(x,y)?l?|V(G)|−1 and l−h(x,y) is even where h(x,y) is the Hamming distance between x and y.     

Maximum number of edges joining vertices on a cube

Let E_d(n) be the number of edges joining vertices from a set of n vertices on a d-dimensional cube, maximized over all such sets. We show that E_d(n)=∑_i=0^r−1(l_i/2+i)2^l_i, where r and l₀>l₁>?>l_r−1 are nonnegative integers defined by n=∑_i=0^r−12^l_i.     

Modified nonlinearly dispersive mK(m,n,k) equations: I. New compacton solutions and solitary pattern solutions

In this paper exact solutions of the modified nonlinearly dispersive KdV equations (simply called mK(m,n,k) equations), u^m−1u_t+a(uⁿ)_x+b(u^k)_xxx=0, are investigated by using some direct ansatze. As a result, abundant new compacton solutions: solitons with the absence of infinite wings, solitary pattern solutions having infinite slopes or cups, solitary wave solutions and periodic wave solutions are obtained.     

A program to compute exact hydrogenic radial integrals and Einstein coefficients

An exact expression for the dipole radial integral of hydrogen has been given by Gordon [Ann. Phys. 2 (1929) 1031]. It contains two hypergeometric functions F(a,b;c;x), which are difficult to calculate directly, when the (negative) integers a, b are large, as in the case of high Rydberg states of hydrogenic ions. We have derived a simple method [D. Hoang-Binh, Astron. Astrophys. 238 (1990) 449], using a recurrence relation to calculate exactly F, starting from two initial values, which are very easy to compute. We present here a numerical code using this method. The code computes exact hydrogenic radial integrals, oscillator strengths, Einstein coefficients, and lifetimes, for principal quantum numbers up to 1000.

Program summary

Program title: ba5.2Catalogue identifier: ADUU_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUU_v2_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1400No. of bytes in distributed program, including test data, etc.: 11 737Distribution format: tar.gzProgramming language: Fortran 77Computer: PC, iMacOperating system: Linux/Unix, MacOS 9.0RAM: Less than 1 MBClassification: 2, 2.2Catalogue identifier of previous version: ADUU_v1_0Journal reference of previous version: Comput. Phys. Comm. 166 (2005) 191Does the new version supersede the previous version?: YesNature of problem: Exact calculation of atomic data.Solution method: Use of a recurrence relation to compute hypergeometric functions.Reasons for new version: This new version computes additional important related data, namely, the total Einstein coefficients, and radiative lifetimes.Summary of revisions: Values of the total Einstein transition probability from an upper level n to a lower level n^′ are computed, as well as the radiative lifetime of a level n.Running time: About 2 seconds     

The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation

We investigate the periodic nature of the positive solutions of the fuzzy difference equation , where k, m are positive integers, A₀, A₁, are positive fuzzy numbers and the initial values x_i, i = −d, −d + 1, … , −1, d = max{k, m}, are positive fuzzy numbers. In addition, we give conditions so that the solutions of this equation are unbounded.     

Online Metric Tracking and Smoothing

We consider the online smoothing problem, in which a tracker is required to maintain distance no more than Δ≥0 from a time-varying signal f while minimizing its own movement. The problem is determined by a metric space (X,d) with an associated cost function c:?→?. Given a signal f ₁,f ₂,…∈X the tracker is responsible for producing a sequence a ₁,a ₂,… of elements of X that meet the proximity constraint: d(f _i ,a _i )≤Δ. To complicate matters, the tracker is on-line—the value a _i may only depend on f ₁,…,f _i —and wishes to minimize the cost of his travels, ∑c(d(a _i ,a _i+1)). We evaluate such tracking algorithms competitively, comparing this with the cost achieved by an optimal adversary apprised of the entire signal in advance. The problem was originally proposed by Yi and Zhang (In: Proceedings of the 20th annual ACM-SIAM symposium on discrete algorithms (SODA), pp. 1098–1107. ACM Press, New York, 2009), who considered the natural circumstance where the metric spaces are taken to be ? ^k with the ? ₂ metric and the cost function is equal to 1 unless the distance is zero (thus the tracker pays a fixed cost for any nonzero motion).

We begin by studying arbitrary metric spaces with the “pay if you move” metric of Yi and Zhang (In: Proceedings of the 20th annual ACM-SIAM symposium on discrete algorithms (SODA), pp. 1098–1107. ACM Press, New York, [2009]) described above and describe a natural randomized algorithm that achieves a O(logb _Δ)-competitive ratio, where b _Δ=max _x∈X |B _Δ(x)| is the maximum number of points appearing in any ball of radius Δ. We show that this bound is tight.

We then focus on the metric space ? with natural families of monotone cost functions c(x)=x ^p for some p≥0. We consider both the expansive case (p≥1) and the contractive case (p<1), and show that the natural lazy algorithm performs well in the expansive case. In the contractive case, we introduce and analyze a novel deterministic algorithm that achieves a constant competitive ratio depending only on p. Finally, we observe that by slightly relaxing the guarantee provided by the tracker, one can obtain natural analogues of these algorithms that work in continuous metric spaces.

     

Construction of high degree resilient S-boxes with improved nonlinearity

We present a simple method to use an [n−d−1,m,t+1] code to construct an n-input, m-output, t-resilient function with degree d>m and nonlinearity ²n−1−²n−⌈(d+1)/2⌉−(m+1)²n−d−1. For any fixed values of parameters n,m,t and d, with d>m, the nonlinearity obtained by our construction is higher than the nonlinearity obtained by Cheon in Crypto 2001.     

Approximate Solutions of Polynomial Equations

In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x)  ∈ Q [ x ], with its constant term equal to 0, such that F(x, y)  = ∑_j = 0ⁿc_j(y − g(x))^jfor some rational numbers c_j, hence, F(x, g(x)  + a)  ∈ Q for all a ∈ Q, or there are at most t distinct polynomials g₁(x),⋯ , g_t(x), t ≤ n, such that F(x, g_i(x))  ∈ Q for 1  ≤ i ≤ t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g₁(x),⋯ , g_t(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).     

Monotone Einschließung von Inversen positiver Elemente durch Verfahren vom Schulz-Typ

The conditional iterationx _n+1 =sup (x _n ,x _n +x _n (e?ax _n )),y _n?1 =inf (y _n ,x _n +y _n (e?ax _n )) generating sequences (x _n ) and (y _n ) is considered in partially ordered spaces. Under certain conditions it is shown, that the inversea ^?1 of a positive elementa≧0 is monotonously enclosed in the sensex _n ≦x _n+1 ≦a ^?1 ≦y _n+1 ≦y _n and that (x _n ) and (y _n ) converge toa ^?1 quadratically.