Position-space renormalization for systems with weak long-range interactions and the breakdown of hyperscaling

Exact renormalization group equations are derived for a position-space renormalization of spin systems with weak long-range forces. It is shown how an apparent dependence of the critical exponents on the choice of the renormalization group can be resolved via the mechanism of dangerous irrelevant variables and that this same mechanism is responsible for the breakdown of hyperscaling. The dimensiond=4 can be seen to be a borderline dimension above which classical critical exponents are expected.     

RATE OF INTRA-MOLECULAR CONTACT FORMATION IN SHORT TWO-DIMENSIONAL COMPACT POLYMERS CHAINS

It is important to know the rate of intra-molecular contact formation in proteins in order to understand how proteins fold clearly. Here we investigate the rate of intra-molecular contact formation in short two-dimensional compact polymer chains by calculating the probability distribution p(r) of end-to-end distance r using the enumeration calculation method and HP model on two-dimensional square lattice. The probability distribution of end-to-end distance p(r) of short two-dimensional compact polymers chains may consist of two parts, i.e. p(r) = p1(r) p2(r), where p1(r) and p2(r) are different for small r. The rate of contact formation decreases monotonically with the number of bonds N, and the rate approximately conforms to the scaling relation of k(N) ∝ N-α. Here the value of α increases with the contact radius a and it also depends on the percentage of H (hydrophobic) residues in the sequences of compact chains and the energy parameters of εHH, εHP and εPP . Some comparisons of theoretical predictions with experimental results are also made. This investigation may help us to understand the protein folding.     

Comparative analysis of affine scaling algorithms based on simplifying assumptions

We analyze several affine potential reduction algorithms for linear programming based on simplifying assumptions. We show that, under a strong probabilistic assumption regarding the distribution of the data in an iteration, the decrease in the primal potential function will be with high probability, compared to the guaranteed(1). ( 2n is a parameter in the potential function andn is the number of variables.) Under the same assumption, we further show that the objective reduction rate of Dikin's affine scaling algorithm is with high probability, compared to no guaranteed convergence rate.Research supported in part by NSF Grant DDM-8922636.     

Finite-size scaling for Potts models

Recently, Borgs and Kotecký developed a rigorous theory of finite-size effects near first-order phase transitions. Here we apply this theory to the ferromagneticq-state Potts model, which (forq large andd2) undergoes a first-order phase transition as the inverse temperature is varied. We prove a formula for the internal energy in a periodic cube of side lengthL which describes the rounding of the infinite-volume jumpE in terms of a hyperbolic tangent, and show that the position of the maximum of the specific heat is shifted by _m (L)=(Inq/E)L ^–d +O(L ^–2d ) with respect to the infinite-volume transition point _t . We also propose an alternative definition of the finite-volume transition temperature _t (L) which might be useful for numerical calculations because it differs only by exponentially small corrections from _t .     

Novel perturbation expansion for the Langevin equation

We discuss the randomly driven systemdx/dt= -W(x) +f(t), wheref(t) is a Gaussian random function or stirring force withf(t)f(t)=2(t–t), andW(x) is of the formgx ¹⁺². The parameter is a measure of the nonlinearity of the equation. We show how to obtain the correlation functionsx(t)f(t)···x(t( n)) _f as a power series in. We obtain three terms in the expansion and show how to use Padé approximants to analytically continue the answer in the variable. By using scaling relations, we show how to get a uniform approximation to the equal-time correlation functions valid for allg and.     

On the Scaling Study for the A_f-A_g Type Free Radical Polymerization

根据Ａｆ－Ａｇ自由基加聚反应的数量分布函数，导出凝胶点附近的渐进分布函数和高分子矩的表示式．进一步应用标度变换，得到了描述溶胶－凝胶相变的广义标度律，从而揭示了Ａｆ－Ａｇ自由基加聚的固化反应是一个相变过程．     

On the ν-dimensional one-component classical plasma: The thermodynamic limit problem revisited

We prove theH-stability property and the existence of the thermodynamic limit of the free energy density of the two-dimensional, one-component classical plasma. We give lower and upper bounds on the free energy density in any dimensionv and draw some consequences.     

The polynomial reconstruction problem: The first 50 years

The problem of reconstructing the characteristic polynomial of a graph of order at least 3 from the collection of characteristic polynomials of its vertex-deleted subgraphs was posed by Cvetkovi? in 1973 as a spectral counter part to the well-known Ulam's reconstruction conjecture. Over the last 50 years, this problem has received notable attention, many positive results have been obtained, but in the general case the problem is still unresolved. In particular, no counter example is found in literature. In this expository paper we survey classical and some more recent results concerning the polynomial reconstruction problem, discuss some related problems, variations and generalizations.     

Performance analysis of linear and nonlinear techniques for automatic scaling of discretized control problems

This paper introduces three (one linear and two nonlinear) automatic scaling techniques for NLPs with states and constraints spread over several orders of magnitude, without requiring complex off-the-shelf external tools. All of these methods have been compared to standard techniques and applied to three problems using SNOPT and IPOPT. The results confirm that the proposed techniques significantly improve the NLP conditioning, yielding more reliable and in some cases, faster NLP solutions.     

A fitting algorithm for real coefficient polynomial rooting

A new algorithm for computing all roots of polynomials with real coefficients is introduced. The principle behind the new algorithm is a fitting of the convolution of two subsequences onto a given polynomial coefficient sequence. This concept is used in the initial stage of the algorithm for a recursive slicing of a given polynomial into degree-2 subpolynomials from which initial root estimates are computed in closed form. This concept is further used in a post-fitting stage where the initial root estimates are refined to high numerical accuracy. A reduction of absolute root errors by a factor of 100 compared to the famous Companion matrix eigenvalue method based on the unsymmetric QR algorithm is not uncommon. Detailed computer experiments validate our claims.