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A seminal result of Nisan and Szegedy (STOC, 1992) shows that for any total Boolean function, the degree of the real polynomial that computes the function, and the minimal degree of a real polynomial that point-wise approximates the…
This paper addresses a multi-scale finite element method for second order linear elliptic equations with arbitrarily rough coefficient. We propose a local oversampling method to construct basis functions that have optimal local…
We consider a one-dimensional directed polymer in a random potential which is characterized by the Gaussian statistics with the finite size local correlations. It is shown that the well-known Kardar's solution obtained originally for a…
We propose a scaling description of phase separation of polymer solutions. The scaling incorporates three universal limiting regimes: the Ising limit asymptotically close to the critical point of phase separation, the "ideal-gas" limit for…
We develop a unified scaling framework for the end-position distributions of tethered polymers confined in finite cylindrical geometries. Two observables are analysed: the longitudinal distribution P(x), along the confinement axis, and the…
A self-consistent renormalization group flow equation for the scalar lambda phi^4 theory is analyzed and compared with the local potential approximation. The two prescriptions coincide in the sharp cutoff limit but differ with a smooth…
Direct verification of the existence of an infinite set of multicritical non-perturbative FPs (Fixed Points) for a single scalar field in two dimensions, is in practice well outside the capabilities of the present standard approximate…
We consider polymers in which M randomly selected pairs of monomers are restricted to be in contact. Analytical arguments and numerical simulations show that an ideal (Gaussian) chain of N monomers remains expanded as long as M<<N; its mean…
Explicit solutions are obtained for a class of semilinear radial Schrodinger equations with power nonlinearities in multi-dimensions. These solutions include new similarity solutions and other new group-invariant solutions, as well as new…
It is a classical result in rational approximation theory that certain non-smooth or singular functions, such as $|x|$ and $x^{1/p}$, can be efficiently approximated using rational functions with root-exponential convergence in terms of…
The purpose of the paper is to provide a characterization of the error of the best polynomial approximation of composite functions in weighted spaces. Such a characterization is essential for the convergence analysis of numerical methods…
Renormalization group flow equations for scalar lambda Phi^4 are generated using three classes of smooth smearing functions. Numerical results for the critical exponent nu in three dimensions are calculated by means of a truncated series…
The conformational properties of flexible polymers in d dimensions in environments with extended defects are analyzed both analytically and numerically. We consider the case, when structural defects are correlated in \varepsilon_d…
We investigate critical $N$-component scalar field theories and the spontaneous breaking of scale invariance in three dimensions using functional renormalisation. Global and local renormalisation group flows are solved analytically in the…
We present fully polynomial approximation schemes for a broad class of Holant problems with complex edge weights, which we call Holant polynomials. We transform these problems into partition functions of abstract combinatorial structures…
We introduce a fast implementation of the pivot algorithm for self-avoiding walks, which we use to obtain large samples of walks on the cubic lattice of up to $33 \times 10^6$ steps. Consequently the critical exponent $\nu$ for…
We discuss possible sources of systematic errors in the computation of critical exponents by renormalization-group methods, extrapolations from exact enumerations and Monte Carlo simulations. A careful Monte Carlo determination of the…
We study the problem of adsorption of self-interacting linear polymers situated in fractal containers that belong to the three-dimensional (3d) Sierpinski gasket (SG) family of fractals. Each member of the 3d SG fractal family has a fractal…
A recently proposed renormalization group technique, based on the hierarchical structures present in theories with fluctuating geometry, is implemented in the model of branched polymers. The renormalization group equations can be solved…
In order to study long chain polymers many lattice models accommodate a pulling force applied to a particular part of the chain, often a free endpoint. This is in addition to well-studied features such as energetic interaction between the…