Classification of Possible Finite-Time Singularities by Functional Renormalization

Starting from a representation of the early time evolution of a dynamical system in terms of the polynomial expression of some observable f (t) as a function of the time variable in some interval 0 < t < T, we investigate how to extrapolate/forecast in some optimal stability sense the future evolution of f(t) for time t>T. Using the functional renormalization of Yukalov and Gluzman, we offer a general classification of the possible regimes that can be defined based on the sole knowledge of the coefficients of a second-order polynomial representation of the dynamics. In particular, we investigate the conditions for the occurence of finite-time singularities from the structure of the time series, and quantify the critical time and the functional nature of the singularity when present. We also describe the regimes when a smooth extremum replaces the singularity and determine its position and amplitude. This extends previous works by (1) quantifying the stability of the functional renormalization method more accurately, (2) introducing new global constraints in terms of moments and (3) going beyond the ``mean-field'' approximation.Comment: Latex document of 18 pages + 7 ps figure

Self-Similar Factor Approximants

The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving a novel type of approximants. The derivation is based on the self-similar approximation theory, which presents the passage from one approximant to another as the motion realized by a dynamical system with the property of group self-similarity. The derived approximants, because of their form, are named the self-similar factor approximants. These complement the obtained earlier self-similar exponential approximants and self-similar root approximants. The specific feature of the self-similar factor approximants is that their control functions, providing convergence of the computational algorithm, are completely defined from the accuracy-through-order conditions. These approximants contain the Pade approximants as a particular case, and in some limit they can be reduced to the self-similar exponential approximants previously introduced by two of us. It is proved that the self-similar factor approximants are able to reproduce exactly a wide class of functions which include a variety of transcendental functions. For other functions, not pertaining to this exactly reproducible class, the factor approximants provide very accurate approximations, whose accuracy surpasses significantly that of the most accurate Pade approximants. This is illustrated by a number of examples showing the generality and accuracy of the factor approximants even when conventional techniques meet serious difficulties.Comment: 22 pages + 11 ps figure

Critical Indices as Limits of Control Functions

A variant of self-similar approximation theory is suggested, permitting an easy and accurate summation of divergent series consisting of only a few terms. The method is based on a power-law algebraic transformation, whose powers play the role of control functions governing the fastest convergence of the renormalized series. A striking relation between the theory of critical phenomena and optimal control theory is discovered: The critical indices are found to be directly related to limits of control functions at critical points. The method is applied to calculating the critical indices for several difficult problems. The results are in very good agreement with accurate numerical data.Comment: 1 file, 5 pages, RevTe

Self-Similar Bootstrap of Divergent Series

A method is developed for calculating effective sums of divergent series. This approach is a variant of the self-similar approximation theory. The novelty here is in using an algebraic transformation with a power providing the maximal stability of the self-similar renormalization procedure. The latter is to be repeated as many times as it is necessary in order to convert into closed self-similar expressions all sums from the series considered. This multiple and complete renormalization is called self-similar bootstrap. The method is illustrated by several examples from statistical physics.Comment: 1 file, 22 pages, RevTe

Self-Similar Interpolation in Quantum Mechanics

An approach is developed for constructing simple analytical formulae accurately approximating solutions to eigenvalue problems of quantum mechanics. This approach is based on self-similar approximation theory. In order to derive interpolation formulae valid in the whole range of parameters of considered physical quantities, the self-similar renormalization procedure is complimented here by boundary conditions which define control functions guaranteeing correct asymptotic behaviour in the vicinity of boundary points. To emphasize the generality of the approach, it is illustrated by different problems that are typical for quantum mechanics, such as anharmonic oscillators, double-well potentials, and quasiresonance models with quasistationary states. In addition, the nonlinear Schr\"odinger equation is considered, for which both eigenvalues and wave functions are constructed.Comment: 1 file, 30 pages, RevTex, no figure

A Comparative Study of the Magnetization Process of Two-Dimensional Antiferromagnets

Plateaux in the magnetization curves of the square, triangular and hexagonal lattice spin-1/2 XXZ antiferromagnet are investigated. One finds a zero magnetization plateau (corresponding to a spin-gap) on the square and hexagonal lattice with Ising-like anisotropies, and a plateau with one third of the saturation magnetization on the triangular lattice which survives a small amount of easy-plane anisotropy. Here we start with transfer matrix computations for the Ising limit and continue with series in the XXZ-anisotropy for plateau-boundaries using the groundstates of the Ising limit. The main focus is then a numerical computation of the magnetization curves with anisotropies in the vicinity of the isotropic situation. Finally, we discuss the universality class associated to the asymptotic behaviour of the magnetization curve close to saturation, as observed numerically in two and higher dimensions.Comment: 21 pages plain TeX (with macro package included), 7 PostScript figures included using psfig.st

Production of simian virus 40 large tumor antigen in bacteria: altered DNA-binding specificity and dna-replication activity of underphosphorylated large tumor antigen

A bacterial expression system was used to produce simian virus 40 large tumor antigen (T antigen) in the absence of the extensive posttranslational modifications that occur in mammalian cells. Wild-type T antigen produced in bacteria retained a specific subset of the biochemical activities displayed by its mammalian counterpart. Escherichia coli T antigen functioned as a helicase and bound to DNA fragments containing either site I or the wild-type origin of replication in a manner identical to mammalian T antigen. However, T antigen purified from E. coli did not efficiently bind to site II, an essential cis element within the simian virus 40 origin of replication. It therefore could not unwind origin-containing plasmids or efficiently replicate simian virus 40 DNA in vitro. The ability of protein phosphorylation to modulate the intrinsic preference of full-length T antigen for either site I or site II is discussed

Critical properties of 1-D spin 1/2 antiferromagnetic Heisenberg model

We discuss numerical results for the 1-D spin 1/2 antiferromagnetic Heisenberg model with next-to-nearest neighbour coupling and in the presence of an uniform magnetic field. The model develops zero frequency excitations at field dependent soft mode momenta. We compute critical quantities from finite size dependence of static structure factors.Comment: talk given by H. Kr{\"o}ger at Heraeus Seminar Theory of Spin Lattices and Lattice Gauge Models, Bad Honnef (1996), 20 pages, LaTeX + 18 figures, P

Analysis of the effect of clinical and laboratory parameters on survival in patients with metastatic renal cell cancer with intermediate prognosis according to IMDC (International mRCC Database Consortium)

Introduction. Most patients with metastatic renal cell carcinoma (mRCC) who received systemic therapy are classified into as intermediate risk group according to the International mRCC Database Consortium (IMDC) assessment. However, survival differs in patients with one and two unfavourable prognostic risk factors.Objective. To analyze the impact of possible clinical and laboratory parameters on survival in mRCC patients with intermediate prognosis according to IMDC in the presence of one or two unfavourable prognostic risk factors.Materials &amp; methods. A retrospective analysis of data from 316 mRCC patients with intermediate prognosis receiving systemic therapy was carried out. The presence and effect on survival of the following laboratory parameters were compared: hemoglobin, neutrophil count, LDH, platelet count, alkaline phosphatase, serum calcium level, ESR, and emphasis was also placed on the time of metastases appearance. Overall survival (OS), post-progression survival (PPS), and 3- and 5-year survival were evaluated.Results. The overall 3- and 5-year survival rate for subgroups with one and two unfavourable prognostic risk factors were 85.2%  ±  1.8% and 58.1  ±  1.6%; 79.1%  ±  1.7% and 35.6 1.6%, respectively (p &lt; 0.001). Three- and 5-year PPS for both subgroups amounted to 66.1%  ±  1.6% and 21.8%  ±  1.4%; 45.2%  ±  1.5% and 12.2%  ±  1.3%, respectively (p &lt; 0.001). Median for OS was 61 and 51 months and PPS was 50 and 32 months respectively (p &lt;0.001). No statistically significant difference was found in the frequency of gradations of the investigated laboratory indicators with a predictive effect on prognosis, depending on the number of a unfavourable prognostic risk factors. There were also no significant differences in survival rates when laboratory parameters deviated from the normal, except for hemoglobin: OS in patients with one prognostic factor is significantly higher with normal hemoglobin (p &lt; 0.05). In the future, considering the absence of influence of the selected factors on survival rates with their obvious significant differences in patients with one and two prognostic factors, the influence of the time of appearance of metastases (synchronous / metachronous) was analysed: all survival parameters were significantly worse in patients with synchronous metastases. Furthermore, OS in patients with metachronous metastases with the same relapse-free period was significantly better in patients with one prognostic factor according to IMDC.Conclusions. The presence of one or two prognostic factors significantly affects the indicators of 3 and 5-year overall survival and PPS in patients with an intermediate prognosis of mRCC, while laboratory parameters do not affect survival, except for hemoglobin levels, the time of metastases appearance has a significant effect, and the time of metastases appearance has a significant effect

Interactions in vivo between the Vif protein of HIV-1 and the precursor (Pr55GAG) of the virion nucleocapsid proteins

The abnormality of viral core structure seen in vif-defective HIV-1 grown in PBMCs has suggested a role for Vif in viral morphogenesis. Using an in vivo mammalian two-hybrid assay, the interaction between Vif and the precursor (Pr55GAG) of the virion nucleocapsid proteins has been analysed. This revealed the amino-terminal (aa 1–22) and central (aa 70–100) regions of Vif to be essential for its interaction with Pr55GAG, but deletion of the carboxy-terminal (aa 158–192) region of the protein had only a minor effect on its interaction. Initial deletion studies carried out on Pr55GAG showed that a 35-amino-acid region of the protein bridging the MA(p17)–CA(p24) junction was essential for its ability to interact with Vif. Site-directed mutagenesis of a conserved tryptophan (Trp21) near the amino terminus of Vif showed it to be important for the interaction with Pr55GAG. By contrast, mutagenesis of the highly conserved YLAL residues forming part of the BC-box motif, shown to be important in Vif promoting degradation of APOBEC3G/3F, had little or no effect on the Vif–Pr55GAG interaction