One Dimensional Behavior of Singular N Dimensional Solutions of Semilinear Heat Equations

We consider u(x,t) a solution of u _t =Δu+|u| ^{p − 1} u that blows up at time T, where u:ℝ ^N ×[0, T)→ℝ, p>1, (N−2)p<N+2 and either u(0)≥ 0 or (3N−4)p<3N+8. We are concerned with the behavior of the solution near a non isolated blow-up point, as T−t→ 0. Under a non-degeneracy condition and assuming that the blow-up set is locally continuous and N−1 dimensional, we escape logarithmic scales of the variable T−t and give a sharper expansion of the solution with the much smaller error term (T−t)^{1, 1/2−η} for any η>0. In particular, if in addition p>3, then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow-up set. Finally, we prove that the mere hypothesis that the blow-up set is continuous implies that it is C ^{1, 1/2−η} for any η>0. Received: 20 June 2001 / Accepted: 6 October 2001     

Exact solutions for electromagnetic waves in inhomogeneous media

Exact solutions of the wave equation for the propagation of electromagnetic waves in some inhomogeneous media are found. The first solution corresponds to the barometric model of the atmosphere, whose index of refraction can be expressed by the formulaN ²(z)=1+(N ₀ ² -1) exp (–z/z ₀). The other cases correspond toN ²(z)=1+(N ₀ ² -1) ch^–2(z) andN ²(z)=a-b. [1+exp(z/L)]^–1.Dedicated to Academician Vladimír Hajko on the occasion of his 65th birthday.     

Transition from pulled to pushed premixed turbulent flames due to countergradient transport

The influence of countergradient transport on the speed of a statistically stationary, planar, 1D premixed flame that propagates in frozen turbulence is studied theoretically and numerically by considering the normalised magnitude N_B of the countergradient flux to be an input parameter. Spectra of admissible flame speeds are analytically determined and explicit travelling wave solutions are found for two algebraic relations widely used to close the mean rate of product creation. A problem of selecting the physically relevant solution that is approached for sufficiently steep initial conditions is addressed. It is argued that, if N_B is larger than an analytically determined critical number N^cr_B, then the type of the physically relevant solution is drastically changed. If N_B < N^cr_B, the physically relevant solution is of pulled wave type, i.e. its speed is controlled by processes localised to the leading edge of the flame brush and can be determined within the framework of a linear analysis at the leading edge. If N_B > N^cr_B, the physically relevant solution is of pushed wave type, i.e. its speed is controlled by processes in the entire flame brush. Analytical expressions for the speed of the physically relevant solution as a function of N_B and the density ratio are obtained. For N_B > N^cr_B, the mean flame brush thickness and the spatial profile of the Favre-averaged combustion progress variable are also determined analytically. These results are validated by numerical simulations. Both analytical expressions and numerical data indicate that (i) both turbulent flame speed and thickness are decreased when N_B is increased and (ii) the direction of total scalar flux (i.e. the sum of countergradient and gradient contributions) is strongly affected not only by N_B, but also by the shape of the dependence of the mean rate of product creation on the mean combustion progress variable.     

Exact Analytical Solution of the Schrödinger Equation for an N-Identical Body-Force System

A mathematical method is presented for solving the Schr?dinger equation for a system of identical body forces. The N-body forces are more easily introduced and treated within the hyperspherical harmonics. The problem of the N-body potential has been used at the level of both classical and quantum mechanics. The hypercentral interacting potential is assumed to depend on the hyperradius x = (ξ₁² + ξ₂² + ⋯ + ξ_N−1²)^1/2 only, where ξ₁,ξ₂,…,ξ_N−1 are Jacobi relative coordinates which are functions of N-particle relative positions r₁₂,r₂₃,…,r_N1. The problem of the harmonic oscillator and the Coulomb-type potential has been widely studied in different contexts. Using the N-body potential V(x) = ax² + bx − (c/x) as an example, and assuming an ansatz for the eigenfunction, an exact analytical solution of the Schr?dinger equation for an N-body system in three dimensions is obtained. This method is also applicable to some other types of potentials for N-identical interacting particles.     

Landauer resistance in the case of multichannel scattering

The Landauer resistance ρ _N ^L has been generalized to the case of multichannel scattering of a particle by the system of N nonoverlapping random potentials that are localized at the points x _i (i = 1, 2,..., N) and depend on x − x _i and y. It has been shown that, in this case, a new resistance ρ _N ^L appears, which is an exponential function of N. The recurrence equation for determining the Landauer resistance ρ _N ^L has been derived and its solution in the general case has been obtained.     

Solvent effect on the heat of solution and partial molar volume of magnesium perchlorate

For magnesium perchlorate (MP), the solvent effect on the heat of solution and partial molar volumes (PMV) at 25 °C was studied. Since the complete dissociation of magnesium perchlorate is more difficult to achieve as compared with lithium perchlorate (LP), the concentration dependence of the values of the heat of solution and partial molar volume were noted. Only in highly polar solvents with large donor numbers (DN), such as water, dimethyl sulfoxide, N,N‐dimethyl formamide, and formamide, the differential and integral values of the enthalpies of solution were the same in the range of concentrations studied. In all solvents studied, the values of the heat of solution of MP were highly exothermic and exceed that of LP by more than 30 kcal mol^?1. The values of the partial molar volume of MP were changed from 82.3 cm³ mol^?1 in formamide to ?2.4 cm³ mol^?1 in acetone, and correlate linearly with that of LP (R = 0.975). Taking into account the significant change in the properties of molecules in the solvate shell of cation Mg²⁺, the large increase in the reactivity of reactants, activated by such interaction with Mg²⁺ cation is expected. Copyright © 2010 John Wiley & Sons, Ltd.     

Some comments on the solvable chiral potts model

We present a simple proof of the conjecture produced by Baxter, Perk and Au-Yang on the structure of the normalization factorR(p, q, r) corresponding to their new solution of the star-triangle equation related with the generalized Fermat curve. Some important properties of the underlying curvex ^N y ^N+x ^N+y ^N+1/k ²=0 for theN=3 state case are also established. Particularly, we calculate exactly its matrix of theb-periods for some normalized basis of holomorphic differentials. We also show that associated four-dimensional theta function may be decomposed into a sum containing 12 terms, each term being the product of four one-dimensional theta functions. We also derive Picard-Fuchs equations for the periods of holomorphic differentials of the same curve. The remarkable appearance of the hypergeometric functions in our answers seems to be closely related with an expression for the groundstate energy per site, obtained for the superintegrable case by Albertini, Perk, and McCoy and independently by Baxter, although for a moment the connection is not clear.     

N=2 affine superalgebras and Hamiltonian reduction inN=2 superspace

We constructN=2 affine current algebras for the superalgebrassl(n/n-1)⁽¹⁾ in terms ofN=2 supercurrents subjected to nonlinear constraints and discuss the general procedure of the hamiltonian reduction inN=2 superspace at the classical level. We consider in detail the simplest case ofN=2sl(2/1)⁽¹⁾ and show howN=2 superconformal algebra inN=2 superspace follows via the hamiltonian reduction. Applying the hamiltonian reduction to the case ofN=2sl(3/2)⁽¹⁾, we find two new extendedN=2 superconformal algebras in a manifestly supersymmetricN=2 superfield form. Decoupling of four component currents of dimension 1/2 in them yields, respectively,u(2/1) andu(3) Knizhnik-Bershadsky superconformal algebras. We also discuss how theN=2 superfield formulations ofN=2W ₃ andN=2W ₃ ⁽²⁾ superconformal algebras come out in this framework, as well as some unusual extendedN=2 superconformal algebras containing constrainedN=2 stress tensor and/or spin 0 supercurrents.     

Deformed Bosons: Combinatorics of Normal Ordering

We solve the normal ordering problem for (A A) ⁿ where A and A are one mode deformed ([A,A ] = [N+1] – [N]) bosonic ladder operators. The solution generalizes results known for canonical bosons. It involves combinatorial polynomials in the number operator N for which the generating function and explicit expressions are found. Simple deformations provide examples of the method.     

Temperature Dependence of the CPN-1 Model and the Analogy with Quantum Chromodynamics

The two-dimensional CP^N-1 model — a simple field-theoretic analogue of four-dimensional quantum chromodynamics (QCD) — is analysed and reviewed. The major themes are the temperature dependence of the CP^N-1 model, and the analogy between CP^N-1 and QCD. A detailed treatment of the 1/N approximation of the CP^N-1 model is given. The main results emerging from this approximation are discussed at length. These are: asymptotic freedom, dimensional transmutation, confinement and topological charge nonquantization at zero temperature T = O, screening and topological charge quantization at finite temperature T. The analogy with QCD is explained in detail. A new, qualitative, analysis of the CP^N-1 model at finite temperature is introduced. This approach exploits the conformal invariance of the model to “heat” an arbitrary CP^N-1 field from T = O to finite temperature. This is achieved by conformal-transforming the flat Euclidean space-time of the T = O theory to the cylindrical space-time of the finite temperature theory.     

Simulated quantum computation of global minima

Finding the optimal solution to a complex optimisation problem is of great importance in practically all fields of science, technology, technical design and econometrics. We demonstrate that a modified Grover's quantum algorithm can be applied to real problems of finding a global minimum using modest numbers of quantum bits. Calculations of the global minimum of simple test functions and Lennard-Jones clusters have been carried out on a quantum computer simulator using a modified Grover's algorithm. The number of function evaluations N reduced from O(N) in classical simulation to O(N ^1/2) in quantum simulation. We also show how the Grover's quantum algorithm can be combined with the classical Pivot method for global optimisation to treat larger systems.     

Selection of peptomeric inhibitors of bovine α-chymotrypsin and cathepsin G based on trypsin inhibitor SFTI-1 using a combinatorial chemistry approach

A peptomeric library consisting of 360 monocyclic analogues of trypsin inhibitor SFTI-1 isolated from sunflower seeds was designed and synthesized by a solid-phase approach in order to select chymotrypsin and cathepsin G inhibitors. All peptomers contained a proteinogenic-Phe-mimicking N-benzylglycine (Nphe) at positions 5 and 12. Into the synthesized library, different peptoid monomers were introduced in the 7–10 segment. Deconvolution of the library against both proteinases through an iterative method in solution revealed that the strongest chymotrypsin inhibitory activity was displayed by two analogues, [Nphe^5,12]SFTI-1 (1) and [Nphe^5,12, Naem⁸]SFTI-1 (2), where Naem stands for N-(2-morpholinoethyl)glycine. After deconvolution against a cathepsin G analogue, [Nphe^5,12, Npip^8,9, Nnle¹⁰] SFTI-1 (3) (Npip = N-(3,4-methylenedioxybenzyl)glycine) appeared to be the most potent inhibitor with a high serum stability. It is worth noting that the analogues obtained by a combinatorial approach display high specificity towards one of the experimental enzymes. Another interesting feature is the lack of Pro8 in analogues 2 and 3, the amino acid residue absolutely conserved in the family of Bownan–Birk inhibitors.     

The light σ-meson

In the framework of the dispersion relation N/D-approach, we restore the low-energy (IJ ^PC = 00^{+ +})-wave amplitude sewing it with the previously obtained K-matrix solution for the region 450-1900 MeV. The restored N/D-amplitude has a pole on the second sheet of the complex-s plane near the threshold, that is the light -meson. Received: 21 February 2000 / Accepted: 20 June 2000     

Free-field realization of theWKP(N) algebra

TheW _KP ^(N) algebra has been identified with the second Hamiltonian structure in theNth Hamiltonian pair of the KP hierarchy. In this Letter, by constructing the Miura map that decomposes the second Hamiltonian structure in theNth pair of the KP hierarchy, we show thatW _KP ^(N) can also be decomposed toN independent copies ofW _KP ⁽¹⁾ algebras, therefore its free-field realization can be worked out by constructing free fields for each copy ofW _KP ⁽¹⁾ . In this way, the free fields may consist ofN + 2n number of bosons, among them, 2n are in pairs, wheren is an arbitrary integer between 1 andN. We also express the currents ofW _KP ^(N) in terms of the currents ofN —n copies of U(1) andn copies of SL(2,R) _k algebras with levelk = 1. By reductions, we give similar results forW ^(N) andW ₃ ⁽²⁾ algebra.     

Study of N*(1440) from J/Ψ decays

For J/Ψ↦ pπ⁰ and pπ⁺π^-, the π⁰ p and pπ⁺π^- systems are limited to be pure isospin-(1/2) due to isospin conservation. This is a big advantage in studying N^* resonances from J/Ψ decays, compared with πN and γN experiments. The process J/Ψ↦ N ^* or p provides a new way to probe the internal structure of the N^* resonances. Here we report a quark model calculation for J/Ψ↦ p, N ^*(1440) and N ^*. The implication for the internal structure of N ^*(1440) is discussed. Received: 1 June 2001 / Accepted: 20 June 2001     

Rational Design of Novel Benzimidazole-Based Sensor Molecules that Display Positive and Negative Fluorescence Responses to Anions

Two novel and neutral benzimidazole derivatives-based anion receptors bearing a 1,10-phenanthroline fluorophore, N,N′-di-(2′-benzimidazolyl-methylene)-1, 10-phenanthroline-2,9-diamide (1) and N,N′-di-[2′-(benzimidazolyl-2′-) ethyl-]-1,10-phenanthroline-2,9-diamide (2), which exhibited turn-on and turn-off fluorescence responses to various anions, were rationally designed and synthesized and their fluorescent response toward anions was investigated in DMSO solution. In the process of anions binding, there were two different fluorescent responses in presence of anions: a quenching of the fluorescence emission for F^- and AcO^- and an enhancement of the fluorescence emission for Cl⁻, Br⁻ and I⁻. Two different luminescent mechanisms of the receptors 1 and 2 resulting from various anions were exploited to rationalize quenching and enhancement of the fluorescence emission: a photo-induced electronic transfer mechanism (PET) and the increase of the rigidity of the host molecules, respectively. In particular, chloride could be recognized selectively from the anions tested according to changes of fluorescence spectrum. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

A tunable green alkaline-earth silicon-oxynitride solid solution (Ca1? x Sr x )Si2O2N2:Eu2+ and its application in LED

A series of (Ca_1−x−y Sr _x )Si₂O₂N₂:yEu²⁺ (x=0.0–0.97, y=0.03) phosphors were synthesized by high-temperature solid-state reaction. The XRD patterns confirm the formation of a solid solution of (Ca_1−x−y Sr _x )Si₂O₂N₂:yEu²⁺. An intense tunable green light is observed with the increasing ratio of Sr/Ca. With an increase in x, the excitation and emission spectra show a redshift and blueshift, respectively, due to large centroid shift and small Stokes shift. The temperature dependent luminescence is also investigated in the temperature range of 77–450 K. The Huang–Rhys factor and the thermal-quenching temperature are determined. Intense green LEDs were successfully fabricated based on the (Ca_1−x−y Sr _x )Si₂O₂N₂:yEu²⁺ phosphor and near-ultraviolet (∼395 nm) GaN/blue (460 nm) InGaN chips. All the results indicate that the solid solution (Ca_1−x−y Sr _x )Si₂O₂N₂:yEu²⁺ is a promising phosphor applicable to near-UV and blue LEDs for solid-state lighting.     

第Ⅰ类两态叠加多模叠加态光场的非线性高阶压缩特性研究

本文在文献20的基础上进一步提出了多模辐射场的N-Y最小测不准态、N-H最小测不准态、N-Y压缩最小测不准态以及N-H压缩最小测不准态等的定义.构造了由多模(q模)相干态|{Z_j}>_q及其相反态|{-Z_j}>_q的线性叠加所组成的第Ⅰ类两态叠加多模叠加态光场|ψ₁⁽²⁾>_q,利用文献20新近提出的有关多模辐射场的两种非线性高阶压缩的定义,首次对态|ψ₁⁽²⁾>_q的N次方Y压缩及N次方H压缩效应进行了详细研究.结果表明,1)当N为偶数时,态|ψ₁⁽²⁾>_q恒处于N-Y最小测不准态;当N为奇数时,态|ψ₁⁽²⁾>_q在一定条件下可呈现出周期性变化的、任意阶的N次方Y压缩效应.2)当q·N为偶数时,态|ψ₁⁽²⁾>_q恒处于N-H最小测不准态;当q·N为奇数时,在另外的条件下,态|ψ₁⁽²⁾>_q则可呈现出周期性变化的、任意阶的N次方H压缩效应.3)N次方Y压缩及N次方H压缩效应的压缩深度与腔模总数q、压缩参数R_j以及压缩阶数N等非线性相关,后者与上述参量的非线性关联程度要比前者的更强.     

D2O‐catalyzed proton transfer selectivity of 4 different NH protons in the same molecule

When N‐benzyl‐N′‐methylacetamidinium hydrochloride (pKa=11.8) is dissolved in D₂O/DCl(1 M), an equilibrium of 2 54:46 stereoisomers in an ~2:1 =(R)N^δ+H(D) D/H ratio is formed. Therefore, 2 R =N‐benzyl (E and Z) and 2 R =N‐methyl (E and Z) groups attached to the corresponding H(D) (Z and E) for a total of 8 ¹H‐NMR signals are observed. Consequently, their rates of H and D transfer to D₂O can be measured by means of the ¹H‐NMR broadness (line shape) of the =(R )N^δ+H doublets and =(R )N^δ+D broad singlets. Acidity selectivity is observed for both processes. In fact, the relative proton and deuterium transfer rates follow the acidity order: =(PhCH₂)N^δ+‐H(E) > =(PhCH₂)N^δ+‐H(Z) > =(Me)N^δ+‐H(E) > =(Me)N^δ+‐H(Z). Proton transfer rates are in the range of 8 to 0.5 s^‐1 with α = .92. This tendency is independently supported by the observed experimental chemical shift deuterium isotopic perturbation. The rate‐limiting step for proton exchange is the breaking of the hydrogen bond due to the fast amidine reprotonation (~10¹¹ s). =(R)N^δ+D/=(R)N^δ+H equilibration is reached at ~80 s, and it can be measured by the relative =(R) N^δ+H versus =(R) N^δ+D signal integrations. The equilibrium of the 4 =(R)N^δ+H(D) centers is shifted toward deuterium, but they are further shifted in the more basic centers. Equilibrium is completely shifted toward D in the 4 centers when OD^? contributes with the exchange process at pD > 3.     

Thermodynamical universality of the Lovelock black holes

The necessary and sufficient condition for the thermodynamical universality of the static spherically symmetric Lovelock black hole is that it is the Nth order pure Lovelock Λ-vacuum solution. By universality we mean the thermodynamical parameters: temperature and entropy always bear the same relation to the horizon radius for d = 2N + 1, 2N + 2 dimensions for all N which is the degree of the Lovelock polynomial. For instance, the entropy always goes in terms of the horizon radius as r _h and r _h ², respectively for the odd and even dimensions. Not only that the universality uniquely identifies the pure Lovelock black hole with Λ, it is the characterizing property of this class of black holes.