A Note on the Prime Spectrums of Character Rings of Finite Groups

0Introduction LetGbeafinitegroupoforder|G|=g,andletR(G)denotethecharacterringofG,whichisgeneratedbyalltheirreduciblecomplexcharactersofG.LetZbetherationalintegerringandNthesetofnatural numbers,andletZ[ω]betheintegralextensiongeneratedbyaprimitiveg throotωofunity.SupposethatSisasubringof thealgebraicnumberfieldsuchthatZ[ω]S.πisasetofra tionalprimenumbersdefinedasfollowsπ={p｜pisarationalprimenumbersuchthatp-1S}.Definition1WecallthataconjugacyclassCofthefinite groupGisaπregularconju…     

Instability of relativistic electron beam with strong magnetic field

Stability criteria for a weak relativistic beam-plasma interaction in a strong magnetic field are found. Two beam modes occur,ω≈k _zC andω≈k _zC-ω_c. The dispersion equation of electrostatic two-stream is exactly solved with analytical method.     

Isotope effect of the stereodynamics in the reactions F+HCl→HF+Cl and F+DCl→DF+Cl

Using a new ground-state ab initio potential energy surface reported by Deskevich et al., the product polarizations in the reactions F+HCl→HF+Cl and F+DCl→DF+Cl were studied by employing the quasi-classical trajectory method. At a collision energy of 10 kcal/mol, the four generalized polarization-dependent differential cross-sections (2π/σ)(dσ₀₀/dω _t ), (2π/σ)(dσ₂₀/dω _t ), (2π/σ)(dσ₂₂₊/dω _t ) and (2π/σ)(dσ₂₁₋/dω _t ) were calculated in the center-of-mass frame. The distribution of the angle between k and j′, P(θ _r ), the distribution of the dihedral angle denoting the k-k′-j′ correlation, P(φ _r ), and the angular distribution of the product rotational vectors in the form of polar plots P(θ _r , φ _r ) were also calculated. The evident influence of isotope substitution on the product polarization is revealed. This effect may result from the different mass factors of the two reactions.     

The classification of statistically recursive sets

The main aim of this paper is to make a classification of random setsK _m (ω) constructed in theorem 2. 1 and theorem 2. 1′ in [1]. We provide five criterions for the classification. Many kinds of random sets such as Hawkes constructed in [2], Graf constructed in [3] and Mauldin constructed in [4] are the special cases ofK _m (ω) constructed in [1], and then these random sets belong to some model respectively according to our classification. Supported by the National Natural Science Fundation and the Doctal Programme Fundation of China Hu Dihe: born in May 1935, Professor     

On the boundary value problem for harmonic maps of the Poincaré disc

The boundary value problem for harmonic maps of the Poincare disc is discussed. The emphasis is on the non-smoothness of the given boundary values in the problem. Let T . be a subspace of the universal Teichmüller space, defined as a set of normalized quasisymmetric homeomorphisms h of the unit circle S onto itself where h admits a quasiconformal extension to the unit disc D with a complex dilatation μ satisfying where ρ(z)|dz|2 is the Poincare metric of D. Let B . be a Banach space consisting of holomorphic quadratic differentials φ in D with norms It is shown that for any given quasisymmetric homeomorphism h : S1→S1∈ T . , there is a unique quasiconformal harmonic map of D with respect to the Poincare metric whose boundary corresponding is h and the Hopf differential of such a harmonic map belongs to B .     

A local asymptotic behavior for ruin probability in the renewal risk model

Let R（t）=u＋ct-∑ I=1^N（t） Xi,t≥0 be the renewal risk model, with Fx（x）being the distribution function of the claim amount X. Let ψ（u） be the ruin probability with initial surplus u. Under the condition of Fx（x） ∈ S^＊（γ）,y ≥ 0, by the geometric sum method, we derive the local asymptotic behavior for ψ（u,u ＋ z] for every 0 （ z （ oo, On one hand, the asymptotic behavior of ψ（u） can be derived from the result obtained. On the other hand, the result of this paper can be applied to the insurance risk management of an insurance company.     

Preparation of (R)- and (S)-2-Allyl-1, 3, 2-dinaphtho (α, β) [d.f] dioxaborepin and their reactions with aldehydes

(R)- and(S)-2-Allyl-1,3,2-dinaphtho (α,β) [d.f] dioxaborepin ((R)-2 and(S)-2) have been first prepared by the reaction of(R)-(+)- or(S)-(−)-1, 1′-bi-2-naphthol and triallylborane in THF at room temperature, respectively.(S)-2 and(R)-2 are sensitive to moisture and oxygen in air and disproportionate easily to triallylborane and 1,1′-bi-2 naphthyl bis (1,1′-bi-2-naphtholborate) at ambient temperature. However, THF is a stabilizer for them. The reactions of(R)-2 or(S)-2 and some aliphetic or aromatic aldehydes in CH₂Cl₂ at −78°C for several hours afforded β-alkylenyl alcohols in up to 84.8%ee. Among them, optically active 1-(3, 5-dichlorophenyl)-3-butenol and 1-(2-methoxyphenyl)-3-butenol were first prepared Foundation item: Supported by the National Natural Science Foundation of China (29972033) Biography: Liu Dejun (1973), male, Ph. D, research direction: asymmetric synthesis     

An Improved Genetic Algorithm for Problem of Genome Rearrangement

In view of the fact that the problem of sorting unsigned permutation by reversal is NP-hard, while the problem of sorting signed permutation by reversal can be solved easily, in this paper, we first transform an unsigned permutation of length n,π （π1 ,… ,πn）, into a set S（π） containing 2^n signed permutations, so that the reversal distance of π is equal to the reversal distance of the optimal signed permutation in S（π）. Then analyze the structural features of S（π） by creating a directed graph and induce a new computing model of this question. Finally, an improved genetic algorithm for solving the new model is proposed. Experimental results show that the proposed model and algorithm is very efficient in practice.     

Characteristic singular integral equations with solutions having singularities of higher order on the real axis

Under the appropriate hypotheses subject to the unknown function and the free term, by means of our Lemma, we prove the rationality of order at x = ∞ on two sides for the characteristic singular integral equations with solutions having singularities of higher order on the real axis X. We transform the equations into solving equivalent Riemann boundary value problems with solutions having singularities of higher order and with additional conditions on X. The solutions and the solvable conditions for the former are obtained from the latter. Biography: ZHONG Shouguo (1941–), male, Professor, research direction: singular integral equations and their applications.     

Singular Integral Equations with Cosecant Kernel in Solutions with Singularities of High Order

0　IntroductionIn Ref. [1 3], Lu Jian ke first discussed singular integralequations with solutions having singularities of order oneK≡a(t0)(t0) b(t0)πi∫L(t)t- t0dt ∫Lk(t0,t)(t)dt= f(t0), t0 ∈L (1)wh     

Integrability near the boundary of the Poisson integral of some singular measures

We consider the Poisson integral u = P＊μ on the half-space R＋^N＋1 （ N 〉 1 ） （or on the unit ball of the complex plane） of some singular measureμ. If μ is an s-measure （0 〈 s 〈 N）, then some sharp estimates of the integration of the harmonic function u near the boundary are given. In particular, we show that fpr p〉1,∫R^Nu^p（x,y）dx- y^-τ （y〉0,τ =（N-s）（p-1） ） （Given for p〉1, RN f 〉 0 and g 〉 0, ＂ f-g ＂ will mean that there exist constants C1 and C2, such that C1f ≤ g ≤ CEf ）.     

Chaos in the sense of Li-Yorke and the order of the inverse limit space

Let I= [0,1] and ω0 be the first limit ordinal number. Assume that f: I→I is continuous , piece-wise monotone and the set of periods of f is { 2': i∈ {0} ∪ N}. It is known that the order of (I, f) is ω0 or ω0 + 1. It is shown that the order of the inverse limit space (I, f) is ω0 (resp. ω0 + 1) if and only if f is not (resp. is) chaotic in the sense of Li-Yorke.     

Preparation by a rheological phase reaction method and thermal decomposition reaction mechanism of nickelous salicylate tetrahydrate

0　IntroductionInrecentyears,studiesonBinghamfluid ,suchasElec trorheologicalfluid ,areactivelypursued[1 6] .Rheologicaltechniqueshavebeenplayinganimportantroleinsyntheticchem istry .Therheologicalphasereactionmethodisthe processofpreparingcompoundsormaterialsfromasolid liquidrheologicalmixture.Thatis,thesolidreactantsarefullymixedinapropermolarratio ,andmadeupbyaproperamountofwaterorothersolventstoaBinghambodyinwhichthesolidparticlesandliquidsubstanceareuniformlydistributed ,sothattheprodu…     

The fourier transformations of positive distributions on lorentz groupSO (3, 1)

With the aid of Plancherel-Godement Theorem, we prove that every positive distributionT onSO (3, 1) which is bi-invariant underSO(3) corresponds to a measure μ on ω=∝σC|s(2-s)>=0∝, and μ can be decomposed intoμ=μ ₁+μ ₂, whereμ ₁ is a bounded measure on 0<=s<=2 andμ ₂ is slowly increasing measure on (sχC|Re(s)=1)} Foundation item: Supported by the National Natural Science Foundation of China (19871065) and Hua Cheng Mathematics Science Foundation. Biography: Zhu Li (1976-), male, Graduate student, research interest: partial differential equation     

A Large Deviation Principle for the Risk Process with Varying Premium

Let u ∈ R ,for any ω 〉 0, the processes X^ε = {X^ε（t）; 0 ≤ t≤ 1} are governed by the following random evolution equations dX^ε（t）= b（X^ε（t）,v（t））dt-εdSt/ε, where S={St; 0≤t≤1} is a compound Poisson process, the process v={v（t）; 0≤t≤1} is independent of S and takes values in R^m. We derive the large deviation principle for{（X^ε,v（.））; ε〉0} when ε↓0 by approximation method and contraction principle, which will be meaningful for us to find out the path property for the risk process of this type.     

Operational properties and matrix representations of quantum measures

Denoted by M(A), QM(A) and SQM(A) the sets of all measures, quantum measures and subadditive quantum measures on a σ-algebra A, respectively. We observe that these sets are all positive cones in the real vector space F(A) of all real-valued functions on A and prove that M(A) is a face of SQM(A). It is proved that the product of m grade-1 measures is a grade-m measure. By combining a matrix M _μ to a quantum measure μ on the power set A _n of an n-element set X, it is proved that μ ≪ ν (resp. μ ⊥ ν) if and only if M _μ ≪ M _ν (resp. M _μ M _ν =0). Also, it is shown that two nontrivial measures μ and ν are mutually absolutely continuous if and only if μ·ν∈QM(A _n ). Moreover, the matrices corresponding to quantum measures are characterized. Finally, convergence of a sequence of quantum measures on A _n is introduced and discussed; especially, the Vitali-Hahn-Saks theorem for quantum measures is proved.     

Non-homogeneous Riemann boundary value problem with radicals

The solution of the non-homogeneous Riemann boundary value problem with radicals (1. 2) together with its condition of solvability is obtained for arbitrary positive integersp andq, which generalizes the results for the casep=q=2. Biography: Lu Jian-ke(1922-), male, Professor, research interests: theory and applications of complex functions.     

分数阶半正边值问题一个正解的存在性定理

考虑分数阶半正边值问题:
D^α₀₊u(t)=λf(t,u(t)),0         u(0)=u(1)=u′(0)=u′(1)=0
正解的存在性. 其中: 3<α≤4是一个实数; D^α₀₊是标准的Riemann-Liouville微分, 非线性项没有数值下界. 应用Krasnosel’skii不动点定理证明该方程一个正解的存在性.     

Quantum chemical studies on tuning the second-order nonlinear optical molecular switching of triarylborane derivatives

In this work, density functional theory (DFT) combined with the finite field (FF) method has been adopted to analyze the second-order nonlinear optical (NLO) properties of the triarylborane (TAB) derivatives obtained by introducing different inductive electron groups into the phenylene ring of the TAB (RTAB, where R=2-C6H5-C2B10H10(1),R=F(2), R=Me(3),R=NO2(4),R=NH2(5)). The static first hyperpolarizabilities (βtot) of the RTAB molecules can be switched by binding one F- to the boron center (RTAB′) or one-electron reduction (RTAB"). The DFT-FF calculations show that the βtot values of 2′, 3′ and 5′ decrease while those of 1′ and 4′ increase compared with the values of their neutral molecules, which was attributed to the fact that the charge transfers of 2, 3 and 5 become smaller and those of 1 and 4 become larger by binding one F- ion to the boron center, according to time-domain DFT (TD-DFT) analysis. However, the incorporation of one electron enhances the second-order NLO properties of the RTAB molecules remarkably, especially for system 1. It is notable that the βtot value of reduced form 1″ is 508.69×10-30 esu, i.e. about 578 times larger than that of system 1. Frontier molecular orbital (FMO) and natural bond orbital (NBO) analyses suggest that the reversal of the charge distribution between the neutral molecules and their reduced forms leads to low HOMO-LUMO energy gaps (E0) and thus large βtot values for the reduced forms.     

On the Pólya conjecture and the weak Weyl-Berry conjecture

The Pólya conjecture and its connection with the weak Weyl-Berry conjecture are studied Specifically let Ω⊆R ⁿ (n≥1) be a non-empty bounded open set with boundary ∂Ω. LetN ₀(λ, −Δ,Ω) be the Dirichlet counting function and φ(λ) the associated Weyl term. If the interior Minkowski dimension of ∂Ω is δ∈[n−1,n], then under certain realisable conditions we prove that for λ sufficiently large the Pólya conjecture φ(λ) −N ₀(λ,−Δ,Ω)≥0 is true. Under related conditions we also prove thatϕ(λ)−N ₀(λ,−Δ, Ω)≈λ^5/2, as λ→+∞. That is, the Weak Weyl-Berry conjecture is true. Similar results are obtained for the Neumann counting function. Partially supported by the National Natural Science Foundation of China and the Royal Society of London Chen Hua: born in March 8, 1956, Professor