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…     

Walsh spectrum properties of rotation symmetric boolean function

Rotation symmetric function was presented by Pieprzyk. The algebraic configuration of rotation symmetric（RotS） function is special. For a Rots n variables function f（x1, x2, …, xn） we have f（ρn^k （x1, x2, …xn））=f（x1, x2, …, xn） for k=0, 1, …, n-1. In this paper, useing probability method we find that when the parameters of RotS function is under circular translation of indices, its walsh spectrum is invariant. And we prove the result is both sufficient and necessary.     

Finite Time Ruin Probability with Variable Interest Rate and Extended Regular Variation

Consider an insurance risk model, in which the surplus process satisfies a recursive equation Un=Un-1(1 rn)-Xn for n≥1, where U0=x≥0 is the initial surplus, {rn；n≥1} the interest rate sequence, {Xn；n≥l} the sequence of i. i. d. real-valued random variables with common distribution function F, which denotes the gross loss during the nth year. We investigate the ruin probability within a finite time horizon and give the asymptotic result as x→∞.     

The Total Irredundance Numbers on Graphs

This paper discusses the total irredundance relations between the graph G and its clone-contraction graph H, that is, let H be the clone-contraction graph of G and v1,v2,...,vk be all contraction vertices ofH. IfS is a maximal total irredundant set of H such that A = S ∩ {V1,V2,…,Vk} contains as few vertices as possible, then S＇= S-A is the maximal total irredundant set of G. Furthermore, we obtain the bound of the total irredundance A（G） number： irt ≤△（G）/2△（G）＋1 n, which n is the order of graph G, and △（G） is maximum degree in G.     

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.     

关于Smarandache LCM函数的方程S(SL(n~(14,36)))=φ_2(n)的可解性

令S(n)为Smarandache函数,SL(n)为SmarandacheLCM函数,φ_2(n)为广义欧拉函数。讨论方程S(SL(n~(14)))=φ_2(n)和S(SL(n~(36)))=φ_2(n)可解性,利用初等方法并结合函数φ_2(n)与函数S(n)的性质,给出了这两个方程的所有正整数解。     

A Novel Degradation Pathway of Zn（dmid） （ phen）2 in Pyridine

The light-oxidation degradation processes of Zn(dmid)(phen)2(dmid = 4, 5-dimercapto-1,3-dithiole-2 one, phen = 1,10-phenanthroaline) in pyridine solvent has been monitored, h has been found under the light, dmid^2- of Zn(dmid)(phen)2 in pyridine solution could generate NCS^- and NCS^-replaces dmid^2- to form Zn(NCS)2 (phen)2 simultaneously. The crystal structure of Zn(NCS)2 (phen)2 has been determined. In the crystal of Zn(NCS)2 (phen)2. two NCS^- ligands are arranged in syn-configuration, and there is strong π-π interaction between the two adjacent parallel phen.     

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.     

Strong stability of linear forms in φ -mixing random variables

In this paper some new results of strong stability of linear forms in φ-mixing random variables are given. It is mainly proved that for a sequence of φ-mixing random variables {x _n , n ⩾ 1} and two sequences of positive numbers {a _n , n ⩾ 1} and {b _n , n ⩾ 1} there exist d _n ∈ R, n = 1,2,L, such that a.s. under some suitable conditions. The results extend and improve the corresponding theorems for independent identically distributed random variables. Foundation item: Supported by the National Natural Science Foundation of China(10671149)     

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     

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     

Generalized Galerkin approximations for pseudoinverses and operator equations of the first kind

0 IntroductionLeqtuiliHnebaer afo rHimlbaenrtd sap baoceun.dLeedtlianaenardfufnbceti oan baolu onndeHd ,serse--spectively. Our goal is to seekv∈Hsuchthata(u,v) =f(u) , u∈ H (1)As a rule, we call the expression (1) a variational equationandva solutiontothis equation.By virtue of the Riesz repre-sentationtheorem,there exists a uniquely determined boundedlinear operatorA:H→Hsuchthata(u,v) =〈u,Av〉, u,v∈ H,where〈·,·〉is the inner product onH.If the sesquilinearformais also with posit…     

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…     

Inorganic-organic hybrids constructed of bis（undecatungstogerma- nate）lanthanates polyoxoanions and oxalate-bridged dinuclear copper complexes and their magnetic properties

A series of inorganic-organic hybrids K₂Na _m H_9−m [{Ln(GeW₁₁O₃₉)₂}{Cu₂(bpy)₂(μ-ox)}]·nH₂O (bpy = 2,2-bipyridine and ox = oxalate; Ln = La, Nd, Sm, Eu, Gd; n = 19, 17, 22, 20, 19; m = 4, 4, 4, 9, 2) were isolated after reacting in a potassium acetate buffer. X-ray structural analyses show that compounds 1–5 are isomorphic and consist of [Ln(GeW₁₁O₃₉)₂]¹³⁻ polyoxoanion building blocks and oxalate-bridged dinuclear copper metalorganic complex with a 1D chain structure. The 1D chain further connects into the 3D framework by π-π interactions with neighboring bpy groups. The magnetic susceptibility data indicate that antiferromagnetic coupling between the neighboring Cu²⁺ ions in the structure and the rare earth ions affects magnetic property of the structure.     

Biotic Resources Abundance and the Corresponding Causes in Panxi Area

0IntroductionThere are luxuriant biotic resources in PanxiArea,where mineral resources,hydraulic re-source and biotic resources are preponderant.Thedevelopment of Panxi Area began withthe exploita-tion of mineral resources,which has established asolidfoundationfor the development of Panxi Area.Withthe consumption of the mineral resources,be-ing a kind of nonrenewable resource,the mineral in-dustry contributes less and less to the developmentof Panxi Area and does more and more harmto en-vir…     

Efficient scalar multiplication for elliptic curves over binary fields

Scalar multiplication [n]P is the kernel and the most time-consuming operation in elliptic curve cryptosystems. In order to improve scalar multiplication, in this paper, we propose a tripling algorithm using Lopez and Dahab projective coordinates, in which there are 3 field multiplications and 3 field squarings less than that in the Jacobian projective tripling algorithm. Furthermore, we map P to（φε^-1（P）, and compute [n]（φε^-1（P） on elliptic curve Eε, which is faster than computing [n]P on E, where φε is an isomorphism. Finally we calculate （φε（[n]φε^-1（P）） = [n]P. Combined with our efficient point tripling formula, this method leads scalar multiplication using double bases to achieve about 23% improvement, compared with Jacobian projective coordinates.     

二维数组的快速排序算法

在一维数组快速排序算法的基础上，给出了二维数组的快速排序算法，理论分析和大量的数值实验结果表明，其算法的平均计算时间仍然是Ｏ（ｎｌｏｇ２ｎ），一般所需的栈空间仍为Ｏ（ｌｏｇ２ｎ），是有效而可靠的快速排序算法。     

A note on characteristic generators of a free cyclic code and its dual

Let C be a free cyclic code over Z _n ^α and dim _p C = k. In the paper, we prove that if the k characteristic generators of C are p-linearly independent then the corresponding nα − k characteristic generators of C ^⊥ are p-linearly independent. We then show that to any trellis that can be constructed from k p-linearly independent characteristic generators of C, there exists a trellis for C ^⊥ with the same state-complexity profile, which generalizes the conjecture of Koetter and Vardy to a free cyclic code over Z _n ^α. Foundation item: Supported by the National Natural Science Foundation of China(60673071)     

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     

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.