A Remark on the Weak Topology and the Degree

ＡＲｅｍａｒｋｏｎｔｈｅＷｅａｋＴｏｐｏｌｏｇｙａｎｄｔｈｅＤｅｇｒｅｅＬｏｔｆｉＬａｓｓｏｕｅｄ（Ｌａｂｏｒａｔｏｉｒｅｄ＇ＡｎａｌｙｓｅＮｕｍéｒｉｑｕｅＵｎｉｖｅｒｓｉｔéＰｉｅｒｒｅｅｔＭａｒｉｅＣｕｒｉｅ４，ｐｌａｃｅＪｕｓｉｅｕ，７５２５...     

A Necessary Condition of the Cardinal

In this paper the auther begins with some known results about η_λ(=the least cardinal K such that K→(λ)~<∞), proving this theorem: If λ is not Ramsey cardinal and η_λ exists, then for every a<η_λ there is a weakly compact cardinal γ, such that λ<γ_α<η_λandγ_α<γ_βwhenever a<β<η_λ, therefore η_λ is the limit of the sequence(γ_α:a<η_λ), i.e. η_λ=limγ_α. The theorem is mainly based on the theory of models with indiscernibles.     

A Correction on the Paper of ＜A Extension of the Mitrinovit－Djokovie Inequality＞

ＡＣｏｒｒｅｃｔｉｏｎｏｎｔｈｅＰａｐｅｒｏｆ＜ＡＥｘｔｅｎｓｉｏｎｏｆｔｈｅＭｉｔｒｉｎｏｖｉｔ－ＤｊｏｋｏｖｉｅＩｎｅｑｕａｌｉｔｙ＞冯慈璜ＡＣｏｒｒｅｃｔｉｏｎｏｎｔｈｅＰａｐｅｒｏｆ＜ＡＥｘｔｅｎｓｉｏｎｏｆｔｈｅＭｉｔｒｉｎｏｖｉｔ－Ｄｊｏ...     

Some remarks concerning matrices of the form A–A′, A−1A′

Matrices and operators of the formA ^–1 A ^* have received a certain amount of attention in recent years. Here some of the literature is surveyed and the caseA ^–1 A is studied for complex matrices withA denoting the transpose ofA. A generalization ofA ^–1 A is introduced.

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Zusammenfassung Matrizen und Operatoren, die in der FormA ^–1 A ^* ausgedrückt werden können, sind in den letzten Jahren häufig studiert worden. Hier wird ein Ueberblick über Teil der relevanten Literatur gegeben und auch der FallA ^–1 A für komplexe Matrizen studiert, wobeiA die Transponierte vonA ist. Es wird auch eine Verallgemeinerung vonA ^–1 A eingeführt.

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Dedicated to Eduard Stiefel by a long time admirer     

A realization theorem in the context of the Schur-Szegő composition

Every real polynomial of degree n in one variable with root ?1 can be represented as the Schur-Szeg? composition of n ? 1 polynomials of the form (x + 1) ^n?1(x + a _i ), where the numbers a _i are uniquely determined up to permutation. Some a _i are real, and the others form complex conjugate pairs. In this note, we show that for each pair (ρ, r), where 0 ? ρ, r ? [n/2], there exists a polynomial with exactly ρ pairs of complex conjugate roots and exactly r complex conjugate pairs in the corresponding set of numbers a _i .     

A Remark on the Fixed point Theorem of the Expansion Mappings

Let X be a Complete metric space with distance function d, anp T a mapping from X into X. T is said to be a first type of the expansion mapping, if it satisfies the following condition     

A Class of the Quasilinear Hyperbolic Equations with the Inhomogenous Terms

In this paper,we discuss a class of the quasillinear hyperbolic equations with the inhomogeneous terms:ur σ(v)x 2α(t)u=0,vr-ux=0 Under the certain of hypothesis,we prove the globally existence theorems of the smooth solutions for is cauchy problem.     

A Dynamical model of the atmospheric turbulencefromtheunstablestratificationtothestablestratification

ＡＤｙｎａｍｉｃａｌｍｏｄｅｌｏｆｔｈｅａｔｍｏｓｐｈｅｒｉｃｔｕｒｂｕｌｅｎｃｅｆｒｏｍｔｈｅｕｎｓｔａｂｌｅｓｔｒａｔｉｆｉｃａｔｉｏｎｔｏｔｈｅｓｔａｂｌｅｓｔｒａｔｉｆｉｃａｔｉｏｎＳｈｉｄａＬＩＵ，ＳｈｉｋｕｏＬＩＵ，ＢｅｎｋｕｉＴＡＮ（Ｄ...     

A Note on Self-Mapping of the Interval

In this note we showed by construct a example that for, the con ditions and are not equivalent. Let, define by, for any We can construct a example as following: Let be identity, define by     

A Note on the Viscous Cahn-Hilliard Equation

In this note, we study the global existence of classical solutions for the viscous Cahn-Hilliard equation with spatial dimension n ≤ 5. Based on the Schauder type estimates and energy estimates, we establish the global existence of classical solutions.     

A Carleson Problem for the Boussinesq Operator

In this paper,we show that the Boussinesq operator B_tf converges pointwise to its initial data.f ∈ H^s(R) as t→0 provided s≥1/4-more precisely-on one hand,by constructing a counterexample in R we discover that the optimal convergence index s_c,1=1/4;on the other hand,we find that the Hausdorff dimension of the divergence set for B_tf is■Moreover,a higher dimensional lift was also obtained for f being radial.     

A Problem on the Spherical Maximal Functions

The classical Lebesgue’s theorem shows that for any locally integrable functionf defined on R~n where B(x;ε)are n-dimensional balls with center x and radius ε. A extendedproblem of the above is,does     

A New Proof for the Interpolating Theorem

ＡＮｅｗＰｒｏｏｆｆｏｒｔｈｅＩｎｔｅｒｐｏｌａｔｉｎｇＴｈｅｏｒｅｍＳｕｎＳｈｕｎｈｕａ（孙顺华）ａｎｄＹｕＤａｈａｉ（余大海）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｉｃｈｕａｎＵｎｉｖｅｒｓｉｔｙ，Ｃｈｅｎｇｄｕ，６１００６４）Ａｂ...     

A method for the numerical solution of the Painlevé equations

A numerical method for solving the Cauchy problem for all the six Painlevé equations is proposed. The difficulty of solving these equations is that the unknown functions can have movable (that is, dependent on the initial data) singular points of the pole type. Moreover, the Painlevé III–VI equations may have singularities at points where the solution takes certain finite values. The positions of all these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Such auxiliary equations are derived for all Painlevé equations and for all types of singularities. Efficient criteria for transition to auxiliary systems are formulated, and numerical results illustrating the potentials of the method are presented.     

A Refinement of the Arithmetic-geometric Means Inequality

We suppose throughout that(1)m,n∈N,a_i,a_ij,q_j,p,x are all positive numbers;∑_j=1~n q_j=1,l≥1,λ>0,(i=1,2,…,m;j=1,2,…,n).     

A note on the α-quantile option

Some properties of a class of path-dependent options based on the α-quantiles of Brownian motion are discussed. In particular, it is shown that such options are well behaved in relation to standard options and comparatively cheaper than an equivalent class of lookback options.