Remarks on a Noble''''s Theorem

Noble proved the following Theorem: If A (?) X with a nonisolated point and B (?) Y, then A × B is bounded in X×Y if and only if the projection map π : X × Y → X is a z-map with respect to A × B and A, A is bounded in X and B is bounded in Y. In this note, we give two examples showing the necessary and sufficient conditions of Noble's theorem are not right.     

Decomposable Multipliers on a Banach Algebra

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Note on a Conjecture of Gopakumar-Vafa

We rephrase the Gopakumar-Vafa conjecture on genus zero Gromov-Witten invariants of Calabi-Yau threefolds in terms of the virtual degree of the moduli of pure dimension one stable sheaves and investigate the conjecture for K3 fibred local Calabi-Yau threefolds.     

Some Notes on a Minimization Problem

Let X[a,b] be a compact set containing at least n+1 points and Kan n-dimensional Haar subspace in c[a,b]. Let F(x,y) be a nonnegativefunction, defined on X×(-∞,∞), satisfying ‖F(·,p)‖<∞ with the L_∞norm forsome∈K, where F(x,p)≡F(x,p(x)). The minimization problem discussed in this paper is to find an elementp∈K such that ‖F(·,p)‖=inf ‖F(·,q)‖, such an element p(if any) is saidto be a minimum to F in K~(q∈K). The author in [1,2] studied this problem and has given the main theoremsin the Cbebyshev theory under the following assumptions: (A) lim F(x,y)=∞, x∈X; (B) lim F(x,u)=F(x,y), x∈X,y; (C)lim F(u,υ)=F(x,y),x∈X,y; (D) For each x∈X there existtwo real numbers f~-(x) and f~+(x),f~-(x)f~+(x). such that F(x,y) is strictlydecreasing with respect to y on (-∞,f~-(x)] and strictly increasing on [f~+(x),∞), and F(x,y)=F(x):=inf F(x,υ) on [f~-(x),f~+(x)]. Denote f_1(x)=inf{y:F(x,y)‖F~*‖},f_2(x)=sup{y:F(x,) ‖F‖},f_1(x)=lim f_1(u),f_2(x)=lim f_2(u), G=(q∈K: f_1qf_2}.For pεK set X_p={     

On a Problem on Chebyshev Centers

Let X be a normed linear space, A, G be subsets of X. Define r_A(G) = inf sup ||α - g||then r_A(G) is called the relative Chebyshev radius of A with respect to G. If     

Remarks on “Quasi-contraction on a cone metric space”

Recently, D. Ili? and V. Rako?evi? [D. Ili?, V. Rako?evi?, Quasi-contraction on a cone metric space, Appl. Math. Lett. (2008) doi:10.1016/j.aml.2008.08.011] proved a fixed point theorem for quasi-contractive mappings in cone metric spaces when the underlying cone is normal. The aim of this paper is to prove this and some related results without using the normality condition.     

Semiclassical spectral series of the Schrödinger operator with a delta potential on a straight line and on a sphere

We describe the spectral series of the Schrödinger operator H = ?(h²/2)Δ + V(x) + αδ(x?x₀), α ∈ ?, with a delta potential on the real line and on the three- and two-dimensional standard spheres in the semiclassical limit as h → 0. We consider a smooth potential V(x) such that lim_|x|→∞V(x)=+∞ in the first case and V(x) = 0 in the last two cases. In the semiclassical limit in each case, we describe the classical trajectories corresponding to the quantum problem with a delta potential.     

关于＂ Comments on a note on three-term recurrence for a tridiagonal matix ＂的进一步标记

在文章中,作者给出了块三对角矩阵行列式的一些关系式,本文在,=三基础之上给出了此类矩阵一些类似的关系式．     

Some notes on a conjecture of Zygmund

In this paper, some classes of differentiation basis are investigated and several positive answers to a conjecture of Zygmund on differentiation of integrals are presented.     

Study on a Cross Diffusion Parabolic System

This paper considers a kind of strongly coupled cross diffusion parabolic system,which can be usedas the multi-dimensional Lyumkis energy transport model in semiconductor science.The global existence andlarge time behavior are obtained for smooth solution to the initial boundary value problem.When the initialdata are a small perturbation of an isothermal stationary solution,the smooth solution of the problem under theinsulating boundary condition,converges to that stationary solution exponentially fast as time goes to infinity.     

Fixed Point Theorems on a Non-convex Set

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Variations on a Theme of Hardy’s

Let θ > 1 and let ϕ : [0,1] → ℂ be such that the two-sided series converges for all x ∊ [0,1], (then necessarily φ(0) = φ(1) = 0).Suppose Define For different classes of functions φ we show that À notre ami, Jean-Louis Nicolas2000 Mathematics Subject Classification: Primary—11B83, 11B99     

A Remark on a Theorem of Szegö

Let be a polynomial with complex coefficients and define, for , where ||P|| is the euclidean norm of the polynomial P. By a theorem of Szegö where is the Mahler measure of F. Recently, J. Dégot proved an effective version of this result. In this paper we sharpen Dégot's result, under the additional hypotheses that F is a square-free polynomial with integer coefficients and without reciprocal factors.     

Note on a Class of Inverse Series Relations

Let Γ≡(Γ, +,·) be the commutative ring of formal power series with real or complex coefficients, in which the ordinary addition and Cauchy multiplication are defined.Forφ,ψ∈Γ the composition φ(ψ(t) just means a formal substitution of u=ψ(t) into φ(u) in which the operations+and·can be performed indefinitely.We have the following     

A Note on a Paper of Dickmeis etc

Let X be a Banach space, X~* be the class of all bounded and sublinear functionals T on X, i .e. (1) |T(f+g)|≤|T f|+|Tg|for all f, gX; (2) |T(af)|=|a||Tf|for all numbers a and f X; (3) ‖T‖_x~·=sup |T f|<∞. In[1] , W.Dickmeis, R.J.Nessel and E. van Wickeren proved the following theorem.     

Notes on a problem of Turán

Let z1,z2, ... ,znbe complex numbers, and write S= z ^j _{1 + ... +} z ^j _n for their power sums. Let R _n= min_{z 1,z2,...,zn} max_1&le;j&le;n &verbar;Sj&verbar; where the minimum is taken under the condition that max_1&le;t&le;n &verbar;z_t&verbar; = 1 Improving a result of Komlós, Sárközy and Szemerédi (see [KSSz]) we prove here that Rn &lt;1 -(1 - ") log log n /log n We also discuss a related extremal problem which occurred naturally in our earlier proof ([B1]) of the fact that Rn &gt;&half;     

Comment on “A note on a study on an integrable system of coupled KdV equations”

We analyze the note by Muraved [A note on A study on an integrable system of coupled KdV equations [1], Commun Nonlinear Sci Numer Simulat. doi:10.1016/j.cnsns.2010.06.018]. Author tried to show that our approach is curious. We demonstrate that, author has not shown any error and presented a curious note.