Invariants of an augmentation of Γ0(n)

Let G _n ^× be the 2-group of primary factors of a positive integer n and fix a direct product decomposition of this group. We define an augmentation of Γ₀(n) based on G _n ^×, paralleling augmentations used by Fricke, Cohn and Knopp, and others. Using the decomposition of G _n ^×, we then define a family of functions based on η-functions and use these functions to construct invariants of the augmented group. Along with proving results analogous to those of Cohn and Knopp, we make a complete determination of the multiplier systems for these new functions.     

Estimation of the Separation of Two Matrices (Ⅱ)

In this paper we give a lower bound of the separation $sep_F(A,B)$ of two diagonalizable matrices A and B. The key to finding the lower bound of $sep_F(A,B)$ is to find an upper bound for the condition number of a transformation matrix Q which transforms a diagonalizable matrix A to a diagonal form. The obtained lower bound of $sep_F(A,B)$ involves the eigenvalues of A and B as well as the departures form the normality $\delta_F(A)$ and $\delta_F(B)$.     

Abbreviations of Names of Serials of China Used in Mathematical Reviews (MR)

Ｆｏｒｎｏｒｍａｌｉｚａｔｉｏｎ ,ｙｏｕａｒｅｓｕｇｇｅｓｔｅｄｔｏｕｓｅｔｈｅｆｏｌｌｏｗｉｎｇａｂｂｒｅｖｉａｔｉｏｎｓｉｎｔｈｅｒｅｆｅｒｅｎｃｅｓｏｆｙｏｕｒｐａｐｅｒｓ.Abbreviations of Names of Serials of China Used in Mathematical Reviews (MR)     

Sets of Minimality of (1 − 1)-Rational Functions

We describe dynamical systems associated to (1 ? 1)-rational functions on the field of p-adic numbers.We focus on sets of minimality of such systems.     

An elementary proof of the irrationality of ζ(3)

An elementary proof of the irrationality of ζ(3) is presented. The proof is based on a two times more dense sequence of Diophantine approximations to this number than the sequence in the original proof of Apery.     

Characterization of submartingales of a new class (Σr)

In this work, we shall consider a new class (Σ^r) of local submartingales of the form X_t = M_t + A_t, where (M_t)_{t ? 0} is a càdlàg (right continuous with left limits) local martingale, (A_t)_{t ? 0} is a càdlàg increasing process, and the measure (dA) is carried by the set {t: X_{t ?} = 0}.

The aim of the present paper is to study the positive and negative parts of processes of this class and establish some martingale characterizations. The formula of relative martingales is derived in terms of last passage time. Finally, by using balayage formula, we calculate predictable compensator.     

A Characterization of a Class of Symmetric(0,1)Matrices

ＡＣｈａｒａｃｔｅｒｉｚａｔｉｏｎｏｆａＣｌａｓｓｏｆＳｙｍｍｅｔｒｉｃ（０，１）ＭａｔｒｉｃｅｓＬｉＷｅｎ（黎稳）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｏｕｔｈＣｈｉｎａＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｇｕａｎｚｈｏｎ，５１０６３１...     

The Normality of Cayley (Di)graphs of Out-valency 3 of Dihedral Group of Order 4p

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＧｂｅａｆｉｎｉｔｅｇｒｏｕｐａｎｄＳａｓｕｂｓｅｔｏｆＧｎｏｔｃｏｎｔａｉｎｉｎｇｔｈｅｉｄｅｎｔｉｔｙｅｌｅｍｅｎｔ１ｏｆＧ．ＷｅｄｅｆｉｎｅｔｈｅＣａｙｌｅｙ（ｄｉ）ｇｒａｐｈＸ＝Ｃａｙ（Ｇ，Ｓ）ｏｆＧｗｉｔ...     

Extensions of Vietoris''s Inequalities (Ⅱ)

It was shown by Vietoris[1] in 1958 that for n ≥ 1 and x ∈ (0, π), n∑k=0ckcos(kx)>0,(1)     

On Strong Wellposedness of (ACP_n)

This paper concerns the following higher order Cauchy problem where A_i (0≤i≤n-1) are densely defined closed linear operators on a Banach space X. Our main aim is to establish, as shown in [1]. the cha-     

Trace forms of abelian extensions of number fields of type (1, 0)

This article is concerned with describing certain bilinear forms associated with finite abelian extensions N|K of an algebraic number field K. These abelian trace forms are described up to Witt equivalence, that is, they are described as elements in the Witt ring W(K). When the base field K has exactly one dyadic prime and no real embeddings, it is shown that the Witt class of every abelian trace form over K is a product of Witt classes of five specified types.     

Existence and regularity of solutions for a class of singular (p(x), q(x))-Laplacian systems

In this paper, we study the existence of positive smooth solutions for a class of singular (p(x), q(x))-Laplacian systems using sub and supersolution methods.     

Characterizations of BMOφ(w)

In this paper we give two characterizations of functions with weighted mean oscillation over cubes controlled by a nonnegative function , that is, functions in BMO(w). The first characterization is formulated in terms of rearrangements, while the second one by means of Riesz transforms and -Lipschitz functions. These results extend those contained in [S] and [J].     

Classification of (fξηρ)-structures

The classification of (f)-structures on differentiable manifolds of even and odd dimensions is given.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 14, pp. 57–72, 1983.     

Computation of K0(RMNS_(θ,h))

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｐｕｒｐｏｓｅｏｆｔｈｉｓｐａｐｅｒｉｓｔｏｃｏｍｐｕｔｅｔｈｅＧｒｏｔｈｅｎｄｉｅｃｋｇｒｏｕｐｏｆＴ＝ＲＭＮＳ（θ，φ）．ＴｉｓｃａｌｌｅｄｔｈｅｒｉｎｇｏｆＭｏｒｉｔａＣｏｎｔｅｘｔｉｎｓｏｍｅｌｉｔｅｒａｔｕｒｅ（ｓｅｅ［１］）．Ｉｎ１９７４，ＱｕｉｌｌｅｎａｎｎｏｕｃｅｄｔｈａｔＫ０（ＲＭＯＳ）Ｋ０（Ｒ）Ｋ０（Ｓ）．ＵｐｔｏｔｈｅｐｒｅｓｅｎｔｍｏｍｅｎｔｔｈｅｒｅｓｔｉｌｌｅｘｉｓｔｓｎｏｉｎｆｏｒｍａｔｉｏｎｃｏｎｃｅｒｎｉｎｇＫ０（Ｔ）ｆ…     

On the solution set of an equation of the type f(t, Φ(u(t)) = 0

We investigate the solution set Γ of an equation of the type f(t, Φ(u(t)) = 0, where Φ is a linear homeomorphism from a topological vector space X onto L ¹(T) and f: T×R → R is a Carathéodory function. More precisely, we characterize the property of Γ of intersecting each closed hyperplane of X.     

On Multiplicities of Maximal Weights of sl ̂ ( n ) -Modules

We determine explicitly the maximal dominant weights for the integrable highest weight \(\widehat {sl}(n)\) -modules V((k ? 1)Λ ₀ + Λ _s ), 0 ≤ s ≤ n ? 1, k ≥ 2. We give a conjecture for the number of maximal dominant weights of V(k Λ ₀) and prove it in some low rank cases. We give an explicit formula in terms of lattice paths for the multiplicities of a family of maximal dominant weights of V(k Λ ₀). We conjecture that these multiplicities are equal to the number of certain pattern avoiding permutations. We prove that the conjecture holds for k = 2 and give computational evidence for the validity of this conjecture for k > 2.     

Automatic quadrature of functions of the formg(¦f(x¦)

The standard routine QAGE from Quadpack is modified to allow the efficient evaluation of integrals of the form _a ^b g(¦f(x)¦)dx, wheref andg are assumed to be smooth. Using function values available during the adaptive integration process, zeros off are found and used as subdivision points. The technique compares favourably with both the original bisection strategy and the nonuniform trisection strategy of Berntsen et al. The modified algorithm extends the class of integrals that can be integrated with an automatic code, since it copes with problem integrals that can not be integrated using the bisection or trisection strategies.