Paranorm <Emphasis Type="Italic">I</Emphasis>-convergent sequence spaces

In this article we introduced the sequence spaces c ^I (p), c ₀ ^I (p), m ^I (p) and m ₀ ^I (p) for p = (p _k ), a sequence of positive real numbers. We study some algebraic and topological properties of these spaces. We prove the decomposition theorem and obtain some inclusion relations.      

Explicit valuations of division polynomials¶of an elliptic curve

In this paper, we estimate valuations of division polynomials and compute them explicitely at singular primes. We show that ν_?(ψ _m (M)) is asymptotically equal to ν_?(m) for a non-torsion point M such that M mod ? is non-zero and non-singular, and it is asymptotically equal to c ₁ m ¹ for some constant c ₁ for a non-torsion point M such that M mod ? is either singular or zero. Furthermore, we show that the common factors of φ _m (M) and ψ _m ²(M) have valuations at ? asymptotically equal to c ₂ m ² for some constant c ₂ when M mod ? is singular, which is a generalization of M. Ayad's result. Received: 10 July 1997 / Revised version: 11 May 1998     

SOME EULER SPACES OF DIFFERENCE SEQUENCES OF ORDER m

Kizmaz [13] studied the difference sequence spaces e∞(△), c(△), and c0(△).Several article dealt with the sets of sequences of m-th order difference of which are bounded, convergent, or convergent to zero. Altay and Basar [5] and Altay, Basar, and Mursaleen [7] introduced the Euler sequence spaces eτ0, eτ0, andeτ∞, respectively. The main purpose of this article is to introduce the spaces eτ0(△(m)), eτc(△(m)), and eτ∞(△(m)) consisting of all sequences whose mth order differences are in the Euler spaces eτ0, eτc, and eτ∞, respectively. Moreover, the authors give some topological properties and inclusion relations, and determine the α-, β-, and γ-duals of the spaces eτ0(△(m)), eτc(△(m)), and eτ∞(△(m)), and the Schauder basis of the spaces eτ0(△(m)), eτc(△(m)). The last section of the article is devoted to the characterization of some matrix mappings on the sequence space eτc(△(m)).     

A characterization of totally <Emphasis Type="Italic">η</Emphasis>-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

We give a characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M ⁿ (c), c ≠ 0, n ⩾ 3, satisfies g(AX, Y) = ag(X, Y) for any X, Y ∈ T ₀(x), a being a function, where T ₀ is the holomorphic distribution on M, then M is a totally η-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of η-umbilical real hypersurfaces.     

Idempotence preserving maps between matrix spaces

Suppose 𝔽 is an arbitrary field of characteristic not 2 and 𝔽?≠?𝔽₃. Let M _n (𝔽) be the space of all n?×?n full matrices over 𝔽 and P _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n idempotent matrices and GL _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n invertible matrices. Let Φ_𝔽(n,?m) denote the set of all maps from M _n (𝔽) to M _m (𝔽) satisfying A???λB?∈?P _n (𝔽)???φ(A)???λφ(B)?∈?P _m (𝔽) for every A,?B?∈?M _n (𝔽) and λ?∈?𝔽, where m and n are integers with 3?≤?n?≤?m. It is shown that if φ?∈?Φ_𝔽(n,?m), then there exists T?∈?GL _m (𝔽) such that φ(A)?=?T?[A???I _p ?⊕?A ^t ???I _q ?⊕?0]T?^?1 for every A?∈?M _n (𝔽), where I ₀?=?0. This improves the results of some related references.     

Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on the Unit Circle

We study the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product on the unit circle

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where f(Z)=(f(z₁), …, f^(l₁)(z₁), …, f(z_m), …, f^(l_m)(z_m)), A is a M×M positive definite matrix or a positive semidefinite diagonal block matrix, M=l₁+…+l_m+m, dμ belongs to a certain class of measures, and |z_i|>1, i=1, 2, …, m.     

Spectral Criteria of Abstract Sequences; Application to Difference Equations

We suppose that M is a closed subspace of l ^∞(J, X), the space of all bounded sequences {x(n)} _n?J ? X, where J ? {Z⁺,Z} and X is a complex Banach space. We define the M-spectrum σ_M (u) of a sequence u ? l ^∞(J,X). Certain conditions will be supposed on both M and σ_M (u) to insure the existence of u ? M. We prove that if u is ergodic, such that σ_M (u,) is at most countable and, for every λ ? σ_M (u), the sequence e^?iλnu(n) is ergodic, then u ? M. We apply this result to the operator difference equationu(n + 1) = Au(n) + ψ(n), n ? J,and to the infinite order difference equation Σ ^r _k=1 a_k (u(n + k) ? u(n)) + Σ _{s ? Z}?(n ? s)u(s) = h(n), n?J, where ψ?l ^∞(Z,X) such that ψ| _J ? M, A is the generator of a C ₀-semigroup of linear bounded operators {T(t)} _t>0 on X, h ? M, ? ? l ¹(Z) and a_k ?C. Certain conditions will be imposed to guarantee the existence of solutions in the class M.     

On approximation of meromorphic functions having index-pair (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

The purpose of this paper is to generalize the results obtained by Winiarski (Ann. Polon. Math. 29:259–273, 1970) and Kasana and Kumar (Publ. Mat. 38:255–267, 1994) for the M ₀(C) of all entire functions onto the class M _m (C), m ≥ 0 of all meromorphic functions with exactly m poles on the complex plane C.     

On some new paranormed euler sequence spaces and Euler Core

In this paper, the sequence spaces e0^τ（u, p） and ec^τ（u, p） of non-absolute type which are the generalization of the Maddox sequence spaces have been introduced and it is proved that the spaces e0^τ（u,p） and ec^τ（u,p） are linearly isomorphic to spaces co（p） and c（p）, respectively. Furthermore, the α-, β- and γ-duals of the spaces 0^τ（u,p） and ec^τ（u,p） have been computed and their bases have been constructed and some topological properties of these spaces have been investigated. Besides this, the class of matrices （e0^τ）（u, p） ： μ） has been characterized, where μ is one of the sequence spaces l∞, c and co and derives the other characterizations for the special cases of μ. In the last section, Euler Core of a complex-valued sequence has been introduced, and we prove some inclusion theorems related to this new type of core.     

The Dual Spaces of Sets of Difference Sequences of Order <Emphasis Type="Italic">m</Emphasis> and Matrix Transformations

Let p = （pk）k=0^∞ be a bounded sequence of positive reals, m C N and u be s sequence of nonzero terms. If x = （xk）k=0^∞ is any sequence of complex numbers we write Δ（m）x for the sequence of the m th order differences of x and Δu^（m）X = {x=（x）k=0^∞ uΔ（m）x ∈ X} for any set X of sequences. We determine the α-, β- and γ-duals of the sets Δμ^（m）X for X=co（p）,c（p）,l∞（p） and characterize some matrix transformations between these spaces Δ^（m）X.     

Scale‐free graphs of increasing degree

We study a scale‐free random graph process in which the number of edges added at each step increases. This differs from the standard model in which a fixed number, m, of edges are added at each step. Let f(t) be the number of edges added at step t. In the standard scale‐free model, f(t) = m constant, whereas in this paper we consider f(t) = [t^c],c > 0. Such a graph process, in which the number of edges grows non‐linearly with the number of vertices is said to have accelerating growth. We analyze both an undirected and a directed process. The power law of the degree sequence of these processes exhibits widely differing behavior. For the undirected process, the terminal vertex of each edge is chosen by preferential attachment based on vertex degree. When f(t) = m constant, this is the standard scale‐free model, and the power law of the degree sequence is 3. When f(t) = [t^c],c < 1, the degree sequence of the process exhibits a power law with parameter x = (3 ? c)/(1 ? c). As c → 0, x → 3, which gives a value of x = 3, as in standard scale‐free model. Thus no more slowly growing monotone function f(t) alters the power law of this model away from x = 3. When c = 1, so that f(t) = t, the expected degree of all vertices is t, the vertex degree is concentrated, and the degree sequence does not have a power law. For the directed process, the terminal vertex is chosen proportional to in‐degree plus an additive constant, to allow the selection of vertices of in‐degree zero. For this process when f(t) = m is constant, the power law of the degree sequence is x = 2 + 1/m. When f(t) = [t^c], c > 0, the power law becomes x = 1 + 1/(1 + c), which naturally extends the power law to [1,2]. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 396–421, 2011     

Fuzzy complex Grassmannians and quantization of line bundles

We construct by purely representation-theoretic methods fuzzy versions of an arbitrary complex Grassmannian M=Gr _n (ℂ ^n+m ), i.e., a sequence of matrix algebras tending SU(n+m)-equivariantly to the algebra of smooth functions on M. We also show that this approximation can be interpreted in terms of the Berezin-Toeplitz quantization of M. Furthermore, we use branching rules to prove that the quantization of every complex line bundle over M is given by a SU(n+m)-equivariant truncation of the space of its L ²-sections.     

Matlis duals of local cohomology modules and their endomorphism rings

Let ${(R, \mathfrak{m})}Let (R, \mathfrakm){(R, \mathfrak{m})} denote a local ring. Let I ì R{I \subset R} be an ideal with c =  grade I. Let D(·) denote the Matlis duality functor. In recent research there is an interest in the structure of the local cohomology module H^c_I : = H^c_I(R){H^c_I := H^c_I(R)}, in particular in the endomorphism ring of D(H^c_I){D(H^c_I)}. Let E _R (k) be the injective hull of the residue field R/\mathfrakm{R/\mathfrak{m}}. By investigating the natural map H^c_I ?D(H^c_I) ? E_R(k){H^c_I \otimes D(H^c_I) \to E_R(k)} we are able to prove that the endomorphism rings of D(H^c_I){D(H^c_I)} and of H^c_I{H^c_I} are naturally isomorphic. This natural homomorphism is related to a quasi-isomorphism of a certain complex. As applications we show results when the endomorphism ring of D(H^c_I){D(H^c_I)} is naturally isomorphic to R generalizing results known under the additional assumption of Hⁱ_I(R) = 0{H^i_I(R) = 0} for i 1 c{i \not= c}.     

Invariant means

Let m and M be symmetric means in two and three variables, respectively. We say that M is type 1 invariant with respect to m if M(m(a,c),m(a,b),m(b,c))≡M(a,b,c). If m is strict and isotone, then we show that there exists a unique M which is type 1 invariant with respect to m. In particular, we discuss the invariant logarithmic mean L₃, which is type 1 invariant with respect to L(a,b)=(b−a)/(logb−loga). We say that M is type 2 invariant with respect to m if M(a,b,m(a,b))≡m(a,b). We also prove existence and uniqueness results for type 2 invariance, given the mean M(a,b,c). The arithmetic, geometric, and harmonic means in two and three variables satisfy both type 1 and type 2 invariance. There are means m and M such that M is type 2 invariant with respect to m, but not type 1 invariant with respect to m (for example, the Lehmer means). L₃ is type 1 invariant with respect to L, but not type 2 invariant with respect to L.     

The modulator of the local time

Let x(t), 0 ≦ t ≦ 1, be a real measurable function having a local time α(x, t) which is a continuous function of t for almost all x. It is also assumed that, for some m ≧ 2 and some real interval B, α^m(x, 1) is integrable over B. The modulator is a function M_m(t, B), t > 0, denned in terms of α. It is shown that the modulator serves as a measure of the smoothness of the L_m(B)-valued function α(., t) with respect to t. Then it is shown that the modulator plays a central role in precisely describing certain irregularity properties of x(t). The results are applied to the case where x(t) is the sample function of a real stochastic process. In this way new results are obtained for large classes of Gaussian and Markov processes.     

On the iteration of holomorphic self-maps of ℂ*

Letf be a holomorphic self-map of the punctured plane ℂ^*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI ₀(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞} andI _∞(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞}. We try to find the relation betweenI ₀(f),I _∞(t) andJ(f). It is proved that both the boundary ofI ₀(f) and the boundary ofI _∞)f) equal toJ(f),I ₀(f) ∩J(f) ≠ θ andI _∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI ₀(f) andI _∞(f) are not doubly-bounded. Supported by the National Natural Science Foundation of China     

Additive Set of Idempotents in Rings

Let R be a ring with identity 1, I(R) be the set of all nonunit idempotents in R, and M(R) be the set of all primitive idempotents and 0 of R. We say that I(R) is additive if for all e, f ∈ I(R) (e ≠ f), e + f ∈ I(R), and M(R) is additive in I(R) if for all e, f ∈ M(R)(e ≠ f), e + f ∈ I(R). In this article, the following points are shown: (1) I(R) is additive if and only if I(R) is multiplicative and the characteristic of R is 2; M(R) is additive in I(R) if and only if M(R) is orthogonal. If 0 ≠ ef ∈ I(R) for some e ∈ M(R) and f ∈ I(R), then ef ∈ M(R), (2) If R has a complete set of primitive idempotents, then R is a finite product of connected rings if and only if I(R) is multiplicative if and only if M(R) is additive in I(R).     

Remarks on ideal boundedness, convergence and variation of sequences

We answer the questions asked by Faisant et al. (2005) [2]. The first main result states that for every admissible ideal I⊂P(N) the quotient space l^∞(I)/c₀(I) is complete. The second main result states that consistently there is an admissible ideal I⊂P(N) such that the sets W(I), of all real sequences with finite I-variation, and c^?(I), of all restrictively I-convergent sequences, are equal.     

Pappus type theorems for hypersurfaces in a space form

In order to get further insight on the Weyl’s formula for the volume of a tubular hypersurface, we consider the following situation. Letc(t) be a curve in a space formM _λ ⁿ of sectional curvature λ. LetP ₀ be a totally geodesic hypersurface ofM _λ ⁿ throughc(0) and orthogonal toc(t). LetC ₀ be a hypersurface ofP ₀. LetC be the hypersurface ofM _λ ⁿ obtained by a motion ofC ₀ alongc(t). We shall denote it byC ^PorC ^Fif it is obtained by a parallel or Frenet motion, respectively. We get a formula for volume(C). Among other consequences of this formula we get that, ifc(0) is the centre of mass ofC ₀, then volume(C) ≥ volume(C),^P),and the equality holds whenC ₀ is contained in a geodesic sphere or the motion corresponds to a curve contained in a hyperplane of the Lie algebraO(n−1) (whenn=3, the only motion with these properties is the parallel motion). Work partially supported by a DGES Grant No. PB97-1425 and a AGIGV Grant No. GR0052.     

On the Riesz means of coefficients of <Emphasis Type="Italic">m</Emphasis>th symmetric power <Emphasis Type="Italic">L</Emphasis>-functions

Let c _n be the Fourier coefficients of L(sym ^m   f, s), and Δρ(x; sym ^m   f) be the error term in the asymptotic formula for ∑ _n≪x c _n . In this paper, we study the Riesz means of Δρ(x; sym ^m   f) and obtain a truncated Voronoi-type formula under the hypothesis Nice(m, f).