On the 2 k-th power mean of $$ \frac{{L'}} {L}(1,\chi ) $$ with the weight of Gauss sums

The main purpose of this paper is to study the hybrid mean value of $ \frac{{L'}} {L}(1,\chi ) $ \frac{{L'}} {L}(1,\chi ) and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value $ \sum\limits_{\chi \ne \chi _0 } {|\tau (\chi )||\frac{{L'}} {L}(1,\chi )|^{2k} } $ \sum\limits_{\chi \ne \chi _0 } {|\tau (\chi )||\frac{{L'}} {L}(1,\chi )|^{2k} } of $ \frac{{L'}} {L} $ \frac{{L'}} {L} and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.     

The generalized Roper-Suffridge extension operator in Banach spaces (III)

Suppose that X is a complex Banach space with the norm ‖·‖ and n is a positive integer with dim X ⩾ n ⩾ 2. In this paper, we consider the generalized Roper-Suffridge extension operator $ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f) $ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f) on the domain $ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } $ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } defined by

Imaginary powers of a Laguerre differential operator

Imaginary powers associated to the Laguerre differential operator $ L_\alpha = - \Delta + |x|^2 + \sum _{i = 1}^d \frac{1} {{x_i^2 }}(\alpha _i^2 - 1/4) $ L_\alpha = - \Delta + |x|^2 + \sum _{i = 1}^d \frac{1} {{x_i^2 }}(\alpha _i^2 - 1/4) are investigated. It is proved that for every multi-index α = (α₁,...α _d ) such that α _i ≧ −1/2, α _i ∉ (−1/2, 1/2), the imaginary powers $ \mathcal{L}_\alpha ^{ - i\gamma } ,\gamma \in \mathbb{R} $ \mathcal{L}_\alpha ^{ - i\gamma } ,\gamma \in \mathbb{R} , of a self-adjoint extension of L _α, are Calderón-Zygmund operators. Consequently, mapping properties of $ \mathcal{L}_\alpha ^{ - i\gamma } $ \mathcal{L}_\alpha ^{ - i\gamma } follow by the general theory.     

On the shift semigroup on the Hardy space of Dirichlet series

We develop a Wold decomposition for the shift semigroup on the Hardy space $ \mathcal{H}^2 $ \mathcal{H}^2 of square summable Dirichlet series convergent in the half-plane $ \Re (s) > 1/2 $ \Re (s) > 1/2 . As an application we have that a shift invariant subspace of $ \mathcal{H}^2 $ \mathcal{H}^2 is unitarily equivalent to $ \mathcal{H}^2 $ \mathcal{H}^2 if and only if it has the form $ \phi \mathcal{H}^2 $ \phi \mathcal{H}^2 for some $ \mathcal{H}^2 $ \mathcal{H}^2 -inner function φ.     

Real moments of the restrictive factor

Let λ be a real number such that 0 < λ < 1. We establish asymptotic formulas for the weighted real moments Σ _n≤x R ^λ (n)(1 − n/x), where R(n) =$ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu - 1} } $ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu - 1} } is the Atanassov strong restrictive factor function and n =$ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } } $ \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } } is the prime factorization of n.     

Determination of Riesz bounds for the spline basis with the help of trigonometric polynomials

The problem of determining the upper and lower Riesz bounds for the mth order B-spline basis is reduced to analyzing the series $ \sum\nolimits_{j = - \infty }^\infty {\frac{1} {{(x - j)^{2m} }}} $ \sum\nolimits_{j = - \infty }^\infty {\frac{1} {{(x - j)^{2m} }}} . The sum of the series is shown to be a ratio of trigonometric polynomials of a particular shape. Some properties of these polynomials that help to determine the Riesz bounds are established. The results are applied in the theory of series to find the sums of some power series.     

On the hierarchies of higher order mKdV and KdV equations

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $ \hat H_s^r \left( \mathbb{R} \right) $ \hat H_s^r \left( \mathbb{R} \right) defined by the norm

Boundedness and compactness of the embedding between spaces with multiweighted derivatives when 1 ⩽ <Emphasis Type="Italic">q</Emphasis> < <Emphasis Type="Italic">p</Emphasis> < ∞

We consider a new Sobolev type function space called the space with multiweighted derivatives $ W_{p,\bar \alpha }^n $ W_{p,\bar \alpha }^n , where $ \bar \alpha $ \bar \alpha = (α ₀, α ₁,…, α _n ), α _i ∈ ℝ, i = 0, 1,…, n, and $ \left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|} $ \left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|} ,

Phase portraits for quadratic homogeneous polynomial vector fields on $$ \mathbb{S}^2 $$

Let X be a homogeneous polynomial vector field of degree 2 on $ \mathbb{S}^2 $ \mathbb{S}^2 . We show that if X has at least a non-hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on $ \mathbb{S}^2 $ \mathbb{S}^2 is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16 ^th Hilbert’s problem restricted to this class of polynomial vector fields to the study of two particular families. Moreover, we present two criteria for studying the nonexistence of periodic orbits for homogeneous polynomial vector fields on $ \mathbb{S}^2 $ \mathbb{S}^2 of degree n.     

Two operators related to Marcinkiewicz integrals

In this paper, the authors consider the boundedness of generalized higher commutator of Marcinkiewicz integral μΩ^b, multilinear Marcinkiewicz integral μΩ^A and its variation μΩ^A on Herz-type Hardy spaces, here Ω is homogeneous of degree zero and satisfies a class of L^s-Dini condition. And as a special case, they also get the boundedness of commutators of Marcinkiewicz integrals on Herz-type Hardy spaces.     

Σ-definability of uncountable models of <Emphasis Type="Italic">c</Emphasis>-simple theories

We show that each c-simple theory with an additional discreteness condition has an uncountable model Σ-definable in ℍ$ \mathbb{H} $ \mathbb{H} ($ \mathbb{L} $ \mathbb{L} ), where $ \mathbb{L} $ \mathbb{L} is a dense linear order. From this we establish the same for all c-simple theories of finite signature that are submodel complete.     

Existence of planar curves minimizing length and curvature

We consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $ \smallint \sqrt {1 + K_\gamma ^2 ds} $ \smallint \sqrt {1 + K_\gamma ^2 ds} , depending both on the length and curvature K. We fix starting and ending points as well as initial and final directions. For this functional we discuss the problem of existence of minimizers on various functional spaces. We find nonexistence of minimizers in cases in which initial and final directions are considered with orientation. In this case, minimizing sequences of trajectories may converge to curves with angles. We instead prove the existence of minimizers for the “time-reparametrized” functional $ \smallint ||\dot \gamma (t)||\sqrt {1 + K_\gamma ^2 dt} $ \smallint ||\dot \gamma (t)||\sqrt {1 + K_\gamma ^2 dt} for all boundary conditions if the initial and final directions are considered regardless of orientation. In this case, minimizers may present cusps (at most two) but not angles.     

Fourier-Feynman transforms of unbounded functionals on abstract Wiener space

Huffman, Park and Skoug established several results involving Fourier-Feynman transform and convolution for functionals in a Banach algebra S on the classical Wiener space. Chang, Kim and Yoo extended these results to abstract Wiener space for a more generalized Fresnel class $ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } $ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } _A1,A2 than the Fresnel class $ \mathcal{F} $ \mathcal{F} (B)which corresponds to the Banach algebra S. In this paper we study Fourier-Feynman transform, convolution and first variation of unbounded functionals on abstract Wiener space having the form

Integral-type operators from a mixed norm space to a bloch-type space on the unit ball

Let $ \mathbb{B} $ \mathbb{B} be the unit ball in ℂ ⁿ and let H($ \mathbb{B} $ \mathbb{B} ) be the space of all holomorphic functions on $ \mathbb{B} $ \mathbb{B} . We introduce the following integral-type operator on H($ \mathbb{B} $ \mathbb{B} ):

On equivalences of derived and singular categories

Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → $ \mathbb{A}^1 $ \mathbb{A}^1 , g:Y → $ \mathbb{A}^1 $ \mathbb{A}^1 . Assuming that there exists a complex of sheaves on X × $ \mathbb{A}^1 $ \mathbb{A}^1 Y which induces an equivalence of D ^b (X) and D ^b (Y), we show that there is also an equivalence of the singular derived categories of the fibers f ⁻¹(0) and g ⁻¹(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class.     

On $$ \mathfrak{F}_h $$-normal subgroups of finite groups

Let G be a finite group, and let $ \mathfrak{F} $ \mathfrak{F} be a formation of finite groups. We say that a subgroup H of G is $ \mathfrak{F}_h $ \mathfrak{F}_h -normal in G if there exists a normal subgroup T of G such that HT is a normal Hall subgroup of G and (H ∩ T)H _G /H _G is contained in the $ \mathfrak{F} $ \mathfrak{F} -hypercenter $ Z_\infty ^\mathfrak{F} $ Z_\infty ^\mathfrak{F} (G/H _G ) of G/H _G . In this paper, we obtain some results about the $ \mathfrak{F}_h $ \mathfrak{F}_h -normal subgroups and then use them to study the structure of finite groups.     

On stably -monotone Banach couples

The $ \vec E $ \vec E = (E, E(2^-k )) is a stably $ \vec E $ \vec E is $ \vec E $ \vec E is $ \vec E $ \vec E is -monotone, then either E = l _p (1 ≤ p < ∞) or E = c ₀.     

On set-valued cone absolutely summing maps

Spaces of cone absolutely summing maps are generalizations of Bochner spaces L ^p (μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L ¹(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] , and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map. We also describe these set-valued cone absolutely summing maps as those that map order-Pettis integrable functions to integrably bounded set-valued functions.     

Multidimensional analog of the Hardy condition for power series

In the middle of the 20th century Hardy obtained a condition which must be imposed on a formal power series f(x) with positive coefficients in order that the series f ⁻¹(x) = $ \sum\limits_{n = 0}^\infty {b_n x^n } $ \sum\limits_{n = 0}^\infty {b_n x^n } b _n x ⁿ be such that b ₀ > 0 and b _n ≤ 0, n ≥ 1. In this paper we find conditions which must be imposed on a multidimensional series f(x ₁, x ₂, …, x _m ) with positive coefficients in order that the series f ⁻¹(x ₁, x ₂, …, x _m ) = $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } satisfies the property b _{0, …, 0} > 0, $ bi_1 ,i_2 , \ldots ,i_m $ bi_1 ,i_2 , \ldots ,i_m ≤ 0, i ₁² + i ₂² + … + i _m ² > 0, which is similar to the one-dimensional case.     

On the 2<Emphasis Type="Italic">m</Emphasis>-th power mean of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions with the weight of trigonometric sums

Let p be a prime, χ denote the Dirichlet character modulo p, f (x) = a ₀ + a ₁ x + ... + a _k x ^k is a k-degree polynomial with integral coefficients such that (p, a ₀, a ₁, ..., a _k ) = 1, for any integer m, we study the asymptotic property of