Results in Mathematics最新文献

We present several formulas involving the classical Bernoulli numbers and polynomials. Among others, we extend an identity for Bernoulli polynomials published by Wu et al. (Fibonacci Quart 42:295-299, 2004).

Let (Lambda ) denote the von Mangoldt function, and (n, q) be the greatest common divisor of positive integers n and q. For any positive real numbers x and y, we shall consider several asymptotic formulas for sums of sums involving the von Mangoldt function; ( S_{k}(x,y):=sum _{nle y}left( sum _{qle x}right. left. sum _{d|(n,q)}dLambda left( frac{q}{d}right) right) ^{k} ) for (k=1,2).

In this paper we study isometric immersions (f:M^n rightarrow {mathbb {C}^{prime }}!P^n ) of an n-dimensional pseudo-Riemannian manifold (M^n) into the n-dimensional para-complex projective space ({mathbb {C}^{prime }}!P^n ). We study the immersion f by means of a lift (mathfrak {f}) of f into a quadric hypersurface in (S^{2n+1}_{n+1}). We find the frame equations and compatibility conditions. We specialize these results to dimension (n = 2) and a definite metric on (M^2) in isothermal coordinates and consider the special cases of Lagrangian surface immersions and minimal surface immersions. We characterize surface immersions with special properties in terms of primitive harmonicity of the Gauss maps.

In 1939 Turán raised the question about lower estimations of the maximum norm of the derivatives of a polynomial p of maximum norm 1 on the compact set K of the complex plain under the normalization condition that the zeroes of p in question all lie in K. Turán studied the problem for the interval I and the unit disk D and found that with (n:= deg p) tending to infinity, the precise growth order of the minimal possible derivative norm (oscillation order) is (sqrt{n}) for I and n for D. Erőd continued the work of Turán considering other domains. Finally, in 2006, Halász and Révész proved that the growth of the minimal possible maximal norm of the derivative is of order n for all compact convex domains. Although Turán himself gave comments about the above oscillation question in (L^q)norms, till recently results were known only for D and I. Recently, we have found order n lower estimations for several general classes of compact convex domains, and proved that in (L^q) norm the oscillation order is at least (n/log n) for all compact convex domains. In the present paper we prove that the oscillation order is not greater than n for all compact (not necessarily convex) domains K and (L^q)norm with respect to any measure supported on more than two points on K.

We introduce the Banach spaces (ell ^p_{a,b}) and (c_{0,a,b}), of analytic functions on the unit disc, having normalized Schauder bases consisting of polynomials of the form (f_n(z)=(a_n+b_nz)z^n, ~~nge 0), where ({f_n}) is assumed to be equivalent to the standard basis in (ell ^p) and (c_0), respectively. We study the weighted backward shift operator (B_w) on these spaces, and obtain necessary and sufficient conditions for (B_w) to be bounded, and prove that, under some mild assumptions on ({a_n}) and ({b_n}), the operator (B_w) is similar to a compact perturbation of a weighted backward shift on the sequence spaces (ell ^p) or (c_0). Further, we study the hypercyclicity, mixing, and chaos of (B_w), and establish the existence of hypercyclic subspaces for (B_w) by computing its essential spectrum. Similar results are obtained for a function of (B_w) on (ell ^p_{a,b}) and (c_{0,a,b}).

Let ({textbf{A}}={A_{1},...,A_{n}}) and ({textbf{B}}={B_{1},...,B_{n}}) be two finite sequences of strictly positive operators on a Hilbert space ( {mathcal {H}}) and f, (h:{mathbb {I}}rightarrow {mathbb {R}}) continuous functions with (h>0).. We consider the generalized Csiszár f-divergence operator mapping defined by

where

is introduced for every strictly positive operator A and every self-adjoint operator B, where the spectrum of the operators

are contained in the closed interval ({mathbb {I}}). In this paper we obtain some lower and upper bounds for ({textbf{I}}_{fDelta h}({textbf{A}},{textbf{B}})) with applications to the geometric operator mean and the relative operator entropy. We verify the information monotonicity for the Csisz ár f-divergence operator mapping and the generalized Csiszár f-divergence operator mapping.

Given Banach space operators A, B, let (delta _{A,B}) denote the generalised derivation (delta (X)=(L_{A}-R_{B})(X)=AX-XB) and (triangle _{A,B}) the length two elementary operator (triangle _{A,B}(X)=(I-L_AR_B)(X)=X-AXB). This note considers the structure of m-symmetric operators (delta ^m_{triangle _{A_1,B_1},triangle _{A_2,B_2}}(I)=(L_{triangle _{A_1,B_1}} - R_{triangle _{A_2,B_2}})^m(I)=0). It is seen that there exist scalars (lambda _iin sigma _a(B_1)), (1le ile 2), such that (delta ^m_{lambda _1 A_1,lambda _2 A_2}(I)=0). Translated to Hilbert space operators A and B this implies that if (delta ^m_{triangle _{A^*,B^*},triangle _{A,B}}(I)=0), then there exists ({overline{lambda }}in sigma _a(B^*)) such that (delta ^m_{(lambda A)^*,lambda A}(I)=0=delta ^m_{{overline{lambda }}B,lambda B^*}(I)). We prove that the operator (delta ^m_{triangle _{A^*,B^*},triangle _{A,B}}) is compact if and only if (i) there exists a real number (alpha ) and finite sequnces (i) ({a_j}_{j=1}^nsubseteq sigma (A)), ({b_j}_{j=1}^nsubseteq sigma (B)) such that (a_jb_j=1-alpha ), (1le jle n); (ii) decompositions (oplus _{j=1}^n {mathcal {H}}_j) and (oplus _{j=1}^n{texttt {H}_J}) of ({mathcal {H}}) such that (oplus _{j=1}^n{(A-a_j I)|_{ H_j}}) and (oplus _{j=1}^n{(B-b_j I)|_{texttt {H}_j}}) are nilpotent. If (delta ^{m}_{triangle _{A^*,B^*},triangle _{A,B}}(I)=0) implies (delta _{triangle _{A^*,B^*},triangle _{A,B}}(I)=0), then A and B satisfy a (Putnam-Fuglede type) commutativity theorem; conversely, a sufficient condition for (delta ^{m}_{triangle _{A^*,B^*},triangle _{A,B}}(I)=0) to imply (delta _{triangle _{A^*,B^*},triangle _{A,B}}(I)=0) is that ({lambda }A) and ({overline{lambda }}B) satisfy the commutativity property for scalars (overline{lambda} in sigma _a(B^*)). An analogous result is seen to hold for the operators (triangle ^m_{delta _{A^*,B^*},delta _{A,B}}) and (triangle ^m_{delta _{A^*,B^*},delta _{A,B}}(I)). Perturbation by commuting nilpotents is considered.

In the first part of this paper, we study several Bohr radii for holomorphic mappings with values in the unit polydisc (mathbb {U}^N) in (mathbb {C}^{N}). In particular, we obtain the new Bohr radius (r_{k,m}^{***}) for holomorphic mappings with lacunary series. Further, we show that when (mge 1), (r_{k,m}^{***}) is asymptotically sharp as (Nrightarrow infty ). Note that when (mge 1), (r_{k,m}^{***}) is completely different from the cases with values in the unit disc (mathbb {U}) and in the complex Hilbert balls with higher dimensions. In the second part of this paper, we obtain the Bohr type inequality for holomorphic mappings F with values in the unit ball of a JB(^*)-triple which is a generalization of that for holomorphic mappings F with values in the unit ball of a complex Banach space of the form (F(z)=f(z)z), where f is a (mathbb {C})-valued holomorphic function.

The existence of a Fourier basis with frequencies in (mathbb {R}^d) for the space of square integrable functions supported on a given parallelepiped in (mathbb {R}^d), has been well understood since the 1950s. In a companion paper, we derived necessary and sufficient conditions for a parallelepiped in (mathbb {R}^d) to permit an orthogonal basis of exponentials with frequencies constrained to be a subset of a prescribed lattice in (mathbb {R}^d), a restriction relevant in many applications. In this paper, we investigate analogous conditions for parallelepipeds that permit a Riesz basis of exponentials with the same constraints on the frequencies. We provide a sufficient condition on the parallelepiped for the Riesz basis case which directly extends one of the necessary and sufficient conditions obtained in the orthogonal basis case. We also provide a sufficient condition which constrains the spectral norm of the matrix generating the parallelepiped, instead of constraining the structure of the matrix.

We give different integral representations of the Lommel function (s_{mu ,nu }(z)) involving trigonometric and hypergeometric (_2F_1) functions. By using classical results of Pólya, we give the distribution of the zeros of (s_{mu ,nu }(z)) for certain regions in the plane ((mu ,nu )). Further, thanks to a well known relation between the functions (s_{mu ,nu }(z)) and the hypergeometric ( _1F_2) function, we describe the distribution of the zeros of (_1F_2) for specific values of its parameters.