Let $Z(t)=\chi ^{-1/2}(1/2+it)\zeta (1/2+it)=e^{i\theta (t)}\zeta (1/2+it)$ be Hardy’s function and $g(n)$ the $n$th Gram point defined by $\theta (g(n))=\pi n$. Titchmarsh proved that $$ \sum _{n \leq N} Z(g(2n)) =2N+O(N^{3/4}\log ^{3/4}N) \

In this paper, we study generalized Douglas-Weyl Finsler metrics. We find some conditions under which the class of generalized Douglas-Weyl (&alpha, &beta)-metric with vanishing S-curvature reduce to the class of Berwald metrics.

Let φ : M → Rn+p(c) be an n-dimensional submanifold in an (n + p)dimensional space form Rn+p(c) with the induced metric g. Willmore functional of φ is W (φ) = ∫M (S − nH2)n/2dv, where S = ∑ α,i, j (h α i j ) 2 is the square of the length of the second fundamental form, H is the mean curvature of M . The Weyl functional of (M, g) is ν(g) = ∫M |Wg|n/2dv, where |Wg|2 = ∑ i, j,k,l W 2 i jkl and Wi ...

Abstract In [Sa], it was proved that the Selberg zeta function for SL2(Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL2(Z). As applications, we study the Brun-Titc...

In the present paper, we prove spectral mapping theorem for (m,n)-paranormal operator T on a separable Hilbert space, that is, f (?w(T)) = ?w(f(T)) when is an analytic function some open neighborhood of ?(T). We also show T, Weyl?s holds, ?(T)-?w(T) ?00(T). Moreover, if algebraically (m,n)-paranormal, then and hold.

A direct, bundle-theoretic method for defining and extending local isometries out of curvature data is developed. As a by-product, conceptual direct proofs of a classical result of Singer and a recent result of the authors are derived. A classical result of I. M. Singer [7] states that a Riemannian manifold is locally homogeneous if and only if its Riemannian curvature tensor together with its ...

Abstract We generalize a theorem by Titchmarsh about the mean value of Hardy’s $$Z$$ Z -function at Gram points to Hecke $$L$$ L -functions, which in turn implies weak law for them. Instead proceeding analogously with an approximate functional equation we employ different method using contour...

we have applied the method of integration of the heisenberg equation of motion proposed by bender and dunne, and m. kamella and m. razavy to the potential v(q) = v q - µ q with linear and nonlinear dissipation. we concentrate our calculations on the evolution of basis set of weyl ordered operators and calculate the mean position , velocity , the commutation relation [q, p], and the energy of pa...

The construction of the ∗-product proposed by Fedosov is implemented in terms of the theory of fibre bundles. The geometrical origin of the Weyl algebra and the Weyl bundle is shown. Several properties of the product in the Weyl algebra are proved. Symplectic and abelian connections in the Weyl algebra bundle are introduced. Relations between them and the symplectic connection on a phase space ...