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) \

objectives: autism spectrum disorder (asd) is a group of neurodevelopment syndromes with impairments in social communications and restrictive or repetitive behaviors. neurofeedback is a new method that may regulate neuro and metabolic function in brain. this study is semi-experimental, with pre and post test.  methods: in 26 children with asd, 40 sessions of neurofeedback were done. brain waves...

Let $D$ be the open disk $|x|< R$ of a complete ultrametric algebraically closed field $\mathbb K$. We define growth order $\rho(f)$, type $\sigma(f)$ cotype $\psi(f)$ and another expression $\theta(f)$ an analytic function in we show relations between number zeros $f$ ``maximum modulus'' involving $\rho,\sigma,\psi$. Then $\rho(f)$ lies $[\theta(f)-1, \theta(f)]$. Moreover, if $0< \rho(f)< +\i...

In this paper, we prove that it is always possible to define a realization of the Laplacian $$\Delta _{\kappa ,\theta }$$ on $$L^2(\Omega )$$ subject nonlocal Robin boundary conditions with general jump measures arbitrary open subsets $${\mathbb {R}}^N$$ . This made by using capacity approach an admissible pair $$(\kappa allows associated form $${\mathcal {E}}_{\kappa be closable. The generates...

For $X(n)$ a Rademacher or Steinhaus random multiplicative function, we consider the polynomials $$ P_N(\theta) = \frac1{\sqrt{N}} \sum_{n\leq N} X(n) e(n\theta), and show that $2k$-th moments on unit circle \int_0^1 \big| \big|^{2k}\, d\theta tend to Gaussian in sense of mean-square convergence, uniformly for $k \ll (\log N / \log N)^{1/3}$, but contrast case i.i.d. coefficients, this behavior...

&lrm;The problem of shrinkage testimation&nbsp;(test-estimation) for the Rayleigh scale&lrm; &lrm;parameter &theta; based on censored samples under the reflected&lrm; &lrm;gamma loss function is considered&lrm;. We obtain the minimum risk&lrm; &lrm;estimator among a subclass and compute its risk&lrm;. &lrm;A shrinkage&lrm; &lrm;testimator based on acceptance or rejection of a null hypothesis&lr...

Quantum theta functions were introduced by the author in [Ma1]. They are certain elements in the function rings of quantum tori. By definition, they satisfy a version of the classical functional equations involving shifts by the multiplicative periods. This paper shows that for a certain subclass of period lattices (compatible with the quantization form), quantum thetas satisfy an analog of ano...