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Assuming the Generalized Riemann Hypothesis, it is known that at least half of the central values $L(\frac{1}{2},\chi)$ are non-vanishing as $\chi$ ranges over primitive characters modulo $q$. Unconditionally, this is known on average over…
We describe some recent applications of Tsallis statistics in fully developed hydrodynamic turbulence and high energy physics. For many of these applications nonextensive properties arise from spatial fluctuations of the temperature or the…
In this paper, we construct the spectral expansion for the one dimensional non-self-adjoint Dirac operator L(Q) with a complex-valued periodic matrix potential Q. To this end, we study in detail asymptotic formulas for the Bloch eigenvalues…
This paper presents a new approach to evaluating the special values of the Dirichlet beta function, $\beta(2k+1)$, where $k$ is any nonnegative integer. Our approach relies on some properties of the Euler numbers and polynomials, and uses…
We derive new all-purpose methods that involve the Dirac Delta distribution. Some of the new methods use derivatives in the argument of the Dirac Delta. We highlight potential avenues for applications to quantum field theory and we also…
One of the main tools used to understand both qualitative and quantitative spectral behaviour of periodic and almost periodic Schr\"odinger operators is the method of gauge transform. In this paper, we extend this method to an abstract…
While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants…
In this paper, we found new q-binomial formula for Q-commutative operators. Expansion coefficients in this formula are given by q-binomial coefficients with two bases (q,Q), determined by Q-commutative q-Pascal triangle. Our formula…
The present work stemmed from the study of the problem of harmonic analysis on the infinite-dimensional unitary group U(\infty). That problem consisted in the decomposition of a certain 4-parameter family of unitary representations, which…
In this research, we present a Python-based solution designed to simulate a one-dimensional quantum system that incorporates multiple Dirac $\delta -$% potentials. The primary aim of this research is to investigate the scattering problem…
An admixture of 10-20 % of qqqq\bar q components in the Delta(1232) resonance is shown to reduce the well known underprediction for the decay width for Delta(1232)->N gamma decay by about half and that of the corresponding helicity…
A new definition of a fractional derivative has recently been developed, making use of a fractional Dirac delta function as its integral kernel. This derivative allows for the definition of a distributional fractional derivative, and as…
Fermi-Dirac and Bose-Einstein integral functions are of importance not only in quantum statistics but for their mathematical properties, in themselves. Here, we have extended these functions by introducing an extra parameter in a way that…
We address the question of representativeness of a single long unstable periodic orbit for properties of the chaotic attractor it is embedded in. Y. Saiki and M. Yamada [Phys. Rev. E 79, 015201(R) (2009)] have recently suggested the…
In this paper, an integral identity for twice differentiable functions is generalized. Then, by using convexity of |f''| or q-th power of |f''| and with the aid of power mean and Holder's inequalities we achieved some new results. We also…
In the framework of distributionally generalized quantum theory, the object $H\psi$ is defined as a distribution. The mathematical significance is a mild generalization for the theory of para- and pseudo-differential operators (as well as a…
We revisit the q-deformed counterpart of the Zassenhaus formula, expressing the Jackson $q$-exponential of the sum of two non-$q$-commuting operators as an (in general) infinite product of $q$-exponential operators involving repeated…
Possibilities for defining the radial derivative of the delta distribution $\delta(\underline{x})$ in the setting of spherical coordinates are explored. This leads to the introduction of a new class of continuous linear functionals similar…
Chaudhry and Qadir obtained new identities for the gamma function by using a distributional representation for it. Here we obtain new identities for the Riemann zeta function and its family by using that representation for them. This also…
It is known that the nonextensive statistics was originally formulated for the systems composed of subsystems having same $q$. In this paper, the existence of composite system with different $q$ subsystems is investigated by fitting the…