We study the distribution of spacings between squares in Z/QZ as the number of prime divisors of Q tends to infinity. In [3] Kurlberg and Rudnick proved that the spacing distribution for square free Q is Poissonian, this paper extends the result to arbitrary Q.

we prove that every r-quadratic metric of scalar flag curvature with a dimension greater than twois of constant flag curvature. then we show that generalized douglas-weyl metrics contain r-quadraticmetrics as a special case, but the class of r-quadratic metric is not closed under projective transformations

in this paper, we obtain the upper exponential bounds for the tail probabilities of the quadratic forms for negatively dependent subgaussian random variables. in particular the law of iterated logarithm for quadratic forms of independent subgaussian random variables is generalized to the case of negatively dependent subgaussian random variables.

The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics but also in insurance, finance, and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the Itō and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingal...

Given positive integers $n\ge s$, we let ${\mathrm{ex}}(n,s)$ denote the maximum number of edges in a triangle-free graph $G$ on $n$ vertices with $\alpha(G)\le s$. In early sixties Andr\'{a}sfai conjectured that for $n/3<s<n/2$ function ${\mathrm{ex}}(n, s)$ is piecewise quadratic critical values at $s/n={k}/({3k-1})$. We confirm this indeed case whenever $s/n$ slightly larger than value, thus...

The Generalized Uncertainty Principle (GUP) has emerged in numerous attempts to a theory of quantum gravity and predicts the existence minimum length Nature. In this work, we consider two cosmological models arising from Friedmann equations modified by GUP (in its linear quadratic formulations) compare them with observational data. Our aim is derive constraints on parameter discuss viability ph...

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This paper presents an implementation of open loop and closed loop control of Quadratic boost converter using PID controller.  Quadratic boost converter consists of boost converter and fly back converter driven by a single switch. Quadratic boost converter QBC is designed especially for regulating the DC interface between various micro sources and dc-ac inverter to electricity grid. Quadratic b...