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Authors
Andrea Pulita
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The main local invariants of a (one variable) differential module over the complex numbers are given by means of a cyclic basis. In the $p$-adic setting the existence of a cyclic vector is often unknown. We investigate the existence of such a cyclic vector in a Banach algebra. We follow the explicit method of Katz, and we prove the existence of such a cyclic vector under the assumption that the matrix of the derivation is small enough in norm.
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We call $n$ a cyclic number if every group of order $n$ is cyclic. It is implicit in work of Dickson, and explicit in work of Szele, that $n$ is cyclic precisely when $gcd(n,phi(n))=1$. With $C(x)$ denoting the count of cyclic $nle x$, ErdH{o}s proved that $$C(x) sim e^{-gamma} x/logloglog{x}, quadtext{as $xtoinfty$}.$$ We show that $C(x)$ has an asymptotic series expansion, in the sense of Poincare, in descending powers of $logloglog{x}$, namely $$frac{e^{-gamma} x}{logloglog{x}} left(1-frac{gamma}{logloglog{x}} + frac{gamma^2 + frac{1}{12}pi^2}{(logloglog{x})^2} - frac{gamma^3 +frac{1}{4} gamma pi^2 + frac{2}{3}zeta(3)}{(logloglog{x})^3} + dots right). $$
In this article we explicitly describe irreducible trinomials X^3-aX+b which gives all the cyclic cubic extensions of Q. In doing so, we construct all integral points (x,y,z) with GCD(y,z)=1, of the curves X^2+3Y^2 = 4DZ^3 and X^2+27Y^2=4DZ^3 as D varies over cube-free positive integers. We parametrise these points using well known parametrisation of integral points (x,y,z) of the curve X^2+3Y^2=4Z^3 with GCD(y,z)=1.
Let $G$ be a finite abelian group. We say that $M$ and $S$ form a textsl{splitting} of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $min M$ and $sin S$, while $0$ has no such representation. The splitting is called textit{purely singular} if for each prime divisor $p$ of $|G|$, there is at least one element of $M$ is divisible by $p$. In this paper, we mainly study the purely singular splittings of cyclic groups. We first prove that if $kge3$ is a positive integer such that $[-k+1, ,k]^*$ splits a cyclic group $mathbb{Z}_m$, then $m=2k$. Next, we have the following general result. Suppose $M=[-k_1, ,k_2]^*$ splits $mathbb{Z}_{n(k_1+k_2)+1}$ with $1leq k_1< k_2$. If $ngeq 2$, then $k_1leq n-2$ and $k_2leq 2n-5$. Applying this result, we prove that if $M=[-k_1, ,k_2]^*$ splits $mathbb{Z}_m$ purely singularly, and either $(i)$ $gcd(s, ,m)=1$ for all $sin S$ or $(ii)$ $m=2^{alpha}p^{beta}$ or $2^{alpha}p_1p_2$ with $alphageq 0$, $betageq 1$ and $p$, $p_1$, $p_2$ odd primes, then $m=k_1+k_2+1$ or $k_1=0$ and $m=k_2+1$ or $2k_2+1$.
In [13], K. Roth showed that the expected value of the $L^2$ discrepancy of the cyclic shifts of the $N$ point van der Corput set is bounded by a constant multiple of $sqrt{log N}$, thus guaranteeing the existence of a shift with asymptotically minimal $L^2$ discrepancy, [11]. In the present paper, we construct a specific example of such a shift.
We study the initial value problem for the wave maps defined on the cyclic spacetime with the target Riemannian manifold that is responsive (see definition of the self coherence structure) to the parametric resonance phenomena. In particular, for arbitrary small and smooth initial data we construct blowing up solutions of the wave map if the metric of the base manifold is periodic in time.
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