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Register a new userA generalization of Levin-Schnorrs theorem
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Authors
Keita Yokoyama
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In this paper, we will generalize the definition of partially random or complex reals, and then show the duality of random and complex, i.e., a generalized version of Levin-Schnorrs theorem. We also study randomness from the view point of arithmetic using the relativization to a complete $Pi^0_1$-class.
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Caratheodory showed that $n$ complex numbers $c_1,...,c_n$ can uniquely be written in the form $c_p=sum_{j=1}^m rho_j {epsilon_j}^p$ with $p=1,...,n$, where the $epsilon_j$s are different unimodular complex numbers, the $rho_j$s are strictly positive numbers and integer $m$ never exceeds $n$. We give the conditions to be obeyed for the former property to hold true if the $rho_j$s are simply required to be real and different from zero. It turns out that the number of the possible choices of the signs of the $rho_j$s are {at most} equal to the number of the different eigenvalues of the Hermitian Toeplitz matrix whose $i,j$-th entry is $c_{j-i}$, where $c_{-p}$ is equal to the complex conjugate of $c_{p}$ and $c_{0}=0$. This generalization is relevant for neutron scattering. Its proof is made possible by a lemma - which is an interesting side result - that establishes a necessary and sufficient condition for the unimodularity of the roots of a polynomial based only on the polynomial coefficients. Keywords: Toeplitz matrix factorization, unimodular roots, neutron scattering, signal theory, inverse problems. PACS: 61.12.Bt, 02.30.Zz, 89.70.+c, 02.10.Yn, 02.50.Ga
Let $n$ be a positive integer. In 1915, Theisinger proved that if $nge 2$, then the $n$-th harmonic sum $sum_{k=1}^nfrac{1}{k}$ is not an integer. Let $a$ and $b$ be positive integers. In 1923, Nagell extended Theisingers theorem by showing that the reciprocal sum $sum_{k=1}^{n}frac{1}{a+(k-1)b}$ is not an integer if $nge 2$. In 1946, ErdH{o}s and Niven proved a theorem of a similar nature that states that there is only a finite number of integers $n$ for which one or more of the elementary symmetric functions of $1,1/2, ..., 1/n$ is an integer. In this paper, we present a generalization of Nagells theorem. In fact, we show that for arbitrary $n$ positive integers $s_1, ..., s_n$ (not necessarily distinct and not necessarily monotonic), the following reciprocal power sum $$sumlimits_{k=1}^{n}frac{1}{(a+(k-1)b)^{s_{k}}}$$ is never an integer if $nge 2$. The proof of our result is analytic and $p$-adic in character.
In this paper we first prove a version of $L^{2}$ existence theorem for line bundles equipped a singular Hermitian metrics. Aa an application, we establish a vanishing theorem which generalizes the classical Nadel vanishing theorem.
In 1972, Chvatal gave a well-known sufficient condition for a graphical sequence to be forcibly hamiltonian, and showed that in some sense his condition is best possible. Nash-Williams gave examples of forcibly hamiltonian n-sequences that do not satisfy Chvatlas condition for every n at least 5. In this note we generalize the Nash-Williams examples, and use this generalization to generate Omega(2^n/n^.5) forcibly hamiltonian n-sequences that do not satisfy Chvatals condition
We prove a new duality theorem for the category of precontact algebras which implies the Stone Duality Theorem, its connected version obtained in arXiv:1508.02220v3, 1-44 (to appear in Topology Appl.), the recent duality theorems of Bezhanishvili, G., Bezhanishvili, N., Sourabh, S., Venema, Y. and Goldblatt, R. and Grice, M, and some new duality theorems for the category of contact algebras and for the category of complete contact algebras.
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