To say that R is a root set modulo n means that R is a subset of Z n , the ring of integers modulo n , and there is a polynomial the roots of which modulo n are exactly the elements of R . Note that [ and Z n are always root sets modulo n . It seems that only two papers have appeared which mention the nature of root sets modulo n , and then only at a very basic level : Sierpin ́ ski [3] and Choj...

We prove that every two-player nonzero–sum stopping game in discrete time admits an ε-equilibrium in randomized strategies for every ε > 0. We use a stochastic variation of Ramsey's theorem, which enables us to reduce the problem to that of studying properties of ε-equilibria in a simple class of stochastic games with finite state space. 1. Introduction. The following optimization problem was p...

In this note, we establish $L^p$-bounds for the semigroup $e^{-tq^w(x,D)}$, $t \ge 0$, generated by a quadratic differential operator $q^w(x,D)$ on $\mathbb{R}^n$ that is Weyl quantization of complex-valued form $q$ defined phase space $\mathbb{R}^{2n}$ with non-negative real part $\operatorname{Re} q 0$ and trivial singular space. Specifically, show $e^{-tq^w(x,D)}$ bounded from $L^p(\mathbb{R...

Proof of Observation 1. Before the proof, we remind that what we want to prove is the following statement: (∀ε ∈ R) ε > 0⇒ ((∃N ∈ N)(∀n ∈ N) n ≥ N ⇒ |an − x | < ε). To this end, choose arbitrary positive real ε > 0 and pick a positive rational ε′ with 0 < ε′ < ε by utilizing the Archimedean property. Then there exists N (depending on ε′ and hence on ε) such that j, k ≥ N ⇒ |a j − ak | < ε′. Now...

For positive integers r, k0, k1, ..., kr−1, the van der Waerden number w(k0, k1, ..., kr−1) is the least positive integer n such that whenever {1, 2, . . . , n} is partitioned into r sets S0, S1, ..., Sr−1, there is some i so that Si contains a ki-term arithmetic progression. We find several new exact values of w(k0, k1, ..., kr−1). In addition, for the situation in which only one value of ki d...