In this article we consider the linear stability of the spherically-symmetric stationary solutions of the Schrödinger-Newton equations. These have been found numerically by Moroz et al [9] and Bernstein et al [3]. The ground state, characterised as the state of lowest energy , turns out to be linearly stable, with only imaginary eigenvalues. The (n + 1)-th state is linearly unstable having n qu...

Let H be a 4-uniform hypergraph on an n-element vertex set V containing no 4-book of 3 pages, i.e., a hypergraph of 4 quadruples with vertices {1, 2, . . . , 7} and edges {1234, 1235, 1236, 4567}. Then for n>n0 e(H) ( n/2 2 )( n/2 2 ) . Moreover, here equality is possible only if V (H) can be partitioned into two sets A and B so that each quadruple of H intersects A (and B) in exactly two verti...

A set of m positive integers is called a Diophantine m-tuple if the product of any two of them is one less than a perfect square. It is known that there does not exist a Diophantine sextuple and that there are only finitely many Diophantine quintuples. On the other hand, there are infinitely many Diophantine m-tuples for m = 2, 3 and 4. In this paper, we derive asymptotic extimates for the numb...

We study question answering as a machine learning problem, and induce a function that maps open-domain questions to queries over a database of web extractions. Given a large, community-authored, question-paraphrase corpus, we demonstrate that it is possible to learn a semantic lexicon and linear ranking function without manually annotating questions. Our approach automatically generalizes a see...

A complex diophantine quadruple with the property D (z), where z ∈ Z[√−2], is a subset of Z[√−2] of four elements such that the product of its any two distinct elements increased by z is a perfect square in Z[ √−2]. In the present paper we prove that if b is an odd integer, then there does not exist a diophantine quadruple with the property D(a + b √−2). For z = a + b√−2, where b is even, we pr...

We take the point of view that building a one-way bridge from experimental data to mathematical models instead other way around avoids running into controversies resulting attaching meaning symbols used in latter. In particular, we show adopting this offers new perspectives for constructing and interpreting results Einstein–Podolsky–Rosen–Bohm experiments. first prove Bell-type inequalities con...

The Greek mathematician Diophantus of Alexandria noted that the numbers x, x + 2, 4x + 4 and 9x + 6, where x = 1 16 , have the following property: the product of any two of them increased by 1 is a square of a rational number (see [4]). Fermat first found a set of four positive integers with the above property, and it was {1, 3, 8, 120}. Later, Davenport and Baker [3] showed that if d is a posi...

We introduce and study a modal logic wK4F which is closely related to the logic S4F that is important in the context of epistemic logics for representing and reasoning about an agent’s knowledge. It is obtained by adding the axiom F to the modal logic wK4, or dropping from S4F the T axiom. We show that wK4F is sound and complete with respect to the class of all minimal topological spaces i.e. t...

Necessary and sufficient conditions will be found for quadruples of lattices to be isomorphic to lattices of weak, relative, strong subalgebras and initial segments, respectively, of one partial unary algebra. To this purpose we will start with a characterization of pairs of lattices that are weak and strong subalgebra lattices of one partial unary algebra, respectively. Next, we will describe ...