Related papers: Pairs of diagonal quartic forms: the non-singular …

We establish a microscopic convexity principle for nonlinear elliptic and parabolic partial differential equations in general form.

We use the invariant theory of binary quartics to give a new formula for the Cassels-Tate pairing on the $2$-Selmer group of an elliptic curve. Unlike earlier methods, our formula does not require us to solve any conics. An important role…

In this paper we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be extended to solve simultaneous homogeneous polynomial diophantine…

We establish an analytic Hasse principle for linear spaces of affine dimension m on a complete intersection over an algebraic field extension K of Q. The number of variables required to do this is no larger than what is known for the…

Working on Berkovich analytic curves, we propose a geometric approach to the study of the Hasse principle over function fields of curves defined over a complete discretely valued field. Using it, we show the Hasse principle to be verified…

In this paper we establish a Hasse principle concerning the linear dependence over $\Z$ of nontorsion points in the Mordell-Weil group of an abelian variety over a number field.

Let F be a field of characteristic different from 2. The u-invariant and the Hasse number of a field F are classical and important field invariants pertaining to quadratic forms. These invariants measure the suprema of dimensions of…

We obtain an essentially optimal estimate for the moment of order 32/3 of the exponential sum having argument $\alpha x^3+\beta x^2$. Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine…

Let $F$ be a global field, let $\vp \in \Fx$ be a rational map of degree at least 2, and let $\a \in F$. We say that $\a $ is periodic if $\vpn (\a) = \a$ for some $n \geq 1$. A Hasse principle is the idea, or hope, that a phenomenon which…

When all ternary cubic forms over $\mathbb Z$ are ordered by the heights of their coefficients, we show that a positive proportion of them fail the Hasse principle, i.e., they have a zero over every completion of $\mathbb Q$ but no zero…

We establish improved mean value estimates associated with the number of integer solutions of certain systems of diagonal equations, in some instances attaining the sharpest conjectured conclusions. This is the first occasion on which…

We investigate pairs of diagonal cubic equations with integral coefficients. For a class of such Diophantine systems with 11 or more variables, we are able to establish that the number of integral solutions in a large box is at least as…

A method for designing variational principles for the dynamics of a possibly dissipative and non-conservatively forced chain of particles is demonstrated. Some qualitative features of the formulation are discussed.

We investigate the entanglement properties in the symmetric subspace of $N$-partite $d$-dimensional systems (qudits). For diagonal symmetric states, we show that there is no bound entanglement for $d = 3,4 $ and $N = 3$. Further, we present…

In this paper, we derive a formula to express the maximum number of non-intersecting diagonals of arbitrary length that can be drawn in n x n square arrays, where n is a multiple of l+1.

In this work, we give a variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type. We cover several cases of differential equations with deviated…

We show that the number of lines contained in a supersingular quartic surface is 40 or at most 32, if the characteristic of the field equals 2, and it is 112, 58, or at most 52, if the characteristic equals 3. If the quartic is not…

We consider equations of the form $a_{1}x_{1}^{k}+...+a_{s}x_{s}^{k}$ and when they have solutions in the primes. We define an analogue of the Hasse principle for solubility in the primes (which we call the prime Hasse principle), and prove…

In this article, we obtain an upper bound for the number of integral solutions, of given height, of system of two quadratic forms in five variables. Our bound is an improvement over the bound given by Henryk Iwaniec and Ritabrata Munshi in…

A method for constructing the Lax pairs for nonlinear integrable models is suggested. First we look for a nonlinear invariant manifold to the linearization of the given equation. Examples show that such invariant manifold does exist and can…