Related papers: Linear estimates for solutions of quadratic equati…

We provide polynomial upper bounds on the size of a shortest solution for quadratic equations in a free group. A similar bound is given for parametric solutions in the description of solutions sets of quadratic equations in a free group.

We prove that in a torsion-free hyperbolic group $\Gamma$, the length of the value of each variable in a minimal solution of a quadratic equation $Q=1$ is bounded by $N|Q|^3$ for an orientable equation, and by $N|Q|^{4}$ for a…

Let $G$ be a non-trivial torsion free group and $t$ be an unknown. In this paper we consider three equations (over $G$) of arbitrary length and show that they have a solution (over $G$) provided two relations among their coefficients hold.…

We prove that the Diophantine problem for orientable quadratic equations in free metabelian groups is decidable and furthermore, NP-complete. In the case when the number of variables in the equation is bounded, the problem is decidable in…

Let Q be a non-singular diagonal quadratic form in at least four variables. We provide upper bounds for the number of integer solutions to the equation Q=0, which lie in a box with sides of length 2B, as B tends to infinity. The estimates…

An optimum solution free from degeneration is found to the system of linear algebraic equations with empirical coefficients and right-hand sides. The quadratic risk of estimators of the unknown solution vector is minimized over a class of…

Let $s \geq 3$ be a natural number, let $\psi(x)$ be a polynomial with real coefficients and degree $d \geq 2$, and let $A$ be some large, non-empty, finite subset of real numbers. We use $E_{s,2}(A)$ to denote the number of solutions to…

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…

We derive an upper bound for the least number of variables needed to guarantee that a system of t quadratic forms (t>=2) over a field F has a nontrivial zero. In particular, if F is a local field, then 2t^2+3 variables insure the existence…

We give an asymptotic for the number of prime solutions to $Q(x_1,\dots, x_8) = N$, subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$. The argument initially proceeds via the circle method, but this does not…

We establish an asymptotic formula for the number of integral solutions of bounded height for pairs of diagonal quartic equations in $26$ or more variables. In certain cases, pairs in $25$ variables can be handled.

Let c(d) denote the minimal composition length of all finite solvable groups with solvable (or derived) length d. We prove that c(d) equals 0,1,2,4,5,7,8,13,15 when d=0,1,2,3,4,5,6,7,8.

We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete.

For $g\geq1$ denote by $F_{2g}=\langle x_1, y_1,\dots,x_g,y_g\rangle$ the free group on $2g$ generators and by $B_g=[x_1,y_1]\dots[x_g,y_g]$. For $l,c\geq 1$ and elements $w_1,\dots,w_l\in F_{2g}$ we study orientable quadratic equations of…

For a finite group generated by involutions, the involution width is defined to be the minimal $k\in\mathbb{N}$ such that any group element can be written as a product of at most $k$ involutions. We show that the involution width of every…

In this paper, we consider linear quadratic team problems with an arbitrary number of quadratic constraints in both stochastic and deterministic settings. The team consists of players with different measurements about the state of nature.…

We prove that, for every integer $n \ge 2$, a finite or infinite countable group $G$ can be embedded into a 2-generated group $H$ in such a way that the solvability of quadratic equations of length at most $n$ is preserved, i.e., every…

A word equation with one variable in a free group is given as $U = V$, where both $U$ and $V$ are words over the alphabet of generators of the free group and $X, X^{-1}$, for a fixed variable $X$. An element of the free group is a solution…

Analytical solutions of variable coefficient nonlinear Schr\"odinger equations having four-dimensional symmetry groups which are in fact the next closest to the integrable ones occurring only when the Lie symmetry group is five-dimensional…

Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of non-abelian simple groups. The minimum number of nonsolvable factors, attained on all possible such series in $G$, is called…