Related papers: The Hasse principle for systems of diagonal cubic …
Let $L$ be a finite extension of $\mathbb{F}_q(t)$. We calculate the proportion of polynomials of degree $d$ in $\mathbb{F}_q[t]$ that are everywhere locally norms from $L/\mathbb{F}_q(t)$ which fail to be global norms from…
The maximum principle for SPDEs is established in multidimensional $C^{1}$ domains. An application is given to proving the H\"older continuity up to the boundary of solutions of one-dimensional SPDEs.
The paper deals with some generalizations of Fibonacci and Lucas sequences, arising from powers of paths and cycles, respectively. In the first part of the work we provide a formula for the number of edges of the Hasse diagram of the…
Let $f(\mathbf x)$ be a non-singular quadratic form with sufficiently many mixed terms and $t$ an integer. For a sequence of weights $\mathcal A$ we study the number of weighted solutions to $f(\mathbf x) = t$. In particular, we give…
We present a finite-order system of recurrence relations for a permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix (for k = 1, 2, and 3) as well as the method for deriving such recurrence…
Denote by $s_0^{(r)}$ the least integer such that if $s \ge s_0^{(r)}$, and $F$ is a cubic form with real coefficients in $s$ variables that splits into $r$ parts, then $F$ takes arbitrarily small values at nonzero integral points. We bound…
We describe a practical algorithm for computing Brauer-Manin obstructions to the existence of rational points on hyperelliptic curves defined over number fields. This offers advantages over descent based methods in that its correctness does…
In this paper, we consider the convergence rate with respect to the Wasserstein distance in the invariance principle for sequential dynamical systems. We utilize and modify the techniques previously employed for stationary sequences to…
An avoidance problem of configurations in 4-cycle systems is investigated by generalizing the notion of sparseness, which is originally from Erd\H{o}s' r-sparse conjecture on Steiner triple systems. A 4-cycle system of order v, 4CS(v), is…
We prove a Tb theorem on quasimetric spaces equipped with what we call an upper doubling measure. This is a property that encompasses both the doubling measures and those satisfying the upper power bound \mu(B(x,r)) \le Cr^d. Our spaces are…
We show that the d-dimensional Haar system H^d on the unit cube I^d is a Schauder basis in the classical Besov space B_{p,q,1}^s(I^d), 0<p<1, defined by first order differences in the limiting case s=d(1/p-1), if and only if 0<q\le p. For…
We prove that for any t in Q, the curve 5 x^3 + 9 y^3 + 10 z^3 + 12((t^12-t^4-1)/(t^12-t^8-1))^3 (x+y+z)^3 = 0 in P^2 is a genus 1 curve violating the Hasse principle. An explicit Weierstrass model for its Jacobian E_t is given. The…
It is classically known that the circle method produces an asymptotic for the number of representations of a tuple of integers $(n_1,\ldots,n_R)$ by a system of quadratic forms $Q_1,\ldots, Q_R$ in $k$ variables, as long as $k$ is…
We focus on the second part of Hilbert's 16th problem and provide an upper bound on the number of limit cycles that a polynomial, differential, planar system may have, depending exclusively on the degree $n$ of the system. Such a bound…
Working on doubling metric spaces, we construct generalised dyadic cubes adapting ultrametric structure. If the space is complete, then the existence of such cubes and the mass distribution principle lead into a simple proof for the…
In this paper we formulate and prove a combinatorial version of the section conjecture for finite groups acting on finite graphs. We apply this result to the study of rational points and show that finite descent is the only obstruction to…
Takens Theorem for a partially hyperbolic dynamics provides a normal linearization along the center manifold. In this paper, we give the nonautonomous version of Takens Theorem under non-resonance conditions formulated in terms of the…
K. Kato's conjecture about the cohomological Hasse principle for regular connected schemes $\mathfrak X$ which are flat and proper over the complete discrete valuation rings $\mathcal O_N$ of higher local fields $F_N$ is proven. This…
A general sufficient condition for the convergence of subsequences of solutions of non-autonomous, nonlinear difference equations and systems is obtained. For higher order equations the delay sizes and patterns play essential roles in…
In this paper, we investigate the convergence rate of the averaging principle for stochastic differential equations (SDEs) with $\beta$-H\"older drift driven by $\alpha$-stable processes. More specifically, we first derive the Schauder…