Explicit Bound Research Articles

Nonexistence of Large Nuclei in the Liquid Drop Model

We give a simplified proof of the nonexistence of large nuclei in the liquid drop model and provide an explicit bound. Our bound is within a factor of 2.3 of the conjectured value and seems to be the first quantitative result.

Nonultralocality and causality in the relational framework of canonical quantum gravity

The relational framework of canonical quantum gravity with nonultralocal constraints is explored. After demonstrating the absence of anomalies, a spatially discretized version of the relational framework is introduced. This allows the application of Lieb-Robinson bounds to on-shell monotonic gauge flow when there is a continuous external ``time'' parameter. An explicit Lieb-Robinson bound is derived for the differential on-shell evolution of the operator norm of the commutator of discretized Dirac observables, demonstrating how a local light conelike causal structure emerges. Ultralocal constraints do not permit such a structure to arise via Lieb-Robinson bounds. Gauge and ($3+1$)-diffeomorphism invariance of the light cone is discussed along with the issues of quantum fluctuations, the nature of the nonlocalities, the spatial continuum limit, and the possible links to noncommutative geometry.

Möbius bilipschitz homogeneous arcs on the plane

A mobius bilipschitz mapping is an η-quasimobius mapping with the linear distortion function η(t) = Kt. We show that if an open Jordan arc γ ⊂ C with distinct endpoints a and b is homogeneous with respect to the family F K of mobius bilipschitz automorphisms of the sphere C with K specified then γ has bounded turning RT(γ) in the sense of Rickman and, consequently, γ is a quasiconformal image of a rectilinear segment. The homogeneity of γ with respect to F K means that for all x, y ∈ γ {a, b} there exists f ∈ F K with f(γ) = γ and f(x) = y. In order to estimate RT(γ) from above, we introduce the condition BR(δ) of bounded rotation of γ, and then the explicit bound depends only on K and δ.

Courant-sharp eigenvalues of the three-dimensional square torus

In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus ( R / Z ) 3 (\mathbb {R}/\mathbb {Z})^3\, , all the eigenvalues having an eigenfunction which satisfies the Courant nodal domain theorem with equality (Courant-sharp situation). Following the strategy of Å. Pleijel (1956), the proof is a combination of an explicit lower bound of the counting function and a Faber–Krahn-type inequality for domains in the torus, deduced, as in the work of P. Bérard and D. Meyer (1982), from an isoperimetric inequality. This inequality relies on the work of L. Hauswirth, J. Pérez, P. Romon, and A. Ros (2004) on the periodic isoperimetric problem.

Comparison of Offset Approximation Methods of Conics with Explicit Error Bounds

In this paper the approximation methods of offset curve of conic with explicit error bound are considered. The quadratic approximation of conic(QAC) method, the method based on quadratic circle approximation(BQC) and the Pythagorean hodograph cubic(PHC) approximation have the explicit error bound for approximation of offset curve of conic. We present the explicit upper bound of the Hausdorff distance between the offset curve of conic and its PHC approximation. Also we show that the PHC approximation of any symmetric conic is closer to the line passing through both endpoints of the conic than the QAC.

Universal Components of Random Nodal Sets

We give, as $L$ grows to infinity, an explicit lower bound of order $L^{n/m}$ for the expected Betti numbers of the vanishing locus of a random linear combination of eigenvectors of $P$ with eigenvalues below $L$. Here, $P$ denotes an elliptic self-adjoint pseudo-differential operator of order $m\textgreater{}0$, bounded from below and acting on the sections of a Riemannian line bundle over a smooth closed $n$-dimensional manifold $M$ equipped with some Lebesgue measure. In fact, for every closed hypersurface $\Sigma$ of $\mathbb R^n$, we prove that there exists a positive constant $p\_\Sigma$ depending only on $\Sigma$, such that for every large enough $L$ and every $x\in M$, a component diffeomorphic to $\Sigma$ appears with probability at least $p\_\Sigma$ in the vanishing locus of a random section and in the ball of radius $L^{-1/m}$ centered at $x$. These results apply in particular to Laplace-Beltrami and Dirichlet-to-Neumann operators.

Indivisibility of class numbers of real quadratic function fields

In this paper we work on indivisibility of the class numbers of real quadratic function fields. We find an explicit expression for a lower bound of the density of real quadratic function fields (with constant field F) whose class numbers are not divisible by a given prime ℓ. We point out that the explicit lower bound of such a density we found only depends on the prime ℓ, the degrees of the discriminants of real quadratic function fields, and the condition: either |F|≡1(modℓ) or not.

Explicit upper bound for an average number of divisors of quadratic polynomials

Consider the divisor sum \({\sum_{n\leq N} \tau(n^2+2bn+c)}\) for integers b and c which satisfy certain extra conditions. For this average sum, we obtain an explicit upper bound, which is close to the optimal. As an application we improve the maximal possible number of \({D(-1)}\)-quadruples.

Stability of nonlocal fractional Langevin differential equations involving fractional integrals

In this paper, we investigate Ulam-Hyers stability, Practical-Ulam-Hyers stability and Practical-Ulam-Hyers-Rassias stability of nonlocal fractional Langevin equations involving Riemann-Liouville fractional integral with positive constant coefficients. By utilizing an explicit bound of Mittag-Leffler function with two parameters and analytical methods, the practical stability results are presented via a generalized Gronwall inequality. Finally, two examples are given to illustrate our theoretical results.

An Explicit upper Bound of the Argument of Dirichlet <i>L</i>-functions on the Generalized Riemann Hypothesis

We prove an explicit upper bound of the function ( ) , St χ , defined by the argument of Dirichlet L-functions attached to a primitive Dirichlet character χ (mod q > 1). An explicit upper bound of the function S(t), defined by the argument of the Riemann zeta-function, have been obtained by A. Fujii (1). Our result is obtained by applying the idea of Fujii's result on S(t). The constant part of the explicit upper bound of ( ) , St χ in this paper does not depend on χ . Our proof does not cover the case q = 1 and indeed gives a better bound than the one of Fujii that covers the case q = 1.

Parity sheaves and tilting modules

We show that tilting modules and parity sheaves on the affine Grassmannian are related through the geometric Satake correspondence, when the characteristic is bigger than an explicit bound.

Time-optimal control of a self-propelled particle in a spatiotemporal flow field

ABSTRACTWe address a minimum-time problem that constitutes an extension of the classical Zermelo navigation problem in higher dimensions. In particular, we address the problem of steering a self-propelled particle to a prescribed terminal position with free terminal velocity in the presence of a spatiotemporal flow field. Furthermore, we assume that the norm of the rate of change of the particle's velocity relative to the flow is upper bounded by an explicit upper bound. To address the problem, we first employ Pontryagin's minimum principle to parameterise the set of candidate time-optimal control laws in terms of a parameter vector that belongs to a compact set. Subsequently, we develop a simple numerical algorithm for the computation of the minimum time-to-come function that is tailored to the particular parametrisation of the set of the candidate time-optimal control laws of our problem. The proposed approach bypasses the task of converting the optimal control problem to a parameter optimisation problem, which can be computationally intense, especially when one is interested in characterising the optimal synthesis of the minimum-time problem. Numerical simulations that illustrate the theoretical developments are presented.

Explicit lower bound of blow-up time in a fully parabolic chemotaxis system with nonlinear cross-diffusion

This paper detects the lower bounds of blow-up time of smooth solutions for the chemotaxis model{ut=Δu−χ∇⋅(u(u+1)m−1∇v),x∈B1(0),t>0,vt=Δv−v+u,x∈B1(0),t>0, under homogeneous Neumann boundary conditions in a unit ball B1(0)⊂R3 centered at the origin, with positive constant χ and parameter m∈R. Under the assumption that (u(x,0),v(x,0))=(u0(|x|),v0(|x|))∈C0(B¯1(0))×W1,∞(B1(0)), it is shown that whenever m∈[23,2], the blow-up time of a classical solution to the corresponding initial–boundary problem has an explicit lower bound measured in terms of χ, ∫B1(0)u0p and ∫B1(0)|∇v0|2q for appropriate p>1 and q>1. Here we underline that the global classical solution exists and is bounded if m<23, which leads to the assumption m≥23 for addressing the properties of blow-up solutions. However, the question of lower bounds of blow-up time for the case m>2 remains open due to technical reasons.

Newton polygons of L functions of polynomials xd + ax

Let p be a prime number and q=ph. For f(x)=xd+ax∈Fq[x](a≠0), we obtain the slopes of the Newton polygons of the L-functions of the exponential sums associated to f(x) for any nontrivial finite character χ. For χ of order p, our result recovers Zhu's genericity result [10] by giving p an explicit bound. The general χ case is based on improvement of results of Davis–Wan–Xiao [2].

Degrees of irreducible characters of the symmetric group and exponential growth

We consider sequences of degrees of ordinary irreducible S n S_n - characters. We assume that the corresponding Young diagrams have rows and columns bounded by some linear function of n n with leading coefficient less than one. We show that any such sequence has at least exponential growth and we compute an explicit bound.

An Integer Programming-Based Bound for Locally Repairable Codes

The locally repairable code (LRC) studied in this paper is an $[n,k]$ linear code of which the value at each coordinate can be recovered by a linear combination of at most $r$ other coordinates. The central problem in this work is to determine the largest possible minimum distance for LRCs. First, an integer programming based upper bound is derived for any LRC. Then by solving the programming problem under certain conditions, an explicit upper bound is obtained for LRCs with parameters $n_1>n_2$, where $n_1 = \left\lceil \frac{n}{r+1} \right\rceil$ and $n_2 = n_1 (r+1) - n$. Finally, an explicit construction for LRCs attaining this upper bound is presented over the finite field $\mathbb{F}_{2^m}$, where $m\geq n_1r$. Based on these results, the largest possible minimum distance for all LRCs with $r \le \sqrt{n}-1$ has been definitely determined, which is of great significance in practical use.

Consensus tracking in sensor networks with periodic sensing and switching connected topologies

This paper studies the consensus tracking of general linear dynamical target in periodically sensing networks with switching and connected sensing topologies. Allowable upper bounds of sensing period are investigated for both time-invariant sensing and time-varying sensing networks. For tracking system with constant sensing period, discretization approach is applied and the tracking problem under switching connected topologies is converted to the robust stability problem of a discrete-time uncertain system. Then an allowable upper bound of sensing period is given by solving optimal H∞ control problem, and an explicit bound, which is composed of the eigenvalues of target’s dynamic matrix and topology matrix, is further provided. For tracking system with time-varying sensing period, input-delay approach is applied and a bound is given by using Lyapunov–Krasovskii functional analysis and solving the feasibility of three LMIs. Numerical examples are given to illustrate the results.

Explicit estimates for the number of rational points of singular complete intersections over a finite field

Let V⊂Pn(F‾q) be a complete intersection defined over a finite field Fq of dimension r and singular locus of dimension at most 0≤s≤r−2. We obtain an explicit version of the Hooley–Katz estimate ||V(Fq)|−pr|=O(q(r+s+1)/2), where |V(Fq)| denotes the number of Fq-rational points of V and pr:=|Pr(Fq)|. Our estimate improves all the previous estimates in several important cases. Our approach relies on tools of classical algebraic geometry. A crucial ingredient is a new effective version of the Bertini smoothness theorem, namely an explicit upper bound of the degree of a proper Zariski closed subset of (Pn)s+1(F‾q) which contains all the singular linear sections of V of codimension s+1.

Convergence rate of CLT for the estimation of Hurst parameter of fractional Brownian motion

In this paper, we consider the convergence of the estimator of the Hurst parameter H of a fractional Brownian motion with Hurst parameter H∈(0,1). The main goal of our work is to derive an explicit upper bound of Kolmogorov distance in order to find the information about the rate of convergence of the central limit theorems (CLT) for the estimator of the Hurst parameter.

An explicit upper bound of the number of negative eigenvalues associated to an elliptic operator

The purpose of the present article is to provide explicitly an upper bound of the number of negative eigenvalues of an elliptic operator defined on a compact manifold.