Search results for "lower bounds"
showing 10 items of 259 documents
On Inverse Distance Weighting in Pollution Models
2011
When evaluating the impact of pollution, measurements from remote stations are often weighted by the inverse of distance raised to some nonnegative power (IDW). This is derived from Shepard's method of spatial interpolation (1968). The paper discusses the arbitrary character of the exponent of distance and the problem of monitoring stations that are close to the reference point. From elementary laws of physics, it is determined which exponent of distance should be chosen (or its upper bound) depending on the form of pollution encountered, such as radiant pollution (including radioactivity and sound), air pollution (plumes, puffs, and motionless clouds by using the classical Gaussian model),…
Pollution models and inverse distance weighting: some critical remarks
2013
International audience; When evaluating the impact of pollution, measurements from remote stations are often weighted by the inverse of distance raised to some nonnegative power (IDW). This is derived from Shepard's method of spatial interpolation (1968). The paper discusses the arbitrary character of the exponent of distance and the problem of monitoring stations that are close to the reference point. From elementary laws of physics, it is determined which exponent of distance should be chosen (or its upper bound) depending on the form of pollution encountered, such as radiant pollution (including radioactivity and sound), air pollution (plumes, puffs, and motionless clouds by using the cl…
Upper bounds for the zeros of ultraspherical polynomials
1990
AbstractFor k = 1, 2, …, [n2] let xnk(λ) denote the Kth positive zero in decreasing order of the ultraspherical polynomial Pn(λ)(x). We establish upper bounds for xnk(λ). All the bounds become exact when λ = 0 and, in some cases (see case (iii) of Theorem 3.1), also when λ = 1. As a consequence of our results, we obtain for the largest zero xn1(λ)0.. We point out that our results remain useful for large values of λ. Numerical examples show that our upper bounds are quite sharp.
Quantum Property Testing for Bounded-Degree Graphs
2011
We study quantum algorithms for testing bipartiteness and expansion of bounded-degree graphs. We give quantum algorithms that solve these problems in time O(N^(1/3)), beating the Omega(sqrt(N)) classical lower bound. For testing expansion, we also prove an Omega(N^(1/4)) quantum query lower bound, thus ruling out the possibility of an exponential quantum speedup. Our quantum algorithms follow from a combination of classical property testing techniques due to Goldreich and Ron, derandomization, and the quantum algorithm for element distinctness. The quantum lower bound is obtained by the polynomial method, using novel algebraic techniques and combinatorial analysis to accommodate the graph s…
A note on the Schur multiplier of a nilpotent Lie algebra
2011
For a nilpotent Lie algebra $L$ of dimension $n$ and dim$(L^2)=m$, we find the upper bound dim$(M(L))\leq {1/2}(n+m-2)(n-m-1)+1$, where $M(L)$ denotes the Schur multiplier of $L$. In case $m=1$ the equality holds if and only if $L\cong H(1)\oplus A$, where $A$ is an abelian Lie algebra of dimension $n-3$ and H(1) is the Heisenberg algebra of dimension 3.
A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient
2019
Let $\Omega \subset \mathbb R^N$, $N \geq 2$, be a smooth bounded domain. We consider a boundary value problem of the form $$-\Delta u = c_{\lambda}(x) u + \mu(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(\Omega)\cap L^{\infty}(\Omega)$$ where $c_{\lambda}$ depends on a parameter $\lambda \in \mathbb R$, the coefficients $c_{\lambda}$ and $h$ belong to $L^q(\Omega)$ with $q>N/2$ and $\mu \in L^{\infty}(\Omega)$. Under suitable assumptions, but without imposing a sign condition on any of these coefficients, we obtain an a priori upper bound on the solutions. Our proof relies on a new boundary weak Harnack inequality. This inequality, which is of independent interest, is established in the gener…
Devroye Inequality for a Class of Non-Uniformly Hyperbolic Dynamical Systems
2005
In this paper, we prove an inequality, which we call "Devroye inequality", for a large class of non-uniformly hyperbolic dynamical systems (M,f). This class, introduced by L.-S. Young, includes families of piece-wise hyperbolic maps (Lozi-like maps), scattering billiards (e.g., planar Lorentz gas), unimodal and H{\'e}non-like maps. Devroye inequality provides an upper bound for the variance of observables of the form K(x,f(x),...,f^{n-1}(x)), where K is any separately Holder continuous function of n variables. In particular, we can deal with observables which are not Birkhoff averages. We will show in \cite{CCS} some applications of Devroye inequality to statistical properties of this class…
Global fixed point proof of time-dependent density-functional theory
2011
We reformulate and generalize the uniqueness and existence proofs of time-dependent density-functional theory. The central idea is to restate the fundamental one-to-one correspondence between densities and potentials as a global fixed point question for potentials on a given time-interval. We show that the unique fixed point, i.e. the unique potential generating a given density, is reached as the limiting point of an iterative procedure. The one-to-one correspondence between densities and potentials is a straightforward result provided that the response function of the divergence of the internal forces is bounded. The existence, i.e. the v-representability of a density, can be proven as wel…
On the slope of hyperelliptic fibrations with positive relative irregularity
2016
Let $f:\, S \to B$ be a locally non-trivial relatively minimal fibration of hyperelliptic curves of genus $g\geq 2$ with relative irregularity $q_f$. We show a sharp lower bound on the slope $\lambda_f$ of $f$. As a consequence, we prove a conjecture of Barja and Stoppino on the lower bound of $\lambda_f$ as an increasing function of $q_f$ in this case, and we also prove a conjecture of Xiao on the ampleness of the direct image of the relative canonical sheaf if $\lambda_f<4$.
Notes on the subspace perturbation problem for off-diagonal perturbations
2014
The variation of spectral subspaces for linear self-adjoint operators under an additive bounded off-diagonal perturbation is studied. To this end, the optimization approach for general perturbations in [J. Anal. Math., to appear; arXiv:1310.4360 (2013)] is adapted. It is shown that, in contrast to the case of general perturbations, the corresponding optimization problem can not be reduced to a finite-dimensional problem. A suitable choice of the involved parameters provides an upper bound for the solution of the optimization problem. In particular, this yields a rotation bound on the subspaces that is stronger than the previously known one from [J. Reine Angew. Math. (2013), DOI:10.1515/cre…