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When programs feature a complex control flow, existing techniques for resource analysis produce cost relation systems (CRS) whose cost functions retain the complex flow of the program and, consequently, might not be solvable into closed-form upper bounds. This paper presents a novel approach to resource analysis that is driven by the result of a termination analysis. The fundamental idea is that the termination proof encapsulates the flows of the program which are relevant for the cost computation so that, by driving the generation of the CRS using the termination proof, we produce a linearly-bounded CRS (LB-CRS). A LB-CRS is composed of cost functions that are guaranteed to be locally bounded by linear ranking functions and thus greatly simplify the process of CRS solving. We have built a new resource analysis tool, named MaxCore, that is guided by the VeryMax termination analyzer and uses CoFloCo and PUBS as CRS solvers. Our experimental results on the set of benchmarks from the Complexity and Termination Competition 2019 for C Integer programs show that MaxCore outperforms all other resource analysis tools.
Ecological inference (EI) is the process of learning about individual behavior from aggregate data. We relax assumptions by allowing for “linear contextual effects,” which previous works have regarded as plausible but avoided due to nonidentification, a problem we sidestep by deriving bounds instead of point estimates. In this way, we offer a conceptual framework to improve on the Duncan–Davis bound, derived more than 65 years ago. To study the effectiveness of our approach, we collect and analyze 8,430
$2\times 2$
EI datasets with known ground truth from several sources—thus bringing considerably more data to bear on the problem than the existing dozen or so datasets available in the literature for evaluating EI estimators. For the 88% of real data sets in our collection that fit a proposed rule, our approach reduces the width of the Duncan–Davis bound, on average, by about 44%, while still capturing the true district-level parameter about 99% of the time. The remaining 12% revert to the Duncan–Davis bound.
We obtain the best possible upper bounds for the moments of a single-order statistic from independent, nonnegative random variables, in terms of the population mean. The main result covers the independent identically distributed case. Furthermore, the case of the sample minimum for merely independent (not necessarily identically distributed) random variables is treated in detail.
We consider a two-node queue modeled as a two-dimensional random walk. In particular, we consider the case that one or both queues have finite buffers. We develop an approximation scheme based on the Markov reward approach to error bounds in order to bound performance measures of such random walks. The approximation scheme is developed in terms of a perturbed random walk in which the transitions along the boundaries are different from those in the original model and the invariant measure of the perturbed random walk is of product-form. We then apply this approximation scheme to a tandem queue and some variants of this model, for the case that both buffers are finite. The modified approximation scheme and the corresponding applications for a two-node queueing system in which only one of the buffers has finite capacity have also been discussed.
The present paper deals with non-real eigenvalues of singular indefinite Sturm–Liouville problems with limit-circle type endpoints. A priori bounds and the existence of non-real eigenvalues of the problem associated with a special separated boundary condition are obtained.
Pesaran, Shin, and Smith (2001) (PSS) proposed a bounds procedure for testing for the existence of long run cointegrating relationships between a unit root dependent variable (
$y_{t}$
) and a set of weakly exogenous regressors
$\boldsymbol{x}_{t}$
when the analyst does not know whether the independent variables are stationary, unit root, or mutually cointegrated processes. This procedure recognizes the analyst’s uncertainty over the nature of the regressors but not the dependent variable. When the analyst is uncertain whether
$y_{t}$
is a stationary or unit root process, the test statistics proposed by PSS are uninformative for inference on the existence of a long run relationship (LRR) between
$y_{t}$
and
$\boldsymbol{x}_{t}$
. We propose the long run multiplier (LRM) test statistic as a means of testing for LRRs without knowing whether the series are stationary or unit roots. Using stochastic simulations, we demonstrate the behavior of the test statistic given uncertainty about the univariate dynamics of both
$y_{t}$
and
$\boldsymbol{x}_{t}$
, illustrate the bounds of the test statistic, and generate small sample and approximate asymptotic critical values for the upper and lower bounds for a range of sample sizes and model specifications. We demonstrate the utility of the bounds framework for testing for LRRs in models of public policy mood and presidential success.
In the present communication, we introduce quantile-based (dynamic) inaccuracy measures and study their properties. Such measures provide an alternative approach to evaluate inaccuracy contained in the assumed statistical models. There are several models for which quantile functions are available in tractable form, though their distribution functions are not available in explicit form. In such cases, the traditional distribution function approach fails to compute inaccuracy between two random variables. Various examples are provided for illustration purpose. Some bounds are obtained. Effect of monotone transformations and characterizations are provided.
In this paper, we obtain gradient estimates of the positive solutions to weighted p-Laplacian type equations with a gradient-dependent nonlinearity of the form
0.1
$${\rm div }( \vert x \vert ^\sigma \vert \nabla u \vert ^{p-2}\nabla u) = \vert x \vert ^{-\tau }u^q \vert \nabla u \vert ^m\quad {\rm in}\;\Omega^*: = \Omega {\rm \setminus }\{ 0\} .$$
Here,
$\Omega \subseteq {\open R}^N$
denotes a domain containing the origin with
$N\ges 2$
, whereas
$m,q\in [0,\infty )$
,
$1<p\les N+\sigma $
and
$q>\max \{p-m-1,\sigma +\tau -1\}$
. The main difficulty arises from the dependence of the right-hand side of (0.1) on x, u and
$ \vert \nabla u \vert $
, without any upper bound restriction on the power m of
$ \vert \nabla u \vert $
. Our proof of the gradient estimates is based on a two-step process relying on a modified version of the Bernstein's method. As a by-product, we extend the range of applicability of the Liouville-type results known for (0.1).
The purpose of this paper is to provide further understanding into the structure of the sequential allocation (“stochastic multi-armed bandit”) problem by establishing probability one finite horizon bounds and convergence rates for the sample regret associated with two simple classes of allocation policies. For any slowly increasing function g, subject to mild regularity constraints, we construct two policies (the g-Forcing, and the g-Inflated Sample Mean) that achieve a measure of regret of order O(g(n)) almost surely as n → ∞, bound from above and below. Additionally, almost sure upper and lower bounds on the remainder term are established. In the constructions herein, the function g effectively controls the “exploration” of the classical “exploration/exploitation” tradeoff.
It is well known that a weak solution φ to the initial boundary value problem for the uniformly parabolic equation
$\partial _t\varphi - {\rm div}(A\nabla \varphi ) +\omega \varphi = f $
in
$\Omega _T\equiv \Omega \times (0,T)$
satisfies the uniform estimate
$$\Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi\Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{q,\Omega_T}, \ \ \ c=c(N,\lambda, q, \Omega_T), $$
provided that
$q \gt 1+{N}/{2}$
, where Ω is a bounded domain in
${\open R}^N$
with Lipschitz boundary, T > 0,
$\partial _p\Omega _T$
is the parabolic boundary of
$\Omega _T$
,
$\omega \in L^1(\Omega _T)$
with
$\omega \ges 0$
, and λ is the smallest eigenvalue of the coefficient matrix A. This estimate is sharp in the sense that it generally fails if
$q=1+{N}/{2}$
. In this paper, we show that the linear growth of the upper bound in
$\Vert f \Vert_{q,\Omega _T}$
can be improved. To be precise, we establish
$$ \Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi_0 \Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{1+{N}/{2},\Omega_T} \left(\ln(\Vert f \Vert_{q,\Omega_T}+1)+1\right). $$
In some cases, navigation of aircraft or spacecraft may need to be conducted in a Global Navigation Satellite System (GNSS)-denied environment. So, additional sources of navigation information may need to be used to increase navigation precision and resilience. Such sources can include visual navigation systems such as visual shoreline navigation. The main feature of visual shoreline navigation is the severe variability of navigation errors depending on the shape of the observed shoreline, the distance and the view angle of the observation. Such variations are so great that it is not possible to use average values of errors. So, each measurement of an aircraft or spacecraft position should be accompanied with an estimation of the error covariance matrix in real time. It is proposed to use the Cramer-Rao lower bound of visual shoreline navigation errors as such a matrix. The method for constructing the Cramer-Rao lower bound is described in this paper.
In this paper, we discuss new bounds and approximations for tail probabilities of certain discrete distributions. Several different methods are used to obtain bounds and/or approximations. Excellent upper and lower bounds are obtained for the Poisson distribution. Excellent approximations (and not bounds necessarily) are also obtained for other discrete distributions. Numerical comparisons made to previously proposed methods demonstrate that the new bounds and/or approximations compare very favorably. Some conjectures are made.
Regularity criteria for solutions of the three-dimensional Navier-Stokes equations are derived in this paper. Let
$$\Omega(t, q) := \left\{x:|u(x,t)| > C(t,q)\normVT{u}_{L^{3q-6}(\mathbb{R}^3)}\right\} \cap\left\{x:\widehat{u}\cdot\nabla|u|\neq0\right\}, \tilde\Omega(t,q) := \left\{x:|u(x,t)| \le C(t,q)\normVT{u}_{L^{3q-6}(\mathbb{R}^3)}\right\} \cap\left\{x:\widehat{u}\cdot\nabla|u|\neq0\right\},$$
where
$$q\ge3$$
and
$$C(t,q) := \left(\frac{\normVT{u}_{L^4(\mathbb{R}^3)}^2\normVT{|u|^{(q-2)/2}\,\nabla|u|}_{L^2(\mathbb{R}^3)}}{cq\normVT{u_0}_{L^2(\mathbb{R}^3)} \normVT{p+\mathcal{P}}_{L^2(\tilde\Omega)}\normVT{|u|^{(q-2)/2}\, \widehat{u}\cdot\nabla|u|}_{L^2(\tilde\Omega)}}\right)^{2/(q-2)}.$$
Here
$$u_0=u(x,0)$$
,
$$\mathcal{P}(x,|u|,t)$$
is a pressure moderator of relatively broad form,
$$\widehat{u}\cdot\nabla|u|$$
is the gradient of
$$|u|$$
along streamlines, and
$$c=(2/\pi)^{2/3}/\sqrt{3}$$
is the constant in the inequality
$$\normVT{f}_{L^6(\mathbb{R}^3)}\le c\normVT{\nabla f}_{L^2(\mathbb{R}^3)}$$
.
This article offers a modern perspective that exposes the many contributions of Leray in his celebrated work on the three-dimensional incompressible Navier-Stokes equations from 1934. Although the importance of his work is widely acknowledged, the precise contents of his paper are perhaps less well known. The purpose of this article is to fill this gap. We follow Leray's results in detail: we prove local existence of strong solutions starting from divergence-free initial data that is either smooth or belongs to
$$H^1$$
or
$$L^2 \cap L^p$$
(with
$$p \in (3,\infty]$$
), as well as lower bounds on the norms
$$\| \nabla u (t) \|_2$$
and
$$\| u(t) \|_p$$
(
$$p\in(3,\infty]$$
) as t approaches a putative blow-up time. We show global existence of a weak solution and weak-strong uniqueness. We present Leray's characterisation of the set of singular times for the weak solution, from which we deduce that its upper box-counting dimension is at most 1/2. Throughout the text we provide additional details and clarifications for the modern reader and we expand on all ideas left implicit in the original work, some of which we have not found in the literature. We use some modern mathematical tools to bypass some technical details in Leray's work, and thus expose the elegance of his approach.
An A1−A∞ estimate, improving on a previous result for [b, TΩ] with
$\Omega \in L^{infty}({\open S}^{n - 1})$
and b∈BMO is obtained. A new result in terms of the A∞ constant and the one supremum Aq−A∞exp constant is also proved, providing a counterpart for commutators of the result obtained by Li. Both of the preceding results rely upon a sparse domination result in terms of bilinear forms, which is established using techniques from Lerner.
We present a simple proof of the Chebotarev density theorem for finite morphisms of quasi-projective varieties over finite fields following an idea of Fried and Kosters for function fields. The key idea is to interpret the number of rational points with a given Frobenius conjugacy class as the number of rational points of a twisted variety, which is then bounded by the Lang–Weil estimates.