31 research outputs found
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Optimal Solution of Nonlinear Equations Satisfying a Lipschitz Condition
For a given nonnegative e we seek a point x* such that if(x*)[ l) satisfying a Lipschitz condition with the constant K and having a zero in B. The information operator on f consists of n values of arbitrary linear functionals which are computed adaptively. The point x* is constructed by means of an algorithm which is a mapping depending on the information operator. We find an optimal algorithm, i.e., algorithm with the smallest error, which uses n function evaluations computed adaptively. We also exhibit nearly optimal information operators, i.e., the linear functionals for which the error of an optimal algorithm that uses them is almost minimal. Nearly optimal information operators consists of n nonadaptive function evaluations at equispaced points xj in the cube B. This result exhibits the superiority of the T. Aird and J. Rice procedure ZSRCH (IMSL library [1]) over Sobol's approach [7] for solving nonlinear equations in our class of functions. We also prove that the simple search algorithm which yields a point x*=x k such that If(Xk)]= min If(xj)l is nearly optimal. The complexity, i.e., the minimal cost of solving our problem is roughly equal to (K/e)m
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Minimal number of function evaluations for computing topological degree in two dimensions
A lower bound n-min roughly equal log-2 (diam(T)/m) is established for the minimal number of function evaluations necessary to compute the topological degree of every function f in a class F. The class F consists of continuous functions f = (f1, f2) defined on a triangle T, f: T → R squared, such that the minimal distance between zeros of f1 and zeros of f2 on the boundary of T is not less than m, m > 0. Information is exhibited which permits the computation of the degree for every f in F with at most 2n-min function evaluations. An algorithm, due to Kearfott, uses this information to compute the degree. These results lead to tight lower and upper complexity bounds for this problem
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Study of Linear Information for Classes of Polynomial Equations
Linear adaptive information for approximating a zero of f is studied where f belongs to the class of polynomials of unbounded degree. A theorem on constrained approximation of smooth functions by polynomials is established
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For Which Error Criteria Can We Solve Nonlinear Equations?
For which error criteria can we solve a nonlinear scalar equation f(x) = 0 where f is a real function on the interval [a,b]
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Complexity of Computing Topological Degree of Lipschitz Functions in N Dimensions
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Asymptotic Optimality of the Bisection Method
The bisection method is shown to possess the asymptotically best rate of convergence for infinitely differentiable functions having zeros of arbitrary multiplicity. If the multiplicity of zeros is bounded methods are known which have asymptotically at least quadratic rate of convergence
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Can We Approximate Zeros of Functions with Non-zero Topological Degree?
The bisection method provides an affirmative answer for scalar functions. We show that the answer is negative for bivariate functions. This means, in particular, that an arbitrary continuation method cannot approximate a zero of every smooth bivariate function with non-zero topological degree
Data Mining of Atherosclerotic Plaque Transcriptomes Predicts STAT1-Dependent Inflammatory Signal Integration in Vascular Disease
Atherosclerotic plaque development involves multiple extra- and intra-cellular signals engaging cells from the immune system and from the vasculature. Pro-inflammatory pathways activated by interferon gamma (IFNγ) and toll-like receptor 4 (TLR4) ligands are profoundly involved in plaque formation and have been shown to involve cross-talk in all atheroma-interacting cell types leading to increased activation of signal transducer and activator of transcription-1 (STAT1) and elevated expression of pro-inflammatory mediators. Here we demonstrate that in Gene Expression Omnibus repository (GEO) deposited microarray datasets, obtained from human coronary and carotid atherosclerotic plaques, a significant increase in expression of pro-inflammatory and immunomodulatory genes can be detected. Moreover, increased expression of multiple chemokines, adhesion molecules and matrix-remodeling molecules was commonly detected in both plaque types and correlated with the presence of putative STAT1 binding sites in their promoters, suggesting strong involvement of STAT1 in plaque development. We also provide evidence to suggest that STAT1-nuclear factor kappa-light-chain-enhancer of activated B cells (NFκB) or STAT1-interferon-regulated factor (IRF) regulatory modules are over-represented in the promoters of these inflammatory genes, which points to a possible contribution of IFNγ and TLR4 cross-talk in the process of atherogenesis. Finally, a subset of these genes encodes for secreted proteins that could serve as a basis of a non-invasive diagnostic assay. The results of our in silico analysis in vitro provide potential evidence that STAT1-dependent IFNγ-TLR4 cross-talk plays a crucial role in coronary and carotid artery plaque development and identifies a STAT1-dependent gene signature that could represent a novel diagnostic tool to monitor and diagnose plaque progression in human atherosclerosis