890 research outputs found
Uniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials
We survey developments, over the last thirty years, in the theory of Shape
Preserving Approximation (SPA) by algebraic polynomials on a finite interval.
In this article, "shape" refers to (finitely many changes of) monotonicity,
convexity, or q-monotonicity of a function (for definition, see Section 4). It
is rather well known that it is possible to approximate a function by algebraic
polynomials that preserve its shape (i.e., the Weierstrass approximation
theorem is valid for SPA). At the same time, the degree of SPA is much worse
than the degree of best unconstrained approximation in some cases, and it is
"about the same" in others. Numerous results quantifying this difference in
degrees of SPA and unconstrained approximation have been obtained in recent
years, and the main purpose of this article is to provide a "bird's-eye view"
on this area, and discuss various approaches used.
In particular, we present results on the validity and invalidity of uniform
and pointwise estimates in terms of various moduli of smoothness. We compare
various constrained and unconstrained approximation spaces as well as orders of
unconstrained and shape preserving approximation of particular functions, etc.
There are quite a few interesting phenomena and several open questions.Comment: 51 pages, 49 tables, survey, published in Surveys in Approximation
Theory, 6 (2011), 24-7
Possible use of repeated cold stress for reducing fatigue in chronic fatigue syndrome: a hypothesis
<p>Abstract</p> <p>Background</p> <p>Physiological fatigue can be defined as a reduction in the force output and/or energy-generating capacity of skeletal muscle after exertion, which may manifest itself as an inability to continue exercise or usual activities at the same intensity. A typical example of a fatigue-related disorder is chronic fatigue syndrome (CFS), a disabling condition of unknown etiology and with uncertain therapeutic options. Recent advances in elucidating pathophysiology of this disorder revealed hypofunction of the hypothalamic-pituitary-adrenal axis and that fatigue in CFS patients appears to be associated with reduced motor neurotransmission in the central nervous system (CNS) and to a smaller extent with increased fatigability of skeletal muscle. There is also some limited evidence that CFS patients may have excessive serotonergic activity in the brain and low opioid tone.</p> <p>Presentation of the hypothesis</p> <p>This work hypothesizes that repeated cold stress may reduce fatigue in CFS because brief exposure to cold may transiently reverse some physiological changes associated with this illness. For example, exposure to cold can activate components of the reticular activating system such as raphe nuclei and locus ceruleus, which can result in activation of behavior and increased capacity of the CNS to recruit motoneurons. Cold stress has also been shown to reduce the level of serotonin in most regions of the brain (except brainstem), which would be consistent with reduced fatigue according to animal models of exercise-related fatigue. Finally, exposure to cold increases metabolic rate and transiently activates the hypothalamic-pituitary-adrenal axis as evidenced by a temporary increase in the plasma levels of adrenocorticotropic hormone, beta-endorphin and a modest increase in cortisol. The increased opioid tone and high metabolic rate could diminish fatigue by reducing muscle pain and accelerating recovery of fatigued muscle, respectively.</p> <p>Testing the hypothesis</p> <p>To test the hypothesis, a treatment is proposed that consists of adapted cold showers (20 degrees Celsius, 3 minutes, preceded by a 5-minute gradual adaptation to make the procedure more comfortable) used twice daily.</p> <p>Implications of the hypothesis</p> <p>If testing supports the proposed hypothesis, this could advance our understanding of the mechanisms of fatigue in CFS.</p
- …