Partition (number theory): Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older editNewer edit →
en>David Eppstein
m Reverted edits by 95.46.85.162 (talk) to last version by 75.0.185.131
→Approximation formulas: new record computation
Line 1: Line 1:
{{Refimprove|date=March 2013}}
A BMI examination result of 25 to 29.9 signifies which anyone is obese or overweight. Getting chubby poses many wellbeing risks. A few of the well-being difficulties which are widespread to overweight folks are: Diabetes, Asthma, Heart Complications, Cancer plus Hypertension. Each one of these ailments is frequently prevented by shedding off some kilos and preserving a healthy body body weight that can be reached through correct eating habits and exercise. A person's bodyweight can greatly decrease by just taking away sweets plus fatty foods off their diet regime. Frequent walking may even support shed those fats.<br><br><br><br>The BMI or the body mass index is a valuable tool of measuring body fat utilizing the height and weight of a individual. While there are calories burned walking calculators for children, this article can focus found on the 1 utilized for adults 18 to 65 years of age. calories burned walking calculators are accessible online. If you don't have access to the Internet, nevertheless, you are able to nevertheless manually use the formula given under and interpret the BMI based on what a scores mean.<br><br>Clients are needed to have a 15-minute personal in-person or telephone consultation weekly with Jenny Craig "counselors" (whom lack formal nutrition/behavior training) to help clients meet their weight reduction goals. A range of online help tools including [http://safedietplans.com/calories-burned-walking calories burned while walking], interactive diary, menu planner, fat tracker, chat area and e-newsletters are accessible to strengthen clients' need to achieve desired weight reduction objectives.<br><br>Women are notoriously recognized for being busy bodies. This really is the cause why we can easily find yourself doing over one thing at when without even understanding it. When this happens, you tend to forget regarding the truth which you need to take care of your body. Because using this calculator won't actually take up so much of the time, you'll be motivated to use it because we are well aware of the truth that it will allow you to receive the body mass index in really a couple of seconds. Once we are aware of the result, you'll be capable to face the truth that something may be wrong with you since the test is truly objective plus accurate.<br><br>A common fad diet is not generally meant for lasting weight loss considering it refuses to deal with the principal issue, which may be physical or psychological. The diets generally last from 24 hours to one week and serve standard reasons, including, wanting to fit into a specific dress or a favorite jeans.<br><br>Be that as it can, it's nonetheless the simplest technique of determining whether an individual is underweight, obese or obese- conditions that necessitate a visit to one's healthcare provider to determine what changes is created in terms of life-style plus nutritional practices.<br><br>Like every rule of thumb, there are some exceptions. Body builders usually have a BMI above 25, or even 25. In that case, an elevated Body Mass Index could reflect the fact that body builders sometimes have more muscle mass, without a excess of fat.
The '''Whittaker–Shannon interpolation formula''' or '''sinc interpolation''' is a method to construct a [[continuous-time]] [[bandlimited]] function from a sequence of real numbers.  The formula dates back to the works of [[E. Borel]] in 1898, and [[E. T. Whittaker]] in 1915, and was cited from works of [[J. M. Whittaker]] in 1935, and in the formulation of the [[Nyquist–Shannon sampling theorem]] by [[Claude Shannon]] in 1949. It is also commonly called '''Shannon's interpolation formula''' and '''Whittaker's interpolation formula'''. E. T. Whittaker, who published it in 1915, called it the '''Cardinal series'''.
 
==Definition==
[[Image:bandlimited.svg|thumb|right|240px|Fourier transform of a bandlimited function.]]
Given a sequence of real numbers, x[n], the continuous function''':'''
 
:<math>x(t) = \sum_{n=-\infty}^{\infty} x[n] \cdot {\rm sinc}\left(\frac{t - nT}{T}\right)\,</math>
 
has a [[Fourier transform]], X(f), whose non-zero values are confined to the region''':''' |f| ≤ 1/2T.&nbsp;  When parameter T has units of seconds, the '''bandlimit''', 1/2T, has units of cycles/sec ([[hertz]]). When the x[n] sequence represents time samples, at interval T, of a continuous function, the quantity ''f''<sub>''s''</sub> = 1/T is known as the [[sample rate]], and ''f''<sub>''s''</sub>/2 is the corresponding [[Nyquist frequency]]. When the sampled function has a bandlimit, B, less than the Nyquist frequency, x(t) is a '''perfect reconstruction''' of the original function.  (''See [[Sampling theorem]].'')  Otherwise, the frequency components above the Nyquist frequency ''fold'' into the sub-Nyquist region of X(f), resulting in distortion.  (''See [[Aliasing]].'')
 
==Equivalent formulation: convolution/lowpass filter==
The interpolation formula is derived in the [[Nyquist–Shannon sampling theorem]] article, which points out that it can also be expressed as the [[convolution]] of an [[Dirac comb|infinite impulse train]] with a [[sinc function]]''':'''
 
:<math> x(t) = \left( \sum_{n=-\infty}^{\infty} x[n]\cdot \delta \left( t - nT \right) \right) *
{\rm sinc}\left(\frac{t}{T}\right). </math>
 
This is equivalent to filtering the impulse train with an ideal (''brick-wall'') [[low-pass filter]].
 
==Convergence==
The interpolation formula always converges [[absolute convergence|absolutely]] and [[uniform convergence|locally uniformly]] as long as
 
:<math>\sum_{n\in\Z,\,n\ne 0}\left|\frac{x[n]}n\right|<\infty.</math>
 
By the [[Hölder inequality]] this is satisfied if the sequence <math>\scriptstyle (x[n])_{n\in\Z}</math> belongs to any of the <math>\scriptstyle\ell^p(\Z,\mathbb C)</math> [[Lp space|spaces]] with 1&nbsp;<&nbsp;''p''&nbsp;<&nbsp;∞, that is
 
:<math>\sum_{n\in\Z}\left|x[n]\right|^p<\infty.</math>
 
This condition is sufficient, but not necessary.  For example, the sum will generally converge if the sample sequence comes from sampling almost any [[stationary process]], in which case the sample sequence is not square summable, and is not in any <math>\scriptstyle\ell^p(\Z,\mathbb C)</math> space.
 
==Stationary random processes==
If ''x''[''n''] is an infinite sequence of samples of a sample function of a wide-sense [[stationary process]], then it is not a member of any <math>\scriptstyle\ell^p</math> or [[Lp space|L<sup>p</sup> space]], with probability 1; that is, the infinite sum of samples raised to a power ''p'' does not have a finite expected value. Nevertheless, the interpolation formula  converges with probability 1.  Convergence can readily be shown by computing the variances of truncated terms of the summation, and showing that the variance can be made arbitrarily small by choosing a sufficient number of terms. If the process mean is nonzero, then pairs of terms need to be considered to also show that the expected value of the truncated terms converges to zero.
 
Since a random process does not have a Fourier transform, the condition under which the sum converges to the original function must also be different. A stationary random process does have an [[autocorrelation function]] and hence a [[spectral density]] according to the [[Wiener–Khinchin theorem]].  A suitable condition for convergence to a sample function from the process is that the spectral density of the process be zero at all frequencies equal to and above half the sample rate.
 
==See also==
* [[Aliasing]], [[Anti-aliasing filter]], [[Spatial anti-aliasing]]
* [[Fourier transform]]
* [[Rectangular function]]
* [[Sampling (signal processing)]]
* [[Signal (electronics)]]
* [[Sinc function]], [[Sinc filter]]
 
{{Use dmy dates|date=September 2010}}
 
==References==
<references />
*http://www.stanford.edu/class/ee104/shannonpaper.pdf
 
{{DEFAULTSORT:Whittaker-Shannon Interpolation Formula}}
[[Category:Digital signal processing]]
[[Category:Signal processing]]
[[Category:Fourier analysis]]

Revision as of 04:45, 3 March 2014

A BMI examination result of 25 to 29.9 signifies which anyone is obese or overweight. Getting chubby poses many wellbeing risks. A few of the well-being difficulties which are widespread to overweight folks are: Diabetes, Asthma, Heart Complications, Cancer plus Hypertension. Each one of these ailments is frequently prevented by shedding off some kilos and preserving a healthy body body weight that can be reached through correct eating habits and exercise. A person's bodyweight can greatly decrease by just taking away sweets plus fatty foods off their diet regime. Frequent walking may even support shed those fats.



The BMI or the body mass index is a valuable tool of measuring body fat utilizing the height and weight of a individual. While there are calories burned walking calculators for children, this article can focus found on the 1 utilized for adults 18 to 65 years of age. calories burned walking calculators are accessible online. If you don't have access to the Internet, nevertheless, you are able to nevertheless manually use the formula given under and interpret the BMI based on what a scores mean.

Clients are needed to have a 15-minute personal in-person or telephone consultation weekly with Jenny Craig "counselors" (whom lack formal nutrition/behavior training) to help clients meet their weight reduction goals. A range of online help tools including calories burned while walking, interactive diary, menu planner, fat tracker, chat area and e-newsletters are accessible to strengthen clients' need to achieve desired weight reduction objectives.

Women are notoriously recognized for being busy bodies. This really is the cause why we can easily find yourself doing over one thing at when without even understanding it. When this happens, you tend to forget regarding the truth which you need to take care of your body. Because using this calculator won't actually take up so much of the time, you'll be motivated to use it because we are well aware of the truth that it will allow you to receive the body mass index in really a couple of seconds. Once we are aware of the result, you'll be capable to face the truth that something may be wrong with you since the test is truly objective plus accurate.

A common fad diet is not generally meant for lasting weight loss considering it refuses to deal with the principal issue, which may be physical or psychological. The diets generally last from 24 hours to one week and serve standard reasons, including, wanting to fit into a specific dress or a favorite jeans.

Be that as it can, it's nonetheless the simplest technique of determining whether an individual is underweight, obese or obese- conditions that necessitate a visit to one's healthcare provider to determine what changes is created in terms of life-style plus nutritional practices.

Like every rule of thumb, there are some exceptions. Body builders usually have a BMI above 25, or even 25. In that case, an elevated Body Mass Index could reflect the fact that body builders sometimes have more muscle mass, without a excess of fat.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Partition_(number_theory)&oldid=224214"