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In [[mathematics]], '''Welch bounds''' are a family of [[inequality (mathematics)|inequalities]] pertinent to the problem of evenly spreading a set of unit [[vector space|vectors]] in a [[vector space]]. The bounds are important tools in the design and analysis of certain methods in [[telecommunication]] engineering, particularly in [[coding theory]]. The bounds were originally published in a 1974 paper by L. R. Welch.
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==Mathematical statement==
 
If <math>\{x_1,\ldots,x_m\}</math> are unit vectors in <math>\mathbb{C}^n</math>, define <math>c_\max = \max_{i\neq j} |\langle x_i, x_j \rangle|</math>, where <math>\langle\cdot,\cdot\rangle</math> is the usual [[inner product]] on <math>\mathbb{C}^n</math>. Then the following inequalities hold for <math>k=1,2,\dots</math>:
 
: <math>(c_\max)^{2k} \geq \frac{1}{m-1} \left[ \frac{m}{\binom{n+k-1}{k}}-1 \right]</math>
 
==Applicability==
 
If <math>m\leq n</math>, then the vectors <math>\{x_i\}</math> can form an [[orthonormal set]] in <math>\mathbb{C}^n</math>. In this case, <math>c_\max=0</math> and the bounds are vacuous. Consequently, interpretation of the bounds is only meaningful if <math>m>n</math>. This will be assumed throughout the remainder of this article.
 
==Proof for ''k'' = 1==
 
The "first Welch bound," corresponding to <math>k=1</math>, is by far the most commonly used in applications. Its proof proceeds in two steps, each of which depends on a more basic mathematical inequality. The first step invokes the [[Cauchy-Schwarz inequality]] and begins by considering the <math>m\times m</math> [[Gram matrix]] <math>G</math> of the vectors <math>\{x_i\}</math>; i.e.,
 
: <math>G=\left[ \begin{array}{ccc} \langle x_1, x_1 \rangle & \cdots & \langle x_1, x_m \rangle \\ \vdots & \ddots & \vdots \\  \langle x_m, x_1 \rangle & \cdots & \langle x_m, x_m \rangle \end{array}\right]</math>
 
The [[trace (linear algebra)|trace]] of <math>G</math> is equal to the sum of its eigenvalues. Because the [[rank (linear algebra)|rank]] of <math>G</math> is at most <math>n</math>, and it is a [[positive-semidefinite matrix|positive semidefinite]] matrix, <math>G</math> has at most <math>n</math> positive [[eigenvalue]]s with its remaining eigenvalues all equal to zero. Writing the non-zero eigenvalues of <math>G</math> as <math>\lambda_1,\ldots,\lambda_r</math> with <math>r\leq n</math> and applying the Cauchy-Schwarz inequality to the inner product of an <math>r</math>-vector of ones with a vector whose components are these eigenvalues yields
 
: <math>(\mathrm{Tr}\;G)^2 = \left( \sum_{i=1}^r \lambda_i \right)^2 \leq r \sum_{i=1}^r \lambda_i^2 \leq n \sum_{i=1}^m \lambda_i^2 </math>
 
The square of the [[Frobenius norm]] (Hilbert&ndash;Schmidt norm) of <math>G</math> satisfies
 
: <math> ||G||^2 = \sum_{i=1}^{m} \sum_{j=1}^m |\langle x_i , x_j \rangle|^2 = \sum_{i=1}^m \lambda_i^2</math>
 
Taking this together with the preceding inequality gives
 
: <math>\sum_{i=1}^m \sum_{j=1}^m |\langle x_i , x_j \rangle|^2\geq \frac{(\mathrm{Tr}\;G)^2}{n}</math>
 
Because each <math>x_i</math> has unit length, the elements on the main diagonal of <math>G</math> are ones, and hence its trace is <math>\mathrm{Tr}\;G = m</math>.  So,
 
: <math>\sum_{i=1}^{m} \sum_{j=1}^m |\langle x_i , x_j \rangle|^2 = m+\sum_{i\neq j} |\langle x_i , x_j \rangle|^2 \geq \frac{m^2}{n}</math>
 
or
 
: <math>\sum_{i\neq j} |\langle x_i , x_j \rangle|^2 \geq \frac{m(m-n)}{n}</math>
 
The second part of the proof uses an inequality encompassing the simple observation that the average of a set of non-negative numbers can be no greater than the largest number in the set. In mathematical notation, if <math>a_{\ell}\geq 0</math> for <math>\ell=1,\ldots, L</math>, then
 
: <math>\frac{1}{L}\sum_{\ell=1}^L a_{\ell} \leq \max a_{\ell}</math>
 
The previous expression has <math>m(m-1)</math> non-negative terms in the sum,the largest of which is <math>c_\max^2</math>.  So,
 
: <math>(c_\max)^2\geq \frac{1}{m(m-1)}\sum_{i\neq j} |\langle x_i , x_j \rangle|^2\geq\frac{m-n}{n(m-1)}</math>
 
or
: <math>(c_\max)^2\geq \frac{m-n}{n(m-1)}</math>
 
which is precisely the inequality given by Welch in the case that <math>k=1</math>
 
==Achieving Welch bound equality==
 
In certain telecommunications applications, it is desirable to construct sets of vectors that meet the Welch bounds with equality. Several techniques have been introduced to obtain so-called '''Welch Bound Equality''' (WBE) sets of vectors for the ''k''&nbsp;=&nbsp;1 bound.
 
The proof given above shows that two separate mathematical inequalities are incorporated into the Welch bound when <math>k=1</math>. The Cauchy&ndash;Schwarz inequality is met with equality when the two vectors involved are collinear. In the way it is used in the above proof, this occurs when all the non-zero eigenvalues of the Gram matrix <math>G</math> are equal, which happens precisely when the vectors <math>\{x_1,\ldots,x_m\}</math> constitute a [[tight frame]] for <math>\mathbb{C}^n</math>.
 
The other inequality in the proof is satisfied with equality if and only if <math>|\langle x_i, x_j \rangle|</math> is the same for every choice of <math>i\neq j</math>. In this case, the vectors are [[equiangular lines|equiangular]]. So this Welch bound is met with equality if and only if the set of vectors <math>\{x_i\}</math> is an equiangular tight frame in <math>\mathbb{C}^n</math>.
 
==References==
{{refbegin}}
*{{cite journal |first1=S. |last1=Datta |first2=S.D. |last2=Howard |first3=D. |last3=Cochran |title=Geometry of the Welch Bounds |journal=Linear Algebra and its Applications |volume=437 |issue=10 |pages=2455–70 |year=2012 |doi=10.1016/j.laa.2012.05.036 |url=http://www.sciencedirect.com/science/article/pii/S0024379512004405 |arxiv=0909.0206v1}}
*{{cite journal |first=L.R. |last=Welch |title=Lower Bounds on the Maximum Cross Correlation of Signals |journal=IEEE Trans. on Info. Theory |volume=20 |issue=3 |pages=397–9 |date=May 1974 |doi=10.1109/TIT.1974.1055219 |url=http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=1055219}}
{{refend}}
 
[[Category:Inequalities]]

Latest revision as of 18:48, 29 November 2014

Kershaw Ken Onion Leek - The leek collection is one of the hottest among the many brand of Kershaw. The onion has a very glossy design made by Ken Onion. It encompasses a very good Pace Protected system, which the proprietor can easily and simply deploy the blade with one hand. That hunting knife is the perfect knife for everyday carry, it has a “reliable” clip which seems to be arduous to seek out in most folding knives. It’s simple to say, you received’t find a folding knife at this worth with such a sleek design, good blade and overall a stable-made knife. $36.29

The Rockwell scale is a measurement system used by business producers to find out the relative hardness of objects primarily based on the depth of indentation from a heavy object. In different words, they drop a heavy ball or diamond cone onto the metal and measure the scale of indentation. For knives they use the “C” scale in items labeled “HRC.” High carbon metal kitchen knives typically run from HRC fifty six-fifty eight. HRC sixty two is a very arduous knife. The Victorinox SwissChamp is the heaviest knife in our take a look at, but additionally probably the most versatile. When assessing portability we checked out weight and size, as well as choices for safe yet helpful carriage.

Lastly, a few notes on blade form and cutting edge design. All but the Smith and Wesson Extreme Ops have some type of drop-point shaped blade. This is probably the most versatile blade form. The tanto-shaped blade of the Smith And Wesson sacrifices some versatility in exchange for a slight increase in energy and significant enhance in the intimidating appearance. Also, word that blades in our test and elsewhere might be either straight or serrated. Neither is in any method better than the other. Serrated blades minimize tough supplies more simply while straight blades are easier to sharpen. The OutdoorGearLab crew prefers, usually, straight blades.

It isn't simple to discover a completely pure face and physique on the pink carpet nowadays because cosmetic surgery has grow to be increasingly more fashionable. It isn't only for the wealthy and famous, but also for the center class. Most of us suppose that well-known people are more excellent and attractive than the typical particular person. Nonetheless, plainly each Hollywood star has had a little bit work completed and so they look extra attractive than us as a result of plastic surgery. The nicely-recognized stars reminiscent of Jennifer Aniston, Melanie Griffith, Megan Fox, Lil\' Kim, Joan Rivers and so forth are rumored to go under the knife.

There are particular exceptions, akin to where the knife is to be used in a "lawful occupation, for lawful recreational functions, or as a recognized spiritual follow.") The County of Los Angeles has an analogous rule, which makes it unlawful to overtly carry, in public, "any knife having a blade of three inches or extra in size; any spring-blade, swap-blade or snap-blade knife; any knife any blade of which is routinely launched by a spring mechanism or different mechanical device; any ice choose or similar sharp stabbing device; any straight-edge razor or any razor blade fitted to a handle."

Camping. There are tons of different issues you could possibly use a pocket knife for whereas tenting akin to creating tinder, gutting a fish, cutting rope, and slicing up your shirt to make a sling while you slip off the sting of a hill while hiking and break your arm. You know that’s going to happen to someone Best fillet knives! By the late sixteenth Century, pocket knives started to regain popularity. They had been especially standard within the American colonies. From farmers to attorneys, males from all walks of life began to carry them. The final word portable instrument, the pocket knife was a useful assist in various duties, together with eating on the go.

I recall wanting a Swiss Army Knife. My father appeared to have a place in his heart for the Previous Timer line by Schrade. He probably also was anxious all of those tools would add to the potential for harm—not that three blades on the stockman was a safe bet either. I highly recommend a Swiss Military Knife as the primary knife for a child. It is internationally acknowledged as a knife of quality and has set the usual for all other pocket knives on this planet at present. Never put stress on the back of a folding blade that lacks a lock mechanism—the knife might immediately shut on your hand.

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