Hypergeometric identity: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
en>RussBot
m Robot: Editing intentional link to disambiguation page in hatnote per WP:INTDABLINK (explanation)
 
en>BD2412
m minor fixes, mostly disambig links using AWB
 
Line 1: Line 1:
{{redirect-distinguish|Variational theorem|variational principle}}
Earlier than you resolve whether or not stainless-steel cookware is value buying, lets first talk about what stainless steel cookware is. Chrome steel is made from an alloy, or a mix of metals. Mostly, primary iron with chromium, nickel or another minor metals. The chromium provides rust safety and provides your cookware sturdiness. The nickel provides rust safety as well, and adds a cultured look. Most effectively made chrome steel cookware has copper or aluminum added to the bottom of the pan or pot. This is achieved to increases the power of the pot or pan to conduct warmth.<br>The most effective stainless steel cookware is the principle class, however nonetheless it's divided into a number of subcategories primarily based on the standard and the worth vary. It can be complicated to decide on the very best stainless-steel cookware out of the categories that may meet your necessities. That is the place we took a step ahead to explain you all the knowledge that will probably be useful so that you can understand how to decide on the perfect stainless-steel cookware. One of the best stainless-steel cookware set is manufactured from cheap to costly and quality constructed pots and pans. <br>You'll find magnetic stainless steel in the layer on the skin of some high quality pieces of chrome steel. That is to make it appropriate with induction stovetops, which involve the usage of a rapidly charging electromagnetic subject to heat cookware. Excessive-quality stainless-steel, like All-Clad , uses three layers of metallic—the austenite layer of steel on the inside, ferrite metal on the surface, and a layer of aluminum sandwiched between the 2 for optimal warmth conductivity (steel alone doesn't conduct heat evenly). Lesser-quality stainless steel is normally only one layer of austenitic chrome steel.<br>Aesthetically speaking, stainless-steel is a clever alternative should you prefer to display or cling pots or pans. The clean, crisp look of all stainless-steel kitchenware can transform a mishmash of cookware into a sophisticated décor statement. Chrome steel kettles, such because the Cuisinart Tea Kettle will combine individual kitchenware into a cohesive and nice entity. Contemplate purchasing chrome steel utensils as effectively. For more about [http://cookwarehq.drupalgardens.com/best-stainless-steel-cookware-top-reviews-2014 stainless steel cookware reviews] stop by our web-page. Already acquired a stunning stainless-steel cookware collection? The Cuisinart Chef’s Assortment stainless pot rack could be the final touch for a kitchen, releasing up space and making those pots and pans readily accessible. Get the chrome steel cookware of your culinary desires at Macy’s!<br>Hard-anodized aluminum cookware is among the most popular forms of materials, despite the fact that many people do not quite perceive the construction. Arduous-anodized aluminum is apparent aluminum that has been processed in a collection of chemical baths charged with an electrical current. The result's a material that has the identical superior heat conductivity as aluminum but is non-reactive with acidic meals, resembling tomatoes, and twice as hard as stainless-steel. Two drawbacks to onerous-anodized cookware are that it's not dishwasher-secure and, because it isn't magnetic, it is not going to work with induction vary tops.<br>The enamel over metal technique creates a piece that has the heat distribution of carbon metal and a non-reactive, low-stick floor. Such pots are a lot lighter than most other pots of comparable measurement, are cheaper to make than stainless-steel pots, and do not have the rust and reactivity issues of forged iron or carbon metal.  quotation wanted  Enamel over steel is good for giant stockpots and for other massive pans used largely for water-primarily based cooking. Due to its light weight and straightforward cleanup, enamel over steel is also fashionable for cookware used whereas camping. Clad aluminium or copper  edit<br>Distinctive specialty cookware items served a la carte to go with any cookware set are constructed of a sturdy Stainless Steel with a brushed exterior finish. Designed with an impact bonded, aluminum disk encapsulated base which distributes warmth shortly and evenly to permit precise temperature management. Handles are riveted for sturdiness and efficiency. The New Specialty Cookware is compatible for all range varieties together with induction. In addition to the multi use function, another unique feature is bottom to top interior volume markings in each quarts and metric measurement; and each bit comes with a tempered glass lid, oven safe to 350°F.<br>Whether you are a cooking fanatics, a professional chef or just cooking for your family you understand the importance of getting a totally stocked kitchen. Not solely do you need the proper elements, however you also need the proper instruments to get the job performed. In any type of fundamental cooking coaching lesson, you'll study that stainless steel is your new best buddy on the subject of kitchen cookware. What additionally, you will learn is that quality cooking gear does not normally come at a discounted worth. For that reason, it is important to take good care of your cookware! Listed here are some fundamentals for chrome steel care. <br>To fight the uneven heating downside, most stainless steel pans are laminations of aluminum or copper on the underside to spread the warmth round, and stainless steel inside the pan to supply a cooking floor that's impervious to no matter you would possibly put inside. In my expertise, this chrome steel floor continues to be too sticky to fry on, and in the event you ever burn it you get a everlasting bother spot. However, generally a chrome steel cooking surface comes in handy when you possibly can't use aluminum (see under) so I keep some around. Choose one thing with a fairly thick aluminum layer on the underside.<br>Well, except you’re a metals skilled and go inspect the factory where the metal is made to see whether or not their manufacturing process creates a pure austenite without corrosive materials shaped, you’re not going to know for sure whether or not the craftsmanship of your stainless is of the best quality. I feel your finest guess is to easily purchase high-quality chrome steel from the beginning, from a model with a status for good quality. However, I think I've figured out a technique that you could determine if the stainless cookware you have already got is probably reactive.
{{Refimprove|date=November 2011}}
 
In [[linear algebra]] and [[functional analysis]], the '''min-max theorem''', or '''variational theorem''', or '''Courant&ndash;Fischer&ndash;Weyl min-max principle''', is a result that gives a variational characterization of eigenvalues of [[Compact operator on Hilbert space|compact]] Hermitian operators on Hilbert spaces. It can be viewed as the starting point of many results of similar nature.
 
This article first discusses the finite dimensional case and its applications before considering compact operators on infinite dimensional Hilbert spaces. We will see that for compact operators, the proof of the main theorem uses essentially the same idea from the finite dimensional argument.
 
The min-max theorem can be extended to [[self adjoint operator]]s that are bounded below.
 
== Matrices ==
 
Let ''A'' be a ''n'' &times; ''n'' [[Hermitian matrix]]. As with many other variational results on eigenvalues, one considers the [[Rayleigh quotient|Rayleigh&ndash;Ritz quotient]] ''R''<sub>''A''</sub>: '''C'''<sup>''n''</sup> \{0} &rarr; '''R''' defined by
 
:<math>R_A(x) = \frac{(Ax, x)}{(x,x)}</math>
 
where (&middot;, &middot;) denotes the Euclidean inner product on '''C'''<sup>''n''</sup>. Clearly, the Rayleigh quotient of an eigenvector is its associated eigenvalue. Equivalently, the Rayleigh&ndash;Ritz quotient can be replaced by
 
:<math>f(x) = (Ax, x), \; \|x\| = 1.</math>
 
For Hermitian matrices, the range of the continuous function ''R''<sub>''A''</sub>(''x''), or ''f''(''x''), is a compact subset [''a'', ''b''] of the real line. The maximum ''b'' and the minimum ''a'' are the largest and smallest eigenvalue of ''A'', respectively. The min-max theorem is a refinement of this fact.
 
=== Min-max Theorem ===
 
Let ''A'' be a ''n'' &times; ''n'' [[Hermitian matrix]] with eigenvalues ''&lambda;''<sub>1</sub> &ge; ... &ge; ''&lambda;<sub>k</sub>'' &ge; ... &ge; ''&lambda;<sub>n</sub>'' then
:<math>
\lambda_k = \max \{ \min \{ R_A(x) \mid x \in U \text{ and } x \neq 0 \} \mid \dim(U)=k \}
</math>
and
:<math>
\lambda_k = \min \{ \max \{ R_A(x) \mid x \in U \text{ and } x \neq 0 \} \mid \dim(U)=n-k+1 \}
</math>
in particular,
:<math>
\lambda_n \leq R_A(x) \leq \lambda_1 \quad\forall x \in \mathbb{C}^n
</math>
and these bounds are attained when ''x'' is an eigenvector of the appropriate eigenvalues.
 
=== Proof ===
 
Since the matrix ''A'' is Hermitian it is diagonalizable and we can choose an orthonormal basis of eigenvectors {''u''<sub>1</sub>,...,''u''<sub>''n''</sub>} that is, ''u''<sub>''i''</sub> is an eigenvector for the eigenvalue ''&lambda;''<sub>''i''</sub> and such that (''u''<sub>''i''</sub>, ''u''<sub>''i''</sub> ) = 1 and (''u''<sub>''i''</sub>, ''u''<sub>''j''</sub> ) = 0 for all ''i'' &ne; ''j''.
 
If ''U'' is a subspace of dimension ''k'' then its intersection with the subspace
:<math>
\text{span}\{ u_k, \ldots, u_n \}
</math>
isn't zero (by simply checking dimensions) and hence there exists a vector ''v≠0'' in this intersection that we can write as
:<math>
v = \sum_{i=k}^n \alpha_i u_i
</math>
and whose Rayleigh quotient is
:<math>
R_A(v) = \frac{\sum_{i=k}^n \lambda_i \alpha_i^2}{\sum_{i=k}^n \alpha_i^2} \leq \lambda_k
</math>
and hence
:<math>
\min \{ R_A(x) \mid x \in U \} \leq \lambda_k
</math>
And we can conclude that
:<math>
\max \{ \min \{ R_A(x) \mid x \in U \text{ and } x \neq 0 \} \mid \dim(U)=k \} \leq \lambda_k
</math>
And since that maximum value is achieved for
:<math>
U = \text{span}\{u_1,\ldots,u_k\}
</math>
We can conclude the equality.
 
In the case where ''U'' is a subspace of dimension ''n-k+1'', we proceed in a similar fashion: Consider the subspace of dimension ''k''
:<math>
\text{span}\{ u_1, \ldots, u_k \}
</math>
Its intersection with the subspace ''U'' isn't zero (by simply checking dimensions) and hence there exists a vector ''v'' in this intersection that we can write as
:<math>
v = \sum_{i=1}^k \alpha_i u_i
</math>
and whose Rayleigh quotient is
:<math>
R_A(v) = \frac{\sum_{i=1}^k \lambda_i \alpha_i^2}{\sum_{i=1}^k \alpha_i^2} \geq \lambda_k
</math>
and hence
:<math>
\max \{ R_A(x) \mid x \in U \} \geq \lambda_k
</math>
And we can conclude that  
:<math>
\min \{ \max \{ R_A(x) \mid x \in U \text{ and } x \neq 0 \} \mid \dim(U)=n-k+1 \} \geq \lambda_k
</math>
And since that minimum value is achieved for
:<math>
U = \text{span}\{u_k,\ldots,u_n\}
</math>
We can conclude the equality.
 
== Applications ==
 
=== Min-max principle for singular values ===
 
The [[singular value]]s {''&sigma;<sub>k</sub>''} of a square matrix ''M'' are the square roots of eigenvalues of ''M*M'' (equivalently ''MM*''). An immediate consequence of the first equality from min-max theorem is
 
:<math>\sigma_k ^{\uparrow} = \min_{S_k} \max_{x \in S_k, \|x\| = 1} (M^* Mx, x)^{\frac{1}{2}}=
\min_{S_k} \max_{x \in S_k, \|x\| = 1} \| Mx \|.
</math>
 
Similarly,
 
:<math>\sigma_k ^{\downarrow} = \max_{S_k} \min_{x \in S_k, \|x\| = 1} \| Mx \|.</math>
 
=== Cauchy interlacing theorem ===
 
Let ''A'' be a symmetric ''n'' &times; ''n'' matrix. The ''m'' &times; ''m'' matrix ''B'', where ''m'' &le; ''n'', is called a '''[[compression (functional analysis)|compression]]''' of ''A'' if there exists an orthogonal projection ''P'' onto a subspace of dimension ''m'' such that ''P*AP'' = ''B''. The Cauchy interlacing theorem states:
 
'''Theorem''' If the eigenvalues of ''A'' are ''&alpha;''<sub>1</sub> &le; ...  &le; ''&alpha;<sub>n</sub>'', and those of ''B'' are ''&beta;''<sub>1</sub> &le; ... ''&beta;<sub>j</sub>'' ... &le; ''&beta;<sub>m</sub>'', then for all ''j'' < m+1,
 
:<math>\alpha_j \leq \beta_j \leq \alpha_{n-m+j}.</math>
 
This can be proven using the min-max principle. Let ''&beta;<sub>i</sub>'' have corresponding eigenvector ''b<sub>i</sub>'' and ''S<sub>j</sub>'' be the ''j'' dimensional subspace ''S<sub>j</sub>'' = span{''b''<sub>1</sub>...''b<sub>j</sub>''}, then
 
:<math>\beta_j = \max_{x \in S_j, \|x\| = 1} (Bx, x) = \max_{x \in S_j, \|x\| = 1} (P^*APx, x) \geq \min_{S_j} \max_{x \in S_j, \|x\| = 1} (Ax, x) = \alpha_j.</math>
 
According to first part of min-max,
 
:<math>\alpha_j \leq \beta_j.</math>
 
On the other hand, if we define ''S''<sub>''m''&minus;''j''+1</sub> = span{''b''<sub>''j''</sub>...''b<sub>m</sub>''}, then
 
:<math>\beta_j = \min_{x \in S_{m-j+1}, \|x\| = 1} (Bx, x) = \min_{x \in S_{m-j+1}, \|x\| = 1} (P^*APx, x)= \min_{x \in S_{m-j+1}, \|x\| = 1} (Ax, x) \leq \alpha_{n-m+j},</math>
 
where the last inequality is given by the second part of min-max.
 
Notice that, when ''n''&nbsp;&minus;&nbsp;''m'' =&nbsp;1, we have
 
:<math>\alpha_j \leq \beta_j \leq \alpha_{j+1}.</math>
 
Hence the name ''interlacing'' theorem.
 
== Compact operators ==
 
Let ''A'' be a [[Compact operator on Hilbert space|compact]], [[Hermitian]] operator on a Hilbert space ''H''. Recall that the [[spectrum (functional analysis)|spectrum]] of such an operator form a sequence of real numbers whose only possible [[cluster point]] is zero. Every nonzero number in the spectrum is an eigenvalue. It no longer makes sense here to list the positive eigenvalues in increasing order. Let the positive eigenvalues of ''A'' be
 
:<math>\cdots \le \lambda_k \le \cdots \le \lambda_1,</math>
 
where [[Multiplicity (mathematics)|multiplicity]] is taken into account as in the matrix case. When ''H'' is infinite dimensional, the above sequence of eigenvalues is necessarily infinite. We now apply the same reasoning as in the matrix case. Let ''S<sub>k</sub>'' &sub; ''H'' be a ''k'' dimensional subspace, and ''S' '' be the closure of the linear span ''S' '' =&nbsp;span{''u<sub>k</sub>'',&nbsp;''u''<sub>''k''&nbsp;+&nbsp;1</sub>,&nbsp;...}. The subspace ''S' '' has codimension ''k''&nbsp;&minus;&nbsp;1. By the same dimension count argument as in the matrix case, ''S' '' &cap; ''S<sub>k</sub>'' is non empty. So there exists ''x''&nbsp;&isin;&nbsp;''S'&nbsp;''&nbsp;&cap;&nbsp;''S<sub>k</sub>'' with ||''x''|| =&nbsp;1. Since it is an element of ''S' '', such an ''x'' necessarily satisfy
 
:<math>(Ax, x) \le \lambda_k.</math>
 
Therefore, for all ''S<sub>k</sub>''
 
:<math>\inf_{x \in S_k, \|x\| = 1}(Ax,x) \le \lambda_k</math>
 
But ''A'' is compact, therefore the function ''f''(''x'') = (''Ax'', ''x'') is weakly continuous. Furthermore, any bounded set in ''H'' is weakly compact. This lets us replace the infimum by minimum:
 
:<math>\min_{x \in S_k, \|x\| = 1}(Ax,x) \le \lambda_k.</math>
 
So
 
:<math>\sup_{S_k} \min_{x \in S_k, \|x\| = 1}(Ax,x) \le \lambda_k.</math>
Because equality is achieved when ''S<sub>k</sub>'' = span{''&u;''<sub>1</sub>...''&u;<sub>k</sub>''},
 
:<math>\max_{S_k} \min_{x \in S_k, \|x\| = 1}(Ax,x) = \lambda_k.</math>
 
This is the first part of min-max theorem for compact self-adjoint operators.
 
Analogously, consider now a ''k''&nbsp;&minus;&nbsp;1 dimensional subspace ''S''<sub>''k''&minus;1</sub>, whose the orthogonal compliment is denoted by ''S''<sub>''k''&minus;1</sub><sup>&perp;</sup>. If ''S' '' =&nbsp;span{''u''<sub>1</sub>...''u<sub>k</sub>''},
 
:<math>S' \cap S_{k-1}^{\perp} \ne {0}.</math>
 
So
 
:<math>\exists x \in S_{k-1}^{\perp} \, \|x\| = 1, (Ax, x) \ge \lambda_k.</math>
 
This implies
 
:<math>\max_{x \in S_{k-1}^{\perp}, \|x\| = 1} (Ax, x) \ge \lambda_k</math>
 
where the compactness of ''A'' was applied. Index the above by the collection of (''k''&nbsp;&minus;&nbsp;1)-dimensional subspaces gives
 
:<math>\inf_{S_{k-1}} \max_{x \in S_{k-1}^{\perp}, \|x\|=1} (Ax, x) \ge \lambda_k.</math>
 
Pick ''S''<sub>''k''&minus;1</sub> = span{''u''<sub>1</sub>...''u''<sub>''k''&minus;1</sub>} and we deduce
 
:<math>\min_{S_{k-1}} \max_{x \in S_{k-1}^{\perp}, \|x\|=1} (Ax, x) = \lambda_k.</math>
 
In summary,
 
'''Theorem (Min-Max)''' Let ''A'' be a compact, self-adjoint operator on a Hilbert space ''H'', whose positive eigenvalues are listed in decreasing order:
 
:<math>\cdots \le \lambda_k \le \cdots \le \lambda_1.</math>
 
Then
 
:<math>\max_{S_k} \min_{x \in S_k, \|x\| = 1}(Ax,x) = \lambda_k ^{\downarrow},</math>
 
:<math>\min_{S_{k-1}} \max_{x \in S_{k-1}^{\perp}, \|x\|=1} (Ax, x) = \lambda_k^{\downarrow}.</math>
 
A similar pair of equalities hold for negative eigenvalues.
 
== See also ==
* [[Courant minimax principle]]
* [[Max–min inequality]]
 
== References ==
*M. Reed and B. Simon, ''Methods of Modern Mathematical Physics IV: Analysis of Operators'', Academic Press, 1978.
 
[[Category:Functional analysis]]
[[Category:Articles containing proofs]]
[[Category:Theorems in functional analysis]]

Latest revision as of 05:33, 2 May 2014

Earlier than you resolve whether or not stainless-steel cookware is value buying, lets first talk about what stainless steel cookware is. Chrome steel is made from an alloy, or a mix of metals. Mostly, primary iron with chromium, nickel or another minor metals. The chromium provides rust safety and provides your cookware sturdiness. The nickel provides rust safety as well, and adds a cultured look. Most effectively made chrome steel cookware has copper or aluminum added to the bottom of the pan or pot. This is achieved to increases the power of the pot or pan to conduct warmth.
The most effective stainless steel cookware is the principle class, however nonetheless it's divided into a number of subcategories primarily based on the standard and the worth vary. It can be complicated to decide on the very best stainless-steel cookware out of the categories that may meet your necessities. That is the place we took a step ahead to explain you all the knowledge that will probably be useful so that you can understand how to decide on the perfect stainless-steel cookware. One of the best stainless-steel cookware set is manufactured from cheap to costly and quality constructed pots and pans.
You'll find magnetic stainless steel in the layer on the skin of some high quality pieces of chrome steel. That is to make it appropriate with induction stovetops, which involve the usage of a rapidly charging electromagnetic subject to heat cookware. Excessive-quality stainless-steel, like All-Clad , uses three layers of metallic—the austenite layer of steel on the inside, ferrite metal on the surface, and a layer of aluminum sandwiched between the 2 for optimal warmth conductivity (steel alone doesn't conduct heat evenly). Lesser-quality stainless steel is normally only one layer of austenitic chrome steel.
Aesthetically speaking, stainless-steel is a clever alternative should you prefer to display or cling pots or pans. The clean, crisp look of all stainless-steel kitchenware can transform a mishmash of cookware into a sophisticated décor statement. Chrome steel kettles, such because the Cuisinart Tea Kettle will combine individual kitchenware into a cohesive and nice entity. Contemplate purchasing chrome steel utensils as effectively. For more about stainless steel cookware reviews stop by our web-page. Already acquired a stunning stainless-steel cookware collection? The Cuisinart Chef’s Assortment stainless pot rack could be the final touch for a kitchen, releasing up space and making those pots and pans readily accessible. Get the chrome steel cookware of your culinary desires at Macy’s!
Hard-anodized aluminum cookware is among the most popular forms of materials, despite the fact that many people do not quite perceive the construction. Arduous-anodized aluminum is apparent aluminum that has been processed in a collection of chemical baths charged with an electrical current. The result's a material that has the identical superior heat conductivity as aluminum but is non-reactive with acidic meals, resembling tomatoes, and twice as hard as stainless-steel. Two drawbacks to onerous-anodized cookware are that it's not dishwasher-secure and, because it isn't magnetic, it is not going to work with induction vary tops.
The enamel over metal technique creates a piece that has the heat distribution of carbon metal and a non-reactive, low-stick floor. Such pots are a lot lighter than most other pots of comparable measurement, are cheaper to make than stainless-steel pots, and do not have the rust and reactivity issues of forged iron or carbon metal. quotation wanted Enamel over steel is good for giant stockpots and for other massive pans used largely for water-primarily based cooking. Due to its light weight and straightforward cleanup, enamel over steel is also fashionable for cookware used whereas camping. Clad aluminium or copper edit
Distinctive specialty cookware items served a la carte to go with any cookware set are constructed of a sturdy Stainless Steel with a brushed exterior finish. Designed with an impact bonded, aluminum disk encapsulated base which distributes warmth shortly and evenly to permit precise temperature management. Handles are riveted for sturdiness and efficiency. The New Specialty Cookware is compatible for all range varieties together with induction. In addition to the multi use function, another unique feature is bottom to top interior volume markings in each quarts and metric measurement; and each bit comes with a tempered glass lid, oven safe to 350°F.
Whether you are a cooking fanatics, a professional chef or just cooking for your family you understand the importance of getting a totally stocked kitchen. Not solely do you need the proper elements, however you also need the proper instruments to get the job performed. In any type of fundamental cooking coaching lesson, you'll study that stainless steel is your new best buddy on the subject of kitchen cookware. What additionally, you will learn is that quality cooking gear does not normally come at a discounted worth. For that reason, it is important to take good care of your cookware! Listed here are some fundamentals for chrome steel care.
To fight the uneven heating downside, most stainless steel pans are laminations of aluminum or copper on the underside to spread the warmth round, and stainless steel inside the pan to supply a cooking floor that's impervious to no matter you would possibly put inside. In my expertise, this chrome steel floor continues to be too sticky to fry on, and in the event you ever burn it you get a everlasting bother spot. However, generally a chrome steel cooking surface comes in handy when you possibly can't use aluminum (see under) so I keep some around. Choose one thing with a fairly thick aluminum layer on the underside.
Well, except you’re a metals skilled and go inspect the factory where the metal is made to see whether or not their manufacturing process creates a pure austenite without corrosive materials shaped, you’re not going to know for sure whether or not the craftsmanship of your stainless is of the best quality. I feel your finest guess is to easily purchase high-quality chrome steel from the beginning, from a model with a status for good quality. However, I think I've figured out a technique that you could determine if the stainless cookware you have already got is probably reactive.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Hypergeometric_identity&oldid=293412"