text:
Lattice model (physics)

In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are quite popular in theoretical physics, for many reasons. Some models are exactly solvable, and thus offer insight into physics beyond what can be learned from perturbation theory. Lattice models are also ideal for study by the methods of computational physics, as the discretization of any continuum model automatically turns it into a lattice model. The exact solution to many of these models (when they are solvable) includes the presence of solitons. Techniques for solving these include the inverse scattering transform and the method of Lax pairs, the Yang–Baxter equation and quantum groups. The solution of these models has given insights into the nature of phase transitions, magnetization and scaling behaviour, as well as insights into the nature of quantum field theory. Physical lattice models frequently occur as an approximation to a continuum theory, either to give an ultraviolet cutoff to the theory to prevent divergences or to perform numerical computations. An example of a continuum theory that is widely studied by lattice models is the QCD lattice model, a discretization of quantum chromodynamics. However, digital physics considers nature fundamentally discrete at the Planck scale, which imposes upper limit to the density of information, aka Holographic principle. More generally, lattice gauge theory and lattice field theory are areas of study. Lattice models are also used to simulate the structure and dynamics of polymers.

Mathematical description
A number of lattice models can be described by the following data:   A lattice $$\Lambda$$, often taken to be a lattice in $$d$$-dimensional Euclidean space $$\mathbb{R}^d$$ or the $$d$$-dimensional torus if the lattice is periodic. Concretely, $$\Lambda$$ is often the cubic lattice. If two points on the lattice are considered 'nearest neighbours', then they can be connected by an edge, turning the lattice into a lattice graph. The vertices of $$\Lambda$$ are sometimes referred to as sites.  A spin-variable space $$S$$. The configuration space $$\mathcal{C}$$ of possible system states is then the space of functions $$\sigma: \Lambda \rightarrow S$$. For some models, we might instead consider instead the space of functions $$\sigma: E \rightarrow S$$ where $$E$$ is the edge set of the graph defined above.  An energy functional $$E:\mathcal{C}\rightarrow\mathbb{R}$$, which might depend on a set of additional parameters or 'coupling constants' $$\{g_i\}$$.

Examples
The Ising model is given by the usual cubic lattice graph $$G = (\Lambda, E)$$ where $$\Lambda$$ is an infinite cubic lattice in $$\mathbb{R}^d$$ or a period $$n$$ cubic lattice in $$T^d$$, and $$E$$ is the edge set of nearest neighbours (the same letter is used for the energy functional but the different usages are distinguishable based on context). The spin-variable space is $$S = \{+1,-1\} = \mathbb{Z}_2$$. The energy functional is
 * $$E(\sigma) = -H\sum_{v\in\Lambda}\sigma(v) - J\sum_{\{v_1,v_2\}\in E}\sigma(v_1)\sigma(v_2).$$

The spin-variable space can often be described as a coset. For example, for the Potts model we have $$S = \mathbb{Z}_n$$. In the limit $$n\rightarrow \infty$$, we obtain the XY model which has $$S = SO(2)$$. Generalising the XY model to higher dimensions gives the $$n$$-vector model which has $$S = S^n = SO(n+1)/SO(n)$$.

Solvable models
We specialise to a lattice with a finite number of points, and a finite spin-variable space. This can be achieved by making the lattice periodic, with period $$n$$ in $$d$$ dimensions. Then the configuration space $$\mathcal{C}$$ is also finite. We can define the partition function
 * $$Z = \sum_{\sigma \in \mathcal{C}}\exp(-\beta E(\sigma))$$

and there are no issues of convergence (like those which emerge in field theory) since the sum is finite. In theory, this sum can be computed to obtain an expression which is dependent only on the parameters $$\{g_i\}$$ and $$\beta$$. In practice, this is often difficult due to non-linear interactions between sites. Models with a closed-form expression for the partition function are known as exactly solvable.

Examples of exactly solvable models are the periodic 1D Ising model, and the periodic 2D Ising model with vanishing external magnetic field, $$H = 0,$$ but for dimension $$d>2$$, the Ising model remains unsolved.

Mean field theory
Due to the difficulty of deriving exact solutions, in order to obtain analytic results we often must resort to mean field theory. This mean field may be spatially varying, or global.

Global mean field
The configuration space $$\mathcal{C}$$ of functions $$\sigma$$ is replaced by the convex hull of the spin space $$S$$, when $$S$$ has a realisation in terms of a subset of $$\mathbb{R}^m$$. We'll denote this by $$\langle\mathcal{C}\rangle$$. This arises as in going to the mean value of the field, we have $$\sigma \mapsto \langle \sigma \rangle := \frac{1}{|\Lambda|}\sum_{v\in\Lambda}\sigma(v)$$.

As the number of lattice sites $$N = |\Lambda|\rightarrow \infty$$, the possible values of $$\langle \sigma \rangle$$ fill out the convex hull of $$S$$. By making a suitable approximation, the energy functional becomes a function of the mean field, that is, $$E(\sigma)\mapsto E(\langle \sigma \rangle).$$ The partition function then becomes
 * $$Z = \int_{\langle\mathcal{C}\rangle}d\langle\sigma\rangle e^{-\beta E(\langle\sigma\rangle)}\Omega(\langle\sigma\rangle) =: \int_{\langle\mathcal{C}\rangle}d\langle\sigma\rangle e^{-N\beta f(\langle\sigma\rangle)}.$$

As $$N\rightarrow \infty$$, that is, in the thermodynamic limit, the saddle point approximation tells us the integral is asymptotically dominated by the value at which $$f(\langle\sigma\rangle)$$ is minimised:
 * $$Z \sim e^{-N\beta f(\langle\sigma\rangle_0)}$$

where $$\langle\sigma\rangle_0$$ is the argument minimising $$f$$.

A simpler, but less mathematically rigorous approach which nevertheless sometimes gives correct results comes from linearising the theory about the mean field $$\langle\sigma\rangle$$. Writing configurations as $$\sigma(v)=\langle\sigma\rangle + \Delta\sigma(v)$$, truncating terms of $$\mathcal{O}(\Delta\sigma^2)$$ then summing over configurations allows computation of the partition function.

Such an approach to the periodic Ising model in $$d$$ dimensions provides insight into phase transitions.

Spatially varying mean field
Suppose the continuum limit of the lattice $$\Lambda$$ is $$\mathbb{R}^d$$. Instead of averaging over all of $$\Lambda$$, we average over neighbourhoods of $$\mathbf{x}\in\mathbb{R}^d$$. This gives a spatially varying mean field $$\langle\sigma\rangle:\mathbb{R}^d\rightarrow \langle\mathcal{C}\rangle$$. We relabel $$\langle\sigma\rangle$$ with $$\phi$$ to bring the notation closer to field theory. This allows the partition function to be written as a path integral
 * $$Z = \int \mathcal{D}\phi e^{-\beta F[\phi]}$$

where the free energy $$F[\phi]$$ is a Wick rotated version of the action in quantum field theory.

Condensed matter physics

 * Ising model
 * ANNNI model
 * Potts model
 * Chiral Potts model
 * XY model
 * Classical Heisenberg model
 * n-vector model
 * Vertex model
 * Toda lattice
 * cellular automata

Polymer physics
 Bond fluctuation model 2nd model

High energy physics
 QCD lattice model

title: Lattice model (physics)


text:
Zero sum (disambiguation)

Zero sum is a situation in which a participant's gain or loss is exactly balanced by the losses or gains of the other participants.

Zero sum may also refer to:
 * "Zero Sum" (The X-Files episode)
 * Monthly Comic Zero Sum, a monthly shōjo manga published by Ichijinsha
 * "Zero-Sum", a song by Nine Inch Nails from Year Zero
 * Zero-sum problem
 * Zero-sum thinking
 * Zero Sum Game, a 2018 novel by S. L. Huang
 * Zero-Sum: Stories, a 2023 collection of short stories by Joyce Carol Oates

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Zero sum (disambiguation)


text:
Laodicea

Laodicea may refer to:

Turkey

 * Laodicea on the Lycus, in Phrygia
 * Laodicea Pontica, in the Pontus
 * Laodicea Combusta, in Pisidia

Syria

 * Laodicea ad Libanum, near Homs
 * Laodicea ad Mare, the present city of Latakia

Other countries

 * Laodicea (Arcadia), in Greece
 * Laodicea (Mesopotamia), in Iraq
 * Laodicea in Media, former name of Nahavand, Iran
 * Laodicea in Phoenicia, former name of Beirut, Lebanon

Biology

 * Laodicea (cnidarian), a genus of hydrozoans in the suborder Conica

title: Laodicea


text:
Talk:Anthony of Sourozh

Untitled
What is Sourozh? RickK 01:15, 8 Aug 2003 (UTC)

 * Explained here --rbrwr

 * Could we include this information in the article? RickK 19:46, 9 Aug 2003 (UTC)

 * Done. – Kaihsu 21:33, 2005 Jan 22 (UTC)

Can I suggest that it is proper to entitle this article 'Metropolitan Anthony of Sourozh' and not 'Anthony Bloom'? It is customary in the Orthodox Church to title such articles about hierarchs by their episcopal names and titles - to not do so generally insinuates to an Orthodox a lack of respect for the hierarch.

 * He's more widely known by Anthony Bloom due to his publishing under that name and other public activities. Wikipedia articles are usually named by whatever is the most common reference for their subject.  — A.S. Damick  talk contribs 20:09, 15 June 2006 (UTC)

Well, not as far as I'm aware he's not. I don't think I've ever heard anyone call him 'Anthony Bloom' rather than 'Metropolitan Anthony' (now of blessed memory). He was ordained a Bishop since 1957, and I'm not sure many of the Wikipedia community remember him from his pre-episcopal days. Again, to refuse to list him primarily under his episcopal title is seen in the Orthodox Church as a lack of respect both to the man and to his episcopacy. Active disrespect is hardly compatible with neutrality?

Would it not be better to refer references to 'Anthony Bloom' to an entry entitled 'Metropolitan Anthony of Sourozh'?

External links modified
Hello fellow Wikipedians,

I have just modified 4 one external links on Anthony of Sourozh. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20071110014146/http://www.metropolit-anthony.orc.ru:80/eng/index.htm to http://www.metropolit-anthony.orc.ru/eng/index.htm
 * Added archive https://web.archive.org/web/20140516190750/http://masf.org.uk/ to http://www.masf.org.uk
 * Added archive https://web.archive.org/web/20041204154617/http://www.metropolit-anthony.orc.ru:80/eng/ to http://www.metropolit-anthony.orc.ru/eng/
 * Added archive https://web.archive.org/web/20041204102113/http://sourozh.org:80/memorial/funeral.htm to http://www.sourozh.org/memorial/funeral.htm

When you have finished reviewing my changes, please set the checked parameter below to true or failed to let others know (documentation at ).

Cheers.— InternetArchiveBot  (Report bug) 12:43, 15 October 2016 (UTC)

title: Talk:Anthony of Sourozh


text:
Talk:Binghamton, New York

Wiki Education Foundation-supported course assignment
This article is or was the subject of a Wiki Education Foundation-supported course assignment. Further details are available on the course page. Student editor(s): 5u5tudent555.

Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 15:44, 16 January 2022 (UTC)

Merge Binghamton Police Department
The article Binghamton Police Department is almost entirely uncited and largely consists of trivia about the department's uniforms and equipment. Other similar-sized cities in New York do not have a separate page for their police department: compare, for example, Utica, New York or Schenectady, New York. The list of fallen officers should be moved to List of law enforcement officers killed in the line of duty in the United States or an appropriate related page, and if any cited and notable facts remain they should be merged here. -Apocheir (talk) 20:55, 24 September 2021 (UTC)
 * No opposition, so I went through with the merge. I say merge, but there wasn't really anything worth including here. I also removed some trivia from this page about how many employees are in each department, which appear to have not been updated in a decade.
 * The notability of Binghamton Fire Department is also dubious, but that one at least has some citations. -Apocheir (talk) 18:35, 3 October 2021 (UTC)
 * Yeah, redirected Binghamton Fire Department here too. All the noteworthy content there is covered in 1913 Binghamton Factory fire. -Apocheir (talk) 21:46, 3 October 2021 (UTC)

Edit warring over unsourced content
You have twice reverted my edit where I removed the following unsourced content: "Binghamton and its surrounding area also support a variety of youth sports. Major youth sports in the area include football, lacrosse, ice hockey, baseball, basketball."

There is no mention of lacrosse in the article, and nothing to support the rest of the edit as well. Could you please revert your edit, or find a source to support it. Thank you. Magnolia677 (talk) 14:06, 15 July 2023 (UTC)

 * I did not edit war at all. I've lived in Binghamton for over 20 years. You've continued to target all my contributions personally since you had a minor disagreement with me in the Vestal article. You even went as far to threaten to have me blocked, which you've yet to do. The entire point of my additions were to introduce new information to the "youth sports" sub heading. UpstateCops (talk) 14:48, 15 July 2023 (UTC)
 * Please see WP:USCITIES, specifically, "do not include amateur teams or semi-pro teams." High school sports teams do not belong on a city article; you are welcome to move your edit to the appropriate school article.  Thank you. Magnolia677 (talk) 17:41, 15 July 2023 (UTC)
 * I didn't even add the "youth sports" subheading, somebody else did and I just elaborated on exactly what youth sports are major in this area. You'd need to delete the whole thing then if that is the case. UpstateCops (talk) 18:25, 15 July 2023 (UTC)

title: Talk:Binghamton, New York


text:
Forbidden Forest

Forbidden Forest may refer to:

 * The Forbidden Forest, a fictional woodland in the Harry Potter series
 * The Forbidden Forest, a 1955 novel by Mircea Eliade
 * Forbidden Forest (video game), a Commodore 64 computer game
 * Forbidden Forest (film), a 2004 Canadian documentary film
 * Forbidden Forest is some sacred forest in Indonesian culture
 * The Forbidden Forest, a location in Smurfs: The Lost Village and associated media.

title: Forbidden Forest


text:
Audience (meeting)

An audience is a formal meeting that takes place between a head of state and another person at the invitation of the head of state. Often, the invitation follows a request for a meeting from the other person. Though sometimes used in republics to describe meetings with presidents, the term is more usually associated with monarchs and popes.

Holy See
In the past, rigid dress codes had to be followed by those granted a papal audience. For a general audience, smart business professional attire (technically, informal wear, that is, a suit and tie or equivalent for ladies) was acceptable. The suit was to be dark, and ladies would ideally not wear slacks anywhere in the Vatican. Private audiences, however, were a much more formal affair. Ladies were to wear a long (well below the knees), black dress, with a matching hat or other head covering and veil. Exceptions were made for Catholic queens who by dispensation could wear white (privilège du blanc). Gentlemen were to wear a correct morning suit, either black or very dark gray with the traditional striped trousers, patent leather shoes, and cutaway. An ascot or equivalent formal tie would be worn with a gray waistcoat over a white dress shirt with arrow collars. If any part of the audience were to occur outdoors, a black top hat would be worn. If the function were entirely indoors, then the hat was optional for men. Both gentlemen and ladies were to wear dress gloves. Evening functions were usually white tie.

Formal dress is now normally reserved for diplomatic audiences. In the 1990s, an Irish Catholic priest provoked a controversy by claiming that the president of his country, Mary Robinson, had breached protocol by wearing jewellery and by not wearing black nor a mantilla for an audience with Pope John Paul II. The Vatican subsequently pointed out that the traditional form of dress worn for papal audiences was no longer obligatory.

Modern popes grant large audiences to crowds in St. Peter's Square or the Paul VI Audience Hall.

United Kingdom
In the United Kingdom, audiences with the British monarch are usually listed in the Court Circular, which is published daily by the broadsheet press. The Prime Minister has a weekly audience with the monarch usually every Wednesday, during parliamentary time at Buckingham Palace.

Denmark
In the Kingdom of Denmark, public audiences with the King usually take place every other Monday at Christiansborg Palace. An invitation is not required for public audiences. Private audiences of the King with heads of state, heads of government, parliamentary delegations, leaders of international organisations and military leaders are usually held at the residence palace of the Monarch at Amalienborg Palace.

title: Audience (meeting)


text:
Talk:Piano Sonata No. 14 (Beethoven)

Schnabel recording of Sonata #14
Can it be ascertained that this recording is definitely out of copyright? I ask because in 2010 I posted a sound recording from 1953, and was quickly reverted by XLinkBot for copyvio. Milkunderwood (talk) 05:36, 12 January 2021 (UTC)
 * I am not the uploader of the file, but it was recorded in 1934, and the specific source (archive.org) marks it as Public Domain. I'm not a copyright expert though. intforce (talk) 12:11, 12 January 2021 (UTC)

Year of Rellstab's comparison
The article says that Rellstab compared the opening to moonlight on Lake Lucerne in 1832. However the source itself, found here, doesn't mention a year at all. The Encyclopedia Britannica only says 1830s as do many other sources like Timothy Jones' book Beethoven, the Moonlight and other sonatas, op. 27 and op. 31. Tripler06 (talk) 20:59, 23 December 2021 (UTC)

Second movement score example
There are score examples showing the beginning of each movement. However, while the first and third movements have their examples typeset in embedded LilyPond, the second movement's example is in the form of an image made using SCORE. I request that someone who knows how to write in LilyPond make the second example consistent with the other two. Wilh3lmGo here to trout me if I do a stupid 18:17, 30 December 2021 (UTC)
 * Yes, either that or we go back to using the SCORE images for all three movements. I prefer the latter, because the SCORE engravings look more neat and professional. There's a reason why many music publishers still use SCORE and none use LilyPond. intforce (talk) 01:24, 31 December 2021 (UTC)
 * The Japanese article has a LilyPond score for the 2nd movement, plus a few more for significant other sections in the sonata. I suggest to copy them here. The advantage of LilyPond scores is of course that they give an immediate opportunity to listen to the passage, if so enabled. -- Michael Bednarek (talk) 01:50, 31 December 2021 (UTC)

Interpretation of First Movement
I am astonished to find this "dirge" interpretation as basically the only and primary one. I had always understood this to be a morphine trip. As you know, Beethoven suffered from stomach ulcers all his life as a result of his fathers abuse. It was these ulcers that killed him. So Beethoven often had recourse to morphine, as it was the most common analgesic of the age. Apparently he took too much sometimes and this is the result. A study of this works' structure in relation to that described in the thesis "Born Under Saturn" reveals that this work does indeed conform to overall structure of a morphine trip as depicted in all spheres of art from etchings of the New Jerusalem to Alice in Wonderland. Not only that, but as you read the score, each note achieves relevance with this interpretation, up to the climax where you feel you are floating in space, which is indeed the effect this piece has if it is played well. And this is obviously Beethoven's intention because he has so explicitly laid out descriptions of the main phases.

I am concerned that we are in the 21st century and this has not been mentioned. Has no one written about it ? Well if someone writes the thesis there is a Doctorate in it.150.107.175.246 (talk) 07:55, 29 January 2022 (UTC)

PS Beethoven also touches this subject in the Waldstein and the final Aria and variations. — Preceding unsigned comment added by 150.107.175.246 (talk) 08:06, 29 January 2022 (UTC)
 * Do you have any reliable sources for this? intforce (talk) 14:08, 29 January 2022 (UTC)

Commons files used on this page or its Wikidata item have been nominated for speedy deletion
The following Wikimedia Commons files used on this page or its Wikidata item have been nominated for speedy deletion: You can see the reasons for deletion at the file description pages linked above. —Community Tech bot (talk) 00:08, 8 March 2022 (UTC)
 * Klaviersonate Nr. 14 c-moll op.27 Nr.2 Mondschein - I. Adagio sostenuto.ogg
 * Klaviersonate Nr. 14 c-moll op.27 Nr.2 Mondschein - II. Allegretto & Trio.ogg
 * Klaviersonate Nr. 14 c-moll op.27 Nr.2 Mondschein - III. Presto Agitato.ogg

title: Talk:Piano Sonata No. 14 (Beethoven)


text:
Talk:Matrix norm

Old and unsigned
What is sup(x) ???? Dr. Universe (talk) 00:08, 8 June 2015 (UTC)

> sup(x) stands for the supremium or the least upper bound property of x. Themumblingprophet (talk) 01:43, 16 April 2020 (UTC)

I removed the condition that the matrix be square for the induced norm (when p = 2) to be equivalent to the largest singular value. Indeed, this equivalence is true for non-square matrices too.

The following page will be replaced by a table.--wshun 01:34, 8 Aug 2003 (UTC)

The most "natural" of these operator norms is the one which arises from the Euclidean norms ||.||2 on Km and Kn. It is unfortunately relatively difficult to compute; we have
 * $$\|A\|_2=\mbox{ the largest singular value of } A$$

(see singular value). If we use the taxicab norm ||.||1 on Km and Kn, then we obtain the operator norm
 * $$\|A\|_1=\max_{1\le j\le n} \sum_{i=1}^m |a_{ij}|$$

and if we use the maximum norm ||.||∞ on Km and Kn, we get
 * $$\|A\|_\infty=\max_{1\le i\le m} \sum_{j=1}^n |a_{ij}|$$

The following inequalities obtain among the various discussed matrix norms for the m-by-n matrix A:

\frac{1}{\sqrt{n}}\Vert\,A\,\Vert_\infty \leq \Vert\,A\,\Vert_2 \leq \sqrt{m}\Vert\,A\,\Vert_\infty $$

\frac{1}{\sqrt{m}}\Vert\,A\,\Vert_1 \leq \Vert\,A\,\Vert_2 \leq \sqrt{n}\Vert\,A\,\Vert_1 $$

\Vert\,A\,\Vert_2 \leq \Vert\,A\,\Vert_F\leq\sqrt{n}\Vert\,A\,\Vert_2 $$

German Link
This site needs to be linked to http://de.wikipedia.org/wiki/Matrixnorm

--91.113.18.247 (talk) 19:03, 5 January 2011 (UTC)

What's wrong with Frobenius norm?
Why does the article say that Frobenius norm is not sub-multiplicative? It does satisfy the condition $$\|A B\|\leq \|A\| \|B\|$$, which can be easily proved as follows: $$ \|A B\|^2_F = \sum_{i,j=1}^n |\sum_{k=1}^n a_{i,k} b_{k,j}|^2 \leq \sum_{i,j=1}^n \Big(\sum_{k=1}^n |a_{i,k}|^2\Big) \Big(\sum_{l=1}^n |b_{l,j}|^2\Big) = $$ $$=(\sum_{i,k=1}^n |a_{i,k}|^2) (\sum_{j,l=1}^n |b_{l,j}|^2) = \|A\|^2_F \|B\|^2_F $$. --Igor 21:21, Feb 18, 2005 (UTC)

Is it true that the Frobenius norm is $$\|A\|_p $$ when p=2. It seems to me that it is the $$ \|A\|_2 $$ norm that is mentioned earlier in the article. $$ \|A\|=\sqrt{\lambda_{max}A^HA} $$. Also it is also called the Hilbert-Schmidt norm, because the page for Hilbert-Schmidt norm says that it is only analogous to the Frobenius norm.--kfrance 13:40, Oct 9, 2007 (MST)

 * @KFrance, That is not true. The Frobenius norm is the Hilbert-Schmidt norm, but it is not the same as $$ \|A\|=\sqrt{\lambda_{max}A^HA} $$ (this is the 'spectral norm'). For vectors, $$ \|a\|_2 $$ is the Euclidean norm which is the same as the Frobenius norm if the input vector is treated like a matrix, but when the input is a matrix, the notation $$ \|A\|_2 $$ usually denotes spectral norm, which is not the Frobenius norm. @Igor, that is true. Lavaka (talk) 17:54, 9 July 2014 (UTC)

What happened to the article?
The above discussion suggests that the article used to be more extensive. However, the revision history of the current article shows only one edit, by CyborgTosser on 25 Feb 2005. Did something drastic happen to the article? -- Jitse Niesen 11:36, 2 Mar 2005 (UTC)

 * I'm not quite sure what happened. Apparently there used to be an article here, but the content must have been moved.  I'm not sure where and I'm not sure why, but a lot of articles link here, so I figured we needed the article.  Hopefully whoever moved the content will replace whatever is relevant. CyborgTosser (Only half the battle) 03:21, 11 Mar 2005 (UTC)

 * I don't know either. I couldn't find the old page on wikipedia with google, but I've put a copy (from a wikipedia clone) at Matrix norm/old. Lupin 13:50, 11 Mar 2005 (UTC)

 * It seems that User:RickK deleted this page after it had been vandalised. Idiot. I've asked him to restore it with edit history to a subpage if possible. Lupin 14:10, 11 Mar 2005 (UTC)

Induced norm
I'm a little confused where the article says that "any induced norm satisfies the inequality ...". Is the intended meaning that the operator norm satisfies that inequality, or are there other norms which are also known as induced norms which satisfy that inequality? If the former, it should be rephrased as "the induced norm satisfies..." and if the latter, an explanation of what is meant by an induced norm should be given. Lupin 01:24, 11 May 2005 (UTC)

 * The terms "induced norm" and "operator norm" are synonymous. I used "any induced norm" instead of "the induced norm" because there are several operator norms. One example is the spectral norm, another example arises when one takes the ∞-norm on Kn, defined by
 * $$ \|v\|_\infty = \max_i |v_i|; $$
 * the resulting operator norm is
 * $$ \|A\|_\infty = \max_i \sum_j |a_{ij}|. $$
 * I hope this resolves the confusion; feel free (of course) to edit the article to make it clearer. -- Jitse Niesen 10:23, 11 May 2005 (UTC)

Submultiplicativity
I feel that this article is quite unclear about when submultiplicativity applies. In particular, it should be made clear that for matrix norms based on vectors p-norms that for $$A\in{\mathbb{C}}^{m\times n}$$ and $$B\in{\mathbb{C}}^{n\times q}$$ that $$\|AB\|_p\leq\|A\|_p\|B\|_q$$. This is shown in Proposition 2.7.2 on the following page.

 * You are right that this could be added. So, why don't you change the article to include this? You can edit the article by clicking on "edit this page", see How to edit a page for details. Don't worry about making mistakes; you will be corrected if necessary. I look forward to your contributions, Jitse Niesen (talk) 11:24, 12 August 2005 (UTC)

 * It took me two days time to figure out that the statement on Wikipedia about submultiplicative property was misleading. As said, the submultiplicative property also holds for consistent p-norms, be it that in this case you are actually splitting $$\|Ax\|$$ in two different norms. That is probably the reason why it is mentioned that the submultiplicative property holds for square matrices only. However, in the special case of the 2-norm the definition this is wrong. But even without the special case it is misleading for the reader, as the "submultiplicative" definition is used in a much wider range than a norm that only splits in two equal norms. See page 5 of . — Preceding unsigned comment added by 94.210.213.220 (talk) 14:00, 20 September 2016 (UTC)

Update: I edited the page. Can somebody check? Does it need references? — Preceding unsigned comment added by 94.210.213.220 (talk) 14:50, 20 September 2016 (UTC)

Bad Notation
Moreover, when m = n, then for any vector norm | · |, there exists a unique positive number k such that k| · | is a (submultiplicative) matrix norm. A matrix norm || · || is said to be minimal if there exists no other matrix norm | · | satisfying |A|≤||A|| for all |A|.

Doesn't |A| specify the absolute value? Using the correct notation yields ||A||≤||A|| for all ||A||. Isn't that self evident? Furthermore m and n are not specified. Therefore I have removed this section till someone can clarify this content. It looks as if though someone partially moved content such that it's meaning was lost. —The preceding unsigned comment was added by ANONYMOUS COWARD0xC0DE (talk • contribs) 02:53, 24 December 2006 (UTC).

So sorry; don't know what I was thinking. I will just change |A| to ||A||_q and ||A|| to ||A||_p, it's clear from the sentence what |A| refereed to. I was reading a book earlier and |A| was refereed to as the determinant of A. More-over I will just add these statements back in and reword them. --ANONYMOUS COWARD0xC0DE 01:06, 29 December 2006 (UTC)

Matrix Norm not Vector Norm
Resolved *$$\|A\|_1\le \sqrt n \|A\|_2$$ *$$\|A\|_1\le n \|A\|_\infty$$ *$$\|A\|_2\le \sqrt n \|A\|_1$$ *$$\|A\|_2\le \sqrt n \|A\|_\infty$$ *$$\|A\|_\infty \le n \|A\|_1$$ *$$\|A\|_\infty \le \sqrt n \|A\|_2$$

These are properties of vectors of the form $$A\in\mathbb{R}^{n}$$ and not of the form $$A\in\mathbb{R}^{m\times n}$$. --ANONYMOUS COWARD0xC0DE 03:38, 24 December 2006 (UTC)

equivalence of norms
article is not really clear about the equivalence of norms: since we are talking about matrices of finite size, all vector norms should be equivalent. the bunch of inequalities in the bottom could (mis)lead the reader into thinking otherwise. if, in addition, submultiplicativity is required, does this change? (apparently so, the article seems to imply the Banach algebra topology is not unique.) Mct mht 14:08, 13 February 2007 (UTC)

trace norm vs. Frobenius norm
it isn't true that the trace norm, sum(sigma), is <= the Frob. norm, sum(sigma^2); e.g. suppose all sigma<1. Lpwithers 16:34, 8 October 2007 (UTC)

trace norm
The article doesn't explain why the "trace norm" is an "entry-wise norm". sattath (talk) 14:49, 23 July 2008 (UTC) Fixed. --sattath (talk) 13:02, 27 April 2011 (UTC)

Gradient of the Norm
I'm interested in learning about the gradient of the matrix norm but I can't seem to find this information within wikipedia. I guess I'm requesting a new article and I don't know where to do that, but it seems logical for this article to point me to the gradient of the norm (maybe under see also). —Preceding unsigned comment added by Arbitrary18 (talk • contribs) 01:00, 23 September 2008 (UTC)

Matrix Norm Definition
Matrix norm on the set of all nxn matrices is a real value function, ||.|| defined on this set, satisfying for all nxn matrices A and B and all real number $$\alpha$$:

 * $$\|A\|> 0$$ if $$A\ne0$$ and $$\|A\|= 0$$ if and only if $$A=0$$
 * $$\|\alpha A\|=|\alpha| \|A\|$$ for all $$\alpha$$ in $$K$$ and all matrices $$A$$ in $$K^{m \times n}$$
 * $$\|A+B\| \le \|A\|+\|B\|$$ for all matrices $$A$$ and $$B$$ in $$K^{m \times n}.$$
 * $$\|AB\| \le \|A\|\|B\| $$

@@@@

Matrix Norm Example
The two following functions are two examples of matrix norm:

$$ \left \| A \right \| _1 = \max \limits _{1 \leq j \leq n} \sum _{i=1} ^m | a_{ij} |, $$ and

 * $$ \left \| A \right \| _\infty = \max \limits _{1 \leq i \leq m} \sum _{j=1} ^n | a_{ij} |, $$

For examples: With matrix A:

$$      \begin{bmatrix} 3 & 5 & 7 \\          2 & -6 & 4 \\           0 & 2 & 8 \\        \end{bmatrix} $$

We have: $$  \left \| A \right \| _1 = |7|+|4|+|8|=19    $$

And: $$ \left \|A \right \| _\infty $$ = |3|+|5|+|7|=15

Note: In the above example $$ \left \| A \right \| _1$$ is the maximum absolute column sum of the matrix, and $$ \left \|A \right \| _\infty $$ is the maximum absolute row sum of the matrix. In addition both $$ \left \| A \right \| _1$$ and $$ \left \|A \right \| _\infty $$ are the special norm of a general norm called p-norm for vectors

@@@@

max?
In some of the definitions I wasn't sure if max should actually be the supremum. I thought a maximum is guaranteed to exist for compact sets of real numbers, but not necessarily for open sets. In the case of linear, finite-dimensional operators(open sets are mapped to open sets) wouldn't this be equivalent to the domain being compact? In the case of the induced norm that would imply (from my perspective) max in the case abs(x)<=1 and supremum in the case x not equal to zero. I am not sure if it is actually an issue or not because at least in case of the induced 2 norm, the supremum is actually part of the range. That in turn implies to me that the supremum is reached for any similarly defined induced norm because of the equivalence of norms in finite dimensional spaces. Can someone with experience maybe point out the disconnect I seem to be having? —Preceding unsigned comment added by 79.235.159.125 (talk) 18:49, 19 July 2010 (UTC)

 * The domain is usually a sphere. These are closed and bounded, and thus compact by Heine-Borel. — Preceding unsigned comment added by 79.131.226.245 (talk) 17:06, 1 August 2011 (UTC)

spectral radius
There is a statement in the article: "For a symmetric or hermitian matrix A, we have equality for the 2-norm, since in this case the 2-norm is the spectral radius of A"

I guess the equality actually holds for more general case: It holds for any diagonalizable A. (Note that symmetric/hermitian is a special case of diagonalizable matrices when the diagonalizing matrix are unitary, which in turn, is a special case of normal matrices. All these are diagonalizable.)

Trivial proof: Let A = P D P-1. Then $$\|A\| = \sqrt{\lambda_{max} (A^{*} A) } = \sqrt{\lambda_{max} ((P^{-1})^{*} D^{*} P^{*} P D P^{-1}) } = \sqrt{\lambda_{max} ((P^{-1})^{*} P^{*} D^{*} D P P^{-1}) } = \sqrt{\lambda_{max} (D^{*} D) }$$ (since the set of eigenvalues of AB is same as the set of eigenvalue of BA)

Does anybody see any problem with this argument? - Subh83 (talk | contribs) 18:47, 7 February 2013 (UTC)

 * That argument was wrong. If $$\lambda(A)$$ gives the set of eigenvalues of matrix $$A$$, then $$\lambda(A_1 A_2 \cdots A_n) = \lambda(A_{\sigma(1)} A_{\sigma(2)} \cdots A_{\sigma(n)})$$ if $$\sigma$$ is a cyclic permutation. - Subh83 (talk | contribs) 04:23, 8 February 2013 (UTC)

Thank you
This article was very useful. I was getting confused with that double-meaning notation and this article clarified it. Sorry for my English.--147.83.79.107 (talk) 15:31, 19 October 2013 (UTC)

Centralized discussion on proofs
See WT:MATH — Arthur Rubin  (talk) 17:58, 29 September 2015 (UTC)

Poor article layout
It would be much clearer if the definitions of the norms and their properties was more clearly demarcated. At present, being sub-multiplicative is defined at the top, but the fact that all induced norms are sub-multiplicative is just mentioned in passing in the discussion of induced norms. Contrast this to consistency, for which the fact that induced norms are consistent is mentioned next to the definition of consistency.

I would suggest one of the following two layouts:
 * 1) Start with the definition of a matrix norm, and the formal definition of each property that such a norm might have. Then go through the definitions of induced, Frobenius etc. norms, with clear results for each norm on which properties it does (not) possess.
 * 2) Start with the definition of a matrix norm, then go through the definitions of induced, Frobenius etc. norms as examples. Then go through the definitions of each property matrix norms might have, with clear results on which norms (do not) possess the given property.

In either approach, a table of norms and properties might help the presentation.

--cfp (talk) 20:56, 14 November 2015 (UTC)

Article contains no motivations or applications
I came here looking for an introduction to the concept of matrix norms and an understanding of why they are important and what their applications are. The article lacks any of this information - it would be very useful to have here. 36.53.254.212 (talk) 14:07, 22 December 2015 (UTC)

External links modified (January 2018)
Hello fellow Wikipedians,

I have just modified one external link on Matrix norm. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20160304053759/https://cs.uwaterloo.ca/~watrous/CS766/LectureNotes/02.pdf to https://cs.uwaterloo.ca/~watrous/CS766/LectureNotes/02.pdf

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Cheers.— InternetArchiveBot  (Report bug) 15:43, 21 January 2018 (UTC)

Frobenius norm - corrected definition
Dear fellow Wikipedians,

Previously, the definition

 * $$\|A\|_{\rm F} = \sqrt{\operatorname{trace}\left(A^\textsf{T} A\right)},$$

was given for the Frobenius norm, which only holds for real matrices (without any reference to this restirction). I now changed this, adding the correct definition (using the notation $$A^*$$ for the conjugate transpose of $$A$$, which was used in other sections in this article), and I also changed the other parts of this section accordingly. One more thing: the inequality between the induced 2-norm and the Frobenius norm is mentioned before the Frobenius norm section, so probably we should change this.

Zimboras (talk) 12:19, 10 August 2019 (UTC)

Horrible clashing notations
The clashing notations here are so confusing. I see people use ||T||_p for the Schatten norms all the time, but I don't see this notation meaning something else. For the sake of having a readable article, I would suggest we use different notations for the other ones. Do other people think the other kinds of norms take precedence for this notation?

Sam W

2607:9880:1A18:10A:64C9:2106:FDEB:3FFD (talk) 06:58, 5 June 2021 (UTC)

Hölder's inequality for matrices
The text as of 2013-03-29 claimed that $$ |A|_F \leq \sqrt{|A|_1 |A|_\infty} $$ as a matrix generalization of Hölder's inequality. It turns out this was for Schatten norm, not for induced p-norm. So I moved it to the Schatten norm section with a hint about how to derive it.

correction to the correction
All induced vector norms upper bound the spectral radius, in particular,

$$ \|A\|_2 = \sqrt{\rho(A^{*}A)}\leq\sqrt{\|A^{*}A\|_\infty}\leq\sqrt{\|A\|_1\|A\|_\infty} $$

This is an important inequality, so I think it should be re-included on this page.

User:Jfessler

Submultiplicativity?
The section on the Frobenius norm contains this sentence:

"The Frobenius norm is sub-multiplicative and is very useful for numerical linear algebra. The sub-multiplicativity of Frobenius norm can be proved using Cauchy–Schwarz inequality."

It would be a useful improvement to this article if the meaning of this submultiplicativity were to be also stated in mathematical notation.

I hope someone knowledgeable about this subject can add the appropriate inequality to the article.

title: Talk:Matrix norm


text:
Talk:Sugar

Willful and Deliberate Deception in reported Data
The footnote states "The carbohydrate figure is calculated in the USDA database and does not always correspond to the sum of the sugars, the starch, and the dietary fiber."

The problem with this is that the data still doesnt add up and is stilltherefore clearly erroneous, completely desyroying all reasonable credibillity of all the data in the table.

I would submit that if it doesnt add up the whole table should be outright removed from wikipedia in any event becauswle it clearly, by admission of the footnote, isn't factual data in any event regardless of the claimed source, unless there is a reasonable explanation as to exactly why the data is miscalculated in the table as to show that the miscalculation was not reasonably avoidable and was not intended by any party to be a willful and deliberate deception. Otherwise, I would be in favor of removing the entire table as clearly false data as to maintain a factual article and ensure strict WP:NPOV adherance. 2600:8804:6F0F:6D00:10C4:EC80:B6EF:5F7E (talk) 18:32, 27 September 2023 (UTC)

Allergy
Is there a way to add potential allergies from sugar? I have hives whenever I go above 75 grams of sugar content per day FyoCascade (talk) 05:21, 24 July 2022 (UTC)
 * The section about health effects of sugar is missing a lot of details; it doesn't even mention the effects of sugar on the immune system. Jarble (talk) 22:48, 6 April 2023 (UTC)
 * This is most probably not a reaction to sugar but a reaction to something else that is with the sugar, given that even table sugar has other things in it such that chemically isolated pure sugar is hard to come by outside the lab, and is what would be needed to establish that the reaction is to sugar and nothing else.2600:8804:6F0F:6D00:10C4:EC80:B6EF:5F7E (talk) 18:59, 27 September 2023 (UTC)

Sugar and CVD
Has the connection of sugar and CVD been discussed here before? I just came across this based on this study.

Here is a not too old editorial discussing the issue. The AHA cautions to reduce sugar intake, while more up-to-date studies found much higher thresholds for sugar intake (~ sugar being less harmful).

I did not do rigorous research on this but only browsed through some pages.

I think the possibility of a connection is in many people‘s minds and it would therefore be a welcome addition if the article could sum up the current evidence (or lack thereof). CarlFromVienna (talk) 15:44, 30 December 2022 (UTC)

 * The study you cite there really doesnt mean anything other than maybe a positive correlation, which is not enough to demonstrate a cause and effect relationship within proper application of the scientific method. In general studies using a cohort design must be viewed as particularly suspect, meaning that this needs way more research before it is due to even be considered to appear on Wikipedia, if wikipedia is to have any reasonable credibillity whatsoever.. 2600:8804:6F0F:6D00:10C4:EC80:B6EF:5F7E (talk) 18:55, 27 September 2023 (UTC)

Split proposal: sugar (product) vs. sugar (biochemistry)?
I know there's been some previous merging and splitting related to specific products like white sugar, but has anyone considered splitting off much of the chemistry into its own topic?

My thinking is that while there's a carbohydrate article, sugars are technically only a subset of carbohydrates. I know the article specifically says oligosaccharides aren't usually considered sugars, but I would think the actual biochemistry of sugars might be more adjacent to that than the other parts of this article, which are oriented more towards human use. Zar2gar1 (talk) 17:36, 26 November 2023 (UTC)

 * See Alcohol for one possible treatment. Certes (talk) 17:43, 26 November 2023 (UTC)
 * That's actually a perfect example of what I had in mind. I've noticed articles for other products / commodities don't always have the cleanest separation from their source, but this one stood out. Zar2gar1 (talk) 18:17, 26 November 2023 (UTC)
 * It would need to be Sugar and Sugar (biochemistry). I'm not convinced there's enough on the second to justify a split, but I might be wrong. The article is long, but not so long the need for a split is urgent. Johnbod (talk) 19:25, 26 November 2023 (UTC)
 * There's also precedent for that approach: Salt and Salt (chemistry). It's a question of whether the product is a primary topic. Certes (talk) 19:54, 26 November 2023 (UTC)
 * I'm fine with leaving the product the primary topic. I'm used to disambiguating most things by default, but that's just a habit, not a preference. I also agree article length isn't really an issue.
 * My rationale is more compositional, that even though all food sugars are obviously chemical sugars, they're different enough in aspect and audience to justify separate articles. Besides flowing better, distinct articles might be easier to categorize, infobox, and wikify, plus more focus might attract more future edits. Zar2gar1 (talk) 20:20, 26 November 2023 (UTC)
 * WP:TITLEDAB has useful guidance on article titles. The base name ("Sugar" rather than "Sugar (whatever)") belongs to its primary topic if any, or to the disambiguation page if there is no primary topic. Certes (talk) 20:48, 26 November 2023 (UTC)

"No previous war in history"
The quote "No previous war in history has been fought so largely on sugar and so little on alcohol" that has been somewhat nonsensically anonymized here and referred to Chris Otter, actually origins from Edwin Emery Slosson's Creative Chemistry: Descriptive of Recent Achievements in the Chemical Industries (New York: Century, 1920), p. 175. (Otter renames Slosson into "Edward" and repeats it in the index, although the book reference is correct; he also calls Slosson a "journalist", ignoring the fact that Slosson was Wyoming state chemist who had a PhD in organic chemistry, and his "journalism" was postly just popular science. It might be best to name the author of the quote and replace the reference with the original source. --Ehitaja (talk) 18:53, 26 December 2023 (UTC)

Glycation end products
This article does not mention the role of glycation end products in the health effects of sugar. Are these effects relevant to this article? Jarble (talk) 18:55, 22 May 2024 (UTC)

Cancer
There seems to be some contradiction in this article. The top section says "Excessive consumption of free sugar is associated with obesity, diabetes, cardiovascular disease, cancer and tooth decay". Then the "Cancer" section says "There is an indirect relationship between sugar consumption and obesity-related cancers". Yet the first sentence of the "Cancer" section plainly states "Sugar does not cause cancer". I've marked that sentence as contradictory, please discuss. Herpesklaus (talk) 09:05, 24 June 2024 (UTC)

 * I have deleted one source that failed. The other source points out the *indirect* link between sugar and cancer: obesity. Rephrasing the paragraph accodingly may be sufficient. CarlFromVienna (talk) 10:31, 26 June 2024 (UTC)

There is no scientific evidence that sugar causes cancer. The source doesn't fail. Here is what it says:

"Can eating sugar cause cancer? You might be wondering whether the sugar in the foods you eat can cause cancer cells to develop. The short answer is no. No studies in people have shown that reducing sugar intake prevents or treats cancer. Furthermore, no studies have shown that eating too much sugar causes cancer. In other words, there is not a direct link between sugar and cancer."

https://web.archive.org/web/20231001021316/https://www.cancer.net/blog/2021-11/does-sugar-cause-cancer

 * Hi Psychologist Guy, the source does fail for me as it redirects me to a page without the quote you posted above. Here's a screenshot of what I get. Maybe it's some geo-based redirect? CarlFromVienna (talk) 05:54, 27 June 2024 (UTC)
 * Yes the weblink no longer exists, it was removed a few months ago but this isn't an issue as we have the archive so any reader can just click on that to see what it says . Usually links after a while may be moved or removed, the internet archive is helpful here. There is sadly tonnes of online misinformation about sugar from the low-carb/carnivore diet community. In reality there is no evidence it causes cancer. There is only an indirect relationship through obesity. Psychologist Guy (talk) 15:22, 27 June 2024 (UTC)

 * Herpesklaus, there is major difference between association and cause. There is no contradiction in the article. The reference cited in the lead that says sugar is associated with cancer is this umbrella review . The data was from mostly sugar-sweetened beverages. There is no direct link between cancer and sugar. Sugar is not a carcinogen. We have very good WP:RS on this, , ,  Psychologist Guy (talk) 11:11, 26 June 2024 (UTC)

title: Talk:Sugar


text:
Wikipedia talk:Main Page/Temp2

So grey and blue. --Menchi 01:53, Aug 8, 2003 (UTC)

 * I like it that way. -- 212.229.115.84 (23:04, Aug 15, 2003 (BST))

title: Wikipedia talk:Main Page/Temp2


text:
Tancred of Hauteville

Tancred of Hauteville (c. 980 – 1041) was an 11th-century Norman petty lord about whom little is known. He was a minor noble near Coutances in the Cotentin. Tancred is primarily known by the achievements of his twelve sons.

Various legends arose about Tancred which have no supporting contemporary evidence that has survived the ages.

Ancestors
The Hauteville family was from one of the places called Hauteville (Latin Altavilla) near Coutances. This cannot be identified with certainty, and some modern scholarship favours Hauteville-la-Guichard.

Family and descendants
Between his two wives, he had twelve sons and several daughters, almost all of whom left Normandy for Southern Italy and acquired some prominence there.

With his first wife, Muriella, he had five sons and one daughter:
 * Beatrix (d. 1101), married first to Armand de Mortain, son of Robert, Count of Eu, and second to a Roger (family unknown)
 * William Iron Arm, count of Apulia and Calabria (d. 1046)
 * Drogo, count of Apulia and Calabria (d. 1051)
 * Humphrey, count of Apulia and Calabria (d. 1057)
 * Serlo (stayed in Normandy)
 * Geoffrey, lord of Hauteville, count of Loritello (d. 1063)

According to the Italian chronicler of the Norman feats in the south, Amatus of Montecassino, Tancred was a morally upright man, who would not carry on a sinful relationship and being unable also to live out his life in perfect celibacy, he remarried. With his second wife, Fressenda (or Fredesenda), he had seven more sons and at least one daughter:
 * Robert Guiscard de Hauteville, count of Apulia and Calabria (1057), then duke of Apulia, Calabria and Sicily (d. 1085)
 * Mauger, (d. 1064), count of the Capitanate (part of the Province of Foggia, within Apulia)
 * William, count of the Principate (d. 1080)
 * Aubrey (Alberic or Alvared, Alveredus in Latin, sometimes called Alvred or Alfred) (stayed in Normandy)
 * Humbert (Hubert) (stayed in Normandy)
 * Tancred (stayed in Normandy)
 * Roger de Hauteville, count of Sicily from 1062 (d. 1101)
 * Fressenda, who married Richard I (dead in 1078), count of Aversa and prince of Capua

Other Tancred of Hauteville
Tancred's great-grandson, also bearing the same name, Tancred, Prince of Galilee, was a leader in the First Crusade. The line of descent was:
 * Tancred the elder
 * son Robert Guiscard (Duke Robert d'Hauteville)
 * granddaughter Emma of Hauteville
 * great-grandson Tancred of Hauteville, who became Prince of Galilee and regent of the Principality of Antioch

title: Tancred of Hauteville


text:
Samantha Fox

Samantha Karen Fox (born 15 April 1966) is an English pop singer and former glamour model from Crouch End in North London. She has also appeared on reality television shows, and has occasionally worked as a television presenter.

Fox rose to fame as a Page 3 model in The Sun tabloid newspaper, where she appeared regularly from 1983 until 1986, becoming one of the most photographed British women of the 1980s and a prominent sex symbol of the era. She left Page 3 in 1986 to focus on her singing career. Released in March of that year, her debut single for the Jive Records label, "Touch Me (I Want Your Body)", became a top-10 hit across Europe, North America, and Australia, reaching number one in several countries.

She released her debut studio album Touch Me in July 1986, and followed it with the studio albums Samantha Fox in 1987 and I Wanna Have Some Fun in 1988, each of which produced international hit singles. At the Brit Awards 1988, she was nominated in the British Female Solo Artist category. Fox's subsequent studio albums Just One Night (1991), 21st Century Fox (1997), and Angel with an Attitude (2005) were less commercially successful, but she has continued to tour and has announced a forthcoming seventh studio album, produced by Ian Masterson.

Following rumours about her sexual orientation, Fox came out as a lesbian in 2003. She was in a long-term relationship with her former manager Myra Stratton from 2003 until the latter's death in 2015. She became engaged to Linda Birgitte Olsen in 2020 and the couple married in 2022.

Early life
Fox was born on 15 April 1966, the eldest child of John Patrick Fox, a builder, and Carole Ann Wilken, an actress and former dancer on the 1960s pop music programme Ready, Steady, Go! She grew up in Crouch End in North London, later describing her background as "a working-class market-trader family". She had one full sibling, a younger sister Vanessa, who died in March 2023, aged 50.

Fox's parents separated in 1988. Fox claims that her father, who abused cocaine and alcohol, violently assaulted her in 1991, leaving her with cuts, bruises, and two broken ribs. Prior to 1991 Fox had her father manage her career. After firing her father in 1991, Fox hired accountants to trace over £1 million that she believed he had embezzled from her accounts. Her father had moved her money into offshore bank accounts, and had not paid tax on Fox's earnings for 10 years. She sued and was awarded a £363,000 court settlement in 1995. However, the funds from the court settlement were used to settle her tax bill.

Following the legal dispute over her finances, she spoke to her father only once during the last decade of his life; he died in 2000. She has two half-siblings, Frederica and Frankie, from her father's second marriage. Fox took an interest in the theatre from an early age, attending the Anna Scher Theatre School from age five, and subsequently attending Mountview Theatre School. She also attended St Thomas More Catholic School, Wood Green. She formed her first pop band when she was 14 and signed her first record deal with Lamborghini Records at age 15.

Modelling
Fox's glamour modelling career began at age 16 when her mother took photographs of her wearing lingerie and submitted them to The Sunday People 's "Face and Shape of '83" amateur modelling contest. After Fox placed second in the contest, a photographer for The Sun invited her to pose topless for Page 3, becoming the youngest person at the time to topless model for Page 3. British publications could then legally feature topless models aged 16 and older, although the minimum age (for both topless modelling & commercial pornography) was later raised to 18 in May 2004. Fox later said she was grateful for the opportunity to break into glamour modelling, since many people in her area were unemployed or had "rough, low-paying jobs”. She had previously considered joining the police force, but at 155 cm tall, did not meet the height requirement then in place of 163 cm for female officers. Having become comfortable with sunbathing topless on European beaches, she later said that she did not consider nudity a "big deal".

Fox's first Page 3 photograph appeared in The Sun on 22 February 1983, under the headline "Sam, 16, Quits A-Levels for Ooh-Levels". She signed a four-year modelling contract with The Sun, and was named "Page 3 Girl of the Year" for three consecutive years from 1984 to 1986. She also made modelling appearances in multiple men's magazines. During her glamour modelling career, she insured her breasts for £1 million, inspired by Betty Grable insuring her legs for $1 million. In 1985, she appeared in a David Cassidy music video and modelled for an accompanying photoshoot; she later claimed that Cassidy sexually assaulted her during the shoot and again afterwards in the toilets of a restaurant.

Fox left Page 3 in 1986, aged 20, to focus on her singing career. She made occasional glamour modelling appearances later in her career, including a nude photoshoot for the October 1996 issue of Playboy magazine when she was aged 30. She is recognised today as the most popular pin-up girl of her era, as well as one of the most photographed British women of the 1980s, alongside Diana, Princess of Wales, and Margaret Thatcher.

Singing
Fox attempted to launch a music career in 1983. Her first single, credited to S.F.X., was a cover version of the 1981 Lesley Jayne song "Rocking with My Radio", released on the Lamborghini Records label. It was produced by Ian Gillan Band and Spencer Davis Group member Ray Fenwick, who also wrote the B-side, "My Old Man". Fox continued her collaboration with Fenwick, releasing her second single, "Aim to Win", under her own name in 1984. Neither of these singles was successful. Fox later said that "[Lamborghini Records] didn't really have a clue in those days and I didn't either," but said that performing live gave her useful experience for her later musical career. Songs from her early Lamborghini Records singles were included as bonus tracks on the 2012 deluxe reissue of her album Touch Me.

Two years later, Fox was invited to an open audition for Jive Records, which was seeking a "British Madonna" to record the single "Touch Me (I Want Your Body)". It offered Fox a five-album deal. Released in March 1986, the single became an instant hit, reaching the top 10 across Europe and topping the charts in multiple countries. In the UK, the song reached number 3 in the charts, and in the US it reached number 4 on the Billboard Hot 100. Her second single, "Do Ya Do Ya (Wanna Please Me)", also reached the top 10 in the UK. Her debut album, Touch Me, peaked at number 17 in the UK and charted all over Europe, reaching number one in Finland. Fox hit back at critics who gave the album mixed reviews, claiming that critics routinely disapproved of attractive artists; she called female reviewers "plain, drab women without any sex appeal" and said that many male music critics "look like Elvis Costello".

Her second album, Samantha Fox, released in 1987, featured the hit singles "Nothing's Gonna Stop Me Now" (UK #8) and "Naughty Girls (Need Love Too)" (US #3). The former song was produced by Stock, Aitken and Waterman, while the latter was remixed by US production duo Full Force. At the Brit Awards 1988, she was nominated in the British Female Solo Artist category alongside Kate Bush, Alison Moyet, Sinitta, and Kim Wilde. The award went to Moyet. Her third album, I Wanna Have Some Fun, released later in 1988, featured the hits "I Only Wanna Be With You" (UK #16) and "I Wanna Have Some Fun" (US #8). The album also charted in both the UK and US. By the end of the decade, Fox had achieved three top-10 singles in both the UK and US and three gold albums in the latter. She continued to have notable success in Australia and Europe.

Fox's record sales declined sharply after the 1980s. She released three further studio albums, Just One Night in 1991, 21st Century Fox in 1997, and Angel with an Attitude in 2005, but none produced any hits. A Greatest Hits album was issued in 2009, both in single and double CD formats. In 2012, her first four albums were remastered and reissued as double deluxe CDs by Cherry Red, with the addition of remixes and previously unreleased tracks. In 2022, Fox announced that she was working on a seventh studio album, produced by Ian Masterson, that would include a song she co-wrote with Steve Strange as well as songs co-written with Ricky Wilde, brother of Kim Wilde.

In addition to her solo career, Fox collaborated with Cris Bonacci (former lead guitarist for the all-female rock band Girlschool) and Lauraine McIntosh to form the short-lived band Sox and enter A Song for Europe, the UK's pre-selection competition for the 1995 Eurovision Song Contest. Their entry, "Go for the Heart," came in fourth place out of eight songs. Sox released the song as a single before disbanding. Fox collaborated on "Santa Maria" by DJ Milano, a 1997 cover version of the original 1995 song by Croatian-Dutch model and singer Tatjana. She collaborated with Swedish singer Günther, who recorded a cover version of her 1986 hit "Touch Me (I Want Your Body)" for his 2004 album Pleasureman. His version features new vocals by Fox, who also appeared in the music video. In August 2023, she embarked on a multi city US tour along with Bad Boys Blue and Boney M, which included stops in Boston, Los Angeles, Chicago, New York City and San Jose. The shows were put together by Los Angeles-based promoters LA Concert Group.

Film and television
Fox and Mick Fleetwood co-presented the Brit Awards 1989. Televised live by the BBC, the broadcast has been described as "chaotic" and an "epic shambles". The presenters repeatedly missed cues and made incorrect introductions, including failing to show a recorded message from Michael Jackson and introducing Boy George as the Four Tops. Fox later claimed that the presenters were given cards containing incorrect information, that the autocue was malfunctioning, that Bros fans disrupted the show with continuous screaming, and that her co-presenter was "out of it".

In 1990, Fox appeared on the American sitcom Charles in Charge as Samantha Steele, a fictional rock star. She also featured in The Match (1999), 7 Cases (2015), and the comedy horror film Sharknado 5: Global Swarming (2017). In 1995 she guest-starred in the Hindi film Rock Dancer. In 2008, Fox and her then-partner Myra Stratton took part in Celebrity Wife Swap, exchanging with Freddie Starr and his wife Donna. In November 2009, she participated in ITV's I'm a Celebrity...Get Me Out of Here!; she was voted out on day 16. In 2010, she appeared in a celebrity episode of Come Dine with Me with Calum Best, Janice Dickinson, and Jeff Brazier. In 2016, she featured in Celebrity Big Brother 18, finishing in seventh place. In 2023, she appeared on Celebrity MasterChef.

Charitable activity
In 2008, Fox donated her favourite bra to a charity auction. In 2011, she appeared as part of a campaign for LGBT charity The Albert Kennedy Trust.

Personal life
Fox had several relationships with men. She dated Australian fraudster Peter Foster but turned down his marriage proposal, later saying: "My parents had split and here was a man who was clever, manipulative and domineering. I came close to marrying him because I was so vulnerable". She was also romantically linked to journalist Kit Miller, Spanish bullfighter Rafi Camino, and Paul Stanley, the rhythm guitarist and lead vocalist of American rock band Kiss.

Fox's sexual orientation was publicly called into question in 1999, after she judged a lesbian beauty pageant. At the time, she was living with Australian-born guitarist and musical collaborator Cris Bonacci, who was rumoured to be her lover. Bonacci later confirmed that she and Fox were secretly in a relationship for four and a half years. In 2003, Fox came out as a lesbian and stated that she was in a relationship with her then-manager Myra Stratton. Fox later said she had realised she was gay by her mid-20s but had been reluctant to come out because, having dealt with stalkers, she feared the potential reactions of some fans. Fox remained in a relationship with Stratton until the latter died from cancer in 2015, aged 60. The following year, Fox began dating her tour manager Linda Birgitte Olsen, a native of Norway. The couple became engaged on Valentine's Day in 2020, after Olsen proposed to Fox, but postponed their wedding due to COVID-19 restrictions. They married on 18 June 2022 in a Eurovision-themed ceremony at King's Oak Hotel in Epping Forest, Essex.

Police arrested Fox at Heathrow Airport on 3 December 2023 after an incident that prevented the departure of an evening flight to Munich. Released on bail the following day, Fox appeared at Uxbridge Magistrates’ Court in April 2024, where she pleaded guilty to drunk and disorderly behaviour on a plane and to threatening behaviour towards a female police officer. She denied a charge of assaulting her wife during the incident, which was reported as a "drunken row" between the two women. She will stand trial on 2 September 2024.

Discography
Studio albums
 * Touch Me (1986)
 * Samantha Fox (1987)
 * I Wanna Have Some Fun (1988)
 * Just One Night (1991)
 * 21st Century Fox (1997)
 * Angel with an Attitude (2005)

Recording Industry Association of America (RIAA)
Source:

title: Samantha Fox


text:
Single-photon emission computed tomography

Single-photon emission computed tomography (SPECT, or less commonly, SPET) is a nuclear medicine tomographic imaging technique using gamma rays. It is very similar to conventional nuclear medicine planar imaging using a gamma camera (that is, scintigraphy), but is able to provide true 3D information. This information is typically presented as cross-sectional slices through the patient, but can be freely reformatted or manipulated as required.

The technique needs delivery of a gamma-emitting radioisotope (a radionuclide) into the patient, normally through injection into the bloodstream. On occasion, the radioisotope is a simple soluble dissolved ion, such as an isotope of gallium(III). Usually, though, a marker radioisotope is attached to a specific ligand to create a radioligand, whose properties bind it to certain types of tissues. This marriage allows the combination of ligand and radiopharmaceutical to be carried and bound to a place of interest in the body, where the ligand concentration is seen by a gamma camera.

Principles
Instead of just "taking a picture of anatomical structures", a SPECT scan monitors level of biological activity at each place in the 3-D region analyzed. Emissions from the radionuclide indicate amounts of blood flow in the capillaries of the imaged regions. In the same way that a plain X-ray is a 2-dimensional (2-D) view of a 3-dimensional structure, the image obtained by a gamma camera is a 2-D view of 3-D distribution of a radionuclide.

SPECT imaging is performed by using a gamma camera to acquire multiple 2-D images (also called projections), from multiple angles. A computer is then used to apply a tomographic reconstruction algorithm to the multiple projections, yielding a 3-D data set. This data set may then be manipulated to show thin slices along any chosen axis of the body, similar to those obtained from other tomographic techniques, such as magnetic resonance imaging (MRI), X-ray computed tomography (X-ray CT), and positron emission tomography (PET).

SPECT is similar to PET in its use of radioactive tracer material and detection of gamma rays. In contrast with PET, the tracers used in SPECT emit gamma radiation that is measured directly, whereas PET tracers emit positrons that annihilate with electrons up to a few millimeters away, causing two gamma photons to be emitted in opposite directions. A PET scanner detects these emissions "coincident" in time, which provides more radiation event localization information and, thus, higher spatial resolution images than SPECT (which has about 1 cm resolution). SPECT scans are significantly less expensive than PET scans, in part because they are able to use longer-lived and more easily obtained radioisotopes than PET.

Because SPECT acquisition is very similar to planar gamma camera imaging, the same radiopharmaceuticals may be used. If a patient is examined in another type of nuclear medicine scan, but the images are non-diagnostic, it may be possible to proceed straight to SPECT by moving the patient to a SPECT instrument, or even by simply reconfiguring the camera for SPECT image acquisition while the patient remains on the table.

To acquire SPECT images, the gamma camera is rotated around the patient. Projections are acquired at defined points during the rotation, typically every 3–6 degrees. In most cases, a full 360-degree rotation is used to obtain an optimal reconstruction. The time taken to obtain each projection is also variable, but 15–20 seconds is typical. This gives a total scan time of 15–20 minutes.

Multi-headed gamma cameras can accelerate acquisition. For example, a dual-headed camera can be used with heads spaced 180 degrees apart, allowing two projections to be acquired simultaneously, with each head requiring 180 degrees of rotation. Triple-head cameras with 120-degree spacing are also used.

Cardiac gated acquisitions are possible with SPECT, just as with planar imaging techniques such as multi gated acquisition scan (MUGA). Triggered by electrocardiogram (EKG) to obtain differential information about the heart in various parts of its cycle, gated myocardial SPECT can be used to obtain quantitative information about myocardial perfusion, thickness, and contractility of the myocardium during various parts of the cardiac cycle, and also to allow calculation of left ventricular ejection fraction, stroke volume, and cardiac output.

Application
SPECT can be used to complement any gamma imaging study, where a true 3D representation can be helpful, such as tumor imaging, infection (leukocyte) imaging, thyroid imaging or bone scintigraphy.

Because SPECT permits accurate localisation in 3D space, it can be used to provide information about localised function in internal organs, such as functional cardiac or brain imaging.

Myocardial perfusion imaging
Myocardial perfusion imaging (MPI) is a form of functional cardiac imaging, used for the diagnosis of ischemic heart disease. The underlying principle is that under conditions of stress, diseased myocardium receives less blood flow than normal myocardium. MPI is one of several types of cardiac stress test.

A cardiac specific radiopharmaceutical is administered, e.g., 99mTc-tetrofosmin (Myoview, GE healthcare), 99mTc-sestamibi (Cardiolite, Bristol-Myers Squibb) or Thallium-201 chloride. Following this, the heart rate is raised to induce myocardial stress, either by exercise on a treadmill or pharmacologically with adenosine, dobutamine, or dipyridamole (aminophylline can be used to reverse the effects of dipyridamole).

SPECT imaging performed after stress reveals the distribution of the radiopharmaceutical, and therefore the relative blood flow to the different regions of the myocardium. Diagnosis is made by comparing stress images to a further set of images obtained at rest which are normally acquired prior to the stress images.

MPI has been demonstrated to have an overall accuracy of about 83% (sensitivity: 85%; specificity: 72%) (in a review, not exclusively of SPECT MPI), and is comparable with (or better than) other non-invasive tests for ischemic heart disease.

Functional brain imaging
Usually, the gamma-emitting tracer used in functional brain imaging is Technetium (99mTc) exametazime. 99mTc is a metastable nuclear isomer that emits gamma rays detectable by a gamma camera. Attaching it to exametazime allows it to be taken up by brain tissue in a manner proportional to brain blood flow, in turn allowing cerebral blood flow to be assessed with the nuclear gamma camera.

Because blood flow in the brain is tightly coupled to local brain metabolism and energy use, the 99mTc-exametazime tracer (as well as the similar 99mTc-EC tracer) is used to assess brain metabolism regionally, in an attempt to diagnose and differentiate the different causal pathologies of dementia. Meta-analysis of many reported studies suggests that SPECT with this tracer is about 74% sensitive at diagnosing Alzheimer's disease vs. 81% sensitivity for clinical exam (cognitive testing, etc.). More recent studies have shown the accuracy of SPECT in Alzheimer's diagnosis may be as high as 88%. In meta analysis, SPECT was superior to clinical exam and clinical criteria (91% vs. 70%) in being able to differentiate Alzheimer's disease from vascular dementias. This latter ability relates to SPECT's imaging of local metabolism of the brain, in which the patchy loss of cortical metabolism seen in multiple strokes differs clearly from the more even or "smooth" loss of non-occipital cortical brain function typical of Alzheimer's disease. Another recent review article showed that multi-headed SPECT cameras with quantitative analysis result in an overall sensitivity of 84-89% and an overall specificity of 83-89% in cross sectional studies and sensitivity of 82-96% and specificity of 83-89% for longitudinal studies of dementia.

99mTc-exametazime SPECT scanning competes with fludeoxyglucose (FDG) PET scanning of the brain, which works to assess regional brain glucose metabolism, to provide very similar information about local brain damage from many processes. SPECT is more widely available, because the radioisotope used is longer-lasting and far less expensive in SPECT, and the gamma scanning equipment is less expensive as well. While 99mTc is extracted from relatively simple technetium-99m generators, which are delivered to hospitals and scanning centers weekly to supply fresh radioisotope, FDG PET relies on FDG, which is made in an expensive medical cyclotron and "hot-lab" (automated chemistry lab for radiopharmaceutical manufacture), and then  delivered immediately to scanning sites because of the natural short 110-minute half-life of Fluorine-18.

Applications in nuclear technology
In the nuclear power sector, the SPECT technique can be applied to image radioisotope distributions in irradiated nuclear fuels. Due to the irradiation of nuclear fuel (e.g. uranium) with neutrons in a nuclear reactor, a wide array of gamma-emitting radionuclides are naturally produced in the fuel, such as fission products (cesium-137, barium-140 and europium-154) and activation products (chromium-51 and cobalt-58). These may be imaged using SPECT in order to verify the presence of fuel rods in a stored fuel assembly for IAEA safeguards purposes, to validate predictions of core simulation codes, or to study the behavior of the nuclear fuel in normal operation, or in accident scenarios.

Reconstruction
Reconstructed images typically have resolutions of 64×64 or 128×128 pixels, with the pixel sizes ranging from 3–6 mm. The number of projections acquired is chosen to be approximately equal to the width of the resulting images. In general, the resulting reconstructed images will be of lower resolution, have increased noise than planar images, and be susceptible to artifacts.

Scanning is time-consuming, and it is essential that there is no patient movement during the scan time. Movement can cause significant degradation of the reconstructed images, although movement compensation reconstruction techniques can help with this. A highly uneven distribution of radiopharmaceutical also has the potential to cause artifacts. A very intense area of activity (e.g., the bladder) can cause extensive streaking of the images and obscure neighboring areas of activity. This is a limitation of the filtered back projection reconstruction algorithm. Iterative reconstruction is an alternative algorithm that is growing in importance, as it is less sensitive to artifacts and can also correct for attenuation and depth dependent blurring. Furthermore, iterative algorithms can be made more efficacious using the Superiorization methodology.

Attenuation of the gamma rays within the patient can lead to significant underestimation of activity in deep tissues, compared to superficial tissues. Approximate correction is possible, based on relative position of the activity, and optimal correction is obtained with measured attenuation values. Modern SPECT equipment is available with an integrated X-ray CT scanner. As X-ray CT images are an attenuation map of the tissues, this data can be incorporated into the SPECT reconstruction to correct for attenuation. It also provides a precisely registered CT image, which can provide additional anatomical information.

Scatter of the gamma rays as well as the random nature of gamma rays can also lead to the degradation of quality of SPECT images and cause loss of resolution. Scatter correction and resolution recovery are also applied to improve resolution of SPECT images.

SPECT/CT
In some cases a SPECT gamma scanner may be built to operate with a conventional CT scanner, with coregistration of images. As in PET/CT, this allows location of tumors or tissues which may be seen on SPECT scintigraphy, but are difficult to locate precisely with regard to other anatomical structures. Such scans are most useful for tissues outside the brain, where location of tissues may be far more variable. For example, SPECT/CT may be used in sestamibi parathyroid scan applications, where the technique is useful in locating ectopic parathyroid adenomas which may not be in their usual locations in the thyroid gland.

Quality control
The overall performance of SPECT systems can be performed by quality control tools such as the Jaszczak phantom.