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# Representation Theory - Science topic

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Suggest me some very new qualitative frameworks and Coding Frames for textual analysis so that these can be used to analysis data using the concepts of Representation Theory.
From my perspective, thick description approach for textual analysis is still need to apply based on the Clark and Brown method (Thimatic Analysis).
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Let $p$ be a prime number, and let $\mathbb{F}_{p}$ be a finite
field of order $p$. Let $GL_{n}(\mathbb{F}_{p})$ denote the general linear
group and $U_{n}$ denote the unitriangular group of $n\times n$ upper
triangular matrices with ones on the diagonal, over the finite field $% \mathbb{F}_{p}$. In fact $U_{n}$ is a Sylow $p$-subgroup of $GL_{n}(\mathbb{F}_{p})$ of order $p^{\frac{n(n-1)}{2}}$. Given $n_{p^{2}}$ be the number of elementary abelian p-subgroups of rank $2$ in $U_{n}$. How can we deduce the number of elementary abelian p-subgroups of rank $2$ in the whole linear group $GL_{n}(\mathbb{F}_{p})$?.
Conversely, given $N_{p^{2}}$ be the number of elementary abelian p-subgroups of rank $2$ in $GL_{n}(\mathbb{F}_{p})$. Is there a criterion deduces the number of elementary abelian p-subgroups of rank $2$ in $U_{n}$?.
In other words, what is the relationship between $N_{p^{2}}$ and $n_{p^{2}}$?.
Any help would be appreciated so much. Thank you all.
What you can certainly do is to determine the number of elementary abelian p-subgroups of rank 2 in GL(n,p) by means of computation. -- For example using the following GAP function:
NumberOfElementaryAbelianpSubgroupsOfRank2InGLnp := function ( n, p )
local G, S, cclS, cclG;
G := Image(IsomorphismPermGroup(GL(n,p)));
S := SylowSubgroup(G,p);
cclS := Filtered(ConjugacyClassesSubgroups(S),
cl->Size(Representative(cl))=p^2
and not IsCyclic(Representative(cl)));
cclG := List(EquivalenceClasses(cclS,
function(cl1,cl2)
return IsConjugate(G,Representative(cl1),
Representative(cl2));
end),
Representative);
cclG := List(cclG,cl->Representative(cl)^G);
return Sum(List(cclG,Size));
end;
Do you have a reason to believe that there is a relationship between your values N_{p^2} and n_{p^2} which can be described in some 'nice' way?
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For ThCr2Si2 structure (space group I4/mmm) sample, e.g. CaCo2As2, Co atoms occupy the 4d site - site position (0, 0.5, 0.25). Assume that the propagation vector as (0,0,0) and that a antiferromagnetic structure occurs along the c-axis. According to irreducible representation analysis using BasIreps in FullProf, there is only one possible model as shown in Fig1. However, one journal paper presented another kind of antiferromagnetic structure as shown in Fig2 which is not included in the model from irreducible representation.
So my question is that a magnetic structure model must agree with a model from irreducible representation theory?
Interesting question Dear Prof. Qingyong Ren
I guess that if the crystal obeys traslational invariance, that is, the crystal is not in an inconmensurable phase, it should follow the irreducible representation theory.
1. Crystal Symmetries: Shubnikov Centennial Papers edited by B. K. Vainshtein, I. Hargittai.
"DETERMINATION OF MAGNETIC STRUCTURES USING THE LANDAU THEORY OF SECOND ORDER PHASE TRANSITIONS" by J. Sólyom
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Representations are about things other than themselves and are intentional in the sense of being about ‘this’ or ‘that’. Because the mental representations have content, which is related to belief, intention, thought, and action, they are also intentional in the sense of being purposive. Now we may ask: What is it that distinguishes items that serve as representations from other objects or events? And what distinguishes the various kinds of symbols from one another? As for the first question, there has been general agreement that the basic notion of a representation involves things like ‘standing for’, ‘being about’, ‘referring to’, and ‘un denoting’ something else. Some theorists have maintained that it is only the use of symbols that exhibits or indicates the presence of mind and mental states. Mental representation, like beliefs and thoughts, constitutes the broad domain of cognitive science. They explain how cognition takes place in the human mind. Cognitive science (including cognitive linguistics and cognitive psychology) has brought about a cognitive revolution in the study of mind. Here, we can undertake two important developments in cognitive science. One is the representational theory of mind. For, to accept the representational theory of mind is to accept that mental representations are very much like the inter representational states of a digital computer. The other is the adoption of a computational model of mind or computational theory of mind.
In turn, two questions have to be answered in this connection: What kinds of representational systems are employed in cognition? What is machine intelligence or artificial intelligent? Fodor has answered these questions in his computational representational theory of mind (CRTM in short). The computational representational theory of mind makes a strong assumption about mental processes: Mental processes are computational processes, i.e., formal operations defined over symbols. In Fodor’s view, “computational processes are both symbolic and formal. They are symbolic because they are defined over representations, and they are formal because they apply to representations in virtue of (roughly) the syntax of the representations.”[i] The theory purports to offer a solution to the problem raised by the compositionality of propositional attitudes like beliefs, thoughts, etc.; secondly, it proposes to vindicate the strong reading of the intentional realist casual thesis regarding the mental phenomena. Again, it may be noted that the CRTM is consequently based on two fundamental assumptions; the first is Fodors’ Language of Thought (LoT) hypothesis, and the second is the view that psychological explanation that is both intentional and nomological.
[i] Fodor, J. A., Representations: Philosophical Essay on the Foundation of Cognitive Science, The Harvester University Press, Sussex, 1981, p.226.
Thanks. I shall read all these articles.
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Let R be a commutative ring with unity. Given two algebras A and A' over a field K, both with R-grading form. Prove that B:= A \otimes_R A' (the tensor product of A and A') has a R²-grading.
Any tip of how to prove this?
Dear Kelvin,
I think you meant: B=A\otimes _K A' (i.e. tensor product over the field K). Next, we should prove this by definition: if a\in A and deg(a)=r\in R, a'\in A' and deg(a')=r'\in R are two R-homogeneous elements then we assign deg(a\otimes a')=(r,r')\in R2. It remains to show that this grading  is compatible with multiplication in B=A\otimes _K A' (which is not hard).
Best regards, Yuri
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What would be the main importance you see on the study of Artinian/Noetherian Modules?
The Noetherian condition makes many results much easier to prove (or in fact  a necesary condition!) especially if the ring over which the module is defined is Noetherian.  Perhaps the single most important technical device is the following lemma:
Let M be a Noetherian module over a Noetherian ring R, then M is of finite presentation, i.e. there is an exact sequence:
R^m --> R^n --> M --> 0
in other words the module is given by a finite number of generators AND a finite number of relations.
Surprisingly, Artinian modules are much more specialised. They tend to occur when working locally at a prime ideal p  (i.e. the module is killed by some power of p, often by starting with a module M and modding out p^n M giving a filtration and exact sequences
0--> p^(k +1) M --> p^k M --> p^k M / p^(k-1) M -> 0
where the latter is a vector space over R_p/pR_p ). The Artinian module is then a stepping stone for working over the completion of R_p at p,
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The GDP per capita was as follows for these countries, expressed in current $US : Syria: 2,080/ cap in 2007 Yemen : 1,408/ cap in 2013 Burundi : 286/ cap in 2014 Belgium : 47,517/ cap in 2014 U.S.A. : 54,629/ cap in 2014 So clearly, if we just bomb Syria, Yemen and Burundi off of the map, the data would be a lot stronger for suggesting that there is no relation between poverty and violence, in keeping with your hypothesis. • asked a question related to Representation Theory Question 18 answers The only papers I found have Matroid Representations limited to matrix structures written in special occasions. I want advice on readings. Relevant answer Answer Write to Federico Ardila: • asked a question related to Representation Theory Question 12 answers When we consider two different representations of a C* algebra, physicists generally bother about whether they are "unitarily equivalent". Now, if the representations are different, the Hilbert spaces on which they act are also different. So I think the right thing they worry about should be called isometric isomorphism instead of unitarity. (A linear operator which preserves the inner product in the sense that one is careful to confirm to the way the inner product is defined in the appropriate vector space considered!) Relevant answer Answer The idea of wondering about a "unitary" operator was an attempt to see the fact that quasi free states may generate representations which are unitarily equivalent, but not always! • asked a question related to Representation Theory Question 13 answers Electronic voting is often presented as an advance for democracy, but it also raises a number of issues (secrecy, security...). Is a faster way to aggregate individual preferences a safer way? Relevant answer Answer I feel evoting is a good idea. but citizens have little trust in machines. I am a leader programmer on the evoting platform at IUIU but we need to tackle more of people's thoughts to believe in the system. • asked a question related to Representation Theory Question 12 answers I have looked at the ideas of gender schema and Stuart Hall's ideas on 'representation theories'. But I have not come across references to contemporary feminist theories in this context. Relevant answer Answer There is of course also Laura Mulvey's famous theoretical approach. Try this to analyze the wolf in Tex Avery's Little Red Riding Hood ;o) • asked a question related to Representation Theory Question 35 answers Unit vectors i, j, k are defined as the basis for orthogonal 3-space. My view is that these unit vectors are man-made and that Nature does not know about them. That is to say, the information they contain is in the minds of mathematicians, not in the machinery of Nature. I am therefore looking for a self consistent mathematical system, that asserts the orthogonal information of 3-space. Is it possible that this man-defined orthogonality is false representation that enters General Relativity which then obstructs progress in quantising gravity? Relevant answer Answer Quaternions again, and again ... Ok, let as form Clifford basis, take scalar part (grade 0) and bivector part (grade 2) and we have even multivectors basis for quaternions (just check multiplication table) . With my Mathematica code for Clifford algebra I can do any calculation on quaternions. Any! Bivectors have natural geometric interpretation and they are consequence of vector multiplication. Quaternions are ingenious step forward from 19 century, they are very suitable for rotations in 3D, but the same formalism doesn't apply to higher dimensions, there is no 5D quaternions. Clifford algebra deals easily with rotations in any dimension, using the same formalism as in 3D. For complex quaternions just use even part of 3D Clifford algebra over complex field (but Clifford 3D algebra possess complex and hyper-complex structure over real field). "Four dimensional dynamics" (van Leunen) follows naturally and easily from 3D Clifford algebra (Baylis, Sobczyk, Chappell, ...). There is no need for time dimension. Question here, as I understand it, was about Pauli matrices and vectors in 3D, and, again, quaternions are not vectors in 3D, they are squared to -1. Geometric (Clifford) algebra gives amassing possibility to express great extent of mathematical physics in one unique language and even simplify it, with many surprising consequences. Rotations are extremely simple, spinors are there, geometry is intuitive and powerful, one can forget about matrices, tensors ... Fundamental theorem of calculus in geometric algebra contains all well known theorems from vector calculus in one formula and is richer then all them together, etc. All You need is to start with is simple and intuitive: 1) there is a vector product and it is not commutative 2) vectors are squared to reals. This is legacy from 19-th cent., from Hamilton, Grassman and Clifford. Clifford died young, Gibbs published his famous book about vectors and mathematica started to grow as wild tree without proper care: many branches with too many languages. Geometric algebra contains all that branches and much more, but within just one powerful language. It is very easy to implement it on computer. Forget about nonintuitive matrices, forget about coordinates, they are easy to include after your calculations. So, dear physicists, especially young students, we all should to think seriously about our choice: to follow complicated mathematics with so many languages, or just rethink about vectors multiplication and just one powerful language. Hestenes and his followers are done so many nice work since 1960s. Here is may challenge to readers: give me any "branch" from mathematical physics and I will give You an answer within geometric algebra, or at least suggest You were to find answer, as a rule, simpler one and with unexpected consequences. For example, what is origin of a spin in quantum mechanic? Well, look at rotations in 3D geometric algebra. Spin formulas appear as geometrical consequence of properties of rotations. Famous physicist Pauli introduced his matrices to describe spin in quantum mechanics. But there is no need for matrices, just learn how to properly multiply vectors and use them: ordinary well known unit vectors are suited here much better than Pauli's matrices. This is proven mathematical fact! Multiply all base unit vectors from 3D and find interesting object: commutes with all elements of algebra, squares to -1, so, it is nice imaginary unit, but this time with geometrical interpretation. Now enjoy in quantum mechanic formalism, every appearance of "imaginary unit" will give You something interesting and geometrically clear. My prophecy for future is use of Clifford product of vectors (and Gibbs's book will be known as sideway in history of physics). • asked a question related to Representation Theory Question 17 answers Thanks to Number Theory, we had been studying numbers and their properties since a long time now. Dealing with numbers usually involves trying to find out the existence of the certain special magical powers they possess, if any. My question rotates around some of the immediate clinical aspects: Is there a general, generic, genetic manner in which numbers can be used as a memory storage unit? Is there a measure of how much information can be stored in numbers and representations of them? Is it possible to find how many numbers are there? Relevant answer Answer Also, if you are not aware of Chaitin's constant, it's something I'd recommend to you: it's almost the reverse of what you asked for: a number about which we know almost nothing! • asked a question related to Representation Theory Question 13 answers Representation theory? What else? Key words: user generated content, collective identity, self-representation I am doing my thesis on nationalism in social media. I have ideas on how to conduct my study (methods), but I don't have a concrete study framework yet. What theoretical framework can you suggest? My focus is on the posts of people that they deem as "nationalistic" and how it affects national identity in social media. I'm planning to do content analysis. Relevant answer Answer Depends highly on your specific research questions, I d say. In any case I expect that lit on nationalism is important for this kind of phd i.e. the role of the media in constructing/reproducing the nation i.e. Billig's banal nationalism, Anderson's Imagined communities etc. Sorry I cant advice on lit associating in particular social media and nationalism. But there should be recent publications on sth similar. Maybe search in journals such as 'new media and society'. • asked a question related to Representation Theory Question 9 answers Whenever we invoke a representation for a group --using a matrix group, or group of differential operators -- are we converting a problem in group variables to one in arithmetical variables (field variables, actually)? Relevant answer Answer One can always realize usual symmetry groups in the language of algebra, but forgetting the geometry of the symmetry group is not always a good idea. • asked a question related to Representation Theory Question 11 answers I have been observing the connections of differential geometry, representation theory and lie theory in general relativity, and was wondering what's essential in the view of a course on general relativity. Relevant answer Answer IMHO it is highly unlikely that RT would be an essential part of a graduate course in GR. Solid background in classical differential geometry ((pseudo)Riemannian manifolds, affine connections, tensor calculus, geodesics, etc.) and some familiarity with the calculus of variations (the Lagrangian and the Euler--Lagrange equations) are far more important. • asked a question related to Representation Theory Question 1 answer Let$G$be a finite group of even order has only one non-principal irreducible character$\chi$of degree$d$,$d\in \mathbb{N}$, with the following property (we name it$(*_d)$):$(*_d)$: There exist$x,y\in G$such that$o(x)=o(y)=2$and$\chi(x)\neq \chi(y)$. I know that$A_n$($n\geq 8$) has the property$(*_{n-1})$and$S_n$($n\geq 4$) has the property$(*_1)$. Now, for a suitable$d\in \mathbb{N}$, I have a question: Is there any other solvable example of groups$G$with the property$(*_d)$? Relevant answer Answer Notice that the group$S_4$also has property$(*_2)$, because there is a unique irreducible representation$\chi$of degree 2, which takes value 0 on the conjugacy class of transpositions, and value 2 on the conjugacy class of the product of 2 disjoint transpositions. I'm sure that there are plenty of other examples. • asked a question related to Representation Theory Question 8 answers a) G has at least two distinct conjugacy classes which include at least two [non-involution] elements of group and inverses of them. In other words, there exist two conjugacy classes of G, C_1 and C_2, such that: If a \in C_1, b \in C_2 then a^{-1} \in C_1, b^{-1} \in C_2 where the generated subgroups from these elements have the same cardinality. In other words: Card( < a , a^{-1} > )= Card( < b , b^{-1} > ). OR b) G has at least two distinct conjugacy classes of involutions. Relevant answer Answer Ok, so you really need a group G that does not satisfy neither condition a) nor b). Consider the nonabelian group of order 21. This group has a presentation$G=<xy|x^7=1=y^3, y^{-1}xy=x^2>. You can easily see that G has the following 5 conjugacy classes: {1}, {x, x^2, x^4}, {x^3,x^5,x^6}, {yx^j|j=,0...,6}, {y^2x^j|j=,0...,6}.