Science topics: MathematicsAlgebra

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# Algebra - Science topic

For discussion on linear algebra, vector spaces, groups, rings and other algebraic structures.

Questions related to Algebra

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Greeting everyone, I introduce myself as a chemist and fine artist with a strong interest in the Theory of Groups, particularly in the history of this theory. I will be delighted with the opportunity to learn some digital publishing on the subject.

To share the happiness and the other issues apredizado unavoidable already receiving thank this group.

Bruno Monteiro.

See also a recent book by Ian Stewart, Symmetry: A Very Short Introduction (http://books.google.com/books?id=BBdG76882IUC) which discusses groups in the broader context of symmetries.

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Let G be a group. and H is a finite subset of G which is closed under the binary operation of G.

Can anyone suggest a book?

Every element $x$ of $H$ induces by left multiplication an injective (thus bijective as $H$ is finite) map of $H$ into itself. As $x$ lies in $H$, one can find an element $y$ of $H$ such that $xy=x$, so $y=1$, and thus $H$ contains the identity element. Similarly, one can find $z$ in $H$ such that $xz=1$, thus $z=x^{-1}$ lies in $H$.

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It is known that a semisimple Banach algebra A whose multiplication is continuous (in both variables) with respect to the weak topology, is finite dimensional (see M. Akkar, E. Albrecht, L. Oubbi; A further characterization of finite dimensional Banach algebras; Preprint 1997). It is also known that, in a radical Banach algebra, the multiplication may be weakly continuous (any Banach space with the trivial multiplication). It may also happen that the multiplication in such an algebra is not weakly continuous (Take any infinite dimensional radical Banach algebra without any maximal ideals. see L. Oubbi, Weak topological algebras and P-algebra property; Mathematics Studies 4, Proceedings of ICTAA 2008, Estonian Mathematical Society, Tartu 2008, pp.73-79). Therefore the following question occurs : Which radical Banach algebras have a weakly continuous multiplication?

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Some physicists say”everything is quantum”? Why would they say so? And what is the meaning of this sentence? No one doubts that quantum theory is successful. But from this statement it does not follow that everything is quantum! Therefore these physicists are making logically unjustified conclusions. Do they use quantum logic to ascertain conclusions that are only probable?

The essence of the quantum formalism is algebra. A generic algebra, for instance a von Neumann algebra, has a nontrivial center – consisting of those elements that commute with other elements. The elements of this center correspond to what we may call „classical observables”. Algebras with trivial center are special; they are called „factors”. Why should we assume that algebra is governing our world, if there is such has a nontrivial center? What is the basis of such a bold assumption?

It is true that every algebra can be decomposed into factors. It is true that every algebra can be factored by its center. But it is not true that such a quotient contains all the information contained in the original algebra. Some information is lost. Why should we lose information?

Or, in easier terms: wave functions in quantum theory depend on parameters: space, time, and other numbers. These parameters are classical, not quantum. Of course operators of multiplication by functions depending on these parameters belong to the quantum formalism, but not the parameters themselves. Can a theory be constructed that has no classical parameters at all? No space, no time, no structure, no „nothing?” In such a theory nothing would ever be deduced.

If so, why not accept that once the dream of „everything is quantum” is contradictory and self-destructive, why not to start with a more reasonable assumption that not everything is quantum and draw the consequences of such an assumption? If not everything is quantum, the what exactly is it that is not quantum? Space? Time? Group? Homogeneous space? Some geometry that organizes the algebra structure?

That is a nice story, Marcoen. I am reasonably happy that quantum theory is the final answer to at least some of the fundamental questions. One might query whether 'thing' is the right word, since it brings to mind 'objects', which are very dubious entities. James Ladyman said Every Thing Must Go and I tend to agree.

Maybe all there is is dynamic connections, and experiences. And maybe dynamic connections are experiences and experiences are dynamic connections, depending on the point of view or choice of protagonist. If so, I suspect they are all quantised. And maybe this is even an analytic truth if to be A connection is to be something discrete and therefore in some way discontinuous from the rest - quantised?

A researcher

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Explicitly.

Having trouble with the algebra homework?

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GAMMA NEAR-RING is an algebraic system which is a generalization of the algebraic systems GAMMA RING and NEAR-RING.

Γ- Near- Fields and bi-ideals, Commutativity in Prime Gamma Near-Rings with Permuting Tri-derivations, GENERALIZED PSEUDO COMMUTATIVE GAMMA NEAR-RINGS, PRIME GAMMA-NEAR-RINGS WITH (σ, τ)-DERIVATIONS, Fuzzy Ideals , and many more

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P ≡ 1 (mod 48),

2^(p-1)/4 ≡ −1 (mod p), and

6^(p-1)/4 ≡ 1 (mod p).

Humm the best person to answer this question is Pieter Moree : he is on this site.

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In the definition of a group, several authors include the Closure Axiom but several others drop it. What is the real picture? Does the Closure Axiom still have importance once it is given that 'o' is a binary operation on the set G?

The definition of a group is a set G together with a function "o" from GxG to G satisfying the properties of associativity, existence of a neutral element, existence of inverses. Saying that o maps GxG to G is called "closure".

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I know that in the "Earliest known uses" website the origin is traced back to Viete in the 16th/17th century. The given source, (Cajori 1919, page 139) which is Florian Cajori (1919): "A history of Mathematics" (available at archive.org) is correct, however, nothing is substantiated about the context.

* Does someone have access to the 1591 text to confirm the correctness of the Cajori reference?

Around 1585, Stevins in a french text uses "multinom*" as umbrella notion for "binom*" and "trinom*" in the glossary. These are used exclusively in the meaning "sum of algebraic terms", as found inside a root or power. It can be guessed that Viete knew about the text of Stevin and altered "multinom*" to "polynom*" for essentially the same usage, roots and powers of aggregated algebraic terms.

*Can anyone shed light on the question if Viete used "polynomial" outside of a glossary or how widespread the use of "polynom*" was in texts before the 19th century?

At some time in the late 18th or early 19th century, the "polynomial" began to be used in the modern sense. Perhaps via multinomial and polynomial coefficients. In algebra, as relates to the fundamental theorem of algebra, "polynomial" was not or rarely used until the end of the 19th century. Instead, "entire (rational) (algebraic) function" or just "function" or "equation" with an indication of the degree is used. For instance, Weierstrass in 1891 still used "entire (rational) function" ("ganze (rationale) Function") for polynomials.

* Is there one document that started the shift in usage, or was it gradual?

* As a related question, when did "entire function" stop to indicate a polynomial and start to indicate a power series with infinite convergence radius?

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I want to read about Pythagorean trees, does anybody suggest good books or related literature.

Basic literature which is helpful for the masters project

A researcher

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I wanted to know how many distinct number of subgroups are there for a group containing countable number of elements. Z under addition is countable and it has countable number of subgroups.

There must be at least countably many non-trivial subgroups. But there could also be uncountably many subgroups -- e.g. if G is the set of all finite subsets of N equipped with the operation of symmetric difference.

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In the attached file I have managed to show using the Wronskian the value of A_1B_2 -A_2B_1. How we can now find the value of A_1, A_2, B_1 and B_2?

My main interest is to find the value of A1,A2,B1 and B2. I have been given a Wronskian only and by using this I will have to find these values.

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Boolean reasoning appears to get much less attention than Boolean minimization in system design work. Are there any standard, classical examples of Boolean reasoning that are used to show their application in system and logic design?

Robert..

Thanks for the information.

I have downloaded the books and will take a look when I get a chance.

My specific interest at this time is the evaluation and exploration of Boolean functional consequents. This type of Boolean reasoning has received very little attention in the past.

It appears to me that some techniques for Boolean reasoning that were used in a manual process may be converted over to a computer based application.

Further, my current research indicates that the this type of logic may be applied with out computer languages designed specifically for logic computation (Prolog) or the use of rule based engines written in Lisp, Scheme, or other language.

This approach makes Boolean reasoning computing feasible with standard computing tools.

I am using SAGE math and Javascript at this time for the calculations and the initial results are good.

Have fun,

Joe

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Where Zr is the ring of integers modulo r.

You are asking about the order $Aut(G)$, where $G$ is a direct product of $n$ copies of $Z_r$. It is easy to see that $Aut(G)$ is a direct product of $Aut(G_p)$, where $G_p$ is the p-component of $G$, so without loss of generality, one may assume that $G$ is a p-group, or equivalently, that $r$ is a prime power (say $r=p^m$, for some k). The order of such an automorphism group is known to be $p^{n^2(m-1} (p^n-1) (p^n-p)...(p^n-p^{n-1})$ (see C. Hillar and D. Rhea, Automorphisms of an abelian p-group, Amer. Math.

Monthly, 114, 917–922 (2007)).

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I know only the solution using Tailor series

Have you tried a Pade approximant?

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I have attached my definition and some results about this.

Dear Akbar Sadighi,

The definition you gave is known in Group Theory. In fact, in the article by Kiran S. Kedlaya, Product-Free Subsets of Groups (Amer. Math. Monthly, 105, No. 10 (1998), 900-906), are studied the cardinality of the largest ucs subsets (there, these subsets are called "product-free") in finite abelian groups and compact topological groups. In this article you can also find some open problems. The subject seems to be interesting. It is related to some combinatorial questions (Schur Theorem, Ramsey Theory) and I encourage you to continue working on it.

Best wishes,

Alessio Russo

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By using comparison tests, how we can explain the convergence and divergence of these two integrals?

First one obviously diverges as far as the function under the integral is equivalent to \frac{constant}/{1 - w} as w \to 1-0.

The second one converges iff 0 \neq [\alpha_1; \alpha_2]

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" There is no x ∈ A such that f(x,a)<g(x,a) for all a ∈ B. "

Hello again,

In your new statements, you force the inequality for all a in B... But this is not needed, since it suffices only one a in B with f(x,a) >= g(x,a) for the given x in order for the original proposition to hold.

Best regards,

Domingo

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What would be the best option for reduction of 6 variable Boolean functions?

Quine Mccluskey is one of the feasible solution for 6 variable solutions.

you can read more here

or watch this

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Let F:R

^{2 }--> R^{2}be a homogeneous function of degree k in Z_{+}(i.e., F(tx)=t^{k}F(x), t>0) such that F^{-1}(0) = {0}. What conditions we should impose on F susc that F|_{R^2\ {0}}: R^2\ {0} --> R^2\ {0} has topological degree less than or equal to k?This is true if F is a homogeneous polynomial!

An example that fails is $F(r\,\cos(\theta),r\,\sin(\theta))=(r\cos(n\theta),r\,\sin(n\theta))$!

I guess it is sufficient that F is smooth everywhere, in particular at 0. Then its first nonvanishing Taylor polynomial at 0 must be homogeneous of degree k (in particular k > 0), and F is equal to that polynomial.

Best regards

Jost

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Let group G act transitively on a set X. Let x\in X and H=Stab(x). Let G act on X\times X via g(x_1,x_2)=(gx_1,gx_2) for any g\in G. Prove that all G-orbits in X\times X are in a bijective correspondence with all H-orbits in X.

Here's a hint of where to map the orbit G(x_1,x_2) for (x_1,x_2) € X x X:

Since the action is transitive there exists a g € G with gx_1 = x, So the orbits G(x_1,x_2) and G(x, x_2*) are equal for x_2*=g^{-1}x_2. Now define a map mapping the G-orbit G(x_1,x_2) to the H-orbit Hx_2*. Prove that it is well-defined, injective and surjective.

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There has been various generalizations to rings, like P rings, near rings, Von Neumann rings, Boolean semirings, Boolean like rings, Boolean lke semi rings and so on. So my question is how far these generalized systems are crucial especially for applications, if any, that many people have been struggling on these theories.

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This problem looks so easy and classic. Just search a cuboid with every side, face diagonals, and inner diagonals are integers. But, I have still not getting any cause why it still unsolved when the hardest problem like Poincare Conjecture has been solved? Is this problem too easy for professional mathematicians or what?

Because you have to find some numbers with very rare properties. Take the easiest case possible: a rectangular triangle (this is by far no perfect cuboid, but it is a very simple example that shows the problem). If you try to find such an triangle, where Pythagoras holds, a² + b² = c² there are not so much numbers that solve this equation as integers.

For your perfect cuboid you also need to solve:

a² + b² = d²

a² + c² = e²

b² + c² = f²

a² + b² +c² = g²

You have to solve all of these equations within the set of integers, but by adding one of those formulars you filter out more and more elements of your integers. Hence you reach higher and higher numbers which are very hard to handle even by supercomputers.

The final problem is then to prove whether you have a very small set of solutions or if there is no solution at all.

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Let Ext(C,A) be the group of extensions of A by C and Pext(C,A) the subgroup of Ext(C,A) containing pure extensions. Is there a proper subgroup of Ext(C,A) containing Pext(C,A)?

See the linked to PDF for reference.

Proposition 5.6 gives you the information to build an example that it is indeed possible.

From Proposition 5.6.1, you get a way of writing $Pext$ as an intersection of kernels of maps $Ext$ and another group, so if you can create an example where this intersection is non-trivial, then you've got your answer.

On the other hand, maybe you're more familiar with inverse limits. Then in Proposition 5.6.3 there is a short exact sequence (See 5.7). If the inverse limit on the right hand side is non-zero, then you have an example where $Pext$ can be identified with a proper subgroup of $Ext.$

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The structure of a lattice of varieties of lattices is considered. In particular lattice of subvarieties of variety of modular lattices.

Dear prof.

should you please explain your question for me. What do you want about the lattice of subvarieties?

best wish.

o.zahiri

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What is the best reference on algebraic theory of linear and nonlinear systems that addresses topics such as manipulations of "augmented systems", decompositions, similarity transformation, and manipulations of systems matrices

Dear Ibrahim,

This is on Nonlinear System while my question is about the algebraic manipluation of the multivariable linear systems like e.g. augmented matrices ...etc

A researcher

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Given the values matrix B and variables constrained vector e, find vector e that B.e=0

If the constraints are as you say, then e=0 is indeed a solution. If you don't like it , I'm afraid the universe won't change - you need to change the question.

As to "the linear algebra solution", it is "e = 0", as given. In general it will be anywhere inside the intersection of spaces which seem to amount to your constraints. Since each of the constraints you have given is a full vector subspace (not affine subspace .. i.e. it has 0 and the sum and constant multiplier of any of its contents), the intersection is another full vector subspace, and you can describe the solution as "any linear combination of a covering set of vectors". If you can find n independent vectors that solve the problem, then any linear combination of those n is "the answer"

If, by some chance, you didn't mean to create a question that has that as answer, then you need to say something else :-). It strikes me that you are perhaps not expressing yourself well enough.

Do you mean perhaps that you want a closed form solution in terms of the coefficients of B and whatever is the constraint matrix? Uh, uh, that's not going to fly, in general.

If you have some physical insight, that is much more likely to yield you a result. You say you have an "incidence matrix", for example - that sounds promising. What is that? Do you mean "adjacency matrix"? What does a solution to Be=0 mean, when B is an "incidence matrix"?

As it stands, my original answer stands - simple linear constraint problem. So what's the question?

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I want to know how can I arrive at commutative relations (16), by starting from killing vector (14) and its constraint (15) ?

With some guessing one finds out that:

- the generators P^i (i=1,2), P^{+}, P^{-} correspond to parallel translations (i.e., when the matrix Q in (14) is zero)

- the generator J_z corresponds to rotations of <P^1, P^2>, i.e. C=0 and Q_{ij} = i \epslion_{ij}

- the generators T_i correspond to C=0 and nilpotent Q sending P^{-} to -i P_i and P_j to - i \delta_[ij} P^{+}.

The elements J_z and T_i form a 3-dimensional subalgebra of the Lorentzian Lie algebra so(1,3), which annihilates a null-vector b_\mu. This is exactly what the constraint (15) says.

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the general form is Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0 where A,........J are constants. I need to solve the equation for x, y and z.

if the equation take the form A(x-x0)^2+B(y-y0)^2+C(z-z0)^2+D(x-x0)(y-y0)+E(x-x0)(z-z0)+F(y-y0) (z-z0)+G(x-x0)+H(y-y0)+I(z-z0)+J=0

how can get the value of x0, y0 and z0 in case I have the values of all constants at several points (x,y,z)?

This equation defines a surface in 3-dimentional space. Depending on coefficients, it may have infinite number of solutions, unique solution, or no solution at all.

First, you have to find out, what type of surface it might be. Is it a sphere, ellipsoid, hyperbolloid, paraboloid, etc? There are 17 types of such quadratic surfaces. Some of them are so called imaginary. Yes, you have to find the canonical form of the surface, as it was mentioned in a previous answer. Please, clarify the second part of your question. In your original equation you already have x0=y0=z0=0.

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What are the current research interests in linear algebra? Can anyone tell me specifically?

My main interest is in matrix partial orders in Linear Algebra and a good amount of activity is going on in this area, I can help you with people who are currently working here.

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I am looking for a reference giving the decomposition of a prime p in the maximal real subfield of a cyclotomic number field Q(zeta_m). Something along these lines: If p is a prime not dividing m, then let f be the smallest positive integer such that p to the f is congruent to 1 or -1 modulo m. Write f.g=phi(m)/2. Then (p) splits into g different prime ideals of norm f (plus what happens if p divides m).

Dear Pieter,

I think you can find your answer in the book of Daniel A. Marcus, "Number Fields", Springer Universitex, 19977, chapter 4, exercise 12, p.118 (see also ex. 14)

Best, T. NQD

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Special interest when A and X are over non-commutative linear polynomials.

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If true, what is the evidence?

Let F be a field and R = F^{2\times 2} two by two matrix ring over F. Then R is simple ring, therefore semiprime(= I is an ideal and I^2 = 0, then I=0). Consider a to be the matrix in R having 1 at the entry (1,2) and zeros elsewhere. Then a is nonzero but a^2 = 0 . I think it is OK. Yours. A.Harmanci.

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Definition of expectation.

Let us begin from a field σ(ρ) generated by one function, called ρ, on a measure space with a measure μ. We always consider σ(ρ) complete with respect to μ, that is, with all zero measure sets added. Denote the restriction of μ to σ(ρ) by μ_ρ. Denote further on the indicator of set E by 1_E.

By the very definition, the conditional expectation f_ρ of a function f with respect to σ(ρ) is the Radon-Nikodym derivative f_ρ = dfμ / dμ_ρ. How to compute it?

First note that σ(ρ) is generated by sets E_{ (a,b) } = {a<ρ<b} for all real numbers a<b, that is, σ(ρ) is the μ-completed pull back of the Borel filed on the line, under the set map ρ^{-1}. All functions measurable with respect to σ(ρ) are pointwise limits of simple functions which are linear combinations of indicators 1_{ E_{ (a,b) } } of sets E_{ (a,b) }. However,

1_{ E_{ (a,b) } } (x) = 1_{ (a,b) } ( ρ(x) )

So all functions measurable with respect to σ(ρ) are μ-a.e. pointwise limits of Borel functions of ρ, that is, of functions of the form g( ρ ), where g is a Borel function. This is the form of f_ρ = G(ρ): a composition of a scalar function with ρ.

Now let y be a number such that for all ε >0, set E_ε(y) = { |ρ-y| < ε } is of a positive measure, that is, μ( E_ε(y) ) > 0.

Since μ (f 1_{ E_ε(y) } ) = μ (f_ρ 1_{ E_ε(y) } ) it follows that

G(y) = lim_{ ε -> 0} μ (f 1_{ E_ε(y) } ) / μ (E_ε(y) )

For y such that μ( E_ε(y) ) = 0 for some ε >0, we may define G(y) to be anything

(say, zero) since it will not change the function G(ρ).

Some argument necessary to show that G is Borel, follows from approximating ρ with step functions. I can elaborate on this if necessary.

With a field generated by several functions you proceed similarly, replacing an interval

(a,b) with a rectangular (a_1,b_1)\times (a_2,b_2).....\times(a_n,b_n).

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I would be really grateful if someone could do this in a succinct manner that I, as a non statistician, can easily understand.

I never came across Cartesian sets although I did mathematics all my life. Do you mean Cartesian products of sets?

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Their quotients in the calculator are both "error". Math books show they are both infinity.

Dear Eddie,

you should not consider the output of a calculation programme as mathematical "truth".

Mathematically spoken 1/0 and 0/0 are not the same: While 1/0 can be obtained as limes from 1/x with lim(x) --> 0 = infinity, the term 0/0 is not defined in mathematics - hence it cannot be stated that is also infinity. You could also say that the solution set S of the equation 0/0 = x is the empty set S = {}.

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Textbook suggestions on quantum algebra for an elementary course on the topic.

Hi dear.

See for example:

1.An Introduction to Quantum Algebras and Their Applications

2. Introduction to Quantum Algebras

Search them by google

Best wishes

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I'd like to know where to learn more about this. I just was presented some piece of mathematical theory without any names being mentioned, only that it's applied in string theory:

I assume you can multiply the sequences with a scalar and you also have componentwise addition compatible with it in the sense of a vector-space. now define the equivalence relation implicitly: "for all sequences x of whole nonzero numbers: x~y if y contains zeroes and after their removal would become the same as x". please note that here no "if-and-only-if" was used, since for compatibility with vector-space operations you need some flexibility. in this setting I was shown the following proof for the statement in my question above:

the sequence <(-1)^n>_n=0,1,2,... == <1,-1,1,-1,...> I denote with S1. S2 is the sequence <(n+1)*(-1^n)>_n=0,1,2,... == <1,-2,3,-4,...>. S2 with one zero in front of it(thereby fulfilling above partial definition of the equivalence, and thereby being equivalent to the original S2), added to S2 itself results in S1. so modulo our equivalence relation we get "2*S2~S1". the sequence of natural numbers <n> denoted here by N plus the additive inverse of S2 results in something that is equivalent (with above relation) to 4*N. the actual result is <0,4,0,8,0,12,...> so the equivalence again is obvious. this gives us the second equation "N-S2~4*N". the value of S2 we get from the first equation, in terms of S1. so in the 2nd equation subtract the N and divide by 3 to get "-(S1/2)/3~N". so in terms of algebra, the sequence of Natural number is linear dependent on S1, the factor here is -1/6.

in case you forgot: my question is, which area of mathematics is at work here? what literature is there? how many dimensions does the vector space have, and what is the actual generic definition of the equivalence relation? I suspect all these questions already have been investigated somewhere, I just never encountered above proof and these ideas before. so, where should I look for that?

You might want to have a look at this: https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B

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I got one real and 2 imaginary roots. But by Descartes' Rule of signs, there are three sign changes in this equation and we are suppose to have 3 real roots. However, MATLAB gives one real and two imaginary roots. How can it be possible or how can it be derived from Descartes' rule?

Actually, Descarte's Rule of signs gives you the number of positive and negative roots.

So, if you have a function with three sign changes you may have either three positive roots OR (and that is also important) one positive root. This is also a possibility of Descartes: You can have also a number of positive roots which is your number of sign changes minus an even natural number, so in your example: 3 - 2 = 1

You can go now one step further and try to find negative roots by making f(x) --> f(-x) and reapply descartes rule and you get your number of negative roots.

If you know the number of negative and positive roots, you can check for the number of complex roots with the following method:

you know that an n-th degree polynomial (yours has n = 3) must have n roots. Thus you obtain the number of complex roots by taking the total number of roots and substract the positive and negative ones (but check before if there is no root at x = 0)!

So, n - (p+q) = number of complex roots

Here p = number of positive roots and q = number of negative ones.

For your function that means:

number of complex roots = 3 - (1 + 0) = 2 complex roots (as there are no negative ones and there is none at 0.

And this is in agreement with your solution from matlab.

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I am looking for some comprehensive, basic text about quasivarieties answering questions like "what properties of varieties also hold/does not hold in quasivarieties" etc. Ideally available online. Does anyone have some suggestion?

I would recommend you:

Algebraic Theory of Quasivarities (Gorbunov). It is a good reference for prevarieties and Quasivarieties, mainly for those results that we know are valid but we don't know where to find them.

Algebraic Theories (Manes). Good reference for categorical-related issue on classes of algebras in general.

Of course these list is not complete, there are several books on Universal algebra that you might find interesting, but I guess you can find them easily with some websearch engine.

Best,

Leo

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I know the Milnor and Tjurina numbers reflect in certain way the complexity of

a singular point for a variety defined by one polynomial V=v(<f(x_1,...,x_n>)).

Now I know there exists some generalizations (or at least attempts have been made to generalize) similar quantities for a variety defined by more than one polynomial V(f_1,...,f_n).

For example the Milnor number should generalize to object called 'Milnor Class' in Chow's group.

I have mainly 2 questions regarding these possible generalizations:

1) Does anybody know a good reference book in which such problems would have been investigated, discussed and possibly rigorously defined.

2) Are there any known algorithms to compute such objects ? And if so does anyone know an article or a book where such algorithms could have been described ? Or at least partial results ?

Hi Rogier

Thank you for very much for your answer and book references are always appreciated! Yes I know these are implemented somehow in Singular. In fact Singular is one the programs that I mainly use. Unfortunately the cases we are dealing with are not always complete intersections, although in most cases they are and are generated by codim number of generators.

Also are you aware about the Hilbert-Samuel function/multiplicity number for singularities ? Do you remember any good books where the algorithms and def.s for these would be introduced in . I would really appreciate all book tips.

Also about a question about the 'intersection multiplicity': If two varieties intersect

transversally do you remember if there are kind of 'straight' or sequence of commands to verify this with Sage, Singular, Macaulay2 or CoCoa without coding the algorithms completely yourself ? Just to save time.

Also if you remember good books where these problems are tackled it

would also be good to know !

But thanks again for you response !

regards

Samuli Piipponen

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I wonder if the existence of such examples is a conjecture or not. I believe they should exist.

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Let R be a regular ring with 1.

prove that for any two idempotents e,f€R, there exist an idempotent g€R such that Re+Rf=Rg.

It looks like a homework problem.

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Intrinsic criteria for 3X3 idempotent matrix.

what do you mean by ``intrinsic''? clearly, all eigenvalues of an idempotent matrix are either 1 or 0, so if you take any block diagonal matrix A with diagonal blocks that are either the identify or zero, then the product B . A . B-inverse is idempotent, and i suspect that a version of the converse is also true

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I'm undergraduate student and I have big interest in algebra and (especially) in mathematical logic. I got task from my lecturer to prepare a research topic for my final task. Can anyone help me to give me some research topic on these topics?

A researcher

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It is known that (e_6, sp(8,C)) forms a complex symmetric pair. However, how is sp(8,C) embedded into e_6 concretely? Or, what is the involutive automorphism for this symmetric pair?

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M. I. Kuznetsov refers to a 1-graded Lie algebra in "Simple modular Lie algebras with a solvable maximal subalgebra, Math. USSR Sb30 (1976), 68–76". Does he mean Z-graded?

I think that "1-graded Lie algebra" just means "$\mathbb{Z}$-graded Lie algebra of depth 1". This seems to be consistent with the terminology used by the same author in other papers. For instance, in the paper "M.I. Kuznetov: Lie algebras with a subalgebra of codimension p . Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 6, 1224–1247, 1439" the more general notion of q-graded Lie algebra is used with this meaning.

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The Wikipedia articles on these two subjects contradict one another.

Note: I refer to *the* Clifford algebra Cℓ_{0,3}(R), not to *a* Clifford algebra. The one with starting point R^3, the most common one in Geometric Algebra.

If I understand it correctly they are different. Both contain, as it were, two copies of the quaternions, but the multiplication tables are different.

The Clifford algebra has the right to be called a non-commutative algebra, as the phrase is usually understood, since multiplication is associative.

The algebra of the octonions is not associative but only satisfies a weaker property. Am I right? What is the connection of either with S^7 ?

I really don't know what this means, but I've retired from physics and maths, by and large. I don't see why something of order 8 is automatically octonionic. It's also interesting to me how often spacetimes are treated as fundamental. I don't think of them that way. Spinor space is my starting point, and out of this falls spacetime, the groups and particles of the standard model, an explanation for why our universe is constructed of matter (where are the antimatter frogs and asteroids?), and a whole lot more. I don't need to "try" anything, or seek elsewhere for the pieces of the standard model. The mathematics speaks; I listen. Or I used to. I no longer have any faith that it matters. Consent is now being manufactured on an industrial scale.

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Let R be the ring of integers in a number field and p an odd rational prime. In the paper "Higher class groups of orders" Math Zeitchrift 228, (1998) 229-246 , M. Kolster and R. Laubenbacher proved that p-torsion in odd-dimensional class groups of R-orders A can only occur for primes p which lie under the prime ideals of R at which A is not maximal --with consequences for group rings RG (G finite). The question arises if a similar result can be proved for even-dimensional class groups of arbitrary orders and grouprings. In the papers, "Higher class groups of generalized Eichler orders" , Communications in algebra, 33, 709-718 (2005); Higher class groups of locally triangular orders over number fields (Algebra Colloquim, 16, 1 (2009) 79 - 85, X Guo and I obtained similar characterization for p-torsion in even-dimensional higher class groups for generalized Eichler orders, e.g Eichler orders in quaternion algebras, hereditary orders as well as locally triangular orders.. Also, X Guo obtained further results along these lines in the paper " Even dimensional Higher class groups of orders" Math Zeitchrift ,261, 617-624 (2009). It is still open to obtain such a characterization(if it exists) for arbitrary R-orders and hence groups rings RG for even dimensional higher class groups.

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This question arises from my paper " Higher Algebraic K-theory of p-adic orders and twisted polynomial and laurent series rings over p-adic orders" Communications in Algebra, 39, 3801-3812 , 2011 where a partial answer is given for a more general question involving p-adic orders in p-adic semi-simple algebras. In that paper, I proved that the answer is afirmative if the p-adic semi-simple algebra splits as a product of matrix algebras over p-adic fields. For n = 1, the answer is in the afirmative and is classical (due to C. T .C Wall). When G is a finite p-group, the answer is also in the affirmative for all n >1 (See A. O. Kuku "Finiteness of Higher K-groups of orders and group-rings" K-theory 36, 51-58 2005.

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Suppose that G is a complex semisimple Lie group, then every finite-dimensional representation of G is completely reducible by Weyl's unitary trick. But how about the infinite-dimensional representation? Is it still completely reducible?

Given G is complex semi simple Lie Group. Then G can be expressed as intersection of Gi for all i = 1,2,3..... upto infinity so that no Gi is in common and in all Gi G is a subset i.e., G is completely reducible by Weyls unitary trick.

Infinite dimensional representation as you come to know from topological spaces every open cover has a finite sub cover i.e., G is subset of union of Gi i=1,2,3,...,n and is subset of Gj for all j= 1,2,3,......, up to infinity so Gj j= 1,2,3,....,infinity is of dimension infinite definitely infinite dimensional Gj j= 1,2,......, up to infinity is completely reducible why because every compact subspace of a compact set is compact in compactness.

In similar manner, Suppose that G is a complex semisimple Lie group, then every finite-dimensional representation of G is completely reducible by Weyl's unitary trick. But the infinite-dimensional representation Is completely reducible. so that G = H cap K is a complex semi simple Lie Group take it H as Hi i=1,2,3,......,infinity and Gi i=1,2,3,.....infinity so that we can explain infinite-dimensional representation Is completely reducible.

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While reading, some algebra, an idea came to my mind that why don't we study the relationship between Mathematics and Music? Especially the notes? Can anybody help or shall we work?

This is a seminar dedicated on this topic

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An urn has N balls, each marked with an integer number from 1 to N. For each number in [1,N], there would be exactly one ball marked in the urn. If we randomly pick n balls with replacement from the urn, then we know that there would be N^n (N power n) possible ways of picking n balls. The question is, how many of such ways we’ll have at least one occurrence of c or more successive picks having consecutive increasing numbers?

Example:

For N=9 (range of numbering 1-9, also the number of balls in the urn is 9), n=7 (number of picks), and c=3(number of successive picks with consecutive increasing numbers)

Some of the sequences that are included for answer to this specific example:

i) 1,2,3,1,2,3,1 (2 occurrences of 3 consecutive numbers)

ii) 1,2,3,3,4,5,6 (1 occurrence of 3 consecutive numbers and 1 occurrence of 4 consecutive numbers)

iii) 8,9,9,9,4,5,6 (1 occurrence of 2 consecutive numbers and 1 occurrence of 3 consecutive numbers)

Some of the sequences that are NOT included for answer:

i) 1,2, 4,5, 7,8, 8 (3 occurrences of 2 consecutive numbers. We need at least one occurrence of c=3 or more consecutive numbers)

There are (n+1-c) possible locations for a c-sequence, there are (N+1-c) different c-sequences and N^(n-c) possibilities for the other numbers. However, cases with c-sequences in two locations have been counted twice, so we have to subtract these. Then cases with c-sequences in three locations must be added etc. Inclusion/exclusion is the name of this procedure. The formula is long but simple and for your example it gives the answer 210969.

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I know Cramer, Gauss-Jordan and a few others, but i want to know wich one is the best.

For big linear systems (if that is what you mean by "solving a matrix") stay away from the direct methods like Gauss, Cramer, etc and go for the iterative ones, like GMRes.

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What does prime ideal in ternary semi ring mean?

A subset I of a semiring R is an ideal if a,b ∈ I and r ∈ R implies

a + b ∈ I and ra ∈ I.

An ideal I of a semiring R is prime if for every a and b in R, aRb is a subset of I implies either a is in I or b is in I.

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Ehrenfeucht's conjecture is one of the subsection on the lecture notes of semigroups theory by tero harju

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I am deeply interested in partial algebras and have advanced knowledge in this topic. I would like to start discussion of the possibility of using the theory of partial algebras in any discipline.

Partial operations play a fundamental role in the theory of natural dualities. See my text with David Clark, "Natural Dualities for the Working Algebraist". The most recent results on the role of partial operations in the theory of natural dualities can be found in "B. A. Davey, J. G. Pitkethly and R. Willard, The lattice of alter egos, Internat. J. Algebra Comput. 22 (2012), 1250007, 36 pp."

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When reading about field theory the only example of a skew field I have ever come across are the quaternions. Are there other examples?

The topic of division rings is very interesting and there are many deep results and construction around it.

Roughly speaking there are two kinds of division rings:

-The ones that are finite dimensional over its center (e. g. the quaternions). Here one has to point out that there is a famous theorem of Wedderburn stating that any finite division ring is commutative. On the other hand one of the goals of "the Brauer group" is to study finite dimensional division rings with a given center k. This is a very classical topic, that was one of the starting points of the non-commutative algebra and some of the fundamental results are due to E. Noether. A very interesting computation is the Brauer group of the rationals which shows that one can construct infinitely many non-isomorphic finite dimensional division rings with center Q that are not isomorphic to quaternion algebras.

- Forgetting about the finite dimensionality, one has a theory of field of fractions for non necessarily commutative domains that produces construction of division rings. In the non commutative setting it is not true that any domain has a field of fractions. The naive approach of construct a field of fractions copying the commutative setting was followed by Ore already in the thirties, the outcome was the theory of Ore domains. Nice examples of Ore domains are the so called skew-polynomial rings, that is a ring of polynomials such that the coefficients and the variable commute up to some automorphism and/or derivation. For example, if C denotes the complex numbers and \alpha is the complex conjugation the skew-polynomial ring R=C[x,\alpha] is a ring with elements the polynomials of the form z_0+\dots +z_nx^n , z_i in C (coefficients always on the left!) and with the rule xz=\alpha (z)x for any z\in C. R is an Ore domain and this means that has a field of fractions which is going to be a non-commutative one!

This is really a wide topic, I think a good source for some constructions and examples (and to have a better introduction to the subject than the one I am doing here!) is the book by T. Y. Lam

A First Course in Noncommutative Rings, Graduate Texts in Mathematics, Vol. 131, Springer-Verlag, 1991 (Second Edition, 2001).

It is impossible to talk about division rings without mentioning P.M. Cohn who developed a complete theory of localization that, between other things, solved the so called Artin problem: There is a division ring extension D subset F such that the dimension of F as a left D vector space does not coincide with the dimension of F as a right D vector space. Cohn has written many interesting books (e.g. Skew Fields, for the Encyclopedia of Mathematics)

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I know that Euler solved this somehow for a=1 and came to the value of e. I also found a series expansion that Euler developed. However, is there someone who could explain the derivation ending up in exp(a) using "high school math"? Maybe one needs to be genius or have high math skills to understand. But if this is not the case I would be happy for an understandable explanation, probably just framing the idea...

o(f(x)) means "something that tends towards 0 when x tends to 0 quicker than f(x)" --- hence, o(a) means something tending to 0 quicker than the a constant, which simply means it tends to 0 as x tends to 0. Here, 1/n tends to 0 as n tends to infinity, hence the usage.

Taylor expansion is a way to replace, in a vicinity of a point, a complex (but "regular") function by a polynomial expression, in such a way that the error tends to 0 quicker than the polynome itself. This leads to the expression

f(x) = f(x0) + f'(x0)(x - x0) + f"(x0) (x-x0)^2/2 + ... + f(n)(x0) (x-x0)^n/n! + o( (x-x0)^n )

provided of course different conditions, which here are met. It is a generalisation of the usual linearisation, using the tangent instead of the curve itself, which corresponds to n = 1 in the previous expression. the o(...) expression is in fact the most important, in a way, since it ensures that the error by using the polynomial part to compute the function is controlled, and the closer x is from x0, the smaller this error.

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Commutative reduced ring

Hello,

My proposal is formal power series, not polynomials.

On the other hand, a question: Set K=C={Complex numbers} in your reply. Then following your reply Spec(C[x]) is extremelly disconnected.

But Spec(C[x]) is the affine one-dimensional space. Surely I've missed something ... somewhere.

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If in a ring R for any two ideals (two-sided) I and J, rann(I)+rann(J)=rann(I\cap J), does lann(I)+lann(J)=lann(I\cap J)?

Note that we means of rannI={r\in R: Ir=0} and lannI={r\in R

: rI=0}. (I\cap J) is the intersection of I and J.

Dear Ali Taherifar,

Let R be a Ring and I and J are both left and right ideals in R so is in R annihilator of I and J are subrings of R implies rann(I),rann(J) sub rings of R and sum of sub rings are again sub ring implies a ring again. Now rann(I)+rann(J) are rings also ideals in R as well defined Cap J is also ideal hence part of I is quotient of Cap J denoted by quotient ideal I\Cap J (or difference of two ideals I and Cap J) is R which is either I or Cap J Now if you consider with respect to Ideal I of R Let us consider , annihilator of I for a in I , (a) = ann(I) subset of I w.r.t I is said to be Iann(I) similarly consider J, Iann(J) implies sum of Iann(I) + Iann(J) is w.r.t. I ideal in R consider part of ideal capJ in I i.e., quotient ideal in I only so w.r.t I, Iann(I/Cap J) is subset of Iann(I)+Iann(J) and we can prove other part i.e., Iann(I\Cap J) is subset of Iann(I)+Iann(J) ideal in I (w.r.t I) if you take w.r.t J, result will be Jann(I)+Jann(J) = Jann(J\Cap I) interchanges its position since both are left and right idelas of R. Hence, it si true that Iann(I)+Iann(J) = Iann(I\Cap J).......N V Nagendram

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I am trying to figure out if the following expression:

(n^2 - n)! / ( n! ((n-1)!)^n ) is an integer for all positive integer n.

I tried the induction, but the induction case is running into problems. So, I was looking at permutation/combination problems, but so far, I couldn't come up with a convincing argument.

Does anybody has any thoughts on this to share ?

If you are working with integers, integers squared and factorials (of integers), the result will always be an integer.

A researcher

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The standard proof runs as follows: You start with the tensor algebra of the vector space in question, say V. Then you take the quotient by the ideal generated by elements of the form

v \otimes v - q(v) 1 , for all v in V

Here q is the bilinear form on V. That works for me, the problem comes when I try to see that this ideal is not so big (the ideal is proper and non trivial). The standar reference [H.B. Lawson , M.L. Michelson, Spin Geometry] says; well calm down, this is not a big deal. Take an element in the intersection of this ideal with V, then in one hand this element has the standard expresion as a sum of products of elements in the ideal with elements in the tensor algebra. On the other hand, is an element of V, so all the term of maximal degree should be zero, and then comes the obscure trick I can't figure out. He takes a weird contraction with q, and concludes by induction that the previous fact (elements of maximal degree vanish) imply that the element itself is equal to zero.

Please, if you can clarify this last part (or gave me a fresh reference where I can find another proof scheme) I'll be grateful.

This is proved in Chapter 20 of Fulton and Harris "Represention Theory: A first course". One can think of the Clifford algebra as a deformation of the exterior algebra (which is the Clifford algebra with q=0). One can filter elements of the Clifford algebra by degree, and then see that the graded quotient is isomorphic to the exterior algebra, which gives that the algebra has dimension 2^n (I'm guessing this is what you're referring to by the existence of the Clifford algebra, meaning the non-triviality, or as you say, the properness of the ideal).

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No mathematical treatment is required

When you throw a ball against ground it will bounce back but need not be in same ( i.e opposite) direction. If a ball hit a horizontal ground with angle @ it will bounce back with angle of 180-@. The amount of regained momentum of the ball depends on the coefficient of restitution(COR) of the collision. Neglect the gravity. So here the COR can be considered the 2D scalar field of the collision problem i.e A=(1, 0 || 0, -COR)

The momentum of the ball is considered as some vector(V) in a xy plan where y axis is normal to the ground. When the vector V is multiplied with the scalar matrix(A) the V is reflected about the y axis in xy plan. The new vector V2 will be differs from the V. When @ becomes 90(V=k*[0,1]), the ball will bounce back in same path but opposite direction with V2= k*COR*[0,-1]. Here the V2 and V are parallel but opposite. V direction not changed by A.So V can be considered as eigenvector of the A. COR is eigenvalue.

(This explanation can be wrong but still i tried to explain with my knowledge)

classical examples: principal stress, natural frequency of the spring mass system

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It is known that this orbit is an immersed submanifold of C^n, so the question becomes: When is this orbit locally closed?

can there be a sheaf with GL(n,C) and the projection of this sheaf onto a C^n manifold have orbits which are closed and thus have a natural identification as a manifold.

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Is there are paper discussing Magic Squares as a Vector Space and, for example, the dimension of the set of 5x5 Magic Squares?

This is a paper computing the dimension of the vector space (over any field) of magic squares for any matrix size

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I have attached a PDF which more elegantly describes this (hopefully) simple problem. In the course of developing a new theory as part of my graduate research I have encountered an expression U* (U^{T} * S * U)^{-1} *U^{T} where U contains random orthonormal vectors and S is positive-definite. Through brute force I have demonstrated that the product is invariant to the particular vectors one chooses to populate U, but I cannot prove why. Does anyone have any ideas?

The answer is best viewed from the perspective of a projection matrix. Solution is in the attached PDF.

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Can you give an example of regular sequence and system of parameters?

Dear Premkumar AK, concerning your question:

Let $(R,\fm)$ be a Noetherian local ring. It is known that every $R$-sequencecould be a part of system of parameter of $R$. Every weak $R$-sequence of length at most $\dim R$ is a part of system of parameter of $R$.

Note that weak $R$-sequence is a generalization of $R$-sequence defined as follows:

$x_1, ..., x_t \in \fm$ is weak sequence of $R$ if for every $i=1,...,t$

(x_1,...,x_{i-1}):x_i=(x_1,...,x_{i-1}):\fm.

This leads to the definition of the so- called Buchsbaum rings.

A researcher

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Every cyclic group is isomorphic to Zn. Now what is the total number of isomorphic direct product groups of the form Zn1*Zn2*Zn3*........Where n1*n2*n3*.....=n. It is known that (Zp*Zq) is cyclic if and only if p and q are relatively prime.

@Bhupesh Dutt Sharma phi(n1*n2*...)=phi(n) since n=n1*n2*n3.... so let us consider this example when n is a prime p, in this case Zp has a unique decomposition as Zp itself but according to your claim it should be phi(p)=p-1, which contradicts the fact that Zp has unique such decomposition.

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There are mainly 4 difficulties

language difficulty, prerequisite skill ...... I need clear view

Once the concept is studied i think students have to try the problems in the respective concept so that he concept will be more understanding and interesting...one can suggest herstein or fraleigh for the reference

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Consider the operation of G is (multiplication).

From (ab)^3 = ababab = aaabbb we get that

baba = aabb, so (ba)^4 = (aabb)(aabb)=(a^2 b^2) (a^2 b^2) = (a^2 b^2)^2 = b^4 a^4=

bbbbaaaa=bb(abab)aa=b (ba)^3 a

So babababa=bbababaa.

Now we multiply this equality by b^-1 on the left and a^-1 on the right, we get

ababab=bababa => (ab)^3 = (ba)^3 hence by injectivity of f we obtain ab=ba so group is commutative.

Note that you must require that f in injective: Take group of nine elements {1, x, y, xx, yy, xy, xxy, xxyy, xyy} with multiplication defined by yx = xyy.

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I'm interested in a simple explanation of the Van Kampen theorem. What would be a "toy example" for applying it?

A nice example is the fundamental group of the torus. View it as a rectangle with opposite sides identified. Let U be the interior of the rectangle, and let V be a neighborhood of the boundary of the rectangle. pi(U) is trivial. V deformation retracts to the boundary, which is the one-point union of two circles. It has fundamental group Z*Z (by Van Kampen's theorem!). U intersect V is an annulus, with fundamental group Z. Now Van Kampen's theorem says the torus is the product of Z*Z with the trivial group amalgamated over the image of the fundamental group of the annulus. If a and b are the generators of Z*Z, then the image of the annulus is generated by aba^-1b^-1, the commutator of a and b. So the torus has fundamental group presentation <a,b : ab=ba>, which is just the free abelian group on two generators.

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I have two real matrices, A and B. A is idempotent (in fact it is a matrix that redistributes payoff or money with some properties) and B is a transition probability matrix. Both are non symmetric.

According to Gelfand's formula this should be the case when they commute. There might be other instances that escape me.

On the other hand, if I read your question correctly, then you could replace rho(A)rho(B) by 1, as the spectral radius of a non-zero idempotent matrix is one. The same can be said for a stochastic matrix.

Assuming for a moment that both matrices are diagonalisable (which is always the case for the idempotent matrix, but not necessarily for the stochastic one), then for any vector we would have that \|Bx\| \leq \|x\| and \|Ax\| \leq \|x|, and thus \|ABx\|\leq \|x\|. Which gives \|AB\| \leq 1. I might be mistaken, but that should show that \rho(AB) \leq \|AB\| \leq 1 = rho(A)rho(B).

I hope this helps you.

Rob

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related to the paper "Square Root as Homorphism" by W. C. Waterhouse

Correction: If p is congruent to 3 mod 4 then either a or -a is a square in Z/pZ. This is because the Legendre symbol (a/p) (which is 1 for a square and -1 for a non-square) is multiplicative, and (-1/p)=-1. However, when p is congruent to 1 mod 4, -1 is a square. So either a and -a are both squares or neither is. The smallest example is Z/5, where 2 and -2 are not squares.

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As matters arising from my joint paper with G. Tang, " Higher K-theory of group-rings of virtually infinite cyclic groups" Math Annalen 325 (2003) 711-725, I am curious to know if anyone has since proved that the Nil groups of Bass and those of Waldhausen are isomorphic for all n > 1. Such an isomorphism is well known for n less than equal to 1 and we used it to show the vanishing of lower Waldhausen nil groups for regular rings.

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I have a 5th order algebraic equation. I want to know the number of positive or negative or imaginary roots of that equation. Apart from Descarte's rule of sign's can anyone give an idea to find the number of negative roots of the equation.

Use Sturm's theorem (see Wikipedia) with a=-infinity and b=0.

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If any, please provide explanation (examples are best welcomed) and/or references.

Hi Gro,

Thank you for your answer. 2n × 2n real skew-symmetric matrix could be decomposed to something like symplectic matrix, what would it be if order of matrix is 2n+1?

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Full text of a paper...?

Could anyone possibly provide me with any help finding the full text of the following paper:
R. S. Pierce, Distributivity and normal completion of Boolean algebras, Pacific J. Math. 8 (1958) 133-140.
I`ve looked everywhere and could not find it for the life of me.

Thanks men! May the Algebra be with you...!

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This happens for some classes which contains only elements of An.

Let G=S_n and g an element of A_n. The conjugacy class of g in G splits into 2 classes in A_n if and only if the centralizer C_G(g) is contained in A_n.

The more concrete criteria is: the conjugacy class of g in G splits if and only if g is the product of cycles of distinct odd length.

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Firstly, l define what I mean by structure equation. The solution of an algebraic equation is a number. The solution of a functional equation is a function, for instance, differential equations. The solution of a structure equation would be a structured set ,say, a geometry, a differentiable manifold, an algebra etc.

As far as I know, the only structure equation that has been widely investigated is the Einstein one, the solution of which is a differentiable manifold determined by its connection coefficients.

A learning system, modifies its own structure as he is learning new topics and handling new information. Thus, the unknown of a learning system theory is its structure sequence determined by the acquired information. I think that this topic must be investigated.

Dear Cj Nev

I have voted up your answer, because you have pointed out the core of the intelligent system structure, but this fundational matter is extended by experience together with learning processes via neuron conections and conditioned reflexes- (Paulov).

These conections form a scalable struture. Finding out those equations ruling these processes, in my opinion, is a fundational question.

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I know that for isomorphism between two groups we check one to one, onto mapping and homomorphism between them. I want to know how to find the number of isomorphic subgroup of two groups.

i assume that the question you are asking can be formalized in the following way: given two finite groups G and H, how many pairs (A, B) are there with A subgroup G and B subgroup H and A is isomorphic to B?

the first clear limitation is that the subgroup orders |A| = |B| must divide gcd(|G|, |H|), but that is a very weak condition (though it can be enough, e.g., when the gcd is 1)

a simpler first cut might be the case G = H, which is to find the number of isomorphic pairs of subgroups of a given finite group, but i don't think this is even known for all p-groups (groups G with |G| = power of a prime p)

another special case is for abelian (commutative) groups, for which the problem can be reduced to the various p-groups, and, i suspect, a complete answer can be written

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Module theory is generalization of ring theory.

One can think of a module as an abelian group acted upon by a ring of homomorphisms of that group into itself. Thus, for example, an ordinary vector space (an example of a module) is a group of vectors acted upon by the ring formed by the homomorphisms of the form v--->c*v for some scalar.

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What is basic difference between Algebra and Analysis?

"algebra deals with equality but analysis deals with inequality and error terms" - sorry, but that is not correct: error terms originate from limes-considerations, but you cannot say that analysis deals only with these terms or inequalities.

For example the one of the most fundamental equations of analysis is not an inequaltity: the definition of the differential quotient.

Also analysis does not deal only with individual objects. As a counter-example: you can have a set consisting of only one individual object, but it is still algebra or a series with an infinite number of terms and it is still analysis...

Furthermore your definition is unfortunately not very rigid in a mathematical sense, you can look up better definitions in wikipedia (just as an example):

A researcher

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What is the answer when 0 is divided by 0 ?

Since division of a by b means breaking a into b equal parts and so the answer is : one part. For example : 8/2 means breaking 8 into 2 equal parts so the answer is 4

Black holes are where God divided by zero. :-)

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How can we solve an n^th degree polynomial in general or in any other softwares? I need a symbolic solution in terms of the co-efficients.

If you are asking about polynomials in 1 variable, there are well-known solution expressions built from the coefficients using radicals (square and cube roots of expressions) for degrees up to 4 - no such expressions exist for n>=5 (this is a famous result of niels abel, and also follows from work of galois in the 1800's) - these expressions get increasingly complicated as n goes from 2 (the quadratic formula) up to 4

this does not mean that your question has no answer, however: for n >= 5, there may be expressions to find solutions in terms of the coefficients, but they are not based on simple or even common mathematical functions that are familiar to most - part of the difficulty is that even when the coefficients are simple rationals or even integers, the expressions have to account for the fact that the solutions need not be real numbers (there are many theorems that provide restrictions on the solutions based on the coefficients, but they are not sufficient to solve the equations - they do often help for finding good starting points to use with numerical methods, but that isn't what you asked)

if you were not asking about one variable polynomials only, then the answers are different - one polynomial in more than one variable usually has solutions in a complex space of dimension one less than the number of variables, with even more and stranger complications than the ones with one variable, and multiple polynomials have even more possibilities, with one variable or more

hope this helps; sorry for the bad news

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Simple question: we are trying to compare absolute and relative exponential decay rates of a chromogenic decay assay. In this assay the chromogenic compound (and thus absorption) decays over time in a biphasic way. I am trying to fit the data to the ExpDec2 function in Origin (y = A1*exp(-x/t1) + A2*exp(-x/t2) + y0). I would like to compare the fast and slow decaying rates under different conditions. In one example where I can clearly see a two-step decay with a continued negative slope, I get a negative t1 and a positive t2 (y0 94.27757, A1 -0.28311, t1 -30.39113, A2 6.00467, t2 8.08285). How can A1 be negative? Is it common practice to write the respective reaction rates (k1 = 1/t1 and k2 = 1/t2) as absolute numbers (positive) for comparison? I'm not sure what the best way is to compare the rates in this case.

Maybe each decay process has a time domain in which it is applicable or dominant. So rather than making the effective decay equal to the sum of the two processes, make it a function in which there are different domains. For example, in the first time domain, only process A is applicable; then in a second domain, perhaps use the sum of the two processes, A+B, and in a third domain only proces B is applicable. See if this makes sense mathematically and in your scientific application.

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Rings and Categories of Modules, Anderson, Fuller.

Let Q denote the field of rationals. Let S=Z_{(p)} the ring of integers localized at the prime ideal generated by a prime p.

The triangular matrix ring R=\left( Q&Q\\0&S\right) has Jacobson radical J(R)=\left( 0&Q\\0&pS\right) and R/J(R)\cong Q\times Z/pZ. So that R/J(R) is semisimple artinian and the idempotents can be lifted modulo the Jacobson radical. This shows that the ring is semi-perfect.

Notice

Soc ( {}_RR)=\left( Q&Q\\0&0\right) =I

because the lattice of submodules of ${}_RI$ is the lattice of submodules of ${}_Q(Q\times Q)$. However on the right the socle of R is zero. Indeed, one easily sees that if R_R has a simple submodule it must be inside de ideal

J=\left( 0&Q\\0&S\right)

But J_R behaves like (Q\times S)_S, which contains no simple submodules (because it is torsion free).

Taking the ring \left( S&Q\\0&Q\right) one has a similar situation but exchanging sides. That is, the ring is semiperfect with nonzero right socle and zero left socle.

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We study about some laws for group theory and ring theory in algebra but where it is used.

Dear Rahul,

Group and ring theory are central in many ways. Conservation laws of physics are reflections of the principle of least action. Once you have one of these laws in place, then your immediate concern is what actions can you take which preserve the law, and that set of actions is generally a group. This was a big understanding arrived at by Emmy Noether.

Ring theory has many uses as well. As Rama Bandi mentioned above, it is useful in coding theory, and number theory in general, e.g., cryptography.

Semigroups are to do with actions that preserve partial symmetries, and can be used to model plant growth or the growth of quasi-crystals, and etc..

Basically, these algebraic structures are useful for understanding how one can transform a situation given various degrees of freedom, and as this is a fundamental type of question, these structures end up being essential.

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Recall that recursive definitions are those defining statements involving the defined object.

The defined object can occur either explicitly or implicitly in the recursive definition.

For instance, the definition.

x = positive number that coincides with x^2.

This recursive definition determines de number 1. The defined object x occurs implicitly. By contrast, in the following

definition

x = positive number that coincides with its powers.

contains x implicitly; because of, in this definition, the expression "its powers" is equivalent to x^1, x^2, x^3 ….

Another example is

x = set of all sets.

This definition is also recursive, because defines x as a set, therefore and the expression "all sets" involves x.

In fact, a recursive definition fits into the pattern

x = p(x)

where p(x) stands for any predicate involving x.

In general, recursive definitions are stated in several metamathematical topics without proving its consistence.

Nevertheless, if the pattern x = p(x) is regarded as an equation instead of a definition, then it is required

to show that such an equation has at least one solution. Why do not require also the existence proof in recursive definitions?

@Ivan Suskov Suppose the domain is the positive integers and f(∅)=∅, and otherwise f(S) is the subset obtained when the smallest integer in S is deleted. In this case, would your "least solution" be ∅?

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Derivative of cyclomatic

There is no such x when p=5 so what about p ≥7

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Ring theory

You have at least three different objects that may be called "p-adic rings".

The first one is the ring of p-adic integers, usually denoted by $\mathbb Z_p$ or by $\mathbb J_p$. There is plenty of different (but of course all equivalent) definitions of this ring, I list here some of these:

1- $\mathbb Z_p$ is the inverse limit of the finite rings $\mathbb Z(p^n)$ with $n\in \mathbb N_+$ (the transition map from $\mathbb Z(p^{n+1})$ to $\mathbb Z(p^n)$ is the obvious projection). Another way to say this is to say that $\mathbb Z_p$ is the completion of the ring $\mathbb Z$ with respect to the maximal ideal $p\mathbb Z$, that is, the completion of $\mathbb Z$ in its $p$-adic topology;

2- $\mathbb Z_p$ is the endomorphism ring of the Abelian $p$-Prüfer group $\mathbb Z(p^{\infty})$. (the equivalence with the above definition can be seen using the fact that $\mathbb Z(p^{\infty})$ is the direct limit of the Abelian groups $\mathbb Z(p^n)$)

3- $\mathbb Z_p$ is the ring of formal sums $\sum_{i=0}^{\infty}n_i p^i$, with $n_i\in \{0,\dots,p-1\}$.

The second object I want to describe is the field $\mathbb Q_p$ of the $p$-adic numbers, which is just the field of fractions of $\mathbb Z_p$. Really, all what you need to obtain $\mathbb Q_p$ from $\mathbb Z_p$ is to make invertible the operation "multiplication by $p$". So $\Q_p$ can be defined to be the direct limit of following direct system

$$\mathbb Z_p\overset{p}{\longrightarrow}\mathbb Z_p\overset{p}{\longrightarrow}\dots\overset{p}{\longrightarrow}\mathbb Z_p\overset{p}{\longrightarrow}\dots\, ,$$

that is, $\mathbb Q_p=\bigcup_{n\to\infty}1/p^n \mathbb Z_p$. Using the description 3 of $\mathbb Z_p$ I gave above you can see $\mathbb Q_p$ as the ring of all formal sums $\sum_{i=-k}^{\infty}n_i p^i$, with $n_i\in \{0,\dots,p-1\}$ and $k\in \N_+$.

One can also introduce the $p$-adic norm in $\mathbb Q$ and define $\mathbb Q_p$ as the completion with respect to $\mathbb Q_p$.

FInally, the last object that may be called a $p$-adic ring is a (topological) ring which has properties that are similar to the properties of $\mathbb Z_p$. More precisely, a commutative ring $R$, equipped with the $p$-adic topology, that is, the linear ring topology which has a base of neighborhoods of $0$ of the form

$$\ldots \subset p^3 R \subset p^2 R \subset p R\, ,$$

is said to be a $p$-adic ring if the following conditions are satisfied:

(a) the residue ring $\overline{k}=R/pR$ is a perfect ring of characteristic $p$;

(b) the ring $R$ is Hausdorff and complete (for the $p$-adic topology).

(c) $p$ is not a zero-divisor of $R$.

(In general, a commutative ring with a linear ring topology induced by a filtration by ideals $...\subset I_2\subset I_1$, is said to be a $p$-ring if it satisfies the above conditions (a) with $\overline{k}=R/I_1$ and (b) for this topology. In particular any $p$-adic ring is a $p$-ring).

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When n = 2 or 3, we can solve this integral by reducing the denominator to the form (x + a)^2 + B. What is the general procedure for evaluating such integrals for n > 3?

if p(x) is a polynomial with real coefficients, it can be factorized into linear and quadratic factors with real coefficients. if l1(x), l2(x)..., q1(x), q2(x).... are these factors, you can write 1/p(x) as b1/l1(x) + b2/l2(x) + ... + (c1 x + d1)/q1(x) + (c2 x + d2)/q2(x) + ..., then integrate each term separately, getting a sum of a whole bunch of log and arctan functions.

NB : p(x) can also be factorized into linear factors with complex coefficients, then your real integral 1/p(x) becomes a complex integral b1/l1(x) + b2/l2(x) + ... (no quadratic terms), which gives a bunch of log functions, some with complex arguments. The ones with complex arguments could be combined (in pairs) into a bunch of arctangent functions. Try this method too, and see which is easier.

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As we know (a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2. I wish to have such an identity for quartics: (a^4+b^4)(c^4+d^4)=A^4+B^4, but I do not know how I take A,B.

I have solved the quartic Diophantine equation x^4+y^4=2(u^4+v^4). Now I want to consider x^4+y^4=n(u^4+v^4) with n=c^4+d^4. Then we have x^4+y^4=(c^4+d^4).(u^4+v^4). Indeed, I wish to write (c^4+d^4)(u^4+v^4)=A^4+B^4, hence in this case the equation x^4+y^4=n(u^4+v^4) can be solved by Euler's parametrization as well.

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When is every cosingular right module, projective over a right H-ring?

Y. Talebi, N. Vanaja, The torsion theories cogenerated by M-small modules. Comm. Algebra, 2002.

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Please see attached file for question:

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Is every inective cosingular module zero?

Y. Talebi and N. Vanaja, The torsion theories cogenerated by M-small modules

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$G$ is a groupoid but not semigroup. On $G$ we can define so many binary operations. Is there any way we can show that at least two binary operations are the same?

Sorry, the last formula in my previous post should read f(x*y) = f(x) o f(y).

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"Intelligence agencies hire lots of mathematicians, but would-be employees must realise that their work is misused to snoop on everyone, says Tom Leinster"

New Scientist has published an article recently, where Tom Leinster asks mathematicians to stay away from supporting NSA, CIA, GCHQ, (former) KGB and all the other organizations that spy on us. I even don't know the name of their Chinese colleagues' organization.

What are your thoughts on this?

All of us are afraid of surveillance based Orwellian societies. All of us are worried that some of our democratically elected government are secretly putting in place a surveillance infrastructure and that most of the information coming to us comes from delators, conscience objector such as Snowden. We are told by our democratically governments that yes a minimal surveillance infrastructure is being put in place for our protection agains terrorist. For our own good! Hmm! I do not like the way it is being done presently and the total lack of transparency regarding the real objective, who have access , ect.

That being said we have to look at this topic from other perspectives. Orwell warned us against this possibility because of the rise of the Marxism authoritarian states and what they could do if equipped with surveillance technologies, not only human surveillance.

But there is another totally different perspective one can adopt towards this question of surveillance and it is not the perspective of Orwell , nor that of the prevention against terrorist. It is a perspective on what is about the nature of the homo sapiens sapiens, why it evolved and where its civilisation is evolving towards. To make a potential long story short, lets say that since the evolution of mammals the trend has been towards the formation of collective actions and the key to the coordination of action has been the outer expressing of internal state. Human language evolved for that, writing allowed the broadcasting and stabilisation and extended message. Internet has permitted instant communication between all of us. We are rapidly moving towards a collective coordination of our actions by electronic communication on a global scale. The next step is the creation of a monitoring core of all human actions so that automated tools of human coordination could be available to all of us for all inimaginable kind of action cooperation which would lead even closer not to a big brother but to a Gaia with a brain, the vision of multiple visionaries, a collective type of stellar being (The expression of Theilard).

Now how do we get there without this tremendous power of monitoring be highjack by all the potential big brothers lurking out there. Here is the task of the mathematicians. How to create a monitoring core whose access is fully controled for the common good and cannot be breached by malevolent big brother in the waiting.