Science topics: Mathematical SciencesAlgebra

Science topic

# Algebra - Science topic

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

Questions related to Algebra

Updated information of my thoughts and activities.

This is meant to be a one-way blog, albeit you can contribute with your recommendations and comments.

The strings in matrix B predict this statement.

As a numerical example, the sum of the entire series 0.99 + 0.99^2 + 0.99^3 + . . . . +0.99^N increases to 190 as N goes to infinity.

Additionally, B-matrix chains provide rigorous physical proof.

The question arises whether a pure mathematical proof can also be found?

Since the generally accepted theory of gravity is GRT, one could try to derive it from some algebraic theory. The basic thing in this theory should be the Dirac algebra, in which the Minkowski space is naturally embedded. Thus, we need to find some dynamical algebra that would be locally equivalent to the Dirac algebra. Where does such a Miracle live?

Giuseppe Piano was the first to establish the axioms of vector space, where the group axioms were linked to the axioms proposed regarding the second law, “number multiplication,” with a basic concept of compatibility.

From your cognitive and scientific point of view, can you give an explanation of this compatibility , which is the secret of the composition of this structure ''vector space'', in a way different from the usual explanation ?

Thanks in advance.

Article Topic: Some Algebraic Inequalitties

I have been collecting some algebraic inequalities, soonly it has been completed and published on Romanian Mathematical Magazine.

In the study of quantum deformations of Witt and Virasoro algebras, the notion of a Hom-Lie algebra first appeared, so is deformation theory insufficient to justify how algebraic structures such as Lie-algebras, ... are deformed, which requires the intervention of the concept of ''Hom'' to define the concepts that have recently appeared, such as Hom-groups...with reasoning ?

Following the findings in the field of deformable algebra, can deformation of an n-dimensional vector space contribute to changing its initial dimension?

How will this contribute to finding the appropriate deformations for infinite-dimensional algebra on the basis that algebra is a vector space with an additional structure, with the hope that this will lead to new physical systems with an infinite number of symmetries, and thus complementarity in the hope of enhancing one of what the theory of strings aspires to ?

Unbelievable is make believable in the world we connected prime number in matrix algebra form and I change history of math
Please see it

The leibniz rule reads

D(xy)=xdy +dx y

But a subtraction of two slightly

Different rectangles gives an extra term dx dy.

People have ddbated ever since

Tje smallest positive number doed not exist.

We will fix this with an ideal in algebra

he, with e a nilpotent element ee=0?

Methods: interviews w teachers, focus groups, observation) Leaning towards social constructivist, but was told I should probably broaden my search. Help!

I am starting a new area of research that is Algebraic topology. Kindly suggest some latest problems and related publications

Dear All,

I am a MSc in Theoretical Physics and I am finishing my master thesis on Algebraic geometry over Lie algebras in Tabriz University . I am searching a PhD position.

If anyone is interested in a PhD student, please feel free to contact me.

Thanks in advance,

Sona Samaei

My name is Sona and I have received my master degree in Lie Algebra from Public University of Tabriz (PUT).

In my master thesis, we have studied the algebraic geometry over Lie algebras. Also, I have passed Advanced algebra , finite group, manifold geometry and Real analysis and received IELTS certificate.

If there is an open position and I will be ready for an in-person interview if it is needed.

Best Regards,

Sona Samaei

Dear ResearchGate Community,

I am currently conducting research in the field of algebraic geometry, and I am in need of the Magma (computer algebra system) for my analysis. Unfortunately, my university does not have access to this software, and it seems unlikely that it will be made available in the near future.

I would be incredibly grateful if anyone could provide me with a student version of Magma, as it would be crucial for my research progress. If anyone has a copy, I would be very appreciative of your assistance.

Thank you in advance for your help and support in advancing the field of algebraic geometry.

It would be very interesting to obtain a database of responses on this question :

What are the links between Algebra & Number Theory and Physics ?

Therefore, I hope to get your answers and points of view. You can also share documents and titles related to the topic of this question.

I recently read a very interesting preprint by the mathematician and physician Matilde Marcolli : Number Theory in Physics. In this very interesting preprint, she gave several interesting relations between Number Theory and Theoretical physics. You can find this preprint on her profile.

We know that ideals of a lattice are dual of its filters. Moreover, Heyting Algebra is a special class of lattice. Can we define an ideal of a Heyting Algebra dually to its filter?

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?

I noticed that there is a structural similarity between the syntactic operations of Bealer's logic (see my paper "Bealer's Intensional Logic" that I uploaded to Researchgate for my interpretation of these operations) and the notion of non-symmetric operad. However for the correspondence to be complete I need a diagonalisation operation.

Consider an operad P with P(n) the set of functions from the cartesian product X^n to X.

Then I need operations Dij : P(n) -> P(n-1) which identify variables xi and xj.

Has this been considered in the literature ?

I see the study of Drapeau et al. (2016), The algebra of conditional sets and the concepts of conditional topology and compacness, Journal of Mathematical Analysis and Applications. They bring the new concept of 'conditional set' (Definition 2.1):

A conditional set

**X**of a non-empty set*X*and a complete Boolean algebra*A*is a collection of objects*x*|*a*for*x*in*X*and*a*in*A*such that- if

*x*|*a*=*y*|*b*, then*a*=*b*;- if

*x*,*y*in*X*and*a*,*b*in*A*with*a*<=*b*, then*x*|*b*=*y*|*b*implies*x*|*a*=*y*|*a*;- if (

*ai*) in*p*(1) and (*xi*) is a family of elements in*X*, then there exists exactly one element*x*in*X*such that*x*|*ai*=*xi*|*ai*for all*i*.In my naive reading, it seems that the traditional axioms of ZFC set theory fit just fine this definition, e.g. axiom schema of separation is defined on the common properties of set elements, for these 'properties' refer (to me) as the 'conditions' in the Drapeau et al. (2016) study.

The thing is that I could combine a set of 'properties/conditions' and the basic set operations (union, inclusion,...), so to create an algebra of conditions satisfying the usual properties.

Is there something I do not see in the Drapeau et al. (2016) study?

Why do algebraic multiplicity of eigenvalues of skew-symmetric matrix pencils are even?

That an n degree polynomial CAN always be factored up into n factors containing Complex roots has been shown true, but there are things not explained in the phrase.

Teachers will not usualy explain alternatives.

Say (Z-x)(Z-x) =0 CAN have the solution Z=x, but can also have Z=x+ey where ee=0 is an

nilpotent element.

Just as i can be made real in 2 by 2 matrices, so can e. Just as we use i or -i dupliciously,

without knowing which, we can use e or e(T) , the transpose of e. One can define i=e-e(T)

using ee=0 , e(T) e(T) =0 and e e(T)+e(T)e =I to show ii=-1. Now substitute this everywhere you see i, and the factorization carries out the same.

Defining ww=1, one can also claim w=e +e(T).

Reading attentively the fundamental theorem of algebra, we find the funny phrase that it is not derivable from algebra...well no wonder.

Dont really have a good name for these elements, i plus whatever else there is. The supra real? Hyper real is already occupied by non standard elements. Neither have they seemingly been well studied well beyond quadratic, though they do exist...

This means a gap in knowledge? Or stuborn historical usage?

I did not understand from the author how he arrived at these specific equations ?

what is phi^1 ,phi^2 ,...

Let $A=(a_{ij})\in M_{m \times n}(\mathbb{R_{+}})$ and $B=(b_{ij}) \in M_{n \times l}(\mathbb{R}_{+}).$ The product of $A$ and $B$ in max algebra is denoted by $A\otimes B,$ where $(A\otimes B)_{ij}=\displaystyle\max_{k=1,\ldots,n} a_{ik}b_{kj}.$

A set $\mathcal{X}_{n \times k} \subset M_{n \times k}(\mathbb{R}_{+})$ is defined by

$$\mathcal{X}_{n \times k}= \{X \in M_{n \times k}(\mathbb{R}_{+}): X^{t}\otimes X = I_{k}\}.$$

It is known that for the case $k = n,$ $\mathcal{X}_{n \times n}$ is equal to $\mathcal{U}_{n},$ where $\mathcal{U}_{n}$ is a unitary matrix in max algebra.

Irrational numbers are uncomputable with probability one. In that sense, numerical, they do not belong to nature. Animals cannot calculate it, nor humans, nor machines.

But algebra can deal with irrational numbers. Algebra deals with unknowns and indeterminates, exactly.

This would mean that a simple bee or fish can do algebra? No, this means, given the simple expression of their brains, that a higher entity is able to command them to do algebra. The same for humans and machines. We must be able also to do quantum computing, and beyond, also that way.

Thus, no one (animals, humans, extraterrestrials in the NASA search, and machines) is limited by their expressions, and all obey a higher entity, commanding through a network from the top down -- which entity we call God, and Jesus called Father.

This means that God holds all the dice. That also means that we can learn by mimicking nature. Even a wasp can teach us the medicinal properties of a passion fruit flower to lower aggression. Animals, no surprise, can self-medicate, knowing no biology or chemistry.

There is, then, no “personal” sense of algebra. It just is a combination of arithmetic operations.There is no “algebra in my sense” -- there is only one sense, the one mathematical sense that has made sense physically, for ages. I do not feel free to change it, and did not.

But we can reveal new facets of it. In that, we have already revealed several exact algebraic expressions for irrational numbers. Of course, the task is not even enumerable, but it is worth compiling, for the weary traveler. Any suggestions are welcome.

Our answer is YES. Irrationals, since the ancient Greeks, have had a "murky" reputation. We cannot measure physically any irrational, as one would require infinite precision, and time. One would soon exhaust all the atoms in the universe, and still not be able to count one irrational.

The set of all irrationals does not even have a name, because there seems to be no test that could indicate if a member belongs to the set or not. All we seem to know is it is not a rational number -- but what is it?

The situation is clarified in our book Quickest Calculus, available at lowest price in paper, for class use. See https://www.amazon.com/dp/B0BHMPMMTY/

There, Instead of going into complicated values of elliptic curves, and infinite irrationals, algebra allows us to talk about "x".

No approximating rational numbers need to be used, nor Hurwitz Theorem.

Thus, one can "tame" irrationals by algebra, with 0 (zero) error. For example, we know the value of pi. It is 2×arcsin(1) exactly, and we can calculate it using Hurwitz Theorem, approximately.

GENERALIZATION: Any irrational number is some function f(x), where x belongs to the sets Z, or Q -- well-defined, isolated, and surrounded by a region of "nothingness". The set of all such numbers we call "E", for Exact. It is an infinite set.

What is your qualified opinion?

Hello I start my research work recently. I ask that how we find the unknown From system of algebric eq of nonlinear PDS of higher order which was obtained by applying differaent f

Transformation or methods using Mathematica I use the comment

Solve[{experision 1, ,,,,},{unknowns}] but I can not find the values of them .

I define the omega-th Cayley-Dickson Algebra as the union of all the finite-dimensional Cayley-Dickson algebras (over the reals).

Let k be a field of characteristic zero and let E=E(x,y) be an element of k[x,y].

Define t_x(E) to be the maximum among 0 and the x-degree of E(x,0).

Similarly, define t_y(E) to be the maximum among 0 and the y-degree of E(0,y).

The following nice result appears in several places:

Let A,B be two elements of k[x,y] having an invertible Jacobian (= their Jacobian is a non-zero scalar); such A,B is called a Jacobian pair.

Assume that the (1,1)-degree of A is >1 and the (1,1)-degree of B is >1.

Then the numbers t_x(A),t_y(A),t_x(B),t_y(B) are all positive.

**Question:**Is the same result holds in the first Weyl algebra over k, A_1(k)? where instead of the Jacobian we take the commutator.

Of course, we must first define t_x(A),t_y(A),t_x(B),t_y(B) in A_1(k); it seems to me that the same definition holds for A_1(k), or am I missing something? Perhaps it is not possible to consider E(x,0), where E is an element of A_1(k)?

Thank you very much! Please see https://mathoverflow.net/questions/334897/a-non-commutative-analog-of-a-result-concerning-a-jacobian-pair

As is well known, camera calibration in photogrammetry and with the use of Bundle Adjustment with self-calibration, the coordinates of the principal points cannot be recovered from parallel images. This situation calls for convergent images to recover the coordinates of the principal point. A common explanation is attributed to the algebraic correlation between the exterior orientation parameters and the calibration parameters. Now the question in other words, Is there is any deep explanation about the nature or the type of this algebraic correlation? Is there is any analytical proof for this correlation? or we have to accept this empirical finding (we need convergent images for camera calibration)

I am primarily interested in 2-player combinatorial games with perfect information. Useful wiki links are below.

How to linearize any of these surface functions (separately) near the origin?

I have attached the statement of the question, both as a screenshot, and as well as a PDF, for your perusal. Thank you.

*This paper is a project to build a new function. I will propose a form of this function and I let people help me to develop the idea of this project, and in the same time we will try to applied this function in other sciences as quantum mechanics, probability, electronics …*

I don't need this anymore

Here the text:

Do these two algebraic modeling languages for optimization totally similar? Or they have some differences? Any comparison of similarities and differences would be highly appreciated.

Thanks!

If I wanted to link algebra and topology in order to specialize in algebraic topology (mathematics), what researches would you recommend me to start reading with?

Here we discuss about one of the famous unsolved problems in mathematics, the Riemann hypothesis. We construct a vision from a student about this hypothesis, we ask a questions maybe it will give a help for researchers and scientist.

Most current methods of NP vs P solution are solved by reduction of one problem to another problem. Is it possible to proof NP vs P using only algebra transformation, e.g. in how we solve quadratic equation using completing the square and prove of Trigonometric Identities?

- One may define the scalar product in a vector space in terms of vectors. Alternatively, one may first define the space of co-vectors and then define the scalar product in terms of vectors and co-vectors. The first mentioned way is simpler. Then for what reasons do some prominent authors, e.g., Van der Waerden in his "Algebra", choose the second mentioned way?

In fact I want to express expm(D * M) as a product exponential matrix.

Thanks for your tips

I need some suggestions what are the growing topics in algebraic combinatorics and graph theory for research? Thank you in advance to everyone who will answer.

Hello all researcher,

i have an idea to learn about fuzzy and algebra, how can we deal with algebras in any fuzzy theorem?

In general in mathematics we work with groups that includ numbers, generally we work with constants.

So for exempel: For any

**x**from**R**we can choose any element from**R**we will find it a constant, the same thing for any complex number, for any**z**from**C**we will find it a constant.My question say: Can we find a group or groups that includ variabels not constants for exemple, we named

**G**a group for any**x**from**G**we will find an**x'**it's not a constant it's another variabel ?and if it exisit can you give me exempels ?

**Thank you !**

In 2010, Dr. Khmelnik has found the suitable method of resolving of the Navier-Stokes equations and published his results in a book. In 2021, already the sixth edition of his book was released that is attached to this question for downloading. Here it is worce to mention that the Clay Mathematics Institute has included this problem of resolving of the Navier-Stokes equations in the list of seven important millennium problems. Why the Navier-Stokes equations are very important?

I have confirmed that the Hessenberg determinant whose elements are the Bernoulli numbers $B_{2r}$ is negative. See the picture uploaded here. My question is: What is the accurate value of the Hessenberg determinant in the equation (10) in the picture? Can one find a simple formula for the Hessenberg determinant in the equation (10) in the picture? Perhaps it is easy for you, but right now it is difficult for me.

I have drived a formula of computing a special Hessenberg determinant. See the picture uploaded here. My question is: Can this formula be simplified more concisely, more meaningfully, and more significantly?

What lessons and topics are prerequisites for algebraic number theory and analytic number theory?

Please tell me the exact topic of each lesson.

More precisely, if the Orlik-Solomon algebras A(A_1) and A(A_2) are isomorphic in such a way that the standard generators in degree 1, associated to the hyperplanes, correspond to each other, does this imply that the corresponding Milnor fibers $F(A_1)$ and $F(A_2)$ have the same Betti numbers ?

When A_1 and A_2 are in C^3 and the corresponding line arrangements in P^2 have only double and triple points, the answer seems to be positive by the results of Papadima and Suciu.

See also Example 6.3 in A. Suciu's survey in Rev. Roumaine Math. Pures Appl. 62 (2017), 191-215.

I got to know about

**Mliclos schweitzer Competition(named after the brilliant Hungarian mind who unfortunately left us in World War-II)**.These problems are so lively and motivating,and*AOPS*contains problems upto its 2020 edition,which means it is still going on. I wanted to know any website for

**Mliclos schweitzer competition,2021**or its past editions;and*how to enroll*to sit for this,*And who all are eligible*. Kindly respond if you have any information. Thanks in advance!

P.S. Feel free to have discussion over the commen section regarding this,but I hereby declare this to be closed(notifications) as of now.The suggested book in the comment section is really interesting,and highly reccomended.

**The aim of this Conference was more far-reaching than the presentation of the latest scientific results. It consisted of finding connections between this fundamental theoretical branch of mathematics and other fields of mathematics, applied mathematics, and science in general, as well as the introduction of top scientists with paragraded structures, which would lead to the connection and cooperation of scientists working in various fields of abstract algebra and algebraic theory of numbers, ultrametric and p-adic analysis, as well as in graph theory and mathematical logic.**

**SARAJEVO JOURNAL OF MATHEMATICS, Vol. 12 (25), No.2-Suppl.**

This issue is dedicated to the memory of Professor Marc Krasner, Officier des Palmes de l'Academie des Sciences de Paris on the occasion of the 30th anniversary of his death.

All manuscripts of this issue were presented at

**International Scientific Conference "Graded Structures in Algebra and their Applications" held in Inter University Center, Dubrovnik Croatia, September 22-24, 2016****Contents of Vol. 12, No. 2-Suppl. DOI: 10.5644/SJM.12.2.00**

**Professor Marc Krasner**-

**photos DOI: 10.5644/SJM.12.3.01**

**Mirjana Vuković**, Remembering Professor Marc Krasner

**DOI:**

**10.5644/SJM.12.3.02**

**Alain Escassut,**Works involving Marc Karsner and French mathematicians

**DOI: 10.5644/SJM.12.3.03**

**Emil Ilić-Georgijević, Mirjana Vuković,**A note on radicals of paragraded rings

**DOI: 10.5644/SJM.12.3.04 Emil Ilić-Georgijević, Mirjana Vuković,**A note on general radicals of paragraded rings

DOI: 10.5644/SJM.12.3.05

**Mirna Džamonja,**Paragraded structures inspired by mathematical logic

**DOI: 10.5644/SJM.12.3.06**

**Vlastimil Dlab,**Towers of semisimple algebras, their graphs and Jones index

**DOI: 10.5644/SJM.12.3.07**

**Elena Igorevna Bunina, Aleksander Vasilevich Mikhalev,**Elementary equivalence of linear groups over graded rings with finite number of central idempotents

**DOI: 10.5644/SJM.12.3.08**

**Nadiya Gubareni,**Tensor algebras of bimodules and their representations

**DOI: 10.5644/SJM.12.3.09 Dušan Pagon,**On codimension growth of graded PI-algebras

**DOI: 10.5644/SJM.12.3.10**

**Smiljana Jakšić, Stevan Pilipović, Bojan Prangoski, Spaces of ultradistributions of Beurling type over ℝ**

^{d}_{+}through Laguerre expansions**DOI: 10.5644/SJM.12.3.11**

**Alexei Panchishkin,**Graded structures and differential operators on nearly holomorphic and quasimodular forms on classical groups

**DOI: 10.5644/SJM.12.3.12**

**Siegfried Böcherer,**Quasimodular Siegel modular forms as p-adic modular forms

**DOI: 10.5644/SJM.12.3.13**

**Alain Escassut, Kamal Boussaf, Abdelbaki Boutabaa,**Order, type and cotype of growth for p-adic entire functions

**DOI: 10.5644/SJM.12.3.14**

1. Functions

2. Matrix algebra & eigenvectors

3. Vector algebra

4. Complex numbers

I was wondering is there any

- model theory of number theory ,hence are there model theorists working in number theory
- the development of arithmatic geometry ,does it have anything to do with questions in logic;and is there any group studying this interaction.
- Anyone is welcome and up for collaboration
- I am interested in finding interaction between algerraic and arithmatic number theory with logic,and to study it to answer logical questions about Arithmatic

Last year (August 2020) I uploaded the algebraic approach to the construction of a regular nonagon DOI 10.13140/RG.2.2.26286.95044/1 which in fact was the trisection of an angle of 60 degrees and so the tangent of the angle of 20 degrees was found to be 0.36387622008. Some years ago (2014) I published with IOSR Journal of Mathematics a paper for the trisection of every angle equal or less than 180 degrees the "Angle Trisection by Straighedge and Compass Only". A trisection, done by the intersection of two circumferences. Now, taking for trisection an angle of 60 degrees, for the intersection of the two circumferences we get the system of equations:

x^2 + (y + sqrt27)^2 = 36

(8 - x)^2 + y^2 =49

and the tangent of the angle of 20 degrees to be: y/(3-x) = 0.36345134567644, differing from the one calculated last year with the algebraic approach to the construction of the regular nonagon.

Where do you think that the difference of the two calculations of the same tangent is due?

We have seen a

**stability in the supply chains of goods, food in particular, during the current pandemic of Covid19**continue, mostly undisturbed.It is very reassuring at a time of uncertainty and macro-risks falling onto societies.

How much do we owe to the optimised management and supervision of Container transport, and multimodal support to it with deep sea vessels, harbour feeder vessels, trains and trucks/lorries?

What is the granularity involved? Hub to hub, regional distribution, local delivery?

Do we think that the connectivity models with matrices, modelling the transport connections, the flows per category (passengers, freight, within freight: categories of goods), could benefit from a synthetic model agreggation of a single matrix of set federating what has been so far spread over several separate matrices of numbers?

What do you think?

Below references on container transport, and on matrices of sets

**REF**

**A) Matrices of set**

**[i] a simple rationale**

**[ii] use for containers**

**[iii] tutorial****B) Containers**

*[1] Generating scenarios for simulation and optimization of container terminal logistics by Sönke Hartmann, 2002*

*[2] Optimising Container Placement in a Sea Harbour, PhD thesis by by Yachba Khedidja*

*[3] Impact of integrating the intelligent product concept into the container supply chain platform, PhD thesis by*

*Mohamed Yassine Samiri*

- The
**Chu-Construction**allows to obtain a *-autonomous category from the data of a closed symmetric monoidal category and a dualizing element. - The
**Cayley-Dickson-construction**builds an algebra B = A + A with involution from the data of an algebra A with involution *. Applied to the field of real numbers it gives successively the field of complex numbers, then the skew-field of quaternions, then the non-associative algebra of octonions, etc.

Due to closeness of the expressions of multiplication m: B \otimes B -> B for the multiplicative unit B we believe that there is an intimate

**link between both notions**.**Has such a link been described in a reference text ?**

**Bibliography:**Can we check the positive definiteness of a multivariable polynomial using MATLAB functions? Of course in literature, there are few complex methods that use tensor algebra. However, before going to such methods if you have some simple suggestions that will be appreciated. In a stability analysis of systems, It is usually required to check that whether the obtained Lyapunov function is positive or not for x>0. For example, if we obtained the following Lyapunov function with these parameters:

a1 = 0.689
a6 = 0.368
a7 = -0.238
a8=-0.336

That particular example is in a homogeneous form. But if the obtained polynomial is in a nonhomogeneous form then what will be the way to proceed? If we have more than two variables (e.g x1, x2, x3, x4,...)? Please correct me if I am wrong in statements.

- We know ,euclidean algorthm is feasible in the set of integers.
- Taking motivation from this,we define an Euclidean Domain(E.D.) as follows:

R is an E.D. if it is a domain,andwe have a map d:R->non-zero integers,

such that, for given a,b(non-zero),

there exists q,r with

a=bq+r and d(q)<d(r).

- Now,this d function ,for any abstract ED,is the counterpart of |.| function,in case of intgers,

we have ,a=bq+r with,0 \le r < |b|.

*Now the question is:*

*[] It turns out (q,r) that exists for a given (a,b) in case of intgers ,is unique .**[A simple proof would be:if another (q',r') exists for same (a,b),*

*if q'=q,we have r'=r,and we are done.*

*and if,q' and q are distinct,bq+r=bq'+r' implies r' and r differ by multiplies of b,thus if*0 \le r < |b| holds,it is clear it won't hold for r',and thus we can never have (q',r') and (q,r) distinct,and we are done.]

*[] But,one now would ask,is it true as well that for given (a,b) in any E.D. the existing (q,r) would be unique ? If yes,we need a proof,and clearly,the same proof does not work,as d(.) is much more generalized than |.| .Or,we need a counter eg!*

*__________________________________________________________________________________*If $A$ is a G-graded algebra then one can define on it a color involution, i.e. a bijective linear map preserving the grading such that the image of a product of two homogeneous elements is defined through a bicharacter of the group G. Color algebras are strictly related with color Lie and Jordan algebras.

Does exist a classification of simple (associative) algebras with color involution? Is there at least a classification of color involutions on matrix algebras?

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?

I am doing a project with PV array in Simulink. When I run small section of the total system i do not get an error. But when i integrate full system(my system model`s are completely okay) I get an algebraic error saying

"An error occurred while running the simulation and the simulation was terminated

Caused by:

Simulink cannot solve the algebraic loop containing 'system_approach_first/PV Array/Diode Rsh/Product5' at time 0.0 using the TrustRegion-based algorithm due to one of the following reasons: the model is ill-defined i.e., the system equations do not have a solution; or the nonlinear equation solver failed to converge due to numerical issues."

Can anyone say me the route to solve this irritating problem?

I have made simulation sample time very small but the problem still on. Also i reduced tolerance to a very small amount

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

I have attached my definition and some results about this.

I am looking for references on Heisenberg-Weyl algebras, namely I am interested in the action of algebra elements on some basis states, i.e. matrix elements in a certain basis.

**Quantum programming**

As we know, old computers work according to the rules of Boolean logic and classical set theory . On the other hand there is a discussion about quantum computers nowadays. If we accept that (the hardware of ) these computers work according to the rules of quantum mechanics (QM), is it plausible that their software must obey the rules of quantum logic and mathematical description of QM (like c*-algebras)? It is wellknown that the distributive law of intersection over union fails in QM though there are some rules in lattice theory which are true in both Boolean and quantum logics.

Do we need to rethink about computer programming or at least in the ways we interact with quantum machines of future?

Can we say that the future programmers or software developers of such machines need some basic skills in mathematical theory of QM like c-*algebras? What would be the role of such mathematics in quantum computers?

As we know, computational complexity of an algorithm is the amount of resources (time and memory) required to run it.

**If I have algorithm that represents mathematical equations , how can estimate or calculate the computational complexity of these equations, the number of computation operations, and the space of memory that are used.**

To build a matrix of differences between the elements of a vector, you could simply get the vector in column form and substract its transposed (row) vector. Similarly to what you would do in matrix multiplication but by subtracting instead of multiplying:

--------1-------2-------3------4

1---(1-1) (1-2)

2---(2-1) ... et cetera

3---(3-1)

4---(4-1)

Giving as a result the matrix of differences between the elements of the vector.

However, this very simple operation is never defined in basic introductory courses or texts to matrix algebra. Is there a name for this operation, like "transposed vector subtraction" or "col-row vector subtraction"?. Why isn't this very simple operation defined in general matrix algebra?

Moreover, not all software packages allow it. For instance, I can perform it in MatLab simply by coding:

X - X' ;

where X is a column vector and X' is its transpose

(while, if I wanted to get the element-by-element difference, the code should be: X .- X')

...But, if I do the same in R, it simply doesn't do it. I had to create a for-loop to code it.

I have proved some property.

1. Idempotency,

2. Commutativity,

3. Associativity,

4. Absorption law,

5. Distributivity and

6. De Morgan's laws over complement.

what is the structure of this property?

I have tried to link some topics in mathematics which included the word " Rational" , I have got many references which used " Rational" in Group theory and Probability and number theory and algebraic geometry and Topology,Chaos theory and so on , Now I'm confused and I have asked my self many times why that "Rational" occurs so much in all topics of mathematics probably informatic and physics ? Why this word interesting in mathematics ? According to the below linked reference I ask why always we investigate to get things in mathematics to be rational ?What is the special of that word rational in mathematics ?

**List of Linked reference include word " Rational"**:

[Regularization of Rational Group Actions](https://arxiv.org/abs/1808.08729)

[Rational Points on Rational Curves](https://arxiv.org/abs/1911.12551)

[Automatic sets of rational numbers](https://arxiv.org/abs/1110.2382)

[Rational homology 3-spheres and simply connected definite bounding](https://arxiv.org/abs/1808.09135)

[Rational Homotopy Theory](https://link.springer.com/book/10.1007/978-1-4613-0105-9)

[A Rational Informatics-enabled approach to the Standardised Naming of Contours and Volumes in Radiation Oncology Planning](https://www.academia.edu/7430350/A_Rational_Informatics-enabled_approach_to_the_Standardised_Naming_of_Contours_and_Volumes_in_Radiation_Oncology_Planning)

[Is Science Rational?](https://link.springer.com/chapter/10.1007/978-94-010-2115-9_36)

[A note on p-rational fields and the abc-conjecture](https://arxiv.org/abs/1903.11271)

[Remarks on rational vector fields on CP1](https://arxiv.org/abs/1909.09439)

[Rational Analysis](https://www.sciencedirect.com/topics/computer-science/rational-analysis)

[Rational probability measures](https://www.sciencedirect.com/science/article/pii/030439758990042X)

[A trace formula for the distribution of rational G-orbits in ramified covers, adapted to representation stability](https://arxiv.org/abs/1703.01710)

[Rational cobordisms and integral homology](https://arxiv.org/abs/1811.01433)

[Conditioned invariant subspaces, disturbance decoupling and solutions of rational matrix equations](https://www.tandfonline.com/doi/abs/10.1080/00207178608933450)

[Why study unirational and rational varieties?](https://mathoverflow.net/q/287364/51189)

[Degree of rational maps via specialization](https://arxiv.org/abs/1901.06599)

[Minimum rational entropy fault tolerant control for non-Gaussian singular stochastic distribution control systems using T-S fuzzy modelling](https://www.tandfonline.com/doi/abs/10.1080/00207721.2018.1526984)

[On the generic nonexistence of rational geodesic foliations in the torus, Mather sets and Gromov hyperbolic spaces](https://link.springer.com/article/10.1007/BF01377597)

[Irrationality Measure of Pi](https://arxiv.org/abs/1902.08817)

[Rational Unified Process](https://arxiv.org/abs/1609.07350)

[Rational Computations of the Topological K-Theory of Classifying Spaces of Discrete Groups](https://arxiv.org/abs/math/0507237)

[On Periodic and Chaotic Orbits in a Rational Planar System](https://arxiv.org/abs/1405.3124)

**Note** I do not investigate about the meaning of the word "Rational" in each topic but I want to know why its were dominated why it is interesting whatever the kind of its meaning ?

Prove that if W is a diagonal matrix having positive diagonal elements and size (2^n – 1)x(2^n – 1), K is a matrix with size (2^n – 1)xn, then:

A = K'*(inv(W) - K*inv(K'*W*K)*K')*K

is a positive definite matrix.

Where:

K '- transpose of a matrix K

inv (W) is the inverse matrix of the matrix W

Using the Monte-Carlo method, I find that the matrix inv(W) - K*inv(K'*W*K)*K' can be negative definite.

Thank you so much for reading my question

I am looking forward to getting your response!

I'm reviewing a lot of papers where the authors take a 3-D autonomous chaotic system (think Lorenz) and add a fourth variable bidirectionally coupled to the other three and then report its unusual properties which typically include lines of equilibria, initial conditions behaving like bifurcation parameters, and sometimes hyperchaos. Usually these systems have two identical Lyapunov exponents (often two zeros) and a Kaplan-Yorke dimension ~1.0 greater than the dimension determined by other methods. Thus it seems clear that the system has a constant of the motion such that it is actually 3-dimensional with an extraneous variable nonlinearly dependent on the other three. Are there algebraic or numerical methods for demonstrating this by finding a constant of the motion?

Over here we are attempting at emulating high quality algebra books (like Euler's and others in order to generate a course for university studients. In Puerto Rico we get extremely able students (not as good at Berkeley's thouch) and they do dismally in the entrance math course which is a precalculus type course. We would like to generate activities, notes and compiuter suport for such a course. Your project sounds very challenging and interesting. We have been reading your papers on this topic. JMLópez