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

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

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?

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.

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?

**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 ?

Are there any relationship between matroids and algebraic structures? I believe that we find this by study on a link between graphs and codes!

Of course, by prof P.Sole opinion : some Australian mathematicians have worked on this method, but we are looking for a geometric study of this concept with algebraic and matrix bases.

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

Hello... ِDear researchers

When i evaluate textbooks with using standards,

To what extent should educational content address any of the following topics?

Problem solving, representation, reasoning, communication, connections

And number and operations, geometry, algebra, probability and data analysis, and measurement?

i want to know replying these questions in preschool at 4-6 years old specially with NCTM standards.

Hi i am planning to investigate the applications of semiring in decision making.i wish to have some clues/clarifications of basic queries as follows:

Q.1 what are the examples of semiring structure's real life situation? how do we describe some real life situation that could be modeled into a semiring structure?

Q.2 how can we use boolean lattices/boolean logic/ boolean search etc to solve certain practical problems in semiring structure to arrive at decision making?

Q.3 can we use graphs/vectors/matrices etc as tools in " Application of semiring in decision making" ?

Q.4 how to link semirings to :

(a) graph theory?

(b) vectors and matrices?

(c) boolean algebra and boolean logic etc ?

Is it possible to find a non-amenable group G, an elementary C*-algebra A (i.e A=K(H) for some Hilbert space) with an action of G on A, such that the corresponding semi-direct bundle (or called C*-dynamical system) is amenable (i.e the homomorphism from the full crossed product to the reduced crossed product is injective)?

The following idea is my attempt: let G be a non-amenable group, and let G act on itself by left multiplication, i. e t \mapsto s^{-1}t for each s in G. This action is amenable, and we can use this action to define an action of G on K(L_2(G)) by natural way. Is the semi-direct bundle consisting of this G, K(L_2(G)) and the action amenable?

Given a second-order differential equation，for example

md

^{2}r/dt^{2}=fsin(wt)Convert the above formula to the first-order form

dr/dt=v

mdv/dt=fsin(wt)

Given the initial and final positions

r(0)=0.r(1)=10

Using the differential quadrature rule, express dr / dt and dv / dt in summation form

Ay=z

But I found that under Dirichlet boundary conditions, the coefficient matrix A is not full rank, so the algebraic equation cannot be solved. I don't know if I applied the wrong boundary conditions. Can anyone help me check if there is a problem with the derivation of the attachment?

Hello scientists,

We are proposing to develop a potential fault diagnostic model of Power Transformers according to the approach based on Hedge Algebra. Following this approach, we need to have many realistic DGA dataset samples to train the diagnostic model.

So, can someone share with me the DGA dataset?

Many thanks.

Hello, I have a matrix M of dimensions m * n, and I have the decomposition into singular values M = U * S * V, I want to know if it's possible to recover the matrix M without having U and V, but having only the S ? Thank you so much!

If B(H) is the algebra of bounded linear operators acting on an infinite dimensional complex Hilbert space, then which elements of B(H) that can't be written as a linear combination of orthogonal projections ?

It is well known that spacetime metric is connected with the Dirac algebra. However, the question of the connection of space-time curve metric with algebra has not yet received wide coverage.

If you are curious about the origin of the pseudo-Riemannian metric, then look to the chapter "On some applications of vector field algebra" of the book "Mathematical Notes...", where it is argued that the reason for the curvature of space-time is the curvature of the vacuum vector field of accelerations of particles of matter moving along the surface of a 7-dimensional sphere and perceived as a local algebra of vector fields.

Research Proposal MATHEMATICAL NOTES ON THE NATURE OF THINGS

I'm a retired applied mathematician seeking a volunteer job using my (math/EE) education and experience in (disc drive/airplane/space) industries. A calculus-level compiler provides a quick way to solve most continuous equations. Know where such a job may be found?

Hark back 11th century, we find algebra shackled by geometry. So, there was no honour in the negative number. When algebra was released, it opened a new world for us. Before then, geometry had brought so much joy to generations and established its omnipotence. Today, conics is a bounty of joy. It is a loyal tool as we observe the dynamics of the universe. Yet, our social world exhibits cubics but we have hardly explored that world. Does conics shackle cubics?

I am studying PG final year. I wanna do my final year proje

A detailed consideration of the First and Second Universal Enveloping Algebras of a semi-simple Lie algebra and their contributions to infinite-dimensional representations of the group are recently undertaken. Hopefully, the second Universal Enveloping Algebra of a semi-simple Lie algebra would make the classification of these representations complete.

The question is simple:

Let A be associative real algebra and A

^{(+)}the related jordan algebra with product A * B= 1/2(AB+BA). The centre of A^{(+)}denoted by z(A^{(+)}) (set of ''operator commute'' i.e. J_a (x ) = x * a ) is it equal to A^{c}?where A

^{c}is the set of elements belongs to A such that ab=ba for all b in A.We have that A

^{c}is contained in z(A^{(+)}), but I dont know when it is equal.we recall that if A is associative algebra of bounded linear operator on Hilbert space, for each a,b self adjoint operators we have (see Topping 1965 prop.1)

ab=ba if and only if J_a J_b = J_b J_a

Thank You

The world is a variety. When we remove the noises around us, we see the beauty of that variety. That is the message of the Euler-Lagrange equation. But where is that beautiful message in ordinary life? Did algebra steal Euler-Lagrange?

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?

A ring R is simple if it has no two-sided ideal. A ring R is Abelian if each idempotent in R is central and R is domain if for each a,b in R, ab=0 implies that a=0 or b=0.

such as the book written by Hartshorn

I am a theoretical physicist and I sometimes use Mathematica to algebraically manipulate large equations. I though use it heuristically and I know a lot of researchers use Mathematica for symbolic computation.

What are the best ways to learn it.

Are there any books or any online course to understand it

What are good practices.

As the algebric sum of the all oxidations states in a compound is zero. There it looks like C has zero oxidation state. What is your opinion about this?

In the context of drug analysis, cliques are very useful to extract the main dense connections between elements that aloud to make partitions to minimal interactions. My question focus on know that, if are you using cliques on your analysis. Im currently working on an experimental project about cliques and its operands as an algebraic structure.

We know that mathematicians study different mathematical spaces such as Hilbert space, Banach space, Sobolev space, etc...

but as engineers, is it necessary for us to understand the definition of these spaces?

I wonder how this projective geometric algebra with arbitrarily introduced metric can be recasted in mother neutral algebra ( https://en.wikipedia.org/wiki/Universal_geometric_algebra) aproach which embeds all metrics in one coherent structure?

My interest is to use such approach for description of fluid essence (aether) flows of this wonderful Creation of our Creator.

Such flows would define space time metric and all other phenomena.

Not established at all ! who is the first founder of algebra in mathematical history ? Websites below are very interesting but not satisfactory !

I have problem with simulation in Simulink, it cuts it after few microsecund due to algebric loop. Problem starts when i try to implement more than 2 surge arresters on overhead line. I' ve tried with different type of solver, but it wasn't better. Also I've tried to change algebric solver from trust to line search, and that didn't help either. Thanks in advance..

Let us have Minkowski space-time, which must be curved so that its metric does not change, and the coordinates cease to be straight lines. How can I do that? In this matter, a hint can be found in the mathematical apparatus of quantum mechanics. Indeed, if we take the Pauli matrices and the Pauli matrices multiplied by the imaginary unit as the basis of the Lie algebra sl

_{2}(**C**), then the four generators of this algebra can be associated with the coordinates of Minkowski space-time not only algebraically, but also geometrically through the correspondence of the elements of the algebra sl_{2}(**C**) and linear vector fields of the 4-dimensional space. Then the current lines of the vector fields of space-time become entangled in a ball, which, when untangled, surprisingly turns into Minkowski space-time.Compute nontrivial zeros of Riemann zeta function is an algebraically complex task. However, if someone able to prove such an iterative formula can be used to get all approximate nontrivial using an iterative formula, then its value is limitless.How ever to prove such an iterative formula is kind of a huge challenge. If somebody can proved such a formula what kind of impact will produce to Riemann hypothesis? . Also accuracy of approximately calculated non trivial accept as close calculation to non trivial zeros ?

Here I have been calculated and attached first 50 of approximate nontrivial using an iterative such formula that I have been proved. Also it is also can be produce millions of none trivial zeros. But I am very much voirie about its appearance of its accuracy !!. Are these calculations Is ok?

Let the linear conformal transformations (homotetics and orthogonal transformation) act on the real plane (x,y) by the real matrix

a b

-b a

If such a matrix acts on a pair of planes, we will talk about the pair conformal mapping. In the case where there is both a pair conformal map and a conformal map in each plane, we obtain an algebra isomorphic to the algebra of quaternions from these maps.

Similarly, if such a matrix acts on a pair of pairs of planes, then we will talk about a two-pair conformal map. Then the algebra of octonions is interpreted by us as the algebra of simultaneously two-pair, pair and simply conformal maps.

It is now clear that the Lie algebras of the octonion algebra g_2 is simply a 14-dimensional algebra of pair rotations of an 8-dimensional Euclidean space.

It would be possible to write out here and generators of this algebra, but we will not clutter the screen. However, if you wish, I will show the generators of another exceptional algebra e_8 that perform rotation in a 16-dimensional space with Euclidean and neutral metric.