Science topic

# Algebra - Science topic

For discussion on linear algebra, vector spaces, groups, rings and other algebraic structures.
Questions related to Algebra
Question
I am starting a new area of research that is Algebraic topology. Kindly suggest some latest problems and related publications
Algebraic topology is a mathematics branch that uses abstract algebra tools to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
Question
I define the omega-th Cayley-Dickson Algebra as the union of all the finite-dimensional Cayley-Dickson algebras (over the reals).
Are they not equal?
Question
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 Fawaz Raad Jarullah for all the references!
Question
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)
Question
I am primarily interested in 2-player combinatorial games with perfect information. Useful wiki links are below.
Question
algebraic geometry
The projective plane satisfies the following axioms:
A. Any two distinct points are contained in a unique line.
B. Any two distinct lines are intersected in a unique point.
C. There exists four distinct points no three of them are collinear.
Question
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.
It seems the linearization is accomplished by replacing x1, for x1^2. And separately by replacing x2, for x2^2 & x2^4.
In this way, the surface function is linearized about the origin (0,0), it means we can find f1(x1,x2)=a*x1+b*x2, whilst a and b are calculable in terms of the algebraic parameters, k and c.
But my question transforms to another level. How, we can find a compact algebraic expression for f1(x1,x2), and f2(x1,x2), close enough to the origin. This algebraic expression, need NOT be necessarily linear (it could be a nonlinear function).
Question synopsis:
1--How to find another compact analytical expression equivalent to f1(x1,x2), f2(x1,x2)? (with fair accuracy)
2-- Is it possible to find an approximation near the origin (0,0), for f1(x1,x2), f2(x1,x2), as a function of only one of the two variables (either x1, or x2)?
Regarding the second synopsis, I am to cite another ResearchGate question linked below:
However, the gist of the idea in this link is not clear to me.
Question
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 …
Are you sure you have defined your function correctly?
1. Usually z=x+iy. But in your function z is in the limit, thus being in both the arguments and what the integral is computed against. If z is not x+iy, the function is not a function of (x,y).
2. What do you mean by limit? Do you want to compute the case when z->0?
Question
I don't need this anymore
Here the text:
I am not specialist in Maths, but as you mentioned, it has a new solution. I recommend to check the journal finder to select a suitable journal for your paper.
Question
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!
I think totally these two languages are similar In terms of modeling algebraic.
Question
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?
Hamed Sadaghian Thank you, this is very useful for me.
Question
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.
I put together a solution of the RH myself. While it can't be considered a complete proof while not vetted by experts, it presents various strong arguments and a real breakthrough, which is the inversion formula for Dirichlet series. Given any Dirichlet F(s), you know a(n) from F(s). Unfortunately, it's impossible to have an integral representation for a(n) usually, it's a Taylor power series. Please head to my page for the paper.
Question
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?
That's a very hard problem. Questions of the type you asked are somewhat hopeless.
Question
• 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?
Such things are done in order to show advanced readers new connections of concepts and, consequently, branches of mathematics.
Question
In fact I want to express expm(D * M) as a product exponential matrix.
These links might help, have a look:
Kind Regards
Qamar Ul Islam
Question
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.
（enhanced） power graphs of groups
Question
Hello all researcher,
i have an idea to learn about fuzzy and algebra, how can we deal with algebras in any fuzzy theorem?
10.1016/j.heliyon.2018.e00863
Question
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 !
Question
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 finally could check the PDF, Prof. Aleksey Anatolievich Zakharenko
Dr. Khmelnik uses a variational principle to solve the NS equation, which is very powerful indeed.
He also discusses and gives examples & a reason for turbulence.
I know that the solution of NS is a non-linear problem that involves several modes and that it depends on the source.
However, my knowledge of the foundations of NS is very limited to a few linear/non-linear problems on non-equilibrium gas dynamics& MHD solved by the method, Prof. Miguel Hernando Ibanez had.
Thank you for sharing the link. I recovered my account.
Question
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.
Question
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?
Till now, I do not get the book
J. M. Hoene-Wro\'nski, \emph{Introduction \a la Philosophie des Math\'ematiques: Et Technie de l'Algorithmie}, Paris, 1811.
Question
What lessons and topics are prerequisites for algebraic number theory and analytic number theory?
Please tell me the exact topic of each lesson.
It would help if you studied advanced abstract algebra, topology, mathematical analysis besides the introductory courses in general number theory.
Regards
Question
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.
Regards and the best wishes,
Mirjana
Question
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.
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.
Interesting topic.
Question
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 ℝd+ 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
Dear Sajda,
A good conference ... please offer other conferences.
I 'll invite you to the next conference if you can fit in.
All the best for you,
Sincerely Mirjana
Question
1. Functions
2. Matrix algebra & eigenvectors
3. Vector algebra
4. Complex numbers
Dear Prof. Halim,
It is difficult to categorize papers, but in general you can go for papers dealing with:
Fuzzy sets ( Equivalent to defining membership functions)
Can i put sets (defined through characteristic functions)
In fact, if you go by the latest definition of Mathematics (It is the study of sets, functions and their properties) there are substantial portion of Mathematics dealing with functions only.
Rough sets (Rough membership functions)
Soft sets and its variants (through characteristic function approach and membership function approach)
Matrices are also functions (In fact transformations)
Keeping my above observations in view, can you please be more specific!!
Question
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
As far as I know, in the entire history of mankind, only two philosophers have seriously dealt with logic, this is Aristotle and Hegel. Of these, only Hegel did mathematics. Nobody else dealt with this problem.
Sincerely, Alexander
Question
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?
Angle trisection is the division of an arbitrary angle into three equal angles. It was one of the three geometric problems of antiquity for which solutions using only compass and straightedge were sought. The problem was algebraically proved impossible by Wantzel (1836).
Weisstein, Eric W. "Angle Trisection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AngleTrisection.html
Question
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
 Generating scenarios for simulation and optimization of container terminal logistics by Sönke Hartmann, 2002
 Optimising Container Placement in a Sea Harbour, PhD thesis by by Yachba Khedidja
 Impact of integrating the intelligent product concept into the container supply chain platform, PhD thesis by Mohamed Yassine Samiri
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Question
• 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:
Due to the close Connection between the Chu constuction and the Caley-Dickinson construction, as I found in the question that the two notions are
Question
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.
Thanks for the suggestion.
Question
• 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!
__________________________________________________________________________________
Let K={|z|<=1} be the closed unit circle (a connected compact) in C, R=O(K) be the ring of holomorphic functions f on K with d(f)=#{zeroes of f in K}, R is a ED.
Let a=a(z)=z^2+2z+1, b=b(z)=z (hence d(a)=2, d(b)=1). Let h=h(z)=sin(z)/z, h(0)=1.
Then
1) z^2+2z+1=(z+2)z+1, q=z+2, r=1 (hence d(r)=0)
2) z^2+2z+1=(z+2+h)z+1-zh, q'=z+2+h, r'=1-zh=1-sin(z) (hence d(r')=0,
since sin(z)=1 iff z=\pi/2 +2\pi*k, k\in Z).
Question
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 think you can use soft aggregates and include them in algebra
Question
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 models 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
Dear Zarzisur Rahman Rony, it should be enough to add a "memory" block (you can find it in Simulink's Discrete toolbox) where algebraic loop occurs (e.g. after "system_approach_first/PV Array/Diode Rsh/Product5") then try with/without setting "inherit sample time" option. This allows breaking algebraic loop and it should work.
Question
We study about some laws for group theory and ring theory in algebra but where it is used.
an application of ring theoty is geometry, for example check the geometrical properties of complex numbers ring
or neutrosophic numbers
Question
it is not clear for me
Question
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.
Question
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?
What actually happens is that, closure property is a common one to almost all structures (systems). Therefore, authors who drop it in their texts assume that it is automatically embedded in the structure. Others who include it want to be vivid in their texts for clarity sake.
So, those who drop this important property do not truncate it entirely or cancel it.
Question
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?
Linear Algebra :
Matrices multiplication , Inner Products, Tensor Products, etc.
Question
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.
Question
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.
Awesome, what you sent me in the files makes perfect sense... I did not know that answer, thanks a lot.
However saying " new operators should be introduced only if we are not able to reach desired effect by existing ones " ... would mean that, in basic arithmetic, the multiplication and exponentiation operations are not needed because:
4 x 3 = 4 + 4 + 4
and
4^2 = 4+4+4+4
You must agree that your answer is quite complicated. I'm thinking about myself at 11 years old, on my first introductory class to matrix algebra... thinking, what is going on!!!!... Why the first opperation is not a sum??!?!?!
Question
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 think it is called Demorgan,s Algebra
Question
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)
[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)
[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 ?
In elliptic curves theory Rational is derived from the firleld which the curve is built over
It is very impotant to study the rational points (rational coordinates) since the rational field has many algebraic extensions and this is an important property
Question
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.
Question
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 appreciate the answer of Dr. Peter Breuer .
Question
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?
Dear Professor Sprott,
x = 3u^2+4u+5y-z
Question
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
The three books I have written for high school teachers will be published before the end of 2020 by the American Math Society:
Rational Numbers to Linear Equations
Algebra and Geometry
Pre-Calculus, Calculus, and Beyond.
The first volume is scheduled to appear in August. It will be easier to discuss the question about school materials once these volumes have appeared.
Question
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.
Following
Question
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 ?
You can link semiring with graph theory by giving weights to vertices and edges of a graph G by the elements from a given semiring. Such a study have been initiated in the year 2015 by me with my scholars. The theory developed is called $S$ valued (Semiring valued) graphs. Lot of work is going on in this area, which you can use to study for decision making problems by using shortest path problem or network analysis.
Question
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?
Question
Given a second-order differential equation，for example
md2r/dt2=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?
Question
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.
You can contact the specialized laboratories in universities or professors in this specialization
Question
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!
In the Singular value decomposition of M= U*S*V', each of the three component matrices U, S and V are providing separate pieces of information which together is used to generate M.
A Real Matrix M represent a correlation beetween two bases, one in its Column space and the other in its Row space, this is represented by the Full Rank factorization of M :
M(m by n)= P(m by r)*Q(r by n), here P is the Basis matrix in Column space of M, Q' is the Basis matrix in Row space of M, correlated to P, here r is the rank of M.
By multiplying with an arbitrary Invertible matrix X(r by r) and invX(r by r) as P*X*invX*Q = P1*Q1 = M, we see that the bases themselves are not unique but the correlation is preserved and so is M, under such Basis transformations, here inv X stands for inverse of X.
However, if we take Y(r by r) as a different invertible matrix, Y not inverse of X, then P*X*Y*Q = N( m by n), here M and N are two different matrices having all four Fundamental subspaces ( Column space, Left Nullspace, Row space, Nullspace ) identical.
In general for any real M, we have M(m by n)= Q(m by r)*C(r by r)*R( r by n), where Q and R' are Orthonormal bases of Column space and Row space respectively of M, interrelated by the Invertible, non diagonal matrix C, it is a special case of the above full rank factorization, this special case becomes the SVD decomposition when we construct U and V in terms of eigenvectors of MM' and M'M respectively, which are real, symmetric and positive semidefinite/positive definite, and hence are Diagonalizable in Orthonormal eigenbases, since Column space of M = Column space of MM' and Rowspace of M = Column space of M'M, these orthonormal eigenbases can be used as Q and R matrices appropriately, with C matrix now becoming diagonal, the set of singular values of M.
Starting with S, unless you provide correct U and V, you will end up with a different matrix M1 having same set of singular values but different set of four Fundamental subspaces, likewise if you choose correct U and V but wrong S, you will get a matrix M1 with Identical set of the four Fundamental subspaces as that of M, but different set of singular values. Unless there is degeneracy in singular values, and that they are to be conventionally arranged in non increasing sequence from top to bottom in S, there is no flexibility in choosing the U and V matrices arbitrarily and still generating correct M.
Question
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 ?
I found something more interesting here https://arxiv.org/pdf/1608.04445.pdf
Question
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.
The line element field is a dynamical object independent of the Riemannian metric. There is no physical requirement that either objects exist, they just do in a Lorentzian spacetime. The physics comes out from what the combination of those objects explains. For example, general relativity is explainable from the Riemannian metric as long as it is Minkowskian in free fall. The line element fields in MGR provide for the description of dark energy and dark matter that GR cannot explain by itself. It must be emphasized that Einstein's equation is the same in both GR and MGR, but the latter contains a tensor that describes the e-m of the gravitational field. Please carefully go through the papers so that we can get on the same wavelength.
Question
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?
You are in the right place, the RG platform for students and researchers as well.
Wish you good luck
Question
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?
That is very good
Question
I am studying PG final year. I wanna do my final year proje
Question
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.
Infinitesimal theory of representations of semisimple Lie groups by V. S. Varadarajan
Question
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 Ac ?
where Ac is the set of elements belongs to A such that ab=ba for all b in A.
We have that Ac 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
Dear Muna S. Kassim
I'm sorry, but i don't know, maybe there will be a easy prove for finite dimensional algebras...that I have not found.
Question
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?
Euler- Lagrange equation is one of the most important equations that provides an optimum solution of some integral functional equation. It raised in several applications. It is the most famous equation in the calculus of variation regardless of the advances in all branches of mathematics.
Algebra may show some numerical algorithms that help in solving the partial differential Euler Lagrange equation.
So, we can say algebra supports and not ignore such an elegant original equation.
Best regards
Question
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?
If $A$ is a trivial Banach algebra, i.e., a Banach space endowed with the trivial multiplication xy=0, for every x and y, then $A$ has obviously weakly continuous multiplication. In this case, the algebra is even the algebraic direct sum of radical one dimensional (trivial) algebras. This can be shown using a Hamel basis of $A$.
Then the question now is the following : Is a radical Banach algebra with weakly continuous multiplication always a subalgebra of the algebraic direct sum of finite dimensional radical algebras ?
Question
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.
Yes, I didn't pay attention to the condition on the ring.
If R is a simple abelian ring, then R should be a field and hence an integral domain.
Therefore, there is no example of a simple abelian ring which is not a domain.
See
en.wikipedia.org › wiki › Simple_ring
Simple ring - Wikipedia
Question
such as the book written by Hartshorn
see references
Algebraic geometry
SpringerRobin HartshorneYear:1977
Introduction to Algebraic Geometry
AMSSteven Dale CutkoskyYear:2018
Question
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.
I have used Maple for years and absolutely love it! I've also used Mathematica and the Wolfram web site is the boss! I can really appreciate these resources because I learned higher math in the stone age, when we had to do everything by hand. I derived equations that went on for a dozen pages. One tiny mistake along the way ruined the outcome. At least I had a pencil and didn't have to chisel equations into stone. Be very thankful for the technology but don't neglect the theory. Knowing why and how is as important as what (getting an answer)!
Question
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?
Yes, it has an oxidation number of 0. Since CH2Cl2 is neutral, each H contributes +1, and each Cl contributes -1, we have:
0 = oxidation number(C) + 2(1) + 2(-1)
--> the oxidation number of C is 0.
Question
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.
Below is an interesting and essential article about clique analyzing: www.analytictech.com › borgatti › papers › analyzing_clique_overlap
Best regards
Question
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?
Yes, we do have to know about the spaces, at least during our university studies, to enables us to expand our mind into abstact level, that will be very usefull for design activities
Question
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.
In the mother algebra, or any 2-up model, one can easily create null vectors (e_+ + e_-), such a null vector can then be used to projectivize. The R_{n,0,1} used for projectivisation in PGA is simply the smallest (and hence most efficient) algebra that offers one of these null vectors. When one is dealing with geometry in spaces of constant curvature, the 1-up approach can be considerably faster. (and in this scenario, the degenerate metric is needed as it corresponds to zero curvature).
One of the common ways to work in the mother algebra is by using R4,4 - and then using 4 zero vectors in combination with a projective model for geometry.
Question
Not established at all ! who is the first founder of algebra in mathematical history ? Websites below are very interesting but not satisfactory !
The scientist is characterized in that they is interested in everything (“how?” Or “who?”). But the next questions are: "why?" and "for what?".
Question
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..
Please check for any signs of circular dependency of block outputs and inputs in the same time-step. If the Trust-Region Method and Line Search Method do not work, then you may need to manually break the algebraic loop.
Question
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 sl2(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 sl2(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.
In fact, in the previous post, the path from Dirac's quantum geometry to Einstein's geometry was indicated, and the mathematical apparatus for successfully passing this path must be found in the mechanism of local algebras of vector fields1.
1)
Question
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?
In a paper that can be found on arXiv or at , LeClair gives a reasonably accurate algorithm to estimate the non-trivial zeros up to 10^200=Googol^2. My paper that can be found at Cogent Mathematics, on arxiv or RG
gives an estimate that bounds the n'th zero and checks LeClairs result for the number Googol. Although both these are not iterative, and work only for non-trivial zeros that sit on the critical line, they are predictive and easily calculated. Once a zero is estimated, or bounded, it's accurate value can then be found from formula given.
Question
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.
The generators of e_8 Lie algebra:
28 type generators
(1_{i j } - 1_{j i}) (x_1,...,x_16)[\partial x_1,...,\partial x_16]
where i,j = 1,...,8
28 type generators
(1_{i+8 j+8} - 1_{j+8 i+8})(x_1,...,x_16)[\partial x_1,...,\partial x_16]
where i,j = 1,...,8
64 type generators
(1_{i j+8} - 1_{j+8 i})(x_1,...,x_16)[\partial x_1,...,\partial x_16] where i,j = 1,...,8
64 type generators
(1_{i j+8} + 1_{j+8 i})(x_1,...,x_16)[\partial x_1,...,\partial x_16] where i,j = 1,...,8
64 type generators
(1_{i i} - 1_{j+8 j+8})(x_1,...,x_16)[\partial x_1,...,\partial x_16] where i,j = 1,...,8
Total: 28 + 28 + 64 + 64 +64 = 248
where 28 + 28 + 64 = 120 and 64 + 64 = 128