Algebra - Science topic

Comparing to elusive truth, how much is an irrational falsity worth?

A sound, archetypical, immemorial, cross-species, well-known, and logical rule is that it is better to reject 100 truths than to accept one falsity. A rejected truth can be accepted tomorrow, but an accepted falsity contaminates the thought immediately.

Truth is worth 1:100 a falsity, maybe more. This applies to all discussions at RG.

Here is a brief response, first to thank our fellow Bulgarian author for his helpful and accurate response and second to shed more light on the question and his search for an answer.

We assume that all mathematicians and physicists know that mathematics is the language of physics, but not all mathematicians and physicists accept that modern physics (classical physics supplemented by the strings of the B matrix) can be the language of mathematics and replace it in some special cases, such as the case of this question.

We now have two similar answers for the same question:

i- The response of modern physics offered by B-matrix statistical physical chains,

the infinite integer series [(1+x)/2]^N is equal to (1+x)/(1-x), ∀x∈[0,1]

ii-The mathematical response of our fellow Bulgarian author which presents almost the same thing as physics.

The question is valid,

Is there a difference between the mathematical answer and the physical answer or are both equivalent?

I will try to explain on my fingers how a pseudo-Riemannian manifold is obtained from the local algebra of vector fields. First, it must be understood that every algebra of linear vector fields is equivalent to the corresponding matrix algebra, and the Dirac matrix algebra is equivalent to the algebra of linear vector fields of an 8-dimensional space with a neutral metric (+4, -4). Secondly, it should be understood that the local algebra of linear vector fields is generated by curvilinear coordinates of an 8-dimensional space satisfying generalized Cauchy-Riemann conditions that provide local preservation of the Dirac algebra. In this case, the generators of the Dirac algebra are locally transformed, thereby generating a pseudo-Riemannian manifold. Note also that the curvilinear coordinates are generated by the current line of the vacuum vector field of matter.

The notion "vector space" simply extends, from a geometrical point of view, the behaviour of vectors in the Euclidean plane. What can we do with vectors in the Euclidean plane? We can add two vectors, having all properties of this addition to can say that a structure of additive group can be identified.

We have also the scalar product of two vectors! But here we have a problem! The scalar product of two vectors in the plane is a real number, not a vector! So such operation is not useful, is not internal! Not compatibility with normal properties of operations on algebraic structures!

But we have also the product of an vector with a real number(called scalar) and this product is a vector too! Good job! Now we have compatibility!

Generalizing, we obtain this new structure, called vector space, which is useful both geometrically and algebraically! This is "the secret", nothing more!

On a set E (of vectors) we have addition and we have multiplication with scalars from fields R or C , obtaining a general structure (VECTOR SPACE) extremely appropriate both for linear algebra and for analysis and its branches (functional analysis, operator theory, ...)

Certainly some exotic collection.

Often inequalities are better

Understood as comming from

Some identity, if you suppress

some always positive or negative

term.

Of course questions like this continue to be trendy in mathematical physics,

I dont see any central importance in algebra or physics yet. Lie algebra is central in the

non commutivity of operators in QM, and far reaching hard things like indefinitness.

Things like Virasoro algebra stand out as exotic

so far, irrelevant in normal QM.

The essential structure on noncommutivity, Lie algebra, stands on its own.

Its just a natural property, no exterior aide.

We can say, that the concept of deforming an n-dimensional vector space can indeed change its initial dimension, and this idea extends to exploring deformations in infinite-dimensional algebras, particularly in the context of theoretical physics. These deformations can impact symmetries in physical systems, offering alternative descriptions that may complement the insights from theories like string theory. Overall, studying deformed algebras is an ongoing area of research with the potential to uncover new mathematical structures and enhance our understanding of fundamental physics.

Budee U Zaman , You must also be interested, and in parallel, in non-primes. As I do here with ultimates and non-ultimates.

Im rather disturbed with the

Synthetical differential crowd

now that I read.

All their results could easily been drawn

from Lagrange and or Taylor.

This old wine from new bottles...

One theory that is often considered relevant for English Language Learners (ELLs) in an Algebra inclusion classroom is the Socio-Cultural Theory. This theory, developed by Russian psychologist Lev Vygotsky, emphasizes the role of social interaction and cultural context in cognitive development.

From the perspective of ELLs, the Socio-Cultural Theory suggests that positive impacts on their learning in the Algebra inclusion classroom can be achieved through:

1. Collaborative Learning: Encouraging opportunities for ELLs to work together with peers and engage in collaborative activities can promote language development, problem-solving skills, and understanding of mathematical concepts.

2. Scaffolding: Providing appropriate support and guidance to ELLs, such as visual aids, manipulatives, or simplified explanations, can help them grasp complex Algebraic concepts while simultaneously building their English language proficiency.

3. Cultural Relevance: Incorporating culturally relevant examples, contexts, and real-life applications within the Algebra curriculum can increase ELLs' engagement and motivation, as they can connect mathematical concepts to their own experiences and cultural backgrounds.

4. Language Support: Offering explicit language support strategies, such as vocabulary instruction, sentence frames, or explicit instruction in academic language, can help ELLs better comprehend and express mathematical ideas in English.

Sure dear professor

can contact me @ romeopg@cust.ac.in

You can use Magma online for free at: http://magma.maths.usyd.edu.au/calc/

Computations are limited to 120 seconds, but otherwise it's the full version.

Hi Michel,

Good question!!!

What about this?

My best wishes....

Muhammad Shahnewaz Bhuyan

In the context of Heyting algebras, an ideal is a subset of the algebra that is closed under taking lower bounds (infima) and is also closed under implication.

More formally, let H be a Heyting algebra. An ideal of H is a subset I ⊆ H that satisfies the following conditions:

1) For any a, b ∈ I, the infimum (greatest lower bound) a ∧ b is also in I.

2) For any a ∈ I and b ∈ H, if a → b ∈ I, then b ∈ I.

Intuitively, this means that an ideal is a subset of H that contains all the "smaller" elements (i.e., lower bounds) of any pair of elements in the ideal, and that is closed under implication, meaning that if a is "smaller" than b, and a belongs to the ideal, then b should also belong to the ideal.

Ideals are an important concept in Heyting algebras because they generalize the notion of a prime ideal in a Boolean algebra. In a Boolean algebra, a prime ideal is an ideal that is also closed under taking complements.

However, in a Heyting algebra, not all ideals are prime, and the concept of a prime ideal needs to be generalized to reflect the algebra's more general structure.

The statement "everything is quantum" is often used as a shorthand to express the idea that quantum mechanics is a fundamental theory that underlies all physical phenomena, including classical mechanics. This idea is based on the observation that classical mechanics can be derived as a limiting case of quantum mechanics when certain conditions are met.

However, as you correctly point out, this statement is not logically justified and is more of a philosophical position than a scientific one. Physicists do not use quantum logic to draw conclusions that are only probable, but rather rely on the mathematical formalism of quantum mechanics to make precise and testable predictions.

The assumption that algebra governs our world is based on the success of quantum mechanics in describing a wide range of phenomena, including atomic and subatomic physics, solid-state physics, and even cosmology. The idea that the world is described by algebra is not a new one, as it has been explored in the context of algebraic geometry and algebraic topology. The center of an algebra, which corresponds to classical observables, is a natural mathematical structure that arises in the context of quantum mechanics.

It is true that some information may be lost when an algebra is factored by its center, but this does not necessarily imply that the resulting theory is incomplete or inadequate. Rather, it suggests that the full richness of the algebraic structure may not be captured by the classical observables alone. In fact, the study of noncommutative geometry and operator algebras has shown that many interesting geometric and topological properties can be extracted from the algebraic structure of quantum mechanics.

In category theory, one can describe the relationship between the algebra of quantum mechanics and classical mechanics using the concept of a functor. A functor is a mapping between categories that preserves the structure and relationships between objects and morphisms.

One can define a functor from the category of classical mechanical systems to the category of quantum mechanical systems. This functor maps classical observables, which are represented by real-valued functions on the phase space of a classical system, to self-adjoint operators on a Hilbert space, which represent quantum observables.

Furthermore, the functor maps classical phase space points to density matrices on the Hilbert space, which represent quantum states. The functor also preserves the Poisson bracket algebra of classical observables, which is mapped to the commutator algebra of quantum observables.

This functor can be seen as a way of quantizing classical mechanical systems, and it provides a bridge between classical and quantum mechanics. However, it is important to note that this functor is not surjective, meaning that not all quantum systems can be obtained from classical systems in this way.

In summary, category theory provides a framework for understanding the relationship between classical and quantum mechanics in terms of functors between categories.

The idea of diagonalization in operad theory has been studied in the literature, although it is typically formulated in terms of "partial compositions" rather than "variable identification" as in your proposed Dij operation.

One approach to diagonalization in operads is to define a "partial composition" operation that takes two elements of an operad P and produces a new element by composing them along a diagonal. More precisely, given elements f ∈ P(m) and g ∈ P(n), we define their diagonal composition f ∘g ∈ P(m+n-1) as follows:

(f ∘g)(x1,...,xm+n-1) = f(x1,...,xi,...,xm,g(i-m+1),...,g(n))

where i is the unique index such that i-m+1 ≤ j < i for all j ∈ {1,...,m+n-1}.

This partial composition operation satisfies some important algebraic properties and has been extensively studied in the context of operad theory. However, it may not be directly applicable to your specific problem of identifying variables in an operad.

Another approach to diagonalization in operad theory involves the use of "modular operads", which are operads that allow for the composition of operations in a non-symmetric fashion. Modular operads provide a powerful framework for studying algebraic structures that arise in geometry, topology, and mathematical physics, and they have been used to study a wide range of phenomena, including Feynman diagrams, string field theory, and knot invariants.

In summary, while the specific operation you propose (Dij : P(n) -> P(n-1)) may not have been studied in the literature, there are related concepts in operad theory that may be useful for your purposes, such as diagonal composition and modular operads. I would recommend exploring these ideas further to see if they can be adapted to your specific problem.

The concept of a conditional set introduced by Drapeau et al. (2016) is based on the notion of a dependent type in type theory. It is a generalization of the traditional notion of a set in set theory, and it allows for the definition of sets in a more flexible and expressive way.

In traditional set theory, a set is defined as a collection of distinct objects, and it is often denoted using set-builder notation. For example, the set of all even natural numbers can be defined as {n | n is a natural number and n is even}.

In the conditional set theory introduced by Drapeau et al. (2016), a set is defined as a type, and it is often denoted using dependent type notation. The dependent type notation allows for the definition of a set in terms of a condition that the elements of the set must satisfy. For example, the set of all even natural numbers can be defined using dependent type notation as {n : N | n mod 2 = 0}, where N is the type of natural numbers and n mod 2 = 0 is the condition that the elements of the set must satisfy.

Therefore, the definition of a conditional set cannot be reduced to the traditional definition of a set in set theory. However, the two concepts are related, and the traditional notion of a set can be seen as a special case of the more general notion of a dependent type in conditional set theory.

See Linear Algebra & its Applications(LAA) vol.438 (2013) p.4625-4653 for the answer. An even stronger result is proved there; not just eigenvalues, but all elementary divisors have even multiplicity, and this is true for skew-symmetric matrix polynomials of any degree, not just pencils. See LAA vol.147 (1991) p.323-371 for an earlier canonical form result for just the pencil case.

Juan, re: TT = T. Yes, my ep operator, ep(x, y) = (y, x) was idempotent under composition: ep o ep = ep. But when treated as a number, ep x ep = 1, as in the split-complexes. My point was Not about ep, but about the resulting system when you added Z to C, where ZxZ= 0. New algebraic ystem < C u {Z} > is no longer a field, and we don't have an alternative Fundamental Theorem in C, but a new theorem in the New system. That's all I am trying to say.

Indeed, a step would be missing (at least).

I have the same idea

We need to be optimistic, because that is the lesson from nature. An animal can self-medicate, obeying natural laws in chemistry that are unknown to animals. A tree grows when pruned, so we can see this pandemic as an opportunity. Let's grow, nature is not a zero-sum game!

Irrational numbers and mathematical real-numbers are uncomputable, with probability 1.

But irrational numbers can be calculated exactly in algebra a and that is how animals are able to calculate-- in a network of thoughts.

Irrational numbers are uncomputable with probability one. In that sense, numerical, they do not belong to nature.

But algebra (this question) can deal with irrational numbers.

Algebra deals with unknowns and indeterminates, exactly.

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

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

(Debug) In[1]:= Solve[{-24 (k - 1)^3 A2 X + 12 (k - 1) A2^2 ==

0, -6 (k - 1)^3 A1 X + 18 (k - 1) A1 A2 == 0,

8 (k - 1)^3 A2 + 12 (k - 1) A0 A2 + 6 (k - 1) A1^2 + 2 w A2 ==

0}, {A0, A1, A2}]

(Debug) During evaluation of In[1]:= Solve::svars: Equations may not give solutions for all "solve" variables. >>

(Debug) Out[1]= {{A1 -> 0,

A2 -> 0}, {A0 -> (4 - 12 k + 12 k^2 - 4 k^3 - w)/(6 (-1 + k)),

A1 -> 0, A2 -> 2 (X - 2 k X + k^2 X)}}

Are they not equal?

Thank you Fawaz Raad Jarullah for all the references!

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.

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).

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.

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?

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.

I think totally these two languages are similar In terms of modeling algebraic.

Thank you so much. Christian Neurohr

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.

That's a very hard problem. Questions of the type you asked are somewhat hopeless.

Such things are done in order to show advanced readers new connections of concepts and, consequently, branches of mathematics.

Dear James Larrouy

These links might help, have a look:

1. https://yutsumura.com/the-matrix-exponential-of-a-diagonal-matrix/

2. https://cpb-us-w2.wpmucdn.com/sites.uml.edu/dist/e/98/files/2020/03/expon3e.pdf

3. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5826581/

Kind Regards

（enhanced） power graphs of groups

10.1016/j.heliyon.2018.e00863

see

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.

See some approaches in this question:

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.

Dear Amirali Fatehizadeh

It would help if you studied advanced abstract algebra, topology, mathematical analysis besides the introductory courses in general number theory.

Regards

Your question is very interesting,

Regards and the best wishes,

Mirjana

Interesting topic.

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

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!!

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

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

Follow

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

linked.

Thanks for the suggestion.

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).

I think you can use soft aggregates and include them in algebra

I think no need to include closure property, if it is given that 'o' is a binary operation on the set G.

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.

an application of ring theoty is geometry, for example check the geometrical properties of complex numbers ring

or neutrosophic numbers

it is not clear for me

I hope these two refs may help you

Linear Algebra :

Matrices multiplication , Inner Products, Tensor Products, etc.

Muhammad Ali Thank you for answer

Muhammad Ali thanks a lot!

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??!?!?!

I think it is called Demorgan,s Algebra

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

I appreciate the answer of Dr. Peter Breuer .

Dear Professor Sprott,

x = 3u^2+4u+5y-z