Science topic

# Algebra - Science topic

For discussion on linear algebra, vector spaces, groups, rings and other algebraic structures.
Questions related to Algebra
Question
Updated information of my thoughts and activities.
This is meant to be a one-way blog, albeit you can contribute with your recommendations and comments.
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.
Question
The strings in matrix B predict this statement.
As a numerical example, the sum of the entire series 0.99 + 0.99^2 + 0.99^3 + . . . . +0.99^N increases to 190 as N goes to infinity.
Additionally, B-matrix chains provide rigorous physical proof.
The question arises whether a pure mathematical proof can also be found?
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?
Question
Since the generally accepted theory of gravity is GRT, one could try to derive it from some algebraic theory. The basic thing in this theory should be the Dirac algebra, in which the Minkowski space is naturally embedded. Thus, we need to find some dynamical algebra that would be locally equivalent to the Dirac algebra. Where does such a Miracle live?
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.
Question
Giuseppe Piano was the first to establish the axioms of vector space, where the group axioms were linked to the axioms proposed regarding the second law, “number multiplication,” with a basic concept of compatibility.
From your cognitive and scientific point of view, can you give an explanation of this compatibility , which is the secret of the composition of this structure ''vector space'', in a way different from the usual explanation ?
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, ...)
Question
Article Topic: Some Algebraic Inequalitties
I have been collecting some algebraic inequalities, soonly it has been completed and published on Romanian Mathematical Magazine.
Certainly some exotic collection.
Often inequalities are better
Understood as comming from
Some identity, if you suppress
some always positive or negative
term.
Question
In the study of quantum deformations of Witt and Virasoro algebras, the notion of a Hom-Lie algebra first appeared, so is deformation theory insufficient to justify how algebraic structures such as Lie-algebras, ... are deformed, which requires the intervention of the concept of ''Hom'' to define the concepts that have recently appeared, such as Hom-groups...with reasoning ?
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.
Question
Following the findings in the field of deformable algebra, can deformation of an n-dimensional vector space contribute to changing its initial dimension?
How will this contribute to finding the appropriate deformations for infinite-dimensional algebra on the basis that algebra is a vector space with an additional structure, with the hope that this will lead to new physical systems with an infinite number of symmetries, and thus complementarity in the hope of enhancing one of what the theory of strings aspires to ?
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.
Question
Unbelievable is make believable in the world we connected prime number in matrix algebra form and I change history of math Please see it
Budee U Zaman , You must also be interested, and in parallel, in non-primes. As I do here with ultimates and non-ultimates.
Question
D(xy)=xdy +dx y
But a subtraction of two slightly
Different rectangles gives an extra term dx dy.
People have ddbated ever since
Tje smallest positive number doed not exist.
We will fix this with an ideal in algebra
he, with e a nilpotent element ee=0?
Im rather disturbed with the
Synthetical differential crowd
All their results could easily been drawn
from Lagrange and or Taylor.
This old wine from new bottles...
Question
Methods: interviews w teachers, focus groups, observation) Leaning towards social constructivist, but was told I should probably broaden my search. Help!
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.
Question
I am starting a new area of research that is Algebraic topology. Kindly suggest some latest problems and related publications
1. "Simplicial Complexes and Homology Spheres" by Jeff Cheeger and Michael Gromov (1985)This article explores the relationship between simplicial complexes and homology spheres, which are important objects in algebraic topology. It introduces key concepts and provides insights into the combinatorial and topological properties of simplicial complexes.
2. "Cohomology of Groups" by Daniel Quillen (1968)Quillen's article focuses on group cohomology, which is a powerful tool in algebraic topology. It provides an accessible introduction to the subject, presenting the foundations and basic techniques of group cohomology.
3. "Euler Characteristic of Discrete Groups and Finite Buildings" by Pierre-Emmanuel Caprace and Bruno Duchesne (2012)This article investigates the Euler characteristic of certain classes of discrete groups and finite buildings. It discusses the connection between algebraic topology and group theory, offering a deeper understanding of both fields.
4. "Singularities of Smooth Maps and the Whitney Conditions" by Hassler Whitney (1936)Whitney's seminal article establishes foundational results in the theory of singularities of smooth maps. It introduces the concept of Whitney conditions and provides insights into the structure of singularities, making it a classic in algebraic topology.
5. "Homotopy Groups of Spheres" by Michael Hopkins, Alexei A. Ananyevskiy, and Nitu Kitchloo (2021)This recent article examines the homotopy groups of spheres, a fundamental topic in algebraic topology. It presents a comprehensive survey of the state of knowledge about these groups, with a focus on recent developments and open questions.
Question
Dear All,
I am a MSc in Theoretical Physics and I am finishing my master thesis on Algebraic geometry over Lie algebras in Tabriz University . I am searching a PhD position.
If anyone is interested in a PhD student, please feel free to contact me.
Sona Samaei
Sure dear professor
Question
My name is Sona and I have received my master degree in Lie Algebra from Public University of Tabriz (PUT).
In my master thesis, we have studied the algebraic geometry over Lie algebras. Also, I have passed Advanced algebra , finite group, manifold geometry and Real analysis and received IELTS certificate.
If there is an open position and I will be ready for an in-person interview if it is needed.
Best Regards,
Sona Samaei
can contact me @ romeopg@cust.ac.in
Question
Dear ResearchGate Community,
I am currently conducting research in the field of algebraic geometry, and I am in need of the Magma (computer algebra system) for my analysis. Unfortunately, my university does not have access to this software, and it seems unlikely that it will be made available in the near future.
I would be incredibly grateful if anyone could provide me with a student version of Magma, as it would be crucial for my research progress. If anyone has a copy, I would be very appreciative of your assistance.
Computations are limited to 120 seconds, but otherwise it's the full version.
Question
It would be very interesting to obtain a database of responses on this question :
What are the links between Algebra & Number Theory and Physics ?
Therefore, I hope to get your answers and points of view. You can also share documents and titles related to the topic of this question.
I recently read a very interesting preprint by the mathematician and physician Matilde Marcolli : Number Theory in Physics. In this very interesting preprint, she gave several interesting relations between Number Theory and Theoretical physics. You can find this preprint on her profile.
Hi Michel,
Good question!!!
My best wishes....
Question
We know that ideals of a lattice are dual of its filters. Moreover, Heyting Algebra is a special class of lattice. Can we define an ideal of a Heyting Algebra dually to its filter?
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.
Question
Some physicists say”everything is quantum”? Why would they say so? And what is the meaning of this sentence? No one doubts that quantum theory is successful. But from this statement it does not follow that everything is quantum! Therefore these physicists are making logically unjustified conclusions. Do they use quantum logic to ascertain conclusions that are only probable?
The essence of the quantum formalism is algebra. A generic algebra, for instance a von Neumann algebra, has a nontrivial center – consisting of those elements that commute with other elements. The elements of this center correspond to what we may call „classical observables”. Algebras with trivial center are special; they are called „factors”. Why should we assume that algebra is governing our world, if there is such has a nontrivial center? What is the basis of such a bold assumption?
It is true that every algebra can be decomposed into factors. It is true that every algebra can be factored by its center. But it is not true that such a quotient contains all the information contained in the original algebra. Some information is lost. Why should we lose information?
Or, in easier terms: wave functions in quantum theory depend on parameters: space, time, and other numbers. These parameters are classical, not quantum. Of course operators of multiplication by functions depending on these parameters belong to the quantum formalism, but not the parameters themselves. Can a theory be constructed that has no classical parameters at all? No space, no time, no structure, no „nothing?” In such a theory nothing would ever be deduced.
If so, why not accept that once the dream of „everything is quantum” is contradictory and self-destructive, why not to start with a more reasonable assumption that not everything is quantum and draw the consequences of such an assumption? If not everything is quantum, the what exactly is it that is not quantum? Space? Time? Group? Homogeneous space? Some geometry that organizes the algebra structure?
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.
Question
I noticed that there is a structural similarity between the syntactic operations of Bealer's logic (see my paper "Bealer's Intensional Logic" that I uploaded to Researchgate for my interpretation of these operations) and the notion of non-symmetric operad. However for the correspondence to be complete I need a diagonalisation operation.
Consider an operad P with P(n) the set of functions from the cartesian product X^n to X.
Then I need operations Dij : P(n) -> P(n-1) which identify variables xi and xj.
Has this been considered in the literature ?
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.
Question
I see the study of Drapeau et al. (2016), The algebra of conditional sets and the concepts of conditional topology and compacness, Journal of Mathematical Analysis and Applications. They bring the new concept of 'conditional set' (Definition 2.1):
A conditional set X of a non-empty set X and a complete Boolean algebra A is a collection of objects x|a for x in X and a in A such that
- if x|a = y|b, then a = b;
- if x,y in X and a,b in A with a <= b, then x|b = y|b implies x|a = y|a;
- if (ai) in p(1) and (xi) is a family of elements in X, then there exists exactly one element x in X such that x|ai = xi|ai for all i.
In my naive reading, it seems that the traditional axioms of ZFC set theory fit just fine this definition, e.g. axiom schema of separation is defined on the common properties of set elements, for these 'properties' refer (to me) as the 'conditions' in the Drapeau et al. (2016) study.
The thing is that I could combine a set of 'properties/conditions' and the basic set operations (union, inclusion,...), so to create an algebra of conditions satisfying the usual properties.
Is there something I do not see in the Drapeau et al. (2016) study?
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.
Question
Why do algebraic multiplicity of eigenvalues of skew-symmetric matrix pencils are even?
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.
Question
That an n degree polynomial CAN always be factored up into n factors containing Complex roots has been shown true, but there are things not explained in the phrase.
Teachers will not usualy explain alternatives.
Say (Z-x)(Z-x) =0 CAN have the solution Z=x, but can also have Z=x+ey where ee=0 is an
nilpotent element.
Just as i can be made real in 2 by 2 matrices, so can e. Just as we use i or -i dupliciously,
without knowing which, we can use e or e(T) , the transpose of e. One can define i=e-e(T)
using ee=0 , e(T) e(T) =0 and e e(T)+e(T)e =I to show ii=-1. Now substitute this everywhere you see i, and the factorization carries out the same.
Defining ww=1, one can also claim w=e +e(T).
Reading attentively the fundamental theorem of algebra, we find the funny phrase that it is not derivable from algebra...well no wonder.
Dont really have a good name for these elements, i plus whatever else there is. The supra real? Hyper real is already occupied by non standard elements. Neither have they seemingly been well studied well beyond quadratic, though they do exist...
This means a gap in knowledge? Or stuborn historical usage?
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.
Question
I did not understand from the author how he arrived at these specific equations ?
what is phi^1 ,phi^2 ,...
Indeed, a step would be missing (at least).
Question
‎Let $A=(a_{ij})\in M_{m \times n}(\mathbb{R_{+}})$ and $B=(b_{ij}) \in M_{n \times l}(\mathbb{R}_{+}).$ The product of $A$ and $B$ in max algebra is denoted by $A\otimes B,$ where $(A\otimes B)_{ij}=\displaystyle\max_{k=1,\ldots,n} a_{ik}b_{kj}.$
A set $\mathcal{X}_{n \times k} \subset ‎M_{n \times k}(\mathbb{R}_{+})$ is defined by‎
‎$$\mathcal{X}_{n \times k}= \{X \in M_{n \times k}(\mathbb{R}_{+})‎: ‎X^{t}\otimes X = I_{k}\}.$$‎
‎It is known that ‎for the case $k = n,$ $\mathcal{X}_{n \times n}$ is ‎equal to $\mathcal{U}_{n},$‎ where $\mathcal{U}_{n}$ is a unitary matrix in max algebra.
I have the same idea
Question
Irrational numbers are uncomputable with probability one. In that sense, numerical, they do not belong to nature. Animals cannot calculate it, nor humans, nor machines.
But algebra can deal with irrational numbers. Algebra deals with unknowns and indeterminates, exactly.
This would mean that a simple bee or fish can do algebra? No, this means, given the simple expression of their brains, that a higher entity is able to command them to do algebra. The same for humans and machines. We must be able also to do quantum computing, and beyond, also that way.
Thus, no one (animals, humans, extraterrestrials in the NASA search, and machines) is limited by their expressions, and all obey a higher entity, commanding through a network from the top down -- which entity we call God, and Jesus called Father.
This means that God holds all the dice. That also means that we can learn by mimicking nature. Even a wasp can teach us the medicinal properties of a passion fruit flower to lower aggression. Animals, no surprise, can self-medicate, knowing no biology or chemistry.
There is, then, no “personal” sense of algebra. It just is a combination of arithmetic operations.There is no “algebra in my sense” -- there is only one sense, the one mathematical sense that has made sense physically, for ages. I do not feel free to change it, and did not.
But we can reveal new facets of it. In that, we have already revealed several exact algebraic expressions for irrational numbers. Of course, the task is not even enumerable, but it is worth compiling, for the weary traveler. Any suggestions are welcome.
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.
Question
Our answer is YES. Irrationals, since the ancient Greeks, have had a "murky" reputation. We cannot measure physically any irrational, as one would require infinite precision, and time. One would soon exhaust all the atoms in the universe, and still not be able to count one irrational.
The set of all irrationals does not even have a name, because there seems to be no test that could indicate if a member belongs to the set or not. All we seem to know is it is not a rational number -- but what is it?
The situation is clarified in our book Quickest Calculus, available at lowest price in paper, for class use. See https://www.amazon.com/dp/B0BHMPMMTY/
There, Instead of going into complicated values of elliptic curves, and infinite irrationals, algebra allows us to talk about "x".
No approximating rational numbers need to be used, nor Hurwitz Theorem.
Thus, one can "tame" irrationals by algebra, with 0 (zero) error. For example, we know the value of pi. It is 2×arcsin(1) exactly, and we can calculate it using Hurwitz Theorem, approximately.
GENERALIZATION: Any irrational number is some function f(x), where x belongs to the sets Z, or Q -- well-defined, isolated, and surrounded by a region of "nothingness". The set of all such numbers we call "E", for Exact. It is an infinite set.
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.
Question
Hello I start my research work recently. I ask that how we find the unknown From system of algebric eq of nonlinear PDS of higher order which was obtained by applying differaent f
Transformation or methods using Mathematica I use the comment
Solve[{experision 1, ,,,,},{unknowns}] but I can not find the values of them .
(Debug) In:= 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:= Solve::svars: Equations may not give solutions for all "solve" variables. >>
(Debug) Out= {{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)}}
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?
Thank you so much. Christian Neurohr
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
Follow
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
In the definition of a group, several authors include the Closure Axiom but several others drop it. What is the real picture? Does the Closure Axiom still have importance once it is given that 'o' is a binary operation on the set G?
I think no need to include closure property, if it is given that 'o' is a binary operation on the set G.
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
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
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!