Pure Mathematics - Science topic

In quantum fluids the phase of a wavefunction is smooth and can be represented by a topological manifold of genus 0, the velocity creates a vector field over this manifold. Then can the hairy ball theorem be directly applied to state that there must exist a point where the vector field creates a vortex, showing a purely mathematical reason for the formation of vortices?

Can we apply the theoretical computer science for proofs of theorems in Math?

If it does, give me the name of the topic with its Software.

I am Ziad Rabea, a high school student with a deep passion for pure math and computer science, I'd like to introduce myself as the founder of LBFCT (Linguistic Barriers Free Coding Tech), the developer of a new multilingual programming language and the author of a book about it.

Back in my primary school years when I was self-learning calculus for the first time, my mind struggled with concepts like infinity and continuity, especially the idea of infinitesimals but I somehow connected these concepts to the definition of Zero, I thought of developing a framework that allows division by Zero and develop theories that help us understand the behavior of some paradoxes that involve Zero and infinity.

I recently launched a pre-print about it called Zerodivics and I would be happy to get your valuable feedback ❤, as a high schooler, I accept that my academic skills may not be that good, but I am willing to get your valuable advice.

we think of a point as a spot in an Euclidean plane

indicating a position.Intuitively we assign arbitrary

framework.The confusion (for me) is when

we consider a line segment.It is made of

infinite points aligned in a certain direction.

So a segment results when all these “ directions”

are added vectorial, but a zero vector is

conceptually an infinite direction entity

so how is this vector direction specified for

a point nathematically. physically of course

we have a visible dot with visible dimension,

Any insight or explanation from all sources

is welcome.

In general relativity, singularities, like those theorized to exist at the centres of black holes or at the origin of the Big Bang, present unique challenges to both mathematical and physical understanding. These singularities are characterized by conditions where physical laws as currently understood cease to be predictable or observable. This raises a fundamental question: Are singularities purely mathematical constructs that exist beyond the Planck scale, and thus beyond the scope of empirical validation?

General relativity predicts the existence of singularities, regions of infinite density where the gravitational field becomes infinite. Notably, the mathematical representation of these singularities involves values that approach division by zero, which is undefined and non-physical. For example, the Friedman-Robertson-Walker (FRW) solution to Einstein's field equations, which underpins the standard Big Bang model, indicates a singularity at the time of the universe's inception.

These singularities occur at scales smaller than the Planck scale, where the effects of quantum gravity are hypothesized to become significant, yet remain unquantified by existing theories. As such, singularities are not observable with current technology or provable by existing physical laws, which are based on empirical evidence. This limitation leads to the interpretation of singularities as mathematical abstractions rather than physical entities.

Given these considerations, should singularities be viewed solely as theoretical constructs within the mathematical frameworks of cosmology and black hole physics? How might advances in theoretical physics, particularly in quantum gravity, change our understanding of these enigmatic features? Whether and how singularities might bridge the gap between current mathematical theory and physical reality.

I want to do phd on mathematical modeling of infectious diseases (eg. Covid 19, maleria, denge). I am also interested in pure mathematics as well like Nonlinear Analysis, Variational Inequalities so my question is can i get any connection between this two part. Need suggestions thank you.

Hellow.

Honestly I am so proud of "my guys", i.e. the young generation of Physicists that engaged with my thesis. They grasped the main ideas so fast and actually took on the establishment grizzled veterans. What took me years to figure out, they got it in couple of months.

I think this generation, my guys, can challenge the entire establishment with what they can do now. The Establishment too stiff and rigid, and is composed mainly by dogmatic thinking and irrelevant figures who refuse to think ambitious . As one Physicist, who was considered an influential:

"one of the reasons I was able to achieve something, Is because I was not too ambitious".

Terrible message. I think that the young generation should be as ambitious, as daring, as creative as possible, and not be scared to take on the entire establishment if needed, even If they stand alone, by themselves.

Physics is not only about equations, it's also about influence and Leadership. Influencing the young generation to push forward at all costs, is a much greater contribution, In my view, than any individual achievement.

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?

Hellow.

I want to know what are the leading journals in Pure mathematics, and what would be the cost are the publishing fees estimations for a proof that is less than five pages, assuming it will be accepted ?

Hellow.

I have been wondering why some people claim that the new theory proposed is not experiment tested. To those one ask:

1. when the theory produces the principle that govern the right measurement values, it does not count as experimentally tested ?

2. When the theory produces a the correct number at the 26 position of an infinite series along side two first numbers which are correct, The only implication is that the continuation is correct, as those are dependent on one another.

3. The Physics community Not yet accepted the full framework despite It is fully principle governed. Therefore it is a bit disappointing.

1) There is no publications of mine in areas other than pure mathematics.

2) there is only 1 article of mine with P. Coulton: "Besicovitch triangles cover unit arcs"

published in 2006 by Geometric Dedicata

Is there compatable between plagiarism program with research of pure mathematics ?

Response to “Did AI prove our proton model WRONG?" https://www.youtube.com/watch?v=TbzZIMQC6vk

Your video brought two things to mind - the book “The Grand Design” by Stephen Hawking and Leonard Mlodinow, and the picture of the TARDIS – Doctor Who’s blue “telephone box” that travels anywhere in space and time. Combining these with the early experimental hints of a charm quark existing inside the proton suggests there may be experimental support for intergalactic and time travel, as proposed by the following interpretation of the Riemann hypothesis and inverse-square law.

Owing to a phenomenon called Colour Confinement, quarks can never exist in isolation – they’re found combined within protons, neutrons, and mesons. “The Grand Design” states,

“The question of whether it makes sense to say quarks really exist if you can never isolate one was a controversial issue in the years after the quark model was first proposed.” “It is certainly possible that some alien beings with seventeen arms, infrared eyes and a habit of blowing clotted cream out their ears would make the same experimental observations that we do, but describe them without quarks.”

If the quark model is used, a charm quark existing inside a proton reminds me of the repeated comment on Dr. Who shows that the Tardis is bigger (has more volume) on the inside than the outside. The charm quark in the proton is a bigger MASS. This can make us wonder if the charm quark in the proton is, like the Tardis, associated with travel anywhere in space and time. The bigger volume or mass inside can be exaggerated to equal infinity while the smaller volume or mass viewed from the outside can be exaggerated to equal zero.

What happens if we don’t use the quark model but rely on the aliens’ model, which may be purely mathematical? (The final paragraph speaks of particles' centres occupying identical space-time coordinates. Current knowledge says this is impossible for fermions but in a purely mathematical universe which is as malleable and flexible as any image on a computer screen, the quark model doesn't necessarily apply and a future theory of quantum gravity could unite bosons and space itself with fermions in a kind of supersymmetry.)

The Riemann hypothesis doesn’t just apply to the distribution of prime numbers but can also apply to the fundamental structure of the mathematical universe’s space-time. In mapping the distribution of prime numbers, the Riemann hypothesis is concerned with the locations of “nontrivial zeros” on the “critical line”, and says these zeros must lie on the vertical line of the Complex Number Plane (CNP). Besides having a real part, zeros in the critical line (the y-axis) have an imaginary part (an equally real part described with “imaginary” numbers). This is reflected in the real +1 and -1 of the x-axis in the CNP, as well as by the imaginary +i and -i of the y-axis. In the upper half-plane of the CNP, a quarter rotation plus a quarter rotation equals a half – both quadrants begin with positive values and ¼ + ¼ = ½. (The Riemann hypothesis states that the real part of every nontrivial zero must be 1/2.) While in the lower half-plane, both quadrants begin with negative numbers and a quarter rotation plus a negative quarter rotation equals zero: 1/4 + (-1/4) = 0.

The inverse-square law states that the force between two particles becomes infinite if the distance by which they’re separated goes to zero. Remembering that gravitation partly depends on the distance between the centres of objects, the separation only goes to zero when those centres occupy identical space-time coordinates (not merely when the objects’ sides are touching). That is – infinity equals the total elimination of distance, or zero. Remembering that the holographic-universe theory - https://www.youtube.com/watch?v=klpDHn8viX8- hints at the possibility of deletion of the 3rd dimension, the cosmos could possess this absence of distance in space and time via the electronic mechanism of binary digits which would make the universe as malleable and flexible as any image on a computer screen. If infinity is the total deletion of distance in space-time, there is nothing to rule out instant intergalactic travel or time travel to the past and remote future. Infinity does not equal nothing – nor does zero. Zero would be something if it’s paired with one to form the binary digits used in electronics.

Importantly for quantum computing [1], software can use only the set Q, while the hardware can use the sets N or B for better results. Computers do not find imaginary numbers in the Boolean set of {0,1}; only 0 and 1, and yet can calculate a reactive current! Imaginary numbers have no physical meaning [2], so we can discard "non-abelian anyons" in attempts for quantum computing.

The numbers to use in all sciences must be based on Boolean numbers (set B), natural numbers (set N), integer numbers (set Z), and rational numbers (set Q). Other sets can be added, for human benefit, such as set Q*, but are not needed by computers -- that use only set B.

There is no reason to use the set ℂ in mathematics, physics, or engineering, because there is no physical meaning [2] -- and hides physical meaning.

One can totally avoid using the set ℂ and can calculate, numerically and in algebra, all that one needs [2]. We, thus, proved a negative -- the set ℂ is not needed in mathematics, physics, or engineering [2].

We just have to enlarge our view of the set Q, with a possible rotation included (i.e., not added) for each number, and resulting expressions using arithmetic. This new set, to be published elsewhere, is still called by the same letter Q in the set Q*, but the letter now stands for "Quantum".

We have a new terminology in the 21st century. Are we aware of more dimensions? We propose at least 6 dimensions.

Nature seems holographic, providing the connectedness of disjoint regularities -- i.e., connecting Number Theory In "pure" mathematics with quantum physics, continued fraction with the Schrödinger equation for bound states in quantum computing [1].

Regularities such as amplitude and phase can be taken naturally into account together, not just amplitude. The well-known double-slit experiment], the laser effect on targets, and holograms, e.g., depend on this (in publication).

In addition, as in the FFT and now in the FT, one does not need to calculate trigonometric functions (in publication).

Then, using the cartesian construction, the new set Q* applies to higher-dimensional spaces nD, where n is in N, such as 4D and spacetime.

In a next publication, we will show that the reality we live in is described by at least 6 dimensions, including the Euler-Lagrange equation and the Calabi-Yau manifold, but also using the set Q*. This proposal also includes SR and QM. A preview is available in [3].

As a consequence, reality becomes quantum ontically, as evidenced by the new set Q* using amplitude and phase as part of a number to describe the dimension nD, where n>=6.

What is your qualified opinion?

REFERENCES

[1] https://www.researchgate.net/publication/370063125/

[2] https://www.researchgate.net/publication/370755777/

[3] https://www.researchgate.net/publication/333677283/

When a set of functions with a property is a prevalent subset of some other set of functions, almost all functions with that property belong to that set of functions.

Is it true the set of all measurable functions (using the uniform probability measure for sets measurable in caratheodory sense) with infinite or undefined expected values form a prevalent subset of the set of all measurable functions?

Suppose A is a set measurable in the Caratheodory sense such for n in the integers, A is a subset of Rⁿ, and function f:A->R

After reading the preliminary definitions in section 1.2 of the attachment where, e.g., a pre-structure is a sequence of sets whose union equals A and each term of the sequence has a positive uniform probability measure; how do we answer the following question in section 2?

Does there exist a unique extension (or method constructively defining a unique extension) of the expected value of f when the value’s finite, using the uniform probability measure on sets measurable in the Caratheodory sense, such we replace f with infinite or undefined expected values with f defined on a chosen pre-structure depending on A where:

How do we answer this question?

(See sections 3.1 & 3.3 in the attachment for an idea of what an answer would look like)

Edit: Made changes to section 3.5 (b) since it was nearly impossible to read. Hopefully, the new version is much easier to process.

How to write formally within the context of mathematics that: "given two series S1 and S2 and they are subtracted each other coming from a proved identity that is true and the result of this subtraction is a known finite number (real number) (which is valid) the two series S1 and S2 are convergent necessarily because the difference could not be divergent as it would contradict the result of convergence? I need that definition within a pure mathematical scenario ( I am engineer).

" Given S1- S2 = c , if c is a finite and real number, and the expression S1-S2 = c comes from a valid deduction, then, S1 and S2 are both convergent as mandatory."

Best regards

Carlos

What are the most frequent challenges which undergraduate students encounter in understanding the concept of the continuity of real functions and do educators address these challenges?

I am aware of the facts that every totally bounded metric space is separable and a metric space is compact iff it is totally bounded and complete but I wanted to know, is every totally bounded metric space is locally compact or not. If not, then give an example of a metric space that is totally bounded but not locally compact.

Follow this question on the given link

I am trying to solve the differential equation. I was able to solve it when the function P is constant and independent of r and z. But I am not able to solve it further when P is a function of r and z or function of r only (IMAGE 1).

Any general solution for IMAGE 2?

Kindly help me with this. Thanks

Is the reciprocal of the inverse tangent $\frac{1}{\arctan x}$ a (logarithmically) completely monotonic function on the right-half line?

If $\frac{1}{\arctan x}$ is a (logarithmically) completely monotonic function on $(0,\infty)$, can one give an explicit expression of the measure $\mu(t)$ in the integral representation in the Bernstein--Widder theorem for $f(x)=\frac{1}{\arctan x}$?

These questions have been stated in details at the website https://math.stackexchange.com/questions/4247090

Hello

Can someone help me to solve this?

Because I really don't know about these problems and still can't solve it until now

But I am still curious about the solutions

Hopefully you can make all the solutions

Sincerely

Wesley

Many proposals for solving RH have been suggested, but has it been splved? What do you

A careful reading of THE ABSOLUTE DIFFERENTIAL CALCULUS, by Tullio Levi-Civita published by Blackie & Son Limited 50 Old bailey London 1927 together Plato's cosmology strongly suggest that gravity is actually a real world mathematics or in another words is gravitation a pure experimental mathematics?

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.

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

By dynamical systems, I mean systems that can be modeled by ODEs.

For linear ODEs, we can investigate the stability by eigenvalues, and for nonlinear systems as well as linear systems we can use the Lyapunov stability theory.

I want to know is there any other method to investigate the stability of dynamical systems?

Please share your Opinion.

Mathematics is the queen of Sciences. It deals with the scientific approach of getting useful solutions in multifarious fields. It is the back bone of modern science. Ever since its inception it is going into manifold directions. Now in these days of advanced development, it is interlinked with every important branch of technical and modern science. Pure mathematics and Applied mathematics are two eyes of Mathematics. Both are having and playing an equal and significant role in the field of research.

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?

What are your opinions and suggestions regarding my research (Development of Mathematical Model for accounting system in the Stocks)

Is our use of pure mathematics in this research a shift in the field of modern accounting

And continuous development of analytical accounting curricula

Is there a difference between pure and applied mathematics?

In Wikipédia, we can find the following definition :

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers. But Number Theory is mostly applided, for example, in modern Data Encryption Techniques (cryptography).

Dear All,

I am hoping that someone of you have the First Edition of this book (pdf)

The other editions are already available online. I need the First Edition only.

It would be a great help to me!

Thank you so much in advance.

Sarah

I'm teacher and suffering a lot to complete my MS. I need to write an MS level research thesis. I can work in Decision Making (Preference relations related research work), Artificial Intelligence, Semigroups or Γ-semigroups, Computing, Soft Computing, Soft Sets, MATLAB related project etc. Kindly help me. I would be much grateful to you for this. Thanks.

The function assumes a direct and reverse law. What do we know about the inverse function? Never mind. This is just the shadow of the direct function. Why don't we use the inverse function, as well as direct? ------------- I propose the concept of an unrelated function as extended concept of reverse function. ------------ There is a sum of intervals, on each of which the function is reversible (strictly monotonic) -nondegenerate function. ---------- For any sum of intervals, there is an interval where the function is an irreversible-degenerate function.

Mathematics has been always one of the most active field for researchers but the most attentions has gone to one or few subjects in one time for several years or decades. I'd like to know what are the most active research areas in mathematics today?

People usually say that the number greater than any assignable quantity is infinity and probably same in the case of -ve ∞.

We are dealing with infinity ∞ in our mathematical or statistical calculations, sometimes we assume, sometimes we come up with it. But whats the physical significance of infinity.

Or

Anyone with some philosophical comments?

Today, every educational field or domain contains several branches. You first choose one branch of your field to prepare your Master and Ph.D degrees. What is preferable system for you:

1- study the same branch at both Master and Ph.D degrees

or

2-study different branch at Ph.D degrees from Master degree

And, Why?

In QFT, computations are done with plane-wave free solutions of the Dirac equations: if one were to consider full solutions of the Dirac equation in interaction with its own electrodynamic field, even without field quantization, what would one obtain? Does anybody know of full solutions, even at a classical level?

NOTE: the question is of purely mathematical interest, so I am not interested in reading that we commonly do not do that in standard computations, I would like to know what would happen if considering the problem with mathematical rigour.

All of us have a different view of point when we prepare study on some topics. Some of us try to study only the depth results in the work, whereas other think it must to investigate all results that associated to the topic regardless its difficulty or ease. On the other hand, some scholars focus on both.

What is your opinion?

it is pure mathematics.

Given a₁, a₂, ..., a_n positive real numbers, and defined

p_k = a_k \prod { i = 1 to n, i <> k} [(a_i)² - (a_k)²]

how to prove that \sum \frac{1}{p_i} is positive?

I had an idea of proof, but not sure it would work....

I have the idea written in the attached .png file.

EDIT: See the .png file here.

In order to get a homogeneous population by inspecting two conditions and filtering  the entire population (all possible members) according these two conditions, then used the all remaining filtered members in the research, Is it still population? or it is a sample ( what is called?).

working on mathematical equation by adding other part to it then find the solution and applying it on the real world. can we generalize its result to other real world?

It will be of immense help for me if you can suggest me some papers and books related to the same.

(1)  How  can we   find    the partial sum    of   n¹⁰⁰⁰  instantly ?  

(2)  is  there  is  a  simple method to find  partial sum  of the  sequence  f(n)  ?

(3)  Any general method   to compute partial   sum of sequence   ?

(4)  What is the  value  of   Method , if we have   good  approximation  for all differentable  sequence   ?

In basic numerical analysis, it is shown that Aitken's method improves on the basic iteration method in speed of convergence in the asymptotic sense (see detail below if desired). Now it seems that this should be meaningless in practice, giving no guarantees of 'faster' for any finite number of iterations. I found that this is not only my feeling, but that this concern is echoed in the related Wikipedia article:

My questions then are

1. Barring empirical evidence, is there ANY formal way of turning the asymptotic result into a result in terms of finite iterations? Even at least probabilistically? Even when conditions are added? This would be an example of what I have in mind: Given a function with condition such and such (smooth, etc.), the convergence is indeed faster in nn iterations with probability p(n)p(n).

2. If there are such results, can you point me to some of them? 

3. If there are no such results, should there be no interest in trying to find them? If not why not?

4. Doesn't this state of affairs 'bother' numerical analysts? If not shouldn't it?

5.Do people reading this have their own 'intuitions' about when the speed of convergence holds in practice? What are these intuitions? Why not try to formalise them?

When the series {x_n} generated using x_i=f(x_i−1) converges under the usual conditions, Aitken's method, generating the series {x′_n} using 

x′n=xn−(xn+1−xn)^2 / (xn+2−2xn+1+xn)

converges faster in the sense that, with s being the solution f(s)=sf(s)=s, we have

(x′n−s) / (xn−s) → 0, n→∞.

 

Dear professors:

Good afternoon. I am researching about teaching triangle inequality. Are there papers about theorems´production (or formulation) by secondary level's students? Which theoretical framework) in Mathematics Education) may be suitable to study conditions to construct triangle with three segments?

Best regards of Peru!

Luis

It is possible to write a set of quaternionic partial differential equations that are similar to Maxwell equations. For example:

The quaternionic nabla ∇ acts like a multiplying operator. The (partial) differential ∇ ψ represents the full first order change of field ψ. 

      ϕ = ∇ ψ = ϕᵣ + 𝟇 = (∇ᵣ + 𝞩 ) (ψᵣ + 𝟁) = ∇ᵣ ψᵣ − ⟨𝞩,𝟁⟩ + ∇ᵣ 𝟁 + 𝞩 ψᵣ ±𝞩 × 𝟁

The terms at the right side show the components that constitute the full first order change.

They represent subfields of field ϕ and often they get special names and symbols.

𝞩 ψᵣ is the gradient of ψᵣ

⟨𝞩,𝟁⟩ is the divergence of 𝟁.

𝞩 × 𝟁 is the curl of 𝟁

The equation is a quaternionic first order partial differential equation.

      ϕᵣ = ∇ᵣ ψᵣ − ⟨𝞩,𝟁⟩ (This is not part of Maxwell equations!)

      𝟇 = ∇ᵣ 𝟁 + 𝞩 ψᵣ ±𝞩 × 𝟁

      𝜠 = −∇ᵣ 𝟁 − 𝞩 ψᵣ

      𝜝 = 𝞩 × 𝟁

From the above formulas follows that the Maxwell equations do not form a complete set.

Physicists use gauge equations to make Maxwell equations more complete.

      χ = ∇* ∇ ψ = (∇ᵣ − 𝞩 )(∇ᵣ + 𝞩 ) (ψᵣ + 𝟁) = (∇ᵣ ∇ᵣ + ⟨𝞩,𝞩⟩) ψ

and

      ζ = (∇ᵣ ∇ᵣ − ⟨𝞩,𝞩⟩) ψ

are quaternionic second order partial differential equations. 

      χ = ∇* ϕ

and

      ϕ = ∇ ψ

split the first second order partial differential equation into two first order partial differential equations.

The other second order partial differential equation cannot be split into two quaternionic first order partial differential equations. This equation offers waves as parts of its set of solution. For that reason it is also called a wave equation.

In odd numbers of participating dimensions both second order partial differential equations offer shape keeping fronts as part of its set of solutions.

After integration over a sufficient period the spherical shape keeping front results in the Green’s function of the field under spherical conditions.

      𝔔 = (∇ᵣ ∇ᵣ − ⟨𝞩,𝞩⟩) is equivalent to d'Alembert's operator.

      ⊡ = ∇* ∇ = ∇ ∇* = (∇ᵣ ∇ᵣ + ⟨𝞩,𝞩⟩ describes the variance of the subject

Maxwell equations must be extended by gauge equations in order to derive the second order partial wave equation.

Maxwell equations use coordinate time, where quaternionic differential equations use proper time. In terms of quaternions the norm of the quaternion plays the role of coordinate time. These time values are not used in their absolute versions. Thus, only time intervals are used.

The quaternionic nabla obeys some other pure mathematical relations:

      ⟨𝞩 × 𝞩, 𝟁⟩=0

      𝞩 × (𝞩 × 𝟁) = 𝞩⟨𝞩,𝟁 ⟩ − ⟨𝞩,𝞩⟩ 𝟁

       (𝞩𝞩) ψ = (𝞩 × 𝞩) ψ − ⟨𝞩,𝞩⟩ ψ = (𝞩 × 𝞩) 𝟁 − ⟨𝞩,𝞩⟩ ψ = 𝞩⟨𝞩,𝟁 ⟩ − 2 ⟨𝞩,𝞩⟩ ψ + ⟨𝞩,𝞩⟩ ψᵣ

The term (𝞩 × 𝞩) ψ indicates the curvature of field ψ.

The term ⟨𝞩,𝞩⟩ ψ indicates the stress of the field ψ.

(𝞩 × 𝞩) ψ + ⟨𝞩,𝞩⟩ ψ = 𝞩⟨𝞩,𝟁 ⟩ − ⟨𝞩,𝞩⟩ ψᵣ

The advantage of writing equations with nabla based operators instead of with the help of tensors is that these PDE's are more compact and therefore easier comprehensible. The disadvantage is that the quaternionic PDE's enforce you to work in an Euclidean space-progression structure instead of in a spacetime structure that has a Minkowski signature.

Personally I consider the Euclidean structure as an advantage, but the Minkowski signature is more in concordance with mainstream physics.

I have triangle mesh and calculate normal of triangles then calculate vertex normal and do some calculations on it and want to calculate vertex coordinates from this vertex normal after do calculations.

In this figure called concentric circles, the circumference on A less than circumference on B. and so on. i.e, The distance between neighbor circles are the same, s.

circA < circB < circC < circD < circE

When XA finishes moving round the circumference A, it moves (transits) to the next circle, B, to help XB complete moving round the circumference. When XA and XB complete the movement round the circumference B, they now make a transition to circumference C and help XC complete the movement round the circumference C. After the completion, they also transit to join XC on circumference C and so on.

The question is this. How can this be presented mathematically (arithmetically)?

I am interested in prime number generation. Apart from the 2^p-1 formula generated so many years ago by the French mathematician are there known formula for determining the next prime?

Suppose $u(n)$ is the Lie algebra of the unitary group $U(n)$, why the dual vector space of $u(n)$ can be identified with $\sqrt{-1}u(n)$?

Any mathematical expert can see my attachment I have highlighted few mathematical symbols , what that symbol signifies  how to understand that can anyone tell

As we know, an elliptic curve defined over Fq with a rational 2-torsion subgroup can be expressed in the special form (up to twists). Accordingly a natural question arises about the number of distinct (up to isomorphism) elliptic curves over Fq in the family.

Let HT denotes the statement of Hindman’s theorem. Within RCA₀ one can prove that:

1. HT implies ACA₀

2. HT can be proved in ACA₀⁺.

An open question is the strength of Hindman’s theorem.

Is HT equivalent to ACA₀⁺ , or to ACA₀, or does it lie strictly between them?

Let f(x)+g(x) = h(x). Here, h(x) is minimum at the points(a1,a2,...,ak). For which condition , we can say that f(x) is also minimum at the points(a1,a2,...,ak)?

Thanks in advance for your idea and please give any reference

.

Let $p(.)$ be an equivalent norm to the usual norm on $\ell_1$ such that

$$\limsup\limits_{n\to\infty} p(x_n+x)=\limsup\limits_{n\to\infty}p(x_n)+p(x)$$ for every $w^*-$null sequence $(x_n)$ and for all $x\in\ell_1,$ moreover, let $$\rho_{k}(x)=p(x)+\lambda\gamma_{k}\sum\limits_{n=k}^{\infty}|x_n|,$$ where, $(\gamma_{k})$ be any non-decreasing sequence in $(0,1)$ and $\lambda >0$. I'd like to prove for every $w^*-$null sequence $(x_n)$ and for all $x\in\ell_1,$

$\limsup\limits_{n\to\infty}\rho_k(x_n+x)=\limsup\limits_{n\to\infty}\rho_k(x_n)+\rho(x)$ .

**My attempt is the following**

\begin{align}

\limsup\limits_{n\to\infty}\rho_k(x_n+x)

=\limsup\limits_{n\to\infty} p(x_n+x) +\limsup\limits_{n\to\infty}\lambda\gamma_{k}\sum\limits_{n=k}^{\infty}|x_n+x| \\

=\limsup\limits_{n\to\infty}p(x_n)+p(x) +\limsup\limits_{n\to\infty}\lambda\gamma_{k}\sum\limits_{n=k}^{\infty}|x_n+x|\\

\end{align}

Now I could not proceed to prove, any ideas or hints would be greatly appreciated.

Thanks in advance

Every natural number n can be written as:

n= a_0 + a_1 *(10)^1 + a_2 * (10)^2 +... a_i between 0 and 9. how can we generate a new method to find the divisors of n apart from the well known method of prime factorization. If so, we can provide a new method to calculate the sum of divisors function.

In some cases, learners find it easy to deal with decimal fractions than proper and improper fractions. Looking at the complex formation of fractions when adding or subtracting seems harder and almost impossible.

e.g.

0.5 + 3.3 = 3.8

1/2 + 33/10 =  38/10

A mathematical colleague and I are working on an article which uses the pure mathematical analysis for equilibrium in equity crowdfunding. We inspired from the model of consumer-product (brand) preference to build up a mathematical model for investor-project preference on a crowdfunding platform. A common point in consumer-product and investor-project relations is that an agent has to choose among different options with optimal efficiency.

We presented a first draft recently at a conference.Non mathematician researchers had difficulty to follow and understand our paper.How can we make such a mathematical reasoning more understandable for non-mathematicians? Do you any article as model? What is your adive? 

By giving evidence or reference,

Who first discovered the base of the natural logarithm: e?

A well-known result of B.M. Levitan and T.V. Avadhani asserts that the Riesz-summability of order k of the eigenfunction expansion f (P) of f (P) from L² (D) at the point P =P_o from D depend only on the behaviour of f (P) in the neighbourhood of P_o if k > (n-1)/2, i. e. is a local property of f(P) at the considered point P_o if k > (n-1)/2. Is it possible to prove (applying Parseval’s formula) the analogue of Avadhani's theorem for Avakumovic’s G - method of summability. A crucial step in the proof of this theorem is to find a function g that lead us to the core of Avakumovic's summability which is more complex than the core of Riesz’s summability.

Since it is difficult to write mathematical formulae please consider the attached file.

We define a factoriangular number (Ft_n) as the sum of a factorial and its corresponding triangular number, that is, Ft_n = n! + n(n+1) / 2. If both n and m are natural numbers greater than or equal to 4, is there an Ft_n that is a divisor of Ft_m? Please also see the article provided in the link below, specifically Conjecture 2 on pp. 8-9.

My function is nonlinear with respect to a scalar \alpha .

However, the calculation of objective function is very time consuming, making optimization also very time consuming. Also, I have to do it for 1/2 millon voxels (3d equivalent of pixels). I plan to do it using “lsqnonlin” of matlab. 

Rather than optimizing over all possible real values, I plan to search over preselected 60 values. My variable \alpha  (or flip angle error) could be anything between 0-35%; but, I want to pass only linearly spaced points as candidates (i.e. 0:005:0.35). In other words, I want lsqnonlin to choose possible solution only from (0:005:0.35). Since I can pre-calculate objective values for these, it would be very fast. In other words, I need to restrict search space.

Here, I am talking about single voxel; though I performs lsqnonlin over multivoxel and corresponding \alpha is mapped accordingly to a column vector. 

I can not do grid search over preselected value as I plan to perform spatial smoothing in 3D. Some guidance would be highly appreciated.

Regards, Dushyant

For example,

{2},

{3,5,7},

{11,13},

{17,19}, etc.

`A second interested question would be that if any such patterns terminate at any level then "Does the cardinality of such sets follow any pattern?" 

It is believed that there are the bijection relationships between Infinite Natural Number Set and Infinite Rational Number Set, but following simple story tells us that Infinite Rational Number Set has far more elements than that of Infinite Natural Number Set:

The elements of a tiny portion of rational numbers from Infinite Rational Number Set (the sub set ： 0, 1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) map and use up (bijective) all the numbers in Infinite Natural Number Set (0,1, 2, 3, 4, 5, 6, …, n …); so，infinite rational numbers (at least 2,3,4,5,6,…n,…) from Infinite Rational Number Set are left in the “one—to—one element mapping between Infinite Rational Number Set and Infinite Natural Number Set (not the integer set )------- Infinite Rational Number Set has infinite more elements than Infinite Natural Number Set.

This is the truth of a one-to-one corresponding operation and its result between two infinite sets: Infinite Rational Number Set and Infinite Natural Number Set. This is the business just between the elements’ quantity of two infinite sets and it can be nothing to do with the term of “proper subset, CARDINAL NUMBER, DENUMERABLE or INDENUMERABLE”.

Can we have many different bijection operations (proofs) with different one-to-one corresponding results between two infinite sets? If we can, what operation and conclusion should people choose in front of two opposite results, why?”

Such a question needs to be thought deeply: there are indeed all kinds of different infinite sets in mathematics, but what on earth make infinite sets different?

There is only one answer: unique elements contained in different infinite sets -------the characteristics of their special properties, special conditions of existence, special forms, special relationships as well as very special quantitative meaning! However, studies have shown that, due to the lack of the whole “carriers’ theory” in the foundation of present classical infinite theory, it is impossible for mathematicians to study and cognize those unique characteristics of elements operationally and theoretically in present classical set theory. So, it is impossible to carry out effectively the quantitative cognitions to the elements in various different infinite set scientifically -------a newly constructed Quantum Mathematics.

The article《On the Quantitative Cognitions to “Infinite Things” (IX) ------- "The Infinite Carrier Gene”, "The Infinite Carrier Measure" And "Quantum Mathematics”》 has been up loaded onto RG introducing the working ideas. https://www.researchgate.net/publication/344722827_On_the_Quantitative_Cognitions_to_Infinite_Things_IX_---------_The_Infinite_Carrier_Gene_The_Infinite_Carrier_Measure_And_Quantum_Mathematics

Dear RG friends:

In two weeks time, I am all set to conduct a technical session on Analysis.

I plan to deliver a long lecture on "Fixed Point Theorems". Of course, Banach fixed point theorem is useful to establish the local existence and uniqueness of solutions of ODEs, and contraction mapping ideas are also useful to develop some simple numerical methods for solving nonlinear equations. Are there any other interesting science / engineering applications?

Kindly let me know! Thank you for the kind help.

With best wishes,

Sundar