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# Pure Mathematics - Science topic

Explore the latest questions and answers in Pure Mathematics, and find Pure Mathematics experts.
Questions related to Pure Mathematics
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If it does, give me the name of the topic with its Software.
For universal algebra you can search the software by writing jre-6-windows-i586 for UA
Question
Is there compatable between plagiarism program with research of pure mathematics ?
Some universities and officials have special protocols for themselves that keeps researches from plagiarism..
So this related for them..
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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.
The short answer is, No. It's known that protons are bound states of three, so-called, valence quarks, that are bound by gluons. The video mixes too many things up. It is possible to interpret excited states of the proton in terms of pairs of a quark and an antiquark, which can be of all possible quark kinds, among them the charm quark. This doesn't change the model for the proton, however.
Question
Importantly for quantum computing , 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 , 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  -- and hides physical meaning.
One can totally avoid using the set ℂ and can calculate, numerically and in algebra, all that one needs . We, thus, proved a negative -- the set ℂ is not needed in mathematics, physics, or engineering .
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 .
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 .
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.
REFERENCES
First, abandon the set C. 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 , so we can discard "non-abelian anyons" in attempts for quantum computing.
Question
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?
With regard to your questions, I have some issues: Just considering functions [0,1]->R,
1. You talk of a set of functions having positive 2-D Lebesgue measure. The graph of a function R->R has a 2-D Lebesgue measure; how is this extended to a set of functions R->R? This leads to questions about constructing a measure over functions or subsets of RxR.
2. In what sense is the set of totally disconnected functions (or graphs of functions) [-n,n]->[-n,n] a subspace? Do you mean a vector space (I doubt this interpretation)? A measure space? Can you define the sigma-algebra?
3. To apply this, you need a measure on sets of measurable functions, which again is a problem.
This is why prevalence was formulated as a concept. Function spaces usually do not have measures defined on them (which is why the Wiener measure is so important). There is no need to create measures over large function spaces. Without a measure, mathematicians either turn to Baire category theory for complete metric spaces or something like prevalence which uses finite-dimensional Lebesgue measure.
So it's not that your intuition is so bad, it's that it is so hard to formulate what you are thinking into something that is provable.
Question
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:
1. The expected value of f on each term of the pre-structure is finite
2. The pre-structure converges uniformly to A
3. The pre-structure converges uniformly to A at a linear or superlinear rate to that of other non-equivalent pre-structures of A which satisfies 1. and 2.
4. The generalized expected value of f on the pre-structure (an extension of def. 3 to answer the full question) satisfies 1., 2., and 3. and is unique & finite.
5. A choice function is defined that chooses a pre-structure from A that satisfies 1., 2., 3., and 4. for the largest possible subset of RA.
6. If there is more than one choice function that satisfies 1., 2., 3., 4. and 5., we choose the choice function with the "simplest form", meaning for a general pre-structure of A (see def. 2), when each choice function is fully expanded, we take the choice function with the fewest variables/numbers (excluding those with quantifiers).
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.
Einstein was also determined to answer questions he found worth pursuing, so, continue studying , reading, writing, and , sharpen your mind by studying published refereed papers as well, and then, after maybe quite some time, you really know, whether what you are doing is really worthwhile , and then, knock on the door of a professor.....and YOU should tackle your questions......and, also very important, when you write a research paper, introduce your problem carefully, in such a way, that your paper triggers the minds of those, who are reading your paper, and, do not write , in the beginning , in a terse style.
Good luck, and take your time, as did Einstein and Gödel or Hilbert!
Question
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.
Nonlinear analysis and variational inequalities are mathematical tools that can be used to model and analyze a wide range of phenomena, including those related to infectious diseases. In particular, nonlinear analysis and variational inequalities can be used to study the dynamics of infectious disease outbreaks, such as the spread and control of the disease in a population.
Question
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
The matter here is the S2 which computationallu converges by resting even 10.000 terms of expansion. But it is a series without an analytical comprehension. I can’t use some expression lemma like “for large enough n.... the expansion gives us a convergent series S2...”?
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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?
If the topic of continuity of functions is explained theoretically and by solving some examples numerically, students face difficulties in understanding the issue of continuity of functions. In my opinion, the best way to learn continuity of functions is to use engineering applications. Through shapes, the concept of continuity of functions deepens in students' minds.
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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.
Metric space A is said to be a totally bounded if every Cauchy sequence in A has convergent sub sequence
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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
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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
It seems that a correct proof for this question has been announced at arxiv.org/abs/2112.09960v1.
Qi’s conjecture on logarithmically complete monotonicity of the reciprocal of the inverse tangent function
Question
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
Hi In what area was the issue raised? Euclidean space, Hilbert space, Banach space?
Question
Many proposals for solving RH have been suggested, but has it been splved? What do you
Interesting question for possible future discussions.
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1. I would like to post this question to clarify my doubts as there were two different answers seems to be correct. Two different experts (1. faculty members in applied mathematics department and (ii) and faculty in pure mathematics department ) have different opinion . Question: Find the limits of integration in the double integral over R , where R is the region in the first quadrant (i) bounded by x=1, y=1 and y^2=4x Problem 2 (ii) bounded by x=1, y=0 and y^2=4x.
If we consider your problems as MCQ problems (I) and (ii) for undergraduate students, the correct answers are :
(a) for problem(i) and (d) for problem(ii).
PS. Observe that we have several correct choices in each case, but they are not included in the offered choices.
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
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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
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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?
An alternative method of demonstrating stability is given by Vasile Mihai POPOV, a great scientist of Romanian origin, who settled in the USA.
The theory of hyperstability (it has been renamed the theory of stability for positive systems) belongs exclusively to him ... (1965).
See Yakubovic-Kalman-Popov theorem, Popov-Belevitch-Hautus criterion, etc.
If the Liapunov (1892) method involves "guessing the optimal construction" of the Liapunov function to obtain a domain close to the maximum stability domain, Popov's stability criterion provides the maximum stability domain for nonlinearity parameters in the system (see Hurwitz , Aizerman hypothesis, etc.).
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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?
Sorry for the delay.Good question. I think this is a matter for the future.
Greetings,
Sergey Klykov
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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.
Mathematics is a backbone of all branches of knowledge.
Question
Compute nontrivial zeros of Riemann zeta function is an algebraically complex task. However, if someone able to prove such an iterative formula can be used to get all approximate nontrivial using an iterative formula, then its value is limitless.How ever to prove such an iterative formula is kind of a huge challenge. If somebody can proved such a formula what kind of impact will produce to Riemann hypothesis? . Also accuracy of approximately calculated non trivial accept as close calculation to non trivial zeros ?
Here I have been calculated and attached first 50 of approximate nontrivial using an iterative such formula that I have been proved. Also it is also can be produce millions of none trivial zeros. But I am very much voirie about its appearance of its accuracy !!. Are these calculations Is ok?
In a paper that can be found on arXiv or at , LeClair gives a reasonably accurate algorithm to estimate the non-trivial zeros up to 10^200=Googol^2. My paper that can be found at Cogent Mathematics, on arxiv or RG
gives an estimate that bounds the n'th zero and checks LeClairs result for the number Googol. Although both these are not iterative, and work only for non-trivial zeros that sit on the critical line, they are predictive and easily calculated. Once a zero is estimated, or bounded, it's accurate value can then be found from formula given.
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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
best wishes
Question
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).
So, Is there really a difference between pure and applied mathematics?
The division of “Mathematics” into two “kinds” is fictitious and misleading. A particular mathematical technique may turn out to be useful in practical situations and, conversely, the application of mathematics to the solution of a practical problem may lead to new mathematical discoveries. In either case, the adjectives “pure” and “applied” are not descriptors of the technique itself, but to its context.
Question
Dear All,
I am hoping that someone of you have the First Edition of this book (pdf)
Introduction to Real Analysis by Bartle and Sherbert
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
Search by the elements of real analysis by Bartle and Sherbert.
(For its first edition )
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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 answers to the question for this thread are excellent. There is a bit more to add.
Before starting either a M.Sc. or Ph.D. thesis, it is very important to read published theses by others. Here are examples:
Source of M.Sc. and Ph.D. theses:
Another source of theses:
Question
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.
@ Vasiliy Knyshev ,
Kindly state and explain clearly the Newton's Second Law of the third order.
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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?
Yes, mathematics has been always not only one of the most active field of the researchers, it was for a long time along with philosophy one of the first sciences. But, it’s hard to say what are the most active research areas in mathematics today or what are the most important scientifically explored in mathematics. Less and less support is provided for purely fundamental mathematics is nowadays, and more and more is required to solve specific problems by "someone else" i.e. mathematics turns into a servant of other sciences
Question
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.
Igor, is it possible that you missed the Volkov solutions? These are used routinely to study matter-field interaction in the strong e.m. field limit.
Question
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
Mathematicians have a precise definition of infinity that can be used to prove theorems about it. A set S has infinite size if it is possible to create a 1-to-1 correspondence of the elements of S with the elements of a proper subset of S. For example, the positive integers S = {1,2,3,...} is an infinite set because there is a 1-to-1 correspondence between S and the even integers in S: 1 <-> 2, 2 <-> 4, 3 <-> 6, 4 <-> 8, ... Such a 1-to-1 correspondence is impossible for finite sets.
Question
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?
I believe that the continuation of research begun in the Masters preferred.
Question
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.
This is an important question. In addition to the excellent answer given by @ Md. Sarfaraz Alam , there is a bit more to add.
Quality is the most important feature of good papers. A paper with very high quality will stand the test of time and will be remembered long after it is written. By quality, I mean that precise definitions and detailed examples are given. Appropriate definitions lead to theorems. The need for the highest quality is needed in proofs of theorems. The quality of proofs of theorems is measured in terms of concise but accurate deductions from the definitions,
relations and other theorems that precede each proof.
Quantity is definitely not an appropriate goal in writing good papers. It is better to let the size of a paper be a function of the need for narrative and examples that illustrate abstract ideas in a paper.
Question
it is pure mathematics.
<geometers don't like groups>:
In his influential 'Erlanger Programm' Felix Klein characterized geometries as the theorie of associated group invariants.
<algebrists don't like fields>:
Galois solved the most urgent algebra problem of his time (general solvability of polynomial equations of order n) by studying properties of fields.
So my rudimentary ideas concerning the history of mathemathics suggest just the contrary of what Claude cames up with.
Question
Given a1, a2, ..., an positive real numbers, and defined
p= a\prod { i = 1 to n, i <> k} [(ai)- (ak)2]
how to prove that \sum \frac{1}{pi} 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.
Thanks Viera answer I've got that the proof for even  n  is quite sufficient. Thank you Viera for keeping calm while giving interesting answers to interesting questions:)
Best regards, Joachim
PS. Meanwhile I have realized, that replacing   ak2  by  ck  and and assuming increasing ordering (without losses, as Viera has noticed) , we are getting for the sum S of the question the following expression with the use of divided difference of order  n-1:
(n-1)! (-1)n-1 S = (n-1)! [ c1, c2, . . . , cn :  f(c) ],
which equals the value   f(n-1)(b)  of the derivative  of  f  of  order  n-1  at some point  b  from [c1 ,  cn ],  where  in this case   f(c) = 1/\sqrt{c}.
For the calculus of divided differences and their reprepresentation see e.g.
Having this and the sign changes of the derivative of the inverse square root one gets positive value of   S, for any choice of   positive   ck -s .
Further conclusion is the  this holds for every negative power of  c  put in place of  f(c),  and also for every Laplace transform  of a positive measure on  R+  since then the derivatives of order  n  have  sign equal   (-1)n  (cf. the William Feller Bible on probability about the completely monotone functions)  JoD
Question
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?
Rula -
I am not sure I understand your process, but if your 'sample' is really just a census of a special part of your population, then you can get descriptive statistics on it, but you cannot do inference to the entire population from it.
You might find the following instructive and entertaining.  I think it is quite good.
Ken Brewer's Waksberg Award article:
Brewer, K.R.W. (2014), “Three controversies in the history of survey sampling,” Survey Methodology,
(December 2013/January 2014), Vol 39, No 2, pp. 249-262. Statistics Canada, Catalogue No. 12-001-X.
Cheers - Jim
Question
It will be of immense help for me if you can suggest me some papers and books related to the same.
Dear Alka Munjal,
Perhaps you can see
Summability Theory And Its Applications
Author(s): Feyzi Basar
From the cover
''The theory of summability has many uses throughout analysis and applied mathematics. Engineers and physicists working with Fourier series or analytic continuation will also find the concepts of summability theory valuable to their research. The concepts of summability have been extended to the sequences of fuzzy numbers and also to the theorems of ergodic theory. This ebook explains various aspects of summability and demonstrates applications in a coherent manner. The content can readily serve as a useful series of lecture notes on the subject. This ebook comprises of 8 chapters starting from classical sequence spaces and covering matrix transformations and fuzzy numbers. An accompanying bibliography with extensive references makes this a valuable source of information for readers interested in summability theory as well as other branches of science.''
Question
(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   ?
@ Juan Weisz
That  is more than that   .  It  is   a little secret   have little secret     of  analytic continuity   and discreetness  of  integers
Question
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:
Although strictly speaking, a limit does not give information about any finite first part of the sequence, this concept is of practical importance in dealing with a sequence of successive approximations for an iterative method, as then typically fewer iterations are needed to yield a useful approximation if the rate of convergence is higher. This may even make the difference between needing ten or a million iterations insignificant.
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?
Detail
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→∞.

Hi, I think in practice there is a real interest in using this kind of acceleration scheme. I have implemented in the past Richardson scheme (an alternative of Aitken) and the convergence speed was improved in practice. I think the asymptotic methematical result cannot be converted in a probabilistic condition. In order to really measure the benefit of using an acceleration scheme you can perform Monte Carlo runs, changing the initial conditions in ordrer to quantify the acceleration of convergence.
Question
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
A good paper on the straightforward formulation of a theorem and its proof (useful for those starting out in getting comfortable with theorems and their proofs) is given in
The beauty of this paper on a simple proof of a well-known algebraic curves theorem is its piecewise approach with the use of a number of lemmas.   For related papers, see also:
For an introduction to theorem proving paradigms, see
A new proof of the Pythagorean theorem is given in
An elementary proof of the converse of the mean value theorem is given in
A very good discussion of the discovery (and invention) of theorems is given in
Question
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
Einstein equations for general relativity use the curvature tensor and the stress tensor. Above is shown that some terms of the partial differential equations relate to terms in Einstein's equations.
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.
Stefano,
I undertake the conversion enterprise because for me the PDE's are better comprehensible than tensor equations. For similar reasons I use quaternions instead of Clifford algebras.
Question
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.
Look at this doc it may be helpful for your topic. Good luck.
Question
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)?
Your statement of the problema is not clear. One can interpret it in two ways:
If you have only one person (or object) moving and n circles, the answer is:
D(n) = n(n+1)pi + (n-1)s.
If you have n persons moving (one on each circle), the answer is:
D(n) = (ns/2){[(n+1)(2n+1)pi/3] + n-1}.
I got this by doing D(n) = 2pi.s (sum of i squared from 1 to n) + s .(sum of i from 1 to n-1).
Question
I am interested in prime number generation. Apart from the 2p-1 formula generated so many years ago by the French mathematician are there known formula for determining the next prime?
Question
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 $\sqrt{-1}u(n)$?
Hi Pan,
$u(n)$ is a real Lie algebra, and in particular a real vector space. Using a non-degenerate symmetric bilinear form on $u(n)$, you can identify $u(n)$ with its dual vector space. The $\sqrt{-1}$ is not that important in a sense, and probably comes from using a non-degenerate pairing between skew-hermitian and hermitian matrices (instead of a non-degenerate symmetric bilinear form). $u(n)$ is the space of skew-hermitian $n$ by $n$ matrices, and $\sqrt{-1} u(n)$ is the space of hermitian $n$ by $n$ matrices by the way.
Question
Any mathematical expert can see my attachment I have highlighted few mathematical symbols , what that symbol signifies  how to understand that can anyone tell
Naveen, you probably did not have good teachers who explain basic ideas of hydrodynamics "on fingers". Imagine water over rigid bottom. If it is incompressible, Laplace equation is valid in every internal point. \psi is velocity potential, and \eta is deviation of free surface from equilibrium. Due to non-compressivility, the total volume of water is preserved, and thus an integral of deviation function over unperturbed surface is zero. The condition on bottom shows that the velocity is locally parallel to the bottom (given by differentiable function); its normal component is zero.
To understand notations. Recall that a symbol resembling Euro symbol means: element belongs to a set. R is a set of real numbers, while other sets in your case are some subsets of R1(real line) or R2 (plane).
You can look at equations (1) in my attached article to see what is what and what one can do with that: https://www.researchgate.net/publication/275581998_Evolution_of_long_nonlinear_waves_on_shelves
Question
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.
Dear Davood,
Here are links and attached files in subject.
-On the number of distinct elliptic curves in some families - ResearchGate
-On the number of distinct elliptic curves in some families | SpringerLink
-On the isomorphism classes of Legendre elliptic curves ... - Springer
Best regards
Question
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?
This is a very good question.
A good place to start in answering this question is
Unfortnately, this paper is not available on RG but it is available a University of Connecticut web page at
where the issue of the strength of Hindman's theorem is raised.
Question
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)?
.
My answer from above gives a sufficient condition (on g) for the desired conclusion on f, and it is independent on any differentiability hypothesis. It follows by simple handling inequalities. Hence this implication holds and one can say something on this subject.
Question
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.
First of all, I suppose that you consider $\ell_1$ as dual to $c_0$, so by $w^*-$null sequence in $\ell_1$ you mean such a sequence $x_n$ that is norm-bounded and goes to zero coordinate-wise. In your question you are using letter $x_n$ in two different senses: as elements of $\ell_1$ and as coordinates of an element $x$. This leads to a confusion. If you denote coordinates of $x$ as $x_n$ and consider vectors $y_n = (y_{n,1}, y_{n,2}, \ldots)$, then the formula you are going to demonstrate is $\limsup\limits_{n\to\infty}\rho_k(y_n+x)=\limsup\limits_{n\to\infty}\rho_k(y_n)+\rho(x)$
The hint to this exercise is: if a sequence of vectors $y_n \in \ell_1$ is norm-bounded, goes to zero coordinate-wise and if there exists the limit $\lim_{n \to \infty} \|y_n\|$, then there exists $\lim_{n \to \infty} \|x +y_n\|$, and
$$\lim_{n \to \infty} \|x +y_n\| = \lim_{n \to \infty} \sum_{j=1}^\infty |x_j + y_{n,j}| = \|x\| + \lim_{n \to \infty} \|y_n\|.$$
Question
Yes. Pure mathematics is useful for theoretical physics. Mathematics is nothing but  logical expression of physics and physical things. It is much more clear than physical logic if the methodology is right. A physical concept has to be converted to mathematical equation and the mathematics will drag it to a final equation which can explain a new concept which is not visible to the conceptual logic. Conceptual logic must provide a physical phenomena which can be observed and verified physically. Otherwise that mathematics will create complicated and chaotic outputs in theoretical physics..
Question
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.
Dear Hanifa;
Could you please write down an example
Question
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
Since children had an intensive experience with numbers, hence from my experience working with children in the primary level, adding decimal numbers is not a difficult concept for them to comprehend.  The challenge is just to help children extend their understand regarding place value of, one tenth, one hundreth etc... However, fraction is a relatively new concept for children to grasp.  Hence, I feel that operations with fractions should be taught later, once their are comfortable with the concept of fractions itself.
Question
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?
following paper and links may be useful for u
Question
By giving evidence or reference,
Who first discovered the base of the natural logarithm: e?
Does anyone know something about first sciences, especially mathematics.               1. My question is: when and which science have emerged as first ?                     2. The place and role of mathematics among the oldest sciences.
1. Already its paradigmatic place in the domain of human knowledge, independent of all other valid reasons, mathematics deserves a special place.
2. The oldest known thinkers of antique civilization have been characteristic way of mathematical form of knowledgeand since then it deserves as a model of scientific value and measures of  exactness of the overall knowledge.
3. Already in the middle ages, mathematics in his former division accounted for two of the seven skills which was dedicated to the study of the traditional University (geometry and arithmetic) in quadriviumu. And the third one-seventh, logic, the trivium of today would be the relevant part, in the form of mathematical logic, also regarded as one of the domains of mathematics.
Question
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 =Po from D depend only on the behaviour of f (P) in the neighbourhood of Po if k > (n-1)/2, i. e. is a local property of f(P) at the considered point Po 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.
Dear Cenap Özel,
If you have a problem with russian presentaton of my work  "About one application Parseval's formula to the Avakumović's G - method of summability of eigenfunction expansion" I want inform you that it soon publish in Sarajevo Journal of Mathematics in number dedicated to the memory of my professor Academician Mahmud Bajraktarevic,
Sincerely, Mirjana Vukovic
Question
Since it is difficult to write mathematical formulae please consider the attached file.
${F_{{5^k}n}}(q) \equiv 0\bmod {[{5^k}]_q}$ is equivalent with ${F_{{5^k}}}(q) \equiv 0\bmod {[{5^k}]_q}$
because ${q^{{5^k} + n}} \equiv {q^{{5^k}}}\bmod {[{5^k}]_q}$ and therefore  ${F_{{5^k}(n + 1)}}(q) \equiv {F_{{5^k}n}}(q){F_{{5^k}n + 1}}(q) \equiv 0\bmod {[{5^k}]_q}.$
Therefore it suffices to show that
${F_{{5^k}}}(q) \equiv 0\bmod {[{5^k}]_q}.$
Question
We define a factoriangular number (Ftn) as the sum of a factorial and its corresponding triangular number, that is, Ftn = n! + n(n+1) / 2. If both n and m are natural numbers greater than or equal to 4, is there an Ftn that is a divisor of Ftm? Please also see the article provided in the link below, specifically Conjecture 2 on pp. 8-9.
Let m=Ftn if Ftn is odd, and m=2Ftn if Ftn is even. Then Ftn divides Ftm.
Question
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
My two cents to the question:
1) What you want to do is the discrete optimization (see the link). The discrete coded (binary) genetic algorithm, e.g., could do the trick.
2) You could perform the factorial design of experiments (factorial d.o.e.) with a number of levels for each parameter you want, but the fraction should be very small to calculate it (see below, why).
1/2 millon voxels as optimization variables is somewhat difficult to optimize, even assuming varying them on 60 levels. It is probably possible to run on supercomputer, but first consider if the problem is formulated correctly and the results would be significant, if it is worth trying.
Question
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?"
Dear U. Dreher,
about the conjecture on the 'twin numbers' let me inform you that I proved this conjecture as a particular case of the 'de Polignac’s conjecture'.
You can find solutions of all the Landau problems in the following my papers:
These solutions could answer also to the Shahid's question.
My best regards
Agostino,
Question
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 Geng
Lets use your own reasoning in a different way:
Just take a tiny portion of the Natural Number Set (2,4,6, ... ,2n, ...) and they map very well on the set of all Natural Numbers  (1,2,3,...,n,...). So a lot of natural numbers are left behind in this one -to-one mapping (2n onto n) from the Natural Numbers onto the Natural Numbers, Hence your conclusion should be: the Natural Number Set has far more elements than the Natural Number set. Think about this.
Best regards, Joseph.
Question
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
As applications :   logic programming and fuzzy logic programming
Nash equilibrium and game theory.
Question
Let q be an odd positive integer, and let Nq denote the number of integers a such that 0 < a < q/4 and gcd(a, q) = 1. How do I see that Nq is odd if and only if q is of the form pk with k a positive integer and p a prime congruent to 5 or 7 modulo 8?
I see that you, too, follow the Putnam Exam.
Question
As the title suggests, how do i see that for any n, the covering map S^{2n} → RP^{2n} induces 0 in integral homology and cohomology, except in dimension 0?
Maybe you would like to see geometrically how the n-th homology of  S^n can be killed when projected to RP^n? Take a triangulation of the n-sphere which is invariant under the antipodal map  -I , e.g. the generalized octahedron (which exists in every dimension). The projection is a triangulation of the projective space RP^n, and each of its simplices is hit twice by the projection map S^n \to RP^n. However, when n is even, the antipodal map -I on R^{n+1} reverses orientation (determinant -1) but preserves the exterior normal vector field of S^n, thus it reverses the orientation on S^n. Hence each simplex contributes with both signs and you get extinction, while in odd dimensions you obtain a factor 2 (the mapping degree of the projection map).
Best regards
Jost
Question
Say a definition to be self-referential provided that contains either an occurrence of the defined object or a set containing it. For instance,
Example 1) n := (n∈ℕ)⋀(n = n⁴)⋀(n > 0)
This is a definition for the positive integer 1, and it is self-referential because contains occurrences of the defined object denoted by n.
Example 2) Def := "The member of ℕ which is the smaller odd prime."
Def is a self-referential definition, because contains an occurrence of the set ℕ containing the defined object.
Now, let us consider the following definition.
Def := "The set K of all non-self-referential definitions."
If Def is not a self-referential definition, then belongs to K, hence it is self-referential. By contrast, if Def is self-referential does not belong to K, therefore it is non-self-referential. Can you solve this paradox?
Take into account that non-self-referential definitions are widely used in math.
Juan-Esteban,
That's a nice way to avoid the so-called Russell paradox at any finite stage of the process. It feels as if you avoid the paradox by rejecting infinity. If you read my post carefully, you will see that the problem is solved in a more fundamental way with or without  considerations of infinite sets. Formal logic tells you that there cannot be x such that
(all y) ( not P(y,y) <--> P(y,x) ),
whatever you mean by P(y,x) and whatever your universe of discourse may be. If you take the "barber definition" (anyone  shaving all those people not shaving themselves), even in an imaginary infinite society of humans, it turns out that there is no such barber. In fact, it is not a definition because it deals with nothing. Let me illustrate this with a more obvious contradiction:
Remarkable_Weather := weather with rain and yet without rain.
Such a "definition" is verbosity with (literally) no subject, hence with no meaning.
I prefer Webster's definition of "paradox":
A tenet or proposition contrary to received opinion; an
assertion or sentiment seemingly contradictory, or opposed to
common sense; that which in appearance or terms is absurd,
but yet may be true in fact.
The only difference between having a Remarkable_Weather and the Russell paradox is, that the latter involves a slightly more hidden logical contradiction.
Question
Two numbers a and b are elements of the set of real numbers exclusive of the set of rational numbers; they are irrational.  Are there cases where  a times b, or where a divided by b, yield a member of the set of integers? How rare or commonplace is the condition of irrational number products yielding rational values?
If a is irrational and n is an integer, then b:=n/a is irrational (!) and ab =n.
Question
I was working on 2 papers on statistics when I recalled a study I’d read some time ago: “On ‘Rethinking Rigor in Calculus...,’ or Why We Don't Do Calculus on the Rational Numbers’”. The answer is obviously trivial, and the paper was really in response to another suggesting that we eliminate certain theorems and their proofs from elementary collegiate calculus courses. But I started to wonder (initially just as a thought exercise) whether one could “do calculus” on the rationals and if so could the benefits outweigh the restrictions? Measure theory already allows us to construct countably infinite sample spaces. However, many researchers who regularly use statistics haven’t even taken undergraduate probability courses, let alone courses on or that include rigorous probability. Also, even students like engineers who take several calculus courses frequently don’t really understand the real number line because they’ve never taken a course in real analysis.
The rationals are the only set we learn about early on that have so many of the properties the reals do, and in particular that of infinite density. So, for example, textbook examples of why integration isn’t appropriate for pdfs of countably infinite sets typically use examples like the binomial or Bernoulli distributions, but such examples are clearly discrete. Other objections to defining the rationals to be continuous include:
1) The irrational numbers were discovered over 2,000 years ago and the attempts to make calculus rigorous since have (almost) always taken as desirable the inclusion of numbers like pi or sqrt(2). Yet we know from measure theory that the line between distinct and continuous can be fuzzy and that we can construct abstract probability spaces that handle both countable and uncountable sets.
2) We already have a perfectly good way to deal with countably infinite sets using measure theory (not to mention both discrete calculus and discretized calculus). But the majority of those who regularly use statistics and therefore probability aren’t familiar with measure theory.
The third and most important reason is actually the question I’m asking: nobody has bothered to rigorously define the rationals to be continuous to allow a more limited application of differential and integral calculi because there are so many applications which require the reals and (as noted) we already have superior ways for dealing with any arbitrary set.
Yet most of the reasons we can’t e.g., integrate over the rationals in the interval [0,1] have to do with the intuitive notion that it contains “gaps” where we know irrational numbers exist even though the rationals are infinitely dense. It is, in fact, possible to construct functions that are continuous on the rationals and discontinuous on the reals. Moreover, we frequently use statistical methods that assume continuity even though the outcomes can’t ever be irrational-valued. Further, the Riemann integral is defined in elementary calculus and often elsewhere as an integer-valued and thus a countable set of summed "terms" (i.e., a function that is Riemann integrable over the interval [a,b]  is integrated by a summation from i=1 to infinity of f(x*I)Δx, but whatever values the function may take, by definition the terms/partitions are ordered by integer multiples of i). As for the gaps, work since Cantor in particular (e.g., the Cantor set) have demonstrated how the rationals “fill” the entire unit interval such that one can e.g., recursively remove infinitely many thirds from it equal to 1 yet be left with infinitely many remaining numbers. In addition to objections mostly from philosophers that even the reals are continuous, we know the real number line has "gaps" in some sense anyway; how many "gaps" depends on whether or not one thinks that in addition to sqrt(-1) the number line should include hyperreals or other extensions of R1. Finally, in practice (or at least application) we never deal with real numbers anyway (we can only approximate their values).
Another potential use is educational: students who take calculus (including multivariable calculus and differential equations) never gain an appreciable understanding of the reals because they never take courses in which these are constructed. Initial use of derivatives and integrals defined on the rationals and then the reals would at least make clear that there are extremely nuanced, conceptually difficult properties of the reals even if these were never elucidated.
However, I’ve been sick recently and my head has been in a perpetual fog from cold medicines, so the time I have available to answer my own question is temporarily too short. I start thinking about e.g., the relevance of the differences between uncountable and countable sets, compact spaces and topological considerations, or that were we to assume there are no “gaps” where real numbers would be we'd encounter issues with e.g., least upper bounds, but I can't think clearly and I get nowhere: the medication induced fog won't clear. So I am trying to take the lazy, cowardly way out and ask somebody else to do my thinking for me rather than wait until I am not taking cough suppressants and similar meds.
David: We actually do integrate over the rational numbers. Probably the most essential integration formula is that of the integral of x^n over the interval [0,1]. The value of this can be established entirely over the rationals. You can have a look at my Famous Math Problems10 video at my channel (user njwildberger).
There are some of us that don't believe in the infinite-precision dream which supports the real numbers'. If you are interested in why, my recent seminar: `A Socratic look at logical weaknesses in modern pure mathematics' gives some reasons. Also at my YouTube channel.
Question
I have come out with my own equation(I have no idea whether it is new) for pie : π = √2/2 x n x √((1-cos(dΘ)), where n is the number of triangles in the circle, and  dΘ is the angle of the triangle that is -->0. Ok, now, say, I put n = 1440, so dΘ will be 360/1440 = 0.25, and put it into the equation, i will get  π= 3.141590118…; if I put n=2880, so dΘ will be 0.125, putting it into the equation, I will get π=3.141591603.., if I put n=5760, so dΘ will be 0.0625, putting it into the equation, i will get π= 3.141592923..., we know π= 3.141592654..,but i can never really get the n and dΘ to give that answer. Anyone can come out with some good idea??
Dear Mason,
I dont know. I took this from you.
Best Regards,
Henri.
Question
If a polynomial P(z) of degree n omits w in |z|<1, show that P(z)+(1-e^{ih})zP'(z)/n also omits w in |z|<1 for every real h. I know at least two proofs of this result, one follows by using Laguerre's theorem concerning the polar derivative of a polynomial. I want to find the direct proof of this result with out using any known Theorem.
It seems that you should give more details. Is this known
result? Is the hypothesis that h is real? For example, the set
$1-e^{ih}: h \in R$ is the circle $K(1;1)$.
One can try to use induction  or Bernstein inequality with Argument Principle.
Question
If a1,a2,...,aare given positive integers in strictly increasing order, what would be the best possible lower bound of
|1+za1+za2+...+zan| for |z|>1?
It seems that 0 is exactly this lower bound in the general case.
Indeed, the product of all roots of the polynomial 1+...+z^{a(m)} is 1. Therefore, it is impossible that all these roots lie within the unit circle.
Hence some roots are at least on |z|=1 or even outside the unit circle.
Question
A (single variable) function is differentiable iff some other function is continuous. Can we get similar characterizations for (different kinds of) multifunctions? How to express: a multifunction is X-differentiable iff some other (multi)function is continuous?
Hello dear friends
You know we can put such type questions in pure classification field and there is no reason for  thinking about any applications (may be it could be find many applications in future). But, it is not bad we could get some sights about the question and we could get some imagines. Anyway, I interest more answers from members of this group. Finally, may be it will be interest for somebody that some researchers are investigating some systems of fractional differential inclusions.
Question
PARITY is about whether a unary predicate of a structure has even numbers of elements in it.
If a kind of logic can define PARITY, then there is a formula of this logic so that:
PARITY return True on a structure iff this structure is a model of this formula.
We have known that logics with counting can easily define PARITY.
But what about others without counting?