Science topics: Mathematics
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# Mathematics - Science topic

Mathematics, Pure and Applied Math
Questions related to Mathematics
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See enclosed
The most interesting peculiarity of this theorem is that the estimate does not depend on the number of vectors (which can be very large), but only on the dimension of the space.
By the way, analogous statements (basically known to E.Steinitz in the beginning of 20th century) were addressed and re-discovered by people working in mathematical economics. See Shapely-Folkman Theorem in appendix 2 of the paper by Ross M. Starr
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How is Error Analysis used as pedagogical tool to teach mathematics?
Which theories support it?
First you must understand the concept of error. Second you have to konw how many types of errors can be found and which are the main characteristics of each of them.
There are two main types of errors. These are:
Random Error: A random error is one which is di?erent after repeated measurements, and are distributed randomly around true value. The scatter or uncertainty due to random errors will produce values which are di?erent from measurement to measurement. A special case of random errors are counting measurements. If you are counting something that is truly random, the uncertainty in the number of counts is the square root of the number of counts. That is, δν = ±√ν (2) where ν is the number of counts. If you again count for the same amount of time, it is unlikely that you will get the same number of counts. But it is likely that the number of counts will fall within the range ν +√ν to ν ?√ν.
Systematic Error: A systematic error will be the same after repeated measurement and is a consistent deviation from the true value. Incorrect calibration of an instrument will cause systematic error. Experimental circumstances that always “push” the value in the same direction, such as friction, will be systematic errors.
An experiment will always contain both random and systematic errors. For example, suppose we measurement the time it takes a ball to fall from the roof of Meyer and compare it to the time estimate from the acceleration of gravity. We will start a stopwatch just as we drop it o? the roof, and stop the watch when we see it bounce. One source of error is our reaction time. This is a random error: we may delay too long in starting the watch, or delay too long in stopping the watch. In practice, the reaction times will not be the same in repeated measurements; our measured times will be randomly distributed around the true time. Another source of error will be air resistance. This will always cause the time of the ball’s fall to increase. This is a systematic error since it will always add an error in the same direction.
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Dear George,
Your idea still works if the cardinality of K is greater than the degree of P. Namely, take r = 1. The derivative of F(x) = xP(x) has at most n zeros. Denote A the set of zeros of the derivative of F, then there is a q in the field which do not belong to F(A). Take Q = -q. Then xP(x) + Q = F(x) - q? has only simple zeros. Indeed, if x is a zero of F(x) - q, then F(x) = q, so according to our selection of q, x is not a root of (F - q)' = F'.
Taking in account the fact, that for every polynomial P over a finite field K of cardinality N there is another polynomial P1?of degree smaller than N, such that P(x) = P1(x) for all x in K, I would say that the case of deg P > card K looks a little bit artificial. Maybe this is because I am not an algebraist, and for me a polynomial is not a formal expression but a function.
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Let an be a decreasing sequence towards 0; give explicit examples of Riemann integrable functions on [0, 1] such that the difference between?its Riemann sum of order n and?its Riemann integral is greater or equal than an for every n.
Vladimir gave us a good hint on how to proceed. The construction is simple,but a little boring.?In fact, we can ?define f inductively in a countable set formed by all uniform grids as follows:
?f(k/n) = 1 + an, for the smallest point of the form k/n which still were not took into account ?for previous values of n. This construction is possible because, for each n, f ?was ?only defined previously for those k/n such that k and n are not coprime. Therefore, there always is at least one vaue of k that fits this purpose. ?
Let M be the union of all these grids.?Then set f(x) = 1 for all other numbers in [0,1] which are not in M. Notice that f is continuous in every point x outside M (but this is only true because the given sequence an converges to zero). ?Hence, the set of the points of discontinuity of f is a subset of M and, therefore, have measure zero (because M is countable.) Therefore, f is Riemann integrable.
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I recalled the following problem (from my youth), but not its solution. Maybe you can help me.
Let f:?→? be a function such that?|f(x) + 2f(y) + 2f(z)| ≤?|2f(x/2 + y + z)| for all x,y,z∈?.?Prove that f(x+y) = f(x) + f(y) for all x,y∈?.
Perfect team!
Dear George, there′s intruder: the last identity is?f(λx)=λf(x). Or not? If yes, the answer is:??Yes, dear Hanifa, it remains?true at least for λ>0.
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Let μ be a Borel measure on ?. Then μ?is σ-finite on ??if and only if, for every a∈?, there exists an ε>0 such that μ([a, a+ε)) < ∞.
Dear Peter,
the first part of your post? is not clear for me. You are trying to demonstrate that, for a sigma-finite measure m, if a sequence An , n = 1,2, ... of measurable? subsets decreases, m(An) is infinite and their intersection is a single point (say, zero) , then m({0}) is also infinite.
This statement is not true even for the Lebesgue measure on R. Namely, take as An the union of {0} and [n, +\infty).
Moreover, the "easy" implication (sigma-finiteness implies local finiteness) is not correct. Consider the following disjoint sequence of sets Dk=(1/(k+1), 1/k), k = 1,2, ...
Define a positive Borel measure m? in such a way that the measure on each Dk is? a multiple of the Lebesgue measure such that m(Dk)=1,? k = 1,2, ... and put the measure of the remaining? part of R equal to zero. This m is sigma-finite, but not locally finite at zero.
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Thank you Peter.
I understand?where was my wrong consideration.
But I don't understand, which of my points are dubious.
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I am currently doing research on how the rise of modern science has affected the way we teach the sciences and mathematics. I would like to know some of the best practices you have teaching the subject and the reason why you used such strategies.
You might want to read the article/book "Eight Ways to Promote Generative Learning" by Logan Fiorella and Richard Mayer. They review research on?various proven effective instructional strategies (e.g. teaching to others) that l end themselves for science and mathc topics.
As for William Roger Nelson's comment, I would advice you to steer clear from learning styles, because this is a commonly believed educational myth that severely lacks scientific evidence.
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I have a curve in R^3 (lets say X(t) = (x(t),y(t),z(t)) with t in [-1,1]) and I am looking for its convex hull, i.e. conv( { X(t) | t in [-1,1] } ) = { Y in R^3 | there exists a,b in [0,1], a+b = 1, t_1,t_2 in [-1,1] such that Y = a*X(t_1)+b*X(t_2) }
The special curve that I am investigating for has the form X(t) = (t, max(t,0)^2, min(t,0)^2). Some of the bounds on the convex hull are rather simple, e.g. y+z>= x^2 and y+z<=1.
However, there are more which I cannot write down explicitly, so any help is highly appreciated.
This is a great question.
Here is an article that may be of interest to followers of this thread:
Consider, for example, the convex hull for Steiner's Roman surface shown in Fig. 2 on page 6.
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Dear All
Best regards
Feng Qi (F. Qi)
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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.
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I am currently doing research on this topic for my thesis. Any help would be greatly appreciated especially in relation to articles etc. I have found numerous articles on each of the different topics i.e. Group work/collaboration, Mathematics in Primary school and Multi Grade classes but have found very little incorporating all three. Perhaps this could be one of my reasons for?my rationale.
Hi,
Most of time student show less interest and feels challenging to solve the problem. Team activity and collaborative learning helps them to understand the approach of their peers. Knowledge sharing will be better. Increases their confidence level and enthusiasm. Especially in the beginning of the course enhances their positive approaches.?
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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: http://www.fondpageant.com/publication/275581998_Evolution_of_long_nonlinear_waves_on_shelves
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Every closed subset of [0, 1] has a unique representation as the union of three?disjoint sets: one is open, another one is?perfect & nowhere dense, and the last one is at most countable.
Peter's counterexample shows that uniqueness fails.? Requiring the open set to be regular (that is, equal to the interior of its closure) would give both existence and uniqueness, unless I've made a silly mistake somewhere.
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I am?doing the study of Local fractional derivatives and integral. so please suggest me some good book of this topics.
This is good question with many possible answers.
In response to @Vikash Pandey's question about the local nature of fractional calculus, see
and?
Also, see the many related papers available in?Hari Mohan Srivastava's RG page at?
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For calculating mean of x1, x2 and x3, instead of simply adding them and divide by 3, I?have adopted another method similar to inverse distance weighing function.
μx = (x1 + x2 + x3)/3 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
wi = 1/(μx ? xi)^2 = 1/(di)^2......i ∈ [1, 3] ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
μ(wx)=wi ?xi/(w1+?w2+w3)....i∈[1,3]
where, wi is the weight of each observation and?μ(wx) is the final mean which I have used in my research.
Dear Rajkamal Kumar,
I like your idea of diminishing the role of outstanding data, which is an especially important feature for measurements/observations exposed to extraordinary random perturbations. However, there are some places, where we cannot aply the term of the 'true' mean value, unless a particular model of the perturbation is assumed and then analysed. Seemingly, a simle and directly correponding to your idea example can be given as follows:
The statistical procedure measures a rv $X$ with pd being a convex combination of the PROPER pd (with pr. p) and the FALSE pd Q (with pr. q = 1- p), in symbols:
$P_X = p \cdot P + q \cdot Q$
Now, for being more specific, let both be normal
${\cal N}(m,v)$ and ${\cal N}(M,V)$, respectively,
where the expectations are different ( $m \ne M$ ) and with variances satisfying $0 < v << V$. I think, that simulations will show in this case better approximation of $m$ with your weighted mean that the usual arithmetic mean.
Best wishes and good luck in continuing the proposal.
PS. The following is a sketch of a reason to ASSUME, that Kumar's average $K(x)$ of $d$ entries
$x = ( x_1, . . . x_d)$
takes value zero, at points, where at least one coordinate equals the arithmetic average, denoted further by $A(x)$:
0.The average $K$ is defined only for sequences where none coordinate equals the average $A(x)$.
1. For these $x$-s, denoting
$y_j := 1 / (x_j - A(x))$ and $\sum$ for $\sum_{j=1}^d$
we have horeover,
$|K(x) - A(x)| \le |(\sum_{j=1}^d y_j ) / ( \sum y_j^2 ) | \le (\sum_{j=1}^d |y_j| ) / ( \sum y_j^2 ) \le ( \sum_{j=1}^d |y_j| ) / ( [ max{ |y_j| : over j} ]^2 ) \le ( d \cdot max{ |y_j| : over j} ) / ( [ max{ |y_j| : over j} ]^2 ) \le d \cdot [ min{ |x_j - A(x)| : over j} ]$
Joachim Domsta
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i want to research about the quantitative reasoning in high school students.
I did a comparative study of learning methods to the quantitative reasoning abilities of students.
How to measure the the quantitative reasoning ability?
What are the components measuring the quantitative reasoning ability?
Hopefully you can help me.
Conceptual understanding is the ability to reason in settings involving the careful application of concept definitions, relations, or representations of either.
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Hi to all,
When simulating faults in a transmission line using Matlab/Simulink, we can change inception angle by varying the time in the three phase fault breaker bloc.
My question is about the mathematical relationship between time and inception angle?
Regards
Dear Toufik,
Here are links and attached files in subject.
-Fault Detection in Transmission Lines Using Instantaneous Power with ...
-A novel transmission line relaying scheme for fault detection and ...
-Fault location technique using GA-ANFIS for UHV line - De Gruyter
-Transmission Line - ResearchGate
-Detection and Classification of Faults on Six Phase Transmission Line ...
Best regards
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Can I get the parameters (value) for all the components in the figure below and clear mathematical expression for the optical fields ?and optical intensity , to enable me write code for simulation of the fig.1. please?
Have you tried contacting the authors directly?
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Looking to discuss an intervention that more closely aligned semester long smaller assessments with exams in a college level mathematics class. Perhaps something on summative and formative assessments having same format or expectations of students being the same for multiple types of assessment.
Jessica,
I do not have citations to share with you, but I learned in my doctoral program years ago that tests should be predictable, so students know what to study. ?
In some of my classes, I give students study guides for each chapter and/or give them quizzes. ?I promise them that the tests will be based on the quizzes/study guides, plus what we have talked about in class. ?In this way, the tests never contain items we have not addressed in class in some way.
I greatly dislike it when tests ask about some obscure statement in the book that has never been touched on in class.?
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It is well known that the equation x3+y3+z3-3xyz=1 has infinitely many rational solutions (x,y,z). Is the same true for the equation x3+ny3+n2z3 -3nxyz=1, where n≥2?is a given natural number? Can we determine the integer solutions?
Dear George, dear Peter Breuer,
in fact, the following algebraic number theory will help us.
Let K be the number field generated by the 3rd root of n; let's call this root t.
The factorisation that George gave shows that x3 + ny3 + n2 z3 - 3n xyz is the norm of x + ty + t2z, where x,y,z are rationals. I speak here about the algebraic norm from K down to Q, which is in fact the determinant of the matrix of multiplication by the element whose norm we want, evaluated on the basis 1,t,t2.
But every element of K is of this form, so the real question is whether K contains infinitely many elements of norm 1, and which of these have x,y,z integral.
The number field K has signature (1,1), having one real and one pair of complex embeddings. Thus, its unit rank is 1; this means that the multiplicative subgroup of every order inside K (that is a rank 3 lattice that is also a ring) has the group structure Z/2 x Z. The Z/2 comes from the fact that K has a real embedding, so its only roots of unity are +/-1. Any generator of the Z part is called a fundamental unit.
In particular, this is true of the order Z[t], generated over Z by 1,t,t2. It doesn't matter whether Z[t] is the full ring of integers inside K, or not.
Let eta = a + b t + c t2 be such a fundamental unit; then either eta or -eta has norm 1 and is thus an integral solution to George's equation. But then also every power of eta has this property, giving us infinitely many integral solutions, and there are no other ones.
The multiplicative structure of K is much richer; one just needs to find two elements of the same norm in K; their quotient will have norm 1, and will hence yield a rational solution to the equation. However, there might exist values of n for which the integral solutions are the only ones.
One example: for n = 3, the fundamental unit is -2 + t2, giving the integral solution (-2,0,1). Its square is 4 + 3t - 4t2, giving the solution (4,3,-4), and so on.
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I plan to design educational material to teach students with learning difficulties in mathematics using blended learning, It would be great if is there any suggestion.
Dear Jawhara Abueita
Perhaps this could be the useful reference: "Blended Learning, Research Perspectives" Edited by Anthony G. Picciano and Charles D. Dziuban.
Regard Asmuni
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A category is unital in the sense of Bourn if it has a zero object (is a pointed category), admits finite limits and for all objects X,Y ?the pair of maps (idX,0):X→X×Y ?and (0,idY):Y→X×Y is (jointly) strongly epimorphic.
Being unital is incompatible with being coherent in the sense of topos theory: only the trivial category is unital and coherent at once (Proposition 2.10 in Cigoli-Gray-Van der Linden: Algebraically coherent categories). Hence the category of pointed objects in a non-trivial topos is never unital. It is probably easy to find further examples along these lines.
On the other hand, most pointed categories “occurring naturally” in algebra are unital. A pointed variety of algebras is unital if and only if it is a Jonsson-Tarski variety (Theorem 1.2.15 in Borceux-Bourn: Mal’cev, protomodular, homological and semi-abelian categories). This means that its theory contains a unique constant 0 and a binary operation + satisfying 0+x = x = x+0. One simple example of a variety of algebras which is not unital is the variety of subtraction algebras (which occurs for instance in Zurab Janelidze’s work on subtractivity). A subtraction algebra is a triple (X,-,0) where X is a set, 0 is an element of X and - is a binary operation such that x-0 = x and x-x = 0.
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You may try the same problem assuming that f is twice differentiable on R.
Dear George,
It seems to me that your demonstration is correct. Great!
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Hello,
How To Write ARIMA / SARIMA model mathematically? I am trying to generte the mathematical structure of ARIMA(2,1,3)? and ARIMA(1,0,1)(0,1,1)12?
Dear Nema
Please have a look on the attachment.
Best regards.
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Dear folllowers,
Imagine you are in 1905, where experiments known today were not done.
Does aberration of light exist for a source horizontal to a moving frame?
Such a phenomenon (which mathematical reinterpretation should be done) wouldn't it explain c constancy in case of galilean addition of speed?
(The source being independant, every photon evolves in the detection frame where information is delayed relative to the real position giving c constant instead of galilean addition).
NG: I mean a change of measurement. The diagonal is what is seen by the observer. The distance seen of the cloud is not the same as the distance of the cloud.
It depends on your method of measurement. For example if you use parallax then you can calculate the difference in the aberration for your two observations at the ends of the baseline and see how it affects the derived distance.
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g (h(?X?X ,Y),?Y) = g (?h(X ,Y),h(?X,Y))+(X( ln f ))2 g(Y ,Y)- ??X ?X (ln f ) g(Y ,Y)
where? ?2X =?X +η(X)ξ
If this is a condition of the space then you cant prove it, rather you can find an example or model of the space where this one is true.
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What is the criteria for initial guess for initial conditiond in shooting technique whenever we are solving third order non linear diff. Equations.goven f and f' at zero is zero and f'(infinity)=1)
f' equal infinity is a real BC
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I have a derived block formulae and I want to use it to solve second order system of equations using maple. So, I need a sample of maple code which will serve as guide for me
Solving Second Order Differential Equations
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I am looking for a collaborative research on mathematics competence among engineers in Asian countries. For this purpose I have prepared a survey questionnaire and interested to have it filled from other countries too. Could anybody help me in this regards? Please text me. We can make our research group in this regards.
Thank you all for replying. The purpose of this research is to measure mathematics competency among graduated/ professional electrical/ electronic engineers. Thus to research for the deficiencies in the mathematics teaching and learning at graduate level and compare the mathematics learning potential among different Asian countries (a comparative study).I have prepared a questionnaire to be filled or interviewed from professional engineers. I can surly share that with you all (those who have replied and those following). And though I have already collected few responses from my country, Pakistan but any changes in that questionnaire are welcomed.??
I think it will be a better idea if we can formally make this research group with all the interested members, target a conference or indexed journal and work accordingly and productively.? We need to make this group private. I don't know how to do this at research gate but we can easily do this if we have email contacts of each other and formally know the members. My idea is that we can do that using google groups. I have once used it before. my gmail id is ikramekhuda@gmail.com. If you all have any other idea then plz do share.
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Referring to the attached set of equations,
Wave function of the "Schrodinger equation for two-particles [equation 1]"?can be separated using separation of variable method, when they are distinguishable. And two separate "one particle" equations must be [equation 3]. In this form, the physical meaning must be, the particle 1 is in the potential field "V", and particle 2 feels no potential at all, unless I assume V=V1+V2 where particle 1 is in facing?V1 and particle 2 is facing?V2 potential.?I mean V1 and V2 could be any real value that I choose!!!! But is it?
However for distinguishable particles this mathematical derivation is fairly simple. But how people come up with the composite wave function for indistinguishable particles [equation 4]? Is it just a guess or there is any formal mathematical derivation?
Thanks a lot James and Christian?
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We know that Pythagorean Identities and so his famous theorem about the relations between lengths of the right-angled triangles, and other famous theorems of Euclid, are still used till now. My question is about non-famous, ancient mathematical theorems that still used this time?
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Whenever we have a solution with square root of negative one, we always dismissed it to be unphysical, whenever we used complex numbers its always argued that it is just a mathematical trick or tool in order to obtain the Real number. Always we think of what is physically meaningful variable or quantities are those expressed in real numbers. What if we have to subscribe a physical meaning to those solutions or variables expressed in complex or imaginary numbers? What if their simplest physical meaning is this: they physically exists but can never be "fundamentally" observe? Then we will eventually asks if we deal with these things, Will it still be science? ?
A complex number has a physical meaning; it means two orthogonal quantity. ??
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if we use iteration methods like successive over relaxation, we should use a suitable value of the weight factor and this value will be updated through?iteration to force iteration variables to get rapid convergence. my question?is how to update this value? what is mathematics we use to do that?
Find the relationship between the parameters and use this equation to uddate the factor.
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Let m and n be nonzero integers. It is known, but the proof is rather long and tedious, that the equation m arctan (1/x) + n arctan (1/y) = π/4 has only 4 solutions (x,y) in nonzero natural numbers. Can one give a simple(r) argument?
Dear KURT, I'm really sorry I missed?the most important:
I APPRECIATE THAT YOU DID NOT SIMPLY DOWN-VOTED, but instead YOU OPENLY EXPRESSED YOUR VIEWS. Great!
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I am trying to justify the use of AT instead of UTAUT for my paper on teacher challenges faced when using technology to teach mathematics...
Thank you so much for the feedback. It has been very helpful.
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?How?Tor_1(M,F) = 0 for all finitely presented modules F implies M is flat?
Every module is a colimit of finitely presented modules. Thus the condition implies that Tor_1(M,F)=0 for all modules F -- since Tor_1(M,?) commutes with colimits. Thus, tensor-product with M is an exact functor, i.e., M is flat.
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in context with group theory I came across the term 'invariant subalgebra' which says: some generators of group G which goes either into itself or zero under commutation with any element of the whole algebra.
Can anyone please elucidate an example?
See,for example:
and
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Let X be a smooth curve over k, S a noetherian scheme over k, F a vector bundle on X\times_k S, L a line bundle X\times_k S which is the pullback of a line bundle on X. The theta function associated to L is a section on the determinant line bundle on S of a representative for the cohomology of F\otimes L on S. How exactly does this generalize classical theta functions?
@Cenap's answer is a bit cryptic, so lets ?first do the easier case of F = linebundle.
W.l.o.g. S is connected. Then for fixed s (i.e. for a closed point of S), F_s is a linebundle of degree d independent of S i.e. the degree of the linebundle is well defined. Over the complex numbers this just means that the topological type determined by ?by c_1(F) \cap [point]\times [X] is constant.?Now the?line bundle over X \times S induces a map \phi: ?S \to Pic_d(X), ?the degree d component of the Jacobian (aka the Picard Variety) which is a (principally polarised) Abelian variety. The map \phi is such that a point s is mapped to the isomorphism class [F_s]. In a slightly more abstract (and correct) language: there is a universal line bundle \F over X\times J_d, a line bundle L_S ?over S such that F \iso \phi^* \F \tensor L_S. ?This means that we can basically concentrate on the case S = J_d and ?pull everything back from there up to the unavoidable (and easy) possibility to twist with a line bundle on S. ?Now over ?the complex numbers the Jacobian is just a torus.?To simplify further?lets us assume ?d > 2g - 2, (which in any case we can do by twisting with the fixed linebundle ?L on X of suitable degree) so that there are exactly d - (2g - 2) sections over X ?which of course depend on the point Z on Pic_d. ?In other words the sheaf \pi_* \F ?over Pic_d is actually a vectorbundle of rank \chi = d ?+ 1 - g, each point of which parametrises a a section in the linebundle Z over which it lies, or equivalently, a meromorphic function on X. If we pull the vectorbundle back to Cg?the bundle trivialises in a more or less canonical way with factors of holomorphy (this is explained well by Mumford in his book on Abelian varieties) and you get d + 1 - g dimensional space of meropmorphic functions on X depending holomorphically on z in?Cg . ?This is the anlogue of the classical theta functions.?
Incidentally, there is also ?a period matrix \Tau in ?a complex period domain H?(the higher dimensional analogue of the upper halfplane) in C^(g \times g), satisfying the Riemann period relations acted upon by the group Sp(2g, Z) (the analogue of SL(2, Z). Note that in our case this period actually is determined by (and by Torelli determines) the curve X, so we can do this over the moduli space of curves and pull back to Teichmüller space, to get functions that also depend holomorphically on Tau but the story gets more complicated.?
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Let p≥5 be a prime number. Prove that there exists an unique triple (x,y,z) of non-zero natural numbers such that x3?+ y3?+ z3?= p + 3xyz.
p≥5 ?is a prime number and p =?x3 + y3 + z3?- 3xyz ? ?(1)
1. p is odd,?therefore?exactly one?of x,y,z?is odd. Without loss of generality, suppose x,y are even and z is odd.
2. (x,y,z)=(2,2,1) satisfies for p=5, since?5 = 23+23+13-3*2*2*1,?
and (2,2,3), (4,4,3), (4,4,5), (6,6,5), (6,6,7) ... are solutions for p=7,11,13,17,19,... respectively.
1. Notice that any prime number p≥5 has the form 6k-1 or 6k+1 for k natural.
2. The steps 2. and 3. together with the proposal of George lead to the hypothesis:
For any prime number p=6k+(-1)m, where m=0 or m=1, the equation (1) has a solution (2k,2k,2k+(-1)m).?This solution is unique up to permutations.
A straightforward computation gives (2k)3+(2k)3+(2k+1)3-3*2k*2k*(2k+1) = 6k+1, similarly for x=y=2k and?z=2k-1 we obtain 6k-1.
The uniqueness remains open at the moment.?I hope somebody will finish my approach, it is too warm here :-)?
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Let E, K be a finite extensions of Q_p and V a finite dimensional E vector space which is also a potentially semistable G_K module (that is, V is semistable over a finite extension of K). Let K' be the maxl unrammified extension of K. The period ring D_{pst}(V) is taken to be the direct limit of the semistable period rings of all finite extensions of K is a G_K module and has dimension dim_{Q_p} V as a K' vector space. As an?R=K'\otimes_{Q_p} E module it turns out that D_{pst}(V) is free, the heuristic being?that the frobenius is a bijection on the potentially semistable period ring, I'm not sure how to finish the argument.
Please send me the detail in my massage as PDF or TeX formats.
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For example, in the MLS method, the error at the points near the boundary are more than the points in the domain under study. How can we manage the error at points near the boundary?
Hi
Prof B.Rath
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HI everyone,
I need to know the latest researches and problems that still unsolved in area of Complex differential equations in The Unit Disc, with applying Nevanlinna Theory.?
Best regards
Dear? Mohamed,
See for example? Solutions of complex linear differential equations in the unit disc? by? Saleh Omran,???
MATHEMATICAL COMMUNICATIONS 205
Math. Commun. 16(2011), 205-214.
Complex oscillation of differential polynomials generated by
analytic solutions of differential equations in the unit disc by?
Ting-Bin Cao, Lei-Min Li, Jin Tu and Hong-Yan Xu,
and
G.G. Gundersen, E.M. Steinbart, S. Wang,
The possible orders of solutions of linear differential equations with polynomial coeficients, Trans. Amer. Math. Soc. 350(3) (1998),1225-1247.
There are also? a major breakthrough concerning the initial Schoen Conjecture related to Complex differential equations (for hypebolic harmonic) in The Unit Disc.
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As we have cardinal numbers of a set and cardinal arithmetic, partition technique with the cardinal numbers, is it possible to generalize this concept in fuzzy set also?
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See the attachment.
?Dear all
Just now I accidently saw this question and your discussions. Then I recalled a general result obtained very recently by me. See the picture. When letting $\lambda\to0$ on both sides of the equations respectively, then the integral you discussed can be derived immediately. Is it okay?
I must go, see you later.
Best regards
Feng
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Any mathematical or practical based method
This is a very good question. ? ?In addition to the excellent answer by @Kelly Cohen, there is a bit more to consider.
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We have at least two different definitions of “infinite” (“actual infinite” and “potential infinite”) since antiquity, and these two different “infinites” with different natures unavoidably become the foundation of present classical infinite related science theory system dominating all the infinite related contents as well as all of our infinite related cognizing activities since then.
When one faces an infinite related content in mathematical analysis, is it in ?“actual infinite mathematical?analysis” or “potential infinite mathematical?analysis”?
The infinite small (few) related errors and paradox families (such as the newly discovered Harmonic Series Paradox) and the infinite big (many) related errors and paradox families (such as the newly discovered Cantor’s ideas and operating process of mistaken diagonal proof of “the elements in real number set are more infinite than that in natural number set”) are typical examples of “master pieces of confusing potential infinite and actual infinite”.
Dear Geng Ouyang,
one possible answer is the following:
What is meant by 1 + 1/2 + 1/3 + 1/4 + ...
It is the limes of the sequence
1
1+ 1/2
1 + 1/2 + 1/3
...
Each of the elements of this sequence is a finite sum. And therefore in every pair of brackets you can place there can only be finitely many summands. Perhaps this is not an answer to the question you intended, but I was not yet able to read the papers I found on your profile.
Best regards
Johann Hartl
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I taught an introductory stat class and one of the subtopics involved confidence intervals( CI).? I decided to focus my examples on the meaning and applications of CI.? For example, students? were? involved? obtaining the confidence interval for the population mean from a random sample of data that they themselves collected. I discovered from students' work that? interpreting the meaning of confidence interval was truly challenging to many. How have others approached this topic in introductory statistics?
.
Actually, neither of those definitions is correct. The correct interpretation of a 95% confidence interval is: if you repeat the experiment infinite times (i.e., if you take infinity random samples from the population), 95% of those samples' CIs will contain the population mean. There is some R code at the end of this post which you can run to view simulations showing that this definition is accurate, whereas the definition you mentioned (that 95% of the sample means fall within the interval) is not.
Note that in many cases it is safe to use the real definition of a CI to infer the "classical" interpretation you mentioned. That is to say: if, out of inifinity random samples of the population, 95% of those sample CIs contain the population mean and 5% of the samples don't, then it is usually reasonable to assume that the sample you got in your experiment is more likely to be from the 95% of sample CIs that do contain the population mean rather than from the 5% that don't. But it's definitely not the same as saying that 95% of replication means will also fall within the CI (if your experiment sample happened to be un-representative, e.g. drawn mostly from one tail of the distribution, then actually very few replication means will fall within your observed CI, as you can observe from the simulations below).
Also note that the purpose of a CI, at least in applications I am familiar with, is not "to determine if the sample is representative of the population"; rather, it is to form a reasonable estimate of where the population mean might fall, given an observed sample. This could, in turn, be used to see if the sample is unlikely to have come from a certain population (e.g., performing a hypothesis test by checking whether the CI doesn't include 0 is the same thing as asking whether the sample is unlikely to have come from the null distribution [which is a population with mean 0]) but there's nothing about the CI itself that inherently tells you how representative of the population a certain sample is.
I highly recommend the Hoestra et al. and Belia et al. papers (which I mentioned and linked in my above posts), they have good explanations of the interpretation of CIs.
####################
### CI SIMULATIONS ###
####################
# Create a normally distributed population
population <- rnorm( 1000000 )
### INCORRECT DEFINITION OF CI
### The following code chunk uses simulations to show
### that it is not true that 95% of hypothetical samples
### will fall within the original observed CI
# A function to simulate one "real" experiment and multiple hypothetical experiments
simulate_experiments_wrong <- function( n_simulation, n_subj ){
# Do an experiment: sample n_subj subjects from the population
realsample <- sample( population, n_subj, replace=T )
# Get the 95% CI of this sample
CI.lengths <- sd( realsample ) / sqrt(n_subj) * qt( .975, n_subj-1 )
realCI <- c( mean(realsample)-CI.lengths, mean(realsample)+CI.lengths )
# Get n_simulation samples and their means, to compare against the "real" experiment
newsamplemeans <- unlist( lapply( 1:n_simulation, function(x){ mean( sample( population, n_subj, replace=F ) ) } ) )
# Make a histogram showing the original sample CI and the hypothetical sample means
samplehist <- hist( newsamplemeans, main="Sample means" )
lines( realCI, rep( max(samplehist$counts)/2, 2), col="red", lwd=4 ) points( mean(realsample), max(samplehist$counts)/2, col="red", cex=2, pch=16 )
legend( "topleft", col="red", lwd=4, legend="Observed CI", pch=16 )
# Get the proportion which are within the original CI
proportion_in_original_CI <- length( which( newsamplemeans>=realCI[1] & newsamplemeans<=realCI[2] ) ) / n_simulation
# Show the results
print( paste0( proportion_in_original_CI*100, "% of simulated experiments had a mean that fell within the sample CI (wrong definition)" ) )
}
# If you run this simulation over and over again, you will get a lot of values quite far from 95%
simulate_experiments_wrong( n_simulation=1000, n_subj=24)
### CORRECT DEFINITION OF CI
### The following code chunk uses simulations to show that 95% of
### the hypothetical samples' CIs contain the population mean
# A function that simulates one hypothetical experiment and tests whether
# its confidence interval contains the real population mean
experiment <- function(population, n, conf.level){
# Draw a sample from the population
exp_sample <- sample( population, n )
# Calculate the conf.level% CI
CI.lengths <- sd( exp_sample ) / sqrt(n) * qt( mean(c(1,conf.level)), n-1 )
CI <- c( mean(exp_sample)-CI.lengths, mean(exp_sample)+CI.lengths )
# Return the sample CI)
return( CI )
}
# A function to repeat that experiment many times and show how many of the
# sample CIs contained the population mean
simulate_experiments <- function( n_simulation ){
# Simulate a bunch of experiments and get their sample CIs
sampleCIs <- lapply( 1:n_simulation, function(x){experiment(population, 24, .95)} )
# see which sample CIs include the population mean
pop_mean_in_sample_CI <- unlist( lapply( sampleCIs, function(x){x[1]<=mean(population) & x[2]>=mean(population)} ) )
# Create a plot of all the sample CIs.
# Sample CIs that include the population mean will be in blue; sample CIs
# that don't include the population mean will be in red.
lowerbounds <- unlist( lapply( sampleCIs, function(x){ x[1] } ) )
upperbounds <- unlist( lapply( sampleCIs, function(x){ x[2] } ) )
matplot( rbind(lowerbounds,upperbounds), rbind( 1:length(lowerbounds), 1:length(lowerbounds) ), type="l", col=ifelse( pop_mean_in_sample_CI, "blue", "red"), lty=1, lwd=1, ylab="Samples", xlab="CI" )
lines( rep( mean(population),2 ), c(0, length(lowerbounds)+5 ), col="black", lwd=2 )
legend( "top", col="black", lwd=2, legend="Population mean", xpd=NA, inset=-.1 )
# Get the proportion of sample CIs that include the population mean, and show it on the screen
proportion_CIs_including_popmean <- length(which(pop_mean_in_sample_CI))/length(pop_mean_in_sample_CI) * 100
print( paste0( proportion_CIs_including_popmean, "% of simulated experiments contained the real population mean in their sample CIs (right definition)" ) )
}
# If you run this simulation over and over again (by pressing the 'up' key and running the following line again, repeatedly), you will repeatedly get values around 95%.
# The larger the value of n_simulation you use, the closer the results will be to 95% (but it
# will get slow to run)
simulate_experiments( n_simulation=250 )
Question
Transform the arithmetic of this exercise into algebra.
I used the term "partially correct" since there were a good idea in your solution, but the indicated terms were not all negative. Consequently, it was wrong to ignore the terms after kx - hx.
Now?mathematical induction:
in your solution,?only the cases (k,h) where k=h+1 are considered. Why? Because you began with k=2, h=1 and the subsequent computations were made for the couples (k+1,h+1) ?hence for (3,2), (4,3) ...
If you want to?get a result for any (k,h) where k>h and both are natural numbers,?try to?use mathematical induction twice, for 2 independent cases
• first step (k,h)=(2,1), induction step (k,h) => (k,h+1) if h+1<k
• first step as above, i.s (k,h) => (k+1,h)
the points (k,h) will spread in?a whole domain (h>0, k>h), not only on the line k = h+1.
Good luck! Viera
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i want best approximate? for f(x)=g(x).u(x)
As regards the question for a product as the corresponding question for a sum, I just would like to remark that one can map the first to the second by taking logarithms, the second to the first by exponentiation.? - Besides thinking in error norms, one can also try to tackle the problem in terms of function theory (analyticity assumptions), power series or asymptotic expansions. Also, orthogonal expansions may be a point of view, that may be related to Hilbert spaces. One could also tackle the question by algebra methods like formal power series, that is, without regard to convergence. Using power series and/or asymptotic expansions, one then has an approximation by truncation of the series/expansion. One may also use something like the tau method of Lanzcos. - As regards the question which type of product, one may use convolution of functions here. Then, there is a mapping by Fourier transformation from multiplication of two functions to convolution of their transforms and vice versa . - A further interesting question might be, in what sense the product of two distributions (that is not defined easily if at all) may be approximated by a further distribution, and in what sense. - In view of all these possibilities, I strongly suspect, that there are probably as many answer as there are? possibilities or even more. Thus, one gains quite a lot by studying specific examples.
Question
I tried a purely algebraic solution - unsuccessfully.
Dear George,
I understood what?you?meant by "algebraic solution ", you meant algebraic method (calculation, tools), it is ok. ?I meant if??the result of the article is for all real numbers as the author said, then you can take it, but if the result related only to algebraic numbers, then ?we can not ?take it.
Really, I looked for Tijdeman's article to check?if it was for all real numbers or algebraic ones only, but I could not get it.
Best, Hanifa
Question
There are three number line in mathematics:
1. Real Number Line.
2. Imaginary Number Line.
3. Circular Number Line.
1 subset 2 subset 3.
If X is any set, then Card (X ) less than of Card (P(X)) less than of?Card(P(P(X))) less than of ...
Question
Let two positive integers n and m be given. How many pairs of integers (x,y) can one find in the range 1..n such that (i) all x's are different, (ii) all y's are different, (iii) all x+y's are different, and (iv) x+y <= m.
I came across this problem (with n=7, m=11) in a safety discussion of a stream cypher (trinomials that have to fit into an opportunity window). I can only solve it by an extremely lengthy analysis, while I'm hoping there is an elegant (or at least a not too lengthy) solution.
If a third coordinate z?is defined as m+1-x-y, the problem becomes a kind of magic-square thing: fill each of three rows with distinct numbers in 1..n, such that each column has constant sum m+1. How long can the rows be? This problem (which is the actual translation of the cryptological question) is not exactly equivalent with the pairs-problem; it may have slightly smaller solutions.
Can this perhaps be linked to a known problem?
?Good easy fix. So start from k+1 instead of k for 2nd coordinate and it is enough to get n=2k different pairs, using otherwise the same formula. Thanks I.E. Kaporin. Seems completely settled now.
A) n=2k. Use?(2k-(2i-2),k+i) for i=1 to i=k, then use (2k-(2i-1),i) for i=1 to k.?
B) n=2k+1. Use (2k-(2i-3),k+i) for i=1 to k+1, then use (2k-(2i-2),i) for i=1 to k.
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Three Egyptian texts come to mind, the Akhmim Wooden Tablet, 1950 BCE, the Kahun Papyrus,18000 BCE and the Rhind Mathematical Papyrus, 1650 BCE, discussed by: https://www.academia.edu/24746984/ABSTRACT_SCRIBAL_MULTIPLICATION_AND_DIVISION_An_Update
Any other examples out there? Thanks.
Abstact math was first defined by rational numbers n/p scaled by LCM m/m to mn/mp so that the best divisors of mp summed to man recorded in red auxiliary numbers. Final unit fraction series were recorded form right to left ciphered onto hieratic sound symbols, Ionian and Dorian alphabets when the abstract notation was adopted by Greeks 1500 years later. The EMLR and RMP 2/n table selections of LCMs are included in the linked paper, written in 2011.
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I have seen lot of papers on incomplete hypercube architecture on 1991-2000 Why there is no recent works on that?
This is an interesting question.
For very ?recent consideration of incomplete hypercubes, see Section 4 in
and Section 3 in
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It is easy to solve non linear problems with initial condition.Feeling it difficult to solve non linear problems with two boundary conditions.Any guidance or direction will be appreciated.
There are various ways to solve nonlinear BVP numerically. If the BVP problem is ?an ODE one can use a collocation method, Finite Difference method, Finite Element method and a shooting method. All all cases you will end up with a system of nonlinear algebraic equations that have to be solved. And Newton's method ?at this point is normally your best choice, especially if you have a reasonable guess for the solution.
Spectral Collocation Method:
Shooting Method:
Finte Difference Method:
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In general, I am interested in when for any positive number d, that for any given trajectory in phase space and any point P1, there exists another point on the trajectory, P2, that satisfies |P1-P2| = d.
Thank you for the clarification.I also see that I goofed. The original question stated |P1-P2|=d but I wasn't reading it correctly. Instead, I was reading |P1-P2|<d.
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FEM
Can you explain mixed formulation?
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The attached sequence?expands on?the sequence 'created' by Pythagoras to depict the Modal System.
Because only the top and Base alpha-values were provided in the Dodecachordon (Glaeranus, 1547), I had to sequence what is in between.
Whether g11 is 45.562 or 48 makes a big difference here, and looking at the three final interlocking sequences, I am not sure that all numerical values are correctly attributes to the corresponding alpha value?(representing a Tone).
Please consult the legend for details and the provenance of the values. You may also consult the Control equation posted on my profile for more information.
You are the only person I have talked to that is trying to understand the relation of pitch and frequency.? But you are using an archaic method that can be characterized as archimedan, meaning you are trying to make an arithmetic of pitch.? As Gauss said,? we need to be careful about attaching meaning to a simple magnitude like frequency.? In fact, frequency is not even by itself continuous because we cannot define fractions of wavelengths without assuming the frequency curve has a certain shape.? So frequency is not compact:? Between 440 and 441 Hz there are no defined points. We get out of this problem because there is always a function from the musical system to frequency.? The frequency can be eliminated and the music structure is still there.
You work reminds me of the summary of Euler's music theory in "Leonhard Euler Mathematic Genus in the Enlightenment" by Ronald S Calinger.? (as an aside, unfortunately the author left out Euler's Tonnetz.)? This discussion continues endlessly, mindlessly on the internet and Wikipedia, reaching the stage of total confusion with the "Geometry of Music" and "Topos of Music" books.
I think the Pythagorean ratio you are refering to are something like this; "the general term 2nx3mx5p covers almost all pleasantly perceived notes" [Euler] or at least some notion that intends to explain the reason for consonance and dissonace using frequency.? This is coupled with the idea that there are "no clear borders between notes C and D."
I think that I understand what you are saying is you get different scales by using a different constant.? I suppose then you are using an inductive scheme in which each note n then generates the next note n + 1.? That is, how integers are generated in number theory.
The cardinal and ordinal numbers in music measure size and order.? They are what remains after the pitch quantifier that you use to build the set is eliminated.? This method was introduced around 1900 and is called quantifier elimination.? You know how in music they have the chords in the key represented by roman numerals.? That means the frequency (pitch) has been eliminated since the key no longer depends on pitch.
This same problem occurs with colors and light: We understand that light is not a theory of frequency.? So why do you think frequency can explain the usage of letters in the alphabet.
Look, no one says the reason e is the most common letter in English is because E is the 5th letter and there is something special about the ratio 5/26.
So why do we look for a ratio to explain why a 5th is a probable interval based on frequency? Intervals never depend on the end points! Intervals are independent of pitch, and since what ever is true for notes is also true for intervals it must also be true that notes are defined independent of pitch.? Perhaps the logic of this seems to be unusual, but that is because you are in a pre-modern mathematic form where you still think that music theory is explained by pitch.? Then you get confused because you have Russell's paradox.? You are trying to make a set of all sets using? real numbers and fractions which are approximations.
I would like to see if I can get a better idea of the formulas that you use to make the frequency value list and the formula for the Pythagorean ratios.? Do you have formulas or do you just have a list of numbers without a generating rule? Do you have to calculate them by hand?
The basic question is where does the unit 1 come from.?
I wonder if we could start by agreeing that the fundamental mode of vibration is 0 and the octave is exactly 1 unit because it is exactly twice the fundamental?? I would like to know what do you think about Z/12Z?? Do you know log 2 arithmetic?
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I'm searching for mathematical foundation of this model.
@Peter Enders:?What is bad with Minkowski's 1908 talk?
Minkowski's 1908 talk was indeed very important. ? An excellent, detailed overview of Minkowski's views (including his 1908 talk) is given in
This talk includes what Minkowski wrote in German as well as the English translation (there are many helpful drawings showing how Minkowski viewed hypebolic space and Raum und Zeit (Space and Time), e.g., Fig. 2 on page xxvii, page 27 in the pdf file). ? Minkowski's portrait plus signature are given in the attached image (p. 13).?
A particularly important part of this book is chapter 7, starting on page 200, on quantum space-times. ? Minkowski fused space and time into a 4D continuum. ? Of particular interest in this chapter is the discussion about big bang, black holes and information loss. ?A discussion about the quantum nature of the big bang is given in Section 7.2, starting on page 203. ? ?See the key questions in Section 7.3.
In his September?21, 1908 Hermann Minkowski included in?his talk at the 80th Assembly of German Natural Scientists and Physicians the following observations:
A point of space at a point of time, that is, a system of values, x, y, x, t, I will call a world-point. The multiplicity of all thinkable x, y, x, t systems of values we will christen the world... Not to leave a yawning void anywhere, we will imagine that everywhere and everywhen there is something perceptible. To avoid saying "matter" or "electricity" I will use for this something the word "substance". We fix our attention on the substantial point which is at the world-point x, y, x, t, and imagine that we are able to recognize this substantial point at any other time. Let the variations dx, dy, dz of the space co-ordinates of this substantial point correspond to a time element dt. Then we obtain, as an image, so to speak, of the everlasting career of the substantial point, a curve in the world, a world-line, the points of which can be referred unequivocally to the parameter t from - oo to + oo. The whole universe is seen to resolve itself into similar world-lines, and I would fain anticipate myself by saying that in my opinion physical laws might find their most perfect expression as reciprocal relations between these world-lines. [1, p. 76]
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Both functions are defined on [0, 1] and take real values.
take a triangle (equilateral, base at zero ?one. divide edges to two equal parts keep going. in the limit you get continuous and nowhere differentiable (L C Young , Calculus of variations etc) or look at Natanson real analysis (Andre Kolmogorov told met that in 1958)
Have fun?
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What are the dominant factors that enable students to solve higher order thinking skills mathematics problem solving questions from your experience?
Read the works of Prof. Carol Dweck of implicit theory.
Implicit theory means the one believes about his own ability: either fixed for developed. If he believes that his ability (intelligence) is fixed, then he will see no meaning to face challenging tasks more than his "fixed ability". This will grow to make him avoid new tasks after any successful challenge (including learning new things).....
Please refer to Prof. Carol Dweck's work. Her works are full of supporting evidences, in specifically in mathematics.
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i need a clear solution provided by mathematical calculation?
I agree with Mr. Pandey since there are friction loses
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I am presently working on multipliers on Banach algebra and i found out that most result discovered so far are been establish on the fact that the algebra is faithful (without order). Please can anyone help me with the main reason why the algebra must be faithful (without order).
A few years back I had also come across the curiosity of the role of faithfulness in the study of multipliers. As a result, I prepared some detailed notes on this topic which are attached herewith. I hope these would serve to supplement the useful information already given above.?
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--
Try to use the characteristic polynomial of A+B:
C(t)=tn -tr(A) tn-1 +....+(-1)n det(A+B), then take an eigenvalue ?λ of A+B, you get:
(-1)n det(A+B)=λn -tr(A)?λ n-1+...+c λ =?λ (λn-1 +...+c), where c is the sum of all ( n-1) products of eigenvalues of A+B.
Here you can take n=6.?Now , it is to you to have energy and do calculations. :)
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In general, we can represent a nite partial ordering?set (S, <) using this procedure: Start with the directed graph for this relation. Because a partial ordering is reexive, a loop (a,a) is present at every vertex a. Remove these loops. Next, remove all edges that must be in the partial ordering because of the presence of other edges and transitivity. That is, remove all edges (x,y) for which there is an element z ∈ S such that x ? z and z ? x. Finally, arrange each edge so that its initial vertex is below its terminal vertex. Remove all the arrows on the directed edges, because all edges point “upward” toward their terminal vertex. These steps are well dened, and only a nite number of steps need to be carried out for a nite partial ordering?set. When all the steps have been taken, the resulting diagram contains sufficient information to end the partial ordering. The resulting diagram is called the Hasse diagram of?(S,< ), named after the twentieth-century German mathematician Helmut Hasse who made extensive use of them.
Is there another application of Hasse Diagram?
In ring theory the Hasse diagram of ideals ordered by inclusion is used often. In particular the attached Moebius function is used to compute the so-called homogenous weight in Coding Theory.
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Why the modern mathematics uses ancient representation about countable infinity such that:
there is a set which intuitively has infinitely many elements
0,{0},{{0}},...etc.
Jaykov's question is not very precise, so a precise answer is not possible. Proof by contradiction (which dates back to Zeno of Elea) is an important tool for showing a set is infinite: e.g. proving the reals are uncountable using Cantor's?diagonalization argument, or the primes are infinite as per Euclid. Cantor's ideas were revolutionary and were not accepted quickly by the mathematical community. Even today some authors of popular articles tacitly assume all infinite sets are countable. A notable (living) mathematician (Zeilberg) questions the ontology of Cantor and infinite sets, in http://www.math.rutgers.edu/~zeilberg/Opinion108.html . The mathematical establishment, in general, eschews such philosophical matters.
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I want to create mathematics quizzes making use of Moodle and WIRIS and replace the paper and pencil assessments, which is problematic for Deaf and Hard of Hearing learners. Anyone ever tried it or have any suggestions?
Numbas is free.? Numbas is an easy way to create online tests. Our free web-based system helps you build the exams you need to challenge your students, complete with videos and interactive diagrams.
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I would like to know the mathematical derivation for fuzzy clustering optimisation.
The proof is available in this book. Cited in the same page, which you have posted.
J. C. Bezdek (1981), Pattern Recognition with Fuzzy Objective Function Algorithms
Please have a look at the 5th chapter, page 170-190.
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What are the additional conditions as needed in order to establish the
existence of a unique fixed point satisfying the condition for any α ∈ (0, 1] there exists
β ∈ (0, α) such that dα(x, z) ≤ s[dβ(x, y) + dβ(y, z)] in Gbq-family (X, dα).
Related information can be found in below paper.
Kumari, Panda S., and Dinesh Panthi. "Cyclic contractions and fixed point theorems on various generating spaces." Fixed Point Theory and Applications 2015.1 (2015): 153.
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Do you usually use the long procedures given in the standard textbooks in calculus or have you tried using some derived formulas like the one given in the link below? Perhaps, you can also try deriving other formulas for solving some other optimization problems and if you have already, may be you can also share your formulas with us.
In the particular case when the distance is attained, mentioned above, the hyperplanes H_1, H_2 are tangent ?respectively to the boundaries of A, B at the points where the distance d(A,B) is attained.
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Dear researcher in this world!
Please help me! How to reduce NP-Hard bin packing problem not using mathematic solutions? if there is mathematic solutions, can teach me how?
thanx jose...
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In standard form?x^2(d^2y/dx^2)+x(dy/dx)+(x^2-n^2)y=0 last term is (x^2-n^2). How to tackle the problem if instead of minus sign we have positive sign. i.e:?(x^2+n^2)
Dear mohamed Ali,
the original form of Bessel differential equation is:
x2(d2y/dx2)+x(dy/dx)+(x2-n2)y=0
if you put n =i m ---> ?n2=-m2, thus Bessel differential equation transforms to the form:
?x2(d2y/dx2)+x(dy/dx)+(x2+m2)y=0
which has the same solution, but with imaginary argument ( i m)
See the attached Mathematica Solution.
Best Luck.
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A minimal spanning path in a graph is a path that contains all the vertices of a graph whose weight is the least among the spanning paths.
This is a question much related to finding a Hamiltonian path - where a Hamiltonian cycle would be a feasible solution to the famous traveling salesperson problem. The current standing on the Hamiltonian path question is that it is unknown whether it is hard or not - no-one has yet found a polynomial algorithm for the problem.
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The choice to be made is similar to the choice between set theory and category theory.
@Tadeusz Ostrowski:?In her [Mathematics] palace there are rooms for all theories.
In addition to each of the incisive (and interesting) answers, already given, the answer given by @Tadeusz Ostrowski (and @George Stoica) makes perfect sense: rather than choosing between set theory and topos theory as a foundation of mathematics, we can view Mathematics as a form of Hilbert Hotel (there is always room for one more guest). ? In terms of this thread, a new guest would be a fundamentally important theory in the foundations of mathematics.
I myself favour axiomatic set theory, especially if we start considering various forms of topology. ??
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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 knowledge,?and 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.
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There have been two opposite phenomenon about the definition for infinite in present classical infinite related science system: (1) plenty of different mathematical definitions for infinite and (2) no mathematical?definition for infinite.
On the one hand, we really have had many different mathematical definitions for just half of the infinite: infinities (infinite bigs) in set theory such as “super (higher, more, less…) infinite, super (higher, more, less…) unbounded, super (higher, more, less…) unlimited, super (higher, more, less…) endless,…”, the plenty of non-self-justification “half definition” for infinite which is nothing to do with infinitesimals strongly prove that we human in fact are unable to know how to have mathematical?definition for “infinite” at all------it is impossible to have mathematical definition for “infinite” in present classical infinite related science system.
On the other hand, we really have had many different mathematical definitions for another half of the infinite: non-number infinitesimal variables, actual infinitesimals,… in standard and non-standard analysis, but those suspended infinitesimals related paradoxes (the 2500—year suspended black cloud over mathematical sky), especially the newly discovered Harmonic Series Paradox in present classical infinite related mathematics make us have to admit that till now we human in fact are unable to know how to have mathematical?definition for “infinite” at all------it is impossible to have mathematical definition for “infinite” in present classical infinite related science system.
It is the time for us human to work at the integration of mathematical?definition for “infinite” now?!
One should bring in two definitions for infinite. One should be called relative infinite with a different symbol, and the other the absolute infinite.When we connect the abstract truth to the physical truth we can cognize, there is always a convenience factor varying? to the stretch of human imagination. Nice question to dwell on to stumble on the mysterious to unlock further secrets.
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It would be interested to know if there are programming environments for teaching and learning of mathematics for high school, outside the Italian territory.
I would consider MATLAB, which has many resources for this purpose (assuming college level from calculus on). Octave is the "free" version.
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It could even be a book or paper that deals with that kind of approach.
If we make a graph of the pitch values and the piano key numbers then we have what seems to be a graph of the line x = y.? If we graph the strings of the guitar then it seems the strings do not intersect.? In fact, the graph does not lie in the R2 plane.? The guitar strings actually do intersect at the system fundamental by gluing, which makes a directed multigraph.?? This means music is defined in projective space and not in affine space.? In affine theory all guitar tunings are the same, but that is obviously not true.
You may think the guitar fret board has two dimensions (string, fret) but in fact there are 3 directions of tone movement.? To demonstrate this consider the isotonic line that connection notes with the same pitch, starting at the top string, zero fret.? Along this line the fret and string numbers move in opposite directions but pitch does not change.? The isotonic line is not movement by string or by fret because in these directions the pitch rises or falls in the same direction as the string or frets.? That is, pitch goes up, so does the string or fret number.
In the same way, it is easily shown in the graph of pitch and piano keys that the graph is not correctly drawn because there is a a triangle that is 1 octave in length on three sides and has at least one right angle.? Therefore the pitch position triangle does not lie in a plane.
Similarly, on piano there are 2 directions of movement: by pitch value and by musical key.? But the transposing piano with a lever to shift the keyboard shows there is a third degree of freedom.
Tablature is an algebraic system that represents guitar music.? Sequences in tablature are written using a typed alphabet of fret numbers on a horizontal staff representing the string lines.? Since sequences in tablature are written using integers, the central question is how the integers are induced by the system fundamental using basic set theory operations.? Briefly, construction starts with the null set and then defines the system fundamental by a key function that maps F to the frequency domain.? This defines a point (0, 1) which is a new direction and a unit of measure.? Then the octave point (1, 1) makes an identity (an a filter) in a metric space.
The guitar strings are the open cover of the guitar model because every point on the guitar is contained in at least one string.? Therefore the guitar is convex,? compact, and complete.? Convex means there is no line between any 2 points on guitar that lies outside the guitar.? Complete means that it is possible to construct every possible sequence.? The union of the strings is the interior of the guitar and the intersection is the closure.?? The numbers in tablature are ordinals and the numbers in the guitar tuning ring signature are a cardinals.?? So if the guitar tuning EADGBE is expressed as the intervals (0 5 5 5 4 5), then we have a Zariski topology that is a? decidable 6-tupple.? In this theory the piano have at most 1 tonal center while the guitar has 5.? This makes the guitar tuning a Baire metric space.
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In the traffic route guidance models:
1- What are the widely used method and?mathematical formation of this problem?
2- What are the controlling parameters and the decision variables that should be solved to obtain simultaneous user equilibrium and social optimum?
3- Is there known exact prevailng generalized solution of this ITS application?
I have not exact idea what is route guidance problem but i think this may be correlated with transportation problem(operation research ) and also with depth search problem or network stheory problems
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I am looking to estimate the diameter (nm) of a variety of double stranded plasmids (pUC19, pMAL pIII, pKLAC2, etc.)?when they are natively supercoiled and when they are relaxed.
If someone could point me towards a formula it would be much appreciated! Thanks.?
By using the tools in stochastic geometry, you can estimate the diameter of R2, if estimating them is enough for you, please send figures or illustration to see the shape of them. Thank you. Mehmet
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My request is primarily directed towards the optics to be a good user and not towards an expertise of the mathematical dimensions
Dear James
Thank you very much for all these documents and for your help
best regards
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Two related questions: Are the seven independent axes (dimensions) of the octonian space relatable to the adult capacity of STM (short term memory – number of independent objects simultaneously retainable in the mind)?
Can octonions serve as a mathematical Rosetta stone for cognitive and brain modeling by linking the world of algebra (operators, e.g. for mental rotation) to the world of geometry (relations, membership, and clustering), by exploiting the fact that octonions are composed of algebraic structures (7 quaternion sets) that also map onto a simple projective geometry structure (the Fano plane) that is organized as a system of 7 lines and 7 points?
?I'm glad that the material on quaternions is meaningful for you and turning you on.
I see that your primary background and interest is developmental psych, along with neuropsych. That is a strong part of my underlying interests. I met Piaget during a conference in 1971 and he told me that used quaternion ideas in his research "right from the beginning," probably meaning polarities, inverses, noncommutativity, systems of relationships, etc.?
My knowledge of quaternion possibilities grew much further this year when Martin Hay in England, a pharma patent agent and developer of a "social exchange" model using chiral tetrahedral quaternions, contacted me after reading the main PDF I sent you.
We would up collaborating with a third person I knew and introduced him to, Terry Marks-Tarlow, a woman psychotherapist and Jung-oriented researcher in California interested in quaternions and fractals as analogies for developmental processes. We three wrote and published a paper together on the potential for quaternion models of brain coordination processes, attached here.
I'm also attaching my follow-on presentation for this June of the presentation I gave a year ago (that main PDF). The current one, in draft form, deals with quaternion connections to evolutionary-viewed compassion and aesthetics in 1843 (quaternion invention year) and in 2016 (with a focus on empathic social robotics). I know you have a lot of material to deal with already, but you may want to browse these so as to broaden and deepen your picture of what is there in quaternions --more than meets the eye.
Thanks for the interest that you and your mentor have shown.?
By the way, here is a link that Martin found and sent me today on birdsong research. He suggests it gives?us a clue as to how quaternion models might in some way underlie language.
?The abstract states: ?Thus, compositional syntax is not unique to human language but may have evolved independently in animals as one of the basic mechanisms of information transmission.
What strong interest do you have that is driving you toward systems and processes like quaternions? Please share it if you like.
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I am trying to understand more about knots in higher dimensions. I understand that in general one wants co-dimension 2 so that, for example, S^2 can be knotted in S^4.
What I don't understand is to what extent the notions and theorems relating to knot complements and knot groups carry over to higher dimensions. Specifically I would like to know if the Gordon-Luecke theorem which states that "a knot is determined by its complement" is valid in general or only for S^1 knots in S^3.
I have also come across (twisted) spun knots and am wondering if there exists a one-to-one mapping between ordinary S^1 knots and spun knots. Presumably one could have higher dimensional knots that are not spun knots....?
Any insight would be greatly appreciated!
Dear James,
Thank you for the excellent response. I will take a look at the links you provided!
Best regards,
Niels
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I don't know how to character the predual of Morrey space with variable exponent
One of the predual space of Morrey space with variable exponents is the block space with variable exponent, see?
Cheung K.,and Ho, K.-P. Boundedness of Hardy-Littlewood maximal operator on block spaces with variable exponent, Czechoslovak Math. J. 64(139), 155-171 (2014).
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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? ${F_{{5^k}(n + 1)}}(q) \equiv {F_{{5^k}n}}(q){F_{{5^k}n + 1}}(q) \equiv 0\bmod {[{5^k}]_q}.$
${F_{{5^k}}}(q) \equiv 0\bmod {[{5^k}]_q}.$