Science topics: Mathematics
Science topic

# Mathematics - Science topic

Mathematics, Pure and Applied Math
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Martingale property
No, but a supermartingale, since f^2 is a submartingale.
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In its modern form, mathematical research is distinguished by the attribute of rigor: a claim is not accepted as a "theorem" even if experience suggests it is quite accurate for practical purposes. In particular, an argument supporting a claim does not constitute a "proof" unless it is completely airtight in relation to the assumptions. Does this practice advance or hinder progress of Science in general, and can rigor be useful or even implemented in other fields?
A mathematician belives nothing until it is proven
A physicist believes everything until it is proven wrong
A chemist doesn't care
biologist doesn't understand the question.
--------------------------
Biologists think they are biochemists,
Biochemists think they are Physical Chemists,
Physical Chemists think they are Physicists,
Physicists think they are Gods,
And God thinks he is a Mathematician.
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It was always thought impossible to have a closed-form formula that can calculate an arbitrary Nth digit of pi, until Borwein produced a formula in base 16 in the mid-1990s. My question is why is it only possible in base 16 and what is so special about "16"?
Have formulas for the Nth digit of other transcendental numbers (eg. e) been produced yet? Are these always in base 16, or do they require other bases? What is the status of this research on transcendental numbers? How many now have formulas for their Nth digit and in what base?
yes, the Chinese version was better than averaging Archimedes' guess
22/7 < pi < 223/71
anyone know how to solve pi from R, in A = pi(R^2), such that Egyptian and Greek square root method can be applied in that manner?
Bet some nice solutions came from that ... like 22/7 and better ...
thanks..
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May we generalize a formula to determine the cube roots of any arithmetic number?
Arithmetically how we calculate cube root? For example, how to Cube roots of 17, as square root?
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It is interesting how maths is useful for describing the physical world. But are there any branches of mathematics that are totally useless for physics? Why? Could it be that we perhaps anthropocentrically chose to follow branches of math that are interesting to us (ie. could have possible application)? To prove a point, could we invent a branch of math that is totally useless?
Could we come up with a sophisticated group theory for the game of chess? Is the reason no one has attempted that because it would be in fact utterly useless with an unexciting loss of generality?
If mathematics is of no use today it will be of use tomorrow. A good example is the geometrical concepts developed by the German mathematician Bernhard Riemann. A century later, Einstein used them to develop his general theory of relativity.
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I would like to know the reason that a person who is interested in studying a language by means of computers should study matrices and linear algebra.
A fundamental strategy to characterize meaning is by what is called "distributional semantics". The idea is that you represent linguistic units in terms of (co-)occurrence with other linguistic units. For example, you can represent documents in terms of the words that they contain (which is the standard "vector space model" in Information Retrieval). Or you can represent words in terms of the contexts in which they occur in corpora, which gives you "semantic space models" of words meaning. For example, both "apple" and "orange" occur frequently with "fruit", "eat", "peel", but infrequently with "car", "fast", ...
These representations can be understood naturally as high-dimensional vectors; vectors for multiple units form a vector space (=matrix). In short, that's why it's helpful to know some linear algebra. Look at these overview papers to learn more:
Baroni, M., & Lenci, A. (2010). Distributional Memory : A General Framework for Corpus-Based Semantics. Computational Linguistics.
Clark, S. (to appear). Vector Space Models of Lexical Meaning. Wiley-Blackwell Handbook of Contemporary Semantics — second edition, edited by Shalom Lappin and Chris Fox.
Erk, K. (2012)."Vector space models of word meaning and phrase meaning: a survey", Language and Linguistics Compass. 6(10), 635-653.
Turney, P. D., & Pantel, P. (2010). From Frequency to Meaning: Vector Space Models of Semantics. Journal of Artificial Intelligence Research, 37(1), 141–188.
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All our applied mathematics has evolved under the assumption of continuity, while all basic processes in Nature appear to be 'discrete', meaning non-continuous. (The prototypical example of continuous model is the inner-product vector space, of which the Hilbert space of QM is a special case.)
However, instead of looking for a fundamentally new formalism, or formal language, explicating the observed and formally unfamiliar discreteness, for some *irrational* (but quite human) reasons most physicists believe that one can somehow 'save' the conventional formalism by "discretizing" it . Well, the bad news is that, from a formal point of view, this does not make sense: you cannot do this to any formalism without destroying its integrity (that is how formalisms are structured). Heisenberg has a paper on this topic.
From the experimental point of view, since *all* our measurement instruments are discrete, we have no obvious way to verify the continuity hypothesis.
So to compete with the continuous formalism, we need a fundamentally new formalism to tell us how to interpret and to deal with the discovered ubiquitous 'discreteness'. Obviously, the new formalism must offer some radically new insights into the nature of reality, together with the new formal tools that should allow us to see physical processes in a completely new light, eliminating, in particular, the unacceptable wave-particle duality.
(Nevertheless, it may turn out later that some 'surrogate' form of *spatial* continuity is valid but not as a basic *underlying* model.)
*****************************************************************************************
For convenience, I will be collecting here just some of the relevant quotes by prominent scientists from the answers below:
1. "If you envisage the development of physics in the last half-century, you get the impression that the discontinuous aspect of nature has been forced upon us very much against our will. We seemed to feel quite happy with the continuum. Max Planck was seriously frightened by the idea of a discontinuous exchange of energy ... Twenty-five years later the inventors of wave mechanics indulged for some time in the fond hope that they have paved the way of return to a classical continuous description, but again the hope was deceptive. Nature herself seemed to reject continuous description …
The observed facts (about particles and light and all sorts of radiation and their mutual interaction) appear to be repugnant to the classical ideal of continuous description in space and time. ... So the facts of observation are irreconcilable with a continuous description in space and time …" (Schrödinger, 1951)
[I want to comment that we don't "feel quite happy with the continuum": we simply have not had any other formalism to compete with it.]
2. "As is well known, physics became a science only after the invention of differential calculus. It was after realizing [rather postulating] that natural phenomena are continuous that attempts to construct abstract models were successful. …
True basic [physical] laws can only hold in the small and must be formulated as partial differential equations. Their integration provides the laws for extended parts of time and space." (Riemann)
[But now it seems most likely that "the basic laws" do not "hold in the small", and hence the logic of the continuous model is broken]
3. "You have correctly grasped the drawback that the continuum brings. . . . The problem seems to me how one can formulate statements about a discontinuum without calling upon a continuum (space-time) as an aid; the latter should be banned from the theory as a supplementary construction not justified by the essence of the problem, [a construction] which corresponds to nothing "real". But we still lack the mathematical structure unfortunately. How much have I already plagued myself in this way!" (Einstein, 1916 letter, quoted from the paper by John Stachel mentioned in my Dec. 6 answer)
4. "Newton thought that light was made up of particles---he called them "corpuscles"---and he was right . . . We know that light is made of particles because we can take a very sensitive instrument that makes clicks when light shines on it, and if the light gets dimmer, the clicks remain just as loud---there are just fewer of them. Thus light is something like raindrops---each little lump of light is called a photon---and if the light is all one color, all the "raindrops" are the same size.
I want to emphasize that light comes in this form---particles. It is very important to know that light behaves like particles, especially for those of you who have gone to school, where you were probably told something about light behaving like waves. I'm telling you the way it *does* behave---like particles." (Feynman, QED, 1985)
Lev,
"All our mathematics has evolved under the assumption of continuity"
The natural numbers and aritmetics does not assume continuity but discreteness. It took a long time for mathematicians, the history of the invention of the real numbers, to find a way to express continuity in mathematics.
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I have an infinite set of functions F = {f1, f2, ...} mapping Rn into itself. I know that this set has a group structure with respect to composition, i.e. for every fi and fj in F there exist fk in F, such that fi * fj = fk (* stands for the composition). There is unique fe which corresponds to the group unity and for every fi there is inverse: fi * fj = fj * fi = fe.
I guess that this set of functions defines a Lie group, however I don't know the number of its parameters and the indeces have no topological meaning. Is there any way to find the number of parameters and to introduce them so that the family of functions F would be smooth with respect to those parameters? My first guess was to introduce a metric on F, but I don't know how to do that. Any help would be highly appreciated.
Dear Lee,
Take for a example a group of rigid transformations of a plane, i.e. translations and rotations. Imagine now that you have a set of this functions, say, a finite set of functions defined by the image of a regular grid of points.. Now I would like to determine the dimensionality of this set of functions and to parametrize the set I have, so that the parametrization would preserve the topology. For that I would like to introduce a metric on this functions and then use dimension reduction techiques, like isomap, to find the dimension of the corresponding manifold. This can be easily done for translations, but becomes less trivial for rotations. One idea is to define a metric based on a mapping of a bounded domain in the plane, e.g. rho(f1, f2) = ||f2*f1^-1|| + ||f1*f2^-1|| and ||f|| = max_{x in X} ||x-f(x)||, where X is a bounded domain in plane.
The problem with this definition is that I cannot be sure that there always exist such X that for every two different mappings the metric defined this way will not vanish.
Best,
Alex
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In some countries (including some universities in India) a "higher" form of doctoral work was awarded a D.Sc degree. It was considered in general to be "superior" degree to Ph.D. as it was presumed that the work was done by the candidate unaided by a supervisor. Are there similar programs in other countries?
William what you say may be true but it depends on the discipline. For example in Mathematics it is unlikely that one would complete a post-doctoral assignment at laboratory. It may not be even desirable that one do so. But this has little to do with
what was asked. In the US there is essential no post-Ph.D. program, such as the "Doctor of Science" degree. I expect as we continue to over produce Ph.D.s that
there will eventually be such a program. Nowadays in order to obtain a position (in Mathematics) after receiving a Ph.D. it seems that one must have had a post-doctoral assignment and a dozen publications. It was not the case when I fished
my degree 25 years ago.
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Math can generate material reality as follows :
1- Through geometry: in his book " The Shape of Inner Space: The Universe's Hidden Dimensions", Shing-Tung Yau shows how pure geometry can generate material universes
2- Proof by contradiction: why did a physical universe arise in the first place: break down the question into 2 separate questions:
2a- First, if there is absolute nothingness, i.e. if there had never been a universe the way we know it to begin with, could it (nothingness) have gone on unimpeded,
and
2b- Given that something now exists, or equivalently has existed at some point, can it ever be that full nothingness can re-establish itself at some point.
Tackling 2b first - There is no credible mechanism for full dissipation of what is (at any arbitrary point in time when something happens to be).
The only way to reach full nothingness would be to dilute everything to a fully vanishing point, but then because of infinitesimal residuals all the conditions extant to create at least one element term of the Heisenberg equation would be there - thus immediately giving rise to quantum foam, which would by definition not be pure nothingness. It's too late for pure nothingness to ever exist
Which leads us to 2a: Could there ever have been full nothingness.
Mathematics demonstrates that the Heisenberg relationship is inescapable in any universe, i.e. including a void universe (its ensemble of conjugate attributes being a perhaps infinite collection of 2-element sets). Heisenberg will however always give rise to 'quantum foam' - i.e., to something.
Therefore the question can be equivalently reformulated as : Can it be that abstract mathematics cannot exist independently of some material support.
Or equivalently: Can it be that abstract mathematics is not fully abstract. This seems to be a logical inconsistency, and therefore it must be answered in the negative.
Therefore, going back up the chain, we can safely conclude that pure nothingness cannot possibly exist.
Our Universe may be regarded as experimental proof
Question: is there any other mechanism, NOT reducible to pure math (e.g., not a quantum fluctuation), capable of generating a material universe ex nihilo ?
Louis,
You'd have to admit that this is probably not very satisfactory.
As far back as September 15th, 2007, Newscientist ran a cover story on the big and mounting body of evidence that math (or rather our human apprehension of it) constitutes *the* ultimate reality.
Since then, evidence has kept heaping up and become well-nigh overwhelming. There are very many 1st-rate Physicists out there (oft-cited Tegmark, who featured prominently in the Newscientist coverage, and many others) who take this view, whereas others take the view that math is but a language.
I believe that evidence for the former view is simply overwhelming (even Einstein had owned up to being baffled that nature 'obeyed' math so fastidiously), to address the subject properly however a book-length treatment is needed.
By the way - you can actually quite easily demonstrate that you love your dog - all you need to do is a MRI scan when you think of your dog.
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It is said that if you want to make high quality softwares than you must know data structures
Data are bases for getting information, and information is basis for making decisions - it can be considered as an axiom in data analysis. Only an appropriate structure of data can enable us to get information hidden in the data. Especially, data structures are relevant in situation of longitudinally collected data, e.g. in health care, where one patient can visit his/her doctor ones, twices or many times (or never). In such cases you have at least two entities you should consider: patient and visit (one patient, more visits, 1:n relationship). The database is then the appropriate data structure. In case you have only one entity (e.g. status of somthing (or someone) in a moment) the appropriate data structure is just a file, or table (e.g. made by using Excel). Data structuring is the crucial question in probvlem solving, in data analysis, for results which could be get by using either statistics or artificial intelligence methods.
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I am new to chaotic systems and have a question about Lyapunov exponents as a measurement for quantifying chaos. It is mentioned in chaos text books that positive Lyapunov exponent means chaos in the system. While this seems not exactly true, since for example an unstable system also can lead to positive Lyapunov exponent (other than positive eigen values). For example both logistic systems {x(n+1)=r*x(n)*(1-x(n)) with r=1.9} and an unstable system {for example x(n+1)=r*x(n) with r>1} lead to positive Lyapunov exponent.
Can anybody explain the difference between these two? Is there a better measurement tool than lyapunov exponent for chaotic systems?
Hello Mohsen:
I believe the short answer is "No". As reflected in many of the other posted responses, positive Lyapunov exponents, by themselves, do not always indicate "chaos". Additional information about the system, and / or texts need to be performed to conclusively diagnose "chaos" in most systems.
The Lyapunov exponent itself merely quantifies the degree of "sensitivity to initial conditions" (i.e., local instability in a state space). Such local instability can arise for a variety of reasons in different types of systems. For example, in nearly all experimental systems, where Lyapunov exponents are estimated from recorded time series data, these data include some level of stochastic noise (which itself might include measurement noise, as well as system noise, or both, etc.). Noise, by itself, will create "sensitivity to initial conditions" that can trigger findings of positive Lyapunov exponents even for systems that are not remotely "chaotic".
For a more detailed discussion, I wrote a book chapter on Lyapunov exponents a few years ago:
You can download a PDF of the chapter from the "Publications" section of my web site:
(Scroll to the bottom to see the link)
This is a complex issue, as indicated by the many responses already posted. So the book chapter is in no way intended to be "exhaustive", but I hope it is informative.
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Curvature is the main determinant of ripening processes in nanoscale materials. Morphologies with mimimal and constant mean curvature can be ripening resistant.
The sign of the mean curvature has no intrinsic meaning - it depends on your choice of unit normal to the surface. Consider for example a sphere. If you choose the inward normal then the mean curvature is positive while if you choose the outward normal it is negative (and constant in both cases). Does that answer your question, or have I not understood properly?
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I'm working on a problem where I would need an error correcting code of length at least 5616 bits (and preferably not much longer) that could achieve correct decoding even if 40% of the received word is erroneous. I have looked into some basic coding theory textbooks, but I have not found anything that would suit my purpose. Does such a code exist? If it does, what kind of code is it and can it be efficiently realized? If it doesn't, why not?
Any insight to the problem will be much appreciated.
I agree with Patrick. 40% correction if errors are random cannot be achieved for that length. If errors are not random, that is a different story, but in that case we need a model for the channel (like a bursty channel), as other people are pointing out.
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The approximate solution that satisﬁes the boundary conditions is:
The below equation should be minimized:
So we have this integral:
Initially, for no deﬂection, for w = 0, the value of λ from the above equation, is found as:
I can not understand, how λ is calculated.
This is the qustion, how λ is calculated?
Hi,
I think, the approach, which has been shown in the document, is called the weighted residual method or the weak form of the strong form ( i.e. the partial differential equation (PDE) ). Generally, the weight (or W(x,y) which is multiplied in the left-side of the PDE as shown in the document) and the domain of integration, which means -a to +a, can be chosen in different ways or arbitrary, by regarding some limitations on the smoothness of the weight function. The most general approach is called Local Petrov-Galerkin method. The simplified, and maybe the most common, version is called Bubnov-Galerkin or Galerkin method. Noting that the function W(x,y) has been chosen the same for trial and test functions in this solution, the approach is the Galerkin method. As a result, you should apply the integration by part technique to this integral equation and extract the lambda value. In other words, the divergence theorem should be applied to this integral, as you may be familiar with. Just be careful when you define the boundary conditions for the boundary integral which stems from the divergence theorem. I hope this would be helpful.
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Boolean Function simplification required (to minimum no of literals)!
F=w'x(z'+yz)+y(ww+w'yz)..?
F=w'x(z'+yz)+y(ww+w'yz)
=w'xz'+w'yz+yww+yw'yz
=w'xz'+w'yz+yw+w'yyz
=w'xz'+w'yz+yw+w'yz
=w'xz'+w'yz+w'yz+yw
=w'xz'+(w'yz+w'yz)+yw
=w'xz'+w'yz+yw
=w'(xz'+yz)+yw
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While demonstrating importance of infinity he gave an example of a hotel with infinite rooms. This Hotel is called Hilbert’s hotel. One night a guest, who is a mathematician, appears in the reception of the hotel and requests for a room. The receptionist tells him that all the rooms in the hotel are full. Generally hotel room numbers are natural numbers and start with 1, 2, 3 etc. The mathematician guest thinks for a while and tells the receptionist about possibility of giving him a room without asking any of the occupants to leave the hotel. Receptionist wondered and asked the mathematician to explain. The mathematician says that it is simple. You can ask occupants to shift to next room. For instance you can shift occupant of room no. 1 to room no. 2, room no. 2 to room no. 3 ….so on….and occupant of room no. (n) to room no. (n+1). Give me room no. 1. Since there are infinitely many rooms in Hilbert’s hotel this works.
Now, what is two guests appear in the Hilbert's hotel which is full and ask to get accommodated without asking previous occupants to move out?
What if a group of (n) guests appear in Hilbert's hotel which is full and ask to get accommodated without asking previous occupants to leave the hotel?
The occupant in the room no. 1 will be shifted to the room no. n+1, the occupant in the room no. n+1 will be shifted to the room no. n+n+1=2n+1 and so on. The occupant in the room no. 2 will be shifted to the room no. n+2. The occupant in room no. n+2 will be shifted to n+n+2=2n+2 and so on. The occupant in the room no. n will be shifted to the room no. n+n=2n. The occupant in room no. 2n will be shifted to room no. n+2n=3n and so on. Thus the first n rooms will be vacant and the n gusts could be accommodated.
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We are working on coupled fibonacci sequences of higher orders.
Applications:
1-Biology
Some of the best-known difference equations have their origins in the attempt to model population dynamics. For example, the Fibonacci numbers were once used as a model for the growth of a rabbit population.
2-Digital signal processing
In digital signal processing, recurrence relations can model feedback in a system, where outputs at one time become inputs for future time. They thus arise in infinite impulse response (IIR) digital filter.
3-Economics
Recurrence relations, especially linear recurrence relations, are used extensively in both theoretical and empirical economics.
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Binomial Theorem.
As I understand it, you are asking for an expression with respect to the number of permutations of n items taken r at a time; i.e., your index should be r instead of k because otherwise it is not bound in the expression ∑ⁿᵣ₌₀P(n,r) xʳ a⁽ⁿ⁻ʳ⁾ that is analogous the Binomial Theorem (x+a)ⁿ = ∑ⁿᵣ₌₀ (ⁿᵣ) xʳ a⁽ⁿ⁻ʳ⁾. (I'd love to be able to use Mathjax to present a better expression, but Unicode will have to do for now.)
P(n,r) = (ⁿᵣ) r! = (n)ᵣ, also known as Pochhammer's symbol, or falling factorial is the coefficient of each term xʳ a⁽ⁿ⁻ʳ⁾. Hence, (x+a)⁽ⁿ⁾ is the simplest closed-form expression that I can think of, where the exponent n is replaced by (n), with the subscript r removed (and understood as the falling factorial in context), as it is the index for the coefficients P(n,r).
The first four polynomials are:
(x+a)⁽¹⁾ = (1)₀x+(1)₁a = x + a
(x+a)⁽²⁾ = (2)₀x²+(2)₁ax+(2)₂a² = x² + 2ax + 2a²
(x+a)⁽³⁾ = (3)₀x³+(3)₁ax²+(3)₂a²x+(3)₃a³ = x³ + 3ax² + 6a²x + 6a³
(x+a)⁽⁴⁾ = (4)₀x⁴+(4)₁ax³+(4)₂a²x²+(4)₃a³x+(4)₄a⁴ = x⁴ + 4ax³ + 12a²x² + 24a³x + 24a⁴
BTW, the Umbral Taylor series would give the same expression, but the overarching theory that abstracts all of the above would probably be related to generalized hypergeometric, Fox-Wright, and elliptic hypergeometric functions. If the recent proof of the ABC conjecture by Mochizuki holds, then we can expect much deeper connections to surface.
- Marshall Mayberry
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I am defining algebra over smooth functions.
So what is the importance of leibniz rule over there?
I hope not to sound to technical, nor abstract, but i want to recall another way of looking the chain rule that turns out to be powerfull and "algebraically elegant". From the point of view of "abstract nonsense" (i.e. category theory) the chain rule implies that the derivative behaves as a covariant functor. Let me explain this. First, a category is made out of objects and arrows between them (and that satisfy certain properties). A functor is the categorical analog of the concept of function in set theory, namely, a functor assigns to every object in a category (domain) an object in another "range" category, and to each arrow between a pair of objects (tip and tail for short), it assigns another arrow that corresponds to the functor's value on the the tip and tail of the original arrow. One says that the functor is covariant if preserves the orientation of the arrow and contravariant otherwise. (sorry for that digression).
Once said that, you might be asking, what kind of objects constitute the category on wich the derivative is a functor. Well, objects are smooth manifolds (finite or not in dimension) and arrows are called differentiable functions. This way of looking derivatives is too powerfull, for example, it allows to compute certain things in a coordinate free fashion. This is desirable when one has manifolds with complicated coordinate descriptions or simply for doing general proofs without too many computations.
Greetinga
Raúl
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I tried some ready package as Mathematica and Maple but the results were not satisfactory
Thank you indeed dear Prof. Singh. I'll check this reply.
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Newton introduced differential equations to physics, some 200 years ago. Later Maxwell added his own set. We also have Navier-Stokes equation, and of course - Schroedinger equation. All they were big steps in science, no doubts. But I feel uneasy, when I see, for example in thermodynamics,
differentiation with respect to the (discrete!) number of particles. That's clear abuse of a beautiful and well established mathematical concept - yet nobody complains or even raises this question. Our world seems discrete (look at STM images if you don't like XIX-th century Dalton's law), so perhaps we need some other mathematical tool(s) to describe it correctly? Maybe graph theory?
One of my math professors used to dream about the universal differential equation that would describe everything in the universe. This notion is of course utterly ridicoulus, as present modern theories gives us computable limits on predictability. But the Liebnitz ideas die slowly.
Math is in my opinion best wieved as a modeling language that enables us to make models of reality which allow us to extrapolate and make predictions in the physical world. The models have to be simple enough to make a mathematical solution possible, and this in turn sets limits on how far the model can extrapolate known physical results.
Wether a differential equation is the best model depends on the real world phenomena that it is going to model, It is going to describe reality with varying accuracy, but never fully. But it may or may not exactly describe the simplified description of reality made. Sometimes other models are better descriptions of reality. For large ensembles of particles, such as gases, differential equations are quite good. The 3 body problem on the other hand is a classical example of a seemingly simple problem that is very dificult to solve.
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Is there any relationship between Walsh functions and rationalized Haar functions?
Can we express a Walsh functions in terms of Haar functions?
Dear Lakhdar Chiter . The answer yes, becous the Rademacher functions defined as some sums of Haar functions. More you can find in the book " Kaczmarz, S., Steinhaus H. Theorie der Orthogonalreihen "
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Consider the basic arithmetical operations of addition, subtraction, multiplication and division.
Can the sum H_n = 1+1/2+1/3+..+1/n be expressed in number of these basic operations that does not depend on n ? Supposedly this can not be done but what is the proof ?
For example the sum:
S_n = 1+2+3+...+n with n-1 additions can be expressed compactly by Gauss formula as n*(n+1)/2 where the number of operations is three and (so it does not depend on n).
I need a similar thing for H_n, or a proof that this can not be done.
The question is not sufficiently precise: What are the possible inputs of your computation? If anything can be the input, then one can compute H_n with no operations from H_n. If only n is allowed as input (which your example suggests), then the answer is no, because H_n is no rational function in n. This is clear by Edward Smart's remark, since H_n grows logarithmically, and no rational function does.
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I am trying to cover a sphere (surface of a ball in 3D) with identical unilateral triangles. At first step an octahedron is inscribed into a (unit) sphere. Its faces make the first generation of 8 triangles, obviously unilateral. Next generation triangles are created as follows: centers of edges of selected "old" triangle are found and projected onto the sphere. This replaces an old triangle with 4 new, smaller ones. The procedure is repeated for all "old" triangles, producing in effect the new
generation of triangles. Unexpectedly, new triangles are no longer unilateral, only 2 edges are of equal length (after, say, 5th generation is created). The observed length differences, of order of 6-10%, seem far too large to blame rounding errors as their source. So, what am I doing wrong?
All calculations are done exclusively in Cartesian coordinates.
Are you trying to construct a Platonic solid with more than 20 triangles? This is provably impossible :) (http://mathworld.wolfram.com/PlatonicSolid.html)
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The equations are like this: 1-X=exp(-A*Y^B)
I need to derive the parameters A and B and use the same model to predict the values, which should be compared with the experimental observations.
The solved equation is: Ln (X) = {Ln(-Ln(1-X))-Ln (A) }/ B
Linear regression applied to transformed equation only looks easy. In fact, any nonlinear transformation introduces following undesirable effects:
1. when the original data are not correlated then their transformed version may not enjoy this property,
2. experimental uncertainties are usually symmetric around measured values; after non-linear(!) transformation this property is lost.
The lack of correlations and symmetric (Gaussian) distribution of of uncertainties are the main assumptions on which the linear regression is based.
The two features of nonlinear transformations, shown above, conspire to deform the desired result. In some cases also rounding errors are the source of severe distortion of the results obtained this way (think of functions like tanh(x) or 1/x computed for large |x|).
Conclusion: linear regression on transformed data often produces only very crude approximations to true values. Nevertheless, such results are useful as a start point for more advanced (nonlinear) fitting routine, but should never be regarded as final ("best fit").
The phenomenon you observed was extensively researched for the data fitted to power law (y ~ a*x^b), often encountered as the tails of various distributions Zipf's law, for example). Here, only the parameter 'b' is interesting but its value appears very unreliable after 'linearized fit'.
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In terms of computation, what is the difference between a diagonal matrix (for which the nonzero terms are in the diagonal: tridiagonal, pentadiagonal, ... and all the other elements are equal to zero) and a hollow matrix (for which almost all the elements are null exept a few ones?
Which is the best matrix to use for computation? Which one performs well for calculus?
Rather than saying "hollow", the most common term is "sparse matrix". If you look for this in Wikipedia you'll find lots of information.
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y= -x/(a^2-x^2)
what is dy/dx?
where a is a constant
Given how unlikely it is to get a useful explanation here, I suggest just using www.wolframalpha.com for cheating instead, you would've gotten Pathak's solution with for instance "dy/dx of y= -x/(a^2-x^2)" or "derivative of y = -x/(a^2-x^2)". An added plus is that it won't judge you for being too lazy to execute the straightforward derivation algorithm yourself. And also that it's very likely to be correct, so you don't have to rely on majority voting.
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What kind of problems we can solve with this numbers?
The use of prime numbers in addition to their practical application in cryptography, has big contributions in the development of mathematics in abstract form in number of fields in algebra and analysis, where irreducible factorization is of great importance.
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Design and sample surveys.
Coding theory and Cryptography etc.
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What is a genetic algorithm?
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Can you suggest some (real-life) applications of Computable Analysis?
Why don't we just use classical analysis?
Dear Henk,
As much as I would like to agree with you that computability in analysis has no application whatsoever, I feel the need to think about it further, thank you for that.
Here is a question to ponder: What is the use of recursion theory in mathematics ?
I think the answer to this question will illuminate the need to do "computable mathematics". We CAN do without computability of course, to answer my second question.
But for me the answer is this: It is not about the computable (this is the easy part) but about the UNCOMPUTABLE.
For me, when something is uncomputable, the road does not end there, it is a statement about what we should do next in mathematics.
It means that every time you encounter a problem of this uncomputable type, you must invent a new algorithm for it (or if you wish, a new theory).
Simply put, there is NO one algorithm to solve them ALL.
And this is an opportunity for mathematicians to invent new methods, add new assumptions, invent new theories, new types of objects, etc. for solving each instance of this problem.
Let me give you an example:
The Halting Problem is undecidable right ? Yes, it is.
But can we restrict the type of programs that we consider ? Again, the answer is YES.
There is certainly is a subset of the Halting Problem that is decidable (for example a set of all programs without any loops), in fact there are many such subsets.
Each of them gives a new ground for mathematical investigation how to solve this "restricted" Halting problem.
So, to answer my first question, computable analysis is actually recursion theory in disguise, telling us what parts of analysis are computable and what are not. With computable things we already know how to deal, and the "uncomputables" are actually an invitation to deal with them (usually each being a new problem by itself) and make future progress in mathematics.
Kind Regards,
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Lot of discussion "what x is used to denote an unknown?' Lot things are supposed hypothetically before their existence is proved, so we have to name the 'unknown' to prove its true existence and let it be x (or else).
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The work studies the distribution of primes via multiplication modulo n where n is a primorial. Instead of studying the set members I examine the member spacing and show it meets the Hardy Littlewood gap conjectures. It seems straight forward to me but I would like to know if it is actually correct and worth further pursuit. Thank you in advance.
Yes, with the condition that n is a primorial
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Can a non-constant function V exist such that for every N-tuple t there exists a weight w such that V(t, w) is at least as great as V(j, w) for every other N-tuple j?
The idea for an application of such a function is for devising a system in which every "applicant" (represented by an N-tuple) has at least one instance (represented by a weight) where they are a "best choice." In short, it would be useful in a "no losers" system.
If there is nothing more, define V(j,w) = - | w - ||j|| |, where for a given t you will choose w = ||t||.
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y = 1=(x²+c) is a one-parameter family of solutions of the fi rst-order Di erential Equation; y+ 2xy² = 0. Find a solution of the fi rst-order Initial Value Problem(IVP) consisting of this di fferential equation and the given initial condition. Give the largest interval I over which the solution is de fined. y(2) =1/3
Hii Adrien........ The largest interval over which solution is defined is (-infinity,-1) U (-1,1) U (1,infinity). As the solution of This IVP is y=1/(x^2 -1), so the only points where it is not defined are x=-1 and x= 1. So the interval is whole real line except these two points.
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I am a student of Computer Science and I'm just loving the story on Linear Algebra.
I wonder if there is a good indication of books regarding this subject?
Linear Algebra by Gilbert Stang ,,,,,,,,,Read online MIT LECT. NOTES
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What is really a fractional derivative? When can we affirm it is? Shouldn't we require that they recover the classic when the order becomes integer?
The word fractional appears in a lot of contexts. It became like a fashion. It is clear that Grunwald-Letnikov, Riemann-Louville, Caputo, or Riesz are fractional derivatives, but it is clear for me that the so called fractional Fourier transform is not really fractional. And like this one there are others. Personally, I believe that the most correct way of going into the fractional derivative is the Grunwald-Letnikov, because all the others can be deduced from it. Besides it is the most suitable for numerical computations.
The sixth FDA workshop, FDA'2013 (FDA : Fractional differentiation and
its Applications) will be held in Grenoble, France, February 4-6, 2013
in the SSSC joint conferences (Track D of this conference). All the
important dates of this workshop are in the attached file Papers
submission deadline June 15th, 2012) . As for all IFAC events, I remind
that all the papers will be peer-reviewed. The reviews will be done on a
6 pages draft (full paper, no abstract).
Invited sessions within the technical scope of the conference are also
solicited. These proposals should contain six (6) full papers, or five
(5) including a survey paper limited to 12 pages
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Does anyone have any recommendations for programs that evaluate the Lyapunove exponents for time delay dynamical systems?
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Does anybody know some concrete applications of matrix factorizations in Computer Science?
Exist some applications with images processing. For instance, eigenfaces and images compression. You may to search this in Google. You should find some articles.
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Does anyone know how many different versions of the Non Negative Matrix factorization exist and what they are ? I did a Google search, but couldn't figure out any definitive answer.
>> Nonnegative matrix and tensor factorizations : applications to exploratory multiway data analysis
and blind source separation / Andrzej Cichocki . . . [et al.]. 2009 John Wiley & Sons, Ltd
Maybe you may find that you want in this book.
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Suppose {an}, {bn} are convergent sequences of real numbers such that an > 0 and bn > 0 for all n.
Suppose lim an = a and lim bn = b. Let cn = an/bn.
n -> oo
Then which of the following is/are correct?
1. {cn} converges if b > 0
2. {cn} converges only if a = 0
3. {cn} converges only if b > 0
4. lim sup cn = infinity if b = 0
I guess (1) is correct. and (2), (3) are not correct. Am I right? and What about option (4)? please explain.
1. true, by algebraic limit theorems
2. false. Counter example: let a_n = b_n = 1 for all n so that a = 1 and b = 1 as well. Clearly c_n = 1 for all n as well so that lim c_n = 1.
3, 4. false. Counter example: let b_n = a_n = 1/n, so that b = 0. Then c_n = 1 for all n, so that limsup c_n = lim c_n = 1.
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How the no of generator relate with the order of a cyclic group?
An ifinite cyclic group has only two generaters and a finite group of order n has phi(n) generators
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I have a number of Lorentzian curves ("reference curves"), all having the same half width at half maximum, and I know each curve's location parameter. However, the reference curves are multiplied by various different factors, so most heights will differ. I would now like to calculate those factors from the sum over all curves.
I have already written code that performs the calculation; it uses the sum's value at the location of each curve's maximum, and each reference curve's value at this location. The method I'm employing is singular value decomposition, and it seems to work fine. I would, however, like to know whether this approach makes sense, and whether the task can really be solved in this way in all cases, or whether I might just have been looking at a few examples where it happened to work (that it doesn't work when curves have the same location parameter isn't a problem).
I am not a mathematician, so I don't know how to tell whether this seemingly sensible approach is actually misguided. Also, in case you've been wondering, I'm aiming to use it for working with NMR spectra (where I'll also have to deal with random noise, and non-Lorentzian curves), but I would like to check whether the theoretical approach isn't wrong-headed.
This is a classic linear estimation problem for which SVD will give the classic solution.
$y_j = n_j+\sum_i c_i f_\lambda(x_j-x_i)$
where $$n_j$$ is a white (uncorrelated) random noise variable. Rewrite this as a linear matrix equation
$y = Ac + n$
The least-squares estimate minimizes $$||y-A\hat{c}||$$, which SVD can compute for you.
If you think your noise is correlated you must estimate its correlation matrix. It's likely your "noise" won't be a random process at all but model error (wrong curves, wrong location parameters, etc). Estimating location parameters and widths from measured data will be a non-linear optimization problem that SVD won't solve.
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Domination graph theory is the most popular topic for research.
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I know parallel lines can't touch but what if they share all of the same points? Just wondering if there are any exceptions.
@Deepak Anand: The problem is the use of the word "Euclidean". This implies many things, including Euclid's 5th Postulate wherein parallel lines do not intersect. If you want to investigate other geometries (there are many) in which there are no parallel lines or there are many parallel lines, all passing through the same point. These are important and just as valid as Euclidean Plane Geometry. They just have different assumptions and contexts. You might want to do a Google search on Vanishing Point or on Projective Geometry which are probably relevant to the point you would like to make.
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Few examples are FFT,wavelet ect
See the transforms and applications handbook
crc press and ieee press
all the books on signals and systems have several chapters on transforms
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Can anybody suggest me the problems where research could be done in the field of Linear Integral equation.
Thanks.
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Once you understand what PvsNP problem is actually all about, you might as well try and solve it.
In loose terms, the P vs. NP problem actually seeks an answer to this simply stated question:
"Is finding a solution to a math problem equally hard in comparison to verifying that it IS a solution ?"
Math guys usually "search" for a solution to their problem (e.g. solving some equation), but this can apply to "searching" any data set.
Imagine a program that searches for a solution to some equation. That program will most certainly consist of two major parts: a searching part (the solver) and a verifying part (the verifier). The solver tries to construct a solution by some rules and a verifier checks that it actually *is* a solution.
This solution constructing part is like when you do all sorts of manipulations (factoring, cancelling common terms, ...) to solve an equation, and this verifying part is more like when you plug in some values for your solution back to the original equation to check if both sides turn out equal.
The first part will usually take up much time, as finding a solution to some equations is sometimes hard, but once the right solution is constructed, the verifier will take only a fraction of that time to check if that actually IS a solution. The PvsNP asks if those two parts are actually the same thing, because it would be nice of course, that solving an equation is as easy as checking the result.
Another way to look at it, it's basically a question about searching trough (potentially large) sets of data. In that context the PvsNP asks this:
"Is there a systematic way of searching trough a large data set ?"
(a large data set means for example, a data set not completely searchable in the course of one persons lifetime, for example the whole Internet)
Of course, people have been trying to answer this for decades ever since the computer era started, but with no luck, in my opinion because of the way the final solution needs to be presented.
It is widely believed that P is not equal to NP, because otherwise it would have baffling implications for say cryptography and code breaking. As there is a huge number of potential passwords that one can make up, a positive answer to PvsNP means that a brute force search is not necessary when trying to guess someone's password and there is also a systematic way how to obtain it. On the other hand, if P is not equal to NP than it means that there is no such thing.
Also, in this digital age, when almost everything is stored on a computer (music, pictures, texts, ...) if P = NP is true then we could generate any piece of music, any picture, anything ... by means of a computer program that would solve P vs NP, we just "search" for it, provided we have a computer program that recognizes that something is "a piece of music".
Finally, the PvsNP can be restated in terms of creativity as: "Can creativity be effectively automated ?"
The hardest thing about solving the problem is actually proving that either case is true. There are of course up till now many false starts and dead ends, and people today that are still trying are trying to prove that in fact P does not equal NP. Richard Karp, one of the most renowned computer scientists once said that this problem will someday be solved (either way) by someone under thirty using a completely new method. So, until then, you might try and solve it for yourself.
What if it's only solvable as long as the solution is known to exist? Can't run a successful search for music that doesn't exist. I think the only factual solution would be (P=NP) as long as (P-NP>0).
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Pls mention the book names
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I have three data sets,A,B and C.A is dependent on B and C therefore has (C*B number of data points).
Do you have some references on how i can do this in matlab,excel or mathcad?
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I have these equations with parameters. I have to plot graph for numerical solutions
Thnx.... I have the trial version. I m working on models , so i need the graphs to see the stability directly with given parameter values in the system of 3 equations stated earlier.
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C= inv( Trans(A) * inv(B) * A )
A is a rectangular matrix, and B is a large square matrix.
You can indeed avoid the calculation of inv(B) if some orthonormality conditions hold among the columns of A. See "Moore–Penrose pseudoinverse".
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Assume the integral of "z*h(z,p,q)" over all values of the scalar z is equal to that of "z*h(z,p)", where both of the scalar-valued h(.) functions respectively integrate to 1 taken over all values of z. So then, is this true iff h(z,p,q) = h(z,p) for all z, p, and q? (Bonus points if you can also let me know the same for discrete z.) Thanks in advance!
Before giving bonus, first you have to decide if h is a function of 2 or 3 variables.
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Here y can be assumed as a function of any independent variable ! I want the differentiate its present value of f(x(n)) with respect to its past value f(x(n-1)
Sir i was working with neural network where i encountered such a problem when i backpropagated the error ! I did find out that my calculations were wrong ! I am deeply sorry that i had troubled u all this much ! I figured out that i never encountered the above situation(my question) ! Thank you all !
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Number theory and computer science question
Here is my interpretation of your question:
"Continued fractions are very useful and as a novice I'm very impressed playing around with them. Hey, we can find patterns even on pi, wikipedia says! And periodic expansion of sqrt(2) is a miracle! Why nobody shares my enthusiasm?"
Correct?
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What are the various fields
John, so if you are suggesting that institutions/ academia have been fooled for so long in the area of calculus does that mean that all the actuaries around the world and the statistician who use calculus in their modelling have been coming up to wrong conclusions overall, thus resulting in wrong hypothesis.... and the scientific research modelling and statistical analysis carried out in healthcare sector was predominantly wrong????? please explain this
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The following list includes free math software and tools together with the corresponding descriptions and download sites.
Operating systems:
Scientific Linux: A linux distribution put together by Fermilab and CERN. Freely available from
Ubuntu Linux: A Linux distribution, easy to install and freely available from
Debian: Perhaps the best Linux distribution.
DesktopBSD:
A freeBSD distribution easy to use which can be tested through a live DVD. Freely available from
BSD: Several Unix distributions.
Applications for symbolic calculus.
wxMaxima:
Calculus with a graphic interface. Freely available from
Axiom: Similar to the preceding one.
Euler: id.
Scilab: id.
octave: id.
Gap: Computational discrete algebra,
R: Statistics
PSPP: Statistics
haskell: Pure and lazy functional programming language with an interpreter.
Astronomy:
Stellarium: Free astronomy appl.
Star charts: Free star charts PDF files.
Math graphics:
Gnuplot: To build any graphic in 2D or 3D. Freely available from
DISLIN: A graphical library, easy to use.
Word processors:
TexMacs: WYSIWYG editor with a graphical interface, by means of which one can type scientific texts, and export them in PDF, PS, HTML, LaTeX formats. Freely available from
Lyx: Similar to the preceding one.
Miktex: A complete LaTeX distribution for Windows. Freely available from
TexMaker: A LaTeX editor: Freely available from
TeXniccenter: Another powerful LaTeX editor for Windows OS.
Kile: Another LaTeX editor.
TexShop: A LaTeX distribution and editor for Mac OS X. Freely available from
Texlive: A LaTeX distribution for Linux and Unix OS'.
Open Office: A package similar to Microsoft Office:
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Write what you want
Dear Hanspeter,
It is the metamorphosis of another that degenerated.
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I have prepared a paper on Graph theory but I don't know how or where to publish it.
Whoever it may concern I am N V Nagendram working as Assistant Professor in Mathematics and i got published 14 papers in the academic year 2010 - 2011 after continuous and constant work done in Near Rings under Algebra of Mathematics. please visit lbrce.ac.in CSS Dept. Faculty / publications you can have my profile.
For your answer after writing an article to how it is to be sent ?
Ans: please go through international journal for graph theory topic you can find many journal titles go through each and every journal inside till you get the publication on your specialization of topic if you find then you goto submission online from there only. it will ask you title, author, co-author if any, Abstract , Key words and Subject specification/classification code mention thereof. upload your paper in the form of either MSWORD / LATEX or required form there mentioned by publisher.
So for this you can not ask every time to where my paper / article to be sent what you need to do is on regular basis you must have to had habit of reading journal by opening site international journal of topic name then you will find and read increase your reading capacity about journals. it is a good habit for a researcher on any topic. ok!
Now i am going to give you one journal name here you please send this article to that journal.
"International Journal of Mathematics Archive (IJMA)." or "International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN(Print):2249-6955 ISSN(Online): 2249-8060 " like this you have to thorough with net web site on your selected topic. bye yours .............N V Nagendram
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I know here f '(1) =0. I found in some texts if f ' (x) = 0 for some x then we can't apply N-R method. Is there any other technique to find first approximation for x^3-3x+1 = 0 taking initial approximation as 1? Which is correct ? a) 1 b) 0.5 c) 1.5 d) 0
write it as x=1/3+x^3/3. Since the derivative of the rhs is <1 for all |x|<1, it follows that the repetative method converges. So peak any x in (0,1)-{0}. Btw, the three roots are -1.87939, 0.347296 and 1.53209. So guess what happens for these three initial values of yours.
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Notice, that if f:K --> M is an injective map which can be defined by a finite statement, then
for every y in img(f) there is an x in K satisfying the relation y = f(x), which can be regarded as a definition for y. Thus, either both x and y can be defined by finite statements, or no one of them can be defined finitely.
Dear Carrol
Of course, a rational number must be always defined by means of a finite sequence of figures. For instance in your example 0.0999.... the dots in the tail means that the following figures are constantly equal to "9". If by means of the dots ...... you do not denote that the following figures are equal to "9" it is not possible to guess whether the 1000-th figure can be "5" or "9".
If I impose the rationality either the number must be defined as a quotient of two integers n/m or in decimal expressions the mantissas must contain a repeating pattern. If you do not define the pattern, the number is undefined.
The requirement of being rational imposes to be defined as a quotient or by means of a repeating pattern. In both cases is possible to be identified by a finite sequence of figures,
I am tired explaining such an elementary questions. I have under consideration to ignore others similar questions.
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Moments Descriptors are invariant under RST ( Rotation, Scaling, Translation) in PR.
Is Moment Descriptor almost invariant under Optimization?
Is Moment Descriptor almost invariant under Guassian Noise?
Can the reduction of pixels to draw an optimal shape of a planar curve shape using mouse on computer affect recognition rate? If yes then upto which extent ?
I'm not sure about your question, but an important factor if you are considering using moment descriptors is to make sure that moments exist in your domain. (For example Lorentzian curves, which are common in my work, have no defined moments).
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Hi i send you one problem.
If for (4) you don't need that it is 'strictly' increasing then you can just use Q(t)=t/3
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(a+b) (a+b) = a2+2ab+b2
(a+b) (a+b) = a2 +ab +ba +b2 for the non commutative case.
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May i know a book which gives a basic results or informations about Matrix theory?
- Matrix Theory, Joel N. Franklin‏
- Elementary matrix theory, Howard Whitley Eves
- Introduction to matrix analysis, Richard Ernest Bellman
- Matrix theory, James M. Ortega
- Matrix Theory, David W. Lewis‏
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For instance, pi = 3.141592... can be defined as the ratio between the length and the diameter of a circle. Any integer can be defined by a finite sequence of figures, and so on.
Dear Steven,
Every mathematical paper contains always some definitions and notations introduced by the corresponding author. In general, these definitions must be only considered in the paper context. Unfortunately, there are a limited symbol set to be used, and this fact oblige us to term different objects by the same symbols. This inconvenient does not matter whenever the author takes care of defining them.
The great french mathematician Henri Poincaré says: "Mathematics is the art of denoting different things by the same name". Of course, he was thinking in equivalence classes and analogies. Analogies are also particular cases of equivalences.
The father of normed spaces, Banach, wrote the following:
A mathematician is a person who can find analogies between theorems;
a better mathematician is one who can see analogies between proofs
and the best mathematician can notice analogies between theories.
One can imagine that the ultimate mathematician is one who can
see analogies between analogies.
(Stefan Banach 1892 - 1945)
Best regards.
Juan Esteban
P.S. I have sent you my paper about Cantor's theorem via e-mail.
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I would like to send my publications to this publisher. I would like to ask if someone has an experience with this publisher. Thank you!
Thanks Zafar for your response! I am not sure but it seems it publishes dissertations and monographs. What is your experience with Lambert Academic Publishing, is it well-known?
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Homotopy continuation method provide an useful aproach to find zeros of nonlinear equation systems in a globally convergent way. Homotopy methods transform a hard problem into a simpler one and solve it, then gradually deform this simpler problem into the original one. I usually solve equations from nonlinear circuits: diode, bipolar, MOS. Now, i want to solve other kind of equations with applications, specially if the equation is multivaluated. ¿Somebody want to collaborate with me?
Yeh we can work together
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The symbols we use in mathematics to form equations are just an aid in clearly forming an argument and communicationg it to others. We are clearly restricted when we use this formal language. If we could only cast out any mention of this language and symbols when doing mathematics, then we would be on the right track in truly understanding reality's ways.
The notion of quantity, form, change, space, shape, order, etc. are all independent of their symbolic representation. The language can easily change trough time, but these notions will not.
Computation as we know it, is merely a formal manipulation or transformation of symbols. It can be done by hand or by a computer. Either way, there is always a notion of a conciever and an executor present, when talking about computation. These two are usually one and the same, but I like to think about them as separate entities. The executor, follows a fixed set of rules to transform given string of symbols, that a conciever has conceived having some end goal in mind. The executor blindly follows these rules and eventually, (if he's in luck and didn't get stuck somewhere blindly following the rules),he will get a transformed string of symbols representing the final result.
And the conciever is the one that anticipates this result, again as a string of symbols.
So, when doing computation, the main assumption is that, when we manipulate symbols, we manipulate the notions that they represent. Just like in the primitive times, when people practiced magic, they believed that the symbols they use in their spells represent objects from the real world.
They believed that drawing these symbols in some special sequence will result in a spell being cast, and as a result something in the real world will change according to the spell's intention. So, in an amusing way, doing mathematics can be regarded as "doing magic", not in the real world, but in the world of ideas.
Computers process strings of symbols by following a fixed set of rules that we call a program. The conciever is the programmer, and the executor is of course the computer. The processing by a computer is usually done in a one-by-one
fashion, but is much faster that doing it by hand. Computers can be seen as manipulators of symbols, or executors of programs, but the acctual thing we are after is the "manipulated" idea after the computer has done millions and millions of manipulations on it (that would be too tedious to do by hand).
So "ideas" are the ones that we are after when doing computation, because we hope that this mechanical grinding away of symbols will tell us something new and interesting about reality and nature, although this point of view was refuted a hundred years ago by Godel's famous incompleteness theorems. These theorems show that there is definately something more to mathematics and computation than just "symbol grinding". Remarkably, Godel showed this using only using some basic facts from NUMBER THEORY, nothing fancy.
And what about nature and reality ?
What are nature's rules, and what "language" is used to set these rules ? Nature is the executor, but who is the conciever ? And what is the final result ? Is it LIFE maybe ?
The answers to these questions are certainly beyond human comprehension, but there is, as always a lot if speculation about it! But, when we finally find this out, only then we can make a significant progress in truly understanding this "manipulation of ideas" notion and and "reality's ways" in general that mathematicians are still desperately and vaguely trying to capture by the notion of "computation".
Dear Alana, thank you for your comment :) it's always nice to see someone from high school reading and commenting posts on Research Gate, the youth is our future :) Cheers
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When I read about Fourier transform, there are several definitions about Fourier transform. It is because there are several conventions about it. Which kind of definition should I refer to? Because it is a little bit confusing.
Thank you, Henk Smid.
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Like hyperbolic and circular trigonometric functions can we able to generalize trigonometric ratios with respect to a general curve?
Thanks......
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Anyone can help me in find solution to problems like
P * x + Q * y + ......... >= some constant
Capitals(P,Q) are constants while lower case letters(x,y) are variable
Problem states that we have to find solution to this problem while keeping solution minimized such that it should be greater than a given constant.
solution must contain the two variables (x and y ). So that you can define the function f(x,y)=0 after knowing the constant. then try to find the local points( min , max , saddle) by the known method. you can allocate the min point (x,y) ; the solution you needed.
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Solve this Indices problem friends....
Your ans. Right.........and good explanation thanx
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Need to know.
D^3+d+1=0 solve it
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Is Zeta[2+n^2]-1 a Normal[mu,sigma] ?? (Zeta is Zeta Riemann funtion)
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In this link you will find discussions on the proof:
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Given three vectors x,y,z., how do i plot the magnitude[sqrt(x^2+y^2+z^2)] and show it in 3D using matlab or mathematica?
If you have any other math package i can use and how-that would be great too.
Thanks to all of you!
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What are the main differences between finsler spaces and riemann spaces
In Riemannian geometry the metric tensor depends only on the points x of tha manifold M (g=g(x)), whereas in Finsler geometry the metric tensor depends on both a point x of M and a tangent vector y to M at x (g=g(x,y)).
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There are various implementations and variations of the LLL-algorithm, depending on the specific scope. Different "editions" have differet input variables and so on.. Has anyone experience of any of these implementations?
I shall have a look at these in detail!
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I need help to understand the computer science application of Algebra (rings, fields, groups, etc.)
See also "algebra of programming". These ideas live in the functional programming, category theory and type theory communities. One thinks about programming problems as algebraic specifications.
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Abstract algebra
Total no. Of homomorphisms will be gcd(m,n)... I wl find the no. Of onto homomorphisms...
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Integration
no
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I am at present in need of help with the mathematical package bifurcation XPPAUTO
Actually, I have 3 diff. eqns and when I apply XPP I get results, some of which I cannot interpret. If anyone is interested I can give the eqns etc.
give me
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Inverse matrix on PPU and on SPU using SIMD instructions.
This article will talk about how to convert some scalar code to SIMD code for the PPU and SPU using the inverse matrix as an example.
Most of the time in the video games, programmers are not doing a standard inverse matrix. It is too expensive. Instead, to inverse a matrix, they consider it as orthonormal and they just do a 3x3 transpose of the rotation part with a dot product for the translation. Sometimes the full inverse algorithm is necessary.
The main goal is to be able to do it as fast as possible. This is why the code should use SIMD instructions as much as possible.
A vector is an instruction operand containing a set of data elements packed into a one-dimensional array. The elements can be fixed-point or floating-point values. Most Vector/SIMD Multimedia Extension and SPU instructions operate on vector operands. Vectors are also called Single-Instruction, Multiple-Data (SIMD) operands, or packed operands.
SIMD processing exploits data-level parallelism. Data-level parallelism means that the operations required to transform a set of vector elements can be performed on all elements of the vector at the same time. That is, a single instruction can be applied to multiple data elements in parallel.