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A fascinating question in theoretical physics is whether it is possible to extend Einstein's ideas beyond gravitation to all aspects of physics. The energy-momentum tensor is usually defined extrinsically over the space-time manifold. But could it rather be derived from the geometry alone ? Likewise our local subjective notion of time is given by a local orientation which need not be globally consistent as in Gödel's famous model.
It has been proposed that space-time may have a foam- or sponge-like fine-grained structure (possible involving extra dimensions) which explains energy and matter and the other fundamental forces in a Kaluza-Klein style. That is, "microlocally" the topology of the space-time manifold is highly complex and there may be even a direct relationship between mass, energy and cohomology complexes in an appropriate derived category. At this fine scale there may even be non-local wormholes that connect distant regions of space-time and explain quantum entanglement.
But why not consider the universe as a Thom-Mather stratified space (one can think of this as a smooth version of analytic spaces or algebraic varieties) rather than a manifold ? In this case "singularities" would be "natural" structures not pathologies as in black holes. It is difficult not to think of matter (or localised energy) as corresponding to a singular region of this stratified space. Has this approach been considered in the literature ?
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Clarence Lewis Protin I had a look on Wikipedia to find out about Thom-Mather stratified spaces:
It mentions its use in the study of singularities. My comment about trying to apply this approach to cosmology is based on the Spacetime Wave theory:
From this worldview, singularities do not exist in physics and the laws of physics apply everywhere and for all time. Also the idea that spacetime may have a sponge like or fine grained structure (quantum fluctuations in empty space) is ruled out by the adoption of the Einstein equations of GR as the fundamental equations of spacetime at all scales. This means that if the Mass Energy tensor is identically zero then spacetime curvature must be identically zero.
This the idea of quantum fluctuations in empty space from quantum theory has to give way.
Richard
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Consider the powerful central role of differential equations in physics and applied mathematics.
In the theory of ordinary differential equations and in dynamical systems we generally consider smooth or C^k class solutions. In partial differential equations we consider far more general solutions, involving distributions and Sobolev spaces.
I was wondering, what are the best examples or arguments that show that restriction to the analytic case is insufficient ?
What if we only consider ODEs with analytic coeficients and only consider analytic solutions. And likewise for PDEs. Here by "analytic" I mean real maps which can be extended to holomorphic ones. How would this affect the practical use of differential equations in physics and science in general ? Is there an example of a differential equation arising in physics (excluding quantum theory !) which only has C^k or smooth solutions and no analytic ones ?
It seems we could not even have an analytic version of the theory of distributions as there could be no test functions .There are no non-zero analytic functions with compact support.
Is Newtonian physics analytic ? Is Newton's law of gravitation only the first term in a Laurent expansion ? Can we add terms to obtain a better fit to experimental data without going relativistic ?
Maybe we can consider that the smooth category is used as a convenient approximation to the analytic category. The smooth category allows perfect locality. For instance, we can consider that a gravitational field dies off outside a finite radius.
Cosmologists usually consider space-time to be a manifold (although with possible "singularities"). Why a manifold rather than the adequate smooth analogue of an analytic space ?
Space = regular points, Matter and Energy = singular points ?
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For a function describing some physical property, when complex arguments and complex results are physically meaningful, then often the physics requires the function to be analytic. But if the only physically valid arguments and results are real values, then the physics only requires (infinitely) smooth functions.
For example, exp(-1/z^2) is not analytic at z=0, but exp(-1/x^2) is infinitely smooth everywhere on the real line (and so may be valid physically).
One place where this happens is in using centre manifolds to rigorously construct low-D model of high-D dynamical systems. One may start with an analytic high-D system (e.g., dx/dt=-xy, dy/dt=-y+x^2) and find that the (slow) centre manifold typically is only locally infinitely smooth described by the divergent series (e.g., y=x^2+2x^4+12x^4+112x^6+1360x^8+... from section 4.5.2 in http://bookstore.siam.org/mm20/). Other examples show a low-D centre manifold model is often only finitely smooth in some finite domain, again despite the analyticity of the original system.
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Hi, Not sure if I am understanding this correctly?
In attached file I do not get how the red marked values came?
shouldn't the projections of unit vector's value be like what I wrote in purple???
Please give an explanation.
???
Attached is extracted from Electromagnetic Field Theory Fundamentals / Bhag Guru (Page 25)
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to find the values shown in red boxes, you just need to use triangles shown in the diagrams and note that the hypotenuse of these triangles are unit vectors with the length of 1. in the diagram, the downward direction is defined as positive x and right direction is positive t. Now you just need to use one of the angles of the triangles (phi) and use cos(phi)=adjacent/hyp and sin(phi)=opposite/hype to find the expressions in the red boxes. Don’t forget to consider the sign in front of then based on their direction.
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The methods of zeta function regularization and Ramanujan summation assign the series a value of −1/12, which is expressed by a famous formula:
1 + 2 + 3 + 4 + 5 + 6 + . . . and so on to infinity is equal to . . . −1/12.
Is it correct?.
Can you suggest some work related to this formula?, application?...
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the sum of positive integers is not equal to -1/12.
the sum of positive integers diverges.
What equals -1/12 is zeta(1). However, note that zeta(s)=sum(1/N^s) only for Re(s)>1. Hence, zeta(1) is not equal to the sum of positive integer. In this region, we have to use the reflection formula and doing so gives -1/12.
An interesting topic to see is the following: Ramanujan was the first to state that the sum of positive integer is -1/12. He also state such expression for other divergent series. However, these series are not equal to these values. With the help of Hardy, Ramanujan develop a rigorous version of this. Every divergent series can be associated with a Ramanujan constant. This constant associated with the sum of positive integers is -1/12. But this sum is not equal to -1/12. The purpose of Ramanuja’s idea was to have a finite value associated with divergent series.
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What are the current trends in Commutative Algebra and its interactions with Algebraic Geometry? What are the important articles that one should read for starting to research in this area?
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For Cryptographic purposes.
I hope you find the attached article is helpful.
Best regards
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I know that the 3 vectors x,y,z in Rn , where angles between them are 120`are coplanar .
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Indeed it is an interesting problem. It is more profound than it seems. Your note about my counterexample is true. I have proved that your claim is valid in R3, but I am not sure about higher dimensions.
Regards
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I am interested in studying the curves on a projective surface in P^3 and came across Mumford's work "Lectures on Curves of an Algebraic Surface." Hence, I would want to explore more of the same. Specifically, the linear equivalence of curves on the surface is interesting. So any direction for this would be helpful.
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Dear Snehal Indukumar Bhayani,
"Projective Spheres in Three Geometries", DOI: 10.13140/RG.2.2.20665.19043
Probably this is not exactly the material that you are looking for, but it might be of interest to see a different approach to the concepts and problems that are rooted in projective geometry. Best regards, István Lénárt.
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The pitch value set in music and guitar group in tablature are connected by adjunction of tangent-cotangent bundles. Tuning g is the tangent gradient to the flow of pitch on guitar. It determines the directional derivative at every point in tablature. Intonation f is cotangent. It connects every point on guitar to a pitch. Tuning g:(set→ group), which might be 0 5 5 5 4 5 (Standard tuning), is a left adjoint pullback vector used by guitarist as an algorithm to construct tablature by the principle of least action. The right adjoint f:(group→ set), respectively 0 5 10 15 19 24, is a forgetful vector transforming fret number vectors to the codomain pitch number vectors by intonation at a specific pitch level. When the tablature is played, the frequency spectrum observed seems to forget the tablature group, but it can be proven an efficient Kolmogorov algorithm for learning the tuning exists.
The symmetry of (0 5 5 5 4 5) and (0 5 10 15 19 24) is obvious. The second vector is just the summation of the first. The first vector gives the intervals between strings. It points in the direction of steepest pitch ascent. The second vectors gives the pitch values of the open strings. When added to the fret vector, 0 5 10 15 19 24 gives the pitch vector.
The tablature is pitch-free and the music is tablature-free. These vectors form a Jacobian matrix on the transformation.
I want to know if a mathematician can see the tangent-cotangent relation of these two vectors. If not, then what is required to convince?
Does it help to know that addition and multiplication are the same? That the vectors are open subsets of the octave intervals? Do I need to prove a partition exists, or is it obvious?
Is the tensor notation clear?
Is there any mathematician out there that can say something useful about tablature?
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Yes, exactly. The string manifold is identically zero because it has a fundamental value. The guitar manifold has the lowest string identical to zero.
Every pitch value and fret value can be zero.
The vibrating string retracts upon the string at rest. The guitar string retract on to the lowest string which is constant for all tunings.
That is why the equation of motion for guitar has a zero for the first coordinate, like (0, 5, 7, 5, 4, 3) for Open G tuning. The property of zero extends to each string by this connecting map.
Tablature always seems like nonsense when you do not understand.
But the joke is on math and physics which cannot explain guitar theory. I do not see that there is any mathematician or physicist who has written anything about tablature.
Tablature is surely a form of mathematics, is it not?
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I want know about an algorithm or formula to find the asymptote from coordinates obtained from Machine Learning. Like the ML will always give points precise & closer to the Asymptote if I ran it, but won't ever reach the asymptote value. The normal methods are only created for humans like: limit tends to zero -> 0 and by graphing but there is not an algorithms way for computers which can compute an asymptote. If anybody knows about this can give me a direction on this topic.
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The approach creates a hyper cube around test point effectively.
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I have two points in an (x,y) coordinate system, the only information i have is the distance of these points from the origin and the distance between these two points. I need to find the coordinates of these two points. As these points form a triangle so i use the properties of triangles to find these coordinates. Using the sides length information i can find out the three angles inside. But still i haven't been able to find the coordinates. The attached pictures can help to visualize the problem.
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The solution is not unique: for every solution, if you turn the whole picture around zero for some fixed angle, you obtain another solution. So, you may assume that the point $(x_1, y_1)$ equals $(5, 0)$ and obtain the remaining solutions by the above argument. For this particular case you obtain two equations for unknown point $(x_2, y_2)$: $(x_2)^2 + (y_2)^2 = 16$ (the distance from the point to zero equals 4) and $(x_2 - 5)^2 + (y_2)^2 = 36$ (the distance from the point to $(5, 0)$ equals 6). Solving this system of 2 equations with 2 unknowns you obtain the solution (or more precisely, two solutions, above and below the x axes).
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The function f(x,y) also contains other constants.
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In general, the answer is not simple. There are many techniques to tackle this problem based on the terms of F(x,y)= 0.
If it splits into y = P(x) a polynomial, then it has no loops.
If not, we assume that F(x,y) = 0 is continuous curve and
I suggest to transform the rectangular coordinates into the polar coordinates
or other suitable parametric representation r(t) = x(t)i + y(t)j
then, if r(t) = r(t1 ) implies t = t1 , then no cycles or loops available, otherwise
we have a potential existence of loops.
The study of variation of the vector dr/dt provides the phase portrait
which is useful to detect closed loops.
Examples: 1.The cardioid r - 1- cosθ = r - 1- cosθ1
we obtain cos θ = cosθ1 and θ = θ1 +2k(pi), k is in Z.
therefore, it is a closed loop.
2. The ellipse x2 /9 + y2 /4 -1=0
solve (rcosθ)2 /9 + (rsinθ)2 /4 - 1= (rcosθ1)2 /9 + (rsinθ1)2 /4 - 1
we obtain 4(cosθ)2 + 9(sinθ)2 = 4(cosθ1)2 + 9(sinθ1)2
then 4(cosθ)2 + 9(1- (cosθ)2 ) = 4(cosθ1)2 + 9(1- (cosθ1)2)
finally, we obtain (cosθ)2 = (cosθ1)2
this is true for θ = θ1 +2k(pi) k is in Z.
therefore it is a closed path.
OR r(t) = (3cost)i + (2sin(t)j
solve r(t) = r(t1 ) we obtain an infinite number of solutions.
3. Parabola : y = x2 polynomial, hence no loops.
4. x2 +xy +y2 +x + y -1 = 0
The discriminant b2 -4ac < 0 , it is an ellipse which is a closed path.
we can do it by using parametric representations.
5. x2 +xy - y2 +x + y -1 = 0
b2 -4ac > 0 , hyperbola, no loops.
and more.
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How to find the (arriving angles) αb,βb ?
If we know the values of two sides of triangle b, c and angles between them αa, βa . The angles α and β are representing the azimuth and elevation angles. I am not understanding how to implement the law of sines and cosines ( or any other method ) for this 3D problem.
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Dear Muhammad Urd Sher Qaisrani,
it is difficult, to write calculations in plain text. So I added two pdf-documents.
With best wishes
Johann Hartl
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(Kindly think your idea before looking at details file )
1. How to find the distances between P0 to P1 in this 3 dimensional ellipsoid?
2. Is any of the method 1 or 2 is correct ?
3: How to find the value of D from equation of ellipsoid? As in case of 2D ellipse the distance between two foci can be found using the D^2=a^2-b^2 . Where D is the distance between two foci. And a and b are the semi major axis and semi minor axis of ellipse respectively.
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Me and Viera Čerňanová showed you a good answer to your question"
Given in 3D Ellipsoid semi major axis a=5 and b =4 ,c=3 . How can we find the distance D between two focal points (F1 and F2)? "
posted 9 days ago?
I printed a pdf file to show you how to use the Euclidean distance.
But you got silence, didn't respond to answers, and now you are repeating the same question with a little bit change!!
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Given in 3D Ellipsoid semi major axis a=5 (along axis) and semi minor axis b =4 ,c=3 (along y and z axis) . How can we find the distance D between two focal points (P0 and P2) ? Also how to find the distance from focal point P0 to P1 ? Also will the distance from P0 to P1+ P1 to P2 be equal to 2a like in ellipse ?
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Hi Muhammad,
You give 3 different values for the semi-major axes so there are many different focal points, one pair for each cut plane through the center of the ellipsoid.
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Suppose an element of even subalgebra of geometric algebra over 3D is written in two ways: a+b1B1+b2B2 +b3B3 and exp(acos(a)B). I want to differentiate it, say, by b1. It looks like differentiation of geometrical sum and exponent give very different results. Where is possible error?
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One just needs to accurately use the rule of differentiation in the direction of vector b: (f(..., b+dtb,...)-f(..., b,...))/dt when dt approaches 0.
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Mathematics has been always one of the most active field for researchers but the most attentions has gone to one or few subjects in one time for several years or decades. I'd like to know what are the most active research areas in mathematics today?
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Mathematics is a science that creates, models, describes, explains, applies, and of course its areas of research are always new. Ask about a branch in recent development and of interest to the scientific community is to prepare to find countless answers. Of course, the investigator's self-interest will guide him to appropriate topics.
I have read some answers to this question, published in this medium and I am surprised. They do not do science a favor with them.
What is the reason for writing " Physics will beat mathematics — look at my reform! " . What reform? Somebody knows about that reform?
Physics is a great science, and from its observations mathematics has developed very serious theories (Fourier - Heat, Gauss - sound). They were times of illustration. And conversely, Physics has found that without the development of mathematics there are many observations that could not be modeled.
But, returning to the initial question, I think you should look for an answer, and in that I agree with @ Mirjana Vukovic , if your interest is in pure or applied mathematics.
Greetings from Venezuela
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Algebraic geometry studies zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, for solving geometrical problems about these sets of zeros. Representatives: Riemann, A.Grothendieck. [From wiki]
Geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by Hermann Minkowski (1910). [From wiki]
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Algebra takes a more abstract view. This yields more structure and more (and more abstract) results.
One might question to what extent abstract results are applicable to concrete original questions. However, mathematics is not engineering, and problems are often perceived more as inspiration than as a task.
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Let H1 and H2 be two convex hulls defined by sets of point P1 and P2.
Let H be the convex hull defined by the sets of points {p1 + p2 with p1 in P1 and p2 in P2}
Is H the Minkowski's sum of two convex hulls H1 and H2?
(Minkowski'sum of two convex hull H1, H2= convex hull : {a + b with a in H1 and b in H2} )
Or at least, is it true in 3D?
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This is a very interesting question.
The Minkowski sum of convex hulls is one of the highlights of the following paper:
Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes
Recall that a convex polytope is the convex hull of a finite set of points. For more about this, see page 4. For the details concerning the Minkowski sum of polytopes, see page 20.
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Please I would like your help on getting an e-copy of the book entitled "The theory of fixed point classes" by Tsai-Han Kiang.
Best regards,
M.S. Abdullahi
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James F Peters, Artur Sergyeyev Thank you very much Professors for your king suggestions. However, I am looking for where I can get a free e-copy of the book.
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I am a graduate student in Mathematics and interested in Algebraic Geometry , In particular questions on Moduli Space . Now to start thinking about some problem for research what kind of question we may ask?
Are there any paper that will be very helpful?
How to start thinking about it.
Looking forward for help and suggestion.
I got problem posted on internet , like compatifying moduli space and motivic structure.
But I am not sure about those problem in initial research.
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Please visit this link you will get enough information
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It is easy to verify that the product of two linear forms gives a quadratic form, my question is the reciprocal if it is right for any quadratic form?
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To tackle the problem in general we must first define precisely what we mean by a quadratic form. On a vector space V of dimension n over a field K of characteristic not 2, a quadratic form Q(.) is characterized in a unique way by its associated symmetric bilinear form B(.,.), which is such that 2B(x,y)=Q(x+y)-Q(x)-Q(y). Recall that the "radical" Rad (V)=Rad(V,B) is then defined as the subspace consisting of the vectors x in V s.t. B(x,y)=0 for all y in V. There are 2 extreme cases: Rad(V)=V, in which case B is identically 0 ; and Rad(V)=0, in which case B is called "non degenerate".
Back to your problem, suppose that B is the product of 2 linear forms, i.e. B(x,y)=L1 (x)L2 (y). The point will be the determination of the dimension of Rad(V), which is here the subspace generated by the kernels Hi of Li . Each Hi is a hyperplane (of dimension n-1) of V, hence Rad(V) is not 0, i.e. V is degenerate. Besides, it is classically known from linear algebra that H1=H2 iff L1 and L2 are proportional. Thus Rad(V)=V (resp. is a hyperplane H) iff L1 and L2 are not proportional (resp. are proportional). In the first case B is 0 and there is nothing to add. In the second case, incorporating the factor of proportionality into the linear form, we can write B(x,y)=L(x)L(y) for all x,y in V, where the kernel of L is a hyperplane H. Introduce the quotient vector space V=V/H, which is a line (i.e. has dimension 1), generated say by the class a of a vector a in V, not in H. Denoting by x in V the class of x in V, we can define Q(x)=Q(x) because H=Rad(V), and L(x)=L(x) because H=ker(L), so that Q(x)=L(x)2 . Choosing a to be a basis of V , we can write the quadratic form as Q(x)=k2 Q(a), k in K, or B(x,x')=kk' Q(a). Conversely, knowing H=Rad(V), such a Q or B can obviously be lifted from V to V.
In conclusion : on V , a non null symmetric bilinear form B is the product of 2 linear forms L1 and L2 iff these are proportional and Rad(V,B)=ker(Li ).
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Given a bivariate homogeneous polynomial p(x,y), how can we verify if all the roots (even complex roots) are distincts, or if there is some multiple roots ?
Is there a way to do it without computing all the roots, or without the factorized form of the polynomial ?
Thanks in advance.
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Dear Hamza,
You are right. This is one of the trivial cases: (0,n) , (m,0) , (0,0) appeared as a multiple roots where P(x,y) has xr, ys or (xy)t as a common factor. We remove this term before proceeding the test.
Best regards
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Who is the scientist or researcher who used the topological term for the first time? and when it was specifically ?
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Dear Baravan A. Asaad
Thank you for your excellent answer
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Let (X,0) be an irreducible plane curve singularity and let n:(C,0) \to (X,0) be the normalization morphism, with C=complex line. It is known that the delta-invariant of (X,0) is the dimension of the quotient $O_{(C,0}/n^*(O_(X,0))$.
Some special cases suggest that the delta-invariant of (X,0) is also the dimension of the quotient $\Omega^1_{(C,0}/n^*(\Omega^1_(X,0))$, where $\Omega^1$ denotes the corresponding sheaves of 1-forms. Do you have a reference or a simple proof for this claim?
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Thm 8 on page 177 in M. Rosenllicht, Equivalence relations on algebraic curves, Ann. of Math., 56 (1952), 169-191 says that there is a nonsingular pairing
f_*O_X'/O_X \times \omega_X/f_*\Omega^1_X' \to k
So \delta=dim f_*O_X'/O_X = dim \omega_X/f_*\Omega^1_X'
where \omega_Xis the dualizing sheaf of X. I use this fact in my paper Polar classes of singular varieties (p.261) and also discuss related issues in the paper Ideals associated to a desingularization.
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The Fibonacci numbers can be obtained through the recursivity 
f_k=f_{k-1}+f_{k2} (with f_1=1 and f_2=1)
This is one type of a Lucas sequence, and can be written in matrix form as
(f_{k+2},f_{k+1})=({1,1},{1,0})(f_{k+1},f_k)
(the left-hand-side and the rightmost product of the right-hand-side product are column vectors). 
The matrix ({1,1},{1,0}) is called the Fibonacci matrix.
Now, I would like to be able to write similar expressions for other Lucas sequences, such as
a_n=a_{n-1}+a_{n-2}+a_{n-3}, a_1=1, a_2=1, a_3=1
and also for rational recursions, such as
a_n=a_{n-1}+a_{n-2}*(1+b/d), a_1=1, a_2=1,b and d natural numbers different of zero and b and d are fixed, for all times.
I would sincerely appreciate, and be thankful about any concrete reference where the process for obtaining such matrices is being explained.
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@Issam
from AXn =Xn+1, you get An X0 =Xn and there you can use diagonalization for the matrix A and your initial conditions X0 or X1  to derive what you claim. I don't understand why this nervous discussion. I didn't say any of low graduation or up graduation, but it is clear that the questioner does not know much about the relation of diagonalization of a matrix and its application in linear recurrence sequences,   linear differential equations linear difference equations. It is not bad if someone do not know something, all us need to know more in other fields of Maths :) 
I am so sorry if someone here understand my answer as "bother", it never come to my mind to bother anyone, only answer the question by what I know and seek the better.
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  • Let x=(x1,x2,...,xn) be standart coordinates of a Cartesian (n+1)-space and y=(y1,y2,...,yn) another one, where yi=aijxj, i,j=1,2,...,n.
  • Denote Hess(z(x)) the Hessian of a real-valued function z=z(x1,x2,...,xn).
  • We obtain that
  1.  Hess(z(x))=(A^{T}).Hess(z(y)).A, 
  • where A=(aij), A^{T} the transpose of A, "." matrix multiplication.
  • The question is: the equality in (1) is new one? Or is well-known?
  • Many thanks in advance for your interest and comments...
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Dear Professors Bradly and Low, many thanks for your answers. I agree with both of you. 
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I have triangle mesh and calculate normal of triangles then calculate vertex normal and do some calculations on it and want to calculate vertex coordinates from this vertex normal after do calculations.
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Look at this doc it may be helpful for your topic. Good luck.
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In Riemannian geometry, we know that every 2d manifold is locally conformally flat thanks to local existence of isothermal coordinates; what could we say about surfaces for which these coordinates exist globally ?
Are they "Riemann surfaces of parabolic type" (-> uniformisation thm) ? The reason would be that they are conformally equivalent to the complex plane C.
And can they have negative curvature ?
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I think that every oriented Riemannian 2-manifold M admits a global conformal map into the Riemann sphere. Just pass to the universal covering and patch the local isothermic systems together. However, this is one-to-one only on the universal covering.
A global conformal coordinate system means that M is an open subset of the complex plane. Whenever the image not the full plane, it is biholomorphic to the unit disk, i.e. it is hyperbolic, not parabolic.
Best regards
Jost
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Consider a parameter space Rn, comprising points a=[a1, a2,...an]'∈Rn. Suppose we've got two constrained sets within the space.
The first is linearly constrained: f(a1, a2,...an)=0, with f() being a linear function. So it's a linear subspace.
The second is nonlinearly constrained: g(a1, a2,...an)=0, with g() being a nonlinear function including only polynomial operations i.e. addition and multiplication. So it's a nonlinear manifold.
If the dimensions of the two sets are identical, then the intersection must be a null set of Lebesgue measure 0, comprising only countable crossovers with lesser dimension.
Am I right? How to prove it?
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 I afraid, there should be some additional restrictions on the polynomial. For example, the following non-linearly constrained set
(a1)- (a2)2 = 0
consists of two linear subspaces
a1+a2 = 0  and a1 - a2 = 0,
so it intersects with the linear subspace a1+a2 = 0 by the whole subspace. So, you must exclude such a possibility.
The branch of mathematics that works with intersections of zero sets of polynomials (which are called "algebraic manifolds") is algebraic geometry. You should ask advise from  people that work in that area or from algebraists what the condition on the polynomial can ensure that its null set does not to contain a linear subspace of the same dimension.
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The string is a real closed field minus finitely many points.
If pitch is a continuous function defined on the string interval [0, 1] then there must be a fixed point in the interval.
Each mode of vibration is represented by an n-tuple that is a subset of Rn so that the waves are intervals between nodal points in projective space.
The null set is just the endpoints, no string, which are the points (-1, 0) and (1, 0) represented by S0 .
The string is connected like a circle missing a single point which can be used to glue the circle into the origin of RP3 - 0.
There is a curve-lifting equation that proves the string at rest and in the perturbed state match point for point, which proves the force acting upon the string is orthonormal to the string axis.  Without a force in the direction the string axis there can be no traveling wave! They kinetic and potential energy go to zero at the endpoints.
The string assumes the same shape regardless of how it is struck and clearly the fundamental has the lowest frequency and therefore the lowest engergy.  Why would several energy levels co-exist?  Why wouldn't higher modes degenerate to lower modes if they can co-exist.  Clearly the modes are singletons that are all 1 step away from the fundamental.  Isn't it obvious that the string is given a subspace topology?
The string has an atomic structure that is the union of wave and not waves, and so on.
Now my question is, if the sound wave that radiates from the string is a purely algebraic object that is a  function of one continuous variable pitch(frequency) and the wave is a polyhedron with n + 1 vertices, n edges, and 1 face in R2 , then why isn't it clear already that the string is a semialgebariac ring, and not just a frequency transducer.
We have a graph (the string with nodes and wave) and a tuning function f and intonation function g.  What else is needed here?  Why isn't clear that each n-tuple is a different system and the n-tuples cannot just add?  Pitch cannot be divided into 0 and 1 without an algorithm. Clearly the string is partitioned into a finite number of points and intervals in the real closed field.  Isn't that enough, by itself, to make a new theoretic model for the string that makes sense?
It is just astonishing that people believe in things like traveling waves reflect to make standing waves (clearly the boundary condition for this (1, 0, 1) cannot exist); or that  the string can just be divided into smaller and smaller fundamentals that all co-exist in the string as independent modes.
Nodes and waves cannot co-exist. Period. If evidence shows they can, the evidence needs to be re-examined.  The monochord proves that string can have only one mode at a time according to a fixed point on the string and also that higher modes exist only when the string driven in a higher energy state.  But each fulcrum point is a different system. 
Notice the monochord string has the property that it recognizes when [0, 1] is equal to a simple multiple the string fundamental. That is a boundary condition for harmonic motion.  Recognition is a property of a finite state machine.  The string recognizes it's finite modes. Finite state machines have only one state at a time.
I cannot understand why such fallacy as classic string theory is allowed to persist in science.  Such a profound illusion that no one gets this! 
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You probably need to look at some figures to understand this so I added some sketch to illustrate the string modes, boundary conditions, and identity.  The last file is a set of equations for the string.
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hello,
I have a vector (x,y,z) and q certain number of vectors (x1,y1,z1) and (x2,y2,z2)............
I want to define a metric to now which vector is the nearest to (x,y,z)
for example
if my vector is (9,9,9) and my candidate vectors are (3,3,0), (0,3,6) and (9,9,9)
so the nearest vector is (9,9,9). 
Any idea please?
Cordially
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If you want to just define things, just do the obvious thing:  
closest((x,y,z), {(x_1, y_1, z_1), ....., (x_n, y_n, z_n)} = argmin_i(d((x,y,z), (x_i, y_i, z_i)).
If your problem is finding such a  point in a massive set of datapoints there are tree based algortithms like the Vantage Tree algorithm which speeds things up dramatically 
See more generally 
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Dear all,
     How can we obtain the analytical solution (in the complex n-space C^n) to the following 2nd order system of polynomial equation with n variables?
a_{11}x_1x_1+a_{12}x_1x_2+...+a_{1n}x_1x_n=b_1
a_{21}x_2x_1+a_{22}x_2x_2+...+a_{2n}x_2x_n=b_2
.
a_{k1}x_kx_1+a_{k2}x_kx_2+...+a_{kn}x_kx_n=b_k
.
a_{n1}x_nx_1+a_{n2}x_nx_2+...+a_{nn}x_nx_n=b_n
where a_{ij} and b_k, i,j,k=1,2,...n are constant complex values, and x_i, i=1,2,...n are unknowns.
Kind regards,
Chao
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You have n^2 + n parameters.  Unless n is rather small, this is a hopelessly large problem.  
The Dixon resultant is far superior to Groebner bases when there are parameters, but your system is very generic.  Google "Dixon resultant".
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I need  geometrical interpretation of runge kutta nystrom method for solving 2nd order differential equation? any body who can help me in this regard. 
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i am very to you for this help
Regards
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The phrase "continuous variation" appears several times in Titchmarsh's books and is repeated verbatim by those who cite his work. But nowhere can I find this phrase defined. The earliest reference I have found (1939) occurs on page 132 of his book "The Theory of Functions". From the context it seems to describe a particular way of defining the value of a cut function. Can anyone provide a (citable) reference to a formal definition of this phrase, or, better yet, an explanation of its meaning?
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@michael.  The continuous variation TIchmarsch means is a so called homotopy of (closed) paths \gamma in the complement of the zero's and poles of  the holomorphic function f(z). We can then define
d\log(f(z)) = (f'(z) / f(z) ) dz .
The theorem that Titschmarsch refers to is
\int_\gamm d\log(f(z)) = \int_\gamma (f'(z))/f(z) dz = 2\pi i (#zero's - #poles) encircled by \gamma. 
Now let D \subset C be the complement of the zero's and poles.  A closed path is just a continuous map  of the interval
      \gamma: [0,1] \to D
such that \gamma(0) = \gamma(1) . A continuous variation  of gamma is what is called a homotopy, Two closed path's \gamma_0 and \gamma_1 are homotopic if there is continuous map
      \Gamma : [0,1] \times [0,1]  \to D
such that
     \Gamma(t, 0) = \gamma_0(t); \Gamma(t, 1) = \gamma_1(t)
and
    \Gamma(0, s) = \Gamma(1,s)
i.e. if write \gamma_s(t) = \Gamma(t,s), then the family of paths \gamma_s continuously deform the path \gamma_0 into \gamma_1 (all the time avoiding the zero's and poles of f!). Now what Titchmarse says is that if \gamma_0 and \gamma_1 are homotopic then 
\int_{\gamma_0} d\log (f(z)) = \int_{\gamma_1} d\log (f(z))
i.e. the integral only depends on the homotopy class  of the path. 
Remark: it so happens that the integral even only depends on the homology  class of gamma. This is a bit harder to explain but here  it  means that as far as contour integration is concerned, if two (possibly disconnected) contours bound a surface not meeting the singularities they are equivalent.  In practice this allows you to reduce a contour integration of a meromorphic function to  integrating  over small circles encircling the singularities (which for a function of the form f'(z)/f(z) are the zero's and poles of f).  This a modern formulation of a classical fact and may well be what Titchmarsch actually uses. 
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For example, if I have a grain with blue color in IPF map of my sample, obtained in x direction, it means that 111 of this grain is parallel to x direction, according to IPF triangle. For grains having yellow, orange or other colors inside the triangle, I want to say that 111 of these grains have .... degree difference relative to x direction. How can I have the value of this angle, with a open-source EBSD software, online calculators or a mathematical operation?
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Hi, Thomas. The link for the free ebsd software is http://www.atom-software.eu/
I have not tried it in detail, but it has friendly interface.
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I know the following result:
Suppose C and D are nonempty disjoint convex sets. The hyperplane {x | a^T x = b} separates these sets provided a=d−c, b= (∥d∥^2 −∥c∥^2)/2, where c and d are objects lying C and D, respectively, that minimizes the distance.
I wonder if is there another result that ensures a better separation than this?
Thank you!
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Hi,
If you search with google on the string "separating hyperplane theorem border" it leads you to a paper on the topic; Theorems 6 and 12 are quite good for your purposes.
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In the question given above, H is the mean curvature of the immersed real projective space in the Euclidean m-space.
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Professor Malkoun:
I shall mention that the most general result on total mean curvature is the following:
For every n-dimensional compact submanifold M immersed in a Euclidean m-space, the total mean curvature satisfies
M |H|n dV  ≥ cn,
where cn is the volume of n-sphere. 
You can find other results on total mean curvature in my books  "Total mean curvature and submanifolds of finite type", 1984 and 2015 editions.
Sincerely,
B.-Y. Chen
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Please can someone give me a reference on the distribution of Pisot numbers on the real line.
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Communication protocol  with  high  level Scurry  type   supported   distribution  
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In imaginary quadratic fields we have:
* ELL(O_K) : = {elliptic curves E/C with End(E) ∼= O_K}/{isomorphism over C}
∼= {lattices L with End(L) ∼= OK}/{homothety}∼=ideal class group CL(K)
* #CL(K)=#ELL(O_K)
this notation at the papper :A SUMMARY OF THE CM THEORY OF ELLIPTIC CURVES
JAYCE GETZ
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As you say, the theory of "complex multiplication" (looking at modular invariants of elliptic curves with CM) solves the problem of determining the maximal abelian extension of an imaginary quadratic field. But as yet no analogous theory is known for real quadratic fields, although class field theory affords  a complete theoretical (but not explicit) description of the abelian extensions of a given number field. You can  take the measure of our ignorance by  listening to the short negative answer given by Serre at the end of his talk on abelian Galois theory at the occasion of Galois' bicentenary (the talk can be found on You Tube, I think).
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In the image I show what I figured out so far and where I'm stuck... I would be more than happy if anyone could help me with this.   Thanks! :-)
Hello RG people,
It's been too long since I last solved such an equation and I'm sure a high school student would be able to solve it easily. However, I'm stuck and can't figure out the coordinates of a point C. I know the following things:
  • the coordinates of A and B
  • if A and B are on a straight line f, then B and C are on a straight line g
  • f and g are perpendicular 
  • the euclidean distance of A to B is double the euclidean distance of B to C, which is defined as e
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Number of isosceles triangles drawn on the longest side = infinitely large.
Let BC = 2a  be the longest side. Draw its perpendicular bisector (say LMN) through the mid point M of AB. Choose any point (say  A) on LMN and complete the isosceles triangle ABC. Since base BC of the triangle is given longest no side AB (=AC) should be larger than 2a.  
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An n_k ctheorem (configurational theorem) is a set of n points and n hyperplanes with k points on each hyperplane and k hyperplanes through each point, all embedded in (k-1)-dimensional space.  (The type of space could be e.g. a projective (or affine) space over a general commutative field (type (0)), over a general possibly non-commutative field (type (1)), or over a general field of prime characteristic p (type (p)).  If the existence of n-1 hyperplanes implies the existence of the n'th hyperplane then it is called a "ctheorem".
Those known are:
Desargues 10_3 (type (1)) discovered about 1650 CE
Pappus 9_3 (type (0)) discovered about 300 CE
Moebius 8_4 (type (0)) discovered 1828 by A.F. Mobius
Glynn 8_4 (type (0)) discovered 2010 by  D.G. Glynn (Theorems of points and planes ...)
Glynn 9_4 (type (1)) discovered 2010 by D.G. Glynn (same paper)
Fano 7_3 (type (2)) known to geometers in the late19th century (the matroid dual is a 7_4 ctheorem of type (2) also)  Could be called "anti-Fano" since Fano's axiom proscribed it in the geometry.  It is also called PG(2,2), the projective geometry of dimension 2 over the finite field GF(2).
Note that the matroid dual of an n_k ctheorem is also a ctheorem n_{n-k} (if type (0) or (p)), so Pappus gives a ctheorem in 5-d space.  The two 8_4's (they are unique) are self-matroid-dual.  (Sometimes the matroid dual is a bit degenerate, as in the cases 10_3 and 9_4.)
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Thanks for the 19_4 information. The question is for geometries over any field.  Type (0) is for general fields commutative of characteristic 0, type (1) is for general non-commutative fields (including the commutative ones), type (p) is for characteristic p commutative fields.  I do not know of examples of such n_k ctheorems that restrict to more than this: e.g. just to rationals.  Fano (or its matroid dual 7_4) is the only one I know that works only for a certain characteristic (2).  Instead of searching for n_k with n large and k fixed it could be more likely to find new ctheorems for k about n/2.  Maybe looking at the (p) case is easiest.  Perhaps Fano is the only such ctheorem.
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Foliation
Geo diff
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At a singular point, there are more than 2 prongs. A singularity has angle n \pi around it, where n equals the number of prongs. Hence metrically it is not a smooth manifold point (for a smooth manifold point the angle necessarily is 2 \pi). If the map is a diffeomorphism, then there would be compatible smooth structures on the surface. Hence a compatible smooth metric. Hence the angle needs to be 2 \pi.
A beautiful book that deals with the theory is  Fathi, Laudenbach, Poenaru's Travaux de Thurston sur les surfaces. Dan Margalit has recently translated this into English and the translation has appeared in the Princeton Math Notes series.
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Any expert working in the area of computational geometry.
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Thank you. The problem is, i know only chord length. Now an ellipse should be cpnstructed. 
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Is it possible to give an elementary proof of the aforementioned result? This was proved by Robert Penner (at "A construction of pseudo-Anosov homeomorphisms". Robert C. Penner. Trans. Amer. Math. Soc., 310(1):179–197, 1988) under the techniques of measured-foliations.
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The answer is given by Penner at the bottom of pg 196 of the paper you cite:
"On each of the projective plane, Klein bottle, and torus-sum-projective-plane there are unique simple curves with special properties."
Thus diffeomorphisms preserve the free homotopy classes of such curves and are therefore reducible (in particular not pseudo Anosov).
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A hypersurface is called a Dupin hypersurface if the multiplicities of its principal curvatures are constant and moreover each principal curvature isconstant along its principal directions.
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Unfortunately it does not fall within the sphere of my close interest and scientific work.
With my best regards and wishes for a Happy New Year,
Mirjana Vukovic
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A compact submanifold lying in a hypersphere Sm of a Euclidean (m+1)-space Em+1 is call mass-symmetric if the center of mass of M is the center of the hypersphere.
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Unfortunately, I can not help to you, because it does not fall within the sphere of my close interest and scientific work.
Mirjana Vukovic
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Dear all,
     I have a question about the existence of real solutions to a sixth order polynominal system (equations) with two variables (i.e., f(x,y)=0 and g(x,y)=0 each equation is a  sixth order polynominal equation). Is there a criterion to determine the existence of real solutions? I know this is a problem in Algebraic Geometry:)  
Kind regards,
Chao
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Chao,
Sorry, right now I'm out of time - I'm under a sharp deadline in completing one work until Dec. 18th. After that will try to sketch the underlying idea  (I didn't care as yet in its  rigorous justification, but it yields practical results...) 
AU
PS Drop me a mail  (alexeiuteshev[at]gmail.com)
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Attack angle and sideslip angle have math definitions based on arcsin and arctan trigonometric functions (e.g., in J. N. Nielsen' MISSILE AERODYNAMICS). How to choose between one or another definition? Can someone provide a formal reference or proof on that? (textbook or paper)
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Hi Adolfo! Your comment is part of the answer, but it is not just that. See the quotes below:
(1) " Therefore, for attack and sideslip angles, which have low values that do not require a reduction to the first quadrant of the trigonometric functions, we can use the relations indicated in [3], [2]: α = -arctan(w/u); β∗ = arcsin(v/V) " [1];
(2) "The angles of attack and sideslip have been defined in at least three ways. (...) small angle definitions (...) sine definitions (...) tangent definitions (...) For small angles, the angles of attack and sideslip do not depend on which definition is used. For large angles, it is necessary to know which definitions have been adopted. Frequently, the sine definition is used for one quantity and the tangent definition for another. (...) It is noteworthy that Eqs. (1-8) and (1-10) would be used to set a sting-mounted model in a wind tunnel to previously selected values of αs,βs and αt,βt . " [2]
(3) "The definition for the angle of attack and angle of sideslip is a standard
one (ref. 11). An alternate definition for the angle of sideslip, which
is sometimes used in the helicopter technical community, is: β=tan^-1(v/u)" [3]
Furthermore, what would the choice of the definition (sin or tan) have with application (missile x airplane) and computational aspects?
REFERENCES
[1] T.-V. Chelaru, V. Panã, A. Chelaru, "Dynamics and Flight control of the UAV formations"
[2] J. N. Nielsen, "Missile Aerodynamics"
[3] R. T. N. Chen, J. A. Jeske, "Kinematic Properties of the Helicopter in Coordinated Turns"
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One knows that a collection of k<n points on the veronese curve with multiplicities m_1,...,m_k (m_1+...+m_k=n+1) give a secant projective space to the veronese curve, and one can compute the dual space of this secant and it is the intersection of the (m_i)-1 osculating hyperplanes of the curves on the relative points. My question is: it is possible that any projective subspace of any dimension on CP^n can be a dual of some secant to the veronese curve?
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Not yet, i only know that the set of secants for some divisors (linear combinations of points in the Veronese curve with multiplicity) it is a subvariety of the grassman variety of some degree.
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Suppose given an m-sequence M over a field Fq of size q (a prime power) and of order s. In other words: there is a primitive polynomial f(x) of degree s over Fq serving as the characteristic polynomial for a recurrently defined sequence. If alpha is a primitive root of f(x), then M is essentially the (trace of the) sequence of integer powers of alpha (which may be reduced modulo f). I am thinking of rather large-sized sequences (with s many hundreds or even thousands). 
Given r < s and given r linearly independent powers of alpha with exponents i1<i2<..<ir (there is an abundance of them with successive distances ik+1-ik < s), I am looking for information on how dependent powers of alpha are distributed.  Assuming that r is much smaller than s, and knowing that there are "only" qr vectors in an r-dimensional linear subspace, dependent vectors are likely to be far apart in M. What is known about this?
I find the case q=2 most interesting, but quite often results on binary m-sequences extend to general powers of a prime.
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Actually if q=2 and r>=3 and n=2^s-1 then this probability is directly linked to the weight distribution A_i of the dual. Thus p(r)=(A_r/2^k), where k is the dimension of Hamming code: k=n-s.
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I have found a (maybe the) faithful and full representation
of quaternions in the ordinary real three dimensional Euclidean space.
A non constant quaternion is the roto-dilation, usually associated
to it, endowed with a verse ("clockwise" or "counter-clockwise")
for the underlying rotation.
A negative constant  -c  is the dilation associated to the positive
constant  c  endowed with the rotation of  2π  of the space, independent
of the axis and of the verse of the rotation.
A positive constant  c  is simply the the dilation having ratio  c, 
endowed, if you like, with the identity rotation.
I'm asking if this simple faithful and full interpretation
of quaternions, in the usual real three dimentional Euclidean space,
is new or if it is well known.
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Dear Christian and Ronald
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Suppose I have two matrices A=[3 1;1 4] and B=[5 -2;-2 4]. Where A and B represent covariance ellipses in 2D. Now, if I want to combine it in 4D, that is, C=[A 0;0 B]. C is a four dimensional ellipsoid but the cross correlation between A and B is zero. I want to rotate A and B such that, they are highly correlated.
I have the following questions:
1) How can I rotate the A and B (either in 4D or 2D separately) such that they have maximum cross correlation with each others?
2) Is there any 4D rotation(either Euler or Quaternion) matrix in explicit form(I couldn't found any)?
Kindly guide me.
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The answer to 2) can be found in the english wikipedia
see section "Relation to quaternions". Moreover, you can find some references there.
1) In my opinion this really depends on the properties of the two ellipses, and therefore, it will be hard to find a general approach.
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There is a twin prime conjecture that for any given integer N>0,
there is a twin primes p, p+2 such that p>N.
What about triplets of primes p, p+2, p+4. Although there is a result that
there is an integer A such that for any integer N>0, there are primes p, p+B, p+C
such that 1. p>N and 2. B<C<A,
this could not really be a progress toward the triplet primes
conjecture since any triple integers p, p+2, p+4 has one of them which is dividable by 3.
Therefore, the triplet prime conjecture as it stated above is wrong.
There could be a modified triplet prime conjecture that for any integer N,
there is a triplet primes p, p+B, p+C such that 1. p>N and 2. B<C=6.
The examples are: A. lower twin triplets: (5, 7, 11); (11, 13, 17); (17, 19, 23); (41, 43, 47);
(101, 103, 107); (191, 193, 197).
B. upper twin triplets: (7, 11, 13); (13, 17, 19); (37, 41, 43); (67, 71, 73); (97, 101, 103);
(013, 107, 109); (197, 193, 199).
Similarly, for the quadruplets, we have the twin twins p, p+2, p+6, p+8:
(5, 7, 11, 13); (11, 13, 17, 19); (101, 103, 107, 109); (191, 193, 197, 199);  (821, 823, 827, 829).
My question is: Could we find all this kind of quadruplets or prove that the quadruple conjecture is true?
Of course we expect that these kind of multiplets with minimal length are rare just as the human multiplets.
Notice that there is a eight-let: (3, 5, 7, 11, 13, 17, 19, 23) and a six-let (97, 101, 103, 107, 109, 113).
Notice that we could not have a six-let (p, p+2, p+6, p+8, p+12, p+14).
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Thank you for all the answers. My field is not number theory. So, I do not really understand the more abstract stuff. However, it is very interesting that when
H={1, 3, 7, 9} and {7, 11, 13, 17, 19, 23}.
Dear Patrick, what does the Hardy-Littlewood Conjecture say in these two cases?
Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862 proved in the second case, there are at least two primes for infinite many n under an assumption of another conjecture. Even though they are still short of proving all of them are primes.
I am wondering that if one just restrict his or her attention to these two cases, possibly he or she could get some useful results.
By the way, the proof for either of these two cases implies the twin prime conjecture. On the other hand, if someone could prove the second case is wrong, that would be also very interesting!
For the case H={0, 2}, the best result is due to Chen, Jingrun.
He proved that for any even h, {p, p+h} with p a prime hit  infinite many p+h
which is either a prime or an almost prime, i.e., the product of two primes.
Also, it seems to me that the Goldbach Conjecture is much harder than the twin
prime conjecture. Chen's result for that is seemly only:
There is an integer N such that for any even integer n>N there is a prime p<n
such that n-p is either a prime or an almost prime.
Chen's Conjecture has the form:
For any even number n, there is a p<n such that n-p is either a prime or an almost prime. I am not sure that Chen's Conjecture was proven.
To prove Chen's Conjecture, one have to find a concrete N_0 first and then
prove that Chen's Conjecture is true for n<=N_0.
Best,
DG
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I consider an application a "true" one if it does not come as a reformulation of an optimization problem. I already know about applications of the second order and positive semidefinite cone-complementarity problems which are reformulations of optimization problems (for example related to Nash equilibrium). I also know about true practical applications where the cone is either the nonnegative orthant or the direct product of the nonnegative orthant with a Euclidean space. However I don't consider the later cones essentially different from the nonnegative orthant. I am mostly interested in practical applications, but I am also interested in possible applications of cone-complementarity to another field of mathematics. I would be grateful if you could point me to any papers, books, links or other materials in this topic.
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Dear Sandor
I think there is a nice application in applied mechanics, where friction and impacts are formulated as a complementarity problem. Please have a look at a book on Multibody Dynamics with Unilateral Contacts by Friedrich Pfeiffer and Christoph Glocke. 
Best wishes
Marian.
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Hello,
I am wondering whether the area enclosed by an ellipse drawn around data points on an x - y graph where both axes are % (or in my case involving stable isotopes, ‰) takes a unit or is dimensionless.
From what I have seen most researchers report these areas sans unit. However, I came across a thesis recently with ‰2 which started me thinking - why not? 
I am sure someone knows and can explain the answer to this for me. I would greatly appreciate any thoughts! And thanks in advance. 
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I would say both of your axis are dimensionless. '%' means nothing else than divide by 100 or multiply with 1/100. I never came across '%2', but the meaning would be clear:
1 %2 = 1 * (1/100)2 = 1 * (1/100) * (1/100) = 0.01 % = 0.0001
So you could give the result in %2 or in % or without 'unit'. Depends a bit on what you want to say with the result. If you want to compare the area of the ellipse to the total area I would give the result in %.
Hope this helps!
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One would like a good definition of etale cohomology for non-commutative rings A with corresponding Chern characters from Higher Algebraic K-theory (Quillen type) of A.  In particular, one would like a non-commutative analogue of Soule's definition  of etale cohomology for rings of integers in a number field with Chern characters from the K-theory of such rings. A possibly accessible setting is to define such a theory for maximal orders in  semi-simple  algebras  over number fields and then extend this to arbitrary orders in semi-simple algebras over number fields.  The goal in this case is to be able to understand such theories for non-commutative integral group-rings i.e group-rings of finite  non-abelian groups over integers in number fields.
REMARKS: Geometrically, Soule's construction translates into etale cohomology of affine  and related schemes and so the envisaged construction should translate into etale cohomology for  a suitably defined 'non-commutative' scheme.
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Dear Aderemi,
I believe a positive answer to your question can be derived for Azumaya algebras using the results of Dwyer and Friedlander in their paper:  Étale K-theory of Azumaya algebras.  Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983).  J. Pure Appl. Algebra 34 (1984), no. 2-3, 179-191.  
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where p is a prime number greater than or equal to 3, and n is a natural number between 2+p and 2p.
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I tried to use sin(x)/cos(x) but i can't calculate the  product (0,2); please see my attachment
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I'm working in a project I would know if there is some references about the algorithms we can use to test if a given ideal is trivial in a given ring.
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An ideal of a unitary ring is trivial if and only if it contains 1. For polynomial rings and local rings (and some other structures), ideal membership can be tested using Groebner bases. For this and other algorithms, I can recommend A Singular Introduction to Commutative Algebra by Greuel and Pfister, see http://www.singular.uni-kl.de/index.php/publications/a-singular-introduction-to-commutative-algebra.html
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For example we have rule 90 in cellular automaton which can produce Sierpinski triangle, which can be made algorithmically by removing triangles, or also through applying fractal formulas, like what we have in XaoS fractal software's example. So is it impossible, theoretically, to produce any fractal which we can produce from any system, in another one? And the answer must lead to say how we can?
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 ``Computability and Julia Sets" by Braverman and Yampolsky may be helpful
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Some Riemannian manifolds are expressed as a product manifold. Recently, I have read two articles about space-times. In both articles, the authors prove that a Riemannian manifold \bar{M}^n is expressed as a product of the form I×M^{n−1}. Both authors use similar techniques, namely integrable distribution, in this decomposition. Really, I do not understand this technique. But it is enough to know a characterization of Riemannian manifolds which we can express it as a product manifold M^1\times M^2.
Q1 Does this characterization exist?(if yes, a reference is required)
Q2 What conditions and proof hints could one think of to characterize these manifolds? 
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Dear Professor Shenawy, 
Please search for your answer in the following references:
1. Bang Yen Chen:  Differential geometry of semiring of immersions I: general theory, Bull. Inst. Mathemat. Academia Sinica, 21 (1993)
2. I. Mihai, L. Verstraelen : Introduction to tensor products of  submanifolds, Geometry and Topology of Submanifolds VI, World Scientific Publ. 1994. 
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Let E in P3be a real elliptic normal curve with two non-null-homotopic components. Is there a parametrization (R/Z) x (Z/2Z) -> E such that any four points on E are coplanar precisely when their corresponding parameters sum to zero? As an example one could take E to be the complete intersection of the quadrics XY + ZW = 0 and -X2 + Y2 - 2Z2 + ZW + 2W2 = 0.
For more details (and better formatting), see the corresponding question on MathOverflow: http://mathoverflow.net/questions/197848/ .
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An answer to this question was provided by Alex Degtyarev on MathOverflow. To paraphrase: If such a parametrization exists, the image of zero would have to be a hyperflex point on one of the real connected components, in which the curve has 4-fold intersection with a plane. Identify the complement of this plane with R3. Since the curve has degree four, this plane cannot intersect the other component, which is therefore contained in R3. This other component is therefore null-homologous, contradicting our initial assumption.
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Part of the Bernstein's theorem (in algebraic geometry, about polynomial equations) says for n polynomial equations with n variables, if the coefficients are generic, the number of solutions must be finite and equal to the mixed volume.
But here is a counter example. (Sorry, it seems I cannot use La TeX symbols in this field so please excuse the ugliness.)
Consider a set of equations
sum_i {h_{i,j}*u_i*v_i}=0,
where the summation is from i=1 to N. j runs from 1 to 2N. h_{i,j} are the generic coefficients. u_i and v_i are variables. So we have 2N variables and 2N equations.
The solution is: u_i=a_i, i=1,...N, u_i=0, i=N+1,...2N
v_i=0, i=1,...,N, v_i=b_i, i=N+1,...2N and a_i, b_i can take any value. Therefore, we have infinite number of solutions.
This seems to violate the Bernstein's theorem.
Anyone have an explanation?
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These system of polynomials is not a generic one. Even more, these polynomials are linearly dependent, because you use just n monomials, but the number of equations is 2n. Say, for N=1 you get 2 equations with just  1 monomial u1v1.
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Mesh has 4.2 million cells. Geometry is a cylinder of 122mm diameter, 200mm length. 
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If you want a fast convergence, probably yes. Just a technicality, if you are using OpenFOAM implementation of GAMG, you should set the nCellsCoarsestLevel more or less equal to the square root of the number of cells.
If you are looking at scalability, GAMG is not the best choice and conjugate gradient methods tend to scale better.
For a large case, I think GAMG is still the way to go, even if it scales less because the reduction of the number of iterations is typically very large compared to other methods.
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Has anyone actually compared them on a specific problem?
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One should also add CoCoA (see lin below) to the list of the freely available computer algebra systems specifically dedicated to commutative algebra (and non-free one could also add Magma). As written in the thread mentioned by Artur, there is not real point in asking which of the three is the "best" one, as this depends where much on what is important for you: breadth of the library, speed of computation, ease of programming. The next question is whether you have some specific problem in mind (for which perhaps only one of the three has some code available) or whether you are asking generally. But even for a very specific problem there is usually no simple answer. We developed recently a new algorithm for computing resolutions and implemented it in CoCoa. Then we run a lot of benchmarks comparing it with Singular and Macaulay2: each system had some examples where it was faster than the other one. Computational commutative algebra is a very complex field and there is usually not a "best" solution to a particular class of problems.
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Suppose I have an affine algebraic variety, and I suspect that a certain element of its coordinate ring can be represented as a product of two elements having some nice form. Are there computer algorithms for such factoring?
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I have now made some progress; the program the helped me was Singular.
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What happens geometrically when one multiplies a matrix A by the inverse of a matrix B?
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Consider the rows of B as a basis b_1, … b_n, and the rows of A as vectors (not necessarily a basis) a_1, .. , a_n. Then AB^{-1} is the unique linear map that sends 
b_i --> a_i  for all i =1, … , n. 
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Jaconbson's Lemma to MP-inverse is not right.
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In general, it isn't true on MP-invertibility for Jacoson Lemma in rings. So I want to find some conditions which can ensure the statement is true, for example it is true for the ring who has the property : 1+a*a is inverttible. Next we will ask if there are some other rings  can make the statement is true
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I found in the book of Kashiwara-Schapira a precise description of this construction but I want to know some illustrative applications maybe a clearer motivation. Thanks a lot!
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The analogue you're looking for is the notion of tangent groupoid. You can look it up in the book of A. Connes. Also in several other places -- it is central in Noncommutative Geometry. As for stratified pseudo-manifolds, it is used there as well (see work of Claire Debord and Jean-Marie Lescure).
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Given an algebraic system based on polarities on the sphere. A pair of opposite points and their equator constitute a basic element. (Add the equator to Riemann's unification of opposite points in elliptic geometry.) Two elements determine a resulting element of the same set. This is a partial binary operation with two axioms: ab = ba; (ab)(ac) = a. I call any set of this type a projective sphere. (Cf. Baer's finite projective planes and Devidé's plane pre-projective geometries.) From these axioms a number of important properties can be deduced. For example, if the set has at least two elements a and b, then xx cannot be properly defined for the whole set, because (ab)(ab)=a=(ba)(ba)=b contradiction. This means that in the general case xx must remain undefined, as with the case of division by zero in fields. However, if a smooth curve is given on the sphere, or an oval in a finite set, then the xx operation CAN partially be defined for the elements of the curve or of the oval as the tangent to the given point. Example: given the oval of the four reflexive (self-conjugated) elements in a 13-element finite sphere; the derivative consists of the same four elements. Another example: Given the basic elements on the sphere with homogeneous coordinates. Take the circle with center (1,0,0) and radius pi/4, given by elements (1,√(1-c^2 ),c); its derivative is the curve given by elements (-1,√(1-c^2 ),c). In this interpretation, the derivative does not represent the number indicating the slope of a straight line, but a set of the same type of geometric objects out of which the original curve is made. Also, this gives that every smooth curve evokes a geometry of its own, defined specifically for the given curve.
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Dear Geng, thank you for your empathic words and wishes. I am stubborn and ignorant enough to keep on my way (but look forward to any comment or correction), and leave to posterity to decide if there's a method in the madness.
Dear Anatolij, at the moment I try to focus on curves on the sphere and their tangents. I want to determine the homogeneous coordinates of a tangent, and to express some of its properties in the language of the algebraic system above. On the first picture attached there is a bigger yellow circle of radius 60° with tangents and a smaller yellow circle of 30°, its polar circle, which represents the derivative of the first circle. The green circle in the middle is of radius 45°. It coincides with its derivative, but its elements are not self-conjugated. I added the respective expressions for the circles in homogeneous coordinates. For the second picture, I begin with a definition: In this algebraic system, elements a and b are conjugated if at least one element x exists for which ax=b. It can be proved that in this case element y also exists for which by=a (symmetric relation), and there is more than one x or y with this property. The expressions on the second picture describe the property of polar reciprocity (if a pole is conjugated to another polar, then the other pole is conjugated to the other polar): If there is x for which (ab)x=(cc)(dd), then there is y for which (cd)y=(aa)(bb). Here aa, bb, cc, dd represent the tangents to elements a, b, c, d, resp.. If we define spherical ellipse with constant spherical sums from the foci (ellipse is the same as hyperbola, parabola is a special case), then I think that the same statement is valid for ellipses. Can this statement be generalized for other spherical curves or at least certain arcs of these? Another question: Can we replace the definition of aa as tangent to element a with another: a*a* is the normal line to element a? Can we save some form of polar reciprocity in this case? For circles we get degenerate case with all normals being concurrent in the center; but what happens with non-circle ellipses?
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something on simplicial surfaces and  regular and mean valence on simplicial surfaces.
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For a concrete approach to triangulation have a look at books on computer graphics e.g. this one:
@book{de2000computational,
title={Computational geometry},
author={De Berg, Mark and Van Kreveld, Marc and Overmars, Mark and Schwarzkopf, Otfried Cheong},
year={2000},
publisher={Springer}
}