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# Algebraic Topology - Science topic

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I noticed that there is a structural similarity between the syntactic operations of Bealer's logic (see my paper "Bealer's Intensional Logic" that I uploaded to Researchgate for my interpretation of these operations) and the notion of non-symmetric operad. However for the correspondence to be complete I need a diagonalisation operation.
Consider an operad P with P(n) the set of functions from the cartesian product X^n to X.
Then I need operations Dij : P(n) -> P(n-1) which identify variables xi and xj.
Has this been considered in the literature ?
I guess operads are not so popular.
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I am starting a new area of research that is Algebraic topology. Kindly suggest some latest problems and related publications
@peterkepp
I have started reading algebraic topology from three book
Topology by J Munkres
Algebraic Topology by A. Hatcher
Basic of Algebraic topology by A. SHASTRI
But I can confused from where did I find the research problem and the recent papers are too advance for me
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How can I find a list of open problems in Homotopy Type Theory and Univalent Foundations ?
A list of open problrems for hpmotopy type theory is presented in HISTORIC.
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Is the Poincaré Conjecture relatively easy to prove for the case of simply connected real analytic 3- manifolds ?
Stiefel Manifolds and Grassmann manifolds are two examples of Smooth/analytic Riemannian Manifolds, with its far reaching applications in data analysis, such as classification, clustering and object tracking etc.
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How can I find a list of open problems in Homotopy Type Theory and Univalent Foundations ?
Regards,
Germán Benitez Monsalve
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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 ?
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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It is well known the theoretical applications of generalized open sets in topological spaces, for example we can by them define various forms of continuous maps, compact spaces, separation axioms, etc. My question is: what the practical (reality) applications of generalized open sets such as semi-open and pre-open sets?.
Are these sets used to modeling some phenomena or problems?
Agreed with J. G. von Brzeski
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If I wanted to link algebra and topology in order to specialize in algebraic topology (mathematics), what researches would you recommend me to start reading with?
Hamed Sadaghian Thank you, this is very useful for me.
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The version I came across gives the upper and lower bounds in terms of simple binomial coefficients expressed in terms of the degree of the Conway polynomial and the exponent of z. Any other known references for bounds on the Conway (or other skein) polynomial coefficients related to braid classes would also be of interest.
see
1-A C n -MOVE FOR A KNOT AND THE COEFFICIENTS OF THE CONWAY POLYNOMIAL
Journal:
Journal of Knot Theory and Its Ramifications
Year:
2008
2-Concordance invariance of coefficients of Conway's link polynomial
Tim D. Cochran
Journal:
Inventiones mathematicae
Year:
1985
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In general in mathematics we work with groups that includ numbers, generally we work with constants.
So for exempel: For any x from R we can choose any element from R we will find it a constant, the same thing for any complex number, for any z from C we will find it a constant.
My question say: Can we find a group or groups that includ variabels not constants for exemple, we named G a group for any x from G we will find an x' it's not a constant it's another variabel ?
and if it exisit can you give me exempels ?
Thank you !
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Hamilton defined the Ricci flow in 1982 to prove the famous three-dimensional sphere theorem. How singularities of solutions of the Ricci flow could identify the topological data predicted by William Thurston's geometrization conjecture?
How did Thurston's geometrization conjecture eventually lead to the proof of the Poincaré conjecture?
congratulations
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The problem is:
Let Z denote the set of all integers.
Consider Z/nZ = Zn as trivial G-module.
Show that there is a isomorphism between the First Homology Group H1(G, Zn) and the factor G/G'Gn;
Where G' is the commutator (derived) subgroup G'=[G,G] and Gn is the subgroup of G generated by all its n-th Powers.
• With this isomorphism proved, we want to conclude that if G is a finite p-Group, then H1(G,Fp) is a Fp-module (vector space) with dimension equal the number of least generators of G.
The difficulty depends on your view of the (co)homology groups. Generally speaking, explicit formulas involving (co)cycles are not of great use. The best approach is to go back to the fundamental property of the (co)homology functor, as is done in Serre's "Local Fields", chap.VII, which is in my opinion the best introduction to the subject.
Let us look e.g. at homology. Given a group G and its group algebra A = Z[G], an exact sequence of Z[G]-modules o --> A --> B --> C --> 0 gives rise to an exact sequence of co-invariants ... --> AG --> BG --> CG --> 0, where H0 (G,M) := (M)G denotes the maximal quotient of M on which G acts as identity. But this sequence is not exact on the left, where it extends to an a priori infinite exact sequnce of homology groups ... --> H1(G,A) --> H1(G,B) --> H1(G,C) --> H0(G,A) --> ... The philosophy is that the homology functor is an automatic machinery which transforms a short exact sequence into an infinite exact sequence on the left. Think of the Taylor development of a function f at a point M, which you can write down without thinking, but the terms of the expansion give you valuable information on f itself in the neighbourhood of M.
1) Let us apply this to your specific problem. Denote by IG the augmentation ideal of A , i.e. the ideal generated by all the elements 1-g when g runs through G. Obviously H0(G, M) = M/IGM . Consider the natural exact sequence 0 --> IG --> A --> Z --> 0, where G acts by conjugation on G and as identity on Z, and the rightmost map consists in taking the sum of the coefficients of an element of the group algebra A. Taking homology gives ... --> H1(G, A) --> H1(G, Z) --> H0(G, IG) --> H0(G,A) -->... But A is a free A-module, hence H1(G, A) = 0, and the image of H0(G, IG) --> H0(G,A) is null practically by definition, so that H1(G, Z) = IG/ IG2 , and one can check "by hand" that the map g --> 1-g induces an isomorphism H1(G, Z) = G/G' (remember that G acts by conjugation on itself).
Consider now the exact sequence 0 --> Z --> Z --> Z/n --> 0, where the leftmost map is multiplication by n , hence is injective.Taking homology gives H1(G, Z) --> H1(G, Z) --> H1(G, Z/n) --> Z --> Z . Applying our preliminary result H1(G, Z) = G/G' , we get immediately that G/G'Gn = H1(G, Z/n) as desired.
2) Suppose now that G is a finite p-group, where p is a prime, and write Fp instead of Z/p. After 1), we know that G/G'Gp = H1(G, Fp). Both sides are finite abelian groups killed by p, so they can be viewed (when written additively) as Fp finite - dimensional vector spaces. Let d(G) denote the minimal number of generators of G. Passing to the quotient, we get obviously d(G) >= d(G/G'Gp ), and this last quantity is = dim G/G'Gp by elementary linear algebra.
The reverse inequality holds true, but its proof is somewhat independent from the previous homological considerations. Write G*=G/G'Gp. A theorem of M. Hall in group theory implies that a homomorphism f : G1 --> G2 is surjective iff the induced map f* : G1* --> G2* is surjective. The equality d(G) = dim G/G'Gp is known as the Burnside basis theorem.
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Let Z denote the set of all integers, and let G be a finite cyclic Group.
For every ZG-module A, and n=1,2,3...
Show that:
Hn(G,A) is isomorphic to Hn+1(G,A).
I think that you can use the article
FROM HOMOLOGICAL ALGEBRA TO GROUP COHOMOLOGY
Semester Project By
Maximilien Holmberg-Péroux
Responsible Professor
prof. Jacques Thévenaz
Supervisor
Rosalie Chevalley
Spring Semester
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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
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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There are many different forms of vortexes in the physical world. Here are two common examples:
Tornado vortex. These are common atmospheric disturbance in the Midwest in North America (see, e.g., the attached image).
Seashell vortex. Seashell sometimes are vortex-shaped (see, e.g., the attached image).
I am hoping that followers of this thread can post or point to other examples of vortexes in the physical world.
Here are several more examples more examples:
Photon vortex and electromagnetic vortexes.
This interest in vortexes is related to photon vortexes and electromagnetic vortexes.
Shape Vortexes [Geometry and Algebraic Topology].
From a shape theory perspective, it seems that many physical shapes are non-concentric, nesting cycles. In homology theory, the analogue of a physical vortexes is a collection of non-concentric, nesting 1-cycles. A 1-cycle is a collection of connected edges or surfaces that form a connected path so that one can find a path between any pair of vertices in the 1-cycle.
The most common examples of vortical structures are found in classical fluids; at the laboratory scale, we can mention
• Kelvin-Helmholtz billows
• Hexagonal vortical patterns in fluids heated at the lower boundary
• Von Karman vortex wakes
In the rotating stratified fluids in geophysical systems, we have a multitude of vortices at different scales; they include
• tropical cyclones
• Vorticity tubes associated with tornadoes
• Rotors on the lee side of the mountain ranges
• Lagrangian coherent vortical structures in the oceanic flow
The main characteristic of all these structures is that they do not keep their identity for a very long time, the main exception being perhaps the red spot of Jupiter which has preserved its integrity over the last few hundred years.
The completely new reality of vortex systems is in quantum fluids where they can last indefinitely.
On a larger scale, in space, practically all forms of matter organization come in the form of swirls with the most spectacular examples provided by rotating galaxies. It is fairly easy to conclude that the vortex movement is ubiquitous in all physical systems, from the smallest to the largest. There are even some attempts to connect the scales proposed in the framework of a Cantorian Superfluid Vortex hypothesis; these attempts are very similar, in many respects, to Lord Kelvin's old hypothesis explaining atoms as vortex structures.
For an in-depth discussion of all these fascinating topics, please see a very interesting article by Frank Wilczek entitled: "Beautiful Losers: Kelvin's Vortex Atoms". The text is available at:
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A compact topological model which relates displacement and rotation isotropically leads to diverging geodesics. A volume of points which individually follow these geodesics becomes expanded during the transport.
The volume expansion red shifts electromagnetic energy transported with the volume. Receiving the enlarged volume after a long transport  leads to the impression of a time dilatation.
What is the impact of this volume expansion on particles or matter transported along cosmic distances? Is there a loss of kinetic energy? Does this mean that a very weak static gravitational potential exists, which extracts energy from movements over cosmic distances?
Attachment: Description of the model
The model relates an orientation O=(x/R, y/R, z/R) to a point P=(x,y,z) and also relates a rotation vector V=(dx/R,dy/R,dz/R) to a displacement D=(dx,dy,dz).The axes of the rotation operation is perpendicular to the orientation O and the rotation vector V. R is in the range of  about 14 billion light years.
The model has the following properties:
The coupling properties of the three axis rotation operations cause the divergence of the geodetic lines. The rotation operation takes place in an image space. A displacement operation in the native space includes a transformation to the image space, a rotation and a back transformation. The Quaternion formalism allows finding the diverging properties of the geodesics in a straight forward way.
The existence or non-existence or transitions between different kinds of particles may have small scale geometrical implications. This has nothing to do with a possible existence of gravitational impact of a curved large scale geometry.
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Most of the modern texts use category theory for algebraic topology rather than set theory. What are the pros and cons of both the set theory and the category theory in this formulation.
Moreover, is it necessary to use Category theory or set theory suffices for all the concepts.
@Romeo P.G
For sure this is true. There are many examples from algebraic geometry. I often work with kinds of locally ringed spaces - think of just generalising the sheaf of smooth functions on a manifold -  and these are not set theoretical objects' in the sense they are not simply a set of points (together with a topology or something like that).
But of course, as ringed spaces morphisms can be defined as so can their category. The point is that the objects are not sets, though in the cases I deal with, the Hom sets are genuine sets.
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I am looking for an algorithm that constructs Cech Complex for point clouds.
I am specifically looking for some thing in Mathematica but any resource would be very helpful.
I am looking for speeded up implementations as the traditional approach is very computationally intensive.
Zubair
This is an important question with many possible answers.
A good overview of computation for persistent homology is given in
An approach to constructing Cech complexes is given in Section 2.2.1, starting on page 5 in
An approach to approximating the Cech complex in Euclidean space is given in Section 4, starting on page 13, in
One of the most interesting ways to compute Cech complexes is introduced in Section 4 (especially, 4.3.1, starting on page 75) in
More to the point, this thesis includes a hefty survey of Mathematica implementations useful in computing Cech complexes.
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Some easy to understand references from the literature will be much appreciated. Thanks in advance for all the answer.
Regards,
Zubair
See pgs 102-104 of Hatcher's "Algebraic Topology" for a precise definition of Delta complex. It roughly says that the interior of any simplex is mapped homeomorphically to its image in the space, though interiors of different faces may be identified.
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What is quantitative topology and what quality of a topological space does it aim to quantify?
This is a great question.
The beginning of an answer to this question can be found in
Real Analysis, Quantitative Topology, and Geometric Complexity
Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities
Typical problems tackled by quantitative topology are briefly described in
DIRECTIONS IN QUANTITATIVE TOPOLOGY
A detailed but narrow study of one aspect of quantitative topology is given in
Centralities in Simplicial Complexes
Another aspect of quantitative topology is given in
Analysis and prediction of protein folding energy changes upon
mutation by element specific persistent homology
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Kindly suggest good books and other references to the shape theory using a notion of fibrations.
Good reviews are also welcome.
I thank all of you in advance.
for fibration the best book is Dale Husemoller "Fiber bundles " book
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The Inscribed Square Problem (ISP) is difficult because the general Jordan curve (J) may take countless forms. So we should seek out its invariant properties and apply our knowledge of analytic geometry (equations of parallel and perpendicular lines and Pythagoras’s Theorem) and Euclidean geometry (similar or almost similar triangles with common diagonals, Law of sines and the Law of cosines, angular rotation about a point and translation from a point to another point).
We believe the first step is to inscribe the general Jordan curve (J) in a square, S (please see 'Jordan Curve (2).docx' attached file below for some details of S),  with a diagonal whose endpoints are the two points on J that are farthest apart. And we define the Center Point (Cp) of J as the intersection of the two diagonals of S. And the work of analysis can begin there… And we believe the result should be affirmative!
Let J’ be the complement object of J such that S = J’ + J.
Let’s fill our square (S) with unit squares (s1), and let’s assume our  exceptional inscribed Jordan curve (J) does not have four points on it that form an inscribed square of any size from s1 to sn less than the size of S = J’ + J. We should expect a mix of square sizes, orientations, and shapes in and about J without violating our prime assumption.
Moreover, we need to formulate boundary conditions between J and J’ and search for a contradiction to our assumption about J. It could be a limiting process or some kind of process of elimination.  We are not sure what will work initially.   But we have hope in our formulation of the boundary conditions between J and J’ that we shall discover the correct process or procedure and solve our problem affirmatively.
Notes: We can arrange and rearrange (translate or rotate) our unit squares (s1) appropriately without affecting our general exceptional Jordan Curve (J).
"Jordan Curve Theorem:  A simple closed curve, J, partitions the plane into exactly two faces, each having as boundary." -- (http://www.ti.inf.ethz.ch/ew/lehre/GT06/lectures/PDF/lecture13.pdf).
Furthermore, we believe we can have a tentative proof of ISP in less than a week. Wish ourselves luck! :-)
This is a great question.
For an answer to this interesting question, see
arXiv:1611.07441v1
at
For periodic variants of the square peg problem, see Section 4, starting on page 12.
For the algebraic square peg problem in a 2014 M.Sc. thesis, see
arXiv:1403.5979v1
also
at
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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.
Look at this doc it may be helpful for your topic. Good luck.
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We call local homeomorphism of R a function f:U--->R with: U open neighborhood of 0 \in R, f(0)=0, f injective continuous and open.
We identify two local homeomorphisms of R if and only if they coincide in a suitable neighborhood of 0.
We denote by LH the set of all equivalence classes of local homeomorphisms of R.
LH becomes a group when endowed with the operation of composition of functions.
I am interested in results describing the structure of the group LH.
It seems to me "germs of homeomorphisms at 0"
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Fix a positive integer n.
We call local homeomorphism of R^n a function f:U--->R^n with: U open neighborhood of 0 \in R^n, f(0)=0, f injective continuous and open.
We identify two local homeomorphisms of R^n if and only if they coincide in a suitable neighborhood of 0.
We denote by LO the set of all equivalence classes of local homeomorphisms of R^n.
We call local diffeomorphism of R^n a local homeomorphism of R^n which is smooth and has a smooth inverse.
We identify two local homeomorphisms of R^n if and only if they coincide in a suitable neighborhood of 0.
We denote by LD the set of all equivalence classes of local diffeomorphisms of R^n. LO is a group with the compiosition of functions, and LD is a subgroup of LO.
Is LD a normal subgroup of LO? If yes then what is known about the quotient group LO/LD? References are very welcome.
A little comment. In the definition of local homeomorphism  f:U--->R^n there is no need to say that f is open. Openness follows from continuity and injectivity, according to  the Brower invariance of domain principle. See the exposition written by Terry
Tao in his blog:
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It is known that homeomorphism between topological spaces implies homotopy but the converse is not true. And if I'm not mistaken, some topological properties such as separablility are not preserved under homotopy. My question is:
Let X be separable, completely metrizable topological space and let Y be a topological space. If X and Y are homotopic, does that mean that Y is also separable, and completely metrizable, necessarily?  I searched the basic algebraic topology books but they just repeat the same piece of information and  I couldn't find explicit answer.
A lot of good answers, I'll try to explain intuitively why this is the case, but please take this with a grain of salt, as I'll be arguing heuristically.
At a very fundamental intuitive level, a homeomorphism h: X -> Y, is the requirement that there is a correspondence between open sets in the topological space Y correspond precisely to open sets in the topological space X (and vice versa).
If you think back to the definition of a topological space (from which all of the topological properties arise), it is entirely dependent on which open sets exist in your given space, and what properties arise as a consequence, such as compactness or connectedness. Consider the two extreme cases - If you were to look at |R under the discrete topology (every possible subset is declared an open set) this is non-compact, and separable while |R under the trivial topology (only |R and the empty set are considered open sets) this is compact and connected.
In your specific question, the metric topology can be induced on space X... and hence you'd like to know if a homotopy is sufficient to induce metrizability on space Y. However, we would need some correspondence between the induced open sets of Y and the open sets of X to make sure that we don't somehow "reduce/increase the number" or somehow fundamentally disrupt the behaviour of open sets (very loosely speaking, since we're dealing with issues of countability here).
This condition necessarily means we have a homeomorphism from X to Y...and since we can't guarantee this with homotopy alone, we cannot guarantee X and Y have the same topological properties.
And just to add an explicit example, as Vladimir said above, let X be D^2, the unit 2-D Disk, and {x} be the origin of \R^2. There exists a homotopy that maps X to {x} in a continuous manner (imagine X 'shrinking' forever until a single point remains in the limiting case). But clearly, no homeomorphism exists as these are very different topological spaces. (Such spaces X are said to be contractible)
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The string detained between two points can sustain harmonic motion.  At the string midpoint, the potential and kinetic energy are inverse, so that when the string is in the mid-line, the kinetic energy is 1 and the potential energy is 0. When the string is at the boundary the string stops for an instant, so the kinetic energy goes to zero while the potential energy is 1.
A node is defined as a point on a string where there is no movement possible, so that both the kinetic and potential energy are zero.  The fixed-point theorem says if pitch is a real function defined on [0, 1], then there must be a fixed point on the interval.
Since the kinetic and potential energy is just the result of basic trig functions sin and cos, it seems clear to me then that wave reflection cannot occur at the string endpoints.  The endpoints are fixed points which are in effect fulcrums with a fixed-point position so that length L = 1 is a bound variable.  The fulcrum allows the fundamental in the monochord to drive the string on the other side of a node, but the condition for wave reflection does not exist at the node.
When a sin wave crosses zero there is no requirement that the point is fixed, so the boundary cannot simply be added to the wave function arbitrarily without changing the nature of Fourier analysis.
If the waves reflect at endpoints, do they also reflect at nodes that are not enpoints? Of course not!  But then, in the 1/3 mode, what makes the middle wave where there are 4 nodes and 3 waves?  Is the middle wave the reflection of 2 traveling waves between the two non-endpoint nodes?
The boundary condition for traveling wave reflection is 1, 0 , 1 which is clearly a false statement.
Significantly, the frequency and the wave length are bound by the string and not free variables subject to real analysis as continuous variables.  Nodes and waves cannot add at the same point.
If physicists really think there are two traveling waves on a string moving in opposite directions that make a standing wave, but no one can see or demonstrate these waves, then maybe they have action-at-a-distance wrong, too.  After all, the basic error is assuming frequency is continuous, and then using functions with free variables adjoined with arbitrary integers.
But in the typical reflection the kinetic energy goes to zero at the wall while the potential energy is at a maximum so there is a force that explains the reflection,  If a wave travels down a funnel it does not reflect.  So I say these imaginary traveling ways are nonsense.  Where is the equation of motion if the wave is standing?  Where is the stroboscopic image of these two waves that some how reflect to make one.  There is a curve lifting function between the string at rest and the string in wave form that is point-for-point so the only force vectors are orthogonal to the string axis.
Statements about the string without quantifiers are provable.  Can you prove there is a mode lower than the fundamental with wavelength 2L? Of course not.
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I want a way to avoid the Mersenne Theory notion that the wavelength of the fundamental is 2L, not L.  The problem is that assuming 2L means the fundamental is not the lowest mode in the system, which of course is absurd.
So in a previous question I asked if there is a curve-lifting map that sends the interval 2L to L.
The reference I am using here is Algebraic Topology by CRF Maunder, Dover 1996. Page 84, Example 3.3.21
"Consider the real projective space RP3 again.  By Proposition (ref) this space is S1 U E2, where f:(cos x, sin x) = (cos 2x, sin 2x)" where E is Euclidean space and I replaced the angle theta with x for typing purposes.
There follows a figure showing alpha as the generator of the group Z2.
This seems to describe the boundary of the string vibration modes which have the boundary condition (0, 1, 0) for the fundamental n = 1, (0, 1, 0, 1, 0) for the first overtone.  Each mode has an n x n square matrix where the determinant is 1 because the matrix values are all zero except 1 where the wave is on the diagonal.
The string is the union of waves and nodes so it is Z2.  Then there is one generator, not 2 (as in the torus).
This does not result in the same model of the string as Mersenne.
Dear  Terence B Allen,
I need to understand better your question.
At the moment I note only if we apply the algebraic rule for complex multiplication we have
f(z)= z^2 =(x + iy)(x + iy)= x^2 - y^2 +2ixy. This mapping maps the circle S^1 onto itself. But if z travels along the circle S^1 one times then the image
f(z)= z^2 goes two times around S^1. We can also write f(e^{i\theta})= e^{2i\theta}, where \theta is the measure of the angle.
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Bounded sets are defined on general topological vector spaces, topological modules, topological rings and topological groups. But, I could not find a suitable definition of a bounded set in a topological field.
Any nice material regarding the question will be appreciated.
Boundedness is a topic of bornology. You can introduce many bornologies on your spaces, so you will get many notions of a bounded set.
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In "A Geometry of Music" by Tymoczko the author describes a lattice that "lives in what mathematicians call 'the interior of a twisted triangular two-torus,' otherwise known as a triangular doughnut."
Clearly the torus is isomorphic to a closed, bound rectangle in  a plane. I am assuming that the twisted torus is a torus that has a composition function that forms a line that winds around the torus.
It seems to me then that the twisting torus is really a projection of a sphere in two dimensions.  The winding number implies there is a curve lifting function.
My assertion is the torus T is an approximation of the sphere because the inclusion map i: Sn→ (R3 – 0) induces i: Sn/T→ RP3 and the map f:(Rn+1 – 0)→ Sn induces f:RPn→ Sn/T.
This seems to be what the author implies anyway but does not say: "Pitch-class space is formed out of pitch space when we choose to ignore, or abstract away from, octave information.  As a result, many properties of linear space are transferred to circular pitch-class space."
Doesn't this last quote indicate that the pitch line is automatically a circle in RP3? Then the manifold is a sphere and not a torus, right?
My goal here is to show that music spaces are projective but have an affine cover with at single face in R1.
The torus is absurd: it cannot reduce to a point.
Dear Terence, my limited knowledge doesn't permit me to speak about those "music spaces". I am just surprised by a statement you made: """Clearly the torus is isomorphic to a closed, bound rectangle in a plane""". Not at all. Consider a rectangle ABCD. To get a torus, one has to identify AB with DC, and then identify the two circles such that A  is identified with B. So you get ABCD / rel, where rel is an equivalence relation, and the result of this factorisation of a topological space is not at all an isomorphic (homeomorphic) topological space. For example, let M be the middle of AB translated with epsion to the right side and N be the middle of DC translated with epsilon to the left side. The rectangle \setminus MN is no more a connected topological space, while the torus \setminus MN is still connected. Of course, there are thousands of other reasons why they are not isomorphic. Similarly, rectangle, torus, sphere and projective space are pairwise not isomorphic (not homeomorphic).
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The terminology in this question is taken from "Algebraic Topology" by CRF Maunder. See Problem 9 on Page 59.
"Let H be the abstract 1-dimensional simplicial complex with vertices a0, a1, a2, a3, a4, a5, each pair of vertices being an abstract 1-simplex. Show H has no realization in R2."
Note that I have added a5 to the author's text.
The reason I ask is this. We have the guitar tuning as the union of 6 string intervals defined by a 6-tuple representing a point such as (0, 5, 5, 5, 4, 5) or EADGBE, so each string is itself a 1-simplex.  Note that because each tuning interval is already defined on the system fundamental the secondary string spectrum is already inside the system fundamental spectrum.  When the strings are subsets of the system fundamental we say the guitar is "in tune".
If the vertices are the fundamentals of six guitar strings and the each string is defined by an interval between the fundamental state of system and the string, so the interval is always a known as a whole prime number defined on the system fundamental, then does not the solution to the above problem show that guitar music cannot be realized in R2?
I would like to prove in general the structure of music is 3-fold and not 2-fold as Euler thought.
The answer is No. It is a standard fact that the complete graph K_5 on 5 vertices (i.e.a_0, ...,a_4 with all pairs of vertices joined by edges) is non-planar, i.e. it does not admit an embedding in the plane. This is also true for the complete bi-partite (3,3) graph.  It is a non-trivial theorem of Kuratowski that any non-planar graph contains one of these two as a subgraph. See the wikipedia article https://en.wikipedia.org/wiki/Kuratowski%27s_theorem and references therein.
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We like to think that music is mathematical but according to Wikipedia there is no axiomatic basis for music.  How are musical sets constructed using basic set theoretic tools of union, intersection, and complimentation?
I propose using sequences in tablature music for guitar to study how polyphonic objects are constructed by adding point-wise limits to the continuous function of pitch.  Tablature music is a rich algebraic language that has substantial archival, educational, cultural, and economic significance but no mathematic theory of tablature exists.
I have mastered reading and writing tablature.  Anyone familiar with my study of multiple parathyroid tumors in the Journal of Theoretic Biology back in 1985 can see I have substantial training in mathematics.
I need a mathematician who knows model theory or algebraic topology to review my work. Please see the attached manuscript if you are interested in making a modern mathematic model for music theory.
The basic problem: Are music sets constructible?
I am a professional musicologist and music theorist, and I must confess that what you people write is Chinese to me. Music can be notated in staff notation, or in various kinds of tablature, or in various types of alphanumeric notations. Passing from the one to the other (which is called "transcription") can now be automated, especially with the help of MusicXML and MEI (Music Encoding Initiative).
BUT this all is valid only provided that the music can be notated at all. And the transcriptions exclusively concern the notated aspects of the music. Notation is utterly unable to notate everything. What it notates is called "notes", which represent a combination of pitch and duration. But, because notations (like music itself) are semiotic systems, the notated pitches and durations are but abstract categories escaping any attempt at quantification. Notes are distinctive elements of a semiotic system. And, in addition, not every music can be notated, which probably means that not every music is semiotic.
The so called pitch classes of musical set theory are abstractions. If they are considered classes properly speaking (which I doubt they are), then the members of these classes must themselves be considered abstractions, and so also are the intervals between them. And none of this can be evaluated in terms of "true/false". Musical set theory is not a set theory in the sense of Zermelo-Frankel.
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I need to implement various kind of topological chain. Arbitrary dimension and efficient traveling in the data structure is a necessity, space efficiency is less crucial. Construction of complexes from other complexes is appreciated (suspension, product, quotient, etc.). I am mostly interested in the combinatorial aspects, not in the embedding in an euclidean space.
Take a look at the GUDHI library: http://gudhi.gforge.inria.fr/doc/1.2.0/
based on Simplex Tree: An Efficient Data Structure
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Let (X,d) be a metric space, f: X->X be Borel measurable. Let (K(X),d_H) be a space of compact subsets of X with Hausdorff metric. Is f:K(X)->K(X), where f(A)=\cup_{x\in A}{f(x)} Borel measurable? Is there needed separability of X?
Instead of an answer I have an example. Let A be nonmeasurable in X, f:X->X, f(X)=Z f(x)=x, for x in Z, f(x)=z, z in Z. Then f^{-1}(Z) =X. The preimage of nonmeasurable set is measurable.
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Dear Professor,
I have some questions about the labelled configuration space in homotopy theory. I post my questions at: http://mathoverflow.net/questions/225990/questions-about-configuration-spaces-in-homotopy-theory.
When I use the definition of smash product (i.e. the quotient space of Cartesian product by identifying the wedge product), I always get something wrong, different from the equations. Could you give a help?
Thanks!
Best regards,
Ren Shiquan
Hello.
Handbook of Algebraic Topology edited by I.M. James, pages 561-566 may have what you want.
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Omega groupoids are equivalent to certain crossed complexes in Algebraic Topology. Is there a publication that defines the 3- or n- cube groupoids, not the n-groupoids as a particular case of
n-category ?
Dear Ion C.Baianu
For me a groupoid is a small category in which all morphisms are invertible. Would you please provide the explicit definition of omega groupoid?
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As the title suggests, how do i see that for any n, the covering map S^{2n} → RP^{2n} induces 0 in integral homology and cohomology, except in dimension 0?
Maybe you would like to see geometrically how the n-th homology of  S^n can be killed when projected to RP^n? Take a triangulation of the n-sphere which is invariant under the antipodal map  -I , e.g. the generalized octahedron (which exists in every dimension). The projection is a triangulation of the projective space RP^n, and each of its simplices is hit twice by the projection map S^n \to RP^n. However, when n is even, the antipodal map -I on R^{n+1} reverses orientation (determinant -1) but preserves the exterior normal vector field of S^n, thus it reverses the orientation on S^n. Hence each simplex contributes with both signs and you get extinction, while in odd dimensions you obtain a factor 2 (the mapping degree of the projection map).
Best regards
Jost
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Specifically, let M be a connected closed orientable n-manifold, n > 1, such that there is a map f : S^n → M of nonzero degree, i.e. for which the image of the generator of H_n(S^n) is equal to a nonzero multiple of the generator of H_n(M). How do I see that π_1(M) is finite?
Since n>1, we have that π_1(S^n)=0. Thus, by the lifting criterion for covering spaces, the map f : S^n → M lifts to a map f~ : S^n --> M~, where M~ is the universal cover of M.
Now, if π_1(M) were to be infinite, the universal cover of M would be non-compact, in which case H_n(M~)=0.  But this would force the degree of f to be zero, a contradiction.
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I want to accelerate my research in topological quantum field theories, I know that in N=1 super Yang-Mills there is conformal twist, but how to get to obtain topological twist in this case is actually my problem, does somebody have any idea?
Dear Nounahon,
I would suggest you take a look at my papers on Haag Theorem and Politcial Environment and Stock Markets papers on my RG page. They might be helpful. Although I do not know what exactly you mean by topological twist is it conformal flows or heterotic flows that you are looking for? SKM QC FEPS(D)
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Please explain digitalbtopology, digital manifold , and digital algebraic topolog.
This is a good question that taps into cutting edge research on applications of the usual notion of homotopy considered in the context of sources of digital information sources such as digital images.
A good place to start in answering this question is in
Sang-Eon Han, KD-(k0, k1)-HOMOTOPY EQUIVALENCE AND ITS APPLICATIONS, J. Korean Math. Soc. 47 (2010), No. 5, pp. 1031–1054:
Digital homotopy is defined in the context of the Khalimsky digital topological category defined by
1.  class of objects
2.  KD-(k0, k1)-continuous maps as morphisms.
A map f : X → Y is KD-(k0, k1)-continuous at a point x0 ∈ X, provided
(1) f is continuous at the point x0; and
(2) for any Nk1 (f(x0), ε) ⊂ Y , there is a neighbourhood Nk0(x0, δ) ⊂ X such that f(Nk0(x0, δ)) ⊂ Nk1 (f(x0), ε), where ε, δ ∈ N.
The map f : X → Y is KD-(k0, k1)-continuous on X into Y, provided it is continuous at every point x in X.   From this, we obtain the notion of a digital homotopy.  Then we say that F is a digital KD-(k0, k1)-homotopy between f and g, and f and g are KD-(k0, k1)-homotopic in Y.
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Condition (7) of Theorem 4.54 in the book Axiom of Choice" by Horst Herrlich is the sentence: Each second countable topological space is separable.
Theorem 4.54 of this book says that (7) is equivalent to CC(R) where CC(R) is the axiom of countable choice for the real line R. I am sure that this is not true in a model for ZF because I can prove that (7) is equivalent to CC where CC is the axiom of countable choice. I  believe that I can prove that CC(R) is equivalent in ZF to the statement that every second countable T_0-space is separable. I would be grateful if you could tell me whether someone else has ever noticed this mistake of H. Herrlich and published it somewhere. If the answer is YES, please, tell me where and by whom the correction of this Theorem 4.54 was published.
Dear Eliza,
Regarding your question about  Horst's mistake I believe that it was just a typo. The reason I believe this is that  I have proved this result and I have communicated it to him. See the the reference ker 200? in the references in his book. Unfortunately the status of the paper is still unknown. Recently Paul Howard ask for copy of this paper and I have prepapred a pdf file which I will forward to you via e-mail. I think that Horst told me once that some guy from Brazil also notice this typo. However, I do not know whether he has publish anything.
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What is the Euler-Poincare principle and how to apply it to (1.4) and get (1.6)?
What is meant by the Euler-Poincare principle in the text is that in a complex of finite dimensional vector spaces the alternating sum of the dimensions of the spaces coincides with the alternating sum of the dimensions of the homology groups. (The standard application of this is that the Euler characteristic of a finite CW-complex coincides with the alternating sum of the number of cells in each dimension.
If you complex is not just a complex of vector spaces but a complex of representations of a semi-simple Lie algebra $\mathfrak g$ with $\mathfrak g$-equivariant differentials, then you can look at weight spaces in the representations. Since all differentials map weight spaces to weight spaces, you can apply the above argument for each fixed weight (provided that all weight spaces are finite dimensional). So for each weight, the alternating sum of the dimensions of the weight spaces in each homology group coincides with the alternating sum of the dimensions of the weight spaces in each chain group. Since the character is just the formal sum of the dimensions of the weight spaces, it follows that the alternating sum of the characters of the homology groups coincides with the alternating sum of the characters of the chain groups.
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A polyhedron mesh consists convex polyhedrons. Is this formula right: F+1=E+K, F=number of the interior faces, E=number of the interior edges, K=number of the polyhedrons in the mesh? If yes answer, references?
Many thanks.
Eric is right. I am a finite element person and don't know much about geometry which is tooooo hard for me.  What we are doing is to divide a 3D convex polyhedron domain to many small convex polyhedrons shared faces. Then define piecewise polynomials associated with these partitions (or meshes) to approximate solutions of PDE.
Let me make it straight that for my situation, the formula is
n_0 − n_1 + n_2 − n_3 = −1.
with
n_0=total  number of interior vertices,
n_1= total number interior edges,
n_2=total number interior faces
n_3=total number polyhedrons (not just interior).
Is the above statement right?
Many thanks to Stam and Eric.
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how can i prove the following statement? In other words, how can i prove existence of the following norm?
Let K be any totally disconnected local field. Then there is an integer q=pr, where p is a fixed prime element of K and r is a positive integer, and a norm ∣⋅∣ on K such that for all x∈K we have ∣x∣≥0 and for each x∈K other than 0, we get ∣x∣=qk for some integer k.
I would like to also recommend chapter 2 ("Global Fields") of the classical book 3Algebraic Number Theory", Cassels & Fhröhlich ed., Acad. Press 1969, which gives a compact and complete account of the subject. Concerning your specific question, see especially §7, p.50
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Let C be closed Jordan curve in R 2 and for arbitrary circle L from property that intersection C ∩L contains 3 points follows, that C ∩L contains no less then 4 points. Is it true, that C also be a circle?
Dear Yurii, I am not sure I understood your question since the number of intersection points between any 2 closed curves in R^2 (counted with multiplicities) is even. Thus your condition is automatically satisfied.
Yours B.Shapiro
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Suppose G is a discrete group (may be infinite). Does there exist a topological space, say W, with finite dimension as a CW complex, such that G is the fundamental group of W?
But your topological  consideration  on G is  a motivation to ask the following question:
Question: Assume that G is  a topological group. Is there a CW complex X wich fundamental group is isomorphic to G and  the  standard  action of G on $\tilde{X}$, the universal covering space of X,  is jointly continuous in two variables,  that is the action as a map \tilde{X} \times G to  \tilde{X}  would be continuous.
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Suppose X is a simplicial complex and A its subcomplex. So, we can consider relative cocycles, in particular, relative coboundaries in dimension 2, for concreteness. Also, we can consider absolute cocycles in dimension 2, i.e. such that are not obliged to vanish on A.
And if we factor the space of these absolute cocycles by the space of those relative coboundaries - what is the generally accepted name for the resulting space?
Just a remark. (I write it for singular cohomology, but it should be similar for simplicial homology).
On the cochain level we have:  0-> C^*(X,A)->C^*(X)->C^*(A)->0, (where C^*(X,A)=Hom(Delta_*(X)/Delta_*(A),\Z) etc.). A relative cocycle (from C^*(X,A)) is represented in C^*(X) as an absolute cocycle that vanishes on A.
So what is 'absolute cocycles modulo relative boundaries'?
A special case gives an idea of what the result should be: If A is a component of  X, every absolute cocycle can be uniquely written as the sum of (the image of) of a relative cocycle and cocycle on A; taken the quotient with respect to the relative co-boundaries, we get an element of H^*(X,A) + Z^*(A),  i.e. the sum of a relative cohomology group and a cocycle group.
In general (if A is not a component of X) I would expect something like H^*(X,A) + some subgroup of Z^*(A) (didn't think about it, though).
Thus the quotient you are looking for is not a true 'cohomology'; in the setting of simplicial cohomology it would depend on the cell decomposition.
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Assume that a Lie group G acts on a  manifold  M, effectively. So the Lie  algebra g of  G is embedded in $\chi^{\infty}(M)$ in a natural way. (effective action: if x.g=x for  all x then g=e)
Under what dynamical conditions this embedding is  an "Ideal embedding"?
That is : The image of  g is  an ideal in the Lie algebra of  smooth vector fields on M.
By  dynamical  conditions I mean the dynamical properties arising from the action of G on M.
The Lie algebra of smooth vector fields on a manifold has no finite dimensional ideals.
Proof. Assume that g is a finite dimensional linear subspace in the Lie algebra of vector fields over the manifold M. Then M has a finite number of points p_1,\dots,p_k such that for any X\in g, X\neq 0, at least one of the vectors X(p_1),\dots,X(p_k) is not zero. (This follows from a compactness argument on the unit sphere of g with respect to an arbitrary positive definite inner product.) Choose a vector field X\in g, X\neq 0 and a point q, such that q is different form  p_1,...,p_k and  X(q)\neq 0. Then there is a vector field Y on M such that [Y,X](q)\neq 0. Choose a smooth function h such that h vanishes on an open subset containing the points p_1,...,p_k and h is constant 1 on an open neighborhood  of q. Put Z=hY. Then the Lie bracket [Z,X] is not zero, since
[Z,X](q)=[Y,X](q) \neq 0, on the other hand, [Z,X](p_i)=0 for i=1,...,k, thus, [Z,X] cannot be in g by the choice of the points p_1,...,p_k. This means that g cannot be an ideal.
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Let f and g be continuous self-maps on a compact metric space X, and the product system of f and g be mean sensitive. Is f or g mean sensitive?
The sensitive dependence on initial conditions.
Definition. A metric S-system (X,d) is sensitive if it satisfies the following condition: there exists a (sensitivity constant) c > 0 such that for all x∈X and all δ>0 there are some y∈Bδ(x) and s∈S with d(sx,sy) > c. We say that (S,X) is non-sensitive otherwise.
In my opinion, if the product system of f and g be mean sensitive then f or g is need not sensitive.
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Let f and g be continuous self-maps on a compact metric space X, and the product system of f and g be snydetically sensitive. Is f or g snydetically sensitive?
It the product system of f and g are strongly sensitive then both the systems f and g will be so. However, I doubt the result in case of snydetically sensitive. It would be interesting to see an example here.
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What is an example of a simple $C^{*}$ algebra which is acted by $\mathbb{Z}$ but this action can not be extended  to  an $\mathbb{R}$-action?\Are there such examples for compact operators as nonunital example and reduced C* algebra of F2, as unital example?
The motivation for this question is the following question in dynamical system:
"What diffeomorphisms of  a compact manifold can be considered as time-one flow map of  a vector field"?
In this classic case the orientation reversing is the first obstruction. Now my question is that "What type of non commutative obstruction can be introduced in the context of NC C* algebras?
Take any automorphism of a compact topological space X that has a non trivial action on homology, cohomology or any other homotopy invariant algebraic topological invariant.  Then the free group generated by the automorphism acts on X and so it acts on C^0(X). But clearly the Z action cannot be extended to an R action, because that R action must come from an R action on X [1]. Such an R action would be homotopically trivial contradicting that the generator of the Z action gives a non trivial algebraic topological invariant.
Example: the action of the mappingclass group on a two dimensional oriented surface Sigma  which give non trivial elements of  Sp(2g,Z). In particular the so called Dehn twists, give examples with non trivial actions on H^1(Sigma).
Now obviously this is not a non commutative example, but if I understand correctly the equivariant case (a topological space with a toplogical group action) gives rise to non commutative C^* actions that are related to the space with a a group action in much the same way as above, i.e look for a space with an equivariant automorphism that gives rise to a non trivial element in equivariant (co) homology or other equivariant invariants.
[1]  The Gelfand Naimark theorem is not just that the statement about maximal ideals of commutative C^* algebras correspond to points, but also that continuous *-homomorphisms correspond to continuous maps. Its a functorial equivalence of categories!
[2]  the Z- rank of H^1(Sigma) is 2g which comes with non degenerate symplectic form defined by cup product and evaluation on the fundamental class
[Sigma] \in H_2(Sigma).
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In the semialgebraic context, Delfs and Knebusch defined in 1985 their "locally semialgebraic spaces" and later (only Knebusch) "weakly semialgebraic spaces" as some infinite gluings of semialgebraic spaces. But the majority of model theory seems to be carried out in Mn, where (M,...) is a structure (a kind of "affine" situation).
Do model theorists need to pass to infinite gluings from time to time?
M^eq is not affine; it is an expansion of the vocabulary to guarantee that quotients are present. For definition of terminology see http://en.wikipedia.org/wiki/Imaginary_element
The examples there of elimination of imaginary' are dull; more interesting is algebraically closed fields.
The compactness theorem is sometimes used to reduce infinite coverings to finite.  See for example Poizat's stable groups for the interaction between definable converings and algebraic geometry.
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Note that a k- mean on a topological space $X$ is a continuous function $f:X^{k}\to X$ which is identity on the diagonal and is invariant under permutations.
In the literature is there an appropriate analogy of concept "mean" in the context of commutative Banach algebra? That is a morphism $\alpha:A\to A\otimes A$ which is invariant under flip-operator and its composition with mutiplicative operator would be identity. (with respect to an appropriate norm tensor product). (And the generalization to k-fold tensor product). Please see the attached file for a topological mean.
In [Hilton: A new look at means of topological spaces (http://www.emis.de/journals/HOA/IJMMS/Volume20_4/584598.pdf)], the dual notion of co-mean would provide such a morphism in the category of commutative R-algebras (the co-product being the tensor product in this category.)  I don't know whether this carries over to commutative  Banach algebras (is the coproduct in this category  the completion of the tensor product?)
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Let smooth maps f,g:N→S, where N and S are compact manifolds of the same dimension, be such that the generic point of S has an even number of preimages for f and an odd number for g. It can be shown that if S is a sphere, then f(x)=g(x) for some x, for arbitrary N. What other pairs (N,S) have this property? For what manifolds S may N be arbitrary?
Yes, and you can smooth out the map to count preimages of regular values, and connectivity needs to be assumed. Not clear, though, whether S can have aspherical type over Z2 or Z, as this may take more than algebraic topology even in 3D.
Pozdrawiam, Witold.
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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.
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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If polynomial of degree n with zeros z1,z2,...zn assumes maximum at w on |z|=1 and ak=1/|w-zk|. Is the following inequality true
a1+a2+....+an<=n.
@ Muhammad Sirajo; If |z_j|<1 and |w|=1, it is not necessary that |w-z_j|<1. For example if w=1 and z_j=-1/2 then |w-z_j|=3/2>1.
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How to determine and generate all different topologies of full binary trees? By different topology is meant that the tree cannot be obtained by some transformation that is composed by switch of subtrees of particular node
Dear sir,
The number you look for is related to Catalan number. You should tag your question with : "Combinatorics" and/or "Combinatorics on words". You will find more people that will help you to get the right algorithm.
Best wishes.
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I saw this sentence "The spin structure of the Fe sites is of the GxAyFz type in Bertaut's notation" in the literature(DOI: 10.1038/NMAT2469). Does anybody has the illustration of GxAyFz ??
You're welcome. Take a look at the article from http://www.iucr.org/__data/assets/pdf_file/0006/13929/final_23.pdf
Besides having nice visualizations, he seems to put his graphical routines at public disposal. His author, de Baer also wrote a book with McHenry: Introduction to Crystallography, Diffraction and Symmetry (see https://books.google.ch/books?id=NMUgAwAAQBAJ&pg=PA223&lpg=PA223&dq=visualization+of+magnetic+groups&source=bl&ots=VQ3MagtpF8&sig=oEbp78MRrjxPfdt2D1WUpDY5cwM&hl=en&sa=X&ei=ucjTVIjrFI3SaJjLgKgB&ved=0CEwQ6AEwBw#v=onepage&q=visualization%20of%20magnetic%20groups&f=false).  You might also want to look at http://www.cryst.ehu.es/
Good luck!
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In GRP, as in any category, isomorphisms are bimorphisms. How can I prove the converse, using the fact of being in GRP?
You cn also look the definitions of Balanced and regular categories. Infact GRP is a regular category and so bimorphisms are isomorphisms.
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We need an example
thanks.
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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!
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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I am looking for a homomorphism in a topological group that which uses by a single and fixed point. I mean for a fixed element of G, how we can construct a group homomorphism. For a fixed g, I tried the definition x \to gx, by this is not a group homomorphism. How we can do it?
We can add some hypothesis to G, like being locally compact or connected.
we can define a map Fg:G--->G by Fg(x)=gxg-1.
then it will be a homomorphism and also continuous map .
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Let G be a topological group and U be a fixed neighborhood of identity. For a fixed element g, consider gU=\{gu, u\in U\}. Do we have the following property in an Abelian topological group.
For each neighborhood V of identity there is a positive integer n such that gU is a subset of V^n?
In your previous question you asked if every neighborhood of the neutral element is absorbing. You have received a number of answers with various counterexamples. Every such a counterexample works also as a counterexample to your new question: if g is not absorbed by any Vn then gU is not absorbed neither.
Moreover, for your new question the answer is negative even for locally convex Hausdorff topological vector spaces  (considered as additive groups). Namely, let X be the space. Take U = X, then g+U = X, and surely there is a convex neighborhood of 0 V such that for every n  the set nV is not equal to the whole X.
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Closed 1-forms are well studied in foliation topology, algebraic geometry, and theory of manifolds. What are their applications in physics?
I would just like add a few remarks to the first part of M. Amiri's answer:
Perhaps one should explain which quantities in classical physics can be represented by 1-forms at all, because at first sight they should look like vector fields. Dual vectors and vectors are related by a pairing. Given a physical quantity that is modelled as a vector field, and given a pairing with another quantity, the latter should be modelled as a 1-form.  E.g. velocites are modelled as vector fields. There is a pairing between velocities and forces, yielding the power needed (or obtained)  when you move with velocity v in the force F. If you prefer not to express this pointwise you might say that there is a pairing between differentiable paths and force fields F, yielding the energy (used or obtained) on the path in F.     With respect to this energy/power-pairing force fields are a typical example of physical quantities that are represented by 1-forms in a natural way. Closed 1-forms are 'locally conservative force fields',  exact 1-forms are 'conservative force fields'.
That's the nice part of the story, but the geometrical meaning of quantities is often obliterated in the physics idiom:
Of course, given a 1-form \alpha and a non-degenerate bilinear form (e.g. a (semi-) Riemannian metric g(.,.)) one can define a vector field w that represents \alpha via g(w,v)=\alpha(v) for all v.
Phycicists would normally say that under coordinate changes the coordinate vectors of vectors  behave 'contravariantly', say via x'=Ax, and those of dual vectors 'covariantly' via l'=(A^T)^{-1} l.  The possibility to represent a dual vector (a 1-form) by a vector (a vector field), given such a nondegenerate bilinear form g, goes like this in the physics language:   The same quantity has contravariant and covariant coordinates.
In Euclidean 3-space 'the coordinates of a force' are the coordinates of a 1-form \alpha with respect to the dual basis (e^*_1,...,e^*_3)  to some reference orthonormal basis (e_1,...,e_3) .  The exterior derivative d\alpha is a two form which has again three coordinates with respect to  the basis (e^*_2 \wedge e^*_3, -e^*_1\wedge e^*_3, e^*_1\wedge e^*_2).   With respect to these choices of basis exterior differentiation is the 'curl'.
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Dillon Scofield gives an important example of a closed  (exact) 2-form in physics, not a 1-form. The electromagnetic field F is a 2-form on 4-d-space.
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Let $F=g\circ H$, where $H:\mathbb{C}^n\to \mathbb{C}^n$ is a homeomorphism such that $H(tz)=tH(z)$ for $t>0$ and $g$ is a homogeneous polynomial of degree $k$. Let $L$ be a complex line such that $(g|_H(L))^{−1}(0)=0$. Is it true that $F|_{L\setminus\{0\}}:L\setminus\{0\}\to \mathbb{C}\setminus\{0\}$ has topological degree $r$, such that $|r|≤k$?
For example, this is true when H is $\mathbb{R}$-linear!
Hi Edson. You are right, in 1) and 2) instead of
$r = ... = topdeg(g \circ id|_{L'}) \leq topdeg(g|_{L'}) = algdeg(g) = k$
there must be
$r = ... = topdeg(g \circ id|_{L'}) = topdeg(g|_{L'}) \leq algdeg(g) = k$
On the other hand $r$ can be not equal to $k$, that is it may happens that
$r = topdeg(g|_{L'}) < algdeg(g) = k$.
It seems that in this case we always have that $r=0$, but I can not produce at least just now an example when $0<r<k$.
For $r=0$ an example can be constructed as follows.
As in 2a) suppose that $L$ is contained in a subspace $\{ z_1 = 0 \}$.
Take $g(z_1,z_2,...,z_n) = z_1^2$.
Then $g|_{L'} = 0$, and so $0 = topdeg(g|_{L'}) < algdeg(g) = 2$.
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For example Ivan Kolar worked on this issue but on functeurs with general fibers.
As a first iteration, I would try this http://dml.cz/bitstream/handle/10338.dmlcz/107982/ArchMathRetro_042-2006-1_7.pdf ... It took me to this one http://journals.impan.pl/cgi-bin/doi?ap82-3-6 where is a reference [6]. This might be what you are looking for. However, the fiber preserving only.  When more general, this should be mentioned in [6] (Kolar, Mikulski DGA 1999. I guess the biggest group working in this area contains Doupovec, Kurka, Miukulski and Kolar (they should be active in this field...as far as I know). Of course, it depends in waht extent functors you want to classify ... and how fine and explicit is the classification-- often thse are done using the algebraic (Weil algebra)  or vector space data.
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i.e.,  Is a convex set in R^n with the fixed point property always compact ?
It does not hold when you drop the assumption of convexity. The first planar counter-example was found by Klee:
V. L. Klee, Some topological properties of convex sets. Trans. Amer. Math. Soc. 78, (1955). 30–45.
V. L. Klee, An example related to the fixed-point property. Nieuw Arch. Wisk. (3) 8 1960 81–82.
Connell found other examples:
E. H. Connell, Properties of fixed point spaces. Proc. Amer. Math. Soc. 10 1959 974–979.
In particular, he showed that FPP is not closed under finite products or taking closures.
However, for convex sets in Rn your question has an affirmative answer. See p. 35 in Klee's paper from TAMS and p. 958 in Connell's paper from PAMS.
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(A) Let X\neq\emptyset and \tau\in P(x), where P(X) is the power set of X. Then if we have that:
i) X,\emptyset\in\tau
ii) i\in I, A_{i}\in\tau \Rightarrow \cup_{i\in I}{A_{i}} \in \Tau
iii) j\in J, J finite, A_{j}\in \tau \Rightarrow \cap_{j\in J}{A_{j}}\in \tau
We say that the collection \tau is a Topology in X and that (X,\tau) is a topological space.
(B) The minimum requirement for every algebraic structure is closure under the defined binary operation.
What is the essential difference between Algebra & Topology?
Both of them are guided by the concept of closure.
So, why have we defined two branches that are almost of the same philosophy?
What about motivating intuitions? Topology was developed basically to deal with intuitions about "space," "connectivity, "continuity," notions of "near" and "far," etc. Algebra came about in order to deal with notions of "finitary manipulation," especially in connection with equalities. The historical developments happened quite organically, without any particular "guiding hand," so the boundaries are fuzzy indeed...
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In thermodynamics (like other areas of physics), work is defined as a function that assigns a real number to each thermodynamic process. Some of the latter (called quasi static) are modeled as continuous curves on certain smooth manifold. I want to know what conditions must work (as a function) to satisfy in order to be represented by the integral of certain 1-form along a quasi static process.
@ Ulrich Mutze: Infinitesimal changes of state would define vector tangent to the state space. Alternatively one might define these such a vector as a local derivation on the set of state functions. Work is pointwise a linear form on the tangent space, i.e. the space of 'state change vectors'. For this 1-form (or covector) to be defined one would have to consider sufficiently many state changes to span the tangent space. In principle one can change all quantities, that can be considered as the coordinates of the state manifold. If we have extensive quantities V, N, S as state variables, only changes in V would contribute to mechanical work. The 1-covector pdV is zero when evaluated along submanifolds defined by chaninge S and N only. I think that even the practically minded physicist somehow notices that pdV says 'how much work is done along a process' and so implicitly acknowledges work to be defined by a 1-form / a covector field.
I don't know whether there is a natural Riemannian structure in thermodynamics which would allow to give an interpretation of this 1-form as a vector field.
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We know the greatest feature of Clifford algebra is coordinate-free. One can do vector operations without knowing the representation of vectors. And due to this very characteristic, Clifford or geometric algebra is believed to be a reinterpretation of differential geometry as suggested mainly by Hestenes and Doran.
But as far as I know, many manifold-related theorems depend on the topology of the manifold such as connectedness, compactness, boundaryless or not. I want to know how Clifford algebra behaves in different topologies?
A Clifford algebra is defined for a quadratic vector space - so it's a purely algebraic object not depending on a manifold. But if you have a spin manifold M, then, depending on the topological structure, it can admit several spin structures (they are classified by H^1(M;Z_2)), and each spin structure yields a spinor bundle, i.e., a bundle over M that looks *locally* as the spin representation constructed via the associated Clifford algebra, but globally not. For details, look at Th. Friedrich's book
"Dirac operators in Riemannian geometry" (AMS, GTM vol. 25).
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During the review of Eilenberg-Montgomery fixed point theorem, I have faced a term acyclic space, map which needs a background in algebraic topology. Can any one simply tell what are those terms and if a contractible set valued map is an acyclic map?
To Behrad Mahboobi: Just to reinforce my response to your question that we cannot say that an acyclic space is necessarily contractible, I thought I should let you know there is a result which says that a space X is contractible if and only if it is acyclic and its fundamental group is trivial. So an acyclic space is not contractible unless it has a trivial fundamental group.
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I am interested in manifolds with finitely generated homotopy groups. As I remember I saw somewhere that the homotopy groups of a compact manifold are finitely generated, but I am not sure if this is true.
There is an injective immersion (i.e. embedding) $\phi: \partial M\to R^{2n+1}$ by the (weak form of) Whitney embedding theorem (2(n-1)<2n+1). Extend this map to an embedding $\Phi:\partial M\times [-\epsilon,\epsilon] \to R^{2n+2}$ by $\Phi(x,t)=\phi(x)+t e_{2n+2}$.
Identify (by means of a Riemannian metric) $\partial M\times [-\epsilon,\epsilon]$ with a collar neighbourhood $N$ of $\partial M$ in $2M$. Thus we obtain an injective immersion $N\to R^{2n+2}$ which we extend to a smooth map $\Psi: 2M \to R^{2n+2}$. The extension can be chosen in $E_+$ for $M_1\setminus N$ and in $E_-$ for $M_2\setminus N$, where $M_1$ and $M_2$ denote the two copies of $M$ in $2M$. $\Psi(2M)$ intersects $\partial E_+$ in $\Psi(\partial M)$.
By the weak Whitney embedding theorem there is an injective immersion (i.e. embedding) arbitrarily close to $\Psi$ which is identical to $\Psi$ on $N$ (as 2n<2n+1$. This map is an embedding as required. If there is no error in this argument it should work for$2M\to R^{2n+1}$. I don't know whether the claim would be correct for$2M\to R^{2n}\$, but I suppose the Whitney trick would work in each of the halves.