DrRocket

a) You should know a bit more that just that L is 1-1 and onto. What is L(x) ? What is L^-1(L(x)) ? b) Writing it down would be considered "stating" it.

I know scads of people who have done engineering management and executive management, including me. Several of them, including me, have PhDs in science, mathematics or engineering. I don't know anyone with an MBA in engineering management. My experience with degrees is that they get you in the door. After that people forget about your degree and focus on what you produce.

You are correct in stating that owl is wrong. But your explanation is also wrong in this context. You are treating "infinite" as meaning an infinite point-set. In the context of a space-like slice of spacetime, aka "space", a manifold is "finite" if it is compact as a topological space -- or equivalently has finite volume. It is "infinite, or "open" otherwise. In either case there is no boundary (aka edge). The terms "infinite" and "finite" in cosmology are a bit unfortunate since they have other meanings in related disciplines.

OK you have correctly identified the crux of the matter. Here is a sketch of a proof. I will let you fill in the details. f is strictly increasing therefore injective. g and h must agree on the image of f. Since f has a monotone left inverse (g or h) f cannot be bounded below or above. So either f is surjective or else it is not continuous. If f is surjective we are done since g and h agree on the image of f. Since f is strictly increasing, left-hand and right-hand limits exist everywhere and any point of discontinuity of f is a "jump discontinuity". Since g and h are non-decreasing they must agree in the gap corresponding to any jump discontinuity of f. (Think about it graphically then translate into an analytical argument.)

A closed manifold is a compact manifold without boundard. A compact space is one for which every open cover admits a finite subcover. If this is not clear please refer to any book on introductory topology. Lecture Notes on Elementary Topology and Geometry by Singer and Thorpe would be a good one.

Incorrect. Shor's algorithm is a quantum algorithm, and quantum algorithms were specifically excluded in the OP.

The traveling salesman problem is in fact NP-complete, so finding a fast algorithm comes under the heading of "really hard'. (Add as many "really"s as you like).

This problem is not so difficult. If it is not a homework problem, it should be one. What have vyou done so far in attempting to solve it ?

I was always perfectly willing to let students do 2 x 3 on their calculators. It did not seem to give them much advantage over the students who could do it in their heads. I personally favor HP calculators and RPN notation. Mine is an HP 48, and if I were buying one now I would probably opt for the HP 58. But any good calculator should suffice if you learn to use it. Once upon a time back in the near dark ages (1970 or so) a young physics prof bought an HP 35 calculator for about $400 (that is 1970 $). He figured that he could then sell his Pickett slide rule to some lesser light, someone perhaps not quite such a star clearly representing the future of physics. So he put a for sale notice on the departmental bulletin board. Along came a much older fellow who figured he might be able to get by with the rather nice Pickett slide rule, so he bought it. That older gentleman, that "lesser light", was Eugene wigner.

Works for me. I would certainly be willing to tell a threatening crook that I pray that he will cease and desist -- so that I don't have to shoot him. I presume that Darwin and a bit of the science of psychology will help me from there.

A plot of frequency vs frequency is a line of slope 1 so that cannot possibly be what you mean. The "discrete Fourier transform" usually means the Fourier transform on a finite number of points, or in group terms [math] \mathbb Z_n = \dfrac {\mathbb Z}{n \mathbb Z} [/math]. Now the Fourier transform on an Abelian group is taken with respect to the "essentially unique" translation invariant measure (Haar measure) on the group. "Essentially unique" means unique up to a positive real multiple -- which could well be your normalization constant. The natural invariant measure on a finite group is "counting measure" -- the measure of a subset is the number of points in the set. On the other hand in the theory of Fourier transforms on groups one usually chooses the Haar measure of a compact group to be 1 (and finite groups are compact. This is done to make the Fourier inversion theorem simple. So in that case "counting measure is normalized by dividing by the number of elements in the group -- [math]n[/math] in the case of [math]\mathbb Z_n[/math]. If this is what you mean by a normalized DFT the normalization merely assures that you are using the same measure for both the Fourier transform and the inverse Fourier transform. It is very convenient, but not essential to adopt such a convention. For a very nice discussion of Fourier analysis on groups, the natural setting, you might want to take a look at Walter Rudin's classic book Fourier Analysis on Groups. The beauty is that all of the different variations on the theme of "Fourier transform" and "Fourier series" that you see in engineering and physics texts are seen as just examples of one overall concept. Oh, yeah, another useful point -- Rudin will not state theorems that are not true, in contrast to some engineering and physics books.

I know of no proof that factorization in polynomial time is impossible. But no existing algorithm will do the trick either. Sieve methods are up against what is known about the distribution of prime numbers. http://en.wikipedia.org/wiki/Integer_factorization http://en.wikipedia.org/wiki/Prime_number_theorem

Herein "manifold" means differentiable manifold of class [math]C^\infty[/math]. Intrinsic curvature or just plain curvature, refers to any of several curvature tensors, most commonly the Riemannian curvature tensor, or in the case of GR the Einstein tensor, that are defined in terms of a connection or metric on the manifold without reference to any embedding of the manifold in any larger larger space (Euclidean space or vector space with metric of apppropriate signature). The key concept is that of a manifold, which is defined independently of any embedding space. There are (difficult) theorems that show that any Riemannian or pseudo-Riemannian manifold can be realized as an embedded manifold (Nash embedding theorem and later generalizations) in a space of suitably high dimension, but these theorems are not useful in GR and actually detract from a deep understanding. The general definition of a manifold is intrinsic, and that, rather than curvature, is what is really intrinsic. Cuirvature (perhaps 0 curvature) is just part of the package. Thinking of curvature in terms of an embedding takes you down the wrong path. It has nothing to do with the curvature of a manifold. For instance, all smooth curves (1-dimensional manifolds) are flat when arc length provides the metric.

mathematics, physics, chemistry, statistics, electrical engineering, mechanical engineering, aeronautical engineering, chemical engineering, metallurgy, materials science, ... A few specific subjects within those academic fields: real and complex analysis, functional analysis, classical mechanics, electrodynamics, circuit design, fluid mechanics, continuum mechanics, control theory, communication theory, thermodynamics, organic chemistry, polymer chemistry, structural analysis, visco-elastic materials, composite materials, aerodynamics, orbital mechanics, penetration mechanics, etc, etc, etc, In addition it helps to know about project management, scheduling and engineering economics.

It all depends on the individual. I took almost no undergraduate mathematics (except calculus and introductory linearbalgebra and on semester of introductory real analysis) but did manage to step from a master's program in electrical engineering into a PhD program in pure mathematics. Pre-requisites are there to be waived.

Things are ok until you take the limit. The limit doesn't make sense. You ought to also be getting significant shear near the point loads and particularly as d becomes very small. While I can see situations in which this notion would be useful, it also seems to me that there are potential pitfalls. As I recall in continuum mechanics, as usually formulated "point moments" are not allowed -- basically by fiat. If you permit point moments (torsion) then the stress tensor may not be symmetric. So, it seems to me that you need to be very careful in applying this notion to structural analysis. I suspect that this would be very upsetting to most finite element codes. I have no idea what this would do to the common failure criteria (e.g. Von Mises stress) , but as they are based on the usual formulation of continuum mechanics, I would be leery of applying them without further investigation.

Is signal analysis one sometimes needs to consider not only the delta function but also its derivative. To make sense of the derivative of something like that you need to use the theory of Schwartz distributions. More generally, distribution theory is important in the theory of partial differential equations. At the fundamental level it is sometimes simply important to know what you are doing and not just mindlessly push symbols around and hope that you make enough errors to compensate for one another. Physics and engineering texts sometimes make statements that are simply false. Usually they avoid getting into trouble because there are unstated hypotheses that limit the applicability of the statements made to regimes in which the more generally invalid statement does hold, but the reader/student is left unaware of the limitations. You generally need to be rigorous in the more mathematically sophisticated engineering disciplines such as information theory and control theory (particularly stochastic control theory).

You spell just like most engineers. Remember, I have probably had as much engineering education as have you, and I have quite a bit of experience in industry as well, with all kinds of engineering and science. It is not an are with which I lack familiarity.

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Yes. It is just like the Cartesian plane, but stretchy.

1. You need a better inagination. 2. Not true. Just stretch it everywhere. 3. See 1. 4. No. I am suggesting that the expansion is describable by a scale factor --- Google FLWR metric.

Yes it can. Quite easily. Imagine an infinite sheet of rubber. Now stretch it. Or imagine an infinite sheet of paper with grid lines. Now look at it through a magnifying glass.

More often it is mathematicians who advance engineering. Solomon Lefshetz and L.S. Pontryagin are major figures in the development of modern control theory. Norbert Weiner developed the methods for automatic aiming and of anti-aircraft guns and invented the field of cybernetics. Abraham Robinson did pioneering work on airfoils. Claude Shannon invented information theory. Raoul Bott did seminal work in network synthesis.

I'd say Electra is definitely hot.

a_n = 1/n for n odd a_n=1/(2^n) for n even