Help with understanding Hermitian symmetry & inner product of 2 complex functions in L^2

Hello all,

So I’m reading this (Stanford lecture notes on Fourier analysis) and I’m stumped by the fact that when you take the inner product of 2 complex-valued functions f and g it yields an integral of the product with a conjugate sign on g. Would someone please enlighten me on that (found on page 30)? Also I don’t seem to understand Hermitian symmetry, i.e. (f,g)=conj((g,f)), and I guess a link to something that proves all these properties would be nice, or an intuitive statement (again also on page 30).

Thanks to those who at least took the time to read this

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Advice for a kiddo

I’ll be a senior in high school when summer vacation ends, and I want to further my mathematical knowledge. I don’t dislike math, but Physics is my passion. The main reason I want to further my knowledge on mathematics is so I can further my knowledge on Physics. I’m aware Calculus is a large portion of Physics, so I’m wondering if precalc is truly necessary. My school offers precalc and calc, but you have to take precalc before calc. As it happens, I’ll be in precalc rather than Calc next year. I’ve been told by the teacher of these classes that precalc is basically just a review of algebra 2 and that it’s not really necessary. Is this true? I have a calc textbook, so should I just be going through it now?

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2 Questions I had while reading Goedel’s proof by Negel and Newman

Hey There!

1:The book begins by introducing the reader to what axioms and the axiomatic thinking mean, and I came across Euclid’s postulates. Something which i find very strange about the postulates is that they include: all right angles are congruent.. Why is this statement important as a big-P Postulate, when it is apparent by the very fact that they are all named right angles, that is, a 90degree angle.. it is inherent in their very definition that they are congruent! Was is not too obvious? I mean, as obvious as the fact that, for instance, all points are congruent? Why didn’t he include this in the postulates too? Is there some historical background that made him write it?

2: I don’t understand the jump in reasoning by which Goedel’s theorem is understood to be talking about the completeness of a system. Let me clarify what I mean: The line of thought goes: We translate (or map) the statement “This theorem is not provable using the axioms and rules of inference of the available formal system” into the system itself. Then, we show that this is a well-formed string of the system which is true yet not derivable, and hence the system is incomplete. What I don’t understand is why does this result generalise to include theorems other than the very particular (and peculiar, and naughty) statement “This theorem is not provable using the axioms and rules of inference of the available formal system”. For example, months ago I read the novel Uncle Petros and Goldbach’s Conjecture in which one of the characters (namely Uncle Petros) claims desperately that Goldbach’s Conjecture is one of those naughty theorems that are true and yet not formally provable when he reads Goedel’s paper. How can somebody make such a claim? The context and meaning of Goldbach’s conjecture has nothing to do with terrible, self-referencing statements, right? Yet I know that this is not just a baseless product of the novel’s author’s mind because in the concluding paragraphs of Goedel’s Proof they explicitly say that some theorems, though true, might, for all we know, be unprovable using the formal methods. What am I missing?

I hope I have made my questions clear enough. Thank you.

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Could someone explain a bit on a paper about symbolic representation in math?

Article: https://onlinelibrary.wiley.com/doi/epdf/10.1111/j.1467-9744.2008.00921.x

page 361 is where the most math stuff is that I don’t get. It is also the key criticism of the paper, so I want to understand it.

The use of symbolic algebra in physics would therefore appear problem-atic prima facie, because the science of physics claims to be “about” thephysical world itself rather than about a symbolic entity. Indeed, for acentury or so after Newton, mathematical physicists resisted the use ofalgebra, none more so perhaps than Newton himself, and for the very rea-son of the felt need to keep symbolic and physical quantities conceptuallydistinct. Mathematical physics instead employed the traditional languageof ratio and proportion. At some point in the nineteenth century, the equa-tions of physics generally ceased to be understood as abbreviated propor-tions and began to be taken instead as direct assertions about the physicalworld. Thus we today think of a body’s energy as itself “being” mc2, eventhough we have no intuitive conception of the product of mass and veloc-ity. The training of scientists undoubtedly encourages this kind of reifica-tion of symbolic mathematical entities, and Weil is therefore right to see init the danger of a science that does not actually think. Beyond that, how-ever, Weil seems to assume that any valid formula of physics should inprinciple be redeemable as a proportion involving intuitable physical quantities. “Science,” she maintains, “has as its object the study and the theoretical reconstruction of the order of the world—the order of the world in relation to the mental, psychic, and bodily structure of man. Contrary to the naïve illusions of certain scholars, neither the use of telescopes and microscopes, nor the employment of the most unusual algebraic formulae . . . will allow it to get beyond the limits of this structure” (1951, 169).But if this assumption was true for mathematical physics at least well into the nineteenth century, it is no longer true.

this is where it starts to lose me

In Hermann Minkowski’s “spacetime” formulation of Einstein’s specialtheory of relativity, first set forth in 1908, an invariant quantity designated the “spacetime interval” is defined for any two events: s2 = (ct)2 – x2 (wheres is the spacetime interval, t is the time interval between the events, x is the distance between the events, and c is a constant, the velocity of light in empty space).13 Of course, since ct has units of distance, the literal mean-ing of c2t2 – x2 is simply the difference between the distance separating two events and the distance light would travel in the time interval between the two events.14 But if we represent time in terms of ct, the expression can be understood as a “spacetime interval” via an analogy with a rotation of co-ordinate axes in Euclidean space, where the distance between two pointson a Cartesian plane is given by the Pythagorean theorem: s = x2+ y2.Representing ct and x on respective axes of a Cartesian plane, we can per-form a “hyperbolic rotation,” in which angles are measured on arcs of ahyperbola rather than arcs of a circle as in regular trigonometry (Fig. 1).The hyperbolic rotation does in fact yield the relativistic expressions2 = (ct)2 – x2.15 If we then set c equal to unity and drop its units (distanceper unit time), such that ct = 1t or simply t, substitution into the original expression yields t – x2 as the “spacetime interval.”

Clearly, because t now designates time, the subtraction is intuitively meaningless. The operation can still be carried out, however, if we perform it on symbolic, dimensionless numbers and then “plug” the result back into units of “spacetime.” Observe that to carry out the “spacetime” sub-traction, we first had to convert c, the velocity of light, into the symbolicand dimensionless number “1,” and then drop the dimensions of t and xrespectively.16 Consequently, save for the symbolic conception of number uncovered by Klein, the very mathematical operations by which the spacetime interval is defined would be impossible. Here, that is to say, the symbolic form of representation cannot be intuitively redeemed even indi-rectly. Rather, the “spacetime interval” is an irreducibly symbolic entity.

If four-dimensional “spacetime” is devoid of intuitive sense, by contrast with the three-dimensional space and one-dimensional time of experience,why should it be accepted as real? The answer, in short, is that the concept of space time is an inevitable result of recognition in mathematical physics of the relativity of particular reference frames for measuring space and time.Thus Minkowski characterizes the theory of space time as the “postulate of the absolute world” (1952, 83). Such reasoning, we shall see, cannot simply be dismissed as thoughtless manipulation of algebraic symbols. At the same time, there is no denying that the Minkowski approach leaves us with the sense that “things” have somehow been replaced by relations among symbols.

Weil evinces a clear sense for this aspect of modern science when shecomments, for instance, that in modern science order as expressed in alge-braic symbols has become “a thing instead of an idea” (Weil 1965, letter toAlain of 1933), and that algebra “puts everything on the same level,” since“things, once translated into letters, play an equal role in equations” (Weil1968, 54). Indeed, for Weil, the deleterious effects of “algebraic conscious-ness” extend beyond mathematical physics per se. For in the contemporary world such consciousness has become socially reified, as it were, such that thought itself is now essentially “without a thinker”:

“In all spheres, thought, the prerogative of the individual, is subordinated to the vast mechanisms which crystallize collective life, and that is so to such an extent that we have almost lost the notion of what real thought is . . . signs, words, and algebraic formulae in the field of knowledge, money and credit symbols in eco-nomic life, play the part of realities of which the actual things themselves consti-tute only the shadows, exactly as in Hans Anderson’s tale in which the scientistand his shadow exchange roles. . . . (Weil 1958, 93)”

The reification of method, so compelling in the lone Cartesian thinker or ego cogito, and so productive in mathematical physics, evidently has destructive consequences when embodied in social structures. The architects of seventeenth-century mechanics knew that algebraic formula could not be simply read into nature. However, to the degree that a symbolic system of thought becomes socially reified, such that the thoughts of individual thinkers themselves are essentially constituted by the system itself,there is no truly individual thought remaining by which such a reification of technique could be even recognized.

For Weil, the reification of technique represents an abdication of indi-vidual thought and responsibility. But she further suggests that algebra could be relegated in science itself to the status of a “mere instrument”(Weil 1965, 3), an aid to the imagination in conceiving analogies (propor-tions). In other words, if algebra is reified technique, we should dereify it.Here, however, Weil underestimates the hurdles for a reorientation of science along the lines of Pythagorean number mysticism. As we have seen,in the context of modern mathematical science, there is no way of dereifying algebraic technique, at least not in the way Weil suggests. That could bedone only by abandoning modern science per se, since in modern science algebraic entities are indeed “the thing.” To be sure, in some sense Weil did wish for contemporary science to be, if not abandoned, at least conceptually reformulated at its very root. Whether that desire is justifiable is a question to which we presently turn.

There are some images included on the actual document.

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A fun problem about number theory (I think?) you might enjoy.

Here’s the question as I thought of it initially: given an integer n, is it possible to predict which a, b such that a+b=n form the biggest ab?

For example, for n=5 we have 6 possibilities: 05, 14, 23, 32, 41, 50. In this case, the biggest ab is a=3, b=2, because it is bigger than any other number in the list.

I’ve written a script in Python for the first 100 integers, and the results are in this Pastebin.

Maybe this is kind of trivial but at least I had some fun with it. I don’t have much of an idea on why some a’s appear more often than others (4 to 5 times for n decently big), maybe someone does? I don’t know.

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Help figuring out percentage contribution to a change in f(x,y)

We have a function f(x,y) = xy where x and y are independent variables.

We have two points on f and we take the difference Δf = f2-f1 = Δxy = x2y2-x1y1

I want to identify the percentage contribution (to the change in f) of the change in x and the change in y respectively

In other words, if x increased by a and y increased by b, causing f to increase by c. What percentage of the increase c was caused by a and b respectively?

My thought process: if only x increases then the change is 100% due to x and is equal to = Δx•y1 If only y increases then the change is 100% due to y and is equal to = Δy•x1 If both change then surely the above equations still detail changes due to x and y respectively, but a third term is still missing: Δx•Δy How do I attribute that part of the change?

Anyway, any help is appreciated!

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Any recommendations for Complex Analysis video lectures?

Taking a complex analysis course this fall, looking to get a little bit of a head start and since it’s the summer I’d much rather watch some lectures than try and slog through a textbook. Any recommendations? The Harvard abstract algebra series was great when I was learning algebra, so something along those lines would be ideal.

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Request: Recommendations for Logic Courses, Lectures, Seminal Papers, and Textbooks

Recently I’ve taken an interest in metamathematics and logic. I’m looking for good self-study sources, such as textbooks and lecture series, as well as primary sources to get an understanding of the historical context of major results.

In essence, my current heuristic is trying to connect the dots between Cantor’s set theory (ca 1880) all the way up to Cohen’s Independence Proof (ca 1963).

Non-technical publications are good too–right now I’m reading Gödel, Escher, Bach.

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