Riemann Definition and 618 Threads

Apologies in advance for the TeX in this post, I'm new and having difficulty with the formatting. Homework Statement I'm trying to understand the logic my professor uses to derive a second linearly independent solution to the hypergeometric DE: z(1-z)\frac{d^2 w}{dz^2} + (\gamma - z(1+\alpha...

Consider the separation of the Riemann Zeta function in two terms \begin{flalign*} \zeta(s) &= 1^{-s} + 2^{-s} + 3^{-s} + 4^{-s} + 5^{-s} + 6^{-s} + ... = & \\ &=(1^{-s} + 3^{-s} + 5^{-s} + 7^{-s} + 9^{-s}+ ... ) + ( 2^{-s} + 4^{-s} + 6^{-s} + 8^{-s} + ...)&=& \\ &= (1 - 2^{-s}) \zeta(s) +...

Hi, I'm stuck on this problem here about composite function, help is appreciated: Let g : [a,b] -> [c,d] be Riemann integrable on [c,d] and f : [c,d] -> R is Riemann integrable on [c,d]. Prove that f o g is Riemann integrable on [a,b] if either f or g is a step function I was able to solve...

Sean Carroll: Lecture Notes on GR (2:20): Presumable to be a coordinate system it would have to exist at more than one point! Does he mean to define a Riemann normal coordinate system as a chart such that g_{\mu\nu} takes its canonical form and the first derivatives \partial_\sigma...

[b]1. If abs(f) is Riemann integrable on [a,b], then f is Riemann integrable on [a,b]. True or false (show work) [b]2. A function f is Riem Int iff f is bounded on [a,b], and for every epsilon>0 there is a partition P of [a,b] s.t. U(f,P)-L(f,P)<epsilon [b]3. I believe that this...

(Dis)proof of Riemann hypothesis,Goldbach,Polignac,Legendre conjecture I'm just an amateur and not goot at english ^^;

Background: Math: An affine parameter provides a metric along a geodesic but not a metric of the space, for example between geodesics. A connection provides an affine parameter, and a non-trivial connection gives rise to Riemann curvature. Given the existence of a connection with Riemann...

Homework Statement Hello …….. I have a question about a statement mentioned in the book “Introduction to tensor calculus and continuum mechanics” . it is :- Where the space (Vn) is Riemann space . Is this statement really true ? Homework Equations The Attempt at a...

This is my first time posting & I am not familiar with how to get all the correct math symbols or how to use Latex, so I did the best I could. Homework Statement Suppose f is bounded on [a,b] and there is a partition P* of [a,b] for which S(f,P*)=S(f,P*). Is f Riemann integrable on...

HERE http://vixra.org/pdf/1007.0005v1.pdf is my proposed proof of an operator whose Eigenvalues would be the Imaginary part of the zeros for the Riemann Hypothesis the ideas are the following* for semiclassical WKB evaluation of energies the number of levels N(E) is related to the integral of...

Basically from what I understand the integral of a function, say ∫x^2dx from say 0 to 1, can be represented as the supremum/infimum of the function values within each of a countably infinite number of vanishingly small intervals in the domain created by a countably infinite number of partition...

http://en.wikipedia.org/wiki/Riemann-Lebesgue_lemma Have I made a mistake when it looks to me that the Wikipedia proof on Riemann-Lebesgue lemma looks like nonsense? How are you supposed to use dominated convergence theorem there?

Let f be a Riemann integrable function defined on an interval [a,b], and let P = \{a = x_0 < x_1 < \ldots < x_n = b\} be a partition of [a,b]. I don't understand why the definition of (say) the upper Riemann sum of f associated with P is always given as U(f,P) = \sum_{i=1}^n M_i (x_i -...

If I have a function c(x,Δx) that gives the area between x and x + Δx of a function. The area under the function can be given by: Sum from j = 0 to n-1 of c(b/n j,c/b) As n tends to infinity and b is the upper limit of integration. How can I convert this from a sum into a integral? I'm not...

Hi, thanks for the attention and excuse for my bad english. I'm studying general relativity and I have a doubt about the number of indipendent component of the riemann curvature tensor. We have two kind of riemann tensor: type (3,1) Rikml type (4,0) Rrkml There are also some symmetry...

Let f_n : [0,1] → [0,1] be a sequence of Riemann integrable functions, and f : [0, 1] → [0, 1] be a function so that for each k there is N_k so that supremum_(1/k<x≤1) of |f_n(x) − f(x)| < 1/k , for n ≥ N_k . Prove that f is Riemann integrable and ∫ f(x) dx = lim_n→∞ ∫ f_n(x) dx I am really...

Homework Statement If a<c<d<b and f is integrable on (a,b), show that f is integrable on (c,d) Homework Equations The Attempt at a Solution I know that f is integrable on (a,b) iff for all e>0 there exists step functions g and h such that g \leq f1(a,b) \leq h and I(g-h) <e (...

Homework Statement We have a corollary that if f(x) is in the set of Riemann Integrable functions and g(x) is continuous, then g(f(x)) is also a riemann integrable function Show that if g(x) is piecewise continuous then this is not true Homework Equations Hint: take f to be a ruler...

suppose f and g are bounded functions on [a,b] such that f+g is in R[a,b] Then, does it follow that f and g are also in R[a,b]? i wanto to prove whether it is or not

Hello. Does anyone know of a group that has used 3D printing techniques such as laser sintering to create Riemann surfaces of some simple functions? For example, just \sqrt{z}? Actually I would be interested in more complex function and preferable color-code various components of the surface...

I'm having a lot of trouble getting my head around this topic.. I am currently trying to create a numerical model of traffic flow, which has worked out to be a nonlinear hyperbolic PDE. If anyone could explain the concept of the Riemann problem to me in lamen terms, it would be greatly...

When Riemann tensor = 0, spacetime is flat. Is the geometry of this flat spacetime that of special relativity?

How do you mathematically equate a Riemann sum as area under the curve to an anti-derivative? How do you prove that, theoreticlly, the one is equalent to the other? Assuming the function is continuous between points a and b, there is always a Riemann sum and thus the function is integrable...

I saw a documentary recently that talked about the distribution of prime numbers and their similarity to vibrations in a sphere of quartz when struck by metal ball bearings. I tried to look up Riemann online and was overloaded with advanced math. Is there a resource where I can find out more...

Homework Statement Some proofs I've looked at vary, but they generally follow the format show here: http://en.wikibooks.org/wiki/Real_Analysis/Riemann_integration#Theorem_.28Cauchy_Criterion.29 This isn't a question about an exercise, but rather a request for a clarification or a way of...

This is essentially a "homework question", but I'm not looking for an explicit solution so I have posted it here. 1. Homework Statement Find a simplified expression for the Riemann tensor in terms of the connection in normal coordinates. 2. Homework Equations Riemann tensor =...

Homework Statement Find a simplified expression for the Riemann tensor in terms of the connection in normal coordinates. Homework Equations Riemann tensor = (derivative of connection term) - (derivative of connection term) - (connection term)(connection term) - (connection...

Homework Statement Homework Equations The Attempt at a Solution Right now, I'm still trying to understand why the hint is true. This is what I've got so far... Let ||f||∞= sup{|f(x)|: x E [a,b]} M_i(f,P) = sup{f(x): x_{i - 1} ≤ x ≤ x_i} m_i(f,P) = inf{f(x): x_{i - 1} ≤ x ≤ x_i}...

Does anyone know where to find this paper? Formule de trace en géométrie non-commutative et hypothèse de Riemann = Trace formula in noncommutative geometry and the Riemann hypothesis http://cat.inist.fr/?aModele=afficheN&cpsidt=2561461 The purchase link is broken there.. it gets stuck...

There are three kinds of light-cone string diagrams for four closed string interactions. As displayed by fig. 26.10, 26.11 and 26.12 of section 26.5 of Zwiebach's book. For each light-cone string diagram, it is characterized by two parameters, the time difference of the two interaction...

Does anybody know where I can find the proof that an infinite number of zeros of the riemann zeta function exist when re(s) = 1/2?

I was wondering how do you calculate the Riemann value, of a Riemann Zeta Function, for example the riemann zeta function for n = 0, is -1/2, which envolves a bernoulli number (what is a bernoulli number and what roll does it play in the Riemann Zeta Function), can anyone explain that to me...

Homework Statement a) Let A in R^n be compact, and let f: A -> R be continuous and also non-negative. Show that if there exists some a in A with f(a) > 0, then \int_{A}f > 0 b) Now let A in R^n be a closed rectangle, and suppose f: A -> R be bounded and integrable. Show that if f(x) > 0 for all...

Let f = x, for 0<=x<=1 1, for 1<x alpha = x^2, for 0<=x<=1 1, for 1<x Find Integral (f) d(alpha) -- from 0 to 23 pls help!

Could anyone tell me what is the Riemann zeta function. On Wikipedia , the definition has been given for values with real part > 1 , as : Sum ( 1 / ( n^-s) ) as n varies from 1 to infinity. but what is the definition for other values of s ? It is mentioned that the zeta function is the...

Clearly I am missing something obvious here, but how is it that negative even numbers are zeros of the Riemann zeta function? For example: \zeta (-2)=1+\frac{1}{2^{-2}}+\frac{1}{3^{-2}}+...=1+4+9+.. Which is clearly not zero. What is it that I am doing wrong?

Hi, The divisor summatory function, D(x), can be obtained from \zeta^{2}(s) by D(x)=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty}\zeta^{2}(w)\frac{x^{w}}{w}dw and I was trying to express \zeta^{2}(s) in terms of D(x) but I didnt succeed, could someone help?

http://arxiv1.library.cornell.edu/PS_cache/math/pdf/0102/0102031v10.pdf and http://arxiv1.library.cornell.edu/PS_cache/math/pdf/0102/0102031v1.pdf what do you think ? Author defines 2 operators D_{+} and D_{-} so they satisfy the properties D_{+} = D^{*}_{-} D_{-} = D^{*}_{+}...

Where is the use of the "tangents at every point on the curve" in the Riemann sum? Riemann sum allows us to find the area of under the curve, and this involves only the height of each rectangle (i.e. the function f(x) at each x), and the width (i.e. the x), and the two are multiplied together...

Hello dear forum members I wanted to know where are the research on the Riemann hypothesis , the latest advances ,who are the currently leading experts and is now known that mathematics it requires for its resolution

I am having trouble describing the Riemann surface of log(z) + log(z-a)

Homework Statement How do you solve the surface area of a sphere using Riemann Sums?Homework Equations The Attempt at a Solution I started out with 2 * (lim n->∞ [ (i=1 to n) ∑ [ 2*pi*(√(r^2 - (i/rn)^2))*(r/n) ] ]) where the summation is the surface area of the cylinders (or discs) inside a...

Find the Riemann tensor of the 2-sphere of radius r S^{2}_{r}={(x,y,z) \in\Re^{3}|x^{2} + y^{2} + z^{2} = r^{2}} with metric g obtained as the pull-back of the Euclidean metric gR3 by the inclusion map S^{2} \hookrightarrow\Re^{3}. Any help would be appreciated. Thanks

Is it true that a function is Riemann integrable on a bounded interval only if it's equal to a continuous function almost everywhere? I'd imagine this is the case, given the Riemann-Lebesgue lemma, which says that a function is RI iff its set of discontinuities has measure zero. (So the...

Hi Everyone, A problem I have here. I am trying to solve a problem involving Ito Integrals and Riemann interals. Homework Statement Prove \int^{T}_{0} tdW(t) = TW(T) -\int^{T}_{0} W(t)dt Homework Equations I want to solve this question WITHOUT using Ito's Lemma directly.The Attempt at a...

Hello everyone, I have passed my integral calculus class and it's been a little while so I don't really remember everything. Can anyone help me out with this? Homework Statement Let f(x) = sqrt(x), x E [0,1] and P=\left \{ 0,\left ( \frac{1}{n} \right )^{2}, \left ( \frac{2}{n} \right...

Homework Statement This is somewhat a repost... except I have figured out some of it and I have cleaned up the question. Your task is to estimate how far an object traveled during the time interval 0<= t >= 8 , but you only have the following data about the velocity of the object...

Homework Statement I really need help starting this problem as I am not sure what to do. Your task is to estimate how far an object traveled during the time interval 0t8 , but you only have the following data about the velocity of the object. time (sec) 0 1 2 3 4 5 6 7 8...

Homework Statement The following table shows the power produced by a 600kW wind turbine at the given wind speed and the number of hours the wind blows at that speed. a) Plot the power characteristic as a function of wind speed. b) Plot the wind duration curve as a function of wind speed...

Hello, for my FORTRAN class it wants me to use the method of Riemann's sums to find the area under the curve for the function f(x) = -(x-3)**2 +9, and stop when successive iterations yield a change of less than 0.0000001. I know I am going to have to used double precision. I am just confused...