Riemann Definition and 586 Threads

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

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...

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 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...

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...

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...

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...

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...

Hi: ____________________________________________________________________ Added Nov.3, 2009 (For anyone who can't read the formula below (probably everyone) and who might have an interest in the subject: - the derivation of two simple equations that locate all the zeros of the zeta...

Riemann sum help! Homework Statement Use Riemann sum with ci= i3/n3 f(x)= \sqrt[3]{x} +12 from x=0 to x=18 n= 6 subintervals Approximate the sum using Riemann's Sum Homework Equations \Sigma f(ci) \Delta xi is the equation for riemanns sum i think The Attempt at a Solution i...

http://img156.imageshack.us/i/17818455.jpg/ http://img215.imageshack.us/i/53355598.jpg/ http://img509.imageshack.us/i/11493310.jpg/ If you look at the above, I have underlined the problem that I am having. So, my first question is, where are these inequalities coming from? If you do have...

show that in two dimensions, the Riemann tensor takes the form R_{abcd}=R g_{a[c}g_{d]b}. i've expanded the RHS to get R g_{a[c}g_{d]b}=\frac{R}{2!} [g_{ac} g_{db} - g_{ad} g_{cb}]=\frac{1}{2} R_e{}^e [g_{ac} g_{db} - g_{ad} g_{cb}] but i can't seem to simplify it down. this is problem...

i need to show that R_{abc}{}^{e} g_{ed} + R_{abd}{}^{e} g_{ce}=(\nabla_a \nabla_b - \nabla_b \nabla_a) g_{cd} = 0 ok well i know that R_{abc}{}^{d} \omega_d=(\nabla_a \nabla_b - \nabla_b \nabla_a) \omega_c so i reckon that R_{abc}{}^{e} g_{ed} = (\nabla_a \nabla_b - \nabla_b \nabla_a)...

You always see in books that one advantage of the Lebesgue integral over the Riemann integral is that a sequence of continuous functions f_n does not have to converge unifomly to a function f to have: integral of the limit of the sequence = the limit of the integrals of functions in the...

Let me start off by saying I have not yet had a formal course in Number Thoery and have only read briefly on the subject...hence the question: How close (in terms that would be understood by someone in my position) is the math community to proving the Riemann Hypothesis? I'm assuming there...

(i) show that R_{abcd}+R_{cdab} (ii) In n dimensions the Riemann tensor has n^4 components. However, on account of the symmetries R_{abc}^d=-R_{bac}^d R_{[abc]}^d=0 R_{abcd}+-R_{abdc} not all of these components are independent. Show that the number of independent components is...

Can anyone prove the following formula: R_{abf}^{\phantom{abf}e} \Gamma_{cd}^f = R_{abc}^{\phantom{abc}f} \Gamma_{fd}^e + R_{abd}^{\phantom{abd}f} \Gamma_{cf}^e I found it in "General Relativity" by Wald (in slightly different notation).

Hi, I'm Yr 13 and just wanted to do some further reading/exploring. So i understand that the zeta function is something to do with summing up like this: 1/ (1^s) + 1/(2^s) etc etc Now, I just want to know what are non-trivial zeros and trivial zeros? I just want to be able to understand this...

Dear Friends and Colleagues! I have this practise question:- Show that z(sin(z))(cos(z)) statisfies the Cauchy-Riemann Conditions for analyticity for all values of z. Does 1/[z(sin(z))(cos(z))] statisify simiar conditions? Calculate the derivative of 1/[z(sin(z))(cos(z))] at z=0, +...

I recall reading somewhere that Legendre's conjecture implies the Riemann Hypothesis. But the Wiki article suggests that Legendre imposes lighter bounds on the density of primes than does RH, so I would think the other way around, if anything. Thanks for any enlightenment.

Homework Statement Given the following sum, turn it into an integral: \lim_{n \to \infty}\Sigma^n_{k=1}\dfrac{1}{n\sqrt{1+(k/n)^2}} Homework Equations The answer says =\int^2_1\dfrac{1}{\sqrt{1+x^2}} The Attempt at a Solution I understand how to get the equation, but why...

Accurate Proof verification of Riemann’s Hypothesis Riemann Hypothesis states that \int \frac{1}{ln (x)} has a root at \frac{1}{2} when s=2 The time series expansion of Log function is, [tex] \ln(x) = \frac {[x-1}{[x-2}+ \frac{1){3} \frac{x-3}{x-4} + \frac{1}{5}\frac{x-5}{x-6}+……...

Well we know what matryoshka dolls are? Those nested dolls one inside another. I am a mere laymen and amateur that's why I am using descriptive terms instead of math rigor. So what should the approach be: If RH nest Hilbert-Polya conjecture, then what things nest HP conjecture? And ad...

Homework Statement Suppose α(x) increases on [a,b] a≤ x_0 ≤b, α is continuous at x_0, f(x_0) =1 , at all other x in [a,b] f(x)=0. denote ('x knot' as x_0) Prove that f is Riemann Integrable and that ∫fdα=0. Homework Equations Can anyone check my proof or suggest a good method...

To what degree would proving the Riemann hypothesis facilitate the factoring of large composites? In other words, how much would a complete (as opposed to "hit-or-miss") knowledge of primes help to reduce the operations needed to factor large composites?

Hi All, I would like to present what I believe to be a simple way to convey the essence of the Riemann Hypothesis to High School students. I hope you like it, and reply with suggestions for further improvements. Note for teachers: the rationale behind the graphs lays with the geometric...

Homework Statement (x, f(x)) (2,1) (3,4) (5,-2) (8,3) (13,6) A) Estimate f '(4). Show work. B) Evaluate the Intergral from 2 to 13 of (3 - 5f '(x))dx. show work C) Use left riemann sum with subintervals indicated bye the data in the table to apporoximate the intergral from 2 to 13 of...

Looking for a connection to astronomy for a history of mathematics report on 19th century mathematicians. I don't do astronomy. Would like to know of any mathematical developments of Riemann that are used in astronomy that I can understand (which expressly excludes any thing with the word tensor...

Hi There Everyone I am studying undergraduate calculus in first year. My question regards the rules for identifying a limit sum as a Riemann sum and therefore a definite integral. The book we are using says that when choosing \inline \large c_{i} for some f(x) , if \inline \large x_{i -...

http://www.mth.uct.ac.za/omei/gr/chap6/frame6.html" is a derivation of the components of the riemann curvature tensor. the problem is that i can't understand the transition between eq97 and eq89 . what does "To lowest order " mean ?

Homework Statement Suppose f is integrable for all x in[a,b] and f(x)>C ( C is some constant), Must show that 1/f is also integrable. Homework Equations f is integrable implies Upf-Lpf<\epsilon for some partition in [a,b] The Attempt at a Solution Therefore, I must come up...

Hi everyone, For integrable f,g:\left[a,b\right]\rightarrow\mathbb{R} with f(x)\leq g(x) for all x\in\left[a,b\right], it's a basic property of the riemann integral that \[\int_a^b f(x)\,dx \leq \int_a^b g(x)\,dx\] My question is whether the strict version of this inequality holds...

Homework Statement Assume S contained in R2 is bounded. Prove that if S is Riemann measurable, then so are its interior and closure 2. The attempt at a solution Proof: If S is Riemann measurable, its boundary is a zero set. Since the boundary of each open U in the int(S) is part of...