Property Definition and 635 Threads

I would like to show that if a prime number P mod 8 is a) 1 or 7 or b) 3 or 5 then a) \frac{(P+1)}{2}(1-sqrt{2})(3+sqrt{8})^\frac{P-1}{2}+ \frac{(P+1)}{2}(1+sqrt{2})(3-sqrt{8})^\frac{P-1}{2} = (\frac{P-3}{2} + 2) mod P b) \frac{(P+1)}{2}(1-sqrt{2})(3+sqrt{8})^\frac{P-1}{2}+...

Hi Does anybody know if the irrational numbers have the least upper bound property?

Homework Statement I am using the time differentiation property to find the Fourier transform of the following function: Homework Equations f(t)=2r(t)-2r(t-1)-2u(t-2) The Attempt at a Solution f'(t)=2u(t)-2u(t-1)-2δ(t-2) f''(t)=2δ(t)-2δ(t-1)-?? Can somebody explain what the...

I'm guessing that if z\in \mathbb C, then we have \left| z^{-1/2} \right|^2 = |z^{-1}| = |z|^{-1} = \frac{1}{|z|}. Proving this seems to be a real headache though. Is there a quick/easy way to do this?

How can I calculate degrees of freedom of a rank (o,3) tensor, Aabc, that is mixed symmetry and antisymmetric in the first 2 indices? By mixed symmetry I mean this: Aabc+Acab+Abca=0.

OK - this one has been argued to death in several different threads, but the answers have been less than satisfactory... so someone provide a reason why I am wrong here: 48 / 2(9 + 3) = 2. Why? Because the distributive property of multiplication means that 2(9+3) = (2*9+2*3). For...

As a preface to a different question, it is valid to think of the property of spin of elementary and related particles as basically just tiny magnets, right?

Could you please give me a hint on how to show that a set of operators with a property P is closed under addition? In other words, how one could prove that a sum of any two operators from the set still possesses this property P. The set is assumed to be infinite. Any references, comments...

Dirac Matrix Property? Possible Book mistake? Derive KG from Dirac I got a copy of QFT demystified and on pg. 89 he says he can write \gamma_{\nu} \gamma^{\mu} = g_{\nu \sigma} \gamma^{\sigma} \gamma^{\mu} = g_{\nu \sigma} \frac{1}{2} (\gamma^{\sigma} \gamma^{\mu} + \gamma^{\mu}...

Homework Statement Given Pressure=P=700 KPa Specific Entropy = s = 7.6953 KJ / ( K Kg ) Find the Specific Enthalpy (h) Homework Equations No equation Am using the property tables at the end of the following book : Thermodynamics, an engineering approach, by Yunus A. Cengel and...

Hi, I'm working through a paper and I am quite stupid so some things that maybe obvious are not obvious to me. Say you have some have some complex analytic function that is defined on some simply closed curve, and the index of this function defined on this curve is zero, \int_C...

Hello, I am looking at the proof (Theorem 2.5 (b) Apostol) of $$ \phi (mn) = \phi(m) \phi(n) \frac{d}{\phi(d)} $$ where $$ d = (m, n) $$. Can someone explain how they go from $$ \prod_{p|mn} \left( 1 - \frac{1}{p} \right) = \frac{\prod_{p|m} \left( 1 - \frac{1}{p} \right) \prod_{p|n}...

I have already solved it, but I need confirmation: Are there other ways of proving this? Thanks in advance!

A power tower (x^^n) is a variable raised to the power of itself n amount of times. x^^4 = x^x^x^x x^^3 = x^x^x x^^2 = x^x x^^1 = x I was wondering if an associative property for power towers exists. Does x^(x^x) equal the same thing as (x^x)^x? Is x^(x^^n) equal to x^^(n + 1)? If anybody...

Suppose S\subsetℝn is compact, f: S-->R is continous, and f(x)>0 for every x \inS. Show that there is a number c>0 such that f(x) ≥ c for every x\inS. Attempt: Since S is contained in Rn is compact, then S is closed and bounded. By the extreme value thm there exists values a,b that are an...

Hi guys, I've been playing around with structure generated patterns and have come across one which has caught my attention. I am only just starting to learn about number theory and so I am sure someone might be able to provide an explanation for this. Let me describe what I did then show you the...

Suppose a space X has the intermediate value property (f: X->Y continuous, Y has the order topology), then X is connected. How would you show this? This is just the converse of the intermediate value property.

Homework Statement Let y(t) be the convolution of x(t) with h(t), show that the area under y(t) is the product of the areas under x(t) and h(t) Homework Equations Convolution definition The Attempt at a Solution I found a derivation but it skips a step, uploaded it here: htt...

Doesn't a linear change of coordinates preserve complete intersection for a set of homogeneous polynomials, all of the same degree, in a polynomial ring? That is, apply a change of coordinates to a set of homogeneous polynomials {f_1,... f_k} in C[x_1,...,x_M] to obtain {h_1,..., h_k}. Suppose...

Hello team! I saw the other day in a textbook that the Dirac delta function of the form d(x-a) can be written as d(a-x) but the method was not explained. I was wondering if anyone know where this comes from. I've been googling but can seem to find it out. Any help would be appreciated...

Hi, All: I'm a chemistry ignorant. Please bear a bit. Just curious as to the chemical explanation for why the mixture of bakind soda and (kitchen/cooking) vinegar is an effective disinfectant. Is this a neutralization reaction, with vinegar as the acid and bakind soda as the base? If...

Playing around with logarithms I found an interesting property that "log b^n(a^n) = log b(a)". Then I tried to find some kind of proof that this is right and not only a coincidence. Ι made a gereral formula for any value of both n's (α and β) so that "log b^β(a^α) = x". Therefore "a^α = b^(β*x)"...

Homework Statement use the sifting property of the dirac delta function to evaluate the following integrals. a) integral from -inf to inf sin(t) delta(t-pi/2)dt b) integral from 0 to 2 e^(2t) delta(t-1)dt c) integral from 0 to pi e^tan(theta) delta(theta- 3pi/4)d(theta) d)...

Hi. I have the following sentence: \begin{array}{l} A,B \in {M_{nxn}}\\ A \ne 0\\ B \ne 0\\ {\rm{if }}AB = 0{\rm{ then}}\\ {\rm{|A| = 0 or |B| = 0}} \end{array} I know this is true but how can I realize? Just thinking about an example? Thanks!

Hi, I know that if I have a monomorphism f:X\rightarrow Y then for any arrows g,h:A \rightarrow X we have f \circ g = f \circ h \; \Rightarrow \; g=h However in a topological space, if I have f to be an injection but now have f \circ g \simeq f \circ h (where \simeq denotes homotopic) then...

Homework Statement How can i prove this property of Delta, http://e1204.hizliresim.com/w/d/4bw2j.png Homework Equations This is my homework from John David Jackson's Classical Elektrodynamics book The Attempt at a Solution I can't an attempt. Some properties is proved at wiki and...

Homework Statement Hello friends, i couldn't find a solution for the question below. Can you help me? Thank you very much. Let α-norm and β-norm be two different norms on ℝn. Show that f:ℝn->ℝm is Lipschitz in α-norm if and only if it is Lipschitz in β-norm Homework Equations...

I am reading through Kiselev's Geometry: Book I. It is a plane geometry textbook and in the introduction it says the following "One can superimpose a plane on itself or any other plane in a way that takes one given point to any other given point, and this can also be done after flipping the...

I can't for the life of me figure out why the underlined sentence in red is true: Could someone clarify?

Homework Statement Let A \subset \mathbb{R}^{2} be an open connected set, and g: A → ℝ a C2 function. Show that if g is harmonic, i.e. \frac{\partial ^{2} g}{\partial {x_{1}}^{2}} + \frac{\partial ^{2} g}{\partial {x_{2}}^{2}} = 0, then g(x) = \frac{1}{2 \pi r} \int_{\partial B_{r}(x)}gds...

How do I show that Maslov index satisfies the next property (product property). Let \Lambda : \mathbb{R} / \mathbb{Z} \rightarrow \mathcal{L}(n), and \Psi : \mathbb{R}/\mathbb{Z} \rightarrow Sp(2n) be two loops, then if \mu is defined as the Maslov index then \mu(\Psi \Lambda )=...

Homework Statement Prove the following: Let \rho be the correlation coefficient. Prove: \rho(X, Y) = 1 \iff P(Y=aX+b) = 1Homework Equations \rho(X, Y) = cov(X,Y)/\sigma_x\sigma_y The Attempt at a Solution I have no idea how to prove this, I know that rho must lie between 1 and -1 (inclusive)...

Proving the "triangle inequality" property of the distance between sets Here's the problem and how far I've gotten on it: If you are unfamiliar with that notation, S(A, B) = (A \ B) U (B \ A), which is the symmetric difference. And D(A, B) = m^*(S(A, B)), which is the outer measure of...

I feel like I'm missing something obvious, but anyway, in the text it states: lim as n→∞ of an+bn = ( lim as n→∞ of an ) + ( lim as n→∞ of bn ) But say an is 1/n and bn is n. Then the limit of the sum is n/n = 1, but the lim as n→∞ of bn doesn't exist and this property doesn't work...

Prove that in any convex hexagon there is a diagonal which which cuts off a triangle with area no more than one sixth of the area of the hexagon.

The definition of 'Bounded above' states that: If E⊂S and S is an ordered set, there exists a β∈S such that x≤β for all x∈E. Then E is bounded above. The 'Least Upper Bound Property' states that: If E⊂S, S be an ordered set, E≠Φ (empty set) and E is bounded above, then supE (Least Upper...

The Order Divisibility Property states that if an = 1 (mod p), then the order ep(a) of a (mod p) divides n. How can I go about proving this? Additionally, if a is relatively prime to p, when does the congruence am = an (mod p) hold? Is there a proof for this as well? Thanks!

Homework Statement If x is a real number, show that there is an integer m such that: m≤x<m+1 Show that m is uniqueHomework Equations Archimedean Property: The set of natural numbers has no upper boundThe Attempt at a Solution I'm having trouble with showing that m is unique. If x is a real...

I have Recently Buyed a new Farm House and Taxes on that is too high according to the actual price of that property, so to get reduction on my property tax. Do I need a Property tax lawyer? Please Help…..

Hi, I have a quick question about certain algebraic properties of convolution. If I have 3 functions x(f), y(f) and z(f), is the following true? [x(f) . g(f)] * z(f) = [x(f) * z(f)].g(f) I looked on Wikipedia but there's only a property like this if one of the terms is a scalar, so most...

Homework Statement two hundred seventy grams of argon at a pressure of 160kpa and a volume of 1.3 m^3. Homework Equations pv=nrt The Attempt at a Solution (160^3)(1.3) = (.27)(8.31)(t) 208=2.2437t t=92.7

Homework Statement a gas in 3-L container at a pressure of 300kpa and a temperature of 700 degrees celsius, and with a mass of .66 g. Homework Equations pv=nrt The Attempt at a Solution (300)(v)= (.00066 kg)(8.31)(191906 k) = 3.51 im not sure what conversion needs to be...

Hey! So when I enter this in Mathematica In[246] = ZTransform[Sum[f[k] g[k - n], {k, 0, n}], n, z] InverseZTransform[%, z, n] I get: Out[246] = ZTransform[f[n], n, z] ZTransform[g[-n], n, z] Out[247] = InverseZTransform[ZTransform[f[n], n, z] ZTransform[g[-n], n, z], z, n]...

Homework Statement As an example, if we find a DFT of x[n]={1,1,0,1} the result will be X(m)={3,1,-1,1} Homework Equations My Question is that as we know DFT holds symmetry property, why this answer does not void for that property?

I'm trying to prove eA eB = eA + B using the power series expansion eXt = \sum_{n=0}^{\infty}Xntn/n! and so eA eB = \sum_{n=0}^{\infty}An/n! \sum_{n=0}^{\infty}Bn/n! I think the binomial theorem is the way to go: (x + y)n = \displaystyle \binom{n}{k} xn - k yk = \displaystyle...

Is color an intrinsic property of a substance? I thought that if a red object is in an enclosed space, so that no light gets in, will no longer be red - and what makes the object "red" is that electrons absorb all colors of light and then reemit (reflect) the red light. Therefore, if no light...

Homework Statement Show: sinh(z + i2(pi)) = sinh(z) using sinh(z) = (ez - e-z)/2 Homework Equations The Attempt at a Solution So far I have (ex + i(2∏+y) - e-(x+i(2∏+y))/2. Need help proceeding from here. My thoughts were to define a z' = x + i(2∏+y) but I don't think that I can then say...

Homework Statement If X is a metric space then a map f: X \rightarrow \mathbb{R} is called lower semi-continuous if for each ray of the type (a, + \infty) , the inverse image of the ray under f is also open. If X is a compact metric space, prove that f is bounded below, and that f...

Referring to following video, if I magnetize my screw driver having a magnetic property in this way, I would like to reverse the process, is there any approach to completely remove this magnetic property? so there is no magnetic property from my screw driver. Does anyone have any suggestions...

I'm interested in the proper way to give a mathematical definition of a certain geometric property exhibited by certain maps from points to sets. Consider mappings from a n-dimensional space of real numbers P into subsets of an m-dimensional space S of real numbers. For a practical...