Finite Definition and 1000 Threads

Homework Statement Assume F is a field of size p^r, with p prime, and assume f \in F[x] is an irreducible polynomial with degree n (with both r and n positive). Show that a splitting field for f over F is F[x]/(f). Homework Equations Not sure. The Attempt at a Solution I know from...

Homework Statement The formulation of the problem confused me a little, so just to check. No T1 space has a locally finite space unless it is discrete. The Attempt at a Solution This means that, if X is a discrete T1 space, it has a locally finite basis, right? Btw, for the...

In a noetherian ring, why is it true that there are only a finite number of minimal prime ideals of some ideal? (And is it proven somewhere in the Atiyah-mcdonald book?)

A "countable basis" vs. "countably locally finite" problem Homework Statement Sometimes it's fairly difficult to name a thread for a specific problem. :smile: So, one needs to show that, if X has a countable basis, a collection A of subsets of X is countably locally finite of and only if...

Homework Statement I'm not especially good at creating examples, so I'd like to check this one. One needs to find a point-finite open covering of R which is not locally finite. (A collection is point-finite if each point of R lies in only finitely many elements of that collection) The...

do suggest a good book for finite element method?some books have left me confused over the various methods

Finite Dimensional Inner-Product Space Equals its Dual!? Let V be a finite dimensional inner-product space. Then V is 'essentially' equal to its dual space V'. By the Reisz Representation theorem, V is isomorphic to V'. However, I've been told that V=V', which I am having a hard time...

Homework Statement Construct a finite field of order 16. And find a primative element. Homework Equations The Attempt at a Solution What I did was find an irreducible polynomial in Z/<2> of degree 4. I used f(x)=x^4+x+1. Then I took a to be a root of f(x) and set a^4=a+1...

Homework Statement A is compact and B is an open covering of A. Each a in A is contained in at least 2 subsets of B. Show that B has a finite sub-covering where A is still contained in at least 2 members of this finite sub-covering. Homework Equations I just posted the general idea of my...

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I'm trying to understand the proof given in the last 10 minutes or so of this video lecture, but after some struggle, it occurs to me that I may be misinterpreting what the theorem says. According to this, Cantor's finite intersection principle states the following. Given a metric space (X,d)...

Homework Statement We have a solenoid of radius a, length L, with ends at z = +/- L/2. The problem is to use Ampere's law to show that the longitudinal magnetic induction just outside the coil is approximately B_z (\rho=a^+, z) \approx \left(\frac{2 \mu_0 N I a^2}{L^2} \right) \left(1+...

Homework Statement 1. Let G and H be finite groups and let a: G → H be a group homomorphism. Show that if |G| is a prime, then a is either one-to-one or the trivial homomorphism. 2. Let G and H be finite groups and let a : G → H be a group homomorphism. Show that if |H| is a prime, then a...

Find the volume of the finite region enclosed by the surfaces z = 0 and x2 + y2 + z = 1 I know I have to do triple integration on dV to accomplish this but do not know where to start and what limits to use for x, y and z? Cheers guys

Homework Statement Solve 3x + 50 = 11 in F53 Homework Equations Extended Euclidean AlgorithmThe Attempt at a Solution To find 3-1 mod 53 using the euclidean algorithm: gcd(53,3) 1)53/3 = 17 + 2R 53 = 17 * 3 + 2 2 = 53 - 17 * 3 3/2 = 1 + 1R 3 = 1 * 2 + 1 1 = 3 - 1 * 2 = 3 - 2Now...

Homework Statement If E has finite measure and \epsilon>0, then E is the disjoint union of a finite number of measurable sets, each of which has measure at most \epsilon. Homework Equations The Attempt at a Solution I proceeded by showing that by definition of measure, there is a...

Let x>0 be a random variable with some distribution with finite mean and let E denote the expectation with respect to that distribution. By Jensen's inequality we have Elog(x) =< logE(x) < +inf But, does this imply that -inf < Elog(x) too? Or is it possible that Elog(x) = -inf Sorry if my...

Homework Statement Find all the subgroups of Q* (set of all non-zero rational #s) under multiplication. Explain how you know that Q* has no other finite subgroups.Homework Equations The subgroups must satisfy the properties of association, closure, inverse, and identity. The Attempt at a...

Homework Statement A thin rod extends along the z-axis from z=-d to z=d, carrying uniformly distributed charge along it's length with charge density lambda. Calculate the potential at P1 on the z-axis with coordinates (0,0,2d). Then find an equal potential at point P2 somewhere on the x-axis...

Are magnetic field lines around a finite current carrying straight conductor concentric circles in plane perpendicular to length of wire? I have seen texts derive an expression for it : B = μ0.i/4πd [cos Φ1-cosΦ2] where d is perpendicular distance of separation of the point...

Hi, We all know that the finite difference formulae for the derivatives are given by: \frac{dy}{dx}_{i}=\frac{y_{i}-y_{i-1}}{\delta x} and \frac{d^{2}y}{dx^{2}}=\frac{y_{i-1}-2y_{i}+y_{i+1}}{\delta x^{2}} What would be the formulae for the boundary terms? when i=1? I think I can...

Let's say I have this statement. {a^p | p is prime and p < N} a is considered a string so so a^2 = aa, a^3 = aaa and so on... anyway, in this case, since it says that p< N, then is mean that p will be finite right??

Homework Statement I have to program a three component decay chain using finite difference approximation. I understand finite difference and have written my code, but I have an error I can not find which is giving me an erroneous answer. The curve is correct, but the magnitude of the...

Prove that the collection F(N) of all finite subsets of N (natural numbers) is countable.

A field K is called algebraically closed field if any no-zero polynomial has at least one root in K. Given finite field F_q, q=p^m, p is a prime and m is non-negative integer. A famous property of finite field is any element in F_q satisfies: x^q=x. Then I have such an assumption...

Homework Statement As the title says Homework Equations Definitions of "open" and "closed" The Attempt at a Solution Suppose a finite set S is not closed. Then Sc is not open, and there exists an element x of Sc, so that for all µ > 0, either x + u, or x - u, is an element of S...

I know this is really stupid and it looks like i haven't tried at all but i am genuinely confused about this so any guidance at all would help big time. so here is the question. A steel bar, 70mm long is struck at one end by a heavy mass moving at 20m/s. The impact causes a compression wave...

Why is it enough to prove that a set of vectors is a BASIS to a FINITE DIMENSIONAL Vector Space, it is enough to show that it is Linearly Independent. No Need to prove that it spans the whole vector space?

I know the union can be, but how about the intersection? I am trying to prove that: Suppose (X,T) is a finite topological space, n is a positive integer and U_i\in T for 1<= i <= n. Use mathematical induction to prove \bigcap U_i \in T, where the intersection goes from i=1 to n.

Hiya. :) While doing an assignment I ran into this little problem. We are working in the set of natural numbers \mathbb{N}. If i collect each natural number in a set S_1 = \{1\}, S_2 = \{2\},\ldots, S_n = \{n\},\ldots What happens when I take the countable union of all these? S =...

a group or a cyclic group of finite order can i just repeatedly write down the repeated elements and form a very large even infinite group?

Homework Statement Suppose the random variable X has finite exponential moment. Show by comparison to the Taylor series for EXP[x] that X has finite nth moment (E|X|n<inf) for all positive integers nHomework Equations ex=\sum(\frac{x^n}{(n!)}, n,0,inf) The Attempt at a Solutionwe know that...

Is it true that if a finite group G contains a subgroup of index 2, then there is an element of G with order 2?

Homework Statement Let T be a family of finite subsets of the natural numbers N = {1, 2, 3,...} such that if A and B are any members of T, then the intersection of A and B is nonempty. (a) Must N contain a finite subset F such that the intersection of A, B and F is nonempty for any sets A...

Homework Statement I'm supposed to prove that an improper fraction with a finite binary expansion also can be written as a decimal. Homework Equations Obviously my fraction a/b, where a>b, will look like p1/21 + p2/22 + ... + pn/2n The Attempt at a Solution And I have no idea...

Homework Statement Suppose a uniformly charged wire starts at point 0 and rises vertically along the positive y-axis to a length L. Determine the components of the electric field Ex and Ey at point (x,0). That is, calculate \vec{}E near one end of a long wire, in the plane perpendicular to...

I cannot figure out the sum of this finite series: |ysin(x)|+|y2sin(2x)|+...+|ynsin(nx)| where y is real. so I want to listen any opinion may help me>

Hi, I'm here for help and hope somebody could give a hand on this because I'm noob in this. I'm now constructing a MATLAB program to find Electrical field and potential within a square grid mesh with square cavity inside. like the picture above. I only manage up to this...

Is the Jellium model only suitable for an electron gas of infinite volume? If I confined a gas to a finite volume using an infinite potential well, is there still a way to cancel out the infinities in the coulomb interactions between electrons?

Homework Statement a) Find the potential of an axial quadrupole: point charges q, -2q, and q placed on the z-axis at distances L, 0, and -L from the origin. b) Find the potential only at distances r>>L. c) Show that this potential is proportional to one of the zonal harmonics.Homework...

Homework Statement In a finite group, show that the number of non-identity elements that satisfy the equation: x^5 = e = identity element of multiplication mod n = 1 is a multiple of 4. (Also need to show: if the stipulation that the group be finite is omitted, what can you say...

Can Ampère's circuital law be used to find electric field for a finite (say length l) current carrying this conductor at a finite point away from it? If yes, then what will be Magnetic field due to a wire extending from (0,-a/2) to (0,a/2) carrying current “I” at a point (b,0) from it, if I...

Homework Statement Homework Equations <see above>The Attempt at a Solution I'm a bit unsure how to set this up to solve for a solution. Any advice? Its obviously a separable differential equation. But I'm unsure what it is I'm looking for. This looks different then some population...

Homework Statement Show that the set of all finite subsets of N is a countable set. The Attempt at a Solution At first I thought this was really easy. I had A = {B1, B2, B3, ... }, where Bn is some finite subset of N. Since any B is finite and therefore countable, and since a union of...

Hi, In a paper I have v_{n,k} = \Delta^K ( (-1)^n n^k y_n ) with n = K, \dots , N-1, k = 0, \dots, K and N = 2K where \Delta^K is the Kth finite difference operator. As you can see, all v_{n,k} consistute an (N-K) \times (K+1) matrix. So without the \Delta's, each v_{n,k} would be a...

Homework Statement Use the midpoint rule to estimate the area under the graph of f(x) = 7/x and above the graph of f(x) = 0 from [1,25] using two rectangles of equal width. Homework Equations N/A The Attempt at a Solution So first I found \Deltax by using (b-a) / n and got (25 - 1)...

The Lagrangian finite strain tensor is defined as: E_{i,j}=\frac{1}{2}\left(\frac{\partial x_k}{\partial X_i}\frac{\partial x_k}{\partial X_j}-\delta _{i,j}\right) Is it in Einstein Notation so that there is a summation symbol missing, i.e. would it be the same thing if one wrote it as...

Can someone give some clarification of why this would be the case: "A group with less then 1000 elements can be generated by less than 10 elements" Clearly this is the case for some groups, but is it really the case for any group with less than 1000 elements?

24. Apply the boundary conditions to the finite square-well potential at x=0 to find the relationships between the coefficients A, C, and D and the ratio C/D. I understand the wave equations in the three separate regions. For this question I need to only consider I, II. The wave equations need...

It's simple for you mathematicians, but I'm a physician, I don't know much about set theory or logic and such, so it's difficult for me. Let M be the set of all integers that can be described in English in, say, ten lines of text. For example, "fourteen" or "seventy minus eight" or...