-2
votes
0answers
16 views

Gauss Chebyshev formula [on hold]

Use Gauss Chebyshev formula with $n=3$ to approximate the value of the integral. $$\int \frac{x^4}{\sqrt{1-x^2}}dx$$ from -1 to 1. Also compare the result with true value, where the zeros and the ...
0
votes
0answers
17 views

Existence of nice exhaustion - Rudin.

This is taken from Rudin's Complex Analysis/Real Analysis Can someone tell me why $K_n \subset \Omega$? I agree it is compact, but why does it follow that it is a subset of $\Omega$? WLOG, I ...
0
votes
1answer
29 views

Matrix representation of a transformation

We have a linear transformation $T: M_{2\times 2}(F) \to F$ by $T(A) = tr(A)$. We want to compute the matrix representation $[T]$ from $\alpha$ to $\gamma$ coordinates. $M_{2\times 2}$ has the ...
3
votes
2answers
57 views

Equality of a quadratic function

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ an arbitrary function and $g: \mathbb{R}\rightarrow \mathbb{R} $ a quadratic function with the following property: For any $m$ and $n$ the equation ...
0
votes
0answers
41 views

The derivative of $D_x$, the differential operator?

I was thinking about how the derivative could also be an operator and I came upon the question: What is the derivative of the differential operator? I'm very much interested in what the answer to the ...
3
votes
1answer
15 views

A question about biorthogonal basis composed of eigenvectors of a finite-dimensional non-self-adjoint matrix

The non-self-adjoint matrix M has non-degenerate eigenvalues, that is $M \psi_i = e_i \psi_i$, and its adjoint matrix satisfies $M^\dagger \chi_j= e_j^* \chi_j$. I know that $(\chi_j, \psi_i) = ...
0
votes
0answers
7 views

Find the clamped spline with the initial and final derivative are equal, and the derivative at $x = 3$ is set to $0$.

The set of $x$ and $y$ are $x=\{0,1,2,3,4,5\}, y=\{0,1,2,3,4,5\}$ I want to do a clamped spline with the two end derivatives equal and the derivative at $x=3$ is $0$. I tried splitting the data into ...
0
votes
2answers
15 views

Every tournament conatins a hamiltonian path - question about the proof

There is a proof that every tournament contains a Hamiltonian path and it goes as follows: Let $P$ be a path of greatest length in a tournament $T$, say $P = (v_1,v_2,...,v_k)$. Let's say that there ...
1
vote
1answer
16 views

Pure death poisson process

I have a pure death process $X=\{X(t) : 0 \leq t < \infty\}$ with parameters $\lambda_n=0$ and $\mu_n=\mu$ and if $X(0)=N$ and I'm supposed to determine $P_n(t)=P\{X(t)=n\}$ for $n=0,1,2,\ldots,N$ ...
0
votes
0answers
5 views

Are derivatives of generic representations generic?

I am trying to learn about the Bernstein-Zelevinsky derivatives. If $\pi$ is a generic representation of $GL_{n}$, then will the $k$-th derivative $\pi^{(k)}$ be generic? Thanks.
1
vote
2answers
23 views

Can there exist $3$, $4$ and $5$-faceted shapes with congruent flat sides in $\mathbb{R}^3$?

I rose this question in my discrete math class (the unit on probability) not too long ago. For instance, a two-sided shape (like a coin) can be one with any geometrical shape as its "side," such as a ...
-2
votes
0answers
17 views

Need help identifying a classification of tetrahedra. [on hold]

Does anyone know what type of tetrahedra the below would be classified under?
0
votes
1answer
18 views

Conditional probability distribution notation versus conditional probabilities of a single sample space?

When writing conditional marginal probabilities, the following seems to be the notation: $$p_{i|Y=y_{j}} = P(X=x_{i}|Y=y_{j}) = \frac{P(X=x_{i},Y=y_{j})}{P(Y=y_{j})}=\frac{p_{ij}}{p_{+j}}$$ This is ...
0
votes
1answer
70 views

To prove $\sin(x) > x - \frac{x^3}{6}$ is strictly increasing [on hold]

How do I prove this is strictly increasing? $ \sin(x) > x - \frac{x^3}{6} $ where $ x>0 $ What I did so far, I first rearrange the inequality, Let $ f(x)=\sin(x) - x + \frac{x^3}{6} ...
-1
votes
0answers
34 views

Expected costs, benefits of a test [on hold]

The president of a firm in a highly competitive industry believes that an employee of the company is providing confidential information to the competition. She is 90% certain that this informer is the ...
0
votes
0answers
18 views

Creating a sequence convergent to zero with special characteristic

Let $\{a_k\}$ and $\{b_k\}$ be positive sequences in $\mathbb{R}$ that both converge to zero. Can we choose $\{c_k\}$ such that it converges to zero and $$ 0<\lim_{k \to \infty} \frac{a_k}{c_k} = ...
0
votes
1answer
23 views

Problems on Divergence theorem

I am struggling in the following problem: $ S \subset R^3$ is a region in divergence theorem. $\vec{n}$ is outward normal to the surface of $S$. Then, what does $div \vec{F}=0$ mean in the ...
0
votes
0answers
9 views

Unbiased estimator for geometric distribution parameter p

I believe that the MLE of parameter $p$ in the geometric distribution, $\hat p = 1/(\bar x +1)$, is an unbiased estimator for $p$ and would like to prove it. So far, I have: $E[\bar x + 1] = E[\bar ...
0
votes
1answer
18 views

How do I find a basis for the following subspace?

I'm unsure how to do the following problem: Find a basis of the following subspace of $R^4$. W = all vectors of the form $(a,b,c,d)$ where $a+b-c+d=0$. Any help would be great, many thanks :)
0
votes
0answers
20 views

Compute the solutions of the following equation in Fourier space:

$$\frac{d^3u}{dx^3} − αxu = 0, x ∈ R, $$ where $ α > 0$ is some constant and $u(x)$ is assumed to satisfy $\int_R u(x) dx = π.$ I know this is a ODE so this is what I came up with so far: ...
0
votes
1answer
17 views

Existence of Solutions to PDEs - How do I know I've got them all?

I'm taking a very computational course in partial differential equations. Because of this emphasis, I'm feeling very underwhelmed by the course, and have a lot of questions that really aren't ...
1
vote
1answer
18 views

Points of scheme with residue field $k$ vs $k$-point

Let $X$ be a scheme over a field $k$. Consider the following definitions. The residue field of a point $x\in X$ is $k(x)=\mathcal{O}_{X,x}/\mathfrak{m}_x$. The $k$-point of $X$ is the morphism of ...
0
votes
2answers
15 views

Graph Theory Edge-Disjoint Spanning Trees

I have the problem: Show that if a graph $G$ contains $k$ edge-disjoint spanning trees, then for each partition $(V_1, V_2, . . . , V_n)$ of $V(G)$, the number of edges of $G$ which have ends in ...
1
vote
1answer
28 views

Cardinality of a UFD [on hold]

Why can we be sure that if A is a Unique Factorization Domain and has at least one irreducible element, then A is infinite? I can't see how to prove this. Does it have any connection with primes? ...
0
votes
1answer
27 views

Calculating the residue of a function

Let $f(z) = \frac{1+z}{1-\cos(z)}$ I wish to calculate the residue of $f$ at $0$, $2\pi$ and $-2\pi$. I believe this can be done by the following since $f$ has simple poles at these points $Res(f, ...
2
votes
1answer
28 views

If E is measurable, then $\delta E$ is measurable.

Problem: If $\delta =(\delta_1,\delta_2,\cdots,\delta_d)$ is a d-tuple of positive numbers $\delta_i>0$, and $E$ is a subset of $\mathbf{R^d}$, we define $\delta E$ by $\delta E = ...
1
vote
1answer
13 views

$_Linear regression for polynomial fitting

I am doing some curve fitting. The theoretical curve is hyperbolic and have the form $(x-x_0)(y-y_0)=c$. This is not linear, so the normal linear least square regression is not apply immediately. ...
0
votes
1answer
22 views

which choice is better?

Lets say you were writing a program to play checkers. Im simplifying the numbers, but the gist should be obvious. Your program calculates the odds of Move A to have a 100 chances to win the game and ...
2
votes
3answers
30 views

Sum of power series using derivation or integration

could anyone help with this question? $$\sum_{n=1}^{\infty}\frac{(x-\frac{1}{2})^{n+1}}{n(n+1)}$$ I have to find sum of this power series using differentiation or integration. Thanks a lot!
1
vote
1answer
23 views

Why Riemann Mapping Theorem is not valid for $U=\mathbb{C}$

If we take simple connected domain $U=\mathbb{C}$, in the statement of Riemann mapping theorem, then why is it not valid. What is the proper justification?
2
votes
1answer
48 views

How to prove that $|a_{1}+2a_{2}+…+na_{n}| \leq 1.$

Let $$f(x)=\sum_{k=1} ^{n}a_{k}\sin(kx)$$ where $n \in \mathbb{Z^+}$ and $a_{k} \in \mathbb{R}$ for each $k=1,…,n.$ Suppose that $ \vert f(x)\vert \leq \vert \sin(x) \vert $ for every $x.$ Prove ...
1
vote
1answer
28 views

Projection Mappings are Quotient Mappings?

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Prove that if $X=X_1\times X_2$ is a product space, then the first coordinate projection is a ...
1
vote
3answers
64 views

Bob starts with \$20. Bob flips a coin. Heads = Win +\$1 Tails = Lose -\$1. Stops if he has \$0 or \$100. Probability he ends up with \$0?

I'm working on the extra credit for my Discrete Structures homework, but so far I have been unable to get a handle on the problem, even with help from 3rd parties, so I've decided to turn to you guys. ...
0
votes
2answers
39 views

divisibility of complex numbers

I want to show that $(a + bi)|(c + di)$ is equivalent to the statement that the ordinary integers $(a^2 + b^2)|(ac + bd)$ and $(a^2 + b^2)|(-ad + bc)$. I also want to show that $(a + bi)|(c + di) ...
1
vote
0answers
21 views

When does the quotient metric is equivalent to the quotient topology?

Suppose that we have an equivalence relation $\sim$ in a topological metrizable space $(X,d).$ Then we can endow $X/\sim$ with the quotient topolgy. Also, under certains circunstances, there exists a ...
1
vote
2answers
23 views

Changing order of double integral

I have a double integral with the integral with respect to x on the inside between 0 and y^2 and the outer integral with respect to y between 0 and 1. If i change the order of the integrals what would ...
-6
votes
0answers
39 views

Uses of numbers in unusual ways [on hold]

I am trying to find examples of either mathematical proofs or real world applications of numbers in fairly strange ways and places. Examples like $${x}^{1/\pi}$$ or $$\ln^ex$$
0
votes
3answers
20 views

Proving kerT is a subspace of V. and rangeT is a subspace of W.

My question is as follows: Suppose $V$ and $W$ are vector spaces, and let $T: V \longrightarrow W$ be a linear transformation. Show that $\ker T$ is a subspace of $V$. Show that ...
2
votes
1answer
28 views

Some relation between parallel vector field and Jacobi field along a geodesic

Cross posted from my question: http://mathoverflow.net/questions/204097/parallel-transport-along-a-geodesic-and-the-related-jacobi-field This is a formula/theorem (written below) that I found ...
2
votes
2answers
60 views

Why do we need to rationalize fractions? [duplicate]

Teachers often take off points from students who write 1/sqrt(2) instead of sqrt(2)/2. Why do we need to write it as sqrt(2) / 2 ? Where did that convention come from? Do we need to even do it? Why do ...
6
votes
1answer
47 views

Irreducibility of $x^{2n}+x^n+1$

I want to know for what $n$, $$x^{2n}+x^n+1$$ is irreducible modulo 2. I think for $n=3^k$ but have no idea how to prove it.
0
votes
0answers
26 views

Growth rate of an infinite product

I have an (infinite) Blaschke product in the upper half plane, $$ f(z) = \prod_k \frac{z-z_k}{z-z_k^*}. $$ The zeros $z_k$ of the function are complex numbers in the upper half plane. Suppose the ...
2
votes
2answers
32 views

Number Theory Simple Proof

I am looking at a solution for a problem where the following line is stated but not explained, and I can not seem to make sense of it: If a prime $p\equiv 3\pmod 4$ then why is $\frac{p(p+1)}{2}$ ...
2
votes
2answers
19 views

How do you prove that T and U are the same linear transformation on an inner product space V?

Is it enough to show that $<T(x),y>$ = $<U(x),y>$ for any x and y in V?
0
votes
1answer
16 views

Solving a system of equations containing complex numbers - Gaussian elimination

Problem: Determine the solutions in $\mathbb{C}^3$ of the following system over $\mathbb{C}$: \begin{align*} \begin{cases} 2x+iy-(1+i)z &=1 \\ x-2y+ iz &= 0 \\ -ix +y -(2-i)z &= 1 ...
0
votes
0answers
16 views

continuity and discontinuity of an equation

I tried to solve this question by using algebraic factorization but it did not work. Thus I want to know how to go about solving $y=5x^4-3x+7$ where values of x is discontinuous.
0
votes
2answers
37 views

Probability of segment lying in circle

Given a circle of radius R: $x^2+y^2\le R$, find probability of horizontal segment with length $\frac{R}{2}$ lie whole inside this circle. Position of segment's center has uniform distribution in ...
0
votes
1answer
7 views

Determine rank and nullity of linear transformation between polynomial of degree $\leq$ 5 to $R^6$

Define the mapping $T$to be the one that maps a polynomial $f(x) \in V$ to the vector $(f(0), f(1),f(2),f(3),f(4),f(5))^t$, where $V$ is the vector space of all real polynomials of degree 5 or less. ...
0
votes
1answer
19 views

X and Y Joint Density Question - Compute the Density of X and Y

I simply cannot understand this. My TAs aren't available until tomorrow and I really do not want to put this off until then. I'd like to have some idea of how to do this beforehand. The question is: ...
0
votes
0answers
9 views

Limits of this parametrisation

Use Stokes’ theorem to solve the following integral (each time the curve is oriented counterclockwise when viewed from above). $$\int \limits_C (x+2y)dx+(2z+2x)dy+(z+y)dz$$ where $C$ is the ...

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