# All Questions

60 views

### Logic and Generalizations

As as logic student, I often encounter the following: a situation where a formal system can't see a generalization is true, but I can. The usual case is that of $\omega$-incompleteness. It is often ...
51 views

### Leibniz integral rule - any connection to the Lebesgue dominated convergence theorem?

If a scalar-valued function f(x,y,z) were smooth and decreases rapidly to zero at infinity (so it is bounded), we're able to differentiate under the integral sign - applying the Leibniz integral rule ...
77 views

38 views

### Trouble Understanding Pinocchio (Verifiable Computing) Sparse Polynomials

I hope I'm asking the question properly. I've never asked anything on this exchange before, but I didn't know where else to ask. The paper in question I've almost got all the pieces to understand can ...
72 views

22 views

### Non linear ordinary differential system

Is there an analytical solution (in the general case) to the following differential system (Cauchy Problem) : $\dot{f}=\frac{Af}{(f^2+g^2)^{1/2}}$ $\dot{g}=\frac{Bg}{(f^2+g^2)^{1/2}}$ with the ...
41 views

### elementry properties of closure

Definition: Let X⊂R and let x'∈R, we say that x' is an adherent point of X iff ∀ε>0 ∃x∈X s.t.d(x′,x)≤ε. the closure of X is denoted as \overline(X) and is defined to be the set of all the adherent ...
39 views

### Integral of erfc

I need to calculate this integral: $$\int{dx(erfc(\sqrt{x^2+a^2}))^2 e^{ixc}}$$ where a and c are parameters. Is there any way to calculate this integral or approximate it?
57 views

### Mathcad doesn't calculate an expresion

could anyone help how to force Mathcad to calculate an expression. After some steps I got such result: http://i.imgur.com/7FtprAI.png but how can I get the result of such expression? Thank you!
31 views

26 views

### Normal operators and computing them

H is an inner product space with inner product $( . , . )$ over the complex numbers, and $T∈L(H,H)$. Let $R=T+T^*$, $S=T-T^*$ . Supposing that T is normal and $T(\alpha)=(x+iy)\alpha$, how do I ...
47 views

### I am struggling in calculationg Fourier transform?

I need to find the Fourier transformation of $f(x)=\frac{x}{(x^2+1)^2}$ by two different methods. one is using the property of Fourier transformation, another is computing the integral by definition. ...
143 views

### Spring Mass Damp - Differential Eq

The molecular bond due to intermolecular forces is flexible. A diatomic molecule like oxygen ($O_2$), if disturbed, will oscillate to and fro the equilibrium position ( $R_0$ or $x = 0$ minimum ...
51 views

### Hausdorff measures and densities

I've been stuck on this one for a while now. It's problem 2.4 from Falconer's "The geometry of fractals" Given an $\mathcal{H}^{s}$ measurable subset $E\subset \mathbb{R}^n$ with ...
35 views

### Contrapositive verifications

The contrapositive of: The product of an irrational number and a non-zero rational number is irrational. is: If the product of two numbers is rational, then it cannot be the case that one ...
39 views

### Does an inequality between kernels imply an inequality between the norms of integral operators?

Assume that $g(x,y)$ and $h(x,y)$ are two positive functions such that $0<g<h$ and assume that $$T_g, T_h : L^2(B^n,R)\to L^2(B^n,R)$$ are integral operators defined by T_k[f](x)=\int_{B^n} ...
20 views

### Finding Relations algebraically

I have selfstudy on this subject and want to know if I am grasping the concept well. Here is the question: Let $A=Z^+$, all integers that are positive; Let $R$= relation defined by $aRb$ iff there ...
56 views

### A variant of factorial

Given the definition of a function f as f(n)=1^1 * 2^2 * 3^3 * ... * (n-1)^(n-1) * n^n. Another function g is defined as g(n,r)=f(n)/(f(r)*f(n-r)) Given an n,r,m we are to output g(n,r)%m where m is ...
106 views

### Von Neumann and Hausdorff continuous dimensions are related?

Von Neumann in his book Continuous Geometry introduced (in a suitable lattice) a dimension function that has a continuous range. The definition of a dimension function is axiomatic: see Continuous ...
110 views

### Rotational invariance and distributions

Let $k\leqslant n$ denote two positive integers, $A$ an $n \times k$ matrix with $A'A = I_k$, and $X$ and $Y$ two independent random variables on $\mathbb R^n$, each rotationally invariant (that ...
160 views

### Implicit function theorem to prove tangent plane to the surface

Let $\Phi$ be the regular surface at $(u_o,v_o)$ (ie., $\Phi$ is of class $C^1$ and $T_u\times T_v\ne 0$ a)Use the implicit function theorem to show that the image of $\Phi$ near $(u_o, v_o)$ is the ...
34 views

### Show that $T$ is a sufficient statistic.

Suppose $X_1,\ldots,X_n$ is a sample from a population with parameter $\theta$. Prove that if $T$ is a sufficient statistic for $\theta$, and $\theta=h(\eta)$ where $h$ is differentiable, then $T$ is ...