3
votes
0answers
234 views

Closed-form expression for sum of Vandermonde matrix elements

Given the Vandermonde matrix: $$\begin{pmatrix}1^0 & 1^1 & 1^2 & ... & 1^n \\ 2^0 & 2^1 & 2^2 & ... & 2^n \\ \vdots & \vdots & \vdots & \ddots & ...
0
votes
1answer
98 views

Why is not language $L=\{w_i \mid w_{2i} \notin M_i\}$ recursively enumerable?

Why is not language $$L=\{w_i \mid w_{2i} \notin M_i\}$$ recursively enumerable? I need to show that by diagonalization, but dont know how? Its quite obvious for $L=\{w_i \mid w_i \notin M_i\}$, but ...
21
votes
1answer
816 views

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$? Recall the definition: Let $S = \mathbb{N^{<N}} = \bigcup_{n = 1}^\infty \mathbb{N}^n$ be the set of ...
0
votes
3answers
176 views

Need help starting a Numerical Differentiation problem

Derive the following difference approximation for the first derivative: $f'(x_0) = (f'(x_0 + 2h) - f(x_0 - h))/3h$ I really just need some pointers in how to start this out. If I were to guess, it ...
1
vote
1answer
155 views

Let $L_p$ be the complete, separable space with $p>0$.

Let $L_p$ be the complete, separable space with $p>0$. $\mathbf{J}=\{I = (r,s] \}$ where $r$ and $s$ are rational numbers. $\mathbf{A}$ is the algebra generated by $\mathbf{J}$, with ...
1
vote
1answer
86 views

On equivalent definitions of Gorenstein local rings

I'm studying these notes, in particular page 17 theorem 1. I have some problems in the implication iii $\Rightarrow$ ii: Let $A$ be a local ring of dimension $n$ and maximal ideal $m$ ii) $A$ is ...
1
vote
1answer
3k views

Eigenvector of matrix that reduces to identity

Let's say for the eigenvector equation, $ (\lambda I - A)X = 0 $, some eigenvalue of $A$, $ \lambda_1 $, is found and $ \lambda_1 I - A $ is reduced to solve for its respective eigenvector $ X_1 $. If ...
0
votes
1answer
46 views

Functions with complex value

Let be $f:\mathbb{R}\rightarrow \mathbb{C}$. This $$\operatorname{Re} \int_a^b {{e^{ - i\theta }}f(x)dx = } \int_a^b {\operatorname{Re} ({e^{ - i\theta }}f(x))dx}$$ is True? Why?
0
votes
2answers
590 views

Radius word problem help?

It's about the sprockets and chain of a bicycle. The pedal sprocket has a radius of 4 in., the wheel sprocket has a radius of 2 in., and the wheel a radius of 13 in. The cyclist pedals at 36 rpm. I ...
15
votes
4answers
974 views

An Inequality problem relating $\prod\limits^n(1+a_i^2)$ and $\sum\limits^n a_i$

Let $(a_1,\space a_2,\space \cdots, \space a_n) \in \mathbb R^n_+$ such that $\displaystyle \prod^n_{i=1 }a_i = 1$. Prove that $$\displaystyle \prod^n_{i=1} (1+a_i^2) \le \cfrac ...
3
votes
1answer
63 views

Show there is no measure on $\mathbb{N}$ such that $\mu(\{0,k,2k,\ldots\})=\frac{1}{k}$ for all $k\ge 1$

For $k\ge1$, let $A_k=\{0,k,2k,\ldots\}.$ Show that there is no measure $\mu$ on $\mathbb{N}$ satisfying $\mu(A_k)=\frac{1}{k}$ for all $k\ge1$. What I have done so far: I am trying to apply ...
3
votes
3answers
2k views

Determinant of the sum of matrices: $\det (A + B^T) = \det(A^T + B)$

How can you show that $$ \det(A + B^T) = \det(A^T + B)$$ for any $n\times n$ matrices $A$ and $B.$
4
votes
2answers
642 views

Finding the image of a mapping over a region.

I'm having a very hard time understanding the concept of images and mappings in the complex plane. Considering the map $w=e^{z}=e^{x}e^{iy}$, find the image of the region $\left\lbrace x+iy:x\geq 0, ...
1
vote
1answer
162 views

Let $a,b \in \Bbb Z$, $p$ a prime and $p \gt 2$, given the following

Let $a\in \Bbb Z$, $b\in \Bbb Z$ such that $p \nmid b$, and $p$ a prime where $p \gt 2$. If for all $x \in \Bbb Z$ such that $p \nmid x$ and $\operatorname{ord}_p(x) \ne p-1$, $p$ satisfies ...
0
votes
2answers
139 views

Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$

Let $C_c^\infty$ denotes the set of real valued function with compact support. Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$. If ...
4
votes
2answers
247 views

Finding $\text{ker}(Tr)$

Defining the trace in the usual way as a function $Tr: F^{n\times n} \rightarrow F$, where $F$ is some field. I want to show that $\text{ker}(Tr)=\text{span}_F(\{AB-BA|A,B\in F^{n\times n}\})$. So ...
2
votes
1answer
110 views

Homomorphism $\mathbb{Z}^2 \to \mathbb{Z}^2$

I want to find a group homomorphism $f: \mathbb{Z}^2 \to \mathbb{Z}^2$ which satisfies $f(1,0) = (2,6)$. Can any such homomorphism be made into an isomorphism?
1
vote
0answers
62 views

Confused about easy integration by substitution

I want to integrate $\int \nabla f(x)$ by substitution. Let $x = g(y)$. Then is $\int \nabla f(x)$ equal to $$\int \nabla [f(g(y))] |\det Dg| $$ or $$\int \nabla f|_{g(y)} |\det Dg| ?$$ I am ...
3
votes
1answer
132 views

How to turn a series representation into a formula

I need to turn the following into something that doesn't use summation notation. Can someone help me figure out how to do that? I would know how to do it were it a simpler case but this one is ...
1
vote
2answers
146 views

Unexpected results when integrating definite integral with variable bounds

When I plug something like this into Mathematica: $$\int_0^{x^2-1} k y \, dy$$ I get exactly what I would expect: $$\frac{k}2 (x^2-1)^2 $$ However, when I change my bounds ever so slightly, from ...
2
votes
0answers
41 views

Which filter is the most suitable if I know points with zero noise amplitude

I've got observed data $Y_1,\ldots, Y_n $ which consists of real values $X_1,\ldots, X_n$ and additive high-frequency noise $e_1,\ldots, e_n$, so $Y_i=X_i+e_i$. I know, that indicies $i_1,\ldots, i_m, ...
2
votes
2answers
611 views

How to prove positive definiteness?

$B_{(n+1)(n+1)}$ = $ \begin{bmatrix} A & u \\ u^T & 1 \\ \end{bmatrix}$ is given, and $A$ is a positive definite matrix where its Cholesky factorization is given ...
1
vote
0answers
73 views

Trying to price options using infinite series

If you are trying to price an option if the stock surges you can reap a very large return, but most of the time the return is $-p_1$ where $p_1$ is the amount you invested The problem i'm running ...
5
votes
1answer
615 views

Non-trivial homomorphism between multiplicative group of rationals and integers

Let $\mathbb{Q}^{\times}$ be the multiplicative group of non-zero rationals. Is there a non-trivial homomorphism $\mathbb{Q}^{\times} \to \mathbb{Z}$? In the same spirit, is there a homomorphism ...
1
vote
2answers
133 views

Limit points question

Let $X$ be any topological space and let $C \subseteq X$ be any subset. If $C$ is a closed subset of $X$ does it follow that the set of all limit points of $C$ is closed as well?
1
vote
3answers
3k views

20 options. 10 must be chosen. how many combinations exist

I have a list of twenty options and each option is different. If ten options must be chosen, how many combinations exist? Can someone show me how the math on this? Example, if i have 20 different ...
0
votes
3answers
67 views

Is $Z$ also independent??

In a problem I am asked to find $\Bbb P(X=1|\frac{X+Y}{2}=2)$, and $X$ and $Y$ are independent random variables. In a previous part of the problem I defined $X+Y$ to be $Z$. So I simplified the ...
0
votes
1answer
42 views

How many times does this expression give the value 0 as modulus?

$S=\{1,2,3\ldots,19\}$ $(5k + 5) \mod 20$ $\gcd(20,5) = 5$ $20$ and $5$ are divisible by $5$ and $1$. thus the expression gives the value $0$ $2$ times between $1$ and $19$?
1
vote
1answer
4k views

What is the probability one card from each suit will be represented when 5 cards are dealt?

If each suit is represented then we will have two cards from one suit and one card each from the remaining suits. So I am counting the ways this can happen like so - ${52 \choose 1}$ ways of ...
1
vote
1answer
96 views

Analytic function on convergent sequence

Let $f:U\to\mathbb{C}$ analytic function, where $U$ is a region. $x_n \to x_0 \in U$ is a real convergent sequence, it is known that $f(x_n)$ is real for all $n$. Is it true $f^{(n)}(x_0)$ is real for ...
1
vote
2answers
2k views

Intuitively explaining the difference between a combination and permutation

I'm having a hard time trying to determine when to use combination and when to use permutation with a problem. Can someone offer a clear and concise explanation or general rules to follow so I don't ...
3
votes
1answer
188 views

Does $\lim_{n\rightarrow \infty} \int_X f_n - \int_X f\gt 0$ implies that convergence of $f_n$ to $f$ a.e. fails?

I've come across this problem as a part of another proof that I'm writing and I want to know if this is a right conclusion: Let $X$ be a finite measure space and $\{f_n\}$ be a sequence of ...
1
vote
1answer
45 views

How to determine a in r - in a function of relations

I'm pretty stuck on the following question $f$ on $\mathbb{R}$ given by $xfy\Leftrightarrow (y(2x-3)-3x=y(x^2-2x)-5x^3)$ is a function. Let $g$ be the restriction of $f$ to $\mathbb{Z}^+$, implying ...
1
vote
2answers
190 views

What is the cardinality of a transcendence basis of $\mathbb{C}$ over $\mathbb{Q}$?

What is the cardinality of a transcendence basis of $\mathbb{C}$ over $\mathbb{Q}$? Is it that of the continuum? Proof?
0
votes
1answer
228 views

Calculating the Fundamental group of $\Bbb R P^2$

The fundamental group of $\Bbb R P^2$ is $\Bbb Z \times \Bbb Z$. I cannot understand why though, since $\Bbb RP^2$ is a disc with a Möbius strip and the disc is contractible so wouldn't it have ...
1
vote
2answers
114 views

How do we show that this modular function is one-to-one?

$$f(x) = {(x^2+5)}\bmod{9}$$ $${(x^2 + 5)} \bmod {9} = (x^2 + 5)\bmod 9$$ $$(x^2 + 5) = (x^2 + 5)$$ $$x^2 = x^2$$ $$x = x$$ Is this the correct way to do this? I have no idea how to manipulate the ...
1
vote
1answer
494 views

Find all Laurent series of the form…

Find all Laurent series of the form $\sum_{-\infty} ^{\infty} a_n $ for the function $f(z)= \frac{z^2}{(1-z)^2(1+z)}$ There are a lot of problems similar to this. What are all the forms? I need to ...
0
votes
2answers
151 views

Finding connected components of two given spaces

Suppose that in the Cartesian plane $\mathbb{R}^2$ we let $X$ denote the union of all lines through the origin with rational slope. Would that make $X$ connected, since all such lines are connected ...
4
votes
1answer
74 views

Smooth Monotone $\mathbb{R}^3$ curve with constant (nontrivial) curvature

So I was trying to construct a closed curve in $\mathbb{R}^3$ with constant positive curvature and non-trivial torsion. To do this I tried to glue two helices together in a smooth way with a curve ...
0
votes
3answers
1k views

Help finding coterminal angles?

I'm trying to find an angle between 0 and 2π that is coterminal with -4π/3 in terms of pi. How do I go about doing this?
1
vote
1answer
398 views

Prove the Hyberbolic identity

(Note: The ? is where I have to fill in something) Prove: $$ \begin{align} \sinh2x & = 2\sinh x\cosh x\\ \sinh2x & = \sinh (x + ?)\\ & = \sinh x(?) + \cosh x \sinh x\\ & =\space?\\ ...
4
votes
4answers
411 views

Proving $(A \triangle B)\cup C = (A\cup C)\triangle (B\setminus C)$ using set algebra

I tried to prove this equation $(A\bigtriangleup B)\cup C=(A\cup C)\bigtriangleup(B\setminus C)$ by elementhood and set algebra but with no result. I can see that equality stands in Venn's diagrams, ...
3
votes
1answer
493 views

Integral of a curve with respect to its curvature?

I've been struggling with this one for about $3$ weeks: What is the integral of a $\mathbb{R}^3$ curve with respect to its curvature? I though about approaching it with the Ferret-S formulas, and ...
1
vote
1answer
237 views

Related Rates calculus problem

For some reason I keep getting this question wrong. Suppose a 6 feet tall man is walking away from a 15 foot tall lamp post at 5ft/s. What is the rate at which the man's shadow is moving when he is ...
3
votes
1answer
103 views

Summation over a Vector

I am trying to find the Fourier series of a 3D function, $e^{-\alpha(x^2 + y^2 + z^2)}$ with bounds $-\ell_1 < x < \ell_1$, $-\ell_2 < y < \ell_2$, $-\ell_3 < z < \ell_3$. I have ...
3
votes
1answer
970 views

How can a turing machine solve the element distinctness problem?

I am reading an example in Sipser's famous book on the theory of computation. In this example, Sipser creates a turing machine M to solve the element distinctness problem. M is given a list of strings ...
0
votes
1answer
71 views

Use of integration and difference between two equations in monograph

I am reading a monograph on seismic filtering techniques (link). The author rewrites Equation (7.3) in this monograph as Equation (7.4). By adding integrals to the equation as in Equation (7.4), ...
8
votes
3answers
553 views

What is the probability on rolling $2n$ dice that the sum of the first $n$ equals the sum of the last $n$?

The Question What is the probability, rolling $n$ six-sided dice twice, that their sum each time totals to the same amount? For example, if $n = 4$, and we roll $1,3,4,6$ and $2,2,5,5$, adding them ...
0
votes
1answer
81 views

showing G is abelian

If $G$ group of order $52$ includes a normal group of order $4$ then $G$ is abelian. I did like this $$ |G|=52=2^2\cdot 13 $$ let $H$ be normal group of order $4$. $n_{13}=1$ thus $G$ has a $K$ ...
3
votes
4answers
136 views

Prove that if $5$ divides $a^2$, then $5$ divides $a$

Ok so my teacher said we can use this sentence: If $a$ is not a multiple of $5$, then $a^2$ is not a multiple of $5$ neither. to prove this sentence: If $a^2$ is a multiple of $5$, then $a$ itself is ...

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