3
votes
1answer
147 views

Is there a theorem that can help us count easily the number of subgroups of any given finite group?

Is there a theorem that can help us count easily the number of subgroups of any given finite group?
1
vote
1answer
379 views

Quadratic Matrix Equation

Consider real diagonal (known) matrices $A$, $B$, and $C$, and a comforming matrix $\Pi$. $C + \Pi B + \Pi A \Pi = 0$ I have been trying to solve this system using elementary algebra. I have two ...
5
votes
2answers
2k views

Examples of measurable and non measurable functions

I'm a new in measure theory and I want to understand measurable functions. As I expect measurable function is the function that maps one set to another where preimage of measurable subset is ...
0
votes
1answer
29 views

how to show that g(t) is continuous?

how to show that g(t) is continuous? $g(t) = piecewise(t \neq 0, 0, t=0, 1)$ $\lim_{t \rightarrow 0} g(t) =1$? So it should be continuous at $t=1$? Well it seems this question doesn't meet you ...
1
vote
1answer
109 views

characterization of compactness

When $X$ is an arbitrary topological space, I need to know which of the followings are true or false $1$. If $X$ is compact, then every sequence in $X$ has a convergent subsequence. $2$. If every ...
3
votes
4answers
195 views

Showing that the equivalence class of partial orders order isomorphic to $(X, \leq)$ is not a set

I'm trying to do the following (original image): EXERCISE 25(R):(a) Argue on the grounds of the architect's view of set theory (which was outlined in the Introduction) that $\mathbf{PO}$ cannot be ...
2
votes
1answer
258 views

A Borel-Cantelli lemma related question

I am learning about the Borel-Cantelli lemmas, which I think allow us to conclude that things converge "almost surely" to other things. I am having trouble with a homework question: Show: If ...
3
votes
1answer
806 views

Linearly Independent Dot Product Proof

Let $v_1,...,v_n$ and $w_1,...w_n$ be two sets of linearly independent vectors in $\mathbb{R^n}$. Show that all their dot products are the same, so $v_j \dot\ v_i = w_i \dot\ w_j$ for all $i,j ...
0
votes
1answer
92 views

What technique of integration is applicable for this?

What technique should i use to integrate $$\int(4-x)^{1/4}e^{\frac{1}{2}x}dx?$$ Ive tried to use algebraic manipulation and integration by parts but it just became complicated.
2
votes
1answer
496 views

Can someone explain this non-noetherian subring example?

I'm trying to find an example showing that subrings of noetherian rings are not necessarily noetherian. So I just searched the net and this site, and came upon this: "A common example showing that a ...
0
votes
1answer
52 views

Prove K is a normal subgroup of An for some integer n

I am given a set K with some given values and want to show that it is a normal subgroup of An for some given integer n. Is this how i prove it? First prove K is a subgroup of An Second prove that An/K ...
8
votes
1answer
189 views

Integrating $\frac{x dx}{\sin x+\cos x}$

I am trying to carry out this integration but I seem to be going wrong: $$I=\int_{0}^{\frac{\pi}{2}}\frac{x dx}{\sin x+\cos x}=\int_{0}^{\frac{\pi}{2}}\frac{(\frac{\pi}{2}-x) ...
2
votes
1answer
133 views

Count possible traversals in an undirected graph

A graph of $n$ nodes is given. We have to visit each node twice. How many such traversals are there? It's a complete graph and it's not possible to visit the nodes in a consecutive order. Example: ...
3
votes
5answers
250 views

Please explain this step in proving the square root of 3 is irrational

Assume that $$3 = \frac{p^2}{q^2}$$ So, $$ 3 q^2 = p^2$$ So $p^2$ is divisible by $3$. How we can conclude this?
7
votes
2answers
254 views

Representation of a linear functional in vector space

In the book Functional Analysis,Sobolev Spaces and Partial Differential Equations of Haim Brezis we have the following lemma: Lemma. Let $X$ be a vector space and let $\varphi, \varphi_1, \varphi_2, ...
1
vote
1answer
72 views

$\mathbb{Q}$-dimension of f. g. $\mathbb{Q}[\mathbb{Z}/p^l]$-modules.

My question arises from the previous question Let $M$ be a finitely generated $\mathbb{Q}[\mathbb{Z}/p^l]$-module, where $p$ is a prime number. Is it true that \begin{equation}\dim_\mathbb{Q} ...
0
votes
1answer
976 views

Orders of the Normal Subgroups of A4

Prove that $A_4$ has no normal subgroup of order $3.$ This is how I started: Assume that v has a normal subgroup of order $3$ for e.g. $K.$ I take the Quotient Group $A_4/K$ with $4$ distinct cosets, ...
0
votes
3answers
2k views

Logical Reasoning: Identify next figure

I came across this logical reasoning/ abstract reasoning question in my logical reasoning book. Can somebody please help me identify the next pattern: Thanks!
0
votes
1answer
41 views

Two compactness questions

I'm stuck on how to show one of these is compact, and I want to verify my method for the other. This one I am stuck on: Proposition Let $(\mathbb{Q},d)$ be a metric space with $d(a,b)=|a-b|$. ...
2
votes
4answers
686 views

Is $x^n$ Cauchy in $(C[0, 1], ||\cdot||_{\infty})$?

Consider the sequence of functions \begin{equation} f_n(x) = x^n, \quad x \in [0, 1]. \end{equation} Is this sequence Cauchy in $(C[0, 1], ||\cdot||_{\infty})$? The pointwise limit is not ...
2
votes
1answer
35 views

Rolle-like equality

Let $f$ be a ${\cal C}^2$ function $[a,b] \to {\mathbb R}$, (i.e. the second derivative $f''$ is continuous). Let $g$ be the unique affine map agreeing with $f$ on $a$ and $b$ : $$ ...
0
votes
1answer
2k views

Solving differential equation for simple harmonic motion. Finding k?

A 1 lb weight is suspended from a spring. Let y give the deflection (in inches) of the weight from its static deflection position, where “up” is the positive direction for y. If the static deflection is ...
1
vote
1answer
42 views

Describing open neighborhoods

Let $d$ be a metric on $\mathbb{R}^2$ defined as $$d((x_1,y_1),(x_2,y_2))=\begin{cases} |y_1-y_2| \mbox{ if } x_1=x_2 \\ 1+|y_1-y_2| \mbox{ if } x_1 \neq x_2 \end{cases}$$. Let $N((x,y),\epsilon)$ be ...
1
vote
1answer
143 views

Lemonade, Sandwiches, and Biscuits

Here is the example case. You might recognize this to be from Lewis Carroll's "A Tangled Tale". If you're already familiar, you can scroll down to the question. Example Given that one glass of ...
12
votes
1answer
668 views

Constructing a bijection from (0,1) to the irrationals in (0,1).

How does one construct a bijection from (0,1) to the irrationals in (0,1)? Or if I am getting my notation right, can you provide an explicit function $f:(0,1)\rightarrow(0,1)\backslash\mathbb{Q}$ such ...
1
vote
1answer
30 views

Economics simplification of stochastic transition of capital

I'm taking an macro-econ paper and I can't seem to work out the following simplification. Basically somehow by combining equation 4.14 and ...
1
vote
3answers
103 views

Prove that for all positive reals $k,j$ with $2k>3j$, there exists positive integers $s,t$ such that $k > |\sqrt{s}-\sqrt{t}| > j$

Prove that for all positive reals $k,j$ with $2k>3j$, there exists positive integers $s,t$ such that $k > |\sqrt{s}-\sqrt{t}| > j$
0
votes
2answers
510 views

Prove that $f$ is uniformly continuous on the interval $(a,d)$

Let $a<b<c<d$ and $f: (a,d) \rightarrow \mathbb{R}$. Assume $f$ is uniformly continuous on $(a,c)$ and $(b,d)$. Prove that $f$ is uniformly continuous on the interval $(a,d)$. My proof ...
2
votes
1answer
518 views

Orthogonal Basis

Let $\mathbb{R^2}$ have the inner product definied by the positive definite matrix $K=\pmatrix{2&-1\\-1&3}$ a.) Show that $v_1 = (1,1)^T$, $v_2 = (-2,1)^T$ form an orthogonal ...
2
votes
1answer
156 views

Describe in terms of congruence class all of the odd primes $p = 2m+1$ such that $p \mid 10^m - 1$

Describe in terms of congruence class all of the odd primes $p = 2m+1$ such that $p \mid10^m - 1$. $p=2m+1 \iff 2m \equiv 1 \pmod p$ $p \mid 10^m - 1 \iff 10^m \equiv 1 \pmod p$ So, I have $2m = ...
1
vote
1answer
189 views

Weak derivative of $\operatorname{sgn}(x_1)$

Let $x\in \mathbb{R}^{n}, x = (x_1,\ldots,x_n)$, and $f(x) = \operatorname{sgn}(x_{1})$. Is $f$ weakly differentiable on $U = B(0,1)$, i.e. unit ball in $\mathbb{R}^{n}$, and what is the weak ...
4
votes
1answer
412 views

Spectral radius and positive definite of matrices

Denote $ \rho(A)$ to be the spectral radius of a matrix $A,$ that is the maximal eigenvalue of $A.$ We say that a matrix $M$ is positive definite, respectively positive semidefinite, if $x^TMx>0$ ...
0
votes
1answer
852 views

Difference between rotation and pure rotation

Hi i am trying to understand my teacher's assignment. I have 2 write 2 Matlab functions ...
0
votes
3answers
298 views

Why does this cyclic group have 8 elements?

The cyclic group of $\mathbb{C}- \{ 0\}$ of complex numbers under multiplication generated by $(1+i)/\sqrt{2}$ I just wrote that this is $\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i$ making a polar angle ...
1
vote
2answers
119 views

Sur- in- bijections and cardinality.

I think about surjection, injection and bijections from $A$ to $B$ as $\ge$, $\le$, and $=$ respectively in terms of cardinality. Is this correct? And extrapolating from that, are these theorems ...
0
votes
2answers
170 views

Finding a basis containing v

Let $w = \{ (a,b,c,d)^T \mid 2a+3b+c+2d=0 \}$ in the vector space $\mathbb{Z}_5^4$. $(2,2,1,2)$ is in $w$. ($w(2,2,1,2) = 15$, $15\,\mbox{mod}\,5=0$) How do I find a basis for $w$ that contains ...
1
vote
0answers
50 views

Having difficulty understanding cosets. [duplicate]

Possible Duplicate: What is a quotient ring and cosets? I am taking an algebra class and having trouble understanding the elements (cosets) and ideals of quotient rings. Could someone ...
1
vote
1answer
190 views

N piles of hidden cards of known marginal probability distribution, then a card is revealed in one of the piles.

I am currently trying to use probability theory to help solve a programming problem involving Monte Carlo Tree Search with Information Sets and have hit a roadblock. The problem can be described as ...
2
votes
1answer
152 views

connectedness of two particular set of matrices

I need to know whether the $1)$ The set of all symmetric positive definite matrices are connected or not? Well I guess, This set is convex set, Let $M$ be a symmetric positive definite so ...
1
vote
2answers
41 views

Solve equation for alpha

$E_1 = Ch_1^\alpha$ How can I solve for the exponent alpha? Is it just $\frac{\frac{E_1}{C}}{ln(\alpha)}$?
1
vote
0answers
75 views

Maximising the correlation coefficient

How do I show that the correlation coefficient $\rho(Y,f(X))$, where $f$ is any measurable function, is maximized in absolute value when $f(X)$ is linear in $E[Y|X]$. I know that for 2 r.v.s X and Y ...
1
vote
1answer
283 views

Min,max distances between point and spherical surface

I have a spherical surface defined by four points on an ellipsoid centered at (0,0,0). That is, the four points define a bounding box projected onto the ellipsoid. I have another point, P at some ...
1
vote
0answers
2k views

Rewriting triple iterated integrals

I'm currently learning multivariate calculus, and one of the problems I had for homework is: Rewrite this integral as an equivalent iterated integral in the five other orders. $$ \int_0^1 ...
0
votes
2answers
321 views

The subset of $C^{\infty}$ functions with compact support in $\mathbb{R}$ in the space of bounded real valued continuos function on $\mathbb{R}$

could any one tell me the following statement is true or false? and any reference for proof or counter examples? The subset of $C^{\infty}$ functions with compact support in $\mathbb{R}$ in the space ...
1
vote
0answers
184 views

Simultaneous Eigenvalue Problem

I have what I think is a simultaneous eigenvalue problem in three parameters: $$\alpha A_1x + \beta B_1x + \gamma C_1x + D_1x = 0$$ $$\alpha A_2x + \beta B_2x + \gamma C_2x + D_2x = 0$$ $$\alpha A_3x ...
2
votes
1answer
347 views

The meaning of the Jacobi Symbol and its efficient evaluation

Are the following jacobi symbol evaluations correct?: $(\frac{35}{53}) = -1$ $(\frac{68}{233}) = -1$ $(\frac{126}{509}) = 1$ $(\frac{672}{1297}) = 1$ $(\frac{1235}{3499}) = -1$ Also what is the ...
2
votes
1answer
475 views

Showing a metric space is bounded.

This is from a review packet: Let $d:\mathbb{R} \to \mathbb{R}$ be defined as $$d(x,y)=\frac{|x-y|}{1+|x-y|}.$$ i) Show that $(\mathbb{R},d)$ is a bounded metric space. ii) Show that $A=[a,\infty)$ ...
0
votes
1answer
166 views

extension of analytic to meromorphic function

could any one help me to show that any $f$ analytic in the annulus $1\le |z|\le 2$ and $|f|$ is a constant on $|z|=1$ and on $|z|=2$ (not necessarily the same constant) admits a meromorphic extension ...
0
votes
3answers
221 views

Prove dot product will get maximum

If we have two identical sets $A_1 = A_2 $, and we were asked to get the maximum sum of multiplying one distinct element from $A_1$ by another distinct element of $A_2$, for all elements in $A_1$. ...
4
votes
2answers
507 views

Proving the 3-dimensional representation of S3 is reducible

The 3-dimensional representation of the group S3 can be constructed by introducing a vector $(a,b,c)$ and permute its component by matrix multiplication. For example, the representation for the ...

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