-1
votes
1answer
35 views

Is this homomorphism in general surjective?

Let $R$ be a commutative ring and $I$ an ideal of $R$. Pick a fix $0 \not= a \in I$ and consider the map $\phi: R \to I$ given by $r \mapsto ra$. Is this map surjective?
-1
votes
2answers
25 views

To test following system of linear equation for equivalency

Let F be field of complex numbers I have two system of equations $x_1 - x_2 =0 $ $2x_1 + x_2 =0$ And $3x_1 + x_2 -0$ $x_1 + x_2 =0$ The definition says that each if equation in first system ...
2
votes
1answer
82 views

Spivak: basic question on the existence of multiplicative identity

The properties explained up to this point are: $$ \begin{matrix} \text{(P1)} & \text{(Associative law for addition)} & a+(b+c)=(a+b)+c. \\ \text{(P2)} & \text{(Existence of an additive ...
-4
votes
1answer
86 views

How can I find the solutions of this third degree equation?

I can't find the solutions of the following third degree equation: $$4 \lambda^3 + 4 \lambda^2 - \lambda -1 =0$$ with Ruffini's rule. Can someone help me find $\lambda_{1,2,3}$? Thank you for any ...
0
votes
0answers
96 views

Factorising $x^7-7x^6+21x^5-35x^4+35x^3-21x^2+20x+14$ into $\Bbb Q$-irreducible factors (Eisenstein's Criterion)

Factorize $x^7-7x^6+21x^5-35x^4+35x^3-21x^2+20x+14$ into $\Bbb Q$-irreducible factors. I've made the substitution $y=x-1$. So I get $y^7+13y+28$ which satisfies Eisenstein's Criterion for $p=13$. ...
3
votes
3answers
259 views

Prove the following using the Wilson's Theorem

Given a prime number, $p$, prove that $(p-1)!\equiv p-1\,\,\left(\text{mod }\frac{p(p-1)}2\right)$ How do we modify the Wilson's theorem into modulo $p(p-1)/2$ ? I can't get any clue (original ...
0
votes
1answer
80 views

Integrate $\int \frac{\sin(3x)}{\cos x}~dx$

An answer is $\ln|\cos x| - \cos2x$. I'm trying to get the answer but I'm getting something different $$\int \frac{\sin(3x)}{\cos x}~dx = \int\frac{3\sin x - 4\sin^3x}{\cos x}~dx=3 \int \tan x ~dx +...
0
votes
1answer
119 views

Proving set is dense

Let $A$ be a dense set of real numbers in $[0,1]$. I need to prove that $B=\{na : a \in A, n \in \mathbb{N}\}$ is dense in $[0,\infty)$. This is very intuitive but I fail to prove it. Any tips?
0
votes
1answer
245 views

To prove in a Group Left identity and left inverse implies right identity and right inverse

Let $G$ be a nonempty set closed under an associative product, which in addition satisfies : A. There exists an $e$ in $G$ such that $a \cdot e=a$ for all $a \in G$. B. Given $a \in G$, ...
1
vote
3answers
152 views

Determine if the following series is convergent or divergent:

$S=\sum_{k=1}^\infty (-1)^{k+1} \frac{k}{k^{2}+1}$ Now, I started by saying: consider, $\sum_{k=1}^\infty \left\lvert \frac{k}{k^{2}+1} \right\rvert$ , if this converges, that means S ...
1
vote
1answer
65 views

Paths and connectivity of graphs

I am trying to show that for a graph on $n\ge 3$ vertices with minimum degree of all vertices $\ge k/2$, G connected that G has a path of length k. I know if n is greater than k but n/2 is less than ...
0
votes
1answer
47 views

Logarithmic question

In the following question I fail to understand why the A option is correct. I understand that D is wrong, and that B and C are correct, but why is A correct? If $3^x=4^{x-1}$, then $x $cannot be ...
0
votes
1answer
45 views

For what value of $k$ is the vector field solenoidal

Problem: Let $\mathbf{r} = x \hat{i} + y \hat{j} + z \hat{k}$ be the position vector of a general point in $3$-space, and let $s= |\mathbf{r}|$ be the length of $\mathbf{r}$. For what value of the ...
1
vote
2answers
178 views

Prove that the set of all integers $>0$ is the smallest inductive set

An inductive set is a set $I$ such that $1 \in I$ and if $x \in I$ then $x+1 \in I.$ Some authors define the set of all integers $>0$ as the smallest inductive set, say Apostol's Analysis. But I ...
3
votes
3answers
105 views

Choosing new teammates

My sister gave me a combinatorical riddle. It doesn't appear to be hard, but I ask you if my thoughts are right, just for certainty. Here it is. Assume you belong to a group of $100$ people, and ...
0
votes
1answer
55 views

Non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i,$ for two consecutive integers.

Given an example of a non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i, \forall a, b \in G$ for two consecutive integers. This is question 5 from Herstein Page 35. I have proved that ...
2
votes
1answer
126 views

Pseudo-Surreal numbers are analogous to?

I've been exploring surreal numbers. Real equivalent of the surreal number {0.5|} I see that pseudo-surreal numbers seem to have an interesting branch of game theory. Still having a form of {x|y}, ...
4
votes
1answer
59 views

Integration of Exponential and Logarithms, $\int_{z-1}^z \log(\frac{1}{z-y}) \exp (-| y| ^{3}) \, dy$

The integral I am dealing with is: $$\frac{3}{2 \Gamma \left(\frac{1}{3}\right)}\int_{z-1}^z \log \left(\frac{1}{z-y}\right) \exp \left(-\left| y\right| ^{3}\right) \, dy$$ where $z\in \mathbb{R}$ ...
0
votes
1answer
15 views

can we say in finite measure that $f_n \to f$ in measure iff every subsequence of $f_n$ coverge almost every where to f

Assume $(X, \mu )$ be finite measure space then can we say that $f_n \to f$ in measure iff every subsequence of $f_n$ coverge almost every where to f . My tries : I know a theorem that states ...
0
votes
0answers
81 views

show that if $M \times \mathbb{R}^{n}$ is orientable than so is $M$

I need to show if $M \times \mathbb{R}^{n}$ is orientable than so is $M$, where $M$ is connected manifold. $R^{n}$ has standard orientation (determined by standard basis ) and by the assumption $M \...
0
votes
2answers
55 views

Hamiltonian Graphs

I am trying to decide whether it is possible to have two graphs with the same degree sequence where both are connected, but only one has a Hamiltonian cycle. Can anyone give me an example? It is ...
1
vote
1answer
61 views

Every nilpotent left ideal is contained in a nilpotent 2 sided ideal.

Question: Let $R$ be a ring and let $J$ be a left ideal of $R$. Assume that $J$ is nilpotent. Prove that $J$ is contained in a nilpotent 2-sided ideal of $R$. Comments: I have found lots of ...
0
votes
1answer
174 views

Summation and minimal value function

I am working on a summation problem that is asking me to find the sum of an expression with the minimum value function in the exponent. I'm not sure about the rules when working with sums and min-...
0
votes
2answers
128 views

Find all solutions to the polynomial

Find all solutions to $$x^4-3x^3-5x+1 \equiv 0\pmod{1125}.$$ I was thinking of solving the system separately so we would have $x^4-3x^3-5x+1 \equiv 0\pmod{3^2}$ and $x^4-3x^3-5x+1 \equiv 0\pmod{5^3}$...
1
vote
2answers
78 views

Identity about a Functor

I'm moving my first steps in CT and suddenly after reading about functors, this question came up in my mind: Let $F \colon A \to B$ a functor between two categories $A$ and $B$, is it true that for ...
0
votes
1answer
256 views

Proving that two expressions are equivalent

So, I'm working through some proof exercises, and one of the questions is about the following regular expression: (a|b)* = a*(a|b)* if they are equivalent, prove ...
1
vote
1answer
108 views

Irreducible elements in $\mathbb{Z} [ \sqrt{-3}]$

I am having trouble this problem in my textbook. It ask us to determine whether the following elements are irreducible in $\mathbb{Z} [ \sqrt{-3} ] = \{a+b\sqrt{-3} : a,b \in \mathbb{Z}\}.$ $\sqrt{-3}...
1
vote
2answers
145 views

Proving a property of a Logic Formal Language

I am stuck at this problem: Let $\Sigma = \{\lnot,\lor,\land,\rightarrow,\leftrightarrow,(,),P_1,...,P_n\}$ be an alphabet. Now let's define the set of logical expressions $\mathscr{L} \subseteq \...
0
votes
1answer
42 views

nilpotent subgroup of finite index in finitely generated abelien by nilpotent

let $N$ be nilpotent subgroup of finite index in finitely generated abelien by nilpotent(i.e there exist normal subgroup $M$ abelien such that $G/M$ is nilpotent ), proceed by induction on the order ...
1
vote
1answer
50 views

Hadamard's product of Fibonacci generating functions.

$F(s) = \frac{1}{1-s-s^2}=\sum_{n\geq0}F_ns^n$. I want to calculate $F(s) \circ F(s) = \sum_{n\geq0}F_{n}^2s^n$. I have tried using Binet"s formula, but problem remains unsolved.
-1
votes
2answers
110 views

Prove that if $G$ is an Abelian group, then for all $a,b$ in $G$, $(ab)^{n} = a^{n}b^{n}$

This question have already been asked on this site, but i could not understand the details so i ask it again. Also what i have done is that first for $n=1$ its trivial, for $n=2$ we have $a(ba)b=a(ab)...
2
votes
2answers
62 views

Sum of arithmetic progression

While solving discrete math problem I've got the sequence of positive whole numbers defined like this (I've looked up simplification to arithmetic progression in the answers): $$ (n-2) + (n-3) + ... = ...
3
votes
2answers
124 views

Product of repeated cosec.

$$P = \prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$$ I realize that there must be some sort of trick in this. $$P = \csc^2(1)\csc^2(3).....\csc^2(89) = \frac{1}{\sin^2(1)\sin^2(3)....\sin^2(89)}$$ I ...
0
votes
1answer
93 views

What points help identify a cubic and an exponential graph?

What are the different points on the graph we need to know which would help determine the equation of the curve. For example in a quadratic graph, we can determine the equation if we know either any ...
2
votes
4answers
133 views

How to prove that the equation $3(a^2+b^2)=7(c^2+d^2)$ has no solution

Question 1: Let $a,b,c,d$ be positive integers, show that $$3(a^2+b^2)=7(c^2+d^2)$$ has no solution. question 1 is from Mathlove, Curious How prove it? My question 2: Find the least ...
1
vote
1answer
141 views

Quiz: people and hats

I've created this quiz, but I'm not sure if the answer that I've found is correct or not. Three people meet at a pub, each of them has a blue or a red hat on his\her head. Nobody knows the colour of ...
0
votes
3answers
74 views

If G is a group such that $(a.b)^{2}=a^{2}.b^{2}$ for all a and b,Then show that G is abelian

This is problem from I.N Herstein Page 35 Q3 .How should i start doing this ?Hints ? Thanks
0
votes
1answer
75 views

Expected value of $\max(x,y) $ given that $x<y$

What is the expected value of $\max(X,Y)$ given that $X<Y$? $X$ and $Y$ are independent and exponential random variables.
0
votes
1answer
31 views

Question about an invertible matrix and elementary matrices

Say we have an invertible matrix $A$, so if we want to find $A^{-1}$ we can do the following process: $(A|I)\to(A'|B_1)\to(A''|B_2)\to ... \to (I|A^{-1})$$\star$. We also know that $A^{-1}=E_1E_2......
6
votes
1answer
471 views

How to prove that the exponential function is the inverse of the natural logarithm by power series definition alone

The exponential function has the well-known power series representation/definition: $e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$ And the natural logarithm has the less well-known power series ...
3
votes
2answers
368 views

Ball and urn method (counting problems)

How many ordered triples $(a, b, c)$ of positive integers exist with the property that $abc = 500$? Since, $500 = 2^2 5^3$ I believe this can be solved using Ball and Urn let $a = 2^{x_1}5^{y_1}$ ...
1
vote
0answers
24 views

On Procedure Constrained Matrix Factorization

Given rank $r$ $A\in\Bbb R^{n\times m}$, is there a procedure to find $XY=A$ such that $X\in\Bbb R^{n\times r}$, $Y\in\Bbb R^{r\times m}$ with property that $\max_{i,j}|x_{i,j}|$, $\max_{i,j}|y_{i,j}|$...
1
vote
1answer
62 views

Distribution equation (x-a)T=0

I have to solve the equation $(x-a)T=0$ , T is a distribution. By definition : $(x-a)\int T(x)\varnothing (x)=0$ I know if I pose $X=x-a$ I find $XT(X)=0$ and $T(X)=\delta(X)$. But I stuck to find ...
1
vote
1answer
50 views

How to proof: The set A × B is infinite if either A or B is infinite and the other set is not the empty set.

I'm having some difficulty trying to prove that. Suppse $A$ is an infinite set and $B$ is the non empty set, I thoght of showing that the cartesian product of both sets have the same cardinality as $...
4
votes
2answers
71 views

Related rate questions and the intuition behind them

I am trying to do the following question from the Schaum Calculus book. Gas is escaping from a spherical balloon at the rate of $2$ ft$^3$/min. How fast is the surface area shrinking when the radius ...
0
votes
1answer
43 views

Is this expression $7x – 2y + 4z = 7x + 2(-y) + 4z$ right?

$7x – 2y + 4z = 7x + 2(-y) + 4z$ Is above expression right I am stuck with a problem ,by using this(above) expression I am getting ambiguous answers but when I use different way i.e $7x – 2y + 4z =...
19
votes
1answer
367 views

What is the intuition behind / How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices?

Given a set of points $x_1,x_2,\dots,x_m$ in the euclidean space $\mathbb{R}^n$, we can form a $m\times m$ Euclidean Distance Matrix $D$ where $D_{ij}={\|x_i-x_j\|}^2$. We know a little bit about ...
0
votes
1answer
48 views

Cauchy sequence under a uniform continuous function

Let , $f:(1,4)\to \mathbb R$ be uniformly continuous and $\{a_n\}$ be a Cauchy sequence in $(1,2)$. Consider: $x_n=a_n^2f(a_n^2)$ and $y_n=\frac{1}{1+a_n^2}f(a_n^2)$ Then which is correct ?...
1
vote
1answer
41 views

Closed form of recurrence equation

I am solving warm up problem 1.2 from Concrete Mathematics book. I've got the right answer by induction: $$ f(0) = 0\\ f(n) = 3f(n-1) + 2, $$ But I can not figure how to simplify it to the closed ...
1
vote
0answers
61 views

Normalizer of a transitive subgroup in the symmetric group

Let $G$ be a finie group, and $H$ be a core-free subgroup of $G$ (that is to say, there is no nontrivial normal subgroup of $G$ contained in $H$). Denote by $\Omega$ the set of right cosets of $H$ in $...

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