-2
votes
1answer
479 views

Injectivity and Surjectivity of a piecewise defined function

Let there be a function $f\colon\def\R{\mathbb R}\R \to \R$ given by $$ f(x)= \begin{cases} 5x+2 &x\ge 1 \\ x-1 &x<1 \end{cases}, \qquad x \in \R. $$ Prove that $f$ is not surjective, also ...
3
votes
1answer
90 views

comparing bit lengths of binary numbers

Suppose I have two binary numbers x and y that have bit lengths of nx and ...
0
votes
0answers
39 views

the division $14^{256}$ by $17$ [duplicate]

What the rest of the division $14^{256}$ by $17$? $$14^2\equiv9\pmod{17}\\14^4\equiv13\pmod{17}\\14^8\equiv16\equiv-1\pmod{17}\\(14^8)^{32}\equiv(-1)^{32}\equiv1\pmod{17}$$The rest is $1$, ...
1
vote
2answers
92 views

Functional equation, find functions

Find all of the functions defined on the set of integers and receiving the integers value, satisfying the condition: $$f(a+b)^3-f(a)^3-f(b)^3=3f(a)f(b)f(a+b)$$ for each pair of integers $(a,b)$.
1
vote
1answer
114 views

Any 'odd unit fraction' whose denominator is not $1$ can be represented as the sum of three different 'odd unit fractions'?

Let us call a fraction whose denominator is odd 'odd fraction'. Also, let us call an odd fraction whose numerator is 1 'odd unit fraction'. Then, here is my question. Question : Is the following ...
0
votes
3answers
1k views

Matlab - finding gradient

Given the function: $$v(x,y) = x + e^{-((x-1)^2 + (y-1)^2)}$$ I am supposed to calculate the gradient of this expression in Matlab for x defined in the interval -1:0.1:0.9 and y defined in the ...
1
vote
3answers
59 views

Congruences doubt!

What the rest of the division $2^{100}$ by $11$? $$2^5=32\equiv10\equiv-1\pmod{11}\\(2^5)^{20}=2^{100}\equiv-1^{20}\;\text{or}\; (-1)^{20}$$??
1
vote
2answers
76 views

Prove the following statement…

So, how can I prove this limit: $$\lim_{x \to x_0} \frac{1}{(x-x_0)^2}=+\infty$$ What I've tried is to multiply both numerator and denominator with (x+x0), but I guess that's wrong, what can I do to ...
0
votes
1answer
59 views

prove $| \cup_n A_n| <\mathfrak{c}$

If $\{A_n: n \in \mathbb{N} \}$ is a sequence of subsets of $\mathbb{R}$ and $|A_n| < \mathfrak{c}$ for all $n$. Prove $| \cup_n A_n| <\mathfrak{c}$ with $\mathfrak{c}$ the cardinality of ...
1
vote
1answer
100 views

Find pattern in recursion

this is a example from my book Write several elements of the recursion, and see if you can find a pattern. T(n) = T(n – 1) + n T(n –1) = T(n – 2) + (n –1) T(n –2) = T(n – 3) + (n –2) T(n –3) ...
6
votes
1answer
116 views

Hypergeometric Function Differential Equation

Is there some nice obvious way to see that the hypergeometric function $$_2F_1(a,b;c:z) = \sum_{i=0}^\infty \tfrac{(a)_n(b)_n}{(c)_n}\tfrac{z^n}{n!}$$ should satisfy the differential equation ...
0
votes
1answer
43 views

Discrete and combinatorial mathematics

Suppose we have a relation on a set $A$, i.e. $A \times A$, where $|A| = n; \;n$ a positive integer. How can we count the number of relations on set $A$ which are reflexive, symmetric, transitive and ...
0
votes
1answer
87 views

marching band conductor

Let $f(x)$ be the unique polynomial that satisfies: $f(n)=\sum_{i=1}^{n} i^{101}$, for all positive integers $n$. The leading coefficient of $f(n)$ can be expressed as $\frac {a}{b}$, where $a$ and ...
0
votes
1answer
32 views

Find an infinite set?

$X$ is the union of the intervals $[\frac{1}{n^2},2-\frac{3}{\sqrt{n}}]$ for $n=1$ to infinity. Find $X$. My question is about the first few intervals in the union. When $n=1$, the interval is ...
0
votes
1answer
146 views

When f is absolutely integrable and contiunous prove that $\sqrt{f}$ is absolutely integrable.

If $f: (a ,b) \rightarrow [0,\infty)$ is continuous and absolutely integrable on (a,b), then prove that $\sqrt{f}$ is absolutely integrable on (a,b). I have that $\sqrt{f}$ is locally integrable. I ...
0
votes
1answer
41 views

Proof that $P(\cup_{n=1}^\infty\cap_{m=n}^{\infty}A_m^c)=1$

Let $A_1,A_2,...$ be events and P be probability measure. If $\Sigma_{n=1}^{\infty}P(A_n)<\infty$ then prove that $P(\cup_{n=1}^\infty\cap_{m=n}^{\infty}A_m^c)=1$ ie. ...
1
vote
1answer
180 views

Prove the included formula relating cos(nx) and cos(x)

I'm struggling with the below problem. Can anyone shed some light on it? Show that the below formula is a correct relation between $y = \cos n\theta$ and $x = cos \theta$ for all $n$: $$ x = \frac 12 ...
1
vote
2answers
686 views

Is any differentiable function $f : (0,1)\rightarrow [0,1]$ is uniformly continuous

Question is to check if : any differentiable function $f : (0,1)\rightarrow [0,1]$ is uniformly continuous. I know that any continuous function on compact subset of $\mathbb{R}$ is uniformly ...
1
vote
1answer
451 views

Is this definition of limit at infinity of complex functions correct?

In my book (Churchill), a limit of a function at infinity is defined as: $$ \lim\limits_{z \to \infty}f(z) \equiv \lim\limits_{z \to 0}f(\frac{1}{z}) $$ But why can't you define the point at infinty ...
3
votes
1answer
127 views

How can these be the weights of the adjoint representation?

This is perhaps a stupid question. We consider $G =\text{SU}(3)$ and $\pi : G \to \textrm{GL}(\mathfrak{g})$ the adjoint representation that sends $g \in G$ to $Ad_g$ that acts on the Lie algebra ...
6
votes
1answer
155 views

Formal proof $\binom{n}{k}$ is an integer

In mathematics one defines: $\left(\begin{array}{c}n\\k\end{array}\right)=\displaystyle\frac{n!}{k!\cdot (n-k)!}$ This is the number of combinations of $k$ elements from a collection of $n$ ...
1
vote
2answers
56 views

$H=E-k \alpha \alpha ^T$ is orthogonal matrix

Suppose $\alpha$ is n-dim vector, $\alpha ^T\alpha =1$. Solve for $k$ such that $H=E-k \alpha \alpha ^T$ is orthogonal matrix. I guess that $\alpha \alpha ^T$ is a identity matrix, then k is $0$ or ...
1
vote
1answer
34 views

A certain ideal of a valuation ring

A question from Frohlich and Taylor's book 'Algebraic Number Theory', page 60. Let $\mathfrak o$ be a Dedekind domain with quotient field $K$ and let $v$ be a discrete valuation on $K$. Let ...
0
votes
2answers
229 views

“Point P lies on the sphere described a cube.”

Point P lies on the sphere described a cube. Show that the sum of squared distances of the point P of the vertices of the cube does not depend on the choice of P. I cannot found any logical ...
2
votes
1answer
68 views

Compute $\displaystyle\int\frac{e^{f(x)}}{f(x)}\,dx$

I am wondering if there is a non-elementary function in literature that describes the integration of $e^{f(x)}/f(x)$ with respect to $x$, where $f(x)$ is: $$\sqrt{A + Bx + Cx^2}$$ or ...
0
votes
1answer
125 views

How to approach this Secret Sharing scheme?

Suppose that I want to break up a secret into shares such that any set of k people can recover the secret, but I’m also worried that some people might be dishonest and may lie about the secrets they ...
0
votes
1answer
276 views

Writing a permutation as products of transpositions

If a can write a permutation $\sigma$ as a product like $\Delta \alpha \beta$, where $\Delta$ is a product of transpositions (in fact, anything) and $\alpha$ and $\beta$ are two disjoint ...
1
vote
1answer
36 views

A certain ideal of a valuation ring

This is a question from Frohlich and Taylor's book 'Algebraic Number Theory', page 60. Let $\mathfrak o$ be a Dedekind domain with quotient field $K$ and let $v$ be a discrete valuation on $K$. Let ...
1
vote
1answer
78 views

Question about trees

When must an edge for a connected simple graph appear in every spanning tree for this graph? I would have thought it was the midpoint of the longest simple path in the graph. However, there would ...
0
votes
1answer
87 views

Is $f : [0,\infty]\rightarrow [0,\infty]$ which is continuous and bounded has a fixed point

Question is to check if : $f : [0,\infty]\rightarrow [0,\infty]$ which is continuous and bounded has a fixed point. I have first of all considered boundedness. So, $f(x)$ should not have $x$ as ...
3
votes
1answer
256 views

Where to take Real Analysis and Linear Algebra?

I am undergraduate in economics. As you may know, most prestigious departments in economics now require their aspirants to have taken Real Analysis (and Linear Algebra, too) before entering their ...
3
votes
1answer
64 views

Infinite primes represented by a power function?

Can we find infinite many primes of the form $3\cdot 2^n-1$ or $2\cdot 3^n-1$ for some $n$?
9
votes
2answers
373 views

How far do I need to drive to find an empty parking spot?

A parking lot consists of an infinite row of bays. Cars arrive at random intervals (mean interval $T_a$) and stay for a random time (mean stay $T_s$). The time intervals are memoryless (negative ...
3
votes
2answers
1k views

How to create number six using three zeroes?

How to create number 6 using only three 0, any arithmetic operation is allowed? I know it is possible, but I don't know how...
2
votes
1answer
191 views

Solve a system of equations with $3$ unknown powers?

I'm trying to solve this, knowing $X$ $$\begin{cases}1^x*2^y*3^z=X\\x+y+z=13\end{cases}$$ So for example, if $X=2048$ , we have $$x=2\\y=11\\z=0$$ I barely have memories from high school ...
0
votes
0answers
344 views

difference quotient cube root?

I am trying to figure out how to get the difference quotient of $f(x)= \sqrt[3]{x}$ I can only put in the ${h}$ $\frac{\sqrt[3]{x + h} - \sqrt[3]{x}}{{h}}$ and have no idea what to do next. I have ...
1
vote
2answers
171 views

What is the product and coproduct of Morphism category(Arrow category)?

Given category C, Its morphism category D means a category that has 1) "morphisms of C" as its objects 2) "pair (f,g) s.t. the diagram(square) commutes" as its morphisms The above definition is ...
2
votes
1answer
115 views

RSA-210 factorization

RSA-210 has been factored ...
1
vote
0answers
408 views

Find the minimum,maximum, infimum and supremum of sets?

If $X$ is the intersection of all the intervals $(1-\frac{1}{n^2},1+\frac{5}{n^3}]$ for $n=1$ to infinity, what is the minimum, maximum, supremum and infimum of $X$? If $Y$ is the intersection of all ...
0
votes
1answer
46 views

Two discrete valuations on a Dedekind domain

A question from Frohlich and Taylor's book 'Algebraic Number Theory', page 60. Let $\mathfrak o$ be a Dedekind domain with quotient field $K$ and let $v$ be a discrete valuation on $K$. Let ...
1
vote
0answers
42 views

Confirm my working for the conditional posterior of $\beta$

So I have the following question from my textbook, the answer I get is slightly different from the book's answer, which I think may be wrong, could someone please confirm? Question: Suppose $y_{1:T} ...
1
vote
1answer
63 views

Does this inequality hold for $r>1$?

Does this inequality hold? Let $r>1$ and $i<j$ where $i$ and $j$ are positive integers, then ...
4
votes
1answer
104 views

Are complete intersection prime ideals of regular rings regular ideals?

Let $(R, \mathfrak{m})$ be a regular local ring and let $\mathfrak{p}$ be a prime ideal of $R$ which is a complete intersection, i.e. the minimal number of generators of $\mathfrak{p}$ equals its ...
1
vote
2answers
141 views

Accurate computation of $\exp(a x^2) Q(x)$ for big values of $x$?

I was wondering how one can accurately compute the value of $\exp(a x^2) Q(b x)$ for large values of $$x \left(Q(x) \triangleq \frac{1}{\sqrt{2\pi}}\int_x^{\infty} e^{-\frac{u^2}{2}} du \right)?.$$ ...
5
votes
5answers
116 views

What is this limit equal to:

What is the following limit equal to and how do I prove it? $$\lim_{x\to 0^+} \frac{1}{1-\cos(x^2)}\cdot \sum_{n=4}^\infty{n^5x^n} $$ I've tried l'hospital but it doesn't seem to help since I don't ...
1
vote
0answers
133 views

Every torsion free divisible abelian group $D$ is a direct sum of copies of the rationals $\Bbb{Q}$.

From Hungerford's Algebra Every torsion free divisible abelian group $D$ is a direct sum of copies of the rationals $\Bbb{Q}$. Hint: If $0 \not= n \in \Bbb{Z}$ and $a \in D$, then $\exists ! ...
2
votes
2answers
94 views

Limit of sequences ($\lim x_n = a > 0 \Rightarrow \lim x_n ^{1/k} = a^{1/k}$)

I need to show that if $(x_n)_{n \in \mathbb{N}}$ is a sequence such that $\lim x_n = a>0$, then $\lim \sqrt[k]{x_n} = \sqrt[k]{a}$. It was suggest to use the equality $(x-a) = (x^{1/k} - ...
0
votes
1answer
84 views

Showing the limit of a flow must be an equilibrium point under certain restrictions.

I'm stumped on how to approach this one: Consider the autonomous ODE $\dot{x} = f(x)$, $x \in \mathbb{R}^n$ with initial condition $x(0) = x_0$ and $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ (at ...
1
vote
1answer
228 views

Tan Binomial formulas from a set S and its k-subsets

Working around, I found some Tan Binomial formulas. Let's $S$ be a set such that: $$ S=\left\{\text{ }\tan ^2\left(\frac{1\pi }{n}\right), \tan^2\left(\frac{2\pi }{n}\right), ...
4
votes
1answer
59 views

Winning Percentages

A friend and I both play in an NFL pick league. His requires that he only pick 5 games per week. So far there have been 76 total games this year. His record is 18 for 25. My league requires me to ...

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