1
vote
1answer
60 views

For a tournament T of order n, let $Δ=max \{od v:v∈V(T)\}$ And $δ=min\{od v:v∈V(T)\}$ Prove that if $Δ-δ< \frac n2$, then T is strong

For a tournament $T$ of order $n$, let $Δ=max \{od v:v∈V(T)\}$ And $δ=min\{od v:v∈V(T)\}$ Prove that if $Δ-δ< \frac n2$, then $T$ is strong Here is my final attemp. Prove this by contrapositive. ...
0
votes
1answer
451 views

Alpha and Omega limit sets (dynamical systems)

What are all the possible $α$- and $ω$-limit sets of points for the linear system of equations $x'=Ax$ given by the following matrices $A$: I guess I don't really know how to get started, Could I ...
1
vote
1answer
238 views

determine the revenue, cost and profit functions

The demand of gloves is $$x(p)=20,000-2000p,$$ where p denotes price per pair. The total cost of $x$ pairs of gloves is $$c(x)=30,000+1.50x$$ dollars. Determine the revenue and cost functions in ...
1
vote
1answer
40 views

Multiplying algebraic terms problem

Today I have this problem to solve (I already know the answer though, because I cheated by looking it up): The area of a rectangle is equal to x^2 + 9x + 18. If the length of one side is 2x+6 , what ...
3
votes
3answers
167 views

Prove the condition of the score sequence make the tournament strong

Prove the theorem 4.19: A non-decreasing sequence $\pi:s_1,s_2,\ldots,s_n$ of nonnegative integers is a score sequence of a strong tournament if and only if $$\sum_{i=1}^ks_i > \binom k 2 $$ for ...
0
votes
2answers
46 views

Show that there are no $f_1, f_2$ such that $f_1f_2 = f$.

Let $\mathbb{F}$ a finite field. Show that there's $f\in\mathbb{F}[x]$ such that $\deg(f)=2$ and there are no linear polynomials $f_1,f_2\in \mathbb{F}[x]$ such that $f_1f_2 = f$. Hint: Define $\...
7
votes
3answers
184 views

In logic, do the $\Longrightarrow$ and $\rightarrow$ signify different things?

In logic, do the $\Longrightarrow$ and $\rightarrow$ signify different things? Are there contexts where one is more appropriate than the other? I had believed that the $\Longrightarrow$ was for ...
1
vote
1answer
24 views

Proposotional logic derivation

Show that (φ ∧ ψ) ↔ ¬(φ → ¬ψ) is derivable. I have derived ¬(φ → ¬ψ) from (φ ∧ ψ) by assuming (φ → ¬ψ) and (φ ∧ ψ) and deducing a contradiction. By cancellation of the hypotheses I can then conclude ...
1
vote
1answer
198 views

Why is an autocorrelation matrix always positive(semi)definite?

Can someone help me understand why an auto-correlation matrix is always positive definite or positive semidefinite? Can adding some value down the main diagonal convert it from a semi definite to a ...
0
votes
1answer
68 views

Not Quite Metrization

Let's say I have a space $X$ with a function $d\colon X \times X \to \mathbb R$ that has the following 2 properties: $d(x,y)\ge 0$ for all $x$, $y \in X$ and $d(x,x) = 0$ for all $x$, $y \in X$. ...
3
votes
2answers
56 views

Find minima of the multivariable function $\frac{4}{x^2+y^2+1}+2 xy$

$$f(x,y)=\frac{4}{x^2+y^2+1}+2 xy \\ \text{within the domain: }1/5\leq x^2+y^2\leq 4$$ I am able to find the maximum of the function at $x^2+y^2=4$ by substituting x,y for $\cos(t)$ and $\sin(t)$ ...
0
votes
5answers
110 views

If H a subgroup, and the left and right cosets are equal, does this mean that H is the center?

Theorem: If $H$ is a normal subgroup of $G$, then $aH=Ha$ for every $a \in G$. If H a subgroup, and the left and right cosets are equal, does this mean that H is the center? Isn't the definition of ...
-1
votes
1answer
26 views

Probability of getting 3 different CDS from 3 different boxes.

Boxes of Cheerios contain a CD. There are 3 different CD's. A person buys 3 boxes of cheerios. Of course there is the possibility that there will be duplicates among the prizes. What is the ...
2
votes
2answers
127 views

Convolution with delta function

I am merely looking for the result of the convolution of a function and a delta function. I know there is some sort of identity but I can't seem to find it. $\int_{-\infty}^{\infty} f(u-x)\delta(u-...
0
votes
1answer
29 views

Is my solution proper proof for theese two constants making the piecewise function continuous?

If we have the piecewise defined function $$f(x) = \begin{cases} x^2+ax+b &,\ x\le0 \\ e^x(x+1) &,\ x> 0 \end{cases}$$ Solution: $$\lim_{x\rightarrow0} f(x) = f(0) \implies \lim_{x\...
1
vote
2answers
79 views

Projections and open maps

I have the projection $\pi_x : X \times Y \to X$ where $\pi_x$ is defined as: $\pi_x(x,y) = x$. How can I show this is an open map? I know that a map $f: X \to Y$ is an open map if whenever $U$ is ...
2
votes
1answer
52 views

Real Analysis. The Mean Value Theorem

Suppose $f:[0,1] \to [0,1]$ is continuous on $[0,1]$ and differentiable on $(0,1)$. If $f'(x)\neq 1$ for all $x\in(0,1)$. Prove there is at most one $c\in[0,1]$ such that $f(c)=c$.
1
vote
2answers
62 views

Solving the Diff. Eq: $y''+9y=36x\cos(3x)$

I'm stuck on this differential equation: $$y''+9y=36x\cos(3x), \quad \text{with }y(0)=-3, y'(0)=4$$ I know the homogenous equation is: $y_H(x)=A\cos(3x)+B\sin(3x)$ Now to find the particular ...
0
votes
2answers
37 views

How to extract factor when expression is with a power

$$f(x) = x^2(2x-3)^3$$ I tried to extract the 2 from the parenthesis. $$f(x) = 2x^2(x-\frac{3}{2})^3$$ But the graphic from this function is different. What should I consider when doing this kind ...
0
votes
1answer
82 views

Algebraic invariant theory

Deal all, I am looking for a gentle introduction to algebraic invariant theory (for a Bachelor project) with some simple (but interesting) applications in representation theory (of finite groups, of ...
0
votes
1answer
26 views

Limits of functions and sequences

Suppose that $f$ is defined on a deleted nbh of $x_0$. Denote by $D(f)$ the domain of $f$. Show that the following statements are equivalent: (i) $\lim_{x\to x_0} f(x) = \ell$ (ii) whenever $\{u_k\}$...
0
votes
0answers
47 views

Prove (x3−2) is maximal ideal of Q[x] using isomorphism theorem for rings. [duplicate]

Prove $(x^3−2)$ is a maximal ideal of $\mathbb{Q}[x]$ using isomorphism theorems for rings. I tried using the second isomorphy theorem for rings, to use that $( x ^ 3-2)$ is maximal if and only if $\...
0
votes
1answer
311 views

Minimizing a function of a complex variable

Given complex numbers $z_1,z_2,z_3,\ldots,z_n \in \mathbb{C}.$ Does there exist a $z \in \mathbb{C}$, for which the function $$f(z) = \sum_{j=1}^n |z-z_j|$$ achieves a global minimum? If yes, then ...
1
vote
1answer
30 views

Evaluating a line integral ( vector calculus)

Evaluate the line integral, $$\int_C |y|\, {\rm d}s,$$ Where the curve $C$ is the is given by the equation $(x^2+y^2)^2=29^2(x^2-y^2)$ What I tried. I recognize this as a scalar valued function, ...
1
vote
1answer
25 views

convergence of the derivatives

I am trying to solve the question: Let $u_n$ a sequence converging uniformly to $u$ where $u_n\in C^3(\Omega)$ for each $n$ and $\Omega$ is a subset limited of $\mathbb{R^n}$. Suppose $u_n=0$ on $\...
45
votes
2answers
2k views

A strange integral: $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi.$

While browsing on Integral and Series, I found a strange integral posted by @Sangchul Lee. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind ...
5
votes
3answers
123 views

prove that $2^n+2^{n-1}+2^{n-2}+8^n-8^{n-2}$ is a multiple of 7

Prove that a number $2^n+2^{n-1}+2^{n-2}+8^n-8^{n-2}$ is a multiple of 7 for every natural $n\ge2$. I am not sure how to start.
0
votes
0answers
34 views

Determining the joint density of W and Z when W = X/(1-X) and Z = Y/(1-Y)

The exercise relates to joint density $$f(x,y)= Cx^{\alpha-1}y^{\beta-1}(1-x-y)^{\gamma -1}$$ for $x > 0, y > 0, x + y \leq 1$ and $$C = \frac{\Gamma(\alpha+\beta+\gamma)}{\Gamma(\alpha)\Gamma(\...
5
votes
1answer
451 views

How can I solve this equation analytically. $\sqrt{x+\sqrt{2x+\sqrt{3x…}}}-100x\sin(x)=0$

How can I solve this equation analytically. $$\sqrt{x+\sqrt{2x+\sqrt{3x...}}}-100x\sin(x)=0$$
1
vote
2answers
159 views

Picking 3 random books probability problem

So the question is: Suppose a bookcase holds 6 chemistry, 5 math, 3 physics, and 8 computer science texts. if 3 books are selected, find the probability that none of the math texts are selected. My ...
2
votes
1answer
74 views

Riemann Integrals and Periodic Functions

Consider the following result, where integrals are, say, Riemann integrals: Let $f:\mathbb{R}\to\mathbb{C}$ be a periodic function of period $L$. Then for all $a\in\mathbb{R}$ $$ \int_0^Lf(x)\,...
0
votes
2answers
71 views

Prove that $I$ is a maximal ideal

I have a question. To show that the ideal $I=\langle f(x)\rangle $ is a maximal ideal of $K[x]$ do I have to show that $f(x)$ is irreducible in $K[x]$? Or is there an other way to prove that $I$ is a ...
1
vote
2answers
40 views

A question on differentiation of exponential function

Recently,one of my friend asked me a question but I'm not able to answer this question .His question about differentiation of exponential function .Here is his question:
2
votes
3answers
158 views

Solve for positive integers: $\frac{4}{13}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ [closed]

Solve for $x,y,z\in\mathbb{N}$ $\frac{4}{13}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ I tried by some general methods but they didn't help me.
2
votes
1answer
258 views

Proof regarding Robin's inequality (RI).

Let $\sigma$ be the divisor sum function, $\gamma$ the Euler-Mascheroni constant and $n>5040$. Robin showed that if the inequality$$\displaystyle \sigma(n)<e^{\gamma}n\log\log n$$ ever fails, it ...
3
votes
1answer
181 views

Show that the set of all positive L-formulas is consistent.

I am given the following definition of L-formulas: "Positive formulas are defined with the following properties: (i) Every atomic formula is positive. (ii) If $\phi,\psi$ are positive that $\phi\...
1
vote
0answers
33 views

$\frac{4}{13}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ solve for positive integers. [duplicate]

Solve for positive integers $x,y,z$ $$\frac{4}{13}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$ I tried to solve it by some generel work but it didn't help.
0
votes
1answer
45 views

One set dominating another in tournament

Consider a tournament with $799$ contestants. Each contestant plays against all other contestants exactly one; there are no draws. Prove that there exist two disjoint groups $A,B$, of $7$ contestants ...
1
vote
1answer
482 views

Number of Necklaces of Beads in Two Colors

I was reading this paper, and came across an equation which gives an expression for the number of necklaces of beads in two colors, with length n. $Z_n = \dfrac{1}{n} \displaystyle \sum \limits_{d \...
0
votes
2answers
33 views

Proving a surjection. Clarification

I just want to make sure this is all correct. So my definition of a function $f:A\to B$ being a surjection is: For all $b \in B$, there exists an $a \in A$ such that $f(a) = b$. Now the ...
1
vote
1answer
108 views

calculating the taylor term of an integral

an exercise ask me to calculate the Taylor term at $x = 0$ and degree four. I know how to take a derivative of an integral, but I'm having doubts about this one. The function: $$\int_0^x e^{-t^2} ...
0
votes
1answer
40 views

Basic question: $H^1$ and $H^{0,1}$

Please could you explain why for a smooth projective variety over $\mathbb{C}$ (or - if you prefer the analytic world - compact complex manifold) $T$ we have $H^1(\mathcal{O}_T)\simeq H^{0,1}(T)$ as ...
0
votes
1answer
53 views

Fourier Differentiation Property

I have been given this problem to solve: Define the function f(t) by $$ f(t) =\begin{cases} e^{-kt},& t \geq 0 \\ 0,& \text{otherwise}\end{cases} $$ where $k > 0$ is a real number. ...
1
vote
2answers
134 views

Fun results from modular arithmetic

I'm trying to go through various bits of neat, fun math with some junior-high-school students in my local area, and am thinking of doing a short unit on modular arithmetic/finite groups. I'm looking ...
2
votes
0answers
56 views

Why is $H^1(X, \mathcal{O}) \neq 0$ for $X = (B(0, 1)\times B(0, 2)) \cup (B(0, 2)\times B(0, 1))$?

In this MathOverflow answer, David Speyer says that \begin{align*} X &= \{ (z,w) \in \mathbb{C}^2 : (|z|, |w|) \in [0,1) \times [0,2) \cup [0,2) \times [0,1) \}\\ &= (B(0, 1)\times B(0, 2)) \...
1
vote
1answer
144 views

An analytic function with minimum and maximum at the boundary

Suppose there is a complex valued function analytic on some open connected set U and continuous on the boundary of that set. Then the maximum of $|f|$ is attained at some point on the boundary. Say $...
3
votes
3answers
101 views

$f$ an analytic function on $\mathbb{C}$ which takes values in $\mathbb{C}\backslash(-\infty,0]$ implies constant

Let $f$ be an analytic function on $\mathbb{C}$ which takes values in $\mathbb{C}\backslash(-\infty,0]$, i.e. takes values in the complement of the nonpositive part of the real axis. Show that $f$ is ...
3
votes
1answer
107 views

epsilon-delta limit with multiple variables

I am getting confused with this epsilon delta proof of the limit for this particular case. Prove that $\lim_{(x,y)\to(0,0)} (2x^2+3y^2)=0.$ if the limit is equal to zero, then for any given positive ...
0
votes
1answer
36 views

Limits and functions. Real Analysis

Let $x_0, l$, and $m$ be in $\mathbb R$. Assume that $f$ is defined on a deleted nbh of $x_0$ with $\lim_{x\to x_0} f(x) = l$. let $g$ be defined on a nbh of $l$. (a) Carefully show that $g(f)$ is ...
0
votes
2answers
36 views

Finding the volume when the region between $y=x^2$ and $y=4-x^2$ is rotated about the $x$-axis.

So I want to find the volume when the region between $y=x^2$ and $y=4-x^2$ is rotated about the $x$-axis. So I start by finding the roots where they meet, so I find : $$\int_{ -\sqrt{2} }^{\sqrt{2}}...

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