0
votes
2answers
59 views

Solutions of $2^x 7^{1/x}\le 14$

The solution is supposed to be $(-\infty,0)$ and $[1,\log_2 7]$. What I get when solving the problem is $(-\infty, \log_2 7]$. Where did I get it wrong? I start by dividing both sides by 14, then ...
0
votes
0answers
20 views

Seminar concearning Spectral Theory of Differential Operators?

I must prepare a seminar about spectral theory of linear partial differential operators. However, I'm at a loss as to a nice reference. I'm looking for something that fits in a graduate spectral ...
1
vote
1answer
10 views

Expressing problems in canonical form for solving with simplex

The Picnic Hamper Company has a store containing 10,000kg of nuts, 4000 packs of smoked salmon, 2000 bottles of wine and 1500 Victoria sponges. It intends to use these goods to make up three different ...
0
votes
0answers
20 views

2 Definitions of Holomorphic functions on Riemann surfaces

In a lecture that I currently attend we defined Riemann surfaces and holomorphic mappings on it somewhat different than in another lecture that I attended a year ago. My question is: Are these ...
2
votes
1answer
17 views

Sequences formed by integer evaluations of polynomials modulo $ p^{k} $, where $ p $ is a prime number and $ k \in \Bbb{N} $.

I have the following question. Let $ p $ be a prime number and $ k $ a positive integer. Let $ (a_{n})_{n \in \Bbb{Z}} $ be a two-way sequence in $ \Bbb{Z} / p^{k} \Bbb{Z} $. Then is it true that ...
0
votes
0answers
9 views

Higher dimensional SDEs cannot be compressed into a one dimensional SDE.

This question comes from quantitative finance but I think it's true in general outside that setting. I'm trying to make sense of the idea that if a process depends on at least two noises there ...
0
votes
0answers
24 views

complex variable inequality

Let $B$ and $C$ be nonegative real numbers and $A$ a complex number. Suppose that $$ 0\leq B-2Re(\overline{\lambda}A) + |\lambda|^2 C \ \forall \ \lambda \in \mathbb{C} $$ Conclude that $|A|^2 \leq ...
3
votes
1answer
41 views

weird trig problem $\tan(\theta)=-\sqrt{2}\sin(\theta)$ on the interval $0 \leq \theta \leq 2\pi$

$\tan(\theta)=-\sqrt{2}\sin(\theta)$ on the interval $0 \leq \theta \lt 2\pi$ I started off with $[(\sin(\theta)/\cos(\theta)] \times (1/\sin(\theta) )= - \sqrt 2$, then after simplification i got ...
-2
votes
2answers
67 views

$\int _{ 0 }^{ 1 }{ \frac { { x }^{ t }-1 }{ \ln { x } } dx } $ [duplicate]

How do I solve the following integral: $$\int _{ 0 }^{ 1 }{ \frac { { x }^{ t }-1}{ \ln { x } } dx } $$
1
vote
0answers
16 views

Envelope of a family of lines. When does it exist?

Given a family of lines $(\gamma_t): a(t)x+b(t)y+c(t)=0,\ t\in I$ (interval in $\mathbb{R}$), where $ a,b,c:I\to\mathbb{R},\ a^2(t)+b^2(t)\neq 0,\ \forall t\in I$ , what conditions should we impose ...
0
votes
1answer
26 views

Differentiating expression involving summation

My problem seemed very simple at glance but I keep missing one term from the answer. Any suggestions? This is the problem: We have $$x_i^* + \xi_i + \frac{\alpha_i}{p_i} \left[ y - \sum_{j=1}^n ...
-5
votes
2answers
31 views

proving f is continuous iff it takes limits to limits [on hold]

How to show that iff $x_i\to x$ implies $f(x_i)\to f(x)$ then f is continous? For metric Spaces. Continuit definition the standard epsilon delta
3
votes
1answer
24 views

General solution to a system of non linear equations with a specific pattern

I am seeking a general solution to a system of non linear equations with a specific pattern: Order 1: $$ x_0 = a^2 + b^2 $$ $$ x_1 = 2ab $$ Order 2: $$ x_0 = a^2 + b^2 + c^2 $$ $$ x_1 = 2ab + 2bc ...
0
votes
0answers
23 views

Where does the $kΔt$ go?

Frame 142 out of Quick Calculus, 2nd Edition: Suppose the position of an object is given by $S=f(t)=kt^2+lt+S_0$, where $k$, $l$ and $S_0$ are constants. Find $v$. I've worked the problem to: ...
2
votes
2answers
72 views

Showing $r\in\mathbb Q\setminus \{0\}\implies e^r\notin \mathbb Q$

For a given $n>0$, let $\displaystyle J_n:x\to \frac{1}{n!}\int_{-x}^x(x^2-t^2)^ne^tdt$ a. Prove that there exists $A_n,B_n\in \mathbb R_n[X]$ such that $\forall x\in \mathbb R^+, ...
1
vote
1answer
25 views

How can I find these two limits?

How can I find these two limits? I've no idea how to improve or continue now. Can someone give me a hint? 1)$$\lim_{x \to 0^+}\left(\frac{\cos^{\pi}(25x)} {\tan^3(x)}\right)=\lim_{x \to 0^+} ...
0
votes
2answers
38 views

Is the group $(\Bbb Z,+)$ isomorphic to the the group $(\Bbb Q\setminus\{0\},\cdot)$? [duplicate]

Is the group $(\Bbb Z,+)$ isomorphic to the the group $(\Bbb Q\setminus\{0\},\cdot)$?
0
votes
0answers
10 views

The difference between weakly ordered set and partially ordered set?

I need a reference that discusses the difference between a weakly ordered set and a partially ordered set. My understanding is that a weakly ordered set $<X, \succsim>$ is one where the ...
2
votes
2answers
97 views

Understanding why $a+b\sqrt {2}\neq \sqrt {3} $

I want to intuitively understand why $a+b\sqrt {2}\neq \sqrt {3} $ for $a, b \in \mathbb Q $ I really have no intuition regarding this matter, and have to deal with similar concepts regularly while ...
0
votes
0answers
4 views

multivariate convergence in distribution for Skorohod topology

If $X_n$ is a $R^d$-valued sequence of stochastic processes and $X$ is a $R^d$-valued the limiting process, all having Cadlag paths. I know what it means by $X_n \Rightarrow X$, i.e. $X_n$ converges ...
0
votes
1answer
40 views

How to proof the $\displaystyle \lim_{x \to \infty}\frac{\log(x)}{x}=0$? [duplicate]

I will soon make a math exame where one can't use L'Hôpital's rule, integrals concept neither the formal limit definition. The most I can use is the derivative definition and the algebraic ways to ...
2
votes
1answer
38 views

If $1+x+x^2+\cdots+x^{y-1}$ is a prime number then how prove that y is also a prime number?

If $1+x+x^2+\cdots+x^{y-1}$ is a prime number then how prove that $y$ is also a prime number? $x$ and $y$ are natural numbers
1
vote
1answer
32 views

How can I expand this

How can I expand $\dfrac{\pi \csc(z\pi)}{(2z+1)^3}$? so then I can find the residue ? thanks
4
votes
2answers
43 views

Probability when cutting the stick twice

Given a stick of length $l$. We cut the stick twice. Let $X$ be the random variable defined by the length of the stick after the first cut, and $Y$ be the random variable defined by the length of ...
0
votes
1answer
14 views

is it possible to decompose nonperiodic sinusoidal signal?

Using Fourier series we can decompose any any signal into it's elementary signals but condition is that signal should be periodic and sinusoidal one. Now, is it possible to decompose nonperiodic ...
0
votes
1answer
17 views

Triangle in circumference of circle

Points $A$, $B$, and $C$ are on the circumference of a circle with radius 2 such that $\angle BAC = 45^\circ$ and $\angle ACB = 60^\circ$. Find the area of $\triangle ABC$. How would I start this ...
1
vote
1answer
16 views

Is it necessary to use the Hahn-Banach theorem to show that $(X/M)^*\simeq M^\perp$?

Let $X$ be a Banach space with dual space $X^*$, and let $M$ be a closed subspace of $X$. Then $M^\perp=\{x^*\in X^*: x*(m)=0 \text{ for all } m\in M\}$ is a closed subspace in $X^*$. I read the ...
1
vote
1answer
23 views

What is the probability of arrive either A or B at starting point K?

There are two points which are A and B. The distance between A and B is 50meter.One person goes to A with probability 1/6, he goes to B with probability 3/6. And he goes nowhere with probability 2/6. ...
1
vote
1answer
19 views

Coupon Collector's Problem — Expected Value of each item

So I guess my problem is based on the famous coupon collector's problem, which is, if you should not be familiar with it, the following: Given N different coupons from which coupons are being drawn ...
1
vote
3answers
45 views

Prove that $f(x,y)$ is continuous in $(0,0)$

Prove that $f(x,y)$ is continuous in $(0,0)$, where \begin{equation} f(x,y) = \begin{cases} \frac{x^2y}{x^4+y^2}, & (x,y)\neq 0\\ 0, & (x,y) = (0,0) \end{cases} \end{equation} The solution I ...
2
votes
0answers
12 views

Why is $m \infty$ the conductor of $K = \mathbb{Q}(\zeta_m)/\mathbb{Q}$?

Wouldn't this be saying that for all $p$ dividing $m$, $1 + p^{\operatorname{ord}_p(m)} \mathbb{Z}_p$ is contained in the group of local norms $N_{\mathfrak p/p}(K_{\mathfrak p})$, where $\mathfrak p$ ...
3
votes
2answers
61 views

Finding line that divides an area into equal halves.

My question is simple, but I am not getting the answers for some reason. The question is: Consider the area enclosed between the graph of $y = 1 - x^2 $and the $x$ axis. Which line parallel to the ...
0
votes
0answers
9 views

Matching student-company at a fair (A variation of The Marriage Problem)

This problem is connected to the famous http://en.wikipedia.org/wiki/Stable_marriage_problem#Algorithm We have $s$ students and $c$ companies, where $s<c$. (Roughly speaking, $c \approx 20$ and $s ...
-3
votes
1answer
35 views

How to find $E(X\mid X+Y=k)$ for binomial distribution? [on hold]

Let $X = \mathrm{Bin}(n,p)$ ; $Y = \mathrm{Bin} (m,p)$. How do I find $E(X\mid X+Y=k)$ for the binomial distribution?
0
votes
2answers
45 views

How do you find the value of n in this example

$$n^{n-2} = 16$$ I know $n = 4$ through trial and error but how do you find $n$ in a conventional manner? I'm basically trying to solve how many nodes are in a tree that has $16$ spanning trees ...
0
votes
1answer
22 views

The multiplication of a smooth function and a distribution

Let $f$ be a smooth function on $\mathbb{R}$ and let $g$ be a distribution. Then $f\cdot g$ is a well defined distribution. Suppose $$ f\cdot g=\delta_0, $$ where $\delta_0$ is a dirac function. ...
-1
votes
1answer
30 views

Do all n x n matrices over the reals represent linear transformations?

Do all $v \in M_n (\mathbb{R})$ represent linear transformations? To add to that a bit to further clarify for myself: Looking up the def. of a transformation it is any function $f$ mapping a set $X$ ...
1
vote
0answers
22 views

On Zero-Free Regions for $\zeta(s)$ and $L(s,\chi)$ with $|t| \le 2$

I'm reading the proof from Hildebrand that for some $c_1 > 0$, the Riemann zeta function $\zeta(s)$ has no zero in the region $\sigma > 1-c_1$, $|t| \le 2$. (Here $s = \sigma + it$ per ...
1
vote
1answer
14 views

Can a field extension still have “non-separability” above its maximal purely inseparable subextension?

Question 1 Let $E/F$ be an algebraic field extension. Let $K$ be the set of all elements of $E$ that are purely inseparable over $F$. Then, $E/K/F$ is a tower of fields, and $K/F$ is purely ...
0
votes
1answer
33 views

Question regarding a loan from a bank for my friend

I have a question please, I'm having a difficult times calculating the right way this one... I took a loan on July 3 2014 of \$50,000 from my bank for my friend, split it on 12 installments (about ...
4
votes
1answer
34 views

Find the range of a $4$th-degree function

For the function $y=(x-1)(x-2)(x-3)(x-4)$, I see graphically that the range is $\ge-1$. But I cannot find a way to determine the range algebraically?
1
vote
1answer
19 views

A question about $\gcd$ and divisibility

Let $\sigma$ be the classical sum-of-divisors function. Suppose that I have the following equations: $$2n^2 - \sigma(n^2) = \frac{\sigma(n^2)}{q^k}\cdot{\sigma(q^{k-1})}$$ $$2n^2 - \sigma(n^2) = ...
0
votes
1answer
21 views

Prove the surjectivity of this injective linear map

I am working on the following problem. Let $g : V\to V$ be linear and injective, where $V$ is a vector space over the field K. Prove that, if $V$ is finite-dimensional, then $g$ is surjective. In an ...
1
vote
1answer
46 views

An awkward Functional Equation

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(y)+x^2+1)+2x=y+(f(x+1))^2$$ for all $x,y \in \mathbb{R}.$ I proved that $f$ is bijective, but I am stuck there. Any help please?
1
vote
1answer
11 views

Rootspace decomposition of a Lie algebra

$ \DeclareMathOperator{\ad}{ad}$ Let $L$ be a non-zero Lie algebra which is semi simple. Then $L$ contains a toral element and hence a non-trivial toral subalgebra. Let $H$ denote a maximal toral ...
6
votes
2answers
78 views

Elementary question in Group Theory with less prerequisite

Here I am posing a problem, which my beginning students of algebra were discussing for long time. Question: Without using theorem of Cauchy or Sylow, can we show that a group of order $15$ contains ...
0
votes
0answers
10 views

formula to establish correlation between multiple library functions

I am trying to predict the change in timeliness of holds delivery relative to number of owned Bestsellers, number of holds and the checkout window. (yes, it really is a library question). To do this ...
0
votes
1answer
41 views

logarithmic Series

I'm aware that by properties of logarithm $$\sum_{k=1}^n \ln (k) = \ln (n!)$$ My question is if $$\sum_{k=1}^n \ln^2 (k) = \ln^2 (n!)?$$ Because when I am verifying the value where $n = 5$, I get ...
0
votes
0answers
25 views

Mastermind Probability Distribution

I was thinking of the game MasterMind, a game with 6 different colored pegs and a 4 part code using those colors. There can be more than one peg of the same color in the code. For example ...
0
votes
4answers
41 views

How many three digit numbers with increasing digits can be formed from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$?

Suppose we pick 3 numbers $x,y,z \in \{1,2,3,4,5,6,7,8\}$ and form a 3 digit number $xyz$ how many possible combinations numbers can we create such that $x < y < z$. For example $357$ ...

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