6
votes
1answer
417 views

Continuous right derivative implies differentiability

A book of mine says the following is true, and I am having some trouble proving it. (I've considered using the Lebesgue differentiation theorem and absolute continuity, as well as elementary analysis ...
1
vote
1answer
781 views

Normalizing an exponential function

Given the equation $a^\frac yx + a^x=b$ is there a way to normalize this function into a form where $y=$...? In short can I express $y$ in terms of $x$ if $a$ and $b$ are constants?
1
vote
2answers
108 views

derivative, the way to show in a graph

http://www.wolframalpha.com/input/?i=derivative+x%5E2%2C+x%5E2 Just reminding myself some math.. Is it ok to show the derivative in such a way like was shown in the link above for $x^2$ function? A ...
2
votes
1answer
54 views

Embedding into a morphism of distinguished triangles

Everything in this question happens in a triangulated category $\mathbf{D}$. I am trying to prove that in a diagram like this $$ ...
4
votes
3answers
282 views

Prove $x^{n}-5x+7=0$ has no rational roots

This question arises in STEP 2011 Paper III, question 2. The paper can be found here. The first part of the question requires us to prove the result that if the polynomial ...
5
votes
1answer
114 views

Closed form formula for the following sum

Does anyone know of a closed-form formula for the sum $\sum_{n = 1}^\infty x^{2^n-1}$? We can assume that $0<x<1$. Thanks!
3
votes
2answers
105 views

Inertial Frames of Refereence

I am told that in Newtonian mechanics, no coordinate system is "superior" to any other. Also, all inertial frames are in a state of constant, rectilinear motion with respect to one another. So am I ...
2
votes
2answers
68 views

Find “opposite” point in square

My first post here, I’m a programmer but my geometry knowledge is very limited. I have a fairly simple question: How can I calculate the opposite point in a square? Here is an image: Say I have ...
2
votes
1answer
82 views

Two variable function continuity

$$f(x,y)= \left\{ \begin{array}{c} x^2y \over x^3+y & \mbox{if } x^3+y \neq 0 \\ 0 & \mbox{if } x^3+y = 0 \end{array} \right. $$ So how do I find out whether this function is continuous? ...
3
votes
1answer
206 views

Make $21$ out of $1,5,6,7$ (challenge).

Here is a small mathematical challenge. You have to use once and only once each of the four numbers $1,5,6,7$ in order to obtain, via the help of the usual operators ($+,-,/,*$ together with ...
0
votes
1answer
5k views

Magnitude of a Matrix?

Consider a vector V. The magnitude of this vector (if it describes a position in euclidean space) = distance from the origin is simply: $(V^TV)^{1/2} $ aka the square root of the dot product... ...
3
votes
2answers
176 views

Are real numbers also hyperreal? Are there hyperreal $\epsilon$ between $-a$ and $a$ for any positive real $a$?

The set of all hyper-real numbers is denoted by $R^*$. Every real number is a member of $R^*$, but $R^*$ has other elements too. The infinitesimals in $R^*$ are of three kinds: positive, negative ...
0
votes
1answer
55 views

A reference to learn about duality

I am interested in learning about duality in convex optimization. I am looking for something to read which is: Reasonably short. Fairly self-contained (if it is a chapter in a textbook, I would ...
4
votes
2answers
137 views

Need explanation of passage about Lebesgue/Bochner space

From a book: Let $V$ be Banach and $g \in L^2(0,T;V')$. For every $v \in V$, it holds that $$\langle g(t), v \rangle_{V',V} = 0\tag{1}$$ for almost every $t \in [0,T]$. What I don't ...
2
votes
0answers
208 views

Unambiguous expression for binary strings containing some substring

Is there some systematic way for finding an unambiguous expression for a binary string which contains a certain substring? For finding expressions not containing a substring, it is sometimes easy to ...
1
vote
1answer
165 views

Homework problem on identifying a sequence

I had this problem in my discrete math/modular arithmatic course where I had to find the first 10 terms of a series F(r), starting from F(3). The given information is: F(3)=1 F(4)=13 F(10) % ...
0
votes
1answer
135 views

Doubly periodic functions

Consider the following translations: $T_{\lambda_1}: x+ \lambda_1$ and $T_{\lambda_{2}}: x+ \lambda_2$ acting on the complex plane. More specifically, let us look at $T: x \to x+a \lambda_1 + b ...
9
votes
9answers
3k views

What is sum of all positive odd integers less than $1000$?

If the sum of all positive even integers less than $1000$ is $ A $ , what is the sum of all positive odd integers less than $1000$?
0
votes
1answer
72 views

Solid angle definition - can it be seen shown using an image?

How do we show on a picture that a solid angle equals to this equation i found on a Wikipedia: \begin{align} \Omega =\!\!\!\int\limits_{\vartheta ...
2
votes
0answers
170 views

Number of solutions for $x$ of such form that $\frac{(x-1)^y}{x} = z,\;y>2,\;z \neq 0$

Consider $$\frac{(x-1)^y}{x} = z,\;y>2,\;z \neq 0$$ where $y$ is an integer. In relation to solutions for $x$; How could one prove that: $(1)$: There are $y$ solutions for $x$, in total. ...
2
votes
1answer
155 views

Pythagorean theorem proof by dissection

I have this proof of the Pythagorean theorem, but in the last two lines of the fourth paragraph I can't seem to find geometrically how the congruency between $1$, $2$, $3$, $4$ and $1'$, $2'$, $3'$, ...
1
vote
1answer
79 views

continuity of a function with 2 variables x,y

i ran into this question: check if this function is continuous: $f(x,y)= (1+xy^2)^{\frac{1}{x^2+y^2}}$ when $(x,y) \neq (0,0)$ $f(x,y)= 1$ when $(x,y) = (0,0)$ thanks in advance, yaron.
1
vote
1answer
48 views

distance between a convex set and a point

Let's look at the following famous theorem: Let $\mathcal H$ be a Hilbert space and let $C< \mathcal H$ be a (proper) closed CONVEX set. If $x_0\in\mathcal H\setminus C$ and $\eta:=d(x_0, ...
0
votes
2answers
58 views

How do I find all real numbers of Y with a nontrivial solution?

My problem is: Find all real values of Y, if any, for which the system has a nontrivial solution. $$2X_1 + 3X_2 = YX_1$$ $$4X_1 + 3X_2 = YX_2$$ Thank you.
9
votes
3answers
2k views

Self teaching Galois Theory

At uni, I did a module in group theory which I really enjoyed. I also did one on number theory which had aspects of rings and fields in it and I enjoyed learning this. Now that I have finished uni, I ...
2
votes
1answer
146 views

Is there any binary relation operator that has these properties in any objects?

Consider binary relation operators d b q p (with a direct correspondence by generalization of: < > ≮ ≯ these are a ...
1
vote
2answers
391 views

Finding all the points (x,y) where $f(x,y)=\sqrt{|x^3y|}$ is differentiable.

Finding all the points (x,y) where $f(x,y)=\sqrt{|x^3y|}$ is differentiable. I started off with this: If a function is differentiable then: $f(x_0+\Delta x,y_0+\Delta y)=f(x_0,y_0)+\frac {\partial ...
1
vote
1answer
103 views

upper bound on product of distances from points on a circle

Let $C$ be a circle of radius $1$ in the complex plane with $n$ points on the boundary. Provide an upper bound on the product of the distances of a given point on the circle to the other $n$ points. ...
1
vote
1answer
167 views

applying Neyman-Pearson Lemma

Suppose that we have a random sample $X_1,X_2,...,X_n$ from a prob function with density: $$ f(x) = 3\theta^3x^{-4} $$ given that $x\geq \theta$ Now the question is use Neyman-Pearson's Lemma ...
1
vote
2answers
76 views

Confusion regarding direct sum decomposition of representations from Serre's book

Sorry if the question is dumb. I am trying to learn representation theory of finite groups from J.P.Serre's book by myself. In section 2.6 on canonical decomposition, he says that let V be a ...
1
vote
3answers
490 views

Can a graph with 7 vertices and 17 edges have an isolated vertex?

The question is: Show or disprove that a graph with 7 vertices and 17 edges can have an isolated vertex. I know what is an isolated vertex, but don't know how to connect it with the concrete ...
4
votes
1answer
89 views

Elementary lower bounds for $n^{1/n}$

I can show that $n^{1/n} > 1+1/n$ for integer $n \ge 3$ by completely elementary means - no logs, exponentials, or calculus. Are there better bounds that can be proved in an elementary way? Here ...
2
votes
0answers
67 views

Abelian extensions with squarefree discriminant

Is it true that for all $2 \leq n \in \mathbb N$ we can find an abelian extension of the rationals with squarefree discriminant and degree n? Are there even infinitely many? Some even degrees may be ...
8
votes
3answers
529 views

Smooth Pac-Man Curve?

Idle curiosity and a basic understanding of the last example here led me to this polar curve: $$r(\theta) = \exp\left(10\frac{|2\theta|-1-||2\theta|-1|}{|2\theta|}\right)\qquad\theta\in(-\pi,\pi]$$ ...
2
votes
4answers
84 views

Finding the ratio of two persons time spent driving to a meeting

Mark and pat drive separately to a meeting. mark's average driving speed is $1/3rd$ greater than pat's and mark drives twice as many miles as pat. What is ratio of number of hours mark spends driving ...
2
votes
0answers
58 views

Does this monoid have a name?

Let $S$ be the set of all real sequences $x_i$ that are equal to zero for sufficiently high indices. Consider the set of random variables $X$ with image in $S$ such that there exists an $n$ for which ...
1
vote
1answer
240 views

proving that a certain function is absolutely continuous

Let's consider positive real numbers $\alpha,\beta>0$. Then let's define the function: $ f(x)= x^{\alpha} sin(\frac{1}{x^\beta})$ if $x\in (0,1]$. $f(0)=0$. Prove that if $\alpha>\beta>0$ ...
1
vote
1answer
96 views

Given a positive definite matrix $A$, how can I find a matrix $S$ such that $A=S^TS$ and $S$ has following restrictions?

My former question on this problem was not that popular, I was asking to prove the existence and uniqueness of a upper triangular matrix $S$ with positive entries on diagonal such that $A=S^TS$ for a ...
1
vote
2answers
66 views

Name of function that changes $\mathbb{R}$

Is there a specific function that takes the real number line $\mathbb{R}$ and converts it into a helix?
1
vote
0answers
62 views

Characteristic polynomial of the tree

How can one show that a coefficient of $\lambda^{n-2k}$ in characteristic polynomial of the tree is a number of matchings of size k in this tree. $n$ is a number of vertexes in the tree.
3
votes
3answers
545 views

Exponential function and uniform convergence of polynomials.

How can I prove that no sequence of polynomials converges uniformly to the exponential function? Thanks in advance for any help.
1
vote
1answer
92 views

Proving $\alpha+(\beta+\gamma) = (\alpha+\beta)+\gamma$ for ordinals

I am following Jech's construction, by definition $\alpha+0 = \alpha, \alpha+(\beta+1)=(\alpha+\beta)+1$, and for limit $\beta$ we define $\alpha+\beta = \cup\{\alpha+\xi: \xi<\beta\}$. Jech's ...
1
vote
0answers
60 views

Help with an asymptotic proof?

I am having difficulty with a proof of an asymptotic result in maximum likelihood estimation theory. I don't believe any statistical theory however is needed to solve this problem, rather I think ...
1
vote
2answers
109 views

If $X$ is the set of positive integers, how can some topology on it not satisfy the first axiom of countability?

Let $X$ be the set of integers. Obviously, $X$ satisfies the second axiom of countability, which implies it also implies the first axiom of countability. Regardless of the topology! Let $X$ be ...
2
votes
0answers
36 views

Commutator of two “special” conformal Killing fields

Theorem 1.7 of Schottenloher's "Mathematical Intro to CFT" says that every conformal Killing field on a connected open subset $M$ of $\mathbb{R}^{p,q}$ for $n=p+q>2$ is of the form $$X^{\mu}(x) = 2 ...
4
votes
0answers
38 views

Intuition on continuty in probability/mean square of a process

How to explain that a process is continuous in probability? I know the definition, but what does it mean? The same with continuity in mean square.
6
votes
2answers
315 views

Wikipedia's definition of isolated point.

Wikipedia defines an isolated point of a subset $S \subseteq X$ to be a point $x \in S$ such that there exists a neighborhood $U$ of $x$ not containing any other points of $S$. Furthermore, it claims ...
2
votes
0answers
82 views

Motivating the definition of right derived functors in the context of derived categories.

Let $A$ and $B$ be abelian categories and let $F : A \to B$ be an additive functor. Let $K^+(F) : K^+(A) \to K^+(B)$ be the induced functor on the corresponding homotopy categories of left bounded ...
1
vote
0answers
28 views

signature of pseudo-Riemannian metric made of Newton polynomials

Given a polynomial with roots $x_1,\ldots,x_n$ and real coefficients, it can be written $$ P(x)=\prod_{i=1}^n \left( x-x_i \right);$$ define Newton polynomials $$s_k(x_1,\ldots,x_n):=\sum_{i=1}^n ...
0
votes
0answers
120 views

Bilinear form signature and properties

So my professor gave me this question and I do not know how to deal with it. Let $B:\mathbb{R}^{8}$x$\mathbb{R}^{8}->\mathbb{R}$ be a symmetric Bilinear form with the signature (5,3). Regarding ...

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