0
votes
3answers
448 views

Sum of floor function

I have the following question: Show that $\displaystyle\sum_{k=1}^n a_k=na_n-\displaystyle\sum_{k=1}^{n-1} k(a_{k+1}-a_k)$ then evaluate $\displaystyle\sum_{k=1}^n \lfloor \log_2(k) ...
5
votes
1answer
695 views

Tensor product and Kronecker Product

Is there any difference between tensor product and Kronecker Product?
2
votes
1answer
222 views

English to Predicate Logic (Imply and AND)

The question is: If Bob is happy, then all his friends are happy My attempt looks like: $happy(bob) \Rightarrow (\forall x(friend(x, bob) \wedge happy(x)))$ The answer is $happy(bob) ...
5
votes
2answers
178 views

Help showing subadditivity of a map

I'm stuck with the following problem. Show that the map: $$ r(x)=\inf\limits_{k\in\mathbb{N}}\limsup\limits_{m\to\infty}\frac{1}{k}\sum\limits_{j=0}^{k-1}S^j(x)(m) $$ is subadditive on ...
2
votes
0answers
491 views

exp(ab) decomposition

How can one write $e^{a(x) \cdot b(x)} = c(x) e^{ b(x) }$ with $c(x)$ not implicitly depending on $b(x)$. I do not believe this is generally possible so alternatively one can use an infinite series or ...
2
votes
1answer
308 views

Collapsing the boundary of a Möbius Strip to a point

I'm strunggling to prove that when we collapse the boundary of a Möbius strip we obtain the RP² thanks
2
votes
0answers
260 views

Determining the probability density function from an equation

I have the following (for me quite interesting) densities for which I am completely stuck. I only hope that you can provide me some help. Let me introduce my problem. I have two probability ...
0
votes
1answer
69 views

Definition of $\operatorname{arg}(z)$ choosing value of $\operatorname{Arg}(z)$

On what curve is $\arg (z)$ discontinuous if it is defined as the value of $Arg(z)$ satisfing the inequality: $$|z|-2\pi<\operatorname{Arg}(z)\leq|z|$$ would it be a ray from the origin with ...
1
vote
1answer
39 views

How to prove that the infimum of the sum of two sequences is at most the sum of their infima?

I'm trying to show: $\inf(x_k+y_k) \leq \inf(x_k)+\inf(y_k)$. So I did: Let $r>0$, there is a $k_1$ such that $x_{k_1}\leq \inf(x_k)+r/2$ and there is a $k_2$ such that $y_{k_2}\leq ...
3
votes
4answers
281 views

Limit of difference of two irrational functions

Firstly, this is not a homework. I just want to solve this limit for my own curiosity and self-learning. I have tried to solve this limit for 5-6 hours with no luck. Then I tried to read information ...
0
votes
4answers
623 views

Trigonometric equations: $ 2 + \cot\theta = \csc\theta$

Solve $$2+\cot\theta = \csc\theta $$ where $$ 0 \leq \theta \lt 2\pi $$ The suggested answer is $2.21$ only (in rad, corr to $3$ sig. fig.) My reasonable guess is there are at least two solutions. ...
0
votes
1answer
19 views

why the optimized point always appear in the interception in LP problem

As the topics, why the optimized point always appear in the interception in LP problem? I think there should be a proof but i am not sure about it.
0
votes
1answer
82 views

$\sum_{i=1}^n \frac{p(a_i)}{q'(a_i)}=p_{a_{n-1}}$

In a question in Fuhrmann's "A polynomial approach to linear algebra" it is stated that $$\sum_{i=1}^n \frac{p(a_i)}{q'(a_i)}=p_{a_{n-1}},$$ where $p,q$ are polynomials over a field with $deg(p)=n-1$ ...
1
vote
0answers
67 views

The exponentiated sum of the number of success before a single failure over multiple trials

I roll two $k$-sided die, and if the same faces appear on both die, which happens with probability $p$, I roll again and again until the faces disagree with probability $(1-p)$ (implying a negative ...
1
vote
1answer
95 views

Evaluate $\sum_{n=1}^{1024}\left \lfloor \log_2n \right \rfloor$.

Evaluate $\sum_{n=1}^{1024}\left \lfloor \log_2n \right \rfloor$. I thought the answer is $1+1*2+2*2^2+3*2^3+4*2^4+5*2^5+6*2^6+7*2^7+8*2^8+9*2^9+2^{10}=9219$, but the answer should be 8204. What ...
1
vote
2answers
662 views

How to prove error function $\mbox{erf}$ is entire (i.e., analytic everywhere)?

How do I prove the error function $$ \mbox{erf}(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^{2}} dt. $$ is entire? Could you give me some scratch proof?
9
votes
3answers
2k views

Combinatorial argument to prove the recurrence relation for number of derangements

Give a combinatorial argument to prove that the number of derangements satisfies the following relation: $$d_n = (n − 1)(d_{n−1} + d_{n−2})$$ for $n \geq 2$. I am able to prove this ...
1
vote
0answers
36 views

I need to describe a behavior of a curve depending one one parameter in z-plane

$|z^{2}-1|=\lambda$ My aprouch: taking $z=x+iy$ $$z^{2}=(x^{2}-y^{2})+i(2xy)$$ Then : $$|z^{2}-1|^{2}=x^{4}-2x^{2}y^{2}+2y^{2}-2x^{2}+y^{4}+1+4x^{2}y^{2}=\lambda^{2}$$ Now taking ...
2
votes
3answers
3k views

Composition of functions that are onto or one-to-one

I found part of my answer here: If g(f(x)) is one-to-one (injective) show f(x) is also one-to-one (given that...); however I wanted to flesh out the last two statements I had in a proposition in my ...
0
votes
3answers
562 views

How to solve $\cos(\pi/2+t)\ge 0$?

I have a trig question. How do you I solve this. I appreciate much if you could show it step by step. Find all the value of in the interval $[0,2\pi]$ for which $\cos(\pi/2+t)\ge 0$.
1
vote
2answers
153 views

Listing all derangements for a given n

How to find all the derangements of [n]. Specifically,if n is 4? What is the process to get all the derangments in general?
1
vote
1answer
3k views

How to find formula for a series of numbers

Can you tell me how to find logically a formula if series of numbers is given. for example $$2,4,8,16$$ the formula for this series is $2^n$ where $n$ is the $n$-th position. This is an easiest series ...
16
votes
2answers
543 views

If $G$ is biconnected and $\delta(G) \geq 3 \Rightarrow \exists v: G-v$ is also biconnected.

If $G$ is biconnected and $\delta(G) \geq 3 \Rightarrow \exists v: G-v$ is also biconnected. Where $\delta (G) - $ minimum degree of all vertices, $G-v$ is equal to if we remove this vertex from $G$ ...
2
votes
1answer
196 views

About Sobolev Embedding Theorem

I want to know that how the statement below holds. The statement : There exists a constant $C = C(s)$ such that the continuous embedding of $W^{s,2}$ into the space of uniformly bounded, continuous ...
0
votes
1answer
52 views

How is it simplified?

1 ) I have this equation and don't know how $(1 + 2e^y)$ and $(1 + 2e^x)$ from each side of equation cancelled each other out and get the final answer $x = y$. ...
31
votes
7answers
20k views

How many triangles

I saw this riddle today, it asks how many triangles are in this picture . I don't know how to solve this (without counting directly), though I guess it has something to do with some recurrence. ...
1
vote
2answers
126 views

On a Surjection

For any set $B$ let $\mathcal{P}(B)$ denote the set of all subsets of $B$. Let $A$ be an infinite set and suppose there exists a surjection $f : A \mapsto \mathcal{P}(A)\setminus A$. Consider the ...
1
vote
3answers
2k views

How many of the 30 vertices in G have degree 3 and how many have degree 4?

A graph G has 50 edges and 30 vertices. Each vertex in G has either degree 3 or degree 4. How many of the 30 vertices in G have degree 3 and how many have degree 4?
1
vote
2answers
3k views

How many ways you can make change for an amount [duplicate]

I am looking for a formula or at least something to use when trying to compute how many ways I can make change for an amount. Example: there are $3$ ways to give change for $4$ if you have coins ...
7
votes
1answer
284 views

Total mean curvature in $L^2$ and minimal surfaces in spaces with non-positive sectional curvature

Let's suppose we have a Riemannian $n$-manifold $(N,g)$ and an immersed surface $f:\Sigma\rightarrow N$, with genus zero, equipped with the induced metric. Let's further assume that the ambient space ...
2
votes
1answer
2k views

Ideas on the ways to integrate $\int \tan^2( x)\sec^3( x) dx$

I would proceed by thus , let $y = [\sec (x)]^2 $ then $$dy = 2 \cdot \sec(x) \cdot \sec(x) \cdot \tan(x) \cdot dx = 2 \cdot ( \sec (x))^2 \cdot \tan(x) \cdot dx $$ so, $$ 2 \tan^2(x) \sec^2 (x) ...
3
votes
2answers
210 views

Why for number of leaves in a tree (all types of trees) is it true

I have to prove the following claim, given the tree $T=(V,E)$, $|V|\geq3$: $$|V_1| \leq \frac { |V| \times (\Delta (V) - 2) + 2 }{ \Delta (V) - 1 } $$ where $|V_1| - $ number of leaves in a tree, and ...
8
votes
1answer
136 views

Show that $8^{1/\pi}$ has infinitely many values.

Show that $8^{1/\pi}$ has infinitely many values. If it were possible to plot all its values, what would the picture look like. How do I go about solving this.
4
votes
1answer
660 views

Why is the number of subgroups of a finite group G of order a fixed p-power congruent to 1 modulo p?

At page (24) of the book Finite Group Theory by I.Martin.Issacs, one finds the statement: In the situation of Corollary 1.25, the number of subgroups of $G$ having order $p^b$ is congruent to 1 ...
1
vote
1answer
219 views

Proving Schwarz inequality for complex numbers using calculus

I am working on this problem for real analysis and it's rather baffling. I've never really seen any of this kind of notation before which is part of it I think? But also it just seems utterly ...
0
votes
1answer
43 views

General formula for calculating $\prod_{i \in I}(1-a_i)$

Does anyone know the general formula for calculating $\prod_{i \in I}(1-a_i)$ where $I$ is a set of indices? I suspect it is $1+\sum_{\emptyset \subsetneq J \subseteq I}(-1)^{|J|}\prod_{j \in J}a_j$. ...
2
votes
6answers
314 views

Solving for x with exponents (algebra)

So I am trying to help a friend do her homework and I am a bit stuck. $$8x+3 = 3x^2$$ I can look at this and see that the answer is $3$, but I am having a hard time remembering how to solve for $x$ ...
0
votes
1answer
90 views

Probability Density function vs Mass function

I am a bit confused by using Probability Mass Function and Probability Density Function. I understand that for discrete case like Bernoulli or Binomial, we call it pmf. For continuous case like ...
5
votes
3answers
1k views

Enumerating number of solutions to an equation

How do you find the number of solutions like this? $$x_1 + x_2 + x_3 + x_4 = 32$$ where $0 \le x_i \le 10$. What's the generalized approach for it?
0
votes
0answers
116 views

integer solutions to $a^m+nx^2 = y^n$ with various conditions

I consider the following equation with conditions of obtaining solutions $$a^m+nx^2 = y^n$$ This equation has solution when $a$ is an even prime and $x, y, m$ are positive integers with $(nx, y) = ...
-1
votes
3answers
202 views

Possible Cardinality of a Field

The following question struck me as pretty interesting: Let $\Bbb F$ be a field of characteristic $p$ (a prime, of course). I'm then asked to show that $|\mathbb{F}| = p^n$ for some $n\geq 1$. ...
2
votes
0answers
266 views

Problem regarding Annihilator of a matrix and the minimal polynomial.

I have some doubt regarding annihilator of an matrix. Please help me to understand the subject. Let $T$ be a n by n Complex matrix. Let $m(x)$ be the minimal polynomial for $T$. Now suppose $f(T)=0$ ...
55
votes
14answers
4k views

What is $x^y$? How to understand it?

$x+y=z$ I have a pen. He has a pen. Total is two pen. This is plus. $x-y=z$ I had two pens. A pen was lost. So, I have a pen. Total remaining is one. This is minus. $x\cdot y=z$ I have two pens. ...
1
vote
1answer
80 views

Help in finding Jacobian

I have $$\begin{aligned}x_{1}&=r\sin(\theta_{1}),\\ x_{2}&=r\cos(\theta_{1})\sin(\theta_{2})\\ x_{3}&=r\cos(\theta_{1})\cos(\theta_{2}). \end{aligned} $$ I know how to compute the ...
0
votes
2answers
60 views

Sequential Limit of Line Integral Is The Same As The Usual Limit of Line Integral? (Gamma Function Related)

Let $\epsilon > 0$, and $ n \in \mathbb{Z}^{+} $. Let $C_{n}$ be a positively oriented polygonal line that is from $-n + 1/2 - i \epsilon$ to $ 1/2 - i \epsilon$ and from $ 1/2 - i \epsilon$ to $ ...
2
votes
1answer
237 views

Counting multidimensional structures (Chomp game states)

The game Chomp is described as follows on Wikipedia: Chomp is a 2-player game of strategy played on a rectangular "chocolate bar" made up of smaller square blocks (rectangular cells). The ...
1
vote
1answer
54 views

Finding probabilities of $X_{(N)}$ (order statistics)

Draw N samples from unif(a,b) and consider the largest realization $X_{(N)}$. What's the probability that this value falls within a certain subset of (a, b)? What expression has to be integrated ...
1
vote
2answers
74 views

Verifying a bijection

Let $V$ be vector space over $\mathbb{F}$, and $W\subseteq V$ a subspace. Let $p:V\rightarrow V/W$ be the canonical projection. Let $X$ be the set of all subspaces containing $W$ and $Y$ be the sets ...
2
votes
0answers
96 views

Searching for prime candidates

For some additional excitement, I've been searching for primes $p \gg q = 104729$, where $q$ is of course the ten-thousandth prime. It seems that the best way to search for prime candidates $p$ is to ...
1
vote
1answer
66 views

Generalization of groups of the form (R-{0}, regular multiplication)

Result: Fix $a \in \mathbb{R}$. Then $(\mathbb{R} \backslash \{a\}, *)$ is a group, where our group operation is defined by $x*y = (x-a)(y-a) + a$. One consequence of this is the standard fact that ...

15 30 50 per page