0
votes
1answer
119 views

Integers Positioned Around a Circle [duplicate]

Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of n and the product of any two adjacent numbers in the ...
0
votes
4answers
238 views

When does the two cars meet

At 10:30 am car $A$ starts from point $A$ towards point $B$ at the speed of $65$ km/hr, at the same time another car left from point $B$ towards point $A$ at the speed of $70$ km/hr, the total ...
1
vote
0answers
46 views

Counting distinct positive valued k-tuples that sum to n where each entry can be no greater than some value.

This is motivated by the desire to count the number of ways two dice can form the sums 2,3,4,...,12 respectively. We can safely use the stars and bars method for 2,3,4,...,7 where the number of ways ...
0
votes
1answer
11 views

Area with different unit measurements

What is the area of a rectangle, in square meters, with a length of 108 meters and a width of 300 millimeters? I think it could be 324 sqm.
3
votes
1answer
105 views

Counterexample for Maschke's lemma

I'm trying to relax conditions and come up with a counterexample to Maschke's lemma in such a case. For example, with $k = \mathbb Z/p\mathbb Z$, I'm considering the two dimensional representation of ...
3
votes
0answers
93 views

Is this proof of a mathematical olympiad problem correct?

I'm quite sure about the exactness of my proof, but I'd like to hear (constructive) criticism about my writing. This is the problem: Every non-negative integer is coloured white or red, so that: 1) ...
0
votes
1answer
20 views

Mathematical presentation of a problem

The issue that I am dealing with now ends up with the solution of a second order equation. The solutions are the Z positions of a point in 3D. So, basically I have two points with the Z positions of ...
2
votes
1answer
88 views

Does $\wp(A \cap B) = \wp(A) \cap \wp(B)$ hold? How to prove it?

I'm currently working on some discrete mathematics work and I've encountered a question I'm not sure how to answer exactly. Precisely, I'm trying to prove that two power, intersected sets statements ...
2
votes
1answer
52 views

Series does not converge [closed]

How would I go about showing that the series$$\sum_{n + m\tau \in \Lambda} {1\over{{|n + m\tau|}^2}}$$does not converge, where $\tau \in \mathbb{H}$?
0
votes
1answer
19 views

Determining at what points multiple variable functions are continuous

With a two variable function what is the procedure to figure out at what points it is continuous? Do I basically just look at what points it would be undefined and anywhere between those points it is ...
2
votes
1answer
73 views

Proof Using Lagrange's Theorem

I am working on a problem in Kurzweil & Stellmacher's introductory finite group theory that looks like this: Let $A, B$, and $C$ be subgroups of the finite group $G$. Prove that if $B \leq A$, ...
0
votes
1answer
36 views

Denumerable partition of a denumerable set where each set in the partition is denumerable. [duplicate]

Suppose that a set $A$ is denumerable. Prove that there is a partition $P$ of $A$ where $P$ is denumerable and every $X \in P$ is also denumerable. I can see that this can be done but I cannot figure ...
0
votes
1answer
58 views

Is it possible to prove dot product by the law of cosines?

It seems many people prove the geometric definition of dot product by the law of cosines. However, i think this is incomplete because the law of cosines is for a triangle, which means we can't use it ...
2
votes
1answer
26 views

Fourier series: Show that $f$ is a trigonometric polynomial

Let $N\in\mathbb{N}$ and $f_m:\mathbb{R}\to\mathbb{R}$, continuous functions and periodic, $T=2\pi$. Let's assume that $f_m \to f$ uniformly and for all $m\ge 1$: $$\left| \hat{f_m}(n)\right| \le ...
0
votes
2answers
75 views

Binary relation of composite function

Suppose S is a binary relation on a set X. If S ◦ S is reflexive, Is S is reflexive? can we prove this with example too and by definition "Let U be a non-empty set and let R be a binary relation ...
0
votes
1answer
50 views

Summation notation with ambiguous subscripts

I'm reading a paper which has the following description; Say we have a time series of correlated sequential observations of the random variable $X$ denoted $\{x_n\}_{n=1}^N$ from a stationary, time ...
0
votes
2answers
35 views

Find equation of a plane throw two given point and orthogonal to another space

I have $\pi:4y-3z-4=0 \quad A=(2,4,4) \quad B=(2,-2,-4)$. I have to calculate the equation of the plane $\sigma$ throw $A$ and $B$ and orthogonal to $\pi$. What is the solution? Thanks in advice!
2
votes
0answers
69 views

Prove that $\underline{\int_{a}^b} f \leq 0 \leq \overline{\int_{a}^{b} f}$

My question is just to make sure my proof is on the right track. Problem: Suppose that the bounded function $f\colon [a,b]\rightarrow \mathbb{R}$ has the property that for each $x\in \mathbb{Q}$, ...
2
votes
1answer
76 views

improbable sum of random variables

Let $U$ be a uniform random variable on the interval $[0,1]$. It is exceedingly unlikely that $U$ can be written as a sum $U = X + Y$ where $X$ and $Y$ are independent identically distributed random ...
1
vote
2answers
71 views

Diophantine equation with division

How can I find all the cases where y is positive integer in the next equation: $$\frac{ax + b}{c-x} = y$$ $a,b,c,x$ are not negative integers $a,b,x < c$ $ax + b = 0$ is a trivial solution
1
vote
1answer
131 views

difference between poisson and exponential distributions in the context of client server systems?

I am studying client's request arrival patterns on web and application servers. About web server's request arrival pattern I read that "The request arrival rate on web server follows Poisson ...
1
vote
0answers
53 views

Using Stirling's formula to uniformly bound Bernoulli success probabilities (part 2)

In this paper, the authors say that for any $\gamma \in [1/2,1)$, there is a positive constant $A=A(\gamma)$ such that for any $n$, $$ \sum_{n\gamma\leq k \leq n} \binom{n}{k} \geq A n^{-1/2}2^{n ...
1
vote
1answer
94 views

is it possible to project any triangle on a plane as a right triangle on another plane?

I scratching my head over this problem from my projective geometry book (C. R. Wylie, Jr). Given a triangle in the plane $z = 0$, is it possible to find a viewing point, $C$, from which the triangle ...
0
votes
1answer
77 views

Problem of understanding transitive relations

I would like to understand the transitive property in relations...I just cant get it in my brain. I mean the definition is crystal clear. However I still struggle. For example: Given the set ...
1
vote
1answer
65 views

Set of Numbers when added in any combination always produce unique result

What I'm looking for is a set of numbers that when added in any combination they always have a unique sum? Is this called something? I have searched on google for hours and I'm having a hard time ...
1
vote
1answer
50 views

Analyze the continuity of the following function

Here in my book I have such an exercise with the explanation given below, but still there is something the authors didn't add, but simply put "...after some operations...". Here is such an exercise: ...
1
vote
0answers
34 views

Showing that a subrepresentation is isomorphic to the trivial representation

I'm considering $V$ to be the regular representation of a group $G$ and $W$ to be the 1-dimensional subspace of $V$ generated by the element $x=\sum_{s\in G} e_s$. I'm trying to show that $W$ is a ...
2
votes
2answers
73 views

[Proof Verification]Prove that if f is differentiable at $c \in I$ and $f'(c) = 0$, then g is not differentiable at $d:=f(c)$.

Proposition. Let I be an interval, and let $f: I \to \mathbb{R}$ be a strictly monotone and continuous on I. Let $J := f(I)$ and let $g:J \to \mathbb{R}$ be the inverse function of f. Prove that if f ...
4
votes
2answers
182 views

integral ring extension, maximal ideals

Let $\varphi:A\rightarrow A'$ be an integral ring extension. 1) Show that for every maximal ideal $m'\subset A'$ the ideal $\varphi^{-1}(m')\subset A$ is maximal. 2) and that for every ...
2
votes
2answers
34 views

Solve $\textbf y'=A\textbf y$ with $\textbf y\in \mathbb R^4$ and $A\in \text{Mat}(4\times 4,\mathbb R)$

We consider $$\textbf y'(t)=A\textbf y(t)$$ with $\textbf y(0)=\textbf y_0\in \mathbb R^4$ and $A\in \text{Mat}(4\times 4,\mathbb R)$. Let $\textbf y_1,\textbf y_2,\textbf y_3,\textbf ...
1
vote
1answer
65 views

Diophantine Equation: $a^3=a(b^2+c^2+d^2)+2bcd$

Let $a,b,c,d\in \mathbb{Z}$. Solve $$a^3=a(b^2+c^2+d^2)+2bcd$$ I've tried everything but I haven't been able to find a general solution. Note: We may assume $\gcd(a,b,c,d)=1$ because of homogeneity. ...
0
votes
1answer
37 views

Defining suitable predicate and function symbols.

I am really struggling to build intuition with regards to how to do this sort of question. The abstract: A graph is a set (whose elements are called nodes) together with a symmetric relation on that ...
0
votes
1answer
111 views

Prove that $nCr = n(n-1)(n-2)\cdots(n-r+1)/ 1\cdot2\cdot3 \cdots r$ is an integer for all positive integral $n$ and for all integers $r \geq 0$.

Prove that $nCr =\frac{ n(n-1)(n-2)\cdots(n-r+1)}{ 1\cdot2\cdot3 \cdots r}$, is an integer for all positive integral values of $n$ and for all integers $r \geq 0$. Can someone please explain it to ...
0
votes
0answers
28 views

Conditional expectation binomial

Toss $n$ coins. What is the conditional expectation of the number of heads given the number of heads among the first x tosses? I let $X \sim Bin (x, 0.5)$ and $Y \sim Bin (n-x , 0.5)$ where $X$ is ...
2
votes
1answer
53 views

Prove that the edge coloring number is smaller than or equal to two times the maximum degree

Let G be a graph with maximum degree ∆(G) and χ’(G) the edge coloring number. Prove that χ’(G) ≤ 2∆(G) without using Vizing's theorem. I really don't have a clue on how to tackle this problem. Can ...
2
votes
1answer
207 views

Checking the IBAN and dividing large numbers mod 97. Why does it work?

What's the reason (or is there an easy explanation) of why it is possible to calculate the division mod $97$ of a large number by first calculating it for the first $9$ (or $6$?) leftmost digits and ...
0
votes
1answer
40 views

Help solving simultaneous equation with powers

I am trying to solve the following equations: $0.5 = exp(-(3*c)^k)$ and $0.99 = exp(-(29*c)^k)$ I have used MATLAB to get the answers of $c = 0.21487$ and $k = 0.83471$ but I'd really like to know ...
8
votes
3answers
273 views

Intuition behind functional dependence

What is the intuition behind functional independence ? (This is defined in the following way: Let $k\leq n$. The $C^1$ functions $F_1,\ldots,F_k:\mathbb{R}^n\rightarrow \mathbb{R}$ are functionally ...
0
votes
1answer
103 views

Prove that linear system has no solution

$A$ is a matrix $4\times3$, $rank(A)=3$ Also known that all elements of $A$ are nonzero values, $a_{ij}\neq0$ $c_i$ columns of $A$ ,$A=[c_1;c_2;c_3]$ $F$ is a diagonal matrix $4\times4$, all the ...
2
votes
1answer
41 views

Solve $y'(x)=y(x)f(x)+y^2(x)$

We consider the equation $$y'(x)=y(x)f(x)+y^2(x)$$ with $y(0)=y_0\in\mathbb R$ and $f\in\mathcal C^0(\mathbb R)$. Give a necessary and sufficient condition on $f$ and $y_0$ for that the ...
1
vote
1answer
43 views

Angles in Inner Product Spaces

In inner product spaces, angle is defined to be the only $0 \leq \theta \leq \pi$ satisfies: $$ cos(\theta)=\frac{<v,u>}{||v||\cdot||u||}$$ where $u,v\in V$ - an inner product space. I ...
0
votes
1answer
55 views

Combinatorics deck of card question

In a deck of cards there are $4$ sets, each set has $13$ cards. you choose a series of n cards like this : you choose a card, you write it as the next in the series, you put it back in and you ...
0
votes
2answers
29 views

Normalization in $L^{p}$ and $L^{q}$

Given a function $f$ in $L^{p}\cap L^{q}$ where $0<p,q<\infty$, can $f$ always be normalized such that $\| f \|_p=\| f \|_q=1$?
0
votes
1answer
86 views

Constructing idele from a rational number.

I am a novice to concept of idele, despite the fact that I have gone through all its expositions in standard literature. Excusing my ignorance, suppose I take $q=396000$. Does it mean that the idele ...
1
vote
1answer
49 views

Are 'vectors' vectors?

Let us say I have a 'vector' $\vec v$ for which I can do the following operation on $A\vec v$ where $A$ is a matrix. Now most people (i think) would say that $\vec v \in R^n$ however $\vec v$ is not a ...
-1
votes
2answers
49 views

What is right solution for this probability problem?

This drug can cure $90$% of all diseases. What is probabilty of successful healing at least $18$ people of $20$ people, who have taken the drug? What is the right solution and why? From my point of ...
3
votes
0answers
75 views

Voltera equation

Consider the Voltera integral equation: $$ψ(x)=e^{-x}\cos(x)-\int_{0}^{x}e^{-(x-t)}\cos(x)ψ(t)dt$$ How can I solve this equation by converting it to a differential equation? The solution is ...
0
votes
1answer
37 views

legitimate proof of complex arguments?

If $\operatorname{Re}(z_1)>0, \operatorname{Re}(z_2)>0$, then $\operatorname{Arg}(z_1z_2) = \operatorname{Arg}(z_1) + \operatorname{Arg}(z_2)$ My question is if the proof i used is legitimate. ...
2
votes
0answers
8 views

Differential Equations of Transformed System

At the moment I'm struggling with a problem I found in a script to one of my lectures: Let $\phi \in C^\infty(\mathbb{R}^{2n})$ have the property that the system $p_i=\frac{\partial}{\partial ...
-1
votes
4answers
72 views

The next number in series

What is the next number in the following series: 12, 35, 81, 173, 357 , ___ I am not able to find the answer. Please explain.

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