1
vote
2answers
199 views

Find the locus of points whose distances from the line $y=\sqrt3x$ and x-axis are equal.

Find the locus of points whose distances from the line$\hspace{0.2cm}$ $y=\sqrt3x$$\hspace{0.2cm}$ and x-axis are equal. My solution:I start with the following ...
0
votes
3answers
563 views

How is the set of all closed intervals countable?

I am trying to figure out the answer to the problem: Show that the set of all closed intervals $[a,b]$ with $a,b \in \mathbb{Q}$ is countable. Now I know that the interval $[0,1)$ for example is ...
1
vote
1answer
50 views

Show Inequality Below

Let m and n be positive integers. Show that $$\frac{(m+n)!}{(m+n)^{(m+n)}} <\frac{m!n!}{m^m n^n}$$ I know $m,n> 0$, the inequality can be rewritten as, $$(m^m)(n^n)((m+n)!) < ...
2
votes
0answers
124 views

Derivation of Steepest Descent Direction used in Line Search Methods

In the numerical optimization text I am reading, the Steepest Descent Direction was derived by considering $$ \min_{||p||_2\leq 1} p^T\nabla f(x_k) $$ This resulted in $$ p_k=-\frac{\nabla ...
-1
votes
1answer
69 views

A large number divisible by 4 [closed]

Let $S=\displaystyle x! + \sum_{k=0}^{2013} k!$, where $x$ is a one-digit non-negative integer. How many possible values of $x$ are there so that $S$ is divisible by 4?
0
votes
1answer
136 views

Calculate maximum possible velocity, given D, V0, A, J

I'm writing a program that controls space ship. I needed to determine total travel time for the following scenario: ship has some initial velocity V0 ship needed to travel distance D with maximum ...
0
votes
1answer
40 views

An easy number choosing problem

Suppose we have $k$ identical sets $A$ consisting of $n$ integers. How many ways are there to choose $k$ numbers $(a_1,...,a_k)$ such that $a_i\in A$? The order of numbers matter. At first I ...
0
votes
1answer
29 views

Characterize the following sets as closed/open in the space of M2(R)

Characterize the following sets as closed/open in the space of $M_2(R)$(topologized by considering it as a subset of euclidean space of dimension $4$ in the obvious way ) Set of matrices of the ...
2
votes
1answer
86 views

In additive category product and co-product over finite family of objects are isomorphism.

Let C be a additive category.Show that the co-product and product over finite family of objects are isomorphism.
1
vote
4answers
60 views

Linear algebra: Matrix identity

I am trying to understand a derivation and there is a matrix manipulation which I do not understand. So, there is the following derivative: $$ \frac{d}{dx} (x^T\Sigma^{-1}x) $$ Here x is a D ...
1
vote
0answers
57 views

Evaluating convoluted integrals of complex exponentional and rational

I want to evaluate the following integral: \begin{equation} f_{abcd}(t) = \int_{-\infty}^{\infty}d\lambda\int_0^{t-\lambda} d\tau \frac{e^{i a \tau}}{ (b+i \tau)^{5/2} } \int_0^{t-\lambda} d\tau ...
2
votes
1answer
129 views

Is an orthogonal projector bounded in $L_p$-spaces?

Let $P$ be an orthogonal projector on $C^\infty([0,1])$. For $0<p<\infty$, we define for $f \in C^\infty$ the norm (quasi-norm if $p<1$) $\lVert f \rVert_p$ in the usual way. Moreover, we ...
1
vote
2answers
67 views

Finding the element of the quotient ring $Z[i]/\langle 2+2i\rangle$

First, I'm writing an element to confirm whether I understood this quotient ring correctly. $$(5 + 7i) + \langle 2+2i \rangle = 2(2+2i) + (1+3i) + \langle 2+2i \rangle = ...
2
votes
1answer
48 views

Optimize $f(x,y,z) = xyz$ restricted to $g(x,y,z)= x^2+2y^2+3z^2= 6$

I'm stuck doing this problem. Optimize $f(x,y,z) = xyz$ restricted to $g(x,y,z) = x^2+2y^2+3z^2 = 6$ First, I found ${\nabla}f$ and ${\lambda}{\nabla}g$, and for Lagrange Multipliers, I got these ...
0
votes
1answer
26 views

solve for the max of the sum of two points on a function a given distance apart?

I just thought of this concept and am not very experienced in math, so I'm assuming there's an easy solution I'm overlooking. For a given function y = f(x), how can one find the maximum value for the ...
0
votes
3answers
55 views

Maximum value for $x(t)$ in $x(t) = -\frac{1}{2}gt^2 + vt$

In a book I am reading it says that the maximum value of $x(t)$ in $$x(t) = -\frac{1}{2}gt^2 + vt$$ is $\frac{v^2}{2g}$ and that this happens when $t=\frac{2v}{g}$ I cannot derive this though. When I ...
0
votes
4answers
64 views

Prove that condition is rational

I tried to solve this about hour, but I can't... $$\begin{align} \sqrt{7+4\sqrt{3}} - \sqrt{3} \end{align}$$ Answer should be 2. I don't need to solve this for me, I just need explanation how to ...
0
votes
0answers
53 views

Finding the appropriate bounds of integration for this joint probability.

Let $f(x_1, x_2) =\begin{cases} 4x_1x_2, \ 0 < x_1 < 1, \ 0 < x_2 < 1 \\ \\ 0, \text{elsewhere} \end{cases}$, be the pdf of $X_1$ and $X_2$. Find $P(X_1 = X_2)$. I understand that ...
0
votes
1answer
318 views

Prove ${2n\choose n}=\sum\limits_{k=0}^n {n\choose k}^2$ [duplicate]

Prove ${2n\choose n}=\sum\limits_{k=0}^n {n\choose k}^2$ My Approach: I will be making use of $$\tag 1\quad{m+n\choose r} = {m\choose 0}{n \choose r} + {m\choose 1}{n\choose r- 1} + ...
1
vote
0answers
81 views

Moment generating function of a piecewise function

I am struggling like crazy with question b. Here is what I have come up with: But if I then test $M_X'(0)$, I don't get the expected value I calculated in part a: Where have I gone wrong?
0
votes
1answer
37 views

what is the relationship between joint and marginal probability of dependent random variables?

If $x$ and $(y_1,y_2)$ are independent random variables, then we know that $$f(x,y_1,y_2)=f(x)f(y_1,y_2)$$ now if they are not independent is it true that $$f(x,y_1,y_2) < f(x)f(y_1,y_2)$$ or not?
4
votes
2answers
744 views

$\bar S$ is the smallest closed subsets of $X$ which contains $S$

There is indeed a very similar question asked [See Proving closure of S is the smallest closed set containing S. but the content is too advance for a starter in analysis like me and the focus of ...
10
votes
1answer
282 views

Maxima of the function $\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$

I am looking for extrema of the function $$g(a,b):=\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$$ where $a,b >0$ are real parameters. I already plotted this function and got the ...
1
vote
0answers
58 views

simplex in MAPLE

I am trying to solve a large system of linear inequalities (about 500 variables) subject to the nonnegativity condition on the variables.Call this system500 for future reference. I do not need to ...
1
vote
1answer
104 views

Solving differential equation.

In my research work I need to find the solution of the following differential equation. $\displaystyle y'(x)=\frac{y(x)+1}{2 \sqrt{x y(x)}-x},$ $y(0)=0$, where the solution must satisfies the ...
0
votes
3answers
80 views

How to find the integral of $(1-|\tau|)\cos(\omega\tau)e^{-j\omega\tau}$

I have a function that need calculate the integral. Could you help me to find it. Thank you so much $$f(\omega)=\int_{-1}^1(1-|\tau|)\cos(\omega\tau)e^{-j\omega\tau}d\tau$$ where $\omega$ is constant. ...
0
votes
1answer
82 views

Proof of an interesting matrix property

Suppose you have a square matrix $M$ with $n$ rows and $n$ columns. Suppose $M$ enjoys a property $p$ defined as follows: $M_{i,j} = 0$ if $i + j$ is odd and non zero otherwise. Question: if square ...
0
votes
2answers
30 views

Let $X,Y$ be random variables in uniform distribution, $0\leq X\leq 3$,$0\leq Y\leq 4$, the probability of $X\leq Y$

Let $X,Y$ be random variables in uniform distribution, $0\leq X\leq 3$,$0\leq Y\leq 4$, I want to compute the probability of $X\leq Y$ For each $X$, the probability of $X\leq Y$ is ...
0
votes
0answers
66 views

Synthetic proof of curvature formula

The radius of curvature of a curve $\gamma:I \to \mathbf{R}$ parametric by arc length is $||\ddot \gamma||^{-1}$. I want to demonstrate this using synthetic geometry. Let $A$, $B$ and $C$ be three ...
2
votes
3answers
164 views

Sum of squares of two integers divisible by five [closed]

Supposing $x,y$ are natural numbers, what is the probability that the sum of their squares are divisible by 5? I am getting $1/3$ as squares can only end with $0,1,4,5,6,9$. So $36$ pairs are ...
4
votes
2answers
112 views

For a complete truth-set $T$ is a countable transitive model satisfying $T$ unique?

Let $T$ be a maximal (in the sense that either $\phi \in T$ or $\phi \not \in T$ for all $\phi \in \mathcal{L}_\in$) set of sentences consistent with $ZFC$. Question For a countable transitive model ...
0
votes
2answers
48 views

General solution for $U_{xy}+U_y=e^{-x}$

Let $$U_{xy}+U_y=e^{-x}$$ I followed the substitution mentioned here. Let $V_x+V=e^{-x}$. So now we have $(e^{x}V)_x=e^{-x}$. Integrating w.r.t $x$ we get $$V=-e^{-2x}+e^{-x}c_1(y).$$ Then ...
1
vote
1answer
141 views

Proof that random walk visits zero infinitely many times

Since the Green function $G(x,1)=\sum\limits_{n\in \mathbb{N}_0}P(S_n=x), x\in\mathbb{Z}^d$ gives the expected number of visits to $x$ in a random walk, I'm asked to prove the following: I have to ...
1
vote
1answer
140 views

Show that $\infty$-norm and $C^1$-norm are not equivalent.

Show that $\infty$-norm and $C^1$-norm are not equivalent. For the $C^1([a,b],\mathbb{R})$ space, show that $\displaystyle ||g||_\infty=sup_{a\leq t\leq b}|g(t)|$ and $\displaystyle ...
1
vote
3answers
93 views

Why is the polynomial $f(x)=x^3+x^2+x+1$ monotonic?

I have to argue why the polynomial $f(x)=x^3+x^2+x+1$ has a reverse function $f^{-1}$ which is defined in on the whole of $\mathbb R$. I'm certain the argument would simply be that because $f(x)$ is ...
2
votes
0answers
99 views

Branched coverings of unit disk

Is there a classification of branched coverings of the closed unit disk $\mathbb{D} =\{z\in \mathbb{C} \ | \ |z| \leq 1 \}$? Here we consider only branched covering projections which restrict to ...
0
votes
1answer
152 views

# of ways to place books on shelf

This is a question about the the # of ways I can place the books on shelves. I have to place a book on n number of shelves with m number of books; m >= n >= 1. But I have to have atleast 1 book on ...
3
votes
3answers
259 views

Using a given identity to solve for the value of an expression

This problem caught my eye in the book yesterday. Till now I still get stuck. Here it is: If $$\frac{x}{x^2+1}=\frac{1}{3},$$ what is the value of $$\frac{x^3}{x^6+x^5+x^4+x^3+x^2+x+1}?$$ The ...
1
vote
1answer
68 views

Tough Subgroup problem

Let $G$ be a group, and $S \subseteq G$ a subset. Define subsets of $G$ by putting $T_0=\{e\}$, $T_1=\{x\in G | {x\in S} \ \text{or} \ {x^{-1} \in S\}}$, and for $n>1$, let $T_n=\{xy| x\in T_{n-1} ...
1
vote
1answer
116 views

Two definations of the dynkin system. [closed]

In this article, there are two definitions of the Dynkin system. I want to prove they are equivalent. How to use the second definition to prove the second item in the first definition?
1
vote
2answers
58 views

Seeking proof using mathematical induction

\begin{equation}a: \mathbb N ×\mathbb N \to \mathbb R \end{equation} where for all \begin{equation}x,y\in\mathbb N\end{equation}\begin{equation}a(x,y) =a(y,x)\end{equation} How do I show that the ...
0
votes
1answer
260 views

Practice Examples of Proofs by Induction, Direct/Indirect Method

I'm learning about proofs in school, quite a few different sorts (but not geometry ones), but the teacher is teaching by slides mainly, not books. The main ones are proof by ...
4
votes
2answers
504 views

Is this extension of ZFC known to be outright inconsistent?

Is the following first-order theory known to be outright inconsistent? Adjoin to ZFC a unary function $U$ together with the following axioms. If $\alpha$ is an ordinal, then there exists an ...
1
vote
3answers
448 views

difference between linear transformation and its matrix representation .

I can't understand this: Given a matrix T =$\small \{T_{ij}\} \in \mathbb M_{mn}$,define a transformation $\small T:\mathbb R^n \rightarrow \mathbb R^m$ as follows: If $~~\small T(x_1,\ldots ...
0
votes
1answer
32 views

Showing that an $(S,T)$-bimodule is a right $S^{op} \otimes_\mathbb{Z} T$-module

The questions I'm trying to answer is as follows (all rings are unital): Let $_{S}B_{T}$ be an $(S,T)$-bimodule, and let $R = S^{op} \otimes_\mathbb{Z} T$. Show that $B$ can be made into a right ...
2
votes
0answers
135 views

Definition and derivation of conditional expectation/probability

I read quite a few books introducing the notion of conditional probabilities/expectation by putting a formula out there coming from what they call "intuition". Can someone provide me a good measure ...
0
votes
2answers
53 views

Proving the congruency of triangles

How can I prove the congruency? I have tried, $\angle{CDA}=\angle{DCB}$ But, which rule should I use?
0
votes
0answers
75 views

Help in Understanding the Formula for The Lattice Point Counting in Triangles with Rational Coordinates

Yesterday I have found this paper while searching Google. However, since the author of this paper gave no examples of implementing the following formula, I don't understand how to implement it in ...
1
vote
1answer
52 views

Compactness, Convexity, Convex Hull of Sets including sequences

Is the following set compact, is it convex and what is the convex hull? $V = \{(x_1, x_2,...,x_n) \in \mathbb{R}^n :\frac{1}{1 + i} \leq x_i \leq \frac{1}{i}, i=1,2,...,n\}$ My thoughts: I was ...
1
vote
0answers
62 views

Triangulations of the concave polygon

It is known that the amount of possible triangulations of the convex polygon by disjoint diagonals is the Catalan number. But can we somehow know possible amount of the triangulations of the concave ...

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