0
votes
0answers
2 views

Riemann sum and partitions

If f is riemann integrable and if $(P_n)$ is any sequence of tagged partitions of [a,b] such that $\lVert P_n \rVert$ -> 0, prove that $\int_a^b f = lim_n S (f;P_n)$. I am confused as to how to ...
0
votes
0answers
2 views

Diagonal decomposition, square root and eigenvector / eigenvalue of a matrix

I have encountered a problem of finding eigenvector and eigen value of a matrix of type $$ A = \dfrac{1}{2} \begin{pmatrix} 4&1&-2\\ -4&1&6\\ 2&0&-2 \end{pmatrix} $$ Also I ...
0
votes
0answers
2 views

Still stuck on simplifying terms while doing linear combinations

So I'm currently trying to wrap my head around finding gcd through the Euclidean Algorithm in order to write the integers as a linear combination. For example, a problem is to express the ...
0
votes
2answers
15 views

Show $ex \leq e^x$ for all $x \in \mathbb{R}$

So far all I have is this: Let $f$ be a function where $f(x)=ex-e^x\leq 0$ $f'(x)=e-e^x \leq 0$, so $f$ is decreasing. I'm stuck here. Can someone help me with the next steps?
0
votes
0answers
7 views

determine whether a combination number is odd or even

Let $k$ be a given positive integer (fixed). I want to determine whether $$ 2n-k\choose n $$ is even or odd, for each positive integer $n$. Is there any general result? My attempt: Case (1). ...
0
votes
0answers
11 views

Why are structures with no relations called algebras?

Elementary algebra has at least one relation: the equality (or identity) relation, signalized by the symbol "=" The equality relation is quintessential to linear algebra and algebraic equations, such ...
1
vote
1answer
19 views

Simplify $\frac{(\cos \frac{π}{7}-i\sin\frac{π}{7})^3}{(\cos\frac{π}{7}+i\sin\frac{π}{7})^4}$

Simplify $$\frac{(\cos \frac{π}{7}-i\sin\frac{π}{7})^3}{(\cos\frac{π}{7}+i\sin\frac{π}{7})^4}$$ I used de Morvre's theorem to get to $$\frac{(\cos ...
0
votes
1answer
18 views

Example of Abelian Group of order 2014

What are some examples of Abelian Groups of order $2014$ ?
1
vote
1answer
10 views

Show that if $x\equiv 1 \pmod {m^k}, $then $x^m \equiv 1\pmod{m^{k+1}}$.

Let $k\ge 1, m\ge 1.$ Show that if $x\equiv 1 \pmod {m^k}, $then $x^m \equiv 1\pmod{m^{k+1}}$. First I noticed that the assumption would imply $x^m \equiv 1 \pmod{m^k}$, but that doesn't seem to ...
-2
votes
2answers
21 views

How do I write this equation as an algorithm?

how do I write "from 3 to 6" or > 2 but < 7 in an algorithm?
1
vote
0answers
10 views

Is $\mathbb{Z}_{(p)}$ a Dedekind ring?

Is $\mathbb{Z}_{(p)}$ a Dedekind ring? (for a prime number $p$) By $\mathbb{Z}_{(p)}$ i mean the localization of $\mathbb{Z}$ at $p$. I know that one must check a couple of conditions, like it being ...
0
votes
0answers
13 views

Limits and sequences question

Let $$a_n=n^x(n^{1/n^2}−1).$$ Show that $$\lim_{n \to \infty} \frac{a_n}{\ln(n)/(n^{2-x})} = 1. $$ It is on the study guide for my final exam, which is tomorrow so I am trying to figure it out. ...
0
votes
0answers
4 views

Show an absolute minimum and positive/negative derivative of function

Let $f : \mathbb R \to \mathbb R$ be defined by $f(x) := 2x^4+x^4\sin(1/x)$ for $x \neq 0$ and $f(0) = 0$. Show that f has an absolute minimum at x = 0, but that its derivative has both positive and ...
-1
votes
1answer
7 views

Scaling the subtraction of volume

Phil is making a fruit drink. He has one large jar filled with $96$oz of water. However, this is too much water, and he needs to get rid of some to make room for other ingredients. He removes $25$oz ...
0
votes
0answers
4 views

Verify combination of disjoint subsets $C$ and $D$ is onto

Let $C$ and $D$ be disjoint subsets of set $A$ and $f:C→B$ and $g:D→B$. Define a function $h(x)$ as follows: $$ h(x)=\left\{ \begin{array}{c} f(x) \textrm{ if } x∈C \\ g(x) \textrm{ if } x∈D ...
2
votes
1answer
15 views

What did I do wrong with this combinatorics question?

I was given the following problem. "A teacher wants to choose a captain and vice-captain among 12 volleyball players. In how many ways can she do so?" I tried to solve it by multiplying 12 by 11 ...
1
vote
0answers
15 views

How is $K(x)$ pronounced, where $K$ is a field?

Let $K$ be a field. The ring of polynomials $K[x]$ is pronounced "$K$ adjoin $x$", right? How is the field of rational functions $K(x)$ pronounced? (Sorry if this is a silly question. I am ...
2
votes
1answer
12 views

Describing polar coordinates in a window

I'm having trouble with the following problem and have no idea what to do. I tried drawing a horizontal and vertical line down the middle of the window but got nowhere. A window is in the shape of a ...
0
votes
1answer
5 views

Probability of choosing one of two desired boxes out of five on the second try.

I've got a problem that I'm not understanding how a given probability is being found. If you have five boxes, three of which are empty and two of which have items you want, if you choose boxes in a ...
0
votes
0answers
6 views

Geometric distribution converges to exponential distribution

For $n\in \mathbb{N}$ let $X_n$ be geometric with parameter $p_n \in (0,1)$, that means $\mathbf{P}[X_n = k] = p_n(1-p_n)^k$, $k\in\mathbb{N}_0$. How must the sequence $(p_n)_{n\in\mathbb{N}}$ ...
0
votes
0answers
7 views

Differentiability & continuity proof?

Prove that if the absolute value of f is differentiable at a and f is continuous then f is differentiable at a. I've started off by writing down the formal definition of differentiability and the ...
0
votes
1answer
12 views

What is a function which differs when differentiated with respect to x and then with y to function differentiated with respect to y and then x?

What is a function which differs when differentiated with respect to x and then with y to function differentiated with respect to y and then x? Even if it differs at one particular point.
0
votes
0answers
14 views

Functions and range

$$ a \colon \mathbb{R}\setminus\{0\} \to \mathbb{R} \;\text{ defined by }\; a(x)= 6/x \\ b \colon \mathbb{Z} \to \mathbb{R} \;\text{ defined by }\; b(x) = 3x + 1 $$ a) State the range of ...
1
vote
0answers
4 views

Let X_1,..,X_n i.i.d negative binomial. Find the best unbiased estimator for P(x<=3)

I am not sure where I should even start with this problems. I know that the sum of negative binomial random variables is itself a negative binomial random variable. I am sure that I can show that the ...
-1
votes
0answers
8 views

For X,Y random variables, with pdfs that are symmetric around 0, does $V(X)\geq V(Y) \Rightarrow E(|X|)\geq E(|Y|)$?

I need to show the following thing. Consider two continuous random variables $X,Y$ which take values in $[-1,1]$ and are have cdc's that are symmetric around zero. How can I show that $V(X)\geq V(Y) ...
0
votes
0answers
9 views

Solutions to a laplacian preservation

I'm trying to write an impementation to that paper over here http://www.cs.jhu.edu/~misha/Fall07/Papers/Sorkine04.pdf The main idea is that i have a series of points, and i displace some of them. ...
0
votes
0answers
3 views

Find the dual of the given primal linear programming problem

The primal problem is as followed: Minimize z=4x-5y Subject to y<=10-x y<=2+3x x,y>=0 Write out its dual and solve it geometrically. ...I have found its dual and graphed out the ...
0
votes
0answers
15 views

Basis of a Z-module

I think I might know how to start this problem but I'm not sure how to finish. Here is the statement: Determine a basis for the ℤ-module of integer solutions to the following system of equations: ...
1
vote
1answer
15 views

Show that there exists sets $A, B$ in $R$ such that $(A \cup B)^o \neq A^0 \cup B^o$

$\newcommand{\closure}{\operatorname{closure}}$ Show that there exists sets $A, B$ in $R$ such that 1) $(A \cup B)^\circ \neq A^\circ \cup B^\circ$ and $2)$ $\operatorname{closure}(A \cap B) \neq ...
1
vote
2answers
36 views

What is $p+4 \pmod {13}$

$p=\sum_{m=1}^{12} (m \cdot m!)$, and the question is just the title. I tried applying Wilson's theorem i.e $(p-1)!+1\equiv0 \pmod p$ but did not get much help. Thanks for helping
1
vote
1answer
18 views

$a,b,c,p$ are rational number and $p$ is not a perfect cube

Given that $a,b,c,p$ are rational number and $p$ is not a perfect cube, if $a+bp^{1\over 3}+cp^{2\over 3}=0$ then we have to show $a=b=c=0$ I concluded that $a^3+b^3p+c^3p^2=3abcp$ but how can I go ...
0
votes
0answers
4 views

tridiagonal matrix with a corner entry from upper diagonal

I am trying a construct a matlab code such that it will solve an almost tridiagonal matrix. The input I want to put in is the main diagonal (a), the upper diagonal (b) and the lower diagonal and the ...
1
vote
1answer
25 views

If $n\equiv 4 \pmod 9$ then $n$ cannot be written as sum of three cubes?

Show that if $n\equiv 4 \pmod 9$ then $n$ cannot be written as sum of three cubes. This might be a silly question but I really don't see it? The thing I ended up was: let $n=a^3 + b^3 + c^3$, ...
0
votes
0answers
8 views

Oscillatory of Differential Equation

Show that the DE $$y'' +q(x)y = 0 \; (*)$$ is oscillatory if one of the following condition is satisfied: i) $q(x) \geq m^2 > 0$ eventually ii) $q(x) = 1 + \phi(x)$ where $\phi(x) \to 0 \; as ...
0
votes
2answers
25 views

Simplest way to prove that $e^{ix}$ is an open mapping into $S^1$

Let $S^1$ be the unit circle in $\mathbb R^2$ and give it the subspace topology. What's the simplest way to prove that $f:R\to S^1$, $f(x)=e^{i x}$ is an open mapping, that is $f(U)$ is open when $U$ ...
1
vote
1answer
14 views

Using mean value theorem for multiple inequalities

Use the Mean Value Theorem to prove that $\frac{(x-1)}{x} < \ln x < x-1$ for $x > 1$. I was thinking of breaking up the inequality into \frac{(x-1)}{x} < \ln x$, and $\ln x < x-1$ and ...
2
votes
0answers
9 views

Iterative method for a positive definite matrix

Given A-> a positive definite matrix, if W(x)= 1/2- is a functional form and x-> x + a_i * v_i is the i-th step iteration( v_i is the ith eigen vector of A) 1. how do I find the step size a_i 2. Show ...
1
vote
1answer
16 views

Boundary Value Problem and solutions

The linear ordinary differential equation $y'' + y = 0$ has the family of solutions $y = A \sin(x) + B \cos(x)$ Determine whether $y(0)=2, y'(\pi/2)=3$ is a unique solution. If not, does it have no ...
0
votes
0answers
8 views

Relationship between ball speed and distance traveled [on hold]

What is the relationship between the speed of a thrown baseball, and the distance it travels? In other words, if a baseball is thrown 300 feet, what does its speed need to be? What about 325 feet? ...
1
vote
0answers
33 views

Integrable but not differentiable function

Suppose $f(x)$ is continuous on $[a, b]$ except at a point $c$ in $(a, b)$ at which $f(x)$ has a jump discontinuity. For $x$ in $(a,b)$, set $F(x)=\int_a^xf(t) \, dt$. Show that $F(x)$ is continuous ...
1
vote
0answers
8 views

Linearising equations about a base state.

Consider a shallow-water system with mean depth H, where the base state consists of the flow (u,v)=($u_{0},$0), with a sloped water surface $\eta_{0}$(x,y) = - $\gamma y$, where u$_{0}$ and $\gamma$ ...
0
votes
0answers
17 views

Series and sequences- Help

Let $a_n=n^x(n^(1/n^2)-1)$ for n in natural numbers and assume that lim(n goes to infinity)ln(n)/n^r = 0 for any r>0. Let ln(x)=integral(from t=1 to x)dt/t for x>0. Prove the inequality h/1+h < ...
0
votes
0answers
7 views

Construct a triangulated adjunction from a right triangulated adjoint functor?

In the Lemma 40. of the note on triangulated categories by Daniel Murfet, one finds the construction of a triangulated adjunction from a left (triangulated) adjoint triangulated functor, whose proof ...
1
vote
0answers
14 views

Methods of solving nonlinear systems of equations derived from combinatorial problem

I'm trying to find a way to generalize the expression of polynomials of degree $n-1$ such that $$ k_1+k_2x+k_3x^2+k_4x^3+\dots+k_nx^{n-1}=\frac ...
2
votes
3answers
28 views

$300$ dice rolls, at most $42$ “$2$'s”.

If a fair die is rolled $300$ times, what is the probability of rolling at most $42$ "$2$'s"? I plug this into my calculator: binomcdf($300, .166667, 42$) and get $.121$ as the solution. This is not ...
2
votes
3answers
43 views

Series Proof $\sum_{k=1}^n (1/k) > \ln(n+1)$

Prove that $\sum_{k=1}^n (1/k) > \ln(n+1)$. I have been trying to do this for some time now, but I cannot figure it out. It is on the study guide for my final exam, which is tomorrow so I am trying ...
0
votes
0answers
25 views

How can I prove that

Let $μ^*$ be an outer measure on $P(X)$ and $m^*$ be the class of measurable sets with respect to the outer measure $μ^*$. How can I prove that the class of measurable set $m^*$ with respect to the ...
2
votes
1answer
17 views

How to express the oscillator equation $y'' + 3y'+2y=\cos(t)$ as a first order system?

Can someone please help me express this oscillator equation $y'' + 3y'+ 2y= \cos{(t)}$ as a first order system? I also need to plot an approximate solution curve for the initial condition $x_0 = 5, ...
2
votes
1answer
13 views

An idempotent bounded linear operator has eigenvalues $0,1$

I am thinking of the following problem: suppose $T$ is an idempotent bounded linear operator on a Banach space $X$ over the complex field. Of course, suppose $T$ is not zero map or identity map to ...
4
votes
1answer
52 views

Zeilberger's potential proof of Fermat's last theorem.

Doron Zeilberger suggested the following potential proof for Fermat's last theorem: Let's define: $$W(n,a,b,c) \equiv (a^n + b^n - c^n)^2$$ I am almost sure that there exists a polynomial, ...

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