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If n is a positive integer, how many 5-tuples of integers from 1 through n can be formed in which the elements of the 5-tuple are written in decreasing order but are not necessarily distinct? In other words, how many 5-tuples of integers (h,i, j,k,m) are there with n ≥h ≥i ≥ j ≥k ≥ m ≥1?
I thought in consider this 5-tu... |
I wish to show that
for $(a_k)$ a sequence of numbers, $a_k \in \mathbb{R}$
then claim :
$|\sum\limits_{k = n+1}^m a_k | \leq ||\sum\limits_{k = n+1}^\infty
a_k| - |\sum\limits_{k = m+1}^\infty a_k ||$
I tried to show this by using:
$|\sum\limits_{k = n+1}^m a_k | \leq |\sum\limits_{k = n+1}^\infty a_k - \sum\limit... |
In school I was taught to memorize this formula.
$ \frac {d} {dx} (\cos x) = - \sin x $
However, recently I found out a proof using the first principle (under the "Derivatives" chapter), but could not understand the proof.
The formula for first principle is as follows:
$\ f'(x) = \lim_{h\to 0} \frac {f(x+h)-f(x)} {... |
Let $L:\mathbb{R}[X]\rightarrow\mathbb{R}[X]$ be an operator given by the following formula
$L\left(\sum\limits_n a_nX^n\right)=\sum\limits_n a_{2n}X^{2n}$.
We assume that on $\mathbb{R}[X]$ we have a norm $\|p\|=\int\limits_{-1}^{1}|p(t)|dt$.
Is this operator bounded ? If yes what is its norm ?
I tried to do it in a d... |
Let $(M,g),(N,h)$ be two Riemannian manifolds.
Assume $f:M \to N$ is an isometric immersion.
Is $f$ harmonic? (i.e a critical point of the energy functional)
(I know this is true when $(N,h)=(\mathbb{R}^n,e)$ is the standard Euclidean space).
Edit: I assume also $M$,$N$ are manifolds of the same dimension.
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My task is to implement the inner product and vector triad forms for a dense $A$ in single and double precision. I have successfully implemented the inner product and vector triad form although, I am not sure if by the way I make a dense matrix $A$ if it is in fact Dense or Sparse.
What I did was essentially just make ... |
I am trying to solve this equation my self with my knowledge about characteristic polynomials, etc but I have placed it here earlier because I'm not a mathematician and maybe you give me ideas to solve the problem faster.
Consider the following matrices:
$C=\begin{bmatrix}C_{11}&C_{12}&C_{13}\\C_{12}^*&C_{22}&C_{23... |
I ask this question as a subsequent of following one. Suppose that $f$ and $g$ are two analytic functions defined on $\mathbb C$ and that $(a_n)_{n \in \mathbb N}$ is a sequence of complex numbers such that $\lim\limits_{n \to \infty} \vert a_n \vert = \infty$ and $f(a_n)=g(a_n)$ for all $n \in \mathbb N$. Is it true t... |
In a symmetric group $S_n$ find number of permutations $P$ such that in the disjoint cycle decomposition of $P$ , length of cycle containing $1$ is $k$ . Here's my attempt at this .
I found number of cycles containing $1$ and of length $k$ as $\frac{(n-1)!}{(n-k)!}$. Rest of them can be permutated in $(n-k)!$ ways. So ... |
Problem
What is the last digit of $235!^{69}$?
It's been far too long since I did any modulo calcuations, and even then, the factorial would set me back.
My initial thought goes to the last digit $5$ in $235$. If $235!$ can be shown to have a $5$ as last digit, raising it to any natural number shouldn't change that, ri... |
Having trouble (again) with this DE can someone help me find the general solution for it? I feel like my biggest problem is doing the algebraic manipulations to identify what kind of DE it is.
$$y' + \frac2x y\ = y ^3/ x^3$$
Thank you in advance
|
I've read this statement in a presentation slide, but it isn't obvious to me on why this is true:
Forgetting about BCH codes, the question is: if an alement $\beta$ has even order ($2k$ is always even for $k \in \mathbb{N}$), why can't such an element exist in a field with characteristic 2 (e.g. in $\mathbb{F}_2$)? I ... |
Find the first five solutions for,
$$x(x+1)(x+2) \equiv 0 \pmod{221}$$
I am very confused. By CRT,
$x(x+1)(x+2) \equiv 0 \pmod{13}$ and $x(x+1)(x+2) \equiv 0 \pmod{17}$
But these two congruences are also reckless.
|
i am trying calculate $$ \int_{C} \frac{1}{\sin(z)} \, dz $$ when $C=C(0,1)$ by complex methods, its said, by residues, some one could help me?
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A dragon is green if at least one of its parents is green
I have
∀X⋅dragon(X) ∧ green (X) ⇒ ∃Y⋅childOf(Y,X)∧green(X)
Is this correct?
|
I'm trying to answer a question where I have an ergodic and covariance stationary process $\{x_t\}$, and without imposing further moment conditions need to prove $\frac{1}{n} \sum\limits_{t=1}^n x_t^2 \rightarrow_p \mathbb{E}[x_t^2]$.
So far I have since $g(x) = x^2$ is a Borel measurable transformation, $x_t^2$ is erg... |
The problem of determining whether a recursively enumerable language is empty or infinite cannot be solved. The proof goes by reduction to the problem of decidability, which is known to be unfeasible for recursively enumerable languages.
However, the recursive languages are decidable. What about the problem of emptines... |
Proof that $\mathcal{T}:=\{\,\left]a,\infty\right [\mid a\in\mathbb{R}\,\}\cup\{\mathbb{R} \}\cup\{\emptyset \}$ is topology of $\mathbb{R}$.
I have slight trouble on writing this down.. I'll first write what I have now.
Obviously $\mathbb{R},\emptyset\in\mathcal{T}$.
Let $U_{i}\in\mathcal{T}$ $(i\in I)$.
Let $x\in\bi... |
I really need your help to solve this exponential equation.
It looks so simple, but I haven't been able to find a solution so far:
$$ {A_1 + A_2 \over 2} = A_1 \exp\left({-x^2 \over c_1^2}\right) + A_2 \exp\left({-x^2 \over c_2^2}\right)$$
$$ x = \cdots$$
|
There is a very long, straight highway with $N$ cars placed somewhere along it, randomly. The highway is only one lane, so the cars can’t pass each other. Each car is going in the same direction, and each driver has a distinct positive speed at which she prefers to travel. Each preferred speed is chosen at random. Eac... |
This is how I've resolve this ODE-1 :
$$(x^2 +y^2 +x) \, dx + xy \, dy=0$$
Check if the eq is exact:
$${\partial M \over \partial y}={\partial \over \partial y}(x^2 +y^2 x)=2y$$
$${\partial N \over \partial x}={\partial \over \partial x}(xy)=y$$
$$A=2y \neq y=B$$
The equation is not exact. Trying to find the integrati... |
Show that $\phi (x,y)=\arctan{\frac{2x}{x^2+y^2-1}}$ is harmonic by considering $w(z)=\frac{i+z}{i-z}$.
I know that if $\phi$ is harmonic then it satisfies Laplace's equation but I don't see how considering $w(z)$ will help me.. I've tried looking at the real and imaginary parts of $z$ but they don't seem to match up. ... |
A random variable $Y_i$ is given such that, $\mid$Y$_i\mid$$\leq$ $c_i$ where i ranges from 1,.....,t and t is some constant. Now, $Y_i$ is expressed as :
$Y_i = ((Y_i - c_i) + (Y_i + c_i))/2$
$= -c_i(1-Y_i/c_i)/2 + c_i(1+Y_i/c_i)/2$
Using the convexity of $e$kYi where k is some constant,
$e$kYi $\leq$ $0.5*(1 - Y_i/c... |
What is the last digit of $7^{2015}$?
|
Let $k$ be a field and suppose $$k(X)=\text{field of fractions of }\ k[X]=\left\{ \frac{f(X)}{g(X)}\mid f,g\in k[X], g\neq 0\right\}.$$
Show that $k(X)$ is not a finitely generated $k$-algebra.
|
In Bernt Øksendals stochastic differential equations he says that if we have a random variable $X:\Omega\rightarrow\mathbb{R}^d$. He defines the expectation:
$E[X]=\int_\Omega X(\omega)dP(\Omega)=\int_{\mathbb{R}^d}xd\mu_X(x)$.
But what is the definition of an integral over a vector-valued function? I've tried to find ... |
Let $(\Omega,\mathcal{F},P)$ be a probability space and $(X_n)$ i.i.d. random variables with uniform distribution on $(-1,1)$. If $Y_n=X_1\ldots X_n$, prove that $(Y_n)$ converges to $0$ a.s.
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Let $K := \{x \in C[0,1] : x(0) \in [-3,4], |x(t)-x(s)| \leq d
|t^2-s^2|, \forall t,s \in C[0,1]\}$. Let $y \in C[0,1]$ and $f : K
\to \mathbb{R}$ defined as $f(x)= \int_0^1 x(t)y(t)dt$. Show that $f$
is uniformly continuous.
Partial proof :
$f(x)= \int_0^1 x(t)y(t)dt$ is a dot product, noted $<x,y>$.
$||f||_2 ... |
Prove that $$a^ab^bc^c\geq \left(\frac{a+b}{2}\right)^{\frac{a+b}{2}} \left(\frac{b+c}{2}\right)^{\frac{b+c}{2}}\left(\frac{c+a}{2}\right)^{\frac{c+a}{2}}\geq \left(\frac{a+b+c}{3}\right)^{a+b+c}$$ where $a,b,c$ are positive real numbers.
I am able to prove the $\left(\frac{a+b}{2}\right)^{\frac{a+b}{2}} \left(\frac{... |
I think this is a basic question, but it's hard to wrap my head around.
|
The question goes:
Let $A,B,C,D$ be sets. Prove that $\big(A\times B\big)\bigcup \big(C\times D\big)=\big(A\bigcap C\big)\times \big(B\bigcap D\big)$
I started with the definition of the cartesian product $A \times B=\left\{\big(a,b\big): a\in A \wedge b \in B\right\}$
The union of $A\times B$ and $C \times D$ would be... |
Here's a sample problem:
Before each workout, I either drink a cup of coffee, a gatorade, or a cup of water.
The probability of coffee is P(C) = 0.6,
the probability of gator is P(G) = 0.3,
the probability of water is P(W) = 0.1.
If I drink coffee, the probability that the my workout ends early is 0.2.
If I drink g... |
My problem is this:
"Show that the function represented by the power series, $$y=\sum_{n=0}^{\infty} \frac{x^{2n}}{2^nn!}=1+ \frac{x^2}{2}+ \frac{x^4}{8}+ \frac{x^6}{48}+...$$ is a solution of the differential equation, $y''-xy'-y=0$."
This older question (Series solution to $y''-xy'-y=0$) looks at this same different... |
Suppose I have the set: $$A=\{0, 1, 2, ... 224, 225\}$$
I want to find a triple that sums to $225$ (where a triple is a set of 3 unique values from the set).
No Repetition Version:
There are many such triples including: $$(0, 1, 224), (0, 2, 223), ... , (74, 75, 76)$$ Note: $(0, 0, 225)$ is not a valid triple as $0$ ... |
Let $f(x)$ be a decreasing function defined over the interval $[0,a]$, with $f(0)=b$. The first derivative of $f(x)$, which is negative, is such that $f^\prime(x) > g(x)$, or equivalently $|f^\prime(x)| < |g(x)|$, where $g(x)=c_1x+c_2$.
For $x_2 >x_1$, can we say something like $f(x_1) > k f(x_2)$ (or $f(x_1) < k f(x_2... |
I think I have this question figured out almost completely, but I'm a little worried about using a certain notation.
Suppose $\mathbb{Z} \stackrel{\phi}{\longrightarrow} R$ is a ring homomorphism. Observe that $\phi(1) = \phi(1\cdot1) = \phi(1)\phi(1) = \phi(1)^2$. Thus, $\phi(1)-\phi(1)^2 = 0$ and moreover $\phi(1)(\... |
I know that I have to find a positive integer $x$ that I can multiply with $602$ and then divide the result by $784$ so that the remainder of that
integer division is $308$. I am sure that this is quite a trivial
question, but I simply don't get it. How do I efficiently calculate $x$ in its residue class? Obviously tri... |
We have 2 dice. One is fair. The other one lands by the following probabilities:
6: 1/2
5: 1/10
4: 1/10
3: 1/10
2: 1/10
1: 1/10
We roll both dice.
What is the probability that the sum is 9 given that exactly one die lands on 6?
Here's what I have so far:
Cases where sum is 9:
Case1: Loaded = 6 | Fair = 3 -> 1/2 * 1/6... |
Let $M, N$ be normal subgroups of $G$ with $M∩N=\{e\}$.
I'm trying to prove that $M\times N$ is isomorphic to $G$. I proved that $nm=mn$ for all $n$ in $N$ and $m$ in $M$. So now I'm trying to take any fixed $g$ in $G$ and represent it in terms of $m$ and $n$. I've been doing this for a while but still no answer. Any h... |
Let $P(x)$ be a real polynomial with degree $n$ such that $\sqrt{1-x^2}|P(x)|\le 1$ for all $x\in [-1,1]$.
Prove that $|P(x)|\le n+1$ for all $x\in [-1,1]$.
This question was posted some years ago at Mathlinks forum and I'm pretty sure it came from some old AMM (American Mathematical Monthly).
I believe this proble... |
The following is a question on a passage in the book A Course in the Theory of Groups by Derek Robinson. There he constructs the induced module $M^G$ from a $FH$-module for some subgroup $H \le G$ of finite index. The proof shows in some sense that this module is $F$-vector space isomorphic to $M^r$ with $r = |G : H|$,... |
If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar?
So I read that it's true only if $A,B$ are diagonalizable, but why? if the characteristic polynomial is the same, then $A,B$ have exactly the same eigenvalues and the same amount of each eigenvalue, so they both have the same s... |
I am curious about the following problem:
Let the diffusion process $\{X_t\}_{t\ge 0}$ be defined as
$$dX_t=c(1-X_t)X_td B_t$$
where $X_0\in (a,b)\subseteq (0,1)$, $c>0$, and $B_t$ is the standard Brownian motion. Define the (first) hitting time $\tau\equiv\inf\{t\ge 0\,|\,X_t\notin(a,b)\}$.
My question is: How to deri... |
Let $A$ be a Noetherian ring, and let $P$ be a prime ideal. How come we know that $P \setminus P^2$ is non-empty? Thank you!
|
The velocity $\displaystyle\vec{v}$ of a particle $=\frac{d\vec{x}}{dt}$. So surely this means that $\vec{v}$ is dependent on the position of the particle?
|
The integral I have to evaluate is $\int_Czdz$, where $C$ is the line from 0 to $1+i$, and then from $1+i$ to 2.
My work:
$z_1(t)=(1+i)t$ and $z_2(t)=(t+1)+i(1-t)=t(i-1)+(1+i)$, $t\in[0,1]$.
$z_1'(t)=1+i$ and $z_2'(t)=1-it$.
$$\int_Czdz=\int_0^1z_1(t)\cdot z_1'(t)dt+\int_0^1z_2(t)\cdot z_2'(t)dt$$
$$=\int_0^1(1+i)t(1+... |
Let a sequence be defined as $$a_n=\lim_ {x \to 0}{1-\cos (x)\cos(2x).....\cos (nx)\over x^2}$$
a)prove that the given sequence is monotonic and that it is not bounded above.
b)calculate $$\lim_{n \to \infty} ({6a_n \over n^3})^{n^2 \over n+1}$$
NOTE:i am a high school student in the 12th grade
|
This is an assignment my sister in primary school got. I think they're supposed to solve it via trial and error, but I was wondering if there's a clever way to solve it.
Lay the tiles of a double-six domino set in a a square shape such that adjacent tiles touch with matching values, and that the sum of the values on ea... |
Let $A:H\rightarrow H$ be a bounded linear operator on Hilbert space $H$. Suppose we have $x\in H$ and $r>\mathrm{dist}(x,A(H))=\inf\{\|x-Ah\|,\ h\in H\}$. How to prove that then there exist exactly one $y_0\in H$ such that $\|x-Ay_0\|\le r$ and $\|y_0\|=\inf\{\|y\|,\ y\in H, \|x-Ay\|\le r\}$? Is it possible to replace... |
A complex number is the sum of a real number and an imaginary number. Is there a collective name for the two parts comprising a complex number, such that when used, it is (pretty) clear that the context is complex numbers?
Usage example: "Write down the (two parts) of that number" - can the (two parts) be replaced by a... |
How many times will the innermost loop be iterated when the algorithm segment is implemented and run? Assume $n$, $m$, $k$, and $j$ are positive integers.
for k :=1 to n
for j :=k to n
for i := j to n
[Statements in the body of the inner loop, none containing branching statements that lead outside the loop]
... |
Let's say I have a Hilbert style system, with a few axioms and rules of inference, and I want to find a proof for some formula $\varphi$, is it possible to create an algorithm that would find a proof for it with the given system (guaranteeing it'd be in finite time)?
I know a bit of programming, but I think answering t... |
I have a heat equation for which in the solution I need to evaluate the following integral:
$$\int_{-\infty}^{\infty}\sin(k_0 \xi)e^{-\frac{(x-\xi)^2}{4a^2 t}}d\xi$$
Except of the common gaussian integral ($\int_{-\infty}^\infty exp(-\alpha s^2 +\beta s)ds=\sqrt{\pi/\alpha}e^{\beta^2/4\alpha}$) by break the sine functi... |
Three messages in binary format are sent $$ a_0 a_1 a_2 a_3 $$ and coded in binary format $$b_0 b_1 b_2 b_3 b_4 b_5 b_6$$ Symbols $$b_0,b_1,b_2,b_3,b_4,b_5,b_6$$
are the coefficients of the Boolean polynomial
$$B(x) = b_0\oplus b_1x\oplus b_2x^2\oplus b_3x^3\oplus b_4x^4\oplus b_5x^5\oplus b_6x^6$$ which is calculated... |
I have a piece of steel, cylindrical (hollow), 200mm outside diameter with 160mm inside diameter (20mm wall thickness), and I need to screw a bolt through the outside of the cylinder towards the center.
I chose a 6mm thick bolt, which requires a 12mm washer (outside diameter).
This means I need to use a flat sander (or... |
For the first part of the above problem, I copied an example from my book and I got the answer to be
$$t(t-1)+t(t-2)=t^2-3t$$
considering that the integral is the sum of the area of the rectangles in the graph. Does this make sense? I don't quite understand how this answer would work or if it's right and I'm not sure ... |
The Chebyshev functions are defined as $\psi(x) = \sum_{p^m \leq x} \log n$ and $\theta(x) = \sum_{p\leq x} \log p$, where $p$ is a prime, $m\geq 1$ is an integer and $n=p^m$ in $\psi(x)$. It is known that there exist positive constants $c_{1}$ and $c_2$ such that
... |
I've an homework problem that i'm unable to find the right answer.
The problem is:
The line $tx + sy = 2$ goes through point $(2,1)$ and is parallel to line $y = 8 -3x$, find the value of $t^2 + s^2$.
$ A. {32\over49}$ $B.{18\over49}$ $C.{36\over49}$ $D.{25\over49} $ $E.{40\over49} $
I was able to find a parallel line... |
I have a question about cluster points that would like to ask you, this question is one of the exercises in my textbook.
Question: If $A$ is any subset of $R^d$, then $der(der($A$))$ is the set of all cluster points of $der$ ($A$)
(a) Given an example of a subset $A$ of $R$ for which der(der($A$))= empty set and der(A)... |
We want to compute the following quantity
\begin{align}
\int E[V|W=u] f_U(u) du
\end{align}
For random variables $ V,W,U$ where $V$ and $U$ are independent and $U$ is absolutely continuous r.v.
Is the following chain of equalities correct
\begin{align}
\int E[V|W=u] f_U(u) du = E_U[ E_{V|W}[V|W] ]=E_V[V]
\end{align}... |
Let $X$ be a normed space and $A \subset X$. Prove that:
$$\sup(f(A))=\sup(f(cl(A))=\sup(f(conv(A)),$$ where $f \in X^{*}$.
For the first equality, I thought to prove it by double inequality:
Let $a=\sup(f(A))$, $b=\sup(f(cl(A))$ and $c=\sup(f(conv(A))$; we have that $A\subset cl(A)$, so $f(A)\subset f(cl(A))$, so $a \... |
I couldn't find the answer neither on Google, nor this website, so decided to ask.
The Heine definition of limit: from Wikipedia
$\lim_{x\to a}f(x)=L$ if and only if for all sequences $x_n$ (with $x_n$ not equal to $a$ for all $n$) converging to $a$ the sequence $f(x_n)$ converges to $L$.
Does it work with limit of ... |
Let's say I have $3$ events with probabilities $P(A) = 0.5, P(B) = 0.5$ and $P(C)= 0.5,$ and I need to find if
$$P(A \cap B \mid C) = P(A \mid C)P(B \mid C)$$
I am tying to prove this by expanding the formula above to:
$$P(A \cap B \mid C) = \frac{P(A \cap C)}{P(C)}\frac{P(B \cap C)}{P(C)}$$
Is this a correct assumpti... |
Let $c_0=\{ (x_n) : x_n\in \Bbb{R}, x_n \to 0\}$ and $M=\{(x_n)\in c_0 : x_1+x_2+\cdots + x_{10}=0\}$. Then, what is dim($c_0/M$) ?
|
I have the problem:
$$a_{n} = \frac{e^{n+5}}{\sqrt{n+7}(n+3)!}$$
I am told to use the ratio test to determine convergence or divergence (or the test could be inconclusive).
So I take the limit:
$$\lim_{n\to \infty} \frac{a_{n+1}}{a_{n}}$$
$$\lim_{n\to \infty} a_{n+1}\frac{1}{a_{n}}$$
$$\lim_{n\to \infty} \frac{e^{n+6}}... |
I have a 15 dimensional space (corresponding to 15 joints of a robot, 5 joints for each of two legs, and another 5 joints for the right arm). I'll call this space A. Bear in mind that each of the 15 values in space A is bounded (i.e. there's a minimum possible, and a maximum possible value corresponding to the joint li... |
Having $f(z)=\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n$
I had to find what $\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n\sum\limits_{n=0}^{+\infty} \frac{D_n}{n!}z^n=\sum\limits_{n=0}^{+\infty} z^n$ was and it seems that it is
$$e^zf(z)=\frac{1}{1-z}$$
I understand the left size of the equation (even though I only fe... |
In the textbook that I'm using the standard inner product is defined as
$$\langle x,y\rangle = \sum_{i=1}^{n}a_{i}\overline{b_{i}}$$
where $x=(a_{1}, a_{2}, {...}, a_{n})$ and $y=(b_{1}, b_{2}, {...}, b_{n})$ and the bar denotes complex conjugation.
In one the practice problems I was doing they used the fact that
$\... |
Let p1, p2, . . . , pn be n points in the plane such that the distance between any two points is
at least one. Let G = (V, E) be the graph such that V = {p1, p2, . . . , pn} and E = {pipj
|
distance between pi and pj
is exactly one}. Show that ∆(G) = 6.
|
Imagine we have the following problem;
$$\cos(x) = \cos(a) \Rightarrow x=a+k\times 2\pi\\ or \\x=-a+k\times 2\pi$$
And we have the following answers.. :
$$a=\frac{\pi}{3} \\or \\a=-\frac{\pi}{3}$$
Should I write:
$$a=-\frac{\pi}{3}$$
as:
$$a=\frac{5\pi}{3}$$
My book sometimes does this but I can't figure out why. Is t... |
Knowing the definition of convergence of a sequence in $\Bbb R$: ($x_n$) converges to x iff $\forall \epsilon \gt 0$ $\exists N$ such that for every $n\gt N$, $d(x_n,x)\lt \epsilon$.
Consider a new definition, say $C-convergence$: ($x_n$) C-converges to x iff $\exists N$ such that for every $\epsilon \gt 0$ and every ... |
I thought that solving a 2x2 linear system of equations using either the substitution method or the addition method (adding the two equations to eliminate a variable, then using the substitution method to get the value of the other variable) would lead to the same result even if the equations contained parameters (whic... |
Let $\bar{B}$ be a set in a vector space $E$. Then $\bar{B}$ is the closed unitary ball for a norm iff
$\mathbb{N} . \bar{B} = E$
$\lambda x + (1-\lambda)y \in \bar{B}$ for all $x \, ,y \, \in \bar{B}$ and $\lambda \in [0,1]$
$\lambda . \bar{B} \subseteq \bar{B}$ for all $|\lambda|=1$
If $\lambda x \in \bar{B}$ for a... |
The question I'm having troubles with is as follows:
Evaluate $\int_{-r}^r\sqrt{r^2-t^2}\,dt$
(Hint: substitute $t=r\sin x$)
So, immediately I did $dt=r\cos x\,dx$ and substitute it all in...
$\int r\cos x\sqrt{r^2-r^2\sin^2 x}\,dx$
And here's where I'm having a problem: I'm not sure what the new boundaries would be... |
If the characteristic polynomial $f_A(x)$ has multiples of the same product, for example $f_A(x)= (x+2)^2(x-1)$ so $(x+2)$ has a multiple of $2$, then is there a condition on $A$ such that we know for sure that $m_A(x)= (x+2)(x-1)$ or $(x-1)$ or $(x+2)$ ? i.e, no multiples.
|
Is there a simple way to show that the least square solution of an overdetermined linear system is equal to the right singular vector of the coefficient matrix corresponding to the smallest singular value?
|
Bertrand's postulate states that for any integer $n>3$, there's always a prime $p$ between $n$ and $2n-2$.
That result sets a reasonable 'lower bound' on how often we can expect primes to show up, and there are even better estimates (such as the Prime Number Theorem, although that one is an asymptote, and doesn't prove... |
For $n\in{\mathbb{N}}$ let $$f_n(x)=nx(1-x^2)^n\qquad(0\le x\le 1).$$
Show that $\{f_n\}_{n=1}^\infty$ converges pointwise to $0$ on $[0,1]$.
Show that $\{\int_0^1f_n\}_{n=1}^\infty$ converges to $\frac12$.
I've already shown both of these statements to be true. What I don't understand is this: how can $f_n$ converge p... |
Hi I am trying to workout the solution to this propositional logic formula using the below semantic equivalence formula but I am stuck. Could someone please help me out.
These are the rules
A ∧ A ≡ A, A ∨ A ≡ A idempotence
A ∧ B ≡ B ∧ A, A ∨ B ≡ B ∨ A ... |
Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be a entire function. Suppose that there are $M$, $r>0$ and $n\in \mathbb{N}$ such that $\left|f(z)\right|<M\left|z\right|^n$ for all $z \in \mathbb{C}$ with $\left|z\right|\geq r$.
Show that $f$ polynomial of degree at most $n$.
Remark: I tried to follow the proof of Liuville... |
Why does $$\DeclareMathOperator{arccot}{arccot}\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$$ diverge? In my textbook it says that from the positive side it's zero, and from the negative side it's $\pi$. However, when entering it into Symbolab the solution is zero (according to them, it converges). I also got ... |
I have a local ring $A$ and suppose I localized this ring at prime $P$.
Is the localized ring $A_P$ a local ring? I was wondering if it requires additional properties on $A$. Thank you very much!
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Task at hand:
Let $a_n$ and $b_n$ be nonzero complex numbers for $n=1,2,3...$ . Suppose $\lim_{n\to \infty} \left|\frac{a_n}{b_n}\right|=l$ exists, and $l\neq0,\infty.$ show that if one of the series $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ converges absolutely, then so too does the other. In particular, the... |
Any decimal number between 0 and infinity that multiplied by itself or another decimal number. Like 0.4 times 0.25, 2.425 times 2.425 and so on.
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I need to show that $\Bbb Z^*_8$ is not isomorphic to $\Bbb Z^*_{10}$.
$\Bbb Z^*_n$ means integers up to $n$ coprime with $n$
I do not know how to do this. I have difficulties doing proofs involving isomorphisms. A methodological answer would be highly appreciated.
Thanks in advance!
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if $E$ is a metric space nd $B\neq E$ how to prove that:
$$\overset{\circ}{B}=\bigcup_{n=1}^{\infty} (\{x\in E, d(x, E\setminus B)\geq \frac1n\})$$
i don't know how to start
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I have a problem like, "to graduate you must choose 6 out of 20 courses, but at least 2 out of the 6 courses must be a math course. 8 out of the 20 offered courses are math courses. How many choices are there for which 6 courses to take?"
My reflex is to say $\binom{8}{2} \binom{18}{4}$, but that was given as an exampl... |
Let $(X_{1},d_{1})$ and $(X_{2},d_{2})$ be metric spaces and $f: X_{1} \to X_{2}$ be a continuous onto map such that
$$ d_{1}(x,y) \leq d_{2}(f(x),f(y)) \hspace{2mm} \forall\phantom{i}x,y \in X_{1}$$
(a) Prove or disprove: If $(X_{1},d_{1})$ is complete, then $(X_{2},d_{2})$ is complete.
My attempt, but I lose confid... |
In one paper I have read a note "Thus, unlike approaches which make use of full first order logic, unprovability of a formulae with respect to a agent specification can be shown by each of two approaches automatically without reliance on modeltheoretic consideration outside the actual proof method".
note means that th... |
I am looking for some project ideas for beginning math students in the topic of game theory. I am not very knowledgeable in the topic so it would be great if I could get an good introductory source to learn the material with these students and then a project idea as a semester end-goal. It would be fun if it was a proj... |
Is there a general formula to solve the Diophantine equation $$a^2+b^2+c^2=d^2+e^2+f^2?$$ If so, can I please have the reference? Thanks.
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Suppose we have $f:\mathbb{R} \rightarrow \mathbb{R}$ which is of bounded variation. I would like to show that it can be presented as a sum of left and right continuous functions of bounded variation. Actually I have no idea how to prove that since I think it can't be proven that Jordan decomposition has that propertie... |
Let $y(t)$ be a continuous function on $[0,\infty)$. If $$ y(t)= t\left(1-4 \int^t_0 y(x) dx\right) +4 \int^t_0 xy(x) dx$$ then what is the value of $\int^{\frac{\pi}{2}}_0 y(t) dt\,$?
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Let $\mathbb{F} = \{ \mathcal{F}_t \}_{t \geq 0}$ be a continuous time filteration. $\tau : \Omega \to [0, \infty]$ is called an $\mathbb{F}$-stopping time if $\{ \tau \leq t \} \in \mathcal{F}_t$ for all $t\geq 0$.
Let $\tau_n$, $n \geq 1$ be $\mathbb{F}$-stopping times. We know that $sup_{n \geq 1} \tau_n$ is also a ... |
Find the disc of convergence: $$\sum_{n=3}^\infty \left(1-\frac{1}{n^2}\right)^{-n^3}z^n$$
I have been manipulating the power series and I am pretty sure it has something to do with $e$ but I cannot seemed to get it manipulated in the right form.
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I've often read that, whereas top extensions increase the height of the cumulative hierarchy, forcing extensions increase its width (see, e.g., the answer to this question where's it's said that forcing extensions builds sideways instead of upwards; or see the last paragraph of John Burgess's review of a number of arti... |
$$f\in C[a,b] $$If given that for every $x\in[a,b]\text{ there exists } y\in[a,b]$ such that:
$$|f(y)| \le \frac{1}{2}|f(x)| $$
Prove that there exists $ c\in[a,b]$ such that :
$$f(c)=0$$
What I've done so far :
We will build a sequence in this way:
$$\text{take } x_1\in[a,b]$$
for this $x_1$ there exists $y_1$ such t... |
Let $X$ be a Hausdorff topological vector space over $\mathbb{K}$. Suppose $W$ is a convex subset of its topological dual $X'$. How to prove that if for any $x\in X\setminus\{0\}$ set $\{f(x):f\in W\}$ is dense in $\mathbb{K}$ then $\overline{W}^{\sigma(X',X)}=X'$ ?
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Is the following statement true or not?
Let $X$, $Y$, $Z$ be $3$ events in the same sample space such that $p(X)$, $P(Y)$, $p(Z) > 0$ and every pair of these events is independent. Then $p(X \cap Y \cap Z) > 0$.
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I'm trying to solve this integral with trigonometric substitution but am having a ton of trouble:
$$\int\limits_{0}^{a}{\frac{dx}{(a^2+x^2)^{\frac{3}{2}}}}$$
I tried $x=a\tan{\theta}$ and thus $dx=a\sec^2{\theta}$ but I cannot get anywhere.
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