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Consider the matrix
$$
A =
\begin{pmatrix}
0.1 & 0.3 & 0.4 & 0.2 \\
0.2 & 0.4 & 0.0 & 0.4 \\
0.0 & 0.3 & 0.5 & 0.2 \\
0.5 & 0.3 & 0.2 & 0.0
\end{pmatrix}.
$$
Note since
$$
A^2 =
\begin{pmatrix}
0.17 & 0.33 & 0.28 & 0.22 \\
0.30 & 0.34 & 0.16 & 0.20 \\
0.16 & 0.33 & 0.29 & 0.22 \\
0.11 & 0.33 & 0.30 & 0.26
\end{pmat... |
In the book "Word processing in groups" by Epstein et al. (p.28-29), the definition of generators begins with the following sentence:
Let $G$ be a group, $A$ an alphabet and $p \colon A \rightarrow G$ a map, which need not be injective.
My question is whether or not he assumes $A \subset G$. The confusion arises beca... |
I am trying to solve the following exercise:
Let $f$ be integrable. Assume that $\int_A f d\mu = 0$ for every measurable set $A$. Prove that $f = 0$ a.e. [$\mu$].
I have the following proof but it seems to me too simple to be true. What is wrong with it?
For any $a>0$, define $W_a:=\{w|f(w)\geq a\}$. Now we have:
$$\in... |
We consider a r.v. $X$ that satisfies $0 \leq X \leq 1$ a.s. and a sample of $n$ i.i.d. random variables $X_1,\dots, X_n$ with the same distribution as $X$. We denote by $\mu= E[X]$ and we let $\lambda >0$.
I need to use convexity of the exponential function to show that the following inequality holds a.s.: $$e^{\lambd... |
I have the following problem. I'm struggling a little bit with the expression $P(X \in A)$. My problem is that $A$ is a set, whereas $X$ is a function. I can not really related this two items.
Here are some related definitions from the textbook I use.
Sample space: $\Omega$
Outcome: $\omega$
Event: $A, (A \subset \... |
Suppose that $X$ and $Y$ are independent and identically distributed:
$$P (X = k) = P (Y = k) = ρ (1 − ρ)^k$$
for $k = 0, 1, \dots$ and let $Z := X + Y$. Find the joint distribution of $(X, Z)$ and find the conditional distribution of $X$, given $Z = n$.
I need a little help just setting this up.
If I understand ... |
When I look up why $\mathbb{R}$ and $\mathbb{Q}$ are not homeomorphic, almost all the answers just say something along the line of "Because, Cardinality" and then ends there.
Can someone provides the reason or the proof as to why cardinality would matter?
|
Let $k$ be a field (finite if you'd like), and let $f:A\to B$ be a map of graded, commutative $k$-algebras. Suppose further that $A$ is finitely generated and choose a presentation
$$A:=k[x_1,\ldots,x_n]/I$$
with $\deg{x_i}=d_i$. If $B'\subset B$ is a graded subalgebra, is there an algorithm for determining a presentat... |
How does one prove that $A⊆B$ means that $A∪B=B$ ? I can understand it in my head but I don't know how you'd put down in logic notation.
|
Let $A$ be a unital C*-algebra. Then we know that $M_n(A)$ is also a C*-algebra.
Let $x=[x_{ij}]\in M_n(A)$. I want to prove that if for every state $\phi$ on $A$ and for every $\{y_1,...,y_n\}\subset A$ we have $$\phi\big(\sum_{i,j}y_i^*x_{ij}y_j\big)\geq 0,$$ then this implies that $x=[x_{ij}]\geq 0$ in $M_n(A)$.
S... |
How can I show the points of continuity of the following function
$$f(x) =
\begin{cases}
2x, & \text{if $x \in \Bbb Q$} \\[2ex]
x+3, & \text{if $x \in \Bbb I$ }
\end{cases}$$
I am having problems trying to solve this kind of exercise
:C
|
Given the set $\mathbb{C}^*\times \mathbb{C}$ with the group structure given by $(x,y)\cdot (z,w) =(xz, zy+wx) $. How can I check that this is reductive or not? ?
I think that its maximal normal solvable subgroup is itself, how can I see the identity component of this group?
Thanks
|
I have the following limit
$$\lim_{n \to \infty} 2^{2n+3}\left(\sqrt[3]{8^n+3}-\sqrt[3]{8^n-3}\right)$$
I don't know with what I should multiply the term.
I know that if I have for example
$$\lim_{n \to \infty} \sqrt{n-1}-\sqrt{n+3} $$ one way to solve it is to multiply the term with $\frac{\sqrt{n-1}+\sqrt{n+3}}{\sqrt... |
Given monoid ($\Bbb{Z}$124, ⋅ ) and a parametric equation with parameter $a$ where
$ax+2=5(x+a)-1$
Give the number of parameters $a$ belong $\Bbb{Z}$124 for which the above equation has precisely one solution ?
i tried to solve it by using the euclidean algorithm but i dont know how to calculate it with the parameter $... |
I came across the following question in a calculus book: For the function $$f(x)=1-\sqrt{1-x^2}$$ show that it is continuous on the interval $$-1≤x≤1$$ The solution in the book showed that the one sided limits were equivalent to f(x) at the points x= -1 and 1. It also showed that there exists a limit at the point c tha... |
How would you find the limit of this function:
The n value is always even (this is for a problem and it would not make sense for n to be an odd number)
Also, how could I plot this with n as the x value? I've tried to do it with an excel spreadsheet, but I just can't find the way to do it. I've done the first 10 even v... |
Suppose that $Y$ has the binomial distribution, $Bin(20, 0.25)$ and conditioned on $Y$, a random variable $X$ that has the binomial distribution, $Bin(Y, 0.5)$.
How can I derive the $k$th moment of $X$?
I know the general case:
$E(X)=∑_{x∈Ω_X}x^kP[X=x]$, and I know that just for one binomial random variable, $E[X]$ is... |
So the question is: given that you roll $10$ dice, what is the probability of the sum of the total dice rolls adding up to $57$?
I know that there are three ways to do this:
Seven die rolls must be $6$ with three $5$s
Eight die rolls must be $6$, one die roll must be $5$ and one must be $4$
Nine die rolls must be $6$... |
I have two equations:
$$N_sc_s + p_s L =N_s \gamma^sw$$
and
$$\frac{(1+g)N_s c_s}{\gamma} + \frac{(1+h)p_s L}{(1+n)\gamma} = N_s \gamma^s w$$
Where the variables are $c_s, p_s$. Obviously One way for both these equations to be true is for:
$$ \tag{1} N_sc_s = \frac{(1+g)N_s c_s}{\gamma} \text{ and } p_s L = \frac{(1+h)... |
Recall that $p \rightarrow \sim q$ is equivalent to $p \land \sim q$, how can this equivalence be used as an explanation for how to use proof by contradiction.
I'm having a hard time answering this question because I do not understand the role of $p \land \sim q$. So i know $p \land \sim q$ means $p$ and not $q$ but ho... |
Let $\sigma$ be a permutation. Prove that there exists some permutation $\rho$ such that $\sigma^{-1} = \rho \sigma \rho^{-1}$
I tried playing with it a little but to no avail.
|
I don't really know where to start. I had this approach in mind:
take any integer, let's name it $k$, and show that the limit on the left side of $k$ equals to $k-1$, and that the limit on the right side of $k$ equals to $k$, thus the limits are different, and so the floor function has no limit.
|
I wish to find the inverse of $\dfrac{x}{\|x\|}$, where $x \in \mathbb{R}^2$
Let's do this.
Let $$y_1 = \dfrac{x_1}{\sqrt{x_1^2+x_2^2}}$$
$$y_2 = \dfrac{x_2}{\sqrt{x_1^2+x_2^2}}$$
Then $$y_1 = \dfrac{x_1}{\sqrt{x_1^2+x_2^2}} \Rightarrow \dfrac{x_2^2}{x_1^2} = \dfrac{1 - y_1^2}{y_1^2}$$
Similarly
$$y_2 = \dfrac{x_2}{\sq... |
So I came up with this example when I was teaching:
consider the IVP
$$
y'(x) = xy-x-5y+5, y(0)=1.
$$
The standard approach is to separate variables: $y'(x) = (x-5)(y-1)$, which allows me to integrate $\int \frac{dy}{(y-1)} = \int (x-5) dx$, at this point I have $y-1 = e^C e^{x^2/2-5x}$ for some constant $C$, so to sat... |
The cargo space of a bulk carrier is 60m long. The shaded part of the diagram represents the uniform cross-section of this space. It is shaped like a parabola with equation ${{1\over 4}x^2, - 6 \le x \le 6}$ between the lines ${y = 1}$ and ${y = 9}$. Find the area of the cross section and hence find the volume of ca... |
I have been trying to evaluate
$$\lim_{x\to 0}{\frac{\sin^2x}{2x^2}}$$
Finally, I used the L'Hospital's Theorem and I got the answer $1/2$, but I wonder if there is a way to solve this without this.
I also tried using Squeeze Theorem, but my boundaries were approaching different numbers.
|
I know how to calculate a standard deviation when given a set of values. You must calculate the mean, then square the difference of each value between the mean to get the variance. Lastly, the standard deviation is the square root of the variance. However, I have a problem where I am given the following values:
Mean: 3... |
Let $k$ be a field (finite if you'd like), let $(I,\le)$ be a finite directed poset with $|I|=n$, and let $(A_i,f_{ij})_{i\le j\in I}$ be an inverse system of finitely generated, graded, commutative $k$-algebras. Suppose further that we fix a presentation for each $A_i$:
$$A_i:=k[x_{i1},\ldots,x_{in_i}]/I_i$$
Is there ... |
I have a linear map $A:\Bbb{R}[X]\mapsto \Bbb{R}$ such that $A(P)=P(1)$, for the $p-$norm : $\Vert P\Vert=\bigl(\sum_{i=1}^n\vert a_i\vert^p\bigr)^{1/p}$ where $p\in[1,+\infty].$
For the cas where $p=1$ it's continuous (triangle inequality), anyway for the case $p>1$ I would like to prove that is not continuous.
For ... |
My teacher asked us to illustrate, using a scientific poster the search for the right proportion that symbolizes the golden ratio. The problem statement is the following:
What about today that aspiration to the agreement between reason and things, between reason and men? What correspondence, quantifiable or not, what m... |
For every $\alpha\in{\rm Aut}(G)$, I've defined $A:H\rightarrow H$ by $$A(h)=\phi(\alpha(\phi^{-1}(h)))$$ where $\phi$ is an isomorphism from G onto H, and I've shown that $A\in {\rm Aut}(H)$. What I can't figure out is how to show that every $B\in {\rm Aut}(H)$ can be expressed as $\phi\beta\phi^{-1}$ for some $\beta... |
Is there a good database of unsolved problems in graph theory?
|
Let $A$ be C*-algebra then we know that $M_n(A)$ is also a C*-algebra.
Let $\rho:M_n(A)\rightarrow B(K)$ be a *-representation of $M_n(A)$ on some Hilbert space $K$. Then there exists a *-representation $\pi:A\rightarrow B(H)$ of $A$ on a Hilbert space $H$ such that
1) $H^{(n)}=K$, that is, the $H\oplus ... \oplus H$... |
I am trying to prove this statement with strong induction, but I'm a little lost on the inductive step.
Proposition: Let P(n) be the sentence ‘any collection of n coins can be obtained using a combination of 3¢ and 5¢ coins.’
P(n) is true for all integers where n ≥ 14.
Proof:
Prove P(14) is true:
P(14) is true, because... |
I solved an indefinite integral that gave me the following result:
\begin{equation}
f(u) = -\sum_{k=0}^{n}\binom{n}{k}\frac{\mathrm{sgn}(u)^{k+1}\, b^{n-k}}{2\, a^{\frac{k+1}{2}}}\, \Gamma\left(\frac{k+1}{2},\, au^{2}\right)+C
\end{equation}
where, $u\in (-\infty, \infty)$, $a\geq0$, $b\in (-\infty, \infty)$, and $n$ i... |
A function $f$ is defined as a set of ordered pairs $(x, y)$ such that $(x,b), (x,c) \in f \Longrightarrow b=c$. Since $y$ is determined uniquely by $x$, it is customarily denoted $f(x)$.
One difficulty I am having with this definition is that it does not specify the codomain. In particular, $f:\mathbb{R} \rightarrow ... |
If $z$ is constructible, then its minimal irreducible polynomial has a degree a power of $2$. Does the polynomial have to be defined over the rationals?
I am asking this because we can equivalently define a constructible number if the field extension $\mathbf{Q}(z)/\mathbf{Q}$ has dimension a power of $2$. Why do we ... |
I want to prove that $n! = O(n^n)$ based on the definition of big-$O$.
I find it pretty easy to show that $n! = O(n^n)$ by simply showing that $n (n-1) < n \cdot n \ldots$ etc. However I can't figure out how to prove this based on the definition of big-$O$. How can I prove this based upon the definition?
|
I, I am trying solve the following integral $$\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$$
Teachers teached me that I can solve the integral $$\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}=\frac{d^2}{d\lambda^2} \int_{0}^{\infty} \frac{x^{\lambda}}{1+x^2}$$
I get to the second integral that $$ \int_{0}^{\infty} \frac{x^{\l... |
Let $e_1 = (1, 0)$ and $e_2 = (0, 1)$. Assume that $f : \Bbb R^2 → \Bbb R^2$ is an isometry of the plane fixing $(0, 0)$.
Let $f(e_1) = (a, b)$ and $f(e_2) = (c, d)$, and let $A = \begin{vmatrix}
a & c \\
b & d \\
\end{vmatrix}$. Prove that $\det A=±1$.
So far from what I can gather, since $f$ is an isometry, it pr... |
Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. H... |
I'm not sure if this is true or not. I was going to prove it via induction. Here is what I have so far.
Base Case (x=0): $\lfloor 0 + 0\rfloor = 0 = 0 + \lfloor 0 \rfloor$
Induction Hypothesis (IH): $\lfloor x + x\sqrt3\rfloor = x + \lfloor x\sqrt3 \rfloor$
Induction Step: $\lfloor x + 1 + (x + 1)\sqrt3\rfloor = \lfloo... |
It is well known that any positive linear functional $L$ on the spase $C_c([a,b])$ of functions continuous on an interval $[a,b]$ with compact support can be written as
\begin{align*}
L(x)=\int_a^bx(t)d\mu(t),\qquad x\in C_c([a,b]),
\end{align*}
where $\mu$ is some proper Radon-measure. The question for me now arises,... |
Consider all finite binary words that have the same number of zeroes as ones (ex: 0101). How can we show that this is countable?
I have tried listing some words in lexicographic order, but I don't see any pattern. Any recommendations on how to approach this problem?
\begin{array}{|c|c|c|c|}
\hline
1& 2 & 3 & 4 & 5 & 6 ... |
I know that in order to find a subsequence that is a basis of a subspace is to check whether the given vectors are linearly independent and whether they span the subspace.
However how can I find the dimensions of the subspaces of a vector space K^n ?
|
I've been given a task that reads:
Prove that given formulae is correct with the use of set theory
axioms: $(\forall a)(\exists b)(\forall c)((c \in b) \iff (\exists d \in a)(c \subset d))$
Now, how I interpret it: For every set a there exists a set b, which is a power set of some element from set a. Some friends ... |
Let $X$ be a random variable. Show that $\frac{X}{n}$ converges to zero in probability and almost surely, as $n \rightarrow \infty$.
I am sort of confused by this question since I only learnt a definition of convergence for a sequence of random variables. Am I reading something wrong ?
|
Why is this true? $$\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$$
|
I want to prove or disprove that $3n^3 +n^2\log(n) = Θ(n^3)$. I'm aware that I will need to either prove or disprove both big-o and big-Ω to prove or disprove Θ. I am simply struggling to do so. Help appreciated.
|
Could someone help me out for a second, please? See here's the problem:
9 greeks, 17 finns, 7 russians, 11 chinese and 8 swedish students are studying in groups. A group can consist of one or more persons. If a group has two or more persons of the same nationality, it must also have at least one person representing ano... |
Let I have a sequence as follows,
$$a_1,a_2,a_3,a_4,a_5...$$
where $(a_2-a_1),(a_3-a_2),(a_4-a_3),... $ are in arithmetic progression.
How can I find the Nth element ($a_N$) of the series $a_1,a_2,a_3,a_4,...,a_N,..$
I am new in AP Series. Please help.
|
I am reading about Ridge Regression in Machine Learning (in particular, the calculation of the empirical risk w.r.t. the square loss function) and do not understand the following step:
$$\frac{1}{n}\sum_{i=1}^{n}(\langle\vec{x}_i,\vec{w}\rangle-y_i)^2 = \frac{1}{n}(X\vec{w}-\vec{y}_i)^T (X\vec{w}-\vec{y}_i)$$
I tried t... |
Let us say that x is a set of binary numbers $$x = \{0, 1, 1001\}$$ Am I correct that $x^R$ is equal
$$x^R = \{0,1,1001\}$$
or is it:
$$x^R = \{1,0,0110\}$$
What I mean by that is: do we create a reversal for binary by a switch between zeros and ones, or do we do it by putting the whole set in diverted order?
|
I'm trying to compute the minimal bounding box of an arc segment so when it's time to render it, I only have to examine pixel coordinates within a minimal rectangular region. The code below covers the case when $0 \leq \theta_0 \lt \pi/2$. Was wondering if there is some sort of rotational symmetry so that I don't nee... |
We have that $N$ and $X_1, X_2, \dots$ are all independent. We also have $\operatorname{E} [X_j] = \mu$ and $\operatorname{Var}[X_j] = σ^2$.
Then, we introduce an integer–valued random variable, $N$, which is the random sum such that:
$$Z = \sum_{j=1}^{N+1}X_j.$$
Assuming that $N$ is distributed $\sim\mathrm{Poisson}(\... |
I have the following problem I am confused about:
Let $x,y\in \Bbb R, x\ast y=xy+x+y$
Is $\Bbb R$ a group? I wrote $(x\ast y)\ast z= x\ast (y\ast z)$, then calculated it, associativity did not hold. Is this true?
My mistake apparently is that I wrote $xy\ast z+ x\ast z+ y\ast z$.
Secondly, I am asked to prove that $\Bb... |
In the framework of my science fair project I am working on conic sections in different metric spaces. What simple topological properties/operations and so can I explore on them?
Edit:
To clarify, I am considering these spaces - Chebyshev, discrete, maybe a Paris space (wiki).
I define conic sections via their distan... |
So I've got a bijection. It clearly has an inverse, but how exactly do I prove that the inverse is a linear map as well?
Suppose that the linear map $T:U\to V$ is a bijection. So $T$ has an inverse map $T^{-1}:V\to U$. Prove that $T^{-1}$ is a linear map.
|
I have an exercise in my book that states
Show that the set of positive elements in $\mathbb{Z}$ can be identified with $I\mathbb{(N)}$
we finally define a mapping $I$ from $\mathbb{N}$ to $\mathbb{Z}$ by $$ I(a) := [(a+1,1)]$$
The mapping $I$ defined above is an order preserving injection, i.e. it is injective and ... |
My question is very simple: Suppose $u,v:(a,b)\subset \mathbb{R} \to \mathbb{R}^+$ solve
\begin{equation}
(p(x)u'(x))'=-q(x)f(u(x))
\end{equation}
\begin{equation}
(p(x)v'(x))'=-r(x)g(v(x))
\end{equation}
where $p>0$ and sufficiently smooth and $qf \geq rg \geq 0$ are sufficiently smooth as well. Suppose moreover that ... |
Suppose you have a real random variable $X$ that has probability distribution $f_X$ meaning
$$ P(\alpha \le X \le \beta) = \int_{\alpha}^{\beta} f_X(x) \ dx $$
Now assume $\Phi(f_X)$ is also a probability distribution where $\Phi$ is some function-of-functions of $f_X$. How does one find a $Q$ such that $Q(x)$ is a... |
I was recently thinking about prime numbers, and at the time I didn't know that they had to be greater than $1$. This got me thinking about negative prime numbers though, and I soon realized that, for example, $-3$ could not be prime because $3 \cdot (-1) = -3$. In some sense $-1$ could be though because its only facto... |
Let's say there is this conditional statement:
If I am in Paris, then I am in France.
So, p = 'I am in Paris', and q = 'I am in France'
I do not understand when p and q are false, how would that translate to?
Would it not be the same as inverse of conditional statement?
I would be grateful if you guide me on my misinte... |
An operation $*$ is defined on the set $\Bbb{Z} \times \Bbb{Z}$, ie. the set containing all pairs of integers by:
$$
(u,v) * (x,y)=(u+x,v \cdot y)
$$
if $(\Bbb{Z} \times \Bbb{Z}, *)$ is not a group what is an equation that does not have a solution?
i have found the the identity element (0,1) and the inverse element (-u... |
So I have been wondering. I have heard many times statements like "if we find the end of $\pi$ then we might be in a virtual reality" or "new computer can calculate $X$ digits of $\pi$".
My question: $\pi$ is irrational so how exactly does one find more digits to it? I understand that it is a ratio.... But it's not l... |
Exercise 2.3 from "Introduction to Mathematical Cryptography"
Let $p$ be a prime and $g$ an element in $\mathbb{F}_p^*$ of order $r$.
(a) Suppose that $x = a$ and $x = b$ are both integer solutions to the equation $g^x\equiv_p h$. Prove that $a\equiv_r b$.
(b) Prove that $\log_g{(h_1h_2)}\equiv_r \log_g{(h_1)} + \log_... |
Find all the numbers $a$ such that the number $an(n+2)(n+4)$ is an integer for all $n \in \mathbb{N}$
It's trivial to see that if $a$ is irrational, we get no solution.
Thus $a \in \mathbb{Q} \Rightarrow a = \dfrac{p}{q} ~ (*)$ where $p, q \in \mathbb{Z}$ and $\gcd(p,q)=1$. Then
$$\begin{split} an(n+2)(n+4)=k &\stac... |
Let $X_1, X_2, \ldots$ be a sequence of independent random variables with uniform distribution on $[0, e]$. Let $R_n:=\prod_{k=1}^n X_k$. By Kolmogorov's zero–one law $(R_n)$ converges with probability $0$ or $1$. I want to determine this probability.
Observe that $\log(R_n)=\sum_{k=1}^n \log(X_k)$ with $\mathbb{E}[\lo... |
Is there any known explicit bijection between these two sets?
I know it can be proved that such bijection exists using two injections and Schröder–Bernstein theorem, but I wanted to know whether some explicit bijection is known. I failed to find any except ones constructed awkwardly from the Schröder–Bernstein theorem... |
Consider a vector space $V \cong \mathbb{R}^n$ with an operator $I \in O(n)$ satisfying the property $I^2 = -Id_{V}$. See Linear Complex Structure for context. I want to show that $V$ has real dimension $2n$ (even dimension).
To do this, I was given the hint of working with $W = span\{x,Ix\}$ for a vector $x \in V$. It... |
My question arises from this definition: Poincare group is the group of Minkowski space-time isometries. Which means that it leaves the space-time intervals unchanged.
Now here is my understanding:
Isometry is a (diffeomorphic) map (from a manifold to itself) that preserves the metric (which accepts two vectors as the ... |
For any given $k$, show that an integer $n$ can uniquely be represented as: $$n={m_1 \choose 1} + {m_2 \choose 2} + \cdots + {m_k \choose k}$$ where $0 < m_1 < m_2 < \cdots < m_k$.
My hunch is to find what the $m_i$'s must be, since the representation is unique.
Any tips would be greatly appreciated.
|
Why is it that when a vector-valued function $r(t)$ is continuous at some time $t$ then $\|r(t)\|$ is also continuous at that time $t$, but the converse is not true? That if $\|r(t)\|$ is continuous at time $t$, then $r(t)$ is discontinuous at time $t$.
|
What is the Laurent series for the function $f(z)=\dfrac{1}{\sin(\frac{1}{z})}$ at the point $z=\dfrac{1}{\pi}$ and $z=0$?
|
Integrate $$\frac{e^x -2}{e^{x/2}}$$
This is my calculation:
but it is wrong....
|
...where k is a positive integer. The Big Oh case is not so hard. But how do I show that $1^k+2^k+\cdots+n^k$ is $\Omega (n^{k+1})$?
|
I have the inequality
$\frac{n}{n+1} < \frac{n+1}{n+2}$
I'm not sure how to go about proving it. I've started by testing with n = 1, which results in
$\frac{1}{2} < \frac{2}{3}$ which is true
I then assume true for n = k and have to prove that it is true for n = k + 1, but I don't know how to start manipulating
$\frac{... |
I'm writing some notes on Spectral theory and I would like to make a simple example finding the generalized fourier coefficient of a function in respect of two different bases. I was thinking about Hermitian functions and the usual trigonometric functions, i.e.
For the Hermitian series the coefficients are labelled $h... |
Evaluate the Surface Integral $xyz$ $dS$ where $S$ is the surface defined by $2y=\sqrt{9-x}$, $x>0$, and between $z=0$ and $z=3$.
Don't even know where to start with this question.
|
I'm self-learning multivariable calculus and am using double integration to compute the average value of $f(x,y)$ over some region. I'm trying to solve the following simple problem using two different approaches, with double integration and with basic intuition only, but am getting slightly different results and am won... |
I have trouble finding the value of the following limit:
$$\lim_{n \to \infty} \sqrt{n} \sin\left({\sqrt{n+3}-\sqrt{n-2}}\right)$$
For now I have rewritten the term into:
$$ \lim_{n \to \infty} \dfrac{\sin\left({\sqrt{n+3}-\sqrt{n-2}}\right)}{\large \frac{1}{\sqrt{n}}}$$
Now I have a limit of type $\large \frac{0}{0}$ ... |
I'm using the proof on this page, except with $q$ instead of $p$ on the left side. The misconception of the proof is that $q$ has to be a prime number. I found this using $n = 6$, which gets me $q = 1 + (2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13)$, which results in 30,031, which is not prime. Why doesn't $q$ have to ... |
Let $S$ be a collection of subsets of $X$ and $\mu : S \to [0, \infty]$ a set function. Is every set in $S$ measurable with respect to the outer measure induced by $\mu$
Here is how we defined outer measure induced by $\mu$
Let $S$ be a collection of subsets of set $X$ and $\mu: S \to [0,\infty]$ a set function. Define... |
I'm faced with the following nonlinear 2nd order system of ODEs:
$$
\phi''(r)+\frac{4r^3-1}{r^4-r}\phi'(r)+\frac{r^2 h(r)^2+2r(r^3-1)}{(r^3-1)^2}\phi(r)=0, \\
h''(r)+\frac{2}{r}h'(r)-\frac{2r\phi(r)^2}{r^3-1}h(r)=0.
$$
Here, $\phi$ and $h$ are real-valued functions of $r\in[1,\infty)$ and $h(1)=0$ and $\phi(1)$ is fini... |
Let $s$ be a collection of subsets of $X$. Assume that $\mu$ and $\nu$ are two measures on $M(s)$. Prove that if $\mu(A) = \nu(A)$ for all $A \in s$, then this also holds for all $A \in M(s)$, i.e., $\mu=M$.
I think that I can start the solution with the monotone class lemma and continuity measure. Do you have any sugg... |
Is $\Bbb R_{+}$ under multiplication cyclic? Would it suffice to show that this group is isomorphic to a cyclic group? For example, $\Bbb Z_+$ is cyclic under addition, $\Bbb R_{+}$ is isomorphic to it, then $\Bbb R_{+}$ is cyclic too?
Thanks!
|
This was inspired by this post. Define,
$$a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$$
$$b_n = \sum_{k=0}^n \binom{-\tfrac{1}{2}}{k}\big(-\tfrac{1}{2}\big)^k$$
where $_2F_1$ is the hypergeometric function and binomial $\binom n k$. The first few numerators $N$ are,
$$N_1(n) = \color{brown}{1, \... |
I was doing my calculus homework, and one of the questions asked for the first and second derivative of $ye^y=x$, I did the computations and arrived at $-(x+1)^{-2}$, which was a lot neater and simpler than I expected, I am particularly perplexed by the fact that this expression doesn't have a $y$ in it, while my first... |
I recently learned that one can think of regression as a projection of a vector in a high dimension space onto the other vector. I tried implementing this and got it to work:
C=(dot(A,B)/norm(B)^2)*B; %projecting A onto B (A is the independent variable)
coef=regress(A,b);
D=B*coef; trying to obtain the same result as C... |
The problem statement is,
Find a linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satisfying $4x_1-3x_2+x_3=0$ is the (i) null space of $T$ (ii) range of $T$.
For (i),
I found out the basis of the null space for the system,
$$\begin{pmatrix}4&-3&1\end{pmatrix} \begin{pmatrix}x_1\\... |
I am reading MacPherson's paper "Chern Classes for singular varieties".
Proposition1 : There is a unique covariant functor from compact complex algebraic variety to abelian groups whose value on a variety is the group of constructible functions from that variety to the integers and whose value $f_*$ on a map $f$ satis... |
Does anyone know a counterexample to show that a weighted sum of convex sets is not necessarily convex, unless our coefficients are positive?
A weighted sum for me is defined as: $$\alpha C_1 + \beta C_2 = \{ y : y = \alpha x_1 + \beta x_2, x_1 \in C_1, x_2 \in C_2, \alpha , \beta \in \mathbb{R} \}$$.
It is easy to pro... |
I'm watching a video about the Jacobi Theta Function on YouTube.
At around 6:14 in the video, he has shown that...
$$\vartheta (x) =
\sum_{n\in \mathbb Z} e^{-\pi {n^2} x} =
\sum_{k\in \mathbb Z} {e^{-\pi {k^2} {1\over x}}}
\int_{-\infty}^{\infty} {e^{-\pi x \big( y+ i {k\over x}\big)^2}} \space dy$$
I understand the s... |
if $-a\leq b\leq a$, then $|b|\leq a$
I started by showing that $-|b|\leq b\leq |b|$, but then I can only see that $a$ can be equal to $|b|$, how do I show $|b|\leq a$?
|
I am having trouble with my homework problem, it says:
Suppose that $n$ is a positive integer and that $x > 0$. Show that
$$\left(1+\frac{x}{n}\right)^n < e^x.$$
I have proved the base case, but I am unsure how to proceed with the induction step. The question continues:
and, if $x < n$, then
$$e^x < \left(1-\... |
Let $M = \mathbb{N} \ \mathbb{x} \ \mathbb{N}$. We define the following relation on $M$. Let $(a,b)R(a',b')$ iff $a + b'=a'+b$
We define the set of intergers $\mathbb{Z}$, to be the set of equivalence classes of $M$ under the equivalence relation above. We try to define the arithmetic operations of addition and mult... |
Let $k \leq n$ denote a pair of fixed but arbitrary natural numbers.
Definition 0. Write $\varphi$ for the unique $\mathbb{R}$-linear function $$\Lambda^k\mathbb{R}^n \rightarrow \mathbb{R}$$ such that if $e$ is an element of the standard basis for $\Lambda^k\mathbb{R}^n,$ then $\varphi(e) = 1.$
Definition 1. Given a ... |
I need help understanding and solving this problem.
Prove or give a counterexample: If $A$ is a nonsingular matrix, then $\|A^{-1}\| = \|A\|^{-1}$
Is this just asking me to get the magnitude of the inverse of Matrix $A$, and then compare it with the inverse of the magnitude of Matrix $A$?
|
Define: $S^1 = \{(x,y): x^2+y^2 = 1\}$
Then wish to show $S^1$ is complete
Attempt:
Let $(z_n)$ be a Cauchy sequence on $S^1$, $(z_n) = (x_n, y_n)$
Then $S^1$ is complete if $\forall \epsilon > 0, \exists N, d(z_n,z_m) = \sqrt{(x_n - x_m)^2 + (y_n - y_m)^2}< \epsilon, $for $n, m \geq N$
However, $\sqrt{(x_n - x_m)^... |
Let $M$ be an $R$-module. I want to show only using the definition of noetherian that if $N$ is a noetherian submodule of $M$ such that $M/N$ is noetherian, then $M$ is noetherian.
I know that if $M_1 \subset M_2 \subset \cdots$ is a chain of submodules in $M$, then since $N$ and $M/N$ are noetherian, the chains
$$M... |
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