text stringlengths 83 79.5k |
|---|
H: Functions satisfying differential equation of the Weierstrass elliptic function $\wp$
The Weierstrass elliptic function $\wp$ satisfies the following differential equation: $${\wp'}^2 = 4 \wp^3 - g_2 \wp - g_3$$
Which other functions do?
Are the solutions always elliptic functions?
AI: Given theorem on existence an... |
H: How to solve $x^2 + \ln(x) = 0$
I was just investigating $y = f(x) = e^{-x^2}$ and then went ahead to plot $x=f(y), y=-f(x), and x=-f(y)$, and what I got was interest rounded square shape, and I think we can calculate this area using integration.
However to get the bounds I must solve the equation $x^2 + \ln(x) = 0... |
H: $L^p$-norm diverges for a sequence of functions
Let $\{f_n \}_n$ real valued functions such that:
\begin{cases}
f_n \geq 0 \\
\int_{\mathbb{R}} f_n =1\\
\forall \delta >0: \lim_{n \rightarrow +\infty } \int_{|t|>\delta}f_n(t)dt = 0
\end{cases}
Show that, for every $p>1$:
$$\lim_{n \rightarrow + \infty} ||f_n||_{L^... |
H: What is the motivation for sequences to be defined on natural numbers?
Possible duplicate: Sequences with Real Indices
I am trying to understand the motivation for various definitions in real analysis. Take for instance the definition of a sequence where it is defined as a function from natural numbers to that of r... |
H: Finding curvature for $y = \sin ( -2x )$ at $x =\pi/4$
The answer is 4, but I got 1. I said $r(t)=<t,\sin(-2t)>$ and the $|r^\prime(\frac{\pi}{4})|$ is equal to 1. I also got 1 for $T^\prime(\frac{\pi}{4})$ but none of this information matters because I got the question wrong, can someone help me out please? thanks... |
H: Split area under the curve into two halves
Draw a vertical line say $x=t$ on curve $y= \sqrt{x}$ between $x=0$ and $x=a$ such that area under the curve from $x=0$ to $x=t$ equals area under the curve from $x=t$ to $x=a$.
I am looking for a generalized approach for other functions such as $y = \ln(x)$
AI: Hint
Find ... |
H: Limit of sum of probability distributions
Suppose that $\lbrace X_i:i=1,2,...\rbrace$ is i.i.d. with density function $f$, finite $\sigma$ and mean $\mu <0$. Prove or disprove that
$$\mathbb{P}(\sum_{i=1}^n X_i \geq 0) \to 0 $$
as $n$ tends to infinity.
Note that
$$\mathbb{P}(\sum_{i=1}^n X_i \geq 0) = \mathbb{P}(\... |
H: What is the opposite side of convergence of function?
Assume we define $f(x)$ on the domain of $x\ge 0$. Given the following statement,
$$\lim_{x\rightarrow\infty} f(x) =0.$$
It is known that it is equivalent to
$\forall \epsilon>0, \exists X>0$, if $x\ge X$, $|f(x)| \le \epsilon$.
I want to know the negation state... |
H: uniformly converge of a serie with given conditons
i have this question about series.
So i came along a question that started with :
$(f_{n})_{n}$:[-1,1]->$R^{+}$ is a row of continuous functions. The serie $\sum{f_{k}(x)}$ from k=0 to $\inf$ will converges and thefunction f:[-1,1]->$R^{+}$:x->$\sum(f_{k}(x))$ is c... |
H: Writing $0$ with $6$ significant figures concisely
I have some calculations that result in exactly $0$, calculated to 6 significant figures. However, I do not want to write $0.00000$ for each of these calculations, as it makes it look messier. How can I write it more concisely, while still retaining the number of s... |
H: What is the condition to split integral in Lebesgue theory?
Let $f:[0,1]^2\rightarrow \Bbb R$ be defined by
$$f(x,y)=1/y^2 \text{ if }0\leq x <y<1$$
$$f(x,y)=-1/x^2\text{ if }0\leq y <x \leq 1$$
$$f(x,y)=0 \text{ otherwise}$$
If can show that $$\int_{0}^{1}\int_{0}^{1}f(x,y)dxdy \neq \int_{0}^{1}\int_{0}^{1}f(x,y)d... |
H: Finding the upper bound of a linear functional
I have a linear functional $\phi_n:(C[0,1],\Vert{\cdot}\Vert_\infty) \to \mathbb{R}, n\in\mathbb{N}$ defined by $$\phi_n(x)=\int_0^1t^nx(t)dt$$
I have to calculate an upper bound for $\Vert\phi_n\Vert$ which is independent from $n$.
I thought about using the Cauchy–Sch... |
H: Is countable infinity part of the natural numbers?
A hopefully relatively simple question: is countable infinity an element of the set of natural numbers? A related question here has been answered with the statement that (classic) infinity is not part of the natural numbers, but what about countable infinity?
Cont... |
H: Given that $f(x)$ is a polynomial of degree $3$, and its remainders are $2x - 5$ and $-3x + 4$ when divided by $x^2 - 1$ and $x^2 - 4$ respectively.
So here is the Question :-
Given that $f(x)$ is a polynomial of degree $3$, and its remainders are $2x - 5$ and $-3x + 4$ when divided by $x^2 - 1$ and $x^2 - 4$ respe... |
H: How to properly write a solution to the following inequality?
Trying after many years to refresh my calculus and stack at very basic.
I'm following Thomas' Calculus and here is the problem:
Find a domain and a range for the following function:
$$f(t)=\frac{2}{t^2-16}$$
Domain I've found:
$D\in{\{t|(-\infty,-4)\cup(... |
H: What is $q$ in the factorized form $a(x+p)^2 + q$ of a quadratic equation?
I can't put the following equation into $a(x+p)^2 + q$ form:
$$-4x^2 - 2x + 3$$
I've gotten as far as: $-4(x + (\frac{1}{4})^2)$, but what would $q$ be? I got $\frac{1}{4}$ by the fact that $p = \frac{b}{2}.$
AI: Take out the factor of $-4$ ... |
H: Fourier transform definition on $L^2$.
I'm trying to prove that $C^\infty_0(R^n)$ as the set of those smooth functions with compact support, is included in the set $X=\{f\in L^1(R^n) \ : \ \hat{f}\in L^1(R^n)\}$.
All because our professor said that one could define the Fourier transform on $L^2(R^n)$ by density of ... |
H: If $E(X)=15$, $P(X\le11)=0.2$, and $P(X\ge19)=0.3$, what can be $V(X)$?
If $E(X)=15$, $P(X\le11)=0.2$, and $P(X\ge19)=0.3$, which of the following is impossible ?
$V(X)\le7$
$V(X)\le8$
$V(X)>8$
$V(X)>7$
I know $V(X)=E(X^2)-E(X)^2$.
Please, any other hints??
AI: Notice that you have $P(|X-E(X)|\geqslant 4)$ and... |
H: Finding perpendicular lines in $\mathbb R^4$
Let $$g=\begin{pmatrix}2\\-5\\-3\\-3\end{pmatrix}+\mathbb R\begin{pmatrix}1\\2\\3\\4\end{pmatrix}$$ and $$h=\begin{pmatrix}1\\-3\\0\\-1\end{pmatrix}+\mathbb R\begin{pmatrix}2\\3\\4\\5\end{pmatrix}.$$
Find all lines that are perpendicular to both $g$ and $h$.
Find the sm... |
H: Circle with points on coordinate axis.
If a circle has points $A$ and $B$ which lie on the coordinate axis and $AB$ is the diameter of the circle, would it always form a perfect circle where the midpoint of $AB$ is the centre of the circle?
The question states the line $y = -3x + 12$ meets the coordinate axis at $A... |
H: Finding the joint probability density function of two independent random variables
Is there a way of determining the joint probability density function of two random variables? If we have two independent random variables, $X$ and $Y$ that both are uniform on [0,1], then how do one calculate the joint probability de... |
H: Evaluate $\lim\limits_{x \to \infty} \sqrt[n]{(1+x^2)(2+x^2)...(n+x^2)}-x^2 $
I'm trying to calculate:
$$T = \lim\limits_{x \to \infty} \sqrt[n]{(1+x^2)(2+x^2)...(n+x^2)}-x^2$$
Here is my attempt.
Put $x^2=\dfrac{1}{t}$ so when $x\to \infty, t \to 0$ and the limit become
\begin{align*}
T &= \lim\limits_{t \to 0} \s... |
H: Velocity damping using acceleration yielding incorrect results
(I am using Python to demonstrate this question)
I have two functions, x and y. They should apply a damping factor of k to velocity v over the time t. The first function works fine, and you can see that I am using the power operator to achieve an accura... |
H: Parabolic Bootstrapping
I started studying parabolic pdes. Often I come across an integral solution where the regularity is proven by a standard bootstrap argument or by standard parabolic results, but it is never explained how this works in detail.
For example, let $\Omega$ be an open, bounded subset of $\mathbb{R... |
H: Countably infinite Cartesian product of finite sets is infinite
Say that $A \neq \emptyset$, but $\emptyset \in A$. With the term "infinite set" I mean uncountable or countably infinite. I want to prove that
$A \times A \times \cdot \cdot \cdot A \times \cdot \cdot \cdot$
is an infinite set either if $A$ is finite ... |
H: How to study the monotonicity of $c_{n+1} = \sqrt{2+\frac{c_n}{3}}$ given that $c_1 =5$
I am studying the monotonicity of $c_{n+1} = \sqrt{2+\frac{c_n}{3}} = \sqrt{\frac{6+c_n}{3}}$, given than $c_1 = 5$
I proved by induction that $c_n > 0$.
My attempt
Lets make the hypothesis that:
$$ \frac{c_{n+1}}{c_n} = \sqr... |
H: Sum of two independent random variables (density)
I remember seeing the statement the sum of two independent random variables has a density if one of them has a density somewhere but forgot where it is. Is this statement true? If it is can someone provide me with a hint or a proof? Thanks!
AI: Let $X,Y$ be random v... |
H: 2 seemingly isomorphic groups
Take the following two groups:
$G_1$
$$\begin{array}{c|c|c|c|c}
\cdot & e
& a & b& c\\\hline
e & e & a & b & c \\\hline
a &a & e & c& b\\\hline
b & b & c & e ... |
H: Write the polynomial equation given information about a graph
With the following information, I am to write the equation of the polynomial:
Degree 3, zeros at $x=-2$, $x=1$, $x=3$, y intercept: $0,-4$
I know that the answer is: $f(x)=\frac{-2}{3}(x+2)(x-1)(x-3)$
If you look at my post history you can see that I nea... |
H: Proving that $ \sum_{k=1}^\infty \frac{k^{8} + 2^{k} }{3^{k} - 2^{k}} $ converges by the comparison test
I would like to prove that the following series converges:$ \sum_{k=1}^\infty \frac{k^{8} + 2^{k} }{3^{k} - 2^{k}} $ by comparing it with a series that I already know converges. One such series could be the geom... |
H: question about sequences convergent to $e$
Let $(x_n)$ be an increasing divergent sequence of positive reals. Then $(1+\frac{1}{x_n})^{x_n}\to e$, it is a standard theory, it is proven by comparing $x_n$ with $[x_n]$ and $[x_n]+1$ and we use a known fact that $(1+\frac{1}{n})^n$ is increasing and $e$ is its limit.
... |
H: Definition of geodesic not as critical point of length $L_\gamma$ [*]
Context of this question:
This question follows from a post Decomposition of a function and chain rule. and discusses on something different. Using calculation of variation we can find critical points of a function of a vaiable curve $\gamma$ wit... |
H: Is there a convention to interpret equalities of functions as series?
I am wondering about whether there is a default or standard interpretation of statements such as $$\sum_{n=1}^\infty f_n(x) = f(x)$$ or equivalently
$$\sum_{n=1}^\infty f_n = f$$
In some cases these statements can mean 'uniformly convergent to $f... |
H: Confusion regarding usage of Lambert function
I stumbled upon an equation that goes like:
$$e^{\pi x} - \frac{x}{k} = -1$$
I learnt that Lambert function is useful when dealing with such equations where it can take the form $f(x) = xe^x$.
So, the equation essentially becomes:
$$ x = \frac{1}{\pi} \ln\Big(\frac{x}{k... |
H: Notation: setting multiple variables to zero
I would like to set multiple variables to zero in an latex algorithm environment. Is there a more pleasing way than to state simply
$a=b=c=d=e=f=0$ ?
This looks cluttered. Maybe someone with more knowledge in math notation knows a better (shorter) way to express this. Th... |
H: Confusion in solutions of logarithm.
An expression $log_2 x^2=2$ can be written as $2log_2 x=2$ but leads to loss of a root. I am having difficulty is recognizing the expressions in which above property is applicable. Another expression $log_2^3 x=log_2 x^3$ can be written as $log_2^3 x=3log_2 x$ and doesn't leads ... |
H: Probability density of $R=\sqrt{X^2+Y^2}$ when $(X,Y)$ is distributed in a disk
I am struggling with the following question:
Let $\left(X, Y\right)$ be a pair of random variables with joint density function $\mathrm{g}\left(x,y\right) = \frac{1}{2}xy$ if $\left(x,y\right)\in D$, $0$ else. Here, $D$ denotes the fir... |
H: Gram matrix of $A$ is equal to $A^T \bar{ A }$
$A$ is an $m \times n$ matrix, $( \vec{ a_1 }, \vec{ a_2 }, \ldots, \vec{ a_n } )$ and $A'$, the Gram matrix of $A$, is a matrix having $\vec{ a_i } \cdot \vec{ a_j }$ as $(i,j)$ element.
How can we get $A'= A^T \bar{ A }$ ? ($A^t$ is the transpose of $A$ and $\bar{ A... |
H: Let $R$ be a $\mathbb Z$ graded ring and let $f \in R_1$ where $f \ne 0$. Show that $R[f^{-1}]$ is a $\mathbb Z$ graded ring.
I've defined $R[f^{-1}]_i = \{ \frac{r}{f^k} \mid r \in R_{i+k}\}$ and I've shown that each $R[f^{-1}]_i$ is an abelian group.
I've shown that $R[f^{-1}]_i R[f^{-1}]_j \subset R[f^{-1}]_{i+j... |
H: Weak Convergence in $W^{1,2}(D)$ implies strong convergence in $L^{2}(D)$
Let $D$ is some bounded domain in $\mathbb{R^n}$. At the bottom of page 20/top of page 21 in this book, they observe that a sequence of functions $\{u_{\epsilon}\} \in W^{1,2}(D)$ is uniformly bounded so that $ u_{\epsilon} \rightarrow u$ wea... |
H: In a metric space is a dense subset of a dense subspace dense in the space itself?
In a metric space is a dense subset of a dense subspace dense in the space itself?
I think that must be false, but I couldn't think of any examples of the contrary.
AI: It is true. Suppose $(X,d)$ is a metric space and we have inclus... |
H: Convergent or Divergent Series? $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt[n+1]{10}}$
I need to find whether this series is convergent or divergent:
$$
\sum_{n = 1}^{\infty}\frac{\left(-1\right)^{ n + 1}}
{\,\sqrt[n + 1]{\, 10\, }\, }
$$
(1) Alternating series test does not provide any additional informati... |
H: How to prove or intuitively understand that $\operatorname{P}(\max X_i > \varepsilon) = \operatorname{P}(\bigcup {X_i > \varepsilon})$
I am trying to at least get a feeling of how this equality works. I have a basic understanding of probability theory. However I could not wrap my head around this equality. Any expl... |
H: Prove that $ |A| = \lim_{t\rightarrow \infty}| A \cap (-t,t)|$ for all $A \subset \mathbb{R}$
Problem taken from the books sheldon Axler Measure , integration Real analysis
Prove that $ |A| = \lim_{t\rightarrow \infty}| A \cap (-t,t)|$ for all $A \subset \mathbb{R}$
My attempt : $\lim_{t\rightarrow \infty}| A \cap ... |
H: Adjunction isomorphisms imply full and faithful
Let $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G:\mathcal{D}\rightarrow\mathcal{C}$ be two functors such that $\alpha:1_{\mathcal{B}}\cong F\circ G$ and $\beta:G\circ F\cong 1_{\mathcal{C}}$. I want to show that $F$ is full and faithful.
It can be deduced that $F\circ... |
H: Is this an equivalent way of stating Cauchy's convergence test for series?
Let $(x_n)_{n\in \mathbb{N}}$ be a sequence of elements from $\mathbb{R}^p$. Then Cauchy's test states that $\sum\limits_{i=0}^\infty x_i$ converges if and only if for every $\epsilon >0$ there is a natural number $N$ such that $||x_{n+1}+x_... |
H: Find the complete solution set of the equation ${\sin ^{ -1}}( {\frac{{x + \sqrt {1 - {x^2}} }}{{\sqrt 2 }}} ) = \frac{\pi }{4} + {\sin ^{ - 1}}x$ is
The complete solution set of the equation ${\sin ^{ - 1}}\left( {\frac{{x + \sqrt {1 - {x^2}} }}{{\sqrt 2 }}} \right) = \frac{\pi }{4} + {\sin ^{ - 1}}x$ is
(A)[-1,0]... |
H: When trying to prove some $M$ is an $A$-module, with $A$ being an algebra, how do I deal with the set of generators of $A$?
Suppose I want to find out if $M$ is an $A$-module, with $A$ being an algebra with a set of generators $S$ and $R$ a set of equations satisfied by the generators of $A$.
For the sake of simpli... |
H: Calculating the $p$-value
How do I calculate the $p$ value of the following?
Students' height is approximately normal with
s.d = $4$ inches,
sample = $10$,
mean height = $68$ inches.
Calculate the $p$ value corresponding to the following null hypotheses.
$H_o$ = Avg. height is $70$ inches
$H_1$ = Avg. height is not... |
H: Order Preserving Bijection
Consider the posets $(\mathbb{Z}^+,\leq)$ and $(\mathbb{Z}^-,\leq)$. Is the bijection $f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^-$ not order preserving?
I am new to set theory and I don't have an idea on how to show whether $f$ is order preserving. Any help will be highly appreciated.
AI... |
H: Equal number of $n$th roots of unity within character values $f_1(a), \dots, f_m(a)$. (Apostol exercise 6.12 Intro to ANT)
This problem is taken from Exercise 6.12 from Apostol's "Introduction to Analytic Number Theory".
Verbatim, the problem states
Let $f_1, \dots, f_m$ be the characters of a finite group $G$ of ... |
H: Prove that the circumference of an ellipse is given by this infinite series
Prove that the circumference of an ellipse is given by :
$$2\pi a\left[1-\sum_{n=1}^{\infty}\left(\frac{\left(2n-1\right)!!}{\left(2n\right)!!}\right)^{2}\frac{e^{2n}}{2n-1}\right]$$
The parametric of an ellipse is :
$$x=a\cos(\theta)$$
$$... |
H: Is $X$ a Borel subset of $\beta X$?
Consider a Tychonoff space $X$ and $\beta X$ its Stone-$\check{\rm C}$ech compactification. I'm currently studying the existence of certain types of regular Borel measures on $X$. Since it's much simpler to obtain regular Borel meaures for a compact, I'd like to obtain them for $... |
H: Mathematical proof regarding circular permutations
The question here is in how many ways can $n$ people stand in order to form a ring.
Now I understand the concept behind it. I have understood it like this:
$n$ people can be arranged in $n!$ ways. Now in each ring, if the ring is broken from a unique point and is s... |
H: Let $X$ be a set and ${\rm Sym}(X)$, the symmetric group on $X$. If $x,y\in X$, is there guaranteed to be an $f\in{\rm Sym}(X)$ such that $f(x)=y$?
Let $X$ be a set and $\operatorname{Sym}(X)$ be the symmetric group on $X$. If $x, y \in X$, is there guaranteed to be an $f \in \operatorname{Sym}(X)$ such that $f(x)... |
H: Isomorphism equivalence relation
I am reading through Real Analysis by Fomin and Kolmogorov, and the book makes the statement that:
"Isomorphism between partially ordered sets is an equivalence relation as defined in Sec. 1.4, being obviously reflexive, symmetric, and transitive".
So I have a couple questions regar... |
H: Are algebraic varieties strictly more general than (differentiable) manifolds?
I have read that every non-singular algebraic variety is a smooth manifold. However, I was wondering if every smooth manifold can be expressed as a non-singular algebraic variety, or even just a general algebraic variety; such that algeb... |
H: Let $f$ be a bounded linear functional on a Hilbert space. Show the function $\|x\|^2+f(x)$ achieves a minimum.
Let $f$ be a bounded linear functional on a Hilbert space. Show the function $\|x\|^2+f(x)$ achieves a minimum, describe the point and the minimum value.
The only way I know of solving something of the so... |
H: Are "forgetful" and "mindful" Turing machines equivalent?
Premise:
Define a "mindful" Turing machine (MTM) to be a Turing machine (TM) with a log that records the configuration of the head (i.e. current state, symbol being read, next state, symbol to write, and shift) at each step while the machine is running. In a... |
H: Complete projection matrix
I am trying to solve some exercise, which seems to be beyond me.
Let $P$ be a projection. "A projection $P$ is a linear map, that has the following property"1: $P*P=P$, where as $P \in \mathbb{R}^{n\times n}$.
Furthermore, $P$ is defined as
$$P =
\begin{Bmatrix}
\frac{3}{2} & -1 ... |
H: Weaker Choice of the Real Numbers
In set theory, a set $A$ is a projective set if for any other sets $B, C$ and for any function $f:A\rightarrow B$ and surjective function $g:C\rightarrow B$, there exists a function $h:A\rightarrow C$ such that $g \circ h = f$. The Axiom of Choice is the statement that all sets are... |
H: Greatest Common Divisor Problem: Prove that $\gcd(\frac{a^3+b^3}{a+b}, a+b) = \gcd(a+b, 3ab)$
I've been stuck in this problem for some time now. Currently what I have accomplished is, using the propriety $\gcd(a,b) = \gcd(b,a \bmod(b))$ to get in the equation
$$ \gcd(a+b, \frac{a^3+b^3}{a+b}\bmod(a+b)) $$
but I do... |
H: Corestriction of a full and faithful functor
Let $F:\mathcal{C}\rightarrow\mathcal{D}$ be a full and faithful functor. Consider the corestriction $F:\mathcal{C}\rightarrow F(\mathcal{C})$ of $F$ to its image. Note that for $D\in F(\mathcal{C})$, we have $D=F(C)$ for some $C\in\mathcal{C}$: i.e. $D\cong F(C)$. This ... |
H: Is a function increasing if the derivative is positive except at one point of an interval?
Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable on $(a,b)$. Suppose $f' > 0$ on $(a,b)$ except at a point $c \in (a,b)$ (that is, $f'(c) \leq 0$).
Is $f$ increasing on $(a,b)$?
Must $f'(c)$ be zero, or can it be negativ... |
H: Finding a polynomial $f(x)$ of degree 5 such that $f(x)$ is divisible by $x^3$ and $f(x)+2$ is divisible by $(x+1)^3.$
There is some polynomial $f(x)$ of degree $5$ such that both of these properties hold:
$f(x)$ is divisible by $x^3$.
$f(x)+2$ is divisible by $(x+1)^3.$
Find that polynomial.
I know that because $f... |
H: How to solve for a specific gradient in an implicit relationship, when no points are known?
The problem
Consider the following relation:
$$x^2-3xy+y^2=7$$
I'm struggling with what is essentially the following task:
Find all coordinates of all points where the gradient of the tangent of the curve is ${2\over3}$.
U... |
H: Hartog's lemma: does a proof for $\mathbb N$ generalise?
I'm reading some online notes taken from Imre Leader's Cambridge lectures on logic and set theory. I find the notes very clear on the whole, but one particular proof - the proof of Hartog's Lemma on page 21 - strikes me as odd.
Hartog's lemma: For any set $... |
H: Show that the transformations are linear
I'm a little confused with this question:
Let $W=V \bigoplus U$. where U and V are subspaces of W. Let $P_{1}$ and $P_{2}$ The transformations of W in W such that $w=u+v$ of W (u $\in$ U and v $\in$ V) associate, respectively, u and v, that is, $P_{1}(w)=u$ and $P_{2}(w)=v$... |
H: is the integral of any polynomial of the form $a+bt+ct^2......+dt^n$ from zero to t always not zero.
For instance,
I am looking at a question in linear algebra,
Define $T : P_{3}\to P_{4}$ such that
$T(p)=\int_{0}^{t}p(x)dx$
find the Nul(T)
This ends up being = {0}. Since this is true I was wondering how no polyn... |
H: Smallest closed ideal containing an element in a $C^*$-algebra
Let $A$ be a $C^*$-algebra and $a \in A$. I want to describe the smallest closed ideal containing $a$. If the algebra is unital, I think this ideal will be $\overline{AaA}$. But can we describe this ideal when $A$ is non-unital? Maybe something like
$$\... |
H: Question about whether function is odd or even
If x is an odd number, would the function $2020^x + x^{2020} $ be odd or even?
I'm currently having trouble figuring out the answer to this question. Currently I believe the result would be odd, since an even number raised to an odd power results in an even number, wh... |
H: Finding the limit of $\frac{N_n}{\ln(n)}$ where $N_n$ is the number of digits of $n$
I came across this question in an entrance exam of a local college where we are asked to evaluate the limit : $$\lim\limits_{n\to\infty} \frac{N_n}{\ln(n)}$$
Where $N_n$ denotes the number of digits of $n$, with the latter being a ... |
H: Explicit description of $\mathcal{O}_{\Bbb{P}^1}(-1)$ as a line bundle
I understand the construction of $\mathcal{O}_{\Bbb{P}^1}(-1)$ as a sheaf on $\Bbb{P}_\Bbb{C}^1$, but I'm trying to understand how exactly does this define a line bundle and why people call this the "tautological line bundle".
Following the sugg... |
H: Justify the equation $\sum_{n=0}^{\infty} \frac{(-1)^n}{m+nk} = \int_{0}^1 \frac{t^{m-1}}{1+t^k} dt$, $m > 0$.
I am trying to solve this problem from my exam prep under real analytic function section.
Justify the equation $\sum_{n=0}^{\infty} \frac{(-1)^n}{m+nk} = \int_{0}^1 \frac{t^{m-1}}{1+t^k} dt$, $m > 0$.
I am... |
H: How do I compute my limits of integration for a density function?
$ f(x,y) = 6x^2y \ $ if $0 \leq x \leq 1 , 0\leq y \leq 1$ and $0$ in other case.
How to compute $ P(X+Y>1) $ ?
$$ P(X+Y>1) = P(X>1-Y) = \int_{0}^{1} \int_{y}^{1-y} 6x^2y \ dx \ dy $$
Is this correct?
AI: No. It is $\int_0^{1}\int_{1-y}^{1} 6x^{2}y d... |
H: Prove the inequality $1 - \tanh(xy) \leq \cosh(x)^{-y}$
Using some tricks in statistical mechanics I came across the inequality.
$$
1 - \tanh(xy) \leq \cosh(x)^{-y}
$$
for all $x,y >0$.
Do you have a proof (or counterexample)?
AI: Let $\alpha$ be a positive real number greater than or equal to $1$.
Since $f_\alpha... |
H: How to write recursive relationship as a summation of matrices/vectors
I have a recursive relationship, where for a given iteration $k$,
\begin{align*}
k=0, f(k) &= v_1\\
k=1, f(k) &= v_1 - 2Av_2\\
k=2, f(k) &= v_1 - 3Av_2 + 3ABv_1 - ABAv_2\\
k = 3, f(k) &= v_1 - 4Av_2 + 6ABV_1 - 4ABAv_2 + ABABv_1\\
\end{align*}
wh... |
H: Let $S_1$ and $S_2$ be the symmetric closures of $R_1$ and $R_2$, respectively. Prove that $S_1 \subseteq S_2$.
This is an exercise from Velleman's "How To Prove It".
Suppose $R_1$ and $R_2$ are relations on $A$ and $R_1 \subseteq R_2$. Let $S_1$ and $S_2$ be the symmetric closures of $R_1$ and $R_2$, respectivel... |
H: Show if $\int_U g$ exists, so does $\int_U f$ (extended integral question)
Let $U \subseteq \mathbb{R}^n$ be open(not necessarily bounded), let $f,g: U \rightarrow \mathbb{R}$ be continuous, and suppose that $|f(x)| \leq g(x)$ for all $ x \in U$. Show if $\int_U g$ exists, so does $\int_U f$
Definition: Let $f:S ... |
H: Idempotent linear operators are projections
I'm working on this problem:
Suppose $V$ is a Hilbert space and $P : V \to V$ is a linear map such that $P^2= P$ and $\Vert Pf \Vert \leq \Vert f \Vert$ for every $f \in V$. Prove that there exists a closed subspace $U$
of $V$ such that $P = P_U$.
I did do some searching ... |
H: A multiplication problem: $OUI \times OUI = OOUUI + OOUUI$
Here's a $6$th-grade exam problem:
$$OUI \times OUI = OOUUI + OOUUI$$
$O, U$ and $I$ are digits.
e.g $365 \times 365 = 33665 + 33665$ (not the case)
Thus, because the digits are into account I believe there's no algebraic equation. Also, $6$th graders don'... |
H: First Variation of $L_2$ with Linear operator
Let $\Omega \subset \mathbb{R}^2$ be open and bounded, $P:C^1(\Omega) \to L_2(\mathbb{R})$ be a linear and bounded operator. I want to calculate the first variation of $\|Pu - f\|^2_{L_2}$, $u \in C^1(\Omega), f \in L_2(\Omega)$.
${d \over dt}\|P(u+tv) - f\|^2_{L_2} = {... |
H: Prove $D \in \mathcal{L}(\mathcal{P}(\mathbf{R}),\mathcal{P}(\mathbf{R})) : \text{deg}(D(p)) = \text{deg}(p) - 1$ is surjective
Suppose $D \in \mathcal{L}(\mathcal{P}(\mathbf{R}),\mathcal{P}(\mathbf{R}))$ is such that $\deg(D(p)) = \deg(p) - 1$ for every nonconstant polynomial $p \in \mathcal{P}(\mathbf{R})$. Prov... |
H: Advantages of Each Coordinate System
I am currently learning about the spherical coordinate system in class, but I do not know its advantages or even if it is advantageous in using this coordinate system over another.
I am very comfortable in using the rectangular coordinate system and the cylindrical coordinate sy... |
H: Is $\oint_{\left | z \right |=2} \frac{e^{\frac{1}{z}}}{z(z^{2}+1))}dz$ equal to zero?
I based my analysis on the fact that the only residue that's outside the curve is the reside in $\infty$ that's equal to zero, so all the other resides inside the curves must add to zero too.
Am I correct?
AI: The substitution $w... |
H: Correct way to integrate $\int x(x^2-16)dx$
Evaluate:
$$\int x(x^2-16)dx$$
I have noticed that this integral can be solved using two different methods, but I am not sure which one is the correct one.
Way 1: Using $u$-subtitution
Let $u=x^2-16, du = 2xdx$
Then, we have $$\int x(x^2-16)dx$$ $$= \frac{1}{2}\int udu$... |
H: Finding $dy/dx$ from $ y^2 = \sin^4{2x} + \cos^4{2x} $ using implicit differentiation
Problem:
Find $\frac{dy}{dx}$ by implicit differentation for the following:
$$ y^2 = \sin^4{2x} + \cos^4{2x} $$
Answer:
\begin{align*}
2y \frac{dy}{dx} &= 4(2) \sin^3(2x) \cos(2x) - 4(2) \cos^3(2x) \sin(2x) \tag1\\
y \frac{dy}{d... |
H: Proving that a certain definition of neighborhoods forms a neighborhood topology
This is Exercise 2 from Section 2.1 on page 21 of Topology and Groupoids, by Brown.
Exercise:
Let $\leq$ be an order relation on the set $X$. Let $x \in X$ and $N \subseteq X$. We say that $N$ is a neighborhood of $x$ if there is an
o... |
H: What are the unit elements in $\Bbb{Z}[i]$?
What are the unit elements in $\Bbb{Z}[i]$, where $\Bbb{Z}[i]$ is defined to be the set of Gaussian integers ?
My progress: My mentor gave this problem and he said to use determinants and scalefactor.
Then I was able to proceed and got the unit elements in $\Bbb{Z}[i]=1,-... |
H: Uncountable sets - Why is the following proof false?
Let $S$ be any subset of the natural numbers. Then the sum
$$
\sum_{n \in S} \frac{1}{2^n}
$$
converges against a unique value for each subset $S$.
This sum yields a computable number because it is possible to compute it digit by digit. Therefore, these sums map ... |
H: $\pi(n)$ is always more than the sum of the prime indices of the factors of composite $n \geq 12$
Let $n=p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} \geq 12$ be any composite integer.
Then it seems that this is true: $$\pi(n) > \sum_{i=1}^{k}{\pi(p_i)a_i}\ .$$
You get equality instead iff $n$ is prime.
I also assume that ... |
H: How to prove these four propostions about Borel-Cantelli Lemma in Shriyaev's book 'Probability'?
The original problems are shown in picture linked below. Thank you.
Problem $\mathbf{2.10.19}$. (On the second Borel-Cantelli lemma.) Prove the following variants of the second Borel-Cantelli lemma: given an arbitrary... |
H: What is the correct choice of the contour in the case of undamped forced harmonic oscillator?
I am interested in finding the Green's function (GF) for the undamped forced harmonic oscillator equation: $$\Big(\frac{d^2}{dx^2}+\omega_0^2\Big)x(t)=f(t).$$ In order to find the GF, start by define it: $$\Big(\frac{d^2}{... |
H: confusion about uniform probability distribution expectation value
I read in my statistics book about Discrete Uniform Probability Distribution and the Expected Value
A discrete random variable $X$ with $k$ possible outcomes $x_1, x_2, ...,x_k$ is said to follow a discrete uniform distribution if the probability ma... |
H: The intersection assumption in the definition of direct sums
Let $W_1,W_2,\ldots,W_k$ be subspaces of a vector space $V$. We define the sum of these spaces to be the set
$$\{v_1+v_2+\cdots+v_k:v_i\in W_i\text{ for }1\leq i\leq k\},$$
which we denote by $W_1+W_2+\cdots+W_k$ or simply by $\sum_{i=1}^k W_i.$ If, in ad... |
H: Why does the negative of the direction of steepest ascent result in the direction of steepest descent?
The gradient is a vector composed of partial derivatives and is a vector that gives the direction of the steepest ascent.
In a YouTube video that I was watching, it talked about how the gradient descent of the fun... |
H: Represent $f(x) - f(y)$ as an integral
Description
I've come across the following transition in a textbook of Convex Optimisation. I couldn't figure out what's going on so that I'd appreciate if anyone hits me with any hint!
Problem
Suppose $x, y \in \mathbb{R}^n$ and $f$ be a $\beta$-smooth convex function on $\ma... |
H: Simplify the radical $\sqrt{x-\sqrt{x+\sqrt{x-...}}}$
I need help simplifying the radical $$y=\sqrt{x-\sqrt{x+\sqrt{x-...}}}$$
The above expression can be rewritten as $$y=\sqrt{x-\sqrt{x+y}}$$
Squaring on both sides, I get $$y^2=x-\sqrt{x+y}$$
Rearranging terms and squaring again yields $$x^2+y^4-2xy^2=x+y$$
At th... |
H: When to use ds vs dy and dx for line integrals
Are $\int_{C}^{} f(x,y) ds $ and $\int_{C}^{} F(x,y) = \int_{C}^{} P(x,y) dx + \int_{C}^{} Q(x,y) dy$ equivalent to each other for some curve C? If not, when are they not interchangeable? And if so, when is there an advantage to using one over another?
Edit: it seems t... |
H: Evaluate the given limit $\lim_{n\to \infty} \frac{e^n}{(1+\frac 1n)^{n^2}}$
The procedure I am about to write is wrong, and I know why it is wrong yet it was the only one I could think of, so I will put it up anyway
$$\lim_{n\to \infty} \frac{e^n}{\left(1+\frac 1n\right)^{n^2}}$$
For the denominator
$$\begin{align... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.