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H: Calculating a limit with exponent and trig function
I got this limit to calculate:
$$
\lim_{x\to\frac{\pi}{2}}(\tan x)^\frac{1}{x-\frac{\pi}{2}}
$$
I'm trying to solve it with De L'Hopitals rule and the first step should be this, I guess:
$$
\lim_{x\to\frac{\pi}{2}}e^\frac{\ln(\tan x)}{x-\frac{\pi}{2}}
$$
Then I'm ... |
H: Explanation of a scalar product calculation
Let $f_1, f_2, \ldots, f_n$ be continuous real valued functions on
$[a,b]$. Show that the set $\{f_1,\ldots,f_n\}$ is linearly dependent
on $[a,b]$ if and only if $$\det\left(\int\limits_a^b
f_i(x)f_j(x)dx\right)=0$$
I can't understand the solution to the sufficient co... |
H: Probability selecting three cards out of a deck
In Introduction to Probability by Blitzstein & Hwang, Chapter 2 Problem 5:
Three cards are dealt from a standard, well-shuffled deck. The first two cards are flipped
over, revealing the Ace of Spades as the first card and the 8 of Clubs as the second card.
Given this ... |
H: Calculate $\int x^m \ln(x)\,dx,\,\,\, m \in \mathbb{Z}$
Calculate
$$\int x^m \ln(x)\,dx,\,\,\, m \in \mathbb{Z}$$
My attempt:
First suppose $m\ne-1$
$$\int x^m \ln(x)\,dx=\left[\frac{1}{(m+1)}x^{m+1}\ln(x)\right]-\int \frac{1}{(m+1)}x^{m+1}x^{-1}\,dx$$
$$=\left[\frac{1}{(m+1)}x^{m+1}\ln(x)\right]-\int \frac{1}{(m+... |
H: Does this situation indicate that $g(x)>f(x)$?
I have two functions $f(x)$ and $g(x)$ for $0<x<5$. By calculating the first and second derivatives I see that both functions are increasing and concave (not strictly concave), i.e. $f''<0$ and $g''<0$. Then If I have $$f(0)=g(0)=0$$ and $$f(5)=10\quad ,\quad g(5)=11.$... |
H: Bounded idempotent on Hilbert space has closed range.
Let $H$ be a Hilbert space and $p\in B(H)$ an idempotent on $H$, i.e. a continuous linear map satisfying $p^2 = p$. Is it true that $p(H)$ is a closed subspace of $H$?
Attempt:
Consider a sequence $p(h_n) \to h $. We show $h \in p(H)$. By continuity, $$p(h_n) = ... |
H: Solution to trigonometric equation
I wonder that the solution to the following equation:
If $0\le x,y\le\pi$, solving the equation $\cos(x-y)=\cos(2x)$ for $x=?$.
My attempted solutions are $x=\frac{y+2k\pi}{3}$ for $k=0,1$.
Am i right? I'm not sure about that ;(
Give some comment. Thank you!
AI: Use https://math... |
H: What is the largest number of row operations required to row reduce an $n \times n$ matrix efficiently?
Given an $n \times n$ matrix, and row operations shear, switch, and scale, what is the largest number of efficient row operations required to row reduce to echelon form?
For reference, a shear is defined as $R_... |
H: High dimensional "eigenvector"
I have come across an oddity in the solution to a problem and I am unsure with how to proceed.
For an $n\times n$ (symmetric and real-valued, usually poorly conditioned and low-rank) matrix $A$, and an $n\times m$ matrix $B$ (with $m \leq n$), and a scalar $c \neq 0$, the solution com... |
H: If $\sum a_{n} x^{n}$ converges in $( -1,1)$, then $\sum a^{k}_{n} x^{n}$ also converges in $( -1,1)$
I have to prove or disprove the following statement:
If $\sum a_{n} x^{n}$ converges in $( -1,1)$, then $\sum a^{k}_{n} x^{n}$ also converges in $( -1,1)$ , for $k\in \mathbb{N} $.
I actually already proved it.... |
H: If $f\cdot g$ is differentiable at $x_0$, what conditions on $f$ guarantee the differentiability of $g$ at $x_0$?
Justify: If $f\cdot g$ is differentiable at $x_0$, what conditions on $f$ guarantee the differentiability of $g$ at $x_0$?
I tried coming up with an example, like $f(x)=x$ and $g(x)=|x|$. $g$ is not d... |
H: Determine whether the given function $F(x,y) = x/y$ is as binary operation on the given set $A$
The given function is $F(x,y) = x/y$ for all $x$ and $y$ in the set $A$.
where $A = \{a+b \sqrt[]{2} : a,b \in \mathbb{Q}\}$
and $a$ and $b$ cant be both $0.$
So what I need is to prove $F(x,y) \in A$ for every $x,y \in ... |
H: Let $f(x) \in \mathbb{Q}[x]$ be an irreducible polynomial of degree $4$ with exactly $2$ real roots.
Show that the Galois group of $f$ over $\mathbb{Q}$ is either $S_4$ or the dihedral group of order $8$.
So I'm studying from quals remotely, in a state hundreds of miles away from my university that is also quaranti... |
H: Rescaling multivariable normal pdf and normalizing constant
I am trying to understand change of variables for a random variable and how it changes the pdf and the normalizing constant.
Let $\mathbf{w}$ be $N$-dimensional normal variable and let $\mathbf{S}$ be covariance matrix. I can diagonalize that matrix: $\mat... |
H: The signature of the tensor product of skew-symmetric non-singular matrices
Let $X$ be a non-singular (real) symmetric matrix which is the tensor product of two (real) $n\times n$ skew symmetric non-singular matrices, i.e. $X=A\otimes B$. Then how to see the number of negative eigenvalues of $X$ equals to the numbe... |
H: Function with a parameter controlling its growth
I am looking for a mathematical function with growth controlled by a parameter.
It would have two inputs:
A growth scale, further called $w$
An input ranging from $0$ to $1$, further called $x$
The function $f(x, w)$ should behave according to the following pattern... |
H: What does it mean to say that a measure is supported on a set?
What does it mean to say that a measure $\mu$ is supported on set $E$ ? I keep coming across this phrase, but its meaning is always assumed which can get somewhat confusing.
Does it simply mean that for each (open) neighborhood $N \subset E$, we have t... |
H: $\omega$ a continuous $m$-form in $M$, $\dim M=m$, $M$ is orientable $\iff \omega \neq 0$
Let $M$ a surface, $\dim M=m$, and let $\omega$ a continuous $m-$form in $M$. I want to prove that $M$ is orientable $\iff \omega(x)\neq 0\; \forall x\in M$.
The $(\Leftarrow)$ part is ok to me, since I say this post. However... |
H: How is the "diagonal proof" valid?
Years ago I was taught a proof that there are more irrational numbers than rational numbers, i.e. that it is a bigger infinity, that went like this:
Take the set of all rational numbers. Now construct a new number as follows: Take the first rational number, and choose a digit for ... |
H: Stein and Shakarchi, Complex Analysis, Chapter 2, Problem 1(b)
Fix $\alpha$ with $0 < \alpha < \infty$. Show that the analytic function $f$ defined by $f(z) = \sum_{n=0}^\infty 2^{-n\alpha} z^{2^n}$ for $|z| < 1$ extends continuously to the unit circle, but cannot be analytically continued past the unit circle.
The... |
H: Finding the Centre of Gravity (Using Single Integral) of an Arc of the Astroid $x^{2/3} + y^{2/3} = a^{2/3}$ in the First Quadrant
My Attempted Solution: First off, the curve will intersect the x and y axis at $(a, 0)$ and $(0, a)$. We can also rewrite the function: $y = (a^{2/3}-x^{2/3})^{3/2}$. To find the y-coor... |
H: solve matrix equation
Given symmetric positive definite (SPD) matrices $A$ and $B$, I was wondering there is a solution for
$XAX=B$,
where $X$ is also a SPD matrix.
Thank you!
AI: Your question can be rewritten as
$$ (A^{1/2} X A^{1/2})^2 = A^{1/2} B A^{1/2}. $$
So
$$ X = A^{-1/2} (A^{1/2} B A^{1/2})^{1/2} A^{-1/2}... |
H: Why are these dependent and independent events respectively.
Case #1.
P(A) is the probability of rolling a fair standard dice and getting numbers less than or equal to 4.
P(B) is the probability of rolling it and getting even numbers.
Case #2.
P(A) is the probability of rolling a fair standard dice and getting ... |
H: How to determine solutions to homogenous systems Ax = 0?
I have a 4x4 matrix which I need to determine the solutions for Ax = 0 as well as one basis of the space of solutions.
What I've done so far was to reduce the matrix to the echelon form and I ended up with:
Then I wrote:
(x = -2y)
(y = z)
(w = 0)
How do I fi... |
H: How many different ways are there to write $2016$ as the difference of two squares?
How many different ways are there to write $2016$ as the difference of two squares?
So the question is asking for the integer solutions for $2016 = x^2-y^2$. The prime factorization of $2016$ is $2016=2^5\cdot3^2\cdot 7^1$. I’m no... |
H: Show that for every integer $n$ there is a multiple of $n$ that has only $0s$ and $1s$ in its decimal expansion.
Can anyone please explain this example as I tried a lot to understand it but I can't!
The problem:
Show that for every integer n there is a multiple of n that has only
0s and 1s in its decimal expansion... |
H: calculate the Integer solutions of an equation
How does anyone calculate the "integer solutions" of an equation in this example:
$$
x(x+1) + 62 = y^2
$$
https://www.wolframalpha.com/input/?i=x*%28x%2B1%29%2B62%3Dy%5E2
here the Integer solutions are :
$x = -62$, $y= 62$
$x = -2$, $y= 8$
$x = 1$, $y= 8$
$x = 61$, $y=... |
H: Are fully faithful functors stable under pullback?
Let $A$, $B$ and $C$ be locally small categories, let $I:A\to C$ be a fully faithful functor and let $F:B\to C$ be any functor. Is the pullback $F^*I: A \times_C B\to B$ still fully faithful?
If yes, is there an underlying orthogonal factorization system?
Any refer... |
H: The geometry meaning of Unitary matrix/operator
After studying well about the unitary,self adjoint and normal matrices and operator, I can say that they have pretty intresting characteristics, but I do not know how to visualize them in low dimensions spaces, I have seen some youtube clips about visualize a linear t... |
H: Fibrantly generated model category
An important concept in the study of model categories is that of "cofibrantly generated model categories". These are nice because all morphisms can be obtained from a small subset of them and in general these specific categories are often easier to work with.
Now I was wondering w... |
H: What is the conjunctive normal form for $(\neg Q\land P) \lor (\neg Q\land R) \lor (\neg P \land \neg R)$
$(\neg Q\land P) \lor (\neg Q\land R) \lor (\neg P \land \neg R)$
i have calculated this using wolframalpha and the output of CNF was $(\neg Q \lor\neg P) \land (\neg Q \lor\neg R) $
but all i can reach ou... |
H: Smoothness of quotient map
I'm a beginner in differential topology and was studying: Introduction to Smooth Manifolds by John M. Lee. And came across an example (Example 2.13 e ) :
Could someone please explain how one could deduce that the quotient map, used to define the projective space, is smooth based on the ... |
H: Inequality involving two convergent sequences
Let $(x_n)$ and $(y_n)$ be sequences of real numbers that satisfy $\lim x_n =a$ and $\lim y_n=b$.
Prove that if $a<b$ then $\exists n_0 \in \mathbb{N}$ such that $\forall n > n_0$, $x_n < y_n$
I found this exercise at Elon Lages Lima's book Real Analysis vol. 1, chapt... |
H: How to show that $(3,x-1)\not=(3,x+1)$ as ideals in $Z[x]$
How to show that $(3,x-1)\not=(3,x+1)$ as ideals in $Z[x]$
Both are maximal.
I think all i need to do is to show is that $x+1 \not \in (3,x-1)$ but I do not know how to show that. I wanted to assume towards contradiction that $3f(x)+(x-1)g(x)=x+1$. It is cl... |
H: Prove that there is a point $c \in (0,1)$ such that $f(c) \int_0^1 g(x)dx =g(c) \int_0^1 f(x)dx$.
Let $I = [0,1]$ and $f,g : I \to \Bbb{R}$ be real-valued functions such that $f$ and $g$ are continuous on $I$ and differentiable on $(0,1)$. Prove that there is a point $c \in (0,1)$ such that
$$
f(c) \int_0^1 g(x)dx ... |
H: If the symmedian at $A$ of $\triangle ABC$ meets the circumcircle at $K$, then $KA$ is a symmedian of $\triangle KBC$.
Let the symmedian of $\triangle ABC$ at $A$ pass through the circumcircle of $\triangle ABC$ at $K$. Prove that $KA$ is a symmedian of $\triangle KBC$.
The book says that it is very obvious, but ... |
H: Is there a constant $c$ such that $\sigma_1(A)\leq c\rho(A)$ for all $n\times n$ matrices $A$?
Let $A$ be a $n\times n$ matrix. Denote by $\sigma_1(A)$ the largest singular value of $A$, and $\rho(A):=\max_\mu\{{|\lambda_\mu|}\}$ the radius of the eigenvalues $\lambda_\mu$ of $A$.
As is well-known, there is
\begin{... |
H: Notation for annihilator with ring and module interchanged?
Let $M$ be an $R$-module. For a subset $S\subseteq R$, the set
$$\{m\in M\mid (\forall s\in S)[sm=0]\}$$
is clearly a submodule of $M$. Is there a name or notation for it? It is essentially the definition of annihilator with ring and module interchanged.
A... |
H: $f_{*}$ is surjective if and only if $f$ is surjective
I’m having trouble proving the following theorem. I’ll show my proof so far. I would really appreciate if you could help me, so I can improve my proof and correct some mistakes.
Theorem: Let $f:A \rightarrow B$ be a map. Think of this map as inducing the map $... |
H: Can I get the derivation for the geodesic equation in Euclidean 3D?
Where can I find the derivation of the following formula for a geodesic in 3D Euclidean space expressed as:
Calling $\vec u$ the tangent unit vector $\mathrm dr/\mathrm d s$ along the geodesic, and $\vec n$ the unit vector normal to the surface, t... |
H: How many five letter words can be made from the letters in SUCCESSFUL?
Question from this video: https://www.youtube.com/watch?v=nU2NrXOCdwk
Actually the word there was SUCCESSES, I will just change it to SUCCESSFUL.
So let me partially do the solution because I don't have a problem doing exactly the entire thing. ... |
H: Prove $A \setminus (A \setminus (A \setminus B)) = A \setminus B.$
$\renewcommand{\backslash}{\setminus}$
The question is as is in the title. I am able to show that $A \setminus (A \setminus (A \setminus B)) \subseteq A \setminus B$ but I am stuck on showing that $A \setminus B \subseteq A \setminus (A \setminus (... |
H: Twice differentiable but not infinitely differentiable
I want an example of a function satisfying:
(1) $f:\mathbb R\to\mathbb R$ twice continuously differentiable
(2) $f''(x)$ $\ge 0$ for every x$\in \mathbb R$
(3)$\int\limits_0^x{f(t)dt}$ is NOT infinitely differentiable with respect to x.
I find it very difficult... |
H: Finding the equation of a circle given three points on the circle.
Problem:
Find an equation for the circle though the points $(2,3)$, $(3,2)$ and $(-4,3)$
Answer:
One of the general form of a circle is:
$$ x^2 + y^2 + ax + by + c = 0 $$
Hence we have:
\begin{align*}
4 + 9 + 2a + 3b + c &= 0 \,\,\, \text{ This is ... |
H: Theorem $8.38$ - Elliptic Partial Differential Equations by Gilbarg and Trudinger
I am trying understand why if $u$ is an eigenfunction for the first eigenvalue $\sigma_1$, then $|u|$ is one also.
Theorem 8.38. Let $L$ be a self-adjoint operator satisfying $(8.5)$ and $(8.6)$. Then the minimum eigenvalue is simple... |
H: Error in Kleene "Introduction to Metamathematics" parentheses lemmas?
In the classic book, Introduction to Metamathematics by Steven Kleene, Lemma 2 of Section 7 (chapter 2), seems to me to be false. I am wondering if I am missing something. Here is the context:
The following definitions are used:
Proper pairing - ... |
H: Other way to evaluate $\int \frac{1}{\cos 2x+3}\ dx$?
I am evaluating
$$\int \frac{1}{\cos 2x+3} dx \quad (1)$$
Using Weierstrass substitution:
$$ (1)=\int \frac{1}{\frac{1-v^2}{1+v^2}+3}\cdot \frac{2}{1+v^2}dv =\int \frac{1}{v^2+2}dv \quad (2) $$
And then $\:v=\sqrt{2}w$
$$ (2) = \int \frac{1}{\left(\sqrt{2}w\ri... |
H: Which irrationals become rational for some positive integer power?
Related to Irrationals becoming rationals after being raised to some power. Let $r \in \mathbb{R} \setminus \mathbb{Q}$. True or false: there exists an $n \in \mathbb{N}$ (positive integers) such that $r^n = r \cdot \dots \cdot r \in \mathbb{Q}$. Th... |
H: Prove that $ab$ is a cube of a positive integer.
This is a question from Advanced Math Examination of Vietnam:
Let $a, b, c$ be the three positive integers such that $c+\frac{1}{b}=a+\frac{b}{a}$. Prove that $ab$ is a cube of a positive integer.
First solution I thought about is form the hypothesis to a more simila... |
H: Probability questions involving percentages
Lola is obsessed by the colour of her hair. On any given day there is
an 80% chance she will change the colour of her hair for the next day.
Her hair is blond 40% of the time, brown 30% , red 20% and purple for
the remainder. Given Lola has red hair on Friday, what is the... |
H: about prisoners and selection of numbers
Each of the three prisoners had a natural number written on their foreheads: 1, 2 or 3. Numbers can be repeated. The prisoners see all numbers except their own. After that, everyone tries to guess their number. If someone succeeds, the prisoners will be released, otherwis... |
H: Length of coefficient vector of linear combination of unit lengths vector is 1
In Prasolov's "Geometry", problemset 1, problem 1.10 in the solution the author states:
Suppose that set of vectors of unit length $a_1 ... a_n$ is linearly dependent. Then, there exist coefficients $y_1 ... y_n$ such that $\sum_{k=1}^{... |
H: If you know 2 of 8 questions but the test has only 4 questions. What is the probability you will know at least one of the questions on the test?
Background:
This is a hypothetical question that and my friend and I have been arguing over all day. We're both upper-year mathematics students and should reasonably be ab... |
H: Construct a triangle, given the angle at the vertex, the altitude, and the ratio in which its foot divides the base.
The problem is from Kiselev's Geometry exercise 392:
Construct a triangle, given the angle at the vertex, the altitude, and
the ratio in which its foot divides the base.
The chapter is about homoth... |
H: Sets of measure zero on the real line
Is there anything wrong with what I have written here?
Let $D$ be a non-empty subset of the real numbers, with Lebesgue measure $0$.
Say $x$ is in $D$. Since $D$ has measure $0$, it cannot contain any intervals. Therefore,
for any $d > 0$, we know $(x - d,x + d)$ is not contai... |
H: Given $A,B$ symmetric positive definite matrices, is $B^{-1}A$ always symmetric positive definite?
Given $A$ and $B$ to be symmetric positive definite matrices. Is the product $B^{-1}A$ always positive definite and symmetric?
AI: No for symmetry. Symmetry is not true because:
$(B^{-1} A)^T = A^T B^{{-1}^T} = A B^{T... |
H: Tautology propositions
Anyone knows, considering $a$, $b$ and $c$ as propositions, if those propostions below are both tautologies? What I found out yet is that they are, but im not totally sure. $((a∨b)∧((a→c)∨(b→c)))→c$
$a→((¬b→c)∨(¬b→¬c))$
AI: The first statement is not necessarily true, consider when $a$ is fa... |
H: Is there a one-word term for "grows by square root"?
We can describe a growth relationship $y = x$ as linear (e.g. "linear growth").
We can describe $y = \log(x)$ as logarithmic (e.g. "logarithmic growth").
We can describe $y = x^2$ as quadratic (e.g. "quadratic growth").
We can describe $y = 2^x$ as exponential (e... |
H: existence of a function such that it is equivalent to cosine.
Prove there exist continuous functions $Cos : \mathbb{R} \to \mathbb{R}$ and $Sin : \mathbb{R} \to \mathbb{R}$ such that $Cos(q) = \cos(q)$ and $Sin(q) = \sin(q)$ for all $q \in \mathbb{Q}$.
I have attempted this by noting that cosine and sine are contin... |
H: X uniformly distributed, is X and X to the power of n got correlation or not
Basically I got this question where:
$X$~$U[-C,C]$ where C is natural > 1.
And the question that asks, name the correct answer:
$X$ and $X^2$ are independent
$X$ and $X^7$ are uncorrelated
$X^5$ and $X^6$ are uncorrelated
And I got an op... |
H: How to solve polynomial rational relations for $y$ (e.g $\sqrt{4-3y-y^2} = x(y+4)$)?
From time to time, I struggle to solve polynomial relations for $y$.
A trivial example is :
$$ \frac{y}{x} = x \iff y = x^2$$
Easy.
But consider this relation:
$$ \sqrt{4-3y-y^2} = x(y+4)$$
No matter how much I mess around it, see... |
H: Find the domain and range of $f(x)=\sqrt{(16-x^2)}$
I came up with a wrong solution, can someone tell me where I went wrong?
AI: $\sqrt{x}$ is the principal square root function, it takes nonnegative value.
Hence $y \ge 0$.
I don't see how you get $16-y^2 \ge 16$.
We have $$-4 \le x \le 4$$
$$0 \le x^2 \le 16$$
$$1... |
H: Definition for Hermitian inner product?
I have found two different papers (both are well cited) Classification of Self-Orthogonal Codes over $\mathbb{F}_3$ and $\mathbb{F}_4$ (pg.3) and Convolutional and Tail-Biting Quantum Error-Correcting Codes (end of pg.4) that give different definitions for Hermitian inner pro... |
H: Forming a 3x3 magic square with digits 1-9, subject to the constraint that sum of digits in each row, column and diagonal must be equal.
The Problem (Chapter 0, Problem 16 from Fomin's Mathematical Circles):
Form a magic square with the digits 1-9; that is, place them in the boxes of a 3x3 table so that all the su... |
H: Suppose that $N$ and $r$ are positive integers. Prove or disprove that if $N$ is an even integer and $r$ is odd, then $\binom{N}{r}$ is even.
Suppose that $N$ and $r$ are positive integers. Prove or disprove that if $N$ is an even integer and $r$ is odd, then $\binom{N}{r}$ is even.
My attempt:
Let $N=2m$ and $r=... |
H: Conjugate in $S_n$ do not imply conjugate in its subgroup.
$(12)(34)$ and $(13)(24)$ are conjugate in $S_4$, but they are not conjugate in $\mathbf V_4$, the Klein four group, since $\mathbf V_4$ is abelian.
I know that
$\alpha ,\beta \in S_n , \alpha \text{ and } \beta \text{ are conjugate in $S_n$}\iff$ $\alpha... |
H: why are $e^{2x}$ and $e^{x^2}$ inequal?
enter image description here
From the index rules I learned from school, $a^{x^2}=a^{2x}$
Does it work the same for the natural constant?
Why is it?
AI: There is a difference between ${(a^b)}^c$ and $a^{(b^c)}$.
The first simplifies to ${(a^b)}^c=a^{bc}$ but the second does n... |
H: The equivalence relation defined by normal form on Von Neumann algebra, its support and the it's representation .
Let M be a von Neumann algebra and $\varphi$ a positve normal form on M.
$N = \lbrace x\in M | \varphi(x^*x)=0\rbrace $ . We denote $M_{\varphi} := M/N$ as the pre-Hilbert space defined by the inner p... |
H: Differentiation Product Rule Question on $x^2(x-1)^3$
I have a question:
$x^2(x-1)^3$
Which I am supposed to solve using the product rule but it seems that I have hit a dead end with the algebraic part.
Here's my working so far:
$u = x^2$
$u' = 2x$
$v = (x-1)^3$
$v' = 3(x-1)^2$
$(uv)'= (x-1)^32x+x^23(x-1)^2$
This... |
H: Solve a optimization problem with Lagrange multipliers
Trying to solve next issue $$ \begin{cases} \sum_{i=1}^n x_i(ln_{x_i} - c_i) - min
\\
\sum_{i=1}^n x_i = 1
\end{cases}$$
where
$$
x_i, c_i \in R^n, c_i = const > 0 \forall i
$$
I tried Lagrange method:
$$
L(x,\lambda) = \sum_{i=1}^n x_i(ln_{x_i} - c_i) + \l... |
H: Unbounded on every interval except null set but finite a.e
This question originates from Folland's problem 2.25. In this problem, first given $f(x)=x^{-1/2}$ when $0<x<1$, and $0$ otherwise.Then consider $g(x)=\sum_{n}2^{-n}f(x-r_{n})$, where sequence $r_{n}$ is the all the rational number. Then it requires us to p... |
H: Critical points of a nonnegative quadratic form on a subspace
Let $Q(x)=x^tAx$ for some square symmetric matrix $A\in R^{n\times n}$, such that $Q(x)\geq0$ for each $x\in R^n$. Let $S$ be an affine subspace of $R^n$. How can I show that if $y$ is a critical point of $Q$ on $S$, then $y$ is a point of global minim... |
H: Jordan normal form of $\;\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & a & b \end{pmatrix},\; a,b\in\mathbb{R}$
If possible, compute the Jordan normal form of
$\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & a & b \end{pmatrix}\in\mathbb{R}^{3\times 3}$ with $a,b\in\mathbb{R}$.
In the case that $a,b=0$ the matrix alr... |
H: Derivative of random normal times indicator function
I have to find $\frac{dE[f(X)]}{dX}$ where $f(X) = X1_{X>a}$ where $X \sim N(0,1)$ , $1_{X>a}$ is an indicator function taking value 1 if $X>a$ and $0$ otherwise, and $a$ is some constant. I have trouble understanding how to differentiate a random variable (stand... |
H: Vector notation for $k>n$ vectors in $\Bbb{R^n}$
I'm reading Advanced calculus of several variables by Edwards, C.Henry. On page 6, the author is proceeding with the proof that any $n+1$ vectors in $\Bbb{R^n}$ are linearly dependent and writes :
"Suppose that $v_1,...,v_k$ are $k>n$ vectors in $\Bbb{R^n}$ and write... |
H: Definition of addition and multiplication on $ℕ$ using recursion
My book, Classic Set Theory: For Guided Independent Study, gives me this theorem:
"let $y_0$ be any element of $ℕ$ and $h"ℕ×ℕ×→ℕ$ a function on pairs $(x,y)∈ℕ×ℕ$. Then there exists a unique function $f:ℕ→ℕ$ such that $f(0)=y_0$ and $f(n^+)=h(n,f(n))$... |
H: Define a linear functional $T$ on $V$ by $Tv$ = $\langle v, u\rangle$. What is $T^∗ (\alpha)$ for a scalar $\alpha$ where $T^*$ is the adjoint.
Now I do understand that the question might involve using Riesz representation as it involves a linear functional and we know that it can be written using an inner product.... |
H: How to show given function in $L^r(\Omega)$?
I wanted to show that $h(x-y)=|x-y|^{n(\mu-1)}\in L^r(\Omega)$ where $\Omega $ is bounded .\
$\Omega\in B(x,R)$ for some $R$.
So I have to show that $\int_{B(x,R)}|h|^r\leq\infty $.
$\int_{B(x,R)}|h|^r=\int_0^R\int_{\partial B(x,k)}|k|^{nr(\mu-1)}dSdk=\int_0^R k^{nr(\mu... |
H: Solving the quadratic equation $a X^2 + a Y^2 + 2 b X + 2 c Y + d = 0$.
I'm trying to solve the following quadratic equation, but I have no idea how:
$$a X^2 + a Y^2 + 2 b X + 2 c Y + d = 0,$$
where the coefficients are real numbers. Can someone give me a hint?
AI: Find solutions of $ax^2+ay^2+2bx+2cy+d=0$, where $... |
H: To find supremum of this
How do I find supremum of set ?
$ (0,1) \cap\Bbb{Q}$ , where $\Bbb Q$ is set of al rationals. How will answer change if rationals are replaced by irrationals?
I know supremum of $(0,1)$ is $1$. How do I do?
Thank you.
AI: $(0,1)\cap Q$ is the set of rational numbers between 0 and 1 (not ... |
H: Value of $\frac{\partial }{\partial x}\left(f\left(x,y\right)\right)$ at $(0,1)$
$f\left(x,y\right)=e^{x+y}\left(x^{\frac{1}{3}}\left(y-1\right)^{\frac{1}{3}}+y^{\frac{1}{3}}\left(x-1\right)^{\frac{2}{3}}\right)$
What's the value of $\frac{\partial }{\partial x}\left(f\left(x,y\right)\right)$ at $(0,1)$? There tw... |
H: How to compute the series: $\sum_{n=0}^{\infty} (-1)^{n-1}\binom{1/2}{n}$
I'm wondering about how to show compute this series: $$\sum_{n=0}^{\infty}(-1)^{n-1}\binom{1/2}{n}$$
My approach was to use the general formula of the binomial series, which is: $$(1+z)^r=\sum_{k=0}^{+\infty}z^{k}\binom{r}{k}$$
Yet this can't... |
H: $F= \bigcap_{i=1}^{\infty} F_i$ isn't necessarily connected where $F_{i+1} \subseteq F_i$ and $F_i \subseteq \mathbb{R}^2$ are closed and connected
In my attempt, I first show that $F$ is closed, this is since we can write $F= \bigcap_{i=1}^{\infty} F_i = (\bigcup_{i=1}^{\infty} F_i^C)^C$ and $\bigcup_{i=1}^{\infty... |
H: Why does $\gcd(c_1,c_2,\ldots ,c_k)=1\Rightarrow \gcd(c_1-c_2,c_2, \ldots ,c_k)=1$?
Let $c_i\in \mathbb{N}$ and $c_1\geq c_2$. Why does $\gcd(c_1,c_2,\ldots ,c_k)=1\Rightarrow \gcd(c_1-c_2,c_2, \ldots ,c_k)=1$?
AI: Hint: Suppose $gcd(c_1-c_2,c_2,\cdots ,c_k)>1,$ then there exists a prime $p$ such that $p| c_1-c_2$ ... |
H: plot of $\sin(x) + \sin(y)= \cos(x) + \cos(y)$
I was playing arround with implicit plots of the form $f(x,y) = g(x,y)$, and I noticed that if you plot in the plane the following equation: $\sin(x) + \sin(y)= \cos(x) + \cos(y)$ you get the following graph:
My question is why does this trigonometric functions give u... |
H: Convolution: Integral vs. Discrete sum
I recently stumbled across a question which really confused my understanding of convolution. It's the relation between the continuous integral and the discrete counterpart I don't get.
What I learned in school was:
$\int_{a}^{b} f(x) dx = \lim_{\Delta x\to 0} \sum_{i=1}^{n} f(... |
H: Convergence of $\sum\limits_{n=1}^\infty\left\{\frac{1\cdot 3\dots 2n-1 }{2\cdot 4\dots 2n}\cdot\frac{4n+3}{2n+2}\right\}^2$
Show if the inf series
$\sum\limits_{n=1}^\infty\left\{\frac{1\cdot3\dots2n-1 }{2\cdot 4\dots2n}\cdot\frac{4n+3}{2n+2}\right\}^2$
converges.
My thought:
When $2n=2^k$, $\frac{1\cdot 3\dots2n-... |
H: Convergence of martingales is a martingale
I am reading the book "Introduction to Stochastic Integration (Second Edition)" by K.L. Chung and R.J. Williams. I have a question about the proof of Proposition 1.3 (on page 13) in that book. First, here are two definitions just so we are on the same page:
Definition. A c... |
H: Analytic and bounded in a region with two holes is constant
Let $p,q\in\mathbb{C}, p\neq q$ and $G = \mathbb{C}\setminus\left\{p,q\right\}$ and
$f: G\to \mathbb{C}$ analytic and bounded. Show that $f$ is constant.
Hi. How could I start from this problem? I suspect that with Cauchy's modification ...
actualization.
... |
H: A variant version of Euler's phi function
The original Euler's phi function goes like this.
$$\phi(n)=n\prod_{p|n} (1-1/p)$$
But I want to prove a modified version of it.
$\psi(n) $ : The number of $x$s when $1\le x \le n$ , $x\bot n$ and $(x+1) \bot n$. Then, for $n \ge 3 $
$$\psi(n)=n\prod_{p|n} (1-2/p)$$ wher... |
H: What am I misunderstanding about this isomorphism?
From Rotman's Algebraic Topology:
If $K$ is a connected simplicial complex with basepoin $p$, then $\pi(K,p) \simeq G_{K,T}$, where $T$ is a maximal tree in $K$ and $G_{K,T}$ represents the group with presentation $$ \langle \text{all edges $(p,q)$ in $G$} | (p,q)... |
H: Evaluate $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{i-3 n}$
Evaluate $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{i-3 n}$
Here $\sum_{i=1}^{n} \frac{1}{i-3 n}=\frac{1}{1-3 n}+\frac{1}{2-3 n}+\cdots+\frac{1}{n-3 n}$.
I tried to make the sum squeezed
between Convergent Sequences, but failed by get... |
H: How to prove something is a metric.
Let $X$ be a set and show that the following function defines a metric.
$f(x, y) = (0 \text{ if } x = y \text{ and } 1 \text{ if } x \neq y)$
I'm especially having trouble with the symmetry and triangle inequality steps. Thanks so much!
AI: Let's do symmetry first. We just need t... |
H: What's the best alternative to using the $\sin^2x$ notation?
Although I understand that $\sin^2x$ refers to $(\sin x)^2$, and not $\sin(\sin(x))$, I find this notation to be confusing—often it can hamper my thinking. For instance, if I was solving the following equation:
$$
\sin^2x-\frac{7}{2}\sin x-2=0
$$
I might ... |
H: What is the number of ways you can build a sequence of positive integers with each number divisible by all the previous numbers?
I am looking for a solution to count the number of ways I can create a sequence of $n$ numbers, repetition allowed, where each number is divisible by all the previous numbers in the seque... |
H: need help solving a quartic equation
The question I am asking is to solve the equation $x^4-4x-1=0$,
I need an exact answer. What I have done was found out that it equals $(x^2+1)^2 - 2(x+1)^2 =0$.
Anybody help me, please?
AI: So you know
$$\begin{align}
x^4-4x-1&=(x^2+1)^2-2(x+1)^2\\
&=(x^2+1)^2-[\sqrt2(x+1)]^2.
\... |
H: Convergence of $\sum \limits_{n=1}^{\infty}\sqrt{n^3+1}-\sqrt{n^3-1}$
Hello I am a high school student from germany and I am starting to study math this october. I am trying to prepare myself for the analysis class which I will attend so I got some analysis problems from my older cousin who also studied maths. But ... |
H: Recover $f$ if we know that $\frac{d}{dx} \log f(x)$ and $f(x) \to 0$ as $x \to \infty$
Let me set up the question by stating something that is well-known. Suppose $f$ is an unknown function, but we are given $f'$ and the fact that
\begin{align}
\lim_{x \to \infty} f(x)=0.
\end{align}
Then, by using fundamental t... |
H: Is every function on the plane that satisfies this mixed convexity condition convex?
$\newcommand{\half}{\frac{1}{2}}$
Let $g:\mathbb R^2\to \mathbb R$ be a smooth symmetric function, i.e. $g(x,y)=g(y,x)$ for every $x,y$. Suppose that
$$
g\left( \half(x,y)+\half(y,x) \right) \le \half g(x,y)+\half g(y,x) =g(x,y), \... |
H: Finding the parametric equation of a line in linear algebra
I was given the above question to answer, and I do not know how to do it. If t is the parameter, then I suppose we can create a relationship between a and b using t, but I don't see how that is helpful. Any guidance?
Thank you!
AI: $l:\Bbb R \to \Bbb R^2$... |
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