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H: Integration of an exponential function with an extra constant
I have an integral as shown below.
$$ \int_{-a}^a e^{jkx(sin\theta cos\phi-\alpha) }dx$$
Normally I would define the $sin\theta co\phi$ as $x$ and solution of the integral would become
$$a\dfrac{sinX}{X}$$
What does $\alpha$ term change in the calculatio... |
H: Is every invertible matrix a composition of elementary row operations?
If $A$ is invertible that means we can multiply it on the left by matrices $E_1, ..., E_k$ that correspond to elementary row operations until we get the identity matrix $I$. In other words, we get $E_1 \cdots E_k A = I$. Hence the inverse of $A$... |
H: Trying to solve $\frac{f(x) f(y) - f(xy)}{3} = x + y + 2$ for $f(x)$
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
$$\frac{f(x) f(y) - f(xy)}{3} = x + y + 2$$ for all $x,y \in \mathbb{R}$. Find $f(x)$.
I started by multiplying both sides by $3$, which gets
$$f(x)f(y)-f(xy)=3x+3y+6.$$
I tried to find... |
H: How should I calculate $||\underline{u}-\underline{w}||_{2}$?
I'm trying to calculate $||\underline{u}-\underline{w}||_{2}$ where:
$$
u=\begin{bmatrix}1 & 3\\
2 & 2\\
3 & 1
\end{bmatrix},\,\,\, w=\begin{bmatrix}3 & 1\\
2 & 2\\
1 & 3
\end{bmatrix}
$$
I'm not familiar with the $|| \cdot ||_2$ operator and I'm not sur... |
H: Question about notation: linear map
For a linear map $A\in Hom(U,V)$ and a linearly independent subset $L\subseteq U$ where U and V are vector spaces, what does the following statement mean:
$A$ is injective $\left.\Rightarrow A\right|_{\text {Span } L}$ is injective $\left.\Leftrightarrow A\right|_{L}$ is injectiv... |
H: Proving a given set is not a vector space
Let $V$ denote the set of ordered pairs of real numbers. If $(a_1,a_2)$ and $(b_1,b_2)$ are elements of $V$ and $c\in \mathbb{R}$, define $$(a_1,a_2)+(b_1,b_2)=(a_1+b_1,a_2b_2)$$ and $$c(a_1,a_2)=(ca_1,a_2)$$
Is $V$ a vector space over $\mathbb{R}$ with these operations?... |
H: How many kinds of average are there?
I have learned that the mean, median and mode are three kinds of average.
Are there other kinds of average or are there just these three ?
AI: There are two main categories of averages. One, mathematical averages focus on using mathematical tools to find the "average". These are... |
H: Finding $x$ such that $2^{4370} \equiv x \ (\mathrm{mod} \ 31)$
How to find $x$ such that $2^{4370} \equiv x \ (\mathrm{mod} \ 31)$?
The task is to compute $2^{4370} \ (\mathrm{mod} \ 4371$).
I know it's $4371=3 \cdot 31 \cdot 47$, so it's $2 \equiv -29 \ (\mathrm{mod} \ 31)$.
With Fermat's little theorem it's $-29... |
H: How do I obtain the pdf of a random variable, which is a function of random variable.
A random variable, $X$, has a value of zero with probability $1/3$, and follows a uniform distribution over $[-1, 1]$ with probability $2/3$. How can I derive the pdf of $X$?
In my opinion, $X$ can be formulated as $$X=\cases{0, &... |
H: If I have two consecutive Integers and I have the following formula $n(m+1)^2$ is it even of odd?
I am helping my sister study for the praxis exam of this study book, and I reviewed a question based on number theory. I see it involves constant integers my question is:
If $m$ and $n$ are consecutive integers, which ... |
H: Prove that $\forall x \in \mathbb{R}:f(x + 2\pi) = f(x)$
I'm trying to prove that if $f : \mathbb{R} \rightarrow \mathbb{R} \phantom{2}$ is a function that verifies :
$\exists\, K \in \mathbb{R^+}, \phantom{1}\forall\, x,y \in \mathbb{R}: \lvert f(y)-f(x) \rvert \le K\lvert \cos y - \cos x \rvert$
then $\forall x... |
H: Finding all zeros of $f(z)=\sin(\frac{z}{\pi})$
I'm trying to find the zeros of the function $f(z)=\sin(\frac{z}{\pi})$.
I began by noting that we can define $g(z)=\sin(z)$ as $g(z)=\frac{1}{2i}\left(e^{iz}-e^{-iz}\right)$, so:
$$f(z)=\frac{1}{2i}\left(e^{i\frac{z}{\pi}}-e^{-i\frac{z}{\pi}}\right)$$
Now we need to ... |
H: Finding a diagonal matrix $B$ and a unitary matrix $C$ that satisfy $B=C^{-1}AC$.
The matrix $A$ is given as $$A=\frac{1}{9} \begin{bmatrix} 4+3i & 4i & -6-2i \\ -4i & 4-3i & -2-6i \\ 6+2i & -2-6i & 1 \end{bmatrix}$$
Find a diagonal matrix $B$ and a unitary matrix $C$ that satisfy $B=C^{-1}AC$.
Could anyone hel... |
H: why $x_m$ converges weakly to $x_\infty$?
Let $(X,\|.\|)$ be reflexive Banach space and $Y$ be a closed separable subspace of $X$ $\big((Y ,\|.\|)$is clearly a separable reflexive Banach space$\big)$, then the dual space $Y^*$ of $Y$ is separable. Let $\{y_n^*\}$ be a countable dense subset of $Y^*$.
Let $\{x_m\}$ ... |
H: Prove $\sqrt{a^2 + ab + b^2} + \sqrt{b^2 + bc + c^2} + \sqrt{c^2 + ac + a^2} \ge \sqrt{3}(a + b + c)$
Prove $\sqrt{a^2 + ab + b^2} + \sqrt{b^2 + bc + c^2} + \sqrt{c^2 + ac + a^2} \ge \sqrt{3}(a + b + c)$
So, using AM-GM, or just pop out squares under square roots we can show:
$$\sqrt{a^2 + ab + b^2} + \sqrt{b^2 + ... |
H: Is $f$ differentiable at $0$? and if it is what is the value of $f'(0)$
I'm studing that if $f : \mathbb{R} \rightarrow \mathbb{R} \phantom{2}$ is a function that verifies :
$\exists\, K \in \mathbb{R^+}, \phantom{1}\forall\, x,y \in \mathbb{R}: \lvert f(y)-f(x) \rvert \le K\lvert \cos y - \cos x \rvert \Rightarr... |
H: Find $\lim_\limits{x\to 0}\frac{1-(\cos x)^{\sin x}}{x}$ using little o
$\lim_\limits{x\to 0}\frac{1-e^{\sin(x) \ln(\cos x)}}{x}=\lim_\limits{x\to 0}\frac{1-e^{(x+o(x))\ln(1-\frac{x^2}{2}+o(x^2))}}{x}=\lim_\limits{x\to0}\frac{1-e^{(x+o(x))(-\frac{x^2}{2}+o(x^2))}}{x}$. If this is correct, what will happen with $o()... |
H: How to find such U and V such that U$\cap$V=$\emptyset$
Assume that A and B are closed disjoint subsets. Then there exist open sets U$\supset$A and V$\supset$B with U$\cap$V =$\emptyset$
I am said to deduce it from (Urysohn's Lemma). Let A, B be two disjoint closed subsets of a metric space. There exists a continuo... |
H: Would symmetry positive semi-definite matrix always decomposable?
Given symmetry positive semi-definite matrix $A \in R^{n\times n}$. And $Det(A) \geq 0$.
Would there always exist real matrix $B$, such that $A = B \cdot B^T$?
If so why? Or why not?
AI: The answer is positive. There exists several decompositions l... |
H: Limit problem $\lim_{T \to 0} \frac{1}{T} \int_0^T S_u du$
Consider $dS_t = \mu S_t dt + \sigma S_t dW_t$ with initial $S_0 > 0$. We may obtain that $S_t = S_0 \exp\left[(\mu - \frac{1}{2}\sigma^2) t +\sigma W_t\right]$. Hence we may consider average value of $S_T$. $A(T) = \displaystyle \frac{1}{T}\int_0^T S_0 e^{... |
H: If $a$ is relatively prime to $m$ and $a \equiv b\ (\textrm{mod}\ m)$, is $b$ relatively prime to $m$?
If $a$ is relatively prime to $m$ and $a \equiv b\ (\textrm{mod}\ m)$, is $b$ relatively prime to $m$?
Hint:. Recall that $a \equiv b\ (\textrm{mod}\ m)$ if and only if $a$ and $b$ differ by a multiple of $m$.
S... |
H: Help with Conditional probability for a future event
I have the following problem, in a statement I am given the following conditional probabilities
$$P(x_i | x_{i-1}) = 0.7$$
$$P( \overline{x_i} | x_{i-1}) = 0.3$$
$$P(x_i | \overline{x_{i-1}}) = 0.4$$
$$P(\overline{x_i} | \overline{x_{i-1}}) = 0.6$$
These indica... |
H: Inequality involving ranks
I'm trying to prove the following inequality,
$$
\rho(AB) + \rho(BC) \le \rho(B) + \rho(ABC)
$$
where $A, B, C \in L(V)$, $V$ is a finite-dimensional vector space and $\rho(A)$ means the rank of the linear operator $A$.
I know that $\min(\rho(AB), \rho(BC)) \le \rho(B)$. So I tried to pr... |
H: Modified Newton method and contraction principle
I am studying Newton's method modified by the book Zorich, Mathematical analysis II, page 39,40:
It seems to me, if I make no mistakes, that there is a problem in the derivative of $ A (x) $. The author says that $ | A '(x) | = | [f' (x_0)] ^ {- 1} \cdot f '(x) | $... |
H: Show that $K_{r, s}$ is planar if and only if $\min$ {r, s} ≤ 2.
So I've done some draws and this is true, but How can I argument to prove that, by the maximum number of edges in $K$ ?
Or by $d(v)$
Any help?
AI: This basically boils down to a special case of Kuratowski's theorem. In the special case of bipartite g... |
H: Range of the function $f:\mathbb{Z} \to (\mathbb{Z}/4\mathbb{Z},\mathbb{Z}/6\mathbb{Z})$
Let $f:\mathbb{Z} \to (\mathbb{Z}/4\mathbb{Z},\mathbb{Z}/6\mathbb{Z})$ be he function given by $f(n)=(n$ mod 4,$n $ mod $6)$.Then
$(1)(0$ mod $ 4 ,3$ mod $6)$ is in the image of $f$
$(2)(a$ mod $ 4 ,b$ mod $6)$ is in the image... |
H: $\frac{w_k}{x-x_k}$ expansion into decreasing powers of $x$
How can $\dfrac {(w_k)}{(x-x_k)}$ becomes:$$\dfrac {w_k}x+\dfrac {w_kx_k}{x^2}+\dfrac {w_kx_k^2}{x^3}+...$$
I couldnt figured out the process.
AI: Hint:
You can use this:
$$\dfrac 1 {1-x}=\sum_{n=0}^\infty x^n$$
For $|x| <1$ and note that you have:
$$\dfra... |
H: Proof that the set of polynomial in $Z[x]$ with linear term coefficient equal to $0$ is a domain
Can anybody help me on that? I'm having trouble
AI: Hint: Let $R$ be the set in question. It is enough to prove $R$ is a subring of $\mathbb Z[x]$. It'll automatically be a domain. To prove that $R$ is a subring, it is ... |
H: write down functions in terms of complex coordinate $z=x+iy$
Indentifying $\Bbb R^2$ with the complex plane $\Bbb C$ via the map $(x,y)→ x+iy$, write down the following functions in terms of complex coordinate $z = x + iy$.
i) the translation by the vector (1,2)
ii) a rotation anticlockwise by $\theta$
iii) a refle... |
H: How to prove that $\lim_{n\to\infty}\int_{0}^{2}\frac{x^n}{x+1}=\infty$
I'm asked to prove that $$\lim_{n\to\infty}\int_{0}^{2}\frac{x^n}{x+1}=\infty$$ I've tried using the fact that for every $$x\in[0,2], \frac{x^n}{x+1}\ge\frac{x^n}{3}$$ but I can't seem to be able to calculate $\int_{0}^{2}\frac{x^n}{3}$ using o... |
H: Is there an infinite set with a discrete cyclic order?
Let's call a cyclic order of a set discrete if every cut of the order is a jump.
A cut of a cyclic order is a linear order $<$ such that $x < y < z \implies (x, y ,z)$ for any elements $x$, $y$, $z$ of the set.
A cut of a cyclic order is a jump if it has the le... |
H: $f_n:(0,\infty) \to\mathbb{R}, f_n(x)=\frac{1}{1+nx}$ uniform convergence
If $(f_n)_{n\in \mathbb{N}}$ is pointwise convergent: Is the limit function continuous? Is it uniform convergent?
$f_n:\mathbb{R} \to\mathbb{R}, f_n(x)=xsin(nx)$
$f_n:(0,\infty) \to\mathbb{R}, f_n(x)=\frac{1}{1+nx}$
diverges
$\lim_{n\to\i... |
H: Trouble getting to an explicit solution
Given the ODE: $(y^2-1)\frac{dy}{dx} =4xy^2$
I can get to an implicit solution easily enough: $y+\frac{1}{y} = 2x^2$+c.
However, I've been given an explicit solution: $y(x)=x^2-c_2\pm\sqrt{(c_2-x^2)^2-1}$
and I can't figure out how to get there. I'm teaching myself ODEs so ... |
H: Finding the general formula for the sequence with $d_0=1$, $d_1=-1$, and $d_k=4 d_{k-2}$
Suppose that we want to find a general formula for the terms of the sequence
$$d_k=4 d_{k-2}, \text{ where } d_0=1 \text{ and } d_1=-1$$
I have done the following:
\begin{align*}d_k=4d_{k-2}&=2^2d_{k-2} \\ &=2^2\left (2^2d_{(... |
H: Find maximum $\theta$ such that $|x + \theta a| \leq b$
I have an optimisation problem:
$$
\max_{\theta} \quad \theta \\
\text{such that} \qquad |x + \theta a| \leq b
$$
where $x, a \in \mathbb{R}^{n}$. We know that $|x| \leq b$. The norm referred to here is the $\ell_{1}$-norm.
Is there a simple way to solve this... |
H: Is Banach–Tarski paradox false without axiom of choice?
I know that you need axiom of choice to prove Banach–Tarski paradox. But what happens with paradox when we remove axiom of choice? Does theorem become false? Or is there just no proof of it without axiom of choice?
AI: As explained in the last paragraph here, ... |
H: What is the time complexity of the function $5^{\log_3(n)}+n^{1.5}\sum_{j=0}^{log_3n-1}\left(\frac{5}{3^{1.5}}\right)^j$?
I need to find the $\Theta$ complexity of this function:
$$5^{\log_3(n)}+n^{1.5}\sum_{j=0}^{log_3n-1}\left(\frac{5}{3^{1.5}}\right)^j$$
It shouldn't be too hard, and I already have simplified it... |
H: If $f(x)$ is a polynomial in $\mathbb{Z}$ and $f(a)\equiv k\pmod{n}$, prove that, for all integer $m$, $f(a+mn)\equiv k\pmod{n}$
I'm trying to solve this excercise:
If $f(x)$ is a polynomial in $\mathbb{Z}$ and $f(a)\equiv k\pmod{n}$
. Prove that for all integer m
$f(a+mn)\equiv k\pmod{n}$.
I know that if $f(a)\equ... |
H: Probabilistic Proof of a Hausdorff-Young Type Inequality
Let $1 \leq p <2$ and let $q$ be the Holder conjugate of $p$ so that $\frac{1}{p} + \frac{1}{q} = 1$. Show that for any $\epsilon >0$, there exists a Schwartz function $f \in S(\mathbb{R}^d)$, such that:
$$
\|\hat{f}\|_{L^{q}(\mathbb{R}^d)} \leq \epsilon \|f\... |
H: Proving that $I_{n}+\lambda C^{T}C$ is a positive defined matrix
I'm trying to prove that the matrix $A=I_{n}+\lambda C^{T}C$ is positive defined (PD) for $\lambda >0$ and some $C_{n\times m}$. I have already proved that the matrix $A$ is symmeric and of order $n\times n$. I'm trying to prove that for every $\under... |
H: Combinatorics - how many ways to divide balls in two groups
Suppose I have:
8 black balls
3 white balls
5 blue balls
how many ways there are to divide those balls into two different groups (note that there is no need to divide into two groups with even number of balls, one group could have 1 e the other 15).
In ... |
H: Linear map triangulizable
What's the definition of a linear map that is triangulazible? I can't find it anywhere.
In addition, I was asked to find a linear map that doesn't have any invariant sub-spaces. I know that if a map is triangulazible it does have invariant sub-spaces, from there my request on the exact def... |
H: Need help understanding statement "By linear algebra we know $\left|A,B,C\right|=-(A\times C)\cdot B=-(C\times B)\cdot A$
I am reading a paper for a famous ray-triangle intersection procedure https://cadxfem.org/inf/Fast%20MinimumStorage%20RayTriangle%20Intersection.pdf
They use Cramer's rule to solve a set of equa... |
H: Solution of the following integral Equation $\varphi(x) - \lambda\int\limits_{-1}^1 x e^t\varphi(t) \: dt=x$
Consider that the following equation is solvable then analyze with respect to $\lambda$
$$\varphi(x) - \lambda\int\limits_{-1}^1 x e^t\varphi(t) \: dt=x$$
Can someone tell me how can I solve it ?
AI: If ther... |
H: Does $f_{n}(x)=\frac{x^{2n}}{1+x^{2n}}$ converge pointwise / uniformly?
Since our lectures were cancelled because of the ongoing situation, I have to essentially self-study for my analysis exam in two months. Understandably this comes with a great deal of trouble, so I would like someone to help me with the followi... |
H: Show that $\Gamma(x) \sim \sqrt{2 \pi} e^{-x}x^{x-\frac12}$.
The gamma function is defined by
$$\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} dt $$
where $x > 0$. Show that $\Gamma(x) \sim \sqrt{2 \pi} e^{-x}x^{x-\frac12}$.
$\sim$ denotes that the ratio between the left and the right side tends to $1$.
I thin... |
H: Every neighborhood is an open set proof question
Theorem: Every neighborhood is an open set.
Proof: Consider a neighborhood $E=N_r(p)$, and let $q$ be any point of $E$. $$\text{There is a positive real number $h$ such that $d(p,q)=r-h$} \tag 1$$
$$\text{For all points s such that $d(q,s) \lt h$, we have $d(p,s) \... |
H: prove that function mapping is injective iff ker (f) = {e}
Heyall, would appreciate some help with abstract algebra because my undergrad brain is fried from doing all the proofs my prof asked me to do. I've hit a bit of a wall with this one; it involves group homomorphisms - super basic but the proof has got to be ... |
H: finiteness of Koszul groups
A basic question about Koszul homology from Matsumura's Commutative Ring Theory
In Theorem 16.5(ii) it is assumed that $(A,m)$ is a local ring and $x_1,\ldots,x_n \in m$, and $M$ is a finite $A$-module. Then it is claimed without much explanation that the Koszul homology groups $H_p(X,M)... |
H: Limit of the exponent of Random Variables
Suppose $X_1, X_2,\ldots$ are i.i.d. normal ($\mu=0, \sigma^2=1$) random variables and let $S_n$ denote the sum of first $n$ $X_i$'s. Show that $$\lim_{n\to \infty} \exp(2S_n - 2n)=0$$
I think I am supposed to use the Martingale Convergence Theorem here, where $M_n=\exp(2S_... |
H: How to come up with a set of three linearly dependent vectors in a systematic way
Give an example of three linearly dependent vectors in $\mathbb{R}^{3}$ such that none of the three is a multiple of another.
Three vectors that satisfy this property are the vectors :$\{(-1,2,1), (3,0,-1),(-5,4,3)\}$.
Now that's all... |
H: Good reference that discusses NP hardness in the context of optimization?
Sometimes I read a book on optimization and the author states (without proof) that finding a certain solution to the (non-convex) optimization problem is NP hard.
I've learned about complexity theory in the past through a course in CS, and I ... |
H: In a triangle, G is the centroid of triangle ADC. AE is perpendicular to FC. BD = DC and AC = 12. Find AB.
G is the centroid of the triangle ADC. AE is perpendicular to FC. BD = DC
and AC = 12. Find AB.
According to the solution manual, we can let the midpoint of AC be H. D, G, and H are collinear as G is the cent... |
H: Entire function $f(\frac{1}{p})=\frac{1}{1+p}$ for all prime $p$.
I'm working on this problem
"Find all entire functions $f(\frac{1}{p})=\frac{1}{1+p}$ for all prime $p$."
My approach is using identity theorem. But in this case it does not seem good. We have $$f(1/p)=\frac{1/p}{1+1/p}$$
So naturally, I set $g(z)=f(... |
H: Generate a Poisson random variable from a standard uniform random variable.
I can't solve the following exercise:
A random number generator generates random values $U \sim \text{U}(0,1)$ from the standard uniform distribution. Use $U$ to generate a random variable $P \sim \text{Pois}(\lambda = 5)$ from a Poisson ... |
H: Show that if $f:X\to\textbf{R}$ is a continuous function, so is the function $|f|:X\to\textbf{R}$ defined by $|f|(x) = |f(x)|$.
Show that if $f:X\to\textbf{R}$ is a continuous function, so is the function $|f|:X\to\textbf{R}$ defined by $|f|(x) = |f(x)|$.
MY ATTEMPT
According to the definition of continuity, for ev... |
H: In the isosceles triangle, the two squares (white) both have an area of four. Find the area of the shaded.
In the isosceles triangle below, the two squares (white) both have an area
of four. Find the area of the shaded.
According to my answer key, the answer is $9\sqrt{2}$ square units. How can I show the solution ... |
H: Spivak's Calculus Q 1-20
Question 1-20: Prove that if $|x-x_0| < \frac{\epsilon}{2}$ and $|y-y_0| < \frac{\epsilon}{2}$ then $|(x+y) - (x_0 + y_0)| < \epsilon$ and $|(x-y) - (x_0 - y_0)| < \epsilon$.**
I have proven the first inequality by expanding the absolute value into $x_0 -
\frac{\epsilon}{2} < x < x_0 + \fr... |
H: Obtain for n > 0 a relation of the form $I(m, n) = kI(m, n − 1)$
The function $I(m, n)$, where $m ≥ 0$ and $n ≥ 0$ are integers, is defined by
$$I(m, n) = \int_{0}^{1} x^m(-\ln x)^n dx$$
Obtain for $n > 0$ a relation of the form $I(m, n) = kI(m, n − 1)$, where $k$ is
to be found. Hence obtain an explicit formula f... |
H: Can the proof about direct sum decomposition of the inner product space be generalize to infinitely dimension space
There is a theorem about the finite-dimensional inner product space.
Suppose a finite-dimensional inner product space $V$ with a subspace $W$, then $V=W\bigoplus W^{\bot}$.
And the proof is as follo... |
H: Show that the direct sum $f\oplus g:X\to\textbf{R}^{2}$ defined by $f\oplus g(x) = (f(x),g(x))$ is uniformly continuous.
Let $(X,d_{X})$ be a metric space, and let $f:X\to\textbf{R}$ and $g:X\to\textbf{R}$ be uniformly continuous functions. Show that the direct sum $f\oplus g:X\to\textbf{R}^{2}$ defined by $f\oplus... |
H: How can $z = xa + x$ be differentiated with only chain rule?
I am trying to put some rigour to my understanding of the Chain Rule (with Leibniz Notation). I came across this question and the second answer there (David K's) states,
$\frac{dz}{dx} = a + 1 + x\frac{da}{dx}$, as you surmised, though you could also hav... |
H: Why is $f$ differentiable on each straight line through the origin because on the straight line $y = m x$, it has the value $\frac{m x}{m^2 + x^2}$?
I am reading "Analysis on Manifolds" by James R. Munkres.
Let $f(0, 0) := 0$ and $f(x, y) := \frac{x^2 y}{x^4 + y^2}$ for $(x, y) \ne (0, 0)$.
There are the foll... |
H: Understand significance of theorem related to normal subgroups
I found in an online book about abstract algebra the following theorem:
The following theorem is fundamental to our understanding of normal subgroups.
Theorem 10.3. Let $G$ be a group and $N$ be a subgroup of $G$. Then the following statements are equ... |
H: Is my limit development correct?
I have this limit to find:
$\lim\limits_{x\to0}\frac{\sqrt[n]{1+x}-1}{x}$
My development was:
Let $\large{u^n = 1+x}$, from here if $x\to 0$ implies that $\large{u^n \to 1}$
And i got: $\Large{\lim_{u^n \to 1}\frac{u-1}{u^n - 1}}$ and using that $\Large{u^n - 1 = (u-1)\sum_{j=0}^{... |
H: the family of analytic functions with positive real part is normal.
I'm reviewing Complex Analysis and I don't quite understand the concept of normal family. There is an exercise in Ahlfors' Complex Analysis: Prove that in any region the family of analytic functions with positive real part is normal.
I believe this... |
H: A sequence $a_1=f'(0),a_2=f''(0),...$
I am working on this problem from my past Qual
"Give a sequence s.t. there is no analytic function $f:D\to \mathbb{C}$ s.t. $a_1=f'(0),a_2=f''(0),...$" where $D$ is the unit disk."
The only thing I can think of the Cauchy's integral formula$$f^{(n)}(0)=\frac{n!}{2\pi i} \int \f... |
H: Let $S \subset \mathbb{R^2}$ consisting of all points $(x,y)$ in the unit square $[0,1] × [0,1]$ for which $x$ or $y$, or both, are irrational.
Full question: Let $S$ be the subset of $\mathbb{R^2}$ consisting of all points $(x,y)$ in the unit square $[0,1] × [0,1]$ for which $x$ or $y$, or both, are irrational. Wi... |
H: Injective function from unit circle
Let $S$ denote the set of points on the unit circle centred at $(0,0)$. Does there exist an injective function $f : S \rightarrow S \setminus \{(1,0)\}$?
AI: Let $v_1=(1,0), v_2=(\cos 1, \sin 1), v_3=(\cos \frac1 2, \sin \frac 1 2 ),v_4=(\cos \frac 1 3, \sin \frac 1 3 ),....$... |
H: Where is $f(x) = |x^2(x+1)|$ differentiable? And where are they $C^1$ and $C^2$?
I'm a maths student taking a real-analysis paper and I'm currently working down my problem sheet. I've been asked the above question.
First I define a piece-wise function to describe the absolute function above.
$$f(x)= \begin{cases}
... |
H: Is the restriction of an inverse function invertible?
Let $f:A\rightarrow B$, where $A$ and $B$ are open sets of $\mathbb{R^n}$ for some n, be invertible. Let $C$ and $D$ be open subsets of $A$ and $B$ respectively. Is $f:C \rightarrow D$ invertible?
AI: One can discuss this question more generally, without restric... |
H: What's the relation between $\mathbf E(X)$ and $\mathbf E(e^X)$?
Given $\mathbf E(X)=0$ and $-1\le X\le 1$, show that $\mathbf E(\text{exp}(\sqrt{2}X))\le e^{\sqrt{2}}-\sqrt{2}$.
It seems that Jensen's inequality will help, but I have no idea.
Thanks in advance.
AI: Hint: Since $-1 \le X \le 1$, we have $X^k \le ... |
H: What tools i need to show the following?
Let $X,Y$ be NLS. Let $T: X\rightarrow Y$ be a linear map. Prove that $T$ is continuous iff it is continuous at $0\in$X.
Honestly I don't understand this question if $T$ is continuous on $X$ then it is continuous at each point of $X$ and since $0\in X$ then it is continuous ... |
H: What is the minimum score after which all numbers can be scored?
There are two types of scores present in a game 4,7. What is the minimum score after which all numbers can be scored?
I found the answer '18' without any accurate logic. This a math problem of Olympiad. Can you please give me a method?
AI: The link ... |
H: Show that $|\sin(0.1) - 0.1| \leq 0.001$ with the lagrange remainder
Show that $|\sin(0.1) - 0.1| \leq 0.001$
I know that's a basic exercise on taylor polynomial but I have made a mistake somewhere that I don't find out. Anyway, here's my attempt :
Because the function $f: \mathbb{R} \rightarrow \mathbb{R}$, $x \... |
H: How to find the pre-image of a relation given the interval?
$$
\begin{aligned}
&\begin{array}{l}
\text { 2) Given the following relations: } \\
\qquad f=\left\{(x, y) \text { , } x, y \in Z, y=x^{4}+4\right\}, \text { a relation from } Z \text { to } Z \text { . }
\end{array}\\
&\begin{array}{l|l}
\mathrm{g}=\{(x,... |
H: What will be the remainder when 7^2020 is divided by 4?
Problem: "What will be the remainder when $7^{2020}$ is divided by $4$?"
I can't get a step to approach such type of question but all I know is the answer is $1$.
AI: $7\equiv -1\pmod{4}$
So, $7^{2020}\equiv (-1)^{2020}\equiv 1\pmod{4}$ |
H: Expectation of score function (partial derivative of the log-likelihood function)
according to the Wikipedia: https://en.wikipedia.org/wiki/Score_(statistics), expected value of a score function should equals to zero and the proof is following:
\begin{equation}
\begin{aligned}
\mathbb{E}\left\{ \frac{ \partial }{ \... |
H: Sum of arithmetic progression formula
The question that is asked is to find the series described, and than calculate the sum of the first n terms.
Now I have done some research and I found a formula I thought might work, which is
$$ s_n = \frac{n}{2}(2a_1 + (n-1)d)$$
where the series is of the form
$a_n = a_1 + (... |
H: How to prove the Riemann Zeta fuction tends to infinity when $x$ tends to $1$
The Riemann Zeta Function is convergent over the interval $(1,\infty)$, and $\sum_{n=1}^{\infty}\frac{1}{n^x}$ tends to infinity when $x\rightarrow 1^ {+} $, it seems one can feel it is right because the when $x=1$ the function is infinit... |
H: sigma algebra created by partition
So there is a countable (disjoint) partition of $\Omega = \bigcup_{i \in \mathbb{N}}B_i$ and now I'm interested in the $\sigma$-algebra created by this partition $\sigma(\{B_i:i\in\mathbb{N}\})$.
I've been wondering whether this is just the Powerset $\mathcal{P(\Omega)}$, because ... |
H: If $Z_n = X_n + Y_n$ for $X_n\in M$ and $Y_n\in N$ then $(X_n)$ and $(Y_n)$ converge
Let $H$ be a Hilbert space (infinite dim) with $M,N\subset H$ being closed subspaces satisfying $N\subset M^\perp$.
I'm trying to show that $M+N$ is closed.
If $(Z_n)_{n=1}^\infty \subset M+N$ is a sequence then $Z_n = X_n+Y_n$ for... |
H: Prove that $\det(B^TB) \neq 0$
I have a matrix $A_{N \times M}$ such that $$A=U^T_{N \times N} \cdot B_{N\times M} \cdot V_{M \times M},$$ where $U,V$ orthogonal and $B_{ij}$ may has nonzero values only for $i\le j \le i+1$ and $A$ is full order matrix. How to prove that $B^TB$ also has $\det(B^TB) \neq 0$.
I thin... |
H: whats the simplest way to find this circle's center if known its tangent line
the circle has a tangent line $y = 2x + 1$ at $(2,5)$ and its center on the line $y = 9 - x$. If that's circle intersect the $x$ -axis at $x_1, x_2$ what's $x_1 + x_2$ ?
i understand than $x_1 + x_2 = 2x_0$ when $x_0$ is the circle's cent... |
H: Prove that the function $f :\Bbb R \to \Bbb R$ defined by $f(x) = e^{-\cos(x)^2}$, for all $x \in\Bbb R$, has a unique fixed point on $\Bbb R$.
Hint: some arguments might be simpler if you recall the trigonometric formula $2\sin(x)\cos(x) = \sin(2x)$. Remember also that $\cos$ and $\sin$ are $2\pi$-periodic functio... |
H: Monotonicity and strict order relations
Suppose we have a function $g$ that is differentiable (and hence continuous) and monotonically increasing on the interval $[P,Q]$.
I know that this alone is not enough to imply that if $a,b\in [P,Q]$ and $a>b$, then $g(a)>g(b)$, because this is a strict order relation and we ... |
H: contour integration complex variables
Using Contour integration, evaluate
$$
\int_{-\infty}^{\infty} \frac{\cos x}{ (x^2 +1)^2}\, dx
$$
AI: Consider integrating the function
$$f(z) = \frac{e^{iz}}{(z^2+1)^2}$$
along the contour $C_R$, the curve from -R to R along the real line and then along the half circle in the... |
H: Continuous function on $[0,1]\to [0,1]$ that is not Lipschitz Continuous?
Continuous function on $[0,1]\to [0,1]$ that is not Lipschitz Continuous?
One example I could perhaps think of is $f(x)=sin(\frac{1}{x})$ where we define $f(0)=0$.
Then this function has the required domain and range. Now, I was wondering by... |
H: Find inverse of $[x+1]$ in factor ring $\mathbb{Q}[x]/\left\langle x^3-2 \right\rangle$
Find inverse of $[x+1]$ in factor ring $\mathbb{Q}[x]/\left\langle x^3-2 \right\rangle$. I remember that I need to use the extended Euclidean algorithm, but it has been some time, so I am a bit rusty. Thanks in advance!
Edit: I ... |
H: How many possible passwords are there for 8-100 characters?
Requirements/Restrictions: Minimum of 8. Maximum of 100. At least 1 letter from the latin alphabet (capitalisation doesn’t matter—g is same as G, 26 letters), at least 1 number (0-9, 10 numbers) and it may also include special characters (33 special charac... |
H: Uniform Convergence Infinite Series
I have been given the following series:
$$
\sum_{n=1}^\infty e^{-n^2x^2}
$$
Let now $a>0$. Argue that the series converge uniformly on the interval $[a,\infty)=\{x\in\mathbb{R}:x\geq a\}$. To do this i have been using Weierstrass' M-test. First i have said that as the exponential... |
H: Using definition of derivative at an inequality
The question is really simple but I'm not sure how can I prove it.
Let $f : \mathbb{R} \rightarrow \mathbb{R} \phantom{2}$ a function that verifies :
$\exists\, K \in \mathbb{R^+}, \phantom{1}\forall\, x,y \in \mathbb{R}: \lvert f(y)-f(x) \rvert \le K\lvert \cos y -... |
H: Calculating positive elements of $a_n$ with formula for $a_{n+1}$
I know this is a very simple high-school problem, but there is one detail that won't let me sleep.
The question:
For all $n\in \mathbb{N}_+$, the sequence $\{a_n\}$ satisfies following equations:
$$
a_n+a_{n+1}=\frac{-n^2+3n+17}{n^2+1}\\
a_n-a_{n+1}=... |
H: How to compute $\parallel f \parallel_{L_2(\mathbb{R}^2)}$ for $f(x,y)=\frac{1}{1+(x-y)^2}$?
So I want to compute $$\int\limits_\mathbb{R} \int\limits_\mathbb{R} \frac{1}{(1+(x-y)^2)^2} dxdy.$$
As I understand, I cannot reduce it to $1$-dimensional integrals, since Fubini's theorem requires measure of whole space t... |
H: How to solve $ x^\top A x = 0$?
We could assume $A$ is a positive-definite matrix, if that makes a difference.
How does one solve the equation $x^\top A \; x = 0$ ?
Is there a name to call such an $x$ which is a solution to the above equation?
AI: $x$ has to be the zero vector (in $\mathbb{R^n}$ or $\mathbb{C}^n$) ... |
H: How to solve this integral with transformation to polar coordinates?
How do I determine new limits when transforming to polar coordinates. I have this example, and I don't know how to solve it correctly.
$$
\iint_D \frac{\ln\left(x^2+y^2\right)}{x^2+y^2}\,dx\,dy
$$
where $D: 1\leq x^2+y^2\leq e^2.$
So I transforme... |
H: Find $f(x,y)$ when $f(x)$ and $f(y)$ are known
I have a problem related to the combination of 2 relations. I know the relation between the diffusion coefficient and the temperature (say D(T)) and I know the relation between the diffusion coefficient and the humidity (say D(H)). Now, I would like to write a function... |
H: Wrong proof of $TM$ is diffeomorphic to $M\times \mathbb{R^m}$
I want to see what's wrong in here:
Let $M$ be a smooth manifold with dimension $m$. I will show $TM$ is diffeomorphic to $M\times \mathbb{R^m}$.
proof) Define $F:TM\rightarrow M\times \mathbb{R^m}$ by $F(p,v)=(p,v^1,...,v^m)$ where $v=v^i\frac{\partial... |
H: $\Lambda(f) = f(N)$ for each $x$ is a bounded linear functional on $N^*$ of norm $||x||$.
In this case $N$ is a normed linear space and $N^*$ is the dual space with norm
$$\|f\| = \sup_{\|x\|\leq 1} \{ |f(x)| \} $$
I am required to show that the mapping $\Lambda : f \to f(x)$ for each $x\in N$ is a bounded linear f... |
H: Does $[−2, 3]\subset \operatorname{Im} f'$ for the defined function?
I'm trying to prove that if a function
$$
f : [−1, 1] \rightarrow \mathbb{R}$$
is continuous in
$[−1, 1],\phantom{2}$ differentiable in
$(−1, 1)$ and verifies
$$
f(−1) = 1,\phantom{1} f(0) = −1, \phantom{1} f(1) = 2
$$
Then the interval $[−2, 3]... |
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