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H: Why not generalize the concept of prime numbers?
A prime number is a natural number N > 1, that is divisible only by 1 and N (N/1).
The number 1 is given special treatment in this definition, and this can be generalized by extending the set of the specially treated numbers, for example to also include number 2, let... |
H: Would I be correct to assume that the minimum amount of vertices required to have an object with 3 dimensions is 4?
The question seems so simple to me but often the simplest ones are the most complex. My assumption is that the minimum amount of points required to create an object with 3 dimensions is 4?
You would g... |
H: Let $f: [0, +\infty] \rightarrow \mathbb{R}$ such as f' is decreasing and positive. Prove that $\sum_ {n=1}^\infty f'(n) $ converges iff f is bounded
It seems a lot like this question, but I wasn't able to prove with the same tips
AI: As $f'$ is decreasing, we have
$$f(n)-f(n-1)=\int_{n-1}^nf'(x)\,\mathrm dx \ge \i... |
H: Property of Lebesgue measure in $\mathbb{R}^2$
Let $I=[0,1]\times [0,1]$ and $E\subset \mathbb{R}^2,$ be a set of zero Lebesgue measure. Is it true that $$\overline{I\setminus E}=I?$$
I guess that the counterexample will be some form space filling curve.
AI: Yes, it is true.
Proving that $\overline{I\setminus E}\su... |
H: Is a column-wise and row-wise Gaussian matrix, jointly Gaussian?
Let $A\in\mathbb{R}^{n\times m}$ be a random matrix that each of its rows or columns is a Gaussian vector with iid components. More formally we have $A_{i,\cdot}\sim\mathcal{N}(0,I_m),A_{\cdot,j}\sim\mathcal{N}(0,I_n)$ for all $i\in[n],j\in[m]$, and f... |
H: How to show that the distance of the points of tangency along a tangent line on two tangent circles with radius $a$ and $b$ is equal to $2\sqrt{ab}$?
How to show that the distance of the points of tangency along a tangent line on two tangent circles with radius $a$ and $b$ is equal to $2\sqrt{ab}$?
Please see the i... |
H: arrow category and functor category
Let A be an abelian category and D the category having two objects and only one nonidentity morphism between them.
The functor category A$^D$ is also abelian and it is called an arrow category with objects morphisms in A and morphisms commutative squares.
I cannot see the equival... |
H: Property of Lebesgue measure in $\mathbb{R}^2$, part 2
Let $I=A\times B,$ where $A,B\subset \mathbb{R}$ are closed sets of positive Lebesgue measure, and $E\subset \mathbb{R}^2,$ be a set of zero Lebesgue measure. Is it true that $$\overline{I\setminus E}=I?$$
When $I=[0,1]\times [0,1]$ the answer is true and one c... |
H: Compute $\iint (x+y)\,dx\, dy$ with circle constraint $x^{2}+y^{2}=x+y$
I have a double integral:
$$\iint (x+y)\,dx\, dy$$
with circle constraint:
$$x^{2}+y^{2}=x+y$$
I tried to calculate it with transition to polar coordinates:
$$x^{2}+y^{2}=x+y$$
$$\left(x-\frac{1}{2}\right)^{2}+\left(y-\frac{1}{2}\right)^{2}=\fr... |
H: Intersection point of two curves with errors - covariance matrix
I have measured the parameters for a hyperbola and an ellipse, let us call them
$$
\begin{cases} a^2x^2 + b^2y^2 = 1 \\ c^2x^2 + d^2y^2 = 1 \end{cases}
$$
and I have errors associated with each parameter a, b, c and d ($\pm\delta a$, $\pm\delta b$, $\... |
H: Show $\mathbb{P}[X-m>\alpha]\leq \frac{\sigma^2}{\sigma^2+\alpha^2}$
I found this problem in an old statistics book:
Suppose $X$ is a square integrable random variable with mean $m$ and variance $\sigma^2$. For any $\alpha>0$, show
$$
\mathbb{P}[X-m>\alpha]\leq\frac{\sigma^2}{\sigma^2 +\alpha^2}
$$
At first I thou... |
H: Reference books on the Baum Connes conjecture
Do there exist readable reference books about Baum-Connes Conjecture for beginners ?
AI: Alain Valette, who is an active user on mathoverflow, has a book called Introduction to the Baum-Connes connecture. I remember reading parts of it a few years ago and finding it qui... |
H: Prove if infinite product of $f(x)$ is $0$ then so is infinite product of $f(x\varphi)$
Prove or disprove that if $$\prod\limits_{x=2}^{\infty} f(x)=0$$ and $f(x)\neq0$ for any $x\geq0$ then $$\prod\limits_{x=2}^{\infty} f(x\varphi)=0$$ for any constant $\varphi\geq2$
This seems true but I'm not quite sure how to p... |
H: Determining analytically the number of times a line intersects a general 3D surface
Consider a general surface and a line in $\mathbb{R}^3$. Given equations for both the surface and line, is there a way to analytically determine the number of times the line intersects the surface?
I am only interested in the number... |
H: Intersecting diameter and chord
A diameter $AB$ and a chord $CD$ of a circle $k$ intersect at $M.$ $CE$ and $DF$ are perpendiculars from $C$ and $D$ to $AB$. $(A,E,M,F,B$ lie on AB in that order$)$. What is the length of $CD$ if $AE=1,FB=49$ and $MC:MD=2:7$?
How do I approach the given problem? I would be very g... |
H: Proof $\exists\alpha$ s.t. $P(X>\alpha)>0$ if $P(X>0)>0$
For probability triple $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mu)$ prove that for a random variable $X$, if $\mu(X>0)>0$, there must be $\alpha>0$ s.t. $\mu(X>\alpha)>0$.
So if $X$ is a random variable with that property, it means that $\exists $ event $ A \... |
H: Show that $\sin\theta \tan\theta <2(1-\cos 3\theta )$
Show that $$\sin\theta \tan\theta <2(1-\cos 3\theta)$$ for $0<\theta<\pi/2$
MY ATTEMPT :
Let $$ t = \cos \theta $$
$$t \times E= (1-t^2) -2t(1-4t^3+3t) $$
$$t \times E= 8t^4-7t^2-2t+1 $$
After this, I'm confused about how to proceed !
Any help ? thank you
AI: ... |
H: Equality of an Inequality
I encountered the following problem in my textbook-
Let $a, b, c$ be three arbitrary real numbers. Denote $$ x = \sqrt{b^2-bc+c^2}, y = \sqrt{c^2-ca+a^2}, z = \sqrt{a^2-ab+b^2} $$ Prove that $$ xy+yz+zx \ge a^2+b^2+c^2 $$
Textbook's Solution:
Rewrite x,y in the following forms $$ x = \sq... |
H: The meaning of definition written in the form ... is...
I am studying mathematics logic and i am now criticising all the things...
I met some definitions which are in the form of ... is ....
For example, A construction sequence for an expression $\alpha$ is a finite sequence $\alpha_1,...\alpha_n=\alpha$ such that ... |
H: Rootspaces are $\mathop{ad}$ nilpotent
$ \DeclareMathOperator{\ad}{ad}$
Let $L$ be a semisimple Lie algebra with root space $L=H \oplus \bigoplus_{\alpha \in \Phi}L_\alpha$. Let $x\in L_\alpha$ with $\alpha\neq 0$. I want to show
Then $\ad x $ is nilpotent.
I know that if $\alpha, \beta\in H^*$ then $[L_\alpha,L_... |
H: If $F, K$ are fields, $F$ algebraically closed, and $F \subseteq K$ then $K = F$?
I want to show that an algebraically closed field $F$ cannot be contained in a larger field $K$. So if $F \subseteq K$ then $F = K$ for all fields $K$.
Here's my attempt at a proof:
For contradiction, Say $F \subsetneq K$. Hence there... |
H: For what values of $k$ is the following matrix diagonalizable?
The matrix is:
$$\begin{bmatrix}k&1\\0&k^2\end{bmatrix}$$
I am aware that a square matrix is diagonalizable if there exists a matrix $P$ such that $D = P^{-1 }* A * P$ is a diagonal matrix.
AI: You can see that the set of the eigenvalues of the matrix ... |
H: Difference in eigenvalue equations
I came across a strange equation in the solution to a problem. It looks like this:$$My - Mx = \lambda x$$
In this problem $M$ is an $n\times n$ (full rank) matrix, and $x$ and $y$ are vectors. ($y$ is given in the problem and I am looking for $x$). Is there a solution for $x$ here... |
H: Preservation of convergence in measure by absolutely continuous measures
In a paper on Risk theory that I am reading, it is stated that, unlike convergence in $L_p$, $1\leq p<\infty$, convergence in measure is preserved within a collection of probability measures that are absolutely continuous. That is,
Suppose $\... |
H: Example which proves that a closed subset of an incomplete metric space need not be complete.
We know that a closed subset of a complete metric space is complete.
But I want to find a closed subset $A$ of an incomplete metric space $(X,d)$ such that $A$ is not complete.
AI: Let $x$ be an irrational number and use t... |
H: 'Classical' Infinitesimals and Tangent Spaces
I do not know much differential geometry, and was led to this question from complex dynamics. It seems that it is often possible to reason 'infinitesimally' about maps between tangent spaces. For example, quasiconformal maps are typically motivated and thought of as sen... |
H: construct matrix group in GAP
I am having trouble construct the following group in GAP. It is a solvable primitive linear group acting on V where |V|=5^8. We know the Fitting subgroup is of order 2^6*4 (central product of extra special group E of order 2^7 with a cyclic group of order 4). On top of E/Z(E) we have a... |
H: Rolling Dice Game, Probability of Ending on an Even Roll
The game is described as follows. $A$ and $B$ take turns rolling a fair six sided die. Say $A$ rolls first. Then if $A$ rolls {1,2} they win. If not, then $B$ rolls. If $B$ rolls {3,4,5,6} then they win. This process repeats until $A$ or $B$ wins, and the gam... |
H: Example of absolutely continuous function $f$ with $\sqrt{f}$ not absolutely continuous
I'm looking for an example of a function $f$ that is absolutely continuous, but $\sqrt{f}$ is not absolutely continuous.
I've been playing around with the Cantor-Lebesgue function, but I feel like there should be something simpl... |
H: What is the problem with this method while integrating $(e^x-(2x+3)^4)^3$?
What is the problem with this method while integrating $(e^x-(2x+3)^4)^3$?
I already know what is its integration. I collected the answers from Quora (black ones) and WolframAlpha website (the red one).
Alright, then I tried to solve the in... |
H: Find all functions $f$ that satisfy the following
Let $\Omega $ be an open, bounded, and connected subset of $\mathbb{C}$
Find all functions $f:\bar{\Omega }\rightarrow \mathbb{C}$ that satisfy the following conditions simultaneously :
$f$ is continuous
$f$ is holomorphic on $\Omega $
$f(z)=e^z$ for all $z\in \p... |
H: Assume $T_n,T$ are bounded bijective linear operators $T_n \to T$ pointwise. Show $T_n^{-1}\to T^{-1}$ pointwise $\iff$ $\|T_n^{-1}\|\leq C$
Assume $T_n,T$ are bounded bijective linear operators $X \to Y$ and $T_n \to T$ pointwise. Show $T_n^{-1}\to T^{-1}$ pointwise $\iff$ $\|T_n^{-1}\|\leq C$
Note: $X,Y$ are bana... |
H: Two sequences $f_n$ and $g_n$ such that $\int_{[0,1]}f_n g_n$ does not go to $0$ as $n\rightarrow\infty$, with these conditions on $f_n$ and $g_n$
Question: Suppose $f_n, g_n:[0,1]\rightarrow\mathbb{R}$ are measurable functions such that $f_n\rightarrow 0$ a.e. on $[0,1]$ and $\sup_n\int_{[0,1]}|g_n|dx<\infty$.
Gi... |
H: Line Integral gives no work done?
For the following question,
$$
\mathbf{F}=\langle-y, x\rangle
$$
For this field:
Compute the line integral along the path that goes from (0,0) to (1,1) by first going along the $x$ -axis to (1,0) and then going up one unit to (1,1) .
I got an answer of $0$, by doing:
But the answ... |
H: Question involving ratios and Greatest Common Divisors
Consider 6 variables $a,b,c,x,y,z\in\mathbb Z$. We have two ratios, $a:b:c=1:2:3$ and $x:y:z=1:2:3$. We also have that $\gcd(a,x)=2$. What is $\gcd(a+b+c,x+y+z)$? I know that $\gcd(a,x)=2$, which means $a$ and $x$ are even, but I'm not too sure how to use this ... |
H: Is the barrier problem for a linear program a convex problem?
By applying the barrier method to the linear programming problem min $c^T x, Ax ≥
b$, we can formulate:
$$\underset{x}{\text{minimize}} \hspace{0.5cm} c^T x - \sum_{i=1}^{m} \log(e_i - d_i^T x)$$
But is the sequence of optimization problems solved in the... |
H: Area between parabola and a line that don't intersect? 0 or infinity
Came across a problem on social media,
Find the area of the region bounded by a parabola, $y = x^2 + 6$ and
line a line $y = 2x + 1$.
I tried to draw it on paper and they didn't seem to intersect. So I drew them online (attached screenshot). My ... |
H: Why is identity map on a separable Hilbert space not compact? False proof.
Why is identity map on a separable Hilbert space not compact? False proof.
Let $e_n$ be the orthonormal basis. Then the projection map onto $H$ is defined by $\sum(x,e_k)e_k$. What is stopping us from taking a finite part of this series. Thi... |
H: Finding a general way to construct least degree polynomial having rational coefficient having irrational roots
Let p(x) be the least degree polynomial equation having $\sqrt[3]{7}$ + $\sqrt[3]{49}$ as one of it's roots, Then product of all roots of p(x) is ?
Following from the 'irrational roots occurring in pair... |
H: Are $\mathbb{C}-\mathbb{R}$ imaginary numbers?
Background
I am teaching senior high school students about the structure of numbers.
Start from defining $\mathbb{Q}$ and $\mathbb{R}$ as the rational and real numbers respectively, we can define $\mathbb{R}-\mathbb{Q}$ as the irrational numbers.
I am trying to use the... |
H: For a function $f: X \to Y$, if $Y-V$ is finite, when is $X - f^{-1}(V)$ finite?
I apologize if this is a silly question but I just do not know enough set theory (i.e., sizes) to understand if it's even silly.
My question is
Let $f: X \to Y$ be a function of sets. Suppose $V \subset Y$, and that $Y - V$ is finite ... |
H: Root space decomposition of $C_n=\mathfrak{sp}(2n,F)$
I want to find the root space decomposition of the symplectic lie algebra $\mathfrak{sp}(2n,F)=C_n$.
I use the notation from Humphreys.
The root space decomposition of a semisimple lie algebra $L$ is $L=H\oplus \bigoplus_{\alpha \in \Phi} L_\alpha$. Where $H$ is... |
H: Conditions on inequalities $a>b$ and $b
If $a>b$ and $b<c$, and $a$ and $c$ are positive, under what conditions is $a<c$? I am just curious to know. I know that the following are true $\frac ab>1$, $\frac bc<1$, and $\frac ac<1$. Any ideas on what next?
AI: This may be what you are looking for.
$$\begin{cases} a>b ... |
H: Finding the local extrema of $f(x, y) = \sin(x) + \sin(y) + \sin(x+y)$ on the domain $(0, 2 \pi) \times (0, 2 \pi)$
I am trying to find the relative extrema of
$$f(x, y) = \sin(x) + \sin(y) + \sin(x+y), \text{ where } (x, y) \in (0, 2 \pi) \times (0, 2 \pi)$$
Setting the partial derivatives equal to zero gives
$$... |
H: Find all integer values of $a$ such that $a^2 - 4a$ is a perfect square
I am trying to determine a systematic way to find all the integer vales of $a$ such that $a^2 - 4a$ is a perfect square. If it helps, I already know that the two solutions to this equation are 0 and 4.
Furthermore, I wonder how I can prove tha... |
H: Show that $\|uv^T-wz^T\|_F^2\le \|u-w\|_2^2+\|v-z\|_2^2$
Show that $\|uv^T-wz^T\|_F^2\le \|u-w\|_2^2+\|v-z\|_2^2$, assuming $u,v,w,z$ are all unit vectors.
AI: Another approach: note that for orthogonal matrices $U,V,$ we have
$$
\|uv^T - wz^T\|_F^2 = \|U(uv^T - wz^T)V\|_F^2 = \|(Uu)(Vv)^T - (Uw)(Vz)^T\|_F^2.
$$
So... |
H: Are all complex functions onto?
I am not sure whether this question even makes sense. But I was just wondering whether all inverse operations of functions defined in complex numbers will stay inside complex numbers. (i.e. we don't have to extend the complex number system):
$x^2$ is a well-defined function of real n... |
H: Suppose a pair of random variable is independent from another pair, does it mean that each random variable is independent from the other?
Let $(X_1, Y_1)$, and $(X_2, Y_2)$ be two pairs of random variables, and they are assumed to be independent.
Does it mean that:
$X_1$ is independent from $X_2$?
$X_1$ is indepen... |
H: Evaluate $\int_0^1\frac{\mathrm{e}^{12x}-\mathrm{e}^{-12x}}{\mathrm{e}^{12x}+\mathrm{e}^{-12x}}\,\mathrm{d}x$
My work so far
$${\displaystyle\int_0^1}\dfrac{\mathrm{e}^{12x}-\mathrm{e}^{-12x}}{\mathrm{e}^{12x}+\mathrm{e}^{-12x}}\,\mathrm{d}x$$
using substitution with $u=\mathrm{e}^{12x}+\mathrm{e}^{-12x}$
$$={\disp... |
H: Prove this tabulated integral $\int_0^\infty x^ne^{-\alpha x} \, dx=\frac{n!}{\alpha^{n+1}}$
I ran into this problem where I needed to use the following integral equality in my physics textbook.
$$\int_0^\infty x^ne^{-\alpha x} \, dx=\frac{n!}{\alpha^{n+1}}$$
where $n$ is a positive integer and $\alpha$ is a positi... |
H: About holomorph of a finite group being the normalizer of regular image
Here is part of Exercise 5.5.19 in Dummit & Foote's Abstract Algebra:
Let $H$ be a group of order $n$, let $K=\operatorname{Aut}(H)$ and $G=\operatorname{Hol}(H)=H\rtimes K$ (where $\varphi$ is the identity homomorphism). Let $G$ act by left m... |
H: Prove that the functional in $C_c^0(\Omega)$ is a Radon measure
Let $\Omega \subset \mathbb{R}^n$ be an arbitrary open set and $(x_n)_{n \in\mathbb{N}} \subset \Omega$ a sequence. Let $(a_n)_{n \in\mathbb{N}} \subset \mathbb{C}$ be a sequence such that
$$\sum_{j=1}^{\infty} |a_j| < \infty.$$
I want to prove that t... |
H: How can I evaluate ${\lim_{h\to 0}\frac{\cos(\pi + h) + 1}{h}}$?
I'm supposed to evaluate the following limit using the cosine of a sum and one of the "special limits" which are ${\lim_{x\to 0}\frac{\sin(x)}{x}=1}$ and ${\lim_{x\to 0}\frac{1-\cos(x)}{x}=0}$.
The limit is : ${\lim_{h\to 0}\frac{\cos(\pi + h) + 1}{h}... |
H: Is eigenvalue multiplied by constant also an eigenvalue?
Let $A$ be an $n × n$ matrix.
If $\lambda$ is an eigenvalue of $A$ and $c$ is a nonzero scalar, then $c\lambda$ is another eigenvalue of $A$.
I found this on "Linear Algebra and its applications (Jim Defranza)", summary of Chapter 5.
It is acceptable, that ... |
H: If $f ∈ C^∞(M)$ has vanishing first-order Taylor polynomial at $p$, is it a finite sum of $gh$ for $g, h ∈ C^∞(M)$ that vanish at $p$?
This is 11-4(a) in Lee's "Introduction to Smooth Manifolds":
Let $M$ be a smooth manifold with or without boundary and $p$ be a point of $M$. Let $\mathcal{I}_p$ denote the subspace... |
H: Evaluate the following series $ \sum_{n=0}^{\infty} \big(e^{(4n+1)\pi\sqrt{3}}+2+e^{-(4n+1)\pi\sqrt{3}}\big)^{-1} $
I found this convergent series while solving a calculus problem $$ \sum_{n=0}^{\infty} \big(e^{(4n+1)\pi\sqrt{3}}+2+e^{-(4n+1)\pi\sqrt{3}}\big)^{-1}=A $$How can I evaluate A?
My Attempt
$$\sum_{n=0... |
H: Transformation of function $y = x^2 $
I want to transform function $y = x^2$ to $y = 4 x^2 $. Now this transformed function can be thought of as $ y/4 = x^2$ or $ y = (2x)^2$. If it is $ y/4 = x^2$, then this is a vertical stretch by a factor of 4 and if it is $ y = (2x)^2 $, then this is a horizontal compression b... |
H: Specific Probability Question
So I had come up with this problem during my spare time and was wondering if my answers to the questions were correct? The problem is:
Let's say an individual does not go out that much from their house and lives in the basement, and so they decide to check the outside weather every tim... |
H: Evaluating $\iint dx\,dy$ over the region bounded by $y^2=x$ and $x^2+y^2=2x$ in the first quadrant
Identify the region bounded by the curves $y^2=x$ and $x^2+y^2=2x$, that lies in the first quadrant and evaluate $\iint dx\,dy$ over this region.
In my book the solution is like:
$$\begin{align}\\
\iint dx\,dy &=\... |
H: Property of a positive Lebesgue measure set in $\mathbb{R}^n$
Let $E\subset\mathbb{R}^n$ be a positive Lebesgue measure set. Can we always find a $x\in E$ such that for any $r>0$, $B(x,r)\cap E$ is positive Lebesgue measure in $\mathbb{R}^n$?
($B(x,r)$ denotes the ball of radius $r$ with centre at $x$ in $\mathbb{R... |
H: In factoring consecutive numbers, how soon do we expect to see the smallest prime not yet seen?
For $A\subseteq\{1,2,3,\ldots\}$ one can take $\Pr(A)$ to mean $\displaystyle \lim_{n\,\to\,\infty} \frac{|A\cap\{1,\ldots,n\}|} n,$ if one doesn't insist on probability being countably additive.
Suppose we have a long s... |
H: Inherited Riemannian metric on a submanifold
I am a beginner in differential geometry and I am reading chapter 1 of Differential Geometry of Loring Tu. For a smooth manifold $M$, a Riemannian metric on $M$ is an assignment that assigns $p\in M$ to an inner product on $T_pM$, such that for any smooth vector fields $... |
H: Proof of equation
Suppose, $E_1, E_2, E_3$ and $E_4$ are subsets of a universal set $E$ such that $E_1\subseteq E_2$ and $E_4\subseteq E_3$. Then, $(E_1\cap E_3) \cup (E_2\cap E_4)=E_2\cap E_3$.
It is observed that any counter examples we take sayisfy this equation, but unable to establish a formal proof of this... |
H: True or false questions about divisibility and non-divisibility in the integers. I'm confused.
For $a$, $b$, c $\in \mathbb{Z}$.
True or false that:
If $a$ doesn't divide $b$ and $b$ doesn't divide $c$ then $a$ doesn't divide $c$
If $a$<$b$ then $a$ divide $b$
If $a$ doesn't divide $b$ and $b$ doesn't divide $c$... |
H: The number of ways to represent a natural number as the sum of three different natural numbers
Prove that the number of ways to represent a natural number $n$ as the sum of three different natural numbers is equal to $$\left[\frac{n^2-6n+12}{12}\right].$$
It was in our meeting a year ago, but I forgot, how I proved... |
H: Probability of picking buttons from a bag
A bag contains $30$ buttons that are colored either blue, red or yellow. There are the same number of each color ($10$ each). A total $4$ buttons are drawn from the bag. Compute the followings:
Find $n(\Omega)$.
The probability that at least $3$ of them are red?
The probab... |
H: To show a function with a singularity is holomorphic by proving it is bounded.
If $f\in H(A)$, where $A=\text{Ann($z_0;r_1,r_2$)}$.
Let $z\in \text{Ann($z_0;s_1,s_2$)}$, where $r_1<s_1<s_2<r_2$, define $g$ by
$$g(\zeta)=\begin{cases}
\frac{f(\zeta)-f(z)}{\zeta-z} & \text{if }\zeta\neq z \\
\ f'(z) & \text{if }\zet... |
H: What is the deep meaning of this quotes acording to Sir David Hilbert logics?
One of the famous mathematician David Hilbert quotes:
“Wir müssen wissen. Wir werden wissen."
(We must know. We will know.)
What is the deep meaning of this quotes acording to Sir David Hilbert logics ?
AI: That was in Hilbert's radio sp... |
H: Let $h:[0,1] \times [0,1] \rightarrow \mathbb{R}$ be the function $h(x,y)=f(x)g(y)$. Show h is integrable.
Let $f,g:[0,1] \rightarrow R$ be bounded, nonnegative, and nondecreasing $f(x_1) \leq f(x_2)$ for all $x_1 \leq x_2$ functions. Let $h:[0,1] \times [0,1] \rightarrow \mathbb{R}$ be the function $h(x,y)=f(x)g(... |
H: Deriving the Cauchy integral formula from the residue theorem
I'm currently going through complex analysis, and I'm trying to grasp the concept of the whole residue theorem and so on.
I followed the derivation of the residue theorem from the Cauchy integral theorem, and I think I kind of understand what is going on... |
H: Proof Verification: $(x^n)_{n=1}^{\infty}$ diverges when $x>1$
Proof
Suppose for the sake of contradiction that $(x^n)_{n=1}^{\infty}$ converges to some limit $L$. Consider the identity $(1/x)^n (x^n) = 1$. Since this holds for all $n \in \mathbb{N}$, $\lim_{n \rightarrow \infty}(1/x)^n \lim_{n \rightarrow \infty} ... |
H: If $p,q$ are prime, $pq\pm 2$, $pq\pm 4$, $pq\pm 6$ cannot be all primes
Let $p,q$ be distinct primes. Prove that the six integers $pq-2$, $pq+2$, $pq-4$, $pq+4$,$pq-6$, $pq+6$ cannot be all primes.
This is Exercise 5.60 in Chartrand's Mathematical Proofs. The claim is intriguing as for $p=3$ and $q=5$, the four ... |
H: Evaluate $\int \frac{2-x^3}{(1+x^3)^{3/2}} dx$
Evaluate:
$$\int \frac{2-x^3}{(1+x^3)^{3/2}} dx$$
I could find the integral by setting it equal to $$\frac{ax+b}{(1+x^3)^{1/2}}$$
and differentiating both sides w.r.t.$x$ as
$$\frac{2-x^3}{(1+x^3)^{3/2}}=\frac{a(1+x^3)^{3/2}-(1/2)(ax+b)3x^2(1+x^3)^{-1/2}}{(1+x^3)}$$$... |
H: Show the function $f(x)=\begin{cases}|x|^x, &x\neq0 \\ 1, &x=0\end{cases}$ is not differentiable at zero.
By the definition of differentiability, we need to show the following limit doesn't exist:
$$\lim_{x \to 0}{\frac{|x|^x-1}{x}}$$
I've shown $\displaystyle \lim_{x \to 0}{|x|^x}=1$, but couldn't proceed since L'... |
H: $4$-element subset of $\{1..6\}$ that includes $1$ or $4$, and $2$ or $5$, and $3$ or $6$?
I am stuck on the following question which was asked in our combinatorics exam:
Let us consider the set $S=\{1,2,3,4,5,6\}$.
We want to find a set $A\subset S$ such that $A$ must have the following property:
either $1$ or $... |
H: Compress a three digit number into a single number
If I have a three digit number like 293 Is there a method to compress it into one digit. Can any three digit number be rewritten into a single digit with some formula and then back to its original form?
AI: No, assuming by three digit number you mean $100\leq n\leq... |
H: How to represent the given information correctly to solve for a particular solution to a differential equation?
Lupita's lawn is left unattended so that an infestation of weeds begins to take over. The rate of growth of the weeds is proportional to the area of lawn not yet invaded by weeds.
a) If $W \space m^2$ is ... |
H: In how many ways can a group of six people be divided into: 2 equal groups? 2 unequal groups, if there must be at least one person in each group?
In how many ways can a group of six people be divided into:
a) two equal groups
I have $^6C_3 \times \space ^3C_3 = 20$
So, to choose the first group I have $6$ possibi... |
H: Is this series $\sum_{n=0}^\infty(\frac{n}{2n^2-1})^2$ convergent?
Is the series $\sum_{n=0}^\infty(\frac{n}{2n^2-1})^2$ convergent?
My attempt:
The series is convergent iff $\sum_{n=0}^\infty 2^ka_{2^k}$ convergent.
Then we need to show that $\sum_{n=0}^\infty 2^k (\frac{2^k}{2\times2^k-1})^2$ converges.
I want to... |
H: If $φ(p) > φ(k)$ for $k
So I was playing around with the Euler totient function on desmos, and found that whenever the function "spikes", we can add $1$ to it and I always found a prime number. With very powerful computers or software why can't we use this for finding prime numbers?
It's my first time on this site ... |
H: Trivial Fundamental Group and Orientation
Maybe it is an easy question but I cannot figure out.
If the fundamental group of (you may assume compact) an $n$-dimensional manifold $M$ is trivial, i.e., $$\pi_1(M)=0,$$ then can we conclude that $M$ is orientable?
AI: If $M$ is non-orientable, the orientable double cove... |
H: Proving that $g\circ f$ is injective if $f$ and $g$ are injective. [Verification]
Let $f:X\rightarrow Y, g:Y\rightarrow Z$. Show that if $f$ and $g$ are injective, then $g \circ f$ is also injective.
My attempt:
$x \neq x’ \implies f(x) \neq f(x’)$
$f(x) \neq f(x’) \implies g(f(x)) \neq g(f(x’))$
$(A\implies B)... |
H: Let $f(x)=\lim_{n\to \infty} (1-\sin x + e^{\frac 1n} \sin x)^n)$. If ....
Let $f(x)=\lim_{n\to \infty} (1-\sin x + e^{\frac 1n} \sin x)^n)$. If $\lim_{u\to 0} (1+u \log (1+k^2))^{\frac 1u}=2k\log^2 (f(x))$, for $k>0$ and $x\in (0,\pi)$, find $x+k$
Log is to the base $e$
$$f(x)=e^{\sin x}$$
And $$\lim_{u\to 0} (1... |
H: Prove that if $f$ is bijective, then $f^{-1}$ is bijective. [Verification]
Let $f: X \to Y$ be bijective, and let $f^{-1}: Y \to X$ be it's inverse. Conclude that $f^{-1}$ is also invertible.
Suppose that $f^{-1}(f(x)) = f^{-1}(f(x')) \nRightarrow x=x'$ (not injective), then $x=x' \nRightarrow x=x'$ which is a co... |
H: Intuitive explanation of what happens when we remove functions from integrals by exploiting bounds
Suppose I have an integral like,
$$ P = \int_{0}^{\pi} x \sin x \cos^4 x dx$$
by the property,
$$ \int_{0}^{a} f(a-x) dx = \int_{0}^{a} f(x) dx$$
And, we do
$$ P = \int_{0}^{\pi} (\pi -x) \sin x \cos^4 x dx $$
Now if... |
H: Means of neighboring squares
mn squares of equal size are arranged to form a rectangle of dimension m by n where m and n are natural numbers. Two squares will be called 'neighbours' if they have exactly one common side. A natural number is written in each square such that the number in written any square is the ari... |
H: What is the minimal Hamming distance of concatenation of some word and a part of it encoded using Hamming codes?
A word $M$ is of $n^2$ bits, $n>3$ is arranged in a $n\times n$ matrix. $A$ is the main diagonal of the matrix (that is elements $a_{i,j}$, $i=j$) and is encoded with Hamming code, resulting in $B$. The... |
H: Random Walk Around A Circle
I am having difficulties in solving the following problem
I guessed that the chain is irreducible when $\gcd(n,s)=1$. But I'm unable to proceed. Can someone help me? Any hint will be appreciated.
AI: You are right thinking the chain is irreducible iff $n$ and $s$ are coprime:
Suppose $n... |
H: Find the value of $a$ for which the two lines are on the same plane
Find the value(s) of $a$ for which the two lines $$L_1:=\left\{\left(-1,0,-1\right)+t\left(a,2,0\right):t \in \mathbb R\right\}$$
$$L_2:=\left\{\left(1,2a,1\right)+t\left(2,3,2\right):t \in \mathbb R\right\}$$
place on the same plane in $\mathbb R^... |
H: Generating subgroup of $\langle \mathbb{Q} \setminus \{0\},\cdot\rangle$
Does such a subgroup even exist?
My guess is That it is $\mathbb{Z} \setminus \{0\}$
Any help?
AI: A proper subgroup cannot generate the whole group. The set of nonzero integers is a generating set for the group $(\mathbb{Q}\setminus\{0\},{\cd... |
H: Number of wrong answers
In a certain test $a_i$ students gave wrong answers to at least i questions, where $i= 1,2,3......k$ No student gave more than k wrong answers. The total number of wrong answers is
I wasn't able to start solving this. I tried subtracting 2-1 for exactly 1 answer but I didn't understand what... |
H: Show that a certain projection is rank one.
Consider the following fragment from Murphy's "$C^*$-algebras and operator theory", namely a part of the proof of Theorem 2.4.8.
Can someone explain why $q_e$ is a rank-one projection? (see marked text). Thanks in advance.
AI: Let $V$ be a closed subspace of a Hilbert sp... |
H: Exercise about a limit with greatest integer function
Let:
$$f(x) := \begin{cases} x+3 &\text{, if } x\in (-2,0) \\ 4 &\text{, if } x=0 \\ 2x+5 &\text{, if } 0<x<1\end{cases}\;.$$
Then find $\lim_{x\to 0^-} f([x-\tan x])$, where $[\cdot]$ is greatest integer.
Since $x$ is approaching zero from the negative side, ... |
H: Show that the solution of the equation $x^5-2x^3-3=0$ are all less than 2 (using proof by contradiction).
The question is "Show that the solution of the equation $x^5-2x^3-3=0$ are all less than 2."
I have attempted to answer this question using proof by contradiction and I think my answer is either wrong or not ... |
H: how to find convolution between $f(x)=e^{-2x^2}$ and $g(x)=e^{-2x^2}$?
I wanted to find the convolution between the two function. By definition I get,
$$(f*g)(x)=\int_{\mathbb R}e^{-2(x-y)^2}e^{-2x^2}dy=\int_{-\infty}^{\infty}e^{-4x^2+4xy-2y^2}dy $$
But I am not able to calculate the given integral.
AI: Note that $... |
H: Convergence in distribution to normal random random variable
Let $X_n$ be a sequence of random variables on $\mathbb R$ such that $X_n$ converges in distribution to a Gaussian variable $N(0,\sigma^2)$.
Is it true that
$$\mathbb E[e^{-\frac{X_n}{\sqrt n}}]\to 1?$$
Can you help me a bit?
AI: Let $Z\sim\mathcal{N}(0,\... |
H: Probability density function of s = $u^2 + v^2$ with uniformly distributed u and v
I have a question on the section "Polar form" in the Wikipedia article of the Box-Muller-Scheme. There it is said that if u and v are two independent uniformly distributed random variables in the interval [-1,1] each, then if you sam... |
H: Visualizing a lemma in metric spaces and possibly relaxing requirements
I know that the following statement can be proved with relative ease:
Let $ K \subset \mathbb{R}^n $ be a compact (closed and bounded) set in real Euclidean space. Assume also that $ K \subset U $ where $ U \subset \mathbb{R}^n $ is an open se... |
H: How to estimate the number of loop given that the inner loop index always greater than outer loop for Big O complexity estimation?
I have run into a program where there are two nested dependent loops in this manner:
for k=1:K-1
for ell=k+1:K
do something
end
end
How do I calculate the number of tot... |
H: Solution of differential equation ${x^2}\frac{{dy}}{{dx}} + {y^2}{e^{\frac{{x\left( {y - x} \right)}}{y}}} = 2y\left( {x - y} \right)$
Solutions of differential equation
$$
{x^2}\frac{{dy}}{{dx}} + {y^2}{e^{\frac{{x\left( {y - x} \right)}}{y}}} = 2y\left( {x - y} \right)
$$
are given by
(A) $x(x + y) = y \ln (Ce^x... |
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