text stringlengths 83 79.5k |
|---|
H: The Broken Calculator Problem
So here is the Problem :-
Tom has a specific calculator . Unfortunately, all keys are broken except for one row$: 1,2,3,+,-$. Tom presses a sequence of $5$ random keys; where at each stroke, each key is equally likely to be pressed. The calculator then evaluates the entire expression, ... |
H: What is value of this integral? $\int_{0}^{\infty}\frac{\log(1+4x^2)(1+9x^2)(9+x^2)+(9+x^2)\log(4+x^2)(10+10x^2)}{(9+x^2)^{2}(1+9x^{2})}dx$
What is value of this integral $$I=\int_{0}^{\infty}\frac{\log(1+4x^2)(1+9x^2)(9+x^2)+(9+x^2)\log(4+x^2)(10+10x^2)}{(9+x^2)^{2}(1+9x^2)}dx$$
My work :
\begin{align*}I&=\int_{... |
H: Is it always possible to get local basis elements of a vector field module on a manifold?
I'm studying about vector fields from Schuller's lectures on youtube. Given a smooth manifold $M$, if we have a smooth module $\Gamma(TM)$ consisting of all smooth vector fields on the manifold, it's clear to me that we can't ... |
H: Show that the elements of the sequence are divisible by $2^n$
I am trying to prove the following:
Consider the sequence defined by $A_{n+2}=6A_{n+1}+2A_n, A_0=2, A_1=6$. Show that $2^n|A_{2n-1}$ but $2^{n+1}\nmid A_{2n-1}$.
The first terms of this sequence are 2, 6, 40, 252, 1592, 10056, 63520.
In fact, the maxim... |
H: How is notation $\{f_n(x)\}\nearrow x$ interpreted?
In the context of continuity of probabilities we define $\{A_n\}\nearrow A$ to mean that $A_1 \subseteq A_2 \subseteq A_3 \subseteq ...$ and $\cup_n A_n=A$, where $A, A_1, A_2, A_3, ... \in \mathcal{F}$ for some probability triplet $(\Omega, \mathcal{F}, P)$. I ha... |
H: Proof that $\lim_{x\rightarrow \infty} f(x+a) =\lim_{x\rightarrow \infty} f(x) $ for constant $a$
It is easy to understand why the following statement is true for variable $x$ and constant $a$:
$\lim_{x\rightarrow \infty} f(x+a) =\lim_{x\rightarrow \infty} f(x) $
However, is there a proof for it, or is it true by d... |
H: Combination to find integers satisfying a condition
Let $n$ and $k$ be positive integers such that $n\ge\frac{k(k+1)}{2}$.
The number of solutions $(x_1,x_2,\dots,x_{k})$, with $x_1\ge1$, $x_2\ge2$,..., $x_{k}\ge k$ for all integers satisfying $x_1+x_2+\dots+x_{k}=n$ is?
I substituted the last equation in the fir... |
H: Evaluate $\lim_{x\to -\infty} \frac{x^4\sin( \frac 1x )+ x^2}{1+|x|^3}$
Let $y=-x$
$$\lim_{y\to \infty} \frac{y^4 \sin (-\frac 1y) + y^2}{1+|y|^3}$$
$$=\lim_{y\to \infty} \frac{y^3 ((-y \sin \frac 1y) +\frac 1y)}{y^3(1+\frac{1}{y^3})}$$
At $y\to \infty$, $y\sin \frac 1y=0$ (What I think is happening)
Therefore the ... |
H: If entropy says the number of bits an information needs, why in this case it's less than one?
If,$3/4$ of the times, is raining in a city and $1/4$ is not, the entropy $(-\log(3/4))$ would say we need almost $0.415$ of a bit to say it's raining, and $2$ bits $(-\log(1/4) )$ to say it is not, right?? How can we make... |
H: Can we estimate the profuct $(1+x_1)(1+x_2)\ldots(1+x_n)$ by the term $x_1\ldots x_n$?
Let $x_1, x_2, \ldots x_n$ be postive integers with $x_i\ge 2(1\le i\le n)$. Can we have the following inequality:
$$(1+x_1)(1+x_2)\cdots(1+x_n)\le f(n)x_1\cdots x_n,$$
where $f(n)$ is a term dependent on $n$ and I hope it is in... |
H: Local $\mathbb{k}$-algebra homomorphism
Let $(A,m)$ and $(B,n)$ be local commutative rings that are also $\mathbb{k}$-algebras. Let $\phi :B \rightarrow A$ be a local $\mathbb{k}$-algebra homomorphism.
Suppose that $A/m \cong \mathbb{k}$. I want to show that $B/n \cong \mathbb{k}$.
If we call the maps
$\tau : \math... |
H: What is the proof that $(a+b)^2 >a^2 + b^2$?
I would like to know if there is a theorem that proves that
$$(a+b)^2>a^2+ b^2$$
where $ab>0$
I am also wondering whether there is a name associated with this inequality.
AI: By distributing out, you get that
$${(a+b)^2 = a^2 + 2ab + b^2 = (a^2 + b^2) + 2ab}$$
Now - s... |
H: Triangle tangent to 3 Parabolas, finding the common area
Triangle $ABC$, $AB=4$, $BC=15$, $AC=13$.
Two sides are tangents to the respective Parabolas.
We have to find the area shaded.
My approach-
I tried finding the area of the quadratures(Archimedes) formed but it
doesn't help as- I have to find the area betwee... |
H: Hilbert function is not eventually equal to a polynomial function
Consider the ring $S=k[x_1,\ldots,x_r]$, graded by $\mathbb N$ with each variable in degree $1$. Let $M$ be finitely generated graded $S$-module. The Hilbert function is defined by $H_M(d)=\dim_k M_d$. It is a well-known theorem of Hilbert that if $M... |
H: Confusion about probability space associated with infinite coin flips
Let $\Omega = \{ \omega = (\omega_1, \omega_2, \ldots) : \omega_j = 1 \text{ or } 0 \}$. For each positive integer $n$, let $\Omega_n = \{ \omega = (\omega_1, \ldots, \omega_n) : \omega_j = 1 \text{ or } 0 \}$. We can consider $\Omega_n$ as a pr... |
H: How to solve $\int_0^1dx\int_0^1\frac{x^2-y^2}{(x^2+y^2)^2}\,dy$
The original question is:
Prove that:$$\begin{aligned}\\
\int_0^1dx\int_0^1\frac{x^2-y^2}{(x^2+y^2)^2}\,dy\neq\int_0^1dy&\int_0^1\frac{x^2-y^2}{(x^2+y^2)^2}\,dx\\
\end{aligned}\\$$
But I can't evaluate the integral $$\int_0^1dx\int_0^1\frac{x^2-y^2}... |
H: If the largest positive integer is n such that $\sqrt{n - 100} + \sqrt{n + 100}$ is a rational no. , find the value of $\sqrt{n - 1}$ .
So here is the Problem :-
If the largest positive integer is n such that $\sqrt{n - 100} + \sqrt{n + 100}$ is a rational no. , find the value of $\sqrt{n - 1}$ .
What I tried :- I ... |
H: How to prove that zeros of two polynomials interlace implies the Wronskian is nonnegative or nonpositive?
Given two real polynomials $f,g$ with the roots are all real. We say the zeros of $f,g$ interlace if
$$\alpha_1\leq\beta_1\leq\alpha_2\leq\beta_2\leq\ldots$$ or
$$\beta_1\leq\alpha_2\leq\beta_2\leq\alpha_2\leq\... |
H: Finding unknown constants
As $\lim_{x\rightarrow 0}\frac{\tan 2x - 2\sin ax}{x-x\cos 2x} = b$ where $a$ and $b$ are real constants. Then what is the value of $a$ and $b.$ Can someone help me with idea on how to approach this problem.
AI: Hint:
If $a\ne 1$, we have $\tan 2x-2\sin ax\sim_0 2(1-a)x$, whereas $1-\cos 2... |
H: Questions on the proof of the strong law of large numbers
Here is the proof of the strong law of large numbers presented in the textbook I'm currently using.
Let $X_1,X_2,...$ be a sequence of independent identically distributed random variables and assume that $E[(X_i)^{4}] < \infty$. Prove the strong law of large... |
H: Question regarding the proof that every non-negative measurable function is the limit of simple functions
$f$ be a non-negative measurable function. For each $x$ define $k_n(x)$ to be the unique integer satisfying $$k_n(x)\leq 2^nf(x)<k_n(x)+1$$ Define $$s_n(x)=\begin{cases}\frac{k_n(x)}{2^n},\text{ if }f(x)<n
\\n,... |
H: Inequality involving integral and second derivative
UCLA basic exam spring 2017 problem 8
Show that there is a constant $C$ so that $$\left| \frac{f(0)+f(1)}{2}-\int_0^1 f(x)\, \mathrm dx \right| \leq C \int_0^1 \lvert f''(x)\rvert \, \mathrm dx$$ for every $C^2$ function $f:\mathbb{R} \rightarrow \mathbb{R}$
The... |
H: How to solve the ODE $y' = \frac{x+y-2}{y-x-4}$?
I am trying to solve the ODE $$y' = \frac{x+y-2}{y-x-4} \tag1 $$
This is a homogeneous special form ODE.
Let $x = u -1$ and $y=v+3$ in order to transform it to a homogeneous ODE. Hence,
$$ (1) \iff v'(u) = \frac{u+v(u)}{v-u} \tag 2$$
At last let $v(u) = z(u)u \iff ... |
H: find bound on condition number of matrix given matrix norm
Suppose $A$ is a $202 \times 202$ matrix with $\|A\|_2 = 100$ and $\|A\|_f = 101$. Give the sharpest lower bound on the 2-norm condition number $k(A)$.
I know $k(A) = \|A\|\cdot\|A^{-1}\| = 100 \|A^{-1}\|$
I also know that $\|A\| = \sup_x \frac{\|Ax\|}{\|x\... |
H: Show that $\left \langle A^tx, y \right \rangle = \left \langle x, Ay \right \rangle$
Let $V = \mathbb{R}^n$ and $A\in\mathbb{M}_{n\times n}(\mathbb{R})$ show that $\left \langle A^tx, y \right \rangle = \left \langle x, Ay \right \rangle$.
I managed to demonstrate the exercise in the case of the standard inner pro... |
H: The set of all [group/ring/module/etc.] structures on a cardinal number
Given a cardinal number $\kappa$, how can one construct the set of the (algebraic) structures of a certain kind (group, ring, module, etc.) whose base set is a subset of $\kappa$?
For example, using the axioms of ZFC, how can one justify the ex... |
H: Cantor's theorem and countable additivity of Lebesgue measure
In this book, with more than 300 pages, the author claims that Cantor was wrong:
Transfinity, Wolfgang Mückenheim
https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf
In particular, the author lists in more than 100 pages authors that had con... |
H: Let $f$ be a differentiable function with $f ′(x) = 4$. If $f(0) = 2$, find $f(3)$.
So obviously this is some anti derivative stuff, and I the anti derivative would've been $2x^2$, with $f(x)=4x$ and then just plug in $x=3$ to get the answer. That's not the case tough, because with $f(x)=4x$, I see now that $f(0)$ ... |
H: Finding the Last Eigenvalue for a Matrix
$K$ is a $3 \times 3$ real symmetric matrix such that $K = K^3$. Furthermore, we are given that:
\begin{align*}
K(1, 1, 1) \ \ & = \ \ (0, 0, 0) \\
K(1, 2, -3) \ \ & = \ \ (1, 2, -3)
\end{align*}
So we know that $0, 1$ are two of the eigenvalues of $K$. What can I do to as... |
H: projective space minus a closed point
Let $k$ be an algebraically closed field and let $\mathbb P^n_k=\text{Proj}(k[x_0,x_1,...,x_n])$ .
If $n\ge 2$, and $p\in \mathbb P^n_k$ is a closed point, then can $\mathbb P^n_k\setminus \{p\}$ be a Projective variety ?
Considering global section ring doesn't give any contrad... |
H: Practical algorithm to calculate power subgroup of a polycyclic group
I am looking for a practical algorithm to calculate the power subgroup $G^n := \langle g^n \mid g \in G \rangle$ of a (possibly infinite) polycyclic group $G$. A theoretical algorithm is given in [1], but it does not appear to be practical, as it... |
H: You can prove that compacts in the weak topology are limited without the Eberlein-Šmulian theorem
You can prove that compacts in the weak topology are bounded without the Eberlein-Šmulian theorem?
With Smuliam's Theorem this is immediate, because if you assume by contradiction that $K$ is unborded in the norm topol... |
H: Prove that the diagonals of a rhombus are orthogonal.
I'm trying to solve some of the problems in Ahlfors' Complex Analysis book. On the section about analytic geometry, the following problem is stated:
Prove that the diagonals of a rhombus are orthogonal.
Since the idea was to use complex analysis tools to solve... |
H: Second order derivative of a chain rule (regarding reduction to canonical form)
I've been stuck on this for a couple of days. So this is from this book ("Partial Differential Equations in Mechanics 1", page 125).
Section 4.2 Reduction to canonical forms, which leads to the development of the Laplace equation.
In t... |
H: how to calculate this problem please help me with steps
Person A can finish one task in 10 days. What if he work 10% faster for first day how much time it takes to finish the work now?
AI: Since he finishes $1$ task in $10$ days, he finishes $\frac{1}{10}$ of a task in $1$ day. $10%$ more than $\frac{1}{10}$ is jus... |
H: Nilpotents of $\Bbb{Z}_n$
I am trying to find the nilpotents of the ring $\Bbb{Z}_n$.
Let $\bar{a}$ $\in$ $\Bbb{Z}_n$ be a nilpotent.
Then by definition, $\exists$ $m \in \Bbb{N}$ such that $\bar{a}^m$ = $\bar{0}$.
From here, we get that $a^m$ $\in$ $n\Bbb{Z}$.
So we have that $n|a^m$.
Let $p$ be any prime divisor ... |
H: Identities with respect to composition
In my abstract algebra textbook, when introducing category it says that morphisms should satisfy several properties and two of them are:
For every object $A$ of $C$, there exists (at least) one morphism $1_A \in \text{Hom}_C(A,A)$, the ‘identity’ on A.
and
The identity morp... |
H: If $f: \mathbb{Z} × \mathbb{N} \to \mathbb{Z}, f(x,y)=x^2+y$. Prove if is an injection and prove if is an surjection.
If $f:\mathbb{Z} × \mathbb{N} \to \mathbb{Z}$.
$f(x,y)=x^2+y$.
$f:\mathbb{Z} × \mathbb{N}$ is a relation
Prove if is an injection and prove if is an surjection.
Help, I do not know how to proof it.
... |
H: Personal proof to the notion that the set of rational numbers is countable
I am not a student of core mathematics and hence as a result, all of my educational (engg.physics) background is based on the notion of applied mathematics rather than core mathematics, but I am attending a lecture in probability theory wher... |
H: Find all functions $f:\Bbb R^+\to\Bbb R^+$ s.t. for all $x\in \Bbb R^+$ the following is valid: $f\bigg(\frac{1}{f(x)}\bigg)=\frac{1}{x}$
Find all functions $f:\Bbb R^+\to\Bbb R^+$ s.t. for all $x\in \Bbb R^+$ the following is valid:
$$f\bigg(\frac{1}{f(x)}\bigg)=\frac{1}{x}$$
I tried to substitute $\frac{1}{x... |
H: Prove that if $F$ is a family of sets and $A\in F$, then $\bigcap F\subseteq A$
Question is from "How to Prove it" by Vellenman.
I am struggling to even understand how this theorem could ever be true?
I get that the first step is to assume x is an arbitrary element of $\bigcap F$, as the definition of $\bigcap F\su... |
H: Inverse Laplace transform of $\exp(-s)F(s)$
$$y(s)=1/s - 1/s^2 + \exp(-s)\cdot(1/s^2)$$
I'm struggling with $$\exp(-s)\cdot(1/s^2)$$
formulas: $$f(t-1)u(t-1)\to \exp(-s)F(s)$$
But $$f(t)=t\cdot u(t)$$
is this true:
inverse Laplace transform of $$\exp(-s)F(s)=(t-1)u(t-1)u(t-1)$$
AI: It's simply:
$$\mathcal{L^{-1}} \... |
H: Proof of Morera's Theorem for Triangular Contours
Sorry if this has been proven previously on MSE but I cannot find an obvious duplicate. I am attempting to prove the stronger version of Morera's theorem namely:
If $f:U\mapsto\mathbb{C}$ is a continuous function on an open set $U$ such
that $\int_\gamma f(z)\,\mat... |
H: Find $a,b,c$ if ${(1+3+5+.....a)}+{(1+3+5+....b)}={(1+3+5+.....c)}$
If $${(1+3+5+.....a)}+{(1+3+5+....b)}={(1+3+5+.....c)}$$
and $$(a+b+c)=21, a\gt6$$
We have to find $a,b,c$
My attempt
I use a little fact that the sum of first $n$ odd numbers is $n^2$
From that i get $$a^2+b^2=c^2$$
Which means that the solutions ... |
H: Number of non-zero solutions of an equation in $F \times F$ where $F$ is a field
Let $F$ be a field of order $32$.
Then I need to find the number of non-zero solutions $(a,b)$ $\in$ $F\times F$ of the equation $x^2+xy+y^2 = 0$.
I know that characteristic of a ring with unity $1$ is the order of $1$ in the group $(R... |
H: Finding coefficients of a quadratic with roots having certain intervals
If the roots of the quadratic equation $$(4p−p^2 −5)x^2 −(2p−1)x+3p=0$$ lie on either side of unity, then the number of integral values of $p$ is?
Okay so I'm having a hard time what the question means by both sides of unity. Does it mean o... |
H: Integral in $3D$
Let $R = [0,1]\times [0,1]$ and $f(x,y) = x.$ Note that $f$ is uniformly continuous on $R$. Use the definition of an integral to show that $\displaystyle\int_R f(v)dv = 0.5.$
Drawing this out, it is easy to see that this integral is the volume of a triangular prism with base area $0.5$ and height... |
H: Prove that $m(\{x\in[0,1]:\lim \sup_{j\rightarrow\infty}f_j(x)\geq\frac{1}{2}\})\geq\frac{1}{2}$ under these conditions...
Question: Suppose for each $j\in\mathbb{N}, f_j:[0,1]\rightarrow\mathbb{R}$ is Lebesgue measurable such that $0\leq f_j\leq\frac{3}{2}$ and $\int_0^1 f_j dm=1$. Prove that $m(\{x\in[0,1]:\lim ... |
H: Probability two uniform distribution(0,1) = 2/9
Two numbers are independently and uniformly chosen from the interval (0,1). What is the probability that the sum of the numbers is less than 1 and the product of the numbers is less than 2/9? (Note that both conditions hold simultaneously.)
Given than $n_1\sim\operato... |
H: Upper bound for amount of intervals in intersection of interval sets
I have two sets of numbers which are unions of disjoint intervals, and I have to find an upper bound for how many of such intervals can there be in the intersection of the two sets.
Here's a diagram of how that would look like. After trying for a ... |
H: Normality is transitive along a chain of index 2 subgroups
Let $G$ be a finite group. Let $G_1$ be a subgroup of $G$ such that $[G:G_1]=2$. Now suppose that $H$ is a normal subgroup of $G_1$ with odd order. Prove that $H$ is a normal subgroup of $G$.
Some remarks/partial progress:
I have shown that if $G$ is a gro... |
H: Dense Subspace of Extremally Disconnected Space is Extremally Disconnected
Problem 15G of Willard is -
Every dense subspace and every open subspace of an extremally disconnected space is extremally disconnected.
I've been able to prove the 'open subspace' part of the problem, but the result for 'dense subspace' h... |
H: Given $N, b$ find the largest value of $k$ such that $N=a \times b^k$. $a,b,k,N \in \mathbb{Z}$
I'm trying to find a function that produces $k$ in this problem using $N$ and $b$ as inputs.
Given $N, b$ find the largest value of $k$ such that $N=a \times b^k$. $a,b,k,N \in \mathbb{Z}$
My instinct is that this is not... |
H: Surface integrals: why do we use two integrals to find the area of a surface?
Why is it a double sum of rectangles and not a single sum?
Recently I found another way to calculate the area of a surface on my course of differential geometry and really confused me because its one.
$r_u=$ the partial with respect of $u... |
H: Alteration in Binary Strings Question
Question:
An alteration in a binary string is said to occur when the string encounters one of the following two patterns -
“01” or “10”.
For example, the string 1101001 has exactly 4 alterations in it - which occur at positions 3,4,5
and 7. Count the total number of n bit stri... |
H: Prove that the axiom of choice is necessary in order to prove something else.
My mathematical background is perhaps a little lacking on this topic, but I've been searching and haven't come up with a satisfactory answer to this question. I have no idea how to approach the problem or if it has been answered.
I have s... |
H: Proof of equivalence of two statements about relationship between two generating functions
I am trying to prove the equivalence of the following two statement:
$(a_i)_{0}^{\infty}$ and $(b_i)_{0}^{\infty}$ are infinite sequences of numbers, such that their elements are related in the following manner: $b_n=\sum_{k... |
H: 3 new Points lying on Jerabek Hyperbola?
R is the circumcenter, H is the orthocenter of the triangle ABC.
Then points F,G,E are the points of intersection of the altitudes with the sides of the triangle ABC. U, V, W are the intersection points of cevians AR, BR, CR with the sides of the triangle ABC.
X is defined a... |
H: Integrate ${\sin(x)\cos(x)}$ by parts, by letting ${u=\cos(x),dv=\sin(x)dx}$
I was able to integrate by parts using $u=\sin(x)$ but I'm trying to do it the other way.
With $\int \sin\left(x\right) \cos\left(x\right) dx$
$u = \cos\left(x\right)$
$dv = \sin\left(x\right) dx$
$v = -\cos\left(x\right)$
Then,
$I = uv - ... |
H: Inverse of matrix plus identity
I have a matrix $A$ which is symmetric and positive definite, and I am curious about the properties of $(I+A)^{-1}$.
I can tell that the matrix will exist (that is, $I+A$ will be symmetric and invertible), and thus that $(I+A)^{-1}$ will also be symmetric. I am curious if there are a... |
H: Help with Independent Probability Problem
There are n dice of different colors being rolled simultaneously. The numbers that show up on the faces are
added up to compute X. How many different die rolls result in X being divisible by 3? (basically how many
of the n tuples with each number being between 1 and 6 are s... |
H: Can inflection points be determined from the local extrema of the first derivative?
The local extrema of the first derivative determining the inflection points makes sense; for $x = c$ to be a local extremum for the first derivative, the first derivative's derivative (i.e., the second derivative) would need to chan... |
H: Prove that there exists a sequence $(a_n)_{n=1}^{\infty}$
I have the following question that,
Let $A$ be a nonempty set of real numbers with a lower bound. Prove that there exists a sequence $(a_n)_{n=1}^{\infty}$ such that $a_n \in A $ for all $n$ and $\lim_{n \to \infty} a_n = \inf(A)$.
Adopting solution from t... |
H: Chess Board counting problem
Consider a n x n chess board. Count the number of shortest paths from the position (0,0) to the position (100,100) if each move can either be a horizontal step or a vertical step. You may assume that n is larger than 100.
AI: Each move can be either $(x,y)\to(x+1,y)$ (we call it "h") or... |
H: $2^x$ is irrational if $x$ is irrational?
Prove/Disprove that if $x$ is irrational, then $2^x$ is also irrational.
My attempt for the proof:
Suppose $2^x>0$ is a rational number, then $2^x=\frac{a}{b}$ for some natural numbers $a$ and $b$. Taking logarithm with base $2$ on both sides to get, $x=\log_2 \frac{a}{b}$.... |
H: Relationship between convexity of geometric figures and functions.
This is a rather simple question that made me curious while studying basic geometry. In Hardy's number theory book, a region is convex if it is possible, through every point $P$ in the boundary - to draw at least one line $l$ such that the whole of ... |
H: Question about the definition of chain homotopy
I recently learned about the definition of chain homotopy.
If $f^\bullet, g^\bullet\colon C^\bullet\to D^\bullet$ are chain maps, then the definition is the following.
A chain homotopy between $f^\bullet$ and $g^\bullet$ is a family of
morphisms $T^n\colon C^n\to D... |
H: Deck of Cards combinatorics
{A,2,3,4,5,6,7,8,9,10,J ,Q ,K} and four suits - {Hearts, Diamonds, Spades, Clubs} . A Hand is a set of 5 cards picked up from the standard deck. How many different hands contain at least one of the following two cards : {K of Hearts, Q of Diamonds} ?
I'm having a little trouble findin... |
H: Let $E$ a normed vector space and $L \neq E$ a vector subspace of $E$. Can $L$ contain any open ball of $E$?
I was trying to solve a problem of the book Elementos de Topologia Geral by Elon Lages and I found an exercise that I don't have success. The exercise is: prove that any vector subspace $L≠E$ of a normed vec... |
H: Percentage question on playing a game
I was doing some past papers of AMC when I got stuck on this question:
Two people were playing a video game. One morning each of them won 70% of their games. That afternoon, they played the same number of games as each other and each won them all. One of the player's winning pe... |
H: Mapping from $Re(z)\leq0$ onto the disk $|z|\leq1$
How am I supposed to do it? I really have no idea at all.
AI: Hint: Recall the Möbius transformation
$$
f\colon z\mapsto\frac{z+1}{z-1}
$$
maps the open left half plane onto the open unit disc. This almost does what you want: $f(\{z\in\mathbb{C}\mid\operatorname{R... |
H: Proving the function is always positive in the interval $(0,1)$
Let $f_0(x)=\frac{1}{1-x}$ and define $f_{n+1}(x)=xf'_n(x)$ prove that
$f_{n+1}(x)>0$ for $0<x<1$ and $n\in \mathbb{N}$
my attempt was as following. Since the expression invloves the natural numbers, induction will be a good method of proof. After eval... |
H: What is the value of $L$ and the arcs $AG$ and $BH$?
What is the value of $L$ and the arcs $AG$ and $BH$, as a function of $d, R$ and $r$?
Note that $G, H, C, D$ are collinear and $AB$ tangent to both circles.
The problem was inspired by a question from the ACT exam.
AI: Hint: Please draw a line from center $C$ to... |
H: Analogue of Turing recognizable languages
A language $S$ is called Turing recognizable if for some Turing machine $S$ is exactly the set of inputs when the machine halts. How can we call the language which is the set of outputs for some Turing machine? How are these two classes related?
AI: Any language "generated"... |
H: Why is the general solution to linear homogeneous differential equation with constant coefficients different if roots are distinct or repeating?
Consider the second order linear homogeneous differential equation:
$$ax'' + bx' + c = 0$$
If we start from the assumption that the solution has the form
$$ x(t) = e^{rt}$... |
H: Finding values using the equation of $x$ that satisfies $\left\{ x \right\} + \left\{ {\frac{1}{x}} \right\} = 1$
Let the real number $x$ satisfies $\left\{ x \right\} + \left\{ {\frac{1}{x}} \right\} = 1$ and $k$ denotes the value of $\left\{ {{x^3}} \right\} + \left\{ {\frac{1}{{{x^3}}}} \right\} = k$ (where $\{\... |
H: Determining the leading coefficient of Vandermonde's Determinant.
On Shilov's Book "Linear Algebra", when calculating the Vandermonde's Determinant,
the author concludes the leading coefficient of the product of the roots of the determinant (seen as a polynomial) is
$$ W(x_1, ..., x_{n-1}) $$
Shilov proceded to see... |
H: Polygon Diagonal Combinatorics
A diagonal for a polygon is defined as the line segment joining two non-adjacent points. Given an n-sided polygon, how many different diagonals can be drawn for this polygon?
I know that the number of diagonals is C(n,2). However, I don't know how to account for the fact that you can'... |
H: Isomorphism Definition - Error in Hoffman Kunze?
I am Referring to the book Linear Algebra by Hoffman and Kunze $2$e, page $84$ section $3.3$.
It defines Isomorphism as follows:
If $V$ and $W$ are vector spaces over a field $\mathbb{F}$, then any one-one $T$ $\in$ $\mathcal{L}(V, W)$ of $V$ onto $W$ is called an ... |
H: What are some ways to prove that a k-partite graph is nonplanar?
I am reading papers on graph theory and I encountered one work that talked about the planarity of a certain graph. In one of the proofs of a theorem, the author stated that the graph $K_{1,2,3}$ is nonplanar. Is this a known result? How can one check ... |
H: Prove the inequality $u^{\alpha}v^{1-\alpha} \leq \alpha u + (1-\alpha)v $
I'm trying to prove the inequality highlighted by the red line in the picture, but I do not know how. Please provide detail as much as you can, thank you!
For a minimal family, the sufficient statistic $\mathbf{T}$ is also minimal sufficien... |
H: Independence between fractional parts of consecutive sums of independent uniforms
Let $X_1,X_2$ be independent $\text{Uniform}(0,1)$ random variables. Define $U_1 = X_1 - \lfloor X_1 \rfloor$ and $U_2 = X_1 + X_2 - \lfloor X_1 + X_2 \rfloor$ where $\lfloor a \rfloor$ is the largest integer less or equal to $a \in \... |
H: Evaluating $\int _0^1\frac{\ln \left(x^3+1\right)}{x+1}\:dx$
What methods would work best to find $\displaystyle \int _0^1\frac{\ln \left(x^3+1\right)}{x+1}\:dx$
As usual with this kind of integral i tried to differentiate with the respect of a parameter
$$\int _0^1\frac{\ln \left(ax^3+1\right)}{x+1}\:dx$$
$$\int _... |
H: How to express a Python loop and a condition mathematically?
I have the following Python code that I want to express mathematically.
W = 0
for i in range(5):
if (A > i_1) and ( (A < i_2) or (A < i_3) ):
W = W + 1
This is a generic problem that I am trying to achieve; looping over some values and checki... |
H: Doubts on a game that two players take turns to take an element and xor it to their sums.
This is originally a question from https://codeforces.com/contest/1383/problem/B, but this is more of a math problem than an algorithmic one, so I decided to post here.
There is a game where two players take turns to take an ... |
H: If $X$ is a Banach space any capable conditions to make sure that $X$ is also Hilbert?
Let $X$ be a Banach space and suppose that $X$ is isometric with a Hilbert space.Is it true to say that $X$ is also a Hilbert space?If I assume that $X$ is isomorphic with a Hilbert space is it still true?
AI: YES for the first q... |
H: Given $k, a \in \mathbb{R}$, find a polynomial $P$ such that $P(k) = a$
You are given two real numbers $k,a \in \mathbb{R}$, and you are promised that there is a polynomial with integer coefficients $P \in \mathbb{Z}[X]$ such that evaluating it on $k$ yields $a$, i.e.
Promise: $\ \exists P \in \mathbb{Z}[X] \ \ \ ... |
H: Solving method for fractions having Prime numbers as Denominator.
I have a question , like
$30 = 1.18x$ , and $30 = 0.82 y$ , find $x+y$.
when solving, $x=1500/59$, $y=1500/41$. Since the denominator is involving prime numbers I was wondering is there any method or some logic is there to solve this question under... |
H: (Verification) If $g \circ f$ is injective, then $f$ must also be injective.
Prove that if $g \circ f$ is injective, then $f$ must also be injective.
$g(f(x))$ is injective.
Then, $x\neq x’ \implies (g\circ f)(x) \neq (g\circ f)(x’)$.
Suppose for the sake of contradiction: $f(x)=f(x’) \nRightarrow x=x’$
Then t... |
H: Why is my integrand wrong?
I am given the following integral. $$\iiint\limits_E \sqrt{x^2+z^2} \ dV$$ bound by the paraboloid $y=x^2+z^2$ and the plane $y=9$.
Half way through the problem, it says the given integral can be simplified to $$\int_{-3}^3\int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} {\left(9-x^2-z^2\right)\sqrt{x... |
H: When is a Lie group action is a local diffeomorphism?
My whole question is actually in the title. Suppose a Lie group $G$ acting smoothly on a smooth manifold $M.$ I am looking for the conditions on the group action or the Lie group $G$ that makes $M$ and $g(M)$ are locally diffeomorphic for all $g\in G.$ Perhaps t... |
H: Given the position vectors of the vertices of a triangle, prove that another point is the orthocentre of the triangle.
The original problem is:
If a, b, c, d are the position vectors of points A, B, C, D respectively such that $$(\vec{a}-\vec{d}). (\vec{b}-\vec{c})= (\vec{b}-\vec{d}). (\vec{c}-\vec{a})= 0$$then pro... |
H: The region of convergence of the series $\sum_{n=1}^{\infty} \frac{(-1)^n n (x-1)^n}{(n^3+1)(3^n+1)}$.
I am trying to find the region of convergence of the series
$$
\sum_{n=1}^{\infty} \frac{(-1)^n n (x-1)^n}{(n^3+1)(3^n+1)}.
$$
I know that, by the limit comparison test,
$$
\sum_{n=1}^{\infty} \frac{n (x-1)^n}{(n^... |
H: Find the limit: $\lim_{x\to 3^-} \dfrac{(e^{(x+3)\ln 27})^{\frac{x}{27}} -9}{3^x-27}$
The expression simplifies to
$$\lim_{x\to 3^-}\dfrac{(27^{x+3})^{x/27} -9}{3^x-27}=\lim_{h\to 0} \frac{27^{(6-h)(3-h)/27} -9}{3^{3-h}-27}.$$
I simplified the denominator as follows: $\;27(3^{-h}-1)=-27h \ln 3$.
How should I simpli... |
H: Continuity of a map between a product space equipped with product topology
Consider $[n]=\{1, 2, \cdots, n\}$ with the discrete topology and let $X=\prod_{n\geq 1}[n]$ be the product space with the product topology. For $x=(a_1, a_2, \cdots)$, define $T(x)=(1, a_1, a_2, \cdots)$. Then how to show the following?
Th... |
H: How to find the critical index $a$ of $x^af(x)$?
Let $f\in C^1(0,+\infty)$, $f(x)> 0$ in $(0,+\infty)$, $f(0+)=+\infty$, and $f$ is decrased in $(0,+\infty)$. Consider $g(x)=x^af(x)$, $(a>1)$. Now one can show that $\liminf_\limits{x\rightarrow0^+}g(x)=\limsup_\limits{x\rightarrow0^+}g(x)$, so we write $\lim_\limit... |
H: What is the supremum, infimum, maximum and minimum of the set $B:=\{x \in\mathbb{R}, \left||x-1|-|x-2|\right|<1\}$?
What is the supremum, infimum, maximum and minimum of the set
$$B:=\{x \in\mathbb{R}, \left|\left|x-1\right|-\left|x-2\right|\right|<1 \}?$$
I am not sure how to find any of them. I thought I have ... |
H: Derive $\mathbf b$ from $\mathbf a = \mathbf b× \mathbf c$
I have an equation:
$$\mathbf a = \mathbf b × \mathbf c,$$
where $\mathbf a$ $\mathbf b$ and $\mathbf c$ are 3-vectors.
How could I derive $b$ from the equation and express it in terms of $\mathbf a$ and $\mathbf c$?
AI: You can't. Given $c$ and a vector $... |
H: Convergence of a Real number sequence
I want to prove if the following statement is true:
Let a sequence of real numbers $x_{m}\left(j\right)\to x\left(j\right)$ as $m \to \infty$, $ \forall j\in \mathbb{N}$. Then:
\begin{equation*}
\sum_{j \in \mathbb{N}} |x_{m}\left(j\right) - x\left(j\right)| \to 0 \text{ as } m... |
H: $H$ and $K$ are normal subgroups of a group $G$ with $K \le H$, then $aK =H \iff a \in H$
Let $H$ and $K$ are normal subgroups of a group $G$ with $K \le H$.
How can I prove that $aK =H \iff a \in H$?
I‘m stuck with the part when $a \in H$ then $aK$ contains $H$.
Given $u$ in $H$, how can I show that $u \in aK$ ？... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.