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H: $x^4-4=y^2+z^2$ prove that it has no integer solution
$x^4-4=y^2+z^2$ prove that it has no integer solution
I tried to check mod$4$ , mod $3$ ...
It doesn't give anything.
I want to solve this problem
by supposing that I'm finding the smallest solution and then prove that a smaller one exists. But for that I have... |
H: Evaluating the indefinite Harmonic number integral $\int \frac{1-t^n}{1-t} dt$
It is well-known that we can represent a Harmonic number as the following integral:
$$H_n = \int_0^1 \frac{1-t^n}{1-t} dt$$
The derivation of this integral doesn't need you to derive the indefinite integral first, so now I'm wondering w... |
H: How this kind of permutation is called in math?
Assume we have a set $\{a_0,a_1,a_2,...,a_{n-1}\}$. Our permutation maps each element $a_i$ to $a_{ki\bmod n}$, where n and k are relatively prime.
Geometrically it looks like if take a regular $n$-polygon, which vertices are numbered from $0$ to $n - 1$ clockwisely, ... |
H: Solutions of $e^x-1-k\cdot \arctan{x}=0$,
consider $h(x)=e^x-1-k\cdot\arctan{x}$,then find on which condition on $k$ there will be two solutions for $h(x)=0$ ($k$ is real).
I let $f(x)=e^x-1$ and $g(x)=k.\arctan{x}$ and let $f(x)-g(x)=0$ has roots $0$ and $y$. I found that if $y$ tends to $0$ then $k$ tends to $1$... |
H: Find all continuous function $ \frac{1}{2} \int_{0}^{x}(f(t))^{2} d t=\frac{1}{x}\left(\int_{0}^{x} f(t) d t\right)^{2} $
Find all continuous function $f:(0, \infty) \rightarrow(0, \infty) \ni f(1)=1$ and
$$
\frac{1}{2} \int_{0}^{x}(f(t))^{2} d t=\frac{1}{x}\left(\int_{0}^{x} f(t) d t\right)^{2}
$$
My approach:-
Le... |
H: How can I construct a nilpotent matrix of order 100 and index 98?
I know to construct a nilpotent matrix of order $n$ with index of nilpotency $n$, but how to construct a nilpotent matrix of order $n$ but index of nilpotency $(n-2)$? Is there any general rule for the same?
AI: The other answer gives you a very gene... |
H: Can the vertices of $20$-gon be labeled with $1, 2, ..., 20$ in such a way that sum of any $4$ consicutive numbers is less than $43$?
Can the vertices of a regular $20$-gon be labeled with numbers $1, 2, ..., 20$ in such a way that each label is used exactly once and for every four consecutive vertices the sum of t... |
H: Example of a set with empty boundary in $\mathbb{Q}$
I was dealing with a problem, if subset of a metric space has empty boundary then it is open as well as closed in the space. The proof is easy. But I am wondering for a nontrivial example of such set (which has empty boundary).
Since $\Bbb{R}$ is connected, so in... |
H: What is the solution of this summation?
$$S(x) = \frac{x^4}{3(0)!} + \frac{x^5}{4(1)!}+\frac{x^6}{5(2)!}+.....$$
If the first term was $$x^3$$ and the next terms were $$x^{3+i}$$ then differentiating it would have given $$x^2.e^x$$ and then it was possible to integrate it. But how to solve this one?
AI: I'm assumin... |
H: Finding the value of $\sin^{-1}\frac{12}{13}+\cos^{-1}\frac{4}{5}+\tan^{-1}\frac{63}{16}$
Find the value of $\sin^{-1}\frac{12}{13}+\cos^{-1}\frac{4}{5}+\tan^{-1}\frac{63}{16}$.
My attempt: $$\sin^{-1}\frac{12}{13}+\cos^{-1}\frac{4}{5}+\tan^{-1}\frac{63}{16}$$
$$=\tan^{-1}\frac{12}{5}+\tan^{-1}\frac{3}{4}+\tan^{-... |
H: Definition of subsequence used in defining accumulation points
According to the definition given on this page, a number $a$ is an accumulation point of a sequence $(a_n)$ if there is a subsequence $(a_{n_k})$ that converges to $a$ in the $\lim_{k\to \infty}$.
What does the word subsequence mean in this definition?
... |
H: Check function uniform continuity
The task is to check function uniform continuity in terms of the following set $\{(x,y): x^2+y^2 \geq 2\}$:
$$f(x,y)=(x^2+y^2)\cdot \sin\left(\frac{1}{x^2+y^2}\right)$$
Can you help me with this one?
I have been solving with only single point to check, however here is the whole set... |
H: Compliment probability
Two events A and B have the following probabilities: P[A] = 0.4, P[B] = 0.5, and P[A ∩ B] = 0.3
Calculate the following:
(a) P[A ∪ B] =0.6
(b) P[A ∩ '] = 0.1
(c) P[A' ∪ '] =0.7
I get this which is incorrect:
P[' ∪ '] = P(A')+p(B')-P(A ∩ B) =0.6
What is the correct formula.
AI: a) $P(A \cu... |
H: Hausdorff and locally compact
Theorem A space $X$ is locally compact and Hausdorff if and only if it is
homeomorphic to an open subset of a compact Hausdorff space.
Can any one give me hint to prove this result. I want the strategy of proof of this particular theorem.
Thanks in advance.
AI: Hint: if $X$ is compact,... |
H: Differential Equation: Cauchy-Euler Boundary Value Problem
I'm a bit confused with how my text finds the constants:
$$
x^{2} y^{\prime \prime}-3 x y^{\prime}+3 y=24 x^{5}, \quad y(1)=0, \quad y(2)=0
$$
Auxiliary / Characteristic Equation: $m(m-1)-3 m+3=(m-1)(m-3)=0$. General solution of the associated homogeneous e... |
H: Solution to a differential equation using equations that have no analytic solution
So I have the following question here.
Suppose that $y_1$ solves $2y''+y'+3x^2y=0$ and $y_2$ solves $2y''+y'+3x^2y=e^x$. Which of the following is a solution of $2y''+y'+3x^2y=-2e^x$?
(A) $3y_1-2y_2$
(B) $y_1+2y_2$
(C) $2y_1-y... |
H: If $f(x) = x^4 - x^2 + 1$, find the values of $x$ such that $f(f(f(x))) \le x^8$
If $$f(x) = x^4 - x^2 + 1$$ find the values of $x$ such that $f(f(f(x))) \le x^8$
I noticed that $f(\pm 1) = 1 \implies \underbrace{f(f(f(... f(\pm 1)..)))}_{\text{n times}} = 1$. thus, $f(f(f(x))) = x^8$ at $x = \pm 1$. Fortunately,... |
H: Binomial coefficient expansion question
I'm trying to follow this expansion (linked below) for one of my classes, but nothing I have tried is proving successful.
Any hints or help would be very appreciated.
Thanks :)
Binomial coefficient expansion
AI: There is a mistake, here is the correct expansion
$${n\choose r}... |
H: Eigen values and vector norm
$A:\mathbb R^2\to\mathbb R^2$ is a 2 by 2 matrix whose eigen values are 2/3 and 9/5. Prove that there exists a non zero vector v such that |Av|=|v|.
It’s not given that A is symmetric. So I can not conclude that A is positive definite. How do I proceed?? Please help.
AI: Hint : Let $S$ ... |
H: Closed set that is not complete
Can anyone help me out with an example of a closed set that is not complete? I have read up on the set of irrational numbers with the euclidean metric is such an example on other web pages, but that does not make any sense to me since the set of irrational numbers with the euclidean ... |
H: $\sigma(\xi)$ is independent of a fixed $\sigma$-algebra
I want to prove that if $\lim_{n->\infty}\xi_n = \xi$ pointwise and each $\xi_n$ is independent from a fixed sigma-algebra $F$ then $\xi$ is independent from $F$.
I understand that $\sigma(\xi)$ will be independent from $F$ too.
How can it be proved: if $\sig... |
H: Correspondence between time and the level of water while filling a hemisphere with water at a constant rate
I am trying some question in quantitative aptitude section and I couldn't reason for those question.
A hemispherical bowl is being filled with water at a constant volumetric rate. The level of water in the bo... |
H: Determine a basis in $R^4$ containing the vectors $u$, $v$ and $ w$.
Let $u = (2, 3, 4, 4)^T$, $v = (0, 1, 2, 6)^T$
, $w = (0, 0, 1, 1)^T$.
Determine a basis in $\mathbb{R^4}$
containing the vectors $u$, $v$ and $ w$.
I thought the basis will contain only $u,v, w$ . However in answer-sheet there is another fourth v... |
H: Showing that $f$ is always cohomologous to $f_m$, for some $m$.
I am working through problem 10.16 of Morandi's Field and Galois Theory, which is a guided computation of the second cohomology of a cyclic group of order $n < \infty$.
Let $G =\langle\sigma\rangle$ be cyclic of order $n$, let $G$ act on an Abelian gro... |
H: Evaluate the integral using Euler integrals
I have the following integral:
$$\int_{0}^\infty \frac{\sqrt{x}}{7+x^7} \ dx$$
I want to evaluate this using the Euler integral. What I have tried:
I tried to make a substitution, because I want to evaluate it via gamma integrals. But I can not find the substitution. Can ... |
H: Show that T is a regular distribution
I am given a Distribution $$T:D(\mathbb{R}) \rightarrow \mathbb{C}$$ and need to show that it is in fact a regular distribution.
Let $T(\varphi)=\varphi(-1)+\varphi'(1)$.
How can I show that this is in fact a regular distribution ? By definition I would need to find a locally... |
H: What test should I use for this problem? (assessing the significance of a change in a categorical variable between two different sized populations)?
I have 2 high schools, School A and School B.
For the first school, I have 5 classes of students; for the second, I have 3 classes (so 8 classes in total).
Within each... |
H: Alternating series estimation test proof
The first part of the proof of the error estimate theorem in integral calculus is confusing me. It states that $$\biggr\vert \sum_{n=0}^{\infty}(-1)^nb_n-\sum_{n=0}^N(-1)^nb_n\biggr\vert=\biggr\vert \sum_{n=N+1}^{\infty}(-1)^nb_n)\biggr\vert$$
I don't understand why the lowe... |
H: Is every closed operator a bijection?
Let $H$ be a $\mathbb R$-Hilbert space and $A$ be a closed linear operator on $H$. Can we conclude that $A$ is an isomorphism between $\mathcal D(A)$ and $H$?
My intuition is that this is clearly wrong, but this seems to be what's been claimed in this book after the proof of Le... |
H: Let $f$ and $g$ be the functions defined by f$(t)=2t^2$ and $g(t)=t^2+5t$
I have solved the following:
$f′(t)=4x$
$g′(t)=2x+5$
To solve the next step:
Let $p(t)=2t^2(t^2+5t)$ and observe that $p(t)=f(t)⋅g(t)$. Rewrite the formula for p by distributing the $2t^2$ term. Then, compute p′(t) using the sum and constant ... |
H: A basic question on Probability in quantitative aptitude
I am trying some aptitude questions and this question asks for use of probability . I am not able to find the right answer.
A cupboard is filled with a large number of balls of 6 different
colours. You already. have one batl of each colour. If you are
blind-... |
H: A zero divisor which is not an element of a minimal prime ideal
Iam looking for a ring R in whiche there is a zero divisor which is not an element of a minimal prime ideal.
In rings which I have checked I couldn't find such an element...
AI: et $K$ be a field, and $R=K[X,Y]/(X^2, XY)$. This ring has a single minima... |
H: Solve the equation $\operatorname{arcsinh}=\operatorname{arcsech}(x)$ analytically
I am trying to obtain an analytical solution of the equation.
$$\operatorname{arcsinh}(x) = \operatorname{arcsech}(x)$$
Equating the logarithmic definitions leads to the rather unwieldy equation
$$x^4+x^3\sqrt{x^2+1} +x^2 -1.0 -\sqrt... |
H: Question on vector cross product.
Show that $\big((\mathbf{u} \times (\mathbf{u}\times \mathbf{v})) \times \mathbf{v}\big) \times (\mathbf{u} \times \mathbf{v})=0$.
From wolfram, it gives zero but there's no details. How to prove this?
AI: Using the identity
$$\textbf{A}\times (\textbf{B}\times \textbf{C})= (\text... |
H: What can we say about a group $G$ if for all $a,b,c,d$ in the group, $ab=cd\implies ba=dc$?
Let $G$ be a group. Suppose that, for elements $a, b, c, d$ of $G$, we have $ab = cd \implies ba = dc$. Can we derive anything from this, or are there any conditions that result in such a property? I.e. what does it imply, a... |
H: Does this formula defines a definable subset?
I am studying definable subsets in Introduction to O-minimal geometry, M. Coste, and I have recently seen some formulas named: first order formulas: if $\phi$ is a first order formula, the set $ \{ x \in R^n : \phi(x) \}$ is definable in $R^n$, where $R$ is a real clos... |
H: A confusion about complex measure
The 3.13 proposition from the book "Real Analysis" by Folland:
Let $\nu$ be a complex measure on $(X,\mathcal{M})$.
a.$\left| \nu \left( E \right) \right|\le \left| \nu \right|\left( E \right) $ for all $E\in\mathcal{M}$.
b.$\nu \leqslant \left| \nu \right|$,and $d\nu/d\lvert ... |
H: How do I prove that eigenspaces and root subspaces are invariant for A?
So the eigenspace is $Ker(A-λI)$ where $λ$ is an eigenvalue of A and the root subspace is $Ker(A-λI)^r$ where $r$ is the exponent of $(x-λ)$ in the minimal polynomial for $A$. My professor stated that both of these are invariant for $A$, but di... |
H: when $n=12$ and $k=7$, can we generate a sub-group?
$\def\llg{\langle} \def\rrg{\rangle}$
There is a theorem that says if $|\llg a\rrg|=n$, then for each positive divisor $k$ of $n$, the group $\llg a\rrg$ has exactly one subgroup of order $k$—namely, $\llg a^{n/k}\rrg$.
Assume $\llg a\rrg$ is a cyclic group of ord... |
H: Exercise 3.4.7 from Tao Analysis I (Set of all partial functions)
This is an exercise you can find here, but I recall the context:
Let $X, Y$ be sets. Define a partial function from $X$ to $Y$ to be
any function $f: X' \rightarrow Y'$ with $X' \subseteq X$ and $Y'\subseteq Y$.
Show that the collection of all parti... |
H: Coin toss probability - With two variables
My question is -
If I toss a fair coin $3$ times,
$X$ - The number of heads in the first two tosses.
$Y$ - The number of heads in the last two tosses.
$Z$ is $Z = X + Y$. What is $V(Z)$?
So I'm thinking if I just should decide that X is the probability to get heads in the ... |
H: Probability of winning a lot
So there's this game that I'm analysing, in which out of $45$ numbered balls ( numbered from $1$ to $45$ ), I choose $8$ balls.
$6$ out of the $45$ balls are drawn in the end of round by the organiser, and who get's $6$ out of $8$ balls of his draws, matching with the winning $6$ drawn ... |
H: Proving that $f_n(\alpha_{n+1}) > 0$
We have that : $ f_{n}(x) = 2x - 2 + \frac{\ln(x^2+1)}{n}$
and : $ f_{n}(\alpha_{n}) = 0 $
Where : $ 0 < \alpha_{n} < 1$
and they asked us to prove that : $f_{n}(\alpha_{n+1}) > 0$
and to prove that $ \alpha_{n} $ is a geometric series.
AI: $f_n$ is differentiable on $[0,1]$ an... |
H: Isomorphism between a principal bundle and a pullback bundle.
I have seen in many texts on the classification of main bundles that, given two homotopically equivalent X and Y spaces, this equivalence being the function $f: Y \rightarrow X$, given a group G, if $k_{G}(X), k_{G}(Y)$ represents the set of isomorphism ... |
H: How AM-GM is applied here
I don't understand how AM-GM is applied in the last part of the picture. This is on the the $16^{th}$ page of the book in chapter $1$ about $AM-GM$.
AI: We need to prove that:
$$\frac{a+c}{b+c}+\frac{a+c}{a+d}\geq\frac{4(a+c)}{a+b+c+d}$$ or
$$(a+b+c+d)\left(\frac{1}{b+c}+\frac{1}{a+d}\rig... |
H: A ratio of red beads to black is $r:b=4:3$. Why doesn't this translate to $4r=3b$ instead of $3r=4b$?
Suppose that for every 4 red beads ($r$), there are 3 black beads ($b$)
The ratio of red beads to black beads is $r:b=4:3$
But then why is this ratio not equal to $4r=3b$ and instead $3r=4b$?
I have that 4 red be... |
H: for what values of $a,b$, $\int_{-1}^{1}((x^{2}+3 x+1)-(a x+b))^{2} \sqrt{1-x^{2}} d x$ is minimal?
Let $V=\mathbb{R}_{\leq 3}[X]$ I need to find $a,b \in \mathbb{R}$ such that the below expression is minimal.
$$\int_{-1}^{1}\left(\left(x^{2}+3 x+1\right)-(a x+b)\right)^{2} \sqrt{1-x^{2}} d x$$
I got a hint to show... |
H: If a polynomial is irreducible and nonconstant over a finite field, it has a multiple root iff it is in the variable $x^p$
I am a very basic field theory question. I must be mixing up a theorem here, but I am unsure which.
My goal here is to determine if there exists an inseparable, irreducible polynomial in a fini... |
H: What is the sum of the $n^2$ terms obtained this way?
We multiply each entry of an $n × n $ matrix A by the cofactor belonging to it. What is the sum of the
$n^2$ terms obtained this way?
I dont understand how a cofactor can belong it entry. Does it mean one cofactor is same for each entry of according row or colum... |
H: Compound Poisson distribution and infinitely divisible probability generating function
Let $(X_j)$ be a sequence of r.v. with common distributions $(f_j)$, $N$ be a r.v. having a Poisson distribution with mean $\lambda t$. Let $S_N = X_1 + \cdots + X_N$. Then, $S_N$ has the compound Poisson distribution with a gene... |
H: A question on Triangle Inequality in $\mathbb{R}^n$
I'm reading a textbook on Topology. We know that $(\rho,\mathbb{R}^n)$ is a metric space, where
$$\rho(x,y)=\sqrt{\sum_{i=1}^n(x_i-y_i)^2}$$for any $x=(x_1,x_2,\ldots,x_n),y=(y_1,y_2,\ldots,y_n)\in\mathbb{R}^n$. When proving that $\rho(x,z)\le \rho(x,y)+\rho(y,z)$... |
H: difference of operators defined through 2 different functional calculi.
Lets say, we have a fixed function $f$ and two normal/self-adjoint operators $\mathcal{L} ,\tilde{\mathcal{L}} $ with discrete spectrum. Are there some conditions (on $f$ or the normal operators , such as lipschitz continuity of f) that ensure:... |
H: Existence of complementary subspace
Let $E$ be a real vector space. If $E$ has finite dimension, then for any subspace $F\subset E$ there is always some subspace $G\subset E$ such that
$$E = F \oplus G$$
In infinite dimension, I know that the axiom of choice allows to construct such a $G$ for any subspace $F\subset... |
H: How to show that $\lim_n \beta_n|f(\alpha_n x)|=0 \, \, a.e.$
Let $f\in L^1(\mathbb{R})$, $\{\beta_n\}$ a positive sequence and $\{\alpha_n\}$ such that $\sum_n \beta_n/|\alpha_n|<\infty. $ Prove that
$$\lim_{n\to \infty} \beta_n|f(\alpha_n x)|=0 \, \, a.e.$$
AI: Hint: This follows from the a priori stronger fa... |
H: Counterexample for 2-22 of Spivak's Calculus on Manifolds?
If $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ and $D_2f = 0$, show that $f$ is independent of the second variable.
I was thinking of ways to show this, when I came across what I think might be a counterexample.
Possible counterexample: Consider the function $... |
H: Location of new point in new rectangle
I am stuck on a problem that maybe trivial but I am stumped. Suppose that there is a rectangle with points $(5,0), (20,0), (5,5) \text{and} (20,5)$. Inside the rectangle, there is a point $(6,1)$. Doing modifications to the rectangle, we get a new rectangle with points $(10,1... |
H: For about real eigenvector.
Let $A$ be a real $n×n$ matrix. If $A$ has a complex eigenvalue, then is there any possibility to have a real eigenvector corresponding to this complex eigenvalue?
And my second question is if there is no real eigenvector for this complex eigenvalue, then is it not violating the definit... |
H: Let R = $\{ (n+4,n) \mid n \in \Bbb Z^+\}$, Find $R^2$
I found that this relation is not transitive, does this mean that $R^2$ does not exists?
Any help is appreciated thanks!
AI: The relation $R$ on the set $\mathbb{Z}^+$ is given by
$$
R \colon= \left\{ \, (n+4, n) \colon n \in \mathbb{Z}^+ \, \right\}.
$$
This $... |
H: Find the values of $a$ and $b$ if $\lim_{x\to -\infty}$ $\sqrt{x^2-x+1} + ax - b = 0$?
I took out x from the square root and reached the following expression,
$$\lim_{x\to -\infty} x\sqrt{1-\frac{1}{x}+\frac{1}{x^2}} + ax - b = 0$$
then I separated the part of the expression which contains x from b and tried evalua... |
H: Double integration over a Region $D$ enclosed by circle and lines
If $D=\{(x,y):x^2+y^2\geq 1, 0\leq x\leq 1,0\leq y\leq 1\}$, then find $$\iint_{D}\frac{1}{(x^2+y^2)^2}\,dA$$
What i try: I have draw diagram of region $D$
Using polar coordinates $x=r\cos\theta,y=r\sin\theta$
And $x^2+y^2=r^2$ and $dA=rdrd\theta$... |
H: Proof that greatest entry of a unit vector is $\leq 1$ and $\geq \frac{1}{\sqrt{n}}$
I have a real orthogonal matrix so the column vectors form an orthogonal system and thus the vectors have length one.
I now want to show that for an arbitrary column vector $v_k \in \mathbb{R^n}$ the absolute value of the greatest ... |
H: Differentiability using differentiation
We use this method for piecewise functions to determine differentiability at the point where function changes its definition.
For example-
$f(x)$ =
\begin{cases}
x+1, & x<1 \\
2x^2, & x\geqslant 1
\end{cases}
First we check the continuity at the value of $x=1$. Then we calcul... |
H: Deduce that $ e < \left({1+ \frac 1n}\right)^{n+\frac12} < e^{1+\frac {1}{2n (n+1)}} $ from the result obtained.
Prove that $$ 2x < \log{\frac {1+x}{1-x}} < 2x \left[{\frac {1+x^2}{3 (1-x^2)}}\right] $$ where $0 < x <1.$ Hence, deduce that $$ e < \left({1+ \frac 1n}\right)^{n+\frac12} < e^{1+\frac {1}{2n (n+1)}} $... |
H: Is local minimum/maximum necessarily global when it's the only stationary point of a continuous & differentiable function?
Couldn't find this theorem even though it feels very intuitive to me.
If the $f:R^n \to R$ is continuous, and has only one stationary point - a local minimum/maximuma. Doesn't it necessarily m... |
H: Choose balls out of boxes
Part (a) How many ways are there to put $4$ balls into $3$ boxes, given that the balls can all be distinguished and so can the boxes? (For instance, perhaps each ball is a different color, and each box is a different color as well.)
Part(b) How many ways are there to put $4$ balls into $3$... |
H: Simple tensors and projective modules
For a (unital) ring $R$, let $M$ and $N$ be a projective right, and left, $R$-module respectively.
Does it hold that $0 \neq m \otimes n \in M \otimes_R N$ for all non-zero $m \in M$, $n\in N$?
If so, then why?
AI: It does not work even for projective, or free, modules, and the... |
H: 7 digit number combinations where the first 3 digits must be equal to another 3.
Seven-digit telephone numbers are not allowed to begin with $0$ or $1$.
I can only remember a seven-digit telephone number if the first three digits (the "prefix") are equal to either the next three digits or the last three digits. For... |
H: Eigenvalues and operator norm
$A: \mathbb R^2 \to \mathbb R^2$ is $2 \times 2$ matrix with eigenvalues $\frac{2}{3}$ and $\frac{9}{5}$. Prove that there exists
a non-zero vector $v$ with $\|Av\|> 2\|v\|$, and
a non-zero vector $v$ with $\|Av\|<\frac{1}{2} \|v\|$.
By defining a continuous function from the unit... |
H: Is it true that this entry can always be chosen in such a way that the matrix obtained has 0 determinant?
All the entries of an $n × n$ matrix are fixed with the exception of one entry. Is it true that this entry
can always be chosen in such a way that the matrix obtained has 0 determinant?
I thought in order to $d... |
H: For any finitely-generated abelian group, show M(G,n) exists
Let $G$ be a finitely-generated abelian group. Prove there is a CW complex $M(G,n)$ which has $\tilde H_k(M(G,n))$ equals $G$ if $k=n$ or zero otherwise.
Here's what I have so far:
By the fundamental theorem for finitely generated abelian groups, $G \cong... |
H: Sum of roots of trigonometric equation
This is the hardest problem on Georgian (country) high school math exam.
Find all values for parameter $a$ for which the sum of all the roots of the equation:
$$\sin\left(\sqrt {ax-x^2}\right)=0 $$
equal to $100$.
Note that you can't use calculus and we assume only real roots!... |
H: Two homomorphisms $f\colon A\to A',\ g\colon B\to B'$ induce a homomorphism $f \otimes g: A\otimes B \to A'\otimes B'$
Yesterday I was working on an exercise-sheet, given the following definition
For two Abelian groups $A$ and $B$ we define their tensor product
$A\otimes B$ as the quotient of the free Abelian grou... |
H: Proof of an IF - THEN statement
Let $S \subseteq \mathbb{R}$. IF $\exists m \in \mathbb{R}$ such that $\forall n \in S, m \geq n$, THEN $\exists m \in \mathbb{R}$ such that $\forall n \in S, m > n$
Now I know that we get to pick $m$ since its an existence statement, we can pick it as $m = n + 1$ or I can pick $m$ s... |
H: Which is the standard notation for an infinite summation (or any summation-like operator) without indexes?
An infinite indexed summation is written as $\sum\limits_{i=1}^\infty i$.
A summation of items in a set, finite or not, is $\sum\limits_{c\in C} c$.
How should I represent an infinite sum of the same thing ove... |
H: Prove that there exists and angle $\alpha$ and $r \in \Bbb R$ such that $a\cos x + b\sin x = r\cos\alpha$
Let's say that we have an expression $a\cos x + b\sin x$ where $a \in \Bbb R$ and $b \in \Bbb R$.
I was learning about finding the minimum and maximum values of an expression of this form for some given value o... |
H: Did I find the right function $f(x) = mx+n?$
I have the following task:
Let $(x_1, y_1)$ and $(x_2, y_2)$ be two points in the plane. We want to determine a
straight line given by the function $f$, i.e. $f(x) = mx + n$, such that $f(x_k) = y_k$
($k = 1,2$).
Find $m$ and $n$.
I solved it like this:
$$ m = \frac{y_... |
H: Evaluate $\int_0^1 \{\ln{\left(\frac{1}{x}\right)}\} \mathop{dx}$
$$\int_0^1 \left\{\ln{\left(\frac{1}{x}\right)}\right\} \mathop{dx}$$
Where $\{x\}$ is the fractional part of x. I was wondering if this integral converges and has a closed form but I dont know how to calculate it. I tried $u=\frac{1}{x}$ to get $$\... |
H: Solve $y'(x)=\begin{pmatrix}1 & 1 \\ 4 & 1\end{pmatrix}y(x)$
Find the general solution for $y'(x)=\begin{pmatrix}1 & 1 \\ 4 & 1\end{pmatrix}y(x)$, for $y:\mathbb{R}\to \mathbb{R}^2$.
I've tried to solve this component-wise, that is I've tried to solve $y_1'-y_1=y_2$ and $4y_1+y_2=y_2'$ by plugging the first equati... |
H: Find generating function for $F_{2n}$
Given that $F(x)=\sum_{n=0}^\infty F_nx^n= \frac{x}{1-x-x^2}$, where $F_n(x)$ is the $n^{th}$ term of the Fibonacci series, and $F(x)$ is the generating function associated to it, find the generating function associated to $F_{2n}$
I know that $F_{2n}=F^2_n+2F_nF_{n-1}$ but t... |
H: The existence of group isomorphism between Euclidean space.
Is there any group isomorphism for addition $\mathbb{R}^n$ to $\mathbb{R}^m$ where $n\neq m$? I could prove that there exists any vector space isomorphism or smooth map, but I still could not know that if we consider only abelian group structure for the ad... |
H: Change (or increment) raised to some power
After some shallow research, I've found no results of anyone asking the same question as me. Please feel free to refer me to wherever this has been discussed previously.
When expressing a change with the letter delta: $\Delta x$ for example and raise it, let's say, to the ... |
H: Sum of standard deviations
Suppose I have two series:
$$A = \{a_1,...,a_n\}$$
$$B = \{b_1,...,b_n\}$$
And I define the series C as:
$$C = \{a_1,...,a_n,b_1,...,b_n\}$$
I am wondering if the standard deviation of $C$, $\sigma_C$, could be greater than the sum of the standard deviations of $A$ and $B$, that is, if it... |
H: Given three IID random variables $X_1$, $X_2$, and $X_3$, what is the probability that $X_1X_3$?
Does the answer to this question depend on the distribution of the IID random variables? If so, what are the answers if we assume their distributions are $\bf{N}(\mu, \sigma^2)$ or $\textbf{Unif}(a, b)$?
AI: By symmetry... |
H: Integrate $\int_0^{\frac{\pi}{2}} \frac{dx}{{\left(\sqrt{\sin{x}}+\sqrt{\cos{x}}\right)}^4} $
I found a challenge problem and am confused$$\int_0^{\frac{\pi}{2}} \frac{dx}{{\left(\sqrt{\sin{x}}+\sqrt{\cos{x}}\right)}^4} $$
$u=\frac{\pi}{2}-x$ is no good and square or 4th power the denominator does not help? Sugges... |
H: Prove that $\frac{1}{a^2}+\frac{1}{(a+1)^2}+\frac{1}{(a+2)^2}+\dotsm\infty=\frac{1}{a}+\frac{1}{2a(a+1)}+\frac{2!}{3a(a+1)(a+2)}+\dotsm\infty$
Question:- Prove that $$\frac{1}{a^2}+\frac{1}{(a+1)^2}+\frac{1}{(a+2)^2}+\dotsm\infty=\frac{1}{a}+\frac{1}{2a(a+1)}+\frac{2!}{3a(a+1)(a+2)}+\dotsm\infty$$
Nothing is menti... |
H: Does continuous and strictly increasing implies convex function?
Let $f:[0,\infty)\to [0,\infty)$ be a continous and strictly monotone increasing function and $f(0)=0$. Then prove or disprove that $f$ is a convex function.
My initial guess that, $f$ is a convex function, I want to prove it.
I am unable to proceed... |
H: Are there examples of continuous, non-differentiable functions whose "rational derivative" exists?
Define the operator $\Delta_n$ according to the equation
$$\Delta_nf(x)=f\left(x+\frac1n\right)-f(x)$$
Observe that for differentiable $f:\Bbb{R}\to\Bbb{R}$
$$\frac{df}{dx}=\lim_{n\to\infty}n\Delta_nf$$
(Note: The lim... |
H: How do I find $f(1)$ and $f'(1)$ if $2x+3y=5$ is the tangent of $f(x)$ at $x=1$?
Find $f(1)$ and $f'(1)$ if $2x+3y=5$ is the tangent of $f(x)$ at $x=1$.
Is this correct:
$$2x+3y=5$$
$$3y=5-2x$$
$$y=5/3-(2/3)x$$
From here I get that $f'(1) = -2/3$. Here I am not sure how to continue for $f(1)$? If I just replace i... |
H: What is the meaning of this symbol $\ll_d$?
I apologize for the simple question but I'm reading a paper "On the Convex Hull Of The Integer Points In A Disc" and I'm confused by some notation. They say
$$\# \textrm{ vertices of }P \ll_{\;d} (vol P)^{\frac{d-1}{d+1}}$$
And I'm unfamiliar with the meaning of $\ll_{\;d... |
H: Calculating maximum area of trapezoid
With a $40\mathrm{m}$ long fence, it is desired to create a trapezoidal region with a base wall. What is the largest area that can be created? How can I calculate this question?
I've tried all $3$ sides of trapezoid $\frac{40}3$ and creating $3$ triangles. And that : For only $... |
H: If $\sum_{i=k}^n {n \choose i} p^{i}(1-p)^{n-i} \approx 0.05$, how can we find $k$?
Let $n$ be any natural number, let $k\in\{0, \dots, n\}$, and let $p \in [0, 1]$.
If $\sum_{i=k}^n {n \choose i} p^{i}(1-p)^{n-i} \approx 0.05$, how can we find $k$ (in terms of $n$ and $p$)?
AI: In some cases you can use a normal a... |
H: Eigenvalues of $p(A$)
Let $\lambda$ be an eigenvalue of a matrix $A$. I am trying to show that all the eigenvalues of $p(A)$ are $p(\lambda)$ where $p(x)$ is any polynomial.
I have been able to show that $p(\lambda)$ is an eigenvalue of $p(A)$. But how to show that these will be the only eigenvalues?
For instance, ... |
H: Integrate $\int_{-\infty}^{\infty} \frac{e^{2020x}-e^{x}}{x\left(e^{2020x}+1\right)\left(e^x+1\right)} \mathop{dx}$
A challenge problem says integrate $$\int_{-\infty}^{\infty} \frac{e^{2020x}-e^{x}}{x\left(e^{2020x}+1\right)\left(e^x+1\right)} \mathop{dx}$$
I thought $u=-x$ helps but I get $I$ so it is even. I al... |
H: Prove that negation of the continuum hypotheses implies existence of subset of R such that...
Prove that the negation of the continuum hypothesis implies that there exist $A⊂R$ such that $ℵ_0<|A|<|R|$.
The negation of the hypotheses implies existence of a set B such that $ℵ_0<|B|<|R$|, but how can I create a subset... |
H: Looking for a symmetric matrix
Do you know a method to find a particular $2 \times 2$ symmetric matrix $M$ with rational coefficients knowing that $\lambda = \sqrt 2$ is an eigenvalue of $M$?
Many thanks.
AI: Let's go for a $2 \times 2$ matrix. If $A$ has rational coefficients it's minimal polynomial $\mu$ will hav... |
H: Prove that $\int_{0}^{\fracπ2}(\log(\tan x))^{2n}dx=\left (\frac {π} {2} \right )^{2n+1} \left ( \frac {d^{2n} \sec(z)}{d z^{2n}} \right ) _{z=0}$
Question: Prove that for $n\in Z^{+}$
$$\int_{0}^{\fracπ2}(\log (\tan x))^{2n}dx=\left (\frac {π} {2} \right )^{2n+1} \left ( \frac {d^{2n} \sec(z)}{d z^{2n}} \right ) ... |
H: getting formula about finding height of a vertical line in different positions in coordinate system.
I want to find height of a vertical line in different positions. I need a formula for calculating. please help me.
AI: $$H=mx+\frac{1}{2}h$$
where m is the slope of the line. |
H: Pushforward of a Vector Field by a Diffeomorphism
A question concerning when the pushforward of a vector field is well-defined. if $ F:N \rightarrow M $ is a smooth map between manifolds, the pushforward of a tangent vector $ X_p \in T_PN $ is given by $ F_{*,p}:T_pN \rightarrow T_{F(p)}M $, where $ \big(F_{*,p}(X_... |
H: Divisibility with factorials
Find all positive integers $n$, less than 17, for which $n!+(n+1)!+(n+2)!$ is an integral multiple of 49.
I tried to factor the expression, but I am not having any luck.
AI: $\textbf{Hint:}$
$n!+(n+1)!+(n+2)!=n! \times (n+2)^2$
After that I think you can quite easily find the answers wh... |
H: Diagonalizable matrix is similar to a diagonal matrix with its eigenvalues as the diagonal entries
My book defines a diagonalizable matrix as follows:
A matrix $A$ is diagonalizable if it is similar to a diagonal matrix say $D$. So there exists an invertible matrix $P$ such that $A =PDP^{-1}$.
Now let eigen values ... |
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