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H: Convergence of Matrix Series
I would just like a quick sanity check. If I have a matrix $ M $, then the series $ 1 + M + M^2 + M^3 \cdots $ converges to $ (1-M)^{-1} $ if the operator norm $ \lVert M \rVert_{\mathrm{op}} < 1$. Is it sufficient to show that each column vector $ v $ of $ M $ has norm $ \lVert v\rVert... |
H: Discrete Math Question: arithmetic progression
A lumberjack has $4n + 110$ logs in a pile consisting of n layers. Each layer has two more logs than the layer directly above it. If the top layer has six logs, how many layers are there? Write the steps to calculate the equation for the problem and state the number of... |
H: Basis for a $R$-Module $R$, with $R$ the ring of endomorphisms
I got a doubt with this problem:
Let $M=\{f:\mathbb{N}\to \mathbb{Z}|\text{$f$ is a function}\}$, defining the sum in $M$ as $(f+g)(n)=f(n)+g(n)$, $M$ is an abelian group. Let $R=\{\phi :M\to M : \phi\ \text{is a morphism}\}$. $R$ is a ring with the po... |
H: Confusion about an example in Miles Reid Undergraduate Algebraic Geometry
He is giving examples of lines at infinity and how they correspond to asymptotes (pg. 14). So he says:
"The hyperbola $(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1)$ in $\mathbb{R}^2$ corresponds in $\mathbb{P}^2\mathbb{R}$ to $\mathrm{C}:\left(\fr... |
H: Quotient Spaces that are $T_0$, and the Quotient Space $x \sim y$ iff $\overline{\{x\}}=\overline{\{y\}}$.
Let $X$ be an arbitrary topological space; verify that by letting $xE_0y$ whenever $\overline{ \{ x \} } = \overline{ \{ y \} }$, we define an equivalence relation $E_0$ on $X$ and that $X/E_0$ is a $T_0$-spac... |
H: Another doubt about real functions on manifolds
Well, some days ago I've asked here how do we describe functions on manifolds. My idea was that it could be done using the coordinate functions of a chart: if $(x,U)$ is a chart for a manifold $M$ then we can define one function $f : U \to \Bbb R$ as a combination of ... |
H: Preparing for Mathematics Olympiad
I am preparing for Mathematics Olympiad , can any one suggest me some books to prepare for olympiad ? The topics that usually come up involve:
congruence modulo $n$,
inequalities ,
number system, elementary number theory, etc.
Please help me!
Thanks
Kushashwa
AI: I'd recommend... |
H: Multisets with Exact Number of Repeated Integers
Given a multiset that contains 5 numbers where the numbers are from 0 to 5 inclusive, and the numbers can be repeated:
a) In how many ways can you have a multiset with exactly four 4s?
b) In how many ways can you have a multiset with exactly three 3s?
c) In how many ... |
H: Probablity of finding A or B or both A and B
In a jungle, the probability of an animal being a mammal is 0.6, a nocturnal is 0.2. What is the probability that an animal found in this jungle is either a mammal, or nocturnal or both. Assume that these are independent traits.
AI: Hint: The probability that $A$, $B$, o... |
H: Mathematicians talking about their identity as a person and as a mathematician?
I was wondering if any of you know of any books, articles, interviews, youtube videos, ... (etc) where a mathematician talks about his or her identity as a person and as a mathematician? Thank you for any sources!
AI: A mathematician's ... |
H: For what $x\in[0,2\pi]$ is $\sin x < \cos 2x$
What's the set of all solutions to the inequality $\sin x < \cos 2x$ for $x \in [0, 2\pi]$? I know the answer is $[0, \frac{\pi}{6}) \cup (\frac{5\pi}{6}, \frac{3\pi}{2}) \cup (\frac{3\pi}{2}, 2\pi]$, but I'm not quite sure how to get there.
This is what I have so far: ... |
H: Johann Bernoulli did not fully understand logarithms?
This wikipedia article makes the claim:
"Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand logarithms."
This is found under "History" . What does wikipedia mean here? Surely, Bernoulli underst... |
H: Show $s(s(a))=s(b)$ implies $s(a)=b$
Let us have a first order language $L=\{0,s\}$, where $0$ is a constant, $s$ is a function symbol of arity $1$. The first-order theory $T$ is axiomatized as follows:
$\forall x \neg( s(x) = 0)$
$\forall x \exists y(x =0 \vee s(y)=x)$
How could I prove the following statement?
... |
H: Non-convergent series of convergent integral
I'm trying to find a series representation for a integral, but I think there's something I'm missing, as even though the algebraic manipulations I'm doing are valid (I think!), the series representation of the integral (which I know to converge) ends up diverging. Here'... |
H: If the points $x_1,x_2,\ldots,x_n$ are distinct,then...
I am stuck on the following problem that says:
If the points $x_1,x_2,\ldots,x_n$ are distinct,then for arbitrary real values $y_1,y_2,\ldots,y_n$, prove that the degree of the unique interpolating polynomial $p(x)$ such that $p(x_i)=y_i,\,\,(1 \le i \le n)... |
H: Determining Convergence of Power Series
Flip a fair coin until you get the first "head". Let X represent the number of flips before the first head appears. Calculate E[X].
So I solved this problem and you get a power series:
$E[X] = 1*0.5 + 2*0.5^2 + 3*0.5^3+ ...$
This is basically of the form $\sum\limits_{i=0... |
H: Matrices with trace zero.
I would like to show that every trace zero square matrix is similar to one with zero diagonal elements.
This question has been asked before, and has had an answer by Don Antonio.
And my problem is that I cannot understand the cited paper.
In the cited paper, (proof 4), one finds the senten... |
H: a matrix with determinant $1$, what can be said about the column $(a \space c)^T$?
$\begin{pmatrix}a&b\\c&d\end{pmatrix}$ be a matrix with determinant $1$, what can be said about the column $(a \space c)^T$?
from the condition we have $ad-bc=1$, so do I have to conclude something from this condition only?
Thank yo... |
H: Is $\alpha^{\beta}$ a cardinal?
Let $\alpha , \beta$ cardinals. Is $\alpha ^{\beta}$, defined as the set of all functions $f:\beta\to \alpha$, a cardinal?
I do this question because an autor of a text book says that the exponentiation of cardinals $\alpha$ raised to $\beta$ is defined as the cardinal of the set $\a... |
H: Analytic extension of functions in Hardy spaces
This is a problem I came across in a direct scattering problem. I have a function $a(s)$ that
is of the form$$
a(s)=\int_0^{\infty}e^{is\xi}A(\xi)d\xi
$$ where $A(\xi)\in L^1\cap L^2$. Then is it possible to extend this function to a bounded analytic function in $\ma... |
H: Research and application of causal inference
I have been reading Pearl's book to understand how Bayesian networks and causal discovery might work. Other than Pearl, I haven't yet found a rigorous, systematic approach to causal inference from observational data. The theorems and algorithms he presents (e.g. Inductiv... |
H: Question regarding a Jacobian
Suppose I have these two pairs of variables:
\begin{equation}
u = g_1(x,y), \qquad v = g_2(x,y),
\end{equation}
\begin{equation}
x = h_1(u,v), \qquad y = h_2(u,v).
\end{equation}
If my jacobian of $x$ and $y$, $J(x,y)$ is the determinant of the partial derivatives of the functions $g_1... |
H: Integral of $\frac{2}{x^3-x^2}$
How can I integrate $\dfrac2{x^3-x^2}$? Can someone please give me some hints? Thanks a lot!
AI: HINT:
$$\frac2{x^3-x^2}=\frac2{x^2(x-1)}=\frac{A}x+\frac{B}{x^2}+\frac{C}{x-1}$$
for what values of $A,B$, and $C$? |
H: How many distinct copies of $P_m$ are in $K_n$?
Let $K_n$ be the complete graph of order $n$ and $P_m$ a path with $m$ distinct vertices, $1 \leq m \leq n$.
Question: How many distinct copies of $P_m$ are contained in $K_n$?
Given that a permutation maps a path to a different path it seems like there will always be... |
H: How to solve a system of equations
I have this system of equations:
3x²+7y²=55
and
2x²+7xy=60
Is there a method of solving [x,y] without using x²=t, y²=z?
AI: Yes, since $x$ is nonzero (because of the second equation), we can eliminate
$y=(60-2x^2)/(7x)$ and substitute this in the first equation, which gives
$$
25(... |
H: Perfect matching in a graph
Assuming I have a bipartite graph with the following property:
for each subgroup of nodes $s \subseteq {V} $ :
$$
\sum_{v\epsilon N(S),z\epsilon N(N(S)) }{} {(v,z) \geq 2\left \| S \right \|}
$$
Where $N(S)$ is the neighbourhood of $A$.
i.e. if you go over each of $S$ neighbors and co... |
H: What does $\mathrm d^2 x$ exactly mean?
I am learning radiometry and one of the equation is radiance which is given as the radiant flux per unit projected area per unit solid angle. In equation:
$$L = {d^2\Phi \over {cos(\theta)dAd\omega}} (eq. 1)$$
Now further in the book I read they use intensity which is the ang... |
H: why $\frac{d}{dy}$ can pass through integral w.r.t. $x$?
When I calculate integration of multivariables, many books use the following step without proofing. I want to know that why it is true:
$$\frac{d}{dy}\left[\int^a_b f(x,y)dx\right]_{y=k}=\int^a_b \frac{\partial}{\partial y} \left[f(x,y)\right]_{y=k}dx$$
I als... |
H: Explicitly writing out a differential 2-form
In Tu's An Introduction to Manifolds, one question asks:
At each point $p\in \mathbb{R}^3$, define a bilinear function $\omega_p$ on $T_p(\mathbb{R}^3)$ by:
$$\omega_p(\underline{a},\underline{b})=\omega_p((a^1,a^2,a^3),(b^1,b^2,b^3))=p^3(a^1b^2-a^2b^1)$$
For tangen... |
H: Definition of the fundamental group
Why are the elements of the fundamental group of a space equivalence classes? Why isn't the group defined to be the set of all possible loops at a base point with the product operation of paths? What would go wrong if it was defined so? Or is it simply not useful?
AI: We need the... |
H: In an inner product space over $\mathbb R$, prove $ (u,w)=0 \Leftrightarrow \left \| u+w \right \|=\left \| u-w \right \| $
Let $V$ be an inner product space over field $F$ and $u,w\in V$.
Prove that if $F=\mathbb{R}$ then:
$$ (u,w)=0 \Leftrightarrow \left \| u+w \right \|=\left \| u-w \right \| $$
Is it also true ... |
H: Show mapping involving tensor product is well defined.
Let $R$ be a subring of $S$, let $N$ be a left $R$-module and let $\iota : N \to S \otimes_RN$ be the $R$-module homomorphism defined by $\iota(n) = 1 \otimes n$. Suppose that $L$ is any left $S$-module and that $\varphi : N \to L$ is an $R$-module homomorphism... |
H: Solve the equation : $2013x+\sqrt[4]{(1-x )^7}=\sqrt[4]{(1+x )^7}$.
Solve the equation : $2013x+\sqrt[4]{(1-x )^7}=\sqrt[4]{(1+x )^7}$.
Show that it has percisely one root: $x=0$.
AI: Let $f(x)=2013x+\sqrt[4]{(1-x )^7}-\sqrt[4]{(1+x )^7}$. We want to solve the equation $f(x)=0$. Observe that the domain of $f$ is $[... |
H: Need pointers on how to do this trigonometric proof
$$ \cos x = \cos y + \cos^3 y$$
$$\sin x = \sin y - \sin^3 y$$
Prove that $\sin {(x - y)} = \pm \frac{1}{3}$.
I need a little hint, not a complete answer.
AI: HINT: here $\sin x=\sin y(1-\sin^2y)=\sin y\cos^2y$
$\sin^2x+\cos^2x=1\implies \sin^2y\cos^4y+\cos^2y+\c... |
H: Why do $UU^* = I$ and $U^*U = I$ hold on different spaces for the unitary matrix $U$ of a polar decomposition?
The following is from Lang $SL_2$.
Consider the polar decomposition of a matrix A. We let $P_A = (A^*A)^{1/2}$ and set $U$ s.t. we have $$UP_Av = Av.$$
Then, it follows that $U\colon \operatorname{im} P_A ... |
H: What is the maximum value of $\frac{2x}{x + 1} + \frac{x}{x - 1}$, if $x \in \mathbb{R}$ and $x > 1$?
What is the maximum value of
$$f(x) = \frac{2x}{x + 1} + \frac{x}{x - 1},$$
if $x \in \mathbb{R}$ and $x > 1$?
A 2-D plot of of $f$ for $x \in (\infty, \infty)$ is here.
Lastly, note that WolframAlpha cannot find a... |
H: $O\in M_{3}(\mathbb{R})$ is orthogonal and $\det O=-1$ then $\lambda=-1$ is an eigenvalue of $O$
I was asked the following question on a test:
If $O\in M_{3}(\mathbb{R})$ is orthogonal
and $\det O=-1$
then $\lambda=-1$
is an eigenvalue of $O=(o_{ij})$
.
I tried building equations using $OO^{t}=I\ \Rightarrow... |
H: What is the minimum value of $\sqrt{\frac{2(x - 1)}{x}} + \frac{x + 1}{x}$, if $x > 1$?
What is the minimum value of
$$f(x) = \sqrt{\frac{2(x - 1)}{x}} + \frac{x + 1}{x},$$
if $x \in \mathbb{R}$ and $x > 1$?
Note that $f$ has a global minimum value of
$$f(1) = 2$$
if we allow $x \geq 1$. (The WolframAlpha verific... |
H: What is a harmonic complex function?
So, as far as I have learned, a complex function $f(u)$ is considered harmonic if and only if it satisfies the undermentioned equation:
$$
\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} = 0
$$
for any given complex number $x+iy$.
Is that right ?
AI: I ... |
H: Differentiability Question
I have the following true/false claim:
There exists a function $f(x,y)$ which is differentiable function at $(x_0,y_0) $ and its directional derivatives at each direction $(\cos \theta, \sin \theta )$ for $0\leq \theta <2\pi $ equal $\cos^2 \theta + 2\sin \theta $ .
I am almost sure th... |
H: A strange trigonometric identity in a proof of Niven's theorem
I can't understand the inductive step on Lemma A in this proof of Niven's theorem. It asserts, where $n$ is an integer:
$$2\cos ((n-1)t)\cos (t) = \cos (nt) + \cos ((n-2)t)$$
I tried applying the angle subtraction formula to both sides, but all that doe... |
H: Why $\omega$ can't be bijectively mapped to $\omega +1$
Let $\omega$ be the order type of the totally ordered set $\mathbb{N}$, and $\omega +1$ the set $\Bbb{N}\cup \{0\}$.
$0$ is greater than all the natural numbers as per this ordering.
My question is why can't $\Bbb{N}$ be bijectively mapped to $\Bbb{N}\cup\{... |
H: value of $\sum_{k=0}^{49}(k+1)(1.06)^{k+1}$
How do I calculate the value of $\sum_{k=0}^{49}(k+1)(1.06)^{k+1}$? I do not know the way to solve this type of a summation. Any guidance will be much appreciated
AI: For $x\in\mathbb{R}\setminus\{1\}$,
$$\begin{align*}
\sum_{k=0}^{49}(k+1)x^{k+1}&=x\sum_{k=0}^{49}(k+1)x^... |
H: I found that $\frac{dx}{dt} \cdot x = x$. What did I do wrong?
I found the following while fiddling around with the product rule.
$$
\frac{dx}{dt} \cdot x
= \frac{1}{2} \left( \frac{dx}{dt} \cdot x + x \cdot \frac{dx}{dt}\right)
= \frac{1}{2} \frac{d}{dt} x^2
= x
$$
Which is wrong iff $\frac{dx}{dt} \neq 1$. What ... |
H: How do I solve the following difference differential equation
While studying a particular physical system, I arrived at the following difference differential equation:
$$\frac{dx_n(t)}{dt} = -g \left\{\sqrt{(n + 1)(n + 2)}x_{n+1}(t) - (2n +1)x_n(t)\right\},$$
where $g$ is a constant and the initial conditions are i... |
H: Prove: $\|\lambda v\| = |\lambda| \cdot \|v\| $
Prove: $\|\lambda v\| = |\lambda| \cdot \|v\| $, for a vector space $V$ with an inner product and $\lambda \in F$,
How do we prove this?
I understand the geometric meaning is that if you multiply a vector by a scalar then you make it length greater by that scalar ti... |
H: boundary of the boundary of a set is empty
I am learning some stuff about the interior, closure and boundary of sets $A\subset\mathbb R^n$ and I am wondering about the following:
1) $\partial\partial A=\partial A$ ?
2) $\partial\partial\partial A=\partial A$ ?
3) $\partial\partial A=\emptyset$ ?
So 1) is false for... |
H: Holomorphic function zeros on the circle
I'm learning to use some methods of complex analysis, solving some problems.
Could you give me a hint to solve the following problem?
$f$ is holomorphic in $D^2=\{z: |z|<1\}$ and continious in $\partial D^2\cup D^2$. Also, there is an open subset $U$ of $\partial D^2$ such a... |
H: Can it happen that an object will not cast any shadow at all?
I am puzzled by a question in Trigonometry by Gelfand and Saul on p. 57.
Can it happen that an object will not cast any shadow at all? When and where? You may need to know something about astronomy to answer this question.
I have drawn a diagram with... |
H: Invertibility of a linear operator on a Hilbert space.
Let $H$ be an infinite dimensional Hilbert space over $\mathbb C$, $T$ be a continuous linear operator of $H$, $r(T)=\sup_{||x||=1}|(Tx|x)|$ be the numerical radius of $T$, and $z\in \mathbb C$, such that $|z|<1$.
Assume that $r(T)\leqslant 1$.
Clearly $\ker (I... |
H: Division between two numbers of the form $u + v\sqrt 2$
I need to do a division $a/b$, where $a$ and $b$ are numbers of the form $u + v\sqrt 2$, and $u$ and $v$ are integers (I'll write $a = u + v\sqrt 2$ and $b = u' + v'\sqrt 2$).
What is an effective way of computing that division? That is, how can I compute that... |
H: Minimal polynomial over an extension field divides the minimal polynomial over the base field
I need help proving this theorem:
Given the field extension: $\mathbf{K} \subseteq \mathbf{L}$, for $\alpha \in \mathbf{L}$ and $g(x) \in \mathbf{K}[x]$, $\alpha$'s minimal polynomial over $K$,
and $f(x) \in \mathbf{L}[... |
H: Partial fractions for $\frac{t+1}{2\sqrt{t}(t-1)}$
How do I use partial fractions for the expression $\dfrac{t+1}{2\sqrt{t}(t-1)}$? Because I have to find the integral of it... Thank you
AI: Your given integrand isn't a rational function (yet) to use partial fractions, we must first obtain a ratio of polynomials. W... |
H: Restriction of a lower semi-continuous functional again lower semi-continuous?
Let $F: [a,b]\times \mathbb R \times \mathbb R \rightarrow \mathbb R$ be continuous, $J(u) = \int_{[a,b]} F(x, u, u') dx$ be a functional over $W^{k,p}([a,b])$. We assume that for any uniformly convergent sequence $(u_{r})_{r\in\mathbb N... |
H: If $\alpha$ is an ordinal, proving that $\alpha\cup\{\alpha\}$ is an ordinal.
I refer to pg.4 of this article.
Assuming $\alpha$ is an ordinal, we have to prove $\alpha\cup \{\alpha\}$ or $\alpha +1$ is an ordinal.
Isn't this obvious from the construction of ordinals? As per the construction given in the article... |
H: Unique root to a function
Let $f:[a,\infty)\rightarrow \mathbb{R}, \ \ f\in C^2[a,\infty)$ such that $$ \\ f(a)>0 , \ \ f'(a)<0, \ \ f''(x)\leq 0 \ \ \forall x\in [a,\infty)$$ Prove that
$$ \exists !~t\in (a,\infty):f(t)=0$$
AI: From the mean value theorem, $f'(x)-f'(a)=(x-a)f''(\xi)$ for some $\xi\in(a,x)$, hen... |
H: What is the definition of $H^{-k}(\mathbb{R}^n)$ and its norm?
What is the definition of $H^{-k}(\mathbb{R}^n)$ and its norm? How can I understand the fact $$\|f\|_{H^{-k}(\mathbb{R}^n)}=\|(I-\triangle)^{-k}f\|_{H^{k}(\mathbb{R}^n)}.$$
AI: By definition, $\|f,H^s\| =\|(1+|\xi|^2)^{s/2}\hat f(\xi),L_2\| $, where $\h... |
H: Prove/disprove the following theories regarding operators in inner product spaces
Prove/disprove:
(I) Let $V$ be a vector space with an inner product upon field $F$. Given an operator $T:V\to V$, which is invertible. Is $T^{*}$ invertible?
(II) Let $v_1, ..., v_k$ be eigenvectors of $T$ that are correspondent to d... |
H: About irreducibility of a particular class of bihomogeneous polynomials
Is the polynomial
$$
x_0^2y_0+x_0x_1y_1+x_1^2y_2+x_1x_2y_3+x_2^2y_4+x_0x_2y_5 \in \mathbb K[x_i, y_j]
$$
reducible over an algebrically closed field $\mathbb K$?
I've noticed that the polynomial is bi-homogeneous of degree $(2,1)$ and I... |
H: How to solve the following problems with exponent?
If $9^{x+2}= 240+9^x$ then x= ?
$10^x = 64$ what is the value of $10^{(x/2)+1} = ?$
$x/x^{1.5} = 8*x^{-1}$ and x > 0 , then x = ?
$x^{-2} = 64$, then $x^{1/3} + x^0$ = ?
$4^x - 4^{x-1} = 24 $ then $(2x)^x = ?$
AI: $1: 9^x(9^2-1)=240\implies 9^x=3=9^{\frac12}\impli... |
H: What texts do you recommend to study calculus?
I've studied calculus 2 years from Arabic text . It was great text , which is supported with huge amount of examples and exercises , Now , I find it's a good step to study the material in English as my future studies will be in English , So i search for a text which co... |
H: How do I prove that $\int_{0}^{1} \frac{1}{\log(x)}dx$ diverges?
This is an exercise of a book I'm using to study... The book gives a hint: compare to $f(x)=\frac{1}{1-x}$. However I was not able to realize how could I compare these two functions. I tried changing the variable $x$ to $(1-u)$ in the integral, but it... |
H: Help understanding a property of modulus
If it is given that $|b|>1$ and $|ab|=1$
can someone please explain what should be the value of $|a|$ ?
AI: One of basic properties of $|\cdot|$ is that $$|ab|=|a||b|,$$
so we can write $$1=|ab|=|a||b|.$$ Clearly $b\ne 0$. Then we divide both parts by $|b|$:
$$|a|=\frac{1}{... |
H: What is the height of a regular polygon?
I have three small circles forming a pyramid. I would like to centre that group in a square but have spent a couple of hours trying to calculate the height of the pyramid. I just can't seem to get them vertically centred.
Given a square, a large circle filling the square an... |
H: Is $\lim_{t\to 0}\frac{e^{xt}-1}{x}=0$ uniformly on $x>0$?
I would like to determine whether the following limit is uniform on $x\in (0,\infty)$: $$\lim_{t\to 0}\frac{e^{xt}-1}{x}.$$
By "uniform" here, we mean $\exists \delta_\epsilon>0$ such that $\frac{e^{xt}-1}{x}<\epsilon$ for all $0<t<\delta_\epsilon$, for all... |
H: Outer measure discontinuous from below
I was trying to find an example of an outer Measure which is not continuous from below.
These are the definitions I use
An outer measure on $X$ is a function $\mu^\ast: \mathcal{P}(X)\to [0,\infty]$ if
it fulfills
$\mu^\ast(\emptyset)=0$
$\mu^\ast\Big( \bigcup_{j=1}^\inf... |
H: Showing that $U(2^n)$ is not cyclic for $n \ge 3$
I'd appreciate a hint on how to solve the following problem:
Prove that $U(2^n) (n \ge 3)$ is not cyclic. ($U(m)$ is the group of positive integers $j \le m$ such that $\gcd(j,m)=1$, under multiplication $\mbox{mod}\,\,m$)
Since elements in $U(2^n)$ are coprime to $... |
H: Prime ideals and epimorphism
Let $\phi$:$R$$\rightarrow$$S$ be a ring epimorphism.
Show that if $P\triangleleft S$ is a prime ideal (of S), then $\phi^{-1}(P)\triangleleft R$ is a prime ideal (of R).
Can someone help me with that?
Thanks in advance
AI: Expanding on @Daniel Fischer's comment:
$P\subseteq S$ is prime... |
H: Quadratic ternary forms
What is the difference between solubility, local solubility and global solubility when it comes to solving quadratic ternary normal forms, i.e a equation of the form $ax^2 + by^2 + cz^2 =0$?
Thanks in advance.
AI: Global solvability means solvable over a global field (such as a number fiel... |
H: Geometric progression — Sum of terms, sum of terms' cubes, sum of terms' squares
Consider the infinite geometric progression $a_1, a_2, a_3, \dots$. Given the sum of its terms, as well as the sum of the terms' cubes
$a_1 + a_2 + a_3 + \cdots = 3\\
a_1^3 + a_2^3 + a_3^3 + \cdots = \frac{108}{13}$
find the sum of the... |
H: Proof by induction that the sum of terms is integer
I'm having some trouble in order to solve this induction proof.
Proof that $\forall{n} \in \mathbb{N}$ the number $\frac{1}{5}n^5+\frac{1}{3}n^3 + \frac{7}{15}n$ is an integer.
I've tried proving this by induction, but I've not succeed so far. What I did was:
... |
H: Programming language to learn mathematics
I am a computer scientist who programs since 3 years. I am currently in my 4th semester and I struggle with some math classes, not because they are extremely difficult but they are taught extremely boring and I get no feedback at all.
Because I am totally in love with progr... |
H: Intuition Behind an Identity
I'm currently studying for a complex analysis prelim. exam in August, so I'm working through some of the exercises in Titchmarsh. One of the exercises has us evaluate the integrals $$\int_0^\infty\frac{1}{1+x^4}\,dx\quad\text{and}\quad\int_0^\infty\frac{x^2}{1+x^4}\,dx.$$After evaluatin... |
H: The value of $w$ also has a max error of $p\%$
Suppose $\frac{1}{w}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ where each variable $x,y,z$ can be measured with a max error of $p\%$
Prove that the calculated value of $w$ also has a max error of $p\%$
I guess I need to take its derivative i.e $-\frac{1}{w^2}dw=-\frac{1}{... |
H: Simple limit problem: $\lim_{x\to2}(\frac{1}{x-2}-\frac{4}{x^2-4})$
While trying to help my sister with her homework she gave me the next limit: $$\lim_{x\to2}(\frac{1}{x-2}-\frac{4}{x^2-4})$$
I know the conventional way of solving it would be (That's what i showed her):
$$\lim_{x\to2}(\frac{1}{x-2}-\frac{4}{x^2-4}... |
H: Metric Space (Elementary Analysis)
Let $d: X \times X \to \Bbb R$ is a function satisfying all properties of a metric space but $d(x,y)=0 \implies x = y$.
If we define $\sim$ on $X$ by $x\sim y \iff d(x,y) = 0$,
prove that $D([x], [y]) = d(x,y)$ where $[x] = \{z \in X \mid z\sim x\}$ is well-defined on equivalence... |
H: Weighted uniform convergence of Taylor series of exponential function
Is the limit
$$
e^{-x}\sum_{n=0}^N \frac{(-1)^n}{n!}x^n\to e^{-2x} \quad \text{as } \ N\to\infty \tag1
$$
uniform on $[0,+\infty)$?
Numerically this appears to be true: see the difference of two sides in (1) for $N=10$ and $N=100$ plott... |
H: Determinant of a Matrix Proof: $\;\det(qA) = q^n(\det A)$
I am required to show that:
$\det(qA) = q^n(\det A)$, where $A$ is a real $n\times n$ Matrix, and $q$ is a constant
I believe that this claim is true after doing few examples. However, but I do not know how to start the proof.
AI: Andrea Mori's method is th... |
H: Dynamics of a linear map
Let $F : \mathbb{R^2} → \mathbb{R^2}$ be any map and, given a point
$(x_0, y_0)$ in $\mathbb{R^2}$ define $x_n$ and $y_n$ by $(x_{n+1}, y_{n+1}) = F(x_n, y_n)$. We study the dynamics of the map $F$ by
studying the limiting behaviour of the sequence $(x_n, y_n)$ as $n→∞$
for different... |
H: Numerical computation of continuous Fourier transform
Are there any algorithms that numerically compute the continuos Fourier transform of a given function f?
I find plenty of implementations of the discrete Fourier transform, using FFT,
but, if I´m not mistaken, DFT is not a discrete approximation of the continuou... |
H: Prove that $x^2<\sin x \tan x$ as $x \to 0$
$$x^2<\sin x \tan x \quad as \; x \to 0$$
I made the substitution $x \to \arctan x$ .
$\arctan^2 x<x\sin (\arctan x)$
$\arctan x < \large \frac{x}{(x^2+1)^{\frac 14}}$
There are two functions $f(x)$ and $g(x)$ . $f(0)=g(0)$ . If $f'(x)>g'(x)$ on the interval $(0, a)$ , t... |
H: What does $f \in H^\infty$ mean?
I am reading this research paper about polynomials with non-negative coefficients. Can some one tell what does the notation $f \in H^\infty$ mean so that I can research about this function class?
AI: $H^\infty$ is the class of bounded holomorphic functions on the open unit disc (or ... |
H: Factorizations of $x^2+x$ in $\mathbb Z_6[x]$
So I was looking through my old algebra book and found a question that I can't seem to answer.
Find two Factorizations of $x^2+x$ as the product of nonconstant polynomials that are not associates of $x$ or $x+1$.
I found $(x+3)(x+4)$, can anyone find the other one?
I w... |
H: singular (co)homology over various fields of same characteristic
Is the following true: if $K$ and $F$ are fields with the same characteristic and $X$ is a topological space, then for any $n$ there holds $$\dim_K H_n(X;K) = \dim_F H_n(X;F)\text{ and }\dim_K H^n(X;K) = \dim_FH^n(X;F),$$ where $H_n(-;-)$ and $H^n(-;-... |
H: Basic statistics - Calculate distribution of winning
I have a 100 sided fair dice with each side labelled 1 thru 100. I win if the number rolled is 49 or higher (1% advantage).
1. What is the probability of me winning exactly 500 rolls if the dice is rolled 1000 times?
What is the general formula for calculating... |
H: Proving irreducibility of $x^6-72$
I have the following question:
Is there an easy way to prove that $x^6-72$ is irreducible over $\mathbb{Q}\ $?
I am trying to avoid reducing mod p and then having to calculate with some things like $(x^3+ax^2+bx+c)\cdot (x^3+dx^2+ex+f)$ and so on...
Thank you very much.
AI: I ha... |
H: Find All Points on a Paraboloid where Tangent Plane is Parallel to a Given Plane
Find all points on the paraboloid $z=x^2+y^2$ where tangent plane is parallel to the plane $x+y+z=1$ and find equations of the corresponding tangent planes. Sketch the graph of these functions.
I have its answer. I don't really unde... |
H: Proving and understanding the Fixed point lemma (Diagonal Lemma) in Logic - used in proof of Godel's incompleteness theorem
http://en.wikipedia.org/wiki/Diagonal_lemma
I am wondering about the proof of the "Fixed-Point Lemma"
$\text{Mod } \Sigma$ is the class of all models of $ \Sigma$. $\text{Th Mod } \Sigma$ is t... |
H: Recursively Defined Entities
So I am having some trouble understanding how one is to come up with the recursive definition to the following problem...
We are given a rectangle of width $2$ and length $n$. Suppose we have dominoes of size $2\times 1$. What is the number of different ways we can cover the $2\times n... |
H: Find all invertible elements of $ \Bbb{Q}[x]/(x^{600}) $.
I know that invertible elements of $\Bbb{Q}[x]$ are constants, so $\Bbb{Q}$. But in $\Bbb{Q}[x]/(x^{600})$, I suppose there are more invertible elements. How to find all of them?
AI: HINT:
Show first that if $R$ is a commutative ring with identity with a uni... |
H: Prove that $\arctan\left(\frac{2x}{1-x^2}\right)=2\arctan{x}$ for all $|x|<1$, directly from the integral definition of $\arctan$
I would like to show that for $A(x) = \int_{0}^{x}\frac{1}{1+t^2}dt$, we have $A\left(\frac{2x}{1-x^2}\right)=2A(x)$, for all $|x|<1$.
My idea is to start with either $2\int_0^x\frac{1}{... |
H: Find the coefficient of $x^{20}$ in $(x^{1}+⋯+x^{6} )^{10}$
I'm trying to find the coefficient of $x^{20}$ in
$$(x^{1}+⋯+x^{6} )^{10}$$
So I did this :
$$\frac {1-x^{m+1}} {1-x} = 1+x+x^2+⋯+x^{m}$$
$$(x^1+⋯+x^6 )=x(1+x+⋯+x^5 ) = \frac {x(1-x^6 )} {1-x} = \frac {x-x^7} {1-x}$$
$$(x^1+⋯+x^6 )^{10} =\left(\dfrac {x-... |
H: A definition of metric space
Can you please help me solve the question below?
I have no idea how to prove this one.
Define the set
$$X:=\{K\subset\mathbb C:K\text{ is bounded and closed}\}$$
Define a function $d\colon X \times X \to \mathbb{R}$ via
$$ d(K_1,K_2)=\inf\{\delta>0:K_1\subset N_\delta(K_2)\text{ and }K_... |
H: Rotor Identity $ \frac{1+ba}{|a+b|} = e^{-B\theta /2} $
To prove:the identity given above where $ a, b $ are vectors, $ B $ is the unit bivector in the $ a\wedge b $ plane and $\theta $ is the angle between $ a$ and $ b$. (From "Geometric Algebra for Physicists" by Doran and Lasenby).
Expanding the L.H.S i get $$ ... |
H: Eulerian graph in two color
How can we prove the Eulerian Map can be color in 2 colors. I know the Eulerian graph can be colored at most 4, which is Four color problem. But I have no idea how to prove into 2 colors. Anyone can help me do this? Thanks!
The Eulerian map at here is mean the Eulerian planar graph (so a... |
H: Finding the multiples of a number that satisfy the question.
Two numbers multiply to equal 200. Find the numbers such that the difference between the square root of one number and the reciprocal of the other is minimized.
Having a tough time working around this problem, I'm having some trouble.
AI: If one number is... |
H: If $X$ and $Y$ are independent then $f(X)$ and $g(Y)$ are also independent.
Knowing that if you have two independent $X$ and $Y$, and $ f $ and $ g $ measurable functions, how to show that then $ U = f (X) $ and $ V = g (Y) $ are still independent.
AI: You said measurable so I am going to assume you want a measure-... |
H: Find $m, n$ such that $\frac{n^2 + 1}{m^2 + 1 }$ is an integer multiple of a perfect square
I'm trying to find $n,m \in \mathbb{N}$ such that $\sqrt{ \frac{n^2+1}{2(m^2+1)}}$ is rational.
I see that if $a,b$ are relatively prime $\sqrt{ \frac{a}{b}}$ is rational if and only if $a,b$ are perfect squares. $n^2+1$ ca... |
H: (Revisited$_2$) Injectivity Relies on The Existence of an Onto Function Mapping Back to Its Preimage
QUEST:
For any sets $X$ and $Y$, there exists an injective function $f:X\rightarrow Y$ if and only if there exists a surjective function $g:Y\rightarrow X$.
QUESTION$_1$:
How do you people approach this problem.... |
H: Solve the equation about matrix
The equation is $x^2 = x$, which $x$ is a $2\times2$ matrix. Anyone can give me some hint? Thanks!
AI: Two different hints:
1) This is a question with 2-by-2 matrices... why not just write down good ol' $x=\begin{pmatrix}a & b\\c & d\end{pmatrix}$, compute $x^2-x$, and see what happe... |
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