Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Prove: $null ( A - \lambda I)^{a_{\lambda}} = a_{\lambda}$ The problem here is straight forward. Let $a_{\lambda}$ be the algebraic multiplicity corresponding to $\lambda$. Prove $null (A - \lambda I)^{a_{\lambda}} = a_{\lambda}$
I know the following bits: $a_\lambda$ is the highest power of $(x-\lambda)$ that divides... | First, each matrix has a unique structure of Jordan normal form. It consists of Jordan cells of the form $$J_\lambda=\begin{pmatrix}\lambda&1&0&\dots&\dots\\0&\lambda&1&0&\dots\\ \vdots&0&\ddots&\ddots&\ddots\\0&\dots&\dots&\lambda&1\\0&\dots&\dots&0&\lambda\end{pmatrix}$$
The size of this cell is called its order - de... | {
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MENSA IQ Test and rules of maths In a Mensa calendar, A daily challenge - your daily brain workout. I got this and put a challenge up at work.
The Challenge starts with..
Assume you are using a basic calculator and press the numbers in the order shown,
replacing each question mark ...continues... What is the highest nu... | It seems like a fair assumption that the rules of math should be applied to a mathematical expression.
Btw, it is not stated (but I guess it's implied) that the operators must be put in place of the question marks, or else
-5/4+7*32 = 222.75
| {
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"question_score": "17",
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"answer_id": 10
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Linear Algebra - Can vector $v$ be expressed as a linear combination of $u_1$ and $u_2$ I have a question: Can the vector $v = (1,2)$ be expressed as a linear combination of $u_1 = (1,3)$ and $u_2 = (4,1)$?
What I have tried:
$a + 4b = 1$
$3a + b = 2$
$a = 1 - 4b$
$3(1 - 4b) +b = 2$
$3 - 12b + b = 2$
$3 -11b=2$
$3 -2 =... | Your answer is indeed correct. You can (and should) always check your solution directly:
$$\frac{7}{11}(1,3) + \frac{1}{11}(4,1) = (\frac{7}{11},\frac{21}{11})+(\frac{4}{11},\frac{1}{11}) = (\frac{11}{11},\frac{22}{11}) = (1,2)$$
as desired.
It's generally true that as long as $u_2$ isn't a scalar multiple of $u_1$, yo... | {
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Proof for the formula of sum of arcsine functions $ \arcsin x + \arcsin y $ It is known that
\begin{align}
\arcsin x + \arcsin y =\begin{cases}
\arcsin( x\sqrt{1-y^2} + y\sqrt{1-x^2}) \\\quad\text{if } x^2+y^2 \le 1 &\text{or} &(x^2+y^2 > 1 &\text{and} &xy< 0);\\
\pi - \arcsin( x\sqrt{1-y^2} + y\sqrt{1-x^2}) \\\quad\t... | Using this, $\displaystyle-\frac\pi2\leq \arcsin z\le\frac\pi2 $ for $-1\le z\le1$
So, $\displaystyle-\pi\le\arcsin x+\arcsin y\le\pi$
Again, $\displaystyle\arcsin x+\arcsin y=
\begin{cases}
\\-\pi- \arcsin(x\sqrt{1-y^2}+y\sqrt{1-x^2})& \mbox{if } -\pi\le\arcsin x+\arcsin y<-\frac\pi2\\
\arcsin(x\sqrt{1-y^2}+y\sqrt{1... | {
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An inequality for completely positive maps. Let $f\colon A\to B$ be a contractive completely positive, ${}^*$-preserving map between C*-algebras and take $a\in A$. How one can prove that
$$0\leqslant f(a)f(a^*)\leqslant f(aa^*)?$$
Some authors take it for granted without any explanation.
| One of the most important results about completely positive maps is Stinespring's Dilation Theorem.
Suppose that $f:A \to B$ is a completely positive map, where $A$ and $B$ are $C^*$-algebras. Then we can find a Hilbert space $H$ such that $B \subseteq B(H)$.
Stinespring's Theorem then states that there exists a Hilber... | {
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Proof that a matrix is nonsingular Let $A$ be $n \times n$ matrix. Show that if $A^2 = 0$, then $I - A$ is non-singular and $(I-A)^{-1} = I+A$.
The second part is easy for me, but how can I show that if $A^2 = 0$, then $I - A$ is non-singular. I found in Wolfram Alpha that "A matrix is singular iff its determinant is 0... | Comparing $|I-A|$ to the characteristic polynomial of $A$:
$|I-A| = 0 \implies \lambda=1$ is an eigenvalue of $A$. But $A$ nilpotent necessitates that the only eigenvalue of $A$ is $0$, contradiction.
| {
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i^i^i^i^... Is there a pattern? I was messing around with $i$ and I (haha) noticed that certain progressions arise when I keep on raising $i$ to $i$ to $i$ and so forth. Though, I am not really quite sure what is going on (and I don't have time to explore further).
In other words, is there an interesting pattern in the... | Actually the limit exists.
Define $a_0=i$, $a_{n+1}=i^{a_n}$, $\lim_{n\to\infty}a_n=\frac{W(-\ln(i))}{-\ln(i)}\approx0.4383+0.3606i$, where $W(z)$ is the Lambert W function, $\ln(z)$ is the principle branch of $\log(z)$.
More generally, for each $z\in\mathbb{C}$, we can define such sequence $a_n(z)$, the limit exists o... | {
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Prove: The product of any three consecutive integers is divisible by $6$. I'm new to number theory and was wondering if someone could help me with this proof.
Prove: The product of any three consecutive integers is divisible by $6$.
So far I have $\cfrac{x(x+1)(x+2)}{6}$; How would I go about proving this? Should I r... | Of $n$, $n +1$, $n +2$, one must be even, so divisible by 2 (why?). One must be divisible by 3 (why?). So their product must be divisible by $2 \times 3$ (why?) ...
| {
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Can an open ball have just one point. As per my understanding it cannot. Please clarify. I am new to functional analysis and am just learning. To my understanding an open ball must have at least 2 points else its definition will not be satisfied.
Now if I have just an empty set and this open ball, why cannot it constit... | You say that you are studying functional analysis, so perhaps you are mainly interested in Banach spaces. But even in that context, you are only almost correct - any open ball of a non-trivial Banach space is infinite. The trivial Banach space $V=\{0\}$ consists of just its zero vector $0$, and thus for any $r>0$,
$$B_... | {
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A problem about Lebesgue measurable set I'm doing exercise in "REAL ANALYSIS" of Folland and got stuck on this problem. I got no clue on how to find the set $I$. Hope someone can help me solve this. Thanks so much
Suppose $m$ is Lebesgue measure and $L$ is its domain. If $E \in L$ and $m(E) \gt 0$, for any $\alpha < ... | Let $E$ be a Lebesgue measurable set with $m(E)>0$, and let $f = \mathbf{1}_{E}$ be the indicator function of this set. For some $x\in E$, we have by the Lebesgue differentiation theorem that
$$\lim_{{I\ni x}\atop{\left|I\right|\rightarrow 0}}\dfrac{m(E\cap I)}{m(I)}=\lim_{{I\ni x}\atop{\left|I\right|\rightarrow 0}}\df... | {
"language": "en",
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Find all functions $f$ satisfying a certain property How can we find all real-valued functions $f$ such that $f^{(n+1)}(x) = f^{(n)}(x)$? The question gives a hint which says: "If $n=2$, then we have $f(x) = ae^x+be^{-x}$, for some $a,b \in \mathbb{R}$. After proving this proceed by induction on $n$." I can see how to ... | Take $f(x) = ae^x + be^{-x}$. If $n$ is odd, then $f^{(n)}(x) = ae^x - be^{-x}$. If $n$ is even, then $f^{(n)}(x) = ae^x + be^{-x}$, which shows that $b = 0$ (since the derivative of $e^{-x}$ depends on its order). Therefore, $f(x) = ae^{x}$, where $a$ is any value.
I don't think there are any sophisticated method t... | {
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How many distinct factors can be made from the number $2^5*3^4*5^3*7^2*11^1$? How many distinct factors can be made from the number $2^5*3^4*5^3*7^2*11^1$?
Hmmm... So I didn't know what to do here so I tested some cases for a rule.
If a number had the factors $3^2$ and $2^1$, you can make $5$ distinct factors: $2^1$, ... | If $\begin{equation}x = a^p \cdot b^q\cdot c^r+...\end{equation}$
then there are $(p+1)(q+1)(r+1)...$ numbers that divde $x$.
Any number that divides $x$ will be of the form $a^\alpha\cdot b^\beta\cdot c^\gamma$ .
So we have p p+1 options for $\alpha$ because we need to consider $\alpha = 0$ also. Similarly, we have $q... | {
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Finding a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$. I have to find a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$.
I determined that $\sqrt{2}+\sqrt{3}$ satisfies the equation $(x^2-5)^2-24$ in $\Bbb{Q}$.
Hence, the basis should be $1,(\sqrt{2}+\sqrt{3}),(\sqrt{2}+\sqrt{3})^2$ and $(\sqrt{2}+\sq... | A basis of $\mathbb Q\big[\sqrt{2},\sqrt{3}\big]$ consists of the elements $\{1,\sqrt{2},\sqrt{3},\sqrt{6}\}$, and hence its dimension over $\mathbb Q$ is equal to $4$.
Clearly, all the above elements belong to $\mathbb Q\big[\sqrt{2},\sqrt{3}\big]$, and hence it remains to show that they are independent over $\mathbb ... | {
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Linear Algebra - Show that $V$ is not a vector space Let $V = \{(x,y,z) | x, y, z \in R\}$. Define addition and scalar multiplication on $V$ as follows:
$$(x_1, y_1, z_1) + (x_2, y_2, z_2) = (x_1,y_1+y_2,z_1+z_2)$$
$$c(x_1,y_1) = (2cx_1,cy_1)$$
where $c$ is any real number.
Show that $V$, with respect to these operatio... | Hint: One of the axioms is that $av + bv = (a+b)v$ for scalars $a,b$ and vector $v$. See how that works with your scalar multiplication.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Pole-zero cancellation Paradox Suppose we have an open-loop transfer function $$G(s) = \frac{1}{s(s+a)(s+b)}$$
If we plot the root locus for the closed-loop system we will get roughly something like this :
Now the question is when I add a new zero to the system which is at $-a$ then the book says that we should plot t... | The 2nd plot is indeed fundamentally different from the 1st, because the systems are different - the zero changes its character. The system with relative degree (number of poles minus number of zeros) 2 is more stable than the one with relative degree 1. If you think about the Bode diagram and the Nyquist plot, it has ... | {
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Finding a point on a plane closest to another point I have the point $(1,1,1)$ and the plane $2x+2y+z = 0$. I want to find a point that is closest to my point on the plane.
In other words, I want to find a point along the line $(1,1,1)+t(2,2,1)$ but on my plane. Notice that the vector $(2,2,1)$ is my normal vector and ... | Hint: A general point on the line has the form $(x,y,z) = (1 + 2t, 1 + 2t, 1 + t)$, and for this to be on the plane, it must satisfy the equation of the plane. Plug those in and solve for $t$.
| {
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Combinatorial proof of sum of numbers Does anyone have any insight on showing that $\sum_{i=1}^n i = {n+1\choose 2}$, through a combinatorial argument (i.e., not an algebraic argument)?
| There is a way to see this inductively. Suppose you knew this formula up till $n-1$. Then, $n + 1$ choose $2$ is equal to $n$ choose $2$ plus the number of ways to choose a pair of distinct elements from $\{1,\cdots, n + 1\}$ such that one of them is the 'new' element $n+1$. This is equal to $n$ choose $1$, i.e. $n$, a... | {
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evaluation of $\int\frac{1}{\sin^3 x-\cos^3 x}dx$ Evaluation of $\displaystyle \int\frac{1}{\sin^3 x-\cos^3 x}dx$
$\bf{My\; Try::}$ Given $\displaystyle \int\frac{1}{\sin^3 x-\cos^3 x}dx = \int\frac{1}{(\sin x-\cos x)\cdot (\sin^2 x-\sin x\cos x+\cos^2 x)}dx$
$\displaystyle = 2\int\frac{(\sin x-\cos x)}{(\sin x-\cos x)... | You're missing a minus sign at one point, but other than that I think you're OK. Next, use partial fractions:
$$
\frac{1}{(2-t^2)(3-t^2)} = \frac{A}{\sqrt{2}-t} + \frac{B}{\sqrt{2}+t} + \frac{C}{\sqrt{3}-t} + \frac{D}{\sqrt{3}+t}
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Determine if $\beta = \{b_1, b_2, b_3\}$ is linearly independent. Let $\beta = \{b_1, b_2, b_3\}$.
Suppose that all you know is that:
*
*$b_2$ is not a multiple of $b_3$.
*$b_1$ is not a linear combination of $b_2, b_3$.
Can you determine if $\beta$ is linearly independent?
From the second assumption, I think... | Three vectors can be linearly independent without each being strictly a multiple of one other. You are correct that the vectors are linearly independent.
Assuming $b_1, b_2, b_3$ are non-zero:
We start with the set $\{b_2, b_3\}$ where we know that $b_2$ and $b_3$ must be linearly independent, since one is not a multip... | {
"language": "en",
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Optimization of cake pan volume from area of pan It was difficult to accurately word this question, so hopefully a bit of context will clear that up.
Context:
I have a cake dish that is made by cutting out squares from the corners of a 25cm by 40 cm rectangle of tin.
40cm
_... | Hint: Suppose we remove $x\times x$ squares. The "finished" pan will have length $40-2x$, width $25-2x$, and depth $x$. So its volume $V(x)$ is given by
$$V(x)=(40-2x)(25-2x)(x).$$
You want to choose $x$ that maximizes $V(x)$. Note that we will need $0\lt x$ and $2x\lt 25$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/674168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Trying to show derivative of $y=x^\frac{1}{2}$ using limit theorem I am trying to understand why the derivative of $f(x)=x^\frac{1}{2}$ is $\frac{1}{2\sqrt{x}}$ using the limit theorem. I know $f'(x) = \frac{1}{2\sqrt{x}}$, but what I want to understand is how to manipulate the following limit so that it gives this res... | Hint: Simplify
$$
\lim_{h\to0}\frac{\sqrt{x+h}-\sqrt{x}}{h} = \lim_{h\to0}\frac{\sqrt{x+h}-\sqrt{x}}{h}\cdot\frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/674259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How does $\tbinom{4n}{2n}$ relate to $\tbinom{2n}{n}$? I got this question in my mind when I was working on a solution to factorial recurrence and came up with this recurrence relation: $$(2n)!=\binom{2n}{n}(n!)^2$$ which made me wonder: is there also a recurrence relation for $\tbinom{4n}{2n}$ in terms of $\tbinom{2n}... | Here is an estimate that gives a good approximation of $\binom{4n}{2n}$ in terms of $\binom{2n}{n}$.
Using the identity
$$
(2n-1)!!=\frac{(2n)!}{2^nn!}\tag{1}
$$
it is straightforward to show that
$$
\frac{\binom{4n}{2n}}{\binom{2n}{n}}=\frac{(4n-1)!!}{(2n-1)!!^2}\tag{2}
$$
Notice that
$$
\begin{align}
\frac{(2n-1)!!}{... | {
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is there a formula for modulo I have been trying to find a formula for modulo for a long time now. I was wondering, is this even possible? I know there are lot's of solutions for this problem in computer science but is there a solution for this problem in arithmetics? I mean is there a function that uses only arithmeti... | For positive integers $x$ and $n$, a solution for $x$ modulo $n$ is
$$\bmod \left( {x,n} \right) = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n - 1} \mathop \sum \limits_{k = 0}^{n - 1} i\exp \left( {j\left( {x - i} \right)\frac{{2\pi k}}{n}} \right)$$
where ${j^2} = - 1$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Equivalence relation and subgroup I am taking abstract algebra now, and there's a lemma: Let $H$ be a subgroup of group $G$, for $a,b \in G$,define $a\sim b$ if $ab^{-1}\in H$, then it is an equivalence. I know how to prove it and how to use it in the prove of Lagrange theorem, but can anyone give me a more mathematica... | If $H$ is what is called a normal subgroup then the intuition is the following: There exists a homomorphism (that is, a map which preserves the group structure) $\phi: G\rightarrow K$ such that $a\sim b$ if and only if $\phi(a)=\phi(b)$. The subgroup $H$ is precisely the elements of $G$ which map to the identity of $K$... | {
"language": "en",
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How can you find the cubed roots of $i$? I am trying to figure out what the three possibilities of $z$ are such that
$$
z^3=i
$$
but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you solve this geometrically? Any help would be greatly appreciated.
| The answer of @Petaro is best, because it suggests how to deal with such questions generally, but here’s another approach to the specific question of what the cube roots of $i$ are.
You know that $(-i)^3=i$, and maybe you know that $\omega=(-1+i\sqrt3)/2$ is a cube root of $1$. So the cube roots of $i$ are the numbers ... | {
"language": "en",
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Exact sequence induces exact sequences for free parts and torsion parts? Let $A$ be a PID and consider the exact sequence of finitely generately modules over$A$:
$$0\longrightarrow M' \overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}M''\longrightarrow 0 \tag{1}.$$
Denote the free part and torsion part by $F(M)$... | The sequence
$$
0\rightarrow \mathbb Z\xrightarrow{n} \mathbb Z\rightarrow \mathbb Z_n\rightarrow 0
$$
is exact in $\mathbb Z\text{-}\mathsf{Mod}$. Passng to torsion we have
$$
0\rightarrow 0\rightarrow 0\rightarrow \mathbb Z_n\rightarrow 0
$$
which is not exact. Passing to free parts we have
$$
0\rightarrow\mathbb Z\x... | {
"language": "en",
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Strong Induction: Finding the Inductive Hypothesis Consider this claim:
Every positive integer greater than 29 can be written as a sum of a non-negative multiple of 8 and a non-negative multiple of 5.
Assume you are in the inductive step and trying to prove P(n+1) using strong induction. What would be the inductive hyp... | 3) is the right one. In general strong induction means in fact you do not have $P(n)$ as hypothese. But 'more strongly' that $\forall k\leq n\; P\left(k\right)$ is your hypothese. Notice that
$P\left(n\right)$ is a consequence of this hypothese. Here $P(n)$ is the statement: $$n>29\Rightarrow\exists i\geq0\exists j\geq... | {
"language": "en",
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Is the prove correct for: If both ab and a + b are even then both a and b are even Show:
If both $ab$ and $a + b$ are even, then both $a$ and $b$ are even
Proof:
Assume both $ab$ and $a + b$ are even but both $a$ and $b$ are not even
Case1: one is odd
$a=2m+1$, $b=2n$
Hence $a+b = (2m+1) + 2n = 2(m+n) + 1$
Case2: both ... | You might also argue that as $a + b$ is even, both are odd or both are even. But if $a$ and $b$ are odd, then $a b$ is odd, contradicting the premises. So both are even.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/674973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How to calculate radius of convergence of the following series? How can I calculate radius of convergence of the following series?
$$\Large
\sum\limits_{n=0}^\infty \frac{5^{n+1}}{\sqrt[n]{(2n)!}}z^{n}
$$
I tried using D'alembert convergence test but cannot figure out how to calculate.
I know the answer is $\LARGE\frac... | This is what I have got.
$$
1\le((2n)!)^{\frac{1}{n^2}}\le((2n)^{2n})^{\frac{1}{n^2}} = (2n)^{\frac{2}{n}}＝2^{\frac{2}{n}}n^{\frac{2}{n}}\xrightarrow{\scriptscriptstyle n\to\infty}1
$$
Therefore
$$
\sqrt[n]{\frac{5^{n+1}}{\sqrt[n]{(2n)!}}}
=\frac{5^{\frac{n+1}{n}}}{((2n)!)^{\frac{1}{n^2}}}\xrightarrow{\scriptscriptsty... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/675041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
need help in complex numbers argument Any help appreciated please,
use an argand diagram to find, in the form a+bi, the complex numbers which satisfy the following pairs of equations.
arg(z+2)=1/2π, argz=2/3π
Thanks
| $z= -2 +2\sqrt{3} i$ -- to see why, first note that $arg(z+2)=\frac{1}{2}\pi$ means $z+2$ is on the positive imaginary axis, so $z$ is (in rectangular terms) 2 units left of that, so $a=-2$ (where $z=a+bi$). Now thinking of $z$ in polar form, the given condition $arg(z)=\frac{2}{3}\pi$ means $z=r e^{\frac{2}{3}\pi i}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/675162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Coin based subtraction game I'm having a problem in Game Theory where I am trying to understand how a subtraction game can be interpreted by a coin based game.
From my book:
The problem I'm having is if I have 9 coins and the subtraction set $ \{\ 1,2,3 \}\ $, say, and 3 of them are heads, let's say positions 5, 6 and... | If you have three heads coins $5$, $6$, $7$, this is equivalent to three piles of size $5$, $6$ and $7$, not a single pile of $18$ !
If you remove $2$ from the $7$-pile, you obtain three piles of size $5$, $5$ and $6$. But any impartial combinatorial game combined with itself is irrelevant (the second player can always... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/675230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Is the cube root of a prime number rational? The question is: if $P$ is prime, is $P^{1/3}$ rational?
I have been able to prove that if $P$ is prime then the square root of $P$ isn't rational (by contradiction) how would I go about the cube root?
| Suppose $\sqrt[3]{P} = \dfrac{a}{b}$ where $a$ and $b$ have no common factors (i.e. the fraction is in reduced form). Then you have
$$
b^3 P = a^3.
$$
Both sides must be divisible by $a$ (if they're both equal to $a^3$). We already know that $a$ does not divide $b$ (when we assumed the fraction is reduced). So then $a$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/675327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Expected Value of 10000 coin flips We toss a fair coin 10000 times and record the sequence of the results. Then we count the number of times that a sequence of 5 heads in a row followed immediately by 5 tails in a row has occurred among these results. (Of course, this number is a random variable.) What is the expected ... | $$N=10000,\ n=10\implies\frac{N-n+1}{2^n}=\frac{9991}{1024}\equiv9.7568359375$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/675459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the value of $\lim_{x \to - \infty} \left( \sqrt{x^2 + 2x} - \sqrt{x^2 - 2x} \right)$ I am stuck on this. I would like the algebraic explanation or trick(s) that shows that the equation below has limit of $-2$ (per the book). The wmaxima code of the equation below.
$$
\lim_{x \to - \infty} \left( \sqrt{x^2 + 2x} ... | For $x>0$: For brevity let $A=\sqrt {x^2+2 x}.$ We have $(x+1)^2=A^2 +1>A^2>0$ so $x+1>A>0 . $ ..... So we have $$0<x+1-A=$$ $$=(x+1-A)\frac {x+1+A}{x+1+A}=\frac {(x+1)^2-A^2}{x+1+A}=\frac {1}{x+1+A}<1/x.$$ Therefore $$(i)\quad \lim_{x\to \infty} (x+1-A)=0.$$ For $x>2$: For brevity let $B=\sqrt {x^2-2 x}.$ We have $(x-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/675516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Automorphisms on a field $F$ I am trying to understand this proposition with respect to algebraic closures of a field $F$
Prop: If $F$ is a finite field, then every isomorphism mapping $F$ onto a subfield of an algebraic closure $\bar{F}$ of $F$ is an automorphism of $F$
Does this mean that if we have some subfields $K... | Yes, it does mean that for every field homomorphism $\Phi \colon F \to \overline{F}$ we have $\operatorname{im}\Phi = F$.
Every $x \in F$ satisfies the relation $x^n = x$, where $n$ is the number of elements of $F$, thus is a zero of the polynomial $P(X) = X^n - X$. The degree of $P$ is $n$, so $P$ has exactly $n$ zero... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Using continued fractions to prove a bijection. Prove that $\mathbb{N}^{\mathbb{N}} \equiv_c \mathbb{R}$
$\textbf{My Attempt:}$
Let \begin{equation*} \mathbb{N}^{\mathbb{N}}=\{f: \text{all functions} \mid f: \mathbb{N} \to \mathbb{N} \} \end{equation*} Let
\begin{equation*} g:= f \to f(\mathbb{N}) \end{equation*} In th... | I wouldn't bother with decimal expansions here. If $\mathbb N$ means $\{1,2,3,\ldots\}$, then a function $f:\mathbb N\to\mathbb N$ corresponds to a smiple continued fraction
$$
f(1) + \cfrac{1}{f(2)+\cfrac{1}{f(3)+\cfrac{1}{\ddots}}}
$$
This gives a bijection from the set of all functions $f:\mathbb N\to\mathbb N$ to ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Solve differential equation $ (dy/dx)(x^2) + 2xy = \cos^2(x)$ $(dy/dx)(x^2) + 2xy = \cos^2(x)$
$(dy/dx) + 2y/x = \cos^2 x$
I multiplied both sides by $e^{2\ln\ x + c}$, then rewrote the equation as
$(d/dx)(y* e^{2\ln\ x + c}) = (\cos^2(x)/x^2)*(e^{2\ln\ x + c})$
Now when I try to integrate, the right side becomes compl... | The left-hand side is the derivative of $x^2y$. Let $u=x^2y$. Solving the differential equation $\frac{du}{dx}=\cos^2 x$ is a routine integration.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/675758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Incongruent solutions to $7x \equiv 3$ (mod $15$) I'm supposed to find all the incongruent solutions to the congruency $7x \equiv 3$ (mod $15$)
\begin{align*}
7x &\equiv 3 \mod{15} \\
7x - 3 &= 15k \hspace{1in} (k \in \mathbb{Z}) \\
7x &= 15k+3\\
x &= \dfrac{15k+3}{7}\\
\end{align*}
Since $x$ must be an integer, we mus... | We can solve this congruence equation in elementary way also.
We shall write $[15]$ to denote the word mod 15. Fine?
Note that \begin{align*}
&7x\equiv 3[15]\\
-&15x +7x\equiv 3-15[15] ~~\text{because}~~ 15x\equiv 0\equiv 15[15]\\
-&8x\equiv -12[15]\\
&2x\equiv 3\left[\frac{15}{\gcd(15, -4)}\right]~~\text{since}~~ax\e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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If $A$ and $B$ are sets of real numbers, then $(A \cup B)^{\circ} \supseteq A^ {\circ}\cup B^{\circ}$ I have a proof for this question, but I want to check if I'm right and if I'm wrong, what I am missing.
Definitions you need to know to answers this question: $\epsilon$-neighborhood, interior points and interiors. No... | Your proof is good- there is no problem with it. I have reworded a bit to make things a little more clear, specifically, where you say:
Then there is $\epsilon>0$, where $J_\epsilon (x)\subseteq A$ or $J_\epsilon (x)\subseteq B$.
Instead it would be more appropriately stated as: If $x\in A^0\cup B^0$ then $x\in A^0$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to denote a set of functions Say there is an unknown function $h(x)$ $$\int_A^B h(x) = c$$ $A$, $B$ and $c$ are known. So $h(x)$ can have various forms on the range $[A,B]$. I want to know how to denote the set of functions for $h(x)$. I know the notation for a set is $\{...\}$.
So would it be: $\{h(x)|\int_A^B h(x... | Summarizing the comments: try
*
*$\{h \in C([A,B])\vert\int_A^B h(x)\,dx = c\}$ or
*$\{h \in L^1([A,B])\vert\int_A^B h(x)dx = c\}$
depending on what kind of functions you consider. Simply saying functions with integral equal to $3$ is usually ambiguous: there are different kinds of integrals.
Avoid writing $h(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/676089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can a closed set in $\Bbb R$ be written in terms of open sets Is it possible to write any non-empty closed set in $\Bbb R$ as a combination of unions / intersection of open sets. Note that I don't demand just one union / intersection. I am happy with any combination (finite) of unions / intersection, but elements in ea... | For any set $A\subset \mathbb R$ define $B(A, r)=\left\{x\in\mathbb R:\text{dist(x,A) < r}\right\}$, where $\,\text{dist(x,A)} = \inf_{y\in A}\rho(x,y)$ with metric $\rho$. $B(A,r)$ is open for any $A$ and $r$. Now for every closed set $F\subset\mathbb R$
$$F = \bigcap_{n\in\mathbb N}B(F, \frac 1 n)$$
So every closed s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/676173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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calculate x,y positions in circle every n degrees I am having trouble trying to work out how to calculate the $(x,y)$ point around a circle for a given distance from the circles center.
Variables I do have are:
constant distance/radius from center ($r$)
the angle from $y$ origin
I basically need a point ($x$ and $y$) a... | x = radius * cos(angle)
y = radius * sin(angle)
Inverse Y-axis:
x = radius * sin(angle)
y = radius * -cos(angle)
If radians is used then
radian = angle * 0.0174532925
and
x = radius * cos(radian)
y = radius * sin(radian)
Radian is the standard unit of angular measure, any time you see angles, always assume the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/676249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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Prove that if $\operatorname{rank}A=n$, then $\operatorname{rank}AB=\operatorname{rank}B$
Let $A \in M_{m\times n}(\mathbb{R})$ and $B \in M_{n\times p}(\mathbb{R})$.
Prove that if $\operatorname{rank}(A)=n$ then $\operatorname{rank}(AB)=\operatorname{rank}(B)$.
I tried to start with definitions finding that $n \le m... | Remember, if:
(a) $rk(A + B) \leq rk(A) + rk(B)$ for any two $mxn$ matrices $A,B$;
(b) $rk(AB) \leq \min (rk(A),\ rk(B))$ for any $k\times l$ matrix $A$ and $l\times m$ matrix $B$;
(c) if an $n\times n$ matrix $M$ is positive definite, then $rk(M) = n$.
So that, for your question: Prove that if $rk(A)=n$ then $rk(AB)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Smoothness of the Picard group of a smooth curve Let $X$ be a smooth projective curve over $k=\bar{k}$ and denote its Picard group by $\operatorname{Pic}(X)$, with the usual scheme structure coming from the representability of the relative Picard functor.
It's well known that $\operatorname{Pic}(X)$ is smooth of dimens... | The Picard group splits as the product of $\mathbb{Z}$ and the Jacobian variety of $X$, and so each connected component of $\mbox{Pic}(X)$ is (non-canonically) isomorphic to the Jacobian of $X$ which is smooth.
Edit: To see that the Picard group splits, consider the exact sequence
$$0\to\mbox{Pic}^0(X)\to\mbox{Pic}(X)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/676403",
"timestamp": "2023-03-29T00:00:00",
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Why aren't these two integration methods yielding the same answer? I'm trying to solve this (not homework, if it matters), and both u-substitution and integration by parts are both yielding two different answers. Where am I going wrong?
Equation: $$\int \frac{(4x^3)}{(x^4+7)}dx$$
u-substitution answer: $$=\ln\big|(x^4+... | I don't understand it. For $u = x^4+7, du = 4x^3dx$ so
$$
\int \frac{4x^3}{x^4+7} dx = \int du/u = \ln |x^4+7| + C.
$$
Show work for your by parts results and it will become clear where the error is...
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find value of $r$ and the limit For some $r \in \mathbb Q$, the limit
$$\lim_{x \rightarrow \infty}x^r.\frac{1}2.\frac{3}4.\frac{5}6......\frac{2x-1}{2x}$$ exists and is non zero
What is that value of $r$ and what is that limit equal to?
I rewrote the product $\frac{1}2.\frac{3}4.\frac{5}6......\frac{2x-1}{2x}$ =... | Let $x^r.\frac{1}{2}.\frac{3}{4} \dots \frac{2x-1}{2x} = A \tag{1}$.
Clearly $A < x^r. \frac{2}{3} \tag{2}. \frac{4}{5}.\frac{6}{7} \dots \frac{2x}{2x+1}$ and $A > x^r. \frac{1}{2}.\frac{2}{3}.\frac{4}{5} \dots \frac{2x-2}{2x-1} \tag{3}$
Multiplying $(1)$ and $(3)$,
$A^2 > x^{2r}.\frac{1}{2(2x)} \text{ or }, A > \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/676574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do I put $\sqrt{x+1}$ into exponential notation? I think $\sqrt{x+1} = x^{1/2} + 1^{1/2}$. Is this incorrect? Why or why not?
| Remember the formula for fractional exponents:
$$x^\frac{m}{n} = \sqrt[n]{x^m}$$
It can also be written as:
$$x^\frac{m}{n} = (\sqrt[n]{x})^m$$
$\sqrt{x+1}$ can be rewritten as $\sqrt[2]{(x+1)^1}$
Using our formula, we can say that:
$$\sqrt[2]{(x+1)^1} = (x+1)^\frac{1}{2}$$
Also remember that:
$$x^m + y^m \neq (x+y)^m$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/676634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is ZF${}-{}$(Axiom of Infinity) consistent? Godel's theorem implies that Con(ZF) is not provable in ZF since it contains the axiom of infinity.
So is it consistent if the Axiom of infinity is removed?
| Your question is unclear.
It is true that $\sf ZF$ cannot prove its own consistency. But $\sf ZF$ can prove the consistency of $\sf ZF-Infinity$, simply by verifying that the set of hereditarily finite sets satisfies all the axioms of $\sf ZF$ except the axiom of infinity.
This set, often denoted by $HF$ or $V_\omega$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Under what conditions on $f$ does $\|f\|_r = \|f\|_s$ for $0 < r < s < \infty$. Question:
If $f$ is a complex measurable function on $X$, such that $\mu(X) = 1$, and $\|f\|_{\infty} \neq 0$ when can we say that $\|f\|_r = \|f||_s$ given $0 < r < s \le \infty$?
What I know:
Via Jensen's inequality that,
$\|f\|_r \le \|... | A look at the proof of Jensen's inequality is all you really need; there is no need for (more sophisticated) Hölder's inequality.
For simplicity, scale $f$ so that $\|f\|_r=1$. Let $g=|f|^r$. Jensen's inequality says $\int_X g^{p}\ge 1$ (for $p=s/r>1$), which is just the result of integrating the pointwise inequality... | {
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"source": "stackexchange",
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Newton's Function Confusion "Suppose that $r$ is a double root of $f(x) = 0$, that is $f(x)=f'(x)=0$, $f''(x) \neq 0$, and suppose that $f$ and all derivatives up to and including the second are continuous in some neighborhood of $r$. Show that $\epsilon_{n+1} \approx \frac{1}{2}\epsilon_{n}$ for Newton's method and th... | This solution was shown to me by a friend. I understand now!
The solution is as follows:
Let us state the statement of Newton's Method: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$
We know that (our initial guess) is $x_{n} = r + \epsilon$ where $r$ is our double-root and $epsilon$ is our $\textit{very}$ small error. Then ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Statistics: If $X_1$ and $X_2$ are both normally distributed then explain why $X_1 - X_2$ can be standardized with mean 0 and standard deviation of 1 I am currently studying hypothesis testing for two populations and I would like a math major or someone experienced to explain to me why this particular statistic has a m... | A linear combination of independent normal RVs is normal. The mean of $X=\sum_i a_iX_i$ is $\mu =\sum_i a_i \mu_i$, and the variance is $\sigma^2=\sum_i a_i^2\sigma_i^2$. So you have just standardized a normal RV $X \sim N(\mu,\sigma^2)$, and a standardized normal RV has distribution $N(0,1)$.
See this link for proofs:... | {
"language": "en",
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Is $f(a)\!=\!0\!=\!f(b)\Rightarrow (x\!-\!a)(x\!-\!b)\mid f(x)\,$ true if $\,a=b?\ $ [Double Factor Theorem] I encountered this proving problem, I can do the proof but my question is why in the condition/premise we need $a$ to be unequal to $b$? My guess is that even $a=b$, the statement is still true, is it correct? I... | There are obvious counterexamples, e.g. $\,f = x-a.\,$ Less trivially see the remark below. Here is the theorem you seek.
Bifactor Theorem $\ $ Let $\rm\,a,b\in R,\,$ a ring, and $\rm\:f\in R[x].\:$ If $\rm\ \color{#C00}{a\!-\!b}\ $ is cancellable in $\rm\,R\,$ then
$$\rm f(a) = 0 = f(b)\ \iff\ f\, =\, (x\!-\!a)(x\!-\... | {
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Values of square roots Good-morning Math Exchange (and good evening to some!)
I have a very basic question that is confusing me.
At school I was told that $\sqrt {a^2} = \pm a$
However, does this mean that $\sqrt {a^2} = +2$ *and*$-2$ or does it mean:
$\sqrt {a^2} = +2$ *OR*$-2$
Is it wrong to say 'and'? What are the ... | Even though quite a bit has been said already, i wanted to add something.
The numbers which you normally use in school (-1, $\frac{2}{3}$, $ \pi$, etcetera) are called the real numbers. The set of real numbers is denoted by $\mathbb{R}$.
Now the square root of any number $b$ is normally considered to be any number $x$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to calculate $\theta$ when we know $\tan \theta$. Hej
I'm having difficulties calculating the angle given the tangent.
Example:
In a homework assignement I'm to express a complex variable $z = \sqrt{3} -i$ in polar form. I know how to solve this except for when I get to calculating the angle $\theta$.
I know that $... | You shouldn't use the tangent for this kind of problems; compute
$$
|z|=\sqrt{z\bar{z}}=\sqrt{(\sqrt{3}-i)(\sqrt{3}+i)}=
\sqrt{3+1}=2
$$
Then you have $z=|z|u$, where
$$
u=\frac{\sqrt{3}}{2}-i\frac{1}{2}
$$
and you need an angle $\theta$ such that
$$
\cos\theta=\frac{\sqrt{3}}{2},\quad\sin\theta=-\frac{1}{2}.
$$
Since ... | {
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"answer_count": 3,
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} |
Find the contour integral. Let $f(z)=π \exp(π\cdot\overline{z})$. Let $C$ be the square whose vertices are $0,1,(1+i)$, and $i$.
How can I evaluate the contour integral of $f(z)$ over $C$?
| We have
$$
\int_C\pi e^{\pi\bar{z}}\,\mathrm{d}\bar{z}=0
$$
On $[0,1]$ and $[1+i,i]$, $\mathrm{d}\bar{z}=\mathrm{d}z$, and on $[1,1+i]$ and $[i,0]$, $\mathrm{d}\bar{z}=-\mathrm{d}z$. Thus,
$$
\begin{align}
\int_C\pi e^{\pi\bar{z}}\,\mathrm{d}z
&=\int_C\pi e^{\pi\bar{z}}\,\mathrm{d}\bar{z}+2\int_{[1,1+i]\cup[i,0]}\pi e^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/677302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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nonlinear diophantine equation $x^2+y^2=z^2$ how to solve a diophantine equation $x^2+y^2=z^2$ for integers $x,y,z$
i strongly believe there is a geometric solution ,since this is a pythagoras theorem form
or a circle with radius $z$
$x^2+y^2=z^2$
$(\frac{x}{z})^2+(\frac{y}{z})^2=1\implies x=y=\pm z$ or $0$
so we consi... | Euclid's Formula says that in essence, $(m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2$ for all positive integers $m > n$.
This is basically a parametrization of Pythagorean Triplets with two parameters.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/677384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Can we say $TT^{*}=T^{2}$ implies $T=T^{*}$? Let $A$ be a $C^{*}$-algebra, Can we say $TT^{*}=T^{2}$ implies $T^{*}=T$? for $T\in A$
I am looking for a counterexample!
Thanks
| Via a faithful state, we can think of $A$ as represented in some $B(H)$. We have $$ H=\ker T\oplus \overline{\text{ran} T^*}. $$
For $x\in\ker T$, we have $Tx=0$ and then
$$
\|T^*x\|^2=\langle T^*x,T^*x\rangle=\langle TT^*x,x\rangle=\langle T^2x,x\rangle=0;
$$
so $T^*x=0$ and $T=T^*$ on $\ker T$.
Taking adjoints on $T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/677425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Logic puzzle: Which octopus is telling the truth? King Octopus has servants with six, seven, or eight legs. The servants with seven legs always lie, but the servants with either six or eight legs always tell the truth.
One day, four servants met.
The blue one says, “Altogether, we have 28 legs.”
The green one says, “A... | This is more fundamental than it seems. The way the octopod nervous system works is different from ours. They have no body image, and their legs send no signals back to their brain. So an octopus can only know how many legs other octopuses have. So they can only be sure about what the others say. This means that the on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/677495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "106",
"answer_count": 12,
"answer_id": 5
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Can any piecewise function be represented as a traditional equation? In "Fundamentals of Electrical Engineering" we learned about piecewise functions for the "unit-step" and "ramp" which are represented by
$f(x)= \begin{cases}0, & \text{if }x< 0 \\ 1, & \text{if }x>0\end{cases}$ and
$f(x)= \begin{cases}0, & \text{if }... | The crucial step is to come up with an acceptable way to describe indicator functions, i.e. for certain subsets $S\subseteq \mathbb R$ to replace the piecewise definition
$$1_S(x)=\begin{cases}1&\text{if }x\in S\\0&\text{if }x\notin S\end{cases} $$
with something not involving piecewise, but only "traditional" definiti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/677618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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How many different elements can we obtain by multiplying all element in a group? Let $G$ be a finite group.
How many different elements can we obtain by multiplying all element in a group?
Of course, if $G$ is abelian the answer is one but when G is non-abelian, changing the order of the multiplication may produce ne... | The answer to your question is even more subtle. The set of all the possible products is always a coset of the commutator subgroup.
Theorem Let $G$ be a finite group of order $n$, say $G=\{g_1, \dots, g_n\}$ and let $P(G)=\{g_{\sigma(1)}\cdot g_{\sigma(2)} \dots g_{\sigma({n-1})} \cdot g_{\sigma(n)}: \sigma \in S_n\}$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/677688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
"answer_id": 1
} |
"reversing" non-linear equation system I'm not a mathematician and I'm facing a problem with those equations that I found in a book of history of colorscience.
The equations were created by MacAdam to transform the classical colorimetric diagram of the CIE into something better.
The CIE diagram plots chromaticity with ... | Hello and welcome to the site!
The question you are asking has no simple solution. Basically, you have some mapping of pairs of real numbers into some other pair of real numbers, $(D,M)=F(x,y)$ and are asking to find an inverse of $F$. There are many problems with this, the main two being:
*
*In general, $F^{-1}$ ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/677846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove this matrix bound Let an $m$ by $n$ matrix $A\in\mathbb C^{m\times n}$. Denote $M=\max_i\sum_{j=1}^n|A_{ij}|$ and $N=\max_j\sum_{i=1}^m|A_{ij}|$. Prove for any two vectors $x\in\mathbb C^m$ and $y\in\mathbb C^n$, we have
$$\left\vert x^TAy\right\vert\leq\sqrt{MN}|x||y|$$
Here's what I think:
$$|x^TAy|=\le... | Let $\left\Vert X\right\Vert$ denote the spectral norm of $X$. Also, let $\left\Vert \cdot\right\Vert_{p\to p}$ denote the matrix norm induced by the $\ell_p$-norm:
$$\left\Vert X\right\Vert_{p\to p}=\max_{v\neq 0} \frac{\left\Vert Xv\right\Vert_p}{\left\Vert v\right\Vert_p}.$$
Let $v$ be the principal eigenvector of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/677929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Inequality $\left|\,x_1\,\right|+\left|\,x_2\,\right|+\cdots+\left|\,x_p\,\right|\leq\sqrt{p}\sqrt{x^2_1+x^2_2+\cdots+x^2_p}$ I could use some help with proving this inequality:
$$\left|\,x_1\,\right|+\left|\,x_2\,\right|+\cdots+\left|\,x_p\,\right|\leq\sqrt{p}\sqrt{x^2_1+x^2_2+\cdots+x^2_p}$$ for all natural numbers p... | This is an application of Jensen's Inequality:
$$
\left(\frac1p\sum_{k=1}^p|x_k|\right)^2\le\frac1p\sum_{k=1}^p|x_k|^2
$$
since $f(x)=x^2$ is convex.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/678044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 0
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Series representation of $\sin(nu)$ when $n$ is an odd integer? So, out of boredom and curiosity, today I came up with a series representation for $\sin(nu)$ when $n$ is an even integer:
$$\sin(nu) = \sum_{k=1}^\frac n2 \left(\left(-1\right)^{k-1}\binom{n}{-\left|2k-n\right|+n-1}\sin\left(u\right)^{2k-1}\cos\left(u\rig... | Using complex methods and the binomial theorem,
$$\eqalign{\sin(nu)
&={\rm Im}(\cos u+i\sin u)^n\cr
&={\rm Im}\sum_{m=0}^n \binom nm (\cos u)^{n-m}(i\sin u)^m\ .\cr}$$
As only the terms for odd $m$ contribute to the imaginary part we can take $m=2k-1$ to give
$$\sin(nu)=\sum_{k=1}^{(n+1)/2}(-1)^{k-1}\binom n{2k-1}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/678112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Direct proof. Square root function uniformly continuous on $[0, \infty)$ (S.A. pp 119 4.4.8) (http://math.stanford.edu/~ksound/Math171S10/Hw8Sol_171.pdf)
Prove for all $e > 0,$ there exists $d > 0$ : for all $x, y \ge 0$, $|x - y| < d \implies |\sqrt{x} - \sqrt{y}| < e$.
(a) Given $\epsilon>0$, pick $\delta=\epsilon^{... | 2. You know that $\sqrt{x}$ and $\sqrt{y}$ are non negative. The sum of two non-negative numbers is always at least as great as their difference. Alternatively, $|x| + |y| = |x + y|$ for non negative $x$ and $y$. Thus $|\sqrt{x} - \sqrt{y}| \le |\sqrt{x}| + |\sqrt{y}| = |\sqrt{x} + \sqrt{y}|$.
3. Practise! The solutio... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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A conjecture about Pythagorean triples I noticed for the integer solutions of $a^2 + b^2 = c^2$, there don't seem to be cases where both a and b are odd numbers. In fact, I saw this property pop up on a nice question, which required you to prove it. So I have tried proving it, but I have failed so far. A reductio ad ab... | The square of an odd number is always a multiple of four plus one: $(2n+1)^2=4(n^2+n)+1$.
So the sum of two odd squares cannot be a multiple of four.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/678397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Extremum of functional of a complex function consider functional $E$ defined by
$$E[z]=\int F(x,z(x))dx$$
where $F$ is a complex-valued nonlinear function.
How can we find the function $z(x)$ so that
$$G=|E|^2=EE^*=\iint F(x_1,z(x_1))F^*(x_2,z(x_2))dx_1dx_2$$
takes its maximum?
| Find $z$ maximizing $$G(z):=\iint_\Omega F\big(x_1,z(x_1)\big) F^*\big(x_2,z(x_2)\big) \,dx_1 \,dx_2$$
We are given $F$ -- I assume it is twice differentiable, but there are methods (mentioned below) in nonlinear optimization that work for functions which are not. I assume there are no restrictions on $z$; if it must ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/678473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Showing that $\sinh(\mathrm{e}^z)$ is entire I am attempting to show that $\sinh(\mathrm{e}^z)$, where $z$ is a complex number, is entire. The instructions of the problem tell me to write the real component of this function as a function of $x$ and $y$, which I used algebra to do; this function is $u(x, y)=\cos(\mathrm... | You have real and imaginary part $ u(x,y )$ and $v(x,y)$, so you can control if the Cauchy-Riemann equations are verified: $$u_{x} = v_{y} \ \ \ u_{y} = -v_{x}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/678551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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If $\sum a_n$ converges, so does $\sum a_n^{\frac{n}{n+1}}$ Let $a_n$ be a positive real sequence such that the series $\sum a_n$ converges.
I was asked to prove that under such circumstances $\sum a_n^{\frac{n}{n+1}}$ converges.
The previous sum can be rewritten as $\sum \frac{a_n}{a_n^{1/(n+1)}}$.
I can't prove the c... | Let $A=\sum_{n=1}^\infty a_n$, and
set $$S=\big\{n: a_n^{n/(n+1)}\le 2a_n\big\},$$ and $$T=\big\{n: a_n^{n/(n+1)}> 2a_n\big\}.$$
If $n\in T$, then
$$
a^{n/(n+1)}_n> 2a_n\quad\Longrightarrow\quad a_n^n>2^{n+1}a_n^{n+1}\quad\Longrightarrow\quad
a_n<2^{-n-1}.
$$
$$
\sum_{n=1}^\infty a^{n/(n+1)}_n=
\sum_{n\in S} a^{n/(n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/678628",
"timestamp": "2023-03-29T00:00:00",
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Prove that $(n!)^2$ is greater than $n^n$ for all values of n greater than 2. This problem , I assume can be proved using induction, however I am trying to find another way.
Is there a simple combinatorial approach? One notices that $(n!)^2$ is equal to the number of permutations of size n squared, and that $n^n$ is t... | divide $(n!)^2 > n^n$ by $n!$ to get
$$n! = 1 \times 2 \times \ldots \times (n-1) \times n > \frac{n}{n} \times \frac{n}{n-1} \times \ldots \times \frac{n}{2} \times \frac{n}{1}$$
It is a bit of simple algebraic manipulation to show that each term on the lhs is greater or equal to the corresponding term on the rhs, or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/678682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is this process a martingale I was solving some practice problems in stochastics and faced the following exercise:
Given Brownian motion $W(t)$ and a stochastic process $B(t)$ defined as:
$$B(t) =
\begin{cases}
W(t), & \text{if $0 \le t < 1$} \\
tW(1), & \text{if $1 \le t < \infty$} \\
\end{cases}$$
Answer the follow... | Hi your first intuition is correct.
Formally you could write for example to hsow the statement that for $t>s>1$ :
$E[W_t | \mathcal{F}_s]\not=W_s$
For your second question it is more a direct application of your course if you want my opinion. For $t<1$ Have you seen what the Quadratic variation a Brownian motion is ?... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/678764",
"timestamp": "2023-03-29T00:00:00",
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Prove $f(x,y) > g(x,y)$ for all $x,y \in [0,1]$ I'm trying to prove the following:
$$ 4xy + 4(1-x)(1-y) < \max\{8xy,8(1-x)(1-y),3\} \qquad \forall x,y \in [0,1] $$
In the language from the class, I'm trying to show that: $m_2 < \max\{m_1,m_3,m_4\}$
I think that's written correctly. It's derived from a microeconomics g... | Say that $a=4xy$ and $b=4(1-x)(1-y)$. If $a<b$, we know that $a+b<b+b=2b=8(1-x)(1-y)$. If $a>b$, we know that $a+b<a+a=2a=8xy$. If $a=b$, we have $xy=(1-x)(1-y)$. We then want to show that $a+b<3$, since $a+b=2a=2b$. Equality holds if and only if $x=1-y$. Then, the maximum of
$$
4xy+4(1-x)(1-y)=8y(1-y)
$$
occurs at $y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/678867",
"timestamp": "2023-03-29T00:00:00",
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Minimal answer possible. Are there a finite or infinite set of solutions that would satisfy this equation? If there are a finite set of solutions, what would it be? $$(2x+2y+z)/125\leq9.5$$ where $x$, $y$ and $z$ $\leq$ 250. Also $x$, $y$, and $z$, have to be integers.
| Let a = 250 - x, b = 250 - y, c = 250 - z when the equation becomes: 2(250 - a) + 2(250 - b) + 250 - c < 125(9.5) ===> 4a + 4b + 2c > -875/2 and a, b, c are now whole numbers. So if this condition was now in effect then there are infinitely many solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/678943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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intuition for matrix multiplication not being commutative I want to have an intuition for why AB in matrix multiplication is not same as BA. It's clear from definition that they are not and there are arguments here (Fast(est) and intuitive ways to look at matrix multiplication?) that explain that this is in order to ma... | I know this was awhile ago, but I think I have a nice answer. Remember that matrices are linear transformations, so we can rephrase the question: "Is $AB$ the same as $BA$?" as "Does the order of two linear transformations matter?"
The answer is clearly yes. To see why, consider two transformations of the $xy$-plane. T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/679002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $a = \mathrm {sup}\ B$, how to show that the following holds? Let $B \subseteq \mathbb R$ be a nonempty set. If $a = \mathrm {sup}\ B$, then it will be the case that for all $n \in \mathbb N$ that an upper bound of $B$ is $$a +\frac {1}{n}$$ while $$a - \frac {1}{n}$$ won't be an upper bound of $B$.
I attempted to p... | If $a + \frac 1 n$ is not an upper bound of $B$, then there is an element $a'$ in $B$ such that $a' \gt a + \frac 1 n \gt a$ leading to a contradiction since $a$ is an upper bound of $B$.
Similarly, if $a - \frac 1 n$ is an upper bound of $B$ then $a - \frac 1 n \lt a$ contradicting the fact that $a$ is the least upp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/679109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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local minimum of $|f|$ Suppose $f \in H(\Omega)$, where $\Omega\subset\mathbb C$ is an open set. Under what condition can $|f|$ have a local minimum?
Here $|f| = u^2 +v^2 = g$ say. We assumed $f(x,y)= u(x,y) +i v(x,y)$.
Then $g$ has local minimum if $g_{xx} > 0$ and $g_{xx}= 2[u_x^2 +uu_{xx} +v_x ^2 +vv_{xx}]$. So as s... | If $f$ has a zero on $\Omega$, then clearly $|f|$ has a local minimum at those points. Otherwise, the open mapping principle prevents $|f|$ from having a local minimum. (If $a \in \Omega$ then $f$ maps open discs centered at $a$ to open sets. In particular if $f(a) \neq 0$ then there are nearby points $z$ where $|f(z)|... | {
"language": "en",
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Limit of cosine function Can I evaluate the following limit
$\lim_{(x, y) \to (0, 1)}\cos (x)$
as below
$\lim_{(x, y) \to (0, 1)}\cos (x)=\lim_{x \to 0}\cos (x)=\cos (0)=1?$
Can I further explain why I can evaluate the limit in that way as follows?
As $(x, y) \to (0, 1)$, $x \to 0$ and $y \to 1$. The cosine function is... | It is always true that:
$$
\lim_{(x_1,\dots,x_n)\rightarrow (x^0_1,\dots,x^0_n)}f(x_i)=\lim_{x_i\rightarrow x^0_i}f(x_i)
$$
without any hypothesis on $f$. It is just not a function of other variables.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Combinatorial interpretation of Euclid's form for even perfect numbers Euclid showed that if $p$ is a prime such that $2^{p}-1$ is also a prime, then the number $n=2^{p-1}.(2^{p}-1)$ is perfect. Much later, Euler proved that every even perfect number is of this form. Thinking of proper divisors of a number as analogues... | Comment converted to an answer, so that this question does not remain in the unanswered queue.
If $m$ is (even) perfect, then $m = 2^{p-1} (2^p - 1)$, for some Mersenne prime $2^p - 1$. So taking $k = p$, the answer to your second question is yes for even perfect numbers. As for odd perfect numbers $M$, it is curren... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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The reason behind the name "Orthogonal transformation". An orthogonal transformation is a linear transformation such that $(Tx,Ty)=(x,y)$.
Orthogonality is suggestive of perpendicularity. What might have been the reason for naming a distance preserving linear transformation on a vector space "orthogonal"?
Thanks!
| I think it's due to the fact that, in a finite dimensional vector space, the columns of the associated matrix of the transformation are mutually orthogonal.
They don't just preserve distance, they also preserve inner product. In more familiar examples, you can think of this as preserving the angle between two vectors. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Easier way to show $(\mathbb{Z}/(n))[x]$ and $\mathbb{Z}[x] / (n)$ are isomorphic $$(\mathbb{Z}/(n))[x] \simeq \mathbb{Z}[x] / (n)$$
I've shown this by showing that the map that sends $\overline{1} \mapsto [1+(n)]$ (where the bar denotes the congruence class mod $n$) and $x \mapsto [x+(n)]$ is a homomorphism that is in... | The homomorphism in the other direction is maybe easieer to see.
From the $\mathbb Z\to \mathbb Z/(n)\hookrightarrow \mathbb Z/(n)[x]$ (canonical projection and canonical inclusion) and $x\mapsto x$ we obtain a ring homomorphism $\mathbb Z[x]\to \mathbb Z/(n)[x]$ (universal property of polynomial ring). The kernel is ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Value of $\sum\limits_{n= 0}^\infty \frac{n²}{n!}$
How to compute the value of $\sum\limits_{n= 0}^\infty \frac{n^2}{n!}$ ?
I started with the ratio test which told me that it converges but I don't know to what value it converges. I realized I only know how to calculate the limit of a power/geometric series.
| The idea for $$\frac{n^r+b_{r-1}n^{r-1}+\cdots+b_1\cdot n}{n!},$$ we can set this to $$\frac{n(n-1)\cdots(n-r+1)+a_{r-1}\cdot n(n-1)\cdots(n-r+2)+\cdots+a_2n(n-1)+a_1\cdot n+a_0}{n!}$$
$$\frac1{(n-r)!}+\frac{a_{r-2}}{(n-r+1)!}+\frac{a_1}{(n-2)!}+\frac{a_1}{(n-1)!}+\frac{a_0}{(n)!}$$
where the arbitrary constants $a_is,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/679790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
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Example of a domain in R^3, with trivial first homology but nontrivial fundamental group Let $\Omega \subset \mathbb{R}^3$ be a domain. Is it true that if $H_1(\Omega)$ = 0, then $\pi_1(\Omega) = 0$? For a counterexample, the group $\pi_1(\Omega)$ needs to be a perfect group and so I was trying with the smallest one i.... | First, suppose that you have a compact connected submanifold $C$ with nonempty boundary in 3d sphere. If some boundary components are spheres, you add the 3-ball which they bound $C$ without changing 1st homology or fundamental group. Suppose the result, which I will still call $C$, still has nonempty boundary. Recall ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/679910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Showing set linear independence How do I show that the set $\{ e^x , ... ,e^{nx} \}$ is linearly independent?
I tried using induction as the base case of $\{ e^x \}$ and even $\{ e^x, e^{2x} \}$ is easy, but I can't use the I.H. to go further. What I try to do with the induction step is plug in values of x and try to ... | Hint: You can use the Wronskian. See here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/680017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Trigonometry, knowing 3 sides how to find the height? I have a mathematician problem where, I knew the 3 sides of a triangle, with these sides I can figer out what type of type of triangle is.
What I realy want to find is the height of the triangle and another one "side".
Let me explain what I want, with the above pic... | Or from Heron's formula for triangle area http://en.wikipedia.org/wiki/Heron%27s_formula
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/680262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Trigonometric Series Proof I am posed with the following question:
Prove that for even powers of $\sin$:
$$ \int_0^{\pi/2} \sin^{2n}(x) dx = \dfrac{1 \cdot 3 \cdot 5\cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots 2n} \times \dfrac{\pi}{2} $$
Here is my work so far:
*
*Proof by induction
$P(1) \Rightarrow n = 2 \Rightarrow... | For $k=1$, it's straightforward to verify$$\int_0^{\pi/2}\sin^2x~dx=\int_0^{\pi/2}\frac{1-\cos 2x}2dx=\frac\pi4$$
Assume $k=n$ we have
$$I_n=\int_0^{\pi/2}\sin^{2n}x~dx=\frac{(2n-1)!!}{(2n)!!}\frac\pi2$$
Then for $k=n+1$,
$$\begin{align}I_{n+1}&=\int_0^{\pi/2}\sin^{2n}x(1-\cos^2x)dx\\
&=I_n-\int_0^{\pi/2}\sin^{2n}x\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/680340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If $a$ and $b$ are odd, prove $\gcd(a,b) = \gcd(\frac {\left| {a-b} \right |} {2}, b)$ Honestly I don't have a strong idea. I don't know where to even begin, I have considered that the $\gcd(a,b)$ is somehow less than $a-b$, but I'm not even sure why that would be the case.. Any help would be great!
| Some hints. Let $g=\gcd(a,b)$. We want to show that $g\mid{\rm RHS}$.
(1) $g$ is a factor of $a-b$ because. . .
(2) therefore $g$ is a factor of $|a-b|$ because. . .
(3) therefore $g$ is a factor of $\frac{1}{2}|a-b|$ because. . . (this is the hard bit)
(4) and it is obvious that $g$ is a factor of $b$, so
(5) $g$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/680418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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List of functions not integrable in elementary terms When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the examples I give mostly come from "folklore" or guesswork.
Can anyone... | Liouville's theorem in fact exactly characterizes functions whose antiderivatives can be expressed in terms of elementary functions.
However, the only proof I have seen is not exactly suitable for teaching beginning calculus students. In fact, the proof of the impossibility of solving a general 5th degree polynomial by... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/680478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 3,
"answer_id": 2
} |
Why isn't a t test used when comparing two proportions? All the examples I've seen say to use a z test to compare two proportions. For example, n=13, x=0.22 versus n=10, x=0.44. Then all the examples warn that the z test doesn't work with low sample sizes.
So why can't we just use a t test, which provides p values appr... | Stefan may have already addressed your concerns, but here is the basic rationale for the normal approximation for comparing proportions and why the t-test is not useful:
Although sample proportions are not normally distributed for any finite sample size, they approach a normal distribution as the sample size approaches... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/680587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Convolution of convolution Let us write a convolution
$\int_{0}^{t} A(t-\tau) \mathrm{d}x(\tau)$ as
$A \star \mathrm{d}x$
I would like to write down the expression for the double convolution
$A \star \mathrm{d}x \star \mathrm{d}x $
Following the definition I obtain
$ \int_{0}^{t} \int_{0} ^{t-\tau} A(t-\tau-s) \mat... | When the function is differentiable and you can write the operation as a regular convolution, you can use the fact that $\dot x\ast \dot x $ makes sense, differently from $dx\star dx$, which is not defined.
In this case you would have $A\star dx\star dx = A\ast \dot x \ast \dot x = A\ast (\dot x \ast \dot x)$: $$\int_0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/680683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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} |
how to find this line integral and what is its answer evaluate the line integral $$\int_C (xy^2 dy-x^2y dx), $$ taken in the counter-clockwise sense along the cardioid $$r= a(1+\cos\theta)$$
here putting the parametric form of cardioid $x=a(2\cos t-\cos2t), y= a(2\sin t-\sin2t) $ and taking $\theta$ , $0 $ to $2\pi $ t... | I think this problem can be solved by Green's identity. Let $D$ denote the area of shape enclosed by $C$. By using polar substitution, we have:
\begin{equation}
\int_C(xy^2dy-x^2ydx)=\iint_Dx^2+y^2dxdy
\\=\int_0^{2\pi}\int_0^{a(1+\cos(\theta))}r^3drd\theta\\
=\frac{35}{16}\pi a^4
\end{equation}
I ignore the trigonometr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/680787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
Distance between two sets in a metric space is equal to the distance between their closures Let $A,B \subseteq \mathbb{R}^d$ be non-empty sets. Define their distance to be
$$ d(A,B) = \inf \{ ||x-y|| : x \in A, \; \; y \in B \} $$
For any $A,B$, do we have that $d(A,B) = d( \overline{A}, \overline{B} ) $.
Is the follow... | The proof is correct. Just to have a shorter proof of the same fact (the distance between sets is equal to the distance between their closures), here goes:
Let $\rho = d(A,B)$. The inequality $d(\overline{A},\overline{B})\le \rho$ holds because the infimum on the left is over a larger set. In the opposite direction... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/680855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Ordinary Differential Equation - Boundary Conditions Question The following problem has brought up some misunderstandings for me -
Find the eigenvalues λ, and eigenfunctions u(x), associated with the following homogeneous ODE problem:
$$ {u}''\left ( x \right )+2{u}'\left ( x \right )+\lambda u\left ( x \right )=0\; ,\... | Since $\sinh(0)=0$, the first condition is obviously $0=u(0)=C\cosh(0)=C$. The sum applies to the exponential representation, i.e., $0=u(0)=Ae^0+Be^0=A+B$.
At the point $1$ you get to
$$0=u(1)=C\cosh(−1−\sqrt{1−λ})+D\sinh(−1−\sqrt{1−λ})=D\sinh(−1−\sqrt{1−λ})$$
where neither factor reduces to zero, but $C=0$ can be inse... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/680919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Which of the following collections of subsets of $\mathbb{R}$ form a topology on $\mathbb{R}$? 1) $T_3 =\{ \emptyset, \mathbb{R}, [−a,a] : a \in \mathbb{R},a>0 \}$;
2) $T_4 = \{ \emptyset, \mathbb{R}, [−n,n], (−a,a) : a \in \mathbb{R}, a > 0, n \in \mathbb{N}^{>0} \}$.
I have that $T = \{ \emptyset, \mathbb{R}, (−a,a) ... | Apologies for the blunder earlier. Hope this is correct.
Say $A = (0, 10)$ Then, $ \bigcup_{a\in A} [-a, a] = (-10, 10) \not \in T_3$. Therefore, $T_3$ is not closed under arbitrary unions and hence does not form a Topology.
Now to $T_4$. Consider $\bigcup_{n \in \Bbb N \setminus \{1\} } ( - \frac 2 3 + \frac 1 n, \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/681005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Finding Arc Length using First Fundamental Form Let $v=ln(u+\sqrt{u^2+1})+C$ be the curve given on the right helicoid $x=u\cos(v),y=u\sin(v),z=2v$. Calculate the arc lengths of this curve between the points $M_1(0,0)$ and $M_2(1,ln(1+\sqrt{2}))$
So I am trying to find $r_u(u,v)$ and $r_v(u,v)$ to construct the first fu... | We usually use partial derivatives to get the metric
$$g_{uu}=(\cos v,\sin v,0)\cdot (\cos v,\sin v,0)=1,
\\ g_{vv}=(-u\sin v,u\cos v,2)\cdot (-u\sin v,u\cos v,2)=u^2+4
\\ g_{uu}=(\cos v,\sin v,0)\cdot (-u\sin v,u\cos v,2)=0$$
The the metric is given by
$$(g)= \begin{pmatrix} 1 & 0\\ 0 & u^2+4 \end{pmatrix} $$
the cur... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/681095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Structure of the functional space $\int_ {- \infty} ^ \infty f (x) dx = 1 $ Please, help me with studying of useful practical features of the following functional space:
$$\int_{-\infty}^\infty f(x) \, dx = 1$$
For example:
1) What basis types are most convenient for representation an element from the space?
2) How to ... | The integral condition says very little about $f$; it selects an affine subspace of codimension $1$, which is not any more manageable than the space you began with. For any $f\in L^1(\mathbb R)$, the function $$f-c\chi_{[0,1]},\quad \text{where } c = \int_{-\infty}^\infty f -1 $$ satisfies your condition. So, if you h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/681170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Can we conclude that a distribution is a $L^2$ function by testing with $L^2$? Let $T\colon \mathcal{D}\to\mathbb{R}$ be a distribution. Does $|T(f)|\leq\|f\|_2 \forall f\in\mathcal{D}$ imply $T=T_g$ for some $g\in L^2$? What if $T$ is tempered?
| Your hypothesis leads to $$T:D\to\mathbb C,$$ linear continuous functional in the sense of $L^2(\mathbb R)$. $D$ is dense in $L^2$, therefore by continuity we can extend $T$ to the whole $L^2$. By Riesz representation theorem, there exists an $L^2$ function $g$ such that $T(f) = (g,f)_{L^2}$. Moreover, it implies that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/681271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is the Center of G the same as the Centralizer of g in G? Is the center, $Z(G)$, of a group $G$ the same as the centralizer, $C(g)$, of an element $g\in G$?
I have proven that $C(g)\leq G\forall g\in G$ but my homework, in a later problem, asks me to prove that $Z(G)\leq G$. This confuses me, because I thought $C(g)$ i... | Consider a group (G, *).
Let 'g' be a fixed element of G.
The set of all the elements in G that commute with 'g' is known as the centralizer of 'g'. It is denoted by C(g).
The set of all the elements in G that commute with every element of G is known as the center of G. It is denoted by Z(G).
There can be many element... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/681326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
An inequality for sides of a triangle Let $ a, b, c $ be sides of a triangle and $ ab+bc+ca=1 $. Show
$$(a+1)(b+1)(c+1)<4 $$
I tried Ravi substitution and got a close bound, but don't know how to make it all the way to $4 $.
I am looking for a non-calculus solution (no Lagrange multipliers).
Do you know how to do it?
| Solving $ab+bc+ca=1$ for $c$ gives
$$
c=\frac{1-ab}{a+b}\tag{1}
$$
The triangle inequality says that for non-degenerate triangles
$$
|a-b|\lt c\lt(a+b)\tag{2}
$$
Multiply $(2)$ by $a+b$ to get
$$
|a^2-b^2|\lt1-ab\lt(a+b)^2\tag{3}
$$
By $(3)$, we have $(a+b)^2-1+ab\gt0$; therefore,
$$
\begin{align}
(a+b+1)(a+b+ab-1)
&=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/681433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Invertible Matrix to Higher power I'm working on showing if A is invertible, that for any positive integer $n$, $(AMA^{-1})^n=(AM^nA^{-1})$
My first idea is induction on $n$ but is there a property of $A$ that explans why its power remains 1 or -1? Thanks in advance.
| Hint: $AA^{-1}=A^{-1}A=I$ where $I$ is the identity matrix. Induction would be a good idea.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/681499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What does $\prod_{n\geq2}\frac{n^4-1}{n^4+1}$ converge to?
What does $\prod_{n\geq2}\frac{n^4-1}{n^4+1}$ converge to?
As far as I can tell, this has no closed-form solution (not saying much, I don't know much math), but a friend of mine swears he saw a closed-form solution to this in some text he doesn't remember.
Ru... | Since your post was interesting and the answers really nice, just for personal curiosity, I looked at the more general function $$\prod_{n=2}^\infty\frac{n^q-1}{n^q+1} $$
where $q$ in an integer. I report below some results I found interesting
$$\prod_{n=2}^\infty\frac{n^2-1}{n^2+1}=\pi \text{csch}(\pi )$$
$$\prod_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/681619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
} |
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