Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Integral of exponential, polynomial and gamma function I am trying to solve the following integral
$$ \int_{0}^\infty x^d \left( \frac{e^{- a x} (b+cx)^{d}}{\left( 1- \frac{\Gamma(d, b+cx)}{\Gamma(d)} \right)}\right)^s dx $$
where $a>0$, $b>0$, $c>0$, $d \in \mathbb{Z}^+$, $s\in \mathbb{R}$ and $\Gamma(\cdot, \cdot)$ ... | Be $\,a,b,c,d,s>0\,$ , $\,d\,$ an integer.
$(A)\hspace{1cm}$ Only an approximation for large positive $\,s\,$ .
Be $\,cd<ab\,$ . Be $\,s\,$ large enough so that $\,\displaystyle \left(1+\frac{x}{s}\right)^s\approx e^x$ and $\,\displaystyle \Gamma\left(x,y+\frac{z}{s}\right)\approx \Gamma(x,y)\,$ .
It's $\,\displa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2421684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $\int_0^1 f_n dx \leq a_n $ where $\sum_{n=1}^{\infty}a_n < \infty$ show $f_n \to 0$
Suppose $\{f_n\}_{n=1}^{\infty}$ is a sequence of nonnegative measurable functions on $[0,1]$ with
$$\int_0^1 f_n \, dx \leq a_n \qquad \text{for all} \quad n\geq 1$$
where $$\sum_{n=1}^{\infty}a_n < \infty.$$ Prove that $f... | Convergence of the integrals $\int_0^1 f_n \to 0$ does, in general, not imply $f_n \to 0$; therefore your approach doesn't work.
Hints:
*
*Recall the Borel-Cantelli lemma: Let $\mu$ be a finite measure. If $(A_n)_{n \in \mathbb{N}}$ is a sequence of measurable sets such that $$\sum_{n \geq 1} \mu(A_n)<\infty$$ then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2421775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Why does an integral extension over a ring has the same Krull dimension as the ring? The Krull dimension of a ring R is defined as the supremum of the lengths of chains of prime ideals contained in R. I heard that an integral extension over a ring R has the same Krull dimension as R, however, I don't really see why thi... | The wiki article hints that an integral extension $R\subseteq S$ satisfies going-up, lying-over and incomparability, the Krull dimensions of $R$ and $S$ are the same.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2421885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
De Moivre's Theorem to calculate the fourth roots of 8 I was given this question:
Use de Moivre's theorem to derive a formula for the $4^{th}$ roots of 8.
As far as my understanding of this theorem goes, it is only applicaple to complex numbers. How am I supposed to use it for 8?
My initial thought was use 8 to make $z... | Even though you were told to use the estimable Mr. de Moivre, it’s easiest without:
You know that if $\rho$ is one fourth root of a number, the others are $-\rho$ and $\pm i\rho$. Since one fourth root of $8$ is the real number $2^{3/4}$, you have them all.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2421989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Expected value of a marginal distribution is a function of $x$?
Let $f$ be the joint PDF of the random vector $(X, Y)$.
$f(x, y) = \displaystyle\frac{x(x-y)}{8}$ for $0 < x < 2$ and $-x < y < x$, otherwise it's zero.
Calculate the correlation between $X$ and $Y$.
The problem I'm struggling with here is that to comput... | No, indeed the expectation of $Y$ should not be a function over $X$ (nor over $x$). It shall be a constant.
Be careful. You are using the support for the conditional pdf for $Y$ given $X=x$, not that for the marginal pdf for $Y$.
Examine the support for the joint pdf:
Because $\{(x,y): 0\leqslant x\leqslant 2, -x\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2422096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Formalizing the Mathematical statement in Graph theory This question is in Diestel - Graph Theory (5th Edition) Exercise 1.15.
Let $\alpha, \beta$ be two graph invariants with positive integer values.
Formalize the two statements below, and show that each implies the other:
(i) $\beta$ is bounded above by a functi... | Formalize as follows:
(i)
There exists a function $f: \mathbb{N} \to \mathbb{N}$ such that $\beta(G) \le f(\alpha(G))$ for every graph $G$.
(ii)
For every graph $G$ there is a positive integer $N$ such that for all graphs $G'$ with $\beta(G') > N$ we have $\alpha(G') > \alpha(G)$.
Suppose (i) holds.
Let $G$ be any grap... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2422204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Meaning of $\mathbb K\in\{\mathbb R,\mathbb C\}$? What is the meaning of $\mathbb K\in\{\mathbb R,\mathbb C\}$ in the following?
Let $A$ be a non-empty open subset of $\mathbb K\in\{\mathbb R,\mathbb C\}$ and let $f:A\rightarrow \mathbb K$ be a continuous function.
Is $\mathbb K$ a set of real numbers and the complex... | It means that $\Bbb K$ is $\Bbb R$ or $\Bbb C$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2422332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Evaluating $\int^\infty _1 \frac{1}{x^4 \sqrt{x^2 + 3} } dx$ I was evaluating
$$\int^\infty _1 \frac{1}{x^4 \sqrt{x^2 + 3} } dx$$
My work:
I see that in the denominator, there is the radical $\sqrt{x^2 + 3}$. This reminds me of the trigonometric substitution $\sqrt{u^2 + a^2}$ and letting
$ u = a \tan \theta$. With ... | Hint:
You have done a good substitution so here you are
$$\int \frac{1}{9 (\tan \theta)^4 \sqrt{3} \sec \theta } \sqrt{3} (\sec \theta )^2 d\theta=\dfrac{1}{9}\int\dfrac{\cos^2\theta}{\sin^4\theta}\cos\theta\,d\theta=\dfrac{1}{9}\int\dfrac{1-\sin^2\theta}{\sin^4\theta}\cos\theta\,d\theta$$
now let $\sin\theta=u$ and c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2422416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
How to define the exponential function without calculus? For fun, I would like to define the complex exponential function from these two properties:
*
*$\exp(0) = 1$
*$\exp(z + w) = \exp(z) \exp(w)$
From here, I would like to find a way to compute values of $\exp(z)$, or at least to compute $\exp(1)$.
So far, I ... | You won't be able to do derive
$exp(1)=e$
from your definition, since your definition works for the exponential function for any base $b$:
$b^0=1$
and
$b^{x+z}=b^x\cdot b^z$
are true for any $b$!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2422499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 12,
"answer_id": 3
} |
Is $x^j+x^k+2$ irreducible whenever $j+k$ is odd? Let $j,k$ be positive integers with $j>k$ and consider the polynomial
$$f(x)=x^j+x^k+2$$
I want to prove the conjecture :
$f(x)$ is irreducible in $\mathbb Q[x]$, whenever $j+k$ is odd. This is true for $j\le 300$ as I checked with PARI/GP.
If $f$ has real roots, the... |
Can we use the bound of the absolute values of the roots and that the constant coefficient is prime to show that $f(x)$ must be irreducible in $\mathbb Q[x]$
Yes, that is indeed exactly how we can prove the irreducibility of $x^j+x^k+2$, by more closely inspecting its (complex!) roots.
Firstly, no root $\alpha \in \m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2422567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 0
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How to prove this statement with "first" principle of Mathematical induction and not strong Mathematical induction?
Define a sequence $s_0,s_1,s_2,...$ as follows :
$$s_0=0, s_1=4, s_k=6s_{k-1} - 5s_{k-2} \; \forall \; \text{integers} \; k\ge 2.$$ Prove by Principle of strong mathematical induction that $$s_n=5^n-1 ... | It seems I have got my answer. I saw a proof that "first" principle of mathematical induction and the strong principle of mathematical induction are equivalent. So by that proof I have formulated a proof for this one too. Here it goes :
Let $P(n) : s_n=5^n-1$
Let $Q(n) : P(j) \; \text{is true} \; \forall \; 0\le j\le n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2422666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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non-normal covering of wedge of three circles How might I systematically approach the task of finding a three-fold, non-normal covering of a wedge of three circles? My instinct is to find a non-normal subgroup of the free group on 3 generators and try to sketch a space whose loops realize that subgroup. As this is in... | I think that if you can do it with a bouquet of two circles, you can
do it with a bouquet of three. Just hang an extra circle at the preimages
of the base-point.
Let $B$ be a bouquet of two circles, and $x$ the point where
they meet. The fundamental group $\pi_1$ is free on two generators, $g$ and $h$
corresponding to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2422747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that $ \frac{{d}}{dx}\sin(x) = \cos(x)$, using summation forms. It is well-known that: $ \frac{{d}}{dx}\sin(x) = \cos(x)$. [statement (1)]
Given the definitions:
$$\sin(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{(2n-1)!} x^{2n-1}$$
And:
$$\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n)!} x^{2n}$$
Can you show t... | Expanding the first one:
$$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}...$$
Calculate its derivative:
$$1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}...$$
Then calculate the summation form. You're done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2422816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Prove $a_0=a_1=\dots =a_{p-1}$ If $w=\cos \frac{2\pi}{p}+i\sin \frac{2\pi}{p}$ and $p$ is a prime and $a_0,a_1,\dots ,a_{p-1}$ are non zero integers and $a_{p-1}w^{p-1}+\dots +a_1w+a_0=0$ Prove $a_0=a_1=\dots =a_{p-1}$
I got a solution somewhere but don't know how it works:
"The thing is that $\Phi_p (X)$ is irreducibl... | $\mathbb Z[x]$ is the ring of polynomials with integer co-efficients. Definition: Any $f(x)\in Z[x]$ is irreducible in $Z[x]$ iff whenever $g(x), h(x)\in \mathbb Z[x]$ and $f(x)=g(x)h(x)$ then at least one of $g(x),h(x)$ is a constant.
$(\bullet)$. Let $f(x)\in \mathbb Z[x]$ be irreducible in $Z[x]$ with deg$(f(x))>0$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2422881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Determine all local minimums, maximums and saddle points for $f(x,y)= x^2y +4/x +4/y$ $$f_x=2xy-\frac{4}{x^2}$$ $$f_y=x^2-\frac{4}{y^2}$$
And now when i'm trying to solve that i get somethink like this
$$ x^2-x^6=0$$ $\to x=0, x=1, x=-1$ and when i'm putting this into this equation
$$ y=\frac{2}{x^3}$$
i get $y=2$ or $... | Hint
See that $(0,0)$ is not in the domain. You then got, $x\ne0$ and $y\ne0$:
$$xy=\frac{2}{x^2}\to (xy)^2=\frac{4}{x^4}\quad (1)$$
and
$$x^2=\frac{4}{y^2}\to (xy)^2=4\quad (2)$$
and then
$$4=\frac{4}{x^4}\to x=\pm1$$
and backing to $(1)$ we get $y=\pm2$ what give us the pairs $(1,2)$ and $(-1,-2)$ as candidates.
Can ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2422993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the meaning of the following expressions? Let $A_1,A_2,\ldots,A_n,\ldots$ be a sequence of events. What is the meaning of these events:
$$A^*=\bigcap_{k=1}^{\infty}\bigcup_{n=k}^{\infty}A_n$$
$$A_*=\bigcup_{k=1}^{\infty}\bigcap_{n=k}^{\infty}A_n$$
My attempt: I think $A^*$ means that $$\forall N\in \mathbb{N}: ... | I suspect you mean something like:
$$
A^* = \{x \mid (\forall N\in\mathbb{N})(\exists n\ge N)\,x\in A_n\}
$$
and
$$
A_* = \{x \mid (\exists N\in\mathbb{N})(\forall n\ge N)\,x\in A_n\}.
$$
These sets are commonly called $\limsup_n A_n$ and $\liminf_n A_n$, respectively. It can be useful to describe these sets in words:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2423177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Question regarding Gambler's Ruin Consider a gambling process $(X_n)_{n∈\mathbb{N}}$ on the state space $S = {0, 1, . . . , N}$, with probability
$p$, resp. $q$, of moving up, resp. down, at each time step. For $x = 0, 1, . . . , N$, let $τ_x$
denote the first hitting time, $τ_x := \inf\{n ≥ 0 : X_n = x\}$
Let $p_x := ... | Let us write $\mathbb P_x(\cdots) = \mathbb P(\cdots \mid X_0 =x)$. Let $$A_{x+1} = \{\tau_{x+1} < \tau_0 \}=\{\text{hit $x+1$ before 0}\}$$ be the desired event. We have
\begin{align*}
p_x = \mathbb P_x(A_{x+1}) &= \mathbb P_x(X_1 = x+1) \mathbb P(A_{x+1} \mid X_1 = x+1)+
\mathbb P_x(X_1 = x-1) \mathbb P(A_{x+1} \mid ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2423304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Finding the integral of $\frac{8t^3 +13}{(t+2)(4t^2+1)} dt.$ I was looking for the integral of $\frac{8t^3 +13}{(t+2)(4t^2+1)} dt.$
My work:
Dividing the $\frac{8t^3 +13}{(t+2)(4t^2+1)}$, I get $2 + \frac{-16t^2 -2t + 9}{4t^3 + 8t^2 + t + 2}$ or $2 + \frac{-16t^2 -2t + 9}{(t+2)(4t^2+1)}$
Using the partial fraction dec... | You know that
$$
\int \frac{1}{t^2+1}dt=\arctan t + C.
$$
Thus, let $u=2t$, then
\begin{align}
\int \frac{6}{4t^2+1}dt &= \int \frac{6}{u^2+1}\frac{1}{2}du\\
&=\int \frac{3}{u^2+1}du\\
&=3\arctan u + C\\
&=3\arctan 2t + C.
\end{align}
Thus $\int \frac{6}{4t^2+1}dt = 6\arctan (2t) + c$, you found, is wrong.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2423402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
$\Big| \dfrac{df}{dx}(x)\Big|\leq 5$ for all $x$
$f:\mathbb{R}\rightarrow \mathbb{R}$ is such that $f(0)=0$ and $\Big| \dfrac{df}{dx}(x)\Big|\leq 5$ for all $x$. We can conclude that $f(1)$ is in
*
*$(5,6)$
*$[-5,5]$
*$(-\infty,-5)\cup (5,\infty)$
*$[-4,4]$
The answer would be 2. (You can also find a solution... | Mean value Theorem:
$\dfrac{f(1) -f(0)}{1-0} = f'(t)$, with $t \in (0,1).$
Hence:
$|f(1)| = |f'(t)| \le 5.$
$\Rightarrow: f(1) \in [-5,5].$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2423514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Find the solution of partial differential equation about $x\left( t \right) $ and $t$
Let $J$ is a function about $x\left( t \right) $ and $t$, such that
$$\frac{\partial J}{\partial t}=\frac{1}{4}\left( \frac{\partial J}{\partial x} \right) ^2-x^2-\frac{1}{2}x^4,\qquad \text{where}\, J\left[ x\left( 1 \right) ,1 \r... | Hint:
Let $\begin{cases}p=t-1\\q=x\end{cases}$ ,
Then $\dfrac{\partial J}{\partial t}=\dfrac{\partial J}{\partial p}\dfrac{\partial p}{\partial t}+\dfrac{\partial J}{\partial q}\dfrac{\partial q}{\partial t}=\dfrac{\partial J}{\partial p}$
$\dfrac{\partial J}{\partial x}=\dfrac{\partial J}{\partial p}\dfrac{\partial p}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2423639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Difficulty with rewriting DNF's
Is the following formula in DNF?
$$((P \land Q) \land ( \neg R \land \neg S)) \lor ((r \lor s) \land (\neg p \lor \neg q))$$
I think it is not a DNF, because in the second part there are OR's inside an AND. Is this correct?
How should I rewrite this to a proper DNF?
| You must simply expand it performing the AND's in the second part (following the same rule as if AND were a multiplication and OR an addition), so obtaining:
$$(p\land q\land \neg r\land\neg s)\lor(\neg p\land s)\lor(\neg p\land r)\lor(\neg q\land s)\lor(\neg q\land r)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2423763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Get zero, poles and gain from state space model? I'm going to transform a state space model:
$$\dot{x} = Ax + Bu \\ y = Cx + Du$$
Into a transfer function:
$$G(s) = \frac{Y(s)}{U(s)}$$
What I need is to find the zeros, poles and gain. Finding poles are really easy. I just find the eigenvalues of the matrix $A$.
$$det(s... | This is a standard problem of finding the transfer function from a state-space model of a linear system. In particular, $\dot{x}=Ax+Bu \implies X(s)=(sI-A)^{-1}B U(s)$, and $y=Cx+Du \implies Y(s)=CX(s)+DU(s)$. Consequently,
$$Y(s) = CX(s) + D U(s) = (C(sI-A)^{-1}B +D)U(s) \implies G(s) = C(sI-A)^{-1}B +D.$$
Once you ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2423847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Binomial Expansion - Simple application of the formula I found this one question in an old book:
What is the coefficient of $x^{n+1}$ in the expansion of $(x+2)^n \cdot x^3$?
Answer (according to my book): $(n^2-n) \cdot 2^{n-3}$
Here is my work: Since $\ T_{k+1} = \binom{n}{k}\cdot a^k\cdot b^{n-k}$, we can obtain a... | The binomial theorem yields
$$
(x+2)^n = \sum_{k=0}^n \binom nk x^k 2^{n-k},
$$
and so the coefficient of $x^{n+1}$ in $(x+2)^nx^3$ is
$$
\binom n{n-2}2^{n-(n-2)}) = \binom n2 2^2 = 2n(n-1).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2423935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Understanding why the empty set is closed
Definition. A set is called closed if its complement in $\mathbb{R}$ is open.
In my lecture notes it says: $\emptyset$ is closed because $\emptyset = \emptyset \setminus \mathbb{R}$ and $\mathbb{R}$ is open. I think there is a typo because $\emptyset \neq \emptyset \setminus ... | Yes, it is probably a typo. It should be $\emptyset=\mathbb{R}\setminus\mathbb R$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2424032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 2
} |
Simplify a combinatorial expression I came a cross a kind of combinatorial expression in my research. I'm wondering if there is a way to simplify or rewrite it. The expression is pretty simple. So I'm posting it here instead of MO. It is the following.
$\displaystyle \sum\limits_{i=0}^n (-1)^i{n \choose i} {x-i \choose... | Just to show another way to solve it.
The backward Delta (finite difference) is defined as
$$
\eqalign{
& \nabla _{\,x} \,f(x) = f(x) - f(x - 1) \cr
& \nabla _{\,x} ^n \,f(x) = \nabla _{\,x} \,\left( {\nabla _{\,x} ^{n - 1} \,f(x)} \right) = \sum\limits_{\left( {0\, \le } \right)\,k\,\left( { \le \,n} \right)} {... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2424156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Calculate the differential at a point I'm to calculate the differential to $f(x_1,x_2)=e^{x_1}+x_2$ at $x=(2,-1)$
I get the differential to be: $e^{x_1}dx_1+dx_2$
I'm ready to plug in the coordinates (only $x_1$) but I'm put off by the "$dx_1$" and "$dx_2$", how do I deal with them if I only want a value as an output?
| For $f\colon \mathbb R^n\to \mathbb R$, and $x\in \mathbb R^n$, you can compute the differential of $f$ at $x$, $df_x \colon \mathbb R^n\to \mathbb R$ as follows.
If $\nabla f$ is the gradient of $f$, then $df_x$ is given by $$d f_{x}(v)= (\nabla f)_{x}\cdot v$$
Then, for $f(x_1,x_2)=e^{x_1}+x_2$ and $x=(2,-1)$ you ha... | {
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"url": "https://math.stackexchange.com/questions/2424253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof check for "every neighbourhood is an open set" I have been goind through Rudin for the past month and I have arrived at this theorem in the topology chapter. Now I understand Rudin's proof but I was trying to come up with my own and I am not sure if it is correct or not.
Theorem: Every neighbourhood is an open se... | I think your proof is incorrect. There is a problem with your statement: "Since $N_q$ is not in $N$, then $p$ is not in $N$".
It is true that $N_q$ is not a subset of $N$, but it is certainly possible that some elements (points) of $N_q$ are also elements of $N$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2424360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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domain and range of function $y= 1+\sum_{n=2}^\infty x^n$ My friend give me a question to find domain and range of
$y= 1+\sum_{n=2}^\infty x^n$
there is no more description about the problem, so I think the domain of that function is all $\mathbb{R} $ and range of that function is $ \mathbb{R}$ too
but he told me th... | If all you've got is $\text{“} 1 + x^2+x^3+x^4+\cdots\text{''},$ then the question of what the domain is is problematic in several respects.
The simplest of those may be whether $x$ is supposed to be a real number or a complex number. And perhaps it could also be a matrix or any of a variety of other things. Next there... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find an element $\theta \in \mathbb{R}$ such that $\mathbb{Q}(\sqrt{2},\sqrt[3]{5}) = \mathbb{Q}(\theta)$
Find an element $\theta \in \mathbb{R}$ such that
$\mathbb{Q}(\sqrt{2},\sqrt[3]{5}) = \mathbb{Q}(\theta)$
I must find one element $\theta$ that
$$\mathbb{Q}(\sqrt{2},\sqrt[3]{5}) \subseteq \mathbb{Q}(\theta)$$
... | Take $\theta=\sqrt{2}{\sqrt[3]{5}}$
We have that $(\sqrt{2}\sqrt[3]{5})^2=2\sqrt[3]{5^2} \Rightarrow \sqrt[3]{5^2} \in \mathbb{Q}(\theta)$
Therefore $\sqrt{2}=\frac{(\sqrt{2}\sqrt[3]{5})\sqrt[3]{5^2}}{5} \in \mathbb{Q}(\theta) \Rightarrow \sqrt{2}\in \mathbb{Q}(\theta)$
Now with the same way we can prove that $\sqrt[3]... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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About $f(x) = \frac{1}{1+x^{2}}$ I realize that this function has a horizontal asymptote $y=0$. And that the range of this function is $(0, 1]$
Is the function $f: \mathbb{R} \rightarrow \mathbb{R}$ since for every $x \in \mathbb{R}$ $\exists$ a $f(x) \in \mathbb{R}$.
i.e. can I say $f: \mathbb{R} \rightarrow \mathb... | Yes, it's ok to say that. Technically, the definition of a function includes a description of its domain, so you are right to wonder. The second $\mathbb{R}$ is fine in any case, since it represents the codomain and not the range.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Product of two sequences convergence proof Suppose that a sequence $\{a_n\}$ converges to a nonzero number and a sequence $\{b_n\}$ is such that $\{a_nb_n\}$ converges. Prove that $\{b_n\}$ must also converge.
| hint: let $a_n \to A \ne 0, a_nb_n \to B$, we have: $\left|b_n - \dfrac{B}{A}\right|= \dfrac{1}{|A|}\left|Ab_n-B\right|\le \dfrac{1}{|A|}\left(\left|b_n||a_n-A| \right|+ |a_nb_n - B|\right)$. Secondly you can write: $|b_n| = \dfrac{|a_nb_n|}{|a_n|}< \dfrac{M}{\frac{|A|}{2}}= \dfrac{2M}{|A|}, n \ge N_0$. Can you finish ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solution of the PDE $yu_x+xu_y=0$ subject to the initial condition $u(x,0) = \exp \left(-\frac{x^2}{2}\right)$
Consider the following first-order PDE $$yu_x+xu_y=0$$
subject to the initial condition $$u(x,0) = \exp \left(-\frac{x^2}{2}\right)$$ Show that the above problem has
*
*a unique solution in a neighbourhood ... | Using separation of variables, let
$$u (x,y) = X (x) Y (y)$$
The PDE can then be broken into $2$ uncoupled ODEs
$$\begin{array}{rl} \dfrac{X ' (x)}{X (x)} &= \,\,\,\,\gamma \, x\\\\ \dfrac{Y ' (y)}{Y (y)} &= -\gamma \, y \end{array}$$
Thus, the general solution is of the form
$$u (x,y) = u_0 \exp \left( \frac{\gamma}{2... | {
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Inequalities involving geometry but I can't post a picture yet How do I show that
$$ \frac 12 \left(\frac 1 {3^2}+\frac 1{4^2}+ \frac 1{5^2}+\dots\right) < \frac 1 {3^2} + \frac 1{5^2} + \frac1{7^2} +\dots \quad ?$$
| After moving the odd terms from the LHS to the RHS, we obtain the following equivalent inequality,
$$\frac 12 \left(\frac 1{4^2}+ \frac 1{6^2}+ \frac 1{8^2}+\dots\right) < \left(1-\frac 12\right)\left( \frac 1 {3^2} + \frac 1{5^2} + \frac1{7^2} +\dots\right).$$
Then note that for all positive integer $n$, each term $\d... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Example of an optimum FJ point which is not a KT point The Kuhn-Tucker conditions talk about what locally optimum points in a non linear program satisfy WHEN the gradient of the active restrictions in said points are linearly independent.
However, this opens the possibility of an optimum showing up which is not a KT po... | Consider the problem
$$\min f(x)=x,\;s.t \; h_1(x)= x^2=0,\;h_2(x)=x^4=0.$$ The obvious minimum is $x=0.$ This point is not KT, however it is certainly FJ.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Find value of $\sum\limits_{k=0}^{10}{(-1)^k C(10,k)/(2^k)}$
Find value of $\sum\limits_{k=0}^{10}{(-1)^k\dbinom{10}{k}\dfrac{1}{2^k}}$
Do I have to open the factorials of all combinations, or is there any other way? please help.
| You have $$(x-1)^{10} = \sum_{k=1}^{10}C(10,k)x^k(-1)^{10-k}$$.
Note that $(-1)^{10-k} = (-1)^k$, for all $0 \leq k \leq 10$.
Taking $x=\frac{1}{2}$, you have a result.
| {
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Show that if $b>0; \mathrm{cor}(x,z)=\mathrm{cor}(x,y)$. Did I take a wrong turn somewhere? I don't know where to go from here... Can I not do the division in step 6? Can standard deviation or cor(x,y) ever be zero?
*Let x and y be jointly distributed numeric variables and let z=a+by, where a and b are constants. ... | Your step 6 is fine, because when any of the standard deviations is zero, correlation is undefined.
So you need to prove that $b\cdot\mathrm{sd}(y)=\mathrm{sd}(a+by)$.
Your next line is odd, because you have $x_i$ on both sides, but:
$$\mathrm{sd}(y)=\sqrt{\sum (y_i-\mu(y))^2}\\
\mathrm{sd}(a+by)=\sqrt{\sum (a+by_i - \... | {
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Density of $X-Y$ where $X,Y$ are independent random variables with common PDF $f(x) = e^{-x}$?
$X,Y$ are independent random variables with common PDF $f(x) = e^{-x}$ then density of $X-Y = \text{?}$
I thought of this let $ Y_1 = X + Y$, $Y_2 = \frac{X-Y}{X+Y}$, solving which gives me $X = \frac{Y_1(1 + Y_2)}{2}$, $Y ... | I already posted an answer involving no integrals of functions of more than one variable; here's another approach.
\begin{align}
\text{First assume } u >0. \text{ Then} \\
\Pr( X-Y > u) & = \int_0^\infty \left( \int_{y+u}^\infty f_{X,Y} (x,y) \, dx \right) \,dy \\[10pt]
& = \int_0^\infty \left( \int_{y+u}^\infty e^{-x}... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 4
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What is this equation formally? Having fun with the calculator, I realized that :
(a^c) and (a^b)
with c > b and c > 4 and b = 2
a^c / a^b = a^(c-2)
So, for example:
3^5 / 3^2 = 27 is same that 3^(5-2) => 27
I know it's basic, but how is this happening?
What is this formally called? I realized thinking "How often is t... | A fraction with five factors equal to $3$ in the numerator, and two factors $3$ can be simplified by the elementary rules on fractions.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to determine the $\lim_{n\to \infty} \frac{1+2^2+\ldots+n^n}{n^n}=1$. I stuck to do this,
$$\lim_{n\to \infty} \frac{1+2^2+\ldots+n^n}{n^n}=1.$$
The only thing I have observed is $$ 1\le \lim_{n\to \infty} \frac{1+2^2+\ldots+n^n}{n^n}$$ I am unable to get its upper estimate so that I can apply Sandwich's lemma.
| $$1\le \frac{1+2^2+\ldots+n^n}{n^n}\le \frac{n+n^2+\ldots+n^n}{n^n} = \frac{n\frac{n^n-1}{n-1}}{n^n} = \frac{n^{n+1}-n}{n^{n+1}-n^n}\xrightarrow{n\to\infty} 1$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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Uniform Convergence Preserves Continuity Briefly, the definitions of point-wise convergence (PWC) and uniform convergence (UC) for a sequence of functions $f_n:[a,b]\to\mathbb{R}$ in my mind are recorded as
\begin{align*}
&\text{Point Wise Convergent on $[a,b]$} \iff \\
&\forall x\in [a,b]\,\forall\epsilon\gt0\,\exist... | Your notations and proof seem great, and why the condition PWC is not sufficient is that under this you cannot choose your $\mathcal{N}(\epsilon_2)$ feasible for any $x$ in your domain. (Maybe for arbitrarily large $N$ there always exist some $x$ near $x_0$ making your argument fail.)
| {
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What methods can I use to show that $2^{50} < 3^{33}$, without a calculator How would I show that $2^{50} < 3^{33}$, without a calculator, and what different methods are there of doing this?
Any help would be much appreciated.
Thanks.
P.S sorry if the tag on this post is wrong. I wasn't sure what to put.
| Note that
$$
3^{34}=(2^3+1)^{17}=2^{51}+17\cdot 2^{48}+C>(2+\frac{17}{4})\cdot 2^{50}>3\cdot 2^{50}
$$
| {
"language": "en",
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Determine whether $A (2, 2, 3)$, $B(4, 0, 7)$, $C (6, 3, 1)$ and $D (2, −3, 11)$ are in the same plane.
a) Compute a suitable volume to determine whether $A (2, 2, 3)$, $B
(4, 0, 7)$, $C (6, 3, 1)$ and $D (2, −3, 11)$ are in the same plane.
b) Find the distance between the line $L$ through $A$, $B$ and the
line ... | Hint:) Idea is, the volume made by three vectors $u$, $v$ and $w$ is
$$u.(v\times w)$$
so with four points we can make three vectors and if this volume be zero, then these vectors and correspondence, four points lie on a plane.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Some formula related with factor of (a+b+c+d) I am looking for some math formula
For example
\begin{align}
& a^2 -b^2 = (a+b)(a-b) \\
&a^3 +b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2 - ab-bc-ca)
\end{align}
First one related with factor a+b and the second one related with factor a+b+c
then
How about some formula relat... |
$$\begin{align}
& a^2 -b^2 = (a+b)(a-b) \\
&a^3 +b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2 - ab-bc-ca)
\end{align}$$
The two relations are not quite "alike" since the second one is symmetric in $\,a,b,c\,$ (i.e. stays invariant if you permute the variables), while the first one is not (both sides change sign). Maybe a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2426135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$ 3^{2^n }- 1 $ is divisible by $ 2^{n+2} $ Prove that if n is a positive integer, then $ \ \large 3^{2^n }- 1 $ is divisible by $ \ \large 2^{n+2} $ .
Answer:
For $ n=1 \ $ we have
$ \large 3^{2^1}-1=9-1=8 \ \ an d \ \ 2^{1+2}=8 $
So the statement hold for n=1.
For $ n=2 $ we have
$ \large 3^{2^2}-1=81-1=80 \ \... | $$\begin{array}{} &3^{2^m} -1 &=x 2^{m+2} & \text{assumed and checked by } m=1 \text{with $x$ odd}\\
&3^{2^{m+1}} -1 &= 3^{2 \cdot 2^m} -1 & \text{general step in induction} \\
& &= (3^{2^m})^2 -1\\
& &= (3^{2^m} -1)(3^{2^m} +1) \\
& &=x 2^{m+2}(3^{2^m} +1) & \text{ with $x$ odd }\\
& &=x 2^{m+2}(3 \cdot 3^{2^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2426197",
"timestamp": "2023-03-29T00:00:00",
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How to Solve Non-Homogeneous Recurrence Relations : $r_n = 2\left(r_{n-1} - \binom{n-1}{2}\right) + \binom{n-1}{2}$? $$r_n = 2\left(r_{n-1} - \binom{n-1}{2}\right) + \binom{n-1}{2}$$
which is equal to $$r_n - 2r_{n-1} = -\frac{n^2-3n+2}{2}$$
This given recurrence relation is derived from the question "How many regions ... | Using generating functions
$$f(x)=\sum\limits_{n=0}^{\infty}r_nx^n=r_0+r_1x+r_2x^2+\sum\limits_{n=3}^{\infty}r_nx^n=r_0+r_1x+r_2x^2+\sum\limits_{n=3}^{\infty}\left(2r_{n-1}-\binom{n-1}{2}\right)x^n=\\
r_0+r_1x+r_2x^2+2\left(\sum\limits_{n=3}^{\infty}r_{n-1}x^n\right)-\sum\limits_{n=3}^{\infty}\binom{n-1}{2}x^n=\\
r_0+r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Extension of continuous map in topological space In the book Simmons, George F., Introduction to topology and modern analysis, page no- 98, question no- 2, the problem is : Let $X$ be a topological space and a $Y$ be metric space and $f:A\subset X\rightarrow Y$ be a continuous map. Then $f$ can be extended in at most... | To follow your idea: Suppose we have $f_1$ and $f_2$ that are both continuous extensions of $f: A \to Y$ to $\overline{A}$. Let $p \in \overline{A}$ and so we have a net $a_i, i \in (I,\le)$ from $A$ such that $a_i \to p$.
The continuity of $f_1$ implies that $f_1(a_i) \to f_1(p)$.
The continuity of $f_2$ implies that ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that $AB = 3AD$
Given that $AF=EF$ and $BE=CE$. Show that $AB=3AD$.
This question was given during my exams today and it surprised my whole class. No one knew how to start and any tips would be helpful!
| Let $X$ be a point on $BD$ such that $EX\parallel CD$. Then $$\frac{BX}{XD}=\frac{BE}{EC}=1$$ and $$\frac{AD}{DX}=\frac{AF}{FE}=1.$$ It follows that $AD=DX=XB$, so $AB=3AD$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2426610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can you apply XOR to the following expression? So I know that
$$\bar{x}y + x\bar{y} = x \oplus y$$
Can this be applied to something like
$$ \bar{a}\bar{b}cd + ab\bar{c}\bar{d} $$
to get
$$ab \oplus cd$$
| (Assuming I'm interpreting your notation correctly: $\overline{x}$ is "not $x$", etc.)
No, because the negation of $ab$ isn't $\overline{a}\overline{b}$, it's $\overline{a}+\overline{b}$. So you could write $$(\overline a+\overline b)cd+ab(\overline c+\overline d)=ab\oplus cd.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2426726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving system of equation in two variables If $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$ are the real solutions of two equation $$x^3-3xy^2=2005$$ and $$y^3-3x^2y=2004$$ then find value of
$$\frac{y_1 y_2 y_3}{2(y_1-x_1)(y_2-x_2)(y_3-x_3)}.$$
I added and subtracted the two equation to get $$(x-y)(x^2+4xy+y^2)=1$$ and $$(x... | Hint: by brute force, let $t=x/y\,$, then:
$$
\begin{align}
t^3 - 3 t = \frac{2005}{y^3} \\[5px]
-3t^2 + 1 = \frac{2004}{y^3}
\end{align}
$$
Dividing the two:
$$
\frac{t^3-3t}{-3t^2+1} = \frac{2005}{2004} \quad\iff\quad 2004 t^3 + 6015 t^2 -6012 t - 2005 = 0
$$
Now $\;\displaystyle \frac{{y_1 \cdot y_2 \cdot y_3}}{{2(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $G$ is connected if $δ(G) ≥ (v−1)/2$ Suppose $G$ is a graph has v vertices and $δ(G) ≥ (v−1)/2$. Prove that $G$ is connected.
| Take any two $A$ and $B$. If they are connected we are done. Suppose they are not.
Let $N(X)$ be a set of neighbors of vertex $X$. Then $N(X)\geq (n-1)/2 $ for each vertex $X$ by assumption. Remember we have $$|N(A)\cup N(B)| = |N(A)|+|N(B)|- |N(A)\cap N(B)|$$
If $|N(A)\cap N(B)| =0$ we have $$n-2\geq |N(A)\cup N(B)|\... | {
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What's a "deleted neighborhood"? (other than very very confusing) My text has the following definitions:
3.4.1 DEFINITION Let $x\in\mathbb{R}$ and let $\epsilon>0$. A neighborhood of $x$ (or an $\epsilon$-neighborhood of $x$)† is a set of the form $$N(x; \epsilon) = \{y\in\mathbb{R} : |x-y|<\epsilon\}.$$
3.4.2 DEFI... | If a set $N$ is a neighborhood of a point $p$, then $N-\lbrace p \rbrace$ will be a deleted neighborhood of $p$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2427061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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What is the name of this special function? Consider a special function defined as:
$$f(a_1,a_2,a_3;b_1,b_2,b_3;c;x_1,x_2,x_3,y_1,y_2,y_3,z_1,z_2,z_3)=\\
=\sum_{i_1,i_2,i_3,\\j_1,j_2,j_3,=0\\k_1,k_2,k_3}^\infty \frac{(a_1)_{i_1+i_2+i_3}(a_2)_{j_1+j_2+j_3}(a_3)_{k_1+k_2+k_3}(b_1)_{i_1+j_1+k_1}(b_2)_{i_2+j_2+k_2}(b_3)_{i_... | After some search I found that this function is actually the so called hypergeometric function of type $(4,8)$. Series and integral definitions can be found around eq. (3.24) in the book "Theory of Hypergeometric Functions" by Aomoto and Kita.
| {
"language": "en",
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Riemann Integrability and Jordan Measure It can be proven that if a function, $f : [a,b] \rightarrow \mathbb{R}$ is Riemann-integrable then its graph is measurable and has Jordan measure $0$. Is the converse, for a function true?
That is;
If for a function $f : [a,b] \rightarrow \mathbb{R}$, the set $\{(x,f(x)) | x \i... | Initially I was going to answer your question in the affirmative but I really could not find any flaw in the answers already given so I stopped for a while to figure out why I wanted to answer in the affirmative. This is what I found out.
Let us start with the following theorem
Theorem: Let $A$ be a non empty and bo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Formula for b ... b ( b ( a ) + c ) + c) + ... c 1) Let say we have a number A.
2) We multiply it by B and add C to it.
3) We repeat this action for N times.
For example for N = 3, A = 1, B = 3, C = 1, We have:
3 ( 3 ( 3 ( 1 ) + 1 ) + 1 ) + 1 ) = 40
What kind of progression do we have? Is it arithmetic or geometric?... | We want to look at the recurrence,
$$x_{n+1}=bx_{n}+c \tag{1}$$
With some initial value $x_0=a$.
We may notice that there is a number $L$ such that,
$$L=bL+c \tag{2}$$
In the case $b \neq 1$, this number is $\frac{c}{1-b}$. We shall assume that, otherwise the solution is easy.
Subtracting the second equation from the ... | {
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"source": "stackexchange",
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Multi variable function and its corresponding range? How do we find the range of
$z=f(x,y)=1/\ln(4−x^2−y^2)$
1) Replace $t=\ln(4−x^2−y^2)$ and $t\in (−\infty,\ln4)$ and there is a DNE at ln('')=0
2) Find range of $1/t$ ; How is this done?
Answers: $(−\infty,0)∪(1/\ln4,\infty)$
| We know the domain of $f(x,y)=\dfrac{1}{\ln(4-x^2-y^2)}$ is
$$\color{blue}{D_f=\{(x,y)\in\mathbb{R}^2:x^2+y^2<4~,~x^2+y^2\neq3\}}$$
and in this domain $0\leq x^2+y^2<4$ so
$$0< 4-x^2-y^2\leq4~,~(x^2+y^2\neq3)$$
the function $\ln x$ is increasing then
$$-\infty< \ln(4-x^2-y^2)\leq\ln4~,~(x^2+y^2\neq3)$$
with reciprocati... | {
"language": "en",
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Find the integral $\int_0^1 \frac{x^2 - 1}{ \ln(x)} dx$ I tried substituting $\ln (x)$ as $t$, but it led to no standard integral for me to move further.
Substituting $\ln (x)$ as $-t$ gives
$$
\int_0^\infty\frac{e^{-t}-e^{-3t}}{t}\,\mathrm{d}t
$$
| $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
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Is function $f$ continuous? $$f(x) = \begin{cases} x-1.5 & \text{ if } x<1\\\\ \dfrac{1-x}{x^2-1} & \text{ if } x>1\end{cases}$$
Is this function continuous or not? Some people say it is because the value $x=1$ is not defined for the function while others say it is not continuous because it has a hole at $x=1$. Which i... | The function, as it stands now with definitions for $x<1$ and $x>1$ and neither $x\leq1$ nor $x\geq1$, is undefined at $x=1$. Thus it is neither continuous nor discontinuous for $x=1$; it simply isn't. At all other points, it is continuous.
It is also worth noting that if you fill in the gap with $f(1)=-0.5$, then the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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The probability to obtain a $3$ is $p$, and the probability that there is at least one $3$ in four tosses is $0.9375$. Find $p$. I have this simple problem but, I have some problems with it:
A die is biased and the probability to obtain a $3$ is $p$. The probability that there is at least one $3$ in four tosses is $0.9... | $$\begin{array}{lcl} 1−(1−p)^4 & = & 0.9375 \\ (1−p)^4 & = & 1-0.9375 \\ (1−p)^4 & = & .0625 \\ 1−p & = & .0625^\frac{1}{4} \\ p & = & 1-.0625^\frac{1}{4} \\ p & = & 0.5\end{array}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A function is real iff the coefficients of complex fourier series $c_{-n}=\overline{c_n}$ Given $f$ be a $2\pi$-periodic complex-valued function which is integrable on $[−\pi, \pi]$. Write $$f(x) \sim \sum_{n=-\infty}^{\infty}c_ne^{inx}$$ and
$$\overline {f(x)} \sim \sum_{n=-\infty}^{\infty}d_ne^{inx}$$
But even if I c... | Suppose $c_{-n} = \overline{c_n}$. Then \begin{align*}
\sum_{n = -\infty}^\infty c_n \mathrm{e}^{\mathrm{i} n x} &= c_0 \mathrm{e}^{\mathrm{i} 0 x} + \sum_{n = 1}^\infty c_n \mathrm{e}^{\mathrm{i} n x} + c_{-n} \mathrm{e}^{-\mathrm{i} n x} \\
&= c_0 + \sum_{n = 1}^\infty c_n \mathrm{e}^{\mathrm{i} n x} + \overline{c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $x\in {\rm span}(S\setminus \{x\})$, then ${\rm span}(S\setminus \{x\})={\rm span}S$. Let $S$ be a set of vectors from the vector space $V$. Let ${\rm span} S$ denote the set of all linear combinations of subsets of $S$. It is understood that $S$ can be finite or infinite, depending on how "big" $V$ is. I tried to p... | As you said, it is clear that the span of $S\smallsetminus x$ is contained in that of $S$. Suppose now that we can write $x = \sum \lambda_i v_i$ where none of the $v_i$ are $x$, and take $z\in \langle S\rangle$. If $z$ is a linear combination of vectors of $S$ where $x$ does not appear, then $z$ is certainly in $\lang... | {
"language": "en",
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Prove that the golden ratio is irrational by contradiction I am struggling to see where the contradiction lies in my proof. In a previous example, $1/\phi = \phi-1$ where $\phi$ is the golden ratio $\frac{\sqrt{5} + 1}{2}$.
Since I am proving by contradiction, I started out by assuming that $ϕ$ is rational. Then, by d... | Another way:
Let´s assume that φ is rational.
a/b=φ is completely reduced (we can do this when it is rational)
b<a (by definition)
a/b = (1+ √5)/2 < ( 1 + √9) / 2 = 2 → a < 2 b → a-b < b
From 1/φ = φ - 1
→ b/a = a/b -1 = (a - b)/b
b/a is completely reduced but is equal to another fraction with both a smaller ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2428163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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"answer_id": 5
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what is the meaning of weighting in mathematics? What is the mathematical meaning of weighted by a Gaussian for numbers or vectors or Weighting by bilinear and weighted vectors?
Regards and thanks in advance!
| Let's say that you're interested in computing some stuff about data. However, you don't think that each data point should be counted the same. You can weigh the data by giving more value/strength/weight/whatever you wanna call it to some data points than others. The "weight" of a particular data point is how much it co... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Minimize function
$f$ is a density function given by $f(x|\theta)= (1/2)e^{-|x-\theta|}.$ The function $h$ is defined by $h(\theta) = \sum_{i=1}^n |x_i-\theta|.$ Show that if $n$ is even $h$ minimizes for all $[x_{n/2},x_{n/2+1}]$, and if $n$ is odd then $h$ minimizes in $x_{n/2+1}$.
I'm considering the even case, bu... | Observe that $$\frac{d}{dx}|x|=\mbox{sgn}(x),$$ where $\mbox{sgn}(x)=1$ when $x>0$, $\mbox{sgn}(x)=-1$ when $x<0$, and $\mbox{sgn}(0)=0$ by definition.
Therefore, $$h'(\theta)=\sum_{i=1}^{n}\mbox{sgn}(\theta-x_i).$$
Let $y_i$ be the sorted values of $x_i$ such that $y_1 \le y_2 \le \cdots \le y_n$. Then,
$$h'(\theta)=... | {
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How many ways to choose teams $30$ people want to play capture the flag. There are two teams. Each team has ten people. How many ways to choose the teams? I figure there are $30\choose 10$$20\choose 10$ ways to choose members for the team. I am told that this is "double counting", and the answer should be half of $30\c... | The question is not 100% clearly phrased. You could also consider your answer correct.
Say there is a red and a blue team. Then you correctly figured that there are $\binom{30}{10}\binom{20}{10}$ ways to pick 10 players for 10 red and 10 players for the blue team.
But now you can argue that swapping the team colors doe... | {
"language": "en",
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Time taken to complete a lap on a circular track Suppose I am running around a circular track of radius $1/10$ mi and I know my speed in miles per hour is given as $v(\theta)=5+3\cos \theta$. I want to know how long it takes me to travel one lap. What is wrong with the following logic...
The average speed on the inter... | When you integrated the speed function to get the average, I assume you integrated $v$ with respect to $\theta$. This is the "average of $v$ with respect to $\theta$" in a sense. To use your logic, what you would want to use is the "average with respect to time".
Think of the values of the speed function $v(\theta)$ a... | {
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Topology of subspace Suppose that $X$ is a topological space and that $Y$ is a subset of $X$.
A subset $V$ of the set $Y$ is said to be open in the space $Y$ when there exists an open subset $U$ of the space $X$ such that $V = Y \cap U$.
Let's suppose that I encounter a space $Y$ that is a subset of another space $X$.... | If $(X, \mathcal{T})$ is a topological space and $Y$ is a subset of $X$, then by default (if nothing is said), $Y$ is seen as a topological space with the subspace topology induced from $\mathcal{T}$.
This is similar to when we have a group $(G,\cdot)$, and we have $H \subseteq G$, we wonder whether $H$ is a group und... | {
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"source": "stackexchange",
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Gradient of Matrix Functions Suppose there is a matrix function $$f(w)=w^\top Rw.$$ Where $R∈ℝ^{mxm}$ is an arbitrary matrix, and $w∈ℝ^m$. The gradient of this function with respect to $w$comes out to be $Rw$.
I have looked at different formulas and none of them give me this answer. What is the procedure of solving su... | Have a look at this Wikipedia article of the Gâteaux-Derivative.
So using a small increment $ε$ and a direction $δw$ we yield
\begin{align*}f(w,εδw) &= (w+εδw)^\top R(w+εδw)\\
&= w^\top Rw + ε(δw)^\top Rw + εw^\top R(δw) + ε^2(δw)^\top R(δw)
\end{align*}
Applying the derivative w.r.t. $ε$:
\begin{align*}
\frac{\mathr... | {
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How do you prove you've found all symmetries of an object in 3D? In group theory class we studied the example of rotational symmetries of the regular tetrahedon. The teacher showed us 12 symmetries and then said "if you stare long and hard you can convince yourself that those are all symmetries". Is there a way to rigo... | A linear transformation in $\mathbb R^3$ is uniquely determined by its action on any set of $3$ linearly independent vectors, or a superset of such a set. In particular, it is determined by its action on the vertices of the tetrahedron. So it suffices to consider permutations of the vertices.
There are $4!=24$ differen... | {
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In how many ways can $k$ distinct objects be placed in $n$ distinct boxes? In how many ways can $k$ distinct objects be placed in $n$ distinct boxes? Allegedly the correct answer is $n^k$ times, I just don't know how to arrive to this answer. In the first box I believe I have $\sum_{i=0}^{k}\binom{k}{i}$ options no? Th... | Consider functions from set $A=\{1,2,3, \ldots , k\}$ to set $B=\{1,2,3, \ldots, n\}$. Think of your objects as members of set $A$ and your bins as members of set $B$. Number of ways to do this assignment will be simply the number of function from $A$ to $B$.
To count the latter, the first element $1$ has $n$ choices,... | {
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Proof verification: show that $x - a = b$ can be rewritten as $x = b + a$. I am beginning an introductory college math course to catch up from my bad high school education. This is one my first proofs.
Prove that $x - a = b$ can be rewritten as $x = b + a$.
We have been given the properties of the operations of the s... | To your second question.
When you have two terms $t_1$ and $t_2$ which are equal
$$ t_1 = t_2 $$
then since a function $f$ is well-defined we also have
$$f(t_1) = f(t_2)$$
or in other words
$$t_1 = t_2 \quad\Rightarrow \quad f(t_1) = f(t_2)$$
Your particular function is $f:\mathbb{R} \to \mathbb{R},\; x \mapsto x+a$. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Dimension of vector space of real-valued functions over $R$ I'm trying to prove that the space of real-valued functions on the closed interval $[a,b]$ where $a < b$ is an infinite dimensional vector space over $\mathbb{R}$.
$V$: The space of real-valued functions on the closed interval.
$W$: Subset of $V$ where all t... | This does not make sense to talk about a basis for $W$ since $W$ is not a subspace (not stable by sum). You don't need to introduce anything new, simply to show that $x^{n+1}$ is not in the span of $\{1,x, \dots, x^n\}$, this will shows that a basis of $V$ can't be finite.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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uniform convergence of series at endpoints I am studying the convergence of a power series, when I encounter this theorem:
Let $\sum a_nx^n$ be power series with a convergence radius of $R$, then for all $0<r<R$ the series converges uniformly on $[-r,r]$.
Moreover, if the series converges at $x=R$ then it converges ... | The answer is yes, iff the series converges at end points.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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GCD of two numbers divided by their greatest common divisor is 1 Im trying to prove that, given $a,b$ with at least one of $a,b \neq 0$,
$$
\gcd\left(\frac{a}{\gcd(a,b)},\frac{b}{\gcd(a,b)}\right)=1
$$
I have tried to prove the identity
$$
\gcd(c\cdot a, c\cdot b) = c\cdot \gcd(a,b)
$$
with $c = \dfrac{1}{\gcd(a,b)}$... | Let g be a common divisor of both a and b, and let g' be a common divisor of both a/g and b/g. Then (a / g) / g' = a / (gg') is an integer, and so is (b / g) / g' = b / (gg'). Therefore g*g' is a common divisor of a and b.
If g is the greatest common divisor of a and b, then g*g', which is also a common divisor, is no... | {
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Singular Value Decomposition on covariance matrix for multivariate normal distribution Suppose $x$ is MVN($0_n$, $I_n$), how to find $a$ and $B$ such that $a+Bx$ is MVN($\mu$, $\Sigma$)?
Here is what I try:
$a$ is easy to find: $$a = \mu$$
for B:
$$Cov(Bx) = BI_nB^T = \Sigma$$
The problem is to find matrix $B... | Let $\Sigma = UDU'$ is the SVD decomposition of a positive definite matrix $\Sigma$. Then $a = \mu$ and $B = U D^{1/2}$.
When $\Sigma$ is only semi-positive definite, then $\Sigma = UDV'$, possibly with $V \neq U$, but can still take $B = U D^{1/2}$.
Alternatively you can perform the pivoted Cholesky decomposition of... | {
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"timestamp": "2023-03-29T00:00:00",
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Solving system of equations (3 unknowns, 3 equations) So, I've been trying to solve this question but to no avail. The system of equations are as follows:
1) $x+ \frac{1}{y}=4$
2) $y+ \frac{1}{z}=1$
3) $z + \frac{1}{x}=\frac{7}{3}$
Attempt:
Using equation 1, we can rewrite it as $\frac{1}{y}=4-x \equiv y=\frac{1}{4-x}$... | Multiply (1) by $y$ to get:
$$xy + 1 = 4y \implies y = \frac{1}{4-x}$$
Multiply (2) by $z$ to get:
$$yz + 1 = z \implies z = \frac{1}{1-y}$$
Multiply (3) by $x$ to get:
$$zx + 1 = \frac{7z}{3} \implies x = \frac{3}{7-3z}$$
Plug the equation for $z$ into the equation for $x$. This will give you $x$ in terms of $y$. Now ... | {
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"timestamp": "2023-03-29T00:00:00",
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Probability that no team in a tournament wins all games or loses all games. Five teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $1/2$ probability of winning any game it plays. Find the probability that no team wins/loses all the games.
My try:
Each te... | Yes, you are right.
Moreover, the similar method can be used to solve a more general problem:
$N$ teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $\frac{1}{2}$ probability of winning any game it plays. Find the probability that no team wins/loses all ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2430133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Why is $\lim_{x\to 0^+} \Biggr(\tan^{-1}{\Big(\dfrac{b}{x}\Big)} - \tan^{-1}{\Big(\dfrac{a}{x}\Big)}\Biggr)= 0$? For $$f(x) = \tan^{-1}{\Big(\frac{b}{x}\Big)} - \tan^{-1}{\Big(\frac{a}{x}\Big)}$$
where $a$ and $b$ are differently valued constants, why is $\lim_{x\to0^+}= 0$? I understand that separately both expression... | Note that the result holds only if $ab>0$ (ie $a, b$ are of same sign). And your argument is correct. Both the terms tend to $\pi/2$ (or to $-\pi/2$) hence their difference tends to $0$.
One can not expect that a function tending to $0$ should be identically equal to $0$. For example $\lim_{x\to 0}x^{n}=0$, yet $x^{n} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2430274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Using Lagrange multiplier to find the shortest distance from the origin to a given Set An exercise of an old exam wants me to find the point with the shortest distance to the origin which is in $M=\{(x,y): x^2y=2, x>0\}$.
So I think the function I have to minimize is $f(x,y)=\sqrt{(x^2+y^2)}$ with the condition $g(x,y)... | If I were you, I will use a simpler way. Since $y=\frac{2}{x^2}$, the distance is
$d(x)^2=x^2+\frac{4}{x^4}$. Applying the AM-GM inequality, we have
$$
d(x)^2=\frac{x^2}{2}+\frac{x^2}{2}+\frac{4}{x^4}\geq 3 \sqrt[3]{\frac{x^2}{2}\cdot \frac{x^2}{2}\cdot \frac{4}{x^4}}\geq 3.
$$
The identity holds when $x^6=8$ and he... | {
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Total derivative not unique? I probably did something wrong as I get two different total derivatives, but I don't see what. I use this definition: https://en.wikipedia.org/wiki/Total_derivative#The_total_derivative_as_a_linear_map
Let $f(x, y) = (x + y, x^2 + y^2, xy)$. Then $f(\xi + h) - f(\xi) = (h_1 + h_2, 2\xi_1h_1... | Let $h=\pmatrix{\xi \\ \eta}.$
Write the increment of $f:$
$$f(x+\xi,\, y+\eta)-f(x,\,y) = \\
= \Big(x+\xi + y+\eta,\; (x+\xi)^2 + (y+\eta)^2,\; (x+\xi)(y+\eta)\Big) -(x + y,\; x^2 + y^2,\; xy) = \\
= (\xi + \eta, \; 2\xi x + \xi^2 + 2\eta y +\eta^2 , \; \xi y + \eta x + \xi\eta)= \\
=(\xi + \eta, \; 2\xi x + 2\eta y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2430472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Equation of a "tilted" sine I would like to know what's the equation of a "tilted" sine, that looks like this (no idea how to show it better).
I remember first seeing this waveform in some kind of sound synthesizer, where one of the knobs for controlling shape of the sine was doing just what im looking for - graduall... | Let's free-hand Fourier transform! First, we sketch what we want on top of an actual sine function. Draw one period. We notice that in that period, the sine function is first too low, then too high, then too low, then too high again. That means that the difference between what we have and what we want has two tops and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2430564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "54",
"answer_count": 8,
"answer_id": 7
} |
Find the coordinates of the points on $y=-(x+1)^2+4$ that have a distance of $\sqrt {14}$ to $(-1,2)$ Create a function that gives the distance between the point $(-1,2)$ the graph of $$y=-(x+1)^2+4.$$ Find the coordinates of the points on the curve that have a distance of $\sqrt {14}$ units from the point $(-1,2)$. ... | Your formula for the distance is correct. So if you want this distance to be equal to $\sqrt{14} $ then you need to solve
$$\sqrt{(x+1)^2 + \big(-(x+1)^2 + 2\big)^2}=\sqrt{14}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2430658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 3
} |
Proving that if $a,b \in \mathbb{N}^*$ then $\frac{\pi a^n}{n!} \int ^{1}_{0}x^n(1-x)^n \sin (\pi x) dx \in \mathbb{N}$ I have an exercise, whose aim is to show that $\pi^2$ is irrational by contradiction. We suppose that $\frac{a}{b} = \pi ^2$ with $a,b \in \mathbb{N} ^*$. We put $$N_n = \frac{\pi a^n}{n!} \int ^{1}_{... | You missed some crucial details (or perhaps your book author is trying to be smart by missing out on these and expecting you to generate it yourself). The same exercise is given in Apostol's Mathematical Analysis (problem 7.33 page 180) in a much better fashion and describes Ivan Niven's proof of irrationality of $\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2430857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Decide if the statement " $n^2-1$ is multiple of $4$ if and only if $n-1$ is multiple of $4$" is true or false. The statement is biconditional ( P $\Longleftrightarrow$ Q )
$n-1$ is multiple of 4 $\Longleftrightarrow n^2-1$ is multiple of 4
The statement is true if $\,$ P $\Rightarrow$ Q and Q$\Rightarrow$ P are true... | 1)$n^2-1= (n-1)(n+1)$,
Let $n$ be odd: $n+1, n-1$ are even:
$n+1= 2r$, $n-1= 2s$.
$n^2-1= 4rs$ is divisible by $ 4$.
Choose $n=7:$
$n-1 = 6$, is not divisible by $4$.
2) let $n-1$ be divisible by $4$, I.e. $n-1 =4k$.
Since $n^2-1=(n-1)(n+1)=$
$4k(n+1)$, hence
$n^2-1 $ divisible by $4$.
Recapping:
False:
1) $n^2-1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2430945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 6
} |
Fourier transform of a triangle Consider a 2-dim regular n-gon whose vertices lie on the unit circle.
Let $\chi_n$ denote the characteristic function of this polygon and
$\widehat{\chi}_n$ its Fourier transform.
The special case n = 4 lends itself particularly well to calculation.
Namely, without much loss of generalit... | For a $n$-gon $P$ whose boundary in positive orientation is $p_1 \to p_2 \to\ldots \to p_n \to p_1, \ \ p_j = (x_j,y_j)$ and indicator function $\displaystyle\chi(x,y) = 1_{(x,y) \in P}$, then the distributional derivative $\partial_x\chi$ is the distribution $$\partial_x \chi = -\sum_{j=1}^n a_j \delta_{[p_j \to p_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2431048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Is the partition function 5-adically differentiable at 1/24? Ramanujan's congruences, as extended by Watson and Atkin, show that in the $\ell$-adic metric for $\ell\in\{5,7,11\}$, the partition function is continuous at $\frac{1}{24}$, having a limit $\lim_{n\to_\ell1/24}p(n)=0$. And for $\ell\neq7$, the speed of conve... | Naively, let $n_1=4, n_2=4+4\cdot 5,n_3=4+4\cdot 5+3\cdot 5^2,\dots$ where $n_k\cdot 24\cong 1 \pmod {5^k}$ and $p(n_k)\to0\;$ $5$-adically.
Then it seems $p(n_k)/(n_k-1/24)\cong \pm1\pmod{5}$ depending on if $k\cong 0,1\pmod4$ or $k\cong2,3\pmod4$. Thus the difference quotient does not approach a limit.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2431173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving $3^n < n!$ for some $n\in \mathbb{N}$ Prove:
There exists $N\in\mathbb{N}$ such that for all $n\in\mathbb{N}$ with $n\geq N$, we have $$3^n \leq n!$$
This is what I did (I had to use a calculator).
Suppose $n\geq 7$. Then
$$n! = n(n-1)\ldots 7\cdot 6\cdot ... \cdot 1\\ \geq 7^{n-7}\cdot 7! \\ \geq 3^{n-7}\cdo... | If $n!>3^n$ thus, $(n+1)!=(n+1)n!>(n+1)3^n>3\cdot3^n=3^{n+1}$ for all $n>2$.
Thus, it remains to make a base induction and $n=7$ is valid because $7!>3^7$.
Thus, for all $n\geq7$ we have: $n!\geq3^n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2431231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Probability of getting at least one of each balls from an urn Consider an urn containing $4$ red balls, $4$ blue balls, $4$ yellow balls, and $4$ green balls. If 8 balls are randomly drawn from these 16 balls, what is the probability that it will contain at least one ball of each of the four colors?
Attempted Solution:... | OP answer and method is not correct.
still does not count correctly.
this can be easily seen if draws 4 thru 13 are done (the distribution)
(left to the reader of course)
-Inclusion-exclusion method gets it right-
I used this:
(C(16,8)^-1) * sum((-1)^k * C(4,k) * C(4*(4-k),8)) , k=0 to 3
in Wolfram Alpha
returns
1816/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2431289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
Prove $f(x)$ is quadratic if $f(2x)=4f(x)$ and $f(x)$ is increasing over positive $x$ The problem arose in the context of kinetic energy, where it can be proven from symmetry principles that $E(2v)=4E(v)$ without assuming $E=mv^2/2$ (see e.g. physics stackexchange).
While one may do further physics from this point to p... | Hint :$$f(x)=ax^2+bx+c \\f(2x)=4ax^2+2bx+c \\if \space f(2x)=4f(x)\to c=0 $$so we have
$$f(2x)=4f(x) \to 2f'(2x)=4f'(x) \\2(2a(2x)+b)=4(2ax+b) \to 8ax+2b=8ax+4b \\\implies b=0 $$so $f(x)$ is in form of $ax^2$ and finally $a>0$ because
$$f'>0 (\forall x>0) \implies f=2ax>0 \\a>0$$ now plug your physical information i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2431450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Finding the inverse elements of $\mathbb{Z}[i] = \{ a + ib | a,b \in \mathbb{Z} \}$ So I am asked to find the inverse elements of this set $\mathbb{Z}[i] = \{ a + ib | a,b \in \mathbb{Z} \}$ (I know that this is the set of Gaussian integers).
I was pretty much doing the same thing the correction suggested. Suppose $x ... | Let $c=a^2+b^2$ and $c'=a'^2+b'^2$. Then $c$ and $c'$ are both non-negative integers, and $cc'=1$. So the non-negative integer $c$ has an "inverse" (reciprocal) in $\Bbb N$
namely $c'$. What must $c$ (and $c'$) be?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2431575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Mathemmatical notation for the summation of the sets based on some other sets values. Here are the example set values.:
X = [x1,x2,x3,x4,x5]
P = [x1,x1,x2,x3,x4,x1,x2,x5,x3,x2]
Here x1,x2,x3,x4,x5 are some numeric values.
What I am trying to do is:
Adding the respective values and creating a set according to th... | This might be a job for the Kronecker delta function. If $x = y$, then $\delta_y^x = 1$, but if $x \neq y$, then $\delta_y^x = 0$.
Then, if $\mathcal L_P$ is how many elements $P$ has and $\mathcal L_X$ is how many elements $X$ has, then, given $0 < n \leq \mathcal L_X$, we have $$C_n = \sum_{k = 1}^{\mathcal L_P} \del... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2431649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Why can an element in a vector space with an infinite basis always be in the span of a finite set? I'm working on a practice problem that says the following:
Let $V$ be an $F$-vector space and $X, Y$ infinite bases for $V$. Show that for any $x \in X$ you can find a finite $Y_0 \subset Y$ with $x$ in the subspace gene... | When you have a look at the proof of the fact that every theorem has a basis you see that every element can be written as a finite sum of basis vectors. So let's have a look at the outline of the proof.
Let $P:=\{S\subset V | S \text{ is linearly independent}\}$ be the set of all linearly independent subsets with parti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2431755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
distance between 2 lines in 3d
Calculate the distance between the lines
$$L_1:x=4+2t,y=3+2t,z=3+6t$$
$$L_2: x=-2+3s ,y=3+4s ,z=5+9s$$
I tried subtraction $L_1$ from $L_2$ then multiplying the resting vector by the $t$'s and $s$'s original values and trying to find value for $t$ to or $s,$ but I found $t=\frac{19}... | $ \hat a = 4\hat i + 3\hat j+3\hat k$
$ \hat b = -2\hat i + 3\hat j+5\hat k$
$\hat t =2\hat i + 2\hat j+6\hat k$
$\hat s = 3\hat i + 4\hat j+9\hat k$
$L1: \hat a+t\hat t$
$L2: \hat b+s\hat s$
So the distance $$d = \dfrac{\left|(\hat a - \hat b).(\hat t \times \hat s)\right|}{|\hat t \times \hat s|}$$
$\hat t \times \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2431843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
$a,b,c$ are positive reals and distinct with $a^2+b^2 -ab=c^2$. Prove $(a-c)(b-c)<0$
$a,b,c$ are positive reals and distinct with $a^2+b^2 -ab=c^2$.
Show that $(a-c)(b-c)<0$.
This is a question presented in the "Olimpiadas do Ceará 1987" a math contest held in Brazil. Sorry if this a duplicate.
Given the assumptions,... | Observe triangle with sides $a,b,c$ and angles $\alpha,\beta, \gamma$.
By cosine theorem angle $\gamma = 60^{\circ}$, thus $\alpha +\beta =120^{\circ}$.
So we can assume $\alpha \leq 60^{\circ}$ and $\beta \geq 60^{\circ}$. So $b\geq c\geq a$ and thus conclusion:
$$(a-c)(b-c)\leq 0$$
with equality iff $a=b=c$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2431928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 1
} |
Integrating modular function $f(x,y)=|xy|$ in the area of the circle $x^2+y^2=a^2$ I need to solve the integral $$\iint_\omega |xy|\,dx\,dy$$ where $\omega:x^2+y^2=a^2$ I created the following integral and I have no idea how can I integrate modular function:
$$\int_{-a}^a dy\int_{-\sqrt{a^2-y^2}}^\sqrt{a^2-y^2} |xy|\,d... | You can break your integral into four parts (corresponding to four quadrants) and can integrate easily using polar coordinates.
$$\iint_\omega |xy|dxdy = \int_{r=0}^a\int_{\theta = 0}^{2\pi} f(r, \theta) \ r\ dr\ d\theta$$
$$= \int_{r=0}^a\int_{\theta = 0}^{\pi/2 } r^3 cos\theta \ sin \theta \ dr \ d\theta \ + $$
$$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2432157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
How would one calculate $\lim_{n\to\infty}n((1+\frac{1}{n})^n-e)$? What I have thought about this is:
*
*we may use L'Hopstal's rule to calculate $\lim_{n\to\infty}\frac{((1+\frac{1}{n})^n-e)}{\frac{1}{n}}$, both the numerator and denominator goes to 0 as n goes to infinity. But calculating the derivative of $(1+1/n... | hint
$$f (x)=(1+x)^\frac 1x=e^{\frac {1}{x}\ln (1+x)} $$
$$\ln (1+x)=x-\frac {x^2}{2}(1+\epsilon (x)) $$
$$f (x)=e.e^{-\frac {x}{2}(1+\epsilon (x))} $$
$$=e\Bigl (1-\frac {x}{2}(1+\epsilon (x)\Bigr)$$
The limit will be $$-\frac {e}{2} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2432249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$RP^1$ is not a regular level surface of any $C^1$ map from $RP^2$ into $R$ Since $RP^2$ is compact and connected, its continuous image in $R$ is a closed interval. Let $f$ be this map. Suppose $RP^1 = f^{-1}(c)$. If $c$ is in the interior of $[a,b]$ then $RP^2$ \ $RP^1$ is not connected. Hence $c=a$ or $c=b$. WLO... | If a submanifold $N\subset M$ is the preimage of a regular value for some smooth mapping $U\to\Bbb R^k$ ($N\subset U\subset M$, $U$ open), then the normal bundle of $N$ is $M$ must be trivial. But you can easily check that the normal bundle of $\Bbb RP^1$ in $\Bbb RP^2$ is not a trivial line bundle. (You can check this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2432420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How robust is the AM-GM inequality? Suppose we pick $\lambda$ with a constant probability distribution in the interval $[0,1]$ and $x>0$ and $y>0$, also with uniform distribution in the first quadrant up to distance $R$ from the origin (a fourth of circle). Then what's the probability that
\[
\lambda x+(1-\lambda)y\geq... | I believe your expression is correct for the case of N=2 except for the bounds of integration. When you divide by "X-Y" and don't flip the inequality you are assuming X-Y is positive or X>Y so you only need to integrate over the lower diagonal of the first quadrant i.e. theta=(0,pi/4).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2432523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to find the sufficient statistics for the shifted exponential distribution $f_{\theta, k}(y) = \theta e^{-\theta (y - k)}, y \ge k, \theta \gt 0$? How to find the sufficient statistics for the shifted exponential distribution $f_{\theta, k}(y) = \theta e^{-\theta (y - k)}, y \ge k, \theta\gt 0$?
If
a) $k$ is known
... | \begin{align}
& f(y_1,\ldots,y_n) \\[6pt]
= {} & \prod_{i=1}^n e^{-\theta(y_i-k)} 1_{\min\{y_1,\ldots,y_n\}\ge k} \\[6pt]
= {} & \exp\left( -\theta\sum_{i=1}^n (y_i-k)\right) 1_{\min\{y_1,\ldots,y_n\}\ge k}.
\end{align}
The factor $1_{\min}$ does not depend on $\theta.$ The other factor, the exponential function, depen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2432575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Is $Y\cap(X-C) = Y\cap{X} - Y\cap{C}$? Is $Y\cap({X-C}) = (Y\cap X) - (Y\cap C)$? I tries using the definitions of intersections and difference of sets but am not able to prove it.
| Yes:
\begin{align}
(Y \cap X) - (Y \cap C) &= (Y \cap X) \cap (Y \cap C)^C &\text{ (By definition of difference)}\\
&= (Y \cap X) \cap (Y^C \cup C^C) &\text{ (By de Morgan)}\\
&= (Y \cap X \cap Y^C) \cup (Y \cap X \cap C^C) &\text{ (By distributive law)}\\
&= \emptyset \cup (Y \cap X \cap C^C) \\
&= (Y \cap (X \cap C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2432788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Where is the absolute value function differentiable? I have that $f(x) = |x^2 -4x|$
What I've done is trying to define $f(x)$ with the zero values being 0 and 4. But not really sure if that's how I'm supposed to go about solving this.
| Note that
$$\lim_{x\to 0^+}\frac{f(x)-f(0)}{x-0}=\lim_{x\to 0^+}\frac{|x^2-4x|}{x}=\lim_{x\to 0^+}\frac{-(x^2-4x)}{x}=-\lim_{x\to 0^+}(x-4)=4$$
and
$$\lim_{x\to 0^-}\frac{f(x)-f(0)}{x-0}=\lim_{x\to 0^-}\frac{|x^2-4x|}{x}=\lim_{x\to 0^-}\frac{(x^2-4x)}{x}=\lim_{x\to 0^+}(x-4)=-4$$
What may we conclude about the differe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2432914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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