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I have this problem, I would appreciate any hints.
Let V be a vector space over a field $\mathbb F$. Let W be a subspace of V and let W' be a complement of W in V. Let $\alpha:W\to W'$ be a linear transformation. Show that W is isomorphic to the subspace $Y=${$w+\alpha (w)|w\in W$} of V. | Hint:
Show the linear map $W\to V$, $w\mapsto w+\alpha(w)$ is injective. |
In a binomial experiment we know that every trial is is independent and that the probability of success, $p$ is the same in every trial. This also means that the expected value of any individual trial is $p$. So if we have a sample of size $n$, by the linearity property of the expectation, the expected value of the sam... | We have a population of $N$ balls, or which $R$ are red, and extract a sample of $n$ balls (without replacement).
Suppose we line the balls in a row as we extract them. Let $X_k$ be the indicator that the $k$-th ball drawn is red. That means: $X_k=1$ if it is, $X_k=0$ if it is not. It is a Bernouli random variab... |
Let $a=4j+5k$ and $b=-5i+5j+2k$. Find $a\cdot(a\cdot b).$
And I tried to compute $(a.b).a$ as $$(0i+20j+10k)\cdot(4j+5k)=80j+50k$$ but this answer is wrong. So what wrong in my calculation? | You compute the dot product which is a scalar
$$ a \cdot b = \pmatrix{a_x \\ a_y \\ a_z} \cdot \pmatrix{b_x \\ b_y \\ b_z} = a_x b_x + a_y b_y + a_z b_z $$
and multiply with the vector
$$ ( a\cdot b) a = (a_x b_x + a_y b_y + a_z b_z) \pmatrix{a_x \\ a_y \\ a_z} = \pmatrix{a_x (a_x b_x + a_y b_y + a_z b_z)\\ a_y (a_x... |
Tomorrow, April 6, 2010, André Néron will have been dead for 25 years.
In spite of the weight of his work on abelian varieties, I've only been able to ascertain the following information:
His birth and death dates (born November 30, 1922)
In 1943 he graduated from the École Normale. He got his doctorate in 1951 and h... | There might be a little more information in an 1986 article published by the alumni magazine of ENS: google would only reveal that he was born in the small village of La Clayette and died of cancer aged 62.
Concerning the wartime hiatus: in february 1943 it was announced by french authorities that people born in eithe... |
Lets say I am given
$$m \cdot \frac{dv}{dt} = mg - \beta v^2 $$
Where $m$ is mass, $g$ is gravity, $\beta$ is unknown units, $v$ is velocity.
Let
Length - $L$
Mass - $M$
Time - $T$
So using units we write as
$$M\cdot(\frac{L}{T^2}) = M\cdot \frac{L}{T^2} - [\beta] \cdot \frac{L^2}{T^2}$$
But then I get $[\bet... | You can't add thing that don't have the same dimension so:
$$[mg]=[\beta v^2]=M \frac{L}{T^2}$$
and $$[\beta]=\frac{M}{L}$$ |
$$\sum _{n=0}^{\infty }\:2^{2n}\cdot \frac{\left(n!\right)^2}{\left(2n\right)!}x^n$$
Using ratio test we can see that radius of convergence is $R = 1$. Though I'm not sure how to find the exact function in a closed form, like $\frac{1}{1-x}$ with $\sum _{n=0}^{\infty }\:x^n$. My idea is to find some relation with othe... | You can investigate the function that your series represents as follows:
Let $$f(x) = \sum_{n\geqslant 0} \frac{(2^n n!)^2}{(2n)!}x^n $$ and define the function $$\begin{align}g(t) = \frac{f(t^2)-1}{t}, \tag 1 \end{align}$$ with $ g(0)=0 $. The power series for $g$ is easily deduced, $$ g(t)=\sum_{n \geqslant 1} \fr... |
It's a try to find a closed form to the minimum of the function:
Let $0 then define:
$$g(x)=x^{2(1-x)}+(1-x)^{2x}$$
Denotes $x_0$ the abscissa of the minimum.
Miraculously using Slater's inequality for convex function I have found that:
Define $f(x)=x^{2(1-x)}$ then:
$$\lim_{x\to x_0}\Bigg(0.5+\frac{(x-1)f'(x)-xf'(1... | Because $x_0$ the abscissa of the minimum we know that $g'(x_0)=0$ then you can calculate the function $g'(x)$:
$$g'(x) = \bigr(x^{2(1−x)} \bigl)'+\bigr((1−x)^{2x}\bigl)' = \bigr(e^{\ln{(x^{2(1−x)})}} \bigl)'+\bigr(e^{\ln{((1−x)^{2x})}}\bigl)' = \bigr(e^{{2(1−x)} \ln{(x)}} \bigl)'+\bigr(e^{{2x} \ln{(1−x)}}\bigl)' =
\b... |
Where can I find "Groups of homotopy spheres: II", the sequel to "Groups of homotopy spheres: I"? | «Sadly, the promised paper "Groups of Homotopy Spheres: II" never appeared» (wikipedia) |
In high school algebra, natural number exponents are defined as
$$\begin{aligned}a^1 &= a\\
a^{(n + 1)} &= a^n a\end{aligned}$$
With these I used induction to prove the 3 exponent laws. I also proved some ancillary theorems such as the nth power of a positive is positive, distribution of powers over fractions, and so... | Here's another approach.
Lemma If $a$ and $b$ are real numbers such that $0 < a < b$, and $n$ is a positive integer, then $0 < a^n < b^n$.
You should try to prove this by induction!
Claim If $a$ and $b$ are positive real numbers and $n$ is a positive integer such that $a^n = b^n$, then $a = b$.
Proof. Let $a$ and $... |
I'm an electrical engineer and I recently came across an unforeseen issue in my masters thesis because I lack a deeper mathematical education.
I want to know for which positive real $x$ the following inequality is an equality:
$$
n \log(1 + \tfrac{x}{n}) - \log(1 + x) \leq a\,, \;\;\;\; (\ast)
$$
i.e.
$$
n \log(1 + \... | As noted, your equation is the same as $$\left(1+\frac{x}{n}\right)^n=e^a(1+x)$$ Take a look at the graphs of $y=\left(1+\frac{x}{n}\right)^n$ and $y=e^a(1+x)$. The first is a polynomial with a single negative repeated root. The other is a line with slope $e^a$ and root at $-1$. When $n$ is even, there are two solution... |
I have the ODE:
$$x^2 y'' + x y’ + y = f(x) ~.$$ I am trying to find a way to express the solution $$u(x)=u(\text{particular})+u(\text{homogeneous})$$ to the boundary value problem $$u(e^{-\frac{\pi}{2}}) = u_1$$ and $$u(e^{\frac{\pi}{2}}) = u_2$$
I’ve found $~u(\text{homogeneous})~$ to be $A\cos{(\ln(t))} $.
But ... | Referring to Jack Crawford's comment above, there are three possible ways to get $x^4$:
$$\underbrace{1\times1\times\cdots\times 1}_{9}\times x^4\\
\underbrace{1\times1\times\cdots\times 1}_{8}\times x\times x^3\\
\underbrace{1\times1\times\cdots\times 1}_{6}\times x\times x\times x \times x$$
The first is the combinat... |
My answer is $\frac{(6x^2)^{1/4}}{ 2x^2}$
However the book's answer is $\frac{(24x^2)^{1/4}}{2x}$
Where did the book get that from?
Here's my work:
$$\sqrt[4]{\frac{3}{2x^2}}= \frac{\sqrt[4]{3}}{\sqrt[4]{2x^2}}\cdot \frac{\sqrt[4]{2x^2}}{\sqrt[4]{2x^2}}$$
$$= \frac{\sqrt[4]{6x^2}}{2x^2}$$
Since the product of tw... | I don't know how to answer this question for abelian groups, but I can answer it over some rings which have a very simple classification of all modules. In particular, suppose $k$ is a field and $X$ and $Y$ are $k$-vector spaces such that $\operatorname{Hom}(X,Z)\cong\operatorname{Hom}(Y,Z)$ (as $k$-vector spaces) for... |
I have the following two equations ($a,b,c,d,e,f,g$ are constants)
$$\frac{dx_{1}}{dt}=-a-b\sin(c-x_{2})$$
$$(d+x_{1}^{3})\frac{d^{2}x_{2}}{dt^{2}}+ex_{1}^{2}\frac{dx_{1}}{dt}\frac{dx_{2}}{dt}=f(g-x_{1}^{2})\cos (x_{2})$$
and want to integrate it numerically as an initial value problem.
Is the right way to convert ... | $2^{-52}+2^{-53} = 2\cdot2^{-53} + 2^{-53} = 3\cdot 2^{-53}.$ |
I've come across a function from the set of integer partitions to the natural numbers which I don't recognise but which probably ought to be familiar; it arises in the homogeneous Garnir relations for graded Specht modules (see Kleshchev, Mathas & Ram: "Universal graded Specht modules", Proc. LMS 105). I hope someone w... | Okay, I agree that this answer comes actually a little late...
If you type some values of your function into FindStat, you will see David Speyer's answer automatically generated, since it is obtained as a natural statistic obtained after applying a natural combinatorial map.
To have some values, for size up to $4$ it... |
I need a hint for this problem.
Let the vertices of a square ABCD represent on the Argand diagram the complex numbers a,b,c, and d respectively. A,B,C,D are taken anti-clockwise in the order named.
If $$a = 3 + i, b = 4 - 2i$$, find c and d.
For a different problem, where square was at the origin, I used the idea th... | A rotation in the complex plane around point $b$ with angle $\theta$ has the expression $z \mapsto b+(z-b)\cdot e^{i\theta}$, where $e^{i\theta}=\cos \theta+i\sin \theta$. |
Determine number of real roots of $f(x)=x^{5}+x^{3}-2x+1$.
I used Descartes's rule of sign to determine the number of positive and negative roots; there is only 1 negative (hence minimum 1 real root) and two positive roots. These 2 positive roots can be anything- a complex pair, or two real roots- but what about the o... | Descarte's rule doesn't give you the actual number of positive or negative roots. It just gives you an upper bound for them. As there are $2$ sign-changes there are either $2$ positive roots or the number of positive roots is less than $2$ by an even number ($0$ is the only possible number of positive roots in this sit... |
$Question$
Prove that there is exactly one function $f,$ continuous on the positive real axis, such that
$$f(x) = 1+ \frac 1 x \int_1^x f(t) \, dt \text{ for all } x> 0$$ and find this function.
$Attempt$
Differentiating the about equation, we get
$$f'(x) = \frac{-1}{x^2}\int_1^x f(t) \, dt + \frac 1 x f(x)$$
... | Since $f(x)=1+\frac{1}{x}\int_1^xf(t) \, dt$, therefore we can use this in the equation $f^{'}(x)=\frac{-1}{x^2}\int_1^xf(t) \, dt+\frac{1}{x}f(x)$ to get
\begin{align*}
f^{'}(x) & =\frac{x(1-f(x))}{x^2}+\frac{1}{x}f(x)\\
& =\frac{x(1-f(x))}{x^2}+\frac{1}{x}f(x)\\
&=\frac{1}{x}.
\end{align*}
So $f(x)=\ln x+C$. Now we c... |
Question: Find the factors of the expression $x^{10}+x^5+1$.
My work:
$x^5=t$ so we have: $t^2+t+1$ now we know $x^3-1=(x-1)(x^2+x+1)$ so $t^2+t+1=\dfrac{t^3-1}{t-1}$
now what do i do? | Note that $x^2+x+1$ divides $f(x)$. Well, how did I see this? This is because if $\omega$ is an imaginary cube root of unity, note that $f(\omega)=0$. And a property of $\omega$ is that $\omega^2+\omega +1=0$.
Can you take it from here? |
I want to color a subset of the edges of my graph with a different color.
My graph currently has a large number of edges, so the following is just a toy example
s = {1->2,1->3}
g = {1->2, 1->3,3->4,4->1}
GraphPlot[g,VertexLabeling->True]
Here s is a subset of g I would like to tell GraphPlot to plot the edges
in s ... | Using Graph with options EdgeStyle:
Graph[g, GraphStyle -> "VintageDiagram",
EdgeStyle -> {_ -> Directive[Arrowheads[0], Blue],
## & @@ Thread[s -> Directive[Opacity[1], Thick, Red]]}]
Same result with a combination of options EdgeStyle and EdgeShapeFunction:
EdgeStyle -> Prepend[Thread[s -> Directive[Opacity... |
I've been reading Velleman's How to Prove it and I'm having a good time with the book up to now. However, I've been really stuck in the exercice 10 (Ch 3, Sec 3.2):
Prove that for any real numbers $x$ and $y$ if $x \neq 0$, then if $y=\frac{3x^2+2y}{x^2+2}$ then $y=3$.
I've tried to prove the theorem by conditional, ... | Just simplify the equation:
If we have
$$
y = \frac{3x^2 + 2y}{x^2 + 2},
$$
then by multiplying through by $x^2 + 2$ this simplifies to
$$
yx^2 + 2y = 3x^2 + 2y.
$$
Can you finish it off from here? |
Today a student came to me with the following exercise from a course in topology:
Suppose $(X,\mathcal{T})$ is first-countable. Show that the following are equivalent:
$X$ is Hausdorff
Every convergent sequence in $X$ converges to precisely one point
$X$ is T1 and every compact subset of $X$ is closed
I was able to ... | To prove $3. \Rightarrow 2.$, consider a convergent sequence $(x_n)_{n\in\mathbb{N}}$, and let $x_\ast$ be any one of its limit points (we don't know yet that there is only one). The set
$$K = \{x_\ast\} \cup \{ x_n: n \in \mathbb{N}\}$$
is compact.
By $3.$, $K$ is closed. But any limit point of the sequence is cont... |
In Tao's Book on Analysis I, we are asked to prove that the axiom of replacement implies the axiom of specification.
This implication seems to be true even outside of the environment Tao sets up in his book, as I've seen it mentioned elsewhere.
During my own quest into the foundations-of-mathematics jungle, I've come... | The abridged version of Replacement in [1] is $\forall X (\;[\forall x\in X\exists! y (F(x,y))]\implies [\exists Z\forall x\in X\forall y( F(x,y)\implies y\in Z]\;)$ where $F(x,y)$ is a formula whose free variables, if it has any, are $x$ or $y$ or both.
We can then take the set $Z$ and apply Comprehension to get $Y=\... |
In a triangle $ABC$ with point $I$ inside it such that $\overrightarrow{IA}+2\overrightarrow{IB}+3\overrightarrow{IC}=0$. If $S$ is the area of $\triangle ABC$ and $T$ is the area of $\triangle AIC$ then prove that $S=3T$
By the given we could get:
\begin{equation}
-\overrightarrow{AI}+2\overrightarrow{IB}+3\overright... | Actually, every odd number satisfies this property, since
$$1^2=1 \equiv 1 \,[8]$$
$$3^2=9 \equiv 1 \,[8]$$
$$5^2=25 \equiv 1 \,[8]$$
$$7^2=49 \equiv 1 \,[8]$$ |
A doctor has appointments at 9 and 9:30. The amount of time each appointment lasts is exponential with mean 30 minutes. What is the expected amount of time after 9:30 until the 2nd patient has completed his appointment?
My idea here is to condition on the first appointment taking more than (and less than) 30 minutes. ... | Why all the craze with conditionings? These mainly seem like occasions of going wrong... so, let us keep things simple and let us start from the fact that you are after the mean of $$T=(t_1-30)^++t_2$$ where $t_i$ is the length of appointment $i$, $t^+=\max(t,0)$, and $t_1$ and $t_2$ are both exponential with parameter... |
I have trouble following the algebraic simplification between step 2 and 3.
How do I simplify $\frac{bH}{a\sqrt{a^2+H^2}}$ to $\frac{b}{a\sqrt{a^2H^{-2}+1}}$? | There you go
$$\frac{bH}{a\sqrt{a^2+H^2}}$$
$$=\frac{bH}{a\sqrt{(H^2)(\frac{a^2}{H^2}+1)}}$$
$$=\frac{bH}{aH\sqrt{\frac{a^2}{H^2}+1}}$$
$$=\frac{b}{a\sqrt{a^2H^{-2}+1}}$$ |
OK this is a general question. How do I determine if a relation given to me is transitive or not?
Let the following be sets $A=\{1, 2, 3\}$ and $R=\{(1, 1),(2, 2), (1, 2), (2, 1), (1, 3)\}$ then is $R$ transitive or not and why? Please explain with more examples. | R is a transitive relation if for every $a,b,c\in A$, $$(a,b),(b,c)\in R$$then $$ (a,c)\in R$$
When $A=\{1, 2, 3\}$ and $R=\{(1, 1),(2, 2), (1, 2), (2, 1), (1, 3)\}$
You have $(2,1),(1,3)\in R$, but $(2,3)\notin R$, so R is not transitive. |
If we suppose that we want $-p'(x)q(x) = f(x)$ for a given $f(x)$, and
$$q(x)p'(x)+2p(x)q'(x)=0$$
Can we get $p(x)$ and $q(x)$? | Since no answer has been submitted to complete the result.
$$
p=Cq^{-2}
$$
Then using the expression for q(x) we find
$$
p(x) = \frac{C}{\int f(x) dx}
$$
We can immediately obtain q(x). |
Convert
$$((p \wedge q) → r) \wedge (¬(p \wedge q) → r)$$
to DNF.
This is what I've already done:
$$((p \wedge q) → r) \wedge (¬(p \wedge q) → r)$$
$$(¬(p \wedge q) \vee r) \wedge ((p \wedge q) \vee r)$$
$$((¬p \vee ¬q) \vee r) \wedge ((p \wedge q) \vee r)$$
And from this point I'm not sure how to proceed. Help ... | You can continue by using Distributivity of the boolean algebra:
$((¬p \vee ¬q) \vee r) \wedge ((p \wedge q) \vee r)$
$ \Leftrightarrow (¬p \vee ¬q \vee r) \wedge ((p \wedge q) \vee r)$
Here we apply distributivity:
$ \Leftrightarrow (¬p \wedge p \wedge q) \vee (¬q \wedge p \wedge q) \vee (r \wedge p \wedge q) \vee... |
Taylor series for $\ln(\frac{1-z^2}{1+z^3})$
I've tried to $$\ln(1-z^2)-\ln(1+z^3)=\sum (-1)^{3n-1}z^{2n}-\sum(-1)^{4n-1}z^{3n}$$
I didn't manage to make it one series any help is good | Starting from @J.G.'s answer, all symplify to
$$\log\left(\frac{1-z^2}{1+z^3}\right)=\sum_{n=2}^\infty \frac{2 \cos \left(n\frac{\pi }{3}\right)-1}{n} z^n$$ |
I have a problem finding the Inverse Fourier transform of the
\begin{equation}
\dfrac{1}{k^2+a^2}\dfrac{1}{k^2}e^{i\textbf{k}\cdot \bf{r}}(\textbf{k}\cdot \bf{v}) \bf{k}
\end{equation}
where $\bf{r}$ is the radius vector, $\bf{v}$ is the velocity,$\bf{k}$ is the variable in Fourier space and $a$ is a contant.
Any sug... | Hint: If $g(x)=f(x)f(1/x)$ then $g(x)=g(1/x)$. |
I'm trying to answer this question which has a hint: think about $\mathbb Z_{3600}$.
I tried to set up a linear equation,$\mod{3600},$ without any success. Not even the factorization of $3600$ gives me any ideas on how to set the problem.
Any help? | A (different) hint: notice that
$$ \gcd(3600,x) = 9 \iff \gcd(400,x/9) = 1.$$ |
Hi everyone: Suppose we have for two power series
$$0\leq\sum_{n=0}^{\infty}a_{n}(x-\alpha)^{n}\leq \sum_{n=0}^{\infty}b_{n}(x-\alpha)^{n}$$ for all $|x-x_{0}|0$. The left hand series converges for $|x-x_{0}|r$. Can we conclude that the left hand series also converges for $|x-x_{0}| | $\frac 1{1+x^2} = \sum (-1)^{n} x^{2n}$ conveges when $r<1$ and diverges outside this radius.
$\sum (-1)^{n} x^{2n} < 1$ for all $x$ in $(-1,1)$
$\sum \frac {e x^n}{n!} = e^{x+1}$ converges for all $x$
$\sum \frac {e x^n}{n!} = e^{x+1} > 1$ for all $x\in (-1,1)$
No, your premise is not true. |
I have a plane i reprensent by its normal $n$ and a point on it $p$. I calculate $d = -n^Tp$, and i consider "point $x$ is above the plane"
if $n$ is pointing at $x$.
How to determine if a point $x$ is above or below the plane?
I thought that if $n^Tx-d>=0$ then $x$ is above the plane but that seem to not hold to a... | Let's pretend $\vec{xp}$ is a vector representing the point on the plane and $\vec{x0}$ is a vector representing the point you want to determine the location in respect to the plane.
You could simply check if $(\vec{x0}-\vec{xp})\cdot \vec{n} \geq 0$ (where $\cdot$ is dot product of vector).
If it is greater than 0, ... |
I understand that the definition of a Laplace transform of a function $f(t)$ is $$F(s)= \int_0^{+\infty} e^{-st}f(t) \,dt$$
Is there an easy way to find the Laplace transform of $$f(t)=7t \cdot \mathrm{e}^{-3t}\cdot\sin(3t)$$
I saw some Laplace transform tables online but this does not fall into one of the scenarios... | Hints: Use formal rules $\mathcal{L}(t f(t)) = -F^{\prime}(s)$, and combine it with the following table result:
$$
\mathcal{L}\left(\mathrm{e}^{-\lambda t} \sin(\mu t) \right) = \Im \int_0^\infty \mathrm{e}^{-(s + \lambda)t} \cdot \mathrm{e}^{i \mu t} \mathrm{d} t = \Im\left( \frac{1}{s+\lambda - i \mu}\right) = \fra... |
I am working on an optimization problem related to Hölder continuous functions:
A function $f:I\subset \mathbb{R}\to\mathbb{R}$ is Hölder continuous if there are constants $\alpha$ and $K$ such that $$|f(x)-f(y)|\leq K|x-y|^\alpha$$ for all $x,y\in I$.
For my algorithm to work the constant $K$ must be known in advanc... | Using the topological isomorphism $S^1 \cong \mathbb{R}/\mathbb{Z}$ and lifting the situation to $\mathbb{R}$, you are asking that, if $\tilde{G}$ is a subgroup which intersects $[0,1]$ infinitely, then is $\tilde{G}$ dense in $\mathbb{R}$?
The answer is yes, since for such $\tilde{G}$, one can always find elements $g... |
Prove the following:
$$\frac{1}{1+k}=\frac{\frac{1}{k}}{1+\frac{1}{k}}\leq \ln(1+\frac{1}{k})\leq\frac{1}{k}$$
I know I can prove it with induction if the values were naturals. However, the "problem" for me is that they're real. | Set $f:[k,k+1]\to\mathbb{R}$ given by $f(x)=\log(x)$
Then, $f$ is continuous in $[k,k+1]$ and differentiable in $(k,k+1)$. Thus, there is $\xi\in(k,k+1)$ such that $\frac{f(k+1)-f(k)}{k+1-k}=f^\prime(\xi)$.
That is $\log(k+1)-\log(x)=\frac{1}{\xi}$ for some $\xi\in(k,k+1)$. Then
$$\frac{1}{k+1}<\frac{1}{\xi}=\log(k... |
I needed to calculate $5^{5^5}$. This took about $5$ minutes or so using the standard trick of building the answer out of smaller answers. My method was this:
Calculate $5^{25}=5^{23} 5^2$, which gives $10$, using Fermat's Little Theorem
Calculate $10^{25}$, using the same trick, finding that it equals $11$
Calculate ... | Since $22|5^5-1$, by FLT $23|5^{5^5}-5^1$. |
I'm familiar with rigorous linear algebra, and I've had a very elementary course on modern algebra. I'm not interested in algebra but I need to learn more about it. Hence I'm looking for a concise and self-contained book on abstract algebra which covers what is needed for applications in the parts of mathematics releva... | Elements of Abstract Algebra by Allan Clark is the shortest book I've ever seen. This book is a little strange in that it covers Field and Galois Theory before ring theory if I remember correctly. It's also not a "hand holding" book and it expects you to do some work to read through it. This can be good or bad dependin... |
I have read definitions in my PDE book as follows: If $M$ is a smooth paracompact manifold, the space of all linear functional on $C^\infty(M)$ is denoted by $E'$ and the space of all linear functional on $C^\infty_0(M)$ is denoted by $D'$.
I have already known the topologies of these 4 spaces when $M$ is $R^n$, can y... | You can also look at Dieudonne's treatrise, chap. XVII, where it's also explained for differential forms (obviously, you're just looking for the gluing argument). |
I want to find out if these sets have a supremum and/or infimum:
$\{ \frac{|x|}{1+|x|}:x\in \mathbb{R} \}$
and
$\{ x+x^{-1}:0
Regarding the first one I would say that all elements are non-negative and since $0 \in \mathbb{R}$ it follows that $0$ is minimum as well as infimum.
But is that enough?
Regarding the sec... | If $S_1$ is your first set, then $0=\min S_1$ because $0\in S_1$ (you did not state this) and because $(\forall s\in S_1):0\leqslant s$. On the other hand, every element of $S_1$ is smallar than $1$ and therefore $1$ is an upper bound of $S_1$. In fact, $1=\sup S_1$ (and $S_1$ has no maximum) since $\lim_{x\to\infty}\f... |
In Theorem 3.1 of this paper, the following result is formulated:
Suppose $f:S_1 \to \mathbb{R}$ is a continuous function where $S_1$ is a compact subset of $S(\mathcal{V}^p(J,E))$. Then for any $\epsilon > 0$, there exists a linear functional $L \in T((E))^*$ such that for every $a \in S_1$,
$$
|f(a)-L(a)| \leq \ep... | Let $E=\mathbb{R}^d$. Each $E^{\otimes n}= (\mathbb{R}^d)^{\otimes n}$ can be endowed with the usual distance, e.g. when $a^2,b^2 \in (\mathbb{R}^d)^{\otimes 2} = \mathbb{R}^{d\times d}$, we can define $$ d_2(a^2,b^2) = \max_{1\le i,j \le d} |a^2_{i,j}-b^2_{i,j}|. $$ Then for $a = (1,a^1,a^2,\ldots),b=(1,b^1,b^2,\ldots... |
Let $f=u+i v$ be a complex function. Then the real part $u=u(x,y)$ is a multivariable real-valued function. It is well known that if $f$ is holomorphic at $z=x+iy$, then $u$ is $C^1$ near $(x,y) \in \mathbb{R}^2$.
I was wondering if $u$ is differentiable at $(x,y)$ when $f$ is complex-differentiable at a point $z=x+iy... | Let $z_0=x_0+iy_0$ and suppose that $f$ is complex differentiable at $z_0$. Let $h=h_1+ih_2 \ne 0$ and $f'(z_0)=a+ib$.
We then have that
$Q(h):= \frac{f(z_0+h)-f(z_0)-hf'(z_0)}{|h|} \to 0$ as $h \to 0$.
An easy computation shows that
$R(h):=Re (Q(h))$ is given by
$R(h)=\frac{u(x_0+h_1,y_0+h_2)-u(x_0,y_0)-(h_1a-h_... |
I have a question regarding a combinatorics problem. A college offers $8$ subjects for the undergraduate course. A student can opt for any three subjects. So there are total in total $\binom{8}{3}$ subjects combinations. However there are certain subject restrictions. A student can not opt Geography and History, Anthr... | Hint:
If $(x_1,y_1);(x_2,y_2)$ are the two vertices of hypotenuse of an isosceles right triangle
and $(h,k)$ be third vertex
$$\dfrac{k-y_1}{\cos t}=\dfrac{h-x_1}{\sin t}=r$$
where $2r^2=(x_1-x_2)^2+(y_1-y_2)^2$ where $t=\pm45^\circ$ |
Frenkel, Lepowsky, Meurman constructed the vertex operator algebra (VOA) $V^\natural$ as a chiral orbifold CFT whose target space is $\mathbb{R}^{24}/\Lambda/\mathbb{Z}_2$. (Here the last $\mathbb{Z}_2$ is a group of order two which acts by sending $\vec x\to -\vec x$, and not the ring of 2-adic integers. Sorry for usi... | I have not studied FLM in complete detail, but don't think the involution $\sigma$ is sufficiently natural to make the generation of $G$ an automatic process. That is to say, even if it does always exist as an automorphism of the VOA, it may be difficult to find without explicit combinatorial information about the lat... |
$$\eqalign{ x &\equiv 5 \mod 15\cr
x &\equiv 8 \mod 21\cr}$$
The extended Euclidean algorithm gives $x≡50 \bmod 105$.
I understand now that if we combine the two it implies $15a-21b = 3$ but I don't understand how to use the extended GCD to go from there to finding $x$ and the corresponding modulus.
Th... | Since the GCD of the moduli is $(15,21)=3$, it is necessary that $x$ be the same thing in both equations mod $3$. That is,
$$
x\equiv5\pmod{15}\implies x\equiv2\pmod{3}
$$
and
$$
x\equiv8\pmod{21}\implies x\equiv2\pmod{3}
$$
If we didn't get that $x\equiv2\pmod{3}$ from both equations, a solution would not be possible.... |
How would you simplify this?
$$\left(a^{0.5} + b^{0.2}\right)^{-0.4}$$
This isn't the normal $(a + b)^2$ or $(a+b)^3$.
So, I'm a little confused. | Just like Cornman had said, it is unwise to "simplify" the said expression $(a^(0.5)+b^(0.2))^(-0.4)$. In fact, if you define "simplify" as to condense the expression as much as possible, then it is already simplified.
Using Wolfram Alpha to test it out, this is the result. As you can see, "simplifying" (ie breaking t... |
http://en.wikipedia.org/wiki/Euclid's_theorem
I just read Euclid's proof for the existence of infinitely many primes (I have never used his proof earlier to prove this). It seems to me that he assumes it exist a finite number and he constructs the number $x=p_1 \cdot... \cdot p_n + 1$, then he says that $x$ is either ... | Theorem: Any $n\in N$ with $n>1$ is divisible by a prime. Proof: Among the divisors of $n$ that are greater than $1$ (noting that $n$ itself is a divisor of $n$), there is a LEAST one, $M.$ Now $M$ is prime because $11
Theorem (Euclid).If $p_1,...,p_k$ are primes then there is a prime $p$ not equal to any of $p_1,..... |
In a typical course (proof-based, for the context of this question) I have about 50 exercises that I might want to use when I teach the course again. Each exercise might be used in the same way as before, or perhaps in a modified form (for difficulty or dependence on course material/pacing). Sometimes, I might even hav... | I have a colleague who TeXed up a whole bunch of questions as \newcommands in one file. This way he can make his homeworks/quizzes/exams by including this source file, and quickly calling the questions. With appropriate naming of the questions, it becomes easy to look back and see what you've used and haven't used alre... |
Let $X$ be the set of all nonempty compact nowhere dense subsets of the real line.
Is there a theorem that describes the form of the elements of $X$?
Context
For open subsets of the line, such a result is well-known: every open set is the disjoint union of open intervals. But compact sets can be substantially more com... | It is the Mazurkiewicz–Sierpiński theorem combined with the fact that every compact perfect metric space contains a copy of the Cantor set. So in your case, the Mazurkiewicz–Sierpiński theorem applies and all countable, successor ordinals exhaust the homeomorphism type of non-empty compact nowhere-dense subsets of the... |
Let $B$ be a flat algebra over a Dedekind domain $A$. Let $f\in B$ be such that for every maximal ideal $\mathfrak m$ of $A$, the image of $f$ in $B/\mathfrak mB$ is not a zero divisor. How can I show that $B/fB$ is flat over $A$? (Liu, Algebraic Geometry and Arithmetic Curves, Exercise 2.12.) | Since the flatness is a local property one can assume that $A$ is local, so $A$ is a DVR. Let $\mathfrak m$ be the maximal ideal of $A$, and $a\in A$ such that $\mathfrak m=aA$.
For DVRs flatness is equivalent with torsion-free.
Then it's enough to prove that $B/fB$ is a torsion-free $A$-module. (Note that this is equi... |
Compute the indefinite integral
$$
\int \cos (2x)\cdot \ln \left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)\,dx
$$
My Attempt:
First, convert
$$
\frac{\cos x+\sin x}{\cos x-\sin x} = \frac{1+\tan x}{1-\tan x} = \tan \left(\frac{\pi}{4}+x\right)
$$
This changes the integral to
$$
\int \cos (2x)\cdot \ln \left(\tan ... | Let's recall the definition of the Inverse Hyperbolic Tangent:
$$\begin{cases}\operatorname{arctanh}(x)=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)\\\frac{d}{dx}\operatorname{arctanh}(x)=\frac{1}{1-x^2}\end{cases}; x\in(-1,1)$$
So, the integral can be rewritten as:
$$\require{cancel}\begin{align}\int\cos(2x)\log\left(\... |
It asks to prove the above statement, and I was given a hint to use the AM$\ge$GM inequality, i.e. the geometric mean is always less than or equal to the arithmetic mean: $(a_1a_2· · · a_n)^\frac{1}{n} ≤
\frac{a_1 + a_2· · · + a_n}{n}$, with equality if and only if $a_1=a_2=...=a_n$.
I tried to use induction to prove ... | We can simply apply the $AM -GM$ inequality for first n natural numbers as they are all positive
$GM ≤ AM$
Now $GM \to \sqrt[n]{\prod^n_{i=1} i} = \sqrt[n]{n!}$
$AM \to \frac{\sum_{r=1}^n r}{n} = \frac{\frac{n(n+1)}{2}}{n}= \frac{(n+1)}{2} $
Now $AM-GM$ gives $\sqrt[n]{n!} ≤ \frac{(n+1)}{2}$
But as $n≥ 2$ and as yo... |
Can you verify my proof if it is right?
Let $A$ and $B$ be sets.
(a) Show that $A$ is a subset of $B$ if and only if for any set $C$, one has $A$ union $C$ is a subset of $B$ union $C$.
(b) Show that $A$ is a subset of $B$ if and only if for any set $C$, one has $A$ intersect $C$ is a subset of $B$ intersect $C$.
Ed... | For the second let x be in A, since for any set C, AUC is subset of BUC we can take C as the complement of the single element set containing x then it follows that x must be in B |
Assume we're taking about simple graphs here.
What is the exact definition of a non-hamiltonian graph?
Is it true if I say that:
Adding one single edge to the max non-hamiltonian graph, will make it hamiltonian? | Currently I am reading How Not To Be Wrong: The Power of Mathematical Thinking (Amazon Link). That sounds like what you are looking for, and so far I'm quite enjoying it. |
Is the function $x\mapsto \sqrt{x^2}$ even or odd?
Mentioning the square root does not have negative sign,
$\sqrt{x^2} = \pm x$
As it is clear LHS is even and RHS is odd for both sign, which one is true? | The symbol $\sqrt y$, when $y>0$, means explicitly the positive square root. If you want to express "either square root" you would need to write $\pm\sqrt y$. This is why the quadratic formula is written as $$\frac{-b\pm\sqrt{b^2-4ac}}{2a}:$$ you need the $\pm$ in order for it to cover both roots.
Therefore $f(x)=\sq... |
Question is precisely the title. I've been trying to come up with an example for an hour now and haven't found one. Do such matrices, $A$ and $B$ exist? | The birthday probability can be written as:
$$\prod_{k=1}^{n-1}(1-k/365)=\frac{364!/(365-n)!}{365^{n-1}}=\frac{\binom{365}{n}}{365^{n}/n!}.$$
To use binomial coefficients, you can reason as follows. If you have $n$ people, you need to choose $n$ distinct birthdays, which can be done in $\binom{365}{n}$ ways. The tota... |
Does $\sum_{i=1}^nx_i=0$ and $\sum_{i=1}^nc_ix_i=0$ implies $c_1=\cdots=c_n$ where $c_i$ and $x_i$ are elements of a field $F$?
If so, why? | No: with either $F=\mathbb{Q}$ or $\mathbb Z/7\mathbb Z$,
$$
1+2+(-3)=0,\qquad 5\cdot1+(-1)\cdot 2+1\cdot(-3)=0,\qquad 5\neq -1\neq 1
$$ |
Suppose that $f(x)$ has a horizontal asymptote. Must it be the case that the derivative approaches $0$ as $x$ tends to infinity? I do not think so, and I think I have a counter example, but I have yet to prove it.
Of course, I know that the converse is not true (a derivative approaching $0$ need not come from a funct... | Suppose $f(x)=\frac{\sin(x^2)}{x}$. Then, surely we have $\lim_{x\to \infty}\frac{\sin(x^2)}{x}=0$.
However, the derivative $f'(x)$ is given by $f'(x)=2\cos(x^2)-\frac{\sin(x^2)}{x^2}$ and $\lim_{x\to\infty }\left(2\cos(x^2)-\frac{\sin(x^2)}{x^2}\right)$ does not even exist. |
$$\int{\frac{1}{\sqrt{3-2x-x^2}}\,dx}$$
I tried to do it by substitution with no sucess. Anyone can solve it? | The thing you should think of instantly when you see a thing like that is that completing the square is the standard method in algebra for reducing a problem involving a quadratic polynomial with a first-degree term to a problem involving a quadratic polynomial with no first-degree term.
In this case $3-2x-x^2 = 4 - (... |
I am having trouble finding the first derivative of $R(P) = (P\textrm{e}^r)(1-\frac{P}{K})$
I am told to use the product rule for this. The first part of it $(P\textrm{e}^r)$ I believe will remain the same for the derivative of it. I am struggling on the $(1-P/K)$. | So we have $|a_n-a_m|\le\sum_{k=m+1}^{n} |a_k-a_{k-1}| \leq \sum_{k = m+1}^nr^{k-1}\ell = \ell (r^m - r^n)/(1-r)$ where $\ell = |a_1 - a_0|$ is some constant. To satisfy Cauchy condition, we need for every $\epsilon > 0$ an $N$ such that if $m \geq N$, then $\ell (r^m - r^n)< \epsilon(1-r)$. We can always find such $N$... |
Let $a_n = \arctan ( \ln n )$, find $\lim_{ n \to \infty} a_n$ if any. I think we can't apply the theorem (i.e $\lim_{ n \to \infty} \arctan ( \ln n ) = \arctan (\lim_{ n \to \infty} \ln n) = \pi /2 )$. The condition of theorem is $a_n \to L$ but here $a_n$ diverges to infinity. Although, we know that $\pi / 2$ is the... | I believe you are correct and that you cannot use that theorem (which would require arctan to be continuous at infinity. You have touched on another way to do this in the end of your comment; In particular, using the sequential characterization of limits.
Since $\lim_{x \to \infty} \arctan(x) = \pi/2$, we know that $\... |
In the following notes: http://www.math.toronto.edu/mgualt/courses/MAT1300F-2016/docs/1300-2016-notes-5.pdf
during the proof of proposition 3.2, why is the $ker(Df(p))=T_pK$? | The vectors in $T_p K$ are the vectors at $p$ which are tangent to $K$. In the proof, $K$ is assumed to be (locally) a level set of the function $f$, meaning the function $f$ is constant along $K$. If you take a vector $v$ based at $p$, it represents a "directional derivative" operator on functions. Specifically, for a... |
It is easy to see that $7$ isn't the sum of the squares of two integers. However, I want to show why it isn't. Can someone show me a proof as to why $7$ is not the sum of the squares of two rational numbers. | Alternative answer: the equation $a^2+b^2=7c^2$ implies
$$a^2+b^2+c^2\equiv0\pmod4\.$$
Since squares modulo $4$ are $0$ and $1$ only, this has no solution unless $a,b,c$ are all even. But then they have a common factor (namely, $2$), which without loss of generality is not so. |
Among 100 coins one is defective: it has two heads. One chooses a coin (a good or bad one) and tosses it 10
times. It turns out that the head comes out all 10 times. What is the probability that the head comes out again
when the coin is tossed one more time? | Please note that Gambler's fallacy does not apply here, because not all coins are equal.
The formula for Bayes's theorem expresses the relationship between $\Pr(A\mid B)$ and $\Pr(B\mid A)$. In our context it can be written as
$$\Pr(\text{Defective}\mid 10H) = \frac{\Pr(10H\mid\text{Defective})\Pr(\text{Defective})}{... |
$$
2y''+y'-y=e^{3t}; \text{ with } y(0)=2,\ y'(0)=0
$$
I got to this point:
$$
L(y)=\frac{1}{(s-3)^2}\cdot\frac{1}{(2s-1)(s+1)}
$$
but now I'm not sure what to do with these polynomials. I know that the inverse Laplace of $\dfrac{1}{(s-3)^2}$ would be $\dfrac{te^{3t}}{1!}$, but because it is being multiplied by anot... | In general, when you do Laplace transforms, you need to do partial fraction decomposition to separate all the terms:
$$L(y)=\frac{1}{(s-3)^2}\cdot\frac{1}{2s-1}\cdot\frac{1}{s+1}$$
$$\frac{1}{(s-3)^2}\cdot\frac{1}{2s-1}\cdot\frac{1}{s+1}=\frac{A}{s-3}+\frac{B}{(s-3)^2}+\frac{C}{2s-1}+\frac{D}{s+1}$$
Clear the denominat... |
$ab$ is:
A. $18$
B. $36$
C. $54$
D. $72$
E. $90$
Would like to know if there is a way other than testing out each option to get the answer. | $$\begin{array}{rcl}
\dfrac1a + \dfrac1b &=& \dfrac7{18} \\
\dfrac b{ab} + \dfrac a{ab} &=& \dfrac7{18} \\
\dfrac{b+a}{ab} &=& \dfrac7{18} \\
\dfrac{a+b}{ab} &=& \dfrac7{18} \\
\dfrac{21}{ab} &=& \dfrac7{18} \\
\dfrac{ab}{21} &=& \dfrac{18}7 \\
ab &=& 54
\end{array}$$ |
Let $f:[a,b] \to \mathbb{R}$ be a continuous function and let $f(a)
$(1) f([a,b])=[f(a),f(b)]\\$
$(2)f([a,b])\supseteq [f(a),f(b)]\\ $
$(3) f([a,b])\subseteq [f(a),f(b)]\\$
$(4) f([a,b])\ne [f(a),f(b)]$
For example if I take the function $f(x)=x^2$ on $[-1,2]$ then $f([-1,2])=[0,4]$ which is not equal to $[f(-1),f(... | IVT says that $f$ take all value between $f(a)$ and $f(b)$. In other word that $$[f(a),f(b)]\subseteq f([a,b]).$$ |
I am trying to figure out whether the following statement is true.
If $f$ is such that $f^2$ is holomorphic on an open set $\Omega \subset \mathbb{C}$ then $f$ itself is holomorphic on $\Omega$.
My feeling would be that since the function mapping $z$ to $\sqrt{z}$ is multivalued that this should not be the case, but... | Pick any discontinuous $f: \Bbb C \to \{-1;1\}$.
$f^2$ is constant, hence holomorphic, but $f$ clearly isn't. |
Hello everyone,
This is my first question in math.stackexchange. I want to know how the minute hand of the clock "gains" 55 minutes in 60 minutes. I have read this article-: Wiki Answers
about how an hour hand gains 55 mins in 60 mins. But my understanding of it is that the hour hand of the clock only m... | 55 min. spaces are gained by minute hand in 60 min period.
to find how many spaces it has actually gained, we need to fix a standard point first..!!
with respect to it, we need to see the difference by how much is it actually varying..!!
so let us assume the standard point to be the place where the minutes hand and ... |
Let $K_n$ be the complete graph on $n$ vertices, where $n$ is even and greater than 2. Let $G$ be a spanning subgraph of $K_n$, and let $G'$ be its complement in $K_n$. The edges of $G$ and $G'$ can be thought of as two color classes in an edge 2-coloring of $K_n$.
We are looking for the simplest proof of the follo... | Some leading questions
For which values of $u$ is $H(u) = 1$?
For which values of $f(x) -t $ is $H(f(x) - t))= 1$?
Fixing $x$ for the moment, for which values of $t$ is $H(f(x) - t) = 1$?
The integrand on the left hand side is always either $0$ or $1$. On which interval is it $1$?
What is the length of that in... |
With the triangle
angle bisector theorem
and
Morley's trisector theorem
as background,
are there any pretty theorems known for triangle $n$-sectors,
$n > 3$?
For example, angle quadrisectors?
The images below suggest a THEOREM which I'm hesitant to believe,
but illustrates what I seek:
Q1. Do the $\tfrac{1}{4}$ ra... | Q1:
The "low" angle quadrisectors coming from $B$ and $C$ (i.e. the ones closer to $\overline{BC}$) meet on the angle bisector coming from $A$ iff $ABC$ is isosceles (with $AB = AC$).
Proof:
The difficult proof is the "only if". Let $O$ be the incenter of $ABC$, where the bisectors meet. Then the angle quadrisector... |
Give example (if exists) of an infinite abelian group having a non-cyclic finite subgroup. Please help | Just take your favourite non-cyclic finite abelian group, and take the direct product with the group of integers. |
I am curious about at what conditions the expectation and a mapping could exchange their operation.
Say, X is some random variable, and $f:R\rightarrow R$ is a continuous function. Does
$$f[E(X)] = E[f(X)]$$
hold?
What if X is absolutely continuous (i.e., the density function g(x) of X exists)? Does it make any differ... | No it doesn't. In fact, if $f$ is strictly convex (e.g. $f(x)=x^2$), and $X$ is not a constant with probability 1, then Jensen's inequality tells us that $$E[f(X)] > f(E(X))$$
One important class of functions where equality does hold is linear functions (e.g. $f(x)=ax+b$) |
some more questions about conditional probability:
You have five coins, of which two have heads on both sides, one has tails on both on both sides, and the other two are fair. You flip a randomly chosen coin. a) What is the probability that the lower side is heads? b-i) You open your eyes and see that the upper side i... | Since both $x^TA^Tb$ and $b^TAx$ are scalars, you can transpose any of them.
You can transpose $x^TA^Tb$ and get
$$x^TA^Tb + b^TAx = b^TAx + b^TAx = 2 b^TAx$$
or you can transpose $b^TAx$ and get
$$x^TA^Tb + b^TAx = x^TA^Tb + x^TA^Tb = 2 x^TA^Tb$$
and since both results are scalar, they are equal to their respective t... |
Prove that $${ a }^{ 2 }+2ab+{ b }^{ 2 }\ge 0,\quad\text{for all }a,b\in \mathbb R $$
without using $(a+b)^{2}$.
My teacher challenged me to solve this question from any where. He said you can't solve it. I hope you can help me to solve it. | Without loss of generality suppose $0<-b
Multiplying a positive number by the positive scale factor $r=\frac a c>1$ has more effect on large positive numbers than on small positive numbers. Thus
$$ra-a>rc-c$$
in other words
$$a^2/c-a>a-c$$
multiplying by $c$ yields
$$a^2-ac>ac-c^2$$
or
$$a^2+2ab+b^2>0$$
(the in... |
Is there a way to resolve probability of an event, given another event that never happens? Mathematically speaking the problem is:
Given that $P(B) = 0$,
$$P(A|B)=\frac{P(A \cap B)}{P(B)} = \frac{0}{0}$$
Is this probability vacuously $0$ of $1$? Can we show that it's one or the other? | The comment by Dilip Sarwate points to conditioning on the level of densities which can be interpreted as conditioning on a family of events of probability zero. It is a probabilistic version of Radon-Nikodym derivative.
One can also condition on an individual event of probability zero, if that event admits a natural... |
Let $G$ be finite group of order $8$ of the form: $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$.
The elements are $\left\lbrace 1, a, a^2, a^3, b, ab, a^2b, a^3b\right\rbrace$.
Question 1: Show that the subgroup $H=\left\langle a^2 \right\rangle$ is a normal subgroup of $G$
$H$ is a normal su... | Since
$$\Bbb Z=\langle 2,3\rangle\implies \Bbb Z/5\Bbb Z=\langle 2+5\Bbb Z,\,3+5\Bbb Z\rangle$$
Everything, of course, additive. Of course, you don't need two generators of either group, but you can use two generators. |
Here is why I think the limit does not exists.
link to image if mathjax doesn't work
$\lim \limits_{x \to ∞ } \frac{3|x|+2x}{7|x|-5x}$ == $\lim \limits_{1/h \to 0 } \frac{3|1/h|+2(1/h)}{7|1/h|-5(1/h)}$
$\lim \limits_{1/h \to 0^+ } \frac{3|1/h|+2(1/h)}{7|1/h|-5(1/h)}$ == $\lim \limits_{1/h \to 0^+ } \frac{3+2}{7-5}$ ... | $$\underset{x\to +\infty }{\text{lim}}\frac{3 \left| x\right| +2 x}{7 \left| x\right| -5 x}=\underset{x\to +\infty }{\text{lim}}\frac{3 x +2 x}{7x -5 x}=\underset{x\to +\infty }{\text{lim}}\frac{5x}{2x}=\frac{5}{2}$$
$$\underset{x\to -\infty }{\text{lim}}\frac{3 \left| x\right| +2 x}{7 \left| x\right| -5 x}=\underset{x... |
In reference to method shown at 4:00 of this video,
Consider,
$$f(x) = x^x$$
Then,
$$ f(x+h) = (x+h)^{x+h} = x^{x+h} ( 1 + \frac{h}{x})^{x+h}$$
Now, consider right most term in brackets,
$$ ( 1 + \frac{h}{x})^{x+h} = 1+ (x+h) \frac{h}{x} + \frac{ (x+h)(x+h-1)}{2!} ( \frac{h}{x})^2...= 1+h+ stuff$$
This suggests ... | Hint:
$$x^2y''-7xy'+7y=13\sqrt{x}$$
This is Cauchy-Euler's differential equation:
$$y''-7\dfrac {(xy'-y)}{x^2}=\dfrac {13}{x\sqrt{x}}$$
$$y''-7\left (\dfrac {y}{x}\right)'=\dfrac {13}{x\sqrt{x}}$$
Integrate:
$$y'-\dfrac {7y}{x}=13\int \dfrac { \ dx}{x\sqrt{x}}$$
$$y'-\dfrac {7y}{x}=- \dfrac {26 }{\sqrt{x}}+C_1$$ |
If i suppose that $\overline{A}$ is not connected then $\begin{cases} \overline{A}=U\cup V\\ U\neq\emptyset, V\neq\emptyset\\U\cap V=\emptyset\end{cases}$ where $U,V$ are open in $\overline{A}$
We know that $A$ is connected and $A\subset \overline{A}$ then $A\subset U$ or $A\subset V$
How to continue?
Thank you | Writing form $\left|\dfrac{f(x)}{g(x)}\right|$ makes impossible to consider function $g$, for which point $a$(from $x\rightarrow a$) is a limit point for $g$'s zeros.
The big majority of books, where I saw big-O, contains 1'st definition from wikipedia, not last with limit superior.
If somebody want to create unify d... |
$$a_n=\frac{4-e^n}{6+5e^{n+1}}$$
As $a_n$ approaches infinity
Since there is euler number in here, it really confuses me... | Seems fine, but you got a typo at the last part. It should be $I(b) = \pi/4$ not $\pi/2$. Also we could do this quicker:
$$
I = \int_0^{\pi/2} \frac {\mathrm dx} {1+\tan(x)^\pi} = \int_0^{\pi/2} \frac {\mathrm dx} {1 + \cot(x)^\pi} = \int_0^{\pi/2} \frac {\tan(x)^\pi \mathrm dx}{1+ \tan(x)^\pi} \implies 2I = \frac \pi ... |
How can I graph $x + y + z = 1$ without using graphing devices?
I equal $z = 0$ to find the graph on the xy plane. So I got a line, $y = 1-x$
But when I equal 0 for either the $x$ or the $y,$ I get $z = 1-y$ or $z = 1-x$, and those are two different lines from different angles. Different graphing websites were telling... | In 2D, a graph
$$\frac xa +\frac yb=1$$ is a straight line connecting $(a,0)$ and $(0, b)$.
This can be confirmed easily by substitution.
Similarly, in 3D,
$$\frac xa +\frac yb+\frac zc=1$$
is a plane which goes through $(a,0,0), (0, b, 0), (0,0,c)$.
The question above refers to the case where $a=b=c=1$. |
I am stuck on this problem:
If the lines $y=x+\sqrt{2}$ and $y=x-2\sqrt{2}$ are two tangents of a
circle and $(0,\sqrt{2})$ lies on this circle then what is the equation of the circle?
I found out the distance between the two tangents $y=x+\sqrt{2}$ and $y=x-2\sqrt{2}$ is $3$. The radius is $3/2$ but I don't kno... | Assume the center to be (a,b). Write the general equation of the circle $(x-a)^2+(y-b)^2=(3/2)^2$ and substitute the point $(0, \sqrt{2})$. This is the first equation.
It is also true that the point (a,b) lies on the the line given by $y = x - \dfrac{\sqrt{2}}{2}$. Substitute for 'a' and 'b'. This is the second equati... |
I don't know what the right terminology is so I am using some programming jargon, sorry.
But for example can we say that a function accepts reals and returns naturals? Or accepts complex and returns reals? More generally accepting one type and returning another?
For example $f(x) = \sqrt{x}$ defined as having domain ... | Yes and no.
Yes, a function can map from any set $A$ to any set $B$ ($f:A\to B$). The sets don't have to have any intrinsic relations to each other. So we can easily have $f: \mathbb R\to \mathbb N$ or from $g:\mathbb N \to \mathbb R$ or $h:\{$mathetical symbols$\} \to \{$elephants whose middle name is Fred$\}$. (E... |
This is what I have tried so far. I am just not sure whether I got it right or not.
We have $y^2+z^2=1$, then $r=1,\quad x=\sqrt{2-y^2-z^2}\quad\text{and}\quad x=\sqrt{2-r^2}$
$$\frac{\partial{x}}{\partial{y}}=-\frac{y}{\sqrt{2-y^2-z^2}}\quad\text{and}\quad
\frac{\partial{x}}{\partial{z}}=-\frac{z}{\sqrt{2-y^2-z^2}}
... | The calculus part has already been answered. Your calculations are correct, and you have been offered a different way of arriving to the solution using spherical coordinates. Another way of checking your final answer is using a "visual" aid and the formula for spherical caps.
By visual I mean that I actually construct... |
I have two lists:
u = {{1, 3}, {2, 6}, {3, 9}}
v = {0, 4}
and I want to obtain this list from them:
z = {{{0, 1}, {4, 3}}, {{0, 2}, {4, 6}}, {{0, 3},{4, 9}}}
I guess the solution will make use of Map, Thread, or even MapThread but I've tried every combinaison I can think of with no luck. How can I do it? | Another possibility using Outer:
Outer[Composition[Transpose, List], {v}, u, 1][[1]]
(* {{{0, 1}, {4, 3}}, {{0, 2}, {4, 6}}, {{0, 3}, {4, 9}}} *)
I also like using Riffle for this:
Riffle[v, #] ~Partition~ 2 & /@ u |
Use Lagrange Multipliers to find the absolute extrema (if any) of:
$f(x,y) = 4x^2 + 9y^2$; subject to $2x +3y = 6$.
Using Lagrange I end up with one point: $(\frac{3}{2}, 1)$
I'm just not sure how to show if that point is a max or a min? | Another way to check the nature of the extremum in this situation is to substitute the constraint into the function. Thus, with $ \ 3y \ = \ 6 \ - \ 2x \ $, we have
$$ f(x,y) \ = \ 4x^2 \ + \ 9y^2 $$
$$ \rightarrow \ \ \phi(x) \ = \ 4x^2 \ + \ (6 \ - \ 2x)^2 \ = \ \ 4x^2 \ + \ 36 \ - \ 24x \ + \ 4x^2 \ = \ 4 \ ( \ 2... |
I want to prove whether the sequence $\{a_n\} = \left \{ \dfrac{n}{n+1}\sin\dfrac{n\pi}{2} \right \}$ (defined for all positive integers $n$) is divergent or convergent.
I suspect that it diverges, because the $\sin\frac{n\pi}{2}$ factor oscillates between -1, 0 and 1.
Here is my attempt to prove that it is divergent... | Here is another approach:
show that there exists infinitely many terms within $\delta$ neighbourhood of 0,1 and -1.
in other words 0,1 and -1 are all limits so there is no single limit for this sequence. |
This question looked pretty easy at the start, but I did it in a different method as to what the Answerbook has done. And I couldnt get it right. And the answer in the book gave the correct value but Why isn't mine giving the right answer?
Here is the method used by the answerbook:
Firstly, I know this method even th... | You haven't applied the length of the curve formula correctly...
Length of the arc is given by
$\int _{l1} ^{l2} \sqrt {1+(dy/dx)^2} dx$ |
So I have been working on a homework assignment and I'm just beyond stuck and can't seem to figure out where to start. We are suppose to prove some proofs with either direct proof, proof by contrapositive, proof by contradiciton, or a proof by cases.
The statement I've been trying to prove is
If a group of 8 kids hav... | That's not correct. What you should do is to assume that none of them won more that $8$ trophies and to deduce that the $8$ kids together cannot have won $65$ trophies, |
If $|G|=2n$ where $n$ is odd and $H$ is a subgroup of order $n$. Then $\underset{g\in G}{\prod}g\notin H$
pf: Since $G$ has even order, it must have a element of order $2$, then $\underset{g\in G}{\prod}g$ has even order. So $\underset{g\in G}{\prod}g\notin H$ as the order of everything in $H$ must divide $n$. Is this... | That answer is not a valid proof. The product of the elements does not need to have the order related to one element in the product. For example, you can multiply two elements with order 2 to get an element of order 3 or 5 or any number.
Hint: This fact I think must have come up before this exercise: if a subgroup H t... |
I have been using a licenced mathematica, provided by my research institute, for a while. Weeks ago it happened to me that at the startup of my mathematica (while initializing Kernel), mathematica froze and it couldn't proceed. Then, I found a useful post instructing how to get rid of it and solve the issue by deleting... | This has happened to me twice now with Mathematica 11.3, and unfortunately, the second time I could not recall the solution and went into the "uninstall-reboot-reinstall" rinse cycle for several frustrating hours.
The clue is in these error messages that flashed too-fast-to-read in the Messages window when in desperat... |
I have always wondered how the digits of π are calculated. How do they do it?
Thanks. | From one of my favorite mathematicians,
$$
\frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{k \ge 0} \frac{(4n)!(1103+26390n)}{(n!)^4396^{4n}}.
$$
Looking here, we see some many formulae. Some of particular interest are:
$$
\pi=\sum_{k \ge 0}\frac{3^k-1}{4^k}\zeta(k+1),\quad \zeta(s)=\sum_{k\ge 0}k^{-s}.
$$
$$
\frac{1}{6}... |
Throughout this post I'm assuming that the cone is pointing "up" in the $Z$ axis.
It's known that the intersection of a plane and cone is an ellipse. But what about the projection of this ellipse onto the $X,Y$ plane - is it always a circle? Or equivalently, does every intersection of a plane and cone lie in some cyli... | The integral
$$f(a) = \int_0^\infty\frac{\cos(a x)}{1+x^N}\;dx$$
with $N= 1, 2,...$ and $a\ge 0$ can be calculated explicitly in terms of known functions as follows.
Partial fraction decomposition leads to the representation
$$\frac{1}{1+ x^N} = \sum_{k=1}^{N} \frac{c_k}{z_k - x}\tag{1}$$
where
$$z_k=\exp\left( \... |
I would like to understand how to remove the root squares from this expression:
$$x = \frac 1{\sqrt{2} + \sqrt{3} + \sqrt{5}}$$
How to do it? | Its all about rationalization,
\begin{align}
\frac 1{\sqrt{2} + \sqrt{3} + \sqrt{5}} &= \frac 1{\sqrt{2} + \sqrt{3} + \sqrt{5}}
\cdot \frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{\sqrt{2} + \sqrt{3} - \sqrt{5}}
\\[10pt] &=\frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{(\sqrt{2}+\sqrt{3})^2-(\sqrt{5})^2}
\\[10pt] &=\frac{\sqrt{2} + \s... |
I have this idea for a graph but don't know what function could describe it better.
The idea is something like the "squared" function turned $90$ degrees to the right, so that possible values for $x$ are always positive and $y$ may be both positive and negative.
The graphs of $\sqrt x$ and $-\sqrt x$ combined look go... | If you want the square function
$$ y = x^2 $$
turned 90 degrees to the right, you get the inverse
$$ x = y^2 $$
or written in another way (like you already mentioned)
$$ y = \pm \sqrt{x}. $$
I'm not actually sure what you mean by writing it as a single function as it has two branches (the positive and the negative one... |
What is the number of solution for this equation?
$2x + 3y + z = 10 $?
(For non-negative integers) | L = {14}. There are 14 ways for non-negative integers to solve 2x+3y+z=10.
We are explicetely given non-negative numbers, so let us think through the $2^3=8$ ways to have our three variables (x,y,z) equal zero or not.
These 8 arrangements for (x,y,z) are (where "." means 0 and "-" means integer):
of the form (...) ... |
The question is the title of a 2013 publication in the Notices of the American Mathematical Society, by twelve authors (of which I am one). The contention is that traditional history of mathematics is based on the assumption of an inevitable evolution toward the real continuum-based framework as developed by Cantor, D... | (This is meant as a response to a comment by Pete L. Clark on whether the history of analysis was a "linear progression". Due to its length I decided to post it as a separate answer) I agree that focusing on the term "linear" is not the issue. What does seem to be a meaningful issue is the following closely related que... |
The study of matrix quantum group coactions on the noncommutative disk algebra turns up the following series, which is a $q$-deformation of the negative binomial series, for integer $t\ge 0$, complex $z$ and $q\in[0,1]$:
$$\sum_{n\ge 0}(-1)^n q^{n(n-1)/2}\frac{[n+t]_q!}{[t]_q![n]_q!}\,z^n =
\sum_{n\ge 0} (-1)^n q^{n(... | "Basic hypergeometric series are series $\sum c_n$,
with $c_{n+1}/c_n$ a rational function of $q^n$, for a fixed parameter $q$,
which is usually taken to satisfy $|q|<1$, but at other times is a power of a prime."
This quote is from the Forward (by Richard Askey) to "Basic Hypergeometric Series", by Gaspar and Rahman... |
The identities
$1+2=3$
$4+5+6=7+8$
$3^2+4^2=5^2$
$10^2+11^2+12^2=13^2+14^2$
$21^2+22^2+23^2+24^2=25^2+26^2+27^2$
are examples of consecutive identities. Are there any consecutive identities except $3^3+4^3+5^3=6^3$ for exponents greater than $2?$
The identities should be built upon additions only.
For consecuti... | $$n^2+(n+1)^2+...+(n+k)^2=(n+k+1)^2+...+(n+2k)^2$$
Solutions are easy. $n=(2k+1)k$ |
End of preview. Expand in Data Studio
Math SFT Solutions No CoT V3
Math SFT Solutions No CoT V3 is a large-scale mathematics supervised fine-tuning (SFT) dataset designed for instruction tuning and mathematical capability adaptation.
Version 3 substantially expands mathematical coverage while improving dataset quality through stronger filtering, cleaning, and supervision refinement.
Unlike reasoning-heavy datasets, this release focuses on clean instruction → response pairs without hidden chain-of-thought style supervision.
Dataset Summary
This dataset provides concise mathematical supervision for training and adapting language models.
Dataset format:
{
"instruction": "...",
"response": "..."
}
Features
- Broad mathematical domain coverage
- Instruction → response format
- No hidden reasoning traces
- Deduplicated and cleaned
- Synthetic augmentation included
- Optimized for supervised fine-tuning
- Suitable for mathematical adaptation
What's New in Version 3
Expanded Mathematical Coverage
Version 3 significantly increases mathematical diversity and domain coverage.
Included domains:
- Arithmetic
- Pre-Algebra
- Algebra
- Geometry
- Trigonometry
- Calculus
- Number Theory
- Probability
- Statistics
- Combinatorics
- Discrete Mathematics
- Symbolic Manipulation
- Competition Mathematics
- Multi-step Problem Solving
- Mixed Difficulty Mathematical Reasoning
The objective is broader mathematical supervision and improved generalization.
Cleaner Supervision Targets
Version 3 continues the cleanup introduced in Version 2.
Processing improvements include:
- removal of reasoning artifacts
- removal of thinking blocks
- response normalization
- formatting cleanup
- duplicate removal
- near-duplicate filtering
- quality-oriented preprocessing
Responses contain only intended supervised targets.
This dataset is not intended for hidden chain-of-thought supervision.
Increased Diversity
Version 3 expands beyond earlier GSM8K-style and MATH-style distributions.
Data construction emphasizes:
- wider mathematical structures
- varied instruction styles
- broader solution distributions
- multiple difficulty ranges
Benchmark improvements are not guaranteed and depend on training setup.
Dataset Structure
Data Fields
| Field | Type | Description |
|---|---|---|
| instruction | string | Mathematical problem or instruction |
| response | string | Expected target solution |
Example
{
"instruction": "Solve for x: 3x + 9 = 24",
"response": "Subtract 9 and divide by 3. Final answer: x = 5."
}
Data Sources
This dataset contains transformed and augmented mathematical examples derived from:
- GSM8K-style arithmetic tasks
- MATH-style mathematics tasks
- synthetic mathematical augmentation pipelines
- transformed instruction–response datasets
- expanded multi-domain mathematical supervision
Coverage includes:
- Arithmetic
- Algebra
- Geometry
- Calculus
- Number Theory
- Combinatorics
- Symbolic Manipulation
- Mathematical Reasoning
Intended Uses
Recommended for:
- Supervised Fine-Tuning (SFT)
- Instruction Tuning
- Mathematical Adaptation
- LoRA
- QLoRA
- Response Generation
- Small Model Specialization
Compatible with:
- Qwen
- Llama
- Gemma
- SmolLM
- Mistral
- other decoder-only language models
Limitations
- contains synthetic augmentation
- mathematical correctness is not guaranteed for every sample
- not intended for theorem verification
- not intended for hidden chain-of-thought training
- benchmark performance depends on training setup
- may contain residual distribution artifacts
Version History
V1
Initial release.
Contained intermediate reasoning-format contamination.
V2
Introduced:
- removal of thinking contamination
- cleaner supervision targets
- response augmentation
V3
Introduced:
- expanded mathematical coverage
- stronger preprocessing
- broader supervision
- improved filtering
- cleaner instruction tuning targets
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