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How can I find nicely behaving functions which are easy to compute and which fit well to the upper limits of the (2-logarithm) of the nCr function?
What I am interested in in practice is to be able to model "how many binary digits (bits) will a digital representation of nCr(a,b)" have? A guess of 1 or 2 too many bit... |
How can I find numerically nice-to-compute upper limits of nCr? |
How can I find nicely behaving functions which are easy to compute and which fit well to the upper limits of the (2-logarithm) of the nCr function?
What I am interested in in practice is to be able to model "how many binary digits (bits) will a digital representation of nCr(a,b)" have? A guess of 1 or 2 too many bit... |
How can I find wedge product of tensors? |
For $k$-cycle $(1, 2, ..., k)$, do the elements in it need to be taken in order?
Is $(3,1,5,2,4)$ a $5$-cycle? Or the elements in the rotation must satisfy a law of order? i.e. $5$-cycle just are $(1,2,3,4,5)$ or $(5,4,3,2,1)$.
For $k$-cycle, do the elements in them need to be in order? |
For $k$-cycle $(1, 2, ..., k)$, do the elements in it need to be taken in order? |
We know that an additive full subcategory $\mathcal{S}$ of a triangulated category $\mathcal{T}$ is called a triangulated subcategory if it is closed under isomorphism, shift and if any two objects in a distinguished triangle in $\mathcal{T}$ are in $\mathcal{S}$, then so is the third. See Neeman's book section 1.5. Wh... |
There is the following problem in my math book, but the author did not include a solution.
>Let $a_i$ and $b_{ij}$ be expressions dependent on i and j. Prove that: $$\sum_{i=1}^n a_i \sum_{j=1}^m b_{ij} = \sum_{i=1}^n \sum_{j=1}^m a_ib_{ij}$$
I tried doing this the following way:
$$\sum_{i=1}^n a_i \sum_{j=1}... |
Is this a correct proof for the following summation problem? |
Note that for a manifold $X$, isomorphism classes of principal $SO(2)$-bundles over $X$ are classified by their Euler classes in $H^2(Z;\Bbb Z)$. Now consider a lens space $L(p,q)$; we have $H^2(L(p,q);\Bbb Z)=\Bbb Z_p$. Suppose there is a principal $SO(2)$-bundle (or equivalently a principal $S^1$-bundle) $P\to L(p,q)... |
I know that this is a very general fact about limits and colimits, but I would like to prove it directly for ends and coends. If $\mathcal V$ is a closed braided monoidal category, $V$ an object in $\mathcal V$ and $P\colon\mathcal M^\mathsf{op}\otimes\mathcal M\to \mathcal V$ is a $\mathcal V$-functor, I can define it... |
I'm reading The Princeton Companion to Mathematics, and in it I see this passage:
> If S is a geometrical shape, then a rigid motion of S is a way of moving S in such a way that the distances between the points of S are not changed—squeezing and stretching are not allowed. A rigid motion is a sym- metry of S if, aft... |
What is a symmetry? |
For question 1., assuming that $g$ is extended to the powerset of $C$ via
$$\tilde{g}(A)=\bigcup_{c\in A}g(c),$$
the answer is yes - this function will preserve all operations. This is most easily seen by noting that this $\tilde{g}$ is in fact *almost* a preimage function in disguise - namely of the partial function... |
Let $X$ be some normed space with norm $||\cdot||_X$. Additionally, consider a continuous increasing function $\eta: [0,\infty) \to (0,\infty)$ and set $\psi(x) := \eta(||x||_X)$. We refer to $\psi$ as the "weight function". Equipped with $X$ and $\psi$ we may define the following vector space
$$F_\psi(X):= \left\{ f:... |
For a fixed $p \in (1,\infty)$, consider $L^p(\Omega)$ for a bounded domain $\Omega$ in the Euclidean space and a sequence $\{ f_{n} \} \subset L^p(\Omega)$ converging weakly. That is, for each $g \in L^{p'}(\Omega)$ with $p^{-1} + p'^{-1}=1$, the limit
\begin{equation}
\lim\limits_{n \to \infty} \int_\Omega f_n g
... |
First I made this question for myself just out of curiosity. But got stuck in the middle of solving it.
**The question:**
Say we have a probability density function of a continuous stochastic variable X as
$$f_X(x)=\begin{cases}
\frac{2}{\pi^2}x & 0\leq x\leq\pi \\
0 & \text{otherwise}
\end{cases... |
I've been trying to fit some data with polynomials using a pseudo-inversion library (numpy.linalg). There are two real functions $f$ and $g$ such that $f=-\alpha g$, $\alpha > 0$, and
$$
f=f(t_1,\ldots,t_5),
$$
$$
g = g(t_1,\ldots,t_5),
$$
where $t_1,\ldots,t_5$ are real parameters. The parameter $\alpha$ ha... |
I got stuck when reading Apendix II of *G-Induced Flows* by L. Auslander and L. Green and hopefully someone could help me out. The article can be found at https://www.jstor.org/stable/2373046 and let me describe the setting of my problem.
Let $S_2$ a three dimensional sovable Lie group whose Lie algebra has a basis ... |
A potential problem of your proof: the meaning of "$\cdots$" is *intuitively* obvious but *formally* unclear.
How rigurous is your book?
How is $\sum$ defined in your book? With "$\cdots$"? Maybe using induction?
My suggestion: start proving
$$a\sum_{j=1}^m b_{ij} = \sum_{j=1}^m a b_{ij}.$$ |
Let $f : (-\pi, \pi)\setminus\{0\} \longrightarrow \mathbb{R}$ be $2\pi$-periodic and defined by
$$f(x) = -\log\left(\left|2\sin\left(\frac{x}{2}\right)\right|\right).$$
I have to find the Fourier coefficients of $f$.
There is a suggestion: it can be useful to prove that the integral
$$I_k = \int_0^\pi \cot\left(... |
How to deal with this 3-dimensional global optimization problem with singular points? |
I know that this is a very general fact about limits and colimits, but I would like to prove it directly for ends and coends. If $\mathcal V$ is a closed braided monoidal category, $V$ an object in $\mathcal V$ and $P\colon\mathcal M^\mathsf{op}\otimes\mathcal M\to \mathcal V$ is a $\mathcal V$-functor, I can define it... |
let $A = ( a+bi: \in C : a > 4)$ and $B = ( a+bi: \in C : a >= 4)$
and let $d(z,w)$ be defined as
$$ 0 ,z =w $$
$$ |z| +|w| , z\neq w$$
my questions are:
1) Is A and b open under (C,de) ? where de is the usual metric under C
2) Is A and B open under (C,d(z,w))?
my solution:
1) from my understanding, A i... |
Is it possible to prove the measurability of the following map
$\Phi_n \colon (C(\mathbb{R}^d), \sigma(\mathcal{C})) \to (\mathbb{R}, \mathcal{B}(\mathbb{R}))$, $n \in \mathbb{N}$, defined by
$$ \Phi_n(f) := \int_{[-n,n]^d} |f(x)|^2 \, dx, \quad f \in C(\mathbb{R}^d), $$
where $\mathcal{C}$ is the collection of fin... |
Measurability of an integral operator?! |
As far as teaching the calculus is concerned, infinitesimals are useful in explaining concepts such as derivative, integral, and even limit. That's why Kathleen Sullivan's controlled study of infinitesimal and epsilontic methodologies in the 1970s revealed that students taught using infinitesimals possess better concep... |
$\require{AMScd}\newcommand{\op}{{^\mathsf{op}}}\newcommand{\M}{\mathcal{M}}\newcommand{\V}{\mathscr{V}}\newcommand{\D}{\mathscr{D}}\newcommand{\C}{\mathscr{C}}$Why can you only define (co)ends in the ordinary way for $\V$-valued functors?
Let me say $\V$ is a closed *symmetric* monoidal category. I stipulate that ... |
I read about Infinitesimal Differential Geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf, page 4) and there was case with Cauchy sequence for infinitesimal functions:
If $x(t)=y(t)+o(t)$ for $t$->$0$ then $x=y$ ($x$ and $y$ are numbers which corresponds for Cauchy sequence for functio... |
How to correctly choose number for Cauchy sequence for infinitesimal functions? Will it be $x=y=[sin(t)]$ or $x=y=[t]$? |
What is the definition of $L^{1,\infty}$ in the sense of Lebesgue functions? |
What does $L^{1,\infty}$ mean? |
$\require{AMScd}\newcommand{\op}{{^\mathsf{op}}}\newcommand{\M}{\mathcal{M}}\newcommand{\V}{\mathscr{V}}\newcommand{\D}{\mathscr{D}}\newcommand{\C}{\mathscr{C}}$Why can you only define (co)ends in the ordinary way for $\V$-valued functors?
Let me say $\V$ is a closed *symmetric* monoidal category. I stipulate that ... |
In the topic of divisibility. Suppose I starts with $a \mid n$, then I manipulate it into $a \mid m$, then $a \mid s$, and finally $a \mid t$. Should I write my train of thought as
$$
\begin{align}
a \mid n
&\implies a \mid m \\
&\implies a \mid s \\
&\implies a \mid t.
\end{align}
$$
Or can I just shortha... |
Let $\pi:(M,g)\to (N,h)$ be a surjective Riemannian submersion, i.e. $\forall p\in M, D\pi_p$ is surjective between the respective tangent spaces. We know that, $T_pM=H_pM \oplus V_pM$ (orthogonal direct sum), and that $H_pM$ is isometric to $T_qN, q:=\pi(p)$ via $D\pi_p.$
Define $S:=exp_p(H_pM)\subset M.$
My qu... |
Isometric copy of the quotient $N$ embedded in the domain $M$ when $\pi:M\to N$ is a surjective Riemannian submersion? |
Let $\pi:(M,g)\to (N,h)$ be a surjective Riemannian submersion, i.e. $\forall p\in M, D\pi_p$ is surjective between the respective tangent spaces. We know that, $T_pM=H_pM \oplus V_pM$ (orthogonal direct sum), and that $H_pM$ is isometric to $T_qN, q:=\pi(p)$ via $D\pi_p.$
Define $S:=exp_p(H_pM)\subset M.$
**My ... |
Let $K\subset L$ be field extension.
Are the following equivalent?
(1) L/K is purely transcendental. Namely, there is subset $S\subset L$ such that $S$ is algebraically independent over $K$ and $L=K(S)$.
(2) Forall $x\in L\backslash K$, $x$ is **not** algebraic over $K$.
Please tell me relationship (1) a... |
It seems to me that the sum of the continued fraction sequence minus one can serve as a metric on positive rational numbers.
Given a positive rational number $q=[a_0, a_1, ...]=a_0+\frac{1}{a_1+\dots}$, let us define the following "distance" from 1:
$$d(q,1)=\sum_k a_k -1$$
Note that for $q=\frac{a}{b}$, this numb... |
Is the sum of the continued fraction sequence / number of steps in Euclid algorithm a metric? |
Let $\pi:(M,g)\to (N,h)$ be a surjective Riemannian submersion, i.e. $\forall p\in M, D\pi_p$ is surjective between the respective tangent spaces. We know that, $T_pM=H_pM \oplus V_pM$ (orthogonal direct sum), and that $H_pM$ is isometric to $T_qN, q:=\pi(p)$ via $D\pi_p.$
Define $S:=exp_p(H_pM)\subset M.$
**My ... |
> Let $(X_n)_n$ be a sequence of square-integrable RVs and let $\mathcal{F}$ be given by $\mathcal{F}_n = \sigma (X_1, \ldots, X_n)$. Suppose that $S_n := \sum_{i=1}^n X_i$ is an $\mathcal{F}$-martingal. Show that
>
> $$Cov[X_m, X_n] = \mathbb{E}[X_m X_n]$$
I know that by definition we have
$$Cov[X_m, X_n] = ... |
Let $\pi:(M,g)\to (N,h)$ be a surjective Riemannian submersion, i.e. $\forall p\in M, D\pi_p$ is surjective between the respective tangent spaces. We know that, $T_pM=H_pM \oplus V_pM$ (orthogonal direct sum), and that $H_pM$ is isometric to $T_qN, q:=\pi(p)$ via $D\pi_p.$
Define $S:=exp_p(H_pM)\subset M.$
**My ... |
I recently learned about the formula for finding the surface area of a function rotated about the x-axis:
$$S = \int 2\pi y \sqrt{1+\bigg[ \frac{dy}{dx} \bigg]^2}dx$$
I tried to apply this formula to a problem involving the function $y = x^2$ rotated around the x-axis. The standard approach seems to involve ta... |
I need to perform some heavy computations involving polynomials in many variables. I don't need any fancy features, just adding and multiplying polynomials and substituting variables by other polynomials. Ideally, I would need to define variables with multiple indices, such as $x_{3, 1, 4}$. I've tried SageMath but the... |
I am wondering if a kind of theorem of tangents from a point to a circumference holds in the sphere.
The situation that I have in mind is the following.
Take two geodesics $\gamma_1$ and $\gamma_2$ starting from a point $p\in S^2$ in direction $v_1$ and $v_2$ respectively ($\gamma_i$ parametrised by arc-lenght). fo... |
Does the Tangent theorem for circumferences hold in the sphere $S^2$? |
Let $\pi:(M,g)\to (N,h)$ be a surjective Riemannian submersion, i.e. $\forall p\in M, D\pi_p$ is surjective between the respective tangent spaces. We know that, $T_pM=H_pM \oplus V_pM$ (orthogonal direct sum), and that $H_pM$ is isometric to $T_qN, q:=\pi(p)$ via $D\pi_p.$
Define $S:=exp_p(H_pM)\subset M.$
**My ... |
I am attending a introductory model theory course this semester. The professor started talking about formulas and sentences without paying much attention to the formalism of the syntax. However, as the course progressed, I began to notice that we need to put the formulas in a set because Zorn's lemma is needed to prove... |
For $k$-cycle $(1, 2, ..., k)$, do the elements in it need to be taken in order?
Is $(3,1,5,2,4)$ a $5$-cycle? Or the elements in the rotation must satisfy a law of order? i.e. $5$-cycle just are $(1,2,3,4,5)$ or $(5,4,3,2,1)$.
For $k$-cycle, do the elements in them need to be in order?
When we say "$\sigma=(... |
Let $\pi:(M,g)\to (N,h)$ be a surjective *Riemannian* submersion,
* i.e. $\forall p\in M, D\pi_p$ is surjective between the respective tangent spaces and that .
* $T_pM=H_pM \oplus V_pM$ ( $g_p$-orthogonal direct sum), and
* $H_pM$ is isometric to $T_qN, q:=\pi(p)$ via $D\pi_p, i.e. h_q(D\pi_p(v), D\pi_p(w)=g_... |
I would like to show that the function
$$f=\frac{c-z}{1-\bar c z}$$
with $|c| < 1$ is holomorphic and an automorphism from the complex disk $\{z: |z|<1\}$ to itself? I'm looking for an elementary proof of this (e.g without using that we can characterize the rotations of the unit disk as in <https://math.stackexch... |
I encountered an interesting problem in “Practical Methods of Optimization” by R. Fletcher, which utilizes the penalty method to solve the following constrained optimization problem\
\begin{align}
{\mathop{\text{minimize} }} \quad & \left \| Ay \right \|^2 \\
\text{s}\text{.t}\text{.} \quad\quad\quad\,\,&
y ... |
Let $\Omega \subset \mathbb{R}^{N} = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$. Suppose that $\Omega \subset A_1 \times A_2$, where $A_i \subset \mathbb{R}^{N_i}$. If $u \in C^\infty_0(\Omega)$, I know that, for a fixed $x \in A_1$, the function $v_x(y) := u(x,y)$ is in $C^{\infty}_0(A_2)$. I wonder if the same happen... |
I'm self studying Billingsley Probability and Measure (3rd edition). I'm having trouble with exercise 2.11 b the reverse direction:
Show that if $\Omega$ is countable, then the sigma-algebra $\mathscr{F}$ consisting of the countable and cocountable subsets of $\Omega$ are countably generated.
The book already pro... |
Let $\theta_1, \theta_2$ and $\theta_3$ be nonnegative parameters with the constraint $\theta_1+\theta_2+\theta_3=1$. We observe $X_{i 1}=\theta_1+\epsilon_{i 1}, X_{i 2}=\theta_2+\epsilon_{i 2}, X_{i 3}=\theta_3+\epsilon_{i 3}$ for $i=1,2, \ldots, n$, where $\epsilon_{i k} \sim N(0,1)$ are independent normal random va... |
A teacher leaves his home everyday at 8:00A.M to come to school. If he comes 95 meters per minute. He will arrive one minute late for his class. If he walks 105 meters per minute he will arrive one minute early for his class. At what speed should he walk to arrive perfectly on time?
Is this the correct way to approa... |
> I need to calculate the area using a double integral (converting to polar system), which are
> bounded by the following curves: $(x^2+y^2)^2=2a^2(x^2-y^2)$ and $x^2+y^2 \ge a^2$.
So I plotted both of these curves, assuming that a = 1. I converted it to the polar system and found $r=\pm\sqrt{2a^2cos2\theta}$. But I ... |
how do I find the intersection points of two curves? |
Let V be the following vector space.
> $V = \{ f : \mathbb{R} \longrightarrow \mathbb{R} : \text{f = infinitely differentiable at all x, } f^{(k)}(0) = 0 \;\; (\forall \;k \in \mathbb{N}) \}$.
How would one show that this vector space is infinite dimensional?
----------
To show that a vector space is ... |
Does there exist a non-commutative field of a characteristic different from zero? |
A function similar to Minkiwski's question-mark [function][1] pairs periodic 2-adic numbers with quadratic irrationals, and pairs these with rectangle-shapes.
It is likely that periodic 5-adic numbers can be paired with quintics and with 5-hyperrectangles.
Are the 5-hyperrectangles known to break some complexity ... |
Does the 5-hyperrectangle break any complexity thresholds analogous to quintics? |
> I need to calculate the area using a double integral (converting to polar system), which are
> bounded by the following curves: $(x^2+y^2)^2=2a^2(x^2-y^2)$ and $x^2+y^2 \ge a^2$.
So I plotted both of these curves, assuming that a = 1. I converted it to the polar system and found $r=\pm\sqrt{2a^2cos2\theta}$. But I ... |
I've been trying to fit some data with polynomials using a pseudo-inversion library (numpy.linalg). There are two real functions $f$ and $g$ such that $f=-\alpha g$, $\alpha > 0$, and
$$
f=f(t_1,\ldots,t_5),
$$
$$
g = g(t_1,\ldots,t_5),
$$
where $t_1,\ldots,t_5$ are real parameters. The parameter $\alpha$ ha... |
Not true. Consider the identity map $\operatorname{id}\colon \mathbb{R} \rightarrow \mathbb{R}$ where the first $\mathbb{R}$ is endowed with the discrete topology and the second, the euclidean topology. This map is continuous and surjective.
$\mathbb{Q}$ is dense in $\mathbb{R}$ under the Euclidean topology. Its in... |
how to simplify this expression so that I get an expression only?
when $t = 1, 2,.... $
$$y = \sum _{t=1}^{\infty }\:\frac{\left(1+g\right)^{ \lceil t/2\rceil}}{\left(1+k\right)^t}$$
I know that for t = 1, $\lceil \frac{t}{2}\rceil$ = 1
for t = 2, $\lceil \frac{2}{2}\rceil$ = 1
for t = 3, $\lceil \... |
Let $a_1,a_2,a_3 \in (0,1]$ and $a_1 \geq a_2 > a_3$. And let $b_1,b_2,b_3$ be defined by elliptical integrals:
\begin{equation}
b_j = a_1 a_2 a_3 \int^\infty_0 \frac{dx}{(a_j^2+x)\sqrt{a_1^2+x}\sqrt{a_2^2+x}\sqrt{a_3^2+x}}.
\end{equation}
Useful identity can be $b_1+b_2+b_3 = 2$. I want to prove the following... |
I've been trying to find the control points for a NURBS curve that I want to trace the first octant (1/8) of a unit circle using a second-degree rational bezier. I'm not entirely sure whether such is possible, but I have no reason to assume not, which is why I am trying anyways. I'm very confused and I believe I'm misu... |
I have a problem: I want to find a method which, given *a set of centroids* as `x,y` coordinates (the number of the centroids does not matter, more than one) it returns me a tessellation of the plane such that **each region has the center of mass exactly at one of the given centroids.**
It is similar to the Centroid... |
Let $a_{i}$ and $b_{i}$, $i = 1,\dotsb,n$ be real numbers. Denote $S_{pq}:=\sum_{i=1}^{n}a_{i}^{p}b_{i}^{q}$.
I want to show that,
$$ (S_{20}S_{02} - S_{11}^2)S_{00} - (S_{10}S_{02} - S_{01}S_{11})S_{10} + (S_{10}S_{11} - S_{01}S_{20})S_{01} \geq 0. $$
and find its equivalence condition.
I thought about it in s... |
Let $f$ be a real function with $f(0)=0 ,$ differentiable for $x \neq0 $ and $f'(x)= \frac{e^x}{3f^2(x)+2f(x)+1}$. How can I show that there exists a $x_0, 0<x_0<2 $ s.t. $f(x_0)=1$?
----------
I’ve already shown that $f^3(x)+2f(x)+f(x)=e^x-1$ from multiplying the given fraction with the denominator and intergrat... |
First I made this question for myself just out of curiosity. But got stuck in the middle of solving it.
**The question:**
Say we have a probability density function of a continuous stochastic variable X as
$$f_X(x)=\begin{cases}
\frac{2}{\pi^2}x & 0\leq x\leq\pi \\
0 & \text{otherwise}
\end{cases... |
how to simplify this expression so that I get an expression only?
when $t = 1, 2,.... $
$$y = \sum _{t=1}^{\infty }\:\frac{\left(1+g\right)^{ \lceil t/2\rceil}}{\left(1+k\right)^t}$$
I know that for t = 1, $\lceil \frac{t}{2}\rceil$ = 1
for t = 2, $\lceil \frac{2}{2}\rceil$ = 1
for t = 3, $\lceil \... |
Let $a_{i}$ and $b_{i}$, $i = 1,\dotsb,n$ be real numbers. Denote $S_{pq}:=\sum_{i=1}^{n}a_{i}^{p}b_{i}^{q}$.
I want to show that,
$$ (S_{20}S_{02} - S_{11}^2)S_{00} - (S_{10}S_{02} - S_{01}S_{11})S_{10} + (S_{10}S_{11} - S_{01}S_{20})S_{01} \geq 0. $$
and find its equivalence condition.
I thought about it in s... |
Problem: Let $S=\{v_1,v_2,...,v_r\}$ be a linearly independent subset of a vector space $V(F)$. If $w\in V$ such that $w\notin L(S)$, then prove that the set $S_1=\{w,v_1,v_2,...,v_r\}$ is also linearly independent.
Attempt: Let $\alpha_1w+\alpha_2v_1+\alpha_3v_2+...+\alpha_{r+1}v_r=0$ where $\alpha_i\in F,1\leq i\l... |
Let $\displaystyle f(x)= e^{-Ae^x} e^{Bx} \left( 1+ \mathcal{O} \left( e^{-x} \right)\right)$, where $A$ and $B$ are constants.
**Claim:** $|f(x)| = \mathcal{O}\left( \exp(-e^x)\right)$ as $x \rightarrow \infty$. <br>
Note that $$\frac{1}{e^{Ae^x}} \leq \frac{e^{Bx}}{e^{Ae^x}} \leq \frac{e^{x}}{e^{Ax}}\leq \fr... |
I was looking the solution given by Simply Beautiful Art of the following problem: https://math.stackexchange.com/a/2029616/952348
In a part of the solution the author claims that [![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/8GRTw.png
How I can prove the left side of the inequality?, ... |
How to prove $e^{-x}-\frac{1}{n}< \lim_{n \rightarrow \infty} \left (1-\frac{x}{n} \right )^{n}$? |
Let $f$ be a real function with $f(0)=0 ,$ differentiable for $x \neq0 $ and $f'(x)= \frac{e^x}{3f^2(x)+2f(x)+1}$. How can I show that there exists a $x_0, 0<x_0<2 $ s.t. $f(x_0)=1$?
----------
I’ve already shown that $f^3(x)+f^2(x)+f(x)=e^x-1$ from multiplying the given fraction with the denominator and intergra... |
Problem: Let $S=\{v_1,v_2,...,v_r\}$ be a linearly independent subset of a vector space $V(F)$. If $w\in V$ such that $w\notin L(S)$, then prove that the set $S_1=\{w,v_1,v_2,...,v_r\}$ is also linearly independent.
Attempt: Let $\alpha_1w+\alpha_2v_1+\alpha_3v_2+...+\alpha_{r+1}v_r=0$ where $\alpha_i\in F,1\leq i\l... |
Suppose that $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is continuous and monotone, i.e.,
\begin{align}
(\forall x,y \in \mathbb{R}^{n}) \quad \langle F(x) - F(y), x-y \rangle \geq 0.
\end{align}
Show that
\begin{align}
\text{zer }F := \{x\in\mathbb{R}^{n}\mid F(x) = 0\}
\end{align}
is convex.
Let $x,y\in \text{... |
$y_{0}=0$
$y_{1}=\frac{-1}{2}x^2$
$y_{2}=\frac{-1}{2}x^2 - \frac{1}{10}x^5$
$y_{3}=\frac{-1}{2}x^2 - \frac{1}{10}x^5- \frac{1}{80}x^8$
$y_{4}=\frac{-1}{2}x^2 - \frac{1}{10}x^5- \frac{1}{80}x^8- \frac{1}{880}x^{11}$
Now I am trying to figure out a general pattern. I can see that the powers follow an arith... |
Some problematic points in your proof:
> Proof: let $\{a_1,a_2,...,a_i,v_1,v_2...v_n\}$ be basis for $W_1$, then $\dim(W_1)= i+n$ and $\{b_1,b_2,...,b_j,v_1,v_2...v_n\}$ be basis for $W_2$, then $\dim(W_2)= j+n$ .
You are missing some fundamental information. What are $i,j,n$? Why do both bases contain the sam... |
Recently, while using GeoGebra, I was able to generalize the idea of a parabolic vertex-kissing circuit I have achieved a wonderful result, but I cannot prove it. If anyone can prove it, please do so
Let $P$ be a parabola whose focus is $F$, whose directory is $L$, whose vertex is $O$, and whose axis of symmetry i... |
The statement is as follow:
> Suppose $(S_n)$ is a 2d simple random walk, and $(S^1_n)_n, (S^2_n)_n$ are two independent random walks of the same type. Let $R_n = \sum_{j=1}^n\mathbf{1}\{S_j = 0\},$ and $ L_n = \sum_{j=1}^n \mathbf{1}\{ S^1_j = S^2_j\}$. Show that
$$
L_n = R_{2n} \enspace \text{ in distribution}.... |
Recently, while using GeoGebra, I was able to generalize the idea of a parabolic vertex-kissing circuit I have achieved a wonderful result, but I cannot prove it. If anyone can prove it, please do so
Let $P$ be a parabola whose focus is $F$, whose directory is $L$, whose vertex is $O$, and whose axis of symmetry i... |
I appreciate your assistance!
I’m working on locating a point that minimizes the sum of squared distances from a set of circles. The objective function to minimize is:
$$
E(x,y) = \sum_i \big( (x-w_i)^2 + (y-f_i)^2 - m_i^2 \big)^2
$$
With three circles in my scenario, I aim to determine this point. Despite cal... |
Consider the three categories $\mathbf{Top}$ (of topological spaces with continuous maps), $\mathbf{Graph}$ (of simple graphs with maps that preserve or contract edges) and $\mathbf{Pros}$ (of pre-ordered sets with monotone maps). These three categories, albeit rather different in feel, share a common property: They co... |
Why does this specific adjoint chain come up for different categories? |
Some problematic points in your proof:
> Proof: let $\{a_1,a_2,...,a_i,v_1,v_2...v_n\}$ be basis for $W_1$, then $\dim(W_1)= i+n$ and $\{b_1,b_2,...,b_j,v_1,v_2...v_n\}$ be basis for $W_2$, then $\dim(W_2)= j+n$ .
You are missing some fundamental information. What are $i,j,n$? Why do both bases contain the sam... |
How to prove $e^{-x}-\frac{1}{n}< \left (1-\frac{x}{n} \right )^{n}$? |
Given a family of curves, an envelope is defined as a curve that it is tangent to every curve in the family of curves at some point on it.
To derive the equation for it, the first step is to parameterize the envelope with parameter $t$, which also indexes which curve in the family.
But what if there is a curve ta... |
I have tried many ways, including denote a tangent point $F'$ and prove $F'\equiv F$ or denote the points at which the incircle touches the sides but I can't go further. Can someone help me in this problem?
The incircle (its center is point $I$) of $\Delta ABC$ touches the side $AB$ at point $D$. Point $E$ is the re... |
I got this problem in my home exersice at the University. I think I solved it, but I am not sure, that my solution is right. Can somebody check it?
Let $A$ and $B$ be two events on a probability space $(\Omega, \mathbb{P})$, with $\mathbb{P}(A)=\frac{1}{3}$ and $\mathbb{P}(B)=\frac{1}{4}.$Determine the value of $\ma... |
Using your notions, we have that $$x=B\cdot (9-\frac{30}{A+B}),\tag1$$
because $\frac{30}{A+B}$ is the time they travelled, so $9-\frac{30}{A+B}$ days were spent by the second traveler to get from $\Gamma$ to $\Delta$. Putting $A=\frac{30-x}8$ and $B=\frac{30+x}{18}$ in $(1)$, we get
$$x= \frac{30+x}{18}\cdot (9-... |
Recently, while using GeoGebra, I was able to generalize the idea of a parabolic vertex-kissing circuit I have achieved a wonderful result, but I cannot prove it. If anyone can prove it, please do so
Let $P$ be a parabola whose focus is $F$, whose directory is $L$, whose vertex is $O$, and whose axis of symmetry i... |
The title of the question [Solving $(2n)^{\log 2}=(5n)^{\log 5}$][1] asks how to solve (in some western conventions) base 10 for $(2n)^{\log 2}=(5n)^{\log 5}$. However that question's title does not seem to indicate an arbitrary base. For instance, one convention for "$\log$" is $\log c = \log_{10} c$.
Also the bas... |
> Let$f$ be holomorphic on $\mathbb{C}$. Prove that there exists a
> holomorphic function $g$ on $\mathbb{C}$ s.t $g'=f$.
My attempt is not too much, at least I have tried to think, but can't go further:\
If $f$ is bounded on $\mathbb{C}$, then according to Liouville's theorem, $f$ is constant. Let $f=a\in\mathbb... |
I'm working with convolution operations and encountered a theoretical question that I hope someone here can help clarify.
Consider a Gaussian function $g(x)$ and a delta distribution $\delta(x - (a+bi))$ where $a,b > 0$ and $i$ is the imaginary unit. The delta distribution is centered at a complex location $(a+bi)$... |
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