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Imaging if I have n sequares where the first one is fixed at the origin. The subsequent blocks may be stacked either on the right on top of the previous block. If one configuration can **rotate** into another, they are counted as the same configuration. If one can just be mirrored into another, but not rotated, then th... |
Let $f \in \mathcal{C}^1(\mathbb{R})$ a function such that $f(1)=0$ and let $0 < x_0 < 1$. Prove that the Cauchy problem
$\begin{cases} x'=xf(x^2+y^2)-y \\ y'=yf(x^2+y^2)+x \\ x(0) = x_0 \\ y(0) = 0 \end{cases}$
Has a solution, and it is defined in $\mathbb{R}$.
So I know I can write the problem as
$\begin{ca... |
When self studying a text where it is not expected to be able to solve all (or most) of the problems, what are the appropriate criteria to use for advancement?
> A word about the problems. There are a great number of them. It would be an extraordinary student indeed who could solve them all.... Many are introduced n... |
My question is to do with parametrisation of arclengths.
As part of a course on mechanics (with mathematical focus), I have covered intrinsic coordinates in a plane curve. In class and homeworks etc, we covered ideas such as determining the shape of a chain hanging under its own weight.
We also learned arclength... |
I recently stumbled upon the Gaussian $q$-binomail coefficients. The generalization is obtained using the following formula:
$$\binom{n}{k}_q = \frac{(1-q^n)(1-q^{n-1})\cdots(1-q^{n-k+1})}{(1-q)(1-q^2)\cdots(1-q^k)}$$
and these $q$-binomails obey many of the nice properties that we expect from anything which we... |
I have a time series of log prices that looks like a random walk. I want to denoise this series using wavelet denoising.
I care about predicting future returns (so predicting the difference in the series).
So far I've found it's possible to denoise the prices (not the differences), and then from that denoised ser... |
Let $\bigwedge^{p,q}V:=\bigwedge^{p}V^{1,0}\wedge \bigwedge^{p}V^{0,1}$ and $\bigwedge {V}^*_{\mathbb{C}}=\bigoplus_k \bigwedge^{k} V_{\mathbb{C}} $.
Then one defines the natural projection $$ Π^{p,q}:\bigwedge {V}^*_{\mathbb{C}} \to \bigwedge^{p,q}V$$
Is the projection defined as: $$(v_1,w_1, v_1\wedge w_1, \ld... |
I was playing around with some equations at desmos and, after finding out about ```floor``` function, I played with it a bit, but nothing caught my attention except for the last equation I tried: ```y = floor(x/y)```([Screenshot][1]). It looks like some recursion function with increasing by 0.5 step which is confusing ... |
Below, the notation $\max(t,0)\equiv \max(t)$ is used, and the function $\max$ is seen as a distribution. From the theory of distributions, we have that $\max'(t)=H(t)$ (in the distributional sense) where $H$ is the Heaviside distribution. Now, I am considering the one-parameter distribution $\max(at)$ where $a$ is the... |
I was experimenting with some equations on Desmos and stumbled upon the `floor` function. I tried using it in different equations, but nothing caught my attention until I tried `y = floor(x/y)` ([Screenshot][1]). This graph looks like a recursive function with an increasing step of 0.5, which is confusing to me. I aske... |
I'm trying to figure out the mathmatical explaination for placing the area of which a siren covers within a certain area. I'm working on a Math AA SL IA right now and I'm very stuck. I sense that there is an explaination for my work but I guess I'm not at that level yet. I have a rudimentary understanding of calculus, ... |
I was reading a proof on the lower bound of the mixing time of the lazy $(p − q)$-biased walk, which remains in its current position with probability 1/2, moves clockwise with probability $p/2$, and moves counterclockwise with probability $q/2$. Here $p + q = 1$. The proof goes as follows
[![enter image description he... |
In an old exam there was the exercise to show whether
$\sum\limits_{n\geq 0}\mathrm{e}^{\mathrm{i}n}$
converges or not.
Wolframalpha told me, that it diverges, but how can I show it?
The root test gives me $1$, so it does not help. |
I was experimenting with some equations on Desmos and stumbled upon the `floor` function. I tried using it in different equations, but nothing caught my attention until I tried `y = floor(x/y)` ([Screenshot][1]). This graph looks like a step function with an increasing step of 0.5, which is confusing to me. I asked Mat... |
I'm having trouble understanding what does the covariant derivative do in a coordinate system where we have changing basis vectors. I always thought it was giving us the change in coordinates while taking into account the change in basis vectors $\vec{a}_i(\xi^1,\xi^2,...,\xi^n)$. I tried to visualize it with the follo... |
So I'd seen a bit of amenability from the functional-analytic point of view, but I was looking for some motivation. I found the following set of notes which I think is doing quite a nice job [see here](https://www.math.uni-duesseldorf.de/~garrido/amenable.pdf). These notes however are mostly looking at discrete groups ... |
Let's assume there is some non-standard model of the reals containing a number $N$ that is larger than any real number.
> Suppose $\exists N\in {^*}\mathbb{R} ( \forall r\in\mathbb{R}: r<N).$
Now I know I can find even larger unlimited hyper-reals $2N,3N...$ which are smaller than $N^2,N^3...$ which are still hyp... |
How big do hyper-reals get? |
Informally, the silhouette of a 3D shape is a viewpoint-dependent 2D projection of it. You might imagine looking at several silhouettes and attempting to construct the overall shape. My question is theoretical: if you have access to the set of all silhouettes of a convex figure —unordered, that is to say, you kn... |
I'm reading Miroslav Repicky's paper "A proof of the independence of the Axiom of Choice from the Boolean Prime Ideal Theorem" and I'm confused by the claim that an ideal $I$ on a Boolean algebra $B$ being prime in $OD^{V[G]}[A,f\frown h]$ implies that it must be a maximal ideal of $B$ in $OD^{V[G]}[A,f\frown h]$. For ... |
Let the *multiplicative closure* operator $h$ be defined on $\mathcal{P}(\mathbb{N})$ by setting $hA$ equal to the smallest set containing $A$ that is closed under multiplication.
In L. F. Meyers' solution to Problem E $1941$ on pp. $408$-$9$ of the April $1968$ *American Mathematical Monthly* (https://www.jstor.org/s... |
Reading Liu, I got quite confused about what the "canonical map" $T_{f,x}:T_{X,x}\to T_{Y,f(x)}\otimes_{k(f(x))}k(x)$ should be for a map of general schemes $f:X\to Y$. Reading several MSE posts, I get the impression this map is only well-defined if $T_{Y,f(x)}$ is finite dimensional over $k(f(x))$.
My question is v... |
How do we define the map on Zariski tangent space in the non-finite case? |
> Let $f(x) = ax^2+bx+c \in \Bbb{R}[x]$ be an irreducible polynomial. Prove that the field $\Bbb{R}[x] \mathbin{/} \langle f(x) \rangle$ is isomorphic to the field of complex numbers.
Knowing that
$$
\Bbb{R}[x] \mathbin{/} \langle f(x) \rangle = \{h + gv \mid h, g \in \Bbb{R};\, av^2+bv+c = 0 \},
$$
how can I d... |
Simple question. Let $(M,g)$ be a flat manifold (i.e. a Riemannian manifold with vanishing Riemannian curvature). Is it true that the metric $g_{ij}$ in Riemannian coordinates around each point is constant and equal to the identity, i.e. $g_{ij}\equiv \delta_{ij}$?
It looks the Taylor expansion of $g_{ij}$ can be ex... |
Is the metric in Riemannian coordinates constant for flat manifolds? |
Okay, I'm not very comfortable with either complex geometry or minimal surfaces, so bear with me. I've encountered the following theorem, but I can't find a proof:
Suppose $V\subset \mathbb{C}^n$ is a properly embedded complex submanifold, say of complex-dimension $k$. Here, $\mathbb{C}^n$ is equipped with the usual... |
I have two sets: $I=\{1,2,\dots,n\}$ and $J=\{1,2,\dots,m\}$ and a set of subsets as $P=\{p_1, p_2, \dots, p_d\}$ that each member of $P$ is a subset of $I$, for example, $p_d=\{3,4\}$.
This is my optimization problem where $x_j$ is the decision variable and $z_{ij}$, $LC_i$, $T_{p_d}$, $U_j$ and $c_j$ are the paramet... |
Simple question. Let $(M,g)$ be a flat manifold (i.e. a Riemannian manifold with vanishing Riemannian curvature). Is it true that the metric $g_{ij}$ in normal coordinates around each point is constant and equal to the identity, i.e. $g_{ij}\equiv \delta_{ij}$?
It looks the Taylor expansion of $g_{ij}$ can be expres... |
Is the metric in normal coordinates constant for flat manifolds? |
Can we find three random variables $X_1, X_2, X_3$ such that $E[X_2 \mid X_1] = X_1$ and $E[X_3 \mid X_2] = X_2$ but $E[X_3 \mid X_1, X_2] \neq X_2$?
I'm stuck on how to choose such random variables; in particular how to define $X_3$ in relation to the other two. At first, I was thinking of setting $X_n = \frac{S_n}... |
I've been researching some operator spaces and have stumbled upon the following problem. Suppose $x,y\in\mathcal{B}(\mathcal{H})$ (for $\mathcal{H}$ separable) such that $0\leq x\leq y\leq 1$. Further let $t\in[0,1]$ and $\rho$ be a density matrix (i.e. $\rho\geq 0$, $\mathrm{Tr}(\rho)=1$). Is it true that
$$
\mathrm... |
I am reading Milnor's lectures on characteristic classes. He defines the canonical line bundle $\gamma_n^1$ as the set of points $(\pm x, v) \in \mathbb{R}P^n \times \mathbb R^{n+1}$, where $v = tx$ for some $t$. Later, he defines the orthogonal complement (within $\epsilon^{n+1}$), $\gamma^\top$, of $\gamma_n^1$ (on ... |
I am reading Milnor's lectures on characteristic classes. He defines the canonical line bundle $\gamma_n^1$ as the set of points $(\pm x, v) \in \mathbb{R}P^n \times \mathbb R^{n+1}$, where $v = tx$ for some $t$. Later, he defines the orthogonal complement (within $\epsilon^{n+1}$), $\gamma^\perp$, of $\gamma_n^1$ (on... |
I am taking a course on Markov chains which is intended to be accessible without a rigorous understanding of probability measure, but I'm not grasping even basic definitions and it is hindering my progress. Specifically, I want to precisely understand the definition of a discrete random variable. The following is an ex... |
Reading Liu, I got quite confused about what the "canonical map" $T_{f,x}:T_{X,x}\to T_{Y,f(x)}\otimes_{k(f(x))}k(x)$ should be for a map of general schemes $f:X\to Y$. Reading several MSE posts, I get the impression this map is only well-defined if $T_{Y,f(x)}$ is finite dimensional over $k(f(x))$ (or if $k(x)$ is a f... |
How do we define the map on Zariski tangent space in the infinite case? |
I was doing exercises from Joel Franklin's Methods of Mathematical Economics out of personal interest and found myself a bit conflicted as to how I approached problem 8 in the Multiobjective Linear Programming Section. The question is find all efficient points $x$ and all their weight vectors $w$ for the following opti... |
Here:
https://math.stackexchange.com/questions/3146697/proof-that-n-planes-cut-a-solid-torus-into-a-maximum-of-frac16n33n28n
an intuitive idea of understanding the formula of Torus Slicing is given. Nevertheless I wanted to ask if there is any concrete Literature with a formal proof given?
In this book:
Gardner... |
A person in my class asked today if $\mathfrak{A}$ is a unital $C^{*}$-algebra and $x,y \in \mathfrak{A}$ commute and are normal, then is it always the case there exists $\alpha \in C^{*}(x,1)$ and $\beta \in C^{*}(y,1)$ such that $C^{*}(x,y,1) = C^{*}(\alpha+\beta,1)$?
I don't think this could be true, as it would ... |
$C^{*}$-algebras generated by commuting normal elements? |
**Background:**
In Peter Smith's *[Introduction to Gödel's Theorems][1]* §36.1, he introduces the sentence
$$\mathsf{Prf^\circ(x,y)} =_\text{def} \mathsf{Prf(x, y) \wedge \forall v \leq x \neg Prf(v, \ulcorner \bot \urcorner)}. $$
He then defines
\begin{align}
\mathsf{Prov^\circ(y)} &:= \mathsf{\exists v P... |
**Background:**
In Peter Smith's *[Introduction to Gödel's Theorems][1]* §36.1, he introduces the sentence
$$\mathsf{Prf^\circ(x,y)} =_\text{def} \mathsf{Prf(x, y) \wedge \forall v \leq x \neg Prf(v, \ulcorner \bot \urcorner)}. $$
He then defines
\begin{align}
\mathsf{Prov^\circ(y)} &:= \mathsf{\exists v P... |
Suppose X is a continuous random variable, and Y = 2X is a deterministic transformation of it. Then the following identities hold, according to Cover & Thomas:
$$
I(X; Y) = h(X) - h(Y|X) \\
I(X; Y) = h(Y) - h(X|Y),
$$
where $h$ is differential entropy. Also:
* $Y$ is a deterministic function of $X$, and vice... |
Suppose X is a continuous random variable, and Y = 2X is a deterministic transformation of it. Then the following identities hold, according to Cover & Thomas:
$$
I(X; Y) = h(X) - h(X|Y) \\
I(X; Y) = h(Y) - h(Y|X),
$$
where $h$ is differential entropy. Also:
* $Y$ is a deterministic function of $X$, and vice... |
There is no right answer for this question. And the answer I am about to give is solely based on my personal experience as a Mathematics student.
Self studying Mathematics is not an easy task. I was lucky enough to have great professors at university. Most of them had a vast teaching experience. After teaching somet... |
**Background:**
In Peter Smith's *[Introduction to Gödel's Theorems][1]* §36.1, he introduces the sentence
$$\mathsf{Prf^\circ(x,y)} =_\text{def} \mathsf{Prf(x, y) \wedge \forall v \leq x \neg Prf(v, \ulcorner \bot \urcorner)}. $$
He then defines
\begin{align}
\mathsf{Prov^\circ(y)} &:= \mathsf{\exists v P... |
For $f,g\in L^1(\mathbb{R}^d)$, their convolution is defined as $(f*g)(x)=\displaystyle\int_{\mathbb{R}^d}f(y)g(x-y){\rm d}y$, and it is standard that $f*g\in L^1(\mathbb{R}^d)$, so $*$ is an operator $L^1(\mathbb{R}^d)\times L^1(\mathbb{R}^d)\to L^1(\mathbb{R}^d)$. It would be interesting to know some basic properties... |
Is the sum of convolutions of $L^1$ functions also a convolution? |
**Background:**
In Peter Smith's *[Introduction to Gödel's Theorems][1]* §36.1, he introduces the sentence
$$\mathsf{Prf^\circ(x,y)} =_\text{def} \mathsf{Prf(x, y) \wedge \forall v \leq x \neg Prf(v, \ulcorner \bot \urcorner)}. $$
He then defines
\begin{align}
\mathsf{Prov^\circ(y)} &:= \mathsf{\exists v P... |
How do I get the sample deviation from population deviation when sampling is done without replacement from a finite population?
[enter image description here][1]
[1]: https://i.stack.imgur.com/kfZL7.png |
Going through Spivak's Calculus, I have stumbled on the following function
$$\displaystyle{f(x) = \sqrt{|x|}}$$
What I find curious is that when you work towards a derivative at $0$ you get the following
$
\frac{f(h) - f(0)}{h} =
\begin{cases}
\dfrac{\sqrt{h}}{h} = \dfrac{1}{\sqrt{h}}, & \text{if } h > 0,\... |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
If a function $G(x)$ is defined by
$$G(x)=\int_a^xf(t)dt$$
Then by the [Fundamental Theorem][1] I can see that $G(x)$ is an antiderivative of $f(x)$ and thus
$$G'(x)=f(x)=\frac{dG(x)}{dx}$$
My confusion starts when I look at the integrand $f(t)$ and its variable $t$. The integrand $f(t)$ must have an antiderivative... |
How do I get the sample deviation from population deviation when sampling is done without replacement from a finite population? The deriving step by step was needed.
[enter image description here][1]
[1]: https://i.stack.imgur.com/kfZL7.png |
First of all, $GL(n, {\mathbb Z})$ contains $SL(n, {\mathbb Z})$ as index 2 subgroup. Therefore, for $G= GL(n, {\mathbb R})$, $GL(n, {\mathbb Z})\backslash G$ is compact if and only if $SL(n, {\mathbb Z})\backslash G$ is compact. To see that the latter quotient is noncompact, observe that $G$ splits as the direct produ... |
I know how to do this question if the rooms can be differentiated, which has multiple posts in Math Exchange. Thise is done by knowing that each of the 10 person can pick either room 1 or 2, and hence there is $2^{10}$ ways. However, how to do this if the room cannot be differentiated? I am confused because in the case... |
If any of the following exposition is unclear, please write a comment.
In essence, I am looking at the graph $G$ that is generated by the polynomial $q(x) = x^2+ax+b$ ($a,b \in \mathbb{Z}$) via the edge set
$$\{(n, q(n) \text{ mod } p) : n =0,\ldots,p-1\}$$
for some prime $p \in \mathbb{P}$. After some thought, it... |
I am looking for the Fourier transform of $x \tanh(x)$ in $\mathbb{R}$. The Fourier transform on $\mathbb{R}\setminus\{0\}$ is $$2\sum_{n=0}^{\infty}(2n+1)\exp\left(-(2n+1)\frac{|t|}{2}\right).$$ But there is singularity at $0$. How can I get
the Fourier transform of $x \tanh(x)$ in $\mathbb{R}$?
|
Does the "physicist common knowledge" that "solenoidal vector fields have closed integral curves" have any mathematical foundation? |
I would like to verify my thinking logic on this question.
For the case where the room can be differentiated, I take the sample space to be $3^{9}$, which is the number of ways the 9 people can go inside the 3 rooms. If we would like to arrange such that the first room has 4 person, second room 3 person, and third r... |
For two arbitrary multivectors $\mathbf u$ and $\mathbf v$, what are the definitions of
* the scalar (or interior) product $\mathbf u\cdot \mathbf v$,
* the cross product $\mathbf u\times \mathbf v$ (if it exist; I'm not sure it's defined for general multivectors), and
* the wedge (or exterior) product $\mathbf u\we... |
Scalar, cross and wedge product between two multivectors? |
Let's start with question 2, because it's easier.
In one sense, an angle between two line segments in higher dimensions does make sense, because the two line segments (typically) form a two-dimensional plane. We can always embed a plane in $\Bbb{R}^n$ (or any real inner product space) into $\Bbb{R}^2$ isometrically,... |
Let $\Omega$ be a two dimensional bounded set and $f$ be an integrable function. It is well known that the Dirichlet problem for the Poisson equation
$\partial \bar{z} \partial_z u(z) = f (z)\quad \mbox{in} \quad \Omega\quad $ such that $\quad u(z) = h(z) \quad on \quad \partial\Omega$
is uniquely solveable, wh... |
If any of the following exposition is unclear, please write a comment.
In essence, I am looking at the graph $G$ that is generated by the polynomial $q(x) = x^2+ax+b$ ($a,b \in \mathbb{Z}$) via the edge set
$$\{(n, q(n) \text{ mod } p) : n =0,\ldots,p-1\}$$
for some prime $p \in \mathbb{P}$. After some thought, it... |
>> Henry recorded the number of students absent everyday for a total of $40$ days. Here is the frequency table:
[enter image description here][1]
[1]: https://i.stack.imgur.com/uisXk.png
Also it is given that,on exactly half of the days, No more than one student was absent. Find the values of $x,y$.
... |
>> Henry recorded the number of students absent everyday for a total of $40$ days. Here is the frequency table:
[enter image description here][1]
Also it is given that,on exactly half of the days, No more than one student was absent. Find the values of $x,y$.
My try:
Obviously $$1+2+x+10+y+12=40 \Righta... |
**Edit:** This long question was basically answered by a quick comment! I'll accept an answer if someone posts one, but it's basically answered already.
***
**Background:**
In Peter Smith's *[Introduction to Gödel's Theorems][1]* §36.1, he introduces the sentence
$$\mathsf{Prf^\circ(x,y)} =_\text{def} \math... |
>> Henry recorded the number of students absent everyday for a total of $40$ days. Here is the frequency table:
![enter image description here][1]
Also it is given that,on exactly half of the days, No more than one student was absent. Find the values of $x,y$.
My try:
Obviously $$1+2+x+10+y+12=40 \Right... |
For two arbitrary multivectors $\mathbf u$ and $\mathbf v$, what are the definitions of
* the interior (or scalar) product $\mathbf u\cdot \mathbf v$,
* the cross product $\mathbf u\times \mathbf v$ (if it exist; I'm not sure it's defined for general multivectors), and
* the exterior (or wedge) product $\mathbf u\we... |
Interior, cross and outer products between two multivectors? |
I don't think this is correct. When the rooms cannot be differentiated, the answer shouldn't simply be $\binom {9}{3} \binom {6}{4} \binom {2}{2}$ divided by $3^9$, because you're assuming that the rooms are distinguishable. For example, you're counting $2, 2, 5$ and $5, 2, 2$ as distinct cases. To clarify, the reasoni... |
For two arbitrary multivectors $\mathbf u$ and $\mathbf v$, what are the definitions of
* the interior (or scalar) product $\mathbf u\cdot \mathbf v$,
* the cross product $\mathbf u\times \mathbf v$ (if it exist; I'm not sure it's defined for general multivectors), and
* the exterior (or wedge) product $\mathbf u\we... |
>> Henry recorded the number of students present everyday for a total of $40$ days. Also there are a total of $29$ students. Here is the frequency table:
![enter image description here][1]
Also it is given that,on exactly half of the days, No more than one student was absent. Find the values of $x,y$.
My t... |
Let $\Omega$ be a two dimensional bounded set and $f$ be an integrable function. It is well known that the Dirichlet problem for the Poisson equation
$\partial_ \bar{z} \partial_z u(z) = f (z)\quad \mbox{in} \quad \Omega\quad $ such that $\quad u(z) = h(z) \quad on \quad \partial\Omega$
is uniquely solveable, w... |
Let $\Omega$ be a two dimensional strictly convex bounded set and $f$ be an integrable function. It is well known that the Dirichlet problem for the Poisson equation
$\partial_ \bar{z} \partial_z u(z) = f (z)\quad \mbox{in} \quad \Omega\quad $ such that $\quad u(z) = h(z) \quad on \quad \partial\Omega$
is uniqu... |
What I mean is that the number 3 comes up a lot in these fields as sort of a boundary between decidability and undecidability, or chaos and order.
Examples:
1. Quadratic Diophantine equations are always decidable, but cubic ones are not.
2. Time independent second order ODEs cannot admit chaotic solutions, but ... |
Why is the number 3 come up so often in Chaos theory and Undecidability as a boundry? Is it just a coincedence? |
Let us take a look at this equation.
$$e^{e^x}=1$$
when $e$ is the base of the natural logarithm.
It may seem intuitive that there are no solutions in any field. After all, $e^0=1$, so there can not be anything $x$ such that $e^x=0$.
Is this true? Are there absolutely no solutions? |
Let us take Euler's identity, where $i^2=-1$
$$e^{i\pi}+1=0$$
Subtracting $1$ from both sides, we get
$$e^{i}=-1$$
What happens when we square both sides?
$$e^{i2}=1$$
This means that i2 is *also* a natural logarithm of $1$.
$$e^x=i2$$
Taking the natural log of both sides:
$$x=\ln(i2)=\ln(2)... |
So, I (first year undergraduate mathematics student) was looking around the internet and found an [interesting enough piece of C++][1]. It contained the definition of something they called a 'recomplex' number so I'm assuming it's some kind of hypercomplex construction, which it appears to be. It also obeys the field a... |
What is this field in $\mathbb{R}^4$ that contains both the real and complex numbers called? |
Suppose $f$ is a function from a complete metric space $X$ to a metric space $Y$, and suppose $Y$ has points $y_{0}$, $y_{1}$ such that the subsets $f^{-1}(y_{0})$ and $f^{-1}(y_{1})$ are both dense in $Y$.
- Show that $f$ is not a pointwise limit of continuous functions $X \to
Y$.
- Show that in the above s... |
Consider the set of all the rectangles with dimensions $2^a\times 2^b\,a,b\in \mathbb{Z}^{\ge 0}$. We want to tile an $n\times n$ square with rectangles from this set (you can use a rectangle several times). What is the minimum number of rectangles we need?
If $f(n)$ is the sum of digits of $n$ in base $2$, I think ... |
I have two finite probability mass functions (pmfs) $P(x)$ and $Q(x)$ on the same support $(0,1,\ldots,n)$. Let $(p_0,p_1,\ldots,p_n)$ and $(q_0,q_1,\ldots,q_n)$ be the probability vectors from the two functions, $0<p_i<1$, $0<q_j<1$, $\sum_i^np_i=1$, $\sum_j^nq_j=1$.
It is known that the mean of $P(x)$ is less than... |
*Edit: In this answer I'm assuming the well-order of the reals is fixed at the outset and not somehow "chosen randomly", though, as commenter on another answer has woken me up to, you phrased the question as if were part of a random process. This would be a whole other can of worms (What distribution does this come fro... |
So, I (first year undergraduate mathematics student) was looking around the internet and found an [interesting enough piece of C++][1]. It contained the definition of something they called a 'recomplex' number so I'm assuming it's some kind of hypercomplex construction, which it appears to be. It also obeys the field a... |
Let us take Euler's identity, where $i^2=-1$
$$e^{i\pi}+1=0$$
Subtracting $1$ from both sides, we get
$$e^{i}=-1$$
What happens when we square both sides?
$$e^{i2}=1$$
This means that i2 is *also* a natural logarithm of $1$.
$$e^x=i2$$
Taking the natural log of both sides:
$$x=\ln(i2)=\ln(2)... |
I've been reviewing some basic optimal transport concepts ([ref Peyré and Cuturi][1]). I like the notation of the Kantorovich optimal transport problem.
$$ L_C(a,b) = \min_{ P \in U(a,b) } \langle P , C \rangle = \sum_{i,j} C_{i,j}P_{i,j} $$
The optimal transport cost between discrete probability measures $a$ and... |
I'm new to proof-writing in Real Analysis, and came across the following problem:
**Prove $lim(a_nb_n)=lim(a)lim(b)$**. **Let $lim(a), lim(b)$ exist and $lim(a)=L_a, lim(b)=L_b$**
The first thing I was tempted to do was the following manipulation...
$$|a_nb_n-L_aL_b|=|a_n(b_n-\frac{L_aL_b}{a_n})|\asymp L_a|b_n... |
Let $X$ and $Y$ be random variables taking values in the same finite abelian group $G$, so that we can define their sum $X+Y$. Let $(X_1, Y_1)$ and $(X_2, Y_2)$ be conditionally independent copies of $(X,Y)$ relative to $X+Y$; that is, for all $z\in G$, the random variables $(X_1, Y_1 | X+Y = z)$ and $(X_2, Y_2 | X+Y =... |
Note: this question is wrong – this is not a field, though it is not obvious why it wouldn't be.
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So, I (first year undergraduate mathematics student) was looking around the internet and found an [interesting enough piece of C++][1]. It contained the definition of something they called a 'recomplex' number so I'... |
Let us take Euler's identity, where $i^2=-1$
$$e^{i\pi}+1=0$$
Subtracting $1$ from both sides, we get
$$e^{i}=-1$$
What happens when we square both sides?
$$e^{i2}=1$$
This means that i2 is *also* a natural logarithm of $1$.
$$e^x=i2$$
Taking the natural log of both sides:
$$x=\ln(i2)=\ln(2)... |
Suppose I have a group of $n$ employees. Each one is expected to perform a certain task, and the result of this task is either Pass or Fail. As a part of their reviews, I have to tally up their totals, and each employee should have > 80% passing results.
But suppose there is an unknown (but constant) chance that the... |
I'm learning some ML algorithm online, which talks about the use of entropy as a measurement of impurity of the dataset. Suppose the dataset contains two types of objects: objects with positive label and objects with negative label. The entropy of the dataset is defined as $H(p)=-plog_2(p)-(1-p)log_2(1-p)$, where $p$ i... |
why the entropy of a data set is not defined as H(p)=p(1-p)? |
In this [paper][1] they talk about Sqrt-LASSO which is simply just trying to minimize $\|Ax-b\|_2 + \lambda\|x\|_1$ rather than the regular LASSO $\|Ax-b\|_2^2 + \lambda\|x\|_1$.
Can anyone point out the theoretical differences between the two in terms of whether one is more robust to outliers, do we still have sp... |
**Let f and g be functions such that for all real numbers x and y,
(( + )) = () + ( + )() . The value of (0) + (1)+. . . (2024) is?**
My attempt:
When x=0 and y=0 then, g(f(0))=f(0)..........(i).
When x=0 and y=1 then, g(f(1))=f(0)+g(1)
or, g(f(1)) =g(f(0))+g(1) [because f(0)=g(f(0)) from equation (i)]
... |
**Equivalent Problem**
Isn't this simply solving for the number of integer solutions of
$$
x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=7
$$
subject to $x_{i}\leq 3$? Here $x_{i}$ is the number of white balls in bowl $i$.
**Solution**
Because $4+4>7$, it's not possible to have more than one $x_{i}\geq 4$. Therefore, ... |
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