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Suppose that $(X,\mathcal{A})$ is a measurable space and let $\mathcal{A}^0$ be an algebra generating $\mathcal{A}$. For each $\epsilon>0$ and $A\in \mathcal{A}$, [one can find][1] $A^0\in\mathcal{A}^0$ such that $\mu(A\Delta A^0)<\epsilon$, where $\mu$ is a finite measure on $(X,\mathcal{A})$. Now, if we're given anot... |
I'm having trouble following how the authors got from equation 1 to equation 2.
We have a score $\nabla log p(y)$, measurement density, $p(y|x)$ (which is gaussian), and a posterior density $p(x|y)$. Using the chain rule, the probability density can be written:
\begin{equation}
p(y) = \int p(x) p(y|x) dx
\end{e... |
**Assuming the measures are non-negative:**
Consider the non-negative measure $\eta$ given by $\eta = \mu + \nu$. Do you see how you can proceed from here?
In full detail, given $A\in \mathcal{A}$ we can find $A^{0} \in \mathcal{A}^{0}$ such that _this new measure_ is small on $A\Delta A^{0}$: for $\varepsilon = ... |
So I know how to calculate percentile on a simple data set:
lets say student grades are 38, 47, 49, 58, 60, 65, 70, 79, 80, 92
Number of scores below 70 = 6
Using the percentile formula,
Percentile = (Number of Values Below “x” / Total Number of Values) × 100
Percentile of 70
= (6/1... |
percentile calculation on accumulated data? |
It is a known result that an infinite one-dimensional random walk almost surely crosses every point (according to Wikipedia at least). Can this result be generalized to random walks with probabilities depending on the current position?
Formally speaking, given a function from the integers to the reals $p_n$, does a ... |
When does a random one-dimensional walk with varying probabilities almost surely hit every value? |
Can someone help me prove the equation:
$$\sum_{k=0}^n \binom{2n}{2k} = 2^{2n-1}$$
I know that via binomial theorem for even k:
$$2^{2n} = (1+1)^{2n} = \sum_{k=0}^n \binom{2n}{k}$$
and for odd k: $$0 = (1-1)^{2n} = \sum_{k=0}^n \binom{2n}{k}\cdot(-1)^k$$
That's how i solve the equation:
$$2^{2n-1}=\frac{2^{2n}}... |
Let $V$ be a vector space with an increasing filtration $V_j, j\in \mathbb Z$, we assume the filtration is Hausdorff $\bigcap_j V_j=0$ and exhaustive $\bigcup_j V_j=V$. Consider the associated graded space $\text{gr}V=\bigoplus_j V_j/{V_{j-1}}$, when is $\text{gr}V\cong V$ as a vector space?
In the case of $V_j$ is ... |
## Context
Both the Fulton and Cox/Little/Schenck books on Toric Varieties include an exercise to show that if an inclusion $\tau\subset \sigma$ of cones induces an open embedding $U_\tau \hookrightarrow U_\sigma$, then $\tau$ is a face of $\sigma$.
[This answer](https://math.stackexchange.com/questions/4703972/if-... |
I'm having trouble following how the authors got from equation 1 to equation 2.
We have a score $\nabla \log p(y)$, measurement density, $p(y|x)$ (which is gaussian), and a posterior density $p(x|y)$. Using the chain rule, the probability density can be written:
\begin{equation}
p(y) = \int p(x) p(y|x) dx
\end{... |
Ironically, you can get a closed, [quantile](https://en.wikipedia.org/wiki/Quantile) special function, inverse using [this special case](https://www.wolframalpha.com/input?i=1%2F16+%2812+x+%2B+8+sin%282+x+%29+%2B+sin%284+x+%29%29+vs+beta%28sin%28x%29%5E2%2C1%2F2%2C5%2F2%29) of [Incomplete Beta function](https://mathwor... |
I'm trying to understand how to calculate the Hausdorff dimension of some profinite groups and came across the following paper ([here][1]) which defines the Hausdorff dimension for a closed subgroup $H \leq G$ as
$$dim(H)=\liminf_{n \rightarrow \infty} \frac{\log{ \vert H:H \cap G_n \vert }}{\log{\vert G:G_n \vert}}$... |
I have trouble understanding the first part of proof of the Theorem 10.25 of Bondy and Murty's textbook. The Theorem states "Overlapping bridges are either skew or equivalent 3-bridges", where skew bridges are defined as bridges with two couples of distinct vertices that alternate in cyclic order on the cycle where the... |
**Special Function Solution:**
Ironically, you can get a closed, [quantile](https://en.wikipedia.org/wiki/Quantile) special function, inverse using [this special case](https://www.wolframalpha.com/input?i=1%2F16+%2812+x+%2B+8+sin%282+x+%29+%2B+sin%284+x+%29%29+vs+beta%28sin%28x%29%5E2%2C1%2F2%2C5%2F2%29) of [Incompl... |
I am currently studying dynamic programming using the Bersketas book: Dynamic programming and optimal control, volume 1. The question is the following:
[![enter image description here][1]][1]
The solutions to certain 'selected' problems are provided online by the author, and this one has me perplexed. In order ... |
Suppose that f(x)g(x) is in $L^1([a,b])$ for every f(x) in $L^2([a,b])$, I understand that this implies that g(x) is in $L^2([a,b])$. Is this assertion true and, if so, how do I prove it?
I have seen a more general version of this question [here][1] but I do not fully follow the reasoning and I wonder if there is a ... |
Evaluating the infinite sum of the below trigonometric series yields a result of zero. However, when I calculate the finite sum up to $n$ and subsequently take the limit as $n$ approaches infinity on the result, it seems to be undefined.
We can explicitly perform the infinite sum for $x \in [0 , \frac{\pi}{2})$:
... |
Why is the infinite sum of this trigononmetric series =zero yet when I evaluate the finite sum and take the limit $n -> \infty$ it not well defined? |
Is there a question that both the knight **and** the knave contradicts? |
Let us assume we have linear elements $X = \{x_i\}_i^n$ on the d-sphere. Depending on the number of elements, we may find a configuration that is in expectation, orthogonal, subject to the minimization of $\Vert XX^T - I\Vert^2_F$, where $E_{x,y}[x^Ty] = 0$ and $E_{x,y}[(x^Ty)^2]$ is minimal.
In the case where we a... |
I am currently studying dynamic programming using the Bersketas book: Dynamic programming and optimal control, volume 1. The question is regarding the notation used, but is the following:
[![enter image description here][1]][1]
The solutions to certain 'selected' problems are provided online by the author, and ... |
I have a PDE and wanted to solve it using Lagrange's method
$(y+u) u_x + yu_y=x-y$
so the characteristic equation is
$\frac{dx}{y+u}=\frac{dy}{y}=\frac{du}{x-y}$
First, I want to work with
$\frac{dx}{y+u}=\frac{dy}{y}$
to find the $C_1$ to make the general solution of PDE : $F(C_1,C_2)=0$
However I... |
Suppose that $f(x)g(x)$ is in $L^1([a,b])$ for every $f(x)$ in $L^2([a,b])$, I understand that this implies that $g(x)$ is in $L^2([a,b])$. Is this assertion true and, if so, how do I prove it?
I have seen a more general version of this question [here][1] but I do not fully follow the reasoning and I wonder if there... |
I need to find the limit of a function $f(x,y)$ as $(x,y)\rightarrow (0,0)$. The only method I know of is to consider all paths through $(0,0)$ and do polar coordinate substitution to make it into a limit of $r\rightarrow0$.
$$\lim_{(x,y)\rightarrow(0,0)}f(x,y)=\frac{x^3-x^2y}{x^2+y^6}$$
So with polar coordinate subs... |
Let $H_n$ and $T_n$ be the number of heads and tails, respectively, in $n$ flips. Since $H_n+T_n=n$ for all positive integers $n$, we have
$$ P[H_n\geq T_n] = P[H_n\geq n/2] \quad \forall n \in \{1, 2, 3, ...\}$$
Let $X_i$ be an indicator function that is 1 if flip $i$ is heads, and 0 else, so
$$H_n = \sum_{i=1}^n... |
Why is the infinite sum of this trigonometric series =zero yet when I evaluate the finite sum and take the limit $n -> \infty$ it not well defined? |
I am learning about connections using Lee's Intro to Riemannian Manifolds, and I am confused regarding the coefficients of the pullback connection. For the setting, let $(M,g) $ be a Riemannian manifold and let $\nabla$ be a connection on $M$. From Prop 4.7 in Lee's book, if we have two local frames $(E_i)$ and $(\tild... |
Let $V$ be a vector space with an increasing filtration $V_j, j\in \mathbb Z$, we assume the filtration is Hausdorff $\bigcap_j V_j=0$ and exhaustive $\bigcup_j V_j=V$. Consider the associated graded space $\text{gr}V=\bigoplus_j V_j/{V_{j-1}}$, when is $\text{gr}V\cong V$ as a vector space?
In the case of $V_j$ is ... |
Let us assume we have linear elements $X = \{x_i\}_i^n$ on the d-sphere. Depending on the number of elements, we may find a configuration that is in expectation, orthogonal, subject to the minimization of $\Vert XX^T - I\Vert^2_F$, where $E_{x,y}[x^Ty] = 0$ and $E_{x,y}[(x^Ty)^2]$ is minimal.
In the case where we a... |
For humans, in general, it's far easier to do an addition than a multiplication. One might argue this is just an effect of the particular way brains are wired.
But from what I find, multiplication is also more expensive on computers, where one could certainly arrange chips in an arbitrary way, or at least without ... |
Can any characterized property give a solid account of why multiplication is a harder computation operation than addition? |
Let $V$ be a vector space with an increasing filtration $V_j, j\in \mathbb Z$, we assume the filtration is Hausdorff $\bigcap_j V_j=0$ and exhaustive $\bigcup_j V_j=V$. Consider the associated graded space $\text{gr}V=\bigoplus_j V_j/{V_{j-1}}$, when is $\text{gr}V\cong V$ as a vector space?
In the case of $V_j$ is ... |
Is there a question that both the knight **and** the knave contradicts? |
I am told relations are (sets of?) ordered pairs. How do these relate to ordered sets? Are ordered sets ordered by ordered pairs (i.e., relations)? |
What’s the difference between an ordered pair and an ordered set? |
Is there a simpler expression for $\sum_{j=0}^\infty 2^{I(j\neq0)} \cos j\theta \Gamma(2j+1) {\mathbb I}_j(\kappa) $, for large $\kappa$? |
How economists came up with the Cobb Douglas function? How can I deduce it by myself? |
In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs (particularly physical capital and labor) and the amount of output that can be produced by tho... |
**Context**
I understand from [1] that to define the contour integral, let $f:\mathbb {C} \to \mathbb {C} $ be a continuous function on the directed smooth curve $\gamma$. Then the integral along $\gamma$ is denoted
$$\int _{\gamma }f(z)\,dz.$$
For a research project that I am working on in pseudo-Riemannia... |
Euclid's proof of the infinitude of primes goes like this:
Let $p_i$ be the $i$-th prime number.
Then for every $n$, $1+\prod_{1\leq i\leq n}p_i$ is not divisible by any of the first $n$ primes, so its prime divisors must be greater than $p_n$. Thus, we have shown that for every prime, there is a larger prime.
... |
Does the Cauchy Integral Formula by itself proves that the Dirichlet Problem is solvable for any simply connected region?
I was looking for the proof of the Poisson Integral Formula and one uses the fact that if $u$ is harmonic in the closed disk $\bar D$ then (because the disk is simply connected) there is $f=u+iv... |
Why is the infinite sum of this trigonometric series =zero yet when I evaluate the finite sum and take the limit $n \to \infty$ it not well defined? |
I was studying polynomial rings over commutative rings from the book Topics in Algebra by I.N Herstein.
From there what I understood was, that:
If $R$ be a commutative ring with a unit element then $R[x]$ is defined to be the set of all symbols $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ where, $a_0,a_1,...,a_n\in R$ and... |
This question is part of the original question: https://math.stackexchange.com/posts/4875789/edit.
For each ${n}$, let ${A_n = (a_{ij,n})_{1 \leq i,j \leq n}}$ be an ${n \times n}$ random sign matrix (i.e. the entries ${a_{ij,n}}$ of ${A_n}$ are jointly independent in ${i,j}$ and take values in ${\{-1,+1\}}$ with a ... |
I'm planning to use an algorithm called [Harmony](https://github.com/immunogenomics/harmony), designed for data normalization, particularly in the context of single cell data analysis. Harmony operates by taking principal components (*PC*s) as input and outputting corrected principal components (*PC'*).
Typically, H... |
In Borevich & Shafarevich's *Number Theory*, Theorem 6.2.3 claims that
> Let $p \neq 2$ and $0<r<n$. The form $$F = F_0+pF_1$$ represents zero in the field $\mathbb{Q}_p$ iff at least one of the forms $F_0$ or $F_1$ reprensents zero, where $$F_0 = \epsilon_1 x_1^{2} + \dots + \epsilon_r x_r^{2}, F_1 = \epsilon_{r+1} ... |
Let $f:[0,\infty)\to[0,\infty)$ be a smooth, strictly monotonically increasing function. Consider the inequality
$$
\qquad \quad f(n) \cdot \frac{n-1}{n^2} > a \qquad (n\in\{2,3,4,...\};\ a>0). \qquad (*)
$$
Assume that inequality $(*)$ holds for $n=2$. Notice that $\frac{n-1}{n^2}$ is strictly monotonically decr... |
I'm having trouble understanding Edward/Pierre's argument showing that the distance function is subanalytic. Specifically, "Remarks 3.11" in Semianalytic and subanalytic sets (https://link.springer.com/article/10.1007/BF02699126). Below is a direct copy of "Remarks 3.11".
$\textit{Remarks}\; \mathbf{3.11.}$ Let $X$ ... |
Let $\mathbb{R}^n$ be $n$-dimensional Euclidean space and $S^n$ be $n$-sphere.
Then, it is well-known that $S^n$ is the one-point compactification of $\mathbb{R}^n$.
Now consider $C^\infty(S^n)$, the Frechet space of smooth functions on $S^n$.
Then, my question is
> Which function space on $\mathbb{R}^n$ is... |
Which function space on $\mathbb{R}^n$ is identified with $C^\infty(S^n)$? |
At the beginning of "Analytic Combinatorics" by Flajolet and Sedgewick, there's some discussion about the "usefulness" of the book's techniques in the analysis of algorithms and computational biology. However, I've struggled to find recent papers that employ techniques from analytic combinatorics, as well as to find an... |
Suppose that $X$ is a well-pointed topological space (i.e. the basepoint is cofibered, not necessarily closed). Is the Moore loop space/space of measured looks $\Omega_MX$ well-pointed?
To recap, the Moore loop space is the subspace $\Omega_MX$ of $X^{[0,\infty)}\times[0,\infty)$ consisting of pairs $(\gamma,t)$ s.t... |
Is the Moore Loop space of a well-pointed space well-pointed? |
Suppose that $X$ is a well-pointed topological space (i.e. the basepoint is cofibered, not necessarily closed). Is the Moore loop space/space of measured looks $\Omega_MX$ well-pointed? I have seen this claimed in multiple places, but have not been able to locate or produce a proof.
To recap, the Moore loop space is... |
Let $f(x)$ be a continuous strictly increasing smooth function on $[0,1]$ with $f(0)=0$ and $f(1)=1$. In addition, $F(t)=\int_0^t f(x)dx$ and $\phi>1$. Prove that the following function $h(t)$ is strictly increasing with respect to $t$ on $(0,1)$, where
$$h(t)=\frac{1-t+F(t)+t(1-f(t))}{(\phi-t)(1-f(t))}-\frac{F(1)}{... |
How to prove a function is strictly increasing? |
Name for maximum size of common neighbors between all the pairs of vertices? |
Let $f(x)$ be a continuous strictly increasing smooth function on $[0,1]$ with $f(0)=0$ and $f(1)=1$. In addition, $F(t)=\int_0^t f(x)dx$ and $\phi>1$. Prove that the following function $h(t)$ is strictly increasing with respect to $t$ on $(0,1)$, where
$$h(t)=\frac{1-t+F(t)+t(1-f(t))}{(\phi-t)(1-f(t))}-\frac{F(1)}{... |
Let $G$ be a finite group.
In M. Isaacs "Character theory of finite groups", chap. 2, we cover the fact that $\mathbb{C}[G] = \bigoplus_{i = 1}^{k} M_i(\mathbb{C}[G])$, where $M_i(\mathbb{C}[G])$ is the sum of all submodules of $\mathbb{C}[G]$ that are isomorphic to an irreducible $\mathbb{C}[G]$-module $M_i$. The rea... |
Name for maximum size of common neighborhoods between all the pairs of vertices? |
Let $F$ be a cdf with associated pdf $f$ on support $[0,b]\subseteq R_+$. Suppose $f$ is log-concave, which implies that $F$ and $1-F$ are also log-concave. Specifically, we have that $\frac{f'}{f}$ and $\frac{1-F}{f}$ are both decreasing functions, and $\frac{F}{f}$ is an increasing function. Is either $\frac{F}{f}\cd... |
Is this a property of a log-concave distribution? |
It is known that
$\pi_7(\mathbb S) = \mathbb Z/240$
and
$\pi_{11}(\mathbb S) = \mathbb Z/504$.
Is this connected to the normalized Eisenstein series,
$E_4(\tau) = 1 + 240\sum\sigma_3(n)q^n$
and
$E_6(\tau) = 1 - 504\sum\sigma_5(n)q^n$
in some direct way?
|
Let $\mathsf{ZFC_2}$ denote $\mathsf{ZFC}$ with a second-order replacement axiom. It has been discussed in some [other][1] [answers][2] that every model of $\mathsf{ZFC_2}$ is isomorphic to $V_\kappa$ for some inaccessible cardinal $\kappa$. But it seems to me that the intended "model" of $\mathsf{ZFC}$ is supposed to ... |
Is $\mathsf{ZFC_2}$ "class categorical"? |
Let $f(x)$ be a continuous strictly increasing smooth function on $[0,1]$ with $f(0)=0$ and $f(1)=1$. In addition, $F(t)=\int_0^t f(x)dx$ and $\phi>1$. Prove that the following function $h(t)$ is strictly increasing with respect to $t$ on $(0,1)$, where
$$h(t)=\frac{1-t+F(t)+t(1-f(t))}{(\phi-t)(1-f(t))}-\frac{F(1)}{... |
Let $\mathsf{ZFC_2}$ denote $\mathsf{ZFC}$ with a second-order replacement axiom. It has been discussed in some [other][1] [answers][2] that every model of $\mathsf{ZFC_2}$ is isomorphic to $V_\kappa$ for some inaccessible cardinal $\kappa$. But it seems to me that the intended "model" of $\mathsf{ZFC}$ is supposed to ... |
I wanted to ask some clarification on one of the proofs in Rudin's PMA, specifically Theorem 3.11 (c). It states that
>In $\mathbb R^k$, every Cauchy sequence converges.
The proof for it is as follows:
> Let $\{\mathbf x_n\}$ be a Cauchy sequence in $\mathbb R^k$. Define $E_N$ to be the set $\mathbf x_N, \mathbf x... |
Let $\mathsf{ZFC_2}$ denote $\mathsf{ZFC}$ with a second-order replacement axiom. It has been discussed in some [other][1] [answers][2] that every model of $\mathsf{ZFC_2}$ is isomorphic to $V_\kappa$ for some inaccessible cardinal $\kappa$. But it seems to me that the intended "model" of $\mathsf{ZFC}$ is supposed to ... |
I'm trying to find all integral solutions to the equation
$ \frac{1}{x} + \frac{1}{y}= \frac{1}{4} $
Is there an easier method than just brute-forcing it? |
Let $\{\mathcal{F}_t\}_{0\le t\le T}$ be a filtration on $(\Omega,\mathcal{F},P)$, to which the process $\lambda_t$ is adapted. Let $\{W(t),\mathcal{F}_t,0\le t\le T\}$ be a Brownian motion and $\mathcal{G}_t\subset\mathcal{F}_t$ be a sub-filtration, suppose further that $W(t)$ is also adapted to $\mathcal{G}_t$.
Co... |
I have the following system of equations
$$X R a - c_1 = 0$$
$$X a - c_2 = 0$$
$$X X^T=Identity$$
where $X\in\mathbb{R}^{3x3}$, $a,c_1,c_2\in\mathbb{R}^{3x1}$, $R\in\mathbb{R}^{3,3}$ is a rotation matrix.
For some $a,R,c_1,c_2$ the system has two possible $X$'s as a solution. In cases where there is no solutio... |
Let $G$ be a finite group.
In M. Isaacs "Character theory of finite groups", chap. 2, we cover the fact that $\mathbb{C}[G] = \bigoplus_{i = 1}^{k} M_i(\mathbb{C}[G])$, where $M_i(\mathbb{C}[G])$ is the sum of all submodules of $\mathbb{C}[G]$ that are isomorphic to an irreducible $\mathbb{C}[G]$-module $M_i$. The rea... |
I have the following system of equations
$$X R a - c_1 = 0$$
$$X a - c_2 = 0$$
$$X X^T=Identity$$
where $X\in\mathbb{R}^{3x3}$, $a,c_1,c_2\in\mathbb{R}^{3x1}$, $R\in\mathbb{R}^{3,3}$ is a rotation matrix.
For some $a,R,c_1,c_2$ the system has two possible $X$'s as a solution. In cases where there is no solutio... |
Let $G$ be a finite group.
In M. Isaacs "Character theory of finite groups", chap. 2, we cover the fact that $\mathbb{C}[G] = \bigoplus_{i = 1}^{k} M_i(\mathbb{C}[G])$, where $M_i(\mathbb{C}[G])$ is the sum of all submodules of $\mathbb{C}[G]$ that are isomorphic to an irreducible $\mathbb{C}[G]$-module $M_i$. The rea... |
Let $G$ be a finite group.
In M. Isaacs "Character theory of finite groups", chap. 2, we cover the fact that $\mathbb{C}[G] = \bigoplus_{i = 1}^{k} M_i(\mathbb{C}[G])$, where $M_i(\mathbb{C}[G])$ is the sum of all submodules of $\mathbb{C}[G]$ that are isomorphic to an irreducible $\mathbb{C}[G]$-module $M_i$. The rea... |
My professor gave us the following question:
> Refute (with a simple example): Let ,:ℝ→ℝ be two convex functions.
> The composition ℎ≜∘ (that is, ℎ()=(())) is also a convex
> function.
But from what I can read online the composition of two convex functions is convex as well, what am I missing here?
https://m... |
The composition of Convex functions? |
I'm trying to find all integral solutions to the equation
$$ \frac{1}{x} + \frac{1}{y}= \frac{1}{4} $$
Is there an easier method than just brute-forcing it? |
I'm trying to find all integral solutions to the equation I found in a competition math packet
"Find the number of integer solutions (x, y) to
if, for instance, (x, y) = (2, −4) and (x, y) = (−4, 2) are counted as different integer
solutions."
$ \frac{1}{x} + \frac{1}{y}= \frac{1}{4} $
Is there an easier m... |
Somehting like this is explored in [Hopkins: Algebraic Topology and Modular Forms](https://arxiv.org/pdf/math/0212397.pdf) (very similar looking piece at PDF page 14, internal page 296). Skimming (so take with a grain of salt), it looks like only "$c_4$" (a.k.a. $E_4$) and "$c_6$" ($E_6$) are used and (if there are an... |
I am looking at repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integers $a$ and $b$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
For exponentiat... |
I was fiddling around with Arithmetic Progressions and I noticed this pattern.
\begin{align}
n*T_n = m*T_m \implies T_\text{n+m} = 0
\end{align}
where $n, m \in \{0, \mathbb{Z}^{+}\}$ and $n \neq m$.
I could prove that it's true like so:-
\begin{align*}
& m*T_m = n*T_n \\
& \Rightarrow m\{a + ... |
I am looking at repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integers $a$ and $b$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
For exponentiat... |
I am looking at repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integers $a$ and $b$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
For exponentiat... |
I am looking at repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integers $a$ and $b$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
For exponentiat... |
Assume that $c$ is an arbitrary non-vanishing antisymmetric bilinear form on $\mathbb{R}^2$ and view $\mathbb{R}^2$ as an Abelian Lie algebra. Can $c$ define a non-trivial real valued two-cocycle on $\mathbb{R}^2$? |
> If $p(x) \in \mathbb{Q}[X]$ is irreducible and has two roots $r,s$ such that $rs = 1$, then $p(x)$ is of even degree.
I'm not sure how to solve this problem. My initial idea was to consider the field tower $\mathbb{Q}(r, r+s) \supseteq \mathbb{Q}(r+s) \supseteq \mathbb{Q}$. We know that $\mathbb{Q}(r, r+s) = \math... |
I am looking at repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integers $a$ and $b$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
For exponentiat... |
I am looking at repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integers $a$ and $b$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
For exponentiat... |
I am looking at repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integers $a$ and $b$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
For exponentiat... |
Using the Banach fixed-point theorem, I can prove that there exists exactly one continuous function $f:[0,1]→\mathbb{R}$ that satisfies
$$f(x) = \frac{1}{2}f(3x)+\cos x.$$
I currently don't know how to solve this $f$ precisely. My attempt is similar to Picard iteration. Define
$$Tf = \frac{1}{2}f(3x)+\cos x.$$
Le... |
How to solve $f(x)=\frac{1}{2} f(3x) + \cos x$? |
> If $p(x) \in \mathbb{Q}[X]$ is irreducible and has two roots $r,s$ such that $rs = 1$, then $p(x)$ is of even degree.
I'm not sure how to solve this problem. My initial idea was to consider the field tower $\mathbb{Q}(r, r+s) \supseteq \mathbb{Q}(r+s) \supseteq \mathbb{Q}$. We know that $\mathbb{Q}(r, r+s) = \math... |
I just came to know that it is possible to define a _generalized base_ as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\in \mathbb N$ can be uniquely expressed as
$$m=\overline{x_n\dots x_1}^\mathbf b := x_1 + x_2\cdot b_1 + x_3\cdot b_1b... |
Let $X=[X_1,\dots, X_n]$ be a random vector with $X_i \in (0,2)$ and constant vector $a=[a_1,\dots, a_n]$ with $a_i \in [0,1]$ and $\sum_{i=1}^n a_i = 1$. Consider the variance
$$
v(K):={\rm var}( \log a^\top X)
$$
Is it possible to tell if the function $v(K)$ is convex or concave in $K$?
My attempts: We know th... |
Using the Banach fixed-point theorem, I can prove that there exists exactly one continuous function $f:[0,1]→\mathbb{R}$ that satisfies
$$f(x) = \frac{1}{2}f(3x)+\cos x.$$
I currently don't know how to solve this $f$ precisely. My attempt is similar to Picard iteration. Define
$$Tf = \frac{1}{2}f(3x)+\cos x.$$
Le... |
Let $X:=[X_1,\dots, X_n]$ be a random vector with $X_i \in (0,2)$ and take a constant vector $a:=[a_1,\dots, a_n]$ with $a_i \in [0,1]$ and $\sum_{i=1}^n a_i = 1$. Consider the variance
$$
v(K):={\rm var}( \log a^\top X)
$$
Is it possible to tell if the function $v(K)$ is convex or concave in $K$?
My attempts: W... |
Let $A$ be a $m\times n$ matrix and $\boldsymbol{v}$ be a $n$-dimensional vector and $\boldsymbol{u}$ be a $m$-dimensional vector. I have difficulty understanding $\boldsymbol{u}\cdot (A\boldsymbol{v})=\boldsymbol{u}^TA\boldsymbol{v}$, especially $\boldsymbol{u}^TA$. Based on my understanding $A\boldsymbol{v}$ returns ... |
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