instruction stringlengths 12 30k |
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First, is this question well-defined? That is, the homomorphisms are determined by the order n.
Since $R/Z$ is the multiplicative unit circle, all finite subgroups are cyclic, so the image of the homomorphisms must be cyclic. My intuition is there exists bijection between cyclic subgroups of $G$, but I don't know ho... |
Let $G$ be a finite abelian group of order n. How many distinct group homomorphisms are there from $G$ to $R/Z$? |
*A standard $6$-sided fair die is rolled until the last $3$ rolls are ascending. What is probability that the last $3$ digits start with $1,2,3,4$ respectively?*
**My attempt**
We can investigate the $3-$roll (when there are exactly $3$ rolls), the $4-$roll, the $5-$roll, the $n-$roll.
I did that via SQL, basica... |
Is this proof vigorous enough?
Also, is the question complete? In the textbook that I'm using, a $k$-connected graph $G$ means at least $k$ vertices must be removed to disconnect $G$. But after removing $2$ vertices from a triangle graph, isn't a graph with one vertex connected?
(UPDATED) Proof:
Let $G$ be a ... |
Is this proof vigorous enough?
Also, is the question complete? In the textbook that I'm using, a $k$-connected graph $G$ means at least $k$ vertices must be removed to disconnect $G$. But after removing $2$ vertices from a triangle graph, isn't a graph with one vertex connected?
(UPDATED) Proof:
Let $G$ be a ... |
Let $M$ be a compact metric space and $\mathcal C^0(M)=\{f:M\to\mathbb R; f\ \text{is continous}\}$. Let $T:\mathcal C^0(M)\to \mathcal C^0(M)$ be a bounded linear transformation.
Suppose that for every $f\in\mathcal C^0(M)$ we have that $Tf$ is $1/2$-Hölder. Does this imply that $T$ is compact?
----------
It ... |
Let $R \subset S$ be an injective map of commutative rings, $M \subset N $ an inclusion of $R$-modules. We identify $M, N$ respectively as $M \otimes_R 1_S \subset N \otimes_R 1_S \subset N \otimes S$.
In [this answer][1] Haran elaborated that if $R \subset S$ a faithfully flat extension, then the intersection o... |
An answer, which needs at the end a complementary study to be fully rigorous.
First of all, it is clear that setting $y=2x$ in the implicit equation gives $0$. Therefore, it is not surprizing that the following graphical representation of the implicit equation produced by GeoGebra has 2 branches : the straight line ... |
Define $f:A \to A$ where $A$ contains $n$ distinct elements. How many functions exist such that $ \forall x \in A, f^m(x)=x$, $(m<n)$ where $f^m$ represents the $m^{th}$ composition of $f(x)$?
The key idea I recognized here was that there are only two ways in which this rule can be followed; assigning elements in ... |
I am reading this paper "[Choosability and fractional chromatic numbers][1]"
It concerns the fractional chromatic numbers of a graph $G$: let $\mathcal{S}(G)$ be the collection of independent sets in $G$. A fractional coloring is a function $f:\mathcal{S}(G)\rightarrow\mathbb{R}_{\ge 0}$ such that $\sum_{S\in\mathca... |
Given $a^2-b^2=37$ and also a and b are integers, can we evaluate $a^2+b^2$ possible values? Are those many or just some? I found that $a^2+b^2$ can be only $685$. But how to proove it? I just guessed, but can we somehow evaluate it? |
TL;DR
Given a smooth projective variety $X\subseteq \mathbb{P}^m$ (I guess this means that the homogeneous coordinate ring $S(X)$ is regular whenever localizing at a non-maximal graded prime ideal). Is there an another projective variety $Y\subseteq \mathbb{P}^n$ isomorphic to $X$ such that the homogeneous coordinat... |
My textbook has the following problem:
> There is 52 cards in a deck and 13 cards of each type/color. You are drawing 5 cards. Whats the probability of all these 5 cards being the same type?
My solution:
There is a $\frac{52}{52}$ probability of drawing the first card, then a $\frac{12}{51}$ chance of drawing ... |
What is the probability of drawing 5 cards of the same type from a deck of 52 cards? |
Given $a^2-b^2=37$ and also a and b are integers, can we evaluate $a^2+b^2$ possible values? Are those many or just some? I found that $a^2+b^2$ can be only $685$. But how to prove it? I just guessed, but can we somehow evaluate it? |
In the book ["A Graduate Course on Statistical Inference"][1] (as you can see in this link) there's the following passage: "[a function] $f(\theta )$ is differentiable at $\theta _0$ if and only if, for every $\theta$ in a neighborhood $\theta_0$, the function $f((1-t)\theta_0+t\theta )$ is differentiable with respect ... |
It sounds like you want to know why it is “helpful” or “intuitive” for a function to have a codomain.
Just to be clear (I don’t think this is what you’re asking about), it’s not possible to make a function with *no idea* of what the codomain is. How could you specify a rule for computing the function without giving ... |
Let $f$ be a twice differentiable function over real line. Let $a\in \mathbb{R}$ and $h\in \mathbb{R}^{+}$. Let $P_f$ denote the interpolating polynomial of degree $2$ of $f$, i.e.,
$$f(a)=P_f(a), \qquad f(a-h)=P_f(a-h), \qquad f(a+h)=P_f(a+h).$$
Let $g$ be any polynomial of degree $d$ such that
$$g''(a)=P_g''(a).$... |
I am following [this][1] proof (actually, the "second" one) about the independence of sample mean and sample variance in the normal case.
I understand the computations but still a (big) element is missing in my understanding: why is the vector $\begin{pmatrix} \bar{X}, X_1 - \bar{X}, \dots, X_n - \bar{X}\end{pmatri... |
Prove that normally marginally distributed variables are also jointly distributed? |
I have a universal first order universal theory T such that:
-T is universal over a finite language L
-T has a model completion T ⊆ T*
-T* is countable, has quantifier elimination (QE) and is complete.
Now considering a theory T ⊆ T' over L, suppose T' has QE. Can we say that T' is (at most) countable? If s... |
I am following [this][1] proof (actually, the "second" one) about the independence of sample mean and sample variance in the normal case.
I understand the computations but still a (big) element is missing in my understanding: why is the vector $\begin{pmatrix} \bar{X}, X_1 - \bar{X}, \dots, X_n - \bar{X}\end{pmatri... |
From the [Mosek cookbook][1], it is clear that by using the log-sum-exp transformation and some extra variable $u_k$, a geometric programing problem can be transform into a exponential cone presentation.
Therefore, my question is "Can geometric programming be considered as a special case of second order cone program... |
Can geometric programming be considered as a special case of second order cone programming? |
I understand probability and statistics as a two way route. Either I begin from a random variable (r.v.) $X$ and sample from its probability distribution, or I observe a sample of values $x_1,...,x_n$ drawn from that distribution and I infer something about the distribution of the r.v. $X$. I have a conceptual problem ... |
In the book ["A Graduate Course on Statistical Inference"][1] (as you can see in this link) there's the following passage: "[a function] $f(\theta )$ is differentiable at $\theta _0$ if and only if, for every $\theta$ in a neighborhood $\theta_0$, the function $f((1-t)\theta_0+t\theta )$ is differentiable with respect ... |
Let $K$ be a number field and $K_v$ be the completion of $K$ at the place $v$. Consider an elliptic curve $E/K$ over $K$.
Let $f: H^1(G_K,E)\to \bigoplus_v H^1(G_{K_v},E)$ be a natural map given by
$f \mapsto (f\mid_{G_{K_v}})_v$
For $M$: profinite abelian group $M$, we let $\widehat{M}=\text{Hom}(M, \Bbb{Q}/\... |
Wieferich-prime question about the current year :
I found only $4$ base-$2024$-Wieferich primes , in other words prime numbers $p$ satisfying $$2024^{p-1}\equiv 1\mod p^2$$
gp > forprime(p=1,10^6,if(Mod(2024,p^2)^(p-1)==1,print1(p," ")))
3 5 17 2311
gp >
The only really interesting case is the p... |
Is another base-$2024$-Wieferich prime known? |
Let $\overline{M}=M \cup \partial M$ be a manifold with boundary, where $\partial M$ is the boundary of $\overline{M}$ and $M$ is the interior of $\overline{M}$.
Let $P$ be an injective elliptic partial differential operator of order $2$ on $\overline{M}$.
For $k\geq 0$, let $u \in H^{2+k}$, the $L^2$-Sobolev space... |
I am aware that related topics have been discussed in other threads. However, not being an expert, I would like to know the answers to the following questions:\
Let $E$ be a set having n elements ; $ |E|=n.$\
$A_p$ is a subset of $E$ with $p$ elements $(1 \le p \le n)$, chosen uniformly at random from the set of $\d... |
Let $\alpha_{i},\beta_{i} \in \mathbb{C}$, $\forall i= 1,2$ and $0 < a < b < \infty$.
\begin{align}
&f^{\prime \prime} + \alpha_{1} f = \alpha_{2} g^{\prime} \quad \text{in} \quad [a,b] \\
&g^{\prime \prime} - \beta_{1} g = \beta_{2}f^{\prime} \quad \text{in} \quad [a,b].
\end{align}
with $f(b) = g(a)=g(b)=0$.
... |
$f(x) = \sum_{j=1}^{4}c_{j}e^{r_{j}x} \quad \text{and} \quad g(x) = \sum_{j=1}^{4}d_{j}e^{s_{j}x}$. How to determine $c_{j},d_{j}$? |
It is relatively well-known that any arrangement of points that can be constructed with a straightedge and compass can also be constructed with an unstretchable string (of arbitrary length, negligible thickness/stiffness, etc.) However, a string is capable of generating a much larger array of curves (or surfaces) than ... |
- My understanding about fractional order system is that it can have poles on the right hand side of imaginary axis in s-plane and yet being stable. (statement 1)
There are other theorems about stability that cannot be true at the same time with statement 1.
- The second understanding is that Region Of Convergenc... |
This question is designed for high school students, so advanced linear algebra solution might not be needed.
Suppose I have a $\vec{G}$ as a meet point between vector $\vec{DB}$ and $\vec{CE}$ as illustrated here:
[![enter image description here][1]][1]
If $\vec{AD}:\vec{DC}=3:1$ and $\vec{AE}:\vec{EB}=1:2$, ... |
How to find a vector from the intersection of 2 vectors? |
warning: This may be a stupid question, because I am poor at differential euqations
$v_p\in T_p\mathbb{R}^n$ is a linear function over $C_p^\infty (\mathbb{R}^n)$, where $C_p^\infty (\mathbb{R}^n)$ denote the functions over $\mathbb{R}^n$ who are smooth at $p$. And $T_p\mathbb{R}^n$ has basis of $\left\{ \frac{\parti... |
Can $C_p^\infty(\mathbb{R}^n)$ be considered as a dual space of $T_p\mathbb{R}^n$, where $p\in \mathbb{R}^n$? |
All values that satisfy $x^2=2^x$ would satisfy $ln(2)x^3=xln(2)e^{xln(2)}$, and would therefore satisfy the relationship$W(ln(2)x^3)=xln(2)$. The problem is that when I graph these functions, there are four real solutions to this relation: 4, 2, ~-.77, and 0. The first three are solutions to $2^x=x^2$, but 0 is not. S... |
Proof using the Lambert W function that 1 = 0 - What went wrong? |
This question is designed for high school students, so advanced linear algebra solution might not be needed.
Suppose I have a $\vec{G}$ as a meet point between vector $\vec{DB}$ and $\vec{CE}$ as illustrated here:
[![enter image description here][1]][1]
If $\vec{AD}:\vec{DC}=3:1$ and $\vec{AE}:\vec{EB}=1:2$, ... |
f(x)=0.7x^2
g(x)=2.8x^2
Interval: 4<=x<=8
How do I find the average rate of change over f(x) and g(x) over the Interval? |
How do you find the average rate of change over an interval that is a compound inequality? |
warning: This may be a stupid question, because I am poor at differential euqations
$v_p\in T_p\mathbb{R}^n$ is a linear function over $C_p^\infty (\mathbb{R}^n)$, where $C_p^\infty (\mathbb{R}^n)$ denote the functions over $\mathbb{R}^n$ who are smooth at $p$. And $T_p\mathbb{R}^n$ has basis of $\left\{ \frac{\p... |
I am exploring Taylor's Classical Mechanics and I am unsure about the mathematics when he derives the minimum of the action. We define this function $f(x(t),\dot{x}(t),t)$ to be a Lagrangian. We define the action to be
$$ S = \int_{t_{1}}^{t_{2}} f(x(t),\dot{x}(t),t)dt$$
Let some function $x$ satisfy the condition ... |
I’m preparing my thesis on the paley wiener theorem and I’m stuck at the beginning, can you please provide some references for Fourier analysis on the multiplicatif groupe R* and on the paley wiener theorem there? |
Find references on the paley wiener theorem on the multiplicatif group R*? |
How to solve for the angle $x$ ?($\tan(x)+ 4 \sin(x)=\sqrt 3$) |
$\newcommand{\T}{\mathscr{T}}\newcommand{\C}{\mathscr{C}^\infty}$I've been enjoying Iversen's book on sheaf cohomology. He briefly mentions De Rham cohomology and a sheafy perspective on it but he doesn't mention if the canonical isomorphism is functorial; everyone says that it is, but with the sheaf perspective I'm no... |
How is the sheafy De Rham cohomology functorial? |
A general question:
I came across a vague statement that 'a uniform version of a theorem is one where the existence of objects concerned only depends on few parameters’, but that gives little intuition.
What exactly does one call a *uniform theorem*? |
It is relatively well-known that any arrangement of points that can be constructed with a straightedge and compass can also be constructed with an unstretchable string (of arbitrary length, negligible thickness/stiffness, etc.) However, a string is capable of generating a much larger array of curves (or surfaces) than ... |
How do you solve this integral? |
Let $G=K_{m,\ldots,m}$ be a complete $r$-partite graph. How to show that for any $(r-1)$-partition $V_1, \ldots, V_{r-1}$ of $V(G)$, the number of edges within a part is at least $m^2$ (i.e., $\sum_{1\leq i\leq r-1}e_G(V_i)\geq m^2$)?
If we consider a $k$-partition for certain $k\leq r-1$, what can we get? |
I'm currently starting to learn K-homology and something bothers me about the definition of Fredholm modules. I looked in 5/6 papers and each time a different definition is given... For example in [1] and [2]:
**Definition (Fredholm module)** : Let $\mathcal{A}$ be a separable $C^*$-algebra. A **Fredholm module** ov... |
This question is designed for high school students, so advanced linear algebra solution might not be needed.
Suppose I have a $\vec{G}$ as a meet point between vector $\vec{DB}$ and $\vec{CE}$ as illustrated here:
[![enter image description here][1]][1]
If $\vec{AD}:\vec{DC}=3:1$ and $\vec{AE}:\vec{EB}=1:2$, ... |
Let $f(x)=0.7x^2$ and $g(x)=2.8x^2$.
How do I find the average rate of change over $f(x)$ and $g(x)$ over the interval $[4,8]$? |
Let \(S\) be the set of all positive integers such that each element \(n\) in \(S\) satisfies the following conditions:
1. The sum of the digits of \(n\) is a prime number.
2. The sum of the digits of \(n^2\) is a perfect square.
Prove or disprove the existence of an infinite subset \(T\) of \(S\) such that ... |
A general question:
I came across a vague statement that 'a uniform version of a theorem is one where the existence of objects concerned only depends on few parameters’, but that gives little intuition.
Is it only about using a principle multiple times?
What exactly does one call a *uniform theorem*?
Examples. To... |
Here is the setting
X:
Positive semi-definite;
Non-negative entries;
||X||_∞ = λ (largest absolute row sum is λ).
G:
Diagonal matrix;
Non-negative diagonal entries g_i;
All g_i <= τ (for some positive τ).
XG:
all XG’s eigenvalue are within unit circle.
I want to bound the largest element of ... |
Is infinity norm submulicative? What about its power? |
I’m trying to check that $\mathfrak{su}(m,n), \mathfrak{sp}(n,\mathbb R), \mathfrak{so}^*(2n)$ are closed under the conjugate transpose as described in Anthony Knapp’s “Lie Groups Beyond an Introduction”. On page 60, Knapp asserts that this follows from the definitions of these groups as
$$\mathfrak{su}(m,n) = \{X\i... |
Let $S$ be the set of all positive integers such that each element $n$ in $S$ satisfies the following conditions:
1. The sum of the digits of $n$ is a prime number.
2. The sum of the digits of $n^2$ is a perfect square.
Prove or disprove the existence of an infinite subset $T$ of $S$ such that for every $n$ in $... |
This question is designed for high school students, so advanced linear algebra solution might not be needed.
Suppose I have a $\vec{G}$ as a meet point between vector $\vec{DB}$ and $\vec{CE}$ as illustrated here:
[![enter image description here][1]][1]
Note that the intersection point is labeled with G and not ... |
Consider the following decision problem:
> We have a set of vectors $S \subseteq \mathbb Z^2$, and a target $k$. Is there a subset $S' \subseteq S$ such that $\sum_{v \in S'} v$ has both coordinates greater than or equal to $k$?
This question came up naturally in the course of doing some (hobby) research on anoth... |
Is $\operatorname{CRing}^\mathrm{op}$ a regular category, where $\operatorname{CRing}$ is the category of commutative rings (with unity, as usual)?
To be regular, a category must have finite limits, coequalizers of kernel pairs, and pullback-stable regular epimorphisms.
In the case of $\operatorname{CRing}^\mathr... |
I proved the claim that for every $m\in\mathbb{Z^+}$ The Fibonacci sequence repeats itself for modulo m. But how can i move on and make connections with it to prove that for every $m\in\mathbb{Z^+}$ There exists a Fibonacci number such that it's last m digits are 0. Should i show it like using the modulo $10^m$ and sho... |
Let $p = 2 m {q^2} - 1$ be a prime, where $m$ is a smooth integr and $q$ be any odd prime, and ${g_1},{g_2},h$ be primitive elements of ${\mathbb{F}_p}$ where are related by
$${g_1}^{{x_1}} = h\,\,\,,\,\,\,\,{g_2}^{{x_2}} = h,$$
for some unknown integers ${x_1},{x_2}$. We know that $${x_1} \equiv {x_2}\left( {\bm... |
I proved the claim that for every $m\in\mathbb{Z^+}$ The Fibonacci sequence repeats itself for modulo m. But how can i move on and make connections with it to prove that for every $m\in\mathbb{Z^+}$ There exists a Fibonacci number such that it's last m digits are 0. Should i show it like using the modulo $10^m$ and sho... |
Here is the setting
$X$:
Positive semi-definite;
Non-negative entries;
$||X||_\infty = \lambda$ (largest absolute row sum is $\lambda$).
$G$:
Diagonal matrix;
Non-negative diagonal entries $g_i$;
All $g_i \leq \tau$ (for some positive $\tau$).
$XG$:
all $XG$’s eigenvalue are within unit circl... |
Here is the setting
$X$:
Positive semi-definite;
Non-negative entries;
$||X||_\infty = \lambda$ (largest absolute row sum is $\lambda$).
$G$:
Diagonal matrix;
Non-negative diagonal entries $g_i$;
All $g_i \leq \tau$ (for some positive $\tau$).
$XG$:
all $XG$’s eigenvalue are within unit circl... |
Is infinity norm submultiplicative? What about its power? |
I proved the claim that for every $m\in\mathbb{Z^+}$ the Fibonacci sequence repeats itself for modulo $m$. But how can I move on and make connections with it to prove that for every $m\in\mathbb{Z^+}$ there exists a Fibonacci number such that its last $m$ digits are $0$. Should I show it like using the modulo $10^m$ an... |
Let $ M $ be a manifold and let $ X $ be a _complete_ vector field on $ M $. A $ k $-form $ \omega $ on $ M $ is said to be _invariant_ under the flow of $ X $ (or just _$ X $-invariant_) if $ (\Phi_t)^*\omega = \omega $, where $ \Phi^X\colon \mathbb R\times M\to M $ is the flow of $ X $.
I was wondering if the noti... |
I am wondering about the following generalization of the group $B_{1,2}=\langle a,b\, |\, bab^{-1}=a^2\rangle$:
$$
G_k=\langle a_1,a_2,\ldots,a_{k+1}\, |\, a_{i+1}a_ia_{i+1}^{-1}=a_i^2, i=1,2,\ldots,k\rangle.
$$
These groups, probably, were studied before; is there a name for these groups? One specific question:
... |
I've been trying to prove that the initial value problem
$$y'=y^2+t^2,\quad y(0)=0$$
has a maximal solution in $[0,2[$. From the uncountable simulations I've run it is clear to me that it is the case. The solution moreover seems to be asymptotic to $\frac1{2-t}$ as one can check by doing the change of variables $u=(2... |
I have a question regarding convex functions and compositions that I'm hoping someone could provide some insight on.
For a given non-convex function g(x), I'm wondering if it's possible to find another function h such that the composition h(g(x)) results in a convex function. In other words, does there exist a funct... |
[convex optimization]Does there exist a function h such that its composition with a non-convex function g, h(g(x)), is a convex function? |
* $f(x) = 2[x]-x$,
* $[x]$ is the floor function,
* the domain of the function is $[-1,2]$,
* its codomain is $\Bbb R$
$$
f(x) = \begin{cases}
-2-x & \text{if } -1 \leq x < 0 \\
-x & \text{if } 0 \leq x < 1 \\
2-x & \text{if } 1 \leq x < 2 \\
4-x & \text{if } x = 2 \\
\end{cases}
$$
a diagonal line is forme... |
Is there a general way to invert (solve for $u$) this?
$$\sum_{ij}R_{ijk}a_iu_j = -x_k$$
With $a,u,x \in \mathbb{R}^N$. $R_{ijk}$ is symmetric in the last two indices. |
Let $p = 2 m {q^2} + 1$ be a prime, where $m$ is a smooth integr and $q$ be any odd prime, and ${g_1},{g_2},h$ be primitive elements of ${\mathbb{F}_p}$ where are related by
$${g_1}^{{x_1}} = h\,\,\,,\,\,\,\,{g_2}^{{x_2}} = h,$$
for some unknown integers ${x_1},{x_2}$. We know that $${x_1} \equiv {x_2}\left( {\bm... |
Here's the approach I learned, with differentials. One only needs the product rule. I'll respond to the final part of the question, "Or specifically, how do you get from $x^TAx$ to $A^Tx + Ax$?"
Let $f(x) = x^T A x$. Then, to make the application of the production rule absolutely clear, let $g(x) = x^T$ and $h(x) = ... |
Let's contemplate the two possible definitions of triangulated subcategory that OP is hinting at. For simplicity, the translation functor of a triangulated category (that I shall denote as $[1]$) is always assumed to be an automorphism. The class of distinguished triangles of a (pre)triangulated category $\mathcal{D}$ ... |
The category of commutative rings is not balanced (the inclusion $\mathbb{Z} \to \mathbb{Q}$ is monic and epic but not an isomorphism).
So, is there a category $C$ and a functor $\mathbf{CRing} \to C$ that is faithful but not conservative (isomorphism-reflecting)? If so, then there should be such a functor that maps... |
Is there a functor from the category of commutative rings that is faithful but not conservative? |
I am working with a problem from a statistics book as follows:
You have 20 questions and 20 answers. You are supposed to pair correct answer to the correct question. In the book they say that the probability of guessing and getting everything right is 1/400. However doesn`t order matter here? Is the answer not 1/20... |
Let $M$ be a non-compact contractible bounded submanifold of the Euclidean space. Is it possible to find a compact submanifold $S$ of the Euclidean space such that $M$ is a submanifold of $S$?
As a positive example, let $M = \{(x,y) \in \mathbb{R}^2: (x,y) \in S^1, x>0, y>0\}$ be the portion of the 1-sphere $S^1$ ly... |
Can a non-compact contractible bounded submanifold of the Euclidean space be extended to a compact submanifold? |
**Background**
In the <a href="https://math.stackexchange.com/questions/170747/sum-equals-integral-identities-similar-to-int-01-fracdxxx-sum-n%20">following question</a> and references therein, a number of "Sum equals integral" identities are described.
For instance, we have $$\int_{-\infty}^{+\infty} {\rm sinc}... |
Here is the setting
$X$:
Positive semi-definite;
Non-negative entries;
$||X||_\infty = \lambda$ (largest absolute row sum is $\lambda$).
$G$:
Diagonal matrix;
Non-negative diagonal entries $g_i$;
All $g_i \leq \tau$ (for some positive $\tau$).
$XG$:
all $XG$’s eigenvalue are within unit circl... |
I'm trying to understand [the converse][1] of theorem 1.3 in Robert's Bieri Homological dimension of discrete groups which says that a $\Lambda$-module $A$ is of type $FP_n$ if and only if for every direct product $\Pi \Lambda$ the natural map $Tor_k(\Pi\Lambda,A) \to \Pi Tor_k(\Lambda,A)$ is an isomorphism for $k\in[|... |
Do you know [Eisenstein's theorem](https://en.wikipedia.org/wiki/Eisenstein's_theorem)? It states that if you have any algebraic power series
$$f(z) = \sum_n a_n z^n$$
with $a_n \in \mathbb Q$, then there is a number $A \in \mathbb N$ such that $a_n A^n \in \mathbb Z$ for all $n$. In other words, the power series
$$... |
Let $A=A_R+iA_I$ be a $n\times n$ complex matrix. If we want to solve a linear system with regard to $A$ and do not want to take complex arithmetic, then we often generate the following real matrix:
\begin{equation}
\widehat{A} = \begin{bmatrix}A_R&-A_I\\ A_I&A_R \end{bmatrix}.
\end{equation}
Furthermore, when $A$ ... |
It is a bit strange to speak of the ‘probability’ of a single event that has already occurred. The decimal expansion of $\pi$ either does, or does not begin with a palindromic sequence, so the probability is either $0$ or $1$, we just don't know which. Probability is an expression of uncertainty about _future_ events... |
$
\newcommand{\real}{\mathbb{R}}
$
A $\color{red}{\text{(strictly)}}$ positive definite kernel $k: \real^d\times \real^d \to \real$ satisfies for all $x_i \in \real^d$, $a=(a_1,\dots, a_n)\in \real^d$
$$
\sum_{i=1}^n a_i a_j k(x_i, x_j) \ge 0 \quad \color{red}{>0 \iff a\neq 0}
$$
The reproducing kernel hilbe... |
(Why) is the norm of a RKHS positive definite? |
This question was asked in my assignment of linear algebra and I am not able to make any significant progress on it:
> Question: For arbitrary subspaces $U, V,W$ of a finite dimensional vector space, which of the following hold:
(a) $U \cap (V+W) \subset U \cap V + U \cap W$
(b)$ U \cap V + U \cap W \subset U ... |
How to approach this question: For arbitrary subspaces of $U , V ,W$ of a finite dimensional vector space, which of the following hold? |
Let $M$ be a vector space of all $3\times 3$ real matrices and let $$A=\begin{pmatrix}2&3&1\\0&2&0\\0&0&3\end{pmatrix}.$$
I am struck on the following problem: Show that $\{X\in M:\text{det}(AX)=0\}$ is not a subspace of M.
Can you please let me know why it can't be a subspace of M? What would be the contrad... |
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