instruction stringlengths 12 30k |
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**Context:**<br>
Hi, I started by calculating the chance of an event happening with a chance of 1 in x after n times. Next I wanted to flip this approach to be able to say how many n attempts are needed to have a given chance P(n) of an event happening.
**Examples:**<br>
1. With a chance of 1 in 4.2 billion km, h... |
For any $m \in \mathbb{N}$, are there primes $p_1, \dots, p_n$ such that $\gcd(p_1 + 1, \dots, p_n + 1) = m$? |
**Hint:** Do the same trick for $T_0 = T - \lambda I$ and $\lambda_0 = -\lambda$ and consider degrees. More concretely, if $p_0$ is minimal polynomial of $T_0$ different from $q$, what's the relation between $q_0$ and $p$ and what does it tell you about minimality of $p$? |
I want to solve the following problem:
A real estate agent has 8 master keys to open several new homes. Only one master key will open any of the houses. If 40% of these are usually left open, what is the probability that the real estate consultant will open any of the houses?
So, we can define the events:
A:Th... |
We know that, If $f$ and $g$ are functions then their convolution is defined as,
$(f*g)(x) =\int_{-∞}^{∞} f(t)g(x-t) dt$
(This is the convolution structure for Fourier transform)
**But, what if $f$ and $g$ are functions of two variables?** say $f(x, y)$ & $g(x, y)$ then how we can define the convolution in t... |
How do I prove that any even number, divided by 2 a number of times, would eventually go down to an odd number or 1? |
How to prove that any even number, divided by 2 a number of times, would eventually turn into an odd number or 1? |
Hopefully a quick one!
--------------------
Let $R$ be a compact noetherian ring and $T,T^\prime$ be compact $R$-modules. Does it then follow that the $R$-module $T+T^\prime$ is compact too (where $T+T^\prime$ is the $R$-module generated by $T \cup T^\prime$)?
I know that this is false for general modules, but... |
Sum of compact modules over compact ring is compact? |
Given the trajectories of two objects
$$r(t)=(4\cos(t),2\sin(t))$$
$$s(t)=(2\sin(2t),3\cos(2t))$$
I have to evaluate
$$\left.\frac{dD}{dt}\right|_{t=\pi}$$
where $D$ is the distance between them. May I say
$$D(t)=r(t)-s(t)?$$
Or should it be
$$D(t)=\sqrt{r^2(t)-s^2(t)}?$$ |
Hopefully a quick one!
--------------------
Let $R$ be a compact noetherian ring and $S$ be an $R$-module. If $T,T^\prime$ are compact submodules of $S$, does it then follow that the module $T+T^\prime$ is compact too (where $T+T^\prime$ is the module generated by $T \cup T^\prime$)?
I know that this is false ... |
I have a solid understanding of the basics of expander graphs and their properties and the recent development of High-Dimensional Expanders and their application to Random Walks, along with other stuff like Local-to-Global Theorem. As they progress, I feel there is ever strengthening connection between Representation T... |
> Is it still guaranteed that the remaining $N-km$ characters (`abcdd`) can be arranged into a string that has no repeating characters?
No - if some character is a large enough proportion of the whole, what's left can have more of that character than half its length. For example, applying this algorithm to `abcddd`,... |
What is $\left|\bigcup\limits_{n=0}^{10}\{n\}\right|$ supposed to be? |
Theorem:
If $L/K$ is Galois and $E$ is an intermediate field, then the subspace topology on $\text{Gal}(L/E)$ as a subset of $\text{Gal}(L/K)$ equals the Krull topology on $\text{Gal}(L/E)$.
I want to prove that the open set of $\text{Gal}(L/E)$ under krull topology is also open set of subspace topology of $\text{G... |
It is commonly defined as every real number that let the expression make sense. Is there a more precise and rigorous way to define it?
Actually I have some idea about this:
First, define basic functions like +,-,*($\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$),/($\mathbb{R}\times\mathbb{R}-\{0\}\rightarrow\ma... |
How to precisely define the natural domain of a function? |
> and $R \subseteq \bigcup_{j=1}^n V_j$ is a finite (sub)cover (as $R$ is compact).
Subcover of what? $U_i$ are subsets of $S$, not $R$. They do not cover $R$ at all, to pick a "subcover".
> Then $T+T^\prime = R \cdot (T \cup T^\prime)$
No, those subsets are hardly ever equal. The subset on the left consists o... |
We have two languages $L_1, L_2 \subseteq \{{a, b}\}^{*}$.
According to the following formulas find $L_1$ and $L_2$:
$L_1 = \{\lambda\} \cup \{a\}.L_1 \cup \{b\}.L_2 $
$L_2 = \{\lambda\} \cup \{b\}.L_2 $
I am trying to solve this question. extra explaination would be great. Thank you in advance for your hel... |
For neutral elements in a lattice, the definition is typically stated as follows:
*An element a of a lattice L is neutral, iff every triple {a, x, y} generates a distributive sublattice of L.*
Which now leads to my question: What is this "distributive sublattice generated by {a, x, y}" and how do you compute it? ... |
> and $R \subseteq \bigcup_{j=1}^n V_j$ is a finite (sub)cover (as $R$ is compact).
Subcover of what? $U_i$ are subsets of $S$, not $R$. They do not cover $R$ at all, to pick a "subcover".
> Then $T+T^\prime = R \cdot (T \cup T^\prime)$
No, those subsets are hardly ever equal. The subset on the left consists o... |
Let $U$ be an open subset of $\mathbb{R}^n$ and $g$ be a $C^\infty$ Riemannian metric on $U$. Given a point $x_0\in U$, does there exist a local neighborhood $x_0\in V\subset U$ and new coordinates thereon so that the components of $g$ with respect to the new coordinates are real analytic (i.e. is it possible to change... |
Smooth Riemannian metric is locally real analytic? |
We have two languages $L_1, L_2 \subseteq \{{a, b}\}^{*}$.
According to the following formulas find $L_1$ and $L_2$:
$L_1 = \{\lambda\} \cup \{a\}.L_1 \cup \{b\}.L_2 $
$L_2 = \{\lambda\} \cup \{b\}.L_2 $
I am trying to solve this question. extra explaination would be great. Thank you in advance for your hel... |
I am currently working through Cormen et al's classic book on algorithms and data structures. I am trying to follow an inductive substitution proof that the following time complexity function is $O(n\circ lg(n))$:
$T(n) = 2T([\frac{n}{2}]) + \Theta(n)$.
I am not quite following the details of his proof. His bas... |
I am new to math. stack exchange, I am really not sure how I am supposed to ask this but I need a logical explanation and a way to logical way to approach questions like these. I tried doing it myself. Thank you.
The question is as follows:
$$\text{Let } A = \{a, b, c, d, f, g, h, k, u, x, z,y\}. \\$$
... |
I think I ve found a solution for "c)" in all the cases I will be happy to have your feedback on what I ve writen.
c)
To prove that $\vec{\phi} (r ; \theta) $ is a bijection from $r > 0 , 0 \leq \theta < 2\pi $ to $S$ I need to demonstrate that $\vec{\phi} (r ; \theta) $ is injective and surjective.
1- *Inj... |
We have two languages $L_1, L_2 \subseteq \{{a, b}\}^{*}$.
According to the following formulas find $L_1$ and $L_2$:
$L_1 = \{\lambda\} \cup \{a\}.L_1 \cup \{b\}.L_2 $
$L_2 = \{\lambda\} \cup \{b\}.L_2 $
I am trying to solve this question. extra explaination would be great. Thank you in advance for your hel... |
Can someone guide me in the right direction on this question?
Prove that every $n$ in $\mathbb{N}$ can be written as a product of an odd integer and a non-negative integer power of $2$.
For instance: $36 = 2^2(9)$ , $80 = 2^4(5)$ , $17 = 2^0(17)$ , $64 = 2^6(1)$ , etc...
Any hints in the right direction are ap... |
I have the following optimization
\begin{align}
\max_{\|x\|^2\le1} \|Lx - y\|
\end{align}
where $L$ is lower triangular and $y$ is a given sequence. Does it admit a closed-form solution?
The motivation is to understand the standard least squares method but as a game from the "adversarial" point of view. |
We have two languages $L_1, L_2 \subseteq \{{a, b}\}^{*}$.
According to the following formulas find $L_1$ and $L_2$:
$L_1 = \{\lambda\} \cup \{a\}.L_1 \cup \{b\}.L_2 $
$L_2 = \{\lambda\} \cup \{b\}.L_2 $
I am trying to solve this question. extra explaination would be great. Thank you in advance for your hel... |
I was solving a electrostatic problem which involved solving Laplace equation in two dimension. The solution is (in the form of Fourier sine series):
$$V(x,y)= \frac{4V_0}{\pi} \sum_{n=1}^{\infty} \frac{1}{2n-1}exp(-(2n-1)\pi \frac xa)\sin ((2n-1)\pi \frac ya)$$
Now it asked to find the surface charger density ... |
Alice and Bob are playing a game. They throw a fair coin, and write out the results of the throws. Each of the players has a victory criterion, the one whose criterion comes first wins. Alice wins at the moment when the results of the last three throws are equal to 001. Bob wins when the results of the last three throw... |
We have two languages $L_1, L_2 \subseteq \{{a, b}\}^{*}$.
According to the following formulas find $L_1$ and $L_2$:
$L_1 = \{\lambda\} \cup \{a\}.L_1 \cup \{b\}.L_2 $
$L_2 = \{\lambda\} \cup \{b\}.L_2 $
I am trying to solve this question. extra explaination would be great. Thank you in advance for your hel... |
Let $f$ be a continuous function over the interval $[a, b]$ such that $f(a)f(b) \leq 0$. Suppose $f(x)=0$ has greater than $1$ root in this interval. Which root does the bisection method eventually converge to, given that we don't stop at any point given an $\epsilon$ precision has already been established and that ins... |
Which root does the bisection method prefer in case of multiple roots existing in the given interval? |
I'm a masters student writing my dissertation on forcing. I am reading Jech and following his Boolean algebra models approach to it. I'm wondering if I've got the right idea here.
Let $M \models \mathrm{ZFC}$ and $B \in \mathbf{CompBoolAlg}$. We construct $M^B$ as follows:
- $M^B_0 = \emptyset$
- $M^B_{\alph... |
How to get the gradient of this function? |
How do I call a matrix $A\in \mathbb{C}^{m\times n}$ satisfying $A^*A=\mathbf{1}$? Maybe orthogonal?
Also I'd like to confirm that the following equivalences are true:
> Let $A\in \mathbb{C}^{m\times n}$. Suppose that
> $\langle\cdot,\cdot\rangle_k $ is the canonical internal product in
> $\mathbb{C}^k$ for $... |
How do I call a matrix $A\in \mathbb{C}^{m\times n}$ satisfying $A^*A=\mathbf{1}$? |
The question I have is styled in the following:
"Consider the natural isomorphism of commutative rings:"
$\varphi:\mathbb{Z}/(1088) \rightarrow \mathbb{Z}/(17) \oplus \mathbb{Z}/(64)$
Write down a formula for the inverse isomorphism $\varphi^{-1}$ and calculate $\varphi^{-1}(\bar{1},\bar{17}$).
The only exa... |
Would it be correct to imply from Euler's function
$x = \frac {(\log x)^{\pi(x)}}{\prod \log p}$
where $\pi(x)$ is prime counting function?
$(\sum_{i=0}^x i^2) = (1 + \frac {1}{2} + \frac {1}{2^2} + \frac {1}{2^3} + \dots)(1 + \frac {1}{3} + \frac {1}{3^2} + \frac {1}{3^3} + \dots)(1 + \frac {1}{5} + \frac {1}{... |
I'm reading Jech and following his Boolean algebra models approach to it. I'm wondering if I've got the right idea here.
Let $M \models \mathrm{ZFC}$ and $B \in \mathbf{CompBoolAlg}$. We construct $M^B$ as follows:
- $M^B_0 = \emptyset$
- $M^B_{\alpha+1} = \{ \text{partial functions} \ f : M^B_\alpha \to B \... |
(**Question:**) Suppose that $H$ is a Hilbert space over the field $\mathbb{C}$ and that $(v_n)_{n=1}^\infty$ is a Hilbert basis for it, that is a sequence of orthonormal vectors whose span is dense in $H$. I am wondering whether we can turn the sequence of real and imaginary parts of every $v_n$ also to a Hilbert basi... |
If $(v_n)_n$ is a Hilbert basis for $H$ over $\mathbb{C}$, can you turn $(\mathrm{Re}(v_n), \mathrm{Im}(v_n))_n$ also to a Hilbert basis for $H$? |
Product of two quotient maps need not be a quotient map.
However,
given two quotient maps $$p : A → B\\ q : C → D$$
if B, D are locally compact Housdorff then $$p × q : A × C → B × D$$
is a quotient map.
Does this still hold for product of arbitrary number of maps?
Given collection of quotient maps $\{... |
Topology - Is infinite product of quotient maps a quotient map? |
As part of a personal project, I found the paper ["On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg][1], which discusses deriving equations for a 3D surface I want to parameterize. However, despite my efforts to decipher the paper, I can't follow everything it is describing. The ... |
Would it be correct to imply from Euler's function
$x = \frac {(\log x)^{\pi(x)}}{\prod \log p}$
where $\pi(x)$ is prime counting function?
$(\sum_{i=0}^x i^2) = (1 + \frac {1}{2} + \frac {1}{2^2} + \frac {1}{2^3} + \dots)(1 + \frac {1}{3} + \frac {1}{3^2} + \frac {1}{3^3} + \dots)(1 + \frac {1}{5} + \frac {1}{... |
# Background
I have two parametric curves, and I want to find the parameter values of their intersection point closest to zero under certain conditions.
The first curve is an involute of a circle with parameter theta:
$$
x_1(θ) = \frac{Z}{2} ⋅ \cos(α) ⋅ (\cos(θ − \tan(α) + α) + θ ⋅ \sin(θ − \tan(α) + α)),
$$... |
Can one interpret the closure of a set inside an inverse limit as the closure of its individual components? I have not been able to find a source confirming or denying this claim. I have only been able to show that there exist closed sets in the inverse limit that have an open component, making me believe that it is no... |
I've read about that $E(Y|X)=\underset{f(x)\in \mathcal{F}}{argmin} E(Y-f(X))^2$, where $\mathcal{F}$ is the set of all square integrable functions in $x$. The proof of this result is simple and does not seem to use square integrability of function $f(x)$. I'm wondering why do we restrict $f(x)$ to be square integrable... |
Why do we need square integrability in showing $E(Y|X)$ minimizes expected quadratic loss? |
I'm taking a course in topology and we just defined the connected sum of connected surfaces (2-manifolds) as follows:
Given $S_1,S_2$ surfaces and $D_i\subseteq S_i$ open disks with $\varphi : \partial D_1 \rightarrow \partial D_2$ a homeomorphism the connected sum of $S_1,S_2$ is $S_1\#S_2=(S_1-D_1\cup S_2-D_2)/\va... |
Why is the connected sum of surfaces definition well posed? |
The result can be derived from the theorem below on surface integrals, by setting the integrand function $f$ to unity.
I have used the following conventions below :
1. The physics convention $(r, \theta, \varphi)$ for spherical coordinates, with $\theta$ the polar angle and $\varphi$ the azimuthal angle [SPH]. ... |
I've read about that $E(Y|X)=\underset{f(x)\in \mathcal{F}}{\arg\min} E(Y-f(X))^2$, where $\mathcal{F}$ is the set of all square integrable functions in $x$. The proof of this result is simple and does not seem to use square integrability of function $f(x)$. I'm wondering why do we restrict $f(x)$ to be square integrab... |
I am trying to wrap my head around the different ways to describe commutative monoids in categorical algebra.
1. The Lawvere Theory of commutative monoids can be described as the opposite of the full subcategory $\mathsf{FreeCMon}\subseteq\mathsf{CMon}$ on the commutative monoids of the form $\Bbb N^{\oplus k}$ for ... |
Product of two quotient maps need not be a quotient map.
However,
given two quotient maps $$p : A → B\\ q : C → D$$
if A, B, C, D are locally compact Housdorff then $$p × q : A × C → B × D$$
is a quotient map.
Does this still hold for product of arbitrary number of maps?
Given collection of quotient ma... |
Product of two quotient maps need not be a quotient map.
However,
given two quotient maps $$p : A → B\\ q : C → D$$
if A, B, C, D are locally compact Housdorff then $$p × q : A × C → B × D$$
is a quotient map.
Does this still hold for product of arbitrary number of maps?
Given collection of quotient ma... |
$\DeclareMathOperator{\sech}{sech}$I've been trying to understand a question in Barrett O'Neill's Elementary Differential Geometry that goes as follows:
"Show that the geometric surface in Exercise 2 is isometric to a helicoid. (Hint: use
$$
x(u,v) = \bigl(f(u)\cos v, f(u)\sin v, v\bigr)
$$
for a suitable funct... |
> Does this mean that for any *particular* "ordinary" natural number
> that can be obtained by writing $S$ sufficiently many times in my
> formula, I can deduce $P(S(S(...S(0)))$ without appealing to the axiom
> of induction?
That is actually the logical equivalent the axiom of induction.
In general, if $f: X\... |
I've come across the Dyson Series solution of the Schrödinger Equation arising in the interaction picture when dealing with a time dependent Hamiltonian. Since then I've been looking for a rigorous definition of the integral of operators which appears repeatedly in the formula but I didn't find much. Is there perhaps a... |
I'm reading Jech and following his Boolean algebra models approach to it. I'm wondering if I've got the right idea here.
Let $M \models \mathrm{ZFC}$ and $B \in \mathbf{CompBoolAlg}$. We construct $M^B$ as follows:
- $M^B_0 = \emptyset$
- $M^B_{\alpha+1} = \{ \text{partial functions} \ f : M^B_\alpha \to B \... |
sum of piecewise functions - twice continously differentiable? |
I'm reading Jech and following his Boolean algebra models approach to it. I'm wondering if I've got the right idea here.
Let $M \models \mathrm{ZFC}$ and $B \in \mathbf{CompBoolAlg}$. We construct $M^B$ as follows:
- $M^B_0 = \emptyset$
- $M^B_{\alpha+1} = \{ \text{partial functions} \ f : M^B_\alpha \to B \... |
Would it be correct to imply from Euler's function
$x = \frac {(\log x)^{\pi(x)}}{\prod \log p}$
where $\pi(x)$ is prime counting function?
Edit: I didn't notice i was doing a typo (my bad, sorry guys)
$(\sum_{i=0}^x 1/i) = (1 + \frac {1}{2} + \frac {1}{2^2} + \frac {1}{2^3} + \dots)(1 + \frac {1}{3} + \frac... |
Let a f(n) denote nth number in the fibonnaci sequence. find such two digit prime that satisfies that p divides f(n) + f(n+100) for all n.
Well to be honest I have no idea how to solve this problem so I dont really have anything to show for my attempts.
Well I did try to play around with the recurssion a bit but it... |
Like many coloring variants, this problem can be phrased as a [graph homomorphism](https://en.wikipedia.org/wiki/Graph_homomorphism) problem. Let $H_{N,d}$ be the graph whose vertices are subsets of $\{1,\dots,N\}$, with an edge between two subsets if and only if the symmetric difference between them has size at least ... |
$ABC$ is an isosceles triangle so $AB=BC$, $D$ is the midpoint of $BC$ so $BD=DC$ and $BED$ and $DFC$ are isosceles triangles such that $BE=ED=DF=FC$.
We know that $BED$ and $DFC$ are isosceles triangles but $BD=DC$ then $BED$ and $DFC$ are equilateral triangles where $BE=ED=DF=FC=BD=DC$.
Since $ABC$ is an isosce... |
> May someone help me understand the following inductive definition of concatenation of strings
You would have an *inductive definition* of "string", which typically has two constructors, one for the "empty" string, and one for the "compound" string with one head character followed by the rest of the string. (In fa... |
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 was looking for a proof of the fact that the forgetful functor $U : \mathbf{Grp} \to \mathbf{Set}$ preserves reflexive coequalizers, but couldn't find one. Here is what I have:
---
Let $A {\rightrightarrows}^f_g B$ be a reflexive pair of groups with common section $s$.
The coequalizer in $\mathbf{Grp}$ is $A \... |
> Does this mean that for any *particular* "ordinary" natural number
> that can be obtained by writing $S$ sufficiently many times in my
> formula, I can deduce $P(S(S(...S(0)))$ without appealing to the axiom
> of induction?
That is actually the logical equivalent the axiom of induction.
In general, if $f: X\... |
Let $\xi^*,\varsigma,\eta:\mathbb{R}\rightarrow (0,\infty)$. You may assume any degree of smoothness required of these functions.
Assume that $\xi^{*}(t)$ and $\varsigma(t)$ have finite positive limits as $t\rightarrow\infty$ and as $t\rightarrow -\infty$.
Define $\delta:\mathbb{R}\rightarrow\mathbb{R}$ by:
$$\d... |
Let $\mathcal{A}$ be an algebra over a set $E$, and $\mu$ be a pre-measure over $\mathcal{A}$. We can define an outer measure $\mu^*$ over $2^E$, and a $\sigma-$algebra $\mathcal{M}$ over $E$ (by Caratheodory theorem). In fact $\mu^*\mid_{\mathcal{M}}$ is a measure $\mu$. (cf. Real Analysis by Folland P29, 30, 31).
... |
This weird question was given by my professor as a part of my assignment :
$\textbf{Question :}$
“Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that it satisfies $f(x + y) = x.f(\frac{1}{y}) + y.f(\frac{1}{x})$, whenever $x$ and $y$ are both irrational numbers. Then prove that $f(0)$ is always $0$.”
$\... |
I am struggling to understand how to use the definition:
The vector d is said to be a tangent (or tangent vector) to 6 at a point x if there are a
feasible sequence ${z_k}$ approaching x and a sequence of positive scalars $t_k$ with $t_k\rightarrow 0$ such that
$\frac{z_k-x}{t_k}=d$.
I am trying to use this one t... |
Let a f(n) denote nth number in the Fibonacci sequence. find such two digit prime that satisfies that p divides f(n) + f(n+100) for all n.
Well to be honest I have no idea how to solve this problem so I dont really have anything to show for my attempts.
Well I did try to play around with the recurssion a bit but it... |
Let $\xi^*,\varsigma,\eta:\mathbb{R}\rightarrow (0,\infty)$. You may assume any degree of smoothness required of these functions.
Assume that $\xi^{*}(t)$ and $\varsigma(t)$ have finite positive limits as $t\rightarrow\infty$ and as $t\rightarrow -\infty$.
Define $\delta:\mathbb{R}\rightarrow\mathbb{R}$ by:
$$\d... |
This shape is formed by 3 'small' circles on the surface of a sphere, each touching the other 2.
[Region bounded by 3 small circles][1]
On a plane, the shape is called a 'circular triangle' (refer to Wikipedia). In this particular example it is called a 'circular horn triangle'.
[Circular Triangle in a plan... |
Does this shape have a name? (A 'spherical circular triangle' ???) |
To find all $\displaystyle f\ \mathbb{N}\rightarrow \mathbb{N} \ $such that
\begin{gather}
\frac{f( x+y) \ +\ f( x)}{2x\ =\ f( y)} \ =\ \frac{2y\ +\ f( x)}{f( x\ +\ y) \ +\ f( y)} \notag\\
\notag\\
x\ =\ 1,\ y\ =\ 1 \notag\\
f( 2) \ +\ f( 1) \ =\ 2\ +\ f( 1) \notag\\
f( 2) \ =\ 2\\
x\ =\ 1,\ y\ =\ 2 \notag\\
... |
Is my solution to find $f(x)$ exhaustive? |
Find the best dimensions of a tent with horizontal ridge pole. Assume the two ends closed by isosceles triangles. There is no floor.
The total surface area of the tent is
\begin{align*}
A & = 2 \times \text{Area of the rectangular sides} + 2 \times \text{Area of the isosceles triangles at the ends}.
\end{align... |
Let $R$ be a commutative unital ring and $S$ an associative rng (nonunital) that is an $R$-module (= $(R,R)$-bimodule with $\forall s \in S, r \in R$: $rs = sr$). The Dorroh extension is a well-known construction that "adjoins unity to $S$ using $R$", i.e. constructs a unital ring $T$ with $S \lhd T$ and $R \subset T$.... |
How to prove the characteristic function of normal distribution? |
If *S* = {0,1,2,3,4} and *(a,b)* is an arbitrary ordered pair such that *a ∈ S* and *b ∈ S*, which of the matchings in Exercises 12-15 are binary operations on *S*? Construct operation tables.
12. *(a,b)* ----> *a * b* = *2a - b*
13. *(a,b)* ----> *a ∇ b* is the maximum of *a* and *b*
14. *(a,b)* ----> *a Δ b* ... |
What do the symbols for these binary operations on a set mean? |
I have a doubt in the last step of the derivation of the second variation formula for the length functional (Lee's intro to riemannian geometry) (theorem 10.22 see the proof below)
**Why is it true that**
$Rm(\gamma',\gamma',\cdot,\cdot)=Rm(\cdot,\cdot, \gamma',\gamma')=0$**?**
The first one seems to be inte... |
How does the riemann tensor symplify to normal components in the second variation formula? |
This weird question was given by my professor as a part of my assignment :
$\textbf{Question :}$
“Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that it satisfies $f(x + y) = x f(\frac{1}{y}) + y f(\frac{1}{x})$, whenever $x$ and $y$ are both irrational numbers. Then prove that $f(0)$ is always $0$.”
$\... |
Considering the function $f(x,y)=\frac{x^{5}}{sin^{2}x\: +\, cos^{2}y }$, I found that its domain $D_f$ is:
$D_f = \mathbb{R}^{2}\backslash \{k\cdot pi \;|\; k \in \mathbb{Z} \} \times\{k\cdot \frac{pi}{2} \;|\; k \in \mathbb{Z}^{*}\}$.
I am looking for all the points $a \in \mathbb{R}^{2}\backslash D_f$ to whic... |
To what points can $f(x,y)=\frac{x^{5}}{sin^{2}x\: +\, cos^{2}y}$ be extended by continuity? |
This shape is formed by 3 'small' circles on the surface of a sphere, each touching the other 2.
![Region bounded by 3 small circles][1]
On a plane, the shape is called a 'circular triangle' (refer to Wikipedia). In this particular example it is called a 'circular horn triangle'.
![Circular Triangle in a pla... |
Consider the following function:
\begin{equation}
Y_t = A_t \left[\delta K^{\frac{\sigma -1}{\sigma}}_t + (1-\delta)L^{\frac{\sigma -1}{\sigma}}_t \right]^{\frac{\sigma}{\sigma -1}}
\end{equation}
where $\sigma$ and $\delta$ are two constants, and $\sigma \in [0, \infty]$ and $\delta \in (0,1)$.
Now, I want to f... |
I have a doubt in the last step of the derivation of the second variation formula for the length functional (Lee's intro to riemannian geometry) (theorem 10.22 see the proof below)
**Why is it true that**
$Rm(\gamma',\gamma',\cdot,\cdot)=Rm(\cdot,\cdot, \gamma',\gamma')=0$**?**
The first one seems to be inte... |
Considering the function $f(x,y)=\frac{x^{5}}{\sin^{2}x + \cos^{2}y }$, I found that its domain $D_f$ is:
$D_f = \mathbb{R}^{2}\backslash \{k\cdot \pi \;|\; k \in \mathbb{Z} \} \times\{k\cdot \frac{\pi}{2} \;|\; k \in \mathbb{Z}^{*}\}$.
I am looking for all the points $a \in \mathbb{R}^{2}\backslash D_f$ to whic... |
To what points can $f(x,y)=\frac{x^{5}}{\sin^{2}x + \cos^{2}y}$ be extended by continuity? |
For a matrix $A$, let $\|A\|_2$ denote the spectral norm (operator norm with euclidean norm used in both spaces). And for a vector $x$, let $\|x\|_2$ denote the euclidean norm.
***
Is the following inequality correct?
$
\|Ax\|_2 \leq \|A\|_2 \|x\|_2
$
using the first property listed [here](https://en.wiki... |
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