instruction stringlengths 12 30k |
|---|
$p$ - is the order of the finite field
$n$ - is the order of the group. Private keys can range from $1$ (the generator point $G$) to $n - 1$.
All the private keys ($Priv$) lie in certain ranges of 2.
$2^{a} < Priv < 2^{b}$
If we know the $a$ then after converting our private key to the public key ($Pub$), ... |
How to know the number of digits in the decimals place in division results (Ellitptic curves over Fp)? |
$$
f(x) \text{ is the “integrand" of} \int\limits_{a}^{b} f(x) \thinspace dx\\
\ \\
\ \\
f(x) \text { is the “_____" of }\lim_{x \to a} f(x) \\
$$
Is there a name for the operand of a limit operation? |
Is there a name for the operand of a limit operation? |
**Background**
In the following <a href="https://mathoverflow.net/questions/418486/along-what-curve-does-an-ellipse-roll-down-the-fastest">MO question</a>, I asked along what curve an ellipse rolls down fastest. It was pointed out that one first needs to find *any* curve the ellipse rolls down from, without losing c... |
Can an ellipse roll down a tilted sine curve without jumping? |
I have the following Definition with I struggle with. Namely it is the definition of a open set of class $C^1$.
We say an open set $O$ in $\mathbb{R}^d$ is of class $C^1$ iff for sets
1) $S_1:=\lbrace (x_1,\ldots,x_{d})\in \mathbb{R}^d : (x_1,\ldots,x_{d-1}) \text{ is an element of the open unit ball in }\mathbb{... |
How to know the number of digits in the decimals place in division results (Ellitptic curves over $\mathbb F_p$)? |
In the book [1] the covariance of an SDE is derived. I am not sure about a particular step. Let me describe it in a TLDR version, then in a longer version.
We have an SDE
$$dx = f(x,t) dt + L(x,t) d\beta $$
where $x$ is a vector, $\beta$ is a Brownian motion and $L$ is a matrix.
In the derivation of the va... |
It is a well-know observation that, given two points $x_1,x_2 \in \mathbb{R}^d$, the path signature associated to their linear interpolation is given by the tensor exponential. Precisely, if $\Delta x$ denotes the linear interpolation $x_1 \rightarrow x_2$, then
$$S(\Delta x) = \left(1 , \Delta x, \frac{\Delta x^{\ot... |
I've found [this][1] demonstration (for [this][2] problem taken from "*Mathematical methods of Classical Mechanics*" by V. I. Arnol'd), and I could not decode this particular step:
> A $k$-dimensional plane can be uniquely identified by the value of its
> symplectic form. The symplectic form takes values in the int... |
I believe the answer is False, but only based on an intuitive feeling right now as I cannot manage to definitively prove why, ie, I've been trying to find a counterexample but can't. Also, does the fact that $U$ is a "subspace" rather than a vector-space change anything? (I don't think it does).
My current reasoning... |
If the set of vectors $U$ is linearly independent in a subspace $X$, then can vectors be removed from $U$ to create a basis for $X$? |
The answer is 9 ways. There are i pennies and 8-i nickels. i can be 0,1,...,8.
Why is the answer not 8C0+8C1+8C2+8C4+8C5+8C6+8C7+8C8?
|
How many ways are there to choose 8 coins from 100 identical pennies and 100 identical nickels? |
In the title, $\overset{.}{x}^\mu$ is meant to mean the derivative of the geodesic with respect to the curve parameter, that is, the tangent to the geodesic. This is another of my general relativity exercises, so Einstein summation is again implied. The title basically says it all - if we have a metric that is independ... |
**Def**. A topological space $X$ is *locally euclidean* iff $\forall x\in X$, $\exists U$ an open neighbourhood of $x$ in $X$ such that $U\cong_{hom} D^n$ (where $D^n$ is the $n$-dimensional ball) for some $n\in \mathbb{N}$.
**Def**. A topological space $X$ is *locally $n$-euclidean* iff $\forall x\in X$, $\exists... |
The tensor product of a space with itself once is $V^{\otimes1}$, but what is $V^{\otimes0}$? Since it is an empty tensor product, it is - _a fortiori_ - an empty product. So I'm looking for a "1" of some sort, just not sure what that would mean in this context. "If I take the tensor product of a vector space with itse... |
What is the base case of a k-power of a tensor product (i.e., a k-fold tensor product of a space with itself)? |
I am considering the following problem：
>Let $ A $ be a commutative Noetherian local ring, $ I $, $ I' $ be two ideals of $ A $ satisfying that $ \dim(A/I) = \dim(A/I') $. Is it true that $ \dim(A/I)=\dim(A/I\cap I') $ always holds? (Here $ \dim $ means Krull dimension.)
Any ideas would be thankful. |
I have the following PDE (with small Parameter $0<\epsilon<<1$):\
$\frac{\partial u(x,t)}{\partial t}+\frac{\partial ^2u(x,t)}{\partial x^2}-\epsilon u(x,t)^2=0, x\in (0,1), t>0$\
$u(x,0)=u_0(x), x\in (0,1)$\
$u(0,t)=u(1,t)=0, t>0$
I now have to find the equations for the first three terms of the formal asymptoti... |
Let $(R, m_R, \kappa_R) $ a discrete valuation ring (DVR) with quotient field $K$, and $K \subset L$ a finite field extension. Let $ B$ be the integral closure of $R$ in $L$, (which in general is not a DVR), let denote by $\frak{P}_1, \frak{P}_2, ..., \frak{P}_m$ it's maximal primes lying over $m_R$. Then we obtain ram... |
Let $G=Sp(2)$ be a symplectic group over a $p$-adic field and $P$ be the Siegel parabolic subgroup of $G$. Let $\text{Ind}_P^G$ be the normalized induction functor from smoothe representations of $P$ to that of $G$. Let $J_P$ be the unnormalized Jacquet functor from smooth representations of $G$ to that of $P$.
Let... |
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 of ... |
I am a bit confused about modular arithmetic.
We say that two integers are equal modulo n if their difference
is an integer multiple of n.
So let's take as example: $5(mod2) = 1$, since $5-1 = 4$ which is an integer ($k$) multiple of 2.
But isn't this also true then: $5(mod2) = 3$? Because $5-3 = 2$ which is a... |
Is any bounded open set which is of class $C^1$ automatically a uniform Lipschitz domain?
This question is motivated from here:
https://math.stackexchange.com/questions/4886632/relation-between-sets-of-class-c1-and-uniformly-lipschitz-open-sets
It seems that this notions do have plenty of different definitions ... |
Is any bounded open set which is of class $C^1$ automatically a uniform Lipschitz domain? |
The answer is $9$ ways. There are $i$ pennies and $8-i$ nickels. $i$ can be $0,1,...,8$.
Why is the answer not ${8 \choose 0}+{8 \choose 1}+{8 \choose 2}+{8 \choose 3}+{8 \choose 4}+{8 \choose 5}+{8 \choose 6}+{8 \choose 7}+{8 \choose 8}$?
|
Let $R$ be discrete valuation ring with field of fractions $K$, and $\phi:R \to A$ a *finite, local* (i.e., $A$ dominates $R$ in the sense that its maximal ideal contains the image of maximal ideal of $R$) map of local rings.
**#Edit:** As Giorgio Genovesi pointed out, essentially the "interesting" case appear where... |
Finite map of local rings $R \to A$ with $R$ a DVR. When is target $A$ a DVR, too? |
I am trying to find bounds for the following quantity. Take two functions $f,g \in L^{1} \cap L^2$ but $\|h \|_{\infty} = \infty$. Is there a way to obtain a bound of the following type:
$$ | (h * f^2) (x)| \leq \| f\|^2_2 g(h)$$
where $g$ is some function of $h$? This would be the case if we could take the sup n... |
Reading about graph theory, I am presented with the case in which a square incidence submatrix of a
directed graph has, in each column, only 2 non-zero entries (from the definition of the incidence matrix it is known that there will only be 2 entries for each column then these entries are $1$ and $−1$).
Here is so... |
¿$det(\bar B)=0$ for $B$ columned matrix with only $2$ non-null entries equal to $1$ and $-1$? |
$\det(\bar B)=0$ for $B$ columned matrix with only $2$ non-null entries equal to $1$ and $-1$? |
I am trying to find bounds for the following quantity. Take two functions $f,h \in L^{1} \cap L^2$ but $\|h \|_{\infty} = \infty$. Is there a way to obtain a bound of the following type:
$$ | (h * f^2) (x)| \leq \| f\|^2_2 g(h)$$
where $g$ is some function of $h$? This would be the case if we could take the sup n... |
$\det(\bar B)=0$ for $\bar B$ columned matrix with only $2$ non-null entries equal to $1$ and $-1$? |
I have an old math exam question with the solution included, but there is a certain step of the solution I don't understand.
Task: Determine the order of $7$ in $\mathbb{Z}_{44}^*$
Solution: From the Chinese Remainder Theorem, $\mathbb{Z}_{44}^*$ $\cong$ $\mathbb{Z}_4^*\times \mathbb{Z}_{11}^*$. Therefore, it su... |
Why must the order of $7$ either be $5$ or $10$ in $\mathbb{Z}_{11}^*$? |
#### Background
I am following this MIT OCW course on mathematics for computer science.
In one of the recitations they come to the below result:
#### Official solution
###### Task:
A planar graph is a graph that can be drawn without any edges crossing. Also, any planar graph has a node of degree at most 5. Now... |
What do we count here? What do we count with (2 choose 1) in this particular situation of taking the derivative of x^2? The ratio of the speed at the growth of the area of a square and the growth of its side. As in the "art of counting", why nCr appears in calculus? In the rate of change of the area of a growing square... |
There are 4 different event, $A$, $B$, $C$, and $D$, and the probability are 0.3, 0.4, 0.6, and 0.7, respectively, but we don't know the joint probability of any of them.
Additionally, we already know that there are exact 2 events happen among those 4.
Is the information given above enough to calculate $P(B|¬A)$?... |
There are 4 different event, $A$, $B$, $C$, and $D$, and the probability are 0.3, 0.4, 0.6, and 0.7, respectively, but we don't know the joint probability of any of them.
Additionally, we already know that there must be exact 2 events occur among those 4.
Is the information given above enough to calculate $P(B|¬A... |
How to assign optimal coefficients to the time-derivative terms so that the PDE will quickly evolve into a time independent one? |
What do we count here? What do we count with (2 choose 1) in this particular situation of taking the derivative of x^2? The ratio of the speed at the growth of the area of a square and the growth of its side. As in the "art of counting", why nCr appears in calculus? In the rate of change of the area of a growing square... |
The well known proof of Cantor's theorem (stating that $A<2^A$ for any set $A$) does not make any use of the axiom of choice. I have now spent some time wondering if the analogous result $A^2<3^A$ can be proved by similarly explicit means. It is straightforward enough to produce an injection from $A^2$ into $3^A$ (just... |
**Background**
Assume that we have a solid ellipse with uniform density, and that it rolls along a curve without any friction.
In the following <a href="https://mathoverflow.net/questions/418486/along-what-curve-does-an-ellipse-roll-down-the-fastest">MO question</a>, I asked along what curve an ellipse rolls dow... |
What do we count here? What do we count with (2 choose 1) in this particular situation of taking the derivative of x^2? The ratio of the speed at the growth of the area of a square and the growth of its side. As in the "art of counting", why nCr appears in calculus? In the rate of change of the area of a growing square... |
#### Background
I am following this MIT OCW course on mathematics for computer science.
In one of the recitations they come to the below result:
#### Official solution
###### Task:
A planar graph is a graph that can be drawn without any edges crossing. Also, any planar graph has a node of degree at most 5. Now... |
#Here graph will be create most likely same but how Q1. will be Unbounded and Q2. will be Bounded
Q$1$
How is it an Unbounded LPP?
$$\max z = 30 x + 15 y$$
Subject to
$$4 x + 6 y \ge 12$$
$$4 x + 1 y \ge 4$$
$$x,y \ge 0$$
Q$2$
How is it a Bounded LPP?
$$\max z = 2 x + 3 y$$
Subject to
$$2 x + 2 ... |
By [this](https://math.stackexchange.com/a/4257701/736008) answer I knkow that the equality
$$
(\omega+1)\cdot\omega=\omega^3
$$
holds whereas by the definition of ordinal multiplication I know that the equality
$$
\omega\cdot(\omega+1)=\omega^2+\omega
$$
holds: so it seems to me that the inequality
$$
\omega... |
Does the equality $\omega\cdot(\omega+1)=(\omega+1)\cdot\omega$ hold? |
Firstly, you need to define an inner product in $T^1(V)$. As far as I am aware this is defined by summing all of the component-wise inner products for each tensor power; but, this is only guaranteed to converge in the range of the signature thanks to the factorial decay. Ilya's work that you link doesn't define the inn... |
A permutation $\phi:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\}$ is called a well-mixed if $\phi(\{1,2, \ldots, k\}) \neq\{1,2, \ldots, k\}$ for each $k<n$.
What is the number of well-mixed permutations of $\{1,2, \ldots, 15\}$?
$\textbf{Progress and Technical Insights:}$
- $\textbf{Decomposition Insight... |
I know that compact self-adjoint operators [have a countable, orthonornal eigendecomposition][1], and the spectrum is real, positive and the only cluster point is zero.
What sorts of algorithms does one use to compute the first K eigenvalues of a compact self-adjoint operator? Is it similar to computing the eigenva... |
It is known that for the Jordan block
$$ J_p = \left(\overbrace{\begin{array}{cccc}
\lambda & 1& \ldots & 0\\
0 & \lambda& & 0\\
0 & 0 & \ddots & 1 \\
0 &0 &\ldots & \lambda \\
\end{array}}^{p}\right)$$
the following formulas are valid:
$$
e^J= e^{\lambda} \sum_{r=0}^{p-1} \frac1{r!} I_r^{(p)};
\qquad
{... |
What is the reason for endowing the positive semidefinite cone with the Affine Invariant Riemannian Metric instead of a Euclidean metric? |
Let $\mathscr{F}=\{0,1\}$ and suppose that addition and multiplication are defined on $\mathscr{F}$ $not$ in the usual way but by means of the tpeculiar’ addition and multiplication tables below:
$\begin{array}{c|ccc}+&0&1\\\hline0&0&1\\1&1&0\end{array}$
$ \begin{array}{c|cc}\times&0&1\\\hline0&0&0\\1&0&1\end{array... |
I know that compact self-adjoint operators [have a countable, orthonornal eigendecomposition][1], and the spectrum is real and positive, and the only cluster point is zero.
What sorts of algorithms does one use to compute the first K eigenvalues of a compact self-adjoint operator? Is it similar to computing the eig... |
I'm currently studying the topic of the continuity of the integral $\displaystyle \int_a^x f(t) dt$, where $a$ is an arbitrary constant.
I understand that if $f$ is a continuous function on the interval $[a, b]$, then $\displaystyle F(x) = \int_a^x f(t) dt$ is continuous on $[a, b]$ and differentiable on $(a, b)$.
... |
How to prove the continuity of [integral with variable bounds of non-continuous function] at essential (but not infinite) discontinuity point? |
>Find a fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$ of $\mathbb Q(\sqrt{141})$
I have different corollaries for different numbers, the most appropriate for $141$ is the one below.
![enter image description here][1]
I used an algorithm (don't know if you know this, but
$\b... |
You have to form a group of r people. In your class, there are $m$ men, $w$ women
and $n$ non-binary individuals. Suppose a group is selected uniformly at random, and
assume $m, w, n ≥ r ≥ 5$. Find a closed-form probability function $f(x)$ which provides the probability that a given group has $x$ women, and $x + 5$ ... |
Is my logic valid for this probability function? |
This is probably a silly question, but I am interested in looking at limits of multivariable functions, such as \begin{equation}\lim_{x_1\to\infty}\lim_{x_2\to\infty}\cdots\lim_{x_m\to\infty}f(x_1,\cdots,x_m).\end{equation}
Is it notationally acceptable to simply write the above as \begin{equation}\lim_{\substack{x_i\... |
**Background**
Assume that we have a solid ellipse with uniform density, and that it rolls along a curve without any friction.
In the following <a href="https://mathoverflow.net/questions/418486/along-what-curve-does-an-ellipse-roll-down-the-fastest">MO question</a>, I asked along what curve an ellipse rolls dow... |
I am tasked with proving the following statement:
*"Every cut is the union of edge-disjoint minimal cuts"*
The only information given, is the existance of the cut-set subspace $W_S(G)$. It seems I have to say something about the fact that $W_S(G)$ is a subspace, thus closed under addition, and then proceed to sho... |
How can I prove the existence or not of a surjective morphisms of Riemann surfaces $\Bbb P^1 \to \Bbb C$ ? and $\Bbb C \to \Bbb P^1$? In case that it exists, how can I prove if there's any that can be extended to a morphism $\Bbb P^1\to \Bbb P^1$?
It should be enough to find and example to prove the existance but I... |
Do surjective morphism exists between the Riemann surfaces $\Bbb P^1$ and $\Bbb C$ ? Can they be extended to a morphism $\Bbb P^1 \to \Bbb P^1$? |
Is there any specific reason for which integrals start at zero? Logically when we think of an intgral as a summation of area, it must start from negative infinity right? Why do indefinite integrals start at zero?
I expect an intuitive reason rather than a rigorous proof. |
Why don't integrals start at -∞? |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...-83,-53,-23,7,37,67...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf... |
Why don't integrals start at $-\infty$? |
Going through Thomas Calculus, question 90 in the chapter on Chain Rule:
> Suppose that $f(x) =x^{2}$ and $g(x) =|x|$. Then the composites
>
>$$
( f\circ g)( x) =|x|^{2} =x^{2} \ \ \ \ \ \ and\ \ \ \ \ ( g\circ f)( x) =|x^{2} |=x^{2}
$$
> are both differentiable at $x=0$ even though $g$ itself is not differen... |
> Let $\mathscr{F}=\{0,1\}$ and suppose that addition and multiplication
> are defined on $\mathscr{F}$ *not* in the usual way but by means of
> the "peculiar" addition and multiplication tables below:
>
> $$\begin{array}{c|ccc}+&0&1\\\hline0&0&1\\1&1&0\end{array} \qquad
\begin{array}{c|cc}\times&0&1\\\hline0&0&0... |
Explain why there's no ordering of the field? |
I have tried to reach a contradiction by working with the definition:
> $M$ is locally Euclidean of dimension $\boldsymbol{n}$ : each point of $M$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^n$.
Consider the following: I assumed that there exists an open set around $(0,0)$, let’s cal... |
For example:
$$\begin{array}{l}
y = {x^{11}}\\
\Delta x = 0.1 \cdot x\\
\Delta y = {\left( {\frac{{df}}{{dx}}\Delta x} \right)^2} = 1.1 \cdot {x^{10}}\\
y \pm \Delta y = {x^{10}} \pm 1.1 \cdot {x^{10}}\\
\pm \frac{{\Delta y}}{y} = \pm 110\%
\end{array}$$
This above looks incorrect because more than 100%.
... |
Can a large error be calculated using logarithm? |
Roger has 101 dalmatian dogs. Each dog has a unique number of black spots from the set {1, 2, 3, . . . , 101}. We choose any 52 of the 101 dogs. We want to prove that any set of 52 dogs satisfies the following addition property:
Addition property: The numbers of spots on two of the dogs add up to exactly the number ... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/PCb21.png
Can someone explain the first assignment and implied? We prove bottom to up and I don't follor after the $(1=x+1)$ if-Statement. This is what my book says about the assignment rule:
> , if we wish to show that ψ holds in the ... |
I understand that, for the set $\{0,1\}$ and any finite number of constants and operations on that set, the resulting algebraic structure has a finite basis of identities. But, does there exist a finite algebraic structure on $\{0,1\}$ whose quasi-equational theory is not finitely based? If not, is there such a structu... |
Smallest cardinality of finite algebraic structure whose quasi-equational theory is not finitely based? |
to solve this I first solve the characteristic equation $r+2=0$ and since $r = -2$ I have $y_{1} = e^{-2x}$. Then I get a particular solution $y_{p} = C$ because number on the right side is just a constant 1 (what is this "number on the right side" called?)
plugging $y_{p}=C$ into the original equation I get $2C = 1... |
to solve this I first solve the characteristic equation $r+2=0$ and since $r = -2$ I have $y_{1} = e^{-2x}$. Then I get a particular solution $y_{p} = C$ because number on the right side is just a constant 1 (what is this "number on the right side" called?)
plugging $y_{p}=C$ into the original equation I get $2C = 1... |
I have a question with my reasoning (or lack thereof) -- I understand what the correct answer is, but I do not see the gap in my logic.
**Premise:**
We have n balls and we randomly allocate them to n buckets with equal probability. What is the probability that exactly one bucket remains empty?
**My Attempt:**
L... |
I'm looking for good exopository books and resources for probability concentration inequalities that covers both the theoretical aspects and practical applications of these inequalities. My interest primarily lies in understanding how they can be applied to fields such as analysis, geometry, and random phenomena... |
How can I prove the existence or not of a surjective morphisms of Riemann surfaces $\Bbb P^1 \to \Bbb C$ ? and $\Bbb C \to \Bbb P^1$? In case that it exists, how can I prove if there's any that can be extended to a morphism $\Bbb P^1\to \Bbb P^1$? $\Bbb P^1$ is the complex projective space. Morphism here means a holo... |
>A group of 8 people is to be chosen from three countries, K, L, and M. K has 3 people, L has 4 people, and M has 5 people. What are the number of ways in which the group can be chosen if it includes at least 1 person from each country?
I understand that a solution is to take the complement through
**all possibi... |
Why couldn't the duplicate cases be "divided out"? |
It is known that for the Jordan block
$$ J_p = \left(\overbrace{\begin{array}{cccc}
\lambda & 1& \ldots & 0\\
0 & \lambda& & 0\\
0 & 0 & \ddots & 1 \\
0 &0 &\ldots & \lambda \\
\end{array}}^{p}\right)$$
the following formulas are valid:
$$
e^{J_p}= e^{\lambda} \sum_{r=0}^{p-1} \frac1{r!} I_r^{(p)};
\qqua... |
Is there any specific reason for which integrals start at zero? Logically when we think of an intgral as a summation of area, it must start from negative infinity right? Why do indefinite integrals start at zero? Like when I plug in a value in the indefinite integral, we get the area from 0 to the plugged in value. The... |
This is probably a silly question, but I am interested in looking at limits of multi-variable functions, such as \begin{equation}\lim_{x_1\to\infty}\lim_{x_2\to\infty}\cdots\lim_{x_m\to\infty}f(x_1,\cdots,x_m).\end{equation}
Is it notationally acceptable to simply write the above as \begin{equation}\lim_{\substack{x... |
The naive way I think about rotations is by imagining a finger sliding the sphere and in doing so it traces a spherical curves. Let's start now with a spherical curve which I assume to be rectifiable. Can I recover the rotation I performed to produce the curve?
I think we should impose the condition that when the curv... |
The naive way I think about rotations is by imagining a finger sliding the sphere and in doing so it traces a spherical curve. Let's start now with a spherical curve which I assume to be rectifiable. Can I recover the rotation I performed to produce the curve?
I think we should impose the condition that when the curve... |
For example:
$$\begin{array}{l}
y = {x^{11}}\\
\Delta x = 0.1 \cdot x\\
\Delta y =\sqrt {\left( {\frac{{df}}{{dx}}\Delta x} \right)^2} = 1.1 \cdot {x^{10}}\\
y \pm \Delta y = {x^{10}} \pm 1.1 \cdot {x^{10}}\\
\pm \frac{{\Delta y}}{y} = \pm 110\%
\end{array}$$
This above looks incorrect because more than 100... |
For example:
$$\begin{array}{l}
y = {x^{11}}\\
\Delta x = 0.1 \cdot x\\
\Delta y =\sqrt {\left( {\frac{{dy}}{{dx}}\Delta x} \right)^2} = 1.1 \cdot {x^{10}}\\
y \pm \Delta y = {x^{10}} \pm 1.1 \cdot {x^{10}}\\
\pm \frac{{\Delta y}}{y} = \pm 110\%
\end{array}$$
This above looks incorrect because more than 100... |
*This question has been extensively edited to meet site requirements.*
As is well-known, the Diophantine equation $a^n+b^n=c^n$ has many solutions when $n=2$ (Pythagorean triples) but none when $n>2$ (the Fermat-Wiles Theorem). If one includes in the equation an extra term $\pm k$, yielding the equation:
$$a^n \p... |
For a nondecreasing function $F:\mathbb{R}\to\mathbb{R}$ the generalized inverse function is defined as $$F^{-}(y):=\inf\{x\in\mathbb{R}:F(x)\ge y\},$$where $F(-\infty):=\inf\{F(x):x\in\mathbb{R}\}<y<\sup\{F(x):x\in\mathbb{R}\}=:F(\infty)$. Is there anything, that can be said about the limits $\lim\limits_{y\to F(-\inf... |
I have the following PDE (with small parameter $0<\epsilon\ll 1$):\
$\frac{\partial u(x,t)}{\partial t}+\frac{\partial ^2u(x,t)}{\partial x^2}-\epsilon u(x,t)^2=0, x\in (0,1), t>0$\
$u(x,0)=u_0(x), x\in (0,1)$\
$u(0,t)=u(1,t)=0, t>0$
I now have to find the equations for the first three terms of the formal asympto... |
I am trying to solve an HJB equation in a very simple case that put me in trouble.
For a function $v(t,x)$ regular enough defined on $[0,T]$ with condition $v(T,x) = 0$ , I want to find a solution of
$$
\partial_{t}v(t,x) -\frac{1}{2}\left(\partial_{x}v(t,x)\right)^2 + 2x\partial_{x}v(t,x) + \frac{1}{2}x^2 = 0... |
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