instruction stringlengths 12 30k |
|---|
What is the importance of prime numbers in cryptographic algorithms ? |
I am trying to understand why $1^{x}=1$ for any $x\in\mathbb{R}$
Is this even legal to write this? As the base 1 should not equal 1 for $1^{x}$ to be an exponential function?
Is $1^{x}=1$ just because it is defined to be so?
If possible please refer me to a book or article that discusses this topic.
|
I am trying to understand why $1^{x}=1$ for any $x\in\mathbb{R}$
Is it OK to write $1^{x}$? As the base 1 should not equal 1 for $1^{x}$ to be an exponential function?
Is $1^{x}=1$ just because it is defined to be so?
If possible please refer me to a book or article that discusses this topic.
|
Yes, $\mathbb{Z}$ is one.
(**Note**: As pointed out in the comments, $\mathbb{Z}$ is _technically_ locally Euclidean as it is discrete. If you are bothered by such things, replace $\mathbb{Z}$ by something like $\mathbb{Q}$, or the union of copies of the Cantor set. I think that, despite this flaw, $\mathbb{Z}$ is... |
**HINT**
Given $\varepsilon>0$ can you measure the set $\{x:f(x)>\varepsilon\}$? |
Re: "What's the best way to explain to a non-mathematician that complex numbers are necessary and meaningful, in the same way that real numbers are?"
I am a non-mathematician...so fully qualified to answer this question...
* first explain imaginary numbers
* then explain that complex numbers are simply numbers... |
If I have term -5 and I want to change its sign to opposite can I say that I'm negating it's sign? |
**HINT**
Given $\varepsilon>0$, can you measure the set $\{x:f(x)>\varepsilon\}$? |
FInding index of a fibonacci number:: any mathematical solution possible ? |
If a Diophantine equation is: exists v, p(x,v) = 0 (where v is a vector of finitely many integers) for some polynomial p, is there a Turing machine which prints out all values of x?
|
A while ago I was dreaming up point-set topology exam questions, and this one came to mind:
Is $\mathbb Q\setminus \{0\}$ homeomorphic to $\mathbb Q$? (Where both sets have the subspace topology induced from the standard topology on $\mathbb R$.)
However, I couldn't figure this out at the time, and I'm curious to... |
Are the rationals minus a point homeomorphic to the rationals? |
There is a direct relationship between equivalence relations and partitions.
Is there a way to simply use an equivalence relation's definition to get the matching partition? And what about the other way around? |
I have been looking for a proof of the statement:
"Assume the Generalized Riemann Hypothesis. Let $d$ be a fundamental discriminant and $\chi_d$ the associated primitive quadratic character. Then, $$L(\tfrac{1}{2},\chi_d)\geq 0."$$
Can anyone point me in the right direction or give a reference? Thanks a lot! |
Finding index of a Fibonacci number: any mathematical solution possible? |
What's the thing with $\sqrt{-1} = i$? Do they really teach this in the US? It makes very little sense, because $-i$ is also a square root of $-1$, and the choice of which root to label as $i$ is arbitrary. So saying $\sqrt{-1} = i$ is plainly false!
So why do people say $\sqrt{-1} = i$? Is this how it's taught in t... |
When asked to evaluate $\int_{a}^{\infty}f(x)dx$, you split the interval based on the improper points.
If there is another improper point other than $\infty$, at $b$, we will write: $\int_{a}^{\infty}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx+\int_{c}^{\infty}f(x)dx$ and ask whether each of the integrals on the ri... |
When asked to evaluate $\int_{a}^{\infty}f(x)dx$, you split the interval based on the improper points.
If there is another improper point other than $\infty$, at $b$, we will write: $\int_{a}^{\infty}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx+\int_{c}^{\infty}f(x)dx$ and ask whether each of the integrals on the ri... |
Let's say I have two numbers that I know to have been normalized- 39, 12. How do I calculate the percentage difference between them? Is it the same as if they hadn't been normalized? ie: (12/39) * 100 |
There is a large literature about how to take a drawing in the plane which tries to give a 3-dimensional view of a polyhedron and extract a polyhedron from this drawing, in contrast to applying Steinitz's Theorem which gives necessary and sufficient conditions that a graph be the vertex-edge graph of a convex 3-dimensi... |
I just read through a proof of the Compact Uniformization Theorem, and I follow it up to the very last line. The proof is:
Compact Uniformization Threorem. If $X$ is a compact regular space, then the neighborhood filter $\mathcal{N}_\Delta$ of the diagonal $\Delta\subset X\times X$ is a uniform structure on $X$.
... |
Shouldn't the percentage difference be [(12-39)/39]*100 or [(39-12)/12]*100? Note that you get different answers, not just the negative-when the two numbers are unequal the percentage difference depends upon which you regard as the base.
Whether normalization changes the percentage depends upon what you mean by nor... |
This is a good question. One way to think about the issue is that the "convergence" of your integral is not just about whether the value is finite, but also about whether the finite value is well defined. This is somewhat related to [indeterminant forms](http://en.wikipedia.org/wiki/Indeterminate_form) where $+\infty +... |
Is $\sum \sin{\frac{\pi}{n}}$ convergent? |
I just read through a proof of the Compact Uniformization Theorem, and I follow it up to the very last line. The proof is:
Compact Uniformization Threorem. If $X$ is a compact regular space, then the neighborhood filter $\mathcal{N}_\Delta$ of the diagonal $\Delta\subset X\times X$ is a uniform structure on $X$.
... |
How can one find if a function $f$ is infinitely differentiable? |
If I have term -5 and I want to change its sign to opposite can I say that I'm negating its sign? |
By differentiating it an infinite number of times?
But seriously, that's what you do. Just that usually you can infer what the higher order derivatives will be, so you don't have to compute it one by one.
**Example** To see that $\sin(x)$ is infinitely differentiable, you realize the following: $\frac{d}{dx}\si... |
Define a **quadtope** P as a polytope which is the convex hull of a set Q of quadrilaterals, any pair of which intersect in at most one vertex, and are in maximally general position, or, a k-fold pyramid over such a convex hull. (So, a pair of quadrilaterals from Q with a common vertex subtends a 4-dimensional subpolyt... |
How many combinatorial types of pure quadtopes of dimension 5 are there? |
Here's another way to think about the "line at infinity" and the "points at infinity"...
Think of the usual $XY$-plane as sitting inside of $3$-space, but instead of it sitting in its usual place, $\{(x,y,0) : x,y\in\mathbb{R}\}$, shift it up by $1$ so that it sits as the $z=1$ plane.
Now, you are sitting at the ... |
Does $\lim_{n \to \infty}\frac{\mathrm{d}^n}{\mathrm{d}x^n} f(x)$ converge? |
Does $\lim_{n \to \infty}\frac{\mathrm d^n}{\mathrm dx^n} f(x)$ converge? |
I know this is a silly question, but I've tried to find an answer using my TI-89 calculator, Maple and wolframalpha but none of those could tell me whether
$$\int_0^\infty |\cos(x^2)| \mathrm dx$$
converges or diverges.
Thus, I'd be very happy if someone could help me out and tell me, whether the given integra... |
Does $\int_0^\infty |\cos(x^2)| \mathrm dx$ converge? |
Most of the time, an equivalence relation is hiding an "equality" somewhere; that is, $x\sim y$ if and only if $x$ and $y$ have *something* that you are trying to isolate which is "equal". With integers, the equivalence $a\equiv b\pmod{m}$ means that $a$ and $b$ have **equal** *remainder when divided by $m$*. We usuall... |
If you sum an expression over an uncountable set
$\sum_{x\in \mathbb{R}}f(x)$, then do we need $f(x)=0$ on all but a countable subset in order for the sum to have a finite value?
If not can you give an example of a function everywhere nonzero that has a transfinite sum with a finite value?
Possible keywords: Tra... |
Here's another way to think about the "line at infinity" and the "points at infinity"...
Think of the usual $XY$-plane as sitting inside of $3$-space, but instead of it sitting in its usual place, $\{(x,y,0) : x,y\in\mathbb{R}\}$, shift it up by $1$ so that it sits as the $z=1$ plane.
Now, you are sitting at the ... |
If this is homework, it's a pretty strange one. :-? Anyway, do you like the following system?
$$
\begin{align}
x &= 1 \\\
y &= 2 \\\
z &= 3
\end{align}
$$
-Too much easy, I see.
So, try this: your system must fulfil the following equation:
$$
A
\begin{pmatrix}
1 \\\
2 \\\
3
\end{pma... |
Given an elliptic curve over a finite field $p$, $y^2 = x^3 + ax + b$
How can I retrieve the value of $y$ given the value of $x$?
My knowledge in this area is quite limited, so I understand this question might seems silly.
I'm currently working on an application that uses an elliptic curves library and I saw ... |
Expanding on Jack's answer, there is a whole exciting field known as "chromatic homotopy theory". I don't know much about it, I'll say a few slogans since I noticed you also asked the question "geometry or topology". Hopefully none of them are wrong. (If so, somebody please correct me!)
<s>Chromatic homotopy theo... |
I am attempting to write a function (programming, not mathematics) which calculates the shortest distance between two points on a toroid. I got the following formula from [StackOverflow][1], but I don't recognize the `min` portion. Can someone explain that to me, please?
sqrt(min(|x1 - x2|, w - |x1 - x2|)^2 + ... |
Can someone tell me what "min" means in this context? |
Define a **quadtope** P as a polytope which is the convex hull of a set Q of convex quadrilaterals, any pair of which intersect in at most one vertex, and are in maximally general position, or, a k-fold pyramid over such a convex hull. (So, a pair of quadrilaterals from Q with a common vertex subtends a 4-dimensional s... |
This is cool. Are you doing chemistry?
So, a slightly more standard notation might be $G_1\backslash SO(3)/G_2$ -- this indicates that you're left-multiplying by elements of $G_1$ and right-multiplying by elements of $G_2$.
A "properly discontinuous" action of a group $G$ on a space $X$ is one where every point ... |
For real $s$, $L(s,\chi_d)$ is real, and it is certainly positive for
large $s$. It is nonzero for $s>1$ by the Euler product and $L(1,\chi_d)\ne0$.
If $L(1/2,\chi_s)<0$ what would happen for some $1/2<s<1$?
|
I'm doing a multiple choice exercise and I'm stuck at a question, where there are two possible answers (may be both or none correct).
> Consider the differential equation
>
> $$y''+y'+y=0$$
>
> for $y(x)$. Which of the following statements are true?
>
> - The set of complex solutions forms a two-dimensional co... |
What kind of vector space does the set of (complex/real) solutions of the following differential equation form? |
I am trying to figure out the fundamental group (actually simply connected or not will suffice) of the following quotient space of $SO(3)$:
Let $X = SO(3)/E$, where $E$ is the equivalence relation defined as follows:
$E \equiv M \sim {S_{A}}^{i} * M * {S_{B}}^{j}$ where ${S_{A}}^{i} \in $ Crystallographic point g... |
I am trying to figure out the fundamental group (actually simply connected or not will suffice) of the following quotient space of $SO(3)$:
Let $X = SO(3)/E$, where $E$ is the equivalence relation defined as follows:
$E \equiv M \sim {S_{A}}^{i} * M * {S_{B}}^{j}$ where ${S_{A}}^{i} \in $ Crystallographic point g... |
Does Slater condition hold trivially (because there are no inequality constraints) for the problem:
$$\min_{x,y} \:\: cx+dy$$
s.t.
$$e^x + e^y = 1.$$
Can I conclude there is a zero duality gap here? |
Does Slater's condition hold for the following problem? |
Given an elliptic curve $y^2 = x^3 + ax + b$ over a finite field $\mathbf{F}_p$, how can I retrieve the value of $y$ given the value of $x$?
My knowledge in this area is quite limited, so I understand this question might seems silly.
I'm currently working on an application that uses an elliptic curves library an... |
Anyone know of an online tool available for making graphs (as in graph theory - consisting of edges and vertices)? I have about 36 vertices and even more edges that I wish to draw.
(why do I have so many? It's for pathing in a game)
Only tool available to me right now is MS Paint and it would be very messy.
|
Online tool for making graphs (vertices and edges)? |
**Motivation**
I have a prior on a random variable $X\sim \beta^\prime(\alpha,\beta)$ that I would like to use in JAGS. Although JAGS does not support the $\beta^\prime$ distribution, [Wikipedia] states:
> if $X\sim\beta(\alpha,\beta)$ then $\frac{X}{1-X} \sim\beta^\prime(\beta,\alpha)$
and that
> if $X\s... |
Are these transformations of the $\beta^\prime$ distribution to $\beta$ and $F$ correct? |
It seems so. $\mathbb{Q}_p(\zeta_p)$ is a $p-1^{th}$ extension of $\mathbb{Q}_p$ which doesn't extend the residue field; and so is $\mathbb{Q}_p(p^{\frac{1}{p-1}})$. However I can't see how to express $\zeta_p$ in $\mathbb{Q}_p(p^{\frac{1}{p-1}})$ or how to express $p^{\frac{1}{p-1}}$ in $\mathbb{Q}_p(\zeta_p)$.
Can... |
Is $\mathbb{Q}_p(\zeta_p)$ the same as $\mathbb{Q}_p(p^{\frac{1}{p-1}})$? |
**Motivation**
I have a prior on a random variable $X\sim \beta^\prime(\alpha,\beta)$ that I would like to use in JAGS. Although JAGS does not support the $\beta^\prime$ distribution, [Wikipedia][1] states:
> if $X\sim\beta(\alpha,\beta)$ then $\frac{X}{1-X} \sim\beta^\prime(\alpha,\beta)$
and that
> if $... |
**Motivation**
I have a prior on a random variable $X\sim \beta(\alpha,\beta)$ but I need to transform the variable to $Y=\frac{X}{1-X}$, for use in an analysis and I would like to know the distribution of $Y$.
[Wikipedia][1] states:
> if $X\sim\beta(\alpha,\beta)$ then $\frac{X}{1-X} \sim\beta^\prime(\alpha,\... |
**Motivation**
I have a prior on a random variable $X\sim \beta(\alpha,\beta)$ but I need to transform the variable to $Y=\frac{X}{1-X}$, for use in an analysis and I would like to know the distribution of $Y$.
[Wikipedia][1] states:
> if $X\sim\beta(\alpha,\beta)$ then $\frac{X}{1-X} \sim\beta^\prime(\alpha,\... |
**Motivation**
I have a prior on a random variable $X\sim \beta(\alpha,\beta)$ but I need to transform the variable to $Y=\frac{X}{1-X}$, for use in an analysis and I would like to know the distribution of $Y$.
[Wikipedia][1] states:
> if $X\sim\beta(\alpha,\beta)$ then $\frac{X}{1-X} \sim\beta^\prime(\alpha,\... |
I am Computer science student so pardon me if I am asking this in a wrong place! I am trying to work on my rusty math since last two years but until now I am not much confident enough the main problem that I am facing is that I no matter whatever I do I am making silly mistakes,I will be appearing a college level math... |
I am Computer science student so pardon me if I am asking this in a wrong place!My inspiration of learning maths is mainly due to algorithms and its feels to me that without mathematics I am not justifying my computer science degree,I am trying to work on my rusty math since last two years but until now I am not much ... |
I am Computer science student so pardon me if I am asking this in a wrong place!
My inspiration of learning maths is mainly due to algorithms and its feels to me that without mathematics I am not justifying my computer science degree,I am trying to work on my rusty math since last two years but until now I am not m... |
I am Computer science student so pardon me if I am asking this in a wrong place!
My inspiration of learning maths is mainly due to algorithms and its feels to me that without mathematics I am not justifying my computer science degree,I am trying to work on my rusty math since last two years but until now I am not m... |
Suppose I know the values of a continuous function on $[a,b]$ in some finite number of points $x_0,x_1 \ldots x_n$. I can form the Lagrange interpolating polynomial, $p$. I am curious if there is any interesting estimate of the expression $|f(x)-p(x)|$ for an arbitrary point $x$ that does not assume $f$ is smooth? (Per... |
The best way to develop "ingenuity" (really, ability to solve certain kinds of problems) is practice. Practice, practice, practice. And then, more practice. The vast majority of problems you will encounter are problems that have something in common with other problems, that can be analyzed with the same kind of ideas, ... |
Advice: How can one prepare for a maths entrance exam? How can one develop mathematical thinking? |
Whether or not "ingenuity" is "required to be able to crack" (do well?) the exam you will present depends enormously on the kind of exam it is. Ingenuity is not particularly required, say, in the United States' GRE examination (at least, when I took it, even the Mathematics Subject test).
The ability to concentrate... |
**Motivation**
I have a prior on a random variable $X\sim \beta(\alpha,\beta)$ but I need to transform the variable to $Y=\frac{X}{1-X}$, for use in an analysis and I would like to know the distribution of $Y$.
[Wikipedia][1] states:
> if $X\sim\beta(\alpha,\beta)$ then $\frac{X}{1-X} \sim\beta^\prime(\alpha,\... |
I am Computer science student so pardon me if I am asking this in a wrong place!
My inspiration of learning maths is mainly due to algorithms and its feels to me that without mathematics I am not justifying my computer science degree, I am trying to work on my rusty math since last two years but until now I am not ... |
Whether or not "ingenuity" is "required to be able to crack" (do well?) the exam you will present depends enormously on the kind of exam it is. Ingenuity is not particularly required, say, in the United States' GRE examination (at least, when I took it, even the Mathematics Subject test).
The ability to concentrate... |
Are these transformations of the $\beta^\prime$ distribution from $\beta$ and to $F$ correct? |
Given a collection of points $P \subset \mathbb{R}^3$, a crude
characterization of the "shape" of $P$ is sometimes given by the
principal components. We construct a covariance matrix, e.g., if $P$
is discrete, $C = \sum_{p\in P} (p - \mu)(p - \mu)^\intercal$, where
$\mu = \frac{1}{|P|}\sum_{p\in P}$ is the centre o... |
Given a collection of points $P \subset \mathbb{R}^3$, a crude
characterization of the "shape" of $P$ is sometimes given by the
principal components. We construct a covariance matrix, e.g., if $P$
is discrete, $C = \sum_{p\in P} (p - \mu)(p - \mu)^\intercal$, where
$\mu = \frac{1}{|P|}\sum_{p\in P}\, p$ is the cent... |
Given a collection of points $P \subset \mathbb R^3$, a crude characterization of the "shape" of $P$ is sometimes given by the principal components. We construct a covariance matrix, e.g., if $P$ is discrete, $C = \displaystyle\sum_{p\in P} (p - \mu)(p - \mu)^\intercal$, where $\mu = \displaystyle\frac1{|P|}\sum_{p\in ... |
I am writing a program to solve a Rubik's cube, and would like to know the answer to this question.
There are 12 ways to make one move on a Rubik's cube. How many ways are there to make a sequence of six moves?
From my project's specification: up to six moves may be used to scramble the cube. My job is to write a... |
How many ways can I make six moves on a Rubik's cube? |
What is the value of $1^x$ ? |
A question I recall from Carl Linderholm's "Mathematics Made Difficult", Chapter 3 Exercise 8.
A farmer acquires and algebraically closed field by extending his field finitely.
What can be said about the original field?
What can be said? I immediately see two fields with this property, ${\Bbb R}$, and $... |
If E is lebesgue measureable in R^n and I=[a,b] how do I show that EXI is measureable in R^(n+1) ?
|
If $E$ is lebesgue measureable in $\mathbb{R}^n$ and $I=[a,b]$ how do I show that $E\times I$ is measureable in $\mathbb{R}^{n+1}$ ?
|
[a sequence on twin primes][1] In the diagram, How do the stripes come from? Can the prime numbers also interfere like light and wave? When zoom in or zoom out the diagram in Mathematica, the stripes are changing.
[1]: http://my.unix-center.net/~martian/math%20album/%E6%95%B0%E5%AD%A6%E4%B8%93%E8%BE%91%202%20-%... |
![complex plot of the zeros][1]
(Diagram and setup from UCSMP _Precaluclus and Discrete Mathematics_, 3rd ed.)
Above is a partial plot of the zeros of $p_c(x)=4x^4+8x^3-3x^2-9x+c$. The text stops at showing the diagram and does not discuss the shape of the locus of the zeros or describe the resulting curves. Ar... |
If $E$ is lebesgue measureable in $\mathbb{R}^n$ and $I=[a,b]$ how do I show that $E\times I$ is measureable in $\mathbb{R}^{n+1}$ ?
Jonas:
Im using Outer_measure(E)=inf { Sum Vol(I_k) : U I_k contains E} and for every epsilon >0 there exists an open set G containing E such that Outer_Measure(G-E)<epsilon
i ... |
If $E$ is Lebesgue measurable in $\mathbb{R}^n$ and $I=[a,b]$ how do I show that $E\times I$ is measurable in $\mathbb{R}^{n+1}$?
Jonas:
I'm using $\mu^*(E)=\inf \{ \sum \mathrm{Vol}(I_k) \mid E\subseteq \cup I_k\}$ and for every $\epsilon \gt 0$ there exists an open set $G$ containing $E$ such that $\mu^*(G-E)\l... |
What does 'general' in 'general topology' really refer to? |
What does 'general' in 'general topology' really refer to? |
Here's another way to think about the "line at infinity" and the "points at infinity"...
Think of the usual $XY$-plane as sitting inside of $3$-space, but instead of it sitting in its usual place, $\{(x,y,0) : x,y\in\mathbb{R}\}$, shift it up by $1$ so that it sits as the $z=1$ plane.
Now, you are sitting at the ... |
To distinguish [(general) topology][1] from more specific branches of topology? -[Differential topology][2], [algebraic topology][3], [geometric topology][4]...
[1]: http://en.wikipedia.org/wiki/General_topology
[2]: http://en.wikipedia.org/wiki/Differential_topology
[3]: http://en.wikipedia.org/wiki/Algeb... |
I've noticed when sweeping the floor that dust gathers particularly in the corners. I assume there is a fluid mechanics reason for this. Does anyone know what it is? |
Why does dust gather in corners? |
I've noticed when sweeping the floor that dust gathers particularly in the corners. I assume there is a fluid mechanics reason for this. Does anyone know what it is?
-----------------
**Edit:** No, really, this is a mathematical question. Air blows around the room, which constitutes a vector field. Let's say it b... |
I was thinking about Feynman integrals the other day and in particular about discretizing the paths.
Does anyone know the lay of the land about what happens when you do path integrals over, say, a lattice or graph?
Sorry this is not a very specific question. |
[another sequence on twin primes][1] The maximal prime factor of average of twin prime pair:
n = 100000;
averageList = Select[Prime[Range[n]], PrimeQ[# + 2] &] + 1;
mpfList = FactorInteger[#][[-1, 1]] & /@ averageList;
ListPlot[%]
Why does this diagram look like a series of trajectories?
![alt text][2]
... |
The Sum of 2 sets of Measure zero might well be very large for example, the sum of $x$-axis and $y$-axis, is nothing but the whole plane. Similarly one can ask this question about Cantor sets:
If $C$ is the cantor set, then what is the measure of $C+C$? |
**The Problem:**
$$\frac{1}{1 \cdot 2 \cdot 3 \cdot 4} + \frac{4}{3 \cdot 4 \cdot 5 \cdot 6} + \frac{9}{5 \cdot 6 \cdot 7 \cdot 8} + \frac{16}{7 \cdot 8 \cdot 9 \cdot 10} + \cdots $$
My solution:
I started off with trying to find the $T_n$ th term
$$T_n = \frac{n^2}{(2n-1)(2n)(2n+1)(2n+2)}$$
$$ = \frac{1}{... |
How to do this problem quickly? |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.