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I'm studying graphs in algorithm and complexity,
(but I'm not very good at math) as in title:
Why a complete graph has $\frac{n(n-1)}{2}$ edges?
And how this is related with combinatorics?
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I'm always needing to learn more about GAP.
Can anyone help me on the following question?
Question:
I have a list of 500 sublists where each sublist consists of six groups.
Is there a command to display or renumber only those sublists with trivial intersection?
Is there a function in GAP can solve this?
This questionin... |
What is the maximum number of times a piece of paper (of non-zero thickness) can be folded in half (mathematically)?
Edit:
I totally forgot to mention the "non-zero thickness" part; I've now included it.
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I've just spent past 5 days trying to get a solution to this:
Find all whole numbers, for whom the fraction
$\frac{n^3 + 2010}{n^2 + 2010}$
yields another whole number.
Before asking, please note that this is indeed an exercise from a math competition. However, this competition is already over, so don't worry; you're n... |
Suppose I have a stack and I want to find the permutations of numbers 1,2,3,...n.
I can push and pop. e.g. if n=2: push,pop,push,pop 1,2 and push,push,pop,pop 2,1
if n=4 I can only get 14 from the 24 permutations using the stack.. Does anyone know any function F(n) that could produce the number of permutations the sta... |
I have two square matrices: $A$ and $B$. $A^{-1}$ is known and I want to calculate $(A+B)^{-1}$. Are there theorems that help with calculating the inverse of the sum of matrices? In general case $B^{-1}$ is not known, but if it is necessary then it can be assumed that $B^{-1}$ is also known.
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I've got a couple of theorems that I must prove, but I'm completely stumped. Could someone help me out?
Prove: If P(x) can be divided by Q(x), then it can be divided by cQ(x), where c - any non zero number.
Prove: If the degrees of P and Q are the same ( equal), then P(x) can be divided by Q(x) then and only then, if ... |
the problem I'm stuck on is the following:
Suppose that S is a countably infinite subset of $\ell_2$ with the property that the linear span of S′ is dense in $\ell_2$ whenever S\S′ is finite. Show that there is some S′ whose linear span is dense in $\ell_2$ and for which S\S′ is infinite.
I have tried repeatedly to sol... |
In respect to a larger proof I need to prove that $(n-m) \mid (n^r - m^r) $ (where $\mid$ means divides, i.e., $a \mid b$ means that $b$ modulus $a$ = $0$). I have played around with this for a while now but to no avail and would be grateful if you could help me.
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Let $(X, \mathscr{O})$ be a ringed space and $\mathscr{F}, \mathscr{G}$ be sheaves of $\mathscr{O}$-modules on $X$.
Define $\mathscr{H}(U) = \mathscr{F}(U) \otimes_{\mathscr{O}(U)} \mathscr{G}(U)$. I am stuck trying to prove that $\mathscr{H}_p \cong \mathscr{F}_p \otimes_{\mathscr{O}_p} \mathscr{G}_p$ as $\mathscr{O... |
In dynamics, they talk about Abelian differentials on surfaces, are they the same as holomorphic differentials?
Quadratic differentials are multiple valued and can change sign as you move around a zero.
|
I have a terrible hand-writing and I'm very good at typing, so I had an idea about taking my math lecture notes using a computer.
I've tried using a simple syntax (using purely ASCII) but it's getting harder and harder, so I need something a bit more sophisticated.
Friend suggested Latex, but said that I probably won't... |
how can we generate random numbers using skew normal distribution in multivariate case
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Given an equation in the form $M(x)dx + N(y)dy = 0$ we test that the partial derivative of $M$ with respect to $y$ is equal to the partial derivative of $N$ with respect to $x$. If they are equal, then the equation is exact. What is the geometric interpretation of this?
Further more to solve the equation we may integra... |
Let $X$ be a completely regular space and let $T$ a topological space such that $X \subseteq T \subseteq \beta X$. Then $\beta T = \beta X$, where $\beta$ denotes the Stone-Cech compactification.
Solution: Let $f: T \mapsto [0,1]$. Then it suffices to show that the restriction of f to $X$ can be extended to $\beta X$".... |
I have learned trigonometry, in school but never understood clearly what it is..just mugged up formula's and theorems to get clear the exams. But now i want to know what exactly, is trigonometry from the basics so that i can understand.
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I know that following is a tautology because I've checked its truth table. I am now attempting to prove that it is a tautology by using the rules of logic, which is more difficult. How should I proceed?
$(p\land(p\implies q))\implies q$
$(p\land(\lnot p \lor q))\implies q$
$(p\land \lnot p) \lor (p\land q) \implies q$ ... |
Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then
$$ \det(I_m+AB) = \det(I_n+BA)$$
where $I_m$ and $I_n$ denote the $m \times m$ and $n \times n$ identity matrices, respectively.
Could you sketch a proof for me, or point to an accessible reference?
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here is a good math movie. what is your favorite software to make math movies? I used Mathematica, GeoGebra, Sketchpad. But they are not perfect. Also, could you recommend some interesting math movies?
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how to prove it?
I am talking about matrixes which satisfy:
$$( Ax , x ) > 0\quad \text{ for any}\quad \;x \neq 0.$$
How to prove that $A^T\;$ is also positive?
$$x^T A x = ( x^T A x )^T$$
and what?
|
Working in $F_q^n$. How many different ways do we have to choose $l$ vectors such that every subset of size $k$ of them is linearly independent.
(Assume n is large)
My Progress: For the first k vectors, just keep out of the subspace spanned so far, so for $1 \leq i \leq k$ we have $q^n-q^{i-1}$ ways to choose the $i^{t... |
Given canonical cone equation, how to find equation of cone's generatrix?
|
Looking at an earlier post Finding the power series of a rational function, I am trying to get a closed formula for the n'th coefficient in the Taylor series of the rational function (1-x)/(1-2x-x^3). Is it possible to use any of the tricks in that post to not only obtain specific coefficients, but an expression for th... |
$A = r \int_{0}^{\pi}\int_{0}^{2\pi} e^{i (\alpha+\theta)} d\alpha d\theta
= r \int_{0}^{\pi} [\frac{-i}{\alpha+\theta} e^{i(\alpha + \theta)}]_{0}^{2\pi} d\theta
= ...
= r \int_{0}^{\pi} e^{2i\alpha} ( \frac{1}{\alpha} - \frac{1}{\alpha+2\pi} ) d\theta$
and according to wolfram it is wrong... |
If A and B are two different nonempty sets, how many distinct sets can be formed
with these sets using as many unions,intersections,complements and parentheses as desired.
Four sets are fundamental:$A$,$B$,$A \cup B$,$A \cap B$.
Other sets are $A \cup B'$,$A' \cup B$,$A \cap B'$,$A' \cap B$,$A' \cup B'$,$A' \cap B'$.
A... |
Recently I became interested in trying to visualize the fact that $\pi_1(\text{SO}(3)) = \mathbb{Z}/2\mathbb{Z}$. For whatever reason, the plate trick doesn't do it for me, so I've been looking for something else. I know that $\text{SO}(3)$ is homeomorphic to $P^3(\mathbb{R})$, and one concrete visualization of this ... |
I'm a computer science major and like many of us we have to take two additional sciences. These two additional science courses are in addition to three semesters of calculus,two semesters of physics, and one semester each of linear algebra and discrete math.
I could take the easy way out and take an algebra based astro... |
Is there an odd prime integer $\displaystyle q$ such that $\displaystyle p= \frac{q^3+1}{2}$ is also prime?
A quick search did not find any, nor a pattern in the prime factorization of p. This is a possible quick solution to the unitary and Ree cases of ME.16954.
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I was just trying out some problems, when i couldn't solve this question:
Does there exist $n \times n$ matrices $A$ and $B$ with real entries such that $$ \Bigl(I - (AB-BA)\Bigr)^{n}=0?$$
I really don't know how to proceed for this question. What i did was to find some examples, but that didn't quite work. Any ideas... |
Let's suppose there are $n$ real numbers $a_0 < ... < a_n$ uniformly selected from interval [0, 1). If one knows $k$ numbers on consecutive positions $a_i < ... < a_{i+k-1}$ how good is $(k - 1) / (a_{i+k-1} - a_i)$ an estimator for $n$? What other estimators are possible/better?
NOTE: $n >> k$.
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I had referred to this structure earlier in a previous question which went unanswered.
If $u$ and $v$ are in $T_pM$ and $n \in (T_pM)^\perp$ then the extrinsic curvature form $K$ be defined as,
$K(u,v) = g(n,\nabla _u V)$
(where $V$ is a local extension of the vector $v$)
Similarly use say $U$ as a local extension of ... |
It is my understanding that one-time pad encryption is the only unbreakable encryption, but suffers from the management of huge keys, and the secure distribution of those keys.
Could one-time pads be used without key distribution to create secure communication?
Example Scenario:
Adam encrypts his message to Bob with ... |
For a problem such as:
$ \sum_{n=1}^\infty \frac{lnn}{n^3}$
How do you tell if it is decreasing or increasing? I know that you can compute the derivative and test if it is less than or greater than zero, but the way my solutions manual does it, I'm puzzled by.
The derivative is:
$\frac{1-3lnx}{x^4} < 0$
Solving for x g... |
If
$$A = \frac{1}{2} \times \begin{pmatrix}
-1& -\sqrt3 & 0 \\
-\sqrt3& 1 &0 \\
0 & 0 & 0
\end{pmatrix} \text{ and } E = \frac{1}{2} \times \begin{pmatrix}
1& \sqrt3 & 0 \\
-\sqrt3& 1 &0 \\
0 & 0 & 1
\end{pmatrix}$$
then show that $E^{-1}AE$ is diagonal matrix.
Can we prove this without actu... |
There is a book about rational points on elliptic curves. What about algebraic points?
|
What exactly does it mean by "half a unit in the nth place"? I am reading numerical analysis and this term is popping out very frequent in the first chapter itself,my first encounter with this is while reading the rounding-off error rules.
From few solved examples I can understand (or probably guess) it mean $5$ but wh... |
I'm having some difficulty with the following question:
Let $(f_{n}(x))$ be a uniformly convergent functions sequence in $(a,b)$ (where b can be $\infty$) such that $(f_{n}(x)) \to f(x)$. Suppose that for almost all $n$, the limit $\displaystyle \lim_{x\to b^{-}}f_{n}(x)$ exists.
Prove that $\displaystyle \lim_{x\to b... |
In reference to Stanley's Enumerative Combinatorics Vol. 2: right after he has defined Kostka numbers (section 7.10), he mentions that it is unlikely that a general formula for $K_{\lambda\mu}$ exists, where $K_{\lambda\mu}$ is the number of semistandard Young tableaux of shape $\lambda$ and type $\mu$ with $\lambda\vd... |
You can use the proof of the Schwarz reflection principle to show that if $D$ is a domain in the complex plane, where $D^+$ denotes the subset of D above the real line, and $D^-$ the subset below, if $f:D\to C$ which is analytic on $D^+$ and $D^-$, and continuous on $D$, then $f$ is analytic on $D$.
I want to generaliz... |
Axis Angle to quaternion and quaternion to Axis angle question
Greetings All (matlab / octave code)
Link to text file in case formatting gets messed up
http://db.tt/nVv8Ivj
I created two functions one to convert axis angle to quaternion
and another one to convert quaternion to axis angle as far as I can tell the formul... |
The set of rational numbers $\mathbb{Q}$ is an unbounded subset of $\mathbb{R}$ with Lebesgue outer measure zero. In addition, $\mathbb{R}$ is an unbounded subset of itself with Lebesgue outer measure $+\infty$. Therefore the following question came to my mind: is there an unbounded subset of $\mathbb{R}$ with positive... |
When one has an expression in terms of $i$, one can send $i$ to $-i$ and, if the expression remains unchanged, one can conclude that the expression is, in fact, real. Analogous statements hold for expressions involving radicals. Why is this?
One admittedly trivial example is the expression
$$\frac{1}{x-i}+\frac{1}{x+i}... |
Which computer algebra system allows me to compute the homology of a complex of finitely presented abelian groups which are not necessarily free?
Sage and Magma apparently don't: see here and here.
Edit: I should specify that the main obstacle for me of doing this by hand with a CAS is the lack of an algorithm for comp... |
What conditions are needed on the matrices A and B so that the following is true:
$A^{T}BA=B$. Clearly, if B is invertible, the determinant of A must be 1 or -1. What other conditions are needed?
|
I am looking for a term that describes function that is "localized". What I mean is a function that is non zero in a bounded range and zero else where, such as the a rectangle pulse function. But would also like that the term cover functions such as gaussian that approach zero as you get away from the peak. If you have... |
Some background: I was in class today, and the professor was proving that given a matrix $A$ that is $n$ by $m$ and a $B$ such that $AB=I$ where $B$ is obviously $m$ by $n$ and $I$ is $n$ by $n$, if it is also the case that $BA=I$, then $A$ must be a square matrix (that is, $m=n$). We didn't manage to get through howev... |
How does one compute the cardinality of the set of functions $f:\mathbb{R} \to \mathbb{R}$ (not necessarily continuous)?
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Possible Duplicate:
Highest power of a prime $p$ dividing $N!$
How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
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How do you demonstrate that a local minimum of a function has its derivative equal to zero with the definition of the derivative.
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A jar contains some red and some yellow jelly beans. If a child ate 1 red jelly bean, 1/7 of the remaining candies would be red. If instead the child ate 5 yellow jelly beans, 1/6 of the remaining candies would be red. How do I found out the number of jelly beans that are in the jar?
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When you take the derivative of an expression with respect to x, does x have to be a variable, or is it allowed to be a polynomial, a term, a vector, or anything else? It doesn't seem to make sense to me if x is not a variable.
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This is my first question in mathSE, hope that it is suitable here!
I'm currently self-studying complex analysis using the book by Stein & Shakarchi, and this is one of the exercises (p.67, Q14) that I have no idea where to start.
Suppose $f$ is holomorphic in an open set $\Omega$ that contains the closed unit disc, e... |
How well accepted are proofs where critical parts depend on software of which we have no access to the source, e.g Mathematica, Maple?
More generally how much can we take on faith with regards to software. Do we have to check the compiler for the Four Color Theorem?
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I'm interested in (unity-preserving) homomorphisms $f: S \to T$ between (commutative, with-unity) rings $S$ and $T$ so that if $f(x)$ is a unit, then $x$ was a unit to start with. For example, an inclusion of fields has this property, but a nontrivial localization like $\mathbb{Z} \to \mathbb{Q}$ does not; it sends $2... |
If $\mathbf{x}$ is a vector and $Q$ a symmetric Matrix;
Does $Q\mathbf{x} = \mathbf{x}^T Q$ ?
EDIT :
I have this:
$$\frac{1}{2}\left(Q\mathbf{x} + \mathbf{x}^TQ\right) + \mathbf{c},$$
where $\mathbf{c}$ is also a vector, and I want to obtain
$Q\mathbf{x}+\mathbf{c}$.
I thought that was the answer to my question, but ... |
Let $V,U$ vector subspaces of $\mathbb{C}^{n+1}$ of dimension $r+1$, $s+1$.
Let $\pi : \mathbb{C}^{n+1}/\{0\} \longrightarrow \mathbb{CP}^n$ the projection map to complex projective space defined by $\pi:(x_0, ... ,x_n)\mapsto(x_0:...:x_n)$.
This is a question that came up while doing homework and I wanted to check som... |
Show that if A has a right-inverse, then $Ax = b$ has at least one solution for every choice of b in $R^n$.
|
I was telling someone about the smash product and he asked whether it was the categorical product in the category of based spaces and I immediately said yes, but after a moment we realized that that wasn't right. Rather, the categorical product of $(X,x_0)$ and $(Y,y_0)$ is just $(X\times Y,(x_0,y_0))$. (It seems lik... |
Suppose I choose two rational functions, say,
$$u = \frac{t(4+t)^5}{(1+4t)^5}, \qquad v = \frac{t^5 (4+t)}{(1+4t)}.$$
Then I know that $K(X):= \mathbf{C}(u,v)$ is the function field of the projective line (Proof: If $K(Y):=\mathbf{C}(t)$, then there is an inclusion $K(Y) \subseteq K(X)$, and hence a surjection $Y \sim... |
Is there a compactly supported and smooth (except at one point) function $f : \mathbb{R} \to \mathbb{R}$ where (non smooth point) it is is exactly k times differentiable ?
|
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series:
$$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl( \frac{ 2 a + (n-1)\cdot d}{2}\biggr)$$
There is a slight difference in... |
Dear all, next year, I will probably teach a one-semester course of Differential Geomtry of curves and surfaces. Its content must be something along the lines of the first four chapters of Do Carmo's famous book and my plan is indeed to follow it.
But I would like to know some other, more modern, references. Particular... |
Numerically solving $y'=f(x,y)$ with $k_1=f(x_n,y_n),
k_2=f(x_n+c_2*h, y_n+c_2*h*k_1),
y_{n+1}=y_n+h(b_1*k_1+b_2*k_2)$
what would the local truncation error be?
Also, how would we perform a Taylor expansion on this to show that it is second order, provided $b_1+b_2=1$ and $b_2*c_2=\frac{1}{2}$?
Many thanks.
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I have two sets of ranked data which are ranked on different scales, one fixed and one variable, and I would like to convert one of the data sets ranking values to match the other.
Set 1 is ranked 10 - 1 with 10 being the highest rank.
Set 2 is ranked 1 - x with 1 being the highest rank and x being the number of items ... |
On the number line, starting from 0. There is a probability of $p$ of moving 1 unit to the positive direction, and $1-p$ of moving 1 unit to the negative direction.
What is the probability the walk never reaches $-1$ after $n$ steps?
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When searching a number on Wolfram Alpha, one of the results is its representation.
For example, for 8549:
8549 has the representation 8549 = $5·2^6·3^3-91$.
Similarly for 75290:
75290 has the representation 75290 = $3·2^9·7^2+26$.
What is the significance of these representations?
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Is it possible to write $\int_{-\infty}^\infty e^{-iqu}\left( \frac{A}{\sqrt{1-iau}}+\frac{B}{\sqrt{1-ibu}}\right)^Ndu$ in this form $\int\frac{e^{-iu}}{(\sqrt{1-iau})^n(\sqrt{1-ibu})^m}du$
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Let V be the vector space of all real-valued bounded sequences. Then for $a,b \in V$ $\langle a,b \rangle :=\sum _{n=1}^{\infty } \frac{a(n) b(n)}{n^2}$ defines a dot product. Find a subspace $U \subset V$ with $U \neq 0, U \neq V, U^\bot=0$.
I couldn't find anything that works, thank you in advance. $U^\bot$ is the or... |
Does there exist a continuous function $f : [0, 1] → [0, 1]$ such that the pre-image $f^{−1}(y)$ of any point $y \in [0, 1]$ is uncountable?
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What is an Improper Random Variable? I know the definition in terms of the CDF like, F(∞) - F(-∞) <1. Could any one explain it more clearly, specifically I am looking for an example of an improper random variable. Is Cauchy distributed Random Variable improper? (Since it has a long tail)
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I have an equation
$$x = \csc(\theta) - \cot(\theta).$$
As $\theta$ approaches zero, $x$ approaches zero. However, trying to solve the equation at zero yields an undefined result.
How do I rewrite the equation to be continuous at 0?
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Suppose there is a random walk starting at the origin such that the probability to move right is $\frac13$ and the probability to move left is $\frac23$.
What is the probability to return to the origin?
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Can we find two functions $f$ and $g$ that are reasonably defined nontrivial (not everywhere zero, $f\neq g$, not linear polynomials) functions such that the following condition is satisfied?
$$ f \left(\int_{0}^{t} g(x) \ \text{d}x\right) = g \left(\int_{0}^{t} f(x) \ \text{d}x\right) $$
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I have to demonstrate this formulae:
Prove $A = (A\setminus B) ∪ (A ∩ B)$
But it seems to me that it is false.
$(A\setminus B) ∪ (A ∩ B)$
$X \in A\setminus B \implies { x ∈ A \text{ and } x ∉ B }$
or
$X ∈ A ∩ B \implies { x ∈ A \text{ and } x ∈ B }$
so:
$x ∈ A ∩ B$
so:
... |
I have come across an artificial, simulated, stock-market type of situation, whose rules, I find, create a rather interesting problem. I want to know if there is a mathematically optimal solution for "trading" on this simplified market, and if not, what may be a good approximation of this optimal solution.
Here are th... |
A set is a collection of distinct objects.
Suppose I sample some parameter of some population and the results are $\{35,35,36,36,37 \}$.
Do I need to say (i) the multiset of the samples is $\{35,35, \dots \}$ instead of (ii) the set of samples is $\{35,35, \dots \}$?
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Question: Let $A \in M(n,n,\mathbb{R}), a_{ij} = 1$ for all $i,j$. Calculate the real eigenvalues and the affiliated eigenspace of $A$.
So first of all what I would be trying to calculate are values of $\lambda$ such that $0=\text{det}\left( \begin{array}{cccc} 1- \lambda & 1 & \cdots & 1 \\ 1& 1- \lambda & & \vdots\\ ... |
I have the matrix $A$ which is of size $m \times n$, with $m > n$.
Let $Q = A^T A$.
How do I determine, if $Q$ is Semi-Positive-definite?
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As most of us are aware the cost for solving a linear system ("exactly") with Gauss Elimination and other similar methods with a few right hand side and where the matrix has no structure is $\mathcal{O}(N^3)$ where $N$ is the system size.
I am wondering about the lower bound for solving a linear system. An obvious lowe... |
Using basic operators, derive the additional operator intersection.
$R \cap S = ?$
I thought it might be: $(R \cup S) - ((R -S) \cup (S -R))$ but I also think there might be an easier way of doing it?
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My questions are about a sequence or function with several variables.
I vaguely remember some while ago one
of my teachers said taking limits of
a sequence or function with respect to
different variables is not
exchangeable everywhere, i.e.
$$ \lim_n \lim_m a_{n,m} \neq \lim_m \lim_n a_{n,m}, \quad \lim_x \lim_y f(x,... |
There are 6 of x and 4 of y. A group of 5 is selected. What is the probability of there being 3x and 2 y? I completely forgot how to do this and my textbook doesn't have any information on it.
|
$U$ and $W$ are two subspaces of vector space $V$.
If $U \oplus W = V$, then $\forall v \in V$, there exist two unique vectors $u \in U$ and $w \in W$ such that $v = u + w$.
Is the reverse true? That is, if any vector can has such unique decomposition, do we have $U \oplus W = V$?
Can the above statements be generaliz... |
Exercise 1.8.5 of Berenstein-Gay "Complex variables" asks to count the number of zeroes in $(0,2\pi)$ of certain trigonometric polynomial.
Towards the exercise, the book suggests to first show the following:
Given $0\le a_0<\dots<a_n$, the roots of $$p(z)=a_0+a_1z+\dots+a_nz^n$$ lie in the unit disk.
Do you see how... |
I was thinking about cardinality of all symmetric relations, for example in $\mathbb{Z}$. I know, that if I have finite set (which contains $n$ elements), there are $2^{\frac{n(n+1)}{2}}$ symmetric relations. But what happens when we are talking about infinite sets ($\mathbb{Z}$ for example)? How many symmetric relatio... |
In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient
Ker d / Im d
where d as usual denotes the differentials, indexes skipped for simplicity.
How can this be generalized to a general Abelian Category? Do I have the notion of a Quotient there?
|
Let $f \in L^1_{loc}$. We call $x \in \mathbb R^n$ an Lebesgue point, if
$$\lim \limits_{R \rightarrow 0} \frac{1}{m(B_R(x))}\int_{B_R(x)} f \quad \text{ exists} \tag{1}$$
or
$$\lim \limits_{R \rightarrow 0} \;\; \sup \limits_{ B, B' \in \mathcal B_R(x)} \left\vert \dfrac{1}{m(B)}\int_{B} f - \dfrac{1}{m(B')}\int_{B'} ... |
The aim is to find a formula for the question.
For $n=2$ i get $2^{2^n}=16$ possible functions.
This is the solution for a boolean function with 2 attributes:
01) a -> separable
02) b -> separable
03) not a -> separable
04) not b -> separable
05) a and b -> separable
06) a or b -> separable
07)... |
i had a geometry/trignometry problem come up at work today, and i've been out of school too long: i've lost my tools.
i'm starting with a rectangle of known width (w) and height (h). For graphical simplification i can convert it into a right-angle triangle:
i'm trying to find the coordinates of that point above which ... |
Im having trouble with this derivative: $3^{x/2}$
I know that the derivative of $c^x$ is $c^x\ln c$.
Is $3^{x/2} = 3^{x/2}\ln 3$ correct?
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$g(n)=n^a$, $f(n)=n^b$, where $a$ and $b$ are real numbers such that:
$0 \lt a \lt b$.
I want to use L'Hopital's technique to prove that $g(n)$ belongs to $f(n)$.
My attempt:
$$ \lim_{n\to\infty}\frac{n^a}{n^{a+1}}.$$
I'm using $a+1$ to state that $b \gt a$.
The problem is that I don't know if using $a+1$ is correct.
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I am trying to understand the justification of step (*) below,
which I have seen used to find the integral of the Error Function Er(f); the
cdf of the standard-normal normal .
Here is the derivation I know:
We set $I:= \int_{-\infty}^\infty e^{-x^2}dx$
Then we use:
$$I^2= \left( \int_{-\infty}^\infty e^{-x^2} dx\right)... |
What is the derivative of $f(x)= \log^{k}x$ where $k$ is a positive integer?
Update:
Hello I have found that $\log^k n$ = $(\log n)^k$
(Thomas H. Cormen, Introduction to Algorithms (2ed) p. 53 )
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My question refers to functions with values in Banach spaces and under what conditions the limit of a sequence of measurable functions is also measurable.
But first, let me recall some well-known results from real analysis. Let $(X,\mathcal{F})$ be a measurable space, $\bar{\mathbb{R}}$ the extended real line, and $\m... |
Can someone help me with the formula needed to obtain the following. I need to discount products on a sliding scale. If a customer purchases 2 products, he must get a 15% discount. If he purchases 10, he must get a 30% discount. I don't want a linear slope, but rather as the number of products increases, the increase i... |
I read the result that every compact $n$-manifold is
a compactification of $\mathbb{R}^n$.
Now, for surfaces, this seems clear: we take
an n-gon, whose interior (i.e., everything in
the n-gon except for the edges) is homeo. to $\mathbb{R}^n$,
and then we identify the edges to end up with
a surface that is closed and bo... |
I need assistance with the following question:
The particle starts from the origin on the number line .
It makes n steps at total.
At each step it jumps with the probability 1/2 to the origin and with the uniform probability jumps to the one of its neighbors.I have to prove that the expectation of its maximal distance ... |
In Physics, the Simple Harmonic Oscillator is represented by the equation $d^2x/dt^2=-\omega^2x$ .
By using the characteristic polynomial, you get solutions of the form $x(t)=Ae^{i\omega t} + Be^{-i\omega t}$. I get that you use Euler's formula $e^{i\theta}=\cos\theta + i\sin\theta$, but I can't seem to find my way all... |
This may be a stretch, but are there examples of proofs that prove that a proof exists for a theorem.
For example, if A is a theorem, and it is too tedious to prove that, is it possible to show that a proof exists for theorem A.
|
I am wondering if anyone could prove the following equivalent definition of recurrent/persistent state for Markov chains:
1) $P(X_n=i,n\ge1|X_0=i)=1$
2) Let $T_{ij}=min\{n: X_n=j|X_0=i\}$, a state $i$ is recurrent if for $\forall j$ such that $P(T_{ij}<\infty)>0$, one has $P(T_{ij}<\infty)=1$
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