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We can show that the dual bundle $E^*$ with fibers $(E^*)_x = (E_x)^*$ for all $x \in B$ have transition functions $g_{\alpha\beta} = (f_{\alpha\beta}^T)^{-1} : U_\alpha \cap U_\beta \to GL(r,\mathbb R)$.
However, I can't seem to show that the $g_{\alpha\beta}$ satisfy the cocycle conditions, that is $g_{\alpha\alph... |
I begin by three examples showing that the solutions of
$$AL+L^T=0\tag{1}$$
can depend on parameters.
a) If we take
$$A=\pmatrix{1&0&0\\0&1&1\\0&0&-1},$$
then the general solution of (1) is :
$$L=\pmatrix{0&a&0\\-a&b&-2b\\0&-2b&4b},$$
with $\det(L)=4a^2b$.
b) If $$A=\pmatrix{1&0&0&0\\0&0&1&0\\... |
How can you construct a right triangle, ABC with a right angle in C, if you are given the hypotenuse, c, and the altitude of the point C? |
I begin by three examples showing that the solutions of
$$AL+L^T=0\tag{1}$$
can depend on parameters.
a) If we take
$$A=\pmatrix{1&0&0\\0&1&1\\0&0&-1},$$
then the general solution of (1) is :
$$L=\pmatrix{0&a&0\\-a&b&-2b\\0&-2b&4b},$$
with $\det(L)=4a^2b$.
b) If $$A=\pmatrix{1&0&0&0\\0&0&1&0\\... |
So, I'm revisiting Complex Analysis mainly focused on the application of conformal maps in PDE's. Using Churchill's book (Complex Variables and Applications), the way to solve was using the fact that harmonic functions still harmonic after composed with (complex) conformal maps, the same for the boundary conditions. Al... |
This is the Sylvester-Transpose matrix equation sometimes called the T-Sylvester equation, analyzed in [The solution of the equation $AX + X^* B = 0$](https://pdf.sciencedirectassets.com/271586/1-s2.0-S0024379513X00035/1-s2.0-S0024379512008191/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEJj%2F%2F%2F%2F%2F%2F%2F%2F%2F... |
I am reading about General Number Field Sieve (GNFS) algorithm and its implementations, and the literature seems to have two different versions of the algorithm, but I can't find how these two are related. Is one an improvement over another? Or are they both used in practice? I just can't find any article or anything t... |
What is the relation between these two descriptions of General Number Field Sieve? |
find values of n for which the integral converges
$ \int_0^1 \ln\left(\frac{1}{x}\right)^n $ dx
i am able to understand that Integral needs to be broken in two parts so that convergence can be checked at x=1 and x=0 separately.
for the first part if n<0 then its proper integral
similarly for the second half n... |
To provide context, the continued fraction in the form $\frac{a_0}{1-\frac{a_1}{1+a_1-\frac{a_2}{1+a_2-...}}}$ evaluated to the $n$th denominator equals $\sum_{k=0}^{n}\prod_{j=0}^{k}a_j$. If one wants to write $e^x$ as a continued fraction, you can take the Maclaurin series of it which is $1+\sum_{n=1}^{\infty}x^n/n!$... |
I am reading about General Number Field Sieve (GNFS) algorithm and its implementations, and the literature seems to have two different versions of the algorithm, but I can't find how these two are related. Is one an improvement over another? Or are they both used in practice? I just can't find any article or anything t... |
$\textbf{Background.}$ I'm trying to apply Lojasiewicz's inequality to a specific function, $f$, and the Euclidean distance function $d$. Hence, I need to prove that $f$ and $d$ are subanalytic. Moreover, let $Q = [0,1]^2$. Specifically, $f:Q \to \mathbf{R}$ is real-analytic and non-negative. Moreover, $f$ is bounded o... |
So, I'm revisiting Complex Analysis mainly focused on the application of conformal maps in PDE's. Using Churchill's book (Complex Variables and Applications), the way to solve was using the fact that harmonic functions still harmonic after composed with (complex) conformal maps, the same for the boundary conditions. Al... |
To provide context, the continued fraction in the form $\frac{a_0}{1-\frac{a_1}{1+a_1-\frac{a_2}{1+a_2-...}}}$ evaluated to the $n$th denominator equals $\sum_{k=0}^{n}\prod_{j=0}^{k}a_j$. If one wants to write $e^x$ as a continued fraction, you can take the Maclaurin series of it which is $1+\sum_{n=1}^{\infty}x^n/n!$... |
I am an undergraduate student taking Numerical Analysis. I’m having a hard time understanding some of the material because it feels as though my instructor is jumping all over the place. When it comes to the applied problems I understand them and can complete them. But when he does a proof in class it’s hard to follow.... |
How can I fix my understanding of Numerical Analysis? |
$K$ is a nonempty subset of $R$n I have to show that k is compact if and only if every continuous function f:$K$ -> $R$ is bounded.
I know that if f is continuous and $K$ is compact then f is uniformly continuous does it mean the function is bounded ?? no
I have no idea how to connect compact and continuity. |
> Find the possible remainders of $(1-3i)^{2009}$ when divided by $13+2i$ in $\mathbb{Z}[i]$.
I'm having a hard time understanding remainders in $\mathbb{Z}[i]$. I'm gonna write my solution to the problem above just to give some context, but you can skip this part and read the question below if you want. We first st... |
> Find the possible remainders of $(1-3i)^{2009}$ when divided by $13+2i$ in $\mathbb{Z}[i]$.
I'm having a hard time understanding remainders in $\mathbb{Z}[i]$. I'm gonna write my solution to the problem above just to give some context, but you can skip this part and read the question below if you want. We first st... |
We have that the endomorphism bundle of a smooth vector bundle $E$ is $\text{End}(E) = \text{Hom(E,E)}$. Is it necessarily true that $\det(\text{End}(E))$ is always trivial for any smooth vector bundle $E$?
I'm unsure on how to prove this. I have seen and proved that a smooth $n$-dimensional manifold is orientable i... |
> Find the possible remainders of $(1-3i)^{2009}$ when divided by $13+2i$ in $\mathbb{Z}[i]$.
I'm having a hard time understanding remainders in $\mathbb{Z}[i]$. I'm gonna write my solution to the problem above just to give some context, but you can skip this part and read the question below if you want. We first st... |
> Find the possible remainders of $(1-3i)^{2009}$ when divided by $13+2i$ in $\mathbb{Z}[i]$.
I'm having a hard time understanding remainders in $\mathbb{Z}[i]$. I'm gonna write my solution to the problem above just to give some context, but you can skip this part and read the question below if you want. We start by... |
Given $Ax = b$, $x \ge 0$, and the covariance matrix $Cov(b)$, what is variance of $\min x_i$? Could there be an analytical solution?
$x_i$ is the $i$-th element of $x$. $A$ is a $m \times n$ matrix where $m < n$, an $A$ is $m$-rank. Also, we can assume the variance of $b_i$ and $b_j$ are independent, so $Cov(b)$ i... |
Error Propagation from Linear Programming Constraints. Given $Ax = b$, $x \ge 0$, and $Cov(b)$, what is variance of $\min x_i$? |
> Find the possible remainders of $(1-3i)^{2009}$ when divided by $13+2i$ in $\mathbb{Z}[i]$.
I'm having a hard time understanding remainders in $\mathbb{Z}[i]$. I'm gonna write my solution to the problem above just to give some context, but you can skip this part and read the question below if you want. We start by... |
When I entered a test at my school, I stuck this problem (it is also posted [here][1])
> Let $x,y,z$ be positive real numbers such that $a+b+c=3$, prove that
> $$\frac{1}{x+yz}+\frac{1}{y+zx}+\frac{1}{z+xy} \le
\frac{9}{2(xy+yz+zx)}.$$
I have tried AM GM
$$\frac{1}{x+yz}\le \frac{1}{2\sqrt{xyz}}.$$
But it le... |
How to prove $\sum\limits_{cyc} \frac{1}{x+yz} \le \frac{9}{2(xy+yz+zx)}$ for all $x,y,z >0:x+y+z=3.$? |
Indeed as mentioned in the comments and by you, this often just means almost surely: for almost every $\omega\in \Omega$ we have
$$|X_{t}(\omega)-X_{s}(\omega)|\leq C_{1}(\omega), \forall |t-s|\leq C_{2},$$
or in probability terms
$$P[|X_{t}-X_{s}|\leq C_{1}, \forall |t-s|\leq C_{2}]=1$$
$$\Leftrightarrow ... |
Some time ago I came across the following integral on the internet.
\begin{equation}
\int_0^1 \frac{\ln^2(1-x)}{x}\, dx =2 \zeta(3)
\end{equation}
I think I can generalise this integral to the following...
\begin{equation}
\boxed{\int_0^1 \frac{\ln^k(1-x)}{x}\, dx = (-1)^k\Gamma(k+1)\zeta(k+1)\,, \quad k \i... |
> Find the possible remainders of $(1-3i)^{2009}$ when divided by $13+2i$ in $\mathbb{Z}[i]$.
I'm having a hard time understanding remainders in $\mathbb{Z}[i]$. I'm gonna write my solution to the problem above just to give some context, but you can skip this part and read the question below if you want. We start by... |
In Fujino's book *Foundations of the minimal model program*, the definition of semi-ample $\mathbb{R}$-divisors is as follows
[![enter image description here][1]][1]
Then he claims that
[![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/U1IXr.png
[2]: https://i.stack.imgur.com/y... |
find values of n for which the integral converges
$ \int_0^1 (\ln\left(\frac{1}{x}\right))^n $ dx
i am able to understand that Integral needs to be broken in two parts so that convergence can be checked at x=1 and x=0 separately.
for the first part if n<0 then its proper integral
similarly for the second half... |
Let $f$ and $g$ be real functions. I quite like the notations $f \ll g$ to mean $f \in O(g)$ and $f \sim g$ to mean $\lim \frac fg = 1$. Are there similar notations which are commonly understood to mean $f \in o(g)$ or $f \in \Theta(g)$? For instance $f \lll g$ seems like a natural candidate for the former but I hav... |
Vinogradov-like notation for little-o or Theta? |
Let $f$ and $g$ be real functions. I quite like the notations $f \ll g$ to mean $f \in O(g)$ and $f \sim g$ to mean $\lim \frac fg = 1$. For example it is nice to be able to write out chains of asymptotic inequalities like "$f \ll g \ll h$" rather than $f \in O(g) \subset O(h)$. The latter also doesn't have a natura... |
I am working on a problem involving the maximization of a function subject to a normalization constraint, specifically in the context of quadratic forms and eigenvalues. The objective function is defined as:
$$ F(x) = **x**^T(w**Q_1** + (1-w)**Q_2**)**x** $$
where $**x**$ is the vector of decision variables subje... |
A sequence $M_i$ of length 100 is consisting of 50 "-1"s and 50 "+1"s randomly permuted (like balls drawn from an urn without replacing). Then a path $S_i$ is generated by the partial sums of this sequence, i.e. $S_k = \sum_{i=1}^k M_i$. The problem is to find the probability of the path not hitting +1, in other words,... |
Probability of Some Discrete Walk Hitting a Boundary? |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/bdxWx.png
I am having trouble wrapping my head around the second equality in the proof, namely $ \max_{\eta = g(\theta)} L(\eta) = \max_{\eta} L(g^{-1}(\eta))$.
So as far as I understand, $\hat{\theta}$ maximizes the function $L(\theta)... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/bdxWx.png
I am having trouble wrapping my head around the second equality in the proof, namely $ \max_{\eta = g(\theta)} L(\eta) = \max_{\eta} L(g^{-1}(\eta))$.
So as far as I understand, $\hat{\theta}$ maximizes the function $L(\theta)... |
I have a multi-variate convex function, and I want to find its global minimum. We know there is one and only one minimum.
Is there any algorithm which is guaranteed to converge to the minimum starting from an arbitrary initial point?
How about the Newton method?
ps. The function is smooth. |
I am working on a problem involving the maximization of a function subject to a normalization constraint, specifically in the context of quadratic forms and eigenvalues. The objective function is defined as:
$$ F(\mathbf{x}) = \mathbf{x}^T(w\mathbf{Q_1} + (1-w)\mathbf{Q_2})\mathbf{x} $$
where $\mathbf{x}$ is the ... |
Is this a valid way to prove a premise false? |
> Find the possible remainders of $(1-3i)^{2009}$ when divided by $13+2i$ in $\mathbb{Z}[i]$.
I'm having a hard time understanding remainders in $\mathbb{Z}[i]$. I'm gonna write my solution to the problem above just to give some context, but you can skip this part and read the question below if you want. We start by... |
If one has $\neg p \vee q$ and $\neg q$, does one need to apply the Commutative Law to $\neg p \vee q$ to obtain $q \vee \neg p$, before concluding that $\neg p$ ? |
The equivalence transformation says that any sequence of non-zero complex numbers satisfy the general continued fraction in the following manner. Here is the link: https://en.wikipedia.org/wiki/Generalized_continued_fraction#The_equivalence_transformation
Wikipedia says that this can be proved via induction but give... |
The Equivalence Transformation (??) for Generalized Continued Fractions |
I am working on a problem involving the maximization of a function subject to a normalization constraint, specifically in the context of quadratic forms and eigenvalues. The objective function is defined as:
$$ F(\mathbf{x}) = \mathbf{x}^T(w\mathbf{Q_1} + (1-w)\mathbf{Q_2})\mathbf{x} $$
where $\mathbf{x}$ is the ... |
The equivalence transformation says that any sequence of non-zero complex numbers satisfy the general continued fraction in the following manner. Here is the link: https://en.wikipedia.org/wiki/Generalized_continued_fraction#The_equivalence_transformation
Wikipedia says that this can be proved via induction but give... |
> Find the possible remainders of $(1-3i)^{2009}$ when divided by $13+2i$ in $\mathbb{Z}[i]$.
I'm having a hard time understanding remainders in $\mathbb{Z}[i]$. I'm gonna write my solution to the problem above just to give some context, but you can skip this part and read the question below if you want. We start by... |
What is (-1)^(2/10)? If we do ((-1)^2)^(1/10) than it is 1 but if we do (-1)^(1/5) then it is -1. So how to solve it? |
How to calculate this expression (-1)^(2/10)? |
Let $G$ be a finite group. When does the automorphism group $\text{Aut}(G)$ act transitively on $n$-tuples of generators of $G$?
I'm interested in this question in relation to the triangle groups. For example, when $G=\text{PSL}(2,7)$, there are exactly 336 triples of generators of orders 2, 3, and 7 which multiply ... |
When does $\text{Aut}(G)$ act transitively on $n$-tuples of generators of $G$? |
Can this proof that
$\sqrt{2}$
is irrational
be rewritten using only integers?
Most proofs that
$\sqrt{2}$ is irrational
start with assuming that
$2=\dfrac{a^2}{b^2}$
and derive a contradiction.
For a nice collection,
see my question here:
https://math.stackexchange.com/questions/1311228/what-is-the-most... |
Can this proof that $\sqrt{2}$ is irrational be rewritten using only integers? |
What is the maximal possible variance of a random variable that takes values in the set
[−1, 1]?
Let's say X is our random variable. I assumed maximum variance occurs when X uniformly distributed over this interval.
Thus as we know $\text{Var}(X) = \frac{(b - a)^2}{12}$ which evaluates in this case to $\frac{1... |
What is the maximal possible variance of a random variable that takes values in the set
$[−1, 1]$?
Let's say $X$ is our random variable. I assumed maximum variance occurs when $X$ is uniformly distributed over this interval.
Thus as we know $\text{Var}(X) = \frac{(b - a)^2}{12}$ which evaluates in this case to... |
According to the comments to [this question][1],
> Truth is ordinarily defined by reference to models.
If so, even axioms and theorems are not true without reference to a model.
However, when mathematicians say "this is true," they are usually not referring to a specific model. Is he saying "true in all model... |
When mathematicians say "true" do they mean "true in all models"? |
I'm trying to make a sinusoidal function that dilates as it progresses between two linear functions so that its peaks and troughs are always in contact with the linear functions. To do this I have worked with the function;$$\frac{y_{3}+y_{2}}{2}-\frac{y_{3}-y_{2}}{2}\sin x $$ where y_3 and y_2 are linear functions buil... |
What is happening in this 'multi'-sinusoidal function? |
I've been struggling with this exercise (Exercise 8.2.28, *Introduction to Abstract Algebra* by Nicholson) for a few hours, and I haven't made much progress.
> Let $G$ be a group of order $p^n$ and let $H_1, \dots ,H_m$ be the distinct subgroups of $G$ of index $p$. If $N = H_1 \cap \cdots \cap H_m$, show that $N \t... |
$K$ is a nonempty subset of $\mathbb{R}^n$. I have to show that $K$ is compact if and only if every continuous function $f: K \to \mathbb{R}$ is bounded.
I know that if $f$ is continuous and $K$ is compact then $f$ is uniformly continuous does it mean the function is bounded ?? no
I have no idea how to connect compac... |
> Find the possible remainders of $(1-3i)^{2009}$ when divided by $13+2i$ in $\mathbb{Z}[i]$.
I'm having a hard time understanding remainders in $\mathbb{Z}[i]$. I'm gonna write my solution to the problem above just to give some context, but you can skip this part and read the question below if you want. We start by... |
I understand how this proof works geometrically. Considering the unit square with corners (0,0),(0,1),(1,1), and (0,0), taking the integral of f(x) and then adding its inverse would produce an area equivalent to this unit square. However, I am having a hard time trying to prove this algebraically or anyway that is not ... |
I understand how this proof works geometrically. Considering the unit square with corners (0,0),(0,1),(1,1), and (0,0), taking the integral of f(x) and then adding its inverse would produce an area equivalent to this unit square. However, I am having a hard time trying to prove this algebraically or anyway that is not ... |
For any positive integer $d$ define the box $B_d$ as all points $(x_1, ..., x_d) \in \mathbb{R}^d$ with $|x_i| < 1.$
Suppose I have some invertible $n \times n$ matrix $M_1$, and for an integer $k<n$ I have the $k \times n$ projection matrix $P$ which projects a vector from $\mathbb{R}^n$ to $\mathbb{R}^k$ by remov... |
We all agree that the provided answer is incorrect for the reasons you state and others. Then the fun starts. Here is how I see it:
First, choose the leaders, eight out of 16 students C(16,8) ways.
Then assign a partner for each leader from the remaining 8 students 8! ways.
Finally, choose team A from the ... |
I understand how this proof works geometrically. Considering the unit square with corners $(0,0),(0,1),(1,1)$, and $(0,0)$, taking the integral of $f(x)$ and then adding its inverse would produce an area equivalent to this unit square. However, I am having a hard time trying to prove this algebraically or anyway that i... |
$K$ is a nonempty subset of $\mathbb{R}^n$. I have to show that $K$ is compact if and only if every continuous function $f: K \to \mathbb{R}$ is bounded.
I know that if $f$ is continuous and $K$ is compact then $f$ is uniformly continuous. Does it mean the function is bounded ? I have no idea how to connect compact an... |
How to show a continuous function is bounded when the domain is compact? |
I've been trying to develop an intuition for stochastic processes and would like some help clearing up the following situation.
Suppose $\{X_t\}_{t \in \mathbb{R}}$ is a real valued stochastic process defined on the probability space $(\Omega, \mathcal{F}, P)$. Suppose all the $X_t$ are identically distributed for ... |
Are paths in a stochastic process $\{X_t\}_{t \in \mathbb{R}}$ constant if $X_t$ is identically distributed for all $t$? |
The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan:
[![enter image description here][1]][1]
I understand everything in the proof except for:
How existence of $c$ such that $(c,q)=1$ and $cm \equiv n$ (mod $q$) is guaranteed?
[1]: https://i.sta... |
Dong's lemma: if $A(z),B(z),C(z)\in \operatorname{End}(V)[[z^{\pm 1}]]$ pairwise local, then $:\mathrel{A(z)B(z)}:$ and $C(z)$ are local.
I'm trying the following approach for proving the lemma:
We have the following characterization for formal power series that "vanish outside diagnal": if $(z-w)^k\sum_{n,m} f_... |
Are there "brute force" method for Dong's lemma? |
The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan:
[![enter image description here][1]][1]
I understand everything in the proof except for:
How $m' \equiv n'$ (mod $d$)?
I tried every elementary rule I knew but failed. For example I concluded ... |
The equivalence transformation says that any sequence of non-zero complex numbers satisfy the general continued fraction in the following manner. Here is the link: https://en.wikipedia.org/wiki/Generalized_continued_fraction#The_equivalence_transformation
Wikipedia says that this can be proved via induction but give... |
Consider a sequence of polynomials $P(x)=x^n+x^{n-1}+x^{n-2}+...+x^2+x-1, n>2$.
(i) Prove that it has a unique positive real root $x_n$
(ii) Find $\lim_{x \to ∞} 2^n (x_n - 1/2)$
The first part was relatively simple; upon differentiating $P(x)$, it is easy to see that for all $x>0$, the function is strictly... |
When solving a parabolic PDE with boundary conditions (e.g. the reflection method of the barrier option), we can change the parabolic PDE into a heat equation by applying some substitution. For example, changing:
$$\left\{\begin{array}{cc}
\frac{\partial u}{\partial t}+\frac{1}{2} \sigma^{2} \frac{\partial^{2} u}{\pa... |
While trying to solve a probability question about a coin flip game, I arrived at the expression $\sum_{k=0}^n {n\choose k}\sum_{j=0}^{k-1}{{n-1}\choose j}.$ Computing this sum for several low values of $n$ suggested that it summed to $2^{2n-2}$, suggesting there is an identity:
$$\sum_{k=0}^n {n\choose k}\sum_{j=0}... |
Given a smoothly time-dependent density matrix (positive-semidefinite matrix with trace 1) $\rho(t)$, its von Neumann entropy is defined as
$$
S(t)=\mathrm{Tr}(f(\rho(t))),\ f:[0,\infty)\ni x\mapsto -x\ln x\in\mathbb{R}\ (0\ln0=0).
$$
I want to know when $S(t)$ is differentiable.
If $\rho(t)$ has the same rank, th... |
When is von Neumann entropy differentiable? |
> Let $n$ be a nonnegative integer. Evaluate $\sum\limits_{i=0}^{n-1}\sum\limits_{j=i+1}^{n+1} {n+1\choose j}{n\choose i}$.
Below is a summary of a solution based off of a problem in the summation chapter of the book Problem Solving Through Problems by Loren Larson.
Multiply both sides of the sum by $\dfrac{1}{2... |
Given a smoothly time-dependent density matrix (positive-semidefinite matrix with trace 1) $\rho(t)$, its von Neumann entropy is defined as
$$
S(t)=\mathrm{Tr}(f(\rho(t))),\ f:[0,\infty)\ni x\mapsto -x\ln x\in\mathbb{R}\ (0\ln0=0).
$$
I want to know when $S(t)$ is differentiable.
If $\rho(t)$ has the same rank, th... |
Is it possible to find a spatial curve $\alpha(s) = (x(s), y(s), z(s))$ such that the lines containing the normal vector $\hat{n}(s)$ at each point on the curve $\alpha(s)$ intersect the $z$-axis at a constant angle $\theta$?
My attempts:
For any point $(r(s) \cos \phi(s), r(s) \sin \phi(s), h(s))$, the unit normal... |
Existence of a Spatial Curve with normal vector constant angle to the $z$-Axis? |
From my lecture notes I have that:
>The Lie algebra is introduced as a set of matrices $\theta$ that gave rise to elements $M$ of the corresponding matrix group through $M=e^{i\theta}$. However, just like a group can be thought of as an abstract set of instead of matrices, we can also give an abstract definition of th... |
I just found this kind of “conjecture” recently. Define the set of positive prime numbers $P$. Suppose $$ n \in P, n\geq 3. $$ Define $S := \{1,p_1,\dots,p_{x-1}\}$ that are primes and $1$ which are smaller than $n$.
(in which $p_{x-1}$ is the prime directly before n)
1. Can you find at most 2 distinct $a, b \in S$ (... |
When I entered a test at my school, I stuck this problem (it is also posted [here][1])
> Let $x,y,z$ be positive real numbers such that $x+y+z=3$, prove that
> $$\frac{1}{x+yz}+\frac{1}{y+zx}+\frac{1}{z+xy} \le
\frac{9}{2(xy+yz+zx)}.$$
I have tried AM GM
$$\frac{1}{x+yz}\le \frac{1}{2\sqrt{xyz}}.$$
But it le... |
Is Wikipedia's inductive proof of the Binomial Theorem acceptable? |
### problem statement
Zagreb and Yerevan play a restricted version of nim as follows: They start with 2 possibly empty (but not both empty) piles of rocks, and take turns Z-Y-Z-Y-Z... until the last player who has a move wins. On a turn, each must take at least one rock out of exactly one pile, so that the total numb... |
I was thinking about constructing functions that have arbitrary values at some integers.
One way to do that is by taking the functions
$$ f_n = \sum_{0 \leq i \leq m, i \not = n} \frac{x - i}{n-i} $$
This is $ 1 $ at $ x = n $ and $ 0 $ at all other integers $ 1 \leq x \leq m $. One can add these together with the ... |
>Let $a$ and $b$ be positive numbers such that $a>1$ and $b<a$. Let $P$ be a point in the first quadrant that lies on the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$. Suppose the tangent to hyperbola at $P$ passes through the point $(1,0)$, and suppose the normal to hyperbola at $P$ cuts off equal intercepts on the... |
Let $A\in\mathbb R^{n\times n}$ be a symmetric matrix. Consider the function
$$f: u\in\mathbb R^n\mapsto f(u)= Au\cdot u\in\mathbb R.$$
As an exercise, I have to prove that
$$f(u+v)\le f(u) +f(v)\quad\forall u,v\in\mathbb R^n.$$
I am stuck at this point. I write
$$f(u+v) = A(u+v)\cdot (u+v) = A(u+v)\cdot u + A... |
Let $A\in\mathbb R^{n\times n}$ be a symmetric and positive definite matrix. Consider the function
$$f: u\in\mathbb R^n\mapsto f(u)= Au\cdot u\in\mathbb R.$$
As an exercise, I have to prove that
$$f(u+v)\le f(u) +f(v)\quad\forall u,v\in\mathbb R^n.$$
I am stuck at this point. I write
$$f(u+v) = A(u+v)\cdot (u+... |
Suppose that $X_1,\dots, X_n$ are iid $P$ on $\mathcal{X}$. The empirical measure $\mathbb{P}_n$ is defined by $$\mathbb{P}_n:=\frac{1}{n}\sum_{i=1}^n\delta_{X_i}$$
For a real-valued function $f$ on $\mathcal{X}$ we write $$\mathbb{P}_n(f)=\int f d\mathbb{P}_n=\frac{1}{n}\sum_{i=1}^n f(X_i).$$
If $\mathcal{F}$ is... |
I came across the following problem while studying series of functions:
> Prove that $S(x)=\sum_{n=1}^\infty \frac{\sin(2^n x)}{2^n}$ is not differentiable at integers.
I am aware that this series is commonly known as the [Weierstrass function](https://en.wikipedia.org/wiki/Weierstrass_function), and according to... |
My question is taking place more specifically in a tensor category $(\mathcal{C},\otimes,\phi,\psi)$ (I will denote this category with $\mathcal{C}$ going forward) where $\otimes:\mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is a *bifunctor*, $\phi$ is the *associator* and $\psi$ is the *commutator*.
Now, let $U$ ... |
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