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I have a matrix of the form $$\mathbb{A}_{i,j}(\omega) = \sum_n \frac{\sin(nx_i)\sin(nx_j)}{-\omega^2+C(n)}+\sum_{r_x}\sum_{r_y}\frac{\sin(r_xx_i)\sin(r_xx_j)\sin(r_y\alpha)}{-\omega^2+D(r_x,r_y)}.$$ In truth, the matrix formulation is much more complex that this, but I have tried to make things simpler by introduci... |
For a much simpler verification, double count the angle sum.
Consider all of the vertex angles of the triangles. What is their sum?
- Each triangle has an angle sum of $180^\circ$.
- The 1999 points each contribute $360^\circ$ (Why?). Add in the 4 corners which have total of $360^\circ$.
- Hence, ther... |
A [Maclaurin spheroid](https://en.wikipedia.org/wiki/Maclaurin_spheroid) can become a [Jacobi ellipsoid](https://en.wikipedia.org/wiki/Jacobi_ellipsoid#Jacobi_formula) but I don't understand how this happens. From the plot I see that it must be spinning fast enough and from the text there seems to be some viscous dampi... |
**Question**
Given a sequence $a_n$ when it is possible to express such sequence as a recurrence relation? *i.e.* when
$$
a_n= \sum_{i=1}^{n-1} f_i(n) a_{i} + g(n)
$$
where $f_i(n)$ are generic function of $n$ that does not depend from $a_i$.
I'm looking for a characterization for such sequences, but also s... |
Lets say I have $f$, $g$ both analytic functions on some domain (open and connected set) D.
Suppose also I was able to prove that $f = 0$ on the entire domain.
is it legitimate to then claim that $\frac{f(z)}{g(z)}$ is identically zero on that domain?
In particular, g(z) may assume a zero value in that domain, bu... |
if I know an analytic function is identically zero in a domain, can I always divide it by another analytic function in that domain? |
When a sequence satisfies $a_n= \sum_{i=1}^{n-1} f_i(n) a_i$? |
**Question**
Given a sequence $a_n$ when it is possible to express such sequence as a "linear recurrence relation with not constant coefficients"? *i.e.* when
$$
a_n= \sum_{i=1}^{n-1} f_i(n) a_{i} + g(n)
$$
where $f_i(n), g(n)$ are generic function of $n$ that does not depend from $a_i$.
I'm looking for a c... |
When a sequence satisfies $a_n= \sum_{i=1}^{n-1} f_i(n) a_i+ g(n)$? |
In my introductory differential geometry class, we learnt that isometries preserve the torsion $\tau$ and curvature $\kappa$ of curves. I was wondering if the converse of this statement is true: if a map $F$ preserves $\kappa$ and $\tau$ for all curves, is $F$ an isometry?
Thanks for any help. |
Is a map that preserves torsion and curvature of all smooth curves an isometry? |
**Question**
Given a sequence $a_n$ when it is possible to express such sequence as a "linear recurrence relation with not constant coefficients"? *i.e.* when
$$
a_n= \sum_{i=1}^{n-1} f_i(n) a_{i} + g(n)
$$
where $f_i(n), g(n)$ are generic function of $n$ that does not depend from $a_i$.
I'm looking for a c... |
Currently reading the first edition "Introduction to Graph Theory" by Douglas B. West. Exercise 1.1.9 states the following:
**1.1.9.** Suppose that $G$ is a simple graph having no vertex of degree 0 and no induced subgraph with three edges. Prove that $G$ has at most four vertices.
**Counter-example:** Let $V(G) ... |
$\DeclareMathOperator \erf{erf}$
Wolfram Alpha gives the following [$\erf^{-1}(z)$ series](https://www.wolframalpha.com/input?i=series+expansion+inverf%28x%29):
>$$\sum_{n=1}^\infty\frac{z^n}{2\pi n}\int_0^{2\pi}e^{it}\erf(e^{it})^{-n}dt$$
which can be derived via [Lagrange inversion](https://en.m.wikipedia.org/... |
Let $M$ be a rectangular matrix. Suppose that $x$ and $y$ are vectors so that $$x^T M = (x^T My) y.$$
Does this imply that $x$ and $y$ are singular vectors of $M$? If M is square and $x=y$ then indeed this implies that $x$ must be an eigenvector of $M$ but not sure about this case for singular vectors. |
Currently reading the first edition "Introduction to Graph Theory" by Douglas B. West. Exercise 1.1.9 states the following:
**1.1.9.** Suppose that $G$ is a simple graph having no vertex of degree 0 and no induced subgraph with three edges. Prove that $G$ has at most four vertices.
**Counter-example:** Let $V(G) ... |
Let $M$ be a rectangular matrix. Suppose that $x$ and $y$ are vectors so that $$x^T M = (x^T My) y^T.$$
Does this imply that $x$ and $y$ are singular vectors of $M$? If M is square and $x=y$ then indeed this implies that $x$ must be an eigenvector of $M$ but not sure about this case for singular vectors. |
Let $f: X \to Y$ be a proper birational morphism of Noetherian quasi-separated schemes. We have the derived pull-back $Lf^*: D(QCoh(Y))\to D(QCoh(X))$ (https://stacks.math.columbia.edu/tag/06YI) and similarly the derived push-forward $Rf_*: D(QCoh(X))\to D(QCoh(Y))$ functor. I have the following questions:
(1) Is $... |
This is a question I ended up with while trying to make a program that would find an (at least locally-) minimal-energy configuration for a piecewise linear 1-dimensional object in $\mathbb{R}^3$ in relation to a practical knot theory type of question. If it helps at all for the purposes of context or intuition, the se... |
In the matrix ring $M_3(\mathbb{Z}_6)$, what is the determinant of the matrix diag($3,3,1$)? Is it $9$ with respect to the integers or is it $3$ modulo $6$? How about its trace? |
Where should we compute the determinant and trace of a matrix? |
A bracelet contains at least 1 and at most 4 beads of identical size on a loop of string. Anna is making a bracelet and has 3 green, 3 blue and 3 red beads. How many distinct bracelets can she possibly make?
The way I attempted this question was by counting up the number of distinct bracelets case by case, and summe... |
How many distinct bracelets can be made? |
Let $M$ be a rectangular matrix. Suppose that $x$ and $y$ are vectors so that $$x^T M = (x^T My) y^T.$$
Does this imply that $x$ and $y$ are singular vectors of $M$?
If M is square and $x=y$ then indeed this implies that $x$ must be an eigenvector of $M.$ This is because then $Mx = (x^T M x) x,$ where $x^TMx$ is... |
A cylinder flattened out is a rectangle (ignore the top and bottom faces).
But what about a cylinder where the top face is smaller than the bottom face...what shape is that flattened out? (again ignoring the faces themselves)
See here for an image: https://imgur.com/a/5FlHGrZ
I'm asking because I'd like to 3d ... |
What shape is an irregular cylinder flattened out? |
I know very little about analysis or PDEs, so apologies if this is an elementary question.
Let $g \in C^0(\mathbf{R}^2, \mathbf{R})$ and suppose that there exists $f \in C^2(\mathbf{R}^2, \mathbf{R})$ with $\Delta f = g$. Here $\Delta = \partial^2/\partial x^2 + \partial^2/\partial y^2$ is the Laplacian on $\mathbf{... |
If the area integrals of a function are zero, is it in the image of the Laplacian? |
I am interested in integer sequence A254828 (https://oeis.org/A254828), but from the link it seems to have a recursive formula $a(n) = a(n-1) + a(n-30) - a(n-31)$.
They are **the number of positive integers up to n+1 divisible by 2, 3 or 5.**
However, is there a concise closed-form expression?
7 6... |
I am interested in integer sequence A254828 (https://oeis.org/A254828), but from the link it seems to have a recursive formula $a(n) = a(n-1) + a(n-30) - a(n-31)$.
As joriki said, they are **the number of positive integers up to n+1 divisible by 2, 3 or 5.** We can check by the following code
```
//A254828... |
In this [answer][1] is it mentioned that
$$\frac{1}{(u+z)^2}\in\Theta(1/N^2),$$
I know that
$$\frac{1}{2\pi i}\int_C\frac{1}{z^2}dz=0$$
where $C$ is any circle centered at the origin. But what about
$$\frac{1}{2\pi i}\int_C\frac{1}{(u+z)^2}dz?$$
Can we make a change of variables to get that this integral is... |
Why is $\frac{1}{(u+z)^2}\in\Theta(1/N^2)$? |
I know very little about analysis or PDEs, so apologies if this is an elementary question.
Let $g \in C^0(\mathbf{R}^2, \mathbf{R})$ and suppose that there exists $f \in C^2(\mathbf{R}^2, \mathbf{R})$ with $\Delta f = g$. Here $\Delta = \partial^2/\partial x^2 + \partial^2/\partial y^2$ is the Laplacian on $\mathbf{... |
In the Bayesian setting, suppose that we know the PDF of $X$ given $\theta$ is $p_X(x \vert \theta)$ and the PDF of $Y$ given $\theta$ is $p_Y(y \vert \theta)$. In the standard fashion, we may assume some (continuous) prior distribution of $\theta$, say $g(\theta)$, which we can then use to find the posterior distribut... |
Let $m^*$ be an outer measure and $\Delta$ denote the symmetric difference.
- Define a semi-metric $d: \mathscr{P}(\mathbb{R}) \times \mathscr{P}(\mathbb{R})\to [0, \infty]$ by $d(A, B)=m^* (A \Delta B)$.
- Denote by $\mathscr{E}$ the set of all elementary subset of $\mathbb{R}$ where an elementary subset of $\math... |
Is $\bar{\mathscr{E}}=\mathscr{E}\cup F$, where $\mathscr{E}$ is the set of all elementary subsets of $\mathbb{R}$ and $F$ is finite or countable? |
I'm trying to show that
$$e^{-\mu}\left(\left(\frac{e}{1+\delta}\right)^{(1+\delta)\mu} + \left(\frac{e}{1-\delta}\right)^{(1-\delta)\mu}\right) \le 2e^{-C\mu\delta^2},$$
for an absolute constant $C > 0$, where $\delta \in (0,1]$ and $\mu > 0$.
Using the idea in [this answer][1], I wrote
$$\log(1+\delta) = 0 + \... |
Circular metal tokens are used to operate a washing machine in a laundromat. The diameters of the tokens are known to be normally distributed. Only tokens with diameters between 1.94 cm and 2.06 cm will operate the machine.
Find the probability that less than two tokens out of 20 will not operate the machine.
... |
This is a question I ended up with while trying to make a program that would find an (at least locally-) minimal-energy configuration for a piecewise linear 1-dimensional object in $\mathbb{R}^3$ in relation to a practical knot theory type of question. If it helps at all for the purposes of context or intuition, the se... |
I'm trying to show that
$$e^{-\mu}\left(\left(\frac{e}{1+\delta}\right)^{(1+\delta)\mu} + \left(\frac{e}{1-\delta}\right)^{(1-\delta)\mu}\right) \le 2e^{-C\mu\delta^2},$$
for an absolute constant $C > 0$, where $\delta \in (0,1]$ and $\mu > 0$.
Using the idea in [this answer][1], I wrote
$$\log(1+\delta) = 0 + \... |
This is similar to https://math.stackexchange.com/questions/1214874/parameterizing-equilateral-polygons but for 3 dimensions.
I have $n$ points in $\mathbb{R}^3$ ($n \geq 3$), labeled $1$, $2$, ..., $n$. Point $1$ is unit distance from point $2$, which is unit distance from point $3$, ..., and point $n$ is unit dist... |
A cylinder flattened out is a rectangle (ignore the top and bottom faces).
But what about a cylinder where the top face is smaller than the bottom face...what shape is that flattened out? (again ignoring the faces themselves)
See here for an image: 
I'm asking because I'd li... |
From [Wikipedia](https://en.wikipedia.org/wiki/Shattered_set):
>The class [of sets] $C$ shatters the set $A$ if for each subset $a$ of $A$, there is some element $c \in C$ such that
>$$\{c\cap A: c\in C\} = P(A)$$
In other words, $C$ shatters $A$ if for every subset $S\subset A$, it's possible to find a set $... |
Can a triangle up to isometry shatter seven points? |
Let $P(X,Y)$ be a non zero polynomial of $\mathbb C[X,Y]$. Let $\{(x_i,y_i)\in\mathbb C^2\mid i\in\mathbb N\}$ be an infinite subset of $\mathbb C^2$ with the property that $x_i\ne x_j$ if $i\ne j$. Does there exist an index $i_0\in \mathbb N$ such that there exists $n_0\in\mathbb N$ with the property that $P(x^n_{i_0}... |
This is a question I ended up with while trying to make a program that would find an (at least locally-) minimal-energy configuration for a piecewise linear 1-dimensional object in $\mathbb{R}^3$ in relation to a practical knot theory type of question. If it helps at all for the purposes of context or intuition, the se... |
Does the Fourier Transform method to prove the spectrum of the Laplace operator generalize to other multiplication operators?: https://math.stackexchange.com/questions/790401/how-to-prove-the-spectrum-of-the-laplace-operator
As a concrete example: Let $T_a:f(x)\to f(x-a),f\in L^2(\mathbb{R})$. The translation operat... |
$K_n$ be a complete undirected graph, then it has $\frac {n(n-1)}{2}$ edges. Each of these must be present in the digraph of $K_n$ and we have 2 choices for its direction. Then there are $2^{\frac{n(n-1)}{2}}$ digraphs. Now for each n to 3 vertices there could exist a cycle. There could be multiple cycles in a graph wi... |
Given a complete undirected graph on n vertices, in how many ways can you choose a direction for the edges so that there are no directed cycles? |
Let f:[0,1]->R be a continous function on [0,1] and differentiable at (0,1). Prove there exist $c\in (0,1)$ so that $f(c)+\frac{1}{e-1}=f'(c)$ where e is euler number.
I tried to subs $f(x)=e^{x}g(x)$ so that $f'(x)-f(x)=e^{x}g'(x)$ .Then i tried to use mean value theorem(to ensure the existence of $r \in (0,1)... |
$\sum_{n=1}^{\left\lfloor \delta/2 \right\rfloor}$
What does this symbol ${\left\lfloor \delta/2 \right\rfloor}$ mean? Any reference on understanding its usage in binomial theorem would be helpful. |
I have doubt on Stefan Problem with time boundary conditions, does this problem has an analytical solution or not? For example in the following article
<cite authors="Tao, L. N.">_Tao, L. N._, [**The Stefan problem with arbitrary initial and boundary conditions**](https://doi.org/10.1090/qam/508769), Q. Appl. Math.... |
Let f:[0,1]->R be a continous function on [0,1] and differentiable at (0,1).$f(0)=0,f(1)=1$ Prove there exist $c\in (0,1)$ so that $f(c)+\frac{1}{e-1}=f'(c)$ where e is euler number.
I tried to subs $f(x)=e^{x}g(x)$ so that $f'(x)-f(x)=e^{x}g'(x)$ and define $e^{x}g'(x)=w(x)$.Then i tried to use mean value theo... |
How to prove this (geometrically correct) inequality? |
Let $f:[0,1]\to \mathbb R$ be a continous function on [0,1] and differentiable at (0,1).$f(0)=0,f(1)=1$ Prove there exist $c\in (0,1)$ so that $f(c)+\frac{1}{e-1}=f'(c)$ where e is euler number.
I tried to subs $f(x)=e^{x}g(x)$ so that $f'(x)-f(x)=e^{x}g'(x)$ and define $e^{x}g'(x)=w(x)$.Then i tried to use mea... |
Suppose we have a sequence $(x_n)_{n=1}^\infty\subseteq X$ with $(X,d)$ a metric space, and we have the following property:
> $$\exists x\in X\forall\epsilon>0[|B_\epsilon(x)\cap (x_n)_{n=1}^\infty|=\aleph_0]$$
(That is to say there exists a point such that for any $\epsilon>0$ there are a (countably) infinite nu... |
Let $f:[0,1]\to \mathbb R$ be a continous function on $[0,1]$ and differentiable on $(0,1)$, $f(0)=0, f(1)=1$. Prove that there exists $c\in (0,1)$ so that $f(c)+\frac{1}{e-1}=f'(c)$ where $e$ is the Euler number.
I tried to subs $f(x)=e^{x}g(x)$ so that $f'(x)-f(x)=e^{x}g'(x)$ and define $e^{x}g'(x)=w(x)$. Then... |
I am trying to get the average rate of growth across a series of numbers that will lead to the same sum as the discrete rates of growth.
For example, if I have 1 unit at $t=1$ that grows at 2%, then 3%, then 2% again the resulting sum of units would be
$$1+1\cdot(1.02)+1\cdot(1.02)\cdot(1.03)+1\cdot(1.02)\cdot(1.... |
I have a question regarding the definition of 'Markov process' given by Shreeve, book II (continuous models).
It says (see below) this: "Assume that for all $0 \leq s \leq t \leq T$ and for every nonnegative Borel-measurable function $f$, there is another Borel-measurable function $g$ such that $\mathbb E [ f(X(t))... |
I believe there are several definitions for Radon measure, which are equivalent under certain conditions. The one in the Wikipedia says that a measure $m$ on the $\sigma$-algebra of Borel sets of Hausdorff space $X$ is called a **Radon measure** if both
- $m$ is **inner regular**, that is, for any open set $U$, $m(U... |
I have found this tough combinatorial geometry question in some Soviet math competition circa $1972$. It looks very simple (and rather natural), yet I cannot find the proof for quite some time now.
*$2n$ points are given on the plane. Prove that there exist $n$ non-overlapping circles, each containing exactly two gi... |
This is a question I ended up with while trying to make a program that would find an (at least locally-) minimal-energy configuration for a piecewise linear 1-dimensional object in $\mathbb{R}^3$ in relation to a practical knot theory type of question. If it helps at all for the purposes of context or intuition, the se... |
What does the symbol ${\left\lfloor \delta/2 \right\rfloor}$ mean here?
$$\sum_{n=1}^{\left\lfloor \delta/2 \right\rfloor}$$
Any reference on understanding its usage in binomial theorem would be helpful. |
In $\sum_{n=1}^{\left\lfloor \delta/2 \right\rfloor}$, what does ${\left\lfloor \delta/2 \right\rfloor}$ mean? |
Currently I am reading the first edition "Introduction to Graph Theory" by Douglas B. West.
Exercise 1.1.9 states the following:
**1.1.9.** Suppose that $G$ is a simple graph having no vertex of degree $0$ and no induced subgraph with three edges. Prove that $G$ has at most four vertices.
**Counter-examp... |
I have found this tough combinatorial geometry question in some Soviet math competition circa $1972$. It looks very simple (and rather natural), yet I cannot find the proof for quite some time now.
*$2n$ points are given on the plane. Prove that there exist $n$ non-overlapping circles, each containing exactly two gi... |
Let $ρ_A = \sum_{i=1}^r p_i|e_i⟩⟨e_i|$, where $p_i$ are the nonzero eigenvalues
of $ρ_A$ and $|e_i⟩$ corresponding orthonormal eigenvectors. If some eigenvalue appears more
than once then this decomposition is not unique.
Show that, nevertheless, any purification $|\psi_{AB}⟩$ of $ρ_A$ has a Schmidt decomposition ... |
In the comment of [this post][1], it says that $(\Bbb Z/p^k\Bbb Z)\otimes \Bbb F\simeq \Bbb F$, where $\Bbb F$ is a characteristic $p$ field. But I don't understand why it's true. Could somebody explain about this?
[1]: https://math.stackexchange.com/q/4244843/922120 |
Let's say I want to saturate an argument number `x` to a lower-bound value `lb` if it is lower than `lb` and otherwise return `x` itself. Then I can use `max(lb, x)`. Analogously, I can use `min(x, ub)` if I have an upper bound `ub`.
Is there a function `f`, such that if I write `f(lb, x, ub)`, one understands that ... |
Simplified notation for max(lb, min(x, ub))? |
I was trying to make an IBP problem for an integration bee.
I came up with the simple arctan integral with the square roots just for an easy round.
However, WolframAlpha challenged me a suggestion, which was adding the bounds -1 to 1.
I was so confused because I thought it was undefined, divergent, or complex. But i... |
If $E \to B$ is a smooth vector bundle, then the tangent bundle $T E$ is a vector bundle over both $E$ and $T B$. (If you like, the latter structure is the derivative of the vector bundle structure on $E$.) If $F \to B$ is another smooth vector bundle and $\pi : T B \to B$ is the obvious projection, I have seen it writ... |
Let $ρ_A = \sum_{i=1}^r p_i|e_i⟩⟨e_i|$, where $p_i$ are the nonzero eigenvalues
of $ρ_A$ and $|e_i⟩$ corresponding orthonormal eigenvectors. If some eigenvalue appears more
than once then this decomposition is not unique.
Show that, nevertheless, any purification $|\psi_{AB}⟩$ of $ρ_A$ has a Schmidt decomposition ... |
Suppose $\Gamma$ is a (necessarily connected) weighted tree with $k$ vertices $v_1,\dots,v_k$. Suppose the weight of $v_i$ is $n_i\in \Bbb Z$. Let $A$ be the $k\times k$ symmetric matrix defined by $a_{ii}=n_i$ and for $i\neq j$, $$a_{ij}=\begin{cases}1 & \textrm{if there is an edge from $v_i$ to $v_j$}, \\ 0 & \textr... |
Is this a programming question? I am unsure of a mathematical trick that would yield the result, but here is a quick python script
```
from sympy import integer_nthroot
for x in range(100000):
if all(integer_nthroot(x,k)[0]<integer_nthroot(x,k-1)[0] for k in range(2,9)):
print(x)
break
```
... |
It is well-known that the following is one of the equivalent definitions of "jointly normal":
**Definition 1.** Let $(X_1,X_2)$ be a random vector. Then $X_1$ and $X_2$ are said to be *jointly normal* if and only if there exist real numbers $\mu_1, \mu_2, \alpha_1, \ldots, \alpha_k, \beta_1,\ldots, \beta_k$ and ind... |
The problem occurs in the [Convex Optimization II lecture by Stephen Boyd][1]. There we want to choose $\alpha_i$'s to minimize $$\frac{R^2+G^2 \sum_{i=1}^k \alpha_i^2}{2\sum_{i=1}^k \alpha_i}$$
where $\alpha_i>0,R>0,G>0$. The professor argued this in the [video 20:00][2] with the statement that a quadratic over a p... |
I read in Wiki (https://en.wikipedia.org/wiki/Bivector) that antisymmetric part of geometric product can be represent as $(a \land b)$ and $(a \cdot b)^2 - (a \land b)^2 = a^2b^2$
I have 2 questions:
1) How to derive $(a \cdot b)^2 - (a \land b)^2 = a^2b^2$ ? (solved: $(ab)(ba) = (a \cdot b)^2 - (a \land b)^2$)
... |
Suppose $E_1$ and $E_2$ are two vector bundles over a base $B$ such that $E_1 \oplus E_2 \cong \epsilon$, i.e., the sum is trivial. After taking intersections, we get a cover $U_i$ of $B$ such that $E_1$ and $E_2$ are trivial over the $U_i$, giving transition functions $f_{ij}$ and $g_{ij}$. Thus, possible transition f... |
I read in Wiki (https://en.wikipedia.org/wiki/Bivector) that antisymmetric part of geometric product can be represent as $(a \land b)$ and $(a \cdot b)^2 - (a \land b)^2 = a^2b^2$
I have 2 questions:
1) How to derive $(a \cdot b)^2 - (a \land b)^2 = a^2b^2$ ? (solved: $(ab)(ba) = (a \cdot b)^2 - (a \land b)^2$)
... |
How to obtain $(a \cdot b)^2 - (a \land b)^2 = a^2b^2$ and why $(a \land b)=|a||b||\sin(\phi)|$ if $(a \land b)^2=-|a|^2|b|^2\sin^2(\phi)$? (GA) |
is $sin^2$(x) the same as sin(sin(x))? |
is $\sin^2 (x)$ the same as $\sin(\sin(x))$? |
In a chemical reaction that follows the logistic equation, if the initial velocity is 0, it becomes a steady solution. However, in the equation of motion, the fact that the initial velocity is 0 does not necessarily mean that it is a steady solution.
For example,for the $$\frac{dx}{dt}(t) = 4(1-0.5x(t))x(t)$$,
unde... |
Is there a concise closed-form expression for the integer sequence A254828? |
I am reading a paper https://arxiv.org/abs/1910.07143
I am unable to understand the construction of table III, for the section Orthogonality and completeness of matrix elements in group space.
The theory behind it is given as
[![enter image description here][1]][1]
They have constructed 24x24 table stating th... |
> Let $f$ be holomorphic on $\mathbb{D}$, continuous on
> $\overline{\mathbb{D}}$ s.t $|f(z)|=1$ whenever $|z|=1$. Prove that if
> $f(z)\neq 0$ for all $z\in\mathbb{D}$ then $f$ is constant on
> $\mathbb{D}$.
I already proved the following result:
Let $f$ be non-constant, holomorphic on $\mathbb{D}$, continu... |
I don't have enough reputation to comment and I know that this is quite an old question, but this video gives an excellent explanation of the composite hypothesis issue: https://www.youtube.com/watch?v=9RvLy554NnI. I hope this helps future visitors to this question. |
In my complex analysis course, I am given a function $f(z)=\frac{1}{z^2-z}$ and a domain on which it is holomorphic $f(z)=\mathbb{C}\setminus\{-1,0,1 \}$. I am then tasked with `calculating' the difference
$$\int_{\gamma_1}f(z)dz-\int_{\gamma_0}f(z) dz $$
where $\gamma_1$ and $\gamma_0$ are arbitrary open curves indi... |
So I have the following question
> Let $V$ be the set of all $2\times2$ matrices $A=(a_{ij})$ with complex entries and trace zero. Show that $V$ is a vector space over the field of reals. Let $W$ be the set of all matrices $A=(a_{ij})$ in $V$ such that $a_{21}=-{\bar{a_{12}}}$. Prove that $W$ is a subspace of $V$ a... |
Any copula to force one variable larger than the other in the joint? |
**Disclaimer:** I know the Residue Theorem could be used for this, but we cannot use it as we have not proven it in class.
In my complex analysis course, I am given a function $f(z)=\frac{1}{z^2-z}$ and a domain on which it is holomorphic $f(z)=\mathbb{C}\setminus\{-1,0,1 \}$. I am then tasked with `calculating' the... |
So I had an assignment due yesterday, regarding the probability of the attackers second highest dice roll, being strictly bigger than the defenders second highest dice roll, in the game Risk.
In the game, the attacker has three dices and the defender has two. The highest dice of the attacker is compared with the hi... |
Did my TA make a mistake regarding probabilities? |
I want to know the behavior of the solution without solving the ordinary differential equation for the following enzyme-substrate reactions.
[![enter image description here][1]][1]
Here, $k_1,k_2,k_3>0$, and [S] is the concentration of S, and the others similarly represent the concentrations of substances sandwiche... |
I am interested in integer sequence A254828 (https://oeis.org/A254828), but from the link it seems to have a recursive formula $a(n) = a(n-1) + a(n-30) - a(n-31)$.
As joriki said, they are **the number of positive integers up to n+1 divisible by 2, 3 or 5.** We can check them by the following code:
```
//A2... |
I have a real-world acoustics problem that I'm trying to solve for some work related R&D. Hopefully you big brained folks can explain down to me. :) I'm good at visualizing math concepts, but I lack any substantial experience, save perhaps g/FooBar. So, sans elegance:
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Edit: Using the below descriptions, I'... |
**Disclaimer:** I know the Residue Theorem could be used for this, but we cannot use it as we have not proven it in class.
In my complex analysis course, I am given a function $f(z)=\frac{1}{z^2-z}$ and a domain on which it is holomorphic $f(z)=\mathbb{C}\setminus\{-1,0,1 \}$. I am then tasked with `calculating' the... |
How can i calculate limits of $\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{x \to + \infty } \cos \left( {a + i\varepsilon } \right)x = a$, $\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{x \to - \infty } \sin \left( {a + i\varepsilon } \right)x = 1$, $\mathop {\lim }\limit... |
I have recently been reading Werner Ballmann's Introduction to Geometry and Topology, and when I get to section 7 of chapter 3 (where the author wants to define the oriented integral), he says "Let $A \subset M$ be Lebesgue-measurable...", where $M$ is a smooth manifold. I have studied some basic measure theory, mainly... |
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