1
vote
0answers
19 views
+50

How to express a Poisson regression equation as a quasi-Poisson

I was reading this paper: http://www.researchgate.net/publication/13878515_A_simple_non-linear_model_in_incidence_prediction and was wondering whether the equations 1 and 3 presented in the paper to ...
7
votes
2answers
193 views
+100

What is the value of $x$ such that $\frac{\text{d}^2y}{\text{d}x^2}=0$ where $\frac{\text{d}y}{\text{d}x}=-ae^{-bx}y-cy+d$?

How can you find the values of $x$ such that $$\frac{\text{d}^2y(x)}{\text{d}x^2}=0$$ where $$\frac{\text{d}y}{\text{d}x}=-ae^{-bx}y-cy+d$$ with $$y(0)=y_0$$ and $$a,b,c,d>0$$ If it helps I can ...
3
votes
0answers
86 views
+50

Is the inverse of a bijective connectedness preserving map , on a complete real inner product space , also connectedness preserving?

Let $X$ be a complete real inner-product space and $f:X \to X$ be a bijection which maps connected sets to connected sets ; then is it necessarily true that $f^{-1}$ also maps connected sets to ...
3
votes
1answer
91 views
+50

Mathematical structures textbook recommendation

I am busy doing an undergraduate course called "Fundamentals of Mathematics". It is not well-defined as there is no syllabus nor recommended textbook (there are lectures and notes), but the course ...
2
votes
0answers
56 views
+50

Hessian Matrix of an Angle in Terms of the Vertices

I am attempting to derive the analytical formula for the Hessian matrix of a the second derivatives of the value of an angle in terms of the (9) coordinates of the 3 3D points that define it. While I ...
11
votes
1answer
76 views
+50

Configurations of eleven (or more) points in the Euclidean plane, such that out of any four there is a pair at unit distance.

Inspired by this question, I was wondering the following: What is the maximal size of a subset $C$ of the Euclidean plane such that out of any four points in $C$ there are two at unit distance ...
18
votes
1answer
218 views
+50

Integers $n$ for which the digit sum of $n$ exceeds the digit sum of $n^5$

This question is strongly inspired by The smallest integer whose digit sum is larger than that of its cube? by Bernardo Recamán Santos. The number $n=124499$ has digit sum $1+2+4+4+9+9=29$ while its ...
19
votes
2answers
161 views
+50

For all $n$ there exists $x$ such that $\varphi(x)<\varphi(x+1)<\ldots<\varphi(x+n)$

Let $\varphi$ be the Euler's function, i.e. $\varphi(n)$ stands for the number of integers $m \in \{1,\ldots,n\}$ such that $\text{gcd}(m,n)=1$. Let $n\ge 2$ be a positive integer. Show that there ...
4
votes
1answer
95 views
+100

Help with a DCT problem

The question is Let ${p_t}\left( x \right) = \frac{1}{{\sqrt {2\pi t} }}{e^{ - \frac{{{x^2}}}{{2t}}}},t > 0,x \in \Bbb{R}$. It is known that $\int_R {\frac{1}{{\sqrt {2\pi t} }}{e^{ - ...
5
votes
2answers
120 views
+50

Understanding implicit differentiation with concepts like “function” and “lambda abstraction.”

In high school, we learned to reason like so: $$(*) \qquad \frac{d}{dx}(x^2+x) = \frac{d}{dx}(x^2)+\frac{d}{dx}(x) = 2x+1$$ Now that I know more, I can "reanalyze" this chain of reasoning using ...
4
votes
1answer
407 views
+200

The image of a map $H^2 \setminus \Delta \to \mathbf{R}^4$ where $H$ is the upper half-plane and $\Delta$ is the diagonal

Let $H = \{z \in \mathbf{C}: \operatorname{Im} z > 0\}$ be the upper half-plane, and let $D = H^2 \setminus \Delta$, where $\Delta = \{(u,u) \in H^2\}$ is the diagonal. Define $\varphi: D \to ...
1
vote
2answers
90 views
+100

How can I indicate that n and k are natural numbers in ∀n[(∀k < n P(k)) → P(n)].

$∀\, x \, \{x\in\mathbb N\rightarrow P(x)\}$ can be abbreviated to $∀ \hspace{.1cm} x∈ℕ[P(x)].$ But, I am not sure how I can indicate "concisely" that n and k are natural numbers in ∀n[(∀k < n ...
4
votes
0answers
51 views
+50

Bounding 2nd-smallest eigenvalue of the Laplacian of the binary tree

I am reading on my own the notes of this lecture series from 2012: http://www.cs.yale.edu/homes/spielman/561/2012/lect04-12.pdf. In section 4.7.2 (page 8) it's mentioned that we can prove a lower ...
2
votes
0answers
20 views
+50

Invariant points and lines under homography

Given a matrix representation of an homography in a real projective space $P(\mathbb{R^3})$, what is the general procedure to calcule the invariant subspaces? A brief description would be enough.
26
votes
1answer
311 views
+50

Conjectured value of a harmonic sum $\sum_{n=1}^\infty\left(H_n-\,2H_{2n}+H_{4n}\right)^2$

There is a known asymptotic expansion of harmonic numbers $H_n$ for $n\to\infty$: $$\begin{align}H_n&=\gamma+\ln n+\sum_{k=1}^\infty\left(-\frac{B_k}{k\cdot n^k}\right)\\ &=\gamma+\ln ...
2
votes
0answers
276 views
+50

Multivariate B-Spline Derivatives

To construct a multivariate B-spline, we simply take the Kronecker tensor product between the univariate basis functions constructed for each individual dimension. What I'd like to know is how do you ...
3
votes
2answers
101 views
+100

find a group of lowest N numbers so that no 2 pairs have the same bitwise or

I am trying to find the lowest group of N numbers (i.e. N=1000) so that no 2 pairs from the group have the same bit-wise or. more specific need to find a group $A = \{a_1,a_2,a_3,..,a_N\} $ such ...
6
votes
0answers
68 views
+50

Siegel's Theorem for Genus 0 Curves over Function Fields

Let $V$ be a quasi-projective algebraic curve over a field $k$. Are there finitely many morphisms from $S$ to the triply-punctured sphere $\mathbb{P}^1 -\{0,1,\infty\}$? This is true if $k = ...
13
votes
0answers
657 views
+100

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers ...
3
votes
2answers
116 views
+200

Differential algebra and differential-algebraic equations

Could you give me some information about differential algebra? What is it about? Differential-algebraic equations (DAEs) are polynomials with complex coefficients and the unknown variables are $z, ...
2
votes
0answers
67 views
+50

When a vector space will be a complete lattice?

Let $E$ a vector space, and let $P$ a strict cone in $E$ (i.e) $P\subset E$ verify: $$ \mathbb{R}^+ P\subset P \\ P+P\subset P\\ P\cap (-P)=\{0\} $$ So we can easily construct a partial order on $E$ ...
2
votes
1answer
87 views
+50

Showing existences of biholomorphic maps.

Define a complex polynomial $p:\mathbb{C}\longrightarrow\mathbb{C}$ where $\deg p=n\in\mathbb{N}\setminus\{2\}$. Define the set $\mathcal{S}=\{z\in\mathbb{C}:\text{Re }p(z) < ...
2
votes
0answers
128 views
+50

Conjectured primality test for specific class of $N=k\cdot 6^n-1$

How to prove that this conjecture is true ? Definition : $\text{Let}~ P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)~ , \text{where}~ m ~\text{and}~ x ...
4
votes
1answer
66 views
+50

Variation of the Josephus problem

Suppose we have a circle of $2n$ people, where the first $n$ people are good guys and the people $n+1$ to $2n$ are bad guys. Can we always choose an integer $q$, such that if we execute successively ...
0
votes
1answer
38 views
+100

simplify/solve nonlinear equations for constrained least squares problem

I am trying to find a simple, ideally closed form formula for the (not necessarily unique) unit vector $\vec{x}$ minimizing total squared cosine distance from a collection of unit vectors $\vec{v_i}$. ...
3
votes
0answers
40 views
+150

Solving a Matrix DE involving the KL divergence

If we let $U_\mu$ be a vector field that associates a direction vector $U_\mu(\pi)$ with each $\pi \in $ unit simplex. Each such vector field is associated with a system of ODEs: $$ \pi'(u) = ...
1
vote
0answers
36 views
+250

Injury-free proof of Cof being $\Sigma^0_3$-complete

How can I prove, without using priority argument, that Cof, the set of indices of cofinite c.e. sets, is $\Sigma^0_3$-complete? I know an injury-free proof of Rec being $\Sigma^0_3$-complete, where ...
1
vote
1answer
58 views
+50

How can I solve this stochastic system of equation?

$(B_1(t),B_2(t))$ is a 2-dimensional standard Brownian motion. $\alpha , \beta$ are constant. The system of equations is: $$dX_1(t)=X_2(t)dt+\alpha dB_1(t)\\dX_2(t)=-X_1(t)dt+\alpha dB_2(t)$$ I tried ...
7
votes
2answers
196 views
+50

Research in algebra.

First of all, I don't know if this is the right place to ask about this. If not, please direct me somewhere I can get more help. I like algebra a lot as a mathematics undergraduate student on his 3rd ...
3
votes
2answers
114 views
+100

Weierstrass's M-test example for uniform convergence and switching Sum and Integral.

How would I go about finding $M_n$ in \begin{equation} \sum_{n=1}^{\infty} \int_{0}^\infty x^{\frac{s}{2}-1}e^{-\pi n^{2}x}dx \end{equation} to show that it is uniformly convergent? UPDATE: Just ...
6
votes
1answer
76 views
+50

Element of Grothendieck group is eigenvector of operator

Let $K_\mathbb{C}(G)$ be the Grothendieck group (over $\mathbb{C}$) of finite dimensional representations of a finite group $G$. Associated with any such representation $V$, there is a linear ...
5
votes
0answers
33 views
+50

Closed formula for Poincaré series in terms of adjacency matrix.

Let $Q$ be a finite quiver with vertex set $I$. For each $n = 0, 1, 2, \dots,$ let $k^{(n)}Q \subset kQ$ be the $k$-linear span of all paths of length $n$, in particular, we have$$k^{(0)}Q = ...
1
vote
0answers
17 views
+300

Proof of multivariate regression plane maximizes correlation in normals

I am doing a homework sheet as practice for an upcoming course in multivariate statistics and been stuck on the following problem: Let ...
0
votes
1answer
66 views
+50

A question about an infinite sequence of elementary row operations

Do there exist matrices $A$ and $B$ such that $B$ can be transformed into $A$ only if an infinite number of elementary row operations are performed on $B$? "What can we multiply the top equation by ...
4
votes
0answers
86 views
+100

Solving ODE rigorously

I am given the ODE $$(f''(r)+\frac{f'(r)}{r})(1+f'(r)^2)-f'(r)^2f''(r)=0$$ and want to solve it rigorously for $r>0.$ So especially, I don't want to loose any solutions. $\textbf{Derivation of ...
0
votes
1answer
64 views
+50

What is the probability that no two consecutive boxes have blue balls.?

There are red and blue balls which can be filled in 5 boxes.All balls are similar except color. what is the probability that no two consecutive boxes have blue balls. Assume:A ball can be either ...
2
votes
0answers
88 views
+100

How much does convolution with a C^m kernel increase the order of continuity.

EDIT: I rewrote the question, see the edit history for my previous attempt. My questions are: Is the following proof/reasoning essentially correct? How do I make it more precise, concise, correct? ...
6
votes
0answers
83 views
+50

Is the space of $G$-maps $G/H \to X$ naturally homeomorphic to $X^H$?

Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$, define$$X^H = \{x : hx = x \text{ for all }h \in H\} \subset X;$$$X^H$ is the $H$-fixed point subspace of $X$. ...
0
votes
1answer
41 views
+50

Comparing Coefficients

If I have the equation: $4m(m-1)x^m .\sum_{i\geq 0}a_ix^i+x^m.\sum_{i\geq 0}a_ix^i=0$ ; $a_0\neq 0$ why am I able to say that $4m(m-1)+1=0$? I would understand if the equation rather than being an ...
1
vote
1answer
54 views
+50

What rotation rules can be applied to stacked cubes to make a 3D spirograph?

If you arrange building blocks for example toy cubes so that every next cube is tilted over its base by 30 degrees and rotated to it's right by 12 degrees, it would wind through space in a helical ...
2
votes
1answer
60 views
+100

solving 2nd order pde with dirac delta

I want to find the functional form of the Green function G(x,t) for a parabolic differential equation: $$ ...
1
vote
1answer
148 views
+100

Product of 2 dirac delta functions

Find the functional form of the Green function G(x,t) for a parabolic differential equation (i.e. heat diffusion etc): $$ ...
0
votes
0answers
44 views
+50

construct a triangle with a compass and a ruler, given $a, B, t_a$

$a,b,c$ the sides of the triangles; $A,B,C$ the angles of the triangles; $t_a, t_b, t_c$ the internal bisectors of the angles $A,B,C$. How to construct a triangle with a compass and a ruler(a ...
0
votes
2answers
62 views
+50

Proving the Ideal Generated by the Coefficients of $f(X)\cdot g(X)\in R[X]$ is $R$.

Let $R$ be a commutative ring with unity, and let $f(X),g(X)\in R[X]$. Assume the ideals generated by the coefficients of $f(X),g(X)$ are both $R$. Prove that the ideal generated by the ...
5
votes
1answer
158 views
+100

Correlation between an event and a time series

I have a time series, e.g. the daily number of visitors on my blog. I have a set of events of some class, like the days when I made a new posting. I want to measure the effect of a new posting on the ...
1
vote
1answer
55 views
+200

3D projection coordinates onto 2D plane to determine transformation matrix?

I'm not sure if there is an actual solution to this problem or not, but thought I would give it a shot here to see if anyone has any ideas. So here goes: I basically have three vertices of a rigid ...
0
votes
0answers
19 views
+50

Kernel, Green function and the functional derivative.

I am pretty new to the subject of differential equations and am reading about Green functions and Kernels for the first time. I am more familiar with functional differentiation and am comfortable with ...
14
votes
1answer
212 views
+200

Prove Euler characteristic is a homotopy invariant without using homology theory

I was flipping through May's Concise Course in Algebraic Topology and found the following question on page 82. Think about proving from what we have done so far that $\chi(X)$ depends only on ...
10
votes
0answers
240 views
+100

Cauchy induction: are there examples of cases where choosing an integer other than $2$ is a better strategy?

Cauchy induction, sometimes called backwards induction, works as follows: show that $p(1)$ is true show that $p(n)$ implies $p(2n)$ (which inductively implies $p(2^n)$ is true) show that $p(n)$ ...
4
votes
2answers
73 views
+50

Pentagonal Numbers

I recently was passing some time on Project Euler, when I came across this question. It deals with finding Pentagonal Numbers $P_j$ and $P_k$ such that $P_j+P_k$ and $P_j-P_k$ are also pentagonal ...

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