ag.algebraic geometry – Distinguishing ample divisors by minimally intersecting curves on a projective simplicial toric variety

My question has an easily formulated generalization, which I will state first. Let $sigma subseteq mathbf{R}^n$ be a strongly convex polyhedral cone. For each minimally generating lattice point $m in sigma^o cap mathbf{Z}^n$ of the interior cone $sigma^o subseteq sigma$, let $S(m) subseteq sigma^{vee} cap mathbf{Z}^n$ denote the set of lattice points $u$ with $langle u,m rangle = 1$. My question is:

Does $S(m) = S(m’)$ imply that $m = m’$?

As a special case, assume that $sigma$ is the nef cone of a simplicial projective toric variety $X_{Sigma}$. Then my question seems to amount to the following:

If $D_1$ and $D_2$ are two ample divisors minimally generating in the ample cone, then does $D_1 cdot C = 1 Leftrightarrow D_2 cdot C = 1$ for all effective curves $C$ imply that $D_1 = D_2$?

This is the case I am most interested in.

modular forms – L functions of Symmetric power of elliptic curves

Let $E$ be an elliptic curve over the raional field with conductor $N$, which corresponds to the eigenform $f(z)=sum a_nq^n$. Let $L(Sym^2E,s)$ be the L function of the symmetric power of $E$.I am trying to understand a formula in Zagier’s paper “Classical and elliptic polylogarithms and special values of L series” on Page 36 which can be used to compute $L(Sym^2E,3)$.

Let $L_2(f,s):=zeta(s-1)L(Sym^2E,s)=sum_{n}b_n/n^s$. By Rankin’s method we have the functional equation $L^{*}_2(f,s)=L^{*}_2(f,3-s)$, where $L^{*}(f,s)=(2pi)^{-2s}N^sGamma(s)Gamma(s-1)L_2(f,s)$.

The Gamma factor $Gamma(s)Gamma(s-1)$ has a relation with the Bessel function:

$int_{0}^{infty}K_v(y)y^sfrac{dy}{y}=2^{s-2}Gamma(frac{s+v}{2})Gamma(frac{s-v}{2})$ See page 106 of Bump’s book Automorphic forms and representations.

Then Zagier gives the following formula

$$L_2(f,3)=C(A-frac{A^2}{2})+frac{1}{16}sum_{n=1}^{infty}frac{b_n}{n^3}G_1(4pisqrt{frac{nA}{N}})+frac{2^8pi^6}{N^3}sum_{n=1}^{infty}b_nG_2(4pisqrt{frac{n}{NA}}),$$

Where $$C=2pi^2N^{-1}L(Sym^2E,2)$$ and

$$G_1(x):=int_x^{infty}t^4K_1(t)dt$$ and $$G_2(x):=int_x^{infty}t^{-2}K_1(t)dt$$

Zagier mentioned that the formula $L_2(f,3)$ is obtained by splitting up the integral of the Mellin transform of the K-Bessel function into two pieces in the usual way. I would like to know how this can be done by choosing some $A$ in the formula of $L_2(f,s)$

homotopy theory – Kontsevich’s A-infty cohomology classes of the moduli space of curves

In his paper “Feynman diagrams and low-dimensional topology,” Kontsevich attaches to each $A_infty$ algebra a cohomology class (with complex coefficients) on the moduli space of smooth, complex curves of genus g with n marked points.

He does this by noting that the moduli space of ribbon graphs with metric (where each vertex has degree at least 3) of genus g and n boundary components, is isomorphic to the moduli space of curves above. He uses the given $A_infty$ algebra to produce a partition function $Z(Gamma)$ for a ribbon graph $Gamma$.
Finally, he says that $sum_Gamma{Z(Gamma).Gamma}$ is a closed cochain, and that it is closed “due to the higher associativity conditions on the algebra.”
Would someone be able to explain why 1. the cochain is closed; 2. why the sum is finite; 3. why this gives you a cochain rather than a chain?

I’m tried several of the references and tried to work it out myself but I can’t see what Kontsevich is saying here. (I am aware that to answer this question, one probably needs to already be aware of Kontsevich’s idea – and I’m hoping someone here is.)

In particular, I’ve looked at Harer’s “The cohomology of the moduli of curves” to understand the ribbon graph moduli isomorphism as Kontsevich mentions this as an “exposition” of the result (to not much success). I’ve looked at Kontsevich’s “Notes on $A_infty$-algebras, $A_infty$-categories, and noncommutative geometry,” but this is even more incomprehensible to me (since he vastly generalizes the above construction). I’ve also looked at various papers referenced in Kontsevich’s paper.

How to do boolean operations on two planar parametric curves?

Assume that the parametric equations of the two curves is given and the curves are simple one-piece, closed, planar and, in principle, we have the intersection points.

Do curves in GIMP have "targeted adjustments" like in Photoshop?

Is there a targeted adjustment feature in the curves settings of GIMP. To set colour curves. Like in Photoshop?

Photoshop curves

Is there a targettted adjustment feature in the curves settings of gimp

Is there a targettted adjustment feature in the curves settings of gimp. To set colour curves. Like in Photoshop?enter image description here

Is there a colour picker in the curves settings of gimp

Is there a colour picker in the curves settings of gimp. To set colour curves. I’m trying to make a colour curve to my picture but I can’t do it. Instead, I can make it done if the colour is red,green, or blue but not a specific colour

real analysis – What is the maximum number of intersections of two sigmoid curves?

I have two specific sigmoid functions with domain in $(0,infty)$ that depend on a common set of parameters. I was able to show that they have the same limit at infinity (please see figure below). I now need to show that they do not intersect. I was wondering if there are any general results on the maximum number of intersections of two sigmoid curves? Such a result will definitely be a good start for me. Any suggestion is much appreciated.
enter image description here

curves – Find the equation of the osculating circle of Cycloid

Find the equation of the osculating circle of the curve defined parametrically by :

$${displaystyle mathbf{r}(t),=,{begin{pmatrix}rleft(t-sinleft(tright)right)\rleft(1-cosleft(tright)right)end{pmatrix}}}$$

AKA cycloid.


The center $C$ of the osculating circle is given by:

$$C= mathbf{r}(t)+frac{1}{kappa } mathbf{N}(t)$$

Where $kappa $ is the curvature of the curve computed by :

$$kappa = frac{|mathbf{r}'(t)timesmathbf{r}”(t)|}{|mathbf{r}'(t)|^3}$$

And $mathbf{N}(t)$ is the unit normal vector defined by:

$$mathbf{N}(t) = frac{mathbf{r}'(t) times left(mathbf{r}”(t) times mathbf{r}'(t) right)}{left|mathbf{r}'(t)right| , left|mathbf{r}”(t) times mathbf{r}'(t)right|}$$


$${displaystyle mathbf{r}'(t),=,{begin{pmatrix}rleft(1-cosleft(tright)right)\rsinleft(tright)end{pmatrix}}};;;;,;;;;{displaystyle mathbf{r}”(t),=,{begin{pmatrix}rsinleft(tright)\rcosleft(tright)end{pmatrix}}}$$

$${displaystyle mathbf{r}'(t)timesmathbf{r}”(t) =,{begin{pmatrix}0\0\r^{2}left(cosleft(tright)-1right)end{pmatrix}}}$$

$$left|mathbf{r}'(t) right|
=rsqrt{2}sqrt{1-cosleft(tright)}$$

$$left|mathbf{r}'(t) times mathbf{r}”(t) right|
=r^{2}left|cosleft(tright)-1right|$$

$${displaystyle mathbf{r}'(t)times(mathbf{r}”(t) times mathbf{r}'(t))={begin{pmatrix}r^{3}left(sinleft(tright)-frac{sinleft(2tright)}{2}right)\r^{2}4sin^{4}left(frac{t}{2}right)\0end{pmatrix}}}$$

So the center of the circle is :

$$C= mathbf{r}(t)+frac{ left|mathbf{r}'(t) right|^3}{left|mathbf{r}'(t)times mathbf{r}”(t)right| }frac{ mathbf{r}'(t)times(mathbf{r}”(t) times mathbf{r}'(t)}{left|mathbf{r}'(t) right|left|mathbf{r}”(t)times mathbf{r}'(t)right| }$$

$$=mathbf{r}(t)+left(frac{ left|mathbf{r}'(t) right|}{left|mathbf{r}'(t)times mathbf{r}”(t)right| }right)^2mathbf{r}'(t)times(mathbf{r}”(t) times mathbf{r}'(t)$$

$$=,{begin{pmatrix}rleft(t-sinleft(tright)right)\rleft(1-cosleft(tright)right)end{pmatrix}}+left(frac{rsqrt{2}sqrt{1-cosleft(tright)}}{r^{2}left|cosleft(tright)-1right|}right)^{2}{begin{pmatrix}r^{3}left(sinleft(tright)-frac{sinleft(2tright)}{2}right)\r^{2}4sin^{4}left(frac{t}{2}right)end{pmatrix}}$$

The center simplifies to:

$$,{begin{pmatrix}rleft(t-sinleft(tright)right)\rleft(1-cosleft(tright)right)end{pmatrix}}+frac{csc^{2}left(frac{t}{2}right)}{r^{2}}{begin{pmatrix}r^{3}left(sinleft(tright)-frac{sinleft(2tright)}{2}right)\r^{2}4sin^{4}left(frac{t}{2}right)end{pmatrix}}$$

Let $frac{csc^{2}left(frac{t}{2}right)}{r^{2}}=v$.

Implies the equation of the circle:

$$left(x-color{blue}{left(r^{2}left(t-sinleft(tright)right)vleft(sinleft(tright)-frac{sinleft(2tright)}{2}right)right)}right)^{2}+left(y-color{blue}{left(r^{3}left(1-cosleft(tright)right)vleft(4sin^{4}left(frac{t}{2}right)right)right)}right)^{2}=left(frac{left(rsqrt{2}sqrt{1-cosleft(tright)}right)^{3}}{r^{2}left(1-cosleft(tright)right)}right)^{2}$$


I’m not sure if my work is right,can someone check that?

Non-algebraic holomorphic maps between algebraic curves

Let $V$ be a connected smooth complex projective curve of negative Euler characteristic. Can there exist a connected smooth complex algebraic curve $U$ such that there is a non-constant holomorphic map $Uto V$ but no non-constant holomorphic map from the compactification of $U$ to $V$?