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I work on the solution of the Van Karman equations system and, in the process of solving this nonlinear set of PDE, I arrived at linear PDE with a biharmonic operator, and now I want to solve these PDE of forth order in rectangular domain and homogeneous boundary conditions as analytically or semi-analytically as possiblث.Read more

I am trying to understand the CM method for elliptic curves. Suppose we fix a discriminant $ D<0$ and a prime $ p$ . In the CM method, we look for integer solutions $ (t,y)$ to the norm equation $ 4p = t^2 -Dy^2$ . If these solutions exist, then we can construct an ellipticRead more

Consider divergence form elliptic pde in smooth boundary domain D $ $ Au:=\sum_{i,j}\partial_{i}(a_{ij}(x)\partial_{i}u(x)), $ $ with boundary data $ u|_{\partial D}:=1_{A}$ for $ A\subset \partial D$ . By Riesz representation there is a measure called the L-measure s.t. $ $ u(x)=\int_{\partial D} \phi(y)d\omega_{x}^{L}(y)$ $ if $ u|_{\partial D}=\phi\in C(\partial D)$ . For the harmonic caseRead more

The comments to this answer seem to make the following claim. Claim. Let $ K$ be the maximal abelian extension of $ \mathbb Q$ that is unramified away from $ p$ (more generally, away from a finite set $ S$ of primes). Then for every elliptic curve $ E$ over $ K$ , the groupRead more

Let $ X=E_1\times\cdots\times E_n$ , where $ E_i$ is the elliptic curve $ E_i=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\alpha_i)$ . In Grothendieck’s “The Hodge Conjecture is False for Trivial Reasons,” $ X$ is stated to fail the original formulation of the Generalized Hodge Conjecture, for appropriate choices of $ \alpha_i$ . Namely, there exist classes in $ F^1H^n(X,\mathbb{C})\cap H^n(X,\mathbb{Q})$ thatRead more

Consider a binary mask as Image[DiskMatrix[All,{100,150}]]; Now I wish to have a strip like this: such that I can manipulate the inner radius, and the outer radius (here major and minor axes) without changing the original image size. How can I do this? Edit 1: I just figured out that I can do this inRead more

Maybe this is a trivial question. But I do not find any thing about it in my searching in the net. Let us consider the elliptic curve $ $ y²=x³+ax+b$ $ where $ a,b$ are integers. My question is about the existence of a single singular bijective function $ f(t,a,b)$ verifying $ $ x=f^{(1/3)}(t,a,b)$ $Read more

I am stuck with one question in my research. I found some mentions that bitcoin(or blockchain) is a messeging system, I played with couple curve encoding algorithms and it was like: Curve params, priv key, pub key or coordinates + messege = signature. Its easy to read about it but I stuck to figure outRead more

Let (E) be an elliptic curve $ y^2=x^3-x-1$ over $ \mathbb{F}_3$ , $ a_0=1$ , $ a_n$ is the number of positive divisor of degree $ n\geq 1$ . $ a_1$ in this case is the number of points of E, i.e., $ a_1=1$ only a point $ \infty$ . The Zeta function $ Z(t)=\sum_{n=0}^{\infty}a_nRead more

I want to implement an elliptic curve point multiplication (Q=dG) operation with libgcrypt (gcrypt). I write the following code: #include <stdio.h> #include <string.h> #include <stdlib.h> #include <gcrypt.h> //select and insert key and curve parameters char *key521 = “(key-data\n” ” (public-key\n” ” (ecc\n” ” (curve NIST P-521)\n” ” (q #00F2271E305679EBF9D7673FBF75EEA22E90D100A9E0B5DE30C500411E91710D05826F17F2C33A9527CC6799553C626C015ED666F63A4D5CEB27CDBE61F34DB0AF#)\n” ” )\n” ” )\n” ” (private-key\n”Read more