We present a new algorithm based on binary quadratic forms to factor integers of the form N = pq. Its heuristic running time is exponential in the general case, but becomes polynomial when special (arithmetic) hints are available, which is exactly the case for the so-called NICE family of public-key cryptosystems based on quadratic fields introduced in the late 90s. Such cryptosystems come in t...

In this article, we define the set H of quaternion numbers as the set of all ordered sequences q = 〈x, y,w, z〉 where x,y,w and z are real numbers. The addition, difference and multiplication of the quaternion numbers are also defined. We define the real and imaginary parts of q and denote this by x = R(q), y = I1(q), w = I2(q), z = I3(q). We define the addition, difference, multiplication again...

We prove that QCD in the ǫ-regime of chiral Perturbation Theory is equivalent to chiral Random Matrix Theory for zero and both non-zero real and imaginary chemical potential μ. To this aim we prove a theorem that relates integrals over fermionic and bosonic variables to super-Hermitian or super-Unitary groups also called superbosonisation. Our findings extend previous results for the equivalenc...

Brain signals recorded from the primary motor cortex (M1) are known to serve a significant role in coding the information brain-machine interfaces (BMIs) need to perform real and imagined movements, and also to form several functional networks with motor association areas. However, whether functional networks between M1 and other brain regions, such as these motor association areas, are related...

We present a new algorithm based on binary quadratic forms to factor integers of the form N = pq. Its heuristic running time is exponential in the general case, but becomes polynomial when special (arithmetic) hints are available, which is exactly the case for the so-called NICE family of public-key cryptosystems based on quadratic fields introduced in the late 90s. Such cryptosystems come in t...

When a planar dielectric medium has a permittivity profile that is an analytic function in the upper or lower half of the complex position plane x = x′+ ix′′ then the real and imaginary parts of its permittivity are related by the spatial Kramers-Kronig relations. We find that such a medium won’t reflect radiation incident from one side, whatever the angle of incidence. Using the spatial Kramer...

Numerous microscopic models [1] for price fluctuations on stock or currency markets have been invented within the last decade [2, 3] in the physics literature. An alternative are more phenomenological differential equations for the price itself [4, 5, 6, 7, 8], with some noise as in Langevin equations. The present note follows Sornette and Ide [5] but uses a complex instead of a real variable. ...