In this note I would like to introduce a new approach to (or rather a new language for) representation theory of groups. Namely, I propose to consider a (complex) representation of a group G as a sheaf on some geometric object. This point of view necessarily leads to a conclusion that the standard approach to (continuous) representations of algebraic groups should be modified. Let us start with...

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence a1x1 + · · · + akxk ≡ b (mod n), with gcd(xi, n) = ti (1 ≤ i ≤ k), where a1, t1, . . . , ak, tk, b, n (n ≥ 1) are arbitrary integers. Some special cases of this problem have been already studied in many ...

The exact theory of phase separation in a two-dimensional wedge is derived from the properties of the order parameter and boundary condition changing operators in field theory. For a shallow wedge we determine the passage probability for an interface with endpoints on the boundary. For generic opening angles we exhibit the fundamental origin of the filling transition condition and of the proper...

Galaxy structures represent one of the most challenging observations in cosmology. Whether or not these structures are found to be compatible with the standard model of galaxy formation crucially depends on the a-priori and assumptions encoded in the statistical methods employed to characterize the data and on the a-posteriori hypotheses made to interpret the results. We present strategies to t...

Autoencoders are unsupervised machine learning circuits, with typically one hidden layer, whose learning goal is to minimize an average distortion measure between inputs and outputs. Linear autoencoders correspond to the special case where only linear transformations between visible and hidden variables are used. While linear autoencoders can be defined over any field, only real-valued linear a...

It is shown that delta hedging provides the optimal trading strategy in terms of minimal required initial capital to replicate a given terminal payoff in a continuous-time Markovian context. This holds true in market models where no equivalent local martingale measure exists but only a square-integrable market price of risk. A new probability measure is constructed, which takes the place of an ...

Spintronics relies on the ability to transport and utilize the spin properties of an electron rather than its charge. We describe a spin rachet at the singleelectron level that produces spin currents with no net bias or charge transport. Our device is based on the ground state energetics of a single electron transistor comprising a superconducting island connected to normal leads via tunnel bar...

Although approximate Bayesian computation (ABC) has become a popular technique for performing parameter estimation when the likelihood functions are analytically intractable there has not as yet been a complete investigation of the theoretical properties of the resulting estimators. In this paper we give a theoretical analysis of the asymptotic properties of ABC based parameter estimators for h...