We show how to use duality theory to construct minimized versions of a wide class of automata. We work out three cases in detail: (a variant of) ordinary automata, weighted automata and probabilistic automata. The basic idea is that instead of constructing a maximal quotient we go to the dual and look for a minimal subalgebra and then return to the original category. Duality ensures that the mi...

In this paper, we introduce the concept of x-maximal flatness. A control system is x-maximally flat if the number of new states gained by each successive derivation of the flat output is the largest possible. Firstly, we show that the only control-linear systems that are xmaximally flat are those that are static feedback equivalent to the m-chained form. Secondly, we generalize that result from...

Let G be a simple complex Lie group, g be its Lie algebra, K be a maximal compact form of G and k be a Lie algebra of K. We denote by X → X the anti-involution of g which singles out the compact form k. Consider the space of flat g-valued connections on a Riemann sphere with three holes which satisfy the additional condition A(z) = −A(z). We call the quotient of this space over the action of th...

The theory of Whittaker functors for GLn is an essential technical tools in Gaitsgory’s proof of the Vanishing Conjecture appearing in the geometric Langlands correspondence ([5]). We define Whittaker functors for GSp 4 and study their properties. These functors correspond to the maximal parabolic subgroup of GSp 4 , whose unipotent radical is not commutative. We also study similar functors cor...

We propose a recurrent extension of the Ladder networks [22] whose structure is motivated by the inference required in hierarchical latent variable models. We demonstrate that the recurrent Ladder is able to handle a wide variety of complex learning tasks that benefit from iterative inference and temporal modeling. The architecture shows close-to-optimal results on temporal modeling of video da...

Let p be a prime. It is a fundamental problem to classify the absolute Galois groups GF of fields F containing a primitive pth root of unity. In this paper we present several constraints on such GF , using restrictions on the cohomology of index p normal subgroups from [LMS]. In section 1 we classify all maximal p-elementary abelian-by-order p quotients of these GF . In the case p > 2, each suc...

Let p be a prime. It is a fundamental problem to classify the absolute Galois groups GF of fields F containing a primitive pth root of unity ξp. In this paper we present several constraints on such GF , using restrictions on the cohomology of index p normal subgroups from [LMS]. In section 1 we classify all maximal p-elementary abelian-by-order p quotients of these GF . In the case p > 2, each ...

Let the special linear group G := SL 2 act regularly on a Q-factorial variety X. Consider a maximal torus T ⊂ G and its normalizer N ⊂ G. We prove: If U ⊂ X is a maximal open N-invariant subset admitting a good quotient U → U/ /N with a divisorial quotient space, then the intersection W (U) of all translates g·U is open in X and admits a good quotient W (U) → W (U)/ /G with a divisorial quotien...

We prove sharp power-weighted L, weak type and restricted weak type inequalities for the heat and Poisson integral maximal operators, Riesz transform and a Littlewood-Paley type square function, emerging naturally in the harmonic analysis related to Bessel operators.

Let U (N) denote the maximal length of arithmetic progressions in a random uniform subset of {0, 1}N . By an application of the Chen-Stein method, we show that U −2 log N/ log 2 converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W (N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we ...