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## A combinatorial formula for affine Hall–Littlewood functions via a weighted Brion theorem

We present a new combinatorial formula for Hall–Littlewood functions associated with the affine root system of type (Formula presented.), i.e., corresponding to the affine Lie algebra (Formula presented.). Our formula has the form of a sum over the elements of a basis constructed by Feigin, Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation. Our formula can be viewed as a weighted sum of exponentials of integer points in a certain infinite-dimensional convex polyhedron. We derive a weighted version of Brion’s theorem and then apply it to our polyhedron to prove the formula. © 2016 Springer International Publishing

Antistochastic strings are those strings that do not have any reasonable statistical explanation. We establish the follow property of such strings: every antistochastic string *x* is “holographic” in the sense that it can be restored by a short program from any of its part whose length equals the Kolmogorov complexity of *x*. Further we will show how it can be used for list decoding from erasing and prove that Symmetry of Information fails for total conditional complexity.

he aim of this study was to investigate the unique find from medieval Novgorod the Great—an almost complete skull of a young Eurasian beaver (*Castor fiber* L.). Comparisons of the craniometry of this skull with the skulls of the autochthonous and reintroduced populations of beavers from the same and adjacent regions suggest that a type of large beaver once inhabited the Volkhov basin. Further studies are necessary to accept or reject this hypothesis

We present a new structural lemma for deterministic con- text free languages. From the first sight, it looks like a pumping lemma, because it is also based on iteration properties, but it has significant distinctions that makes it much easier to apply. The structural lemma is a combinatorial analogue of KC-DCF-Lemma (based on Kolmogorov complexity), presented by Li and Vit ́anyi in 1995 and corrected by Glier in 2003. The structural lemma allows not only to prove that a language is not a DCFL, but discloses the structure of DCFLs Myhill-Nerode classes.

The paper [Harry Buhrman, Michal Kouck ́, Nikolay Vereshcha- y gin. Randomized Individual Communication Complexity. IEEE Con- ference on Computational Complexity 2008: 321-331] considered com- munication complexity of the following problem. Alice has a bi- nary string x and Bob a binary string y, both of length n, and they want to compute or approximate Kolmogorov complexity C(x|y) of x conditional to y. It is easy to show that deterministic communica- tion complexity of approximating C(x|y) with precision α is at least n − 2α − O(1). The above referenced paper asks what is random- ized communication complexity of this problem and shows that for r- round randomized protocols its communication complexity is at least Ω((n/α)1/r ). In this paper, for some positive ε, we show the lower bound 0.99n for (worst case) communication length of any random- ized protocol that with probability at least 0.01 approximates C(x|y) with precision εn for all input pairs.