Homology torsion growth and Mahler measure

We prove a conjecture of K. Schmidt in algebraic dynamical system theory on the growth of the number of components of fixed point sets. We also generalize a result of Silver and Williams on the growth of homology torsions of finite abelian covering of link complements. In both cases, the growth is expressed by the Mahler measure of the first non-zero Alexander polynomial of the corresponding modules. We use the notion of pseudo-isomorphism, and also tools from commutative algebra and algebraic geometry, to reduce the conjectures to the case of torsion modules. We also describe concrete sequences which give the expected values of the limits in both cases. For this part we utilize a result of Bombieri and Zannier (conjectured before by A. Schinzel) and a result of Lawton (conjectured before by D. Boyd).Comment: 25 pages. Minor mistakes and typos correcte

Sequential Convolutional Neural Networks for Slot Filling in Spoken Language Understanding

We investigate the usage of convolutional neural networks (CNNs) for the slot filling task in spoken language understanding. We propose a novel CNN architecture for sequence labeling which takes into account the previous context words with preserved order information and pays special attention to the current word with its surrounding context. Moreover, it combines the information from the past and the future words for classification. Our proposed CNN architecture outperforms even the previously best ensembling recurrent neural network model and achieves state-of-the-art results with an F1-score of 95.61% on the ATIS benchmark dataset without using any additional linguistic knowledge and resources.Comment: Accepted at Interspeech 201

Is there the color-flavor locking in the instanton induced quark-antiquark pairing in QCD vacuum?

By means of the functional integral method we show that in the case of the quark-antiquark pairing at zero temperature and zero chemical potential (in the vacuum) the singlet pairing is more preferable than that with the color-flavor locking (CFL)Comment: 6 pages, 5 figure

Bifurcation set, M-tameness, Asymptotic critical values and Newton polyhedrons

Let $F=(F_1, F_2, ..., F_m): \mathbb{C}^n \to \mathbb{C}^m$ be a polynomial dominant mapping with $n>m$. In this paper we give the relations between the bifurcation set of $F$ and the set of values where $F$ is not M-tame as well as the set of generalized critical values of $F$. We also construct explicitly a proper subset of $\mathbb{C}^m$ in terms of the Newton polyhedrons of $F_1, F_2, ..., F_m$ and show that it contains the bifurcation set of $F$. In the case $m= n-1$ we show that $F$ is a locally $C^{\infty}$-trivial fibration if and only if it is a locally $C^0$-trivial fibration.Comment: 12 pages; accepted for publication in Kodai M. J. Add comment in Remark 2.1, Remark 3.5 and correct Definition 3.

Quantum Teichm\"uller spaces and quantum trace map

We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum Teichm\"uller space of a marked surface, defined by Chekhov-Fock (and Kashaev) in an abstract way, can be realized as a concrete subalgebra of the skew field of the skein algebra.Comment: Final version. To appear in Journal of the Institute of Mathematics of Jussie

Quantum groups and ribbon G-categories

For a group G, the notion of a ribbon G-category was introduced by the second author in a previous work with a view towards constructing 3-dimensional homotopy quantum field theories (HQFT's) with target K(G,1). We discuss here how to derive ribbon G-categories from a simple complex Lie algebra g where G is the center of g. The construction is based on a study of representations of the quantum group $U_q(g)$ at a root of unity. Under certain assumptions on the root of unity, the resulting G-categories give rise to numerical invariants of pairs (a closed oriented 3-manifold M, an element of $H^1(M;G)$) and to 3-dimensional HQFT's.Comment: 13 page

On the structure of distance sets over prime fields

Let $\mathbb{F}_q$ be a finite field of order $q$ and $\mathcal{E}$ be a set in $\mathbb{F}_q^d$. The distance set of $\mathcal{E}$, denoted by $\Delta(\mathcal{E})$, is the set of distinct distances determined by the pairs of points in $\mathcal{E}$. Very recently, Iosevich, Koh, and Parshall (2018) proved that if $|\mathcal{E}|\gg q^{d/2}$, then the quotient set of $\Delta(\mathcal{E})$ satisfies $$\left\vert\frac{\Delta(\mathcal{E})}{\Delta(\mathcal{E})}\right\vert=\left\vert \left\lbrace\frac{a}{b}\colon a, b\in \Delta(\mathcal{E}), b\ne 0\right\rbrace\right\vert\gg q.$$ In this paper, we break the exponent $d/2$ when $\mathcal{E}$ is a Cartesian product of sets over a prime field. More precisely, let $p$ be a prime and $A\subset \mathbb{F}_p$. If $\mathcal{E}=A^d\subset \mathbb{F}_p^d$ and $|\mathcal{E}|\gg p^{\frac{d}{2}-\varepsilon}$ for some $\varepsilon>0$, then we have $$\left\vert\frac{\Delta(\mathcal{E})}{\Delta(\mathcal{E})}\right\vert, ~\left\vert \Delta(\mathcal{E})\cdot \Delta(\mathcal{E})\right\vert \gg p.$$ Such improvements are not possible over arbitrary finite fields. These results give us a better understanding about the structure of distance sets and the Erd\H{o}s-Falconer distance conjecture over finite fields.Comment: 8 pages. Submitted for publicatio

On positivity of Kauffman bracket skein algebras of surfaces

We show that the Chebyshev polynomials form a basic block of any positive basis of the Kauffman bracket skein algebras of surfaces.Comment: 11 pages. Typos corrected. Details added. Theorem 3.2 is strengthened. To appear in IMR

Strong Integrality of Quantum Invariants of 3-manifolds

We prove that the quantum SO(3)-invariant of an arbitrary 3-manifold $M$ is always an algebraic integer, if the order of the quantum parameter is co-prime with the order of the torsion part of H_1(M,\BZ). An even stronger integrality, known as cyclotomic integrality, was established by Habiro for integral homology 3-spheres. Here we generalize Habiro's result to all rational homology 3-spheres.Comment: 19 pages. Minor typos correcte

Growth of homology torsion in finite coverings and hyperbolic volume

We give an upper bound for the growth of homology torsions of finite coverings of irreducible 3-manifolds with tori boundary in terms of hyperbolic volume.Comment: 25 pages. The case of closed 3-manifolds is added back, with new proof. Final version to appear in prin