In the last five years, deep learning methods and particularly Convolutional Neural Networks (CNNs) have exhibited excellent accuracies in many pattern classification problems. Most of the state-of-the-art models apply data-augmentation techniques at the training stage. This paper provides a brief tutorial on data preprocessing and shows its benefits by using the competitive MNIST handwritten digits classification problem. We show and analyze the impact of different preprocessing techniques on the performance of three CNNs, LeNet, Network3 and DropConnect, together with their ensembles. The analyzed transformations are, centering, elastic deformation, translation, rotation and different combinations of them. Our analysis demonstrates that data-preprocessing techniques, such as the combination of elastic deformation and rotation, together with ensembles have a high potential to further improve the state-of-the-art accuracy in MNIST classification.

We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Previous work in the complex plane by Goldberg and Guo focused on q = 2, which corresponds to the case of the Ising model; for q > 2, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane.Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing #P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all q ≥ 2 and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lovász and Welsh in the context of quantum computations.

No abstract

The n th cyclotomic polynomial Φn(x) is the minimal polynomial of an n th primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas which are valid for all natural numbers n. In these a host of famous number theoretical objects such as Bernoulli numbers, Stirling numbers of both kinds and Ramanujan sums make their appearance, sometimes even at the same time! In this paper we present a survey of these formulas which until now were scattered in the literature and introduce an unified approach to derive some of them, leading also to shorter proofs as a by-product. In particular, we show that some of the formulas have a more elegant reinterpretation in terms of Bell polynomials. This approach amounts to computing the logarithmic derivatives of Φn at certain points. Furthermore, we show that the logarithmic derivatives at ±1 of any Kronecker polynomial (a monic product of cyclotomic polynomials and a monomial) satisfy a family of linear equations whose coefficients are Stirling numbers of the second kind. We apply these equations to show that certain polynomials are not Kronecker. In particular, we infer that for every k ≥ 4 there exists a symmetric numerical semigroup with embedding dimension k and Frobenius number 2k + 1 that is not cyclotomic, thus establishing a conjecture of Alexandru Ciolan, Pedro García-Sánchez and the second author. In an appendix Pedro García-Sánchez shows that for every k ≥ 4 there exists a symmetric non-cyclotomic numerical semigroup having Frobenius number 2k + 1. 1 arXiv:1805.05207v2 [math.NT] 22 Aug 2018Proof. The result follows on invoking (5.5) and noting that for every positive integer j we haved|n J j (d) − J j (n) = (2 j − 1)(n j − J j (n)).The following corollary is a consequence of the previous identity and its proof is similar to that of Theorem 5.8.

We introduce the concept of isolated factorizations of an element of a commutative monoid and study its properties. We give several bounds for the number of isolated factorizations of simplicial affine semigroups and numerical semigroups. We also generalize α-rectangular numerical semigroups to the context of simplicial affine semigroups and study their isolated factorizations. As a consequence of our results, we characterize those complete intersection simplicial affine semigroups with only one Betti minimal element in several ways. Moreover, we define Betti sorted and Betti divisible simplicial affine semigroups and characterize them in terms of gluings and their minimal presentations. Finally, we determine all the Betti divisible numerical semigroups, which turn out to be those numerical semigroups that are free for any arrangement of their minimal generators.