Motivic zeta functions of abelian varieties, and the monodromy conjecture

We prove for abelian varieties a global form of Denef and Loeser?s motivic monodromy conjecture, in arbitrary characteristic. More precisely, we prove that for every tamely ramified abelian variety A over a complete discretely valued field with algebraically closed residue field, its motivic zeta function has a unique pole at Chai?s base change conductor c(A) of A, and that the order of this pole equals one plus the potential toric rank of A. Moreover, we show that for every embedding of Q? in C, the value exp(2πic(A)) is an ?-adic tame monodromy eigenvalue of A. The main tool in the paper is Edixhoven?s filtration on the special fiber of the Néron model of A, which measures the behavior of the Néron model under tame base change.     

Stable reduction of curves and tame ramification

We study stable reduction of curves in the case where a tamely ramified base extension is sufficient. If X is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring, there is a criterion, due to Saito, that describes precisely, in terms of the geometry of the minimal model with strict normal crossings of X, when a tamely ramified extension suffices in order for X to obtain stable reduction. For such curves we construct an explicit extension that realizes the stable reduction, and we furthermore show that this extension is minimal. We also obtain a new proof of Saito’s criterion, avoiding the use of ℓ-adic cohomology and vanishing cycles.     

The Néron component series of an abelian variety

We introduce the Néron component series of an abelian variety A over a complete discretely valued field. This is a power series in ${\mathbb{Z}}\left[\left[T\right]\right]$ , which measures the behaviour of the number of components of the Néron model of A under tame ramification of the base field. If A is tamely ramified, then we prove that the Néron component series is rational. It has a pole at T = 1, whose order equals one plus the potential toric rank of A. This result is a crucial ingredient of our proof of the motivic monodromy conjecture for abelian varieties. We expect that it extends to the wildly ramified case; we prove this if A is an elliptic curve, and if A has potential purely multiplicative reduction.     

A New Family of Ternary Sequences with Ideal Two-level Autocorrelation Function

Let be a primitive element of . Letd=3^2k -3 ^k +1 wheren=3k. We show that the ternary sequence s(t) given by has a two-level idealautocorrelation function.     

A logarithmic interpretation of Edixhoven's jumps for Jacobians

Let A be an abelian variety over a discretely valued field. Edixhoven has defined a filtration on the special fiber of the Néron model of A that measures the behavior of the Néron model under tame base change. We interpret the jumps in this filtration in terms of lattices of logarithmic differential forms in the case where A is the Jacobian of a curve C, and we give a compact explicit formula for the jumps in terms of the combinatorial reduction data of C.     

Alternating control tree search for knapsack/covering problems

The Multidimensional Knapsack/Covering Problem (KCP) is a 0–1 Integer Programming Problem containing both knapsack and weighted covering constraints, subsuming the well-known Multidimensional Knapsack Problem (MKP) and the Generalized (weighted) Covering Problem. We propose an Alternating Control Tree Search (ACT) method for these problems that iteratively transfers control between the following three components: (1) ACT-1, a process that solves an LP relaxation of the current form of the KCP. (2) ACT-2, a method that partitions the variables according to 0, 1, and fractional values to create sub-problems that can be solved with relatively high efficiency. (3) ACT-3, an updating procedure that adjoins inequalities to produce successively more constrained versions of KCP, and in conjunction with the solution processes of ACT-1 and ACT-2, ensures finite convergence to optimality. The ACT method can also be used as a heuristic approach using early termination rules. Computational results show that the ACT-framework successfully enhances the performance of three widely different heuristics for the KCP. Our ACT-method involving scatter search performs better than any other known method on a large set of KCP-instances from the literature. The ACT-based methods are also found to be highly effective on the MKP.     

A phase transition from monoclinic C2 with Z′ = 1 to triclinic P1 with Z′ = 4 for the quasiracemate l‐2‐aminobutyric acid–d‐methionine (1/1)

Racemates of hydrophobic amino acids with linear side chains are known to undergo a unique series of solid‐state phase transitions that involve sliding of molecular bilayers upon heating or cooling. Recently, this behaviour was shown to extend also to quasiracemates of two different amino acids with opposite handedness [Görbitz & Karen (2015). J. Phys. Chem. B, 119 , 4975–4984]. Previous investigations are here extended to an l ‐2‐aminobutyric acid–d ‐methionine (1/1) co‐crystal, C₄H₉NO₂·C₅H₁₁NO₂S. The significant difference in size between the –CH₂CH₃ and –CH₂CH₂SCH₃ side chains leads to extensive disorder at room temperature, which is essentially resolved after a phase transition at 229 K to an unprecedented triclinic form where all four d ‐methionine molecules in the asymmetric unit have different side‐chain conformations and all three side‐chain rotamers are used for the four partner l ‐2‐aminobutyric acid molecules.