Cohomology of locally closed semi-algebraic subsets

Let k be a non-Archimedean field, let ? be a prime number distinct from the characteristic of the residue field of k. If χ is a separated k-scheme of finite type, Berkovich’s theory of germs allows to define étale ?-adic cohomology groups with compact support of locally closed semi-algebraic subsets of χ ^an . We prove that these vector spaces are finite dimensional continuous representations of the Galois group of k ^sep /k, and satisfy the usual long exact sequence and Künneth formula. This has been recently used by E. Hrushovski and F. Loeser in a paper about the monodromy of the Milnor fibration. In this statement, the main difficulty is the finiteness result, whose proof relies on a cohomological finiteness result for affinoid spaces, recently proved by V. Berkovich.     

Finite p-groups whose nonnormal subgroups have orders at most p 3

We classify finite p-groups all of whose nonnormal subgroups have orders at most p3, p odd prime. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.     

Finite 2-groups whose nonnormal subgroups have orders at most 23

In this paper, we classify finite 2-groups all of whose nonnormal subgroups have orders at most 2³. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.     

Finite p-groups with a minimal non-abelian subgroup of index p(Ⅱ)

We classify completely three-generator finite p-groups G such that Ф(G)≤Z(G)and|G′|≤p2.This paper is a part of the classification of finite p-groups with a minimal non-abelian subgroup of index p,and solve partly a problem proposed by Berkovich.     

Tropical hyperelliptic curves

We study the locus of tropical hyperelliptic curves inside the moduli space of tropical curves of genus g. We define a harmonic morphism of metric graphs and prove that a metric graph is hyperelliptic if and only if it admits a harmonic morphism of degree 2 to a metric tree. This generalizes the work of Baker and Norine on combinatorial graphs to the metric case. We then prove that the locus of 2-edge-connected genus g tropical hyperelliptic curves is a (2g?1)-dimensional stacky polyhedral fan whose maximal cells are in bijection with trees on g?1 vertices with maximum valence 3. Finally, we show that the Berkovich skeleton of a classical hyperelliptic plane curve satisfying a certain tropical smoothness condition is a standard ladder of genus g.     

A proof of the S-genus identities for ternary quadratic forms

In this paper we prove the main conjectures of Berkovich and Jagy about weighted averages of representation numbers over an S-genus of ternary lattices (defined below) for any odd squarefree S∈?. We do this by reformulating them in terms of local quantities using the Siegel–Weil and Conway–Sloane formulas, and then proving the necessary local identities. We conclude by conjecturing generalized formulas valid over certain totally real number fields as a direction for future work.     

<Emphasis Type="Italic">q</Emphasis>-Hypergeometric Proofs of Polynomial Analogues of the Triple Product Identity,Lebesgue’s Identity and Euler’s Pentagonal Number Theorem

We present alternative, q-hypergeometric proofs of some polynomial analogues of classical q-series identities recently discovered by Alladi and Berkovich, and Berkovich and Garvan.Work supported by the Australian Research Council     

Dynamics of Commuting Rational Maps on Berkovich Projective Space over C_p

We study the dynamics of commuting rational maps with coefficients in Cp. By lifting the dynamics from P1(Cp) to Berkovich projective space P1 Berk, we prove that two nonlinear commuting maps have the same Berkovich Julia set and the same canonical measure. As a consequence, two nonlinear commuting maps with coefficient in Cp have the same classical Julia set. We also prove that they have the same pre-periodic Berkovich Fatou components.     

Differentiability of the volume of a region enclosed by level sets

The level of a function f on Rn encloses a region. The volume of a region between two such levels depends on both levels. Fixing one of them the volume becomes a function of the remaining level. We show that if the function f is smooth, the volume function is again smooth for regular values of f. For critical values of f the volume function is only finitely differentiable. The initial motivation for this study comes from Radiotherapy, where such volume functions are used in an optimization process. Thus their differentiability properties become important.     

LOCAL CLASSICAL SOLUTION OF FREE BOUNDARY PROBLEM FOR A COUPLED SYSTEM

This paper considers a two-phase free boundary problem for coupled system including one parabolic equation and two elliptic equations. The problem comes from the discussion of a growth model of self-lnaintaining protocell in multidimensional case. The local classical solution of the problem with free boundary F : y = g(x,t) between two domains is being seeked. The local existence and uniqueness of the problem will be proved in multidimensional case.     

On the Schur multiplier of special p-groups

Let G be a special p-group minimally generated by $d\ge 3$ elements and having derived subgroup of order $p^{\frac{1}{2}d(d?1)}$. Berkovich asked to find the Schur multiplier and covering groups of such groups G Berkovich and Janko (2011) [1]. We try to give an answer to this question in this article.     

On inequalities and linear relations for 7-core partitions

Recently, Ramanujan’s modular equations have been applied by N.D. Baruah and B.C. Berndt to obtain a linear relation for 5-core partitions and by A. Berkovich and H. Yesilyurt to obtain inequalities for 7-core partitions. In this paper, we generalize their results by using the theory of modular forms. In particular, we prove conjectures of Berkovich and Yesilyurt.     

Some sufficient conditions for a finite group to be supersolvable

This paper represents an attempt to extend and improve the following result of Berkovich: Let G be a group of odd order. Let G=G ₁ G ₂ such that G ₁ and G ₂ are subgroups of?G. If the Sylow p-subgroups of G ₁ and of G ₂ are cyclic, then G is p-supersolvable.     

Polynomial approximation of Berkovich spaces and definable types

We investigate affine Berkovich spaces over maximally complete fields and prove that they may be approximated by simpler spaces when the only functions we need to evaluate are polynomials with bounded degrees. We derive applications to semi-algebraic sets and recover a result of E. Hrushovski and F. Loeser claiming that points of Berkovich spaces give rise to definable types (a model-theoretic notion of tameness).     

Radius of Convergence of p-adic Connections and the p-adic Rolle Theorem

We apply the theory of the radius of convergence of a p-adic connection [2] to the special case of the direct image of the constant connection via a finite morphism of compact p-adic curves, smooth in the sense of rigid geometry. We detail in sections 1 and 2, how to obtain convergence estimates for the radii of convergence of analytic sections of such a finite morphism. In the case of an étale covering of curves with good reduction, we get a lower bound for that radius, corollary 3.3, and obtain, via corollary 3.7, a new geometric proof of a variant of the p-adic Rolle theorem of Robert and Berkovich, theorem 0.2. We take this opportunity to clarify the relation between the notion of radius of convergence used in [2] and the more intrinsic one used by Kedlaya [16, Def. 9.4.7.].     

Enumerative geometry of elliptic curves on toric surfaces

We establish the equality of classical and tropical curve counts for elliptic curves on toric surfaces with fixed j-invariant, refining results of Mikhalkin and Nishinou–Siebert. As an application, we determine a formula for such counts on P² and all Hirzebruch surfaces. This formula relates the count of elliptic curves with the number of rational curves on the surface satisfying a small number of tangency conditions with the toric boundary. Furthermore, the combinatorial tropical multiplicities of Kerber and Markwig for counts in P² are derived and explained algebro-geometrically, using Berkovich geometry and logarithmic Gromov–Witten theory. As a consequence, a new proof of Pandharipande’s formula for counts of elliptic curves in P² with fixed j-invariant is obtained.     

A Poincaré lemma for real-valued differential forms on Berkovich spaces

Mathematische Zeitschrift - Real-valued differential forms on Berkovich analytic spaces were introduced by Chambert-Loir and Ducros in (Formes différentielles réelles et courants sur les...     

Every minor-closed property of sparse graphs is testable

Suppose G is a graph of bounded degree d, and one needs to remove ?n of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G^′. In fact, a similar result is proved for any minor-closed property of bounded degree graphs.The main motivation of the above result comes from theoretical computer-science. Using our main result we infer that for any minor-closed property P, there is a constant time algorithm for detecting if a graph is “far” from satisfying P. This, in particular, answers an open problem of Goldreich and Ron [STOC 1997] [20], who asked if such an algorithm exists when P is the graph property of being planar. The proof combines results from the theory of graph minors with results on convergent sequences of sparse graphs, which rely on martingale arguments.     

Ad hoc heuristic for the cover printing problem

We address an NP-hard combinatorial optimization problem arising in a printing shop. An impression grid is composed by a set of plates. The cover printing problem consists in designing the composition of impression grids, and determining the number of times each grid is to be printed in order to fulfill the demand of different book covers at minimum total printing cost; the latter comes from three fixed costs: for printing one sheet, for producing one plate, and for composing one impression grid. For each cover an unlimited number of plates can be made. To deal with this challenging problem we present an ad hoc heuristic that outperforms all previously proposed approaches, including genetic algorithms, GRASP, and simulated annealing.     

On maximal abelian subgroups in finite 2-groups

We determine here up to isomorphism the structure of any finite nonabelian 2-group G in which every two distinct maximal abelian subgroups have cyclic intersection. We obtain five infinite classes of such 2-groups (Theorem 1.1). This solves for p = 2 the problem Nr. 521 stated by Berkovich (in preparation). The more general problem Nr. 258 stated by Berkovich (in preparation) about the structure of finite nonabelian p-groups G such that A ∩ B = Z(G) for every two distinct maximal abelian subgroups A and B is treated in Theorems 3.1 and 3.2. In Corollary 3.3 we get a new result for an arbitrary finite 2-group. As an application of Theorems 3.1 and 3.2, we solve for p = 2 a problem of Heineken-Mann (Problem Nr. 169 stated in Berkovich, in preparation), classifying finite 2-groups G such that A/Z(G) is cyclic for each maximal abelian subgroup A (Theorem 4.1).