Recent Preprint Research Articles

Inflexibility, Weil–Petersson distance, and volumes of fibered 3-manifolds

A recent preprint of S. Kojima and G. McShane [KM] observes a beautiful explicit connection between Teichmuller translation distance and hyperbolic volume. It relies on a key estimate which we supply here: using geometric inflexibility of hyperbolic 3-manifolds, we show that for S a closed surface, and ψ ∈ Mod(S) pseudo-Anosov, the double iteration Q(ψ−n(X),ψn(X)) has convex core volume differing from 2nvol(Mψ ) by a uniform additive constant, where Mψ is the hyperbolic mapping torus for ψ . We combine this estimate with work of Schlenker, and a branched covering argument to obtain an explicit lower bound on Weil-Petersson translation distance of a pseudo-Anosov ψ ∈Mod(S) for general compact S of genus g with n boundary components: we have vol(Mψ )≤ 3 √ π/2(2g−2+n)‖ψ‖WP. This gives the first explicit estimates on the Weil-Petersson systoles of moduli space, of the minimal distance between nodal surfaces in the completion of Teichmuller space, and explicit lower bounds to the Weil-Petersson diameter of the moduli space via [CP]. In the process, we recover the estimates of [KM] on Teichmuller translation distance via a Cauchy-Schwarz estimate (see [Lin]).

The super period matrix with Ramond punctures in the supergravity formulation

In a very recent preprint, Witten showed how to construct a g|r×g|r super period matrix for super Riemann surfaces of genus g with 2r Ramond punctures, which is symmetric in the Z2 graded sense. He also showed how it can be applied to analyze supersymmetry breaking in string compactifications which are supersymmetric at tree-level. Witten's construction is in the purely holomorphic formulation of super Riemann surfaces. In this paper, a construction is given in the formulation of two-dimensional supergravity. The variations of the super period matrix with respect to supermoduli deformations are also given, as well as an explicit illustration of how the super period matrix with two Ramond punctures would emerge from a degeneration of the super period matrix without punctures in higher genus.

The n-point correlation of quadratic forms

In this paper we investigate the distribution of the set of values of a quadratic form Q, at integral points. In particular we are interested in the n-point correlations of the this set. The asymptotic behaviour of the counting function that counts the number of n-tuples of integral points $$\left( v_{1},\ldots ,v_{n}\right) $$ , with bounded norm, such that the $$n-1$$ differences $$Q\left( v_{1}\right) -Q\left( v_{2}\right) ,\ldots Q\left( v_{n-1}\right) -Q\left( v_{n}\right) $$ , lie in prescribed intervals is obtained. The results are valid provided that the quadratic form has rank at least 5, is not a multiple of a rational form and n is at most the rank of the quadratic form. For certain quadratic forms satisfying Diophantine conditions we obtain a rate for the limit. The proofs are based on those in the recent preprint (Distribution of values of quadratic forms at integral points. http://www.math.uni-bielefeld.de/sfb701/files/preprints/sfb13003.pdf , 2013) of Gotze and Margulis, in which they prove an ‘effective’ version of the Oppenheim conjecture. In particular, the proofs rely on Fourier analysis and estimates for certain theta series.

Comment on “Linear Wave Dynamics Explains Observations Attributed to Dark Solitons in a Polariton Quantum Fluid”

In a recent preprint (arXiv:1401.1128v1) Cilibrizzi and co-workers report experiments and simulations showing the scattering of polaritons against a localised obstacle in a semiconductor microcavity. The authors observe in the linear excitation regime the formation of density and phase patterns reminiscent of those expected in the non-linear regime from the nucleation of dark solitons. Based on this observation, they conclude that previous theoretical and experimental reports on dark solitons in a polariton system should be revised. Here we comment why the results from Cilibrizzi et al. take place in a very different regime than previous investigations on dark soliton nucleation and do not reproduce all the signatures of its rich nonlinear phenomenology. First of all, Cilibrizzi et al. consider a particular type of radial excitation that strongly determines the observed patterns, while in previous reports the excitation has a plane-wave profile. Most importantly, the nonlinear relation between phase jump, soliton width and fluid velocity, and the existence of a critical velocity with the time-dependent formation of vortex-antivortex pairs are absent in the linear regime. In previous reports about dark soliton and half-dark soliton nucleation in a polariton fluid, the distinctive dark soliton physics is supported both by theory (analytical and numerical) and experiments (both continuous wave and pulsed excitation).

Computing Bases of Twisted Cohomology Groups for Sparse Polynomials

We present an algorithm to construct bases of twisted cohomology groups for the complement of a generic sparse polynomial over a generic local system of rank 1. The algorithm is based on the theory given in a recent preprint [1].

Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation

We construct two families of non-localized standing waves for the hyperbolic cubic nonlinear Schrodinger equation \[iu_t+u_{xx}-u_{yy}+|u|^2u=0.\] The first family of standing waves consists of solutions which correspond to some generalized breathers for each fixed time $t$, while solutions in the second family are periodic both in $x$ and $y$. The second family of solutions were numerically observed by Vuillon, Dutykh and Fedele in a recent preprint [17].

On ampleness and pseudo-Anosov homeomorphisms in the free group

We use pseudo-Anosov homeomorphisms of surfaces in order to prove that the first-order theory of non-Abelian free groups, T_{fg}, is n-ample for any n \in \omega. This result adds to the work of Pillay, which proved that T_{fg} is non-CM-trivial. The sequence witnessing ampleness is a sequence of primitive elements in F_{\omega}. Our result provides an alternative proof to the main result of a recent preprint by Ould Houcine and Tent.

$F$-pure thresholds of binomial hypersurfaces

In this note, we use estimates given in the recent preprint [Her11b]to deduce a formula for the F-pure threshold of a binomial hypersurface over a field of prime characteristic. These formulas are given in terms of the associated splitting polytope, and remain valid over any characteristic.

About the Young measures associated with Y. Brenier's ABI model

The Born–Infeld system is a nonlinear model of electromagnetism, intended to serve in high-energy physics. It has been embeded by Y. Brenier into a quasilinear system of conservation laws endowed with a convex entropy [3]. The latter system, denoted as ABI, governs the evolution of the electromagnetic field (B,D), together with the energy density h and the Poynting vector P.The recent preprint [6] stresses that the convex hull in the (h,B,D,P)-space R10 of the equilibrium manifold defined byP=D×B,h=1+|B|2+|D|2+|P|2 plays a key role in the approach by convex integration. In this article, S. Müller and M. Palombaro point out a mistake in the description of this hull given in our previous paper [8].We give below the correct description of the convex hull and show that it is not positively invariant under the evolution of ABI. Its points are mass centers of the Young measures permitted by the ABI system. We give additional restrictions on these measures.

A spanning tree cohomology theory for links

In their recent preprint, Baldwin, Ozsváth and Szabó defined a twisted version (with coefficients in a Novikov ring) of a spectral sequence, previously defined by Ozsváth and Szabó, from Khovanov homology to Heegaard–Floer homology of the branched double cover along a link. In their preprint, they give a combinatorial interpretation of the E3-term of their spectral sequence. The main purpose of the present paper is to prove directly that this E3-term is a link invariant. We also give some concrete examples of computation of the invariant.

Gelfand models for classical Weyl groups

In a recent preprint Kodiyalam and Verma give a particularly simple Gelfand model for the symmetric group that is built naturally on the space of involutions. In this manuscript we give a natural extension of Kodiyalam and Verma's model to a Gelfand model for Weyl groups of type Bn and D2n+1. Then we define an explicit isomorphism between this Gelfand model and the polynomial model using a technique we call telescopic decomposition.

Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

Suppose that X={X t :tÂ?0} is a supercritical super Ornstein-Uhlenbeck process, that is, a superprocess with an Ornstein-Uhlenbeck process on $\mathbb{R}^{d}$ corresponding to $L=\frac{1}{2}\sigma^{2}\Delta-b x\cdot\nabla$ as its underlying spatial motion and with branching mechanism Â?(Â?)=Â?Â?Â?+AŸÂ? 2+Â?(0,+Â?)(e Â?Â?x Â?1+Â?x)n(dx), where Â?=Â?Â?Â?(0+)>0, AŸÂ?0, and n is a measure on (0,Â?) such that Â?(0,+Â?) x 2 n(dx)<+Â?. Let $\mathbb{P} _{\mu}$ be the law of X with initial measure μ. Then the process W t =e Â?Â?t Â?X t Â? is a positive $\mathbb{P} _{\mu}$ -martingale. Therefore there is W Â? such that W t Â?W Â?, $\mathbb{P} _{\mu}$ -a.s. as tÂ?Â?. In this paper we establish some spatial central limit theorems for X. Let $\mathcal{P}$ denote the function class $$ \mathcal{P}:=\bigl\{f\in C\bigl(\mathbb{R}^d\bigr): \mbox{there exists } k\in\mathbb{N} \mbox{ such that }|f(x)|/\|x\|^k\to 0 \mbox{ as }\|x\|\to\infty \bigr\}. $$ For each $f\in\mathcal{P}$ we define an integer Â?(f) in term of the spectral decomposition of f. In the small branching rate case Â? 2Â?(f)b. Our central limit theorems in the small and critical branching rate cases sharpen the corresponding results in the recent preprint of Miloś in that our limit normal random variables are non-degenerate. Our central limit theorems in the large branching rate case have no counterparts in the recent preprint of Miloś. The main ideas for proving the central limit theorems are inspired by the arguments in K. Athreya's 3 papers on central limit theorems for continuous time multi-type branching processes published in the late 1960's and early 1970's.

Monotony based inclusion detection in EIT for realistic electrode models

In electrical impedance tomography (EIT), there is a monotony relation between the conductivity of an imaging subject and the EIT measurements. Such monotony relations can be used for inclusion detection methods. However, up to now, monotony-based methods were only known to reconstruct an upper or lower bound of inclusions.In a recent preprint, the authors showed that, for continuous boundary data, monotony based methods are in fact capable of reconstructing the exact shape of inclusions. In this work, we discuss how to extend our results to the practically relevant complete electrode model setting and study the possibility of rigorous resolution guarantees.

Remark on the Off-Diagonal Expansion of the Bergman Kernel on Compact Kähler Manifolds

In this short note, we compare our previous works on the off-diagonal expansion of the Bergman kernel and the recent preprint of Lu-Shiffman (arxiv.1301.2166). In particular, we note that the vanishing of the coefficient of p^{-1/2} is implicitly contained in Dai-Liu-Ma's work (J. Differential Geom. 72 (2006), no. 1, 1-41) and was explicitly stated in our book (Holomorphic Morse inequalities and Bergman kernels, Progress in Math., vol. 254, Birkh\"auser, 2007).

ON A CONJECTURE OF IGUSA

In his Tata Lecture Notes, Igusa conjectured the validity of a strong uniformity in the decay of complete exponential sums modulo powers of a prime number and determined by a homogeneous polynomial. This was proved for non-degenerate forms by Denef–Sperber and then by Cluckers for weighted homogeneous non-degenerate forms. In a recent preprint, Wright has proved this for degenerate binary forms. We give a different proof of Wright's result that seems to be simpler and relies upon basic estimates for exponential sums mod as well as a type of resolution of singularities with good reduction in the sense of Denef.

MAGNITUDE GAP STATISTICS AND THE CONDITIONAL LUMINOSITY FUNCTION

In a recent preprint, Hearin et al. (2012,H12) suggest that the halo mass-richness calibration of clusters can be improved by using the difference in the magnitude of the brightest and the second brightest galaxy (magnitude gap) as an additional observable. They claim that their results are at odds with the results from Paranjape & Sheth (2012, PS12) who show that the magnitude distribution of the brightest and second brightest galaxies can be explained based on order statistics of luminosities randomly sampled from the total galaxy luminosity function. We find that a conditional luminosity function (CLF) for galaxies which varies with halo mass, in a manner which is consistent with existing observations, naturally leads to a magnitude gap distribution which changes as a function of halo mass at fixed richness, in qualitative agreement with H12. We show that, in general, the luminosity distribution of the brightest and the second brightest galaxy depends upon whether the luminosities of galaxies are drawn from the CLF or the global luminosity function. However, we also show that the difference between the two cases is small enough to evade detection in the small sample investigated by PS12. This shows that the luminosity distribution is not the appropriate statistic to distinguish between the two cases, given the small sample size. We argue in favor of the CLF (and therefore H12) based upon its consistency with other independent observations, such as the kinematics of satellite galaxies, the abundance and clustering of galaxies, and the galaxy-galaxy lensing signal from the Sloan Digital Sky Survey.

A note on Frobenius divided modules in mixed characteristics

If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if $\sD_X$ is the sheaf of differential operators on $X$ [EGAIV], it is well known that giving an action of $\sD_X$ on an $\sO_X$-module $\sE$ is equivalent to giving an infinite sequence of $\sO_X$-modules descending $\sE$ via the iterates of the Frobenius endomorphism of $X$. We show that this result can be generalized to any infinitesimal deformation $f : X \to S$ of a smooth morphism in characteristic $p$, endowed with Frobenius liftings. We also show that it extends to adic formal schemes such that $p$ belongs to an ideal of definition. In a recent preprint, dos Santos used this result to lift $\sD_X$-modules from characteristic $p$ to characteristic 0 with control of the differential Galois group.

Left adequate and left Ehresmann monoids II

This article is the second of two presenting a new approach to left adequate monoids. In the first, we introduced the notion of being T-proper, where T is a submonoid of a left adequate monoid M. We showed that the free left adequate monoid on a set X is X ⁎ -proper. Further, any left adequate monoid M has an X ⁎ -proper cover for some set X, that is, there is an X ⁎ -proper left adequate monoid M ˆ and an idempotent separating epimorphism θ : M ˆ → M of the appropriate signature. We now show how to construct a T-proper left adequate monoid P ( T , Y ) from a monoid T acting via order-preserving maps on a semilattice Y with identity. Our construction plays the role for left adequate monoids that the semidirect product of a group and a semilattice plays for inverse monoids. A left adequate monoid M with semilattice E has an X ⁎ -proper cover P ( X ⁎ , E ) . Hence, by choosing a suitable semilattice E X and an action of X ⁎ on E X , we prove that the free left adequate monoid is of the form P ( X ⁎ , E X ) . An alternative description of the free left adequate monoid appears in a recent preprint of Kambites. We show how to obtain the labelled trees appearing in his result from our structure theorem. Our results apply to the wider class of left Ehresmann monoids, and we give them in full generality. Indeed this is the right setting: the class of left Ehresmann monoids is the variety generated by the quasi-variety of left adequate monoids. This paper, and the two of Kambites on free (left) adequate semigroups, demonstrate the rich but accessible structure of (left) adequate semigroups and monoids, introduced with startling insight by Fountain some 30 years ago.

I Teoremi di Campbell, Baker, Hausdorff e Dynkin. Storia, Prove, Problemi Aperti

The aim of this lecture is to provide an overview of facts and references about past and recent results on the Theorem of Campbell, Baker, Hausdorff and Dynkin (shortcut as the CBHD Theorem), following the recent preprint monograph [13]. In particular, we shall give sketches of the following facts: A historical precis of the early proofs (see also [1]); the statement of the CBHD Theorem as usually given in Algebra and that employed in the Analysis of linear PDE’s; a review of proofs of the CBHD Theorem (as given by: Bourbaki; Hausdorff; Dynkin; Varadarajan) together with a unifying demonstrational approach; an application to the Third Theorem of Lie (in local form). Some new results will be also commented: The intertwinement of the CBHD Theorem with the Theorem of Poincare-Birkhoff-Witt and with the free Lie algebras (see [12]); recent results on optimal domains of convergence.

DARK MATTER SEARCH USINGCHANDRAOBSERVATIONS OF WILLMAN 1 AND A SPECTRAL FEATURE CONSISTENT WITH A DECAY LINE OF A 5 keV STERILE NEUTRINO

In a recent preprint entitled "Searching for dark matter in X-rays: how to check the dark matter origin of a spectral feature" [arXiv:1001.0644v1], the authors have claimed that some archival X-ray data could be used to rule out dark matter in the form of 5-keV sterile neutrinos at the level of 20 sigma. Unfortunately, the limit was derived incorrectly. We point out the shortcomings of this analysis and show that the tentative detection of a spectral feature consistent with a 5-keV sterile neutrino is not in contradiction with existing limits. Future observations of dwarf spheroidal galaxies will test this hypothesis.