Overview of the T2K long baseline neutrino oscillation experiment

Neutrino oscillations were discovered by atmospheric and solar neutrino experiments, and have been confirmed by experiments using neutrinos from accelerators and nuclear reactors. It has been found that there are large mixing angles in the $\nu_e \to \nu_\mu$ and $\nu_\mu \to \nu_\tau$ oscillations. The third mixing angle $\theta_{13}$, which parameterizes the mixing between the first and the third generation, is constrainted to be small by the CHOOZ experiment result. The T2K experiment is a long baseline neutrino oscillation experiment that uses intense neutrino beam produced at J-PARC and Super-Kamiokande detector at 295 km as the far detector to measure $\theta_{13}$ using $\nu_e$ appearance. In this talk, we will give an overview of the experiment.Comment: To be published in the proceedings of DPF-2009, Detroit, MI, July 2009, eConf C09072

(Relative) dynamical degrees of rational maps over an algebraic closed field

The main purpose of this paper is to define dynamical degrees for rational maps over an algebraic closed field of characteristic zero and prove some basic properties (such as log-concavity) and give some applications. We also define relative dynamical degrees and prove a "product formula" for dynamical degrees of semi-conjugate rational maps in the algebraic setting. The main tools are the Chow's moving lemma and a formula for the degree of the cone over a subvariety of $\mathbb{P}^N$. The proofs of these results are valid as long as resolution of singularities are available (or more generally if appropriate birational models of the maps under consideration are available). This observation is applied for the cases of surfaces and threefolds over a field of positive characteristic.Comment: 24 pages. This paper incorporates Section 3 in the paper arXiv:1212.109

Absolutely superficial sequences

Absolutely superficial sequences was introduced by P. Schenzel in order to study generalized Cohen-Macaulay (resp. Buchsbaum) modules. For an arbitrary local ring, they turned out to be d-sequences. This paper established properties of absolutely superficial sequences with respect to a module. It is shown that they are closely related to other sequences in the theory of generalized Cohen-Macaulay (resp. Buchsbaum) modules. In particular, there is a bounding function for the Hilbert-Samuel function of every parameter ideal such that this bounding function is attained if and only if the ideal is generated by an absolutely superficial sequence

Note on potential theory for functions in Hardy classes

The purpose of this note is to show that the set functions defined in \cite{trong-tuyen} can be suitably extended to all subsets $E$ of the unit disk $\mathbb{D}$. In particular we obtain uniform nearly-optimal estimates for the following quantity D_p(E,\epsilon, R) = \sup \{\sup_{|z| \leq R}|g(z)|: g\in H^p, ||g||_{H^p}\leq 1, (1-|\zeta |)|g(\zeta)| \leq \epsilon \forall \zeta\in E\}.Comment: 3 page

Coarse categories I: foundations

Following Roe and others (see, e.g., [MR1451755]), we (re)develop coarse geometry from the foundations, taking a categorical point of view. In this paper, we concentrate on the discrete case in which topology plays no role. Our theory is particularly suited to the development of the_Roe (C*-)algebras_ C*(X) and their K-theory on the analytic side; we also hope that it will be of use in the strictly geometric/algebraic setting of controlled topology and algebra. We leave these topics to future papers. Crucial to our approach are nonunital coarse spaces, and what we call _locally proper_ maps (which are actually implicit in [MR1988817]). Our_coarse category_ Crs generalizes the usual one: its objects are nonunital coarse spaces and its morphisms (locally proper) coarse maps modulo_closeness_. Crs is much richer than the usual unital coarse category. As such, it has all nonzero limits and all colimits. We examine various other categorical issues. E.g., Crs does not have a terminal object, so we substitute a_termination functor_ which will be important in the development of exponential objects (i.e., "function spaces") and also leads to a notion of_quotient coarse spaces_. To connect our methods with the standard methods, we also examine the relationship between Crs and the usual coarse category of Roe. Finally we briefly discuss some basic examples and applications. Topics include_metric coarse spaces_,_continuous control_ [MR1277522], metric and continuously controlled_coarse simplices_,_sigma-coarse spaces_ [MR2225040], and the relation between quotient coarse spaces and the K-theory of Roe algebras (of particular interest for continuously controlled coarse spaces).Comment: 70 pages; citation/reference added, minor corrections, changed formatting; up-to-date version before major overhau

Observability of a 1D Schr\"odinger equation with time-varying boundaries

We discuss the observability of a one-dimensional Schr\"odinger equation on certain time dependent domain. In linear moving case, we give the exact boundary and pointwise internal observability for arbitrary time. For the general moving, we provide exact boundary observability when the curve satisfies some certain conditions . By duality theory, we establish the controllability of adjoint system.Comment: 20 page

Complex Monge-Ampere operators via pseudo-isomorphisms: the well-defined cases

Let $X$ and $Y$ be compact K\"ahler manifolds of dimension $3$. A bimeromorphic map $f:X\rightarrow Y$ is pseudo-isomorphic if $f:X-I(f)\rightarrow Y-I(f^{-1})$ is an isomorphism. Let $T=T^+-T^-$ be a current on $Y$, where $T^{\pm}$ are positive closed $(1,1)$ currents which are smooth outside a finite number of points. We assume that the following condition is satisfied: {\bf Condition 1.} For every curve $C$ in $I(f^{-1})$, then in cohomology $\{T\}.\{C\}=0$. Then, we define a natural push-forward $f_*(\varphi dd^cu\wedge f^*(T))$ for a quasi-psh function $u$ and a smooth function $\varphi$ on $Y$. We show that this pushforward satisfies a Bedford-Taylor's monotone convergence type. Assume moreover that the following two conditions are satisfied {\bf Condition 2.} The signed measure $T\wedge T\wedge T$ has no mass on $I(f^{-1})$. {\bf Condition 3.} For every curve $C$ in $I(f^{-1})$, the measure $T\wedge [C]$ has no Dirac mass. Then, we define a Monge-Ampere operator $MA(f^*(T))=f^*(T)\wedge f^*(T)\wedge f^*(T)$ for $f^*(T)$. We show that this Monge-Ampere operator satisfies several continuous properties, including a Bedford-Taylor's monotone convergence type when $T$ is positive. The measures $MA(f^*(T))$ are in general quite singular. Also, note that it may be not possible to define $f^*(T^{\pm})\wedge f^*(T^{\pm})\wedge f^*(T^{\pm})$.Comment: 13 pages. Some materials added. Typos and minor inaccuracies are corrected. The introduction and references to relevant literature will be added later, when this and arXiv:1403.5235 will be combine

A mathematical model for measurements in Quantum Mechanics

Let $V=\mathbb{C}^N$, and $H$ (an observable) a Hermitian linear operator on $V$. Let $v_1,..., v_n$ be an orthonormal basis for $V$. Let $\mathcal{M}$ be a measurement apparatus prepared to measure a state of an observed system and collapses the state to one of the $v_j$'s. Here we propose a simple model which explains the Born rule and is compatible with entanglement.Comment: 7 pages. Largely revised and extended. Main change: Add that the model can be extended to be compatible with quantum entanglemen

The local criteria for blowup of the Dullin-Gottwald-Holm equation and the two-component Dullin-Gottwald-Holm system

We investigate wave breaking for the Dullin-Gottwald-Holm equation and the two-component Dullin-Gottwald-Holm system. We establish a new blow-up criterion for the general case $\gamma+c_0\alpha^2 \geq 0$ involving local-in-space conditions on the initial data.Comment: 17 page

Evaluations of initial ideals and Castelnuovo-Mumford regularity

This paper characterizes the Castelnuovo-Mumford regularity by evaluating the initial ideal with respect to the reverse lexicographic order