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

The inverse pp-maxian problem on trees with variable edge lengths

We concern the problem of modifying the edge lengths of a tree in minimum total cost so that the prespecified $p$ vertices become the $p$-maxian with respect to the new edge lengths. This problem is called the inverse $p$-maxian problem on trees. \textbf{Gassner} proposed efficient combinatorial alogrithm to solve the the inverse 1-maxian problem on trees in 2008. For the problem with $p \geq 2$, we claim that the problem can be reduced to finitely many inverse $2$-maxian problem. We then develop algorithms to solve the inverse $2$-maxian problem for various objective functions. The problem under $l_1$-norm can be formulated as a linear program and thus can be solved in polynomial time. Particularly, if the underlying tree is a star, then the problem can be solved in linear time. We also devised $O(n\log n)$ algorithms to solve the problems under Chebyshev norm and bottleneck Hamming distance, where $n$ is the number of vertices of the tree. Finally, the problem under weighted sum Hamming distance is $NP$-hard.Comment: 9 page

Some inequalities for the matrix Heron mean

Let $A, B$ be positive definite matrices, $p=1, 2$ and $r\ge 0$. It is shown that \begin{equation*} ||A+ B + r(A\sharp_t B+A\sharp_{1-t} B)||_p \le ||A+ B + r(A^{t}B^{1-t} + A^{1-t}B^t)||_p. \end{equation*} We also prove that for positive definite matrices $A$ and $B$ \begin{equation*}\label{det} \Dt (P_{t}(A, B)) \le \Dt (Q_{t}(A, B)), \end{equation*} where $Q_t(A, B)= \big(\frac{A^t+B^t}{2}\big)^{1/t}$ and $P_t(A, B)$ is the $t$-power mean of $A$ and $B$. As a consequence, we obtain the determinant inequality for the matrix Heron mean: for any positive definite matrices $A$ and $B,$ \Dt(A+ B + 2(A\sharp B)) \le \Dt(A+ B + A^{1/2}B^{1/2} + A^{1/2}B^{1/2})). These results complement those obtained by Bhatia, Lim and Yamazaki (LAA, {\bf 501} (2016) 112-122).Comment: Any comments are welcom

Simple t-designs: A recursive construction for arbitrary t

The aim of this paper is to present a recursive construction of simple t-designs for arbitrary t. The construction is of purely combinatorial nature and it requires finding solutions for the indices of the ingredient designs that satisfy a certain set of equalities. We give a small number of examples to illustrate the construction, whereby we have found a large number of new t-designs, which were previously unknown. This indicates that the method is useful and powerful.Comment: 12 page

Strong submeasures and several applications

A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By Hahn-Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. We give several applications of strong submeasures in various diverse topics, thus illustrate the usefulness of this classical but largely overlooked notion. The applications include: - Pullback and pushforward of all measures by meromorphic selfmaps of compact complex varieties. - The existence of invariant positive strong submeasures for meromorphic maps between compact complex varieties, a notion of entropy for such submeasures (which coincide with the classical ones in good cases) and a version of the Variation Principle. - Intersection of every positive closed (1,1) currents on compact K\"ahler manifolds. Explicit calculations are given for self-intersection of the current of integration of some curves $C$ in a compact K\"ahler surface where the self-intersection in cohomology is negative. All of these points are new and have not been previously given in work by other authors. In addition, we will apply the same ideas to entropy of transcendental maps of $\mathbb{C}$ and $\mathbb{C}^2$.Comment: 43 pages. Exposition improved, typos fixed, many new results adde