Abstract:
Background: Methotrexate, vinblastine, doxorubicin, and cisplatin regimen, and gemcitabine and cisplatin regimen are widely used for advanced or metastatic urothelial carcinomas (UCs). However, a standard treatment for patients who fail these firstline chemotherapies is unavailable. We examined the efficacy and safety of secondline paclitaxel, gemcitabine, and cisplatin (PCG) chemotherapy in Japanese patients. Methods: Between 2004 and 2010, 25 patients with metastatic UCs who failed to respond to platinumbased regimens were treated with PCG. They received intravenous paclitaxel (60 mg/m^{2}) and gemcitabine (1000 mg/m^{2}) on days 1 and 8, and cisplatin (70 mg/m^{2}) on day 2 of every 21 day course. We retrospectively collected patients’ clinical and pathological data and evaluated adverse effects and survivals. Results: Patients underwent 95 PCG cycles in all (average, 3.8 cycles per patient). One patient (4%) achieved complete response, 5 (20%) showed partial response, 8 (42%) had disease stabilization, and 5 (26%) had disease progression. Median overall survival was 8.5 months. Neutropenia and thrombocytopenia of grade ≥ 3 were observed in 68% and 56% of patients, respectively. No treatment related death occurred. Multivariate analysis revealed that hemoglobin levels < 10 g/dL and estimated glomerular filtration rate < 60 mL/(min1.73 m^{2}) were significant risk factors for overall survival. Conclusion: PCG chemotherapy in the secondline setting potentially contributed to good prognosis in selected patients with relatively significant but tolerable toxicity.

Abstract:
The Maximum Degree and Diameter Bounded Subgraph Problem (MaxDDBS) asks: given a host graph G, a bound on maximum degree \Delta, and a diameter D, what is the largest subgraph of the host graph with degree bounded by \Delta and diameter bounded by D? In this paper, we investigate this problem when the host graph is the k-dimensional mesh. We provide lower bounds for the size of the largest subgraph of the mesh satisfying MaxDDBS for all k and \Delta > 3 that agree with the known upper bounds up to the first two terms, and show that for \Delta = 3, the lower bounds are at least the same order of growth as the upper bounds.

Abstract:
The Sunyaev-Zel'dovich effect provides a useful probe of cosmology and structure formation in the Universe. Recent years have seen rapid progress in both quality and quantity of its measurements. In this review, we overview cosmological and astrophysical implications of recent and near future observations of the effect. They include measuring the evolution of the cosmic microwave background radiation temperature, the distance-redshift relation out to high redshifts, number counts and power spectra of galaxy clusters, distributions and dynamics of intracluster plasma, and large-scale motions of the Universe.

Abstract:
Twisted Alexander invariants of knots are well-defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the invariants coincide with sign-determined Reidemeister torsion in a normalized setting, and refine the duality theorem. We further obtain necessary conditions on the invariants for a knot to be fibered, and study behavior of the highest degrees of the invariants.

Abstract:
We study two sorts of actions on the space of conjugacy classes of irreducible $SU_2$-representations of a knot group. One of them is an involution which comes from the algebraic structure of $SU_2$ and the other is the action by the outer automorphism group of the knot group. In particular, we consider them on an 1-dimensional smooth part of the space, which is canonically oriented and metrized via a Reidemeister torsion volume form. As an application we show that the Reidemeister torsion function on the 1-dimensional subspace has symmetry about the metrization.

Abstract:
Given a circle-valued Morse function of a closed oriented manifold, we prove that Reidemeister torsion over a non-commutative formal Laurent polynomial ring equals the product of a certain non-commutative Lefschetz-type zeta function and the algebraic torsion of the Novikov complex over the ring. This paper gives a generalization of the result of Hutchings and Lee on abelian coefficients to the case of skew fields. As a consequence we obtain a Morse theoretical and dynamical description of the higher-order Reidemeister torsion.

Abstract:
The purpose of this paper is to give an explicit dimension formula for the spaces of vector valued Siegel cusp forms of degree two with respect to a certain kind of arithmetic subgroups of the non-split Q-forms of Sp(2,R). We obtain our result by using Hashimoto and Ibukiyama's results in [HI80],[HI83] and Wakatsuki's formula in [Wak]. Our result is a generalization of formulae in [Has84,Theorem 4.1] and [Wak,Theorem 6.1].

Abstract:
We study algebraic structures of certain submonoids of the monoid of homology cylinders over a surface and the homology cobordism groups, using Reidemeister torsion with non-commutative coefficients. The submonoids consist of ones whose natural inclusion maps from the boundary surfaces induce isomorphisms on higher solvable quotients of the fundamental groups. We show that for a surface whose first Betti number is positive, the homology cobordism groups are other enlargements of the mapping class group of the surface than that of ordinary homology cylinders. Furthermore we show that for a surface with boundary whose first Betti number is positive, the submonoids consisting of irreducible ones as 3-manifolds trivially acting on the solvable quotients of the surface group are not finitely generated.

Abstract:
We study an invariant of a 3-manifold which consists of Reidemeister torsion for linear representations which pass through a finite group. We show a Dehn surgery formula on this invariant and compute that of a Seifert manifold over $S^2$. As a consequence we obtain a necessary condition for a result of Dehn surgery along a knot to be Seifert fibered, which can be applied even in a case where abelian Reidemeister torsion gives no information.