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We developed a new liquid phase epitaxy (LPE) growth method for growing large GaN single crystals. In this method, the homo-epitaxial growth of the GaN crystal proceeds by nitrogen dissolution against a Na-Ga solution film coating the GaN thin-film substrate. A growth rate of 4 µm/h was achieved though nitrogen pressure was relatively low at 9.5 atm. Furthermore, the GaN film grown by this method showed a flat surface.

Growth of large and transparent GaN single crystals was carried out by applying the liquid phase epitaxy (LPE) method in a Ca-Na mixed flux system. We have previously reported LPE growth of GaN in a Na flux system, and that GaN crystals grown by LPE have extreme low dislocations and show excellent photoluminescence characteristics. In this study, use of a Ca-Na mixed flux system enabled us to grow transparent GaN crystals under low nitrogen pressure and to further improve the photoluminescence (PL) characteristic. The dislocation density of this crystal is very low (2 ×105 cm-2 in highest point).

We found the yield of GaN grown in a Li-Na flux system to be much higher than that grown in Na flux. Nitrogen dissolution seemed to increase with Li in solution, which promotes the growth of GaN crystals. In this paper, we report that the yield of GaN and habit modifications greatly depend on the composition of the Li-Na flux system and that GaN can be grown in a pure Li flux at about 50 atm.

In this chapter we recollect the preliminary materials from topology and number theory. In particular, we give a summary about fundamental groups and Galois theory for topological spaces and arithmetic rings, together with the basic concepts and examples in 3-dimensional topology and number fields. We also review class field theory as arithmetic duality theorems in Galois, etale cohomology groups.

In this chapter we discuss close parallels between Alexander-Fox theory and Iwasawa theory such as the Alexander polynomial and the Iwasawa polynomial, based on the analogy between the infinite cyclic covering of a knot exterior and the cyclotomic ℤ p -extension of a number field. In particular, we give asymptotic formulas on the orders of the 1st homology groups (p-ideal class groups) of cyclic ramified coverings (extensions).

In this chapter we review Hilbert theory which deals with, in a group-theoretic manner, the decomposition of a prime in a finite Galois extension of number fields. Based on the analogies in Chap. 3, we give a topological analogue of Hilbert theory for a finite Galois covering of 3-manifolds.

In this chapter we introduce the differential module for a group homomorphism and show the Crowell exact sequence associated to a short exact sequence of groups. Applying these constructions to the Abelianization map of a link group, we obtain the Alexander module of a link and the exact sequence relating the Alexander module with the link module. We then discuss analogous constructions for pro-finite (pro-l) groups to obtain the complete differential module and the complete Crowell exact sequence. Applying these constructions to a homomorphism from a Galois group with restricted ramification, we obtain the complete Alexander module for primes and the exact sequence relationg the complete Alexander module with a Galois (Iwasawa) module.

In this chapter we discuss the analogy between Galois groups with restricted ramification and link groups. In particular, we shall see the close analogy between Milnor’s theorem on the structure of a link group and a theorem by H. Koch on the structure of a pro-l Galois group over the rational number field with restricted ramification.

In this chapter we take up the Milnor invariants, higher order linking numbers, of a link in the 3-sphere, together with the Fox free differential calculus as a basic tool. By the analogy between a link group and a Galois group with restricted ramification in Chap. 7, we introduce arithmetic analogues of the Milnor invariants for prime numbers by using the pro-l Fox differential calculus, and show that they may be regarded as multiple generalization of the power residue symbol and the Redei triple symbol.

In this final chapter, as an extension of the analogies in Chap. 13 to 2-dimensional representations, we discuss intriguing analogies between the family of holonomy representations associated to deformations of hyperbolic structures (due to W. Thurston) and the family of Galois representations associated to deformations of p-adic ordinary modular forms (due to H. Hida and B. Mazur).

In this chapter, in view of the analogy between a knot group and a prime group, we discuss some analogies between the moduli spaces of 1-dimensional representations of knot and prime groups, and the associated higher Alexander and Iwasawa invariants.

In this chapter we first study the 2-part of the 1st homology group of a double covering of the 3-sphere ramified over a link, introducing the higher order linking matrices which are defined by using the Milnor numbers of the link in Chap. 8. Imitating the method for a link, we study the 2-part of the narrow ideal class group of a quadratic extension of the rationals, using the arithmetic Milnor numbers introduced in Chap. 8. Our theorem may be regarded as a higher order generalization of Gauss’ and Redei’s theorems on the 2-rank and 4-rank of the ideal class group.

In this chapter we explain the basic analogies between knots and primes, 3-manifolds and number rings and present a dictionary of these analogies, which will be fundamental and used throughout subsequent chapters.

In this chapter we discuss the close analogy between the linking number and the Legendre symbol, based on the analogies between knots and primes in Chap. 3. An analogy between Gauss’ linking integral and Gauss sum is also pointed out.

In this chapter we review Gauss’ genus theory from the link-theoretic point of view. We shall see that the notion of genera is defined by using the idea analogous to the linking number. We also present, vice versa, a topological analogue of genus theory.

Following the analogies between knots and primes, we introduce arithmetic analogues of higher linking matrices for prime numbers by using the arithmetic Milnor numbers. As an application, we describe the Galois module structure of the l-class group of a cyclic extension of ℚ of degree l (l being a prime number) in terms of the arithmetic higher linking matrices. In particular, our formula generalizes the classical formula of Redei on the 4 and 8 ranks of the 2-class group of a quadratic field.

The ${N}^{*}(1440)\ensuremath{\rightarrow}N\ensuremath{\pi}\ensuremath{\pi}$ decay is studied by making use of the chiral reduction formula. This formula suggests a scalar-isoscalar pion-baryon contact interaction that is absent in the recent study of Hern\'andez et al. [Phys. Rev. C 66, 065201 (2002)]. The contact interaction is introduced into their model and is found to be necessary for the simultaneous description of ${g}_{\mathit{RN}\ensuremath{\pi}\ensuremath{\pi}}$ and the $\ensuremath{\pi}\ensuremath{\pi}$ and $\ensuremath{\pi}N$ invariant mass distributions.

The purpose of this paper is to investigate analogues for 3-manifolds after the model of capitulation theorems for number fields. This will give one of the foundational analogies between 3-dimensional topology and algebraic number theory.

Following the conceptual analogies between knots and primes, 3-manifolds and number fields, we discuss an analogue in knot theory after the model of the arithmetical theory of genera initiated by Gauss. We present an analog for cyclic coverings of links following along the line of Iyanaga-Tamagawa's genus theory for cyclic extentions over the rational number field. We also give examples of $\mathbf{Z} / 2\mathbf{Z} \times \mathbf{Z} / 2\mathbf{Z}$-coverings of links for which the principal genus theorem does not hold.

This is a résumé of the author's recent work on certain analogies between primes and links. The purpose of this article is to introduce a new invariant, called Milnor invariant, in algebraic number theory, based on an analogy between the structure of a certain Galois group over the rational number field and that of the group of a link in three dimensional Euclidean space. It then turns out that the Legendre, Rédei symbols are interpreted as our link invariants. We expect that this is a tip of an arithmetical theory after the model of link theory which may give a new insight in algebraic number theory. The details will be published elsewhere.

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The structural analysis for the cobalt oxyhydroxide has been done by using high-energy synchrotron X-ray diffraction (XRD) and X-ray absorption fine structure (XAFS) analysis. The relationship between the structure and the electrical conductivity for cobalt oxyhydroxide was investigated. The structural refinement for cobalt oxyhydroxide heat-treated in the temperature range of 80-160°C has been done successfully on the basis of two phase models (L and S phase) with large and small c lattice constants. With increasing treatment temperature, the phase abundance for the L phases was increased, whereas the one for the S phases was decreased. By heat-treatment above 100°C, the cobalt ions for the cobalt oxyhydroxide were oxidized to the higher oxidation state over 3. The electrical resistivity was extremely decreased by treatment temperatures above 100°C. The increase in the electrical conductivity for the cobalt oxyhydroxide could be explained by the increase in cobalt oxidation state.

As an interpretation and a generalization of Gauss’ genus theory on binary quadratic forms in the language of arithmetic of algebraic tori, Ono [02] established an equality between a kind of “Euler number E(K/k)” for a finite Galois extension K/k of algebraic number fields and other arithmetical invariants associated to K/k. His proof depended on his Tamagawa number formula [01] and Shyr’s formula [Sh] which follows from the analytic class number formula of a torus. Later, two direct proofs were given by Katayama [K] and Sasaki [Sa].

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We grew a 12×5×0.8 mm GaN single crystal using the liquid phase epitaxy (LPE) method with Na flux. This GaN single crystal was grown on a 3 µm-thick GaN thin film synthesized on sapphire using the metal organic chemical vapor deposition (MOCVD) method. It grew in a polyhedral form larger than the MOCVD-GaN substrate. Results indicate that a MOCVD-GaN thin film on a sapphire substrate functions as a seed crystal in Na flux. The use of mixed nitrogen gas containing 40% ammonia instead of pure N2 gas also enabled the growth of a 10 µm-thick GaN homo-epitaxial film on an MOCVD-GaN film under 5 atm, the lowest reported pressure for growing GaN in Na flux. In this paper, we describe the liquid phase epitaxy (LPE) technique for growing bulk GaN single crystals, as well as the results of photoluminescence (PL) measurements. We also compare the PL intensity of the bulk GaN obtained in this study and the MOCVD-GaN. PL measurements revealed that the peak intensity of GaN single crystal grown by LPE indicates 40 times larger than MOCVD-GaN film. Also, dislocation density of bulk GaN crystals could be drastically reduced by the LPE growth technique.