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We introduce mod $l$ Milnor invariants of a Galois element associated to Ihara's Galois representation on the pro-$l$ fundamental group of a punctured projective line ($l$ being a prime number), as arithmetic analogues of Milnor invariants of a pure braid. We then show that triple quadratic (resp. cubic) residue symbols of primes in the rational (resp. Eisenstein) number field are expressed by mod $2$ (resp. mod $3$) triple Milnor invariants of Frobenius elements. For this, we introduce dilogarithmic mod $l$ Heisenberg ramified covering ${\cal D}^{(l)}$ of $\mathbb{P}^1$, which may be regarded as a higher analog of the dilogarithmic function, for the gerbe associated to the mod $l$ Heisenberg group, and we study the monodromy transformations of certain functions on ${\cal D}^{(l)}$ along the pro-$l$ longitudes of Frobenius elements for $l=2,3$., Comment: 33 pages, 1 figure

We study the twisted knot module for the universal deformation of an SL 2 \textrm {SL}_2 -representation of a knot group and introduce an associated L L -function, which may be seen as an analogue of the algebraic p p -adic L L -function associated to the Selmer module for the universal deformation of a Galois representation. We then investigate two problems proposed by Mazur: Firstly we show the torsion property of the twisted knot module over the universal deformation ring under certain conditions. Secondly we compute the L L -function by some concrete examples for 2 2 -bridge knots.

Lithium ion batteries for electric vehicle (EV) applications require high energy density and high safety. For this reason, layered oxides such as LiNiCoMnO2 (NCM) and LiNiCoAlO2 (NCA) are selected as positive electrode materials because of their large charge / discharge capacity. In recent years, capacity and energy density have been increasing due to an increase in the Ni content. However, an increase in the Ni content tends to cause heat generation and oxygen release from the crystal structure during charge / discharge, which leads to a decrease in battery safety such as the battery being easily ignited. On the other hand, olivine-type cathode materials such as LiFePO4 (LFP) and LiFeMnPO4 (LFMP) are widely known as highly safe cathode materials that generate less heat and do not cause oxygen release from the crystal structure during charge / discharge. However, these olivine-type materials have smaller charge / discharge capacities and powder densities than layered oxides, and thus have low energy densities and are therefore difficult to apply to batteries for electric vehicles. In this study, we investigated the effects of mixing layered oxides and olivine-type materials to compensate for the advantages and disadvantages of layered oxides and olivine-type materials, and to achieve cathode materials with both high energy density and high safety. As the cathode electrode material, NCM532 was used as a layered oxide, and LFP or LFMP produced by a hydrothermal synthesis method were used as an olivine type material. These materials were mixed at a predetermined weight ratio when preparing the electrode paste. The prepared paste was applied to an Al foil and dried to obtain a mixed electrode of NCM / olivine type material. The mixed state of NCM / olivine type material in the electrode was observed by SEM. As a result, a high dispersion state was obtained regardless of the mixing ratio of both materials. For charge / discharge tests, 50 mAh Laminate cells were prepared using the NCM / olivine type material electrode for cathode and natural graphite for anode. As a result, NCM / LFP mixed electrode showed a slight decrease in energy density due to an increase in the capacity of the 3.1 V region derived from the change in Fe valence as the amount of LFP mixed increased. On the other hand, in the NCM / LFMP mixed electrode, the capacity in the 3.9 V region derived from the change in the Mn valence in LFMP increased as the amount of LFMP mixed increased. Since the Fe content of LFMP was small, the capacity in the 3.1 V region was hardly observed due to the increase in the amount of LFMP. For a safety test, a 2 Ah laminate cells were prepared and an overcharge test was performed. Overcharging was performed up to a voltage of 30 V by applying a current of 20 A. As a result of evaluating the cell using the NCM single electrode, the surface temperature rose to 195.2 oC and the cell was flamed. On the other hand, in the cell using the NCM / LFP electrode, the surface temperature during overcharging was about 50 oC, and although the cell swelled, no leakage or flame occurred. In the case of the cell using the NCM / LFMP electrode, the cell surface temperature increased to 155.4 oC and leakage occured at a weight ratio of 80/20. At the weight ratio of 70/30 or less, the cell surface temperature decreased to around 50 oC, and the cells only swelled and did not flame. In conclusion, we were able to obtain a positive electrode material that has both high safety and high energy density by mixing NCM with an olivine type positive electrode material. In the future, we will consider mixing olivine-type cathode materials with higher energy density oxides such as NCA. Figure 1

The Li pre-doping free cells using tin-phosphate glass (GSPO)-silicon composite anodes and lithium-rich layered oxide (LR-NMC) cathode were developed, in which the irreversible capacity of the cathode was utilized in order to compensate that of the anode. The irreversible capacity was equalized to that of the cathode at a Si/(GSPO + Si) mass ratio, r , of 0.8. The cell using the composite anode of r = 0.8 displayed a discharge capacity of 187 mAh g −1 as the cathode capacity at 30 °C and the 0.1C-rate. The long cycle performance of over 100 charge–discharge cycles was achieved. The cells also showed an excellent high- and low-temperature performance and a good rate performance. A stable performance was observed for over 50 cycles at 60 °C and the 0.5C-rate, and a high capacity of ca. 100 mAh g −1 as the cathode capacity even at −20 °C and the 0.2C-rate. The cell could discharge even at the 10C-rate. Furthermore, the Li pre-doping free cell showed an outstanding safety performance during the nail penetration test. When the 1500 mAh cell using the composite anode was penetrated by a nail, no thermal runaway was observed during the test.

Based on the analogies between knot theory and number theory, we study a deformation theory for SL_2-representations of knot groups, following after Mazur's deformation theory of Galois representations. Firstly, by employing the pseudo-SL_2-representations, we prove the existence of the universal deformation of a given SL_2-representation of a finitely generated group Pi over a field whose characteristic is not 2. We then show its connection with the character scheme for SL_2-representations of Pi when k is an algebraically closed field. We investigate examples concerning Riley representations of 2-bridge knot groups and give explicit forms of the universal deformations. Finally we discuss the universal deformation of the holonomy representation of a hyperbolic knot group in connection with Thurston's theory on deformations of hyperbolic structures., Comment: 25 pages, to appear in Tohoku Math. J; corrected typos

Olivine lithium manganese phosphate (LiMnPO 4 ) becomes research focus because of its high energy density and improved thermal stability. However, its application in lithium ion batteries suffers severely from poor electrochemical activity due to low conductivity and structural instability upon the charge and discharge process. By applying a high-energy ball-milling method we succeed in improving the capacity delivery and rate capability. LiMnPO 4 materials ball-milled without or with acetylene black are able to deliver a high capacity of 135 and 127 mAh g −1 , respectively, more than 50% greater than the pristine one. Particularly, the latter also shows an improved discharge plateau and stable cyclability. High-energy synchrotron radiation X-ray diffraction (XRD), scanning electron microscopy (SEM), Raman spectroscopy, laser particle analysis, and galvanostatic charge and discharge are employed to understand the effect of ball-milling on the LiMnPO 4 material.

Lithium vanadium metasilicate (LiVSi 2 O 6 ) with pyroxene structure has been exploited as an electrode material for Li-ion batteries. Galvanostatic charge and discharge tests show that LiVSi 2 O 6 is able to deliver a capacity of 85 mAh g −1 at 30 °C, and a high capacity of 181 mAh g −1 at 60 °C. The high capacity is believed to be due to the reactions of V 3+ /V 4+ and V 2+ /V 3+ redox couples, accompanied by the excess 0.42 Li + insertion into the lattice forming a Li-rich phase Li 1.42 VSi 2 O 6 . High-energy synchrotron XRD combined with the Rietveld refinement analysis confirms that the electrochemical delithiation–lithiation reaction proceeds by a single phase redox mechanism with an overall volume variation of 1.9% between LiVSi 2 O 6 and its delithiated state, indicating a very stable framework of LiVSi 2 O 6 for Li + ions extraction–insertion.

Well-crystallized LiFePO 4 nanoparticles have been directly synthesized in a short time via hydrothermal process in the presence of organic acid, e.g. citric acid or ascorbic acid. These acid-mediated LiFePO 4 products exhibit a phase-pure and nanocrystal nature with size about 50–100 nm. Two critical roles that the organic acid mediator plays in hydrothermal process are recognized and a rational mechanism is explored. After a post carbon-coating treatment at 600 °C for 1 h, these mediated LiFePO 4 materials show a high electrochemical activity in terms of reversible capacity, cycling stability and rate capability. Particularly, LiFePO 4 mediated by ascorbic acid can deliver a capacity of 162 mAh g −1 at 0.1 C, 154 mAh g −1 at 1 C, and 122 mAh g −1 at 5 C. The crystalline structure, particle morphology, and surface microstructure were characterized by high-energy synchrotron X-ray diffraction (XRD), transmission electron microscopy (TEM) and scanning electron microscopy (SEM), and Raman spectroscopy, respectively. And the electrochemical properties were thoroughly investigated by galvanostatic test and electrochemical impedance spectroscopy (EIS).

The detailed structural change in the charge–discharge process was investigated for the 20 mol% aluminum-substituted α-nickel hydroxide by using high-energy synchrotron X-ray diffraction (XRD), X-ray absorption fine structure (XAFS) and transmission electron microscopy (TEM) analysis. The 20 mol% aluminum-substituted α-nickel hydroxide showed the α-Ni(OH) 2 /γ-NiOOH phase transformation in the charge–discharge process. The structural refinement has been done successfully on the basis of two phase models of the ideal phases and the fault ones, including the occupation sites for the potassium ions and H 2 O (OH − ) molecules. The substituted aluminum ions were situated on both nickel sites and 18h sites in the interlayer. The α-Ni(OH) 2 structure would be stabilized by the presence of the aluminum ions on the 18h sites. The α-Ni(OH) 2 phases were transformed to the β-Ni(OH) 2 phases after 50 cycles because the aluminum ions on the 18h sites would be migrated from the bulk to the surface.

We introduce a multiple generalization of the tame symbol, called {\em polysymbols}, associated to meromorphic functions on a Riemann surface as the Massey products in Deligne cohomology, and also give a geometric construction of polysymbols using Chen's iterated integrals. We then deduce some basic properties of polysymbols using our holonomy formula, and show that trivializations of polysymbols give variations of mixed Hodge structure.

The detailed phase transformation in the charge-discharge process for the nickel hydroxide electrode have been examined by using the high-energy synchrotron X-ray absorption fine structure (XAFS) and X-ray diffraction (XRD). The structural refinement for the β-Ni(OH)2 and the β-NiOOH have been done successfully on the basis of two phase models of the ideal and fault phases. At 0% of SOC, it was found that the sample consisted of 80% of the ideal β-Ni(OH)2 phases and 20% of the fault ones. At 100% of SOC, the ideal and fault β-Ni(OH)2 phases were transformed to the ideal and fault β-NiOOH phases, respectively. It was found that the fault phases contained a larger amount of intercalated potassium ions and H2O (OH−) than the ideal ones. At 150% of SOC, the only ideal β-NiOOH phases was transformed to the ideal γ-NiOOH phases, accompanied by further intercalation of the potassium ions and H2O (OH−) into the interlayer

Ihara initiated to study a certain Galois representation which may be seen as an arithmetic analogue of the Artin representation of a pure braid group. We pursue the analogies in Ihara theory further, following after some issues and their inter-relations in the theory of braids and links such as Milnor invariants, Johnson homomorphisms, Magnus-Gassner cocycles and Alexander invariants, and study relations with arithmetic in Ihara theory., Comment: 66pages, to appear in Publ. Res. Inst. Math. Sci., corrected typos

We observed the memory effect, the working voltage lowering, which occurred due to the repeated partial charge-discharge cycling of a Ni-capacity-limited alkaline cell, for example 50–70% state of charge, and γ-NiOOH in the Ni electrode was identified as the cause of the memory effect. During this condition, γ-NiOOH is proposed to be formed by partial overcharging due to the heterogeneity of the conductive materials such as cobalt compounds in the Ni electrode. Using nano-sized conductive materials homogeneously dispersed in the electrode leads to a significant effect on the suppression of the memory effect occurring due to the repeated partial charge-discharge cycling.

Following the analogy between primes and knots, we give an arithmetic analogue of a theorem by E. Hironaka on the Alexander stratification in the character variety of a knot group.

In the growth of GaN single crystal using the Na flux method, we succeeded in clarifying the role of Na in promoting nitrogen dissolution in the Ga-Na melt system above 900 K. At the gas-liquid interface of high temperature Ga-Na melt, Na functions to ionize the nitrogen gas. The ionization of the nitrogen gas results in drastic increase of nitrogen dissolution in the melt. In consequence, these synthesize GaN single crystals easily. On the other hand, the addition of a minor amount of Ca or Li to the Ga-Na melt system also increases the nitrogen dissolution. However, the additives function to maintain the nitrogen dissolved in the Ga-Na melt, which results in drastic increase in the nitrogen concentration. In the present study, we report the solubility of GaN in the Ga-Na system and the threshold pressure of nitrogen gas to grow GaN. On the basis of these data, we propose a growth mechanism of GaN and the role of additives to maintain nitrogen.

Following the analogy between primes and knots, we introduce the refined Milnor invariants for prime numbers and establish their connection with certain Massey products in Galois cohomology. This generalizes the well-known relation between the power residue symbol and cup product and gives a cohomological interpretation of L. Redei's triple symbol.

We introduce mod 3 triple Milnor invariants and triple cubic residue symbols for certain primes of the Eisenstein number field $\mathbb{Q}(\sqrt{-3})$, following the analogies between knots and primes. Our triple symbol generalizes both the cubic residue symbol and R\'{e}dei's triple symbol, and describes the decomposition law of a prime in a mod 3 Heisenberg extension of degree 27 over $\mathbb{Q}(\sqrt{-3})$ with restricted ramification, which we construct concretely in the form similar to R\'{e}dei's dihedral extension over $\mathbb{Q}$. We also give a cohomological interpretation of our symbols by triple Massey products in Galois cohomology., Comment: 42 pages. The previous version entitled "Arithmetic Milnor invariants and multiple power residue symbols in number fields" by F. Amano and M. Morishita contained some mistakes concerning Theorem 2.8. This is a new version for the Eisenstein number field, which revises largely the previous one, with the collaboration of Y. Mizusawa. To appear in Research in Number Theory

In 1939, L. Rédei introduced a certain triple symbol in order to generalize the Legendre symbol and Gauss' genus theory. Rédei's triple symbol $[a_1,a_2, p]$ describes the decomposition law of a prime number $p$ in a certain dihedral extension over $\mathbb{Q}$ of degree 8 determined by $a_1$ and $a_2$. In this paper, we show that the triple symbol $[-p_1,p_2, p_3]$ for certain prime numbers $p_1, p_2$ and $p_3$ can be expressed as a Fourier coefficient of a modular form of weight one. For this, we employ Hecke's theory on theta series associated to binary quadratic forms and realize an explicit version of the theorem by Weil-Langlands and Deligne-Serre for Rédei's dihedral extensions. A reciprocity law for the Rédei triple symbols yields certain reciprocal relations among Fourier coefficients.

Let G be a connected linear algebraic group and X an algebraic variety, both defined over Q, the field of rational numbers. Suppose that G acts on X transitively and the action is defined over Q. Suppose that the set of rational points X(Q) is non-empty. Choosing x ∈ X(Q) allows us to identify G/Gx and X as varieties over Q, there Gx is the stabilizer of x.

We introduce a nilpotent Hurewicz homomorphism for a topological manifold, following the Lie algebra method in nilpotent class field theory for local and number fields by H. Koch et al. Using Labute-Anick’s results on the determination of the Lie algebra attached to the lower central series of a group, we present explicitly nilpotent class field theory for some two and three dimensional manifolds.

We propose an approach to study non-Abelian Iwasawa theory, using the idea of Johnson homomorphisms in low dimensional topology. We introduce arithmetic analogues of Johnson homomorphisms/maps, called the p-Johnson homomorphisms/maps, associated to the Zassenhaus filtration of a pro-p Galois group over a Z_p-extension of a number field. We give their cohomological interpretation in terms of Massey products in Galois cohomology., Comment: 38 pages, to appear in J. of Algebra, changed title from "p-Johnson homomorphisms in non-Abelian Iwasawa theory", added references in section 5, corrected typos

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.

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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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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