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The Delta-Notch system plays a vital role in a number of areas in biology and typically forms a salt and pepper pattern in which cells strongly expressing Delta and cells strongly expressing Notch are alternately aligned via lateral inhibition. Although the spatial arrangement of the cells is important to the Delta-Notch pattern, the effect of cell rearrangement is not often considered. In this study, we provide a framework to analytically evaluate the effect of cell mixing and proliferation on Delta-Notch pattern formation in one spatial dimension. We model cell rearrangement events by a Poisson process and analyze the model while preserving the discrete properties of the spatial structure. We find that the homogeneous expression pattern is stabilized if the frequency of cell rearrangement events is sufficiently large. We analytically obtain the critical frequencies of the cell rearrangement events where the decrease of the pattern amplitude as a result of cell rearrangement is balanced by the increase in amplitude due to the Delta-Notch interaction dynamics. Our theoretical results are qualitatively consistent with experimental results, supporting the notion that the heterogeneity of expression patterns is inversely correlated with cell rearrangement \textit{in vivo}. Our framework, while applied here to the specific case of the Delta-Notch system, is applicable more widely to other pattern formation mechanisms., Comment: 35 pages, 5 figures and Supplementally material (12 pages, 6 figures)

For an odd prime number $p$, we study the number of generators of the unramified Iwasawa modules of the maximal multiple $\mathbb{Z}_p$-extensions over Iwasawa algebra. In a previous paper of the authors, under several assumptions for an imaginary quadratic field, we obtain a necessary and sufficient condition for the Iwasawa module to be cyclic as a module over the Iwasawa algebla. Our main result is to give methods for computation and numerical examples about the results. We remark that our results do not need the assumption that Greenberg's generalized conjecture holds.

A new method to derive an essential integral kernel from any given reaction-diffusion network is proposed. Any network describing metabolites or signals with arbitrary many factors can be reduced to a single or a simpler system of integro-differential equations called "effective equation" including the reduced integral kernel (called "effective kernel" ) in the convolution type. As one typical example, the Mexican hat shaped kernel is theoretically derived from two component activator-inhibitor systems. It is also shown that a three component system with quite different appearance from activator-inhibitor systems is reduced to an effective equation with the Mexican hat shaped kernel. It means that the two different systems have essentially the same effective equations and that they exhibit essentially the same spatial and temporal patterns. Thus, we can identify two different systems with the understanding in unified concept through the reduced effective kernels. Other two applications of this method are also given: Applications to pigment patterns on skins (two factors network with long range interaction) and waves of differentiation (called proneural waves) in visual systems on brains (four factors network with long range interaction). In the applications, we observe the reproduction of the same spatial and temporal patterns as those appearing in pre-existing models through the numerical simulations of the effective equations.

We found evidence of dynamic scaling in the spreading of MDCK monolayer, which can be characterized by the Hurst exponent ${\alpha} = 0.86$ and the growth exponent ${\beta} = 0.73$, and theoretically and experimentally clarified the mechanism that governs the contour shape dynamics. During the spreading of the monolayer, it is known that so-called "leader cells" generate the driving force and lead the other cells. Our time-lapse observations of cell behavior showed that these leader cells appeared at the early stage of the spreading, and formed the monolayer protrusion. Informed by these observations, we developed a simple mathematical model that included differences in cell motility, cell-cell adhesion, and random cell movement. The model reproduced the quantitative characteristics obtained from the experiment, such as the spreading speed, the distribution of the increment, and the dynamic scaling law. Analysis of the model equation revealed that the model could reproduce the different scaling law from ${\alpha} = 0.5, {\beta} = 0.25$ to ${\alpha} = 0.9, {\beta} = 0.75$, and the exponents ${\alpha}, {\beta}$ were determined by the two indices: $\rho t$ and $c$. Based on the analytical result, parameter estimation from the experimental results was achieved. The monolayer on the collagen-coated dishes showed a different scaling law ${\alpha} = 0.74, {\beta} = 0.68$, suggesting that cell motility increased by 9 folds. This result was consistent with the assay of the single-cell motility. Our study demonstrated that the dynamics of the contour of the monolayer were explained by the simple model, and proposed a new mechanism that exhibits the dynamic scaling property., Comment: 11 pages, 6 figures, and supplemental materials

For a CM-field $K$ and an odd prime number $p$, let $\widetilde K'$ be a certain multiple $\mathbb{Z}_p$-extension of $K$. In this paper, we study several basic properties of the unramified Iwasawa module $X_{\widetilde K'}$ of $\widetilde K'$ as a $\mathbb{Z}_p[[{\rm Gal}(\widetilde K'/K)]]$-module. Our first main result is a description of the order of a Galois coinvariant of $X_{\widetilde K'}$ in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic $\mathbb{Z}_p$-extension of $K$ under a certain assumption on the splitting of primes above $p$. Second one is that if $K$ is an imaginary quadratic field and $p$ does not split in $K$, we give a necessary and sufficient condition for which $X_{\widetilde K}$ is $\mathbb{Z}_p[[{\rm Gal}(\widetilde K/K)]]$-cyclic under several assumptions on the Iwasawa $\lambda$-invariant and the ideal class group of $K$, where $\widetilde K$ is the $\mathbb{Z}_p^2$-extension of $K$., Comment: v2: 24pages. replace Lemma 3.1 and the proof, typos corrected

We propose a new mechanism to produce a fermion mass hierarchy dynamically in a model with a singlet generation of fermions. A five dimensional gauge theory on an interval with point interactions (zero-width branes) takes responsibility for realizing three generations and each massless zero mode localizes at boundaries of the segments on the extra dimension. An extra-dimension coordinate-dependent vacuum expectation value of a scalar field makes large differences in overlap integrals of the localized zero modes and then an exponential fermion mass hierarchy can appear. The positions of the point interactions control the magnitude of the fermion mass hierarchy and are determined by the minimization condition of the Casimir energy. As a result of the minimization of the Casimir energy, an exponential mass hierarchy appears dynamically. We also discuss the stability of the extra dimension., Comment: 33 pages, 11figures

We classify the combinations of parameters which lead three generations of quarks and leptons in the framework of magnetized twisted orbifolds on $T^2/Z_2$, $T^2/Z_3$, $T^2/Z_4$ and $T^2/Z_6$ with allowing nonzero discretized Wilson line phases and Scherk-Schwarz phases. We also analyze two actual examples with nonzero phases leading to one-pair Higgs and five-pair Higgses and discuss the difference from the results without nonzero phases studied previously., Comment: 28 pages (main body and references) + 65 pages (full list of classification), 22 tables (v1); typos corrected, problem in sentence fixed (v2)

We study chromo-natural inflation in the axiverse. More precisely, we investigate natural inflation with two axions coupled with a SU(2) gauge field. Assuming a hierarchy between the coupling constants, we find that for certain initial conditions, conventional natural inflation commences and continues for tens of e-foldings, and subsequently chromo-natural inflation takes over from natural inflation. For these solutions, we expect that the predictions are in agreement with observations on CMB scales. Moreover, since chromo-natural inflation occurs in the latter part of the inflationary stage, chiral primordial gravitational waves are produced in the interesting frequency range higher than $10^{-10}$Hz, which might be detectable by future gravitational wave observations., Comment: 16 pages, 3 figures

We discuss an effective way for analyzing the system on the magnetized twisted orbifolds in operator formalism, especially in the complicated cases $T^{2}/Z_{3}$, $T^{2}/Z_{4}$ and $T^{2}/Z_{6}$. We can obtain the exact and analytical results which can be applicable for any larger values of the quantized magnetic flux M, and show that the (non-diagonalized) kinetic terms are generated via our formalism and the number of the surviving physical states are calculable in a rigorous manner by simply following usual procedures in linear algebra in any case. Our approach is very powerful when we try to examine properties of the physical states on (complicated) magnetized orbifolds $T^{2}/Z_{3}$, $T^{2}/Z_{4}$, $T^{2}/Z_{6}$ (and would be in other cases on higher-dimensional torus) and could be an essential tool for actual realistic model construction based on these geometries., Comment: 41 pages, 1 figure

We study the relationship between the family number of chiral fermions and the Wilson line phases, based on the orbifold family unification. We find that flavor numbers are independent of the Wilson line phases relating extra-dimensional components of gauge bosons, as far as the standard model gauge symmetry is respected. This feature originates from a hidden quantum-mechanical supersymmetry., Comment: 17 pages, 1 figure

We study the dynamics of $SU(2)_L$ times $U(1)_Y$ electroweak gauge fields during and after Higgs inflation. In particular, we investigate configurations of the gauge fields during inflation and find the gauge fields remain topologically non-trivial. We also find that the gauge fields grow due to parametric resonances caused by oscillations of a Higgs field after inflation. We show that the Chern-Simons number also grows significantly. Interestingly, the parametric amplification gives rise to sizable magnetic fields after the inflation whose final amplitudes depend on the anisotropy survived during inflation., Comment: 16 pages, 8 figures

We address a question whether the recently observed Higgs mass $M_{H} = 126$ GeV, of the order of the weak scale $M_{W}$, is calculable as a finite value in the scnenario of gauge-Higgs unification. In the scenario formulated on a flat 5-dimensional space-time, the Higgs mass is calculable, being protected under the quantum correction by gauge invariance, though the predicted Higgs mass is generally too small compared with $M_{W}$. In the 6-dimensional SU(3) model, however, a suitable orbifolding is known to lead to a mass of the order of $M_{W}$: $M_{H} = 2M_{W}$ at the tree level, which has some similarity to the corresponding prediction by the MSSM, $M_{H}$ leq (cos beta) $M_{Z}$. We demonstrate first by a general argument and secondly by explicit calculations that, even though the quantum correction to the quartic self-coupling of the Higgs field is UV-divergent, its deviation from that of $g^{2}$ is calculable, and therefore two observables, $M_{H}^{2}$ and Delta equiv $(M_{H}/2M_{W})^{2}-1$, are both calculable in the gauge-Higgs unification scenario. The implication of the precise value 126 GeV to the compactification scale and the bulk mass of the matter field in our model is also discussed., Comment: 24 pages, 2 figures, v2: Calculations of corrections to gauge kinetic terms added, v3: final version to be published

We propose new backgrounds of extra dimensions to lead to four-dimensional chiral models with three generations of matter fermions, that is $T^2/Z_N$ twisted orbifolds with magnetic fluxes. We consider gauge theory on six-dimensional space-time, which contains the $T^2/Z_N$ orbifold with magnetic flux, Scherk-Schwarz phases and Wilson line phases. We classify all the possible Scherk-Schwarz and Wilson line phases on $T^2/Z_N$ orbifolds with magnetic fluxes. The behavior of zero modes is studied. We derive the number of zero modes for each eigenvalue of the $Z_N$ twist, showing explicitly examples of wave functions. We also investigate Kaluza-Klein mode functions and mass spectra., Comment: 34 pages, 4 figures

We study the possibility of family unification on the basis of SU(N) gauge theory on the 6-dimensional space-time, $M^4\times T^2/Z_N$. We obtain enormous numbers of models with three families of SU(5) matter multiplets and those with three families of the standard model multiplets, from a single massless Dirac fermion with a higher-dimensional representation of SU(N), through the orbifold breaking mechanism., Comment: 26 pages

We propose a mechanism to obtain the generation of matter in the standard model. We start from the analysis of the $T^2/Z_N$ shifted orbifold with magnetic flux, which imposes a $Z_N$ symmetry on torus. We also consider several orbifolds such as $(T^2\times T^2)/Z_N$, $(T^2 \times T^2 \times T^2)/(Z_N \times Z_{N'})$ and $(T^2 \times T^2 \times T^2)/(Z_N \times Z_{N'} \times Z_{N"})$. On such orbifolds, we study the behavior of fermions in two different means, one is the operator formalism and the other is to analyze wave functions explicitly. For an interesting result, it is found that the number of zero-mode fermions is related to $N$ of the $Z_N$ symmetry. In other words, the generation of matter relates to the type of orbifolds. Moreover, we find that shifted orbifold models to realize three generations are, in general, severely restricted. For example, the three-generation model on the type of $M^4 \times (T^2 \times T^2)/Z_N$ is unique. One can also construct other types of three-generation orbifold models with rich flavor structure. Those results may bring us a realistic model with desired Yukawa structure., Comment: 32 pages, 1 figure

We classify the standard model fermions, which originate from bulk fields of the $\bf{27}$ or $\bar{\bf{27}}$ representation after orbifold breaking, in $E_6$ grand unified theories on 5 or 6-dimensional space-time, under the condition that $q$, $e^c$ and $u^c$ survive as zero modes., Comment: 24 pages, typos corrected, to appear in IJMPA

We study the generation of $\mu$ parameter from the dynamical rearrangement of local U(1) symmetry in a five-dimensional model and discuss phenomenological implications on the $\theta$ parameter, under the assumption that supersymmetry is broken by the Scherk-Schwarz mechanism., Comment: 16 pages

We propose a mechanism that the soft supersymmetry breaking masses and $\mu$ parameter can be induced from the dynamical rearrangement of local U(1) symmetries in a five-dimensional model. It offers to a solution of $\mu$ problem if there is a large hierarchy among the relevant U(1) charge of Higgsinos and that of hidden fields which stabilize the extra-dimensional component of U(1) gauge boson., Comment: 17 pages

We study the possibility of family unification on the basis of SO(2N) gauge theory on the five-dimensional space-time, $M^4\times S^1/Z_2$. Several SO(10), $SU(4) \times SU(2)_L \times SU(2)_R$ or SU(5) multiplets come from a single bulk multiplet of SO(2N) after the orbifold breaking. Other multiplets including brane fields are necessary to compose three families of quarks and leptons., Comment: 28 pages

We study equivalence classes of boundary conditions in an SU(N) gauge theory on six-dimensional space-time including two-dimensional orbifold. For five kinds of two-dimensional orbifolds $S^1/Z_2 \times S^1/Z_2$ and $T^2/Z_m$ $(m=2,3,4,6)$, orbifold conditions and those gauge transformation properties are given and the equivalence relations among boundary conditions are derived. The classification of boundary conditions related to diagonal representatives is carried out using the equivalence relations., Comment: 18 pages, 4 figures, revised version for publication in PTP

We derive sum rules among scalar masses for various boundary conditions of the hidden-visible couplings in the presence of hidden sector dynamics and show that they still can be useful probes of the MSSM and beyond., Comment: 22 pages