A Lemma for a Strong Comparison Principle of Nonlinear Parabolic Equations

In this note, we prove a lemma for a strong comparison principle of nonlinear parabolic equations. We shall prove a function which is a viscosity subsolution minus a viscosity supersolution of the equation becomes a viscosity subsolution of a parabolic equation which may not coincide with the original equation. Thanks to a strong maximum principle of nonlinear parabolic equations we have a strong comparison principle

Reinterpretation of the Starobinsky model

The Starobinsky model of inflation, consistent with Planck 2015, has a peculiar form of the action, which contains the leading Einstein term $R$, the $R^2$ term with a huge coefficient, and negligible higher-order terms. We propose an explanation of this form based on compactification of extra dimensions. Once tuning of order $10^{-4}$ is accepted to suppress the linear term $R$, we no longer have to suppress higher-oder terms, which give nontrivial corrections to the Starobinsky model. We show our predictions of the spectral index, its runnings, and the tensor-to-scalar ratio. Finally, we discuss a possibility that quantum gravity may appear at the scale $\Lambda \gtrsim 5 \times 10^{15}$ GeV.Comment: 13 pages, 2 figures; published version, references and discussion on the uncertainties of the predictions adde

Josephson effect in a multi-orbital model for Sr2_{2}RuO4_{4}

We study Josephson current between s-wave/spin-triplet superconductor junctions by taking into account details of band structures in Sr$_{2}$RuO$_{4}$ such as three conduction bands, spin-orbit interaction in the bulk and that at the interface. We assume five superconducting order parameters in Sr$_{2}$RuO$_{4}$: a chiral p-wave symmetry and four helical p-wave symmetries. We calculate current-phase relationship $I(\varphi)$ in these junctions, where $\varphi$ is the macroscopic phase difference between two superconductors. The results for a chiral p-wave pairing symmetry show that $\cos(\varphi)$ term appears in the current-phase relation due to time-reversal symmetry (TRS) breaking. On the other hand, $\cos(\varphi)$ term is absent in the helical pairing states which preserve the TRS. We also study the dependence of maximum Josephson current $I_c$ on an external magnetic flux $\Phi$ in a corner junction. The calculated results of $I_c(\Phi)$ show a relation $I_{c}(\Phi) \neq I_{c}(-\Phi)$ in a chiral state and $I_{c}(\Phi)=I_{c}(-\Phi)$ in a helical state. We calculate $I_c(\Phi)$ in a corner and a symmetric SQUIDs geometry. In a symmetric SQUID geometry, the relation $I_{c}(\Phi)=I_{c}(-\Phi)$ is satisfied for all the pairing states and it is impossible to distinguish chiral state from helical one. On the other hand, results for a corner SQUID always show $I_{c}(\Phi) \neq I_{c}(-\Phi)$ and $I_{c}(\Phi)=I_{c}(-\Phi)$ for a chiral and a helical states, respectively. Experimental tests of these relations in a corner junctions and SQUIDs may serve as a tool for unambiguous determination of the pairing symmetry in Sr$_{2}$RuO$_{4}$

Space-Time and Matter in IIB Matrix Model - gauge symmetry and diffeomorphism -

We pursue the study of the type IIB matrix model as a constructive definition of superstring. In this paper, we justify the interpretation of space-time as distribution of eigenvalues of the matrices by showing that some low energy excitations indeed propagate in it. In particular, we show that if the distribution consists of small clusters of size $n$, low energy theory acquires local SU(n) gauge symmetry and a plaquette action for the associated gauge boson is induced, in addition to a gauge invariant kinetic term for a massless fermion in the adjoint representation of the SU(n). We finally argue a possible identification of the diffeomorphism symmetry with permutation group acting on the set of eigenvalues, and show that the general covariance is realized in the low energy effective theory even though we do not have a manifest general covariance in the IIB matrix model action.Comment: 25 page

Atiyah-Patodi-Singer index on a lattice

We propose a non-perturbative formulation of the Atiyah-Patodi-Singer(APS) index in lattice gauge theory, in which the index is given by the $\eta$ invariant of the domain-wall Dirac operator. Our definition of the index is always an integer with a finite lattice spacing. To verify this proposal, using the eigenmode set of the free domain-wall fermion, we perturbatively show in the continuum limit that the curvature term in the APS theorem appears as the contribution from the massive bulk extended modes, while the boundary $\eta$ invariant comes entirely from the massless edge-localized modes.Comment: 14 pages, appendices added, details of key equations added, typos corrected, to appear in PTE

Scaling Behaviors of Branched Polymers

We study the thermodynamic behavior of branched polymers. We first study random walks in order to clarify the thermodynamic relation between the canonical ensemble and the grand canonical ensemble. We then show that correlation functions for branched polymers are given by those for $\phi^3$ theory with a single mass insertion, not those for the $\phi^3$ theory themselves. In particular, the two-point function behaves as $1/p^4$, not as $1/p^2$, in the scaling region. This behavior is consistent with the fact that the Hausdorff dimension of the branched polymer is four.Comment: 17 pages, 3 figure

Sarcoidosis-associated hepatocellular carcinoma

Sarcoidosis is a systemic granulomatous inflammation of unknown etiology, and seems to involve the liver parenchyma in most cases. However, sarcoidosis-associated hepatocellular carcinoma is rare. We report here a case in which a hepatocellular carcinoma occurred within the liver, which was probably involved as a result of systemic sarcoidosis. A 57-year-old Japanese man had been followed up for 2 years because of diabetic nephropathy and sarcoidosis. On admission for pneumonia, imaging studies revealed an unexpected hepatic tumor. Histology revealed a hepatocellular carcinoma accompanied by T-lymphocytic infiltration and marked granulomatous inflammation, which was surrounding some tumor nodules. The background liver parenchyma exhibited a moderate degree of fibrosis with granulomatous inflammation. The patient had no other apparent liver disease such as viral hepatitis, steatohepatitis, or primary biliary cirrhosis. Therefore, in the present case, sarcoidosis may be considered the probable background etiology for hepatocarcinogenesis