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The discrete power function on the hexagonal lattice proposed by Bobenko et al is considered, whose defining equations consist of three cross-ratio equations and a similarity constraint. We show that the defining equations are derived from the discrete symmetry of the Garnier system in two variables., 22 pages

In this paper, we consider the discrete power function associated with the sixth Painlev\'e equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with $\widetilde{W}(3A_1^{(1)})$ symmetry. By constructing the action of $\widetilde{W}(3A_1^{(1)})$ as a subgroup of $\widetilde{W}(D_4^{(1)})$, i.e., the symmetry group of P$_{\rm VI}$, we show how to relate $\widetilde{W}(D_4^{(1)})$ to the symmetry group of the lattice. Moreover, by using translations in $\widetilde{W}(3A_1^{(1)})$, we explain the odd-even structure appearing in previously known explicit formulas in terms of the $\tau$ function., Comment: 18 pages, 3 figures

We present the τ-functions for the hypergeometric solutions to the q-Painleve system of type \(E_{8}^{(1)}\) in a determinant formula whose entries are given by Rahman’s q-hypergeometric integrals. By using the symmetry of the q-hypergeometric integral, we can construct 56 solutions and describe the action of \(W(E_{7}^{(1)})\) on the solutions.

The classical transcendental solutions to the Painleve III equation are derived from a family of solutions to the anti-self-dual Yang–Mills equation. It is also shown that the affine Weyl group symmetry of PIII is recovered from the symmetry of Yang's equation.

The ground motion and damage caused by the 2015 Gorkha, Nepal earthquake can be characterized by their widespread distributions to the east. Evidence from strong ground motions, regional acceleration duration, and teleseismic waveforms indicate that rupture directivity contributed significantly to these distributions. This phenomenon has been thought to occur only if a strike-slip or dip-slip rupture propagates to a site in the along-strike or updip direction, respectively. However, even though the earthquake was a dip-slip faulting event and its source fault strike was nearly eastward, evidence for rupture directivity is found in the eastward direction. Here, we explore the reasons for this apparent inconsistency by performing a joint source inversion of seismic and geodetic datasets, and conducting ground motion simulations. The results indicate that the earthquake occurred on the underthrusting Indian lithosphere, with a low dip angle, and that the fault rupture propagated in the along-strike direction at a velocity just slightly below the S-wave velocity. This low dip angle and fast rupture velocity produced rupture directivity in the along-strike direction, which caused widespread ground motion distribution and significant damage extending far eastwards, from central Nepal to Mount Everest.

We study the q-difference analogue of the sixth Painleve equation (q-P VI) by means of tau functions associated with the affine Weyl group of type D 5. We prove that a solution of q-P VI coincides with a self-similar solution of the q-UC hierarchy. As a consequence, we obtain in particular algebraic solutions of q-P VI in terms of the universal character which is a generalization of the Schur polynomial attached to a pair of partitions.

We present a determinant expression for a family of solutions to the SL(2, C) anti-self-dual Yang-Mills equation that corresponds to the classical transcendental solutions of the Painleve II and Painleve IV equations.

We present a simple heuristic method to derive the Painleve differential equations from the corresponding geometry of rational surfaces.

A determinant formula for algebraic solutions to the Noumi-Yamada system of type A5(1) is presented. This expression is regarded as a special case of the universal characters. The entries of the determinant are given by the Laguerre polynomials. Degeneration to the rational solutions to the Painleve IV equation is discussed.

We present a determinant expression for a family of classical transcendental solutions of the Painleve V and the Painleve VI equation. Degeneration of these solutions along the process of coalescence for the Painleve equations is discussed.

A τ function formalism for Sakai's elliptic Painleve equation is presented. This establishes the equivalence between the two formulations by Sakai and by Ohta–Ramani–Grammaticos. We also give a simple geometric description of the elliptic Painleve equation as a non-autonomous deformation of the addition formula on elliptic curves. By using these formulations, we construct a particular solution of the elliptic Painleve equation expressed in terms of the elliptic hypergeometric function 10E9.

We consider the discrete power function introduced by Bobenko, which is associated with the circle patterns of Schramm type. We first present an explicit formula in terms of the hypergeometric \(\tau \) functions for the sixth Painleve equation. There is a restriction on the exponent and the domain by geometric reasons. Based on the formula, we next extend the value of the exponent to arbitrary complex numbers except even integers and the domain to a discrete analogue of the Riemann surface of the power function.

We give an explicit determinant formula for a class of rational solutions of the Painlevé V equation in terms of the universal characters.

We prove analytically the Sasa-Satsuma conjecture which states that their solution of bilinear form of the Ernst equation gives the Neugebauere-Kramer solution in particular cases. This proof relates Hirota's direct method with the Backlund transformation method and opens the way towards the comprehensive interpretation of the Ernst equation.

The Kitakami massif, which is located in the eastern part of Northern Honshu, Japan, is composed of two geological units. The northern Kitakami terrane is characterized as a Jurassic accretionary complex, while the southern Kitakami terrane consists of pre-Silurian basement and Silurian-lower Cretaceous marine sediments. The boundary region of these two units, called the Hayachine tectonic belt (HTB), is composed of mafic to ultramafic rocks. The Kitakami massif experienced intense granitic intrusions in the Cretaceous. We present a detailed crustal structure model for the eastern part of the massif derived from an extensive seismic refraction experiment conducted on a 194-km N-S line. The uppermost crust is covered with a very thin (0.5–1 km) surface layer with a velocity of 3.1–5.4 km/s. The velocity structure below this layer shows remarkable lateral variation. In the northern Kitakami terrane the P wave velocity and Vp/Vs at the top of the basement are 5.85–5.95 km/s and 1.68–1.70, respectively. The seismic attenuation in this region is high (Qp = 150–200 and Qs = 70–100). In contrast, the uppermost crust in the southern Kitakami terrane is characterized by a high P wave velocity (6.05–6.15 km/s) and Vp/Vs (1.74–1.77). The Qp and Qs also show high values of 300–400 and 150–200, respectively. Such a structural difference persists to 14-to 16-km depth, at which the P wave velocity increases to 6.45 km/s. The low velocity and high attenuation in the northern Kitakami terrane represent a highly deformed structure of the accretionary complex. The high P wave velocity and Vp/Vs in the southern Kitakami terrane indicate the relatively mafic crustal composition, which may result from the fragment of the oceanic crust incorporated by the accretion process or the uplifting in the latest Jurassic-early Cretaceous. A midcrustal interface determined from wide-angle reflections shows an abrupt southward depth decrease from 25 to 20 km under the HTB. The P wave velocity and Vp/Vs between 14- and 16-km depth and the midcrustal interface are 6.45–6.55 km/s and 1.74–1.78, respectively. The Moho depth under the northern Kitakami terrane decreases southward from 34 to 32 km. In the southern Kitakami terrane the Moho dips slightly southward. The P wave velocity and the Vp/Vs ratio in the lower crust are 6.9–7.0 km/s and 1.75–1.76, respectively. The P wave velocity in the uppermost mantle is not well resolved but is probably less than 7.7 km/s. The S wave velocity derived from relatively clear Sn is 4.35–4.40 km/s. Our results show that the HTB is a prominent structural boundary extending to the Moho. The crust of Kitakami massif was not homogenized by the Cretaceous granitic intrusions, and the original structural difference remains in the upper crust.

An extensive seismic refraction experiment with the use of explosive sources was conducted in the Kitakami region, northern Honshu, Japan, on November 1, 1990. The experiment area is divided into two geological units by the Hayachine Tectonic Belt (HTB). The southern terrane consists of pre-Silurian basements and Silurian-lower Cretaceous marine sediments, while the northern one is characterized by a Jurassic accretionary complex. Both of the units were intruded by Cretaceous granitic rocks. An almost N-S seismic refraction profile of 194-km length was extended from Kuji City, Iwate Prefecture to Ishinomaki City, Miyagi Prefecture, on which 4 shots with a charge size of 450-700 kg were fired

Neugebauer-Kramer solutions on a flat background are extracted from other series of exact solutions to Ernst equation by imposing asymptotic flatness and reality of metrics. Because of this result, we partially verify the conjecture that Neugebauer's solutions are most comprehensive in physically meaningful solutions to Ernst equation.

A $q$-deformation of discrete dynamical systems associated with the Weyl group of type $A$ is proposed. This is a natural generalization of the birational Weyl group action for the $q$-Painleve equation of type $A_4^{(1)}$. A determinant formula of Jacobi–Trudi type for a family of Laurent polynomials arising from the action of the Weyl group is also presented.

We give an explicit determinant formula for a class of rational solutions of a q-analogue of the Painlevé V equation. The entries of the determinant are given by the continuous q-Laguerre polynomials.

Hypergeometric solutions to seven q-Painleve equations in Sakai's classification are constructed. Geometry of plane curves is used to reduce the q-Painleve equations to the three-term recurrence relations for q-hypergeometric functions.

In December, 1979, a detailed explosion seismic experiment was conducted by Research Group for Explosion Seismology in the Mishima-Shimoda profile in Izu Peninsula, Central Japan, where activities such as large earthquakes with damage, earthquake swarms, anomalous crustal movement have recently occurred. The analysis of travel time data of good quality was made mainly by the time term method. The crustal structure derived revealed significant lateral heterogeneity especially around the middle of the profile. In the northern part of the profile, there are two or three layers with a total thickness of about 2km above the homogeneous granitic layers, while in the southern part one or two layers with a total thickness of about 1km exist above the granitic layer. In the granitic layer in the southern part a velocity-depth function which gives 5.4km/sec at the top and 6.0km/sec around a depth of 4km could be accurately obtained.The comparison of the crustal structure with seismicity, gravity anomaly, crustal movement, and geology along the profile gives important results. Most earthquakes occurred in the granitic layer, especially in the southern part of the profile where a velocity-depth function was found. The distribution of Bouguer gravity anomaly is concordant with the crustal structure. The area of anomalous ground uplift coincides with the southern half of the profile where the structure of surface layers becomes complicated and a velocity-depth function was found in the granitic layer. Significant lateral changes of the crustal structure take place near the area where several active faults such as Sano, Himenoyu are present and the 4.1-4.2km/sec layer is identified with the Yugashima group from geological data.

Under the Fourth Earthquake Prediction Project of Japan starting from 1979, the Research Group for Explosion Seismology conducted a series of explosion seismic experiments to reveal detailed structure of the shallow crust. In this paper, results from two experiments in the Izu Peninsula are shown. Structure of the shallow crust in the peninsula is very complex and is quite valuable for the other geophysical and geological investigations. For example, time terms of the basement layer show a good correlation with Bouguer anomalies when we assume a reduction density of about 2.7 g/cm3. A layer with a velocity of about 4.1 km/s observed in the whole Izu Peninsula may correspond to the Yugashima Group. Our structural models are also useful for discussion of abnormal crustal uplift and seismic activities observed around the Izu Peninsula in recent years. We believe that the middle-size seismic experiment like these Izu experiments may be the most appropriate one in geologically complex regions such as Japan.

Accurate estimation of the physical parameters of earthquake sources requires an objective basis for determining the spectral parameters SO, the low-frequency level, fc, the corner frequency, and γ, the high-frequency decay rate. Knowledge of the appropriate Q value is also necessary. In most cases, however, the estimation of these parameters is rather subjective, being accomplished usually by visual fitting of some schematic spectral shape to the observations, and the average of Q over possible regional variations is sometimes used without sufficient evidence as representative of the local value. An objective method of simultaneous inversion is presented in this study for the estimation of source parameters and local Q values. According to theoretical predictions as well as to many observational results, an observed spectrum may be reasonably regarded as a sample from an ensemble of the form O(f)=SO[(f/fc)a+1]−bexp(− πftQ−1), where a and b are positive constants and γ = ab. For earthquakes occurring in a confined region and observed at a nearby station, the constants a and b are common and a single value may be used as the local value of Q. The observed spectra of K earthquakes at L frequency points give K × L data, while the number of unknowns to be estimated is 2 K + 3, that is, the SO and c values for the K earthquakes, and the values of a, b, and Q. For K × L > 2K +3, least-squares estimators are obtained so as to minimise s2=ϵϵ[ln(Okl)−ln(Skl)+πfltkQ−1]2. A numerical test using an artificial data set indicates that a stable solution is obtained, and the errors in the parameters SO and fc are less than 20%, for a data standard deviation of 0.2. This method has been applied to actual data sets for small earthquakes occurring near Miyako on the Pacific coast of the Northeastern part of Japan which were observed at a nearby station. The solution converges sufficiently at the third or fourth iteration step, the data standard deviation being 0.2. The source parameters and the Q value thus estimated are in good accordance with those obtained by another, independent method. The present method gives reliable data for an accurate scaling model, because the source parameters are obtained by a standard technique for a variety of earthquake sizes.

A new method for estimating source extent parameters has been applied to the Imperial Valley earthquake of October 15, 1979, and the Victoria earthquake of June 9, 1980. From the stations of the World-Wide Standard Seismograph Network and Canadian Seismic Network, estimates of the duration of far-field long-period SH waves were made by computing the variance or second-central moment τ2 of instrument-corrected, attenuation-corrected pulses. Seventeen measurements for the Imperial Valley event and 25 for the Victoria event were made. We inverted for the source duration T, the fault length L, and the directivity parameter D by least squares. We find that for the Imperial Valley, T = 9.0 ± 0.5 s, D = 130 ± 43 km s, and L = 46 ± 13 km, and for the Victoria event, T = 10.9 ± 0.6 s, and D = 251 ± 71 km s. L is not well constrained for this second event; the inversion gives 23 km, the lower bound obtained by constraining τ2 to be positive everywhere on the focal sphere, although a value as high as 40 km would still be within the uncertainties. The positive values of D for both events indicate that rupture propagation was predominantly to the northwest. In order to estimate the degree of “unilateralness” we have computed the rupture mode index defined by R = |D|/LT, which is 1 for a unilateral rupture and zero for a bilateral rupture. We find a value of R = 0.31 ± 0.14 for the Imperial Valley event, significantly less than unity, which indicates that a southern component of rupture is also present. Assuming uniform slip, we estimate the northern and southern fault segments to be 26 and 20 km, respectively. The presence of a bilateral component is supported by near-field studies which have made use of stations along the southern half of the Imperial fault. For the Victoria event, due primarily to the uncertainty in L, the value of R can vary from 0.6 to 1 with the value of 1 being most probable. Assuming that the rupture is unilateral, we can obtain an estimate of fault length from the relation L = |D|/T, which yields L = 23 ± 6 km. Geodetic data and the distribution of aftershocks are consistent with a unilateral rupture mode and the fault length that we have obtained. Values of seismic moment from measurements of the zeroth moment of the displacement pulse are found to be 5.8±0.4 for Imperial Valley and 4.8±0.4 for Victoria, in units of 1018 N m. The presence of a southern component of rupture for the Imperial Valley event would suggest a closer and perhaps causal relationship to the Victoria event, which occurred 8 months later and about 60 km to the south. In a tectonic framework we interpret these two earthquakes as the failure of adjacent transform faults and consider the possibility that a concurrent spreading event may have occurred in the vicinity of Cerro Prieto.