The use of Hilbert transform is essential when we are dealing with bandwidth limited signals and causal functions. However, the Hilbert transform is originally defined for functions of continuous arguments and is not of immediate use for discrete (sampled) functions which are the common form of data today. In this paper we propose two kinds of 'discrete Hilbert transforms'; one is defined on the discrete time domain and the other on the frequency domain of sampled functions. It will be shown that the essential properties of the conventional Hilbert transform are conserved in the new definitions.

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The regions of intense seismic activity off the Pacific coast of Japan are classified into several types according to the pattern of occurrence of large earthquakes (M≥6) in space, time, and magnitude. In type A regions, great earthquakes occur at fairly regular intervals. In type B regions, large earthquakes of magnitude up to about 7.5 tend to occur in swarms. In type A' regions, large earthquakes of magnitude up to about 7.5 occur sporadically. There are regions labeled as type AB, A'B, A(B), (A)B, etc. Type AB means a mixed type of A and B. Type A(B) represents a type A with slight indication of the character of type B.
Investigations of the pattern of occurrence in regions of each type provide useful information for the statistical prediction of large earthquakes. For great earthquakes in type A regions, foreshock activities are usually weak, but the identification of foreshocks, if they are observed, may be easier, because a remarkable seismicity gap is developed before a great earthquake. In regions of types B, (A)B, and AB, the absence of a seismicity gap for large earthquakes does not assure the non-occurrence of a large earthquake in near future. The probability of the successive occurrence of large earthquakes is high after the occurrence of large earthquake in regions of types B and (A)B. A seismic gap for great earthquakes found in regions of types A' A'B, and B is not an indication of a future great earthquake. Swarms in regions of types B, (A)B, and A'B usually have low b-values. Therefore a low b-value is not a criterion for foreshocks in these regions. A remarkable increase in number of destructive earthquakes in the land area of Japan on the inner side of the source regions of great earthquakes along the Nankai trough is recognized during a 50-year period prior to the occurrence of each great earthquake.

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The numerical parameters in the flow equation of the power law creep for ultrabasic rocks are evaluated based upon a critical evaluation of the published experimental data on the steady state deformation. The power law creep controls the deformation of ultrabasic rocks under the condition of (deviatoric stress/shear modulus)<2×10^-3, and the result is expressed by
ε=6.3×10³σ^2.7exp(-(1.2×10⁵-1.15P_H2O)/RT)
where ε is the strain rate in sec^-1, σis the deviatoric stress in bar, and P_H2O is the partial pressure of H₂O in bar. The value of exponent n=2.7, which is considerably smaller than that accepted up to the present, is found to be reasonable for ultrabasic rocks. Quantity V^*_H2O(=-1.15cal/bar·mole= -48cm³/mole) is regarded as the activation volume for H₂O pressure, and it is named hydrolytic activation volume. Based on the idea that the activation energy of creep is linearly related to the melting temperature of the material, the above value of the hydrolytic activation volume is shown to be consistent with that calculated by using a value of the reduction of the solidus of olivine due to P_H2O.

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The thermal diffusivities along three crystallographic orientations are measured for olivine and pyroxene up to 1250K. The result shows that the anisotropy amounts to about 70%. The anisotropy is attributed mostly to anisotropy of mean free path of phonon. The anisotropy of thermal conductivity and thermal diffusivity in dunite is measured and is discussed on the basis of preferred orientation and the anisotropy of constituent minerals.

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Crack in material has two main roles: the concentration of stress near its edges and the regional relase of strain energy. In this paper, we are concerned with such roles in an inelastic material with strain-hardening. The material used here is the punchcard whose stress-strain relation is given by σ∝ε^1/2. The experimental results show that the stress concentrated is proportional to the fourth root of crack length. The strain energy released is proportional to the second power of it as in a linear elastic material, but the proportional coefficient is about twice as large as in the latter. It is also shown that a minimum length is necessary to transfer high stress. This critical length depends on the structural property of material and determines the thickness of fracture zone which is formed during crack propagation.

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The fault-slip motion of an earthquake is described on the basis of the recognition that frictional characteristics of the fault surfaces are important factors to be considered insofar as an earthquake source is a shear-type dislocation accompanying fracture or stick-slip. Expressions for the fault-slip velocity and acceleration are derived in terms of the stress drop (or the stress defined by the difference between the critical stress necessary to initiate slippage or fracture and the kinetic frictional stress in solid friction), the rigidity, the shear wave velocity, a parameter p standing for the magnitude of viscous friction on the fault plane and another parameter c, whose magnitude may depend upon the surface conditions such as roughness, of a natural fault and upon the bounded nature of the fault. It is shown that whether the stress after the earthquake dislocation is larger than the average kinetic frictional stress or not depends upon p and c.

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