Kriesel and Wittgenstein

Georg Kreisel (15 September 1923 - 1 March 2015) was a formidable mathematical logician during a formative period when the subject was becoming a sophisticated field at the crossing of mathematics and logic. Both with his technical sophistication for his time and his dialectical engagement with mandates, aspirations and goals, he inspired wide-ranging investigation in the metamathematics of constructivity, proof theory and generalized recursion theory. Kreisel's mathematics and interactions with colleagues and students have been memorably described in Kreiseliana ([Odifreddi, 1996]). At a different level of interpersonal conceptual interaction, Kreisel during his life time had extended engagement with two celebrated logicians, the mathematical Kurt Gödel and the philosophical Ludwig Wittgenstein. About Gödel, with modern mathematical logic palpably emanating from his work, Kreisel has reflected and written over a wide mathematical landscape. About Wittgenstein on the other hand, with an early personal connection established Kreisel would return as if with an anxiety of influence to their ways of thinking about logic and mathematics, ever in a sort of dialectic interplay. In what follows we draw this out through his published essays—and one letter—both to elicit aspects of influence in his own terms and to set out a picture of Kreisel's evolving thinking about logic and mathematics in comparative relief.Accepted manuscrip

The Radiated Energy of the 2004 Sumatra-Andaman Earthquake

We use several independent methods to estimate the radiated energy E_R of the Sumatra-Andaman earthquake (M_w=9.0 to 9.3), and investigate whether the difference in the rupture patterns between north and south is reflected in the difference in the energy budget. First, we used a finite source model and estimated E_R to be 1.38 x 10^(17) J for a frequency band f (frequency) ≤ 0.1 Hz. Since this method is relatively free from many assumptions commonly made in energy estimation, this value is considered robust. To estimate E_R for a frequency band 0.1 ≤ f ≤ 1 Hz, we used a frequency-domain analysis and obtained E_R = 1.6 x 10^(17) J for this frequency band. This estimate is somewhat uncertain because of the energy attenuation during propagation and the effect of the near-source structure. We also estimated E_R relative to the 2001 Bhuj, India, earthquakes for which a reliable estimate of E_R has been obtained. The total E_R thus estimated is 3.0 x 10^(17) J. The energy-moment ratio, 0.46 x 10^(-5), is slightly smaller than that for other large subduction-zone earthquakes. The radiation efficiency defined by η_R = (2µ /Δr)(E_R/M_0) (µ=rigidity, M_0=seismic moment, Δr =static stress drop) is 0.16 which is smaller than that of many large earthquakes, and is between the values of regular earthquakes and slow tsunami earthquakes. The values of η_R for the Nicobar segment, the Nicobar-Andaman segments combined, and the Sumatra segment are 0.053, 0.11, and 0.21, respectively, which suggests that the slip in the northern segments involves a large amount of energy dissipation associated with water-filled thick sediments

Mechanics of earthquakes

An earthquake is a sudden rupture process in the Earth’s crust or mantle caused by tectonic stress. To understand the physics of earthquakes it is important to determine the state of stress before, during, and after an earthquake. There have been significant advances in seismology during the past few decades, and some details on the state of stress near earthquake fault zones are becoming clearer. However, the state of stress is generally inferred indirectly from seismic waves which have propagated through complex structures. The stress parameters thus determined depend on the specific seismological data, methods, and assumptions used in the analysis,and must be interpreted carefully. This paper reviews recent seismological data pertinent to this subject,and presents simple mechanical models for shallow earthquakes. Scholz(1989), Brune (1991), Gibowicz (1986), and Udias (1991) recently this subject from a different perspective, and we will try to avoid duplication with these papers as much as possible. Because of the limited space,available, this review is not intended to be an exhaustive summary of the literature, but reflects the author’s own view on the subject

Real-time seismology and earthquake damage mitigation

Real-time seismology refers to a practice in which seismic data are collected and analyzed quickly after a significant seismic event, so that the results can be effectively used for postearthquake emergency response and early warning. As the technology of seismic instrumentation, telemetry, computers, and data storage facility advances, the real-time seismology for rapid postearthquake notification is essentially established. Research for early warning is still underway. Two approaches are possible: (a) regional warning and (b) on-site (or site-specific) warning. In (a), the traditional seismological method is used to locate an earthquake, determine the magnitude, and estimate the ground motion at other sites. In (b), the beginning of the ground motion (mainly P wave) observed at a site is used to predict the ensuing ground motion at the same site. An effective approach to on-site warning is discussed in light of earthquake rupture physics

Rupture Process of Subduction-Zone Earthquakes

This review is primarily concerned with the rupture process of large subduction-zone earthquakes determined by various seismological methods, and with its interpretation in terms of an asperity model. It is not possible to make a thorough and extensive review on the subject because of the limited length. Consequently, this review is inevitably biased toward the works in which I was directly involved through collaborations with various investigators. The distribution of large earthquakes along subduction zones has a distinct pattern. Great earthquakes occur in South America, Alaska, the Aleutians, and Kamchatka. In contrast, earthquakes along the Marianas are smaller. The seismicity in other subduction zones is intermediate between these two groups (see Figure 1). Although this regional variation now generally accepted, it was not until an appropriate method for quantification of large earthquakes was developed that the regional variation was clearly recognized. In view of its fundamental importance in seismology, we first review the quantification method

Mode of Strain Release Associated with Major Earthquakes in Japan

It is now widely accepted that most shallow earthquakes are caused by a sudden release of tectonic stress, and therefore strain, in the form of faulting (56, Chap. 14). The elastic rebound theory explains remarkably well the overall nature of major earthquakes. However, it is now evident that the strain release by actual earthquakes takes place in a variety of ways. For example, some earthquakes arc reported to have been preceded by significant premonitory crustal deformations with or without foreshocks. Most major earthquakes are followed by aftershocks and, in some cases, by creep-like postseismic deformations. This variety obviously results from the difference in the nature of the crust and the stress field in the respective epicentral area. Thus, detailed study on the mode of the strain release for individual earthquakes is important for understanding the physics of earthquakes and for predicting the mode of occurrence of future major earthquakes. The latter problem is of course very important for designing a practical method of earthquake prediction. The crustal deformation represents the low frequency spectrum of the strain release, and seismic waves represent the high frequency spectrum. To understand better the mode of the strain release, it is important to study it over a wide frequency range, from geodetic to seismic. From this point of view, this paper discusses the mode of strain release in several major earthquakes in Japan for which both geodetic and seismic data are available, and we will focus our attention on the relation between the geodetic slip and the seismic slip. Figure 1 shows the locations of the earthquakes discussed in this paper. We do not intend to make an exhaustive review of the literature on the subject that can be referred to (19, 21, 36, 39, 50, 63, 71)

Lattice study of two-dimensional N=(2,2) super Yang-Mills at large-N

We study two-dimensional N=(2,2) SU(N) super Yang-Mills theory on Euclidean two-torus using Sugino's lattice regularization. We perform the Monte-Carlo simulation for N=2,3,4,5 and then extrapolate the result to N = infinity. With the periodic boundary conditions for the fermions along both circles, we establish the existence of a bound state in which scalar fields clump around the origin, in spite of the existence of a classical flat direction. In this phase the global (Z_N)^2 symmetry turns out to be broken. We provide a simple explanation for this fact and discuss its physical implications.Comment: 24 pages, 13 figure

Seismological Aspects of the December 2004 Great Sumatra-Andaman Earthquake

The 2004 Great Sumatra-Andaman earthquake had an average source duration of about 500 sec. and a rupture length of 1,200–1,300 km. The seismic moment, M0, determined with a finite source model, was 6.5×1022 N-m, which corresponds to Mw=9.18. Allowing for the uncertainties in the current M0 determinations, Mw is in the range of 9.1 to 9.3. The tsunami magnitude Mt is 9.1, suggesting that the overall size of the tsunami is consistent with what is expected of an earthquake with Mw=9.1 to 9.3. The short-period body-wave magnitude m-hatb is 7.25, which is considerably smaller than that of large earthquakes with a comparable Mw. The m-hatb versus Mw relationship indicates that, overall, the Great Sumatra-Andaman earthquake is not a tsunami earthquake. The tectonic environment of the rupture zone of the Great Sumatra-Andaman earthquake is very different from that of other great earthquakes, such as the 1960 Chile and the 1964 Alaska earthquakes. This difference may be responsible for the unique source characteristics of this earthquake. The extremely large size of the Great Sumatra-Andaman earthquake is reflected in the large amplitude of the long-period phase, the W phase, even in the early part of the seismograms before the arrival of the S wave. This information could be used for various early warning purposes

Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

SUSY WT identity in a lattice formulation of 2D N=(2,2)\mathcal{N}=(2,2) SYM

We address some issues relating to a supersymmetric (SUSY) Ward-Takahashi (WT) identity in Sugino's lattice formulation of two-dimensional (2D) $\mathcal{N}=(2,2)$ $SU(k)$ supersymmetric Yang-Mills theory (SYM). A perturbative argument shows that the SUSY WT identity in the continuum theory is reproduced in the continuum limit without any operator renormalization/mixing and tuning of lattice parameters. As application of the lattice SUSY WT identity, we show that a prescription for the hamiltonian density in this lattice formulation, proposed by Kanamori, Sugino and Suzuki, is justified also from a perspective of an operator algebra among correctly-normalized supercurrents. We explicitly confirm the SUSY WT identity in the continuum limit to the first nontrivial order in a semi-perturbative expansion.Comment: 15 pages, uses elsart.cls, the final version to appear in Phys. Lett.