Traces of nonsmooth functions on polyhedral domains

introduced in this paper are the definitions of the traces for a class of nonsmooth functions on polyhedral domains. By analyzing their properties we get the structures of these traces.

Complex stress functions and plastic zone sizes for notch cracks subjected to various loading conditions

In this paper, the conformal mapping method is used to solve the plane problem of an infinite plate containing a central lip-shaped notch subjected to biaxial loading at a remote boundary or a surface uniform pressure on the notch. The stress intensity factors KI and KII are obtained by the derived complex stress functions. The simple analytical expressions can be applied to the situation of cracks originating from a circular or an elliptical notch. The plastic zone sizes for such notch cracks are subsequently evaluated in light of the Dugdale strip yield concept. The results are consistent with available numerical data.

A passive scalar field convected by turbulence

Classical statistical mechanics is applied to the study of a passive scalar field convected by isotropic turbulence. A complete set of independent real parameters and dynamic equations are worked out to describe the dynamic state of the passive scalar field. The corresponding Liouville equation is solved by a perturbation method based upon a Langevin–Fokker–Planck model. The closure problem is treated by a variational approach reported in earlier papers. Two integral equations are obtained for two unknown functions: the scalar variance spectrum F(k) and the effective damping coefficient (k). The appearance of the energy spectrum of the velocity field in the two integral equations represents the coupling of the scalar field with the velocity field. As an application of the theory, the two integral equations are solved to derive the inertial-convective-range spectrum, obtaining F(k)=0.61 −1/3 k−5/3. Here is the dissipation rate of the scalar variance and is the dissipation rate of the energy of the velocity field. This theoretical value of the scalar Kolmogorov constant, 0.61, is in good agreement with experiments.

The thermal and mechanical structure of a two-dimensional plume in the earths mantle

On the condition that the distribution of velocity and temperature at the mid-plane of a mantle plume has been obtained (pages 213–218, this issue), the problem of determining the lateral structure of the plume at a given depth is reduced to solving an eigenvalue problem of a set of ordinary differential equations with five unknown functions, with an eigenvalue being related to the thermal thickness of the plume at this depth. The lateral profiles of upward velocity, temperature and viscosity in the plume and the thickness of the plume at various depths are calculated for two sets of Newtonian rheological parameters. The calculations show that the precondition for the existence of the plume, δT/L 1 (L = the height of the plume, δT = lateral distance from the mid-plane), can be satisfied, except for the starting region of the plume or near the base of the lithosphere. At the lateral distance, δT, the upward velocity decreases to 0.1 – 50% of its maximum value at different depths. It is believed that this model may provide an approach for a quantitative description of the detailed structure of a mantle plume.

Interim report on the progress of an inventory of artesian wells in Florida: leading to the enforcement of sections 370.051 - 370.54, Florida statues

One of the causes of lower artesian pressure, water waste and aquifer contamination is the misuse and insufficient care of artesian wells. In 1953, Senate Bill No. 57, entitled "An Act to Protect and Control the Artesian Waters of the State" (see Appendix) became a law. This law was passed through the efforts exerted by leading members of the Senate and the House of Representatives, who understood the need for a wise and controlled expenditure of our most valuable natural resource. The State Geologist and his authorized representatives were designated by this law to enforce this conservation measure; however, no financial provision was included for the 1953-55 biennium. The proposed program of the Florida Geological Survey for this biennium did not include the funds nor provide any full-time personnel for the enforcement of this statute. As a result, little actual work was accomplished during these two years, although much time was given to planning and discussion of the problem. Realizing that this program could provide additional basic data needed in the analysis of the water-supply problem, the State Geologist sought and was granted by the 1955 Legislature adequate funds with which to activate the first phase of the enforcement of Florida Statute No. 370.051-054. Enumerated below is a summary of the progress made on this investigation as outlined previously: 1. Data have been collected on 967 wildly flowing wells in 22 counties. 2. Chloride determinations have been run on 850 of the 967 wells. 3. Of the 967 wells, 554 have chlorides in excess of the 250 ppm, the upper limit assigned by the State Board of Health for public consumption. 4. Water escapes at the rate of 37, 762 gallons per minute from these 967 wells. This amounts to 54, 377, 280 gallons per day. The investigation is incomplete at this time; therefore, no final conclusions can be reached. However, from data already collected, the following recommendations are proposed: 1. That the present inventory of wildly flowing wells be completed for the entire State. 2. That the current inventory of wildly flowing wells be expanded at the conclusion of the present inventory to include all flowing wells. 3. That a complete statewide inventory program be established and conducted in cooperation with the Ground Water Branchof the U.S. Geological Survey. 4. That the enforcement functions as set down in Sections 370.051/.054, Florida Statutes, be separated from the program to collect water-resource data and that these functions be given to the Water Resources Department, if such is created (to be recommended by the Water Resources Study Commission in a water policy law presented to the 1957 Legislature). 5. That the research phase (well inventory) of the program remain under the direction of the Florida Geological Survey. (PDF contains 204 pages.)

The Relationship between Decision Trees and the Scale of Train Data Set

Decision Trees need train samples in the train data set to get classification rules. If the number of train data was too small, the important information might be missed and thus the model could not explain the classification rules of data. While it is not affirmative that large scale of train data set can get well model. This Paper analysis the relationship between decision trees and the train data scale. We use nine decision tree algorithms to experiment the accuracy, complexity and robustness of decision tree algorithms. Some results are demonstrated.

Basic equations for ferroelectrics

A set of new formula of energy functions for ferroelectrics was proposed, and then the new basic equations were derived in this paper. The finite element formulation based on the new basic equations was improved to avoid the equivalent nodal load produced by remnant polarization. With regard to the fundamentals of mathematics and physics, the new energy functions and basic equations are reasonable for the material element of ferroelectrics in finite element analysis.