TRAVELTIME CURVES FOR REFLECTIONS IN DIPPING LAYERS

Geophysics ◽  
1976 ◽  
Vol 41 (3) ◽  
pp. 425-440 ◽  
Author(s):  
Anthony F. Gangi ◽  
Sung Jin Yang
Keyword(s):  
Second Order ◽  
Computationally Efficient ◽  
Third Order ◽  
Reflected Waves ◽  
First Order ◽  
The Third ◽  
Order Curve ◽  
Standard Deviations ◽  
Tracing Techniques ◽  
Source Array

The traveltimes for reflected waves from plane, dipping layer interfaces for split‐spread arrays and CGP gathers are determined by a computationally efficient method. The computational efficiency is obtained by (1) interpolating the traveltimes for particular source‐receiver distances from least‐squares curves fitted to traveltime data rather than by using iterative ray‐tracing techniques, and (2) using the traveltime curves for a fixed source to determine those for other source locations and for CGP gathers over the same dipping layer interfaces. The additional computations necessary in this latter case are minimal when this approach is used. The traveltime curves for other source locations and CGP gathers are obtained by taking advantage of the fact that the traveltimes and travel distances for parallel rays are simply related for plane, dipping layer interfaces. For the fixed source array, standard deviations between 0.95 msec for the fourth interface (in a four layer model) and 9.5 msec for the first interface were obtained when the traveltimes t(x) were fitted to a second order curve: [Formula: see text]. These standard deviations decreased to 0.81 and 3.5 msec, respectively, when a third order curve [Formula: see text] was used. The standard deviations became 2.97 and 0 msec, respectively, for a second order curve in [Formula: see text]; and 0.95 and 0 msec, respectively, for the third order curve: [Formula: see text]. For a CGP gather over the same layers, the standard deviations were 0.84 and 0 msec for the fourth and first interfaces, respectively, when the traveltime data were fitted to a first order curve in [Formula: see text]. These standard deviations became 0.10 and 0 msec, respectively, when the curve fitted was: [Formula: see text]. Errors in the traveltimes for a fitted curve, [Formula: see text], in a CGP gather for the third interface at a depth of about 5000 ft were less than 0.5 msec for source‐receiver separations up to 5000 ft. For the same interface, errors were much less than 0.1 msec over the same separations for the fitted curve: [Formula: see text].

Perturbation treatment of the Hartree–Fock equations for the ground states of the boron-like ions

1969 ◽  
Vol 47 (7) ◽  
pp. 699-705 ◽  
Author(s):  
C. S. Sharma ◽  
R. G. Wilson
Keyword(s):  
Ground State ◽  
Atomic System ◽  
Ground States ◽  
Great Accuracy ◽  
Second Order ◽  
Expectation Values ◽  
Third Order ◽  
First Order ◽  
Hartree Fock ◽  
The Third

The first-order Hartree–Fock and unrestricted Hartree–Fock equations for the ground state of a five electron atomic system are solved exactly. The solutions are used to evaluate the corresponding second-order energies exactly and the third-order energies with great accuracy. The first-order terms in the expectation values of 1/r, r, r2, and δ(r) are also calculated.

scholarly journals The "third"-order barrier for technology-integration instruction: Implications for teacher education

2012 ◽  
Vol 28 (6) ◽  
Author(s):  
Chin-Chung Tsai ◽  
Ching Sing Chai
Keyword(s):  
Technology Integration ◽  
Design Thinking ◽  
Second Order ◽  
Access Time ◽  
Pedagogical Beliefs ◽  
Third Order ◽  
Education Practice ◽  
First Order ◽  
The Third ◽  
Technology Beliefs

<span>Technology integration is a major trend in contemporary education practice. When undertaking technology integration in classrooms, a first-order barrier and a second-order barrier, as proposed by Ertmer (1999), can hinder its implementation. The first-order barrier is external, such as lack of adequate access, time, training and institutional support. The second-order barrier includes teachers' personal and fundamental beliefs such as teachers' pedagogical beliefs, technology beliefs, willingness to change. This paper argues that the lack of design thinking by teachers may be the "third"-order barrier for technology integration.</span>

scholarly journals Consensus Conditions for High-Order Multiagent Systems with Nonuniform Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Mengji Shi ◽  
Kaiyu Qin ◽  
Ping Li ◽  
Jun Liu
Keyword(s):  
Multiagent Systems ◽  
Time Delays ◽  
Second Order ◽  
High Order ◽  
Sixth Order ◽  
Multiple Time ◽  
State Variables ◽  
Third Order ◽  
First Order ◽  
The Third

Consensus of first-order and second-order multiagent systems has been wildly studied. However, the convergence of high-order (especially the third-order to the sixth-order) state variables is also ubiquitous in various fields. The paper handles consensus problems of high-order multiagent systems in the presence of multiple time delays. Obtained by a novel frequency domain approach which properly resolves the challenges associated with nonuniform time delays, the consensus conditions for the first-order and second-order systems are proven to be nonconservative, and those for the third-order to the sixth-order systems are provided in the form of simple inequalities. The method revealed in this article is applicable to arbitrary-order systems, and the results are less conservative than those based on Lyapunov approaches, because it roots in sufficient and necessary criteria of stabilities. Simulations are carried out to validate the theoretical results.

scholarly journals 1. A proof of Dupin's theorem with some simple illustrations of the method employed. 2. Two methods of obtaining Cayley's condition that a family of surfaces may form one of an orthogonal triad. 3. An extension of Dupin's theorem to the case in which a family of surfaces is cut orthogonally by two other families which intersect at a constant angle, with the condition that a family may be capable of being cut in this manner

1910 ◽  
Vol 29 ◽  
pp. 41-64
Author(s):  
F. E. Edwardes
Keyword(s):  
Small Angle ◽  
Second Order ◽  
Finite Radius ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Constant Angle ◽  
Continuous Curvature

A proof of Dupin's theorem with some simple illustrations of the method employed.Before plunging into Dupin's theorem, I think it well to speak of certain infinitesimal rotations which play a part in the proof. By an infinitesimal angle of the first order is meant an angle subtended at the centre of a circle of finite radius by an arc whose length is an infinitesimal of the first order. If we neglect infinitesimals of the second order, equal infinitesimal rotations of the first order about axes which meet and are separated by a small angle of the first order are identical. For instance, if AB and BC be elements of a curve of continuous curvature, an infinitesimal rotation about AB may, if we prefer it, be regarded as taking place about BC; and again, if OA, OB, OC be a set of rectangular axes, small rotations about OA, OB, OC may be regarded as taking place in any order. For if P be a point on a sphere of finite radius, and PQ, PR be the displacements of P due to equal infinitesimal rotations of the first order about two diameters separated by a small angle of the first order, the angle QPR is the angle of separation of the axes, and it follows that QR is an infinitesimal of the second order. Further, if the radius of the sphere is an infinitesimal of the first order, QR is of the third order of small quantities.

scholarly journals 02. Social Pathologies, Reflexive Pathologies, and the Idea of Higher-Order Disorders

2015 ◽  
Author(s):  
Arto Laitinen
Keyword(s):  
Social Reality ◽  
Social Criticism ◽  
Higher Order ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
The Social ◽  
Order Reflection

This paper critically examines Christopher Zurn’s suggestion mentioned above that various social pathologies (pathologies of ideological recognition, maldistribution, invisibilization, rationality distortions, reification and institutionally forced self-realization) share the structure of being ‘second-order disorders’: that is, that they each entail ‘constitutive disconnects between first-order contents and secondorder reflexive comprehension of those contents, where those disconnects are pervasive and socially caused’ (Zurn, 2011, 345-346). The paper argues that the cases even as discussed by Zurn do not actually match that characterization, but that it would be premature to conclude that they are not thereby social pathologies, or that they do not have a structure in common. It is just that the structure is more complex than originally described, covering pervasive socially caused evils (i) in the social reality, (ii) in the first order experiences and understandings, (iii) in the second order reflection as discussed by Zurn, and also (iv) in the ‘third order’ phenomenon concerning the pre-emptive silencing or nullification of social criticism even before it takes place 

scholarly journals ON THE TRANSFORMATION OF THE LINEAR DIFFERENTIAL EQUATION INTO A SYSTEM OF THE FIRST ORDER EQUATIONS

2019 ◽  
pp. 71-75
Author(s):  
M.I. Ayzatsky
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Linear Equation ◽  
Second Order ◽  
Dimensional System ◽  
Third Order ◽  
Order Linear ◽  
First Order ◽  
The Third ◽  
Linear Differential

The generalization of the transformation of the linear differential equation into a system of the first order equations is presented. The proposed transformation gives possibility to get new forms of the N-dimensional system of first order equations that can be useful for analysis of the solutions of the N-th-order differential equations. In particular, for the third-order linear equation the nonlinear second-order equation that plays the same role as the Riccati equation for second-order linear equation is obtained.

Benzenesulfonyl chloride with primary and secondary amines in aqueous media — Unexpected high conversions to sulfonamides at high pH

2005 ◽  
Vol 83 (9) ◽  
pp. 1525-1535 ◽  
Author(s):  
James F King ◽  
Manjinder S Gill ◽  
Petru Ciubotaru
Keyword(s):  
Sodium Hydroxide ◽  
Sulfonyl Chloride ◽  
Second Order ◽  
Secondary Amines ◽  
Third Order ◽  
Benzenesulfonyl Chloride ◽  
Aqueous Sodium ◽  
First Order ◽  
The Third ◽  
Primary And Secondary Amines

We have determined pH–yield profiles under pseudo-first-order conditions of the reactions of benzenesulfonyl chloride with a set of primary and secondary water-soluble alkylamines, and have found with certain amines, such as dibutylamine, a profile taking the form of a sigmoid pH&#150yield curve with relatively high yields of the sulfonamide persisting with increasing basicity up to and including 1.0 mol/L sodium hydroxide. This behaviour is quantitatively accounted for by invoking, in addition to the usual second-order reaction of the sulfonyl chloride with the amine, two third-order terms (i) one first-order in sulfonyl chloride, amine and hydroxide anion, and (ii) another first-order in sulfonyl chloride and second-order in the amine. The importance of the third-order terms correlates approximately with the total number of alkyl carbon atoms in the amine, and this in turn is regarded as related to the hydrophobic character of the amine. Experiments to test this picture included: (i) observation of a bell-shaped curve with bis(2-methoxyethyl)amine, (ii) in the reaction of dibutylamine in THF&#150H2O (1:1), and also (iii) in the reaction of dibutylamine in 1.0 mol/L tetrabutylammonium bromide, and (iv) increase in the contributions of the third-order terms in 1.0 mol/L aqueous sodium chloride. Preparative reactions with dibutylamine, 1-octylamine, and hexamethylenimine in 1.0 mol/L aqueous sodium hydroxide with a 5% excess of benzenesulfonyl chloride gave, respectively, 94%, 98%, and 97% yields of the corresponding sulfonamides. Key words: sulfonyl chlorides, primary and secondary amines, pH–yield profiles, organic synthesis in water, hydrophobic effects.

scholarly journals Improvement of Performance Analysis of Amplitude-Comparison Monopulse Algorithm in the Presence of an Additive Noise

Electronics ◽  
2021 ◽  
Vol 10 (21) ◽  
pp. 2649
Author(s):  
Hyeong-Woo Ham ◽  
Joon-Ho Lee
Keyword(s):  
Performance Analysis ◽  
Taylor Series ◽  
Additive Noise ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
The Third

In this paper, it is shown how the performance of the monopulse algorithm in the presence of an additive noise can be obtained analytically. In a previous study, analytic performance analysis based on the first-order Taylor series and the second-order Taylor series was conducted. By adopting the third-order Taylor series, it is shown that the analytic performance based on the third-order Taylor series can be brought closer to the performance of the original monopulse algorithm than the analytic performance based on the first-order Taylor series and the second-order Taylor series.

Coding true arithmetic in the Medvedev and Muchnik degrees

2011 ◽  
Vol 76 (1) ◽  
pp. 267-288 ◽  
Author(s):  
Paul Shafer
Keyword(s):  
Order Theory ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
Closed Sets ◽  
The Third ◽  
First Order Theory ◽  
Medvedev Degrees

AbstractWe prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order theory of true arithmetic are pairwise recursively isomorphic (obtained independently by Lewis, Nies, and Sorbi [7]). We then restrict our attention to the degrees of closed sets and prove that the following theories are pairwise recursively isomorphic: the first-order theory of the closed Medvedev degrees, the first-order theory of the compact Medvedev degrees, the first-order theory of the closed Muchnik degrees, the first-order theory of the compact Muchnik degrees, and the second-order theory of true arithmetic. Our coding methods also prove that neither the closed Medvedev degrees nor the compact Medvedev degrees are elementarily equivalent to either the closed Muchnik degrees or the compact Muchnik degrees.

Morphology and longevity of different-order fine roots in poplar (Populus × euramericana) plantations with contrasting forest productivities

2018 ◽  
Vol 48 (6) ◽  
pp. 611-620 ◽  
Author(s):  
Wanrui Zhu ◽  
Ya Lin Sang ◽  
Qiliang Zhu ◽  
Baoli Duan ◽  
Yanping Wang
Keyword(s):  
Fine Roots ◽  
Inorganic Nitrogen ◽  
Second Order ◽  
Soil Conditions ◽  
Third Order ◽  
Populus Euramericana ◽  
High Productivity ◽  
First Order ◽  
The Third ◽  
Different Order

Fine roots play important roles in the allocation of forest net primary productivity. Here, to provide insights into the intrinsic and extrinsic factors affecting the root growth, we examined the morphology and longevity of different-order fine roots in two poplar (Populus × euramericana (Dode) Guinier) plantations with contrasting forest productivities. The results indicated that the biomass of fine roots decreased significantly as the root orders increased (P < 0.05). In the plantation with relatively high productivity, the specific root length of the first- and second-order roots was significantly greater than that of the third- to fifth-order roots (P < 0.05). Observations of minirhizotrons showed that roots of the third-order and higher lived longer than the first- and second-order roots. Compared with those in the plantation with relatively low productivity, the cumulative survival rate of the first-order roots was significantly higher in the plantation with relatively high productivity, while that of the third-order or higher roots was lower. The life-span of the first-order roots correlated negatively with soil available inorganic nitrogen, while that of the second-order roots correlated positively with the soil phenolic acid content. A Cox proportional hazards analysis revealed that soil conditions, root orders, season of root birth, and soil depth significantly affected fine-root longevity. These findings suggest that roots of different orders contribute unequally to the poplar plantation productivity.

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