Optimal Trajectories and the Accessory Minimum Problem

1965 ◽  
Vol 16 (3) ◽  
pp. 205-220 ◽  
Author(s):  
D. J. Bell
Keyword(s):  
Necessary Conditions ◽  
Sounding Rocket ◽  
Necessary Condition ◽  
Minimum Problem ◽  
Multiplier Rule ◽  
Variation Of A Function ◽  
Accessory Minimum Problem ◽  
The First Variation ◽  
The Second Variation

SummaryThe necessary conditions of Clebsch and Weierstrass and of the multiplier rule in the calculus of variations, which arise from the study of the first variation of a function, are summarised. A further necessary condition associated with the second variation is stated. The latter condition is applied to two problems: (i) the determination of the thrust-time programme which maximises the altitude of a sounding rocket, (ii) the determination of the thrust direction programme for a rocket with a known propellant expenditure programme which yields a maximum range. In both problems it is found that the additional necessary condition is satisfied.

The Determination of Maximum Range for an Unpowered Atmospheric Vehicle

1964 ◽  
Vol 68 (638) ◽  
pp. 111-116 ◽  
Author(s):  
D. J. Bell
Keyword(s):  
Calculus Of Variations ◽  
Lagrange Multipliers ◽  
Digital Computer ◽  
Necessary Conditions ◽  
Lagrange Equations ◽  
Initial Portion ◽  
End Conditions ◽  
Maximum Range ◽  
Isothermal Atmosphere

SummaryThe problem of maximising the range of a given unpowered, air-launched vehicle is formed as one of Mayer type in the calculus of variations. Eulers’ necessary conditions for the existence of an extremal are stated together with the natural end conditions. The problem reduces to finding the incidence programme which will give the greatest range.The vehicle is assumed to be an air-to-ground, winged unpowered vehicle flying in an isothermal atmosphere above a flat earth. It is also assumed to be a point mass acted upon by the forces of lift, drag and weight. The acceleration due to gravity is assumed constant.The fundamental constraints of the problem and the Euler-Lagrange equations are programmed for an automatic digital computer. By considering the Lagrange multipliers involved in the problem a method of search is devised based on finding flight paths with maximum range for specified final velocities. It is shown that this method leads to trajectories which are sufficiently close to the “best” trajectory for most practical purposes.It is concluded that such a method is practical and is particularly useful in obtaining the optimum incidence programme during the initial portion of the flight path.

Certain Fourier Transforms of Distributions: II

1954 ◽  
Vol 6 ◽  
pp. 186-189 ◽  
Author(s):  
Eugene Lukacs ◽  
Otto Szász
Keyword(s):  
Characteristic Function ◽  
Fourier Transforms ◽  
Necessary Conditions ◽  
Laplace Transforms ◽  
Characteristic Functions ◽  
Necessary Condition ◽  
Multiple Roots ◽  
Not Given ◽  
Class Of Functions

In an earlier paper (1), published in this journal, a necessary condition was given which the reciprocal of a polynomial without multiple roots must satisfy in order to be a characteristic function. This condition is, however, valid for a wider class of functions since it can be shown (2, theorem 2 and corollary to theorem 3) that it holds for all analytic characteristic functions. The proof given in (1) is elementary and has some methodological interest since it avoids the use of theorems on singularities of Laplace transforms. Moreover the method used in (1) yields some additional necessary conditions which were not given in (1) and which do not seem to follow easily from the properties of analytic characteristic functions.

Eccentricity in irregular multistory buildings

1986 ◽  
Vol 13 (1) ◽  
pp. 46-52 ◽  
Author(s):  
V. W.-T. Cheung ◽  
W. K. Tso
Keyword(s):  
Dynamic Analysis ◽  
Necessary Conditions ◽  
Vertical Axis ◽  
Building Code ◽  
Necessary Condition ◽  
Practical Procedure ◽  
Multistory Buildings ◽  
Plane Frame ◽  
Code Procedure ◽  
Code Provisions

To evaluate the seismic torsional effect on multistory buildings, the concept of eccentricity is extended from single-story buildings to multistory buildings by defining the locations of the centers of rigidity at each floor. A practical procedure to locate the centers of rigidity and hence floor eccentricity is introduced. This procedure depends on the use of plane frame computer programs only and is suitable for use in design offices. The seismic torsional provisions in the National Building Code of Canada 1985 (NBCC 1985) explicitly emphasize that the code provisions apply to buildings where the centres of rigidity lie on a vertical axis only. By means of examples, it verifies the claim of NBCC 1985. Also, it shows that, for buildings with centers of rigidity scattered from a vertical axis, the code procedure may or may not apply. Therefore, one should interpret the condition of centers of rigidity located along a vertical axis to be a sufficient, but not a necessary, condition for the NBCC 85 code provisions to be applicable. Until the necessary conditions are known, dynamic analysis remains the most reliable method to assign the torsional effects to various portions of the building. Key words: building code, center of rigidity, dynamic analysis, eccentricity, irregular, multistory, seismic, torsion.

scholarly journals Syarat Perlu dan Cukup untuk Keterbatasan Potensial Riesz di Ruang Morrey Klasik

2013 ◽  
Vol 14 (3) ◽  
pp. 227
Author(s):  
Mohammad Imam Utoyo ◽  
Basuki Widodo ◽  
Toto Nusantara ◽  
Suhariningsih Suhariningsih
Keyword(s):  
Characteristic Function ◽  
Morrey Space ◽  
Necessary Conditions ◽  
Riesz Potential ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Research Results ◽  
Necessary And Sufficient

This script was aimed to determine the necessary conditions for boundedness of Riesz potential in the classical Morrey space. If these results are combined with previous research results will be obtained the necessary and sufficient condition for boundedness of Riesz potential. This necessary condition is obtained through the use of characteristic function as one member of the classical Morrey space.

scholarly journals PEMENUHAN ELEMEN NECESSARY CONDITIONS KECAMATAN DALAM PENYELENGGARAAN PEMERINTAHAN UMUM DI KABUPATEN PANDEGLANG

2020 ◽  
Vol 4 (2) ◽  
pp. 129-146
Author(s):  
Arif Nugroho ◽  
Delly Maulana
Keyword(s):  
Government Regulation ◽  
Necessary Conditions ◽  
Transitional Period ◽  
Necessary Condition ◽  
Research Approach ◽  
Regional Government ◽  
National Unity ◽  
General Government ◽  
The Government ◽  
Technical Basis

 Artikel ini mengulas Pemenuhan Elemen Necessary Conditions Kecamatan dalam penyelenggaraan pemerintahan umum baik secara nasional dan spesifik diperdalam dengan fakta empiris di Kabupaten Pandeglang, hal itu sebagai konsekuansi dari pelaksanaan Undang – Undang Nomor 23 Tahun 2014. Penelitian dilakukan dengan menggunakan pendekatan kualitatif. Hasil penelitian diketahui, penyelenggaraan pemerintahan umum Kecamatan baik fakta secara nasional serta pendalaman fakta empiris di Kabupaten Pandeglang menunjukan belum cukup tertopang oleh elemen necessary condition diantaranya kepastian atas kewenangan legalnya serta anggaran yang menyertainya. Oleh sebab itu dipandang perlu ada kemauan politik baik itu dari Presiden untuk segera mengundangkan Peraturan – Pemerintah sebagai landasan teknis bagi pemerintah daerah selaku kepala wilayah maupun dari Kepala Daerah Kabupaten/Kota untuk melakukan terobosan agar supaya di masa peralihan implementasi Undang – Undang Nomor 23 Tahun 2014 kewenangan – kewenangan pada bidang kesatuan bangsa, keamanan dan keteriban umum dapat dilimpahkan pada Kecamatan serta Elemen Necessary Conditions lain yang menyertainya diperkuat.     This article discusses the fulfillment of the elements of the sub-district's necessary conditions in the administration of general government both nationally and specifically and deepened by empirical facts in Pandeglang Regency, this is a consequence of the implementation of Law Number 23 of 2014. The research approach used is qualitative. The results showed that in the administration of district general government both the facts nationally and the deepening of empirical facts in Pandeglang district were not sufficiently supported by elements of necessary conditions, including certainty of legal authority and budget. Therefore, there needs to be political will, both from the president, to immediately ratify the Government Regulation as a technical basis for the regional government (Territory) as well as from the Head of Regency / City to make breakthroughs so that in the transitional period the implementation of Law Number 23 Year 2014 powers in the areas of national unity, security and public order can be transferred to the District and the accompanying elements of necessary conditions are strengthened.

Necessary Conditions for Multistationarity and Stable Periodicity

1998 ◽  
Vol 06 (01) ◽  
pp. 3-9 ◽  
Author(s):  
El Houssine Snoussi
Keyword(s):  
Differential System ◽  
Jacobian Matrix ◽  
Necessary Conditions ◽  
Formal Proof ◽  
Necessary Condition ◽  
Constant Sign ◽  
Interaction Graph ◽  
Partial Derivatives

We show in this paper that, for a differential system defined by a quasi-monotonous function f (with constant sign partial derivatives) the existence of a positive loop in the interaction graph associated to the Jacobian matrix of f is a necessary condition for multistationarity, and the existence of a negative loop comprising at least two elements is a necessary condition for stable periodicity. This gives a formal proof of R.Thomas's conjectures.

Analytical-Numerical Determination of Stress Distribution around a Tip of Polygon-Like Inclusion

2016 ◽  
Vol 713 ◽  
pp. 94-98
Author(s):  
Ondřej Krepl ◽  
Jan Klusák ◽  
Tomáš Profant
Keyword(s):  
Stress Distribution ◽  
Stress Field ◽  
Numerical Procedure ◽  
Plane Elasticity ◽  
Necessary Condition ◽  
Singular Stress ◽  
Reliable Assessment ◽  
Generalized Stress Intensity Factors ◽  
Stress Behaviour

A stress distribution in vicinity of a tip of polygon-like inclusion exhibits a singular stress behaviour. Singular stresses at the tip can be a reason for a crack initiation in composite materials. Knowledge of stress field is necessary condition for reliable assessment of such composites. A stress field near the general singular stress concentrator can be analytically described by means of Muskhelishvili plane elasticity based on complex variable functions. Parameters necessary for the description are the exponents of singularity and Generalized Stress Intensity Factors (GSIFs). The stress field in the closest vicinity of the SMI tip is thus characterized by 1 or 2 singular exponents (1 - λ) where, 0<Re (λ)<1, and corresponding GSIFs that follow from numerical solution. In order to describe stress filed further away from the SMI tip, the non-singular exponents for which 1<Re (λ), and factors corresponding to these non-singular exponents have to be taken into account. Analytical-numerical procedure of determination of stress distribution around a tip of sharp material inclusion is presented. Parameters entering to the procedure are varied and tuned. Thus recommendations are stated in order to gain reliable values of stresses and displacements.

A Multiplier Rule for a Functional Subject to Certain Integrodifferential Constraints

1969 ◽  
Vol 91 (2) ◽  
pp. 185-189 ◽  
Author(s):  
M. Wittler ◽  
C. N. Shen
Keyword(s):  
Optimal Control ◽  
Integral Equation ◽  
Calculus Of Variations ◽  
Necessary Conditions ◽  
Inequality Constraint ◽  
Fundamental Lemma ◽  
First Order ◽  
Multiplier Rule ◽  
Nuclear Rocket ◽  
Equation Constraint

A problem in the optimal control of a nuclear rocket requires the minimization of a functional subject to an integral equation constraint and an integrodifferential inequality constraint. A theorem giving first-order necessary conditions is derived for this problem in the form of a multiplier rule. The existence of multipliers and the arbitrariness of certain variations is shown. The fundamental lemma of the calculus of variations is applied. A simple example demonstrates the applicability of the theorem.

scholarly journals THE FIRST AND SECOND VARIATION OF THE TOTAL ENERGY OF CLOSED DUPLEX DNA IN PLANAR CASE

2010 ◽  
Vol 24 (05) ◽  
pp. 587-597 ◽  
Author(s):  
XIAO-HUA ZHOU
Keyword(s):  
Total Energy ◽  
Equilibrium Shape ◽  
Variation Method ◽  
Second Variation ◽  
General Shape ◽  
Planar Case ◽  
The First Variation ◽  
The Second Variation ◽  
First And Second Variation ◽  
Classical Variation

DNA's shape mostly lies on its total energy F. Its corresponding equilibrium shape equations can be obtained by classical variation method: letting the first energy variation δ(1)F = 0. Here, we not only provide the first variation δ(1)F but also give the second variation δ(2)F in planar case. Moreover, the general shape equations of DNA are abstained and a mistake in Zhang et al., [Phys. Rev. E70, 051902 (2004)] is pointed out.

Determination of Iodide in Feeds and Plants by Ion-Selective Electrode Analysis

1971 ◽  
Vol 54 (4) ◽  
pp. 760-763
Author(s):  
William L Hoover ◽  
James R Melton ◽  
Peggy A Howard
Keyword(s):  
Ionic Strength ◽  
Solid State ◽  
Necessary Conditions ◽  
Ion Selective Electrode ◽  
Phosphate Solution ◽  
Selective Electrode ◽  
Aoac Method ◽  
Low Levels ◽  
High Concentrations

Abstract A method for determining low levels of iodide in feeds and plants is proposed. The samples are mixed with a 10% phosphate solution to maintain relatively constant ionic strength and pH and analyzed with a solid-state iodide electrode. Ashing is not required and there are no significant interferences by ions commonly found in feeds. The method is accurate in determining iodide content ranging from 10.0 ppm to high concentrations. Necessary conditions for storing and cleaning the electrodes are described. The proposed method is rapid and results compare favorably with AOAC method 7.091.

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