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The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type.


We do this in a setting that closely resembles the numerical analysis setting of Mallat and Zhong and that seems to capture something of the essence of their (practical) reconstruction method.


We do this in a setting that closely resembles the numerical analysis setting of Mallat and Zhong and that seems to capture something of the essence of their (practical) reconstruction method.


Mathematical details and numerical examples are included.


Then, we describe a numerical method to compute the dual function and give an estimate of the error.

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 This paper is a supplement to the author's previous paper "The Constants and Analysis of Rigid Frames", published in the first issue of the Journal. Its purpose is to amplify as well as to improve the method of propagating joint rotations developed, separately and independently, by Dr. Klouěek and Prof. Meng, so that the formulas are applicable to rigid frames with nonprismatic bars and of closed type. The method employs joint propagation factor between two adjacent joints as the basic frame constant and the... This paper is a supplement to the author's previous paper "The Constants and Analysis of Rigid Frames", published in the first issue of the Journal. Its purpose is to amplify as well as to improve the method of propagating joint rotations developed, separately and independently, by Dr. Klouěek and Prof. Meng, so that the formulas are applicable to rigid frames with nonprismatic bars and of closed type. The method employs joint propagation factor between two adjacent joints as the basic frame constant and the sum of modified stiffness of all the barends at a joint as the auxiliary frame constant. The basic frame constants at the left of right ends of all the bars are computed by the consecutive applications of a single formula in a chain manner. The auxiliary frame constant at any joint where it is needed is computed from the basic frame constants at the two ends of any bar connected to the joint, so that its value may be easily checked by computing it from two or more bars connected to the same joint.Although the principle of this method was developed by Dr. Klouěek and Prof. Meng, the formulas presented in this paper for computing the basic and auxiliary frame constants, besides being believed to be original and by no means the mere amplification of those presented by the two predecessors, are of much improved form and more convenient to apply.By the author's formula, the basic frame constants in closed frames of comparatively simple form may be computed in a straightforward manner without much difficulties, and this is not the case with any other similar methods except Dr. Klouěek's.The case of sidesway is treated as usual by balancing the shears at the tops of all the columns, but special formulas are deduced for comput ing those column shears directly from joint rotations and sidesway angle without precomputing the moments at the two ends of all the columns.In the method of propagating unbalanced moments proposed by Mr. Koo IYing and improved by the author, the unbalanced moments at all the barends of each joint are first propagated to the barends of all the other joints to obtain the total unbalanced moments at all the barends, and then are distributed at each joint only once to arrive at the balanced moments at all the barends of that joint. Thus the principle of propagating joint rotations with indirect computation of the barend moments is ingeneously applied to propagate unbalanced moments with direct computation of the barend moments, and, at the same time, without the inconvenient use of two different moment distribution factors as necessary in all the onecycle methods of moment distribution. The basic frame constant employed in this method is the same as that in the method of propagating joint rotations, so that its nearest approximate value at any bar end may be computed at once by the formula deduced by the author. Evidently, this method combines all the main advantages of the methods proposed by Profs.T. Y. Lin and Meng ChaoLi and Dr. Klouěek, and is undoubtedly the most superior onecycle method of moment distribution yet proposed as far as the author knows.Typical numerical examples are worked out in details to illustrate the applications of the two methods.  本文為著者前文“剛構常數與剛構分析”之補充,其目的在將角變傳播法及不均衡力矩傳播法加以改善,以便實用。此二法均只需一個公式以計算剛構中所有各桿端之基本剛構常數(即任何二相鄰結點间之角變傳播係數),將此項公式與柯勞塞克之公式相比較,藉以指出前者較後者為便於應用,並亦可用之以直接分析較簡單之閉合式剛構,此外補充說明此法中之剛構常數與定點法之關係,剛構有側移時計算各結點角變所需之各項公式亦行求出。不均衡力矩傳播法係顧翼鹰同志最近研究所得者,既係直接以桿端力矩為計算之對象,而且只須採用不均衡力矩分配比將各結點作用於各桿端不均衡力矩之總和,一次分配,即得所求各桿端分配力矩之總值,實係力矩一次分配法之一大改進,著者將顧氏之法加以推廣与改善,使其原則簡明而計算便捷,著者認為此法係將林、柯、孟三氏法之所有優點熔冶於一爐,實可稱為现下最優之力矩一次分配法。最後列舉算例,以說明此二法在實際工作中之應用。  An approximate and rapid method for computing the stresses and shape constartts of elastic arches is presented in this paper. The fundamental idea of this method lies in the fact that the variation of the elastic area (ds/EI) of the arch with respect to x can be represented approximately by an elementary algebraic function which can be determined with but little labor. After this function is found, all necessary computations can be done by direct integration instead of the usual tedious arithmetic summation.... An approximate and rapid method for computing the stresses and shape constartts of elastic arches is presented in this paper. The fundamental idea of this method lies in the fact that the variation of the elastic area (ds/EI) of the arch with respect to x can be represented approximately by an elementary algebraic function which can be determined with but little labor. After this function is found, all necessary computations can be done by direct integration instead of the usual tedious arithmetic summation. Formulas reduced to the final forms are presented for use, and two numerical examples are given to show the practical procedure and the degree of approximation.  本文目的在提出一種求拱應力及拱常数的近似算法。主要關鍵在求出一能近似表示拱截面ds/EI變化之代數式,使各項計算均得由直接積分完成,避免分段計和,因而極爲簡便。最後結果以公式表示,其準確程度巳敷實際需要,必要時可予提高。  In this paper a method for computing the influence lines in open rigid frames is presented. This method is based on the MüllerBreslau's principle that every deflection diagram is an influence line. If any section of a given rigid frame, at which the influence llne of any stress function——such as reaction, shear, bending moment and torsion——is desired, is allowed to produce freely a corresponding unit deformation, the deflection diagram of this frame will be the same as the influence of that stress function.The... In this paper a method for computing the influence lines in open rigid frames is presented. This method is based on the MüllerBreslau's principle that every deflection diagram is an influence line. If any section of a given rigid frame, at which the influence llne of any stress function——such as reaction, shear, bending moment and torsion——is desired, is allowed to produce freely a corresponding unit deformation, the deflection diagram of this frame will be the same as the influence of that stress function.The fundamental idea of this method is that the anglechanges at ends of bars due to unit deformation can be determined by propagating joint rotations and that the resulting deflection diagram which is the same as the influence line of the corresponding stress function may be determined by method of conjugate beam.Typical numerical examples are worked out to show the application of this method.  本文提供一種求敞口剛架影響線的方法。依據米勒白斯老(MüllerBres1au)氏“變位線即影響線”的原理,令剛架中某點有與其應力函數相应的單位形变,則剛架因此所產生的變位曲線即為該應力函數的影響線。本文所叙述的方法,係利用角變傳播原理,求出各桿兩端由於上项單位形變所引起的角变,再根據此項角變求出各桿的變位曲線,亦即該應力函數的影響線。举有實例以示此法之應用。   << 更多相关文摘 
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