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  theory
We use the theory of tilting modules for algebraic groups to propose a characteristic free approach to "Howe duality" in the exterior algebra.
      
This study was continued in the paper [FKRW] in the framework of vertex algebra theory.
      
As in the case of Mumford's geometric invariant theory (which concerns projective good quotients) the problem can be reduced to the case of an action of a torus.
      
We also show how to distinguish examples of open subsets with a good quotient coming from Mumford's theory and give examples of open subsets with non-quasi-projective quotients.
      
The theory is applied to the case of cubic hypersurfaces, which is the one most relevant to special geometry, obtaining the solution of the two classification problems and the description of the corresponding homogeneous special K?hler manifolds.
      
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It is pointed out in this paper that the following apparent discrepancies exist in Coulomb's Theory: (1) In any problem in mechanics, a force to be definite must have all the three factors involved under consideration. In Coulomb's Theory, however, the point of application of the soil reaction on the plane of sliding is somehow neglected, thus enabling the arbitrary designation of the obliquity of the earth pressure on the wall to be equal to the friction angle between the wall surface and soil. As a matter...

It is pointed out in this paper that the following apparent discrepancies exist in Coulomb's Theory: (1) In any problem in mechanics, a force to be definite must have all the three factors involved under consideration. In Coulomb's Theory, however, the point of application of the soil reaction on the plane of sliding is somehow neglected, thus enabling the arbitrary designation of the obliquity of the earth pressure on the wall to be equal to the friction angle between the wall surface and soil. As a matter of principle, the point of application should never be slighted while the obliquity of the earth pressure could only have a value that is compatible with the conditions for equilibrium. (2) If the point of application of the soil reaction is taken into account in the problem, the sliding wedge would only tend to slide either on the plane of sliding or on the surface of wall but not on both at the same time, thus frustrating the very conceptidn of sliding wedge upon which Coulomb's Theory is founded. (3) The above discrepancies arise from the fact that the shape of the surface of sliding should be curvilinear in order to make the wedge tend to slide as desired, while Coulomb, however, adopted a plane surface instead. (4) Coulomb, in finding the plane of sliding, made use of the maximum earth pressure on the wall (for active pressure), which refers to the different magnitudes of pressure corresponding to different assumed inclinations of the plane of sliding. But from the relation between the yield of wall and amount of pressure, this maximum value is really the minimum pressure on the wall, which it is the purpose of the theory to find. In engineering books, however, this terminology of maximum pressure has caused considerable confusion, with the result that what is really the minimum pressure is carelessly taken as the maximum design load for the wall. How can a minimum load be used in a design?This paper also attempts to clarify some contended points in Rankine's Theory: (1) It is claimed by Prof. Terzaghi that Rankine's Theory is only a fallacy because of the yield of wall and that of the soil mass on its bed. This charge is unjust as it can be compared with Coulomb's Theory in the same respect. (2) Some books declare that Rankine's Theory is good only for walls with vertical back, but it is proved in this paper that this is not so. (3) It is also generally believed that Rankine's Theory is applicable only to wall surfaces with no friction. This is likewise taken by this paper as unfounded and illustration is given whereby, in this regard, Rankine's Theory is even better than Coulomb's, because it contains no contradiction, as does Coulomb's.

本文從力學觀點對庫隆理論提出下列問題:(1)在解算力學問題時,每個力有三個因素都該同時考慮,但庫隆對土楔滑動面上土反力的施力點竟置之不理,因而才能對擋土墙上土壓力的傾斜角作一硬性假定,使它等於墙和土間的摩阻角,然而施力點是不能不管的,因而土壓力的傾斜角是不能離開平衡條件而被隨意指定的。(2)如果考慮了土反力的施力點,則土楔祇能在滑動面上,或在墙面上,有滑動的趨勢,而不能同時在兩個面上都有滑動的趨勢,因而庫隆的基本概念“滑動土楔”就站不住了。(3)問題關鍵在滑動面的形狀;如要使土楔在滑動面和墙面上同時有滑動趨勢,則滑動面必須是曲形面,然而庫隆採用了平直形的滑動面。(4)庫隆的土楔滑動面是從墙上最大的土壓力求出的(指主動壓力),這裏所謂“最大”是指適應各個滑動面的各個土壓力而言,但對適應墙在側傾時土壓力應有的變化來說,這個最大土壓力却正是墙上極限壓力的最小值。一般工程書籍,以為這土壓力既名為最大,就拿它來用作設計擋土墙的荷載,荷載如何能用最小的極限值呢?本文對朗金理論中的下列問題作了一些解釋:(1)朗金理論在擋土墙的位移問題上所受的限制,是和庫隆理論一樣的,竇薩基教授曾就此問題認為朗金理論是幻想,似乎是無根據的。...

本文從力學觀點對庫隆理論提出下列問題:(1)在解算力學問題時,每個力有三個因素都該同時考慮,但庫隆對土楔滑動面上土反力的施力點竟置之不理,因而才能對擋土墙上土壓力的傾斜角作一硬性假定,使它等於墙和土間的摩阻角,然而施力點是不能不管的,因而土壓力的傾斜角是不能離開平衡條件而被隨意指定的。(2)如果考慮了土反力的施力點,則土楔祇能在滑動面上,或在墙面上,有滑動的趨勢,而不能同時在兩個面上都有滑動的趨勢,因而庫隆的基本概念“滑動土楔”就站不住了。(3)問題關鍵在滑動面的形狀;如要使土楔在滑動面和墙面上同時有滑動趨勢,則滑動面必須是曲形面,然而庫隆採用了平直形的滑動面。(4)庫隆的土楔滑動面是從墙上最大的土壓力求出的(指主動壓力),這裏所謂“最大”是指適應各個滑動面的各個土壓力而言,但對適應墙在側傾時土壓力應有的變化來說,這個最大土壓力却正是墙上極限壓力的最小值。一般工程書籍,以為這土壓力既名為最大,就拿它來用作設計擋土墙的荷載,荷載如何能用最小的極限值呢?本文對朗金理論中的下列問題作了一些解釋:(1)朗金理論在擋土墙的位移問題上所受的限制,是和庫隆理論一樣的,竇薩基教授曾就此問題認為朗金理論是幻想,似乎是無根據的。(2)有些工程書中認為朗金理論是專為垂直的墙?

The analysis of stepped beams on spring foundation is a problem of practical importance. It is shown in the paper that this problem is similar to that of a continuous curved beam on rigid supports and can be most easily solved by method of special slope deflection equations. The formulas for computing load and shape constants being necessarily long, a number of tables of useful functions have been prepared to aid in a quick analysis and a typical example is given. The theory and functions presented in...

The analysis of stepped beams on spring foundation is a problem of practical importance. It is shown in the paper that this problem is similar to that of a continuous curved beam on rigid supports and can be most easily solved by method of special slope deflection equations. The formulas for computing load and shape constants being necessarily long, a number of tables of useful functions have been prepared to aid in a quick analysis and a typical example is given. The theory and functions presented in the paper are also applicable in analyzing other problems with same mathematical nature, such as the problem of axisymmetrical bending of cylinders with nonuniform wall thickness.

本文討論彈性地基上的阶形梁的計算問題。应用結構力学上“桿件常数”的观念,这个問题可以簡捷地获得解决。文中除举例說明計算步驟外,並供給必要的函数表,使这些常数能够迅速求出。这些函数同样可以用来解决数学上类似的問題,例如变截面圓筒的軸对称受弯问題。

Two methods for analyzing caisson-beams are introduced in this paper.One is the well-known method of redundant forces. The author has simplified this methed by using couples of redundant forces to set up a typical equation and pointing out the rule that the matrix of the coefficients of simultaneous linear equations which are organized from the expansion of that typical equation. This method can be easily solvd when the number of unknown redundant forces, or that of equations, is less than 3 or 4; but it will...

Two methods for analyzing caisson-beams are introduced in this paper.One is the well-known method of redundant forces. The author has simplified this methed by using couples of redundant forces to set up a typical equation and pointing out the rule that the matrix of the coefficients of simultaneous linear equations which are organized from the expansion of that typical equation. This method can be easily solvd when the number of unknown redundant forces, or that of equations, is less than 3 or 4; but it will be difficult when the number is more than that. In order to solve this difficulty the author suggests another kind of method of which the essential principle is mentioned in the following.Supposing that the distance between the beams is sufficiently short in comparing with their spans, we can set up a partial differential equation for its deffiection W, as we often do in the theory of elasticity. In this way we can solve it with its boundary conditions of simple supporting by sine series. From this we can easily get the formulas of bending moments, shears and twist moments of each beam by partially differentiating the function of deffiection. The result of the calculation proves that it quite agrees with the method of redundant forces when the distance between beams is no longer than 1/5 of their spans.There are some tables given in this raper for practical use.

本文介紹了計算井字梁的兩种方法。 第一种方法是按冗力送来計算的。本文利用了成对的未知力以建立冗力法的典型方程,指明了由典型方程所組成的联立方程中其系数排列的規律,从而簡化了建立方程的过程和減少錯誤的机会。冗力法仅在联立方程的数目不多于3至4时是相当方便的,若未知冗力过多,解算过程便異常繁重。为此,笔者提出了下述的第二种方法。 当梁的間距比梁的跨度为一較小的数值时,可应用彈性力学所常用的方法,建立一个关于井字梁撓度曲面的偏微分方程。以符合簡支边界条件的正弦级数求出撓度后,便可依微分关系求出各梁的弯矩、切力及扭矩。鈇摩辛柯在其著作中(見[3]§37)討論向異性板的弯曲时,亦曾附帶地提及本法的可能性。本文給出了全部計算公式及为实用的目的而制訂了各种数表。計算的結果表明当梁間距不大于跨度的1/5时,本法与冗力法的結果相差無几,而本法在应用上的簡便笑为任何方法所不及。

 
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