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For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation.
      
The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type.
      
The inverse conductivity problem to the the elliptic equation ${\rm div}((1+(k-1)\chi_D)\nabla u)=0\ {\rm in }\ \Omega$ is considered.
      
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Since the year 1886, the Dupuit-Forchheimer theory and formulas for gravityflows toward wells and galleries have been broadly used in all nations of theworld. In these formulas, assumptions are made for underground flows fromfarther distances in horizontal directions at a constant rate toward wells andgalleries. According to the author's analysis, these assumptions are not consistentwith the actual conditions of flow, hence the formulas obtained therefrom are notrational, and the employment of these formulas...

Since the year 1886, the Dupuit-Forchheimer theory and formulas for gravityflows toward wells and galleries have been broadly used in all nations of theworld. In these formulas, assumptions are made for underground flows fromfarther distances in horizontal directions at a constant rate toward wells andgalleries. According to the author's analysis, these assumptions are not consistentwith the actual conditions of flow, hence the formulas obtained therefrom are notrational, and the employment of these formulas to investigate the general effectof surface drop or well diametre upon yield is devoid of rational foundations. According to the author's analysis, the flows toward wells or galleries areactually supplied vertically by draining the stored water above the free surface inthe course of its descending and enlarging. As the drainage of gravity water fromthe pores of soil particles in order to reduce the water content to that of filmshells takes one to two days, as the capillary water columns are interconnectedand mutually supplied sidewise, this vertical supply of water may maintain quite along time, yet the flow may not be absolutely steady. As regards vertical supply of water with unsteady regimen, equations of freesurface for flow pattern near galleries are deduced, corresponding to the Boussinesqpartial differential equation. Besides, the author has derived simplified equationsfor computing flows into wells and galleries. The latter, in comparison with theDupuit-Forchheimer formulas, gives a higher yield, while the free surface curve isreasonably tangent to the horizontal water table at a point which moves fartheraway as time goes on. J. Kozeny first pointed out the phenomenon that the water depth in the groundon the wall will not be further lowered when it reaches one half of the depthbefore pumping. The author hereby proposes a theoretical proof of it on the basisof theorem of least work. Based upon these theories, formulas are proposed for maximum possible yieldof wells and galleries dug to horizontal impervious strata, to be used in prelimi-nary estimations for hydro-geological workers.

1886年以来,杜布义-福熙罕默(Dupuit-Forchheimer)的井流及沟流的理论与计算用公式被世界各国广泛地应用着。公式假设地下水从远处沿着水平方向以定率流向井内或沟内,按作者分析这种假设并不符合实际情况,因之所得公式也不合理,用这些公式来推论水位降落或井径对於出水率的影响也没有合理的凭据。作者推论,井流或沟流的水实际上是从水面线以上,在其降落并扩大的过程中,排除了存积的水,沿着直垂方向所供应着的。因为从土壤颗粒的空隙间排除重力水,使减为薄膜水,每需时一两天,而水面上的毛细管水又是横向贯通并互相接济着,所以垂直供水可以维持很久,而潜流也决不会绝对稳定。根据这垂直供水的不定汉条件引出了沟流的水面线公式,结果符合蒲薪奈斯克的偏微分方程式。另外,作者又拟具了简化的井流及沟流计算用公式。这些公式和杜氏-福氏公式比较,所得出水率较大,而水面线则合理地切於静水线,切点随着时程向远处移动。柯臣尼(J.Kozeny)最早指出井边地内水深不会低於静水深一半的现象,本文中作者根据最小工作定律试拟了理论的证明以支持之。根据这些理论,引出了从静水中抽水时井流、沟流最大可能出水率的公式,以供水文地质工作者初步估算之用。

The problem of investigating the vibration properties of continuous girders on elastic piers frequently occurs in practical design work. An approximate method of attack is to replace the elastic pier by a spring support. This method simplifies greatly the analysis and is generally adopted in practice. In the first part of this paper, this approximate method is investigated in detail and complete solutions including the effect of initial conditions are given. The computations are made more systematic and simple...

The problem of investigating the vibration properties of continuous girders on elastic piers frequently occurs in practical design work. An approximate method of attack is to replace the elastic pier by a spring support. This method simplifies greatly the analysis and is generally adopted in practice. In the first part of this paper, this approximate method is investigated in detail and complete solutions including the effect of initial conditions are given. The computations are made more systematic and simple by the adoption of a system of constants defined as the dynamic shape constants and load constants of the structure. In the second part of this paper, the more accurate method of analysis is investigated. The exact frequency equations of the system are obtained by writing down the dynamic equations of the girders and the piers simultaneously and the boundary conditions are taken into consideration. It is shown that by taking only the fist one or two terms of the series expansions, the exact frequency equation reduces to the approximate one. It is concluded that the approximate method will give adequate solutions in the case when the piers are rigid and when only the lower frequencies of the natural mode need be considered. Particular solutions of structures under transverse vibrations are worked out and is found more involved than that given by the approximate method.

本文所謂架空結構是指搁置在較高的支墩上的連續梁結構,例如高架橋梁或水電站的架空輸水管道等。這類結構在承受震動荷載(例如地震)時,常會引起較大的變形和应力。如支墩較粗矮,則在計算較低的頻率時,其作用可近似的以一彈簧支座代替之,而忽略支墩本身的結構震動影響。這是一個常用的假定,可以使計算工作充分簡化。本文首先對這一情况作了詳盡討論,給出较完整的解答,包括起始條件影響在内,並提出用“載常數”及“形常數”的觀念來進行計算。然后本文給出考虑支墩的震動影響時的精確解答,分別就縱横方向的震動進行討論,並指出近似計算公式和精確公式间的關係,近似公式只能在支墩較剛固和計算较低階的頻率時才適用。文中列出了結構作横向諧和震動時的特別解答,可見其計算工作量將比近似法增加不少。

Through the use of approximate expessions in the form of quadratic equations for the sum as well as the difference of principal stresses along any definite vertical line in foundation soil subjected to triangular and trapezoidal loading due to earth embankments. the author has derived formulas for critical load corresponding to the condition that plastic zone developed it tangential to the vertical line under consideration. Based on studies of the developed plastic zones in the foundation soil due to such...

Through the use of approximate expessions in the form of quadratic equations for the sum as well as the difference of principal stresses along any definite vertical line in foundation soil subjected to triangular and trapezoidal loading due to earth embankments. the author has derived formulas for critical load corresponding to the condition that plastic zone developed it tangential to the vertical line under consideration. Based on studies of the developed plastic zones in the foundation soil due to such a triangular and trapezoidal loading. it is proposed, that the criterion for determining the critical load on the foundation roil is that the developed plastic zone is tangential to a vertical line at one quater base width from the center line. Special considerations are given to the cases for convenient soil as well at for saturated clay of zero degree of consolidation. Charts convenient for pratical use are also prepared. Finally, numerical examples are presented which show that the proposed formulas are quite in conformity with field observations in engineering practice.

作者以二次方程代替在路堤及土坝的三角形与梯形荷载下地基中沿各铅垂线上的主应力和与主应力差的表示式,得出了相应于塑性区发展至与各指定的铅垂线相切的临界荷载公式.在比较全面地研究了三角形与梯形荷载下特有的塑性区发展规律的基础上,提出了以塑性区与铅垂线X=B/4相切作为确定的基的许可荷载的临界条件.文中还就无粘性土以及固结度为零的饱和粘土的临界荷载问题进行专门的讨论,给出了便于实际应用的图表. 最后举出的两个实际工程例子验证了本文提出的理论与实践的一致性.

 
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