The results of the survey were as follows: sample rate P=1.47%, standard error Sp=0.65%, rate of entrapment 1.47%+0.65%, and 95% confidence interval (0.20%-2.74%).

When a function does not belong to such a space, the sampling series may converge, not to the object function but to an "alias" of it, and an aliasing error is said to occur.

Aliasing error bounds are derived for one- and two-channel sampling series analogous to the Whittaker-Kotel'nikov-Shannon series, and for the multi-band sampling series, and a "derivative" extension of it, due to Dodson, Beaty, et al.

Where possible, the sharpness of the error bounds is discussed.

We investigate the $L_p$-error of approximation to a function $f\in L_p({\Bbb T}^d)$ by a linear combination $\sum_{k}c_ke_k$ of $n$

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

The so-called "truss rigid frames" are those rigid frames with trusses as their horizontal beams, of which the two ends are rigidly connected to columns. Within the author's knowledge, all the methods available at present for analyzing such rigid frames are based on Certain special assumptions such as (1) that the positions of the points of contra-flexure in all the columns are previously known; (2) that the end rotations of a truss may be reprensented by that of its assumed line of axis as in the case of an...

The so-called "truss rigid frames" are those rigid frames with trusses as their horizontal beams, of which the two ends are rigidly connected to columns. Within the author's knowledge, all the methods available at present for analyzing such rigid frames are based on Certain special assumptions such as (1) that the positions of the points of contra-flexure in all the columns are previously known; (2) that the end rotations of a truss may be reprensented by that of its assumed line of axis as in the case of an ordinary beam; or (3) that the end verticals of trusses may be given certain prescribed deformations. Of course, the adoption of any of such assumptions leads to only approximate results inconsistent with the actual deformations of such rigid frames under any loading. Heretofore, the author did not know any correct method for analyzing such rigid frames. In this paper, the author presents two principles of the correct analysis of truss rigid frames. The first principle is that of "moment action on column" for computing the angle change constants of columns, and the second principle is that of "effect of span-change in truss" for computing the angle and span change constants of trusses.As, for computing the angle change constants of a truss, the dummy unit moment is a couple applied to its end verticals, so, for computing the angle change constants of a column, the dummy unit moment must also be a couple applied to the section of column rigidly connected to the end of a truss, in order to effect a consistent deformation at the joint of the two. This is the first principle.A truss just like a curved or gabled beam of which the effect of span-change can not be neglected, so truss rigid frames belong to the same category of what may be called "span-change" rigid frames such as rigid frames with curved or gabled beams. Therefore the span-change constants of trusses should be included besides their angle-change constants for analyzing truss rigid frames. This is the second principle.With the constants of columns and trusses are all computed in accordance with respectively the first and second principles mentioned above, truss rigid frames may be analyzed by any method including the effect of span-change as in the case of rigid frames with curved or gabled beams, and the results thus obtained will be exactly the same as by the method of least work or deflections without any special assumptions.In this paper, after the two principles are described and the formulas for computing the constants of columns and trusses are derived, the correctness of the two principles are then proved by the methods of least work, deflections and slope-deflection. A two-span truss rigid frame is analyzed under the following three conditions:Ⅰ. Applying both of the two principles to obtain the correct results.Ⅱ. Applying only the first principle to show the discrepancies of neglecting the effect of span-change in trusses as born out by comparing the results of Ⅱ with Ⅰ.Ⅲ. Applying neither of the two principles, and the truss rigid frames being analyzed by the special assumption (2) mentioned above with the line of axis at the bottom chord of truss, in order to show the discrepancies of neglecting the moment action on column as born out by comparing the results of Ⅲ with Ⅱ. For the sake of brevity, only the results are given in Tables 1 to 5 without computations in details.Although the discrepancies of neglecting the moment acticn on column are only slight as shown by comparing the results of Ⅲ with Ⅱ in Tables 2, 4 and 5, there is no reason why special assumptions should not be replaced by the correct principle of moment action on column to obtain correct results. As shown by comparing the results of Ⅱ with Ⅰ in Tables 2, 4 and 5, the discrepancies by neglecting the span change in trusses are generally considerable and, in certain particular part, as large as 3000%. Therefore, for the safe and economical design of truss rigid frames, the effect of span-change in trusses should not be neglected in their analysis.Finally, for analyzing co

As in general in all time service works, the problem is to obtain and keep an accurate time, to determine the corrections of standard clock and to transmit time signals. This present report is concerned only with the last two items. At Zi-Ka-Wei Observatory, radio method was introduced into time service in 1914. Some rigorous changes occured in 1926, a number of instruments were installed. Many of them are still in use. In 1940, the observatory began to transmit rhythmic signals twice a day. From then on until...

As in general in all time service works, the problem is to obtain and keep an accurate time, to determine the corrections of standard clock and to transmit time signals. This present report is concerned only with the last two items. At Zi-Ka-Wei Observatory, radio method was introduced into time service in 1914. Some rigorous changes occured in 1926, a number of instruments were installed. Many of them are still in use. In 1940, the observatory began to transmit rhythmic signals twice a day. From then on until December 1950, no further developments were made. The main instruments we had in 1950 were: a 80mm Prin transit with impersonal micrometer, two Leroy pendulums as standard clocks, three astronomical clocks and their slaves, equipments for receiving and recording time signals, etc. Since 1952, we began to make some improvements, especially on rhythmic signals. First, we converted a common clock into a "transmitting clock" to obtain 61 impulses in every minute and second by making use of photoelectric arrangement (Fig. 1), we succeeded to raise the accuracy of our signals. To improve signal accuracy further, we have to deal with various sources of errors which are conditioned by our equipments. The most important among these are: 1. error in prediction of corrections of the standard clocks, 2. error in the adjustment of the transmitting clock and 3. time lag in transmission. In consequence of last three-year's research, these errors are reasonably reduced. The first two are now±0~s.007 and ±0~s.005 respectively and the last is small. Accordingly, the deviation of time signals XSG has been supposed to be±0~s.01. However, this is not enough for the requirement. The more efforts are being made to bring further improvements. But as the requirements for signal accuracy is far above what the present installation can offer, some new equipments are ordered and they will joint in force with the old ones in the nearest future. Since February 1954, Zi-Ka-Wei Observatory has cooperated with the time service departments in Soviet Union. This has been helpful to our works in every way.

The method of complementary I_0/I diagram for simplifying the computations of non-uniform beam constants is presented in this paper. The so-called "complementary I_0/I diagram" is the remaining I_0/I diagram of the haunched or de-haunched (or tapered) parts at the two ends of a beam after the I_0/I diagram of a non-uniform beam has been subtracted from the I_0/I = 1 diagram of a uniform beam.In the method of I_0/I diagram presented previously by the second author, the various momental areas have to be computed...

The method of complementary I_0/I diagram for simplifying the computations of non-uniform beam constants is presented in this paper. The so-called "complementary I_0/I diagram" is the remaining I_0/I diagram of the haunched or de-haunched (or tapered) parts at the two ends of a beam after the I_0/I diagram of a non-uniform beam has been subtracted from the I_0/I = 1 diagram of a uniform beam.In the method of I_0/I diagram presented previously by the second author, the various momental areas have to be computed for the entire length of a beam; in the method of complementary I_0/I diagram, the various momental areas need be computed for the lengths of the non-uniform sections at the two ends of the beam only. Hence the latter method is somewhat simpler than the former and may be considered as its improvement.The angle-change constants are the fundamental constants of a nonuniform beam, and only the coefficients of the angle-change constants need be computed. As any non-uniform beam may be considered as a uniform beam haunched or de-haunched or tapered at its one or both ends, the various anglechange coefficients φ may be computed separately in three distinct parts, viz., of a uniform beam, and φ~a and φ~b of the haunches at its two ends a and b, and then summed up as shown by the following general equation:φ=φ~a-φ~b (A) The values φ~a and φ~b are positive for haunched beams and negative for dehaunched or tapered beams, and either of them is zero for the end which is neither haunched nor de-haunched. To simplify the computations of the values of φ~a and φ~b, the complementary I_0/I diagram at each end of a beam is substituted by a cubic parabola passing through its two ends and the two intermediate points of the abscissas equal to 0.3 and 0.7 of its length. Then the value of φ~a or φ~b is computed with an error of usually less than 1% by the following formula:φ~a or φ~b = K_(0y0)+K_(3y3)+K_(7y7), (B) wherein y0, y3 and y7 are respectively the ordinates at the abscissa equal to 0, 0.3, and 0.7 of the length of the diagram, and the three corresponding values K_0, K_3 and K_7 are to be found from the previously computed tables.A set of the tables of K-values for calculating the values of φ~a and φ~b of the shape angle-changes and the load angle-changes under various loading conditions may be easily computed, which evidently has the following advantages: (1) As indicated by formulas (A) and (B), the computations of φ~a, φ~b and φ with K-values known are very simple; (2) the approximation of the results obtained is very close; (3) A single set of such K-value of the tables is applicable to non-uniform beams of any shape, any make-up, and any crosssection; and (4) as the K-values are by far easier to compute than any other constants, a comprehensive set of the tables of K-values with close intervals and including many loading conditions may be easily computed.Besides, by means of formulas (A), existing tables of constants such as A. Strassner's for beams haunched at one end only may be utilized to compute the shape and load constants for asymmetrical beams with entirely different haunches at both ends.Finally, five simple but typical examples are worked out first by the approximate method and then checked by some precise method in order to show that the approximation is usually extremely close.