text:
## Abstract

A rapid aeroelasticity optimization method based on the stiffness characteristics was proposed in the present study. Large time expense in static aeroelasticity analysis based on traditional time domain aeroelasticity method is solved. Elastic axis location and torsional stiffness are discussed firstly. Both torsional stiffness and the distance between stiffness center and aerodynamic center have a direct impact on divergent velocity. The divergent velocity can be adjusted by changing the correlative structural parameters. The relation between structural parameters and divergent velocity is introduced to aeroelasticity optimization design as a constraint condition. After optimization, the structural and aerodynamic characteristics have a large improvement while satisfying the constraint conditions. The optimization method can be well used in high aspect ratio wing and has great computational efficiency.

Keywords: Aeroelasticity optimization method, divergent velocity, stiffness characteristics, aerodynamic characteristics, computational efficiency

## 1. Introduction

New large transport airplane, long-endurance unmanned aerial vehicles and civilian aircraft, which need to meet the need of large lift and light weight, generally have a large flexible feature. Aeroelasticity is the coupling of aerodynamic and structural elasticity of the aircraft. It can be divided into static aeroelasticity and flutter. The change of aerodynamic and deformation distribution is mainly concerned in the static aeroelasticity, which should lead to the decline of flight performance and may even results in the divergent or control reversal. The stiffness of these aircrafts is often small and brings a serious impact on aeroelasticity characteristics, which will lead to a significant decline in flight qualities. There is a critical speed for each aircraft. It is defined as divergent velocity. Both aircraft layout and mass distribution have a direct impact on divergent velocity. Higher divergent velocity and lighter weight are two key factors in aircraft design. Aeroelasticity optimization design method, which considers both weight and aerodynamic characteristics, is a typical application of multi-disciplinary design optimization (MDO). It has been widely used in aircraft design.

Many scholars dedicated to research in the MDO and a lot of methods were put forward. Wakayama [1] showed an initial structural optimization which has not a proper MDO framework. As the development of computational fluid dynamic models, Guruswamy [2] and Yang [3] developed a particularly true MDO framework for the aerodynamic shape optimization. Unfortunately, the computational time of MDO analysis increased drastically. For reducing the time, Zhang [4] proposed surrogate models. At the same time, many algorithmic methods which reproduce some natural phenomena or physical processes were proposed, such as Genetic and Evolutionary Algorithms [5], Swarm Intelligence [6] and Simulated Annealing [7].

Aeroelasticity optimization design method, as a typical application of MOD, is also studied by many scholars. Mastroddi [8] studied the initial design of wing structures based on the integrated modeling of structures, aerodynamic, flight dynamics and aeroelasticity. In recent years, Su [9] proposed a static aeroelasticity optimization method for an entire aircraft vehicle based on computational fluid dynamics and computational structural dynamics coupling method. Nikbay [10] established a reliability-based multidisciplinary optimization framework by coupling high-fidelity commercial solvers for aeroelasticity analysis and an in-house code developed for reliability analysis. Sleesongsom [11] presented an approach to achieve simultaneous partial topology and sizing optimization of a three-dimensional wing-box structure. Wesley [12] developed a design process which incorporates the object-oriented multidisciplinary design, analysis, and optimization tool and the aeroelastic effect of high fidelity finite element models to characterize the design space. Eric [13] presented an approach to the development of a scaled wind tunnel model for static aeroelastic similarity with a full-scale wing model. Bret [14] demonstrated the feasibility of developing, in an automated manner, detailed structural topologies within the ribs and spars of a wing box, driven by complex aeroelastic physics.Mallik [15] investigated the effect of a novel wing-tip on the aeroelastic behavior of passenger aircraft with flexible truss-braced wings. Stanford [16] carried out the static aeroelastic tailoring of a subsonic transport wing box, as part of NASA's Subsonic Fixed Wing project. Francois [17] studied the aeroelastic tailoring using Rib/Spar orientations based on experimental investigation. It is found that the spar/rib orientations have a significant influence on the structural behavior. Roscher [18] presented a structural optimization and aeroelastic analysis of an optimizaed Troposkein vertical axis wind turbine to minimize the relation between the rotor mass and the swept area. Zuo [19] put forward to optimize the cruise shape directly based on the fact that the cruise shape can be transformed into jig shape, and a methodology named reverse iteration of structural model is proposed to get the aero-structural performance of cruise shape. They all investigated the accuracy and efficiency of aircraft aeroelasticity optimization design and a lot of considerable methods were proposed. But computational efficiency still has an obvious impact on the practical aircraft design.

Stiffness characteristics have an directly impact on aeroelasticity characteristics. Serious aeroelasticity problem will occur under an irrational stiffness characteristic design. In the present study, we aim at investigating stiffness characteristics and its application for a rapid aeroelasticity optimization design method.

## 2. Stiffness characteristics analysis

Stiffness characteristic has a direct impact on aeroelasticity characteristics. Serious aeroelasticity problem will occur under an irrational stiffness characteristics design. Stiffness characteristics of the wing structure are usually consisting of bending stiffness, torsional stiffness and a flexibility influence coefficient matrix. Elastic axis plays an important role in stiffness characteristics. Elastic axis location and stiffness characteristics of a high aspect ratio wing will be discussed in this section firstly. Figure 1 shows the finite element model of a high aspect ratio wing. Its span is 32.14m, aspect ratio is 15.1, while backswept is 6.24°. Four control surfaces are arranged on wing root trailing. They are connected with the main wing structure through some hinges. In the finite element model, skin, web of the spar and rib are simulated by shell elements, while stringers and fillets of the spar are simulated by bar elements. The wing root is extended and fixed on the extended ending, which makes an elastic support on wing root.

Figure 1. Finite element model of a high aspect ratio wing

### 2.1 Elastic axis location

Elastic axis is such a position where only bending happens on wing cross section when the load applies on it. Elastic axis location directly affects stiffness characteristics of wing structure. It needs to have an intuitive understanding before stiffness characteristics analysis. Part of this section has been introduced in detailed in reference [20], here we will not repeat it again and just show some important steps. Elastic axis is composed of many wing cross section stiffness centers. For computation convenience, wing cross section is equivalent to a thin walled structure with single cell closed section where parts of the leading edge and trailing edge are neglected. In the two dimensional cross section model, skin and spar web are equivalent to some line sections while stringer and spar flange to some points which just have an intensive area. Two equivalent processes, area equivalent and material equivalent, are carried out for the model simplifying.

(1) Area equivalent

Cross section area of each component has a direct effect on stiffness center. For skin, all of the cross section areas are effective. Selecting part of the skin which locates [a, b] interval, its effective area can be written as

 ${\displaystyle A_{skin}={\int }_{a}^{b}tds}$
(1)

Where, ${\displaystyle A_{skin}}$ is skin effective area and ${\displaystyle t}$ is its thickness.

For stringer, it needs to be equivalent to an intensive area and just shown as a point in the simplified model. Usually, the point needs to be in the line section obtaining from skin equivalent. So stringer area needs to be reduced, it can be written as

 ${\displaystyle A_{stringer}{}^{\ast }=A_{stringer}{\frac {h_{1}{}^{2}}{h_{0}{}^{2}}}}$
(2)

Where, ${\displaystyle A_{stringer}{}^{\ast }}$ and ${\displaystyle A_{stringer}}$ are reduced and initial area of the stringer, respectively. ${\displaystyle h_{1}}$ is the distance between skin and neutral surface of airfoil cross section, ${\displaystyle h_{0}}$ is the distance between stringer centroid and neutral surface of airfoil cross section.

Spar, top and down flanges are reduced as intensive area just as the method in stringer treatment, while web is treated as the same as skin.

(2) Material equivalent

Skin, stringer and spar are always composed of different materials. It is needed to select one of them as material sample and others equivalent to dummy component of material sample. Reduction factor ${\displaystyle \varphi }$ is introduced and it is written as

 ${\displaystyle \varphi =E_{i}/E_{0}}$
(3)

Where, ${\displaystyle E_{i}}$ is Young’s modulus of practical material and ${\displaystyle E_{0}}$ is Young’s modulus of material sample.

The equivalent area of each wing cross section component and cross section centroid can be obtained from equation 4.

${\displaystyle A=\oint {\varphi }_{i}t_{i}ds+\sum _{i}{\varphi }_{i}A_{i}}$
 ${\displaystyle {\xi }_{0}={\frac {\oint {\varphi }_{i}t_{i}x_{i}ds+\sum _{i}{\varphi }_{i}A_{i}x_{i}}{A}}}$
(4)

${\displaystyle {\eta }_{0}={\frac {\oint {\varphi }_{i}t_{i}y_{i}ds+\sum _{i}{\varphi }_{i}A_{i}y_{i}}{A}}}$

Where ${\displaystyle {\varphi }_{i}}$ is reduction factor, ${\displaystyle t_{i}}$ is the thickness of skin which number is ${\displaystyle i}$ , ${\displaystyle A}$ is the sum area of all component cross section, ${\displaystyle {\xi }_{0}}$ and ${\displaystyle {\eta }_{0}}$ are ${\displaystyle x}$ and ${\displaystyle y}$ coordinate value of centroid, respectively.

Stiffness center computation is carried out in inertia axis, so it is needed to transfer coordinate, which can be implement through subtraction the value of centroid. Inertia moment of reduced cross section to inertia axis can be obtained from equation 5.

${\displaystyle J_{x}=\oint y_{i}{}^{2}{\varphi }_{i}t_{i}ds+\sum _{i}{\varphi }_{i}A_{i}y_{i}{}^{2}}$
 ${\displaystyle J_{y}=\oint x_{i}{}^{2}{\varphi }_{i}t_{i}ds+\sum _{i}{\varphi }_{i}A_{i}x_{i}{}^{2}}$
(5)

And moment of area of reduced cross section to inertia axis can be obtained from equation 6.

${\displaystyle S_{x}=\oint y_{i}{\varphi }_{i}t_{i}ds+\sum _{i}{\varphi }_{i}A_{i}y_{i}}$
 ${\displaystyle S_{y}=\oint x_{i}{\varphi }_{i}t_{i}ds+\sum _{i}{\varphi }_{i}A_{i}x_{i}}$
(6)

Based on structural mechanics theory, stiffness center of cross section can be written as

${\displaystyle X={\frac {1}{J_{x}}}\left(\int _{s}S_{x}\rho ds-\Omega {\frac {\oint {\frac {S_{x}}{Gt}}ds}{\oint {\frac {ds}{Gt}}}}\right)}$
 ${\displaystyle Y={\frac {1}{J_{y}}}\left(\int _{s}S_{y}\rho ds-\Omega {\frac {\oint {\frac {S_{y}}{Gt}}ds}{\oint {\frac {ds}{Gt}}}}\right)}$
(7)

Where ${\displaystyle \rho }$ is the distance between centroid and tangent of each component and ${\displaystyle \Omega }$ is two times area of cross section enclosed. Stiffness centers of all cross sections are obtained based on the above method. Hence, wing elastic axis position can be obtained through a least-squares procedure.

### 2.2 Stiffness characteristics

Both bending and torsional stiffness characteristics of cross section can be obtained based on the stiffness center. Bending stiffness ${\displaystyle D_{stiff}}$ of the wing cross section can be written as

 ${\displaystyle D_{stiff}=\sum _{i}E_{i}I_{i}}$
(8)

Where ${\displaystyle E_{i}}$ is Young’s modulus of the cross section and ${\displaystyle I_{i}}$ is the moment of inertia of each component in stiffness center coordinate system. Torsional stiffness ${\displaystyle C_{tor}}$ of the wing cross section can be written as

(9)

Where ${\displaystyle G_{i}}$ is shear modulus and ${\displaystyle t_{i}}$ is thickness of each component.

Figures 2 and 3 show the bending and torsional stiffness of different cross sections, respectively. It can be seen that both of them decrease along wingspan. It is noteworthy that torsional stiffness value is significantly less than that of bending stiffness at the same nondimensional span.

Figure 2. Bending stiffness of different cross sections

Figure 3. Torsional stiffness of different cross sections

## 3. A rapid aeroelasticity optimization method

The stiffness of large aspect ratio wing is relatively weak, which will brings a serious impact on aeroelasticity characteristics. At the same time, the increment of stiffness will lead to an unwieldy structure design. So, an integrated aeroelasticity optimization method, considering strength, stiffness, stability, static aeroelasticity characteristics and weight of the wing structure, is particularly important. Among these factors, static aeroelastictity characteristics analysis is an iterative process, which spent large time and just can give a range for the divergent velocity based on the traditional time domain method. Large time expense takes new challenges to the aeroelasticity optimization method, which leads to many optimization design of high aspect ratio wing can not be successfully carried out. In this part, divergent velocity analysis is carried out firstly. Then, its influence factors are discussed. Finally, a rapid aeroelasticity optimization method, controlling these influence factors replace the traditional iterative process, is introduced.

### 3.1 Divergent velocity analysis

Both structural deformation and aerodynamic distribution are influenced by static aeroelasticity characteristics. The effect is small when flight speed is slow. And it increased with the increased of flight speed until a critical speed is reached. And then, a divergence phenomenon is occurred. This critical speed is divergent velocity. Take example for a two-dimensional wing [21], bending and torsional deformation of the wing are simulated as sink-float and pitch of the airfoil. Aerodynamic load caused by sink-float is feeble, so just pitch is considered in airfoil aeroelasticity problem. Practical airfoil attack angle ${\displaystyle \alpha }$ is composed of initial attack angle ${\displaystyle {\alpha }_{0}}$ and torsion angle ${\displaystyle \theta }$ under the function of aerodynamic pressure. Moment of aerodynamic pressure on stiffness center ${\displaystyle M_{e}}$ can be written as

 ${\displaystyle M_{e}=q_{d}ScC_{MAC}+q_{d}S\left[{\frac {\partial C_{L}}{\partial \alpha }}({\alpha }_{0}+\right.}$${\displaystyle \left.\theta )\right]e}$
(10)

Where ${\displaystyle q_{d}}$ is dynamic pressure, ${\displaystyle S}$ is airfoil area, ${\displaystyle c}$ is cross section chord, ${\displaystyle C_{MAC}}$ is moment coefficient, ${\displaystyle {\frac {\partial C_{L}}{\partial \alpha }}}$ is slope of lift curve, while ${\displaystyle e}$ is the distance between stiffness center and aerodynamic center.

Constraints of the airfoil are applied by two springs. Equilibrium equation of aerodynamic moment and spring elastic moment is written as

 ${\displaystyle q_{d}S\left[{\frac {\partial C_{L}}{\partial \alpha }}({\alpha }_{0}+\right.}$${\displaystyle \left.\theta )\right]e+q_{d}cSC_{MAC}=k_{\alpha }\theta }$
(11)

Where ${\displaystyle k_{\alpha }}$ is spring elastic coefficient. Torsion angle ${\displaystyle \theta }$ can be obtained and written as

 ${\displaystyle \theta ={\frac {{\frac {q_{d}S}{k_{\alpha }}}(e{\frac {\partial C_{L}}{\partial \alpha }}{\alpha }_{0}+cC_{MAC})}{1-{\frac {q_{d}eS}{k_{\alpha }}}{\frac {\partial C_{L}}{\partial \alpha }}}}}$
(12)

When the denominator of in equation 16 equals to zero, torsion angle drives to infinite. By now, the wing is in a divergent state. And divergent velocity ${\displaystyle V_{D}}$ can be written as

 ${\displaystyle V_{D}={\sqrt {\frac {2k_{\alpha }}{{\rho }_{\alpha }eS\partial C_{L}/\partial \alpha }}}}$
(13)

It also can be rewritten as

 ${\displaystyle V_{D}={\sqrt {\frac {2}{\rho S\partial C_{L}/\partial \alpha }}}\cdot {\sqrt {\frac {k_{\alpha }}{e}}}}$
(14)

It can be seen from equation 14 that divergent velocity is composed of two parts data. The left part is related to wing configuration and flight condition, while the right one is wing structural parameters. In the whole aeroelasticity optimization process, it is only the right part varying. Divergent velocity increases with the increase of torsional stiffness, while decreases with the increase of distance between stiffness center and aerodynamic center. It is proportional to the divergent velocity and ${\displaystyle {\sqrt {k_{\alpha }/e}}}$. So, divergent velocity can be adjusted by changing the torsional stiffness and the distance between stiffness center and aerodynamic center in the aeroelasticity optimization process.

### 3.2 Optimization design method

For general multidisciplinary optimization problem, it can be described by optimum objective function ${\displaystyle F\left(f\left(X_{1}\right),f(X_{2}),\cdots ,f(X_{n})\right)}$ , constraint condition

 ${\displaystyle g_{j}(X)<0}$ ${\displaystyle \left(j=1,2,\cdots ,m\right)}$

or

 ${\displaystyle h_{j}(X)=0}$ ${\displaystyle \left(j=m+1,\cdots ,n\right)}$

and design variable ${\displaystyle X_{L}\leq X\leq X_{U}}$ , ${\displaystyle X={\left(X_{1},X_{2},\cdots ,X_{n}\right)}^{T}}$.

(1) Objective function

The main object of aeroelasticity optimization design is to reduce the weight and improve aeroelasticity characteristics. Lift/drag ratio is selected as the index of aeroelasticity characteristics. The optimum objective function can be written as

 ${\displaystyle F=W_{1}G/G_{m}+W_{2}c_{l}/c_{d}}$
(15)

Where ${\displaystyle G}$ is the weight after optimization, ${\displaystyle G_{m}}$ is the maximum weight in design variable span, ${\displaystyle c_{l}/c_{d}}$ is lift/drag ratio under the function of static aeroelasticity and ${\displaystyle W_{i}}$ is weight coefficient. In the present study, ${\displaystyle W_{1}=0.5}$ and ${\displaystyle W_{2}=-0.5}$. The negative coefficient in ${\displaystyle W_{2}}$ is to make both objective functions have the same trend. The weight coefficient values can be adjusted according actual optimization problems.

(2) Constraint conditions

The constraint conditions in aeroelasticity optimization contain strength, stiffness, stability, continuity and divergent velocity. The following will give detailed presentations about these constraint conditions.

a. Strength

The composite wing structure, which is shown in figure 1, is optimized in the present study. Strain value usually as the reference of strength evaluation in composite structures. The maximum tension, compression and shear strain of the composite wing structure are analyzed under the function of aeroelasticity. These parameters, comparison with the constraint values, reflect whether the wing structure meets strength requirement. The constraint conditions of strength can be seen in equation 16. Where ${\displaystyle \epsilon _{tmax}}$ is the maximum tensile strain, ${\displaystyle \epsilon _{cmax}}$ is the maximum compression strain while ${\displaystyle \epsilon _{xymax}}$ is the maximum shear strain.

 ${\displaystyle \left\vert \epsilon _{tmax}\right\vert \leq 4500}$
(16)

b. Stiffness

Stiffness constraint condition is mainly reflected in bending and torsional deformation. The maximum bending and torsion deformation are usually happened in the wing tip. So, it is just need to extract the results of wing tip and compared them with constraint values. The constraint conditions of stiffness can be seen in equation 17. Where ${\displaystyle \theta _{\max }}$ is the maximum torsion angle, while ${\displaystyle U_{\max }}$ is the maximum elastic axis flexivity.

 ${\displaystyle \theta _{\max }\leq 2.0^{0}}$
(17)

c. Stability

The main part of wing structure is a stiffened plate structure. It is likely to have an axial compression buckling. Critical load of the stiffened plate structure can be expressed as

 ${\displaystyle P_{cr}={\frac {P_{e}}{1+\lambda P_{e}/G{\overline {A}}}}}$
(18)

Where ${\displaystyle P_{e}=c{\pi }^{2}EI/L^{2}}$ , ${\displaystyle \lambda }$ is shape factor, ${\displaystyle G}$ and ${\displaystyle {\overline {A}}}$ are equivalent shear modulus and area of web of the stringer, while ${\displaystyle EI}$ is bending stiffness of the stiffened plate structure.The constrait conditions of stability is that the critical load is larger than working condition.

d. Continuity

A bad wing structure form is easy to be designed when every design variable as independent parameter. So, continuity between adjacent skins or skin and its stringer need to be considered in aeroelasticity optimization design. Continuity between adjacent skins mainly affects manufacture. It is set 0.8-1.2 here. Continuity between skin and its stringer is achieved through the area ratio. In the present study, stringer area is set two-thirds of skin area, which can be adjusted in the range of ±10%.

e. Divergent velocity

It is just a divergent velocity range can be obtained in the traditional time-domain aeroelasticity analysis. At the same time, it spends a lot of time. The time-domain aeroelasticity analysis method is nearly impossible to be used in aeroelasticity optimization design. Divergent velocity increases with the increase of torsional stiffness, while decreases with the increase of distance between stiffness center and aerodynamic center. It is proportional to the divergent velocity and ${\displaystyle {\sqrt {k_{\alpha }/e}}}$. So, the constraint about divergent velocity can be equivalent to the control of ${\displaystyle {\sqrt {k_{\alpha }/e}}}$. In the present study, divergent velocity of the optimized wing structure is set not less than 95% of the initial value.

(3) Design variables

The stress condition of wing different regions is different in the flight state. So, different region has its own thickness value will take advantage. Wing finite element model is divided into many regions and each region has its own property. Thickness is the most concerned design variable in current study.

(4) Optimization process

Whole optimization process is carried out in the software of iSIGHT. Multi-Island genetic algorithm is used to wing optimization design. Elastic axis location and torsional stiffness characteristics of the initial model design are analyzed firstly, which as the first constraint condition. Then, aeroelasiticity analysis of the wing structure is carried out and other constraint conditions are obtained. It can be seen from the results that whether all the parameters can meet constraint conditions. If it did, wing weight and lift/drag ratio are output, if not, design variables need to be modified and computed again. Static strength and stability are analyzed through FEM software of NASTRAN while elastic axis location and torsional stiffness in the FORTRAN code based on above introduction. Figure 4 shows the whole optimization process.

Figure 4. Aeroelasticity optimization process

### 3.3 Results and discussion

Table 1 shows the optimization results of the objects. The whole optimization design costs about 15 hours. Ordinarily, static aeroelasticity analysis for one individual costs more than ten hours based on the traditional time-domain method, which can not be accepted in the optimization design. The optimization method of the present study is very efficient. It can be seen that wing weight decreases 17.9% while lift/drag ratio increases 5.88%. Both optimization objects have ideal results. At the same time, the value of ${\displaystyle {\sqrt {k_{\alpha }/e}}}$ just decreases to 97.34% of the initial one, which has a well control on divergent velocity. The method can be well used in aeroelasticity optimization of wing structure.

Table 1. Optimization results of the objects
 Object Trend Initial value Optimized result Changed rate Weight Decline 346Kg 284Kg 17.9% Lift/drag ratio Increase 10.2 10.8 5.88%

Figures 5 and 6 show the change of bending and torsional stiffness of different cross sections in initial and optimized wing structure, respectively. It can be seen that both stiffness values decrease obviously and the decrement of torsional stiffness is slightly less than that of bending stiffness. It is because that torsional stiffness directly affect lift coefficient and is constrained by the divergent velocity.

Figure 5. Change of bending stiffness

Figure 6. Change of torsional stiffness

Figure 7 shows the change of aerodynamic distribution of different cross sections. It can be seen that the pressure differences between top and down aerofoil section decrease after optimization. Flexibility of wing structure is increased through aeroelasticity optimization design, which leads to the decrement of lift coefficient under the effect of aeroelasticity.

 (a) 25% cross section (b) 45% cross section (c) 65% cross section (d) 85% cross section
Figure 7. Change of aerodynamic distribution

## 4. Conclusions

A rapid aeroelasticity optimization method based on the stiffness characteristics was proposed in the present study. Some useful conclusion can be drawn as follows:

(1) Both torsional stiffness and the distance between stiffness center and aerodynamic center have a direct impact on the divergent velocity. The divergent velocity can be adjusted by changing the correlative structural parameters.

(2) A rapid aeroelasticity optimization method based on the stiffness characteristics is introduced. Large time expense in static aeroelasticity analysis based on the traditional time-domain method is solved. After optimization, both structural and aerodynamic characteristics have a large improvement while satisfying the constraint conditions.

(3) Both bending and torsional stiffness decrease obviously after the optimization, while the pressure differences between top and down aerofoil section have the same trend.

## Acknowledgements

This study is supported by the National Nature Science Foundation of China (No. 51305351).

## References

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[5] J.J. Alonso, P. LeGresley, V. Pereyra. Aircraft design optimization, Math. Comput. Simul. 2009, 79(6): 1948-1958

[6] G. Venter, J. Sobieszczanski-Sobieski. Multidisciplinary optimization of a transport aircraft wing using particle swarm optimization, Structural and Multidisciplinary Optimization, 2004, 26(11): 121-131

[7] Kwon-Su Jeon, Jae-Wo Lee, Yung-Hwan Byun. Multidisciplinary design approach using repetitive response surface enhancement and global optimization, Proceedings of the 2008 Ninth ACIS International Conference on Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed, 2008: 25-28

[8] Franco Mastroddi, Marco Tozzi, Valerio Capanolo. On the use of geometry design variables in the MDO analysis of wing structures with aeroelastic constraints on stability and response, Aerospace Science and Technology, 2011, 15: 196-206.

[9] Su Hua, Gu Liangxian, Gong Chunlin. Static aeroelastic optimization of hypersonic aircraft based on high-fidelity model, Journal of Aerospace Power, 2013, 28(8): 1836-1842.

[10] Nikbay Melike, Kuru Muhammet N. Reliability based multidisciplinary optimization of aeroelastic systems with structural and aerodynamic uncertainties, Journal of Aircraft, 2013, 50(3): 708-715.

[11] Sleesongsom S, Bureerat S. New conceptual design of aeroelastic wing structures by multi-objective optimization, Engineering Optimization, 2013, 45(1): 107-122.

[12] Wesley W. Li, Chan-gi Pak. Aeroelastic optimization study based on the X-56A model, AIAA Paper, AIAA-2014-2052.

[13] Eric Ting, Sonia Lebofsky, Nhan Nguyen, Khanh Trinh. Static aeroelastic scaling and analysis of a sub-scale flexible wing wind tunnel model, AIAA Paper, AIAA-2014-0838.

[14] Bret K. Stanford, Peter D. Dunning. Optimal topology of aircraft rib and spar structures under aeroelastic loads, AIAA Paper, AIAA-2014-0633.

[15] Wrik Mallik, Rakesh K. Kapania, Joseph A. Schetz. Aeroelastic analysis and optimization of flexible wing aircraft with a novel control effector, AIAA SciTech, 5-9 January 2015, Kissimmee, Florida, 56th AIAA Structures, Structural Dynamics, and Materials Conference.

[16] Stanford B., Jutte C. Trim and structural optimization of subsonic transport wings using nonconventional aeroelastic tailoring, AIAA Aviation Conference, Atlanta, GA, June 16-20, 2014.

[17]G. Francois, J.E. Cooper, P.M. Weaver. Aeroelastic tailoring using rib/spar orientations: experimental investigation, AIAA SciTech, 5-9 January 2015, Kissimmee, Florida, 56th AIAA Structures, Structural Dynamics, and Materials Conference.

[18] B. Roscher, H.A. Madsen, D.T. Griffith. Combined structural optimization and aeroelastic analysis of a vertical axis wind turbine, AIAA SciTech, 5-9 January 2015, Kissimmee, Florida, 33rd Wind Energy Symposium.

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[20] S.H. Huo, F.S. Wang, Z. Yuan, Z.F. Yue. Mesh regeneration method for jig-shape optimization design of the high-aspect-ratio wing, Advances in Mechanical Engineering, 2013, 513637: 1-8.

[21] Zhao Yonghui, Hu Haiyan. Structural modeling and aeroelastic analysis of high-aspect-ratio composite wing, Chinese journal of Aeronautics, 2005, 18(1): 25-30.

### Document information

Published on 08/02/19
Accepted on 09/12/18
Submitted on 04/04/18

Volume 35, Issue 1, 2019
DOI: 10.23967/j.rimni.2018.12.002
Licence: CC BY-NC-SA license

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The nine delegates to the Economic Cooperation Conference include $2$ officials from Mexico, $3$ officials from Canada, and $4$ officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, find the probability that exactly two of the sleepers are from the same country.

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Find the probability that each delegate sits next to at least one delegate from another country.

A parking lot has $16$ spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires $2$ adjacent spaces. What is the probability that she is able to park?

Meena writes the numbers $1$, $2$, $3$, and $4$ in some order on a blackboard, such that she cannot swap two numbers and obtain the sequence $1$, $2$, $3$, $4$. How many sequences could she have written?

John walks from point $A$ to $C$ while Mary goes from point $B$ to $D$. Both of them will move along the grid, either right or up, so they take shortest routes. How many different possibilities are there such that their routes do not intersect?

Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?

How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?

Randomly draw a card twice with replacement from $1$ to $10$, inclusive. What is the probability that the product of these two cards is a multiple of $7$?

How many even $4$- digit integers are there whose digits are distinct?

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# Moderate Time, Speed & Distance Solved QuestionAptitude Discussion

 Q. Two boys begin together to write out a booklet containing 535 lines. The first boy starts with the first line, writing at the rate of 100 lines an hour; and the second starts with the last line then writes line 534 and so on, backward proceeding at the rate of 50 lines an hour. At what line will they meet?
 ✖ A. 356 ✖ B. 277 ✔ C. 357 ✖ D. 267

Solution:
Option(C) is correct

Writing ratio = $100:50=2:1$

Since equal quantities are taken,

Therefore in a given time, first boy will be writing the line number

$\dfrac{2}{3}\times 535\text{ or }357^{th}$ line

Hence, both of them shall meet on $357^{th}$ line


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Euler's constant

(Redirected from Euler-Mascheroni constant)

Representations The area of the blue region converges to Euler's constant 0.5772156649015328606065120900824024310421... [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, ...][1]Unknown if periodicUnknown if finite 0.1001001111000100011001111110001101111101... 0.93C467E37DB0C7A4D1BE3F810152CB56A1CECC3A...

Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (γ).

It is defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:

{\displaystyle {\begin{aligned}\gamma &=\lim _{n\to \infty }\left(-\log n+\sum _{k=1}^{n}{\frac {1}{k}}\right)\\[5px]&=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,dx.\end{aligned}}}

Here, ⌊ ⌋ represents the floor function.

The numerical value of Euler's constant, to 50 decimal places, is:[2]

0.57721566490153286060651209008240243104215933593992...
Unsolved problem in mathematics:

Is Euler's constant irrational? If so, is it transcendental?

History

The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations C and O for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations A and a for the constant. The notation γ appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time perhaps because of the constant's connection to the gamma function.[3] For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835[4] and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.[5]

Appearances

Euler's constant appears, among other places, in the following (where '*' means that this entry contains an explicit equation):

Properties

The number γ has not been proved algebraic or transcendental. In fact, it is not even known whether γ is irrational. Using a continued fraction analysis, Papanikolaou showed in 1997 that if γ is rational, its denominator must be greater than 10244663.[8][9] The ubiquity of γ revealed by the large number of equations below makes the irrationality of γ a major open question in mathematics.[10]

However, some progress was made. Kurt Mahler showed in 1968 that the number ${\displaystyle {\frac {\pi }{2}}{\frac {Y_{0}(2)}{J_{0}(2)}}-\gamma }$ is transcendental (here, ${\displaystyle J_{\alpha }(x)}$ and ${\displaystyle Y_{\alpha }(x)}$ are Bessel functions).[11][3] In 2009 Alexander Aptekarev proved that at least one of Euler's constant γ and the Euler–Gompertz constant δ is irrational;[12] Tanguy Rivoal proved in 2012 that at least one of them is transcendental.[13][3] In 2010 M. Ram Murty and N. Saradha showed that at most one of the numbers of the form

${\displaystyle \gamma (a,q)=\lim _{n\rightarrow \infty }\left(\left(\sum _{k=0}^{n}{\frac {1}{a+kq}}\right)-{\frac {\log {(a+nq})}{q}}\right)}$

with q ≥ 2 and 1 ≤ a < q is algebraic; this family includes the special case γ(2,4) = γ/4.[3][14] In 2013 M. Ram Murty and A. Zaytseva found a different family containing γ, which is based on sums of reciprocals of integers not divisible by a fixed list of primes, with the same property.[3][15]

Relation to gamma function

γ is related to the digamma function Ψ, and hence the derivative of the gamma function Γ, when both functions are evaluated at 1. Thus:

${\displaystyle -\gamma =\Gamma '(1)=\Psi (1).}$

This is equal to the limits:

{\displaystyle {\begin{aligned}-\gamma &=\lim _{z\to 0}\left(\Gamma (z)-{\frac {1}{z}}\right)\\&=\lim _{z\to 0}\left(\Psi (z)+{\frac {1}{z}}\right).\end{aligned}}}

Further limit results are:[16]

{\displaystyle {\begin{aligned}\lim _{z\to 0}{\frac {1}{z}}\left({\frac {1}{\Gamma (1+z)}}-{\frac {1}{\Gamma (1-z)}}\right)&=2\gamma \\\lim _{z\to 0}{\frac {1}{z}}\left({\frac {1}{\Psi (1-z)}}-{\frac {1}{\Psi (1+z)}}\right)&={\frac {\pi ^{2}}{3\gamma ^{2}}}.\end{aligned}}}

A limit related to the beta function (expressed in terms of gamma functions) is

{\displaystyle {\begin{aligned}\gamma &=\lim _{n\to \infty }\left({\frac {\Gamma \left({\frac {1}{n}}\right)\Gamma (n+1)\,n^{1+{\frac {1}{n}}}}{\Gamma \left(2+n+{\frac {1}{n}}\right)}}-{\frac {n^{2}}{n+1}}\right)\\&=\lim \limits _{m\to \infty }\sum _{k=1}^{m}{m \choose k}{\frac {(-1)^{k}}{k}}\log {\big (}\Gamma (k+1){\big )}.\end{aligned}}}

Relation to the zeta function

γ can also be expressed as an infinite sum whose terms involve the Riemann zeta function evaluated at positive integers:

{\displaystyle {\begin{aligned}\gamma &=\sum _{m=2}^{\infty }(-1)^{m}{\frac {\zeta (m)}{m}}\\&=\log {\frac {4}{\pi }}+\sum _{m=2}^{\infty }(-1)^{m}{\frac {\zeta (m)}{2^{m-1}m}}.\end{aligned}}}

Other series related to the zeta function include:

{\displaystyle {\begin{aligned}\gamma &={\tfrac {3}{2}}-\log 2-\sum _{m=2}^{\infty }(-1)^{m}\,{\frac {m-1}{m}}{\big (}\zeta (m)-1{\big )}\\&=\lim _{n\to \infty }\left({\frac {2n-1}{2n}}-\log n+\sum _{k=2}^{n}\left({\frac {1}{k}}-{\frac {\zeta (1-k)}{n^{k}}}\right)\right)\\&=\lim _{n\to \infty }\left({\frac {2^{n}}{e^{2^{n}}}}\sum _{m=0}^{\infty }{\frac {2^{mn}}{(m+1)!}}\sum _{t=0}^{m}{\frac {1}{t+1}}-n\log 2+O\left({\frac {1}{2^{n}\,e^{2^{n}}}}\right)\right).\end{aligned}}}

The error term in the last equation is a rapidly decreasing function of n. As a result, the formula is well-suited for efficient computation of the constant to high precision.

Other interesting limits equaling Euler's constant are the antisymmetric limit:[17]

{\displaystyle {\begin{aligned}\gamma &=\lim _{s\to 1^{+}}\sum _{n=1}^{\infty }\left({\frac {1}{n^{s}}}-{\frac {1}{s^{n}}}\right)\\&=\lim _{s\to 1}\left(\zeta (s)-{\frac {1}{s-1}}\right)\\&=\lim _{s\to 0}{\frac {\zeta (1+s)+\zeta (1-s)}{2}}\end{aligned}}}

and the following formula, established in 1898 by de la Vallée-Poussin:

${\displaystyle \gamma =\lim _{n\to \infty }{\frac {1}{n}}\,\sum _{k=1}^{n}\left(\left\lceil {\frac {n}{k}}\right\rceil -{\frac {n}{k}}\right)}$

where ⌈ ⌉ are ceiling brackets. This formula indicates that when taking any positive integer n and dividing it by each positive integer k less than n, the average fraction by which the quotient n/k falls short of the next integer tends to γ (rather than 0.5) as n tends to infinity.

Closely related to this is the rational zeta series expression. By taking separately the first few terms of the series above, one obtains an estimate for the classical series limit:

${\displaystyle \gamma =\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\log n-\sum _{m=2}^{\infty }{\frac {\zeta (m,n+1)}{m}}\right),}$

where ζ(s, k) is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, Hn. Expanding some of the terms in the Hurwitz zeta function gives:

${\displaystyle H_{n}=\log(n)+\gamma +{\frac {1}{2n}}-{\frac {1}{12n^{2}}}+{\frac {1}{120n^{4}}}-\varepsilon ,}$

where 0 < ε < 1/252n6.

γ can also be expressed as follows where A is the Glaisher–Kinkelin constant:

${\displaystyle \gamma =12\,\log(A)-\log(2\pi )+{\frac {6}{\pi ^{2}}}\,\zeta '(2)}$

γ can also be expressed as follows, which can be proven by expressing the zeta function as a Laurent series:

${\displaystyle \gamma =\lim _{n\to \infty }\left(-n+\zeta {\Bigl (}{\frac {n+1}{n}}{\bigr )}\right)}$

Integrals

γ equals the value of a number of definite integrals:

{\displaystyle {\begin{aligned}\gamma &=-\int _{0}^{\infty }e^{-x}\log x\,dx\\&=-\int _{0}^{1}\log \left(\log {\frac {1}{x}}\right)dx\\&=\int _{0}^{\infty }\left({\frac {1}{e^{x}-1}}-{\frac {1}{x\cdot e^{x}}}\right)dx\\&=\int _{0}^{1}{\frac {1-e^{-x}}{x}}\,dx-\int _{1}^{\infty }{\frac {e^{-x}}{x}}\,dx\\&=\int _{0}^{1}\left({\frac {1}{\log x}}+{\frac {1}{1-x}}\right)dx\\&=\int _{0}^{\infty }\left({\frac {1}{1+x^{k}}}-e^{-x}\right){\frac {dx}{x}},\quad k>0\\&=2\int _{0}^{\infty }{\frac {e^{-x^{2}}-e^{-x}}{x}}\,dx,\\&=\int _{0}^{1}H_{x}\,dx,\end{aligned}}}

where Hx is the fractional harmonic number.

The third formula in the integral list can be proved in the following way:

{\displaystyle {\begin{aligned}&\int _{0}^{\infty }\left({\frac {1}{e^{x}-1}}-{\frac {1}{xe^{x}}}\right)dx=\int _{0}^{\infty }{\frac {e^{-x}+x-1}{x[e^{x}-1]}}dx=\int _{0}^{\infty }{\frac {1}{x[e^{x}-1]}}\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}x^{m+1}}{(m+1)!}}dx\\[2pt]&=\int _{0}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}x^{m}}{(m+1)![e^{x}-1]}}dx=\sum _{m=1}^{\infty }\int _{0}^{\infty }{\frac {(-1)^{m+1}x^{m}}{(m+1)![e^{x}-1]}}dx=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{(m+1)!}}\int _{0}^{\infty }{\frac {x^{m}}{e^{x}-1}}dx\\[2pt]&=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{(m+1)!}}m!\zeta (m+1)=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}\zeta (m+1)=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}\sum _{n=1}^{\infty }{\frac {1}{n^{m+1}}}=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}{\frac {1}{n^{m+1}}}\\[2pt]&=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}{\frac {1}{n^{m+1}}}=\sum _{n=1}^{\infty }\left[{\frac {1}{n}}-\log \left(1+{\frac {1}{n}}\right)\right]=\gamma \end{aligned}}}

The integral on the second line of the equation stands for the Debye function value of +∞, which is m! ζ(m + 1).

Definite integrals in which γ appears include:

{\displaystyle {\begin{aligned}\int _{0}^{\infty }e^{-x^{2}}\log x\,dx&=-{\frac {(\gamma +2\log 2){\sqrt {\pi }}}{4}}\\\int _{0}^{\infty }e^{-x}\log ^{2}x\,dx&=\gamma ^{2}+{\frac {\pi ^{2}}{6}}\end{aligned}}}

One can express γ using a special case of Hadjicostas's formula as a double integral[10][18] with equivalent series:

{\displaystyle {\begin{aligned}\gamma &=\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1-xy)\log xy}}\,dx\,dy\\&=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\log {\frac {n+1}{n}}\right).\end{aligned}}}

An interesting comparison by Sondow[18] is the double integral and alternating series

{\displaystyle {\begin{aligned}\log {\frac {4}{\pi }}&=\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1+xy)\log xy}}\,dx\,dy\\&=\sum _{n=1}^{\infty }\left((-1)^{n-1}\left({\frac {1}{n}}-\log {\frac {n+1}{n}}\right)\right).\end{aligned}}}

It shows that log 4/π may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series[19]

{\displaystyle {\begin{aligned}\gamma &=\sum _{n=1}^{\infty }{\frac {N_{1}(n)+N_{0}(n)}{2n(2n+1)}}\\\log {\frac {4}{\pi }}&=\sum _{n=1}^{\infty }{\frac {N_{1}(n)-N_{0}(n)}{2n(2n+1)}},\end{aligned}}}

where N1(n) and N0(n) are the number of 1s and 0s, respectively, in the base 2 expansion of n.

We also have Catalan's 1875 integral[20]

${\displaystyle \gamma =\int _{0}^{1}\left({\frac {1}{1+x}}\sum _{n=1}^{\infty }x^{2^{n}-1}\right)\,dx.}$

Series expansions

In general,

${\displaystyle \gamma =\lim _{n\to \infty }\left({\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+\ldots +{\frac {1}{n}}-\log(n+\alpha )\right)\equiv \lim _{n\to \infty }\gamma _{n}(\alpha )}$

for any α > −n. However, the rate of convergence of this expansion depends significantly on α. In particular, γn(1/2) exhibits much more rapid convergence than the conventional expansion γn(0).[21][22] This is because

${\displaystyle {\frac {1}{2(n+1)}}<\gamma _{n}(0)-\gamma <{\frac {1}{2n}},}$

while

${\displaystyle {\frac {1}{24(n+1)^{2}}}<\gamma _{n}(1/2)-\gamma <{\frac {1}{24n^{2}}}.}$

Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below.

Euler showed that the following infinite series approaches γ:

${\displaystyle \gamma =\sum _{k=1}^{\infty }\left({\frac {1}{k}}-\log \left(1+{\frac {1}{k}}\right)\right).}$

The series for γ is equivalent to a series Nielsen found in 1897:[16][23]

${\displaystyle \gamma =1-\sum _{k=2}^{\infty }(-1)^{k}{\frac {\left\lfloor \log _{2}k\right\rfloor }{k+1}}.}$

In 1910, Vacca found the closely related series[24][25][26][27][28][16][29]

{\displaystyle {\begin{aligned}\gamma &=\sum _{k=2}^{\infty }(-1)^{k}{\frac {\left\lfloor \log _{2}k\right\rfloor }{k}}\\[5pt]&={\tfrac {1}{2}}-{\tfrac {1}{3}}+2\left({\tfrac {1}{4}}-{\tfrac {1}{5}}+{\tfrac {1}{6}}-{\tfrac {1}{7}}\right)+3\left({\tfrac {1}{8}}-{\tfrac {1}{9}}+{\tfrac {1}{10}}-{\tfrac {1}{11}}+\cdots -{\tfrac {1}{15}}\right)+\cdots ,\end{aligned}}}

where log2 is the logarithm to base 2 and ⌊ ⌋ is the floor function.

In 1926 he found a second series:

{\displaystyle {\begin{aligned}\gamma +\zeta (2)&=\sum _{k=2}^{\infty }\left({\frac {1}{\left\lfloor {\sqrt {k}}\right\rfloor ^{2}}}-{\frac {1}{k}}\right)\\[5pt]&=\sum _{k=2}^{\infty }{\frac {k-\left\lfloor {\sqrt {k}}\right\rfloor ^{2}}{k\left\lfloor {\sqrt {k}}\right\rfloor ^{2}}}\\[5pt]&={\frac {1}{2}}+{\frac {2}{3}}+{\frac {1}{2^{2}}}\sum _{k=1}^{2\cdot 2}{\frac {k}{k+2^{2}}}+{\frac {1}{3^{2}}}\sum _{k=1}^{3\cdot 2}{\frac {k}{k+3^{2}}}+\cdots \end{aligned}}}

From the MalmstenKummer expansion for the logarithm of the gamma function[30] we get:

${\displaystyle \gamma =\log \pi -4\log \left(\Gamma ({\tfrac {3}{4}})\right)+{\frac {4}{\pi }}\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {\log(2k+1)}{2k+1}}.}$

An important expansion for Euler's constant is due to Fontana and Mascheroni

${\displaystyle \gamma =\sum _{n=1}^{\infty }{\frac {|G_{n}|}{n}}={\frac {1}{2}}+{\frac {1}{24}}+{\frac {1}{72}}+{\frac {19}{2880}}+{\frac {3}{800}}+\cdots ,}$

where Gn are Gregory coefficients[16][29][31] This series is the special case k = 1 of the expansions

{\displaystyle {\begin{aligned}\gamma &=H_{k-1}-\log k+\sum _{n=1}^{\infty }{\frac {(n-1)!|G_{n}|}{k(k+1)\cdots (k+n-1)}}&&\\&=H_{k-1}-\log k+{\frac {1}{2k}}+{\frac {1}{12k(k+1)}}+{\frac {1}{12k(k+1)(k+2)}}+{\frac {19}{120k(k+1)(k+2)(k+3)}}+\cdots &&\end{aligned}}}

convergent for k = 1, 2, ...

A similar series with the Cauchy numbers of the second kind Cn is[29][32]

${\displaystyle \gamma =1-\sum _{n=1}^{\infty }{\frac {C_{n}}{n\,(n+1)!}}=1-{\frac {1}{4}}-{\frac {5}{72}}-{\frac {1}{32}}-{\frac {251}{14400}}-{\frac {19}{1728}}-\ldots }$

Blagouchine (2018) found an interesting generalisation of the Fontana–Mascheroni series

${\displaystyle \gamma =\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{2n}}{\Big \{}\psi _{n}(a)+\psi _{n}{\Big (}-{\frac {a}{1+a}}{\Big )}{\Big \}},\quad a>-1}$

where ψn(a) are the Bernoulli polynomials of the second kind, which are defined by the generating function

${\displaystyle {\frac {z(1+z)^{s}}{\log(1+z)}}=\sum _{n=0}^{\infty }z^{n}\psi _{n}(s),\qquad |z|<1,}$

For any rational a this series contains rational terms only. For example, at a = 1, it becomes[33][34]

${\displaystyle \gamma ={\frac {3}{4}}-{\frac {11}{96}}-{\frac {1}{72}}-{\frac {311}{46080}}-{\frac {5}{1152}}-{\frac {7291}{2322432}}-{\frac {243}{100352}}-\ldots }$

Other series with the same polynomials include these examples:

${\displaystyle \gamma =-\log(a+1)-\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n}(a)}{n}},\qquad \Re (a)>-1}$

and

${\displaystyle \gamma =-{\frac {2}{1+2a}}\left\{\log \Gamma (a+1)-{\frac {1}{2}}\log(2\pi )+{\frac {1}{2}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n+1}(a)}{n}}\right\},\qquad \Re (a)>-1}$

where Γ(a) is the gamma function.[31]

A series related to the Akiyama–Tanigawa algorithm is

${\displaystyle \gamma =\log(2\pi )-2-2\sum _{n=1}^{\infty }{\frac {(-1)^{n}G_{n}(2)}{n}}=\log(2\pi )-2+{\frac {2}{3}}+{\frac {1}{24}}+{\frac {7}{540}}+{\frac {17}{2880}}+{\frac {41}{12600}}+\ldots }$

where Gn(2) are the Gregory coefficients of the second order.[31]

Series of prime numbers:

${\displaystyle \gamma =\lim _{n\to \infty }\left(\log n-\sum _{p\leq n}{\frac {\log p}{p-1}}\right).}$

Asymptotic expansions

γ equals the following asymptotic formulas (where Hn is the nth harmonic number):

${\displaystyle \gamma \sim H_{n}-\log n-{\frac {1}{2n}}+{\frac {1}{12n^{2}}}-{\frac {1}{120n^{4}}}+\cdots }$ (Euler)
${\displaystyle \gamma \sim H_{n}-\log \left({n+{\frac {1}{2}}+{\frac {1}{24n}}-{\frac {1}{48n^{2}}}+\cdots }\right)}$ (Negoi)
${\displaystyle \gamma \sim H_{n}-{\frac {\log n+\log(n+1)}{2}}-{\frac {1}{6n(n+1)}}+{\frac {1}{30n^{2}(n+1)^{2}}}-\cdots }$ (Cesàro)

The third formula is also called the Ramanujan expansion.

Alabdulmohsin derived closed-form expressions for the sums of errors of these approximations.[32] He showed that (Theorem A.1):

${\displaystyle \sum _{n=1}^{\infty }\log n+\gamma -H_{n}+{\frac {1}{2n}}={\frac {\log(2\pi )-1-\gamma }{2}}}$
${\displaystyle \sum _{n=1}^{\infty }\log {\sqrt {n(n+1)}}+\gamma -H_{n}={\frac {\log(2\pi )-1}{2}}-\gamma }$
${\displaystyle \sum _{n=1}^{\infty }(-1)^{n}{\Big (}\log n+\gamma -H_{n}{\Big )}={\frac {\log \pi -\gamma }{2}}}$

Exponential

The constant eγ is important in number theory. Some authors denote this quantity simply as γ. eγ equals the following limit, where pn is the nth prime number:

${\displaystyle e^{\gamma }=\lim _{n\to \infty }{\frac {1}{\log p_{n}}}\prod _{i=1}^{n}{\frac {p_{i}}{p_{i}-1}}.}$

This restates the third of Mertens' theorems.[35] The numerical value of eγ is:[36]

1.78107241799019798523650410310717954916964521430343....

Other infinite products relating to eγ include:

{\displaystyle {\begin{aligned}{\frac {e^{1+{\frac {\gamma }{2}}}}{\sqrt {2\pi }}}&=\prod _{n=1}^{\infty }e^{-1+{\frac {1}{2n}}}\left(1+{\frac {1}{n}}\right)^{n}\\{\frac {e^{3+2\gamma }}{2\pi }}&=\prod _{n=1}^{\infty }e^{-2+{\frac {2}{n}}}\left(1+{\frac {2}{n}}\right)^{n}.\end{aligned}}}

These products result from the Barnes G-function.

${\displaystyle e^{\gamma }={\sqrt {\frac {2}{1}}}\cdot {\sqrt[{3}]{\frac {2^{2}}{1\cdot 3}}}\cdot {\sqrt[{4}]{\frac {2^{3}\cdot 4}{1\cdot 3^{3}}}}\cdot {\sqrt[{5}]{\frac {2^{4}\cdot 4^{4}}{1\cdot 3^{6}\cdot 5}}}\cdots }$

where the nth factor is the (n + 1)th root of

${\displaystyle \prod _{k=0}^{n}(k+1)^{(-1)^{k+1}{n \choose k}}.}$

This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow using hypergeometric functions.[37]

It also holds that[38]

${\displaystyle {\frac {e^{\frac {\pi }{2}}+e^{-{\frac {\pi }{2}}}}{\pi e^{\gamma }}}=\prod _{n=1}^{\infty }\left(e^{-{\frac {1}{n}}}\left(1+{\frac {1}{n}}+{\frac {1}{2n^{2}}}\right)\right).}$

Continued fraction

The continued fraction expansion of γ begins [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...],[1] which has no apparent pattern. The continued fraction is known to have at least 475,006 terms,[8] and it has infinitely many terms if and only if γ is irrational.

Generalizations

abm(x) = γx

Euler's generalized constants are given by

${\displaystyle \gamma _{\alpha }=\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k^{\alpha }}}-\int _{1}^{n}{\frac {1}{x^{\alpha }}}\,dx\right),}$

for 0 < α < 1, with γ as the special case α = 1.[39] This can be further generalized to

${\displaystyle c_{f}=\lim _{n\to \infty }\left(\sum _{k=1}^{n}f(k)-\int _{1}^{n}f(x)\,dx\right)}$

for some arbitrary decreasing function f. For example,

${\displaystyle f_{n}(x)={\frac {(\log x)^{n}}{x}}}$

gives rise to the Stieltjes constants, and

${\displaystyle f_{a}(x)=x^{-a}}$

gives

${\displaystyle \gamma _{f_{a}}={\frac {(a-1)\zeta (a)-1}{a-1}}}$

where again the limit

${\displaystyle \gamma =\lim _{a\to 1}\left(\zeta (a)-{\frac {1}{a-1}}\right)}$

appears.

A two-dimensional limit generalization is the Masser–Gramain constant.

Euler–Lehmer constants are given by summation of inverses of numbers in a common modulo class:[14]

${\displaystyle \gamma (a,q)=\lim _{x\to \infty }\left(\sum _{0

The basic properties are

{\displaystyle {\begin{aligned}\gamma (0,q)&={\frac {\gamma -\log q}{q}},\\\sum _{a=0}^{q-1}\gamma (a,q)&=\gamma ,\\q\gamma (a,q)&=\gamma -\sum _{j=1}^{q-1}e^{-{\frac {2\pi aij}{q}}}\log \left(1-e^{\frac {2\pi ij}{q}}\right),\end{aligned}}}

and if gcd(a,q) = d then

${\displaystyle q\gamma (a,q)={\frac {q}{d}}\gamma \left({\frac {a}{d}},{\frac {q}{d}}\right)-\log d.}$

Published digits

Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Mascheroni attempted to calculate the constant to 32 decimal places, but made errors in the 20th–22nd and 31st-32nd decimal places; starting from the 20th digit, he calculated ...1811209008239 when the correct value is ...0651209008240.

Published Decimal Expansions of γ
Date Decimal digits Author Sources
1734 5 Leonhard Euler
1735 15 Leonhard Euler
1781 16 Leonhard Euler
1790 32 Lorenzo Mascheroni, with 20–22 and 31–32 wrong
1809 22 Johann G. von Soldner
1811 22 Carl Friedrich Gauss
1812 40 Friedrich Bernhard Gottfried Nicolai
1857 34 Christian Fredrik Lindman
1861 41 Ludwig Oettinger
1867 49 William Shanks
1871 99 James W.L. Glaisher
1871 101 William Shanks
1952 328 John William Wrench Jr.
1961 1050 Helmut Fischer and Karl Zeller
1962 1271 Donald Knuth [40]
1962 3566 Dura W. Sweeney
1973 4879 William A. Beyer and Michael S. Waterman
1977 20700 Richard P. Brent
1980 30100 Richard P. Brent & Edwin M. McMillan
1993 172000 Jonathan Borwein
1999 108000000 Patrick Demichel and Xavier Gourdon
March 13, 2009 29844489545 Alexander J. Yee & Raymond Chan [41][42]
December 22, 2013 119377958182 Alexander J. Yee [42]
March 15, 2016 160000000000 Peter Trueb [42]
May 18, 2016 250000000000 Ron Watkins [42]
August 23, 2017 477511832674 Ron Watkins [42]
May 26, 2020 600000000100 Seungmin Kim & Ian Cutress [42][43]

References

• Bretschneider, Carl Anton (1837) [1835]. "Theoriae logarithmi integralis lineamenta nova". Crelle's Journal (in Latin). 17: 257–285.
• Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton University Press. ISBN 978-0-691-09983-5.
• Ram Murty, M.; Saradha, N. (2010). "Euler–Lehmer constants and a conjecture of Erdos". Journal of Number Theory. 130 (12): 2671–2681. doi:10.1016/j.jnt.2010.07.004. ISSN 0022-314X.

Footnotes

1. ^ a b Sloane, N. J. A. (ed.). "Sequence A002852 (Continued fraction for Euler's constant)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. ^
3. Lagarias, Jeffrey C. (October 2013). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 556. arXiv:1303.1856. doi:10.1090/s0273-0979-2013-01423-x. S2CID 119612431.
4. ^ Bretschneider 1837, "γ = c = 0,5772156649015328606181120900823..." on p. 260.
5. ^ De Morgan, Augustus (1836–1842). The differential and integral calculus. London: Baldwin and Craddoc. "γ" on p. 578.
6. ^ Caves, Carlton M.; Fuchs, Christopher A. (1996). "Quantum information: How much information in a state vector?". The Dilemma of Einstein, Podolsky and Rosen – 60 Years Later. Israel Physical Society. arXiv:quant-ph/9601025. Bibcode:1996quant.ph..1025C. ISBN 9780750303941. OCLC 36922834.
7. ^ Connallon, T., Hodgins, K.A., 2021. Allen Orr and the genetics of adaptation. Evolution 75, 2624–2640. https://doi.org/10.1111/evo.14372
8. ^ a b Haible, Bruno; Papanikolaou, Thomas (1998). Buhler, Joe P. (ed.). "Fast multiprecision evaluation of series of rational numbers". Algorithmic Number Theory. Lecture Notes in Computer Science. Springer. 1423: 338–350. doi:10.1007/bfb0054873. ISBN 9783540691136.
9. ^ Papanikolaou, T. (1997). Entwurf und Entwicklung einer objektorientierten Bibliothek für algorithmische Zahlentheorie (Thesis) (in German). Universität des Saarlandes.
10. ^ a b See also Sondow, Jonathan (2003). "Criteria for irrationality of Euler's constant". Proceedings of the American Mathematical Society. 131 (11): 3335–3344. arXiv:math.NT/0209070. doi:10.1090/S0002-9939-03-07081-3. S2CID 91176597.
11. ^ Mahler, Kurt; Mordell, Louis Joel (4 June 1968). "Applications of a theorem by A. B. Shidlovski". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 305 (1481): 149–173. Bibcode:1968RSPSA.305..149M. doi:10.1098/rspa.1968.0111. S2CID 123486171.
12. ^ Aptekarev, A. I. (28 February 2009). "On linear forms containing the Euler constant". arXiv:0902.1768 [math.NT].
13. ^ Rivoal, Tanguy (2012). "On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant". Michigan Mathematical Journal. 61 (2): 239–254. doi:10.1307/mmj/1339011525. ISSN 0026-2285.
14. ^ a b
15. ^ Murty, M. Ram; Zaytseva, Anastasia (2013). "Transcendence of Generalized Euler Constants". The American Mathematical Monthly. 120 (1): 48–54. doi:10.4169/amer.math.monthly.120.01.048. ISSN 0002-9890. JSTOR 10.4169/amer.math.monthly.120.01.048. S2CID 20495981.
16. ^ a b c d Krämer, Stefan (2005). Die Eulersche Konstante γ und verwandte Zahlen (in German). University of Göttingen.
17. ^ Sondow, Jonathan (1998). "An antisymmetric formula for Euler's constant". Mathematics Magazine. 71 (3): 219–220. doi:10.1080/0025570X.1998.11996638. Archived from the original on 2011-06-04. Retrieved 2006-05-29.
18. ^ a b Sondow, Jonathan (2005), "Double integrals for Euler's constant and ${\displaystyle \log {\frac {4}{\pi }}}$ and an analog of Hadjicostas's formula", American Mathematical Monthly, 112 (1): 61–65, arXiv:math.CA/0211148, doi:10.2307/30037385, JSTOR 30037385
19. ^ Sondow, Jonathan (1 August 2005a). New Vacca-type rational series for Euler's constant and its 'alternating' analog ${\displaystyle \log {\frac {4}{\pi }}}$. arXiv:math.NT/0508042.
20. ^ Sondow, Jonathan; Zudilin, Wadim (2006). "Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper". The Ramanujan Journal. 12 (2): 225–244. arXiv:math.NT/0304021. doi:10.1007/s11139-006-0075-1. S2CID 1368088.
21. ^ DeTemple, Duane W. (May 1993). "A Quicker Convergence to Euler's Constant". The American Mathematical Monthly. 100 (5): 468–470. doi:10.2307/2324300. ISSN 0002-9890. JSTOR 2324300.
22. ^ Havil 2003, pp. 75–8.
23. ^
24. ^ Vacca, G. (1910). "A new analytical expression for the number π and some historical considerations". Bulletin of the American Mathematical Society. 16: 368–369. doi:10.1090/S0002-9904-1910-01919-4.
25. ^ Glaisher, James Whitbread Lee (1910). "On Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 41: 365–368.
26. ^ Hardy, G.H. (1912). "Note on Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 43: 215–216.
27. ^ Vacca, G. (1926). "Nuova serie per la costante di Eulero, C = 0,577...". Rendiconti, Accademia Nazionale dei Lincei, Roma, Classe di Scienze Fisiche". Matematiche e Naturali (in Italian). 6 (3): 19–20.
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29. ^ a b c Blagouchine, Iaroslav V. (2016), "Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only", J. Number Theory, 158: 365–396, arXiv:1501.00740, doi:10.1016/j.jnt.2015.06.012
30. ^ Blagouchine, Iaroslav V. (2014). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". The Ramanujan Journal. 35 (1): 21–110. doi:10.1007/s11139-013-9528-5. S2CID 120943474.
31. ^ a b c Blagouchine, Iaroslav V. (2018), "Three notes on Ser's and Hasse's representations for the zeta-functions", INTEGERS: The Electronic Journal of Combinatorial Number Theory, 18A (#A3): 1–45, arXiv:1606.02044, Bibcode:2016arXiv160602044B
32. ^ a b Alabdulmohsin, Ibrahim M. (2018). Summability Calculus. A Comprehensive Theory of Fractional Finite Sums. Springer. pp. 147–8. ISBN 9783319746487.
33. ^
34. ^
35. ^ Weisstein, Eric W. "Mertens Constant". WolframMathWorld (Wolfram Research).
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38. ^ Choi, Junesang; Srivastava, H.M. (1 September 2010). "Integral Representations for the Euler–Mascheroni Constant γ". Integral Transforms and Special Functions. 21 (9): 675–690. doi:10.1080/10652461003593294. ISSN 1065-2469. S2CID 123698377.
39. ^ Havil 2003, pp. 117–8.
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Yee, Alexander J. "y-cruncher - A Multi-Threaded Pi-Program". www.numberworld.org.
43. ^ "Euler-Mascheroni Constant". Polymath Collector. 15 February 2020.


text:
# solving diophantine equations

1. the factoring method

Solve in positive integers , where and are prime.
Solution: , so .

Considering the positive divisors of we obtain:

yielding the solutions (writing ):
, , , ,
and their equivalents with and swapped. And the special case:

with the solution , i.e. 9 solutions in all.

where , has solutions in positive integers.

The equation is equivalent to and has positive divisors.

(to be continued)


text:
# Functions with the domain having greater cardinality than $\aleph_1$

Are there any examples of an algebra on elements belonging to a set of greater cardinality than that of real numbers. Can a reference to their use be given?

• continuous functions on stone-cech compactification of $\mathbb{N}$. – Norbert Nov 11 '13 at 7:22
• @Norbert Thanks. This would be an example of domain having the cardinality $\aleph_2 / 2^\mathfrak c$ are their examples of even greater cardinalities. – ARi Nov 11 '13 at 7:40

It is consistent that the real numbers have cardinality $\aleph_2$, in which case a function $f\colon\Bbb{R\to R}$ satisfies your requirements.
If you want a set whose cardinality is explicitly $\aleph_2$ then you will have to resort, in one way or another, to $\omega_2$ which is the second uncountable ordinal (i.e. the cardinal $\aleph_2$). The function can be anything, from the successor ordinal, to anything crazy.
However, I feel that you're not talking about $\aleph_2$, but rather $\beth_2$ i.e. $|\mathcal{P(P(}\Bbb R))|$. In that case you can consider the Lebesgue measure as a function whose domain has cardinality $\beth_2$, and its range is $\Bbb R$.
• "It is consistent that the real numbers have cardinality $\aleph_2$...isnt $\aleph_1$ the cardinality of the set of real numbers? – ARi Nov 11 '13 at 8:37
• math.stackexchange.com/a/490420/622 There are probably many other questions and answers about this question. But no. The cardinality of the real numbers is not necessarily $\aleph_1$. – Asaf Karagila Nov 11 '13 at 8:41


text:
# param-medic

## Usage:

crux param-medic [options] <spectrum-file>+

## Description:

Examine the spectra in a file to estimate the best precursor and fragment error tolerances for database search.

## Input:

• spectrum-file+ – File from which to parse fragmentation spectra.

## Output:

The program writes files to the folder crux-output by default. The name of the output folder can be set by the user using the --output-dir option. The following files will be created:

• stdout – the estimated parameter values for precursor mass tolerance (in ppm) and fragment bin size (in Th), as well as the standard deviations of the estimated error distributions for precursor and fragment masses.

## Options:

• ### param-medic options

• --pm-min-precursor-mz <float> – Minimum precursor m/z value to use in measurement error estimation. Default = 400.
• --pm-max-precursor-mz <float> – Minimum precursor m/z value to use in measurement error estimation. Default = 1800.
• --pm-min-frag-mz <float> – Minimum fragment m/z value to use in measurement error estimation. Default = 150.
• --pm-max-frag-mz <float> – Maximum fragment m/z value to use in measurement error estimation. Default = 1800.
• --pm-min-scan-frag-peaks <integer> – Minimum fragment peaks an MS/MS scan must contain to be used in measurement error estimation. Default = 40.
• --pm-max-precursor-delta-ppm <float> – Maximum ppm distance between precursor m/z values to consider two scans potentially generated by the same peptide for measurement error estimation. Default = 50.
• --pm-charges <string> – Precursor charge states to consider MS/MS spectra from, in measurement error estimation, provided as comma-separated values. Default = 0,2,3,4.
• --pm-top-n-frag-peaks <integer> – Number of most-intense fragment peaks to consider for measurement error estimation, per MS/MS spectrum. Default = 30.
• --pm-pair-top-n-frag-peaks <integer> – Number of fragment peaks per spectrum pair to be used in fragment error estimation. Default = 5.
• --pm-min-common-frag-peaks <integer> – Number of the most-intense peaks that two spectra must share in order to potentially be generated by the same peptide, for measurement error estimation. Default = 20.
• --pm-max-scan-separation <integer> – Maximum number of scans two spectra can be separated by in order to be considered potentially generated by the same peptide, for measurement error estimation. Default = 1000.
• --pm-min-peak-pairs <integer> – Minimum number of peak pairs (for precursor or fragment) that must be successfully paired in order to attempt to estimate measurement error distribution. Default = 200.
• ### Input and output

• --verbosity <integer> – Specify the verbosity of the current processes. Each level prints the following messages, including all those at lower verbosity levels: 0-fatal errors, 10-non-fatal errors, 20-warnings, 30-information on the progress of execution, 40-more progress information, 50-debug info, 60-detailed debug info. Default = 30.
• --spectrum-parser pwiz|mstoolkit – Specify the parser to use for reading in MS/MS spectra. The default, ProteoWizard parser can read the MS/MS file formats listed here. The alternative is MSToolkit parser. If the ProteoWizard parser fails to read your files properly, you may want to try the MSToolkit parser instead. Default = pwiz.


text:
### ত্রিভুজক্ষেত্রের ক্ষেত্রফল

##### Score: 2 Points

In $\triangle ABC$, $E$ & $M$ on $AB$, $G$ & $F$ on $BC$, $H$ & $N$ on $AC$ are such points where $MN \parallel BC$, $EF \parallel AC$, $GH \parallel AB$. $EF$, $GH$, $MN$ intersect at point $O$.
Given that $[MOE] = 4$, $[NOH] = 9$, $[GOF] = 49$. $[ABC] = ?$

Geometry

#### Statistics

Tried 109

Solved 89

First Solve @judge_1


text:
# Databases in srvyr

#### 2016-09-25

Srvyr now has experimental support of database backed surveys. It uses dplyr databases as the backend, not the survey package’s database support, which allows for nearly identical consistent syntax when working with database or local data.frame backed surveys. The actual implementation is similar to the survey package’s in that it stores the survey variables (such as weighting, stratum, etc.) in local memory and so will provide the most benefit when dealing with surveys with a large number of variables, rather than surveys with a large number of observations. It is slower than the survey package’s support, because it is more careful about the sorting order when pulling data from the database, in my testing it is about 2X slower than survey.

During development, I have tested using SQLite and MonetDBLite database, but in theory other database backends should work as well.

This vignette shows the basics of how to use srvyr with databases. It is based on analysis from the wonderful resource asdfree ( website and github ). Many thanks to ajdamico and collaborators. Specifically, I have adpated code from American Community Survey - 2011 single year analysis and the associated data preparation scripts.

## Database Setup

In order to focus on srvyr and databases, we start with a prepared dataset. The full code is avaliable on Github TKTKTKT, and the high level description of what it does is:

• Merges the household and person datasets so that we can look at the variables related to each person including those at the household level

• Selects only a few variables that will be used in this analysis to save space, but again it could easily be adapted to keep all of the variables.

For more information on the specifics of the American Community Survey, see the asdfree site. Now, our code loads this prepared dataset, initiates a MonetDB database, and puts the data into the dataset.

suppressMessages({
library(MonetDBLite)
library(survey)
library(srvyr)
library(dplyr)
library(RSQLite)
})

# Load data (Currently only Alaska and Hawaii to keep file size small and with
# limited variables, butcode that downloaded the files is available here TKTKTKT,

# Set up database and table
db <- src_monetdblite()
acs_m_db <- copy_to(db, acs_m, "acs_m", temporary = FALSE)

# Make table read only to reflect real world usage (see advanced topics below).
db_status <- dbSendQuery(acs_m_db$src$con, "ALTER TABLE acs_m SET READ ONLY")

# Or, if the data was already stored in the database, you could do this
# acs_m_data <- tbl(db, sql("SELECT * FROM acs_m"))

Now that we have the data in the database, we can interact with the database directly using sql commands, or we can use dplyr’s functionality to treat it mostly the same as a local data.frame. However, the data is not stored in memory, so we could work with much larger datasets (though in this case, the data is too small for this to be a problem).

# Same results
acs_m %>%
group_by(sex) %>%
summarize(hicov = mean(hicov))

acs_m_db %>%
group_by(sex) %>%
summarize(hicov = mean(hicov))

# But smaller object size
object.size(acs_m)
object.size(acs_m_db)

Note that though many commands behave exactly the same whether on a local data.frame or database, sometimes more advanced / complicated syntax around variable modification allowed in dplyr does not work on a particular database and so it is better to be more explicit. For example, creating a variable inside of a summarize call does not work (at least with MonetDB). .

acs_m %>%
group_by(sex) %>%
summarize(hicov = mean(hicov == 1))

# acs_m_db %>%
#   group_by(sex) %>%
#   summarize(hicov = mean(hicov == 1))
#
# > Error in .local(conn, statement, ...) :
# > Unable to execute statement....
# > ....

# Creating the variable separately works as an integer works though
acs_m_db %>%
group_by(sex) %>%
mutate(hicov = ifelse(hicov == 1, 1L, 0L)) %>%
summarize(hicov = mean(hicov))

Further, sometimes working with variable types can get difficult if you are used to working in R. Notice how in the above, instead of hicov = (hicov == 1), I wrote out the ifelse statement. If I hadn’t MonetDB would be unable to calculate the mean of the boolean variable created.

Finally, a major difference when transitioning from dplyr on local data.frames is that not all R functions are translated to SQL. For example, cut() isn’t implemented in SQL, so you can’t create a new variable in the data.frame using it.

acs_m %>%
group_by(agecat = cut(agep, c(0, 19, 35, 50, 65, 200))) %>%
summarize(hicov = mean(hicov == 1))

# acs_m_db %>%
#   group_by(agecat = cut(agep, c(0, 19, 35, 50, 65, 200))) %>%
#   summarize(hicov = mean(hicov == 1))
#
# > Error in .local(conn, statement, ...) :
# > Unable to execute statement....
# > ...

acs_m_db %>%
mutate(agecat = ifelse(agep < 19, "0-18",
ifelse(agep >= 19 & agep < 35, "19-34",
ifelse(agep >= 35 & agep < 50, "35-49",
ifelse(agep >= 50 & agep < 65, "50-64",
ifelse(agep >= 65, "65+", NA)))))) %>%
group_by(agecat) %>%
summarize(hicov = mean(hicov))

For more information on the specifics of databases with dplyr, see vignette("database", package = dplyr), the DBI package or the specific database packages, like MonetDBLite or RSQLite.

## Srvyr Setup

Srvyr commands are nearly identical to old. The only difference for setup is that you need a variable that uniquely identifieds each row in the database (uid).

acs_m_db_svy <- acs_m_db %>%
as_survey_rep(
weight = pwgtp,
repweights = matches("pwgtp[0-9]+") ,
scale = 4 / 80,
rscales = rep(1 , 80),
mse = TRUE,
type = "JK1",
variables = -c(matches("^pwgtp")),
uid = c(serialno, sporder)
)

acs_m_db_svy

Because srvyr stores the survey variables locally, the srvyr object takes up much more memory than the dplyr one. However, this object would not grow in size if you added more data variables to your survey, so if your survey is very wide, it will save a lot space.

object.size(acs_m_db_svy)

# Analysis

Analysis commands from srvyr are also similar to ones that work on local data.frames. The main differences come from the issues discussed above about explicitly creating variables difficulties in translating R commands, and variable types.

The following anlaysis is based on the asdfree analysis and shows some basi analysis on the total populaiton, insurance coverage, age and sex.

# You can calculate the population of the united states #
# by state
acs_m_db_svy %>%
mutate(one = 1L) %>% # Note that because of weird behavior of MonetDB, need to use 1L not just 1
group_by(st) %>%
summarize(count = survey_total(one))

acs_m_db_svy %>%
summarize(agep = survey_mean(agep, na.rm = TRUE))

# Average age by state
acs_m_db_svy %>%
group_by(st) %>%
summarize(agep = survey_mean(agep, na.rm = TRUE))

acs_m_db_svy %>%
mutate(hicov = as.character(hicov)) %>%
group_by(hicov) %>%
summarize(pct = survey_mean(na.rm = TRUE))

# by state
acs_m_db_svy %>%
mutate(hicov = as.character(hicov)) %>%
group_by(st, hicov) %>%
summarize(pct = survey_mean(na.rm = TRUE))

# 25th, median, and 75th percentile of age of residents of the united states (downloaded states)
acs_m_db_svy %>%
summarize(agep = survey_quantile(agep, c(0.25, 0.5, 0.75), na.rm = TRUE))

# Filter works, so we can restrict the acs_m object to females only
acs_m_db_svy_female <- acs_m_db_svy %>%
filter(sex == 2)

# Now any of the above commands can be re-run using the acs_m_female object
# instead of the acs_m object in order to analyze females only

# This is equivalent to using acs_m, and applying the filter every time.

acs_m_db_svy_female %>%
summarize(agep = survey_mean(agep, na.rm = TRUE))

acs_m_db_svy_female %>%
summarize(agep = survey_median(agep, na.rm = TRUE))

# Note that though some R functions are translated by dplyr into SQL, not
# all of them are. For example, when constructing a new age category
# variable in the dataset neither findIntervals nor cut work on databases,
# so we have to spell out groups with ifelse()
acs_m_db_svy %>%
mutate(agecat = ifelse(agep < 19, "0-18",
ifelse(agep >= 19 & agep < 35, "19-34",
ifelse(agep >= 35 & agep < 50, "35-49",
ifelse(agep >= 50 & agep < 65, "50-64",
ifelse(agep >= 65, "65+", NA)))))) %>%
group_by(agecat) %>%
summarize(pct = survey_mean(na.rm = TRUE))


text:
Fork me on GitHub
#clojure
<
2015-12-07
>
maxt10:12:43

clojure-csv or clojure.data.csv ?

tragiclifestories11:12:06

nice, reminds me of the Python csv lib, which rather spoiled me for others

hoopes12:12:10

hi all - i was looking for a simple example of a udp server - i found a gist (that appears outdated) using aleph, but not much else - am i missing any obvious libraries? (coming from a python background). Thanks for any notes/help/etc!

hoopes12:12:16

and uh - this is my first time in a slack channel, so if there's an accepted way to ask questions, please feel free to correct my style....

maxt12:12:13

@dm3: thanks, yes semantic-csv looks nice, but it still needs a lib for parsing if I understood correctly

maxt12:12:35

clojure.data.csv seems to be most recent

pupeno12:12:49

For my Prerenderer library I want to have some programs that exercise it, to help test/develop it. These would be almost full Leineingen projects. Is there any conventions where to put them? I thinking /example. And is there a way to have leineingen pick the Prerenderer source code instead of the installed one? the checkout trick won’t work as github can’t keep track of symlinks.

rcanepa14:12:40

Anyone is using duct?. I am trying to figure out how to pass the database record to the handler component. Actually, I am just passing the database URI, but I would like to pass the DB record in order to facilitate unit testing. The instantiation of the database records happens inside the same map (system-map) that I use to instantiate the handler component. Should I move the DB record instantiation outside (maybe inside the let bindings)?

rcanepa14:12:23

I put comments on the lines where I create the DB record and the handler component.

rcanepa14:12:56

I want to do this in order to mock up the database for my unit tests.

jstew14:12:10

rcanepa: If you just want to add a mock database record, can't you create the system, then just assoc the mock db record into it before calling start on the system (when testing)?

jstew14:12:45

That way you don't have to muck around with the system while you're building it just to account for a mock.

rcanepa14:12:55

@jstew: Yeah, that sounds much more easy. Thanks for the input!

maxt15:12:21

A parsing question: I have a sequence 1ab2cde3f4ghij which I would like to turn into a map {1 (ab) 2 (cde) 3 (f) 4 (ghij)}. I written one version with recur, which got ugly, and another version with partition-by which I'm not entierly happy with either

(let [partitioned (partition-by (set "1234") "1ab2cde3f4ghij")
ks (map first (take-nth 2 partitioned))
vs (take-nth 2 (rest partitioned))]
(zipmap ks vs))

Other suggestions?

maxt15:12:26

In words, I would like each letter to be classified by the digit preceding it.

andreaslundahl15:12:52

@maxt Are you only dealing with strings? Maybe (re-seq #"(\d)([a-z]+)" "1ab2cde3f4ghij") would produce more convenient partitions, to create a map from?

maxt15:12:15

@andreaslundahl: Right, intresting. No I want to use it on any sequences. But your idea does suggest that it's a finite state machine im looking for

jstew15:12:17

(group-by #(nth % 1) (re-seq #"(\d)([a-z]+)" "1ab2cde3f4ghij")) got me close-ish. Not really a better solution than what you posted though.

I need to save various media (images, video) to a file system. Should I use Java interop, or is there a library that handles this for me?

roelof16:12:38

Anyone who knows a good tutorial about making a website with login , authorisation and so on with clojure

roelof16:12:02

I like to not using things like luminus

@roelof I don't know any full tutorials, but Friend is a good place to start.

jaen16:12:09

IMO Friend is too complicated

jaen16:12:32

I suggest buddy instead - https://github.com/funcool/buddy/ - it's considerably simpler

jaen16:12:06

Here are some nice blogposts on how to use it - http://rundis.github.io/blog/tags/buddy.html

jaen16:12:49

The poster uses it to secure a REST API, but securing a website should be analogous

sarcilav16:12:50

I also like/recommend buddy, @roelof here are some examples from https://github.com/funcool/buddy-auth/tree/master/examples

roelof16:12:28

oke, and are there any good tutorials about routes, template engines and so on

jaen16:12:57

Oh, that's probably somewhat better for starters, since it has cookie based authentication example.

jaen16:12:30

For routes and templating you don't really need more than the documentation of libraries, that's pretty simple.

jaen16:12:50

If you want to template in Clojure then hiccup could be a nice templating language

roelof16:12:10

oke, so I do not have to buy a book like web Essentials or web development with clojure ?

jaen16:12:46

If you want something more classic then something like mustache - https://github.com/davidsantiago/stencil - or selmer - https://github.com/yogthos/Selmer - could work

jaen16:12:41

@roelof: I didn't need something like this, but then again I've been doing Rails for four years already.

jaen16:12:53

For you it might be helpful if you didn't do webdev.

jaen16:12:15

Also, why not Luminus? To me it seems like a good introduction to Clojure way of building webapps.

jaen16:12:34

I wouldn't make all of the same choices, but it's a good overview of what you need.

roelof16:12:50

oke, so Luminus is not the same as Rails

roelof16:12:10

I did not like that because it dicating me how I must do something

jaen16:12:12

Yeah, it's not

jaen16:12:17

It's more like a micro framework

jaen16:12:35

If you don't like something Luminus set up for you, you can change it

jaen16:12:48

It's not far off to call it an application template.

jaen16:12:02

I just think it's fairly representative of how most simple Clojure webdev projects would look.

roelof16:12:30

@jaen : oke, then I will take a second look at luminus

jaen16:12:31

There's also duct, but it uses components, that might be somewhat confusing for a newcomer.

jaen16:12:22

@roelof: Luminus also has a book, but it's a bit outdated and a new edition is supposed to come soon-ish.

roelof16:12:40

oke, then I will concentrate on 4clojure at this moment and try to do all the "easy" challenges

eraserhd16:12:59

Has anybody used JNA with Clojure (or even Java)? Is it a good thing?

bigkahuna16:12:11

@jaen: I think duct may have a slightly steeper learning curve but it's definitely worth the effort especially when deciding the best place to hold state. I went to Clojure X and saw a great overview/rationale by the author: https://skillsmatter.com/skillscasts/7229-duct-covered and here's another tutorial which covers all the basics very well by the author of Luminus : http://yogthos.net/posts/2015-10-01-Compojure-API.html

jaen16:12:08

@bigkahuna: yeah, hence my caveat "for a newcomer". I'm using components myself (though not duct).

roelof16:12:55

and im a newcomer, I try to learn clojure for some 3 - 4 weeks

bigkahuna16:12:04

@jaen: Yep, I agree might be information overload for a newcomer

bigkahuna16:12:57

@roelof: Working through 4clojure is a very useful exercise as you'll internalize how Clojure works. I've been attended a few hackadays as part of the LJC and found the following pages useful: http://www.practical.li/ The author John Stevenson also has a github page with a project that can be loaded into a repl or Lightable that outlines Clojure's features. You can experiment with them to further your understanding: https://github.com/practicalli/clojure-through-code/tree/master/src/clojure_through_code I've also found Living Clojure by Carin Meier an excellent book for newbies. It guides you through the language at just the right pace: http://shop.oreilly.com/product/0636920034292.do

roelof17:12:39

@bigkahuna: thanks for the tips

roelof17:12:44

anyone who knows the web essiantials book where I can learn luminus

jaen17:12:59

There's also an older one by the author of the framework - https://pragprog.com/book/dswdcloj/web-development-with-clojure

jaen17:12:07

But I didn't read either so not sure how good they are.

jaen17:12:20

Though probably the first one would be better since it's newer

roelof17:12:14

I mean that book

roelof17:12:33

I saw it it newer , It's from begin this year

bigkahuna17:12:54

https://pragprog.com/book/dswdcloj2/web-development-with-clojure-second-edition is out next year. First book is still worth reading, just that the libraries used are a tad out of date.

roelof17:12:23

Can I refractor this code so it uses reduce

roelof17:12:35

(defn my-gcd
[number1 number2]
(loop
[n1 number1
n2 number2]
(if (zero? (rem n1 n2))
n2
(recur n1 (rem n1 n2)))))

jeff.terrell18:12:56

Probably. I think there's a way to short-circuit reduce so that at some point you can skip processing the rest of the items. But I'd consider that an anti-pattern and a sign that you should be using loop / recur, so I'd say stick with what you have.

jeff.terrell18:12:43

reduce makes the most sense when you want to process every item in a sequence.

roelof18:12:01

oke, I thought as a beginner that I can better use reduce or map then use loop

jeff.terrell18:12:52

Yeah, it's better to use those when they make sense. But to me, this looks like a good use case for loop. Neither map nor reduce are well-shaped tools for this problem.

roelof18:12:46

oke,thanks

roelof18:12:05

chips, code is somewhere wrong (my-gcd gives as output 5 where it suppose to be 1

roelof18:12:30

maybe try to print both numbers to see where it went wrong

roberto18:12:00

roberto18:12:47

you have them in reverse order

roberto18:12:00

if you enter 7 5 you get the answer you are looking for

roberto18:12:20

which would indicate that you are operating on them in reverse order

roelof18:12:00

oke, but on 2 4 , 10 5 and 1023 858 it works well

roelof18:12:28

so it seems that then I do not matter if the first number bigger is then the second

roberto18:12:30

why would you think that is?

roberto18:12:44

roberto18:12:00

in rem it does matter which one is the first and second argument

roberto18:12:34

(rem 2 4) is not equal to (rem 4 2)

roberto18:12:50

similarly (rem 5 7) is not equal to (rem 7 5)

roelof18:12:32

wait, can it be that on this case both numbers are not even

roelof18:12:10

in all cases that work 1 or 2 of the numbers are even. 2 4 in the first, 10 in the second and 858 in the last one

roelof18:12:23

7 and 5 are both not even

roelof18:12:52

I have to think well how I can make that check in the code I have now

roelof18:12:24

@roberto any tips how I can swap the two values if both are odd ?

roelof18:12:13

it is now a "big" problem that variables are immutable

roberto18:12:03

not really

roberto18:12:16

so, look at your recur portion

roberto18:12:38

try switching the order of the arguments, and see what you get

roelof18:12:41

not good. 1023 858 gives now the wrong answer 165 instead of 33

roelof18:12:06

I do now this ' (recur (rem n1 n2) n1 ))

roberto18:12:18

ah, I see your algo is wrong

roberto18:12:30

so, you don’t want to check that (rem x y ) is zero

roberto18:12:04

if you swap the args, when you do (my-gcd num1 num2)

roberto18:12:14

you want to check that num2 is zero

roelof18:12:03

???? according to the Wikipedia page I have to check if the remainder is zero

roelof18:12:07

maybe I have to use another algoritm

roberto18:12:37

it works for me, make sure what whatever algo you are looking at on wikipedia, is using recursion.

roberto18:12:17

#'user/my-gcd
user=> (my-gcd 5 7)
1
user=> (my-gcd 1023 858)
33
user=>

roberto18:12:38

i can’t show you the definition of my-gcd without giving away the answer

roelof18:12:25

I tried to use the euclids solution on this page : https://en.wikipedia.org/wiki/Greatest_common_divisor

roelof18:12:40

but I think I maybe misunderstood it

sarcilav18:12:40

@roelof your solution works you are just forgetting something that is in the definition

\gcd(a,b) = \gcd(a - b,b)\quad, if a > b
\gcd(a,b) = \gcd(a, b-a)\quad, if b > a

roelof18:12:29

@sarcilav: so I have to make another if then

sarcilav18:12:46

that should work

roelof18:12:17

oke, thanks

roberto18:12:02

you solved it?

roelof18:12:30

yep, your tip was right I had to check if the second number was zero

roberto18:12:51

this is my solution:

roberto18:12:57

(defn my-gcd [n1 n2] (loop [a n1 b n2] (if (zero? b) a (recur b (rem a b)))))

roelof18:12:58

and first a number and then the calculation of the remainder

roelof18:12:32

I have now the same but I use other names

roelof18:12:48

(defn my-gcd
[number1 number2]
(loop
[n1 number1
n2 number2]
(if (zero? n2)
n1
(recur n2 (mod n1 n2) ))))

roberto18:12:20

okay, I’m heading home. The chair in this coffee shop is too hard.

roelof18:12:49

I can image. Drive safe

ustunozgur19:12:19

Hacker Dict Our ClojureCup hackathon entry is a dictionary for programmers by programmers. <Http://hacker-dict.com> We believe the programming community needs such a tool for discussing and gathering chronological data about programming. Basic functionality is there, we will improve upon it once the judging period is over. We used Clojure, ClojureScript and Datomic for the project Please sign up and share: http://hacker-dict.com

jeff.terrell19:12:11

Nice work. One thing you probably already noticed: the browser back button doesn't work.

ustunozgur19:12:47

@jeff.terrell: thank you :) yes aim was to use a client side library but didn't have time to learn so it just updates the Url manually via push state

ustunozgur19:12:06

We will implement properly once they allow us to commit more

ustunozgur19:12:40

This is interesting for Clojure community because most stuff on the net is mutable

ustunozgur19:12:47

Wikipedia is mutable

ustunozgur19:12:00

ustunozgur19:12:02

We aim to create a medium where people ca. Share facts and opinions about various aspects of programming

jeff.terrell19:12:18

Yeah, that's a pretty interesting concept. What data should be mutable vs. immutable?

ustunozgur19:12:31

And others can learn chronologically

ustunozgur19:12:47

So that one can learn historical data and trends

ustunozgur19:12:54

And what is what and who is who

ustunozgur19:12:18

We also want to be able to share subjective facts not objective facts :)

ustunozgur19:12:18

You can also link with []

ustunozgur19:12:24

If it contains spaces

ustunozgur19:12:36

[rich hickey]

dmitrig0120:12:55

in some sense, a wikipedia article is a mutable pointer to its most recent revision, and the revisions are immutable

jeff.terrell20:12:13

Much like, in git, the objects (e.g. commits) are immutable but the references (e.g. branches) are pointers that point to different immutable objects at different times. Datomic has a similar model. The stored data is immutable, but an entity's attribute can take on different values at different times.

dmitrig0120:12:36

yep. or clojure atoms

lumengxi21:12:19

wonder anyone used https://github.com/Factual/drake in production? is this still under active development?

roelof21:12:46

When I do lein ring server on a blank luminus project then I see this error message : 'ring' is not a task. See 'lein help'.

roelof21:12:08

I see that ring is mentioned on the dependencies line

roelof21:12:19

so anyone a idea what is wrong ?

johnjtran21:12:38

@roelof most likely this newer and you'll need lein run instead.

roelof21:12:23

?? you mean I have to use lein run instead of lein ring server

borkdude21:12:25

what's wrong with this expression: (aset (make-array Integer 5 5) 0 0 1)

roelof21:12:54

I have this on a book written in febr. 2015 and now a error in the book ?

borkdude21:12:43

nm, this works: (aset (make-array Long/TYPE 5 5) 0 0 1)

johnjtran21:12:47

@roelof: yes, try it lein run when you pull stuff with lein new luminus project it's actually not reflective of the book instead it pulls the latest. In all likelihood this stuff is evolving very rapidly so what was good then is probably dated now.

roelof21:12:22

@johnjtran: thanks, it worked. When I open my browser I see the startup page

borkdude21:12:15

@roelof: when in doubt, you can check http://www.luminusweb.net/docs also

roelof21:12:57

@borkdude: thanks

johnjtran22:12:09

I have a love/hate relationship with luminus. By all accounts, it appears to be very good -- but multiple layers and lots of wrappers etc. I don't advocate that you pick one over another, but take a look at pedestal` as an option. Less bells and whistles but appears to be slightly less aggressive with their updates -- thus stable.

bronsa22:12:55

@borkdude: integer literals in clojure default to long

bronsa22:12:05

you're trying to shove a Long inside an Integer[][]

borkdude22:12:15

@bronsa: yeah, that was it

hoopes23:12:08

hi, i’m trying to go from byte array to map with some spec - googling brought me to buffy and octet, but i just found gloss . is gloss my best bet for byte array to map of data?

hoopes23:12:21

(if this isn’t the right question for beginner questions, please let me know - thanks!)

hoopes23:12:28

right channel, i mean

hoopes23:12:09

http://construct.readthedocs.org/en/latest/ is the python version of what i'm trying to find

hoopes23:12:11

that looks really similar to gloss, down to the -le and -be stuff - 6 of one, half dozen the other? is there a preferred lib? it’s a little weird coming from python, where there’s usually one best way to do something

hoopes23:12:33

paralysis of analysis...


text:
# Math Help - Square root's inverse right but restrictions?

1. ## Square root's inverse right but restrictions?

I get the answer which is f-1(x) = (-1/9(x+2)^2) + 1 but the solution guide says:

" f-1(x) = (-1/9(x+2)^2) + 1 x >/= 2 OR f-1(x) = (-1/9(x+2)^2) + 1 [-2, infinty["

How do I get such restrictions? This seems to be false when checked with my graphing software but I could be wrong.

Any help would be greatly appreciated!

2. let f(x)=y and the equation with respect to x

$y=3\sqrt{-(x-1)}-2$

$y+2=3\sqrt{-(x-1)}$

$(y+2)^2=9(-(x-1))$

$\frac{(y+2)^2}{9}=-(x-1)$

$-\left(\frac{(y+2)^2}{9}\right)=x-1$

$1-\left(\frac{(y+2)^2}{9}\right)=x$

change the variables

$f(x)=1-\left(\frac{(x+2)^2}{9}\right)$

3. $f(x) = 3\sqrt{-(x - 1)} - 2$

The inverse, as Amer worked out, is
$f^{-1}(x) = -\frac{(x + 2)^2}{9} + 1$
(I like to put the variables first. )

But it wouldn't be right to leave it as it is without stating the restrictions. If we consider this function without knowing beforehand that it was the inverse of some other function, the domain would be all reals and the range would be (-∞, 1], but that's wrong in our case. This function $f^{-1}(x)$ is the inverse of $f(x)$, and since $f(x)$ is one-to-one, the domains and ranges switch.

The domain of $f(x)$ is (-∞, 1] and the range is [-2, ∞). Therefore the domain of $f^{-1}(x)$ is [-2, ∞) and the range is (-∞, 1]. The graph of $f^{-1}(x)$ would be the right half of a parabola, opening downward.

01


text:
### This is embarrasing...

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### This is embarrasing...

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# CSA - N287.6-11

## Pre-operational proof and leakage rate testing requirements for concrete containment structures for nuclear power plants

active, Most Current
 Organization: CSA Publication Date: 1 January 2011 Status: active Page Count: 36
##### scope:

This Standard specifies requirements for pre-operational proof and leakage rate testing of concrete containment structures of a containment system that are designated as class containment components.

Note: Other components and appurtenances of the containment system may be tested coincident with the tests specified in this Standard provided that the intent of this Standard is observed and the requirements of CSA N285.0 are considered.

In CSA standards, "shall" is used to express a requirement, i.e., a provision that the user is obliged to satisfy in order to comply with the standard; "should" is used to express a recommendation or that which is advised but not required; and "may" is used to express an option or that which is permissible within the limits of the standard.

Notes accompanying clauses do not include requirements or alternative requirements; the purpose of a note accompanying a clause is to separate from the text explanatory or informative material.

Notes to tables and figures are considered part of the table or figure and may be written as requirements.

Annexes are designated normative (mandatory) or informative (nonmandatory) to define their application.

### Document History

May 1, 2011
Pre-operational proof and leakage rate testing requirements for concrete containment structures for nuclear power plants
This Standard specifies requirements for pre-operational proof and leakage rate testing of concrete containment structures of a containment system that are designated as class containment components....
N287.6-11
January 1, 2011
Pre-operational proof and leakage rate testing requirements for concrete containment structures for nuclear power plants
This Standard specifies requirements for pre-operational proof and leakage rate testing of concrete containment structures of a containment system that are designated as class containment components....
January 1, 2011
Pre-operational proof and leakage rate testing requirements for concrete containment structures for nuclear power plants
This Standard specifies requirements for pre-operational proof and leakage rate testing of concrete containment structures of a containment system that are designated as class containment components....
January 1, 2011
Pre-operational proof and leakage rate testing requirements for concrete containment structures for nuclear power plants
This Standard specifies requirements for pre-operational proof and leakage rate testing of concrete containment structures of a containment system that are designated as class containment components....
January 1, 1994
Pre-Operational Proof and Leakage Rate Testing Requirements for Concrete Containment Structures for CANDU Nuclear Power Plants
This Standard provides the requirements for pre-operational proof tests and leakage rate tests of concrete containment structures of a containment system designed as Class Containment components....
January 1, 1994
Pre-Operational Proof and Leakage Rate Testing Requirements for Concrete Containment Structures for CANDU Nuclear Power Plants
This Standard provides the requirements for pre-operational proof tests and leakage rate tests of concrete containment structures of a containment system designed as Class Containment components....
January 1, 1994
Pre-Operational Proof and Leakage Rate Testing Requirements for Concrete Containment Structures for CANDU Nuclear Power Plants
This Standard provides the requirements for pre-operational proof tests and leakage rate tests of concrete containment structures of a containment system designed as Class Containment...