text:
# American Institute of Mathematical Sciences

• Previous Article
A note on optimization modelling of piecewise linear delay costing in the airline industry
• JIMO Home
• This Issue
• Next Article
Adaptive large neighborhood search Algorithm for route planning of freight buses with pickup and delivery
July  2021, 17(4): 1795-1807. doi: 10.3934/jimo.2020046

## Network data envelopment analysis with fuzzy non-discretionary factors

 1 Department of International Business, Kao Yuan University, Kaohsiung, 82151, Taiwan 2 Department of Mechanical and Automation Engineering, I-Shou University, Kaohsiung, 84001, Taiwan 3 Department of Applied Mathematics, Tunghai University, Taichung 40704, Taiwan 4 Department of Applied Mathematics, National Chiayi University, Chiayi, 60004, Taiwan

* Corresponding author: C.-F. Hu

Received  January 2019 Revised  September 2019 Published  March 2020

Network data envelopment analysis (DEA) concerns using the DEA technique to measure the relative efficiency of a system, taking into account its internal structure. The results are more meaningful and informative than those obtained from the conventional DEA models. This work proposed a new network DEA model based on the fuzzy concept even though the inputs and outputs data are crisp numbers. The model is then extended to investigate the network DEA with fuzzy non-discretionary variables. An illustrative application assessing the impact of information technology (IT) on firm performance is included. The results reveal that modeling the IT budget as a fuzzy non-discretionary factor improves the system performance of firms in a banking industry.

Citation: Cheng-Kai Hu, Fung-Bao Liu, Hong-Ming Chen, Cheng-Feng Hu. Network data envelopment analysis with fuzzy non-discretionary factors. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1795-1807. doi: 10.3934/jimo.2020046
##### References:

show all references

##### References:
]">Figure 1.  General network systems [12]
]">Figure 2.  Network system discussed in [18]
Data set for assessing IT impact on firm performance
 DMU j IT Fixed No. of Deposits Profit Fraction $\rm {budget}$ ${\mbox{assets}}$ ${\mbox{employees }}$ of loans $({＄ \ \mbox{billions})}$ $({＄ \ \mbox{billions})}$ $({＄ \ \mbox{billions})}$ $({＄ \ \mbox{billions})}$ $({＄ \ \mbox{billions})}$ ${\mbox{recovered}}$ $X_1$ $X_2$ $X_3$ $Z$ $Y_1$ $Y_2$ 1 $0.150$ $0.713$ $13.3$ $14.478$ $0.232$ $0.986$ 2 $0.170$ $1.071$ $16.9$ $19.502$ $0.340$ $0.986$ 3 $0.235$ $1.224$ $24.0$ $20.952$ $0.363$ $0.986$ 4 $0.211$ $0.363$ $15.6$ $13.902$ $0.211$ $0.982$ 5 $0.133$ $0.409$ $18.485$ $15.206$ $0.237$ $0.984$ 6 $0.497$ $5.846$ $56.42$ $81.186$ $1.103$ $0.955$ 7 $0.060$ $0.918$ $56.42$ $81.186$ $1.103$ $0.986$ 8 $0.071$ $1.235$ $12.0$ $11.441$ $0.199$ $0.985$ 9 $1.500$ $18.120$ $89.51$ $124.072$ $1.858$ $0.972$ 10 $0.120$ $1.821$ $19.8$ $17.425$ $0.274$ $0.983$ 11 $0.120$ $1.915$ $19.8$ $17.425$ $0.274$ $0.983$ 12 $0.050$ $0.874$ $13.1$ $14.342$ $0.177$ $0.985$ 13 $0.370$ $6.918$ $12.5$ $32.491$ $0.648$ $0.945$ 14 $0.440$ $4.432$ $41.9$ $47.653$ $0.639$ $0.979$ 15 $0.431$ $4.504$ $41.1$ $52.63$ $0.741$ $0.981$ 16 $0.110$ $1.241$ $14.4$ $17.493$ $0.243$ $0.988$ 17 $0.053$ $0.450$ $7.6$ $9.512$ $0.067$ $0.980$ 18 $0.345$ $5.892$ $15.5$ $42.469$ $1.002$ $0.948$ 19 $0.128$ $0.973$ $12.6$ $18.987$ $0.243$ $0.985$ 20 $0.055$ $0.444$ $5.9$ $7.546$ $0.153$ $0.987$ 21 $0.057$ $0.508$ $5.7$ $7.595$ $0.123$ $0.987$ 22 $0.098$ $0.370$ $14.1$ $16.906$ $0.233$ $0.981$ 23 $0.104$ $0.395$ $14.6$ $17.264$ $0.263$ $0.983$ 24 $0.206$ $2.680$ $19.6$ $36.430$ $0.601$ $0.982$ 25 $0.067$ $0.781$ $10.5$ $11.581$ $0.120$ $0.987$ 26 $0.100$ $0.872$ $12.1$ $22.207$ $0.248$ $0.972$ 27 $0.0106$ $1.757$ $12.7$ $20.670$ $0.253$ $0.988$
 DMU j IT Fixed No. of Deposits Profit Fraction $\rm {budget}$ ${\mbox{assets}}$ ${\mbox{employees }}$ of loans $({＄ \ \mbox{billions})}$ $({＄ \ \mbox{billions})}$ $({＄ \ \mbox{billions})}$ $({＄ \ \mbox{billions})}$ $({＄ \ \mbox{billions})}$ ${\mbox{recovered}}$ $X_1$ $X_2$ $X_3$ $Z$ $Y_1$ $Y_2$ 1 $0.150$ $0.713$ $13.3$ $14.478$ $0.232$ $0.986$ 2 $0.170$ $1.071$ $16.9$ $19.502$ $0.340$ $0.986$ 3 $0.235$ $1.224$ $24.0$ $20.952$ $0.363$ $0.986$ 4 $0.211$ $0.363$ $15.6$ $13.902$ $0.211$ $0.982$ 5 $0.133$ $0.409$ $18.485$ $15.206$ $0.237$ $0.984$ 6 $0.497$ $5.846$ $56.42$ $81.186$ $1.103$ $0.955$ 7 $0.060$ $0.918$ $56.42$ $81.186$ $1.103$ $0.986$ 8 $0.071$ $1.235$ $12.0$ $11.441$ $0.199$ $0.985$ 9 $1.500$ $18.120$ $89.51$ $124.072$ $1.858$ $0.972$ 10 $0.120$ $1.821$ $19.8$ $17.425$ $0.274$ $0.983$ 11 $0.120$ $1.915$ $19.8$ $17.425$ $0.274$ $0.983$ 12 $0.050$ $0.874$ $13.1$ $14.342$ $0.177$ $0.985$ 13 $0.370$ $6.918$ $12.5$ $32.491$ $0.648$ $0.945$ 14 $0.440$ $4.432$ $41.9$ $47.653$ $0.639$ $0.979$ 15 $0.431$ $4.504$ $41.1$ $52.63$ $0.741$ $0.981$ 16 $0.110$ $1.241$ $14.4$ $17.493$ $0.243$ $0.988$ 17 $0.053$ $0.450$ $7.6$ $9.512$ $0.067$ $0.980$ 18 $0.345$ $5.892$ $15.5$ $42.469$ $1.002$ $0.948$ 19 $0.128$ $0.973$ $12.6$ $18.987$ $0.243$ $0.985$ 20 $0.055$ $0.444$ $5.9$ $7.546$ $0.153$ $0.987$ 21 $0.057$ $0.508$ $5.7$ $7.595$ $0.123$ $0.987$ 22 $0.098$ $0.370$ $14.1$ $16.906$ $0.233$ $0.981$ 23 $0.104$ $0.395$ $14.6$ $17.264$ $0.263$ $0.983$ 24 $0.206$ $2.680$ $19.6$ $36.430$ $0.601$ $0.982$ 25 $0.067$ $0.781$ $10.5$ $11.581$ $0.120$ $0.987$ 26 $0.100$ $0.872$ $12.1$ $22.207$ $0.248$ $0.972$ 27 $0.0106$ $1.757$ $12.7$ $20.670$ $0.253$ $0.988$
The system efficiency, $\theta_p^{\ast},$ and the membership degree, $\alpha_p, p = 1, 2, \cdots, 27.$
 DMU j Model (2) $\theta^{\ast}$ ${ \text{Model (6)}}$ DMU j Model (2)$\theta^{\ast}$ ${ \text{Model (6)}}$ $\alpha^{\ast}$ $1-\alpha^{\ast}$ $\alpha^{\ast}$ $1-\alpha^{\ast}$ $1$ $0.6388$ $0.3612$ $0.6388$ $15$ $0.6582$ $0.3418$ $0.6582$ $2$ $0.6507$ $0.3493$ $0.6507$ $16$ $0.6646$ $0.3354$ $0.6646$ $3$ $0.5179$ $0.4821$ $0.5179$ $17$ $0.7177$ $0.2823$ $0.7177$ $4$ $0.5986$ $0.4014$ $0.5986$ $18$ $1.0000$ $0.0000$ $1.0000$ $5$ $0.5556$ $0.4444$ $0.5556$ $19$ $0.8144$ $0.1856$ $0.8144$ $6$ $0.7599$ $0.2401$ $0.7599$ $20$ $0.6940$ $0.3060$ $0.6940$ $7$ $1.0000$ $0.0000$ $1.0000$ $21$ $0.7067$ $0.2933$ $0.7067$ $8$ $0.5352$ $0.4648$ $0.5352$ $22$ $0.7942$ $0.2058$ $0.7942$ $9$ $0.6249$ $0.3751$ $0.6249$ $23$ $0.7802$ $0.2198$ $0.7802$ $10$ $0.4961$ $0.5039$ $0.4961$ $24$ $0.9300$ $0.0700$ $0.9300$ $11$ $0.4945$ $0.5055$ $0.4945$ $25$ $0.6270$ $0.3730$ $0.6270$ $12$ $0.6685$ $0.3315$ $0.6685$ $26$ $1.0000$ $0.0000$ $1.0000$ $13$ $0.9487$ $0.0513$ $0.9487$ $27$ $1.0000$ $0.0000$ $1.0000$ $14$ $0.5880$ $0.4120$ $0.5880$
 DMU j Model (2) $\theta^{\ast}$ ${ \text{Model (6)}}$ DMU j Model (2)$\theta^{\ast}$ ${ \text{Model (6)}}$ $\alpha^{\ast}$ $1-\alpha^{\ast}$ $\alpha^{\ast}$ $1-\alpha^{\ast}$ $1$ $0.6388$ $0.3612$ $0.6388$ $15$ $0.6582$ $0.3418$ $0.6582$ $2$ $0.6507$ $0.3493$ $0.6507$ $16$ $0.6646$ $0.3354$ $0.6646$ $3$ $0.5179$ $0.4821$ $0.5179$ $17$ $0.7177$ $0.2823$ $0.7177$ $4$ $0.5986$ $0.4014$ $0.5986$ $18$ $1.0000$ $0.0000$ $1.0000$ $5$ $0.5556$ $0.4444$ $0.5556$ $19$ $0.8144$ $0.1856$ $0.8144$ $6$ $0.7599$ $0.2401$ $0.7599$ $20$ $0.6940$ $0.3060$ $0.6940$ $7$ $1.0000$ $0.0000$ $1.0000$ $21$ $0.7067$ $0.2933$ $0.7067$ $8$ $0.5352$ $0.4648$ $0.5352$ $22$ $0.7942$ $0.2058$ $0.7942$ $9$ $0.6249$ $0.3751$ $0.6249$ $23$ $0.7802$ $0.2198$ $0.7802$ $10$ $0.4961$ $0.5039$ $0.4961$ $24$ $0.9300$ $0.0700$ $0.9300$ $11$ $0.4945$ $0.5055$ $0.4945$ $25$ $0.6270$ $0.3730$ $0.6270$ $12$ $0.6685$ $0.3315$ $0.6685$ $26$ $1.0000$ $0.0000$ $1.0000$ $13$ $0.9487$ $0.0513$ $0.9487$ $27$ $1.0000$ $0.0000$ $1.0000$ $14$ $0.5880$ $0.4120$ $0.5880$
The results of solving the proposed fuzzy non-discretionary Model (14)
 $\begin{array}{c} \mbox{DMU}\\ j \end{array}$ Fuzzy non-discretionary input $\bar{X}_{1j}^{\ast}$ $\bar{X}_{2j}^{\ast}$ $\bar{X}_{3j}^{\ast}$ $\alpha^{\ast}$ $1-\alpha^{\ast}$ Rank 1 0.1102 0.5236 9.6335 0.2654 0.7346 18 2 0.1260 0.7723 12.4259 0.2589 0.7411 17 3 0.1586 0.8079 16.1328 0.3253 0.6747 25 4 0.1506 0.2564 10.9013 0.2864 0.7136 21 5 0.0921 0.2793 11.2165 0.3077 0.6923 23 6 0.4008 4.6342 45.4289 0.1936 0.8064 10 7 0.0600 0.9180 56.4200 0.0000 1.0000 1 8 0.0485 0.7677 8.0988 0.3173 0.6827 24 9 1.0908 13.1471 64.8529 0.2728 0.7272 20 10 0.0798 1.0715 12.8295 0.3351 0.6649 26 11 0.0797 1.0997 12.7416 0.3358 0.6642 27 12 0.0376 0.6544 9.7883 0.2490 0.7510 14 13 0.3519 5.3291 11.8900 0.0488 0.9512 5 14 0.3116 3.1047 29.5929 0.2918 0.7082 22 15 0.3212 3.2772 30.4969 0.2547 0.7453 16 16 0.0824 0.8943 10.7729 0.2512 0.7488 15 17 0.0413 0.3509 5.8871 0.2202 0.7798 11 18 0.3450 5.8920 15.5000 0.0000 1.0000 1 19 0.1080 0.8154 10.6151 0.1565 0.8435 7 20 0.0421 0.3349 4.4948 0.2343 0.7657 13 21 0.0441 0.3904 4.3686 0.2268 0.7732 12 22 0.0813 0.3043 11.6775 0.1707 0.8293 8 23 0.0853 0.3216 11.9554 0.1802 0.8198 9 24 0.1925 2.4125 18.3176 0.0654 0.9346 6 25 0.0488 0.5448 7.5342 0.2717 0.7283 19 26 0.1000 0.8720 12.1000 0.0000 1.0000 1 27 0.0106 1.7570 12.7000 0.0000 1.0000 1
 $\begin{array}{c} \mbox{DMU}\\ j \end{array}$ Fuzzy non-discretionary input $\bar{X}_{1j}^{\ast}$ $\bar{X}_{2j}^{\ast}$ $\bar{X}_{3j}^{\ast}$ $\alpha^{\ast}$ $1-\alpha^{\ast}$ Rank 1 0.1102 0.5236 9.6335 0.2654 0.7346 18 2 0.1260 0.7723 12.4259 0.2589 0.7411 17 3 0.1586 0.8079 16.1328 0.3253 0.6747 25 4 0.1506 0.2564 10.9013 0.2864 0.7136 21 5 0.0921 0.2793 11.2165 0.3077 0.6923 23 6 0.4008 4.6342 45.4289 0.1936 0.8064 10 7 0.0600 0.9180 56.4200 0.0000 1.0000 1 8 0.0485 0.7677 8.0988 0.3173 0.6827 24 9 1.0908 13.1471 64.8529 0.2728 0.7272 20 10 0.0798 1.0715 12.8295 0.3351 0.6649 26 11 0.0797 1.0997 12.7416 0.3358 0.6642 27 12 0.0376 0.6544 9.7883 0.2490 0.7510 14 13 0.3519 5.3291 11.8900 0.0488 0.9512 5 14 0.3116 3.1047 29.5929 0.2918 0.7082 22 15 0.3212 3.2772 30.4969 0.2547 0.7453 16 16 0.0824 0.8943 10.7729 0.2512 0.7488 15 17 0.0413 0.3509 5.8871 0.2202 0.7798 11 18 0.3450 5.8920 15.5000 0.0000 1.0000 1 19 0.1080 0.8154 10.6151 0.1565 0.8435 7 20 0.0421 0.3349 4.4948 0.2343 0.7657 13 21 0.0441 0.3904 4.3686 0.2268 0.7732 12 22 0.0813 0.3043 11.6775 0.1707 0.8293 8 23 0.0853 0.3216 11.9554 0.1802 0.8198 9 24 0.1925 2.4125 18.3176 0.0654 0.9346 6 25 0.0488 0.5448 7.5342 0.2717 0.7283 19 26 0.1000 0.8720 12.1000 0.0000 1.0000 1 27 0.0106 1.7570 12.7000 0.0000 1.0000 1
 [1] Chih-Chiang Fang. Bayesian decision making in determining optimal leased term and preventive maintenance scheme for leased facilities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020127 [2] Saeed Assani, Muhammad Salman Mansoor, Faisal Asghar, Yongjun Li, Feng Yang. Efficiency, RTS, and marginal returns from salary on the performance of the NBA players: A parallel DEA network with shared inputs. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021053 [3] İsmail Özcan, Sirma Zeynep Alparslan Gök. On cooperative fuzzy bubbly games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021010 [4] Roberto Civino, Riccardo Longo. Formal security proof for a scheme on a topological network. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021009 [5] Haripriya Barman, Magfura Pervin, Sankar Kumar Roy, Gerhard-Wilhelm Weber. Back-ordered inventory model with inflation in a cloudy-fuzzy environment. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1913-1941. doi: 10.3934/jimo.2020052 [6] Xinfang Zhang, Jing Lu, Yan Peng. Decision framework for location and selection of container multimodal hubs: A case in china under the belt and road initiative. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021061 [7] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [8] Jingni Guo, Junxiang Xu, Zhenggang He, Wei Liao. Research on cascading failure modes and attack strategies of multimodal transport network. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2020159 [9] Andrey Kovtanyuk, Alexander Chebotarev, Nikolai Botkin, Varvara Turova, Irina Sidorenko, Renée Lampe. Modeling the pressure distribution in a spatially averaged cerebral capillary network. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021016 [10] Mostafa Ghelichi, A. M. Goltabar, H. R. Tavakoli, A. Karamodin. Neuro-fuzzy active control optimized by Tug of war optimization method for seismically excited benchmark highway bridge. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 333-351. doi: 10.3934/naco.2020029 [11] Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023 [12] Haodong Chen, Hongchun Sun, Yiju Wang. A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2217-2242. doi: 10.3934/jimo.2020066 [13] Ru Li, Guolin Yu. Strict efficiency of a multi-product supply-demand network equilibrium model. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2203-2215. doi: 10.3934/jimo.2020065 [14] Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 [15] Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087 [16] Pascal Noble, Sebastien Travadel. Non-persistence of roll-waves under viscous perturbations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 61-70. doi: 10.3934/dcdsb.2001.1.61 [17] Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1673-1692. doi: 10.3934/dcdss.2020449 [18] Liqin Qian, Xiwang Cao. Character sums over a non-chain ring and their applications. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020134 [19] Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2921-2939. doi: 10.3934/dcdsb.2020211 [20] Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2677-2698. doi: 10.3934/dcds.2020381

2019 Impact Factor: 1.366

## Tools

Article outline

Figures and Tables


text:
Traffic Engineering

 Question 1
In a three-phase signal system design for a four-leg intersection, the critical flow ratios for each phase are 0.18, 0.32, and 0.22. The total loss time in each of the phases is 2s. As per Webster's formula, the optimal cycle length (in s, round off to the nearest integer) is _______________
 A 28 B 46 C 50 D 58
GATE CE 2021 SET-2   Transportation Engineering
Question 1 Explanation:
\begin{aligned} L &=2 \times 3=6 \text { seconds } \\ n &=3 \\ y &=(0.18+0.32+0.22) \\ C_{0} &=\left(\frac{1.5 L+5}{1-y}\right)=\left(\frac{1.5 \times 6+5}{1-(0.18+0.32+0.22)}\right) \\ &=50 \text { seconds } \end{aligned}
 Question 2
Relationship between traffic speed and density is described using a negatively sloped straight line. If $v_{f}$ is the free flow speed then the speed at which the maximum flow occurs is
 A 0 B $\frac{v_{f}}{4}$ C $\frac{v_{f}}{2}$ D $v_{f}$
GATE CE 2021 SET-2   Transportation Engineering
Question 2 Explanation:
Speed at maximum flow $=\frac{V_{f}}{2}$
 Question 3
Spot speeds of vehicles observed at a point on a highway are 40, 55, 60, 65 and 80 km/h. The space-mean speed (in km/h, round off to two decimal places) of the observed vehicles is ______________
 A 56.99 B 25.36 C 47.85 D 96.36
GATE CE 2021 SET-1   Transportation Engineering
Question 3 Explanation:
\begin{aligned} V_{1}=40, V_{2}=55, V_{3}=60, & V_{4}=65, V_{5}=80 \\ V_{S}=&\left(\frac{n}{\frac{1}{V_{1}}+\frac{1}{V_{2}}+\ldots+\frac{1}{V_{n}}}\right)=\frac{1}{40}+\frac{1}{55}+\frac{1}{60}+\frac{1}{65}+\frac{1}{80} \\ =& 56.99 \mathrm{kmph} \end{aligned}
 Question 4
Vehicular arrival at an isolated intersection follows the Poisson distribution. The mean vehicular arrival rate is 2 vehicle per minute. The probability (round off to two decimal places) that at least 2 vehicles will arrive in any given 1-minute interval is _____
 A 0.59 B 0.47 C 0.24 D 0.89
GATE CE 2021 SET-1   Transportation Engineering
Question 4 Explanation:
$P(n,t)=\frac{(\lambda t)^n e^{-\lambda t}}{n!}$
$\lambda =2 \;v/m$
$t = 1 \text{ minute}$
probability of at least 2 vehicles arriving in 1 ? minute is given by,
$1-(P(0,1)+P(1,1))$
\begin{aligned} &= 1-\left \{ \frac{(2 \times 1)^0 e^{-2(1)}}{0!}+\frac{(2 \times 1)^1 e^{-2(1)}}{1!} \right \}\\ &=1-\left \{ \frac{1(e^{-2})}{1}+ \frac{2(e^{-2})}{1} \right \} \\ &= 1-(e^{-2}+2e^{-2})\\ &=1-3e^{-2}=0.59 \end{aligned}
 Question 5
A signalized intersection operates in two phases. The lost time is 3 seconds per phase. The maximum ratios of approach flow to saturation flow for the two phases are 0.37 and 0.40. The optimum cycle length using the Webster's method (in seconds, round off to one decimal place) is ______________
 A 44.8 B 92.6 C 25.8 D 60.9
GATE CE 2021 SET-1   Transportation Engineering
Question 5 Explanation:
\begin{aligned} \mathrm{n} &=2, \mathrm{~L}=3 \times 2=6 \mathrm{sec} \\ \mathrm{y}_{1} &=0.37, \mathrm{y}_{2}=0.40 \\ \mathrm{Y} &=\mathrm{y}_{1}+\mathrm{y}_{2} \\ &=0.37+0.40 \\ &=0.77 \\ \mathrm{C}_{0} &=\left(\frac{1.5 \mathrm{~L}+5}{1-\mathrm{Y}}\right)=\frac{1.5 \times 6+5}{1-0.77} \\ &=60.87 \mathrm{sec} \simeq 60.9 \mathrm{sec} \end{aligned}
 Question 6
The flow-density relationship of traffic on a headway is shown in the figure.

The correct representation of speed-density relationship of the traffic on this highway is
 A A B B C C D D
GATE CE 2020 SET-2   Transportation Engineering
 Question 7
24-h traffic count at a road section was observed to be 1000 vehicles on a Tuesday in the month of July. If daily adjustment factor for Tuesday is 1.121 and monthly adjustment factor for July is 0.913, the Annual Average Daily Traffic (in veh/day, round off to the nearest integer) is _______.
 A 1121 B 1023 C 996 D 1212
GATE CE 2020 SET-2   Transportation Engineering
Question 7 Explanation:
\begin{aligned} T_{24} &=1000 veh \; (Tuesday) \\ DAF&=1.121 \\ AADT&=? \\ MAF&= 0.913\\ T_{week}&=T_{24} \times DAF \\ &=1000 \times 1.121=1121 \\ AADT&=MAF \times ADT \\ &= 0.913 \times 1121\\ &= 1023.473\\ &=1023 VPD \end{aligned}
 Question 8
The traffic starts discharging from an approach at an intersection with the signal turning green. The constant headway considered from the fourth or fifth headway position is referred to as
GATE CE 2020 SET-2   Transportation Engineering
 Question 9
Traffic volume count has been collected on a 2 lane road section which needs upgradation due to severe traffic flow condition. Maximum service flow rate per lane is observed as 1280 veh/h at level of service 'C'. The Peak Hour Factor is reported as 0.78125. Historical traffic volume count provides Annual Average Daily Traffic as 122270 veh/day. Directional split of the traffic flow is observed to be 60 : 40. Assuming that traffic stream consists of 'All Cars' and all drivers are 'Regular Commuters', the number of extra lane(s) (round off to the next higher integer) to be provide, is ________.
 A 2 B 8 C 6 D 4
GATE CE 2020 SET-1   Transportation Engineering
Question 9 Explanation:
Number of the needed given LOS $=\frac{DDHV}{PHF \times MSF \times F_{HV} \times f_p}$
where,
DDHV = Directional distribution hour volume
PHF = Peak hour factor
MSF = Maximum service flow rate of Los
$F_{HV}$ = Heavy vehicle familiarity adjustment factor
$f_{p}$ = Road under familiarity adjustment factor
DDHV = 122270 x 0.6 = 7362 vah/day (0.6 - max. directional distributed)
$F_{HV}=f_p=1$ as per given condition
$N=\frac{7362}{0.78125 \times 1280 \times 1 \times 1} =7.362 = 8 \text{ lanes}$
Number of extra lanes $= 8-2 = 6 \text{ lanes}$
 Question 10
The relationship between traffic flow rate (q) and density (D) is shown in the figure.

The shock wave condition is depicted by
 A flow with respect to point 1 ($q_1 = q_{max}$) B flow changing from point 2 to point 6 ($q_2 \gt q_6$) C flow changing from point 3 to point 7 ($q_3 \lt q_7$) D flow with respect to point 4 and point 5($q_4 = q_5$)
GATE CE 2020 SET-1   Transportation Engineering
Question 10 Explanation:

There are 10 questions to complete.

format: plain


text:
# Tomography of the magnetic fields of the Milky Way?

I have just found this “new” (well 150 years old actually) tomographical method… for measuring the magnetic field of our own galaxy

New all-sky map shows the magnetic fields of the Milky Way with the highest precision"
by Niels Oppermann et al. (arxiv work available here)

We aim to summarize the current state of knowledge regarding Galactic Faraday rotation in an all-sky map of the Galactic Faraday depth. For this we have assembled the most extensive catalog of Faraday rotation data of compact extragalactic polarized radio sources to date. In the map making procedure we use a recently developed algorithm that reconstructs the map and the power spectrum of a statistically isotropic and homogeneous field while taking into account uncertainties in the noise statistics. This procedure is able to identify some rotation angles that are offset by an integer multiple of $$\pi$$. The resulting map can be seen as an improved version of earlier such maps and is made publicly available, along with a map of its uncertainty. For the angular power spectrum we find a power law behavior with a power law index of $$-2.14$$ for a Faraday sky where an overall variance profile as a function of Galactic latitude has been removed, in agreement with earlier work. We show that this is in accordance with a 3D Fourier power spectrum $$P(k)$$ proportional to $$k^{-2.14}$$ of the underlying field $$n_e \times B_r$$ under simplifying geometrical and statistical assumptions.

Selected excerpt:

… One way to measure cosmic magnetic fields, which has been known for over 150 years, makes use of an effect known as Faraday rotation. When polarized light passes through a magnetized medium, the plane of polarization rotates. The amount of rotation depends, among other things, on the strength and direction of the magnetic field. Therefore, observing such rotation allows one to investigate the properties of the intervening magnetic fields.

Mmmm… very interesting, at least for my personal knowledge of the wonderful tomographical problem zoo (among gravitational lensing, interferometry, MRI, deflectometry).

Note: Wow… 16 months without any post here. I’m really bad.

Image credit: Wikipedia


text:
# Math Help - Complex numbers question

1. ## Complex numbers question

I'm completely stuck on this one question:

Find real x and y for which x - y +2(x + y - 1)i = 5 + 4i

Can someone go through it with me or give me a hint..?

2. Originally Posted by blackdragon190
Findd real x and y for which x - y +2(x + y - 1)i = 5 + 4i[/I]
Can someone go through it with me or give me a hint..?
Solve this system: $\left\{ \begin{gathered}
x - y = 5 \hfill \\
x + y - 1 = 2 \hfill \\
\end{gathered} \right.$

3. Thanks But how did you get those equations?

4. You equate the real and imaginary parts of the 2 complex numbers. If you look at
$x - y + 2(x + y - 1)i = 5 + 4i$
On the left side, x - y is the real part (no i), and on the right side, 5 is the real part. So you set them equal to each other:
$x - y = 5$

Look at the imaginary parts. On the left side you have 2(x + y - 1), and on the right side you have 4. So they're equal:
$2(x + y - 1) = 4$

You can simplify a little bit by dividing both sides by 2:
$x + y - 1 = 2$

So now you have the system that Plato wrote above.

5. Oh, I get it now. Thanks!


text:
It is currently 12 Nov 2018, 16:35

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# A “descending number” is a three-digit number, such that the

Author Message
TAGS:
Senior Manager
Joined: 20 May 2014
Posts: 282
Followers: 14

Kudos [?]: 48 [0], given: 220

A “descending number” is a three-digit number, such that the [#permalink]  17 Oct 2017, 22:22
00:00

Question Stats:

0% (00:00) correct 100% (05:59) wrong based on 3 sessions
A “descending number” is a three-digit number, such that the units digit is less than the tens digits and the tens digit is less than the hundreds digit. What is the probability that a three-digit number chosen at random is a “descending number”?

(A) 3/25

(B) 2/9

(C) 2/15

(D) 1/9

(E) 1/10

Kudos for correct solution.
[Reveal] Spoiler: OA
Director
Joined: 03 Sep 2017
Posts: 521
Followers: 1

Kudos [?]: 327 [0], given: 66

Re: A “descending number” is a three-digit number, such that the [#permalink]  18 Oct 2017, 05:50
Three-digit numbers are 999-100+1 = 900.

For any arrangement of three numbers, there is only one in which the numbers are arranged so that the units are less than the tens, which are less than the hundreds: 10x9x8 = 720 are the total different combinations of three different integers. Only one of the six possible arrangements (3! = 6) matches up with the restraints in the problem. Thus, 720/6 = 120.

Then, 120/900, which simplifies to 2/15. Answer C
Director
Joined: 20 Apr 2016
Posts: 741
Followers: 6

Kudos [?]: 492 [0], given: 86

Re: A “descending number” is a three-digit number, such that the [#permalink]  18 Oct 2017, 23:42
Bunuel wrote:
A “descending number” is a three-digit number, such that the units digit is less than the tens digits and the tens digit is less than the hundreds digit. What is the probability that a three-digit number chosen at random is a “descending number”?

(A) 3/25

(B) 2/9

(C) 2/15

(D) 1/9

(E) 1/10

Kudos for correct solution.

The total 3 digit numbers can be formed by FCP i.e = 9*10*10 (we have digit from 0 -9 and there are no restriction then the hundred digit can be from 1-9,as it cannot start from 0 , tens digit can be 0-9 and units can also be 0-9).

Now we have to choose the three digit numbers in descending order -

We can leave out 100 - 200 = as they donot form in descending order

Next from 200 to 200 = we have only one number =210

Next from 300 to 400 = we have 3 nos = 310,320,321

Next from 400 to 500 = we have 6 nos = 410,420,421,430,431,432

Now we find that it is following a pattern and i.e

200 = 1

300 = 3 (2+1 from previous possible ways , 2 is the hundred unit )

400 = 6 (3+3)

500 = 10 (4+6,similarly 4 is the hundred digit and 6 in the value from the previous)

600 = 15(5+10)

700 = 21 (6+15)

800 = 28 (7+21)

900 = 36(8+28)

Therefore no. of possible ways = 36 + 28 + 21 + 15 + 10 + 6 + 3 + 1 =120

Therefore $$probability = \frac{120}{900} = \frac{2}{15}$$
_________________

If you found this post useful, please let me know by pressing the Kudos Button

Re: A “descending number” is a three-digit number, such that the   [#permalink] 18 Oct 2017, 23:42
Display posts from previous: Sort by


text:
# rain1

• User since
• Last active
• Started 2 topics
• Posted 3 times

## Started Escaping

Hello! Here is my blog post about escaping.

# Escaped Text

The fundamental idea of escaping is to provide an injective (and hence invertible) function from arbitrary strings to strings with some pattern not occurring. This could mean that a set of reserved characters do not occur in the output, or that a certain character does not occur alone.

The netstring approach of putting the length before the data is an alternative to escaping. NUL delimited text is another alternative to escaping in cases where text cannot contain \0 (like C strings).

The ability to easily escape data we want to print out and unescape data we are taking in as input is important if we want easy and correct "plumbing" between programs. It seems to be overlooked a lot. One of the things that's amazing to lisp beginners coming from other languages is how easy it is to print out and read back data.

The rest of this document is a review of some of the ways escaping comes up in computing.

## url/uri encoding

Data inside web addresses must avoid the / separator and several other characters, as documunted in the URI RFC. To allow these chars to still be passed to a web application we use percent encoding with hex digits: %XX.

The choice of % as the escape char is good. Since it's not using the more common \ URLs can usually be put into a string literal without another layer of escaping.

Originaly in RFC 2396 the \ was not allowed inside a URL. It was marked as "unwise". In the latest spec, RFC 3986 it is allowed but should be escaped as %5C.

## HTML escaping

HTML and XML allow you to mix arbitrary text and <tags>. So to include in our text the same metacharacters that are used to described tags they will need escaped.

The & char is used to escape special characters in HTML. They call it a character reference. Characters can be referenced by decimal, hex or by name.

According to the rules for parsing data inside HTML tags one only really needs to escape & and <. But it is advised to also escape >. The main restriction on html data is that it must not contain the string "</". Here.

## sql escaping

A lot of websites have been exploited from user data ending up unescaped in part of an SQL query. If a web application forgets to escape something user controlled that ends up they can use things like delimiter collision to perform their own SQL queries and take over the site.

## Command line arguments and shell scripting

Command line arguments starting with - are usually understood as flags. The magic -- flag enables one to pass further arguments which are interpreted literally instead of checked to see if they are flags.

On the ext4 filesystem, file names/paths can contain any byte except 0. In shell scripting it is often useful to use find -print0 to get NUL delimited list of filenames to loop over, although usually people use newline or as a delimited which is not 100% but works almost all the time.

## base64 encoding

As base64 uses such a small alphabet of plaintext characters any data that has been base64 encoded can basically be put in any context without further escaping. In URLs, in HTML, strings inside programming languages, in email. etc.

## string escaping in programming languages

The primary place escaped text happens is in string literals. Double quotes are used to contain and delimit the text.

The syntax of a string literal, in regex is at a very basic level something like: /"([^"]|\"|\\)*"/. wiki gets this. It's not "[^"]*".

• Rust basic. newlines allowed. Newlines + whitespace ahead stripped if preceeded by \. x u and nrt. It also has raw string literals. A bit like lua's long ones.
• Go
• C
• POSIX shell. Shell supports things like glob * and ~ which expand to multiple files or a path. To enter a literal asterisk you need to either escape it \*, \~ or put it in a string.
• bash has weak and strong quoting, supports escapes much like C.
• Python has a few different sorts of string literal.
• Ruby - needs # escaped. Also has %(...)
• Lua - \z is special. \xXX, where XX is a sequence of exactly two hexadecimal digits \u{XXX} is for unicode. brackets mandatory. The long literal strings [==[ ]==] are a unique approach.
• Javascript - Probably the simplest string language: x for hex, u for unicoed. special escape chars '"\bfnrtv. Also supports \0.

Another overview here: codecodex - escape sequences.

## regex escaping

. is usually a metacharacter representing any char, so if you want to match a literal dot you need \..

A very weird choice by sed is that ( and ) are matched literally, and to use parens as grouping you need $$ $$. This is a kind of reverse escaping which makes it difficult to remember. I think it is done for the purpose of brevity, assuming that matching brackets literally is done more often than grouping is done - I find the opposite to be true though.

Regex often falls victim to leaning toothpick syndrome.

## UNIX terminals

UNIX terminals use in-band control . This means just catting a file can completely mess up your terminal if the file contains these control codes. This is a mixed bag of good and bad. Command line programs should escape all terminal control sequences to avoid in band signalling.

## Gopher

Gopher terminates files with a period .. Spec here. Because of this it's a bit of extra work to have text files that start with a period in a gopher text page:

Note:  Lines beginning with periods must be prepended with an extra
period to ensure that the transmission is not terminated early.
The client should strip extra periods at the beginning of the line.

## Started What UNIX shell could have been

Hello. I have written a blog post about an idea for UNIX shell. The canonical link is here https://rain-1.github.io/shell-2 but I have copied it so that people are not forced to click out to a different site. I look forward to your thoughts on the general topics.

# What UNIX shell could have been

The shell (bash or whatever) is an excellent tool that saves people a huge amount of work. Being able to easily script complex jobs together is one of the best things. It does have some weaknesses though that I feel could be improved though.

The two biggest weaknesses in shell, in my opinion, are the quoting and escaping mess and secondly that all the objects are strings. I've talked about the quotation stuff before so I wont cover that here. My idea for improving it would be to make separate (dynamic) data types for strings, paths and command lines flags.

This could be implemented in a simple (but ugly) way by encoding each object into strings with a tag saying what type they are:

• "sfoo" for a string
• "fhelp" for a flag
• "p/dev/random" for a path

Maybe there's a more aesthetic way to do this, open to suggestions.

This change can't be done just by writing a new shell though. Every UNIX tool that we have (ls, cat, grep, jq, …) would need to conform to this protocol. It would probably steamroll over 'dd' (which has an ideosyncratic argument style).

What would be good about this idea? Command line tools would throw an error if given the wrong input, instead of what they currently do: attempt to continue with a misinterpreted string. This could be considered an improvement if you're interested in your scripts correctness. Issues like the recent GPG security problem CVE-2018-12020 would have been avoided.

But the drawback might be that some things take a bit more programming to achieve. For example you would have to explicitly convert a string to a path. You would have to build up paths and apply regex and things to them in a different way than you do it on strings. Maybe the language could be designed to make these things easier.

# The current mess

It's worth mentioning the current "solution". In the current system if something starts with a - it's a flag. This leaves the problem of filenames that start with -. They are very rare so it isn't a big problem: We only encounter it occasionally. Some might never encounter it.

Anyway the answer is that tools can provide the – flag to say that the next argument (or all preceeding) are not flags.

We also have incredibly good documentation about working around the difficulties of coping with paths and spaces and stuff in shell:

Now these are excellent resources that I value, but I believe that it is incidental complexity. And a better designed shell would result in much less documentation and edge cases to worry about.

Some shells have added array data types rc, but these only work internally. Across process boundaries everything is squeezed through the string object.

# Summary

I believe we make all of our lives easier by improving the tools we use. Shell has a couple big problems that can actually be solved. There have been experiments to solve the quotation issues execline and s. And I am proposing the idea of a very basic dynamic type system to solve the 'in-band signalling' issues with command line args.

## Posted in Who's Around?

Hello. I am interested in bootstrapping programming languages from nothing.


text:
# Minimalism - Praised but Rarely Utilized

We constantly praise companies that demonstrate their use of minimalism. Take for example Apple or Tesla: Two companies that make their product visually appealing to their customers by simplifying their design and relying on intuition for the products’ complex power. Very few though, are willing to take that approach to a broader scale. The software side of products, especially when it comes down to customers is bloated. First of all, why should minimalism be considered important in our society. And second of all, how would that affect you?

## First, let’s clarify

Before making a case about the importance of minimalism, let’s first establish what the term means in our context: \begin{center} > Software that is written with limited functionality > at it’s core, is extensible, and is efficiently written to allow for maximum efficiency.

## The horrors of multi-use apps

Minimalist software is often designed according to one aspect of the UNIX philosophy which is: > Software should only do one thing and do it right. With time, a lot of software have begun using versatility as a selling point. Take for example the amount of email apps which state they also have a built in calendar and contacts. The idea seems good at first glance. But if you like the email portion of the app but not the email one, the average consumer will acquire another app for calendars and be stuck with a portion of an app being unused in his device. In a world in which everything is about saving data, that is a flagrant waste.

When you think of a lightweight app (or script). You usually expect it to run crisply. That is only possible if the code is efficiently written. Take for example the following python code (taken from PyBytes):

msg = 'line1\n'
msg += 'line2\n'
msg += 'line3\n'
print(msg)

This code is inefficient as it generates a new string on every line. To obtain the same result, a favorable approach would be:

msg = ['line1', 'line2', 'line3']
print('\n'.join(msg))

This may be a very small example of applied efficiency in code, but to a computer, every operation is a small and short one, only sometimes replicated thousands of times.

## But why keep code small

When code is small, it is easier to digest when it is required to debug it or change it. This allows capable users to optimize code to their use cases without being overwhelmed by the calamity of a huge software. Software which is small is easier to adapt and shape. Even if you don’t know how to code. It is all to your advantage to promote small code as other people can make those changes and you will have the privilege of downloading that changed piece of software.

You may as well call me an extremist. To some extent, I probably am. But consider how optimized software would be with this design philosophy as a general priority.


text:
A man invests a certain amount of money at 2% and $800 more than that amount in another account at 4% interest. At the end of the one year, he earned$112 in interest. How much money was invested in each account?

Oct 24, 2016

$1,333.33 invested at 2% $2,1333.33 invested at 4%

Explanation:

As the time is only 1 year, it does matter whether it is simple or compound interest.

Define each amount of money using a variable.

Let the smaller amount of money (at 2%) be $x$
The larger amount (at 4%) is $x + 800$

The interest on the first amount = $\frac{x \times 2}{100} = \textcolor{red}{\frac{2 x}{100}}$

The interest on the larger amount = $\textcolor{b l u e}{\frac{\left(x + 800\right) \times 4}{100}}$

The total interest is $112 $\textcolor{red}{\frac{2 x}{100}} + \textcolor{b l u e}{\frac{4 \left(x + 800\right)}{100}} = 112$$\frac{2 x + 4 x + 3200}{100} = 112$$\frac{6 x + 3200}{100} = 112 \text{ } \leftarrow \times 100$$6 x + 3200 = 11200$$6 x = 8000$x =$1,333.33

x+ 800= \$2,133.33


text:
Find free online Chemistry Topics covering a broad range of concepts from research institutes around the world.

## Thermodynamic Principle of Metallurgy

As we discussed, the extraction of metals from their oxides can be carried out by using different reducing agents. For example, consider the reduction of a metal oxide MxOy. $$\frac{2}{y}$$MxOy (s) → $$\frac{2x}{y}$$M (s) + O2 (g) ………. (1)

The above reduction may be carried out with carbon. In this case, the reducing agent carbon may be oxidised to either CO or CO2.

C + O2 → CO2 (g) …………. (2)
2C + O2 → 2CO (g) ……………. (3)

If carbon monoxide is used as a reducing agent, it is oxidised to CO2 as follows,

2CO + O2 → 2CO2 (g) ………………. (4)

A suitable reducing agent is selected based on the thermodynamic considerations. We know that for a spontaneous reaction, the change in free energy (ΔG) should be negative. Therefore, thermodynamically, the reduction of metal oxide [equation (1)] with a given reducing agent [Equation (2), (3) or (4)] can occur if the free energy change for the coupled reaction. [Equations (1) & (2), (1) & (3) or (1) & (4)] is negative. Hence, the reducing agent is selected in such a way that it provides a large negative ΔG value for the coupled reaction.

Ellingham Diagram

The change in Gibbs free energy (ΔG) for a reaction is given by the expression.
ΔG = ΔH – TΔS ………….. (1)

where, ΔH is the enthalpy change, T the temperature in kelvin and ΔS the entropy change. For an equilibrium process, ΔG° can be calculated using the equilibrium constant by the following expression
ΔG° = – RT lnKP

Harold Ellingham used the above relationship to calculate the ΔG° values at various temperatures for the reduction of metal oxides by treating the reduction as an equilibrium process.

He has drawn a plot by considering the temperature in the x-axis and the standard free energy change for the formation of metal oxide in y-axis. The resultant plot is a straight line with ΔS as slope and ΔH as y-intercept. The graphical representation of variation of the standard Gibbs free energy of reaction for the formation of various metal oxides with temperature is called Ellingham diagram.

Observations from the Ellingham Diagram

1. For most of the metal oxide formation, the slope is positive. It can be explained as follows. Oxygen gas is consumed during the formation of metal oxides which results in the decrease in randomness. Hence, ΔS becomes negative and it makes the term, TΔS positive in the straight line equation.

2. The graph for the formation of carbon monoxide is a straight line with negative slope. In this case ΔS is positive as 2 moles of CO gas is formed by the consumption of one mole of oxygen gas. It indicates that CO is more stable at higher temperature.

3. As the temperature increases, generally ΔG value for the formation of the metal oxide become less negative and becomes zero at a particular temperature. Below this temperature, ΔG is negative and the oxide is stable and above this temperature ΔG is positive. This general trend suggests that metal oxides become less stable at higher temperature and their decomposition becomes easier.

4. There is a sudden change in the slope at a particular temperature for some metal oxides like MgO, HgO. This is due to the phase transition (melting or evaporation).

Applications of the Ellingham Diagram:

Ellingham diagram helps us to select a suitable reducing agent and appropriate temperature range for reduction. The reduction of a metal oxide to its metal can be considered as a competition between the element used for reduction and the metal to combine with oxygen.

If the metal oxide is more stable, then oxygen remains with the metal and if the oxide of element used for reduction is more stable, then the oxygen from the metal oxide combines with elements used for the reduction. From the Ellingham diagram, we can infer the relative stability of different metal oxides at a given temperature.

1. Ellingham diagram for the formation of Ag2O and HgO is at upper part of the diagram and their decomposition temperatures are 600 and 700 K respectively. It indicates that these oxides are unstable at moderate temperatures and will decompose on heating even in the absence of a reducing agent.

2. Ellingham diagram is used to predict thermodynamic feasibility of reduction of oxides of one metal by another metal. Any metal can reduce the oxides of other metals that are located above it in the diagram. For example, in the Ellignham diagram, for the formation of chromium oxide lies above that of the aluminium, meaning that Al2O3 is more stable than Cr2O3.

Hence aluminium can be used as a reducing agent for the reduction of chromic oxide. However, it cannot be used to reduce the oxides of magnesium and calcium which occupy lower position than aluminium oxide.

3. The carbon line cuts across the lines of many metal oxides and hence it can reduce all those metal oxides at sufficiently high temperature. Let us analyse the thermodynamically favourable conditions for the reduction of iron oxide by carbon.

Ellingham diagram for the formation of FeO and CO intersects around 1000 K. Below this temperature the carbon line lies above the iron line which indicates that FeO is more stable than CO and hence at this temperature range, the reduction is not thermodynamically feasible. However, above 1000 K carbon line lies below the iron line and hence, we can use coke as reducing agent above this temperature. The following free energy calculation also confim that the reduction is thermodynamically favoured.

From the Ellingham Diagram at 1500 K,

2Fe (s) + O2 (g) → 2FeO (g) ΔG1 = – 350 kJ mol-1 ……………. (1)
2C (s) + O2 (g) → 2CO (g) ΔG2 = – 480 kJ mol-1 …………….. (2)

Reverse the reaction (1)

2FeO (s) → 2Fe (s) + O2 (g) – ΔG1 = + 350 kJ mol-1 …………….. (3)

Now couple the reactions (2) and (3)

2FeO (s) + 2C → 2Fe (l, s) + 2CO (g) ΔG3 = – 130 kJ mol-1 …………….. (4)

The standard free energy change for the reduction of one mole of FeO is, ΔG3/2 = – 65 kJ mol-1

Limitations of Ellingham Diagram

1. Ellingham diagram is constructed based only on thermodynamic considerations. It gives information about the thermodynamic feasibility of a reaction. It does not tell anything about the rate of the reaction. More over, it does not give any idea about the possibility of other reactions that might be taking place.

2. The interpretation of ΔG is based on the assumption that the reactants are in equilibrium with the products which is not always true.


text:
### AP Mode

First, you need to configure kernel config:

# cd kernel

After each configuration modification, update defconfig:

# cp .config ./arch/arm/configs/infinity2m_spinand_ssc011a_s01a_minigui_double_net_defconfig -f

After the system is up, first load the WiFi driver:

# /config/wifi/ssw01bInit.sh
# ifconfig wlan0 up

Of course, as an AP hotspot, our WiFi module needs to configure the following hotspot information:

# vi /config/wifi/hostapd.conf

Next, open the AP hotspot:

# /config/wifi/hostapd -B /config/wifi/hostapd.conf

At this point, we can search for our AP hotspot on the mobile phone / computer:

Trying to connect, found that the connection was always connected but did not succeed because the DHCP service was not turned on, and the connection failed due to the failure of assigning the connection device to the IP,Therefore, we also need to start the DHCP service (using the dnsmasq tool):

# vi /config/wifi/dnsmasq.conf

Note that the interface is set to wlan0 here:

Since DHCP range is set to 192.168.1.X, the static IP of WLAN0 is set to 192.168.0.1:

# ifconfig wlan0 192.168.0.1

# /config/wifi/dnsmasq -i wlan0 -C /config/wifi/dnsmasq.conf

At this time, the device can be connected normally, and the assigned IP is within the range of “dhcp-range”:

Now the device can normally connect to the hotspot, but at this time, I want to connect the device to the Internet, that is, use the board as a router, eth0 as a WAN and WLAN0 as a LAN.
First, you need to confirm that eth0 can access the Internet:

# ping www.baidu.com -I eth0

This can be achieved through the brctl bridging tool. This tool is not installed by default. As before, it is obtained from buildroot:

# cd buildroot-2020.05/

# cp .config ./configs/ssd20x_defconfig -f
# make BR2_JLEVEL=4
# cd ../project/image/rootfs
# rm rootfs/* -rf
# cp ../../../buildroot-2020.05/output/images/rootfs.tar ./ -f
# tar -xvf rootfs.tar -C ./rootfs/
# tar -cvf rootfs.tar.gz ./rootfs
# cd ../../../

Recompile builderoot:

# ./Release_to_customer.sh  -f nand -p ssd201  -o 2D06

After updating the firmware, the previous load driver, hostapd service and dnsmasq service need to be re executed, and then execute the following command to establish the bridge:

# brctl addbr br0


text:
# What is the dim(N(T))?

Define ne a linear operator T : Pn $\rightarrow$ Pn by
T(p(x)) = $\frac{d^{k}x}{dx^{n}}$p(x)
where 1 $\leq$k $\leq$n. Show that dim(N(T)) = k.

i. The null space of a linear operator is all vectors v $\in$ V such that T(v) = 0

ii. The dimension of a vector space V, is the number of vectors in any basis of V.

My logical understanding is that the null space of T is any k derivative that gives 0 i.e. k is bigger than n. Thus the dimension has k number of possible vectors. My problem is I don't know how to write the proof. . . I'm assuming that I just have to write the null space as a summation of derivatives from 1 to k thus the dimension of the null is k (the largest possible number).

LCKurtz
Homework Helper
Gold Member
Define ne a linear operator T : Pn $\rightarrow$ Pn by
T(p(x)) = $\frac{d^{k}x}{dx^{n}}$p(x)
where 1 $\leq$k $\leq$n. Show that dim(N(T)) = k.

I assume you mean ##\frac {d^k x}{dx^k}p(x)##. Or for easy display, ##D^kp(x)##.

i. The null space of a linear operator is all vectors v $\in$ V such that T(v) = 0

ii. The dimension of a vector space V, is the number of vectors in any basis of V.

My logical understanding is that the null space of T is any k derivative that gives 0

No. The null space isn't "any k derivative". The null space is a subspace of your nth degree polynomials.

i.e. k is bigger than n. Thus the dimension has k number of possible vectors. My problem is I don't know how to write the proof. . . I'm assuming that I just have to write the null space as a summation of derivatives from 1 to k thus the dimension of the null is k (the largest possible number).

No again, the null space is not a "summation of derivatives". For example if ##n=5## so you are considering polynomials of degree ##5##, and you operate on them with ##D^2##, which ones would map to the zero polynomial?

Awesome! So I've been sitting here and had an epiphany. . . I think. . . so here is what I came up with.

Let B = {1,x,x$^{2}$,...,x$^{n}$} be the standard basis for P$_{n}$. Since p(x) is in the N(T) iff it's degree is < k, then the null space is all the polynomials with degree k-1 or less, thus [1,x,x$^{2}$,...,x$^{k-1}$] is a basis for N(T),

and the dim(N(T)) = k

:) I think I have it what do you think?

LCKurtz
Homework Helper
Gold Member
Awesome! So I've been sitting here and had an epiphany. . . I think. . . so here is what I came up with.

Let B = {1,x,x$^{2}$,...,x$^{n}$} be the standard basis for P$_{n}$. Since p(x) is in the N(T) iff it's degree is < k, then the null space is all the polynomials with degree k-1 or less, thus [1,x,x$^{2}$,...,x$^{k-1}$] is a basis for N(T),

and the dim(N(T)) = k

:) I think I have it what do you think?

I think that's much better.


text:
## On the Theoretical Guarantees for Parameter Estimation of Gaussian Random Field Models: A Sparse Precision Matrix Approach

Sam Davanloo Tajbakhsh, Necdet Serhat Aybat, Enrique Del Castillo; 21(217):1−41, 2020.

### Abstract

Iterative methods for fitting a Gaussian Random Field (GRF) model via maximum likelihood (ML) estimation requires solving a nonconvex optimization problem. The problem is aggravated for anisotropic GRFs where the number of covariance function parameters increases with the dimension. Even evaluation of the likelihood function requires $O(n^3)$ floating point operations, where $n$ denotes the number of data locations. In this paper, we propose a new two-stage procedure to estimate the parameters of second-order stationary GRFs. First, a convex likelihood problem regularized with a weighted $\ell_1$-norm, utilizing the available distance information between observation locations, is solved to fit a sparse precision (inverse covariance) matrix to the observed data. Second, the parameters of the covariance function are estimated by solving a least squares problem. Theoretical error bounds for the solutions of stage I and II problems are provided, and their tightness are investigated.

[abs][pdf][bib]        [code]


text:
## Q. 7.3

Find the centroid and the second moment of area about the horizontal (y) axis of the cross section shown. All dimensions are in millimeters. If a beam is constructed with the cross section shown from steel whose maximum allowable tensile stress is 400 MPa, what is the maximum bending moment that may be applied to the beam?

Given: Dimensions of beam cross section; limiting stress.
Find: Location of centroid; maximum applied moment.
Assume: Hooke’s law applies

## Verified Solution

The symmetry of the cross section shown suggests that the horizontal coordinate of the centroid will be on the vertical centerline, as sketched. $y_c$ is then 20 cm. We need only locate the vertical location of the centroid ($z_c$ ). Several strategies are available to us. We recognize that the cross section is a large rectangle, with an inner rectangular hole. It will thus be possible for us to find the area and second moment of area of the large outer rectangle, and simply subtract off the properties of the inner rectangle. Recall that $z_c=\int z \mathrm{~d} A / \int \mathrm{d} A=\sum z \mathrm{~d} A / \sum \mathrm{d} A,$ where z and $z_c$ are measured from an arbitrarily chosen reference datum, in this case the top of the cross section. For clarity, results are tabulated as follows:

With the location of the centroid known, we can find the second moment of area with respect to that location, starting with the second moment of area of a rectangle about its own centroid from Appendix A.

If the maximum allowable normal stress is 400 MPa, we can find the maximum moment that can be applied using the relationship:

$\sigma_{\max }=\frac{M c}{I}$.

We have found the second moment of area I, and c is the maximum distance from the centroid attainable on the cross section, in this case 31.7 mm. Solving for M:

$M=\frac{\sigma_{\max } I}{c}=\frac{\left(400 \mathrm{~N} / \mathrm{mm}^2\right)\left(655,000 \mathrm{~mm}^4\right)}{31.7 \mathrm{~mm}}=8.26 \mathrm{kNm}$


text:
# Frequencies & Waves

## Homework Statement

1)A stretched wire vibrates in its first normal mode at a frequency of 383 Hz. What would be the fundamental frequency if the wire were one fourth as long, its diameter were quadrupled, and its tension were increased four-fold?

2)A 0.0125 kg, 1.88 m long wire is fixed at both ends and vibrates in its simplest mode under a tension of 218 N. When a tuning fork is placed near the wire, a beat frequency of 5.10 Hz is heard. The beat frequency increase as the the tension of the wire decreases; what is the frequency of the tuning fork?

## Homework Equations

For question 1:
f = 1/2L ( sqrt( T/mu)

## The Attempt at a Solution

I really dont know how to start in these two questions, given with a diameter in part 1, but I know the formula to use. And I'm not sure what equation Im going to use for question 2.

Thanks

## Answers and Replies

Related Introductory Physics Homework Help News on Phys.org
Redbelly98
Staff Emeritus
Homework Helper

## Homework Equations

For question 1:
f = 1/2L ( sqrt( T/mu)
Okay, can you answer this: what would be the fundamental frequency if the wire were one fourth as long, and nothing else changed?

## The Attempt at a Solution

I really dont know how to start in these two questions, given with a diameter in part 1, but I know the formula to use. And I'm not sure what equation Im going to use for question 2.
Your textbook or class lecture notes should have a discussion of beat frequencies, along with an equation or two.

In part one, if only the length is changed, the equation would become

f = $$2/L$$($$\sqrt{T/mu}$$)

In part two, I checked my notes and saw f[beat] = f2- f1
I have f1 but I dont have f2. So I guess I should use v^2*mu = T? then what should I do next? or is there another equation that relates tension with frequency?

Thanks

Redbelly98
Staff Emeritus
Homework Helper
Question #1

In part one, if only the length is changed, the equation would become

f = $$2/L$$($$\sqrt{T/mu}$$)
Okay, so in that case the fundamental frequency would be ____ Hz?

.

Question #2

In part two, I checked my notes and saw f[beat] = f2- f1
Yes, good, though actually there should be an absolute value sign there:
f[beat] = |f2- f1|
I have f1 but I dont have f2.
As far as I can tell, you don't really have f1, do you? fbeat was given, but not f1 or f2.

So I guess I should use v^2*mu = T? then what should I do next?
Well, how exactly do you plan to use that equation? What new information will you get from using it?

or is there another equation that relates tension with frequency?
Not that I'm aware of.

But in question 1, I'm not given with tension or mu. Is it possible to get the f?

In question 2:

I thought I would be able to get lambda with string speed, but when I check the other equation again to get lambda I would need wave speed not string speed...So I guess the equation that I mentioned was wrong.

In question 1 again, Do I need to find the length first using the frequency while ignoring mu and tension? I also know the length of the in the first frequency is the same as the length in the 2nd frequency when the length becomes one fourth as long.

Thanks

Redbelly98
Staff Emeritus
Homework Helper
#1:

You do not need to know the length, tension, or mu.

In the problem statement we are told that (1/2L) · (sqrt( T/mu)) is equal to ____(?)

If the length is one fourth as long, then the new frequency (2/L)·( sqrt( T/mu)) would be ____(?) times as much.

Fill in those blanks.

#1:

You do not need to know the length, tension, or mu.

In the problem statement we are told that (1/2L) · (sqrt( T/mu)) is equal to ____(?)

If the length is one fourth as long, then the new frequency (2/L)·( sqrt( T/mu)) would be ____(?) times as much.

Fill in those blanks.
First blank: 383 Hz

Second blank: 1532 Hz. 4 times as much

Redbelly98
Staff Emeritus
Homework Helper
First blank: 383 Hz

Second blank: 1532 Hz. 4 times as much
Good, it looks like you have the idea. Can you apply that same reasoning to

1. increasing the tension 4-fold?
2. quadrupling the diameter (think about how that would affect mu)?

If the tension is four-fold, then the new frequency would be 766 Hz ( twice as much ).

If the diameter is quadrupled, then the new frequency would be 191.5 Hz ( half as much ).

EDIT: I got the answer! its 766 Hz thanks :D.

Question 2:

f[beat] = | f2 - f1 |

I didnt know that 5.10 Hz was the beat frequency. So I guess I wont use the formula using speed. How should I start with this question?

Thanks again

Last edited:
Redbelly98
Staff Emeritus


text:
# Math Help - Discuss continuity and differentiability of f(x)?

1. ## Discuss continuity and differentiability of f(x)?

A function f is defined by:
$f(x) = \arccos\left(Sgn\left(\frac{2[x]}{3x - [x]}\right)\right)$ where $Sgn(x) = \left\{ \begin{array}{cc}\frac{|x|}{x}, &\mbox{if}\ x\neq 0\\0, &\mbox{if}\ x = 0\end{array}\right.$ and [.] denotes the greatest integer function.

Discuss the continuity and differentiability of $f(x)$ at $x = \pm1$

2. Originally Posted by fardeen_gen
A function f is defined by:
$f(x) = \arccos\left(Sgn\left(\frac{2[x]}{3x - [x]}\right)\right)$ where $Sgn(x) = \left\{ \begin{array}{cc}\frac{|x|}{x}, &\mbox{if}\ x\neq 0\\0, &\mbox{if}\ x = 0\end{array}\right.$ and [.] denotes the greatest integer function.

Discuss the continuity and differentiability of $f(x)$ at $x = \pm1$
For continuity, I'm going to use the limit definition.

At $x=1$, we have:

$\lim_{x\to 1^-}f(x) = sgn\left(\frac{2(0)}{3x-0}\right) = 0. ~~~\arccos(0)=\frac{\pi}{2}$

$\lim_{x\to 1^+} f(x)= sgn\left(\frac{2(1)}{3x-1}\right) = sgn(2/2) = 1. ~~~\arccos(1) = 0$

$f(1) = sgn\left(\frac{2(1)}{3x-1}\right) = sgn(2/2) = 1. ~~~\arccos(1) = 0$

So $f(x)$ is not continuous at $x=1$.

At $x=-1$, we have:

$\lim_{x\to -1^-}f(x) = sgn\left(\frac{2(-2)}{3x-(-2)}\right) = sgn(-4/-1)=1. ~~~\arccos(1)=0$

$\lim_{x\to -1^+}f(x) = sgn\left(\frac{2(-1)}{3x-(-1)}\right) = sgn(-2/-2) = 1. ~~~\arccos(1) = 0$

$f(-1) = sgn\left(\frac{2(-1)}{3x-(-1)}\right) = sgn(-2/-2) = 1. ~~~\arccos(1) = 0$

So $f(x)$ is continuous at $x=-1$.

3. It follows that $f$ is not diffrentiable at $x=1$, because it is not continuous.

At $x=-1$, we have: $\lim_{h\to 0}\frac{sgn\left(\frac{2[-1+h]}{3(-1+h)-[-1+h]}\right)-sgn\left(\frac{2[-1]}{3(-1)-[-1]}\right)}{h} = \lim_{h\to 0}\frac{sgn\left(\frac{2[-1+h]}{3(-1+h)-[-1+h]}\right)-1}{h}$.

Looking at $\lim_{h\to 0^+}$ and assuming $0 (I think this is OK to assume because we want $h\to 0$), we know that $[h-1]=-1$ and $3h-2<0$, so as $h$ becomes less than $\frac{2}{3}$, we have:

$\lim_{h\to 0^+}\frac{sgn\left(\frac{-2}{3h-2}\right)-1}{h} = \lim_{h\to 0^+}\frac{sgn\left(\frac{-2}{<0}\right)-1}{h} = \lim_{h\to 0^+}\frac{1-1}{h} = 0$

Looking at $\lim_{h\to 0^-}$ and assuming $-1, we know that $[h-1]=-2$, and we have:

$\lim_{h\to 0^-}\frac{sgn\left(\frac{-4}{3h-1}\right)-1}{h} = \lim_{h\to 0^-}\frac{sgn\left(\frac{-4}{<0}\right)-1}{h} = \lim_{h\to 0^-}\frac{1-1}{h} = 0$

So I think $f(x)$ is differentiable at $x=-1$, and furthermore that $f'(-1)=0$ and the tangent line is $g(x)=1$.


text:
String Instrument
Shop for string section. Get String Instrument essential facts below. View Videos or join the String Instrument discussion. Add String Instrument to your PopFlock.com topic list for future reference or share this resource on social media.
String Instrument
Numerous stringed instruments of Chinese make on display in a shop.
Viol, fidel and rebec (from left to right) on display at Amakusa Korejiyokan in Amakusa, Kumamoto, Japan

String instruments, stringed instruments, or chordophones are musical instruments that produce sound from vibrating strings when the performer plays or sounds the strings in some manner.

Musicians play some string instruments by plucking the strings with their fingers or a plectrum--and others by hitting the strings with a light wooden hammer or by rubbing the strings with a bow. In some keyboard instruments, such as the harpsichord, the musician presses a key that plucks the string.

With bowed instruments, the player rubs the strings with a horsehair bow, causing them to vibrate. With a hurdy-gurdy, the musician operates a mechanical wheel that rubs the strings.

Bowed instruments include the string section instruments of the Classical music orchestra (violin, viola, cello and double bass) and a number of other instruments (e.g., viols and gambas used in early music from the Baroque music era and fiddles used in many types of folk music). All of the bowed string instruments can also be plucked with the fingers, a technique called "pizzicato". A wide variety of techniques are used to sound notes on the electric guitar, including plucking with the fingernails or a plectrum, strumming and even "tapping" on the fingerboard and using feedback from a loud, distorted guitar amplifier to produce a sustained sound. Some types of string instrument are mainly plucked, such as the harp and the electric bass. In the Hornbostel-Sachs scheme of musical instrument classification, used in organology, string instruments are called chordophones. Other examples include the sitar, rebab, banjo, mandolin, ukulele, and bouzouki.

In most string instruments, the vibrations are transmitted to the body of the instrument, which often incorporates some sort of hollow or enclosed area. The body of the instrument also vibrates, along with the air inside it. The vibration of the body of the instrument and the enclosed hollow or chamber make the vibration of the string more audible to the performer and audience. The body of most string instruments is hollow. Some, however--such as electric guitar and other instruments that rely on electronic amplification--may have a solid wood body.

## History

Bow Harp or Harp Lute, West Africa
Musical bows have survived in some parts of Africa.

### Earliest string instruments

Dating to around c. 13,000 BC, a cave painting in the Trois Frères cave in France depicts what some believe is a musical bow, a hunting bow used as a single-stringed musical instrument.[1][2] From the musical bow, families of stringed instruments developed; since each string played a single note, adding strings added new notes, creating bow harps, harps and lyres.[3] In turn, this led to being able to play dyads and chords. Another innovation occurred when the bow harp was straightened out and a bridge used to lift the strings off the stick-neck, creating the lute.[4]

This picture of musical bow to harp bow is theory and has been contested. In 1965 Franz Jahnel wrote his criticism stating that the early ancestors of plucked instruments are not currently known.[5] He felt that the harp bow was a long cry from the sophistication of the 4th-century BC civilization that took the primitive technology and created "technically and artistically well made harps, lyres, citharas and lutes."[5]

Archaeological digs have identified some of the earliest stringed instruments in Ancient Mesopotamian sites, like the lyres of Ur, which include artifacts over three thousand years old. The development of lyre instruments required the technology to create a tuning mechanism to tighten and loosen the string tension. Lyres with wooden bodies, and strings used for plucking or playing with a bow represent key instruments that point towards later harps and violin-type instruments; moreover, Indian instruments from 500 BC have been discovered with anything from 7 to 21 strings.

### Lutes

Hellenistic banquet scene from the 1st century AD, Hadda, Gandhara. Lute player far right.
Spanish stele of a boy with a pandura.[6]

Musicologists have put forth examples of that 4th-century BC technology, looking at engraved images that have survived. The earliest image showing a lute-like instrument came from Mesopotamia prior to 3000 BC.[7] A cylinder seal from c. 3100 BC or earlier (now in the possession of the British Museum) shows what is thought to be a woman playing a stick lute.[7][8] From the surviving images, theororists have categorized the Mesopotamian lutes, showing that they developed into a long variety and a short.[9] The line of long lutes may have developed into the tamburs and pandura.[10] The line of short lutes was further developed to the east of Mesopotamia, in Bactria, Gandhara, and Northwest India, and shown in sculpture from the 2nd century BC through the 4th or 5th centuries AD.[11][12][13]

During the medieval era, instrument development varied from country to country. Middle Eastern rebecs represented breakthroughs in terms of shape and strings, with a half a pear shape using three strings. Early versions of the violin and fiddle, by comparison, emerged in Europe through instruments such as the gittern, a four stringed precursor to the guitar, and basic lutes. These instruments typically used catgut (animal intestine) and other materials, including silk, for their strings.

### Renaissance to modern

String instrument design refined during the Renaissance and into the Baroque period (1600-1750) of musical history. Violins and guitars became more consistent in design, and were roughly similar to what we use in the 2000s and into the present day. The violins of the Renaissance featured intricate woodwork and stringing, while more elaborate bass instruments such as the bandora were produced alongside quill-plucked citterns, and Spanish body guitars.

In the 19th century, string instruments were made more widely available through mass production, with wood string instruments a key part of orchestras - cellos, violas, and upright basses, for example, were now standard instruments for chamber and smaller orchestras. At the same time, the 19th century guitar became more typically associated with six string models, rather than traditional five string versions.

Major changes to string instruments in the 20th century primarily involved innovations in electronic instrument amplification and electronic music - electric violins were available by the 1920s, and were an important part of emerging jazz music trends in the United States. The acoustic guitar was widely used in blues and jazz, but as an acoustic instrument, it was not loud enough to be a solo instrument, so these genres mostly used it as an accompaniment rhythm section instrument. In big bands of the 1920s, the acoustic guitar played backing chords, but it was not loud enough to play solos like the saxophone and trumpet. The development of guitar amplifiers, which contained a power amplifier and a loudspeaker in a wooden cabinet, let jazz guitarists play solos and be heard over a big band. The development of the electric guitar provided guitarists with an instrument that was built to connect to guitar amplifiers. Electric guitars have magnetic pickups, volume control knobs and an output jack.

In the 1960s, larger, more powerful guitar amplifiers were developed, called "stacks". These powerful amplifiers enabled guitarists to perform in rock bands that played in large venues such as stadiums and outdoor music festivals (e.g., Woodstock Music Festival). Along with the development of guitar amplifiers, a large range of electronic effects units, many in small stompbox pedals were introduced in the 1960s and 1970s, such as fuzz pedals, flangers and phaser enabling performers to create unique new sounds during the psychedelic rock era. Breakthroughs in electric guitar and basses technologies and playing styles enabled major breakthroughs in pop and rock music in the 1960s and 1970s. The distinctive sound of the amplified electric guitar was the centerpiece of new genres of music such as blues rock and jazz-rock fusion. The sonic power of the loudly amplified, highly distorted electric guitar was to key element of the early heavy metal music, with the distorted guitar being used in lead guitar roles, and with power chords as a rhythm guitar.

The ongoing use of electronic amplification and effects units in string instruments, ranging from traditional instruments like the violin to the new electric guitar, added variety to contemporary classical music performances, and enabled experimentation in the dynamic and timbre (tone colour) range of orchestras, bands, and solo performances.[14]

## Types of instruments

### Construction

String instruments can be divided in three groups

Lutes
Instruments that support the strings via a neck and a bout ("gourd"), for instance a guitar, a violin, or a saz
Harps
Instruments that contain the strings within a frame
Zithers
Instruments that have the strings mounted on a body, frame or tube, such as a guqin, a cimbalom, an autoharp, harpsichord, a piano, or a valiha

It is also possible to divide the instruments into categories focused on how the instrument is played.

### Playing techniques

An acoustic guitar being strummed.

All string instruments produce sound from one or more vibrating strings, transferred to the air by the body of the instrument (or by a pickup in the case of electronically amplified instruments). They are usually categorised by the technique used to make the strings vibrate (or by the primary technique, in the case of instruments where more than one may apply.) The three most common techniques are plucking, bowing, and striking. An important difference between bowing and plucking is that in the former the phenomenon is periodic so that the overtones are kept in a strictly harmonic relationship to the fundamental.[15]

#### Plucking

Plucking is a method of playing on instruments such as the veena, banjo, ukulele, guitar, harp, lute, mandolin, oud, and sitar, using either a finger, thumb, or quills (now plastic plectra) to pluck the strings.

Instruments normally played by bowing (see below) may also be plucked, a technique referred to by the Italian term pizzicato.

#### Bowing

Bowing (Italian: arco) is a method used in some string instruments, including the violin, viola, cello, and the double bass (of the violin family), and the old viol family. The bow consists of a stick with a "ribbon" of parallel horse tail hairs stretched between its ends. The hair is coated with rosin so it can grip the string; moving the hair across a string causes a stick-slip phenomenon, making the string vibrate, and prompting the instrument to emit sound. Darker grades of rosin grip well in cool, dry climates, but may be too sticky in warmer, more humid weather. Violin and viola players generally use harder, lighter-colored rosin than players of lower-pitched instruments, who tend to favor darker, softer rosin.[16]

The ravanahatha is one of the oldest string instruments. Ancestors of the modern bowed string instruments are the rebab of the Islamic Empires, the Persian kamanche and the Byzantine lira. Other bowed instruments are the rebec, hardingfele, nyckelharpa, koky?, erhu, igil, sarangi and K'ni. The hurdy-gurdy is bowed by a wheel. Rarely, the guitar has been played with a bow (rather than plucked) for unique effects.

#### Striking

The third common method of sound production in stringed instruments is to strike the string. The piano and hammered dulcimer use this method of sound production. Even though the piano strikes the strings, the use of felt hammers means that the sound that is produced can nevertheless be mellow and rounded, in contrast to the sharp attack produced when a very hard hammer strikes the strings.

Violin family string instrument players are occasionally instructed to strike the string with the stick of the bow, a technique called col legno. This yields a percussive sound along with the pitch of the note. A well-known use of col legno for orchestral strings is Gustav Holst's "Mars" movement from The Planets suite.

#### Other methods

The aeolian harp employs a very unusual method of sound production: the strings are excited by the movement of the air.

Some instruments that have strings have an attached keyboards that the player presses keys on to trigger a mechanism that sounds the strings, instead of directly manipulating the strings. These include the piano, the clavichord, and the harpsichord. With these keyboard instruments too, strings are occasionally plucked or bowed by hand. Modern composers such as Henry Cowell wrote music that requires that the player reach inside the piano and pluck the strings directly, "bow" them with bow hair wrapped around the strings, or play them by rolling the bell of a brass instrument such as a trombone on the array of strings. However, these are relatively rarely used special techniques.

Other keyed string instruments, small enough for a strolling musician to play, include the plucked autoharp, the bowed nyckelharpa, and the hurdy-gurdy, which is played by cranking a rosined wheel.

Steel-stringed instruments (such as the guitar, bass, violin, etc.) can be played using a magnetic field. An E-Bow is a small hand-held battery-powered device that magnetically excites the strings of an electric string instrument to provide a sustained, singing tone reminiscent of a held bowed violin note.

Third bridge is a plucking method where the player frets a string and strikes the side opposite the bridge. The technique is mainly used on electric instruments because these have a pickup that amplifies only the local string vibration. It is possible on acoustic instruments as well, but less effective. For instance, a player might press on the seventh fret on a guitar and pluck it at the head side to make a tone resonate at the opposed side. On electric instruments, this technique generates multitone sounds reminiscent of a clock or bell.

Electric string instruments, such as the electric guitar, can also be played without touching the strings by using audio feedback. When an electric guitar is plugged into a loud, powerful guitar amplifier with a loudspeaker and a high level of distortion is intentionally used, the guitar produces sustained high-pitched sounds. By changing the proximity of the guitar to the speaker, the guitarist can produce sounds that cannot be produced with standard plucking and picking techniques. This technique was popularized by Jimi Hendrix and others in the 1960s. It was widely used in psychedelic rock and heavy metal music.

## Changing the pitch of a vibrating string

There are three ways to change the pitch of a vibrating string. String instruments are tuned by varying the strings' tension because adjusting length or mass per unit length is impractical. Instruments with a fingerboard are then played by adjusting the length of the vibrating portion of the strings. The following observations all apply to a string that is infinitely flexible (a theoretical assumption, because in practical applications, strings are not infinitely flexible) strung between two fixed supports. Real strings have finite curvature at the bridge and nut, and the bridge, because of its motion, are not exactly nodes of vibration. Hence the following statements about proportionality are approximations.

### Length

String fingering is proportional and not fixed,[17] as on the piano

Pitch can be adjusted by varying the length of the string.[15] A longer string results in a lower pitch, while a shorter string results in a higher pitch. The frequency is inversely proportional to the length:

${\displaystyle f\propto {\frac {1}{l}}}$

A string twice as long produces a tone of half the frequency (one octave lower).

### Tension

Pitch can be adjusted by varying the tension of the string. A string with less tension (looser) results in a lower pitch, while a string with greater tension (tighter) results in a higher pitch. A homemade washtub bass made out of a length of rope, a broomstick and a washtub can produce different pitches by increasing the tension on the rope (producing a higher pitch) or reducing the tension (producing a lower pitch). The frequency is proportional to the square root of the tension:

${\displaystyle f\propto {\sqrt {T}}}$

### Linear density

The pitch of a string can also be varied by changing the linear density (mass per unit length) of the string. In practical applications, such as with double bass strings or bass piano strings, extra weight is added to strings by winding them with metal. A string with a heavier metal winding produces a lower pitch than a string of equal length without a metal winding. This can be seen on a 2016-era set of gut strings for double bass. The higher-pitched G string is often made of synthetic material, or sometimes animal intestine, with no metal wrapping. To enable the low E string to produce a much lower pitch with a string of the same length, it is wrapped with many wrappings of thin metal wire. This adds to its mass without making it too stiff. The frequency is inversely proportional to the square root of the linear density:

${\displaystyle f\propto {1 \over {\sqrt {\mu }}}}$

Given two strings of equal length and tension, the string with higher mass per unit length produces the lower pitch.

## String length or scale length

The length of the string from nut to bridge on bowed or plucked instruments ultimately determines the distance between different notes on the instrument. For example, a double bass with its low range needs a scale length of around 42 inches (110 cm), whilst a violin scale is only about 13 inches (33 cm). On the shorter scale of the violin, the left hand may easily reach a range of slightly more than two octaves without shifting position, while on the bass' longer scale, a single octave or a ninth is reachable in lower positions.

## Contact points along the string

The strings of a piano

In bowed instruments, the bow is normally placed perpendicularly to the string, at a point halfway between the end of the fingerboard and the bridge. However, different bow placements can be selected to change timbre. Application of the bow close to the bridge (known as sul ponticello) produces an intense, sometimes harsh sound, which acoustically emphasizes the upper harmonics. Bowing above the fingerboard (sul tasto) produces a purer tone with less overtone strength, emphasizing the fundamental, also known as flautando, since it sounds less reedy and more flute-like.

Bowed instruments pose a challenge to instrument builders, as compared with instruments that are only plucked (e.g., guitar), because on bowed instruments, the musician must be able to play one string at a time if they wish. As such, a bowed instrument must have a curved bridge that makes the "outer" strings lower in height than the "inner" strings. With such a curved bridge, the player can select one string at a time to play. On guitars and lutes, the bridge can be flat, because the strings are played by plucking them with the fingers, fingernails or a pick; by moving the fingers or pick to different positions, the player can play different strings. On bowed instruments, the need to play strings individually with the bow also limits the number of strings to about six or seven strings; with more strings, it would be impossible to select individual strings to bow. (Note: bowed strings can also play two bowed notes on two different strings at the same time, a technique called a double stop.) Indeed, on the orchestral string section instruments, four strings are the norm, with the exception of five strings used on some double basses. In contrast, with stringed keyboard instruments, 88 courses are used on a piano, and even though these strings are arranged on a flat bridge, the mechanism can play any of the notes individually.

Similar timbral distinctions are also possible with plucked string instruments by selecting an appropriate plucking point, although the difference is perhaps more subtle.

In keyboard instruments, the contact point along the string (whether this be hammer, tangent, or plectrum) is a choice made by the instrument designer. Builders use a combination of experience and acoustic theory to establish the right set of contact points.

In harpsichords, often there are two sets of strings of equal length. These "choirs" usually differ in their plucking points. One choir has a "normal" plucking point, producing a canonical harpsichord sound; the other has a plucking point close to the bridge, producing a reedier "nasal" sound rich in upper harmonics.

## Production of multiple notes

A single string at a certain tension and length only produces one note. To produce multiple notes, string instruments use one of two methods. One is to add enough strings to cover the required range of different notes (e.g., as with the piano, which has sets of 88 strings to enable the performer to play 88 different notes). The other is to provide a way to stop the strings along their length to shorten the part that vibrates, which is the method used in guitar and violin family instruments to produce different notes from the same string. The piano and harp represent the first method, where each note on the instrument has its own string or course of multiple strings tuned to the same note. (Many notes on a piano are strung with a "choir" of three strings tuned alike, to increase the volume.) A guitar represents the second method--the player's fingers push the string against the fingerboard so that the string is pressed firmly against a metal fret. Pressing the string against a fret while plucking or strumming it shortens the vibrating part and thus produces a different note.

Some zithers combine stoppable (melody) strings with a greater number of "open" harmony or chord strings. On instruments with stoppable strings, such as the violin or guitar, the player can shorten the vibrating length of the string, using their fingers directly (or more rarely through some mechanical device, as in the nyckelharpa and the hurdy-gurdy). Such instruments usually have a fingerboard attached to the neck of the instrument, that provides a hard flat surface the player can stop the strings against. On some string instruments, the fingerboard has frets, raised ridges perpendicular to the strings, that stop the string at precise intervals, in which case the fingerboard is also called a fretboard.

Moving frets during performance is usually impractical. The bridges of a koto, on the other hand, may be moved by the player occasionally in the course of a single piece of music. Many modern Western harps include levers, either directly moved by fingers (on Celtic harps) or controlled by foot pedals (on orchestral harps), to raise the pitch of individual strings by a fixed amount. The Middle Eastern zither, the qanun, is equipped with small levers called mandal that let each course of multiple strings be incrementally retuned "on the fly" while the instrument is being played. These levers raise or lower the pitch of the string course by a microtone, less than a half step.

## Sympathetic strings

Some instruments are employed with sympathetic strings--which are additional strings not meant to be plucked. These strings resonate with the played notes, creating additional tones. Sympathetic strings vibrate naturally when various intervals, such as the unisons or the octaves of the notes of the sympathetic strings are plucked, bowed or struck. This system is used on the sarangi, the grand piano, the hardanger fiddle and the rubab.

## Sound production

### Acoustic instruments

A vibrating string strung on a very thick log, as a hypothetical example, would make only a very quiet sound, so string instruments are usually constructed in such a way that the vibrating string is coupled to a hollow resonating chamber, a soundboard, or both. On the violin, for example the four strings pass over a thin wooden bridge resting on a hollow box (the body of the violin). The pressure from the strings is supported in part by a small cylinder of wood called the soundpost. The violin body also has two "f-holes" carved on the top. The strings' vibrations are distributed via the bridge and soundpost to all surfaces of the instrument, and are thus made louder. The correct technical explanation is that they allow a better match to the acoustic impedance of the air.

It is sometimes said that the sounding board or soundbox "amplifies" the sound of the strings. Technically speaking, no amplification occurs, because all of the energy to produce sound comes from the vibrating string. What really happens is that the sounding board of the instrument provides a larger surface area to create sound waves than that of the string. A larger vibrating surface moves more air, hence produces a louder sound.

All lute type instruments traditionally have a bridge, which holds the string at the proper action height from the fret/finger board at one end of the strings. On acoustic instruments, the bridge performs an equally important function of transmitting string energy into the "sound box" of the instrument, thereby increasing the sound volume. The specific design, and materials the used in the construction of the bridge of an instrument, have a dramatic impact upon both the sound and responsiveness of the instrument.

Achieving a tonal characteristic that is effective and pleasing to the player's and listener's ear is something of an art and craft, as well as a science, and the makers of string instruments often seek very high quality woods to this end, particularly spruce (chosen for its lightness, strength and flexibility) and maple (a very hard wood). Spruce is used for the sounding boards of instruments from the violin to the piano. Instruments such as the banjo use a drum, covered in natural or synthetic skin as their soundboard.

Acoustic instruments can also be made out of artificial materials, such as carbon fiber and fiberglass (particularly the larger, lower-pitched instruments, such as cellos and basses).

In the early 20th century, the Stroh violin used a diaphragm-type resonator and a metal horn to project the string sound, much like early mechanical gramophones. Its use declined beginning about 1920, as electronic amplification through power amplifiers and loudspeakers was developed and came into use. String instrument players can electronically amplify their instruments by connecting them to a PA system or a guitar amplifier.

### Electronic amplification

Most string instruments can be fitted with piezoelectric or magnetic pickups to convert the string's vibrations into an electrical signal that is amplified and then converted back into sound by loudspeakers. Some players attach a pickup to their traditional string instrument to "electrify" it. Another option is to use a solid-bodied instrument, which reduces unwanted feedback howls or squeals.

Amplified string instruments can be much louder than their acoustic counterparts, so musicians can play them in relatively loud rock, blues, and jazz ensembles. Amplified instruments can also have their amplified tone modified by using electronic effects such as distortion, reverb, or wah-wah.

Bass-register string instruments such as the double bass and the electric bass are amplified with bass instrument amplifiers that are designed to reproduce low-frequency sounds. To modify the tone of amplified bass instruments, a range of electronic bass effects are available, such as distortion and chorus.

## Symphonic strings

The string instruments usually used in the orchestra, and often called the "symphonic strings" or string section are:[18]

• Violins (divided into two sections--first violins and second violins; these sections play exactly the same instruments; the difference is that the first violins play higher-register lines and the second violins play lower-register parts, accompaniment parts or countermelodies)
• Violas
• Cellos
• Double basses

When orchestral instrumentation specifies "strings," it often means this combination of string parts. Orchestral works rarely omit any of these string parts, but often include additional string instruments, especially the concert harp and piano. In the Baroque orchestra from the 1600s-1750 (or with modern groups playing early music) harpsichord is almost always used to play the basso continuo part (the written-out bass line and improvised chords), and often a theorbo or lute or a pipe organ. In some classical music, such as the string quartet, the double bass is not typically used; the cello plays the bass role in this literature.

## References

1. ^ Campen, Ank van. "The music-bow from prehistory till today". HarpHistory.info. Archived from the original on April 2, 2015. Retrieved 2015. A cave-painting in the "Trois Frères" cave in France dating from about 15,000 years ago. The magician-hunter plays the musical bow.
2. ^ "Trois Freres Cave". Archived from the original on March 18, 2015. Retrieved 2015.
3. ^ Dumbrill, 1998 & page 179, 231, 235-236, 308-310
4. ^ Dumbrill, 1998 & page 308-310
5. ^ a b Jahnel, Franz (1965). Manual of Guitar Technology: The History and Technology of Plucked String Instruments (Fachbuchreihe Das Musikinstrument, Bd. 37). p. 15. ISBN 0-933224-99-0. There have been some uncertain presumptions concerning the "invention" of the bowed harp...The "musical bow" conjectured by many music scholars is not definitely recognizable in any cave paintings. The fact that some African negroes held the end of their bow shaped harp in their mouths in order to improve the tone...should not be taken as proof that the first European bowmen were also conversant with the musical bow.
6. ^ https://m.flickr.com/#/photos/julio-claudians/8098646683/ Flicker based photo of the museum information sign for the stele. Circa 2nd century A.D memorial stele from Augusta Emerita in modern Spain for a Roman boy, Lutaia Lupata, showing him with his pandurium, the Roman variant of the Greek Pandura. Kept at the Museo Arqueologico, Merida, Spain.
7. ^ a b Dumbrill, 1998 & page 321
8. ^ "Archived copy". Archived from the original on 2017-07-02. Retrieved . British Museum, Cylinder Seal, Culture/period Uruk, Date 3100BC (circa1), Museum number 41632.
9. ^ Dumbrill, 1998 & page 310
10. ^ Dumbrill, Richard J. (2005). The Archaeomusicology of the Ancient Near East. Victoria, British Columbia: Trafford Publishing. pp. 319-320. ISBN 1-4120-5538-5. The long-necked lute in the OED is orthographed as tambura; tambora, tamera, tumboora; tambur(a) and tanpoora. We have an Arabic Õunbur ; Persian tanbur ; Armenian pandir ; Georgian panturi. and a Serbo-Croat tamburitza. The Greeks called it pandura; panduros; phanduros; panduris or pandurion. The Latin is pandura. It is attested as a Nubian instrument in the third century BC. The earliest literary allusion to lutes in Greece comes from Anaxilas in his play The Lyre-maker as 'trichordos'... According to Pollux, the trichordon (sic) was Assyrian and they gave it the name pandoura...These instruments survive today in the form of the various Arabian tunbar...
11. ^ "Encyclopaedia Iranica - Barbat". Iranicaonline.org. 1988-12-15. Archived from the original on 2015-05-17. Retrieved .
12. ^ "Five Celestial Musicians". LACMA.org. Archived from the original on 10 October 2017. Retrieved 2017. Views 3 & 4 show a musician playing a 4th- to 5th-century lute-like instrument, excavated in Gandhara, and part of a Los Angeles County Art Museum collection of Five Celestial Musicians
13. ^ "Bracket with two musicians 100s, Pakistan, Gandhara, probably Butkara in Swat, Kushan Period (1st century-320)". The Cleveland Museum of Art. Archived from the original on April 2, 2015. Retrieved 2015.
14. ^ Michael Chanan (1994). Musica Practica: The Social Practice of Western Music from Gregorian Chant to Postmodernism. Verso. p. 170. ISBN 978-1-85984-005-4.
15. ^ a b "Oxford Music Online by subscription". www.oxfordmusiconline.com. Archived from the original on 2011-02-24. Retrieved .
16. ^ http://stringsmagazine.com/the-differences-between-dark-and-amber-rosin/
17. ^ Piston, Walter (1955). Orchestration, p.5.
18. ^ The Concise Oxford Dictionary of Music. Oxford University Press. 1964. p. 412. ISBN 0-19-311302-3.


text:
# Inline Functions, Default Arguments & Constant Arguments in C++

38
0

## Inline Functions

In C++, an inline function is a function that is expanded in-line at the point where it is called, rather than being executed through a function call. The main advantage of using inline functions is that they can improve the performance of your program by reducing the overhead of function calls.

To define an inline function, you need to use the inline keyword before the function definition. For example:

inline int add(int a, int b) {
return a + b;
}
C++

When the add function is called, the compiler will replace the function call with the actual code of the function. This can result in faster code execution, but it can also increase the size of the executable file.

It’s important to note that the inline keyword is only a suggestion to the compiler, and it’s up to the compiler to decide whether or not to actually inline the function. Generally, smaller functions that are called frequently are good candidates for inlining.

## Default Arguments

In C++, you can provide default arguments for function parameters. Default arguments are values that are used if no value is provided for a particular parameter when the function is called.

To provide a default argument, you need to specify it in the function declaration. For example:

void print(int x = 0, int y = 0) {
cout << "x = " << x << ", y = " << y << endl;
}
C++

In this example, the print function has two integer parameters with default values of 0. If the function is called without providing values for these parameters, the default values will be used. For example:

print();        // prints "x = 0, y = 0"
print(1);       // prints "x = 1, y = 0"
print(2, 3);    // prints "x = 2, y = 3"
C++

Note that default arguments can only be provided for the rightmost parameters in a function declaration. For example, the following code is not valid:

void print(int x = 0, int y) {
cout << "x = " << x << ", y = " << y << endl;
}
C++

## Constant Arguments

In C++, you can specify that a function parameter should not be modified by the function. This is done by using the const keyword before the parameter type. For example:

void print(const string& s) {
cout << s << endl;
}
C++

In this example, the s parameter is a reference to a string that cannot be modified by the print function. This helps to ensure that the function does not accidentally modify the value of the string.

You can also use the const keyword to specify that a member function does not modify the state of an object. This is done by using the const keyword after the function declaration. For example:

class MyClass {
public:
int getValue() const {
return value;
}
private:
int value;
};
C++

In this example, the getValue function is a const member function that does not modify the state of the MyClass object. This allows you to call the function on const objects, like this:

const MyClass obj;
int x = obj.getValue();
C++
WRITTEN BY

## xalgord

Constantly learning & adapting to new technologies. Passionate about solving complex problems with code. #programming #softwareengineering