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Microscale and Nanoscale Thermal Characterization Techniques | J. Electron. Packag. | ASME Digital Collection
Thermal Issues In Emerging Technologies Theory And Applications, Theta
J. Christofferson,
K. Maize,
Y. Ezzahri,
Y. Ezzahri
J. Shabani,
Christofferson, J., Maize, K., Ezzahri, Y., Shabani, J., Wang, X., and Shakouri, A. (November 13, 2008). "Microscale and Nanoscale Thermal Characterization Techniques." ASME. J. Electron. Packag. December 2008; 130(4): 041101. https://doi.org/10.1115/1.2993145
Miniaturization of electronic and optoelectronic devices and circuits and increased switching speeds have exasperated localized heating problems. Steady-state and transient characterization of temperature distribution in devices and interconnects is important for performance and reliability analysis. Novel devices based on nanowires, carbon nanotubes, and single molecules have feature sizes in 1–100 nm range, and precise temperature measurement and calibration are particularly challenging. In this paper we review various microscale and nanoscale thermal characterization techniques that could be applied to active and passive devices. Solid-state microrefrigerators on a chip can provide a uniform and localized temperature profile and they are used as a test vehicle in order to compare the resolution limits of various microscale techniques. After a brief introduction to conventional microthermocouples and thermistor sensors, various contact and contactless techniques will be reviewed. Infrared microscopy is based on thermal emission and it is a convenient technique that could be used with features tens of microns in size. Resolution limits due to low emissivity and transparency of various materials and issues related to background radiation will be discussed. Liquid crystals that change color due to phase transition have been widely used for hot spot identification in integrated circuit chips. The main problems are related to calibration and aging of the material. Micro-Raman is an optical method that can be used to measure absolute temperature. Micron spatial resolution with several degrees of temperature resolution has been achieved. Thermoreflectance technique is based on the change of the sample reflection coefficient as a function of temperature. This small change in
10−4–10−5
range per degree is typically detected using lock-in technique when the temperature of the device is cycled. Use of visible and near IR wavelength allows both top surface and through the substrate measurement. Both single point measurements using a scanning laser and imaging with charge coupled device or specialized lock-in cameras have been demonstrated. For ultrafast thermal decay measurement, pump-probe technique using nanosecond or femtosecond lasers has been demonstrated. This is typically used to measure thin film thermal diffusivity and thermal interface resistance. The spatial resolution of various optical techniques can be improved with the use of tapered fibers and near field scanning microscopy. While subdiffraction limit structures have been detected, strong attenuation of the signal reduces the temperature resolution significantly. Scanning thermal microscopy, which is based on nanoscale thermocouples at the tip of atomic force microscope, has had success in ultrahigh spatial resolution thermal mapping. Issues related to thermal resistance between the tip and the sample and parasitic heat transfer paths will be discussed.
atomic force microscopy, integrated circuit reliability, nanoelectronics, nanowires, phase transformations, Raman spectroscopy, thermal resistance, thermoreflectance
Imaging, Microscale devices, Nanoscale phenomena, Probes, Resolution (Optics), Temperature, Thermal characterization, Thermoreflectance, Wavelength, Atomic force microscopy, Microscopy, Pumps, Thermocouples, Lasers, Phase transitions, Reflectance, Signals, Heating, Integrated circuits, Nanowires, Thermal resistance, Liquid crystals
Oesterschulze
Thermal Imaging and Measurement Techniques for Electronic Materials and Devices
Dynamic Surface Temperature Measurements in ICs
Selected Contactless Optoelectronic Measurements for Electronic Applications
Thermal Mapping With Liquid Crystal Method
Time-Resolved Pumpprobe Experiments With Subwavelength Lateral Resolution
Quantitative Thermal Imaging by Synchronous Thermoreflectance With Optimized Illumination Wavelengths
Lueerssen
Nanoscale Thermoreflectance With 10 mK Temperature Resolution Using Stochastic Resonance
Proceedings of the 21st IEEE Semi-Therm Symposium
Noncontact Transient Temperature Mapping of Active Electronic Devices Using the Thermoreflectance Method
Thermoreflectance Based Thermal Microscope
Xiofeng
Transient Response of Thin Film SiGe Micro Coolers
Proceedings of the International Mechanical Engineering Congress and Exhibition (IMECE 2001)
, Nov., New York.
Thermal Conductivity Measurement From 30 to 750 K: The 3 Omega Method
Study of Thermomechanical Properties of Si/SiGe Superlattices Using Femtosecond Transient Thermoreflectance Technique
Hurdequint
Elastic Properties of Ultrathin Permalloy/Alumina Multilayers Films Using Picosecond Ultrasonics and Brillouin Light Scattering
Stokes/Anti-Stokes Raman Vibrational Temperatures: Reference Materials, Standard Lamps, and Spectrophotometric Calibrations
Theoretical and Experimental Uncertainty in Temperature Measurement of Materials by Raman Spectroscopy
3D Temperature Measurement in IC Chips Using Raman Spectroscopy
Proceedings of the Material Research Society Meeting
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Heron's_formula Knowpia
In geometry, Heron's formula (sometimes called Hero's formula), named after Hero of Alexandria,[1] gives the area of a triangle when the lengths of all three sides are known. Unlike other triangle area formulae, there is no need to calculate angles or other distances in the triangle first.
A triangle with sides a, b, and c
Heron's formula states that the area of a triangle whose sides have lengths a, b, and c is
{\displaystyle A={\sqrt {s(s-a)(s-b)(s-c)}},}
where s is the semi-perimeter of the triangle; that is,[2]
{\displaystyle s={\frac {a+b+c}{2}}.}
{\displaystyle A={\frac {1}{4}}{\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}}
{\displaystyle A={\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}
{\displaystyle A={\frac {1}{4}}{\sqrt {(a^{2}+b^{2}+c^{2})^{2}-2(a^{4}+b^{4}+c^{4})}}}
{\displaystyle A={\frac {1}{4}}{\sqrt {4(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{2}+b^{2}+c^{2})^{2}}}}
{\displaystyle A={\frac {1}{4}}{\sqrt {4a^{2}b^{2}-(a^{2}+b^{2}-c^{2})^{2}}}.}
Let △ABC be the triangle with sides a = 4, b = 13 and c = 15. This triangle’s semiperimeter is
{\displaystyle s={\frac {a+b+c}{2}}={\frac {4+13+15}{2}}=16}
and so the area is
{\displaystyle {\begin{aligned}A&={\sqrt {s\left(s-a\right)\left(s-b\right)\left(s-c\right)}}={\sqrt {16\cdot (16-4)\cdot (16-13)\cdot (16-15)}}\\&={\sqrt {16\cdot 12\cdot 3\cdot 1}}={\sqrt {576}}=24.\end{aligned}}}
In this example, the side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well in cases where one or more of the side lengths are not integers.
The formula is credited to Heron (or Hero) of Alexandria,[3] and a proof can be found in his book Metrica, written around AD 60. It has been suggested that Archimedes knew the formula over two centuries earlier,[4] and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work.[5]
A formula equivalent to Heron's, namely,
{\displaystyle A={\frac {1}{2}}{\sqrt {a^{2}c^{2}-\left({\frac {a^{2}+c^{2}-b^{2}}{2}}\right)^{2}}}}
was discovered by the Chinese. It was published in Mathematical Treatise in Nine Sections (Qin Jiushao, 1247).[6]
There are many ways to prove Heron's formula, for example using trigonometry as below, or the incenter and one excircle of the triangle,[7] or as a special case of De Gua's theorem (for the particular case of acute triangles).[8]
Trigonometric proof using the law of cosinesEdit
A modern proof, which uses algebra and is quite different from the one provided by Heron, follows.[9] Let a, b, c be the sides of the triangle and α, β, γ the angles opposite those sides. Applying the law of cosines we get
{\displaystyle \cos \gamma ={\frac {a^{2}+b^{2}-c^{2}}{2ab}}}
From this proof, we get the algebraic statement that
{\displaystyle \sin \gamma ={\sqrt {1-\cos ^{2}\gamma }}={\frac {\sqrt {4a^{2}b^{2}-(a^{2}+b^{2}-c^{2})^{2}}}{2ab}}.}
The altitude of the triangle on base a has length b sin γ, and it follows
{\displaystyle {\begin{aligned}A&={\frac {1}{2}}({\mbox{base}})({\mbox{altitude}})\\&={\frac {1}{2}}ab\sin \gamma \\&={\frac {1}{4}}{\sqrt {4a^{2}b^{2}-(a^{2}+b^{2}-c^{2})^{2}}}\\&={\frac {1}{4}}{\sqrt {(2ab-(a^{2}+b^{2}-c^{2}))(2ab+(a^{2}+b^{2}-c^{2}))}}\\&={\frac {1}{4}}{\sqrt {(c^{2}-(a-b)^{2})((a+b)^{2}-c^{2})}}\\&={\sqrt {\frac {(c-(a-b))(c+(a-b))((a+b)-c)((a+b)+c)}{16}}}\\&={\sqrt {{\frac {(b+c-a)}{2}}{\frac {(a+c-b)}{2}}{\frac {(a+b-c)}{2}}{\frac {(a+b+c)}{2}}}}\\&={\sqrt {{\frac {(a+b+c)}{2}}{\frac {(b+c-a)}{2}}{\frac {(a+c-b)}{2}}{\frac {(a+b-c)}{2}}}}\\&={\sqrt {s(s-a)(s-b)(s-c)}}.\end{aligned}}}
Algebraic proof using the Pythagorean theoremEdit
Triangle with altitude h cutting base c into d + (c − d)
The following proof is very similar to one given by Raifaizen.[10] By the Pythagorean theorem we have b2 = h2 + d2 and a2 = h2 + (c − d)2 according to the figure at the right. Subtracting these yields a2 − b2 = c2 − 2cd. This equation allows us to express d in terms of the sides of the triangle:
{\displaystyle d={\frac {-a^{2}+b^{2}+c^{2}}{2c}}.}
For the height of the triangle we have that h2 = b2 − d2. By replacing d with the formula given above and applying the difference of squares identity we get
{\displaystyle {\begin{aligned}h^{2}&=b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}\\&={\frac {(2bc-a^{2}+b^{2}+c^{2})(2bc+a^{2}-b^{2}-c^{2})}{4c^{2}}}\\&={\frac {{\big (}(b+c)^{2}-a^{2}{\big )}{\big (}a^{2}-(b-c)^{2}{\big )}}{4c^{2}}}\\&={\frac {(b+c-a)(b+c+a)(a+b-c)(a-b+c)}{4c^{2}}}\\&={\frac {2(s-a)\cdot 2s\cdot 2(s-c)\cdot 2(s-b)}{4c^{2}}}\\&={\frac {4s(s-a)(s-b)(s-c)}{c^{2}}}.\end{aligned}}}
We now apply this result to the formula that calculates the area of a triangle from its height:
{\displaystyle {\begin{aligned}A&={\frac {ch}{2}}\\&={\sqrt {{\frac {c^{2}}{4}}\cdot {\frac {4s(s-a)(s-b)(s-c)}{c^{2}}}}}\\&={\sqrt {s(s-a)(s-b)(s-c)}}.\end{aligned}}}
Trigonometric proof using the law of cotangentsEdit
Geometrical significance of s − a, s − b, and s − c. See the law of cotangents for the reasoning behind this.
From the first part of the law of cotangents proof,[11] we have that the triangle's area is both
{\displaystyle {\begin{aligned}A&=r{\big (}(s-a)+(s-b)+(s-c){\big )}=r^{2}\left({\frac {s-a}{r}}+{\frac {s-b}{r}}+{\frac {s-c}{r}}\right)\\&=r^{2}\left(\cot {\frac {\alpha }{2}}+\cot {\frac {\beta }{2}}+\cot {\frac {\gamma }{2}}\right)\end{aligned}}}
and A = rs, but, since the sum of the half-angles is π/2, the triple cotangent identity applies, so the first of these is
{\displaystyle {\begin{aligned}A&=r^{2}\left(\cot {\frac {\alpha }{2}}\cot {\frac {\beta }{2}}\cot {\frac {\gamma }{2}}\right)=r^{2}\left({\frac {s-a}{r}}\cdot {\frac {s-b}{r}}\cdot {\frac {s-c}{r}}\right)\\&={\frac {(s-a)(s-b)(s-c)}{r}}.\end{aligned}}}
{\displaystyle A^{2}=s(s-a)(s-b)(s-c),}
Heron's formula as given above is numerically unstable for triangles with a very small angle when using floating-point arithmetic. A stable alternative[12][13] involves arranging the lengths of the sides so that a ≥ b ≥ c and computing
{\displaystyle A={\frac {1}{4}}{\sqrt {{\big (}a+(b+c){\big )}{\big (}c-(a-b){\big )}{\big (}c+(a-b){\big )}{\big (}a+(b-c){\big )}}}.}
Other area formulae resembling Heron's formulaEdit
Three other area formulae have the same structure as Heron's formula but are expressed in terms of different variables. First, denoting the medians from sides a, b, and c respectively as ma, mb, and mc and their semi-sum 1/2(ma + mb + mc) as σ, we have[14]
{\displaystyle A={\frac {4}{3}}{\sqrt {\sigma (\sigma -m_{a})(\sigma -m_{b})(\sigma -m_{c})}}.}
Next, denoting the altitudes from sides a, b, and c respectively as ha, hb, and hc, and denoting the semi-sum of the reciprocals of the altitudes as H = 1/2(h−1
a + h−1
b + h−1
c) we have[15]
{\displaystyle A^{-1}=4{\sqrt {H(H-h_{a}^{-1})(H-h_{b}^{-1})(H-h_{c}^{-1})}}.}
Finally, denoting the semi-sum of the angles' sines as S = 1/2(sin α + sin β + sin γ), we have[16]
{\displaystyle A=D^{2}{\sqrt {S(S-\sin \alpha )(S-\sin \beta )(S-\sin \gamma )}}}
where D is the diameter of the circumcircle: D = a/sin α = b/sin β = c/sin γ.
{\displaystyle K={\sqrt {(s-a)(s-b)(s-c)(s-d)}}}
{\displaystyle s={\frac {a+b+c+d}{2}}.}
Heron's formula is also a special case of the formula for the area of a trapezoid or trapezium based only on its sides. Heron's formula is obtained by setting the smaller parallel side to zero.
Expressing Heron's formula with a Cayley–Menger determinant in terms of the squares of the distances between the three given vertices,
{\displaystyle A={\frac {1}{4}}{\sqrt {-{\begin{vmatrix}0&a^{2}&b^{2}&1\\a^{2}&0&c^{2}&1\\b^{2}&c^{2}&0&1\\1&1&1&0\end{vmatrix}}}}}
Another generalization of Heron's formula to pentagons and hexagons inscribed in a circle was discovered by David P. Robbins.[17]
Heron-type formula for the volume of a tetrahedronEdit
If U, V, W, u, v, w are lengths of edges of the tetrahedron (first three form a triangle; u opposite to U and so on), then[18]
{\displaystyle {\text{volume}}={\frac {\sqrt {\,(-a+b+c+d)\,(a-b+c+d)\,(a+b-c+d)\,(a+b+c-d)}}{192\,u\,v\,w}}}
{\displaystyle {\begin{aligned}a&={\sqrt {xYZ}}\\b&={\sqrt {yZX}}\\c&={\sqrt {zXY}}\\d&={\sqrt {xyz}}\\X&=(w-U+v)\,(U+v+w)\\x&=(U-v+w)\,(v-w+U)\\Y&=(u-V+w)\,(V+w+u)\\y&=(V-w+u)\,(w-u+V)\\Z&=(v-W+u)\,(W+u+v)\\z&=(W-u+v)\,(u-v+W).\end{aligned}}}
^ "Fórmula de Herón para calcular el área de cualquier triángulo" (in Spanish). Spain: Ministerio de Educación, Cultura y Deporte. 2004. Retrieved 30 June 2012.
^ Kendig, Keith (2000). "Is a 2000-year-old formula still keeping some secrets?". The American Mathematical Monthly. 107 (5): 402–415. doi:10.1080/00029890.2000.12005213. JSTOR 2695295. MR 1763392. S2CID 1214184.
^ Id, Yusuf; Kennedy, E. S. (1969). "A medieval proof of Heron's formula". The Mathematics Teacher. 62 (7): 585–587. doi:10.5951/MT.62.7.0585. JSTOR 27958225. MR 0256819.
^ Heath, Thomas L. (1921). A History of Greek Mathematics. Vol. II. Oxford University Press. pp. 321–323.
^ 秦, 九韶 (1773). "卷三上, 三斜求积". 數學九章 (四庫全書本) (in Chinese).
^ "Personal email communication between mathematicians John Conway and Peter Doyle". 15 December 1997. Retrieved 25 September 2020.
^ Lévy-Leblond, Jean-Marc (2020-09-14). "A Symmetric 3D Proof of Heron's Formula". The Mathematical Intelligencer. 43 (2): 37–39. doi:10.1007/s00283-020-09996-8. ISSN 0343-6993.
^ Niven, Ivan (1981). Maxima and Minima Without Calculus. The Mathematical Association of America. pp. 7–8.
^ Raifaizen, Claude H. (1971). "A Simpler Proof of Heron's Formula". Mathematics Magazine. 44 (1): 27–28. doi:10.1080/0025570X.1971.11976093.
^ The second part of the Law of cotangents proof depends on Heron's formula itself, but this article depends only on the first part.
^ Sterbenz, Pat H. (1974-05-01). Floating-Point Computation. Prentice-Hall Series in Automatic Computation (1st ed.). Englewood Cliffs, New Jersey, USA: Prentice Hall. ISBN 0-13-322495-3.
^ William M. Kahan (24 March 2000). "Miscalculating Area and Angles of a Needle-like Triangle" (PDF).
^ D. P. Robbins, "Areas of Polygons Inscribed in a Circle", Discr. Comput. Geom. 12, 223-236, 1994.
^ W. Kahan, "What has the Volume of a Tetrahedron to do with Computer Programming Languages?", [1], pp. 16–17.
J. H. Conway discussion on Heron's Formula
"Heron's Formula and Brahmagupta's Generalization". MathPages.com.
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Babylonian Numbers Converter
Babylonian math: calculating with stylus
The base of Babylonian numbers
How to convert decimal numbers to Babylonian numbers
How to write numbers in Babylonian
How to convert from ancient Babylonian numbers to decimal numbers
How to use our Babylonian numbers converter
Discover the math of one of the first civilizations with our ancient Babylonian numbers converter!
Take a clay tablet and stylus in hand, and dive in with us on a journey in time to the four thousand years old civilization where it (almost) all began. Here you will learn about:
Mesopotamian math;
Babylonian numbers; and
How to convert from decimal numbers to Babylonian, and vice-versa.
If Babylonians aren't enough, you can try our other ancient math tools: visit our Mayan numerals converter for another numerical system with another base; our Mayan calendar converter to learn the Mayans' style of counting days; or our Egyptian fractions calculator to learn how to write decimal numbers like an Egyptian!
Mesopotamia, the fertile crescent, was the perfect ground for the development of math. With the creation of a writing system, and the first material used to support such writing, it was finally possible to compute in an abstract fashion.
The first traces of Babylonian math dates back to 3000 BC. Sumerians mastered pretty complex metrology while Babylonians grew a particular interest in numbers and arithmetic. Their clay tablets are preserved enough to give modern historians a wide and varied repertoire of mathematical notions.
Babylonians used their mathematical clay tables mainly for two reasons:
Help with calculation thanks to computed values; and
Statements and solutions to problems.
They complemented this knowledge with some basic geometry, which included one of the first attempts to compute the value of
\pi
There are two fundamental differences between modern and Babylonian math:
Babylonians wrote in clay tablets 😁
Their numeration system was sexagesimal: the base was
60
Babylonians used a sexagesimal positional numerical system. This means that they counted in base
60
59
different symbols (
60
if we count the zero) and that the position of a digit in a number is nothing but the multiplier of the relative power of
60
🙋 Babylonians had a complicated relationship with zero: they didn't know it "existed" for a long time. In later years, though, zeros started appearing, but only in the middle of a number. Babylonians didn't write zeros at the leftmost end: for them,
1
100
were the same!
To convert a decimal number to a Babylonian number, we must change its base from
10
60
. How to do that? Follow these steps!
Take the number in decimal base and apply an integer division by
60
Save the remainder of this operation: it is the first digit of the base 60 number.
Repeat the two steps before on the quotient of the integer division.
Stop when the quotient is smaller than
60
Let's see this with an example: we will convert the number
19281295
into Babylonian!
19281295/60 = 321354
55
321354/60=5355
54
5355/60 = 89
15
89/60=1
29
1/60=0
1
The Babylonian number is then
1.29.15.54.55
. As you can see, we need to use a period to separate the digits: Arabian numerals are not the best choice to represent numbers in base
60
🙋 Try our binary converter, decimal to hexadecimal converter, or binary to hexadecimal converter to learn more about conversion between numerical bases!
Babylonians didn't use Arabic numerals (the digits so familiar to us): their math used the Babylonian cuneiform numbers. Here you will learn how to write numbers as a Babylonian!
Babylonians used a stylus and clay tablets instead of pen and paper. This system reduced the possible set of characters available to a scribe while at the same time allowing for a quick correction of mistakes and solid and durable support. The wedged stylus's jagged end could leave some distinct triangular markings; hence the name cuneiform writing system.
Babylonian cuneiform numbers used a combination of two symbols plus the symbol for zero. Each digit uses two of those symbols to represent each number from
1
59
The first nine digits correspond to an increasing number of vertical wedges:
Babylonian symbols for the digits from 1 to 9.
The other symbol, an open triangle with a "heavier" tip, represented the tens:
10
20
30
40
50
The numbers 10, 20, 30, 40, and 50 had a particular representation in the Babylonian system.
As you can easily see, combining two of these 14 symbols allow you to create every number from
1
59
. Zero had its specific placeholder:
The symbol for 0 in the Babylonian system
Take the number we saw in the previous example:
1.29.15.54.55
. To write it using Babylonian cuneiform numbers, we have to:
Write the single digits using the Babylonian symbols; then
Join them in the correct order, ensuring that enough separation allows for identifying the various positions in the number.
You can see the result below. Note how numbers like
29
use a combination of the symbols seen in the previous section!
How to write 19281295 in Babylonian.
What about converting Babylonian numbers to decimal numbers?
First thing, in case you have a Babylonian cuneiform number in front of you, we need to translate it into Arabic numerals.
To do so, count the horizontal and vertical wedges: each of them equals, respectively,
10
1
Once your number is written more familiarly, the real power of a positional system comes into play. Each "digit" gets multiplied by the power of
60
corresponding to its position. The rightmost position equals
0
, and the value increases going left. We then sum the results together.
Take the number
12.9.35.0.22
60
. Starting from the right, we have:
\footnotesize \begin{align*} &12.9.35.0.22_{60}=22\cdot60^0+0\cdot 60^1+\\ &+35\cdot 60^2+9\cdot 60^3+12\cdot 60^4=\\ &=22+0+126,000+1,944,000+\\ &+ 155,520,000= 157,590,022 \end{align*}
🙋 Since the root of the decimal system (
10
) is smaller than the one of the sexagesimal system (
60
), the same number occupies more positions.
Whether you want to convert from or to Babylonian cuneiform numbers, our tool can help you! Here we will teach you how to use it.
First, choose if you want to convert from or to the Babylonian numbers. The variable direction serves this purpose.
If you choose to convert from base
10
60
, insert a positive number. Do not insert symbols or negative numbers. Choose if you want to use zero in the Babylonian style or as in the modern numerical system. We will show you both the numerical conversion and the Babylonian cuneiform number.
🔎 You can choose to visualize or not the expansion: this may be helpful to understand the conversion to Babylonian numbers.
If you chose to convert from Babylonian numbers to decimal numbers, insert a number in base
60
. Separate the digits using a period, and remember that each digit can't be bigger than
60
. We will translate the number you inserted in cuneiform and then show you the correspondent in base
10
What are Babylonian numbers?
Babylonian numbers are ancient numbers that used base 60 to perform arithmetic operations. Babylonians developed this numerical system more than four thousand years ago and used them intensively. They were originally written using the Babylonian cuneiform script.
How to convert from base 10 to Babylonian numbers?
The conversion to Babylonian numbers requires converting a number from base 10 to base 60. To do so:
Divide the number by 60, and note the remainder.
Repeat the previous step on the quotient.
Repeat the steps before until the quotient of the division is 0.
The remainder is written from the last to the first, forming the base 60 number.
How do I write 13451 in Babylonian?
To convert the number 13451, follow these steps:
Divide 13451 by 60: 13451/60 = 224 with remainder 11.
Repeat: 224/60 = 3 with remainder 44.
Repeat: 3/60 = 0 with remainder 3.
The resulting base 60 number is 3 44 11.
How do you write numbers in Babylonian?
Babylonians used a combination of two symbols to represent every possible number.
A vertical wedge to indicate the numbers from 1 to 9; and
An open triangle indicates the tens: 10, 20, 30, 40, and 50.
With these symbols, you can write every number from 1 to 59, corresponding to a single base 60 digit.
From decimal to Babylonian
Use this online negative log calculator to calculate the negative logarithm of any number.
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Momentum - Wikipedia @ WordDisk
{\displaystyle \mathbf {p} =m\mathbf {v} .}
This article is about linear momentum. It is not to be confused with angular momentum or moment (physics).
{\displaystyle {\textbf {F}}={\frac {d}{dt}}(m{\textbf {v}})}
This article uses material from the Wikipedia article Momentum, and is written by contributors. Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.
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Standard Equation of a Circle Calculator
How to write a circle equation in standard form?
How to convert the standard equation into parametric form
How to convert the standard equation into general form
How to use this standard equation of a circle calculator
Other related calculators to find the equation of a circle
This standard equation of a circle calculator is helpful to calculate the standard form of a circle equation using its center coordinates and radius, or any other form of the circle equation. With this simple tool in your hand, you can easily find the circle equation in any form you want!
How to convert the standard form into parametric or general form and vice versa.
\left(x-a\right)^2 + (y-b)^2 = r^2
(x,y)
- The coordinates of any point on the circle;
(a,b)
- The coordinates of the center of the circle; and
r
- The radius of the circle.
We can use this equation to find the standard form from its center and radius or vice versa.
The parametric form of a circle equation is given by:
\begin{align*} x &= a + r\cos(\alpha)\\ y &= b + r\sin(\alpha)\\ \end{align*}
(x,y)
(a,b)
- The coordinates of the center of the circle;
r
- The radius of the circle; and
\alpha
- The angle subtended by the point
(x,y)
at the circle's center
(a,b)
It is a straightforward conversion between these two forms; the additional parameter
\alpha
is not essential for this conversion, so long as the other variables are known.
The general form of the circle equation is an expansion of its standard form. It can be expressed as:
x^2 + y^2 + Dx+ Ey+ F = 0
(x,y)
D
- The sum of the coefficients of the x-terms;
E
- The sum of the coefficients of the y-terms; and
F
- The sum of the constant terms.
Note that the right-hand side (RHS) of this equation has to be zero. Bring every term to the left-hand side (LHS) and simplify.
As this is the expansion of the standard form, we can complete the squares of this expanded form to arrive at the standard equation and establish the following relationships between the various parameters in these two forms:
\begin{align*} D &= -2a\\ E &= -2b\\ F &= a^2+b^2-r^2\\ \end{align*}
We can use these equations to convert between the standard form and the general form of a circle equation.
You can use this calculator for more than one thing:
Enter the standard equation of a circle to obtain the center, radius, and circle equation in other forms.
Give the center and radius of a circle to simultaneously determine its equation in all three forms.
Enter the equation of a circle in parametric or general form to calculate its center, radius, and equation in the standard form.
In addition, this calculator will also determine other properties of the circle, like its area and circumference.
What is the equation of a circle with a center (0,0) and radius of 7?
x²+y² = 49. To find this equation, follow these steps:
Insert the center coordinates in the place of (a,b) in the standard form of a circle equation (x-a)² + (y-b)² = r². This gives (x-0)² + (y-0)² = r².
Substitute the value of radius in the place of r in this equation. This gives x²+y² = 7².
Evaluate this equation to get the equation of the circle, x²+y² = 49.
How do you determine if a point lies on a circle?
To determine whether a point P(pₓ,pᵧ) lies on a circle (x-a)² + (y-b)² = r², follow these steps:
Substitute the coordinates of the point P(pₓ,pᵧ) in place of x and y in LHS of the circle equation to get (pₓ-a)² + (pᵧ-b)².
If (pₓ-a)² + (pᵧ-b)² = r², then the point P(pₓ,pᵧ) lies on the circle.
If (pₓ-a)² + (pᵧ-b)² ≠ r², then the point P(pₓ,pᵧ) does not lie on the circle.
Center coordinates and radius
Standard form: (x - A)² + (y - B)² = C
Parametric form: x = A + r cos(α), y = B + r sin(α)
Other circle properties
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\int_{a}^b f(s) \, ds
. Numerical integration is also called quadrature because it computes areas. The other meaning applies to solving ordinary differential equations (ODEs). My article about new methods for solving
Some matrices are so special that they have names. The identity matrix is the most famous, but many are named after a researcher who studied them such as the Hadamard, Hilbert, Sylvester, Toeplitz, and Vandermonde matrices. This article is about the Pascal matrix, which is formed by using elements from
Many discussions and articles about SAS Viya emphasize its ability to handle Big Data, perform parallel processing, and integrate open-source languages. These are important issues for some SAS customers. But for customers who program in SAS and do not have Big Data, SAS Viya is attractive because it is the
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What is latitude/longitude?
The haversine formula or haversine distance
Obtaining the distance between two points on Earth – distance between coordinates.
Examples using the latitude longitude distance calculator
The latitude longitude distance calculator will help you calculate the distance between two points on Earth's surface given their latitude/longitude coordinates.
Find the distance between two latitudes;
Find the distance between coordinates; or
Obtain the distance between two points on Earth.
We will explain everything there is to know about using latitude/longitude to obtain the distance between two coordinates using the haversine formula or haversine distance.
First of all, we need to properly explain what the latitude and longitude coordinates mean.
Longitude, measured in degrees °, is the angle between a point on Earth and the Equator. Earth's Equator is an imaginary line which divides the planet horizontally exactly halfway between the South and North pole.
That way, a point's longitude coordinate must have a value between -90° (towards the South pole) and 90° (towards the North pole), with the zero coordinate being precisely on the equator line.
Latitude, on the other hand, is the angle between a point and an arbitrary, imaginary line, called the prime meridian, which divides Earth vertically.
The prime meridian goes near the Royal Observatory in Greenwich, London and acts as the zero latitude. This coordinate must lay between -180° (west of the prime meridian) and 180° (east of the prime meridian).
Any coordinate pair will be expressed in the form of (latitude, longitude). Sometimes, coordinates are also represented using only positive degrees and a letter to indicate the orientation (E, W, N, S). To convert these to a number-only format, we just need to add a minus (-) sign if the orientation is west or south (you can read more about converting coordinates with our coordinates converter calculator).
So, for example, the Empire State building coordinates are: (40.7484°, -73.9857°), which means this famous place is located 40.7484° up north from the Equator (since it's a positive number), and -73.9857° west from the prime meridian (since this longitude is negative).
🙋 Our latitude longitude distance calculator also supports coordinates in DMS (degrees, minutes, seconds). Check how to convert degrees minutes seconds to decimals degrees within another of our tools!
Assume an arbitrary point located somewhere in a three-dimensional space. Let's call:
θ, the angle between the z-axis and the position vector (r); and
φ, the angle between the x-axis and the projection of the position vector on the xy plane.
These are called polar coordinates.
Polar coordinates on a three-dimensional space. Source: Wikimedia
The haversine is a trigonometric function that is equal to half a versine. And what's a versine? It's 1 minus the cosine of an angle.
\text{hav}(\Theta) = \frac{1 - \cos{\Theta}}{2}
If we apply the haversine formula for a central angle, i.e., the angle between two points along a great circle on a sphere with radius R, and solve for distance, we obtain:
d = 2R⋅sin⁻¹(√[sin²((θ₂ - θ₁)/2) + cosθ₁⋅cosθ₂⋅sin²((φ₂ - φ₁)/2)])
θ₁, φ₁ – First point latitude and longitude coordinates;
θ₂, φ₂ – Second point latitude and longitude coordinates;
R – Earth's radius (R = 6371 km); and
d – Distance between them along Earth's surface.
Using this haversine distance formula, you can translate latitude and longitude to distance, given the coordinates of two points on Earth, although with a minor setback. It can result in an error of up to 0.5% because the Earth is not a perfect sphere.
Of course, the faster way is to use our latitude longitude distance calculator, but it's good to know where things come from.
Distance between Paris and Krakow
Let's assume we want to find the distance between these two European cities. First, we need to find their coordinates:
Paris: Lat: 48.8566° N, Long: 2.3522° E.
Krakow: Lat: 50.0647° N, Long: 19.9450° E.
After typing the coordinates in the calculator, we obtain that the distance between Paris and Krakow is
1275.6 km or 792.6 miles. Keep in mind this is a straight line from Paris to Krakow along Earth's circumference, and therefore the distance by ground transport will be greater.
Distance between Mt. Everest and Empire State Building
Here's another example of calculating the distance with longitude and latitude. Let's look at each point's coordinates:
Mt. Everest: Lat: 27.9881° N, Long: 86.9250° E.
Empire State Building: Lat: 40.7484° N, Long: -73.9857° W. Since the longitude coordinate is directed towards the west, we need to add a minus (-) sign to write the longitude coordinate in a degrees-only format.
From the latitude longitude distance calculator, we see that the distance between Mt. Everest and the Empire State Building is 12,122 km or 7,532 miles.
How do I calculate the distance between two points with longitude and latitude?
To calculate the distance between two points given longitude and latitude coordinates:
Write down each point's coordinates in degrees-only format. We'll call θ and φ to their respective latitude and longitude components.
Input them in the haversine distance formula: d = 2R⋅sin⁻¹(√[sin²((θ₂ - θ₁)/2) + cosθ₁⋅cosθ₂⋅sin²((φ₂ - φ₁)/2)]).
(θ₁, φ₁) and (θ₂, φ₂) – Each point's coordinates;
R – Earth's radius; and
d – Great circle or 'as the crow flies' distance between the points.
What is the distance between two latitudes?
The distance between any two latitudes is approximately 69 miles or 111 km. Latitude lines run parallel to each other. That's why the distance between them remains constant from the South to the North pole. On the other hand, longitude lines are furthest apart at the equator and meet at the poles.
Where is the Prime Meridian located?
5.3 arcseconds or 102 meters east of the Royal Observatory, Greenwich. This location serves as the International Reference Meridian, which also acts as the reference meridian for the Global Positioning System or GPS. The zero longitude line was moved to account for local gravitational effects.
What is the distance between the North and South pole?
The distance between the North and South pole is 20,015.09 km or 12,436.80 miles. The North pole, the northernmost point on the planet, is located at 90 degrees of latitude, while the South pole sits at -90 degrees.
Coordinates in:
If the coordinate includes orientation, add a minus (-) sign if heading west (W) or south (S).
Point A coordinates
Point B coordinates
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Computer_science Knowpia
Epistemology of computer scienceEdit
Paradigms of computer scienceEdit
A number of computer scientists have argued for the distinction of three separate paradigms in computer science. Peter Wegner argued that those paradigms are science, technology, and mathematics.[43] Peter Denning's working group argued that they are theory, abstraction (modeling), and design.[44] Amnon H. Eden described them as the "rationalist paradigm" (which treats computer science as a branch of mathematics, which is prevalent in theoretical computer science, and mainly employs deductive reasoning), the "technocratic paradigm" (which might be found in engineering approaches, most prominently in software engineering), and the "scientific paradigm" (which approaches computer-related artifacts from the empirical perspective of natural sciences[45], identifiable in some branches of artificial intelligence).[46] Computer science focuses on methods involved in design, specification, programming, verification, implementation and testing of human-made computing systems.[47]
Theory of computationEdit
{\displaystyle M=\{X:X\not \in X\}}
Information and coding theoryEdit
Data structures and algorithmsEdit
Programming language theory and formal methodsEdit
{\displaystyle \Gamma \vdash x:{\text{Int}}}
Computer systems and computational processesEdit
Computer architecture and organizationEdit
Concurrent, parallel and distributed computingEdit
Computer security and cryptographyEdit
Databases and data miningEdit
Computer graphics and visualizationEdit
Image and sound processingEdit
Applied computer scienceEdit
Computational science, finance and engineeringEdit
Social computing and human–computer interactionEdit
Programming paradigmsEdit
^ Denning, Peter J. (2007). "Computing is a natural science". Communications of the ACM. 50 (7): 13–18. doi:10.1145/1272516.1272529.
Curriculum and classificationEdit
Bibliography and academic search enginesEdit
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TB67H420FTG Dual/Single Motor Driver Carrier Pololu 2999 – MakerSupplies Singapore
TB67H420FTG Dual/Single Motor Driver ...
TB67H420FTG Dual/Single Motor Driver Carrier Pololu 2999
In dual channel mode, this results in a nominal current limit of 4.5 A per channel. You can lower the limit for each motor channel by adding a resistor between the corresponding VREF pin and GND, or you can apply a voltage (3.6 V max) directly to the VREF pin. The formula for the current chopping thresholds in dual-channel mode is
{I}_{\text{out}}=\text{VREF}×1.25
In single-channel mode, the default 3.6 V reference voltage results in a nominal single-channel current limit of 9 A. The formula for the current chopping threshold in single-channel mode is
{I}_{\text{out}}=\text{VREF}×2.5
1) Without included hardware.
2) Note: Reverse voltage protection only works up to 40 V.
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Halting Problem | Brilliant Math & Science Wiki
Karleigh Moore, Agnishom Chattopadhyay, Ivan Koswara, and
The halting problem is a decision problem in computability theory. It asks, given a computer program and an input, will the program terminate or will it run forever? For example, consider the following Python program:
It reads the input, and if it's not empty, the program will loop forever. Thus, if the input is empty, the program will terminate and the answer to this specific question is "yes, this program on the empty input will terminate", and if the input isn't empty, the program will loop forever and the answer is "no, this program on this input will not terminate".
Halting problem is perhaps the most well-known problem that has been proven to be undecidable; that is, there is no program that can solve the halting problem for general enough computer programs. It's important to specify what kind of computer programs we're talking about. In the above case, it's a Python program, but in computation theory, people often use Turing machines which are proven to be as strong as "usual computers". In 1936, Alan Turing proved that the halting problem over Turing machines is undecidable using a Turing machine; that is, no Turing machine can decide correctly (terminate and produce the correct answer) for all possible program/input pairs.
Halting problem is undecidable
Reductions and Consequences
The decision problem
H
, for Halting problem, is the set of all
\{\langle p, x \rangle | \text{program \$p\$ halts on input \$x\$}\}
, for an appropriate definition of "program" (usually "Turing machine"), and where
\langle \cdot \rangle
denotes some kind of encoding. A Turing machine
A
H
if, given any input
\langle p, x \rangle
, it terminates in an accepting state if
\langle p, x \rangle \in H
, and terminates in a rejecting state otherwise. Note that it doesn't matter whether
p
terminates because it accepts, or because it rejects, or because it gets some kind of error; as long as it terminates and doesn't loop forever,
A
should accept it.
Suppose there exists a Turing machine
A
that decides
H
. Now consider a Turing machine
B
defined as follows: it takes an input
\langle p \rangle
, runs
A
\langle p, \langle p \rangle \rangle
, and halts if and only if
A
rejects. That is,
B
takes a program
p
, runs the supposed Turing machine that decides the halting problem on the input "program
p
\langle p \rangle
"; based on the outcome, if it says "yes,
p
\langle p \rangle
halts", then it loops forever, otherwise if it says "no,
p
\langle p \rangle
doesn't halt", then it terminates. Note that
\langle p \rangle
is the encoding of a program
p
into a suitable input (bit-string or natural number), and hence can be accepted as input by another program.
Consider what happens when
B
receives
\langle B \rangle
as input. It runs
A
\langle B, \langle B \rangle \rangle
. We now have two cases:
A
accepts. This means
B
\langle B \rangle
halts. But by definition of
B
A
accepts we will enter an infinite loop, and so
B
\langle B \rangle
doesn't halt after all.
A
rejects. This means
B
\langle B \rangle
doesn't halt. But by definition of
B
A
rejects we will terminate, and so
B
\langle B \rangle
halts after all.
In both cases, we derive a contradiction. This contradiction happens because we assumed the existence of
A
; its existence allows us to create
B
that behaves incorrectly. Thus
A
cannot exist, and so no Turing machine can decide
H
Real-world Example [1]
One way to visualize this is to think of apps on a phone. Apps are types of Turing machines. Sometimes apps crash your phone because they get caught in a loop and do not halt. Let’s supposes a clever team releases an app to check for this. This app, the Checker app can solve the Halting problem.
The Checker app checks some app
M
M
halts, then
Checker(M)
accepts, this app will not crash your phone. If
M
loops, then
Checker(M)
rejects, this app will crash your phone.
Suppose a diabolical computer scientist decides to create a App called Paradox. This app will effectively reverse the Checker app. And because the Checker app works, this one works too, it's really simple: - It turns on the Checker app, and inputs itself into the Checker app. - If the Checker returns an accept, then the Paradox app forces the phone to loop and crash. - If the Checker app returns a reject, then the Paradox app will halt, meaning that it doesn't deserve that rejection, it's a safe app and doesn't crash your phone.
Effectively to run Paradox means to run Paradox(Checker(Paradox)). If Paradox is a bad app, that crashes your phone, Checker will reject it, and Paradox will return a halt. Paradox(Checker(Paradox)) = Paradox(Checker(loop)) = Paradox(reject) = Halt If Paradox is a good app, that halts, then the Checker app will accept it, and Paradox will crash your phone. Paradox(Checker(Paradox)) = Paradox(Checker(halt)) = Paradox(accept) = Loop So the Checker app says Paradox is good, and then Paradox crashes you, or the Checker app says Paradox is bad and Paradox halts. This is a contradiction.
Now some people still don't see this as a contradiction. To which we'd point out that this is a supertask, which we know to be impossible in the real world. A supertask is an uncountable series of infinite tasks, examples include Thompson's lamp or zeno's paradox. We know that running Paradox actually means Paradox(Checker(Paradox)) which means Paradox(Checker(Paradox(Checker(Paradox)))) which means Paradox(Checker(Paradox(Checker(Paradox(Checker(Paradox)))))) and so on. Either the Paradox app does crash your phone or it doesn't. Those are the only two possibilities. But this series of infinite checks switch back and forth between loop and halt, with no result.
Consider the following algorithm which when fed with another algorithm and an input, tells if the program halts:
Simulate the first step of the algorithm given.
If the program halts, return Yes, the algorithm halts and terminate.
If it does not halt (as of yet), capture a snapshot of the simulation.
Compare the snapshot with the previous snapshots. If it is the same as one of the snapshots previously taken, return No, the algorithm does not halt and terminate.
Simulate the next step.
Now which of the following is true?
A. The construction is correct; The halting problem is only undecidable for computers with infinite memory
B. The construction is wrong; Snapshots of a simulation cannot be captured by a computer program
C. The construction is wrong; The simulation might run into a previously encountered snapshot but still halt.
D. The construction is correct; It solves the halting problem in general for any computer
E. The construction is wrong; The described construction cannot be written as a finite program in a proper computer.
Note: The construction refers to the algorithm described, which is the one that is supposed to solve the halting problem.
A typical way to prove that a problem is undecidable is to use reduction. If a problem can be reduced to the halting problem, it is undecidable.
A
B
if a solution to
B
could be used to solve
A
A
has been proven to be an undecidable problem, to prove that a new problem
B
is undecidable, it is sufficient to show that a solution to
B
could be used to decide
A
. This yields a contradiction since it was already proven that
A
is undecidable, and therefore,
B
is also undecidable.
If we could solve the Busy Beaver problem, we could solve the halting problem.
The Busy Beaver problem is the problem of determining the maximum number of steps an
state Turing machine will take before halting.
If we had a function that could compute the Busy Beaver function,
BB(n)
, we could know the maximum number of steps any Turing machine will take before halting. This means that we could know how long we would have to wait for a machine to halt, and ultimately, we will know if the machine will halt. In other words, if we had a way to compute the Busy Beaver function, we could use it to solve the halting problem. Since the halting problem is uncomputable, it follows that the Busy Beaver function is uncomputable.
If the halting problem could be solved, many other problems could be decided:
Goldbach’s conjecture could be decided. It's easy to construct a Turing machine that tests every even natural number greater than 2 on whether it's the sum of two primes or not; if it encounters any counterexample, it immediately halts and reports that a counterexample has been found, otherwise it will run forever. If Halting problem were decidable, we could decide whether this program would halt or not, and thus give an answer to Goldbach's conjecture.
Kolmogorov complexity would be computable.
The Busy Beaver function would also be computable.
It is useful to know about the kinds of problems that are undecidable because it helps us to understand the limitations of our computation models.
It's important to note that Halting problem depends on what programs we're considering. The halting problem on Turing machines is undecidable. Conversely, the halting problem on finite state automata is easily decidable; all finite state automata halt. Thus it's important to specify the model.
The halting problem on usual computers is also decidable. To see this, note that there are a finite number of bits in the memory, and thus a finite number of possible configurations the computer can be in. If a program ever repeats a configuration, it will never terminate. Thus a Turing machine, with infinite memory, can simulate the program. By Pigeonhole Principle, if there are
N
configurations that the program can be in, but we have simulated the program for
N+1
steps, we must have visited a configuration twice and thus the program will never terminate. So halting problem for usual computers is also decidable using a Turing machine.
Husfeldt, T. The-Freeze-App-Does-Not-Exist. Retrieved April, 24 2016, from https://thorehusfeldt.net/2012/06/25/the-freeze-app-does-not-exist/
Cite as: Halting Problem. Brilliant.org. Retrieved from https://brilliant.org/wiki/halting-problem/
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1 Universidad Publica de El Alto, El Alto, Bolivia.
2 Department of Animal Sciences, University of Arkansas, Fayetteville, AR, USA.
Abstract: This study evaluated sperm viability over time, after dilution and refrigerator storage of fresh semen extended in either synthetic cauda epididymal plasma (CEP2), or in a low sodium medium (CJ2) supplemented with either AlbuMAX or egg yolk. Semen collected weekly for 4 weeks from 4 bulls and assigned within bulls, across treatments. After extension in either CEP2, or CJ2 containing either egg yolk or AlbuMAX, semen was cooled to 4°C, and evaluated for 7 days. A computer assisted sperm analysis (CASA) system was used for sperm evaluation. Particular emphasis was placed on sperm motility since it is the single most important sperm parameter influencing bull fertility. Total and progressive motility of sperm in CEP2 and CJ2-AlbuMAX were similar (P = 0.85 and P = 0.23, respectively), but both were lower (P < 0.01) when compared to CJ2-yolk. Fewer sperm had rapid motility in CEP2 and CJ2-AlbuMAX compared to CJ2-yolk (P < 0.01). Sperm straightness and linearity were greater in CJ2-AlbuMAX and CJ2-yolk than in CEP2 (P < 0.01). Mean velocity (VAP) and linear velocity (VSL) were greater (P < 0.01) in CJ2-AlbuMAX than either CEP2 or CJ2-yolk. The calculated curvilinear velocity (VCL) of spermatozoa in CEP2 was lower than CJ2-AlbuMAX (P = 0.01), but similar with CJ2-yolk (P = 0.54). Overall, every sperm parameter measured by the CASA system was equal to or higher for sperm stored 7 days in CJ2 medium as compared with CEP2. The CJ2 extender supplemented with egg yolk is a viable alternative for storing fresh bovine semen.
Keywords: Semen Extender, Bovine Sperm, Refrigerator Storage
{Y}_{ijk}=\mu +bul{l}_{i}+extende{r}_{j}+da{y}_{k}+day\ast extende{r}_{\left(jk\right)}+{e}_{ijk\left( m \right)}
Cite this paper: Delgado, P. , Lester, T. and Rorie, R. (2018) Effect of Low-Sodium, Choline-Based Semen Diluent on Viability of Bovine Spermatozoa Stored at 4°C. Advances in Reproductive Sciences, 6, 12-21. doi: 10.4236/arsci.2018.61002.
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When scientists talk about populations they often refer to the carrying capacity of species in a particular environment. Carrying capacity is the largest population that an environment can sustain forever.
Suppose the carrying capacity of seals for a particular group of islands is 2900 and that there are currently 1800 seals that inhabit the island. The rate of change of the number of seals is jointly proportional to the number of seals and the difference between the number of seals and the carrying capacity. Let S represent the number of seals on the island after t years.
Write a differential equation that models the rate of change in the number of seals.
Find a general solution to the differential equation.
Recall that S = 1800 when t = 0 because that is the current population. Suppose that after ten years, 2500 seals inhabit the island. Write a formula for S in terms of t.
\frac{dS}{dt}=kS(2900-S)
\frac{dS}{S(2900-S)}=kdt
\frac{1}{S(2900-S)}=\frac{1/2900}{S}+\frac{1/2900}{2900-S}
Another Step (b):
\ln\Big|\frac{S}{2900-S}\Big|=2900kt+C
S=\frac{2900\Big(\frac{18}{11}\Big)e^{2900kt}}{1+\frac{18}{11}e^{2900kt}}
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Problems Handled by Optimization Toolbox Functions - MATLAB & Simulink - MathWorks Benelux
The following tables show the functions available for minimization, multiobjective optimization, equation solving, and solving least-squares (model-fitting) problems.
Scalar minimization
\underset{x}{\mathrm{min}}f\left(x\right)
such that lb < x < ub (x is scalar)
\underset{x}{\mathrm{min}}f\left(x\right)
fminunc,
\underset{x}{\mathrm{min}}{f}^{T}x
such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub
\underset{x}{\mathrm{min}}{f}^{T}x
such that A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub, x(intcon) is integer-valued
\underset{x}{\mathrm{min}}\frac{1}{2}{x}^{T}Hx+{c}^{T}x
\underset{x}{\mathrm{min}}{f}^{T}x
‖A\cdot x-b‖\le {d}^{T}\cdot x-\gamma
, A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub
\underset{x}{\mathrm{min}}f\left(x\right)
such that c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub
Semi-infinite minimization
\underset{x}{\mathrm{min}}f\left(x\right)
such that K(x,w) ≤ 0 for all w, c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub
\underset{x,\gamma }{\mathrm{min}}\gamma
such that F(x) – w·γ ≤ goal, c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, lb ≤ x ≤ ub
\underset{x}{\mathrm{min}}\underset{i}{\mathrm{max}}{F}_{i}\left(x\right)
C·x = d, n equations, n variables
mldivide (matrix left division)
Nonlinear equation of one variable
F(x) = 0, n equations, n variables
Least-Squares (Model-Fitting) Problems
\underset{x}{\mathrm{min}}\frac{1}{2}{‖C\cdot x-d‖}_{2}^{2}
m equations, n variables
\underset{x}{\mathrm{min}}\frac{1}{2}{‖C\cdot x-d‖}_{2}^{2}
such that x ≥ 0
\underset{x}{\mathrm{min}}\frac{1}{2}{‖C\cdot x-d‖}_{2}^{2}
\underset{x}{\mathrm{min}}{‖F\left(x\right)‖}_{2}^{2}=\underset{x}{\mathrm{min}}\sum _{i}{F}_{i}^{2}\left(x\right)
such that lb ≤ x ≤ ub
\underset{x}{\mathrm{min}}{‖F\left(x,xdata\right)-ydata‖}_{2}^{2}
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In the polygons at right, all corresponding angles are the same and the side lengths are as shown.
Since all corresponding angles are the same, the shape must be similar. Find the corresponding matching sides of the shape and compare the side lengths. What do you see?
1:1
Are the two figures similar?
Similar shapes have the same angle measures and proportionate sides.
What is the special name for similar figures with a scale factor of
\frac { 1 } { 1 }
What does this tell you about the side lengths and the angles of the two shapes?
Are the side lengths and angle measures for both shapes the same?
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Digamma - Simple English Wikipedia, the free encyclopedia
In mathematics, the name "digamma" is used in digamma function, which is the derivative of the logarithm of gamma function (that is,
{\displaystyle (\ln \Gamma (z))'}
↑ Weisstein, Eric W. "Digamma Function". mathworld.wolfram.com. Retrieved 2020-10-06.
↑ "Digamma Function | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-10-06.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Digamma&oldid=7513028"
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Feferman–Schütte ordinal - Wikipedia
In mathematics, the Feferman–Schütte ordinal Γ0 is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion. It is named after Solomon Feferman and Kurt Schütte.
It is sometimes said to be the first impredicative ordinal,[1][2] though this is controversial, partly because there is no generally accepted precise definition of "predicative". Sometimes an ordinal is said to be predicative if it is less than Γ0.
There is no standard notation for ordinals beyond the Feferman–Schütte ordinal. There are several ways of representing the Feferman–Schütte ordinal, some of which use ordinal collapsing functions:
{\displaystyle \psi (\Omega ^{\Omega })}
{\displaystyle \theta (\Omega )}
{\displaystyle \varphi _{\Omega }(0)}
{\displaystyle \varphi (1,0,0)}
The Feferman–Schütte ordinal can be defined as the smallest ordinal that cannot be obtained by starting with 0 and using the operations of ordinal addition and the Veblen functions φα(β). That is, it is the smallest α such that φα(0) = α.
^ Kurt Schütte, Proof theory, Grundlehren der Mathematischen Wissenschaften, Band 225, Springer-Verlag, Berlin, Heidelberg, New York, 1977, xii + 302 pp.
^ Solomon Feferman, "Predicativity" (2002)
Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics, vol. 1407, Berlin: Springer-Verlag, doi:10.1007/978-3-540-46825-7, ISBN 3-540-51842-8, MR 1026933
Weaver, Nik (2005), Predicativity beyond Gamma_0, arXiv:math/0509244, Bibcode:2005math......9244W
Large countable ordinals
First infinite ordinal ω
Epsilon numbers ε0
Feferman–Schütte ordinal Γ0
Ackermann ordinal θ(Ω2)
small Veblen ordinal θ(Ωω)
large Veblen ordinal θ(ΩΩ)
Bachmann–Howard ordinal ψ(εΩ+1)
Buchholz's ordinal ψ0(Ωω)
Takeuti–Feferman–Buchholz ordinal ψ(εΩω+1)
Proof-theoretic ordinals of the theories of iterated inductive definitions
Nonrecursive ordinal ≥ ωCK
Retrieved from "https://en.wikipedia.org/w/index.php?title=Feferman–Schütte_ordinal&oldid=1086259386"
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Home : Support : Online Help : Mathematics : Linear Algebra : LinearAlgebra Package : Generic Subpackage : Determinant
compute the determinant of a square Matrix
Determinant[R](A)
Determinant[R](A,method=BerkowitzAlgorithm)
Determinant[R](A,method=MinorExpansion)
Determinant[R](A,method=BareissAlgorithm)
Determinant[R](A,method=GaussianElimination)
The parameter A must be a square (n x n) Matrix of values from R.
The optional argument method=... specifies the algorithm to be used. The specific algorithms are as follows:
method=MinorExpansion directs the code to use minor expansion. This algorithm uses O(n 2^n) arithmetic operations in R.
method=BerkowitzAlgorithm directs the code to use the Berkowitz algorithm. This algorithm uses O(n^4) arithmetic operations in R.
method=BareissAlgorithm directs the code to use the Bareiss algorithm. This algorithm uses O(n^3) arithmetic operations in R but requires exact division, i.e., it requires R to be an integral domain with the following operation defined:
R[Divide]: a boolean procedure for dividing two elements of R where R[Divide](a,b,'q') outputs true if b | a and optionally assigns q the quotient such that a = b q.
method=GaussianElimination directs the code to use the Gaussian elimination algorithm. This algorithm uses O(n^3) arithmetic operations in R but requires R to be a field, i.e., the following operation must be defined:
R[`/`]: a procedure for dividing two elements of R
If the method is not given and the operation R[Divide] is defined, then the Bareiss algorithm is used, otherwise if the operation R[`/`] is defined then GaussianElimination is used, otherwise the Berkowitz algorithm is used.
\mathrm{with}\left(\mathrm{LinearAlgebra}[\mathrm{Generic}]\right):
Z[\mathrm{`0`}],Z[\mathrm{`1`}],Z[\mathrm{`+`}],Z[\mathrm{`-`}],Z[\mathrm{`*`}],Z[\mathrm{`=`}]≔0,1,\mathrm{`+`},\mathrm{`-`},\mathrm{`*`},\mathrm{`=`}:
A≔\mathrm{Matrix}\left([[2,1,4],[3,2,1],[0,0,5]]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{5}\end{array}]
\mathrm{Determinant}[Z]\left(A\right)
\textcolor[rgb]{0,0,1}{5}
Q[\mathrm{`0`}],Q[\mathrm{`1`}],Q[\mathrm{`+`}],Q[\mathrm{`-`}],Q[\mathrm{`*`}],Q[\mathrm{`/`}],Q[\mathrm{`=`}]≔0,1,\mathrm{`+`},\mathrm{`-`},\mathrm{`*`},\mathrm{`/`},\mathrm{`=`}:
A≔\mathrm{Matrix}\left([[2,1,4,6],[3,2,1,7],[0,0,5,1],[0,0,3,8]]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{7}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{8}\end{array}]
\mathrm{Determinant}[Q]\left(A\right)
\textcolor[rgb]{0,0,1}{37}
\mathrm{Determinant}[Q]\left(A,\mathrm{method}=\mathrm{BerkowitzAlgorithm}\right)
\textcolor[rgb]{0,0,1}{37}
LinearAlgebra[Generic][BerkowitzAlgorithm]
LinearAlgebra[Generic][GaussianElimination]
LinearAlgebra[Generic][MinorExpansion]
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Exponents and Powers - Practically Study Material
If a certain number a is multiplied m times in succession, then the continued product so obtained is called the
{m}^{th}
power of a and is written as
{a}^{m}
(read as, a to the power m).
{a}^{m}
= a × a × a × a…….. to m factors.
Here, a is called the base of
{a}^{m}
and m is called the index or exponent of
{a}^{m}
{x}^{5}=x.x.x.x.x
ii) (–3)6 = (–3)(–3)(–3)(–3)(–3)(–3)
In particular, a2 is called the square of a (or,
{a}^{2}
) and a3 is called the cube of a (or
{a}^{3}
If a and x are two real numbers and n is a positive integer such that an = x, then a is called the
{n}^{th}
root of x and is written as
a=\sqrt[n]{x} or a={x}^{1/n}
{n}^{th}
root of x (i.e.,
\sqrt[n]{x}
) is such a number whose
{n}^{th}
power is equal to x i.e.,
\left(\sqrt[n]{x}{\right)}^{n}=x
i) In particular, if
{a}^{2}=x
, then a is called the second root or square root of x and is written as
a=\sqrt[2]{x} or a={x}^{1/2} \text{ or simply }a=\sqrt{x}
ii) If a3 = x, then a is called the third root or cube root of x and is written as
a=\sqrt[3]{x} or a={x}^{1/3}
i) Square Root of 25 is 5 i.e.,
\sqrt{25}=5 \left(\because {5}^{2}=25\right)
ii) Cube Root of 27 is 3 i.e., 3
\sqrt[3]{27}=3 \left(\because {3}^{3}=27\right)
iii) Sixth Root of 64 is 2 i.e., 6
\sqrt[6]{64}=2 \left(\because {2}^{6}=64\right)
{5}^{2}=25 \therefore \sqrt{25}=5
{\left(–5\right)}^{2}=25 \therefore \sqrt{25}=–5
Therefore, it is evident that both 5 and (–5) are square roots of 25.
Hence, by the Square root of a real positive number x we mean
±\sqrt{x}\left(\text{ i.e., }+\sqrt{x}\text{ or }–\sqrt{x}\right)
i) If x > 0 and n is any positive integer, then
\sqrt[n]{x}
ii) If x < 0 and n is any odd integer, then
\sqrt[n]{x}
iii) If x < 0 and n is any positive even integer, then
\sqrt[n]{x}
does not exist in the set of real numbers.
12.2. LAWS OF EXPONENTS
I) Multiplication property :
{a}^{m}×{a}^{n}={a}^{m+n}\left(m, n\in {z}^{+}\right)
(Fundamental Index Law)
For multiplying the power of same base, powers are added.
{a}^{m}×{a}^{n}=\left(a×a×a×\dots \dots .\text{ to }m\text{ factors}\right)
×\left(a×a×a×a\dots \dots \text{ . to }n\text{ factors}\right)
=\left(a×a×a×a\dots ..\text{ to }\left(m+n\right)\text{ facors}\right)
\therefore {a}^{m}×{a}^{n}={a}^{m+n}
\left(i\right) {2}^{3}×{2}^{4}={2}^{3+4}\left[\because {a}^{m}×{a}^{n}={a}^{m+n}\right]
\therefore {2}^{3}×{2}^{4}={2}^{7}
\left(ii\right) \left(x+y{\right)}^{2}×\left(x+y{\right)}^{3}=\left(x+y{\right)}^{2+3}
\left[\because {a}^{m}×{a}^{n}={a}^{m+n}\right]
\therefore \left(x+y{\right)}^{2}×\left(x+y{\right)}^{3}=\left(x+y{\right)}^{5}
II) Division property property :
{a}^{m}÷{a}^{n}={a}^{m–n}\left(m,n\in {z}^{+}\right)
For dividing the powers of same base, we subtract the indices.
{a}^{m}÷{a}^{n}=\frac{{a}^{m}}{{a}^{n}}=\frac{a×a×a×\dots ..\text{ to }m\text{ factors }}{a×a×a×\dots ..\text{ to }n\text{ factors }}
Case – 1 : If
m>n
\frac{{a}^{m}}{{a}^{n}}=a×a×a×a\dots \dots \left(m–n\right)\text{ factors }
\therefore \frac{{a}^{m}}{{a}^{n}}={a}^{m–n}
{3}^{5}÷{3}^{2}=\frac{{3}^{5}}{{3}^{2}}={3}^{5–2}={3}^{3}\left(\because \frac{{a}^{m}}{{a}^{n}}={a}^{m–n}\right)
m<n
\frac{{a}^{m}}{{a}^{n}}=\frac{1}{a×a×a×\dots \dots \left(n–m\right)\text{ factors }}
\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{\left(n–m\right)}}
{3}^{2}÷{3}^{5}=\frac{{3}^{2}}{{3}^{5}}=\frac{3×3}{3×3×3×3×3}=\frac{1}{3×3×3}
{3}^{2}÷{3}^{5}=\frac{1}{{3}^{3}}
III) Power of a power property :
{\left({a}^{m}\right)}^{n}={a}^{mn}\left(m,n\in {z}^{+}\right)
{\left({a}^{m}\right)}^{n}={a}^{m}×{a}^{m}×{a}^{m}×\dots \dots \text{ to n factors }={a}^{m+m+m+\dots \dots }\text{ to }n\text{ factors }
{\left({a}^{m}\right)}^{n}={a}^{mn}
{\left({5}^{3}\right)}^{2}={5}^{3×2}={5}^{6} \left(\because {\left({a}^{m}\right)}^{n}=\left({a}^{mn}\right)\right)
{\left({y}^{2}\right)}^{4}={y}^{2×4}={y}^{8}
\therefore {\left({y}^{2}\right)}^{4}={y}^{8}
IV) Power of a product property :
{a}^{m}·{b}^{m}=\left(ab{\right)}^{m} \left(m,n\in {z}^{+}\right)
{a}^{m}×{a}^{m}=\left(a×a×a×\dots ..\text{ to }m\text{ factors}\right)×\left(b×b×b×\dots ..\text{ to }m\text{ factors}\right)
=\left(ab\right)×\left(ab\right)×\left(ab\right)×\dots \dots \text{ to }m\text{ facors}
\therefore {a}^{m}×{b}^{m}=\left(ab{\right)}^{m}
{x}^{3}·{y}^{3}=\left(x×x×x\right)×\left(y×y×y\right)
=\left(xy\right)×\left(xy\right)×\left(xy\right)=\left(xy{\right)}^{3}
\therefore {x}^{3}.{y}^{3}=\left(xy{\right)}^{3}
V) Power of a divison property :
\frac{{a}^{m}}{{b}^{m}}={\left(\frac{a}{b}\right)}^{m} \left(m,n\in {z}^{+}\right)
\frac{{a}^{m}}{{b}^{m}}=\frac{a×a×a×\dots \text{ to }m\text{ factors }}{b×a×b×\dots \dots \text{ to }m\text{ factors }}=\frac{a}{b}×\frac{a}{b}×\frac{a}{b}×\dots ..\text{ to }m\text{ factors}
\therefore \frac{{a}^{m}}{{b}^{m}}={\left(\frac{a}{b}\right)}^{m}
all the above laws are defind for
\mathrm{m}, \mathrm{n}\in {\mathrm{z}}^{+}
12.3. NUMBERS WITH NON-INTEGER EXPONENTS
Numbers with non-integer exponents
What if m is not a positive integer ?
If m is not a positive integer, then there exist four cases.
Case-1 : either m = 0
Meaning of a° (a
\ne
We know the Fundamental Index law being true for all indices hence,
a°·{a}^{m}={a}^{\circ +m}={a}^{m}
Now Dividing both sides by
{a}^{m}
a°=\frac{{a}^{m}}{{a}^{m}}=1\left(\because a\ne 0\right)
Note : a° has no meaning when a = 0 i.e., 0° has no meaning.
Case-2 : Either
m=\frac{p}{q}
where p and q are positive integers.
{a}^{p/q}
, where p and q are positive integer
Since q is a positive integer, hence,
{\left({a}^{p/q}\right)}^{q}={a}^{p/q}·{a}^{p/q}·{a}^{p/q}\dots \dots ..\text{ to q factors }
={a}^{p/q+p/q+p/q\dots \text{ to }q\text{ factors }}={a}^{p/q·q}={a}^{p}
\therefore {a}^{p/q}=\sqrt[q]{{a}^{p}}
{a}^{p/q}
{q}^{th}
{a}^{p}
Case-3 : Either m is a negative number
{\mathrm{a}}^{–m}
, where m is a positive real number and
a\ne 0
Since the fundamental index law is true for all indices, hence,
{a}^{–m}·{a}^{m}={a}^{–m+m}=a°=1
\to {a}^{–m}=\frac{1}{{a}^{m}}\text{ and }\frac{1}{{a}^{–m}}={a}^{m}
{a}^{–m}
{a}^{m}
m=–\frac{p}{q}
where, p and q are positive integers.
{a}^{–p/q}=\frac{1}{{a}^{p/q}}=\frac{1}{\sqrt[q]{{a}^{p}}}
Equations and identities involving Indices
If a, m, n are three real numbers and
{a}^{m}={a}^{n},\text{ then }m=n\left(a\ne 0,1,\infty \right)
Proof : Since
{a}^{m}={a}^{n}\text{ and }a\ne 0,1,\infty \text{ hence, }\frac{{a}^{m}}{{a}^{n}}=1\text{ or }{a}^{m–n}=a°
\therefore m–n=0⇒m=n\text{ proved}
12.4. RATIONAL EXPONENT
I) Rational Number
\frac{p}{q}
where p and q are integers,
q\ne 0
is called a rational number.
II) Positive Rational Exponent
Let a be a positive real number and n a positve fraction equal to
\frac{p}{q},\text{ i.e., }n=\frac{p}{q}
where p and q are positive integers, the equation
{x}^{q}={a}^{p}
has one and only positive solution for x, given by
x={a}^{\frac{p}{q}}=\sqrt[q]{{a}^{p}}={q}^{th}\text{ root of }{a}^{p}
III) Negative Rational Exponent
If n is a negative rational, i.e.
n=–\frac{p}{q}
q\ne 0
and a is a positive real number, then
{a}^{n}={a}^{\frac{p}{q}}=\frac{1}{{a}^{\frac{p}{q}}}
{x}^{–\frac{p}{q}}=\frac{1}{{x}^{\frac{p}{q}}}\left[\because {x}^{–m}=\frac{1}{{x}^{m}}\right]={\left(\frac{1}{x}\right)}^{\frac{p}{q}}
{x}^{–\frac{p}{q}}\text{ is the reciprocal of }{x}^{\frac{p}{q}}
\text{Thus, if }x=\frac{r}{s}\left(r,s>0\right),\text{ then }{\left(\frac{r}{s}\right)}^{\frac{p}{q}}={\left(\frac{s}{r}\right)}^{\frac{p}{q}}
\therefore {\left(\frac{r}{s}\right)}^{–m}={\left(\frac{s}{r}\right)}^{m}\text{ where }m\in Q
IV) Also, all the laws of indices applicable for integral index are also applicable for rational index.
i.e., if a is a positive real number and m, n are rational numbers, then,
{a}^{m}×{a}^{n}={a}^{m+n}
{\left({a}^{m}\right)}^{n}={a}^{mn}
\left(ab{\right)}^{m}={a}^{m}{b}^{m}
12.5. RADICAL AND RADICAND
y>0\text{ and }{y}^{\frac{1}{q}}=x,
then we can write it as
x= \sqrt[q]{y}
\sqrt[q]{\mathrm{y}}
y is called ‘Radical form’ of
{y}^{\frac{1}{q}}
i) In
\sqrt[q]{y}, \sqrt{}
is called Radical Sign
\sqrt[q]{y}
is called a Radical
iii) q is called Index of the radical
iv) y is called the Radicand
Note : Index of a radical is always positive
\sqrt[n]{{a}^{n}}=a
\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}
\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}
\sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}
\sqrt[n]{{a}^{–m}}={a}^{–\frac{m}{n}}
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The Exact Rational Solutions to a Shallow Water Wave-Like Equation by Generalized Bilinear Method
The Exact Rational Solutions to a Shallow Water Wave-Like Equation by Generalized Bilinear Method ()
School of Information and Statistics, Guangxi University of Finance and Economics, Nanning, China.
A Shallow Water Wave-like nonlinear differential equation is considered by using the generalized bilinear equation with the generalized bilinear derivatives D3,x and D3,t, which possesses the same bilinear form as the standard shallow water wave bilinear equation. By symbolic computation, four presented classes of rational solutions contain all rational solutions to the resulting Shallow Water Wave-like equation, which generated from a search for polynomial solutions to the corresponding generalized bilinear equation.
Rational Solution, Generalized Bilinear Equation, Shallow Water Wave Equation
Wei, M. and Cai, J. (2017) The Exact Rational Solutions to a Shallow Water Wave-Like Equation by Generalized Bilinear Method. Journal of Applied Mathematics and Physics, 5, 715-721. doi: 10.4236/jamp.2017.53060.
In recent years, numerous scientists committed to the research of water waves, the shallow water wave can not only describe the freedom of the shallow surface under the gravitational influence of one-way transmission, but also produce on the bottom of the deep sea. The most important of the shallow water wave in the ocean is tsunami, which generated by the huge initial disturbance, such as earthquake, leading to the large wavelength and small height ocean wave. Therefore, the wave solutions, especially the rational solutions to nonlinear differential equations have attracted more and more attentions in the worldwide. What’s more, rogue wave solutions play an important role in rational solutions, which describe significant nonlinear wave phenomena in oceanography [1] . Wronskian formulation or the Casoratian formulation ( [2] - [6] ) usually deal with integrable equations to find their rational solutions in the literature, such as KdV, Boussinesq, KP, Schrodinger Toda equations and Shallow water wave equations [7] - [11] . In recent years, increasingly nonlinear differential equations are studied through the generalized bilinear equation with the generalized bilinear derivatives, for examples, KdV-like equation [12] , Boussinesq-like equation [13] and KP-like equation [14] .
Rational solutions to the non-integrable (3 + 1)-dimensional KP I [15] [16] and KPII [17] are also considered by different approaches. In particular, rational solutions to the (3 + 1)-dimensional KP II have been transformed into a problem of finding rational solutions to the good Boussinesq equation.
In this article, we introduce a Shallow Water Wave (SWW)-like nonlinear differential equation in terms of a generalized bilinear differential equation of Shallow Water Wave type using three generalized bilinear differential operators
{D}_{3,x}
{D}_{3,t}
. We will search for polynomial solutions to the corresponding generalized bilinear equation by Maple symbolic computation and generate four classes of rational solutions to the resulting Shallow Water Wave-like equation. Four particular rational solutions will be plotted to exhibit different distributions of singularities.
2. A SWW-Like Differential Equation
Let us consider a generalized bilinear differential equation of SWW type:
\begin{array}{l}\left({D}_{3,x}{D}_{3,t}-{D}_{3,t}{D}_{3,x}^{3}+{D}_{3,x}^{2}\right)f\cdot f\\ =2{f}_{xt}f-2{f}_{x}{f}_{t}-6{f}_{xx}{f}_{xt}+6{f}_{xxt}{f}_{x}+2{f}_{xx}f-2{f}_{x}^{2}=0.\end{array}
This is the same type bilinear equation as the SWW equation [10] [18] [19]
\left({D}_{x}{D}_{t}-{D}_{t}{D}_{x}^{3}+{D}_{x}^{2}\right)f\cdot f=0,
with the corresponding nonlinear differential equation
{u}_{t}-{u}_{xxt}-3u{u}_{t}+3{u}_{x}{\int }_{x}^{\infty }{u}_{t}\text{d}x+{u}_{x}=0.
The above differential operators are some kind of generalized bilinear differential operators introduced in [20] [21] :
\begin{array}{c}{D}_{p,x}^{m}{D}_{p,t}^{n}f\cdot f\\ ={\left(\frac{\partial }{\partial x}+{\alpha }_{p}\frac{\partial }{\partial {x}^{\prime }}\right)}^{m}{\left(\frac{\partial }{\partial t}+{\alpha }_{p}\frac{\partial }{\partial {t}^{\prime }}\right)}^{n}{f\left(x,t\right)f\left({x}^{\prime },{t}^{\prime }\right)|}_{{x}^{\prime }=x,{t}^{\prime }=t}\\ =\underset{i=0}{\overset{m}{\sum }}\underset{j=0}{\overset{n}{\sum }}\left(\begin{array}{c}m\\ i\end{array}\right)\left(\begin{array}{c}n\\ j\end{array}\right){\alpha }_{p}^{i}{\alpha }_{p}^{j}\frac{{\partial }^{m-i}}{\partial {x}^{m-i}}\frac{{\partial }^{i}}{\partial {{x}^{\prime }}^{\left(i\right)}}\frac{{\partial }^{n-j}}{\partial {t}^{n-j}}\frac{{\partial }^{j}}{\partial {{t}^{\prime }}^{\left(j\right)}}{f\left(x,t\right)f\left({x}^{\prime },{t}^{\prime }\right)|}_{{x}^{\prime }=x,{t}^{\prime }=t}\\ =\underset{i=0}{\overset{m}{\sum }}\underset{j=0}{\overset{n}{\sum }}\left(\begin{array}{c}m\\ i\end{array}\right)\left(\begin{array}{c}n\\ j\end{array}\right){\alpha }_{p}^{i}{\alpha }_{p}^{j}\frac{{\partial }^{m+n-i-j}f\left(x,t\right)}{\partial {x}^{m-i}\partial {t}^{n-j}}\frac{{\partial }^{i+j}f\left(x,t\right)}{\partial {x}^{i}\partial {t}^{j}},\text{}m,n\ge 0.\end{array}
{\alpha }_{p}^{s}
{\alpha }_{p}^{s}={\left(-1\right)}^{{r}_{p}\left(s\right)},s={r}_{p}\left(s\right)\mathrm{mod}p,
it is necessary to point out that
{\alpha }_{p}^{i}{\alpha }_{p}^{j}\ne {\alpha }_{p}^{i+j},\text{}i,j\ge 0.
p=3
{\alpha }_{3}=-1,\text{}{\alpha }_{3}^{2}=1,\text{}{\alpha }_{3}^{3}=1,\text{}{\alpha }_{3}^{4}=-1,\text{}{\alpha }_{3}^{5}=1,\text{}{\alpha }_{3}^{6}=1.
\begin{array}{l}{D}_{3,x}{D}_{3,t}f\cdot f=2{f}_{xt}f-2{f}_{x}{f}_{t},\\ {D}_{3,t}{D}_{3,x}^{3}f\cdot f=-6{f}_{xx}{f}_{xt}+6{f}_{xxt}{f}_{x},\\ {D}_{3,x}^{2}f\cdot f=2{f}_{xx}f-2{f}_{x}^{2}.\end{array}
p=2
, which is the Hirota case, the following equations are true:
\begin{array}{l}{D}_{2,x}{D}_{2,t}f\cdot f=2{f}_{xt}f-2{f}_{x}{f}_{t},\\ {D}_{2,x}^{2}f\cdot f=2{f}_{xx}f-2{f}_{x}^{2},\\ {D}_{2,t}{D}_{2,x}^{3}f\cdot f=2{f}_{xxxt}f+6{f}_{xx}{f}_{xt}-2{f}_{xxx}{f}_{t},\end{array}
which generates the standard bilinear SWW equation [18] .
Motivated by the introduction on a general Bell polynomial theory [20] , a dependent variable transformation is adopted:
u=2{\left(\mathrm{ln}f\right)}_{x},
and then can directly show that the generalized bilinear Equation (2.1) is linked to a SWW-like scalar nonlinear differential equation
{u}_{t}+{u}_{x}+\frac{3}{2}u{u}_{xt}-\frac{3}{2}{u}_{x}{u}_{t}+\frac{1}{2}{u}^{2}{u}_{t}=0,
from the generalized bilinear Equation (2.1). Through the transformation (2.6), the following equality can be deduced:
\frac{\left({D}_{3,x}{D}_{3,t}-{D}_{3,t}{D}_{3,x}^{3}\right)f\cdot f}{{f}^{2}}={u}_{t}+{u}_{x}+\frac{3}{2}u{u}_{xt}-\frac{3}{2}{u}_{x}u{}_{t}+\frac{1}{2}{u}^{2}{u}_{t},
and thus, f solves (2.1) if and only if
u=2{\left(\mathrm{ln}f\right)}_{x}
presents a solution to the SWW-like Equation (2.7).
In [21] [22] , transcendental functions: exponential functions and trigonometric functions have been considered to find the resonant solutions for generalized bilinear equations. In the next section, we would like to consider finding the rational solutions for SWW-like Equation (2.7), which generated from polynomial solutions.
3. Rational Solutions
By symbolic computation with Maple, we look for polynomial solutions, with degree of
x
t
being less than 3:
f=\underset{i=0}{\overset{3}{\sum }}\underset{j=0}{\overset{3}{\sum }}{c}_{i,j}{x}^{i}{x}^{j},
{c}_{i,j}
’s are constants, and present 4 classes of polynomial solutions to the generalized bilinear Equation (2.1), based on the powers of
x
, the solutions could be divided into three categories: cubic polynomials, linear polynomials and the trivial solutions. Besides the trivial solutions, those solutions, in turn, lead to four classes of rational solutions to the SWW-like (2.5) through the transformation (2.4). We list those classes of rational solutions as follows. The first class of rational solutions to (2.5) is
{u}_{1}=\frac{2p}{q},
\begin{array}{c}p={c}_{10}{c}_{12}+{c}_{11}{c}_{12}t+{c}_{12}^{2}{t}^{2},\\ q={c}_{10}{c}_{12}x+{c}_{11}{c}_{12}xt+{c}_{12}^{2}x{t}^{2}+{c}_{10}\left({c}_{11}+{c}_{02}\right)\\ \text{}+\left({c}_{11}{c}_{02}-{c}_{10}{c}_{12}+{c}_{11}^{2}\right)t+{c}_{02}{c}_{12}{t}^{2}-{c}_{12}^{2}{t}^{3}.\end{array}
The second class of rational solutions to (2.5) is
{u}_{2}=\frac{2\left({c}_{11}{c}_{10}+{c}_{11}^{2}t\right)}{{c}_{11}{c}_{10}x+{c}_{11}^{2}xt+{c}_{10}\left({c}_{01}+{c}_{10}\right)+{c}_{01}{c}_{11}t-{c}_{11}^{2}{t}^{2}},
The third class of rational solutions to (2.5) is
{u}_{3}=\frac{6p}{q},
\begin{array}{c}p={c}_{20}^{2}{x}^{2}+2{c}_{10}{c}_{20}x-2{c}_{20}^{2}xt+{c}_{10}^{2}-2{c}_{20}{c}_{10}t+{c}_{20}^{2}{t}^{2},\\ p={c}_{20}^{2}{x}^{3}+3{c}_{10}{c}_{20}{x}^{2}-3{c}_{20}^{2}{x}^{2}t+3{c}_{10}^{2}x-6{c}_{20}{c}_{10}xt+3{c}_{20}^{2}x{t}^{2}\\ \text{}-3\left({c}_{10}^{2}-6{c}_{20}^{2}\right)t+3{c}_{20}{c}_{10}{t}^{2}-{c}_{20}^{2}{t}^{3}+3{c}_{00}{c}_{10}.\end{array}
The fourth class of rational solutions to (2.5) is
{u}_{4}=\frac{36p}{q},
\begin{array}{c}p=3{c}_{01}{x}^{2}-6{c}_{01}xt+6{c}_{10}x{t}^{2},\\ q={c}_{01}{x}^{3}-3{c}_{01}{x}^{2}t+3{c}_{01}{x}^{2}{t}^{2}+18{c}_{01}t-{c}_{01}{t}^{3}+18{c}_{00}.\end{array}
Figure 1 will present the profiles of all rational solutions of (2.5) with
{c}_{i,j}={i}^{2}+{j}^{2}
We took a SWW-like nonlinear differential equation into consideration by the generalized bilinear equation of SWW type. Furthermore, we constructed two classes of rational solutions to the resulting SWW-like equation. A kind of generalized bilinear differential operators, which introduced in [20] [21] is the key instrument. It is an interesting work to search if there exists any Wronskian solutions and multiple type solutions to the SWW-like nonlinear Equation (2.5). At the same time, a speculation is raised, in which the four classes of rational solutions in Equations (3.2)-(3.5) would contain all rational solutions to the SWW-like nonlinear Equation (2.5), generated from polynomial solutions to the generalized bilinear Equation (2.1) with the transformation (2.4).
It is parallel to the discussion, a kind of generalized tri-linear differential equ-
Figure 1. Pictures of the all rational solutions of (2.5) with
{c}_{i,j}={i}^{2}+{j}^{2}
ations and their resonant solutions was considered in [23] . Their rational solutions, which include singular and non-singular wave solutions, rogue and higher-order rogue wave solutions will be a very interesting topic in generalized tri-linear differential equations in the future.
This work is supported by Guangxi College Enhancing Youths Capacity Project (KY2016LX315) and Guangxi University of Finance and Economics Youth Progresss Project (2016QNB22).
[1] Muller, P., Garrett, C., Osborne, A. and Waves, R. (2005) Oceanography, 18 66-75.
[2] Ma, W.X. and You, Y. (2005) Solving the Korteweg-De Vries Equation by Its Bilinear Form: Wronskian Solutions. Transactions of the American Mathematical Society, 357, 1753-1778.
[3] Zhang, Y.J. (1996) Wronskian-Type Solutions for the Vector K-Constrained KP Hrerarchy. Journal of Physics A, 29, 2617-2626.
[4] Hirota, R., Ohta, Y. and Satsuma, J. (1988) Wronskian Structures for Soliton Equations. Progress of Theoretical Physics Supplement, 94, 59-72.
https://doi.org/10.1143/PTPS.94.59
[5] Matveev, V.B. (1992) Genernalized Wronskian Formula for Solutions of the KdV Equations: First Applications. Physics Letters A, 166, 205-208.
[6] Clarkson, P.A. and Mansfield, E.L. (1994) On a Shallow Water Wave Equation. Nonlinearity, 7, 975.
[7] Nimmo, J.C. and Freeman, N.C. (1983) Rational Solutions of the Korteweg-De Vries Equation in Wronskian Form. Applied Physics Letters, 96, 443-446.
[8] Yuasa, F. (1987) Backlund Transformation of the Two-Dimensional Tada Lattice and Casorati’s Determinants. Journal of the Physical Society, 56, 423-424.
[9] Hirota, R. (1986) Soliton of the Classical Boussinesq Equation and the SPH Boussinesq Equation: The Wronskian Technique. Journal of the Physical Society, 55 2137-2150.
[10] Chen, D.Y. (2006) Introduction to Solitons. In: Liu, Y.Q., Hu, C. and Dai, J., Eds., Journal of Applied Mathematics and Physics. Science Press, Beijing.
[11] Bagchi, B.S. and Das, A. (2010) Ganguly, New Exact Solutions of a Generalized Shallow Water Wave Equation. Physica Scripta, 82, Article ID: 025003.
[12] Zhang, Y. and Ma, W.X. (2015) Rational Solutions to a KdV-Like Equation. Applied Mathematics and Computation, 256, 252-256.
[13] Shi, C.G. and Zhao, B.Z. and Ma, W.X. (2015) Exact Rational Solutions to a Boussinesq-Like Equation in (1+1)-Dimensions. Applied Mathematics Letters, 48, 170-176.
[14] Zhang, Y.F. and Ma, W.X. (2015) A Study on Rational Solutions to a KP-Like Equation. Zeitschrift für Naturforschung A, 70, 263-268.
[15] Sinelshchikov, D.I. (2010) Comment on: New Exact Traveling Wave Solutions of the (3+1)-Dimensional Kadomtsev-Petviashvili (KP) Equation. Communications in Nonlinear Science and Numerical Simulation, 15, 3235-3236.
[16] Khalfallah, M. (2009) New Exact Traveling Wave Solutions of the (3+1) Dimensional Kadomtsev-Petviashvili (KP) Communications in Nonlinear Science and Numerical Simulation, 14, 1169-1175.
[17] Ma, W.X. (2011) Comment on the 3+1 Dimensional Kadomtsev-Petviashvili Equations. Communications in Nonlinear Science and Numerical Simulation, 16, 2663-2666.
[19] Guo, Y.C. (2008) An Introduction to Nonlinear Partial Differential Equation, Tsinghua University Press, Beijing.
[20] Ma, W.X. (2011) Generalized Bilinear Differential Equations. Journal of Nonlinear Science, 2, 140-144.
[21] Ma, W.X. (2013) Bilinear Equations and Resonant Solutions Characterized by Bell Polynomials. Reports on Mathematical Physics, 72, 41-56.
[22] Zheng, H.C., Ma, W.X. and Gu, X. (2013) Hirota Bilinear Equations with Linear Subspaces of Hyperbolic and Trigonometric Function Solutions. Applied Mathematics and Computation, 220, 226-234.
[23] Ma, W.X. (2013) Trilinear Equations, Bell Polynomials, and Resonant Solutions. Frontiers of Mathematics in China, 8, 1139-1156.
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Budget set - WikiMili, The Best Wikipedia Reader
All bundles a consumer can afford based on various conditions
{\displaystyle k}
{\displaystyle \mathbf {x} =\left[x_{1},x_{2},\ldots ,x_{k}\right]}
, also known as consumption plans which should not exceed the income, [1] with associated prices
{\displaystyle \mathbf {p} =\left[p_{1},p_{2},\ldots ,p_{k}\right]}
{\displaystyle m}
{\displaystyle B_{\mathbf {p} ,m}=\left\{\mathbf {x} \in X:\mathbf {p} \mathbf {x} \leq m\right\}}
{\displaystyle X=\mathbb {R} _{+}^{k}}
{\displaystyle \mathbf {p} >>0}
{\displaystyle m\in \mathbb {R} _{+}}
{\displaystyle B}
{\displaystyle k}
{\displaystyle \mathbf {p} \mathbf {x} =m}
{\displaystyle \mathbb {R} _{+}^{k}}
Other sources of wealth, including stocks, savings, pensions, profit shares, etc., are not included in the income described above. The income described above are also known as initial wealth. [1]
{\displaystyle \phi (p,m)}
is the set that the consumer chooses to go with based on the preferences from the budget set. [1]
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one or its length may represent the curvature of the object ; its algebraic sign may indicate sides.
In mathematics, a linear form is a linear map from a vector space to its field of scalars.
In economics, a budget constraint represents all the combinations of goods and services that a consumer may purchase given current prices within his or her given income. Consumer theory uses the concepts of a budget constraint and a preference map as tools to examine the parameters of consumer choices. Both concepts have a ready graphical representation in the two-good case. The consumer can only purchase as much as their income will allow, hence they are constrained by their budget. The equation of a budget constraint is where P_x is the price of good X, and P_y is the price of good Y, and m = income.
The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank and its nullity.
In economics, the marginal rate of substitution (MRS) is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of utility. At equilibrium consumption levels, marginal rates of substitution are identical. The marginal rate of substitution is one of the three factors from marginal productivity, the others being marginal rates of transformation and marginal productivity of a factor.
In mathematics, more specifically in multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function.
In economics, nominal value is measured in terms of money, whereas real value is measured against goods or services. A real value is one which has been adjusted for inflation, enabling comparison of quantities as if the prices of goods had not changed on average. Changes in value in real terms therefore exclude the effect of inflation. In contrast with a real value, a nominal value has not been adjusted for inflation, and so changes in nominal value reflect at least in part the effect of inflation.
In microeconomics, a consumer's Marshallian demand function is the quantity he/she demands of a particular good as a function of its price, his/her income, and the prices of other goods, a more technical exposition of the standard demand function. It is a solution to the utility maximization problem of how the consumer can maximize his/her utility for given income and prices. A synonymous term is uncompensated demand function, because when the price rises the consumer is not compensated with higher nominal income for the fall in his/her real income, unlike in the Hicksian demand function. Thus the change in quantity demanded is a combination of a substitution effect and a wealth effect. Although Marshallian demand is in the context of partial equilibrium theory, it is sometimes called Walrasian demand as used in general equilibrium theory.
There are two fundamental theorems of welfare economics. The first states that in economic equilibrium, a set of complete markets, with complete information, and in perfect competition, will be Pareto optimal. The requirements for perfect competition are these:
There are no externalities and each actor has perfect information.
Firms and consumers take prices as given.
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets. Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard.
In microeconomics, a consumer's Hicksian demand function or compensated demand function for a good is his quantity demanded as part of the solution to minimizing his expenditure on all goods while delivering a fixed level of utility. Essentially, a Hicksian demand function shows how an economic agent would react to the change in the price of a good, if the agent's income was compensated to guarantee the agent the same utility previous to the change in the price of the good—the agent will remain on the same indifference curve before and after the change in the price of the good. The function is named after John Hicks.
In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of -dimensional subspaces of , given a point in the Grassmannian corresponding to a -dimensional vector subspace , the fiber over is the subspace itself. In the case of projective space the tautological bundle is known as the tautological line bundle.
In economics and consumer theory, quasilinear utility functions are linear in one argument, generally the numeraire. Quasilinear preferences can be represented by the utility function where is strictly concave. A useful property of the quasilinear utility function is that the Marshallian/Walrasian demand for does not depend on wealth and is thus not subject to a wealth effect; The absence of a wealth effect simplifies analysis and makes quasilinear utility functions a common choice for modelling. Furthermore, when utility is quasilinear, compensating variation (CV), equivalent variation (EV), and consumer surplus are algebraically equivalent. In mechanism design, quasilinear utility ensures that agents can compensate each other with side payments.
Projective space plays a central role in algebraic geometry. The aim of this article is to define the notion in terms of abstract algebraic geometry and to describe some basic uses of projective space.
In theoretical economics, an abstract economy is a model that generalizes both the standard model of an exchange economy in microeconomics, and the standard model of a game in game theory. An equilibrium in an abstract economy generalizes both a Walrasian equilibrium in microeconomics, and a Nash equilibrium in game-theory.
1 2 3 Böhm, Volker; Haller, Hans (2017), "Demand Theory", The New Palgrave Dictionary of Economics, London: Palgrave Macmillan UK, pp. 1–14, doi:10.1057/978-1-349-95121-5_539-2, ISBN 978-1-349-95121-5 , retrieved 2021-12-09
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Impulse_(physics) Knowpia
In classical mechanics, impulse (symbolized by J or Imp) is the integral of a force, F, over the time interval, t, for which it acts. Since force is a vector quantity, impulse is also a vector quantity. Impulse applied to an object produces an equivalent vector change in its linear momentum, also in the resultant direction. The SI unit of impulse is the newton second (N⋅s), and the dimensionally equivalent unit of momentum is the kilogram meter per second (kg⋅m/s). The corresponding English engineering unit is the pound-second (lbf⋅s), and in the British Gravitational System, the unit is the slug-foot per second (slug⋅ft/s).
newton-second (N⋅s) (kg⋅m/s in SI base units)
pound⋅s
{\displaystyle {\mathsf {L}}{\mathsf {M}}{\mathsf {T}}^{-1}}
A resultant force causes acceleration and a change in the velocity of the body for as long as it acts. A resultant force applied over a longer time, therefore, produces a bigger change in linear momentum than the same force applied briefly: the change in momentum is equal to the product of the average force and duration. Conversely, a small force applied for a long time produces the same change in momentum—the same impulse—as a larger force applied briefly.
{\displaystyle J=F_{\text{average}}(t_{2}-t_{1})}
The impulse is the integral of the resultant force (F) with respect to time:
{\displaystyle J=\int F\,\mathrm {d} t}
Mathematical derivation in the case of an object of constant massEdit
The impulse delivered by the "sad" ball is mv0, where v0 is the speed upon impact. To the extent that it bounces back with speed v0, the "happy" ball delivers an impulse of mΔv=2mv0.[1]
Impulse J produced from time t1 to t2 is defined to be[2]
{\displaystyle \mathbf {J} =\int _{t_{1}}^{t_{2}}\mathbf {F} \,\mathrm {d} t}
where F is the resultant force applied from t1 to t2.
From Newton's second law, force is related to momentum p by
{\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}}
{\displaystyle {\begin{aligned}\mathbf {J} &=\int _{t_{1}}^{t_{2}}{\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}\,\mathrm {d} t\\&=\int _{\mathbf {p} _{1}}^{\mathbf {p} _{2}}\mathrm {d} \mathbf {p} \\&=\mathbf {p} _{2}-\mathbf {p} _{1}=\Delta \mathbf {p} \end{aligned}}}
where Δp is the change in linear momentum from time t1 to t2. This is often called the impulse-momentum theorem[3] (analogous to the work-energy theorem).
As a result, an impulse may also be regarded as the change in momentum of an object to which a resultant force is applied. The impulse may be expressed in a simpler form when the mass is constant:
{\displaystyle \mathbf {J} =\int _{t_{1}}^{t_{2}}\mathbf {F} \,\mathrm {d} t=\Delta \mathbf {p} =m\mathbf {v_{2}} -m\mathbf {v_{1}} }
A large force applied for a very short duration, such as a golf shot, is often described as the club giving the ball an impulse.
F is the resultant force applied,
t1 and t2 are times when the impulse begins and ends, respectively,
v2 is the final velocity of the object at the end of the time interval, and
v1 is the initial velocity of the object when the time interval begins.
Impulse has the same units and dimensions (M L T−1) as momentum. In the International System of Units, these are kg⋅m/s = N⋅s. In English engineering units, they are slug⋅ft/s = lbf⋅s.
The term “impulse” is also used to refer to a fast-acting force or impact. This type of impulse is often idealized so that the change in momentum produced by the force happens with no change in time. This sort of change is a step change, and is not physically possible. However, this is a useful model for computing the effects of ideal collisions (such as in game physics engines). Additionally, in rocketry, the term “total impulse” is commonly used and is considered synonymous with the term “impulse”.
Variable massEdit
The application of Newton's second law for variable mass allows impulse and momentum to be used as analysis tools for jet- or rocket-propelled vehicles. In the case of rockets, the impulse imparted can be normalized by unit of propellant expended, to create a performance parameter, specific impulse. This fact can be used to derive the Tsiolkovsky rocket equation, which relates the vehicle's propulsive change in velocity to the engine's specific impulse (or nozzle exhaust velocity) and the vehicle's propellant-mass ratio.
Wave–particle duality defines the impulse of a wave collision. The preservation of momentum in the collision is then called phase matching. Applications include:
Electron phonon scattering
Dirac delta function, mathematical abstraction of a pure impulse
^ Property Differences In Polymers: Happy/Sad Balls
^ Hibbeler, Russell C. (2010). Engineering Mechanics (12th ed.). Pearson Prentice Hall. p. 222. ISBN 978-0-13-607791-6.
^ See, for example, section 9.2, page 257, of Serway (2004).
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Experimental and Numerical Study of Turbulent Heat Transfer in Twisted Square Ducts | J. Heat Transfer | ASME Digital Collection
Liang-Bi Wang,
School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, PRC
Wen-Quan Tao,
Qiu-Wang Wang,
Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received by the Heat Transfer Division October 3, 2000; revision received March 23, 2001. Associate Editor: M. Faghri.
J. Heat Transfer. Oct 2001, 123(5): 868-877 (10 pages)
Wang, L., Tao , W., Wang , Q., and He, Y. (March 23, 2001). "Experimental and Numerical Study of Turbulent Heat Transfer in Twisted Square Ducts ." ASME. J. Heat Transfer. October 2001; 123(5): 868–877. https://doi.org/10.1115/1.1389464
This paper describes the experimental and numerical study of three mildly twisted square ducts (twisted uniform cross section square duct, twisted divergent square duct and twisted convergent square duct). Experiments are conducted for air with uniform heat flux condition. Measurements are also conducted for a straight untwisted square duct for comparison purpose. Numerical simulations are performed for three-dimensional and fully elliptic flow and heat transfer by using a body-fitted finite volume method and standard
k
−ε turbulence model. Both experimental and numerical results show that the twisting brings about a special variation pattern of the spanwise distribution of the local heat transfer coefficient, while the divergent and convergent shapes lead to different axial local heat transfer distributions. Based on the test data, the thermal performance comparisons are made under three constraints (identical mass flow rate, identical pumping power and identical pressure drop) with straight untwisted square duct as a reference. Comparisons show that the twisted divergent duct can always enhance heat transfer, the twisted convergent duct always deteriorates heat transfer, and the twisted constant cross section duct is somewhat in between.
pipe flow, heat transfer, flow simulation, turbulence, finite volume methods, Channel Flow, Enhancement, Experimental, Heat Transfer, Numerical Methods
Ducts, Flow (Dynamics), Heat transfer, Pressure drop, Heat transfer coefficients, Turbulence
Heat Transfer and Pressure Drop Correction for Twisted-Tape Inserts in Isothermal Tubes: Part I—Laminar Flow
Heat Transfer and Pressure Drop Correction for Twisted-Tape Inserts in Isothermal Tubes: Part II—Turbulent Flow
Yampolsky, J. S., 1983, “Spirally Fluted Tubing for Enhanced Heat Transfer,” Heat Exchangers—Theory and Practice, J. Taborek, G. F. Hewitte, G., Afgan, eds. Hemisphere, Washington, D. C., pp. 945–952.
Obot, N. T., Esen, E. B., Snell, K. H., Rabas, T. J., 1991, “Pressure Drop and Heat Transfer for Spirally Fluted Tubes Including Validation of the Roles of Transition,” in ASME HTD-164, T. J. Rabas, J. M. Chenoweth, eds., New York, pp. 85–92.
Enhancement of Heat Transfer With Swirling Flows Issued Into a Divergent Pipe
Augmented Heat Transfer in Square Channels With Parallel, Crosses, and V-Shaped Angled Ribs
Some Comments on Steady, Laminar Flow Through Twisted Pipes
Discussion—Steady Laminar Flow Through Twisted Pipes: Fluid Flow in Square Tubes and Steady Laminar Flow Through Twisted Pipes: Heat Transfer in Square Tubes
Some Remarks on the Helical-Cartesian Coordinates System and Its Applications
Miller, R. W., 1996, Flow Measurement Engineering Handbook, 3rd ed., McGraw-Hill, New York.
Wang, L. B., 1996, “Experimental and Numerical Study of Turbulent Fluid Flow and Heat Transfer in Sectionally Complex and Twisted Ducts,” Ph.D. thesis, Xi’an Jiaotong University, Xi’an, China.
Comput. Methods in Applied Mechanics and Engineering
Tao, W. Q., 1988, Numerical Heat Transfer, Xi’an Jiaotong University Press, Xi’an, China.
Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Co., New York, pp. 95–96.
A Multi-Surface Method of Coordinate Generation
Coordinate Generation With Precise Controls of Mesh Properties
High Level Continuity for Coordinate Generation With Precise Controls
Analysis of Fluid Flow in Constructed Tubes and Ducts Using Body-Fitted Non-Staggered Grids
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Power transmission system with taut belt connecting two pulleys - MATLAB - MathWorks India
Pulley initial velocity
Pulley B
Pulley A initial state
Pulley B initial state
Wrap angle calculation
Pulley center separation
Pulley A wrap angle
Pulley B wrap angle
Power transmission system with taut belt connecting two pulleys
The Belt Drive block represents a pair of pulleys connected with a flexible ideal, flat, or V-shaped belt. When you set Belt type to Ideal - No slip, the belt does not slip relative to the pulley surfaces.
The block accounts for friction between the flexible belt and the pulley periphery. If the friction force is not sufficient to drive the load, the block allows slip. The relationship between the tensions in the driving and driven branches conforms to the capstan equation, also known as the Euler-Eytelwein equation. The block accounts for centrifugal loading in the flexible belt, pulley inertia, and bearing friction.
The Belt Drive block is a structural component based on the Simscape™ Driveline™ Belt Pulley block and Simscape Translational Spring and Translational Damper blocks.
The Translational Spring and Translational Damper blocks simulate the compliance of the belt. For the equations governing the contact dynamics between the belt and the pulley, see the Belt Pulley block. The figure shows the functional block diagram for the Belt Drive block.
To learn more about the construction of the block, see About Composite Components and the reference pages for the individual component blocks.
The diagrams show the open and crossed belt drive configurations. When you set Drive type to Open belt, both pulleys tend to rotate in the same direction and the larger pulley has a larger belt wrap angle. When you set Drive type to Crossed belt, the pulleys tend to rotate in opposite directions and have the same wrap angle.
Belt Drive Diagrams
The figures and equations refer to these quantities:
θA is the wrap angle of pulley A.
θB is the wrap angle of pulley B.
RA is the effective radius of pulley A.
RB is the effective radius of pulley B.
C is the distance between the centers of pulleys A and B.
When you set Drive type to Open belt, the block calculates the wrap angle of the belt around each pulley as:
\begin{array}{l}{\theta }_{A}=\pi +2\ast {\mathrm{sin}}^{-1}\frac{{R}_{A}-{R}_{B}}{C},\\ {\theta }_{B}=\pi -2\ast {\mathrm{sin}}^{-1}\frac{{R}_{A}-{R}_{B}}{C}\end{array}
The diagram shows the wrap angles and parameters.
When you set Drive type to Crossed belt, the two wrap angles are equal and the wrap angle of the belt around each pulley:
{\theta }_{A}={\theta }_{B}=\pi +2\ast {\mathrm{sin}}^{-1}\frac{{R}_{A}+{R}_{B}}{C}.
The pulleys do not translate.
The friction coefficient and friction velocity threshold between the belt and each of the pulleys is the same. To parameterize separate friction interactions, use two Belt Pulley blocks.
A — Pulley A shaft
Mechanical rotational conserving port associated with the shaft of pulley A.
B — Pulley B shaft
Mechanical rotational conserving port associated with the shaft of pulley B.
Drive type — Belt drive configuration
Open belt (default) | Crossed belt
Belt drive configuration. For more information, see Belt Drive Diagrams.
Belt type selection. The belt type affects slip conditions
Ideal - No slip — Parameterize an ideal belt, which does not slip relative to the pulley.
Compliance — Compliance option
Option to simulate compliance.
No compliance - Suitable for HIL simulation — Simulates a noncompliant belt that does not yield elastically when subjected to a force. To prioritize performance, select this option.
Specify stiffness and damping — Simulates a compliant belt that does yield elastically when subjected to a force. To prioritize fidelity, select this option.
To enable this parameter, set Belt type to Ideal - No slip.
30 deg (default) | nonnegative scalar
The block rounds noninteger values to the nearest integer. Increasing the number of belts increases the friction force, effective mass per unit length, and maximum allowable tension.
Centrifugal force contribution in terms of linear density.
Longitudinal stiffness — Belt effective stiffness
1e4 N/m (default) | nonnegative scalar
Effective stiffness of the belt.
To enable this parameter, set Belt type to Ideal - No slip and Compliance to Specify stiffness and damping or set Belt type to Flat belt or V-belt.
Longitudinal damping — Belt effective damping
Effective damping of the belt.
To enable this parameter, either set
Belt type to Ideal - No slip and Compliance to Specify stiffness and damping, or
Belt type to Flat belt or V-belt
Pre-tension — Resting belt tension
Tension in the belt when the belt and pulleys are at rest. The value must be positive.
Option to specify a maximum tension. If you set this parameter to Specify maximum tension and the belt tension on either end of the belt meets or exceeds the value that you specify for the Belt maximum tension parameter, the simulation stops and generates an assertion error.
Belt type to Ideal - No slip, Compliance to Specify stiffness and damping, and Maximum tension to Specify maximum tension, or
Belt type to Flat belt or V-belt and Maximum tension to Specify maximum tension
Option to parameterize rotational inertia with an initial velocity.
Pulley initial velocity — Initial pulley rotational velocity
Pulley A initial state — Initial state of pulley A
Option to initialize the simulation with pulley A locked or unlocked.
Pulley B initial state — Initial state of pulley B
Option to initialize the simulation with pulley B locked or unlocked.
Wrap angle calculation — Angle calculation type
Specify pulley center separation (default) | Specify wrap angles
Option to calculate the wrap angle by the pulley center separation or to specify those values directly.
Specify pulley center separation — The block calculates the wrap angle of the belt on the pulleys by using the pulley radii, center separation, and drive type.
Specify wrap angles — The block uses the wrap angles that you specify.
Pulley center separation — Pulley center separation distance
Distance between the centers of the pulleys.
To enable this parameter, set Wrap angle calculation to Specify pulley center separation.
Pulley A wrap angle — Pulley wrap angle
145 deg (default) | scalar
Angle of contact between the belt and pulley attached to port A.
To enable this parameter, set Wrap angle calculation to Specify wrap angles.
Pulley B wrap angle — Pulley wrap angle
Angle of contact between the belt and pulley attached to port B.
Velocity threshold — Contact threshold
Belt Pulley | Chain Drive
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Adhesin-Specific Nanomechanical Cantilever Biosensors for Detection of Microorganisms | J. Heat Transfer | ASME Digital Collection
Tzuen-Rong J. Tzeng,
Tzuen-Rong J. Tzeng
, 132 Long Hall, Clemson, SC 29634-0314
e-mail: tzuenrt@clemson.edu
Yunyan R. Cheng,
Yunyan R. Cheng
e-mail: ycheng@clemson.edu
Reza Saeidpourazar,
Reza Saeidpourazar
e-mail: rezas@illinois.edu
Siddharth S. Aphale,
Siddharth S. Aphale
Piezoactive Systems Laboratory, Department of Mechanical and Industrial Engineering,
Tzeng, T. J., Cheng, Y. R., Saeidpourazar, R., Aphale, S. S., and Jalili, N. (September 30, 2010). "Adhesin-Specific Nanomechanical Cantilever Biosensors for Detection of Microorganisms." ASME. J. Heat Transfer. January 2011; 133(1): 011012. https://doi.org/10.1115/1.4002363
Lectins (adhesins) on bacterial surfaces play important roles in infection by mediating bacterial adherence to host cell surfaces via their cognate receptors. We have explored the use of
α
-D-mannose receptors as capturing agents for the detection of Escherichia coli using a microcantilever and have demonstrated that E. coli ORN178, which expresses normal type-1 pili, can interact with microcantilevers functionalized with
α
-D-mannose and can cause shifts in its resonance frequencies. Although E. coli ORN208, which expresses abnormal pili, binds poorly to
α
-D-mannose on the nitrocellulose membrane of a FAST slide, it did cause a detectable shift in resonance frequency when interacting with the
α
-D-mannose functionalized microcantilevers.
biosensors, cantilevers, microorganisms, molecular biophysics
Biosensors, Cantilevers, Microcantilevers, Microorganisms, Resonance, Membranes
Development of a Microcantilever-Based Pathogen Detector
B. N. Z.
Applications of Microarrays in Pathogen Detection and Biodefence
Genomics, Gene Expression and DNA Arrays
Sensing Interactions Between Vimentin Antibodies and Antigens for Early Cancer Detection
Stability and Sensitivity Analysis of Periodic Orbits in Tapping Mode Atomic Force Microscopy
Harmonic Analysis Based Modeling of Tapping-Mode AFM
Processing of Kinetic Microarray Signals
Measuring the Intrinsic Nanomechanics of Molecular Interactions With Microcantilever Sensors
Eur. J. Nanomedicine
Carbohydrates as Future Anti-Adhesion Drugs for Infectious Diseases
Karamanska
Bacterial Detection Using Carbohydrate-Functionalised CdS Quantum Dots: A Model Study Exploiting E. coli Recognition of Mannosides
Synthesis of Highly Water-Soluble Fluorescent Conjugated Glycopoly(p -phenylene)s for Lectin and Escherichia coli
Control of Bacterial Aggregation by Thermoresponsive Glycopolymers
Fabrication of Carbohydrate Microarrays on Gold Surfaces: Direct Attachment of Nonderivatized Oligosaccharides to Hydrazide-Modified Self-Assembled Monolayers
Towards Microcantilever-Based Force Sensing and Manipulation: Modeling, Control Development and Implementation
Safe as Mother’s Milk: Carbohydrates as Future Anti-Adhesion Drugs for Bacterial Diseases
Trinchina
Pinkner
X-Ray Structure of the FimC-FimH Chaperone-Adhesin Complex From Uropathogenic Escherichia coli
Expression Profiling: Opportunities and Pitfalls and Impact on the Study and Management of Allergic Diseases
Arefi-Khonsari
Development of Oligonucleotide Microarray Involving Plasma Polymerized Acrylic Acid
Protein Microarrays: High-Throughput Tools for Proteomics
Expression of Type I Pili Is Abolished in Verotoxin-Producing Escherichia Coli O157
Carbohydrate Arrays as Tools for Glycomics
Measuring Magnetic Susceptibilities of Nanogram Quantities of Materials Using Microcantilevers
Detection of Volatile Organic Compounds (VOCs) With Polymer-Coated Cantilevers
Meriaudeau
Study of Differential Hormone-Sensitive Lipase Concentrations Using a Surface Plasmon Resonance Sensor
Geometrical and Flow Configurations for Enhanced Microcantilever Detection Within a Fluidic Cell
Fluid-Structure Interaction Analysis of Flow and Heat Transfer Characteristics Around a Flexible Microcantilever in a Fluidic Cell
|
Special Angles on Unit Circle | Brilliant Math & Science Wiki
Special Angles on Unit Circle
Andy Hayes and Jason Dyer contributed
The special angles on the unit circle refer to the angles that have corresponding coordinates which can be solved with the Pythagorean Theorem. Each of these angles are measured from the positive
x
-axis as the initial side, and the terminal side is the segment connecting the origin to the terminal point on the unit circle.
\text{The sixteen special angles (measured in radians) on the unit circle, each labeled at the terminal point.}
These angles are commonly given as an argument of a trigonometric function such as the sine or cosine functions. When this is the case, one does not need a calculator to compute the value of these functions; the value is easily memorized by pattern. The derivation of the main values is below; once they are obtained the unit circle can be created by reflecting the lengths.
First consider a right triangle with a
45^\circ
angle and a hypotenuse of 1. The other angle must be
180^\circ - 90^\circ - 45^\circ = 45^\circ ,
and since there are two equal angles the triangle is isosceles.
The congruent sides are marked with
x
on the diagram. Now apply the Pythagorean Theorem:
\begin{aligned} x^2 + x^2 &= 1^2 \\ 2x^2 &= 1 \\ x^2 &= \frac{1}{2} \\ x &= \sqrt{\frac{1}{2}} \end{aligned}
\sqrt{\frac{1}{2}}
is correct, typically some rationalization is done:
\sqrt{\frac{1}{2}} = \frac{\sqrt{1}}{\sqrt{2}} = \frac{1\sqrt{2}}{\sqrt{2}\sqrt{2}} = \frac{\sqrt{2}}{2} .
Now consider a right triangle with a
30^\circ
180^\circ - 90^\circ - 30^\circ = 60^\circ .
Reflect a copy of the triangle such that the adjoining triangles form an equilateral triangle as shown.
\overline{BD} \cong \overline{DC}
ABC
m\overline{BD} = m\overline{DC} = \frac{1}{2} .
m\overline{AD}
can now be worked out via Pythagorean Theorem:
\begin{aligned} (m\overline{AD})^2 + {\frac{1}{2}}^2 &= 1^2 \\ (m\overline{AD})^2 + \frac{1}{4} &= 1 \\ (m\overline{AD})^2 &= 1 - \frac{1}{4} \\ (m\overline{AD})^2 &= \frac{3}{4} \\ m\overline{AD} &= \sqrt{\frac{3}{4}} \end{aligned}
\sqrt{\frac{3}{4}}
is simplified:
\sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{\sqrt{4}} = \frac{\sqrt{3}}{2} .
Cite as: Special Angles on Unit Circle. Brilliant.org. Retrieved from https://brilliant.org/wiki/special-angles-on-unit-circle/
|
Hank is planning his vegetable garden. He has created the scale drawing below. He is planning for the actual area for the tomatoes to be
12
9
feet. All angles are right angles.
How many feet in the garden does each inch on the drawing represent?
If the actual area for the tomatoes is
12
9
feet, and the scaled drawing is
2
inches by
1.5
inches, you can write a ratio comparing the dimensions of the actual garden to the dimensions of the scaled drawing.
\frac{\text{12 feet}}{\text{2 inches}} = \frac{x\text{ feet}}{\text{1 inch}}
x
x=6
What are the length and width of the herb garden on the drawing in inches?
Since you already know the length of the herb garden, all you need to do is find the width.
By examining the diagram, you can tell that the width of the herb garden is the width of the zucchini section subtracted from the width of the tomato section.
What are the length and width of the real herb garden in feet?
In part (b), you found that the dimensions of the herb garden in the drawing are
1.5
in by
1
Using the scale factor found in part (a), what are the dimensions of the real herb garden?
9
6
What is the area of the real herb garden (in square feet)?
The area of a rectangular figure is its length multiplied by its width.
54
Remember that there are
12
inches in one foot. If Hank measures the real herb garden in square inches instead of square feet, what will its area be?
First convert the dimensions of the real herb garden into inches by using the ratio
12
in:1ft.
More (e):
9
6
ft →
108
72
How can you use a figure's dimensions to find its area? See part (d).
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Some New Riemann-Liouville Fractional Integral Inequalities
Jessada Tariboon, Sotiris K. Ntouyas, Weerawat Sudsutad, "Some New Riemann-Liouville Fractional Integral Inequalities", International Journal of Mathematics and Mathematical Sciences, vol. 2014, Article ID 869434, 6 pages, 2014. https://doi.org/10.1155/2014/869434
Jessada Tariboon,1 Sotiris K. Ntouyas,2 and Weerawat Sudsutad1
1Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok, Thailand
In this paper, some new fractional integral inequalities are established.
In [1] (see also [2]), the Grüss inequality is defined as the integral inequality that establishes a connection between the integral of the product of two functions and the product of the integrals. The inequality is as follows.
If and are two continuous functions on satisfying and for all , , then
The literature on Grüss type inequalities is now vast, and many extensions of the classical inequality were intensively studied by many authors. In the past several years, by using the Riemann-Liouville fractional integrals, the fractional integral inequalities and applications have been addressed extensively by several researchers. For example, we refer the reader to [3–9] and the references cited therein. Dahmani et al. [10] gave the following fractional integral inequalities by using the Riemann-Liouville fractional integrals. Let and be two integrable functions on satisfying the following conditions: For all , , , then
In this paper, we use the Riemann-Liouville fractional integrals to establish some new fractional integral inequalities of Grüss type. We replace the constants appeared as bounds of the functions and , by four integrable functions. From our results, the above inequalities of [10] and the classical Grüss inequalities can be deduced as some special cases.
In Section 2 we briefly review the necessary definitions. Our results are given in Section 3. The proof technique is close to that presented in [10]. But the obtained results are new and also can be applied to unbounded functions as shown in examples.
Definition 1. The Riemann-Liouville fractional integral of order of a function is defined by where is the gamma function.
For the convenience of establishing our results, we give the semigroup property: which implies the commutative property From Definition 1, if , then we have
Theorem 2. Let be an integrable function on . Assume that there exist two integrable functions , on such that Then, for , , one has
Proof. From , for all , , we have Therefore Multiplying both sides of (11) by , , we get Integrating both sides of (12) with respect to on , we obtain which yields Multiplying both sides of (14) by , , we have Integrating both sides of (15) with respect to on , we get Hence, we deduce inequality (9) as requested. This completes the proof.
As a special case of Theorem 2, we obtain the following result.
Corollary 3. Let be an integrable function on satisfying , for all and . Then, for and , one has
Example 4. Let be a function satisfying for . Then, for and , we have
Theorem 5. Let and be two integrable functions on . Suppose that holds and moreover one assumes that there exist and integrable functions on such that Then, for , , the following inequalities hold:
Proof. To prove , from and , we have for that Therefore Multiplying both sides of (22) by , , we get Integrating both sides of (23) with respect to on , we obtain Then we have Multiplying both sides of (25) by , , we have Integrating both sides of (26) with respect to on , we get the desired inequality .
To prove , we use the following inequalities:
As a special case of Theorem 5, we have the following corollary.
Corollary 6. Let and be two integrable functions on . Assume that there exist real constants such that Then, for , , we have
Lemma 7. Let be an integrable function on and let , be two integrable functions on . Assume that the condition holds. Then, for , , we have
Proof. For any and , we have Multiplying (31) by , , and integrating the resulting identity with respect to , from to , we get Multiplying (32) by , , and integrating the resulting identity with respect to , from to , we have which implies (30).
If and , , for all , then inequality (30) reduces to the following corollary [10, Lemma 3.2].
Corollary 8. Let be an integrable function on satisfying , for all . Then, for all , , one has
Theorem 9. Let and be two integrable functions on and let , , , and be four integrable functions on satisfying the conditions and on . Then, for all , , one has where is defined by
Proof. Let and be two integrable functions defined on satisfying and . Define Multiplying both sides of (37) by , and integrating the resulting identity with respect to and , from to , we can state that Applying the Cauchy-Schwarz inequality to (38), we have Since and , for , we have Thus, from Lemma 7, we get From (39), (41), and (42), we obtain (35).
Remark 10. If and , , then inequality (35) reduces to See [10, Theorem 3.1].
Example 11. Let and be two functions satisfying and for . Then, for and , we have where
This research was funded by King Mongkut’s University of Technology North Bangkok, Thailand. Project code: KMUTNB-GRAD-56-02. Sotiris K. Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
1/\left(b-a\right){\int }_{a}^{b}f\left(t\right)g\left(t\right)dt-\left(1/{\left(b-a\right)}^{2}\right){\int }_{a}^{b}f\left(t\right)dt{\int }_{a}^{b}g\left(t\right)dt
D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, vol. 61 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1993. View at: MathSciNet
G. A. Anastassiou, “Opial type inequalities involving fractional derivatives of two functions and applications,” Computers & Mathematics with Applications, vol. 48, no. 10-11, pp. 1701–1731, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Z. Denton and A. S. Vatsala, “Fractional integral inequalities and applications,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1087–1094, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Belarbi and Z. Dahmani, “On some new fractional integral inequalities,” Journal of Inequalities in Pure and Applied Mathematics, vol. 10, no. 3, article 86, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet
Z. Dahmani, “New inequalities in fractional integrals,” International Journal of Nonlinear Science, vol. 9, no. 4, pp. 493–497, 2010. View at: Google Scholar | MathSciNet
W. T. Sulaiman, “Some new fractional integral inequalities,” Journal of Mathematical Analysis, vol. 2, no. 2, pp. 23–28, 2011. View at: Google Scholar | MathSciNet
M. Z. Sarikaya and H. Ogunmez, “On new inequalities via Riemann-Liouville fractional integration,” Abstract and Applied Analysis, vol. 2012, Article ID 428983, 10 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. Zhu, W. Yang, and Q. Zhao, “Some new fractional
q
-integral Grüss-type inequalities and other inequalities,” Journal of Inequalities and Applications, vol. 2012, article 299, 15 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet
Copyright © 2014 Jessada Tariboon et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Practice Sheets Assistant - Maple Help
Home : Support : Online Help : Education : Student Packages : Basics : Practice Sheets Assistant
Student Basics Practice Sheets Assistant
Create interactive practice sheets using this application. Select from the various options in order to build a customized practice sheet for a given operation.
Click Generate to create a new practice sheet in a new tab.
AdditionSubtractionMultiplicationDivisionFactorExpandSolve
IntegerFractionFloat
\mathrm{parse}\left(\mathrm{DocumentTools}:-\mathrm{GetProperty}\left('\mathrm{To}','\mathrm{value}'\right)\right)
\textcolor[rgb]{0,0,1}{10}
|
Polylogarithm | Brilliant Math & Science Wiki
Aareyan Manzoor, Pi Han Goh, Jongheun Lee, and
The polylogarithm function is an important function for integration, and finding seemingly complicated sum.
Connection between 2 Polylogarithms
Polylogarithm and Geometric Progression
The polylogarithm function is defined as
\operatorname{Li}_s(x) = \sum_{n=1}^\infty \dfrac{x^n}{n^s}
for all complex
s
|x|\leq 1
and can be computed for
|x|>1
by analytical continuation.
_\square
\operatorname{Li}_s(x)=\int_0^x \dfrac{\operatorname{Li}_{s-1}(u)}{u} du.\ _\square
I=\int_0^x \dfrac{\operatorname{Li}_{s-1}(u)}{u} du=\int_0^x \sum_{n=1}^\infty \dfrac{u^{n-1}}{n^{s-1}} du.
Interchanging the summation and integral sign, we end up with
I=\sum_{n=1}^\infty \int_0^x \dfrac{u^{n-1}}{n^{s-1}}du=\sum_{n=1}^\infty \dfrac{x^n}{n^s} = \operatorname{Li}_s(x).\ _\square
Polylogarithm is connected to the infinite geometric progression sum
\operatorname{Li}_0(x)=\sum_{n=1}^\infty x^n=\dfrac{x}{1-x}.
x
and differentiate with respect to
x
\operatorname{Li}_{-1}(x)=\sum_{n=1}^\infty nx^n=\dfrac{x}{(1-x)^2}.
We can keep doing this to get closed form of
\operatorname{Li}_{-s} (x)
, but if we were given something like
\operatorname{Li}_{-25}(0.5)
, it would become extremely tedious to divide by
x
x
25 times. Similar to the binomial theorem which was a shortcut to the tedious Pascal's triangle, there is an easy way to calculate these:
\begin{array} { | c | c | l | } \hline s & \operatorname{Li}_s(z) & \text{ Algebraic expression } \\ \hline 1 & \operatorname{Li}_1(z) & -\ln(1-z) \\ \hline 0 & \operatorname{Li}_0(z) & \dfrac z{1-z} \\ \hline -1 & \operatorname{Li}_{-1}(z) & \dfrac {z}{(1-z)^2} \\ \hline -2 & \operatorname{Li}_{-2}(z) & \dfrac {z(z+1)}{(1-z)^3} \\ \hline -3 & \operatorname{Li}_{-3}(z) & \dfrac {z(z^2+4z+1)}{(1-z)^4} \\ \hline -4 & \operatorname{Li}_{-4}(z) & \dfrac {z(1+z)(z^2+10z+1)}{(1-z)^5} \\ \hline -5 & \operatorname{Li}_{-5}(z) & \dfrac {z(z^4+26z^3+ 66z^2+26z+1)}{(1-z)^6} \\ \hline \end{array}
In general, we can express
\def\Li{Li\,} \Li_{-n}(z)
\def\Li{Li\,} \Li_{-n}(z) = \left( z \; \dfrac {\partial}{ \partial z} \right)^n \dfrac z{1-z} = \sum_{k=0}^n k! \; S(n+1,k+1) \left( \dfrac z{1-z} \right)^{k+1} \; ,
n
S(n,k)
denote the Stirling's number of the second kind.
_\square
Try to solve the following problem using polylogarithms!
\large \displaystyle \sum_{n=1}^{\infty} \frac {n^5}{2^n}
Let the value of the above summation be
A
\sqrt{A+7}
Cite as: Polylogarithm. Brilliant.org. Retrieved from https://brilliant.org/wiki/polylogarithm/
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Effects of Altitude on Power Performance of Commercial Vehicles
College of Automotive Engineering, Shanghai University of Engineering Science, Shanghai, China.
Abstract: Different atmospheric pressure and ambient temperature of different altitude affect the dynamic performance of vehicle. Subject to laboratory conditions, this paper builds a simulation model of a commercial vehicle based on GT-Power software to study the dynamic performance of a commercial vehicle at different altitudes by acceleration time, maximum gradability and maximum speed of vehicle evidently. Advanced turbocharging technique was deemed to be effective measure to improve dynamic performance of vehicle.
Keywords: High Altitude, Dynamic Efficiency, GT-Power, Simulation
China is a country with a vast territory and complex terrain. It has the largest plateau area in the world, and the area with an elevation of more than 1000 m occupies about 58% of the total area of the country. The high altitude area has bad environment, poor traffic conditions, and the vehicle’s dynamic decline is obvious. In order to maintain good dynamic performance of the vehicle in high altitude area, it is necessary to improve the adaptability of the vehicle.
Current researches almost focus on the high-altitude engines and heavy-duty diesel vehicles’ dynamics and economy. Liu Shengji [1] simulated the combusting regulation and economic impact of diesel engines under atmospheric pressure and ambient temperature at different altitudes using a high-altitude environment test bench and explored the changes of oil-gas mixture processes in high altitude environment through simulation models. K Han et al. [2] established a turbocharged diesel numerical model based on the neural network combustion model, adjusting the fuel injection parameters (such as fuel injection time) at different altitudes to improve engine dynamics. Zhou Guangmeng et al. [3] qualitatively analyzed the effect of the engine’s effective thermal efficiency and circulating fuel injection rate on the vehicle’s dynamic performance under high-altitude environmental conditions based on various dynamic index calculation formulas. Xie Shaofa [4] applied the engine, the power of which is improved, to the whole vehicle test at a certain altitude, increased the rear axle speed ratio at the same time, finally reduced the fuel consumption of the whole vehicle and improved the acceleration performance. However, there are fewer researches on the performance of vehicles at all altitudes. In view of the problems such as single-altitude conditions of high-altitude vehicles and the long cycle cost, we study the variation of dynamical indexes under different altitudes and optimize the dynamic performance of cars by the forward dynamic simulation, which simulates the atmospheric environment under different altitudes based on the GT-Power platform.
2. The Establishment of a Vehicle Simulation Model
First of all, engine models should be set up in GT-Power, including the modules of cylinder, intake and exhaust pipe, crankshaft, injector, intake and exhaust valves and so on. The engine takes the Weber and the woschni as the combustion model and heat transfer model respectively. Then the engine module is connected with the body, transmission system, tires and other modules to establish the vehicle model shown in Figure 1; the vehicle parameters are shown in Table 1; the simulation of the vehicle’s various dynamic performance indicators are shown in Table 2.
According to Table 2, the maximum vehicle speed deviation is 2.3%; the maximum climbing grade deviation is 4.8%; and the 0 - 400 m acceleration time deviation is 4.4%. The maximum deviation is less than ±5%. Therefore, this model is suitable for the simulation of the whole vehicle dynamic performance.
3. Simulation Research on Vehicle Dynamic Performance at Different Elevations
Simulation tests the vehicle dynamic performance at different altitudes (0 m, 1000 m, 2000 m, 3000 m and 4000 m) on the environmental conditions shown in Table 3 according to GB/T20969.1-2007.
In order to study the changes in the traction of cars, the stalls were set to gears 1st, 2nd, 3rd, 4th, and 5th. Figure 2 shows the driving resistance balance diagram at an altitude of 0 m. As the altitude increases, the change in the maximum speed is shown in Figure 3.
Figure 1. GT-Power vehicle model.
Figure 2. Zero elevation vehicle driving force - running resistance balance diagram.
It can been seen From Figure 3 that as the altitude increases, the intake of engine decreases result in the deterioration of combustion, so the maximum speed gradually decreases, especially at an altitude of more than 3000 m, the maximum speed drops by more than 35.43%, the vehicle’s power is severely affected, and the normal running of the vehicle cannot be guaranteed.
2) Start-up acceleration capability
The Start-up acceleration capability is measured by the acceleration time of 0 - 400 m. The shifting rules of the altitudes are kept same in the simulation process, switched from the 1st gear to the 5th gear in turn. The situation of the starting accelerations of the vehicle in situ is shown in Figure 4.
The results showed that as the altitude increased, the on-site start-up acceleration time significantly increased while the power decreased significantly. At an
Table 1. Basic parameters of the vehicle.
Table 2. Comparison of simulation results with trial value of vehicle performance.
Table 3. Correspondence between altitude and atmospheric pressure and ambient temperature.
Figure 3. Maximum speeds at different altitudes.
altitude of 1000 m, the on-site acceleration time of 0 - 400 m increased by 11.55%; at an altitude of 2000 m, the acceleration time of 0 - 400 m on-site started to increase by 25.81%; at an altitude of 3000 m, the on-site acceleration time of 0 - 400 m increased. 43.16%; at an altitude of 4000 m, the on-site acceleration time of 0 - 400 m increased by 66.74%.
Figure 4. Vehicle 0 - 400 m acceleration time chart.
The climbing ability of a car refers to the slope that the car can climb when it overcomes the rolling resistance and air resistance on a good road surface to overcome the gradient resistance [5] . At this time, the acceleration is zero.
{F}_{i}={F}_{t}-\left({F}_{f}+{F}_{w}\right)
i=\mathrm{tan}\alpha
\alpha =\mathrm{arcsin}\frac{{F}_{i}}{G}
In the formula, Ft is the maximum traction of the car, Ff + Fw is the running resistance, i is the slope, and G is the total weight of the car.
From the equation above, we can see that the maximum grade of the car is related to traction, driving resistance, vehicle weight and load. Assuming that the vehicle weight is constant, the traction expresses the most obvious effect on it. Taking the maximum traction of 1st gear as the maximum grade of the vehicle, the maximum traction and running resistance at different altitudes are shown in Table 4 below. Substituting the data in Table 4 into the formula, the maximum grade of climbing at each altitude is shown in Table 5.
The results show that with the increase of altitude, the maximum grade of climbing changes significantly. At an altitude of 4000 m, the maximum climbing degree of the vehicle is only 7%, which is a decrease of 41.67% compared with that of the zero altitude. Cars often have long uphill road conditions at high altitudes and cannot guarantee sufficient gradeability.
4. Turbocharged Power Recovery Scheme
According to the analysis above, the problem of the automobile in the high altitude area is mainly reflected in the decrease of the engine power, so this optimization mainly focuses on the engine power improvement. Compared with the plain area, the air density in the high altitude area is smaller, the quality of the air entering the engine cylinder is lower and the fuel combustion is more
Table 4. Maximum traction and corresponding driving resistance at different altitudes.
Table 5. Maximum climbing grades at different altitudes.
Figure 5. Turbocharger model.
Figure 6. Comparison of maximum speed before and after optimization.
insufficient, which results in insufficient engine power. Matching the turbocharger for the engine can increase the intake air volume and make the cylinder work normally, so as to achieve the purpose of restoring the engine power. Vehicle engine boosters include a mechanical booster system, an exhaust gas turbocharger system, and a combined turbocharger system. This article uses the exhaust gas booster system, and the booster module is added on the basis of the original model as shown in Figure 5. The comparison of the maximum speed of the car before and after boosting and the acceleration time of starting at the same place is shown in Figure 6 and Figure 7.
Figure 7. Comparison of 0 - 400 m acceleration time before and after optimization.
Through optimization at different altitudes, the car’s top speed and start-up acceleration performance have improved to varying degrees. Above an altitude of 3000 m, the maximum speed of the original car drops significantly. After optimization, the maximum speed of the vehicle exceeds 150 km/h, which fully satisfies the driving demand in the highland area. When the supercharged vehicle is at an altitude of no more than 2000 m, the acceleration at the start is faster, and the acceleration time at an altitude of more than 2000 m is obviously increased. However, before the optimization, the vehicle’s ability to start and accelerate at the same time has also been improved.
In this paper, the GT-Power simulation software for vehicle performance is used to study the dynamic characteristics of a commercial vehicle at different altitudes and optimize its dynamic performance. The conclusions are as follows.
With the increase of the altitude, various dynamic indicators of automobiles show a downward trend, but the declines are different. Especially when over 3000 m above sea level, the vehicle’s power deteriorates rapidly and it cannot meet the normal driving needs of the vehicle.
With the exhaust gas turbocharger system, all altitude performances have been improved, which can ensure the vehicle’s plateau driving demand, but it is still affected by altitude.
Cite this paper: Liang, F. and Zhang, L. (2018) Effects of Altitude on Power Performance of Commercial Vehicles. Open Access Library Journal, 5, 1-8. doi: 10.4236/oalib.1104642.
[1] Liu, S.-J., Chen, Y., et al. (2016) Effects of Altitude on Fuel Consumption and Thermal Efficiency of Naturally Aspirated Diesel Engines. Chinese Internal Combustion Engine Engineering, 37, 20-25.
[2] Han, K. (2014) Calibration for Fuel Injection Parameters of the Diesel Engine Working at Plateau via Simulating. Advances in Mechanical Engineering, No. 1, 110-121.
[3] Zhou, G.-M., Liu, R.-L., et al. (2014) Effects of Plateau Environment on Power Performance of Vehicles and Measures to Improve Power Performance in Plateau. Equipment Environmental Engineering, No. 3, 45-51.
[4] Xie, S.-F. and Li, C.-Q. (1980) Modification of Type EQ-140 5-T. Truck for Use in Plateau. Automotive Engineering, No. 3, 31-40 + 2.
[5] Yu, Z.-S. (2009) Automotive Theory. China Machine Press, Beijing.
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Scale-Up and Generalization of Horizontal-Base Pin-Fin Heat Sinks in Natural Convection and Radiation | J. Heat Transfer | ASME Digital Collection
Sahray, D., Ziskind, G., and Letan, R. (August 13, 2010). "Scale-Up and Generalization of Horizontal-Base Pin-Fin Heat Sinks in Natural Convection and Radiation." ASME. J. Heat Transfer. November 2010; 132(11): 112502. https://doi.org/10.1115/1.4002032
This paper provides further insight in heat transfer from horizontal-base pin fin heat-sinks in free convection of air. The main objective is to assess the effect of base size, and this with regard to the effects of fin height and fin population density studied in a previous work (Sahray, D., et al., 2010, “Study and Optimization of Horizontal-Base Pin-Fin Heat Sinks in Natural Convection and Radiation,” ASME J. Heat Transfer, 132(012503), pp. 1–13). To this end, experimental and numerical investigations are performed with sinks of different base sizes. The sinks are made of aluminum, with no contact resistance between the base and the fins, and are heated using foil electrical heaters. In the corresponding numerical study, the sinks and their environment are modeled using the FLUENT 6.3 software. In the experiments, sink bases of
100×100 mm2
200×200 mm2
are used, while in the numerical study sinks of
50×50 mm2
are investigated, too. In addition to the sinks with exposed, free edges (Sahray, D., et al., 2010, “Study and Optimization of Horizontal-Base Pin-Fin Heat Sinks in Natural Convection and Radiation,” ASME J. Heat Transfer, 132(012503), pp. 1–13), the same sinks are explored also with their edges blocked. This is done in order to exclude the edge effect, thus making it possible to estimate heat transfer from a sink of an “infinite” base size. Heat-transfer enhancement due to the fins is assessed quantitatively and analyzed for various base sizes and fin heights. The effect of fin location in the array on its contribution to the heat-transfer rate from the sink is analyzed. By decoupling convection from radiation, a dimensional analysis of the results for natural convection is attempted. Interdependence of the base size and fin height effects on the heat transfer is demonstrated. A correlation that encompasses all the cases studied herein is obtained, in which the Nusselt number depends on the Rayleigh number, which uses the “clear” spacing between fins as the characteristic length, and on the dimensions of the fins and the base.
aluminium, natural convection, pin fin, natural convection, scale-up, correlation
Fins, Heat sinks, Natural convection, Radiation (Physics), Heat transfer, Aluminum, Dimensions, Dimensional analysis
Elmakaes
Effect of Geometry on Heat Transfer From a Pin-Fin Array
,” Heat Transfer Laboratory, Department of Mechanical Engineering, Ben-Gurion University of the Negev, Graduation Project 08-38.
Analysis of Natural Convection and Radiation from Pin-Fin Heat Sinks With Horizontal Base
,” M.Sc. thesis, Ben-Gurion University of the Negev, Israel.
Proceedings of the ASME InterPACK ‘07
, Jacksonville, FL, Aug. 10–14.
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Speed of light - Marspedia
2 Consequences of the Speed of Light
The Speed of Light (or c) is the speed at which a photon (an "energy packet" of electromagnetic radiation) travels in a vacuum.
The exact measurement for the speed of light is 299,792,458 meters per second, in the following values, the speed of light will use values based on the rounded 300,000,000 m/s.
Meters per second:
{\displaystyle c=3\times 10^{8}\mathrm {ms^{-1}} }
Kilometers per second:
{\displaystyle c=3\times 10^{5}\mathrm {kms^{-1}} }
Miles per second:
{\displaystyle c=1.9\times 10^{5}\mathrm {mis^{-1}} }
Consequences of the Speed of Light
Humans have discovered methods to communicate over great distances using electromagnetic (EM) radiation, such as radio waves and lasers. On Earth, distances are short enough that these methods seem instantaneous. On Mars and other parts of the solar system, communication with Earth involves a time delay, based on the time it takes the EM radiation to travel. The delay between Earth and Mars varies from 3 minutes to 22 minutes (between about 180 to 1340 seconds depending on the position of planets and resulting distance). For two way communication (request - answer) time is doubled.
To provide uninterrupted communications, a group of communication satellites will be necessary (placed in the orbit around the sun) - due different orbital speeds of both planets, for a period of time sun will be positioned directly between both planets, blocking direct communications.
The speed of light on Wikipedia
Retrieved from "https://marspedia.org/index.php?title=Speed_of_light&oldid=136501"
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Photodiode - 2D PCM Schematics - 3D Model
Photodiode (14650 views - Electronics & PCB)
A photodiode is a semiconductor device that converts light into an electrical current. The current is generated when photons are absorbed in the photodiode. A small amount of current is also produced when no light is present. Photodiodes may contain optical filters, built-in lenses, and may have large or small surface areas. Photodiodes usually have a slower response time as their surface area increases. The common, traditional solar cell used to generate electric solar power is a large area photodiode. Photodiodes are similar to regular semiconductor diodes except that they may be either exposed (to detect vacuum UV or X-rays) or packaged with a window or optical fiber connection to allow light to reach the sensitive part of the device. Many diodes designed for use specifically as a photodiode use a PIN junction rather than a p–n junction, to increase the speed of response. A photodiode is designed to operate in reverse bias.
PARTcloud - Photodiode
Licensed under Creative Commons Attribution-Share Alike 3.0 (Ulfbastel at the German language Wikipedia).
Three Si and one Ge (top) photodiodes.
A photodiode is a semiconductor device that converts light into an electrical current. The current is generated when photons are absorbed in the photodiode. A small amount of current is also produced when no light is present. Photodiodes may contain optical filters, built-in lenses, and may have large or small surface areas. Photodiodes usually have a slower response time as their surface area increases. The common, traditional solar cell used to generate electric solar power is a large area photodiode.
Photodiodes are similar to regular semiconductor diodes except that they may be either exposed (to detect vacuum UV or X-rays) or packaged with a window or optical fiber connection to allow light to reach the sensitive part of the device. Many diodes designed for use specifically as a photodiode use a PIN junction rather than a p–n junction, to increase the speed of response. A photodiode is designed to operate in reverse bias.[1]
1.3 Other modes of operation
2.1 Unwanted photodiode effects
A photodiode is a p–n junction or PIN structure. When a photon of sufficient energy strikes the diode, it creates an electron-hole pair. This mechanism is also known as the inner photoelectric effect. If the absorption occurs in the junction's depletion region, or one diffusion length away from it, these carriers are swept from the junction by the built-in electric field of the depletion region. Thus holes move toward the anode, and electrons toward the cathode, and a photocurrent is produced. The total current through the photodiode is the sum of the dark current (current that is generated in the absence of light) and the photocurrent, so the dark current must be minimized to maximize the sensitivity of the device.[2]
When used in zero bias or photovoltaic mode, the flow of photocurrent out of the device is restricted and a voltage builds up. This mode exploits the photovoltaic effect, which is the basis for solar cells – a traditional solar cell is just a large area photodiode.
In this mode the diode is often reverse biased (with the cathode driven positive with respect to the anode). This reduces the response time because the additional reverse bias increases the width of the depletion layer, which decreases the junction's capacitance. The reverse bias also increases the dark current without much change in the photocurrent. For a given spectral distribution, the photocurrent is linearly proportional to the illuminance (and to the irradiance).[3]
Although this mode is faster, the photoconductive mode tends to exhibit more electronic noise.[4] The leakage current of a good PIN diode is so low (<1 nA) that the Johnson–Nyquist noise of the load resistance in a typical circuit often dominates.
Avalanche photodiodes are photodiodes with structure optimized for operating with high reverse bias, approaching the reverse breakdown voltage. This allows each photo-generated carrier to be multiplied by avalanche breakdown, resulting in internal gain within the photodiode, which increases the effective responsivity of the device.
A phototransistor is a light-sensitive transistor. A common type of phototransistor, called a photobipolar transistor, is in essence a bipolar transistor encased in a transparent case so that light can reach the base–collector junction. It was invented by Dr. John N. Shive (more famous for his wave machine) at Bell Labs in 1948,[5]:205 but it was not announced until 1950.[6] The electrons that are generated by photons in the base–collector junction are injected into the base, and this photodiode current is amplified by the transistor's current gain β (or hfe). If the base and collector leads are used and the emitter is left unconnected, the phototransistor becomes a photodiode. While phototransistors have a higher responsivity for light they are not able to detect low levels of light any better than photodiodes.[citation needed] Phototransistors also have significantly longer response times. Field-effect phototransistors, also known as photoFETs, are light-sensitive field-effect transistors. Unlike photobipolar transistors, photoFETs control drain-source current by creating a gate voltage.
Materials commonly used to produce photodiodes include:[7]
Unwanted photodiode effects
Any p–n junction, if illuminated, is potentially a photodiode. Semiconductor devices such as diodes, transistors and ICs contain p–n junctions, and will not function correctly if they are illuminated by unwanted electromagnetic radiation (light) of wavelength suitable to produce a photocurrent;[8][9] this is avoided by encapsulating devices in opaque housings. If these housings are not completely opaque to high-energy radiation (ultraviolet, X-rays, gamma rays), diodes, transistors and ICs can malfunction[10] due to induced photo-currents. Background radiation from the packaging is also significant.[11] Radiation hardening mitigates these effects.
In some cases, the effect is actually wanted, for example to use LEDs as light-sensitive devices (see LED as light sensor) or even for energy harvesting, then sometimes called light-emitting and -absorbing diodes (LEADs).[12]
Critical performance parameters of a photodiode include:
The Spectral responsivity is a ratio of the generated photocurrent to incident light power, expressed in A/W when used in photoconductive mode. The wavelength-dependence may also be expressed as a Quantum efficiency, or the ratio of the number of photogenerated carriers to incident photons, a unitless quantity.
The current through the photodiode in the absence of light, when it is operated in photoconductive mode. The dark current includes photocurrent generated by background radiation and the saturation current of the semiconductor junction. Dark current must be accounted for by calibration if a photodiode is used to make an accurate optical power measurement, and it is also a source of noise when a photodiode is used in an optical communication system.
A photon absorbed by the semiconducting material will generate an electron-hole pair which will in turn start moving in the material under the effect of the electric field and thus generate a current. The finite duration of this current is known as the transit-time spread and can be evaluated by using Ramo's theorem. One can also show with this theorem that the total charge generated in the external circuit is well e and not 2e as might seem by the presence of the two carriers. Indeed, the integral of the current due to both electron and hole over time must be equal to e. The resistance and capacitance of the photodiode and the external circuitry give rise to another response time known as RC time constant
{\displaystyle \tau =RC}
. This combination of R and C integrates the photoresponse over time and thus lengthens the impulse response of the photodiode. When used in an optical communication system, the response time determines the bandwidth available for signal modulation and thus data transmission.
(NEP) The minimum input optical power to generate photocurrent, equal to the rms noise current in a 1 hertz bandwidth. NEP is essentially the minimum detectable power. The related characteristic detectivity (
{\displaystyle D}
) is the inverse of NEP, 1/NEP. There is also the specific detectivity (
{\displaystyle D^{\star }}
) which is the detectivity multiplied by the square root of the area (
{\displaystyle A}
) of the photodetector, (
{\displaystyle D^{\star }=D{\sqrt {A}}}
{\displaystyle D^{\star }}
When a photodiode is used in an optical communication system, all these parameters contribute to the sensitivity of the optical receiver, which is the minimum input power required for the receiver to achieve a specified bit error rate.
P–n photodiodes are used in similar applications to other photodetectors, such as photoconductors, charge-coupled devices, and photomultiplier tubes. They may be used to generate an output which is dependent upon the illumination (analog; for measurement and the like), or to change the state of circuitry (digital; either for control and switching, or digital signal processing).
Photodiodes are used in consumer electronics devices such as compact disc players, smoke detectors, and the receivers for infrared remote control devices used to control equipment from televisions to air conditioners. For many applications either photodiodes or photoconductors may be used. Either type of photosensor may be used for light measurement, as in camera light meters, or to respond to light levels, as in switching on street lighting after dark.
Photosensors of all types may be used to respond to incident light, or to a source of light which is part of the same circuit or system. A photodiode is often combined into a single component with an emitter of light, usually a light-emitting diode (LED), either to detect the presence of a mechanical obstruction to the beam (slotted optical switch), or to couple two digital or analog circuits while maintaining extremely high electrical isolation between them, often for safety (optocoupler). The combination of LED and photodiode is also used in many sensor systems to characterize different types of products based on their optical absorbance.
Pinned photodiode is not a PIN photodiode, it has p+/n/p regions in it. It has a shallow P+ implant in N type diffusion layer over a P-type epitaxial substrate layer. It is used in CMOS Active pixel sensor.[14]
Comparison with photomultipliers
High quantum efficiency, typically 60–80% [16]
A one-dimensional array of hundreds or thousands of photodiodes can be used as a position sensor, for example as part of an angle sensor.[17] One advantage of photodiode arrays (PDAs) is that they allow for high speed parallel read out since the driving electronics may not be built in like a traditional CMOS or CCD sensor.
This article uses material from the Wikipedia article "Photodiode", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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Stabilize polynomial - MATLAB polystab - MathWorks
Convert Linear-Phase Filter to Minimum-Phase
Stabilize polynomial
b = polystab(a)
polystab stabilizes a polynomial with respect to the unit circle; it reflects roots with magnitudes greater than 1 inside the unit circle.
b = polystab(a) returns a row vector b containing the stabilized polynomial. a is a vector of polynomial coefficients, normally in the z-domain:
A\left(z\right)=a\left(1\right)+a\left(2\right){z}^{-1}+\dots +a\left(m+1\right){z}^{-m}.
Use the window method to design a 25th-oder FIR filter with normalized cutoff frequency
0.4\pi
rad/sample. Verify that it has linear phase but not minimum phase.
h = fir1(25,0.4);
h_linphase = islinphase(h)
h_linphase = logical
h_minphase = isminphase(h)
h_minphase = logical
Use polystab to convert the linear-phase filter into a minimum-phase filter. Plot the phase responses of the filters.
hmin = polystab(h)*norm(h)/norm(polystab(h));
hmin_linphase = islinphase(hmin)
hmin_linphase = logical
hmin_minphase = isminphase(hmin)
hmin_minphase = logical
hfvt = fvtool(h,1,hmin,1,'Analysis','phase');
legend(hfvt,'h','hmin')
Verify that the two filters have identical magnitude responses.
hfvt = fvtool(h,1,hmin,1);
polystab finds the roots of the polynomial and maps those roots found outside the unit circle to the inside of the unit circle:
v = roots(a);
vs = 0.5*(sign(abs(v)-1)+1);
v = (1-vs).*v + vs./conj(v);
b = a(1)*poly(v);
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Ball_bearing Knowpia
Working principle for a ball bearing; red dots show direction of rotation.
A 4-point angular contact ball bearing
A ball bearing for skateboard wheels with a plastic cage
Wingqvist's self-aligning ball bearing
Common designsEdit
Angular contactEdit
AxialEdit
Deep-grooveEdit
Preloaded pairsEdit
The above basic types of bearings are typically applied in a method of preloaded pairs, where two individual bearings are rigidly fastened along a rotating shaft to face each other. This improves the axial runout by taking up (preloading) the necessary slight clearance between the bearing balls and races. Pairing also provides an advantage of evenly distributing the loads, nearly doubling the total load capacity compared to a single bearing. Angular contact bearings are almost always used in opposing pairs: the asymmetric design of each bearing supports axial loads in only one direction, so an opposed pair is required if the application demands support in both directions. The preloading force must be designed and assembled carefully, because it deducts from the axial force capacity of the bearings, and can damage bearings if applied excessively. The pairing mechanism may simply face the bearings together directly, or separate them with a shim, bushing, or shaft feature.
ConradEdit
Slot-fillEdit
Relieved raceEdit
Relieved race ball bearings are 'relieved' as the name suggests by having either the OD of the inner ring reduced on one side, or the ID of the outer ring increased on one side. This allows a greater number of balls to be assembled into either the inner or outer race, and then press fit over the relief. Sometimes the outer ring will be heated to facilitate assembly. Like the slot-fill construction, relieved race construction allows a greater number of balls than Conrad construction, up to and including full complement, and the extra ball count gives extra load capacity. However, a relieved race bearing can only support significant axial loads in one direction ('away from' the relieved race).
Fractured raceEdit
FlangedEdit
CagedEdit
Hybrid ball bearings using ceramic ballsEdit
Ceramic bearing balls can weigh up to 40% less than steel ones, depending on size and material. This reduces centrifugal loading and skidding, so hybrid ceramic bearings can operate 20% to 40% faster than conventional bearings. This means that the outer race groove exerts less force inward against the ball as the bearing spins. This reduction in force reduces the friction and rolling resistance. The lighter balls allow the bearing to spin faster, and uses less power to maintain its speed.
Fully ceramic bearingsEdit
These bearings make use of both ceramic balls and race. These bearings are impervious to corrosion and rarely require lubrication if at all. Due to the stiffness and hardness of the balls and race these bearings are noisy at high speeds. The stiffness of the ceramic makes these bearings brittle and liable to crack under load or impact. Because both ball and race are of similar hardness, wear can lead to chipping at high speeds of both the balls and the race, which can cause sparking.
Self-aligningEdit
Wingqvist developed a self-aligning ball bearing.
Self-aligning ball bearings, such as the Wingqvist bearing shown in the picture, are constructed with the inner ring and ball assembly contained within an outer ring that has a spherical raceway. This construction allows the bearing to tolerate a small angular misalignment resulting from shaft or housing deflections or improper mounting. The bearing was used mainly in bearing arrangements with very long shafts, such as transmission shafts in textile factories.[6] One drawback of the self-aligning ball bearings is a limited load rating, as the outer raceway has very low osculation (its radius is much larger than the ball radius). This led to the invention of the spherical roller bearing, which has a similar design, but uses rollers instead of balls. The spherical roller thrust bearing is another invention derived from the findings by Wingqvist.
Operating conditionsEdit
Failure modesEdit
{\displaystyle \nu }
If the bearing is used under oscillation, oil lubrication should be preferred.[8] If grease lubrication is necessary, the composition should be adapted to the parameters that occur. Greases with a high bleeding rate and low base oil viscosity should be preferred if possible.[9]
Direction of loadEdit
Avoiding undesirable axial loadEdit
If an axle has two bearings, and temperature varies, axle shrinks or expands, therefore it is not admissible for both bearings to be fixed on both their sides, since expansion of axle would exert axial forces that would destroy these bearings. Therefore, at least one of the bearings must be able to slide.[7]
Avoiding torsional loadsEdit
If a shaft is supported by two bearings, and the center-lines of rotation of these bearings are not the same, then large forces are exerted on the bearing, which may destroy it. Some very small amount of misalignment is acceptable, and how much depends on type of bearing. For bearings that are specifically made to be 'self-aligning', acceptable misalignment is between 1.5 and 3 degrees of arc. Bearings that are not designed to be self-aligning can accept misalignment of only 2–10 minutes of arc (0.033-0.166 degrees) .[7]
Computer fan and spinning device bearings used to be highly spherical, and were said to be the best spherical manufactured shapes, but this is no longer true for hard disk drive, and more and more are being replaced with fluid bearings.
In horology, the Jean Lassale company designed a watch movement that used ball bearings to reduce the thickness of the movement. Using 0.20 mm balls, the Calibre 1200 was only 1.2 mm thick, which still is the thinnest mechanical watch movement.[10]
Many yo-yos, ranging from beginner to professional or competition grade, incorporate ball bearings.
In centrifugal pumps.
Railroad locomotive axle journals. Side rod action of newest high speed steam locomotives before railroads were converted to diesel engines.
Ball screw – Low-friction linear actuator
Bearing Specialists Association – American industry trade group
Linear-motion bearing – Mechanical bearing designed to provide free motion in one direction
^ "Double- Row Angular Contact Ball Bearings". Archived from the original on 11 May 2013.
Suriray, "Perfectionnements dans les vélocipèdes" (Improvements in bicycles), French patent no. 86,680, issued: 2 August 1869, Bulletin des lois de la République française (1873), series 12, vol. 6, page 647.
Louis Baudry de Saunier, Histoire générale de la vélocipédie [General history of cycling] (Paris, France: Paul Ollendorff, 1891), pages 62–63.
^ Bicycle History, Chronology of the Growth of Bicycling and the Development of Bicycle Technology by David Mozer. Ibike.org. Retrieved 1 September 2012.
^ a b Brumbach, Michael E.; Clade, Jeffrey A. (2003), Industrial Maintenance, Cengage Learning, pp. 112–113, ISBN 978-0-7668-2695-3.
^ Sobel, Dava (1995). Longitude. London: Fourth Estate. p. 103. ISBN 0-00-721446-4. A novel antifriction device that Harrison developed for H-3 survives to the present day – ...caged ball bearings.
^ "Manufacturing and sales". SKF. Retrieved 5 December 2013.
^ a b c d e f g h i j k l m n o p q r s t u v w "Leerboek wentellagers", SKF, 1985
^ Maruyama, Taisuke; Saitoh, Tsuyoshi; Yokouchi, Atsushi (4 May 2017). "Differences in Mechanisms for Fretting Wear Reduction between Oil and Grease Lubrication". Tribology Transactions. 60 (3): 497–505. doi:10.1080/10402004.2016.1180469. ISSN 1040-2004. S2CID 138588351.
^ Schwack, Fabian; Bader, Norbert; Leckner, Johan; Demaille, Claire; Poll, Gerhard (15 August 2020). "A study of grease lubricants under wind turbine pitch bearing conditions". Wear. 454–455: 203335. doi:10.1016/j.wear.2020.203335. ISSN 0043-1648.
^ Brunner, Gisbert (1999). Wristwatches – Armbanduhren – Montres-bracelets. Köln, Germany: Könnemann. p. 454. ISBN 3-8290-0660-8.
Look up ball bearing in Wiktionary, the free dictionary.
Ball bearing at Curlie
Bearing Modeling using Wolfram
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Metric considers, that the best metric value for specific original frame is the correct shift. Metric can prefer smaller shift with base metric value X to bigger shift with metric value Y if
X > Y \cdot \mbox{threshold}
, where threshold can be set in metric settings. This helps to avoid random fluctuations.
LQ H264, Time Shift-y=0.0000
Blurring, Time Shift-y=0.0000
Random points, Time Shift-y=0.0000
Luminance shift, Time Shift-y=0.0000
JPEG Q=2, Time Shift-y=0.0000
JPEG Q=10, Time Shift-y=0.0000
This metric was introduced in VQMT 13.
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SignAtBox - Maple Help
Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : SemiAlgebraicSetTools Subpackage : SignAtBox
SignAtBox
return the sign of a polynomial at real point
SignAtBox(p, B, R)
a box object encoding a point with real coordinates
The command SignAtBox(p, B, R) returns the sign of the polynomial p at the point encoded by the box object B.
The box object B is assumed to be returned by the command RealRootIsolate.
The sign at B of the polynomial p is given as -1, 0, or 1 for negative, null, or positive, respectively.
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right):
R≔\mathrm{PolynomialRing}\left([y,x]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
Isolate the real points of a polynomial system and pick one of them.
B≔\mathrm{RealRootIsolate}\left([{x}^{2}-2,y-x],[],[x],[],R\right)[1]
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{box}}
Check the sign of a polynomial at that box.
p≔{x}^{2}+{y}^{2}-4
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}
\mathrm{SignAtBox}\left(p,B,R\right)
\textcolor[rgb]{0,0,1}{0}
Checking a couple other signs.
\mathrm{SignAtBox}\left(p-1,B,R\right)
\textcolor[rgb]{0,0,1}{-1}
\mathrm{SignAtBox}\left(p+1,B,R\right)
\textcolor[rgb]{0,0,1}{1}
The RegularChains[SemiAlgebraicSetTools][SignAtBox] command was introduced in Maple 2020.
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Category:Meta-GGA - Vaspwiki
Category:Meta-GGA
Meta-GGA exchange-correlation functionals depend on the electron density
{\displaystyle n}
{\displaystyle \nabla n}
and the kinetic-energy density
{\displaystyle \tau }
{\displaystyle E_{\mathrm {xc} }^{\mathrm {meta-GGA} }=\int \epsilon _{\mathrm {xc} }^{\mathrm {meta-GGA} }(n,\nabla n,\tau )d^{3}r}
Although meta-GGAs are slightly more expensive than GGAs, they are still fast to evaluate and appropriate for very large systems. Furthermore, meta-GGAs can be more accurate than GGAs and more broadly applicable. Note that as in most other codes, meta-GGAs are implemented in VASP within the generalized KS scheme[1]. The meta-GGA that is currently the most widely used in solid-state physics is SCAN[2]. The meta-GGA functionals using the Laplacian of the electron density,
{\displaystyle \nabla ^{2}n}
, are not yet available in VASP.
A meta-GGA can be used by specifying the tag METAGGA in the INCAR file.
How to do a Band-structure calculation using meta-GGA functionals.
↑ Z.-h. Yang, H. Peng, J. Sun, and J. P. Perdew, Phys. Rev. B 93, 205205 (2016).
↑ J. Sun, A. Ruzsinszky, and J. P. Perdew, Phys. Rev. Lett. 115, 036402 (2015).
Pages in category "Meta-GGA"
Band-structure calculation using meta-GGA functionals
Retrieved from "https://www.vasp.at/wiki/index.php?title=Category:Meta-GGA&oldid=17811"
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Sofdar Ali's Next Day | Toph
Sofdar Ali's Next Day
By Kryptonyte · Limits 1.5s, 512 MB
Sofdor Ali goes to Karwan Bazar every Thursday to eat Jilapi! But he often forgets what day is Thursday! In fact, he struggles to figure out the exact date for any day. He has an alarm clock in his pocket and it only rings at a specific time of the day. But this old clock does not give him information about the date. Sofdor has come up with a solution to this problem. Whenever he needs to know what day it is, he buys a newspaper. Newspapers always have dates written in it, and that solves Sofdor's problem. Well, at least for a day.
But buying newspapers is costly for Sofdor Ali, as he has to spend most of his money for buying important equipment for his experiment. So Sofdor is wondering, how he could reduce the cost of buying newspapers. Then he decided, he would buy a newspaper for one day and then calculate the next date from the newspaper's date. That will reduce the cost of buying newspapers to half, as Sofdor only has to buy one newspaper in two days.
Now you have to write a program that will help Sofdor Ali to calculate the date of the next day, once the current date is given.
The first line of the input is an integer
T (
1 \le T \le 1000000
1≤T≤1000000) denoting the number of test cases. Each of the next
T lines describe 3 integers
D (
1 \le D \le 31
1≤D≤31),
M (
1 \le M \le 12
1≤M≤12) and
Y (
1900 \le Y \le 5000
1900≤Y≤5000) which denotes day, month and year.
You can safely assume that every date that is given as an input is a valid date.
For each date, print the next date in the following format:
Day must be formatted with a leading zero if it is less than 10. Month must be one of the following: Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec.
According to Wikipedia, a leap year can be identified in the following way:
If a year is not exactly divisible by 4 then it is a common year. Otherwise, if a year is not exactly divisible by 100 then it is a leap year. Otherwise, if a year is not exactly divisible by 400 then it is a common year. Else it is a leap year.
Ehsanul_FahadEarliest, Feb '16
mdshadeshFastest, 0.2s
subhashis_cseLightest, 131 kB
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Noncentral F mean and variance - MATLAB ncfstat - MathWorks Australia
Noncentral F mean and variance
[M,V] = ncfstat(NU1,NU2,DELTA)
[M,V] = ncfstat(NU1,NU2,DELTA) returns the mean of and variance for the noncentral F pdf with corresponding numerator degrees of freedom in NU1, denominator degrees of freedom in NU2, and positive noncentrality parameters in DELTA. NU1, NU2, and DELTA can be vectors, matrices, or multidimensional arrays that all have the same size, which is also the size of M and V. A scalar input for NU1, NU2, or DELTA is expanded to a constant array with the same dimensions as the other input.
The mean of the noncentral F distribution with parameters ν1, ν2, and δ is
\frac{{\nu }_{2}\left(\delta +{\nu }_{1}\right)}{{\nu }_{1}\left({\nu }_{2}-2\right)}
where ν2 > 2.
2{\left(\frac{{\nu }_{2}}{{\nu }_{1}}\right)}^{2}\left[\frac{{\left(\delta +{\nu }_{1}\right)}^{2}+\left(2\delta +{\nu }_{1}\right)\left({\nu }_{2}-2\right)}{{\left({\nu }_{2}-2\right)}^{2}\left({\nu }_{2}-4\right)}\right]
[m,v]= ncfstat(10,100,4)
ncfpdf | ncfcdf | ncfinv | ncfrnd
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FormStructureEquations - Maple Help
Home : Support : Online Help : Mathematics : DifferentialGeometry : LieAlgebras : LieAlgebraData : FormStructureEquations
LieAlgebraData[FormStructureEquations] - convert a list of exterior derivative equations to a Lie algebra data structure
LieAlgebraData(FormStructureEquations, Basis, AlgName)
FormStructureEquations - a list of equations of the form
d\left({\mathrm{θ}}_{}^{k}\right) =-{C}_{\mathrm{ij}}^{k}{\mathrm{θ}}^{i}∧{\mathrm{θ}}^{j}
i,j
i <j)
\left[{\mathrm{θ}}_{}^{1}, {\mathrm{θ}}^{2}, ... ,{\mathrm{θ}}_{}^{n}\right]
which defines a basis for the dual 1-forms of the Lie algebra
The command DGsetup is used to initialize a Lie algebra, that is, to define the basis elements for the Lie algebra and its dual and to store the structure constants for the Lie algebra in memory. The first argument for DGsetup is a Lie algebra data structure which contains the structure constants in a standard format used by the LieAlgebras package.
One commonly used format for the structure equations of a Lie algebra
\mathrm{𝔤}
is the set of exterior derivative equations for the dual 1-forms of the Lie algebra. For a 1-form
\mathrm{θ}
in the dual of a Lie algebra, the exterior derivative is the 2-form defined by
d\left(\mathrm{θ}\right)\left(x,y\right) = -{\mathrm{θ}}_{}\left(x,y\right)
x,y ∈\mathrm{𝔤}
. The function LieAlgebraData enables one to create a Lie algebra in Maple from a list of exterior derivative equations.
In this example, we create a Lie algebra data structure for a Lie algebra called Ex1 from a list of structure equations for the exterior derivatives of the dual 1-forms. The structure equations contain arbitrary constants
a,b, c
and we determine for which values of these parameters the Jacobi identities actually hold. First, we create the list of structure equations. The variables t1, t2, and t3 must be unassigned names. They simply serve as placeholders for the purpose of entering in the structure equations.
FormStrEq := [d(t1) = -t2 &w t3, d(t2) = - t1 &w t3, d(t3) = a*t1 &w t2 + b*t2 &w t3 + c*t1 &w t3];
\textcolor[rgb]{0,0,1}{\mathrm{FormStrEq}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{t1}}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{t2}}\textcolor[rgb]{0,0,1}{&w}\textcolor[rgb]{0,0,1}{\mathrm{t3}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{t2}}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{t1}}\textcolor[rgb]{0,0,1}{&w}\textcolor[rgb]{0,0,1}{\mathrm{t3}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{t3}}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{t1}}\textcolor[rgb]{0,0,1}{&w}\textcolor[rgb]{0,0,1}{\mathrm{t2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{t2}}\textcolor[rgb]{0,0,1}{&w}\textcolor[rgb]{0,0,1}{\mathrm{t3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{t1}}\textcolor[rgb]{0,0,1}{&w}\textcolor[rgb]{0,0,1}{\mathrm{t3}}\right]
Basis := [t1, t2, t3];
\textcolor[rgb]{0,0,1}{\mathrm{Basis}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{t1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{t2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{t3}}\right]
L := LieAlgebraData(FormStrEq, Basis, Ex1);
\textcolor[rgb]{0,0,1}{L}\textcolor[rgb]{0,0,1}{:=}\left[\left[\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e2}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\textcolor[rgb]{0,0,1}{,}\left[\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{e2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\textcolor[rgb]{0,0,1}{,}\left[\textcolor[rgb]{0,0,1}{\mathrm{e2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\right]
\textcolor[rgb]{0,0,1}{\mathrm{Lie algebra: Ex1}}
\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{θ1}}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{θ2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ3}}
\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{θ2}}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{θ1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ3}}
\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{θ3}}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ3}}
TF, EQ, SOLN, LIEALG := Query({a, b, c}, "Jacobi");
\textcolor[rgb]{0,0,1}{\mathrm{TF}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{EQ}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{SOLN}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{LIEALG}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\left\{\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{c}\right\}\textcolor[rgb]{0,0,1}{,}\left[\left\{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\right\}\right]\textcolor[rgb]{0,0,1}{,}\left[\left[\left[\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e2}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\textcolor[rgb]{0,0,1}{,}\left[\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{e2}}\textcolor[rgb]{0,0,1}{,}\left[\textcolor[rgb]{0,0,1}{\mathrm{e2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{e1}}\right]\right]
We conclude that the structure equations define a Lie Algebra for arbitrary
a,
b=c =0.
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Retrieving Data from a Computer Practice Problems Online | Brilliant
There are various devices that can store data on a computer. For example, hard disk drives (HDD), solid state drives (SSD), random access memory (RAM) modules, optical disks and drives, etc. We will go over key characteristics that are important from the software’s point of view. (We will not cover how each device is made or works.)
Retrieving Data from a Computer
CDs, DVDs, and Blu-rays are optical disks that can hold
700 \text{MB} - 50 \text{ GB}
of data, and are often used to distribute music, movies, and game software. ROM disks such as DVD-ROM are read-only and the stored data cannot be changed or overwritten. Rewritable disks such as DVD-RAM disks are read-write and the stored data can be changed or overwritten.
Assume that there is a video game software distributed on DVD-ROM disks. From the following, what is correct regarding the typical use of optical disks when distributing video game software?
DVD-ROM disks have a high cost since they cannot be mass produced DVD-ROM disks can be reused as blank media after finishing the game Game results or play data can be saved on other writable media but not on the read-only DVD-ROM disk DVD-ROM disks cannot hold enough content to feasibly store games or videos
HDDs are spinning magnetic disks that can hold up to a few TB of data. They are often used as the main storage on computers to hold files and folders. The data access speed is fast for sequential accesses and becomes slower for random access. Sequential access is reading or writing from a continuous chunk of data. Random access is reading or writing from different places on the disk.
Given an HDD where the sequential access speed is
100 \text{ MB/s}
, which has an overhead of 10 ms when accessing a different place (random access overhead), and assuming that files are placed randomly within the drive and each file is stored continuously without any fragmentation, which of the following is correct regarding the speed of (A) and (B)?
(A) read 200 files,
50 \text{ kB}
each (
10 \text{ MB}
total)
(B) read 20 files,
500 \text{ kB}
10 \text{ MB}
(B) is 7 times faster than (A) (B) is 2 times faster than (A) (A) and (B) take the same amount of time (A) is 2 times faster than (B)
RAM is a semiconductor module that stores data within an integrated circuit that can hold up to a few GB of data. RAM is typically about 100 times faster than HDDs, and about 100 times more expensive for the same amount of data capacity. RAM does not have a significant speed decrease on random access. RAM is a volatile memory which can only hold data when power is provided, and will lose data when power is removed. Devices like HDDs and DVDs are non-volatile since they can hold data when power is removed.
From the following, what is correct regarding volatility of devices?
We do not need to continuously provide power to RAM for it to hold data The OS does not need to load data from HDD to RAM when starting the computer We cannot expect data to remain on HDDs after turning the computer off We cannot expect data to remain on RAM after turning the computer off
Considering characteristics such as data capacity, access speed, volatility, read-only or read-write, the modern computer architecture uses HDDs or SSDs to store permanent data, and RAM for intermediate data. Programs will load files from the HDD or SSD to RAM, use RAM to calculate, and store necessary data in files on the HDD or SSD. RAM is accessed frequently from programs, and therefore it is called the “main memory”. In practical programming environments, when we use the word “memory”, we often mean RAM.
Given RAM and HDD where the access speed is 10 GB/s and 100 MB/s respectively, we want to read a 3MB file 25 times. Ignoring random access overhead and the cost of writing to RAM, which answer is correct regarding the speed of (A) and (B)?
(A) load the file from the HDD to RAM once, and read from RAM 25 times.
(B) read directly from the HDD 25 times.
(A) is 20 times faster than (B) (A) is 2 times faster than (B) (A) and (B) take the same amount of time (A) is slower than (B)
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Atmosphere - Kerbal Space Program Wiki
The pressures for all atmospheres
The atmosphere of a celestial body slows the movement of any object passing through it, a force known as atmospheric drag (or simply drag). An atmosphere also allows for aerodynamic lift. The celestial bodies with atmospheres are the planets Eve, Kerbin, Duna and Jool, as well as Laythe, a moon of Jool. Only Kerbin and Laythe have atmospheres that contain oxygen and thus produce intake air for jet engines to work.
Atmospheres allow aerobraking and easier landing. However, an atmosphere makes taking off from a planet more difficult and increases the minimum stable orbit altitude.
3 Atmospheric height
5 On-rails physics
Atmospheres have a pressure that varies in an exponential way with the increasing altitude:
{\displaystyle p=p_{0}\cdot e^{\frac {-altitude}{H}}}
where p0 is the atmosphere's pressure (Pa) at altitude 0, and H is the scale height (m).
The scale height of an atmosphere define at which rate the pressure drops with altitude. In the case of Kerbin's atmosphere, the scale height is about 5600m and means that the atmospheric pressure will be 2.718 times smaller each time you go 5600m higher.
Atmospheres also vary in temperature. Though this has little effect on gameplay, it affects the atmospheric density and the speed of sound both used in drag calculations. With the Ideal gas law the density (ρ) in kg/m3 is calculated with the equation:
where R is the specific gas constant equal to 287.053 J/kg-K.
The speed of sound (c) is calculated with:
Note that for Kerbin atmosphere, the pressure and temperature are modeled to match Earth atmosphere by using U.S. Standard Atmosphere (USSA) equations (see Kerbin atmosphere).
A Mk1-2 pod with a Mk16-XL parachute being slowed by drag in Kerbin's atmosphere.
In the game, the force of atmospheric drag is dependent on the shape of a part and which of the part's attachment nodes are in use. For example, nose cones reduce drag on the part to which they are attached. A debug-mode option allows you to see the exact drag force on a given part by right-clicking on it.
Generally, the drag works with the following equation:
{\displaystyle F_{D}=0.5\,\rho \,v^{2}\,d\,A}
where ρ is the atmospheric density (kg/m3), v is the ship's velocity (m/s), d is the coefficient of drag (dimensionless), and A is the cross-sectional area (m2).
The atmospheric height depends on the scale height of the celestial body and is where 0.000001th (0.0001%) of the surface pressure remains. Therefore, the atmospheric pressure at the edge of the atmosphere is relative; for example a craft in orbit around Jool can have a lower orbit (relative to the surface) because the surface pressure is higher.
{\displaystyle alt_{\text{atmospheric height}}=-ln\left(10^{-6}\right)\cdot {\text{scale height}}}
{\displaystyle p_{\text{atmospheric height}}=p_{0}\cdot 10^{-6}}
To calculate the atmospheric heights of other celestial bodies:
{\displaystyle alt_{\text{atmospheric height (real)}}=-ln\left({\frac {10^{-6}}{p_{0}}}\right)\cdot {\text{scale height}}}
→ See also: terminal velocity on Wikipedia
The terminal velocity of an object falling through an atmosphere is the velocity at which the force of gravity is equal to the force of drag. Terminal velocity changes as a function of altitude. Given enough time, an object falling into the atmosphere will slow to terminal velocity and then remain at terminal velocity for the rest of its fall.
Terminal velocity is important because:
It describes the amount of velocity which a spacecraft must burn away when it is close to the ground.
It represents the speed at which a ship should be traveling upward during a fuel-optimal ascent.
The force of gravity (FG) is:
{\displaystyle F_{G}=m\cdot a=m\cdot {\frac {GM}{r^{2}}}}
where m is still the ship's mass, G is the gravitational constant, M is the mass of the planet, and r is the distance from the center of the planet to the falling object.
The terminal velocity can be found by finding for what speed FG is equal to FD.
The terminal velocity for an given imaginary body depends from the altitude on the different celestial bodies.
0 58.385 100.13 212.41 23.124 115.62
100 58.783 101.01 214.21 23.162 116.32
1000 62.494 109.30 231.16 23.508 122.83
10000 115.27 240.52 495.18 27.272 211.77
On-rails physics
A ship is "on rails" when it's no longer the primary focus of the simulation, which occurs when it's further than 2.25 km from the actively-controlled ship. If such a ship have its orbit passing through a planet's atmosphere, one of two things will happen based on atmospheric pressure at the ship's altitude:
below 0.01 atm: no atmospheric drag will occur — the ship will be completely unaffected
0.01 atm or above: the ship will disappear
The following table gives the altitude of this 0.01 atm threshold for each celestial body with an atmosphere:
Retrieved from "https://wiki.kerbalspaceprogram.com/index.php?title=Atmosphere&oldid=96050"
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Generate pulses for PWM-controlled three-level converter - Simulink - MathWorks Nordic
Generate pulses for PWM-controlled three-level converter
The PWM Generator (3-Level) block generates pulses for carrier-based pulse-width modulation (PWM) converters using three-level topology. The block can control switching devices (FETs, GTOs, or IGBTs) of three different converter types: single-phase half-bridge (one arm), single-phase full-bridge (two arms), or three-phase bridge (three arms).
The reference signal (Uref input), also called the modulating signal, is naturally sampled and compared with two symmetrical level-shifted triangle carriers.
The following figure shows how the pulses are generated for a single-phase, half-bridge three-level converter.
The converter arm can have three states: +1, 0, or −1. When the reference signal is greater than the positive carrier, the state of the arm is +1; when the reference signal is smaller than the negative carrier, the state of the arm is −1.
Otherwise, the state is 0. Based on the current state of the arm, the appropriate pulses are generated.
One reference signal is required to generate the four pulses of an arm. For a single-phase full-bridge converter, a second reference signal is required to generate the four pulses of the second arm. This signal is internally generated by phase-shifting the original reference signal by 180 degrees. For a three-phase bridge, three reference signals are required to generate the 12 pulses.
The reference signal also can be internally generated by the PWM generator. In this case, specify a modulation index, voltage output frequency, and phase.
Specify the number of pulses to generate. The number of pulses generated by the block is proportional to the number of bridge arms to control.
Select Single-phase half-bridge (4 pulses) to fire the self-commutated devices of a single-phase half-bridge converter. Pulses (1, 2) fire the upper devices and pulses (3, 4) fire the lower devices.
Select Single-phase full-bridge (8 pulses) to fire the self-commutated devices of a single-phase full-bridge converter. Eight pulses are then generated. Pulses (1, 2) and (5, 6) fire the upper devices of the first and second arms. Pulses (3, 4) and (7, 8) fire the lower devices of the first and second arms.
Select Three-phase bridge (12 pulses) (default) to fire the self-commutated devices of a three-phase bridge converter. Pulses (1, 2), (5, 6), and (9, 10) fire the upper devices of the first, second, and third arms. Pulses (3, 4), (7, 8), and (11, 12) fire the lower devices of the three arms.
When set to Unsynchronized (default), the frequency of the unsynchronized carrier signal is determined by the Carrier frequency parameter.
When set to Synchronized, the carrier signal is synchronized to an external reference signal (input wt), and the carrier frequency is determined by the Switching ratio parameter.
Specify to determine the frequency, in hertz, of the two triangular carrier signals. Default is 27*60. The Carrier frequency parameter is visible only when the Mode of operation parameter is set to Unsynchronized.
Determines the frequency (Fc) of the two triangular carrier signals.
{F}_{c}=SwitchingRatio×OutputVoltageFrequency
Default is 27. The Switching ratio parameter is visible only when the Mode of operation parameter is set to Synchronized.
Internal generation of modulating signal (s)
When this check box is selected, the block generates the reference signal. Default is cleared.
The parameter is visible only when the Mode of operation parameter is set to Unsynchronized.
Specify the modulation index to control the amplitude of the fundamental component of the output voltage of the converter. Default is 0.8. The modulation index must be greater than 0 and lower than or equal to 1. The parameter is visible only when the Internal generation of modulating signal (s) check box is selected.
Output voltage frequency (Hz)
Specify the output voltage frequency to control the fundamental component frequency of the output voltage of the converter. Default is 60. The parameter is visible only when the Internal generation of modulating signal (s) check box is selected.
Output voltage phase (degrees)
This parameter controls the phase of the fundamental component of the output voltage of the converter. Default is 0. The parameter is visible only when the Internal generation of modulating signal (s) check box is selected.
Specify the sample time of the block, in seconds. Default is 0. Set to 0 to implement a continuous block.
The vectorized reference signal used to generate the output pulses. The input is visible only when the Internal generation of modulating signal (s) is not selected. Connect this input to:
A single-phase sinusoidal signal when the block controls a single-phase half- or full-bridge converter
A three-phase sinusoidal signal when the PWM Generator block controls a three-phase bridge converter
For linear operation of this block, the magnitude of Uref must be between −1 and +1.
The output contains the 4, 8, or 12 pulses used to fire the self-commutated devices (MOSFETs, GTOs, or IGBTs) of a one-, two- or three-arm three-level converter.
The power_PWMGenerator3Level model uses a simple circuit to show how the PWM Generator (3-Level) operates. Run the simulation and use the FFT Analysis tool of the Powergui block to see the harmonics and the THD value of the voltages produced by the three-phase three-level converter.
The model sample time is parameterized by the Ts variable set to a default value of 2e-6. Set Ts to 0 in the command window and change the Simulation type parameter of the Powergui block to Continuous to simulate the model in continuous mode.
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Every angle in the figure is a right angle, and tick marks mean those segments have equal lengths.
If we know the area of the blue region is
48,
can we find the area of the orange region? Keep reading to see one way we can do this, or skip ahead to the challenge if you feel ready for the next step.
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Saltwater intrusion - Wikipedia
Saltwater intrusion is the movement of saline water into freshwater aquifers, which can lead to groundwater quality degradation, including drinking water sources, and other consequences. Saltwater intrusion can naturally occur in coastal aquifers, owing to the hydraulic connection between groundwater and seawater. Because saline water has a higher mineral content than freshwater, it is denser and has a higher water pressure. As a result, saltwater can push inland beneath the freshwater.[1] In other topologies, submarine groundwater discharge can push fresh water into saltwater.
Certain human activities, especially groundwater pumping from coastal freshwater wells, have increased saltwater intrusion in many coastal areas. Water extraction drops the level of fresh groundwater, reducing its water pressure and allowing saltwater to flow further inland. Other contributors to saltwater intrusion include navigation channels or agricultural and drainage channels, which provide conduits for saltwater to move inland. Sea level rise caused by climate change also contributes to saltwater intrusion.[2] Saltwater intrusion can also be worsened by extreme events like hurricane storm surges.[3]
2.1 Groundwater extraction
2.2 Canals and drainage networks
3 Effect on water supply
4 Ghyben–Herzberg relation
6 Mitigation and management
7 Areas of occurrence
Further information: Salt wedge
Cause and impact of saltwater intrusion
At the coastal margin, fresh groundwater flowing from inland areas meets with saline groundwater from the ocean. The fresh groundwater flows from inland areas towards the coast where elevation and groundwater levels are lower.[2] Because saltwater has a higher content of dissolved salts and minerals, it is denser than freshwater, causing it to have higher hydraulic head than freshwater. Hydraulic head refers to the liquid pressure exerted by a water column: a water column with higher hydraulic head will move into a water column with lower hydraulic head, if the columns are connected.[4]
The higher pressure and density of saltwater causes it to move into coastal aquifers in a wedge shape under the freshwater. The saltwater and freshwater meet in a transition zone where mixing occurs through dispersion and diffusion. Ordinarily the inland extent of the saltwater wedge is limited because fresh groundwater levels, or the height of the freshwater column, increases as land elevation gets higher.[2]
Groundwater extractionEdit
Groundwater extraction is the primary cause of saltwater intrusion. Groundwater is the main source of drinking water in many coastal areas of the United States, and extraction has increased over time. Under baseline conditions, the inland extent of saltwater is limited by higher pressure exerted by the freshwater column, owing to its higher elevation. Groundwater extraction can lower the level of the freshwater table, reducing the pressure exerted by the freshwater column and allowing the denser saltwater to move inland laterally.[2] In Cape May, New Jersey, since the 1940s water withdrawals have lowered groundwater levels by up to 30 meters, reducing the water table to below sea level and causing widespread intrusion and contamination of water supply wells.[5][6]
Groundwater extraction can also lead to well contamination by causing upwelling, or upcoming, of saltwater from the depths of the aquifer.[7] Under baseline conditions, a saltwater wedge extends inland, underneath the freshwater because of its higher density. Water supply wells located over or near the saltwater wedge can draw the saltwater upward, creating a saltwater cone that might reach and contaminate the well. Some aquifers are predisposed towards this type of intrusion, such as the Lower Floridan aquifer: though a relatively impermeable rock or clay layer separates fresh groundwater from saltwater, isolated cracks breach the confining layer, promoting upward movement of saltwater. Pumping of groundwater strengthens this effect by lowering the water table, reducing the downward push of freshwater.[6]
Canals and drainage networksEdit
The construction of canals and drainage networks can lead to saltwater intrusion. Canals provide conduits for saltwater to be carried inland, as does the deepening of existing channels for navigation purposes.[2][8] In Sabine Lake Estuary in the Gulf of Mexico, large-scale waterways have allowed saltwater to move into the lake, and upstream into the rivers feeding the lake. Additionally, channel dredging in the surrounding wetlands to facilitate oil and gas drilling has caused land subsidence, further promoting inland saltwater movement.[9]
Drainage networks constructed to drain flat coastal areas can lead to intrusion by lowering the freshwater table, reducing the water pressure exerted by the freshwater column. Saltwater intrusion in southeast Florida has occurred largely as a result of drainage canals built between 1903 into the 1980s to drain the Everglades for agricultural and urban development. The main cause of intrusion was the lowering of the water table, though the canals also conveyed seawater inland until the construction of water control gates.[6]
Effect on water supplyEdit
Many coastal communities around the United States are experiencing saltwater contamination of water supply wells, and this problem has been seen for decades.[10] Many Mediterranean coastal aquifers suffer for seawater intrusion effects.[11][12] The consequences of saltwater intrusion for supply wells vary widely, depending on extent of the intrusion, the intended use of the water, and whether the salinity exceeds standards for the intended use.[2][13] In some areas such as Washington State, intrusion only reaches portions of the aquifer, affecting only certain water supply wells. Other aquifers have faced more widespread salinity contamination, significantly affecting groundwater supplies for the region. For instance, in Cape May, New Jersey, where groundwater extraction has lowered water tables by up to 30 meters, saltwater intrusion has caused closure of over 120 water supply wells since the 1940s.[6]
Ghyben–Herzberg relationEdit
The first physical formulations of saltwater intrusion were made by Willem Badon-Ghijben [pt] in 1888 and 1889 as well as Alexander Herzberg [de] in 1901, thus called the Ghyben–Herzberg relation.[14] They derived analytical solutions to approximate the intrusion behavior, which are based on a number of assumptions that do not hold in all field cases.
the Ghyben–Herzberg relation[2]
{\displaystyle z={\frac {\rho _{f}}{(\rho _{s}-\rho _{f})}}h}
the thickness of the freshwater zone above sea level is represented as
{\displaystyle h}
and that below sea level is represented as
{\displaystyle z}
. The two thicknesses
{\displaystyle h}
{\displaystyle z}
, are related by
{\displaystyle \rho _{f}}
{\displaystyle \rho _{s}}
{\displaystyle \rho _{f}}
is the density of freshwater and
{\displaystyle \rho _{s}}
is the density of saltwater. Freshwater has a density of about 1.000 grams per cubic centimeter (g/cm3) at 20 °C, whereas that of seawater is about 1.025 g/cm3. The equation can be simplified to
{\displaystyle z\ =40h}
The Ghyben–Herzberg ratio states that, for every meter of fresh water in an unconfined aquifer above sea level, there will be forty meters of fresh water in the aquifer below sea level.
In the 20th century the vastly increased computing power available allowed the use of numerical methods (usually finite differences or finite elements) that need fewer assumptions and can be applied more generally.[15]
Modeling of saltwater intrusion is considered difficult. Some typical difficulties that arise are:
The possible presence of fissures and cracks and fractures in the aquifer, whose precise positions and extents are unknown but which have great influence on the development of the saltwater intrusion
The possible presence of small scale heterogeneities in the hydraulic properties of the aquifer, which are too small to be taken into account by the model but which may also have great influence on the development of the saltwater intrusion
The process known as cation exchange, which slows the advance of a saltwater intrusion and also slows the retreat of a saltwater intrusion.
The fact that saltwater intrusions are often not in equilibrium makes it harder to model. Aquifer dynamics tend to be slow and it takes the intrusion cone a long time to adapt to changes in pumping schemes, rainfall, etc. So the situation in the field can be significantly different from what would be expected based on the sea level, pumping scheme etc.
For long-term models, the future climate change forms a large unknown but good results are possible . Model results often depend strongly on sea level and recharge rate. Both are expected to change in the future.
Mitigation and managementEdit
Catfish Point control structure (lock) on the Mermentau River in coastal Louisiana
Saltwater is also an issue where a lock separates saltwater from freshwater (for example the Hiram M. Chittenden Locks in Washington). In this case a collection basin was built from which the saltwater can be pumped back to the sea. Some of the intruding saltwater is also pumped to the fish ladder to make it more attractive to migrating fish.[16]
As groundwater salinization becomes a relevant problem, more complex initiatives should be applied from local technical and engineering solutions to rules or regulatory instruments for whole aquifers or regions.[17]
Areas of occurrenceEdit
Bou Regreg (Morocco)
ACF River Basin (Florida/Georgia)
Essex County, Massachusetts[18]
Hiram M. Chittenden Locks (Washington)
Hutchinson Island (Georgia)
Oxnard Plain (California)
Sonoma Creek (California)
Western Shore of Lake Superior[19] (Minnesota)
Italy[11][20][21]
Environmental migrant – People forced to leave their home region due to changes to their local environment
Peak water – Concept on the quality and availability of freshwater resources
^ Johnson, Teddy (2007). "Battling Seawater Intrusion in the Central & West Coast Basins" (PDF). Water Replenishment District of Southern California. Archived from the original (PDF) on 2012-09-08. Retrieved 2012-10-08.
^ a b c d e f g h Barlow, Paul M. (2003). "Ground Water in Freshwater-Saltwater Environments of the Atlantic Coast". USGS. Retrieved 2009-03-21.
^ "CWPtionary Saltwater Intrusion yes". LaCoast.gov. 1996. Retrieved 2009-03-21.
^ Johnson, Ted (2007). "Battling Seawater Intrusion in the Central & West Coast Basins" (PDF). Water Replenishment District of Southern California. Archived from the original (PDF) on 2012-09-08. Retrieved 2012-10-08.
^ Lacombe, Pierre J. & Carleton, Glen B. (2002). "Hydrogeologic Framework, Availability of Water Supplies, and Saltwater Intrusion, Cape May County, New Jersey" (PDF). USGS. Retrieved 2012-12-10.
^ a b c d Barlow, Paul M. & Reichard, Eric G. (2010). "Saltwater intrusion in coastal regions of North America". Hydrogeology Journal. USGS. 18 (1): 247–260. Bibcode:2010HydJ...18..247B. doi:10.1007/s10040-009-0514-3. S2CID 128870219. Retrieved 2012-12-10.
^ Reilly, T.E. & Goodman, A.S. (1987). "Analysis of saltwater upconing beneath a pumping well". Journal of Hydrology. 89 (3–4): 169–204. Bibcode:1987JHyd...89..169R. doi:10.1016/0022-1694(87)90179-x.
^ Good, B. J., Buchtel, J., Meffert, D.J., Radford, J., Rhinehart, W., Wilson, R. (1995). "Louisiana's Major Coastal Navigation Channels" (pdf). Louisiana Department of Natural Resources. Retrieved 2013-09-14. {{cite web}}: CS1 maint: multiple names: authors list (link)
^ Barlow, Paul M. (2008). "Preliminary Investigation: Saltwater Barrier - Lower Sabine River" (PDF). Sabine River Authority of Texas. Retrieved 2012-12-09. [permanent dead link]
^ Todd, David K. (1960). "Salt water intrusion of coastal aquifers in the United States" (PDF). Subterranean Water. IAHS Publ. (52): 452–461. Archived from the original (PDF) on 2005-10-25. Retrieved 2009-03-22.
^ a b Polemio, Maurizio (2016-04-01). "Monitoring and Management of Karstic Coastal Groundwater in a Changing Environment (Southern Italy): A Review of a Regional Experience". Water. 8 (4): 148. doi:10.3390/w8040148.
^ Polemio, Maurizio; Pambuku, Arben; Limoni, Pier Paolo; Petrucci, Olga (2011-01-01). "Carbonate Coastal Aquifer of Vlora Bay and Groundwater Submarine Discharge (Southwestern Albania)". Journal of Coastal Research. 270: 26–34. doi:10.2112/SI_58_4. ISSN 0749-0208. S2CID 54861536.
^ Romanazzi A, Polemio M. "Modelling of coastal karst aquifers for management support: Study of Salento (Apulia, Italy)" (PDF). Italian Journal of Engineering Geology and Environment. 13, 1: 65–83.
^ Verrjuit, Arnold (1968). "A note on the Ghyben-Herzberg formula" (PDF). Bulletin of the International Association of Scientific Hydrology. Delft, Netherlands: Technological University. 13 (4): 43–46. doi:10.1080/02626666809493624. Retrieved 2009-03-21. [permanent dead link]
^ Romanazzi, A.; Gentile, F.; Polemio, M. (2015-07-01). "Modelling and management of a Mediterranean karstic coastal aquifer under the effects of seawater intrusion and climate change". Environmental Earth Sciences. 74 (1): 115–128. doi:10.1007/s12665-015-4423-6. ISSN 1866-6299. S2CID 56376966.
^ Mausshardt, Sherrill; Singleton, Glen (1995). "Mitigating Salt-Water Intrusion through Hiram M. Chittenden Locks". Journal of Waterway, Port, Coastal, and Ocean Engineering. 121 (4): 224–227. doi:10.1061/(ASCE)0733-950X(1995)121:4(224).
^ Polemio, Maurizio; Zuffianò, Livia Emanuela (2020). "Review of Utilization Management of Groundwater at Risk of Salinization". Journal of Water Resources Planning and Management. 146 (9): 03120002. doi:10.1061/(ASCE)WR.1943-5452.0001278. ISSN 0733-9496. S2CID 225224426.
^ "Case Studies of Various Water Quality Problems | H2O Care".
^ "In a Pickle: The Mystery of the North Shore's Salty Well Water". www.seagrant.umn.edu. Retrieved 2018-09-27.
^ Vespasiano, Giovanni; Cianflone, Giuseppe; Romanazzi, Andrea; Apollaro, Carmine; Dominici, Rocco; Polemio, Maurizio; De Rosa, Rosanna (2019-11-01). "A multidisciplinary approach for sustainable management of a complex coastal plain: The case of Sibari Plain (Southern Italy)". Marine and Petroleum Geology. 109: 740–759. doi:10.1016/j.marpetgeo.2019.06.031. ISSN 0264-8172. S2CID 197580624.
^ Zuffianò, L. E.; Basso, A.; Casarano, D.; Dragone, V.; Limoni, P. P.; Romanazzi, A.; Santaloia, F.; Polemio, M. (2016-07-01). "Coastal hydrogeological system of Mar Piccolo (Taranto, Italy)". Environmental Science and Pollution Research. 23 (13): 12502–12514. doi:10.1007/s11356-015-4932-6. ISSN 1614-7499. PMID 26201653. S2CID 9262421.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Saltwater_intrusion&oldid=1066074760"
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Phased Array Ultrasonic Measurement of Fatigue Crack Growth Profiles in Stainless Steel Pipes | J. Pressure Vessel Technol. | ASME Digital Collection
L. Satyarnarayan,
Center for Nondestructive Evaluation,
, Chennai 600 036, India; Department of Mechanical Engineering,
D. M. Pukazhendhi,
D. M. Pukazhendhi
, Structural Engineering Research Centre, Taramani, Chennai 600 113, India
Krishnan Balasubramaniam,
e-mail: balas@iitm.ac.in
C. V. Krishnamurthy,
C. V. Krishnamurthy
Satyarnarayan, L., Pukazhendhi, D. M., Balasubramaniam, K., Krishnamurthy, C. V., and Ramachandra Murthy, D. S. (July 24, 2006). "Phased Array Ultrasonic Measurement of Fatigue Crack Growth Profiles in Stainless Steel Pipes." ASME. J. Pressure Vessel Technol. November 2007; 129(4): 737–743. https://doi.org/10.1115/1.2767367
This paper reports experimental sizing of fatigue crack profiles that are initiated from artificially made circumferential starter notches in stainless steel pipes of
169mm
outer diameter and
14.33mm
thickness, which were subjected to cyclic bending loads in a four point bending load arrangement using two nondestractive evaluation (NDE) methods: (a) phased array ultrasonic technique and (b) alternating current potential drop technique. The crack growth estimated using the two NDE techniques were compared with the beach marks that were present in the fracture surface. A simulation study using the ray tracing method was carried out to model the ultrasonic wave propagation in the test specimen, and the results were compared with the experimental results.
bending, fatigue cracks, fracture, pipes, ray tracing, stainless steel, ultrasonic arrays, ultrasonic materials testing, ultrasonic propagation, fatigue cracks, phased array, ACPD, beach marks, crack sizing, RATT, ray tracing, simulation
Fatigue cracks, Fracture (Materials), Pipes, Stainless steel, Simulation, Ray tracing
An Approach for Plant Specific, Risk-Informed Decision Making: In-Service Inspection of Piping
,” USNRC Draft Regulatory Guide DG-1063.
The Influence of Skin Depth on Crack Measurement by the AC Field Technique
Data Interpretation in ACPD Crack Inspection
Long-Term Installations of the DC-Potential Drop Method in Four Nuclear Power Plants and the Accuracies Thereby Obtained for Monitoring of Crack Initiation and Crack Growth
In Situ Ultrasonic Monitoring of Surface Fatigue Crack Initiation and Growth From Surface Cavity
Shear-Wave Time of Flight Diffraction (S-TOFD) Technique
Depth Gauging of Defects Using Low Frequency Wideband Rayleigh Waves
R/D Tech Guideline
R/D Tech’s Technology Information
), http://www.ndt.net/article/v07n05/rdtech/rdtech.htmhttp://www.ndt.net/article/v07n05/rdtech/rdtech.htm
Poguet
Phased Array Technology: Concepts, Probes and Applications
), http://www.ndt.net/article/v07n05/poguet/poguet.htmhttp://www.ndt.net/article/v07n05/poguet/poguet.htm
Campos-Pozuelo
Finite-Difference and Finite-Volume Methods for Nonlinear Standing Ultrasonic Waves in Fluid Media
Ciorau
Contribution to Detection and Sizing Linear Defects by Conventional and Phased Array Ultrasonic Techniques
), www.ndt.net/article/v10n09/ciorau1/ciorau1.htmwww.ndt.net/article/v10n09/ciorau1/ciorau1.htm
Through Weld Inspection of Wrought Stainless Steel Piping Using Phased Array Ultrasonic Probes
16th WCNDT 2004-World Conference on NDT
, www.ndt.net/article/wcndt2004/pdf/array_transducers/460_anderson.pdfwww.ndt.net/article/wcndt2004/pdf/array_transducers/460_anderson.pdf
Advanced Ultrasonic Flaw Sizing Handbook
), http://www.ndt.net/article/1198/davis/davis2.htmhttp://www.ndt.net/article/1198/davis/davis2.htm
Approximative Modeling for the Practical Application at Ultrasonic Inspections
), http://www.ndt.net/article/wsho0597/wuesten2/wuesten2.htmhttp://www.ndt.net/article/wsho0597/wuesten2/wuesten2.htm
Crack Initiation and Growth Behavior of Circumferentially Cracked Pipes Under Cyclic and Monotonic Loading
Simultsonic: A Simulation Tool for Ultrasonic Inspection
J. Korean Soc. NDT
Analysis of a Weld Overlay to Address Fatigue Cracking in a Stainless Steel Nozzle
Thermal-Hydraulic and LBB Evaluations to Justify Short-Term Plant Operation With a CRD Return Line Susceptible to Potential Thermal Stratification
Fatigue Crack Initiation in Hydrogen-Precharged Austenitic Stainless Steel
|
How to use the cube density calculator?
What is density? What is the density formula of a cube?
How to calculate the volume of a cube for density?
More density calculators
How do I calculate the density of a concrete cube?
The cube density calculator is a simple tool with great possibilities. 'How to find the density of a cube?' will never again be an issue for you! Read on to find out:
what is the density formula of a cube,
how to calculate the density of a cube step by step, and
how to calculate the density of a concrete cube.
Using the cube density calculator is simple and intuitive, but check those steps out in case of any doubts.
See the calculator panel on the left side of the page.
The first thing you'll is a cube image with some dimensions highlighted. Thanks to it, you can see what measurements we're talking about.
Input your cube's mass (weight) into the cube density calculator. Remember you can always change the unit so that it suits you.
Input the volume of your cube in a convenient unit. If you don't know the volume, follow the next step. Otherwise, you can skip it.
Input the cube measurement that you know: either side, cube diagonal, or face diagonal length. The calculator will calculate the volume automatically.
You see the result in the Density field immediately!
If you want to know how to find the cube's density step by step, keep on reading the text.
Density is a term used in physics to describe mass per unit volume of space.
\rho = \frac{m}{V}
\rho
m
V
If two objects are the same size, but one is heavier, we can say it has a greater density.
A cube is a regular square prism in three orientations. It has got six square faces with three faces meeting at each vertex.
If we want to know the cube's density, we have to use a volume of a cube in the main density equation.
Volume of a cube can be calculated with the formula
V = a^3
V
- volume,
a
- length of a cube edge.
Combining two equations, we get the density formula of a cube:
\rho = \frac{m}{a^3}
\rho
m
a
Here are three ways of calculating the volume of a cube depending on what measurement you have available. The letters in brackets refer to the picture above the calculator.
You know the side (a) of a cube
Calculate the volume (
V
\qquad V = a^3
You know a cube diagonal (d)
\qquad V =\left(\frac{d}{\sqrt{3}}\right)^3
You know a face diagonal (f)
\qquad V=\left(\frac{f}{\sqrt{2}}\right)^3
We've got more cool density calculators at our site:
Although the density of concrete is already defined (2400 kg/m3), sometimes you need to calculate it yourself.
Find the mass (weight) of your concrete cube.
Then, measure the cube side.
How to calculate the volume of a cube for density? Use the formula: volume = side 3.
Now, divide the mass by the volume. density = mass/volume
Your cube density calculation is ready!
How do I find the density of a sugar cube?
To find the density of a sugar cube:
Take a ruler or another measuring tool, and measure the side of the sugar cube. Let's call it an a.
Calculate the volume of the cube: volume = a3
Now weigh the sugar cube using a kitchen scale.
To find the density, you need to solve the equation:
How do I calculate the density of a cube of wood?
Wood is not a homogeneous material, so if you want to be accurate about the density of a wooden cube, better calculate it yourself:
Weigh your wooden cube.
Measure the cube. The easiest way is to measure the side of it.
Calculate the volume of your thing of wood. volume = (side length)3.
Now, count the density of the cube. Density = mass (weight)/volume.
How do I find the density of a cube with a side of 2 ft and a weight of 5 lbs?
The density of that cube is 0.625 lb/ft3. To calculate the density of that cube:
Calculate the volume of the cube with the formula volume = side3.
Our case, volume = 2 ft3 = 8 ft3.
To calculate density, solve the equation density = mass/volume.
In our case, density = 5 lbs/8 ft3 = 0.625 lb/ft3.
The answer is: density of the cube = 0.625 lb/ft3.
|
Analyzing Elastic Collisions | Brilliant Math & Science Wiki
Rohit Gupta, Sravanth C., Ram Mohith, and
A perfectly elastic collision is one in which conservation of energy holds, in addition to conservation of momentum. As a result of energy's conservation, no sound, light, or permanent deformation occurs. As perfectly elastic collisions are ideal, they rarely appear in nature, but many collisions can be approximated as perfectly elastic.
Newton's cradle exhibits nearly perfect elastic collisions, which is why it (ideally) swings back and forth in perpetuity. [1]
Inertia of a body Conservation of momentum and energy Every action has equal and opposite reaction Force is directly proportional to mass and acceleration of the body
The gif given above demonstrates the working of a Newton’s cradle.
What does this amazing science toy explain to us?
Collision in Slow Motion
To understand a collision, consider the following example:
In this case, a block of mass
{m_1}
is coming towards a stationary mass
{m_2}
u
. A spring of spring constant
k
is attached to mass
{m_2}
and the ground is frictionless. Now, when the moving block hits the spring, it tries to compress the spring. As the spring is compressed, it starts applying force on both the blocks. Due to this force, block
{m_1}
is retarded and block
{m_2}
is accelerated.
What can be said about the momentum of the system? Will it be conserved?
According to conservation of linear momentum, if net external force in any direction is zero, then the total momentum of the system is conserved. As the ground is smooth and the forces involved in the collision are internal forces which will be canceled in pairs, there is no force left in the horizontal direction and the momentum of the system of two blocks will be conserved.
In view of energies, the kinetic energy of block
{m_1}
decreases and the potential energy and kinetic energy of
{m_2}
increases. The speed of block
{m_1}
will keep decreasing until its speed is greater than that of block
{m_2}
. When the speeds of the blocks become equal, the compression in spring is at its maximum. Beyond this point, the speed of block
{m_1}
further decreases and the speed of block
{m_2}
increases. Thus block
{m_2}
starts moving faster and the spring starts recovering from compression.
Eventually, the spring gains its natural length and all the energy stored in the spring during the collision is restored back into the kinetic energy of the block. After the process of collision is over, part of kinetic energy of block
{m_1}
is transferred to
{m_2}
. As the spring is considered ideal and perfectly elastic, no loss of kinetic energy eventually occurs. The kinetic energy during the collisions goes into the deformation energy, but due to perfect elastic nature of the spring, deformations are perfectly recovered and all the kinetic energy is recovered. Such a collision is called an elastic collision.
If the spring is not perfectly elastic and converts some of the potential energy into heat energy, then the final kinetic energy of the system will be less than the initial kinetic energy of the system. Such a collision is called an inelastic collision.
An elastic collision is a collision in which colliding objects are perfectly elastic and the deformations occurring during collisions are fully recovered. Thus the kinetic energy of the colliding objects before collision equals the total kinetic energy after collision.
_\square
Consider the diagram shown below. It shows a collision between two moving objects on a frictionless ground.
{u_1}
{u_2}
are initial velocities of blocks
{m_1}
{m_2}
{v_1}
{v_2}
are their corresponding final velocities after the collision.
The two objects are traveling along the same line and hit head-on with each other. The objects are perfectly elastic and the ground is smooth.
As there is no net external force in horizontal direction, linear momentum is conserved in horizontal direction:
{m_1}{u_1} + {m_2}{u_2} = {m_1}{v_1} + {m_2}{v_2}. \qquad (1)
Since the collision is perfectly elastic, the kinetic energy during the collision equals the kinetic energy after the collision:
\frac{1}{2}{m_1}u_1^2 + \frac{1}{2}{m_2}u_{_2}^2 = \frac{1}{2}{m_1}v_{_1}^2 + \frac{1}{2}{m_2}v_{_2}^2. \qquad (2)
Solving these two simultaneous equations, we get
\begin{aligned} {v_1} &= \left( {\frac{{{m_1} - {m_2}}}{{{m_1} + {m_2}}}} \right)\,{u_1} + \frac{{2{m_2}{u_2}}}{{{m_1} + {m_2}}}\\ {v_2} &= \left( {\frac{{{m_2} - {m_1}}}{{{m_1} + {m_2}}}} \right)\,{u_2} + \frac{{2{m_1}{u_1}}}{{{m_1} + {m_2}}}. \end{aligned}
The velocity of approach is the rate at which the distance between the colliding objects is decreasing. In the above case as
{m_1}
is approaching and
{m_2}
is receding, the velocity of approach is
{u_1} - {u_2}
The velocity of separation is the rate at which the distance between the colliding objects (after the collision) is increasing. In the above case, as
{m_1}
is receding and
{m_2}
is still approaching after the collision, the velocity of separation is
{v_2} - {v_1}
On solving the equations (1) and (2), we can also get
{u_1} - {u_2} = {v_2} - {v_1},
{u_1} - {u_2}
is called the velocity of approach and
{v_2} - {v_1}
is called the velocity of separation.
Important points regarding elastic collision
If a perfectly elastic ball collides with a fixed surface, it rebounds with the same speed. A fixed surface can be treated as an infinite mass object with zero speed. Thus putting
{m_2} \to \infty
{u_2}=0
{v_1} = \mathop {\lim }\limits_{{m_2} \to \infty } \left[ {\left( {\frac{{{m_1} - {m_2}}}{{{m_1} + {m_2}}}} \right)\,{u_1} + \frac{{2{m_2}{u_2}}}{{{m_1} + {m_2}}}} \right],
{v_1} = - {u_1} .
If a perfectly elastic collision takes place between two objects of equal masses, then after the collision their velocities will exchange. That means the initial velocity of the first block will be the final velocity of the second block and vice versa. Thus, for the equal mass, we put
{m_1} = {m_2}
{v_1} = \left( {\frac{{{m_1} - {m_2}}}{{{m_1} + {m_2}}}} \right)\,{u_1} + \frac{{2{m_2}{u_2}}}{{{m_1} + {m_2}}}
{v_1} = {u_2}
{v_2} = {u_1}.
moving with horizontal speed 6 m/sec collides with a block of mass
M
moving with speed 4 m/s in the same direction. The ground is smooth. If
m<<M,
then for one-dimensional elastic collision find the speed of mass
m
after collision.
{v_1}
is the final speed of mass
m,
{v_1} = \left( {\frac{{m - M}}{{m + M}}} \right)\,{u_1} + \frac{{2M{u_2}}}{{m + M}}.
m<<M,
{v_1} = - {u_1} + 2{u_2}= - \,6 + 2(4) = 2 ~(\text{m/s}).
That is, the lighter particle will move in the original direction with a speed of 2 m/s.
_\square
Two balls from the right roll out with speed
v
each and the remaining balls remain stationary One ball from the right rolls out with speed
2v
and the remaining balls remain at rest All six balls roll out with speed
\frac v6
each and the two colliding balls come to rest One ball from left rolls back with the same speed
v
and one from right rolls forward with
v
Six identical balls are lined in a straight groove made on a horizontal, frictionless surface, as shown below. Two similar balls each moving with velocity
v
collide with the row of 6 balls from left.
What will happen after the collision, if the collisions are perfectly elastic?
Toussaint, D. Newtons cradle animation book 2. Retrieved May 25, 2016, from https://commons.wikimedia.org/wiki/File:Newtons_cradle_animation_book_2.gif
Cite as: Analyzing Elastic Collisions. Brilliant.org. Retrieved from https://brilliant.org/wiki/analyzing-elastic-collisions/
|
Complexity Theory | Brilliant Math & Science Wiki
Complexity theory is a central topic in theoretical computer science. It has direct applications to computability theory and uses computation models such as Turing machines to help test complexity. Complexity theory helps computer scientists relate and group problems together into complexity classes. Sometimes, if one problem can be solved, it opens a way to solve other problems in its complexity class. Complexity helps determine the difficulty of a problem, often measured by how much time and space (memory) it takes to solve a particular problem. For example, some problems can be solved in polynomial amounts of time and others take exponential amounts of time, with respect to the input size.
Complexity theory has real world implications too, particularly with algorithm design and analysis. An algorithm can be analyzed in terms of its complexity, this is often described in big-O notation. Often times, programmers want to write efficient algorithms, and being able to tell if an algorithm runs in polynomial time versus exponential time can tell a programmer if his or her algorithm is the best choice or not. Complexity theory has applications for biologists studying neurons, electrical engineers who design hardware, linguists who study languages and grammars, and physicists building quantum computers.[1]
Both in theory and in practice, complexity theory helps computer scientists determine the limits of what computers can and cannot do.
In many cases, problems can be modeled as decision problems which are problems that can be answered with a “yes” or a “no.” For example, “is this number prime?”, “does this graph have a hamiltonian path?” “is there an assignment of variables to the equation such that a set of constraints are satisfied?” Examples of decision problems include the travelling salesperson problem, 3SAT, and primality testing. Decision problems can be simulated on computational models such as Turing machines.
Background topics for complexity theory
Complexity of Important Problems
Complexity theory can be one of the more challenging topics in theoretical computer science since it requires a fair amount of background. To really appreciate complexity theory, one should be familiar with the following topics:
Regular languages, context-free grammars, and context-free languages. These topics provide the vocabulary for describing problems that complexity theory deals with.
Turing Machines are the usual model for testing where a problem belongs in the complexity hierarchy, so you should be familiar with how Turing machines are defined and how they work.
Here is a brief overview of complexity classes, find out more on its wiki page.
Complexity classes are used to group together problems that require similar amounts of resources. For example, the group of problems that can be solved in polynomial time are considered a part of the class P. The group of problems that take an exponential amount of space are in the class EXPSPACE. Some classes are contained within other classes — for example, if a problem can be solved in polynomial time, it can certainly be solved in exponential time too. The image below describes the relationship of many common complexity classes.
Determining if classes are equivalent rather than just contained in one another is a key problem in theoretical computer science.This image shows the relationships of many popular complexity classes.
Complexity is often used to describe an algorithm. One might hear something like “my sorting algorithm runs in oh of
n^2
time” in complexity, this is written as
O(n^2)
and is a polynomial running time. Complexity is used to describe resource use in algorithms. In general, the resources of concern are time and space. The time complexity of an algorithm represents the number of steps it has to take to complete. The space complexity of an algorithm represents the amount of memory the algorithm needs in order to work.
The time complexity of an algorithm describes how many steps an algorithm needs to take with respect to the input. If, for example, each of the
n
inputed elements is only operated on once, this algorithm would take
O(n)
time. If, for example, the algorithm needs to operate on one element of an input (no matter the input size), this is a constant time, or
O(1)
, algorithm since no matter the input size only one thing is done. If an algorithm does
n
operations for each one of the
n
elements inputed to the algorithm, then this algorithm runs in
O(n^2)
In algorithm design and analysis, there are three types of complexity that computer scientists think about: best-case, worst-case, and average-case complexity.
Best, Worst, and Average-case Complexity
Complexity can describe time and space, this wiki will speak in terms of time complexity, but the same concepts can be applied to space complexity.
Let’s say you are sorting a list of numbers. If the input list is already sorted, your algorithm probably has very little work to do — this could be considered a “best-case” input and would have a very fast running time. Let’s take the same sorting algorithm and give it an input list that is entirely backwards, and every element is out of place. This could be considered a “worst-case” input and would have a very slow running time. Now say you have a random input that is somewhat ordered and somewhat disordered (an average input). This would take the average-case running time.
If you know something about your data, for example if you have reason to expect that your list will generally be mostly sorted, and can therefore count on your best-case running time, you might choose an algorithm with a great best-case running time, even if it has a terrible worst and average-case running time. Usually, though, programmers need to write algorithms that can efficiently handle any input, so computer scientists are generally particularly concerned with worst-case running times of algorithms.
Complexity has theoretical applications as well. Many important problems in computer science, such as the P vs NP problem are explained using complexity theory. Many important theoretical computer science problems essentially boil down to
“Are these two problems reducible to each other; does an answer to problem A help us solve problem B?”
“Is complexity class X equivalent complexity class Y?”
Answers to these types of questions often have profound implications for theoretical computer science and real world applications. For example, if P were proven to be equal to NP, most of our security algorithms, like RSA, would be incredibly easy to break.
In the 1970s, Cook and Levin proved that Boolean satisfiability is an NP-Complete problem, meaning that it can be transformed into any other problem in the NP class. In other words, the satisfiability problem can model any other problem in NP. This shows us a powerful consequence of complexity theory: by determining how to solve a satisfiability problem and using the Cook-Levin theorem, we know how to approach all other problems in NP. Additionally, this means that if computer scientists figure out how to solve an NP-compete problem in polynomial time, all other problems in NP could be solved in polynomial time, in other words P would equal NP.
To “test” a problem’s complexity, computer scientists will try to solve the problem on a Turing machine and see how many steps (time complexity) and how much tape (space complexity) it requires to decide a problem.
We can use a Turing machine to solve an instance of a problem or verify a proposed answer for the problem. For example, the problem might be the traveling salesperson problem, and the instance would be a particular graph that the traveling salesperson is traveling. One instance of the traveling salesperson problem could have the salesperson traveling in New York City and another instance could have the salesperson traveling in London. In either case, the goal of the salesperson is the same, though the specific graph may be different.
The running time of a particular problem, like the traveling salesperson problem, may depend on the particular instance. For example, larger instances may require more time to solve. This is why the complexity of a given problem is calculated as a function of the size of the particular instance. Usually, the size of the input is measured in bits. In general, complexity theory deals with how algorithms scale with an increase in the input size.[3]. Instances are encoded as strings of bits that follow particular patterns or rules (similar to regular languages and context free languages. The Turing machine will take this problem, modeled as a language, and feed the input to the problem.
While we saw on the Turing machine wiki that a Turing machine takes in a program and operates on an input according to that program, in complexity proofs, we usually just abstract away the specific Turing machine program. The Church-Turing thesis says that any computable problem can be computed on a Turing machine, so we can safely assume that if a problem is computable, there is a programming for it. The time complexity of a problem is determined by how many steps the Turing machine takes to solve the problem, and the space complexity of the problem is how many spaces on the tape the machine needed.
Scheideler, C. 1 Introduction to Complexity Theory. Retrieved July 10, 2016, from http://www.cs.jhu.edu/~scheideler/courses/600.471_S05/lecture_1.pdf
, . Computational complexity theory. Retrieved July 12, 2016, from https://en.wikipedia.org/wiki/Computational_complexity_theory
, N. File:Turing machine 2b.svg. Retrieved July 12, 2016, from https://en.wikipedia.org/wiki/File:Turing_machine_2b.svg
Cite as: Complexity Theory. Brilliant.org. Retrieved from https://brilliant.org/wiki/complexity-theory/
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Home : Support : Online Help : Mathematics : Fractals : Newton
Newton( n, zbl, zur, expr )
Newton( n, zbl, zur, expr, opts )
algebraic; the univariate expression to which the iterative Newton process will be applied
tolerance : keyword option of the form tolerance=value where value is positive and of type realcons. The iterative process is stopped for each complex input point if convergence is ascertained. The default value is 0.001.
If the unknown variable in expr is, say, w and the derivative with respect to w is taken as dexpr then the iterates are computed by the formula,
\mathrm{inc}=\frac{\genfrac{}{}{0}{}{\mathrm{expr}}{\phantom{w={z}_{i-1}}}|\genfrac{}{}{0}{}{\phantom{\mathrm{expr}}}{w={z}_{i-1}}}{\genfrac{}{}{0}{}{\mathrm{dexpr}}{\phantom{w={z}_{i-1}}}|\genfrac{}{}{0}{}{\phantom{\mathrm{dexpr}}}{w={z}_{i-1}}}
{z}_{i}={z}_{i-1}-\mathrm{inc}
Convergence is accepted if either abs(inc)/abs(z[i])<tolerance or abs(z[i])<tolerance.
The 2-D grayscale Array image returned by supplying the option output=layer1 contains data denoting the number of iterations required for each entry to escape. The grayscale image returned by supplying the option output=layer2 contains the absolute values of the final values for entries which escape. For either layer the real data is scaled to 0.0 .. 1.0 before being returned as an image.
The 3-D color Array image returned by supplying the option output=color contains data where the three layers corresponding to red, green, and blue have been computed using the raw escape data.
The 3-D Array returned by supplying the option output=raw contains the unscaled data of layer1 and layer2. This Array can be used to generate a customized color image using the Colorize command.
\mathrm{with}\left(\mathrm{Fractals}:-\mathrm{EscapeTime}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{BurningShip}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Colorize}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{HSVColorize}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Julia}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{LColorize}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Lyapunov}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Mandelbrot}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Newton}}]
\mathrm{with}\left(\mathrm{ImageTools}\right):
\mathrm{bl},\mathrm{ur}≔-6-6I,6+6I
\textcolor[rgb]{0,0,1}{\mathrm{bl}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{ur}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-6}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}
f≔{t}^{3}-{t}^{2}-12
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{12}
M≔\mathrm{Newton}\left(500,\mathrm{bl},\mathrm{ur},f\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893628717907069996}}
\mathrm{Embed}\left(M\right)
\mathrm{Embed}\left(\mathrm{Newton}\left(500,\mathrm{bl},\mathrm{ur},f,\mathrm{output}=\mathrm{layer1}\right)\right)
\mathrm{Embed}\left([[\mathrm{Newton}\left(200,\mathrm{bl},\mathrm{ur},f,\mathrm{iterationlimit}=18,\mathrm{output}=[\mathrm{layer1},\mathrm{color}]\right)],[\mathrm{Newton}\left(200,\mathrm{bl},\mathrm{ur},f,\mathrm{iterationlimit}=30,\mathrm{output}=[\mathrm{layer1},\mathrm{color}]\right)],[\mathrm{Newton}\left(200,\mathrm{bl},\mathrm{ur},f,\mathrm{iterationlimit}=150,\mathrm{output}=[\mathrm{layer1},\mathrm{color}]\right)]]\right)
\mathrm{bl},\mathrm{ur}≔-2-2I,2+2I
\textcolor[rgb]{0,0,1}{\mathrm{bl}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{ur}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-2}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}
f≔{z}^{4}+2{z}^{3}-100
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{100}
\mathrm{Embed}\left(\mathrm{Newton}\left(500,\mathrm{bl},\mathrm{ur},f\right)\right)
\mathrm{Embed}\left([\mathrm{Newton}\left(500,\mathrm{bl},\mathrm{ur},f,\mathrm{output}=[\mathrm{layer1},\mathrm{layer2},\mathrm{color}]\right)]\right)
\mathrm{bl},\mathrm{ur}≔-2-2I,2+2I
\textcolor[rgb]{0,0,1}{\mathrm{bl}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{ur}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-2}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}
f≔\mathrm{cosh}\left(t\right)+3
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{cosh}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
\mathrm{Embed}\left(\mathrm{Newton}\left(500,\mathrm{bl},\mathrm{ur},f\right)\right)
The Fractals:-EscapeTime:-Newton command was introduced in Maple 18.
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Focal Plane Shift Imaging for the Analysis of Multi-Droplet Jumping | J. Heat Transfer | ASME Digital Collection
Hyeongyun Cha,
Department of Mechanical Science and Engineering, University of Illinois, Urbana, IL 61801
cha10@illinois.edu
Jae Min Chun,
Jae Min Chun
chun35@illinois.edu
Yuehan Xu,
Yuehan Xu
yxu56@illinois.edu
Cha, H., Chun, J. M., Xu, Y., and Miljkovic, N. (January 6, 2017). "Focal Plane Shift Imaging for the Analysis of Multi-Droplet Jumping." ASME. J. Heat Transfer. February 2017; 139(2): 020903. https://doi.org/10.1115/1.4035573
High speed images of coalescence induced three-droplet jumping on a nanostructured superhydrophobic carbon nanotube (CNT) surface are presented (
θaapp
≈ 173º). When two or more droplets coalesce on a nanostructured superhydrophobic surface, the resulting droplet can jump away from the surface due to the release of excess surface energy. To more easily study the jumping phenomena, we have developed focal plane shift imaging (FPSI) to determine both jumping speed and direction. Figure 1(a) shows a schematic of the FPSI concept. A high speed camera was attached to an upright optical microscope, and samples were horizontally mounted on a cold stage. Initial conditions were obtained by moving the focal plane to be coincident with the middle of the droplets prior to coalescence (Figure 1b). Then the focal plane was shifted above the droplets by a known distance (Figure 1c), followed by measurement of the time taken for the jumping droplet to pass through the shifted focal plane (Figure 1d). By analyzing the initial and final conditions of the departing droplet for multiple three-droplet coalescence events, the three-droplet jumping droplet speed was determined (Figure 2b). Experimentally measured jumping speeds determined by the FPSI imaging technique show good agreement with three-droplet inertial capillary scaling (Figure 2b, dotted line, equation inset). The FPSI visualization technique provides a novel imaging platform for the study of complex multi-droplet jumping-droplet phenomena.
Drops, Imaging, Carbon nanotubes, Optical microscopes, Surface energy, Visualization
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Raku rules - Wikipedia
Raku rules
(Redirected from Perl 6 rules)
Raku rules are the regular expression, string matching and general-purpose parsing facility of the Raku programming language, and are a core part of the language. Since Perl's pattern-matching constructs have exceeded the capabilities of formal regular expressions for some time, Raku documentation refers to them exclusively as regexes, distancing the term from the formal definition.
Raku provides a superset of Perl 5 features with respect to regexes, folding them into a larger framework called rules, which provide the capabilities of a parsing expression grammar, as well as acting as a closure with respect to their lexical scope.[1] Rules are introduced with the rule keyword, which has a usage quite similar to subroutine definitions. Anonymous rules can be introduced with the regex (or rx) keyword, or simply be used inline as regexes were in Perl 5 via the m (matching) or s (substitution) operators.
2 Changes from Perl 5
2.1 Implicit changes
3 Integration with Perl
In Apocalypse 5, a document outlining the preliminary design decisions for Raku pattern matching, Larry Wall enumerated 20 problems with the "current regex culture". Among these were that Perl's regexes were "too compact and 'cute'", had "too much reliance on too few metacharacters", "little support for named captures", "little support for grammars", and "poor integration with 'real' language".[2]
Between late 2004 and mid-2005, a compiler for Raku style rules was developed for the Parrot virtual machine called Parrot Grammar Engine (PGE), which was later renamed to the more generic Parser Grammar Engine. PGE is a combination of runtime and compiler for Raku style grammars that allows any parrot-based compiler to use these tools for parsing, and also to provide rules to their runtimes.
Among other Raku features, support for named captures was added to Perl 5.10 in 2007.[3]
In May 2012, the reference implementation of Raku, Rakudo, shipped its Rakudo Star monthly snapshot with a working JSON parser built entirely in Raku rules.[4]
Changes from Perl 5[edit]
There are only six unchanged features from Perl 5's regexes:
Literals: word characters (letters, numbers and underscore) matched literally
Repetition quantifiers: *, +, and ?, but not {m,n}
Minimal matching suffix: *?, +?, ??
A few of the most powerful additions include:
The ability to reference rules using <rulename> to build up entire grammars.
A handful of commit operators that allow the programmer to control backtracking during matching.
The following changes greatly improve the readability of regexes:
Simplified non-capturing groups: [...], which are the same as Perl 5's: (?:...)
Simplified code assertions: <?{...}>
Allows for whitespace to be included without being matched, allowing for multiline regexes. Use \ or ' ' to express whitespace.
Extended regex formatting (Perl 5's /x) is now the default.
Implicit changes[edit]
Some of the features of Perl 5 regular expressions are more powerful in Raku because of their ability to encapsulate the expanded features of Raku rules. For example, in Perl 5, there were positive and negative lookahead operators (?=...) and (?!...). In Raku these same features exist, but are called <before ...> and <!before ...>.
However, because before can encapsulate arbitrary rules, it can be used to express lookahead as a syntactic predicate for a grammar. For example, the following parsing expression grammar describes the classic non-context-free language
{\displaystyle \{a^{n}b^{n}c^{n}:n\geq 1\}}
S ← &(A !b) a+ B
A ← a A? b
B ← b B? c
In Raku rules that would be:
rule S { <before <A> <!before b>> a+ <B> }
rule A { a <A>? b }
rule B { b <B>? c }
Of course, given the ability to mix rules and regular code, that can be simplified even further:
rule S { (a+) (b+) (c+) <{$0.elems == $1.elems == $2.elems}> }
However, this makes use of assertions, which is a subtly different concept in Raku rules, but more substantially different in parsing theory, making this a semantic rather than syntactic predicate. The most important difference in practice is performance. There is no way for the rule engine to know what conditions the assertion may match, so no optimization of this process can be made.
Integration with Perl[edit]
In many languages, regular expressions are entered as strings, which are then passed to library routines that parse and compile them into an internal state. In Perl 5, regular expressions shared some of the lexical analysis with Perl's scanner. This simplified many aspects of regular expression usage, though it added a great deal of complexity to the scanner. In Raku, rules are part of the grammar of the language. No separate parser exists for rules, as it did in Perl 5. This means that code, embedded in rules, is parsed at the same time as the rule itself and its surrounding code. For example, it is possible to nest rules and code without re-invoking the parser:
rule ab {
(a.) # match "a" followed by any character
# Then check to see if that character was "b"
# If so, print a message.
{ $0 ~~ /b {say "found the b"}/ }
The above is a single block of Raku code that contains an outer rule definition, an inner block of assertion code, and inside of that a regex that contains one more level of assertion.
There are several keywords used in conjunction with Raku rules:
A named or anonymous regex that ignores whitespace within the regex by default.
A named or anonymous regex that implies the :ratchet modifier.
A named or anonymous regex that implies the :ratchet and :sigspace modifiers.
An anonymous regex that takes arbitrary delimiters such as // where regex only takes braces.
An operator form of anonymous regex that performs matches with arbitrary delimiters.
Shorthand for m with the :sigspace modifier.
An operator form of anonymous regex that performs substitution with arbitrary delimiters.
Shorthand for s with the :sigspace modifier.
Simply placing a regex between slashes is shorthand for rx/.../.
Here is an example of typical use:
rule phrase { <word> [ \, <word> ]* \. }
if $string ~~ / <phrase> \n / {
Modifiers may be placed after any of the regex keywords, and before the delimiter. If a regex is named, the modifier comes after the name. Modifiers control the way regexes are parsed and how they behave. They are always introduced with a leading : character.
Some of the more important modifiers include:
:i or :ignorecase – Perform matching without respect to case.
:m or :ignoremark – Perform matching without respect to combining characters.
:g or :global – Perform the match more than once on a given target string.
:s or :sigspace – Replace whitespace in the regex with a whitespace-matching rule, rather than simply ignoring it.
:Perl5 – Treat the regex as a Perl 5 regular expression.
:ratchet – Never perform backtracking in the rule.
regex addition { :ratchet :sigspace <term> \+ <expr> }
A grammar may be defined using the grammar operator. A grammar is essentially just a namespace for rules:
regex format_token { \%: <index>? <precision>? <modifier>? <directive> }
token index { \d+ \$ }
token flags { <[\ +0\#\-]>+ }
token precision_count { [ <[1-9]>\d* | \* ]? [ \. [ \d* | \* ] ]? }
token vector { \*? v }
token modifier { ll | <[lhmVqL]> }
token directive { <[\%csduoxefgXEGbpniDUOF]> }
This is the grammar used to define Perl's sprintf string formatting notation.
Outside of this namespace, you could use these rules like so:
if / <Str::SprintfFormat::format_token> / { ... }
A rule used in this way is actually identical to the invocation of a subroutine with the extra semantics and side-effects of pattern matching (e.g., rule invocations can be backtracked).
Here are some example rules in Raku:
rx { a [ b | c ] (d | e) f : g }
rx { (ab*) <{ $1.size % 2 == 0 }> }
That last is identical to:
rx { (ab[bb]*) }
^ Wall, Larry (June 24, 2002). "Synopsis 5: Regexes and Rules".
^ Wall, Larry (June 4, 2002). "Apocalypse 5: Pattern Matching".
^ Perl 5.10 now available - Perl Buzz Archived 2008-01-09 at the Wayback Machine
^ moritz (May 5, 2012). "Rakudo Star 2012.05 released".
Raku Grammars - The reference manual page for grammars.
Grammar tutorial - A tutorial for grammars in Raku
Synopsis 05 - The standards document covering Perl 6 regexes and rules.
Perl 6 Regex Introduction - Gentle introduction to Perl 6 regexes.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Raku_rules&oldid=996007735"
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{\displaystyle \mu \,}
The highway lighting items normally field inspected by Materials are galvanizing of anchor bolts, nuts, washers, polyurethane foam and steel standards, bracket arms, and foundations.
Field determination of weight of coating is to be made on each lot of material furnished. The magnetic gauge is to be operated and calibrated in accordance with ASTM E 376. At least three members of each size and type offered for inspection are to be selected for testing. A single-spot test is to be comprised of at least five readings of the magnetic gauge taken in a small area and those five readings averaged to obtain a single-spot test result. Three such areas should be tested on each of the members being tested. Test each member in the same manner. Average all single-spot test results from all members to obtain the average coating weight to be reported. The minimum single-spot test result would be the minimum average obtained on any one member. Material may be accepted or rejected for galvanized coating on the basis of magnetic gauge. If a test result fails to comply with the specifications, that lot should be re-sampled at double the original sampling rate. If any of the resample members fail to comply with the specification, that lot is to be rejected. The contractor or supplier is to be given the option of sampling for Central Laboratory testing, if the magnetic gauge test results are within minus 15 percent of the specified coating weight.
Additional requirements for bolts, nuts, and wasters are given in EPG 1080.1.3 Bolts for Highway Lighting, Traffic Signals or Highway Signing.
Polyurethane foam used as pole backfill is to be accepted on the basis of manufacturer's certification and random sampling and testing. The manufacturer's certification is to show typical test results representative of the material and certify that the material supplied conforms to all the requirements specified. Random samples are to be taken from approximately 10 percent of the lots offered for use. A sample is to consist of a portion of each component adequate in size to yield 2 cu. ft. of polyurethane foam, after mixing. AASHTOWare is to be used when submitting samples to the Central Laboratory. Very small quantities of polyurethane foam may be accepted on the basis of brand name and labeling, provided satisfactory results are obtained in the field.
Tests for weight of coating are to be reported through AASHTOWare. If a sample is submitted to the Central Laboratory for testing, use AASHTOWare.
Polyurethane foam shall be reported through AASHTOWare. The manufacturer's certification shall be retained in the district office, except when reporting very small quantities accepted by brand name and labeling.
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Contiguous Function Relations for -Hypergeometric Functions
Shahid Mubeen, Gauhar Rahman, Abdur Rehman, Mammona Naz, "Contiguous Function Relations for -Hypergeometric Functions", International Scholarly Research Notices, vol. 2014, Article ID 410801, 6 pages, 2014. https://doi.org/10.1155/2014/410801
Shahid Mubeen,1 Gauhar Rahman,1 Abdur Rehman ,1 and Mammona Naz1
In this research work, our aim is to determine the contiguous function relations for -hypergeometric functions with one parameter corresponding to Gauss fifteen contiguous function relations for hypergeometric functions and also obtain contiguous function relations for two parameters. Throughout in this research paper, we find out the contiguous function relations for both the cases in terms of a new parameter . Obviously if , then the contiguous function relations for -hypergeometric functions are Gauss contiguous function relations.
The hypergeometric function plays an important role in mathematical analysis and its applications. This function allows us to solve many interesting problems, such as conformal mapping of triangular domains bounded by line segments or circular arcs and various problems of quantum mechanics. Most of the functions that occur in the analysis are special cases of the hypergeometric functions. Gauss first introduced and studied hypergeometric series, paying special attention to the cases when a series can be summed into an elementary function. This gives one of the motivations for studying hypergeometric series; that is, the fact that the elementary functions and several other important functions in mathematics can be expressed in terms of hypergeometric functions. Hypergeometric functions can also be described as solutions of special second order linear differential equations, the hypergeometric differential equations. Riemann was the first to exploit this idea and introduce a special symbol to classify hypergeometric functions by singularities and exponents of differential equations. The hypergeometric function is a solution of the following Euler’s hypergeometric differential equation , which has three regular singular points 0, 1, and . The generalization of this equation to three arbitrary regular singular points is given by Riemann’s differential equation. Any second order differential equation with three regular singular points can be converted to the hypergeometric differential equation by changing of variable.
On the other hand, in theory of hypergeometric function, the terminology contiguous function was introduced for the case in which one of the parameters is shifted by ±1. For example, is contiguous to . Gauss defined two hypergeometric functions to be contiguous if they have the same power series variable and if two of the parameters are pairwise equal and if third pair differs by ±1. He proved that between and any two of its contiguous functions, there exists a linear relation with coefficients at most linear in . These relationships are of great use in extending numerical tables of the function, since, for one fixed value of , it is necessary only to calculate the value of the function over two units in , , and , and apply some recurrence relations in order to find the function values over a large range of values of , , and in this particular -plane. Since there are six contiguous functions to a given function , Gauss [1] got a total of fifteen relations. In fact, only four of the fifteen are really independent as all others may be obtained by elimination and use of the fact that the is symmetric in and .
Applications of the contiguous function relations range from the evaluation of hypergeometric series to the derivation of the summation and transformation formulas for such series; these can be used to evaluate the contiguous functions to a hypergeometric function. For this, in a series of three research papers, Lavoie et al. [2, 3] have obtained a large number of very interesting results contiguous to Gauss second, Kummer, and Bailey theorems for the series . For more details about hypergeometric series and their contiguous relations, see [4–10].
In 1996, Morita [11] used Gauss contiguous relations in computing the hypergeometric function . In 1996, Gupta et al. [12] derived contiguous relations, basic hypergeometric functions, and orthogonal polynomials. In 2002, Vidunas [13] generalized the Kummer identity by using the contiguous relations. In 2003, Vidunas [14] defined several properties of coefficients of these general contiguous relations and then used them to propose effective ways to compute contiguous relations. In 2006, Rakha and Ibrahim [15] obtained some interesting consequences of the contiguous relations of . In 2008, Ibrahim and Rakha [16] derived the contiguous relations and their computations for hypergeometric series. They obtained the interesting formula as a linear relation of three shifted Gauss polynomials in the parameters , , and .
Díaz et al. [17–19] introduced -gamma and -beta functions () and proved a number of their properties. They have also studied -zeta functions and -hypergeometric functions based on Pochhammer -symbols for factorial functions. Very recently, many researchers [20–23] followed the work of Díaz et al., they studied and obtained some consequence results of -beta, -gamma, -zeta, and -hypergeometric functions and their properties. In 2009, Mansour [24] obtained the -generalized gamma function by functional equations. In 2012, Mubeen and Habibullah [25] defined -fractional integration and gave its applications regarding fractional integrals. In 2012, Mubeen and Habibullah [26] also gave a useful and simple integral representation of some confluent -hypergeometric functions and -hypergeometric functions . Furthermore, in 2013, Mubeen [27] defined a second order linear -hypergeometric differential equation having one solution in the form of -hypergeometric function .
In this section, we present the definitions of some basic concepts in the term of new parameter .
Definition 1. Two hypergeometric functions are said to be contiguous if their parameters , , and differ by integers. The relations made by contiguous functions are said to be contiguous function relations.
Definition 2. Let ; then the Pochhammer -symbol is defined by , for , , and .
Definition 3. For and , the -gamma function is defined by Its integral representation is also given by The relation between Pochhammer -symbol and -gamma function is given as follows: Furthermore, we can write -gamma function in terms of ordinary gamma function in the following form:
Definition 4. The -hypergeometric function with three parameters , , and , two parameters , in numerator and one parameter in denominator is defined by for all , , , and .
3. Contiguous Functions for -Hypergeometric Function with One Parameter
If we increase or decrease one and only one of the parameters of -hypergeometric function, by , then the resultant function is said to be contiguous to . For simplicity, we use the following notations: Similarly, we can write the notations for , , , and .
Let Then, (7) becomes Now, consider (8) as Since , therefore, by using this result in (12), we obtain Similarly, we can write (9) by using the result as Thus we have the following six contiguous functions for , where : By the help of differential operator , we get the following result: Hence, with the aid of (12), it follows that Similarly, we can also write the following relations as
4. Contiguous Relations for -Hypergeometric Function
Since there are six contiguous functions to a given function , therefore, we have to obtain the following fifteen contiguous function relations for -hypergeometric function , for all and . In fact, only four of the fifteen are really independent as all others may be obtained by elimination and use of the fact that the is symmetric in and .
Relation 1. Consider
Proof. By subtracting (18) from (17), we get This implies the required relation.
Proof. By subtracting (19) from (17), we obtain
Proof. Let us consider By the help of differential operator , we get the following: By shifting the index with , we get Since , , and , therefore, by putting these three results in (27), we have Since , by substituting this result in (28), we obtain This implies that Now, from (17), we have By comparing these two equations, (30) and (31), we get
Proof. Let us consider By applying the differential operator , we get By shifting the index with , we can write Since , , and , by substituting these three results in (36), we obtain the following: This implies that Now, by replacing with in (17), we get By comparing (38) and (39), we get Now, we obtain the remaining contiguous function relations by elimination and use of the fact that the is symmetric in and , for .
Relation 10. Consider
5. Contiguous Functions for -Hypergeometric Function with Two Parameters
In this section, we obtain contiguous functions for -hypergeometric function with two parameters. Since we know , , and , where , therefore, by using these contiguous functions, we obtain the following contiguous function with two parameters Similarly, we can write Now, we have to prove the following contiguous relations.
Relation 16. It show that
Proof. One has to prove that From Relations 4 and 5, respectively, we have Therefore, by substituting the values of and in (55), we get After simplification, we get the required relation. Consider
Relation 17. It shows that
Proof. From Relation 16, we have This can be written as Therefore, by replacing with in (61), we obtain
Proof. By consider Relation 9, This may be written as Now, by replacing with in Relation 9, we obtain
In this research work, we determined the contiguous function relations for -hypergeometric function with one and two parameters. So we conclude that if , we obtain Gauss and Rainville contiguous function relations for hypergeometric functions.
The authors would like to express profound gratitude to referees for their deeper review of this paper and the referee’s useful suggestions that led to an improved presentation of the paper.
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Splay Tree | Brilliant Math & Science Wiki
Alex Chumbley and Karleigh Moore contributed
The splay tree is a type of binary search tree. Unlike other variants like the AVL tree, the red-black tree, or the scapegoat tree, the splay tree is not always balanced. Instead, it is optimized so that elements that have been recently acessed are quick to access again.
This property is similar in nature to a stack. A stack has the Last-In-First-Out (LIFO) property, so the most recent item you've added is the quickest one to access. Splay trees are similar in that when you add a new item, it becomes the root of the tree no matter what. But they take it a step further. Even when an item is simply searched for, it becomes the new root of the tree. The following gif shows how a splay tree would insert the elements 7, 3, and 9 in that order.
Because it is an unbalanced binary search tree, the splay tree cannot guarantee worst-case
O(\log_2(n))
time, like balanced binary search trees can. Balanced binary search trees have a height that is at most
\log_2(n)
n
is the number of nodes in the tree. So, the splay tree can only guarantee
O(\log_2(n))
Splay trees are a lot like other binary search trees. They have nodes, and each node has two children, one left and one right. There is a root node that serves at the begining of the tree. The main difference, though, is that the root node is always the last element that was accessed.
If an element is inserted into the tree, the tree will be rotated so that it appears at the root position. Also, if an element is searched for in the tree, it will move the same way.
Deletion is an operation that is largely left up to the implementer. Oftentimes, however, the element to be deleted is either switched with the left-most node in its right subtree or the right-most node in its left subtree. Then it is simply removed from the tree.
Splay trees are often used for inter-tree operations, such as joins, merges, unions, and other set related mathematical operations because splay trees are efficient at these operations. Also, splay trees are used when queries are highly biased. That is, when a set of queries favors a certain element, splay trees are effective. This is because the elements that are queried for frequently will appear towards the top of the tree. For example, if a splay tree is being used to store a set of names of people working at a store, it might be the case that the manager's name is queried for the most. Because of that, the manager's name will spend most of its time at the very top of the tree, making it easy to find again.
Can you think of an order of insertion operations for the set {1, 2, 3, 4} that would make the splay tree extremely inefficient? Hint: It would make the average case of lookup
O(n)
like a regular binary search tree.
If we inserted the set {1, 2, 3, 4} in ascending order, the tree would have no branching factor at all. The splay tree would look like this.
The same would be true if we inserted them in descending order. The tree would just be mirrored.
Splay trees support all of the typical binary search tree operations - search, insertion, and deletion. However, it is the sub-operation of splaying the tree that makes all of these operations possible.
Splaying is what keeps the splay tree roughly balanced. To splay a node, splaying steps are repeatedly performed on it until it rises to the top. To decide what kind of splaying step to perform, the tree looks at three possibilities:
The node's parent is the root
The node is the left child of a right child (or the right child of a left child)
The node is the left child of a left child (or the right child of a right child)
If the node's parent is the root, we only need one rotation to make it the root. If the node is the left child of the root, we perform a right rotation, and if the node is the right child of the root, we perform a left rotation. This is exactly the same as the rotations in an AVL tree. This is sometimes known as the "zig" case.
If the node is the left child of a right child, we need to perform two rotations. Let N be the node we are trying to splay, P is its parent, and G is its grandparent. It first rotates N and P right, then rotates N and G left. If the node is the right child of a left child, it does the opposite. It first rotates N and P left, then N and G right. This is sometimes referred to as the "zig-zag" case.
If the node is the left child of a left child, there are also two rotations. First, G and P are rotated right. Then X and P are rotated right. If the node is the right child of a right child, G and P are first rotated left followed by X and P being rotated left.
This is an example of the "zig-zag" case. The circles represent nodes in the tree, and the triangles under them represent subtrees. Node 9 is being accessed. First it rotates left with it's parent, 7. Then node 9 rotates right with its original grandparent, 12.
Splay zig-zag rotation
All three cases are used on a node until it becomes the root node (the "zig" case will be the last one performed).
Search is simple once splay is implemented. To search for a node, use binary search down the tree to locate the node. Then, perform splay on that node to bring it to the top of the tree.
To insert a node, find its appropriate location at the bottom of the tree using binary search. Then perform splay on that node.
Deletion is the only operation that has some wiggle room for the implementer. After all, there's no obvious node to splay when you're removing a node.
A typical implementation is the programmer can switch the node to be deleted with the right-most node in its left subtree or the left-most node in its right subtree. Then the node can be deleted without any consequence to the tree (since it has no children).
Another way is the node to be deleted is first splayed, making it a root node. Then it is deleted. We are left with two seprarate trees that are then joined together using the join operation, which we will see later.
Regardless, a typical implementation will eventually splay the parent of the deleted node. This is similarly up to the discretion of the implementer.
In addition to search, insertion, and deletion and the splaying that facilitates all three, splay trees also have additional operations. These operations are much faster in a splay tree than in other trees due to the splay operation. They are also one of the reasons splay trees are so attractive for software.
To join two trees, S and T, such that the elements in S are smaller than all the elements in T, two things must happen. First, splay the largest element in S. The results in the root of S being the largest node in S, and it has no right child. Second, set the right child of the root of S to be the root of T. The resulting tree is a binary search tree.
Given a node, splitting a tree at that node results in two trees. One tree has every element that was less than or equal to the node, and the other has every element greater than the node. First, we'll splay the node to the root. Then, we take the right child of the root and make it its own tree. Now there are two trees that satisfy the split condition.
Remembering that there are cases where the splay tree is very inefficient, the worst case asymptotic complexity of the splay tree is linear time operations. However, this is frequently overlooked due to amortized analysis, and so splay trees are always considered to have
O(\log_2(n))
time operations.
Operation Average Worst
O(n)
O(n)
O(\log_2(n))
O(\log_2(n))
O(n)
O(n)
O(\log_2(n))
O(\log_2(n))
O(\log_2(n))
O(\log_2(n))
The splay tree's main advantage is that it keeps the most queried nodes at the top of the tree, decreasing time for subsequent queries. This locality of reference makes the splay very useful for systems like garbage collection for programming languages or cache systems.
Splay trees also have low memory overhead, similar to scapegoat trees. This makes them attractive for memory-sensitive programs.
The biggest disadvantage of splay trees is that they can be linear, as we observed in the Concept Question. This is extremely bad for performance, though it is also extremely rare for this case to occur.
Cite as: Splay Tree. Brilliant.org. Retrieved from https://brilliant.org/wiki/splay-tree/
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When Utility Jumps: The Value of Having Cash in the Hand
Kurt W. Rotthoff1*, Bentley Coffey2
Different theoretical explanations have been developed for seemingly inconsistent actions that deal with varying levels of risk and time. We propose a simple model of utility that unifies these seemingly separate phenomena, while not departing too far from the standard models of utility maximization already in use. Our driving assumption is that preferences over riskier outcomes discontinuously depart from preferences under certainty; a jump from no risk to some risk is fundamentally different from a movement of some risk to more risk.
Binary Jumps, Utility Theory, Risk
Rotthoff, K. and Coffey, B. (2018) When Utility Jumps: The Value of Having Cash in the Hand. Theoretical Economics Letters, 8, 72-78. doi: 10.4236/tel.2018.81004.
Many different theoretical explanations of decision-making have been developed to characterize and judge experimental findings of bias as violations of standard utility theory. Explanations have come in the form of the certainty effect (also called the Allais Paradox), immediacy effect (also called present-bias, dynamic inconsistency, or diminishing impatience), utility of gambling, non-expected utility, risk aversion, and prospect theory. Work by Andreoni and Sprenger [1] suggests that models of preferences should be adjusted to accommodate a discrete taste for the absence of any sort of risk, which appears to be large enough to be empirically detectable. We attempt to unite these concepts via a simple adaptation of expected utility theory, which posits that agents behave as if maximizing their expected utility as described in the von Neumann and Morgenstern expected utility theorem (otherwise an agent’s choices over uncertain lotteries might violate the independence axiom, implying that the individual would gladly succumb to predatory bets such as Dutch books).
Our value-added to literature is to propose that the presence of risk activates a discrete jump to a frame of mind that evaluates expected utility relative to some reference baseline. We implement this innovative idea in a model resembling a fusion of expected utility with quasi-hyperbolic utility, where the desired discrete jump is achieved via a discontinuity in the objective function at the boundary values of the probabilities. The inclusion of a discrete difference in riskless activities, relative to activities that include some positive level of risk, conveniently provides an efficient explanation of many experimental findings of bias.
We argue that all decisions begin with a binary choice: people either choose to take the action in question or not. To abstract away from the intellectual baggage that we carry from how economists have modelled risk, focus for a moment on the action to consume some good. Consider, for example, the decision to consume alcohol. The individual first decides whether to consume a taste of alcohol. Then, in a second stage, the individual decides how much more alcohol to consume on the margin. Economists might describe this binary decision of whether to consume alcohol is made on a coarse (i.e. discrete) margin; yet, it may be viewed differently by the decision maker from the fine (i.e. continuous) margins of (infinitesimally) tiny tweaks in the quantity consumed. Hence, the decision from none to some may be qualitatively different than the step from some to more. Now apply that same logic to a risk averse decision maker who is considering bearing some risk, the disutility of going from no risk to some risk can be distinctly different than going from some risk to more risk.
The classic example has an individual choosing between $100 at time t and $110 at time t + 1 chooses differently depending on the timing of these payments. When the decision is between $100 today and $110 in one month, people tend to choose $100 today. However, if the decision is between $100 in one year and $110 in one year and one month, most people choose the latter [2] - [7] . While the gap in payments (one month) and the gap in pay ($10) remain constant, the risk level does not. Payment today involves a riskless decision, whereas, all the other options involve some non-zero level of risk (although the experimenters hope that their design makes later payments appear riskless, the fact that the participant leaves without the money in hand means that they likely believe there is a non-zero probability of non-payment).
It is instructive to apply our logic to a practical example. If I hand you $10,000 in cash, then say you can either a) keep it, or b) give it back (but I will give it back to you later with more money), which option do you take? It would depend on how much extra I give you back and your perceived risk of me taking it back. Could I offer you $1 to take a small amount of risk? $10? For most people, there is a minimum level of money that would have to be offered to take on the first level of risk (i.e. for any person to be willing for that money to leave their hand, there is some [non-small, non-linear] payment that would have to occur in order for them to take a positive level of risk). The initial movement from no risk to some risk is fundamentally different than the movement from some risk to more risk. The next section sets up the model and describes how these discrete utilities work. The last section concludes.
We begin with a general specification of the decision maker’s objective, as an (indirect) utility function (V) that is increasing in the wealth (W) owned in each of J states of nature (S):
V\left({\left\{\mathrm{Pr}\left({S}_{j}\right),W\left({S}_{j}\right)\right\}}_{j=1}^{J}\right)
Then we propose the following functional form:
V=U\left({W}_{B}\right)+{\beta }^{1\left\{\underset{j}{\mathrm{max}}\mathrm{Pr}\left({S}_{j}\right)<1\right\}}\left[\underset{j=1}{\overset{J}{\sum }}\mathrm{Pr}\left({S}_{j}\right)U\left(W\left({S}_{j}\right)\right)-U\left({W}_{B}\right)\right]
where U is a state-dependent utility function under certainty, WB denotes the amount of wealth which serves as a baseline for the decision-maker, and
\beta \in \left[0,1\right]
represents the penalty to the decision-maker from the presence of risk.1 The baseline reference is the threshold at which the agent is indifferent between a risky gamble and a certain outcome with the same expected value (so that the agent is risk averse above the baseline reference and risk seeking below it); operationally, WB is just a preference parameter.
We immediately note three desirable properties about this specification.
Observation 1. Our specification nests von Neumann-Morgenstern Expected Utility as a special case when b = 1:
V=\underset{j=1}{\overset{J}{\sum }}\mathrm{Pr}\left({S}_{j}\right)U\left(W\left({S}_{j}\right)\right)\text{if}\beta =1
Observation 2. When the uncertainty distribution is degenerate, our specification neatly collapses to utility under certainty:
V\left(\left\{\mathrm{Pr}\left({S}_{A}\right)=1,W\left({S}_{A}\right);\cdots \right\}\right)=U\left(W\left({S}_{A}\right)\right)
Observation 3. When the present is certain and the future is inherently uncertain, then the time separable version of our preferences conforms to the model of preferences that exhibit quasi-hyperbolic discounting:
{V}_{0}={U}_{0}+\underset{t=1}{\overset{T}{\sum }}{\delta }^{t}\left[{\beta }^{1\left\{\underset{j}{\mathrm{max}}\mathrm{Pr}\mathrm{Pr}\left({S}_{jt}\right)<1\right\}}\left[\underset{j=1}{\overset{J}{\sum }}\mathrm{Pr}\left({S}_{jt}\right)U\left(W\left({S}_{jt}\right)\right)-U\left({W}_{B}\right)\right]\right]
1Where the term “risk” is used to mean that there is uncertainty (i.e. a non-degenerate probability distribution) over outcomes that the decision maker strictly orders (this excludes the uninteresting case of uncertainty over outcomes for which the decision maker is indifferent).
Hence, any phenomenon explained with quasi-hyperbolic discounting also explains our preferences that anchor expected utility to a reference baseline. The hyperbolic discounting parameter appears due to our model of a discrete jump in utility when moving from certainty (at the present) to uncertainty (of the future). In our model, the additional discounting of the future can give an intimately tied intuitive interpretation to disutility due to the mere presence of uncertainty in the future.
Observation 4. By design, our specification produces a discontinuity between certainty and uncertainty at any arbitrary wealth level, W(SA), apart from the baseline, when
\beta \in \left(0,1\right)
V\left(\left\{\mathrm{Pr}\left({S}_{A}\right)=1,W\left({S}_{A}\right);\cdots \right\}\right)\ne \underset{\mathrm{Pr}\mathrm{Pr}\left({S}_{A}\right)\to 1}{\mathrm{lim}}V\left(\left\{\mathrm{Pr}\left({S}_{A}\right),W\left({S}_{A}\right);\cdots \right\}\right)
U\left(W\left({S}_{A}\right)\right)\ne \beta U\left(W\left({S}_{A}\right)\right)+\left(1-\beta \right)U\left(WB\right)
Notice that, even when there is some fleetingly small risk, utility is a convex combination of the utility of a von Neumann Morgenstern Expected Utility maximizer and some baseline frame of reference to which this decision maker is tethered. The strength of that tether is determined by the magnitude of β: the closer the parameter for a decision maker is to 1, the closer the decision-maker is to being a pure von Neumann Morgenstern Expected Utility maximizer. The closer the parameter for a decision maker is to 0, the closer the decision-maker is to appearing somewhat irrational relative to the von Neumann Morgenstern model. Our prior is that likely values for β will tend to be rather close to (albeit just less than) 1. To clearly illustrate the mechanics of this model of preferences, Figure 1 depicts the mechanics for an exaggerated value of the β parameter (β » 0.5).
Figure 1 plots indirect utility in units of utils on the vertical axis versus dollar-denominated wealth on the horizontal axis. The green curve is a standard utility function under certainty. The blue horizontal line is the reference level of utility, which crosses the standard utility function at the point of reference (labeled WB). Above this point, anchoring to the reference point makes the decision maker relatively more risk averse but less risk averse below this reference point. The blue curve is just the weighted average of the green curve and the horizontal blue line. The jump from uncertainty to certainty induces a discrete gain in utility above the reference point but a discrete drop in utility below the reference point. The standard graphical exercises can be conducted with any state dependent utility function (e.g. between an outcome yielding WL versus WH), but one must then anchor it to the reference level of utility.
In Figure 2, we analyze how an agent with these preferences would change their valuations of risky outcomes due to a change in the probability of increasing wealth from WL to WH. When the amounts of wealth in question are above the baseline reference (i.e. WH > WL > WB), then the presence of any uncertainty in the amount of wealth decreases the individual’s valuation. When the amounts of wealth in question are below the baseline reference (i.e. WB > WH > WL), then the presence of some uncertainty in the amount of wealth actually increases the individual’s valuation. When the amounts straddle the baseline (i.e. WB > WH > WL), then the presence of risk contracts the valuations toward that baseline.
Figure 1. Depicting our proposed augmentation of the standard expected utility model with a discrete distaste for extensive risk.
Figure 2. How our proposed discrete distaste for extensive risk relates to the reference point.
Note that this middle case appears to resemble a stylized form of the weighting function proposed in the prospect theory of Kahneman and Tversky [4], which famously used a sigmoidal shape. Thus, in some sense, our model could be seen as proposing a clever weighting function for prospect theory that nicely yields quasi-hyperbolic preferences.
Figure 3 depicts how the preferences would appear in a canonical figure from finance: indifference curves between portfolios of various combinations of risk and return as the mean return versus the variance of returns. The indifference curves resemble what we draw from the standard von Neumann Morgenstern expected utility decision-making model; the difference appears in the discontinuities in the intercept. For amounts in excess of the baseline reference, the presence of any risk clearly generates a discrete drop in utility. For amounts beneath the baseline reference, the presence of some risk can enhance utility. This feature can explain how gambling small amounts of money, so long as the amounts in question fall beneath the baseline reference, can actually enhance utility. Indeed, we intuitively conceptualized the reference level as the level beneath which there exists some risky gamble that would be preferred to a certain
outcome with the same expected value. It is certainly conceivable that this reference baseline may change over time, inducing a source of time inconsistency for a longer run scope than the simpler form captured by hyperbolic discounting (i.e. the present versus the future), for reasons that we do not explore here.
Engaging in risky activities is inevitable. Virtually all decisions entail some level of risk; the ability to eliminate all risk is relatively rare and hence very valuable. We have constructed a parsimonious model that captures the discrete jump in utility from selecting a risk-free option. With this discrete jump achieved via a discontinuity in the objective function at the boundary values of the probabilities, our model includes familiar features of both expected utility and quasi-hyperbolic utility (which are special cases). Our model provides a unifying and consistent explanation for a variety of anomalous behavior associated with behavioral biases: certainty effect, Allais Paradox, immediacy effect, present-bias, dynamic inconsistency, diminishing impatience, the utility of gambling, non-expected utility, and prospect theory. We encourage future research to continue to refine the use of this discrete utility function, consider the additional applications, and pursue further estimations of its parameters. Although there are limitations on this application, as there is for any application of utility theory, the ability to unify these different theories opens the door to many avenues of future research.
*A special thanks to Robert Tollison, Bruce Yandle, Angela Dills, Michael Maloney, Pete Groothuis, Daniel Jones, and Hillary Morgan for helpful comments. Any mistakes made are our own.
[1] Andreoni, J. and Sprenger, C. (2012) Risk Preferences Are Not Time Preferences. American Economic Review, 102, 3357-3376.
[2] Friedman, M. and Savage, L.J. (1948) The Utility Analysis of Choices Involving Risk. The Journal of Political Economy, 56, 279-304.
[3] Phelps, E. and Pollak, R. (1968) On Second-Best National Saving and Game-Equilibrium Growth. Review of Economic Studies, 35, 185-199.
[5] Laibson, D. (1997) Golden Eggs and Hyperbolic Discounting. Quarterly Journal of Economics, 112, 443-477.
[6] O’Donoghue, T. and Rabin, M. (1999) Doing It Now or Later. The American Economic Review, 89, 1030-1124.
[7] Benhabib, J., Bisin, A. and Schotter, A. (2010) Present-Bias, Quasi-Hyperbolic Discounting, and Fixed Costs. Games and Economic Behavior, 69, 205-223.
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Mathematics of Music | Brilliant Math & Science Wiki
Arjen Vreugdenhil contributed
Sound consists of a physical wave-- a sound wave. A musical note corresponds to a periodic sound wave with a specific frequency. And, as we discuss here, musical relations between notes correspond to mathematical relations between their frequencies.
Absolute Pitch, Frequency, and Wavelength
Relative Pitch, Interval, and Frequency Ratio
Intervals and Simple Ratios
Pythagorean and Just Intonation
The absolute pitch of a note describes how high or low it is. We denote absolute pitch with letters like C, D, E, ... Historically, different tunings (assignments of frequencies to pitches) have been used; and even today, different instruments have different tunings. (For instance, a trumpeter calls "D" what a flutist calls "C".) However, a commonly adopted standard ("concert pitch") is that on non-transposing instruments, the note
\text{A}_4
corresponds to a sound wave with a frequency of 440 Hz. This choice also determines the frequencies of the other notes.
Principle 1: Higher pitch corresponds to higher frequency.
Recall that the frequency
f
of a sound wave describes how many vibrations there are per second. This is the most important quantity here, because the frequency remains constant as the sound wave travels from its source through the medium into our ears. The wavelength
\lambda
describes how the sound wave is spread out in space. The relationship between the two quantities is
f\lambda = v,
v
is the speed of sound in a medium. In air at normal temperature,
v \approx 340\ \text{m/s}
\text{A}_4
at concert pitch is played, what is the wavelength of the sound wave as it travels through air?
Solution: From the discussion above, we know
f = 440\ \text{Hz}
v = 340\ \text{m/s}
\lambda = \frac v f = \frac{340\ \text{m/s}}{440\ \text{/s}} = 0.77\ \text{m}.
In practice, relative pitch is more important in music than absolute pitch. The impression of a melody or harmony depends on the pitch of notes compared with notes played previously or simultaneously. For instance, if an orchestra plays a piece with
\text{A}_4
at 450 Hz instead of 440 Hz, and all other notes adjusted accordingly, most people will not even notice the difference.
The relative pitch of one note to another is also called the interval between two notes. On a keyboard instrument, an interval may be characterized by the number of keys that separate two notes.
Principle 2: A musical interval corresponds to a frequency ratio.
For instance, in the song "Are You Sleeping" ("Frère Jacques") the melody starts with an upward interval called "major second". The frequencies of these two notes,
f_1
f_2
, have the ratio
\frac{f_2}{f_1} \approx 1.123.
This frequency ratio makes the melody recognizable. In principle, one could start the melody at any frequency
f_1
The Dutch anthem "Wilhelmus" traditionally begins with a note C at 264 Hz, followed by F at 352 Hz.
If the melody is transposed to a different pitch, starting at 330 Hz instead, what should be the pitch of the second note?
Solution: After transposition, the frequency ratio should still be the same. Thus we write the equation
\frac{f}{330\ \text{Hz}} = \frac{352\ \text{Hz}}{264\ \text{Hz}}\ \ \ \ \therefore\ \ \ \ f = 330\cdot \frac{352}{264} = 440\ \text{Hz}.
Some intervals are more pleasant to listen to (consonant) than others. While there is a strongly cultural component in this assessment, there are universal trends in the choice of basic intervals. One of the remarkable discoveries of the ancient Greeks was:
Principle 3: Consonant intervals correspond to simple frequency ratios.
The most important interval is the octave. Notes that differ precisely one octave have a clear difference in pitch, yet sound remarkable similar. An octave is the distance between the first "do" and the second "do" in the music scale; between
\text{A}_3
\text{A}_4
. On a keyboard instrument, an octave corresponds to the repetition of the pattern of white and black keys.
Principle 4: An interval of one octave corresponds to a frequency ratio of 2:1.
For instance, in concert pitch the note
\text{A}_4
has a frequency of 440 Hz. The note
\text{A}_5
, which is one octave higher, has a frequency of
2\cdot 440 = 880
Hz. The note
\text{A}_3
is one octave lower, and has therefore a frequency of
\tfrac12 \cdot 440 = 220
Insert image: Wave patterns of
\text{A}_3
\text{A}_4
\text{A}_5
, with frequencies.
A typical organ keyboard has a range of about five octaves. If the lowest note it can play has a frequency of
66 \text{Hz}
, what is the highest frequency, approximately?
Solution: Each octave higher corresponds to multiplication of the frequency by two. Thus we get
f = 2^5\cdot 66\ \text{Hz} \approx 2100\ \text{Hz}.
An other interval that is considered consonant in all music traditions is the so-called "perfect fifth". This interval is found, for instance, between the note C and the following G.
Insert: Image of keyboard showing interval C-G.
Principle 5: An interval of a natural perfect fifth corresponds to a frequency ratio 3:2.
Many other intervals can be defined in terms of fifths and octaves. For instance, a major second interval (e.g. from C to D) can be constructed by going up a perfect fifth, then an other perfect fifth, and then down one octave. What is the frequency ratio of this major second interval?
Solution: Simply multiply the ratios. When going down an interval, write the ratio with the smallest value in the numerator.
\frac{3}{2} \cdot \frac{3}{2} \cdot \frac{1}{2} = \frac{9}{8}.
Thus, a major second interval corresponds to a frequency ratio of 9:8.
In Western music, the standard collection of musical notes originally consisted of seven notes per octave; later, five more notes were added, making for a total of twelve. (This corresponds to 7 white keys and 5 black keys per octave on keyboard instruments.) The frequencies of these notes are chosen in such a way, that many consonant combinations can be made-- that is, simple frequency ratios.
Starting with the note C, we can define G to be one perfect fifth higher; F to be one perfect fifth lower. The note D is constructed by stacking two fifths together. But what about the notes A, E and B? Musicians make different choices here, resulting in different intonations for instruments.
The Pythagorean intonation continues to pile fifths on top of each other, as follows:
\begin{array}{|ccccccccccccc|} \hline \text{F}_4 & \\ \downarrow \tiny \div 2 & \\ \text{F}_3 & \xrightarrow{\times\ 3/2} & \text{C}_4 & \xrightarrow{\times\ 3/2} & \text{G}_4 & \xrightarrow{\times\ 3/2} & \text{D}_5 \\ & & & & & & \downarrow \tiny \div 2 & \\ & & & & & & \text{D}_4 & \xrightarrow{\times\ 3/2} & \text{A}_4 & \xrightarrow{\times\ 3/2} & \text{E}_5 \\ & & & & & & & & & & \downarrow \tiny \div 2 & \\ & & & & & & & & & & \text{E}_4 & \xrightarrow{\times\ 3/2} & \text{B}_4 \\ \hline \end{array}
The interval C-E is called "major fifth". What is the frequency ratio for this interval in the Pythagorean intonation?
Solution: Following the diagram above, we get from C to E by going up four perfect fifths, and going down two octaves. Thus we get
\left(\frac{3}{2}\right)^4\cdot \left(\frac{1}{2}\right)^2 = \frac{3^4}{2^6} = \frac{81}{64}.
This is hardly a "simple ratio" anymore-- the main reason why most musicians prefer the just intonation instead.
In the just intonation, priority is given to simple ratios. Thus we define the interval called major third to have a ratio of 5:4, and define
the note E to be a major third higher than C;
the note A to be a major third higher than F;
the note B to be a major third higher than G.
The following table shows the frequencies in both intonations when we start with C = 264 Hz.
\begin{array}{rcccccccc} \hline f\ (\text{Hz}) & \text{C} & \text{D} & \text{E} & \text{F} & \text{G} & \text{A} & \text{B} & \text{C'} \\ \hline \text{Pythagorean} & 264 & 297 & 334.1 & 352 & 396 & 445.5 & 501.2 & 528 \\ \text{just} & 264 & 297 & 330 & 352 & 396 & 440 & 495 & 528 \\ \hline \end{array}
The interval D-A is usually considered to be a perfect fifth. Is this the case in either of the intonations described above?
Solution: Yes, for the Pythagorean intonation; no, for the just intonation.
In the Pythagorean intonation we defined A to be a perfect fifth, and indeed we get
\frac{f_A}{f_D} = \frac{445.5}{297} = \frac{3}{2}.\ \ \ \ (\text{Pythagorean})
But in the just intonation
\frac{f_A}{f_D} = \frac{440}{297} = \frac{40}{27}.\ \ \ \ (\text{just})
This frequency ratio is about 1.5% less than 3:2; in the just intonation, the interval D-A sounds slightly lower ("flat") compared to a natural perfect fifth.
In the Pythagorean intonation, the intervals C-D, D-E, F-G, G-A, and A-B all correspond to the same frequency ratio of 9:8.
In the just intonation it is more complicated. Classify the five intervals as "major tones" and "minor tones" based on their frequency ratios.
Solution: We find
\text{frequency ratio C-D, F-G, A-B} = \frac{9}{8}\ \ \ \ (\text{major tone}); \\ \text{frequency ratio D-E, G-A} = \frac{10}{9}\ \ \ \ (\text{minor tone}).
The Pythagorean approach has the advantage that all frequency ratios between notes can be written in powers of 3 and 2. In particular, the "tones" (whole note steps) in the scale all have the same ration
3:2
. A disadvantage is that intervals such as the major-third (C-E) are not simple-number ratios.
The just intonation has the advantage that the intervals are mostly simple-number ratios. However, certain intervals are not quite what one would except; for instance, D-A falls short of being a natural perfect fifth. Moreover, whole note steps such as C-D and D-E do not have the exact same frequency ratio.
How different are these intonations? Consider the interval C-E: in the Pythagorean intonation it has a ratio of 81:64; in the just intonation it is 5:4, or 80:64. Thus the discrepancy between the intonations is of the order
81:80 = 1.0125
. The frequencies can therefore be expected to differ in the order of 1 percent.
Cite as: Mathematics of Music. Brilliant.org. Retrieved from https://brilliant.org/wiki/mathematics-of-music/
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4 question of picture:
4. In the adjoining figure, AD is median of
△
ABC, MB and CN are perpendiculars drawn from B and C respectively on AD and AD produced. Prove that BM =CN.
Hint .
△\mathrm{BMD}\cong △\mathrm{CND} \mathrm{by} \mathrm{AAS}.
I have not got this statement myself. So, kindly Prove this entire statement given in figure with the help of different instance. ( About equialgular triangle that, how two congruent triangle will be equiangular and two equiangular triangles need not to be congruent)
Q. In any triangle , the sum of medians is greater than what times the perimeter ? Sum of medians
>
______ perimeter?
Omja Dwivedi
In triangle ABC we have angle A=100 and angles B=C=40. The side AB is produced to a point D so that B lies between A and D and AD=BC. Find the angle BCD
In triangle ABC we have angle A=100 and angles B=C=40. The side AB is produced to a point D so that B lies between A and D and AD=AC. Find angle BCD
Dhrusti Jain
In the given figure AB = AC . D is point on AC and E on AB such that AD = ED = EC = BC . prove that angle A : angle B = 1 : 3
2{\mathrm{sin}}^{2}\theta =\sqrt{3}
\theta
the average of two non-congruent angles of an isosceles triangle is 65. find the value of these two non-congruent angles?
Diagonal AC and BD of a quadrilateral ABCD intersect each other at O. Prove that
1) AB+BC+CD+DA>AC+BD
2)AB+BC+CD+DA>2(AC+BD)
99% fail to answer this one
Pls give answer quickly..
Prove that angle bisectors of a paralleogram form a rectangle.
53) In
△ABC
, AC = BC, S is the circum - centre and
\angle ASB=150°. Find \angle CAB.
\left(a\right) 55\frac{1}{2}°\phantom{\rule{0ex}{0ex}}\left(b\right) 52\frac{1}{2}°\phantom{\rule{0ex}{0ex}}\left(c\right) 62\frac{1}{2}°\phantom{\rule{0ex}{0ex}}\left(d\right) 35\frac{1}{2}°
Pl ans q51
Q.51. The sides of a triangle are 2006 cm, 6002 cm and m cm, where m is a positive integer. Find the number of such possible triangles.
Arish Husain
Solve it quickly
Q.30. In the given figure,
\angle B<\angle A and \angle C<\angle D
. Show that AD < BC.
Nishtha Kohli .
Pls.sove ques 30 in writing and pls do not provide any link
|
Software | Bartolomeo Stellato
These are my software projects. The code of my research group is available at github.com/stellatogrp. For more details on my previous projects and personal code please visit my personal github account.
The OSQP (Operator Splitting Quadratic Program) solver is a numerical optimization package for solving convex quadratic programs in the form
\begin{array}{ll} \text{minimize} & \frac{1}{2} x^T P x + q^T x \\[.5em] \text{subject to} & l \leq A x \leq u. \end{array}
Academic users include MIT, Stanford, ETH Zurich, University of Oxford, Berkeley, Imperial College, UCLA, KU Leuven, Lund University.
Industrial users include Lyft, Adobe, LinkedIn, Siemens, Baidu, Quantitative Brokers, Goral Trading, Macquarie.
QDLDL is a free sparse
LDL^T
factorization routine for solving linear systems of the form
A x = b,
A
is quasi-definite. Already used in many numerical solvers including OSQP and SCS.
SwitchTimeOpt.jl is a Julia package to easily define and efficiently solve switching time optimization (STO) problems for linear and nonlinear systems of the form
\begin{array}{ll} \underset{\tau}{\text{minimize}} & \int_{t_0}^{T} x(t)^\top Q x(t)\; \mathrm{d}t + x(T)^\top Q x(T)\\ \text{subject to} & \dot{x}(t) = f_i(x(t)) \quad t\in[\tau_i,\tau_{i+1}) \quad \forall i\\ & x(0) = x_0\\ & \tau_i \leq \tau_{i+1}, \quad \forall i \end{array}
f_i
defines the system dynamics and
\tau_i
are the switching times.
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Generate pulses for SVPWM-controlled three-level converter - Simulink - MathWorks Nordic
Vabc_ref
|Vref|∠Vref
Vα Vβ ref
X (internal)
Vc_P0
Vc_0N
NP_Ctrl_Out
Idc filter cut-off frequency (Hz)
PWM switching frequency (Hz)
Reference vector [ Mag (0<m<1), Phase (degrees), Freq (Hz) ]
Generate pulses for SVPWM-controlled three-level converter
The SVPWM Generator (3-Level) generates pulses for three-phase three-level Neutral-Point-Clamped (NPC) converters. The block generates twelve pulses using the space vector pulse width modulation (SVPWM) technique.
The neutral-point voltage deviation is controlled by a proportional regulator using two DC voltages, as well as the DC current flowing in or out of the DC link. The defining equation is
{V}_{out}=m\cdot \frac{{V}_{dc}}{\sqrt{2}},
Vout is the line-to-line rms voltage generated by the NPC.
m is the modulation index and 0 < m < 1.
Vdc is the DC current flowing in or out of the DC link.
Vabc_ref — Three-phase sinusoidal reference signal
Reference voltage in terms of sinusoidal voltages. Specify three voltages, one per phase, that you want the attached converter to output.
Setting the Voltage reference (Vref) parameter to Three-phase signals exposes this parameter.
|Vref|∠Vref — Reference voltage magnitude and angle
Reference voltage in terms of magnitude and angle, in rad. Specify the magnitude and angle of the voltage that you want the attached converter to output.
Setting the Voltage reference (Vref) parameter to Magnitude-Angle (rad) exposes this parameter.
Vα Vβ ref — Reference voltage alpha and beta components
Reference voltage in terms of alpha and beta components. Specify the alpha and beta components of the voltage that you want the attached converter to output.
Setting the Voltage reference (Vref) parameter to alpha-beta components exposes this parameter.
X (internal) — No input
When the block generates the reference voltage internally, no reference signal is input.
Setting the Voltage reference (Vref) parameter to Internally generated exposes this parameter.
Vc_P0 — Three-level NPC converter positive-to-neutral voltage
Voltage between the + and N terminals of the three-level NPC converter block that is controlled by the SVPWM Generator (3-Level) block.
Vc_0N — Three-level NPC converter neutral-to-negative voltage
Voltage between the N and - terminals of the three-level NPC converter block that is controlled by the SVPWM Generator (3-Level) block.
Idc — DC link current
Current of the DC link flowing out the three-level NPC converter block that is controlled by the SVPWM Generator (3-Level) block
P — Gate control
12-pulse waveforms that determine switching behavior in the attached power converter.
NP_Ctrl_Out — Internal control voltage
Neutral-point deviation voltage of the generator control system.
Voltage reference (Vref) — Parameterization method
Three-phase signals (default) | Magnitude-Angle (rad) | alpha-beta components | Internally generated
Reference-voltage parameterization method.
Setting this parameter to:
Three-phase signals exposes the Vabc_ref port, hides the Reference vector [ Mag (0<m<1), Phase (degrees), Freq (Hz) ] parameter, and hides the |Vref|∠Vref, Vα Vβ ref, and X (internal) ports.
Magnitude-Angle (rad) exposes the |Vref|∠Vref port, hides the Reference vector [ Mag (0<m<1), Phase (degrees), Freq (Hz) ] parameter, and hides the Vabc_ref, Vα Vβ ref, and X (internal) ports.
alpha-beta components exposes the Vα Vβ ref port, hides the Reference vector [ Mag (0<m<1), Phase (degrees), Freq (Hz) ] parameter, and hides the Vabc_ref, |Vref|∠Vref, and X (internal) ports.
Internally generated exposes the Reference vector [ Mag (0<m<1), Phase (degrees), Freq (Hz) ] parameter and the X (internal) port and hides the Vabc_ref, |Vref|∠Vref, and Vα Vβ ref ports.
Proportional gain — Proportional gain
Constant for proportional gain.
Idc filter cut-off frequency (Hz) — Cut-off frequency
Idc filter cut-off frequency, in Hz.
PWM switching frequency (Hz) — Switching frequency
PWM switching frequency, in Hz.
Reference vector [ Mag (0<m<1), Phase (degrees), Freq (Hz) ] — Reference vector
[0.8 -30 60] (default) | vector
Three-element reference vector of magnitude, in m, phase, in degrees, and frequency, in Hz.
Time between consecutive block executions. During execution, the block produces outputs and, if appropriate, updates its internal state.
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Failed Compression
A while back, I was working on a small compression algorithm for high entropy data, specifically for a tiny hobby operating system back in 2017. The idea was that I would use 256 bytes or less for decompression code that would decompress an operating system hiding in a 512 byte boot sector.
Unfortunately the project didn't work out, but a recent post on the Pine forum got me thinking. Specifically:
This file is then lz4 compressed, and written to the spi flash of the pinetime. The pinetime decompresses, processes the blocks, and writes them to the display.
That sounds awesome :) Do you get much savings with lz4? I assume the data you are sending is already pretty high entropy?
It goes from 5046208 bytes to 4099181. Which makes it small enough to fit in the 4 MiB of spi flash
This lz4 lossless compression algorithm sounds awesome. It made me think about my previous attempts and what went wrong.
Compression Attempts
The following are some failed methods I created to try and compress high-entropy data. I thought about these quite some years ago, but it's interesting to look back on them and see so obviously now why they couldn't possibly work 1.
The idea was pretty simple with this one - you take a matrix and then attempt to create some abstract representation of it by performing a calculation.
Defining it better, we can give some example matrix of values
a
b
c
and
\mid \begin{array}{cc}a& b\\ c& d\end{array}\mid
We can then sum each of the rows and columns to 'compress' the data:
\left[[[math-ml]]]a+b[[[/math-ml]]]
c+d
a+c
b+d
\right]
. To recover the data, you simple need to solve the matrix and satisfy the summed values. This could be done trivially and the data is always perfectly recoverable.
Of course, there are no real savings here - you store four values as four different values. The area is
{n}^{2}
and the summed data is
2n
. But we see the possibility of savings when we expand the matrix, for
n=3
we in theory can compress 9 bytes down to 6 bytes:
\mid \begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\mid
And the data would be compressed like so:
\left[[[math-ml]]]a+b+c[[[/math-ml]]]
d+e+f
g+h+i
a+d+g
b+e+h
c+f+i
\right]
. But this should be setting your bullshit sensors off - we can simply apply this to any arbitrary data any arbitrary number of times. Something is seriously wrong here...
Compression count - Firstly, you would need a value to contain the number of times the operation was performed in order to undo it.
Multiple solutions - Of course, by far the largest problem is that the
n=3
matrix has multiple solutions. The only way to solve this problem is to introduce another column in the third dimension in order to break symmetry. This of course adds an additional 3 bytes and leads to no savings. This also appears to be true for all cases where
n\ge 3
Bubble Sort Compression
The idea essentially was to implement a reversible bubble sort algorithm - each sort the algorithm does it leaves a symbol in place to indicate that a sort occurred. Eventually the idea is that all of the symbols would be in order, and then a table simply needs to store the frequencies of each character.
Essentially we had some string we wanted to sort, and a special symbol
z
z
had some magic value that placed it in the middle of the data when fully sorted. If we use the string
\left[[[math-ml]]]a[[[/math-ml]]]
b
c
\right]
, we would initially check if
a>b
. If so, we need to swap them. The string would then become
\left[[[math-ml]]]b[[[/math-ml]]]
z
a
c
\right]
. You would then repeat this until the sorting is completed, and store the index of the last location to be sorted.
To decompress, you then reverse this operation, so
\left[[[math-ml]]]b[[[/math-ml]]]
z
a
c
\right]
\left[[[math-ml]]]a[[[/math-ml]]]
b
c
\right]
z
is removed after reversing the operation.
I implemented this and of course, it's not perfect 2. The main issues with the algorithm are:
Disk requirements - The bubble sort method uses so many characters (over so many sort operations) that the size of the frequency data ends up being the same size or larger than the original data you were trying to compress.
RAM requirements - Even expanding the frequency data would be nowhere near to possible on modern hardware, let alone in 1MB 16 bit mode. I estimated back then that for 512 bytes, you would need
1×{10}^{167}
bytes of RAM.
Compression - Compression time was "reasonable" (compared to everything else), but still also grew exponentially. Compressing 512 bytes would take near to a lifetime to complete.
Decompression - Fortunately, decompression time was much quicker - but still suffered from an exponential curve. For 512 bytes, you couldn't reasonably hope for a computer to finish in your lifetime.
The original article has more details on the issues involved, but it clearly is not so great.
Future Compression
So, how can I proceed with more viable compression methods?
Of course this is a very decent compression algorithm, which is extremely fast and well supported. But it's not particularly suitable for this application for multiple reasons:
Large data - It only appears to perform well on large blocks of data. When I was testing on ~2kB ASCII data I saw no compression improvement, often actually delivering a worse result.
Implementation - There is a surprising amount going on - you really do not want to waste so much code space encoding the problem at this level.
Custom Huffman Coding
I believe I will attempt to incorporate something like Huffman coding as it is very simple. Some characters in the binary are worth representing in smaller coding forms (such as a NULL byte or a jmp instruction), whilst many characters either not represented or appear only once or twice in a single block (512 bytes).
Given that only a small subset of bytes can benefit from the coding process, it therefore makes sense to not build a full Huffman coding tree for all 256 symbols but instead only bother building a tree for the most represented data (say the first 16 or 32 bytes). After this, all remaining unrepresented bytes can be assumed to be encoded in order 3.
Why keep on working on this? Well, I still like the OS project of course - but also a compression algorithm that can work on arbitrarily small blocks of data without inter-lock dependency could also be extremely useful. One example is in the audio space, where Google are testing audio streams below 3kbps. I have also seen audio all the way down to 450bps. Compression that can then be applied on top of this could make a big difference.
One application I have been thinking about in the back of my mind is LoRa communications - at some point I really want to revisit my LoRa-chan project and get some form of audio streamed without absolutely demolishing the network.
If we can look back on our past selves and find improvement, this is a great sign of progress.↩
I believe that initial implementation was wrong, but the idea is so ridiculous that I don't want to invest time into fixing it.↩
This will of course require some thinking for optimal implementation↩
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Research on the Impact of Foreign Direct Investment on Chinese Trade Structure Optimization
—Based on the Strategic Background of the Belt and Road
―Based on the Strategic Background of the Belt and Road
{\text{lnEST}}_{it}={\beta }_{0}+{\beta }_{1}\text{\hspace{0.17em}}of\text{\hspace{0.17em}}di_{1}_{it}++\text{γ}\text{\hspace{0.17em}}{\text{control}}_{it}+{\epsilon }_{it}
Zhang, X. (2019) Research on the Impact of Foreign Direct Investment on Chinese Trade Structure Optimization. Modern Economy, 10, 797-810. https://doi.org/10.4236/me.2019.103053
1. Sui, Y.H. (2010) Dual Foreign Direct Investment and Trade Structure: Mechanism and Evidence from China. Journal of University of International Business and Economics, No. 6, 66-73.
2. Du, X.L. and Wang, W.G. (2007) The Technical Structure of China’s Export Trade and Its Changes: 1980-2003. Economic Research, No. 7, 137-151.
3. Mundell, R.A. (1957) International Trade and the Factor Mobility. American Economic Review, 47, 321-335.
4. Helpman, E. (1984) A Simple Theory of International Trade with Multinational Corporations. Journal of Political Economy, 92, 451-471. https://doi.org/10.1086/261236
5. Kojima, K. (1978) Direct Foreign Investment: A Japanese of Multinational Business Operation. Croom Helm, London.
6. Kim, J.D. and Rang, I.S. (1997) Outward FDI and Exports: The Case of South Korea and Japan. Journal of Asian Economics, 8, 39-50. https://doi.org/10.1016/S1049-0078(97)90004-X
7. Hejazi, W. and Safarian, A.E. (2001) The Complementarity between US Foreign Direct Investment Stock and Trade. Atlantic Economic Journal, 29, 420-437. https://doi.org/10.1007/BF02299331
8. Sui, Y.H. and Zhao, Z.H. (2008) The Formation Mechanism of Export Trade Structure: Based on China’s Empirical Research from 1980 to 2005. International Trade Issues, 3, 9-16.
9. Li, X.L. (2015) The Home Trade Structure Effect of Foreign Direct Investment— Based on China’s Provincial Panel Data Analysis. Economic Issues Exploration, No. 4, 138-144.
10. Chen, Y.Y. (2012) Trade Structure Effect of China’s Foreign Direct Investment. Statistical Research, 29, 44-50.
11. Ma, X. (2016) The Impact of China’s Foreign Direct Investment on Export Trade Structure—Based on Empirical Analysis of Data from 1997 to 2014. Journal of Chifeng College, 32, 88-89.
12. Chen, J.C. and Huang, F.H. (2013) Foreign Direct Investment and Export Technology Complexity. World Economic Research, No. 11, 74-79.
13. Liu, G. and Zhang, Z.B. (2007) An Empirical Study of China’s Foreign Direct Investment and Export Relations—Based on ESM Approach. Reform and Strategy, No. 4, 19-21.
14. Yu, Y. and Wan, L. (2009) Research on the Correlation between China’s Import and Export Commodity Structure and Foreign Direct Investment—Analysis Framework Based on Var Model. International Trade Issues, No. 6, 96-104.
15. Liu, B., Wang, J. and Wei, Q. (2015) Foreign Direct Investment and Value Chain Participation: Division of Labor Status and Upgrade Model. Quantitative Economics and Technology Economics Research, No. 12, 39-56.
16. Hu, B. and Qiao, J. (2013) Trade Effect of China’s Foreign Direct Investment— Based on Dynamic Panel Model System GMM Method. Economic Management, No. 4, 11-19.
17. Hu, W.Z. (2015) The Impact of China’s Foreign Direct Investment on the Optimization of Export Trade Structure. Journal of Wuhan Vocational College of Transportation, No. 1, 16-20.
18. Zhang, C.P. (2012) The Impact of China’s Foreign Direct Investment on Import and Export Trade. Academic Exchanges, No. 7, 85-88.
19. Chen, J.C. and Huang, F.H. (2014) Foreign Direct Investment and Trade Structure Optimization. International Trade Issues, No. 3, 113-122.
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Learnings from refinancing a mortagage
The information and links provided are for general information only and should not be taken as constituting personal financial advice. You should always do your own research when making any financial decisions.
I was nearing the end of a fixed interest portion of my loan, so naturally I wanted to know what my options were going forwards. This post records some of the learning through the process.
Before I continue, it is worthwhile establishing my situation, which will very likely be different to yours!
I live in Melbourne (Victoria, Australia)
I had financed the purchase of a property in 2020
The property I purchased is my principal place of residence (PPOR) 1
When I purchased the property I had a loan to value ratio (LVR) 2 of >80%, which meant I had to pay for lender's mortgage insurace (LMI) 3
Repayments were for the principal and interest
The loan had an initial fixed term of 2 years, rolling over to variable afterwards
Before starting, the lender will need to know following objective information:
Outstanding balance of the loan
If you are rolling over to variable after a fixed period, use the variable rate
Estimated property value, you can use estimators from REA4 and Domain5
From this we can calculate the loan to value ratio (LVR) 2:
{LVR} = \frac{Outstanding Loan Amount}{Estimated Property Value}
The higher the LVR, the more risky the loan. A key threshold is having ≤80%, this opens up opportunities for lower interest rates and typically not needing to pay lender's mortgage insurance anymore.
It it also worth thinking about what kind of loan you want. Consider:
Do you want a fixed interest rate? How long to lock in a fixed term?
There are break-costs6 to ending a fixed term early.
Consider your future plans with the property? Do you intend to sell soon, if so, maybe a variable rate or a short fixed term would be best.
Have an emergency fund 7 in cash? Put it to work against the mortgage!
Most variable loans offer an offset account 8, it behaves like an everyday savings account but instead of earning interest, the money inside the account offsets the loan principal - reducing your interest payment
Most loans also offer a redraw facility9 which appears to be similar to an offset account, but it is not suitable for emergencies as it is technically the bank's money - not yours10
Does the lender let you split the loan?
Most of loan should be fixed at a lower rate, while a portion is variable to take advantage of an offset account
The loan continues with the current lender, typically rolling over to a variable rate
Negotiate with current lender
In most cases, you will rollover to the variable rate the lender is currently offering for your mortgage. Always call the lender to confirm the interest rate, as it may be different to what is on the website!
This is the easiest and not wrong in some cases!
Consider the scenario where you are at the edge of an 80% LVR. In a few more repayments, you will be under that threshold where better refinance options open up. In the meantime, you don't want to lock yourself into a subpar fixed term loan. At the end of the day, you will just need to know your options and do the maths.
Negotiate with the current lender
Most lenders are keen to keep you on board, so don't count then out - talk to them! Discuss what you wan't your loan to look like going forward.
Look for another lender
This is where things get the most interesting.
❗ Lender's mortgage insurance (LMI) for LVR > 80%
This is paid to the new lender
It is not transferrable, so even if you have bought it for your current lender, you will need to buy it again for the new lender
LMI is the most significant cost and can outweight any advantage the new loan offers
For even just an LVR of 81%, 1% off not having to pay it, you will be looking at ~$250011
Avoid this by getting your LVR below 80%
This may need to be paid to the new lender
In a lot of cases, to be competitive, lenders have forgone these fees
I considered this as a few up front costs
Establishment fee - someone needs to cross the t's and dot the i's
Valuation fee to get the estimated value of the property
Break costs6 maybe a applicable when you are exiting a fixed term loan early
This is paid to the previous lender
This is higher the more months left until the end of the fixed term
I was quoted ~$300 with 1 month remaining, but consult your mortgage agreement and lender
Avoid this by coordinating the next loan to start after the current fixed term ends
If that is not possible, balance the cost of the break costs and moving to the new loan after rolling over to a variable rate in the current loan
Discharge fees from the lender
Not all lenders have this
Worth knowing if your prospective lender has this, if you plan to refinance again later down the line
Expect to pay ~$3501213
Government charges for registering and deregistering a mortgage
Paid to Victoria Government to process the paperwork around the mortgage
$121.4014 twice ($242.80)
When the interest rate is low and this is the forever home, it is the perfect time to lock in a competitive fixed term loan
The longer the fixed term the higher the interest rate15
The lower the LVR the lower the interest rate16
Interest rates are higher for investment properties, i.e. not principal place of residence (PPOR)
Only available on variable loans
Whether you can split the loan to have a portion fixed and the remaining as variable in order to get an offset account
Pay attention to any ongoing fees or conditions to have such an account17
Pay attention to any ongoing fees for the loan18
Cash bonuses and incentives
Some loans come with a cash bonus19 or other incentives for refinancing
Be sure to look that this truly outweighs the cost to move, especially if you have to pay lender's mortgage insurance
Unique scenario where someone held onto a loan with a very low LVR, profiting by refinancing it every year to take advantage of cash bonuses20
https://www.ato.gov.au/individuals/capital-gains-tax/property-and-capital-gains-tax/your-main-residence-(home)/ ↩
https://moneysmart.gov.au/glossary/loan-to-value-ratio-lvr?gclid=CjwKCAiA0KmPBhBqEiwAJqKK47ouv6T5PkaBHFfDb7KQxGNhvoU79htbYDS1NwZ21XQCOZ37-USUeBoCU3kQAvD_BwE&gclsrc=aw.ds ↩ ↩2
https://www.mortgagechoice.com.au/guides/home-ownership/lenders-mortgage-insurance/ ↩
https://www.realestate.com.au/property/value-my-property ↩
https://www.domain.com.au/owners ↩
https://www.yourmortgage.com.au/compare-home-loans/thinking-of-breaking-a-fixedrate-contract-heres-what-you-should-know ↩ ↩2
https://moneysmart.gov.au/saving/save-for-an-emergency-fund ↩
https://www.commbank.com.au/articles/home-loans/what-is-an-offset-account.html ↩
https://www.anz.com.au/personal/home-loans/tips-and-guides/redraw-facilities-what-why-how/ ↩
https://www.reddit.com/r/AusFinance/comments/gbse46/a_reminder_about_redraw_vs_offset/ ↩
https://www.yourmortgage.com.au/calculators/mortgage-insurance ↩
https://www.commbank.com.au/content/dam/commbank/personal/apply-online/download-printed-forms/003-750.pdf ↩
https://www.nab.com.au/personal/help-and-guidance/personal-banking-fees-and-charges#:~:text=Mortgage%20Discharge%20Fee ↩
https://www.land.vic.gov.au/land-registration/fees-guides-and-forms ↩
https://www.hsbc.com.au/home-loans/rates/reference-rates/#:~:text=equity%20product%20page.-,Fixed%20Rate%20Loans*,-Owner%20Occupiers ↩
https://www.hsbc.com.au/home-loans/products/rates/#:~:text=equity%20product%20page.-,Fixed%20Rate%20Loans,-Available%20on%20borrowings ↩
https://www.anz.com.au/personal/home-loans/compare-home-loan/offset-account/#:~:text=%2410%20monthly%20account%20fee%20applies%2C%20but%20this%20is%20waived%20if%20you%27re%20on%20an%20ANZ%20Breakfree%20package ↩
https://www.hsbc.com.au/home-loans/#:~:text=To%20be%20eligible%20for%20HSBC,month.%20Excludes%20non%2Dresident%20applications. ↩
https://www.ozbargain.com.au/node/663322 ↩
https://www.reddit.com/r/AusFinance/comments/o5za0x/comment/h2pigxs/?utm_source=share&utm_medium=web2x&context=3 ↩
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Quantum number - Wikipedia
"Q-number" redirects here. For the Q-theory concept, see Q-analog. For the number format, see Q (number format).
Single electron orbitals for hydrogen-like atoms with quantum numbers n = 1, 2, 3 (blocks), ℓ (rows) and m (columns). The spin s is not visible, because it has no spatial dependence.
An important aspect of quantum mechanics is the quantization of many observable quantities of interest.[note 2] In particular, this leads to quantum numbers that take values in discrete sets of integers or half-integers; although they could approach infinity in some cases. This distinguishes quantum mechanics from classical mechanics where the values that characterize the system such as mass, charge, or momentum, all range continuously. Quantum numbers often describe specifically the energy levels of electrons in atoms, but other possibilities include angular momentum, spin, etc. An important family is flavour quantum numbers – internal quantum numbers which determine the type of a particle and its interactions with other particles through the fundamental forces. Any quantum system can have one or more quantum numbers; it is thus difficult to list all possible quantum numbers.
1 Quantum numbers needed for a given system
2 Electron in an atom
3 Total angular momenta numbers
3.1 Total angular momentum of a particle
3.2 Nuclear angular momentum quantum numbers
5 Multiplicative quantum numbers
Quantum numbers needed for a given systemEdit
Main article: Quantum system
The tally of quantum numbers varies from system to system and has no universal answer. Hence these parameters must be found for each system to be analyzed. A quantized system requires at least one quantum number. The dynamics (i.e. time evolution) of any quantum system are described by a quantum operator in the form of a Hamiltonian, H. There is one quantum number of the system corresponding to the system's energy; i.e., one of the eigenvalues of the Hamiltonian. There is also one quantum number for each linearly independent operator O that commutes with the Hamiltonian. A complete set of commuting observables (CSCO) that commute with the Hamiltonian characterizes the system with all its quantum numbers. There is a one-to-one relationship between the quantum numbers and the operators of the CSCO, with each quantum number taking one of the eigenvalues of its corresponding operator. As a result of the different basis that may be arbitrarily chosen to form a complete set of commuting operators, different sets of quantum numbers may be used for the description of the same system in different situations.
Electron in an atomEdit
Four quantum numbers can describe an electron in an atom completely:
Different electrons in a system will have different quantum numbers. For example, the highest occupied orbital electron, the actual differentiating electron (i.e. the electron that differentiates an element from the previous one); , r the differentiating electron according to the aufbau approximation. In lanthanum, as a further illustration, the electrons involved are in the 6s; 5d; and 4f orbitals, respectively. In this case the principal quantum numbers are 6, 5, and 4.
Common terminologyEdit
The model used here describes electrons using four quantum numbers, n, ℓ, mℓ, ms, given below. It is also the common nomenclature in the classical description of nuclear particle states (e.g. protons and neutrons). A quantum description of molecular orbitals require other quantum numbers, because the Hamiltonian and its symmetries are different.
Principal quantum numberEdit
Main article: Principal quantum number
The principal quantum number describes the electron shell, or energy level, of an electron. The value of n ranges from 1 to the shell containing the outermost electron of that atom, that is[1]
n = 1, 2, ...
For example, in caesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron in caesium can have an n value from 1 to 6.
For particles in a time-independent potential (see Schrödinger equation), it also labels the nth eigenvalue of Hamiltonian (H), that is, the energy E, with the contribution due to angular momentum (the term involving J2) left out. So this number depends only on the distance between the electron and the nucleus (that is, the radial coordinate r). The average distance increases with n. Hence quantum states with different principal quantum numbers are said to belong to different shells.
Azimuthal quantum numberEdit
Main article: Azimuthal quantum number
See also: electron shell § Subshells
The azimuthal quantum number, also known as the (angular momentum quantum number or orbital quantum number), describes the subshell, and gives the magnitude of the orbital angular momentum through the relation.
L2 = ħ2 ℓ (ℓ + 1)
In chemistry and spectroscopy, ℓ = 0 is called s orbital, ℓ = 1, p orbital, ℓ = 2, d orbital, and ℓ = 3, f orbital.
The value of ℓ ranges from 0 to n − 1, so the first p orbital (ℓ = 1) appears in the second electron shell (n = 2), the first d orbital (ℓ = 2) appears in the third shell (n = 3), and so on:[2]
ℓ = 0, 1, 2,..., n − 1
A quantum number beginning in n = 3,ℓ = 0, describes an electron in the s orbital of the third electron shell of an atom. In chemistry, this quantum number is very important, since it specifies the shape of an atomic orbital and strongly influences chemical bonds and bond angles. The azimuthal quantum number can also denote the number of angular nodes present in an orbital. For example, for p orbitals, ℓ = 1 and thus the amount of angular nodes in a p orbital is 1.
Shape of orbital is also given by azimuthal quantum number.
Magnetic quantum numberEdit
Main article: Magnetic quantum number
The magnetic quantum number describes the specific orbital (or "cloud") within that subshell, and yields the projection of the orbital angular momentum along a specified axis:
Lz = mℓ ħ
The values of mℓ range from −ℓ to ℓ, with integer intervals.[3]
The s subshell (ℓ = 0) contains only one orbital, and therefore the mℓ of an electron in an s orbital will always be 0. The p subshell (ℓ = 1) contains three orbitals (in some systems, depicted as three "dumbbell-shaped" clouds), so the mℓ of an electron in a p orbital will be −1, 0, or 1. The d subshell (ℓ = 2) contains five orbitals, with mℓ values of −2, −1, 0, 1, and 2.
Spin quantum numberEdit
Main article: Spin quantum number
The spin quantum number describes the intrinsic spin angular momentum of the electron within each orbital and gives the projection of the spin angular momentum S along the specified axis:
Sz = ms ħ.
In general, the values of ms range from −s to s, where s is the spin quantum number, associated with the particle's intrinsic spin angular momentum:[4]
ms = −s, −s + 1, −s + 2, ..., s − 2, s − 1, s.
An electron has spin number s = 1/2, consequently ms will be ±1/2, referring to "spin up" and "spin down" states. Each electron in any individual orbital must have different quantum numbers because of the Pauli exclusion principle, therefore an orbital never contains more than two electrons.
There are no universal fixed values for mℓ and ms. Rather, the mℓ and ms values are arbitrary. The only restrictions on the choices for these constants is that the naming schematic used within a particular set of calculations or descriptions must be consistent (e.g. the orbital occupied by the first electron in a p orbital could be described as mℓ = −1 or mℓ = 0 or mℓ = 1, but the mℓ value of the next unpaired electron in that orbital must be different; yet, the mℓ assigned to electrons in other orbitals again can be mℓ = −1 or mℓ = 0 or mℓ = 1).
Principal quantum number n shell 1 ≤ n n = 1, 2, 3, …
Azimuthal quantum number (angular momentum) ℓ subshell (s orbital is listed as 0, p orbital as 1 etc.) 0 ≤ ℓ ≤ n − 1 for n = 3:
ℓ = 0, 1, 2 (s, p, d)
Magnetic quantum number (projection of angular momentum) mℓ Orbital (orientation of the orbital) −ℓ ≤ mℓ ≤ ℓ for ℓ = 2:
mℓ = −2, −1, 0, 1, 2
Spin quantum number ms spin of the electron (−1/2 = "spin down", 1/2 = "spin up") −s ≤ ms ≤ s for an electron s = 1/2,
so ms = −1/2, +1/2
Example: The quantum numbers used to refer to the outermost valence electrons of a carbon (C) atom, which are located in the 2p atomic orbital, are; n = 2 (2nd electron shell), ℓ = 1 (p orbital subshell), mℓ = 1, 0, −1, ms = 1/2 (parallel spins).
Results from spectroscopy indicated that up to two electrons can occupy a single orbital. However two electrons can never have the same exact quantum state nor the same set of quantum numbers according to Hund's rules, which addresses the Pauli exclusion principle. A fourth quantum number, which represented spin with two possible values, was added as an ad hoc assumption to resolve the conflict; this supposition would later be explained in detail by relativistic quantum mechanics and from the results of the renowned Stern–Gerlach experiment.
Many different models have been proposed throughout the history of quantum mechanics, but the most prominent system of nomenclature spawned from the Hund-Mulliken molecular orbital theory of Friedrich Hund, Robert S. Mulliken, and contributions from Schrödinger, Slater and John Lennard-Jones. This system of nomenclature incorporated Bohr energy levels, Hund-Mulliken orbital theory, and observations on electron spin based on spectroscopy and Hund's rules.[5]
Total angular momenta numbersEdit
Total angular momentum of a particleEdit
Further information: Clebsch–Gordan coefficients
See also: Azimuthal quantum number § Total angular momentum of an electron in the atom
When one takes the spin–orbit interaction into consideration, the L and S operators no longer commute with the Hamiltonian, and their eigenvalues therefore change over time. Thus another set of quantum numbers should be used. This set includes[6][7]
The total angular momentum quantum number:
j = |ℓ ± s|
which gives the total angular momentum through the relation
J2 = ħ2 j (j + 1)
The projection of the total angular momentum along a specified axis:
mj = −j, −j + 1, −j + 2, ..., j − 2, j − 1, j
analogous to the above and satisfies
mj = mℓ + ms and |mℓ + ms| ≤ j
This is the eigenvalue under reflection: positive (+1) for states which came from even ℓ and negative (−1) for states which came from odd ℓ. The former is also known as even parity and the latter as odd parity, and is given by
P = (−1)ℓ
For example, consider the following 8 states, defined by their quantum numbers:
ℓ + s
ℓ − s
mℓ + ms
2 1 1 +1/2 3/2 1/2 3/2
2 1 1 −1/2 3/2 1/2 1/2
2 1 0 −1/2 3/2 1/2 −1/2
2 1 −1 +1/2 3/2 1/2 −1/2
2 1 −1 −1/2 3/2 1/2 −3/2
2 0 0 +1/2 1/2 −1/2 1/2
2 0 0 −1/2 1/2 −1/2 −1/2
The quantum states in the system can be described as linear combination of these 8 states. However, in the presence of spin–orbit interaction, if one wants to describe the same system by 8 states that are eigenvectors of the Hamiltonian (i.e. each represents a state that does not mix with others over time), we should consider the following 8 states:
3/2 3/2 odd coming from state (1) above
3/2 1/2 odd coming from states (2) and (3) above
3/2 −1/2 odd coming from states (4) and (5) above
3/2 −3/2 odd coming from state (6) above
1/2 1/2 even coming from state (7) above
1/2 −1/2 even coming from state (8) above
Nuclear angular momentum quantum numbersEdit
In nuclei, the entire assembly of protons and neutrons (nucleons) has a resultant angular momentum due to the angular momenta of each nucleon, usually denoted I. If the total angular momentum of a neutron is jn = ℓ + s and for a proton is jp = ℓ + s (where s for protons and neutrons happens to be 1/2 again (see note)), then the nuclear angular momentum quantum numbers I are given by:
I = |jn − jp|, |jn − jp| + 1, |jn − jp| + 2, ..., (jn + jp) − 2, (jn + jp) − 1, (jn + jp)
Note: The orbital angular momenta of the nuclear (and atomic) states are all integer multiples of ħ while the intrinsic angular momentum of the neutron and proton are half-integer multiples. It should be immediately apparent that the combination of the intrinsic spins of the nucleons with their orbital motion will always give half-integer values for the total spin, I, of any odd-A nucleus and integer values for any even-A nucleus.
Parity with the number I is used to label nuclear angular momentum states, examples for some isotopes of hydrogen (H), carbon (C), and sodium (Na) are;[8]
I = (1/2)+ 9
I = (3/2)− 20
I = 2+
I = 1+ 10
I = (3/2)+
I = (1/2)+ 11
The reason for the unusual fluctuations in I, even by differences of just one nucleon, are due to the odd and even numbers of protons and neutrons – pairs of nucleons have a total angular momentum of zero (just like electrons in orbitals), leaving an odd or even number of unpaired nucleons. The property of nuclear spin is an important factor for the operation of NMR spectroscopy in organic chemistry,[7] and MRI in nuclear medicine,[8] due to the nuclear magnetic moment interacting with an external magnetic field.
Elementary particlesEdit
For a more complete description of the quantum states of elementary particles, see Standard model and Flavour (particle physics).
Elementary particles contain many quantum numbers which are usually said to be intrinsic to them. However, it should be understood that the elementary particles are quantum states of the standard model of particle physics, and hence the quantum numbers of these particles bear the same relation to the Hamiltonian of this model as the quantum numbers of the Bohr atom does to its Hamiltonian. In other words, each quantum number denotes a symmetry of the problem. It is more useful in quantum field theory to distinguish between spacetime and internal symmetries.
Typical quantum numbers related to spacetime symmetries are spin (related to rotational symmetry), the parity, C-parity and T-parity (related to the Poincaré symmetry of spacetime). Typical internal symmetries[clarification needed] are lepton number and baryon number or the electric charge. (For a full list of quantum numbers of this kind see the article on flavour.)
Multiplicative quantum numbersEdit
Most conserved quantum numbers are additive, so in an elementary particle reaction, the sum of the quantum numbers should be the same before and after the reaction. However, some, usually called a parity, are multiplicative; i.e., their product is conserved. All multiplicative quantum numbers belong to a symmetry (like parity) in which applying the symmetry transformation twice is equivalent to doing nothing (involution).
^ specifically, observables
{\displaystyle {\widehat {A}}}
that commute with the Hamiltonian are simultaneously diagonalizable with it and so the eigenvalues
{\displaystyle a}nd the energy (eigenvalues of the Hamiltonian) are not limited by an uncertainty relation arising from non-commutativity.
^ Many observables have discrete spectra (sets of eigenvalues) in quantum mechanics, so the quantities can only be measured in discrete (often integer) values.
^ Beiser, A. (1987). Concepts of Modern Physics (4th ed.). McGraw-Hill (International). ISBN 0-07-100144-1. [page needed]
^ Atkins, P. W. (1977). Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry. Vol. 1. Oxford University Press. ISBN 0-19-855129-0. [page needed]
^ Eisberg, R.; Resnick, R. (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd ed.). John Wiley & Sons. ISBN 978-0-471-87373-0. [page needed]
^ Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010). Quantum Mechanics. Schuam's Outlines (2nd ed.). McGraw Hill (USA). ISBN 978-0-07-162358-2. [page needed]
^ a b Atkins, P. W. (1977). Molecular Quantum Mechanics Part III: An Introduction to Quantum Chemistry. Vol. 2. Oxford University Press. [ISBN missing][page needed]
^ a b Krane, K. S. (1988). Introductory Nuclear Physics. John Wiley & Sons. ISBN 978-0-471-80553-3. [page needed]
Dirac, Paul A. M. (1982). Principles of quantum mechanics. Oxford University Press. ISBN 0-19-852011-5.
Halzen, Francis & Martin, Alan D. (1984). QUARKS AND LEPTONS: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Quantum_number&oldid=1088306626"
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Schlick's approximation - Wikipedia
In 3D computer graphics, Schlick’s approximation, named after Christophe Schlick, is a formula for approximating the contribution of the Fresnel factor in the specular reflection of light from a non-conducting interface (surface) between two media.[1]
According to Schlick’s model, the specular reflection coefficient R can be approximated by:
{\displaystyle {\begin{aligned}R(\theta )&=R_{0}+(1-R_{0})(1-\cos \theta )^{5}\\\\where\\R_{0}&=\left({\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right)^{2}\end{aligned}}}
{\displaystyle \theta }
is the angle between the direction from which the incident light is coming and the normal of the interface between the two media, hence
{\displaystyle \cos \theta =(N\cdot V)}
{\displaystyle n_{1},\,n_{2}}
are the indices of refraction of the two media at the interface and
{\displaystyle R_{0}}
is the reflection coefficient for light incoming parallel to the normal (i.e., the value of the Fresnel term when
{\displaystyle \theta =0}
or minimal reflection). In computer graphics, one of the interfaces is usually air, meaning that
{\displaystyle n_{1}}
very well can be approximated as 1.
In microfacet models it is assumed that there is always a perfect reflection, but the normal changes according to a certain distribution, resulting in a non-perfect overall reflection. When using Schlick’s approximation, the normal in the above computation is replaced by the halfway vector. Either the viewing or light direction can be used as the second vector.[2]
^ Schlick, C. (1994). "An Inexpensive BRDF Model for Physically-based Rendering" (PDF). Computer Graphics Forum. 13 (3): 233–246. CiteSeerX 10.1.1.12.5173. doi:10.1111/1467-8659.1330233. S2CID 7825646.
^ Hoffman, Naty (2013). "Background: Physics and Math of Shading" (PDF). Fourth International Conference and Exhibition on Computer Graphics and Interactive Techniques.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Schlick%27s_approximation&oldid=1064313445"
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Whitehead's point-free geometry - Wikipedia
In mathematics, point-free geometry is a geometry whose primitive ontological notion is region rather than point. Two axiomatic systems are set out below, one grounded in mereology, the other in mereotopology and known as connection theory. A point can mark a space or objects.
2 Inclusion-based point-free geometry (mereology)
3 Connection theory (mereotopology)
Point-free geometry was first formulated in Whitehead (1919, 1920), not as a theory of geometry or of spacetime, but of "events" and of an "extension relation" between events. Whitehead's purposes were as much philosophical as scientific and mathematical.[1]
Whitehead did not set out his theories in a manner that would satisfy present-day canons of formality. The two formal first-order theories described in this entry were devised by others in order to clarify and refine Whitehead's theories. The domain of discourse for both theories consists of "regions." All unquantified variables in this entry should be taken as tacitly universally quantified; hence all axioms should be taken as universal closures. No axiom requires more than three quantified variables; hence a translation of first-order theories into relation algebra is possible. Each set of axioms has but four existential quantifiers.
Inclusion-based point-free geometry (mereology)[edit]
The axioms G1 to G7 are, but for numbering, those of Def. 2.1 in Gerla and Miranda (2008) (see also Gerla (1995)). The identifiers of the form WPn, included in the verbal description of each axiom, refer to the corresponding axiom in Simons (1987: 83).
The fundamental primitive binary relation is inclusion, denoted by infix "≤", which corresponds to the binary Parthood relation that is a standard feature in mereological theories. The intuitive meaning of x ≤ y is "x is part of y." Assuming that equality, denoted by infix "=", is part of the background logic, the binary relation Proper Part, denoted by infix "<", is defined as:
{\displaystyle x<y\leftrightarrow (x\leq y\land x\not =y).}
Inclusion partially orders the domain.
G1.
{\displaystyle x\leq x.}
(reflexive)
{\displaystyle (x\leq z\land z\leq y)\rightarrow x\leq y.}
(transitive) WP4.
{\displaystyle (x\leq y\land y\leq x)\rightarrow x=y.}
Given any two regions, there exists a region that includes both of them. WP6.
{\displaystyle \exists z[x\leq z\land y\leq z].}
Proper Part densely orders the domain. WP5.
{\displaystyle x<y\rightarrow \exists z[x<z<y].}
Both atomic regions and a universal region do not exist. Hence the domain has neither an upper nor a lower bound. WP2.
{\displaystyle \exists y\exists z[y<x\land x<z].}
Proper Parts Principle. If all the proper parts of x are proper parts of y, then x is included in y. WP3.
{\displaystyle \forall z[z<x\rightarrow z<y]\rightarrow x\leq y.}
A model of G1–G7 is an inclusion space.
Definition (Gerla and Miranda 2008: Def. 4.1). Given some inclusion space S, an abstractive class is a class G of regions such that S\G is totally ordered by inclusion. Moreover, there does not exist a region included in all of the regions included in G.
Intuitively, an abstractive class defines a geometrical entity whose dimensionality is less than that of the inclusion space. For example, if the inclusion space is the Euclidean plane, then the corresponding abstractive classes are points and lines.
Inclusion-based point-free geometry (henceforth "point-free geometry") is essentially an axiomatization of Simons's (1987: 83) system W. In turn, W formalizes a theory in Whitehead (1919) whose axioms are not made explicit. Point-free geometry is W with this defect repaired. Simons (1987) did not repair this defect, instead proposing in a footnote that the reader do so as an exercise. The primitive relation of W is Proper Part, a strict partial order. The theory[2] of Whitehead (1919) has a single primitive binary relation K defined as xKy ↔ y < x. Hence K is the converse of Proper Part. Simons's WP1 asserts that Proper Part is irreflexive and so corresponds to G1. G3 establishes that inclusion, unlike Proper Part, is antisymmetric.
Point-free geometry is closely related to a dense linear order D, whose axioms are G1-3, G5, and the totality axiom
{\displaystyle x\leq y\lor y\leq x.}
[3] Hence inclusion-based point-free geometry would be a proper extension of D (namely D ∪ {G4, G6, G7}), were it not that the D relation "≤" is a total order.
Connection theory (mereotopology)[edit]
A different approach was proposed in Whitehead (1929), one inspired by De Laguna (1922). Whitehead took as primitive the topological notion of "contact" between two regions, resulting in a primitive "connection relation" between events. Connection theory C is a first-order theory that distills the first 12 of the 31 assumptions in chapter 2 of part 4 of Process and Reality into 6 axioms, C1-C6. C is a proper fragment of the theories proposed in Clarke (1981), who noted their mereological character. Theories that, like C, feature both inclusion and topological primitives, are called mereotopologies.
C has one primitive relation, binary "connection," denoted by the prefixed predicate letter C. That x is included in y can now be defined as x ≤ y ↔ ∀z[Czx→Czy]. Unlike the case with inclusion spaces, connection theory enables defining "non-tangential" inclusion,[4] a total order that enables the construction of abstractive classes. Gerla and Miranda (2008) argue that only thus can mereotopology unambiguously define a point.
The axioms C1-C6 below are, but for numbering, those of Def. 3.1 in Gerla and Miranda (2008):
C is reflexive. C.1.
{\displaystyle \ Cxx.}
C is symmetric. C.2.
{\displaystyle Cxy\rightarrow Cyx.}
C is extensional. C.11.
{\displaystyle \forall z[Czx\leftrightarrow Czy]\rightarrow x=y.}
All regions have proper parts, so that C is an atomless theory. P.9.
{\displaystyle \exists y[y<x].}
Given any two regions, there is a region connected to both of them.
{\displaystyle \exists z[Czx\land Czy].}
All regions have at least two unconnected parts. C.14.
{\displaystyle \exists y\exists z[(y\leq x)\land (z\leq x)\land \neg Cyz].}
A model of C is a connection space.
Following the verbal description of each axiom is the identifier of the corresponding axiom in Casati and Varzi (1999). Their system SMT (strong mereotopology) consists of C1-C3, and is essentially due to Clarke (1981).[5] Any mereotopology can be made atomless by invoking C4, without risking paradox or triviality. Hence C extends the atomless variant of SMT by means of the axioms C5 and C6, suggested by chapter 2 of part 4 of Process and Reality. For an advanced and detailed discussion of systems related to C, see Roeper (1997).
Biacino and Gerla (1991) showed that every model of Clarke's theory is a Boolean algebra, and models of such algebras cannot distinguish connection from overlap. It is doubtful whether either fact is faithful to Whitehead's intent.
^ See Kneebone (1963), chpt. 13.5, for a gentle introduction to Whitehead's theory. Also see Lucas (2000), chpt. 10.
^ Kneebone (1963), p. 346.
^ Also see Stoll, R. R., 1963. Set Theory and Logic. Dover reprint, 1979. P. 423.
^ Presumably this is Casati and Varzi's (1999) "Internal Part" predicate, IPxy ↔ (x≤y)∧(Czx→∃v[v≤z ∧ v≤y]. This definition combines their (4.8) and (3.1).
^ Grzegorczyk (1960) proposed a similar theory, whose motivation was primarily topological.
Biacino L., and Gerla G., 1991, "Connection Structures," Notre Dame Journal of Formal Logic 32: 242-47.
Casati, R., and Varzi, A. C., 1999. Parts and places: the structures of spatial representation. MIT Press.
Clarke, Bowman, 1981, "A calculus of individuals based on 'connection'," Notre Dame Journal of Formal Logic 22: 204-18.
------, 1985, "Individuals and Points," Notre Dame Journal of Formal Logic 26: 61-75.
De Laguna, T., 1922, "Point, line and surface as sets of solids," The Journal of Philosophy 19: 449-61.
--------, and Miranda A., 2008, "Inclusion and Connection in Whitehead's Point-free Geometry," in Michel Weber and Will Desmond, (eds.), Handbook of Whiteheadian Process Thought, Frankfurt / Lancaster, ontos verlag, Process Thought X1 & X2.
Gruszczynski R., and Pietruszczak A., 2008, "Full development of Tarski's geometry of solids," Bulletin of Symbolic Logic 14:481-540. The paper contains presentation of point-free system of geometry originating from Whitehead's ideas and based on Lesniewski's mereology. It also briefly discusses the relation between point-free and point-based systems of geometry. Basic properties of mereological structures are given as well.
Grzegorczyk, A., 1960, "Axiomatizability of geometry without points," Synthese 12: 228-235.
Kneebone, G., 1963. Mathematical Logic and the Foundation of Mathematics. Dover reprint, 2001.
Lucas, J. R., 2000. Conceptual Roots of Mathematics. Routledge. Chpt. 10, on "prototopology," discusses Whitehead's systems and is strongly influenced by the unpublished writings of David Bostock.
Roeper, P., 1997, "Region-Based Topology," Journal of Philosophical Logic 26: 251-309.
Simons, P., 1987. Parts: A Study in Ontology. Oxford Univ. Press.
Whitehead, A.N., 1916, "La Theorie Relationiste de l'Espace," Revue de Metaphysique et de Morale 23: 423-454. Translated as Hurley, P.J., 1979, "The relational theory of space," Philosophy Research Archives 5: 712-741.
--------, 1919. An Enquiry Concerning the Principles of Natural Knowledge. Cambridge Univ. Press. 2nd ed., 1925.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Whitehead%27s_point-free_geometry&oldid=1047002061"
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Category:DFT+U - Vaspwiki
Category:DFT+U
The LDA and semilocal GGA functionals often fail to describe systems with localized (strongly correlated)
{\displaystyle d}
{\displaystyle f}
electrons (this manifests itself primarily in the form of unrealistic one-electron energies or too small magnetic moments in the case of systems with
{\displaystyle d}
electrons). In some cases this can be remedied by introducing on the
{\displaystyle d}
{\displaystyle f}
atom a strong intra-atomic interaction in a simplified (screened) Hartree-Fock like manner (
{\displaystyle E_{\text{HF}}({\hat {n}})}
), as an on-site replacement of the semilocal functional (double-counting term
{\displaystyle E_{\text{dc}}({\hat {n}})}
{\displaystyle E_{\text{xc}}^{{\text{LDA/GGA}}+U}(n,{\hat {n}})=E_{\text{xc}}^{\text{LDA/GGA}}(n)+E_{\text{HF}}({\hat {n}})-E_{\text{dc}}({\hat {n}})}
{\displaystyle {\hat {n}}}
is the on-site occupancy matrix of the
{\displaystyle d}
{\displaystyle f}
electrons. This approach is known as the DFT+U method (traditionally called LSDA+U[1] ).
The first VASP DFT+U calculations, including some additional technical details on the VASP implementation, can be found in Ref. [2] (the original implementation was done by Olivier Bengone [3] and Georg Kresse).
More detail about the formalism of the DFT+U method can be found here.
DFT+U can be switched on with the LDAU tag in the INCAR file, while the LDAUTYPE tag determines the DFT+U flavor that is used.
↑ V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B 44, 943 (1991).
↑ A. Rohrbach, J. Hafner, and G. Kresse J. Phys.: Condens. Matter 15, 979 (2003).
↑ O. Bengone, M. Alouani, P. Blöchl, and J. Hugel, Phys. Rev. B 62, 16392 (2000).
Pages in category "DFT+U"
Retrieved from "https://www.vasp.at/wiki/index.php?title=Category:DFT%2BU&oldid=17420"
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Sprague Grundy Theorem | Brilliant Math & Science Wiki
Calvin Lin, Arron Kau, Eli Ross, and
The Sprague-Grundy theorem is a statement about impartial games.
In Combinatorial Games - Winning Positions, we analyzed winning positions of impartial games. Several questions arose in trying to find a general characterization for whether a set of nim piles is a winning position or a losing position. Here, we answer these questions by giving the complete characterization for winning and losing positions of nim.
Suppose we have nim piles of sizes
(a_1, a_2, \ldots, a_n)
b_i
be the base 2 representation of
a_i
. A nim position is losing if over all the numbers in
\{b_i\}_{i=1}^{n}
, the number of 1s in each binary position is even and winning otherwise.
Before we prove this statement, let’s look at an example to see what we mean. If the nim piles have sizes
(3,6,8)
, then we can write these numbers as
11_2, 110_2, 1000_2
. We want to look at each binary position of these numbers, so let’s arrange them in a table:
\begin{array}{cccc} & & 1 &1\\ & 1 &1 &0\\ 1 & 0 & 0 & 0\\ \end{array}
We can construct a number that represents the parity of the number of 1's in each binary position. We call this the nim-sum of these numbers. Looking at each column of the table, we see that the 1's column has a single 1, the 10's column has two 1's, and the 100's column and the 1000's column each have a single 1. Our nim-sum here is
1101_2\ = 13
If the piles have sizes
7,9,14
111_2, 1001_2, 1110_2
\begin{array}{cccc} & 1& 1 &1\\ 1& 0&0 &1\\ 1 & 1& 1 & 0\\ \hline 0 & 0 & 0 & 0 \\ \end{array}
The last line represents the parity of the number of 1's, which gives us that the nim-sum of these numbers is 0.
We will modify our description of a winning position to be one where the nim-sum of the pile sizes is not 0, which is equivalent to the original definition. To verify that this accurately describes the set of winning and losing positions, we check 3 things.
First, it is clear that the game with no stones is losing, and has nim-sum 0.
Second, if we are in a losing position, we have nim-sum 0. If we remove stones from a single pile, we are changing the positions of 1's in a single column, which will make the nim-sum no longer 0.
Third, if we are in a winning position, we have nim-sum of the form
1x_2
x
is a binary string. We choose a pile that has a 1 in the same position as the leftmost 1 in the nim-sum. We change this number by flipping each digit for which there is a 1 in the corresponding position in the nim-sum. The nim-sum of these new numbers will be 0, which is a losing position. It is left as an exercise to show that the new number is less than the original number and thus constitutes a valid move.
_\square
The nim-sum actually gives us more information than determining whether a position is winning or losing; it also assigns a value to that position. If a position has nim-sum
k
, then from that position it will always be possible to move to a position with nim-sum
j
j < k,
and not possible to move to a position with nim-sum
k
. To see this, consider the proof that any winning position can get to a 0 position, and replace 0 with any smaller number. The described method will extend to this.
We have spent a lot of time developing the theory for a single impartial game, Nim. In fact, this is all the theory that we need to understand any general impartial game! We have the following surprising result:
Sprague-Grundy Theorem
_\square
We can assign a nim-sum to the positions in any impartial game by building up sequentially through small cases. The nim-sum of a position is the smallest non-negative integer
k
such that we cannot move to a position with nim-sum
k
. This requires us to calculate the nim-sum of every position that we can move to. Since the game will always end, this calculation is finite. However, it can get computationally intensive, and a computer may be useful.
The proof of the Sprague-Grundy theorem is identical to our previous proof, where we checked 3 statements.
Two players play a game starting with a pile of
n
stones. On each turn, a player removes from the pile
1,2,3,\ldots, \mbox{ or } k
stones. The person who takes the last stone wins. Determine, in terms of
n
k
, the nim-sum of each position.
j \leq k
, the nim-sum of the game with
j
stones will be
j
. This is easy to see inductively starting at
j = 0
j = k+1
, this is a losing position, so the nim-sum will be 0. If we repeat this, we see inductively that the nim-sum of
i
i + k + 1
will always be the same. So the nim-sum of
n
stones where we can remove at most
k
n \pmod{k + 1}.
_\square
Odd Nim is like Nim, except you can only remove an odd number of stones from a pile. Determine the nim-sum for a single pile of
n
Since we can only take an odd number of stones from the pile, the game will always be a losing position if the number of stones is even and a winning position if the number of stones is odd. From a winning position, we can only get to the 0 position, so all winning positions have nim-sum 1. Thus the nim-sum is
n \pmod{2}
_\square
Cite as: Sprague Grundy Theorem. Brilliant.org. Retrieved from https://brilliant.org/wiki/sprague-grundy-theorem/
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Search results for: Mama Nsangou
DFT study of geometrical and vibrational features of a 3′,5′-deoxydisugar-monophosphate (dDSMP) DNA model in the presence of counterions and solvent
Alain Minguirbara, Mama Nsangou
The B3LYP/6–31++G* theoretical level was used to study the influence of various hexahydrated monovalent (Li+, Na+, K+) and divalent (Mg2+) metal counterions in interaction with the charged PO2− group, on the geometrical and vibrational characteristics of the DNA fragments of 3′,5′-dDSMP, represented by four conformers (g+g+, g+t, g−g− and g−t). All complexes were optimized through two solvation models...
Rotational cross sections and rate coefficients of CP(X2Σ+) $\mathrm{CP}(\mathrm{X}^{2}\varSigma ^{+})$ induced by its collision with He(1S) $\mathrm{He}(^{1}S)$ at low temperature
Théophile Tchakoua, Mama Pamboundom, Berthelot Said Duvalier Ramlina Vamhindi, Serge Guy Nana Engo, more
The potential energy surface (PES) for the CP(X2Σ+) $\mathrm{CP}(\mathrm{X}^{2}\varSigma^{+})$- He(1S) $\mathrm{He}(^{1}S)$ complex has been calculated at the RCCSD(T)-F12/VTZ-F12 level of theory. The analytic fit of the PES was obtained by using global analytical method. The fitted PES was used subsequently in the close-coupling approach for the computation of the state-to-state collisional excitation...
Fluorescence Spectroscopy Combined with Chemometrics for the Investigation of the Adulteration of Essential Oils
William Mbogning Feudjio, Hassen Ghalila, Mama Nsangou, Youssef Majdi, more
Food Analytical Methods > 2017 > 10 > 7 > 2539-2548
Artificial neural networks (ANNs) were built with excitation-emission matrix fluorescence (EEMF) spectra of essential oils for the investigation of their adulteration. With self-organized maps (SOMs), the clusters formed by all the types of essential oils were visualized. Pure essential oils were globally separated from their adulterated samples. The nature of the adulterant (vegetable oil, essential...
Electronic structure, stability and spectroscopy of low-lying states of NO−, HNO− and HON− molecular anions
Berthelot Saïd Duvalier Ramlina Vamhindi, Mama Nsangou
Computational and Theoretical Chemistry > 2016 > 1094 > C > 69-81
Accurate post Hartree-Fock methods have been used to investigate electronic structure, spectroscopy, stability and metastability of low-lying electronic states of HNO, HON, NO, NO−, HNO− and HON−. Accurate vertical transition energies between lowest electronic states of HNO and HON isomers have been determined. The HNO vertical transition energy of the X∼1A′-ã3A″ transition is shown to be 0.87eV...
Rotational excitation of AlCl induced by its collision with helium: cross sections and collisional rate coefficients
Mama Pamboundom, Théophile Tchakoua, Mama Nsangou
Astrophysics and Space Science > 2016 > 361 > 4 > 1-10
In this work, inelastic rotational collision of AlCl with helium was studied. The CCSD(T) method was used for the computation of an accurate two dimensional potential energy surface (PES). In the calculation of the PES, Al-Cl bond was frozen at the experimental value 4.02678 a 0
4.02678~\mbox{a}_{0}
. The aug-cc-pVQZ basis sets of Dunning was used throughout the computational process. This...
Rotational excitation induced by collision of AlOH with helium
Théophile Tchakoua, Mama Pamboundom, Mama Nsangou, Ousmanou Motapon
The present paper deals with the calculation of rate coefficients for the 16 first rotational levels of the AlOH molecule in its ground vibrational state, induced by collision with helium. Such data are useful for studies of low-temperature interstellar environments. A new potential energy surface (PES) for the AlOH-He system, calculated at a AlO and OH r-distance frozen at their experimental equilibrium...
Solvent effects on the structures and vibrational features of zwitterionic dipeptides: L-diglycine and L-dialanine
Steve Jonathan Koyambo-Konzapa, Alain Minguirbara, Mama Nsangou
Calculations were done by applying the B3LYP/6-31++G(d) method on the zwitterionic l-diglycine and l-dialanine to study the solvent effects on their structures and vibrational features. Three models of solvation (implicit, explicit, and explicit in implicit) were used and the subsequent resulting values compared. Even though both dipeptides surrounded by 12 water molecules seem sufficient to stabilize...
Excitation-emission matrix fluorescence coupled to chemometrics for the exploration of essential oils
William Mbogning Feudjio, Hassen Ghalila, Mama Nsangou, Yvon G. Mbesse Kongbonga, more
Talanta > 2014 > 130 > Complete > 148-154
Excitation-emission matrix fluorescence (EEMF) coupled to chemometrics was used to explore essential oils (EOs). The spectrofluorometer was designed with basic and inexpensive materials and was accompanied by appropriate tools for data pre-treatment. Excitation wavelengths varied between 320nm and 600nm while emission wavelengths were from 340nm to 700nm. Excitation-emission matrix (EEM) spectra of...
Unusual Quantum Interference in the S 1 State of DABCO and Observation of Intramolecular Vibrational Redistribution †
Lionel Poisson, Raman Maksimenska, Benoît Soep, Jean-Michel Mestdagh, more
Geometrical and vibrational features of phosphate, phosphorothioate and phosphorodithioate linkages interacting with hydrated cations: A DFT study
Zoubeida Dhaouadi, Mama Nsangou, Belén Hernández, Fernando Pflüger, more
Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy > 2009 > 73 > 5 > 805-814
The effect of hexahydrated monovalent and divalent cations on the geometrical and vibrational features of dimethyl phosphate, dimethyl phosphorothioate and dimethyl phosphorodithioate anions (simple suitable model compounds representing the anionic moieties of natural and some modified nucleic acids) was studied. For this purpose, density functional theory (DFT) calculations were carried out at the...
Vibrational Analysis of Amino Acids and Short Peptides in Hydrated Media. IV. Amino Acids with Hydrophobic Side Chains: l -Alanine, l -Valine, and l -Isoleucine
Belén Hernández, Fernando Pflüger, Mama Nsangou, Mahmoud Ghomi
COLLISIONAL RATE COEFFICIENTS (3)
CLOSE COUPLING (2)
EXCITATION-EMISSION MATRIX FLUORESCENCE (2)
INELASTIC ROTATIONAL COLLISION (2)
3′,5′-DEOXYDISUGAR-MONOPHOSPHATE (1)
6-31++G(D) (1)
ALOH-HE (1)
B3LYP/6–31++G* (1)
CCSD(T)-F12 (1)
CCSD(T)/AVQZ (1)
CLOSE-COUPLING (1)
COUNTERIONS (1)
DFT (DENSITY FUNCTIONAL THEORY) (1)
HNO− (1)
HON− (1)
MODE ASSIGNMENT (1)
MONO- AND DIVALENT COUNTERIONS (1)
NO− (1)
PARALLEL FACTOR ANALYSIS (1)
PHOSPHORODITHIOATE (1)
PHOSPHOROTHIOATE (1)
ROTATIONAL EXCITATION (1)
SPECTROSCOPIC CONSTANTS (1)
UNFOLD PRINCIPAL COMPONENT ANALYSIS (1)
VIBRATIONAL FEATURES (1)
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Messier_2 Knowpia
Messier 2 or M2 (also designated NGC 7089) is a globular cluster in the constellation Aquarius, five degrees north of the star Beta Aquarii. It was discovered by Jean-Dominique Maraldi in 1746, and is one of the largest known globular clusters.
Messier 2 by Hubble Space Telescope; 2.5′ view
55,000 ly (17 kpc)[3]
{\displaystyle {\begin{smallmatrix}\left[{\ce {Fe}}/{\ce {H}}\right]\end{smallmatrix}}}
M2 was discovered by the French astronomer Jean-Dominique Maraldi in 1746[8] while observing a comet with Jacques Cassini.[citation needed] Charles Messier rediscovered it in 1760, but thought it a nebula without any stars associated with it. William Herschel, in 1783, was the first to resolve individual stars in the cluster.[citation needed]
M2 is, under extremely good conditions, just visible to the naked eye. Binoculars or a small telescope will identify this cluster as non-stellar, while larger telescopes will resolve individual stars, of which the brightest are of apparent magnitude 13.1.[citation needed]
M2 is about 55,000 light-years distant from Earth. At 175 light-years in diameter, it is one of the larger globular clusters known. The cluster is rich, compact, and significantly elliptical. It is 13 billion years old and one of the older globulars associated with the Milky Way galaxy.[citation needed]
M2 contains about 150,000 stars, including 21 known variable stars. Its brightest stars are red and yellow giant stars. The overall spectral type is F4.[7] M2 is part of the Gaia Sausage, the hypothesised remains of a merged dwarf galaxy.[9]
Data from Gaia has led to the discovery of an extended tidal stellar stream, about 45 degrees long and 300 light-years (100 pc) wide, that is likely associated with M2. It was possibly perturbed due to the presence of the Large Magellanic Cloud.[10]
Map showing location of M2
^ Collaboration, Gaia; Helmi, A; van Leeuwen, F; McMillan, P. J; Massari, D; Antoja, T; Robin, A; Lindegren, L; Bastian, U; co-authors, 445 (2018). "Gaia Data Release 2: Kinematics of globular clusters and dwarf galaxies around the Milky Way". Astronomy and Astrophysics. 616: A12. arXiv:1804.09381. Bibcode:2018A&A...616A..12G. doi:10.1051/0004-6361/201832698. {{cite journal}}: CS1 maint: numeric names: authors list (link)
^ "Messier 2". SEDS Messier Catalog. Retrieved 27 April 2022.
^ a b "M 2". SIMBAD. Centre de données astronomiques de Strasbourg. Retrieved 2006-11-15.
^ Stephen James O'Meara (7 April 2014). Deep-Sky Companions: The Messier Objects. Cambridge University Press. pp. 47–. ISBN 978-1-107-01837-2.
^ Myeong, G. C; Evans, N. W; Belokurov, V; Sanders, J. L; Koposov, S. E (2018). "The Sausage Globular Clusters". The Astrophysical Journal. 863 (2): L28. arXiv:1805.00453. Bibcode:2018ApJ...863L..28M. doi:10.3847/2041-8213/aad7f7.
^ Grillmair, Carl J. (2022). "The Extended Tidal Tails of NGC 7089 (M2)". The Astrophysical Journal. 929 (1): 89. arXiv:2203.04425. Bibcode:2022ApJ...929...89G. doi:10.3847/1538-4357/ac5bd7. S2CID 247318732.
M2 in the Staracle Messier catalog
Messier 2 on WikiSky: DSS2, SDSS, GALEX, IRAS, Hydrogen α, X-Ray, Astrophoto, Sky Map, Articles and images
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Compact Modeling of Fluid Flow and Heat Transfer in Straight Fin Heat Sinks | J. Electron. Packag. | ASME Digital Collection
Contributed by the Electronic and Photonic Packaging Division for publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received May 2003; final revision, December 2003. Associate Editor: B. Sammakia.
Kim, D., and Kim, S. J. (July 8, 2004). "Compact Modeling of Fluid Flow and Heat Transfer in Straight Fin Heat Sinks ." ASME. J. Electron. Packag. June 2004; 126(2): 247–255. https://doi.org/10.1115/1.1756149
In the present work, a compact modeling method based on a volume-averaging technique is presented. Its application to an analysis of fluid flow and heat transfer in straight fin heat sinks is then analyzed. In this study, the straight fin heat sink is modeled as a porous medium through which fluid flows. The volume-averaged momentum and energy equations for developing flow in these heat sinks are obtained using the local volume-averaging method. The permeability and the interstitial heat transfer coefficient required to solve these equations are determined analytically from forced convective flow between infinite parallel plates. To validate the compact model proposed in this paper, three aluminum straight fin heat sinks having a base size of
101.43mm×101.43mm
are tested with an inlet velocity ranging from 0.5 m/s to 2 m/s. In the experimental investigation, the heat sink is heated uniformly at the bottom. The resulting pressure drop across the heat sink and the temperature distribution at its bottom are then measured and are compared with those obtained through the porous medium approach. Upon comparison, the porous medium approach is shown to accurately predict the pressure drop and heat transfer characteristics of straight fin heat sinks. In addition, evidence indicates that the entrance effect should be considered in the thermal design of heat sinks when
Re Dh/L>∼O10.
heat transfer, heat sinks, flow through porous media, forced convection, electronics packaging, permeability, flow simulation, aluminium, temperature distribution
Flow (Dynamics), Fluid dynamics, Heat sinks, Heat transfer, Modeling, Porous materials, Temperature distribution, Permeability, Pressure drop
Adams, V. H., Joshi, Y., and Blackburn, D. L., 1997, “Application of Compact Model Methodologies to Natural Convection Cooling of an Array of Electronic Packages in a Low Profile Enclosure,” Proc. The Pacific Rim/ASME International Intersociety Electronic & Photonic Packaging Conference, EEP-Vol. 19-2, E. Suhir et al., eds., The American Society of Mechanical Engineers, New York, pp. 1967–1974.
Package Thermal Resistance Model: Dependency on Equipment Design
Surface-Mount Plastic Packages-An Assessment of Their Thermal Performance
Joint Electron Device Engineering Council (JEDEC), JC 15.1 Subcommittee, 1995, JEDEC Standard EIA/JESD51-2, “Integrated Circuit Thermal Test Method Environmental Conditions-Natural Convection (Still Air),” developed by Electronic Industries Association (EIA).
IEEE Trans. Compon., Hybrid, Manuf. Technol.
Compact Thermal Models of Packages Used in Conduction Cooled Applications
Ewes, I., 1995, “Modeling of IC-Packages Based on Thermal Characteristics,” Proc. EUROTHERM Seminar No. 45, Sept. 1995, Leuven, Belgium, pp. 11.1–11.12.
Characterization of Compact Heat Sink Models in Natural Convection
Heat Transfer of Microctructures for Integrated Circuits
Shah, R. K., and London, A. L., 1978, Laminar Flow Forced Convection in Ducts, Academic Press, London.
Thermal Optimization of a Circular-Sectored Finned Tube Using a Porous Medium Approach
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Restriction in flow area in thermal liquid network - MATLAB - MathWorks United Kingdom
Minimum restriction area
Maximum restriction area
Restriction in flow area in thermal liquid network
The Local Restriction (TL) block models the pressure drop due to a localized reduction in flow area, such as a valve or an orifice, in a thermal liquid network.
Ports A and B represent the restriction inlet and outlet. The input physical signal at port AR specifies the restriction area. Alternatively, you can specify a fixed restriction area as a block parameter.
The block icon changes depending on the value of the Restriction type parameter.
The restriction is adiabatic. It does not exchange heat with the environment.
The restriction consists of a contraction followed by a sudden expansion in flow area. The fluid accelerates during the contraction, causing the pressure to drop. In the expansion zone, if the Pressure recovery parameter is set to off, the momentum of the accelerated fluid is lost. If the Pressure recovery parameter is set to on, the sudden expansion recovers some of the momentum and allows the pressure to rise slightly after the restriction.
Local Restriction Schematic
The mass balance in the restriction is
0={\stackrel{˙}{m}}_{\text{A}}+{\stackrel{˙}{m}}_{\text{B}},
{\stackrel{˙}{m}}_{\text{A}}
is the mass flow rate into the restriction through port A.
{\stackrel{˙}{m}}_{\text{B}}
is the mass flow rate into the restriction through port B.
The pressure difference between ports A and B follows from the momentum balance in the restriction:
\begin{array}{l}\Delta p=\frac{1}{2\rho }\left(1-\frac{{S}_{\text{R}}^{2}}{{S}^{2}}\right){v}_{R}\sqrt{{v}_{R}^{2}+{v}_{Rc}^{2}}\\ {v}_{R}=\frac{{\stackrel{˙}{m}}_{A}}{{C}_{d}\rho {S}_{R}}\\ {v}_{Rc}=\frac{{\mathrm{Re}}_{c}\mu }{{C}_{d}\rho }\sqrt{\frac{\pi }{4{S}_{R}}}\end{array}
Δp is the pressure differential.
ρ is the liquid density.
μ is the liquid dynamic viscosity.
S is the cross-sectional area at ports A and B.
SR is the cross-sectional area at the restriction.
vR is fluid velocity at the restriction.
vRc is the critical fluid velocity.
Rec is the critical Reynolds number.
If pressure recovery is off, then
{p}_{\text{A}}-{p}_{\text{B}}=\Delta p,
If pressure recovery is on, then
{p}_{\text{A}}-{p}_{\text{B}}=\Delta p\frac{\sqrt{1-{\left(\frac{{S}_{R}}{S}\right)}^{2}\left(1-{C}_{\text{d}}^{2}\right)}-{C}_{d}\frac{{S}_{R}}{S}}{\sqrt{1-{\left(\frac{{S}_{R}}{S}\right)}^{2}\left(1-{C}_{\text{d}}^{2}\right)}+{C}_{d}\frac{{S}_{R}}{S}}.
The energy balance in the restriction is
{\varphi }_{\text{A}}+{\varphi }_{\text{B}}=0,
ϕA is the energy flow rate into the restriction through port A.
ϕB is the energy flow rate into the restriction through port B.
The restriction is adiabatic. It does not exchange heat with its surroundings.
The dynamic compressibility and thermal capacity of the liquid in the restriction are negligible.
AR — Restriction area control signal, m^2
Input physical signal that controls the flow restriction area. The signal saturates when its value is outside the minimum and maximum restriction area limits, specified by the block parameters.
This port is visible only if you set the Restriction type parameter to Variable.
Thermal liquid conserving port associated with the inlet or outlet of the local restriction. This block has no intrinsic directionality.
Restriction type — Specify whether restriction area can change during simulation
Select whether the restriction area can change during simulation:
Variable — The input physical signal at port AR specifies the restriction area, which can vary during simulation. The Minimum restriction area and Maximum restriction area parameters specify the lower and upper bounds for the restriction area.
Fixed — The restriction area, specified by the Restriction area block parameter value, remains constant during simulation. Port AR is hidden.
Minimum restriction area — Lower bound for the restriction cross-sectional area
1e-10 m^2 (default)
The lower bound for the restriction cross-sectional area. You can use this parameter to represent the leakage area. The input signal AR saturates at this value to prevent the restriction area from decreasing any further.
Enabled when the Restriction type parameter is set to Variable.
Maximum restriction area — Upper bound for the restriction cross-sectional area
The upper bound for the restriction cross-sectional area. The input signal AR saturates at this value to prevent the restriction area from increasing any further.
Restriction area — Area normal to flow path at the restriction
Area normal to flow path at the restriction.
Enabled when the Restriction type parameter is set to Fixed.
Cross-sectional area at ports A and B — Area normal to flow path at the ports
Area normal to flow path at ports A and B. This area is assumed to be the same for the two ports.
Discharge coefficient — Ratio of actual mass flow rate to theoretical mass flow rate through restriction
The discharge coefficient is a semi-empirical parameter commonly used to characterize the flow capacity of an orifice. This parameter is defined as the ratio of the actual mass flow rate through the orifice to the ideal mass flow rate.
Pressure recovery — Specify whether to account for pressure recovery
Specify whether to account for pressure recovery at the local restriction outlet.
Critical Reynolds number — Reynolds number for transition between laminar and turbulent regimes
The Reynolds number for the transition between laminar and turbulent regimes. The default value corresponds to a sharp-edged orifice.
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True anomaly - WikiMili, The Best Wikipedia Reader
Parameter of Keplerian orbits
The true anomaly of point P is the angle f. The center of the ellipse is point C, and the focus is point F.
From state vectors
The true anomaly is usually denoted by the Greek letters ν or θ, or the Latin letter f, and is usually restricted to the range 0–360° (0–2πc).
As shown in the image, the true anomaly f is one of three angular parameters (anomalies) that defines a position along an orbit, the other two being the eccentric anomaly and the mean anomaly.
For elliptic orbits, the true anomalyν can be calculated from orbital state vectors as:
{\displaystyle \nu =\arccos {{\mathbf {e} \cdot \mathbf {r} } \over {\mathbf {\left|e\right|} \mathbf {\left|r\right|} }}}
(if r ⋅ v < 0 then replace ν by 2π − ν)
v is the orbital velocity vector of the orbiting body,
e is the eccentricity vector,
r is the orbital position vector (segment FP in the figure) of the orbiting body.
For circular orbits the true anomaly is undefined, because circular orbits do not have a uniquely determined periapsis. Instead the argument of latitude u is used:
{\displaystyle u=\arccos {{\mathbf {n} \cdot \mathbf {r} } \over {\mathbf {\left|n\right|} \mathbf {\left|r\right|} }}}
(if rz < 0 then replace u by 2π − u)
n is a vector pointing towards the ascending node (i.e. the z-component of n is zero).
rz is the z-component of the orbital position vector r
Circular orbit with zero inclination
For circular orbits with zero inclination the argument of latitude is also undefined, because there is no uniquely determined line of nodes. One uses the true longitude instead:
{\displaystyle l=\arccos {r_{x} \over {\mathbf {\left|r\right|} }}}
(if vx > 0 then replace l by 2π − l)
rx is the x-component of the orbital position vector r
vx is the x-component of the orbital velocity vector v.
The relation between the true anomaly ν and the eccentric anomaly E is:
{\displaystyle \cos {\nu }={{\cos {E}-e} \over {1-e\cos {E}}}}
or using the sine [1] and tangent:
{\displaystyle {\begin{aligned}\sin {\nu }&={{{\sqrt {1-e^{2}\,}}\sin {E}} \over {1-e\cos {E}}}\\[4pt]\tan {\nu }={{\sin {\nu }} \over {\cos {\nu }}}&={{{\sqrt {1-e^{2}\,}}\sin {E}} \over {\cos {E}-e}}\end{aligned}}}
{\displaystyle \tan {\nu \over 2}={\sqrt {{1+e\,} \over {1-e\,}}}\tan {E \over 2}}
{\displaystyle \nu =2\,\operatorname {arctan} \left(\,{\sqrt {{1+e\,} \over {1-e\,}}}\tan {E \over 2}\,\right)}
Alternatively, a form of this equation was derived by [2] that avoids numerical issues when the arguments are near
{\displaystyle \pm \pi }
, as the two tangents become infinite. Additionally, since
{\displaystyle {\frac {E}{2}}}
{\displaystyle {\frac {\nu }{2}}}
are always in the same quadrant, there will not be any sign problems.
{\displaystyle \tan {{\frac {1}{2}}(\nu -E)}={\frac {\beta \sin {E}}{1-\beta \cos {E}}}}
{\displaystyle \beta ={\frac {e}{1+{\sqrt {1-e^{2}}}}}}
{\displaystyle \nu =E+2\operatorname {arctan} \left(\,{\frac {\beta \sin {E}}{1-\beta \cos {E}}}\,\right)}
The true anomaly can be calculated directly from the mean anomaly via a Fourier expansion: [3]
{\displaystyle \nu =M+\left(2e-{\frac {1}{4}}e^{3}\right)\sin {M}+{\frac {5}{4}}e^{2}\sin {2M}+{\frac {13}{12}}e^{3}\sin {3M}+\operatorname {O} \left(e^{4}\right)}
{\displaystyle \operatorname {O} \left(e^{4}\right)}
means that the omitted terms are all of order e4 or higher. Note that for reasons of accuracy this approximation is usually limited to orbits where the eccentricity (e) is small.
{\displaystyle \nu -M}
is known as the equation of the center.
The radius (distance between the focus of attraction and the orbiting body) is related to the true anomaly by the formula
{\displaystyle r=a\,{1-e^{2} \over 1+e\cos \nu }\,\!}
where a is the orbit's semi-major axis.
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
In mathematics, the Legendre chi function is a special function whose Taylor series is also a Dirichlet series, given by
In relativistic physics, a velocity-addition formula is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different frames. Accompanying velocity addition is a kinematic effect known as Thomas precession, whereby successive non-collinear Lorentz boosts become equivalent to the composition of a rotation of the coordinate system and a boost.
The main trigonometric identities between trigonometric functions are proved, using mainly the geometry of the right triangle. For greater and negative angles, see Trigonometric functions.
Solution of triangles is the main trigonometric problem of finding the characteristics of a triangle, when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.
↑ Fundamentals of Astrodynamics and Applications by David A. Vallado
↑ Broucke, R., & Cefola, P. 1973, Celestial Mechanics, 7, 388
↑ Roy, A.E. (2005). Orbital Motion (4 ed.). Bristol, UK; Philadelphia, PA: Institute of Physics (IoP). p. 84. ISBN 0750310154.
Murray, C. D. & Dermott, S. F., 1999, Solar System Dynamics, Cambridge University Press, Cambridge. ISBN 0-521-57597-4
Plummer, H. C., 1960, An Introductory Treatise on Dynamical Astronomy, Dover Publications, New York. OCLC 1311887 (Reprint of the 1918 Cambridge University Press edition.)
Federal Aviation Administration - Describing Orbits
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A candy store’s specialty is taffy. Customers can fill a bag with taffy and the price is based on how much the candy weighs. The store charges
\$2
10
ounces of taffy. 4-143 HW eTool (Desmos)
Amount of taffy (ounces)
2
5
10
12
15
20
2
4
Try writing a proportion equation.
\frac{10\text{ ounces}}{2\text{ dollars}}=\frac{2\text{ ounces}}{x\text{ dollars}}
x
Graph the values in the table. Let
x
represent the number of ounces and y represent the price in dollars.
Does the price per ounce stay the same no matter how many ounces of taffy are bought?
Notice that the graph makes a straight line and intersects
(0,0)
Slope can be defined as:
\frac{\text{rise}}{\text{run}}
\frac{\text{change in }y}{\text{change in }x}
0.2
, tells you that the price of taffy per ounce is
\$0.20
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Neural representation in F5: cross-decoding from observation to execution | BMC Neuroscience | Full Text
Neural representation in F5: cross-decoding from observation to execution
Murat Kirtay1,
Vassilis Papadourakis2,3,
Vassilis Raos2,3 &
Erhan Oztop1
Mirror neurons fire during both action execution and observation of a similar action performed by another individual [1]. However, this definition does not account for the existence of representational equivalence between execution and observation. To investigate this issue we recorded 68 neurons from area F5 of a macaque monkey trained either to execute reaching-to-grasp actions towards objects or to observe the experimenter performing the same actions [2], and adopted a decoding framework to find whether neurons effective in decoding the object/grip type in (1) execution and (2) observation conditions do exist, and most critically, to (3) assess whether transfer between execution and observation decoders (i.e. cross-decoding) can be employed. By 'transfer' we mean the application of the decoder parameters estimated using the neural discharge in observation to the neural firing recorded in execution, and vice versa. The success rate of such a decoder indicates the equivalence of representations in the two conditions.
Our analysis indicates that, at the level of single neurons, object/grip-specific decoders can be constructed, i.e. the type of the object/grip employed in either execution or observation can be decoded (success rate: 80%-100%, chance level: 25%). However, only in 10% of the cases (corresponding to the congruent type mirror neurons [1]) the decoder based on the execution discharge was effective when transferred to the observation discharge. The same was true for the reverse transfer. To extend this analysis at the population level we examined all pair performance of a 10-neuron set, consisted of 4 neurons having the best decoding performance and 6 neurons randomly selected. Out of the 45 possible pairs, 7 displayed high success rates (80% on average) in cross-decoding. Remarkably, high performing pairs were constituted only when one of the neurons displaying reliable decoding performance was paired with a randomly selected -poor solo decoder- neuron, which acted as a "helper". These results strongly point to a population based representation where good and poor decoders may cooperate to form a robust recognition system.
Neuronal discharges during each condition were trimmed and represented as 14-bin histogram vectors. In the two neuron analysis, each neuron was reduced to a 7-bin histogram, and their concatenation results in a 14-tuple vector, to ensure similar decoder complexity (constant number of adjustable parameters). Thus, for each condition, ten 14-tuple neural firings (one per trial) made up the rows of the input matrix X, and the corresponding object ids (1-4) made up the output vector Y. We assumed a linear relation between input and output as XW = Yand solved for the weights (the decoder parameters) using the pseudo-inverse solution W = X†Y. Then, given a 14-tuple vector representation, z, of a discharge, the predicted object id is given by
{y}_{pred}=\underset{i=0..5}{\mathrm{arg}\mathrm{min}}\left\{\left[0,1,2,3,4,5\right]-{z}^{T}W\right\}
, where 0 and 5 indicates a definite wrong prediction. For execution-only and observation-only experiments, leave-one-out cross validation was applied to obtain the success rates in decoding. For cross-decoding analysis, the weight vector W obtained in one condition was used to predict the object type in the other condition by using the data from that condition.
Fadiga L, Fogassi L, Rizzolatti G: Action recognition in the premotor cortex. Brain. 1996, 119: 593-609.
Papadourakis V, Raos V: Cue-dependent action-observation elicited responses in the ventral premotor cortex (area F5) of the macaque monkey. Soc Neurosci Abstr. 2013, Program No. 263.08
This work was supported by the grant "OBSERVENEMO" within the framework of the bilateral S&T Cooperation Program between the Republic of Turkey and the Hellenic Republic. Grant no 113S391 funded by TUBITAK and grant ΓΓΕΤ 14ΤUR OBSERVENEMO co- Financed by the European Union and the Greek State, MCERA/GSRT.
Computer Science, Ozyegin University, Istanbul, Turkey
Murat Kirtay & Erhan Oztop
Foundation for Research & Technology- Hellas (FORTH), Heraklion, Greece
Vassilis Papadourakis & Vassilis Raos
Vassilis Papadourakis
Correspondence to Erhan Oztop.
Kirtay, M., Papadourakis, V., Raos, V. et al. Neural representation in F5: cross-decoding from observation to execution. BMC Neurosci 16, P190 (2015). https://doi.org/10.1186/1471-2202-16-S1-P190
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Graph implementation and representation | Brilliant Math & Science Wiki
Thaddeus Abiy, Christopher Williams, Geoff Pilling, and
A graph is a binary relation. It provides a powerful visualization as a set of points (called nodes) connected by lines (called edges) or arrows (called arcs). In this regard, the graph is a generalization of the tree data model. Like trees, graphs come in several forms: directed, undirected and labeled.
Minimal Description of the Graph Data Type
A undirected graph means that the relationship along an edge between two nodes is bidirectional, i.e. it can go either way. A directed graph means that the relationship only goes one way. Undirected graphs are typically represented by a line with no arrows, which imply a bidirectional relationship between node A and node B. Directed graphs use an arrow to show the relationship from A to B.
As an abstract type, we can view a graph as a collection of elements that are stored at the graph's edges and nodes (nodes are also sometimes referred to as vertices). The essential methods we need for dealing with graphs will be the following:
nodes(): Returns all the nodes of the graph
edges(): Returns all the edges of a graph
insert_node(value, x): Inserts a node of value v which as a neighbor of x
remove_node(node): Removes the given node of the graph
As discussed before, there are two standard ways of representing a graph: the adjacency list and the adjacency matrix implementation. We shall consider these representations in this section.
A common way to implement a graph using an adjacency list is to use either a hashtable with an array as values or use a hashtable with linked lists as a value. Since Python combines the idea of arrays and linked lists, we can easily implement this representation using a dictionary with nodes as keys and a list as a value.
"""Node to list of neighbors hashtable (dict/dictionary in python)"""
"""Returns all the nodes of the graph"""
return self.nodes.keys()
"""Returns all the edges of a graph"""
for incident in self.nodes[node]:
edge = (node, incident)
def insert_node(self, value, x):
"""Insert a node with a value which is a neighbor of x"""
self.nodes[node] = [x]
self.nodes[x].append(node)
"""Removes a given node of the graph"""
if node in self.nodes[node]:
self.nodes[node].remove(node)
As we have discussed, the two most common ways of implementing graphs are using adjacency matrices and using adjacency lists. We tend to prefer adjacency matrices when the graphs are dense, that is, when the number of edges is near the maximum possible number, which is
n^2
for a graph of
nodes. However, if the graphs are sparse, that is, if most of the possible edges are not present, then the adjacency list representation can save space.
Show that the statement above 'if most of the possible edges are not present, then the adjacency list representation may save space' is correct.
To see why, note that an adjacency matrix for an
node graph has
n^2
bits, and therefore could be packed into
\frac{n^2}{32}
32
-bit words. The common adjacency list cell will consist of two words, one for the node and one for the pointer to the next cell. Thus, if the number of edges is
a
, we need about
2a
words for the lists, and
n
words for the array of headers. The adjacency list will use less space than the adjacency matrix if
n + 2a < \frac{n^2}{32}.\ _\square
The following table summarizes the preferred representation for various operations:
\begin{array}{l|l|l} \textbf{Operation} & \textbf{Dense Graphs} & \textbf{Sparse Graphs} \\ \hline \text{Lookup edge} & \text{Adjacency matrix} & \text{Either} \\ \text{Find succesor} & \text{Either} & \text{Adjacency lists} \\ \text{Find predecessors} & \text{Adjacency matrix} & \text{Either} \end{array}
A spanning tree for an undirected graph
G
is the nodes of
G
together with a subset of the edges of
G
connect the nodes, that is, there is a path between any two nodes using only edges in the spanning tree;
form an unrooted, unordered tree, that is, there are no (simple) cycles.
Find all the spanning trees of the graph below. image
By simple observation, we find 4 total spanning trees. image
Now suppose the edges of the graph have weights or lengths. The weight of a tree is just the sum of weights of its edges. Obviously, different spanning trees have different lengths. Here is the problem: "How can you find the minimum length spanning tree?"
Find the minimum spanning tree of the weighted graph below. image
Remember that the minimum spanning tree is the spanning tree that minimizes the sum of the edges. By observation we see that the following is the minimum spanning tree: image
There are several algorithms for finding minimal spanning trees. We shall consider a specific one called Kruskal's algorithm. The algorithm is executed as follows:
Set an empty set A={} and F = E, where E is the set of all edges.
If it does, remove e from F.
If it doesn't, move e from F to A.
If F={}, stop and output the minimal spanning tree (V,A).
Kruskal's algorithm is a good example of a greedy algorithm, in which we make a series of decisions, each doing what seems best at the time. The local decisions are which edge to add to the spanning tree formed. In each case, we pick the edge with the least label that does not violate the definition of a spanning tree by completing a cycle. Often the overall effect of a locally optimal solution is not globally optimal. However, Kruskal's algorithm is a case where this is not true.
O(m \log n )
nodes and
m
O(m)
O(m \log m )
O(m \log n)
O\big(m( \log n + \log m ) \big).
m \leq n^2
\frac{n(n-1)}{2}
\log m \leq 2\log n
m(\log n + \log m) \leq 3m \log n
. Since the constants can be neglected, we conclude that indeed takes
O(m \log n )
_\square
Cite as: Graph implementation and representation. Brilliant.org. Retrieved from https://brilliant.org/wiki/graphs-intermediate/
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Professor H. M. Srivastava: man and mathematician | Journal of Inequalities and Applications | Full Text
İsmail Naci Cangül1
Biography of Professor Hari M. Srivastava.
This article is being published in each of the four Special Issues of the SpringerOpen journals, Advances in Difference Equations, Boundary Value Problems, Fixed Point Theory and Applications and Journal of Inequalities and Applications, which are entitled ‘Proceedings of the International Congress in Honour of Professor Hari M. Srivastava’.
1 A brief biographical sketch
Professor Hari Mohan Srivastava was born on July 5, 1940 at Karon in Ballia district of the province of Uttar Pradesh in India. His father, Mr. Harihar Prasad (1900-1985), was a lawyer practicing in the civil and district courts in Ballia. His mother, Mrs. Bela Devi (1910-1989), was remarkably well versed in the Vedic and Hindu religious scriptures, which greatly influenced his childhood and later life spiritually as well as culturally. His father, on the other hand, significantly strengthened his pre-university education, especially in the subjects of English and Mathematics.
Having had hardly any formal education at the primary school level, in 1946 Professor Srivastava was admitted into grade 3 of the government higher secondary school in Ballia at the age of six after his successful performance in the mandatory written and oral entrance examinations. It is at this school where he was awarded a double promotion from grade 4 to grade 6, without having to go through grade 5. This did indeed accelerate his completion of high school (grade 10) in 1953. During the next two years (1953-1955), he studied at B. N. V. College at Rath in Hamirpur district of the province of Uttar Pradesh, where he completed his I. Sc. in 1955, breaking all the existing academic and scholarly records of that College as well as in other colleges in the region.
Professor Srivastava received his university education at the University of Allahabad where he completed his B. Sc. in 1957 and M. Sc. in 1959. Besides being a throughout high first class (right from high school (1953) to M. Sc. (1959)) and meritorious product of the University of Allahabad, and having won a number of merit prizes and scholarships, he was awarded the Allahabad Jubilee Medal in the year 1959. During the period of his four-year stay at the University of Allahabad (1955-1959), Professor Srivastava also published two first-prize-winning short stories in English, which were subsequently translated and published in other Indian languages (especially in Hindi).
Professor Srivastava began his university-level teaching career in 1959, at the age of 19. He taught at D. M. Government College in Imphal (now Manipur University) during the academic year 1959-1960 and at the University of Roorkee (now the Indian Institute of Technology at Roorkee) during the academic years 1960-1963. He then moved to Jodhpur University (now Jai Narain Vyas University) where he earned his Ph. D. degree in 1965 while he was a full-time member of the teaching faculty at Jodhpur University (since 1963).
Currently, Professor Srivastava holds the position of a professor emeritus in the Department of Mathematics and Statistics at the University of Victoria in Canada. He joined the faculty there in 1969 (first as an associate professor (1969-1974) and then as a full professor (1974-2006)). Professor Srivastava has held numerous visiting positions including (for example) those at West Virginia University in the USA (1967-1969), Université Laval in Canada (1975), and the University of Glasgow in the UK (1975-1976), and, indeed, also at many other universities and research institutes in different parts of the world.
Professor Srivastava’s academic as well as personal life has been greatly enriched by the dedicated and whole-hearted support of his wife, Prof. Dr. Rekha Srivastava, who is also a mathematician and a colleague in the same Department of Mathematics and Statistics at the University of Victoria, and by his two children, Sapna Srivastava (who is currently working as a journalist in the Seattle area in the USA after gaining her Master’s degree in Journalism from Fordham University in New York) and Dr. Gautam Mohan Srivastava (who is currently teaching in the Department of Computer Science in the Faculty of Engineering at the University of Victoria). Many of Professor Srivastava’s teachers (especially those at the University of Allahabad), too, deserve to be credited for his choice of teaching career and for his academic and scholarly accomplishments in his chosen profession.
When not fully immersed into his research and writing, Professor Srivastava prefers to pursue one of his main hobbies: watching movies and serials (mostly in Hindi) on large-screen television at home with his family. His continuing interest in sports is exemplified by his active participation in hockey games until recently and by his regular attendance at baseball games - live (especially when his son, who is presently also a successful (and nationally well-recognized) baseball coach, used to play) or on television - with his wife (who incidentally got him deeply interested in baseball games, too). Besides, the spiritual and religious inclinations of Professor Srivastava and his wife, which were implanted in them by the cultural and spiritual environment of their respective families, grew much stronger in their own family life. He and his wife, together with their children, have contributed significantly to the community and other related services of the society.
2 Honors, awards and other accomplishments
Professor Srivastava has published 21 books, monographs and edited volumes, 30 book (and encyclopedia) chapters, 43 papers in international conference proceedings, and over 1,000 scientific research journal articles on various topics of mathematical analysis and applicable mathematics. In addition, he has written forewords to several books by other authors and to several special issues of scientific journals. He has also edited (and contributed to) many volumes dedicated to the memories of famous mathematical scientists. Citations of his research contributions can be found in many books and monographs, Ph. D. and D. Sc. theses, and scientific journal articles, much too numerous to be recorded here. Currently, he is actively associated editorially (that is, as an editor, honorary editor, senior editor, associate editor, or editorial board member) with over 200 international scientific research journals. His biographical sketches (many of which are illustrated with his photograph) have appeared in various issues of more than 50 international biographies, directories, and Who’s Who’s.
Professor Srivastava’s over 50-year career as a university-level teacher and as a remarkably prolific researcher in many different areas of the mathematical, physical, and statistical sciences is highlighted (among other things) by the fact that he has collaborated and published joint papers with as many as 385 mathematicians, physicists, statisticians, chemists, astrophysicists, geochemists, and information and business management scientists who are scattered throughout the world, thereby qualifying for his Erdös number 2, implying that at least one of Professor Srivastava’s co-authors is a co-author of the famous Hungarian mathematician, Paul Erdös (1913-1996). Professor Srivastava’s collaboration distances with other famous scientists include his Einstein number 3, Pólya number 3, von Neumann number 3, Wiles number 3, and so on.
In the leading newspaper, The Globe and Mail (Toronto, March 27, 2012, Page B7 et seq.), Professor Srivastava was listed in the second place among Canada’s top researchers in the discipline of Mathematics and Statistics in terms of productivity and impact based upon a measure of citations of their published works.
Some of the most recent prizes and distinctions awarded to Professor Srivastava include (for example) the following items:
NSERC 25-Year Award: University of Victoria, Canada (2004)
The Nishiwaki Prize: Japan (2004)
Doctor of Science (Honoris Causa): Chung Yuan Christian University, Chung-Li, Taiwan, Republic of China (2006)
Doctor of Science (Honoris Causa): ‘1 Decembrie 1918’ University of Alba Iulia, Romania (2007)
Many mathematical entities and objects are attributed to (and named after) him. These entities and objects include (among other items) Srivastava’s polynomials and functions, Carlitz-Srivastava polynomials, Srivastava-Buschman polynomials, Srivastava-Singhal polynomials, Chan-Chyan-Srivastava polynomials, Erkuş-Srivastava polynomials, Srivastava-Daoust multivariable hypergeometric function, Srivastava-Panda multivariable H-function, Singhal-Srivastava generating function, Srivastava-Agarwal basic (or q-) generating function, and Wu-Srivastava inequality in the field of higher transcendental functions; Srivastava-Owa, Choi-Saigo-Srivastava, Jung-Kim-Srivastava, Liu-Srivastava, Cho-Kwon-Srivastava, Dziok-Srivastava, Srivastava-Attiya and Srivastava-Wright operators in the field of geometric function theory in complex analysis; Srivastava-Gupta operator in the field of approximation theory; the Srivastava, Adamchik-Srivastava and Choi-Srivastava methods in the field of analytic number theory; and so on.
Professor Srivastava has supervised (and is currently supervising) a number of post-graduate students working toward their Master’s, Ph. D. and/or D. Sc. degrees in different parts of the world. Besides, many post-doctoral fellows and research associates have worked with him at West Virginia University in the USA and at the University of Victoria in Canada.
Some of the significant and remarkable contributions by Professor Srivastava are listed below under each of the main topics of his current research interests.
Real and complex analysis: A unified theory of numerous potentially useful function classes, and of various integral and convolution operators using hypergeometric functions, especially in geometric function theory in complex analysis, and several classes of analytic and geometric inequalities in the field of real analysis.
Fractional calculus and its applications: Generalizations of such classical fractional-calculus operators as the Riemann-Liouville and Weyl operators together with their fruitful applications to numerous families of differential, integral, and integro-differential equations, especially some general classes of fractional kinetic equations, and also to some Volterra-type integro-differential equations which emerge from the unsaturated behavior of the free electron laser.
Integral equations and transforms: Explicit solutions of several general families of dual series and integral equations occurring in potential theory; unified theory of many known generalizations of the classical Laplace transform (such as the Meijer and Varma transforms) and of other multiple integral transforms by means of the Whittaker
{W}_{\kappa ,\mu }
-function and the (Srivastava-Panda) multivariable H-function in their kernels.
Higher transcendental functions and their applications: Discovery, introduction, and systematic (and unified) investigation of a set of 205 triple Gaussian hypergeometric series, especially the triple hypergeometric functions
{H}_{A}
{H}_{B}
{H}_{C}
, added to the 14-member set conjectured and defined in 1893 by Giuseppe Lauricella (1867-1913). Unified theory and applications of the multivariable extensions of the celebrated higher transcendental (Ψ- and H-) functions of Charles Fox (1897-1977) and Edward Maitland Wright (1906-2005), and also of the Mittag-Leffler E-functions named after Gustav Mittag-Leffler (1846-1927). Mention should be made also of his applications of some of these higher transcendental functions in quantum and fluid mechanics, astrophysics, probability distribution theory, queuing theory and other related stochastic processes, and so on.
q-Series and q-polynomials: Basic theory of general q-polynomial expansions for functions of several complex variables, extensions of several celebrated q-identities of Srinivasa Ramanujan (1887-1920), and systematic introduction and investigation of multivariable basic (or q-) hypergeometric series.
Analytic number theory: Presentation of several computationally-friendly and rapidly-converging series representations for Riemann’s zeta function, Dirichlet’s L-series, introduction and application of some novel techniques for closed-form evaluations of series involving a wide variety of sequences and functions of analytic number theory, and so on. His applications of (especially) the Hurwitz-Lerch zeta function in geometric function theory in complex analysis and in probability distribution theory and related topics of statistical sciences deserve to be recorded here.
Professor Srivastava’s publications have been reviewed by (among others) Mathematical Reviews (USA), Referativnyi Zhurnal Matematika (Russia), Zentralblatt für Mathematik (Germany), and Applied Mechanics Reviews (USA) under various 2010 Mathematical Subject Classifications (MathSciNet) including (for example) the following general classifications:
General Global Analysis, Analysis on Manifolds
Dean of the Faculty of Arts and Science, Uludağ University, Görükle Campus, Bursa, 16059, Turkey
Correspondence to İsmail Naci Cangül.
Cangül, İ.N. Professor H. M. Srivastava: man and mathematician. J Inequal Appl 2013, 413 (2013). https://doi.org/10.1186/1029-242X-2013-413
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Geometry, kinematics, and mechanism of growth unconformities in the Biertuokuoyi piggyback basin: Implication for episodic growth of the Pamir Frontal Thrust
Detrital chromian spinels in the Cretaceous Sindong Group, SE Korea: Implications for tectonic emplacement of hydrated mantle peridotites
Age and provenance of the Middle Jurassic Norphlet Formation of south Texas: Stratigraphic relationship to the Louann Salt and regional significance
U-Pb age constraints on the protolith, cooling and exhumation of a Variscan middle crust migmatite complex from the Central Iberian Zone: insights into the Variscan metamorphic evolution and Ediacaran paleogeographic implications
δ
127 The Mazon Creek Lagerstätte: a diverse late Paleozoic ecosystem entombed within siderite concretions
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Ratio of shear stress to shear strain
In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:[1]
{\displaystyle G\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\tau _{xy}}{\gamma _{xy}}}={\frac {F/A}{\Delta x/l}}={\frac {Fl}{A\Delta x}}}
{\displaystyle \tau _{xy}=F/A\,}
= shear stress
{\displaystyle F}
is the force which acts
{\displaystyle A}
is the area on which the force acts
{\displaystyle \gamma _{xy}}
= shear strain. In engineering
{\displaystyle :=\Delta x/l=\tan \theta }
, elsewhere
{\displaystyle :=\theta }
{\displaystyle \Delta x}
is the transverse displacement
{\displaystyle l}
is the initial length of the area.
The derived SI unit of shear modulus is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in thousand pounds per square inch (ksi). Its dimensional form is M1L−1T−2, replacing force by mass times acceleration.
3 Shear modulus of metals
3.1 MTS model
3.2 SCG model
3.3 NP model
4 Shear relaxation modulus
Diamond[2] 478.0
Steel[3] 79.3
Iron[4] 52.5
Copper[5] 44.7
Titanium[3] 41.4
Glass[3] 26.2
Aluminium[3] 25.5
Polyethylene[3] 0.117
Rubber[6] 0.0006
Granite[7][8] 24
Shale[7][8] 1.6
Limestone[7][8] 24
Chalk[7][8] 3.2
Sandstone[7][8] 0.4
The shear modulus is one of several quantities for measuring the stiffness of materials. All of them arise in the generalized Hooke's law:
Young's modulus E describes the material's strain response to uniaxial stress in the direction of this stress (like pulling on the ends of a wire or putting a weight on top of a column, with the wire getting longer and the column losing height),
the Poisson's ratio ν describes the response in the directions orthogonal to this uniaxial stress (the wire getting thinner and the column thicker),
the bulk modulus K describes the material's response to (uniform) hydrostatic pressure (like the pressure at the bottom of the ocean or a deep swimming pool),
the shear modulus G describes the material's response to shear stress (like cutting it with dull scissors).
These moduli are not independent, and for isotropic materials they are connected via the equations[9]
{\displaystyle E=2G(1+\nu )=3K(1-2\nu )}
The shear modulus is concerned with the deformation of a solid when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force (such as friction). In the case of an object shaped like a rectangular prism, it will deform into a parallelepiped. Anisotropic materials such as wood, paper and also essentially all single crystals exhibit differing material response to stress or strain when tested in different directions. In this case, one may need to use the full tensor-expression of the elastic constants, rather than a single scalar value.
One possible definition of a fluid would be a material with zero shear modulus.
Shear waves[edit]
Influences of selected glass component additions on the shear modulus of a specific base glass.[10]
In homogeneous and isotropic solids, there are two kinds of waves, pressure waves and shear waves. The velocity of a shear wave,
{\displaystyle (v_{s})}
is controlled by the shear modulus,
{\displaystyle v_{s}={\sqrt {\frac {G}{\rho }}}}
{\displaystyle \rho }
is the solid's density.
Shear modulus of metals[edit]
Shear modulus of copper as a function of temperature. The experimental data[11][12] are shown with colored symbols.
The shear modulus of metals is usually observed to decrease with increasing temperature. At high pressures, the shear modulus also appears to increase with the applied pressure. Correlations between the melting temperature, vacancy formation energy, and the shear modulus have been observed in many metals.[13]
Several models exist that attempt to predict the shear modulus of metals (and possibly that of alloys). Shear modulus models that have been used in plastic flow computations include:
the MTS shear modulus model developed by[14] and used in conjunction with the Mechanical Threshold Stress (MTS) plastic flow stress model.[15][16]
the Steinberg-Cochran-Guinan (SCG) shear modulus model developed by[17] and used in conjunction with the Steinberg-Cochran-Guinan-Lund (SCGL) flow stress model.
the Nadal and LePoac (NP) shear modulus model[12] that uses Lindemann theory to determine the temperature dependence and the SCG model for pressure dependence of the shear modulus.
MTS model[edit]
The MTS shear modulus model has the form:
{\displaystyle \mu (T)=\mu _{0}-{\frac {D}{\exp(T_{0}/T)-1}}}
{\displaystyle \mu _{0}}
is the shear modulus at
{\displaystyle T=0K}
{\displaystyle D}
{\displaystyle T_{0}}
SCG model[edit]
The Steinberg-Cochran-Guinan (SCG) shear modulus model is pressure dependent and has the form
{\displaystyle \mu (p,T)=\mu _{0}+{\frac {\partial \mu }{\partial p}}{\frac {p}{\eta ^{\frac {1}{3}}}}+{\frac {\partial \mu }{\partial T}}(T-300);\quad \eta :={\frac {\rho }{\rho _{0}}}}
where, μ0 is the shear modulus at the reference state (T = 300 K, p = 0, η = 1), p is the pressure, and T is the temperature.
NP model[edit]
The Nadal-Le Poac (NP) shear modulus model is a modified version of the SCG model. The empirical temperature dependence of the shear modulus in the SCG model is replaced with an equation based on Lindemann melting theory. The NP shear modulus model has the form:
{\displaystyle \mu (p,T)={\frac {1}{{\mathcal {J}}\left({\hat {T}}\right)}}\left[\left(\mu _{0}+{\frac {\partial \mu }{\partial p}}{\frac {p}{\eta ^{\frac {1}{3}}}}\right)\left(1-{\hat {T}}\right)+{\frac {\rho }{Cm}}~T\right];\quad C:={\frac {\left(6\pi ^{2}\right)^{\frac {2}{3}}}{3}}f^{2}}
{\displaystyle {\mathcal {J}}({\hat {T}}):=1+\exp \left[-{\frac {1+1/\zeta }{1+\zeta /\left(1-{\hat {T}}\right)}}\right]\quad {\text{for}}\quad {\hat {T}}:={\frac {T}{T_{m}}}\in [0,6+\zeta ],}
and μ0 is the shear modulus at absolute zero and ambient pressure, ζ is a area, m is the atomic mass, and f is the Lindemann constant.
Shear relaxation modulus[edit]
The shear relaxation modulus
{\displaystyle G(t)}
is the time-dependent generalization of the shear modulus[18]
{\displaystyle G}
{\displaystyle G=\lim _{t\to \infty }G(t)}
^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "shear modulus, G". doi:10.1351/goldbook.S05635
^ McSkimin, H.J.; Andreatch, P. (1972). "Elastic Moduli of Diamond as a Function of Pressure and Temperature". J. Appl. Phys. 43 (7): 2944–2948. Bibcode:1972JAP....43.2944M. doi:10.1063/1.1661636.
^ a b c d e Crandall, Dahl, Lardner (1959). An Introduction to the Mechanics of Solids. Boston: McGraw-Hill. ISBN 0-07-013441-3. {{cite book}}: CS1 maint: multiple names: authors list (link)
^ Rayne, J.A. (1961). "Elastic constants of Iron from 4.2 to 300 ° K". Physical Review. 122 (6): 1714–1716. Bibcode:1961PhRv..122.1714R. doi:10.1103/PhysRev.122.1714.
^ Material properties
^ Spanos, Pete (2003). "Cure system effect on low temperature dynamic shear modulus of natural rubber". Rubber World.
^ a b c d e Hoek, Evert, and Jonathan D. Bray. Rock slope engineering. CRC Press, 1981.
^ a b c d e Pariseau, William G. Design analysis in rock mechanics. CRC Press, 2017.
^ [Landau LD, Lifshitz EM. Theory of Elasticity, vol. 7. Course of Theoretical Physics. (2nd Ed) Pergamon: Oxford 1970 p13]
^ Overton, W.; Gaffney, John (1955). "Temperature Variation of the Elastic Constants of Cubic Elements. I. Copper". Physical Review. 98 (4): 969. Bibcode:1955PhRv...98..969O. doi:10.1103/PhysRev.98.969.
^ a b Nadal, Marie-Hélène; Le Poac, Philippe (2003). "Continuous model for the shear modulus as a function of pressure and temperature up to the melting point: Analysis and ultrasonic validation". Journal of Applied Physics. 93 (5): 2472. Bibcode:2003JAP....93.2472N. doi:10.1063/1.1539913.
^ March, N. H., (1996), Electron Correlation in Molecules and Condensed Phases, Springer, ISBN 0-306-44844-0 p. 363
^ Varshni, Y. (1970). "Temperature Dependence of the Elastic Constants". Physical Review B. 2 (10): 3952–3958. Bibcode:1970PhRvB...2.3952V. doi:10.1103/PhysRevB.2.3952.
^ Chen, Shuh Rong; Gray, George T. (1996). "Constitutive behavior of tantalum and tantalum-tungsten alloys". Metallurgical and Materials Transactions A. 27 (10): 2994. Bibcode:1996MMTA...27.2994C. doi:10.1007/BF02663849. S2CID 136695336.
^ Goto, D. M.; Garrett, R. K.; Bingert, J. F.; Chen, S. R.; Gray, G. T. (2000). "The mechanical threshold stress constitutive-strength model description of HY-100 steel". Metallurgical and Materials Transactions A. 31 (8): 1985–1996. doi:10.1007/s11661-000-0226-8. S2CID 136118687.
^ Guinan, M; Steinberg, D (1974). "Pressure and temperature derivatives of the isotropic polycrystalline shear modulus for 65 elements". Journal of Physics and Chemistry of Solids. 35 (11): 1501. Bibcode:1974JPCS...35.1501G. doi:10.1016/S0022-3697(74)80278-7.
^ Rubinstein, Michael, 1956 December 20- (2003). Polymer physics. Colby, Ralph H. Oxford: Oxford University Press. p. 284. ISBN 019852059X. OCLC 50339757. {{cite book}}: CS1 maint: multiple names: authors list (link)
{\displaystyle K}
{\displaystyle E}
{\displaystyle \lambda }
{\displaystyle G,\mu }
{\displaystyle \nu }
{\displaystyle M}
{\displaystyle K=\,}
{\displaystyle E=\,}
{\displaystyle \lambda =\,}
{\displaystyle G=\,}
{\displaystyle \nu =\,}
{\displaystyle M=\,}
{\displaystyle (K,\,E)}
{\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}
{\displaystyle {\tfrac {3KE}{9K-E}}}
{\displaystyle {\tfrac {3K-E}{6K}}}
{\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}
{\displaystyle (K,\,\lambda )}
{\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}
{\displaystyle {\tfrac {3(K-\lambda )}{2}}}
{\displaystyle {\tfrac {\lambda }{3K-\lambda }}}
{\displaystyle 3K-2\lambda \,}
{\displaystyle (K,\,G)}
{\displaystyle {\tfrac {9KG}{3K+G}}}
{\displaystyle K-{\tfrac {2G}{3}}}
{\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}
{\displaystyle K+{\tfrac {4G}{3}}}
{\displaystyle (K,\,\nu )}
{\displaystyle 3K(1-2\nu )\,}
{\displaystyle {\tfrac {3K\nu }{1+\nu }}}
{\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}
{\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}
{\displaystyle (K,\,M)}
{\displaystyle {\tfrac {9K(M-K)}{3K+M}}}
{\displaystyle {\tfrac {3K-M}{2}}}
{\displaystyle {\tfrac {3(M-K)}{4}}}
{\displaystyle {\tfrac {3K-M}{3K+M}}}
{\displaystyle (E,\,\lambda )}
{\displaystyle {\tfrac {E+3\lambda +R}{6}}}
{\displaystyle {\tfrac {E-3\lambda +R}{4}}}
{\displaystyle {\tfrac {2\lambda }{E+\lambda +R}}}
{\displaystyle {\tfrac {E-\lambda +R}{2}}}
{\displaystyle R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}
{\displaystyle (E,\,G)}
{\displaystyle {\tfrac {EG}{3(3G-E)}}}
{\displaystyle {\tfrac {G(E-2G)}{3G-E}}}
{\displaystyle {\tfrac {E}{2G}}-1}
{\displaystyle {\tfrac {G(4G-E)}{3G-E}}}
{\displaystyle (E,\,\nu )}
{\displaystyle {\tfrac {E}{3(1-2\nu )}}}
{\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}
{\displaystyle {\tfrac {E}{2(1+\nu )}}}
{\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}
{\displaystyle (E,\,M)}
{\displaystyle {\tfrac {3M-E+S}{6}}}
{\displaystyle {\tfrac {M-E+S}{4}}}
{\displaystyle {\tfrac {3M+E-S}{8}}}
{\displaystyle {\tfrac {E-M+S}{4M}}}
{\displaystyle S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}}
{\displaystyle \nu \geq 0}
{\displaystyle \nu \leq 0}
{\displaystyle (\lambda ,\,G)}
{\displaystyle \lambda +{\tfrac {2G}{3}}}
{\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}
{\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}
{\displaystyle \lambda +2G\,}
{\displaystyle (\lambda ,\,\nu )}
{\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}
{\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}
{\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}
{\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}
{\displaystyle \nu =0\Leftrightarrow \lambda =0}
{\displaystyle (\lambda ,\,M)}
{\displaystyle {\tfrac {M+2\lambda }{3}}}
{\displaystyle {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}}
{\displaystyle {\tfrac {M-\lambda }{2}}}
{\displaystyle {\tfrac {\lambda }{M+\lambda }}}
{\displaystyle (G,\,\nu )}
{\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}
{\displaystyle 2G(1+\nu )\,}
{\displaystyle {\tfrac {2G\nu }{1-2\nu }}}
{\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}
{\displaystyle (G,\,M)}
{\displaystyle M-{\tfrac {4G}{3}}}
{\displaystyle {\tfrac {G(3M-4G)}{M-G}}}
{\displaystyle M-2G\,}
{\displaystyle {\tfrac {M-2G}{2M-2G}}}
{\displaystyle (\nu ,\,M)}
{\displaystyle {\tfrac {M(1+\nu )}{3(1-\nu )}}}
{\displaystyle {\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}}
{\displaystyle {\tfrac {M\nu }{1-\nu }}}
{\displaystyle {\tfrac {M(1-2\nu )}{2(1-\nu )}}}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Shear_modulus&oldid=1069850943"
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IsWeekend - Maple Help
Home : Support : Online Help : Programming : Date and Time : Calendar package : IsWeekend
determine whether a date falls on a weekend
IsWeekend( d )
IsWeekend( year, month, day )
The IsWeekend( d ) command determines whether the Date object d specifies a date that falls on a weekend. The command may also be used in the form IsWeekend( year, month, day ), with the year, month and day fields individually specified.
In the POSIX locale, a date falls on a weekend if the day of the week is either Saturday (6) or Sunday (1).
\mathrm{with}\left(\mathrm{Calendar}\right):
\mathrm{IsWeekend}\left(2017,11,3\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{IsWeekend}\left(2017,11,4\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
The Calendar[IsWeekend] command was introduced in Maple 2018.
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22.5: Odds Ratios - Statistics LibreTexts
https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_Statistical_Thinking_for_the_21st_Century_(Poldrack)%2F22%253A_Modeling_Categorical_Relationships%2F22.05%253A_Odds_Ratios
We can also represent the relative likelihood of different outcomes in the contingency table using the odds ratio that we introduced earlier, in order to better understand the size of the effect. First, we represent the odds of being stopped for each race:
odds_{searched|black} = \frac{N_{searched\cap black}}{N_{not\ searched\cap black}} = \frac{1219}{36244} = 0.034
odds_{searched|white} = \frac{N_{searched\cap white}}{N_{not\ searched\cap white}} = \frac{3108}{239241} = 0.013
odds\ ratio = \frac{odds_{searched|black}}{odds_{searched|white}} = 2.59
22.5: Odds Ratios is shared under a not declared license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.
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Category:Interface pinning - Vaspwiki
Category:Interface pinning
Interface pinning[1] is used to determine the melting point from a molecular-dynamics simulation of the interface between a liquid and a solid phase. The typical behavior of such a simulation is to freeze or melt, while the interface is pinned with a bias potential. This potential applies an energy penalty for deviations from the desired two-phase system. It is preferred simulating above the melting point because the bias potential prevents melting better than freezing.
The Steinhardt-Nelson[2] order parameter
{\displaystyle Q_{6}}
discriminates between the solid and the liquid phase. With the bias potential
{\displaystyle U_{\text{bias}}(\mathbf {R} )={\frac {\kappa }{2}}\left(Q_{6}(\mathbf {R} )-A\right)^{2}}
penalizes differences between the order parameter for the current configuration
{\displaystyle Q_{6}({\mathbf {R} })}
and the one for the desired interface
{\displaystyle A}
{\displaystyle \kappa }
is an adjustable parameter determining the strength of the pinning.
Under the action of the bias potential, the system equilibrates to the desired two-phase configuration. An important observable is the difference between the average order parameter
{\displaystyle \langle Q_{6}\rangle }
in equilibrium and the desired order parameter
{\displaystyle A}
. This difference relates to the the chemical potentials of the solid
{\displaystyle \mu _{\text{solid}}}
and the liquid
{\displaystyle \mu _{\text{liquid}}}
{\displaystyle N(\mu _{\text{solid}}-\mu _{\text{liquid}})=\kappa (Q_{6,{\text{solid}}}-Q_{6,{\text{liquid}}})(\langle Q_{6}\rangle -A)}
{\displaystyle N}
is the number of atoms in the simulation.
Computing the forces requires a differentiable
{\displaystyle Q_{6}(\mathbf {R} )}
. In the VASP implementation a smooth fading function
{\displaystyle w(r)}
is used to weight each pair of atoms at distance
{\displaystyle r}
for the calculation of the
{\displaystyle Q_{6}(\mathbf {R} ,w)}
order parameter. This fading function is given as
{\displaystyle w(r)=\left\{{\begin{array}{cl}1&{\textrm {for}}\,\,r\leq n\\{\frac {(f^{2}-r^{2})^{2}(f^{2}-3n^{2}+2r^{2})}{(f^{2}-n^{2})^{3}}}&{\textrm {for}}\,\,n<r<f\\0&{\textrm {for}}\,\,f\leq r\end{array}}\right.}
{\displaystyle n}
{\displaystyle f}
are the near- and far-fading distances, respectively. The radial distribution function
{\displaystyle g(r)}
of the crystal phase yields a good choice for the fading range. To prevent spurious stress,
{\displaystyle g(r)}
should be small where the derivative of
{\displaystyle w(r)}
is large. Set the near fading distance
{\displaystyle n}
to the distance where
{\displaystyle g(r)}
goes below 1 after the first peak. Set the far fading distance
{\displaystyle f}
{\displaystyle g(r)}
goes above 1 again before the second peak.
Interface pinning uses the
{\displaystyle Np_{z}T}
ensemble where the barostat only acts along the
{\displaystyle z}
direction. This ensemble uses a Langevin thermostat and a Parrinello-Rahman barostat with lattice constraints in the remaining two dimensions. The solid-liquid interface must be in the
{\displaystyle x}
{\displaystyle y}
plane perpendicular to the action of the barostat.
Set the following tags for the interface pinning method:
OFIELD_Q6_NEAR
Defines the near-fading distance
{\displaystyle n}
OFIELD_Q6_FAR
Defines the far-fading distance
{\displaystyle f}
OFIELD_KAPPA
Defines the coupling strength
{\displaystyle \kappa }
of the bias potential.
OFIELD_A
Defines the desired value of the order parameter
{\displaystyle A}
The following example INCAR file calculates the interface pinning in sodium[1]:
TEBEG = 400 # temperature in K
POTIM = 4 # timestep in fs
IBRION = 0 # run molecular dynamics
ISIF = 3 # use Parrinello-Rahman barostat for the lattice
MDALGO = 3 # use Langevin thermostat
LANGEVIN_GAMMA_L = 3.0 # friction coefficient for the lattice degree of freedoms (DoF)
LANGEVIN_GAMMA = 1.0 # friction coefficient for atomic DoFs for each species
PMASS = 100 # mass for lattice DoFs
LATTICE_CONSTRAINTS = F F T # fix x-y plane, release z lattice dynamics
OFIELD_Q6_NEAR = 3.22 # near fading distance for function w(r) in Angstrom
OFIELD_Q6_FAR = 4.384 # far fading distance for function w(r) in Angstrom
OFIELD_KAPPA = 500 # strength of bias potential in eV/(unit of Q)^2
OFIELD_A = 0.15 # desired value of the Q6 order parameter
↑ a b U. R. Pedersen, F. Hummel, G. Kresse, G. Kahl, and C. Dellago, Phys. Rev. B 88, 094101 (2013).
↑ P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Phys. Rev. B 28, 784 (1983).
Pages in category "Interface pinning"
Retrieved from "https://www.vasp.at/wiki/index.php?title=Category:Interface_pinning&oldid=16105"
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Some properties of meromorphically multivalent functions | Journal of Inequalities and Applications | Full Text
Some properties of meromorphically multivalent functions
Yi-Hui Xu1,
By using the method of differential subordinations, we derive certain properties of meromorphically multivalent functions.
2010 Mathematics Subject Classification: 30C45; 30C55.
Let Σ(p) denotes the class of meromorphically multivalent functions f(z) of the form
f\left(z\right)={z}^{-p}+\sum _{k=1}^{\infty }{a}_{k-p}{z}^{k-p}\phantom{\rule{1em}{0ex}}\left(p\in N=\left\{1,2,3,\dots \right\}\right),
which are analytic in the punctured unit disk
{U}^{*}=\left\{z:z\in C\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{0}}<\left|z\right|<1\right\}=U\\left\{0\right\}.
Let f(z) and g(z) be analytic in U. Then, we say that f(z) is subordinate to g(z) in U, written f(z) ≺ g(z), if there exists an analytic function w(z) in U, such that |w(z)| ≤ |z| and f(z) = g(w(z)) (z ∈ U). If g(z) is univalent in U, then the subordination f(z) ≺ g(z) is equivalent to f(0) = g(0) and f(U) ⊂ g(U).
Let p(z) = 1 + p1z + ... be analytic in U. Then for -1 ≤ B < A ≤ 1, it is clear that
p\left(z\right)\prec \frac{1+Az}{1+Bz}\phantom{\rule{1em}{0ex}}\left(z\in U\right)
\left|p\left(z\right)-\frac{1-AB}{1-{B}^{2}}\right|<\frac{A-B}{1-{B}^{2}}\phantom{\rule{1em}{0ex}}\left(-1<B<A\le 1;z\in U\right)
\mathsf{\text{Re}}p\left(z\right)>\frac{1-A}{2}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(B=-1;z\in U\right).
Recently, several authors (see, e.g., [1–7]) considered some interesting properties of meromorphically multivalent functions. In the present article, we aim at proving some subordination properties for the class Σ(p).
Lemma 1 (see [8]. Let h(z) be analytic and starlike univalent in U with h(0) = 0. If g(z) is analytic in U and zg'(z) ≺ h(z), then
g\left(z\right)\prec g\left(0\right)+\underset{0}{\overset{z}{\int }}\frac{h\left(t\right)}{t}dt.
Lemma 2 (see [9]. Let p(z) be analytic and nonconstant in U with p(0) = 1. If 0 < | z0 | < 1 and
\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}p\left({z}_{0}\right)=\underset{\left|z\right|\le \left|{z}_{0}\right|}{\text{min}}\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}p\left(z\right)
{z}_{0}{p}^{\prime }\left({z}_{0}\right)\le -\frac{{\left|1-p\left({z}_{0}\right)\right|}^{2}}{2\left(1-\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}p\left({z}_{0}\right)\right)}.
\alpha \in \left(0,\frac{1}{2}\right]
and β ∈ (0,1). If f(z) ∈ Σ(p) satisfies f(z) ≠ 0 (z ∈ U*) and
\left|\frac{{z}^{-p}}{f\left(z\right)}\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}+p\right)\right|<\delta \phantom{\rule{1em}{0ex}}\left(z\in U\right),
where δ is the minimum positive root of the equation
\alpha \text{sin}\left(\frac{\pi \beta }{2}\right){x}^{2}-x+\left(1-\alpha \right)\text{sin}\left(\frac{\pi \beta }{2}\right)=0,
\left|\text{arg}\left(\frac{f\left(z\right)}{{z}^{-p}}-\alpha \right)\right|<\frac{\pi }{2}\beta \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(z\in U\right).
The bound β is the best possible for each
\alpha \in \left(0,\frac{1}{2}\right]
g\left(x\right)=\alpha \text{sin}\left(\frac{\pi \beta }{2}\right){x}^{2}-x+\left(1-\alpha \right)\text{sin}\left(\frac{\pi \beta }{2}\right).
We can see that the Equation (2.2) has two positive roots. Since g(0) > 0 and g(1) < 0, we have
0<\frac{\alpha }{1-\alpha }\delta \le \delta <1.
\frac{f\left(z\right)}{{z}^{-p}}=\alpha +\left(1-\alpha \right)p\left(z\right).
Then from the assumption of the theorem, we see that p(z) is analytic in U with p(0) = 1 and α + (1 - α)p(z) ≠ 0 for all z ∈ U. Taking the logarithmic differentiations in both sides of (2.6), we get
\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}+p=\frac{\left(1-\alpha \right)z{p}^{\prime }\left(z\right)}{\alpha +\left(1-\alpha \right)p\left(z\right)}
\frac{{z}^{-p}}{f\left(z\right)}\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}+p\right)=\frac{\left(1-\alpha \right)z{p}^{\prime }\left(z\right)}{{\left(\alpha +\left(1-\alpha \right)p\left(z\right)\right)}^{2}}
for all z ∈ U. Thus the inequality (2.1) is equivalent to
\frac{\left(1-\alpha \right)z{p}^{\prime }\left(z\right)}{{\left(\alpha +\left(1-\alpha \right)p\left(z\right)\right)}^{2}}\prec \delta z.
By using Lemma 1, (2.9) leads to
\underset{0}{\overset{z}{\int }}\frac{\left(1-\alpha \right){p}^{\prime }\left(t\right)}{{\left(\alpha +\left(1-\alpha \right)p\left(t\right)\right)}^{2}}dt\prec \delta z
1-\frac{1}{\alpha +\left(1-a\right)p\left(z\right)}\prec \delta z.
In view of (2.5), (2.10) can be written as
p\left(z\right)\prec \frac{1+\frac{\alpha }{1-\alpha }\delta z}{1-\delta z}.
Now by taking
A=\frac{\alpha }{1-\alpha }\delta
and B = -δ in (1.2) and (1.3), we have
\begin{array}{ll}\hfill \left|\text{arg}\left(\frac{f\left(z\right)}{{z}^{-p}}-\alpha \right)\right|& =\left|\text{arg}p\left(z\right)\right|\phantom{\rule{2em}{0ex}}\\ <\text{arcsin}\left(\frac{\delta }{1-\alpha +\alpha {\delta }^{2}}\right)\phantom{\rule{2em}{0ex}}\\ =\frac{\pi }{2}\beta \phantom{\rule{2em}{0ex}}\end{array}
for all z ∈ U because of g(δ) = 0. This proves (2.3).
Next, we consider the function f(z) defined by
f\left(z\right)=\frac{{z}^{-p}}{1-{\delta }_{z}}\phantom{\rule{2.77695pt}{0ex}}\left(z\in {U}^{*}\right).
\left|\frac{{z}^{-p}}{f\left(z\right)}\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}+p\right)\right|=\left|\delta z\right|<\delta \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(z\in U\right).
\frac{f\left(z\right)}{{z}^{-p}}-\alpha =\left(1-\alpha \right)\frac{1+\frac{\alpha }{1-\alpha }\delta z}{1-\delta z},
\underset{z\in U}{\text{sup}}\left|\text{arg}\left(\frac{f\left(z\right)}{{z}^{-p}}-\alpha \right)\right|=\text{arcsin}\left(\frac{\delta }{1-\alpha +\alpha {\delta }^{2}}\right)=\frac{\pi }{2}\beta .
Hence, we conclude that the bound β is the best possible for each
\alpha \in \left(0,\frac{1}{2}\right]
Theorem 2. If f(z) ∈ Σ(p) satisfies f(z) ≠ 0 (z ∈ U*) and
\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}\left\{\frac{{z}^{-p}}{f\left(z\right)}\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}+p\right)\right\}<\gamma \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(z\in U\right),
0<\gamma <\frac{1}{2\phantom{\rule{2.77695pt}{0ex}}\text{log}\phantom{\rule{2.77695pt}{0ex}}2},
\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}\frac{{z}^{-p}}{f\left(z\right)}>1-2\gamma \phantom{\rule{2.77695pt}{0ex}}\text{log}\phantom{\rule{2.77695pt}{0ex}}2\phantom{\rule{1em}{0ex}}\left(z\in U\right).
The bound in (2.14) is sharp.
p\left(z\right)=\frac{f\left(z\right)}{{z}^{-p}}.
Then p(z) is analytic in U with p(0) = 1 and p(z) ≠ 0 for z ∈ U. In view of (2.15) and (2.12), we have
1-\frac{z{p}^{\prime }\left(z\right)}{\gamma {p}^{2}\left(z\right)}\prec \frac{1+z}{1-z},
z{\left(\frac{1}{p\left(z\right)}\right)}^{\prime }\prec \frac{2\gamma z}{1-z}.
Now by using Lemma 1, we obtain
\frac{1}{p\left(z\right)}\prec 1-2\gamma \phantom{\rule{2.77695pt}{0ex}}\text{log}\left(1-z\right).
Since the function 1 - 2γ log(1 - z) is convex univalent in U and
\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}\left(1-2\gamma \phantom{\rule{2.77695pt}{0ex}}\text{log}\left(1-z\right)\right)>1-2\gamma \phantom{\rule{2.77695pt}{0ex}}\text{log}2\phantom{\rule{1em}{0ex}}\left(z\in U\right),
from (2.16), we get the inequality (2.14).
To show that the bound in (2.14) cannot be increased, we consider
f\left(z\right)=\frac{{z}^{-p}}{1-2\gamma \phantom{\rule{2.77695pt}{0ex}}\text{log}\left(1-z\right)}\phantom{\rule{1em}{0ex}}\left(z\in {U}^{*}\right).
It is easy to verify that the function f(z) satisfies the inequality (2.12). On the other hand, we have
\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}\frac{{z}^{-p}}{f\left(z\right)}\to 1-2\gamma \text{log}2
as z → -1. Now the proof of the theorem is complete.
Theorem 3. Let f(z) ∈ Σ(p) with f(z) ≠ 0 (z ∈ U*). If
\left|\mathsf{\text{Im}}\phantom{\rule{1em}{0ex}}\left\{\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\left(\frac{f\left(z\right)}{{z}^{-p}}-\lambda \right)\right\}\right|<\sqrt{\lambda \left(\lambda +2p\right)}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(z\in U\right)
for some λ(λ > 0), then
\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}\frac{f\left(z\right)}{{z}^{-p}}>0\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(z\in U\right).
Proof. Let us define the analytic function p(z) in U by
\frac{f\left(z\right)}{{z}^{-p}}=p\left(z\right).
Then p(0) = 1, p(z) ≠ 0 (z ∈ U) and
\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\left(\frac{f\left(z\right)}{{z}^{-p}}-\lambda \right)=\left(p\left(z\right)-\lambda \right)\left(\frac{z{p}^{\prime }\left(z\right)}{p\left(z\right)}-p\right)\phantom{\rule{1em}{0ex}}\left(z\in U\right).
Suppose that there exists a point z0(0 < | z0 | < 1) such that
\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}p\left(z\right)>0\phantom{\rule{2.77695pt}{0ex}}\left(\left|z\right|<\left|{z}_{0}\right|\right)\phantom{\rule{1em}{0ex}}\mathsf{\text{and}}\phantom{\rule{1em}{0ex}}p\left({z}_{0}\right)=i\beta ,
where β is real and β ≠ 0. Then, applying Lemma 2, we get
{z}_{0}{p}^{\prime }\left({z}_{0}\right)\le -\frac{1+{\beta }^{2}}{2}.
Thus it follows from (2.19), (2.20), and (2.21) that
{I}_{0}=\mathsf{\text{Im}}\phantom{\rule{1em}{0ex}}\left\{\frac{{z}_{0}{f}^{\prime }\left({z}_{0}\right)}{f\left({z}_{0}\right)}\left(\frac{f\left({z}_{0}\right)}{{z}_{0}^{-p}}-\lambda \right)\right\}=-p\beta +\frac{\lambda }{\beta }{z}_{0}{p}^{\prime }\left({z}_{0}\right).
In view of λ > 0, from (2.21) and (2.22) we obtain
{I}_{0}\ge -\frac{\lambda +\left(\lambda +2p\right){\beta }^{2}}{2\beta }\ge \sqrt{\lambda \left(\lambda +2p\right)}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(\beta <0\right)
{I}_{0}\le -\frac{\lambda +\left(\lambda +2p\right){\beta }^{2}}{2\beta }\le -\sqrt{\lambda \left(\lambda +2p\right)}\phantom{\rule{2.77695pt}{0ex}}\left(\beta >0\right).
But both (2.23) and (2.24) contradict the assumption (2.17). Therefore, we have Rep(z) > 0 for all z ∈ U. This shows that (2.18) holds true.
Ali RM, Ravichandran V, Seenivasagan N: Subordination and superordination of the Liu-Srivastava linear operator on meromorphically functions. Bull Malays Math Sci Soc 2008, 31: 193–207.
Aouf MK: Certain subclasses of meromprphically multivalent functions associated with generalized hypergeometric function. Comput Math Appl 2008, 55: 494–509.
Cho NE, Kwon OS, Srivastava HM: A class of integral operators preserving subordination and superordination for meromorphic functions. Appl Math Comput 2007, 193: 463–474.
Liu J-L, Srivastava HM: A linear operator and associated families of meromorphically multivalent functions J. Math Anal Appl 2001, 259: 566–581.
Liu J-L, Srivastava HM: Classes of meromorphically multivalent functions associated with the generalized hypergeometric function. Math Comput Model 2004, 39: 21–34.
Wang Z-G, Jiang Y-P, Srivastava HM: Some subclasses of meromorphically multivalent functions associated with the generalized hypergeometric function Comput. Math Appl 2009, 57: 571–586.
Wang Z-G, Sun Y, Zhang Z-H: Certain classes of meromorphically multivalent functions. Comput Math Appl 2009, 58: 1408–1417.
Suffridge TJ: Some remarks on convex maps of the unit disk. Duke Math J 1970, 37: 775–777.
Miller SS, Mocanu PT: Second order differential inequalities in the complex plane. J Math Anal Appl 1978, 65: 289–305.
Department of Mathematics, Suqian College, Suqian, 223800, PR China
Yi-Hui Xu
Department of Mathematics, Yangzhou University, Yangzhou, 225002, PR China
Qing Yang & Jin-Lin Liu
Xu, YH., Yang, Q. & Liu, JL. Some properties of meromorphically multivalent functions. J Inequal Appl 2012, 86 (2012). https://doi.org/10.1186/1029-242X-2012-86
meromorphically multivalent function
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Category:Density mixing - Vaspwiki
Category:Density mixing
Density mixing refers to the way of updating, e.g., the charge density with each iteration step in a self-consistent calculation within density-functional theory (DFT). In the case of magnetism and metaGGAs, VASP can also consider the spin-magnetization density and kinetic-energy density. Selecting the optimal procedure enhances the electronic convergence and avoids problems such as charge sloshing. In many cases, VASP automatically selects suitable values, and it is unnecessary to set the tags related to density mixing manually.
2.1 Improve the convergence
2.2 Magnetic calculations
2.3 MetaGGAs
In each iteration of a DFT calculation, we start from a given charge density
{\displaystyle \rho _{in}}
and obtain the corresponding Kohn-Sham (KS) Hamiltonian and its eigenstates, i.e., KS orbitals. From the occupied KS orbitals, we can compute a new charge density
{\displaystyle \rho _{out}}
. Thus, conceptionally VASP solves a multidimensional fixed-point problem. To solve this problem, VASP uses nonlinear solvers that work with the input vector
{\displaystyle \rho _{in}}
and the residual
{\displaystyle R=\rho _{out}-\rho _{in}}
. The optimal solution is obtained within the subspace spanned by the input vectors. The most efficient density-mixing schemes are the Broyden[1] and the Pulay[2] mixing (IMIX=4). In the Broyden[1] mixing, an approximate of the Jacobian matrix is iteratively improved to find the optimal solution. In the Pulay[2] mixing, the input vectors are combined assuming linearity to minimize the residual.
The implementation in VASP is based on the work of Johnson[3]. Kresse and Furthmüller[4] extended on it and demonstrated that the Broyden and Pulay schemes transform into each other for certain choices of weights for the previous iterations. They also introduced an efficient metric putting additional weight on the long-range components of the density (small
{\displaystyle \mathbf {G} }
vectors), resulting in a more robust convergence. Furthermore, VASP uses a Kerker preconditioning[5] to improve the choice of the input density for the next iteration.
Improve the convergence
For most simple DFT calculations, the default choice of the convergence parameters is well suited to converge the calculation. As a first step, we suggest visualizing your structure or examining the output for warnings to check for very close atoms. That can happen during a structure relaxation if VASP performs a large ionic step. If the structure is correct, we recommend increasing the number of steps NELM and only if that doesn't work starting to tweak the parameters AMIX or BMIX; preferably the latter.
Magnetic calculations
For magnetic materials, the charge density and the spin-magnetization density need to converge.
Hence, if you have problems to converge to a desired magnetic solution, try to calculate first the non-magnetic groundstate, and continue from the generated WAVECAR and CHGCAR file. For the continuation job, you need to set
MetaGGAs
For the density mixing schemes to work reliably, the charge density mixer must know all quantities that affect the total energy during the self-consistency cycle. For a standard DFT functional, this is solely the charge density. In the case of meta-GGAs, however, the total energy depends on the kinetic-energy density.
In many cases, the density-mixing scheme works well enough without passing the kinetic-energy density through the mixer, which is why LMIXTAU=.FALSE., per default. However, when the self-consistency cycle fails to converge for one of the algorithms exploiting density mixing, e.g., IALGO=38 or 48, one may set LMIXTAU=.TRUE. to have VASP pass the kinetic-energy density through the mixer as well. It sometimes helps to cure convergence problems in the self-consistency cycle.
↑ a b C. G. Broyden, Math. Comput. 19, 577 (1965)
↑ a b P. Pulay, Chem. Phys. Lett. 73, 393 (1980).
↑ D. D. Johnson, Phys. Rev. B 38, 12807 (1988)
↑ G. Kresse and J. Furthmüller, Comp. Mater. Sci. 6, 15 (1996)
↑ G.P. Kerker, Phys. Rev. B 23, 3082 (1981)
Pages in category "Density mixing"
Retrieved from "https://www.vasp.at/wiki/index.php?title=Category:Density_mixing&oldid=15369"
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If each triangle below is drawn accurately, identify whether it is an obtuse, acute, or right triangle. Name the angle or angles that helped you to decide.
Examine all three angles of the triangle.
Is the measure of any of the angles greater than or equal to
90°
m\angle C
90°
, so this figure is an obtuse triangle.
Since all the angles measure less than
90°
, the figure is an acute triangle.
What does the box on angle Y mean?
The box means that the angle at that point is a right angle.
How can you identify this triangle?
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Mersenne Prime | Brilliant Math & Science Wiki
Beakal Tiliksew, Joel Yip, Pi Han Goh, and
A Mersenne prime is a prime number that can be written in the form
2^{n}-1
31
is a Mersenne prime that can be written as
2^{5}-1
. The first few Mersenne primes are
3, 7, 31, 127, 8191
. There are 50 known Mersenne primes as of June 2018, though we hope that it will change in the future. An interesting thing about Mersenne primes is that they are the easiest natural numbers to prove to be primes, so they make up the largest category on the list of known prime numbers.
The search and curiosity for Mersenne primes came from the study of perfect numbers. A perfect number is a number that can be written as the sum of its positive proper divisors. For example,
6
is a perfect number as it can be written as
6=1+2+3
, and in fact it is the smallest perfect number. The next perfect number is
28=1+2+4+7+14
It can be shown that if a positive integer
a
2^{n-1}(2^{n}-1)
2^{n}-1
a
must be an even perfect number. We have seen that if
2^{n}-1
is a prime number, then it is a Mersenne prime, which creates a one-to-one correspondence between Mersenne primes and even perfect numbers. That is, every Mersenne prime corresponds to exactly one even perfect number! (So far no odd perfect number has been found.)
2^{n}-1
is prime, then
must also be prime.
p
q
be positive integers greater than one such that
n=p\cdot q
. Then using the factorization identity,
{ 2 }^{ pq }-1=\left( { 2 }^{ p }-1 \right) \cdot \left( 1+{ 2 }^{ p }+{ 2 }^{ 2p }+{ 2 }^{ 3p }+\cdots+{ 2 }^{ p(q-1) } \right).
n
is composite and WLOG
1<p<q
, then we have the
2^{n}-1
term to be composite because it is divisible by the
2^{p}-1
_\square
The proof tells us that if
2^{n}-1
is prime, then
is also prime. But it doesn't guarantee that if
n
2^{n}-1
is prime, as we have not considered the second term in the above equation. A typical example of this is
11:
even though it is a prime number,
2^{11}-1=2047
isn't a prime number.
[1]
[1]
[2]
[2]
[3]
[1]
p
2^p-1
[2]
p
2^p-1
[3]
[1]
Lucas–Lehmer primality test is a primality test (an algorithm for determining whether an input number is prime) for Mersenne primes. It's currently known to be the most efficient test for Mersenne primes.
First we start with
n=0
{ a }_{ n }={ a }_{ n-1 }^{ 2 }-2
{ a }_{ 0 }=4.
If you want to test if
{ 2 }^{ k }-1
is prime, you have to check if
{ a }_{ k-2 }\equiv 0 ~\big( \text{mod }~ 2^k-1 \big)
{ 2 }^{ 11 }-1
We could just find if there are other factors in that number, or we can use the Lucas–Lehmer primality test.
For this,
k=11
{ 2 }^{ 11 }-1=2047.
a_0=4 \pmod{2047}
a_1=\big(4^2-2\big) \pmod{2047} =14 \pmod{2047}
a_2=\big(14^2-2\big) \pmod{2047} =194 \pmod{2047}
a_3=\big(194^2-2\big) \pmod{2047} =788 \pmod{2047}
a_4=\big(788^2-2\big) \pmod{2047} =701 \pmod{2047}
a_5=\big(701^2-2\big) \pmod{2047} =119 \pmod{2047}
a_6=\big(119^2-2\big) \pmod{2047} =1877 \pmod{2047}
a_7=\big(1877^2-2\big) \pmod{2047} =240 \pmod{2047}
a_8=\big(240^2-2\big) \pmod{2047} =282 \pmod{2047}
a_9=\big(282^2-2\big) \pmod{2047} =1736 \pmod{2047}.
k-2=9
a_9 \equiv 0 \pmod{2^k-1}
is false. Therefore,
{ 2 }^{ 11 }-1
_\square
51
known Mersenne primes, with the largest known prime being
\large 2^{82,589,933} - 1,
which is over 24 million digits long!
This enormous number was discovered by Patrick Laroche in 2018, as part of the GIMPS (Great Internet Mersenne Prime Search). It is a collaborative effort to find new primes by pooling computing power online. It has 24,862,048 digits in total.
\Large {2}^{74,207,281} - 1.
\bullet
p=23
is prime, is it true that
M_{23} = 2^{23} - 1
Cite as: Mersenne Prime. Brilliant.org. Retrieved from https://brilliant.org/wiki/mersenne-prime/
|
This article is about the general framework of distance and direction. For the space beyond Earth's atmosphere, see Outer space. For the writing separator, see Space (punctuation). For other uses, see Space (disambiguation).
Spherical geometry is similar to elliptical geometry. On a sphere (the surface of a ball) there are no parallel lines.
Number of parallels
Ratio of circumference to diameter of circle
Measure of curvature
Infinite < 180° > π < 0
Main article: Three-dimensional space
Not to be confused with Space (mathematics).
{\displaystyle {\textbf {F}}={\frac {d}{dt}}(m{\textbf {v}})}
Find sources: "Space" – news · newspapers · books · scholar · JSTOR (April 2020) (Learn how and when to remove this template message)
Main article: Theory of relativity
One consequence of this postulate, which follows from the equations of general relativity, is the prediction of moving ripples of spacetime, called gravitational waves. While indirect evidence for these waves has been found (in the motions of the Hulse–Taylor binary system, for example) experiments attempting to directly measure these waves are ongoing at the LIGO and Virgo collaborations. LIGO scientists reported the first such direct observation of gravitational waves on 14 September 2015.[26][27]
See also: Spatial analysis
^ "Space – Physics and Metaphysics". Encyclopædia Britannica. Archived from the original on 6 May 2008. Retrieved 28 April 2008.
^ Refer to Plato's Timaeus in the Loeb Classical Library, Harvard University, and to his reflections on khora. See also Aristotle's Physics, Book IV, Chapter 5, on the definition of topos. Concerning Ibn al-Haytham's 11th century conception of "geometrical place" as "spatial extension", which is akin to Descartes' and Leibniz's 17th century notions of extensio and analysis situs, and his own mathematical refutation of Aristotle's definition of topos in natural philosophy, refer to: Nader El-Bizri, "In Defence of the Sovereignty of Philosophy: al-Baghdadi's Critique of Ibn al-Haytham's Geometrisation of Place", Arabic Sciences and Philosophy (Cambridge University Press), Vol. 17 (2007), pp. 57–80.
^ French, A.J.; Ebison, M.G. (1986). Introduction to Classical Mechanics. Dordrecht: Springer, p. 1.
^ Carnap, R. (1995). An Introduction to the Philosophy of Science. New York: Dove. (Original edition: Philosophical Foundations of Physics. New York: Basic books, 1966).
^ a b Huggett, Nick, ed. (1999). Space from Zeno to Einstein: classic readings with a contemporary commentary. Cambridge, MA: MIT Press. Bibcode:1999sze..book.....H. ISBN 978-0-585-05570-1. OCLC 42855123.
^ Janiak, Andrew (2015). "Space and Motion in Nature and Scripture: Galileo, Descartes, Newton". Studies in History and Philosophy of Science. 51: 89–99. Bibcode:2015SHPSA..51...89J. doi:10.1016/j.shpsa.2015.02.004. PMID 26227236.
^ Dainton, Barry (2001). Time and space. Montreal: McGill-Queen's University Press. ISBN 978-0-7735-2302-9. OCLC 47691120.
^ Dainton, Barry (2014). Time and Space. McGill-Queen's University Press. p. 164.
^ Tom., Sorell (2000). Descartes: a very short introduction. Oxford: Oxford University Press. ISBN 978-0-19-154036-3. OCLC 428970574.
^ Leibniz, Fifth letter to Samuel Clarke. By H.G. Alexander (1956). The Leibniz-Clarke Correspondence. Manchester: Manchester University Press, pp. 55–96.
^ Vailati, E. (1997). Leibniz & Clarke: A Study of Their Correspondence. New York: Oxford University Press, p. 115.
^ Sklar, L. (1992). Philosophy of Physics. Boulder: Westview Press, p. 20.
^ Sklar, L. Philosophy of Physics. p. 21.
^ "Newton's bucket". st-and.ac.uk. Archived from the original on 17 March 2008. Retrieved 20 July 2008.
^ Carnap, R. An Introduction to the Philosophy of Science. pp. 177–178.
^ Lucas, John Randolph (1984). Space, Time and Causality. p. 149. ISBN 978-0-19-875057-4.
^ Carnap, R. An Introduction to the Philosophy of Science. p. 126.
^ Jammer, Max (1954). Concepts of Space. The History of Theories of Space in Physics. Cambridge: Harvard University Press, p. 165.
^ A medium with a variable index of refraction could also be used to bend the path of light and again deceive the scientists if they attempt to use light to map out their geometry.
^ Wheeler, John A. A Journey into Gravity and Spacetime. Chapters 8 and 9, Scientific American, ISBN 0-7167-6034-7
^ Castelvecchi, Davide; Witze, Alexandra (11 February 2016). "Einstein's gravitational waves found at last". Nature News. Archived from the original on 16 February 2016. Retrieved 12 January 2018.
^ Abbott, Benjamin P.; et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2016). "Observation of Gravitational Waves from a Binary Black Hole Merger". Phys. Rev. Lett. 116 (6): 061102. arXiv:1602.03837. Bibcode:2016PhRvL.116f1102A. doi:10.1103/PhysRevLett.116.061102. PMID 26918975. S2CID 124959784.
"Observation of Gravitational Waves from a Binary Black Hole Merger" (PDF). LIGO Scientific Collaboration.
^ "Cosmic Detectives". The European Space Agency (ESA). 2 April 2013. Archived from the original on 5 April 2013. Retrieved 26 April 2013.
^ Stanek, Lukasz (2011). Henri Lefebvre on Space: Architecture, Urban Research, and the Production of Theory. Univ of Minnesota Press. pp. ix.
^ "Time-Space Compression – Geography – Oxford Bibliographies – obo". Archived from the original on 20 September 2018. Retrieved 28 August 2018.
^ Harvey, David (2001). Spaces of Capital: Towards a Critical Geography. Edinburgh University Press. pp. 244–246.
^ W., Soja, Edward (1996). Thirdspace: journeys to Los Angeles and other real-and-imagined places. Cambridge, Mass.: Blackwell. ISBN 978-1-55786-674-5. OCLC 33863376.
^ Lefebvre, Henri (1991). The production of space. Oxford, OX, UK: Blackwell. ISBN 978-0-631-14048-1. OCLC 22624721.
^ Ashcroft Bill; Griffiths, Gareth; Tiffin, Helen (2013). Postcolonial studies: the key concepts (Third ed.). London. ISBN 978-0-415-66190-4. OCLC 824119565.
Spaceat Wikipedia's sister projects
Retrieved from "https://en.wikipedia.org/w/index.php?title=Space&oldid=1083107965"
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Integrals involving Powers of Trigonometric Functions | Brilliant Math & Science Wiki
Integrals involving Powers of Trigonometric Functions
Hobart Pao, Earl Potters, Md Zuhair, and
This wiki will enable you to evaluate integrals like
\displaystyle \int \tan^{4} x \, dx
without splitting it up and doing lots of complicated integration.
Sine and Cosine Reduction Formulas
Tangent and Cotangent Reduction Formulas
Integrands of the Form
\sin^{\alpha} x \cos^{\beta} x
\tan^{\alpha} x \sec^{\beta} x
n \geq 2
n \in \mathbb{R}
\begin{aligned} \int \cos^{n} x \, dx &= \dfrac{1}{n} \cos^{n-1}x \sin x + \dfrac{n-1}{n}\int \cos^{n-2} x \, dx \\ \int \sin^{n} x \, dx &= -\dfrac{1}{n}\sin^{n-1}x \cos x + \dfrac{n-1}{n} \int \sin^{n-2}x \, dx. \end{aligned}
Proof: (Under construction)
\displaystyle \int \sin^{5} x \, dx
n \geq 2
n \in \mathbb{R}
\begin{aligned} \int \tan^{n} x \, dx &= \dfrac{\tan^{n-1}x}{n-1} - \int \tan^{n-2} x \, dx \\ \int \sec^{n} x \, dx &= \dfrac{\sec^{n-2} x \tan x}{n-1} + \dfrac{n-2}{n-1} \int \sec^{n-2} x \, dx. \end{aligned}
\displaystyle \int \sec^{5} x \, dx
\sin^{\alpha} x \cos^{\beta} x
There are a few possible scenarios for the integrand in the form
\sin^{\alpha} x \cos^{\beta} x
\alpha
\beta
\displaystyle \int \sin^{3} x \cos^{2} x \, dx
\displaystyle \int \sin x \sin^{2} x \cos^{2} x \, dx = \int \sin x \big( 1-\cos^{2} x \big) \cos^{2} x \, dx. \qquad (1)
u = \cos x,
\frac{du}{dx} = -\sin x
(1)
\displaystyle -\int \big( 1-u^{2} \big) u^{2} \, du.
This integral is very easy to compute now; you just have to expand the terms and use the power rule for antiderivatives. But don't forget what was
u
_\square
\alpha
\beta
Same idea as "
\alpha
\beta
is even."
\alpha
\beta
In this scenario, there are two different things you could do. You could utilize the following identities:
\cos^{2} x = \frac{ 1+ \cos 2x}{2}
\sin^{2} x = \frac{1 - \cos 2x}{2}.
Or, you could rewrite the integrand only in terms of a single trigonometric function.
\displaystyle \int \sin^{2} x \cos^{2} x \, dx.
Using the aforementioned identities, we can rewrite the given expression as
\begin{aligned} \int \left( \dfrac{1-\cos 2x}{2}\right) \left( \dfrac{1+ \cos 2x}{2} \right) \, dx &= \dfrac{1}{4} \displaystyle \int \big( 1- \cos^{2} 2x \big) \, dx \\ &= \dfrac{1}{4} \displaystyle \int \sin^{2} 2x \, dx. \qquad (1) \end{aligned}
u = 2x,
\frac{du}{dx} = 2
(1)
\dfrac{1}{8} \displaystyle \int \sin^{2} u \, du. \qquad (2)
Remember? We got an identity for this! So
(2)
\dfrac{1}{8} \int \dfrac{1-\cos 2u}{2} \, du = \dfrac{u}{16} - \dfrac{\sin 2u}{32} + C,
C
u?
u=2x,
\dfrac{x}{8} - \dfrac{\sin 4x}{32} + C. \ _\square
Something to consider: What's another way you could solve that integral? Since the powers are relatively nice, you could rewrite everything in terms of cosine or sine, and use one of the reduction formulas from above.
\displaystyle \int \sin^{2} x \cos^{4} x \, dx.
It is possible to use the aforementioned problems to solve this problem, but I'm going to use the reduction formulas approach just to contrast the methods. For this type of problem, I actually prefer the reduction formulas:
\begin{aligned} \int \sin^{2} x \cos^{4} x \, dx & = \int \big( 1- \cos^{2} x \big) \cos^{4} x \, dx \\ & = \int \cos^{4} x \, dx - \int \cos^{6} x \, dx. \qquad (1) \end{aligned}
\displaystyle \int \cos^{6} \, dx,
which is the integral with the highest exponent in
(1),
\int \cos^{6} \, dx=\dfrac{\cos^{5} x \sin x}{6} + \dfrac{5}{6} \displaystyle \int \cos^{4} x \, dx, \qquad (2)
(2)
(1)
\int \sin^{2} x \cos^{4} x \, dx = \dfrac{1}{6} \displaystyle \int \cos^{4} \, dx - \dfrac{\cos^{5} x \sin x}{6}. \qquad (3)
Use the reduction formulas again, then
\displaystyle \int \cos^{4} x \, dx
(3)
\int \cos^{4} \, dx = \dfrac{\cos^{3} x \sin x }{4} + \dfrac{3}{4} \int \cos^{2} x \, dx. \qquad (4)
\cos^{2} x = \frac{1+ \cos 2x}{2}
and we also know how to integrate that, so I'm going to skip that step and find
(4)
to be equivalent to
\int \cos^{4} \, dx = \dfrac{\cos^{3} x \sin x }{4} + \dfrac{3}{4} \left( \dfrac{x}{2} + \dfrac{\sin 2x}{4} \right). \qquad (5)
(5)
(3)
\int \sin^{2} x \cos^{4} x \, dx =\dfrac{\cos^{3} x \sin x}{24} + \dfrac{x}{16} + \dfrac{\sin 2x}{32} - \dfrac{\cos^{5} x \sin x}{6} + C,
C
_\square
\tan^{\alpha} x \sec^{\beta} x
Cite as: Integrals involving Powers of Trigonometric Functions. Brilliant.org. Retrieved from https://brilliant.org/wiki/integrals-involving-powers-of-trigonometric/
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Examine the following series. Use one of the tests you have learned so far to determine if the series converges or diverges. State the test that you used.
\displaystyle\sum _ { n = 1 } ^ { \infty } ( \frac { 1 } { 2 n } - \frac { 1 } { 3 n } )
\frac{1}{2n}-\frac{1}{3n}=\frac{1}{6n}
\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { n ! } { ( 2 n ) ! }
\lim\limits_{n\to\infty}\frac{(n+1)!/(2n+2)!}{n!/(2n)!}
=\lim\limits_{n\to\infty}\Bigg(\frac{(n+1)!}{n!}\cdot\frac{(2n)!}{(2n+2)!}\Bigg)
=\lim\limits_{n\to\infty}\frac{n+1}{(2n+2)(2n+1)}=?
\displaystyle\sum _ { n = 1 } ^ { \infty } n e ^ { - n ^ { 2 } }
ne^{-n^2}=\frac{n}{e^{n^2}}
\lim\limits_{n\to\infty}\frac{(n+1)/e^{(n+1)^2}}{n/e^{n^2}}
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Sine and Cosine Graphs | Brilliant Math & Science Wiki
Pi Han Goh, Mei Li, Raghav Vaidyanathan, and
We started filling in parts of this page, but it is missing several details. Places with "???" would need to be edited.
If you draw a circle with radius 1, and have a ray extending from the origin and intersecting the circle, such that the ray makes an angle
\theta
x
-axis, we can say that the point at which the circle is intersected by the ray is
(x, y)
. We can define
\cos \theta
x
value of the coordinate and
\sin \theta
y
value of the coordinate. Now that we have defined the basic trigonometric functions, we will consider properties of these functions by studying their graphs.
Relationship between Sine and Cosine graphs
In the graph of the sine function, the
x
-axis represents values of
\theta
y
\sin \theta
\sin 0=0,
implying that the point
(0,0)
is a point on the sine graph. If we plot the values of the sine function for a large number of angles
\theta
, we see that the points form a curve called the sine curve:
Similarly, plotting the values of the cosine function for a large number of angles forms a curve called the cosine curve:
We can visualize the relationship between these graphs and the definition of cosine and sine from the unit circle as follows:
Animation courtesy commons.wikimedia.org
How many points of intersection are there between the graphs of
\sin x
\cos x
[0, 2\pi]
From the graphs of sine and cosine, it is evident that the number of intersection points in the given range is
2
_\square
The sine and cosine graphs both have range
[-1,1]
and repeat values every
2\pi
(called the amplitude and period). However, the graphs differ in other ways, such as intervals of increase and decrease. The following outlines properties of each graph:
Properties of Sine:
y
0
x
n \pi,
n
is an integer here on
increasing on intervals
\left[\frac{(4n-1)\pi}{2},\frac{(4n+1)\pi}{2} \right]
decreasing on intervals
\left[\frac{(4n+1)\pi}{2},\frac{(4n+3)\pi}{2} \right]
Symmetry: function is symmetric about the origin.
Maximum of
\sin \theta
is achieved for
\theta=\frac{(4n+1)\pi}{2}
Properties of Cosine:
y
-intercept: 1
x
\left(n + \frac{1}{2} \right) \pi,
n
\left[(2n-1)\pi,2n\pi\right]
\left[2n\pi,(2n+1)\pi \right]
Symmetry: function is symmetric about the
y
\cos \theta
\theta=2n\pi
Find all the
x
-intercepts for
f(x) = 2\cos 3x +2
0\leq x\leq 2\pi
When the curve of
f(x)
x
f(x) = 0
2\cos 3x + 2 = 0 \Rightarrow \cos 3x = -1
0 \leq x \leq 2\pi \Rightarrow 0\leq 3x \leq 6\pi
and the cosine function has a period of
2\pi
3x = \pi, ~\pi+2\pi, ~\pi+4\pi \Rightarrow x=\frac{\pi}{3}, ~\pi, ~\frac{5\pi}{3},
which are the three
x
-intercepts we are looking for.
_\square
What is the number of intersection points between the two curves
f(x) = 5\cos x + 7
g(x) = -6\sin x - 10
0 \leq x \leq 2\pi
f(x)
g(x)
are graphs that are a stretch and have a vertical shift, we first try to find whether there is a common intersection between these two curves.
-1 \leq \cos x \leq 1 \Rightarrow -5 \leq 5\cos x \leq 5 \Rightarrow 2 \leq 5\cos x + 7 \leq 12
, the range of
f(x)
[2, 12]
-1 \leq -\sin x \leq 1 \Rightarrow -6\leq -6\sin x \leq 6 \Rightarrow -16 \leq -6\sin x -10 \leq -4
g(x)
[-16, -4]
f(x)
g(x)
is strictly negative, there is no intersection point between these two curves. Specifically, there is no intersection point for these curves in the interval
0 \leq x \leq 2\pi. \ _\square
The graph of sine has the same shape as the graph of cosine. Indeed, the graph of sine can be obtained by translating the graph of cosine by
\frac{(4n+1)\pi}{2}
x
n
is an integer). Also, the graph of cosine can be obtained by translating the graph of Sine by
\frac{(4n+1)\pi}{2}
units along the negative
x
-axis. In other words:
\cos{\theta}=\sin{\left(\frac{(4n+1)\pi}{2}-\theta\right)}, ~\sin{\theta}=\cos{\left(\frac{(4n+1)\pi}{2}-\theta\right)}.
\sin \left( \frac{9 \pi }{ 2} - \theta \right)
Because the function
\sin \theta
has a period of
2\pi
\sin(-\theta)
2\pi
\sin\left(\frac{9\pi}2 -\theta\right) = \sin\left(\frac{9\pi}2 -\theta - 2\pi - 2\pi \right) = \sin\left(\frac{\pi}2 -\theta\right) = \cos \theta.
Alternatively, we recognize this as the form of
\sin \left( \frac{(4n+1)\pi}{2} - \theta \right)
n = 2
. Hence, it is equal to
\cos \theta
_\square
\cos \left ( \theta + \frac{\pi}{2} \right)
There are several approaches to use:
Using the properties listed above, we have
\cos \left ( \theta + \frac{\pi}{2} \right) = \sin ( - \theta ) = - \sin \theta.
Draw the graph, and compare it to what we already know.
By drawing the graph, we can visually see that it is equal to
- \sin \theta
Expand using the cosine - sum and difference formulas, which gives us
\cos \left ( \theta + \frac{\pi}{2} \right) = \cos \theta \cos \frac{\pi}{2} - \sin \theta \sin \frac{\pi}{2} = - \sin \theta.\ _\square
For complete information, see Graph Transformation.
We can manipulate the basic trigonometric graph by adding constants as follows:
y = a \sin (b x - c) + d.
y = a \cos (b x - c) + d.
a
b
contracts the graph horizontally
c
translates the graph horizontally
d
translates the graph vertically.
Find the amplitude, period, horizontal shift, and vertical shift of the function
y = 2 \cos \left( \frac{\pi}{2} x - \pi \right) + 4.
So we want to find the values of
a,b,c
and
when we compare it to the form of
a \cos(bx-c) + d
It's easy to see that
a = 2, b = \frac\pi2, c =\pi, d = 4
. Interpreting the values shows the following:
The period of this function
y = 2 \cos \left( \frac{\pi}{2} x - \pi \right) + 4
2\pi \div \frac{\pi}2 = 4
The amplitude is equals to the maximum absolute value of the scalar multiple of the trigonometric function. In this case, it is
\left | 2 \cos \left( \frac{\pi}{2} x - \pi \right) \right | \leq 2
c = \pi
, the function has been shifted horizontally by
\pi
d = 4
, the function has been shifted vertically by
4
_\square
y = a \sin (b x - c) + d
a,b,c,
and
, what are the largest and smallest possible values of
y?
-1 \leq \sin \theta \leq 1
-|a| \leq a \sin (bx -c) \leq |a|
. Thus, the largest possible value is
|a| + d
and the smallest possible value is
- |a| + d
_\square
The position of a spring as a function of time is represented by an equation of the form
p(t) = a \cos bt
. If the spring starts at 3 units above its rest point, bounces to 3 units below its rest point and then back to 3 units above its rest point in a total of 2 seconds, find an equation that represents this motion.
From the context "the spring starts at 3 units above its rest point," we can interpret it as
p(0) = 3
3 = a\cos(b\times0) \Rightarrow a = 3
From the context "and then back to 3 units above its rest point in a total of 2 seconds," we can interpret it as the fundamental period of
p(t)
2
2 \pi \div b = 2 \Rightarrow b = \pi
Hence the equation that represent this motion is
p(t) = 3 \cos \pi t.\ _\square
Cite as: Sine and Cosine Graphs. Brilliant.org. Retrieved from https://brilliant.org/wiki/sine-and-cosine-graphs/
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Fabrication, characterization and simulation of Ω-gate twin poly-Si FinFET nonvolatile memory | Nanoscale Research Letters | Full Text
Fabrication, characterization and simulation of Ω-gate twin poly-Si FinFET nonvolatile memory
Mu-Shih Yeh1,
Yung-Chun Wu1,
Min-Feng Hung1,
Kuan-Cheng Liu1,
Yi-Ruei Jhan1,
Lun-Chun Chen2 &
This study proposed the twin poly-Si fin field-effect transistor (FinFET) nonvolatile memory with a structure that is composed of Ω-gate nanowires (NWs). Experimental results show that the NW device has superior memory characteristics because its Ω-gate structure provides a large memory window and high program/erase efficiency. With respect to endurance and retention, the memory window can be maintained at 3.5 V after 104 program and erase cycles, and after 10 years, the charge is 47.7% of its initial value. This investigation explores its feasibility in the future active matrix liquid crystal display system-on-panel and three-dimensional stacked flash memory applications.
Electrically erasable programmable read-only memory (EEPROM), which is a kind of nonvolatile memory (NVM) [1, 2], has been widely used in portable products owing to its high density and low cost [3]. Embedded EEPROM that is based on poly-Si thin film transistor (TFT) has attracted much attention because it can meet the low-temperature process requirement in thin film transistor liquid crystal display applications [4, 5]. However, since the process and physical limitations of the device limit the scaling of the flash NVM that is based on a single-crystalline Si substrate, according to Moore’s law, the three-dimensional (3D) multi-layer stack memory provides a high-density flash memory solution. The poly-Si-based NVM also has great potential for realizing 3D high-density multi-layer stack memory [6–8]. A planar EEPROM that uses twin poly-Si TFTs has also been developed for the above aforementioned applications [4, 9]. The advantages of this twin TFT structure include processing identical to that of a conventional TFT, which is easily embedded on Si wafer, glass, and flexible substrates. Additionally, the low program/erase (P/E) operating voltage of this planar NVM can be easily obtained by increasing the artificial gate coupling ratio (αG).
Recently, several investigations have demonstrated that gate control can be substantially enhanced by introducing a multi-gate with a nanowire (NW) structure [10–12]. In our previous works [13, 14], NWs were introduced into twin poly-Si TFT NVM to increase P/E speed. However, reducing the P/E voltage while ensuring the reliability of this device remains a challenge.
Therefore, in this work, to reduce the P/E voltage, we try to use p-channel devices with band-to-band tunneling-induced hot electron (BBHE) operation compared with Fowler-Nordheim (FN) operation and use a Ω-gate structure to have little deterioration. These p-channel twin fin field-effect transistor (FinFET) EEPROM devices with a Ω-gate structure have excellent retention and endurance.
First, a p-type undoped channel twin poly-Si TFT EEPROM with ten NWs was fabricated. Figure 1a presents the structure of the NW twin poly-Si TFT EEPROM. The gate electrodes of two TFTs are connected to form the floating gate, while the source and drain of the larger TFT (T2) are connected to form the control gate. Figure 1b presents the transmission electron microscopy (TEM) image of the NW EEPROM perpendicular to the gate direction; the NWs are surrounded by the gate electrode as a Ω-gate structure with an effective width of 113 nm.
Schematic, TEM image, and equivalent circuit of twin poly-Si TFT EEPROM. (a) Schematic of the twin poly-Si TFT EEPROM cell with ten NWs. (b) The TEM image of Ω-gate NW twin poly-Si TFT EEPROM. The effective channel width is 113 nm × 10 [(61 nm + 16 nm × 2 + 10 nm × 2) × 10)]. (c) The equivalent circuit of twin poly-Si TFT EEPROM.
These devices were fabricated by initially growing a 400-nm-thick thermal oxide layer on 6-in. silicon wafers as substrates. A thin 50-nm-thick undoped amorphous Si (a-Si) layer was deposited by low-pressure chemical vapor deposition (LPCVD) at 550°C. The deposited a-Si layer was then solid-phase-crystallized at 600°C for 24 h in nitrogen ambient. The device’s active NWs were patterned by electron beam (e-beam) direct writing and transferred by reactive-ion etching (RIE). Then, they were dipped into HF solution for 60 s to form the Ω-shaped structure. For gate dielectric, a 15-nm-thick layer of thermal oxide was grown as tunneling oxide. Then, a 150-nm-thick poly-Si layer was deposited and transferred to a floating gate by electron beam direct writing and RIE. Then, the T1 and T2 self-aligned P+ source/drain and gate regions were formed by the implantation of BF2 ions at a dose of 5 × 1015 cm−2. The dopant was activated by ultrarapid thermal annealing at 1,000°C for 1 s in nitrogen ambient. Then, a 200-nm-thick TEOS oxide layer was deposited as the passivation layer by LPCVD. Next, the contact holes were defined and 300-nm-thick AlSiCu metallization was performed. Finally, the devices were then sintered at 400°C in nitrogen ambient for 30 min.
In programming, the electrons tunnel into T1 through the tunneling oxide. The tunneling oxide of NW-based EEPROM is surrounded by the gate electrode (Figure 1b). Figure 1c shows the equivalent circuit of this twin TFT NVM:
\begin{array}{l}{V}_{\mathrm{FG}}=\left[{C}_{2}/\left({C}_{1}+{C}_{2}\right)\right]×{V}_{\mathrm{G}}\\ \phantom{\rule{2.25em}{0ex}}=\left[{W}_{2}/\left({W}_{1}+{W}_{2}\right)\right]×{V}_{\mathrm{G}}={\alpha }_{\mathrm{G}}×{V}_{\mathrm{G}}.\end{array}
To maximize the voltage drop in the tunnel oxide of T1, the gate capacitance of T2 (C2) must exceed the gate capacitance of T1 (C1). Hence, the NVM device with a high αG exhibits a high P/E speed and can be operated at a low voltage. In this work, the devices were designed to have a coupling ratio of 0.85, which is extremely high for memory applications.
The TEM image in Figure 1b shows the rounded corners of the twin TFT device structure. First, the NW tri-gated structure, formed by e-beam lithography, was dipped into DHF solution, forming rounded corners. Then, thermal oxidation was performed to form the tunneling oxide; the junction of the channel and the tunneling oxide exhibits some rounding, protecting the tunneling oxide against excessive damage when it is written and erased. The P/E speed and reliability are balanced by Ω-gate formation. By technology computer-aided design (TCAD) simulation, Figure 2 shows the electric field of NWs using tri-gate and Ω-gate structures. The result indicates that the Ω-gate structure has more programming sites around the NWs than the tri-gate structure which are only at the upper corners and that the Ω-gate structure also has smoother electric field.
Electric field of NWs. By TCAD simulation, cut from the AA’ line in the (a) schematic, the electric field around the NWs of (b) tri-gate and (c) Ω-gate structures is shown.
Figure 3 compares the P/E speed of the BBHE operation with that of the FN operation. The device was programmed by FN injection at Vgs = 17 V and by BBHE injection at Vgs = 7 V with Vds = −10 V. The BBHE operation exhibits higher programming speed than the FN operation.
Programming and erasing characteristics of the EEPROM cell with devices. The P/E speed of BBHE operation is compared with that of FN operation.
Figure 4a shows the twin poly-Si TFT-based (Weff/W2/L = 113 nm × 10/6 μm/10 μm) EEPROM P/E cycling endurance characteristics by FN and BBHE, respectively, using the same input voltage. As the number of P/E cycles increased, the magnitude of the memory window disappeared. The floating-gate memory device maintained a wide threshold voltage window of 3.5 V (72.2%) after 104 P/E cycles for FN operation. For BBHE operation, the memory window was almost closed after 104 P/E cycles. Figure 4b shows high-temperature (85°C) retention characteristics of NW-based (Weff/W2/L = 113 nm × 10/6 μm/10 μm) EEPROMs. This figure reveals that after 10 years, the memory window was still 2.2 V when using FN operation. For BBHE operation, the device exhibited almost no data retention capacity. The Ω-gate structure has a higher P/E efficiency than the tri-gate structure because the four corners of the channel are all surrounded by the gate structure [13, 14]. The Ω-gate structure contributes to the equal sharing of the electric field and reduces the probability of leakage in the floating-gate devices in the form of stress-induced leakage current, improving the reliability of the device. Also, the extra corners improve the P/E speed.
Endurance and retention characteristics. (a) Endurance characteristics of the twin poly-Si TFT EEPROM by FN and BBHE. (b) Retention characteristics of the twin poly-Si TFT EEPROM at 85°C by FN and BBHE.
Figure 5 displays a TCAD simulation of FN and BBHE operations. The result indicates that the FN operation produces a high average electric field in the tunneling oxide from the source to the drain, programmed by the tunneling effect. FN operation indicates the average wearing of electric field on the tunneling oxide. BBHE operation produces a sudden electric field peak at the source side, programmed using hot electrons with high energy, causing considerable local damage to the tunneling oxide. This result of consistent P/E that is caused by FN operation reveals better endurance and retention than the BBHE operation for floating-gate devices.
TCAD simulation. (a) FN programming. VFG = VCG × αG = 14.9 V. (b) BBHE programming. VFG = VCG × αG = 5.95 V. Both use the same voltage drop. (c) Electric field comparison of FN and BBHE programming.
This work developed a novel Ω-gate NW-based twin poly-Si TFT EEPROM. Experimental results demonstrated that the Ω-gate NW-based structure had a large memory window and high P/E efficiency because of its multi-gate structure and even oxide electrical field at the NW corners. After 104 P/E cycles, ΔVth = 3.5 V (72.2%). The proposed twin-TFT EEPROM with a fully overlapped control gate exhibited good data endurance and maintained a wide threshold voltage window even after 104 P/E cycles. This Ω-gate NW-based twin poly-Si TFT EEPROM can be easily incorporated into an AMLCD array press and SOI CMOS technology without any additional processing.
Su CJ, Tsai TI, Lin HC, Huang TS, Chao TY: Low-temperature poly-Si nanowire junctionless devices with gate-all-around TiN/Al2O3 stack structure using an implant-free technique. Nanoscale Res Lett 2012, 7: 339. 10.1186/1556-276X-7-339
Su CJ, Su TK, Tsai TI, Lin HC, Huang TY: A junctionless SONOS nonvolatile memory device constructed with in situ-doped polycrystalline silicon nanowires. Nanoscale Res Lett 2012, 7: 162. 10.1186/1556-276X-7-162
Park KT, Choi J, Sel J, Kim V, Kang C, Shin Y, Roh U, Park J, Lee JS, Sim J, Jeon S, Lee C, Kim K: A 64-cell NAND flash memory with asymmetric S/D structure for sub-40nm technology and beyond. VLSI Tech Dig 2006, 2006: 19.
Young ND, Harkin G, Bunn RM, MaCulloch DJ, French ID: The fabrication and characterization of EEPROM arrays on glass using a low-temperature poly-Si TFT process. IEEE Trans Electron Device 1930, 1996: 43.
Hung MF, Wu YC, Tsai TM, Chen JH, Jhan YR: Enhancement of two-bit performance of dual-pi-gate charge trapping layer flash memory. Applied Physics Express 2012, 5: 121801. 10.1143/APEX.5.121801
Park B, Cho K, Kim S, Kim S: Nano-floating gate memory devices composed of ZnO thin-film transistors on flexible plastics. Nanoscale Res Lett 2011, 6: 41.
Ichikawa K, Uraoka Y, Yano H, Hatayama T, Fuyuki Y, Takahashi E, Hayashi T, Ogata K: Low temperature polycrystalline silicon thin film transistors flash memory with silicon nanocrystal dot. Jpn J Appl Phys 2007, 46: 661. 10.1143/JJAP.46.L661
Lai EK, Lue HT, Hsiao YH, Hsieh JY, Lu CP, Wang SY, Yang LW, Yang T, Chen KC, Gong J, Hsieh KY, Liu R, Lu CY: A highly stackable thin-film transistor (TFT) NAND-type flash memory. VLSI Tech Dig 2006, 2006: 46.
Chung HJ, Lee NI, Han CH: A high-endurance low-temperature polysilicon thin-film transistor EEPROM cell. IEEE Electron Device Lett 2000, 21: 304.
Wu TC, Chang TC, Chang CY, Chen CS, Tu CH, Liu PT, Zan HW, Tai YH: High-performance polycrystalline silicon thin-film transistor with multiple nanowire channels and lightly doped drain structure. Appl Phys Lett 2004, 84: 19. 10.1063/1.1638883
Gabrielyan N, Saranti K, Manjunatha KN, Paul S: Growth of low temperature silicon nano-structures for electronic and electrical energy generation applications. Nanoscale Res Lett 2013, 8: 83. 10.1186/1556-276X-8-83
Lacy F: Developing a theoretical relationship between electrical resistivity, temperature, and film thickness for conductors. Nanoscale Res Lett 2011, 6: 636. 10.1186/1556-276X-6-636
Wu YC, Su PW, Chang CW, Hung MF: Novel twin poly-Si thin-film transistors EEPROM with trigate nanowire structure. IEEE Electron Device Lett 2008, 29: 1226.
Wu YC, Hung MF, Su PW: Improving the performance of nanowires polycrystalline silicon twin thin-film transistors nonvolatile memory by NH3 plasma passivation. J Electrochem Soc 2011, 158: H578. 10.1149/1.3560576
The authors would like to acknowledge the National Science Council of Taiwan for supporting this research under contract no. NSC 101-2221-E-007-088-MY2. The National Nano Device Laboratories is greatly appreciated for its technical support.
Department of Engineering and System Science, National Tsing Hua University, 101, Section 2 Kuang Fu Road, Hsinchu, 30013, Taiwan
Mu-Shih Yeh, Yung-Chun Wu, Min-Feng Hung, Kuan-Cheng Liu & Yi-Ruei Jhan
Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, 1001, Ta Hsueh Road, Hsinchu, 30013, Taiwan
Lun-Chun Chen & Chun-Yen Chang
Mu-Shih Yeh
Min-Feng Hung
Kuan-Cheng Liu
Yi-Ruei Jhan
Lun-Chun Chen
M-SY and M-FH carried out the device mask layout, modulated the coupling ratio of the device, handled the experiment, and drafted the manuscript. K-CL measured the characteristics of the device and made the simulation plot. Y-RJ and L-CC gave some physical explanation to this work. Y-CW conceived the idea of low-temperature deposition of twin FinFET and their exploitation into devices. He also supervised the work and reviewed the manuscript. C-YC participated in the design and coordination of the study. All authors read and approved the final manuscript.
Yeh, MS., Wu, YC., Hung, MF. et al. Fabrication, characterization and simulation of Ω-gate twin poly-Si FinFET nonvolatile memory. Nanoscale Res Lett 8, 331 (2013). https://doi.org/10.1186/1556-276X-8-331
Twin poly-Si
Ω-gate
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Quiz: What Comes Next?
A popular toy among math lovers and others is made by arranging
27
small cubes into a larger
3 \times 3 \times 3
cube and then painting the outside.
(The outside is usually painted in a variety of colors, but for our purposes in today’s challenge, we can just consider one.)
Consider the smaller cubes — how many of them are painted on none of their faces? On one face, two faces, or three faces? Are any painted on more than three faces? How would our answers change if we made a bigger cube?
For answers to these questions, keep reading. Or, skip to today’s challenge for a bigger cube.
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If each triangle below is drawn accurately, identify whether it is most likely an obtuse, acute, or right triangle. Label the angle or angles that helped you to decide.
Examine all three angles of this figure.
Are the measures of any of the angles greater than or equal to
90°
The box on angle A indicates that angle A is a right angle.
The figure is therefore identified as a right triangle.
More (b):
Can you explain why this is an acute triangle?
This figure is an obtuse triangle because the measure of one of its angles is greater than
90°
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Combat moves - Ring of Brodgar
(Redirected from Combat Actions)
For combat moves to be effective, a player must raise either Unarmed Combat or Melee Combat, preferably both.
Heavy changes to the combat system were made in the update Bumfights (2016-09-21).
In order to learn a combat move, an animal must be defeated in melee combat. The move learned is based on the animal defeated, but otherwise appears to be random. (Minor note: During the first iteration of combat discoveries, prerequisites to use the move had to be met. As of world 10, this is not the case and you can pick up moves at random. Some players have had issues with getting a move like chop without getting a way to gain initiative points.)
There has been a widespread assumption that you need to have a weapon equipped for a chance to learn melee attacks. Currently (w13) this does not seem to be the case.
A note on "μ":
Quote from jorb:
"Spending more points on a combat effect increases the efficiency of said combat effect in that school. The meaning of increasing the weight of some particular combat effect is indicated in the descriptive text for that combat effect by a μ-symbol (Mu). The actual value of μ ranges from 1 to 1.5, depending on your weighting. Usually, a higher weight for some particular combat effect will serve to reduce its cooldown, increase its damage, or the like."
Moves in italics are available for free to hearthling when acquiring 'The Will to Power' skill.
Damage formula may be (confirm?)
{\displaystyle basedmg*movedmg*{\sqrt {{\sqrt {ql*str}}/10}}}
General formula for determining amount of opening seems to be:
{\displaystyle {\sqrt[{3}]{W_{a}/W_{d}}}\cdot O_{b}\cdot (1.0-O_{c})}
{\displaystyle W_{a}}
is attack weight,
{\displaystyle W_{d}}
is defense weight,
{\displaystyle O_{b}}
is base opening % from the move, and
{\displaystyle O_{c}}
is the current opening of that type in the target. Example: A player who has 50 uac, 25% green opening and is in lvl 4 Chin Up defense mode is attacked by a player with 100 uac by level 5 punch. Defense weight will be
{\displaystyle 50\cdot 1.4}
and attack weight will be
{\displaystyle 100*0.8*1.5}
. The opening that will be added to the defending player will thus be
{\displaystyle {\sqrt {\sqrt {120/70}}}\cdot 15\%\cdot (1.0-0.25)\approx 13\%}
and the total opening will be 38%
Some moves can be "used at a distance", meaning you can use them at any time, even when far from your foe, IE fleeing. Moves which cannot be "used at a distance" require you to advance up to your opponent as if you were attacking and touch them before they can be used.
Moves are unique abilites used to give you a special advantage.
Initiative cost
Useable at distance
Dash Opponents' initiative points: +2 80/μ Completely removes your slightest opening Yes
Opportunity Knocks 4 45 Opportunity Knocks increases your opponent's greatest opening by 40%·μ. No
Take Aim 0 (30 + IP·2)/μ Gives +1 initiative and can be carried out at a distance. The cooldown of Take Aim increases by 20% for each initiative point you have. Yes
Think 0+4 40/μ Gives +2 initiative and can be carried out at a distance. The cooldown of Think increases by 20% for each initiative point you have. Yes
Restorations will lower your openings, reducing your vulnerabilities and the damage you take. Some will perform other actions as well.
Artful Evasion
Opponents' initiative points: +1
20%·μ Striking
20%·μ Backhanded
20%·μ Sweeping
20%·μ Oppressive
Feigned Dodge
35 No For every unit of opening reduced by Feigned Dodge, twice that amount is given to the opponent.
0 Unarmed On opponent:
+15% Dizzy
Watch Its Moves
Initiative points: +1
45 No Gains you 1 point of initiative against your opponent.
0 On you:
+10% Reeling
Zig-Zag Ruse
Maneuvers affect your attack weight and which skills are used to calculate it, as well as often giving special buffs for hitting and being hit.
Bloodlust Unarmed·75%·μ 10 When attacked:
Bloodlust is charged by 25%·Δ. When you attack an opponent, your attack weight will be increased by four times the amount that Bloodlust is charged.
Chin Up Unarmed·μ 10
Combat Meditation Unarmed·μ 10 When attacked:
Combat Meditation is charged by 25%·Δ. When you attack an opponent, your cooldown will be decreased by the amount that Combat Meditation is charged.
While Combat Meditation is active, all your attacks will have 25% of their normal attack weight.
Death or Glory Unarmed·75%·μ 10 When attacked:
You gain 0.75·Δ points of Initiative against the opponent.
Oak Stance Unarmed·150%·μ 10 When attacked:
Your greatest opening is reduced by 5%·Δ.
While Oak Stance is active, all your attacks will have 50% of their normal attack weight.
Parry Melee·80%·μ 10 When Attacked:
Openings: +10% Dizzy
You need a sword equipped for Parry to inflict its effect upon your opponents.
Shield Up Melee·250%·μ 10 If Shield Up is used without a shield equipped, its block weight will be 50% instead of 250%.
To Arms Melee·μ 10
Attacks increase your opponent's openings and deal damage.
Grievous damage
1 Melee·μ
+15% Off Balance
According to weapon 40 Weapon:
Any edged weapon
4+2 Melee·μ
+25% Cornered
According to Weapon·150% 80
0 Melee·90%·μ
+5% Cornered
Full Circle attacks your main target and all other opponents in range.
2+2 Unarmed·μ
0 Unarmed·μ
Knock Its Teeth Out
20 30% 50 Gains you 1 Point of Initiative against your opponent.
0 Unarmed·80%·μ
Punch 'em Both
10 7.5% 40 Attacks both your primary target and also one other opponent in range.
According to Weapon·25% 20 Weapon:
If your opponent has more than 25% of Oppressive openings, Quick Barrage also gains you 1 Point of Initiative against that opponent.
According to weapon·110% 40 Weapon:
+7.5% Off Balance
+7.5% Dizzy
+7.5% Reeling
+7.5% Cornered
Very effective when paired with Uppercut that creates opening for this move to deal damage effectively.
None 40 To the extent that it is unblocked, Steal Thunder will take 3 points of initiative from its target and you will gain 2 of them. Usable at a small distance, opponent must have 3 or more IP for you to be able to use this move.
Any pointed weapon
Storm of Swords will attack up to five opponents in range, starting with your main target. The targets will receive 100%, 125%, 150%, 175%, and 200%, respectively, of the weapon's damage.
0 Unarmed·0.8·μ
30 5% 30 Useless as the only offensive move due to mismatching colors, but effective at creating Yellow opening for moves like Sideswipe.
Bagpipe Brawl (2016-06-20) >"Added "Sidestep" move. Costs 5IP to remove 20+ % Defense."
Valhalla Rising (2015-10-02) >"Added "Shield" block."
Valhalla Rising (2015-10-02) >"Added "Parry" block."
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Combat_moves&oldid=93791"
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Strong‐Motion Observations of the 2017 Ms 7.0 Jiuzhaigou Earthquake: Comparison with the 2013 Ms 7.0 Lushan Earthquake | Seismological Research Letters | GeoScienceWorld
Strong‐Motion Observations of the 2017 Ms 7.0 Jiuzhaigou Earthquake: Comparison with the 2013 Ms 7.0 Lushan Earthquake
Yefei Ren;
Yefei Ren
Key Laboratory of Earthquake Engineering and Engineering Vibration of China Earthquake Administration, Institute of Engineering Mechanics, China Earthquake Administration, No. 29 Xuefu Road, Harbin 150080, China, renyefei@iem.ac.cn, whw1990413@163.com, xupeibin13@126.com, ruizhi@iem.ac.cn, maqiang@iem.ac.cn
Peibin Xu;
Yadab P. Dhakal;
National Research Institute for Earth Science and Disaster Resilience, 3‐1, Tennodai, Tsukuba, Ibaraki 305‐0006, Japan, ydhakal@bosai.go.jp
Ruizhi Wen;
Sichuan Earthquake Administration, No 29, South Renmin Road, Chengdu 610000, China, jiang_0057@163.com
Yefei Ren, Hongwei Wang, Peibin Xu, Yadab P. Dhakal, Ruizhi Wen, Qiang Ma, Peng Jiang; Strong‐Motion Observations of the 2017
Ms
7.0 Jiuzhaigou Earthquake: Comparison with the 2013
Ms
7.0 Lushan Earthquake. Seismological Research Letters 2018;; 89 (4): 1354–1365. doi: https://doi.org/10.1785/0220170238
Strong‐motion recordings observed during the
Ms
7.0 Jiuzhaigou earthquake, which occurred on 8 August 2017 in western China, were used to reveal the underlying source, propagation path, and site effects in comparison with observations from the 2013 Lushan earthquake of identical magnitude (
Ms
7.0). The similar
VS30
distributions at the strong‐motion stations considered in both events reflect approximately consistent site effects. Amplitudes of short‐ and intermediate‐period ground motions (e.g., peak ground accelerations, peak ground velocities, and pseudospectral accelerations [PSAs] at
T=0.2
and 2.0 s) observed in the Jiuzhaigou event were smaller than in the Lushan earthquake at sites with a Joyner–Boore distance (
RJB
) of
<200 km
, but their amplitudes were similar at far‐field sites (
RJB>200 km
). However, irrespective of
RJB
, larger amplitudes of long‐period ground motions (e.g., PSAs at
T=5.0 s
) were observed in the Jiuzhaigou earthquake compared with the Lushan event. This study revealed that different fault styles (strike slip for the Jiuzhaigou event and reverse slip for the Lushan event), and different centroid depths could be two reasons for the period‐dependent differences of ground motions between the two earthquakes. Analysis of within‐event residuals suggested that the Jiuzhaigou region exhibited slower anelastic attenuation for short‐ and intermediate‐period ground motions than the Lushan region, but the anelastic attenuations for long‐period ground motions of both regions were similar. The durations of ground motions generated by both earthquakes were found comparable, as were the dependences on distance in both seismogenic regions.
Huya Fault
Lushan earthquake 2013
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Park Transformation - MATLAB & Simulink - MathWorks Nordic
The Park transformation used in Simscape™ Electrical™ Specialized Power Systems models and functions corresponds to the definition provided in [1].
It transforms three quantities (direct axis, quadratic axis, and zero-sequence components) expressed in a two-axis reference frame back to phase quantities.
The following transformation is used:
\begin{array}{c}{V}_{a}={V}_{d}\mathrm{sin}\left(\omega t\right)+{V}_{q}\mathrm{cos}\left(\omega t\right)+{V}_{0}\\ {V}_{b}={V}_{d}\mathrm{sin}\left(\omega t-2\pi /3\right)+{V}_{q}\mathrm{cos}\left(\omega t-2\pi /3\right)+{V}_{0}\\ {V}_{c}={V}_{d}\mathrm{sin}\left(\omega t+2\pi /3\right)+{V}_{q}\mathrm{cos}\left(\omega t+2\pi /3\right)+{V}_{0},\end{array}
where ω = rotation speed (rad/s) of the rotating frame.
The following reverse transformation is used:
\begin{array}{c}{V}_{d}=\frac{2}{3}\left({V}_{a}\mathrm{sin}\left(\omega t\right)+{V}_{b}\mathrm{sin}\left(\omega t-2\pi /3\right)+{V}_{c}\mathrm{sin}\left(\omega t+2\pi /3\right)\right)\\ {V}_{q}=\frac{2}{3}\left({V}_{a}\mathrm{cos}\left(\omega t\right)+{V}_{b}\mathrm{cos}\left(\omega t-2\pi /3\right)+{V}_{c}\mathrm{cos}\left(\omega t+2\pi /3\right)\right)\\ {V}_{0}=\frac{1}{3}\left({V}_{a}+{V}_{b}+{V}_{c}\right),\end{array}
The transformations are the same for the case of a three-phase current; you simply replace the Va, Vb, Vc, Vd, Vq, and V0 variables with the Ia, Ib, Ic, Id, Iq, and I0 variables.
[1] Krause, P. C. Analysis of Electric Machinery. New York: McGraw-Hill, 1994, p.135.
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Fungi - Marspedia
Fungi are eukaryotic organisms closely related to animals. The study of fungi is known as mycology.
Some fungi are edible and could be grown for food in a Martian colony. Fungi, especially some sorts of mildew, are certainly of great use in the artificial ecosystem of a greenhouse, since they are involved in the decaying process.
Some people have an allergy against the spores of mildew, and patients with depressed immune system can suffer from a fungus infection. As yet, the development of the human immune system under Martian gravity and inhouse conditions is unclear.
2.2 Genus Amanita
2.2.2 Amanita citrina
2.2.3 Amanita pantherina
2.2.4 Amanita rubescens
2.2.5 Amanita virosa
2.2.7 Amanita jacksonii
Fungal cells have cell walls which may be composed of several glucose polymers. For example, chitin (chains of acetylglucosamine joined by
{\displaystyle \beta (1\rightarrow 4)}
links) or, as in yeast, callose (which is a glucose chain which differs from cellulose only in that it uses
{\displaystyle \beta (1\rightarrow 3)}
links where cellulose uses
{\displaystyle \beta (1\rightarrow 4)}
links[1].
Like all protozoa, animals and plants, fungi have cytoskeletal structures containing microtubules and microfilaments supporting the cytoplasm inside their cells[1]. In some fungi there are also intermediate filaments, though not to the same extent as in animals[1].
The microtubules can be thought of as analogous to the human skeleton, as they maintain the positions of major organelles within the cell[1] Furthermore, the microfilaments and microtubules are responsible for movement in eukarytic cells. This is achieved when the individual polymers slide across one another to create an overall lengthening or contraction of the structure[1].
It is worth noting that microgravity affects the cytoskeleton[2], so fungi could, in principle, be negatively affected by a journey to Mars. While the matter is of academic interest, fungi would almost certainly be much less harmed by a return to gravity than humans and no mission would therefore be compromised. It is even possible that fungi may grow more readily in microgravity, as some primitive eukaryotes do[2], though that is unlikely as the positive effect is at least mostly due to reduced energy expenditure during movement.
Mushrooms are the "fruiting bodies" of certain fungi or, more formally, the sporulating organs of certain complex fungi. The word toadstool is sometimes used in informal conversation to refer to inedible mushrooms. Mushrooms are complete proteins, and their nutritional value are in some respects (such as protein quality) in-between those of plant and animal foods. This makes them a potentially very valuable food source in spaceflight.
WIP: Explain the terminology used below.
Some mushrooms form inside a structure known as a "universal veil", a little bag which covers the entire mushroom and tears open as it grows. The universal veil is important in identification. Remnants of the base of the universal veil (known as the volval bag) are present in some species. Remnants of the top of the veil may remain as rough spots on the top of the cap, possibly a different colour (as in the culturally iconic white-spotted red mushroom, A. muscaria).
Amanita spp. covers a number of edible mushrooms, as well as some of the most toxic known fungi. The Death Cap alone is responsible for more than 90% of mushroom-related deaths on Earth. Notable members of the genus include:
Common name: Death Cap.
Deadly toxic to the liver and kidneys. (Lethal dose about 30g for adult humans[3].) Symptoms of poisoning include vomiting, diarrhoea, thirst and severe abdominal pain[3][4]. Symptoms will appear between 6 and 24 hours after ingestion[3]. If Amanita poisoning is not identified, the victim may appear to recover and die several days later from liver and/or kidney failure. Treatment includes carbon column dialysis, saline cathartic, repeated doses of activated charcoal and blood transfusions[3][4]. Fatal more often than not[3].
Identifying features[3][4]: Like all Amanitas, the Death Cap has a white spore print and forms inside a universal veil. The cap is smooth and flattens with age, growing to a maximum of about 15cm across. The ring is persistent, white and membranous. The gills are white, free and crowded. Pronounced volval bag. The flesh is white with a faint yellow tinge and the cap appears a slightly yellowish, greenish or smoky-olive white. A. phalloides var. alba is especially notable in that this rarer almost pure-white variety can be easily misidentified. The smell is described as "sickly sweet" to "foetid" and it is reported to have a pleasant taste.
Common name: False Death Cap.
Common name: Panther Cap.
Common name: Blusher.
Edible when cooked. Toxic otherwise.
Description: To do.
Similar species: Non-experts may easily confuse A. rubescens with A. pantherina. To do: describe differences.
Common name: Destroying Angel.
Deadly toxic.
Commonly known as Fly Amanita or Fly Agaric.
Edible if correctly prepared[5], despite the fact that field guides usually[3][4] (incorretly) list it as inedible. Detoxification involves boiling out the water-soluble toxins in water and either vinegar or salt, discarding the water[5]. If not detoxified, A. muscaria is hallucinogenic and causes euphoria similar to alcohol intoxification[4]. Relatively few fatalities have been recorded over several centuries and the lethal dose is not exactly known. The historical evidence collected by Rubel and Arora[5] suggests an adult lethal dose somewhere in the vicinity of 12-20 untreated mushrooms. Symptoms may persist for several days in the most extreme cases[4]
Trivia: A. muscaria was used for centuries to kill flies. After the crushed mushrooms have been mixed with milk, flies drinking the milk will become so intoxicated that they drown[3][5]. The Sami of Lapland sometimes scatter dried A. muscaria for their reindeer, as the intoxicating effect makes them easier to round up[4].
↑ 1.0 1.1 1.2 1.3 1.4 S.L. Wolfe - Mollecular and cellular biology 1993. Wadsworth. ISBN 0-534-12408-9. pp. 17-18, 288-289, 514-517.
↑ 2.0 2.1 M.L. Lewis & M. Hughes-Fulford - Cellular responses to spaceflight 1997. (In S.E. Churchill, ed. - Fundamentals of space life sciences. Krieger. ISBN 0-89464-051-8) Vol 1. pp. 21-36.
↑ 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 M. Branch -- First field guide to mushrooms of Southern Africa 2001. Struik Nature. ISBN 978-1-86872-605-9. pp. 12-15.
↑ 4.0 4.1 4.2 4.3 4.4 4.5 4.6 P. Jordan -- The mushroom guide and identifier. 2010. Hermes House. ISBN 978-1-84038-574-8. pp. 100-113.
↑ 5.0 5.1 5.2 5.3 W. Rubel & D. Arora -- A study of cultural bias in field guide determinations of mushroom edibility using the iconic mushroom, Amanita muscaria, as an example. 2008. Economic Botany vol. 62 no. 3. pp. 223-243. Available here and here.
Retrieved from "https://marspedia.org/index.php?title=Fungi&oldid=127421"
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Minimum loop gain constraint for control system tuning - MATLAB - MathWorks América Latina
TuningGoal.MinLoopGain class
Minimum Loop Gain Tuning Goal
Integral Minimum Gain Specified as Gain Value at Single Frequency
Minimum Loop Gain as Constraint on Sensitivity Function
Minimum loop gain constraint for control system tuning
Use the TuningGoal.MinLoopGain object to enforce a minimum loop gain in a particular frequency band. Use this tuning goal with control system tuning commands such as systune or looptune.
This tuning goal imposes a minimum gain on the open-loop frequency response (L) at a specified location in your control system. You specify the minimum open-loop gain as a function of frequency (a minimum gain profile). For MIMO feedback loops, the specified gain profile is interpreted as a lower bound on the smallest singular value of L.
When you tune a control system, the minimum gain profile is converted to a minimum gain constraint on the inverse of the sensitivity function, inv(S) = (I + L).
The following figure shows a typical specified minimum gain profile (dashed line) and a resulting tuned loop gain, L (blue line). The shaded region represents gain profile values that are forbidden by this tuning goal. The figure shows that when L is much larger than 1, imposing a minimum gain on inv(S) is a good proxy for a minimum open-loop gain.
TuningGoal.MinLoopGain and TuningGoal.MaxLoopGain specify only low-gain or high-gain constraints in certain frequency bands. When you use these tuning goals, systune and looptune determine the best loop shape near crossover. When the loop shape near crossover is simple or well understood (such as integral action), you can use TuningGoal.LoopShape to specify that target loop shape.
Req = TuningGoal.MinLoopGain(location,loopgain) creates a tuning goal for boosting the gain of a SISO or MIMO feedback loop. The tuning goal specifies that the open-loop frequency response (L) measured at the specified locations exceeds the minimum gain profile specified by loopgain.
You can specify the minimum gain profile as a smooth transfer function or sketch a piecewise error profile using an frd model or the makeweight (Robust Control Toolbox) command. Only gain values greater than 1 are enforced.
For MIMO feedback loops, the specified gain profile is interpreted as a lower bound on the smallest singular value of L.
Req = TuningGoal.MinLoopGain(location,fmin,gmin) specifies a minimum gain profile of the form loopgain = K/s (integral action). The software chooses K such that the gain value is gmin at the specified frequency, fmin.
If location is a cell array of loop-opening locations, then the minimum gain goal applies to the resulting MIMO loop.
Minimum open-loop gain as a function of frequency.
You can specify loopgain as a smooth SISO transfer function (tf, zpk, or ss model). Alternatively, you can sketch a piecewise gain profile using a frd model or the makeweight (Robust Control Toolbox) command. For example, the following frd model specifies a minimum gain of 100 (40 dB) below 0.1 rad/s, rolling off at a rate of –20 dB/dec at higher frequencies.
Only gain values larger than 1 are enforced. For multi-input, multi-output (MIMO) feedback loops, the gain profile is interpreted as a lower bound on the smallest singular value of L. For more information about singular values, see sigma.
Frequency of minimum gain gmin, specified as a scalar value in rad/s.
Use this argument to specify a minimum gain profile of the form loopgain = K/s (integral action). The software chooses K such that the gain value is gmin at the specified frequency, fmin.
Value of minimum gain occurring at fmin, specified as a scalar absolute value.
Minimum open-loop gain as a function of frequency, specified as a SISO zpk model.
The software automatically maps the input argument loopgain onto a zpk model. The magnitude of this zpk model approximates the desired gain profile. Alternatively, if you use the fmin and gmin arguments to specify the gain profile, this property is set to K/s. The software chooses K such that the gain value is gmin at the specified frequency, fmin.
Use viewGoal(Req) to plot the magnitude of the open-loop minimum gain profile.
Create a tuning goal that boosts the open-loop gain of a feedback loop to at least a specified profile.
Suppose that you are tuning a control system that has a loop-opening location identified by PILoop. Specify that the open-loop gain measured at that location exceeds a minimum gain of 10 (20 dB) below 0.1 rad/s, rolling off at a rate of -20 dB/dec at higher frequencies. Use an frd model to sketch this gain profile.
The dashed line shows the specified the gain profile. The shaded region indicates where the tuning goal is violated, except that gain values less than 1 are not enforced. Therefore, this tuning goal only specifies a minimum gain at frequencies below 1 rad/s.
Create a tuning goal that specifies a minimum loop gain profile of the form L = K / s. The gain profile attains the value of -20 dB (0.01) at 100 rad/s.
viewGoal confirms that the tuning goal is correctly specified. You can use this tuning goal to tune a control system that has a loop-opening location identified as 'X'. Since loop gain values less than 1 are ignored, this tuning goal specifies minimum gain only below 1 rad/s, with no restriction on loop gain at higher frequency.
Although the specified gain profile (dashed line) is a pure integrator, for numeric reasons, the gain profile enforced during tuning levels off at very low frequencies, as described in Algorithms. To see the regularized gain profile, expand the axes of the tuning-goal plot.
Examine a minimum loop gain tuning goal against the tuned loop gain. A minimum loop gain tuning goal is converted to a constraint on the gain of the sensitivity function at the location specified in the tuning goal.
To see this relationship between the minimum loop gain and the sensitivity function, tune the following closed-loop system with analysis points at X1 and X2. The control system has tunable PID controllers C1 and C2.
Create a model of the control system.
Specify some tuning goals, including a minimum loop gain. Tune the control system to these requirements.
Examine the TuningGoal.MinLoopGain goal against the corresponding tuned response.
The plot shows the achieved loop gain for the loop at X2 (blue line). The plot also shows the inverse of the achieved sensitivity function, S, at the location X2 (green line). The inverse sensitivity function at this location is given by inv(S) = I+L. Here, L is the open-loop point-to-point loop transfer measured at X2.
The minimum loop gain goal Rgain is constraint on inv(S), represented in the plot by the green shaded region. The constraint on inv(S) can be thought of as a minimum gain constraint on L that applies where the gain of L (or the smallest singular value of L, for MIMO loops) is greater than 1.
For TuningGoal.MinLoopGain, f(x) is given by:
f\left(x\right)={‖{W}_{S}\left({D}^{-1}SD\right)‖}_{\infty }.
Here, D is a diagonal scaling (for MIMO loops). S is the sensitivity function at Location. WS is a frequency-weighting function derived from the minimum loop gain profile, MinGain. The gain of this function roughly matches MaxGain for values ranging from –20 dB to 60 dB. For numerical reasons, the weighting function levels off outside this range, unless the specified gain profile changes slope outside this range. This adjustment is called regularization. Because poles of WS close to s = 0 or s = Inf might lead to poor numeric conditioning of the systune optimization problem, it is not recommended to specify gain profiles with very low-frequency or very high-frequency dynamics.
Although S is a closed-loop transfer function, driving f(x) < 1 is equivalent to enforcing a lower bound on the open-loop transfer function, L, in a frequency band where the gain of L is greater than 1. To see why, note that S = 1/(1 + L). For SISO loops, when |L| >> 1, |S | ≈ 1/|L|. Therefore, enforcing the open-loop minimum gain requirement, |L| > |WS|, is roughly equivalent to enforcing |WsS| < 1. For MIMO loops, similar reasoning applies, with ||S|| ≈ 1/σmin(L), where σmin is the smallest singular value.
For an example illustrating the constraint on S, see Minimum Loop Gain as Constraint on Sensitivity Function.
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The lemonade stand at the county fair sells the lemonade at a price of two cups for
\$3.60
. Complete the table at right to find what Paula’s family will pay to buy lemonade for all eight members of the family.
\begin{array}{c|c} {\quad \;\text{# of} \\ \text{Lemonades}} & {\; \\ \quad \text{Price (\$)} \quad } \\ \hline 1 \\ 2 & 3.60 \\ 3 \\ 4 \\ 5 \\ 6 \\ 7 \\ 8 \end{array}
2
cups of lemonade is twice as much as
1
cup of lemonade, to find the price of
1
cup of lemonade, you must find half of the price of
2
cups of lemonade.
You can also solve the problem by setting up a proportion like the one below, then solve for
x
\large\frac{\text{2 cups of lemonade}}{\text{3.60 dollars}} \left( \frac{?}{?} \right) = \frac{\text{1 cup of lemonade}}{x\text{ dollars}}
To find the price of any amount of lemonade, in the equation above substitute ''
1
cup of lemonade'' with the number of lemonades you are looking for.
\begin{array}{c|c} {\quad \;\text{# of} \\ \text{Lemonades}} & {\; \\ \quad \text{Price (\$)} \quad } \\ \hline 1 & \mathbf{1.80}\\ 2 & 3.60 \\ 3 & \mathbf{5.40}\\ 4 & \mathbf{7.20}\\ 5 \\ 6 \\ 7 \\ 8 \end{array}
|
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