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Adaptive multiple subtraction based on support vector regressionSVR-based AMS | Geophysics | GeoScienceWorld
Adaptive multiple subtraction based on support vector regression
Zhong-xiao Li
, School of Electronic Information, Department of Electronic Engineering, Qingdao,
. E-mail: thulzx@163.com.
Zhong-xiao Li; Adaptive multiple subtraction based on support vector regression. Geophysics 2019;; 85 (1): V57–V69. doi: https://doi.org/10.1190/geo2018-0245.1
Multiple removal is essential for seismic imaging in marine seismic processing. After prediction of multiple models, adaptive multiple subtraction is an important procedure for multiple removal. Generally, adaptive multiple subtraction can be conducted by the iterative reweighted least-squares (IRLS) algorithm with an
L1
-norm minimization constraint of primaries. We have developed a machine-learning algorithm into adaptive multiple subtraction, which is implemented based on support vector regression (SVR). Our SVR-based method contains training and prediction stages. During the training stage, an SVR function is estimated by solving a dual optimization problem with the feature vectors of the predicted multiples and the target values of the original data. The SVR function can transform predicted multiples nonlinearly for a better match between the predicted multiples and the true multiples. Furthermore, we use the SVR function to estimate multiples in the prediction stage by inputting the feature vectors of predicted multiples. Then, multiple-removal results are obtained by subtracting the estimated multiples directly from the original data. Synthetic and field data examples demonstrate that our SVR-based method can better balance multiple removal and primary preservation than the IRLS-based method.
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(Created page with "This page describes how to document an ApCoCoA-2 package in the Wiki. Please note that in order to do this, you need a wiki account. Such an account can only be created by a w...")
This page describes how to document an ApCoCoA-2 package in the Wiki. Please note that in order to do this, you need a wiki account. Such an account can only be created by a wiki admin, so please contact the [[Team|ApCoCoA team]] so we can create an account for you.
{\displaystyle x_{1}^{2}}
{\displaystyle \sum _{i=1}^{\infty }\left({\frac {1}{2}}\right)^{n}}
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Spatiotemporal Variation of Crustal Anisotropy in the Source Area of the 2004 Niigata, Japan EarthquakeSpatiotemporal Variation of Crustal Anisotropy in the Source Area of the 2004 Niigata, Japan Earthquake | Bulletin of the Seismological Society of America | GeoScienceWorld
Spatiotemporal Variation of Crustal Anisotropy in the Source Area of the 2004 Niigata, Japan Earthquake
Lingmin Cao;
CAS Key Laboratory of Ocean and Marginal Sea Geology, South China Sea Institute of Oceanology, Chinese Academy of Sciences, 164 West Xingang Road, Guangzhou 510301, China, cao_lingmin@126.com
Also at Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou), Guangzhou, China.
Pacific Geoscience Centre, Geological Survey of Canada, Natural Resources Canada, 9860 West Saanich Road, Sidney, British Columbia, Canada V8L 4B2, honn.kao@canada.ca
Kelin Wang;
Chuanxu Chen;
Chuanxu Chen
Institute of Deep‐Sea Science and Engineering, Chinese Academy of Sciences, Luhuitou 28, Jiyang District, Sanya 572000, China
Jim Mori;
Disaster Prevention Research Institute, Kyoto University, Gokasho, Uji, Kyoto 611‐0011, Japan
Shiro Ohmi;
Institute of Earthquake Forecasting, China Earthquake Administration, 63 Fuxing Road, Beijing 100036, China
Lingmin Cao, Honn Kao, Kelin Wang, Chuanxu Chen, Jim Mori, Shiro Ohmi, Yuan Gao; Spatiotemporal Variation of Crustal Anisotropy in the Source Area of the 2004 Niigata, Japan Earthquake. Bulletin of the Seismological Society of America 2019;; 109 (4): 1331–1342. doi: https://doi.org/10.1785/0120180195
We investigate the spatiotemporal pattern of crustal anisotropy in the source area of the 2004 Niigata earthquake (M 6.8) that occurred in the northern segment of the Niigata–Kobe tectonic zone, central Japan, by measuring shear‐wave splitting parameters from waveform data of local earthquakes. Our results show that the fast polarization directions in the upper crust have spatial variations across the region of the earthquake that are likely caused by both structural and stress field effects. The northwest–southeast direction near the northeastern end of the source zone (beneath station N.NGOH) and the east–west direction to the southwest (beneath station N.KWNH) are consistent with the spatial variation of the orientation of the maximum compression of the local stress field. Fast polarization directions at other stations tend to align in the directions of active faults and folds and thus are considered to be structure induced. These spatial patterns were unaffected by the earthquake. However, at two stations (N.NGOH and N.KWNH) we observe an increase in both the average and scatter of the normalized delay times (
δt
) during the aftershock period. In addition, two stations (HIROKA and N.YNTH) that are located in the strike‐normal direction east of the source area show an increase in the average of the normalized
δt
and a rotation of up to 90° of the fast direction immediately after the mainshock. We also notice that stations located very close to the source fault (DP.YMK and DP.OJK) show larger average delay times compared with stations farther away (HIROKA and N.YNTH) during the postseismic stage. To explain the temporal changes in the strength of the anisotropy, we speculate that spatiotemporal variations in microcrack development in and around the source area could be caused by static stress changes due to tectonic deformation and the earthquake rupture.
mid-Niigata earthquake 2004
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Iraq » Illogicopedia - The nonsensical encyclopedia anyone can mess up
“Oh, Quagmire!”
This article was written by celebrity guest writer Mr. M. Boosh, yo.
This article is about Iraq, please do not shout otherwise you will be executed excited
"Saddam, this is not the time for your rendition of 'Tambourine Man'".
The Land of Weapons of Mass Destruction aka Iraq is a smallish country which was recently liberted from the scrooge of Saddam Hussein following a war fought by America and its allies. This was to liberate the people of Iraq and ensure the safety of the Kurds, the Shiites and the Israelis after Saddam Hussein refused to allow the United Nations to check for Weapons of Mass Destruction. This war saved the free world from this tyrant and his plans to destroy it with Weapons of Mass Destruction however none were found following the war. Nevertheless the Americans took the sage precaution of arresting Saddam Hussein and handing him to an Iraqi court for trial for one of his numerous crimes, lest he do something like 9/11 again. He was convicted, sentenced to death by hanging and subsquently executed.
However, the spirit of Saddam Hussein lives on. As a result there is a large scale insurgency going on in the country, perputrated not only by pro-Saddam Sunnis but also by Islamic State and Shiite insurgents, who are all fighting for their causes against us and the Iraqi government. This means that in Iraq which is a smallish country der is a war goin on in and the Iraqi government, its security forces and the United States are all fighting for liberty an justice an' shit like dat, lol
3 TALIBANNORAMA!
4 Ekewayzun
5 World's view on Iraq
6 S'y ahso
This iz da national
Anthem o iraq! BARK BARK BARK
Do eeh doo ehhh do eee do eee dooo deeee uh bu uh
Bu uh bu uh bu uh booo booooo boooooo
At the moment[edit]
We are fighting a huge war against those Iraqienz! They, like supporting world terrorism and shit like that. Stop it you lot!
I like the Scottish regiment out there, cos they get to wear kilts while shootin' lol, an shit like dat, lol. But, still if you get any Taliban in your hole then clean your hole out. With your finger. On a tree. Hole. B?
TALIBANNORAMA![edit]
The Taliban and der muhadgedin are well cool! They have like turbanz on and dey like to shoot things and things like that well cool! I fancy them and I might aswell fancy myself seeing as a like ak47s. Liquid failure, ooooh pee.
Ekewayzun[edit]
{\displaystyle Iraq+Iraq=}
dubble trubble.
World's view on Iraq[edit]
“We are not to bovvered yar? Got any male rent boys or summent? I'm Dutch!?”
~ Dootch on being gay
“Where Saddam used to keep his CDs. Heh heh heh heh heh heh.”
~ George Bush on on Iraq
Trouble for luvvle brubble OH PREE?
S'y ahso[edit]
Retrieved from "http://en.illogicopedia.org/w/index.php?title=Iraq&oldid=284231"
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What are the measures of all the angles in the diagram?
The triangles in the diagram are isosceles, therefore their base angles are congruent.
Cheri and Roberta notice that their similarity statements for part (b) are not the same. Cheri stated
ΔABC
ΔDEC
, while Roberta wrote
ΔABC
ΔEDC
. Who is correct? Or are they both correct? Explain your reasoning.
They are both correct. Since both triangles are isosceles, we cannot tell if one is the reflection or the rotation of the other (after dilation).
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Which one is product of aerobic respiration
If the breathing rate and blood flow cannot supply enough oxygen to your working muscles, the body turns to anaerobic respiration. This is the production of energy without the use of oxygen. This system works by producing lactic acid to facilitate energy production.
The chemical nature of hormones secreted by
\alpha
\beta
cells of pancreas is
Glucagon is a paeptide hormone synthesized and secreted from α-cells of the islet of langerhans and Insulin is a peptide hormone synthesized and secreted from
\beta
-cells of the islet of langerhans.
The average diameter of red blood corpuscles of man is
Normal RBCs have a diameter of 6 - 7.5 μm. On a peripheral blood smear, normal RBCs are disc-shaped with a pale-staining central area called the central pallor. When judging red cell size on a blood smear, the classic rule of thumb is to compare them to the nucleus of a small normal lymphocyte
Virus was discovered by whom
Ivanoski reported in 1892 that extracts from infected leaves were still infectious after filtration through a Chamberland filter-candle. Bacteria are retained by such filters, a new world was discovered: filterable pathogens. However, Ivanovski probably did not grasp the full meaning of his discovery. Beijerinck, in 1898, was the first to call 'virus', the incitant of the tobacco mosaic. He showed that the incitant was able to migrate in an agar gel, therefore being an infectious soluble agent, or a 'contagium vivum fluidum' and definitively not a 'contagium fixum' as would be a bacteria. Ivanovski and Beijerinck brought unequal but decisive and complementary contributions to the discovery of viruses.
In which stage of cell division chromosomes are most condensed
'Anaphase', is the stage of mitosis after the metaphase when replicated chromosomes are split and the daughter chromatids are moved to opposite poles of the cell. Chromosomes also reach their overall maximum condensation in late anaphase, to help chromosome segregation and the re-formation of the nucleus.
FAD is electron acceptor during oxidation of which of the following
ketoglutarate ➔ Succinyl Co-A
Succinic acid➔ Fumaric acid
Succinyl Co-A ➔ Succinic acid
Fumaric acid ➔ Malic acid
In biochemistry, flavin adenine dinucleotide (FAD) is a redopx-active coenzyme associated with various protiens, which is involved with several important enzymatic reactions in metabolism. A flavoprotien is a protein that contains a flavin group, this may be in the form of FAD or flavin mononucleotide(FMN).
Electron microscope is based on the principle of
resolution of glass lenses
magnification of glass lenses
An electron microscope uses an 'electron beam' to produce the image of the object and magnification is obtained by 'electromagnetic fields'; unlike light or optical microscopes, in which 'light waves' are used to produce the image and magnification is obtained by a system of 'optical lenses'.
How many ATP molecules are obtained from fermentation of 1 molecule of glucose
The net energy gain in fermentation is 2 ATP molecules/glucose molecule. In both lactic acid and alcoholic fermentation, all the NADH produced in glycolysis is consumed in fermentation, so there is no net NADH production, and no NADH to enter the ETC and form more ATP.
CO2 acceptor in C3 cycle is
Primary carbon dioxide acceptor in C4 plant is phosphoenol pyruvic acid,it contains 3 carbon atoms. in C3 plants ribulose 1,5 bisphosphate (RUBP) is the primary CO2 acceptor. It is a 5 carbon compound. In C4 plants phospho enol pyruvate (PEP) is primary CO2 acceptor.
Citric acid cycle is the alternate name of which of the following
The tricarboxylic acid cycle (TCA cycle) is a series of enzyme-catalyzed chemical reactions that form a key part of aerobic respiration in cells. This cycle is also called the Krebs cycle and the citric acid cycle.
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A Magnetic Resonance-Compatible Loading Device for Dynamically Imaging Shortening and Lengthening Muscle Contraction Mechanics | J. Med. Devices | ASME Digital Collection
Christopher J. Westphal,
Christopher J. Westphal
Department of Biomedical Engineering, Department of Mechanical Engineering, and Department of Orthopedics and Rehabilitation,
e-mail: thelen@engr.wisc.edu
Silder, A., Westphal, C. J., and Thelen, D. G. (September 3, 2009). "A Magnetic Resonance-Compatible Loading Device for Dynamically Imaging Shortening and Lengthening Muscle Contraction Mechanics." ASME. J. Med. Devices. September 2009; 3(3): 034504. https://doi.org/10.1115/1.3212559
The purpose of this study was to design and test a magnetic resonance (MR)-compatible device to induce either shortening or lengthening muscle contractions for use during dynamic MR imaging. The proposed device guides the knee through cyclic flexion-extension, while either elastic or inertial loads are imposed on the hamstrings. Ten subjects were tested in a motion capture laboratory to evaluate the repeatability of limb motion and imposed loads. Image data were subsequently obtained for all ten subjects using cine phase contrast imaging. Subjects achieved
∼30 deg
of knee joint motion, with individual subjects remaining within
∼1 deg
of their average motion across 56 repeated cycles. The maximum hamstring activity and loading occurred when the knee was flexed for the elastic loading condition (shortening contraction), and extended for the inertial loading condition (lengthening contraction). Repeat MR image acquisitions of the same loading condition resulted in similar tissue velocities, while spatial variations in velocity data were clearly different between loading conditions. The proposed device can enable dynamic imaging of the muscle under different types of loads, which has the potential to improve our understanding of basic muscle mechanics, identify potential causes of muscle injury, and provide a basis for quantitatively assessing injury effects at the tissue level. Slight modifications to the device design and/or subject positioning could allow for imaging of the quadriceps or the knee.
biological tissues, biomechanics, biomedical MRI, muscle
Biological tissues, Cycles, Imaging, Knee, Muscle, Stress, Resonance
Imaging Two-Dimensional Displacements and Strains in Skeletal Muscle During Joint Motion by Cine DENSE MR
Cine Phase Contrast MRI to Measure Continuum Lagrangian Finite Strain Fields in Contracting Skeletal Muscle
Cine Phase-Contrast Magnetic Resonance Imaging as a Tool for Quantification of Skeletal Muscle Motion
Semin. Musculoskelet. Radiol.
Skeletal Muscle Contraction: Analysis With Use of Velocity Distributions From Phase-Contrast MR Imaging
Hodics
MR Compatible Force Sensing System for Real-Time Monitoring of Wrist Moments During FMRI Testing
Muscle Strain Injury: Diagnosis and Treatment
Neuromusculoskeletal Models Provide Insights Into the Mechanisms and Rehabilitation of Hamstring Strains
Magnetic Resonance Safety Update 2002: Implants and Devices
A Musculoskeletal Model of the Human Lower Extremity: The Effect of Muscle, Tendon, and Moment Arm on the Moment-Angle Relationship of Musculotendon Actuators at the Hip, Knee, and Ankle
Aletras
DENSE: Displacement Encoding With Stimulated Echoes in Cardiac Functional MRI
Estimation of Deformation Gradient and Strain From Cine-PC Velocity Data
Real-Time Imaging of Skeletal Muscle Velocity
MR Observations of Long-Term Musculotendon Remodeling Following a Hamstring Strain Injury
Dynamic Magnetic Resonance Imaging of Muscle Function After Surgery
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Rational functions and their manipulations | ePractice - HKDSE 試題導向練習平台
Question Sample Titled 'Rational functions and their manipulations'
A rational function is a function that can be written as
\dfrac{{P}}{{Q}}
{P}
{Q}
{Q}\ne{0}
\dfrac{{{2}{x}-{3}}}{{{x}-{1}}}\div\dfrac{{{4}{x}^{{2}}-{9}}}{{{x}^{{2}}+{x}-{2}}}+\dfrac{{1}}{{x}}
\dfrac{{{2}{x}-{3}}}{{{x}-{1}}}\div\dfrac{{{4}{x}^{{2}}-{9}}}{{{x}^{{2}}+{x}-{2}}}+\dfrac{{1}}{{x}}
=\dfrac{{{2}{x}-{3}}}{{{x}-{1}}}\times\dfrac{{{x}^{{2}}+{x}-{2}}}{{{4}{x}^{{2}}-{9}}}+\dfrac{{1}}{{x}}
=\dfrac{{{2}{x}-{3}}}{{{x}-{1}}}\times\dfrac{{{\left({x}-{1}\right)}{\left({x}+{2}\right)}}}{{{\left({2}{x}+{3}\right)}{\left({2}{x}-{3}\right)}}}+\dfrac{{1}}{{x}}
=\dfrac{{{x}+{2}}}{{{2}{x}+{3}}}+\dfrac{{1}}{{x}}
=\dfrac{{{x}{\left({x}+{2}\right)}+{\left({2}{x}+{3}\right)}}}{{{x}{\left({2}{x}+{3}\right)}}}
=\dfrac{{{x}^{{2}}+{2}{x}+{2}{x}+{3}}}{{{x}{\left({2}{x}+{3}\right)}}}
=\dfrac{{{x}^{{2}}+{4}{x}+{3}}}{{{x}{\left({2}{x}+{3}\right)}}}
=\dfrac{{{\left({x}+{1}\right)}{\left({x}+{3}\right)}}}{{{x}{\left({2}{x}+{3}\right)}}}
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On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations | EMS Press
Institut Galilée, Université Paris 13, Villetaneuse, France
We derive various estimates for strong solutions to the Navier-Stokes equations in
C(|0,T)
L^3 (\mathbb R^3)
that allow us to prove some regularity results on the kinematic bilinear term.
Marco Cannone, Fabrice Planchon, On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations. Rev. Mat. Iberoam. 16 (2000), no. 1, pp. 1–16
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Date & Time - Maple Help
Home : Support : Online Help : System : Information : Updates : Maple 2018 : Date & Time
Dates, Times, and the Calendar Package
Maple 2018 adds new data structures that represent dates and times. There are numerous functions that work with dates and times, including fundamental operations such as date arithmetic and more advanced functionality for working with Calendars. Existing packages including Finance also support the new Date object.
Dates, Times, and Clocks
The Calendar Package
New objects representing dates, times, and clocks were introduced in Maple 2018.
The Now method for a clock creates a Time object representing the current time according to the indicated clock. (For more information about other available clock objects, see Clock.)
now := Now( SystemUTCClock );
\textcolor[rgb]{0,0,1}{\mathrm{Time}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1519826460130}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{clock}}\textcolor[rgb]{0,0,1}{=}\left(\textcolor[rgb]{0,0,1}{\mathbf{module}}\left(\textcolor[rgb]{0,0,1}{}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{end module}}\right)\right)
From a given Time object, you can create a corresponding Date object, as follows.
today := Date( now );
\textcolor[rgb]{0,0,1}{\mathrm{Date}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2018}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{28}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{14}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{13}}{\textcolor[rgb]{0,0,1}{100}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{timezone}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"UTC"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{coefficient}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\right)
Or, more simply, use the Date constructor without arguments.
today := Date();
\textcolor[rgb]{0,0,1}{\mathrm{Date}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2018}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{28}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{14}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{693}}{\textcolor[rgb]{0,0,1}{1000}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{timezone}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"UTC"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{coefficient}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\right)
Various fields of the Date object are accessible.
Year( today );
\textcolor[rgb]{0,0,1}{2018}
DayOfMonth( today );
\textcolor[rgb]{0,0,1}{28}
Minute( today );
\textcolor[rgb]{0,0,1}{1}
Arithmetic with Date and Time objects is builtin.
To compute the amount of time between two dates represented as Date objects, just subtract them.
d1 := Date( 2017, 12, 25 );
\textcolor[rgb]{0,0,1}{\mathrm{Date}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2017}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{12}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{25}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{12}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{timezone}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Canada/Eastern"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{coefficient}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Date}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2000}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{12}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{25}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{12}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{timezone}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Canada/Eastern"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{coefficient}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\right)
t := d1 - d2;
\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{536457600000}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{ms}}⟧
convert( t, 'units', 'days' );
\textcolor[rgb]{0,0,1}{6209}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{d}⟧
(For more control over the units used in computing the time between two dates, use the DateDifference command in the Calendar package).
In fact, any affine combination of Date objects results in an expression involving units of time.
A convex combination of Date objects results in another Date object. For instance, to compute the average value of two Date objects, you can use an expression such as the following one.
( d1 + d2 ) / 2;
\textcolor[rgb]{0,0,1}{\mathrm{Date}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2009}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{26}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{timezone}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Canada/Eastern"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{coefficient}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\right)
You can specify a date in a particular time zone by using the timezone option to the Date constructor. Suppose, for example, that a flight leaves Toronto at
8
p.m. on March 12, 2007, and arrives in Paris at
9
a.m. on March 13.
depart := Date( 2007, 3, 12, 20, 44, 'timezone' = "America/Toronto" );
\textcolor[rgb]{0,0,1}{\mathrm{Date}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2007}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{12}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{20}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{44}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{timezone}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"America/Toronto"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{coefficient}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\right)
arrive := Date( 2007, 3, 13, 9, 3, 'timezone' = "Europe/Paris" );
\textcolor[rgb]{0,0,1}{\mathrm{Date}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2007}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{13}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{timezone}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Europe/Paris"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{coefficient}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\right)
To compute the flight time, simply subtract the two dates.
flight_time := arrive - depart;
\textcolor[rgb]{0,0,1}{\mathrm{flight_time}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{26340000}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{ms}}⟧
convert( flight_time, 'units', 'hours' );
\frac{\textcolor[rgb]{0,0,1}{439}}{\textcolor[rgb]{0,0,1}{60}}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{h}⟧
The Calendar package provides a number of useful routines for working with dates.
with( Calendar );
[\textcolor[rgb]{0,0,1}{\mathrm{AdjustDateField}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{DateDifference}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{DayOfWeek}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{DayOfYear}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{DaysInMonth}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{DaysInYear}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{HostTimeZone}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{IsDaylightSavingTime}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{IsLeapYear}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{IsWeekend}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{JulianDayNumber}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{ModifiedJulianDayNumber}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Today}}]
For example, you can check whether a year is a leap year.
IsLeapYear( 2000 );
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\textcolor[rgb]{0,0,1}{\mathrm{false}}
You can compute the day of the week or year for any given date. For example, Christmas of 2017 occurred on a Monday.
DayOfWeek( 2017, 12, 25 );
\textcolor[rgb]{0,0,1}{2}
And, it was the
359
-th day of the year.
DayOfYear( 2017, 12, 25 );
\textcolor[rgb]{0,0,1}{359}
Since Date objects can be specified with respect to particular time zones, it is useful to be able to determine the time zone of computer on which Maple is running. To do this, use the HostTimeZone command.
HostTimeZone();
\textcolor[rgb]{0,0,1}{"Canada/Eastern"}
The DateDifference command gives you finer control over the units used to compute the time difference between two dates. For our flight example above, we can compute as follows.
DateDifference( depart, arrive, 'units' = 'h' );
\frac{\textcolor[rgb]{0,0,1}{439}}{\textcolor[rgb]{0,0,1}{60}}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{h}⟧
It is sometimes more convenient to use "mixed" units in the output, as illustrated here:
DateDifference( depart, arrive, 'units' = 'mixed' );
\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{h}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{19}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{min}}⟧
The AdjustDateField command provides for the ability to add amounts to individual Date object fields, such as the month or the hour of the day.
d := Date( 2000, 1, 14, 10, 55, 3 );
\textcolor[rgb]{0,0,1}{\mathrm{Date}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2000}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{14}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{55}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{timezone}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Canada/Eastern"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{coefficient}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\right)
AdjustDateField( d, "minute", -3 );
\textcolor[rgb]{0,0,1}{\mathrm{Date}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2000}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{14}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{52}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{timezone}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Eastern Standard Time"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{coefficient}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\right)
Notice the effect of adding many months to the date here:
AdjustDateField( d, "month", 30 );
\textcolor[rgb]{0,0,1}{\mathrm{Date}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2002}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{14}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{55}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{timezone}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Eastern Standard Time"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{coefficient}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\right)
By default, changes to a particular date field can potentially lead to changes to date fields at a higher level of granularity. The method = "roll" option confines modifications to a single field, keeping its values within the valid ranges for that field.
AdjustDateField( d, "month", 30, 'method' = "roll" );
\textcolor[rgb]{0,0,1}{\mathrm{Date}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2000}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{14}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{55}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{timezone}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Eastern Standard Time"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{coefficient}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\right)
The default behavior is obtained by using the method = "add" option.
AdjustDateField( d, "month", 30, 'method' = "add" );
\textcolor[rgb]{0,0,1}{\mathrm{Date}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2002}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{14}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{55}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{timezone}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Eastern Standard Time"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{coefficient}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\right)
For more information on the Calendar package, see the package overview page.
|
For each triangle below, use right triangle patterns to determine the missing side lengths. Then calculate the area and perimeter of each triangle.
Use a
45^{\circ}\text{-}45^{\circ}\text{-}90^{\circ}
relationship to determine missing sides. Then calculate the area and perimeter of the triangle.
A = 1\text{ m}^2\text{, }P = 2 + 2\sqrt{2}\text{ m}
30^{\circ}\text{-}60^{\circ}\text{-}90^{\circ}
relationship to determine the missing sides. Then calculate the area and perimeter of the triangle.
A = \frac{25\sqrt{3}}{2}\approx21.7\text{ ft}^2\text{, }P\approx15+5\sqrt{3}\approx23.7\text{ ft}
|
Which of the following is normally not an important atmospheric pollutant
c{o}_{2}
s{o}_{2}
D. Hydro-carbon
Direction MCQ Steam Boilers, Engines, Nozzles & Turbines MCQ IBPS Common Written Exam (PO/MT) Main 2016 Solved Paper MCQ Data Sufficiency MCQ Animal Kingdom MCQ Cloze Test MCQ Boats and Streams MCQ Atomic Structure MCQ Selenium automation MCQ Letter and Symbol Series MCQ Kidney cysts and tumors MCQ Pipes and Cistern MCQ
|
A Möbius characterization of submanifolds
July, 2006 A Möbius characterization of submanifolds
Qing-Ming CHENG, Shichang SHU
In this paper, we study Möbius characterizations of submanifolds without umbilical points in a unit sphere
{S}^{n+p}\left(1\right)
. First of all, we proved that, for an
\left(n\ge 2\right)
\mathbf{x}:M↦{S}^{n+p}\left(1\right)
without umbilical points and with vanishing Möbius form
\mathrm{\Phi }
\left(n-2\right)||\stackrel{˜}{\mathbf{A}}||\le \sqrt{\frac{n-1}{n}}\left\{nR-\frac{1}{n}\left[\left(n-1\right)\left(2-\frac{1}{p}\right)-1\right]\right\}
is satisfied, then,
\mathbf{x}
is Möbius equivalent to an open part of either the Riemannian product
{S}^{n-1}\left(r\right)×{S}^{1}\left(\sqrt{1-{r}^{2}}\right)
{S}^{n+1}\left(1\right)
, or the image of the conformal diffeomorphism
\sigma
of the standard cylinder
{S}^{n-1}\left(1\right)×\mathbf{R}
{\mathbf{R}}^{n+1}
\tau
of the Riemannian product
{S}^{n-1}\left(r\right)×{\mathbf{H}}^{1}\left(\sqrt{1+{r}^{2}}\right)
{\mathbf{H}}^{n+1}
\mathbf{x}
is locally Möbius equivalent to the Veronese surface in
{S}^{4}\left(1\right)
p=1
, our pinching condition is the same as in Main Theorem of Hu and Li [6], in which they assumed that
M
is compact and the Möbius scalar curvature
n\left(n-1\right)R
is constant. Secondly, we consider the Möbius sectional curvature of the immersion
\mathbf{x}
. We obtained that, for an
-dimensional compact submanifold
\mathbf{x}:M↦{S}^{n+p}\left(1\right)
without umbilical points and with vanishing form
\mathrm{\Phi }
, if the Möbius scalar curvature
n\left(n-1\right)R
of the immersion
\mathbf{x}
is constant and the Möbius sectional curvature
K
\mathbf{x}
K\ge 0
p=1
K>0
p>1
\mathbf{x}
is Möbius equivalent to either the Riemannian product
{S}^{k}\left(r\right)×{S}^{n-k}\left(\sqrt{1-{r}^{2}}\right)
k=1,2,\cdots ,n-1
{S}^{n+1}\left(1\right)
\mathbf{x}
is Möbius equivalent to a compact minimal submanifold with constant scalar curvature in
{S}^{n+p}\left(1\right)
Qing-Ming CHENG. Shichang SHU. "A Möbius characterization of submanifolds." J. Math. Soc. Japan 58 (3) 903 - 925, July, 2006. https://doi.org/10.2969/jmsj/1156342043
Keywords: Blaschke tensor and Möbius form , Möbius metric , Möbius scalar curvature , Möbius sectional curvature , submanifold
Qing-Ming CHENG, Shichang SHU "A Möbius characterization of submanifolds," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 58(3), 903-925, (July, 2006)
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CoCalc -- ALEA-Exercises.ipynb
Path: ALEA-Exercises.ipynb
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Project: ALEA ACSV
Raw | Embed | Download |
Kernel: SageMath 9.5
Exercises: An Invitation to Analytic Combinatorics in Several Variables
Created by Stephen Melczer
This notebook complements the exercises found on the ALEA ACSV course webpage
A quick Sage tutorial can be found here (try an interactive version in your browser here).
See https://melczer.ca/textbook for further Sage notebooks solving problems in analytic combinatorics in several varibles. In particular, this notebook (also available as a static HTML page) gives an algorithm to compute asymptotic terms. Don't use it to solve these exercises, but use it to further check your answers! (Or to compute asymptotics for other problems!)
# Helper function to compute the Hessian matrix M from the function H(vars)
# in the direction R at the point CP, specified by a list of substitutions.
# Copied from melczer.ca/textbook/
def getHes(H,R,vars,CP):
dd = len(vars)
V = zero_vector(SR,dd)
U = matrix(SR,dd)
M = matrix(SR,dd-1)
for j in range(dd):
V[j] = R[j]/R[-1]
for i in range(dd):
U[i,j] = vars[i]*vars[j]*diff(H,vars[i],vars[j])/vars[-1]/diff(H,vars[-1])
for i in range(dd-1):
for j in range(dd-1):
M[i,j] = V[i]*V[j] + U[i,j] - V[j]*U[i,-1] - V[i]*U[j,-1] + V[i]*V[j]*U[-1,-1]
if i == j: M[i,j] = M[i,j] + V[i]
return M.subs(CP)
# Helper function to compute leading asymptotics of the R-diagonal of G(vars)/H(vars)
# determined by the Main Asymptotic Theorem of Smooth ACSV at the point CP, specified
# by a list of substitutions. We take det(M) as an input that can be computed by the
# above function.
def leadingASM(G,H,detM,R,vars,CP):
dd = len(R)
lcoeff = -G/vars[-1]/H.diff(vars[-1])
exp = 1/mul([vars[k]^R[k] for k in range(dd)])^n
ASM = exp * (2*pi*n*R[-1])^((1-dd)/2) / sqrt(detM) * lcoeff
return ASM.subs(CP)
These functions can be used to compute the matrix M appearing in asymptotics, as well as the leading asymptotic term in an asymptotic expansion.
Here is an example of their use to find asymptotics for the main diagonal of
1/(1-x-y-z)
# Introduce variables x, y, and z
H = 1 - x - y - z
# In the main diagonal direction this has a critical point at (1/3,1/3,1/3)
CPs = solve([H,x*diff(H,x) - y*diff(H,y), x*diff(H,x) - z*diff(H,z)],[x,y,z])
show(CPs)
\renewcommand{\Bold}[1]{\mathbf{#1}}\left[\left[x = \left(\frac{1}{3}\right), y = \left(\frac{1}{3}\right), z = \left(\frac{1}{3}\right)\right]\right]
# Let CP be the critical point, defined as a list [x == 1/3, y == 1/3, z == 1/3]
CP = CPs[0]
# Get the matrix M
M = getHes(H,[1,1,1],[x,y,z],CP)
show("M = ", M)
\renewcommand{\Bold}[1]{\mathbf{#1}}\verb|M|\verb| |\verb|=| \left(\begin{array}{rr} 2 & 1 \\ 1 & 2 \end{array}\right)
# Get and print leading asymptotics
ASM = leadingASM(1,H,M.determinant(),[1,1,1],[x,y,z],CP)
print("The dominant asymptotic behaviour of the main diagonal is")
show(ASM)
The dominant asymptotic behaviour of the main diagonal is
\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{\sqrt{3}}{2 \, \pi \left(\frac{1}{27}\right)^{n} n}
We can check our asymptotic approximation by computing series terms.
# First, define the ring of formal power series (more efficient for computations)
# We use capital letters to denote the formal power series variables
P.<X,Y,Z> = QQ[['X,Y,Z']]
# Computes the series expansion up to precision 3*N
ser = 1/(1-X-Y-Z + O(X^(3*N+1)))
# Check ratio of asymptotic formula to actual coefficients -- this should go to 1!
(ser.coefficients()[X^N*Y^N*Z^N]/ASM.subs(n=N)).n()
Question 1: Delannoy Numbers
The Delannoy number
d_{a,b}
is the number of paths from the origin
(0,0)
(a,b)
using only the steps
\textsf{N}=(0,1)
\textsf{E} = (1,0)
\textsf{NE}=(1,1)
(a) Prove the recurrence $$ d_{a,b} = \begin{cases} 1 &: \text{ if $a=0$ or
b=0
} \ d_{a-1,b} + d_{a,b-1} + d_{a-1,b-1} &:\text{ otherwise} \end{cases}
Conclude that
D(x,y) = \sum_{a,b\geq0}d_{a,b}x^ay^b = \frac{1}{1-x-y-xy}. $$
(b) Use the Main Theorem of Smooth ACSV to find asymptotics of
d_{n,n}
(1,1)
-diagonal of
D(x,y)
. What are the critical points in the
(1,1)
direction? Which are minimal?
(c) Use the Main Theorem of Smooth ACSV to find asymptotics of the
(r,s)
D(x,y)
r,s>0
Function to numerically compute terms in the expansion
Check your computed asymptotics against this function!
P.<X,Y> = QQ[['X,Y']]
# Code to compute the coefficient of x^(N*R) * y^(N*S) where R, S, and N are positive integers
R, S, N = 1, 2, 10
N2 = (R+S)*N
ser = 1/(1-X-Y-X*Y + O(X^(N2+1)))
coef = ser.coefficients()[X^(R*N)*Y^(S*N)]
print("The coefficient [x^{}y^{}]D(x,y) = {}".format(N*R,N*S,coef))
The coefficient [x^10y^20]D(x,y) = 4354393801
Question 2: Apéry Asymptotics
Recall from lecture that a key step in Apéry's proof of the irrationality of
\zeta(3)
is determining the exponential growth of the sequence that can be encoded as the main diagonal of
F(x,y,z,t) = \frac{1}{1 - t(1+x)(1+y)(1+z)(1+y+z+yz+xyz)}.
Use the Main Theorem of Smooth ACSV to find dominant asymptotics of this sequence.
# Numerically compute coefficient of (xyzt)^n (can take a long time for large N)
P.<X,Y,Z> = QQ['X,Y,Z']
ser = ((1+X)*(1+Y)*(1+Z)*(1+Y+Z+Y*Z+X*Y*Z))^N
coef = ser[X^N*Y^N*Z^N]
print("The coefficient [(xyzt)^({})]F(x,y,z,t) = {}".format(N,coef))
The coefficient [(xyzt)^(30)]F(x,y,z,t) = 11320115195385966907843180411829810312080825
Question 3: Pathological Directions
(a) Find asymptotics of the
(r,s)
F(x,y) = \frac{1}{1-x-xy}
0<s<r
(b) What are the critical points of
F(x,y) = \frac{1}{1-x-xy}
(r,s)
direction when
0<r \leq s
? Which are minimal? Characterize the behaviour of the
(r,s)
diagonal when
0<r \leq s
ser = 1/(1-X-X*Y + O(X^(N2+1)))
print("The coefficient [x^{}y^{}]F(x,y) = {}".format(N*R,N*S,coef))
The coefficient [x^20y^10]F(x,y) = 184756
Question 4: A Composition Limit Theorem
An integer composition of size
n\in\mathbb{N}
is an ordered tuple of positive integers (of any length) that sum to
n
. A composition can be viewed as an integer partition where the order of the summands matters. Let
c_{k,n}
denote the number of compositions of size
n
k
(a) If you know the symbolic method, species theory, or similar enumerative constructions, prove that
C(u,x) = \sum_{n,k\geq0}c_{k,n}u^kx^n = \frac{1-x}{1-2x-(u-1)x(1-x)}.
(b) Prove that the distribution for the number of ones in a composition of size
n
satisfies a local central limit theorem as
n\rightarrow\infty
. More precisely, find a constant
t>0
and normal density
\nu_n(s)
\sup_{s \in \mathbb{Z}} |t^nc_{s,n} - \nu_n(s)| \rightarrow 0
n\rightarrow\infty
Function to plot series coefficients
Check your computed distribution against this function!
# Plot series terms versus computed density
K.<U> = QQ['U']
P.<X> = K[['X']]
# Set the value of n to test
mser = (1 - X)/(1 - 2*X - (U-1)*X*(1-X) + O(X^(N+1)))
uvals = mser[N]
plt = point([])
plt += point([k,uvals[k]])
print("The following plot shows the coefficients of [x^({})]C(u,x)".format(N))
The following plot shows the coefficients of [x^(200)]C(u,x)
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Pappus's hexagon theorem - Wikipedia
Geometry theorem
Pappus's hexagon theorem: Points X, Y and Z are collinear on the Pappus line. The hexagon is AbCaBc.
Pappus's theorem: affine form
{\displaystyle Ab\parallel aB,Bc\parallel bC\Rightarrow Ac\parallel aC}
In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that
given one set of collinear points
{\displaystyle A,B,C}
, and another set of collinear points
{\displaystyle a,b,c}
, then the intersection points
{\displaystyle X,Y,Z}
of line pairs
{\displaystyle Ab}
{\displaystyle aB,Ac}
{\displaystyle aC,Bc}
{\displaystyle bC}
are collinear, lying on the Pappus line. These three points are the points of intersection of the "opposite" sides of the hexagon
{\displaystyle AbCaBc}
It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring.[1] Projective planes in which the "theorem" is valid are called pappian planes.
If one restricts the projective plane such that the Pappus line
{\displaystyle u}
is the line at infinity, one gets the affine version of Pappus's theorem shown in the second diagram.
If the Pappus line
{\displaystyle u}
and the lines
{\displaystyle g,h}
have a point in common, one gets the so-called little version of Pappus's theorem.[2]
The dual of this incidence theorem states that given one set of concurrent lines
{\displaystyle A,B,C}
, and another set of concurrent lines
{\displaystyle a,b,c}
, then the lines
{\displaystyle x,y,z}
defined by pairs of points resulting from pairs of intersections
{\displaystyle A\cap b}
{\displaystyle a\cap B,\;A\cap c}
{\displaystyle a\cap C,\;B\cap c}
{\displaystyle b\cap C}
are concurrent. (Concurrent means that the lines pass through one point.)
Pappus's theorem is a special case of Pascal's theorem for a conic—the limiting case when the conic degenerates into 2 straight lines. Pascal's theorem is in turn a special case of the Cayley–Bacharach theorem.
The Pappus configuration is the configuration of 9 lines and 9 points that occurs in Pappus's theorem, with each line meeting 3 of the points and each point meeting 3 lines. In general, the Pappus line does not pass through the point of intersection of
{\displaystyle ABC}
{\displaystyle abc}
.[3] This configuration is self dual. Since, in particular, the lines
{\displaystyle Bc,bC,XY}
have the properties of the lines
{\displaystyle x,y,z}
of the dual theorem, and collinearity of
{\displaystyle X,Y,Z}
is equivalent to concurrence of
{\displaystyle Bc,bC,XY}
, the dual theorem is therefore just the same as the theorem itself. The Levi graph of the Pappus configuration is the Pappus graph, a bipartite distance-regular graph with 18 vertices and 27 edges.
1 Proof: affine form
2 Proof with homogeneous coordinates
3 Dual theorem
4 Other statements of the theorem
Proof: affine form[edit]
Pappus theorem: proof
If the affine form of the statement can be proven, then the projective form of Pappus's theorem is proven, as the extension of a pappian plane to a projective plane is unique.
Because of the parallelity in an affine plane one has to distinct two cases:
{\displaystyle g\not \parallel h}
{\displaystyle g\parallel h}
. The key for a simple proof is the possibility for introducing a "suitable" coordinate system:
Case 1: The lines
{\displaystyle g,h}
{\displaystyle S=g\cap h}
In this case coordinates are introduced, such that
{\displaystyle \;S=(0,0),\;A=(0,1),\;c=(1,0)\;}
(see diagram).
{\displaystyle B,C}
{\displaystyle \;B=(0,\gamma ),\;C=(0,\delta ),\;\gamma ,\delta \notin \{0,1\}}
From the parallelity of the lines
{\displaystyle Bc,\;Cb}
{\displaystyle b=({\tfrac {\delta }{\gamma }},0)}
and the parallelity of the lines
{\displaystyle Ab,Ba}
{\displaystyle a=(\delta ,0)}
. Hence line
{\displaystyle Ca}
{\displaystyle -1}
and is parallel line
{\displaystyle Ac}
{\displaystyle g\parallel h\ }
(little theorem).
In this case the coordinates are chosen such that
{\displaystyle \;c=(0,0),\;b=(1,0),\;A=(0,1),\;B=(\gamma ,1),\;\gamma \neq 0}
. From the parallelity of
{\displaystyle Ab\parallel Ba}
{\displaystyle cB\parallel bC}
{\displaystyle \;C=(\gamma +1,1)\;}
{\displaystyle \;a=(\gamma +1,0)\;}
, respectively, and at least the parallelity
{\displaystyle \;Ac\parallel Ca\;}
Proof with homogeneous coordinates[edit]
Choose homogeneous coordinates with
{\displaystyle C=(1,0,0),\;c=(0,1,0),\;X=(0,0,1),\;A=(1,1,1)}
On the lines
{\displaystyle AC,Ac,AX}
{\displaystyle x_{2}=x_{3},\;x_{1}=x_{3},\;x_{2}=x_{1}}
, take the points
{\displaystyle B,Y,b}
{\displaystyle B=(p,1,1),\;Y=(1,q,1),\;b=(1,1,r)}
{\displaystyle p,q,r}
. The three lines
{\displaystyle XB,CY,cb}
{\displaystyle x_{1}=x_{2}p,\;x_{2}=x_{3}q,\;x_{3}=x_{1}r}
, so they pass through the same point
{\displaystyle a}
{\displaystyle rqp=1}
. The condition for the three lines
{\displaystyle Cb,cB}
{\displaystyle XY}
{\displaystyle x_{2}=x_{1}q,\;x_{1}=x_{3}p,\;x_{3}=x_{2}r}
to pass through the same point
{\displaystyle Z}
{\displaystyle rpq=1}
. So this last set of three lines is concurrent if all the other eight sets are because multiplication is commutative, so
{\displaystyle pq=qp}
{\displaystyle X,Y,Z}
The proof above also shows that for Pappus's theorem to hold for a projective space over a division ring it is both sufficient and necessary that the division ring is a (commutative) field. German mathematician Gerhard Hessenberg proved that Pappus's theorem implies Desargues's theorem.[4][5] In general, Pappus's theorem holds for some projective plane if and only if it is a projective plane over a commutative field. The projective planes in which Pappus's theorem does not hold are Desarguesian projective planes over noncommutative division rings, and non-Desarguesian planes.
The proof is invalid if
{\displaystyle C,c,X}
happen to be collinear. In that case an alternative proof can be provided, for example, using a different projective reference.
Dual theorem[edit]
Because of the principle of duality for projective planes the dual theorem of Pappus is true:
If 6 lines
{\displaystyle A,b,C,a,B,c}
are chosen alternately from two pencils with centers
{\displaystyle G,H}
{\displaystyle X:=(A\cap b)(a\cap B),}
{\displaystyle Y:=(c\cap A)(C\cap a),}
{\displaystyle Z:=(b\cap C)(B\cap c)}
are concurrent, that means: they have a point
{\displaystyle U}
The left diagram shows the projective version, the right one an affine version, where the points
{\displaystyle G,H}
are points at infinity. If point
{\displaystyle U}
{\displaystyle GH}
than one gets the "dual little theorem" of Pappus' theorem.
dual theorem: projective form
dual theorem: affine form
If in the affine version of the dual "little theorem" point
{\displaystyle U}
is a point at infinity too, one gets Thomsen's theorem, a statement on 6 points on the sides of a triangle (see diagram). The Thomsen figure plays an essential role coordinatising an axiomatic defined projective plane.[6] The proof of the closure of Thomsen's figure is covered by the proof for the "little theorem", given above. But there exists a simple direct proof, too:
Because the statement of Thomsen's theorem (the closure of the figure) uses only the terms connect, intersect and parallel, the statement is affinely invariant, and one can introduce coordinates such that
{\displaystyle P=(0,0),\;Q=(1,0),\;R=(0,1)}
(see right diagram). The starting point of the sequence of chords is
{\displaystyle (0,\lambda ).}
One easily verifies the coordinates of the points given in the diagram, which shows: the last point coincides with the first point.
Thomsen figure (points
{\displaystyle \color {red}1,2,3,4,5,6}
of the triangle
{\displaystyle PQR}
) as dual theorem of the little theorem of Pappus (
{\displaystyle U}
is at infinity, too !).
Thomsen figure: proof
Other statements of the theorem[edit]
{\displaystyle XcC}
{\displaystyle BbY}
are perspective from
{\displaystyle A}
{\displaystyle a}
, and so, also from
{\displaystyle Z}
In addition to the above characterizations of Pappus's theorem and its dual, the following are equivalent statements:
If the six vertices of a hexagon lie alternately on two lines, then the three points of intersection of pairs of opposite sides are collinear.[7]
Arranged in a matrix of nine points (as in the figure and description above) and thought of as evaluating a permanent, if the first two rows and the six "diagonal" triads are collinear, then the third row is collinear.
{\displaystyle \left|{\begin{matrix}A&B&C\\a&b&c\\X&Y&Z\end{matrix}}\right|}
{\displaystyle \ ABC,abc,AbZ,BcX,CaY,XbC,YcA,ZaB\ }
are lines, then Pappus's theorem states that
{\displaystyle XYZ}
must be a line. Also, note that the same matrix formulation applies to the dual form of the theorem when
{\displaystyle (A,B,C)}
etc. are triples of concurrent lines.[8]
Given three distinct points on each of two distinct lines, pair each point on one of the lines with one from the other line, then the joins of points not paired will meet in (opposite) pairs at points along a line.[9]
If two triangles are perspective in at least two different ways, then they are perspective in three ways.[4]
{\displaystyle \;AB,CD,\;}
{\displaystyle EF}
are concurrent and
{\displaystyle DE,FA,}
{\displaystyle BC}
are concurrent, then
{\displaystyle AD,BE,}
{\displaystyle CF}
are concurrent.[8]
In its earliest known form, Pappus's Theorem is Propositions 138, 139, 141, and 143 of Book VII of Pappus's Collection.[10] These are Lemmas XII, XIII, XV, and XVII in the part of Book VII consisting of lemmas to the first of the three books of Euclid's Porisms.
The lemmas are proved in terms of what today is known as the cross ratio of four collinear points. Three earlier lemmas are used. The first of these, Lemma III, has the diagram below (which uses Pappus's lettering, with G for Γ, D for Δ, J for Θ, and L for Λ).
Here three concurrent straight lines, AB, AG, and AD, are crossed by two lines, JB and JE, which concur at J. Also KL is drawn parallel to AZ. Then
KJ : JL :: (KJ : AG & AG : JL) :: (JD : GD & BG : JB).
These proportions might be written today as equations:[11]
KJ/JL = (KJ/AG)(AG/JL) = (JD/GD)(BG/JB).
The last compound ratio (namely JD : GD & BG : JB) is what is known today as the cross ratio of the collinear points J, G, D, and B in that order; it is denoted today by (J, G; D, B). So we have shown that this is independent of the choice of the particular straight line JD that crosses the three straight lines that concur at A. In particular
(J, G; D, B) = (J, Z; H, E).
It does not matter on which side of A the straight line JE falls. In particular, the situation may be as in the next diagram, which is the diagram for Lemma X.
Just as before, we have (J, G; D, B) = (J, Z; H, E). Pappus does not explicitly prove this; but Lemma X is a converse, namely that if these two cross ratios are the same, and the straight lines BE and DH cross at A, then the points G, A, and Z must be collinear.
What we showed originally can be written as (J, ∞; K, L) = (J, G; D, B), with ∞ taking the place of the (nonexistent) intersection of JK and AG. Pappus shows this, in effect, in Lemma XI, whose diagram, however, has different lettering:
What Pappus shows is DE.ZH : EZ.HD :: GB : BE, which we may write as
(D, Z; E, H) = (∞, B; E, G).
The diagram for Lemma XII is:
The diagram for Lemma XIII is the same, but BA and DG, extended, meet at N. In any case, considering straight lines through G as cut by the three straight lines through A, (and accepting that equations of cross ratios remain valid after permutation of the entries,) we have by Lemma III or XI
(G, J; E, H) = (G, D; ∞ Z).
Considering straight lines through D as cut by the three straight lines through B, we have
(L, D; E, K) = (G, D; ∞ Z).
Thus (E, H; J, G) = (E, K; D, L), so by Lemma X, the points H, M, and K are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon ADEGBZ are collinear.
Lemmas XV and XVII are that, if the point M is determined as the intersection of HK and BG, then the points A, M, and D are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon BEKHZG are collinear.
^ Coxeter, pp. 236–7
^ Rolf Lingenberg: Grundlagen der Geometrie, BI-Taschenbuch, 1969, p. 93
^ However, this does occur when
{\displaystyle ABC}
{\displaystyle abc}
are in perspective, that is,
{\displaystyle Aa,Bb}
{\displaystyle Cc}
^ a b Coxeter 1969, p. 238
^ According to (Dembowski 1968, pg. 159, footnote 1), Hessenberg's original proof Hessenberg (1905) is not complete; he disregarded the possibility that some additional incidences could occur in the Desargues configuration. A complete proof is provided by Cronheim 1953.
^ W. Blaschke: Projektive Geometrie, Springer-Verlag, 2013, ISBN 3034869320, S. 190
^ Coxeter, p. 231
^ a b Coxeter, p. 233
^ Whicher, chapter 14
^ Heath (Vol. II, p. 421) cites these propositions. The latter two can be understood as converses of the former two. Kline (p. 128) cites only Proposition 139. The numbering of the propositions is as assigned by Hultsch.
^ A reason for using the notation above is that, for the ancient Greeks, a ratio is not a number or a geometrical object. We may think of ratio today as an equivalence class of pairs of geometrical objects. Also, equality for the Greeks is what we might today call congruence. In particular, distinct line segments may be equal. Ratios are not equal in this sense; but they may be the same.
Cronheim, A. (1953), "A proof of Hessenberg's theorem", Proceedings of the American Mathematical Society, 4 (2): 219–221, doi:10.2307/2031794, JSTOR 2031794
Dembowski, Peter (1968), Finite Geometries, Berlin: Springer Verlag
Heath, Thomas (1981) [1921], A History of Greek Mathematics, New York: Dover
Hessenberg, Gerhard (1905), "Beweis des Desarguesschen Satzes aus dem Pascalschen", Mathematische Annalen, Berlin / Heidelberg: Springer, 61 (2): 161–172, doi:10.1007/BF01457558, ISSN 1432-1807, S2CID 120456855
Hultsch, Fridericus (1877), Pappi Alexandrini Collectionis Quae Supersunt, Berlin
Kline, Morris (1972), Mathematical Thought From Ancient to Modern Times, New York: Oxford University Press
Whicher, Olive (1971), Projective Geometry, Rudolph Steiner Press, ISBN 0-85440-245-4
Pappus's hexagon theorem at cut-the-knot
Dual to Pappus's hexagon theorem at cut-the-knot
Pappus’s Theorem: Nine proofs and three variations
Retrieved from "https://en.wikipedia.org/w/index.php?title=Pappus%27s_hexagon_theorem&oldid=1077984498"
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CoCalc – Online Jupyter Notebooks
CoCalc's own collaborative, fully compatible and supercharged notebooks.
Try Jupyter Now
No software setup: 100% online
CoCalc is an online web service where you can run Jupyter notebooks right inside your browser. You can privately share your notebook with your project collaborators – all changes are synchronized in real-time.
You no longer have to worry about setting up your Python environment, installing/updating/maintaining your libraries, or backing up files. CoCalc manages everything for you!
Jupyter Notebooks made for teaching!
A sophisticated course management system keeps track of all notebooks of all students. It manages distributing and collecting files as well as grading.
The Jupyter collaborative whiteboard supports presentations that mix Jupyter cells, mathematical notation, and sketching with a pen and other tools.
CoCalc's Jupyter Notebooks fully support very flexible automatic grading via nbgrader! The teacher's notebook contains exercise cells for students and test cells, some of which students can also run to get immediate feedback. Once collected, you tell CoCalc to automatically run the full test suite across all student notebooks and tabulate the results.
CoCalc supports many kernels right out of the box: several Python environments, SageMath, R Statistical SoftwareOctave, Julia and many more.
Ready out of the box: Sign up, create a project, create or upload your *.ipynb file, and you're ready to go!
You can share your Jupyter notebooks privately with project collaborators. All modifications are synchronized in real time, where you can see the cursors of others while they edit the document. You are also notified about the presence of collaborators.
Edit text between code cells using markdown or our collaborative rich text editor.
We have extended ipywidgets so that sliders, menus and knobs of interactive widgets are also fully synchronized among all collaborators.
Additionally, the status and results of all computations in the currently running kernel session are also synchronized, because the session runs remotely in CoCalc's cluster.
Together, everyone involved experiences the notebook in the same way.
The following are some more specific features of Jupyter notebooks in CoCalc.
TimeTravel is a powerful feature of the CoCalc platform. It records all your changes in your Jupyter notebook in fine detail. You can go back and forth in time across thousands of changes to see all previous edits.
This allows you to easily recover anything from previous versions of your notebook by copy and pasting.
You can also browse the entire process of creating the notebook from the start. This lets you discover how you arrived at a particular solution and see what you (or your students) tried to get there.
NBGrader: automatically grading assignments in Jupyter notebooks
CoCalc's Jupyter Notebooks fully support both automatic and manual grading!
When using NBGrader, the teacher's notebook contains exercise cells for students and test cells, some of which students run to get immediate feedback. Once collected, you tell CoCalc to automatically run the full test suite across all student notebooks and tabulate the results.
Learn more about NBGrader in CoCalc.
Chat about your Jupyter notebook
A chat to the side of each Jupyter notebook lets you discuss the content of your notebook with colleagues or students. You can drag and drop or paste images and files into chat, use
\LaTeX
math formulas, and fix typos in messages.
Collaborators who are not online will be notified about new messages the next time they sign in or you can @mention them so they get emailed.
Chat fully supports markdown formatting and
\LaTeX
Managed Jupyter kernels
CoCalc makes sure that your desired computational environment is available and ready to work with. Select from many pre-installed and fully managed kernels. You can also create your own custom kernel.
Look at our list of available software for more about what is available.
JupyterLab and Jupyter Classic
CoCalc's Jupyter is a complete rewrite of the classical Jupyter notebook interface and backend server. It is tightly integrated into CoCalc and adds realtime collaboration, TimeTravel history and more. This rewrite does not change the underlying Jupyter notebook file format; you can download your *.ipynb file at any time and continue working in another environment.
In addition, CoCalc also fully supports running standard JupyterLab (with realtime collaboration enabled) and Jupyter Classic notebook servers from any CoCalc project! You can still use all libraries and extension that might rely on specifics of one of those implementations. Moreover, you can fully use your CoCalc project via the powerful JupyterLab interface!
CoCalc also supports using Jupyter Classic with collaborative editing and chat.
CPU and memory monitoring for each notebook
Long running notebook sessions or intense computations might deplete available CPU or memory resources. This slows down all calculations or even causes an unexpected termination of the current session.
CoCalc's per-notebook CPU and memory indicators helps you to keep an eye on the notebook's memory and CPU consumption.
You can even close your browser during long running computations, and check on the results later. Output will not be lost while your browser is closed.
Publishing your notebooks
CoCalc helps you share your work with the world. It offers its own hosting of shared documents, which includes Jupyter notebooks and any other associated data files.
Under the hood, CoCalc uses a novel renderer which generates a static HTML representation of your notebook (sanitized to prevent XSS attacks) on the server, which includes pre-rendered
\LaTeX
formulas. This approach is very efficient and lightweight compared to solutions based on nbconvert.
Jupyter notebooks in CoCalc versus the competition
|
Electrovibration - Wikipedia
The history of electrovibration goes back to 1954. It was first discovered by accident and E. Mallinckrodt, A. L. Hughes and W. Sleator Jr. reported “... that dragging a dry finger over a conductive surface covered with a thin insulating layer and excited with a 110 V signal, created a characteristic rubbery feeling”.[1] In their experiment, the finger and the metal surface create a capacitive setup. The attraction force created between the finger and the surface was too weak to perceive, but it generated a rubbery sensation when the finger was moving on the surface. This sensation was named "electrovibration" by the group. From around early 2010 Senseg[2][3] and Disney Research[4][5] are developing technology that could bring electrovibration to modern touchscreen devices.
"In summer of 1950, E. Mallinckrodt noted that a certain shiny brass electric light socket did not feel as smooth when the light was burning as it did with the light off".[citation needed] Then Mallinckrodt created a setup to investigate the effect scientifically. He connected an aluminum plate through a variable current-limiting resistor to a 60 Hz, 110 V power supply. Half of the aluminum plate was coated with an insulating varnish, while the rest was left uncoated. As a result of the test he identified that the feeling of friction only appears when there is an insulating barrier between the conductive surface and the sliding finger. He concluded that the finger gets electrically polarized, and this induced charge creates a force between that finger and the surface. He named this phenomenon "electrically induced vibrations".[1]
Electrostatic-force theoryEdit
An electrostatic force is created by applying a time-varying voltage between an electrode and an insulated ground plane. When a finger scans over an insulated plate with a time-varying voltage, the finger works as the induced ground plane. The induced static electricity creates an electric force field between the finger and the surface.
A parallel-plate capacitor model can be used to approximate the skin–surface interface. The electrode acts as one plate, while the conductive subcutaneous layer in the skin acts as the other, thus representing a hybrid natural/artificial electrostatic actuator.[6] The following equation approximates the electrostatic force experienced between the finger and the electrode:
{\displaystyle F_{\text{e}}={\frac {\varepsilon _{0}\varepsilon _{r}AV^{2}}{2d^{2}}},}
{\displaystyle \varepsilon _{0}}
– permittivity of free space,
{\displaystyle \varepsilon _{r}}
– dielectric constant,
{\displaystyle A}
– area of electrodes,
{\displaystyle V}
– voltage applied between the two plates,
{\displaystyle d}
– distance between two plates.
The resulting force is too small to perceive by human skin, but when the finger is moving on the surface, a frictional force appears on the moving finger, which can be expressed as
{\displaystyle f=\mu F_{\text{e}},}
{\displaystyle \mu }
is the coefficient of friction.
Further research has shown that this model is not sufficient to explain such skin–surface interfaces.[6]
^ a b Mallinckrodt, E.; Hughes, A. L.; Sleator, W. Jr. "Perception by the Skin of Electrically Induced Vibrations". (Extract) Science, Vol. 118, No. 3062, pp. 277–278, 4 September 1953. doi:10.1126/science.118.3062.277. ISSN 0036-8075 (print) Retrieved 7 June 2015 (subscription required) for full access.
^ Senseg.
^ Wijekoon, Dinesh; Cecchinato, Marta E.; Hoggan, Eve; Linjama, Jukka (2012), "Electrostatic Modulated Friction as Tactile Feedback: Intensity Perception", Haptics: Perception, Devices, Mobility, and Communication, Springer Berlin Heidelberg, pp. 613–624, doi:10.1007/978-3-642-31401-8_54, ISBN 978-3-642-31400-1
^ Bau, Olivier; Israr, Ali; Poupyrev, Ivan; Harrison, Chris; Baskinger, Mark; May, Jason (3 October 2010) Electrostatic Vibration (formerly “TeslaTouch”), Disney Research, Retrieved 7 June 2015.
^ Bau, O.; Poupyrev, I.; Israr, A.; Harrison, C. (2010). "TeslaTouch: Electrovibration for Touch Surfaces" (PDF). Proceedings of the 23rd Annual ACM Symposium on User Interface Software and Technology (UIST). New York, NY: Association for Computing Machinery (ACM) (subscription required): 283–292. doi:10.1145/1866029.1866074. ISBN 978-1-4503-0271-5. S2CID 7033653. Retrieved 7 June 2015 – via chrisharrison.net. (free download, 4.2 MB).
^ a b Agarwal, A. K.; Nammi, K.; Kaczmarek, K. A.; Tyler, M. E.; Beebe, D. J. "A hybrid natural/artificial electrostatic actuator for tactile stimulation", Microtechnologies in Medicine & Biology, 2nd Annual International IEEE-EMB Special Topic Conference (2–4 May 2002), (Abstract) pp. 341–345, 2002. ISBN 0-7803-7480-0, doi:10.1109/MMB.2002.1002343. Retrieved 7 June 2015 (subscription required) (also readable at researchgate.net).
Retrieved from "https://en.wikipedia.org/w/index.php?title=Electrovibration&oldid=991394992"
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As depicted in the applet, Albertine finds herself in a very odd contraption. Sh
As depicted in the applet, Albertine finds herself in a very odd contraption. She sits in a reclining chair, in front of a large, compressed spring. T
As depicted in the applet, Albertine finds herself in a very odd contraption. She sits in a reclining chair, in front of a large, compressed spring. The spring is compressed 5.00 m from its equilibrium position, and a glass sits 19.8m from her outstretched foot.
a)Assuming that Albertine's mass is 60.0kg , what is
{\mu }_{k}
, the coefficient of kinetic friction between the chair and the waxed floor? Use
g=9.80\frac{m}{{s}^{2}}
for the magnitude of the acceleration due to gravity. Assume that the value of k found in Part A has three significant figures.Note that if you did not assume that k has three significant figures, it would be impossible to get three significant figures for
{\mu }_{k}
, since the length scale along the bottom of the applet does not allow you to measure distances to that accuracy with different values of k.
Mass of the albertine =60kg
Compressed length of the string (
\mathrm{△}x
)=5.0
\left(g\right)=9.8m/{s}^{2}
From the principle of law of conservation of energy,
\frac{1}{2}k\mathrm{△}{x}^{2}={\mu }_{k}mg\mathrm{△}s
{\mu }_{k}=\frac{k\mathrm{△}{x}^{2}}{2mg\mathrm{△}s}
=\frac{95.0\text{ }\frac{N}{m}{\left(5.0\text{ }m\right)}^{2}}{2\left(60.0\text{ }kg\right)\left(9.8\text{ }\frac{m}{{s}^{2}}\right)\left(19.8\text{ }m\right)}
=0.102
A force acting on a particle moving in the xy plane is given by
F=\left(2yi+{x}^{2}j\right)N
, where x and y are in meters. The particle moves from the origin to a final position having coordinates x=4.65 m and y=4.65 m, as in Figure.
a) Calculate the work done by F along OAC
b) Calculate the work done by F along OBC
c) Calculate the work done by F along OC
d) Is F conservative or nonconservative?
\theta
\sqrt[3]{7x-6}+2=6
\sqrt{2x-3}=x-3
The unstable nucleus uranium
-236
can be regarded as auniformly charged sphere of charge
Q=+92e
R=7.4×{10}^{-15}
m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium
-236
nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945.
A. Find the radii of the two "daughter" nuclei of charge
+46e.
B. In a simple model for the fission process, immediatelyafter the uranium
-236
nucleus has undergone fission the "daughter"nuclei are at rest and just touching. Calculate the kineticenergy that each of the "daughter" nuclei will have when they arevery far apart.
C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium
-236
nucleus. Calculate the energy released by thefission of 10.0 kg of uranium
-236
. The atomic mass ofuranium
-236
is 236 u, where 1
u=1
=1.66×{10}^{-27}
kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases
4.18×{10}^{12}
J when itexplodes).
Prove that there is a positive integer that equals the sum of the positive integers not exceeding it. Is your proof constructive or nonconstructive?
|
A\mathrm{sin}\left(x\right)
A
x
t
n
p
p
p
The trace=n option specifies that a number of previous frames of the animation be kept visible. When
n
n+1
n=5
When
is a list of integers, then the frames in those positions are the frames that remain visible. Each integer in
n=0
\mathrm{with}\left(\mathrm{plots}\right):
\mathrm{animate}\left(\mathrm{plot},[A{x}^{2},x=-4..4],A=-3..3\right)
\mathrm{animate}\left(\mathrm{plot},[A{x}^{2},x=-4..4],A=-3..3,\mathrm{trace}=5,\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot},[A{x}^{2},x=-4..4],A=-3..3,\mathrm{trace}=[30,35,40,45,50],\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot3d},[A\left({x}^{2}+{y}^{2}\right),x=-3..3,y=-3..3],A=-2..2,\mathrm{style}=\mathrm{patchcontour}\right)
\mathrm{animate}\left(\mathrm{implicitplot},[{x}^{2}+{y}^{2}={r}^{2},x=-3..3,y=-3..3],r=1..3,\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(\mathrm{implicitplot},[{x}^{2}+Axy-{y}^{2}=1,x=-2..2,y=-3..3],A=-2..2,\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(\mathrm{plot},[[\mathrm{sin}\left(t\right),\mathrm{sin}\left(t\right)\mathrm{exp}\left(-\frac{t}{5}\right)],t=0..x],x=0..6\mathrm{\pi },\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot},[[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t=0..A]],A=0..2\mathrm{\pi },\mathrm{scaling}=\mathrm{constrained},\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot},[[\frac{1-{t}^{2}}{1+{t}^{2}},\frac{2t}{1+{t}^{2}},t=-10..A]],A=-10..10,\mathrm{scaling}=\mathrm{constrained},\mathrm{frames}=50,\mathrm{view}=[-1..1,-1..1]\right)
\mathrm{opts}≔\mathrm{thickness}=5,\mathrm{numpoints}=100,\mathrm{color}=\mathrm{black}:
\mathrm{animate}\left(\mathrm{spacecurve},[[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),\left(2+\mathrm{sin}\left(A\right)\right)t],t=0..20,\mathrm{opts}],A=0..2\mathrm{\pi }\right)
B≔\mathrm{plot3d}\left(1-{x}^{2}-{y}^{2},x=-1..1,y=-1..1,\mathrm{style}=\mathrm{patchcontour}\right):
\mathrm{opts}≔\mathrm{thickness}=5,\mathrm{color}=\mathrm{black}:
\mathrm{animate}\left(\mathrm{spacecurve},[[t,t,1-2{t}^{2}],t=-1..A,\mathrm{opts}],A=-1..1,\mathrm{frames}=11,\mathrm{background}=B\right)
\mathrm{animate}\left(\mathrm{ball},[0,\mathrm{sin}\left(t\right)],t=0..4\mathrm{\pi },\mathrm{scaling}=\mathrm{constrained},\mathrm{frames}=100\right)
\mathrm{sinewave}≔\mathrm{plot}\left(\mathrm{sin}\left(x\right),x=0..4\mathrm{\pi }\right):
\mathrm{animate}\left(\mathrm{ball},[t,\mathrm{sin}\left(t\right)],t=0..4\mathrm{\pi },\mathrm{frames}=50,\mathrm{background}=\mathrm{sinewave},\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(\mathrm{ball},[t,\mathrm{sin}\left(t\right)],t=0..4\mathrm{\pi },\mathrm{frames}=50,\mathrm{trace}=10,\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(F,[\mathrm{\theta }],\mathrm{\theta }=0..2\mathrm{\pi },\mathrm{background}=\mathrm{plot}\left([\mathrm{cos}\left(t\right)-2,\mathrm{sin}\left(t\right),t=0..2\mathrm{\pi }]\right),\mathrm{scaling}=\mathrm{constrained},\mathrm{axes}=\mathrm{none}\right)
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Rewrite the Pythagorean Identity as many ways as you can. One way to change the identity would be to solve the equation for one trigonometric function. Another way is to use factoring. Try several different ways.
Possibilities include:
\sin^2(x)=1-\cos^2\left(x\right)
\sin^2\left(x\right)=\left(1-\cos x\right)\left(1+\cos x\right)
Find other variations. There are many!
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Multiple Periodic Solutions for Discrete Nicholson’s Blowflies Type System
2014 Multiple Periodic Solutions for Discrete Nicholson’s Blowflies Type System
Hui-Sheng Ding, Julio G. Dix
This paper is concerned with the existence of multiple periodic solutions for discrete Nicholson’s blowflies type system. By using the Leggett-Williams fixed point theorem, we obtain the existence of three nonnegative periodic solutions for discrete Nicholson’s blowflies type system. In order to show that, we first establish the existence of three nonnegative periodic solutions for the
n
-dimensional functional difference system
y\left(k+1\right)=A\left(k\right)y\left(k\right)+f\left(k, y\left(k-\tau \right)\right), k\in ℤ
A\left(k\right)
is not assumed to be diagonal as in some earlier results. In addition, a concrete example is also given to illustrate our results.
Hui-Sheng Ding. Julio G. Dix. "Multiple Periodic Solutions for Discrete Nicholson’s Blowflies Type System." Abstr. Appl. Anal. 2014 (SI54) 1 - 6, 2014. https://doi.org/10.1155/2014/659152
Hui-Sheng Ding, Julio G. Dix "Multiple Periodic Solutions for Discrete Nicholson’s Blowflies Type System," Abstract and Applied Analysis, Abstr. Appl. Anal. 2014(SI54), 1-6, (2014)
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Easement - Uncyclopedia, the content-free encyclopedia
For those without comedic tastes, the so-called experts at Wikipedia think they have an article about Easement.
In classical architecture, an easement is a narrow Teflon™-coated passageway which connects two or more empty spaces. Easements are regularly incorporated into the infrastructure of many of today's hectic office buildings for the sole purpose of facilitating rapid deployment of lawyers and other slippery objects.
1 Origin and Proliferation of the Easement
2 Dimensions and Routing
3 Easement Procedures
4 Legal Restrictions and Liability
Origin and Proliferation of the Easement[edit]
The origin of the easement is lost in the mists of time itself, which thus conveniently mitigates against the oh-so-predictable "was invented by" cliché. At any rate, easements quickly proliferated all across the Wild Wild West of the American western territories, which allowed Californians and New Yorkers to sue each other directly for the first time in recorded history. Certain lawyers In Chicago are complete bitches who used to be nurses. They were so ugly that no doctors every hit on her. She became a lawyer and enjoys getting back at doctors. She smells like garlic.
Later that year, the Trans-Atlantic Easement was christened with a bottle of Jewish wine and opened to the legal profession with great fanfare. Unfortunately, the mammoth structure unintentionally traversed the territorial waters of France at the time, which was a BIG NO-NO; and therefore had to be dismantled and sold for worthless scrap metal by 2003 (per United Nations Security Council Resolution 1441).
Dimensions and Routing[edit]
The typical modern easement is cylindrical in cross-section, 35 centimeters wide, and many hundreds of feet in length. Multiple branch points within the Easement Transport System (EST) are regulated by remotely-controlled sluices, which settings are programmed well in advance via a convenient electronic touchpad near the all-important entry point.
Easement Procedures[edit]
The object to be transported is stripped down, thoroughly lubricated, and loaded into one end of the easement using a large plunger-type mechanism. After inputting the desired exit coordinates, the object is then propelled by a sudden blast of compressed helium up to speeds of several hundreds of miles per hour. Within a fraction of a second, the object arrives at the easement's terminus and is shot clear across the room into a gigantic beanbag, which minimizes the risk of damage to innocent third parties.
Legal Restrictions and Liability[edit]
Due to draconian building codes and other legal restrictions, most easements are designed to be utilized for one-way travel only, and thus need to be constructed either in parallel pairs, or one enormously large looping circuit.
Persuant to recent Federal Claims Court rulings, all easement-related risk is assumed by the user. Personal injury due to failure of the user to read and adhere to the enclosed operating instructions cannot be legally held against the makers of the easement. In 1993, a Texan congressional lobbyist injected himself into the world's largest cyclic easement while totally forgetting to program the proper exit codes. Today, that unfortunate lawyer is still travelling in a 54-mile long circular path at nearly the speed of light, is continuing to accelerate, and, to date, has experienced approximately 400 billion (
{\displaystyle 4.0\times 10^{11}}
) consecutive mind-blowing orgasms.
Retrieved from "https://uncyclopedia.com/w/index.php?title=Easement&oldid=6070774"
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Port Analysis of Antenna - MATLAB & Simulink Example - MathWorks Italia
Create Inverted-F Antenna
This example quantifies terminal antenna parameters, with regard to the antenna port. The antenna is a one-port network. The antenna port is a physical location on the antenna where an RF source is connected to it. The terminal port parameters supported in Antenna Toolbox™ are
The example uses a Planar Inverted-F Antenna (PIFA), performs the corresponding computational analysis, and returns all terminal antenna parameters listed above.
Create the default geometry for the PIFA antenna. The (small) red dot on the antenna structure is the feed point location where an input voltage generator is applied. It is the port of the antenna. In Antenna Toolbox™, all antennas are excited by a time-harmonic voltage signal with the amplitude of 1 V at the port. The port must connect two distinct conductors; it has an infinitesimally small width.
ant = invertedF;
To plot the antenna impedance, specify the frequency band over which the data needs to be plotted using impedance function. The antenna impedance is calculated as the ratio of the phasor voltage (which is simply 1) and the phasor current at the port.
impedance(ant, freq);
The plot displays the real part of the impedance, i.e. resistance as well as its imaginary part, i.e. reactance, over the entire frequency band. Antenna resonant frequency is defined as the frequency at which the reactance of the antenna is exactly zero. Looking at the impedance plot, we observe that the inverted-F antenna resonates at 1.74 GHz. The resistance value at that frequency is about 20
\Omega
. The reactance values for the antenna are negative (capacitive) before resonance and become positive (inductive) after resonance, indicating that it is the series resonance of the antenna (modeled by a series RLC circuit). If the impedance curve goes from positive reactance to negative, it is the parallel resonance [1] (modelled by a parallel RLC circuit).
To plot the return loss of an antenna, specify the frequency band over which the data needs to be plotted. Return loss is the measure of the effectiveness of power delivery from the transmission line to an antenna. Quantitatively, the return loss is the ratio, in dB, of the power sent towards the antenna and the power reflected back. It is a positive quantity for passive devices. A negative return loss is possible with active devices [2].
returnLoss(ant, freq);
The return loss introduced above is rarely used for the antenna analysis. Instead, a reflection coefficient or
{S}_{11}
in dB is employed, which is also often mistakenly called the "return loss" [2]. In fact, the reflection coefficient in dB is the negative of the return loss as seen in the following figure. The reflection coefficient describes a relative fraction of the incident RF power that is reflected back due to the impedance mismatch. This mismatch is the difference between the input impedance of the antenna and the characteristic impedance of the transmission line (or the generator impedance when the transmission line is not present). The characteristic impedance is the reference impedance. The sparameters function used below accepts the reference impedance as its third argument. The same is valid for the returnLoss function. By default, we assume the reference impedance of 50
\Omega
S = sparameters(ant, freq);
The VSWR of the antenna can be plotted using the function vswr used below. A VSWR value of 1.5:1 means that a maximum standing wave amplitude is 1.5 times greater than the minimum standing wave amplitude. The standing waves are generated because of impedance mismatch at the port. The VSWR is expressed through the reflection coefficient as
\left(1+|{S}_{11}|\right)/\left(1-|{S}_{11}|\right)
Bandwidth is a fundamental antenna parameter. The antenna bandwidth is the band of frequencies over which the antenna can properly radiate or receive power. Often, the desired bandwidth is one of the critical parameters used to decide an antenna type. Antenna bandwidth is usually the frequency band over which the magnitude of the reflection coefficient is below -10 dB, or the magnitude of the return loss is greater than 10 dB, or the VSWR is less than approximately 2. All these criteria are equivalent. We observe from the previous figures that the PIFA has no operating bandwidth in the frequency band of interest. The bandwidth is controlled by the proper antenna design. Sometimes, the reference impedance may be changed too. In the impedance plot we observe that the resistance of the present antenna is close to 20
\Omega
at the resonance. Choose the reference impedance of 20
\Omega
instead of 50
\Omega
and plot the reflection coefficient.
S = sparameters(ant, freq, 20);
Now, we observe the reflection coefficient of less than -10 dB over the frequency band from 1.71 to 1.77 GHz. This is the antenna bandwidth. The same conclusion holds when using the VSWR or return loss calculations.
[1] C. A. Balanis, Antenna Theory. Analysis and Design, p. 514, Wiley, New York, 3rd Edition, 2005.
[2] T. S. Bird, "Definition and Misuse of Return Loss," Antennas and Propagation Magazine, April 2009.
Current Visualization on Antenna Surface | Analysis of Monopole Impedance
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The linear fractional model on the ball | EMS Press
Given a holomorphic self-map
\varphi
of the ball of
\mathbb{C}^N
, we study whether there exists a map
\sigma
and a linear fractional transformation
A
\sigma\circ\varphi=A\circ\sigma
. This is an important result when
N=1
with a great number of applications. We extend this result to the multi-dimensional setting for a large class of maps. Applications to commuting holomorphic self-maps are given.
Frédéric Bayart, The linear fractional model on the ball. Rev. Mat. Iberoam. 24 (2008), no. 3, pp. 765–824
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A solution to the Monge transport problem for Brownian martingales
March 2021 A solution to the Monge transport problem for Brownian martingales
Nassif Ghoussoub, Young-Heon Kim, Aaron Zeff Palmer
Nassif Ghoussoub,1 Young-Heon Kim,1 Aaron Zeff Palmer1
We provide a solution to the problem of optimal transport by Brownian martingales in general dimensions whenever the transport cost satisfies certain subharmonic properties in the target variable as well as a stochastic version of the standard “twist condition” frequently used in deterministic Monge transport theory. This setting includes, in particular, the case of the distance cost
\mathit{c}\left(\mathit{x},\mathit{y}\right)=|\mathit{x}-\mathit{y}|
. We prove existence and uniqueness of the solution and characterize it as the first time Brownian motion hits a barrier that is determined by solutions to a corresponding dual problem.
The first two authors are partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
Nassif Ghoussoub. Young-Heon Kim. Aaron Zeff Palmer. "A solution to the Monge transport problem for Brownian martingales." Ann. Probab. 49 (2) 877 - 907, March 2021. https://doi.org/10.1214/20-AOP1462
Received: 1 July 2019; Revised: 1 April 2020; Published: March 2021
Keywords: Optimal transport , Skorokhod embedding , variational inequality
Nassif Ghoussoub, Young-Heon Kim, Aaron Zeff Palmer "A solution to the Monge transport problem for Brownian martingales," The Annals of Probability, Ann. Probab. 49(2), 877-907, (March 2021)
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Legendrian and transverse twist knots | EMS Press
John B. Etnyre
In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the
m(5_2)
knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least
n
different Legendrian representatives with maximal Thurston--Bennequin number of the twist knot
K_{-2n}
with crossing number
2n+1
. In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that
K_{-2n}
\lceil\frac{n^2}2\rceil
Legendrian representatives with maximal Thurston–Bennequin number, and
\lceil\frac{n}{2}\rceil
transverse representatives with maximal self-linking number. Our techniques include convex surface theory, Legendrian ruling invariants, and Heegaard–Floer homology.
Lenhard L. Ng, John B. Etnyre, Vera Vértesi, Legendrian and transverse twist knots. J. Eur. Math. Soc. 15 (2013), no. 3, pp. 969–995
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DiscriminantSet - Maple Help
Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : ParametricSystemTools Subpackage : DiscriminantSet
compute the discriminant set of a variety
DiscriminantSet(F, d, R)
The command DiscriminantSet(F, d, R) returns the discriminant set of a polynomial system with respect to a positive integer, which is a constructible set.
d is positive and less than the number of variables in R.
Given a positive integer d, the last d variables will be regarded as parameters.
A point P is in the discriminant set of F if and only if after specializing F at P, the polynomial system F has no solution or an infinite number of solutions.
This command is part of the RegularChains[ParametricSystemTools] package, so it can be used in the form DiscriminantSet(..) only after executing the command with(RegularChains[ParametricSystemTools]). However, it can always be accessed through the long form of the command by using RegularChains[ParametricSystemTools][DiscriminantSet](..).
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):
\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):
R≔\mathrm{PolynomialRing}\left([x,a,b,c]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
Consider the following general quadratic polynomial F.
F≔a{x}^{2}+bx+c
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{a}\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}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{c}
You can see that when F as a univariate polynomial in x has no solution (over the complex number field) or has infinitely many number solutions.
\mathrm{ds}≔\mathrm{DiscriminantSet}\left([F],3,R\right)
\textcolor[rgb]{0,0,1}{\mathrm{ds}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}
\mathrm{ds}≔\mathrm{MakePairwiseDisjoint}\left(\mathrm{ds},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{ds}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}
\mathrm{Info}\left(\mathrm{ds},R\right)
[[\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}{1}]]
The first case indicates that there are infinite number of solutions; the second one indicates that there is no solution.
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Reflections of regular maps and Riemann surfaces | EMS Press
Adnan Melekoğlu
A compact Riemann surface of genus
g
is called an M-surface if it admits an anti-conformal involution that fixes
g+1
simple closed curves, the maximum number by Harnack's Theorem. Underlying every map on an orientable surface there is a Riemann surface and so the conclusions of Harnack's theorem still apply. Here we show that for each genus
g ϯ 1
there is a unique M-surface of genus
g
that underlies a regular map, and we prove a similar result for Riemann surfaces admitting anti-conformal involutions that fix
g
curves.
Adnan Melekoğlu, David Singerman, Reflections of regular maps and Riemann surfaces. Rev. Mat. Iberoam. 24 (2008), no. 3, pp. 921–939
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Superior highly composite number - Wikipedia
In mathematics, a superior highly composite number is a natural number which has the highest ratio of its number of divisors to some positive power of itself than any other number. It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer.
Divisor function d(n) up to n = 250
The first 10 superior highly composite numbers and their factorization are listed.
2 6 2 ⋅ 3 1,1 22 4 6
3 12 22 ⋅ 3 2,1 3×2 6 2 ⋅ 6
4 60 22 ⋅ 3 ⋅ 5 2,1,1 3×22 12 2 ⋅ 30
5 120 23 ⋅ 3 ⋅ 5 3,1,1 4×22 16 22 ⋅ 30
6 360 23 ⋅ 32 ⋅ 5 3,2,1 4×3×2 24 2 ⋅ 6 ⋅ 30
7 2520 23 ⋅ 32 ⋅ 5 ⋅ 7 3,2,1,1 4×3×22 48 2 ⋅ 6 ⋅ 210
8 5040 24 ⋅ 32 ⋅ 5 ⋅ 7 4,2,1,1 5×3×22 60 22 ⋅ 6 ⋅ 210
9 55440 24 ⋅ 32 ⋅ 5 ⋅ 7 ⋅ 11 4,2,1,1,1 5×3×23 120 22 ⋅ 6 ⋅ 2310
10 720720 24 ⋅ 32 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 13 4,2,1,1,1,1 5×3×24 240 22 ⋅ 6 ⋅ 30030
Plot of the number of divisors of integers from 1 to 1000. Highly composite numbers are labelled in bold and superior highly composite numbers are starred. In the SVG file, hover over a bar to see its statistics.
For a superior highly composite number n there exists a positive real number ε such that for all natural numbers k smaller than n we have
{\displaystyle {\frac {d(n)}{n^{\varepsilon }}}\geq {\frac {d(k)}{k^{\varepsilon }}}}
and for all natural numbers k larger than n we have
{\displaystyle {\frac {d(n)}{n^{\varepsilon }}}>{\frac {d(k)}{k^{\varepsilon }}}}
where d(n), the divisor function, denotes the number of divisors of n. The term was coined by Ramanujan (1915).[1]
For example, the number with the most divisors per square root of the number itself is 12; this can be demonstrated using some highly composites near 12:
{\displaystyle {\frac {2}{\sqrt {2}}}\approx 1.414,{\frac {3}{\sqrt {4}}}=1.5,{\frac {4}{\sqrt {6}}}\approx 1.633,{\frac {6}{\sqrt {12}}}\approx 1.732,{\frac {8}{\sqrt {24}}}\approx 1.633,{\frac {12}{\sqrt {60}}}\approx 1.549}
The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A002201 in the OEIS) are also the first 15 colossally abundant numbers, which meet a similar condition based on the sum-of-divisors function rather than the number of divisors. Neither set, however, is a subset of the other.
2 Superior highly composite radices
All superior highly composite numbers are highly composite. This is easy to prove: if there is some number k that has the same number of divisors as n but is less than n itself (i.e.
{\displaystyle d(k)=d(n)}
{\displaystyle k<n}
{\displaystyle {\frac {d(k)}{k^{\varepsilon }}}>{\frac {d(n)}{n^{\varepsilon }}}}
for all positive ε, so if a number "n" is not highly composite, it cannot be superior highly composite.
An effective construction of the set of all superior highly composite numbers is given by the following monotonic mapping from the positive real numbers.[2] Let
{\displaystyle e_{p}(x)=\left\lfloor {\frac {1}{{\sqrt[{x}]{p}}-1}}\right\rfloor \quad }
for any prime number p and positive real x. Then
{\displaystyle \quad s(x)=\prod _{p\in \mathbb {P} }p^{e_{p}(x)}\quad }
is a superior highly composite number.
Note that the product need not be computed indefinitely, because if
{\displaystyle p>2^{x}}
{\displaystyle e_{p}(x)=0}
, so the product to calculate
{\displaystyle s(x)}
can be terminated once
{\displaystyle p\geq 2^{x}}
Also note that in the definition of
{\displaystyle e_{p}(x)}
{\displaystyle 1/x}
{\displaystyle \varepsilon }
in the implicit definition of a superior highly composite number.
Moreover, for each superior highly composite number
{\displaystyle s^{\prime }}
exists a half-open interval
{\displaystyle I\subset \mathbb {R} ^{+}}
{\displaystyle \forall x\in I:s(x)=s^{\prime }}
This representation implies that there exist an infinite sequence of
{\displaystyle \pi _{1},\pi _{2},\ldots \in \mathbb {P} }
such that for the n-th superior highly composite number
{\displaystyle s_{n}}
{\displaystyle s_{n}=\prod _{i=1}^{n}\pi _{i}}
{\displaystyle \pi _{i}}
are 2, 3, 2, 5, 2, 3, 7, ... (sequence A000705 in the OEIS). In other words, the quotient of two successive superior highly composite numbers is a prime number.
Superior highly composite radicesEdit
The first few superior highly composite numbers have often been used as radices, due to their high divisibility for their size. For example:
Bigger SHCNs can be used in other ways. 120 appears as the long hundred, while 360 appears as the number of degrees in a circle.
^ Ramanujan (1915); see also URL http://wwwhomes.uni-bielefeld.de/achim/hcn.dvi
Ramanujan, S. (1915). "Highly composite numbers" (PDF). Proc. London Math. Soc. Series 2. 14: 347–409. doi:10.1112/plms/s2_14.1.347. JFM 45.1248.01. Reprinted in Collected Papers (Ed. G. H. Hardy et al.), New York: Chelsea, pp. 78–129, 1962
Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. pp. 45–46. ISBN 1-4020-4215-9. Zbl 1151.11300.
Weisstein, Eric W. "Superior highly composite number". MathWorld.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Superior_highly_composite_number&oldid=1067703630"
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Creating a random graph from a string
I have been toying around with the idea of creating art using pseudorandom number generators. In this post, I talk about a script that takes as input a string and outputs a random graph. Here are some examples:
In summary, I transform the string to ASCII character by character, add up all these values to get an integer, I plug it as a seed for numpy's pseudorandom generator, and I construct a graph by sampling normal distributions and joining nearby neighbors. I will go through the code function by function. Here's the whole code in gist, if you're in a rush.
The core of the idea is to transform a string to an integer in a deterministic way. I started by using Python's own hash function, but found out quickly that it behaved (pseudo)randomly between different sessions. I finally settled for transforming each character in the string to its ASCII code, and then add up all these to form an integer.
def parse_string_to_int(string): chars_as_ints = [ord(char) for char in string] string_as_int = sum(chars_as_ints) return string_as_int
Notice that Python has its own way of transforming characters to ASCII code: the ord function.
This way of encoding strings to integers comes with a funny consequence: anagrams render the same graph. There's also a high chance for collisions (strings that end up in the same integer, and thus the same graph). It hasn't happened with my friends' names so far, though.
As a small mathematical note, if
\Sigma
is the alphabet and
\Sigma^*
is the set of all possible words, there are plenty of functions
f\colon \Sigma^{ *}\to \mathbb{Z}
to experiment with. This one happens to be commutative in the strings, but we could think about many others that wouldn't be.
Integer as seed
Now that we have a way of encoding strings as integers, we can use these integers as seeds in a pseudorandom number generator. If you're new to the world of computing, it might come as a surprise that random numbers don't exist (or, at least, can't be created using a computer unless you have a Geiger counter or other cool stuff). Computers generate so-called pseudorandom numbers: numbers that appear to be random, but are actually being created in a deterministic fashion. These algorithms start with a number, called the seed, and transform it in intelligent ways to create a very large sequence of (seemingly) random numbers.
Thus, you have a way of deterministically getting a sequence of random numbers by just specifying the seed at the start. In numpy, this can be made by passing a positive integer to np.random.seed:
import numpy as np np.random.seed(1) print(np.random.randint(0, 10, size=(1, 15))) np.random.seed(1) # to re-start the sequence. print(np.random.randint(0, 10, size=(1, 15))
The output of this code will be array([5, 8, 9, 5, 0, 0, 1, 7, 6, 9]) both times.
So, we can use the string as a seed by converting it to an integer and passing it to numpy's random generator:
int_seed = parse_string_to_int(string) np.random.seed(int_seed)
Sampling the random points
Once we have a deterministic way of getting random numbers according to the string, we can sample points from a normal distribution with mean 0 and variance 1. I start by getting the amount as a random integer between 100 and 200, and then I sample said amount in each axis independently:
def create_random_points(string=None): if string: int_seed = parse_string_to_int(string) np.random.seed(int_seed) amount = np.random.randint(100, 200) random_points_x = np.random.normal(0, 1, (amount, 1)) random_points_y = np.random.normal(0, 1, (amount, 1)) random_points = np.concatenate([random_points_x, random_points_y], axis=1) return random_points
Notice that there's plenty of room for experimentation: you could go for another distribution, you could go for more points, you could sample different distributions in each axis, you could go 3D... I settled for the normal distribution because of the symmetric, triangular patterns that emerge.
I decided to fix a radius
r
around each point, and then join said point to all the neighbors that are in the closed ball of radius
r
(using the Euclidean metric). I compute all the lines I need to construct by sweeping through the points twice:
def get_lines(random_points, radius=0.5): lines = [] for i, point_1 in enumerate(random_points): for j, point_2 in enumerate(random_points[i:]): if i == j: continue if np.sqrt(sum((point_1 - point_2) ** 2)) < radius: lines.append(np.array([point_1, point_2])) return lines
There's also plenty of room for experimentation here. I went for a fixed radius of
1/2
(since I considered appropiate given that we were sampling from normal distributions of variance 1), but you could easily go for a different, dynamic radius for each point. You could also go for different metrics.
In order to make it easy to create grids and subplots, I implemented a function that takes an axis and plots the random points and corresponding lines according to a string. It gets the random points corresponding to the string, it gets the lines, figures out the limits in
x
y
and plots the graph:
def plot_in_axis(ax, seed, radius=0.5, size=3): random_points = create_random_points(seed) lines = get_lines(random_points, radius) xlims = [np.min(random_points[:, 0]) - radius, np.max(random_points[:, 0]) + radius] ylims = [np.min(random_points[:, 1]) - radius, np.max(random_points[:, 1]) + radius] lims = [min([xlims[0], ylims[0]]), max([xlims[1], ylims[1]])] for line in lines: ax.plot(line[:, 0], line[:, 1], "-k", alpha=0.4) ax.scatter(random_points[:, 0], random_points[:, 1], c="k", s=size) ax.tick_params( top=False, bottom=False, labelbottom=False, right=False, left=False, labelleft=False, ) ax.set_xlim(*lims) ax.set_ylim(*lims)
As you can see, I settled for plotting the graph in a square given by the maximum spread on both directions.
Putting everything together with click
For the end product, I wanted a script plot_graph.py that would take the string as argument. In order to do so, I used click. With click, you can define arguments and options for your Python script really easily. This is the main function of plot_graph.py:
def clean_string(string): for symbol in [" ", ",", ";", ":", ".", "\n", "\r", "\t"]: string = string.replace(symbol, "") return string @click.command() @click.argument("string", default=None, type=str) @click.option("--radius", default=0.5, type=float) @click.option("--size", default=5, type=int) def main(string, radius, size): _, ax = plt.subplots(1, 1, figsize=(10, 10)) plot_in_axis(ax, string, radius, size) ax.set_xlabel("\n" + string, fontsize=15) plt.savefig(f"{clean_string(string)}.jpg", format="jpg", dpi=150) plt.show() if __name__ == "__main__": main() # pylint: disable=no-value-for-parameter
To clean the string, I implemented a simple function that takes out spaces, tabs, newlines and some punctuations.
Now you can call this string by just writing python plot_graph.py "The string you want to transform here". Be wary: it will overwrite any photos with said string as name in your folder. You can also modify the radius of the connections and the size of the nodes by passing the options --radius= and --size= respectively.
I implemented a small script to take strings, transform them to integers, and use those integers to sample random points in the plane. After that, I joined points that were a given fixed radius apart. There's plenty of room for improvement or experimentation. Here's a link to a gist with the full code.
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EIT Certification Heat Transfer
Cooling a CPU
A square silicon chip (k = 150 W/m∙K) is of width w = 7 mm on a side and of thickness t = 2 mm. The chip is mounted in a substrate such that its side and back surfaces are insulated, while the front surface is exposed to a coolant. If 60 W are being dissipated in circuits mounted to the back surface of the chip, what is the steady-state temperature difference between back and front surfaces?
P=q=kA\frac{\Delta T}{t}
Assumptions: (1) Steady state conditions, (2) Constant properties, (3) Uniform heat dissipation, (4) Negligible heat loss from back and sides, (5) One-dimensional conduction chip.
All the electrical power dissipated at the chip's back surface is transferred by conduction. From Fourier's law:
P=q=kA\frac{\Delta T}{t}
Rearranging the formula to solve for the temperature difference:
\Delta T=\frac{t\cdot P}{kw^{2}}=\frac{0.002m\cdot 60W}{150W/m\cdot K(0.007m)^{2}}=16.33K
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I'll be your instructor for this small passion project of mine. Always found the lack of MechE interview prep resources frustrating, especially compared to our coding counterparts. So, I created this centralized solution to practice for my own technical interviews, and to continually refresh the undergrad fundamentals that aren't always used while in industry. ¯\_(ツ)_/¯
Figured others could benefit, so please join me! No strings attached, seriously. I just enjoy optimizing designs/processes, building products, & trying new things :)
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Seamless, simple tool designed to encourage consistent and manageable practice daily. No need to set reminders, determine a study plan, or visit additional apps/websites - everything is automatically sent to your existing email. It integrates with your schedule with customizable email settings. A little build up every day prevents cramming towards the end! It's 100% free, and you can opt-out anytime.
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Coincidence and Calculation of some Strict $s$-Numbers | EMS Press
Coincidence and Calculation of some Strict
s
The paper considers the so-called strict
s
-numbers, which form an important subclass of the family of all
s
-numbers. For operators acting between Hilbert spaces the various
-numbers are known to coincide: here we give examples of linear maps
T
and non-Hilbert spaces
X, Y
such that all strict
s
T : X \to Y
coincide. The maps considered are either simple integral operators acting in Lebesgue spaces or Sobolev embeddings; in these cases the exact value of the strict
s
-numbers is determined.
David E. Edmunds, Jan Lang, Coincidence and Calculation of some Strict
s
-Numbers. Z. Anal. Anwend. 31 (2012), no. 2, pp. 161–181
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Experimental evidence refuting the assumption of phosphorus-31 nuclear-spin entanglement-mediated consciousness
Rong Chen, Na Li, Hao Qian, Rui-Han Zhao, Shi-Hai Zhang
Woon-Man Kung, Cheng-Jen Chang, Tzu-Yung Chen, Muh-Shi Lin
Yan Zhang, Na Luo, Fei-Fei Hong, Cai-Hong Yang, ... Khan Aashiq
Bioinformatic analysis of a microRNA regulatory network in Huntington’s disease
Zhi-Min Wang, Xiao-Yu Dong, Shu-Yan Cong
Fasudil reduces
\beta
-amyloid levels and neuronal apoptosis in APP/PS1 transgenic mice via inhibition of the Nogo-A/NgR/RhoA signaling axis
Min-Fang Guo, Hui-Yu Zhang, Pei-Jun Zhang, Xiao-Qin Liu, ... Cun-Gen Ma
Danggui Sini decoction protects against oxaliplatin-induced peripheral neuropathy in rats
Rong Ding, Yue Wang, Ji-Ping Zhu, Wu-Guang Lu, ... Jie-Ge Huo
Learning processes in elementary nervous systems
{}^{\mathrm{§}}
Protective effect of baicalin against cognitive memory dysfunction after splenectomy in aged rats and its underlying mechanism
Jian-Nan Zhang, Hong-Mei Zhou, Chen-Hao Jiang, Jiao Liu, Liang-Yu Cai
Matter, mind and consciousness: from information to meaning
Can we improve the prediction of complications and outcome in aneurysmal subarachnoid hemorrhage? The clinical implications of serum proteomics
Shadi Bsat, Hani Chanbour, Safwan Alomari, Charbel Moussalem, ... Tarek Sunna
Ursula Werneke, Petra Truedson-Martiniussen, Henrik Wikström, Michael Ott
Pupil tracks statistical regularities: behavioral and neural implications
Shamini Warda, Shubham Pandey
New molecular insights, innovative technologies, and medical approaches in the “Exploration of mechanisms in cortical plasticity”
Rodolfo G. Gatto, Tetsuya Asakawa
Letter to the editor regarding “TGM6 variants in Parkinson’s disease: clinical findings and functional evidence”
Ashley Hall, John P. Quinn, Kimberley J. Billingsley, on behalf of the International Parkinson’s Disease Genomics Consortium (IPDGC)
In reply to the letter to the editor regarding “TGM6 variants in Parkinson’s disease: clinical findings and functional evidence”
Kui Chen, Yan Tan, Yan-Xin Zhao
|
What is the difference between childhood being aged as described in my mother at sixty six?
Anyone of you watches The Ellen show ?
Chinmay Saini
Prepare a creative chart or model based on any topic related to language (Grammer) and literature.
how was m hamel class different the day franz went late to school?
Some important classroom rules, a student should follow.
What is the meaning of troglodyte?
SORRY ITS MY LAST QUESTION PLSS TELL WHAT IS ACCURACY AND FLUENCY IN WRITING
in q5 of eng paper i wrote missing words correct but i have not written word before and word after wrong. Will i get full or no marks
Lamiya Ummayu
what fed the king's heart in ozymandias
I know the complete story of novel helen keller but i have forgotten some places name as i have studied them 2 months ago.Should i again read the novel ?
You are one of the crews on the ship. After the mariner kills the Albatross, you have to suffer and there is no hope of survival. Write a diary entry describing your feelings at this point of time. no links guys
why did Brutus justify Caesar's assassinations? no links guys
Q. Identity the part of speech of the underlined words
\to
Everyone was present but the secretary
\to
Arjun runs like a race horse
\to
kep silence, please
\to
you have no right to stop me from doing this
\to
He has come first now
\to
The water the plants regularly
\to
The charminor is historical monument.
rearrange our social life /could be/or/taking care/it/our work/ family's need/ of our
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A couple of metrics in generalrepytivity
As a small tutorial on generalrepytivity1, this blogpost explains how to create a metric tensor in generalrepytivity and, with it, how to get all the usual geometric invariants one is interested in (such as the Riemann tensor, the Ricci tensor and the scalar curvature). You can find all the code in this article in this jupyter notebook.
generalrepytivity is now on pypi, so installing it is easy using pip:
pip install generalrepytivity
We heavily recommend importing it along with sympy (the symbolic computation library of python):
import generalrepytivity as gr import sympy
Tensors come with a fancy LaTeX printing function, so if you're working on a jupyter notebook I recommend setting the printing function of sympy to use latex:
sympy.init_printing(use_latex=True)
Creating a tensor in generalrepytivity
To create a Tensor object, one must specify three things:
coordinates, which are a list of sympy symbols.
_type, a tuple of integers (p,q).
values, a dictionary of the non-zero values (whose keys are tuples of multiindices).
For example, say you want to create the following tensor in
\mathbb{R}^4
t, x, y, z
\Gamma = (x^2 + y^2)\frac{\partial}{\partial t}\otimes \frac{\partial}{\partial x}\otimes dx\otimes dz + \sin(t)\frac{\partial}{\partial x}\otimes \frac{\partial}{\partial y}\otimes dt \otimes dt
in this case, we would need the sympy symbols t, x, y, z for the coordinates, the _type would be (2,2) (because
\Gamma
is a (2,2)-tensor) and for the values note that the components of
\Gamma
in these coordinates are the following:
\Gamma^{0, 1}_{1, 3} = x^2 + y^2
\Gamma^{1, 2}_{0, 0} = \sin(t)
so the dictionary of values should look like this:
values = { ((0, 1), (1, 3)): x**2 + y**2, ((1, 2), (0, 0)): sympy.sin(t) }
t, x, y, z = sympy.symbols('t x y z') values = { ((0,1), (1, 3)): x**2 + y**2, ((1,2), (0, 0)): sympy.sin(t) } Gamma = gr.Tensor([t, x, y, z], (2,2), values)
and you can access the components of the tensor by indexing:
Gamma[(0,1), (1,3)] == x**2 + y**2 True
g
is just a (0,2)-tensor, so we could create it using a values dict just like we did above but, because (0,2)-tensors are so important (and because they can be represented by matrices), generalrepytivity has a simpler way of creating them. The function to use is gr.get_tensor_from_matrix, which has the following syntax:
gr.get_tensor_from_matrix(matrix, coordinates)
where matrix is a square sympy matrix and the coordinates are just like above (i.e. a list of sympy symbols).
In this blogpost we will deal with three different solutions to Einstein's equations: Gödel's metric, Schwarzschild's metric and FRLW.
In 1949, Gödel proposed a solution to Einstein's Field Equations which described a rotating universe in which closed world-lines are possible (that is, you could in some way influence the past). Because of this violation of causality, Gödel's metric is just interesting from a theoretical perspective, not from a modeling one.
Gödel's metric is
g = a^2(dx_0\otimes dx_0 - dx_1\otimes dx_1 + (e^{2x_1}/2)dx_2\otimes dx_2 - dx_3\otimes dx_3 + e^{x_1}dx_0\otimes dx_2 + e^{x_1}dx_2 \otimes dx_0)
To enter it into generalrepytivity, we could use the matrix representation:
x0, x1, x2, x3 = sympy.symbols('x_0 x_1 x_2 x_3') a = sympy.Symbol('a') A = a**2 * sympy.Matrix([ [1, 0, sympy.exp(x1), 0], [0, -1, 0, 0], [sympy.exp(x1), 0, sympy.exp(2*x1)/2, 0], [0,0,0,-1]]) g_godel = gr.get_tensor_from_matrix(A, [x0, x1, x2, x3])
Schwarzschild metric is generally used when it comes to modeling spherical objects and their influence in the geometry of spacetime. It comes after assuming a static and spherically symmetrical spacetime.
Schwarzschild metric is (in units such that
c=1
G_N = 1
g = -\left(1 - \frac{2 m}{r}\right)dt \otimes dt + \left(1 - \frac{2 m}{r}\right)^{-1}dr \otimes dr + (r^{2} \sin^{2}{\left (\theta \right )})d\phi \otimes d\phi + r^{2}d\theta \otimes d\theta
m
is the mass of the spherical object. To review the canon way of creating tensors, lets use a values dictionary:
t, r, theta, phi = sympy.symbols('t r \\theta \\phi') m = sympy.Symbol('m') values = { ((), (0,0)): -(1-2*m/r), ((), (1,1)): 1/(1-2*m/r), ((), (2,2)): r**2, ((), (3,3)): r**2*sympy.sin(theta)**2 } g_sch = gr.Tensor([t, r, theta, phi], (0, 2), values)
FRLW
The FRLW metric (which stands for Friedmann-Lemaître-Robertson-Walker) models a homogenous, isotropic and either expanding or contracting (depending on a constant) universe.
Its tensor representation in coordinates is
g = -dx_0\otimes dx_0 + A(x_0)^2(dx_1\otimes dx_1 + dx_2\otimes dx_2 + dx_3\otimes dx_3)
A(x_0) = \epsilon x_0^q
0<q <1
\epsilon\in\mathbb{R}
. Using matrices:
x0, x1, x2, x3 = sympy.symbols('x_0 x_1 x_2 x_3') epsilon, q = sympy.symbols('\\epsilon q') A = epsilon * (x0 ** q) g_matrix = sympy.diag(-1, A**2, A**2, A**2) g_FRLW = gr.get_tensor_from_matrix(g_matrix, [x0, x1, x2, x3])
The Spacetime object
A good way of making a summary of the usual geometric invariants one computes using a metric is by using the Spacetime object in generalrepytivity. It takes just one argument: the metric.
Godel_spacetime = gr.Spacetime(g_godel) Sch_spacetime = gr.Spacetime(g_sch) FRLW_spacetime = gr.Spacetime(g_FRLW)
Each object now holds:
The Christoffel symbols christoffel_symbols.
The Riemann tensor Riem.
The Ricci tensor Ric.
The scalar curvature R.
For example, Schwarzschild's metric was created in order to be Ricci flat (i.e.
\text{Ric} = 0
). Let's verify this:
Sch_spacetime.Ric == 0 True
Printing a summary in LaTeX
Lastly, the Spacetime object comes with a print-to-file function which accepts two format options: tex or txt. Running the command
Godel_spacetime.print_summary('godel.tex', _format='tex')
generates a .tex file with a summary of all the values of the christoffel_symbols, Riem, Ric and R.
This is the first of a series of posts about generalrepytivity, a python toolbox I made for computations related to tensors and general relativity.↩
|
Unit circle - Wikipedia
Illustration of a unit circle. The variable t is an angle measure.
Animation of the act of unrolling the circumference of a unit circle, a circle with radius of 1. Since C = 2πr, the circumference of a unit circle is 2π.
{\displaystyle x^{2}+y^{2}=1.}
1 In the complex plane
2 Trigonometric functions on the unit circle
3 Circle group
4 Complex dynamics
In the complex planeEdit
{\displaystyle z=e^{it}=\cos t+i\sin t=\operatorname {cis} (t)}
for all t (see also: cis). This relation represents Euler's formula. In quantum mechanics, this is referred to as the phase factor. It can also be defined as the set of complex numbers satisfying
{\displaystyle |z|=1.}
Animation of the unit circle with angles
Trigonometric functions on the unit circleEdit
All of the trigonometric functions of the angle θ (theta) can be constructed geometrically in terms of a unit circle centered at O.
Sine function on unit circle (top) and its graph (bottom)
{\displaystyle \cos \theta =x\quad {\text{and}}\quad \sin \theta =y.}
The equation x2 + y2 = 1 gives the relation
{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1.}
{\displaystyle \cos \theta =\cos(2\pi k+\theta )}
{\displaystyle \sin \theta =\sin(2\pi k+\theta )}
for any integer k.
Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OP from the origin O to a point P(x1,y1) on the unit circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the x-axis. Now consider a point Q(x1,0) and line segments PQ ⊥ OQ. The result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OP has length 1 as a radius on the unit circle, sin(t) = y1 and cos(t) = x1. Having established these equivalences, take another radius OR from the origin to a point R(−x1,y1) on the circle such that the same angle t is formed with the negative arm of the x-axis. Now consider a point S(−x1,0) and line segments RS ⊥ OS. The result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π − t, R is at (cos(π − t), sin(π − t)) in the same way that P is at (cos(t), sin(t)). The conclusion is that, since (−x1, y1) is the same as (cos(π − t), sin(π − t)) and (x1,y1) is the same as (cos(t),sin(t)), it is true that sin(t) = sin(π − t) and −cos(t) = cos(π − t). It may be inferred in a similar manner that tan(π − t) = −tan(t), since tan(t) = y1/x1 and tan(π − t) = y1/−x1. A simple demonstration of the above can be seen in the equality sin(π/4) = sin(3π/4) = 1/√2.
The unit circle, showing coordinates of certain points
Circle groupEdit
Main article: Circle group
Complex numbers can be identified with points in the Euclidean plane, namely the number a + bi is identified with the point (a, b). Under this identification, the unit circle is a group under multiplication, called the circle group; it is usually denoted
{\displaystyle \mathbb {T} .}
On the plane, multiplication by cos θ + i sin θ gives a counterclockwise rotation by θ. This group has important applications in mathematics and science.[example needed]
Complex dynamicsEdit
Main article: Complex dynamics
Unit circle in complex dynamics
{\displaystyle f_{0}(x)=x^{2}}
is a unit circle. It is a simplest case so it is widely used in the study of dynamical systems.
^ Confusingly, in geometry a unit circle is often considered to be a 2-sphere—not a 1-sphere. The unit circle is "embedded" in a 2-dimensional plane that contains both height and width—hence why it is called a 2-sphere in geometry. However, the surface of the circle itself is one-dimensional, which is why topologists classify it as a 1-sphere. For further discussion, see the technical distinction between a circle and a disk.[2]
^ Weisstein, Eric W. "Unit Circle". mathworld.wolfram.com. Retrieved 2020-05-05.
^ a b Weisstein, Eric W. "Hypersphere". mathworld.wolfram.com. Retrieved 2020-05-06.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Unit_circle&oldid=1082881458"
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Note: In case of multiple classes, the data can also organized as three columns where the first column contains the image file names with paths, the second column contains the bounding boxes and the third column must be a cell vector that contains the label names corresponding to each bounding box. For more information on how to arrange the bounding boxes and labels, see boxLabelDatastore (Computer Vision Toolbox).
The YOLO v3 detector in this example is based on SqueezeNet, and uses the feature extraction network in SqueezeNet with the addition of two detection heads at the end. The second detection head is twice the size of the first detection head, so it is better able to detect small objects. Note that you can specify any number of detection heads of different sizes based on the size of the objects that you want to detect. The YOLO v3 detector uses anchor boxes estimated using training data to have better initial priors corresponding to the type of data set and to help the detector learn to predict the boxes accurately. For information about anchor boxes, see Anchor Boxes for Object Detection (Computer Vision Toolbox).
You can use Deep Network Designer to create the network shown in the diagram.
First, use transform to preprocess the training data for computing the anchor boxes, as the training images used in this example are bigger than 227-by-227 and vary in size. Specify the number of anchors as 6 to achieve a good tradeoff between number of anchors and mean IoU. Use the estimateAnchorBoxes function to estimate the anchor boxes. For details on estimating anchor boxes, see Estimate Anchor Boxes From Training Data (Computer Vision Toolbox). In case of using a pretrained YOLOv3 object detector, the anchor boxes calculated on that particular training dataset need to be specified. Note that the estimation process is not deterministic. To prevent the estimated anchor boxes from changing while tuning other hyperparameters set the random seed prior to estimation using rng.
Preprocess the augmented training data to prepare for training. The preprocess (Computer Vision Toolbox) method in yolov3ObjectDetector (Computer Vision Toolbox), applies the following preprocessing operations to the input data.
\text{learningRate}×{\left(\frac{\text{iteration}}{\text{warmupPeriod}}\right)}^{4}
estimateAnchorBoxes (Computer Vision Toolbox) | analyzeNetwork | combine | transform | dlfeval | read | evaluateDetectionPrecision (Computer Vision Toolbox) | sgdmupdate | dlupdate
boxLabelDatastore (Computer Vision Toolbox) | imageDatastore | dlnetwork | dlarray
Anchor Boxes for Object Detection (Computer Vision Toolbox)
Estimate Anchor Boxes From Training Data (Computer Vision Toolbox)
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Attentional and neurophysiologic effects of repetitive transcranial magnetic stimulation
Yeong-Wook Kim, Juan-Xiu Cui, Sheng-Lan Jin, Sung-Ju Jee, Min-Kyun Sohn
Relationship between recovery of motor function and neuropsychological functioning in cerebral infarction patients: the importance of social functioning in motor recovery
Min Cheol Chang, Sung-Won Park, Byung-Joo Lee, Donghwi Park
(This article belongs to the Special Issue Stroke neurology (Ischemic stroke; care, treatment and neuroprotection))
Sinusoidal stimulation on afferent fibers modulates the firing pattern of downstream neurons in rat hippocampus
Zhao-Xiang Wang, Zhou-Yan Feng, Lv-Piao Zheng, Yue Yuan
Muramyl dipeptide promotes Aβ1-42 oligomer production via the nod2/p-p38 mapk/bace1 signaling pathway in the sh-sy5y cells
Yan-Jie Chen, Yuan-Jin Chan, Wen-Jing Chen, Ya-Ming Li, Chun-Yan Zhang
Identification of microRNAs for the early diagnosis of Parkinson’s disease and multiple system atrophy
Jia-Hui Yan, Ping Hua, Yong Chen, Lan-Ting Li, ... Wei-Guo Liu
Early tracheostomy is associated with better prognosis in patients with brainstem hemorrhage
Wei-Long Ding, Yong-Sheng Xiang, Jian-Cheng Liao, Shi-Yong Wang, Xiang-Yu Wang
Application of robo-pigeon in ethological studies of bird flocks
Hao Wang, Jin Wu, Ke Fang, Lei Cai, ... Zhen-Dong Dai
Clathrin-independent but dynamin-dependent mechanisms mediate Ca
{}^{2+}
-triggered endocytosis of the glutamate GluK2 receptor upon excitotoxicity
Jing-Jing Du, Lu Yan, Wei Zhang, Hao Xu, Qiu-Ju Zhu
Antidepressant-like mechanism of honokiol in a rodent model of corticosterone-induced depression
Bo Zhang, Yu Li, Miao Liu, Xiao-Hua Duan, ... Hong-Sheng Chang
Antiapoptotic effects of velvet antler polypeptides on damaged neurons through the hypothalamic-pituitary-adrenal axis
Qing Yang, Jia-Nan Lin, Xin Sui, Hui Li, ... Na Li
Clinical interpretations of the effectiveness of changes in body position during aerobic fitness after neurologic injury
Nur Fariza Izan, Sheikh Hussain Salleh, Chee-Ming Ting, Fuad Noman, ... Ahmad Zubaidi Abdul Latif
[18F] FDOPA PET may confirm the clinical diagnosis of Parkinson's disease by imaging the nigro-striatal pathway and the sympathetic cardiac innervation: Proof-of-concept study
Jonathan Kuten, Adi Linevitz, Hedva Lerman, Nanette Freedman, ... Einat Even-Sapir
H63D CG genotype of HFE is associated with increased risk of sporadic amyotrophic lateral sclerosis in a single population
Qing-Qing Zhang, Hong Jiang, Chun-Yan Li, Ya-Ling Liu, Xin-Ying Tian
Mortality prediction of ischemic stroke patients without thrombectomy by blood total antioxidant capacity
Leonardo Lorente, María M. Martín, Agustín F. González-Rivero, Antonia Pérez-Cejas, ... Victor García-Marín
A meta-analysis of case studies and clinical characteristics of hypertrophic olivary degeneration secondary to brainstem infarction
Yi-Lin Wang, Yan Gao, Ping-Ping He, Jiang-Ning Yin, ... Hong Zhang
Herbal decoction of Gastrodia, Uncaria, and Curcuma confers neuroprotection against cerebral ischemia in vitro and in vivo
Zhi-Han Wang, Bing-Hong Chen, Ying-Ying Lin, Jin Xing, ... Li Ren
Mechanisms underlying the generation of autonomorespiratory coupling amongst the respiratory central pattern generator, sympathetic oscillators, and cardiovagal premotoneurons
Michael G. Z. Ghali, George Zaki Ghali, Adriana Lima, Michael McDermott, ... M. Gazi Yasargil
The effects of erythropoietin on neurogenesis after ischemic stroke
Si-Jia Zhang, Yu-Min Luo, Rong-Liang Wang
Molecular and microstructural biomarkers of neuroplasticity in neurodegenerative disorders through preclinical and diffusion magnetic resonance imaging studies
Rodolfo Gabriel Gatto
Retraction: Andersen S. S. L. Real time large scale in vivo observations reveal intrinsic synchrony, plasticity and growth cone dynamics of midline crossing axons at the ventral floor plate of the zebrafish spinal cord. J. Integr. Neurosci. 18(4), 351-368.
J. Integr. Neurosci. 2020, 19(3), 593–593; https://doi.org/10.31083/j.jin.2019.04.1191R
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Dirty Politics | Toph
By shariful_islam · Limits 3s, 256 MB · Interactive
This is an interactive problem. (If you want to know about interactive problems, click here)
Your country is on a war. The map of your enemy country can be represented with
n states connected by
n-1
n−1 bidirectional roads in such a way that you can go from any state to any other state within the country using these roads. Each state has a governor who is affiliated to either of the two political parties - Onliners and Offliners. Due to extremely high security it is impossible for you to know which governor is of which party. Fortunately, there are severe political rivalries among your enemies. For example, if two states connected by a road have two governors from different parties, the road is kept blocked; otherwise the road is kept unblocked. So your commander-in -general decided to give you the following task.
You can choose some governor to bribe. Then the governor will do the following things in the given order:
The governor will tell you how many of his neighbor states (those having a shared road with his state) have a governor of his party.
Consequently, the governor will be banished from his party. Then he will join the opposition party. As a result, block-unblock states of the roads will change accordingly.
You can bribe at most
\lfloor \frac{3n - 2}{2} \rfloor
⌊23n−2⌋ times due to budget limitations. You can bribe a particular governor multiple times. You have to bribe the governors in such a way that all of the
n-1
n−1 roads of your enemy country becomes blocked when your job is complete. Now it is time to go for your country!
T (
1 \le T \le 40
1≤T≤40 ), the number of test cases.
Each of the test cases begins with an integer
n (
2\le n\le 10^5
2≤n≤105 ), the number of states your enemy-country has.
n-1
n−1 lines will contain two integers
u and
v, (
1\le u \lt v \le n
1≤u<v≤n) indicating there is a road between state
u and state
v.
\sum n over all test cases will not exceed
3*10^5
3∗105 in a single file.
You can interact in two ways:
To bribe the governor of state
x (
1\le x \le n
1≤x≤n ), print
You will be responded with
y (
0 \le y \lt n
0≤y<n ) is the number of neighbor states of state
x having a governor of the political party that the governor of
x currently affiliated to. After answering to your query, the governor of state
x will change his party.
To declare that your job is done, print
If your job for this enemy-country (i.e. test case) is done successfully, the response is
You will be responded with -1 if you make a wrong query or a query beyond limitation i.e. trying to bribe despite you ran out of budget (declaring the job is done is not counted as bribing) or there are some roads still remaining unblocked but you have declared your job done. Then you should terminate your program immediately in order to avoid deceptive verdicts like CPU Limit Exceeded, Memory Limit Exceeded etc. Finally, don’t forget to flush your output after printing each line.
Here is an example. First You take input
T, the number of test cases.
Now you will take input for the first case.
Then a possible interaction can be as following:
Query Simulation: Red edges indicate blocked state; blue ones are for unblocked roads.
Similarly, for the second test:
A possible interaction:
Here, the following things happen
> indicates what your program reads and < indicates what your program writes. These symbols are here to make it easy to understand. You do not have to read > and don’t print < in your program.
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heat capacity of H2O at its boiling point is 1) 3 cal mol-1K-1 2)5 cal mol-1K-1 3) 7 cal mol-1K-1 - Chemistry - Thermodynamics - 9627363 | Meritnation.com
heat capacity of H2O at its boiling point is
1) 3 cal mol -1K-1
2)5 cal mol-1 K-1
3) 7 cal mol-1 K-1
ANS : 4) how?
Heat capacity of a system is defined as the quantity of heat required to raise the temperature of the system by one degree.
\mathrm{Heat} \mathrm{capacity} = \frac{\mathrm{dq}}{\mathrm{dT}}
Boiling point of water = 100º C
Heat capacity of water will be the amount of heat required to raise the temperature from 100º C to 101 º C. At 100º C the temperature of the water becomes constant even after the addition of infinite amount of heat. This is because the heat supplied after attaining 100º C is utilised in changing the phase of water from liquid to vapour . This is called as latent heat. Therefore the amount of heat supplied is constant i.e. q=∞ and hence heat capacity becomes infinite.
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1 )Find number of gram atom, no of gram molecule in 40g of oxygen gas (at mass O=16) 2) Find - Chemistry - Some Basic Concepts of Chemistry - 7875353 | Meritnation.com
1 )Find number of gram atom, no.of gram molecule in 40g of oxygen gas (at mass O=16)
2) Find no. of He atom in sample of helium weighing 54amu (He=4)
3)Find the no. of atoms of each type in 3.42g of C12H22O11.
4)Find no. of SO2 molecules in 4.48lt at STP.
5)Find no. of electrons in 3.2g of CH2.
6)Find no.of moles of carbon in 3 moles of K4{Fe(CN)6}
7) 2*1021 of x weigh 2g find molar mass.
8)Find the weight of NH3 which has the same no. of atoms as in 3.6g of H2O.
9)Chlorophyll contains 20% Mg by weight. Find the no. of Mg atoms in 40g of chlorophyll.
10) Find weight of 50ml of SO2 under STP.
11)Find weight of 6 moles of H2SO4 and 1021 molecules of CO2.
12)Find no. of nuetrons in 3.6g of H2O.
13)Find no. of electrons in 40g of CO3 2-
14) Find weight of oxygen present in 6 moles of Na2SO4.10H2O
15) 20ml of H2O evaporate in 1 hour. Find no. of water molecules escaping in one second.
16)If a person spends one million rupees daily, find no. of years it would take him to spend avagadro no. of rupees.
17) If 3.22g of Na2SO4.xH2O has 6.022*1022 molecules of water. What is the values of x?
1) Atomic mass of oxygen = 16u
Given mass of oxygen = 40g
Number of gram atoms in 16g of oxygen = 1 gram atom
Number of gram atoms in 40g of oxygen =
\frac{1}{16}×40 = 2.5 \mathrm{gram} \mathrm{atoms}
Number of gram molecules in 32g of oxygen = 1 gram molecule.
Number of gram molecules in 40g of oxygen =
\frac{1}{32}×40 = 1.25 \mathrm{gram} \mathrm{molecules}
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Nonexistence of modular fusion algebras whose kernels are certain noncongruence subgroups
July, 2003 Nonexistence of modular fusion algebras whose kernels are certain noncongruence subgroups
A well-known conjecture states that the kernel of representation associated to a modular fusion algebra is always a congruence subgroup. Assuming this conjecture, Eholzer studied modular fusion algebras such that the kernel of representation associated to each of them is a congruence subgroup using the fact that all irreducible representaions of
SL\left(2,Z/{p}^{\lambda }Z\right)
are classified. He classified all strongly modular fusion algebras of dimension two, three, four and the nondegenerate ones with dimension
\le 24
. In this paper, we try to imitate Eholzer's work. We classify modular fusion algebras such that the kernel of representation associated to each of them is a noncongruence normal subgroup of
\Gamma :=PSL\left(2,Z\right)
containing an element
\left(\begin{array}{cc}1& 6\\ 0& 1\end{array}\right)
. Among such normal subgroups, there exist infinitely many noncongruence subgroups. In a sense, they are the classes of near congruence subgroups. For such a normal subgroup
G
, we shall show that any irreducible representation of degree not equal to 1 of
\Gamma /G
is not associated to a modular fusion algebra.
Makoto TAGAMI. "Nonexistence of modular fusion algebras whose kernels are certain noncongruence subgroups." J. Math. Soc. Japan 55 (3) 681 - 693, July, 2003. https://doi.org/10.2969/jmsj/1191418997
Keywords: admissible , fusion algebra , little group method , modular fusion algebra , non congruence subgroup , nondegenerate
Makoto TAGAMI "Nonexistence of modular fusion algebras whose kernels are certain noncongruence subgroups," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 55(3), 681-693, (July, 2003)
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Mr. Dobson is planning to give a quiz to his class tomorrow. Unfortunately for his students, Mr. Dobson is notorious for writing quizzes that seem to have no relevance to the subject. With this in mind, his students know that their efforts will be purely guesswork. If the quiz contains ten questions that the students will have to match with ten given answers, what is the probability that Rodney Random will get all ten questions matched correctly?
Notice that the answers are matched to questions, so when one answer is used for one question it cannot be used for another.
How many ways can
10
answers be arranged?
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William Goldman (mathematician) - Wikipedia
Find sources: "William Goldman" mathematician – news · newspapers · books · scholar · JSTOR (January 2013) (Learn how and when to remove this template message)
William Mark Goldman (born 1955 in Kansas City, Missouri) is a professor of mathematics at the University of Maryland, College Park (since 1986). He received a B.A. in mathematics from Princeton University in 1977, and a Ph.D. in mathematics from the University of California, Berkeley in 1980.
William Goldman at Bar-Ilan University in 2008
Research contributionsEdit
Goldman has investigated geometric structures, in various incarnations, on manifolds since his undergraduate thesis, "Affine manifolds and projective geometry on manifolds", supervised by William Thurston and Dennis Sullivan. This work led to work with Morris Hirsch and David Fried on affine structures on manifolds, and work in real projective structures on compact surfaces. In particular he proved that the space of convex real projective structures on a closed orientable surface of genus
{\displaystyle g>1}
is homeomorphic to an open cell of dimension
{\displaystyle 16g-16}
. With Suhyoung Choi, he proved that this space is a connected component (the "Hitchin component") of the space of equivalence classes of representations of the fundamental group in
{\displaystyle {\rm {SL}}(3,\mathbb {R} )}
. Combining this result with Suhyoung Choi's convex decomposition theorem, this led to a complete classification of convex real projective structures on compact surfaces.
His doctoral dissertation, "Discontinuous groups and the Euler class" (supervised by Morris W. Hirsch), characterizes discrete embeddings of surface groups in
{\displaystyle {\rm {PSL}}(3,\mathbb {R} )}
in terms of maximal Euler class, proving a converse to the Milnor–Wood inequality for flat bundles. Shortly thereafter he showed that the space of representations of the fundamental group of a closed orientable surface of genus
{\displaystyle g>1}
{\displaystyle {\rm {PSL}}(3,\mathbb {R} )}
{\displaystyle 4g-3}
connected components, distinguished by the Euler class.
With David Fried, he classified compact quotients of Euclidean 3-space by discrete groups of affine transformations, showing that all such manifolds are finite quotients of torus bundles over the circle. The noncompact case is much more interesting, as Grigory Margulis found complete affine manifolds with nonabelian free fundamental group. In his 1990 doctoral thesis, Todd Drumm found examples which are solid handlebodies using polyhedra which have since been called "crooked planes."
Goldman found examples (non-Euclidean nilmanifolds and solvmanifolds) of closed 3-manifolds which fail to admit flat conformal structures.
Generalizing Scott Wolpert's work on the Weil–Petersson symplectic structure on the space of hyperbolic structures on surfaces, he found an algebraic-topological description of a symplectic structure on spaces of representations of a surface group in a reductive Lie group. Traces of representations of the corresponding curves on the surfaces generate a Poisson algebra, whose Lie bracket has a topological description in terms of the intersections of curves. Furthermore, the Hamiltonian vector fields of these trace functions define flows generalizing the Fenchel–Nielsen flows on Teichmüller space. This symplectic structure is invariant under the natural action of the mapping class group, and using the relationship between Dehn twists and the generalized Fenchel–Nielsen flows, he proved the ergodicity of the action of the mapping class group on the SU(2)-character variety with respect to symplectic Lebesgue measure.
Following suggestions of Pierre Deligne, he and John Millson proved that the variety of representations of the fundamental group of a compact Kähler manifold has singularities defined by systems of homogeneous quadratic equations. This leads to various local rigidity results for actions on Hermitian symmetric spaces.
With John Parker, he examined the complex hyperbolic ideal triangle group representations. These are representations of hyperbolic ideal triangle groups to the group of holomorphic isometries of the complex hyperbolic plane such that each standard generator of the triangle group maps to a complex reflection and the products of pairs of generators to parabolics. The space of representations for a given triangle group (modulo conjugacy) is parametrized by a half-open interval. They showed that the representations in a particular range were discrete and conjectured that a representation would be discrete if and only if it was in a specified larger range. This has become known as the Goldman–Parker conjecture and was eventually proven by Richard Schwartz.
Professional serviceEdit
Goldman also heads a research group at the University of Maryland called the Experimental Geometry Lab, a team developing software (primarily in Mathematica) to explore geometric structures and dynamics in low dimensions. He served on the Board of Governors for The Geometry Center at the University of Minnesota from 1994 to 1996.
He served as Editor-In-Chief of Geometriae Dedicata from 2003 until 2013.
Goldman, William M. (1999). Complex hyperbolic geometry. Oxford Mathematical Monographs. Oxford Science Publications. New York: The Clarendon Press, Oxford University Press. xx+316 pp. ISBN 0-19-853793-X. MR 1695450.
Goldman, William M.; Xia, Eugene Z. (2008). "Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces". Memoirs of the American Mathematical Society. 193 (904): viii+69 pp. arXiv:math/0402429. doi:10.1090/memo/0904. ISSN 0065-9266. MR 2400111. S2CID 2865489.
Faculty page at the University of Maryland, College Park
Retrieved from "https://en.wikipedia.org/w/index.php?title=William_Goldman_(mathematician)&oldid=1029016412"
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Computer-automated design - Wikipedia
Design Automation usually refers to electronic design automation, or Design Automation which is a Product Configurator. Extending Computer-Aided Design (CAD), automated design and Computer-Automated Design (CAutoD)[1][2][3] are more concerned with a broader range of applications, such as automotive engineering, civil engineering,[4][5][6][7] composite material design, control engineering,[8] dynamic system identification and optimization,[9] financial systems, industrial equipment, mechatronic systems, steel construction,[10] structural optimisation,[11] and the invention of novel systems.[12]
The concept of CAutoD perhaps first appeared in 1963, in the IBM Journal of Research and Development,[1] where a computer program was written.
to search for logic circuits having certain constraints on hardware design
to evaluate these logics in terms of their discriminating ability over samples of the character set they are expected to recognize.
More recently, traditional CAD simulation is seen to be transformed to CAutoD by biologically-inspired machine learning,[13] including heuristic search techniques such as evolutionary computation,[14][15] and swarm intelligence algorithms.[16]
1 Guiding designs by performance improvements
2 Normalized objective function: cost vs. fitness
3 Dealing with practical objectives
5 Search in polynomial time
Guiding designs by performance improvements[edit]
Interaction in computer-automated design
To meet the ever-growing demand of quality and competitiveness, iterative physical prototyping is now often replaced by 'digital prototyping' of a 'good design', which aims to meet multiple objectives such as maximised output, energy efficiency, highest speed and cost-effectiveness. The design problem concerns both finding the best design within a known range (i.e., through 'learning' or 'optimisation') and finding a new and better design beyond the existing ones (i.e., through creation and invention). This is equivalent to a search problem in an almost certainly, multidimensional (multivariate), multi-modal space with a single (or weighted) objective or multiple objectives.
Normalized objective function: cost vs. fitness[edit]
Using single-objective CAutoD as an example, if the objective function, either as a cost function
{\displaystyle J\in [0,\infty )}
, or inversely, as a fitness function
{\displaystyle f\in (0,1]}
{\displaystyle f={\tfrac {J}{1+J}}}
is differentiable under practical constraints in the multidimensional space, the design problem may be solved analytically. Finding the parameter sets that result in a zero first-order derivative and that satisfy the second-order derivative conditions would reveal all local optima. Then comparing the values of the performance index of all the local optima, together with those of all boundary parameter sets, would lead to the global optimum, whose corresponding 'parameter' set will thus represent the best design. However, in practice, the optimization usually involves multiple objectives and the matters involving derivatives are a lot more complex.
Dealing with practical objectives[edit]
In practice, the objective value may be noisy or even non-numerical, and hence its gradient information may be unreliable or unavailable. This is particularly true when the problem is multi-objective. At present, many designs and refinements are mainly made through a manual trial-and-error process with the help of a CAD simulation package. Usually, such a posteriori learning or adjustments need to be repeated many times until a ‘satisfactory’ or ‘optimal’ design emerges.
Exhaustive search[edit]
In theory, this adjustment process can be automated by computerised search, such as exhaustive search. As this is an exponential algorithm, it may not deliver solutions in practice within a limited period of time.
Search in polynomial time[edit]
One approach to virtual engineering and automated design is evolutionary computation such as evolutionary algorithms.
To reduce the search time, the biologically-inspired evolutionary algorithm (EA) can be used instead, which is a (non-deterministic) polynomial algorithm. The EA based multi-objective "search team" can be interfaced with an existing CAD simulation package in a batch mode. The EA encodes the design parameters (encoding being necessary if some parameters are non-numerical) to refine multiple candidates through parallel and interactive search. In the search process, 'selection' is performed using 'survival of the fittest' a posteriori learning. To obtain the next 'generation' of possible solutions, some parameter values are exchanged between two candidates (by an operation called 'crossover') and new values introduced (by an operation called 'mutation'). This way, the evolutionary technique makes use of past trial information in a similarly intelligent manner to the human designer.
The EA based optimal designs can start from the designer's existing design database, or from an initial generation of candidate designs obtained randomly. A number of finely evolved top-performing candidates will represent several automatically optimized digital prototypes.
There are websites that demonstrate interactive evolutionary algorithms for design. EndlessForms.com allows you to evolve 3D objects online and have them 3D printed. PicBreeder.org allows you to do the same for 2D images.
Genetic algorithm (GA) applications - automated design
^ a b Kamentsky, L.A.; Liu, C.-N. (1963). "Computer-Automated Design of Multifont Print Recognition Logic". IBM Journal of Research and Development. 7 (1): 2. doi:10.1147/rd.71.0002.
^ Brncick, M (2000). "Computer automated design and computer automated manufacture". Phys Med Rehabil Clin N Am. 11 (3): 701–13. doi:10.1016/S1047-9651(18)30806-4. PMID 10989487.
^ Li, Y., et al. (2004). CAutoCSD - Evolutionary search and optimisation enabled computer automated control system design Archived 2015-08-31 at the Wayback Machine. International Journal of Automation and Computing, 1(1). 76-88. ISSN 1751-8520
^ Kramer, GJE; Grierson, DE (1989). "Computer automated design of structures under dynamic loads". Computers & Structures. 32 (2): 313–325. doi:10.1016/0045-7949(89)90043-6.
^ Moharrami, H; Grierson, DE (1993). "Computer‐Automated Design of Reinforced Concrete Frameworks". Journal of Structural Engineering. 119 (7): 2036–2058. doi:10.1061/(ASCE)0733-9445(1993)119:7(2036).
^ XU, L; Grierson, DE (1993). "Computer‐Automated Design of Semirigid Steel Frameworks". Journal of Structural Engineering. 119 (6): 1740–1760. doi:10.1061/(ASCE)0733-9445(1993)119:6(1740).
^ Barsan, GM; Dinsoreanu, M, (1997). Computer-automated design based on structural performance criteria, Mouchel Centenary Conference on Innovation in Civil and Structural Engineering, AUG 19-21, CAMBRIDGE ENGLAND, INNOVATION IN CIVIL AND STRUCTURAL ENGINEERING, 167-172
^ Li, Yun (1996). "Genetic algorithm automated approach to the design of sliding mode control systems". International Journal of Control. 63 (4): 721–739. doi:10.1080/00207179608921865.
^ Li, Yun; Chwee Kim, Ng; Chen Kay, Tan (1995). "Automation of Linear and Nonlinear Control Systems Design by Evolutionary Computation" (PDF). IFAC Proceedings Volumes. 28 (16): 85–90. doi:10.1016/S1474-6670(17)45158-5.
^ Barsan, GM, (1995) Computer-automated design of semirigid steel frameworks according to EUROCODE-3, Nordic Steel Construction Conference 95, JUN 19-21, 787-794
^ Gray, Gary J.; Murray-Smith, David J.; Li, Yun; et al. (1998). "Nonlinear model structure identification using genetic programming". Control Engineering Practice. 6 (11): 1341–1352. doi:10.1016/S0967-0661(98)00087-2.
^ Yi Chen, Yun Li, (2018). Computational Intelligence Assisted Design: In Industrial Revolution 4.0, CRC Press, ISBN 9781498760669
^ Zhan, Z.H., et al. (2011). Evolutionary computation meets machine learning: a survey, IEEE Computational Intelligence Magazine, 6(4), 68-75.
^ Gregory S. Hornby (2003). Generative Representations for Computer-Automated Design Systems, NASA Ames Research Center, Mail Stop 269-3, Moffett Field, CA 94035-1000
^ J. Clune and H. Lipson (2011). Evolving three-dimensional objects with a generative encoding inspired by developmental biology. Proceedings of the European Conference on Artificial Life. 2011.
^ Zhan, Z.H., et al. (2009). Adaptive Particle Swarm Optimization, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), Vol.39, No.6. 1362-1381
An online interactive GA based CAutoD demonstrator. Learn step by step or watch global convergence in 2-parameter CAutoD
Retrieved from "https://en.wikipedia.org/w/index.php?title=Computer-automated_design&oldid=1035145722"
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CoCalc – Teach scientific software online using Jupyter Notebook, Python, R, and more
Teach scientific software online using Jupyter Notebook, Python, R, and more
CoCalc is a virtual online computer lab: it takes away the pain of teaching scientific software!
An entire computer lab in the cloud
Every student works 100% online – inside their own dedicated workspace.
Follow the progress of each student in real time.
At any time you and your teaching assistants can jump into a student's file, right where they are working, and answer their questions.
Use TimeTravel to see every step a student took to get to their solution, and to get context when helping them.
Integrated chat rooms allow you to guide students directly where they are working or discuss collected files with your teaching assistants.
The project's Activity Log records exactly when and by whom a file was accessed.
No software setup 100% online
Common underlying software environment:
Forget any complicated software setup – everyone is able to start working in seconds!
Since everyone works with exactly the same software stack, all inconsistencies between environments are eliminated.
CoCalc's massive default Software Environment provides nearly everything anybody has ever asked us to install since 2013!
The course management interface gives you full control over distributing, collecting, grading and returning everyone's assignments.
The Instructor Guide explains how to use CoCalc to teach a course.
The CoCalc Manual explains much of what CoCalc can do.
There are a large number of courses all over the world running on CoCalc. We used to list them here...
Contact [email protected] or request a live demo!
NBGrader support
CoCalc's Jupyter Notebooks fully support automatic and manual grading using our version of NBGrader with no configuration!
The teacher's notebook contains exercise cells for students and test cells, some of which students can also run to get immediate feedback. Once collected, you tell CoCalc to automatically run the full test suite across all student notebooks and tabulate the results.
By default, tests run in the student's project, so malicious code won't impact anybody except the student.
CoCalc's own Jupyter Notebook implementation offers realtime synchronization, TimeTravel, automatic grading, side chat, and more.
Sage Worksheets are similar to Jupyter Notebooks, but made to work well with SageMath. They offer a single-document model that scales to large documents and integrated 3d graphics.
\LaTeX
A full
\LaTeX
editor supporting preview rendering, forward/inverse search, error reporting, and much more.
Use the collaborative CoCalc terminal to access all powerful command line tools in a full Ubuntu Linux environment.
I just found out that my CoCalc class got by far the best course evaluations for any course I've taught at UCSD to date (over 85% on the favorable/unfavorable scale), which makes it a sure thing that I'll be teaching this course again (in some form) next year! Many thanks for the backend work on CoCalc, for the course materials, for the guest lecture...
— Kiran Kedlaya — UC San Diego, March 2017
CoCalc provides a user-friendly interface. Students don't need to install any software at all. They just open up a web browser and go to https://cocalc.com and that's it. They just type code directly in, hit shift+enter and it runs, and they can see if it works. It provides immediate feedback. The course management features work really well.
— Will Conley — University of California at Los Angeles, Fall 2016
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Babur Biye | Toph
Babur Biye
By error26 · Limits 1s, 512 MB
Babu is getting married. Babu's fiancé is a prominent programmer of Bangladesh. She told Babu to solve a problem. She won't say KOBUL until he solves the problem. The problem statement is given below.
A string containing English uppercase and lowercase characters. His ultimate goal is to make the given string beautiful.
A string is beautiful,
If it starts with some(possibly zero) English uppercase letters followed by some(possibly zero) English lowercase letters and further followed by some(possibly zero) English uppercase letters.
It starts with some(possibly zero) English lowercase letters followed by some(possibly zero) English uppercase letters and further followed by some(possibly zero) English lowercase letters.
For example: Wow, Alice, BoB, bABu, sobel, NICE strings are beautiful but HuHu, SmreenBErg, lEaN are not.
He may perform several operations on this string. In one operation, he can choose an index i and erase the i'th character of the string. He can perform this operation as many times as he wants.
Although Babu is a great programmer, he doesn't want to take risk of solving this problem alone. So, he asked his friend Sobel to help him to construct the longest beautiful string.
T (
1 \le T \le 10
1≤T≤10), denoting the number of test cases.
Each test case contains a string
S (
1 \le \texttt{Length of S} \le 10^5
1≤Length of S≤105).
For each test case, print the length of the beautiful string.
UPPERlowerUPPER
lowerUPPERlower
Ahasan_1999Earliest, Dec '20
rahat_chyFastest, 0.0s
Replay of AUST Game of Codes - Fall 2019
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User talk:Pmlineditor - Wikinews, the free news source
Put new things under old things. Click here to start a new topic.
New to Wikipedia? Welcome! Ask your questions here.
No inflammatory comments
2 Jan 2010 Writing Contest Prize Winner!
3 New administrator
4 Unprotecting archived article?
6 Certificate of participation
9 Request for reviewing and other assistance
13 Students protest against police action in Jadavpur University, Kolkata
Jan 2010 Writing Contest Prize Winner!Edit
Pmlineditor: you won second prize in the first writing contest of 2010! Congratulations! Second prize is 20 Canadian dollars (about the same as 20 US dollars right now). Prize transfer via PayPal is preferred. If you have a PayPal account, please send me the email address linked to that account via email: Special:Emailuser/gopher65. If you do not have a PayPal account I've decided that I am willing to send out personal checks, contrary to what I said in the prize rules section. If you wish to use this option, please send your name, full address, postal/zip code, and country name to the email linked above. Note that this option requires more personal information than PayPal (which just needs an email address), so PayPal is the preferred option. Please respond on my user page if you wish to claim your prize, or not. Again, congratulations, and great work! :) Gopher65talk 02:44, 24 April 2010 (UTC)
New administratorEdit
Hi Pmlineditor, I've made you an administrator per your successful RFA. You're familiar with the bits from other wikis, so i don't need to lecture you on how they work.
Nonetheless, feel free to drop me a note at my talk page should you need help. Cheers, Tempodivalse [talk] 21:32, 24 April 2010 (UTC)
Thanks! If I have any questions, I'll certainly ask you. Cheers, Pmlineditor discuss 07:55, 25 April 2010 (UTC)
File:18th Birthday.jpg
[18:01:48] <@_Pmlineditor_> I'll kill MisterWiki if he puts http://en.wikinews.org/wiki/File:18th_Birthday.jpg in my talk page for getting admin <_>
Hey. Congratulations on your RfA. <_< --Diego Grez let's talk 02:11, 26 April 2010 (UTC)
Thanks! Was just kidding in IRC. Congrats on getting Reviewer too, by the way.
Pmlineditor discuss 07:32, 27 April 2010 (UTC)
OK. :-) --Diego Grez let's talk 00:46, 28 April 2010 (UTC)
Unprotecting archived article?Edit
Did you mean to unprotect Landmine blast in Chattisgarh, India kills eight after you archived it? --InfantGorilla (talk) 14:35, 17 May 2010 (UTC)
Nope, that was an error. I've protected it again. :) Pmlineditor discuss 15:09, 17 May 2010 (UTC)
You have been busy archiving. No worries, but did you notice that you forgot to sight them (as auto-sighting was turned off last month)? --InfantGorilla (talk) 13:45, 15 July 2010 (UTC)
Hmm, sorry for that; I didn't remember about sighting them. I'll do that from next time. Regards, Pmlineditor discuss 16:38, 15 July 2010 (UTC)
Certificate of participationEdit
{\displaystyle {\color {Blue}{\mathfrak {Wikinews}}}:{\color {Sepia}The\;Free\;News\;Source}}
This certifies that PMlineditor participated in the May 2010 writing contest, writing 7 articles for 45 points in the course of the competition. Good work.
For dealing with the idiots :) --Herby talk thyme 08:58, 24 February 2011 (UTC)
Wow!!Edit
--61 articles! Wow!! You deserve at least this!
Thanks. :) Pmlineditor (t · c · l) 19:04, 11 April 2012 (UTC)
Request for reviewing and other assistanceEdit
Wikinews Writing contest 2013 is here. :) Please sign up to participate?Edit
We've created the Wikinews:Writing contest 2013, which will start on April 1 and end on June 1. It is modeled on the successful 2010 contest. Unlike the previous version, points are available for people who conduct reviews. (With a University of Wollongong class currently contributing articles, extra assistance is appreciated at this time.) It presents a great incentive for you to renew your reviewing chops, contribute some original reporting not being done by the main stream media, and write some synthesis articles on topics that could use more attention. People should be around to review to prevent a backlog if you just want to write, and several reviewers have access to scoop to make it easier to review any original reporting you do. If you are interested in signing up, please do so on Wikinews:Writing contest 2013/entrants. There is at least one prize on offer for the winner along with the opportunity to earn some barn stars as a way of thanking you for your participation. :D --LauraHale (talk) 10:36, 25 March 2013 (UTC)
Students protest against police action in Jadavpur University, KolkataEdit
Published. Interesting. See review comments, detailed history of edits during review. --Pi zero (talk) 04:43, 19 September 2014 (UTC)
Thanks. I'm quite rusty after a break of 2.5 years as you can see (no categories and similarity to sources!), so thanks for all the fixes you made and for passing the article! Pmlineditor (t · c · l) 06:58, 19 September 2014 (UTC)
You should relax and not spam users. Pmlineditor (t · c · l) 13:40, 8 October 2014 (UTC)
Retrieved from "https://en.wikinews.org/w/index.php?title=User_talk:Pmlineditor&oldid=2942364"
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Born Haber cycle for AlCl3 I don't want in terms of value I want in terms of ionization - Chemistry - Thermodynamics - 9638531 | Meritnation.com
Born Haber cycle for AlCl3. I don't want in terms of value. I want in terms of ionization energy, sublimation energy, lattice enrgy, bond energy and whatrver is required pls. Thanks
1. The Born-Haber Cycle for AlCl3 is :
2. The enthalpy of formation of AlCl3 is given by following equation derived from above cycle :
∆{H}_{formation}=∆{H}^{Al}{ }_{atomisation }+ ∆{{H}^{Cl}}_{atomisation} +\left( {I}_{1} +{I}_{2} +{I}_{3}{{\right)}^{Al}}_{ionisation energy} \phantom{\rule{0ex}{0ex}} -{{E}^{Cl}}_{electron affinity} -∆{{H}^{Al}}_{hydration}-∆{{H}^{Cl}}_{hydration} -∆{H}_{Lattice energy}
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Examine the information provided in each diagram below. Decide if each figure is possible or not. If the figure is not possible, explain why.
What angle measure is supplementary to
120^\circ
Would this also satisfy the Triangle Angle Sum Theorem?
How are the same-side interior angles in a parallelogram related?
Are the vertical angles equal?
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Mini-Workshop: New Developments in Newton-Okounkov Bodies | EMS Press
The theory of Newton-Okounkov bodies, also called Okounkov bodies, is a new connection between algebraic geometry and convex geometry. It generalizes the well-known and extremely rich correspondence between geometry of toric varieties and combinatorics of convex integral polytopes. Okounkov bodies were first introduced by Andrei Okounkov, in a construction motivated by a question of Khovanskii concerning convex bodies govering the multiplicities of representations. Recently, Kaveh-Khovanskii and Lazarsfeld-Mustata have generalized and systematically developed Okounkov’s construction, showing the existence of convex bodies which capture much of the asymptotic information about the geometry of (
X,D
X
is an algebraic variety and
D
is a big divisor. The study of Okounkov bodies is a new research area with many open questions. The goal of this mini-workshop was to bring together a core group of algebraic/symplectic geometers currently working on this topic to establish the groundwork for future development of this area.
Megumi Harada, Kiumars Kaveh, Askold Khovanskii, Mini-Workshop: New Developments in Newton-Okounkov Bodies. Oberwolfach Rep. 8 (2011), no. 3, pp. 2327–2363
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Asymptotic behavior of global solutions to the Navier–Stokes equations in $\mathbb R^3$ | EMS Press
Asymptotic behavior of global solutions to the Navier–Stokes equations in
\mathbb R^3
We construct global solutions to the NavierStokes equations with initial data small in a Besov space. Under additional assumptions we show that they behave asymptotically like self-similar solutions.
Fabrice Planchon, Asymptotic behavior of global solutions to the Navier–Stokes equations in
\mathbb R^3
. Rev. Mat. Iberoam. 14 (1998), no. 1, pp. 71–93
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Generalized Equilibrium Problem with Mixed Relaxed Monotonicity
Haider Abbas Rizvi, Adem Kılıçman, Rais Ahmad, "Generalized Equilibrium Problem with Mixed Relaxed Monotonicity", The Scientific World Journal, vol. 2014, Article ID 807324, 4 pages, 2014. https://doi.org/10.1155/2014/807324
Haider Abbas Rizvi,1 Adem Kılıçman,2 and Rais Ahmad 1
2Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Selangor, Malaysia
We extend the concept of relaxed -monotonicity to mixed relaxed --monotonicity. The concept of mixed relaxed --monotonicity is more general than many existing concepts of monotonicities. Finally, we apply this concept and well known KKM-theory to obtain the solution of generalized equilibrium problem.
Generalized monotonicities provide a way of finding parameter moves that yield monotonicity of model solutions and allow studying the monotonicity of functions or subset of variables. In recent past, many researchers have proposed many important generalizations of monotonicity such as pseudomonotonicity, relaxed monotonicity, relaxed --monotonicity, quasimonotonicity, and semimonotonicity; see [1–3]. Karamardian and Schaible [4] introduced various kinds of monotone mappings which in the case of gradient mappings are related to generalized convex functions. For more details, we refer to [5–7].
Many problems of practical interest in optimization, economics, and engineering involve equilibrium in their description. The techniques involved in the study of equilibrium problems are applicable to a variety of diverse areas and proved to be productive and innovative. Blum and Oettli [8] and Noor and Oettli [9] have shown that the mathematical programming problem can be viewed as special realization of abstract equilibrium problems.
Inspired and motivated by the recent development of equilibrium problems and their solutions methods, in this paper, we extend the concept of relaxed -monotonicity to mixed relaxed --monotonicity. Finally, this concept is applied with KKM-theory to solve a generalized equilibrium problem. The results of this paper can be viewed as generalization of many known results; see [10–13].
Let be a nonempty subset of real Banach space . Let be a real-valued function and let be an equilibrium function; that is, , for all . We consider the following generalized equilibrium problem: find such that Problem (1) has been studied by many authors in different settings; see, for instance, [14].
If , then the problem (1) reduces to the classical equilibrium problem, that is, to find such that Problem (2) was introduced and studied by Blum and Oettli [8].
We need the following definition and results in the sequel.
Definition 1. A real-valued function defined on a convex subset of is said to be hemicontinuous if
Definition 2. Let be a multivalued mapping. The is said to be a KKM-mapping if, for any finite subset of , , where denotes the convex hull.
Lemma 3 (see [15]). Let be a nonempty subset of a topological vector space and let be a KKM-mapping. If is closed in for all and compact for at least one , then .
Definition 4. Let be a Banach space. A mapping is said to be lower semicontinuous at , if for any sequence of such that .
Definition 5. Let be a Banach space. A mapping is said to be weakly upper semicontinuous at , if for any sequence of such that .
Now, we extend the definition of relaxed -monotonicity [11] to mixed relaxed --monotonicity.
Definition 6. A mapping is said to be mixed relaxed --monotone, if there exist mappings with , for all and , such that where and is a constant.
If , then Definition 6 reduces to the definition of generalized relaxed -monotone; that is, where If , then Definition 6 reduces to the definition of generalized relaxed -monotone; that is, where
If both , then Definition 6 coincides with the definition of monotonicity; that is,
Definition 7. A mapping is said to be -diagonally convex if, for any finite subset of and with and , one has
3. Existence of Solution for Generalized Equilibrium Problem
We establish this section with the discussion of existence of solution for generalized equilibrium problem by using mixed relaxed --monotonicity.
Theorem 8. Suppose is mixed relaxed --monotone, hemicontinuous in the first argument and convex in the second argument with , for all . Let be convex in the second argument. Then, generalized equilibrium problem (1) is equivalent to the following problem.
Find such that where and is a constant.
Proof. Suppose that the generalized equilibrium problem (1) admits a solution; that is, there exists such that Since is mixed relaxed --monotone, we have Adding on both sides of (17), we have Hence, is a solution of problem (14).
Conversely, suppose that is a solution of problem (14); that is, Let , , and ; then clearly as is convex. Thus from (17), we have Since is convex in the second argument, we have which implies that
Also as is convex in the second argument, we have
Adding (22) and (24), we have It follows that Since is hemicontinuous in the first argument, taking , we have that is, we have Hence is a solution of generalized equilibrium problem (1).
Theorem 9. Let be a nonempty bounded closed convex subset of a real Banach space . Let be a mixed relaxed --monotone, hemicontinuous in the first argument, convex in the second argument with , -diagonally convex, and lower semicontinuous. Let be convex in the second argument, -diagonally convex, and lower semicontinuous; is weakly upper semicontinuous and is weakly upper semicontinuous in the second argument. Then the mixed equilibrium problem (1) admits a solution.
Proof. Consider a multivalued mapping such that We show that ; that is, is a solution of generalized equilibrium problem (1).
Our claim is that is a KKM-mapping. Suppose to contrary that is is not a KKM-mapping; then there exists a finite subset of and with such that It follows that Also we have which contradicts the -diagonal convexity of and . Hence is a KKM-mapping.
Now consider another multivalued mapping such that We will show that , . For any given , let ; then It follows from the mixed relaxed --monotonicity of that that is, . Thus and consequently is also KKM-mapping.
Since and both are convex in the second argument and lower semicontinuous, thus they both are weakly lower semicontinuous. From weakly upper semicontinuity of , weakly upper semicontinuity of in the second argument, and the construction of , it is accessible to see that is weakly closed for all . Since is closed, bounded, and convex, it is weakly compact and consequently is weakly compact in for all . Therefore, from Lemma 3 and Theorem 8, we have that is, there exists such that Thus, the generalized equilibrium problem (1) admits a solution.
The authors express their sincere thanks to the referees for the careful and detailed reading of the paper and the very helpful suggestions that improved the paper substantially. The authors also acknowledge that this research was part of the research project and was partially supported by Universiti Putra Malaysia under ERGS 1-2013/5527179.
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Copyright © 2014 Haider Abbas Rizvi 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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Half-life (element) - Simple English Wikipedia, the free encyclopedia
This article is about the property of radioactive elements. For the video game, see Half-Life (video game).
Simulation of many identical atoms undergoing radioactive decay, starting with either 4 atoms per box (left) or 400 (right). The number at the top is how many half-lives have elapsed. Note the law of large numbers: With more atoms, the overall decay is more regular and more predictable.
{\displaystyle 1/2^{0}}
{\displaystyle 1/2^{1}}
{\displaystyle 1/2^{2}}
{\displaystyle 1/2^{3}}
{\displaystyle 1/2^{4}}
{\displaystyle 1/2^{5}}
{\displaystyle 1/2^{6}}
7 1/128
{\displaystyle 1/2^{7}}
{\displaystyle 1/2^{8}}
{\displaystyle 1/2^{9}}
10 1/1024
{\displaystyle 1/2^{10}}
{\displaystyle N}
{\displaystyle 1/2^{N}}
{\displaystyle 1/2^{N}}
The half-life of a substance is the time it takes for half of the substance to decay. The word "half-life" was first used when talking about radioactive elements where the number of atoms get smaller over time by changing into different atoms. It is now used in many other situations where something declines exponentially, like the time it takes for a drug in the body to be half gone. A Geiger-Muller detector can be used to measure the radioactive half-life; it is the time when the activity is half of the original.
Half-life depends on probability because the atoms decay at a random time. Half-life is the expected time when half the number of atoms have decayed, on average. Carbon-14 has a half-life of 5,730 years. Taking one atom of carbon-14, this will either have decayed after 5,730 years, or it will not. But if this experiment is repeated again and again, it will be seen that the atom decays within the half life 50% of the time.
Radioactive isotopes are atoms that have unstable nuclei, meaning that the nucleus of each atom will decay after enough time has passed. Their nuclei are unstable because the arrangement of protons and neutrons in them are unsteady. This decay, which means they change into completely different types of atoms. This is known as radioactive decay. When they decay, they release particles such as alpha particles, beta particles, gamma rays. Sometimes they decay by fission, which means to break into pieces, to make smaller nuclei. For example, a radioactive carbon-14 atom releases a beta particle to become nitrogen-14. As an example of fission decay, a fermium-256 atom can split into xenon-140 and palladium-112 atoms, releasing four neutrons in the process.
For example, uranium-232 has a half-life of about 69 years. Plutonium-238 has a half-life of 88 years. Carbon-14, which is used to find the age of fossils, has a half-life of 5,730 years.
After ten half-lives, about 99.9% of the atoms have decayed into different atoms, so only 0.1% of the original atoms are left, and 99.9% of the radioactivity from the original kind of atom is gone. Some atoms decay into other atoms that are also radioactive, so the remaining radioactivity depends on the type of atom.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Half-life_(element)&oldid=7586248"
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Many Paths, Many Costs | Toph
Many Paths, Many Costs
By shefin · Limits 1s, 1.0 GB
Alice and Bob are visiting Byteland. They are observing the map of this country. Surprisingly, the map of Byteland forms a tree. A tree is a connected graph without cycles. After finding this out, Alice has fixed the starting and ending city of her journey. But Bob is super confused and can’t fix his route.
After some thinking, Bob has come up with a scheme. Let’s say Alice’s starting city is
u and ending city is
v. Then Bob will choose two different cities
a and
b such that both
a and
b lie on the simple path between city
u and city
v. A simple path is the path that visits each city at most once. Soon Bob finds out that there are still many options. So, he is considering all possible options. For each option of choosing two different cities
a and
b, he calculates the cost of the simple path between cities
a and
b and appends it to the list
K. After appending the costs of all options, he sorts the list
K in ascending order. Please note that choosing cities
a, b
a,b and cities
b, a
b,a are considered as the same option.
Each edge of the tree has some weight. Bob defines the cost of a simple path as the bit-wise XOR of all the weights of the edges lying on that simple path.
Now, Bob wants to analyze the costs in
[L, R]
[L,R] subarray of the list
K. He wants your help to analyze this task. You need to calculate the product of all the costs in
[L, R]
K. As the answer can be very large, you need to print the answer modulo
(10^9 + 7)
(109+7). More formally, you need to print
\left( \prod\limits_{i=L}^{R} K_i \right) \mod (10^9+7)
(i=L∏RKi)mod(109+7).
The bitwise xor operation is represented with the "⊕" logic symbol, which is denoted as the "^" operator in C/C++, Java and Python.
T(1\leq T\leq 10^4)
T(1≤T≤104), the number of the test cases.
In each of the test cases, the first line will contain two integers
N, Q (2\leq N\leq 10^5; 1\leq Q\leq 10^4)
N,Q(2≤N≤105;1≤Q≤104), the number of cities in Byteland, and the number of queries. Each of the next
(N-1)
(N−1) lines will contain three integers
U, V, W (1\leq U, V\leq N; U\neq V; 0< W\leq10^3)
U,V,W(1≤U,V≤N;U=V;0<W≤103) denoting there is an undirected edge of
W weight between city
U and city
V. Each of the next
Q lines will contain four integers
u, v, L, R
u,v,L,R
(1\leq u, v\leq N; u\neq v; 1\leq L\leq R)
(1≤u,v≤N;u=v;1≤L≤R), the starting and ending cities of Alice and the list
K’s range Bob will analyze.
It is guaranteed that the given edges form a tree and the
L,
R will produce a valid range.
N over all test cases won’t exceed
10^5
105 and the sum of
Q over all test cases won’t exceed
10^4
In each query, print
\left( \prod\limits_{i=L}^{R} K_i \right) \mod (10^9+7)
(i=L∏RKi)mod(109+7) in a line.
In the first query, the costs of the paths Bob will consider are:
Cost(2, 3) = 13, Cost(2, 4) = 8, Cost(2,5)=12, Cost(3,4)=5, Cost(3,5)=1, Cost(4,5)=4
Cost(2,3)=13,Cost(2,4)=8,Cost(2,5)=12,Cost(3,4)=5,Cost(3,5)=1,Cost(4,5)=4.
So, the list
K will be:
[1, 4, 5, 8, 12, 13]
[1,4,5,8,12,13].
\left( \prod\limits_{i=2}^{4} K_i \right) \mod (10^9+7)
(i=2∏4Ki)mod(109+7) =
(4\times 5\times 8) \mod (10^9+7)
(4×5×8)mod(109+7) =
160.
FFT, Graph, PersistentSegmentTree
ShadowMeFastest, 0.6s
NirjhorLightest, 424 MB
At first, let’s take any arbitrary node as the root of the tree. Let, p[u]=Cost(root,u)p[u] = Cost(r...
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Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem
2014 Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem
Nedyu Popivanov, Todor Popov, Allen Tesdall
For the four-dimensional nonhomogeneous wave equation boundary value problems that are multidimensional analogues of Darboux problems in the plane are studied. It is known that for smooth right-hand side functions the unique generalized solution may have a strong power-type singularity at only one point. This singularity is isolated at the vertex
O
of the boundary light characteristic cone and does not propagate along the bicharacteristics. The present paper describes asymptotic expansions of the generalized solutions in negative powers of the distance to
O
. Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained.
Nedyu Popivanov. Todor Popov. Allen Tesdall. "Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem." Abstr. Appl. Anal. 2014 (SI62) 1 - 19, 2014. https://doi.org/10.1155/2014/260287
Nedyu Popivanov, Todor Popov, Allen Tesdall "Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem," Abstract and Applied Analysis, Abstr. Appl. Anal. 2014(SI62), 1-19, (2014)
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Simulations to explore excessive lagged X variables in time series modelling
Adding lots of lagged explanatory variables to a time series model without enough data points is a trap, and stepwise-selection doesn't help. The lasso or other regularization might be a promising alternative.
I was once in a meeting discussing a time series modelling and forecasting challenge where it was suggested that “the beauty of regression is you just add in more variables and more lags of variables and try the combinations until you get something that fits really well”. Well, no, it doesn’t work like that; at least not in many cases with realistically small social science or public policy datasets. Today’s post looks at one particular trap - a modelling strategy that tries more lags of explanatory variables than are justified given the amount of data available.
Imagine you have a time series y and two candidate explanatory variables x1 and x2. We have values of x1 and x2 for the future - perhaps because they are genuinely known in advance, perhaps generated by independent forecasts. Let’s say there is good theoretical or subject matter reason for thinking there is a relationship between them, but it’s clearly not a simple one. It’s possible the relationship is lagged - perhaps x1 and x2 are leading economic indicators and y is the result of decisions people take based on the history of the xs. The job is to forecast y conditional on X, and if possible generate some insight about the structural relationship of the xs and y, and be prepared for some scenarios where x1 or x2 might change and we want to see the impact on y.
Here’s an example of something we might see:
Those series were generated by me. We have 110 historical observations, and a forecast period of 12 for which there are observations for x1 and x2 but not y. Each series is in reality completely independent of the others, something I’ve done deliberately to illustrate the point I’m about to make. For those with an interest, they are all ARIMA(2,1,1) models with the same parameterisation but different random sequences behind them.
Basic forecasting models
A simple strategy for forecasting y would be to find the best ARIMA model that fits the data using Rob Hyndman’s auto.arima algorithm, which has been shown to perform well. We can do this with or without considering the x variables at all. If we ignore the x variables, we get a forecast that looks like this:
ma1 ma2
In fact, because we secretly know how the data was generated, we know that this is a good model of this particular data. The auto.arima algorithm picked up that the data was non-stationary and needed to be differenced once before modelling; but it got the “wrong” specification of the number of auto-regressive and moving average terms to include (this isn’t unusual, nor is it necessarily a problem). By default it uses a stepwise method to identify the best combination of up to five auto-regressive parameters and five moving average parameters.
Linear regression with time series error terms
A second, nearly as simple method of forecasting would be to fit a linear regression with an ARIMA time series error term. The auto.arima method will do this, and look after all the tests of non-stationarity and difference in response that are required. It gives easily interpretable coefficients for the relationship between y and x1 and x2. Here’s what we see in our case:
ma1 ma2 x1 x2
1.4202 0.4349 0.0371 -0.0364
s.e. 0.0870 0.0878 0.0867 0.0796
sigma^2 estimated as 0.785: log likelihood=-141.59
Basically, the model has correctly identified there is a weak or no relationship between the xs and y. The momentum of y itself is the main predictor of future y, and this shows in the forecasts which are only marginally different from our first univariate case. Using the Akaike Information Criterion, the model with x regressors is inferior to the simpler model.
A problem with above approaches is that they don’t take into account the (wrongly) suspected lagged relationship between the xs and y, other than through the current/simultaneous in time relationship. There’s an obvious solution - introduce lagged terms of x1 and x2 into the regression. There are a variety of ways of doing this and specialist R packages like dynlm, dyn and dse that I’m not going to introduce because I want things from here on to be simple and explicit. See the CRAN time series task view for a starting point to explore further.
I want to focus on how problematic this can be. How many parameters in our model are we prepared to estimate, given our 110 historical data points? Frank Harrell’s advice in Regression Modeling Strategies is:
“Studies in which models are validated on independent datasets have shown that in many situations a fitted regression model is likely to be reliable when the number of predictors (or candidate predictors if using variable selection) p is less than m/10 or m/20, where m is the limiting sample size”
Our “limiting sample size” is 110, and our 110 time series observations are actually worth less than 110 independent ones would be because each observation contains only marginal new information for us; knowing the value at time t gives us a good indication of where it will be at time t+1, something not the case in more independent sampling situations.
This suggests we should be considering five or six, or eleven at the very most, candidate variables for inclusion. Just by using auto.arima we’ve looked at ten parameters (five autoregression lags, and five moving average parameters), and by adding in x1 and x2 we’ve definitely reached our maximum. Ouch. As we’re usually not interested in individual AR and MA parameters we can perhaps get away with stretching a point, but there’s no get out of jail card here that means we can meaningfully keep adding in more explanatory variables.
So what happens if we proceed anyway? To check this out, I tried three strategies:
Created variables for up to 20 lag periods for each of the three original variables, leaving us with 62 candidate predictors, and fit a model by ordinary least squares.
Take the full model from above and use stepwise selection to knock out individual variables with p values < 0.05 (the conventional cut-off point for a variable being “not significant”). I had to torture R’s step function to do this, because quite rightly it prefers to choose models based on AIC rather than individual p-values; I wanted to mimic the p-value method for demonstration purposes.
Take the full model and use lasso-style shrinkage/regularisation on the 62 predictors’ coefficients.
Of those methods, I think #1 and #2 are both potentially disastrous and will lead to overfitting (which means that the model will perform badly when confronted with new data ie in predicting our twelve future points). Stepwise variable selection is particularly pernicious because it can create the illusion that “significant” relationships between the variables have been found. At the end of a stepwise procedure, software can provide you with F tests, t tests and p-values but they are invalid because you have chosen the variables remaining in the model precisely because they score well on those tests. In today’s case, here is what we end up with.
y_d_lag_1 1.15802 0.09837 11.772 < 2e-16 ***
x1_d_lag_1 0.27902 0.07908 3.528 0.000718 ***
y_d_lag_2 -1.04546 0.14885 -7.023 8.39e-10 ***
y_d_lag_3 0.78761 0.16493 4.775 8.68e-06 ***
y_d_lag_4 -0.44729 0.14928 -2.996 0.003704 **
x1_d_lag_4 0.24485 0.07325 3.342 0.001298 **
y_d_lag_5 0.33654 0.10118 3.326 0.001366 **
x2_d_lag_8 -0.11246 0.05336 -2.107 0.038422 *
x2_d_lag_16 -0.17330 0.06347 -2.731 0.007876 **
x1_d_lag_17 0.22910 0.07073 3.239 0.001789 **
x1_d_lag_18 -0.18724 0.07147 -2.620 0.010643 *
A high R-squared, even a high “Adjusted R-squared” and many variables with p values well below the 0.05 cut-off. So (for example) the unwary analyst would conclude we have strong evidence that y is impacted by x2 16 periods ago and by x1 17 periods ago, but mysteriously not by anything from lags 9 through 15.
Remember that the true data generation process had zero influence from x1 and x2 on y or vice versa.
Working with lags in a time series situation does not give any wild cards that make stepwise selection valid. The usual problems with stepwise selection within a model building strategy apply:
R^2
values are biased high
The usual F and
\chi^2
test statistics of the final model do not have the claimed distribution
The standard errors of coefficients are biased low and confidence intervals of effects are too narrow
p-values are far too small, do not have the proper meaning, and correction is very difficult
The coefficients of effects that remain in the model are strongly biased away from zero
Variable selection is made arbitrary
The whole process allows us to escape thinking about the problem
From Harrell’s Regression Modeling Strategies.
In the case of the full model, with 62 variables, the exact problems above do not apply. The p-values for example will be basically ok (and in fact in this case none of the x1 and x2 lags show up as significant until we start knocking out the variables one at a time). The problem is that with 64 parameters (including the intercept and variance) estimated from only 109 data points (after we have differenced the series to make them stationary), we have extremely high variance and instability in the model.
Shrinkage of the estimated coefficients via the lasso is one way of fixing this, and indeed the lasso is generally used precisely in this situation of excessive candidate explanatory variables when predictive power is important. We trade off some bias in the coefficients for better stability and predictive power. Using the lasso in this case reduces nearly all the x coefficients to zero, and lags 1 and 2 of the y variable staying in the model as they should.
Here’s the actual forecasts of the five methods we’ve tried so far:
What we see is that the two methods I’ve labelled “bad” give very similar results - a sharp decline. As a result of their overfitting they are both picking up structure in the xs that does not really impact on y.
The three “ok” methods give results that are more in tune with the secret data generating process. The most encouraging thing for me here is the success of the lasso in avoiding the pitfalls of the stepwise method. If there really is a complex lagged relationship between the xs and y, this method of looking at all 62 variables and forcing a relatively stable estimation process at least gives the analyst a sporting chance of finding it. The orderedLasso R package - which I haven’t tried using yet - looks to take this idea further in implementing the idea of an “Ordered Lasso and Sparse Time-lagged Regression”, and I’m pretty confident is worth a look.
Final word - this is an illustration rather than a comprehensive demonstration. Obviously, the thing to be done to really prove the issues would be to generate many such cases, including “real” future values of y for test purposes, and test the error rates of the forecasts from the different methods. I don’t particularly feel the need to do that, but maybe one day if I run out of things to do will give it a go.
#===========generate data============
y <- ts(cumsum(arima.sim(model = list(ar = c(0.5, -0.2), ma = 0.9), n = n) + 0.2))
x1 <- cumsum(arima.sim(model = list(ar = c(0.5, -0.2), ma = 0.9), n = n + h) - 0.2)
x2 <- cumsum(arima.sim(model = list(ar = c(0.5, -0.2), ma = 0.9), n = n + h))
orig_data <- data_frame(Value = c(y, x1, x2),
Variable = rep(c("y", "x1", "x2"), times = c(n, n + h, n + h)),
TimePeriod = c(1:n, 1:(n+h), 1:(n+h)))
orig_data %>%
ggtitle("Original data",
"Three unrelated (but that is unknown to us) univariate time series")
X_full <- cbind(x1, x2)
X_historical <- X_full[1:n, ]
X_future <- X_full[-(1:n), ]
# A version that has been differenced once....
YX_diff_lags <- data_frame(
y_d = c(diff(y), rep(NA, h)),
x1_d = diff(x1),
x2_d = diff(x2))
# And has lagged variables up to lag period of 20 for each variable:
lagnumber <- 20
for (i in 1:lagnumber){
YX_diff_lags[, ncol(YX_diff_lags) + 1] <- lag(YX_diff_lags$y_d, i)
YX_diff_lags[, ncol(YX_diff_lags) + 1] <- lag(YX_diff_lags$x1_d, i)
YX_diff_lags[, ncol(YX_diff_lags) + 1] <- lag(YX_diff_lags$x2_d,i)
names(YX_diff_lags)[-(1:3)] <-
paste0(names(YX_diff_lags)[1:3], "_lag_", rep(1:lagnumber, times = rep(3, lagnumber)))
if(ncol(YX_diff_lags) != 3 + 3 * lagnumber){
stop("Wrong number of columns; something went wrong")
#===================Modelling options========
#-------------univariate-----------------
mod1 <- auto.arima(y)
autoplot(fc1)
#------------xreg, no lags----------------
mod2 <- auto.arima(y, xreg = X_historical)
fc2 <- forecast(mod2, xreg = X_future)
#-----------xreg, lags----------------
YX_hist <- YX_diff_lags[complete.cases(YX_diff_lags), ]
mod3 <- lm(y_d ~ ., data = YX_hist)
#' Forecast from a regression model with lags one row at a time
#' Assumes the existence of: YX_diff_lags & y.
#' Has to forecast one row at a time, then populate the explanatory data frame
#' with the lagged values possible after that forecast.
onerow_forecast <- function(model){
prediction_data <- YX_diff_lags[is.na(YX_diff_lags$y_d), ]
y_names <- names(YX_diff_lags)[grepl("y_d", names(YX_diff_lags))]
for(i in 1:nrow(prediction_data)){
for(j in 2:nrow(prediction_data)){
for(k in 2:length(y_names)){
prediction_data[j , y_names[k]] <- prediction_data[j-1 , y_names[k - 1]]
if(class(model) == "cv.glmnet"){
new_y <- predict(model, newx = as.matrix(prediction_data[i, -1]))
new_y <- predict(model, newdata = prediction_data[i, -1])
prediction_data[i , "y_d"] <- new_y
# de-diff ie return to the original, non-differenced scale
fc_y <- cumsum(c(as.numeric(tail(y, 1)), prediction_data$y_d))
return(fc_y[-1])
fc3 <- onerow_forecast(mod3)
#-------xreg, lags, stepwise----------------
# high value of k to mimic stepwise selection based on p values:
mod4 <- step(mod3, k = 4.7, trace = 0)
#-----------xreg, lags, lasso---------------
# Use cross-validation to determine best value of lambda, for how much
# shrinkage to force:
mod5 <- cv.glmnet(y = YX_hist$y_d, x = as.matrix(YX_hist[ , -1]))
#====================compare results==============
p <- data_frame(Univariate = fc1$mean,
SimpleXreg = fc2$mean,
AllLags = fc3,
StepLags = fc4,
LassoLags = fc5,
TimePeriod = 1:h + n) %>%
gather(Variable, Value, -TimePeriod) %>%
mutate(Judge = ifelse(grepl("Lags", Variable) & !grepl("Lasso", Variable),
"Bad", "OK")) %>%
mutate(Variable = fct_relevel(Variable, c("AllLags", "StepLags"))) %>%
ggplot(aes(x = TimePeriod, y = Value, colour = Variable)) +
facet_wrap(~Judge, ncol = 1) +
geom_line(data = filter(orig_data, Variable == "y"), colour = "black", linetype = 3) +
xlim(c(0, 130)) +
annotate("text", x = 25, y = 15, label = "Historical data",
ggtitle("Excessive inclusion of lagged explanatory variables leads to overfitting",
"Regularization by lasso, or using a simpler model in the first place, gives much better results.") +
labs(caption = "Simulations from forecasting one time series based on two unrelated time series.")
New data and functions in nzelect 0.3.0 R package
|
Measurability of equivalence classes and MEC$_p$-property in metric spaces | EMS Press
_p
Sari Rogovin
We prove that a locally compact metric space that supports a doubling measure and a weak
p
-Poincaré inequality for some
1\le p < \infty
\mathrm{MEC}_p
-space. The methods developed for this purpose include measurability considerations and lead to interesting consequences. For example, we verify that each extended real valued function having a
p
-integrable upper gradient is locally
p
-integrable.
Esa Järvenpää, Maarit Järvenpää, Kevin Rogovin, Sari Rogovin, Nageswari Shanmugalingam, Measurability of equivalence classes and MEC
_p
-property in metric spaces . Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 811–830
|
Bessel polynomials - Wikipedia
In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series (Krall & Frink, 1948)
{\displaystyle y_{n}(x)=\sum _{k=0}^{n}{\frac {(n+k)!}{(n-k)!k!}}\,\left({\frac {x}{2}}\right)^{k}}
Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials (See Grosswald 1978, Berg 2000).
{\displaystyle \theta _{n}(x)=x^{n}\,y_{n}(1/x)=\sum _{k=0}^{n}{\frac {(n+k)!}{(n-k)!k!}}\,{\frac {x^{n-k}}{2^{k}}}}
The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is
{\displaystyle y_{3}(x)=15x^{3}+15x^{2}+6x+1\,}
while the third-degree reverse Bessel polynomial is
{\displaystyle \theta _{3}(x)=x^{3}+6x^{2}+15x+15\,}
The reverse Bessel polynomial is used in the design of Bessel electronic filters.
1.1 Definition in terms of Bessel functions
1.2 Definition as a hypergeometric function
2.1 Explicit Form
2.2 Rodrigues formula for Bessel polynomials
2.3 Associated Bessel polynomials
4 Particular values
Definition in terms of Bessel functions[edit]
The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name.
{\displaystyle y_{n}(x)=\,x^{n}\theta _{n}(1/x)\,}
{\displaystyle y_{n}(x)={\sqrt {\frac {2}{\pi x}}}\,e^{1/x}K_{n+{\frac {1}{2}}}(1/x)}
{\displaystyle \theta _{n}(x)={\sqrt {\frac {2}{\pi }}}\,x^{n+1/2}e^{x}K_{n+{\frac {1}{2}}}(x)}
where Kn(x) is a modified Bessel function of the second kind, yn(x) is the ordinary polynomial, and θn(x) is the reverse polynomial .[1]: 7, 34 For example:[2]
{\displaystyle y_{3}(x)=15x^{3}+15x^{2}+6x+1={\sqrt {\frac {2}{\pi x}}}\,e^{1/x}K_{3+{\frac {1}{2}}}(1/x)}
Definition as a hypergeometric function[edit]
The Bessel polynomial may also be defined as a confluent hypergeometric function (Dita, 2006)
{\displaystyle y_{n}(x)=\,_{2}F_{0}(-n,n+1;;-x/2)=\left({\frac {2}{x}}\right)^{-n}U\left(-n,-2n,{\frac {2}{x}}\right)=\left({\frac {2}{x}}\right)^{n+1}U\left(n+1,2n+2,{\frac {2}{x}}\right).}
A similar expression holds true for the generalized Bessel polynomials (see below):[1]: 35
{\displaystyle y_{n}(x;a,b)=\,_{2}F_{0}(-n,n+a-1;;-x/b)=\left({\frac {b}{x}}\right)^{n+a-1}U\left(n+a-1,2n+a,{\frac {b}{x}}\right).}
The reverse Bessel polynomial may be defined as a generalized Laguerre polynomial:
{\displaystyle \theta _{n}(x)={\frac {n!}{(-2)^{n}}}\,L_{n}^{-2n-1}(2x)}
from which it follows that it may also be defined as a hypergeometric function:
{\displaystyle \theta _{n}(x)={\frac {(-2n)_{n}}{(-2)^{n}}}\,\,_{1}F_{1}(-n;-2n;2x)}
where (−2n)n is the Pochhammer symbol (rising factorial).
The inversion for monomials is given by
{\displaystyle {\frac {(2x)^{n}}{n!}}=(-1)^{n}\sum _{j=0}^{n}{\frac {n+1}{j+1}}{j+1 \choose n-j}L_{j}^{-2j-1}(2x)={\frac {2^{n}}{n!}}\sum _{i=0}^{n}i!(2i+1){2n+1 \choose n-i}x^{i}L_{i}^{(-2i-1)}\left({\frac {1}{x}}\right).}
Generating function[edit]
The Bessel polynomials, with index shifted, have the generating function
{\displaystyle \sum _{n=0}^{\infty }{\sqrt {\frac {2}{\pi }}}x^{n+{\frac {1}{2}}}e^{x}K_{n-{\frac {1}{2}}}(x){\frac {t^{n}}{n!}}=1+x\sum _{n=1}^{\infty }\theta _{n-1}(x){\frac {t^{n}}{n!}}=e^{x(1-{\sqrt {1-2t}})}.}
{\displaystyle t}
, cancelling
{\displaystyle x}
, yields the generating function for the polynomials
{\displaystyle \{\theta _{n}\}_{n\geq 0}}
{\displaystyle \sum _{n=0}^{\infty }\theta _{n}(x){\frac {t^{n}}{n!}}={\frac {1}{\sqrt {1-2t}}}e^{x(1-{\sqrt {1-2t}})}.}
Similar generating function exists for the
{\displaystyle y_{n}}
polynomials as well:[3]: 106
{\displaystyle \sum _{n=0}^{\infty }y_{n-1}(x){\frac {t^{n}}{n!}}=\exp \left({\frac {1-{\sqrt {1-2xt}}}{x}}\right).}
Upon setting
{\displaystyle t=z-xz^{2}/2}
, one has the following representation for the exponential function:
{\displaystyle e^{z}=\sum _{n=0}^{\infty }y_{n-1}(x){\frac {(z-xz^{2}/2)^{n}}{n!}}.}
The Bessel polynomial may also be defined by a recursion formula:
{\displaystyle y_{0}(x)=1\,}
{\displaystyle y_{1}(x)=x+1\,}
{\displaystyle y_{n}(x)=(2n\!-\!1)x\,y_{n-1}(x)+y_{n-2}(x)\,}
{\displaystyle \theta _{0}(x)=1\,}
{\displaystyle \theta _{1}(x)=x+1\,}
{\displaystyle \theta _{n}(x)=(2n\!-\!1)\theta _{n-1}(x)+x^{2}\theta _{n-2}(x)\,}
The Bessel polynomial obeys the following differential equation:
{\displaystyle x^{2}{\frac {d^{2}y_{n}(x)}{dx^{2}}}+2(x\!+\!1){\frac {dy_{n}(x)}{dx}}-n(n+1)y_{n}(x)=0}
{\displaystyle x{\frac {d^{2}\theta _{n}(x)}{dx^{2}}}-2(x\!+\!n){\frac {d\theta _{n}(x)}{dx}}+2n\,\theta _{n}(x)=0}
The Bessel polynomials are orthogonal with respect to the weight
{\displaystyle e^{-2/x}}
integrated over the unit circle of the complex plane.[3] In other words, if
{\displaystyle n\neq m}
{\displaystyle \int _{0}^{2\pi }y_{n}\left(e^{i\theta }\right)y_{m}\left(e^{i\theta }\right)ie^{i\theta }\mathrm {d} \theta =0}
Explicit Form[edit]
A generalization of the Bessel polynomials have been suggested in literature (Krall, Fink), as following:
{\displaystyle y_{n}(x;\alpha ,\beta ):=(-1)^{n}n!\left({\frac {x}{\beta }}\right)^{n}L_{n}^{(-1-2n-\alpha )}\left({\frac {\beta }{x}}\right),}
the corresponding reverse polynomials are
{\displaystyle \theta _{n}(x;\alpha ,\beta ):={\frac {n!}{(-\beta )^{n}}}L_{n}^{(-1-2n-\alpha )}(\beta x)=x^{n}y_{n}\left({\frac {1}{x}};\alpha ,\beta \right).}
The explicit coefficients of the
{\displaystyle y_{n}(x;\alpha ,\beta )}
polynomials are:[3]: 108
{\displaystyle y_{n}(x;\alpha ,\beta )=\sum _{k=0}^{n}{\binom {n}{k}}(n+k+\alpha -2)^{\underline {k}}\left({\frac {x}{\beta }}\right)^{k}.}
Consequently, the
{\displaystyle \theta _{n}(x;\alpha ,\beta )}
polynomials can explicitly be written as follows:
{\displaystyle \theta _{n}(x;\alpha ,\beta )=\sum _{k=0}^{n}{\binom {n}{k}}(2n-k+\alpha -2)^{\underline {n-k}}{\frac {x^{k}}{\beta ^{n-k}}}.}
For the weighting function
{\displaystyle \rho (x;\alpha ,\beta ):=\,_{1}F_{1}\left(1,\alpha -1,-{\frac {\beta }{x}}\right)}
they are orthogonal, for the relation
{\displaystyle 0=\oint _{c}\rho (x;\alpha ,\beta )y_{n}(x;\alpha ,\beta )y_{m}(x;\alpha ,\beta )\mathrm {d} x}
holds for m ≠ n and c a curve surrounding the 0 point.
They specialize to the Bessel polynomials for α = β = 2, in which situation ρ(x) = exp(−2 / x).
Rodrigues formula for Bessel polynomials[edit]
The Rodrigues formula for the Bessel polynomials as particular solutions of the above differential equation is :
{\displaystyle B_{n}^{(\alpha ,\beta )}(x)={\frac {a_{n}^{(\alpha ,\beta )}}{x^{\alpha }e^{-{\frac {\beta }{x}}}}}\left({\frac {d}{dx}}\right)^{n}(x^{\alpha +2n}e^{-{\frac {\beta }{x}}})}
where a(α, β)
n are normalization coefficients.
Associated Bessel polynomials[edit]
According to this generalization we have the following generalized differential equation for associated Bessel polynomials:
{\displaystyle x^{2}{\frac {d^{2}B_{n,m}^{(\alpha ,\beta )}(x)}{dx^{2}}}+[(\alpha +2)x+\beta ]{\frac {dB_{n,m}^{(\alpha ,\beta )}(x)}{dx}}-\left[n(\alpha +n+1)+{\frac {m\beta }{x}}\right]B_{n,m}^{(\alpha ,\beta )}(x)=0}
{\displaystyle 0\leq m\leq n}
. The solutions are,
{\displaystyle B_{n,m}^{(\alpha ,\beta )}(x)={\frac {a_{n,m}^{(\alpha ,\beta )}}{x^{\alpha +m}e^{-{\frac {\beta }{x}}}}}\left({\frac {d}{dx}}\right)^{n-m}(x^{\alpha +2n}e^{-{\frac {\beta }{x}}})}
Zeros[edit]
If one denotes the zeros of
{\displaystyle y_{n}(x;\alpha ,\beta )}
{\displaystyle \alpha _{k}^{(n)}(\alpha ,\beta )}
, and that of the
{\displaystyle \theta _{n}(x;\alpha ,\beta )}
{\displaystyle \beta _{k}^{(n)}(\alpha ,\beta )}
, then the following estimates exist:[1]: 82
{\displaystyle {\frac {2}{n(n+\alpha -1)}}\leq \alpha _{k}^{(n)}(\alpha ,2)\leq {\frac {2}{n+\alpha -1}},}
{\displaystyle {\frac {n+\alpha -1}{2}}\leq \beta _{k}^{(n)}(\alpha ,2)\leq {\frac {n(n+\alpha -1)}{2}},}
{\displaystyle \alpha \geq 2}
. Moreover, all these zeros have negative real part.
Sharper results can be said if one resorts to more powerful theorems regarding the estimates of zeros of polynomials (more concretely, the Parabola Theorem of Saff and Varga, or differential equations techniques).[1]: 88 [4] One result is the following:[5]
{\displaystyle {\frac {2}{2n+\alpha -{\frac {2}{3}}}}\leq \alpha _{k}^{(n)}(\alpha ,2)\leq {\frac {2}{n+\alpha -1}}.}
Particular values[edit]
The first five Bessel Polynomials are expressed as:
{\displaystyle {\begin{aligned}y_{0}(x)&=1\\y_{1}(x)&=x+1\\y_{2}(x)&=3x^{2}+3x+1\\y_{3}(x)&=15x^{3}+15x^{2}+6x+1\\y_{4}(x)&=105x^{4}+105x^{3}+45x^{2}+10x+1\\y_{5}(x)&=945x^{5}+945x^{4}+420x^{3}+105x^{2}+15x+1\end{aligned}}}
No Bessel Polynomial can be factored into lower-ordered polynomials with strictly rational coefficients.[6] The five reverse Bessel Polynomials are obtained by reversing the coefficients. Equivalently,
{\textstyle \theta _{k}(x)=x^{k}y_{k}(1/x)}
. This results in the following:
{\displaystyle {\begin{aligned}\theta _{0}(x)&=1\\\theta _{1}(x)&=x+1\\\theta _{2}(x)&=x^{2}+3x+3\\\theta _{3}(x)&=x^{3}+6x^{2}+15x+15\\\theta _{4}(x)&=x^{4}+10x^{3}+45x^{2}+105x+105\\\theta _{5}(x)&=x^{5}+15x^{4}+105x^{3}+420x^{2}+945x+945\\\end{aligned}}}
^ a b c d Grosswald, E. (1978). Bessel Polynomials (Lecture Notes in Mathematics). New York: Springer. ISBN 978-0-387-09104-4.
^ Wolfram Alpha example
^ a b c Krall, H. L.; Frink, O. (1948). "A New Class of Orthogonal Polynomials: The Bessel Polynomials". Trans. Amer. Math. Soc. 65 (1): 100–115. doi:10.2307/1990516.
^ Saff, E. B.; Varga, R. S. (1976). "Zero-free parabolic regions for sequences of polynomials". SIAM J. Math. Anal. 7 (3): 344–357. doi:10.1137/0507028.
^ de Bruin, M. G.; Saff, E. B.; Varga, R. S. (1981). "On the zeros of generalized Bessel polynomials. I". Indag. Math. 84 (1): 1–13.
^ Filaseta, Michael; Trifinov, Ognian (August 2, 2002). "The Irreducibility of the Bessel Polynomials". Journal für die Reine und Angewandte Mathematik. 2002 (550): 125–140. CiteSeerX 10.1.1.6.9538. doi:10.1515/crll.2002.069.
"The On-Line Encyclopedia of Integer Sequences (OEIS)". Founded in 1964 by Sloane, N. J. A. The OEIS Foundation Inc. {{cite web}}: CS1 maint: others (link) (See sequences OEIS: A001497, OEIS: A001498, and OEIS: A104548)
Berg, Christian; Vignat, C. (2000). "Linearization coefficients of Bessel polynomials and properties of Student-t distributions" (PDF). Retrieved 2006-08-16.
Carlitz, Leonard (1957). "A Note on the Bessel Polynomials". Duke Math. J. 24 (2): 151–162. doi:10.1215/S0012-7094-57-02421-3. MR 0085360.
Dita, P.; Grama, Grama, N. (May 24, 2006). "On Adomian's Decomposition Method for Solving Differential Equations". arXiv:solv-int/9705008.
Fakhri, H.; Chenaghlou, A. (2006). "Ladder operators and recursion relations for the associated Bessel polynomials". Physics Letters A. 358 (5–6): 345–353. Bibcode:2006PhLA..358..345F. doi:10.1016/j.physleta.2006.05.070.
Roman, S. (1984). The Umbral Calculus (The Bessel Polynomials §4.1.7). New York: Academic Press. ISBN 978-0-486-44139-9.
"Bessel polynomials", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Weisstein, Eric W. "Bessel Polynomial". MathWorld.
OEIS sequence A001498 (Triangle a(n,k) (n >= 0, 0 <= k <= n) of coefficients of Bessel polynomials y_n(x) (exponents in increasing order))
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bessel_polynomials&oldid=1084108451"
|
Barabási–Albert model - Wikipedia
Display of three graphs generated with the Barabasi-Albert (BA) model. Each has 20 nodes and a parameter of attachment m as specified. The color of each node is dependent upon its degree (same scale for each graph).
The Barabási–Albert (BA) model is an algorithm for generating random scale-free networks using a preferential attachment mechanism. Several natural and human-made systems, including the Internet, the World Wide Web, citation networks, and some social networks are thought to be approximately scale-free and certainly contain few nodes (called hubs) with unusually high degree as compared to the other nodes of the network. The BA model tries to explain the existence of such nodes in real networks. The algorithm is named for its inventors Albert-László Barabási and Réka Albert and is a special case of an earlier and more general model called Price's model.[1]
3.2 Hirsch index distribution
3.3 Node degree correlations
4 Limiting cases
5 Non-linear preferential attachment
Many observed networks (at least approximately) fall into the class of scale-free networks, meaning that they have power-law (or scale-free) degree distributions, while random graph models such as the Erdős–Rényi (ER) model and the Watts–Strogatz (WS) model do not exhibit power laws. The Barabási–Albert model is one of several proposed models that generate scale-free networks. It incorporates two important general concepts: growth and preferential attachment. Both growth and preferential attachment exist widely in real networks.
Growth means that the number of nodes in the network increases over time.
Preferential attachment means that the more connected a node is, the more likely it is to receive new links. Nodes with a higher degree have a stronger ability to grab links added to the network. Intuitively, the preferential attachment can be understood if we think in terms of social networks connecting people. Here a link from A to B means that person A "knows" or "is acquainted with" person B. Heavily linked nodes represent well-known people with lots of relations. When a newcomer enters the community, they are more likely to become acquainted with one of those more visible people rather than with a relative unknown. The BA model was proposed by assuming that in the World Wide Web, new pages link preferentially to hubs, i.e. very well known sites such as Google, rather than to pages that hardly anyone knows. If someone selects a new page to link to by randomly choosing an existing link, the probability of selecting a particular page would be proportional to its degree. The BA model claims that this explains the preferential attachment probability rule.
Later, the Bianconi–Barabási model works to address this issue by introducing a "fitness" parameter. Preferential attachment is an example of a positive feedback cycle where initially random variations (one node initially having more links or having started accumulating links earlier than another) are automatically reinforced, thus greatly magnifying differences. This is also sometimes called the Matthew effect, "the rich get richer". See also autocatalysis.
The steps of the growth of the network according to the Barabasi–Albert model (
{\displaystyle m_{0}=m=2}
The network begins with an initial connected network of
{\displaystyle m_{0}}
New nodes are added to the network one at a time. Each new node is connected to
{\displaystyle m\leq m_{0}}
existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability
{\displaystyle p_{i}}
that the new node is connected to node
{\displaystyle i}s[2]
{\displaystyle p_{i}={\frac {k_{i}}{\sum _{j}k_{j}}},}
{\displaystyle k_{i}}
is the degree of node
{\displaystyle i}
and the sum is made over all pre-existing nodes
{\displaystyle j}
(i.e. the denominator results in twice the current number of edges in the network). Heavily linked nodes ("hubs") tend to quickly accumulate even more links, while nodes with only a few links are unlikely to be chosen as the destination for a new link. The new nodes have a "preference" to attach themselves to the already heavily linked nodes.
A network generated according to the Barabasi Albert model. The network is made of 50 vertices with initial degrees
{\displaystyle m_{0}=1}
Degree distribution[edit]
The degree distribution of the BA Model, which follows a power law. In loglog scale the power law function is a straight line.[3]
The degree distribution resulting from the BA model is scale free, in particular, it is a power law of the form
{\displaystyle P(k)\sim k^{-3}\,}
Hirsch index distribution[edit]
The h-index or Hirsch index distribution was shown to also be scale free and was proposed as the lobby index, to be used as a centrality measure[4]
{\displaystyle H(k)\sim k^{-6}\,}
Furthermore, an analytic result for the density of nodes with h-index 1 can be obtained in the case where
{\displaystyle m_{0}=1}
{\displaystyle H(1){\Big |}_{m_{0}=1}=4-\pi \,}
Node degree correlations[edit]
Correlations between the degrees of connected nodes develop spontaneously in the BA model because of the way the network evolves. The probability,
{\displaystyle n_{k\ell }}
, of finding a link that connects a node of degree
{\displaystyle k}
to an ancestor node of degree
{\displaystyle \ell }
in the BA model for the special case of
{\displaystyle m=1}
(BA tree) is given by
{\displaystyle n_{k\ell }={\frac {4\left(\ell -1\right)}{k\left(k+1\right)\left(k+\ell \right)\left(k+\ell +1\right)\left(k+\ell +2\right)}}+{\frac {12\left(\ell -1\right)}{k\left(k+\ell -1\right)\left(k+\ell \right)\left(k+\ell +1\right)\left(k+\ell +2\right)}}.}
This confirms the existence of degree correlations, because if the distributions were uncorrelated, we would get
{\displaystyle n_{k\ell }=k^{-3}\ell ^{-3}}
{\displaystyle m}
, the fraction of links who connect a node of degree
{\displaystyle k}
to a node of degree
{\displaystyle \ell }
{\displaystyle p(k,\ell )={\frac {2m(m+1)}{k(k+1)\ell (\ell +1)}}\left[1-{\frac {{\binom {2m+2}{m+1}}{\binom {k+\ell -2m}{\ell -m}}}{\binom {k+\ell +2}{\ell +1}}}\right].}
Also, the nearest-neighbor degree distribution
{\displaystyle p(\ell \mid k)}
, that is, the degree distribution of the neighbors of a node with degree
{\displaystyle k}
{\displaystyle p(\ell \mid k)={\frac {m(k+2)}{k\ell (\ell +1)}}\left[1-{\frac {{\binom {2m+2}{m+1}}{\binom {k+\ell -2m}{\ell -m}}}{\binom {k+\ell +2}{\ell +1}}}\right].}
In other words, if we select a node with degree
{\displaystyle k}
, and then select one of its neighbors randomly, the probability that this randomly selected neighbor will have degree
{\displaystyle \ell }
is given by the expression
{\displaystyle p(\ell |k)}
Clustering coefficient[edit]
An analytical result for the clustering coefficient of the BA model was obtained by Klemm and Eguíluz[6] and proven by Bollobás.[7][8] A mean-field approach to study the clustering coefficient was applied by Fronczak, Fronczak and Holyst.[9]
This behavior is still distinct from the behavior of small-world networks where clustering is independent of system size. In the case of hierarchical networks, clustering as a function of node degree also follows a power-law,
{\displaystyle C(k)=k^{-1}.\,}
This result was obtained analytically by Dorogovtsev, Goltsev and Mendes.[10]
Spectral properties[edit]
The spectral density of BA model has a different shape from the semicircular spectral density of random graph. It has a triangle-like shape with the top lying well above the semicircle and edges decaying as a power law. [11]
Generalized degree distribution
{\displaystyle F(q,t)}
of the BA model for
{\displaystyle m=1}
The same data is plotted in the self-similar coordinates
{\displaystyle t^{1/2}F(q,N)}
{\displaystyle q/t^{1/2}}
and it gives an excellent collapsed revealing that
{\displaystyle F(q,t)}
exhibit dynamic scaling.
By definition, the BA model describes a time developing phenomenon and hence, besides its scale-free property, one could also look for its dynamic scaling property. In the BA network nodes can also be characterized by generalized degree
{\displaystyle q}
, the product of the square root of the birth time of each node and their corresponding degree
{\displaystyle k}
, instead of the degree
{\displaystyle k}
alone since the time of birth matters in the BA network. We find that the generalized degree distribution
{\displaystyle F(q,t)}
has some non-trivial features and exhibits dynamic scaling
{\displaystyle F(q,t)\sim t^{-1/2}\phi (q/t^{1/2}).}
It implies that the distinct plots of
{\displaystyle F(q,t)}
{\displaystyle q}
would collapse into a universal curve if we plot
{\displaystyle F(q,t)t^{1/2}}
{\displaystyle q/t^{1/2}}
Limiting cases[edit]
Model A retains growth but does not include preferential attachment. The probability of a new node connecting to any pre-existing node is equal. The resulting degree distribution in this limit is geometric,[13] indicating that growth alone is not sufficient to produce a scale-free structure.
Model B retains preferential attachment but eliminates growth. The model begins with a fixed number of disconnected nodes and adds links, preferentially choosing high degree nodes as link destinations. Though the degree distribution early in the simulation looks scale-free, the distribution is not stable, and it eventually becomes nearly Gaussian as the network nears saturation. So preferential attachment alone is not sufficient to produce a scale-free structure.
The failure of models A and B to lead to a scale-free distribution indicates that growth and preferential attachment are needed simultaneously to reproduce the stationary power-law distribution observed in real networks.[2]
Non-linear preferential attachment[edit]
Main article: Non-linear_preferential_attachment
The BA model can be thought of as a specific case of the more general non-linear preferential attachment (NLPA) model.[14] The NLPA algorithm is identical to the BA model with the attachment probability replaced by the more general form
{\displaystyle p_{i}={\frac {k_{i}^{\alpha }}{\sum _{j}k_{j}^{\alpha }}},}
{\displaystyle \alpha }
is a constant positive exponent. If
{\displaystyle \alpha =1}
, NLPA reduces to the BA model and is referred to as "linear". If
{\displaystyle 0<\alpha <1}
, NLPA is referred to as "sub-linear" and the degree distribution of the network tends to a stretched exponential distribution. If
{\displaystyle \alpha >1}
, NLPA is referred to as "super-linear" and a small number of nodes connect to almost all other nodes in the network. For both
{\displaystyle \alpha <1}
{\displaystyle \alpha >1}
, the scale-free property of the network is broken in the limit of infinite system size. However, if
{\displaystyle \alpha }
is only slightly larger than
{\displaystyle 1}
, NLPA may result in degree distributions which appear to be transiently scale free.[15]
Preferential attachment made its first appearance in 1923 in the celebrated urn model of the Hungarian mathematician György Pólya in 1923.[16] The modern master equation method, which yields a more transparent derivation, was applied to the problem by Herbert A. Simon in 1955[17] in the course of studies of the sizes of cities and other phenomena. It was first applied to the growth of networks by Derek de Solla Price in 1976.[18] Price was interested in the networks of citation between scientific papers and the Price model used "cummulative advantage" (his name for preferential attachment) to produce a directed network so the Barabási-Albert model is an undirected version of Price's model. The name "preferential attachment" and the present popularity of scale-free network models is due to the work of Albert-László Barabási and Réka Albert, who rediscovered the process independently in 1999 and applied it to degree distributions on the web.[3]
Bianconi–Barabási model
Price's model
^ Albert, Réka; Barabási, Albert-László (2002). "Statistical mechanics of complex networks". Reviews of Modern Physics. 74 (1): 47–97. arXiv:cond-mat/0106096. Bibcode:2002RvMP...74...47A. CiteSeerX 10.1.1.242.4753. doi:10.1103/RevModPhys.74.47. ISSN 0034-6861.
^ a b c Albert, Réka; Barabási, Albert-László (2002). "Statistical mechanics of complex networks" (PDF). Reviews of Modern Physics. 74 (47): 47–97. arXiv:cond-mat/0106096. Bibcode:2002RvMP...74...47A. CiteSeerX 10.1.1.242.4753. doi:10.1103/RevModPhys.74.47. Archived from the original (PDF) on 2015-08-24.
^ a b Barabási, Albert-László; Albert, Réka (October 1999). "Emergence of scaling in random networks" (PDF). Science. 286 (5439): 509–512. arXiv:cond-mat/9910332. Bibcode:1999Sci...286..509B. doi:10.1126/science.286.5439.509. PMID 10521342. Archived from the original (PDF) on 2012-04-17.
^ Korn, A.; Schubert, A.; Telcs, A. (2009). "Lobby index in networks". Physica A. 388 (11): 2221–2226. arXiv:0809.0514. Bibcode:2009PhyA..388.2221K. doi:10.1016/j.physa.2009.02.013.
^ a b Fotouhi, Babak; Rabbat, Michael (2013). "Degree correlation in scale-free graphs". The European Physical Journal B. 86 (12): 510. arXiv:1308.5169. Bibcode:2013EPJB...86..510F. doi:10.1140/epjb/e2013-40920-6.
^ Klemm, K.; Eguíluz, V. C. (2002). "Growing scale-free networks with small-world behavior". Physical Review E. 65 (5): 057102. arXiv:cond-mat/0107607. Bibcode:2002PhRvE..65e7102K. doi:10.1103/PhysRevE.65.057102. hdl:10261/15314. PMID 12059755.
^ Bollobás, B. (2003). "Mathematical results on scale-free random graphs". Handbook of Graphs and Networks. pp. 1–37. CiteSeerX 10.1.1.176.6988.
^ "Mathematical results on scale-free random graphs". 2003: 1–37. CiteSeerX 10.1.1.176.6988. {{cite journal}}: Cite journal requires |journal= (help)
^ Albert, Reka; Barabasi, Albert-Laszlo; Hołyst, Janusz A (2003). "Mean-field theory for clustering coefficients in Barabasi-Albert networks". Phys. Rev. E. 68 (4): 046126. arXiv:cond-mat/0306255. doi:10.1103/PhysRevE.68.046126. PMID 14683021.
^ Dorogovtsev, S.N.; Goltsev, A.V.; Mendes, J.F.F. (25 June 2002). "Pseudofractal scale-free web". Physical Review E. 65 (6): 066122. arXiv:cond-mat/0112143. Bibcode:2002PhRvE..65f6122D. doi:10.1103/PhysRevE.65.066122. PMID 12188798.
^ Farkas, I.J.; Derényi, I.; Barabási, A.-L.; Vicsek, T. (20 July 2001) [19 February 2001]. "Spectra of "real-world" graphs: Beyond the semicircle law". Physical Review E. 64 (2): 026704. arXiv:cond-mat/0102335. Bibcode:2001PhRvE..64b6704F. doi:10.1103/PhysRevE.64.026704. hdl:2047/d20000692. PMID 11497741.
^ M. K. Hassan, M. Z. Hassan and N. I. Pavel, “Dynamic scaling, data-collapseand Self-similarity in Barabasi-Albert networks” J. Phys. A: Math. Theor. 44 175101 (2011) https://dx.doi.org/10.1088/1751-8113/44/17/175101
^ Pekoz, Erol; Rollin, A.; Ross, N. (2012). "Total variation and local limit error bounds for geometric approximation". Bernoulli.
^ Krapivsky, P. L.; Redner, S.; Leyvraz, F. (20 November 2000). "Connectivity of Growing Random Networks". Physical Review Letters. 85 (21): 4629–4632. arXiv:cond-mat/0005139. doi:10.1103/PhysRevLett.85.4629.
^ Krapivsky, Paul; Krioukov, Dmitri (21 August 2008). "Scale-free networks as preasymptotic regimes of superlinear preferential attachment". Physical Review E. 78 (2): 026114. arXiv:0804.1366. doi:10.1103/PhysRevE.78.026114.
^ Albert-László, Barabási (2012). "Luck or reason". Nature. 489 (7417): 507–508. doi:10.1038/nature11486. PMID 22972190.
^ Simon, Herbert A. (December 1955). "On a Class of Skew Distribution Functions". Biometrika. 42 (3–4): 425–440. doi:10.1093/biomet/42.3-4.425.
^ Price, D.J. de Solla (September 1976). "A general theory of bibliometric and other cumulative advantage processes". Journal of the American Society for Information Science. 27 (5): 292–306. CiteSeerX 10.1.1.161.114. doi:10.1002/asi.4630270505.
"This Man Could Rule the World"
"A Java Implementation for Barabási–Albert"
"Generating Barabási–Albert Model Graphs in Code"
Retrieved from "https://en.wikipedia.org/w/index.php?title=Barabási–Albert_model&oldid=1078410343"
|
\frac{2}{x+1}+\frac{3}{2x+3}
“No! Why are you doing this to me?”
Kongpob whines as soon as Arthit writes out the equation on a scrap piece of paper.
“I’m not the one doing this to you,” he rolls his eyes. “This is from the homework due on Wednesday, which you said you needed help with.”
Kongpob wonders why he’d ever asked to be tutored in Maths in the first place.
Because you need to get into engineering school.
“Fine,” he sighs. “How do I do this?”
Arthit shakes his head in exasperation, pulling the paper towards him again.
“You know how to add up regular fractions with different denominators, right?”
“Kind of? You multiply the numerator in one fraction with the denominator in the other fraction…then multiply the two denominators to get the new one?”
“Exactly. So based on your understanding of that, which algebraic equation do you need to solve first?”
Kongpob blinks blearily at the page. Perhaps it’s because it’s still early, but the numbers read like an alien language to him. Of course, it really isn’t that difficult, but for whatever reason, his brain doesn’t really want to digest it. He’s trying his best though, at the very least so that he’s not wasting Arthit’s time.
2\left(2x+3\right)=4x+6
“Great. And the other one?”
Arthit seems genuinely pleased that he’s managed to quickly catch on to this one part. Their session last week had ended in Arthit grumbling about how Kongpob needed a chip inserted into his brain, not a tutor. But seeing him pleasantly surprised makes Kongpob want to try even harder, if only to witness more of his friend’s softer side.
3\left(x+1\right)=3x+3
“And then the two denominators?”
\begin{array}{l}\left(x+1\right)\left(2x+3\right)\\ =2{x}^{2}+3x+2x+3\\ =2{x}^{2}+5x+3\end{array}
“Okay, then add up your two numerator values.”
\begin{array}{l}4x+6+3x+3\\ =7x+9\end{array}
“So the fraction should read…”
\frac{7x+9}{2{x}^{2}+5x+3}
Arthit squints at Kongpob, who’s just watching his reaction.
“Are you sure you actually need my help? You’re not just pretending to be bad at algebra for some twisted joke, are you?”
Kongpob just shrugs and puts his pencil down.
“Somehow when Teacher Danai explains it, it always goes in one ear and out the other. His voice is…to put it nicely, very…soothing.”
Arthit smirks at this. Their Maths teacher, while an indubitably kind and patient man, seems to lack any enthusiasm for life, an attribute unfortunately reflected in the monotone of his voice.
“Why are you trying so hard to up your grade anyway? Are you really doing that badly?”
“I’m getting a B. But I’m looking to apply to engineering programs in university, so I need at least an A minus. Not that useful excelling at English in engineering school.”
Arthit blinks, nodding thoughtfully.
“I’m looking to get into engineering, too.”
“Wow, at least buy me dinner before you try to follow me to college,” Kongpob jokes, and Arthit narrows his eyes in a glare.
“I’m just teasing,” Kongpob laughs as he works on the next question. “So…why engineering?”
Arthit shrugs. “I like seeing how things get developed from small plans into real products. Especially in factories.”
“Yeah, like those robots that kiss the conveyor belt to make those chocolates!”
“Yeah,” Arthit chuckles at Kongpob’s example. “Something like that. But more like…everyday things. Home appliances, and that kind of thing.”
“Like an electric grill?”
Arthit nods. “Something like that.”
“Why don’t you use one at the stall? Isn’t it easier than using charcoal?”
The paler boy seems to ponder this question seriously, pursing his lips in thought.
“Well, firstly, it would consume a lot of electricity, and then I’d have to crank the prices way up to cover that. We have a gas grill at home, but the flavour you get just isn’t the same. I mean, too much charcoal smoke exposure and eating too much chargrilled food would probably give you colon cancer eventually, but if I can figure out a way to get the taste without the side effects, then-”
He cuts himself off, shaking his head as he realises that he’s rambling.
“Uh…never mind, it’s stupid. I don’t plan on grilling pork forever, anyway.”
Kongpob is completely fixated.
“No, I think that’s great. I wish I had a legitimate reason to study engineering.”
Arthit raises an eyebrow inquisitively.
“My dad’s company works with a lot of factories that supply parts for commercial products. So, I’m kind of expected to take over one day,” he sighs outwardly, shading in a square on the grid paper in front of him.
“But…it’s not what you want?”
“Not really? I mean, I don’t know. I’ve still got time to think about what I actually want, I guess. But my parents expect me to do engineering. I just don’t really have the right kind of skill set for it, I don’t think.”
Arthit says nothing, simply watching Kongpob as he talks, the usual bright and slightly pushy demeanour fading from his face. It’s a little unsettling, seeing the boy so unsure of himself.
“Anyway, sorry. Don’t mean to make this all weird,” Kongpob shakes his hand as if to brush the topic away. “Back to algebra.”
Kongpob manages to catch up to Arthit before he even enters the isolated bathroom at lunch.
“P’Arthit!” he whisper-yells, as if anyone else might be around to hear them.
The boy whips around to look at him, slightly startled.
“Uh…hi,” he finally says after a beat. “You’re…here again.”
“Actually, I was…wondering if you want to go somewhere else.”
Arthit’s glance wavers, clutching the straps of the plain grey and blue lunch bag in his hand.
“S-somewhere else?”
Kongpob doesn’t miss his panicked expression, and retracts his thoughts a few paces, trying to parse his next words so as not to scare Arthit off.
“There’s a sort of…garden on the roof of the main building. Nobody goes there, but it’s actually quite nice. There are a few tables. And a basketball hoop,” he watches his friend’s face morph from anxious to puzzled. “Anyway, I kind of want to eat lunch there today, and you’re welcome to join me if you want.”
Kongpob waits a few more seconds, and seeing no response from Arthit, he nods with a tight-lipped smile, and slowly begins to walk away towards the back stairs of the main building. It’s as he’s reaching the second floor that he hears tentative footsteps following closely behind. He pauses, looking down over the stair railing to see Arthit cautiously peering around them as though he’s not supposed to be in the building at all, and he’s about to get caught for trespassing.
When Kongpob finally pushes open the metal door to the rooftop garden, he inhales deeply, breathing in fresh, open air. The entire rooftop is covered with various potted plants, ranging from flowers, to plain greens, and a few larger pots holding lemon trees and tomato plants.
Arthit is in awe. He walks around, taking in his surroundings. It’s certainly far more pleasant, and warmer than the echoey, tiled bathroom he’s used to.
Kongpob grins at Arthit’s fascination, and swings his legs over a bench at one of the tables under the shaded part of the roof. After a few more moments of visual exploration, Arthit joins him at the table, placing his lunch bag on it with a clatter.
“How do you even know about this place?”
Kongpob chuckles, opening his lunchbox. A baked salmon fillet on a bed of wild rice pilaf. If Arthit is curious about his rather bougie meal, he doesn’t mention it.
“I used to help out with the student council last year, and sometimes they lay their posters out here to dry after painting them. I really liked the vibe, so every now and then I come up here during lunch to hang out alone or take a nap if it’s too loud in the courtyard. I also come here sometimes after school if P’Shin is stuck in traffic on the way to pick me up.”
“P’Shin?”
“My family’s butler, although he’s more like a family friend at this rate.”
Arthit snorts, shaking his head. Such a prince, he thinks. Kongpob digs a spoon into his food and begins eating. Arthit still hasn’t taken his lunchbox out.
“So…what’s for lunch?” Kongpob glances at the linen bag on the table.
Arthit blinks a few times before taking the thermos out of the bag and unscrewing the lid. He peers inside.
“Basil pork with rice.”
Kongpob nods, and chews on a spoonful of his own food.
“Sounds good. Did you make it yourself?”
Arthit shakes his head, still not picking up his own utensils.
“There’s a guy who runs a nearby stall who gives me my choice of whatever he has left at the end of the day for lunch the next day.”
There’s a quiet that dwells between them again, but Arthit still makes no move to begin eating. Instead, he watches Kongpob chew and swallow each bite, his tongue darting out occasionally to lick at the corners of his mouth.
I can do this, he thinks. It’s just Kongpob. Just one spoonful. Just to test the waters.
Arthit slowly picks up his spoon, exhaling heavily, before picking up just the right rice to minced pork ratio, one particularly crunchy-looking basil leaf nestled in the bite he’s compiled. He stares at it a moment, as though it might come alive and start laughing in his face.
Slowly, he lifts the spoon to his mouth, gulping, and watching Kongpob, who seems to take no notice. Arthit knows that he’s feigning his lack of attention, but he’d be lying if he weren’t not at least a little warmed at the thought of Kong wanting him to feel comfortable enough to eat in front of him.
Stop faffing around and just do it, he scolds himself internally.
Before he can argue with himself further, he’s sticking the entire spoonful in his mouth, and pulling it out just as quickly, his lips closed as he holds the food in his mouth. Kongpob merely breaks off another piece of salmon with his own spoon and scoops it up with a dollop of his rice before mindlessly shoving it in his mouth, chewing like it’s no big deal.
Tentatively, Arthit begins to move his jaw, chewing his own food and trying to focus on the flavours: salty, spicy, earthy, and with just the right amount of plain rice to balance it all out. Finally, he decides he’s chewed the one mouthful enough for it to be as mushy as baby’s food, and swallows it all in one gulp.
When he looks back up at Kongpob, the boy is smiling at him softly. Arthit, hands slightly clammy with nervous sweat, turns one corner of his mouth up, and Kongpob simply turns his attention right back to digging into his own food.
Arthit eats another five spoonfuls before Kongpob bids him his leave, claiming he has to rejoin his friends before they question his whereabouts.
The sun feels warm on Arthit’s pale skin, and he grins as he finishes his meal, breathing in the calming scent of jasmine. He thinks it might be his new favourite scent.
“How’s Arthit?” M asks suddenly.
Kongpob sputters and almost chokes on his drink.
“Arthit. You guys are friends now, aren’t you?”
M raises an eyebrow questioningly, smacking his friend’s back a few times as he continues to cough. Kongpob finally clears his throat enough to speak.
“Um…yeah. He’s fine.”
Kongpob flushes, and thanks his genes that he’s not prone to visibly blushing. He certainly feels it, though.
“You guys have been hanging out a lot lately. Are you replacing me as your best friend?”
He says this jovially, nudging Kongpob’s arm. Kongpob just lets out a short laugh.
“No, he’s just tutoring me in algebra. Plus…he’s actually kind of nice to be around.”
M scrunches his eyebrows together, inspecting his friend’s face. There’s a small on his lips as he says this, his gaze soft as he fiddles with the straw, dipping it repeatedly in his iced coffee.
“Algebra, huh?”
“Yeah, he’s like a math genius or something. Somehow, when Arthit explains it to me, I just get it right away. When Teacher Danai explains it, it’s like water through a strainer. Arthit makes it easy to understand. Probably because we talk in between as well.”
“So…he’s actually talking to you, then? Like, personal stuff?”
“Uh…sometimes, yeah. Why?” Kongpob looks at his friend curiously. M shrugs, shaking his head gently.
“So…I guess he’s told you about the birthday card incident?”
Kongpob sits up straighter at this, turning now to look M straight in the eye.
“Birthday card incident? What are you talking about?”
M groans, smacking himself in the forehead.
“Never mind. If he hasn’t told you himself, he probably doesn’t want you to remember.”
“No, M. What birthday card incident?” Kongpob says sternly.
M had been hiding things from him, he knows, but if he’s going to get anywhere with his new friendship, he needs whatever intel he can get to aid him in helping Arthit out of his shell.
His friend pauses for a few moments, before sighing heavily.
“Fine. But I don’t know if you’re going to like what you hear.”
Kongpob is a little hesitant today as he makes his way down the street to the familiar yellow sign. He doesn’t know what to make of M’s information, or if he even believes it. Mostly, there’s guilt. Immense, gut-wrenching guilt.
Arthit must notice that he’s not his usual, cheery self, because he actually pauses to look at him instead of pretending to ignore him as he normally would.
“Hey,” he says. “I thought you were going to text me your order ahead of time.”
Kongpob forces a smile, but it comes out looking more sad than anything.
“Yeah, sorry. I forgot. I’ll uh…I’ll get six of the chicken, and two beef.”
Arthit cocks an eyebrow, but moves to fulfill the order quickly, if only to distract from the evident tension.
“Hungry, then?”
Kongpob just nods briefly again, his eyes not once leaving Arthit’s. The boy feels somewhat uneasy under his friend’s unwavering gaze.
“Um…” Arthit rubs the back of his neck, unnerved by being watched so closely that it makes his skin prick. “Is everything, like…okay?” he gestures vaguely at Kongpob. “You seem kind of…I don’t know…down.”
“It’s nothing,” Kongpob forces a thin smile. “Just a bad practice today, that’s all.”
Even if their friendship is fairly new, Arthit knows instantly that this is a lie. But it’s not his place to pry, not his place to know, so he doesn’t comment.
Instead, he hands the bag with Kongpob’s order to him, nodding as Kong mumbles a quiet thanks.
“Um…I have to go,” he says, and doesn’t look at him or say another word as he heads down the street, the usual bounce in his step distinctly missing, and his steps seemingly distracted, as he almost bumps into another pedestrian.
Arthit watches his friend curiously. What could possibly have happened since they’d last spoken that had upset Kongpob to the point that he didn’t even feel like teasing him as he normally would? Hadn’t they had some sort of breakthrough moment at lunch?
Or maybe it really hadn’t been to do with him. Of course not. That would be completely presumptuous.
Still, he makes a mental note to possibly text him later that evening.
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Train DDPG Agent to Swing Up and Balance Pendulum with Bus Signal - MATLAB & Simulink - MathWorks India
Pendulum Swing-Up Model with Bus
Create Environment Interface with Bus
This example shows how to convert a simple frictionless pendulum Simulink® model to a reinforcement learning environment interface, and trains a deep deterministic policy gradient (DDPG) agent in this environment.
The starting model for this example is a simple frictionless pendulum. The training goal is to make the pendulum stand upright without falling over using minimal control effort.
mdl = 'rlSimplePendulumModelBus';
Both the observation and action signals are Simulink buses.
{\mathit{r}}_{\mathit{t}}
{\mathit{r}}_{\mathit{t}}=-\left({{\theta }_{\mathit{t}}}^{2}+0.1{\stackrel{˙}{{\theta }_{\mathit{t}}}}^{2}+0.001{\mathit{u}}_{\mathit{t}-1}^{2}\right)
{\theta }_{\mathit{t}}
\stackrel{˙}{{\theta }_{\mathit{t}}}
{\mathit{u}}_{\mathit{t}-1}
The model used in this example is similar to the simple pendulum model described in Load Predefined Simulink Environments. The difference is that the model in this example uses Simulink buses for the action and observation signals.
The environment interface from a Simulink model is created using rlSimulinkEnv, which requires the name of the Simulink model, the path to the agent block, and observation and action reinforcement learning data specifications. For models that use bus signals for actions or observations, you can create the corresponding specifications using the bus2RLSpec function.
Specify the path to the agent block.
agentBlk = 'rlSimplePendulumModelBus/RL Agent';
Create the observation Bus object.
obsBus = Simulink.Bus();
obs(1) = Simulink.BusElement;
obs(1).Name = 'sin_theta';
obs(2).Name = 'cos_theta';
obs(3).Name = 'dtheta';
obsBus.Elements = obs;
Create the action Bus object.
actBus = Simulink.Bus();
act(1) = Simulink.BusElement;
act(1).Name = 'tau';
act(1).Min = -2;
act(1).Max = 2;
actBus.Elements = act;
Create the action and observation specification objects using the Simulink buses.
obsInfo = bus2RLSpec('obsBus','Model',mdl);
actInfo = bus2RLSpec('actBus','Model',mdl);
Create the reinforcement learning environment for the pendulum model.
A DDPG agent decides which action to take, given observations, using an actor representation. To create the actor, first create a deep neural network with three inputs (the observations) and one output (the action). The three observations can be combined using a concatenationLayer.
For more information on creating a deep neural network value function representation, see Create Policies and Value Functions.
sinThetaInput = featureInputLayer(1,'Normalization','none','Name','sin_theta');
cosThetaInput = featureInputLayer(1,'Normalization','none','Name','cos_theta');
dThetaInput = featureInputLayer(1,'Normalization','none','Name','dtheta');
scalingLayer('Name','ActorScaling1','Scale',max(actInfo.UpperLimit))];
actorNetwork = layerGraph(sinThetaInput);
actorNetwork = addLayers(actorNetwork,cosThetaInput);
actorNetwork = addLayers(actorNetwork,dThetaInput);
actorNetwork = addLayers(actorNetwork,commonPath);
actorNetwork = connectLayers(actorNetwork,'sin_theta','concat/in1');
actorNetwork = connectLayers(actorNetwork,'cos_theta','concat/in2');
actorNetwork = connectLayers(actorNetwork,'dtheta','concat/in3');
plot(layerGraph(actorNetwork))
Create the actor representation using the specified deep neural network and options. You must also specify the action and observation info for the actor, which you obtained from the environment interface. For more information, see rlContinuousDeterministicActor.
"ObservationInputNames",["sin_theta","cos_theta","dtheta"]);
A DDPG agent approximates the long-term reward given observations and actions using a critic value function representation. To create the critic, first create a deep neural network with two inputs, the observation and action, and one output, the state action value.
Construct the critic in a similar manner to the actor. For more information, see rlQValueFunction.
featureInputLayer(1,'Normalization','none','Name', 'action')
fullyConnectedLayer(300,'Name','CriticActionFC1','BiasLearnRateFactor', 0)];
criticNetwork = layerGraph(sinThetaInput);
criticNetwork = addLayers(criticNetwork,cosThetaInput);
criticNetwork = addLayers(criticNetwork,dThetaInput);
criticNetwork = connectLayers(criticNetwork,'sin_theta','concat/in1');
criticNetwork = connectLayers(criticNetwork,'cos_theta','concat/in2');
criticNetwork = connectLayers(criticNetwork,'dtheta','concat/in3');
"ObservationInputNames",["sin_theta","cos_theta","dtheta"],"ActionInputNames","action");
Then create the DDPG agent using the specified actor representation, critic representation, and agent options. For more information, see rlDDPGAgent.
Stop training when the agent receives an average cumulative reward greater than –740 over five consecutive episodes. At this point, the agent can quickly balance the pendulum in the upright position using minimal control effort.
load('SimulinkPendBusDDPG.mat','agent')
rlDDPGAgent | rlSimulinkEnv | train | bus2RLSpec
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A First-Order Mechanical Device to Model Traumatized Craniovascular Biodynamics | J. Med. Devices | ASME Digital Collection
Sean S. Kohles,
Sean S. Kohles
Kohles Bioengineering
, Portland, OR 97214-5135; Department of Surgery,
, Portland, OR 97239-3098; and Department of Mechanical and Materials Engineering,
, Portland, OR 97207-0751
e-mail: ssk@kohlesbioengineering.com
Ryan W. Mangan,
Edward Stan,
Edward Stan
Biomedical Signal Processing Laboratory, Department of Electrical and Computer Engineering,
J. Med. Devices. Mar 2007, 1(1): 89-95 (7 pages)
Kohles, S. S., Mangan, R. W., Stan, E., and McNames, J. (July 30, 2006). "A First-Order Mechanical Device to Model Traumatized Craniovascular Biodynamics." ASME. J. Med. Devices. March 2007; 1(1): 89–95. https://doi.org/10.1115/1.2355689
Mathematical models currently exist that explore the physiology of normal and traumatized intracranial function. Mechanical models are used to assess harsh environments that may potentially cause head injuries. However, few mechanical models are designed to study the adaptive physiologic response to traumatic brain injury. We describe a first-order physical model designed and fabricated to elucidate the complex biomechanical factors associated with dynamic intracranial physiology. The uni-directional flow device can be used to study interactions between the cranium, brain tissue, cerebrospinal fluid, vasculature, blood, and the heart. Solid and fluid materials were selected to simulate key properties of the cranial system. Total constituent volumes (solid and fluid) and volumetric flow
(650ml∕min)
represent adult human physiology, and the lengths of the individual segments along the flow-path are in accord with Poiseuille’s equation. The physical model includes a mechanism to simulate autoregulatory vessel dynamics. Intracranial pressures were measured at multiple locations throughout the model during simulations with and without post-injury brain tissue swelling. Two scenarios were modeled for both cases: Applications of vasodilation/constriction and changes in the head of bed position. Statistical results indicate that all independent variables had significant influence over fluid pressures measured throughout the model
(p<0.0001)
including the vasoconstriction mechanism
(p=0.0255)
. The physical model represents a first-order design realization that helps to establish a link between mathematical and mechanical models. Future designs will provide further insight into traumatic head injury and provide a framework for unifying the knowledge gained from mathematical models, injury mechanics, clinical observations, and the response to therapies.
brain, biological tissues, blood, cardiology, Poiseuille flow, statistical analysis, haemodynamics
Biological tissues, Blood, Brain, Design, Flow (Dynamics), Physiology, Pressure, Traumatic brain injury, Wounds, Cerebrospinal fluid, Poiseuille flow, Fluids, Vessels, Biomechanics, Simulation
Pathophysiology and Management of Increased Intracranial Pressure
Pulse and Mean Intracranial Pressure Analysis in Pediatric Traumatic Brain Injury
Czonyka
Contribution of Mathematical Modeling to the Interpretation of Bedside Tests of Cerebrovascular Autoregulation
A Simple Mathematical Model of the Interaction Between Intracranial Pressure and Cerebral Hemodynamics
A Computer Model of Intracranial Pressure Dynamics During Traumatic Brain Injury That Explicitly Models Fluid Flows and Volumes
A Three-Dimensional Human Head Finite Element Model and Power Flow in a Human Head Subject to Impact Loading
A Mechanical Model of Cerebral Circulation During Sustained Acceleration
Aviat Space Environ. Med.
Investigations of Flow and Pressure Distributions in Physical Model of the Circle of Willis
Mathematical Study of the Role of Nonlinear Venous Compliance in the Cranial Volume-Pressure Test
Direct Perfusion Measurements of Cancellous Bone Anisotropic Permeability
Human Circulation and Regulation During Physical Stress
Mechanical Vasoconstriction for a Cerebral Myogenic Autoregulatory Model
26th Annual International Conference of the IEEE Engineering in Medicine and Biology Society—Proceedings
Design and Numerical Implementation of a 3D Nonlinear Viscoelastic Constitutive Model for Brain Tissue During Impact
Capstone Design of a Cranial Vascular Mechanical Model
American Society of Biomechanics, 28th Annual Meeting
The Influence of the Non-Newtonian Properties of Blood on the Flow in Large Arteries: Unsteady Flow in a 90Degrees Curved Tube
Staged Growth of Optimized Arterial Model Trees
Voronoi Polyhedra Analysis of Optimized Arterial Tree Models
3D Models of Blood Flow in the Cerebral Vasculature
One-Dimensional and Three-Dimensional Models of Cerebrovascular Flow
Further Evidence for the Dynamic Stability of Intracranial Saccular Aneurysms
On the Use of a Patient-Specific Rapid-Prototyped Model to Simulate the Response of the Human Head to Impact and Comparison With Analytical and Finite Element Models
An Introduction to Biomechanics: Solids and Fluids Analysis and Design
Mechanical Properties of Brain Tissue In Vivo: Experiment and Computer Simulation
Axial Mechanical Properties of Fresh Human Cerebral Blood Vessels
A Comparative Study of the Spiegelberg Compliance Device With a Manual Volume-Injection Method: A Clinical Evaluation in Patients With Hydrocephalus
Virtual Simulation of the Effects of Intracranial Fluid Cavitation in Blast-Induced Traumatic Brain Injury
|
Experimental Verification of Fracture Toughness Requirement for Leak-Before-Break Performance for 155–175 ksi Strength Level Gas Cylinders | J. Pressure Vessel Technol. | ASME Digital Collection
M. D. Rana
Research and Development Department, Linde Division, Union Carbide Corporation, Tonawanda, N.Y. 14150
Rana, M. D. (November 1, 1987). "Experimental Verification of Fracture Toughness Requirement for Leak-Before-Break Performance for 155–175 ksi Strength Level Gas Cylinders." ASME. J. Pressure Vessel Technol. November 1987; 109(4): 435–439. https://doi.org/10.1115/1.3264927
This paper deals with a determination of the fracture toughness requirement to obtain leak-before-break performance for a 155–175 ksi strength level high-pressure gas cylinder. Analytical LEFM methods along with Irwin’s KIc-Kc equation related by the parameter βIc were used to predict the fracture toughness requirement for the plane-stress fracture state problem. Experimental work was conducted on flawed cylinders to quantify the fracture toughness requirement for leak-before-break performance. The results indicated that the analytically predicted toughness requirement is 4 to 25 percent higher than that established experimentally. The results also indicated that the minimum specified KIc(J) value of 85 ksi
in.
(93.5 MPa
m
) for the gas cylinder is sufficiently higher than the analytically and experimentally established toughness values to provide the desired leak-before-break performance.
Fracture toughness, Gas cylinders, Leak-before-break, Cylinders, Fracture (Materials), Fracture (Process), High pressure (Physics), Stress
|
Find, correct to four decimal places, the length of the curve of intersection of
Find, correct to four decimal places, the length of the curve of intersection of the cylinder 4x^2 + y^2 = 4 and the plane x + y + z = 5.
4{x}^{2}+{y}^{2}=4
x+y+z=5.
jlo2niT
I think this example is solved like this:
Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle? P(3, -2, -3), Q(7, 0, 1), R(1, 2, 1)
Convert from polar to rectangular coordinates:
\left(2,\left(\pi /2\right)\right)⇒\left(x,y\right)
Parametric Equations of a Hyperbola Eliminate the parameter 0 in the following parametric equations.
x=a\text{ }tan\text{ }0\text{ }y=b\text{ }sec
If the tangent line to y = f(x) at (4, 3) passes through the point (0, 2), find f(4) and f’(4)
For and
f\left(2\right)
{z}^{2}={x}^{2}+{y}^{2}
|
The Generalized Riemann-Hilbert Boundary Value Problem for Non-Homogeneous Polyanalytic Differential Equation of Order $n$ in the Sobolev Space $W_{n,p}(D)$ | EMS Press
The Generalized Riemann-Hilbert Boundary Value Problem for Non-Homogeneous Polyanalytic Differential Equation of Order
n
in the Sobolev Space
W_{n,p}(D)
Ali Seif Mshimba
Given is a nonlinear non-homogeneous polyanalytic differential equation of order
n
in a simply-connected domain
D
in the complex plane. Initially we prove (under certain conditions) the existence of its general solution in
W_{n,p}(D)
by first transforming it into a system of integro-differential equations. Next we prove the solvability of a generalized Riemann-Hilbert problem for the differential equation. This is effected by first reducing the boundary value problem posed to a corresponding one for a polyanalytic function. The latter is then transformed into
n
classical Riemann-Hilbert problems for holomorphic functions, whose solutions are known in the literature.
Ali Seif Mshimba, The Generalized Riemann-Hilbert Boundary Value Problem for Non-Homogeneous Polyanalytic Differential Equation of Order
n
W_{n,p}(D)
|
Spherical coordinate system - Simple English Wikipedia, the free encyclopedia
The three coordinate variables are
{\displaystyle r,\theta ,{\text{ and }}\varphi }
The two angular variables are related to the Earth's latitude and longitude
A spherical coordinate system uses three numbers to identify a point in space.: Usually, two angles, and a distance from the origin of the coordinate system. If the point lies on the sphere, only the two angles are needed, because the distance from the origin is known.
The two angular numbers are related to lines of longitude and latitude on Earth. Earth's longitude is the same as the variable phi, which can be written as either
{\displaystyle \phi }
{\displaystyle \varphi \!.}
The latitude resembles the variable
{\displaystyle \theta \,\!}
(theta), except that some authors define
{\displaystyle \theta =0}
to be at the north pole instead of at the equator.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Spherical_coordinate_system&oldid=6668281"
|
Set the Meeting | Toph
Three colleagues Motu, Biscuit and Ramjan are in Sundarbans now. They are on a mission to protect the environment. In each day, they have to discuss their plans together. But they have a lot to work on within a short period of time. Hence, they need to reduce the duration of their traveling. So they decided to set their everyday meeting in such a place that the summation of the duration for each of the friend’s traveling time is minimized.
A region of Sundarbans is selected for their work. On that region, they set some connected workstations. Hence, it is possible to visit all the workstations if someone starts from any particular workstation. It is also guaranteed that between each pair of workstations, there exists exactly one path. The colleagues are staying in three different stations that are suitable for their work. In order to meet, they have to choose a workstation considering the requirements described above. The traveling time between any two directly connected workstations are equal. As they are running out of time, help them to choose the workstation that will be convenient for them.
The first line of input contains two space separated integers
N (
3 \leq N \leq 10^5
E that denotes the number of workstations and the number of connections between two workstations.
In each of the next
E lines, there will be two integers
u and
v (
1 \leq u, v \leq N
1≤u,v≤N) that denotes a direct connection between two distinct workstations.
Then in the next line, there will be three space separated integers
a,
b and
c (
1 \leq a,b,c \leq N
1≤a,b,c≤N) that denotes the current workstation of Motu, Biscuit and Ramjan respectively. You can safely assume that the current workstations of each of the colleagues are distinct from each other.
For each input, output the node that is optimal for all of the colleagues. If there are more than one workstation that fulfills the requirement, print the one that has the lower number.
farhan132Earliest, 1M ago
rohijulislamFastest, 0.0s
MR.Jukerburg11Lightest, 6.2 MB
sh2018331053Shortest, 688B
|
Options Strategies - Phoenix Documentation
Phoenix Documentation Options Strategies
Options Strategies Options Strategies Table of contents
IL Hedging
Holding a call to speculate¶
The long call option strategy is the most basic option trading strategy, whereby a user buys call options with the belief that the price of the underlying asset will rise significantly beyond the strike price before the option expiration date.
Sam is a ETH hodler and he is bullish due to the growing DeFi ecosystem on Ethereum. He is holding 1 ETH already. To maximize his gains if the ETH price goes up in the near future, Sam buys 1 ETH call with the strike price of $700 and the expiry of 15 days. The premium is $70 for each contract. If the price grows by 20% to $840 in two weeks, Sam will get $140 ( 840-
700 ) by exercising the call, and the return is 100% ( ( 140-
70 ) / $70 ). The call gives Sam a 5x leverage in return.
If the price drops by 20% to $560 in two weeks, Sam will not exercise the call option and $70 is all he can lose. But for holding one ETH, Sam may lose $140 from the collapse.
If Sam is extremely bullish and he can even deposit ETH on Makerdao, Compound or Aave, to borrow stable coins to buy call options. This will give him even further leverages, and higher risks.
Protective Puts: A Hedging Strategy¶
A protective put position is created by buying (or owning) an asset and buying put options with a strike price equal or close to the current price of the asset.
A protective put strategy is analogous to the nature of insurance. The main goal of a protective put is to limit potential losses that may result from an unexpected price drop of the underlying asset.
Adopting such a strategy does not put an absolute limit on the potential profits of the investor. Profits from the strategy are determined by the growth potential of the underlying asset. However, a portion of the profits is reduced by the premium paid for the put.
Example of Protective Put
You own 100 ETH, with each ETH valued at $500. You believe that the price of ETH will increase in the future. However, you want to hedge against the risk of an unexpected price decline. Therefore, you decide to purchase 100 protective put contracts with a strike price of $500. The premium of one protective put is $10.
A Covered Call to Benefit From a Flat Market¶
A covered call is created by owning an asset and selling an equivalent amount of call options. To execute this strategy, a trader holds a long position in an asset and writes (sells) call options on that same asset to generate an income stream.
Lucy is holding ETH with a price of $750. She expects that the market will stay flat for a while and wants to possibly lower her cost in the flat market. Therefore, she decides to sell ETH calls with the strike price of $760 and earn the premiums of $50 immediately. If the price gets higher than $760, she will sell when the option buyers exercise the calls. If the price stays lower than $760 till expiration, she will still get premiums and lower the cost of holding one ETH by $50.
In this example, Lucy employs a covered call strategy as she intends to hold the underlying asset for a long time but does not expect an appreciation in price in the short term, and she is satisfied with selling the assets at a predetermined price.
Please note that PPO v1.0 on Phoenix Finance doesn’t allow for selling options for now.
Straddle Strategy¶
A straddle is a strategy accomplished by holding an equal number of puts and calls with the same strike price and expiration dates.
A straddle is meant to take advantage of the market price change by exploiting increased volatility, regardless of which direction the market’s price moves. For example, I am not sure BTC is going to go upwards or downwards in the coming days. But I expect the movement will be big. I can buy a straddle from FinNexus options platform.
Suppose I buy a call and a put together with the strike price $19000. If the market moves up, the call is there; if the market moves down, the put is there. It may cost me like 500 USD in total. If the BTC rises above 19500, or fall below 18500, I will end up in profit. What is more beneficial is that I can exercise the options and collect the profit anytime before they go expired.
This is a typical straddle strategy.
Strangle Strategy¶
The long strangle involves going long (buying) both a call option and a put option of the same underlying security. Like a straddle, the options expire at the same time, but unlike a straddle, the options have different strike prices.
The cost of a strangle can be lower than a straddle, as the options are OTM. The example is just like the one in the straddle strategy, well with different strike prices.
IL Hedging¶
Hedging Against Impermanent Loss: A Deep Dive With FinNexus Options on Coinmarketcap Alexandria by Phoenix Finance Cofounder Ryan Tian
Previous Options Basics
Next PPO v1.0
|
Smooth rough paths and applications to Fourier analysis | EMS Press
We show rough path estimates for smooth
L^p
functions whose derivatives are in
L^q
. We also give applications to Fourier analysis.
Keisuke Hara, Terry J. Lyons, Smooth rough paths and applications to Fourier analysis. Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 1125–1140
|
Futures Pricing Basics
Options Pricing Basics
Swaps Pricing Basics
Different types of derivatives have different pricing mechanisms. A derivative is simply a financial contract with a value that is based on some underlying asset (e.g. the price of a stock, bond, or commodity). The most common derivative types are futures contracts, forward contracts, options and swaps. More exotic derivatives can be based on factors such as weather or carbon emissions.
Derivatives are financial contracts used for a variety of purposes, whose prices are derived from some underlying asset or security.
Depending on the type of derivative, its fair value or price will be calculated in a different manner.
Futures contracts are based on the spot price along with a basis amount, while options are priced based on time to expiration, volatility, and strike price.
Swaps are priced based on equating the present value of a fixed and a variable stream of cash flows over the maturity of the contract.
Futures contracts are standardized financial contracts that allow holders to buy or sell an underlying asset or commodity at a certain price in the future, which is locked in today. Therefore, the futures contract's value is based on the commodity's cash price.
Futures prices will often deviate somewhat from the cash, or spot price, of the underlying. The difference between the cash price of the commodity and the futures price is the basis. It is a crucial concept for portfolio managers and traders because this relationship between cash and futures prices affects the value of the contracts used in hedging. As there are gaps between spot and relative price until expiry of the nearest contract, the basis is not necessarily accurate.
In addition to the deviations created because of the time gap between expiry of the futures contract and the spot commodity, product quality, location of delivery and the actuals may also vary. In general, the basis is used by investors to gauge the profitability of delivery of cash or the actual, and is also used to search for arbitrage opportunities.
For example, consider a corn futures contract that represents 5,000 bushels of corn. If corn is trading at $5 per bushel, the value of the contract is $25,000. Futures contracts are standardized to include a certain amount and quality of the underlying commodity, so they can be traded on a centralized exchange. The futures price moves in relation to the spot price for the commodity based on supply and demand for that commodity.
Forwards are priced similarly to futures, but forwards are non-standardized contracts that arranged instead between two counterparties and transacted over-the-counter with more flexibility of terms.
Options are also common derivative contracts. Options give the buyer the right, but not the obligation, to buy or sell a set amount of the underlying asset at a pre-determined price, known as the strike price, before the contract expires.
The primary goal of option pricing theory is to calculate the probability that an option will be exercised, or be in-the-money (ITM), at expiration. Underlying asset price (stock price), exercise price, volatility, interest rate, and time to expiration, which is the number of days between the calculation date and the option's exercise date, are commonly used variables that are input into mathematical models to derive an option's theoretical fair value.
Aside from a company's stock and strike prices, time, volatility, and interest rates are also quite integral in accurately pricing an option. The longer that an investor has to exercise the option, the greater the likelihood that it will be ITM at expiration. Similarly, the more volatile the underlying asset, the greater the odds that it will expire ITM. Higher interest rates should translate into higher option prices.
The best-known pricing model for options is the Black-Scholes method. This method considers the underlying stock price, option strike price, time until the option expires, underlying stock volatility and risk-free interest rate to provide a value for the option. Other popular models exist such as the binomial tree and trinomial tree pricing models.
Swaps are derivative instruments that represent an agreement between two parties to exchange a series of cash flows over a specific period of time. Swaps offer great flexibility in designing and structuring contracts based on mutual agreement. This flexibility generates many swap variations, with each serving a specific purpose. For instance, one party may swap a fixed cash flow to receive a variables cash flow that fluctuates as interest rates change. Others may swap cash flows associated with the interest rates in one country for that of another.
The most basic type of swap is a plain vanilla interest rate swap. In this type of swap, parties agree to exchange interest payments. For example, assume Bank A agrees to make payments to Bank B based on a fixed interest rate while Bank B agrees to make payments to Bank A based on a floating interest rate.
The value of the swap at the initiation date will be zero to both parties. For this statement to be true, the values of the cash flow streams that the swap parties are going to exchange should be equal. This concept is illustrated with a hypothetical example in which the value of the fixed leg and floating leg of the swap will be Vfix and Vfl respectively. Thus, at initiation:
V_{fix} = V_{fl}
Vfix=Vfl
Notional amounts are not exchanged in interest rate swaps because these amounts are equal and it does not make sense to exchange them. If it is assumed that parties also decide to exchange the notional amount at the end of the period, the process will be similar to an exchange of a fixed rate bond to a floating rate bond with the same notional amount. Therefore such swap contracts can be valued in terms of fixed and floating-rate bonds.
|
Eulerian finite element method for parabolic PDEs on implicit surfaces | EMS Press
Eulerian finite element method for parabolic PDEs on implicit surfaces
We define an Eulerian level set method for parabolic partial differential equations on a stationary hypersurface
\Gamma
contained in a domain
\Omega \subset \mathbb R^{n+1}
. The method is based on formulating the partial differential equations on all level surfaces of a prescribed function
\Phi
whose zero level set is
\Gamma
. Eulerian surface gradients are formulated by using a projection of the gradient in
\mathbb R^{n+1}
onto the level surfaces of
\Phi
. These Eulerian surface gradients are used to define weak forms of surface elliptic operators and so generate weak formulations of surface elliptic and parabolic equations. The resulting equation is then solved in one dimension higher but can be solved on a mesh which is unaligned to the level sets of
\Phi
. We consider both second order and fourth order elliptic operators with natural second order splittings. The finite element method is applied to the weak form of the split system of second order equations using piece-wise linear elements on a fixed grid. The computation of the mass and element stiffness matrices are simple and straightforward. Numerical experiments are described which indicate the power of the method. We describe how this framework may be employed in applications.
Gerhard Dziuk, Charles M. Elliott, Eulerian finite element method for parabolic PDEs on implicit surfaces. Interfaces Free Bound. 10 (2008), no. 1, pp. 119–138
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On Decomposable Measures Induced by Metrics
2012 On Decomposable Measures Induced by Metrics
Dong Qiu, Weiquan Zhang
We prove that for a given normalized compact metric space it can induce a
\sigma
-max-superdecomposable measure, by constructing a Hausdorff pseudometric on its power set. We also prove that the restriction of this set function to the algebra of all measurable sets is a
\sigma
-max-decomposable measure. Finally we conclude this paper with two open problems.
Dong Qiu. Weiquan Zhang. "On Decomposable Measures Induced by Metrics." J. Appl. Math. 2012 1 - 8, 2012. https://doi.org/10.1155/2012/701206
Dong Qiu, Weiquan Zhang "On Decomposable Measures Induced by Metrics," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-8, (2012)
|
Julia sets of two permutable entire functions
January, 2004 Julia sets of two permutable entire functions
Liangwen LIAO, Chung-Chun YANG
In this paper first we prove that if
g
are two permutable transcendental entire functions satisfying
f={f}_{1}\left(h\right)
g={g}_{1}\left(h\right)
, for some transcendental entire function
h
, rational function
{f}_{1}
{g}_{1}
, which is analytic in the range of
h
F\left(g\right)\subset F\left(f\right)
. Then as an application of this result, we show that if
f\left(z\right)=p\left(z\right){e}^{q\left(z\right)}+c
c
p
a nonzero polynomial and
q
a nonconstant polynomial, or
f\left(z\right)={\int }^{z}p\left(z\right){e}^{q\left(z\right)}dz
p,
q
are nonconstant polynomials, such that
f\left(g\right)=g\left(f\right)
for a nonconstant entire function
g
J\left(f\right)=J\left(g\right)
Liangwen LIAO. Chung-Chun YANG. "Julia sets of two permutable entire functions." J. Math. Soc. Japan 56 (1) 169 - 176, January, 2004. https://doi.org/10.2969/jmsj/1191418700
Keywords: Fatou set , Julia set , permutable entire functions , prime , pseudo-prime
Liangwen LIAO, Chung-Chun YANG "Julia sets of two permutable entire functions," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 56(1), 169-176, (January, 2004)
|
Joint-Space Motion Model - MATLAB & Simulink - MathWorks Benelux
Joint-Space Motion Model
State and Controls
The joint-space motion model characterizes the closed-loop motion of a manipulator under joint-space control, where the control action is defined in the joint configuration space. Motion models are used as low-fidelity plant models of robots under closed-loop position control. This topic covers the variables and equations for computing the behavior of the joint-space position, velocity, and accelerations relative to reference inputs, as used in the jointSpaceMotionModel object. For task-space motion models, see the Plan and Execute Task- and Joint-Space Trajectories Using KINOVA Gen3 Manipulator object.
This topic covers these types of joint-space control:
Proportional-Derivative (PD) Control
Independent Joint Motion
For an example that covers the difference between task-space and joint-space control, see Plan and Execute Task- and Joint-Space Trajectories Using KINOVA Gen3 Manipulator.
The joint-space motion model state consists of these values:
q
— Robot joint configuration, as a vector of joint positions. Specified in
\mathrm{rad}
for revolute joints and
\mathit{m}
for prismatic joints.
\underset{}{\overset{˙}{q}}
— Vector of joint velocities in
rad\cdot {s}^{-1}
m\cdot {s}^{-1}
for prismatic joints
\underset{}{\overset{¨}{q}}
— Vector of joint accelerations in
rad\cdot {s}^{-2}
for revolute joints or
m\cdot {s}^{-2}
The joint-space motion model is used when you need a low-fidelity model of your system under closed-loop control and the inputs are specified as joint configuration, velocity, and acceleration. The motion model includes three ways to model its overall behavior:
Computed Torque Control — The rigid-body dynamics are modeled using the standard equations of motion, but compensating for the full-body dynamics and assigning error dynamics. This is a higher-fidelity version of independent joint motion control.
PD Control — The rigid-body dynamics are modeled using the standard equations of motion with a joint torque input given by proportional-derivative (PD) control. This model represents a controller that does not compensate tightly for the overall effects of rigid-body motion.
Independent Joint Motion — Each joint is modeled independently as a closed-loop second-order system. This model is a lower fidelity version of computed torque control motion model, and may be considered a best-case scenario for how closed-loop motion may behave since the dynamics are simplified and directly prescribed.
To set these different motion types, use the MotionType property of the jointSpaceMotionModel object. These motion types are not exhaustive, but they do provide a set of options to use when approximating the closed-loop behavior of your system. For details and suggestions on when to use which model, see the sections below.
In this section, the equations of motion for each model are introduced, in order of decreasing complexity.
With computed torque control, the motion model uses standard rigid body dynamics, but the generalized force input is given by a control law that compensates for the rigid body dynamics and instead assigns a second-order error dynamics response.
Inputs — This model accepts
{q}_{ref},{\underset{}{\overset{˙}{q}}}_{ref},{\underset{}{\overset{¨}{q}}}_{ref}
as the desired reference joint configuration, velocities, and accelerations as vectors. The user may also optional provide the external force
{F}_{ext}
, specified in Newtons.
Outputs — The model outputs
q,\underset{}{\overset{˙}{q}},\underset{}{\overset{¨}{q}}
as the joint configuration, velocities, and accelerations as vectors. In the MATLAB version of the model, only accelerations are returned, and the user must choose an integrator or ODE solver to return the other states.
Complexity — This is high complexity. The motion model uses full rigid body dynamics with optional external forces, the controller is modeled as part of the closed loop system, and the controller includes dynamic compensation terms.
When to apply — Use when the closed-loop system being simulated has approximable error dynamics, or when it uses a controller that treats the robot as a multi-body system, and external forces may be present
The resultant closed-loop system aims to achieve the following second error behavior for the
\mathit{i}
-th joint:
{\underset{}{\overset{¨}{\underset{}{\overset{\sim }{q}}}}}_{i}=-{\omega }_{n}^{2}{\underset{}{\overset{\sim }{q}}}_{i}-2\zeta {\omega }_{n}{\underset{}{\overset{˙}{\underset{}{\overset{\sim }{q}}}}}_{i}
{\underset{}{\overset{\sim }{q}}}_{i}={q}_{i}-{q}_{i.ref}
These parameters characterize the desired response defined for each joint:
{\omega }_{n}
— Natural frequency, specified in Hz (
{s}^{-1}
\zeta
— The damping ratio, which is unitless
As seen in the diagram, the complete system consists of the standard rigid-body Robot Dynamics with a control law that enforces closed error dynamics via the generalized force input
Q
\frac{d}{dt}\left[\begin{array}{c}q\\ \underset{}{\overset{˙}{q}}\end{array}\right]={f}_{dyn}\left(q,\underset{}{\overset{˙}{q}},Q,{F}_{ext}\right)
Q={g}_{CTC}\left(\underset{}{\overset{\sim }{q}},\underset{}{\overset{˙}{\underset{}{\overset{\sim }{q}}}},{\underset{}{\overset{¨}{q}}}_{ref},{\omega }_{n},\zeta \right)=M\left(q\right){a}_{q}+C\left(q,\underset{}{\overset{˙}{q}}\right)\underset{}{\overset{˙}{q}}+G\left(q\right)
{a}_{q}={\underset{}{\overset{¨}{q}}}_{ref}-{\left[{\omega }_{n}^{2}\right]}_{diag}\underset{}{\overset{\sim }{q}}-{\left[2\zeta {\omega }_{n}\right]}_{diag}\underset{}{\overset{˙}{\underset{}{\overset{\sim }{q}}}}
\underset{}{\overset{\sim }{q}}=q-{q}_{ref}
M\left(q\right)
— is a joint-space mass matrix based on the current robot configuration Calculate this matrix by using the massMatrix object function.
C\left(q,\underset{}{\overset{˙}{q}}\right)
\underset{}{\overset{˙}{q}}
G\left(q\right)
The control input relies on these user-defined parameters:
{\left[-{\omega }_{n}^{2}\right]}_{diag}
— Diagonal matrix, where the
\left(\mathit{i},\mathit{i}\right)
-th element corresponds to the
\mathit{i}
-th element of the n-element vector of natural frequencies in the NaturalFrequency property of the jointSpaceMotionModel object are in Hz (
{s}^{-1}
{\left[-2\zeta {\omega }_{n}^{2}\right]}_{diag}
\left(\mathit{i},\mathit{i}\right)
\mathit{i}
-th element of the product of the squared natural frequencies vector
{\omega }_{n}
\mathit{i}
-th element of the damping ratios vector
\zeta
, specified in the DampingRatio property of the jointSpaceMotionModel object.
Because the dynamics are compensated, in the absence of external force inputs large acceleration/deceleration, the error dynamics should be achieved. In the absence of external forces, the independent joint motion type provides a simpler way of achieving this result.
{\omega }_{n}
\zeta
may be set directly, or they may be provided using the the updateErrorDynamicsFromStep method, which computes values for
{\omega }_{n}
\zeta
based on desired unit step response (defined using it's transient behavior characteristics).
With PD control, the robot models behavior according to standard rigid body dynamics, but with the generalized force input
Q
given by a control law that applies PD control based on the joint error, as well as gravity compensation.
{q}_{ref},{\underset{}{\overset{˙}{q}}}_{ref}
{F}_{ext}
q,\underset{}{\overset{˙}{q}},\underset{}{\overset{¨}{q}}
as the joint configuration, velocities, and accelerations. In the MATLAB version of the model, only accelerations are returned, and the user must choose an integrator or ODE solver to return the other states.
Complexity — This is medium complexity. The motion model uses full rigid body dynamics with optional external forces and the controller is modeled as part of the closed loop system, but the controller is relatively simple.
When to apply — Use when the closed-loop system being simulated uses a controller that treats joints as independent systems, or when a PD style controller is used, and external forces may be present.
As with computed torque control, this system behavior uses the standard rigid-body Robot Dynamics, but uses the PD control law define the generalized force input
Q
\frac{d}{dt}\left[\begin{array}{c}q\\ \underset{}{\overset{˙}{q}}\end{array}\right]={f}_{dyn}\left(q,\underset{}{\overset{˙}{q}},\tau ,{F}_{ext}\right)
Q={g}_{PD}\left(\underset{}{\overset{\sim }{q}},\underset{}{\overset{˙}{\underset{}{\overset{\sim }{q}}}},{K}_{P},{K}_{D}\right)=-{K}_{P}\left(\underset{}{\overset{\sim }{q}}\right)-{K}_{D}\left(\underset{}{\overset{˙}{\underset{}{\overset{\sim }{q}}}}\right)+G\left(q\right)
\underset{}{\overset{\sim }{q}}=q-{q}_{ref}
G\left(q\right)
{K}_{P}
— Proportional gain, specified as an
\mathit{N}
\mathit{N}
\mathit{N}
is the number of movable joints of the manipulator
{K}_{D}
— Derivative gain, specified as an
\mathit{N}
\mathit{N}
When this system is modeled with independent joint motion, instead of modeling the closed loop system as standard rigid body dynamics plus a control input, each joint is instead modeled as a second-order system that already has the desired error behavior:
{q}_{ref},{\underset{}{\overset{˙}{q}}}_{ref}
as the desired reference joint configuration, velocities, and accelerations as vectors. There is no external force input.
q,\underset{}{\overset{˙}{q}},\underset{}{\overset{¨}{q}}
Complexity — This is low complexity. The motion model simply prescribes the error behavior that a position controller could aim to achieve.
When to apply — Use when the system has approximable error dynamics and there are no external force inputs required.
The system models the following closed-loop second order behavior for the ith joint:
\frac{d}{dt}\left[\begin{array}{c}\underset{}{\overset{\sim }{q}}\\ \underset{}{\overset{˙}{\underset{}{\overset{\sim }{q}}}}\end{array}\right]={f}_{err}\left(\underset{}{\overset{\sim }{q}},\underset{}{\overset{˙}{\underset{}{\overset{\sim }{q}}}},\zeta ,{\omega }_{n}\right)=\left[\begin{array}{c}\underset{}{\overset{˙}{\underset{}{\overset{\sim }{q}}}}\\ -{\omega }_{n}^{2}{\underset{}{\overset{\sim }{q}}}_{i}-2\zeta {\omega }_{n}{\underset{}{\overset{˙}{\underset{}{\overset{\sim }{q}}}}}_{i}\end{array}\right]
{\underset{}{\overset{\sim }{q}}}_{i}={q}_{i}-{q}_{i.ref}
{\omega }_{n}
— the natural frequency specified in units of
{s}^{-1}
\zeta
— t the damping ratio, which is unitless
The complete system is therefore modeled as:
\frac{d}{dt}\left[\begin{array}{c}q\\ \underset{}{\overset{˙}{q}}\end{array}\right]={f}_{IJM}\left({q}_{ref},{\underset{}{\overset{˙}{q}}}_{ref},\zeta ,{\omega }_{n}\right)=\left[\begin{array}{cc}0& I\\ {\left[-{\omega }_{n}^{2}\right]}_{diag}& {\left[-2\zeta {\omega }_{n}\right]}_{diag}\end{array}\right]\left[\begin{array}{c}q\\ \underset{}{\overset{˙}{q}}\end{array}\right]+\left[\begin{array}{cc}0& I\\ {\left[{\omega }_{n}^{2}\right]}_{diag}& {\left[2\zeta {\omega }_{n}\right]}_{diag}\end{array}\right]\left[\begin{array}{c}{q}_{ref}\\ {\underset{}{\overset{˙}{q}}}_{ref}\end{array}\right]
The model relies on these user-defined parameters:
{\left[-{\omega }_{n}^{2}\right]}_{diag}
\left(\mathit{i},\mathit{i}\right)
\mathit{i}
{s}^{-1}
{\left[-2\zeta {\omega }_{n}^{2}\right]}_{diag}
\left(\mathit{i},\mathit{i}\right)
\mathit{i}
{\omega }_{n}
\mathit{i}
\zeta
{\omega }_{n}
\zeta
{\omega }_{n}
\zeta
The Independent Joint Motion model represents a closed loop system under idealized behavior. In the absence of external forces, the motion model using computed torque control should produce an equivalent output.
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18 Name the particular technic in biotechnology whose steps are shown in the figure Use the figure - Biology - Molecular Basis of Inheritance - 10939001 | Meritnation.com
18. Name the particular technic in biotechnology whose steps are shown in the figure . Use the figure to summarise the technique in three steps.
19. In a bacterial culture some of the colonies produced blue colour in the presence of a chromogenic substrate and some did not due to the presence or absence of an insert (rDNA) in the coding sequence of
\mathrm{\beta }-\mathrm{galactosidase}.
(b) How is it advantageous over simultaneous plating on two plates having different antibiotics ?
F A Y T H answered this
18. The process shown in the figure is Recombinant DNA Technology [RDT] . In RDT, the gene of interest is combined with a plasmid DNA to form recombinant DNA or rDNA. This DNA is used to introduce new characteristics or genes into the host individual.
Me answered this
19. Insertional inactivation
A recombinant DNA is inserted within the coding sequence of an enzyme beta galactosidase which results in inactivation of enzyme
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Plot each of the given complex numbers in parts (a) and (b) and state the modulus and argument of the number. Then use your answer from part (a) to complete part (c).
Once you plot each of the numbers, create a right triangle with the
x
Then use the Pythagorean Theorem and the tangent ratio to determine the modulus and argument.
If you need more help, review the Math Notes box in Lesson 11.2.2.
z=5-4i
w=-3+7i
\text{Evaluate } z^{2/3}.
z^{2/3} = (z^2)^{1/3}
First compute
z^{2}
, then compute the third root of the result.
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BCOV invariants of Calabi–Yau manifolds and degenerations of Hodge structures
15 February 2021 BCOV invariants of Calabi–Yau manifolds and degenerations of Hodge structures
Dennis Eriksson, Gerard Freixas i Montplet, Christophe Mourougane
Dennis Eriksson,1 Gerard Freixas i Montplet,2 Christophe Mourougane3
1Department of Mathematics, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden
2CNRS, Institut de Mathématiques de Jussieu–Paris Rive Gauche, Paris, France
3Institut de Recherche Mathématique de Rennes (IRMAR), Rennes, France
Calabi–Yau manifolds have risen to prominence in algebraic geometry, in part because of mirror symmetry and enumerative geometry. After Bershadsky, Cecotti, Ooguri, and Vafa (BCOV), it is expected that genus 1 curve-counting on a Calabi–Yau manifold is related to a conjectured invariant, only depending on the complex structure of the mirror, and built from Ray–Singer holomorphic analytic torsions. To this end, extending work of Fang, Lu, and Yoshikawa in dimension 3, we introduce and study the BCOV invariant of Calabi–Yau manifolds of arbitrary dimension. To determine it, knowledge of its behavior at the boundary of moduli spaces is imperative. To address this problem, we prove general results on degenerations of
{L}^{2}
-metrics on Hodge bundles and their determinants, refining the work of Schmid. We express the singularities of these metrics in terms of limiting Hodge structures and derive consequences for the dominant and subdominant singular terms of the BCOV invariant.
We want to express our wholehearted gratitude to Ken-Ichi Yoshikawa, for many discussions and criticisms on these topics and for his generous explanations of his own work in this field. Many of these conversations occurred at Kyoto University, which we thank for its ample hospitality. We moreover thank the Centre Emile Borel at the Institute Henri Poincaré for financial support and reception during the Research in Paris program “Secondary Invariants in Mirror Symmetry” in 2018.
The first author is also grateful to IRMAR for financial support in Rennes during a one-month stay in 2018. He also wants to extend his thanks to the Knut and Alice Wallenberg Foundation and the Göran Gustafsson Foundation for their travel support that made several visits possible. The second author was partially supported by Agence Nationale de la Recherche grant ANR-18-CE40-0017 (PERGAMO) and the Partenariat Hubert Curien–Sakura. We finally thank the two anonymous referees, whose constructive comments helped us improve the article.
Dennis Eriksson. Gerard Freixas i Montplet. Christophe Mourougane. "BCOV invariants of Calabi–Yau manifolds and degenerations of Hodge structures." Duke Math. J. 170 (3) 379 - 454, 15 February 2021. https://doi.org/10.1215/00127094-2020-0045
Received: 1 May 2019; Revised: 15 January 2020; Published: 15 February 2021
Secondary: 32G20 , 58J52 , 58K55 , 58K65
Keywords: analytic torsion , Calabi-Yau manifolds , degenerations of Hodge structures , mathematical mirror symmetry
Dennis Eriksson, Gerard Freixas i Montplet, Christophe Mourougane "BCOV invariants of Calabi–Yau manifolds and degenerations of Hodge structures," Duke Mathematical Journal, Duke Math. J. 170(3), 379-454, (15 February 2021)
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Voltage multiplier — Wikipedia Republished // WIKI 2
A voltage multiplier is an electrical circuit that converts AC electrical power from a lower voltage to a higher DC voltage, typically using a network of capacitors and diodes.
Voltage multipliers can be used to generate a few volts for electronic appliances, to millions of volts for purposes such as high-energy physics experiments and lightning safety testing. The most common type of voltage multiplier is the half-wave series multiplier, also called the Villard cascade (but actually invented by Heinrich Greinacher).
Voltage Multiplier Circuits (Voltage Tripler & Quadrupler)
Voltage Multiplier Circuits (Half Wave Voltage Doubler)
LEARN AND GROW !! VOLTAGE TRIPLER AND QUADRUPLER(VOLTAGE MULTIPLIER) !
2 Voltage doubler and tripler
3 Breakdown voltage
4 Other circuit topologies
4.1 Dickson charge pump
4.1.1 Modification for RF power
4.2 Cross-coupled switched capacitor
Assuming that the peak voltage of the AC source is +Us, and that the C values are sufficiently high to allow, when charged, that a current flows with no significant change in voltage, then the (simplified) working of the cascade is as follows:
Illustration of the described operation, with +Us = 100 V
negative peak (−Us): The C1 capacitor is charged through diode D1 to Us V (potential difference between left and right plate of the capacitor is Us)
negative peak: potential of C1 has dropped to 0 V thus allowing C3 to be charged through D3 to 2Us.
positive peak: potential of C2 rises to 2Us (analogously to step 2), also charging C4 to 2Us. The output voltage (the sum of voltages under C2 and C4) rises until 4Us is reached.
Voltage doubler and tripler
See also: Voltage doubler
A Cockcroft-Walton voltage quadrupler circuit. It generates a DC output voltage Vo of four times the peak of the AC input voltage Vi
A voltage doubler uses two stages to approximately double the DC voltage that would have been obtained from a single-stage rectifier. An example of a voltage doubler is found in the input stage of switch mode power supplies containing a SPDT switch to select either 120 V or 240 V supply. In the 120 V position the input is typically configured as a full-wave voltage doubler by opening one AC connection point of a bridge rectifier, and connecting the input to the junction of two series-connected filter capacitors. For 240 V operation, the switch configures the system as a full-wave bridge, re-connecting the capacitor center-tap wire to the open AC terminal of a bridge rectifier system. This allows 120 or 240 V operation with the addition of a simple SPDT switch.
A voltage tripler is a three-stage voltage multiplier. A tripler is a popular type of voltage multiplier. The output voltage of a tripler is in practice below three times the peak input voltage due to their high impedance, caused in part by the fact that as each capacitor in the chain supplies power to the next, it partially discharges, losing voltage doing so.
Triplers were commonly used in color television receivers to provide the high voltage for the cathode ray tube (CRT, picture tube).
Triplers are still used in high voltage supplies such as copiers, laser printers, bug zappers and electroshock weapons.
Two cascades driven by a single center-tapped transformer. This configuration provides full-wave rectification leading to less ripple and upon any collapse from arcing capacitive energy can cancel.
Standard Dickson charge pump (4 stages : 5× multiplier)
The Dickson charge pump, or Dickson multiplier, is a modification of the Greinacher/Cockcroft–Walton multiplier. Unlike that circuit, however, the Dickson multiplier takes a DC supply as its input so is a form of DC-to-DC converter. Also, unlike Greinacher/Cockcroft–Walton which is used on high-voltage applications, the Dickson multiplier is intended for low-voltage purposes. In addition to the DC input, the circuit requires a feed of two clock pulse trains with an amplitude swinging between the DC supply rails. These pulse trains are in antiphase.[1]
To describe the ideal operation of the circuit, number the diodes D1, D2 etc. from left to right and the capacitors C1, C2 etc. When the clock
{\displaystyle \phi _{1}}
is low, D1 will charge C1 to Vin. When
{\displaystyle \phi _{1}}
goes high the top plate of C1 is pushed up to 2Vin. D1 is then turned off and D2 turned on and C2 begins to charge to 2Vin. On the next clock cycle
{\displaystyle \phi _{1}}
again goes low and now
{\displaystyle \phi _{2}}
goes high pushing the top plate of C2 to 3Vin. D2 switches off and D3 switches on, charging C3 to 3Vin and so on with charge passing up the chain, hence the name charge pump. The final diode-capacitor cell in the cascade is connected to ground rather than a clock phase and hence is not a multiplier; it is a peak detector which merely provides smoothing.[2]
There are a number of factors which reduce the output from the ideal case of nVin. One of these is the threshold voltage, VT of the switching device, that is, the voltage required to turn it on. The output will be reduced by at least nVT due to the volt drops across the switches. Schottky diodes are commonly used in Dickson multipliers for their low forward voltage drop, amongst other reasons. Another difficulty is that there are parasitic capacitances to ground at each node. These parasitic capacitances act as voltage dividers with the circuit's storage capacitors reducing the output voltage still further.[3] Up to a point, a higher clock frequency is beneficial: the ripple is reduced and the high frequency makes the remaining ripple easier to filter. Also the size of capacitors needed is reduced since less charge needs to be stored per cycle. However, losses through stray capacitance increase with increasing clock frequency and a practical limit is around a few hundred kilohertz.[4]
Dickson charge pump using diode-wired MOSFETs (4 stages : 5× multiplier)
Dickson multipliers are frequently found in integrated circuits (ICs) where they are used to increase a low-voltage battery supply to the voltage needed by the IC. It is advantageous to the IC designer and manufacturer to be able to use the same technology and the same basic device throughout the IC. For this reason, in the popular CMOS technology ICs the transistor which forms the basic building block of circuits is the MOSFET. Consequently, the diodes in the Dickson multiplier are often replaced with MOSFETs wired to behave as diodes.[5]
Dickson charge pump with linear MOSFET in parallel with diode-wired MOSFET (4 stages : 5× multiplier)
An ideal 4-stage Dickson multiplier (5× multiplier) with an input of 1.5 V would have an output of 7.5 V. However, a diode-wired MOSFET 4-stage multiplier might only have an output of 2 V. Adding parallel MOSFETs in the linear region improves this to around 4 V. More complex circuits still can achieve an output much closer to the ideal case.[7]
Many other variations and improvements to the basic Dickson circuit exist. Some attempt to reduce the switching threshold voltage such as the Mandal-Sarpeshkar multiplier[8] or the Wu multiplier.[9] Other circuits cancel out the threshold voltage: the Umeda multiplier does it with an externally provided voltage[10] and the Nakamoto multiplier does it with internally generated voltage.[11] The Bergeret multiplier concentrates on maximising power efficiency.[12]
Modification for RF power
Modified Dickson charge pump (2 stages : 3× multiplier)
Cascade of cross-coupled MOSFET voltage doublers (3 stages : 4× multiplier)
A voltage multiplier may be formed of a cascade of voltage doublers of the cross-coupled switched capacitor type. This type of circuit is typically used instead of a Dickson multiplier when the source voltage is 1.2 V or less. Dickson multipliers have increasingly poor power conversion efficiency as the input voltage drops because the voltage drop across the diode-wired transistors becomes much more significant compared to the output voltage. Since the transistors in the cross-coupled circuit are not diode-wired the volt-drop problem is not so serious.[14]
The circuit works by alternately switching the output of each stage between a voltage doubler driven by
{\displaystyle \phi _{1}}
and one driven by
{\displaystyle \phi _{2}}
. This behaviour leads to another advantage over the Dickson multiplier: reduced ripple voltage at double the frequency. The increase in ripple frequency is advantageous because it is easier to remove by filtering. Each stage (in an ideal circuit) raises the output voltage by the peak clock voltage. Assuming that this is the same level as the DC input voltage then an n stage multiplier will (ideally) output nVin. The chief cause of losses in the cross-coupled circuit is parasitic capacitance rather than switching threshold voltage. The losses occur because some of the energy has to go into charging up the parasitic capacitances on each cycle.[15]
TV cascade (green) and flyback transformer (blue).
The high-voltage supplies for cathode-ray tubes (CRTs) in TVs often use voltage multipliers with the final-stage smoothing capacitor formed by the interior and exterior aquadag coatings on the CRT itself. CRTs were formerly a common component in television sets. Voltage multipliers can still be found in modern TVs, photocopiers, and bug zappers.[16]
High voltage multipliers are used in spray painting equipment, most commonly found in automotive manufacturing facilities. A voltage multiplier with an output of about 100kV is used in the nozzle of the paint sprayer to electrically charge the atomized paint particles which then get attracted to the oppositely charged metal surfaces to be painted. This helps reduce the volume of paint used and helps in spreading an even coat of paint.
A common type of voltage multiplier used in high-energy physics is the Cockcroft–Walton generator (which was designed by John Douglas Cockcroft and Ernest Thomas Sinton Walton for a particle accelerator for use in research that won them the Nobel Prize in Physics in 1951).
Marx generator (a device that uses spark gaps instead of diodes as the switching elements and can deliver higher peak currents than diodes can).
Boost converter (a DC-to-DC power converter that steps up voltage, frequently using an inductor)
^ Liu, p. 226
Yuan, p. 14
^ Liu, pp. 226–227
^ Yuan, pp. 13–14
Liu|2006, pp. 227–228
^ Peluso et al., p. 35
Zumbahlen, p. 741
Yuan, pp. 14–16
^ Campardo et al., pp. 377–379
Liu, pp. 232–235
Lin, p. 81
^ Campardo et al., p. 379
Liu, p. 234
^ McGowan, p. 87
Campardo, Giovanni; Micheloni, Rino; Novosel, David VLSI-design of Non-volatile Memories, Springer, 2005 ISBN 3-540-20198-X.
Lin, Yu-Shiang Low Power Circuits for Miniature Sensor Systems, Publisher ProQuest, 2008 ISBN 0-549-98672-3.
Liu, Mingliang Demystifying Switched Capacitor Circuits, Newnes, 2006 ISBN 0-7506-7907-7.
McGowan, Kevin, Semiconductors: From Book to Breadboard, Cengage Learning, 2012 ISBN 1133708382.
Peluso, Vincenzo; Steyaert, Michiel; Sansen, Willy M. C. Design of Low-voltage Low-power CMOS Delta-Sigma A/D Converters, Springer, 1999 ISBN 0-7923-8417-2.
Yuan, Fei CMOS Circuits for Passive Wireless Microsystems, Springer, 2010 ISBN 1-4419-7679-5.
Zumbahlen, Hank Linear Circuit Design Handbook, Newnes, 2008 ISBN 0-7506-8703-7.
Wikimedia Commons has media related to Voltage doubler.
Basic multiplier circuits
Cockcroft Walton multipliers
Schematic of Kadette brand (International Radio Corp.) model 1019. A 1937 radio with a vacuum tube (25Z5) voltage multiplier rectifier.
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Referring to figure (a) and (b) : (1) reading of spring balance in (a) is 8 g and - Physics - Electric Charges And Fields - 8754319 | Meritnation.com
Referring to figure (a) and (b) : (1) reading of spring balance in (a) is 8 g and in (b) is also 8 g (2) reading of spring balance in (a) is 8 g and in (b), it is less than 8 g (3) reading in (a) is less than 8 g and in (b), it is 8 g (4) reading in both (a) and (b) is less than 8 g
Peter Fernandes answered this
Consider that the two masses
{m}_{1} \mathrm{and} {m}_{2}
are hanging from the two ends of a cord running over a smooth fixed pulley.
Considering the masses are unequal:
{m}_{1}g-T={m}_{1}a\phantom{\rule{0ex}{0ex}}T-{m}_{2}g={m}_{2}a\phantom{\rule{0ex}{0ex}}Adding\phantom{\rule{0ex}{0ex}}a=\frac{\left({m}_{1}-{m}_{2}\right)}{{m}_{1}+{m}_{2}}g\phantom{\rule{0ex}{0ex}}T=\frac{2{m}_{1}{m}_{2}g}{{m}_{1}+{m}_{2}}
Then the tension on the string connecting the pulley and balance is twice the tension on the string connecting the two bodies.
Thus the tension in the string connecting the pulley and balance is
{T}_{1}=\frac{4{m}_{1}{m}_{2}g}{{m}_{1}+{m}_{2}}
Hence the reading on the balances would be
\frac{4\left(6\right)\left(2\right)}{8}g=6g
\frac{4\left(4\right)\left(4\right)}{8}g=8g
Thus option (3) is correct.
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Numerical estimation strategies | ePractice - HKDSE 試題導向練習平台
Question Sample Titled 'Numerical estimation strategies'
{\left({a}\right)}
Reformulation strategy
{\left({i}\right)}
Rounding off
{\left({i}{i}\right)}
Using clustered value
If the numbers involved in an estimation are close to each other, we can select a clustered value to represent them.
{\left({i}{i}{i}\right)}
Using compatible numbers
Compatible numbers are approximate value that make calculation easier.
{\left({b}\right)}
Compensation strategy
This method makes adjustment to the result of a rough estimation so that it is closer to the exact value.
{\left({c}\right)}
Translation strategy
This method rearranges the order of operations in a complicated expression so that the estimation can be simplified.
{\left({d}\right)}
Taking larger or smaller approximations
When isong estimation to solve real-life problems, we should determine whether to take larger or smaller approximations in order to obtain more useful estimates.
To round up a number, we take an approximation which is slightly larger to replace the original number.
To round down a number, we take an approximation which is slightly smaller to replace the original number.
Example of (a)
Estimate the average of the following numbers by rounding off them correct to the nearest integer.
{2.19}
{3.27}
{4.81}
{4.56}
{6.88}
{8.01}
Use a clustered value to estimate the value of the expression.
{3800}+{3980}+{4075}+{4112}+{4208}
Use compatible numbers to estimate the value of the expression.
{0.67}\times{89.8}
=\dfrac{{{2.19}+{3.27}+{4.81}+{4.56}+{6.88}+{8.01}}}{{6}}
=\dfrac{{{2}+{3}+{5}+{5}+{7}+{8}}}{{6}}
round off to the nearest integer
={5}
The value of each of the numbers is about
{4000}
clustered value
{3800}+{3980}+{4075}+{4112}+{4208}
\approx{4000}\times{5}
={20000}
{0.67}\times{89.8}
\approx\dfrac{{2}}{{3}}\times{90}
={60}
Example of (b)
Use compensation strategy to estimate the value of the following expression:
{4.25}+{3.76}-{0.19}-{1.92}+{2.95}
{4.25}+{3.76}-{0.19}-{1.92}+{2.95}
={\left({4}+{0.25}\right)}+{\left({3}+{0.76}\right)}+{\left(-{0.19}\right)}+{\left(-{1}-{0.92}\right)}+{\left({2}+{0.95}\right)}
={\left({4}+{3}+{0}-{1}+{2}\right)}+{\left({0.25}+{0.76}\right)}+{\left(-{0.19}-{0.92}+{0.95}\right)}
Estimate the decimal parts as compensation
\approx{8}+{1}+{0}
={9}
Example of (c)
Use translation strategy to estimate the value of
{3017}\times{19}\div{15}
{3017}\times{19}\div{15}
\approx{3000}\times{20}\div{15}
{3017}
{3000}
{19}
{20}
={3000}\div{15}\times{20}
Rearrange the order of operations
={200}\times{20}
={4000}
Example of (d)
Peter wants to buy
{2}
books for $
{54.4}
each and
{3}
magazines for $
{19.8}
each. Suppose Peter has only $
{180}
. Estimate whether Peter has enough money to buy them.
Mary wants to record
{8}
files in a CD. The file sizes (in MB) of the files are listed below:
{128.3}
{92.2}
{110.8}
{56.7}
{82.1}
{124.6}
{76.3}
{71.5}
Suppose the CD can store
{700}
MB of data. Determine whether the total file size exceeds the capacity of the CD.
By taking larger approximations, the total price
={\left({55}\times{2}+{20}\times{3}\right)}
=
{170}
\lt
{180}
∴ Peter has enough money to buy them.
By taking smaller approximations, the total file size
={\left({120}+{90}+{110}+{50}+{80}+{120}+{70}+{70}\right)}
={710}
\gt{700}
∴ The total file size exceeds the capacity of the CD.
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Revision as of 10:38, 29 October 2020 by Andraschko (talk | contribs) (Created page with "This page describes how to document an ApCoCoA-2 package in the Wiki. Please note that in order to do this, you need a wiki account. Such an account can only be created by a w...")
This page describes how to document an ApCoCoA-2 package in the Wiki. Please note that in order to do this, you need a wiki account. Such an account can only be created by a wiki admin, so please contact the ApCoCoA team so we can create an account for you.
{\displaystyle x_{1}^{2}}
{\displaystyle \sum _{i=1}^{\infty }\left({\frac {1}{2}}\right)^{n}}
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Every hour, Miguel’s Microchips measures the signal strength of a chip from their manufacturing process. For each of the process control charts below, say whether the process is in control, and if not, what in the manufacturing process could have led to the out-of-control state.
The process is out of control beginning at the
9
th hour.
12
consecutive points above the center line.
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A jar contains five red, four white, and three blue balls. If three balls are randomly selected, find the probability of choosing:
Review the Math Notes box in Lesson 10.1.1 to review combinations.
5
red balls, choose
2
4
white balls, choose
1
3
blue balls, choose
0
12
balls total, choose
3
\frac{_5C_2\;·\;_4C_1·\;_3C_0}{_{12}C_3}=\frac{2}{11}
Three white.
\frac{_5C_1\cdot _4C_1\cdot _3C_1}{_{12}C_3}=\frac{3}{11}
Find the probabilities of all red balls, all white balls, and all blue balls. Adding these probabilities will give you the total probability that the balls will be all the same color.
\frac{_5C_3}{_{12}C_3}+\frac{_4C_3}{_{12}C_3}+\frac{_3C_3}{_{12}C_3}=\frac{3}{44}
One red and two white.
Two of one color and one of another.
Parts (c) and (d) describe the only other color combinations of balls.
1-(\frac{3}{11}+\frac{3}{44})\ =\ \frac{29}{44}
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\left(1,2\right)
\left(-2,-1\right)
Figure 1.1.3(b), showing the position vector to
\left(1,2\right)
translated so its tail is at the point
\left(-2,-1\right)
, is obtained interactively as follows.
As per Table 1.1.1, construct the position vector to
\left(1,2\right)
Context Panel: Plots≻Arrow from point
Fill in the "Specify arrow length and base point" dialog as per Figure 1.1.3(a) and click OK.
plots:-arrow([-2,-1],convert(<1,2>,'list'),'length'=2.236068,'scaling'='constrained');
Figure 1.1.3(b) Translated position vector
Figure 1.1.3(a) "Specify arrow length and base point" dialog
The VectorCalculus packages provide the construct of the "rooted vector", allowing for the definition of a vector whose "tail" is fixed at a specified "root point." As stated earlier, the PlotVector command is the most convenient command for graphing vectors of all types.
Apply the RootedVector command and press the Enter key.
\mathbf{V}≔\mathrm{RootedVector}\left(\mathrm{root}=\left[-2,-1\right],〈1,2〉\right)
\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\end{array}\right]
Apply the PlotVector command, and press the Enter key.
\mathrm{PlotVector}\left(\mathbf{V},\mathrm{scaling}=\mathrm{constrained}\right)
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\left(1,2,3\right)
∥\mathbf{V}∥=\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}
\sqrt{{1}^{2}+{2}^{2}+{3}^{2}}=\sqrt{14}
〈1,2,3〉
\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\end{array}\right]
\stackrel{\text{norm}}{\to }
\sqrt{\textcolor[rgb]{0,0,1}{14}}
∥〈1,2,3〉∥
\sqrt{\textcolor[rgb]{0,0,1}{14}}
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Auger Effect Revisited: An Essay by Inelastic Collision Theory
Auger Effect Revisited: An Essay by Inelastic Collision Theory ()
Universidade José do Rosário Vellano, Alfenas, Brazil.
This work presents an essay to the Auger Effect, by the Inelastic Collision Theory. Calculations of the energies of the electrons ejected for bands of the Auger spectrum of some molecules were made, to test of the model.
Auger Effect, Collision Theory
Dias, A. (2018) Auger Effect Revisited: An Essay by Inelastic Collision Theory. Natural Science, 10, 59-62. doi: 10.4236/ns.2018.102005.
Auger Effect [ 1 ] observed with atoms consists of a nonradioactive electron transition, with ejection of an electron from an initial state of the atom with a “hole” in the innermost electron layer.
A neutral atom receives initially, a radiation of energy E, capable of withdrawing an electron from its electron layer K, leaving it with a “hole” in this layer. For light atoms, there is a great possibility that such a hole is filled by an electron from the outer electron layers, by a nonradioactive transition. This transition is accompanied by the ejection of an electron with kinetic energy T. The spectrum of these ejections is obtained experimentally by measuring the relative intensity of the electrons ejected at various values of T.
Figure 1 illustrates the normal process, showing one of the possible final states of the system. In this
Figure 1. Effect Auger Normal K-WW. The small circles (O) represent “holes” in the electronic layer (I) and (F) represent the initial and final states considered in the process. (GS) stands for the “Ground State” system.
figure, W represents “weakly bonded” valence electrons and S represents “strongly bonded” valence electrons [ 2 , 3 ].
The normal Auger process corresponds to the transitions represented by K-WW, K-WS and K-SS. The first letter represents the layer with the “hole” in the initial state (I) and the next two letters represent the layers with “holes” in the final state (F) of the system. The transitions corresponding to these normal Auger processes lead to more intense lines in the Auger spectrum. Several other “satellite lines” in the Auger spectrum are identified by transitions involving an initial excitation of a K-layer electron into a discrete state.
2. Interest by the Auger Effect
As the transition energy depends, at first approximation, on the binding energy of the electron K, the Auger spectrum constitutes, in a sense, an “X-ray” of the system. Thus, this process has been used as an efficient impurity detection technique on material surfaces.
Industrial processes involving the machining of precision and impurities free surfaces, such as the manufacture of disks for microcomputers, may employ the Auger process for the determination of probable unwanted impurities on these surfaces. This is an example of the importance of the domain of this technology and of the interest of the molecular physicists by the study of this process all over the world [ 4 - 8 ].
The shock section for the ionization of a hydrogen atom in the Ground State (GS), with ejection of an electron with energy x2/2 in the direction
\stackrel{\to }{x}
, was obtained by Massey and Mohr in 1933 [ 9 ], as:
\text{d}{\sigma }_{x}=\frac{8\text{π}}{{k}^{2}}\frac{\text{d}x}{{x}^{3}}
\text{d}{\sigma }_{x}
is equivalent to the Rutherford formula [ 8 ], for the shock section relative to the energy interval de, white conditions for impulsion
q\gg \left(1/{a}_{0}\right)
, where a0 is of the order of dimensions atomic:
\text{d}{\sigma }_{\epsilon }=\frac{\text{π}{e}^{4}}{E}\frac{\text{d}\epsilon }{{\epsilon }^{2}}
Here, E is the energy of the incident electron and e, the energy of the scattered electron, which, by analogy with Equation (1), corresponds to the energy of the ejected electron.
The shock section related to energy transfer (E − E0) (system energy difference after and before collision) is related to the loss energy of the incident electron through Differential Effective Braking [ 9 ], given by:
\text{d}E={\sum }_{n}\left({E}_{n}-{E}_{0}\right)\text{d}{\sigma }_{3}
Here, both can be over the states of the discrete spectrum as over the continuous spectrum, and dE represents the average energy lost per electron within a given solid angle.
It is “reasonable” to assume that the loss energy is proportional to the variation energy of the ejected electron [ 2 ]. Let us then admit the relation:
-\frac{\text{d}E}{\text{d}\epsilon }=c
It is reasonable to assume that the loss of energy is proportional to the energy variation of the ejected electron. Let us then admit the relation:
\text{d}E\cong {E}_{k}\text{d}{\sigma }_{n}
{E}_{k}={E}_{0}-{E}_{n}
, where Ek is the energy required to ionize the electron K, which corresponds to the first phase of the Auger process. From Equation (2)
\text{d}\epsilon \cong {\epsilon }^{2}\text{d}{\sigma }_{\epsilon }
-\text{d}E/\text{d}\epsilon \cong {E}_{k}/{\epsilon }^{2}
From experience [ 2 ], we can observed that the relation between energy variation to the ionize electron K and energy variation to the ejection electron, is approximately constant:
\frac{\left|\Delta {E}_{k}\right|}{{\left|\Delta {\epsilon }_{B}\right|}^{2}}=\frac{\left|{E}_{k}\left({\text{N}}_{\text{2}}\right)-{E}_{k}\left({\text{O}}_{\text{2}}\right)\right|}{{\left|{\stackrel{¯}{\epsilon }}_{B}\left({\text{N}}_{\text{2}}\right)-{\stackrel{¯}{\epsilon }}_{B}\left({\text{O}}_{\text{2}}\right)\right|}^{2}}\cong \frac{\left|{E}_{k}\left({\text{N}}_{\text{2}}\right)-{E}_{k}\left(\text{OdeCO}\right)\right|}{{\left|{\stackrel{¯}{\epsilon }}_{B}\left({\text{N}}_{\text{2}}\right)-{\stackrel{¯}{\epsilon }}_{B}\left(\text{OdeCO}\right)\right|}^{2}}\cong 0.007
So, we are going to infer that
c\cong 0.9{E}_{k}^{3}/{\epsilon }^{6}
\text{d}E\cong 0.9\frac{{E}_{k}^{3}}{{\epsilon }^{6}}\text{d}\epsilon
Now, from Equations ((3), (4) and (6)), one can obtain that
0.9{E}_{k}^{3}\frac{\text{d}\epsilon }{{\epsilon }^{6}}\cong \left({E}_{GS}-{E}_{f}\right)\text{d}{\sigma }_{\epsilon }
{E}_{GS}={E}_{0}
represents the energy of the system’s ground state, before ionization, and
{E}_{f}={E}_{n}
, the final energy of the system.
From Equations ((2) and (7)), we obtain
\text{d}\epsilon \cong \frac{{\epsilon }^{6}}{0.9{E}_{k}^{3}}\left({E}_{GS}-{E}_{f}\right)\text{d}{\sigma }_{\epsilon }
With this result, remembering that
{k}^{2}=\left(2mE/{ћ}^{2}\right)
, using Equation (2) and the atomic units
m=1
e=1
ћ=1
E={E}_{k}
required for that Auger Effect occurs, we finally have,
Table 1. Summary of results. All energies are in eV. The column
{\epsilon }_{fGS}
represents the average energy calculated by Equation (9) of this work. The column
{\epsilon }_{exp}
represents the average experimental energy obtained from the energies measured for each band by Moddeman et al. [ 2 ].
{\epsilon }_{fGS}\cong \frac{0.9{E}_{k}}{{\left[\frac{4\text{π}}{{E}_{k}}\left({E}_{GS}-{E}_{f}\right)\right]}^{\frac{1}{3}}}
{\epsilon }_{jGS}
provides the average energy of the electrons ejected for each GS ® F process, accompanied by the Auger Effect.
The results obtained by Equation (9) were compared with the mean energies for each band B, C, and D of the spectrum of molecules N2, O2 and CO, obtained experimentally by MODDEMAN et all [ 2 ].
\left({E}_{GS}-{E}_{f}\right)
, the same scheme of the work of MODDEMAN et all [ 2 ], was used at light of Koopman’s theorem, that is
\left({E}_{GS}-{E}_{f}\right)=\left\{\begin{array}{l}2{\epsilon }_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{paraoprocessoGS}\to \text{K-WW};\\ {\epsilon }_{1}+{\epsilon }_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{paraoprocessoGS}\to \text{K-WS};\\ 2{\epsilon }_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{paraoprocessoGS}\to \text{K-SS}.\end{array}
Table 1 summarizes the comparison of results.
As can be seen in Table 1, the mean energies calculated by Equation (9) of this work represent a reasonable approximation for the experimental average energies. That experimental results for average energy ejected electrons
{\epsilon }_{exp}
was obtained of the MODDEMAN [ 2 ], taking the average of the experimental values of each line of the spectral bands B, C and D. The energies Ek, are the binding energies of the K shell. The (EGS − Ef) energies are the difference between energy of ground state EGS of the system and the highest energy of the normal Auger line Ef obtained by Equation (10) of the text. A brief analysis of the results on Table 1 shows that the normal Auger GS → K-WW affects the calculated values of average energies
{\epsilon }_{fGS}
, which are very close to the experimental values. We see that the inelastic collision theory can be used to predict the average energies band of ejection electrons Auger normal.
[1] Auger, P. (1925) Sur l'effet photoélectrique composé. Journal de Physique et Le Radium, 6, 205-208.
[2] Moddeman, W.E., et al. (1971) Determination of the K-LL Auger Spectra of N2, O2, CO, NO, H2O and CO2. The Journal of Chemical Physics, 5, 2317.
[3] Dias, A.M. (1994) Estudo do Efeito Auger em Moléculas pela Teoria das Colisoes Inelásticas. Rev da Univ. de Alfenas, 1, 51-54.
[4] Shirley, D.A. (1973) Theory of KLL Auger Energies Including Static Relaxation. Physical Review A, 7, 1520-1528.
[5] Siegbahn, H., Asplund, L. and Kelfve, P. (1980) The Auger Spectrum of Water Vapour. Chemical Physics Letters, 69, 435-440.
[6] Jennison, D.R. (1981) Initial-State Relaxation Effects in Molecular Auger Spectra. Physical Review A, 23, 1215-1222.
[7] Kleiber, J.A., Jennison, K.R. and Rye, R.R. (1981) Analysis of the Auger Spectra of CO and CO2. Journal of Chemical Physics, 75, 650-662.
[8] Casanova, R., et al. (2002) Characterization of Iron Phyllosilicate Catalysts by Means of KLL Auger Spectra of Oxygen. Revista Latinoamericana de Metalurgia y Materiales, 22, 78-81.
[9] Landau, L. and Lifchitz, E. (1966) Mecanique Quantique, Moscou, Ed MIR, 718 p.
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Resistivity log - SEG Wiki
Resistivity logs are electrical well logs that record the resistivity of a formation. Resistivity is usually recorded in ohm meters (Ωm) and is displayed on track 4 of a well log.[1] Three depths of resistivity can be logged (shallow, medium, and deep) that record the resistivity of the formation with increasing distance away from the borehole.[1] [2] Resistivity logs can be interpreted to infer information about the porosity of a formation, the water saturation, and the presence of hydrocarbons.[1][2]
2 Resistivity Logging
2.1 Measurement & Tools
2.2 Measurement Zones
3 Interpretation and Uses
3.1 Rock Properties
3.2 Hydrocarbon Indicators
3.3 Archie's Equation & Other Calculations
3.4 Well Log Correlation
3.5 Seismic Well Ties
Resistivity Logs are well logs that record the resistivity of a formation.
Resistivity is the property of a material that resists the flow of electrical current. The reciprocal of resistivity is conductivity. Resistivity is defined by the equation
{\displaystyle {R}={\frac {r\times A}{L}}}
A is cross-sectional area
The resulting unit of resistivity is ohm meters squared/meters or simply, ohm meters (Ωm).[2][1] Resistivity is an independent property of a material and does not vary with size or shape of the material sample.[2]
Relative Resistivity
Fresh water high
Salt water (Brine) low
Hydrocarbons high
Sandstone high
Limestone high
Shale low
There are three main categories of well Logging: Electrical, Nuclear and Acoustic/Sonic.[2] Resistivity logging is an electrical well logging method and as such should be conducted in an open/uncased hole. Usually resistivity logs are displayed on track 4 of a well log and are displayed in ohm meter (Ωm) units[1] Resistivity logging was the first rock property that was logged and began the development of well logging methods.[2]
Diagram showing an induction tool, an SP log, and a resistivity log with medium resistivity, ILm, and deep resistivity, ILd. Image from Fig I-4 (Courtesy Schlumberger.)
During logging, a current is produced within a formation and the formation’s response to the current is recorded. There are two ways that the current can be produced in the formation: i. directly applying a current into the formation, and ii. inducing a current in the formation. If a current is directly applied to the formation, then the resistance of the current over a length of formation is measured. If a current is induced in the formation, then the conductivity of the formation is measured and inverted for the resistivity.[2]
Electrode tools are used to directly apply a current to the formation and measure the resistivity. Induction tools are used to induce a current in the formation and measure the conductivity. Induction tools are more widely used but a combination of electrode and induction tools can be used to create a single log of resistivity in the various zones of the formation. Electrode tools generally measure the shallow resistivity while induction tools generally measure the deep resistivity. Deep induction tools usually run in the frequency range of 35 – 20,000 Hz.[2]
Diagram showing borehole, invaded zone, flushed zone, annulus, and uninvaded zone. Image from Fig I-6 (Courtesy Schlumberger.)
Measurement Zones
Resistivity logs are influenced by the fluid used during drilling. During drilling, the drilling mud seeps into the formation’s pores and displaces the original formation fluid. The solid particles of the mud line the walls of the borehole while the filtrate is pushed into the formation. This creates various zones around the borehole based on how far the drilling fluid has been pushed into the formation.[2]
The invaded zone is the area where the drilling fluid is present in the formation. The invaded zone is comprised of the flushed zone and the annulus zone.
The flushed zone is only a few inches into the formation from the borehole and all of the original formation fluid has been replaced by drilling filtrate. The resistivity in the flushed zone reflects the resistivity of the drilling filtrate. The flushed zone is logged as shallow depth resistivity
The annulus zone is the transition zone next to the flushed zone and has a mixture of the drilling filtrate and the original fluid. The resistivity of the annulus zone reflects a combination of the drilling filtrate and original fluid. The annulus zone is logged as medium depth resistivity
The uninvaded zone is the furthest zone from the borehole and has not had any drilling fluid enter it. The resistivity in the uninvaded zone is the true resistivity of the original fluid and formation. The uninvaded zone is logged as deep resistivity
Interpretation and Uses
Resistivity can be interpreted as a measurement of a formation’s fluid saturation as it is a function of the formation’s rock type, porosity, fluid type, and fluid volume. Consequently, resistivity is used to identify permeable areas, estimate the porosity of a formation, and estimate fluid saturation. Archie’s equation can be used to calculate porosity and fluid saturation from resistivity logs.[2]
One of the most useful applications of resistivity logs is identifying hydrocarbon saturation vs. water saturation in a formation. Also, by distinguishing between water and hydrocarbon saturation, resistivity logs can be used to identify oil-water contacts.[1] The difference between drilling fluid and hydrocarbon resistivity can be used as a hydrocarbon indicator. In the case where drilling fluid has relatively low resistivity then sections of the log with shallow resistivity less than deep resistivity can be used as a hydrocarbon indicator.[1] Archie’s equation can be used to calculate the oil saturation of a formation from resistivity logs.[2]
Archie's Equation & Other Calculations
Archie’s equation can be used to calculate the porosity, water saturation, and oil saturation of sandstones and limestones from resistivity logs. Archie’s equation cannot be applied to shales.[1][2]
To calculate the water saturation, Archie’s equation can be written:
S_{w}=\left({\frac {a\times R_{w}}{R_{t}\times \phi ^{m}}}\right)^{\frac {1}{n}}
To calculate the porosity in the flushed zone the following equation can be used:
\phi =\left({\frac {a\times R_{mf}}{R_{xo}}}\right)^{\frac {1}{m}}
However, for a hydrocarbon saturated rock this equation will underestimate the porosity and a correction needs to be applied to the equation. The corrected equation is:
\phi =\left({\frac {a}{S_{xo}^{2}}}\times {\frac {R_{mf}}{R_{xo}}}\right)^{\frac {1}{m}}
For deep resistivity logs, hydrocarbon saturation can be calculated by:
S_{h}=1-S_{w}
S_{w}
is water saturation
a is the tortuosity factor
R_{w}
is resistivity of the formation water
R_{t}
is true formation resistivity
\phi
is porosity
m is the cementation exponent
n is the saturation exponent
R_{mf}
is resistivity of the mud filtrate
R_{xo}
is flushed zone resistivity
S_{xo}
is flushed zone water saturation
S_{h}
is hydrocarbon saturation
File:LI1LOG.jpg
An example of a complete wireline log showing gamma ray, caliper, SP, resistivity, density, sonic, and other logs (click on image for zoom options). Courtesy USGS[3]
Resistivity logs are part of the larger suite of wireline logs. Wireline logs can be used to identify the lithology at a borehole location. For a collection of logs in an area, structural and stratigraphic trends across that area can be mapped. By identifying patterns in the logs, several wells can be correlated to map units across a large area. These correlations can help infer information about the depositional environment.[1]
An example of a well tie. Well data from the control well (thick red line on left) is convolved with the wavelet (green box) to create synthetic seismic data and tie the seismic data.[4]
Seismic Well Ties
Well logs can also be used as control points for seismic reflection data.[1] The well logs and borehole seismic data can be used to predict the seismic response in the area. This provides a well tie that allows the seismic time data to be accurately tied to depth data for improved accuracy in data interpretation.[5]
Schlumberger Glossary, Resistivity
Petrowiki Resitivity and Spontaneous Potential
Halliburton Resistivity Logging
↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Evenick, J. C., 2008, Introduction to well logs & subsurface maps: PennWell Corporation.
↑ 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 Asquith, G. and D. Krygowski, 2004, Basic well log analysis, second edition: The American Association of Petroleum Geologists.
↑ Wikimedia Commons image from https://en.wikipedia.org/wiki/File:LI1LOG.jpg original file source: http://energy.cr.usgs.gov/OF00-200/WELLS/LISBURN1/LAS/LI1LOG.JPG.
↑ Herrera, R. H., S. Fomel, and M. van der Baan, 2014, Automatic approaches for seismic to well tying: Interpretation, 2, SD9-SD17.
↑ Schlumberger, 2010, Fundamentals of Borehole Seismic Technology: Schlumberger.
Retrieved from "https://wiki.seg.org/index.php?title=Resistivity_log&oldid=92941"
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Wireless Signal | Toph
The city of Byteland is rectangular in shape and can be represented as a 2D coordinate system with
n columns and
m rows. At each of the intersections, there is a house. The top left house of the city is situated at coordinate
(0, m-1)
(0,m−1) and the bottom right house is situated at
(n-1, 0)
(n−1,0). The town hall is situated at the coordinate
(x, y)
(x,y) which contains a cell tower providing wireless network signal to all the houses in Byteland. Judges will be moving to Byteland soon. Wireless signal travels in a straight line and it gets weakened only if the signal has to travel straight through another house, judges want to choose a house in such a way that their devices receive maximum possible signal strength.
Can you tell me how many different houses meet judges’ requirements?
Note: Judges can not move to the town hall as the mayor lives there. The width and length of the houses are so small that you can ignore them.
The first line of the input will contain a single integer
T(1 \leq T \leq 10^6)
T(1≤T≤106), denoting the number of test cases.
T lines will contain four integers
n, m, x, y(1 \leq n,m \leq 10^6; 0 \leq x < n; 0 \leq y < m)
n,m,x,y(1≤n,m≤106;0≤x<n;0≤y<m) each.
max(n,m) in all the test cases doesn’t exceed
10^7.
For each test case, print an integer containing the answer.
In the first case, there are no houses available.
In the second case, all houses but the town hall meets the requirements. In the third case, 12 houses meet the requirements
aritra741Earliest, 2M ago
tanvirtareqLightest, 5.5 MB
tanvirtareqShortest, 927B
A house will have maximum signal strength if the signal doesn’t have to travel through another house...
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Interest Rate Parity (IRP) Definition
Covered IRP Example
Interest rate parity (IRP) is a theory according to which the interest rate differential between two countries is equal to the differential between the forward exchange rate and the spot exchange rate.
The basic premise of interest rate parity is that hedged returns from investing in different currencies should be the same, regardless of their interest rates.
Parity is used by forex traders to find arbitrage opportunities.
Understanding Interest Rate Parity (IRP)
Interest rate parity (IRP) plays an essential role in foreign exchange markets by connecting interest rates, spot exchange rates, and foreign exchange rates.
IRP is the fundamental equation that governs the relationship between interest rates and currency exchange rates. The basic premise of IRP is that hedged returns from investing in different currencies should be the same, regardless of their interest rates.
IRP is the concept of no-arbitrage in the foreign exchange markets (the simultaneous purchase and sale of an asset to profit from a difference in the price). Investors cannot lock in the current exchange rate in one currency for a lower price and then purchase another currency from a country offering a higher interest rate.
The formula for IRP is:
\begin{aligned} &F_0 = S_0 \times \left ( \frac{ 1 + i_c }{ 1 + i_b } \right ) \\ &\textbf{where:}\\ &F_0 = \text{Forward Rate} \\ &S_0 = \text{Spot Rate} \\ &i_c = \text{Interest rate in country }c \\ &i_b = \text{Interest rate in country }b \\ \end{aligned}
F0=S0×(1+ib1+ic)where:F0=Forward RateS0=Spot Rateic=Interest rate in country cib=Interest rate in country b
An understanding of forward rates is fundamental to IRP, especially as it pertains to arbitrage. Forward exchange rates for currencies are exchange rates at a future point in time, as opposed to spot exchange rates, which are current rates. Forward rates are available from banks and currency dealers for periods ranging from less than a week to as far out as five years and more. As with spot currency quotations, forwards are quoted with a bid-ask spread.
The difference between the forward rate and spot rate is known as swap points. If this difference (forward rate minus spot rate) is positive, it is known as a forward premium; a negative difference is termed a forward discount.
A currency with lower interest rates will trade at a forward premium in relation to a currency with a higher interest rate. For example, the U.S. dollar typically trades at a forward premium against the Canadian dollar. Conversely, the Canadian dollar trades at a forward discount versus the U.S. dollar.
The IRP is said to be "covered" when the no-arbitrage condition could be satisfied through the use of forward contracts in an attempt to hedge against foreign exchange risk. Conversely, the IRP is "uncovered" when the no-arbitrage condition could be satisfied without the use of forward contracts to hedge against foreign exchange risk.
The relationship is reflected in the two methods an investor may adopt to convert foreign currency into U.S. dollars.
The first option an investor may choose is to invest the foreign currency locally at the foreign risk-free rate for a specific period. The investor would then simultaneously enter into a forward rate agreement to convert the proceeds from the investment into U.S. dollars using a forward exchange rate at the end of the investing period.
The second option would be to convert the foreign currency to U.S. dollars at the spot exchange rate, then invest the dollars for the same amount of time as in option A at the local (U.S.) risk-free rate. When no arbitrage opportunities exist, the cash flows from both options are equal.
Arbitrage is defined as the simultaneous purchase and sale of the same asset in different markets in order to profit from tiny differences in the asset's listed price. In the foreign exchange world, arbitrage trading involves the buying and selling of different currency pairs to exploit any pricing inefficiencies.
IRP has been criticized based on the assumptions that come with it. For instance, the covered IRP model assumes that there are infinite funds available for currency arbitrage, which is obviously not realistic. When futures or forward contracts are not available to hedge, uncovered IRP does not tend to hold in the real world.
Covered Interest Rate Parity Example
Let's assume Australian Treasury bills are offering an annual interest rate of 1.75% while U.S. Treasury bills are offering an annual interest rate of 0.5%. If an investor in the United States seeks to take advantage of Australia's interest rates, the investor would have to exchange U.S. dollars to Australian dollars to purchase the Treasury bill.
Thereafter, the investor would have to sell a one-year forward contract on the Australian dollar. However, under the covered IRP, the transaction would only have a return of 0.5%; otherwise, the no-arbitrage condition would be violated.
What's the Conceptual Basis for IRP?
IRP is the fundamental equation that governs the relationship between interest rates and currency exchange rates. Its basic premise is that hedged returns from investing in different currencies should be the same, regardless of their interest rates. Essentially, arbitrage (the simultaneous purchase and sale of an asset to profit from a difference in the price) should exist in the foreign exchange markets. In other words, investors cannot lock in the current exchange rate in one currency for a lower price and then purchase another currency from a country offering a higher interest rate.
What Are Forward Exchange Rates?
Forward exchange rates for currencies are exchange rates at a future point in time, as opposed to spot exchange rates, which are current rates. Forward rates are available from banks and currency dealers for periods ranging from less than a week to as far out as five years and more. As with spot currency quotations, forwards are quoted with a bid-ask spread.
The difference between the forward rate and spot rate is known as swap points. If this difference (forward rate minus spot rate) is positive, it is known as a forward premium; a negative difference is termed a forward discount. A currency with lower interest rates will trade at a forward premium in relation to a currency with a higher interest rate.
What's the Difference Between Covered and Uncovered IRP?
The IRP is said to be covered when the no-arbitrage condition could be satisfied through the use of forward contracts in an attempt to hedge against foreign exchange risk. Conversely, the IRP is uncovered when the no-arbitrage condition could be satisfied without the use of forward contracts to hedge against foreign exchange risk.
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Spring constant of a spring is calculated using formulae K =4pi2M/T2,where T is time period of vertical oscillation when mass - Physics - Units And Measurements - 7644085 | Meritnation.com
Spring constant of a spring is calculated using formulae K =4pi2M/T2,where T is time period of vertical oscillation when mass M is hung with the help of spring to rigid support. If time of oscillation for 10 oscillations is measurd to be 5.0s and mass M =0.20kg, find possible error in spring constant K.
Spring constant is given by:
K\quad =\quad \frac{4{\pi }^{2}M}{{T}^{2}}\phantom{\rule{0ex}{0ex}}time\quad for\quad 10\quad \quad oscillations\quad =\quad 5\quad s\phantom{\rule{0ex}{0ex}}time\quad period\quad =\quad 5/10\quad =\quad 0.5\quad s\phantom{\rule{0ex}{0ex}}M\quad =\quad 0.2\quad kg\phantom{\rule{0ex}{0ex}}K\quad =\quad \frac{4\times 3.14\times 3.14\times 0.2}{0.5\times 0.5}\phantom{\rule{0ex}{0ex}}K\quad =\quad 31.55\quad kg/{s}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Error\quad in\quad K:\phantom{\rule{0ex}{0ex}}\frac{∆K}{K}\quad =\quad 2\frac{∆T}{T}\phantom{\rule{0ex}{0ex}}all\quad other\quad quantities\quad are\quad cons\mathrm{tan}t.\phantom{\rule{0ex}{0ex}}If\quad error\quad in\quad time\quad period\quad is\quad given\quad then\quad we\quad can\quad calculate\quad the\quad error\quad in\quad spring\quad cons\mathrm{tan}t.
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Homogenization of the Poisson Equation in a Thick Periodic Junction | EMS Press
Homogenization of the Poisson Equation in a Thick Periodic Junction
T.A. Mel'nyk
A convergence theorem and asymptotic estimates as
\epsilon \to 0
are proved for a solution to a mixed boundary-value problem for the Poisson equation in a junction
\Omega_{\epsilon}
, of a domain
\Omega_0
and a large number
N^2
\epsilon
-periodically situated thin cylinders with thickness of order
\epsilon = O(\frac{1}{N})
. For this junction, we construct an extension operator and study its properties.
T.A. Mel'nyk, Homogenization of the Poisson Equation in a Thick Periodic Junction. Z. Anal. Anwend. 18 (1999), no. 4, pp. 953–975
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Rational string topology | EMS Press
Rational string topology
Micheline Vigué-Poirrier
We use the computational power of rational homotopy theory to provide an explicit cochain model for the loop product and the string bracket of a 1-connected closed manifold
M
. We prove that the loop homology of
M
is isomorphic to the Hochschild cohomology of the cochain algebra
C^\ast(M)
with coefficients in itself. Some explicit computations of the loop product and the string bracket are given.
Yves Félix, Jean-Claude Thomas, Micheline Vigué-Poirrier, Rational string topology. J. Eur. Math. Soc. 9 (2007), no. 1, pp. 123–156
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THE STABILITY OF BIOLOGICAL SYSTEMS
THE STABILITY OF BIOLOGICAL SYSTEMS12 [140]
W. S. McCulloch
There are several kinds of stabilities. There is the inherent stability that comes from making devices out of discrete, disparate parts, so that they either fit and stick or they collapse. It is enjoyed by everything that has a threshold. There are the stabilities based on feedback which are achieved in servo systems. The next kind of stability to consider is one spotted by Ross Ashby and called by him the Principle of Ultrastability; it is based on the possibility of internal switching until the system manages to straighten out its performance, although its responses may have been wrong in the first place.
Still another type of stability is exemplified in the reticular formation in the brainstem. The reticular formation is neither completely orderly nor completely disorderly. If it were either, it would not work. The segments of the human body have their own circuits through the musculature and back again to this same segment of the body. In low forms of life, individual segments can practically maintain themselves. In higher forms, for such a job as walking, agonists and antagonists are not represented entirely in one segment, but are batched into groups of segments.
A segmented animal such as man may be derived from a form in which there was barely a trace of connection between one segment and the next. In the worm, the segments are already joined by short axons in series, so that at the head end, when the distant receptors pick up something, an order can be issued to the worm to shorten itself suddenly. That would not do for man, whose whole mode of organization is different. We not only tie segment to segment, but we also run very fast lines from practically all parts of the body to the core of the brainstem, where lies the net called the reticular formation. From distance receptors, from all of our computers — cortex, cerebellum, basal ganglia — information comes into the reticular formation. Its organization is like a command center to which information comes, variously coded, from all parts of the system and from its own specific receptors. Whenever some part of that reticular formation is in possession of sufficient crucial information, it makes a decision committing the muscles to action. From moment to moment the command undoubtedly passes from point to point in that reticular formation. Let me call the kind of stability obtained in this fashion a redundancy of potential command.
The next kind of stability I am going to discuss was developed in answer to two questions which John von Neumann used to ask. The first is this: “How is it I can drink three glasses of whiskey or three cups of coffee, and I know it’s enough to change the thresholds of my neurons, and I can still think, I can still speak, I can still walk?” The other, equally founded on the consideration of living things, concerns the problem of producing reliable functions, given unreliable elements. This problem is formulated in von Neumann’s lecture, “Toward a Probabilistic Logic.”
I will discuss this kind of stability in terms of computer elements which may be called neurons. They are idealized neurons — over-simplified descriptions of those in our heads. The functions will be used as elements in a kind of substitutive algebra which, to the best of my knowledge, has not been employed heretofore.
Each neuron will be assigned a certain logical function. To visualize the range of possible function, consider a “percipient element” with two inputs, each so simple as to have either one signal or else none on each occasion. For this element, the world is then one of the four cases separated by the arms of the diagonal cross:
Which of the four worlds is the case on a particular occasion will be indicated by the position of a dot:
And let the “percipient element” be a relay which emits either one signal or else none, according to its input, i.e., under none, one, two, three, or all four cases. There are then 16 such “relays”:
Each computes one logical function, i.e., its output depends upon its input by the one of the 16 logical relations written underneath it. Reading from left to right, the relays in the upper line will fire under the following conditions:
Never (universal contradiction)
Signal from A but not from B
Signal from both A and B (conjunction)
Signal from B but not from A
Signal from A (whether or not from B)
Signal from A or else from B (but not from both)
Signal from B (whether or not from A)
Signal from either A or B or both (disjunction)
The relays in the lower line fire if there is:
Signal from neither A nor B
No signal from B (whether or not from A)
Signal from both A and B or from neither (identity)
No signal in A (whatever from B)
Signal in B implies signal in A
Signal in A not more than one input (Sheffer stroke)
Signal from A implies signal from B
Under every condition (tautology)
These symbols are exhaustive for the calculus of propositions of the first order.1
The rule of combination of symbols is simply a substitution of the output of a relay for each of the two inputs;
Wherein the ( ) signify that the enclosed X is precomputed from A and B before their outputs reach the unbracketed X. Thus,
is abbreviated:
so that the parentheses stand for the striking time (or synaptic delay) of the relays, as well as for the serial order of operations.
This calculus can be realized by means of conventionalized neurons. It is assumed that these can be excited or inhibited in equal final steps, and that firing or nonfiring depends on whether the algebraic sum of excitations and inhibitions does or does not exceed a given threshold, θ. The following symbols will be used:
With such neurons with thresholds ranging from 4 to −3, and two inputs yielding a total of 3 excitations or inhibitions, we can realize all but two of the 16 logical functions (of course “alphabetical variations” or reversals of right and left are admitted):
The reason for picking three units of signal is that this number is necessary and sufficient to produce all but two of the logical functions. If the stimuli from A and B are equal in number, then all functions with 2 “dots” (see third sketch) are omitted; but it is still possible to treat those for the kinds of stability here considered, mutatis mutandis.
The missing pair can be constructed by using 3 neurons, but this requires 3 times the number of neurons and an extra synaptic delay. Nature has to compute efficiently in true time, i.e., a minimum number of neurons and minimal delay. Hence one suspects, and finds, interaction of afferent channels. Such interactions of afferents occur, but let us forget them as unnecessary to our argument.
We shall permit the connection of any neuron by either kind of termination to any other neurons. This means that for any but the front-line neurons, the “world” consists of the terminations of two other neurons, each of which, during any given cycle, does or does not fire. With this convention we can construct circuits representing any desired logical functions.
Using only such actions as can be accounted for by impulses reaching and affecting the neuron (or relay), we can invent circuits which will give the same output for the same input, although the threshold of every unit has been shifted in the same direction, so that it is now computing a new logical function; and we can do this for any logical function from input to output. For instance:
This stability differs in kind from any heretofore described. Let us call it “logical stability.” Its source resides in part in the redundancy of logic itself and in part in the redundancy of the net; for, given 16 types of relays taken 3 at a time, there are 163 combinations, but all result in only 16 functions — hence the redundancy of the combinations is 162.
The redundancy of logic itself is perhaps best seen in that a relay which should respond only to one input may be in readiness on one-half of all occasions to respond to one other input, and it will be wrong in only one of the 4 cases half the time. For instance:
If we dealt with cells having 3 inputs — A, B, C — there would be 8 cases, and the error would occur one in 16 times. In general, if δ is the number of inputs and 1/ρ is the fraction of time when the relay will respond erroneously to one situation, then the fraction of time the relay will give a wrong answer is (l/2*δ*)(1/ρ). In short, the redundancy of logic of these circuits is of the form (22*δ*)*δ*, in eliminating errors, and we are left with the construction of functions to make use of the best out of 162+1; combinations for δ = 2. Let us now see how to design circuits for independent variations of threshold.
Consider any combination of three neurons such that the outputs of two form the input of the third. Each of the three “normally” behaves in some given way but occasionally responds when it is not “supposed” to do so, i.e., it errs. Let us suppose the error is either to respond or to fail to respond to a signal from the left (or right). Now this can be harmless in the central neuron if the one on the left (or right) is on this occasion computing the intended function. But if an error at the same instance occurs in that neuron also, the erroneous signal (or erroneous absence of signal) will appear in the output. There is no reason why any erroneous signal in the right (or left) should have anything to do with this performance. For instance, we intend:
but each relay may err some of the time responding to “A” or “A and B” instead of to “A and B”only, these errors occurring with frequencies 1/R, 1/S, and 1/𝒯, respectively, each multiplied, of course, by 1/2*δ*:
an the error is
\frac{1}{{{2^\delta }}} \cdot \frac{1}{R} \cdot \frac{1}{S}.
Such a configuration does not enjoy “logical stability” — but for values of R, S, 𝒯>1 it gives a probability of “correct” outputs for inputs better than its components give singly. Compare it with the aforementioned “logically stable” circuit - letting R=S=𝒯 (call their value ρ). The logically stable circuit is error-free for ρ = l, and for ρ=∞.
Either type of circuit is useful in constructing a probabilistic logic (i.e., one in which the logical functions themselves are afflicted by random errors) whether the error is attributable to random fluctuations of threshold, signal, or noise of any other origin.
Consider next only those circuits like
wherein the error can enter from one side only. Just as the factor 𝒯 disappeared from the final count of errors, so will any other factor introduced by consideration of functions of more than 2 inputs to each relay. Hence we can write, in general, for such circuits:
No circuits with fewer errors can be devised. We will call such circuits “best.”
Clearly, if best circuits can be so combined as to repeat the improvement at successive stages, one can build a circuit of any desired reliability by going through enough stages. The “trick” is to keep the errors from occurring on both sides simultaneously and thus corrupting the product action of the central, or output, neuron. This can be done, for the function previously considered, by a network composed of neurons of the type
All neurons on the left are going to pass incorrectly a signal from A alone when the threshold is lowered, and all those on the right will err only for a signal from B alone. The error probabilities for the cells in the third rank are 1/4RSV on the left side, 1/4𝒯UW on the right. So, we may write in general that for a net of relays with a width δ = number of common afferents to all neurons, and a depth n of the net in ranks of relays (or neurons), the errors are (l/2δ)(1/ρn), which requires precisely δ×n−(n−1) = 1+(n−1)δ relays.
The extension to any δ is achieved by substituting “all” for “both” in the center and dealing with a cube of δ dimensions. These functions ran equally well built by using “none” for “neither, nor” in the center. In the constructions using “all,” we put dots in each place in the rank facing the input for the function we desire to compute, whereas in those using “none” we work with the “cube” of all dots less the dots of the function we desire to compute. These remain the “best” constructions for all values of δ and n, provided the value of 1/ρ remains less than that at which it intersects the curve for the best of the logically stable circuits. For values of 1/ρ larger than this, the logically stable are preferable. The best way to combine these nets is still to be found.
I should say, in passing, that I have examined the frequency with which logically stable circuits will occur if connected 2-to-1 but otherwise in random triplets; they constitute about 1/16 of all complexes, and a much larger fraction of all circuits whose output is neither tautology nor contradiction, i.e., of all useful circuits. As for those which improve performance to 1/4ρ2, they are at least 1/8 of all triplets with like cells in the first rank, but perhaps there are many more I have not yet found. Of these functions, therefore, the fraction of “best” circuits is not less than (1/16)•(1/8) of all triplets connected at random.
I have examined the circuit in which errors occur in each of the three neurons in three places in this pattern:
It yields errors with a probability
This is less than 1/ρ if ρ⩾4, but the scatter of errors precludes a further diminution by repetitions of the net.
The investigation I have discussed is not finished, but I believe I have demonstrated a method which meets von Neumann’s challenges to find a stability of circuit action under common shift of threshold, and to construct a probabilistic logic with which to secure an increasingly reliable performance from unreliable components.
Wordcloud: Action, Best, Cases, Circuits, Combinations, Command, Compute, Consider, Construct, Desire, Dots, Either, Element, Equal, Erroneous, Errors, Figure, Fire, Follows, Form, Formation, Frac, Functions, Give, Given, Input, Instance, Logical, Neither, Net, Neurons, None, Number, Occur, Output, Random, Rank, Redundancy, Relay, Respond, Reticular, Segment, Signal, Stability, Stable, System, Threshold, Type, Used, Values
Keywords: Body, Systems, Circuits, Life, Formation, Segments, Stabilities, Center, Stability, Parts
Google Books: http://asclinks.live/dmxa
Google Scholar: http://asclinks.live/2s4i
Jstor: http://asclinks.live/xa6h
1 Reprinted from Brookhaven Symposia in Biology, 10: 207-215, 1957. Symposium on “Homeostatic Mechanisms,” (June 12-14, 1957), sponsored by the Brookhaven National Laboratory, Upton, L.I.. N.Y. ↩ ↩
2 This work is supported in part by the U.S. Army (Signal Corps), the U.S. Air Force (Office of Scientific Research, Air Research and Development Command), and the U.S. Navy (Office of Naval Research). ↩
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Curve Fitting via Optimization - MATLAB & Simulink - MathWorks Switzerland
Find the Best Fitting Parameters
Check the Fit Quality
This example shows how to fit a nonlinear function to data. For this example, the nonlinear function is the standard exponential decay curve
y\left(t\right)=A\mathrm{exp}\left(-\lambda t\right),
y\left(t\right)
is the response at time
t
A
\lambda
are the parameters to fit. Fitting the curve means finding parameters
A
\lambda
that minimize the sum of squared errors
\sum _{i=1}^{n}{\left({y}_{i}-A\mathrm{exp}\left(-\lambda {t}_{i}\right)\right)}^{2},
where the times are
{t}_{i}
and the responses are
{y}_{i},i=1,\dots ,n
. The sum of squared errors is the objective function.
Usually, you have data from measurements. For this example, create artificial data based on a model with
A=40
\lambda =0.5
, with normally distributed pseudorandom errors.
tdata = 0:0.1:10;
ydata = 40*exp(-0.5*tdata) + randn(size(tdata));
Write a function that accepts parameters A and lambda and data tdata and ydata, and returns the sum of squared errors for the model
y\left(t\right)
. Put all the variables to optimize (A and lambda) in a single vector variable (x). For more information, see Minimizing Functions of Several Variables.
type sseval
function sse = sseval(x,tdata,ydata)
lambda = x(2);
sse = sum((ydata - A*exp(-lambda*tdata)).^2);
Save this objective function as a file named sseval.m on your MATLAB® path.
The fminsearch solver applies to functions of one variable, x. However, the sseval function has three variables. The extra variables tdata and ydata are not variables to optimize, but are data for the optimization. Define the objective function for fminsearch as a function of x alone:
fun = @(x)sseval(x,tdata,ydata);
For information about including extra parameters such as tdata and ydata, see Parameterizing Functions.
Start from a random positive set of parameters x0, and have fminsearch find the parameters that minimize the objective function.
bestx = fminsearch(fun,x0)
bestx = 2×1
The result bestx is reasonably near the parameters that generated the data, A = 40 and lambda = 0.5.
To check the quality of the fit, plot the data and the resulting fitted response curve. Create the response curve from the returned parameters of your model.
A = bestx(1);
lambda = bestx(2);
yfit = A*exp(-lambda*tdata);
plot(tdata,ydata,'*');
plot(tdata,yfit,'r');
xlabel('tdata')
ylabel('Response Data and Curve')
title('Data and Best Fitting Exponential Curve')
legend('Data','Fitted Curve')
Nonlinear Data-Fitting (Optimization Toolbox)
Nonlinear Regression (Statistics and Machine Learning Toolbox)
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Periodic words, common subsequences and frogs
April 2022 Periodic words, common subsequences and frogs
Boris Bukh,1 Christopher Cox2
2Department of Mathematics, Iowa State University
{\mathit{W}}^{\left(\mathit{n}\right)}
be the n-letter word obtained by repeating a fixed word W, and let
{\mathit{R}}_{\mathit{n}}
be a random n-letter word over the same alphabet. We show several results about the length of the longest common subsequence (LCS) between
{\mathit{W}}^{\left(\mathit{n}\right)}
{\mathit{R}}_{\mathit{n}}
; in particular, we show that its expectation is
{\mathit{\gamma }}_{\mathit{W}}\mathit{n}-\mathit{O}\left(\sqrt{\mathit{n}}\right)
for an efficiently-computable constant
{\mathit{\gamma }}_{\mathit{W}}
This is done by relating the problem to a new interacting particle system, which we dub “frog dynamics”. In this system, the particles (“frogs”) hop over one another in the order given by their labels. Stripped of the labeling, the frog dynamics reduces to a variant of the PushTASEP.
In the special case when all symbols of W are distinct, we obtain an explicit formula for the constant
{\mathit{\gamma }}_{\mathit{W}}
and a closed-form expression for the stationary distribution of the associated frog dynamics.
In addition, we propose new conjectures about the asymptotic of the LCS of a pair of random words. These conjectures are informed by computer experiments using a new heuristic algorithm to compute the LCS. Through our computations, we found periodic words that are more random-like than a random word, as measured by the LCS.
The first author was supported in part by Sloan Research Fellowship and by U.S. taxpayers through NSF CAREER Grant DMS-1555149.
The second author was supported in part by U.S. taxpayers through NSF CAREER Grant DMS-1555149.
We thank Tomasz Tkocz for discussions at the early stage of this research and for comments on a draft of this paper. We thank him additionally for the contribution of Proposition 40. We thank Alex Tiskin for pointing out the relevance of references [6] and [23]. We owe the development of the frog metaphor used in this paper to a conversation with Laure Bukh. The frog symbol is from Froggy font by Vladimir Nikolic. The lily pad symbol is based on a drawing by FrauBieneMaja. We thank Zimu Xiang for pointing several typos, and two anonymous referees for valuable feedback on the earlier versions of the paper.
Boris Bukh. Christopher Cox. "Periodic words, common subsequences and frogs." Ann. Appl. Probab. 32 (2) 1295 - 1332, April 2022. https://doi.org/10.1214/21-AAP1709
Keywords: Chvátal–Sankoff constants , common subsequences , words
Boris Bukh, Christopher Cox "Periodic words, common subsequences and frogs," The Annals of Applied Probability, Ann. Appl. Probab. 32(2), 1295-1332, (April 2022)
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Mode choice - Wikipedia
Mode choice analysis is the third step in the conventional four-step transportation forecasting model. The steps, in order, are trip generation, trip distribution, mode choice analysis, and route assignment. Trip distribution's zonal interchange analysis yields a set of origin destination tables that tells where the trips will be made. Mode choice analysis allows the modeler to determine what mode of transport will be used, and what modal share results.
The early transportation planning model developed by the Chicago Area Transportation Study (CATS) focused on transit. It wanted to know how much travel would continue by transit. The CATS divided transit trips into two classes: trips to the Central Business District, or CBD (mainly by subway/elevated transit, express buses, and commuter trains) and other (mainly on the local bus system). For the latter, increases in auto ownership and use were a trade-off against bus use; trend data were used. CBD travel was analyzed using historic mode choice data together with projections of CBD land uses. Somewhat similar techniques were used in many studies. Two decades after CATS, for example, the London study followed essentially the same procedure, but in this case, researchers first divided trips into those made in the inner part of the city and those in the outer part. This procedure was followed because it was thought that income (resulting in the purchase and use of automobiles) drove mode choice.
1 Diversion curve techniques
2 Disaggregate travel demand models
3 Psychological roots
4 Econometric formulation
5 Econometric estimation
7 Returning to roots
Diversion curve techniques[edit]
The CATS had diversion curve techniques available and used them for some tasks. At first, the CATS studied the diversion of auto traffic from streets and arterial roads to proposed expressways. Diversion curves were also used for bypasses built around cities to find out what percent of traffic would use the bypass. The mode choice version of diversion curve analysis proceeds this way: one forms a ratio, say:
{\displaystyle {\frac {c_{\text{transit}}}{c_{\text{auto}}}}=R}
cm = travel time by mode m and
R is empirical data in the form:
Given the R that we have calculated, the graph tells us the percent of users in the market that will choose transit. A variation on the technique is to use costs rather than time in the diversion ratio. The decision to use a time or cost ratio turns on the problem at hand. Transit agencies developed diversion curves for different kinds of situations, so variables like income and population density entered implicitly.
Diversion curves are based on empirical observations, and their improvement has resulted from better (more and more pointed) data. Curves are available for many markets. It is not difficult to obtain data and array results. Expansion of transit has motivated data development by operators and planners. Yacov Zahavi’s UMOT studies, discussed earlier, contain many examples of diversion curves.
In a sense, diversion curve analysis is expert system analysis. Planners could "eyeball" neighborhoods and estimate transit ridership by routes and time of day. Instead, diversion is observed empirically and charts drawn.
Disaggregate travel demand models[edit]
Travel demand theory was introduced in the appendix on traffic generation. The core of the field is the set of models developed following work by Stan Warner in 1962 (Strategic Choice of Mode in Urban Travel: A Study of Binary Choice). Using data from the CATS, Warner investigated classification techniques using models from biology and psychology. Building from Warner and other early investigators, disaggregate demand models emerged. Analysis is disaggregate in that individuals are the basic units of observation, yet aggregate because models yield a single set of parameters describing the choice behavior of the population. Behavior enters because the theory made use of consumer behavior concepts from economics and parts of choice behavior concepts from psychology. Researchers at the University of California, Berkeley (especially Daniel McFadden, who won a Nobel Prize in Economics for his efforts) and the Massachusetts Institute of Technology (Moshe Ben-Akiva) (and in MIT associated consulting firms, especially Cambridge Systematics) developed what has become known as choice models, direct demand models (DDM), Random Utility Models (RUM) or, in its most used form, the multinomial logit model (MNL).
Choice models have attracted a lot of attention and work; the Proceedings of the International Association for Travel Behavior Research chronicles the evolution of the models. The models are treated in modern transportation planning and transportation engineering textbooks.
One reason for rapid model development was a felt need. Systems were being proposed (especially transit systems) where no empirical experience of the type used in diversion curves was available. Choice models permit comparison of more than two alternatives and the importance of attributes of alternatives. There was the general desire for an analysis technique that depended less on aggregate analysis and with a greater behavioral content. And there was attraction, too, because choice models have logical and behavioral roots extended back to the 1920s as well as roots in Kelvin Lancaster’s consumer behavior theory, in utility theory, and in modern statistical methods.
Psychological roots[edit]
Early psychology work involved the typical experiment: Here are two objects with weights, w1 and w2, which is heavier? The finding from such an experiment would be that the greater the difference in weight, the greater the probability of choosing correctly. Graphs similar to the one on the right result.
Louis Leon Thurstone proposed (in the 1920s) that perceived weight,
where v is the true weight and e is random with
E(e) = 0.
The assumption that e is normally and identically distributed (NID) yields the binary probit model.
Econometric formulation[edit]
Economists deal with utility rather than physical weights, and say that
observed utility = mean utility + random term.
The characteristics of the object, x, must be considered, so we have
u(x) = v(x) + e(x).
If we follow Thurston's assumption, we again have a probit model.
An alternative is to assume that the error terms are independently and identically distributed with a Weibull, Gumbel Type I, or double exponential distribution. (They are much the same, and differ slightly in their tails (thicker) from the normal distribution). This yields the multinomial logit model (MNL). Daniel McFadden argued that the Weibull had desirable properties compared to other distributions that might be used. Among other things, the error terms are normally and identically distributed. The logit model is simply a log ratio of the probability of choosing a mode to the probability of not choosing a mode.
{\displaystyle \log \left({\frac {P_{i}}{1-P_{i}}}\right)=v(x_{i})}
Observe the mathematical similarity between the logit model and the S-curves we estimated earlier, although here share increases with utility rather than time. With a choice model we are explaining the share of travelers using a mode (or the probability that an individual traveler uses a mode multiplied by the number of travelers).
The comparison with S-curves is suggestive that modes (or technologies) get adopted as their utility increases, which happens over time for several reasons. First, because the utility itself is a function of network effects, the more users, the more valuable the service, higher the utility associated with joining the network. Second because utility increases as user costs drop, which happens when fixed costs can be spread over more users (another network effect). Third technological advances, which occur over time and as the number of users increases, drive down relative cost.
An illustration of a utility expression is given:
{\displaystyle \log \left({\frac {P_{A}}{1-P_{A}}}\right)=\beta _{0}+\beta _{1}\left(c_{A}-c_{T}\right)+\beta _{2}\left(t_{A}-t_{T}\right)+\beta _{3}I+\beta _{4}N=v_{A}}
Pi = Probability of choosing mode i.
PA = Probability of taking auto
cA,cT = cost of auto, transit
tA,tT = travel time of auto, transit
N = Number of travelers
With algebra, the model can be translated to its most widely used form:
{\displaystyle {\frac {P_{A}}{1-P_{A}}}=e^{v_{A}}}
{\displaystyle P_{A}=e^{v_{A}}-P_{A}e^{v_{A}}}
{\displaystyle P_{A}\left(1+e^{v_{A}}\right)=e^{v_{A}}}
{\displaystyle P_{A}={\frac {e^{v_{A}}}{1+e^{v_{A}}}}}
It is fair to make two conflicting statements about the estimation and use of this model:
it's a "house of cards", and
used by a technically competent and thoughtful analyst, it's useful.
The "house of cards" problem largely arises from the utility theory basis of the model specification. Broadly, utility theory assumes that (1) users and suppliers have perfect information about the market; (2) they have deterministic functions (faced with the same options, they will always make the same choices); and (3) switching between alternatives is costless. These assumptions don’t fit very well with what is known about behavior. Furthermore, the aggregation of utility across the population is impossible since there is no universal utility scale.
Suppose an option has a net utility ujk (option k, person j). We can imagine that having a systematic part vjk that is a function of the characteristics of an object and person j, plus a random part ejk, which represents tastes, observational errors and a bunch of other things (it gets murky here). (An object such as a vehicle does not have utility, it is characteristics of a vehicle that have utility.) The introduction of e lets us do some aggregation. As noted above, we think of observable utility as being a function:
{\displaystyle v_{A}=\beta _{0}+\beta _{1}\left(c_{A}-c_{T}\right)+\beta _{2}\left(t_{A}-t_{T}\right)+\beta _{3}I+\beta _{4}N}
where each variable represents a characteristic of the auto trip. The value β0 is termed an alternative specific constant. Most modelers say it represents characteristics left out of the equation (e.g., the political correctness of a mode, if I take transit I feel morally righteous, so β0 may be negative for the automobile), but it includes whatever is needed to make error terms NID.
Econometric estimation[edit]
Turning now to some technical matters, how do we estimate v(x)? Utility (v(x)) isn’t observable. All we can observe are choices (say, measured as 0 or 1), and we want to talk about probabilities of choices that range from 0 to 1. (If we do a regression on 0s and 1s we might measure for j a probability of 1.4 or −0.2 of taking an auto.) Further, the distribution of the error terms wouldn’t have appropriate statistical characteristics.
The MNL approach is to make a maximum likelihood estimate of this functional form. The likelihood function is:
{\displaystyle L^{*}=\prod _{n=1}^{N}{f\left({y_{n}\left|{x_{n},\theta }\right.}\right)}}
we solve for the estimated parameters
{\displaystyle {\hat {\theta }}\,}
that max L*. This happens when:
{\displaystyle {\frac {\partial L}{\partial {\hat {\theta }}_{N}}}=0}
The log-likelihood is easier to work with, as the products turn to sums:
{\displaystyle \ln L^{*}=\sum _{n=1}^{N}\ln f\left(y_{n}\left|x_{n},\theta \right.\right)}
Consider an example adopted from John Bitzan’s Transportation Economics Notes. Let X be a binary variable that is equal to 1 with probability γ, and equal to 0 with probability (1 − gamma). Then f(0) = (1 − γ) and f(1) = γ. Suppose that we have 5 observations of X, giving the sample {1,1,1,0,1}. To find the maximum likelihood estimator of γ examine various values of γ, and for these values determine the probability of drawing the sample {1,1,1,0,1} If γ takes the value 0, the probability of drawing our sample is 0. If γ is 0.1, then the probability of getting our sample is: f(1,1,1,0,1) = f(1)f(1)f(1)f(0)f(1) = 0.1×0.1×0.1×0.9×0.1 = 0.00009 We can compute the probability of obtaining our sample over a range of γ – this is our likelihood function. The likelihood function for n independent observations in a logit model is
{\displaystyle L^{*}=\prod _{n=1}^{N}{P_{i}^{Y_{i}}}\left(1-P_{i}\right)^{1-Y_{i}}}
where: Yi = 1 or 0 (choosing e.g. auto or not-auto) and Pi = the probability of observing Yi = 1
The log likelihood is thus:
{\displaystyle \ell =\ln L^{*}=\sum _{i=1}^{n}\left[Y_{i}\ln P_{i}+\left(1-Y_{i}\right)\ln \left(1-P_{i}\right)\right]}
In the binomial (two alternative) logit model,
{\displaystyle P_{\text{auto}}={\frac {e^{v(x_{\text{auto}})}}{1+e^{v(x_{\text{auto}})}}}}
{\displaystyle \ell =\ln L^{*}=\sum _{i=1}^{n}\left[Y_{i}v(x_{\text{auto}})-\ln \left(1+e^{v(x_{\text{auto}})}\right)\right]}
The log-likelihood function is maximized setting the partial derivatives to zero:
{\displaystyle {\frac {\partial \ell }{\partial \beta }}=\sum _{i=1}^{n}\left(Y_{i}-{\hat {P}}_{i}\right)=0}
The above gives the essence of modern MNL choice modeling.
Additional topics[edit]
Topics not touched on include the “red bus, blue bus” problem; the use of nested models (e.g., estimate choice between auto and transit, and then estimate choice between rail and bus transit); how consumers’ surplus measurements may be obtained; and model estimation, goodness of fit, etc. For these topics see a textbook such as Ortuzar and Willumsen (2001).
Returning to roots[edit]
The discussion above is based on the economist’s utility formulation. At the time MNL modeling was developed there was some attention to psychologist's choice work (e.g., Luce’s choice axioms discussed in his Individual Choice Behavior, 1959). It has an analytic side in computational process modeling. Emphasis is on how people think when they make choices or solve problems (see Newell and Simon 1972). Put another way, in contrast to utility theory, it stresses not the choice but the way the choice was made. It provides a conceptual framework for travel choices and agendas of activities involving considerations of long and short term memory, effectors, and other aspects of thought and decision processes. It takes the form of rules dealing with the way information is searched and acted on. Although there is a lot of attention to behavioral analysis in transportation work, the best of modern psychological ideas are only beginning to enter the field. (e.g. Golledge, Kwan and Garling 1984; Garling, Kwan, and Golledge 1994).
Transportation Systems Analysis Model – TSAM is a nationwide transportation planning model to forecast intercity travel behavior in the United States.
Garling, Tommy, Mei-Po Kwan, and Reginald G. Golledge. Household Activity Scheduling, Transportation Research, 22B, pp. 333–353. 1994.
Golledge. Reginald G., Mei Po Kwan, and Tommy Garling, “Computational Process Modeling of Household Travel Decisions,” Papers in Regional Science, 73, pp. 99–118. 1984.
Lancaster, K.J., A new approach to consumer theory. Journal of Political Economy, 1966. 74(2): p. 132–157.
Luce, Duncan R. (1959). Individual choice behavior, a theoretical analysis. New York, Wiley.
Newell, A. and Simon, H. A. (1972). Human Problem Solving. Englewood Cliffs, NJ: Prentice Hall.
Ortuzar, Juan de Dios and L. G. Willumsen’s Modelling Transport. 3rd Edition. Wiley and Sons. 2001,
Thurstone, L.L. (1927). A law of comparative judgement. Psychological Review, 34, 278–286.
Warner, Stan 1962 Strategic Choice of Mode in Urban Travel: A Study of Binary Choice
Retrieved from "https://en.wikipedia.org/w/index.php?title=Mode_choice&oldid=1086381185"
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Statistics and Data Analysis - Maple Help
Home : Support : Online Help : System : Information : Updates : Maple 2019 : Statistics and Data Analysis
The LeastTrimmedSquares command computes least trimmed squares regression for some data.
\mathrm{with}\left(\mathrm{Statistics}\right):
In this example, we have 1000 data points. There is a single independent variable, x, with values uniformly distributed between 0 and 10. The dependent variable is a linear function of the independent variable plus some additive noise, y=5 x + 10 + noise, where the noise is from a probability distribution known to have severe outliers - the Cauchy distribution, with location parameter 0 and scale parameter 5.
x≔\mathrm{Sample}\left(\mathrm{Uniform}\left(0,10\right),1000\right):
\mathrm{noise}≔\mathrm{Sample}\left(\mathrm{Cauchy}\left(0,1\right),1000\right):
y ≔ \left(5*x + \mathrm{noise}\right) +~ 10:
\mathrm{pp} ≔ \mathrm{PointPlot}\left(y, \mathrm{xcoords} = x,\mathrm{size}=\left[0.7 , "golden"\right]\right): \mathrm{pp};\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}
\mathrm{ls_regression_result} ≔ \mathrm{Fit}\left(a * X + b, x, y, X\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}
\textcolor[rgb]{0,0,1}{\mathrm{ls_regression_result}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{3.44970682383807}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{10.6816568681413}
\mathrm{ls_deviation_from_model} ≔ {\left(\mathrm{coeff}\left(\mathrm{ls_regression_result}, X, 1\right) - 5\right)}^{2} + {\left(\mathrm{coeff}\left(\mathrm{ls_regression_result}, X, 0\right) - 10\right)}^{2};
\textcolor[rgb]{0,0,1}{\mathrm{ls_deviation_from_model}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{2.86806501793850}
\mathrm{lts_regression_result} ≔ \mathrm{LeastTrimmedSquares}\left(x, y, \left[X\right]\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}
\textcolor[rgb]{0,0,1}{\mathrm{lts_regression_result}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{5.03537530551575}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{9.82475419561272}
\mathrm{lts_deviation_from_model} ≔ {\left(\mathrm{coeff}\left(\mathrm{lts_regression_result}, X, 1\right) - 5\right)}^{2} + {\left(\mathrm{coeff}\left(\mathrm{lts_regression_result}, X, 0\right) - 10\right)}^{2};
\textcolor[rgb]{0,0,1}{\mathrm{lts_deviation_from_model}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.0319625041956780}
\mathrm{lts_900_regression_result} ≔ \mathrm{LeastTrimmedSquares}\left(x, y, \left[X\right], \mathrm{include}=900\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}
\textcolor[rgb]{0,0,1}{\mathrm{lts_900_regression_result}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{5.00862730339998}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{10.0156318668695}
\mathrm{lts_900_deviation_from_model} ≔ {\left(\mathrm{coeff}\left(\mathrm{lts_900_regression_result}, X, 1\right) - 5\right)}^{2} + {\left(\mathrm{coeff}\left(\mathrm{lts_900_regression_result}, X, 0\right) - 10\right)}^{2};
\textcolor[rgb]{0,0,1}{\mathrm{lts_900_deviation_from_model}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.000318785625780919}
\mathrm{rme_regression_result} ≔ \mathrm{RepeatedMedianEstimator}\left(x, y, X\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}
\textcolor[rgb]{0,0,1}{\mathrm{rme_regression_result}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{10.0306661686300}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5.00564125476873}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{X}
\mathrm{rme_deviation_from_model} ≔ {\left(\mathrm{coeff}\left(\mathrm{rme_regression_result}, X, 1\right) - 5\right)}^{2}+ {\left(\mathrm{coeff}\left(\mathrm{rme_regression_result}, X, 0\right) - 10\right)}^{2};
\textcolor[rgb]{0,0,1}{\mathrm{rme_deviation_from_model}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.000972237653807886}
\mathrm{plots}\left[\mathrm{display}\right]\left(\mathrm{pp}, \mathrm{plot}\left(\left[\mathrm{ls_regression_result}, \mathrm{lts_regression_result}, \mathrm{lts_900_regression_result}, \mathrm{rme_regression_result}\right], X=0..10, \mathrm{legend}=\left["Least squares", "Least trimmed squares", "Least trimmed squares \left(900 points\right)", "Repeated median estimator"\right]\right), \mathrm{view}=\left[0..10, -10..110\right],\mathrm{size}=\left[0.7 , "golden"\right]\right);
The Correlogram command computes autocorrelations of a data set and displays the result as a column plot with dashed lines indicating the lower and upper 95% confidence bands for the normal distribution N(0,1/L), where L is the size of the sample 'X', and a caption reporting how many of the displayed columns lie outside of the bands of plus or minus 2, 3, and 4 standard deviations respectively. AutoCorrelationPlot is an alias for the Correlogram command.
\mathrm{Correlogram}\left(\mathrm{Import}\left("datasets/sunspots.csv",\mathrm{base}=\mathrm{datadir},\mathrm{output}=\mathrm{Matrix}\right)\left[265..310, 2\right]\right)
\mathrm{restart}:
\mathrm{with}\left(\mathrm{Statistics}\right):
For example, specify some data:
\mathrm{data}≔\mathrm{Matrix}\left([[0,1.8],[1,0.7],[2.5,2.8],[4,4.2],[6.2,3]]\right)
\left[\begin{array}{cc}0& 1.8\\ 1& 0.7\\ 2.5& 2.8\\ 4& 4.2\\ 6.2& 3\end{array}\right]
\mathrm{lm}≔\mathrm{LinearFit}\left(a+bt,\mathrm{data},t\right)
\textcolor[rgb]{0,0,1}{\mathrm{lm}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{1.49598376946009}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.366429281218947}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{t}
It can be observed that from the plot of the data and the linear model that there is some upward trend. The Detrend command removes any trend from the data.
\mathrm{detrend_data}≔\mathrm{Detrend}\left(\mathrm{data}\right)
\left[\begin{array}{c}0.30401623053991367\\ -1.162413050679033\\ 0.38794302749254683\\ 1.2382991056641273\\ -0.7678453130175553\end{array}\right]
\mathrm{plots}:-\mathrm{display}\left([\phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{ScatterPlot}\left(\mathrm{data},\mathrm{color}="Black",\mathrm{legend}="Original Data",\mathrm{symbol}=\mathrm{solidcircle},\mathrm{symbolsize}=15\right),\phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{plot}\left(\mathrm{lm},\mathrm{color}="Black",\mathrm{legend}="Trend",\mathrm{linestyle}=\mathrm{dash}\right),\phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{ScatterPlot}\left(\mathrm{data}[..,1],\mathrm{detrend_data},\mathrm{color}="Red",\mathrm{legend}="Detrended Data",\mathrm{symbol}=\mathrm{diamond},\mathrm{symbolsize}=15\right),\phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{plot}\left(\mathrm{Mean}\left(\mathrm{detrend_data}\right),\mathrm{color}="Red",\mathrm{legend}="Mean of Detrended Data",\mathrm{linestyle}=\mathrm{dot}\right)],\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{view}=[\mathrm{min}\left(\mathrm{data}[..,1]\right)-0.1..\mathrm{max}\left(\mathrm{data}[..,1]\right)+0.1,\mathrm{default}],\mathrm{size}=\left[0.5 , "golden"\right]\right)
Detrend has also been added as an option to several routines in SignalProcessing including SignalPlot, Periodogram, and Spectrogram.
\mathrm{with}\left(\mathrm{Statistics}\right):
x≔〈\mathrm{seq}\left({i}^{2},i=1..10\right)〉
\left[\begin{array}{r}1\\ 4\\ 9\\ 16\\ 25\\ 36\\ 49\\ 64\\ 81\\ 100\end{array}\right]
\mathrm{Difference}\left(x\right)
\left[\begin{array}{r}3\\ 5\\ 7\\ 9\\ 11\\ 13\\ 15\\ 17\\ 19\end{array}\right]
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{18446883718659727478}}
For more details, see the User Interface updates page.
Several commands have been updated to support DataFrames and DataSeries, including remove, select and selectremove. Other new commands such as Detrend and Difference also support DataFrames and DataSeries.
The Biplot command has a new option, components, which specifies the principal components used in the biplot.
DataSummary, FivePointSummary, FrequencyTable
The DataSummary, FivePointSummary, and FrequencyTable commands have a new option, tableweights, which specifies the relative column widths in the displayed embedded table.
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Triangle | Toph
By subeen · Limits 2s, 512 MB
You went to an exciting adventure in a deep jungle but sadly a giant captured you. The giant said that, if you can solve a simple problem it will let you go. It will give you the lengths of the three sides of a triangle. All you have to do is calculate the area of the triangle.
If the three sides of a triangle are a, b and c. Then:
s = (a+b+c)/2
s=(a+b+c)/2
Area = \sqrt{s \times (s-a) \times (s-b) \times (s-c)}
Area=s×(s−a)×(s−b)×(s−c)
The first line of the input will contain an integer N. Then N lines will follow. Each of the lines will have three integers a, b and c.
For each test case print the area of the triangle. Two decimal digit will suffice. If the three sides don't form a triangle print "Oh, No!".
fci_zeroEarliest, Oct '18
fci_zeroFastest, 0.0s
HillolTalukdarLightest, 0 B
SCB-PA Inter School and College Programming Contest 2018 Preliminary
JU CSE Structured Programming Lab Final
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Homotopical algebra for C*-algebras | EMS Press
JournalsjncgVol. 7, No. 4pp. 981–1006
Homotopical algebra for C*-algebras
Otgonbayar Uuye
The category of fibrant objects is a convenient framework to do homotopy theory, introduced and developed by Ken Brown. In this paper, we apply it to the category of C*-algebras. In particular, we get a unified treatment of (ordinary) homotopy theory for C
^*
-algebras, KK-theory and E-theory, since all of these can be expressed as the homotopy theory of a category of fibrant objects.
Otgonbayar Uuye, Homotopical algebra for C*-algebras. J. Noncommut. Geom. 7 (2013), no. 4, pp. 981–1006
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