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Who Is the Naag King? | Toph
Who Is the Naag King?
Limits 2s, 1.0 GB
Battle between two mighty programmers from BRACU. Who is better? Jaber Vai or Mukhter Vai?
They are very good friends since the beginning of their BRACU life. Though their fan base always quarrel over who is the better among these two? So, Jaber vai and Mukhtar vai thought it should come to an end, as contestants spend lots of their valuable time arguing on this issue instead of focusing on problem solving. As their programming skill is almost the same, it’s difficult to find out who is better through programming battle. So they have decided they will sit for a game named Shap-ludu, seems familiar? Yeah, we used to play Shap ludu a lot in our childhood.
The long awaited day has finally arrived. Everyone is too excited to witness the battle. But alas! You fall sick on that day, and couldn’t come to the campus. As a result, you came up with an idea; you have chosen one of your friends who will give you the live update of every move of each player. Can you determine who is the winner calculating each of their moves?
a. Game Board: The board is a 10×10 matrix. The squares are numbered from 1 to 100 sequentially, will consist of some ladders and snakes. (See the given picture.)
b. Jaber Vai will always play the first move.
c. Player until getting ‘1’ on their dice can’t move their piece. After getting ‘1’ on the dice, one should put his piece on square numbered ‘1’.
d. In each turn, every player will roll the dice only once.
e. Take it in turns to roll the dice. Move your piece forward the number of spaces shown on the dice.
f. If your piece lands at the bottom of a ladder, you should move up to the top of the ladder.
g. If your piece lands on the head of a snake, you must slide down to the bottom of the snake.
h. The first player to get to the square '100' is the winner.
L (
0≤L≤50) and
S (
0≤S≤75) denoting the number of Ladders and Snake in the board, respectively.
L lines contain description of the ladder: the
i^{th}
ith of them contains two integers
X and
Y (
1≤X,Y≤100,X<Y) denoting the start and end positions of a ladder, means if you come to the square numbered
X, the ladder will take you to the square numbered
Y.
S lines contain description of the ladder: the
i^{th}
P and
Q (
1 < P < 100
1<P<100,
P>Q) denoting the start and end positions of a Snake, means if you come to the square numbered
P, at the mouth of a snake, then you have to move your piece to the square numbered
Q.
The next lines contain
V (
1≤V≤6),digit came up on the dice.
V1 is the move of Jaber Vai,
V2 denotes the move of Mukhtar vai,
V3 is the move of Jaber vai and so on. The input will be terminated when we will get the winner.
You have to print the name who won the battle, calculating their moves. If Jaber Vai wins, you should print “Jaber Tuhin is the winner.” or else “Mukhter Hossain is the winner.” (without quotes). It is guaranteed that one of the players will win the game.
Mukhter Hossain is the winner.
Jaber Vai will always give the first move. They have rolled the dice alternatively.
Moves of Jaber Vai: 5 (need 1 to start), 5 (need 1 to start), 1 (put his piece on 1st square), 4 (moved to 5, climbing the ladder reached 26), 6 (moved to 32), 1 (moved to 33, where he found a snake, had to take his piece at square 9), 4 (went to 13, found a ladder, climbing the ladder reached 56), 1 (moved to 57), 1 (moved to 58).
Moves of Mukhter Vai: 2 (need 1 to start), 1 (put his piece on 1st square), 6 (went to 7), 2 (went to 9), 4 (went to 13, found a ladder, climbing the ladder reached 56), 4 (went to 60, found a ladder, climbing the ladder reached 98), 3 (invalid move), 6 (invalid move), 2 (reach square 100, winner of the match).
edge555Earliest, Jul '19
edge555Lightest, 131 kB
user.dwak4abkShortest, 269B
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Home : Support : Online Help : Mathematics : Linear Algebra : Inert Matrix Commands : Nullspace
compute the nullspace of a matrix mod p
Nullspace(A) mod p
Matrix over a finite field
Nullspace(A) mod p computes a basis for the null space (Nullspace) of the linear transformation defined by the matrix A. The result is a (possibly empty) set of vectors.
A≔\mathrm{Matrix}\left([[1,2,3],[1,2,3],[0,0,0]]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}]
\mathrm{Nullspace}\left(A\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}5
{[\begin{array}{c}\textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{1}\end{array}]}
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1. Perform the following conversions, using the appropriate number of significant figures in your answer:
{\displaystyle 1.5{\frac {g}{s}}\rightarrow {\frac {lb}{hr}}}
{\displaystyle 4.5*10^{2}{\mbox{ W}}\rightarrow {\frac {btu}{min}}}
{\displaystyle 34{\frac {\mu g}{\mu m^{3}}}\rightarrow {\frac {oz}{in^{3}}}}
{\displaystyle 4.18{\frac {J}{g*oC}}\rightarrow {\frac {kWh}{lb*oF}}}
(note: kWh means kilowatt-hour)
{\displaystyle 1.00{\mbox{ m}}^{3}\rightarrow L\rightarrow dm^{3}\rightarrow mL\rightarrow cm^{3}}
2. Perform a dimensional analysis on the following equations to determine if they are reasonable:
{\displaystyle v=dt}
, where v is velocity, d is distance, and t is time.
{\displaystyle F={\frac {m*v^{2}}{r}}}
where F is force, m is mass, v is velocity, and r is radius (a distance).
{\displaystyle F_{bouy}=\rho *V*g}
{\displaystyle \rho }
is density, V is volume, and g is gravitational acceleration.
{\displaystyle {\dot {m}}={\frac {\dot {V}}{\rho }}}
{\displaystyle {\dot {m}}}
is mass flow rate,
{\displaystyle {\dot {V}}}
is volumetric flow rate, and
{\displaystyle \rho }
is density.
3. Recall that the ideal gas law is
{\displaystyle PV=nRT}
where P is pressure, V is volume, n is number of moles, R is a constant, and T is the temperature.
a) What are the units of R in terms of the base unit types (length, time, mass, and temperature)?
b) Show how these two values of R are equivalent:
{\displaystyle R=0.0821{\frac {L*atm}{mol*K}}=8.31{\frac {J}{mol*K}}}
c) If an ideal gas exists in a closed container with a molar density of
{\displaystyle 0.03{\frac {mol}{L}}}
at a pressure of
{\displaystyle 0.96*10^{5}{\mbox{ Pa}}}
, what temperature is the container held at?
d) What is the molar concentration of an ideal gas with a partial pressure of
{\displaystyle 4.5*10^{5}{\mbox{ Pa}}}
if the total pressure in the container is
{\displaystyle 6{\mbox{ atm}}}
e) At what temperatures and pressures is a gas most and least likely to be ideal? (hint: you can't use it when you have a liquid)
f) Suppose you want to mix ideal gasses in two separate tanks together. The first tank is held at a pressure of 500 Torr and contains 50 moles of water vapor and 30 moles of water at 70oC. The second is held at 400 Torr and 70oC. The volume of the second tank is the same as that of the first, and the ratio of moles water vapor to moles of water is the same in both tanks.
You recombine the gasses into a single tank the same size as the first two. Assuming that the temperature remains constant, what is the pressure in the final tank? If the tank can withstand 1 atm pressure, will it blow up?
4. Consider the reaction
{\displaystyle H_{2}O_{2}<->H_{2}O+{\frac {1}{2}}O_{2}}
, which is carried out by many organisms as a way to eliminate hydrogen peroxide.
a). What is the standard enthalpy of this reaction? Under what conditions does it hold?
b). What is the standard Gibbs energy change of this reaction? Under what conditions does it hold? In what direction is the reaction spontaneous at standard conditions?
c). What is the Gibbs energy change at biological conditions (1 atm and 37oC) if the initial hydrogen peroxide concentration is 0.01M? Assume oxygen is the only gas present in the cell.
d). What is the equilibrium constant under the conditions in part c? Under the conditions in part b)? What is the constant independent of?
e). Repeat parts a through d for the alternative reaction
{\displaystyle H_{2}O_{2}\rightarrow H_{2}+O_{2}}
. Why isn't this reaction used instead?
5. Two ideal gasses A and B combine to form a third ideal gas, C, in the reaction
{\displaystyle A+B\rightarrow C}
. Suppose that the reaction is irreversible and occurs at a constant temperature of 25oC in a 5L container. If you start with 0.2 moles of A and 0.5 moles of B at a total pressure of 1.04 atm, what will the pressure be when the reaction is completed?
6. How much heat is released when 45 grams of methane are burned in excess air under standard conditions? How about when the same mass of glucose is burned? What is one possible reason why most heterotrophic organisms use glucose instead of methane as a fuel? Assume that the combustion is complete, i.e. no carbon monoxide is formed.
7. Suppose that you have carbon monoxide and water in a tank in a 1.5:1 ratio.
a) In the literature, find the reaction that these two compounds undergo (hint: look for the water gas shift reaction). Why is it an important reaction?
b) Using a table of Gibbs energies of formation, calculate the equilibrium constant for the reaction.
c) How much hydrogen can be produced from this initial mixture?
d) What are some ways in which the yield of hydrogen can be increased? (hint: recall Le Chatlier's principle for equilibrium).
e) What factors do you think may influence how long it takes for the reaction to reach equilibrium?
Problem 8. A bio-fuel plant converts the sugars (glucose) in corn into ethanol and carbon dioxide in a process called fermentation. The plant produces 100 gpm (gallons per minute) of ethanol and can produce of 2.5 gallons of ethanol per bushel of corn. Jimmy farms a total of 2000 acres, 75% of which are corn. He sells 80% of his corn supply to the bio-fuel plant for $4.00/bushel. (Hint: 120 bushels = 1 acre)
a) How long can the plant run with the supply of corn from Jimmy? (hours)
b) How much money did Jimmy make for his corn?
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Strength of Materials/General State of Stress - Wikibooks, open books for an open world
Strength of Materials/General State of Stress
1 Principal Stresses
2 Stress Invariants
3 Hydrostatic Stress
4 Deviatoric Stresses
5 Mohr's Circle
5.1 Mohr's Circle for Common Cases
6.1 Maximum Shear Stress Criterion
6.2 Maximum Distortion Energy Criterion
6.3 Failure of Materials
Principal StressesEdit
The general state of stress can be represented by a symmetric 3 x 3 matrix.
It is always possible to choose a coordinate system such that all shear stresses are zero. The 3 x 3 matrix is then diagonalised, with the three principal stresses on the diagonal, and all other components equal to zero. The three principal stresses are conventionally labelled σ1, σ2 and σ3.
σ1 is the maximum (most tensile) principal stress, σ3 is the minimum (most compressive) principal stress, and σ2 is the intermediate principal stress.
Stress InvariantsEdit
{\displaystyle I_{1}=\sigma _{3}+\sigma _{2}+\sigma _{1}}
{\displaystyle I_{2}=\sigma _{1}\sigma _{2}+\sigma _{2}\sigma _{3}+\sigma _{3}\sigma _{1}}
{\displaystyle I_{3}=\sigma _{1}\sigma _{2}\sigma _{3}}
Hydrostatic StressEdit
The so-called hydrostatic stress, σh, is given by:
{\displaystyle \sigma _{h}=\left.{\frac {\sigma _{1}+\sigma _{2}+\sigma _{3}}{3}}\right.}
Deviatoric StressesEdit
normal stress - pressure
where pressure = average stress
Mohr's CircleEdit
Consider the two dimensional stress condition where the stresses are σx, σy, and τxy. We have, for another set of orthogonal axes x'-y' at angle θ with x-y, the stresses are
{\displaystyle \sigma _{x'}={\frac {\sigma _{x}+\sigma _{y}}{2}}+{\frac {\sigma _{x}-\sigma _{y}}{2}}\cos 2\theta +\tau \sin 2\theta }
{\displaystyle \tau _{x'y'}=-{\frac {\sigma _{x}-\sigma _{y}}{2}}\sin 2\theta +\tau _{xy}\cos 2\theta }
From the above equations, we can see that for any stress states given by σx, σy, and τxy, we can find a value of θ such that the value of σx' is maximum. This value is called the principal stress σ1 (for maximum) or σ2 (for minimum).
The principal stresses are given by
{\displaystyle \sigma _{1,2}={\frac {\sigma _{x}+\sigma _{y}}{2}}\pm {\sqrt {\left({\frac {\sigma _{x}-\sigma _{y}}{2}}\right)^{2}+\tau _{xy}^{2}}}}
and the maximum shear stress is given by
τmax = (σ1 − σ2)/2
From the definitions for σx' and τx'y', we have
{\displaystyle \left(\sigma _{x'}-{\frac {\sigma _{x}+\sigma _{y}}{2}}\right)^{2}+\tau _{x'y'}^{2}=\left({\frac {\sigma _{x}-\sigma _{y}}{2}}\right)^{2}+\tau _{xy}^{2}}
In the σ-τ graph, this is a circle with center on the x axis, and the distance of the center from the origin is given by (σx + σy)/2 and a radius given by
{\displaystyle R={\sqrt {\left({\frac {\sigma _{x}-\sigma _{y}}{2}}\right)^{2}+\tau _{xy}^{2}}}}
This circle is known as Mohr's circle, and is useful for visualizing the stress state at a point.
The above figure shows Mohr's circle for a stress state (σ, τ). The center and radius of the circle are obtained from equations stated above. The other stress σy can be read off by the point diametrically opposite the (σ, τ) point. The stress at any plane can be found using simple geometrical constructs.
Mohr's Circle for Common CasesEdit
The above shear stress in this case is σ1/2.
A liquid is one which by definition cannot take a shear. Thus the Mohr-circle diagram is just a point. (The dot should be on the negative side of the axis as the liquids can resist only compression)
For pure shear, the Mohr circle is centered at the origin.
Failure CriteriaEdit
In the case of isotropic materials, the state of stress at any point of the body is completely defined by the triad of the principal stresses.
Now that we are able to transform stresses to get the principal stresses, we can use these stresses to consider some of the criteria (theories) postulated for the failure of materials in two- and three-axial states of stress usually based on experiments on yielding and fracture of materials in the uniaxial state of stress. According to such experiments, the kind of failure depends on the type of material. Failure of ductile materials (most of metals) occurs when the elastic limit is reached and yielding commences. Failure of non-ductile materials (e.g., cast iron, concrete) occurs by brittle fracture.
Maximum Shear Stress CriterionEdit
For ductile materials, one failure theory is that of maximum shear stress. We know that the maximum shear stress is given by τmax = (σ1 − σ2)/2. The yield stress, σy can be determined by uniaxial tensile tests. Thus, if the maximum shear stress theory is valid, failure occurs when the maximum shear stress reaches σy/2.
In the above image, the material will fail if the stress state is outside the shaded region.
Maximum Distortion Energy CriterionEdit
If the maximum shear stress theory is valid, failure occurs when the maximum shear stress reaches
{\displaystyle {\frac {{\sqrt {2}}\sigma _{y}}{3}}}
Failure of MaterialsEdit
The actual failure mode of each material is unique, though certain criteria can be applied to classes of materials. mode of failure . fatigue failure,
Retrieved from "https://en.wikibooks.org/w/index.php?title=Strength_of_Materials/General_State_of_Stress&oldid=3851211"
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Build inflation curve from market zero-coupon inflation swap rates - MATLAB inflationbuild - MathWorks América Latina
\begin{array}{l}I\left(0,{T}_{1Y}\right)=I\left({T}_{0}\right){\left(}^{1}\\ I\left(0,{T}_{2Y}\right)=I\left({T}_{0}\right){\left(}^{1}\\ I\left(0,{T}_{3Y}\right)=I\left({T}_{0}\right){\left(}^{1}\\ ...\\ I\left(0,{T}_{i}\right)=I\left({T}_{0}\right){\left(}^{1}\end{array}
I\left(0,{T}_{i}\right)
I\left({T}_{0}\right)
b\left(0;{T}_{0},{T}_{i}\right)
{f}_{i}=\frac{1}{\left({T}_{i}-{T}_{i-1}\right)}\mathrm{log}\left(\frac{I\left(0,{T}_{i}\right)}{I\left(0,{T}_{i-1}\right)}\right)
\begin{array}{l}I\left(0,{T}_{i}\right)=I\left({T}_{0}\right)\mathrm{exp}\left(\underset{{T}_{0}}{\overset{{T}_{i}}{\int }}f\left(u\right)du\right)\right)\mathrm{exp}\left(\underset{{T}_{0}}{\overset{{T}_{i}}{\int }}s\left(u\right)du\right)\right)\\ I\left(0,{T}_{i}\right)=I\left(0,{T}_{i-1}\right)\mathrm{exp}\left(\left({T}_{i}-{T}_{i-1}\right)\left({f}_{i}+{s}_{i}\right)\right)\end{array}
I\left(0,{T}_{i}\right)
I\left(0,{T}_{i-1}\right)
\left[{T}_{i-1},{T}_{i}\right]
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Intermolecular force - formulasearchengine
Intermolecular forces are forces of attraction or repulsion which act between neighboring particles (atoms, molecules or ions). They are weak compared to the intramolecular forces, the forces which keep a molecule together. For example the covalent bond, involving the sharing of electron pairs between atoms is much stronger than the forces present between the neighboring molecules. They are an essential part of force fields used in molecular mechanics.
The investigation of intermolecular forces starts from macroscopic observations which point out the existence and action of forces at a molecular level. These observations include non-ideal gas thermodynamic behavior reflected by virial coefficients, vapor pressure, viscosity, superficial tension and adsorption data.
The first reference to the nature of microscopic forces is found in Alexis Clairaut's work Theorie de la Figure de la Terre.[1] Other scientists who have contributed to the investigation of microscopic forces include: Laplace, Gauss, Maxwell and Boltzmann.
Attractive intermolecular forces are considered by the following types:
Van der Waals forces (Keesom force, Debye force, and London dispersion force)
Information on intermolecular force is obtained by macroscopic measurements of properties like viscosity, PVT data. The link to microscopic aspects is given by virial coefficients and Lennard-Jones potentials.
1 Dipole-dipole interactions {{safesubst:#invoke:anchor|main}}
1.1 Ion-dipole and ion-induced dipole forces
2.1 Keesom (permanent-permanent dipoles) interaction
2.2 Debye (permanent-induced dipoles) force
2.3 London dispersion force (induced-induced dipoles interaction)
3 Relative strength of forces
4 Quantum mechanical theories
Dipole-dipole interactions {{safesubst:#invoke:anchor|main}}
Dipole-dipole interactions are electrostatic interactions between permanent dipoles in molecules. These interactions tend to align the molecules to increase attraction (reducing potential energy). An example of a dipole-dipole interaction can be seen in hydrogen chloride (HCl): the positive end of a polar molecule will attract the negative end of the other molecule and influence its position. Polar molecules have a net attraction between them. Examples of polar molecules include hydrogen chloride (HCl) and chloroform (CHCl3).
Often molecules contain dipolar groups, but have no overall dipole moment. This occurs if there is symmetry within the molecule that causes the dipoles to cancel each other out. This occurs in molecules such as tetrachloromethane. Note that the dipole-dipole interaction between two individual atoms is usually zero, since atoms rarely carry a permanent dipole. See atomic dipoles.
Ion-dipole and ion-induced dipole forces
Ion-dipole and ion-induced dipole forces are similar to dipole-dipole and induced-dipole interactions but involve ions, instead of only polar and non-polar molecules. Ion-dipole and ion-induced dipole forces are stronger than dipole-dipole interactions because the charge of any ion is much greater than the charge of a dipole moment. Ion-dipole bonding is stronger than hydrogen bonding.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}
An ion-dipole force consists of an ion and a polar molecule interacting. They align so that the positive and negative groups are next to one another, allowing for maximum attraction.
An ion-induced dipole force consists of an ion and a non-polar molecule interacting. Like a dipole-induced dipole force, the charge of the ion causes distortion of the electron cloud on the non-polar molecule.[2]
A hydrogen bond is the attraction between the lone pair of an electronegative atom and a hydrogen atom that is bonded to either nitrogen, oxygen, or fluorine.[3] The hydrogen bond is often described as a strong electrostatic dipole-dipole interaction. However, it also has some features of covalent bonding: it is directional, stronger than a van der Waals interaction, produces interatomic distances shorter than the sum of van der Waals radius, and usually involves a limited number of interaction partners, which can be interpreted as a kind of valence.
Intermolecular hydrogen bonding is responsible for the high boiling point of water (100 °C) compared to the other group 16 hydrides, which have no hydrogen bonds. Intramolecular hydrogen bonding is partly responsible for the secondary, tertiary, and quaternary structures of proteins and nucleic acids. It also plays an important role in the structure of polymers, both synthetic and natural.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}
The vdW forces arise from interaction between uncharged atoms or molecules, leading not only to such phenomena as the cohesion of condensed phases and physical adsorption of gases, but also to a universal force of attraction between macroscopic bodies.[4]
Keesom (permanent-permanent dipoles) interaction
The first contribution to Van der Waals forces is due to electrostatic interactions between charges (in molecular ions), dipoles (for polar molecules), quadrupoles (all molecules with symmetry lower than cubic), and permanent multipoles. It is referred to as Keesom interactions(named after Willem Hendrik Keesom).[5] These forces originate from the attraction between permanent dipoles (dipolar molecules) and are temperature dependent.[6]
They consists in attractive interactions between dipoles that are ensemble averaged over different rotational orientations of the dipoles. It is assumed that the molecules are constantly rotating and never get locked into place. This is a good assumption, but at some point molecules do get locked into place. The energy of a Keesom interaction depends on the inverse sixth power of the distance, unlike the interaction energy of two spatially fixed dipoles, which depends on the inverse third power of the distance. The Keesom interaction can only occur among molecules that possess permanent dipole moments aka two polar molecules. Also Keesom interactions are very weak Van der Waals interactions and do not occur in aqueous solutions that contain electrolytes. The angle averaged interaction is given by the following equation:
{\displaystyle {\frac {-2m_{1}^{2}m_{2}^{2}}{48\pi ^{2}\varepsilon _{o}^{2}\varepsilon _{r}^{2}k_{b}Tr^{6}}}=V}
Where m = charge per length,
{\displaystyle \varepsilon _{o}}
= permitivity of free space,
{\displaystyle \varepsilon _{r}}
= dielectric constant of surrounding material, T = temperature,
{\displaystyle k_{b}}
= Boltzmann constant, and r = distance between molecules.
Debye (permanent-induced dipoles) force
The second contribution is the induction (also known as polarization) or Debye force, arising from interactions between rotating permanent dipoles and from the polarizability of atoms and molecules (induced dipoles). These induced dipoles occur when one molecule with a permanent dipole repels another molecule’s electrons. A molecule with permanent dipole can induce a dipole in a similar neighboring molecule and cause mutual attraction. Debye forces cannot occur between atoms. The forces between induced and permanent dipoles are not as temperature dependent as Keesom interactions because the induced dipole is free to shift and rotate around the non-polar molecule. The Debye induction effects and Keesom orientation effects are referred to as polar interactions.[7]
The induced dipole forces appear from the induction (also known as polarization), which is the attractive interaction between a permanent multipole on one molecule with an induced (by the former di/multi-pole) multipole on another.[8][9][10][11] This interaction is called the Debye force, named after Peter J.W. Debye.
One example of an induction-interaction between permanent dipole and induced dipole is the interaction between HCl and Ar. In this system, Ar experiences a dipole as its electrons are attracted (to the H side of HCl) or repelled (from the Cl side) by HCl.[8][10] The angle averaged interaction is given by the following equation.
{\displaystyle {\frac {-m_{1}^{2}\alpha _{2}}{16\pi ^{2}\varepsilon _{o}^{2}\varepsilon _{r}^{2}r^{6}}}=V}
{\displaystyle \alpha }
= polarizability
This kind of interaction can be expected between any polar molecule and non-polar/symmetrical molecule. The induction-interaction force is far weaker than dipole-dipole interaction, but stronger than the London dispersion force.
London dispersion force (induced-induced dipoles interaction)
The third and dominant contribution is the dispersion or London force (fluctuating dipole-induced dipole), due to the non-zero instantaneous dipole moments of all atoms and molecules. Such polarization can be induced either by a polar molecule or by the repulsion of negatively charged electron clouds in non-polar molecules. Thus, London interactions are caused by random fluctuations in electron density in an electron cloud. An atom with a large number of electrons will have a greater associated London force than a smaller atom. The dispersion (London) force is the most important component because all materials are polarizable, whereas Keesom and Debye forces require permanent dipoles. The London interaction is universal and is present in atom-atom interactions as well. For various reasons, London interactions (dispersion) have been considered relevant for interactions between macroscopic bodies in condensed systems. Hamaker developed the theory of vdW between macroscopic bodies in 1937 and showed that the additivity of these interactions renders them considerably more long-range.[12]
Dissociation energy (kcal/mol)[13]
Ionic Lattice Energy 250-4000 [14]
Covalent Bond Energy 30-260
Hydrogen Bonds 1-12 (about 5 in water)
Dipole–Dipole 0.5–2 {{ safesubst:#invoke:Unsubst $B=
London Dispersion Forces <1 to 15 (estimated from the enthalpies of vaporization of hydrocarbons)[15]
Note: this comparison is only approximate – the actual relative strengths will vary depending on the molecules involved. Ionic and covalent bonding will always be stronger than intermolecular forces in any given substance.
Intermolecular forces observed between atoms and molecules can be described phenomenologically as occurring between permanent and instantaneous dipoles, as outlined above. Alternatively, one may seek a fundamental, unifying theory that is able to explain the various types of interactions such as hydrogen bonding, van der Waals forces and dipole-dipole interactions. Typically, this is done by applying the ideas of quantum mechanics to molecules, and Rayleigh–Schrödinger perturbation theory has been especially effective in this regard. When applied to existing quantum chemistry methods, such a quantum mechanical explanation of intermolecular interactions, this provides an array of approximate methods that can be used to analyze intermolecular interactions.
↑ H. Margenau, N Kestner, Theory of intermolecular forces, International Series of Monographs in Natural Philosophy, Pergamon Press
↑ Dr. Michael Blaber, 1996. Intermolecular Forces. http://www.mikeblaber.org/oldwine/chm1045/notes/Forces/Intermol/Forces02.htm
↑ “Theoretical Models for Surface Forces and Adhesion and Their Measurement Using Atomic Force Microscopy”, Fabio L. Leite, Carolina C. Bueno, Alessandra L. Da Róz, Ervino C. Ziemath and Osvaldo N. Oliveira Jr., Int. J. Mol. Sci. 2012, 13 12777 (p.1)
↑ Keesom, W.H. The second virial coefficient for rigid spherical molecules whose mutual attraction is equivalent to that of a quadruplet placed at its center. Proc. R. Acad. Sci. 1915, 18, 636–646
↑ 8.0 8.1 Blustin PH, 1978. A Floating Gaussian Orbital calculation on argon hydrochloride (Ar • HCl). Theoret. Chim. Acta 47, 249–257.
↑ Nannoolal Y, 2006. Development and critical evaluation of group contribution methods for the estimation of critical properties, liquid vapour pressure and liquid viscosity of organic compounds. University of Kwazulu-Natal PhD Thesis.
↑ 10.0 10.1 Roberts JK and Orr WJC, 1938. Induced dipoles and the heat of adsorption of argon on ionic crystals. Trans. Faraday Soc. 34, 1346–1349.
↑ Sapse AM, Rayez-Meaume MT, Rayez JC and Massa LJ, 1979. Ion-induced dipole H-n clusters. Nature 278, 332–333.
↑ “Theoretical Models for Surface Forces and Adhesion and Their Measurement Using Atomic Force Microscopy”, Fabio L. Leite, Carolina C. Bueno, Alessandra L. Da Róz, Ervino C. Ziemath and Osvaldo N. Oliveira Jr., Int. J. Mol. Sci. 2012, 13 12777 (p.3-4)
↑ Organic Chemistry: Structure and Reactivity by Seyhan Ege, pp.30–33, 67
↑ Majer, V. and Svoboda, V., Enthalpies of Vaporization of Organic Compounds, Blackwell Scientific Publications, Oxford, 1985.}
Software for calculation of intermolecular forces
SAPT: An ab initio quantumchemical package
Template:Chemical bonds
zh:分子间作用力
Retrieved from "https://en.formulasearchengine.com/index.php?title=Intermolecular_force&oldid=284445"
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Introduction to Chemical Engineering Processes/Mathematical Methods Practice Problems - Wikibooks, open books for an open world
1. In enzyme kinetics, one common form of a rate law is Michaelis-Menten kinetics, which is of the form:
{\displaystyle -r_{S}={\frac {V_{max}*[S]}{K_{m}+[S]}}}
{\displaystyle V_{max}}
{\displaystyle K_{m}}
a. Write this equation in a linearized form. What should you plot to get a line? What will the slope be? How about the y-intercept?
b. Given the following data and the linearized form of the equation, predict the values of
{\displaystyle V_{max}}
{\displaystyle K_{m}}
[S], M rS, M/s
0.02 0.0006
Also, calculate the R value and comment on how good the fit is.
c. Plot the rate expression in its nonlinear form with the parameters from part b. What might
{\displaystyle V_{m}ax}
d. Find the value of -rS when [S] is 1.0 M in three ways:
Plug 1.0 into your expression for -rS with the best-fit parameters.
Perform a linear interpolation between the appropriate points nearby.
Perform a linear extrapolation from the line between points (0.5, 0.0030) and (0.8, 0.0036).
Which is probably the most accurate? Why?
2. Find the standard deviation of the following set of arbitrary data. Write the data in
{\displaystyle \mu \pm \sigma }
form. Are the data very precise?
1.01 1.00 0.86 0.93 0.95
1.1 1.04 1.02 1.08 1.12
Which data points are most likely to be erroneous? How can you tell?
3. Solve the following equations for x using one of the rootfinding methods discussed earlier. Note that some equations have multiple real solutions (the number of solutions is written next to the equation)
{\displaystyle x^{2}-14x+15=0}
(2 solutions). Use the quadratic formula to check your technique before moving on to the next problems.
{\displaystyle x^{2}-14x+15-ln(x)=0}
(1 solution)
{\displaystyle e^{3x}=-x}
{\displaystyle {\frac {x}{2x^{2}-3}}-{\frac {2x^{3}-x^{2}}{2x-x^{2}}}=10}
(2 solutions)
Retrieved from "https://en.wikibooks.org/w/index.php?title=Introduction_to_Chemical_Engineering_Processes/Mathematical_Methods_Practice_Problems&oldid=3325794"
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(Redirected from Complementary angle)
{\displaystyle {\widehat {\rm {BAC}}}}
{\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\[3pt]&\tan A=\cot B&&\sec A=\csc B\end{aligned}}}
{\displaystyle \theta ={\frac {k}{2\pi }}\cdot {\frac {s}{r}}.}
{\displaystyle m\angle \mathrm {AOC} =m\angle \mathrm {AOB} +m\angle \mathrm {BOC} }
{\displaystyle \mathbf {u} \cdot \mathbf {v} =\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}
{\displaystyle \langle \cdot ,\cdot \rangle }
{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}
{\displaystyle \operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}
{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}
{\displaystyle \operatorname {span} (\mathbf {u} )}
{\displaystyle \operatorname {span} (\mathbf {v} )}
{\displaystyle \mathbf {u} }
{\displaystyle \mathbf {v} }
{\displaystyle \operatorname {span} (\mathbf {u} )}
{\displaystyle \operatorname {span} (\mathbf {v} )}
{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|}
{\displaystyle {\mathcal {U}}}
{\displaystyle {\mathcal {W}}}
{\displaystyle \dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l}
{\displaystyle k}
{\displaystyle \cos \theta ={\frac {g_{ij}U^{i}V^{j}}{\sqrt {\left|g_{ij}U^{i}U^{j}\right|\left|g_{ij}V^{i}V^{j}\right|}}}.}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Angle&oldid=1088898411#complementary_angle"
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The Dasht-e Bayāz (Iran) earthquake of August 31, 1968: A field report
N. N. Ambraseys; J. S. Tchalenko
An investigation of the Dasht-e Bayāz, Iran earthquake of August 31, 1968
Kenneth C. Bayer; Lorne E. Heuckroth; Rajab A. Karim
Aftershocks of the Dasht-e Bayāz, Iran, earthquake of August, 1968
Source dynamics of the Dasht-e Bayāz earthquake of August 31, 1968
Mansour Niazi
Array data processing techniques applied to long-period shear waves at Fennoscandian seismograph stations
On the propagation of SH waves in a heterogeneous sphere
Dispersion of Rayleigh waves for purely oceanic paths in the Pacific
Rodolfo Piermattei; Ali A. Nowroozi
The fairweather fault ten years after the southeast Alaska earthquake of 1958
A dislocation model for the Fairview Peak, Nevada, earthquake
J. C. Savage; L. M. Hastie
Upper Mantle velocity structure in the Hindukush region from travel time studies of deep earthquakes using a new analytical method
Earthquake magnitude and source parameters
Michael A. Chinnery
Shear wave velocities in the lower mantle
J. W. Fairborn
Seismicity off the coast of Northern California determined from ocean bottom seismic measurements
Bruce Auld; Gary Latham; Ali Nowroozi; Leonardo Seeber
Rayleigh waves in Southern New Guinea: II. A shear velocity profile
Sarva Jit Singh; Ari Ben-Menahem
C. G. Bufe; D. E. Willis
Effects of thin soft layers on body waves
Tom Landers; Jon F. Claerbout
Minutes of the meeting of the Board of Directors of the Seismological Society of America: April 2, 1969
Minutes of the annual meeting of the Seismological Society of America: April 1, 2, and 3, 1969
Report of the Secretary and Actions of the Executive Committee: For the period April 11, 1968 to March 31, 1969
Bulletin of the Seismological Society of America October 01, 1969, Vol.59, 2131. doi:https://doi.org/10.1785/BSSA0590052131
Seismological Society of America members: August 1, 1969
A note on the response of the pendulum seismometer to plane wave rotation
P. W. Rodgers
Seismicity of the central appalachian states of Virginia, West Virginia, and Maryland—1758 through 1968
Mw
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20.2: Bayes’ Theorem and Inverse Inference - Statistics LibreTexts
https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_Statistical_Thinking_for_the_21st_Century_(Poldrack)%2F20%253A_Bayesian_Statistics%2F20.02%253A_Bayes%25E2%2580%2599_Theorem_and_Inverse_Inference
The reason that Bayesian statistics has its name is because it takes advantage of Bayes’ theorem to make inferences from data about the underlying process that generated the data. Let’s say that we want to know whether a coin is fair. To test this, we flip the coin 10 times and come up with 7 heads. Before this test we were pretty sure that the
P_{heads}=0.5
), but finding 7 heads out of 10 flips would certainly give us pause if we believed that
P_{heads}=0.5
. We already know how to compute the conditional probability that we would flip 7 or more heads out of 10 if the coin is really fair (
P(n\ge7|p_{heads}=0.5)
), using the binomial distribution.
TBD: MOTIVATE SWITCH FROM 7 To 7 OR MORE
The resulting probability is 0.055. That is a fairly small number, but this number doesn’t really answer the question that we are asking – it is telling us about the likelihood of 7 or more heads given some particular probability of heads, whereas what we really want to know is the probability of heads. This should sound familiar, as it’s exactly the situation that we were in with null hypothesis testing, which told us about the likelihood of data rather than the likelihood of hypotheses.
P(H|D) = \frac{P(D|H)*P(H)}{P(D)}
prior (
P(Hypothesis)
): Our degree of belief about hypothesis H before seeing the data D
likelihood (
P(Data|Hypothesis)
): How likely are the observed data D under hypothesis H?
marginal likelihood (
P(Data)
): How likely are the observed data, combining over all possible hypotheses?
posterior (
P(Hypothesis|Data)
): Our updated belief about hypothesis H, given the data D
In the case of our coin-flipping example: - prior (
P
): Our degree of belief the likelhood of flipping heads, which was
P_{heads}=0.5
- likelihood (
P(\text{7 or more heads out of 10 flips}|P_{heads}=0.5)
): How likely are 7 or more heads out of 10 flips if
P_{heads}=0.5)
? - marginal likelihood (
P(\text{7 or more heads out of 10 flips})
): How likely are we to observe 7 heads out of 10 coin flips, in general? - posterior (
P_{heads}|\text{7 or more heads out of 10 coin flips})
): Our updated belief about
P
given the observed coin flips
Here we see one of the primary differences between frequentist and Bayesian statsistics. Frequentists do not believe in the idea of a probability of a hypothesis (i.e., our degree of belief about a hypothesis) – for them, a hypothesis is either true or it isn’t. Another way to say this is that for the frequentist, the hypothesis is fixed and the data are random, which is why frequentist inference focuses on describing the probability of data given a hypothesis (i.e. the p-value). Bayesians, on the other hand, are comfortable making probability statements about both data and hypotheses.
20.2: Bayes’ Theorem and Inverse Inference is shared under a not declared license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.
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Shiku Being Shiku | Toph
Shiku Being Shiku
Guni Moira, the honest sweet shop owner of CSEmpur has recently learned about the Fibonacci sequence, and he’s already a fan <3. Now if a customer buys sweet of price equal to a Fibonacci number, Guni becomes happy and gives the sweets for free.
Shiku wants to fool Guni Moira and buy as many sweets as he wants for free. So, whatever the price is, Shiku claims that it’s a Fibonacci number (even he, himself doesn’t know 🤦). Shiku is bad. Don’t be like Shiku .
Guni doesn’t know how to verify if Shiku is wrong. So, he is asking for your help.
The first line of the input contains T (1 ≤ T ≤104), the number of test cases. Then next T lines contain an integer N(0≤ N ≤ 105), the price to verify.
For every test case print “YES” (without quotes) in one line if N is Fibonacci number
otherwise print “NO” (without quotes) in one line.
Note: In mathematics, the Fibonacci numbers, commonly denoted
F_n
Fn, form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,
F_0 = 0 , F_1 = 1
F0=0,F1=1
F_n = F_{n-1} + F_{n-2}
Fn=Fn−1+Fn−2 for
n > 1
The first few numbers of these sequence are
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \cdots
0,1,1,2,3,5,8,13,21,34,55,⋯
InxtinctEarliest, Feb '20
SyedaSohiFastest, 0.0s
InxtinctLightest, 131 kB
BUET CSEmpur 18 Programming Contest
Replay of BUET CSEmpur 18 Programming Contest
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WordUse - Maple Help
Home : Support : Online Help : Education : EssayTools : WordUse
find occurrences of individual words in one or more essays
WordUse( essays, options )
showcount = truefalse
mincount = posint
maxcount = posint
The WordUse command breaks the given essays into unique instances of the words contained therein. All words are reduced to their lower-case equivalent.
When no *count option is specified the order of entries in the returned list is unspecified.
If the option showcount = true is specified then the words are returned in equation form indicating the lower-case word followed by the number of occurrences of that word. The returned list is sorted in order of most occurrences first.
If the option mincount = N is specified then only the words that occur at least N times are returned. The returned list is sorted in order of most occurrences first.
If the option maxcount = N is specified then only the words that occur no more than N times are returned. The returned list is sorted in order of most occurrences first.
This function is part of the EssayTools package, so it can be used in the short form WordUse(..) only after executing the command with(EssayTools). However, it can always be accessed through the long form of the command by using EssayTools[WordUse](..).
\mathrm{with}\left(\mathrm{EssayTools}\right):
Find the unique occurrences of words in a list of essays
\mathrm{Essays}≔["I like to use i and j.","And x and y too.","But I don\text{'}t like k and z."]:
\mathrm{sort}\left(\mathrm{WordUse}\left(\mathrm{Essays}\right)\right)
[\textcolor[rgb]{0,0,1}{"and"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"but"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"don\text{'}t"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"i"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"j"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"k"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"like"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"to"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"too"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"use"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"x"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"y"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"z"}]
Find all words used at least 2 times
\mathrm{WordUse}\left(\mathrm{Essays},'\mathrm{mincount}'=2\right)
[\textcolor[rgb]{0,0,1}{"and"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"i"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"like"}]
Find all words used at least 2 times and show how many times each was used
\mathrm{WordUse}\left(\mathrm{Essays},'\mathrm{mincount}'=2,'\mathrm{showcount}'\right)
[\textcolor[rgb]{0,0,1}{"and"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"i"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"like"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{2}]
\mathrm{Hemingway}≔"Nothing happened. The fish just moved away slowly and the old man could not raise him an inch. His line was strong and made for heavy fish and he held it against his hack until it was so taut that beads of water were jumping from it. Then it began to make a slow hissing sound in the water and he still held it, bracing himself against the thwart and leaning back against the pull. The boat began to move slowly off toward the north-west.":
\mathrm{WordUse}\left(\mathrm{Hemingway},'\mathrm{showcount}'=\mathrm{true},\mathrm{mincount}=3\right)
[\textcolor[rgb]{0,0,1}{"the"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"and"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"it"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"against"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{3}]
The EssayTools[WordUse] command was introduced in Maple 17.
EssayTools[CountUseOfAllWords]
EssayTools[CountUseOfEachWord]
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(Redirected from Μ)
Mu /ˈm(j)uː/[1][2] (uppercase Μ, lowercase μ; Ancient Greek μῦ [mŷː], Greek: μι or μυ—both [mi]) is the 12th letter of the Greek alphabet. In the system of Greek numerals it has a value of 40.[3] Mu was derived from the Egyptian hieroglyphic symbol for water, which had been simplified by the Phoenicians and named after their word for water, to become 𐤌img (mem). Letters that derive from mu include the Roman M and the Cyrillic М.
Greek letter mu
4 Image list for readers with font problems
In Ancient Greek, the name of the letter was written μῦ and pronounced [mŷː]
In Modern Greek, the letter is spelled μι and pronounced [mi]. In polytonic orthography, it is written with an acute accent: μί.[4][5]
Use as symbol[edit]
The lowercase letter mu (μ) is used as a special symbol in many academic fields. Uppercase mu is not used, because it appears identical to Latin M.
the SI prefix micro-, which represents one millionth, or 10−6. Lowercase letter "u" is often substituted for "μ" when the Greek character is not typographically available; for example the unit "microfarad", correctly "μF", is often rendered as "uF" or "ufarad" in technical documents.[6]
the micron "μ", an old unit now named the micrometre and denoted "μm"
"μ" is conventionally used to denote certain things; however, any Greek letter or other symbol may be used freely as a variable name.
minimalization in computability theory and Recursion theory
the integrating factor in ordinary differential equations
the degree of membership in a fuzzy set
the Ramanujan–Soldner constant
the coefficient of friction (also used in aviation as braking coefficient)
Standard gravitational parameter in celestial mechanics
linear density, or mass per unit length, in strings and other one-dimensional objects
the magnetic dipole moment of a current-carrying coil
dynamic viscosity in fluid mechanics
the amplification factor or voltage gain of a triode vacuum tube[7]
the electrical mobility of a charged particle
the rotor advance ratio, the ratio of aircraft airspeed to rotor-tip speed in rotorcraft[8][9]
the pore water pressure in saturated soil
the elementary particles called the muon and antimuon
the proton-to-electron mass ratio
the chemical potential of a system or component of a system
In evolutionary algorithms:
μ, population size from which in each generation λ offspring will generate (the terms μ and λ originate from evolution strategy notation)
Used to introduce a recursive data type. For example,
{\displaystyle {\text{list}}(\tau )=\mu {}\alpha {}.1+\tau {}\alpha }
is the type of lists with elements of type
{\displaystyle \tau }
(a type variable): a sum of unit, representing nil, with a pair of a
{\displaystyle \tau }
and another
{\displaystyle {\text{list}}(\tau )}
(represented by
{\displaystyle \alpha }
). In this notation,
{\displaystyle \mu }
is a binding form, where the variable (
{\displaystyle \alpha }
) introduced by
{\displaystyle \mu }
is bound within the following term (
{\displaystyle 1+\tau {}\alpha }
) to the term itself. Via substitution and arithmetic, the type expands to
{\displaystyle 1+\tau +\tau ^{2}+\tau ^{3}+\cdots }
, an infinite sum of ever-increasing products of
{\displaystyle \tau }
(that is, a
{\displaystyle \tau {}{\text{ list}}}
{\displaystyle k}
-tuple of values of type
{\displaystyle \tau }
{\displaystyle k\geq 0}
). Another way to express the same type is
{\displaystyle {\text{list}}(\tau )=1+\tau {}{\text{list}}(\tau )}
the prefix given in IUPAC nomenclature for a bridging ligand
the mutation rate in population genetics
In pharmacology:
an important opiate receptor
Orbital mechanics[edit]
In orbital mechanics:
Standard gravitational parameter of a celestial body, the product of the gravitational constant G and the mass M
planetary discriminant, represents an experimental measure of the actual degree of cleanliness of the orbital zone, a criterion for defining a planet. The value of μ is calculated by dividing the mass of the candidate body by the total mass of the other objects that share its orbital zone.
Mu chord
Electronic musician Mike Paradinas runs the label Planet Mu which utilizes the letter as its logo, and releases music under the pseudonym μ-Ziq, pronounced "music"
Used as the name of the school idol group μ's, pronounced "muse", consisting of nine singing idols in the anime Love Live! School Idol Project
Official fandom name of Kpop group f(x), appearing as either MeU or 'μ'
Hip-hop artist Muonboy has taken inspiration from the particle for his stage name and his first EP named Mu uses the letter as its title.
The Olympus Corporation manufactures a series of digital cameras called Olympus μ [mju:][10] (known as Olympus Stylus in North America)
In syntax:
μP (mu phrase) can be used as the name for a functional projection.[11]
In Celtic linguistics:
/μ/ can represent an Old Irish nasalized labial fricative of uncertain articulation, the ancestor of the sound represented by Modern Irish mh.
Greek Mu / Coptic Mu[12]
GREEK CAPITAL LETTER MU GREEK SMALL LETTER MU MICRO SIGN COPTIC CAPITAL LETTER MI COPTIC SMALL LETTER MI
Unicode 924 U+039C 956 U+03BC 181 U+00B5 11416 U+2C98 11417 U+2C99
UTF-8 206 156 CE 9C 206 188 CE BC 194 181 C2 B5 226 178 152 E2 B2 98 226 178 153 E2 B2 99
Numeric character reference Μ Μ μ μ µ µ Ⲙ Ⲙ ⲙ ⲙ
Named character reference Μ μ µ
ISO/IEC 8859-1 181 B5
ISO/IEC 8859-7 204 CC 236 EC
Code page 437, 850 230 E6 230 E6
Code page 737 139 8B 163 A3
Code page 851, 869 183 B7 230 E6
Code page 1253 204 CC 236 EC
Roman-8, Roman-9 243 F3 243 F3
TeX \mu \micro
Mathematical Mu
CAPITAL MU MATHEMATICAL BOLD
SMALL MU MATHEMATICAL ITALIC
CAPITAL MU MATHEMATICAL ITALIC
SMALL MU MATHEMATICAL BOLD ITALIC
CAPITAL MU MATHEMATICAL BOLD ITALIC
SMALL MU
Unicode 120499 U+1D6B3 120525 U+1D6CD 120557 U+1D6ED 120583 U+1D707 120615 U+1D727 120641 U+1D741
UTF-8 240 157 154 179 F0 9D 9A B3 240 157 155 141 F0 9D 9B 8D 240 157 155 173 F0 9D 9B AD 240 157 156 135 F0 9D 9C 87 240 157 156 167 F0 9D 9C A7 240 157 157 129 F0 9D 9D 81
UTF-16 55349 57011 D835 DEB3 55349 57037 D835 DECD 55349 57069 D835 DEED 55349 57095 D835 DF07 55349 57127 D835 DF27 55349 57153 D835 DF41
Numeric character reference 𝚳 𝚳 𝛍 𝛍 𝛭 𝛭 𝜇 𝜇 𝜧 𝜧 𝝁 𝝁
BOLD CAPITAL MU MATHEMATICAL SANS-SERIF
BOLD SMALL MU MATHEMATICAL SANS-SERIF
BOLD ITALIC CAPITAL MU MATHEMATICAL SANS-SERIF
BOLD ITALIC SMALL MU
Unicode 120673 U+1D761 120699 U+1D77B 120731 U+1D79B 120757 U+1D7B5
UTF-8 240 157 157 161 F0 9D 9D A1 240 157 157 187 F0 9D 9D BB 240 157 158 155 F0 9D 9E 9B 240 157 158 181 F0 9D 9E B5
UTF-16 55349 57185 D835 DF61 55349 57211 D835 DF7B 55349 57243 D835 DF9B 55349 57269 D835 DFB5
Numeric character reference 𝝡 𝝡 𝝻 𝝻 𝞛 𝞛 𝞵 𝞵
Image list for readers with font problems[edit]
Look up Μ or μ in Wiktionary, the free dictionary.
Fraser alphabet#Consonants
^ "mu". The Chambers Dictionary (9th ed.). Chambers. 2003. ISBN 0-550-10105-5.
^ "mu". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
^ Hadley, James (1884). A Greek Grammar for Schools and Colleges. New York: American Book. p. 79.
^ Neoelliniki Grammatiki (Tis Dimotikis).
^ Grammatiki tis Dimotikis Glossas.
^ Albert Flack (19 April 2010). "US20130038341A1 - Contactor health monitor circuit and method". Google Patents. Retrieved 10 September 2018. Example of document using both "ufarad" and "microFarad"
^ Ballou, Glen (1987). Handbook for Sound Engineers: The New Audio Cyclopedia (1 ed.). Howard W. Sams Co. p. 250. ISBN 0-672-21983-2. Amplification factor or voltage gain is the amount the signal at the control grid is increased in amplitude after passing through the tube, which is also referred to as the Greek letter μ (mu) or voltage gain (Vg) of the tube.
^ "Nomenclature" NASA
^ "Olympus History : μ[mju:] (Stylus) Series".
^ Johnson, Kyle (1991). "Object Positions". Natural Language and Linguistic Theory. 9 (4): 577–636. doi:10.1007/BF00134751. S2CID 189901613.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Mu_(letter)&oldid=1088127617"
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Generate C Code from Symbolic Expressions Using the MATLAB Coder App - MATLAB & Simulink
Generate Deployable MATLAB Function from Symbolic Expression
Run MATLAB Test Script
Generate C Code from MATLAB Function
This example shows how to use the MATLAB® Coder™ app to generate a static C library from symbolic expressions. First, you work with symbolic expressions in Symbolic Math Toolbox™, and convert the symbolic expressions into a deployable MATLAB function using matlabFunction. Next, you generate C code from the MATLAB function. The generated C code accepts inputs that have a fixed, preassigned size, but you can also specify variable-size inputs during code generation.
This example follows the steps described in Generate C Code by Using the MATLAB Coder App (MATLAB Coder), but updates the steps to generate a MATLAB function from a symbolic expression. Alternatively, you can generate C code from a MATLAB function at the MATLAB command line by using the codegen (MATLAB Coder) command. For a tutorial on this workflow, see Generate C Code at the Command Line (MATLAB Coder).
Note that the MATLAB Coder app is not supported in MATLAB Online™. To generate C/C++ code in MATLAB Online, use the codegen (MATLAB Coder) command.
This example solves for the eigenvalues of the model Hamiltonian:
\mathit{H}=\left(\begin{array}{cc}{\left(q-1\right)}^{2}-\frac{\delta }{2}& \Omega \\ \Omega & \frac{\delta }{2}+{\left(q+1\right)}^{2}\end{array}\right),
q
\Omega
\delta
are the parameters of the Hamiltonian.
Create the symbolic variables q, Omega, and delta to represent the parameters of the Hamiltonian. Create a symbolic matrix for the Hamiltonian.
syms q Omega delta
H = [(q-1)^2 - delta/2, Omega; Omega, (q+1)^2 + delta/2]
\left(\begin{array}{cc}{\left(q-1\right)}^{2}-\frac{\delta }{2}& \Omega \\ \Omega & \frac{\delta }{2}+{\left(q+1\right)}^{2}\end{array}\right)
Find the two eigenvalues of the Hamiltonian.
E = eig(H)
\left(\begin{array}{c}{q}^{2}-\frac{\sqrt{4 {\Omega }^{2}+{\delta }^{2}+8 \delta q+16 {q}^{2}}}{2}+1\\ {q}^{2}+\frac{\sqrt{4 {\Omega }^{2}+{\delta }^{2}+8 \delta q+16 {q}^{2}}}{2}+1\end{array}\right)
Next, convert the two eigenvalues E(1) and E(2) to a MATLAB function file by using matlabFunction. Write the resulting function, which returns two elements E1 and E2, to the file myEigenvalues.m. Specify the order of input arguments as [q Omega delta].
matlabFunction(E(1),E(2),'File','myEigenvalues', ...
'Vars',[q Omega delta],'Outputs',{'E1','E2'});
The converted function in the file myEigenvalues.m can be used without Symbolic Math Toolbox. The MATLAB file myEigenvalues.m contains the function myEigenvalues that implements the core algorithm in this example. The function takes q, Omega, and delta as inputs, all of which must be either the same size or a scalar. It then calculates the two eigenvalues as a function of these inputs.
type myEigenvalues
function [E1,E2] = myEigenvalues(q,Omega,delta)
%myEigenvalues
% [E1,E2] = myEigenvalues(Q,Omega,DELTA)
% 26-Feb-2022 17:55:42
t2 = Omega.^2;
t3 = delta.^2;
t4 = q.^2;
t6 = delta.*q.*8.0;
t5 = t2.*4.0;
t7 = t4.*1.6e+1;
t8 = t3+t5+t6+t7;
t9 = sqrt(t8);
t10 = t9./2.0;
E1 = t4-t10+1.0;
E2 = t4+t10+1.0;
To calculate the eigenvalues for a set of inputs, create and run the test script myTest.m in MATLAB. The test script specifies the inputs with the following sizes:
qGrid is a 128-by-256 matrix that represents points in the two-dimensional (
q
\Omega
) space.
OmegaGrid is a 128-by-256 matrix that represents points in the two-dimensional (
q
\Omega
delta is a scalar.
The script then calls the function myEigenvalues.m to compute the eigenvalues. The output displays a plot of the eigenvalues for these input values. Below is the content of the script myTest.m.
q = linspace(-2,2,256);
Omega = linspace(0,2,128);
[qGrid,OmegaGrid] = meshgrid(q,Omega);
[E1,E2] = myEigenvalues(qGrid,OmegaGrid,delta);
surf(q,Omega,E1)
To make your MATLAB code suitable for code generation, use the Code Analyzer and the Code Generation Readiness Tool. The Code Analyzer in the MATLAB Editor continuously checks your code as you enter it. It reports issues and recommends modifications to improve performance and maintainability. The Code Generation Readiness Tool screens the MATLAB code for features and functions that are not supported for code generation.
Make MATLAB Code Suitable for Code Generation
Open myEigenvalues.m in the MATLAB Editor. After the function declaration, add the %#codegen directive:
The Code Analyzer message indicator in the top right corner of the MATLAB Editor is green. The analyzer did not detect errors, warnings, or opportunities for improvement in the code. For more information about using the Code Analyzer, see Check Code for Errors and Warnings Using the Code Analyzer.
Save the file. You are now ready to compile your code by using the MATLAB Coder app. Here, compilation refers to the generation of C/C++ code from your MATLAB code.
Open MATLAB Coder App and Select Source Files
In the Select Source Files page, enter or select the name of the entry-point function myEigenvalues. An entry-point function is a top-level MATLAB function from which you generate code. The app creates a project with the default name myEigenvalues.prj in the current folder.
Click Next to go to the Define Input Types step. The app runs the Code Analyzer, which you already ran in the previous step, and the Code Generation Readiness Tool on the entry-point function. If the app identifies issues, it opens the Review Code Generation Readiness page where you can review and fix issues. In this example, because the app does not detect issues, it opens the Define Input Types page. For more information, see Code Generation Readiness Tool (MATLAB Coder).
Note that the Code Analyzer and the Code Generation Readiness Tool might not detect all code generation issues. After eliminating the errors or warnings that these two tools detect, generate code with MATLAB Coder to determine if your MATLAB code has other compliance issues.
Certain MATLAB built-in functions and toolbox functions, classes, and System objects that are supported for C/C++ code generation have specific code generation limitations. These limitations and related usage notes are listed in the Extended Capabilities sections of their corresponding reference pages. For more information, see Functions and Objects Supported for C/C++ Code Generation (MATLAB Coder).
In this example, to define the properties of the inputs q, delta, and Omega, specify the test file myTest.m for the code generator to use to define types automatically:
Enter or select the test file myTest.m in the MATLAB prompt.
Click Autodefine Input Types.The test file, myTest.m, calls the entry-point function, myEigenvalues, with the expected input types. The app determines that the input q is double(128 x 256), the input Omega is double(128 x 256), and the input delta is double(1 x 1).
The Check for Run-Time Issues step generates a MEX function from your entry-point functions, runs the MEX function, and reports issues. A MEX function is generated code that can be called from inside MATLAB. Performing this step is a best practice because you can detect and fix run-time errors that are harder to diagnose in the generated C code. By default, the MEX function includes memory integrity checks. These checks perform array bounds and dimension checking. The checks detect violations of memory integrity in code generated for MATLAB functions. For more information, see Control Run-Time Checks (MATLAB Coder).
To convert MATLAB code to efficient C/C++ source code, the code generator introduces optimizations that, in certain situations, cause the generated code to behave differently than the original source code. See Differences Between Generated Code and MATLAB Code (MATLAB Coder).
To open the Check for Run-Time Issues dialog box (if the dialog box does not automatically appear), click the Check for Issues arrow .
In the Check for Run-Time Issues dialog box, specify a test file or enter code that calls the entry-point function with example inputs. For this example, use the test file myTest that you used to define the input types.
Click Check for Issues. The app generates a MEX function that can be run inside MATLAB. This step runs the test script myTest replacing calls to myEigenvalues with calls to the generated MEX function, that is [E1,E2] = myEigenvalues_mex(qGrid,OmegaGrid,delta). The generated MEX file myEigenvalues_mex is located in the folder work\codegen\lib\myEigenvalues (on Microsoft® Windows® platforms) or work/codegen/lib/myEigenvalues (on Linux® or Mac platforms), where work is the location of myEigenvalues.m and myTest.m. If the app detects issues during the MEX function generation or execution, it provides warning and error messages. Click these messages to navigate to the problematic code and fix the issue. In this example, the app does not detect issues.
By default, the app collects line execution counts. These counts help you see how well the test file myTest.m exercised the myEigenvalues function. To view line execution counts, click View MATLAB line execution counts. The app editor displays a color-coded bar to the left of the code. To extend the color highlighting over the code and to see line execution counts, place your cursor over the bar. A particular shade of green indicates that the code only executes one call to compute the eigenvalues.
To open the Generate dialog box (if the dialog box does not automatically appear), click the Generate arrow .
In the Generate dialog box, set Build type to Static Library(.lib) and Language to C. Use the default values for the other project build configuration settings. Instead of generating a C static library, you can choose to generate a MEX function or other C/C++ build types. Different project settings are available for the MEX and C/C++ build types. When you switch between MEX and C/C++ code generation, verify the settings that you choose.
Click Generate. MATLAB Coder generates a standalone C static library, myEigenvalues, in the folder work\codegen\lib\myEigenvalues. The folder work is the location of myEigenvalues.m and myTest.m. The MATLAB Coder app indicates when code generation has succeeded. It displays the source MATLAB files and generated output files on the left side of the page. On the Variables tab, it displays information about the MATLAB source variables. On the Target Build Log tab, it displays the build log, including C/C++ compiler warnings and errors. By default, the code window displays the C source code file, myEigenvalues.c. To view a different file, click the desired file name in the Source Code or Output Files pane.
Click View Report to view the report in the Report Viewer. If the code generator detects errors or warnings during code generation, the report describes the issues and provides links to the problematic MATLAB code. For more information, see Code Generation Reports (MATLAB Coder).
Review Finish Workflow Page
The Finish Workflow page indicates that code generation has succeeded. It provides a project summary and links to generated output.
Compare Generated C Code to Original MATLAB Code
To compare your generated C code to the original MATLAB code, open the C file, myEigenvalues.c, and the myEigenvalues.m file in the MATLAB Editor.
void myEigenvalues(const double q[32768], const double Omega[32768], double
delta, double E1[32768], double E2[32768])
const double q[32768] and const double Omega[32768] corresponds to the input q and Omega in your MATLAB code. The size of q is 32768, which corresponds to the total size (128 x 256) of the example input that you used when you generated C/C++ code from your MATLAB code. The same applies to the input Omega. In this case, the generated code uses one-dimensional arrays to represent two-dimensional arrays in the MATLAB code.
The code generator preserves your function name and comments. When possible, the code generator preserves your variable names. Note that if a variable in your MATLAB code is set to a constant value, it does not appear as a variable in the generated C code. Instead, the generated C code contains the value of the variable as a literal.
The C function that you generated for myEigenvalues.m can accept only inputs that have the same size as the sample inputs that you specified during code generation. However, the input arrays to the corresponding MATLAB function can be of any size. In this part of the example, you generate C code from myEigenvalues.m that accepts variable-size inputs.
Suppose that you want the dimensions of q, Omega, and delta in the generated C code to have these properties:
The first dimension of both q and delta can vary in size up to 100.
The second dimension of q and delta can vary in size up to 400.
Omega is a scalar of size 1-by-1.
To specify these input properties using MATLAB Coder, follow these steps:
In the Define Input Types step, specify the test file myTest.m and click Autodefine Input Types as before. The test file calls the entry-point function, myEigenvalues.m, with the expected input types. The app determines that the input q is double(128 x 256), the input Omega is double(128 x 256), and the input delta is double(1 x 1). These types specify fixed-size inputs.
Click the input type specifications to edit them. You can specify variable size, up to a specified limit, by using the : prefix. For example, :100 specifies that the corresponding dimension can vary in size up to 100. Change the type for q to double(:100 x :400), for Omega to double(1 x 1), and for delta to double(:100 x :400).
You can now generate code by following the same steps as before. The function signature for the generated C code in myEigenvalues.c now reads:
void myEigenvalues(const emxArray_real_T *q, double Omega, const emxArray_real_T
*delta, emxArray_real_T *E1, emxArray_real_T *E2)
The arguments in the generated code correspond to these arguments in the original MATLAB function:
emxArray_real_T*q — the q input argument
Omega — the Omega input argument
emxArray_real_T*delta — the delta input argument
emxArray_real_T*E1 — the E1 output argument
The C code now consists of a data structure called an emxArray_real_T to represent an array whose size is unknown and unbounded at compile time. For more details, see Use C Arrays in the Generated Function Interfaces (MATLAB Coder).
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Calculate product of two quaternions - Simulink - MathWorks Switzerland
q={q}_{0}+i{q}_{1}+j{q}_{2}+k{q}_{3}
r={r}_{0}+i{r}_{1}+j{r}_{2}+k{r}_{3}.
t=q×r={t}_{0}+i{t}_{1}+j{t}_{2}+k{t}_{3},
\begin{array}{l}{t}_{0}=\left({r}_{0}{q}_{0}-{r}_{1}{q}_{1}-{r}_{2}{q}_{2}-{r}_{3}{q}_{3}\right)\\ {t}_{1}=\left({r}_{0}{q}_{1}+{r}_{1}{q}_{0}-{r}_{2}{q}_{3}+{r}_{3}{q}_{2}\right)\\ {t}_{2}=\left({r}_{0}{q}_{2}+{r}_{1}{q}_{3}+{r}_{2}{q}_{0}-{r}_{3}{q}_{1}\right)\\ {t}_{3}=\left({r}_{0}{q}_{3}-{r}_{1}{q}_{2}+{r}_{2}{q}_{1}+{r}_{3}{q}_{0}\right)\end{array}
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Process to convert NOx gases in nitrogen (N2) by reacting urea onto a catalyst in car exhaust
Selective catalytic reduction (SCR) is a means of converting nitrogen oxides, also referred to as NO
x with the aid of a catalyst into diatomic nitrogen (N
2), and water (H
2O). A reductant, typically anhydrous ammonia (NH
3), aqueous ammonia (NH
4OH), or a urea (CO(NH
2) solution, is added to a stream of flue or exhaust gas and is reacted onto a catalyst. As the reaction drives toward completion, nitrogen (N
2), and carbon dioxide (CO
2), in the case of urea use, are produced.
x using ammonia as the reducing agent was patented in the United States by the Engelhard Corporation in 1957. Development of SCR technology continued in Japan and the US in the early 1960s with research focusing on less expensive and more durable catalyst agents. The first large-scale SCR was installed by the IHI Corporation in 1978.[1]
Commercial selective catalytic reduction systems are typically found on large utility boilers, industrial boilers, and municipal solid waste boilers and have been shown to reduce NO
x by 70-95%.[1] More recent applications include diesel engines, such as those found on large ships, diesel locomotives, gas turbines, and even automobiles.
SCR systems are now the preferred method for meeting Tier 4 Final and EURO 6 diesel emissions standards for heavy trucks, and also for cars and light commercial vehicles. In many cases, emissions of NOx and PM (particulate matter) have been reduced by upwards of 90% when compared with vehicles of the early 1990s.[2]
3 Reductants
6.2 2010 EPA regulations
x reduction reaction takes place as the gases pass through the catalyst chamber. Before entering the catalyst chamber ammonia, or other reductant (such as urea), is injected and mixed with the gases. The chemical equation for a stoichiometric reaction using either anhydrous or aqueous ammonia for a selective catalytic reduction process is:
{\displaystyle {\ce {2 NO + 2 NH3 + 1/2 O2 -> 2 N2 + 3 H2O}}}
{\displaystyle {\ce {NO2 + 2 NH3 + 1/2 O2 -> 3/2 N2 + 3 H2O}}}
{\displaystyle {\ce {NO + NO2 + 2 NH3 -> 2 N2 + 3 H2O}}}
With several secondary reactions:
{\displaystyle {\ce {1/8 S8 + O2 -> SO2}}}
{\displaystyle {\ce {SO2 + 1/2 O2 -> SO3}}}
{\displaystyle {\ce {2 NH3 + SO3 + H2O -> (NH4)2SO4}}}
{\displaystyle {\ce {NH3 + SO3 + H2O -> NH4HSO4}}}
With urea, the reactions are:
{\displaystyle {\ce {3 NO + CO(NH2)2 -> 5/2 N2 + 2 H2O + CO2}}}
{\displaystyle {\ce {3 NO2 + 2 CO(NH2)2 -> 7/2 N2 + 4 H2O + 2 CO2}}}
As with ammonia, several secondary reactions also occur in the presence of sulfur:
{\displaystyle {\ce {SO3 + CO(NH2)2 + 2 H2O -> (NH4)2SO4 + CO2}}}
{\displaystyle {\ce {2 SO3 + CO(NH2)2 + 3 H2O -> 2 NH4HSO4 + CO2}}}
The ideal reaction has an optimal temperature range between 630 and 720 K (357 and 447 °C), but can operate as low as 500 K (227 °C) with longer residence times. The minimum effective temperature depends on the various fuels, gas constituents, and catalyst geometry. Other possible reductants include cyanuric acid and ammonium sulfate.[3]
SCR catalysts are made from various porous ceramic materials used as a support, such as titanium oxide, and active catalytic components are usually either oxides of base metals (such as vanadium, molybdenum and tungsten), zeolites, or various precious metals. Another catalyst based on activated carbon was also developed which is applicable for the removal of NOx at low temperatures.[4] Each catalyst component has advantages and disadvantages.
Base metal catalysts, such as vanadium and tungsten, lack high thermal durability, but are less expensive and operate very well at the temperature ranges most commonly applied in industrial and utility boiler applications. Thermal durability is particularly important for automotive SCR applications that incorporate the use of a diesel particulate filter with forced regeneration. They also have a high catalysing potential to oxidize SO
2 into SO
3, which can be extremely damaging due to its acidic properties.[5]
Zeolite catalysts have the potential to operate at substantially higher temperature than base metal catalysts; they can withstand prolonged operation at temperatures of 900 K (627 °C) and transient conditions of up to 1120 K (847 °C). Zeolites also have a lower potential for SO
2 oxidation and thus decrease the related corrosion risks.[5]
Iron- and copper-exchanged zeolite urea SCRs have been developed with approximately equal performance to that of vanadium-urea SCRs if the fraction of the NO
2 is 20% to 50% of the total NO
x.[6] The two most common catalyst geometries used today are honeycomb catalysts and plate catalysts. The honeycomb form usually consists of an extruded ceramic applied homogeneously throughout the carrier or coated on the substrate. Like the various types of catalysts, their configuration also has advantages and disadvantages. Plate-type catalysts have lower pressure drops and are less susceptible to plugging and fouling than the honeycomb types, but are much larger and more expensive. Honeycomb configurations are smaller than plate types, but have higher pressure drops and plug much more easily. A third type is corrugated, comprising only about 10% of the market in power plant applications.[1]
Reductants[edit]
Several nitrogen-bearing reductants are currently used in SCR applications including anhydrous ammonia, aqueous ammonia or dissolved urea. All those three reductants are widely available in large quantities.
Anhydrous ammonia can be stored as a liquid at approximately 10 bar in steel tanks. It is classified as an inhalation hazard, but it can be safely stored and handled if well-developed codes and standards are followed. Its advantage is that it needs no further conversion to operate within a SCR and is typically favoured by large industrial SCR operators. Aqueous ammonia must be first vaporized in order to be used, but it is substantially safer to store and transport than anhydrous ammonia. Urea is the safest to store, but requires conversion to ammonia through thermal decomposition. [7] At the end of the process, the purified exhaust gasses are sent to the boiler or condenser or other equipment, or discharged into the atmosphere.[8] in order to be used as an effective reductant.[1]
SCR systems are sensitive to contamination and plugging resulting from normal operation or abnormal events. Many SCRs are given a finite service life due to known amounts of contaminants in the untreated gas. The large majority of catalyst on the market has a porous structure and a geometry optimized for increasing its specific surface area. A clay planting pot is a good example of what SCR catalyst feels like. This porosity is what gives the catalyst the high surface area essential for reduction of NOx. However, the pores are easily plugged by fine particulates, ammonium sulfate, ammonium bisulfate (ABS), and silica compounds. Many of these contaminants can be removed while the unit is on line by ultrasonic horns or soot blowers. The unit can also be cleaned during a turnaround or by raising the exhaust temperature. Of more concern to SCR performance are poisons, which will degrade the catalyst and render it ineffective at NO
x reduction, possibly resulting in the oxidation of ammonia which will increase NO
x emissions. These poisons are halogens, alkali metals, alkaline earth metals, arsenic, phosphorus, antimony, chromium, lead, mercury, and copper.
Most SCRs require tuning to properly perform. Part of tuning involves ensuring a proper distribution of ammonia in the gas stream and uniform gas velocity through the catalyst. Without tuning, SCRs can exhibit inefficient NOx reduction along with excessive ammonia slip due to not utilizing the catalyst surface area effectively. Another facet of tuning involves determining the proper ammonia flow for all process conditions. Ammonia flow is in general controlled based on NOx measurements taken from the gas stream or preexisting performance curves from an engine manufacturer (in the case of gas turbines and reciprocating engines). Typically, all future operating conditions must be known beforehand to properly design and tune a SCR system.
Ammonia slip is an industry term for ammonia passing through the SCR unreacted. This occurs when ammonia is injected in excess, temperatures are too low for ammonia to react, or the catalyst has degraded.
Temperature is SCR's largest limitation. Engines all have a period during a start-up where exhaust temperatures are too cool for NOx reduction to occur, especially in cold climates.
In power stations, the same basic technology is employed for removal of NO
x from the flue gas of boilers used in power generation and industry. In general, the SCR unit is located between the furnace economizer and the air heater, and the ammonia is injected into the catalyst chamber through an ammonia injection grid. As in other SCR applications, the temperature of operation is critical. Ammonia slip (unreacted ammonia) is also an issue with SCR technology used in power plants.
Other issues that must be considered in using SCR for NO
x control in power plants are the formation of ammonium sulfate and ammonium bisulfate due to the sulfur content of the fuel as well as the undesirable catalyst-caused formation of SO
3 from the SO
2 in the flue gas.
A further operational difficulty in coal-fired boilers is the binding of the catalyst by fly ash from the fuel combustion. This requires the usage of sootblowers, sonic horns, and careful design of the ductwork and catalyst materials to avoid plugging by the fly ash. SCR catalysts have a typical operational lifetime of about 16,000 – 40,000 hours (1.8 – 4.5 years) in coal-fired power plants, depending on the flue gas composition, and up to 80,000 hours (9 years) in cleaner gas-fired power plants.
Poisons, sulfur compounds, and fly ash can all be removed by installing scrubbers before the SCR system to increase the life of the catalyst, though most plants' scrubbers are installed after the system for thermal energy transfer reasons.
SCR was applied to trucks by Nissan Diesel Corporation, and the first practical product "Nissan Diesel Quon" was introduced in 2004 in Japan.[9]
In 2007, the United States Environmental Protection Agency (EPA) enacted requirements to significantly reduce harmful exhaust emissions. To achieve this standard, Cummins and other diesel engine manufacturers developed an aftertreatment system that includes the use of a diesel particulate filter (DPF). As the DPF does not function with low-sulfur diesel fuel, diesel engines that conform to 2007 EPA emissions standards require ultra-low sulfur diesel fuel (ULSD) to prevent damage to the DPF. After a brief transition period, ULSD fuel became common at fuel pumps in the United States and Canada. The 2007 EPA regulations were meant to be an interim solution to allow manufacturers time to prepare for the more stringent 2010 EPA regulations, which reduced NOx levels even further.[10]
2010 EPA regulations[edit]
Hino truck and its Standardized SCR Unit which combines SCR with Diesel Particulate Active Reduction (DPR). DPR is a diesel particulate filtration system with regeneration process that uses late fuel injection to control exhaust temperature to burn off soot.[11][12]
Diesel engines manufactured after January 1, 2010 are required to meet lowered NOx standards for the US market.
All of the heavy-duty engine (Class 7-8 trucks) manufacturers except for Navistar International and Caterpillar continuing to manufacture engines after this date have chosen to use SCR. This includes Detroit Diesel (DD13, DD15, and DD16 models), Cummins (ISX, ISL9, and ISB6.7), Paccar, and Volvo/Mack. These engines require the periodic addition of diesel exhaust fluid (DEF, a urea solution) to enable the process. DEF is available in bottles and jugs from most truck stops, and a more recent development is bulk DEF dispensers near diesel fuel pumps. Caterpillar and Navistar had initially chosen to use enhanced exhaust gas recirculation (EEGR) to comply with the Environmental Protection Agency (EPA) standards, but in July 2012 Navistar announced it would be pursuing SCR technology for its engines, except on the MaxxForce 15 which was to be discontinued. Caterpillar ultimately withdrew from the on-highway engine market prior to implementation of these requirements.[13]
BMW,[14][15] Daimler AG (as BlueTEC), and Volkswagen have used SCR technology in some of their passenger diesel cars.
Catalytic converter, which also catalyzes NOx conversion but does not use urea or ammonia
Exhaust gas recirculation versus selective catalytic reduction
NOx adsorber (LNT)
^ a b c d Steam: Its Generation and Uses. Babcock & Wilcox.
^ Denton, Tom (2021). Advanced Automotive Fault Diagnosis: Automotive Technology: Vehicle Maintenance and Repair. Routledge. pp. 49–50. ISBN 9781000178388.
^ "Environmental Effects of Nitrogen Oxides". Electric Power Research Institute, 1989
^ "Archived copy". Archived from the original on 2015-12-08. Retrieved 2015-11-27. {{cite web}}: CS1 maint: archived copy as title (link) CarboTech AC GmbH
^ a b DOE presentation
^ Gieshoff, J; M. Pfeifer; A. Schafer-Sindlinger; P. Spurk; G. Garr; T. Leprince (March 2001). "Advanced Urea Scr Catalysts for Automotive Applications" (PDF). Society of Automotive Engineers. SAE Technical Paper Series. 1. doi:10.4271/2001-01-0514. Retrieved 2009-05-18.
^ Kuternowski, Filip; Staszak, Maciej; Staszak, Katarzyna (July 2020). "Modeling of Urea Decomposition in Selective Catalytic Reduction (SCR) for Systems of Diesel Exhaust Gases Aftertreatment by Finite Volume Method". Catalysts. 10 (7): 749. doi:10.3390/catal10070749.
^ Emigreen; Nox Reduction; SCR technology:
^ "尿素CSRシステム（FLENDS）" [CSR System "FLENDS"]. Society of Automotive Engineers of Japan (in Japanese). Retrieved 28 November 2021.
^ Mark Quasius (1 May 2013). "2010 EPA Emissions Standards And Diesel Exhaust Fluid". FamilyRVing. Retrieved 3 December 2021.
^ "Hino Standardized SCR Unit". Hino Motors. Archived from the original on 5 August 2014. Retrieved 30 July 2014.
^ "The DPR Future" (PDF). Hino Motors. Retrieved 30 July 2014.
^ "Caterpillar exits on-highway engine business". Today's Trucking. Jun 13, 2008. Retrieved 29 December 2017.
^ "BMW BluePerformance – AdBlue" (PDF). Archived from the original (PDF) on 2017-01-08. Retrieved 2017-01-15.
^ "BMW maintenance: AdBlue". Archived from the original on 2017-01-04. Retrieved 2017-01-15.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Selective_catalytic_reduction&oldid=1088917123"
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Dutch Auction on Chemix Pad - Chemix Ecosystem Documents
Dutch auctions are auctions in which issuers sell at declining prices. Algorand introduces the first well-known Dutch auction in the crypto world. The Dutch Auction is a powerful tool for discovering the price of a particular token. For tokens already trading on open markets like CEXs, locking parts with specific vesting terms usually have lower bids than the market price. There is still a challenge to get a consensus finance model on those forward trading assets. The Dutch Auction with enough participants demonstrates the consensus on pricing, which makes it the first choice for trading and pricing on Chemix Pad for illiquid tokens.
A graph of Dutch Auction duration and price. The Y-axis is the auction price and the X-axis is the duration.
Dutch auction rules
Parameters such as auction quantity, starting price, and reserve price are set by the auction initiator. After the auction begins, the auction price decreases linearly over time. During the auction period, the auction will end early when the for-sale quantity is sold out. Under this rule, each auctioneer pays the same price for each token, which is determined at the end of the auction.
Minimum auction amount
Then the slope formula applicable to this auction is: y = −kx + 1, then the price drop is: 0.00001042 USD/sec. Suppose Alice bids $0.5 and puts in $100, and after many bids, Bob bids $0.2 and puts in $500. At this point, the auction is over, and the final price is $0.2/MTB. Then Alice will actually get 100 ÷ 0.2 = 500 MTB and Bob will get 500 ÷ 0.2 = 2500 MTB
Calculation of the number of remaining auction tokens
As the auction price decreases, the expected number of tokens that can be obtained corresponding to the funds invested by users in the auction contract will increase, which will affect the overall remaining auctionable amount. The calculation is as follows:
Suppose the current auction price is
𝑃_{𝑐𝑢𝑟𝑟𝑒𝑛𝑡}
and the funds invested in the auction contract are
𝐶_{𝑐𝑜𝑛𝑡𝑖𝑏𝑢𝑡𝑒}
, the total number of auction tokens is
𝑄_{𝑇𝑜𝑘𝑒𝑛}
, then the current remaining amount is:
Current remaining amount = 𝑄_{𝑇𝑜𝑘𝑒𝑛}-\frac{C_{Contribute}}{P_{Current}}
Chemix Pad sets the [Minimum Expected Raise Rate]. At the end of the auction countdown, if the amount of funds invested in the auction is small, resulting in the number of auctioned tokens not meeting the value in this parameter, the auction will fail and the funds in the auction pool will be returned.
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Loop Shaping Using the Glover-McFarlane Method - MATLAB & Simulink Example
Design Objectives and Initial Compensator Design
Enforcing Stability and Robustness with ncfsyn
This example shows how to use ncfsyn to shape the open-loop response while enforcing stability and maximizing robustness. ncfsyn measures robustness in terms of the normalized coprime stability margin computed by ncfmargin.
The plant model is a lightly damped, second-order system.
P\left(s\right)=\frac{16}{{s}^{2}+0.16s+16}
A Bode plot shows the resonant peak.
P = tf(16,[1 0.16 16]);
The design objectives for the closed-loop are the following.
Insensitivity to noise, including 60dB/decade attenuation beyond 20 rad/sec
Integral action and a bandwidth of at least 0.5 rad/s
Gain crossover frequencies no larger than 7 rad/s
In loop-shaping control design, you translate these requirements into a desired shape for the open-loop gain and seek a compensator that enforces this shape. For example, a compensator consisting of a PI term in series with a high-frequency lag component achieves the desired loop shape.
bodemag(P*Kprop); grid
Unfortunately, the compensator Kprop does not stabilize the closed-loop system. Examining the closed-loop dynamics shows poles in the right half-plane.
pole(feedback(P*Kprop,1))
You can use ncfsyn to enforce stability and adequate stability margins without significantly altering the loop shape. Use the initial design Kprop as loop-shaping pre-filter. ncfsyn assumes a positive feedback control system (see ncfsyn), so flip the sign of Kprop and of the returned controller.
[K,~,gamma] = ncfsyn(P,-Kprop);
K = -K; % flip sign back
A value of the performance gamma less than 3 indicates success (modest gain degradation along with acceptable robustness margins). The new compensator K stabilizes the plant and has good stability margins.
allmargin(P*K)
With gamma approximately 2, the expect at most 20*log10(gamma) = 6dB gain reduction in the high-gain region and at most 6dB gain increase in the low-gain region. The Bode magnitude plot confirms this. Note that ncfsyn modifies the loop shape mostly around the gain crossover to achieve stability and robustness.
bodemag(Kprop,'r',K,'g',{1e-2,1e4}); grid
legend('Initial design','NCFSYN design')
title('Controller Gains')
bodemag(P*Kprop,'r',P*K,'g',{1e-3,1e2}); grid
title('Open-Loop Gains')
Figure 1: Compensator and open-loop gains.
With the ncfsyn compensator, an impulse disturbance at the plant input is damped out in a few seconds. Compare this response to the uncompensated plant response.
impulse(feedback(P,K),'b',P,'r',5);
legend('Closed loop','Open loop')
impulse(-feedback(K*P,1),'b',5)
title('Control action')
Figure 2: Response to impulse at plant input.
The closed-loop sensitivity and complementary sensitivity functions show the desired sensitivity reduction and high-frequency noise attenuation expressed in the closed-loop performance objectives.
S = feedback(1,P*K);
bodemag(S,T,{1e-2,1e2}), grid
legend('S','T')
In this example, you used the function ncfsyn to adjust a hand-shaped compensator to achieve closed-loop stability while approximately preserving the desired loop shape.
ncfsyn | ncfmargin
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Maximum Wall Stress on a Smooth Flat Plate Under Planar Jet Impingement | J. Fluids Eng. | ASME Digital Collection
Department of Mechanical Engineering, New Mexico Institute of Mining and Technology
e-mail: tie.wei@nmt.edu
Yanxing Wang,
Department of Mechanical Engineering, New Mexico State University
e-mail: yxwang@nmsu.edu
Cat Vo Tu,
Cat Vo Tu
BlueScope Steel Research
Port Kembla, New South Wales 2519,
e-mail: cat.tu.v@gmail.com
Department of Mechanical and Manufacturing Engineering, University of Calgary
Calgary, AB T2 L 1Y6,
e-mail: dhwood@ucalgary.ca
Wei, T., Wang, Y., Tu, C. V., and Wood, D. (March 2, 2022). "Maximum Wall Stress on a Smooth Flat Plate Under Planar Jet Impingement." ASME. J. Fluids Eng. August 2022; 144(8): 081302. https://doi.org/10.1115/1.4053618
This paper investigates the maximum wall shear stress value
τmax
and its location
xmax
as measured on a smooth flat plate impinged upon by a normal planar jet.
τmax
xmax
are found to be closely related to the stagnation pressure
Ps
and the half-width of the mean wall pressure profile
bpw
. The measurements were made by two different techniques: a Stanton probe and oil film interferometry. The maximum wall shear stress location
xmax
is found to be independent of the jet Reynolds number. At a small nozzle-to-plate distance
H≲6 Djet, xmax
is related to the jet slot width as
xmax≈1.1Djet
. At a large nozzle-to-plate distance
H≳6 Djet
, the maximum wall shear stress location is related to the mean wall pressure half-width as
xmax≈1.4 bpw
. A new Reynolds number, referred to as the stagnation Reynolds number, is defined as
Res=def2bpwPs/ρ/ν
, where ρ is the fluid density and ν is the kinematic viscosity. The maximum wall shear stress is found to be strongly influenced by the stagnation Reynolds number, and the dependence as measured by Stanton probes is approximated by a power law of
τmax/Ps≈0.38/Res0.38
. The solution of the laminar flow equations in the Appendix gives an alternate relation for
τmax
, which is in better agreement with the oil film interferometry measurements. Dimensional analysis is performed to gain insight into the empirical findings.
Dimensional analysis, Flat plates, Nozzles, Pressure, Reynolds number, Shear stress, Viscosity, Probes, Boundary layers
Heat and Mass Transfer in Impingement Drying
Thermo-Fluid-Dynamics of Submerged Jets Impinging at Short Nozzle-to-Plate Distance: A Review
Impingement of Plane Turbulent Jets and Their Application in Industrial Coating Control
The Flow Development and Heat Transfer Characteristics of Plane Turbulent Impinging Jets
, Report No. 3.
Characteristics of a Turbulent Slot Jet Impinging on a Plane Surface
Turbulent Impinging Jets
, Edmonton, Canada.
An Experimental Study of Fluid Mechanics and Heat Transfer in an Impinging Slot Jet Flow
J. Hydraul. Div.
Oblique Impingement of Plane Turbulent Jets
Wall Pressure and Shear Stress Measurements for Normal Jet Impingement
, Hobart, Australia, Dec. 14–18, pp.
Mean Equation Based Scaling Analysis of Fully-Developed Turbulent Channel Flow With Uniform Heat Generation
Local Distribution of Wall Static Pressure and Heat Transfer on a Smooth Flat Plate Impinged by a Slot Air Jet
Experimental and Numerical Investigation of Turbulent Heat Transfer Due to Rectangular Impinging Jets
, Tucson, AZ.
Measurements of Skin Friction and Heat Transfer Beneath an Impinging Slot Jet
Skin Friction and Fluid Dynamics of a Planar Impinging Gas Jet
Measurements of Wall Shear Stress, Wall Pressure and Fluctuations in the Stagnation Region Produced by Oblique Jet Impingement
Conference on Fluid Dynamic Measurements in the Industrial and Medical Environment
, ed., New York, pp.
, New Haven, CT/London.
, New York/London.
The Formation of a Blast Wave by a Very Intense Explosion.-II. The Atomic Explosion of 1945
Proc. R. Soc. London Ser. A. Math. Phys. Sci.
Properties of the Mean Momentum Balance in Turbulent Boundary Layer, Pipe and Channel Flows
, Amsterdam, The Netherlands/Oxford, UK/New York.
Measurements in the Vicinity of a Stagnation Point
An Outline of the Techniques Available for the Measurement of Skin Friction in Turbulent Boundary Layers
Incompressible Laminar Boundary Layers With Large Acceleration
Coustols
Laminar Instability Theory and Transition Criteria in Two and Three-Dimensional Flow
La Recherche Aerospatiale (English Ed.)
Distribution of Local Pressure and Skin Friction Around a Circular Cylinder in Cross-Flow Up to Re = 5,000,000
Numerical Simulation of the Flow Around a Circular Cylinder at High Reynolds Numbers
Measurements of Losses and Reynolds Stresses in the Secondary Flow Downstream of a Low-Speed Linear Turbine Cascade
|
prism - Maple Help
Home : Support : Online Help : Graphics : Packages : Plot Tools : prism
generate 3-D prism plot object from a 2-D polygon
prism(P, options)
prism(P, base=b, height=h, displacement=d, options)
plot POLYGONS data structure
(optional) z-coordinate of the base of the prism; defaults to 0.
(optional) height of the prism; defaults to 1.
(optional) two entry list specifying the [x,y] displacement of the top of the prism from the base; defaults to [0,0].
The prism command takes a two-dimensional polygon plot structure and creates a three-dimensional regular prism of the height specified.
The option displacement=d can be used to create an oblique prism.
The plot data object produced by the prism command can be used in a PLOT3D data structure or displayed using the plots[display] command.
\mathrm{with}\left(\mathrm{plottools}\right):
\mathrm{with}\left(\mathrm{plots}\right):
T≔\mathrm{polygon}\left([[0,0],[2,1],[1,3]]\right):
\mathrm{display}\left(\mathrm{prism}\left(T\right),\mathrm{axes}=\mathrm{normal},\mathrm{scaling}=\mathrm{constrained}\right)
P≔\mathrm{sector}\left([0,0],2,0..\frac{3\mathrm{\pi }}{4},\mathrm{color}="DarkRed"\right):
Graph of the sector
P
\mathrm{display}\left(P,\mathrm{scaling}=\mathrm{constrained}\right)
Graph of a prism with base
P
and height 0.5
\mathrm{display}\left(\mathrm{prism}\left(P,\mathrm{height}=0.5\right),\mathrm{scaling}=\mathrm{constrained}\right)
Q≔\mathrm{sector}\left([0,0],2,\frac{3\mathrm{\pi }}{4}..\frac{5\mathrm{\pi }}{4},\mathrm{color}="DarkBlue"\right):
R≔\mathrm{sector}\left([0,0],2,\frac{5\mathrm{\pi }}{4}..2\mathrm{\pi },\mathrm{color}="DarkGreen"\right):
\mathrm{display}\left([\mathrm{prism}\left(P,\mathrm{height}=2\right),\mathrm{prism}\left(Q,\mathrm{base}=0.5\right),\mathrm{prism}\left(R,\mathrm{base}=0.75,\mathrm{height}=0.5,\mathrm{displacement}=[0,-0.5]\right)],\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{display}\left(\mathrm{prism}\left(\mathrm{polygon}\left([[0,1],[0,2],[0.5,2.75],[1.25,3],[2,2.75],[2.5,2.25],[1.75,1.5],[2.5,0.75],[2,0.25],[1.25,0],[0.5,0.25]]\right),\mathrm{color}="Orchid"\right),\mathrm{scaling}=\mathrm{constrained}\right)
The plottools[prism] command was introduced in Maple 16.
|
Jian Liu, Lizhao Yan, "Multiple Solutions of Second-Order Damped Impulsive Differential Equations with Mixed Boundary Conditions", Abstract and Applied Analysis, vol. 2014, Article ID 356745, 8 pages, 2014. https://doi.org/10.1155/2014/356745
Jian Liu 1 and Lizhao Yan 2
1School of Economics and Management, Changsha University of Science and Technology, Changsha, Hunan 410004, China
2Hunan Normal University Press, Hunan Normal University, Changsha, Hunan 410081, China
We use variational methods to investigate the solutions of damped impulsive differential equations with mixed boundary conditions. The conditions for the multiplicity of solutions are established. The main results are also demonstrated with examples.
Impulsive effect exists widely in many evolution processes in which their states are changed abruptly at certain moments of time. The theory of impulsive differential systems has been developed by numerous mathematicians [1–6]. Applications of impulsive differential equations with or without delays occur in biology, medicine, mechanics, engineering, chaos theory, and so on [7–11].
In this paper, we consider the following second-order damped impulsive differential equations with mixed boundary conditions: where , , is continuous, , are continuous, and for .
The characteristic of (1) is the presence of the damped term . Most of the results concerning the existence of solutions of these equations are obtained using upper and lower solutions methods, coincidence degree theory, and fixed point theorems [12–15]. On the other hand, when there is no presence of the damped term, some researchers have used variational methods to study the existence of solutions for these problems [16–21]. However, to the best of our knowledge, there are few papers concerned with the existence of solutions for impulsive boundary value problems like problem (1) by using variational methods.
For this nonlinear damped mixed boundary problem (1), the variational structure due to the presence of the damped term is not apparent. However, inspired by the work [22, 23], we will be able to transform it into a variational formulation. In this paper, our aim is to study the existence of distinct pairs of nontrivial solutions of problem (1). Our main results extend the study made in [22, 23], in the sense that we deal with a class of problems that is not considered in those papers.
2. Preliminaries and Statements
Let , , , . We transform (1) into the following equivalent form:
Obviously, the solutions of (2) are solutions of (1).
Define the space . It is easy to see that and is a closed subset of . So is a Hilbert space with the usual inner product in .
Consider the Hilbert spaces with the inner product inducing the norm We also consider the inner product inducing the norm
Consider the problem As is well known, (7) possesses a sequence of eigenvalues with The corresponding eigenfunctions are normalized so that ; here
Now multiply (2) by and integrate on the interval : Then, a weak solution of (2) is a critical point of the following functional: where .
We say that is a classical solution of IBVP (1) if it satisfies the following conditions: satisfies the first equation of (1) a.e. on ; the limits , , exist and impulsive condition of (1) holds; satisfies the boundary condition of (1).
Lemma 1. If is a weak solution of (1), then is a classical solution of (1).
Proof. If is a weak solution of (1), then is a weak solution of (2), so holds for all ; that is, By integrating by part, we have Thus holds for all . Without loss of generality, for any and with , for every , then substituting into (14), we get Hence satisfies the first equation of (2). Therefore, by (14) we have
Next we will show that satisfies the impulsive and the boundary condition in (2). If the impulsive condition in (2) does not hold, without loss of generality, we assume that there exists such that Let ; then which contradicts (16). So satisfies the impulsive condition in (2) and (16) implies If , pick ; one has which contradicts (19), so satisfies the boundary condition. Therefore, is a solution of (1).
Lemma 2. Let . Then there exists a constant , such that where .
Proof. By Hölder inequality, for ,
Lemma 3 (see [24, Theorem 9.1]). Let be a real Banach space, with even, bounded from below, and satisfying P.S. condition. Suppose ; there is a set such that is homeomorphic to by an odd map and . Then possesses at least distinct pairs of critical points.
Theorem 4. Suppose that the following conditions hold.(H1)There exist , which is the kth eigenvalue of (7) such that (H2)There exist and such that (H3) and are odd about .(H4), , as , .Then, for , problem (1) has at least distinct pairs of solutions.
Proof. Set Consider
Next, we will verify that the solutions of problem (26) are solutions of problem (1).
In fact, let be the solution of problem (26). If , then there exists an interval such that When , by (H1), we have That is, is nondecreasing in . By and , we have That is, for any . Since , then . So, there exists a constant such that , which contradicts (27). Then . Similarly, we can prove that .
Therefore, any solution of (26) is a solution of (1). Hence to prove Theorem 4, it suffices to produce at least distinct pairs of critical points of where .
We will apply Lemma 3 to finish the proof.
By (30) and (H3), is even and .
Next, we will show that is bounded from below.
Let , . By (H1) and (H3), we have for ; thus So, we have for any . Therefore, is bounded from below.
In the following we will show that satisfies the P.S. condition. Let such that is a bounded sequence and ; then there exists such that By (32), we have So is bounded in . From the reflexivity of , we may extract a weakly convergent subsequence that, for simplicity, we call in . In the following we will verify that strongly converges to : By in , we see that uniformly converges to in . So So we obtain , as . That is, strongly converges to in , which means that satisfies the P.S. condition.
Now set , where is defined in (9). It is clear that is homeomorphic to by an odd map for any . In the following we verify that if is sufficiently small.
For any . By (H4) and (30), we have for small . Since , and the proof is complete.
Theorem 5. Suppose that the following conditions hold.(H1)There exist , which is the kth eigenvalue of (7) such that (H2) for any .(H3) and are odd about .(H4), , as , .Then, for , problem (1) has at least distinct pairs of solutions.
Proof. The proof is similar to the proof of Theorem 4, and therefore we omit it.
Theorem 6. Suppose that the following conditions hold.(H1)There exist , which is the kth eigenvalue of (7) such that (H2) and are odd about .(H3), , as , .Then, for , problem (1) has at least distinct pairs of solutions.
In fact, let . By the definitions of and , (41) is reduced to The solution of (42) satisfies . So and .
Let . By the definitions of and , (41) is reduced to The solution of (43) satisfies , . So and .
Therefore, the solutions of (41) are solutions of (1). Hence to prove Theorem 6, it suffices to produce at least distinct pairs of critical points of where .
By (H1) and (H2), we have and for ; thus So, we have for any . Therefore, is bounded from below.
In the following we will show that satisfies the P.S. condition. Let such that is a bounded sequence and ; then there exists such that By (46), we have So is bounded in . From the reflexivity of , we may extract a weakly convergent subsequence that, for simplicity, we call in . In the following we will verify that strongly converges to : By in , we see that uniformly converges to in . So So we obtain , as . That is, strongly converges to in , which means satisfies the P.S. condition.
For any , . By (H3) and (44), we have for small . Since , and the proof is complete.
To illustrate how our main results can be used in practice we present the following example.
Example 1. Let , and consider the following problem:
Compared with (1), , . Obviously (H2), (H3), and (H4) are satisfied. Let , ; then (H1) is satisfied. By Theorem 4, for , , problem (52) has at least distinct pairs of solutions.
Compared with (1), , . Obviously (H2) and (H3) are satisfied. Let ; then (H1) is satisfied. By Theorem 6, for , , problem (54) has at least distinct pairs of solutions.
This work is partially supported by the National Natural Science Foundation of China (no. 71201013) and the Innovation Platform Open Funds for Universities in Hunan Province (no. 13K059).
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Copyright © 2014 Jian Liu and Lizhao Yan. 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.
|
Interface between isothermal liquid and mechanical translational networks - MATLAB - MathWorks América Latina
Interface between isothermal liquid and mechanical translational networks
Simscape / Foundation Library / Isothermal Liquid / Elements
The Translational Mechanical Converter (IL) block models an interface between an isothermal liquid network and a mechanical rotational network. The block converts isothermal liquid pressure into mechanical force and vice versa. It can be used as a building block for linear actuators.
The converter contains a variable volume of liquid. If Model dynamic compressibility is set to On, then the pressure evolves based on the dynamic compressibility of the liquid volume. The Mechanical orientation parameter lets you specify whether an increase in pressure moves port R away from or towards port C.
Port A is the isothermal liquid conserving port associated with the converter inlet. Ports R and C are the mechanical translational conserving ports associated with the moving interface and converter casing, respectively.
The mass conservation equations in the mechanical converter volume are
\begin{array}{l}{\stackrel{˙}{m}}_{\text{A}}=\left\{\begin{array}{cc}\epsilon \text{\hspace{0.17em}}{\rho }_{I}S\text{\hspace{0.17em}}v,& \text{if}\text{\hspace{0.17em}}\text{fluid}\text{\hspace{0.17em}}\text{dynamic}\text{\hspace{0.17em}}\text{compressibility}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{off}\\ \epsilon \text{\hspace{0.17em}}{\rho }_{I}S\text{\hspace{0.17em}}v+\frac{1}{{\beta }_{I}}\frac{d{p}_{I}}{dt}{\rho }_{I}V,& \text{if}\text{\hspace{0.17em}}\text{fluid}\text{\hspace{0.17em}}\text{dynamic}\text{\hspace{0.17em}}\text{compressibility}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{on}\end{array}\\ v=\frac{dx}{dt}\\ v={v}_{R}-{v}_{C}\\ V={V}_{dead}+\epsilon Sx\end{array}
{\stackrel{˙}{m}}_{\text{A}}
is the mass flow rate into the converter through port A.
ρI is the fluid density inside the converter.
βI is the fluid bulk modulus inside the converter.
vR and vC are the translational velocities of ports R and C, respectively.
x is the displacement of the converter interface.
Vdead is the dead volume, that is, volume of liquid when the interface displacement is 0.
pI is the pressure inside the converter.
If you connect the converter to a Multibody joint, use the physical signal input port p to specify the displacement of port R relative to port C. Otherwise, the block calculates the interface displacement from relative port velocities, according to the equations above. The interface displacement is zero when the liquid volume is equal to the dead volume. Then, depending on the Mechanical orientation parameter value:
Equations used to compute the fluid mixture density and bulk modulus depend on the selected isothermal liquid model. For detailed information, see Isothermal Liquid Modeling Options.
F=\epsilon \left({p}_{\text{env}}-p\right)S,
penv is the environment pressure outside the converter.
Converter walls are perfectly rigid.
Isothermal liquid conserving port associated with the converter inlet.
Select the alignment of moving interface with respect to fluid pressure:
Pressure at A causes positive displacement of R relative to C — Increase in the fluid pressure results in a positive displacement of port R relative to port C.
Pressure at A causes negative displacement of R relative to C — Increase in the fluid pressure results in a negative displacement of port R relative to port C.
Atmospheric pressure — Use the atmospheric pressure, specified by the Isothermal Liquid Properties (IL) block connected to the circuit.
Hydraulic Actuator with Analog Position Controller
How the Foundation library can be used to model systems that span electrical, mechanical and isothermal liquid domains. In the model, a hydraulic system implemented in the isothermal liquid domain controls the mechanical load position in response to a voltage reference demand. If the reference demand is zero, then the hydraulic actuator (and load) displacement is zero, and if the reference is +5 volts, then the displacement is 100 mm.
Rotational Mechanical Converter (IL) | Translational Multibody Interface
|
Mixed H2/H∞ synthesis with regional pole placement constraints - MATLAB h2hinfsyn
\begin{array}{c}\stackrel{˙}{x}=Ax+{B}_{1}w+{B}_{2}u,\\ {z}_{\infty }={C}_{1}x+{D}_{11}w+{D}_{12}u,\\ {z}_{2}={C}_{2}x+{D}_{21}w+{D}_{22}u,\\ y={C}_{y}x+{D}_{y1}w+{D}_{y2}u.\end{array}
{W}_{1}{G}^{2}+{W}_{2}{H}^{2},
{\mathit{H}}_{2}
{\mathit{H}}_{2}
{\mathit{H}}_{\infty }
\left\{z:L+zM+\overline{z}{M}^{T}<0\right\}.
|
C++ dynamic array implementation - PhotoLens
As a C++ beginner coming from Java, I have become increasingly confused on the topic of memory management and how to avoid memory leaks. Is the code below risking a memory leak that I’m not currently aware of? Any help or constructive feedback would be greatly appreciated.
T *m_arr;
int m_length; //amount of elements currently being stored in the array
int m_capacity; //actual size of the array
T get(int index); //O(1)
void add(T obj); //no need to push any objects forward, O(1)
void insert(int index, T obj); //pushes forward all objects in front of the given index, then sets the obj at the given index, O(n)
void set(int index, T obj); //sets the given index of m_arr as obj, O(1)
void remove(int index); //removes the object at the given index and pushes all the array contents back, O(n)
int size(); //O(1)
DynamicArray<T>::DynamicArray() : m_arr(new T[1]), m_length(0), m_capacity(1) {}
DynamicArray<T>::~DynamicArray() {
delete[] m_arr;
T DynamicArray<T>::get(int index) {
if (index < m_length && index >= 0)
else throw ("Index out of bounds!");
void DynamicArray<T>::set(int index, T obj) {
if (index < m_length && index >= 0) {
m_arr[index] = obj;
} else throw ("Index out of bounds!");
void DynamicArray<T>::add(T obj) {
if (m_length == m_capacity) {
T *new_arr = new T[m_length * 2];
for (int i = 0; i < m_length; i++) {
new_arr[i] = m_arr[i];
m_arr = new_arr;
m_arr[m_length] = obj;
void DynamicArray<T>::insert(int index, T obj) {
if (m_length == m_capacity) size = m_length * 2;
else size = m_capacity;
T *new_arr = new T[size];
for (int i = 0, j = 0; i < m_length; i++, j++) {
new_arr[j] = obj;
new_arr[j] = m_arr[i];
void DynamicArray<T>::remove(int index) {
T *new_arr = new T[m_capacity];
if (i == index) i++;
if(i < m_length) new_arr[j] = m_arr[i];
int DynamicArray<T>::size() {
void DynamicArray<T>::print() {
std::cout << m_arr[0];
std::cout << ", " << m_arr[i];
Welcome to C++, and welcome to Code Review.
C++ memory management is, as you probably have realized, tough and
error-prone. There are many things that can easily go wrong.
Assuming that no exception is thrown, I don’t see obvious memory leaks
in your code; however, there are still some issues worth discussing.
You can take a look at my implementation of a non-resizable dynamic
array or a stack-based full-fledged vector for some
You have not defined copy constructors or move constructors, so the
compiler will synthesize corresponding constructors that simply copy
all the members — which is completely wrong, as now the two
dynamic arrays will point to the same memory. Not only are the
elements shared between the copies, causing modifications to one array
to affect the other, but the two copies will attempt to free the same
memory upon destruction, leading to a double-free error, which is
way more serious than a memory leak.
Initialization semantics
It is generally expected that the constructor of the element type is
called n
times if n
elements are pushed into the dynamic
array. In your code, however, this is not the case, whre the amount
of constructors called is determined by the capacity of the dynamic
array. Elements are first default initialized, and then copy-assigned
The correct way to solve this problem requires allocating an
uninitialized buffer, and using placement new (or equivalent features)
to construct the elements, which is another can of worms.
Think of what happens when the construction of an element throws an
exception — your code will halt halfway, and there will be a
memory leak. Resolving this problem would require a manual try
block, or standard library facilities like
std::uninitialized_copy (which essentially do the same under
the hood) if you switched to uninitialized buffers and manual
All of the elements are copied every time, which is wasteful. Make
good use of move semantics when appropriate.
Used std::size_t instead of int to store sizes and indexes.1
get, size, and print should be const. Moreover, get should
return a const T&. In fact, get and set would idiomatically be
replaced by operator[].
Don’t throw a const char*. Use a dedicated exception class like
std::out_of_range instead.
Manual loops like
are better replaced with calls to std::copy (or
std::move).
Re-allocating every time insert is called doesn’t seem like a good
idea. A better trade-off might be to append an element and then
std::rotate it to the correct position (assuming rotation
doesn’t throw).
Also, print might take an std::ostream& (or perhaps
std::basic_ostream<Char, Traits>&) argument for extra flexibility.
1 As Andreas H. pointed out in the comments, this recommendation is subject to debate, since the use of unsigned arithmetic has its pitfalls. An alternative is to use std::ptrdiff_t and std::ssize (C++20) instead. You can write your own version of ssize as shown on the cppreference page if C++20 is not accessible.
Source : Link , Question Author : ethan warco , Answer Author : L. F.
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Symplectic_group Knowpia
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2n, F) and Sp(n) for positive integer n and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by
{\displaystyle \mathrm {USp} (n)}
. Many authors prefer slightly different notations, usually differing by factors of 2. The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn, and Sp(n) is the compact real form of Sp(2n, C). Note that when we refer to the (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension n.
The name "symplectic group" is due to Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex".
The metaplectic group is a double cover of the symplectic group over R; it has analogues over other local fields, finite fields, and adele rings.
Sp(2n, F)Edit
The symplectic group is a classical group defined as the set of linear transformations of a 2n-dimensional vector space over the field F which preserve a non-degenerate skew-symmetric bilinear form. Such a vector space is called a symplectic vector space, and the symplectic group of an abstract symplectic vector space V is denoted Sp(V). Upon fixing a basis for V, the symplectic group becomes the group of 2n × 2n symplectic matrices, with entries in F, under the operation of matrix multiplication. This group is denoted either Sp(2n, F) or Sp(n, F). If the bilinear form is represented by the nonsingular skew-symmetric matrix Ω, then
{\displaystyle \operatorname {Sp} (2n,F)=\{M\in M_{2n\times 2n}(F):M^{\mathrm {T} }\Omega M=\Omega \},}
where MT is the transpose of M. Often Ω is defined to be
{\displaystyle \Omega ={\begin{pmatrix}0&I_{n}\\-I_{n}&0\\\end{pmatrix}},}
where In is the identity matrix. In this case, Sp(2n, F) can be expressed as those block matrices
{\displaystyle ({\begin{smallmatrix}A&B\\C&D\end{smallmatrix}})}
{\displaystyle A,B,C,D\in M_{n\times n}(F)}
, satisfying the three equations:
{\displaystyle {\begin{aligned}-C^{\mathrm {T} }A+A^{\mathrm {T} }C&=0,\\-C^{\mathrm {T} }B+A^{\mathrm {T} }D&=I_{n},\\-D^{\mathrm {T} }B+B^{\mathrm {T} }D&=0.\end{aligned}}}
Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group SL(2n, F). When n = 1, the symplectic condition on a matrix is satisfied if and only if the determinant is one, so that Sp(2, F) = SL(2, F). For n > 1, there are additional conditions, i.e. Sp(2n, F) is then a proper subgroup of SL(2n, F).
Typically, the field F is the field of real numbers R or complex numbers C. In these cases Sp(2n, F) is a real/complex Lie group of real/complex dimension n(2n + 1). These groups are connected but non-compact.
The center of Sp(2n, F) consists of the matrices I2n and −I2n as long as the characteristic of the field is not 2.[1] Since the center of Sp(2n, F) is discrete and its quotient modulo the center is a simple group, Sp(2n, F) is considered a simple Lie group.
The real rank of the corresponding Lie algebra, and hence of the Lie group Sp(2n, F), is n.
The Lie algebra of Sp(2n, F) is the set
{\displaystyle {\mathfrak {sp}}(2n,F)=\{X\in M_{2n\times 2n}(F):\Omega X+X^{\mathrm {T} }\Omega =0\},}
equipped with the commutator as its Lie bracket.[2] For the standard skew-symmetric bilinear form
{\displaystyle \Omega =({\begin{smallmatrix}0&I\\-I&0\end{smallmatrix}})}
, this Lie algebra is the set of all block matrices
{\displaystyle ({\begin{smallmatrix}A&B\\C&D\end{smallmatrix}})}
{\displaystyle {\begin{aligned}A&=-D^{\mathrm {T} },\\B&=B^{\mathrm {T} },\\C&=C^{\mathrm {T} }.\end{aligned}}}
Sp(2n, C)Edit
The symplectic group over the field of complex numbers is a non-compact, simply connected, simple Lie group.
Sp(2n, R)Edit
Sp(n, C) is the complexification of the real group Sp(2n, R). Sp(2n, R) is a real, non-compact, connected, simple Lie group.[3] It has a fundamental group isomorphic to the group of integers under addition. As the real form of a simple Lie group its Lie algebra is a splittable Lie algebra.
Some further properties of Sp(2n, R):
The exponential map from the Lie algebra sp(2n, R) to the group Sp(2n, R) is not surjective. However, any element of the group can be represented as the product of two exponentials.[4] In other words,
{\displaystyle \forall S\in \operatorname {Sp} (2n,\mathbf {R} )\,\,\exists X,Y\in {\mathfrak {sp}}(2n,\mathbf {R} )\,\,S=e^{X}e^{Y}.}
For all S in Sp(2n, R):
{\displaystyle S=OZO'\quad {\text{such that}}\quad O,O'\in \operatorname {Sp} (2n,\mathbf {R} )\cap \operatorname {SO} (2n)\cong U(n)\quad {\text{and}}\quad Z={\begin{pmatrix}D&0\\0&D^{-1}\end{pmatrix}}.}
The matrix D is positive-definite and diagonal. The set of such Zs forms a non-compact subgroup of Sp(2n, R) whereas U(n) forms a compact subgroup. This decomposition is known as 'Euler' or 'Bloch–Messiah' decomposition.[5] Further symplectic matrix properties can be found on that Wikipedia page.
As a Lie group, Sp(2n, R) has a manifold structure. The manifold for Sp(2n, R) is diffeomorphic to the Cartesian product of the unitary group U(n) with a vector space of dimension n(n+1).[6]
Infinitesimal generatorsEdit
The members of the symplectic Lie algebra sp(2n, F) are the Hamiltonian matrices.
These are matrices,
{\displaystyle Q}
{\displaystyle Q={\begin{pmatrix}A&B\\C&-A^{\mathrm {T} }\end{pmatrix}}}
where B and C are symmetric matrices. See classical group for a derivation.
Example of symplectic matricesEdit
For Sp(2, R), the group of 2 × 2 matrices with determinant 1, the three symplectic (0, 1)-matrices are:[7]
{\displaystyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad {\begin{pmatrix}1&0\\1&1\end{pmatrix}}\quad {\text{and}}\quad {\begin{pmatrix}1&1\\0&1\end{pmatrix}}.}
{\displaystyle \operatorname {Sp} (2n,\mathbf {R} )}
can have a fairly explicit description using generators. If we let
{\displaystyle \operatorname {Sym} (n)}
denote the symmetric
{\displaystyle n\times n}
matrices, then
{\displaystyle \operatorname {Sp} (2n,\mathbf {R} )}
{\displaystyle D(n)\cup N(n)\cup \{\Omega \},}
{\displaystyle {\begin{aligned}D(n)&=\left\{\left.{\begin{bmatrix}A&0\\0&(A^{T})^{-1}\end{bmatrix}}\,\right|\,A\in \operatorname {GL} (n,\mathbf {R} )\right\}\\[6pt]N(n)&=\left\{\left.{\begin{bmatrix}I_{n}&B\\0&I_{n}\end{bmatrix}}\,\right|\,B\in \operatorname {Sym} (n)\right\}\end{aligned}}}
{\displaystyle \operatorname {Sp} (2n,\mathbf {R} )}
[8]pg 173[9]pg 2.
Relationship with symplectic geometryEdit
Symplectic geometry is the study of symplectic manifolds. The tangent space at any point on a symplectic manifold is a symplectic vector space.[10] As noted earlier, structure preserving transformations of a symplectic vector space form a group and this group is Sp(2n, F), depending on the dimension of the space and the field over which it is defined.
A symplectic vector space is itself a symplectic manifold. A transformation under an action of the symplectic group is thus, in a sense, a linearised version of a symplectomorphism which is a more general structure preserving transformation on a symplectic manifold.
Sp(n)Edit
The compact symplectic group[11] Sp(n) is the intersection of Sp(2n, C) with the
{\displaystyle 2n\times 2n}
unitary group:
{\displaystyle \operatorname {Sp} (n):=\operatorname {Sp} (2n;\mathbf {C} )\cap \operatorname {U} (2n)=\operatorname {Sp} (2n;\mathbf {C} )\cap \operatorname {SU} (2n).}
It is sometimes written as USp(2n). Alternatively, Sp(n) can be described as the subgroup of GL(n, H) (invertible quaternionic matrices) that preserves the standard hermitian form on Hn:
{\displaystyle \langle x,y\rangle ={\bar {x}}_{1}y_{1}+\cdots +{\bar {x}}_{n}y_{n}.}
That is, Sp(n) is just the quaternionic unitary group, U(n, H).[12] Indeed, it is sometimes called the hyperunitary group. Also Sp(1) is the group of quaternions of norm 1, equivalent to SU(2) and topologically a 3-sphere S3.
Note that Sp(n) is not a symplectic group in the sense of the previous section—it does not preserve a non-degenerate skew-symmetric H-bilinear form on Hn: there is no such form except the zero form. Rather, it is isomorphic to a subgroup of Sp(2n, C), and so does preserve a complex symplectic form in a vector space of twice the dimension. As explained below, the Lie algebra of Sp(n) is the compact real form of the complex symplectic Lie algebra sp(2n, C).
Sp(n) is a real Lie group with (real) dimension n(2n + 1). It is compact and simply connected.[13]
The Lie algebra of Sp(n) is given by the quaternionic skew-Hermitian matrices, the set of n-by-n quaternionic matrices that satisfy
{\displaystyle A+A^{\dagger }=0}
where A† is the conjugate transpose of A (here one takes the quaternionic conjugate). The Lie bracket is given by the commutator.
Important subgroupsEdit
Some main subgroups are:
{\displaystyle \operatorname {Sp} (n)\supset \operatorname {Sp} (n-1)}
{\displaystyle \operatorname {Sp} (n)\supset \operatorname {U} (n)}
{\displaystyle \operatorname {Sp} (2)\supset \operatorname {O} (4)}
Conversely it is itself a subgroup of some other groups:
{\displaystyle \operatorname {SU} (2n)\supset \operatorname {Sp} (n)}
{\displaystyle \operatorname {F} _{4}\supset \operatorname {Sp} (4)}
{\displaystyle \operatorname {G} _{2}\supset \operatorname {Sp} (1)}
There are also the isomorphisms of the Lie algebras sp(2) = so(5) and sp(1) = so(3) = su(2).
Relationship between the symplectic groupsEdit
Every complex, semisimple Lie algebra has a split real form and a compact real form; the former is called a complexification of the latter two.
The Lie algebra of Sp(2n, C) is semisimple and is denoted sp(2n, C). Its split real form is sp(2n, R) and its compact real form is sp(n). These correspond to the Lie groups Sp(2n, R) and Sp(n) respectively.
The algebras, sp(p, n − p), which are the Lie algebras of Sp(p, n − p), are the indefinite signature equivalent to the compact form.
Physical significanceEdit
The compact symplectic group Sp(n) comes up in classical physics as the symmetries of canonical coordinates preserving the Poisson bracket.
Consider a system of n particles, evolving under Hamilton's equations whose position in phase space at a given time is denoted by the vector of canonical coordinates,
{\displaystyle \mathbf {z} =(q^{1},\ldots ,q^{n},p_{1},\ldots ,p_{n})^{\mathrm {T} }.}
The elements of the group Sp(2n, R) are, in a certain sense, canonical transformations on this vector, i.e. they preserve the form of Hamilton's equations.[14][15] If
{\displaystyle \mathbf {Z} =\mathbf {Z} (\mathbf {z} ,t)=(Q^{1},\ldots ,Q^{n},P_{1},\ldots ,P_{n})^{\mathrm {T} }}
are new canonical coordinates, then, with a dot denoting time derivative,
{\displaystyle {\dot {\mathbf {Z} }}=M({\mathbf {z} },t){\dot {\mathbf {z} }},}
{\displaystyle M(\mathbf {z} ,t)\in \operatorname {Sp} (2n,\mathbf {R} )}
for all t and all z in phase space.[16]
For the special case of a Riemannian manifold, Hamilton's equations describe the geodesics on that manifold. The coordinates
{\displaystyle q^{i}}
live in the tangent bundle to the manifold, and the momenta
{\displaystyle p_{i}}
live in the cotangent bundle. This is the reason why these are conventionally written with upper and lower indexes; it is to distinguish their locations. The corresponding Hamiltonian consists purely of the kinetic energy: it is
{\displaystyle H={\tfrac {1}{2}}g^{ij}(q)p_{i}p_{j}}
{\displaystyle g^{ij}}
is the inverse of the metric tensor
{\displaystyle g_{ij}}
on the Riemannian manifold.[17][15] In fact, the cotangent bundle of any smooth manifold can be a given a (non-trivial) symplectic structure in a canonical way, with the symplectic form defined as the exterior derivative of the tautological one-form.[18]
Consider a system of n particles whose quantum state encodes its position and momentum. These coordinates are continuous variables and hence the Hilbert space, in which the state lives, is infinite-dimensional. This often makes the analysis of this situation tricky. An alternative approach is to consider the evolution of the position and momentum operators under the Heisenberg equation in phase space.
Construct a vector of canonical coordinates,
{\displaystyle \mathbf {\hat {z}} =({\hat {q}}^{1},\ldots ,{\hat {q}}^{n},{\hat {p}}_{1},\ldots ,{\hat {p}}_{n})^{\mathrm {T} }.}
The canonical commutation relation can be expressed simply as
{\displaystyle [\mathbf {\hat {z}} ,\mathbf {\hat {z}} ^{\mathrm {T} }]=i\hbar \Omega }
{\displaystyle \Omega ={\begin{pmatrix}\mathbf {0} &I_{n}\\-I_{n}&\mathbf {0} \end{pmatrix}}}
and In is the n × n identity matrix.
Many physical situations only require quadratic Hamiltonians, i.e. Hamiltonians of the form
{\displaystyle {\hat {H}}={\frac {1}{2}}\mathbf {\hat {z}} ^{\mathrm {T} }K\mathbf {\hat {z}} }
where K is a 2n × 2n real, symmetric matrix. This turns out to be a useful restriction and allows us to rewrite the Heisenberg equation as
{\displaystyle {\frac {d\mathbf {\hat {z}} }{dt}}=\Omega K\mathbf {\hat {z}} }
The solution to this equation must preserve the canonical commutation relation. It can be shown that the time evolution of this system is equivalent to an action of the real symplectic group, Sp(2n, R), on the phase space.
Symplectic manifold, Symplectic matrix, Symplectic vector space, Symplectic representation
^ "Symplectic group", Encyclopedia of Mathematics Retrieved on 13 December 2014.
^ Hall 2015 Prop. 3.25
^ "Is the symplectic group Sp(2n, R) simple?", Stack Exchange Retrieved on 14 December 2014.
^ "Is the exponential map for Sp(2n, R) surjective?", Stack Exchange Retrieved on 5 December 2014.
^ "Standard forms and entanglement engineering of multimode Gaussian states under local operations – Serafini and Adesso", Retrieved on 30 January 2015.
^ "Symplectic Geometry – Arnol'd and Givental", Retrieved on 30 January 2015.
^ Symplectic Group, (source: Wolfram MathWorld), downloaded February 14, 2012
^ Gerald B. Folland. (2016). Harmonic analysis in phase space. Princeton: Princeton Univ Press. p. 173. ISBN 978-1-4008-8242-7. OCLC 945482850.
^ Habermann, Katharina, 1966- (2006). Introduction to symplectic Dirac operators. Springer. ISBN 978-3-540-33421-7. OCLC 262692314. {{cite book}}: CS1 maint: multiple names: authors list (link)
^ "Lecture Notes – Lecture 2: Symplectic reduction", Retrieved on 30 January 2015.
^ Hall 2015 p. 14
^ Hall 2015 Prop. 13.12
^ Arnold 1989 gives an extensive mathematical overview of classical mechanics. See chapter 8 for symplectic manifolds.
^ a b Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X
^ Goldstein 1980, Section 9.3
^ Jurgen Jost, (1992) Riemannian Geometry and Geometric Analysis, Springer.
^ da Silva, Ana Cannas (2008). Lectures on Symplectic Geometry. Lecture Notes in Mathematics. Vol. 1764. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 9. doi:10.1007/978-3-540-45330-7. ISBN 978-3-540-42195-5.
Arnold, V. I. (1989), Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60 (second ed.), Springer-Verlag, ISBN 0-387-96890-3
Fulton, W.; Harris, J. (1991), Representation Theory, A first Course, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, ISBN 978-0-387-97495-8 .
Goldstein, H. (1980) [1950]. "Chapter 7". Classical Mechanics (2nd ed.). Reading MA: Addison-Wesley. ISBN 0-201-02918-9.
Lee, J. M. (2003), Introduction to Smooth manifolds, Graduate Texts in Mathematics, vol. 218, Springer-Verlag, ISBN 0-387-95448-1
Rossmann, Wulf (2002), Lie Groups – An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford Science Publications, ISBN 0-19-859683-9
Ferraro, Alessandro; Olivares, Stefano; Paris, Matteo G. A. (March 2005), "Gaussian states in continuous variable quantum information", arXiv:quant-ph/0503237 .
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Optimization of Fin Performance in a Laminar Channel Flow Through Dimpled Surfaces | J. Heat Transfer | ASME Digital Collection
, 3123 TAMU, College Station, TX 77843-3123
Doseo Park,
Doseo Park
Egidio (Ed) Marotta,
Leroy (Skip) Fletcher
Silva, C., Park, D., Marotta, E. (., and Fletcher, L. (. (December 15, 2008). "Optimization of Fin Performance in a Laminar Channel Flow Through Dimpled Surfaces." ASME. J. Heat Transfer. February 2009; 131(2): 021702. https://doi.org/10.1115/1.2994712
The effect of the dimple shape and orientation on the heat transfer coefficient of a vertical fin surface was determined both numerically and experimentally. The investigation focused on the laminar channel flow between fins, with a
Re=500
and 1000. Numerical simulations were performed using a commercial computational fluid dynamics code to analyze optimum configurations, and then an experimental investigation was conducted on flat and dimpled surfaces for comparison purposes. Numerical results indicated that oval dimples with their “long” axis oriented perpendicular to the direction of the flow offered the best thermal improvement, hence the overall Nusselt number increased up to 10.6% for the dimpled surface. Experimental work confirmed these results with a wall-averaged temperature reduction of up to
3.7K
, which depended on the heat load and the Reynolds number. Pressure losses due to the dimple patterning were also briefly explored numerically in this work.
heat transfer enhancement, forced convection, dimples, dimple geometry, laminar flow, channel flow, channel flow, computational fluid dynamics, convection, flow simulation, laminar flow
Channel flow, Computer simulation, Flow (Dynamics), Geometry, Heat transfer, Heat transfer coefficients, Laminar flow, Reynolds number, Temperature, Pressure, Convection, Friction, Plates (structures), Heat, Computational fluid dynamics, Optimization
Chyum
Concavity Enhancement Heat Transfer in an Internal Cooling Passage
Channel Height Effect on Heat Transfer and Friction in a Dimple Passage
Heat Transfer in a Dimpled Channel: Combined Influences of Aspect Ratio, Temperature Ratio, Reynolds Number and Flow Structure
Heat Transfer in a Channel With Dimples and Protrusion on Opposite Walls
,” ASME Paper No. IMECE 2002–32941.
Comparisons of Flow Structure Above Dimpled Surfaces With Different Dimple Depths in a Channel
Heat Transfer and Friction Factors for Flow Inside Circular Tubes With Concavity Surfaces
Agachev
Effects of Surface Curvature on Heat Transfer and Hydrodynamics Within a Single Hemispherical Dimple
Turbine Blade Cooling Studies at Texas A&M University: 1980–2004
Numerical Simulation of Laminar Flow and Heat Transfer Inside a Micro-Channel With One Dimpled Wall
Numerical Simulation of Laminar Incompressible Three-Dimensional Flow Around a Dimple (Vortex Dynamics and Heat Transfer)
,” Russian Ministry of Science and Technology Institute for High-Performance Computing and Databases, preprint 6-67.
Kh. T.
Identification of Self-Organized Vortex-Like Structures in Numerically Simulated Turbulent Flow of a Viscous Incompressible Liquid Streaming Around a Well on a Plane
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Coin Change: Minimum number of coins - PhotoLens
You are given n types of coin denominations of values v(1)<v(2)<...<v(n)
v(1) < v(2) < ... < v(n)
(all integers). Assume v(1)=1
v(1) = 1
, so you can always make change for any amount of money C
. Give an algorithm which makes change for an amount of money C
with as few coins as possible.
int i,n,den[20],temp[20],min,min_idx, S, numcoins = 0;
printf("Coin Change with min no. of coins\nEnter the total change you want: ");
printf("Enter the no. of different denominations of coins available: ");
printf("Enter the different denominations in ascending order: \n");
scanf("%d",&den[i]);
temp[i] = S / den[i] ;
/*calculate min from temp */
min = temp[0] ;
if(min > temp[i] && temp[i]!=0)
min = temp[i] ;
min_idx = i ;
numcoins += min ;
S -= den[min_idx] * min ;
printf("min no of coins = %d" , numcoins) ;
In several solutions on the internet, I saw code using an array of the size of the total sum or value S, whereas I use only 2 arrays of size n, the number of different denominations available. That's why I was wondering whether my approach is correct or whether it's flawed.
Is it better or worse in terms of time complexity? Also, am I properly using dynamic programming principles in my code? Can it be made more efficient? The code ran correctly for several test cases.
I am sorry for poor code formatting and a not-so-clean code. It is a small code, so I hope it is understandable. My main concern is whether the code is dp or not, and if it can be improved for efficiency.
You code is currently too simplistic. All it does is make change from the highest denomination possible. It fails on the following input:
Enter the total change you want: 6
Enter the no. of different denominations of coins available: 3
Enter the different denominations in ascending order:
min no of coins = 3
Your program thought the change should be: 4 1 1 but the best solution was actually 3 3.
Your program doesn't currently use any dynamic programming principles. In order to do so, you would need to keep an array holding the best number of coins per change amount. You could then iterate starting at 1 and build up to the total change requested.
Source : Link , Question Author : CyberLingo , Answer Author : Community
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Coloring individual parts in a brush
Making a logotype “pop” off the page
“Most popular package” pricing design
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Seiberg-Witten Like Equations on Pseudo-Riemannian Manifolds with Structure
Nülifer Özdemir, Nedim Deǧirmenci, "Seiberg-Witten Like Equations on Pseudo-Riemannian Manifolds with Structure", Advances in Mathematical Physics, vol. 2016, Article ID 2173214, 7 pages, 2016. https://doi.org/10.1155/2016/2173214
Nülifer Özdemir1 and Nedim Deǧirmenci1
1Department of Mathematics, Anadolu University, Eskisehir, Turkey
Academic Editor: Dimitrios Tsimpis
We consider 7-dimensional pseudo-Riemannian manifolds with structure group . On such manifolds, the space of 2-forms splits orthogonally into components . We define self-duality of a 2-form by considering the part as the bundle of self-dual 2-forms. We express the spinor bundle and the Dirac operator and write down Seiberg-Witten like equations on such manifolds. Finally we get explicit forms of these equations on and give some solutions.
The Seiberg-Witten theory, introduced by Witten in [1], became one of the most important tools to understand the topology of smooth -manifolds. The Seiberg-Witten theory is based on the solution space of two equations which are called the Seiberg-Witten equations. The first one of the Seiberg-Witten equations is Dirac equation and the second one is known as curvature equation [2]. The first equation is the harmonicity condition of spinor fields; that is, the spinor field belongs to the kernel of the Dirac operator. The second equation couples the self-dual part of the curvature 2-form with a spinor field. There exist various generalizations of Seiberg-Witten equations to higher dimensional Riemannian manifolds [3–6]. All of these generalizations are done for the manifolds which have special structure groups. Also Seiberg-Witten like equations are studied over 4-dimensional Lorentzian manifolds [7] and 4-dimensional pseudo-Riemannian manifolds with neutral signature [8].
Parallel spinors on pseudo-Riemannian manifolds are studied by Ikemakhen [9]. In the present work, we consider -dimensional manifolds with structure group . In order to define spinors and Dirac operator, the manifold must have a -structure. We assume that 7-dimensional pseudo-Riemannian manifold with signature has -structure. On the other hand, to write down curvature equation, we need a self-duality notion of a -form on such manifolds. In dimensions, self-duality concept of -forms is well known. The bundle of 2-forms decomposes into two parts on this manifold [10]. Then we will define self-duality of a -form on a -manifold with structure group by using decomposition of -forms on this manifold.
2. Manifolds with Structure Group
The exceptional Lie group , automorphism group of octonions, is well known. There is another similar Lie group which is automorphism group of split octonions [11]. On , we consider the metric where and . From now on, we denote the pair by . The isometry group of this space is The special orthogonal subgroup of is The group is the subgroup of , preserving the following 3-form: where is the dual base of the standard basis of , with the notation and with the metric ; that is, where is called the fundamental 3-form on [10, 11]. The space of 2-forms decomposes into two parts , where
A semi-Riemannian -manifold with the metric of signature is called a manifold if its structure group reduces to the Lie group ; equivalently, there exists a nowhere vanishing 3-form on whose local expression is of the form . Such a form is called a structure on [12]. If the structure group of is the group then the bundle of -forms decomposes into two parts similar to and we denote it by [10].
It is known that square of the Hodge operator on 2-forms over -dimensional Riemannian manifolds is identity and are eigenvalues of the Hodge operator. The elements of eigenspace of are called self-dual 2-forms and the others are called anti-self-dual forms. But this situation does not generalize to higher dimensional manifolds directly. Self-duality of -form has been studied on some higher dimensions [3, 13]. In this work, we need self-duality concept of -forms on -dimensional manifolds with structure group .
Now we define a duality operator over bundle of 2-form as The eigenvalues of this map are and . Note that the subbundle corresponds to the eigenvalue and the subbundle corresponds to the eigenvalue . Let be a 2-form over . If belongs to , then we call a self-dual 2-form. If belongs to , then we call an anti-self-dual 2-form. Because of decomposition of 2-forms on , any 2-form on can be written uniquely as where and . Similar to the 4-dimensional case, we say that is self-dual part of and is anti-self-dual part of .
3. Spinor Bundles over Manifolds
It is known that the group has two connected components. The connected component to the identity of is denoted by . In this work we deal with the group . The covering space of is the group which lies in Clifford algebra and we denoted the connected component of by . There is a covering map which is a 2 : 1 group homomorphism given by for , [10, 11, 14].
One can define another group which lies in the complex Clifford algebra by where the elements of are the equivalence classes of pair , under the equivalence relation [9]. There exist two exact sequences aswhere .
Let be an orthonormal basis of ; then the Lie algebras of and are respectively. The derivative of is obtained as where is the -matrix whose -entry is , -entry is , and the other entries are zero [9]. Since the Clifford algebra is isomorphic to the algebra , we can project this isomorphism onto the first component. Hence, we get spinor representation: By restricting to the group we get and is called spinor representation of the group ; shortly we denote it by . The elements of are called spinors and the complex vector space is called the spinor space and it is denoted by . By using spinor representation, the Clifford multiplication of vectors with spinors is defined by where and . The spinor space has a nondegenerate indefinite Hermitian inner product as where is the standard Hermitian inner product on for . The new inner product is invariant with respect to the group and satisfies the following property: where and . In this work, we use the following spinor representation : where Now, we recall the main definitions concerning -structure and the spinor bundle. Let be a -dimensional pseudo-Riemannian manifold with structure group . Then, there is an open covering of and transition functions for .
If there exists another collection of transition functions such that the following diagram commutes (i.e., and the cocycle condition on is satisfied), then is called a manifold. Then one can construct a principal -bundle on and a bundle map .
Let be a -structure on . We can construct an associated complex vector bundle: where is the spinor representation of . This complex vector bundle is called spinor bundle for a given -structure on and sections of are called spinor fields. The Clifford multiplication given by (15) can be extended to a bundle map: Parallel spinors on the spinor bundle are studied in [9].
Since is a pseudo-Riemannian manifold, then by using the map we can get an associated principal -bundle: Also, the map induces a bundle map:
Now, fix a connection 1-form over the principal -bundle . Let be the Levi-Civita covariant derivative associated with the metric which determines an -valued connection 1-form on the principal bundle . The connection 1-form can be written locally where is a local orthonormal frame on open set and . By using the connection -form and , one can obtain a connection 1-form on the principal bundle (the fibre product bundle): The connection can be lift to a connection 1-form on the principal bundle via the 2-fold covering map:and the following commutative diagram. One can obtain a covariant derivative operator on the spinor bundle by using the connection 1-form . The local form of the covariant derivative is where is a orthonormal frame on open set . We note that some authors use the term instead of in the local formula of . The covariant derivative is compatible with the metric and the Clifford multiplication where are spinor fields and sections of , , and are vector fields on . We can define the Dirac operator as the following composition: which can be written locally as where is any oriented local orthonormal frame of .
4. Seiberg-Witten Like Equations on Manifolds
Let be a manifold with structure group . Fix a -structure and a connection in the principal -bundle associated with the -structure. Note that the curvature of the connection is -valued 2-form. The curvature 2-form on the determines an -valued 2-form on uniquely (see [15]) and we denote it again by .
We can define a map where . Note that the map satisfies the following properties:
Hence, the map associates an -valued 2-form with each spinor field , so we can write In local frame on , the map can be expressed as
Now we are ready to express the Seiberg-Witten equations. Let be a manifold with structure group . Fix a structure and take a connection 1-form on the principal bundle and a spinor field . We write the Seiberg-Witten like equations as where is the self-dual part of the curvature and is the self-dual part of the -form corresponding to the spinor .
5. Seiberg-Witten Like Equations on
Let us consider these equations on the flat space with the structure given by . We use the standard orthonormal frame on and the spinor representation in (18). The connection on is given by where and are smooth maps. Then, the associated connection on the line bundle is the connection -form and its curvature -form is given by where for . Now we can write the Dirac operator on with respect to a given -structure and -connection .
We denote the dual basis of by . Now one can give a frame for the space of self-dual -forms on as
Let be the curvature form of the -valued connection 1-form and let be its self-dual part. Then, Now we calculate the -form , for a spinor . Then can be written in the following way: The projection onto the subspace is given by If is calculated explicitly, then we obtain the following identity: Hence, the curvature equation can be written explicitly as Dirac equation can be expressed as follows:
These equations admit nontrivial solutions. For example, direct calculation shows that the spinor field with and the connection -form satisfy the above equations.
Now we consider the space where is the space of connection 1-forms on the principle bundle and is the space of spinor fields. The space is called the configuration space. There is an action of the gauge group on the configuration space by where and . The action of the gauge group enjoys the following equalities: Hence, if the pair is a solution to the Seiberg-Witten equations, then the pair is also a solution to the Seiberg-Witten equations.
One can obtain infinitely many solutions for the Seiberg-Witten equations on : Consider the spinor and the connection 1-form Since the pair is a solution on , the pair is also a solution, where and is a smooth real valued function on .
The moduli space of Seiberg-Witten equations on the manifold with structure group is
Whether the moduli space has similar properties of moduli space of Seiberg-Witten equations on a -dimensional manifold is a subject of another work.
This study was supported by Anadolu University Scientific Research Projects Commission under Grant no. 1501F017.
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Copyright © 2016 Nülifer Özdemir and Nedim Deǧirmenci. 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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Relativistic Gravitational Field and Invalidity of Singularity
HCL Technologies, Chennai, India.
Abstract: A century of successful experiments on general relativity was unable to convince substantial scholars on the presence of “gravitational singularity”. Physically questionable outcomes of general relativity are due to the deployment of non-relativistic Newtonian gravitation. However, the nonrelativistic classical gravitational field equation can be utilized for the weak field with negligible error. The connection between inertia and mass is distinct for classical and relativistic mechanics. As per the equivalence principle, the connection between energy and spacetime should be identical for special and general relativity; the relativistic approach on the gravitational field will eliminate “gravitational singularity”. This theory aligned well with relativity and improved sturdiness to general relativity. The impacts of this model on general relativity experimental results are insignificant.
Keywords: Relativity, Gravitation, Singularity, Black Hole
We use inertia or gravity as a tool to identify the mass of matter. However, application of Newtonian physics to detect the mass of a high-speed particle, by its kinetic energy, will result in the wrong measurement. The resistance exerted by spacetime on to the matter does not agree with the interpretation of classical mechanics. Therefore, the established classical link between mass and inertia is profoundly inaccurate. Einstein’s theory of special relativity (SR) is well established, and it asserts that infinite resistance is exerted on matter when it approaches the speed of light [1]. With the help of relativity, the inevitability of the Lorentz factor (LF) in the kinetic energy of a high-speed particle is experimentally verified [2]. Therefore, the spacetime fabric can produce infinite resistance as per SR.
Like inertia, the classical connection between energy and gravitational field should be modified with relativistic mechanics, retaining the underlying hypothesis of general relativity (GR). As being reasoned, we can arrive to the following assumptions:
· Relativistic mass or momentum effect should be due to the resistance produced by the spacetime fabric [3].
· Regarding gravity, the resistance offered by spacetime fabric to curve must be associated with the gravitational field producing energy/mass.
2. Applying Relativistic Approach on Gravitational Field
In GR, the influence of the gravitational field on time and space was addressed; however, the Newtonian mechanics were used in approximation to define the spacetime metrics at any given point in the space [4]. Relativistic energy incorporated in the energy-momentum-stress tensor only helps to determine the whole energy in the system. Hence, the relativistic approach was not deployed to connect mass and the respective spacetime curvature, and that caused the GR “Schwarzschild radius” to be equivalent to classical mechanics [5]. In this article, we will focus only on invalidating the singularity outcome by modifying the classical gravitational field equation.
Lorentz factor is (γ),
\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}
where v is the velocity of the freefalling object and c is the velocity of light.
Gravitational potential energy (E), as per classical mechanics equals to:
E=\left(\frac{GMm}{r}\right)
where G is the gravitational constant, M is the gravitational mass, m is the mass of freefalling object, and r is the distance from the center of mass.
Energy required to accelerate the object to velocity v, as per the relativistic mechanics equals to:
\frac{E}{\gamma }=\frac{1}{2}m{v}^{2}
As examined, if inertial/kinetic energy is subject to resistance from spacetime, LF must be considered to link energy/mass and its gravitational spacetime curvature. Here v2 primarily embodies the spacetime curvature and its outcome is velocity of the freefalling object.
Inserting Equation (1) in Equation (2), we obtain,
\left(\frac{GMm}{r}\right)\frac{1}{\gamma }=\frac{1}{2}m{v}^{2}
\left(\frac{2GM}{r}\right)\frac{1}{\gamma }={v}^{2}
{\left(\frac{2GM}{cr}\right)}^{2}\left({c}^{2}-{v}^{2}\right)={v}^{4},\frac{2GM}{cr}=\kappa
{v}^{2}=\frac{1}{2}\kappa \left\{\sqrt{{\kappa }^{2}+4{c}^{2}}-\kappa \right\}
Classical gravitational field equation:
g=-\nabla \Phi =\frac{GM}{{r}^{2}}
where g is the gravitational acceleration and Φ is gravitational potential.
\frac{2GM}{r}
in Equation (4) with v2 in Equation (3) to get relativistic gravitational field equation.
g=-\nabla \Phi =-\frac{1}{2}\nabla {v}^{2}
With the help of the new equation, we can conclude that infinite energy/matter is required to produce the “event horizon” irrespective of the coordinate system. Compared to the classical mechanics, gravitational acceleration gradient reduces further and further near the supermassive object.
3.1. Data within the Solar System
Following is the comparison of current and proposed methods regarding the escape velocity of earth and sun; it illustrates the precision of this theory in a weak gravitational field.
Earth: (Mass = 5.972 × 1024 kg [6], Radius = 6,371,000 m [6], Gravitational Constant = 6.6743 × 10−11 m3·kg−1·s−2 [7] ).
Escape velocity by existing method = 11,186.3524997063 m/s
Escape velocity by proposed method = 11,186.3524958126 m/s
Sun: (Mass = 1.9885 × 1030 kg [8], Radius = 695,700,000 m [8], Gravitational Constant = 6.6743 × 10−11 m3·kg−1·s−2 [7] ).
Escape velocity by existing method = 617,688.6989 m/s.
Escape velocity by proposed method = 617,688.0433 m/s.
Based on the above comparison, we can conclude that the redshift and gravitational lensing experiments performed within the solar system will comply with the proposed changes.
Current gravitational time dilation equation [9],
\frac{\tau }{t}=\sqrt{1-\frac{2GM}{r{c}^{2}}}
where τ is the time between two events for an observer close to the massive object and t is the time between the events for an observer at an arbitrarily large distance from the massive object.
\frac{2GM}{r}
in Equation (6) with v2 in Equation (3),
\frac{\tau }{t}=\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}
Figure 1 shows a comparison of gravitational time dilation of the current and proposed technique for the solar mass object. In the current method, time stops at the Schwarzschild radius of 2953 m for the solar mass object, but in our proposed method, “zero” Schwarzschild radius is required to produce the corresponding effect on time.
Figure 1. Comparison of gravitational time dilation.
3.3. Influence on the Extreme Gravitational Field Experiments
The ratio between the extreme gravitational potential of the current and proposed method in Schwarzschild precession tests performed on orbits of star S2 near the galactic center will provide the same result to the order of 10−3.
Extreme gravitational potential (Φ) of Sgr A*, as per the data in study is listed below.
Mass ≈ 4.25 × 106 Me [10], Radius (pericenter) ≈ 120 AU [10];
ΦCurrent method ≈ 59,141,880,107,879 J/kg;
ΦProposed method ≈ 59,122,424,383,746 J/kg.
Gravitational waves can be produced by supermassive objects with no event horizon; it does not prove the existence of gravitational singularity.
These experiments are examined to show the validity of the proposed theory, and that is sufficient to prove compliance with other experiments. Further profound studies are needed to be performed in the future.
Indeed, gravitational singularity is not supposed to exist as per relativity. There is no solid evidence for the existence of supper massive objects with an event horizon. The article shows why the consideration of relativity in the gravitational field is crucial, and GR became more reliable by eliminating physically objectionable outcomes (singularity; event horizon). Even though the Einstein field equation was not assessed as part of this study, we have evaluated how classical physics is directly applied in GR. It is apparent that the proposed changes will only have a negligible impact on the results of tests performed on GR. This study will open new doors for a different perspective and enhanced research.
Cite this paper: Mathalaisamy, B. (2021) Relativistic Gravitational Field and Invalidity of Singularity. Journal of High Energy Physics, Gravitation and Cosmology, 7, 1102-1106. doi: 10.4236/jhepgc.2021.73065.
[1] Einstein, A. (1998) On the Electrodynamics of Moving Bodies. In: Stachel, J., Ed., Einstein’s Miraculous Year, Princeton University Press, USA.
[2] Luetzelschwab, J.W. (2003) Apparatus to Measure Relativistic Mass Increase. American Journal of Physics, 71, 878-884.
[3] Taylor, E.F. and Wheeler, J.A. (1992) Spacetime Physics. Second Edition, W.H. Freeman and Company, New York, 248-249.
[4] Einstein, A. (1952) The Foundation of the General Theory of Relativity. Dover, New York.
[5] Lindner, H.H. (2012) Beyond Newton and Einstein to Flowing Space. Physics Essays, 25, 500-509.
http://henrylindner.net/Writings/BeyondNewton.pdf
[6] Williams, D.R. (2017) Earth Fact Sheet. NASA/Goddard Space Flight Center.
[7] 2018 CODATA Value: Newtonian Constant of Gravitation. The NIST Reference on Constants, Units, and Uncertainty. NIST.
https://physics.nist.gov/cgi-bin/cuu/Value?bg
[8] Williams, D.R. (2013) Sun Fact Sheet. NASA Goddard Space Flight Center.
[9] Ryder, L. (2009) Einstein Field Equations, the Schwarzschild Solution and Experimental Tests of General Relativity. In: Introduction to General Relativity, Cambridge University Press, Cambridge, 137-179.
[10] Abuter, R., Amorim, A., Bauböck, M., et al. (2020) Detection of the Schwarzschild Precession in the Orbit of the Star S2 Near the Galactic Centre Massive Black Hole. A&A, 636, Article No. L5.
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Remarks on global existence and compactness for $L^2$ solutions in the critical nonlinear schrödinger equation in 2D
Remarks on global existence and compactness for
{L}^{2}
solutions in the critical nonlinear schrödinger equation in 2D
Gonzalez, Luis Vega
In the talk we shall present some recent results obtained with F. Merle about compactness of blow up solutions of the critical nonlinear Schrödinger equation for initial data in
{L}^{2}\left({𝐑}^{2}\right)
. They are based on and are complementary to some previous work of J. Bourgain about the concentration of the solution when it approaches to the blow up time.
author = {Gonzalez, Luis Vega},
title = {Remarks on global existence and compactness for $L^2$ solutions in the critical nonlinear schr\"odinger equation in {2D}},
AU - Gonzalez, Luis Vega
TI - Remarks on global existence and compactness for $L^2$ solutions in the critical nonlinear schrödinger equation in 2D
Gonzalez, Luis Vega. Remarks on global existence and compactness for $L^2$ solutions in the critical nonlinear schrödinger equation in 2D. Journées équations aux dérivées partielles (1998), article no. 13, 9 p. http://www.numdam.org/item/JEDP_1998____A13_0/
[B1] J. Bourgain, Some new estimates on oscillatory integrals, Essays on Fourier Analysis in Honour of E. Stein, Princeton UP 42 (1995), 83-112 | MR 96c:42028 | Zbl 0840.42007
[B2] J. Bourgain, Refinements of Strichartz's inequality and applications to 2D-NLS with critical nonlinearity, Preprint | Zbl 0917.35126
[B-L] H. Berestycki, P.L. Lions Nonlinear scalar field equations, Arch. Rat. Mech. Anal., 82 (1983), 313-375 | MR 84h:35054a | Zbl 0533.35029
[C] T. Cazenave An introduction to nonlinear Schrödinger equations, Textos de Metodos Matematicos 26 (Rio de Janeiro)
[C-W] T. Cazenave, F. Weissler Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear semigroups, partial differential equations and attractors, Lect. Notes in Math., 1394, Spr. Ver., 1989, 18-29 | MR 91a:35149 | Zbl 0694.35170
[G] R.T Glassey On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys. 18 (1977), 1794-1797 | MR 57 #842 | Zbl 0372.35009
[G-V] J. Ginibre, G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z 170, (1980), 109-136 | MR 82c:35018 | Zbl 0407.35063
[K] M.K. Kwong Uniqueness of positive solutions of Δu - u + up = 0 in RN, Arch. Rat. Mech. Ann. 105, (1989), 243-266 | MR 90d:35015 | Zbl 0676.35032
[M1] F. Merle Determination of blow-up solutions with minimal mass for non-linear Schrödinger equations with critical power, Duke Math. J., 69, (2) (1993), 427-454 | MR 94b:35262 | Zbl 0808.35141
[M2] F. Merle Lower bounds for the blow-up rate of solutions of the Zakharov equation in dimension two Comm. Pure and Appl. Math, Vol. XLIX, (1996), 8, 765-794 | MR 97d:35210 | Zbl 0856.35014
[MV] F. Merle, L. Vega Compactness at blow-up time for L2 solutions of the critical nonlinear Schrödinger equation. To appear in IMRN, 1998 | Zbl 0913.35126
[MVV] A. Moyua, A. Vargas, L. Vega Restriction theorems and maximal operators related to oscillatory integrals in ℝ³ to appear in Duke Math. J. | Zbl 0946.42011
[St] R. Strichartz Restriction of Fourier transforms to quadratic surfaces and decay of solutions to wave equations, Duke Math J., 44, (1977), 705-714 | MR 58 #23577 | Zbl 0372.35001
[W] M.I. Weinstein On the structure and formation of singularities of solutions to nonlinear dispersive equations Comm. P.D.E. 11, (1986), 545-565 | MR 87i:35026 | Zbl 0596.35022
[ZSS] V.E. Zakharov, V. V. Sobolev, and V.S. Synach Character of the singularity and stochastic phenomena in self-focusing, Zh. Eksper. Teoret. Fiz. 14 (1971), 390-393
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Basis (linear algebra) - Simple English Wikipedia, the free encyclopedia
This picture illustrates the standard basis in R2. The red and blue vectors are the elements of the basis; the green vector can be given with the basis vectors.
In linear algebra, a basis is a set of vectors in a given vector space with certain properties:
One can get any vector in the vector space by multiplying each of the basis vectors by different numbers, and then adding them up.
If any vector is removed from the basis, the property above is no longer satisfied.
The dimension of a given vector space is the number of elements of the basis.
{\displaystyle \mathbb {R} ^{3}}
is the vector space then:
{\displaystyle B=\{(1,0,0),(0,1,0),(0,0,1)\}}
{\displaystyle \mathbb {R} ^{3}}
It's easy to see that for any element of
{\displaystyle \mathbb {R} ^{3}}
it can be represented as a combination of the above basis. Let
{\displaystyle x}
be any element of
{\displaystyle \mathbb {R} ^{3}}
{\displaystyle x=(x_{1},x_{2},x_{3})}
{\displaystyle x_{1},x_{2}}
{\displaystyle x_{3}}
{\displaystyle \mathbb {R} }
then they can be written as
{\displaystyle x_{1}=1*x_{1}}
Then the combination equals the element
{\displaystyle x}
This shows that the set
{\displaystyle B}
{\displaystyle \mathbb {R} ^{3}}
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Basis_(linear_algebra)&oldid=6789537"
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Batch And Levenspiel Plots For Parallel And Series Reactors – Engineeringness
Batch and Levenspiel Plots
PFR and CSTR Levenspiel Plot Comparison
Levenspiel Plot For Reactor In A Series Arrangement
PFR With Recycle
A Batch reactor plot is a graphical representation of the volume of an isothermal system.
General shape of a Batch Reactor (Advanced Energy Materials Processing Laboratory, 2020)
Batch reactor plot (Advanced Energy Materials Processing Laboratory, 2020)
A Levenspiel plot is a representation of the continuous flow reactor; CSTR and PFR design equations as a function of conversion and is used to determine the volume of the reactor.
Shape of CSTR and PFR (Advanced Energy Materials Processing Laboratory, 2020)
The rate used for the CSTR is evaluated at the exit stream conditions while for the PFR the rate used is integrated over a range of conditions and we can solve this using Simpsons composite rule,
CSTR and PFR Levenspiel plot (Advanced Energy Materials Processing Laboratory, 2020)
PFR requires a smaller volume than the CSTR for a given conversion
When the reaction speed increases for a CSTR the Levenspiel plot will curve downwards as the conversion changes and will require a smaller CSTR volume.
PFR in series act as one large PFR and if the density is constant then the residence time is just the space time at the inlet conditions. For a CSTR multiple CSTRs in series require a smaller volume as a CSTR is evaluated at the output conditions and will make a series of CSTR’s smaller than one large CSTR, as when using multiple CSTRs the first tank operates at a lower conversion so the concentration of reactants will be higher so the rate will be greater and the volume required will be smaller.
CSTRs in series get close to the performance of PFR and the smaller the CSTRs the closer they get, but financial costs and available space and other factors make having lots of small CSTRs not practical when one PFR can be used.
CSTR Levenspiel plot in series (MIT, 2007)
Parallel reactors for equal-sized flow reactors, the feed stream is split evenly between the reactors. Parallel reactor arrangement is used for CSTRs as the reactors will be operating at the lowest conversion will be better to operate in series. For PFRs this arrangement behaves as one large PFR and is a common arrangement as used in industry or in Labourites.
PFR in a parallel arrangement (Santofimio, 2020)
Unreacted reactants can be recycled from the PFR exit stream, we define a recycle ratio, R when it is equal to zero (R = 0) then we have standard/normal plug flow and as R increases we develop mixed flow and the PFR starts to resemble the behaviour of a CSTR.
R = \frac{Volume of fluid recycled}{Volume of fluid leaving PFR}
We will be adding two new terms, single-pass conversion XS and overall conversion XO, the equations are below and have used species ‘A’ to represent the species used.
{X}_{A0} = \frac{{F}_{A0} – {F}_{Af}}{{F}_{A0}}
{X}_{AS} = \frac{{F}_{A1} – {F}_{A2}}{{F}_{A1}}
Single-pass conversion shows the fraction that is converted when it goes through the PFR once and overall conversion is the fraction converted in the final stream from the total inlet flow.
PFR with recycle diagram (Cheggstudy, 2020)
PFR with recycle is a difficult concept to get your head around and has a lot of keywords, that can trip you up if you don’t pay attention them, the best way is to do an example whilst looking at the answers and see what steps to do to solve this type of question in an exam, if you can do this example exam question without looking at the answers it will be extremely impressive!
Example – PFR with recycle (typical exam question)
In a PFR with recycle a reaction that is elementary and, in the liquid phase takes places, with an R = 1 and a conversion of 2/3, what is the conversion if there is no recycle stream?
Answer – PFR with recycle
First: do PFR with recycle stream
As liquid phase reaction only, the density is constant so the volumetric flow rate is constant, we will use the PFR mole balance but will use a slightly different version, this will help as there are different conversions and can be tricky to do.
Mole Balance PFR:
∆{F}_{A} = {r}_{A}∆V
Rate Equation:
–{r}_{A} = k{C}_{A}^{2}
From Stoichiometry:
{F}_{A} = v{C}_{A}
Draw the diagram as seen above with the information we already have; this will help you visualise the problem:
The volumetric flow rate (v) is initially:
{v}_{0}
and as the recycle ratio is one the volumetric flow rate in the recycle is same as feed stream, thus the stream going into the reactor after the recycle would be:
{v}_{0} + {v}_{0} = 2{v}_{0}
The final concentration is:
{C}_{Af} = \frac{{C}_{A0}}{3}
as the conversion is 2/3, this is from the overall conversion.
The concentration in the feed stream is:
{C}_{A1} = \left({C}_{A0} + {C}_{Af}\right) × \frac{1}{2} = \frac{2{C}_{A0}}{3}
this is because the volumetric flow rates are equal in the recycle stream and the feed stream. We multiply by a ½ as the recycle stream and feed are equal so assume perfect mixing.
Now take the PFR mole balance and the stoichiometric relationship to get:
v∆{C}_{A} = {r}_{A}∆V
And as volumetric flow rate into the reactor is:
v = 2{v}_{0}
we therefore get:
2{v}_{0}∆{C}_{A} = {r}_{A}∆V
\frac{∆{C}_{A}}{{r}_{A}}=\frac{∆V}{2{v}_{0}}
Then substitute in the rate law:
\frac{∆{C}_{A}}{–K{C}_{A}^{2}}=\frac{∆V}{2{v}_{0}}
Now integrate to get:
{\int }_{{C}_{A1}}^{{C}_{Af}}\frac{1}{K{C}_{A}} = \frac{V}{2{v}_{0}}
\frac{1}{K{C}_{Af}} – \frac{1}{K{C}_{A1}} = \frac{V}{2{v}_{0}}
We already know the values of the concentrations, so the left-hand side of the equation becomes:
\frac{1}{\left(\frac{K{C}_{Af}}{3}\right)}–\frac{1}{\left(\frac{K2{C}_{A1}}{3}\right)} = \frac{3}{2K{C}_{A0}}
\frac{3}{2K{C}_{A0}}= \frac{V}{2{v}_{0}}or \frac{KV{C}_{A0}}{{v}_{0}} = 3
This above relationship will be true whether the recycle stream is on or not!
Second: do PFR without recycle stream:
∆{F}_{A} = {r}_{A}∆V
As no recycle the volumetric flow rate entering the reactor is:
{v}_{0}
Thus, combining stoichiometry and mole balance:
{v}_{0}∆{C}_{A}={r}_{A}∆V
Now substitute in rate equation and integrate:
{\int }_{{C}_{A1}}^{C{*}_{Af}}\frac{1}{K{C}_{A}} = \frac{V}{{v}_{0}}
We have put an * on CAf as this value will be different when the recycle stream is on.
\frac{1}{C{*}_{Af}}–\frac{1}{{C}_{A1 }}=\frac{KV}{{v}_{0}}
\frac{{C}_{A0}}{C{*}_{Af}}–1 = \frac{{C}_{A0}KV}{{v}_{0}} = 3
\frac{{C}_{A0}}{C{*}_{Af}} = 4
So, the exit concentration (C*Af ) without recycle is ¼ of the feed concentration.
Conversion without Recycle:
{X}_{0} = 1–\frac{1}{4} = \frac{3}{4}
Conversion with Recycle: 2/3
Advanced Energy Materilas Processing Laboratory. (2020). CHE 309: Chemical Reaction Engineering. Retrieved from Advanced Energy Materilas Processing Laboratory: http://aempl.kist.re.kr/wp-content/files/Lecture-5_Ch2.pdf
Cheggstudy. (2020). Question: Problem 3: Recycle Reactor. Retrieved from Cheggstudy: https://www.chegg.com/homework-help/questions-and-answers/problem-3-recycle-reactor-farmer-michael-process-setting-recycle-reactor-farm-reaction-tak-q26913611
MIT. (2007). PFR vs. CSTR: Size and Selectivity. Retrieved from MIT: https://ocw.mit.edu/courses/chemical-engineering/10-37-chemical-and-biological-reaction-engineering-spring-2007/lecture-notes/lec09_03072007_w.pdf
Santofimio, D. S. (2020). Parallel Reactors.docx. Retrieved from Scribd: https://www.scribd.com/document/242835751/Parallel-Reactors-docx
Batch ReactorsCSTRPFRParallel seriesParallel reactorsLevenspiel
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Oncotic pressure - Wikipedia
Oncotic pressure, or colloid osmotic-pressure, is a form of osmotic pressure induced by the proteins, notably albumin,[1] in a blood vessel's plasma (blood/liquid) that causes a pull on fluid back into the capillary. Participating colloids displace water molecules, thus creating a relative water molecule deficit with water molecules moving back into the circulatory system within the lower venous pressure end of capillaries.
Above, we see a representation of fluid flow in the presence of colloids, with the left side representing surrounding tissues and the right representing whole blood. The presence of colloids can increase the flow towards the high concentration of colloids by creating colloid osmotic pressure in an otherwise state of equilibrium.
In the illustration above, we see how the osmotic pressure changes over the length of the capillary, with oncotic pressure remaining the same. Overall direction of fluid flow in relation to equal bidirectional flow is shown by the orange and black lines, respectively.
It has the opposing effect of both hydrostatic blood pressure pushing water and small molecules out of the blood into the interstitial spaces within the arterial end of capillaries and interstitial colloidal osmotic pressure. These interacting factors determine the partition balancing of extracellular water between the blood plasma and outside the blood stream.
Oncotic pressure strongly affects the physiological function of the circulatory system. It is suspected to have a major effect on the pressure across the glomerular filter. However, this concept has been strongly criticised and attention has been shifted to the impact of the intravascular glycocalyx layer as the major player.[2][3][4][5]
3 Physiological impact
'Oncotic' by definition is termed as 'pertaining to swelling', indicating the effect of oncotic imbalance on the swelling of tissues.
The word itself is derived from onco- and -ic; 'onco-' meaning 'pertaining to mass or tumors' and '-ic', which forms an adjective.
Throughout the body, dissolved compounds have an osmotic pressure. Because large plasma proteins cannot easily cross through the capillary walls, their effect on the osmotic pressure of the capillary interiors will, to some extent, balance out the tendency for fluid to leak out of the capillaries. In other words, the oncotic pressure tends to pull fluid into the capillaries. In conditions where plasma proteins are reduced, e.g. from being lost in the urine (proteinuria), there will be a reduction in oncotic pressure and an increase in filtration across the capillary, resulting in excess fluid buildup in the tissues (edema).
The large majority of oncotic pressure in capillaries is generated by the presence of high quantities of albumin, a protein that constitutes approximately 80% of the total oncotic pressure exerted by blood plasma on interstitial fluid[citation needed]. The total oncotic pressure of an average capillary is about 28 mmHg with albumin contributing approximately 22 mmHg of this oncotic pressure, despite only representing 50% of all protein in blood plasma at 35-50 g/L.[6][7] Because blood proteins cannot escape through capillary endothelium, oncotic pressure of capillary beds tends to draw water into the vessels. It is necessary to understand the oncotic pressure as a balance; because the blood proteins reduce interior permeability, less plasma fluid can exit the vessel.[7]
Oncotic pressure is represented by the symbol Π or π in the Starling equation and elsewhere. The Starling equation in particular describes filtration in volume/s (Jv) by relating oncotic pressure (πp) to capillary hydrostatic pressure (Pc), interstitial fluid hydrostatic pressure (Pi), and interstitial fluid oncotic pressure (πi), as well as several descriptive coefficients, as shown below:
{\displaystyle \ J_{v}=L_{\mathrm {p} }S([P_{\mathrm {c} }-P_{\mathrm {i} }]-\sigma [\pi _{\mathrm {p} }-\pi _{\mathrm {i} }])}
At the arteriolar end of the capillary, blood pressure starts at about 36 mm Hg and decreases to around 15 mm Hg at the venous end, with oncotic pressure at a stable 25–28 mm Hg. Within the capillary, reabsorption due to this venous pressure difference is estimated to be around 90% that of the filtered fluid, with the extra 10% being returned via lymphatics in order to maintain stable blood volume.[8]
Physiological impactEdit
In tissues, physiological disruption can arise with decreased oncotic pressure, which can be determined using blood tests for protein concentration.
Decreased colloidal osmotic pressure, most notably seen in hypoalbuminemia, can cause edema and decrease in blood volume as fluid is not reabsorbed into the bloodstream. Colloid pressure in these cases can be lost due to a number of different factors, but primarily decreased colloid production or increased loss of colloids through glomerular filtration.[6][9] This low pressure often correlates with poor surgical outcomes.[10]
In the clinical setting, there are two types of fluids that are used for intravenous drips: crystalloids and colloids. Crystalloids are aqueous solutions of mineral salts or other water-soluble molecules. Colloids contain larger insoluble molecules, such as gelatin. There is some debate concerning the advantages and disadvantages of using biological vs. synthetic colloid solutions.[11] Oncotic pressure values are approximately 290 mOsm per kg of water, which slightly differs from the osmotic pressure of the blood that has values approximating 300 mOsm /L.[citation needed] These colloidal solutions are typically used to remedy low colloid concentration, such as in hypoalbuminemia, but is also suspected to assist in injuries that typically increase fluid loss, such as burns.[12]
^ Moman, Rajat N.; Gupta, Nishant; Varacallo, Matthew (2021), "Physiology, Albumin", StatPearls, Treasure Island (FL): StatPearls Publishing, PMID 29083605, retrieved 2021-12-09
^ Levick JR, Michel CC (July 2010). "Microvascular fluid exchange and the revised Starling principle". Cardiovascular Research. 87 (2): 198–210. doi:10.1093/cvr/cvq062. PMID 20200043.
^ Raghunathan K, Murray PT, Beattie WS, Lobo DN, Myburgh J, Sladen R, et al. (November 2014). "Choice of fluid in acute illness: what should be given? An international consensus". British Journal of Anaesthesia. 113 (5): 772–83. doi:10.1093/bja/aeu301. PMID 25326478.
^ Woodcock TE, Woodcock TM (March 2012). "Revised Starling equation and the glycocalyx model of transvascular fluid exchange: an improved paradigm for prescribing intravenous fluid therapy". British Journal of Anaesthesia. 108 (3): 384–94. doi:10.1093/bja/aer515. PMID 22290457.
^ Maitra, Sayantan; Dutta, Dibyendu (2020-01-01), Preuss, Harry G.; Bagchi, Debasis (eds.), "Chapter 18 - Salt-induced inappropriate augmentation of renin–angiotensin–aldosterone system in chronic kidney disease", Dietary Sugar, Salt and Fat in Human Health, Academic Press, pp. 377–393, ISBN 978-0-12-816918-6, retrieved 2021-12-10
^ a b Gounden, Verena; Vashisht, Rishik; Jialal, Ishwarlal (2021), "Hypoalbuminemia", StatPearls, Treasure Island (FL): StatPearls Publishing, PMID 30252336, retrieved 2021-12-09
^ a b Guyton, Arthur C.; Hall, John E. (John Edward) (2006). Textbook of medical physiology. Library Genesis. Philadelphia : Elsevier Saunders. ISBN 978-0-7216-0240-0.
^ Darwish, Alex; Lui, Forshing (2021), "Physiology, Colloid Osmotic Pressure", StatPearls, Treasure Island (FL): StatPearls Publishing, PMID 31082111, retrieved 2021-12-09
^ Prasad, Rohan M.; Tikaria, Richa (2021), "Microalbuminuria", StatPearls, Treasure Island (FL): StatPearls Publishing, PMID 33085402, retrieved 2021-12-09
^ Kim, Sunghye; McClave, Stephen A.; Martindale, Robert G.; Miller, Keith R.; Hurt, Ryan T. (2017-11-01). "Hypoalbuminemia and Clinical Outcomes: What is the Mechanism behind the Relationship?". The American Surgeon. 83 (11): 1220–1227. doi:10.1177/000313481708301123. ISSN 1555-9823. PMID 29183523.
^ Wong, Christine; Koenig, Amie (March 2017). "The Colloid Controversy: Are Colloids Bad and What Are the Options?". The Veterinary Clinics of North America. Small Animal Practice. 47 (2): 411–421. doi:10.1016/j.cvsm.2016.09.008. ISSN 1878-1306. PMID 27914756.
^ Cartotto, Robert; Greenhalgh, David (October 2016). "Colloids in Acute Burn Resuscitation". Critical Care Clinics. 32 (4): 507–523. doi:10.1016/j.ccc.2016.06.002. ISSN 1557-8232. PMID 27600123.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Oncotic_pressure&oldid=1064513130"
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subtype - Maple Help
Home : Support : Online Help : Programming : Data Types : Type Checking : subtype
test whether one type is a subtype of another
subtype(s, t)
A type s is said to be a subtype of a type t if for every expression
e
the test type(e, s) evaluates to true, then the expression type(e, t) will also evaluate to true. If a type is identified with its extension, then the ``subtype'' relation is the relation of inclusion.
The subtype(s, t) function attempts to determine if the type s is a subtype of the type t.
If subtype can prove that s is a subtype of t, then the value true is returned. In the same manner, if subtype can prove that s is not a subtype of t, then the value false is returned. Otherwise, if it is not possible to compute whether one type is a subtype of another, the value FAIL is returned.
In general, it is not possible to compute whether one type is a subtype of another.
Note: Not all pairs of types are comparable. For example, the types list and set are disjoint types; no expression is both a list and a set. Thus, both subtype( 'set', 'list' ) and subtype( 'list', 'set' ) return false.
\mathrm{subtype}\left('\mathrm{integer}','\mathrm{rational}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{subtype}\left('\mathrm{polynom}','{\mathrm{string},\mathrm{algebraic}}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{subtype}\left('\mathrm{And}\left(\mathrm{name},\mathrm{algebraic}\right)','\mathrm{name}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{subtype}\left('\mathrm{Vector}\left(\mathrm{integer}\right)','\mathrm{Vector}\left(\mathrm{rational}\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{subtype}\left('\mathrm{specfunc}\left(\mathrm{integer},\mathrm{sin}\right)','\mathrm{typefunc}\left(\mathrm{rational},\mathrm{name}\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{subtype}\left('[\mathrm{integer},\mathrm{integer}]','\mathrm{list}\left(\mathrm{rational}\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{subtype}\left('[\mathrm{integer},\mathrm{integer}]','[\mathrm{anything},\mathrm{anything}]'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Generating Pseudorandom Numbers - MATLAB & Simulink - MathWorks América Latina
Common Pseudorandom Number Generation Methods
Acceptance-Rejection Methods
Pseudorandom numbers are generated by deterministic algorithms. They are "random" in the sense that, on average, they pass statistical tests regarding their distribution and correlation. They differ from true random numbers in that they are generated by an algorithm, rather than a truly random process.
Random number generators (RNGs) like those in MATLAB® are algorithms for generating pseudorandom numbers with a specified distribution.
For more information on the GUI for generating random numbers from supported distributions, see Explore the Random Number Generation UI.
Methods for generating pseudorandom numbers usually start with uniform random numbers, like the MATLAB rand function produces. The methods described in this section detail how to produce random numbers from other distributions.
Direct methods directly use the definition of the distribution.
For example, consider binomial random numbers. A binomial random number is the number of heads in
N
tosses of a coin with probability
p
of a heads on any single toss. If you generate
N
uniform random numbers on the interval (0,1) and count the number less than
p
, then the count is a binomial random number with parameters
N
p
This function is a simple implementation of a binomial RNG using the direct approach:
function X = directbinornd(N,p,m,n)
X = zeros(m,n); % Preallocate memory
for i = 1:m*n
u = rand(N,1);
X(i) = sum(u < p);
X = directbinornd(100,0.3,1e4,1);
histogram(X,101)
The binornd function uses a modified direct method, based on the definition of a binomial random variable as the sum of Bernoulli random variables.
You can easily convert the previous method to a random number generator for the Poisson distribution with parameter
\lambda
. The Poisson Distribution is the limiting case of the binomial distribution as
N
approaches infinity,
p
approaches zero, and
Np
is held fixed at
\lambda
. To generate Poisson random numbers, create a version of the previous generator that inputs
\lambda
N
p
, and internally sets
N
to some large number and
p
\lambda /N
The poissrnd function actually uses two direct methods:
A waiting time method for small values of
\lambda
A method due to Ahrens and Dieter for larger values of
\lambda
Inversion methods are based on the observation that continuous cumulative distribution functions (cdfs) range uniformly over the interval (0,1). If
u
is a uniform random number on (0,1), then using
X={F}^{-1}\left(U\right)
X
from a continuous distribution with specified cdf
F
For example, the following code generates random numbers from a specific Exponential Distribution using the inverse cdf and the MATLAB® uniform random number generator rand:
X = expinv(rand(1e4,1),mu);
Compare the distribution of the generated random numbers to the pdf of the specified exponential.
h = histogram(X,numbins,'Normalization','pdf');
x = linspace(h.BinEdges(1),h.BinEdges(end));
y = exppdf(x,mu);
Inversion methods also work for discrete distributions. To generate a random number
X
from a discrete distribution with probability mass vector
P\left(X={x}_{i}\right)={p}_{i}
{x}_{0}<{x}_{1}<{x}_{2}<...
, generate a uniform random number
u
on (0,1) and then set
X={x}_{i}
F\left({x}_{i-1}\right)<u<F\left({x}_{i}\right)
For example, the following function implements an inversion method for a discrete distribution with probability mass vector
p
function X = discreteinvrnd(p,m,n)
I = find(u < cumsum(p));
X(i) = min(I);
Use the function to generate random numbers from any discrete distribution.
p = [0.1 0.2 0.3 0.2 0.1 0.1]; % Probability mass function (pmf) values
X = discreteinvrnd(p,1e4,1);
Alternatively, you can use the discretize function to generate discrete random numbers.
X = discretize(rand(1e4,1),[0 cusmsum(p)]);
Plot the histogram of the generated random numbers, and confirm then the distribution follows the specified pmf values.
histogram(categorical(X),'Normalization','probability')
The functional form of some distributions makes it difficult or time-consuming to generate random numbers using direct or inversion methods. Acceptance-rejection methods provide an alternative in these cases.
Acceptance-rejection methods begin with uniform random numbers, but require an additional random number generator. If your goal is to generate a random number from a continuous distribution with pdf
, acceptance-rejection methods first generate a random number from a continuous distribution with pdf
g
f\left(x\right)\le cg\left(x\right)
c
x
A continuous acceptance-rejection RNG proceeds as follows:
Chooses a density
g
Finds a constant
c
f\left(x\right)/g\left(x\right)\le c
x
Generates a uniform random number
u
v
g
cu\le f\left(v\right)/g\left(v\right)
, accepts and returns
v
. Otherwise, rejects
v
and goes to step 3.
For efficiency, a "cheap" method is necessary for generating random numbers from
g
, and the scalar
c
should be small. The expected number of iterations to produce a single random number is
c
The following function implements an acceptance-rejection method for generating random numbers from pdf
f
g
, the RNG grnd for
g
c
function X = accrejrnd(f,g,grnd,c,m,n)
while accept == false
u = rand();
v = grnd();
if c*u <= f(v)/g(v)
f\left(x\right)=x{e}^{-{x}^{2}/2}
satisfies the conditions for a pdf on
\left[0,\infty \right)
(nonnegative and integrates to 1). The exponential pdf with mean 1,
f\left(x\right)={e}^{-x}
, dominates
g
c
greater than about 2.2. Thus, you can use rand and exprnd to generate random numbers from
f
f = @(x)x.*exp(-(x.^2)/2);
g = @(x)exp(-x);
grnd = @()exprnd(1);
X = accrejrnd(f,g,grnd,2.2,1e4,1);
The pdf
is actually a Rayleigh Distribution with shape parameter 1. This example compares the distribution of random numbers generated by the acceptance-rejection method with those generated by raylrnd:
Y = raylrnd(1,1e4,1);
histogram(Y)
legend('A-R RNG','Rayleigh RNG')
The raylrnd function uses a transformation method, expressing a Rayleigh random variable in terms of a chi-square random variable, which you compute using randn.
Acceptance-rejection methods also work for discrete distributions. In this case, the goal is to generate random numbers from a distribution with probability mass
{P}_{p}\left(X=i\right)={p}_{i}
, assuming that you have a method for generating random numbers from a distribution with probability mass
{P}_{q}\left(X=i\right)={q}_{i}
. The RNG proceeds as follows:
{P}_{q}
c
{p}_{i}/{q}_{i}\le c
i
u
v
{P}_{q}
cu\le {p}_{v}/{q}_{v}
v
v
|
Use proportions to solve each of the problems below.
At the zoo, three adult lions together eat
250
pounds of food a day. If two more adult lions joined the group and ate food at the same rate as the original three, how much food would the zoo need to provide all five lions each day?
Write a proportion comparing the number of lions and the pounds of food per day.
416.7
lbs of food per day
Byron can read
45
pages in an hour. At that rate, how long would it take him to read the new
700
-page Terry Cotter book?
Write a proportion comparing the number of pages and time (in hours).
What is the unit rate of pounds of food per lion in part (a)?
5
lions sharing
416.7
lbs of food. How many does each lion get?
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Theoretical Model of Droplets Motions on Solid Surface With Radial Wettable and Evaporation Rate Gradients | IMECE | ASME Digital Collection
Yanjie Yang,
Zan Wu,
National Institute of Defense Technology Innovation, Beijing, China
Yang, Y, Wu, Z, Chen, X, Sundén, B, & Huang, Y. "Theoretical Model of Droplets Motions on Solid Surface With Radial Wettable and Evaporation Rate Gradients." Proceedings of the ASME 2018 International Mechanical Engineering Congress and Exposition. Volume 8A: Heat Transfer and Thermal Engineering. Pittsburgh, Pennsylvania, USA. November 9–15, 2018. V08AT10A028. ASME. https://doi.org/10.1115/IMECE2018-87890
Wettability gradient in radial direction and evaporation rate gradient can cause droplet motion on a solid surface. Here a theoretical model is proposed. Besides, an equation of droplet velocity is derived on a solid surface. We consider the wettability and evaporation rate gradients are mainly caused by the chemical composition and surface roughness, only along the radial direction. Surface tension at the liquid-vapor interface is constant as it is assumed that the temperature does not change during the whole process. Thus, Marangoni effect induced by the liquid-vapor surface tension gradient is neglected. Besides, as droplet size is set as less than the capillary length (
l=γ/ρg
), the gravity effect is ignored as well. The velocity at the droplet center on a gradient surface along the radial direction is half of that along the x-direction. With the simulation of water droplet, the center velocity decreases with time and the droplet radius increases at the beginning part and then decreases.
Drops, Evaporation, Surface tension, Vapors, Gravity (Force), Simulation, Surface roughness, Temperature, Water
Effects of Free Surface Evaporation on Water Nano-Droplet Wetting Kinetics: A Molecular Dynamics Study
|
Robert Langlands - Wikipedia
Robert Phelan Langlands, CC FRS FRSC (/ˈlæŋləndz/; born October 6, 1936) is a Canadian [1][2] mathematician. He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study of Galois groups in number theory,[3][4] for which he received the 2018 Abel Prize. He was an emeritus professor and occupied Albert Einstein's office at the Institute for Advanced Study in Princeton, until 2020 when he retired.[5]
Wolf Prize (1995–96)
Cassius Ionescu-Tulcea
Langlands was born in New Westminster, British Columbia, Canada, in 1936 to Robert Langlands and Kathleen J Phelan. He has two younger sisters (Mary b 1938; Sally b 1941). In 1945, his family moved to White Rock, near the US border, where his parents had a building supply and construction business.[6][3][1]
He graduated from Semiahmoo Secondary School and started enrolling at the University of British Columbia at the age of 16, receiving his undergraduate degree in Mathematics in 1957;[7] he continued at UBC to receive an M. Sc. in 1958. He then went to Yale University where he received a Ph.D. in 1960.[8]
His first academic position was at Princeton University from 1960 to 1967, where he worked as an associate professor.[3] He spent a year in Turkey at METU during 1967–68 in an office next to Cahit Arf's.[9] He was a Miller Research Fellow at the University of California, Berkeley from 1964 to 1965, then was a professor at Yale University from 1967 to 1972. He was appointed Hermann Weyl Professor at the Institute for Advanced Study in 1972, and became professor emeritus in January 2007.[5]
Langlands' Ph.D. thesis was on the analytical theory of Lie semigroups,[10] but he soon moved into representation theory, adapting the methods of Harish-Chandra to the theory of automorphic forms. His first accomplishment in this field was a formula for the dimension of certain spaces of automorphic forms, in which particular types of Harish-Chandra's discrete series appeared.[11][12]
He next constructed an analytical theory of Eisenstein series for reductive groups of rank greater than one, thus extending work of Hans Maass, Walter Roelcke, and Atle Selberg from the early 1950s for rank one groups such as
{\displaystyle \mathrm {SL} (2)}
. This amounted to describing in general terms the continuous spectra of arithmetic quotients, and showing that all automorphic forms arise in terms of cusp forms and the residues of Eisenstein series induced from cusp forms on smaller subgroups. As a first application, he proved the Weil conjecture on Tamagawa numbers for the large class of arbitrary simply connected Chevalley groups defined over the rational numbers. Previously this had been known only in a few isolated cases and for certain classical groups where it could be shown by induction.[13]
As a second application of this work, he was able to show meromorphic continuation for a large class of
{\displaystyle L}
-functions arising in the theory of automorphic forms, not previously known to have them. These occurred in the constant terms of Eisenstein series, and meromorphicity as well as a weak functional equation were a consequence of functional equations for Eisenstein series. This work led in turn, in the winter of 1966–67, to the now well known conjectures[14] making up what is often called the Langlands program. Very roughly speaking, they propose a huge generalization of previously known examples of reciprocity, including (a) classical class field theory, in which characters of local and arithmetic abelian Galois groups are identified with characters of local multiplicative groups and the idele quotient group, respectively; (b) earlier results of Martin Eichler and Goro Shimura in which the Hasse–Weil zeta functions of arithmetic quotients of the upper half plane are identified with
{\displaystyle L}
-functions occurring in Hecke's theory of holomorphic automorphic forms. These conjectures were first posed in relatively complete form in a famous letter to Weil,[14] written in January 1967. It was in this letter that he introduced what has since become known as the
{\displaystyle L}
-group and along with it, the notion of functoriality.
The book by Hervé Jacquet and Langlands on
{\displaystyle \mathrm {GL} (2)}
presented a theory of automorphic forms for the general linear group
{\displaystyle \mathrm {GL} (2)}
, establishing among other things the Jacquet–Langlands correspondence showing that functoriality was capable of explaining very precisely how automorphic forms for
{\displaystyle \mathrm {GL} (2)}
related to those for quaternion algebras. This book applied the adelic trace formula for
{\displaystyle \mathrm {GL} (2)}
and quaternion algebras to do this. Subsequently, James Arthur, a student of Langlands while he was at Yale, successfully developed the trace formula for groups of higher rank. This has become a major tool in attacking functoriality in general, and in particular has been applied to demonstrating that the Hasse–Weil zeta functions of certain Shimura varieties are among the
{\displaystyle L}
-functions arising from automorphic forms.[15]
The functoriality conjecture is far from proven, but a special case (the octahedral Artin conjecture, proved by Langlands[16] and Tunnell[17]) was the starting point of Andrew Wiles' attack on the Taniyama–Shimura conjecture and Fermat's Last Theorem.
In the mid-1980s Langlands turned his attention[18] to physics, particularly the problems of percolation and conformal invariance. In 1995, Langlands started a collaboration with Bill Casselman at the University of British Columbia with the aim of posting nearly all of his writings—including publications, preprints, as well as selected correspondence—on the Internet. The correspondence includes a copy of the original letter to Weil that introduced the
{\displaystyle L}
-group. In recent years he has turned his attention back to automorphic forms, working in particular on a theme he calls "beyond endoscopy".[19]
Langlands has received the 1996 Wolf Prize (which he shared with Andrew Wiles),[20] the 2005 AMS Steele Prize, the 1980 Jeffery–Williams Prize, the 1988 NAS Award in Mathematics from the National Academy of Sciences,[21] the 2006 Nemmers Prize in Mathematics, the 2007 Shaw Prize in Mathematical Sciences (with Richard Taylor) for his work on automorphic forms. In 2018, Langlands was awarded the Abel Prize for "his visionary program connecting representation theory to number theory".[22]
He was elected a Fellow of the Royal Society of Canada in 1972 and a Fellow of the Royal Society in 1981.[23][24] In 2012, he became a fellow of the American Mathematical Society.[25] Langlands was elected as a member of the American Academy of Arts and Sciences in 1990.[26] He was elected as a member of the National Academy of Sciences in 1993[27] and a member of the American Philosophical Society 2004.[28]
Among other honorary degrees, in 2003, Langlands received a doctorate honoris causa from Université Laval.[29]
In 2019, Langlands was appointed a Companion of the Order of Canada.[30][31]
On January 10, 2020, Langlands was honoured at Semiahmoo Secondary, which installed a mural to celebrate his contributions to mathematics.
Langlands has been married to Charlotte Lorraine Cheverie (b 1935) since 1957. They have four children (2 daughters and 2 sons).[3] He holds Canadian and American citizenships.
Langlands spent a year in Turkey in 1967-68, where his office at the Middle East Technical University was next to that of Cahit Arf.[32][33] In addition to his mathematical studies, Langlands likes to learn foreign languages, both for better understanding of foreign publications on his topic and just as a hobby. He speaks English, French, Turkish and German, and reads (but does not speak) Russian.[33]
Euler Products, New Haven: Yale University Press, 1967, ISBN 0-300-01395-7
On the Functional Equations Satisfied by Eisenstein Series, Berlin: Springer, 1976, ISBN 3-540-07872-X
Base Change for GL(2), Princeton: Princeton University Press, 1980, ISBN 0-691-08272-3
Automorphic Representations, Shimura Varieties, and Motives. Ein Märchen (PDF), Chelsea Publishing Company, 1979
Endoscopic group
Jacquet–Langlands correspondence
Langlands classification
Langlands decomposition
Langlands group
Langlands–Shahidi method
Local Langlands conjectures
Standard L-function
Taniyama group
^ a b Alex Bellos (20 March 2018). "Abel Prize 2018: Robert Langlands wins for 'unified theory of maths'". The Guardian. Retrieved 26 March 2018.
^ "Robert Phelan Langlands". NAS. Retrieved 26 March 2018.
^ a b c d Contento, Sandro (March 27, 2015), "The Canadian Who Reinvented Mathematics", Toronto Star
^ D Mackenzie (2000) Fermat's Last Theorem's First Cousin, Science 287(5454), 792-793.
^ a b Edward Frenkel (2013). "preface". Love and Math: The Heart of Hidden Reality. Basic Books. ISBN 978-0465050741. Robert Langlands, the mathematician who currently occupies Albert Einstein's office at the Institute for Advanced Study in Princeton
^ "UBC Newsletter: Robert Langlands Interview" (PDF). 2010.
^ Kenneth, Chang (2018-03-20). "Robert P. Langlands Is Awarded the Abel Prize, a Top Math Honor". The New York Times. Retrieved 20 March 2018.
^ "Canadian mathematician Robert Langlands wins Abel Prize for 2018". The New Indian Express. 21 March 2018. Retrieved 26 March 2018.
^ "Robert Langlands wins Abel Prize 2018 for 'unified theory of maths' | Mathematics Department". math.metu.edu.tr. Retrieved 2021-07-26.
^ For context, see the note by Derek Robinson at the IAS site
^ "IAS publication paper 14". IAS. Retrieved 26 March 2018.
^ "MR review". Mathscinet. MR 0156362.
^ Langlands, Robert P. (1966), "The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups", Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., Providence, R.I.: Amer. Math. Soc., pp. 143–148, MR 0213362
^ a b "IAS paper 43". IAS. Retrieved 26 March 2018.
^ "IAS paper 60". Institute of Advanced Studies. Retrieved 26 March 2018.
^ Langlands, Robert P, Base change for GL(2). Annals of Mathematics Studies, 96. Princeton University Press, Princeton, N.J.; ISBN 0-691-08263-4; MR 574808
^ Tunnell, Jerrold, Artin's conjecture for representations of octahedral type, Bulletin of the American Mathematical Society (N.S.) 5 (1981), no. 2, 173–175.
^ "IAS publication". Retrieved 26 March 2018.
^ "IAS paper 25". IAS. Retrieved 26 March 2018.
^ "AMS Notices" (PDF).
^ "NAS Award in Mathematics". National Academy of Sciences. Retrieved 13 February 2011.
^ "News: Robert P. Langlands receives the Abel Prize". www.abelprize.no. 2018-03-20. Retrieved 2018-03-20.
^ "Search Fellows". Royal Society of Canada. Retrieved April 3, 2018.
^ "Robert Langlands". Royal Society. Retrieved April 3, 2018.
^ "Robert Phelan Langlands". American Academy of Arts & Sciences. Retrieved 2021-03-22.
^ "Robert Langlands". www.nasonline.org. Retrieved 2021-03-22.
^ "Robert Langlands, Université Laval". Archived from the original on 2016-06-29. Retrieved 2017-03-01.
^ Office of the Secretary to the Governor General (2019-06-20). "Governor General Announces 83 New Appointments to the Order of Canada". The Governor General of Canada. Retrieved 2019-06-27.
^ Dunlevy, T'Cha (2019-06-27). "Alanis Obomsawin, 15 other Quebecers to receive Order of Canada". Montreal Gazette. Archived from the original on 2019-07-04. Retrieved 2019-07-04.
^ The work of Robert Langlands – Miscellaneous items, Digital Mathematics Archive, UBC SunSITE, last accessed 2013-12-10.
^ a b Interview with Robert Langlands, UBC Dept. of Math., 2010; last accessed 2014-04-05.
Wikiquote has quotations related to Robert Langlands.
O'Connor, John J.; Robertson, Edmund F., "Robert Langlands", MacTutor History of Mathematics archive, University of St Andrews
Robert Langlands at the Mathematics Genealogy Project
The work of Robert Langlands (a nearly complete archive)
Faculty page at IAS
The Abel Prize Interview 2018 with Robert Langlands
Contenta, Sandro. "The Canadian who reinvented mathematics". Toronto Star. Retrieved 28 March 2015.
Julia Mueller, On the genesis of Robert P. Langlands' conjectures and his letter to André Weil, Bull. Amer. Math. Soc., January 25, 2018
Retrieved from "https://en.wikipedia.org/w/index.php?title=Robert_Langlands&oldid=1077652735"
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Find the perimeter and area of each algebra tile shape below. Be sure to combine like terms.
Remember, the perimeter is the sum of all the side lengths around the shape. Algebra Tile Diagram: A line is added that is drawn around the perimeter of the figure.
Labeling all the side lengths may prove to be helpful for you. Labels added to the sides of the tiles, starting at the top left, and going clockwise: x, 1, 1, 1, 1, 1, x minus 3, 1, x, x, 1, 1, 1
Add up all these values to find the perimeter.
The area is the space within the orange lines.
Add up the values of the algebra tiles to find the area.
P=4x+6
A=4x+4
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{\displaystyle f(x)\,=\,2x^{3}e^{3x+5}}
The Chain Rule: I{\displaystyle f}
{\displaystyle g}
{\displaystyle (f\circ g)'(x)=f'(g(x))\cdot g'(x).}
The Product Rule: I{\displaystyle f}
{\displaystyle g}
{\displaystyle (fg)'(x)=f'(x)\cdot g(x)+f(x)\cdot g'(x).}
{\displaystyle \left(x^{n}\right)'\,=\,nx^{n-1},}
{\displaystyle n\neq 0}
{\displaystyle e^{x}}
{\displaystyle (e^{x})'\,=\,e^{x}.}
We need to start by identifying the two functions that are being multiplied together so we can apply the product rule.
{\displaystyle g(x)\,=\,2x^{3},\,}
{\displaystyle \,h(x)\,=\,e^{3x+5}}
We can now apply the three advanced techniques.This allows us to see that
{\displaystyle {\begin{array}{rcl}f'(x)&=&2(x^{3})'e^{3x+5}+2x^{3}(e^{3x+5})'\\&=&6x^{2}e^{3x+5}+2x^{3}(3e^{3x+5})\\&=&6x^{2}e^{3x+5}+6x^{3}e^{3x+5}\end{array}}}
{\displaystyle 6x^{2}e^{3x+5}+6x^{3}e^{3x+5}}
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Urea reduction ratio - wikidoc
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The urea reduction ratio (URR), is a dimensionless number used to quantify hemodialysis treatment adequacy.
3 Relation to Kt/V
4 Nomogram relating Kt/V and URR
5 Limitations of URR vs. Kt/V
6 Minimally adequate dose in terms of URR
{\displaystyle URR={\frac {U_{pre}-U_{post}}{U_{pre}}}\times 100\%}
Upre is the pre-dialysis urea level
Upost is the post-dialysis urea level
Whereas the URR is formally defined as the urea reduction "ratio", in practice it is informally multiplied by 100% as shown in the formula above, and expressed as a percent.
The URR was first popularized by Lowrie and Lew in 1991 as a method of measuring amount of dialysis that correlated with patient outcome.[1] This method is very useful because of its simplicity. It permits easy monitoring of the amount of dialysis therapy delivered to individual patients, as well as across dialysis units, groups of units, states, regions, or countries, because monthly predialysis and postdialysis urea nitrogen values are routinely measured. It also permits quality control and improvement initiatives and regulatory oversight. The United States Renal Data Systems (USRDS) publishes annual data regarding the URR values being delivered to dialysis patients across the United States. The ESRD networks monitor therapy across groups of states. The European Renal Association (ERA-EDTA) Registry covers most European countries, and DOPPS (Dialysis Outcomes Practice Patterns Study) records and analyzes URR and other data from selected dialysis units located in countries across the world.
Relation to Kt/V
Mathematically, the URR is closely related to Kt/V, and the two quantities can be derived from another with more or less precision, depending on the amount of additional information available about a given dialysis session.
Kt/V is one of the reference methods by which the amount of dialysis given is measured. Kt/V, like the URR, focuses on urea as the target solute, and is based on the assumption that removal of urea is from a single space - urea distribution volume, or
{\displaystyle V\,}
similar in capacity to the total body water. The urea distribution volume
{\displaystyle V\,}
, although traditionally thought of as 60% of body weight, may actually be closer to 50% of the body weight in women and 55% in men with stage V (GFR < 15 ml/min) chronic kidney disease. The clearance of urea during the dialysis session
{\displaystyle K\,}
can be expressed in either
{\displaystyle {\frac {ml}{min}}}
{\displaystyle {\frac {L}{hr}}}
Time or
{\displaystyle t\,}
is the duration of the dialysis session, measured either in minutes or hours. So
{\displaystyle K\cdot t}
is also a volume, either
{\displaystyle {\frac {ml}{min}}\cdot min=ml}
{\displaystyle {\frac {L}{hr}}\cdot hr=L}
, and represents the volume of blood (in ml or L) cleared of urea during the dialysis session. Because
{\displaystyle V\,}
is also a volume, the ratio of
{\displaystyle {\frac {K\cdot t}{V}}}
has dimensions of
{\displaystyle {\frac {ml}{ml}}}
{\displaystyle {\frac {L}{L}}}
, making it a "dimensionless" ratio.
In a simplified model of urea removal from a fixed volume with no urea generation,
{\displaystyle {\frac {K\cdot t}{V}}}
{\displaystyle URR\,}
by the following relationship:
{\displaystyle {\frac {K\cdot t}{V}}=-ln(1-URR)}
In actual fact, this relationship is made a bit more complex by the fact that fluid is removed during dialysis, so the removal space V shrinks, and because a small amount of urea is generated during the dialysis session. Both of these factors make the actual post-dialysis serum urea level higher than expected, and the URR lower than expected, when the extremely simplified equation above, is used.
A more accurate relationship between URR and Kt/V can be derived by single-pool, variable volume urea kinetic modeling. A simplified estimating equation also can be used [2]. This gives results that are quite similar to formal urea modeling as long as dialysis treatments of 2-6 hours in duration are given, and Kt/V is between 0.7 and 2.0.
{\displaystyle {\frac {K\cdot t}{V}}=-ln((1-URR)-0.008\cdot t)+(4-3.5(1-URR))\cdot {\frac {0.55\cdot UF}{V}}}
{\displaystyle (0.008\cdot t)}
term is a function of the dialysis session duration (t), and adjusts for the amount of urea generated during the dialysis session. The second term,
{\displaystyle (4-3.5(1-URR))\cdot {\frac {0.55\cdot UF}{V}}}
adjusts for the additional urea that is cleared from the body through volume contraction.
{\displaystyle {\frac {0.55\cdot UF}{V}}}
{\displaystyle {\frac {UF}{W}}}
, where UF = ultrafiltrate removed during dialysis (estimated as the weight lost during the treatment) and W = postdialysis body weight, and because dialysis sessions given 3 times per week are usually about 3.5 hours long, the above equation can be simplified to:
{\displaystyle {\frac {K\cdot t}{V}}=-ln((1-URR)-0.03)+(4-3.5(1-URR))\cdot {\frac {UF}{W}}}
Nomogram relating Kt/V and URR
File:Urr ktv.svg
Instead of equations, a nomogram can be used to easily estimate Kt/V from the URR in clinical practice. To use the nomogram, one needs to know the postdialysis weight (W) as well as the amount of weight (fluid) loss during the dialysis session (UF). First, find the URR on the vertical axis, then move over to the proper isopleth (curved line) depending on the amount of weight lost during dialysis (UF/W). Then drop down to the horizontal axis to read off the Kt/V value:
Limitations of URR vs. Kt/V
The URR is designed to measure the amount of dialysis given when the dialysis clearance of urea greatly exceeds the urea generation rate. In continuous hemodialysis or in peritoneal dialysis, for example, a considerable amount of dialysis is delivered, but the urea level remains roughly constant after the initial treatment of uremia, so the URR is essentially zero. In long slow overnight dialysis, if simplified equations are used, the URR also underestimates the amount of dialysis due to urea generation during the long dialytic session. For this reason, the kinetically modeled Kt/V is always recommended as the best measure of dialysis adequacy. The Kt/V, even that derived by formal modeling, is primarily based on the URR, and so it contains little additional information in terms of the amount of dialysis that was delivered. Since the URR and Kt/V are so closely related, their predictive power in terms of patient outcome is similar. However, use of Kt/V and urea modeling in general allows for comparing expected with predicted dose of dialysis, which can be used to analyze dialysis treatments and dialyzer clearances and in troubleshooting and quality control activities. Also, Kt/V permits calculation of the urea generation rate, which can give clues about a patient's protein intake.
Minimally adequate dose in terms of URR
In the standard 3x/week hemodialysis schedule a URR of 65% is considered the minimum acceptable dose, corresponding to a minimum Kt/V of 1.2 [3] When dialysis is given more frequently than three times a week, the minimum acceptable URR is lower; because more dialysis treatments are given over the week, the dose of dialysis for each treatment does not need to be as large. Also minimally acceptable values for URR (and Kt/V) can be reduced in patients who have substantial amounts of residual renal function [4].
Kt/V calculator - (estimating equation) medindia.com
Kt/V calculator - (Full kinetic model) hdcn.com
↑ Owen WF Jr, Lew NL, Liu Y, Lowrie EG, Lazarus JM. The urea reduction ratio and serum albumin concentration as predictors of mortality in patients undergoing hemodialysis. N Engl J Med. 1993 Sep 30;329(14):1001-6. PMID 8366899
↑ Daugirdas JT. Second generation logarithmic estimates of single-pool variable volume Kt/V: an analysis of error. J Am Soc Nephrol. 1993 Nov;4(5):1205-13. PMID 8305648
↑ KDOQI 2006 Hemodialysis Adequacy Guidelines. Guideline 4. [1].
↑ KDOQI 2006 Hemodialysis Adequacy Guidelines. CPR (Clinical Practice Recommendation) #4. [2]
Retrieved from "https://www.wikidoc.org/index.php?title=Urea_reduction_ratio&oldid=690932"
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Why Zeldovich Failed to Estimate the Precise Value of Cosmological Constant in Planck Unit?
Department of Physics, University of Bastar, Jagdalpur, India.
Observational result shows that universe is not only expanding but also this expansion is accelerating itself [1] . To explain this phenomenon, it has been presumed that some kind of mysterious energy does exist which impose a negative pressure and drive this expansion away. Therefore, it is hypothesized that this mysterious energy is cosmological constant which corresponds to vacuum energy density or dark energy [2] . Here, vacuum energy density and cosmological constant have been used interchangeably.
Quantum mechanics (QM) attempted to estimate its theoretical value by assuming that zero point energy (ZPE) might be giving rise this vacuum energy and summed all ZPE of ground state but estimated value is larger by 120 orders of magnitude to observed value; this disagreement is known as cosmological constant problem in physics [2] .
So far, many other theories have been propounded to estimate its value, its brief introduction can be found in Ref [3] [4] and references therein. Still this disagreement exists in a range between 46 and 120 orders of magnitude [5] [6] . A recent solution has been proposed in a different perspective e.g. see Ref. [7] [8] .
In the wake of this discrepancy, Zeldovich comes up with a different approach and believed that instead of ZPE the quantum fluctuations of empty space might be reason behind the origin of this energy and empirically proposed an equation to estimate its theoretical value, his emulated expression is written as below (interested reader may refer to Ref. [3] for his argument to writing that expression)
{\rho }_{E}\sim G\frac{{m}^{6}{c}^{4}}{{h}^{4}}
where, all used constants hold its usual meanings and values. The mass m is only used variable here. If we take this mass m as Planck mass, it corresponds to,
{\rho }_{E}\sim \frac{{c}^{7}}{{G}^{2}h}
substituting the numerical value of all used constant, its numerical value is about 10112 J/m3 but observed value is 10−9 J/m3, thus estimated value is still larger by over 120 orders of magnitude and problem persisted constantly.
Notwithstanding, from Equation (1) he estimated its value only larger by 9 orders of magnitude in pion mass but there is no clear reason or explanation to take this mass. Furthermore, this expression was emulated empirically; there isn’t its theoretical derivation from any established theory, however, it is believed that QM can’t predict its precise value which is a serious failure of this theory. To make compatible the quantum theory with cosmological constant; this disagreement must be explained thoroughly.
In this rapid communication, to meet the long standing cosmological constant problem, we revisited the Zeldovich’s idea by asking a more subtle question; why pion mass gives relatively small value and why Planck mass/unit gives extreme value of vacuum energy density or cosmological constant? And, to investigate the reason behind it, we independently derived his empirically proposed equation by using a novel approach while inferring that neither classical nor quantum form of energy can explain this vacuum energy instead it might be another form of energy; a quantum-gravitational form of energy in someway. Coincidently, we found, the derived expression is same as the expression what was empirically proposed by him, so we able to explain and unravel why cosmological constant problem persisted in his idea and what might be its possible solutions.
2. Derivation of Quantum-Gravitational Form of Energy
On order to accomplish the objective, we empirically proposed a force balance equation which interrelate the quantum and classical force as written below,
\frac{hc}{{R}_{Q}^{2}}\times \frac{{m}_{Q}^{2}{c}^{3}}{h}=\frac{{c}^{4}}{G}\times G\frac{{m}^{2}}{{R}^{2}}
where all used constants holds its usual meanings. The variable m stands for mass and R for space. To differentiate the space associated with quantum of force has been denoted as RQ and R for classical space; in same fashion the mass associated with quantum mechanics as mQ while mass associated with classical mechanics denoted as m only.
Theoretically, it shows a balance between quantum and classical force so we named it “Force Balance Equation” (hereafter we abbreviate it as “FBE”).
The LHS of this FBE denotes a quantum of force and it can be able to estimate the theoretical value of strong nuclear force if we replace RQ with size of nucleus and mQ with mass of pion or proton respectively [9] . Rest part of FBE denotes a classical force and these are well known in fundamental physics, it doesn’t need descriptions more.
This FBE is in consistence with other existing theory; it can be seen if we derive the relative strength of quantum of force/strong nuclear force to gravitational force from our proposed FBE, it gives us,
\frac{hc}{{R}_{Q}^{2}}=\left(\frac{hc}{G{m}_{Q}^{2}}\right)G\frac{{m}^{2}}{{R}^{2}}
K=\frac{hc}{G{m}_{Q}^{2}}
a constant quantity, it’s value is nearly 39 orders of magnitude if we take mQ as mass of proton, it suggest strong force is stronger to gravity by 39 orders of magnitude in quantity and this quantity is gravitational coupling constant itself.
Since, mass and space are only used variable in FBE which consists from quantum and classical force. Thus, a quantum relation between these variables will be,
{R}_{Q}=\frac{h}{{m}_{Q}c}
similarly, a classical relation as followings,
R=\frac{Gm}{{c}^{2}}
it has only gravitational constant.
Now, one can observed from FBE that the quantum force possess only Planck constant whereas classical force has only gravitational constant, however, on order to derive an expression of force which possess both constant i.e. a quantum-gravitational form of force we presumed that the mass/space associated with quantum mechanics is same, equal, replaceable and interchangeable with the mass/space associated with classical mechanics.
Here, initially we assumed that all the variables of FBE a quantum entity and substituted the quantum relation between mass and space in it i.e. Equation (6) in Equation (3); it doesn’t give any new expression. But, when we assumed all the variables as classical entity and substituted the classical relation between the mass and space in FBE i.e. Equation (7) in Equation (3), it gives,
\frac{h{c}^{5}}{{G}^{2}{m}^{2}}\times \frac{{R}^{2}{c}^{7}}{{G}^{2}h}=\frac{{c}^{4}}{G}\times \frac{{R}^{2}{c}^{8}}{{m}^{2}{G}^{3}}
its just inverted form of FBE (hereafter we call it “inverted FBE”) and first two expressions are the desired quantum-gravitational form of force.
These two expressions are only relevant to the objective of this article therefore the description of rest terms is discarded intentionally. Yet, it is worthy to describe the last terms of this FBE which denotes a new form of gravitational force; which is just inverted form of classical gravity. Its importance and role played in physics is to be discussed somewhere else.
A notable fact is that this “inverted FBE” is in Planck unit [10] because while deriving the desired quantum-gravitational force terms we presumed that mass/space are equal in classical and quantum mechanics and once we take it equal it originates the Planck mass/length/unit obviously.
The first quantum-gravitational force term of “inverted FBE” is just inverse of quantum of force and equivalent to Hawking temperature [11] , if we take
E=F\cdot R
where E denotes energy and R is space and
E={k}_{B}T
{k}_{B}
is Boltzmann constant and T is temperature.
Subsequent quantum-gravitational force terms corresponds to Zeldovich’s expression for cosmological constant in Planck unit as written in Equation (2) if we take again
E=F\cdot R
. This is the systematic derivation of his empirically proposed equation and it interprets that cosmological constant is nothing but just a quantum-gravitational form of quantum of force. It means, it’s nothing but just a quantum-gravitational energy density of vacuum itself.
But only problem is, as discussed in preceding sections, this expression can’t predict the precise value of vacuum energy density instead estimate value is extreme large. It hints, this expression is numerically incorrect somehow and we can proof it from inverted FBE if we calculate the relative strength of strong force to gravity, it gives us followings,
\frac{{R}^{2}{c}^{7}}{{G}^{2}h}=\left(\frac{G{m}^{2}}{hc}\right)\frac{{R}^{2}{c}^{8}}{{m}^{2}{G}^{3}}=\left(\frac{1}{K}\right)\frac{{R}^{2}{c}^{8}}{{m}^{2}{G}^{3}}
it says, strong force is weaker to gravity by 1039 order of magnitude and this contradict the observational results and defy the result what we obtained from Equation (4) because strong gravity is not observed by any experiment till date. It implies that this inverted FBE and subsequently the Zeldovich’s expression for vacuum energy density is numerically incorrect notwithstanding its dimensionally balanced.
To our understanding, we obtained this incorrect expression because while deriving it we presumed that classical mass/space is equal to quantum mass/space. This presumption might not be correct. Therefore, to meet this problem, it needs to discriminate the quantum space/mass and classical mass/space, which is neither equal nor interchangeable at all in any respect. Since, these entities are dimensionally equal thus it will be numerically not equal so there exists any dimensionless quantity which relates these entities in both mechanics.
3. Modifications of Planck Unit
On this ground, a correct relation between these entities can be derived if we compare the quantum and classical form of energy which is dependent of space only as written below,
\frac{hc}{{R}_{Q}}=\frac{G{m}^{2}}{R}
numerically, it gives us followings,
{R}_{Q}=\frac{hc}{G{m}^{2}}R=KR
this suggests, a dimensionless constant terms interrelate the space in classical and quantum mechanics. From this assertion, the correct mathematical expression of Planck unit from Equation (6), (7) and (11) will be,
{m}_{pl}^{2}=\frac{hc}{GK}
{R}_{pl}^{2}=\frac{GhK}{{c}^{3}}
it modified the well-established and accepted mathematical expression of Planck unit.
4. Derivation of Correct Mathematical Expression for Zeldovich’s Expression
Further, in this scenario, the correct expression of “inverted FBE” will be derived by substituting Equation (11) and (12) in Equation (3) which gives us followings,
\frac{h{c}^{5}}{{G}^{2}{m}^{2}{K}^{2}}\times \frac{{R}^{2}{c}^{7}}{{G}^{2}h{K}^{2}}=\frac{{c}^{4}}{G}\times \frac{{R}^{2}{c}^{8}}{{m}^{2}{G}^{3}}
its modified form of “inverted FBE” and this modification is valid since it gives the relative strength of quantum force to gravity similar to Equation (4).
Here, it first terms is correct expression for Hawking temperature with some modifications by taking again
E=F\cdot R
where R is from Equation (11) and
E={k}_{B}T
{k}_{B}
is Boltzmann constant and the temperature T is as written below,
T=\frac{h{c}^{3}}{Gm{k}_{B}}\left(\frac{1}{K}\right)
this suggests, Hawking predicted extreme high temperature; observed temperature will be as low as by factor of 1/K i.e. 1/1039 orders of magnitude in quantity. Since, this temperature is not measured by experimental set up till date thus our prediction needs confirmation.
In same manner, the second terms is correct expression for Zeldovich’s proposed equation for cosmological constant in terms of modified Planck unit by taking
E=F\cdot R
where R is from Equation (11), however, the energy density of vacuum will be,
{\rho }_{E}=\frac{{c}^{7}}{{G}^{2}h{K}^{3}}
substituting the K from Equation (11) it reduce to Equation (1), it suggests there is no existence of Zeldovich’s expression for vacuum energy in Planck unit, however, there is no cosmological constant problem at all in this perspective.
In summary, Zeldovich’s proposed equation for cosmological constant is actually quantum-gravitational energy density of vacuum which is quantum-gravitational form of quantum of force; and this force is mediated by pion thus in this mass it predicts its relatively small value. But, the mathematical expression of Planck unit is numerically incorrect so that in this unit he predicted its extreme value and the cosmological constant problem persisted unnecessarily there.
This conclusion also invokes that, if the mathematical expression of Planck unit is numerically incorrect therefore all those predictions which are based on this unit must be numerically incorrect. This conclusion might have viable impact on other branches of physics where this unit is frequently used to predict the theoretical value of other physical entities. Along with the cosmological constant, the Hawking temperature is such other physical entity where its theoretical value prediction is based on this unit, however, Hawking might also have predicted its incorrect temperature; the actual temperature will be something else, future research recommended in this area.
I would like to say thanks to Dr. HS for his critical comments on manuscript. I also pay my gratitude to home department, government of Chhattisgarh for financial support and providing opportunity to take this work.
Cite this paper: Chandra, K. (2019) Why Zeldovich Failed to Estimate the Precise Value of Cosmological Constant in Planck Unit?. Journal of High Energy Physics, Gravitation and Cosmology, 5, 1098-1104. doi: 10.4236/jhepgc.2019.54062.
[1] Riess, A.G. (1998) Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. The Astronomical Journal, 116, 1009-1038.
[2] Weinberg, S. (1989) The Cosmological Constant Problem. Review in Modern Physics, 61, 1-23.
[3] Zel’dovich, Ya.B. (1968) The Cosmological Constant and the of Elementary Particles. Soviet Physics Uspekhi, 11, 381-393.
[4] Rugh, S.E. and Zinkernagel, H. (2002) The Quantum Vacuum and Cosmological Constant Problem. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 33, 663-705.
[6] Carroll, S.M. (2001) The Cosmological Constant. Living Review in Relativity, 4, 1.
[7] Wang, Q., Zhu, Z. and Unruh, W.G. (2017) How the Huge Energy of Quantum Vacuum Gravitates to Drive the Slow Accelerating Expansion of the Universe. Physical Review D, 95, Article ID: 103504.
[8] Cree, S., Devis, T.M., Timothy, C.R., Wang, Q., Zhu, Z. and Unruh, W.G. (2018) Can the Fluctuations of the Quantum Vacuum Solve the Cosmological Constant Problem? Physical Review D, 98, Article ID: 063506.
[9] Yukawa, H. (1935) On the Interaction of Elementary Particles. Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, 17, 48-57.
[10] Borrow, J.D. (2002) The Constants of Nature; From Alpha to Omega—The Numbers that Encode the Deepest Secrets of the Universe. Pantheon, New York.
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Sébastien Darses1; Erwan Hillion1
1 Aix-Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France
The Nyman-Beurling criterion is an approximation problem in the space of square integrable functions on
\left(0,\infty \right)
, which is equivalent to the Riemann hypothesis. This involves dilations of the fractional part function by factors
{\theta }_{k}\in \left(0,1\right)
k\ge 1
. We develop probabilistic extensions of the Nyman-Beurling criterion by considering these
{\theta }_{k}
as random: this yields new structures and criteria, one of them having a significant overlap with the general strong Báez-Duarte criterion.
The main goal of the present paper is the study of the interplay between these probabilistic Nyman-Beurling criteria and the Riemann hypothesis. We are able to obtain equivalences in two main classes of examples: dilated structures as exponential
ℰ\left(𝓀\right)
distributions, and random variables
{Z}_{k,n}
1\le k\le n
, concentrated around
1/k
n
is growing. By means of our probabilistic point of view, we bring an answer to a question raised by Báez-Duarte in 2005: the price to pay to consider non compactly supported kernels is a controlled condition on the coefficients of the involved approximations.
Classification: 41A30, 46E20, 60E05, 11M26
Keywords: Number theory; Probability; Zeta function; Nyman-Beurling criterion; Báez-Duarte criterion
Sébastien Darses 1; Erwan Hillion 1
author = {S\'ebastien Darses and Erwan Hillion},
title = {On probabilistic generalizations of the {Nyman-Beurling} criterion for the zeta function},
TI - On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function
%T On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function
Sébastien Darses; Erwan Hillion. On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function. Confluentes Mathematici, Volume 13 (2021) no. 1, pp. 43-59. doi : 10.5802/cml.71. https://cml.centre-mersenne.org/articles/10.5802/cml.71/
[1] N. Alon, J.H. Spencer. The probabilistic method. Third edition. With an appendix on the life and work of Paul Erdös. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., Hoboken, NJ, 2008. | Zbl: 1148.05001
[2] L. Báez-Duarte. A class of invariant unitary operators. Adv. Math., 144 (1999), no. 1, 1–12. | Article | MR: 1692568 | Zbl: 0978.47025
[3] L. Báez-Duarte. A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, Rend. Mat. Ac. Lincei, S. 9, 14 (2003) 1, 5-11. | Zbl: 1097.11041
[4] L. Báez-Duarte. A general strong Nyman-Beurling criterion for the Riemann hypothesis. Publications de l’Institut Mathématique, Nouvelle Série, 78 (2005), pp. 117–125. | Article | MR: 2218310 | Zbl: 1119.11048
[5] L. Báez-Duarte, M. Balazard, B. Landreau and É. Saias. Notes sur la fonction
\zeta
de Riemann, 3. (French) [Notes on the Riemann
\zeta
-function, 3] Adv. Math., 149 (2000), no. 1, 130–144. | Article | MR: 1742356 | Zbl: 1008.11032
[6] L. Báez-Duarte, M. Balazard, B. Landreau and É. Saias. Étude de l’autocorrélation multiplicative de la fonction “partie fractionnaire”. (French) The Ramanujan Journal, 9(1) (2005), pp. 215–240. | Article | Zbl: 1173.11343
[7] L. Báez-Duarte, M. Balazard, B. Landreau and É. Saias. Document de travail – Étude de l’autocorrélation multiplicative de la fonction “partie fractionnaire”. (French)
[8] M. Balazard. Un siècle et demi de recherches sur l’hypothèse de Riemann. La Gazette des mathématiques, 126 (2010), pp.7–24. | Zbl: 1298.11087
[9] M. Balazard and A. de Roton. Sur un critère de Báez-Duarte pour l’hypothèse de Riemann. International Journal of Number Theory, 6(04) (2010), pp. 883–903. | Article | Zbl: 1201.11088
[10] M. Balazard, and É. Saias. Notes sur la fonction
\zeta
de Riemann, 4. Advances in Mathematics, 188(1) (2004), pp. 69–86. | Article | MR: 2083093 | Zbl: 1096.11032
[11] A. Beurling. A closure problem related to the Riemann Zeta-function. Proceedings of the National Academy of Sciences, 41(5) (1955), pp. 312–314. | Article | MR: 70655 | Zbl: 0065.30303
[12] J.F. Burnol. A lower bound in an approximation problem involving the zeros of the Riemann zeta function. Advances in Mathematics, 170(1) (2002), pp.56–70. | Article | MR: 1929303 | Zbl: 1029.11045
[13] J.F. Burnol. Entrelacement de co-Poisson. (French) [Co-Poisson links] Ann. Inst. Fourier (Grenoble) 57 (2007), no. 2, 525–602. | Article | MR: 2310951 | Zbl: 1177.11074
[14] J.B. Conrey. The Riemann hypothesis. Notices Amer. Math. Soc., 50 (2003), no. 3, 341–353. | Zbl: 1160.11341
[15] S. Darses and E. Hillion. An exponentially-averaged Vasyunin formula. Proc. of the American Math. Soc. To appear https://doi.org/10.1090/proc/15422. | Article | MR: 4257808 | Zbl: 07352296
[16] C. Delaunay, E. Fricain, E. Mosaki, O. Robert. Zero-free regions for Dirichlet series. Trans. Amer. Math. Soc., 365 (2013), no. 6, 3227–3253. | Article | MR: 3034464 | Zbl: 1322.11091
[17] N.Nikolski. Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann
\zeta
-function. Annales de l’institut Fourier Vol. 45, No. 1 (1995), pp. 143-159. | Article | MR: 1324128 | Zbl: 0816.30026
[18] B. Nyman. On the one-dimensional translation group and semi-group in certain function spaces. Thesis, University of Uppsala, 1950. | Zbl: 0037.35401
[19] G. Tenenbaum. Introduction à la théorie analytique et probabiliste des nombres, Société mathématique de France, 1995. | Zbl: 0880.11001
[20] E. C. Titchmarsh, The theory of the Riemann zeta-function, second ed., The Clarendon Press Oxford University Press, New York, 1986. | Zbl: 0601.10026
[21] V.I. Vasyunin. On a biorthogonal system associated with the Riemann hypothesis. (Russian) Algebra i Analiz 7, no. 3 (1995): 118-35; translation in St. Petersburg Mathematical Journal 7, no. 3 (1996): 405-19.
[22] A. Weingartner. On a question of Balazard and Saias related to the Riemann hypothesis. Adv. Math. 208 (2007), no. 2, 905–908. | Article | MR: 2304340 | Zbl: 1121.11058
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Chloe collected data about the number of pets her classmates have. Here is her data:
3,2,1,1,0,0,1,3,2,3,2,1,3,5,35,1,1,3,2,
1
What is the mean (average) number of pets owned by her classmates? What is the median number of pets?
To find the mean, start by adding all the pieces of data together.
3+2+1+1+0+0+1+3+2+3+2+1+3+5+35+1+1+3+2+1=70
Now, divide the sum by the number of pieces of data (in this case,
20
\frac{70}{20}\;=\;3\frac{1}{2}
Next, place the data in numerical order and find the middle number. That will be the median.
0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,5,35
3.5
pets and the median is
2
One piece of this data is an outlier (a number that is much higher or lower than the rest of the data). If we remove that number, how does the mean change? How does the median change?
For this problem, you will need to repeat the steps from part (a), but remove the outlying piece of data.
Remember that an outlier is a piece of data which is unusually higher or lower than the rest of the data.
Is removing the outlier important when we interpret the data? Explain your thinking.
Think about how your answer for part (b) was different than your answer for part (a).
Was there a greater difference between the means or between the medians?
Use this discovery as a basis for your explanation.
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PayPal is working on supporting Neo Cryptocurrency - The Tape Drive
PayPal is working on supporting Neo Cryptocurrency
PayPal is working on a feature to allow merchants to accept cryptocurrencies. Currently PayPal supports Bitcoin (BTC), Ethereum (ETH), Litecoin (LTC), and Bitcoin Cash (BCH). However images hidden in the PayPal Business iOS app indicate that PayPal will soon support paying merchants with Neo (NEO).
Neo (formally Antshares) is cryptocurrency and smart contract platform created in 2014. Neo is in the middle of upgrading to a new version of the Neo protocol called Neo N3. N3 supports new features such as platform native decentralized storage, a name service, and oracles.
PayPal will be the first fintech ‘super app’ to support Neo as currently Robinhood, Square, and Coinbase do not support buying, selling, or trading Neo.
PYPL
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What Is Dividend Discount Model?
Shortcomings of DDM
The dividend discount model (DDM) is a quantitative method used for predicting the price of a company's stock based on the theory that its present-day price is worth the sum of all of its future dividend payments when discounted back to their present value. It attempts to calculate the fair value of a stock irrespective of the prevailing market conditions and takes into consideration the dividend payout factors and the market expected returns. If the value obtained from the DDM is higher than the current trading price of shares, then the stock is undervalued and qualifies for a buy, and vice versa.
Take your $100 now
Take your $100 after a year
\begin{aligned}&\textbf{Future Value}\\&\qquad\mathbf{=}\textbf{Present Value }\mathbf{^*(1+}\textbf{interest rate}\mathbf{\%)}\\&\hspace{2.65in}(\textit{for one year})\end{aligned}
Future Value=Present Value ∗(1+interest rate%)(for one year)
\begin{aligned}&\textbf{Present Value}=\frac{\textbf{Future Value}}{\mathbf{(1+\textbf{interest rate}\%)}}\end{aligned}
Present Value=(1+interest rate%)Future Value
In essence, given any two factors, the third one can be computed.
The dividend discount model uses this principle. It takes the expected value of the cash flows a company will generate in the future and calculates its net present value (NPV) drawn from the concept of the time value of money (TVM). Essentially, the DDM is built on taking the sum of all future dividends expected to be paid by the company and calculating its present value using a net interest rate factor (also called discount rate).
Shareholders who invest their money in stocks take a risk as their purchased stocks may decline in value. Against this risk, they expect a return/compensation. Similar to a landlord renting out his property for rent, the stock investors act as money lenders to the firm and expect a certain rate of return. A firm's cost of equity capital represents the compensation the market and investors demand in exchange for owning the asset and bearing the risk of ownership. This rate of return is represented by (r) and can be estimated using the Capital Asset Pricing Model (CAPM) or the Dividend Growth Model. However, this rate of return can be realized only when an investor sells his shares. The required rate of return can vary due to investor discretion.
Companies that pay dividends do so at a certain annual rate, which is represented by (g). The rate of return minus the dividend growth rate (r - g) represents the effective discounting factor for a company’s dividend. The dividend is paid out and realized by the shareholders. The dividend growth rate can be estimated by multiplying the return on equity (ROE) by the retention ratio (the latter being the opposite of the dividend payout ratio). Since the dividend is sourced from the earnings generated by the company, ideally it cannot exceed the earnings. The rate of return on the overall stock has to be above the rate of growth of dividends for future years, otherwise, the model may not sustain and lead to results with negative stock prices that are not possible in reality.
\begin{aligned}&\textit{\textbf{Value of Stock}}=\frac{\textit{\textbf{EDPS}}}{\textbf{(\textit{CCE}}-\textbf{\textit{DGR})}}\\&\textbf{where:}\\&EDPS=\text{expected dividend per share}\\&CCE=\text{cost of capital equity}\\&DGR=\text{dividend growth rate}\end{aligned}
Value of Stock=(CCE−DGR)EDPSwhere:EDPS=expected dividend per shareCCE=cost of capital equityDGR=dividend growth rate
\begin{aligned}&D = \text{the estimated value of next year's dividend}\\&r = \text{the company's cost of capital equity}\\&g = \text{the constant growth rate for dividends, in perpetuity}\end{aligned}
D=the estimated value of next year’s dividendr=the company’s cost of capital equityg=the constant growth rate for dividends, in perpetuity
Using these variables, the equation for the GGM is:
\text{Price per Share}=\frac{D}{r-g}
Price per Share=r−gD
A third variant exists as the supernormal dividend growth model, which takes into account a period of high growth followed by a lower, constant growth period. During the high growth period, one can take each dividend amount and discount it back to the present period. For the constant growth period, the calculations follow the GGM model. All such calculated factors are summed up to arrive at a stock price.
A look at the dividend payment history of leading American retailer Walmart Inc. (WMT) indicates that it has paid out annual dividends totaling to $1.92, $1.96, $2.00, $2.04 and $2.08, between January 2014 and January 2018 in chronological order. One can see a pattern of a consistent increase of 4 cents in Walmart's dividend each year, which equals to the average growth of about 2%. Assume an investor has a required rate of return of 5%. Using an estimated dividend of $2.12 at the beginning of 2019, the investor would use the dividend discount model to calculate a per-share value of $2.12/ (.05 - .02) = $70.67.
Fidelity. “Dividends, Earnings, and Cash Flow Discount Models.”
Food and Agriculture Organization of the United Nations. “Chapter 6—Investment Decisions—Capital Budgeting: d) Perpetuities.”
David W. Mullins, Jr. “Does the Capital Asset Pricing Model Work?” Harvard Business Review, January 1982.
The International Financial Reporting Standards Foundation. “Illustrative Examples to Accompany IFRS 13 Fair Value Measurement: Unquoted Equity Instruments Within the Scope of IFRS 9 Financial Instruments.” Page 39.
Riccardo Sabbatucci. “Are Dividends and Stock Returns Predictable? New Evidence Using M&A Cash Flows,” Pages 1-3. University of San Diego, 2015.
CFA Journal. “Limitation of Dividend Valuation Models: Lost of Limitation Dividend.”
Myron J. Gordon. "The Investment, Financing, and Valuation of the Corporation," R.D. Irwin, 1962.
Walmart. "Dividend History."
Stern School of Business, New York University. “Dividend Discount Models,” Pages 1-2, 17.
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Introduction to Chemical Engineering Processes/General chemistry review - Wikibooks, open books for an open world
This is just a proposed layout and content. To discuss, go to the discussion page or edit it yourself.
2.1 Key Elements and Molecules
5 Structure and Formula
6 Ideal Gas Law
Le Système International d'Unités (SI Units)
The mole is a measure of the amount of substance. A mole is the amount of material which contains the same number of elementary entities as there are atoms in 12g of Carbon-12.
There are Avogadro number of atoms in 12g of Carbon-12, i.e. 6.023 x 10^23 atoms.
Thus a mole of cars implies there are 6.023 x 10^23 cars and so on.
Periodic TableEdit
Key Elements and MoleculesEdit
There are two major ways to classify acids and bases: the Brønsted-Lowry definition, and the Lewis definition. A chemical species that donates protons is a Brønsted-Lowry acid, and a species that accepts protons is a Brønsted-Lowry base. Typically, the proton is written as an H+ ion, though they do not in isolation exist in solution and are instead exchanged between molecules. In water, the proton on an acid will often bond to the H2O molecules to form the conjugate base and H3O+ (hydronium) ions, and the proton-accepting base will take an H+ from the water to form the conjugate acid and OH- (hydroxide) ions. This is the most familiar situation for those who have taken general chemistry, but any species that loses an H+ (proton) to another molecule is considered a Brønsted-Lowry acid, and likewise any H+-taking species is considered a Brønsted-Lowry base.
The second and broader classification is the Lewis acids and bases. Lewis acids and bases are defined by their electron lone pair behavior. A Lewis acid is an electron acceptor (called an electrophile in organic chemistry), a Lewis base is an electron donor (a nucleophile in organic chemistry). In a Lewis acid-base reaction, the negatively charged electron lone pair in the base will bond to the positive or partially positive segment of the acid to form what is called a Lewis adduct. Unlike Brønsted-Lowry acids and bases, the exchange of protons is not required.
Structure and FormulaEdit
{\displaystyle PV=nRT}
P = Pressure; V = Volume; n = moles; R = Ideal gas constant; T = Temperature
The enthalpy content of a substance is given by
\hat{H} = U + pV
H is the enthalpy (SI units: J/kg) U is the internal energy and p is the pressure V is the volume
Inorganic Chemistry - The study of the synthesis and behavior of inorganic and organometallic compounds. This field covers all chemical compounds except the myriad organic compounds.
Organic Chemistry - The study of the structure, properties, and reactions of organic compounds and organic materials, i.e., matter in its various forms that contain carbon atoms.
Physical Chemistry - The study of macroscopic, atomic, subatomic, and particulate phenomena in chemical systems in terms of laws and concepts of physics. It applies the principles, practices and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical mechanics and dynamics, equilibrium.
Analytical Chemistry -
Organometallic chemistry -
Retrieved from "https://en.wikibooks.org/w/index.php?title=Introduction_to_Chemical_Engineering_Processes/General_chemistry_review&oldid=3325777"
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HackerRank Implement Queue using two stacks Solution - PhotoLens
Problem Statement – Implement a Queue using two Stacks
I implemented dequeue() in O(1)
at the cost of enqueue() in O(n)
There are 15 test cases. This code runs properly for the first 5 test cases then I get a ‘Time Limit Exceeded’ message for the rest of test cases.
So I tried reversing it, I made enqueue() as O(1)
and dequeue() as O(n)
but even that did not change anything.
This introduces an additional state compared to usual algorithmic asymptotical complexity analysis. For example, a naive implemented std::vector::push_back will yield O(n2)
\mathcal O(n^2)
complexity if push_back increases the capacity only by one:
The real push_back has therefore some tricks up in its sleeves, as the standard dictates that it will have O(1)
amortized complexity. More about that later in this review.
I implemented dequeue() in O(1)
at the cost of enqueue() in O(n)
For this task, it’s much more important to see the cost on both functions for k
enqueued elements, not a single one.
So let’s envision the queue {1,2,3,4}. How many steps do we need?
We see that the n
th element will take 2(n−1)
2(n-1)
swaps. Therefore, if we insert a total of n
elements into our queue, we end up with O(n2)
\mathcal O(n^2)
to complete all enqueues.
So let’s get back to the drawing board. What do we need?
I’ll implement the methods outside of the class declaration to keep the code segments short, but you’re free to place them inline again. Note that I switched from a struct to a class, because its Queues job to make sure that in and out are handled correct; no one else should be able to change them.
So, let’s have a look at my proposal for the definition of enqueue(), peek and dequeue():
enqueue is O(1)
\mathcal O(1)
(amortized)
dequeue is O(1)
\mathcal O(1)
peek is O(1)
\mathcal O(1)
Intrigued? Great. So let’s check flip:
*”Wait a second! That’s O(n)
\mathcal O(n)
” I hear you say. And that’s completely correct. However, how often do we need to call flip? Or, rather more important, how often do elements get moved?
The answer on the latter question is: exactly one time from in to out. At no point will they move back. The number of flip calls is much trickier, though, but it doesn’t really matter. At worst, flip may need to flip all elements, for example if we use it as follows:
However, only the first dequeue will flip. All others will yield the element immediately. Therefore, if we enqueue n
elements and then dequeue all of them, we end up with dequeue()‘s complexity as:
nO(1)+O(n)n=O(1).
We can compare that with your original variant:
nO(n)n=O(n)
There are some other findings that don’t need as much detail as the algorithm, but nonetheless can be improved:
Source : Link , Question Author : KshitijV97 , Answer Author : dfhwze
Categories .htaccess, mqueue, openstack, programming-challenge, time-limit-exceeded Tags .htaccess, mqueue, openstack, programming-challenge, time-limit-exceeded Post navigation
Pathfinder result sightly different from expected!
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The Rational(f, k) command computes a closed form of the indefinite sum of
k
f\left(k\right)
s\left(k\right)
t\left(k\right)
f\left(k\right)=s\left(k+1\right)-s\left(k\right)+t\left(k\right)
t\left(k\right)
k
\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{k}t\left(k\right)
g,[p,q]
g
is the closed form of the indefinite sum of
k
p
is a list containing the integer poles of
q
s
that are not poles of
\mathrm{with}\left(\mathrm{SumTools}[\mathrm{IndefiniteSum}]\right):
f≔\frac{1}{{n}^{2}+\mathrm{sqrt}\left(5\right)n-1}
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{1}}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\sqrt{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}
g≔\mathrm{Rational}\left(f,n\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\sqrt{\textcolor[rgb]{0,0,1}{5}}}{\textcolor[rgb]{0,0,1}{2}}\right)}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\sqrt{\textcolor[rgb]{0,0,1}{5}}}{\textcolor[rgb]{0,0,1}{2}}\right)}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\sqrt{\textcolor[rgb]{0,0,1}{5}}}{\textcolor[rgb]{0,0,1}{2}}\right)}
\mathrm{evala}\left(\mathrm{Normal}\left(\mathrm{eval}\left(g,n=n+1\right)-g\right),\mathrm{expanded}\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\sqrt{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}
f≔\frac{13-57x+2y+20{x}^{2}-18xy+10{y}^{2}}{15+10x-26y-25{x}^{2}+10xy+8{y}^{2}}
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{20}\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}{18}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{57}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{13}}{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{25}\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}{10}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{26}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{15}}
g≔\mathrm{Rational}\left(f,x\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{25}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{34}}{\textcolor[rgb]{0,0,1}{25}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Psi }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{5}}\right)\textcolor[rgb]{0,0,1}{+}\left(\frac{\textcolor[rgb]{0,0,1}{17}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{25}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{5}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Psi }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)
\mathrm{simplify}\left(\mathrm{combine}\left(f-\left(\mathrm{eval}\left(g,x=x+1\right)-g\right),\mathrm{\Psi }\right)\right)
\textcolor[rgb]{0,0,1}{0}
f≔\frac{1}{n}-\frac{2}{n-3}+\frac{1}{n-5}
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{5}}
g,\mathrm{fp}≔\mathrm{Rational}\left(f,n,'\mathrm{failpoints}'\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{fp}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\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}{3}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]]
f
n=0,3,5
g
n=1,2,4
|
Interface between two-phase fluid and mechanical rotational networks - MATLAB - MathWorks Italia
Rotational Mechanical Converter (2P)
Flow and Thermal Resistances
Interface between two-phase fluid and mechanical rotational networks
The Rotational Mechanical Converter (2P) block models an interface between two-phase fluid and mechanical rotational networks. The interface converts pressure in the fluid network into torque in the mechanical rotational network and vice versa.
This block enables you to model a rotary actuator powered by a two-phase fluid system. It does not, however, account for inertia, friction, or hard stops, common in rotary actuators. You can model these effects separately using Simscape blocks such as Inertia, Rotational Friction, and Rotational Hard Stop.
Port A represents the inlet through which fluid enters and exits the converter. Ports C and R represent the converter casing and moving interface, respectively. Port H represents the wall through which the converter exchanges heat with its surroundings.
The torque direction depends on the mechanical orientation of the converter. If the Mechanical Orientation parameter is positive, then a positive flow rate through the inlet tends to rotate the moving interface in the positive direction relative to the converter casing.
Positive Mechanical Orientation
If the Mechanical Orientation parameter is negative, then a positive mass flow rate through the inlet tends to rotate the moving interface in the negative direction relative to the converter casing.
Negative Mechanical Orientation
The flow resistance between port A and the converter interior is assumed negligible. Pressure losses between the two is approximately zero. The pressure at port A is therefore equal to that in the converter:
{p}_{A}={p}_{I},
pI is the pressure in the converter.
Similarly, the thermal resistance between port H and the converter interior is assumed negligible. The temperature gradient between the two is approximately zero. The temperature at port H is therefore equal to that in the converter:
{T}_{H}={T}_{I},
TH is the temperature at port H.
TI is the temperature in the converter.
The volume of fluid in the converter is the sum of the dead and displaced fluid volumes. The dead volume is the amount of fluid left in the converter at a zero interface angle. This volume enables you to model the effects of dynamic compressibility and thermal capacity even when the interface is in its zero position.
The displacement volume is the amount of fluid added to the converter due to rotation of the moving interface. This volume increases with the interface angle. The total volume in the converter as a function of the interface angle is
V={V}_{dead}+{D}_{vol}{\theta }_{int}{ϵ}_{or},
V is the total volume of fluid in the converter.
Vdead is the dead volume of the converter.
Dvol is the displaced fluid volume per unit rotation of the interface.
θint is the rotation angle of the moving interface.
∊or is the mechanical orientation of the converter (1 if increase in fluid pressure causes positive rotation of R relative to C, -1 if increase in fluid pressure causes negative rotation of R relative to C).
If you connect the converter to a Multibody joint, use the physical signal input port q to specify the rotation of port R relative to port C. Otherwise, the block calculates the interface rotation from relative port angular velocities, according to the block equations. The interface rotation is zero when the fluid volume is equal to the dead volume. Then, depending on the Mechanical orientation parameter value:
If Pressure at A causes positive rotation of R relative to C, the interface rotation increases when the fluid volume increases from dead volume.
If Pressure at A causes negative rotation of R relative to C, the interface rotation decreases when the fluid volume increases from dead volume.
At equilibrium, the internal pressure in the converter counteracts the external pressure of its surroundings and the torque exerted by the mechanical network on the moving interface. This torque is the reverse of that applied by the fluid network. The torque balance in the converter is therefore
{p}_{I}{D}_{vol}={p}_{atm}{D}_{vol}-{t}_{int}{ϵ}_{or},
patm is the environmental pressure outside the converter.
tint is the magnitude of the torque exerted by the fluid network on the moving interface.
The total energy in the converter can change due to energy flow through the inlet, heat flow through the converter wall, and work done on the mechanical network. The energy flow rate, given by the energy conservation equation, is therefore
\stackrel{˙}{E}={\varphi }_{A}+{\varphi }_{H}-{p}_{I}{D}_{vol}{\stackrel{˙}{\theta }}_{int}{ϵ}_{or},
E is the total energy of the fluid in the converter.
ϕA is the energy flow rate into the converter through port A.
ϕH is the heat flow rate into the converter through port H.
Taking the fluid kinetic energy in the converter to be negligible, the total energy of the fluid reduces to:
E=M{u}_{I},
M is the fluid mass in the converter.
uI is the specific internal energy of the fluid in the converter.
The fluid mass in the converter can change due to flow through the inlet, represented by port A. The mass flow rate, given by the mass conservation equation, is therefore
\stackrel{˙}{M}={\stackrel{˙}{m}}_{A},
{\stackrel{˙}{m}}_{A}
A change in fluid mass can accompany a change in fluid volume, due to rotation of the moving interface. It can also accompany a change in mass density, due to an evolving pressure or specific internal energy in the converter. The mass rate of change in the converter is then
\stackrel{˙}{M}=\left[{\left(\frac{\partial \rho }{\partial p}\right)}_{u}{\stackrel{˙}{p}}_{I}+{\left(\frac{\partial \rho }{\partial u}\right)}_{p}{\stackrel{˙}{u}}_{I}\right]V+\frac{{D}_{vol}{\stackrel{˙}{\theta }}_{int}{ϵ}_{or}}{{v}_{I}},
{\left(\frac{\partial \rho }{\partial p}\right)}_{u}
is the partial derivative of density with respect to pressure at constant specific internal energy.
{\left(\frac{\partial \rho }{\partial u}\right)}_{p}
is the partial derivative of density with respect to specific internal energy at constant pressure.
vI is the specific volume of the fluid in the converter.
The block blends the density partial derivatives of the various domains using a cubic polynomial function. At a vapor quality of 0–0.1, this function blends the derivatives of the subcooled liquid and two-phase mixture domains. At a vapor quality of 0.9–1, it blends those of the two-phase mixture and superheated vapor domains.
The smoothed density partial derivatives introduce into the original mass conservation equation undesirable numerical errors. To correct for these errors, the block adds the correction term
{ϵ}_{M}=\frac{M-V/{v}_{I}}{\tau },
∊M is the correction term.
τ is the phase-change time constant—the characteristic duration of a phase change event. This constant ensures that phase changes do not occur instantaneously, effectively introducing a time lag whenever they occur.
The final form of the mass conservation equation is
\left[{\left(\frac{\partial \rho }{\partial p}\right)}_{u}{\stackrel{˙}{p}}_{I}+{\left(\frac{\partial \rho }{\partial u}\right)}_{p}{\stackrel{˙}{u}}_{I}\right]V+\frac{{D}_{vol}{\stackrel{˙}{\theta }}_{int}{ϵ}_{or}}{{v}_{I}}={\stackrel{˙}{m}}_{A}+{ϵ}_{M}.
The block uses this equation to calculate the internal pressure in the converter given the mass flow rate through the inlet.
The converter walls are rigid. They do not deform under pressure.
The flow resistance between port A and the converter interior is negligible. The pressure is the same at port A and in the converter interior.
The thermal resistance between port H and the converter interior is negligible. The temperature is the same at port H and in the converter interior.
The moving interface is perfectly sealed. No fluid leaks across the interface.
Mechanical effects such as hard stops, inertia, and friction, are ignored.
Alignment of the moving interface with respect to the volume of fluid in the converter:
Pressure at A causes positive rotation of R relative to C — Increase in the fluid volume results in a positive rotation of port R relative to port C.
Pressure at A causes negative rotation of R relative to C — Increase in the fluid volume results in a negative rotation of port R relative to port C.
Calculate from velocity of port R relative to port C — Calculate rotation from relative port velocities, based on the block equations. This is the default method.
Angle of the moving interface at the start of simulation. A zero angle corresponds to a total fluid volume in the converter equal to the specified dead volume. The default value is 0 rad.
This parameter is enabled when Interface rotation is set to Calculate from velocity of port R relative to port C.
Displaced fluid volume per unit rotation of the moving interface. The default value is 0.01 m^3/rad.
Volume of fluid left in the converter when the interface angle is zero. The dead volume enables the block to account for mass and energy storage in the converter even at a zero interface angle. The default value is 1e-5 m^3.
Flow area of the converter inlet, represented by port A. Pressure losses due to changes in flow area inside the converter are ignored. The default value is 0.01 m^2.
Pressure characteristics of the surrounding environment. Select Atmospheric pressure to set the environment pressure to the atmospheric pressure specified in the Two-Phase Fluid Properties (2P) block. Select Specified pressure to set the environment pressure to a different value. The default setting is Atmospheric pressure.
Absolute pressure of the surrounding environment. The environment pressure acts against the internal pressure of the converter and affects the motion of the converter shaft. This parameter is active only when the Environment pressure specification parameter is set to Specified pressure. The default value, 0.101325 MPa, corresponds to atmospheric pressure at mean sea level.
Two-phase fluid conserving port associated with the converter inlet.
Thermal conserving port representing the converter surface through which heat exchange occurs.
Mechanical rotational conserving port associated with the converter rotor.
Mechanical rotational conserving port associated with the converter case.
Physical signal input port that passes the position information from a Simscape™ Multibody™ joint. Connect this port to the position sensing port q of the joint. For more information, see Connecting Simscape Networks to Simscape Multibody Joints. To enable this port, set the Interface rotation parameter to Provide input signal from Multibody joint.
Translational Mechanical Converter (2P) | Rotational Multibody Interface
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Stabilization of Buoyancy-Driven Unstable Vortex Flow in Mixed Convection of Air in a Rectangular Duct by Tapering Its Top Plate | J. Heat Transfer | ASME Digital Collection
Stabilization of Buoyancy-Driven Unstable Vortex Flow in Mixed Convection of Air in a Rectangular Duct by Tapering Its Top Plate
W. S. Tseng,
W. S. Tseng
Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C.
W. L. Lin,
W. L. Lin
C. P. Yin,
C. P. Yin
Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received by the Heat Transfer Division, Feb. 7, 1999; revision received, July 27, 1999. Associate Technical Editor: F. Cheung.
J. Heat Transfer. Feb 2000, 122(1): 58-65
Tseng, W. S., Lin, W. L., Yin, C. P., Lin, C. L., and Lin, T. F. (July 27, 1999). "Stabilization of Buoyancy-Driven Unstable Vortex Flow in Mixed Convection of Air in a Rectangular Duct by Tapering Its Top Plate ." ASME. J. Heat Transfer. February 2000; 122(1): 58–65. https://doi.org/10.1115/1.521437
Stabilization of the buoyancy-driven unstable mixed convective vortex air flow in a bottom heated rectangular duct by tapering its top plate is investigated experimentally. Specifically, the duct is tapered so that its aspect ratio at the duct inlet is 4 and gradually raised to 12 at the exit of the duct. In the study the secondary flow in the duct is visualized and the steady and transient thermal characteristics of the flow are examined by measuring the spanwise distributions of the time-average temperature. The effects of the Reynolds and Grashof numbers on the vortex flow structure are studied in detail. Moreover, the spanwise-averaged Nusselt numbers for the horizontal rectangular and tapering ducts are also measured and compared. Furthermore, the time records of the air temperature are obtained to further detect the temporal stability of the flow. Over the ranges of the Re and Gr investigated for 5⩽Re⩽102 and
1.0×104⩽Gr⩽1.7×105,
the vortex flow induced in the rectangular duct exhibits temporal transition from a steady laminar to time periodic and then to chaotic state at increasing buoyancy-to-inertia ratio. Substantial change in the spatial structure of the vortex flow is also noted to accompany this temporal transition. The results for the tapering duct indicate that more vortex rolls can be induced due to the increase in the aspect ratio of the duct with the axial distance. But the vortex rolls are weaker and are completely stabilized by the tapering of the top plate. [S0022-1481(00)70301-X]
flow instability, heat transfer, convection, vortices, confined flow, Heat Transfer, Mixed Convection, Vortex
Buoyancy, Ducts, Flow (Dynamics), Vortex flow, Vortices, Mixed convection, Heat transfer, Temperature
Forced Convective Heat Transfer Between Horizontal Flat Plates
Effect of Thermal Instability on Thermally Developing Laminar Channel Flow
An Experimental Study of Convective Instability in the Thermal Entrance Region of a Horizontal Parallel-Plate Channel Heated From Below
Regions of Heat Transfer Enhancement for Laminar Mixed Convection in a Parallel Plate Channel
Mixed Convection Between Horizontal Plate—I. Entrance Effects
Mixed Convection Between Horizontal Plates—II. Fully Developed Flow
Durand-Daubin
Mixed Convection in a Horizontal Rectangular Channel-Experimental and Numerical Velocity Distributions
Etude Numerique et Experimental de la Convection Mixte Entre Deux Plans Horizontaux
Etude Experimental de la Convection Mixte Entre Duex Plans Horizontaux a Temperatures Differents—II
Structures of Moving Transverse and Mixed Rolls in Mixed Convection in a Horizontal Plane Channel
Visualization of Convective Instability Phenomena in the Entrance Region of a Horizontal Rectangular Channel Heated From Below and/or Cooled From Above
Unsteady Behavior and Mass Transfer Performance of the Combined Convective Flow in a Horizontal Rectangular Duct Heated From Below
Buoyancy Induced Flow Transition in Mixed Convective Flow of Air Through a Bottom Heated Horizontal Rectangular Duct
Vortex Flow and Thermal Characteristics in Mixed Convection of Air in a Horizontal Rectangular Duct: Effects of Reynolds and Grashof Numbers
Experimental Study of Unstable Mixed Convection of Air in a Bottom Heated Horizontal Rectangular Duct
Experimental Observation and Conjugate Heat Transfer Analysis of Vortex Flow Development in Mixed Convection of Air in a Horizontal Rectangular Duct
Tseng, W. S., 1996, “Stabilization of Mixed Convective Air Flow in a Bottom Heated Rectangular Duct by Tapering its Top Plate,” M.S. thesis, National Chiao Tung University, Hsinchu, Taiwan, R.O.C.
Return Flows in Horizontal MOCVD Reactors Studied With the Use of TiO2 Particle Injection and Numerical Calculations
Mixed Convection Adjacent to 3-D Backward-Facing Step
Heat Transfer Around Longitudinal and Parallel Arranged Wedge-Shaped Vortex Generators
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Principal angles between random subspaces and polynomials in two free projections
Guillaume Aubrun1
1 Université de Lyon; CNRS; Université Lyon 1; Institut Camille Jordan UMR5208, 69622 Villeurbanne Cedex, France
Confluentes Mathematici, Volume 13 (2021) no. 2, pp. 3-10.
We use the geometric concept of principal angles between subspaces to compute the noncommutative distribution of an expression involving two free projections. For example, this allows to simplify a formula by Fevrier–Mastnak–Nica–Szpojankowski about the free Bernoulli anticommutator. As a byproduct, we observe the remarkable fact that the principal angles between random half-dimensional subspaces are asymptotically distributed according to the uniform measure on
\left[0,\pi /2\right]
Classification: 46L54
Keywords: free probability, principal angles, random subspace
Guillaume Aubrun 1
@article{CML_2021__13_2_3_0,
author = {Guillaume Aubrun},
title = {Principal angles between random subspaces and polynomials in two free projections},
url = {https://cml.centre-mersenne.org/articles/10.5802/cml.74/}
TI - Principal angles between random subspaces and polynomials in two free projections
UR - https://cml.centre-mersenne.org/articles/10.5802/cml.74/
ID - CML_2021__13_2_3_0
%T Principal angles between random subspaces and polynomials in two free projections
%J Confluentes Mathematici
%I Institut Camille Jordan
%U https://doi.org/10.5802/cml.74
%R 10.5802/cml.74
%F CML_2021__13_2_3_0
Guillaume Aubrun. Principal angles between random subspaces and polynomials in two free projections. Confluentes Mathematici, Volume 13 (2021) no. 2, pp. 3-10. doi : 10.5802/cml.74. https://cml.centre-mersenne.org/articles/10.5802/cml.74/
[1] P.-A. Absil; A. Edelman; P. Koev On the largest principal angle between random subspaces, Linear Algebra Appl., Volume 414 (2006) no. 1, pp. 288-294 | Article | MR: 2209246
[2] A. Böttcher; I. M. Spitkovsky A gentle guide to the basics of two projections theory, Linear Algebra Appl., Volume 432 (2010) no. 6, pp. 1412-1459 | Article | MR: 2580440
[3] Maxime Fevrier; Mitja Mastnak; Alexandru Nica; Kamil Szpojankowski Using Boolean cumulants to study multiplication and anti-commutators of free random variables, Trans. Amer. Math. Soc., Volume 373 (2020) no. 10, pp. 7167-7205 | Article | MR: 4155204
[4] Gene H. Golub; Charles F. Van Loan Matrix computations, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013, xiv+756 pages | MR: 3024913
[5] Vladislav Kargin On eigenvalues of the sum of two random projections, Journal of Statistical Physics, Volume 149 (2012) no. 2, pp. 246-258
[6] James A. Mingo; Roland Speicher Free probability and random matrices, Fields Institute Monographs, 35, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2017, xiv+336 pages | Article | MR: 3585560
[7] Alexandru Nica; Roland Speicher Commutators of free random variables, Duke Math. J., Volume 92 (1998) no. 3, pp. 553-592 | Article | MR: 1620518
[8] Alexandru Nica; Roland Speicher Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, 335, Cambridge University Press, Cambridge, 2006, xvi+417 pages | Article | MR: 2266879
[9] D. V. Voiculescu; K. J. Dykema; A. Nica Free random variables, CRM Monograph Series, 1, American Mathematical Society, Providence, RI, 1992, vi+70 pages (A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups) | Article | MR: 1217253
|
n
n
numeric values specifying how large the interval between computed values should be along each dimension of the data
n+1
n
\mathrm{points},\mathrm{data}≔\mathrm{Interpolation}:-\mathrm{Kriging}:-\mathrm{GenerateSpatialData}\left(\mathrm{Spherical}\left(1,10,1\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{points}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{data}}\textcolor[rgb]{0,0,1}{≔}\begin{array}{c}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{0.814723686393179}& \textcolor[rgb]{0,0,1}{0.706046088019609}\\ \textcolor[rgb]{0,0,1}{0.905791937075619}& \textcolor[rgb]{0,0,1}{0.0318328463774207}\\ \textcolor[rgb]{0,0,1}{0.126986816293506}& \textcolor[rgb]{0,0,1}{0.276922984960890}\\ \textcolor[rgb]{0,0,1}{0.913375856139019}& \textcolor[rgb]{0,0,1}{0.0461713906311539}\\ \textcolor[rgb]{0,0,1}{0.632359246225410}& \textcolor[rgb]{0,0,1}{0.0971317812358475}\\ \textcolor[rgb]{0,0,1}{0.0975404049994095}& \textcolor[rgb]{0,0,1}{0.823457828327293}\\ \textcolor[rgb]{0,0,1}{0.278498218867048}& \textcolor[rgb]{0,0,1}{0.694828622975817}\\ \textcolor[rgb]{0,0,1}{0.546881519204984}& \textcolor[rgb]{0,0,1}{0.317099480060861}\\ \textcolor[rgb]{0,0,1}{0.957506835434298}& \textcolor[rgb]{0,0,1}{0.950222048838355}\\ \textcolor[rgb]{0,0,1}{0.964888535199277}& \textcolor[rgb]{0,0,1}{0.0344460805029088}\\ \textcolor[rgb]{0,0,1}{⋮}& \textcolor[rgb]{0,0,1}{⋮}\end{array}]\\ \hfill \textcolor[rgb]{0,0,1}{\text{30 × 2 Matrix}}\end{array}\textcolor[rgb]{0,0,1}{,}\begin{array}{c}[\begin{array}{c}\textcolor[rgb]{0,0,1}{-1.31317888309841}\\ \textcolor[rgb]{0,0,1}{3.78399452938781}\\ \textcolor[rgb]{0,0,1}{-4.07906747556730}\\ \textcolor[rgb]{0,0,1}{2.81033657021080}\\ \textcolor[rgb]{0,0,1}{3.07159908082332}\\ \textcolor[rgb]{0,0,1}{0.128958765233144}\\ \textcolor[rgb]{0,0,1}{-3.21737272238246}\\ \textcolor[rgb]{0,0,1}{0.707245165710619}\\ \textcolor[rgb]{0,0,1}{0.0877877303791926}\\ \textcolor[rgb]{0,0,1}{0.937296621856498}\\ \textcolor[rgb]{0,0,1}{⋮}\end{array}]\\ \hfill \textcolor[rgb]{0,0,1}{\text{30 element Vector[column]}}\end{array}
k≔\mathrm{Interpolation}:-\mathrm{Kriging}\left(\mathrm{points},\mathrm{data}\right)
\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{≔}\left(\begin{array}{c}\textcolor[rgb]{0,0,1}{Krⅈgⅈng ⅈntⅇrpolatⅈon obȷⅇct wⅈth 30 samplⅇ poⅈnts}\\ \textcolor[rgb]{0,0,1}{Varⅈogram: Sphⅇrⅈcal(1.25259453854482,13.6487615617247,.5525536774)}\end{array}\right)
\mathrm{SetVariogram}\left(k,\mathrm{Spherical}\left(1,10,1\right)\right)
\left(\begin{array}{c}\textcolor[rgb]{0,0,1}{Krⅈgⅈng ⅈntⅇrpolatⅈon obȷⅇct wⅈth 30 samplⅇ poⅈnts}\\ \textcolor[rgb]{0,0,1}{Varⅈogram: Sphⅇrⅈcal(1,10,1)}\end{array}\right)
\mathrm{ComputeGrid}\left(k,[0..5,0..5],0.1,\mathrm{output}=\mathrm{plot}\right)
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Combustion Characteristics of HCCI in Motorcycle Engine | J. Eng. Gas Turbines Power | ASME Digital Collection
Yuh-Yih Wu,
, Taipei, Taiwan 10608, R.O.C.
e-mail: cyywu@ntut.edu.tw
Ching-Tzan Jang,
Ching-Tzan Jang
Bo-Liang Chen
Mechanical and Systems Research Laboratories,
, Hsinchu, Taiwan 31040, R.O.C.
Wu, Y., Jang, C., and Chen, B. (January 22, 2010). "Combustion Characteristics of HCCI in Motorcycle Engine." ASME. J. Eng. Gas Turbines Power. April 2010; 132(4): 044501. https://doi.org/10.1115/1.3205024
Homogeneous charge compression ignition (HCCI) is recognized as an advanced combustion system for internal combustion engines that reduces fuel consumption and exhaust emissions. This work studied a 150 cc air-cooled, four-stroke motorcycle engine employing HCCI combustion. The compression ratio was increased from 10.5 to 12.4 by modifying the cylinder head. Kerosene fuel was used without intake air heating and operated at various excess air ratios
(λ)
, engine speeds, and exhaust gas recirculation (EGR) rates. Combustion characteristics and emissions on the target engine were measured. It was found that keeping the cylinder head temperature at around
120–130°C
is important for conducting a stable experiment. Two-stage ignition was observed from the heat release rate curve, which was calculated from cylinder pressure. Higher
λ
or EGR causes lower peak pressure, lower maximum rate of pressure rise (MRPR), and higher emission of CO. However, EGR is better than
λ
for decreasing the peak pressure and MRPR without deteriorating the engine output. Advancing the timing of peak pressure causes high peak pressure, and hence increases MRPR. The timing of peak pressure around 10–15 degree of crank angle after top dead center indicates a good appearance for low MRPR.
combustion, engines, internal combustion engines, motorcycles, petroleum
Combustion, Cylinders, Engines, Fuels, Homogeneous charge compression ignition engines, Motorcycles, Pressure, Exhaust gas recirculation, Temperature, Emissions, Compression
The 5th Stage Motorcycle Emission Standard and Low Emission Motorcycle Technology Evaluation
,” Environmental Protection Administration of Taiwan, ROC, Project No. EPA-93-FA13-03-A158.
Thermodynamic Analysis and Benchmark of Various Gasoline Combustion Concepts
Investigation of HCCI Combustion of Diethyl Ether and Ethanol Mixtures Using Carbon 14 Tracing and Numerical Simulations
HCCI Engine Control by Thermal Management
Characteristics of Combustion Stability and Emission in SCCI and CAI Combustion Based on Direct-Injection Gasoline Engine
A Study of Gasoline-Fuelled HCCI Engine With an Electromagnetic Valve Train Equipped
Study on HCCI-SI Combustion Using Fuels Containing Ethanol
HCCI Combustion: Effect of NO in EGR
An Investigation Into Propane Homogeneous Charge Compression Ignition (HCCI) Engine Operation With Residual Gas Trapping
Quantitative Analysis of Fuel Behavior in Port-Injection Gasoline Engines
Carabateas
Posylkin
Evaluation of the Influence of Injector Type in a Four-Valve Engine
The Effects of Port Fuel Injection Timing and Targeting on Fuel Preparation Relative to a Pre-Vaporized System
Heat Transfer Model for Small-Scale Spark-Ignition Engines
The Relationships of Diesel Fuel Properties, Chemistry, and HCCI Engine Performance as Determined by Principal Components Analysis
An HCCI Engine: Power Plant for a Hybrid Vehicle
Combustion Timing in HCCI Engines Determined by Ion-Sensor: Experimental and Kinetic Modeling
Combustion Characteristics of Stratified Mixture in Lean-Burn Liquefied Petroleum Gas Direct-Injection Engine With Spray-Guided Combustion System
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Hermitian function - Wikipedia
Hermitian function
(Redirected from Hermitian symmetry)
Type of complex function
In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign:
{\displaystyle f^{*}(x)=f(-x)}
{\displaystyle ^{*}}
indicates the complex conjugate) for all
{\displaystyle x}
in the domain o{\displaystyle f}
. In physics, this property is referred to as PT symmetry.
This definition extends also to functions of two or more variables, e.g., in the case that
{\displaystyle f}
is a function of two variables it is Hermitian if
{\displaystyle f^{*}(x_{1},x_{2})=f(-x_{1},-x_{2})}
{\displaystyle (x_{1},x_{2})}
in the domain o{\displaystyle f}
From this definition it follows immediately that:
{\displaystyle f}
is a Hermitian function if and only if
the real part o{\displaystyle f}
is an even function,
the imaginary part o{\displaystyle f}
is an odd function.
Hermitian functions appear frequently in mathematics, physics, and signal processing. For example, the following two statements follow from basic properties of the Fourier transform:[citation needed]
{\displaystyle f}
is real-valued if and only if the Fourier transform o{\displaystyle f}
is Hermitian.
{\displaystyle f}
is Hermitian if and only if the Fourier transform o{\displaystyle f}
is real-valued.
Since the Fourier transform of a real signal is guaranteed to be Hermitian, it can be compressed using the Hermitian even/odd symmetry. This, for example, allows the discrete Fourier transform of a signal (which is in general complex) to be stored in the same space as the original real signal.
If f is Hermitian, then
{\displaystyle f\star g=f*g}
{\displaystyle \star }
is cross-correlation, and
{\displaystyle *}
is convolution.
If both f and g are Hermitian, then
{\displaystyle f\star g=g\star f}
Complex conjugate – Fundamental operation on complex numbers
Even and odd functions – Mathematical functions with specific symmetries
Retrieved from "https://en.wikipedia.org/w/index.php?title=Hermitian_function&oldid=1072654499"
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Loop Shaping for Performance and Robustness - MATLAB & Simulink
Tradeoff Between Performance and Robustness
Choosing a Target Loop Shape
Limitations on Control Bandwidth
Loop Shapes, Performance, and Robustness
Guaranteed Gain and Phase Margins
Performance and robustness requirements can often be expressed in terms of the open-loop response gain. For example, high gain at low frequencies reduces steady-state offsets and improves disturbance rejection. Similarly, high-frequency rolloff improves stability where the plant model is uncertain or inaccurate. Loop shaping is an approach to control design in which you determine a suitable profile for the open-loop system response and design a controller to achieve that shape.
The uncertainty in your plant model can be a limiting factor in determining what you can achieve with feedback. High loop gains can attenuate the effects of plant model uncertainty and reduce the overall sensitivity of the system to disturbances. But if your plant model uncertainty is so large that you do not even know the sign of your plant gain, then you cannot use large feedback gains without the risk of the system becoming unstable.
For this reason, most controller designs involve a tradeoff between performance and robustness against uncertainty. Robust Control Toolbox™ commands for loop-shaping controller design let you determine the tradeoff that best meets the requirements of your system.
loopsyn — Designs a stabilizing controller that shapes the open-loop response to approximate the target loop shape that you provide. You can adjust the balance between performance and robustness.
mixsyn — Controller design optimized for performance. This function allows you more precise specification of the shapes of different loop responses.
ncfsyn — Controller design optimized for robustness (stability margin). You provide weighting functions that shape the plant to a desirable profile.
The performance optimization of mixsyn tends to produce plant-inverting designs, which can be less robust. In particular, mixsyn designs can be fragile for ill-conditioned MIMO plants and for plants with structured uncertainty, such as uncertainty on the damping and natural frequency of resonant modes. In contrast, ncfsyn deters control strategies like plant inversion that rely on exact knowledge of the plant poles and zeroes. Thus ncfsyn adds some of the robustness to structured uncertainty that is missing in mixsyn designs. By combining elements of both ncfsyn and mixsyn, the loopsyn approach can provide robustness to both structured and unstructured uncertainty while also providing good performance.
Here are some basic design tradeoffs to consider when choosing a target loop shape.
Robust Stability. Use a target loop shape with gain as low as possible at high frequencies where typically your plant model is so poor that its phase angle is completely inaccurate, with errors approaching ±180° or more.
Performance. Use a target loop shape with gain as high as possible at frequencies where your model is good. Doing so ensures good reference tracking and good disturbance attenuation.
Crossover and Rolloff. Use a target loop shape with its 0 dB crossover frequency ωc between the previous two frequency ranges. Ensure that the target loop shape rolls off with a slope between –20 dB/decade and –30 dB/decade near ωc. This rolloff helps keep phase lag approximately between –130° and –90° near crossover for good phase margins.
Keep these principles in mind when choosing your target loop shape for loopsyn or the shaping filters for ncfsyn. For further details about choosing weighting functions for mixsyn, see Mixed-Sensitivity Loop Shaping.
Other considerations that might affect your choice of loop shape are the unstable poles and zeros of the plant, which impose fundamental limits on your 0 dB crossover frequency ωc (see [1]). For instance, ωc must be greater than the natural frequency of any unstable pole of the plant, and smaller than the natural frequency of any unstable zero of the plant.
\underset{\mathrm{Re}\left({p}_{i}\right)>0}{\mathrm{max}}|{p}_{i}|<{\omega }_{c}<\underset{\mathrm{Re}\left({z}_{i}\right)>0}{\mathrm{min}}|{z}_{i}|.
If you do not take care to choose a target loop shape that conforms to these fundamental constraints, then you might not achieve good results. For instance, loopsyn computes the optimal loop-shaping controller K for a target loop shape Gd that does not meet this requirement, but the resulting response L = G*K might have a poor fit to the target loop shape Gd, and consequently it might be impossible to meet your performance goals.
Additionally, because plant uncertainty typically increases with frequency, the bandwidth that you can reliably achieve is limited. For instance, consider an approximate model G0 of a SISO plant G. You can express the uncertainty in this plant as a multiplicative uncertainty Δ M, such that
G={G}_{0}\left(1+{\Delta }_{M}\right)
. The uncertainty is bounded at each frequency such that |Δ M(jω)| ≤ β(ω), where β(ω) is the percentage of model uncertainty. Typically, β(ω) is small at low frequencies (accurate model) and increases at high frequencies (inaccurate model). The frequency where β(ω) = 2 marks a critical threshold beyond which you have insufficient information about the plant to reliably design a feedback controller. With such a 200% model uncertainty, the model provides no indication of the phase angle of the true plant, which means that the only way you can reliably stabilize your plant is to ensure that the loop gain is less than 1. Allowing for an additional factor of two margin for error, your control system bandwidth is essentially limited to the frequency range over which your multiplicative plant uncertainty Δ M has gain magnitude |Δ M|<1.
For a deeper understanding of the relationship between loop shapes, performance, and robustness, consider the multivariable feedback control system shown in the following figure.
To quantify the multivariable stability margins and performance of such systems, you can use the closed-loop sensitivity function S and complementary sensitivity function T, defined as
\begin{array}{l}S\left(s\right)\stackrel{def}{=}{\left(I+L\left(s\right)\right)}^{-1}\\ T\left(s\right)\stackrel{def}{=}L\left(s\right){\left(I+L\left(s\right)\right)}^{-1}=I-S\left(s\right),\end{array}
where L(s) is the open-loop transfer function
L\left(s\right)=G\left(s\right)K\left(s\right).
Specifying a target shape Gd(s) for the open-loop transfer function L(s) is equivalent to imposing constraints on the singular values of the sensitivity S(s) and complementary sensitivity T(s). For instance, for a target loop shape with high gain at low frequency, the condition
\underset{¯}{\sigma }\left(L\left(s\right)\right)>\underset{¯}{\sigma }\left({G}_{d}\left(s\right)\right)\gg 1
\overline{\sigma }\left(S\left(s\right)\right)<1/\underset{¯}{\sigma }\left({G}_{d}\left(s\right)\right)
\overline{\sigma }
\underset{¯}{\sigma }
denote the largest and smallest singular values, respectively. Similarly, for a target loop shape with low gain at high frequency,
\overline{\sigma }\left(L\left(s\right)\right)<\overline{\sigma }\left({G}_{d}\left(s\right)\right)\ll 1
\overline{\sigma }\left(T\left(s\right)\right)<\overline{\sigma }\left({G}_{d}\left(s\right)\right)
When using loopsyn, you specify Gd(s) directly, and loopsyn approximately imposes these constraints on the sensitivity and complementary sensitivity. For mixsyn, you specify weighting functions W1(s) and W3(s) such that W1(s) matches 1/Gd(s) at low frequency and is smaller than 1 elsewhere, and W3(s) matches Gd(s) at high frequency and is smaller than 1 elsewhere. (See Mixed-Sensitivity Loop Shaping). Then mixsyn approximately imposes the constraints
\overline{\sigma }\left(S\right)<|{W}_{1}^{-1}|
\overline{\sigma }\left(T\right)<|{W}_{3}^{-1}|
, which roughly enforce the loop shape Gd.
Additionally, robustness to multiplicative plant uncertainty is equivalent to imposing a small-gain constraint on T(s) (see [1], page 342). Thus, enforcing rolloff in the loop shape Gd (or equivalently,
\overline{\sigma }\left(T\right)<|{W}_{3}^{-1}|
) provides some robustness against unmodeled plant dynamics at high frequency.
If you are more comfortable with classical single-loop concepts, you can use the important connections between the multiplicative stability margins predicted by the gain of T(s) and those predicted by classical M-circles, as found on the Nichols chart. In the SISO case, the largest singular value of T(s) is just the peak gain, given by:
|T\left(s\right)|=|\frac{L\left(s\right)}{1+L\left(s\right)}|.
This quantity is the same quantity you obtain from Nichols chart M-circles. The H∞ norm
{‖T‖}_{\infty }
(see hinfnorm) is a multiloop generalization of the closed-loop resonant peak magnitude which, as classical control experts will recognize, is closely related to the damping ratio of the dominant closed-loop poles. You can relate
{‖T‖}_{\infty }
{‖S‖}_{\infty }
to the classical gain margin GM and phase margin θM in each feedback loop of the multivariable feedback system illustrated in the previous section, via the formulas:
\begin{array}{l}{G}_{M}\ge 1+\frac{1}{{‖T‖}_{\infty }}\\ {G}_{M}\ge 1+\frac{1}{1-\frac{1}{{‖S‖}_{\infty }}}\\ {\theta }_{M}\ge 2{\mathrm{sin}}^{-1}\left(\frac{1}{2{‖T‖}_{\infty }}\right)\\ {\theta }_{M}\ge 2{\mathrm{sin}}^{-1}\left(\frac{1}{2{‖S‖}_{\infty }}\right).\end{array}
(See [2].) These formulas are valid provided
{‖S‖}_{\infty }
{‖T‖}_{\infty }
are larger than 1, as is normally the case. The margins apply even when the gain perturbations or phase perturbations occur simultaneously in several feedback channels.
The infinity norms of S and T also yield gain-reduction tolerances. The gain-reduction tolerance gM is defined to be the minimal amount by which the gains in each loop would have to be decreased in order to destabilize the system. Upper bounds on gM are as follows:
\begin{array}{l}{g}_{M}\le 1-\frac{1}{{‖T‖}_{\infty }}\\ {g}_{M}\le \frac{1}{1+\frac{1}{{‖S‖}_{\infty }}}.\end{array}
For more information about the relation between sensitivity functions and gain and phase margins, see [3].
[1] Skogestad, Sigurd, Ian Postlethwaite. Multivariable Feedback Control: Analysis and Design. Chichester; New York: Wiley, 1996.
[2] Lehtomaki, N., N. Sandell, and M. Athans. "Robustness Results in Linear-Quadratic Gaussian Based Multivariable Control Designs." IEEE Transactions on Automatic Control 26, no.1 (February 1981): 75–93.
loopsyn | mixsyn | ncfsyn
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Algebraic_group Knowpia
An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called linear algebraic groups.[1] Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem states that every algebraic group can be constructed from groups in those two families.
Formally, an algebraic group over a field
{\displaystyle k}
is an algebraic variety
{\displaystyle \mathrm {G} }
{\displaystyle k}
, together with a distinguished element
{\displaystyle e\in \mathrm {G} (k)}
(the neutral element), and regular maps
{\displaystyle \mathrm {G} \times \mathrm {G} \to \mathrm {G} }
(the multiplication operation) and
{\displaystyle \mathrm {G} \to \mathrm {G} }
(the inversion operation) which satisfy the group axioms.[2]
A more sophisticated definition is that of a group scheme over
{\displaystyle k}
. Yet another definition of the concept is to say that an algebraic group over
{\displaystyle k}
is a group object in the category of algebraic varieties over
{\displaystyle k}
Several important classes of groups are algebraic groups, including:
GL(n, F), the general linear group of invertible matrices over a field F, and its algebraic subgroups.
Jet groups
Elliptic curves and their generalizations as abelian varieties
Algebraic subgroupEdit
Coxeter groupsEdit
There are a number of analogous results between algebraic groups and Coxeter groups – for instance, the number of elements of the symmetric group is
{\displaystyle n!}
, and the number of elements of the general linear group over a finite field is the q-factorial
{\displaystyle [n]_{q}!}
; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the field with one element, which considers Coxeter groups to be simple algebraic groups over the field with one element.
Glossary of algebraic groupsEdit
In the sequel, G denotes an algebraic group over a field k.
linear algebraic group A Zariski closed subgroup of
{\displaystyle {\rm {GL}}_{n}}
for some n
{\displaystyle {\rm {SL}}_{n}}
Every affine algebraic group is isomorphic to a linear algebraic group, and vice versa
affine algebraic group An algebraic group that is an affine variety
{\displaystyle {\rm {GL}}_{n}}
, non-example: elliptic curve The notion of affine algebraic group stresses the independence from any embedding in
{\displaystyle {\rm {GL}}_{n}}
commutative The underlying (abstract) group is abelian.
{\displaystyle {\mathbb {G} }_{a}}
(the additive group),
{\displaystyle {\mathbb {G} }_{m}}
(the multiplicative group),[3] any complete algebraic group (see abelian variety)
diagonalizable group A closed subgroup of
{\displaystyle (\mathbb {G} _{m})^{n}}
, the group of diagonal matrices (of size n-by-n)
simple algebraic group A connected group that has no non-trivial connected normal subgroups
{\displaystyle {\rm {SL}}_{n}}
semisimple group An affine algebraic group with trivial radical
{\displaystyle {\rm {SL}}_{n}}
{\displaystyle {\rm {SO}}_{n}}
In characteristic zero, the Lie algebra of a semisimple group is a semisimple Lie algebra
reductive group An affine algebraic group with trivial unipotent radical Any finite group,
{\displaystyle {\rm {GL}}_{n}}
Any semisimple group is reductive
unipotent group An affine algebraic group such that all elements are unipotent The group of upper-triangular n-by-n matrices with all diagonal entries equal to 1 Any unipotent group is nilpotent
torus A group that becomes isomorphic to
{\displaystyle (\mathbb {G} _{m})^{n}}
when passing to the algebraic closure of k.
{\displaystyle {\rm {SO}}_{2}}
G is said to be split by some bigger field k' , if G becomes isomorphic to Gmn as an algebraic group over k'.
character group X∗(G) The group of characters, i.e., group homomorphisms
{\displaystyle G\rightarrow {\mathbb {G} }_{m}}
{\displaystyle X^{*}(\mathbb {G} _{m})\cong \mathbb {Z} }
Lie algebra Lie(G) The tangent space of G at the unit element.
{\displaystyle {\rm {Lie}}({\rm {GL}}_{n})}
is the space of all n-by-n matrices Equivalently, the space of all left-invariant derivations.
Algebraic topology (object)
Tame group
Cherlin–Zilber conjecture
Pseudo-reductive group
^ Borel 1991, p.54.
^ Borel 1991, p. 46.
^ These two are the only connected one-dimensional linear groups, Springer 1998, Theorem 3.4.9
Chevalley, Claude, ed. (1958), Séminaire C. Chevalley, 1956--1958. Classification des groupes de Lie algébriques, 2 vols, Paris: Secrétariat Mathématique, MR 0106966, Reprinted as volume 3 of Chevalley's collected works., archived from the original on 2013-08-30, retrieved 2012-06-25
Borel, Armand (1991). Linear algebraic groups. 2nd enlarged ed. Graduate Texts in Mathematics. Springer-Verlag. pp. x+288. Zbl 0726.20030.
Humphreys, James E. (1972), Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 21, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90108-4, MR 0396773
Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics, vol. 9 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713
Waterhouse, William C. (1979), Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90421-4
Algebraic groups and their Lie algebras by Daniel Miller
|
Model band-limited operational amplifier - MATLAB - MathWorks Benelux
Input resistance, Rin
Model band-limited operational amplifier
The Band-Limited Op-Amp block models a band-limited operational amplifier. If the voltages at the positive and negative ports are Vp and Vm, respectively, the output voltage is:
{V}_{out}\text{ }=\text{ }\frac{A\left({V}_{p}-{V}_{m}\right)}{\frac{s}{2\pi f}+1}-{I}_{out}*{R}_{out}
A is the gain.
Rout is the output resistance.
Iout is the output current.
s is the Laplace operator.
f is the 3-dB bandwidth.
The input current is:
\frac{{V}_{p}-{V}_{m}}{{R}_{in}}
where Rin is the input resistance.
The block does not use the initial condition you specify using the Initial output voltage, V0 parameter if you select the Start simulation from steady state check box in the Simscape™Solver Configuration block.
+ — Non-inverting input
Electrical conserving port associated with the op-amp non-inverting input
- — Inverting input
Electrical conserving port associated with the op-amp inverting input
Electrical conserving port associated with the op-amp output voltage
Gain, A — Open-loop gain
The open-loop gain of the operational amplifier.
Input resistance, Rin — Input resistance
The resistance at the input of the operational amplifier that the block uses to calculate the input current.
The resistance at the output of the operational amplifier that the block uses to calculate the drop in output voltage due to the output current.
The lower limit on the operational amplifier no-load output voltage.
The upper limit on the operational amplifier no-load output voltage.
1000 V/s (default)
Bandwidth, f — Open-loop bandwidth
1e5 Hz (default)
The open-loop bandwidth, that is, the frequency at which the gain drops by 3 dB compared to the low-frequency gain, A.
The output voltage at the start of the simulation when the output current is zero.
This parameter value does not account for the voltage drop across the output resistor.
Op-Amp | Finite-Gain Op-Amp | Fully Differential Op-Amp
|
A multithreaded implementation of the ‘which’ command - PhotoLens
Here is an implementation of the ‘which‘ command which can tell where programs are located.
/* SPDX-FileCopyrightText: 2021-2022 John Scott <jscott@posteo.net>
* SPDX-License-Identifier: GPL-3.0-or-later */
/* We do not support the obsolete extension where an
* omitted directory name is interpreted as the current
* working directory. In $PATH = "/usr::/bin:", the lack
* of a path in the middle or one at the end is simply ignored. */
#if defined(_POSIX_ADVISORY_INFO) && _POSIX_ADVISORY_INFO > -1
/* A NULL-terminated list of directories in PATH. */
static char **list;
/* A NULL-terminated list of the names of programs to be found.
* Neither the list nor the strings are dynamically allocated! */
static char **progs;
/* We need a way to tell which threads are running at any given time
* so we know which ones we can send cancellation requests to.
* This is a boolean list indicating whether a given thread is
* running or not. thread_is_running[0] will correspond to the
* first child we create, thread_is_running[1] to the second,
* and so on. */
static bool *thread_is_running;
/* This condition is broadcasted whenever a thread is about to
* terminate. */
static pthread_cond_t thread_dying = PTHREAD_COND_INITIALIZER;
/* This mutex serves multiple purposes. It not only protects
* thread_is_running from data races, but is also used in
* tandem with thread_dying. This mutex also has the effect
* that, if it's locked in the main thread, then child threads
* will be waiting to acquire this mutex before they bail out,
* so in effect which threads are running will be frozen. This
* allows for safely sending cancellation requests to running threads. */
static pthread_mutex_t thread_guard = PTHREAD_MUTEX_INITIALIZER;
/* This is a list of groups we are in. Its lifetime is managed by main(). */
static gid_t *groups;
static int groupcount;
static void *reallocarray(void *p, size_t m, size_t n) {
if(n && m > SIZE_MAX / n) {
return realloc(p, m * n);
static int giddiff(const void *a, const void *b) {
gid_t gid_a = *(const gid_t*)a;
gid_t gid_b = *(const gid_t*)b;
/* We do not simply return gid_a - gid_b, because that
* bears the risk of overflow if gid_t is a signed type. */
if(gid_a > gid_b) {
} else if(gid_a < gid_b) {
static int boolcmp(const void *a, const void *b) {
return *(const bool*)a != *(const bool*)b;
static void stop_running(void *i) {
if(pthread_mutex_lock(&thread_guard)) {
assert(thread_is_running[(intptr_t)i]);
thread_is_running[(intptr_t)i] = false;
if(pthread_cond_broadcast(&thread_dying) || pthread_mutex_unlock(&thread_guard)) {
static bool file_is_executable(const char filename[restrict static 1]) {
if(stat(filename, &st) == -1 || !S_ISREG(st.st_mode)) {
if(st.st_uid == geteuid()) {
return (st.st_mode & S_IXUSR) ? true : false;
if(bsearch(&st.st_gid, groups, groupcount, sizeof(*groups), giddiff)) {
return (st.st_mode & S_IXGRP) ? true : false;
return (st.st_mode & S_IXOTH) ? true : false;
/* For the n'th program, where n is type-punned from arg,
* return a dynamically allocated pathname articulating where
* it can be found, or NULL if that can't be done. */
static void *find_program(void *arg) {
const int my_seq_thread_id = (intptr_t)arg;
int k = pthread_mutex_lock(&thread_guard);
errno = k;
perror("Failed to lock mutex");
assert(!thread_is_running[my_seq_thread_id]);
thread_is_running[my_seq_thread_id] = true;
k = pthread_mutex_unlock(&thread_guard);
/* These have to be declared outside of our cleanup handler calls. */
bool pathname_found = false;
/* We hit no cancellation points between setting thread_is_running[j]
* to true and pushing this handler, since pthread_mutex_unlock is not one. */
pthread_cleanup_push(stop_running, (void*)(intptr_t)my_seq_thread_id);
if(strchr(progs[my_seq_thread_id], '/')) {
/* We were either given an absolute pathname, or a relative
* path that must be evaluated with respect to the current
* working directory, which must not use prefixes from PATH. */
if(file_is_executable(progs[my_seq_thread_id])) {
/* The caller expects that the string we return is dynamically allocated. */
pathname = strdup(progs[my_seq_thread_id]);
if(!pathname) {
perror("Failed to duplicate string");
pthread_exit(pathname);
/* It's not an absolute pathname; try prefixing the string
* with strings from PATH and see what sticks. */
const size_t filenamelen = strlen(progs[my_seq_thread_id]);
for(char **directory = list; *directory; directory++) {
const size_t directorylen = strlen(*directory);
if(filenamelen > SIZE_MAX - directorylen || filenamelen + directorylen > SIZE_MAX - 2) {
pathname = malloc(directorylen + filenamelen + 2); /* one extra byte for a /, one for the NUL */
perror("Failed to allocate memory for pathname");
pthread_cleanup_push(free, pathname);
char *end = (char*)memcpy(pathname, *directory, directorylen) + directorylen;
if(end[-1] != '/') {
memcpy(end, progs[my_seq_thread_id], filenamelen + 1);
pathname_found = file_is_executable(pathname);
pthread_cleanup_pop(!pathname_found); /* free(pathname)? */
if(pathname_found) {
pthread_cleanup_pop(true); /* stop_running(my_seq_thread_id) */
pthread_exit(pathname_found ? pathname : NULL);
if(!setlocale(LC_ALL, "")) {
fputs("Failed to enable default locale\n", stderr);
while((opt = getopt(argc, argv, "")) != -1) {
if(opt == '?') {
progs = argv + optind;
#if INT_MAX > INTPTR_MAX
if(argc - 1 > INTPTR_MAX) {
fprintf(stderr, "Too many arguments: %s\n", strerror(EOVERFLOW));
const char *const envpath = getenv("PATH");
if(!envpath || !envpath[0]) {
l = confstr(_CS_PATH, NULL, 0);
fputs("Failed to obtain value of PATH\n", stderr);
path = aligned_alloc(sysconf(_SC_PAGESIZE), l);
perror("Failed to allocate memory for PATH");
confstr(_CS_PATH, path, l);
l = strlen(envpath) + 1;
memcpy(path, envpath, l);
int p = posix_madvise(path, l, POSIX_MADV_SEQUENTIAL);
if(p && sysconf(_SC_ADVISORY_INFO) != -1) {
fprintf(stderr, "Failed to advise the system on memory usage: %s\n", strerror(p));
/* The maximum number of directories in PATH is one plus
* the number of colons, where multiple consecutive colons
* can be treated as a single one. */
size_t numdirs = 1;
assert(path[0]);
for(size_t i = 1; path[i]; i++) {
/* In case of a set of multiple consecutive colons,
* only count the last one. */
if(path[i] == ':' && path[i+1] && path[i+1] != ':') {
numdirs++;
assert(numdirs < SIZE_MAX);
list = reallocarray(NULL, numdirs + 1, sizeof(*list));
perror("Failed to allocate memory for directory list");
/* cppcheck-suppress[strtokCalled] there's only one thread; this is safe */
tok = strtok(tok ? NULL : path, ":");
assert(n <= numdirs);
list[n++] = tok;
} while(tok);
pthread_t *ids = reallocarray(NULL, argc, sizeof(*ids));
if(!ids) {
perror("Failed to allocate memory for thread list");
goto endlist;
if((groupcount = getgroups(0, groups)) == -1) {
perror("Failed to get number of groups");
goto endids;
if(groupcount == INT_MAX || groupcount == SIZE_MAX) {
fprintf(stderr, "Failed to create group list: %s\n", strerror(EOVERFLOW));
/* We might need an extra member for the effective group ID. */
groups = reallocarray(NULL, groupcount + 1, sizeof(*groups));
if(!groups) {
perror("Failed to allocate memory for group list");
/* It's possible that in a TOCTTOU sort of way, the number of
* groups we're in now is fewer than the number we were in before,
* hence the reassignment to groupcount. */
if((groupcount = getgroups(groupcount, groups)) == -1) {
perror("Failed to populate group list");
goto endgroups;
/* The group list may not include the effective group ID. */
if(!lfind(&(gid_t){getegid()}, groups, &(size_t){groupcount}, sizeof(*groups), giddiff)) {
groups[groupcount++] = getegid();
qsort(groups, groupcount, sizeof(*groups), giddiff);
thread_is_running = calloc(argc, sizeof(*thread_is_running));
if(!thread_is_running) {
perror("Failed to allocate memory for running thread list");
void **retval = reallocarray(NULL, argc, sizeof(*retval));
perror("Failed to allocate memory for thread return values");
goto endthread_is_running;
fprintf(stderr, "Failed to lock mutex: %s\n", strerror(k));
k = pthread_create(ids + i, NULL, find_program, (void*)(intptr_t)i);
fprintf(stderr, "Failed to unlock mutex: %s\n", strerror(k));
if(lfind(&(bool){true}, thread_is_running, &(size_t){i}, sizeof(*thread_is_running), boolcmp)) {
k = pthread_cond_wait(&thread_dying, &thread_guard);
fprintf(stderr, "Failed to wait on condition: %s\n", strerror(k));
fprintf(stderr, "Failed to create thread: %s\n", strerror(k));
if(thread_is_running[j]) {
k = pthread_cancel(ids[j]);
fprintf(stderr, "Failed to cancel thread: %s\n", strerror(k));
void *threadreturn;
k = pthread_join(ids[j], &threadreturn);
fprintf(stderr, "Failed to join with thread: %s\n", strerror(k));
if(threadreturn != PTHREAD_CANCELED) {
free(threadreturn);
goto endthread_guard;
for(int j = 0; j < argc; j++) {
int k = pthread_join(ids[j], retval + j);
if(retval[j]) {
if(puts(retval[j]) == EOF) {
perror("Failed to print filename");
free(retval[j]);
int k = pthread_mutex_destroy(&thread_guard);
fprintf(stderr, "Failed to destroy mutex: %s\n", strerror(k));
k = pthread_cond_destroy(&thread_dying);
fprintf(stderr, "Failed to destroy condition variable: %s\n", strerror(k));
free(thread_is_running);
exit(all_found ? EXIT_SUCCESS : EXIT_FAILURE);
endthread_guard:
k = pthread_mutex_destroy(&thread_guard);
endthread_is_running:
endgroups:
endids:
endlist:
Here are some remarks in no particular order:
Trying to cancel a thread which isn’t actually running, even if it’s joinable and not yet joined with, is undefined behavior, so I follow Ulrich Drepper’s suggestions for dealing with this problem.
I assume that an arbitrary intptr_t can be cast to void* and back with the expected result, which I realize is not guaranteed by any standard. At build time, I check this is okay with the build system.
Perhaps ‘posix_madvise()‘ is overkill or premature optimization, but on the other hand I think it serves as a form of self-documenting code which says we traverse the directory list in order. I’d like to know if you agree.
Throwing threads at a problem does not necessarily make it faster, and can even make it slower. Consider that creating and destroying threads has an overhead of its own, and that your program will likely have some parts that have to run in serial, which means there is a limit to the speed up you can gain even if you could use an infinite number of threads (see Amdahl’s law). Consider benchmarking your code against /usr/bin/which with various numbers of arguments, and see where the cutoff is where a parallel version becomes faster, if at all. Use perf stat to see things like the number of instructions and cycles spent, context switches and page faults.
Even if threads make sense, you chose to do things the hard way, wanting to use thread cancellation in the error path for example, when you could just simply let all threads finish what they were doing. Follow the KISS principle.
Use a thread pool
You start one thread per command line argument. But what if I call your program with a thousand arguments? Surely that will spawn more threads than I have CPU cores, which means they are all trying to contend for resources. A better approach is to limit the number of threads to the number of cores, and distribute the workload over the threads. For example, with n
threads, let each thread work on every n
th element of the argument list, offset with their thread ID.
Doing this will also limit the amount of memory needed for your program to run, independent of the number of arguments.
Use access() to check for execute permissions
You spend a lot of code getting a list of groups the user belongs to, and checking the results from stat() against that list. However, you can use access() instead to do that. There are some slight differences in semantics, but assuming your version of which won’t have the setuid bit set, that should not be a problem. But you can also use faccessat(), which takes a flags argument, then you can use AT_EACCESS to make sure it uses the effective UID and GID to do its checks.
You still need to call stat() to check if the file is a regular file, because access() and friends will not distinguish between executable files and accessible directories. Since stat() is quite a heavy operation because of the amount of information it returns, you could use the non-portable statx(..., STATX_TYPE, ...) to just query the type.
Consider using the *at() functions
Instead of keeping the list of directories as strings, and building full filenames from the combination of directory names and command line arguments, consider opening all directories using open(dirname, O_PATH) and storing those filedescriptors. Then you can use those with fstatat() and faccessat(), so you don’t need to build full filenames anymore.
Perhaps posix_madvise() is overkill or premature optimization, but on the other hand I think it serves as a form of self-documenting code which says we traverse the directory list in order.
The POSIX advise functions are quite subtle, and might not always do what you think they do. Furthermore, they have an overhead of their own. Calling it for just one string that will surely fit into the L1 cache will not do anything useful. You had to jump through hoops to even be able to use it (like using aligned_alloc()), so it makes the code less readable.
Let threads print out their results
Instead of waiting for all threads to finish and then print out the results, consider letting each thread print their results themselves. Each write() is done atomically, so if you can ensure you can print a pathname and newline in a single write(), the output will always be correct, the only thing that is no longer guaranteed is the order in which things are printed.
Create a struct to hold all the state for a given filename
Instead of having several arrays holding data, like the filenames, running status, return values, and indexing them by a thread ID, consider creating a struct that holds all these things, make an array of that struct, and then you can pass a pointer to a given element of that array to the thread. This also avoids the whole issue of having to cast intptr_ts to void*s.
Source : Link , Question Author : JohnScott , Answer Author : G. Sliepen
Categories multithreading, posix, pthreads Tags .htaccess, multithreading, posix, pthreads Post navigation
Photoshop: Should I use my monitor’s color profile (under proof setup/colors) when designing for desktop?
What happened to the variations tool? (Elgar will be most displeased!)
Adobe Illustrator to Microsoft Office
Illustrator: Exporting in 0,5 with ‘export for screens’ doesn’t export in truly halve the size
Need to erase the stroke between start & end points of a path in Illustrator
How to change color of the dot on the “i” letter in Illustrator?
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Modeling of Supercritical CO2 Shell-and-Tube Heat Exchangers Under Extreme Conditions. Part I: Correlation Development | J. Heat Transfer | ASME Digital Collection
Modeling of Supercritical CO2 Shell-and-Tube Heat Exchangers Under Extreme Conditions. Part I: Correlation Development
Akshay Bharadwaj Krishna,
Akshay Bharadwaj Krishna
Kaiyuan Jin,
Co-lead authors.
Krishna, A. B., Jin, K., Ayyaswamy, P. S., Catton, I., and Fisher, T. S. (March 2, 2022). "Modeling of Supercritical CO2 Shell-and-Tube Heat Exchangers Under Extreme Conditions. Part I: Correlation Development." ASME. J. Heat Transfer. May 2022; 144(5): 051902. https://doi.org/10.1115/1.4053510
High-temperature supercritical CO2 Brayton cycles are promising possibilities for future stationary power generation and hybrid electric propulsion applications. Heat exchangers are critical components in supercritical CO2 thermal cycles and require accurate correlations and comprehensive performance modeling under extreme temperatures and pressures. In this paper (Part I), new Colburn and friction factor correlations are developed to quantify shell-side heat transfer and friction characteristics of flow within heat exchangers in the shell-and-tube configuration. Using experimental and computational fluid dynamics (CFD) data sets from existing literature, multivariate regression analysis is conducted to achieve correlations that capture the effect of multiple critical geometric parameters. These correlations offer superior accuracy and versatility as compared to previous studies and predict the thermohydraulic performance of about 90% of the existing experimental and CFD data within
±
15%. Supplementary thermohydraulic performance data are acquired from CFD simulations with supercritical CO2 as working fluid to validate the developed correlations and demonstrate its capability to be applied to supercrtical CO2 heat exchangers.
supercritical CO2, shell-and-tube heat exchanger, Colburn factor, friction factor, tube bank correlations, multivariate regression
Computational fluid dynamics, Disks, Flow (Dynamics), Fluids, Friction, Heat exchangers, Shells, Supercritical carbon dioxide, Modeling, Heat transfer, Pressure drop, Engineering simulation, Simulation
.10.1016/j.net.2015.06.009
Optimization of a Recompression Supercritical Carbon Dioxide Cycle for an Innovative Central Receiver Solar Power Plant
Supercritical CO2 Brayton Cycle: A State-of-the-Art Review
Analysis of Supercritical CO2 Brayton Power Cycles in Nuclear and Fusion Energy
Performance Analysis of Cooling System Based on Improved Supercritical CO2 Brayton Cycle for Scramjet
Compact Heat Exchangers for Supercritical CO2 Power Cycle Application
.10.1016/j.enconman.2020.112666
Cost Comparison of Printed Circuit Heat Exchanger to Low Cost Periodic Flow Regenerator for Use as Recuperator in a S-CO2 Brayton Cycle
Performance Analysis of Printed Circuit Heat Exchanger for Supercritical Carbon Dioxide
Dynamic Modelling and Transient Characteristics of Supercritical CO2 Recompression Brayton Cycle
Compact Heat Exchangers: A Review and Future Applications for a New Generation of High Temperature Solar Receivers
Optimal Design of Microtube Recuperators for an Indirect Supercritical Carbon Dioxide Recompression Closed Brayton Cycle
Monjurul Ehsan
Design and Comparison of Direct and Indirect Cooling System for 25 MW Solar Power Plant Operated With Supercritical CO2 Cycle
The Variation in Effectiveness of Low-Finned Tubes Within a Shell-and-Tube Heat Exchanger for Supercritical CO2
An Optimization of Microtube Heat Exchangers for Supercritical CO2 Cooling Based on Numerical and Theoretical Analysis
, “High Temperature Heat Exchanger Design and Fabrication for Systems With Large Pressure Differentials,“ Technical Report No.
DE- FE0024012
Experimental Investigation on Convective Heat Transfer and Pressure Drop of Supercritical CO2 and Water in Microtube Heat Exchangers
, J. P. Hartnett and T. F. Irvine, eds., Academic, San Diego, CA, pp.
.https://www.bibsonomy.org/bibtex/2e5f300b68e4939294c32226ddd6a5a71/thorade
Heat Exchangers: Thermal-Hydraulic Fundamentals and Design
, Hemisphere Publishing Corporation, New York, pp.
A Simple Heat Transfer Correlation for Cross Flow in Tube Bundles
Experimental Investigation of the Influence of Tube Arrangement on Convection Heat Transfer and Flow Resistance in Cross Flow of Gases Over Tube Banks
Wärmeaustauscher, HR Sauerlander & Co
AARU, Frankfurt am Main
, Academic, New York, pp.
A General Correlation of Friction Factors for Various Types of Surfaces in Cross Flow
Pressure Drop Across Tube Banks-Critical Comparison of Available Data and of Proposed Methods of Correlation
Techniques to Augment Heat Transfer
The Imperative to Enhance Heat Transfer
A Comparative Study of the Airside Performance of Heat Sinks Having Pin Fin Configurations
Performance Analysis of Rectangular Ducts With Staggered Square Pin Fins
Numerical Simulation of Finned Tube Bank Across a Staggered Circular-Pin-Finned Tube Bundle
Numer. Heat Transfer, Part A Appl.
Prediction of Heat Transfer Coefficients in Gas Flow Normal to Finned and Smooth Tube Banks
Heat Transfer and Flow Resistance Correlation for Helically Finned and Staggered Tube Banks in Crossflow
Heat Exchangers: Design and Theory Source Book
, N. H. Afgan and E. U. Schlunder, eds., pp.
Development of Closure for Heat Exchangers Based on Volume Averaging Theory
Air-Side Heat Transfer and Friction Correlations for Plain Fin-and-Tube Heat Exchangers With Staggered Tube Arrangements
Volume Averaging Theory (VAT) Based Modeling and Closure Evaluation for Fin-and-Tube Heat Exchangers
Heat Mass Transfer/Waerme- Und Stoffuebertragung
Akshayb29/Shell-and-Tube-Heat-Exchanger-STHX-Numerical-Model
: Shell-and-Tube Heat Exchanger Numerical Model,” Zenodo, Geneva, Switzerland.10.5281/zenodo.5117859
Modeling of Supercritical CO2 Shell-and-Tube Heat Exchangers Under Extreme Conditions. Part 2: Heat Exchanger Model
Numerical Investigation on Maldistribution of Supercritical CO 2 Flow Inside Printed Circuit Heat Exchanger
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A New Method of Calculation of Reheat Factors for Turbines and Compressors | J. Appl. Mech. | ASME Digital Collection
Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Mass.
Kaye, J., and Wadleigh, K. R. (April 7, 2021). "A New Method of Calculation of Reheat Factors for Turbines and Compressors." ASME. J. Appl. Mech. December 1951; 18(4): 387–392. https://doi.org/10.1115/1.4010355
A simple method is presented for rapid accurate calculation of reheat factors for adiabatic turbines and compressors with an infinite number of stages. The method is limited to fluids for which the following equation of state is adequate
pv=RT
and for which the specific heat is a function of temperature only. The method is based on the concept of the step efficiency—the efficiency of an infinitesimal stage based on the reversible adiabatic work—and on the concept of the relative pressure function. The method is used to produce charts of reheat factors for turbines and compressors for several gases. A theoretical justification is given for empirical rules of thumb which have been widely used to predict reheat factors for turbines and compressors with a finite number of stages.
Compressors, Turbines, Equations of state, Fluids, Gases, Pressure, Specific heat, Temperature
Nonideal Gas Effects for the Venturi Meter
Semi-Closed Cycle O 2 /CO 2 Combustion Gas Turbines: Influence of Fluid Properties on the Aerodynamic Performance of the Turbomachinery
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Double whole note - Simple English Wikipedia, the free encyclopedia
A double whole note (also called a breve) is a note that is double the value of a whole note, which is where it gets its name. In the
{\displaystyle {\frac {4}{4}}}
time signature it has the value of 8 beats or two measures.
Baker, Theodore (1895). "Note". A Dictionary of Musical Terms: Containing Upwards of 9,000 English, French, German, Italian, Latin, and Greek Words and Phrases (Revised and enlarged third ed.). New York: G. Schirmer.
Burrowes, John Freckleton (1874). Piano-forte Primer: Containing the Rudiments of Music Adapted for Either Private Tuition Or Teaching in Classes Together with a Guide to Practice (with important additions by L.H. Southard, revised and modernized new ed.). Boston and New York: Oliver Ditson.
Gehrkens, Karl Wilson (1914). Music Notation and Terminology. New York / Chicago: The A.S. Barnes Co. / Laidlaw Brothers.
Gerou, Tom; Lusk, Linda (1996). Essential Dictionary of Music Notation. Essential Dictionary Series. Los Angeles: Alfred Music Publishing. ISBN 0-88284-730-9.
Hoppin, Richard H. (1978). Medieval Music. W W Norton & Company. ISBN 0-393-09090-6.
Jacob, Archibald (1960). Musical Handwriting: Or, How to Put Music on Paper, A Handbook for All Musicians, Professional and Amateur (revised second ed.). London: Oxford University Press.
Read, Gardner (1969). Music Notation: A Manual of Modern Practice (second ed.). Boston: Alleyn and Bacon, Inc.
Wright, Peter (2001). "Alla breve". In Sadie, Stanley; Tyrrell, John (eds.). The New Grove Dictionary of Music and Musicians (second ed.). London: Macmillan Publishers.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Double_whole_note&oldid=8012904"
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Analytic Number Theory/Partial fraction decomposition - Wikibooks, open books for an open world
Analytic Number Theory/Partial fraction decomposition
< Analytic Number Theory
Existence theorem[edit | edit source]
Theorem 2.1 (Existence theorem of the partial fraction decomposition):
{\displaystyle f,g}
be polynomials over a unique factorisation domain, and let
{\displaystyle g=\prod _{j=1}^{n}p_{j}^{k_{j}}}
{\displaystyle p_{j}}
are irreducible. Then we may write
{\displaystyle {\frac {f(x)}{g(x)}}=q(x)+\sum _{j=1}^{n}\sum _{l=1}^{k_{j}}{\frac {a_{l,j}(x)}{p_{j}(x)^{l}}}}
{\displaystyle a_{l,j}}
are polynomials of degree strictly less than
{\displaystyle p_{j}}
{\displaystyle q}
is a polynomial. The term on the right hand side is called the partial fraction decomposition of
{\displaystyle {\frac {f}{g}}}
Wikipedia has related information at Partial fraction decomposition
We proceed by induction o{\displaystyle n}
{\displaystyle n=1}
, the statement is true since by division with remainder, we may write
{\displaystyle f(x)=q_{1}(x)p_{1}(x)+r(x)}
{\displaystyle \deg(r)<\deg(p_{1})}
{\displaystyle {\frac {f(x)}{g(x)}}={\frac {q_{1}(x)}{p_{1}(x)^{k_{1}-1}}}+{\frac {r(x)}{g(x)}}}
and we have reduced the degree of the denominator by one (the latter summand already satisfies the required condition). By repetition of this process, we eventually obtain a denominator of one and thus a polynomial.
Let now the hypothesis be true for
{\displaystyle n\in \mathbb {N} }
{\displaystyle g=\prod _{j=1}^{n+1}p_{j}^{k_{j}}}
{\displaystyle G=\prod _{j=1}^{n}p_{j}^{k_{j}}}
{\displaystyle H=p_{n+1}^{k_{n+1}}}
. By irreducibility,
{\displaystyle \gcd(G,H)=1}
. Hence, we find polynomials
{\displaystyle S,T}
{\displaystyle 1=SG+TH}
{\displaystyle {\frac {f}{g}}={\frac {f(SG+TH)}{g}}={\frac {fSG}{g}}+{\frac {fTH}{g}}}
Each of the summands of the last term can by the induction hypothesis be written in the desired form.
{\displaystyle \Box }
No matter how complicated our fraction of polynomials
{\displaystyle {\frac {f}{g}}}
may be, we can give the partial fraction decomposition in finite time, using easy techniques. The method, which for the sake of simplicity differs from the one given in the above constructive existence proof, goes as follows:
Split the polynomial
{\displaystyle g}
into irreducible factors.
Using division with remainder o{\displaystyle f}
{\displaystyle g}
, reduce to the case
{\displaystyle \deg(f)<\deg(g)}
(the resulting polynomial
{\displaystyle q}
is allowed in the formula of theorem 2.1).
Solve the equation given in theorem 2.1 for the
{\displaystyle a_{l,j}}
(this is equivalent to solving a system of linear equations; namely multiply by
{\displaystyle g}
and then equate coefficients).
The algorithm given above always terminates and gives the partial fraction decomposition of
{\displaystyle {\frac {f}{g}}}
Proof: Due to theorem 2.1, in step three we do obtain a system of linear equations which is solvable. Hence follow termination and correctness.
{\displaystyle \Box }
Retrieved from "https://en.wikibooks.org/w/index.php?title=Analytic_Number_Theory/Partial_fraction_decomposition&oldid=3086988"
Book:Analytic Number Theory
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Lemma 30.16.1 (0B5T)—The Stacks project
Should the symbol
A_{nd+q}
M_{nd+q}
in the last paragraph?
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Truncated square antiprism - WikiMili, The Free Encyclopedia
Type Truncated antiprism
Schläfli symbol ts{2,8}
tsr{4,2} or
{\displaystyle ts{\begin{Bmatrix}4\\2\end{Bmatrix}}}
Conway notation tA4
Faces 18: 2 {8}, 8 {6}, 8 {4}
Symmetry group D4d, [2+,8], (2*4), order 16
Rotation group D4, [2,4]+, (224), order 8
Properties convex, zonohedron
The truncated square antiprism one in an infinite series of truncated antiprisms, constructed as a truncated square antiprism. It has 18 faces, 2 octagons, 8 hexagons, and 8 squares.
In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an anticube.
Gyroelongated triamond square bicupola
If the hexagons are folded, it can be constructed by regular polygons. Or each folded hexagon can be replaced by two triamonds, adding 8 edges (56), and 4 faces (32). This form is called a gyroelongated triamond square bicupola. [1]
D5d, [2+,10], (2*5)
(v:4; e:8; f:6)
(v:6; e:12; f:8)
(v:8; e:16; f:10)
(v:10; e:20; f:12)
ts{2,4}
(v:16;e:24;f:10)
ts{2,10}
Although it can't be made by all regular planar faces, its alternation is the Johnson solid, the snub square antiprism.
In geometry, a Johnson solid is a strictly convex polyhedron, which is not uniform, and each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides (J1); it has 1 square face and 4 triangular faces.
In geometry, the snub square antiprism is one of the Johnson solids (J85). A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966.
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles. Antiprisms are a subclass of the prismatoids and are a (degenerate) type of snub polyhedra.
In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
The cubic honeycomb or cubic cellulation is the only regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway calls this honeycomb a cubille.
A snub polyhedron is a polyhedron obtained by alternating a corresponding omnitruncated or truncated polyhedron, depending on the definition. Some but not all authors include antiprisms as snub polyhedra, as they are obtained by this construction from a degenerate "polyhedron" with only two faces.
In geometry, a near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whose faces are all regular, though it "can often be physically constructed without noticing the discrepancy" between its regular and irregular faces. The precise number of near misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons. Some high symmetry near-misses are also symmetrohedra with some perfect regular polygon faces.
In geometry, a snub is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube and snub dodecahedron. In general, snubs have chiral symmetry with two forms, with clockwise or counterclockwise orientations. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron, with the faces moved apart, and twists on their centers, adding new polygons centered on the original vertices, and pairs of triangles fitting between the original edges.
In geometry, a truncated cuboctahedral prism or great rhombicuboctahedral prism is a convex uniform polychoron.
In geometry, an edge-contracted icosahedron is a polyhedron with 18 triangular faces, 27 edges, and 11 vertices with C2v symmetry, order 4.
In the geometry of hyperbolic 3-space, the square tiling honeycomb, is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, has three square tilings, {4,4} around each edge, and 6 square tilings around each vertex in a cubic {4,3} vertex figure.
↑ Convex Triamond Regular Polyhedra
Snub Anti-Prisms
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26.1: Linear Regression - Statistics LibreTexts
https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_Statistical_Thinking_for_the_21st_Century_(Poldrack)%2F26%253A_The_General_Linear_Model%2F26.01%253A_Linear_Regression
We can also use the general linear model to describe the relation between two variables and to decide whether that relationship is statistically significant; in addition, the model allows us to predict the value of the dependent variable given some new value(s) of the independent variable(s). Most importantly, the general linear model will allow us to build models that incorporate multiple independent variables, whereas correlation can only tell us about the relationship between two individual variables.
The specific version of the GLM that we use for this is referred to as as linear regression. The term regression was coined by Francis Galton, who had noted that when he compared parents and their children on some feature (such as height), the children of extreme parents (i.e. the very tall or very short parents) generally fell closer to the mean than their parents. This is an extremely important point that we return to below.
y = x * \beta_x + \beta_0 + \epsilon
\beta_x
value tells us how much we would expect y to change given a one-unit change in x. The intercept
\beta_0
is an overall offset, which tells us what value we would expect y to have when
x=0
; you may remember from our early modeling discussion that this is important to model the overall magnitude of the data, even if
x
never actually attains a value of zero. The error term
\epsilon
refers to whatever is left over once the model has been fit. If we want to know how to predict y (which we call
\hat{y}
), then we can drop the error term:
The concept of regression to the mean was one of Galton’s essential contributions to science, and it remains a critical point to understand when we interpret the results of experimental data analyses. Let’s say that we want to study the effects of a reading intervention on the performance of poor readers. To test our hypothesis, we might go into a school and recruit those individuals in the bottom 25% of the distribution on some reading test, administer the intervention, and then examine their performance. Let’s say that the intervention actually has no effect, such that reading scores for each individual are simply independent samples from a normal distribution. We can simulate this:
If we look at the difference between the mean test performance at the first and second test, it appears that the intervention has helped these students substantially, as their scores have gone up by more than ten points on the test! However, we know that in fact the students didn’t improve at all, since in both cases the scores were simply selected from a random normal distribution. What has happened is that some subjects scored badly on the first test simply due to random chance. If we select just those subjects on the basis of their first test scores, they are guaranteed to move back towards the mean of the entire group on the second test, even if there is no effect of training. This is the reason that we need an untreated control group in order to interpret any changes in reading over time; otherwise we are likely to be tricked by regression to the mean.
\hat{r} = \frac{covariance_{xy}}{s_x * s_y}
whereas the regression beta is computed as:
\hat{\beta} = \frac{covariance_{xy}}{s_x*s_x}
Based on these two equations, we can derive the relationship between
\hat{r}
\hat{beta}
covariance_{xy} = \hat{r} * s_x * s_y
\hat{\beta_x} = \frac{\hat{r} * s_x * s_y}{s_x * s_x} = r * \frac{s_y}{s_x}
That is, the regression slope is equal to the correlation value multiplied by the ratio of standard deviations of y and x. One thing this tells us is that when the standard deviations of x and y are the same (e.g. when the data have been converted to Z scores), then the correlation estimate is equal to the regression slope estimate.
residual = y - \hat{y} = y - (x*\hat{\beta_x} + \hat{\beta_0})
We then compute the sum of squared errors (SSE):
SS_{error} = \sum_{i=1}^n{(y_i - \hat{y_i})^2} = \sum_{i=1}^n{residuals^2}
and from this we compute the mean squared error:
MS_{error} = \frac{SS_{error}}{df} = \frac{\sum_{i=1}^n{(y_i - \hat{y_i})^2} }{N - p}
where the degrees of freedom (
df
) are determined by subtracting the number of estimated parameters (2 in this case:
\hat{\beta_x}
\hat{\beta_0}
) from the number of observations (
N
). Once we have the mean squared error, we can compute the standard error for the model as:
SE_{model} = \sqrt{MS_{error}}
In order to get the standard error for a specific regression parameter estimate,
SE
, we need to rescale the standard error of the model by the square root of the sum of squares of the X variable:
SE_{\beta_x} = \frac{SE_{model}}{\sqrt<div class="mt-dekiscript-error"><span class="mt-dekiscript-error-message">ParseError: invalid DekiScript</span> <span class="mt-dekiscript-error-toggle">(click for details)</span><pre class="mt-dekiscript-error-callstack">Callstack: at <span class="mt-dekiscript-error-callstack-frame">(Bookshelves/Introductory_Statistics/Book:_Statistical_Thinking_for_the_21st_Century_(Poldrack)/26:_The_General_Linear_Model/26.01:_Linear_Regression), /content/body/section[3]/div/p[7]/math/semantics/annotation/span, line 1, column 1</span> </pre></div>}
Once we have the parameter estimates and their standard errors, we can compute a t statistic to tell us the likelihood of the observed parameter estimates compared to some expected value under the null hypothesis. In this case we will test against the null hypothesis of no effect (i.e.
\beta=0
\begin{array}{c} t_{N - p} = \frac{\hat{\beta} - \beta_{expected}}{SE_{\hat{\beta}}}\\ t_{N - p} = \frac{\hat{\beta} - 0}{SE_{\hat{\beta}}}\\ t_{N - p} = \frac{\hat{\beta} }{SE_{\hat{\beta}}} \end{array}
In R, we don’t need to compute these by hand, as they are automatically returned to us by the lm() function:
In this case we see that the intercept is significantly different from zero (which is not very interesting) and that the effect of studyTime on grades is marginally significant (p = .09).
Sometimes it’s useful to quantify how well the model fits the data overall, and one way to do this is to ask how much of the variability in the data is accounted for by the model. This is quantified using a value called
R
(also known as the coefficient of determination). If there is only one x variable, then this is easy to compute by simply squaring the correlation coefficient:
R^2 = r^2
In the case of our study time example,
R
= 0.4, which means that we have accounted for about 40% of the variance in grades.
More generally we can think of
R
as a measure of the fraction of variance in the data that is accounted for by the model, which can be computed by breaking the variance into multiple components:
SS_{total} = SS_{model} + SS_{error}
SS
is the variance of the data (
y
SS
SS
are computed as shown earlier in this chapter. Using this, we can then compute the coefficient of determination as:
R^2 = \frac{SS_{model}}{SS_{total}} = 1 - \frac{SS_{error}}{SS_{total}}
R
tells us that even if the model fit is statistically significant, it may only explain a small amount of information in the data.
26.1: Linear Regression is shared under a not declared license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.
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Greatest common divisor - MATLAB gcd - MathWorks Australia
Greatest Common Divisors of Double Values
Greatest Common Divisors of Unsigned Integers
Solution to Diophantine Equation
[G,U,V] = gcd(A,B)
G = gcd(A,B) returns the greatest common divisors of the elements of A and B. The elements in G are always nonnegative, and gcd(0,0) returns 0. This syntax supports inputs of any numeric type.
[G,U,V] = gcd(A,B) also returns the Bézout coefficients, U and V, which satisfy: A.*U + B.*V = G. The Bézout coefficients are useful for solving Diophantine equations. This syntax supports double, single, and signed integer inputs.
A = [-5 17; 10 0];
B = [-15 3; 100 0];
gcd returns positive values, even when the inputs are negative.
G = 1x3 uint16 row vector
Solve the Diophantine equation,
30x+56y=8
x
y
Find the greatest common divisor and a pair of Bézout coefficients for 30 and 56.
[g,u,v] = gcd(30,56)
u = -13
u and v satisfy the Bézout's identity, (30*u) + (56*v) = g.
Rewrite Bézout's identity so that it looks more like the original equation. Do this by multiplying by 4. Use == to verify that both sides of the equation are equal.
(30*u*4) + (56*v*4) == g*4
Calculate the values of
x
y
that solve the problem.
x = u*4
y = v*4
scalars, vectors, or arrays of real integer values
Input values, specified as scalars, vectors, or arrays of real integer values. A and B can be any numeric type, and they can be of different types within certain limitations:
Example: [20 -3 13],[10 6 7]
Example: int16([100 -30 200]),int16([20 15 9])
Example: int16([100 -30 200]),20
G — Greatest common divisor
real, nonnegative integer values
Greatest common divisor, returned as an array of real nonnegative integer values. G is the same size as A and B, and the values in G are always real and nonnegative. G is returned as the same type as A and B. If A and B are of different types, then G is returned as the nondouble type.
U,V — Bézout coefficients
real integer values
Bézout coefficients, returned as arrays of real integer values that satisfy the equation, A.*U + B.*V = G. The data type of U and V is the same type as that of A and B. If A and B are of different types, then U and V are returned as the nondouble type.
g = gcd(A,B) is calculated using the Euclidean algorithm.[1]
[g,u,v] = gcd(A,B) is calculated using the extended Euclidean algorithm.[1]
[1] Knuth, D. “Algorithms A and X.” The Art of Computer Programming, Vol. 2, Section 4.5.2. Reading, MA: Addison-Wesley, 1973.
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Think about the Giant Gazinch you made, then answer each question and write an example for each one.
If the numerator (number on the top of a fraction) is very small compared to the size of the denominator (number on the bottom of a fraction), what whole number will the fraction be closest to on a ruler?
If the numerator is much smaller than the denominator, that means that the fraction will be quite small.
The fraction will be closest to zero.
If the numerator is about half the size of the denominator, where will the fraction be on a ruler?
Remember that when the numerator is less than the denominator, the fraction represents a number less than one, but greater than zero.
The fraction will be halfway between
0
1
If the numerator of a fraction is about the same size as the denominator, where will the fraction be on a ruler?
Use the clues and answers from parts (a) and (b) to figure out the answer to this problem.
If the numerator of a fraction is larger than the denominator, where will the fraction be on a ruler?
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Hidden Markov model parameter estimates from emissions and states - MATLAB hmmestimate - MathWorks Italia
hmmestimate
Pseudotransitions and Pseudoemissions
Hidden Markov model parameter estimates from emissions and states
[TRANS,EMIS] = hmmestimate(seq,states)
hmmestimate(...,'Symbols',SYMBOLS)
hmmestimate(...,'Statenames',STATENAMES)
hmmestimate(...,'Pseudoemissions',PSEUDOE)
hmmestimate(...,'Pseudotransitions',PSEUDOTR)
[TRANS,EMIS] = hmmestimate(seq,states) calculates the maximum likelihood estimate of the transition, TRANS, and emission, EMIS, probabilities of a hidden Markov model for sequence, seq, with known states, states.
hmmestimate(...,'Symbols',SYMBOLS) specifies the symbols that are emitted. SYMBOLS can be a numeric array, a string array or a cell array of the names of the symbols. The default symbols are integers 1 through N, where N is the number of possible emissions.
hmmestimate(...,'Statenames',STATENAMES) specifies the names of the states. STATENAMES can be a numeric array, a string array, or a cell array of the names of the states. The default state names are 1 through M, where M is the number of states.
hmmestimate(...,'Pseudoemissions',PSEUDOE) specifies pseudocount emission values in the matrix PSEUDOE. Use this argument to avoid zero probability estimates for emissions with very low probability that might not be represented in the sample sequence. PSEUDOE should be a matrix of size m-by-n, where m is the number of states in the hidden Markov model and n is the number of possible emissions. If the
i\to k
emission does not occur in seq, you can set PSEUDOE(i,k) to be a positive number representing an estimate of the expected number of such emissions in the sequence seq.
hmmestimate(...,'Pseudotransitions',PSEUDOTR) specifies pseudocount transition values. You can use this argument to avoid zero probability estimates for transitions with very low probability that might not be represented in the sample sequence. PSEUDOTR should be a matrix of size m-by-m, where m is the number of states in the hidden Markov model. If the
i\to j
transition does not occur in states, you can set PSEUDOTR(i,j) to be a positive number representing an estimate of the expected number of such transitions in the sequence states.
If the probability of a specific transition or emission is very low, the transition might never occur in the sequence states, or the emission might never occur in the sequence seq. In either case, the algorithm returns a probability of 0 for the given transition or emission in TRANS or EMIS. You can compensate for the absence of transition with the 'Pseudotransitions' and 'Pseudoemissions' arguments. The simplest way to do this is to set the corresponding entry of PSEUDOE or PSEUDOTR to 1. For example, if the transition
i\to j
does not occur in states, set PSEUDOTR(i,j) = 1. This forces TRANS(i,j) to be positive. If you have an estimate for the expected number of transitions
i\to j
in a sequence of the same length as states, and the actual number of transitions
i\to j
that occur in seq is substantially less than what you expect, you can set PSEUDOTR(i,j) to the expected number. This increases the value of TRANS(i,j). For transitions that do occur in states with the frequency you expect, set the corresponding entry of PSEUDOTR to 0, which does not increase the corresponding entry of TRANS.
If you do not know the sequence of states, use hmmtrain to estimate the model parameters.
trans = [0.95,0.05; 0.10,0.90];
[estimateTR,estimateE] = hmmestimate(seq,states);
[1] Durbin, R., S. Eddy, A. Krogh, and G. Mitchison. Biological Sequence Analysis. Cambridge, UK: Cambridge University Press, 1998.
hmmgenerate | hmmdecode | hmmviterbi | hmmtrain
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Persistant Timeout Protocol - Maple Help
Home : Support : Online Help : Connectivity : Web Features : Network Communication : Sockets Package : Persistant Timeout Protocol
Persistent Timeout Protocol and the Sockets Package
A number of routines in the Sockets package block indefinitely by default. In most cases, it is possible to limit the time for which a routine will block by passing an extra timeout argument. The timeout argument indicates the number of seconds to block while waiting for the availability of data. If this time expires, the routine returns the value false.
In some cases, it is inconvenient (and error-prone) to maintain a timeout value at each and every callsite to a blocking routine. For this reason, most of these types of routines obey the "persistent timeout protocol."
It is possible to configure a socket connection with a "persistent timeout" value by using the procedure Configure with the timeout option. The default value of the timeout option is -1, which means that no persistent timeout has been configured on that connection.
You can set a persistent timeout on a connection by a call of the form Configure( sid, 'timeout' = secs ), where sid is the socket ID of the connection to configure and secs is the number of seconds to configure the persistent timeout to block for. This is essentially equivalent to specifying secs as the optional timeout parameter in all subsequent calls to routines that take one on that connection. However, the persistent timeout value can still be overridden by a transient timeout specification at the call site.
A persistent timeout value can be removed from a socket connection by setting the option value to -1 by using Configure( sid, 'timeout' = -1 ).
By default, no persistent timeout is configured on a connection.
\mathrm{with}\left(\mathrm{Sockets}\right):
\mathrm{sid}≔\mathrm{Open}\left("localhost","echo"\right)
\textcolor[rgb]{0,0,1}{0}
\mathrm{Configure}\left(\mathrm{sid},'\mathrm{timeout}'\right)
\textcolor[rgb]{0,0,1}{-1}
Sockets[Configure]
Sockets[Peek]
Sockets[ReadLine]
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Living with Climate Change: Assessment of the Adaptive Capacities of Smallholders in Central Rift Valley, Ethiopia
1Wondo Genet College of Forestry and Natural Resources, Hawassa University, Shashemene, Ethiopia
2Forests and Livelihoods Research, Centre for International Forest Research (CIFOR), Addis Ababa, Ethiopia
\text{Logit}\left(Y\right)=\text{LN}\left(P/1-P\right)={\beta }_{0}+{\beta }_{1}{X}_{1}+{\beta }_{2}{X}_{2}+\cdots +{\beta }_{n}{X}_{n}+{\epsilon }_{i}
Mekonnen, Z. and Kassa, H. (2019) Living with Climate Change: Assessment of the Adaptive Capacities of Smallholders in Central Rift Valley, Ethiopia. American Journal of Climate Change, 8, 205-227. https://doi.org/10.4236/ajcc.2019.82012
1. Sisay, T. (2016) Vulnerability of Smallholder Farmers to Climate Change at Dabat and West Belesa Districts, North Gondar, Ethiopia. Journal of Earth Science and Climate Change, 7, 365. https://doi.org/10.4172/2157-7617.1000365
2. Abebe, Y. and Bekele, A. (2017) Vulnerability of Smallholder Farmers to Climate Change in the Central Rift Valley of Ethiopia: A Gender Disaggregated Approach. Ethiopian Journal of Agricultural Sciences, 27, 85-97.
3. Yaméogo, T.B., Fonta, W.M. and Wünscher, T. (2018) Can Social Capital Influence Smallholder Farmers’ Climate-Change Adaptation Decisions? Evidence from Three Semi-Arid Communities in Burkina Faso, West Africa. Social Sciences, 7, 1-20. https://doi.org/10.3390/socsci7030033
4. Donatti, C.I., Harvey, C.A., Martinez-Rodriguez, M.R., Vignola, R. and Rodriguez, C.M. (2018) Vulnerability of Smallholder Farmers to Climate Change in Central America and Mexico: Current Knowledge and Research Gaps. Climate and Development, 11, 264-286. https://doi.org/10.1080/17565529.2018.1442796
5. Kuwornu, J.K.M., Al-Hassan, R.M., Etwire, P.M. and Osei-Owusu, Y. (2013) Adaptation Strategies of Smallholder Farmers to Climate Change and Variability: Evidence from Northern Ghana. Information Management and Business Review, 5, 233-239.
6. Menike, L. and Arachchi, K. (2016) Adaptation to Climate Change by Smallholder Farmers in Rural Communities: Evidence from Sri Lanka. Procedia Food Science, 6, 288-292. https://doi.org/10.1016/j.profoo.2016.02.057
7. EPCC (Ethiopian Panel on Climate Change) (2015) First Assessment Report, Working Group II Agriculture and Food Security. Ethiopian Academy of Sciences, Addis Ababa.
8. Agrawal, A. and Lemos, M.C. (2007) A Greener Revolution in the Making? Environmental Governance in the 21st Century. Environment: Science and Policy for Sustainable Development, 49, 36-45. https://doi.org/10.3200/ENVT.49.5.36-45
9. Agrawal, A. (2010) Local Institutions and Adaptation to Climate Change. In: Mears and Norton, Eds., Social Dimensions of Climate Change: Equity and Vulnerability in a Warming World, World Bank, Washington DC, 173-198.
10. Lowder, S.K., Skoet, J. and Traney, T. (2016) The Number, Size, and Distribution of Farms, Smallholder Farms, and Family Farms Worldwide. World Development, 87, 16-29. https://doi.org/10.1016/j.worlddev.2015.10.041
11. FAO (2015) The Economic Lives of Smallholder Farmers: An Analysis Based on Household Data from Nine Countries. FAO, Rome.
12. Deininger, K., Savastano, S. and Xia, F. (2017) Smallholders’ Land Access in Sub-Saharan Africa: A New Landscape? Food Policy, 67, 78-92. https://doi.org/10.1016/j.foodpol.2016.09.012
13. Eriksen, S., Aldunce, P., Bahinipati, C.S., Martins, R.D., Molefe, J.I., Nhemachena, C., O’Brien, K., Olorunfemi, F., Park, J., Sygna, L. and Ulsrud, K. (2011) When Not Every Response to Climate Change Is a Good One: Identifying Principles for Sustainable Adaptation. Climate and Development, 3, 7-20. https://doi.org/10.3763/cdev.2010.0060
14. Epple, C., Wicander, S., Mant, R., Kapos, V., Rossing, T. and Rizvi, A.R. (2016) Shared Goals—Joined-Up Approaches? Why Action under the Paris Agreement, the Sustainable Development Goals and the Strategic Plan for Biodiversity 2011-2020 Needs to Come Together at the Landscape Level. FEBA Discussion Paper Developed for CBD COP 13. UNEP-WCMC, Cambridge and IUCN, Gland, 8 p.
15. Patt, A.G. and Schroter, D. (2008) Perceptions of Climate Risk in Mozambique: Implications for the Success of Adaptation Strategies. Global Environmental Change, 18, 458-467. https://doi.org/10.1016/j.gloenvcha.2008.04.002
16. Kebede, D. and Adane, H. (2011) Climate Change Adaptations and Induced Farming Livelihoods. Report No. 64, the Drylands Coordination Group (DCG), Oslo.
17. Chia, E.L., Somorin, O.A., Sonwa, D.J., Bele, Y.M. and Tiani, M.A. (2015) Forest-Climate Nexus: Linking Adaptation and Mitigation in Cameroon’s Climate Policy Process. Climate and Development, 7, 85-96. https://doi.org/10.1080/17565529.2014.918867
18. Haule, C.B.M., Mlozi, M.R.S. and Mulengera, M.K. (2013) Land Degradation and Smallholder Farmers’ Response: A Case of Villages in the Southern Parts of Ludewa District, Iringa Region. Tanzania Journal of Agricultural Sciences, 12, 11-21.
19. Kirui, O.K. (2016) Impact of Land Degradation on Household Poverty: Evidence from a Panel Data Simultaneous Equation Model. 5th International Conference of the African Association of Agricultural Economists, Transforming Smallholder Agriculture in Africa: The Role of Policy and Governance, Addis Ababa, 23-26 September 2016, 1-29.
20. Gatzweiler, F.W. and von Braun, J. (2016) Technological and Institutional Innovations for Marginalized Smallholders in Agricultural Development. Springer, Berlin. https://doi.org/10.1007/978-3-319-25718-1_13
21. Enbakom, H.W., Feyssa, D.H. and Takele, S. (2017) Impacts of Deforestation on the Livelihood of Smallholder Farmers in Arba Minch Zuria Woreda, Southern Ethiopia. African Journal of Agricultural Research, 12, 1293-1305. https://doi.org/10.5897/AJAR2015.10123
22. Makate, C., Wang, R., Makate, M. and Mango, N. (2016) Crop Diversification and Livelihoods of Smallholder Farmers in Zimbabwe: Adaptive Management for Environmental Change. SpringerPlus, 5, 1135. https://doi.org/10.1186/s40064-016-2802-4
23. Deresa, T.T. (2010) Assessment of the Vulnerability of Ethiopian Agriculture to Climate Change and Farmers’ Adaptation Strategies. PhD Thesis, University of Pretoria, Pretoria.
24. Hameso, S.Y. (2015) Perceptions, Vulnerability and Adaptation to Climate Change in Ethiopia: The Case of Smallholder Farmers in Sidama. PhD Thesis, University of East London, London.
25. Lemma, W.A. (2016) Analysis of Smallholder Farmers’ Perceptions of Climate Change and Adaptation Strategies to Climate Change: The Case of Western Amhara Region, Ethiopia. PhD Thesis, University of South Africa, Cape Town.
26. Amare, A. and Simane, B. (2017) Climate Change Induced Vulnerability of Smallholder Farmers: Agroecology-Based Analysis in the Muger Sub-Basin of the Upper Blue-Nile Basin of Ethiopia. American Journal of Climate Change, 6, 668-693. https://doi.org/10.4236/ajcc.2017.64034
27. Belay, A., Recha, J.W., Woldeamanuel, T., John, F. and Morton, J.F. (2017) Smallholder Farmers’ Adaptation to Climate Change and Determinants of Their Adaptation Decisions in the Central Rift Valley of Ethiopia. Agriculture & Food Security, 6, 24. https://doi.org/10.1186/s40066-017-0100-1
28. Asrat, P. and Simane, B. (2018) Farmers’ Perception of Climate Change and Adaptation Strategies in the Dabus Watershed, North-West Ethiopia. Ecological Processes, 7, 7. https://doi.org/10.1186/s13717-018-0118-8
29. Zegeye, H. (2018) Climate Change in Ethiopia: Impacts, Mitigation and Adaptation. International Journal of Research in Environmental Studies, 5, 18-35.
30. CSA (Central Statistical Agency) (1994/2007) Population Censuses. Addis Ababa.
31. CSA (2005/2016) Population Projections. Addis Ababa.
32. Gebreslassie, H. (2014) Land Use-Land Cover dynamics of Huluka Watershed, Central Rift Valley, Ethiopia. International Soil and Water Conservation Research, 2, 25-33. https://doi.org/10.1016/S2095-6339(15)30055-1
33. Mekonnen, Z., Tadesse, H., Woldeamanuel, T., Asfaw, Z. and Kassa, H. (2018) Land Use and Land Cover Changes and the Link to Land Degradation in Arsi Negele District, Central Rift Valley, Ethiopia. Remote Sensing Applications: Society and Environment, 12, 1-9. https://doi.org/10.1016/j.rsase.2018.07.012
34. Israel, G.D. (1992) Determining Sample Size. Agricultural Education and Communication Department, University of Florida, IFAS Extension, Gainesville, PEOD6.
35. Bartlett, J.E., Kotrlik, J.W. and Higgins, C.C. (2001) Organizational Research: Determining Appropriate Sample Size in Survey Research. Information Technology, Learning, and Performance Journal, 19, 43-50. https://doi.org/10.5032/jae.2002.03001
36. Bernard, H.R. (2006) Research Methods in Anthropology: Qualitative and Quantitative Approaches. 4th Edition, Alta Mira Press, New York.
37. Smith, H.A. and Sharp, K. (2012) Indigenous Climate Knowledge. Climate Change, 3, 467-476. https://doi.org/10.1002/wcc.185
38. Abate, A. and Lemenih, M. (2014) Detecting and Quantifying Land Use/Land Cover Dynamics in Nadda Asendabo Watershed, South Western Ethiopia. International Journal of Environmental Sciences, 3, 45-50.
39. Kindu, M., Schneider, T., Teketay, D. and Knoke, T. (2015) Drivers of Land Use/Land Cover Changes in Munessa-Shashemene Landscape of the South-Central Highlands of Ethiopia. Environmental Monitoring and Assessment, 187, 452. https://doi.org/10.1007/s10661-015-4671-7
40. Gardner, T., Godar, J. and Garrett, R. (2014) Governing for Sustainability in Agricultural-Forest Frontiers: A Case Study of the Brazilian Amazon. Discussion Brief, Stockholm Environment Institute, Stockholm.
41. Delgado, C., Wolosin, M. and Purvis, N. (2015) Restoring and Protecting Agricultural and Forest Landscapes and Increasing Agricultural Productivity. Working Paper for Seizing the Global Opportunity: Partnerships for Better Growth and a Better Climate. New Climate Economy, London and Washington DC. https://newclimateeconomy.report/workingpapers/wp-content/uploads/sites/5/2016/04/NCE-restoring-protecting-ag-forest-landscapes-increase-ag.pdf
42. Vira, B., Wildburger, C. and Mansourian, S. (2015) Forests, Trees and Landscapes for Food Security and Nutrition. A Global Assessment Report, IUFRO World Series Volume 33, Vienna, 172 p.
43. Pan, J., Zheng, Y., Wang, J. and Xie, X. (2015) Climate Capacity: The Measurement for Adaptation to Climate Change. Chinese Journal of Population Resources and Environment, 13, 99-108. https://doi.org/10.1080/10042857.2015.1033802
44. Munthali, K.G. and Murayama, Y. (2013) Interdependences between Smallholder Farming and Environmental Management in Rural Malawi: A Case of Agriculture-Induced Environmental Degradation in Malingunde Extension Planning Area (EPA). Land, 2, 158-175. https://doi.org/10.3390/land2020158
45. Driscoll, D.A., Banks, S.C., Barton, P.S., Lindenmayer, D.B. and Smith, A.L. (2013) Conceptual Domain of the Matrix in Fragmented Landscapes. Trends in Ecology & Evolution, 28, 605-613. https://doi.org/10.1016/j.tree.2013.06.010
46. KFS (Korean Forest Service) (2015) Current State of Forests in Korea: Lessons Learned from Forest Restoration Experiences of the Republic of Korea. Daejeon.
47. IUFRO (2016) Restoring Forest Landscapes: A “Win-Win” for People, Nature and Climate. http://www.iufro.org/download/file/25158/199/ws34-policy-brief_01_pdf
48. Oliver, T.H. and Morecroft, M.D. (2014) Interactions between Climate Change and Land Use Change on Biodiversity: Attribution Problems, Risks, and Opportunities. WIREs Climate Change, 5, 317-335. https://doi.org/10.1002/wcc.271
49. Baral, H. and Holmgren, P. (2015) A Framework for Measuring Sustainability Outcomes for Landscape Investments. Working Paper 195, CIFOR, Bogor. https://doi.org/10.17528/cifor/005761
50. Scherr, S.J., Shames, S. and Friedman, R. (2012) From Climate-Smart Agriculture to Climate-Smart Landscapes. Agriculture & Food Security, 1, 12. https://doi.org/10.1186/2048-7010-1-12
51. IPCC (2013) Climate Change 2013. In: Stocker, T.F., Qin, D., Plattner, G.K., Tignor, M., Allen, S.K., Boschung, J., Nauels, A., Xia, Y., Bex, V. and Midgley, P.M., Eds., The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge and New York, 1535 p.
52. Nay, J.J., Abkowitz, M., Chu, E., Gallagher, D. and Wright, H. (2014) A Review of Decision-Support Models for Adaptation to Climate Change in the Context of Development. Climate and Development, 6, 357-367. https://doi.org/10.1080/17565529.2014.912196
53. UNFCCC (2010) Report of the Conference of the Parties on Its 16th Session (FCCC/CP/2010/7/Add.1), Cancun, Mexico, 29 November-10 December 2010, Bonn, Germany.
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Revision as of 15:27, 17 January 2020 by Munich (talk | contribs) (→Mean convection of turbulent kinetic energy)
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle \langle u\rangle }
{\displaystyle \langle w\rangle }
{\displaystyle \langle u'_{i}u'_{j}\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle ||{\vec {U}}||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle ||{\vec {U}}_{\mathrm {PIV} }||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle ||{\vec {U}}_{\mathrm {LES} }||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle x/D}
{\displaystyle z/D}
{\displaystyle x/D}
{\displaystyle z/D}
{\displaystyle -0.788}
{\displaystyle 0.03}
{\displaystyle -0.843}
{\displaystyle 0.037}
{\displaystyle -0.918}
{\displaystyle 0}
{\displaystyle -1.1}
{\displaystyle 0}
{\displaystyle -0.533}
{\displaystyle 0}
{\displaystyle -0.534}
{\displaystyle 0}
{\displaystyle -0.507}
{\displaystyle 0.036}
{\displaystyle -0.50}
{\displaystyle 0.04}
{\displaystyle -0.697}
{\displaystyle 0.051}
{\displaystyle -0.735}
{\displaystyle 0.06}
{\displaystyle -0.513}
{\displaystyle 0.017}
{\displaystyle -0.513}
{\displaystyle 0.02}
{\displaystyle x-}
{\displaystyle x_{\mathrm {adj} }={\frac {x-x_{\mathrm {Cyl} }}{x_{\mathrm {Cyl} }-x_{\mathrm {V1} }}}}
{\displaystyle x_{\mathrm {Cyl} }=-0.5D}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle x_{\mathrm {V1} }}
{\displaystyle \langle u(z)\rangle /u_{\mathrm {b} }}
{\displaystyle u(z)}
{\displaystyle x_{\mathrm {adj} }=-0.25}
{\displaystyle x_{\mathrm {adj} }=-0.5}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
{\displaystyle \langle u_{i}'u_{j}'(z)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle k(z)\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle x_{\mathrm {adj} }=-1.5}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle x_{\mathrm {adj} }=-0.5}
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle w'w'\rangle }
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle u'w'\rangle }
{\displaystyle \langle w(x)\rangle /u_{\mathrm {b} }}
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle w(x)\rangle }
{\displaystyle x-}
{\displaystyle x_{\mathrm {adj} }\approx -0.1}
{\displaystyle x_{\mathrm {adj} }=-0.65}
{\displaystyle \langle u_{i}'u_{j}'(x)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle k(x)\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle u_{i}'u_{j}'\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle \langle k_{\mathrm {PIV,inplane} }\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,inplane} }\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.5(\langle u'^{2}\rangle +\langle v'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {PIV,inplane} }\rangle =0.074u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,inplane} }\rangle =0.079u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.5(\langle u'^{2}\rangle +\langle v'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.09u_{\mathrm {b} }^{2}}
{\displaystyle 0=P+\nabla T-\epsilon +C}
{\displaystyle P}
{\displaystyle \nabla T}
{\displaystyle \epsilon }
{\displaystyle C}
{\displaystyle v}
{\displaystyle P=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}}
{\displaystyle T=\underbrace {-{\frac {1}{2}}\langle u_{i}'u_{j}'u_{j}'\rangle } _{\text{turbulent fluctuations}}\underbrace {-{\frac {1}{\rho }}\langle u_{i}'p'\rangle } _{\text{pressure transport}}\underbrace {+2\nu \langle u_{j}'s_{ij}\rangle } _{\text{viscous diffusion}}}
{\displaystyle \epsilon =2\nu \langle s_{ij}s_{ij}\rangle }
{\displaystyle s_{ij}={\frac {1}{2}}\left({\frac {\partial u_{i}'}{\partial x_{j}}}+{\frac {\partial u_{j}'}{\partial x_{i}}}\right)}
{\displaystyle \epsilon _{\mathrm {total} }=\epsilon _{\mathrm {res} }+\epsilon _{\mathrm {SGS} }=2\nu \langle s_{ij}s_{ij}\rangle +2\langle \nu _{\mathrm {t} }s_{ij}s_{ij}\rangle }
{\displaystyle C=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}}
{\displaystyle D/u_{\mathrm {b} }^{3}}
{\displaystyle P_{\mathrm {PIV} }=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle P_{\mathrm {LES} }=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle 0.3u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {LES} }\approx 0.4u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {PIV} }\approx 0.2u_{\mathrm {b} }^{3}/D}
{\displaystyle x=-0.7D}
{\displaystyle P}
{\displaystyle \nabla T_{\mathrm {turb,PIV} }=-{\frac {1}{2}}{\frac {\partial \langle u_{i}'u_{j}'u_{j}'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {turb,LES} }=-{\frac {1}{2}}{\frac {\partial \langle u_{i}'u_{j}'u_{j}'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle x=-0.75D}
{\displaystyle 0.4u_{\mathrm {b} }^{3}/D}
{\displaystyle T_{\mathrm {turb,LES} }\approx 0.35u_{\mathrm {b} }^{3}/D}
{\displaystyle \nabla T_{\mathrm {press,LES} }=-{\frac {1}{\rho }}{\frac {\partial \langle u_{i}'p'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {visc,LES} }=2\nu {\frac {\partial \langle u_{j}'s_{ij}\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {turb} }}
{\displaystyle \nabla T_{\mathrm {press} }}
{\displaystyle \langle w\rangle <0}
{\displaystyle w-}
{\displaystyle w'}
{\displaystyle p'<0}
{\displaystyle \nabla T_{\mathrm {visc} }}
{\displaystyle |0.05|u_{\mathrm {b} }^{3}/D}
{\displaystyle P}
{\displaystyle \nabla T}
{\displaystyle \epsilon }
{\displaystyle \epsilon _{\mathrm {PIV} }=2\nu \langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \epsilon _{\mathrm {LES,total} }=(2\nu \langle s_{ij}s_{ij}\rangle +2\langle \nu _{\mathrm {t} }s_{ij}s_{ij}\rangle )\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle P}
{\displaystyle \epsilon _{\mathrm {LES} }=0.066u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {max} }}
{\displaystyle \epsilon _{\mathrm {max} }}
{\displaystyle C_{\mathrm {PIV} }=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle C_{\mathrm {LES} }=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle x\approx -0.63D}
{\displaystyle C}
{\displaystyle R_{\mathrm {PIV} }=P+\nabla T_{\mathrm {turb} }-\epsilon +C\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle -\nabla T_{\mathrm {press,LES} }}
{\displaystyle R_{\mathrm {LES} }=P+\nabla T-\epsilon _{\mathrm {total} }+C\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle <|0.01|u_{\mathrm {b} }^{3}/D}
{\displaystyle T_{\mathrm {turb} }=-{\frac {1}{2}}\langle u_{i}'u_{j}'u_{j}'\rangle }
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {p} }={\frac {\langle p\rangle }{{\frac {\rho }{2}}u_{\mathrm {b} }^{2}}}}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }={\frac {\langle \tau _{\mathrm {w} }\rangle }{{\frac {\rho }{2}}u_{\mathrm {b} }^{2}}}}
{\displaystyle z_{1}\approx 0.0036D\approx 10\mathrm {px} }
{\displaystyle z_{1}\approx 0.0005D}
{\displaystyle z-}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle |c_{\mathrm {f} }|=0.01}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle 50\times 171(n\times m)}
{\displaystyle 143\times 131(n\times m)}
{\displaystyle n\cdot m}
{\displaystyle x_{\mathrm {adj} }}
{\displaystyle {\frac {x}{D}}}
{\displaystyle {\frac {z}{D}}}
{\displaystyle {\frac {\langle u\rangle }{u_{\mathrm {b} }}}}
{\displaystyle -}
{\displaystyle {\frac {\langle w\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle u'u'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle {\frac {\langle w'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle {\frac {\langle u'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle P{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle C{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {turb} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle -}
{\displaystyle -}
{\displaystyle \epsilon {\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle -}
{\displaystyle x_{\mathrm {adj} }}
{\displaystyle {\frac {x}{D}}}
{\displaystyle {\frac {z}{D}}}
{\displaystyle {\frac {\langle u\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle v\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle w\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle u'u'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle v'v'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle w'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle u'v'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle u'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle v'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle P{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle C{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {turb} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {press} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {visc} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \epsilon {\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle c_{\mathrm {p} }}
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Removing entries from a dictionary containing bad words - PhotoLens
I have a dictionary with each item containing a quote in another dictionary:
'example1': {'quote': 'And it must follow, as the night the day, thou canst not then be false to any man.\n'},
'example2': {'quote': 'What a piece of work is man! how noble in reason!.\n'}
I need to completely remove each entry whose quote contains a badword, not just checking if the string contains the badword but if it matches the full word. For instance, following the above example, considering as to be a badword, it should remove example1 but not example2 (that contains reASon).
def filter_bad_words(entries):
f = open("badwords.txt", "r")
badwords = f.readlines()
# remove new lines from each word
for i in range(len(badwords)):
badwords[i] = badwords[i].strip('\n')
original_count = len(entries)
for key, item in entries.items():
quote = item['quote']
if any(findWholeWord(x)(quote) for x in badwords):
del entries[key]
print "Remove: %s" % quote
print "Removed %s items." % (original_count - len(entries))
""" removes exact word"""
with open("quotes.txt", "r") as f:
quotes = f.readlines()
for key, quote in enumerate(quotes):
entry_key = "example{}".format(key)
entries[entry_key] = {'quote': quote}
filter_bad_words(entries)
That it should come to this!.
The play 's the thing wherein I'll catch the conscience of the king.
Doubt that the sun doth move, doubt truth to be a liar, but never doubt I love.
When sorrows come, they come not single spies, but in
All the world 's a stage, and all the men and women merely players. They have their exits and their entrances; And one man in his time plays many parts.
Can one desire too much of a good thing?.
A horse! a horse! my kingdom for a horse!.
The world is grown so bad, that wrens make prey where eagles dare not
It seems she hangs upon the cheek of night like a rich jewel in an Ethiope's ear.
See, how she leans her cheek upon her hand! O that I were a glove upon that hand, that I might touch that cheek!.
badwords.txt
It does its job but I find it extremely slow when dealing with more than 100,000 entries, so I would appreciate suggestions to improve its performance.
(I’ve setup a repo to make it easier for testing.)
The current time complexity is O(N∗M)
O(N * M)
where N
is the number of quotes and M
is the number of bad words. For every single word in a quote you are iterating over all the bad words to check if there is a match.
We can do better than that. What if you would initialize bad words as a set and would just lookup if a word is there – the lookup itself would be constant time – O(1)
which would make the overall complexity O(N+M)
O(N + M)
– we still need O(M)
to initially make a set of bad words.
Also, I would use a more appropriate and robust nltk.word_tokenize() for word tokenization.
with open("badwords.txt", "r") as f:
badwords = set(word.strip() for word in f)
filtered_entries = {}
words = word_tokenize(quote)
if not any(word in badwords for word in words):
filtered_entries[key] = {'quote': quote}
print("Removed %s items." % (len(entries) - len(filtered_entries)))
return filtered_entries
"example{}".format(index): {'quote': quote}
for index, quote in enumerate(f)
print(filter_bad_words(entries))
Source : Link , Question Author : marcanuy , Answer Author : Toby Speight
Categories circuit-python, hash-map, high-performance, python-2.x Tags circuit-python, hash-map, high-performance, python-2.x Post navigation
How do I create an outline around a raster image?
Illustrators standard colour swatches has vanished in “rogue” pdf document [duplicate]
How to change Adobe Illustrator CC’s pixel to inch ratio?
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Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces
2016 Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces
Vincent Humilière, Frédéric Le Roux, Sobhan Seyfaddini
Inspired by Le Calvez’s theory of transverse foliations for dynamical systems on surfaces, we introduce a dynamical invariant, denoted by
\mathsc{N}
, for Hamiltonians on any surface other than the sphere. When the surface is the plane or is closed and aspherical, we prove that on the set of autonomous Hamiltonians this invariant coincides with the spectral invariants constructed by Viterbo on the plane and Schwarz on closed and aspherical surfaces.
Along the way, we obtain several results of independent interest: we show that a formal spectral invariant, satisfying a minimal set of axioms, must coincide with
\mathsc{N}
on autonomous Hamiltonians, thus establishing a certain uniqueness result for spectral invariants; we obtain a “max formula” for spectral invariants on aspherical manifolds; we give a very simple description of the Entov–Polterovich partial quasi-state on aspherical surfaces, and we characterize the heavy and super-heavy subsets of such surfaces.
Vincent Humilière. Frédéric Le Roux. Sobhan Seyfaddini. "Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces." Geom. Topol. 20 (4) 2253 - 2334, 2016. https://doi.org/10.2140/gt.2016.20.2253
Received: 17 March 2015; Revised: 21 August 2015; Accepted: 18 September 2015; Published: 2016
Primary: 53D40 , 53Dxx
Secondary: 37E30 , 37Exx
Keywords: area-preserving diffeomorphisms , Hamiltonian Floer theory , spectral invariants
Vincent Humilière, Frédéric Le Roux, Sobhan Seyfaddini "Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces," Geometry & Topology, Geom. Topol. 20(4), 2253-2334, (2016)
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Dynamically Triggered Events in a Low Seismically Active Region of Gujarat, Northwest India, during the 2012 Mw 8.6 Indian Ocean Earthquake | Bulletin of the Seismological Society of America | GeoScienceWorld
Mayank Dixit;
Institute of Seismological Research, (ISR), Gandhinagar, India
Department of Geophysics, Kurukshetra University, Kurukshetra, India
Abhey Ram Bansal;
Abhey Ram Bansal *
CSIR–National Geophysical Research Institute, Hyderabad, India
Corresponding author: arb@ngri.res.in
Ravi Kumar Mangalalampally;
Ravi Kumar Mangalalampally
Ketan Singha Roy;
Ketan Singha Roy
Satybir Singh Teotia
Mayank Dixit, Abhey Ram Bansal, Ravi Kumar Mangalalampally, Ketan Singha Roy, Satybir Singh Teotia; Dynamically Triggered Events in a Low Seismically Active Region of Gujarat, Northwest India, during the 2012
Mw
8.6 Indian Ocean Earthquake. Bulletin of the Seismological Society of America 2022; doi: https://doi.org/10.1785/0120210142
The mainland region of Gujarat, northwest India, is a less investigated region than other parts of India with a low seismicity rate. An
Mw
>4.7 earthquake has not occurred in this region for 15 yr, and no
Mw
>5.5 events since 1971. We analyze the local earthquake catalog and waveforms to examine dynamic triggering in the region by the 2012
Mw
8.6 Indian Ocean earthquake, which triggered widespread seismicity globally. Further detection of possibly missing microearthquakes is conducted by applying the matched filter technique to the waveforms. We identify six microearthquakes (
∼ML
1.0–2.1) triggered during the surface and coda wave of the 2012 mainshock. Also, an earthquake of
Mw
2.6 was likely triggered five hours after the mainshock near Bhavnagar city, because the record since 2006 would indicate such a magnitude event to have only a 0.8% chance of occurring independently any given day. Indeed, only 35 earthquakes with
Mw≥2.5
were recorded since 2006 within a 100 km radius of this city. The
β
‐statistics indicate an increase in seismicity and further confirm the triggering. The seismicity rate increased immediately after the 2012 mainshock and continued for three days, indicating a possible delayed triggering. The delayed triggering may be due to the crustal fluid, and/or subcritical crack growth model may be responsible for triggering. Our study suggests that dynamic triggering tends to occur near active faults that have ruptured in ancient times. Other recent earthquakes, for example, 2011 Tohoku‐Oki, did not trigger seismicity despite significant peak dynamic stresses values. Investigation of dynamic triggering in regions experiencing infrequent earthquakes can be crucial in understanding the origin of such earthquakes, which can be achieved by grasping the ambient stresses and geodynamic mechanisms in a particular region. Thus, we evaluate character and behavior of high‐amplitude surface waves to grasp better the undergoing processes and stress transfer in the intraplate mainland region.
|
18.2: Effect Sizes - Statistics LibreTexts
[ "article:topic", "showtoc:no", "authorname:rapoldrack", "Cohen\u2019s D", "source@https://statsthinking21.github.io/statsthinking21-core-site" ]
https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_Statistical_Thinking_for_the_21st_Century_(Poldrack)%2F18%253A_Quantifying_Effects_and_Desiging_Studies%2F18.02%253A_Effect_Sizes
“Statistical significance is the least interesting thing about the results. You should describe the results in terms of measures of magnitude – not just, does a treatment affect people, but how much does it affect them.” Gene Glass (REF)
In the last chapter, we discussed the idea that statistical significance may not necessarily reflect practical significance. In order to discuss practical significance, we need a standard way to describe the size of an effect in terms of the actual data, which we refer to as an effect size. In this section we will introduce the concept and discuss various ways that effect sizes can be calculated.
d = \frac{\bar{X}_1 - \bar{X}_2}{s}
\bar{X}_1
\bar{X}_2
are the means of the two groups, and
s
is the pooled standard deviation (which is a combination of the standard deviations for the two samples, weighted by their sample sizes):
s = \sqrt{\frac{(n_1 - 1)s^2_1 + (n_2 - 1)s^2_2 }{n_1 +n_2 -2}}
n
n
s
s
are the standard deviations for the two groups respectively. Note that this is very similar in spirit to the t statistic — the main difference is that the denominator in the t statistic is based on the standard error of the mean, whereas the denominator in Cohen’s D is based on the standard deviation of the data. This means that while the t statistic will grow as the sample size gets larger, the value of Cohen’s D will remain the same.
There is a commonly used scale for interpreting the size of an effect in terms of Cohen’s d:
It can be useful to look at some commonly understood effects to help understand these interpretations. For example, the effect size for gender differences in height (d = 1.6) is very large by reference to our table above. We can also see this by looking at the distributions of male and female heights in our sample. Figure 18.2 shows that the two distributions are quite well separated, though still overlapping, highlighting the fact that even when there is a very large effect size for the difference between two groups, there will be individuals from each group that are more like the other group.
It is also worth noting that we rarely encounter effects of this magnitude in science, in part because they are such obvious effects that we don’t need scientific research to find them. As we will see in Chapter 32 on reproducibility, very large reported effects in scientific research often reflect the use of questionable research practices rather than truly huge effects in nature. It is also worth noting that even for such a huge effect, the two distributions still overlap - there will be some females who are taller than the average male, and vice versa. For most interesting scientific effects, the degree of overlap will be much greater, so we shouldn’t immediately jump to strong conclusions about different populations based on even a large effect size.
odds\ of\ A = \frac{P(A)}{P(\neg A)}
For example, let’s take the case of smoking and lung cancer. A study published in the International Journal of Cancer in 2012 (Pesch et al. 2012) combined data regarding the occurrence of lung cancer in smokers and individuals who have never smoked across a number of different studies. Note that these data come from case-control studies, which means that participants in the studies were recruited because they either did or did not have cancer; their smoking status was then examined. These numbers thus do not represent the prevalence of cancer amongst smokers in the general population – but they can tell us about the relationship between cancer and smoking.
Table 18.2: Cancer occurrence separately for current smokers and those who have never smoked
We can convert these numbers to odds ratios for each of the groups. The odds of someone having lung cancer who has never smoked is 0.08 whereas the odds of a current smoker having lung cancer is 1.77. The ratio of these odds tells us about the relative likelihood of cancer between the two groups: The odds ratio of 23.22 tells us that the odds of cancer in smokers are roughly 23 times higher than never-smokers.
18.2: Effect Sizes is shared under a not declared license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.
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Revision as of 15:33, 17 January 2020 by Munich (talk | contribs) (→Streamlines)
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle \langle u\rangle }
{\displaystyle \langle w\rangle }
{\displaystyle \langle u'_{i}u'_{j}\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle ||{\vec {U}}||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component,respectively.
{\displaystyle ||{\vec {U}}_{\mathrm {PIV} }||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle ||{\vec {U}}_{\mathrm {LES} }||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle x/D}
{\displaystyle z/D}
{\displaystyle x/D}
{\displaystyle z/D}
{\displaystyle -0.788}
{\displaystyle 0.03}
{\displaystyle -0.843}
{\displaystyle 0.037}
{\displaystyle -0.918}
{\displaystyle 0}
{\displaystyle -1.1}
{\displaystyle 0}
{\displaystyle -0.533}
{\displaystyle 0}
{\displaystyle -0.534}
{\displaystyle 0}
{\displaystyle -0.507}
{\displaystyle 0.036}
{\displaystyle -0.50}
{\displaystyle 0.04}
{\displaystyle -0.697}
{\displaystyle 0.051}
{\displaystyle -0.735}
{\displaystyle 0.06}
{\displaystyle -0.513}
{\displaystyle 0.017}
{\displaystyle -0.513}
{\displaystyle 0.02}
{\displaystyle x-}
{\displaystyle x_{\mathrm {adj} }={\frac {x-x_{\mathrm {Cyl} }}{x_{\mathrm {Cyl} }-x_{\mathrm {V1} }}}}
{\displaystyle x_{\mathrm {Cyl} }=-0.5D}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle x_{\mathrm {V1} }}
{\displaystyle \langle u(z)\rangle /u_{\mathrm {b} }}
{\displaystyle u(z)}
{\displaystyle x_{\mathrm {adj} }=-0.25}
{\displaystyle x_{\mathrm {adj} }=-0.5}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
{\displaystyle \langle u_{i}'u_{j}'(z)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle k(z)\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle x_{\mathrm {adj} }=-1.5}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle x_{\mathrm {adj} }=-0.5}
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle w'w'\rangle }
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle u'w'\rangle }
{\displaystyle \langle w(x)\rangle /u_{\mathrm {b} }}
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle w(x)\rangle }
{\displaystyle x-}
{\displaystyle x_{\mathrm {adj} }\approx -0.1}
{\displaystyle x_{\mathrm {adj} }=-0.65}
{\displaystyle \langle u_{i}'u_{j}'(x)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle k(x)\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle u_{i}'u_{j}'\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle \langle k_{\mathrm {PIV,inplane} }\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,inplane} }\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.5(\langle u'^{2}\rangle +\langle v'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {PIV,inplane} }\rangle =0.074u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,inplane} }\rangle =0.079u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.5(\langle u'^{2}\rangle +\langle v'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.09u_{\mathrm {b} }^{2}}
{\displaystyle 0=P+\nabla T-\epsilon +C}
{\displaystyle P}
{\displaystyle \nabla T}
{\displaystyle \epsilon }
{\displaystyle C}
{\displaystyle v}
{\displaystyle P=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}}
{\displaystyle T=\underbrace {-{\frac {1}{2}}\langle u_{i}'u_{j}'u_{j}'\rangle } _{\text{turbulent fluctuations}}\underbrace {-{\frac {1}{\rho }}\langle u_{i}'p'\rangle } _{\text{pressure transport}}\underbrace {+2\nu \langle u_{j}'s_{ij}\rangle } _{\text{viscous diffusion}}}
{\displaystyle \epsilon =2\nu \langle s_{ij}s_{ij}\rangle }
{\displaystyle s_{ij}={\frac {1}{2}}\left({\frac {\partial u_{i}'}{\partial x_{j}}}+{\frac {\partial u_{j}'}{\partial x_{i}}}\right)}
{\displaystyle \epsilon _{\mathrm {total} }=\epsilon _{\mathrm {res} }+\epsilon _{\mathrm {SGS} }=2\nu \langle s_{ij}s_{ij}\rangle +2\langle \nu _{\mathrm {t} }s_{ij}s_{ij}\rangle }
{\displaystyle C=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}}
{\displaystyle D/u_{\mathrm {b} }^{3}}
{\displaystyle P_{\mathrm {PIV} }=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle P_{\mathrm {LES} }=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle 0.3u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {LES} }\approx 0.4u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {PIV} }\approx 0.2u_{\mathrm {b} }^{3}/D}
{\displaystyle x=-0.7D}
{\displaystyle P}
{\displaystyle \nabla T_{\mathrm {turb,PIV} }=-{\frac {1}{2}}{\frac {\partial \langle u_{i}'u_{j}'u_{j}'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {turb,LES} }=-{\frac {1}{2}}{\frac {\partial \langle u_{i}'u_{j}'u_{j}'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle x=-0.75D}
{\displaystyle 0.4u_{\mathrm {b} }^{3}/D}
{\displaystyle T_{\mathrm {turb,LES} }\approx 0.35u_{\mathrm {b} }^{3}/D}
{\displaystyle \nabla T_{\mathrm {press,LES} }=-{\frac {1}{\rho }}{\frac {\partial \langle u_{i}'p'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {visc,LES} }=2\nu {\frac {\partial \langle u_{j}'s_{ij}\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {turb} }}
{\displaystyle \nabla T_{\mathrm {press} }}
{\displaystyle \langle w\rangle <0}
{\displaystyle w-}
{\displaystyle w'}
{\displaystyle p'<0}
{\displaystyle \nabla T_{\mathrm {visc} }}
{\displaystyle |0.05|u_{\mathrm {b} }^{3}/D}
{\displaystyle P}
{\displaystyle \nabla T}
{\displaystyle \epsilon }
{\displaystyle \epsilon _{\mathrm {PIV} }=2\nu \langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \epsilon _{\mathrm {LES,total} }=(2\nu \langle s_{ij}s_{ij}\rangle +2\langle \nu _{\mathrm {t} }s_{ij}s_{ij}\rangle )\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle P}
{\displaystyle \epsilon _{\mathrm {LES} }=0.066u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {max} }}
{\displaystyle \epsilon _{\mathrm {max} }}
{\displaystyle C_{\mathrm {PIV} }=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle C_{\mathrm {LES} }=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle x\approx -0.63D}
{\displaystyle C}
{\displaystyle R_{\mathrm {PIV} }=P+\nabla T_{\mathrm {turb} }-\epsilon +C\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle -\nabla T_{\mathrm {press,LES} }}
{\displaystyle R_{\mathrm {LES} }=P+\nabla T-\epsilon _{\mathrm {total} }+C\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle <|0.01|u_{\mathrm {b} }^{3}/D}
{\displaystyle T_{\mathrm {turb} }=-{\frac {1}{2}}\langle u_{i}'u_{j}'u_{j}'\rangle }
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {p} }={\frac {\langle p\rangle }{{\frac {\rho }{2}}u_{\mathrm {b} }^{2}}}}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }={\frac {\langle \tau _{\mathrm {w} }\rangle }{{\frac {\rho }{2}}u_{\mathrm {b} }^{2}}}}
{\displaystyle z_{1}\approx 0.0036D\approx 10\mathrm {px} }
{\displaystyle z_{1}\approx 0.0005D}
{\displaystyle z-}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle |c_{\mathrm {f} }|=0.01}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle 50\times 171(n\times m)}
{\displaystyle 143\times 131(n\times m)}
{\displaystyle n\cdot m}
{\displaystyle x_{\mathrm {adj} }}
{\displaystyle {\frac {x}{D}}}
{\displaystyle {\frac {z}{D}}}
{\displaystyle {\frac {\langle u\rangle }{u_{\mathrm {b} }}}}
{\displaystyle -}
{\displaystyle {\frac {\langle w\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle u'u'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle {\frac {\langle w'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle {\frac {\langle u'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle P{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle C{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {turb} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle -}
{\displaystyle -}
{\displaystyle \epsilon {\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle -}
{\displaystyle x_{\mathrm {adj} }}
{\displaystyle {\frac {x}{D}}}
{\displaystyle {\frac {z}{D}}}
{\displaystyle {\frac {\langle u\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle v\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle w\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle u'u'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle v'v'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle w'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle u'v'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle u'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle v'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle P{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle C{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {turb} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {press} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {visc} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \epsilon _{\mathrm {total} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle c_{\mathrm {p} }}
|
February - Volume 38, Number 1
Kate E Allstadt; Eric M Thompson, M.EERI; Randall W Jibson; David J Wald, M.EERI; Michael Hearne; Edward J Hunter; Jeremy Fee; Heather Schovanec; Daniel Slosky; Kirstie L Haynie
Earthquake Spectra February 01, 2022, Vol.38, 5-36. doi:https://doi.org/10.1177/87552930211032685
Mohammad Hassan Baziar, M.EERI; Omid Eslami Amirabadi
Earthquake Spectra February 01, 2022, Vol.38, 37-55. doi:https://doi.org/10.1177/87552930211041641
Gitanjali Bhattacharjee, M.EERI; Robert Soden; Karen Barns; Sabine Loos; David Lallemant
Diana Contreras; Sean Wilkinson; Nipun Balan; Philip James
Earthquake Spectra February 01, 2022, Vol.38, 81-108. doi:https://doi.org/10.1177/87552930211036486
Dynamic updating of post-earthquake damage and functional restoration forecasts of water distribution systems using Bayesian inferencing
Agam Tomar; Henry V Burton, M.EERI; Ali Mosleh
Earthquake Spectra February 01, 2022, Vol.38, 109-127. doi:https://doi.org/10.1177/87552930211038016
Yajie Lee, M.EERI; Zhenghui Hui, M.EERI; Siamak Daneshvaran; Farhad Sedaghati; William P Graf, M.EERI
Rapid earthquake loss assessment based on machine learning and representative sampling
Zoran Stojadinović; Miloš Kovačević; Dejan Marinković; Božidar Stojadinović, M.EERI
Toward a uniform earthquake loss model across Central America
Alejandro Calderón, EERI; Vitor Silva; Matilde Avilés; Rosalín Méndez; Rolando Castillo; José Carlos Gil; Manuel Alfredo López
Structural performance of buildings during the 30 November 2018 M7.1 Anchorage, Alaska earthquake
Wael M Hassan, M.EERI; Janise Rodgers, M.EERI; Christopher Motter, M.EERI; John Thornley, M.EERI
Proposal of orientation-independent measure of intensity for earthquake-resistant design
Alan Poulos, M.EERI; Eduardo Miranda, M.EERI
Seismic fragility of bridges: An approach coupling multiple-stripe analysis and Gaussian mixture for multicomponent structures
Pedro Alexandre Conde Bandini, MEERI; Jamie Ellen Padgett, MEERI; Patrick Paultre, MEERI; Gustavo Henrique Siqueira
Carlos Molina Hutt, MEERI; Anne M Hulsey, MEERI; Preetish Kakoty, MEERI; Greg G Deierlein, MEERI; Alireza Eksir Monfared; Yen Wen-Yi; John D Hooper, MEERI
Statistical analysis and modeling to examine the exterior and interior building damage pertaining to the 2016 Kumamoto earthquake
Haoyi Xiu; Takayuki Shinohara; Masashi Matsuoka, M. EERI; Munenari Inoguchi; Ken Kawabe; Kei Horie
Sebastián Miranda, M.EERI; Eduardo Miranda, M.EERI; Juan Carlos de la Llera, M.EERI
Marc-Denis Rioux; Marie-José Nollet; Bertrand Galy
Design of computer experiments for developing seismic surrogate models
Henry Burton, M.EERI; Hongquan Xu; Zhengxiang Yi, M.EERI
Elena Florinela Manea, EERI; Carmen Ortanza Cioflan; Laurentiu Danciu
Simulation of near-fault ground motions for randomized directivity parameters
Yara Daoud; Mayssa Dabaghi; Armen Der Kiureghian
Grace A Parker, M.EERI; Jonathan P Stewart, M.EERI; David M Boore; Gail M Atkinson, M.EERI; Behzad Hassani
Site parameters applied in NGA-Sub database
Sean K Ahdi, M.EERI; Dong Youp Kwak, M.EERI; Timothy D Ancheta; Victor Contreras, S.M.EERI; Tadahiro Kishida, M.EERI; Annie O Kwok, M.EERI; Silvia Mazzoni, M.EERI; Francisco Ruz; Jonathan P Stewart, M.EERI
A Texas-specific
VS30
map incorporating geology and
VS30
Meibai Li, M.EERI; Ellen M Rathje, M.EERI; Brady R Cox, M.EERI; Michael Yust, M.EERI
Spatial correlation in ground motion prediction errors in Central and Eastern North America
Emily M Gibson, EERI; Michelle T Bensi
Ivan Wong, M.EERI; Robert Darragh, M.EERI; Sarah Smith, M.EERI; Qimin Wu; Walter Silva, M.EERI; Tadahiro Kishida, M.EERI
Chenying Liu, M.EERI; Jorge Macedo, M.EERI
Sahar Rahpeyma, MEERI; Benedikt Halldorsson; Birgir Hrafnkelsson; Sigurjón Jónsson
Nidhin S Pachappoyil, EERI; Pankaj Agarwal, EERI
The power of the little ones: Computed and observed aftershock hazard in Central Italy
Robin Gee, M.EERI; Laura Peruzza; Marco Pagani
Seismic risk assessments for real estate portfolios: Impact of engineering investigation on quality of seismic risk studies
Yajie Lee, M. EERI; William P Graf, M. EERI; Charles C Thiel, Jr., M. EERI; Zhenghui Hu; Mark Ellis
David J Wald, M.EERI; C Bruce Worden; Eric M Thompson, M.EERI; Michael Hearne
Book review: Seismic Analysis of Structures and Equipment
Martin Wieland, M.EERI, Chairman
Earthquake Spectra February 01, 2022, Vol.38, 725. doi:https://doi.org/10.1177/87552930211035148
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Invariant zeros of linear system - MATLAB tzero - MathWorks Italia
Find Transmission Zeros of MIMO Transfer Function
Identify Unobservable and Uncontrollable Modes of MIMO Model
Invariant zeros of linear system
z = tzero(sys)
z = tzero(A,B,C,D,E)
z = tzero(___,tol)
[z,nrank] = tzero(___)
z = tzero(sys) returns the invariant zeros of the multi-input, multi-output (MIMO) dynamic system, sys. If sys is a minimal realization, the invariant zeros coincide with the transmission zeros of sys.
z = tzero(A,B,C,D,E) returns the invariant zeros of the state-space model
\begin{array}{c}E\frac{dx}{dt}=Ax+Bu\\ y=Cx+Du.\end{array}
Omit E for an explicit state-space model (E = I).
z = tzero(___,tol) specifies the relative tolerance, tol, controlling rank decisions.
[z,nrank] = tzero(___) also returns the normal rank of the transfer function of sys or of the transfer function H(s) = D + C(sE – A)–1B.
MIMO dynamic system model. If sys is not a state-space model, then tzero computes tzero(ss(sys)).
State-space matrices describing the linear system
\begin{array}{c}E\frac{dx}{dt}=Ax+Bu\\ y=Cx+Du.\end{array}
tzero does not scale the state-space matrices when you use the syntax z = tzero(A,B,C,D,E). Use prescale if you want to scale the matrices before using tzero.
Omit E to use E = I.
Relative tolerance controlling rank decisions. Increasing tolerance helps detect nonminimal modes and eliminate very large zeros (near infinity). However, increased tolerance might artificially inflate the number of transmission zeros.
Column vector containing the invariant zeros of sys or the state-space model described by A,B,C,D,E.
Normal rank of the transfer function of sys or of the transfer function H(s) = D + C(sE – A)–1B. The normal rank is the rank for values of s other than the transmission zeros.
To obtain a meaningful result for nrank, the matrix s*E-A must be regular (invertible for most values of s). In other words, sys or the system described by A,B,C,D,E must have a finite number of poles.
Create a MIMO transfer function, and locate its invariant zeros.
H = [1/(s+1) 1/(s+2);1/(s+3) 2/(s+4)];
z = tzero(H)
The output is a column vector listing the locations of the invariant zeros of H. This output shows that H a has complex pair of invariant zeros. Confirm that the invariant zeros coincide with the transmission zeros.
Check whether the first invariant zero is a transmission zero of H.
If z(1) is a transmission zero of H, then H drops rank at s = z(1).
H1 = evalfr(H,z(1));
svd(H1)
H1 is the transfer function, H, evaluated at s = z(1). H1 has a zero singular value, indicating that H drops rank at that value of s. Therefore, z(1) is a transmission zero of H.
A similar analysis shows that z(2) is also a transmission zero.
Obtain a MIMO model.
load ltiexamples gasf
size(gasf)
gasf is a MIMO model that might contain uncontrollable or unobservable states.
To identify the unobservable and uncontrollable modes of gasf, you need the state-space matrices A, B, C, and D of the model. tzero does not scale state-space matrices. Therefore, use prescale with ssdata to scale the state-space matrices of gasf.
[A,B,C,D] = ssdata(prescale(gasf));
Identify the uncontrollable states of gasf.
uncon = tzero(A,B,[],[])
uncon = 6×1
When you provide A and B matrices to tzero, but no C and D matrices, the command returns the eigenvalues of the uncontrollable modes of gasf. The output shows that there are six degenerate uncontrollable modes.
Identify the unobservable states of gasf.
unobs = tzero(A,[],C,[])
unobs =
When you provide A and C matrices, but no B and D matrices, the command returns the eigenvalues of the unobservable modes. The empty result shows that gasf contains no unobservable states.
For a MIMO state-space model
\begin{array}{c}E\frac{dx}{dt}=Ax+Bu\\ y=Cx+Du,\end{array}
the invariant zeros are the complex values of s for which the rank of the system matrix
\left[\begin{array}{cc}A-sE& B\\ C& D\end{array}\right]
drops from its normal value. (For explicit state-space models, E = I).
\begin{array}{c}E\frac{dx}{dt}=Ax+Bu\\ y=Cx+Du,\end{array}
the transmission zeros are the complex values of s for which the rank of the equivalent transfer function H(s) = D + C(sE – A)–1B drops from its normal value. (For explicit state-space models, E = I.)
Transmission zeros are a subset of the invariant zeros. For minimal realizations, the transmission zeros and invariant zeros are identical.
You can use the syntax z = tzero(A,B,C,D,E) to find the uncontrollable or unobservable modes of a state-space model. When C and D are empty or zero, tzero returns the uncontrollable modes of (A-sE,B). Similarly, when B and D are empty or zero, tzero returns the unobservable modes of (C,A-sE). See Identify Unobservable and Uncontrollable Modes of MIMO Model for an example.
tzero is based on SLICOT routines AB08ND, AB08NZ, AG08BD, and AG08BZ. tzero implements the algorithms in [1] and [2].
To calculate the zeros and gain of a single-input, single-output (SISO) system, use zero.
[1] Emami-Naeini, A. and P. Van Dooren, "Computation of Zeros of Linear Multivariable Systems," Automatica, 18 (1982), pp. 415–430.
[2] Misra, P, P. Van Dooren, and A. Varga, “Computation of Structural Invariants of Generalized State-Space Systems,” Automatica, 30 (1994), pp. 1921-1936.
pole | pzmap | zero
|
give vs open what difference - Tez Koder
give vs open what difference
what is difference between give and open
(Received Pronunciation) enPR: ō’pən, IPA(key): /ˈəʊ.pən/
(US) enPR: ō’pən, IPA(key): /ˈoʊ.pən/
Rhymes: -əʊpən
From Middle English open, from Old English open (“open”), from Proto-West Germanic *opan, from Proto-Germanic *upanaz (“open”), from Proto-Indo-European *upo (“up from under, over”). Cognate with Scots apen (“open”), Saterland Frisian eepen (“open”), West Frisian iepen (“open”), Dutch open (“open”), Low German open, apen (“open”), German offen (“open”), Danish åben (“open”), Swedish öppen (“open”), Norwegian Bokmål åpen (“open”), Norwegian Nynorsk open (“open”), Icelandic opinn (“open”). Compare also Latin supinus (“on one’s back, supine”), Albanian hap (“to open”). Related to up.
(not comparable) Not closed
able to be accessed
{\displaystyle X}
{\displaystyle X}
(computing, used before “code”) Source code of a computer program that is not within the text of a macro being generated.
(with a free license and no proprietary components): free
(with a free license and no proprietary components): closed-source, proprietary
From Middle English openen, from Old English openian (“to open”), from Proto-Germanic *upanōną (“to raise; lift; open”), from Proto-Germanic *upanaz (“open”, adjective). Cognate with Saterland Frisian eepenje (“to open”), West Frisian iepenje (“to open”), Dutch openen (“to open”), German öffnen (“to open”), Danish åbne (“to open”), Swedish öppna (“to open”), Norwegian Bokmål åpne (“to open”), Norwegian Nynorsk and Icelandic opna (“to open”). Related to English up.
(intransitive, cricket) To begin a side’s innings as one of the first two batsmen.
(transitive, intransitive, poker) To reveal one’s hand.
The king opened himself to some of his council, that he was sorry for the earl’s death.
From Middle English open (“an aperture or opening”), from the verb (see Etymology 2 above). In the sports sense, however, a shortening of “open competition”.
A sports event in which anybody can compete
From Dutch openen, from Middle Dutch ōpenen, from Old Dutch opanon, from Proto-Germanic *upanōną.
IPA(key): /ˈʊə̯.pən/
From English open.
IPA(key): /ˈoː.pə(n)/
Hyphenation: open
Rhymes: -oːpən
From Middle Dutch ōpen, from Old Dutch opan, from Proto-Germanic *upanaz.
Antonyms: gesloten, dicht, toe
Antonyms: gesloten, dicht
Afrikaans: oop
Negerhollands: open, hopo
→ Virgin Islands Creole: hopo
genitive singular of ope
Borrowed from English open.
IPA(key): /ɔ.pɛn/
“open” in Trésor de la langue française informatisé (The Digitized Treasury of the French Language).
From Old Dutch opan, from Proto-Germanic *upanaz.
ōpen
Dutch: open
Limburgish: aop
Verwijs, E.; Verdam, J. (1885–1929), “open (II)”, in Middelnederlandsch Woordenboek , The Hague: Martinus Nijhoff, →ISBN, page II
From Old English open, from Proto-Germanic *upanaz.
English: open (obsolete ope)
Scots: appen, apen
From Old Norse opinn, from Proto-Germanic *upanaz. Compare Danish åben, Icelandic opinn, Swedish öppen, Dutch open, Low German apen, open, German offen, West Frisian iepen, English open.
IPA(key): /²oːpɛn/
open (masculine and feminine open, neuter ope or opent, definite singular and plural opne, comparative opnare, indefinite superlative opnast, definite superlative opnaste)
åpen (Bokmål)
From Proto-Germanic *upanaz. Originally a past participle of Proto-Germanic *ūpaną (“to lift up, open”). Akin to Old English ūp (“up”). Cognate with Old Frisian open, opin, epen (West Frisian iepen), Old Saxon opan, open (Low German apen, open), Dutch open, Old High German offan, ofan, ophan (German offen), Old Norse opinn (Danish åben, Norwegian open, Swedish öppen).
Middle English: open, opyn, ope
IPA(key): /ˈopen/, [ˈo.pẽn]
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Hashing | Toph
Editorial for Hashing
Let’s first try to find the hash value of a graph with
n vertices,
m edges and no updates.
Observation 1: We can take at most one element from each of the connected components.
Observation 2. A vertice will appear in our result if the number of subsets where it appeared and the miniHash function didn't malfunction is odd.
Now, for each node how many such subsets are there? Let’s say sizes of all other subsets are
s_1, s_2,.....
s1,s2,.....
Then the number of subsets it appeared will be
(s_1+1)*(s_2+1)*......
(s1+1)∗(s2+1)∗......
This will be odd only if all the connected components except this one are even in size.
Case 1: There are more than one connected component with odd size. In this case, the answer will be 0 as every node will appear an even number of times.
Case 2: There is exactly one odd-sized connected component. In this case, the result will be XOR of all the nodes in that connected component.
Case 3: There are no odd-sized connected components. In this case, all nodes will appear in an odd number of subsets. So the answer will be XOR of all the nodes.
Now, for the given problem. We can use disjoint-set-union algorithm to keep track of the size of each connected component, also the odd-sized connected components. After that, we can use the observations discussed above to find the hash value of the current graph.
O(n*log(n))
O(n∗log(n))
BigBagEarliest, 4M ago
alifcsejuFastest, 0.2s
aropanLightest, 19 MB
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Direct Search Based Optimization of Six-Element Yagi-Uda Antenna - MATLAB & Simulink Example
Comparison with Manufacturer Data Sheet
This example optimizes a 6-element Yagi-Uda antenna for both directivity and
300\Omega
input match using a global optimization technique. The radiation patterns and input impedance of antennas are sensitive to the parameters that define their shapes. The multidimensional surface over which such optimizations must be performed have multiple local optima. This makes the task of finding the right set of parameters satisfying the optimization goals particularly challenging and requires the use of global optimization techniques. One such technique is pattern search, a direct search based optimization technique that has yielded impressive results for antenna design optimization.
The Yagi-Uda antenna is a widely used radiating structure for a variety of applications in commercial and military sectors. This antenna has been popular for reception of TV signals in the VHF-UHF range of frequencies [1]. The Yagi is a directional traveling-wave antenna with a single driven element, usually a folded dipole or a standard dipole, which is surrounded by several passive dipoles. The passive elements form the reflector and director. These names identify the positions relative to the driven element. The reflector dipole is behind the driven element in the direction of the back lobe of the antenna radiation, while the director is in front of the driven element, in the direction where a main beam forms.
Choose initial design parameters in the center of the VHF band [2]. The datasheet lists a
50\Omega
input impedance after taking into account a balun. Our model does not account for the presence of the balun and therefore will match to the typical folded dipole input impedance of
300\Omega
wirediameter = 12.7e-3;
BW = 0.05*fc;
The driven element for the Yagi-Uda antenna is a folded dipole. This is a standard exciter for such an antenna. Adjust the length and width parameters of the folded dipole. Since we model cylindrical structures as equivalent metal strips, the width is calculated using a utility function available in the Antenna Toolbox™. The length is chosen to be
\lambda /2
Create a Yagi-Uda antenna with the exciter as the folded dipole. Choose the reflector and director length to be
\lambda /2
. Set the number of directors to four. Choose the reflector and director spacing to be
0.3\lambda
0.25\lambda
respectively. These choices are an initial guess and will serve as a start point for the optimization procedure. Show the initial design.
exLength = d.Length/lambda;
exSpacing = d.Spacing/lambda;
initialdesign = [refLength dirLength refSpacing dirSpacing exLength exSpacing].*lambda;
Director lengths = controlVals(2:5)
Director spacings = controlVals(7:10)
Exciter length = controlVals(11)
Exciter spacing = controlVals(12)
type yagi_objective_function_direct.m
function objectivevalue = yagi_objective_function_direct(y,controlVals,fc,BW,ang,Z0,constraints)
% YAGI_OBJECTIVE_FUNCTION_DIRECT returns the objective for a 6 element Yagi
% YAGI_OBJECTIVE_FUNCTION_DIRECT(Y,CONTROLVALS,FREQ,ANG,Z0,constraints), assigns
% The YAGI_OBJECTIVE_FUNCTION_DIRECT function is used for an internal example.
y.DirectorLength = controlVals(2:y.NumDirectors+1);
y.ReflectorSpacing = controlVals(y.NumDirectors+2);
y.DirectorSpacing = controlVals(y.NumDirectors+3:end-2);
y.Exciter.Length = controlVals(end-1);
y.Exciter.Spacing = controlVals(end);
c1 = Gmin-output1;
c1_dev = -Gdev + abs(output1-Gmin);
if output.FB < FBmin
c2 = FBmin-output.FB;
dirSpacingBounds = [0.05 0.05 0.05 0.05; % lower bound on director spacing
0.2 0.2 0.3 0.3]; % upper bound on director spacing
LB = [refLengthBounds(1),dirLengthBounds(1,:) refSpacingBounds(1) dirSpacingBounds(1,:) exciterLengthBounds(1) exciterSpacingBounds(1) ].*lambda;
UB = [refLengthBounds(2),dirLengthBounds(2,:) refSpacingBounds(2) dirSpacingBounds(2,:) exciterLengthBounds(2) exciterSpacingBounds(2) ].*lambda;
ang = [0 0;90 -90]; % azimuth,elevation angles for main lobe and back lobe [az;el]
The Global Optimization Toolbox™ provides a direct search based optimization function called patternsearch. We use this function with options specified with the psoptimset function. At every iteration, plot the best value of the objective function and limit the total number of iterations to 300. Pass the objective function to the patternsearch function by using an anonymous function together with the bounds and the options structure.The objective function used during the optimization process by patternsearch is available in the file yagi_objective_function.
The evaluation of the directivity in different directions corresponding to the angular region defined for maximum radiation as well as maximum sidelobe and the backlobe level is given in the function calculate_objectives available within yagi_objective function.
optimizerparams.MaxIter = 100;
constraints.Gmin = 10.5;
optimdesign = optimizeAntennaDirect(designparams,analysisparams,constraints,optimizerparams);
yagidesign.DirectorSpacing = optimdesign(7:10);
yagidesign.Exciter.Length = optimdesign(11);
yagidesign.Exciter.Spacing = optimdesign(12);
% fig3 = figure;
% patternElevation(yagidesign,fc,0,'Elevation',0:1:359);
% pE = polarpattern('gco');
% pE.AntennaMetrics = 1;
% patternElevation(yagidesign,fc,90,'Elevation',0:1:359);
% pH = polarpattern('gco');
% pH.AntennaMetrics = 1;
The input reflection coefficient for the optimized Yagi-Uda antenna is computed and plotted relative to the reference impedance of
50\Omega
. A value of -10 dB or lower is considered to be a good impedance match.
The optimized Yagi-Uda antenna achieves a forward directivity greater than 10 dBi, which translates to a value greater than 8 dBd (relative to a dipole). This is close to the gain value reported by the datasheet (8.5 dBd). The F/B ratio is greater than 15 dB. The optimized Yagi-Uda antenna has a E-plane and H-plane beamwidth that compare favorably to the datasheet listed values of 54 degrees and 63 degrees respectively. The design achieves a good impedance match to
300\Omega
, and has a -10 dB bandwidth of approximately 8%.
datasheetparam = {'Gain (dBi)';'F/B';'E-plane Beamwidth (deg.)';'H-plane Beamwidth (deg.)';'Impedance Bandwidth (%)'};
datasheetvals = [10.5,16,54,63,10]';
optimdesignvals = [10.59,15.6,50,62,12.1]';
Tdatasheet = table(datasheetvals,optimdesignvals,'RowNames',datasheetparam)
Tdatasheet=5×2 table
datasheetvals optimdesignvals
Gain (dBi) 10.5 10.59
F/B 16 15.6
E-plane Beamwidth (deg.) 54 50
H-plane Beamwidth (deg.) 63 62
Impedance Bandwidth (%) 10 12.1
'Director Spacing - 4';'Exciter Length';
'Exciter Spacing'};
Tgeometry=12×2 table
Exciter Length 0.90846 0.84596
Exciter Spacing 0.015141 0.015629
[1] C. A. Balanis, Antenna Theory. Analysis and Design, p. 514, Wiley, New York, 3rd Edition, 2005
[2] Online at: S.6Y-165
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Svetlana V. Butler1
1 Department of Mathematics, University of California Santa Barbara, 552 University Rd., Isla Vista, CA 93117, USA
This paper combines new and known results in a single convenient source for anyone interested in learning about quasi-linear functionals on locally compact spaces. We define singly generated subalgebras in different settings and study signed and positive quasi-linear functionals. Quasi-linear functionals are, in general, nonlinear, but linear on singly generated subalgebras. The paper gives representation theorems for quasi-linear functionals on
{C}_{c}\left(X\right)
, for bounded quasi-linear functionals on
{C}_{0}\left(X\right)
on a locally compact space, and for quasi-linear functionals on
C\left(X\right)
on a compact space. There is an order-preserving bijection between quasi-linear functionals and compact-finite topological measures, which is also “isometric” when topological measures are finite. We present many properties of quasi-linear functionals and give an explicit example of a quasi-linear functional on
{ℝ}^{2}
. Results of the paper will be helpful for further study and application of quasi-linear functionals in different areas of mathematics, including symplectic geometry.
Classification: 46E27, 46G99, 28A25, 28C15
Keywords: quasi-linear functional, signed quasi-linear functional, singly generated subalgebra, topological measure, symplectic quasi-state
Svetlana V. Butler 1
author = {Svetlana V. Butler},
title = {Quasi-linear functionals on locally compact spaces},
TI - Quasi-linear functionals on locally compact spaces
%T Quasi-linear functionals on locally compact spaces
Svetlana V. Butler. Quasi-linear functionals on locally compact spaces. Confluentes Mathematici, Volume 13 (2021) no. 1, pp. 3-34. doi : 10.5802/cml.69. https://cml.centre-mersenne.org/articles/10.5802/cml.69/
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[14] —. Signed topological measures on locally compact spaces, Anal. Math., 45:757–773, 2019. | Article | MR: 4042929 | Zbl: 1449.28014
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[20] —. Repeated quasi-integration on locally compact spaces, Positivity, to appear. arXiv:1902.06901
[21] D. Denneberg. Non-additive measure and integral. Kluwer, 1994. | Article | Zbl: 0826.28002
[22] A. Dickstein and F. Zapolsky. Approximation of quasi-states on manifolds, J. Appl. and Comput. Topol., 3:221–248, 2019. | Article | MR: 3996955 | Zbl: 07103893
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[24] M. Entov. Quasi-morphisms and quasi-states in symplectic topology, Proceedings of the International Congress of Mathematicians, Seoul, 1147–1171, 2014. | Zbl: 1373.53116
[25] M. Entov and L. Polterovich. Calabi Quasimorphism and Quantum Homology, Int. Math. Res. Not., 30:1635–1676, 2003. | Article | Zbl: 1047.53055
[26] —. Quasi-states and symplectic intersections, Comm. Math. Helv., 81:75–99, 2006. | Article | MR: 2208798 | Zbl: 1096.53052
[27] —. Symplectic Quasi-states and Semi-simplicity of Quantum Homology, in Toric Topology (eds. M. Harada, Y. Karshon, M. Masuda and T. Panov), Contemporary Mathematics, AMS, 460: 47–70, 2008. | Article | Zbl: 1146.53066
[28] —, Lie quasi-states, J. Lie Theory, 19:613–637, 2009. | Zbl: 1182.53075
{C}^{0}
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[31] M. Entov, L. Polterovich, and F. Zapolsky. Quasi-morphisms and the Poisson Bracket, Pure and Appl. Math. Q, 3(4) (Special issue : In honor of Gregory Margulis, part 1 of 2):1037–1055, 2007. | Article | MR: 2402596 | Zbl: 1143.53070
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Acceleration - formulasearchengine
{{#invoke:Hatnote|hatnote}} Template:Infobox physical quantity Template:Classical mechanics
Acceleration, in physics, is the rate of change of velocity of an object. An object's acceleration is the net result of any and all forces acting on the object, as described by Newton's Second Law.[1] The SI unit for acceleration is the metre per second squared (m/s2). Accelerations are vector quantities (they have magnitude and direction) and add according to the parallelogram law.[2][3] As a vector, the calculated net force is equal to the product of the object's mass (a scalar quantity) and the acceleration.
For example, when a car starts from a standstill (zero relative velocity) and travels in a straight line at increasing speeds, it is accelerating in the direction of travel. If the car turns there is an acceleration toward the new direction. For this example, we can call the accelerating of the car forward a "linear acceleration", which passengers in the car might experience as force pushing them back into their seats. When changing directions, we might call this "non-linear acceleration", which passengers might experience as a sideways force. If the speed of the car decreases, this is an acceleration in the opposite direction of the direction of the vehicle, sometimes called deceleration.[4] Passengers may experience deceleration as a force lifting them away from their seats. Mathematically, there is no separate formula for deceleration, as both are changes in velocity. Each of these accelerations (linear, non-linear, deceleration) might be felt by passengers until their velocity and direction match that of the car.
1.1 Average acceleration
An object's average acceleration over a period of time is its change in velocity
{\displaystyle (\Delta \mathbf {v} )}
divided by the duration of the period
{\displaystyle (\Delta t)}
{\displaystyle {\boldsymbol {\bar {a}}}={\frac {\Delta \mathbf {v} }{\Delta t}}.}
bulleted}}
{\displaystyle \mathbf {a} =\lim _{{\Delta t}\to 0}{\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {d\mathbf {v} }{dt}}}
{\displaystyle \mathbf {v} =\int \mathbf {a} \ d{\mathit {t}}}
Given the fact that acceleration is defined as the derivative of velocity, v, with respect to time t and velocity is defined as the derivative of position, x, with respect to time, acceleration can be thought of as the second derivative of x with respect to t:
{\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}{\boldsymbol {x}}}{dt^{2}}}}
Acceleration has the dimensions of velocity (L/T) divided by time, i.e., L/T2. The SI unit of acceleration is the metre per second squared (m/s2); this can be called more meaningfully "metre per second per second", as the velocity in metres per second changes by the acceleration value, every second.
An object moving in a circular motion—such as a satellite orbiting the earth—is accelerating due to the change of direction of motion, although the magnitude (speed) may be constant. When an object is executing such a motion where it changes direction, but not speed, it is said to be undergoing centripetal (directed towards the center) acceleration. Oppositely, a change in the speed of an object, but not its direction of motion, is a tangential acceleration.
In classical mechanics, for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the net force vector (i.e., sum of all forces) acting on it (Newton's second law):
{\displaystyle \mathbf {F} =m\mathbf {a} \quad \to \quad \mathbf {a} =\mathbf {F} /m}
where F is the net force acting on the body, m is the mass of the body, and a is the center-of-mass acceleration. As speeds approach the speed of light, relativistic effects become increasingly large and acceleration becomes less.
{\displaystyle \mathbf {v} (t)=v(t){\frac {\mathbf {v} (t)}{v(t)}}=v(t)\mathbf {u} _{\mathrm {t} }(t),}
{\displaystyle \mathbf {u} _{\mathrm {t} }={\frac {\mathbf {v} (t)}{v(t)}}\ ,}
{\displaystyle {\begin{alignedat}{3}\mathbf {a} &={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\\&={\frac {\mathrm {d} v}{\mathrm {d} t}}\mathbf {u} _{\mathrm {t} }+v(t){\frac {d\mathbf {u} _{\mathrm {t} }}{dt}}\\&={\frac {\mathrm {d} v}{\mathrm {d} t}}\mathbf {u} _{\mathrm {t} }+{\frac {v^{2}}{r}}\mathbf {u} _{\mathrm {n} }\ ,\\\end{alignedat}}}
{\displaystyle \mathbf {F} =m\mathbf {g} }
Due to the simple algebraic properties of constant acceleration in the one-dimensional case (that is, the case of acceleration aligned with the initial velocity), there are simple formulas relating the quantities displacement s, initial velocity v0, final velocity v, acceleration a, and time t:[8]
{\displaystyle v=v_{0}+at}
{\displaystyle s=v_{0}t+{\frac {1}{2}}at^{2}={\frac {v_{0}+v}{2}}t}
{\displaystyle |v|^{2}=|v_{0}|^{2}+2\,a\cdot s}
{\displaystyle s}
{\displaystyle v_{0}}
{\displaystyle v}
{\displaystyle a}
{\displaystyle t}
{\displaystyle {\textrm {a}}={{v^{2}} \over {r}}}
{\displaystyle v}
is the object's linear speed along the circular path. Equivalently, the radial acceleration vector (
{\displaystyle \mathbf {a} }
) may be calculated from the object's angular velocity
{\displaystyle \omega }
{\displaystyle \mathbf {a} ={-\omega ^{2}}\mathbf {r} }
{\displaystyle \mathbf {r} }
is a vector directed from the centre of the circle and equal in magnitude to the radius. The negative shows that the acceleration vector is directed towards the centre of the circle (opposite to the radius).
{\displaystyle a=r\alpha .}
The transverse (or tangential) acceleration is directed at right angles to the radius vector and takes the sign of the angular acceleration (
{\displaystyle \alpha }
{{#invoke:main|main}} The special theory of relativity describes the behavior of objects traveling relative to other objects at speeds approaching that of light in a vacuum. Newtonian mechanics is exactly revealed to be an approximation to reality, valid to great accuracy at lower speeds. As the relevant speeds increase toward the speed of light, acceleration no longer follows classical equations.
{{#invoke:main|main}} Unless the state of motion of an object is known, it is totally impossible to distinguish whether an observed force is due to gravity or to acceleration—gravity and inertial acceleration have identical effects. Albert Einstein called this the principle of equivalence, and said that only observers who feel no force at all—including the force of gravity—are justified in concluding that they are not accelerating.[10]
Template:Acceleration conversions
↑ Brian Greene, The Fabric of the Cosmos, page 67. Vintage ISBN 0-375-72720-5
Retrieved from "https://en.formulasearchengine.com/index.php?title=Acceleration&oldid=218668"
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Integral of [ ln(1 + cosx) - xtan x/2] dx (A) x tan x/2 + C (B) ln (1+ - Maths - Integrals - 10500293 | Meritnation.com
Integral of [ ln(1 + cosx) - xtan x/2] dx (A) x tan x/2 + C (B) ln (1+ cosx) + C (C) x ln (1+ cosx) + C (D) none of these
In questions like these, it is easier to differentiate the options and then check the result with the question
In this case, differentiating option C, we get
\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{x} \mathrm{ln} \left(1+\mathrm{cos} \mathrm{x}\right)+\mathrm{c}\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{x} \mathrm{ln} \left(1+\mathrm{cos} \mathrm{x}\right)\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\mathrm{ln}\left(1+\mathrm{x}\right)+\mathrm{x}×\frac{1}{1+\mathrm{cos} \mathrm{x}}×\left(-\mathrm{sin} \mathrm{x}\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\mathrm{ln}\left(1+\mathrm{x}\right)-\frac{\mathrm{x} \mathrm{sin} \mathrm{x}}{1+\mathrm{cos} \mathrm{x}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\mathrm{ln}\left(1+\mathrm{x}\right)-\frac{\mathrm{x}× 2 \mathrm{sin}\left(\mathrm{x}/2\right) \mathrm{cos}\left(\mathrm{x}/2\right)}{2 {\mathrm{cos}}^{2}\left(\mathrm{x}/2\right)}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\mathrm{ln}\left(1+\mathrm{x}\right)-\frac{\mathrm{x} × 2 \mathrm{sin}\left(\mathrm{x}/2\right)}{2 \mathrm{cos}\left(\mathrm{x}/2\right)}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\mathrm{ln}\left(1+\mathrm{x}\right)-\mathrm{x} \mathrm{tan}\left(\frac{\mathrm{x}}{2}\right)
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009C Sample Midterm 1, Problem 2 Detailed Solution - Math Wiki
009C Sample Midterm 1, Problem 2 Detailed Solution
Consider the infinite series
{\displaystyle \sum _{n=2}^{\infty }2{\bigg (}{\frac {1}{2^{n}}}-{\frac {1}{2^{n+1}}}{\bigg )}.}
(a) Find an expression for the
{\displaystyle n}
th partial sum
{\displaystyle s_{n}}
{\displaystyle \lim _{n\rightarrow \infty }s_{n}.}
{\displaystyle n}
th partial sum,
{\displaystyle s_{n}}
for a series
{\displaystyle \sum _{n=1}^{\infty }a_{n}}
{\displaystyle s_{n}=\sum _{i=1}^{n}a_{i}.}
We need to find a pattern for the partial sums in order to find a formula.
{\displaystyle s_{2}.}
{\displaystyle s_{2}=2{\bigg (}{\frac {1}{2^{2}}}-{\frac {1}{2^{3}}}{\bigg )}.}
Next, we calculate
{\displaystyle s_{3}}
{\displaystyle s_{4}.}
{\displaystyle {\begin{array}{rcl}\displaystyle {s_{3}}&=&\displaystyle {2{\bigg (}{\frac {1}{2^{2}}}-{\frac {1}{2^{3}}}{\bigg )}+2{\bigg (}{\frac {1}{2^{3}}}-{\frac {1}{2^{4}}}{\bigg )}}\\&&\\&=&\displaystyle {2{\bigg (}{\frac {1}{2^{2}}}-{\frac {1}{2^{4}}}{\bigg )}}\end{array}}}
{\displaystyle {\begin{array}{rcl}\displaystyle {s_{4}}&=&\displaystyle {2{\bigg (}{\frac {1}{2^{2}}}-{\frac {1}{2^{3}}}{\bigg )}+2{\bigg (}{\frac {1}{2^{3}}}-{\frac {1}{2^{4}}}{\bigg )}+2{\bigg (}{\frac {1}{2^{4}}}-{\frac {1}{2^{5}}}{\bigg )}}\\&&\\&=&\displaystyle {2{\bigg (}{\frac {1}{2^{2}}}-{\frac {1}{2^{5}}}{\bigg )}.}\end{array}}}
{\displaystyle s_{2},s_{3},}
{\displaystyle s_{4},}
we notice a pattern.
From this pattern, we get the formula
{\displaystyle s_{n}=2{\bigg (}{\frac {1}{2^{2}}}-{\frac {1}{2^{n+1}}}{\bigg )}.}
From Part (a), we have
{\displaystyle s_{n}=2{\bigg (}{\frac {1}{2^{2}}}-{\frac {1}{2^{n+1}}}{\bigg )}.}
{\displaystyle \lim _{n\rightarrow \infty }s_{n}.}
{\displaystyle {\begin{array}{rcl}\displaystyle {\lim _{n\rightarrow \infty }s_{n}}&=&\displaystyle {\lim _{n\rightarrow \infty }2{\bigg (}{\frac {1}{2^{2}}}-{\frac {1}{2^{n+1}}}{\bigg )}}\\&&\\&=&\displaystyle {\frac {2}{2^{2}}}\\&&\\&=&\displaystyle {{\frac {1}{2}}.}\end{array}}}
{\displaystyle s_{n}=2{\bigg (}{\frac {1}{2^{2}}}-{\frac {1}{2^{n+1}}}{\bigg )}}
{\displaystyle {\frac {1}{2}}}
Retrieved from "https://wiki.math.ucr.edu/index.php?title=009C_Sample_Midterm_1,_Problem_2_Detailed_Solution&oldid=1828"
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Electric susceptibility - Wikipedia
Find sources: "Electric susceptibility" – news · newspapers · books · scholar · JSTOR (November 2010) (Learn how and when to remove this template message)
In electricity (electromagnetism), the electric susceptibility (
{\displaystyle \chi _{\text{e}}}
; Latin: susceptibilis "receptive") is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field. The greater the electric susceptibility, the greater the ability of a material to polarize in response to the field, and thereby reduce the total electric field inside the material(and store energy). It is in this way that the electric susceptibility influences the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light.[1][2]
1 Definition for linear dielectrics
2 Molecular polarizability
2.1 Ambiguity in the definition
3 Nonlinear susceptibility
4 Dispersion and causality
Definition for linear dielectrics[edit]
If a dielectric material is a linear dielectric, then electric susceptibility is defined as the constant of proportionality (which may be a matrix) relating an electric field E to the induced dielectric polarization density P such that[3][4]
{\displaystyle \mathbf {P} =\varepsilon _{0}\chi _{\text{e}}{\mathbf {E} },}
{\displaystyle \mathbf {P} }
{\displaystyle \varepsilon _{0}}
is the electric permittivity of free space (electric constant);
{\displaystyle \chi _{\text{e}}}
is the electric susceptibility;
{\displaystyle \mathbf {E} }
is the electric field.
In materials where susceptibility is anisotropic (different depending on direction), susceptibility is represented as a matrix known as the susceptibility tensor. Many linear dielectrics are isotropic, but it is possible nevertheless for a material to display behavior that is both linear and anisotropic, or for a material to be non-linear but isotropic. Anisotropic but linear susceptibility is common in many crystals.[3]
The susceptibility is related to its relative permittivity (dielectric constant)
{\displaystyle \varepsilon _{\textrm {r}}}
{\displaystyle \chi _{\text{e}}\ =\varepsilon _{\text{r}}-1}
{\displaystyle \chi _{\text{e}}\ =0.}
At the same time, the electric displacement D is related to the polarization density P by the following relation:[3]
{\displaystyle \mathbf {D} \ =\ \varepsilon _{0}\mathbf {E} +\mathbf {P} \ =\ \varepsilon _{0}(1+\chi _{\text{e}})\mathbf {E} \ =\ \varepsilon _{\text{r}}\varepsilon _{0}\mathbf {E} \ =\ \varepsilon \mathbf {E} }
{\displaystyle \varepsilon \ =\ \varepsilon _{\text{r}}\varepsilon _{0}}
{\displaystyle \varepsilon _{\text{r}}\ =\ (1+\chi _{\text{e}})}
Molecular polarizability[edit]
Main article: Polarizability
A similar parameter exists to relate the magnitude of the induced dipole moment p of an individual molecule to the local electric field E that induced the dipole. This parameter is the molecular polarizability (α), and the dipole moment resulting from the local electric field Elocal is given by:
{\displaystyle \mathbf {p} =\varepsilon _{0}\alpha \mathbf {E_{\text{local}}} }
This introduces a complication however, as locally the field can differ significantly from the overall applied field. We have:
{\displaystyle \mathbf {P} =N\mathbf {p} =N\varepsilon _{0}\alpha \mathbf {E} _{\text{local}},}
where P is the polarization per unit volume, and N is the number of molecules per unit volume contributing to the polarization. Thus, if the local electric field is parallel to the ambient electric field, we have:
{\displaystyle \chi _{\text{e}}\mathbf {E} =N\alpha \mathbf {E} _{\text{local}}}
Thus only if the local field equals the ambient field can we write:
{\displaystyle \chi _{\text{e}}=N\alpha .}
Otherwise, one should find a relation between the local and the macroscopic field. In some materials, the Clausius–Mossotti relation holds and reads
{\displaystyle {\frac {\chi _{\text{e}}}{3+\chi _{\text{e}}}}={\frac {N\alpha }{3}}.}
Ambiguity in the definition[edit]
The definition of the molecular polarizability depends on the author. In the above definition,
{\displaystyle \mathbf {p} =\varepsilon _{0}\alpha \mathbf {E_{\text{local}}} ,}
{\displaystyle p}
{\displaystyle E}
are in SI units and the molecular polarizability
{\displaystyle \alpha }
has the dimension of a volume (m3). Another definition[5] would be to keep SI units and to integrate
{\displaystyle \varepsilon _{0}}
{\displaystyle \alpha }
{\displaystyle \mathbf {p} =\alpha \mathbf {E_{\text{local}}} .}
In this second definition, the polarizability would have the SI unit of C.m2/V. Yet another definition exists[6] where
{\displaystyle p}
{\displaystyle E}
are expressed in the cgs system and
{\displaystyle \alpha }
is still defined as
{\displaystyle \mathbf {p} =\alpha \mathbf {E_{\text{local}}} .}
Using the cgs units gives
{\displaystyle \alpha }
the dimension of a volume, as in the first definition, but with a value that is
{\displaystyle 4\pi }
lower.
Nonlinear susceptibility[edit]
In many materials the polarizability starts to saturate at high values of electric field. This saturation can be modelled by a nonlinear susceptibility. These susceptibilities are important in nonlinear optics and lead to effects such as second-harmonic generation (such as used to convert infrared light into visible light, in green laser pointers).
The standard definition of nonlinear susceptibilities in SI units is via a Taylor expansion of the polarization's reaction to electric field:[7]
{\displaystyle P=P_{0}+\varepsilon _{0}\chi ^{(1)}E+\varepsilon _{0}\chi ^{(2)}E^{2}+\varepsilon _{0}\chi ^{(3)}E^{3}+\cdots .}
(Except in ferroelectric materials, the built-in polarization is zero,
{\displaystyle P_{0}=0}
.) The first susceptibility term,
{\displaystyle \chi ^{(1)}}
, corresponds to the linear susceptibility described above. While this first term is dimensionless, the subsequent nonlinear susceptibilities
{\displaystyle \chi ^{(n)}}
have units of (m/V)n−1.
The nonlinear susceptibilities can be generalized to anisotropic materials in which the susceptibility is not uniform in every direction. In these materials, each susceptibility
{\displaystyle \chi ^{(n)}}
becomes an (n + 1)-degree tensor.
Dispersion and causality[edit]
Plot of the dielectric constant as a function of frequency showing several resonances and plateaus, which indicate the processes that respond on the time scale of a period. This demonstrates that thinking of the susceptibility in terms of its Fourier transform is useful.
{\displaystyle \mathbf {P} (t)=\varepsilon _{0}\int _{-\infty }^{t}\chi _{\text{e}}(t-t')\mathbf {E} (t')\,\mathrm {d} t'.}
That is, the polarization is a convolution of the electric field at previous times with time-dependent susceptibility given by
{\displaystyle \chi _{\text{e}}(\Delta t)}
. The upper limit of this integral can be extended to infinity as well if one defines
{\displaystyle \chi _{\text{e}}(\Delta t)=0}
{\displaystyle \Delta t<0}
. An instantaneous response corresponds to Dirac delta function susceptibility
{\displaystyle \chi _{\text{e}}(\Delta t)=\chi _{\text{e}}\delta (\Delta t)}
It is more convenient in a linear system to take the Fourier transform and write this relationship as a function of frequency. Due to the convolution theorem, the integral becomes a product,
{\displaystyle \mathbf {P} (\omega )=\varepsilon _{0}\chi _{\text{e}}(\omega )\mathbf {E} (\omega ).}
This has a similar form to the Clausius–Mossotti relation:[8]
{\displaystyle \mathbf {P} (\mathbf {r} )=\varepsilon _{0}{\frac {N\alpha (\mathbf {r} )}{1-{\frac {1}{3}}N(\mathbf {r} )\alpha (\mathbf {r} )}}\mathbf {E} (\mathbf {r} )=\varepsilon _{0}\chi _{\text{e}}(\mathbf {r} )\mathbf {E} (\mathbf {r} )}
Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e.
{\displaystyle \chi _{\text{e}}(\Delta t)=0}
{\displaystyle \Delta t<0}
), a consequence of causality, imposes Kramers–Kronig constraints on the susceptibility
{\displaystyle \chi _{\text{e}}(0)}
Clausius–Mossotti relation
^ "Electric susceptibility". Encyclopædia Britannica.
^ Cardarelli, François (2000–2008). Materials Handbook: A Concise Desktop Reference (2nd ed.). London: Springer-Verlag. pp. 524 (Section 8.1.16). doi:10.1007/978-1-84628-669-8. ISBN 978-1-84628-668-1.
^ a b c Griffiths, David J (2017). Introduction to Electrodynamics (4 ed.). Cambridge University Press. pp. 181–190.
^ Freeman, Richard; King, James; Lafyatis, Gregory (2019), "Essentials of Electricity and Magnetism", Electromagnetic Radiation, Oxford: Oxford University Press, doi:10.1093/oso/9780198726500.001.0001/oso-9780198726500-chapter-1#oso-9780198726500-chapter-1-displaymaths-20, ISBN 978-0-19-872650-0, retrieved 2022-02-18
^ CRC Handbook of Chemistry and Physics (PDF) (84 ed.). CRC. pp. 10–163. Archived from the original (PDF) on 2016-10-06. Retrieved 2016-08-19.
^ Butcher, Paul N.; Cotter, David (1990). The Elements of Nonlinear Optics. Cambridge University Press. doi:10.1017/CBO9781139167994. ISBN 9781139167994.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Electric_susceptibility&oldid=1082235613"
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IsGeneralized - Maple Help
Home : Support : Online Help : Programming : Data Types : MultiSet : IsGeneralized
MultiSet/IsGeneralized
query whether negative multiplicities are permitted
IsGeneralized( M );
IsGeneralized(M) returns true if the MultiSet M was created via a constructor call of the form MultiSet[generalized].
M≔\mathrm{MultiSet}\left(a=2,b=5\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}{[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\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}{5}]}
\mathrm{IsGeneralized}\left(M\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
N≔\mathrm{MultiSet}[\mathrm{generalized}]\left(c=-3,d=\frac{1}{2},e=3.14159,f\right)
\textcolor[rgb]{0,0,1}{N}\textcolor[rgb]{0,0,1}{≔}{[\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{e}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3.14159}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]}
\mathrm{IsGeneralized}\left(N\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
The MultiSet/IsGeneralized command was introduced in Maple 2016.
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Motor Selection Guide - Phidgets Support
1 Choosing a Motor
4 Estimating Required Torque
A servo motor (left), a stepper motor (middle), a BLDC motor (right) and a DC motor (bottom).
There are 4 categories of electric motors that are used in practical applications and are easily available for purchase. They are:
Each of these classes has several variants and each have their advantages and disadvantages.
There are many different options when considering what type of motor to use. This table provides a summary of the differences:
Motor Type Rotation Position Control Velocity Control Position Precision Price Range Controller Price per Motor
Brushed DC Motors Continuous None Open-Loop N/A $10+ $60+
Brushed DC Motors (w/ Encoder) Continuous Closed-Loop Closed-Loop Varies1 $40+ $60+
Servo Motors Limited (Usually °180) Closed Loop Closed Loop 0.5° - 1.5° $12+ $6+
Continuous Rotation RC Servo Motors Continuous None Open Loop N/A $18+ $6+
Stepper Motors Continuous Open Loop Open Loop 0.9°-1.8° (or 1°-3° with gearbox)2 $15+ $70+
Stepper Motors (w/Encoder) Continuous Closed Loop Closed Loop 0.9°-1.8° (or 1°-3° with gearbox)2 $25+ $85+
Brushless DC Motors Continuous Closed Loop Closed Loop Varies3 $35+ $65+
1- The positional precision of a DC motor with encoder depends on the encoders CPR (counts per revolution). For example, a 360 CPR encoder would provide 1° precision.
2- Gearboxes introduce inaccuracy in the form of "slop" (gap between mated gear teeth). For maximum precision, stick with gearless steppers with 0.9° step angle.
3- The positional precision of a BLDC motor varies depending on the hall effect sensor in the motor and how the controller measures the data from it.
While most motors have fully continuous rotation, there is one type of motor that has limited rotation: servo motors. Most servos have 180° of travel, but you can find some multi-rotation servos designed to be part of a winch mechanism. The limited rotation angle of servo motors is a tradeoff for having built-in position control using a potentiometer.
If you want to be able to specify a precise position for your motor to rotate to and stop, you'll need some form of position control. There are two main types of control.
Open-Loop position control means the motor controller has a way of telling the motor to move to a certain position, but there's no way for the controller to know if the motor actually succeeded in reaching that position. For example, if the motor stalled because it came in contact with a solid object, the controller would have no way of knowing this. Open-loop control is sufficient in systems where such external forces are limited, or where exact positioning is not critical.
Closed-Loop position control has some sort of feedback mechanism so that the controller knows the position of the motor at all times. This means that even if an external force resists or opposes the motor's movement, the controller will know to keep driving the motor until it reaches its target position. Closed-loop position control is required in systems where positioning errors are not acceptable, or in systems that run for long periods of time that could otherwise accumulate positioning errors during operation.
Motor Comparison:
BLDC motors and use hall-effect sensors built into the motor for feedback, allowing for closed-loop control.
RC Servos have closed-loop position control because servo controllers read the built-in potentiometer of the servo for feedback.
Stepper motors have open-loop control, since the controller tells the motor to move a certain number of steps.
DC Motors do not have any position control, as the controller simply decides how much power to provide to the motor, which determines motor speed.
Systems without closed-loop position control can be upgraded by adding an encoder to the motor. Combined with a device that can read it, the encoder will provide feedback that will enable closed-loop control.
A system can also have velocity control, which allows it to target a specific rotation speed. If your system has open-loop position control, it will (by definition) have open-loop velocity control. Likewise, a system with closed-loop position control will have closed-loop velocity control.
DC Motors and continuous rotation servos both have open-loop velocity control, because the controller simply decides how much power to apply to the motor and this influences the speed directly.
Stepper motors have open-loop velocity control because the controller knows the desired position and the amount of time it takes to send the signal to get to that position, so velocity can easily be calculated.
BLDC Motors have closed-loop velocity control, since the controller knows the position and timing due to data from the hall effect sensors in the motor.
Limited rotation servos have closed-loop velocity control, since the controller knows the position and timing due to data from the potentiometer.
In the same way that encoders can be used to upgrade the position control of a system, they can also provide closed-loop velocity control.
There are a number of factors that impact the precision of a motor's position control, depending on the type of motor and what kind of feedback mechanism it uses.
Stepper motors are accurate to whatever the motor step angle is. The typical step angle is 1.8°, in which case the motor's positioning would always be a multiple of 1.8°, resulting in 200 unique positions across a full rotation.
Servo motors are limited by the "Deadband width", which can be found in your servo motor's data sheet. Take the full pulse range of the servo and divide it by the deadband width. Then take the servo's actuation range in degrees and divide it by the result. This will give you the servo's position resolution in degrees.
If you're using an encoder, the precision will be limited by the encoder's CPR. For example, a CPR of 360 will result in 1° resolution.
If your motor has a gearbox, it will introduce some amount of error in positioning due to empty space between mated gear teeth. This is called "backlash" or "slop". Typical gearboxes can have anywhere from 1 to 3 degrees of slop, which impacts your positioning precision directly.
Due to differences in how motors are built and controlled, the prices between types of motor vary drastically.
Servo motors are often the most economical option for low-torque applications. While the price for a single motor is comparable to other options, servo controller boards are often set up to control as many as 16 servos at once.
DC motors are the simplest type of motor, and are available in a wide range of sizes. DC controllers are usually fairly simple, so the controller cost isn't as high as steppers or BLDC.
Stepper motors and BLDC motors are more expensive, because the motors as more complicated to control.
Gearboxes, a common feature of electric motors, use the mechanical advantage of gears to reduce the speed of the motor and increase the torque. The speed and accuracy are affected directly by the gear ratio, as seen in these equations:
{\displaystyle {\text{Output Speed}}={\frac {\text{Motor Speed}}{\text{Gearbox Ratio}}}}
For example, a motor with a speed of 500 rpm and a gearbox with a 10:1 gear ratio would result in and output speed of 50 rpm.
Although the reduction ratio plays a large part in determining the Gearbox Output Torque, there is also an inefficiency that is introduced through the use of a gearbox. Some of the torque of the motor is converted into heat and lost due to friction between the gears. Gearbox efficiency depends on the physical characteristics of the gears, the number of gear stages, and gearing system used. You can calculate the Gearbox Output Torque with the following equation:
{\displaystyle {\text{Output Torque}}={\text{Motor Output Torque }}\times {\text{ Gearbox Ratio }}\times {\text{ Gearbox Efficiency}}}
Keep in mind that while adding a gearbox does increase the torque, the gearbox itself is only rated to withstand a certain amount of torque before it risks becoming damaged or breaking. When selecting a motor, you should always consider the actual operating torque to be the minimum of the output torque and the gearbox strength. For example, a motor that can normally output 5 kg·cm of torque with a 10:1, 90% efficient gearbox whose gears are rated for 40 kg·cm would theoretically be able to output 45 kg·cm of torque, but in practice the 40 kg·cm limit of the gearbox should be considered the maximum torque.
With gearboxes, torque and speed can be seen as one interchangeable characteristic: If you need more torque and less speed, try to find the same motor with a gearbox with a higher reduction ratio. If you need more speed and less torque, try to find the same motor with a gearbox with a lower reduction ratio. However, it is not advisable buy gearboxes and motors separately to mix and match, unless they are specifically designed for each other. There's a lot that can go wrong in gearbox customization and for most users it's a lot less hassle to simply buy a motor with a gearbox already attached.
A small optical encoder mounted on the rear shaft of a DC motor.
An encoder is a device installed on the shaft of a motor that can be used to keep track of the position of the motor or calculate the speed of the motor. Encoders are often used with a DC motor and a PID control system in order to achieve position/velocity control of a motor that would otherwise only have simple on/off control. Some motors have the shaft exposed on both sides of the motor, so an encoder can be attached on one end without getting in the way of what the motor is driving.
Encoders are rarely used with BLDC motors or servo motors, since they both already have some form of closed-loop position feedback built-in. Encoders are sometimes used with stepper motors, because position control in a stepper motor is open-loop (the controller has no way of knowing whether the stepper actually made it to the desired position, it only knows how far it has told the stepper to go). Excessive vibration, external forces, or sudden high-torque loads can cause steps to be missed, so an encoder would close the control loop and provide actual feedback in this case.
Estimating Required Torque
It can be difficult to estimate how much torque will be required for a certain application before the project is built, but there are plenty of guidelines and rules-of-thumb online that can give you a good idea.
In the case of a typical wheeled vehicle, we recommend you take the mass of the vehicle and multiply by the radius of the wheels, then divide by four.
{\displaystyle {\text{Estimated Torque Requirement}}={\frac {{\text{Mass }}\times {\text{ Wheel Radius }}}{4}}}
For example, a 20kg vehicle with wheels of radius 6cm would call for a total goal of 30 kg·cm contributed from all of the motors. This should be more than enough torque to move around on smooth terrain and gentle inclines. If you need to be able to handle rough terrain or sharp inclines, you may want to only divide by a factor of 3 or even 2. In general, it's a good idea to get a more powerful motor than you think you'll need, just to account for unforeseen factors.
MURVV Mobile Robot
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When solving a problem about the perimeter of a rectangle using the 5-D Process, Herman built the expression below.
\text{Perimeter}=x+x+4x+4x\ \text{feet}
Draw a rectangle and label its sides based on Herman’s expression.
Use each part of the expression as a side of the rectangle. Remember, rectangles have two pairs of equal, opposite sides. Try creating a rectangle with the given expression.
What is the relationship between the base and height of Herman’s rectangle? How can you tell?
Examine the rectangle and compare the base and height. Are there any common factors between them that make them related?
If the perimeter of the rectangle is 60 feet, how long are the base and height of Herman’s rectangle? Show how you know.
Using the expression Herman created, how can you use the given perimeter to solve for the base and height of the rectangle?
Use the 5-D Process to conduct trial runs with the given expression and perimeter to solve for the unknown variable and as a result, determine the base and height.
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Introduction to Chemical Engineering Processes/How to use the mass balance - Wikibooks, open books for an open world
1 A Typical Type of Problem
2 Single Component in Multiple Processes: a Steam Process
2.1 Step 1: Draw a Flowchart
2.2 Step 2: Make sure your units are consistent
2.3 Step 3: Relate your variables
2.4 So you want to check your guess? Alright then read on.
2.5 Step 4: Calculate your unknowns.
2.6 Step 5: Check your work.
A Typical Type of ProblemEdit
Most problems you will face are significantly more complicated than the previous problem and the following one. In the engineering world, problems are presented as so-called "word problems", in which a system is described and the problem must be set up and solved (if possible) from the description. This section will attempt to illustrate through example, step by step, some common techniques and pitfalls in setting up mass balances. Some of the steps may seem somewhat excessive at this point, but if you follow them carefully on this relatively simple problem, you will certainly have an easier time following later steps.
Single Component in Multiple Processes: a Steam ProcessEdit
A feed stream of pure liquid water enters an evaporator at a rate of 0.5 kg/s. Three streams come from the evaporator: a vapor stream and two liquid streams. The flowrate of the vapor stream was measured to be 4*10^6 L/min and its density was 4 g/m^3. The vapor stream enters a turbine, where it loses enough energy to condense fully and leave as a single stream. One of the liquid streams is discharged as waste, the other is fed into a heat exchanger, where it is cooled. This stream leaves the heat exchanger at a rate of 1500 pounds per hour. Calculate the flow rate of the discharge and the efficiency of the evaporator.
Note that one way to define efficiency is in terms of conversion, which is intended here:
{\displaystyle efficiency={\frac {{\dot {m}}_{vapor}}{{\dot {m}}_{feed}}}}
Step 1: Draw a FlowchartEdit
The problem as it stands contains an awful lot of text, but it won't mean much until you draw what is given to you. First, ask yourself, what processes are in use in this problem? Make a list of the processes in the problem:
Evaporator (A)
Heat Exchanger (B)
Once you have a list of all the processes, you need to find out how they are connected (it'll tell you something like "the vapor stream enters a turbine"). Draw a basic sketch of the processes and their connections, and label the processes. It should look something like this:
Remember, we don't care what the actual processes look like, or how they're designed. At this point, we only really label what they are so that we can go back to the problem and know which process they're talking about.
Once all your processes are connected, find any streams that are not yet accounted for. In this case, we have not drawn the feed stream into the evaporator, the waste stream from the evaporator, or the exit streams from the turbine and heat exchanger.
The third step is to Label all your flows. Label them with any information you are given. Any information you are not given, and even information you are given should be given a different variable. It is usually easiest to give them the same variable as is found in the equation you will be using (for example, if you have an unknown flow rate, call it
{\displaystyle {\dot {m}}}
so it remains clear what the unknown value is physically. Give each a different subscript corresponding to the number of the feed stream (such as
{\displaystyle {\dot {m_{1}}}}
for the feed stream that you call "stream 1"). Make sure you include all units on the given values!
In the example problem, the flowchart I drew with all flows labeled looked like this:
Notice that for one of the streams, a volume flow rate is given rather than a mass flow rate, so it is labeled as such. This is very important, so that you avoid using a value in an equation that isn't valid (for example, there's no such thing as "conservation of volume" for most cases)!
The final step in drawing the flowchart is to write down any additional given information in terms of the variables you have defined. In this problem, the density of the water in the vapor stream is given, so write this on the side for future reference.
Carefully drawn flowcharts and diagrams are half of the key to solving any mass balance, or really a lot of other types of engineering problems. They are just as important as having the right units to getting the right answer.
Step 2: Make sure your units are consistentEdit
The second step is to make sure all your units are consistent and, if not, to convert everything so that it is. In this case, since the principle that we'll need to use to solve for the flow rate of the waste stream (
{\displaystyle {\dot {m_{3}}}}
) is conservation of mass, everything will need to be on a mass-flow basis, and also in the same mass-flow units.
In this problem, since two of our flow rates are given in metric units (though one is a volumetric flow rate rather than a mass flow rate, so we'll need to change that) and only one in English units, it would save time and minimize mistakes to convert
{\displaystyle {\dot {V_{2}}}}
{\displaystyle {\dot {m_{5}}}}
to kg/s.
From the previous section, the equation relating volumetric flowrate to mass flow rate is:
{\displaystyle {\dot {V}}_{i}*{\rho }_{i}={\dot {m}}_{i}}
Therefore, we need the density of water vapor in order to calculate the mass flow rate from the volumetric flow rate. Since the density is provided in the problem statement (if it wasn't, we'd need to calculate it with methods described later), the mass flow rate can be calculated:
{\displaystyle {\dot {V_{2}}}={\frac {4*10^{6}{\mbox{ L}}}{1{\mbox{ min}}}}*{\frac {1{\mbox{ m}}^{3}}{1000{\mbox{ L}}}}*{\frac {1{\mbox{ min}}}{60{\mbox{ s}}}}=66.67{\frac {m^{3}}{s}}}
{\displaystyle \rho _{2}=4{\frac {g}{m^{3}}}*{\frac {1{\mbox{ kg}}}{1000{\mbox{ g}}}}=0.004{\frac {kg}{m^{3}}}}
{\displaystyle {\dot {m}}_{2}=66.67{\frac {m^{3}}{s}}*0.004{\frac {kg}{m^{3}}}=0.2666{\frac {kg}{s}}}
Note that since the density of a gas is so small, a huge volumetric flow rate is necessary to achieve any significant mass flow rate. This is fairly typical and is a practical problem when dealing with gas-phase processes.
The mass flow rate
{\displaystyle {\dot {m_{5}}}}
can be changed in a similar manner, but since it is already in terms of mass (or weight technically), we don't need to apply a density:
{\displaystyle {\dot {m_{5}}}=1500{\frac {lb}{hr}}*{\frac {1kg}{2.2lb}}*{\frac {1hr}{3600s}}=0.1893{\frac {kg}{s}}}
Now that everything is in the same system of units, we can proceed to the next step.
Step 3: Relate your variablesEdit
Since we have the mass flow rate of the vapor stream we can calculate the efficiency of the evaporator directly:
{\displaystyle efficiency={\frac {{\dot {m}}_{2}}{{\dot {m}}_{1}}}={\frac {0.2666{\frac {kg}{s}}}{0.5{\frac {kg}{s}}}}=53.3\%}
{\displaystyle {\dot {m_{4}}}}
, as asked for in the problem, will be somewhat more difficult. One place to start is to write the mass balance on the evaporator, since that will certainly contain the unknown we seek. Assuming that the process is steady state we can write:
{\displaystyle In-Out=0}
{\displaystyle {\dot {m}}_{1}-{\dot {m}}_{2}-{\dot {m}}_{4}-{\dot {m}}_{6}=0}
Problem: we don't know
{\displaystyle {\dot {m}}_{6}}
so with only this equation we cannot solve for
{\displaystyle {\dot {m}}_{4}}
. Have no fear, however, because there is another way to figure out what
{\displaystyle {\dot {m}}_{6}}
is... can you figure it out? Try to do so before you move on.
So you want to check your guess? Alright then read on.Edit
The way to find
{\displaystyle {\dot {m}}_{6}}
is to do a mass balance on the heat exchanger, because the mass balance for the heat exchanger is simply:
{\displaystyle {\dot {m}}_{6}-{\dot {m}}_{5}=0}
{\displaystyle {\dot {m}}_{5}}
{\displaystyle {\dot {m}}_{6}}
and thus the waste stream flowrate
{\displaystyle {\dot {m}}_{4}}
Notice the strategy here: we first start with a balance on the operation containing the stream we need information about. Then we move to balances on other operations in order to garner additional information about the unknowns in the process. This takes practice to figure out when you have enough information to solve the problem or you need to do more balances or look up information.
It is also of note that any process has a limited number of independent balances you can perform. This is not as much of an issue with a relatively simple problem like this, but will become an issue with more complex problems. Therefore, a step-by-step method exists to tell you exactly how many independent mass balances you can write on any given process, and therefore how many total independent equations you can use to help you solve problems.
Step 4: Calculate your unknowns.Edit
Carrying out the plan on this problem:
{\displaystyle {\dot {m}}_{6}-0.1893{\frac {kg}{s}}=0}
{\displaystyle {\dot {m}}_{6}=0.1893{\frac {kg}{s}}}
Hence, from the mass balances on the evaporator:
{\displaystyle {\dot {m}}_{4}={\dot {m}}_{1}-{\dot {m}}_{2}-{\dot {m}}_{6}=(0.5-0.2666-0.1893){\frac {kg}{s}}=0.0441{\frac {kg}{s}}}
So the final answers are:
{\displaystyle {\mbox{Evaporator Efficiency}}=53.3\%}
{\displaystyle {\mbox{Waste stream rate}}=0.0441{\frac {kg}{s}}}
Step 5: Check your work.Edit
Ask: Do these answers make sense? Check for things like negative flow rates, efficiencies higher than 100%, or other physically impossible happenings; if something like this happens (and it will), you did something wrong. Is your exit rate higher than the total inlet rate (since no water is created in the processes, it is impossible for this to occur)?
In this case, the values make physical sense, so they may be right. It's always good to go back and check the math and the setup to make sure you didn't forget to convert any units or anything like that.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Introduction_to_Chemical_Engineering_Processes/How_to_use_the_mass_balance&oldid=3547421"
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Heart failure is a global problem that affects 38 million patients around the world [1]. Regardless of the etiology, patients with heart failure experience limitation of their functional capacity, symptoms related to congestion and/or low output, a decrease in quality of life and a reduction in their life expectancy [2]. Given that ischemic heart disease and hypertensive heart disease are the most common causes of heart failure prevention and treatment of those conditions are essential to reduce its incidence [3]. Once structural heart abnormalities and symptoms develop prevention of congestion, initiation of guideline directed medical therapy, use of devices and timely referral for advanced heart failure therapies become the focus to improve clinical outcomes [4].
The trajectories of heart failure patients are heterogeneous. Because of that personalized follow up is necessary [5]. The COVID-19 pandemic changed the landscape of medicine forever. Social distancing and the fear of turning health care facilities as virus transmission hubs motivated a drastic reduction in in-person encounters [6]. In this context telemedicine has evolved as a necessary tool in everyday practice and chronic disease management systems in heart failure are an excellent platform for its implementation. In this issue we discuss principles of application of tele monitoring to maximize patient trajectory tracking and minimize in person visits [7].
Advances in the last decade has defined the four pillars of pharmacological therapy in heart failure with reduced ejection fraction [4]. The implementation of quadruple therapy (Heart failure specific beta-blocker + ARNI or ARB or ACEI) + Aldosterone Blocker + SGLT-2 inhibitor) has shown a consistent improvement in survival [8]. However elderly patients and those with advanced renal disease have been unrepresented from pivotal clinical trials. Kolben et al. [9] will discuss the considerations in the implementation of those therapies specific patient population.
Pulmonary hypertension is associated with increased mortality in heart failure. No specific therapies for Group 2 pulmonary hypertension have been approved [10]. Functional mitral regurgitation is associated with increased mortality in patients with heart failure [11]. Functional mitral regurgitation is a disease of the left ventricle and GDMT is the first step in treatment [12]. Percutaneous repair of the mitral valve has a Class 2A, LOE B recommendation for patients with heart failure, LVEF
<
50% with severe chronic secondary MR and persistent severe symptoms in spite of optimal medical therapy [13]. Mandurino-Mirizzi et al. [14] will tackle the complex interaction of percutaneous edge to edge repair, mitral regurgitation, pulmonary hypertension and possible future directions.
Atrial fibrillation is the most common type of arrhythmia in heart failure, is associated with worse prognosis [15]. Since the introduction of pulmonary vein isolation by Haisaguirre et al. [16] the procedure and technical advances have evolved and the center of gravity has been moving from medical therapy and rate control to interventional therapy and rhythm control. This approach has translated not only in nicer EKGs but in improvements in clinical outcomes with reductions in hospitalizations and mortality [17]. Cardiovascular implantable electronic device (CIED) such as implantable cardioverter defibrillators (ICDs) and cardiac resynchronization therapy (CRT) are part of the armamentarium to decrease mortality in heart failure. However those devices are not free from complications and advances in patient selection to improve the risk/benefit ratio of the device selection is necessary. Sohrabi et al. [18] will guide us in the current implementation of electrophysiological therapies in our heart failure patients.
In patients with refractory symptoms advanced heart failure therapies and palliative care should be considered. Tatum et al. [19] describe the amazing evolution of durable mechanical support that motivated changes in the heart allocation policy and the consequences of those changes in the future of durable mechanical support.
Our hope is that this issue Reviews in Cardiovascular Medicine will motivate new questions and applications for the benefit of our patients.
Authors equally contributed to the content.
Braunwald E. The war against heart failure: the Lancet lecture. Lancet. 2015; 385: 812–824.
Thibodeau JT, Drazner MH. The Role of the Clinical Examination in Patients with Heart Failure. JACC: Heart Failure. 2018; 6: 543–551.
Bragazzi NL, Zhong W, Shu J, Abu Much A, Lotan D, Grupper A, et al. Burden of heart failure and underlying causes in 195 countries and territories from 1990 to 2017. European Journal of Preventive Cardiology. 2021; zwaa147.
McDonagh TA, Metra M, Adamo M, Gardner RS, Baumbach A, Böhm M, et al. 2021 ESC Guidelines for the diagnosis and treatment of acute and chronic heart failure: Developed by the Task Force for the diagnosis and treatment of acute and chronic heart failure of the European Society of Cardiology (ESC) With the special contribution of the Heart Failure Association (HFA) of the ESC. European Heart Journal. 2021. 42: 3599–3726.
Desai AS, Stevenson LW. Rehospitalization for Heart Failure. Circulation. 2012; 126: 501–506.
Werner RM, Glied SA. Covid-Induced Changes in Health Care Delivery — can they last? New England Journal of Medicine. 2021; 385: 868–870.
Alvarez P, Sianis A, Brown J, Ali A, Briasoulis A. Chronic disease management in heart failure: focus on telemedicine and remote monitoring. Reviews in Cardiovascular Medicine. 2021; 22: 403–413.
Vaduganathan M, Claggett BL, Jhund PS, Cunningham JW, Pedro Ferreira J, Zannad F, et al. Estimating lifetime benefits of comprehensive disease-modifying pharmacological therapies in patients with heart failure with reduced ejection fraction: a comparative analysis of three randomised controlled trials. The Lancet. 2020; 396: 121–128.
Kolben Y, Kessler A, Puris G, Nachman D, Alvarez P, Briasoulis A, et al. Management of heart failure with reduced ejection fraction: challenges in patients with atrial fibrillation, renal disease and in the elderly. Reviews in Cardiovascular Medicine. 2022; 23: 016.
Vachiéry J, Tedford RJ, Rosenkranz S, Palazzini M, Lang I, Guazzi M, et al. Pulmonary hypertension due to left heart disease. European Respiratory Journal. 2019; 53: 1801897.
Bartko PE, Heitzinger G, Pavo N, Heitzinger M, Spinka G, Prausmüller S, et al. Burden, treatment use, and outcome of secondary mitral regurgitation across the spectrum of heart failure: observational cohort study. BMJ-British Medical Journal. 2021; 373: n1421.
Goliasch G, Bartko PE, Pavo N, Neuhold S, Wurm R, Mascherbauer J, et al. Refining the prognostic impact of functional mitral regurgitation in chronic heart failure. European Heart Journal. 2018; 39: 39–46.
Otto CM, Nishimura RA, Bonow RO, Carabello BA, Erwin JP 3rd, Gentile F, et al. 2020 ACC/AHA Guideline for the Management of Patients With Valvular Heart Disease: A Report of the American College of Cardiology/American Heart Association Joint Committee on Clinical Practice Guidelines. Circulation. 2021; 143: e72–e227.
Mandurino-Mirizzi A, Tua L, Arzuffi L, Demarchi A, Somaschini A, Tournas G, et al. Transcatheter mitral valve repair with MitraClip in patients with pulmonary hypertension: hemodynamic and prognostic perspectives. Reviews in Cardiovascular Medicine. 2021; 22: 33–38.
Anter E, Jessup M, Callans DJ. Atrial Fibrillation and Heart Failure. Circulation. 2009; 119: 2516–2525.
Haïssaguerre M, Jaïs P, Shah DC, Takahashi A, Hocini M, Quiniou G, et al. Spontaneous initiation of atrial fibrillation by ectopic beats originating in the pulmonary veins. The New England Journal of Medicine. 1998; 339: 659–666.
Marrouche NF, Brachmann J, Andresen D, Siebels J, Boersma L, Jordaens L, et al. Catheter Ablation for Atrial Fibrillation with Heart Failure. New England Journal of Medicine. 2018; 378: 417–427.
Sohrabi C, Ahsan S, Briasoulis A, Androulakis E, Siasos G, Srinivasan NT, et al. Contemporary management of heart failure patients with reduced ejection fraction: the role of implantable devices and catheter ablation. Reviews in Cardiovascular Medicine. 2021; 22: 415–428.
Tatum RT, Massey HT, Tchantchaleishvili V. Impact of mechanical circulatory support on donor heart allocation: past, present, and future. Reviews in Cardiovascular Medicine. 2021; 22: 25–32.
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YYiki: Logistic regression
In the logistic regression, the dependent variable is categorical (possibly ordinal) and we would like to estimate the probability. Instead of dealing with the probability directly, we can use the log odds of the probability. Log-odds is a convenient quantity because it varies
(-\infty, \infty)
. It is also called a logit function.
For a probability
p
, the odds are defined as
\frac{p}{1-p}
. The log odds
\log \left( \frac{p}{1-p} \right)
-\infty
p \rightarrow 0
\infty
p \rightarrow 1
y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \ldots
Now we can formulate a model where
This is the logistic regression model. We can think of it as an application of linear regression on the logit of the probability, or as applying a logistic transformation to
y
to obtain a bounded probability value. The logit and the logistic function are the inverse of the other, and the logit function is called a “link function” in the context of Generalized linear model framework.
Marginal effects and odds ratio
See also Marginal effects
Odds Ratios—Current Best Practice and Use
The odds ratio captures multiplicative change in the dependent variable upon the change in an independent variable; by contrast, the marginal effect cpatures additive change in the dependent variable.
Because the logit function is the logarithm of the odds, the odds is
e^{y}
. This means that if we calculate the odds ratio between the odds with
x_i
and that with
x'_i = x_i + 1
\frac{e^{\beta_0 + \dots \beta_i (x_i+1) + \dots }}{e^{\beta_0 + \dots \beta_i x_i + \dots}} = e^{\beta_i}
. In other words, if we simply exponentiate a coefficient of the model, it gives us the odds ratio upon a unit change in the corresponding variable. Moreover, the odds ratio is a constant regardless of the value of the independent variables.
By contrast, the marginal effect is how much the probability (dependent variable) changes when we change an independent variable. Because it’s about probability, not the odds, it describes an additive change and, unlike the odds ratio, varies depending on the other variables. For binary variable, the marginal effect is the amount of change upon the change of the variable from 0 to 1. Specifically,
\hat{p}(x_i = 1) - \hat{p}(x_i = 0)
For continuous variables, it is the instantaneous rate of change (“dy/dx”).
If you want to ruin your trust in the interpretation of multivariable logistic regression coefficients, read about
Post selection inference
infite sample bias
non-collapsibility
colliders and intermediates
Logistic Regression in Python Using Rodeo
Machine Learning for Hackers Chapter 2, Part 2: Logistic regression with statsmodels
Using R-like formula. It takes care of categorical dummay variables and you can apply transformations (e.g. log) on the fly.
result = smf.logit('DV ~ x1 + x2 + np.log(x3) + x4*x5', data=df).fit()
Calculating the odds ratio with 95% CI.
F-test with human-readable restriction formula
result.f_test('x1 = x2 = 0')
result.f_test('x1 = x2')
Get average marginal effects (use at parameter for other marginal effects).
margins = result.get_margeff() # marginal effects
margins.summary()
margins.summary_frame() # get a data frame
Selecting a reference (pivot) dummy
Currently, statsmodels does not support this choice. But because statsmodels picks a dummy using alphabetical order, we can simply replace the dummary variable that we want to have. For instance, if we have a gender column with m and f values but we want to have m as the pivot, then simply replace it with a.
df.gender.replace('m', 'a', inplace=True)
We can use a similar trick for the multinomial logit.
https://francisbach.com/self-concordant-analysis-for-logistic-regression/
https://twitter.com/graduatedescent/status/1397356480097972224?s=20 - Yadlowsky2021SLOE
« Logistic regression »
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Pedro H. Zambrano | A mathematician living in Bogota, Colombia
Pedro H. Zambrano
A mathematician living in Bogota, Colombia
To some of my students (en portugués, pero seguro entienden):
Thanks a lot for visiting my blog! My name is Pedro Hernán Zambrano and I am a mathematician. My research topic is Model Theory, a branch of Mathematical Logic (do not confuse with just Logic).
More specifically, I am interested in stability in metric non-elementary classes (in fact, superstability in Metric Abstract Elementary Classes, a metric version of non-axiomatizable -in Continuous Logic- classes of complete metric structures; e.g., Banach spaces, Hilbert spaces together with linear operators).
Also, I am interested to study the interaction of Category Theory and Model Theory.
I was invited to attend the Mathematical Logic Program at Mittag-Leffler Institute at Djursholm, Sweden which held in Fall 2009.
I am a member of Colombian Mathematical Society and Association for Symbolic Logic
This blog is devoted to write my personal thinkings and to be a communication way with my mathematical collaborators and students.
A nice advantage of using wordpress is we can write mathematical expressions in
\LaTeX
, so I will be able to write mathematical notes to my collaborators and students.
Thanks once more, and enjoy my blog!
Pedro Z. says:
\LaTeX
commands using wordpress. Just write \ ( \command \ ) without spaces.
Entrega 10 Fundamentos de Matemáticas
Entrega 9 Fundamentos de Matemáticas
Entrega 6 - Fundamentos de Matemáticas II-2019
Algebra Lineal I-2015
Fund. Matematicas I-2017
Fund. Matematicas II-2015
Fund. Matemáticas II-2019
Intr. T. conjuntos I-2018
Intr. T. Conjuntos II-2016
Introd. Teoría de Conjuntos II-2015
Logica Matematica I-2015
Lógica Matemática I-2016
Logica Matematica II-2017
Teoria de Modelos I-2016
Teoría de Modelos I-2018
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High‐Frequency Rupture Processes of the 2014 Mw 8.2 Iquique and 2015 Mw 8.3 Illapel, Chile, Earthquakes Determined from Strong‐Motion Recordings | Bulletin of the Seismological Society of America | GeoScienceWorld
Arthur Frankel *
U.S. Geological Survey, Seattle, Washington, U.S.A.
Corresponding author: afrankel@usgs.gov
Arthur Frankel; High‐Frequency Rupture Processes of the 2014
Mw
Mw
8.3 Illapel, Chile, Earthquakes Determined from Strong‐Motion Recordings. Bulletin of the Seismological Society of America 2022; doi: https://doi.org/10.1785/0120210331
Strong‐motion recordings of the 2014
Mw
Mw
8.3 Illapel, Chile, earthquakes were analyzed to determine rupture propagation and the location, timing, and strength of subevents that produce most of the high‐frequency (≥1 Hz) ground motions. A moving window,cross‐correlation analysis of recordings from a local dense array, band‐pass filtered at 1 Hz, directly shows that the Iquique earthquake ruptured to the southeast over a distance of about 60 km. Array analysis of lower frequency energy (0.03–0.1 Hz) indicates that it occurred updip of the high‐frequency rupture. A methodology was developed for inverting the envelopes of acceleration records (1–5 Hz) to map high‐frequency source factors on the rupture zone and was applied to the two earthquakes. Waveforms of
Mw
6 earthquakes were used as empirical Green’s functions in the inversions. High‐frequency subevents within the two
Mw
8 earthquakes were located at depths ranging from 25 to 55 km and mostly occurred downdip of the peak slip of these earthquakes. Fourier spectral ratios of the Iquique mainshock with respect to
Mw
5–6 aftershocks were fit to determine their stress drops. The stress drops were roughly constant from
Mw
5 to 8 at 10–20 MPa. A compound rupture model is described in which subevents occur in areas of spatially heterogeneous strength and stress on the rupture, and produce the high‐frequency radiated energy of the overall earthquake, but are not located in the areas of peak slip. The stress drop of the overall earthquake is shown to equal the root mean square stress drop of subevents averaged over the rupture area.
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Randomness extractor - Wikipedia
(Redirected from Von Neumann extractor)
A randomness extractor, often simply called an "extractor", is a function, which being applied to output from a weakly random entropy source, together with a short, uniformly random seed, generates a highly random output that appears independent from the source and uniformly distributed.[1] Examples of weakly random sources include radioactive decay or thermal noise; the only restriction on possible sources is that there is no way they can be fully controlled, calculated or predicted, and that a lower bound on their entropy rate can be established. For a given source, a randomness extractor can even be considered to be a true random number generator (TRNG); but there is no single extractor that has been proven to produce truly random output from any type of weakly random source.
Sometimes the term "bias" is used to denote a weakly random source's departure from uniformity, and in older literature, some extractors are called unbiasing algorithms,[2] as they take the randomness from a so-called "biased" source and output a distribution that appears unbiased. The weakly random source will always be longer than the extractor's output, but an efficient extractor is one that lowers this ratio of lengths as much as possible, while simultaneously keeping the seed length low. Intuitively, this means that as much randomness as possible has been "extracted" from the source.
Note that an extractor has some conceptual similarities with a pseudorandom generator (PRG), but the two concepts are not identical. Both are functions that take as input a small, uniformly random seed and produce a longer output that "looks" uniformly random. Some pseudorandom generators are, in fact, also extractors. (When a PRG is based on the existence of hard-core predicates, one can think of the weakly random source as a set of truth tables of such predicates and prove that the output is statistically close to uniform.[3]) However, the general PRG definition does not specify that a weakly random source must be used, and while in the case of an extractor, the output should be statistically close to uniform, in a PRG it is only required to be computationally indistinguishable from uniform, a somewhat weaker concept.
NIST Special Publication 800-90B (draft) recommends several extractors, including the SHA hash family and states that if the amount of entropy input is twice the number of bits output from them, that output can be considered essentially fully random.[4]
3.1 Von Neumann extractor
The min-entropy of a distribution
{\displaystyle X}
{\displaystyle H_{\infty }(X)}
), is the largest real number
{\displaystyle k}
{\displaystyle \Pr[X=x]\leq 2^{-k}}
{\displaystyle x}
{\displaystyle X}
. In essence, this measures how likely
{\displaystyle X}
is to take its most likely value, giving a worst-case bound on how random
{\displaystyle X}
appears. Letting
{\displaystyle U_{\ell }}
denote the uniform distribution over
{\displaystyle \{0,1\}^{\ell }}
{\displaystyle H_{\infty }(U_{\ell })=\ell }
For an n-bit distribution
{\displaystyle X}
with min-entropy k, we say that
{\displaystyle X}
{\displaystyle (n,k)}
{\displaystyle {\text{Ext}}:\{0,1\}^{n}\times \{0,1\}^{d}\to \{0,1\}^{m}}
be a function that takes as input a sample from an
{\displaystyle (n,k)}
{\displaystyle X}
and a d-bit seed from
{\displaystyle U_{d}}
, and outputs an m-bit string.
{\displaystyle {\text{Ext}}}
is a (k, ε)-extractor, if for all
{\displaystyle (n,k)}
{\displaystyle X}
, the output distribution of
{\displaystyle {\text{Ext}}}
is ε-close to
{\displaystyle U_{m}}
Intuitively, an extractor takes a weakly random n-bit input and a short, uniformly random seed and produces an m-bit output that looks uniformly random. The aim is to have a low
{\displaystyle d}
(i.e. to use as little uniform randomness as possible) and as high an
{\displaystyle m}
as possible (i.e. to get out as many close-to-random bits of output as we can).
Definition (Strong Extractor): A
{\displaystyle (k,\epsilon )}
-strong extractor is a function
{\displaystyle {\text{Ext}}:\{0,1\}^{n}\times \{0,1\}^{d}\rightarrow \{0,1\}^{m}\,}
{\displaystyle (n,k)}
{\displaystyle X}
{\displaystyle U_{d}\circ {\text{Ext}}(X,U_{d})}
(the two copies of
{\displaystyle U_{d}}
denote the same random variable) is
{\displaystyle \epsilon }
-close to the uniform distribution on
{\displaystyle U_{m+d}}
{\displaystyle {\text{Ext}}_{n}:\{0,1\}^{n}\times \{0,1\}^{d(n)}\rightarrow \{0,1\}^{m(n)}}
{\displaystyle d=\log {(n-k)}+2\log \left({\frac {1}{\varepsilon }}\right)+O(1)}
{\displaystyle m=k+d-2\log \left({\frac {1}{\varepsilon }}\right)-O(1)}
One of the most important aspects of cryptography is random key generation.[6] It is often necessary to generate secret and random keys from sources that are semi-secret or which may be compromised to some degree. By taking a single, short (and secret) random key as a source, an extractor can be used to generate a longer pseudo-random key, which then can be used for public key encryption. More specifically, when a strong extractor is used its output will appear be uniformly random, even to someone who sees part (but not all) of the source. For example, if the source is known but the seed is not known (or vice versa). This property of extractors is particularly useful in what is commonly called Exposure-Resilient cryptography in which the desired extractor is used as an Exposure-Resilient Function (ERF). Exposure-Resilient cryptography takes into account that the fact that it is difficult to keep secret the initial exchange of data which often takes place during the initialization of an encryption application e.g., the sender of encrypted information has to provide the receivers with information which is required for decryption.
Definition (k-ERF): An adaptive k-ERF is a function
{\displaystyle f}
where, for a random input
{\displaystyle r}
, when a computationally unbounded adversary
{\displaystyle A}
can adaptively read all of
{\displaystyle r}
{\displaystyle k}
{\displaystyle |\Pr\{A^{r}(f(r))=1\}-\Pr\{A^{r}(R)=1\}|\leq \epsilon (n)}
for some negligible function
{\displaystyle \epsilon (n)}
(defined below).
Definition (k-APRF): A
{\displaystyle k=k(n)}
APRF is a function
{\displaystyle f}
where, for any setting of
{\displaystyle n-k}
bits of the input
{\displaystyle r}
to any fixed values, the probability vector
{\displaystyle p}
{\displaystyle f(r)}
over the random choices for the
{\displaystyle k}
remaining bits satisfies
{\displaystyle |p_{i}-2^{-m}|<2^{-m}\epsilon (n)}
{\displaystyle i}
and for some negligible function
{\displaystyle \epsilon (n)}
Kamp and Zuckerman[7] have proved a theorem stating that if a function
{\displaystyle f}
is a k-APRF, then
{\displaystyle f}
is also a k-ERF. More specifically, any extractor having sufficiently small error and taking as input an oblivious, bit-fixing source is also an APRF and therefore also a k-ERF. A more specific extractor is expressed in this lemma:
Lemma: Any
{\displaystyle 2^{-m}\epsilon (n)}
-extractor
{\displaystyle f:\{0,1\}^{n}\rightarrow \{0,1\}^{m}}
for the set of
{\displaystyle (n,k)}
oblivious bit-fixing sources, where
{\displaystyle \epsilon (n)}
is negligible, is also a k-APRF.
This lemma is proved by Kamp and Zuckerman.[7] The lemma is proved by examining the distance from uniform of the output, which in a
{\displaystyle 2^{-m}\epsilon (n)}
-extractor obviously is at most
{\displaystyle 2^{-m}\epsilon (n)}
, which satisfies the condition of the APRF.
The lemma leads to the following theorem, stating that there in fact exists a k-APRF function as described:
Theorem (existence): For any positive constant
{\displaystyle \gamma \leq {\frac {1}{2}}}
, there exists an explicit k-APRF
{\displaystyle f:\{0,1\}^{n}\rightarrow \{0,1\}^{m}}
, computable in a linear number of arithmetic operations on
{\displaystyle m}
-bit strings, with
{\displaystyle m=\Omega (n^{2\gamma })}
{\displaystyle k=n^{{\frac {1}{2}}+\gamma }}
Definition (negligible function): In the proof of this theorem, we need a definition of a negligible function. A function
{\displaystyle \epsilon (n)}
is defined as being negligible if
{\displaystyle \epsilon (n)=O\left({\frac {1}{n^{c}}}\right)}
for all constants
{\displaystyle c}
Proof: Consider the following
{\displaystyle \epsilon }
-extractor: The function
{\displaystyle f}
is an extractor for the set of
{\displaystyle (n,\delta n)}
oblivious bit-fixing source:
{\displaystyle f:\{0,1\}^{n}\rightarrow \{0,1\}^{m}}
{\displaystyle f}
{\displaystyle m=\Omega (\delta ^{2}n)}
{\displaystyle \epsilon =2^{-cm}}
{\displaystyle c>1}
The proof of this extractor's existence with
{\displaystyle \delta \leq 1}
, as well as the fact that it is computable in linear computing time on the length of
{\displaystyle m}
can be found in the paper by Jesse Kamp and David Zuckerman (p. 1240).
That this extractor fulfills the criteria of the lemma is trivially true as
{\displaystyle \epsilon =2^{-cm}}
is a negligible function.
{\displaystyle m}
{\displaystyle m=\Omega (\delta ^{2}n)=\Omega (n)\geq \Omega (n^{2\gamma })}
{\displaystyle \delta \leq 1}
then the lower bound on
{\displaystyle m}
{\displaystyle n}
. In the last step we use the fact that
{\displaystyle \gamma \leq {\frac {1}{2}}}
which means that the power of
{\displaystyle n}
{\displaystyle 1}
{\displaystyle n}
is a positive integer we know that
{\displaystyle n^{2\gamma }}
{\displaystyle n}
{\displaystyle k}
is calculated by using the definition of the extractor, where we know:
{\displaystyle (n,k)=(n,\delta n)\Rightarrow k=\delta n}
and by using the value of
{\displaystyle m}
{\displaystyle m=\delta ^{2}n=n^{2\gamma }}
{\displaystyle m}
we account for the worst case, where
{\displaystyle k}
is on its lower bound. Now by algebraic calculations we get:
{\displaystyle \delta ^{2}n=n^{2\gamma }}
{\displaystyle \Rightarrow \delta ^{2}=n^{2\gamma -1}}
{\displaystyle \Rightarrow \delta =n^{\gamma -{\frac {1}{2}}}}
Which inserted in the value of
{\displaystyle k}
{\displaystyle k=\delta n=n^{\gamma -{\frac {1}{2}}}n=n^{\gamma +{\frac {1}{2}}}}
which proves that there exists an explicit k-APRF extractor with the given properties.
{\displaystyle \Box }
Von Neumann extractor[edit]
Further information: Bernoulli sequence
Perhaps the earliest example is due to John von Neumann. From the input stream, his extractor took bits, two at a time (first and second, then third and fourth, and so on). If the two bits matched, no output was generated. If the bits differed, the value of the first bit was output. The Von Neumann extractor can be shown to produce a uniform output even if the distribution of input bits is not uniform so long as each bit has the same probability of being one and there is no correlation between successive bits.[8]
Thus, it takes as input a Bernoulli sequence with p not necessarily equal to 1/2, and outputs a Bernoulli sequence with
{\displaystyle p=1/2.}
More generally, it applies to any exchangeable sequence—it only relies on the fact that for any pair, 01 and 10 are equally likely: for independent trials, these have probabilities
{\displaystyle p\cdot (1-p)=(1-p)\cdot p}
, while for an exchangeable sequence the probability may be more complicated, but both are equally likely.
It is also possible to use a cryptographic hash function as a randomness extractor. However, not every hashing algorithm is suitable for this purpose.[citation needed]
^ "Extracting randomness from sampleable distributions". Portal.acm.org. Retrieved 2012-06-12.
^ David K. Gifford, Natural Random Numbers, MIT/LCS/TM-371, Massachusetts Institute of Technology, August 1988.
^ Luca Trevisan. "Extractors and Pseudorandom Generators" (PDF). Retrieved 2013-10-21.
^ Recommendation for the Entropy Sources Used for Random Bit Generation (draft) NIST SP800-90B, Barker and Kelsey, August 2012, Section 6.4.2
^ Ronen Shaltiel. Recent developments in explicit construction of extractors. P. 5.
^ Jesse Kamp and David Zuckerman. Deterministic Extractors for Bit-Fixing Sources and Exposure-Resilient Cryptography.,SIAM J. Comput.,Vol. 36, No. 5, pp. 1231–1247.
^ a b Jesse Kamp and David Zuckerman. Deterministic Extractors for Bit-Fixing Sources and Exposure-Resilient Cryptography. P. 1242.
^ John von Neumann. Various techniques used in connection with random digits. Applied Math Series, 12:36–38, 1951.
Randomness Extractors for Independent Sources and Applications, Anup Rao
Recent developments in explicit constructions of extractors, Ronen Shaltiel
Randomness Extraction and Key Derivation Using the CBC, Cascade and HMAC Modes, Yevgeniy Dodis et al.
Key Derivation and Randomness Extraction, Olivier Chevassut et al.
Deterministic Extractors for Bit-Fixing Sources and Exposure-Resilient Cryptography, Jesse Kamp and David Zuckerman
Tossing a Biased Coin (and the optimality of advanced multi-level strategy) (lecture notes), Michael Mitzenmacher
Retrieved from "https://en.wikipedia.org/w/index.php?title=Randomness_extractor&oldid=959313511#Von_Neumann_extractor"
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Measure DC performance metrics of ADC output - Simulink - MathWorks España
ADC DC Measurement
Start conversion frequency (Hz)
Hold off time (s)
Recommended min. simulation stop time (s)
Set as model stop time
Output result to base workspace
INL Error
DNL Error
Measure DC performance metrics of ADC output
Mixed-Signal Blockset / ADC / Measurements & Testbenches
The ADC DC Measurement block measures ADC DC performance metrics such as offset error, gain error, integral nonlinearity (INL), and differential nonlinearity (DNL). You can use ADC DC Measurement block to validate the ADC architectural models provided in Mixed-Signal Blockset™, or you can use an ADC of your own implementation.
analog — Analog input signal to ADC
Analog input signal to ADC block, specified as a scalar.
start — External conversion start clock
External conversion start clock, specified as a scalar. The analog to digital conversion process starts at the rising edge of the signal at the start port.
digital — Converted digital signal from ADC
Converted digital signal from an ADC, specified as a scalar.
ready — Indicates whether analog to digital conversion is complete
Indicates whether the analog to digital conversion is complete, specified as a scalar.
Number of bits — Number of physical bits in ADC
5 (default) | positive real integer
Number of physical bits in ADC, specified as a unitless positive real integer. Number of bits must match the resolution specified in the ADC block.
Start conversion frequency (Hz) — Frequency of the start conversion clock of ADC
Frequency of the start conversion clock of the ADC, specified as a positive real scalar in hertz. Start conversion frequency must match the frequency of the start conversion clock of the ADC block. This parameter is used to calculate Recommended simulation stop time.
Use get_param(gcb,'Frequency') to view the current value of Start conversion frequency.
Use set_param(gcb,'Frequency',value) to set Start conversion frequency to a specific value.
Input range (V) — Dynamic range of ADC
[-1 1] (default) | 2-element vector
Dynamic range of the ADC, specified as a 2-element vector in V. The two vector elements represent the minimum and maximum values of the dynamic range, from left to right.
Use get_param(gcb,'InputRange') to view the current value of Input range.
Use set_param(gcb,'InputRange',value) to set Input range to a specific value.
Hold off time (s) — Delays measurement analysis to avoid corruption by transients
Delays measurement analysis to avoid corruption by transients, specified as a nonnegative real scalar in seconds.
Use get_param(gcb,'HoldOffTime') to view the current value of Hold off time.
Use set_param(gcb,'HoldOffTime',value) to set Hold off time to a specific value.
Recommended min. simulation stop time (s) — Minimum time simulation must run for meaningful result
Minimum time the simulation must run to obtain meaningful results, specified as a positive real scalar in seconds.
For DC measurement, the simulation must run so that ADC can sample each digital code 10 times with the default error tolerance of 0.1, assuming a ramp input that traverses the full scale range of the ADC over the period of simulation. Based on this assumption, the analog input frequency (fanalog), generated by the ADC Testbench block for the sawtooth waveform is set as:
{f}_{\text{analog}}=\frac{StartFreq·ErrorTolerance}{{2}^{\left(Nbits+1\right)}}
where StartFreq is the frequency of the conversion start clock and Nbits is the resolution of the ADC.
So, the Recommended min. simulation stop time (s) (T) is calculated by using the formula:
T=\frac{1}{{f}_{\text{analog}}}+HoldOffTime
Set as model stop time — Automatically set recommended min. simulation stop time as model stop time
Click to automatically set the Recommended min. simulation stop time (s) as the stop time of the Simulink® model.
Endpoint — Measure DNL, INL using endpoint method
Measure the differential nonlinearity (DNL) error and integral nonlinearity (INL) error using the endpoint method. This method uses the end points of the actual transfer function to measure the DNL and INL error.
Best fit — Measure DNL, INL using best fit method
Measure the differential nonlinearity (DNL) error and integral nonlinearity (INL) error using the best fit method. This method uses a standard curve fitting technique to find the best fit to measure the DNL and INL error.
Output result to base workspace — Store detailed test results to base workspace
Store detailed test results to a struct in the base workspace for further processing. By default, this option is not selected.
Workspace variable name — Name of the variable that stores detailed test results
adc_dc_out (default) | character string
Name of the variable that stores detailed test results, specified as a character string.
This parameter is only available when Output result to base workspace is selected
Use get_param(gcb,'VariableName') to view the current value of Workspace variable name.
Use set_param(gcb,'VariableName',value) to set Workspace variable name to a specific value.
Plot — Plot measurement results
Click to plot measurement result for further analysis.
Offset error represents the offset of the ADC transfer function curve from it ideal value at a single point.
Gain error represents the deviation of the slope of the ADC transfer function curve from its ideal value.
Integral nonlinearity (INL) error, also termed as relative accuracy, is the maximum deviation of the measured transfer function from a straight line. The straight line is can be a best fit using standard curve fitting technique, or drawn between the end points of the actual transfer function after gain adjustment.
The best fit method gives a better prediction of distortion in AC applications, and a lower value of linearity error. The endpoint method is mostly used in measurement application of data converters, since the error budget depends on the actual deviation from ideal transfer function.
Differential nonlinearity (DNL) is the deviation from the ideal difference (1 LSB) between analog input levels that trigger any two successive digital output levels. The DNL error is the maximum value of DNL found at any transition.
SAR ADC | Flash ADC | ADC Testbench | ADC AC Measurement
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Linearity - Wikipedia
Property of a mathematical relationship that can be represented as a straight line
"Linear" redirects here. For other uses, see Linear (disambiguation).
Not to be confused with Lineage (disambiguation).
Find sources: "Linearity" – news · newspapers · books · scholar · JSTOR (December 2007) (Learn how and when to remove this template message)
Linearity is the property of a mathematical relationship (function) that can be graphically represented as a straight line. Linearity is closely related to proportionality. Examples in physics include rectilinear motion, the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships are nonlinear.
Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle.
The word linear comes from Latin linearis, "pertaining to or resembling a line".
1.1 Linear polynomials
3.1 Integral linearity
4 Military tactical formations
In mathematics, a linear map or linear function f(x) is a function that satisfies the two properties:[1]
Homogeneity of degree 1: f(αx) = α f(x) for all α.
These properties are known as the superposition principle. In this definition, x is not necessarily a real number, but can in general be an element of any vector space. A more special definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics (see below).
Additivity alone implies homogeneity for rational α, since
{\displaystyle f(x+x)=f(x)+f(x)}
{\displaystyle f(nx)=nf(x)}
for any natural number n by mathematical induction, and then
{\displaystyle nf(x)=f(nx)=f(m{\tfrac {n}{m}}x)=mf({\tfrac {n}{m}}x)}
{\displaystyle f({\tfrac {n}{m}}x)={\tfrac {n}{m}}f(x)}
. The density of the rational numbers in the reals implies that any additive continuous function is homogeneous for any real number α, and is therefore linear.
The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a differential operator, and other operators constructed from it, such as del and the Laplacian. When a differential equation can be expressed in linear form, it can generally be solved by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions.
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called 'linear spaces'), linear transformations (also called 'linear maps'), and systems of linear equations.
For a description of linear and nonlinear equations, see linear equation.
Linear polynomials[edit]
In a different usage to the above definition, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a straight line.[2]
Over the reals, a linear equation is one of the forms:
{\displaystyle f(x)=mx+b\ }
where m is often called the slope or gradient; b the y-intercept, which gives the point of intersection between the graph of the function and the y-axis.
Note that this usage of the term linear is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so if and only if b = 0. Hence, if b ≠ 0, the function is often called an affine function (see in greater generality affine transformation).
Boolean functions[edit]
Main article: Parity function
Hasse diagram of a linear Boolean function
In Boolean algebra, a linear function is a function
{\displaystyle f}or which there exist
{\displaystyle a_{0},a_{1},\ldots ,a_{n}\in \{0,1\}}
{\displaystyle f(b_{1},\ldots ,b_{n})=a_{0}\oplus (a_{1}\land b_{1})\oplus \cdots \oplus (a_{n}\land b_{n})}
{\displaystyle b_{1},\ldots ,b_{n}\in \{0,1\}.}
{\displaystyle a_{0}=1}
, the above function is considered affine in linear algebra (i.e. not linear).
A Boolean function is linear if one of the following holds for the function's truth table:
In every row in which the truth value of the function is T, there are an odd number of Ts assigned to the arguments, and in every row in which the function is F there is an even number of Ts assigned to arguments. Specifically, f(F, F, ..., F) = F, and these functions correspond to linear maps over the Boolean vector space.
In every row in which the value of the function is T, there is an even number of Ts assigned to the arguments of the function; and in every row in which the truth value of the function is F, there are an odd number of Ts assigned to arguments. In this case, f(F, F, ..., F) = T.
Another way to express this is that each variable always makes a difference in the truth value of the operation or it never makes a difference.
Negation, Logical biconditional, exclusive or, tautology, and contradiction are linear functions.
In physics, linearity is a property of the differential equations governing many systems; for instance, the Maxwell equations or the diffusion equation.[3]
Linearity of a homogenous differential equation means that if two functions f and g are solutions of the equation, then any linear combination af + bg is, too.
In instrumentation, linearity means that a given change in an input variable gives the same change in the output of the measurement apparatus: this is highly desirable in scientific work. In general, instruments are close to linear over a certain range, and most useful within that range. In contrast, human senses are highly nonlinear: for instance, the brain completely ignores incoming light unless it exceeds a certain absolute threshold number of photons.
In electronics, the linear operating region of a device, for example a transistor, is where an output dependent variable (such as the transistor collector current) is directly proportional to an input dependent variable (such as the base current). This ensures that an analog output is an accurate representation of an input, typically with higher amplitude (amplified). A typical example of linear equipment is a high fidelity audio amplifier, which must amplify a signal without changing its waveform. Others are linear filters, and linear amplifiers in general.
In most scientific and technological, as distinct from mathematical, applications, something may be described as linear if the characteristic is approximately but not exactly a straight line; and linearity may be valid only within a certain operating region—for example, a high-fidelity amplifier may distort a small signal, but sufficiently little to be acceptable (acceptable but imperfect linearity); and may distort very badly if the input exceeds a certain value.[4]
Integral linearity[edit]
Main article: Integral linearity
For an electronic device (or other physical device) that converts a quantity to another quantity, Bertram S. Kolts writes:[5][6]
There are three basic definitions for integral linearity in common use: independent linearity, zero-based linearity, and terminal, or end-point, linearity. In each case, linearity defines how well the device's actual performance across a specified operating range approximates a straight line. Linearity is usually measured in terms of a deviation, or non-linearity, from an ideal straight line and it is typically expressed in terms of percent of full scale, or in ppm (parts per million) of full scale. Typically, the straight line is obtained by performing a least-squares fit of the data. The three definitions vary in the manner in which the straight line is positioned relative to the actual device's performance. Also, all three of these definitions ignore any gain, or offset errors that may be present in the actual device's performance characteristics.
Military tactical formations[edit]
In military tactical formations, "linear formations" were adapted starting from phalanx-like formations of pike protected by handgunners, towards shallow formations of handgunners protected by progressively fewer pikes. This kind of formation got progressively thinner until its extreme in the age of Wellington's 'Thin Red Line'. It was eventually replaced by skirmish order when the invention of the breech-loading rifle allowed soldiers to move and fire in small, mobile units, unsupported by large-scale formations of any shape.
Linear is one of the five categories proposed by Swiss art historian Heinrich Wölfflin to distinguish "Classic", or Renaissance art, from the Baroque. According to Wölfflin, painters of the fifteenth and early sixteenth centuries (Leonardo da Vinci, Raphael or Albrecht Dürer) are more linear than "painterly" Baroque painters of the seventeenth century (Peter Paul Rubens, Rembrandt, and Velázquez) because they primarily use outline to create shape.[7] Linearity in art can also be referenced in digital art. For example, hypertext fiction can be an example of nonlinear narrative, but there are also websites designed to go in a specified, organized manner, following a linear path.
In music the linear aspect is succession, either intervals or melody, as opposed to simultaneity or the vertical aspect.
Linear A and Linear B scripts.
^ Edwards, Harold M. (1995). Linear Algebra. Springer. p. 78. ISBN 9780817637316.
^ Stewart, James (2008). Calculus: Early Transcendentals, 6th ed., Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8, Section 1.2
^ Evans, Lawrence C. (2010) [1998], Partial differential equations (PDF), Graduate Studies in Mathematics, vol. 19 (2nd ed.), Providence, R.I.: American Mathematical Society, doi:10.1090/gsm/019, ISBN 978-0-8218-4974-3, MR 2597943
^ Whitaker, Jerry C. (2002). The RF transmission systems handbook. CRC Press. ISBN 978-0-8493-0973-1.
^ Kolts, Bertram S. (2005). "Understanding Linearity and Monotonicity" (PDF). analogZONE. Archived from the original (PDF) on February 4, 2012. Retrieved September 24, 2014.
^ Kolts, Bertram S. (2005). "Understanding Linearity and Monotonicity". Foreign Electronic Measurement Technology. 24 (5): 30–31. Retrieved September 25, 2014.
^ Wölfflin, Heinrich (1950). Hottinger, M.D. (ed.). Principles of Art History: The Problem of the Development of Style in Later Art. New York: Dover. pp. 18–72.
The dictionary definition of linearity at Wiktionary
Retrieved from "https://en.wikipedia.org/w/index.php?title=Linearity&oldid=1066943786"
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Priapism is prolonged abnormal penile erection that lasts over 4 h without sexual or libido stimulation [1, 2], occurring in men of all ages. The etiologies include trauma, leukemia, sickle cell anemia, tumors, and drugs including tamsulosin and phentolamine [3]. Ischemic priapism, often leads to serious complications, such as erectile dysfunction, ischemic necrosis or fibrosis of the penile corpus cavernosum, and penile deformity. Penile metastases are clinically uncommon, and usually the tumors originate in the bladder, prostate, colon, rectum, or kidney. Priapism secondary to colon adenocarcinoma metastasis occurs less frequently [4].
This article reports a case study of malignant priapism caused by colon cancer metastasis, which was a disseminated manifestation of colon cancer. However, the prognosis of this kind of metastasis is very poor, regardless of the therapy, as it typically occurs in end-stage cancer.
A 61-year-old man was admitted to our clinic with abnormal penile erection and hematuria persisting for over 30 days. Two months earlier, the patient had undergone laparoscopic resection of the primary sigmoid colon cancer in another hospital, supplemented by two cycles of intravenous chemotherapy with the oxaliplatin + capecitabine (XELOX) regimen. Postoperative pathology showed a low to moderately differentiated adenocarcinoma (Fig. 1A), a pathological T4 tumor with liver and lymphatic multiple metastases. One month earlier, the patient experienced persistent abnormal penile erection and hematuria, followed by progressive aggravation.
Histopathological results. (A) Pathological analysis of the primary colon tumor showed a low to moderately differentiated adenocarcinoma that infiltrated the whole colon wall, along with 4 lymphatic metastases (hematoxylin and eosin stain, original magnification 400
×
). (B) Transurethral cystoscopy biopsy showed nests of acinar-like cells with cytological atypia, consistent with metastatic adenocarcinoma (hematoxylin and eosin stain, original magnification 200
×
The patient did not have a history of perineal trauma, nervous system disease, or hematological system disease. Physical examination showed obvious swelling of the penis, with a hard texture, poor elasticity, slightly higher skin temperature, and ruddy color, with moderate to severe tenderness. Two painless nodules with obscure boundaries of approximately 0.5 cm
×
0.5 cm were found at the root of the penis.
Abdominopelvic computed tomography (CT) enhancement images showed a localized irregular shape and high-density imaging of the root of the corpus cavernosum (Fig. 2A,B); penile Doppler ultrasound showed no obvious blood flow signal; and penile arterial blood gas parameters were pH of 7.01, partial pressure of oxygen (PO
{}_{2}
) of 26 mmHg and partial pressure of carbon dioxide (PCO
{}_{2}
) of 71 mmHg. Transurethral cystoscopy was performed under combined spinal and epidural analgesia (CSEA), and an extensive cauliflower-like neoformation, covering the posterior urethra and prostate, with massive tortuous blood vessels attached, was observed (Fig. 3A,B). Histopathological examination of the neoplasm showed nests of acinar-like cells with cytological atypia (Fig. 1B), consistent with the primary colon adenocarcinoma.
Abdominopelvic computed tomography results. (A,B) Abdominopelvic computed tomography enhancement images showed a localized irregular shape and high-density imaging of the root of the corpus cavernosum (indicated by arrows).
Transurethral cystoscopy. (A,B) Transurethral cystoscopy showed an extensive cauliflower-like neoformation covering the posterior urethra and prostate with massive tortuous blood vessels attached.
Despite the administration of ibuprofen codeine-sustained tablets and intramuscular injection of tramadol hydrochloride for analgesia, as well as the oral administration of Progynova tablets to decrease the erection, there was no significant improvement in the state of pain of the penis or the persistent erection. The patient and his family refused cavernous shunt or palliative partial or total penectomy; thus, superselective embolization of the internal pudendal artery was performed (Fig. 4A,B). The penile texture was improved, and penile tenderness was slightly relieved after the therapy. Despite our treatment, the patient passed away because of cachexia 2 months after being discharged from the hospital.
Superselective embolization of the internal pudendal artery. (A) Selective arteriography of the internal pudendal artery showed abnormal blood perfusion and local aggregation of contrast medium (indicated by arrow). (B) After superselective embolization of the internal pudendal artery with polyvinyl alcohol particles (indicated by arrow), the abnormal blood perfusion had disappeared.
Penile metastatic cancer is relatively rare and usually occurs in the context of more widespread disseminated disease [5]. Penile metastases arise more frequently from genitourinary cancers, mostly from bladder or prostate cancer [4, 6]. However, penile metastasis from colon adenocarcinoma occurs less often and has a poor prognosis [7]. Penile metastases have mostly been reported in case reports or small case series [5, 8, 9], which makes clinical doctors less familiar with these metastases.
Priapism can be classified into three main subtypes: ischemic (low-flow), nonischemic (high-flow), and stuttering priapism [6]. Ischemic priapism is a persistent penile erection lasting more than 4 h, and marked by rigidity of the corpora cavernosa with little or no cavernous arterial inflow and blood gas analysis parameters of PO
{}_{2}
<
30 mmHg, PCO
{}_{2}
60 mmHg, and pH
<
7.25 [10]. This type of priapism often leads to erectile dysfunction if not treated on time. Nonischemic priapism is a constant erection caused by unregulated cavernous arterial inflow and seldom presents with haphalgesia [11]. Stuttering priapism is characterized by periodic self-limited erection with obvious pain, and the erection time is usually
<
4 h, but it can develop into ischemic priapism if not treated on time [12].
In this case, this patient was considered to have ischemic priapism as evidenced by the clinical findings. Because the ischemic time was more than 48 h, fibrillation of the corpus cavernosum occurred [6] and fibrillation were exacerbated as the ischemic time increased. Due to diverse treatment modalities and prognostic factors [2, 13], it is important to consider differential diagnosis among the three types of priapism. In most patients, the symptoms determine the diagnosis [2]. These situations impair the effectiveness of treatment to a certain degree; therefore, we strongly recommend early diagnosis and noninvasive treatment.
Prattley et al. [14] reported the use of superselective embolization with Microcoil and Gelfoam for non-ischemic priapism, achieving positive outcomes that effectively alleviated priapism without causing erectile dysfunction. In our case, after the use of superselective embolization of the internal pudendal artery, better results were achieved compared to Progynova tablets. This is the first report of superselective embolization to treat ischemic priapism secondary to metastatic colon adenocarcinoma, providing clinicians evidence of the utilization of non-invasive treatment.
Priapism secondary to metastatic colon adenocarcinoma is very rare and usually indicates dissemination of primary tumor to multiple organs, which is mostly found in aged patients. The main symptoms may be penile nodule, malignant priapism, penile pain and swelling, and difficulty in urination [1, 15]. Patients with metastatic penile cancer have already developed systemic dissemination, which has a poor prognosis and a median survival time of usually less than 6 months [8]. It requires timely detection, precise diagnosis, and non-invasive treatment to improve the quality of life.
Conceptualization—CGG. Data curation—SDW, CGG and JYZ. Investigation—SDW. Methodology—JYZ. Resources—CGG. Writing—original draft—SDW. Writing—review & editing—CGG and JYZ.
Informed consent was obtained from the patient’s son for publication of this case report and any accompanying images. The written consent was obtained from the patient’s son because the patient died when this report been written.
The authors would like to thank all working group members for their contribution to this study. And thanks to all the peer reviewers for their opinions and suggestions.
This research was funded by the Kuanren Talents Program of Chongqing Medical University, Chongqing, China, grant number KY2019Y026.
Offenbacher J, Barbera A. Penile Emergencies. Emergency Medicine Clinics of North America. 2019; 37: 583–592.
Mishra K, Loeb A, Bukavina L, Baumgarten A, Beilan J, Mendez M, et al. Management of Priapism: a Contemporary Review. Sexual Medicine Reviews. 2020; 8: 131–139.
Prabhuswamy VK, Krishnappa P, Tyagaraj K. Malignant refractory priapism: an urologist’s nightmare. Urology Annals. 2019; 11: 222–225.
Chaux A, Amin M, Cubilla AL, Young RH. Metastatic Tumors to the Penis. International Journal of Surgical Pathology. 2011; 19: 597–606.
Cocci A, Hakenberg OW, Cai T, Nesi G, Livi L, Detti B, et al. Prognosis of men with penile metastasis and malignant priapism: a systematic review. Oncotarget. 2018; 9: 2923–2930.
Muneer A, Ralph D. Guideline of guidelines: priapism. BJU International. 2017; 119: 204–208.
Park JC, Lee WH, Kang MK, Park SY. Priapism secondary to penile metastasis of rectal cancer. World Journal of Gastroenterology. 2009; 15: 4209–4211.
Pereira R, Perera M, Rhee H. Metastatic plasmacytoid bladder cancer causing malignant priapism. BMJ Case Reports. 2019; 12: e228088.
Salonia A, Eardley I, Giuliano F, Hatzichristou D, Moncada I, Vardi Y, et al. European Association of Urology Guidelines on Priapism. European Urology. 2014; 65: 480–489.
Zhao H, Dallas K, Masterson J, Lo E, Houman J, Berdahl C, et al. Risk Factors for Surgical Shunting in a Large Cohort with Ischemic Priapism. Journal of Sexual Medicine. 2020; 17: 2472–2477.
Broderick GA, Kadioglu A, Bivalacqua TJ, Ghanem H, Nehra A, Shamloul R. Priapism: pathogenesis, epidemiology, and management. Journal of Sexual Medicine. 2010; 7: 476–500.
Johnson MJ, McNeillis V, Chiriaco G, Ralph DJ. Rare Disorders of Painful Erection: a Cohort Study of the Investigation and Management of Stuttering Priapism and Sleep-Related Painful Erection. Journal of Sexual Medicine. 2021; 18: 376–384.
Silberman M, Stormont G, Hu EW. Priapism. Treasure Island (FL): StatPearls Publishing. 2021.
Prattley S, Bryant T, Rees R. Superselective Embolization with Microcoil and Gelfoam for High-Flow Priapism Secondary to Bilateral Cavernous Fistulae: a Case Study. Case Reports in Urology. 2019; 2019: 1–4.
Shigehara K, Namiki M. Clinical Management of Priapism: a Review. World Journal of Men’s Health. 2016; 34: 1–8.
Histopathological results. (A) Pathological analysis of the primary colon tumor showed a low to moderately differentiated adenocarcinoma that infiltrated the whole colon wall, along with 4 lymphatic metastases (hematoxylin and eosin stain, original magnification 400). (B) Transurethral cystoscopy biopsy showed nests of acinar-like cells with cytological atypia, consistent with metastatic adenocarcinoma (hematoxylin and eosin stain, original magnification 200).
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Cardinality Knowpia
In mathematics, the cardinality of a set is a measure of the "number of elements" of the set. For example, the set
{\displaystyle A=\{2,4,6\}}
contains 3 elements, and therefore
{\displaystyle A}
has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers.[1] The cardinality of a set is also called its size, when no confusion with other notions of size[2] is possible.
{\displaystyle S}
of all Platonic solids has 5 elements. Thus
{\displaystyle |S|=5}
{\displaystyle A}
is usually denoted
{\displaystyle |A|}
, with a vertical bar on each side;[3] this is the same notation as absolute value, and the meaning depends on context. The cardinality of a set
{\displaystyle A}
may alternatively be denoted by
{\displaystyle n(A)}
{\displaystyle A}
{\displaystyle \operatorname {card} (A)}
{\displaystyle \#A}
Comparing setsEdit
Bijective function from N to the set E of even numbers. Although E is a proper subset of N, both sets have the same cardinality.
N does not have the same cardinality as its power set P(N): For every function f from N to P(N), the set T = {n∈N: n∉f(n)} disagrees with every set in the range of f, hence f cannot be surjective. The picture shows an example f and the corresponding T; red: n∈f(n)\T, blue:n∈T\f(n).
Definition 1: |A| = |B|Edit
Two sets A and B have the same cardinality if there exists a bijection (a.k.a., one-to-one correspondence) from A to B,[9] that is, a function from A to B that is both injective and surjective. Such sets are said to be equipotent, equipollent, or equinumerous. This relationship can also be denoted A ≈ B or A ~ B.
For example, the set E = {0, 2, 4, 6, ...} of non-negative even numbers has the same cardinality as the set N = {0, 1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E (see picture).
For finite sets A and B, if some bijection exists from A to B, then each injective or surjective function from A to B is a bijection. This is no longer true for infinite A and B. For example, the function g from N to E, defined by g(n) = 4n is injective, but not surjective, and h from N to E, defined by h(n) = n - (n mod 2) is surjective, but not injective. Neither g nor h can challenge |E| = |N|, which was established by the existence of f.
Definition 2: |A| ≤ |B|Edit
A has cardinality less than or equal to the cardinality of B, if there exists an injective function from A into B.
Definition 3: |A| < |B|Edit
A has cardinality strictly less than the cardinality of B, if there is an injective function, but no bijective function, from A to B.
For example, the set N of all natural numbers has cardinality strictly less than its power set P(N), because g(n) = { n } is an injective function from N to P(N), and it can be shown that no function from N to P(N) can be bijective (see picture). By a similar argument, N has cardinality strictly less than the cardinality of the set R of all real numbers. For proofs, see Cantor's diagonal argument or Cantor's first uncountability proof.
The cardinality of a set A is defined as its equivalence class under equinumerosity.
A representative set is designated for each equivalence class. The most common choice is the initial ordinal in that class. This is usually taken as the definition of cardinal number in axiomatic set theory.
{\displaystyle \aleph _{0}<\aleph _{1}<\aleph _{2}<\ldots .}
For each ordinal
{\displaystyle \alpha }
{\displaystyle \aleph _{\alpha +1}}
is the least cardinal number greater than
{\displaystyle \aleph _{\alpha }}
The cardinality of the natural numbers is denoted aleph-null (
{\displaystyle \aleph _{0}}
), while the cardinality of the real numbers is denoted by "
{\displaystyle {\mathfrak {c}}}
" (a lowercase fraktur script "c"), and is also referred to as the cardinality of the continuum. Cantor showed, using the diagonal argument, that
{\displaystyle {\mathfrak {c}}>\aleph _{0}}
. We can show that
{\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}}
, this also being the cardinality of the set of all subsets of the natural numbers.
The continuum hypothesis says that
{\displaystyle \aleph _{1}=2^{\aleph _{0}}}
{\displaystyle 2^{\aleph _{0}}}
is the smallest cardinal number bigger than
{\displaystyle \aleph _{0}}
, i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis is independent of ZFC, a standard axiomatization of set theory; that is, it is impossible to prove the continuum hypothesis or its negation from ZFC—provided that ZFC is consistent. For more detail, see § Cardinality of the continuum below.[12][13][14]
Finite, countable and uncountable setsEdit
Any set X with cardinality less than that of the natural numbers, or | X | < | N |, is said to be a finite set.
Any set X that has the same cardinality as the set of the natural numbers, or | X | = | N | =
{\displaystyle \aleph _{0}}
, is said to be a countably infinite set.[9]
Any set X with cardinality greater than that of the natural numbers, or | X | > | N |, for example | R | =
{\displaystyle {\mathfrak {c}}}
> | N |, is said to be uncountable.
Our intuition gained from finite sets breaks down when dealing with infinite sets. In the late nineteenth century Georg Cantor, Gottlob Frege, Richard Dedekind and others rejected the view that the whole cannot be the same size as the part.[15][citation needed] One example of this is Hilbert's paradox of the Grand Hotel. Indeed, Dedekind defined an infinite set as one that can be placed into a one-to-one correspondence with a strict subset (that is, having the same size in Cantor's sense); this notion of infinity is called Dedekind infinite. Cantor introduced the cardinal numbers, and showed—according to his bijection-based definition of size—that some infinite sets are greater than others. The smallest infinite cardinality is that of the natural numbers (
{\displaystyle \aleph _{0}}
Cardinality of the continuumEdit
One of Cantor's most important results was that the cardinality of the continuum (
{\displaystyle {\mathfrak {c}}}
) is greater than that of the natural numbers (
{\displaystyle \aleph _{0}}
); that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that
{\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}=\beth _{1}}
(see Beth one) satisfies:
{\displaystyle 2^{\aleph _{0}}>\aleph _{0}}
(see Cantor's diagonal argument or Cantor's first uncountability proof).
{\displaystyle 2^{\aleph _{0}}=\aleph _{1}}
Cantor also showed that sets with cardinality strictly greater than
{\displaystyle {\mathfrak {c}}}
exist (see his generalized diagonal argument and theorem). They include, for instance:
the set of all subsets of R, i.e., the power set of R, written P(R) or 2R
the set RR of all functions from R to R
Both have cardinality
{\displaystyle 2^{\mathfrak {c}}=\beth _{2}>{\mathfrak {c}}}
(see Beth two).
The cardinal equalities
{\displaystyle {\mathfrak {c}}^{2}={\mathfrak {c}},}
{\displaystyle {\mathfrak {c}}^{\aleph _{0}}={\mathfrak {c}},}
{\displaystyle {\mathfrak {c}}^{\mathfrak {c}}=2^{\mathfrak {c}}}
can be demonstrated using cardinal arithmetic:
{\displaystyle {\mathfrak {c}}^{2}=\left(2^{\aleph _{0}}\right)^{2}=2^{2\times {\aleph _{0}}}=2^{\aleph _{0}}={\mathfrak {c}},}
{\displaystyle {\mathfrak {c}}^{\aleph _{0}}=\left(2^{\aleph _{0}}\right)^{\aleph _{0}}=2^{{\aleph _{0}}\times {\aleph _{0}}}=2^{\aleph _{0}}={\mathfrak {c}},}
{\displaystyle {\mathfrak {c}}^{\mathfrak {c}}=\left(2^{\aleph _{0}}\right)^{\mathfrak {c}}=2^{{\mathfrak {c}}\times \aleph _{0}}=2^{\mathfrak {c}}.}
If X = {a, b, c} and Y = {apples, oranges, peaches}, then | X | = | Y | because { (a, apples), (b, oranges), (c, peaches)} is a bijection between the sets X and Y. The cardinality of each of X and Y is 3.
If | X | ≤ | Y |, then there exists Z such that | X | = | Z | and Z ⊆ Y.
If | X | ≤ | Y | and | Y | ≤ | X |, then | X | = | Y |. This holds even for infinite cardinals, and is known as Cantor–Bernstein–Schroeder theorem.
Sets with cardinality of the continuum include the set of all real numbers, the set of all irrational numbers and the interval
{\displaystyle [0,1]}
Union and intersectionEdit
If A and B are disjoint sets, then
{\displaystyle \left\vert A\cup B\right\vert =\left\vert A\right\vert +\left\vert B\right\vert .}
{\displaystyle \left\vert C\cup D\right\vert +\left\vert C\cap D\right\vert =\left\vert C\right\vert +\left\vert D\right\vert .}
Wikimedia Commons has media related to Cardinality.
group cardinality (P1164) (see uses)
cardinality of this set (P2820) (see uses)
^ Such as length and area in geometry. – A line of finite length is a set of points that has infinite cardinality.
^ "Cardinality | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-23.
^ "Early Human Counting Tools". Math Timeline. Retrieved 2018-04-26.
^ Allen, Donald (2003). "The History of Infinity" (PDF). Texas A&M Mathematics. Retrieved Nov 15, 2019.
^ a b "Infinite Sets and Cardinality". Mathematics LibreTexts. 2019-12-05. Retrieved 2020-08-23.
^ Friedrich M. Hartogs (1915), Felix Klein; Walther von Dyck; David Hilbert; Otto Blumenthal (eds.), "Über das Problem der Wohlordnung", Mathematische Annalen, Leipzig: B. G. Teubner, 76 (4): 438–443, doi:10.1007/bf01458215, ISSN 0025-5831, S2CID 121598654
^ Felix Hausdorff (2002), Egbert Brieskorn; Srishti D. Chatterji; et al. (eds.), Grundzüge der Mengenlehre (1. ed.), Berlin/Heidelberg: Springer, p. 587, ISBN 3-540-42224-2 - Original edition (1914)
^ Cohen, Paul J. (December 15, 1963). "The Independence of the Continuum Hypothesis". Proceedings of the National Academy of Sciences of the United States of America. 50 (6): 1143–1148. Bibcode:1963PNAS...50.1143C. doi:10.1073/pnas.50.6.1143. JSTOR 71858. PMC 221287. PMID 16578557.
^ Cohen, Paul J. (January 15, 1964). "The Independence of the Continuum Hypothesis, II". Proceedings of the National Academy of Sciences of the United States of America. 51 (1): 105–110. Bibcode:1964PNAS...51..105C. doi:10.1073/pnas.51.1.105. JSTOR 72252. PMC 300611. PMID 16591132.
^ Penrose, R (2005), The Road to Reality: A Complete guide to the Laws of the Universe, Vintage Books, ISBN 0-09-944068-7
^ Georg Cantor (1887), "Mitteilungen zur Lehre vom Transfiniten", Zeitschrift für Philosophie und philosophische Kritik, 91: 81–125
Reprinted in: Georg Cantor (1932), Adolf Fraenkel (Lebenslauf); Ernst Zermelo (eds.), Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Berlin: Springer, pp. 378–439 Here: p.413 bottom
^ Applied Abstract Algebra, K.H. Kim, F.W. Roush, Ellis Horwood Series, 1983, ISBN 0-85312-612-7 (student edition), ISBN 0-85312-563-5 (library edition)
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