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Duke Math. J. 171 (4), (15 March 2022)
Diophantine equations in primes: Density of prime points on affine hypersurfaces
Duke Math. J. 171 (4), 831-884, (15 March 2022) DOI: 10.1215/00127094-2021-0023
KEYWORDS: Hardy–Littlewood circle method, Diophantine equations, primes, 11D45, 11D72, 11P32, 11P55
F\in \mathbb{Z}\left[{x}_{1},\dots ,{x}_{n}\right]
be a homogeneous form of degree
d\ge 2
{V}_{F}^{\ast }
denote the singular locus of the affine variety
V\left(F\right)=\left\{\mathbf{z}\in {\mathbb{C}}^{n}:F\left(\mathbf{z}\right)=0\right\}
. In this paper, we prove the existence of integer solutions with prime coordinates to the equation
F\left({x}_{1},\dots ,{x}_{n}\right)=0
provided that F satisfies suitable local conditions and
n-dim{V}_{F}^{\ast }\ge {2}^{8}{3}^{4}{5}^{2}{d}^{3}{\left(2d-1\right)}^{2}{4}^{d}
. Our result improves on what was known previously due to Cook and Magyar, which required
n-dim{V}_{F}^{\ast }
to be an exponential tower in d.
Khovanov homology detects the trefoils
KEYWORDS: Khovanov homology, instanton Floer homology, contact geometry, trefoil, 57M27, 57R17, 57R58
We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a bypass exact triangle in sutured instanton homology, proved here; and Kronheimer and Mrowka’s spectral sequence relating Khovanov homology with singular instanton knot homology. As a byproduct, we also strengthen a result of Kronheimer and Mrowka on
\mathrm{SU}\left(2\right)
representations of the knot group.
The extremals of Minkowski’s quadratic inequality
Duke Math. J. 171 (4), 957-1027, (15 March 2022) DOI: 10.1215/00127094-2021-0033
KEYWORDS: mixed volumes, Minkowski’s quadratic inequality, Alexandrov–Fenchel inequality, extremum problems, convex geometry, 52A39, 52A40, 58J50
In a seminal paper “Volumen und Oberfläche” (1903), Minkowski introduced the basic notion of mixed volumes and the corresponding inequalities that lie at the heart of convex geometry. The fundamental importance of characterizing the extremals of these inequalities was already emphasized by Minkowski himself, but has to date only been resolved in special cases. In this paper, we completely settle the extremals of Minkowski’s quadratic inequality, confirming a conjecture of R. Schneider. Our proof is based on the representation of mixed volumes of arbitrary convex bodies as Dirichlet forms associated to certain highly degenerate elliptic operators. A key ingredient of the proof is a quantitative rigidity property associated to these operators.
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On the structure of the group of Lipschitz homeomorphisms and its subgroups
July, 2001 On the structure of the group of Lipschitz homeomorphisms and its subgroups
Kōjun ABE, Kazuhiko FUKUI
We consider the group of Lipschitz homeomorphisms of a Lipschitz manifold and its subgroups. First we study properties of Lipschitz homeomorphisms and show the local contractibility and the perfectness of the group of Lipschitz homeomorphisms. Next using this result we can prove that the identity component of the group of equivariant Lipschitz homeomorphisms of a principal
G
-bundle over a closed Lipschitz manifold is perfect when
G
is a compact Lie group.
Kōjun ABE. Kazuhiko FUKUI. "On the structure of the group of Lipschitz homeomorphisms and its subgroups." J. Math. Soc. Japan 53 (3) 501 - 511, July, 2001. https://doi.org/10.2969/jmsj/05330501
Keywords: commutator , Lie group , Lipschitz homeomorphisms , Perfect , principal G-bundle
Kōjun ABE, Kazuhiko FUKUI "On the structure of the group of Lipschitz homeomorphisms and its subgroups," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 53(3), 501-511, (July, 2001)
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Installation and measurements of earth electrodes - Electrical Installation Guide
Earthing schemes
Definition of standardised earthing schemes
Characteristics of TT, TN and IT systems
Selection criteria for the TT, TN and IT systems
Choice of earthing method - implementation
Cables and busways
Harmonic currents in the selection of busbar trunking systems (busways)
External influences (IEC 60364-5-51)
Definition and reference standards
List of external influences
Protection provided for enclosed equipment: codes IP and IK
HomeLV DistributionEarthing schemesInstallation and measurements of earth electrodes
2 Influence of the type of soil
3 Measurement and constancy of the resistance between an earth electrode and the earth
4 Measurement of the earth-electrode resistance
A very effective method of obtaining a low-resistance earth connection is to bury a conductor in the form of a closed loop in the soil at the bottom of the excavation for building foundations.
The resistance R of such an electrode (in homogeneous soil) is given (approximately) in ohms by:
{\displaystyle {\mbox{R}}={\frac {2\rho }{\mbox{L}}}}
L = length of the buried conductor in metres
ρ = soil resistivity in ohm-metres
The quality of an earth electrode (resistance as low as possible) depends essentially on two factors:
Three common types of installation will be discussed:
Buried ring
(see Fig. E20)
This solution is strongly recommended, particularly in the case of a new building.
The electrode should be buried around the perimeter of the excavation made for the foundations. It is important that the bare conductor be in intimate contact with the soil (and not placed in the gravel or aggregate hard-core, often forming a base for concrete). At least four (widely-spaced) vertically arranged conductors from the electrode should be provided for the installation connections and, where possible, any reinforcing rods in concrete work should be connected to the electrode.
The conductor forming the earth electrode, particularly when it is laid in an excavation for foundations, must be in the earth, at least 50 cm below the hard-core or aggregate base for the concrete foundation. Neither the electrode nor the vertical rising conductors to the ground floor, should ever be in contact with the foundation concrete.
For existing buildings, the electrode conductor should be buried around the outside wall of the premises to a depth of at least 1 metre. As a general rule, all vertical connections from an electrode to above-ground level should be insulated for the nominal LV voltage (600-1,000 V).
The conductors may be:
Copper: Bare cable ( ≥ 25 mm2) or multiple-strip (≥ 25 mm2) and (≥ 2 mm thick)
Aluminium with lead jacket: Cable ( ≥ 35 mm2)
Galvanised-steel cable: Bare cable ( ≥ 95 mm2) or multiple-strip ( ≥ 100 mm2 and ≥ 3 mm thick)
The approximate resistance R of the electrode in ohms:
{\displaystyle {\mbox{R}}={\frac {2\rho }{\mbox{L}}}}
ρ = resistivity of the soil in ohm-metres (see Influence of the type of soil )
Fig. E20 – Conductor buried below the level of the foundations, i.e. not in the concrete
For n rods:
{\displaystyle {\mbox{R}}={\frac {1}{\mbox{n}}}\ {\frac {\rho }{\mbox{L}}}}
Vertically driven earthing rods are often used for existing buildings, and for improving (i.e. reducing the resistance of) existing earth electrodes.
The rods may be:
Copper or (more commonly) copper-clad steel. The latter are generally 1 or 2 metres long and provided with screwed ends and sockets in order to reach considerable depths, if necessary (for instance, the water-table level in areas of high soil resistivity)
Galvanised[1] steel pipe ≥ 25 mm diameter or rod ≥ 15 mm diameter, ≥ 2 metres long in each case.
Fig. E21 – Earthing rods connected in parallel
It is often necessary to use more than one rod, in which case the spacing between them should exceed the depth to which they are driven, by a factor of 2 to 3.
The total resistance (in homogeneous soil) is then equal to the resistance of one rod, divided by the number of rods in question.
The approximate resistance R obtained is:
{\displaystyle {\mbox{R}}={\frac {1}{\mbox{n}}}\ {\frac {\rho }{\mbox{L}}}}
if the distance separating the rods > 4L
L = the length of the rod in metres
ρ = resistivity of the soil in ohm-metres (see Influence of the type of soil)
n = the number of rods
For a vertical plate electrode:
{\displaystyle {\mbox{R}}={\frac {0.8\ \rho }{\mbox{L}}}}
Rectangular plates, each side of which must be ≥ 0.5 metres, are commonly used as earth electrodes, being buried in a vertical plane such that the centre of the plate is at least 1 metre below the surface of the soil.
The plates may be:
Copper of 2 mm thickness
Galvanised[1] steel of 3 mm thickness
The resistance R in ohms is given (approximately), by:
{\displaystyle {\mbox{R}}={\frac {0.8\ \rho }{\mbox{L}}}}
L = the perimeter of the plate in metres
Fig. E22 – Vertical plate - 2 mm thickness (Cu)
Influence of the type of soil
Measurements on earth electrodes in similar soils are useful to determine the resistivity value to be applied for the design of an earth-electrode system
Fig. E23 – Resistivity (Ωm) for different types of soil
Mean value of resistivity in Ωm
Swampy soil, bogs 1 - 30
Silt alluvium 20 - 100
Humus, leaf mould 10 - 150
Peat, turf 5 - 100
Soft clay 50
Marl and compacted clay 100 - 200
Jurassic marl 30 - 40
Clayey sand 50 - 500
Siliceous sand 200 - 300
Stoney ground 1,500 - 3,000
Grass-covered-stoney sub-soil 300 - 500
Chalky soil 100 - 300
Limestone 1,000 - 5,000
Fissured limestone 500 - 1,000
Schist, shale 50 - 300
Granite and sandstone 1,500 - 10,000
Modified granite and sandstone 100 - 600
Fig. E24 – Average resistivity values (Ωm) for an approximate sizing of earth-electrode
Average value of resistivity in Ωm
Fertile soil, compacted damp fill 50
Arid soil, gravel, uncompacted non-uniform fill 500
Stoney soil, bare, dry sand, fissured rocks 3000
Measurement and constancy of the resistance between an earth electrode and the earth
The resistance of the electrode/earth interface rarely remains constant
Among the principal factors affecting this resistance are the following:
Humidity of the soil
The seasonal changes in the moisture content of the soil can be significant at depths of up to 2 meters.
At a depth of 1 metre the resistivity and therefore the resistance can vary by a ratio of 1 to 3 between a wet winter and a dry summer in temperate regions
Frozen earth can increase the resistivity of the soil by several orders of magnitude. This is one reason for recommending the installation of deep electrodes, in particular in cold climates
The materials used for electrodes will generally deteriorate to some extent for various reasons, for example:
Chemical reactions (in acidic or alkaline soils)
Galvanic: due to stray DC currents in the earth, for example from electric railways, etc. or due to dissimilar metals forming primary cells. Different soils acting on sections of the same conductor can also form cathodic and anodic areas with consequent loss of surface metal from the latter areas. Unfortunately, the most favourable conditions for low earth-electrode resistance (i.e. low soil resistivity) are also those in which galvanic currents can most easily flow.
Brazed and welded joints and connections are the points most sensitive to oxidation. Thorough cleaning of a newly made joint or connection and wrapping with a suitable greased-tape binding is a commonly used preventive measure.
Measurement of the earth-electrode resistance
There must always be one or more removable links to isolate an earth electrode so that it can be tested.
There must always be removable links which allow the earth electrode to be isolated from the installation, so that periodic tests of the earthing resistance can be carried out. To make such tests, two auxiliary electrodes are required, each consisting of a vertically driven rod.
Ammeter method (see Fig. E25)
Fig. E25 – Measurement of the resistance to earth of the earth electrode of an installation by means of an ammeter
{\displaystyle A=R_{T}+{R_{t1}}={\frac {U_{Tt1}}{i_{1}}}}
{\displaystyle B=R_{t1}+R_{t2}={\frac {U_{t1t2}}{i_{2}}}}
{\displaystyle C=R_{t2}+R_{T}={\frac {U_{t2T}}{i_{3}}}}
When the source voltage U is constant (adjusted to be the same value for each test) then:
{\displaystyle R_{T}={\frac {U}{2}}\left({\frac {1}{i_{1}}}+{\frac {1}{i_{3}}}-{\frac {1}{i_{2}}}\right)}
In order to avoid errors due to stray earth currents (galvanic -DC- or leakage currents from power and communication networks and so on) the test current should be AC, but at a different frequency to that of the power system or any of its harmonics. Instruments using hand-driven generators to make these measurements usually produce an AC voltage at a frequency of between 85 Hz and 135 Hz.
The distances between the electrodes are not critical and may be in different directions from the electrode being tested, according to site conditions. A number of tests at different spacings and directions are generally made to cross-check the test results.
Use of a direct-reading earthing-resistance ohmmeter
These instruments use a hand-driven or electronic-type AC generator, together with two auxiliary electrodes, the spacing of which must be such that the zone of influence of the electrode being tested should not overlap that of the test electrode (C). The test electrode (C) furthest from the electrode (X) under test, passes a current through the earth and the electrode under test, while the second test electrode (P) picks up a voltage. This voltage, measured between (X) and (P), is due to the test current and is a measure of the contact resistance (of the electrode under test) with earth. It is clear that the distance (X) to (P) must be carefully chosen to give accurate results. If the distance (X) to (C) is increased, however, the zones of resistance of electrodes (X) and (C) become more remote, one from the other, and the curve of potential (voltage) becomes more nearly horizontal about the point (O).
In practical tests, therefore, the distance (X) to (C) is increased until readings taken with electrode (P) at three different points, i.e. at (P) and at approximately 5 metres on either side of (P), give similar values. The distance (X) to (P) is generally about 0.68 of the distance (X) to (C).
Fig. E26 – Measurement of the resistance to the mass of earth of electrode (X) using an earth-electrode-testing ohmmeter
[a] the principle of measurement is based on assumed homogeneous soil conditions. Where the zones of influence of electrodes C and X overlap, the location of test electrode P is difficult to determine for satisfactory results.
[b] showing the effect on the potential gradient when (X) and (C) are widely spaced. The location of test electrode P is not critical and can be easily determined.
^ 1 2 Where galvanised conducting materials are used for earth electrodes, sacrificial cathodic protection anodes may be necessary to avoid rapid corrosion of the electrodes where the soil is aggressive. Specially prepared magnesium anodes (in a porous sack filled with a suitable “soil”) are available for direct connection to the electrodes. In such circumstances, a specialist should be consulted
Retrieved from "http://www.electrical-installation.org/enw/index.php?title=Installation_and_measurements_of_earth_electrodes&oldid=27574"
Chapter - LV Distribution
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MasseyProduct - Maple Help
Home : Support : Online Help : Mathematics : DifferentialGeometry : LieAlgebras : MasseyProduct
LieAlgebras[MasseyProduct] - calculate the Massey product of a pair of forms
MasseyProduct(
\textcolor[rgb]{0.407843137254902,0.250980392156863,0.36078431372549}{\mathbf{α}},
\textcolor[rgb]{0.407843137254902,0.250980392156863,0.36078431372549}{\mathbf{β}})
\mathrm{α}
p
-form defined on a Lie algebra
\mathrm{𝔤}
\mathrm{𝔤}
\mathrm{β}
q
\mathrm{𝔤}
\mathrm{𝔤}
The Massey product of a pair of forms
\mathrm{α} ∈ {\mathrm{Λ}}^{2}\left(\mathrm{𝔤}\mathit{,}\mathit{ }\mathrm{𝔤}\right)
\mathrm{β} ∈ {\mathrm{Λ}}^{2}\left(\mathrm{𝔤}\mathit{,}\mathit{ }\mathrm{𝔤}\right)
is the 3-form
\left[\mathrm{α}, \mathrm{β}\right]
\left[\mathrm{\alpha }, \mathrm{\beta }\right]\left(x, y, z\right)= \mathrm{α}\left(\mathrm{β}\left(x, y\right), z\right) + \mathrm{\alpha }\left(\mathrm{\beta }\left(z, x\right), y\right) + \mathrm{\alpha }\left(\mathrm{\beta }\left(y, z\right), x\right)
\mathrm{α} \mathrm{ϵ} {\mathrm{Λ}}^{p}\left(\mathrm{𝔤}\mathit{,}\mathit{ }\mathrm{𝔤}\right)
\mathrm{β} \mathrm{ϵ} {\mathrm{Λ}}^{q}\left(\mathrm{𝔤}\mathit{,}\mathit{ }\mathrm{𝔤}\right),
then the Massey product is the
\left(p+q -1\right)-
form defined by
\left[\mathrm{\alpha },\mathrm{\beta }\right]\left({x}_{1}, ... ,{x}_{p+q-1}\right) = \mathrm{\alpha }\left(\mathrm{\beta }\left({x}_{1},...,{x}_{q}\right),{x}_{q+1} ,... ,{x}_{q+p-1}\right)
+ cyclic permutations.
The Massey product plays an important role in the construction of the deformations of a Lie algebra.
\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):
\mathrm{with}\left(\mathrm{LieAlgebras}\right):
First initialize a Lie algebra from a list of structure equations.
\mathrm{StrEq}≔[[\mathrm{x2},\mathrm{x3}]=\mathrm{x1},[\mathrm{x2},\mathrm{x5}]=\mathrm{x3},[\mathrm{x4},\mathrm{x5}]=\mathrm{x4}]
\textcolor[rgb]{0,0,1}{\mathrm{StrEq}}\textcolor[rgb]{0,0,1}{:=}\left[\left[\textcolor[rgb]{0,0,1}{\mathrm{x2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{x3}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{x1}}\textcolor[rgb]{0,0,1}{,}\left[\textcolor[rgb]{0,0,1}{\mathrm{x2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{x5}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{x3}}\textcolor[rgb]{0,0,1}{,}\left[\textcolor[rgb]{0,0,1}{\mathrm{x4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{x5}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{x4}}\right]
\mathrm{LD}≔\mathrm{LieAlgebraData}\left(\mathrm{StrEq},[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4},\mathrm{x5}],\mathrm{alg}\right)
\textcolor[rgb]{0,0,1}{\mathrm{LD}}\textcolor[rgb]{0,0,1}{:=}\left[\left[\textcolor[rgb]{0,0,1}{\mathrm{e2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{,}\left[\textcolor[rgb]{0,0,1}{\mathrm{e2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e5}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\textcolor[rgb]{0,0,1}{,}\left[\textcolor[rgb]{0,0,1}{\mathrm{e4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e5}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{e4}}\right]
\mathrm{DGsetup}\left(\mathrm{LD}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Lie algebra: alg}}
We define the adjoint representation and use this to construct the corresponding Lie algebra with coefficients.
\mathrm{DGsetup}\left([\mathrm{w1},\mathrm{w2},\mathrm{w3},\mathrm{w4},\mathrm{w5}],V\right)
\textcolor[rgb]{0,0,1}{\mathrm{frame name: V}}
\mathrm{\rho }≔\mathrm{Representation}\left(\mathrm{alg},V,\mathrm{Adjoint}\left(\mathrm{alg}\right)\right):
\mathrm{DGsetup}\left(\mathrm{alg},\mathrm{\rho },\mathrm{algV}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Lie algebra with coefficients: algV}}
Here is a pair of 2-forms on
\mathrm{algV}
and their Massey product.
\mathrm{\alpha }≔\mathrm{evalDG}\left(\mathrm{w1}\mathrm{θ1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{θ2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{α}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{w1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ2}}
algV >
\mathrm{\beta }≔\mathrm{evalDG}\left(\mathrm{w2}\mathrm{θ1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{θ4}\right)
\textcolor[rgb]{0,0,1}{\mathrm{β}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{w2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ4}}
\mathrm{MasseyProduct}\left(\mathrm{\alpha },\mathrm{\beta }\right)
\textcolor[rgb]{0,0,1}{\mathrm{w2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ4}}
\mathrm{algV}
\mathrm{\alpha }≔\mathrm{evalDG}\left(\mathrm{w1}\left(\mathrm{θ1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{θ2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{θ3}\right)
\textcolor[rgb]{0,0,1}{\mathrm{α}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{w1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ3}}
\mathrm{\beta }≔\mathrm{evalDG}\left(\mathrm{w4}\left(\mathrm{θ1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{θ4}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{θ5}\right)
\textcolor[rgb]{0,0,1}{\mathrm{β}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{w4}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ4}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ5}}
\mathrm{MasseyProduct}\left(\mathrm{\alpha },\mathrm{\beta }\right)
\textcolor[rgb]{0,0,1}{\mathrm{w4}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ4}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{⋀}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{θ5}}
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On Period of the Sequence of Fibonacci Polynomials Modulo
İnci Gültekin, Yasemin Taşyurdu, "On Period of the Sequence of Fibonacci Polynomials Modulo ", Discrete Dynamics in Nature and Society, vol. 2013, Article ID 731482, 3 pages, 2013. https://doi.org/10.1155/2013/731482
İnci Gültekin1 and Yasemin Taşyurdu 2
2Department of Mathematics, Faculty of Science and Letters, Erzincan University, 24000 Erzincan, Turkey
It is shown that the sequence obtained by reducing modulo coefficient and exponent of each Fibonacci polynomials term is periodic. Also if is prime, then sequences of Fibonacci polynomial are compared with Wall numbers of Fibonacci sequences according to modulo . It is found that order of cyclic group generated with matrix is equal to the period of these sequences.
In modern science there is a huge interest in the theory and application of the Fibonacci numbers. The Fibonacci numbers are the terms of the sequence , where , , with the initial values and . Generalized Fibonacci sequences have been intensively studied for many years and have become an interesting topic in Applied Mathematics. Fibonacci sequences and their related higher-order sequences are generally studied as sequence of integer. Polynomials can also be defined by Fibonacci-like recurrence relations. Such polynomials, called Fibonacci polynomials, were studied in 1883 by the Belgian mathematician Eugene Charles Catalan and the German mathematician E. Jacobsthal. The polynomials studied by Catalan are defined by the recurrence relation where , . The Fibonacci polynomials studied by Jocobstral are defined by where . The Fibonacci polynomials studied by P. F. Byrd are defined by where , . The Lucas polynomials , originally studied in 1970 by Bicknell and they are defined by where , [1].
Hoggatt and Bicknell introduced a generalized Fibonacci polynomials and their relationship to diagonals of Pascal’s triangle [2]. Also after investigating the generalized -matrix, Ivie introduced a special case [3]. Nalli and Haukkanen introduced -Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci Polynomials and the -Fibonacci number. Also they provided properties for these -Fibonacci polynomials where is a polynomial with real coefficients [1].
Definition 1. The Fibonacci polynomials are defined by the recurrence relation that the Fibonacci polynomials are generated by a matrix , can be verified quite easily by mathematical induction. The first few Fibonacci polynomials and the array of their coefficients are shown in Table 1 [2].
Fibonacci polynomials Coefficient array
A sequence is periodic if, after a certain point, it consists of only repetitions of a fixed subsequence. The number of elements in the repeating subsequence is called the period of the sequence. For example, the sequence , is periodic after the initial element and has period 4. A sequence is simply periodic with period if the first elements in the sequence form a repeating subsequence. For example, the sequence , is simply periodic with period 4 [4]. The minimum period length of sequence is stated by and is named Wall number of [5].
Theorem 2. is an even number for [5].
2. The Generalized Sequence of Fibonacci Polynomials Modulo
Reducing the generalized sequence of coefficient and exponent of each Fibonacci polynomials term by a modulus , we can get a repeating sequence, denoted by where . Let denote the smallest period of , called the period of the generalized Fibonacci polynomials modulo .
Theorem 3. is a periodic sequence.
Proof. Let where is reduction coefficient and exponent of each term in polynomials modulo . Then, we have being finite, that is, for any , there exist natural numbers and By definition of the generalized Fibonacci polynomials we have that and . Hence, , and then it follows that which implies that the is a periodic sequence.
Example 4. For , sequence is , , , , , , , . We have, and then repeat. So, we get .
Given a matrix where ’s being polynomials with real coefficients, means that every entry of is modulo , that is, . Let be a cyclic group and denote the order of where is reduction coefficient and exponent of each polynomial in matrix modulo .
Theorem 5. One has .
Proof. Proof is completed if it is that is divisible by and that is divisible by . Fibonacci polynomials are generated by a matrix , Thus, it is clear that is divisible by . Then we need only to prove that is divisible by . Let . It is seen that . Hence . We get that is divisible by . That is, is divisible by . So, we get .
Theorem 6. where is a prime number.
Proof. It is completed if it is that is divisible by and that divisible by . From Theorem 5 , for . Also, . So, we get . Thus is divisible by . Moreover is divisible by . Since , is divisible by . Therefore .
Theorem 7. is an even number where is a prime number.
Proof. It has been shown that in Theorem 6. If it is stated that is an even number then proof is completed. By Theorem 2, is an even number and is an even number for . Hence is always an even number. That is, is an even number.
Table 2 shows some periods of sequence of coefficient and exponent of Fibonacci polynomials modulo, which is a prime number, by using .
Periods of the sequence of Fibonacci polynomials modulo .
A. Nalli and P. Haukkanen, “On generalized Fibonacci and Lucas polynomials,” Chaos, Solitons and Fractals, vol. 42, no. 5, pp. 3179–3186, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. E. Hoggatt Jr. and M. Bicknell, “Generalized Fibonacci polynomials and Zeckendorf's theorem,” The Fibonacci Quarterly, vol. 11, no. 4, pp. 399–419, 1973. View at: Google Scholar | Zentralblatt MATH | MathSciNet
J. Ivie, “A general
Q
-matrix,” The Fibonacci Quarterly, vol. 10, no. 3, pp. 255–261, 1972. View at: Google Scholar | MathSciNet
S. W. Knox, “Fibonacci sequences in finite groups,” The Fibonacci Quarterly, vol. 30, no. 2, pp. 116–120, 1992. View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. D. Wall, “Fibonacci series modulo
m
,” The American Mathematical Monthly, vol. 67, pp. 525–532, 1960. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Copyright © 2013 İnci Gültekin and Yasemin Taşyurdu. 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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Expression for apparent frequency due to doppler effect — lesson. Science State Board, Class 10.
Let \(S\) and \(L\) represent the source and listener, respectively, moving at velocity
{V}_{S}
{V}_{L}
. Consider the case of the source and the listener approaching each other as shown in the below diagram.
Source and the listener
The apparent frequency will be greater than the actual source frequency as the distance between them decreases.
Let \(n\) and \(n'\) represent the frequency of the sound produced by the source and the sound heard by the listener. The apparent frequency n' is then expressed as
Here, \(v\) is the velocity of sound waves in the given medium. Let us consider different possibilities of motions of the source and the listener. In all such cases, the expression for the apparent frequency is given in the below table.
Position of source and listener
Expression for apparent frequency
Both source and listener move.
They move towards each other.
a) Distance between source and listener decreases.
b) Apparent frequency is more than actual frequency.
They move away from each other.
a) Distance between source and listener increases.
b) Apparent frequency is less than actual frequency.
{V}_{S}
{V}_{L}
become opposite in \(case-1\).
They move one behind the other.
Source follows the listener.
a) Apparent frequency depends on the velocity of the source and the listener.
{V}_{S}
becomes opposite to that in \(case-2\).
The listener follows the source.
{V}_{S}
{V}_{L}
become opposite to that in \(case-3\).
Source at rest.
Listener moves towards the source.
{V}_{S}
\(=\) \(0\) in \(case-1\).
Listener moves away from the source.
{V}_{S}
Listener at rest.
Source moves towards the listener.
{V}_{L}
Source moves away from the listener.
{V}_{L}
{n}^{"}=\left(\frac{V}{V+{V}_{S}}\right)n
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1Department of Geography, Geoinformatics and Climate Sciences, Makerere University, Kampala, Uganda.
2Department of Agricultural Production, Makerere University, Kampala, Uganda.
Chombo, O. , Lwasa, S. and Makooma, T. (2018) Spatial Differentiation of Small Holder Farmers’ Vulnerability to Climate Change in the Kyoga Plains of Uganda. American Journal of Climate Change, 7, 624-648. doi: 10.4236/ajcc.2018.74039.
\text{Index value}\left(\text{normalised value}\right)=\frac{\text{Actual value}-\text{minimam value}}{\text{Maximam value}-\text{minimam value}}
{I}_{j}={\sum }_{i=1}^{k}{b}_{i}\left[\frac{{a}_{ji}-{x}_{i}}{{S}_{i}}\right]
V=\left(E+S\right)–AC
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[24] Vishwa, B., Chandel, S. and Karanjot, K. (2013) Drought in Himachal Pradesh, India: A Historical-Geographical Perspective, 1901-2009. Transactions, 35, 260-273.
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[27] Gbetibouo, G.A. (2009) Understanding Farmers’ Perceptions and Adaptations to Climate Change and Variability: The Case of the Limpopo Basin, South Africa. Intl Food Policy Res Inst., 849.
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[33] Bekele, W. and Drake, L. (2003) Soil and Water Conservation Decision Behavior of Subsistence Farmers in the Eastern Highlands of Ethiopia: A Case Study of the Hunde-Lafto Area. Ecological Economics, 46, 437-451.
[34] Glendinning, A., Mahapatra, A. and Mitchell, C.P. (2001) Modes of Communication and Effectiveness of Agroforestry Extension in Eastern India. Human Ecology, 29, 283-305.
[35] Anley, Y., Bogale, A. and Haile-Gabriel, A. (2007) Adoption Decision and Use Intensity of Soil and Water Conservation Measures by Smallholder Subsistence Farmers in Dedo District, Western Ethiopia. Land Degradation & Development, 18, 289-302.
[36] Tizale, C.Y. (2007) The Dynamics of Soil Degradation and Incentives for Optimal Management in the Central Highlands of Ethiopia. University of Pretoria, Pretoria.
[37] Maddison, D. (2007) The Perception of Adaptation to Climate Change in Africa. The World Bank Development Research Group Sustainable Rural and Urban Development Team, Policy Research Working Paper 4308.
[38] Clay, D., Reardon, T. and Kangasniemi, J. (1998) Sustainable Intensification in the Highland Tropics: Rwandan Farmers’ Investments in Land Conservation and Soil Fertility. Economic Development and Cultural Change, 46, 351-377.
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[40] Ajak, B.J., Kyazze, F.B. and Mukwaya, P.I. (2018) Choice of Adaptation Strategies to Climate Variability among Smallholder Farmers in the Maize Based Cropping System in Namutumba District, Uganda. American Journal of Climate Change, 7, 431.
[41] Gould, B.W., Saupe, W.E. and Klemme, R.M. (1989) Conservation Tillage: The Role of Farm and Operator Characteristics and the Perception of Soil Erosion. Land Economics, 65, 167-182.
[42] Anim, F.D. (1999) A Note on the Adoption of Soil Conservation Measures in the Northern Province of South Africa. Journal of Agricultural Economics, 50, 336-345.
[43] Mugagga, F. (2017) Perceptions and Response Actions of Smallholder Coffee Farmers to Climate Variability in Montane Ecosystems. Environment and Ecology Research, 5, 357-366.
[44] Juma, M., Nyangena, W. and Yesuf, M. (2009) Production Risk and Farm Technology Adoption in Rain-Fed, Semi-Arid Lands of Kenya. Environment for Development Discussion Paper Series, 09-22.
[45] Franzel, S. (1999) Socioeconomic Factors Affecting the Adoption Potential of Improved Tree Fallows in Africa. Agroforestry Systems, 47, 305-321.
[46] Knowler, D. and Bradshaw, B. (2007) Farmers’ Adoption of Conservation Agriculture: A Review and Synthesis of Recent Research. Food Policy, 32, 25-48.
[47] Semenza, J.C., Hall, D.E., Wilson, D.J., Bontempo, B.D., Sailor, D.J. and George, L.A. (2008) Public Perception of Climate Change: Voluntary Mitigation and Barriers to Behavior Change. American Journal of Preventive Medicine, 35, 479-487.
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https://doi.org/10.1093/jae/11.1.39
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[52] Pereira de Herrera, A. and Sain, G. (1999) Adoption of Maize Conservation Tillage in Azuero. CIMMYT, Panama.
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[55] Nkonya, E., Kaizzi, C. and Pender, J. (2005) Determinants of Nutrient Balances in a Maize Farming System in Eastern Uganda. Agricultural Systems, 85, 155-182.
[56] Pattanayak, S.K., Mercer, D.E., Sills, E. and Yang, J.-C. (2003) Taking Stock of Agroforestry Adoption Studies. Agroforestry Systems, 57, 173-186.
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Bohr's Model and Electron distribution or Electronic configuration — lesson. Science CBSE, Class 9.
Different scientists have suggested various atom models. These have led to a better understanding of atomic structure. Among these models, Rutherford proposed that electrons revolve in well-defined orbits. If that was the case, there is an issue.
The motion of the electrons in Rutherford's model is unstable. As electrons revolve in orbit, they accelerate and lose energy. After that, they fall into the nucleus resulting in the atom which is highly unstable.
To overcome these objections, Neils Bohr proposed a new atomic model.
Postulate of Rutherford's model:
The electrons revolve around the nucleus in a specific orbit, and these orbits are associated with definite energies called shells or energy levels.
The electrons do not emit energy when revolving in specific orbits.
These shells or energy levels or orbits are represented by the letters K, L, M, N or by the numbers \(1\), \(2\), \(3\), \(4\).
Bohr's atom model (electron shell diagram)
Distribution of electrons in orbits or shells
Bohr and Bury proposed the distribution of electrons in orbits.
The definite distribution of electrons around the nucleus is called Electronic configuration.
To achieve the electronic configuration, it follows a certain set of rules:
2{n}^{2}
defines the total number of electrons in a shell.
Where, n is energy level or orbit number. \(n = 1, 2, 3, 4,\) etc. Therefore, the maximum number of electrons in different shells are as follows:
Energy levels Shells Maximum electrons Electron capacity
This implies that the first shell (K shell) can have a maximum of two electrons, the second shell (L shell) can have a maximum of eight electrons and so on.
Sodium atom with energy level (\(2, 8, 1\)) not (\(2, 9\))
Unless the inner shells are filled, electrons cannot fill in a given shell. In other words, the shells are gradually filled.
Incorrect and correct filling of electrons in sodium
According to Bohr, the energy of the shell is proportional to its size. The greater the size, the greater the energy. Since the first shell is the smallest, it has the lowest energy, and it gets filled first.
Hence, the energy level or size of the shells are given by:
K < L < M < N
The atomic structure of the first eighteen elements is shown schematically.
Some elements and their electronic configurations
Neils Bohr got the Nobel prize for his work on the structure of the atom in \(1922\).
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T-Stresses for Edge Cracks and Vanishing Ligaments | J. Appl. Mech. | ASME Digital Collection
T-Stresses for Edge Cracks and Vanishing Ligaments
e-mail: jdempsey@clarkson.edu
Manuscript received October 12, 2012; final manuscript received November 4, 2012; accepted manuscript posted November 20, 2012; published online May 31, 2013. Assoc. Editor: Pradeep Sharma.
Dempsey, J. P. (May 31, 2013). "T-Stresses for Edge Cracks and Vanishing Ligaments." ASME. J. Appl. Mech. July 2013; 80(4): 041035. https://doi.org/10.1115/1.4023033
An edge-cracked half-plane 0 < x < A and a half-plane x > 0 with a semi-infinite crack x > a perpendicular to the edge are examined in this paper. Uniform crack-face loading is thoroughly examined, with a thorough exposition of the Koiter Wiener–Hopf approach (Koiter, 1956, “On the Flexural Rigidity of a Beam Weakened by Transverse Saw Cuts,” Proc. Royal Neth. Acad. of Sciences, B59, pp. 354–374); an analytical expression for the corresponding T-stress is obtained. For the additional cases of (i) nonuniform edge-crack crack-face loading
σ(x/A)k
ℜ(k)>-1
), (ii) concentrated loading at the edge-crack crack mouth, the Wiener–Hopf solutions and analytical T-stress expressions are provided, and tables of T-stress results for
σ(x/A)k
σ(1-x/A)k
are presented. A Green's function for the edge-crack T-stress is developed. The differing developments made by Koiter (1956, “On the Flexural Rigidity of a Beam Weakened by Transverse Saw Cuts,” Proc. Royal Neth. Acad. of Sciences, B59, pp. 354–374, Wigglesworth (1957, “Stress Distribution in a Notched Plate,” Mathematika, 4, pp. 76–96), and Stallybrass (1970, “A Crack Perpendicular to an Elastic Half-Plane,” Int. J. Eng. Sci., 8, pp. 351–362) for the case of an edge-cracked half-plane are enhanced by deducing a quantitative relationship between the three different Wiener–Hopf type factorizations. An analytical universal T-stress expression for edge-cracks is derived. Finally, the case of a vanishing uncracked ligament in a half-plane is examined, and the associated Wiener–Hopf solution and analytical T-stress expression are again provided. Several limiting cases are examined.
Limitations to the Small Scale Yielding Approximation for Crack Tip Plasticity
Stress Intensity Factors—T-Stresses—Weight Functions
, (Supplemental Volume IKM 55),
, University of Karlsruhe, Karlsruhe, Germany.
On the Flexural Rigidity of a Beam Weakened by Transverse Saw Cuts
Proc. Royal Neth. Acad. of Sciences
, G. C. Sih, ed.,
Fracture of Baltic and Antarctic First-Year Sea Ice
,” Ph.D. thesis, Clarkson University, Potsdam, NY.
The Use of the Mellin Transform in Finding the Stress Distribution in an Infinite Wedge
Tsamasfyros
The Problem of the Infinite Wedge
Discussion of ‘Rectangular Tensile Sheet With Symmetric Edge Cracks' by O.L. Bowie
A Note on T-Stress Determination Using Dislocation Arrays
Analytical Solutions for Stress Intensity Factor, T-Stress and Weight Function for the Edge-Cracked Half-Space
A Crack Perpendicular to an Elastic Half-Plane
A Cohesive Edge Crack
The Determination of the Elastic T-Term Using Higher Order Weight Functions
Stress Distribution in a Notched Plate
Slepyan
A Semi-Infinite Crack Perpendicular to the Surface of an Elastic Half-Plane
Elastic Equilibrium of a Wedge With a Crack
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Market Cap FAQs
Market capitalization, or "market cap", is the aggregate market value of a company represented in a dollar amount. Since it represents the “market” value of a company, it is computed based on the current market price (CMP) of its shares and the total number of outstanding shares.
Market cap is also used to compare and categorize the size of companies among investors and analysts.
Market cap is calculated by multiplying a company's outstanding shares by the current market price of one share. Since a company has a given number of outstanding shares, multiplying X with the per-share price represents the total dollar value of the company.
\text{Market Cap} = \text{Price Per Share} \times \text{Shares Outstanding}
Market Cap=Price Per Share×Shares Outstanding
For example, as of Q2 2022, technology company Apple (AAPL) has a market cap of $2.9 trillion, making it the most valuable company in the world; while online retailer Amazon.com (AMZN) came in next at $1.6 trillion.
Companies that are considered large-cap have a market cap between $10 billion to $200 billion. For example, in Q2 2022, International Business Machines Corp. (IBM) and General Electric (GE) are large-cap stocks with market caps of $116 billion and $99 billion, respectively.
Some of the companies may or may not be industry leaders, but they may be on their way to becoming one. First Solar (FSLR), is a mid-cap leader in the solar power field, with a market cap of around $8 billion as of Q2 2022.
One example is Bed Bath & Beyond Inc. (BBBY) which has a market cap of $2 billion as of Q2 2022, putting it right on the high-end of small cap stocks. Track records of such companies aren’t as long as those of the mid-to-mega-caps, but they also present the possibility of greater capital appreciation.
Some traders and investors, mostly novices, can mistake a stock's price to be an accurate representation of that company’s worth, health, and/or stability. They may perceive a higher stock price as a measure of a company’s stability or a lower price as an investment available at a bargain. Stock price alone does not represent a company's actual worth. Market capitalization is the correct measure to look at, as it represents the true value as perceived by the overall market.
For instance, Microsoft with a stock price of around $300 per share had a market cap of $2,3 trillion as of Q2 2022, while Berkshire Hathaway (BRK.A), with a much higher stock price of more than $50,000 per share, had a lower market cap of $761 billion. Comparing the two companies by solely looking at their stock prices would not give a true representation of their actual relative values.
Why Are Small-Cap Stocks Often More Volatile?
Small-cap stocks have relatively lower market values because these tend to be younger growth companies. Because of their growth orientation, they may be riskier since they spend their revenues on growth and expansion. Small-cap stocks are therefore often more volatile than those of larger companies. Generally, large-cap stocks experience slower growth and are more likely to pay dividends than faster-growing, small- or mid-cap stocks.
What Is a Market Capitalization-Weighted Index?
Many stock indexes, such as the S&P 500, are weighted by market cap. This means that stocks with larger market capitalizations make up comparatively more of the index.
How Do Stock Splits Affect Market Cap?
When a company undergoes a stock split, it increases the number of shares outstanding while reducing the price of each share by a similar proportion. For instance, in a 2:1 stock split, there will be twice as many shares, but at half the pre-split price. As a result, splits do not inherently influence market cap.
Yahoo Finance. "Apple, Inc."
Yahoo Finance. "Amazon"
Yahoo Finance. "International Business Machines Corporation (IBM)."
Yahoo Finance. "General Electric."
Yahoo Finance. "First Solar."
Yahoo Finance. "Bed Bath & Beyond Inc. (BBBY)."
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EUDML | A -set of natural numbers which is not enumerable by natural numbers. EuDML | A -set of natural numbers which is not enumerable by natural numbers.
\text{Σ}
-set of natural numbers which is not enumerable by natural numbers.
Morozov, A.S.
Morozov, A.S.. "A -set of natural numbers which is not enumerable by natural numbers.." Sibirskij Matematicheskij Zhurnal 41.6 (2000): 1404-1408 (2000); translation in Sib. Math. J. 41. <http://eudml.org/doc/50143>.
@article{Morozov2000,
author = {Morozov, A.S.},
keywords = {-set; -function; natural numbers; admissible set; -set; -function},
title = {A -set of natural numbers which is not enumerable by natural numbers.},
TI - A -set of natural numbers which is not enumerable by natural numbers.
KW - -set; -function; natural numbers; admissible set; -set; -function
\text{Σ}
\text{Σ}
-function, natural numbers, admissible set,
{\Sigma }
{\Sigma }
Computability and recursion theory on ordinals, admissible sets, etc.
|
Gap metric and Vinnicombe (nu-gap) metric for distance between two systems - MATLAB gapmetric - MathWorks India
{P}_{1}={N}_{1}{M}_{1}^{-1}
{P}_{2}={N}_{2}{M}_{2}^{-1}
be right normalized coprime factorizations (see rncf). Then the gap metric δg is given by:
{\delta }_{g}\left({P}_{1},{P}_{2}\right)=\mathrm{max}\left\{{\stackrel{\to }{\delta }}_{g}\left({P}_{1},{P}_{2}\right),{\stackrel{\to }{\delta }}_{g}\left({P}_{2},{P}_{1}\right)\right\}.
{\stackrel{\to }{\delta }}_{g}\left({P}_{1},{P}_{2}\right)
{\stackrel{\to }{\delta }}_{g}\left({P}_{1},{P}_{2}\right)=\underset{\text{stable }Q\left(s\right)}{\mathrm{min}}{‖\left[\begin{array}{c}{M}_{1}\\ {N}_{1}\end{array}\right]-\left[\begin{array}{c}{M}_{2}\\ N2\end{array}\right]Q‖}_{\infty }.
{\delta }_{\nu }\left({P}_{1},{P}_{2}\right)=\underset{\omega }{\mathrm{max}}{‖{\left(I+{P}_{2}{P}_{2}^{*}\right)}^{-1/2}\left({P}_{1}-{P}_{2}\right){\left(I+{P}_{1}{P}_{1}^{*}\right)}^{-1/2}‖}_{\infty },
\mathrm{det}\left(I+{P}_{2}^{*}{P}_{1}\right)
has the right winding number. Here, * denotes the conjugate (see ctranspose). This expression is a weighted difference between the two frequency responses P1(jω) and P2(jω). For more information, see Chapter 17 of [2].
b\left(P,C\right)={‖\left[\begin{array}{c}I\\ C\end{array}\right]{\left(I+PC\right)}^{-1}\left[\begin{array}{cc}I& P\end{array}\right]‖}_{\infty }^{-1}={‖\left[\begin{array}{c}I\\ P\end{array}\right]{\left(I+CP\right)}^{-1}\left[\begin{array}{cc}I& C\end{array}\right]‖}_{\infty }^{-1}.
{W}_{1}^{-1}C{W}_{2}^{-1}
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Lue is rolling a random number cube. The cube has six sides, and each one is labeled with a different number
1
6
What is the probability that he will roll a
5
3
on one roll?
Each number has an equal probability of being rolled. Since there are
6
total sides, what is the probability of getting any number?
Each number has a one-in-six probability of being rolled. Since we can roll either a
5
3
, we can add the probabilities of these two numbers together.
\frac{1}{6}+\frac{1}{6}=?
5
and then a
3
in two rolls?
There is a one-in-six probability of Lue rolling a
5
in the first roll, and there is a one-in-six probability of him rolling a
3
in the following roll. Since the order of the numbers rolled must be taken into account, we must multiply these two numbers together.
\left(\frac{1}{6}\right)\left(\frac{1}{6}\right)=?
What is the probability that he will roll a sum of
12
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Radio Button Menu Item - Maple Help
Home : Support : Online Help : Programming : Maplets : Elements : Menu Elements : Radio Button Menu Item
define a menu item with a radio button in a menu
RadioButtonMenuItem(opts)
RadioButtonMenuItem[refID](opts)
equation(s) of the form option=value where option is one of caption, enabled, group, image, onclick, reference, value, or visible; specify options for the RadioButtonMenuItem element
The RadioButtonMenuItem menu element defines a menu entry with a radio button in a menu.
The RadioButtonMenuItem element features can be modified by using options. To simplify specifying options in the Maplets package, certain options and contents can be set without using an equation. The following table lists elements, symbols, and types (in the left column) and the corresponding option or content (in the right column) to which inputs of this type are, by default, assigned.
A RadioButtonMenuItem element can contain Action or command elements to specify the onclick option and an Image element to specify the image option.
A RadioButtonMenuItem element can be contained in a Menu or PopupMenu element.
The following table describes the control and use of the RadioButtonMenuItem element options.
The caption that appears on the radio button menu item. The caption can have a mnemonic key or access key associated with it. For more information, see Maplets Mnemonic Key.
Whether a radio button menu item can be selected. If enabled is set to false, the radio button menu item is dimmed, and any action associated with it cannot be initiated. By default, the value is true.
The action that occurs when the radio button menu item is selected.
A reference to the RadioButtonMenuItem element.
If the reference is specified by both an index, for example, RadioButtonMenuItem[refID], and a reference in the calling sequence, the index reference takes precedence.
Whether the radio button on the menu item is selected. By default, the value is false.
Whether the radio button menu item is visible to the user. By default, the value is true.
\mathrm{with}\left(\mathrm{Maplets}[\mathrm{Elements}]\right):
\mathrm{maplet}≔\mathrm{Maplet}\left(\mathrm{Window}\left('\mathrm{menubar}'='\mathrm{MB1}',[[\mathrm{Button}\left("OK",\mathrm{Shutdown}\left("Closed from button"\right)\right)]]\right),\mathrm{MenuBar}['\mathrm{MB1}']\left(\mathrm{Menu}\left("File",\mathrm{MenuItem}\left("Close",\mathrm{Shutdown}\left("Closed from menu",['\mathrm{RBMI1}','\mathrm{RBMI2}']\right)\right)\right),\mathrm{Menu}\left("Options",\mathrm{RadioButtonMenuItem}['\mathrm{RBMI1}']\left("1st Radio Option",'\mathrm{group}'='\mathrm{BG1}'\right),\mathrm{RadioButtonMenuItem}['\mathrm{RBMI2}']\left("2nd Radio Option",'\mathrm{group}'='\mathrm{BG1}'\right)\right)\right),\mathrm{ButtonGroup}['\mathrm{BG1}']\left(\right)\right):
\mathrm{Maplets}[\mathrm{Display}]\left(\mathrm{maplet}\right)
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Pulse-position modulation - Wikipedia
{\displaystyle 2^{M}}
{\displaystyle M/T}
bits per second. It is primarily useful for optical communications systems, which tend to have little or no multipath interference.
3 Sensitivity to multipath interference
4 Non-coherent detection
5 PPM vs. M-FSK
6 Applications for RF communications
7 PPM encoding for radio control
An ancient use of pulse-position modulation was the Greek hydraulic semaphore system invented by Aeneas Stymphalus around 350 B.C. that used the water clock principle to time signals.[3] In this system, the draining of water acts as the timing device, and torches are used to signal the pulses. The system used identical water-filled containers whose drain could be turned on and off, and a float with a rod marked with various predetermined codes that represented military messages. The operators would place the containers on hills so they could be seen from each other at a distance. To send a message, the operators would use torches to signal the beginning and ending of the draining of the water, and the marking on the rod attached to the float would indicate the message.
In modern times, pulse-position modulation has origins in telegraph time-division multiplexing, which dates back to 1853, and evolved alongside pulse-code modulation and pulse-width modulation.[4] In the early 1960s, Don Mathers and Doug Spreng of NASA invented pulse-position modulation used in radio-control (R/C) systems. PPM is currently being used in fiber-optic communications, deep-space communications, and continues to be used in R/C systems.
One of the key difficulties of implementing this technique is that the receiver must be properly synchronized to align the local clock with the beginning of each symbol. Therefore, it is often implemented differentially as differential pulse-position modulation, whereby each pulse position is encoded relative to the previous, such that the receiver must only measure the difference in the arrival time of successive pulses. It is possible to limit the propagation of errors to adjacent symbols, so that an error in measuring the differential delay of one pulse will affect only two symbols, instead of affecting all successive measurements.
Sensitivity to multipath interference[edit]
Aside from the issues regarding receiver synchronization, the key disadvantage of PPM is that it is inherently sensitive to multipath interference that arises in channels with frequency-selective fading, whereby the receiver's signal contains one or more echoes of each transmitted pulse. Since the information is encoded in the time of arrival (either differentially, or relative to a common clock), the presence of one or more echoes can make it extremely difficult, if not impossible, to accurately determine the correct pulse position corresponding to the transmitted pulse. Multipath in Pulse Position Modulation systems can be easily mitigated by using the same techniques that are used in Radar systems that rely totally on synchronization and time of arrival of the received pulse to obtain their range position in the presence of echoes.
Non-coherent detection[edit]
One of the principal advantages of PPM is that it is an M-ary modulation technique that can be implemented non-coherently, such that the receiver does not need to use a phase-locked loop (PLL) to track the phase of the carrier. This makes it a suitable candidate for optical communications systems, where coherent phase modulation and detection are difficult and extremely expensive. The only other common M-ary non-coherent modulation technique is M-ary frequency-shift keying (M-FSK), which is the frequency-domain dual to PPM.
PPM vs. M-FSK[edit]
PPM and M-FSK systems with the same bandwidth, average power, and transmission rate of M/T bits per second have identical performance in an additive white Gaussian noise (AWGN) channel. However, their performance differs greatly when comparing frequency-selective and frequency-flat fading channels. Whereas frequency-selective fading produces echoes that are highly disruptive for any of the M time-shifts used to encode PPM data, it selectively disrupts only some of the M possible frequency-shifts used to encode data for M-FSK. On the other hand, frequency-flat fading is more disruptive for M-FSK than PPM, as all M of the possible frequency-shifts are impaired by fading, while the short duration of the PPM pulse means that only a few of the M time-shifts are heavily impaired by fading.
Applications for RF communications[edit]
Narrowband RF (radio frequency) channels with low power and long wavelengths (i.e., low frequency) are affected primarily by flat fading, and PPM is better suited than M-FSK to be used in these scenarios. One common application with these channel characteristics, first used in the early 1960s with top-end HF (as low as 27 MHz) frequencies into the low-end VHF band frequencies (30 MHz to 75 MHz for RC use depending on location), is the radio control of model aircraft, boats and cars, originally known as "digital proportional" radio control. PPM is employed in these systems, with the position of each pulse representing the angular position of an analogue control on the transmitter, or possible states of a binary switch. The number of pulses per frame gives the number of controllable channels available. The advantage of using PPM for this type of application is that the electronics required to decode the signal are extremely simple, which leads to small, light-weight receiver/decoder units (model aircraft require parts that are as lightweight as possible). Servos made for model radio control include some of the electronics required to convert the pulse to the motor position – the receiver is required to first extract the information from the received radio signal through its intermediate frequency section, then demultiplex the separate channels from the serial stream, and feed the control pulses to each servo.
PPM encoding for radio control[edit]
More sophisticated radio control systems are now often based on pulse-code modulation, which is more complex but offers greater flexibility and reliability. The advent of 2.4 GHz band FHSS radio-control systems in the early 21st century changed this further.
Pulse-position modulation is also used for communication with the ISO/IEC 15693 contactless smart card, as well as in the HF implementation of the Electronic Product Code (EPC) Class 1 protocol for RFID tags.
^ K. T. Wong (March 2007). "Narrowband PPM Semi-Blind Spatial-Rake Receiver & Co-Channel Interference Suppression" (PDF). European Transactions on Telecommunications. The Hong Kong Polytechnic University. 18 (2): 193–197. doi:10.1002/ett.1147. Archived from the original (PDF) on 2015-09-23. Retrieved 2013-09-26.
^ Michael Lahanas. "Ancient Greek Communication Methods". Archived from the original on 2014-11-02.
^ Ross Yeager & Kyle Pace. "Copy of Communications Topic Presentation: Pulse Code Modulation". Prezi.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Pulse-position_modulation&oldid=1080193659"
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EUDML | Heegaard diagrams and surgery descriptions for twisted face-pairing 3-manifolds. EuDML | Heegaard diagrams and surgery descriptions for twisted face-pairing 3-manifolds.
Heegaard diagrams and surgery descriptions for twisted face-pairing 3-manifolds.
Cannon, J.W.; Floyd, W.J.; Parry, W.R.
Cannon, J.W., Floyd, W.J., and Parry, W.R.. "Heegaard diagrams and surgery descriptions for twisted face-pairing 3-manifolds.." Algebraic & Geometric Topology 3 (2003): 235-285. <http://eudml.org/doc/123040>.
author = {Cannon, J.W., Floyd, W.J., Parry, W.R.},
keywords = {twisted face pairing; Heegaard; corridor; Brieskorn; Heisenberg; surgery; link},
title = {Heegaard diagrams and surgery descriptions for twisted face-pairing 3-manifolds.},
AU - Floyd, W.J.
AU - Parry, W.R.
TI - Heegaard diagrams and surgery descriptions for twisted face-pairing 3-manifolds.
KW - twisted face pairing; Heegaard; corridor; Brieskorn; Heisenberg; surgery; link
twisted face pairing, Heegaard, corridor, Brieskorn, Heisenberg, surgery, link
3
Articles by Cannon
Articles by Floyd
Articles by Parry
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EUDML | -stability of the maximal term of the Hadamard composition of two Dirichlet series. EuDML | -stability of the maximal term of the Hadamard composition of two Dirichlet series.
B
-stability of the maximal term of the Hadamard composition of two Dirichlet series.
Gajsin, A.M.; Belous, T.I.
Gajsin, A.M., and Belous, T.I.. "-stability of the maximal term of the Hadamard composition of two Dirichlet series.." Sibirskij Matematicheskij Zhurnal 43.6 (2002): 1271-1282 (2002); translation in Sib. Math. J. 43. <http://eudml.org/doc/50153>.
@article{Gajsin2002,
author = {Gajsin, A.M., Belous, T.I.},
keywords = {Dirichlet series; maximal term; Hadamard composition; upper density of a set; Lebesgue measure; inverse function},
title = {-stability of the maximal term of the Hadamard composition of two Dirichlet series.},
AU - Gajsin, A.M.
AU - Belous, T.I.
TI - -stability of the maximal term of the Hadamard composition of two Dirichlet series.
KW - Dirichlet series; maximal term; Hadamard composition; upper density of a set; Lebesgue measure; inverse function
Dirichlet series, maximal term, Hadamard composition, upper density of a set, Lebesgue measure, inverse function
Dirichlet series and other series expansions, exponential series
Articles by Gajsin
Articles by Belous
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d
d
Description: find a polynomial according to the positions of its roots. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, algebra, polynomials, roots,complex_number,complex_plane, complex_plane
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The table shows the object IDs used in the default scenes that are selectable from the Simulation 3D Scene Configuration block. If you are using a custom scene, in the Unreal® Editor, you can assign new object types to unused IDs. If a scene contains an object that does not have an assigned ID, that object is assigned an ID of 0. The detection of lane markings is not supported.
These intrinsic camera parameters are equivalent to the properties of a cameraIntrinsics (Computer Vision Toolbox) object. To obtain the intrinsic parameters for your camera, use the Camera Calibrator app.
For details about the camera calibration process, see Using the Single Camera Calibrator App (Computer Vision Toolbox) and What Is Camera Calibration? (Computer Vision Toolbox).
Focal length of the camera, specified as a 1-by-2 positive integer vector of the form [fx, fy]. Units are in pixels.
This parameter is equivalent to the FocalLength (Computer Vision Toolbox) property of a cameraIntrinsics object.
Optical center of the camera, specified as a 1-by-2 positive integer vector of the form [cx,cy]. Units are in pixels.
This parameter is equivalent to the PrincipalPoint (Computer Vision Toolbox) property of a cameraIntrinsics object.
Image size produced by the camera, specified as a 1-by-2 positive integer vector of the form [mrows,ncols]. Units are in pixels.
This parameter is equivalent to the ImageSize (Computer Vision Toolbox) property of a cameraIntrinsics object.
This model is equivalent to the two-coefficient model used by the RadialDistortion (Computer Vision Toolbox) property of a cameraIntrinsics object.
This model is equivalent to the three-coefficient model used by the RadialDistortion (Computer Vision Toolbox) property of a cameraIntrinsics object.
\begin{array}{l}{x}_{\text{d}}=x×\frac{1+{k}_{1}{r}^{2}+\text{}{k}_{2}{r}^{4}+{k}_{3}{r}^{6}}{1+{k}_{4}{r}^{2}+{k}_{5}{r}^{4}+{k}_{6}{r}^{6}}\\ {y}_{\text{d}}=y×\frac{1+{k}_{1}{r}^{2}+\text{}{k}_{2}{r}^{4}+{k}_{3}{r}^{6}}{1+{k}_{4}{r}^{2}+{k}_{5}{r}^{4}+{k}_{6}{r}^{6}}\end{array}
This parameter is equivalent to the TangentialDistortion (Computer Vision Toolbox) property of a cameraIntrinsics object.
Skew angle of the camera axes, specified as a nonnegative scalar. If the X-axis and Y-axis are exactly perpendicular, then the skew must be 0. Units are dimensionless.
This parameter is equivalent to the Skew (Computer Vision Toolbox) property of a cameraIntrinsics object.
For more details, see What Is Camera Calibration? (Computer Vision Toolbox)
cameraIntrinsics (Computer Vision Toolbox)
What Is Camera Calibration? (Computer Vision Toolbox)
Depth Estimation From Stereo Video (Computer Vision Toolbox)
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Enhanced Explore - Maple Help
Home : Support : Online Help : System : Information : Updates : Maple 17 : Enhanced Explore
The Explore command gives you a quick and easy way to create interactive applications and demonstrations. Explore builds interactive explorations using a collection of embedded components, which can be used to explore an arbitrary expression or a plot. Explore inserts these automatically constructed components into a new worksheet that can easily be saved and shared.
The Explore command has been re-implemented in Maple 17 to provide enhanced and new functionality. The most important changes are:
The exploration components can now be automatically inserted into an existing worksheet or document
The exploration parameters can now be specified programmatically in the call to the Explore command
Previously assigned variables (including data) can now be referenced in the expressions being explored
The Explore command is now used by the plots:-interactive command, including use by the PlotBuilder
Full details on all the options provided by the command are available on the Explore help page.
Parameters and initial values
The new parameters option of the Explore command allows you to specify which of the unknown names in an expression of plot are the parameters to be explored via sliders. In addition, the initialvalues option allows you to specify the parameter values to be used in the first evaluation.
These two new options give the Explore command more programmatic usefulness. Compare the effects of the following two calls to the command.
In the first example, a pop-up dialog window will appear, suggesting end points for parameters A and B. Click on Explore to accept the suggested values.
\mathrm{Explore}\left(\mathrm{plot3d}\left(\mathrm{sin}\left(Ay\right)B{x}^{2},x=-1..1,y=-\mathrm{π}..\mathrm{π},\mathrm{view}=-2..2\right)\right)
\textcolor[rgb]{0,0,1}{A}
\textcolor[rgb]{0,0,1}{B}
The second example allows you to bypass the pop-up dialog altogether:
\mathrm{Explore}\left(\mathrm{plot3d}\left(\mathrm{sin}\left(A\cdot y\right)\cdot B\cdot x^2, x = -1 .. 1, y = -\mathrm{Pi} .. \mathrm{Pi}, \mathrm{view} = -2 .. 2\right), \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{parameters} = \left[A = 0 .. 10, B = 0. .. 10.0\right], \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{initialvalues} = \left[A = 3, B = 4\right]\right)
\textcolor[rgb]{0,0,1}{A}
\textcolor[rgb]{0,0,1}{B}
\mathrm{data}:=\left[\left[0,2\right],\left[\frac{1}{2},1.5\right],\left[0.8,2.5\right],\left[1.4,2.6\right],\left[1.8,1.5\right],\left[2.2,3.1\right],\left[2.6,2.4\right],\left[3.1,2.2\right],\left[3.5,2.2\right]\right]:
\mathrm{with}\left(\mathrm{CurveFitting}\right):
\mathrm{Explore}\left(\mathrm{plots}:-\mathrm{display}\left( \mathrm{plot}\left(\left[\mathrm{data}, \mathrm{Spline}\left(\mathrm{data}, v, \mathrm{degree} = n\right)\right], v = 0. .. 3.5, \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{view} = -1 .. 4, \mathrm{style} = \left[\mathrm{point}, \mathrm{line}\right], \mathrm{color} = \left[\mathrm{black}, \mathrm{blue}\right], \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{symbol} = \mathrm{circle}, \mathrm{symbolsize} = 20\right), \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{plot}\left(\mathrm{Spline}\left(\mathrm{data}, v, \mathrm{degree} = n\right), v = -1.0 .. 0., \mathrm{view} = -1 .. 4, \mathrm{color} = \mathrm{red}\right), \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{plot}\left(\mathrm{Spline}\left(\mathrm{data}, v, \mathrm{degree} = n\right), v = 3.5 .. 4.5, \mathrm{view} = -1 .. 4, \mathrm{color} = \mathrm{red}\right)\right), \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{parameters} = \left[n = 1 .. 7\right]\right)
\textcolor[rgb]{0,0,1}{n}
One powerful way to use the Explore command is to pass it a call to a previously assigned function.
The key here is that the procedure must return the kind of object to be visualized, in this case, a plot.
f:=\mathbf{proc}\left(a\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathbf{local} b;\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}} b≔a^2;\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathbf{return} \mathrm{plot}\left(\mathrm{sin}\left(b*x\right),x=-2\cdot \mathrm{Pi}..2\cdot \mathrm{Pi}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end} \mathbf{proc}:
\mathrm{Explore}\left(f\left(p\right),\mathrm{parameters}=\left[p=-2.0..2.0\right]\right)
\textcolor[rgb]{0,0,1}{p}
The next subsection illustrates this functionality with a more involved example.
Suppose that you have an Initial Value Problem (IVP) with unassigned symbolic parameters. The interactive functionality of dsolve,numeric provides an efficient mechanism for numerically solving the solution at multiple values of the parameters.
You can use the Explore command to interact with such a solution space.
\mathrm{sol} ≔ \mathrm{dsolve}\left(\left\{\mathrm{α}\cdot \left(\mathrm{diff}\left(x\left(t\right), t\right)\right) = .7\cdot \left(y\left(t\right)-x\left(t\right)\right)-\mathrm{sin}\left(x\left(t\right)\right)^2, \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{α}\cdot \left(\mathrm{diff}\left(y\left(t\right), t\right)\right) = .7\cdot \left(x\left(t\right)-y\left(t\right)\right)+z\left(t\right), \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{β}\cdot \left(\mathrm{diff}\left(z\left(t\right), t\right)\right) = -y\left(t\right), \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} x\left(0\right) = 3, \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} y\left(0\right) = 0, \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} z\left(0\right) = -3\right\}, \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \left\{x\left(t\right), y\left(t\right), z\left(t\right)\right\}, \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{numeric}, '\mathrm{parameters}' = \left[\mathrm{α}, \mathrm{β}\right], '\mathrm{output}' = '\mathrm{listprocedure}'\right):
\mathrm{caller}:=\mathbf{proc}\left(a,b\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{sol}\left('\mathrm{parameters}'=\left['\mathrm{α}'=a,'\mathrm{β}'=b\right]\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{sol}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end proc}:
F ≔ \mathbf{proc} \left(A, B\right) \phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{plots}:-\mathrm{odeplot}\left(\mathrm{caller}\left(A, B\right), \left[x\left(t\right), y\left(t\right), z\left(t\right)\right], t = 0 .. 500, \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} '\mathrm{style}' = \mathrm{line}, '\mathrm{numpoints}' = 10000, '\mathrm{labels}' = \left[x, y, z\right], \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} '\mathrm{axes}' = '\mathrm{box}', '\mathrm{color}' = \left('\mathrm{COLOR}'\right)\left('\mathrm{HUE}', \left(1/10\right)*A\right)\right) \phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathbf{end} \mathbf{proc}:
F\left(0.1,0.2\right)
You can also explore the family of such plots as the parameter values change. This is accomplished by calling Explore on a function call.
\mathrm{Explore}\left(F\left(a,b\right),'\mathrm{parameters}'=\left[a=0.5..2.0,b=1.0..10.0\right]\right)
After executing the Explore command below, select the plot with the pointer and run the animation using the top menu. By default, the embedded component will show the animation in continuous loop mode. While the animation is playing, you can move the slider and see a changing instance of the animation.
\mathrm{Explore}\left(\mathrm{plots}:-\mathrm{animate}\left(\mathrm{plot},\left[a+\mathrm{sin}\left(bx\right),x=-2\mathrm{π}..2\mathrm{π},\mathrm{view}=\left[-2\mathrm{π}..2\mathrm{π},-5..5\right]\right],b=-2.0..2.0\right),\phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{parameters}=\left[a=-3.0..3.0\right]\right)
\textcolor[rgb]{0,0,1}{a}
Interactive plots produced by the plots:-interactive command now make use of new Explore functionality.
Issuing the following command will produce a pop-up dialog. When the pop-up dialog appears, select Interactive Plot with 1 parameter from the uppermost drop-down menu (combo-box), then click Plot.
The result will now be in terms of Embedded Components in the current worksheet.
\mathrm{plots}:-\mathrm{interactive}\left(\left\{\mathrm{cos}\left(w\right),\mathrm{tan}\left(w+z\right)\right\},\mathrm{variables}=\left[w,z\right]\right)
\textcolor[rgb]{0,0,1}{z}
{ⅇ}^{x}-a{x}^{3}
The Explore command is the quickest and easiest way for you to create apps in Maple. If you simply follow the steps in any of the examples above in your worksheet, you can simply save your worksheet and you will have a working app!
If you would like to clean up your app a bit, here are some more suggestions:
First, keep in mind that when Explore creates the table of embedded components, this is treated like output in Maple. Similar to other commands in Maple, if you change or remove your original Explore command call, the values in the inserted components will also change or be removed.
To retain the inserted embedded components in a more permanent way, you can copy and paste the whole inserted table containing the embedded components. If you copy and paste the whole inserted table elsewhere in the current worksheet then its components will function just as the original does, even if the original Explore command and its own output table is deleted.
If you would like to try this, simply do the following:
1. Go to one of the above examples and highlight the plot and slider. You will need to highlight the whole table, so select an area before the plot and drag your mouse down to an area after the slider component.
2. Right click on the selection and choose Copy.
3. Paste the contents elsewhere in the worksheet, delete the original call to the Explore command and its output.
4. Move any preliminary code that you want hidden into the Startup region of the worksheet.
5. Save your worksheet and you have created a brand new standalone app!
Explore, Dynamic Applications Worksheet
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Add Base Unit - Maple Help
Home : Support : Online Help : Science and Engineering : Units : Manipulating Units : Add Base Unit
AddBaseUnit(unit, 'context'=unit_context, 'dimension'=dimension_name, opts)
unit_context
symbol; unit context. For information on unit contexts, see Details.
(optional) equation(s) of the form option=value where option is one of 'abbreviation', 'abbreviations', 'check', 'default', 'plural', 'prefix', 'spelling', 'spellings', 'symbol', or 'symbols'; specify options for the unit
The AddBaseUnit(unit, 'context'=unit_context, 'dimension'=dimension_name, opts) calling sequence adds a base unit in conjunction with adding a base dimension to the Units package for the current session.
To add a base unit to all future Maple sessions, add the AddBaseUnit command to your Maple initialization file. For more information, see Create Maple Initialization File.
No new unit name or unit symbol can evaluate to any of the symbols in the following list.
The 'context'=unit_context equation specifies the context of the unit. In this way, two units with the same name but different values can be distinguished.
The 'dimension'=dimension_name equation specifies the name of the dimension added. It is the object returned by procedures such as convert/dimensions.
The opts argument can contain one or more of the following equations that describe unit and dimension options.
'\mathrm{abbreviation}'=
This option sets the default abbreviation of the unit for display.
'\mathrm{abbreviations}'=
symbol or set(symbol)
{\mathrm{atmosphere}}_{\mathrm{technical}}
{\mathrm{at}}_{\mathrm{standard}}
'\mathrm{check}'=
For example, if a user attempts to add a new unit with an abbreviation Ys or a unit with the symbol Ys and the context SI, it conflicts with the symbol for the yottasecond. Unless the 'check'=false option is included, the AddBaseUnit routine returns an error and does not add the unit. However, a unit with the symbol
\mathrm{Ys}
\mathrm{Ys}
{\mathrm{Ys}}_{\mathrm{SI}}
{\mathrm{Ys}}_{\mathrm{new}}
\mathrm{Ys}
'\mathrm{default}'=
'\mathrm{plural}'=
'\mathrm{prefix}'=
prefix_style
This option specifies what type of prefixes the given unit takes. This option can be set to false (explicitly indicating that the unit does not take prefixes), SI, IEC, SI_positive, SI_negative, or a set of symbols that is a subset of either SI prefixes or IEC prefixes.
'\mathrm{spelling}'=
'\mathrm{spellings}'=
'\mathrm{symbol}'=
This option sets the default unit symbol for display. If this option is not given, the default symbol is chosen from the option 'symbols' (if any).
'\mathrm{symbols}'=
A unit symbol can be used in place of a unit name. For units that take SI or IEC prefixes, any associated symbol takes the associated symbol prefix.
\mathrm{with}\left(\mathrm{Units}\right):
\mathrm{AddBaseUnit}\left('\mathrm{individual}','\mathrm{context}'='\mathrm{human}','\mathrm{dimension}'='\mathrm{animal}','\mathrm{spellings}'='\mathrm{individuals}'\right)
\mathrm{AddUnit}\left('\mathrm{company}','\mathrm{context}'='\mathrm{human}','\mathrm{conversion}'=2'\mathrm{individual}','\mathrm{spellings}'='\mathrm{companies}'\right)
\mathrm{AddUnit}\left('\mathrm{crowd}','\mathrm{context}'='\mathrm{human}','\mathrm{conversion}'=3'\mathrm{individual}','\mathrm{spellings}'='\mathrm{crowds}'\right)
\mathrm{convert}\left(9,'\mathrm{units}','\mathrm{individuals}','\mathrm{crowds}'\right)
\textcolor[rgb]{0,0,1}{3}
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Accuracy class - zxc.wiki
The accuracy class of a measuring device defines the maximum expected deviation of a measured value from the true value of the physical quantity to be measured , provided the deviation is caused by the measuring device itself. On the one hand, a measuring device cannot be set exactly; on the other hand, its properties can change due to external influences. With the classification in an accuracy class, a quality feature is provided, to what extent these causes may lead to a measurement deviation.
Standards use the term e.g. B. for current transformers , weighing systems or direct-acting measuring devices with a scale display . Such classes are not known for the widespread current and voltage measuring devices with numeric displays ; see digital multimeter , measuring device deviation , resolution (digital technology) .
Scale of a moving coil measuring device of class 2.5 for vertical operating position (symbols on the right).
Under reference conditions, the limit value of the deviation for this measuring device is 2.5% of the end value of the measuring range 10 A, i.e. 0.25 A.
1.3 Class symbols
2 error limits for direct-acting measuring devices with a scale display
2.1 Self-deviation
2.1.1 Limit value
2.1.2 Reference Conditions
2.2 Influence Effects
2.2.1 Individual influencing effect
2.2.2 Multiple Influence Effects
2.3 Deviating reference conditions and nominal areas of use
2.4 Requirements associated with class assignment
3 accuracy classes for further measuring devices
In DIN 1319 , which is fundamental for metrology , the term accuracy class is defined as a class of measuring devices that meet specified metrological requirements so that the measurement deviations of these measuring devices remain within defined limits .
In EN 60051, the accuracy of a measuring device is defined as the degree of correspondence between the displayed and correct value. The accuracy ... is determined by the limits of the inherent deviation and the limits of the influencing effects . The terms are explained below.
Measuring devices that meet specific accuracy requirements can be assigned to an accuracy class. This class is identified by a class symbol in the form of a number. In the picture above this is 2.5. An addition, e.g. B. a circle that encloses the number can be added.
Error limits for direct-acting measuring devices with a scale display
title Direct-acting indicating electrical measuring devices and their accessories; Measuring instruments with a scale display
Area Measuring device
Regulates Part 1: Definitions and general requirements for all parts of this standard
Part 2: Special requirements for current and voltage measuring devices
Part 3: ... for active and reactive power measuring devices
Part 4: ... for frequency measuring devices
Part 5: ... for phase shift angle Measuring devices, power factor measuring devices and synchronoscopes
Part 6: ... for resistance and conductivity measuring devices
Part 7: ... for multiple measuring devices
Part 8: ... for accessories
Part 9: Recommended test methods
Publishing year German version DIN EN 60051-1: 1999;
-2… -9: 1991… 96
Remarks replaces: DIN 43780; VDE 0410
The EN 60051 issued for this is extremely diverse, so that only the basics are explained here. Older measuring devices were manufactured according to the similar predecessor regulations DIN 43780 or VDE 0410.
In addition, this list is limited to current and voltage measuring devices in the preferred versions according to EN 60051-2.
A manufacturer who qualifies his measuring device by specifying a class symbol guarantees compliance
the limits of intrinsic deviation (previously the basic error ),
the limits of the effects of influence .
Intrinsic deviation
If a measuring device is operated under reference conditions (the same conditions as during adjustment ) and within the measuring range , a measurement deviation that then occurs is called intrinsic deviation .
The intrinsic deviation must not exceed the values given as an example for the class symbol 2.5 (in the sense of an error limit according to the amount)
2.5% of the full scale value if the zero point is at one end of the measuring range,
2.5% of the measuring range end value if the mechanical or electrical zero point is outside the measuring range,
2.5% of the sum (regardless of the sign) of the measuring range end values if the zero point is within the scale .
In the case of an addition to the class symbol, e.g. B. circle, a different reference value applies.
Example: Ammeter with measuring range 0 to 100 mA, linearly divided, class symbol 1
The limit of the intrinsic deviation is = 1% · 100 mA = 1 mA. This limit is a constant over the entire measuring range.
{\ displaystyle G}
Note: The relative error limit of a measured value only has the value = 1% at 100 mA ; it is greater for every other measured value. At 25 mA it is already 4%, since the reference value for the relative error limit of the measured value is the respective measured value.
{\ displaystyle g}
{\ displaystyle g}
{\ displaystyle g}
= = 0.01 = 1%
{\ displaystyle {\ tfrac {1 \ \ mathrm {mA}} {100 \ \ mathrm {mA}}}}
{\ displaystyle g}
{\ displaystyle {\ tfrac {1 \ \ mathrm {mA}} {25 \ \ mathrm {mA}}}}
The definition of the reference conditions (reference value or range) is part of the definition of the intrinsic deviation . Essentially, it is defined:
permissible limits of the reference condition
Ambient temperature 23 ° C (previously 20 ° C) 2 K for class symbols 0.5 or greater, otherwise 1 K
location according to labeling 1 °
External magnetic field complete absence Earth field allowed
External electric field complete absence
Frequency of an alternating quantity 45 ... 65 Hz
Curve shape of an alternating quantity sinusoidal
Ripple of a uniform size zero
Display range 0… 12 A
Measuring range 0.6… 6 A
In class 2.5 Limit value of intrinsic deviation 2.5% · 6 A = 0.15 A
Since the information on the above limit value only applies within the measuring range , the measuring range must be recognizable if it does not match the length of the scale. There are three ways of marking the measuring range on the scale :
No fine division outside the measuring range,
Measuring range limit marked by point,
reinforced (wider drawn) scale arc in the measuring range.
If the measuring device is not operated under reference conditions, further deviations can arise in addition to the intrinsic deviation.
Single influencing effect
In the case of a single influencing variable that is not complied with, the influencing effect caused by it must also not be greater than the limit value specified above by means of the class symbol, but still provided with a correction factor. However, this only applies in a certain nominal range of use :
Limits of the nominal range of use
Ambient temperature Reference temperature ± 10 ° C 100%
location from the reference position 5 ° in each direction 50%
frequency Reference range ± 10% of the respective limit 100%
Multiple influence effects
If two or more influencing variables deviate from their reference conditions up to a value within the nominal range of use, the resulting influencing effect must not be greater than the sum of the permissible individual effects.
Example: The measuring device described above is operated at 28 ° C and inclined by 4 °.
Then the limit value of the measurement deviation = (1 + 1 + 0.5) mA = 2.5 mA
{\ displaystyle G}
(Intrinsic deviation + deviation due to temperature influence + deviation due to position influence).
Example: The measuring device described above is operated at 28 ° C and inclined by 10 °.
No guarantee that the measurement deviation will be adhered to, since the nominal range of use is not adhered to.
Deviating reference conditions and nominal areas of use
It is permissible to deviate from the above specifications of the standard if the deviation is indicated by labeling. For example:
Reference value (range)
27 ° C 27 ° C 17 ... 37 ° C
35… 50 … 60 Hz 50 Hz 35 ... 60 Hz
23… 23 … 37 ° C 23 ° C 23 ... 37 ° C
35… 45… 55 … 60 Hz 45 ... 55 Hz 35 ... 60 Hz
Requirements associated with class assignment
For the classes, not only requirements for accuracy, but also various other specifications such as
Conditions that must be observed when it comes to compliance with the limits,
Electrical and mechanical requirements, e.g. B. Overload capacity , damping ,
Test procedure to determine compliance with standardized behavior.
According to the VDE 0410 rules for electrical measuring devices, which was valid until August 1976 , these devices were divided into the following groups:
Precision measuring devices with the classes 0.1 - 0.2 - 0.5
Industrial measuring devices with classes 1 - 1.5 - 2.5 - 5
Accuracy classes for other measuring devices
Pressure gauges : display ranges for pressure gauges, division spacing and numbering of the scale according to EN 837 (accessed on October 12, 2015)
Current transformers : Simplified explanation: Accuracy classes (accessed on October 12, 2015)
Legal-for-trade measuring devices such as gas meters , energy meters , heat meters , measuring standards :
EU Directive 2014/32 / EU on the provision of measuring devices on the market , appendices from p. 205, accessed on July 26, 2020
Thomas Mühl: Introduction to electrical measurement technology. 4th edition, Springer Fachmedien Wiesbaden, Wiesbaden 2014, ISBN 978-3-8348-0899-8 .
Reinhard Lerch: Electrical measurement technology. Analogue, digital and computer-aided processes, 6th edition, Springer Verlag Berlin, Berlin 2012, ISBN 978-3-642-22608-3 .
This page is based on the copyrighted Wikipedia article "Genauigkeitsklasse" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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The aggregating curves of dynamic network filtering methods. The aggregating curves for fastviz fastviz (black line), rectangular sliding time-window (green dashed), and exponential time-window (blue dotted). The steps of the fastviz fastviz method correspond to consecutive multiplications by the forgetting factor
{C}_{\mathrm{f}}=2/3
performed after each forgetting period
{T}_{\mathrm{f}}
. The rectangular time-window width is set to
{T}_{\mathrm{tw}}=3\phantom{\rule{0.2em}{0ex}}{T}_{\mathrm{f}}
. The exponent of the exponentially decaying time-window corresponds to the forgetting factor of fastvizfastviz. For these values of the parameters, areas under the aggregating curves of both methods are approximately equal, according to Equation 1.
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\frac { 2 } { 5 } - \frac { 1 } { 6 }
A common denominator is often a multiple of both denominators of both fractions that are to be combined. A good one to use for this expression would be
30
\frac { 3 } { 7 } - \frac { 7 } { 14 }
Now combine and simplify the entire expression.
\frac { 5 } { 8 } - \frac { 2 } { 3 }
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Parabola - Simple English Wikipedia, the free encyclopedia
A parabola obtained as the intersection of a cone with a plane parallel to a straight line on its surface
The parabola (from the Greek παραβολή) is a type of curve. Menaechmus (380–320 BC) discovered the parabola, and Apollonius of Perga (262 BC–c190 BC) first named it.
A parabola is a conic section. If a cone is dissected by a plane which is parallel to one of the surfaces of the cone, the result is a parabola. The point where the parabola reaches its maximum or minimum is called the "vertex." At this point the curvature of the parabola is greatest.
Each parabola has a focal point. Any ray that enters the parabola and is parallel to the axis of symmetry will pass through this point after being reflected by the curve. Because of this fact, parabolas are important in devices such as satellite dishes, or magnifying mirrors. Parabolas are often used to approximate curves that are more difficult to model by themselves.
Every parabola uses the equation
{\displaystyle y=ax^{2}+bx+c}
{\displaystyle a}
{\displaystyle b}
{\displaystyle c}
{\displaystyle a}
{\displaystyle 0}
The bouncing of a ball will be parabolic, if the force of friction is ignored
The course of the water in a fountain is parabolic
The Gateway Arch in St. Louis looks like it is shaped like a parabola, but it is actually an inverted weighted catenary
A satellite receiver is shaped like a parabola
The parabola is a member of the family of conic sections.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Parabola&oldid=8025438"
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Home : Support : Online Help : Mathematics : Factorization and Solving Equations : LinearFunctionalSystems : SeriesSolution
return the formal series solution of a linear functional system of equations
SeriesSolution(sys, vars, method)
SeriesSolution(A, b, x, case, method)
SeriesSolution(A, x, case, method)
(optional) name indicating the version of EG-eliminations to use; one of 'quasimodular' or 'ordinary', the latter being the default
The SeriesSolution function returns the initial terms of the formal series solutions from the specified linear functional system of equations with polynomial coefficients. If such a solution does not exist, then the empty list
[]
\mathrm{Ly}\left(x\right)=\mathrm{Ay}\left(x\right)+b
y\left(x\right)
The function computes the matrix recurrence system corresponding to the given system. This matrix recurrence system is represented by its explicit matrix (the matrix n by n*m, where n is the order of the system, with the leading and trailing matrix being of size n by n). Then, the function triangularizes the leading matrix using LinearFunctionalSystems[MatrixTriangularization] in order to bound the number of the initial terms of the solution in such a way that the recurrences for the rest terms' coefficients have an invertible leading matrix and then builds these initial terms.
The solution is the list of series expansions in x, corresponding to vars. The order term (for example
\mathrm{O}\left({x}^{6}\right)
) is the last term in the series.
The solution involves arbitrary constants of the form _c1, _c2, etc.
The solution has an attribute which is a table with the following indices:
the list of initial terms
the formal degree of the initial terms
'recurrence'
the corresponding recurrence
the coefficients of the initial terms in a proper basis (depending on the case)
the leading shift of the recurrence
'trail'
the trailing shift of the recurrence
the independent variable of the given system
'the_case'
'differential', 'difference' or 'qdifference'
'homogeneous'
true if the given system is homogeneous, false otherwise
the index of the last arbitrary constant
'q_par'
the q parameter used
Note: This data is used by LinearFunctionalSystems[ExtendSeries] in order to extend the number of computed initial terms.
The error conditions associated with SeriesSolution are the same as those which are generated by LinearFunctionalSystems[Properties].
This function is part of the LinearFunctionalSystems package, and so it can be used in the form SeriesSolution(..) only after executing the command with(LinearFunctionalSystems). However, it can always be accessed through the long form of the command by using the form LinearFunctionalSystems[SeriesSolution](..).
\mathrm{with}\left(\mathrm{LinearFunctionalSystems}\right):
\mathrm{sys}≔[\mathrm{diff}\left(\mathrm{y1}\left(x\right),x\right)-\mathrm{y2}\left(x\right),\mathrm{diff}\left(\mathrm{y2}\left(x\right),x\right)-\mathrm{y3}\left(x\right)-\mathrm{y4}\left(x\right),\mathrm{diff}\left(\mathrm{y3}\left(x\right),x\right)-\mathrm{y5}\left(x\right),\mathrm{diff}\left(\mathrm{y4}\left(x\right),x\right)-2\mathrm{y1}\left(x\right)-2x\mathrm{y2}\left(x\right)-\mathrm{y5}\left(x\right),\mathrm{diff}\left(\mathrm{y5}\left(x\right),x\right)-{x}^{2}\mathrm{y1}\left(x\right)-2x\mathrm{y3}\left(x\right)-\mathrm{y6}\left(x\right),\mathrm{diff}\left(\mathrm{y6}\left(x\right),x\right)-{x}^{2}\mathrm{y2}\left(x\right)+2\mathrm{y3}\left(x\right)]:
\mathrm{vars}≔[\mathrm{y1}\left(x\right),\mathrm{y2}\left(x\right),\mathrm{y3}\left(x\right),\mathrm{y4}\left(x\right),\mathrm{y5}\left(x\right),\mathrm{y6}\left(x\right)]:
\mathrm{SeriesSolution}\left(\mathrm{sys},\mathrm{vars}\right)
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_c}}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{_c}}}_{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{O}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_c}}}_{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{_c}}}_{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{O}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{\mathrm{_c}}}_{\textcolor[rgb]{0,0,1}{4}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{O}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{_c}}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}\frac{{\textcolor[rgb]{0,0,1}{\mathrm{_c}}}_{\textcolor[rgb]{0,0,1}{4}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{O}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{_c}}}_{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{O}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{_c}}}_{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{O}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)]
\mathrm{sys}≔[\mathrm{y2}\left(x\right){x}^{2}+3\mathrm{y2}\left(x\right)x+2\mathrm{y2}\left(x\right)-2\mathrm{y1}\left(x\right){x}^{2}-4\mathrm{y1}\left(x\right)x+\mathrm{y1}\left(x+1\right){x}^{2}+\mathrm{y1}\left(x+1\right)x,\mathrm{y2}\left(x+1\right)-\mathrm{y1}\left(x\right)]:
\mathrm{vars}≔[\mathrm{y1}\left(x\right),\mathrm{y2}\left(x\right)]:
\mathrm{SeriesSolution}\left(\mathrm{sys},\mathrm{vars}\right)
[{\textcolor[rgb]{0,0,1}{\mathrm{_c}}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_c}}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{_c}}}_{\textcolor[rgb]{0,0,1}{1}}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{O}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_c}}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{O}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\right)]
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15 November 2016 Quantum Loewner evolution
Jason Miller, Scott Sheffield
Duke Math. J. 165(17): 3241-3378 (15 November 2016). DOI: 10.1215/00127094-3627096
What is the scaling limit of diffusion-limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the dielectric breakdown model
\eta
-DBM, a generalization of DLA in which particle locations are sampled from the
\eta
th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider
\eta
-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter
\gamma \in \left[0,2\right]
In this generality, we propose a scaling limit candidate called quantum Loewner evolution,
QLE\left({\gamma }^{2},\eta \right)
QLE
is defined in terms of the radial Loewner equation like radial stochastic Loewner evolution, except that it is driven by a measure-valued diffusion
{\nu }_{t}
derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of
{\nu }_{t}
using a stochastic partial differential equation. For each
\gamma \in \left(0,2\right]
, there are two or three special values of
\eta
for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of
{\nu }_{t}
We also explain discrete versions of our construction that relate DLA to loop-erased random walks and the Eden model to percolation. A certain “reshuffling” trick (in which concentric annular regions are rotated randomly, like slot-machine reels) facilitates explicit calculation.
We propose
QLE\left(2,1\right)
as a scaling limit for DLA on a random spanning-tree-decorated planar map and
QLE\left(8/3,0\right)
as a scaling limit for the Eden model on a random triangulation. We propose using
QLE\left(8/3,0\right)
to endow pure LQG with a distance function, by interpreting the region explored by a branching variant of
QLE\left(8/3,0\right)
, up to a fixed time, as a metric ball in a random metric space.
Jason Miller. Scott Sheffield. "Quantum Loewner evolution." Duke Math. J. 165 (17) 3241 - 3378, 15 November 2016. https://doi.org/10.1215/00127094-3627096
Received: 3 July 2014; Revised: 28 October 2015; Published: 15 November 2016
Keywords: Brownian map , dielectric-breakdown mode , Diffusion-limited aggregation , Gaussian free field , Liouville quantum gravity , Schramm–Loewner evolution
Jason Miller, Scott Sheffield "Quantum Loewner evolution," Duke Mathematical Journal, Duke Math. J. 165(17), 3241-3378, (15 November 2016)
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if the area of two similar triangles are equal then prove that its congruent - Mathematics - TopperLearning.com | a86r8f44
if the area of two similar triangles are equal then prove that its congruent?
\mathrm{The}\quad \mathrm{theorem}\quad \mathrm{on}\quad \mathrm{areas}\quad \mathrm{of}\quad \mathrm{similar}\quad \mathrm{triangles}\quad \mathrm{says}:
\mathrm{The}\quad \mathrm{ratio}\quad \mathrm{of}\quad \mathrm{the}\quad \mathrm{areas}\quad \mathrm{of}\quad \mathrm{two}\quad \mathrm{similar}\quad \mathrm{triangles}\quad \mathrm{is}\quad \mathrm{equal}\quad \mathrm{to}\quad \mathrm{the}\quad \mathrm{square}\quad \mathrm{of}\quad \mathrm{the}\quad \mathrm{ratio}\quad \mathrm{of}\quad \mathrm{their}\quad \mathrm{corresponding}\quad \mathrm{sides}.
\mathrm{Consider}\quad 2\quad \mathrm{similar}\quad \mathrm{triangles}\quad \mathrm{ABC}\quad \mathrm{and}\quad \mathrm{PQR},
\frac{\mathrm{area}\quad \mathrm{of}\quad \mathrm{\Delta ABC}}{\mathrm{area}\quad \mathrm{of}\quad \mathrm{\Delta PQR}}={\left(\frac{\mathrm{AB}}{\mathrm{PQ}}\right)}^{2}={\left(\frac{\mathrm{BC}}{\mathrm{QR}}\right)}^{2}={\left(\frac{\mathrm{CA}}{\mathrm{RP}}\right)}^{2}
\mathrm{As}\quad \mathrm{the}\quad \mathrm{areas}\quad \mathrm{are}\quad \mathrm{equal},\quad {\left(\frac{\mathrm{AB}}{\mathrm{PQ}}\right)}^{2}={\left(\frac{\mathrm{BC}}{\mathrm{QR}}\right)}^{2}={\left(\frac{\mathrm{CA}}{\mathrm{RP}}\right)}^{2}=1
\mathrm{So}\quad \mathrm{AB}\quad =\quad \mathrm{PQ};\quad \mathrm{BC}=\mathrm{QR};\mathrm{CA}=\mathrm{RP}
\mathrm{therefore}\quad \mathrm{triangles}\quad \mathrm{are}\quad \mathrm{congruent}\quad \mathrm{by}\quad \mathrm{SSS}.
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Impact of freezing rate on electrical conductivity of produce | SpringerPlus | Full Text
Impact of freezing rate on electrical conductivity of produce
Francesco Marra1
The aim of this work was to compare the effects of freezing rate on electrical conductivity of potatoes, carrots and apples. Electrical conductivity tests were conducted on a custom ohmic cell while samples texture was measured by means of a universal testing machine. The raw un-pretreated samples were used as control. This study showed that freezing pre-treatments lead to differences in electrical conductivity of considered samples, producing structural damage, the latter being relatively more severe when the tested products undergo ohmic treatment.
While microwave (MW) heating has become popular in both domestic and industrial applications, other electro-heating applications, such as radio frequency (RF) and ohmic (OH) heating are gaining importance in the industrial and scientific community, since they showed to be applicable to a wide of processes, from cooking to thawing (Farag et al.,2010), to sterilization (Sun et al.,2008), improving product quality and reducing processing times (McKenna et al.,2006, Somavat et al.,2012).
The main advantages of OH processing are the rapid and relatively uniform heating achieved by means of the direct passage of electric current through the product. In addition, processing times are substantially reduced in relation to conventional heating which results in higher product quality particularly with respect to product integrity, flavor and nutrient retention (Ozkan et al.,2004; Shirsat et al.,2004).
Other studies conducted in the field of electrical (OH) and dielectric (RF, MW) heating of foods demonstrated that many processing factors influence the heating (Romano and Marra,2008; Wang and Sastry,1993,2000) but also that composition, pretreatments and storage conditions in frozen chain may alter the properties of the product under processing (Lyng et al.,2013; Sarang et al.,2007). As stated by Zaritzky (2000), freezing operations can have marked effects on the structure of foods at a cellular level and freezing rate influences the potential magnitude of cellular disruption, from a low to a high level. During slow freezing conditions, there is a tendency for a small number of relatively large ice crystals to form in the extracellular space, with a consequent major disruptive effect on cellular structure (Farag et al.,2009). On the other hand, fast freezing leads to the formation of a large number of relatively small ice crystals, which form within and between cells. These small crystals have a much less disruptive effect on the cellular structure of foods (Lyng et al.,2013).
The aim of this work was to verify the effects of freezing pretreatment on electric conductivity of fresh solid food products (potatoes, carrots and apple), subjected to constant electrical field strength.
Three types of foods were chosen in this work: potatoes and carrots, as typical vegetables used as basis of soups or to be consumed a side of main dishes; apples as one of most diffused fruits, easy to find on the market and, once processed, to be used as fruit-in-syrup or as ingredient for other food preparations. In details, potatoes (Solanum tuberosum L.) of Arielle variety, carrots (Daucus carota var. sativus) of Flakkee extra variety, and apples (Malus domestic) of Golden Delicious variety were bought in a local market and were stored in a ventilated cooled room, at a temperature between 8°C and 14°C.
For each product and for each OH treatment, ten cylinders (9 mm height, 30 mm diameter) of unfrozen controls were prepared, using a circular cutter made on purpose. Totally, ninety cylinders of unfrozen controls were prepared.
In order to examine the effect of freezing rate on the electric conductivity of considered products, entire vegetables and fruits were prepared that were either frozen slowly by placing in a cold storage room at -18°C for two days or rapidly frozen by immersion in liquid nitrogen for 10 minutes and subsequently were placed in a cold storage room at -18°C for two days.
Prior to the measurement of electrical conductivity foods were defrosted in a chill at 4°C, then samples were shaped (by means of the same circular cutter used for preparing the unfrozen controls) as cylinders (9 mm height, 30 mm diameter) obtained of the inner part of the foods just before undergoing the experiments in a OH cell, as below described, and subsequently equilibrated in an air conditioned laboratory (25°C). For each product and for each OH treatment, twenty frozen samples were prepared. Totally, hundred eighty frozen samples were prepared.
The OH cell used in this work to measure the electrical conductivity is the one described into details by (.,Olivera et al 2013), working at 50 Hz and imposing an electric field strength of 3300 V/m.
Electrical conductivity (σ) was calculated according using the following equation:
\sigma =\frac{I\phantom{\rule{0.12em}{0ex}}L}{A\phantom{\rule{0.12em}{0ex}}V}
where I is the current intensity (measured in A), V is the voltage (V), L is the gap between the electrodes (m) and A is the electrode surface area (m2).
(Olivera et al. 2013) demonstrated that the three products considered in this paper show a relatively low heating rate if submitted to an average electric field strength of 3300 V/m. In any case, in order to monitor the temperature during the measurement of electrical conductivity, K-thermocouples (Tersid, Italy) were inserted into the sample’s core and hold in it. Electrical conductivity of samples has been measured at 25°C, 37°C and 50°C.
According with the procedure presented by Olivera et al., (2013), tissue damage degree was estimated from the firmness disintegration index Z, defined as in the following equation:
Z=\frac{{F}_{i}-F\left(t\right)}{{F}_{i}-{F}_{\infty }}
where F(t) is the measured firmness in N/mm; F i is the firmness of intact tissue (raw); and F ∞ is the firmness of totally destroyed tissue. Conventional (in boiling water for 15 minutes) cooked tissues were used for the determination of the firmness of totally destroyed tissue F ∞ .
Firmness was defined as the force (measured in N) to deformation (in mm) ratio from the steep linear portion of the compression curve obtained by using a universal testing machine Instron 4301 (Instron Inc, Canton, MA), using a 100 N load cell. Uniaxial compression analysis was performed, at room temperature (~25°C). Samples were compressed (65% compression) on a non-lubricated platform using a flat disk probe, with a constant crosshead speed of 20 mm/min. The raw untreated sample was used as control. Ten replicate experiments were conducted and data were statistically analyzed (α = 0.05).
A one-way analysis of variance (ANOVA) was conducted using Matlab (The Mathworks, MA, USA).
Comparison of electrical conductivity values measured for the three food samples is shown in Figure 1, where measured values of electrical conductivity are plotted as a function of the treatment (untreated, slow frozen, fast frozen) and of sample temperature. All samples, in all the investigated cases, exhibited an electrical conductivity below 0.1 S/m: this is consistent with other values available in literature (Castro et al.,2003.,, Olivera et al 2013) as no pretreatments in brine solution were done. As previously observed by (.,Olivera et al 2013), differences among the three samples (control) are confined in 10-2 S/m. The same statement remains valid when samples subjected to freezing are considered. When samples were untreated, potato exhibited the higher values of electrical conductivity, followed by the values measured for apple and then for carrot. When samples were previously frozen, apple exhibited the higher values of electrical conductivity. Overall, previously slow frozen samples resulted in slightly higher values of electrical conductivity. Since the overall composition of samples did not change during the freezing and defrosting processes, changes in electrical conductivity were due to changes in the structure of the foods themselves, as addressed later.
Electrical conductivity (taken at 50 Hz, 3300 V/m) of: a) potato; b) carrot; c) apple.
Samples of the three chosen foods underwent OH treatment, at an electric field strength of 3300 V/m. The OH process was stopped after 120 and 240 seconds and samples underwent measurement of their firmness. Unfrozen and untreated samples were chosen as control: among them, carrot exhibited the higher firmness (32.41 N/mm), followed by the potato (18.17 N/mm) and then by the apple (14.31 N/mm). Firmness of previously frozen samples was measured before and after the OH. Results are reported in Tables 1,2 and3, respectively for potato, carrot and apple.
Table 1 Firmness of potato samples
Table 2 Firmness of carrot samples
Table 3 Firmness of apple samples
As shown in Tables 1,2,3, for an electric field strength of 3300 V/m, while the OH process went on, the firmness of all the considered samples decreased, with a slope slightly more pronounced for apple samples (P < 0.05). Final firmness of unfrozen carrot was the highest (25.12 N/mm), followed by the potato (14.99 N/mm) and then by the apple (8.93 N/mm). For all the considered foods, the previously slow-frozen samples resulted in lowest firmness, at any OH processing times.
The firmness disintegration index for each of the samples was computed considering the firmness of untreated samples at time zero as reference values. Results are reported in Tables 4,5 and6, respectively for potato, carrot and apple. Before OH treatment, all samples showed an increased firmness disintegration index. Particularly, potato exhibited the higher structural damage, both for previously slow frozen (21.09%) and fast frozen samples (16.34%), while the indexes of firmness disintegration for carrot and apple samples were respectively 13.70% and 13.97% (for previously slow frozen ones) and 10.26% and 10.88% (for previously fast frozen ones). When OH treatment was applied, apple showed the higher structural damage, while potato shown the lower structure damage, in all considered cases. The freezing rate had a clear influence on the firmness disintegration index of all the samples, the slow frozen ones being more sensitive to structure damage with respect to fast frozen ones. The combination of freezing - both at slow and faster rates – with the passage of electrical current has induced a damage on the cellular structure of the considered foods.
Table 4 Firmness disintegration index (%) of potato samples
Table 5 Firmness disintegration index (%) of carrot samples
Table 6 Firmness disintegration index (%) of apple samples
The differences observed in the three analyzed foods are similar to those reported by (,Olivera et al. 2013) and by (.,Lebovka et al 2005) who explained the behavior of the analyzed foods in terms of differences in tissue structure, size of cells, and content of air cavities. According to (,Luo et al. 1992), softening is due largely to the breakdown of pectin but also of other cell walls constituents, such as cellulose and hemicelluloses. The apple structure is rich in pectin, that allow the maintain the cellular structure and this explains why all the treated apple samples exhibited a more sever structural damage. The carrot structure is divided into xylem (that is typically made by hard wall cells) and phloem (made by relatively soft-walled cells), for which the OH can cause dissolution of cell wall components and dissolution of protopectin and, thus, softening; the softening is then accelerated by the freezing treatment (at slower or faster rate). The potato structure is characterized by a cellular array presenting smaller cells at the inner core and larger ones in the outer core, all – independently by their position – of same shape (,Konstankiewicz et al. 2002); OH and freezing treatments both cause the walls breakdown of large cells, thus accelerating the softening.
Potato, carrot and apple before and after freezing and defrosting exhibited low electrical conductivity. In any case, higher electrical conductivity was measured after products were frozen and then defrosted. Products undergoing slow freezing exhibited different values of their electrical conductivity compared to fast frozen ones.
Castro I, Teixeira JA, Salengke S, Sastry SK, Vicente AA: The influence of field strength, sugar and solid content on electrical conductivity of strawberry products. J Food Process Eng 2003, 26: 17-29. 10.1111/j.1745-4530.2003.tb00587.x
Farag KW, Duggan E, Morgan DJ, Cronin DA, Lyng JG: A comparison of conventional and radio frequency defrosting of lean beef meats: effects on water binding characteristics. Meat Sci 2009, 83(2):278-284. 10.1016/j.meatsci.2009.05.010
Farag KW, Marra F, Lyng JG, Morgan DJ, Cronin DA: Temperature changes and power consumption during radio frequency tempering of beef lean/fat formulations. Food Bioprocess Tech 2010, 3(5):732-740. 10.1007/s11947-008-0131-5
Konstankiewicz K, Czachor H, Gancarz M, Król A, Pawlak K, Zdunek A: Cell structural parameters of potato tuber tissue. Int Agrophysics 2002, 16: 119-127.
Lebovka N, Ghimi P, Vorobiev E: Does electroporation occur during the ohmic heating of food? J Food Sci 2005, 70(5):308-311.
Luo Y, Patterson ME, Swanson BG: Scanning electron microscopic structure and firmness of papain treated apple slices. Food Struct 1992, 11: 333-338.
Lyng JG, Zhang L, Marra F, Brunton NP: The effect of freezing rate and comminution on the dielectric properties of pork. Czech J Food Sci 2013, 31(5):413-418.
McKenna BM, Lyng J, Brunton N, Shirsat N: Advances in radio frequency and ohmic heating of meats. J Food Eng 2006, 77(2):215-229. 10.1016/j.jfoodeng.2005.06.052
Olivera DF, Salvadori VO, Marra F: Ohmic treatment of fresh foods: effect on textural properties. Int Food Res J 2013, 20(4):1617-1621.
Ozkan N, Ho I, Farid M: Combined ohmic and plate heating of hamburger patties: quality of cooked patties. J Food Eng 2004, 63(2):141-145. 10.1016/S0260-8774(03)00292-9
Romano V, Marra F: A numerical analysis of radio frequency heating of regular shaped foodstuff. J Food Eng 2008, 84(3):449-457. 10.1016/j.jfoodeng.2007.06.006
Sarang S, Sastry SK, Gaines J, Yang TC, Dunne P: Product formulation for ohmic heating: blanching as a pretreatment method to improve uniformity in heating of solid–liquid food mixtures. J Food Sci 2007, 72(5):E227-34. 10.1111/j.1750-3841.2007.00380.x
Shirsat N, Brunton N, Lyng J, McKenna B, Scannell A: Texture, colour and sensory evaluation of a conventionally and ohmically cooked meat emulsion batter. J Sci Food Agric 2004, 84: 1861-1870. 10.1002/jsfa.1869
Somavat R, Kamonpatana P, Mohamed HMH, Sastry SK: Ohmic sterilization inside a multi-layered laminate pouch for long-duration space missions. J Food Eng 2012, 112(3):134-143. 10.1016/j.jfoodeng.2012.03.019
Sun H, Kawamura S, Himoto J, Itoh K, Wada T, Kimura T: Effects of ohmic heating on microbial counts and denaturation of proteins in milk. Food Sci Tech Res 2008, 14(2):117-123. 10.3136/fstr.14.117
Wang W, Sastry S: Salt diffusion into vegetable tissue as a pretreatment for ohmic heating: Electrical conductivity profiles and vacuum infusion studies. J Food Eng 1993, 20(4):299-309. 10.1016/0260-8774(93)90080-4
Wang W, Sastry S: Effects of thermal and electrothermal pretreatments on hot air drying rate of vegetable tissue. J Food Process Eng 2000, 23: 299-319. 10.1111/j.1745-4530.2000.tb00517.x
Zaritzky NE: Factors affecting the stability of frozen foods. In Managing Frozen Foods. Edited by: Kennedy CJ. Cambridge: Woodhead Published Ltd and CRC Press LLC; 2000:111-133.
Dipartimento di Ingegneria Industriale, Università degli studi di Salerno, Fisciano, SA, Italy
Correspondence to Francesco Marra.
The author declares that no competing interest exists about the interpretation of data or presentation of information published in this work and its content has not been influenced by author’s personal or financial relationship with other people or organizations.
Marra, F. Impact of freezing rate on electrical conductivity of produce. SpringerPlus 2, 633 (2013). https://doi.org/10.1186/2193-1801-2-633
Electro-heating
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Copy the dot pattern below and draw Figures
0
4
5
. Explain how you could know the number of dots in any figure if you knew the figure number.
\begin{array}{c c c} \newcommand{blank}{\ } & \bullet\\ \bullet & \bullet & \bullet\\ \newcommand{blank}{\ } & \bullet\\ \end{array}\\ \text{Figure 1}
\begin{array}{c c c c c} \newcommand{blank}{\ } & \newcommand{blank}{\ } & \bullet\\ \newcommand{blank}{\ } & \newcommand{blank}{\ } & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet\\ \newcommand{blank}{\ } & \newcommand{blank}{\ } & \bullet\\ \newcommand{blank}{\ } & \newcommand{blank}{\ } & \bullet \end{array}\\ \quad \;\,\text{Figure 2}
\array{ \newcommand{blank}{\ } & \newcommand{blank}{\ } & \newcommand{blank}{\ } & \bullet\\ \newcommand{blank}{\ } & \newcommand{blank}{\ } & \newcommand{blank}{\ } & \bullet\\ \newcommand{blank}{\ } & \newcommand{blank}{\ } & \newcommand{blank}{\ } & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \newcommand{blank}{\ } & \newcommand{blank}{\ } & \newcommand{blank}{\ } & \bullet\\ \newcommand{blank}{\ } & \newcommand{blank}{\ } & \newcommand{blank}{\ } & \bullet\\ \newcommand{blank}{\ } & \newcommand{blank}{\ } & \newcommand{blank}{\ } & \bullet}\\ \qquad \quad \text{Figure 3}
Could you write an expression to find the number of dots in a figure?
Look at Figure 2 to the right where
4
2
dots (the figure number) are circled and and the middle dot is left over.
An expression representing the number of dots in this figure would be
1 + 4x
x
is the figure number.
Use this to help you draw the next figures.
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!! used as default html header if there is none in the selected theme. Primes
an+b
a
b
n
Description: searching for primes in different ways. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, algebra, arithmetic, number, primes, cryptology
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Complex Numbers And Quadratic Equations, Popular Questions: ICSE Class 11-commerce MATH, Math - Meritnation
{\mathrm{log}}_{a}x
{\mathrm{log}}_{a}3
{\mathrm{log}}_{3}x
\left|\begin{array}{ccc}6\mathrm{i}& -3\mathrm{i}& 1\\ 2& 3& \mathrm{i}\\ 4& 3\mathrm{i}& -1\end{array}\right|=\mathrm{x}+\mathrm{iy}, \mathrm{then}
Chivonne asked a question
Write in standard form: ( 1 - 3i ) the whole raised to -3
Experts!! Please answer!!. How can we find the real values of x & y, if : (x -1 / 3 + i) + (y - 1 / 3 - i) = i????
Anushka Chauhan asked a question
1. if z1 and z2are two complex no.s then show that -
(Z1/Z2) = Z1 / Z2
2.if IzI =1, z not = to -1 ,prove that z-1/z+1 is purely imaginary number .what will you conclude if z=1.
3.if (cos x- isin x )2=x-iy then prove that x2 +y2=1.
Samayak Malhotra asked a question
Find the modulus and argument of complex number z=(1+i)^13/(1-i) ^7
Simplify (1-i) whole cube/(1-i cube)
(1+i)x -2i /3 + i + (2-3i)y + i/3-i = i
Deepanshu Mahor asked a question
Find the square root of (-15 + 8i)?
Pls help with 18th question
Barsha Sharma & 1 other asked a question
what is principal argument? how to find the principal argument of a complex no?
plz answer fast...
Dilpreet Singh asked a question
why value is not defined when denominator is zero? In fractional numbers?
If the roots of the equation qx^2+2px+2q=0 are real and unequal prove that the roots of the equation (p+q)x^2+2qx+(p-q)=0 are imaginary.
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Integrated Optical Addressing of a Trapped Ytterbium Ion
M. Ivory, W. J. Setzer, N. Karl, H. McGuinness, C. DeRose, M. Blain, D. Stick, M. Gehl, L. P. Parazzoli
We report on the characterization of heating rates and photoinduced electric charging on a microfabricated surface ion trap with integrated waveguides. Microfabricated surface ion traps have received considerable attention as a quantum information platform due to their scalability and manufacturability. Here, we characterize the delivery of 435-nm light through waveguides and diffractive couplers to a single ytterbium ion in a compact trap. We measure an axial heating rate at room temperature of
0.78±0.05
q/ms and see no increase due to the presence of the waveguide. Furthermore, the electric field due to charging of the exposed dielectric outcoupler settles under normal operation after an initial shift. The frequency instability after settling is measured to be 0.9 kHz.
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What matrix can you multiply by
M=\begin{bmatrix} { 2 } & { 1 } \\ { 4 } & { 0 } \end{bmatrix}
to get a result of
\left[ \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right]
\begin{bmatrix} { 2 } & { 1 } \\ { 4 } & { 0 } \end{bmatrix} \begin{bmatrix} { a } & { b } \\ { c } & { d } \end{bmatrix} =\begin{bmatrix} { 1 } & { 0 } \\ { 0 } & { 1 } \end{bmatrix}
2a + 1c = 1\\4a +0c=0
Repeat the process in step 2 for the rows of
M
multiplied by the
b-d
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EUDML | Three-dimensional slant submanifolds of -contact manifolds. EuDML | Three-dimensional slant submanifolds of -contact manifolds.
Three-dimensional slant submanifolds of
K
-contact manifolds.
Lotta, Antonio
Lotta, Antonio. "Three-dimensional slant submanifolds of -contact manifolds.." Balkan Journal of Geometry and its Applications (BJGA) 3.1 (1998): 37-51. <http://eudml.org/doc/225064>.
@article{Lotta1998,
author = {Lotta, Antonio},
keywords = {slant submanifold; constant -sectional curvature; -contact manifold; constant -sectional curvature; -contact manifold},
title = {Three-dimensional slant submanifolds of -contact manifolds.},
AU - Lotta, Antonio
TI - Three-dimensional slant submanifolds of -contact manifolds.
KW - slant submanifold; constant -sectional curvature; -contact manifold; constant -sectional curvature; -contact manifold
slant submanifold, constant
\varphi
-sectional curvature,
K
-contact manifold, constant
\phi
K
Articles by Lotta
|
EUDML | Compactifications of C3 with reducible boundary divisor. EuDML | Compactifications of C3 with reducible boundary divisor.
Compactifications of C3 with reducible boundary divisor.
Volume: 286, Issue: 1-3, page 409-432
Müller-Stach, Stefan. "Compactifications of C3 with reducible boundary divisor.." Mathematische Annalen 286.1-3 (1990): 409-432. <http://eudml.org/doc/164644>.
author = {Müller-Stach, Stefan},
keywords = {complex projective variety; contraction of an extremal ray; Fano variety; singular Gorenstein surface; adjunction map; compactification of C(sup n); projective compactifications of },
title = {Compactifications of C3 with reducible boundary divisor.},
AU - Müller-Stach, Stefan
TI - Compactifications of C3 with reducible boundary divisor.
KW - complex projective variety; contraction of an extremal ray; Fano variety; singular Gorenstein surface; adjunction map; compactification of C(sup n); projective compactifications of
complex projective variety, contraction of an extremal ray, Fano variety, singular Gorenstein surface, adjunction map, compactification of C(sup n), projective compactifications of
{ℂ}^{3}
Rational and birational maps
Articles by Stefan Müller-Stach
|
EUDML | Saturated Actions of Finite Dimensional Hopf *-Algebras on C*-Algebras. EuDML | Saturated Actions of Finite Dimensional Hopf *-Algebras on C*-Algebras.
Saturated Actions of Finite Dimensional Hopf *-Algebras on C*-Algebras.
C. Peligrad; W. Szymanski
Peligrad, C., and Szymanski, W.. "Saturated Actions of Finite Dimensional Hopf *-Algebras on C*-Algebras.." Mathematica Scandinavica 75.2 (1994): 217-239. <http://eudml.org/doc/167317>.
@article{Peligrad1994,
author = {Peligrad, C., Szymanski, W.},
keywords = {finite-dimensional Kac algebra; finite-dimensional Hopf -algebras on unital -algebras; crossed product; saturated actions; simplicity; primeness},
title = {Saturated Actions of Finite Dimensional Hopf *-Algebras on C*-Algebras.},
AU - Peligrad, C.
TI - Saturated Actions of Finite Dimensional Hopf *-Algebras on C*-Algebras.
KW - finite-dimensional Kac algebra; finite-dimensional Hopf -algebras on unital -algebras; crossed product; saturated actions; simplicity; primeness
finite-dimensional Kac algebra, finite-dimensional Hopf
*
-algebras on unital
{C}^{*}
-algebras, crossed product, saturated actions, simplicity, primeness
{C}^{*}
{W}^{*}
{C}^{*}
K
Articles by C. Peligrad
Articles by W. Szymanski
|
How to Calculate Bond Duration - wikiHow
2 Calculating Macaulay Duration
3 Calculating Modified Duration
Bond duration is a measure of how bond prices are affected by changes in interest rates. This can help an investor understand a bond's potential interest rate risk. In other words, because bond prices move inversely to interest rates, this measure provides an understanding of how badly the bond's price might be affected if interest rates were to increase. Bond duration is stated in years and higher duration bonds are more susceptible to interest rate shifts.[1] X Research source Use the following steps to calculate bond duration.
Find the price of the bond. The first variable you will need is the bond's current market price. This should be available on a brokerage trading platform or on a market news website like the Wall Street Journal or Bloomberg. Bonds are priced at par, at a premium, or at a discount in relation to their face value (the final payment made on the bond), depending on the interest rate that they provide to investors.[2] X Research source
For example, a bond with a par value of $1,000 might be priced at par. This means that it costs $1,000 to purchase the bond.
Alternately, a bond with a par value of $1,000 might be purchased at a discount for $980 or at a premium for $1,050.
Discounted bonds are generally those that provide relatively low, or zero, interest payments. Bonds sold at a premium, however, might pay very high interest payments.
The discount or premium is based upon the bond's coupon rate versus the current interest paid for bonds of similar quality and term.
Figure out the payments paid by the bond. Bonds make payments to investors known as coupon payments. These payments are periodic (quarterly, semiannual, or annual) and are calculated as a percentage of par value. Read the bond's prospectus or otherwise research the bond to find its coupon rate.
For example, the $1,000 bond mentioned above might pay an annual coupon payment at 3 percent. This would result in a payment of $1000*0.03, or $30.
Keep in mind that some bonds do not pay interest at all. These "zero-coupon" bonds are sold at a deep discount to par when issued, but can be sold at their full par value when they mature.
Clarify coupon payment details. To calculate bond duration, you will need to know the number of coupon payments made by the bond. This will depend on the maturity of the bond, which represents the "life" of the bond, between the purchase and maturity (when the face value is paid to the bondholder). The number of payments can be calculated as the maturity multiplied by the number of annual payments.
For example, a bond that makes annual payments for three years would have three total payments.
Determine the interest rate. The interest rate used in the bond duration calculation is the yield to maturity. The yield to maturity (YTM) represents the annual return realized on a bond that is held to maturity. Find a yield to maturity calculator by searching for one online. Then, input the bond's par value, market value, coupon rate, maturity, and payment frequency to get your YTM.[3] X Research source
YTM will be expressed as a percentage. For the purpose of later calculations, you will need to convert this percentage to a decimal. To do this, divide the percentage by 100. For example, 3 percent would be 3/100, or 0.03.
The example bond would have a YTM of 3 percent.
Calculating Macaulay Duration
Understand the Macaulay duration formula. Macaulay duration is the most common method for calculating bond duration. Essentially, it divides the present value of the payments provided by a bond (coupon payments and the par value) by the market price of the bond. The formula can be expressed as:
{\displaystyle {\text{duration}}={\frac {{\text{SUM}}\left({\dfrac {t*c}{(1+i)^{t}}}\right)+{\dfrac {n*m}{(1+i)^{n}}}}{P}}}
In the formula, the variables represent the following:
{\displaystyle t}
is the time in years until maturity (from the payment being calculated).
{\displaystyle c}
is the coupon payment amount in dollars.
{\displaystyle i}s the interest rate (the YTM).
{\displaystyle n}
is the number of coupon payments made.
{\displaystyle m}
is the par value (paid at maturity).
{\displaystyle P}
is the bond's current market price.[4] X Research source
Input your variables. While the formula might seem complicated, it is quite simple to calculate once you have it filled in properly. To fill out the summed portion of the equation
{\displaystyle {\text{SUM}}\left({\frac {t*c}{(1+i)^{t}}}\right)}
, you'll need to express each payment separately. Once they have all been calculated, add them up.
{\displaystyle t}
variable represents the number of years to maturity. For example, the first payment on the example bond from the "gathering your variables" part would be made three years before maturity.
This part of the equation would be represented as:
{\displaystyle \left({\frac {3*\$30}{(1+0.03)^{3}}}\right)}
The next payment would be:
{\displaystyle \left({\frac {2*\$30}{(1+0.03)^{2}}}\right)}
In total, this part of the equation would be:
{\displaystyle \left({\frac {3*\$30}{(1+0.03)^{3}}}\right)+\left({\frac {2*\$30}{(1+0.03)^{2}}}\right)+\left({\frac {1*\$30}{(1+0.03)^{1}}}\right)}
Combine the sum of payments with the remainder of the equation. Once you have created the first part of the equation, which shows the present value of the future interest payments, you will need to add it to the rest of the equation. Adding this to the rest, we get:
{\displaystyle {\text{duration}}={\frac {\left({\dfrac {3*\$30}{(1+0.03)^{3}}}\right)+\left({\dfrac {2*\$30}{(1+0.03)^{2}}}\right)+\left({\dfrac {1*\$30}{(1+0.03)^{1}}}\right)+{\dfrac {3*\$1,000}{(1+0.03)^{3}}}}{\$1,000}}}
Start calculating Macaulay duration. With the variables in the equation, you can now calculate duration. Start by simplifying the addition within the parentheses on the top of the equation.
{\displaystyle {\text{duration}}={\frac {\left({\dfrac {3*\$30}{(1.03)^{3}}}\right)+\left({\dfrac {2*\$30}{(1.03)^{2}}}\right)+\left({\dfrac {1*\$30}{(1.03)^{1}}}\right)+{\dfrac {3*\$1,000}{(1.03)^{3}}}}{\$1,000}}}
Solve the exponents. Next, solve the exponents by raising each figure to its respective power. This can be done by typing "[the bottom number]^[the exponent] into Google. Solving these gives the following result:
{\displaystyle {\text{duration}}={\frac {\left({\dfrac {3*\$30}{1.0927}}\right)+\left({\dfrac {2*\$30}{1.0609}}\right)+\left({\dfrac {1*\$30}{1.03}}\right)+{\dfrac {3*\$1,000}{1.0927}}}{\$1,000}}}
Note that the result 1.0927 is rounded to three decimal places to make calculation easier. Leaving more decimal places in your calculations will make your answer more accurate.
Multiply the numbers in the numerator. Next, solve the multiplication in the figures on top of the equation. This gives:
{\displaystyle {\text{duration}}={\frac {\left({\dfrac {\$90}{1.0927}}\right)+\left({\dfrac {\$60}{1.0609}}\right)+\left({\dfrac {\$30}{1.03}}\right)+{\dfrac {\$3,000}{1.0927}}}{\$1,000}}}
Divide the remaining figures. Solve the division for:
{\displaystyle {\text{duration}}={\frac {\$82.37+\$56.56+\$29.13+\$2745.49}{\$1,000}}}
These results have been rounded to two decimal places, as they are dollar amounts.
Finalize your calculation. Add up the top numbers to get:
{\displaystyle {\text{duration}}={\frac {\$2,913.55}{\$1,000}}}
. Then, divide by the price to get your duration, which is
{\displaystyle 2.914}
. Duration is measured in years, so your final answer is 2.914 years.
Use Macaulay duration. Macaulay duration can be used to calculate the effect that a change in interest rates would have on your bond's market price. There is a direct relationship between bond price and interest rates, mediated by the bond's duration. For every 1 percent increase or decrease in interest rates there is a (1 percent*bond duration) change in the bond's price.
For example, a 1 percent decrease in interest rates would lead to an increase in the example bond's price of 1 percent*2.914, or 2.914 percent. An increase in interest rates would have the opposite effect.[5] X Research source
Calculating Modified Duration
Start with the Macaulay duration. Modified duration is another measure of duration that is sometimes used by investors. Modified duration can be calculated on its own, but it is much easier to calculate it if you already have the Macaulay duration for the bond in question. So to calculate modified duration, start by using the other part of this article to calculate Macaulay duration.[6] X Research source
Calculate the modifier. The modifier is used to convert Macaulay duration to modified duration. It is defined as
{\displaystyle 1+{\frac {\text{YTM}}{f}}}
, where YTM is the yield to maturity for the bond and
{\displaystyle f}
is the coupon payment frequency in number of times per year (1 for annual, 2 for semiannual, and so on). You should already have the YTM and payment frequency from calculating Macaulay duration.[7] X Research source
For the example bond described in the other parts of this article, the modifier would be
{\displaystyle 1+{\frac {0.03}{1}}}
, or 1.03.
Divide by the modifier. Divide your value for Macaulay duration by the modifier to get modified duration. Using the previous example, this would be 2.914/1.03, or 2.829 years.[8] X Research source
Use modified duration. The modified duration reflects the bond's sensitivity to interest rate fluctuations. Specifically, this duration shows the new duration if interest rates were to increase by one percent. The modified duration is lower than the Macaulay duration because the rising interest rate causes the price to move down.[9] X Research source
Is there a modified duration bond calculator online?
Yes, do an internet search for "modified duration calculators" to identify several.
Say you own an asset that had a total return last year of 16 percent. Assume the inflation rate last year was 5 percent. What was your real return?
Do not calculate duration for very large changes in yields. It will not lead to accurate results.
Duration of a zero-coupon bond is equal to its maturity.
↑ https://www.blackrock.com/investing/resources/education/understanding-duration
↑ http://www.investinganswers.com/yield-maturity-ytm-811
↑ http://thismatter.com/money/bonds/duration-convexity.htm
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{\mathbb{R} }^{n}
{\mathbb{R} }^{n}
Description: find a basis of a vector subspace within given vectors. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, algebra,linear_algebra, vector_space, basis, linear_system, vectors
|
Change datatype of DataSeries Entries - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : DataFrames and DataSeries : DataSeries Commands : Change datatype of DataSeries Entries
change the datatype of a data series
SubsDatatype( DS, newdatatype, options )
type; the new datatype for the data series
conversion : procedure; specifies a procedure to be mapped onto the elements in the given data series. This option is entered in the form conversion = procedure.
The SubsDatatype command changes the datatype of the entries in a DataSeries as well as the indicated datatype of the data series.
\mathrm{genus}≔\mathrm{DataSeries}\left(〈"Rubus","Vitis","Fragaria"〉,\mathrm{labels}=[\mathrm{Raspberry},\mathrm{Grape},\mathrm{Strawberry}],\mathrm{datatype}=\mathrm{string}\right):
\mathrm{energy}≔\mathrm{DataSeries}\left(〈220,288,136〉,\mathrm{labels}=[\mathrm{Raspberry},\mathrm{Grape},\mathrm{Strawberry}],\mathrm{datatype}=\mathrm{integer}\right):
\mathrm{carbohydrates}≔\mathrm{DataSeries}\left(〈11.94,18.1,7.68〉,\mathrm{labels}=[\mathrm{Raspberry},\mathrm{Grape},\mathrm{Strawberry}],\mathrm{datatype}=\mathrm{float}\right):
\mathrm{top_producer}≔\mathrm{DataSeries}\left(〈\mathrm{Russia},\mathrm{China},\mathrm{USA}〉,\mathrm{labels}=[\mathrm{Raspberry},\mathrm{Grape},\mathrm{Strawberry}],\mathrm{datatype}=\mathrm{name}\right):
The individual DataSeries are displayed more compactly in a DataFrame:
\mathrm{berries}≔\mathrm{DataFrame}\left(\mathrm{Energy}=\mathrm{energy},\mathrm{Carbohydrates}=\mathrm{carbohydrates},\mathrm{TopProducer}=\mathrm{top_producer},\mathrm{Genus}=\mathrm{genus}\right)
\textcolor[rgb]{0,0,1}{\mathrm{berries}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{}& \textcolor[rgb]{0,0,1}{\mathrm{Energy}}& \textcolor[rgb]{0,0,1}{\mathrm{Carbohydrates}}& \textcolor[rgb]{0,0,1}{\mathrm{TopProducer}}& \textcolor[rgb]{0,0,1}{\mathrm{Genus}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Raspberry}}& \textcolor[rgb]{0,0,1}{220}& \textcolor[rgb]{0,0,1}{11.9400000000000}& \textcolor[rgb]{0,0,1}{\mathrm{Russia}}& \textcolor[rgb]{0,0,1}{"Rubus"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Grape}}& \textcolor[rgb]{0,0,1}{288}& \textcolor[rgb]{0,0,1}{18.1000000000000}& \textcolor[rgb]{0,0,1}{\mathrm{China}}& \textcolor[rgb]{0,0,1}{"Vitis"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Strawberry}}& \textcolor[rgb]{0,0,1}{136}& \textcolor[rgb]{0,0,1}{7.68000000000000}& \textcolor[rgb]{0,0,1}{\mathrm{USA}}& \textcolor[rgb]{0,0,1}{"Fragaria"}\end{array}]
\mathrm{Datatypes}\left(\mathrm{berries}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{integer}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{float}}}_{\textcolor[rgb]{0,0,1}{8}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{name}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{string}}]
\mathrm{Datatype}\left(\mathrm{carbohydrates}\right)
{\textcolor[rgb]{0,0,1}{\mathrm{float}}}_{\textcolor[rgb]{0,0,1}{8}}
You can change the datatype of the Energy data series to float:
\mathrm{SubsDatatype}\left(\mathrm{energy},\mathrm{float}\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{\mathrm{Raspberry}}& \textcolor[rgb]{0,0,1}{220.}\\ \textcolor[rgb]{0,0,1}{\mathrm{Grape}}& \textcolor[rgb]{0,0,1}{288.}\\ \textcolor[rgb]{0,0,1}{\mathrm{Strawberry}}& \textcolor[rgb]{0,0,1}{136.}\end{array}]
\mathrm{energy}≔\mathrm{SubsDatatype}\left(\mathrm{energy},\mathrm{float}\right)
\textcolor[rgb]{0,0,1}{\mathrm{energy}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{\mathrm{Raspberry}}& \textcolor[rgb]{0,0,1}{220.}\\ \textcolor[rgb]{0,0,1}{\mathrm{Grape}}& \textcolor[rgb]{0,0,1}{288.}\\ \textcolor[rgb]{0,0,1}{\mathrm{Strawberry}}& \textcolor[rgb]{0,0,1}{136.}\end{array}]
\mathrm{Datatype}\left(\mathrm{energy}\right)
{\textcolor[rgb]{0,0,1}{\mathrm{float}}}_{\textcolor[rgb]{0,0,1}{8}}
When working with strings or name conversions, it may be necessary to supply an explicit conversion for the values in the data series:
\mathrm{genus}≔\mathrm{SubsDatatype}\left(\mathrm{genus},\mathrm{name},\mathrm{`=`}\left(\mathrm{conversion},x↦\mathrm{convert}\left(x,\mathrm{name}\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{genus}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{\mathrm{Raspberry}}& \textcolor[rgb]{0,0,1}{\mathrm{Rubus}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Grape}}& \textcolor[rgb]{0,0,1}{\mathrm{Vitis}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Strawberry}}& \textcolor[rgb]{0,0,1}{\mathrm{Fragaria}}\end{array}]
\mathrm{Datatype}\left(\mathrm{genus}\right)
\textcolor[rgb]{0,0,1}{\mathrm{name}}
\mathrm{top_producer}≔\mathrm{SubsDatatype}\left(\mathrm{top_producer},\mathrm{string},\mathrm{`=`}\left(\mathrm{conversion},x↦\mathrm{convert}\left(x,\mathrm{string}\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{top_producer}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{\mathrm{Raspberry}}& \textcolor[rgb]{0,0,1}{"Russia"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Grape}}& \textcolor[rgb]{0,0,1}{"China"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Strawberry}}& \textcolor[rgb]{0,0,1}{"USA"}\end{array}]
\mathrm{Datatype}\left(\mathrm{top_producer}\right)
\textcolor[rgb]{0,0,1}{\mathrm{string}}
\mathrm{berries}≔\mathrm{DataFrame}\left(\mathrm{Energy}=\mathrm{energy},\mathrm{Carbohydrates}=\mathrm{carbohydrates},\mathrm{TopProducer}=\mathrm{top_producer},\mathrm{Genus}=\mathrm{genus}\right)
\textcolor[rgb]{0,0,1}{\mathrm{berries}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{}& \textcolor[rgb]{0,0,1}{\mathrm{Energy}}& \textcolor[rgb]{0,0,1}{\mathrm{Carbohydrates}}& \textcolor[rgb]{0,0,1}{\mathrm{TopProducer}}& \textcolor[rgb]{0,0,1}{\mathrm{Genus}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Raspberry}}& \textcolor[rgb]{0,0,1}{220.}& \textcolor[rgb]{0,0,1}{11.9400000000000}& \textcolor[rgb]{0,0,1}{"Russia"}& \textcolor[rgb]{0,0,1}{\mathrm{Rubus}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Grape}}& \textcolor[rgb]{0,0,1}{288.}& \textcolor[rgb]{0,0,1}{18.1000000000000}& \textcolor[rgb]{0,0,1}{"China"}& \textcolor[rgb]{0,0,1}{\mathrm{Vitis}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Strawberry}}& \textcolor[rgb]{0,0,1}{136.}& \textcolor[rgb]{0,0,1}{7.68000000000000}& \textcolor[rgb]{0,0,1}{"USA"}& \textcolor[rgb]{0,0,1}{\mathrm{Fragaria}}\end{array}]
\mathrm{Datatypes}\left(\mathrm{berries}\right)
[{\textcolor[rgb]{0,0,1}{\mathrm{float}}}_{\textcolor[rgb]{0,0,1}{8}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{float}}}_{\textcolor[rgb]{0,0,1}{8}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{string}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{name}}]
The DataSeries/SubsDatatype command was introduced in Maple 2017.
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!! used as default html header if there is none in the selected theme. Segfrac
Here we ask the inverse problem: given a rational number
r
Description: split a fraction into sum of fractions with smaller denominators. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, algebra, arithmetic, rational_number, fraction
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LMIs in Control/Matrix and LMI Properties and Tools/Ellipsoidal inequality - Wikibooks, open books for an open world
LMIs in Control/Matrix and LMI Properties and Tools/Ellipsoidal inequality
{\displaystyle (x_{i})}
{\displaystyle (x_{c})}
{\displaystyle {\begin{aligned}\ (x-x_{c})^{T}P^{-1}(x-x_{c})<1,\\\ where,P=P^{T}>0\\\end{aligned}}}
{\displaystyle Q(x)=1,R(x)=P,andS(x)=(x-x_{c})^{T}:}
{\displaystyle {\begin{bmatrix}1&&(x-x_{c})^{T}\\(x-x_{c})&&P\end{bmatrix}}}
{\displaystyle {\begin{aligned}\geq 0\end{aligned}}}
Retrieved from "https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/Matrix_and_LMI_Properties_and_Tools/Ellipsoidal_inequality&oldid=4009621"
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EUDML | Lq - Lr estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in Lq spaces. EuDML | Lq - Lr estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in Lq spaces.
Lq - Lr estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in Lq spaces.
Iwashita, Hirokazu. "Lq - Lr estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in Lq spaces.." Mathematische Annalen 285.2 (1989): 265-288. <http://eudml.org/doc/164599>.
@article{Iwashita1989,
author = {Iwashita, Hirokazu},
keywords = {existence; uniqueness; asymptotic behaviours; global strong solutions; exterior nonstationary problem; Navier-Stokes equation; estimates},
title = {Lq - Lr estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in Lq spaces.},
AU - Iwashita, Hirokazu
TI - Lq - Lr estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in Lq spaces.
KW - existence; uniqueness; asymptotic behaviours; global strong solutions; exterior nonstationary problem; Navier-Stokes equation; estimates
P. Maremonti, V. A. Solonnikov, On nonstationary Stokes problem in exterior domains
Hideo Kozono, Hermann Sohr, Global strong solution of the Navier-Stokes equations in 4 and 5 dimensional unbounded domains
existence, uniqueness, asymptotic behaviours, global strong solutions, exterior nonstationary problem, Navier-Stokes equation,
{L}_{q}-{L}_{r}
Articles by Hirokazu Iwashita
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EUDML | A fixed point theorem for C*-crossed products with an Abelian group. EuDML | A fixed point theorem for C*-crossed products with an Abelian group.
A fixed point theorem for C*-crossed products with an Abelian group.
Guy Henrard
Henrard, Guy. "A fixed point theorem for C*-crossed products with an Abelian group.." Mathematica Scandinavica 54 (1984): 27-39. <http://eudml.org/doc/166876>.
@article{Henrard1984,
author = {Henrard, Guy},
keywords = {C-dynamical system; -crossed product; fixed point algebra},
title = {A fixed point theorem for C*-crossed products with an Abelian group.},
AU - Henrard, Guy
TI - A fixed point theorem for C*-crossed products with an Abelian group.
KW - C-dynamical system; -crossed product; fixed point algebra
{}^{*}
-dynamical system,
{C}^{*}
-crossed product, fixed point algebra
{C}^{*}
{W}^{*}
Articles by Guy Henrard
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Johansen cointegration test - MATLAB jcitest - MathWorks 日本
\mathit{p}
h=1×6 table
\mathit{p}
pValue=2×6 table
stat=2×6 table
cValue=2×6 table
{B}^{â²}{y}_{t-1}+{c}_{0}
âT\left[\mathrm{log}\left(1â{\mathrm{λ}}_{r+1}\right)+â¦+\mathrm{log}\left(1â{\mathrm{λ}}_{m}\right)\right].
âT\mathrm{log}\left(1â{\mathrm{λ}}_{r+1}\right).
Both tests assess the null hypothesis H(r) of cointegration rank less than or equal to r. jcitest computes statistics using the effective sample size T ≤ nnumObs and ordered estimates of the eigenvalues of C = AB′, λ1 > ... > λm, where m = numDims.
jcitest displays a tabular summary of test results. The tabular display includes null ranks r = 0:(numDims − 1) in the first column of each summary. jcitest displays multiple test results in separate summaries.
Rows of h correspond to tests specified by the values of the last three variables Model, Test, and Alpha. Row labels are t1, t2, …, tu, where u = numTests.
Variables of h correspond to different, maintained cointegration ranks r = 0, 1, …, numDims – 1 and specified name-value arguments that control the number of tests. Variable labels are r0, r1, …, rR, where R = numDims – 1, and Model, Test, and Alpha.
{A, B, B1, … Bq, c0, d0, c1, d1}
eigVec Eigenvector associated with the eigenvalue in eigVal. Eigenvectors v are normalized so that v′S11v = 1, where S11 is defined as in [3].
\begin{array}{c}\mathrm{Φ}\left(L\right)\left(1âL\right){y}_{t}=A\left(Bâ²{y}_{tâ1}+{c}_{0}+{d}_{0}t\right)+{c}_{1}+{d}_{1}t+{\mathrm{ε}}_{t}\\ =c+dt+C{y}_{tâ1}+{\mathrm{ε}}_{t},\end{array}
\mathrm{Φ}\left(L\right)=Iâ{\mathrm{Φ}}_{1}â{\mathrm{Φ}}_{2}â...â{\mathrm{Φ}}_{q}
The parameters A and B in the reduced-rank VEC(q) model are not identifiable, but their product C = AB′ is identifiable. jcitest constructs B = V(:,1:r) using the orthonormal eigenvectors V returned by eig, and then renormalizes so that V'*S11*V = I [3].
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Expression (mathematics) — Wikipedia Republished // WIKI 2
Formula that represents a mathematical object
In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.
Many authors distinguish an expression from a formula, the former denoting a mathematical object, and the latter denoting a statement about mathematical objects.[citation needed] For example,
{\displaystyle 8x-5}
is an expression, while
{\displaystyle 8x-5\geq 5x-8}
is a formula. However, in modern mathematics, and in particular in computer algebra, formulas are viewed as expressions that can be evaluated to true or false, depending on the values that are given to the variables occurring in the expressions. For example
{\displaystyle 8x-5\geq 5x-8}
takes the value false if x is given a value less than –1, and the value true otherwise.
SHS 1 Core Mathematics | Solving Algebraic Expressions
Mathematics form 1 Algebraic expression 6
INTRODUCTION TO ALGEBRAIC EXPRESSIONS, CONSTANT AND VARIABLES || GRADE 7 MATHEMATICS Q2
KSSM Form 2 Mathematics Algebraic Expressions #cikgootube
2 Syntax versus semantics
2.2 Semantics
2.3 Formal languages and lambda calculus
3 Variables
{\displaystyle 3+8}
{\displaystyle 8x-5}
(linear polynomial)
{\displaystyle 7{{x}^{2}}+4x-10}
(quadratic polynomial)
{\displaystyle {\frac {x-1}{{{x}^{2}}+12}}}
(rational fraction)
{\displaystyle f(a)+\sum _{k=1}^{n}\left.{\frac {1}{k!}}{\frac {d^{k}}{dt^{k}}}\right|_{t=0}f(u(t))+\int _{0}^{1}{\frac {(1-t)^{n}}{n!}}{\frac {d^{n+1}}{dt^{n+1}}}f(u(t))\,dt.}
An expression is a syntactic construct. It must be well-formed: the allowed operators must have the correct number of inputs in the correct places, the characters that make up these inputs must be valid, have a clear order of operations, etc. Strings of symbols that violate the rules of syntax are not well-formed and are not valid mathematical expressions.
For example, in the usual notation of arithmetic, the expression 1 + 2 × 3 is well-formed, but the following expression is not:
{\displaystyle \times 4)x+,/y}
Main articles: Semantics and Formal semantics (logic)
In algebra, an expression may be used to designate a value, which might depend on values assigned to variables occurring in the expression. The determination of this value depends on the semantics attached to the symbols of the expression. The choice of semantics depends on the context of the expression. The same syntactic expression 1 + 2 × 3 can have different values (mathematically 7, but also 9), depending on the order of operations implied by the context (See also Operations § Calculators).
The semantic rules may declare that certain expressions do not designate any value (for instance when they involve division by 0); such expressions are said to have an undefined value, but they are well-formed expressions nonetheless. In general the meaning of expressions is not limited to designating values; for instance, an expression might designate a condition, or an equation that is to be solved, or it can be viewed as an object in its own right that can be manipulated according to certain rules. Certain expressions that designate a value simultaneously express a condition that is assumed to hold, for instance those involving the operator
{\displaystyle \oplus }
to designate an internal direct sum.
Main articles: Formal language and Lambda calculus
In the 1930s, a new type of expressions, called lambda expressions, were introduced by Alonzo Church and Stephen Kleene for formalizing functions and their evaluation. They form the basis for lambda calculus, a formal system used in mathematical logic and the theory of programming languages.
The equivalence of two lambda expressions is undecidable. This is also the case for the expressions representing real numbers, which are built from the integers by using the arithmetical operations, the logarithm and the exponential (Richardson's theorem).
Many mathematical expressions include variables. Any variable can be classified as being either a free variable or a bound variable.
For a given combination of values for the free variables, an expression may be evaluated, although for some combinations of values of the free variables, the value of the expression may be undefined. Thus an expression represents a function whose inputs are the values assigned to the free variables and whose output is the resulting value of the expression.[citation needed]
{\displaystyle x/y}
{\displaystyle \sum _{n=1}^{3}(2nx)}
Algebraic closure
Algebraic expression
Analytic expression
Closed-form expression
Computer algebra expression
Defined and undefined
Expression (programming)
Functional programming
Logical expression
Term (logic)
Well-defined expression
Redden, John (2011). "Elementary Algebra". Flat World Knowledge. Archived from the original on 2014-11-15. Retrieved 2012-03-18.
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MATRIX DEFINITIONS, REVIEWED
Write the
2 × 2
4 × 4
identity matrices.
Recall that an identity matrix has
1
s on the diagonal from upper left to lower right and
0
s elsewhere.
State the dimensions of each of the following matrices:
\left[ \begin{array} { l l } { 1 } & { 3 } \\ { 4 } & { 7 } \\ { 2 } & { 6 } \end{array} \right]
\left[ \begin{array} { c c c c } { 1 } & { - 1 } & { 2 } & { - 2 } \\ { 4 } & { x } & { y } & { 7 } \\ { - 9 } & { 5 } & { \pi } & { 2 } \end{array} \right]
\left[ \begin{array} { l l } { 2 } & { 3 } \end{array} \right]
\text{dimensions: rows} ⨯ \text{columns}
In part (i) above, identify
m_{1,1}, m_{2,1},
m_{1,2}
m_{2,1}
means the entry in the second row, first column.
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The average true range (ATR) is a technical analysis indicator, introduced by market technician J. Welles Wilder Jr. in his book New Concepts in Technical Trading Systems, that measures market volatility by decomposing the entire range of an asset price for that period.
It is typically derived from the 14-day simple moving average of a series of true range indicators.
The ATR was originally developed for use in commodities markets but has since been applied to all types of securities.
Calculating Volatility with Average True Range
The Average True Range (ATR) Formula
The first step in calculating ATR is to find a series of true range values for a security. The price range of an asset for a given trading day is simply its high minus its low. Meanwhile, the true range is more encompassing and is defined as:
\begin{aligned} &TR = \text{Max}[(H\ -\ L), \text{Abs}(H\ -\ C_P),\text{Abs}(L\ -\ C_P)]\\ &ATR=\bigg(\frac1n\bigg)\sum\limits^{(n)}_{(i=1)}TR_i\\ &\textbf{where:}\\ &TR_i=\text{A particular true range}\\ &n=\text{The time period employed} \end{aligned}
TR=Max[(H − L),Abs(H − CP),Abs(L − CP)]ATR=(n1)(i=1)∑(n)TRiwhere:TRi=A particular true rangen=The time period employed
Traders can use shorter periods than 14 days to generate more trading signals, while longer periods have a higher probability to generate fewer trading signals.
For example, assume a short-term trader only wishes to analyze the volatility of a stock over a period of five trading days. Therefore, the trader could calculate the five-day ATR. Assuming the historical price data is arranged in reverse chronological order, the trader finds the maximum of the absolute value of the current high minus the current low, the absolute value of the current high minus the previous close, and the absolute value of the current low minus the previous close. These calculations of the true range are done for the five most recent trading days and are then averaged to calculate the first value of the five-day ATR.
What Does the Average True Range (ATR) Tell You?
Wilder originally developed the ATR for commodities, although the indicator can also be used for stocks and indices. Simply put, a stock experiencing a high level of volatility has a higher ATR, and a low volatility stock has a lower ATR.
The ATR may be used by market technicians to enter and exit trades, and is a useful tool to add to a trading system. It was created to allow traders to more accurately measure the daily volatility of an asset by using simple calculations. The indicator does not indicate the price direction; rather it is used primarily to measure volatility caused by gaps and limit up or down moves. The ATR is fairly simple to calculate and only needs historical price data.
The ATR is commonly used as an exit method that can be applied no matter how the entry decision is made. One popular technique is known as the "chandelier exit" and was developed by Chuck LeBeau. The chandelier exit places a trailing stop under the highest high the stock reached since you entered the trade. The distance between the highest high and the stop level is defined as some multiple times the ATR. For example, we can subtract three times the value of the ATR from the highest high since we entered the trade.
The ATR can also give a trader an indication of what size trade to put on in derivatives markets. It is possible to use the ATR approach to position sizing that accounts for an individual trader's own willingness to accept risk as well as the volatility of the underlying market.
Example of How to Use the Average True Range (ATR)
As a hypothetical example, assume the first value of the five-day ATR is calculated at 1.41 and the sixth day has a true range of 1.09. The sequential ATR value could be estimated by multiplying the previous value of the ATR by the number of days less one, and then adding the true range for the current period to the product.
Next, divide the sum by the selected timeframe. For example, the second value of the ATR is estimated to be 1.35, or (1.41 * (5 - 1) + (1.09)) / 5. The formula could then be repeated over the entire time period.
While the ATR doesn't tell us in which direction the breakout will occur, it can be added to the closing price, and the trader can buy whenever the next day's price trades above that value. This idea is shown below. Trading signals occur relatively infrequently, but usually spot significant breakout points. The logic behind these signals is that whenever a price closes more than an ATR above the most recent close a change in volatility has occurred. Taking a long position is betting that the stock will follow through in the upward direction.
Limitations of the Average True Range (ATR)
There are two main limitations to using the ATR indicator. The first is that ATR is a subjective measure, meaning that it is open to interpretation. There is no single ATR value that will tell you with any certainty that a trend is about to reverse or not. Instead, ATR readings should always be compared against earlier readings to get a feel of a trend's strength or weakness.
Second, ATR only measures volatility and not the direction of an asset's price. This can sometimes result in mixed signals, particularly when markets are experiencing pivots or when trends are at turning points. For instance, a sudden increase in the ATR following a large move counter to the prevailing trend may lead some traders to think the ATR is confirming the old trend; however, this may not actually be the case.
Chart-formations. "J. Welles Wilder, Jr." Accessed Aug. 7, 2020.
Corporate Finance Institute. "Chandelier Exit." Accessed Aug. 8, 2020.
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LMIs in Control/Matrix and LMI Properties and Tools/Dilation - Wikibooks, open books for an open world
LMIs in Control/Matrix and LMI Properties and Tools/Dilation
Matrix inequalities can be dilated in order to obtain a larger matrix inequality. This can be a useful technique to separate design variables in a BMI (bi-linear matrix inequality), as the dilation often introduces additional design variables.
A common technique of LMI dilation involves using the projection lemma in reverse, or the "reciprocal projection lemma." For instance, consider the matrix inequality
{\displaystyle {\begin{bmatrix}{\mathbf {PA}}+{\mathbf {A^{T}P}}-{\mathbf {P}}&{\mathbf {P}}\\*&{\mathbf {-P}}\end{bmatrix}}<0,}
{\displaystyle {\mathbf {P}}\in \S ^{n\times m}}
{\displaystyle {\mathbf {A}}\in \mathbb {R} ^{n\times n}}
{\displaystyle {\mathbf {P}}>0.}
{\displaystyle {\begin{bmatrix}{\mathbf {A^{T}}}&{\mathbf {1}}&{\mathbf {0}}\\{\mathbf {1}}&{\mathbf {0}}&{\mathbf {1}}\end{bmatrix}}{\begin{bmatrix}{\mathbf {0}}&{\mathbf {P}}&{\mathbf {0}}\\*&{\mathbf {-P}}&{\mathbf {0}}\\*&*&{\mathbf {-P}}\end{bmatrix}}{\begin{bmatrix}{\mathbf {A}}&{\mathbf {1}}\\{\mathbf {1}}&{\mathbf {0}}\\{\mathbf {0}}&{\mathbf {1}}\end{bmatrix}}<0.}
{\displaystyle {\mathbf {P}}>0,}
{\displaystyle {\begin{bmatrix}{\mathbf {-P}}&{\mathbf {0}}\\*&{\mathbf {-P}}\end{bmatrix}}<0,}
{\displaystyle {\begin{bmatrix}{\mathbf {0}}&{\mathbf {1}}&{\mathbf {0}}\\{\mathbf {0}}&{\mathbf {0}}&{\mathbf {1}}\end{bmatrix}}{\begin{bmatrix}{\mathbf {0}}&{\mathbf {P}}&{\mathbf {0}}\\*&{\mathbf {-P}}&{\mathbf {0}}\\*&*&{\mathbf {-P}}\end{bmatrix}}{\begin{bmatrix}{\mathbf {0}}&{\mathbf {0}}\\{\mathbf {1}}&{\mathbf {0}}\\{\mathbf {0}}&{\mathbf {1}}\end{bmatrix}}<0.}
These expanded inequalities (1) and (2) are now in the form of the strict projection lemma, meaning they are equivalent to
{\displaystyle {\mathbf {\Phi }}({\mathbf {P}})+{\mathbf {G}}({\mathbf {A}}){\mathbf {VH^{T}}}+{\mathbf {HV^{T}G^{T}}}({\mathbf {A}}),}
{\displaystyle N({\mathbf {G^{T}}}({\mathbf {A}}))=R({\mathbf {N}}_{G}({\mathbf {A}})),N({\mathbf {H^{T}}})=R({\mathbf {N}}_{H}),}
{\displaystyle V\in \mathbb {R} ^{n\times n}.}
{\displaystyle {\mathbf {G}}({\mathbf {A}})={\begin{bmatrix}{\mathbf {-1}}\\{\mathbf {A^{T}}}\\{\mathbf {1}}\end{bmatrix}},{\mathbf {H}}={\begin{bmatrix}{\mathbf {1}}\\{\mathbf {0}}\\{\mathbf {0}}\end{bmatrix}},}
we can now rewrite the inequality (3) as
{\displaystyle {\begin{bmatrix}-({\mathbf {V}}+{\mathbf {V^{T}}})&{\mathbf {V^{T}A}}+{\mathbf {P}}&{\mathbf {V^{T}}}\\*&{\mathbf {-P}}&{\mathbf {0}}\\*&*&{\mathbf {-P}}\end{bmatrix}}<0,}
which is the new dilated inequality.
Some useful examples of dilated matrix inequalities are presented here.
Consider matrices
{\displaystyle {\mathbf {A,G}}\in \mathbb {R} ^{n\times n},{\mathbf {\Delta }}\in \mathbb {R} ^{m\times m},{\mathbf {P}}\in \S ^{n},\delta _{1},\delta _{2},a,b\in \mathbb {R} _{>0},}
{\displaystyle {\mathbf {P}}>0}
{\displaystyle b=a^{-1}.}
The following matrix inequalities are equivalent:
{\displaystyle {\mathbf {AP}}+{\mathbf {PA^{T}}}+\delta _{1}{\mathbf {P}}+\delta _{2}{\mathbf {APA^{T}}}+{\mathbf {P\Delta ^{T}\Delta P}}<0;}
{\displaystyle {\begin{bmatrix}{\mathbf {0}}&{\mathbf {-P}}&{\mathbf {P}}&{\mathbf {0}}&{\mathbf {P\Delta ^{T}}}\\*&{\mathbf {0}}&{\mathbf {0}}&{\mathbf {-P}}&{\mathbf {0}}\\*&*&-\delta _{1}^{-1}{\mathbf {P}}&{\mathbf {0}}&{\mathbf {0}}\\*&*&*&-\delta _{2}^{-1}{\mathbf {P}}&{\mathbf {0}}\\*&*&*&*&{\mathbf {-1}}\\\end{bmatrix}}+He({\begin{bmatrix}{\mathbf {A}}\\{\mathbf {1}}\\{\mathbf {0}}\\{\mathbf {0}}\\{\mathbf {0}}\\\end{bmatrix}}{\mathbf {G}}{\begin{bmatrix}{\mathbf {1}}&-b{\mathbf {1}}&b{\mathbf {1}}&{\mathbf {1}}&b{\mathbf {\Delta }}^{T}\end{bmatrix}})<0.}
{\displaystyle {\mathbf {A,V}}\in \mathbb {R} ^{n\times n},{\mathbf {P,X}}\in \S ^{n},{\mathbf {B}}\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{p\times n},{\mathbf {D}}\in \mathbb {R} ^{p\times m},{\mathbf {R}}\in \S ^{m},}
{\displaystyle {\mathbf {S}}\in \S ^{p},}
{\displaystyle {\mathbf {P,R,S,X}}>0.}
The matrix inequality
{\displaystyle {\begin{bmatrix}-{\mathbf {V}}-{\mathbf {V^{T}}}&{\mathbf {VA}}+{\mathbf {P}}&{\mathbf {VB}}&{\mathbf {0}}&{\mathbf {V}}\\*&-2{\mathbf {P}}+{\mathbf {X}}&{\mathbf {0}}&{\mathbf {C^{T}}}&{\mathbf {0}}\\*&*&-{\mathbf {R}}&{\mathbf {D^{T}}}&{\mathbf {0}}\\*&*&*&-{\mathbf {S}}&{\mathbf {0}}\\*&*&*&*&-{\mathbf {X}}\\\end{bmatrix}}<0}
{\displaystyle {\begin{bmatrix}{\mathbf {PA}}+{\mathbf {A^{T}P}}&{\mathbf {PB}}&{\mathbf {C^{T}}}\\*&-{\mathbf {R}}&{\mathbf {D^{T}}}\\*&*&-{\mathbf {S}}\end{bmatrix}}}
{\displaystyle {\mathbf {A,V}}\in \mathbb {R} ^{n\times n},{\mathbf {Q,X}}\in \S ^{n},{\mathbf {B}}\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{p\times n},{\mathbf {D}}\in \mathbb {R} ^{p\times m},{\mathbf {R}}\in \S ^{m},}
{\displaystyle {\mathbf {S}}\in \S ^{p},}
{\displaystyle {\mathbf {Q,R,S,X}}>0.}
{\displaystyle {\begin{bmatrix}-{\mathbf {V}}-{\mathbf {V^{T}}}&{\mathbf {V^{T}A^{T}}}+{\mathbf {Q}}&{\mathbf {0}}&{\mathbf {V^{T}C}}&{\mathbf {V^{T}}}\\*&-2{\mathbf {Q}}+{\mathbf {X}}&{\mathbf {B}}&{\mathbf {0}}&{\mathbf {0}}\\*&*&-{\mathbf {R}}&{\mathbf {D^{T}}}&{\mathbf {0}}\\*&*&*&-{\mathbf {S}}&{\mathbf {0}}\\*&*&*&*&-{\mathbf {X}}\\\end{bmatrix}}<0}
{\displaystyle {\begin{bmatrix}{\mathbf {AQ}}+{\mathbf {QA^{T}}}&{\mathbf {B}}&{\mathbf {QC^{T}}}\\*&-{\mathbf {R}}&{\mathbf {D^{T}}}\\*&*&-{\mathbf {S}}\end{bmatrix}}}
Projection Lemma - The projection lemma.
Reciprocal Projection Lemma - The reciprocal projection lemma.
Retrieved from "https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/Matrix_and_LMI_Properties_and_Tools/Dilation&oldid=4007006"
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Read the Math Notes boxes in this lesson and in the previous lesson. Use the information to complete the following problems.
The coach of the fifth grade girls basketball team measured the height of each player. Their heights in centimeters were
120
122
126
130
133
147
115
106
120
112
142
Make a stem-and-leaf plot of the players’ heights.
A stem-and-leaf plot displays data by ordering it least to greatest. All of the digits except for the last one form the stem and the last digit forms the leaf. The leaf portion is arranged in order from least to greatest.
\begin{array} {c | c c}10&6\\ 11&2&5\\ 12&0&0&2&5\\ 13&0&3\\ 14&2&7 \end{array}
Make a histogram of the players’ heights.
Describe the shape and spread of the data. That is, is it symmetric or non-symmetric? Does it have more than one peak or only one? Is it tightly packed together or widely spread out?
Is the data symmetric? How many peaks does it have? Is it closely packed or spread out?
Does this data have any outliers? Which measure of center, mean or median, would be appropriate to use to describe the typical heights?
The median is the middle number in a set of data, and the mean is the average of all the numbers in a set of data.
What is the typical height of a player on the team?
What is the median and what is the mean in the given set of data?
The range is the difference between the greatest and least number in a set of data.
Subtract the smallest number from the largest number.
Click the link at right for the full version of the eTool: 7-38 HW eTool.
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!! used as default html header if there is none in the selected theme. Choice of ellipses
a(x-{x}_{0}{)}^{2}+b(y-{y}_{0}{)}^{2}=c
a{x}^{2}+b{y}^{2}+cx+dy+e=0
\left({x}_{0},{y}_{0}\right)
is the center of the ellipse and the ratio between the two axes of the ellipse is described by the ratio between
and
b
(how?).
Description: recognize an ellipse according to its equation, or vice versa. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, geometry, analytic_geometry,ellipse,conics,graphing,curves
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Explicit heat kernels of a model of distorted Brownian motion on spaces with varying dimension
June 2021 Explicit heat kernels of a model of distorted Brownian motion on spaces with varying dimension
Shuwen Lou1
1Department of Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois, USA
In this paper, we study a particular model of distorted Brownian motion (dBM) on state spaces with varying dimension. Roughly speaking, the state space of such a process consists of two components: a 3-dimensional component and a 1-dimensional component. These two parts are joined together at the origin. The restriction of dBM on the 3- or 1-dimensional component receives a strong “push” toward the origin. On each component, the “magnitude” of a “push” can be parametrized by a constant
\mathit{\gamma }>0
. In this article, using the probabilistic method, we get the exact expressions for the transition density functions of dBM with varying dimension for any
0<t<\mathrm{\infty }
Shuwen Lou. "Explicit heat kernels of a model of distorted Brownian motion on spaces with varying dimension." Illinois J. Math. 65 (2) 287 - 312, June 2021. https://doi.org/10.1215/00192082-8939623
Received: 24 January 2020; Revised: 12 November 2020; Published: June 2021
Shuwen Lou "Explicit heat kernels of a model of distorted Brownian motion on spaces with varying dimension," Illinois Journal of Mathematics, Illinois J. Math. 65(2), 287-312, (June 2021)
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Exact Travelling Wave Solutions for Isothermal Magnetostatic Atmospheres by Fan Subequation Method
Hossein Jafari, Maryam Ghorbani, Chaudry Masood Khalique, "Exact Travelling Wave Solutions for Isothermal Magnetostatic Atmospheres by Fan Subequation Method", Abstract and Applied Analysis, vol. 2012, Article ID 962789, 11 pages, 2012. https://doi.org/10.1155/2012/962789
Hossein Jafari ,1,2 Maryam Ghorbani,1 and Chaudry Masood Khalique 2
1Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran
2International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa
The equations of magnetohydrostatic equilibria for a plasma in a gravitational field are investigated analytically. An investigation of a family of isothermal magnetostatic atmospheres with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry is carried out. These equations transform to a single nonlinear elliptic equation for the magnetic vector potential . This equation depends on an arbitrary function of that must be specified. With choices of the different arbitrary functions, we obtain analytical solutions of elliptic equation using the Fan subequation method.
The equations of magnetostatic equilibria have been used extensively to model the solar magnetic structure [1–4]. An investigation of a family of isothermal magnetostatic atmospheres with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry is carried out. The force balance consists of the force ( is the magnetic field induction and is the electric current density), the gravitational force, and gas pressure gradient force. However, in many models, the temperature distribution is specified a priori and direct reference to the energy equations is eliminated. In solar physics, the equations of magnetostatic have been used to model diverse phenomena, such as the slow evolution stage of solar flares, or the magnetostatic support of prominences [5, 6]. The nonlinear equilibrium problem has been solved in several cases [7–9].
Recently, Fan and Hon [10] developed an algebraic method, belonging to the sub-equation method to seek more new solutions of nonlinear partial differential equations (NLPDEs) that can be expressed as polynomial in an elementary function which satisfies a more general sub-equation, called Fan sub-equation, than other sub-equations like Riccati equation, auxiliary ordinary equation, elliptic equation, and generalized Riccati equation. As we know, the more general analytical exact solutions of the sub-equation are proposed, the more general corresponding exact solutions of NLPDEs will be obtained. Thus, it is very important how to obtain more new solutions to the sub-equation. Fortunately, the Fan sub-equation method can construct more general exact solutions to the sub-equation that can capture all the solutions of the Riccati equation, auxiliary ordinary equation, elliptic equation, and generalized Riccati equation. Some works using the Fan's technique are presented in [1, 11–16].
In this paper, we obtain the exact travelling wave solutions for the Liouville and sinh-Poisson equations using the Fan sub-equation method. These two models are special cases of magnetostatic atmospheres model. Also in these cases there is force balance between differents forces.
2. The Basic Idea of Fan Subequation Method
In this section, we outline the main steps of Fan sub-equation method [11].
Step 1. For a given nonlinear partial differential equation we consider its travelling wave solutions , , then (2.1) is reduced to a nonlinear ordinary differential equation where a prime denotes the derivative with respect to the variable .
Step 2. Expand the solution of (2.2) in the form where are constants to be determined later and the new variable satisfies the Fan sub-equation where and are constants.
Thus, the derivatives with respect to the variable become the derivatives with respect to the variable as follows:
Step 3. Determine by substituting (2.3) with (2.4) into (2.2) and balancing the linear term of the highest order with the nonlinear term in (2.2).
Step 4. Substituting (2.3) and (2.4) into (2.2) again and collecting all coefficients of , then setting these coefficients to zero will give a set of algebraic equations with respect to .
Step 5. Solve these algebraic equations to obtain . Substituting these results into (2.3) yields the general form of travelling wave solutions.
Step 6. For each solution to (2.4) which depends on the special conditions chosen for the , and , it follows from (2.3) obtained from the above steps that the corresponding exact solution of (2.2) can be constructed.
The relevant magnetohydrostatic equations consist of the equilibrium equation which is coupled with Maxwells equations where , , , and are the gas pressure, the mass density, the magnetic permeability, and the gravitational potential, respectively. It is assumed that the temperature is uniform in space and that the plasma is an ideal gas with equation of state , where is the gas constant and is the temperature. Then the magnetic field can be written as The form of (3.3) for ensures that and there is no mono pole or defect structure.
Equation (3.1) requires the pressure and density to be of the form [4] where is the scale height. Substituting (3.2)–(3.4) into (3.1), we obtain where Equation (3.6) gives where is constant. Substituting (3.7) into (3.4), we obtain Using transformation , (3.5) reduces to These equations have been given in [2].
4. Applications of the Fan Subequation Method
In this section, we will employ the Fan sub-equation method for solving (3.9) for specific forms of the function .
4.1. Liouville Equation
We first consider Liouville equation, which is a special case of (3.9), namely, In order to apply the Fan sub-equation method, we use the wave transformation , and transform (4.1) into the form
We next use the transformation and obtain the nonlinear ordinary differential equation Using Step 3 given above, we get , therefore the solution of (4.3) can be expressed as Following Step 4, we obtain a system of nonlinear algebraic equations for , , and :
Case 1. When , , , (2.4) admits a hyperbolic function solution Thus (4.4) yields the following new solitary wave solution of (2.1) of bell-type where , , , , and are arbitrary constants. Reverting back to the original variables and , we obtain the solution of (4.1) in the form
Case 2. When , , , , (2.4) admits two hyperbolic function solutions and so (4.4) yields one family of solitary travelling wave solutions of (4.1) given by where , , , , and are arbitrary constants.
Case 3. When , , , , (2.4) has two kinds of exact solutions: and (4.4) yields one family of solitary travelling wave solutions of (4.1) given by where , , , , and are arbitrary constants.
Case 4. When , (2.4) admits three Jacobian elliptic doubly periodic solutions and (4.4), respectively, yields two families of Jacobian elliptic doubly periodic wave solutions with , , , , , and being arbitrary constants. Similarly, from (4.4), respectively, we can obtain two families of Jacobian elliptic doubly periodic wave solutions with , , , , , and being arbitrary constants. Similarly, from (4.4), respectively, we can obtain two families of Jacobian elliptic doubly periodic wave solutions with , , , , , and being arbitrary constants.
4.2. The -Poisson Equation
Secondly, we consider sinh-Poisson equation which plays an important role in soliton model with BPS Bound. Also, this equation is a special case of (3.9) and is given by In order to apply the Fan sub-equation method, we use the wave transformation and convert (4.17) into the form We next use the transformation and obtain the equation Applying Step 3, we get , therefore the solution of (4.19) can be expressed as Then using Step 4, we obtain a system of nonlinear algebraic equations for , , and :
Case 1. When , , , (2.4) admits a hyperbolic function solution and (4.20) yields the following new solitary wave solution of (4.17) of bell-type where , , , and are arbitrary constants.
Case 2. When , , , , (2.4) admits two hyperbolic function solutions and (4.20) yields one family of solitary travelling wave solutions of (4.17) given by where , , , and are arbitrary constants.
Case 3. When , , , , (2.4) has two kinds of exact solutions and (4.20) yields one family of solitary travelling wave solutions solitary travelling wave solutions of (4.17) given by where ,, and are arbitrary constants.
Case 4. When , (2.4) admits three Jacobian elliptic doubly periodic solutions and (4.20), respectively, yields two families of Jacobian elliptic doubly periodic wave solutions with , , , , and being arbitrary constants. Similarly, from (4.20), respectively, we can obtain two families of Jacobian elliptic doubly periodic wave solutions with , , , , , and being arbitrary constants. Likewise, from (4.20), respectively, we can get two families of Jacobian elliptic doubly periodic wave solutions with , , , , , and being arbitrary constants.
In this paper, the Fan sub-equation method has been successfully used to obtain some exact travelling wave solutions for the Liouville and sinh-Poisson equations. These exact solutions include the hyperbolic function solutions, trigonometric function solutions. When the parameters are taken as special values, the solitary wave solutions are derived from the hyperbolic function solutions. Thus, this study shows that the Fan sub-equation method is quite efficient and practically well suited for use in finding exact solutions for nonlinear partial differential equations. The reliability of the method and the reduction in the size of computational domain give this method a wider applicability.
A. H. Khater, M. A. El-Attary, M. F. El-Sabbagh, and D. K. Callebaut, “Two-dimensional magnetohydrodynamic equilibria,” Astrophysics and Space Science, vol. 149, no. 2, pp. 217–223, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH
A. H. Khater, D. K. Callebaut, and O. H. El-Kalaawy, “Bäcklund transformations and exact solutions for a nonlinear elliptic equation modelling isothermal magnetostatic atmosphere,” IMA Journal of Applied Mathematics, vol. 65, no. 1, pp. 97–108, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH
I. Lerche and B. C. Low, “Some nonlinear problems in astrophysics,” Physica D. Nonlinear Phenomena, vol. 4, no. 3, pp. 293–318, 1981/82. View at: Publisher Site | Google Scholar
B. C. Low, “Evolving force-free magnetic elds. I—the development of the pre are stage,” The Astrophysical Journal, vol. 212, pp. 234–242, 1977. View at: Google Scholar
X.-H. Wu and J.-H. He, “Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 966–986, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
W. Zwingmann, “Theoretical study of onset conditions for solar eruptive processes,” Solar Physics, vol. 111, pp. 309–331, 1987. View at: Google Scholar
I. Lerche and B. C. Low, “On the equilibrium of a cylindrical plasma supported horizontally by magneticelds in uniform gravity,” Solar Physics, vol. 67, pp. 229–243, 1980. View at: Google Scholar
G. M. Webb, “Isothermal magnetostatic atmospheres. II—similarity solutions with current proportional to the magnetic potential cubed,” The Astrophysical Journal, vol. 327, pp. 933–949, 1988. View at: Google Scholar
G. M. Webb and G. P. Zank, “Application of the sine-Poisson equation in solar magnetostatics,” Solar Physics, vol. 127, pp. 229–252, 1990. View at: Google Scholar
E. G. Fan and Y. C. Hon, “A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves,” Chaos, Solitons & Fractals, vol. 15, no. 3, pp. 559–566, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
D. Feng and K. Li, “Exact traveling wave solutions for a generalized Hirota-Satsuma coupled KdV equation by Fan sub-equation method,” Physics Letters A, vol. 375, no. 23, pp. 2201–2210, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. A. El-Wakil and M. A. Abdou, “The extended Fan sub-equation method and its applications for a class of nonlinear evolution equations,” Chaos, Solitons and Fractals, vol. 36, no. 2, pp. 343–353, 2008. View at: Publisher Site | Google Scholar
E. Yomba, “The modified extended Fan sub-equation method and its application to the
\left(2+1\right)
-dimensional Broer-Kaup-Kupershmidt equation,” Chaos, Solitons and Fractals, vol. 27, no. 1, pp. 187–196, 2006. View at: Publisher Site | Google Scholar
E. Yomba, “The extended Fan's sub-equation method and its application to KdV-MKdV, BKK and variant Boussinesq equations,” Physics Letters A, vol. 336, no. 6, pp. 463–476, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. Zhang and H.-Q. Zhang, “Fan sub-equation method for Wick-type stochastic partial differential equations,” Physics Letters A, vol. 374, no. 41, pp. 4180–4187, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. Zhang and T. Xia, “A further improved extended Fan sub-equation method and its application to the
\left(3+1\right)
-dimensional Kadomstev-Petviashvili equation,” Physics Letters A, vol. 356, no. 2, pp. 119–123, 2006. View at: Publisher Site | Google Scholar
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Train DDPG Agent to Swing Up and Balance Pendulum - MATLAB & Simulink - MathWorks í•œêµ
{\mathit{r}}_{\mathit{t}}
{\mathit{r}}_{\mathit{t}}=-\left({{\mathrm{θ}}_{\mathit{t}}}^{2}+0.1{\stackrel{Ë}{{\mathrm{θ}}_{\mathit{t}}}}^{2}+0.001{\mathit{u}}_{\mathit{t}-1}^{2}\right)
{\mathrm{θ}}_{\mathit{t}}
\stackrel{Ë}{{\mathrm{θ}}_{\mathit{t}}}
{\mathit{u}}_{\mathit{t}-1}
The interface has a continuous action space where the agent can apply torque values between –2 to 2 N·m to the pendulum.
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!! used as default html header if there is none in the selected theme. Graphic integral
The server will give you the curve of a continuous function
f
[-2,2]
. This function is randomly generated by the server, and its expression is hidden from you. You will then be presented with a certain number of other curves, and have to recognize among them that of an anti-derivative function
x\mapsto F(x)
, (or one of
x\mapsto F(-x)
x\mapsto -F(x)
x\mapsto -F(-x)
Description: recognize the graph of the integral of a function. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, analysis, functions, graphing, integral
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offsetA and offsetB are
0
(start at the very beginning)
segsizeA and numsegsA are
1
(copy one segment of one element)
segsizeB is equal to
\mathrm{segsizeA}
(copy each source segment into a target segment of equal size)
numsegsB is equal to
\frac{\mathrm{segsizeA}\mathrm{numsegsA}}{\mathrm{segsizeB}}
(copy the source block into a target block containing an equal number of elements; numsegsB will be exactly numsegsA if segsizeB is also unspecified.)
As an example, copying the upper-right
p
q
block of an
m
C order Matrix A (where
p\le n
q\le m
) corresponds to accessing the elements
m-q+i-1+m\left(j-1\right)
of the underlying rtable data structure (for
i=1..q
j=1..p
). The source offset would be
m-q
, since the first
m-q
elements are skipped. The source increment would be
m
(the number of columns in the input rtable), since each segment begins exactly one row, or
m
elements, after the previous one. The segment size and number of segments would be
q
p
, respectively. To copy this into a
p
q
C order Matrix B, only skipB would need to be computed, since the default values for the other parameters would copy a block of identical size and shape into B. The command to accomplish this would then be BlockCopy(A,m-q,m,q,p,B,q) (where m, n, p, and q are fixed values corresponding to the problem.)
In contrast, the same operation for an
m
Fortran order rtable must be specified differently. The source offset will now be
n\left(m-q\right)
, since all the values in the first m-q columns are skipped. The source increment will be
n
(the number of rows) and the segment size and number of segments become reversed, giving us
q
segments of
p
elements instead of
p
q
elements. If the destination is a
p
q
Fortran order Matrix, the command to accomplish this is BlockCopy(A,n*(m-q),n,p,q,B,p).
\mathrm{with}\left(\mathrm{ArrayTools}\right):
A≔\mathrm{Matrix}\left([[11,12,13,14],[21,22,23,24],[31,32,33,34],[41,42,43,44]]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{11}& \textcolor[rgb]{0,0,1}{12}& \textcolor[rgb]{0,0,1}{13}& \textcolor[rgb]{0,0,1}{14}\\ \textcolor[rgb]{0,0,1}{21}& \textcolor[rgb]{0,0,1}{22}& \textcolor[rgb]{0,0,1}{23}& \textcolor[rgb]{0,0,1}{24}\\ \textcolor[rgb]{0,0,1}{31}& \textcolor[rgb]{0,0,1}{32}& \textcolor[rgb]{0,0,1}{33}& \textcolor[rgb]{0,0,1}{34}\\ \textcolor[rgb]{0,0,1}{41}& \textcolor[rgb]{0,0,1}{42}& \textcolor[rgb]{0,0,1}{43}& \textcolor[rgb]{0,0,1}{44}\end{array}]
B≔\mathrm{Matrix}\left(3,2\right):
\mathrm{BlockCopy}\left(A,8,4,3,2,B,3\right)
B
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{13}& \textcolor[rgb]{0,0,1}{14}\\ \textcolor[rgb]{0,0,1}{23}& \textcolor[rgb]{0,0,1}{24}\\ \textcolor[rgb]{0,0,1}{33}& \textcolor[rgb]{0,0,1}{34}\end{array}]
C≔\mathrm{Matrix}\left(5,3\right):
\mathrm{BlockCopy}\left(A,8,4,3,2,C,2,5\right)
C
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{13}& \textcolor[rgb]{0,0,1}{14}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{23}& \textcolor[rgb]{0,0,1}{24}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{33}& \textcolor[rgb]{0,0,1}{34}& \textcolor[rgb]{0,0,1}{0}\end{array}]
V≔\mathrm{Vector}\left(6\right):
\mathrm{BlockCopy}\left(A,8,4,3,2,V,0,6,6,1\right)
V
[\begin{array}{c}\textcolor[rgb]{0,0,1}{13}\\ \textcolor[rgb]{0,0,1}{23}\\ \textcolor[rgb]{0,0,1}{33}\\ \textcolor[rgb]{0,0,1}{14}\\ \textcolor[rgb]{0,0,1}{24}\\ \textcolor[rgb]{0,0,1}{34}\end{array}]
A≔\mathrm{Matrix}\left([[11,12,13,14],[21,22,23,24],[31,32,33,34],[41,42,43,44]],\mathrm{order}=\mathrm{C_order}\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{11}& \textcolor[rgb]{0,0,1}{12}& \textcolor[rgb]{0,0,1}{13}& \textcolor[rgb]{0,0,1}{14}\\ \textcolor[rgb]{0,0,1}{21}& \textcolor[rgb]{0,0,1}{22}& \textcolor[rgb]{0,0,1}{23}& \textcolor[rgb]{0,0,1}{24}\\ \textcolor[rgb]{0,0,1}{31}& \textcolor[rgb]{0,0,1}{32}& \textcolor[rgb]{0,0,1}{33}& \textcolor[rgb]{0,0,1}{34}\\ \textcolor[rgb]{0,0,1}{41}& \textcolor[rgb]{0,0,1}{42}& \textcolor[rgb]{0,0,1}{43}& \textcolor[rgb]{0,0,1}{44}\end{array}]
B≔\mathrm{Matrix}\left(3,2,\mathrm{order}=\mathrm{C_order}\right):
\mathrm{BlockCopy}\left(A,2,4,2,3,B,2\right)
B
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{13}& \textcolor[rgb]{0,0,1}{14}\\ \textcolor[rgb]{0,0,1}{23}& \textcolor[rgb]{0,0,1}{24}\\ \textcolor[rgb]{0,0,1}{33}& \textcolor[rgb]{0,0,1}{34}\end{array}]
\mathrm{J1}≔\mathrm{Matrix}\left([[1,0],[0,1]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{J1}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}\end{array}]
\mathrm{J2}≔\mathrm{Matrix}\left([[0,1],[1,0]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{J2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\end{array}]
J≔\mathrm{Matrix}\left(4,4\right)
\textcolor[rgb]{0,0,1}{J}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}]
\mathrm{BlockCopy}\left(\mathrm{J1},0,2,2,2,J,0,4,2,2\right)
\mathrm{BlockCopy}\left(\mathrm{J2},0,2,2,2,J,2,4,2,2\right)
\mathrm{BlockCopy}\left(\mathrm{J2},0,2,2,2,J,8,4,2,2\right)
\mathrm{BlockCopy}\left(\mathrm{J1},0,2,2,2,J,10,4,2,2\right)
J
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}\end{array}]
A≔\mathrm{Matrix}\left(5,3,\left(i,j\right)↦10\cdot i+j,\mathrm{order}=\mathrm{C_order}\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{11}& \textcolor[rgb]{0,0,1}{12}& \textcolor[rgb]{0,0,1}{13}\\ \textcolor[rgb]{0,0,1}{21}& \textcolor[rgb]{0,0,1}{22}& \textcolor[rgb]{0,0,1}{23}\\ \textcolor[rgb]{0,0,1}{31}& \textcolor[rgb]{0,0,1}{32}& \textcolor[rgb]{0,0,1}{33}\\ \textcolor[rgb]{0,0,1}{41}& \textcolor[rgb]{0,0,1}{42}& \textcolor[rgb]{0,0,1}{43}\\ \textcolor[rgb]{0,0,1}{51}& \textcolor[rgb]{0,0,1}{52}& \textcolor[rgb]{0,0,1}{53}\end{array}]
V≔\mathrm{Vector}[\mathrm{row}]\left(9\right)
\textcolor[rgb]{0,0,1}{V}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccccccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}]
\mathrm{BlockCopy}\left(A,0,6,3,3,V,0,9,9,1\right)
V
[\begin{array}{ccccccccc}\textcolor[rgb]{0,0,1}{11}& \textcolor[rgb]{0,0,1}{12}& \textcolor[rgb]{0,0,1}{13}& \textcolor[rgb]{0,0,1}{31}& \textcolor[rgb]{0,0,1}{32}& \textcolor[rgb]{0,0,1}{33}& \textcolor[rgb]{0,0,1}{51}& \textcolor[rgb]{0,0,1}{52}& \textcolor[rgb]{0,0,1}{53}\end{array}]
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Alternating Current, Revision Notes: ICSE Class 12-science PHYSICS, Physics Part I - Meritnation
Alternating current is an electric current whose magnitude changes with time continuously and which reverses its direction periodically. It is mathematically represented as
I={I}_{\mathrm{o}} \mathrm{cos}\omega t \mathrm{or} I={I}_{\mathrm{o}} \mathrm{sin\omega t}
Flexibility of converting from one value to other using transformer
Can be transmitted over long distances economically as well as without much power loss
Mean Value or Average value of AC (…
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CDS 110b: Norms of Signals and Systems - Murray Wiki
CDS 110b: Norms of Signals and Systems
(Redirected from CDS 110b:Norms of Signals and Systems)
This lecture provides an introduction to some of the signals and systems concepts required for the study of robust (
{\displaystyle H_{\infty }}
) control.
Norms of linear systems (con't)
Blackboard lecture; no slides. MP3 lost (technical error)
Lecture Notes on system norms
HW #6 (due 22 Feb)
Q: So you can do pole zero cancellations?
As long as they don't occur in the closed right half plane, pole zero cancellations are OK from the point of view of stability. It is generally not a good idea to rely on exact cancellations even if they are stable cancellations (LHP), but they are relatively benign.
Exercise: try plotting the frequency response for
{\displaystyle P(s)={\frac {s-1}{s-1+\epsilon }}}
Q: I'm not sure if I really understand what "sup" is
Formally, the supremum (sup) of a set is the smallest real number that is larger than or equal to every element in the set. For a real-valued function
{\displaystyle f(x)}
{\displaystyle \sup _{x}f(x)}
is smallest real number
{\displaystyle y}
{\displaystyle f(x)\leq y}
. Here's a pretty good Wikipedia article on supremum.
Retrieved from "https://murray.cds.caltech.edu/index.php?title=CDS_110b:_Norms_of_Signals_and_Systems&oldid=3055"
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\phantom{\rule{thickmathspace}{0ex}}\sigma
{\mathbb{R} }^{n}
n
\phantom{\rule{thickmathspace}{0ex}}\sigma
\phantom{\rule{thickmathspace}{0ex}}\sigma
p\in {\mathbb{R} }^{n}
\phantom{\rule{thickmathspace}{0ex}}\sigma (p)=p
\phantom{\rule{thickmathspace}{0ex}}\sigma
\phantom{\rule{thickmathspace}{0ex}}\sigma
Description: find the fixed point of an affine transformation. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
Keywords: CFAI,interactive math, server side interactivity, linear_algebra, affine_geometry, matrix, translation
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GcdFreeBasis - Maple Help
Home : Support : Online Help : Mathematics : Algebra : Polynomials : PolynomialTools : GcdFreeBasis
compute a gcd free basis of a set or list of polynomials
GcdFreeBasis(S)
The GcdFreeBasis command uses repeated gcd computations to compute a gcd free basis B of the polynomials in S. B satisfies the following properties.
Each polynomial in S can be written as a constant times a product of polynomials from B.
The polynomials in B are pairwise coprime, that is, their gcd is constant.
With respect to cardinality, B is minimal with these properties. (This is equivalent to saying that B can be computed using gcds and divisions only, but not factorization.)
The gcd free basis is unique up to ordering and multiplication of the basis elements by constants.
GcdFreeBasis can handle the same types of coefficients as the Maple function gcd.
If S is a set, then the output is a set as well. If S is a list, then the output is also a list. The ordering of the elements in the result is not determined in either case.
Zero polynomials and constants are ignored. In particular, for a constant const, GcdFreeBasis([const]) returns [ ]. The empty set or list is a valid input and is returned unchanged.
f
is considered constant by GcdFreeBasis if
\mathrm{degree}\left(f\right)
0
\mathrm{with}\left(\mathrm{PolynomialTools}\right):
\mathrm{GcdFreeBasis}\left([abcd,abce,abde]\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}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}]
\mathrm{GcdFreeBasis}\left({{x}^{6}-1,{x}^{10}-1,{x}^{15}-1}\right)
{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\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}\textcolor[rgb]{0,0,1}{+}\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}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\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}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\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}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{8}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}
Bach, Eric.; Driscoll, James.; and Shallit, Jeffrey. "Factor Refinement." Journal of Algorithms Vol. 15(2), (1993): 199-222.
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Assessment of an Exponential Scaling Relationship for Backflow Length in Brain Tissue | SBC | ASME Digital Collection
Alejandro Orozco,
Orozco, A, Smith, JH, & García, JJ. "Assessment of an Exponential Scaling Relationship for Backflow Length in Brain Tissue." Proceedings of the ASME 2013 Summer Bioengineering Conference. Volume 1A: Abdominal Aortic Aneurysms; Active and Reactive Soft Matter; Atherosclerosis; BioFluid Mechanics; Education; Biotransport Phenomena; Bone, Joint and Spine Mechanics; Brain Injury; Cardiac Mechanics; Cardiovascular Devices, Fluids and Imaging; Cartilage and Disc Mechanics; Cell and Tissue Engineering; Cerebral Aneurysms; Computational Biofluid Dynamics; Device Design, Human Dynamics, and Rehabilitation; Drug Delivery and Disease Treatment; Engineered Cellular Environments. Sunriver, Oregon, USA. June 26–29, 2013. V01AT07A007. ASME. https://doi.org/10.1115/SBC2013-14121
Convection enhanced delivery is a protocol to deliver large volumes of drugs over localized zones of the brain for the treatment of diseases and tumors. Brain infusion experiments at higher flow rates showed backflow, in which an annular zone is formed outside the catheter and the infused drug preferentially flows toward the surface of the brain rather than through the tissue in the direction of the area targeted for delivery. The foundational model of Morrison et al. [1] considered the deformation of the tissue around the external boundary of the catheter, the axial flow in the annular gap formed around the cannula, and the radial flow from this annular region into the porous tissue in the development of an exponential correlation for backflow length L:
L∝Q0.6R0.8rc0.8G-0.6μ-0.2,
where Q is the infusion flow rate, R is a tissue hydraulic resistance, rc is the catheter radius, G is the tissue shear modulus, and μ is the fluid viscosity. However, this formula was derived under some limiting assumptions, such as considering the solid phase of the infused tissue as a linearly elastic material under infinitesimal deformations, whereas mechanical testing has shown large deformations under physiological loadings [2, 3].
Biological tissues, Brain, Catheters, Deformation, Flow (Dynamics), Drugs, Axial flow, Convection, Diseases, Fluids, Mechanical testing, Physiology, Radial flow, Shear modulus, Tumors, Viscosity
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A critical phenomenon in the two-matrix model in the quartic/quadratic case
1 June 2013 A critical phenomenon in the two-matrix model in the quartic/quadratic case
Maurice Duits, Dries Geudens
We study a critical behavior for the eigenvalue statistics in the two-matrix model in the quartic/quadratic case. For certain parameters, the eigenvalue distribution for one of the matrices has a limit that vanishes like a square root in the interior of the support. The main result of the paper is a new kernel that describes the local eigenvalue correlations near that critical point. The kernel is expressed in terms of a
4×4
Riemann–Hilbert problem related to the Hastings–McLeod solution of the Painlevé II equation. We then compare the new kernel with two other critical phenomena that appeared in the literature before. First, we show that the critical kernel that appears in case of quadratic vanishing of the limiting eigenvalue distribution can be retrieved from the new kernel by means of a double scaling limit. Second, we briefly discuss the relation with the tacnode singularity in noncolliding Brownian motions that was recently analyzed. Although the limiting density in that model also vanishes like a square root at a certain interior point, the process at the local scale is different from the process that we obtain in the two-matrix model.
Maurice Duits. Dries Geudens. "A critical phenomenon in the two-matrix model in the quartic/quadratic case." Duke Math. J. 162 (8) 1383 - 1462, 1 June 2013. https://doi.org/10.1215/00127094-2208757
Secondary: 15B52 , 30E25 , 31A05 , 42C05
Maurice Duits, Dries Geudens "A critical phenomenon in the two-matrix model in the quartic/quadratic case," Duke Mathematical Journal, Duke Math. J. 162(8), 1383-1462, (1 June 2013)
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Cross-entropy loss for classification tasks - MATLAB crossentropy - MathWorks Switzerland
{\text{loss}}_{j}=-\left({T}_{j}\text{ln}{Y}_{j}+\left(1-{T}_{j}\right)\text{ln}\left(1-{Y}_{j}\right)\right),
\text{loss}=\frac{1}{N}\sum _{j}{m}_{j}{w}_{j}{\text{loss}}_{j},
{\text{loss}}_{j}^{*}={m}_{j}{w}_{j}{\text{loss}}_{j}
\text{loss}=-\frac{1}{N}\sum _{n=1}^{N}\sum _{i=1}^{K}{T}_{ni}\text{ln}{Y}_{ni},
\text{loss}=-\frac{1}{N}\sum _{n=1}^{N}\sum _{i=1}^{K}\left({T}_{ni}\mathrm{ln}\left({Y}_{ni}\right)+\left(1-{T}_{ni}\right)\mathrm{ln}\left(1-{Y}_{ni}\right)\right),
\text{loss}=-\frac{1}{N}\sum _{n=1}^{N}\sum _{i=1}^{K}{w}_{i}{T}_{ni}\text{ln}{Y}_{ni},
\text{loss}=-\frac{1}{N}\sum _{n=1}^{N}\sum _{t=1}^{S}{m}_{nt}\sum _{i=1}^{K}{T}_{nti}\text{ln}{Y}_{nti},
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EUDML | Convergence in incomplete market models. EuDML | Convergence in incomplete market models.
Convergence in incomplete market models.
Kopp, P.Ekkehard; Wellmann, Volker
Kopp, P.Ekkehard, and Wellmann, Volker. "Convergence in incomplete market models.." Electronic Journal of Probability [electronic only] 5 (2000): Paper No. 15, 25 p., electronic only-Paper No. 15, 25 p., electronic only. <http://eudml.org/doc/120952>.
author = {Kopp, P.Ekkehard, Wellmann, Volker},
keywords = {incomplete markets; -convergence; nonstandard analysis; minimal martingale measure; risk minimization; -convergence},
title = {Convergence in incomplete market models.},
AU - Kopp, P.Ekkehard
AU - Wellmann, Volker
TI - Convergence in incomplete market models.
KW - incomplete markets; -convergence; nonstandard analysis; minimal martingale measure; risk minimization; -convergence
incomplete markets,
{D}^{2}
-convergence, nonstandard analysis, minimal martingale measure, risk minimization,
{D}^{2}
Nonstandard measure theory
Articles by Kopp
Articles by Wellmann
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Alphabet (formal languages) — Wikipedia Republished // WIKI 2
Base set of symbols with which a language is formed
In formal language theory, an alphabet is a non-empty set of symbols/glyphs, typically thought of as representing letters, characters, or digits[1] but among other possibilities the "symbols" could also be a set of phonemes (sound units). Alphabets in this technical sense of a set are used in a diverse range of fields including logic, mathematics, computer science, and linguistics. An alphabet may have any cardinality ("size") and depending on its purpose maybe be finite (e.g., the alphabet of letters "a" through "z"), countable (e.g.,
{\displaystyle \{v_{1},v_{2},\ldots \}}
{\displaystyle \{v_{x}:x\in \mathbb {R} \}}
Strings, also known as "words", over an alphabet are defined as a sequence of the symbols from the alphabet set.[2] For example, the alphabet of lowercase letters "a" through "z" can be used to form English words like "iceberg" while the alphabet of both upper and lower case letters can also be used to form proper names like "Wikipedia". A common alphabet is {0,1}, the binary alphabet, and a "00101111" is an example of a binary string. Infinite sequence of symbols may be considered as well (see Omega language).
symbol, alphabet, string | basic definations | TOC | Lec-2 | Bhanu Priya
If L is a formal language, i.e. a (possibly infinite) set of finite-length strings, the alphabet of L is the set of all symbols that may occur in any string in L. For example, if L is the set of all variable identifiers in the programming language C, L’s alphabet is the set { a, b, c, ..., x, y, z, A, B, C, ..., X, Y, Z, 0, 1, 2, ..., 7, 8, 9, _ }.
{\displaystyle \Sigma }
{\displaystyle n}
{\displaystyle \Sigma }
{\displaystyle \Sigma ^{n}}
{\textstyle \bigcup _{i\in \mathbb {N} }\Sigma ^{i}}
of all finite strings (regardless of their length) is indicated by the Kleene star operator as
{\displaystyle \Sigma ^{*}}
{\displaystyle \Sigma }
{\displaystyle \Sigma ^{\omega }}
{\displaystyle \Sigma }
{\displaystyle \Sigma ^{\infty }}
{\displaystyle \Sigma ^{\ast }\cup \Sigma ^{\omega }}
For example, using the binary alphabet {0,1}, the strings ε, 0, 1, 00, 01, 10, 11, 000, etc. are all in the Kleene closure of the alphabet (where ε represents the empty string).
Alphabets are important in the use of formal languages, automata and semiautomata. In most cases, for defining instances of automata, such as deterministic finite automata (DFAs), it is required to specify an alphabet from which the input strings for the automaton are built. In these applications, an alphabet is usually required to be a finite set, but is not otherwise restricted.
When using automata, regular expressions, or formal grammars as part of string-processing algorithms, the alphabet may be assumed to be the character set of the text to be processed by these algorithms, or a subset of allowable characters from the character set.
Combinatorics on words
^ Ebbinghaus, H.-D.; Flum, J.; Thomas, W. (1994). Mathematical Logic (2nd ed.). New York: Springer. p. 11. ISBN 0-387-94258-0. By an alphabet
{\displaystyle {\mathcal {A}}}
^ Rautenberg, Wolfgang (2010). A Concise Introduction to Mathematical Logic (PDF) (Third ed.). Springer. p. xx. ISBN 978-1-4419-1220-6. If 𝗔 is an alphabet, i.e., if the elements 𝐬 ∈ 𝗔 are symbols or at least named symbols, then the sequence (𝐬1,...,𝐬n)∈𝗔n is written as 𝐬1···𝐬n and called a string or a word over 𝗔.
John E. Hopcroft and Jeffrey D. Ullman, Introduction to Automata Theory, Languages, and Computation, Addison-Wesley Publishing, Reading Massachusetts, 1979. ISBN 0-201-02988-X.
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A Summary of State-space Models
1. Bayesian Filter
In this article, I summarize some famous state-space models. Here I won't go into details but focus on the entire map to get an overview. All these state-space models originate from Bayesian filter. In these models, two stochastic processes are considered. The first process is states
x_t
, and the second process is observations or measurements
y_t
t
means "time" but generally the processes are not restricted to time series. We are interested in the true value of states, but we can only observe the value of observations. Therefore, state-space models aim to estimate states based on observations. Two relationships should be addressed:
The state-to-state probability
P(x_t | x_{1:t-1})
The state-to-observation probability
P(y_t | x_t)
No direct relationship exists between any two observations. A common assumption is the Markov property, which assumes that the current state depends only on the previous state, namely
P(x_t | x_{1:t-1})=P(x_t|x_{t-1})
2. Prediction and Updating
State-space model has online algorithms with recursive two steps. Prediction is to estimate the posterior distribution
p(x_t | y_{1:t-1})
based on the distribution
p(x_{t-1}| y_{1:t-1})
, according to the state-to-state probability
P(x_t|x_{t-1})
p(x_t|y_{1:t-1})=\int p(x_t|x_{t-1}) p(x_{t-1}|y_{1:t-1}) dx_{t-1}
Updating is to update the previous distribution based on the latest observation
y_t
p(x_t | y_{1:t})= p(y_t|x_t) p(x_t|y_{1:t-1}) /p(y_t) \propto p(y_t|x_t) p(x_t|y_{1:t-1})
3. Considerations in Modeling
Bayesian filters estimate
x_t
by the posterior distribution
p(x_t | y_{1:t})
. Usually the state-to-state probability and state-to-observation probability cannot be obtained directly when modeling practical problems. Instead, they should be inferred from prediction model
x_t=f(x_{t-1})
and measurement model
y_t=g(x_t)
. And a series of questions must be answered:
x_t
discrete or continuous?
What is the distribution of
x_t
Is the prediction model linear or nonlinear?
Is the measurement model linear or nonlinear? According to different answers to these questions, we have different filters as follows.
4. Classification of Bayesian filters
Based on whether the state
x_t
is discrete or continuous, Bayesian filters are divided into discrete filters and continuous filters. When state
x_t
can only be discrete values, the state-to-state probability can be expressed by transition matrix
A=[a_{i,j}]
a_{i,j}=P(x_t =j|x_{t-1}=i)
Based on whether the distribution of
x_t
is assumed to be a specific format, continuous Bayesian filters are divided into parametric and nonparametric filters. For example , in Gaussian filters, the distribution of
x_t
is assumed to be multivariate normal distribution. With this assumption, the posterior distribution
p(x_t|y_{1:t})
can be expressed in close-form explicitly. On the other side, non-parametric filters don't make any assumptions in the distribution of
x_t
, but use some techniques to approximate the distribution. For example, the distribution of
x_t
can be expressed by a histogram (Histogram filter) or a lot of samples (Particle filter) drawn from the target distribution. Non-parametric filters approximate the distribution, and put no restrictions on prediction model
x_t=f(x_{t-1})
y_t=g(x_t)
, thus flexible in various situations. However, the computation load is heavy since there is no close-form expression, and the better of the approximation, the heavier of the computation burden.
Gaussian filters assume the distribution of
x_t
to be multivariate normal distribution. In classical Kalman filter, the prediction model
x_t=f(x_{t-1})
and the measurement model
y_t=g(x_t)
are assumed linear in order to maintain normality. Specifically,
x_t=A x_{t-1}+ \epsilon_t, y_t=B x_t + \delta_t
. Derivatives of Kalman filter such as Extended Kalman filter and Uncented Kalman filter relax the linear relationship assumption, but approximate by linearization techniques such as Taylor expansion. Information filter and its derivatives are essentially the same to Kalman filter family, with information expression of multivariate normal distribution
\Omega=\Sigma^{-1}, \xi=\Omega \mu
Hybrid filters are mixture of parametric and non-parametric filters, with some dimensions of state assumed to be in specific format and other dimensions to be expressed in non-parametric techniques.
All Right Reserved, Changyue Song ©2020
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N-Body in Unity With Threads – David Joiner – Computational Science Educator
N-Body in Unity With Threads
This model will extend from our previous example using TimestepModel. We will build a class that uses a threaded update method, like before, and will use that update method to integrate forward a system of orindary differential equations.
Our simple example showed performance gains by separating out the computation from the Unity scene loop, but how will a more complicated problem behave?
Let’s build something that will require a great deal more work per step. This example will walk through implementing a solution and visualization of the gravitational N-Body problem.
The gravitational N-Body problem describes the behavior of objects in space attracted to each other via gravity. Each object will feel a force of
For simplicity, we’re going to use a unit system in our simulation such that $G=1$, each mass is equal and $\sum{m_i}=M=1$, and all of our objects start off in a space roughly 1 unit around the origin.
(As an aside, scaling your units helps to keep your numbers at an order of magnitude near unity, which is good for preventing overflow and underflow errors in your computation, and keeps your scene sizes closer to typical in your Unity visualization. The easiest way to scale the gravitational N-Body problem is to set a value of G that matches the other units you want to use. If you want to use specific values in your own choice of units, the easiest thing to do is figure out what $G$ is in your unit system. Since G is typically given in SI, just figure out your units in SI values, and then
G_{scaled}=6.6740831e-11 UnitMass_{SI} UnitTime^2_{SI} UnitLength^{-3}_{SI}
Our overall process will be to create an empty GameObject in our scene, and attach a script to it called Model that will extend TimestepModel. We will create and initialize an array of values for the positions and velocities of all of our objects. We will create a second script that extends Integrator called NBody. In this script, we will create a RatesOfChange routine that implements the forces listed above. In Model, we will have a member variable of type NBody. Model’s TakeStep method will have our member variable of type NBody call the RK4Step method. In Model, we will create a spherical game object via scripting for each body, and in Model’s update routine, we will at each screen update the position of each sphere to the corresponding coordinates of each body in the calculation.
Start by creating a new 3D Unity project, and add in the Unity Modeling Toolkit package, or copy TimestepModel.cs and Integrator.cs from previous blog posts.
Add a empty GameObject in the scene, and name it Model. Add a “New Script” component to Model, also called Model. Save your scene.
Open Model.cs and change the class definition to extend TimestepModel instead of MonoBehaviour. Implement an empty TakeStep routine.
public class Model : TimestepModel {
Let’s also create a new C# script in the project panel, not attached to any object, called NBody. Open this, delete the Update and Start methods, and change the extend keyword to extend from Integrator. Add in an empty RatesOfChange override method.
public class NBody : Integrator {
The NBody model will need a few member variables to describe our problem, such as the number of objects, and the parameters of the system. Also, let’s create a routine to place the setting of initial conditions and to create an array $x$ for the state variables.
Note that we have 6 equations for each 1 body. This is because the state of a single body includes $\vec{r} = (x,y,z)$ and $\vec{v} = (v_x, v_y,v_z)$.
Our initial conditions will be random positions and velocities, and the total mass will be evenly divided over the bodies.
int nBodies;
public double [] x; // public for easy access from Model
double [] m;
public void setIC(int nBodies, double M, double G) {
this.nBodies = nBodies;
x = new double[6 * nBodies]; // create state variables
m = new double[nBodies];
for (int i = 0; i < nBodies; i++) {
m [i] = M / nBodies;
x [i * 6 + 0] = Random.Range (-1.0f, 1.0f);
Init (6 * nBodies); // allocate Integrator work arrays
For the rates of change, we want to populate the array xdot with values of $\dot{x}$, $dot{y}$, $dot{z}$, $dot{v}_x$, $\dot{v}_y$, and $\dot{v}_z$, and to do this for every object–so the arrays x and xdot are both $6 \times nBodies$ long. We could arrange the arrays in one of two ways, we could have all the information for one body in a block of data 6 numbers long, or we could have all of the x values for every object in one block, all of the y values in one block, etc. It’s more likely that for a given computation I need to know everything about an object than it is for me to suddenly need to know all of my x components without y and z, so keeping all of an objects information close together is better for memory efficiency. (Memory gets pulled into cache on the CPU in blocks, so if the second piece of information you need is near the first in memory, it’s already in cache and thus is accessed faster.)
To index our arrays, then, we will use the index $[i*6+j]$ to access the jth component (0-5 for x,y,z,vx,vy,vz respectively) of the ith object.
The first step in setting the rates of change will be to set $\dot{\vec{x}}$. This is, by definition, just $\vec{v}$, which is part of our state variables already, so we can just use the current value of $v$ to set $\dot{x}$.
Next, we need to set the accelerations, $\dot{v}$. This will be done one force at a time in summative fashion, so begin by zeroing out all of the accelerations. Then, we will loop over all of the possible interactions between objects, and use the fact that forces are equal and opposite to set the acceleration term for each object affected by each force.
Note that our force terms as vectors all contained the term
If $\Delta x = x_j - x_i$, $\Delta y = y_j - y_i$, and $\Delta z = z_j - z_i$, then we can write $\Delta r = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$. With that,
This makes the calculation and computation of the vector terms of the acceleration very straightforward.
// set xdot
xdot [i * 6 + 0] = x [i * 6 + 3];
// zero out vdot
xdot [i * 6 + 3] = 0;
// force terms
for (int j = i + 1; j < nBodies; j++) {
double dx = x [j * 6 + 0] - x [i * 6 + 0];
double dy = x [j * 6 + 1] - x [i * 6 + 1];
double dz = x [j * 6 + 2] - x [i * 6 + 2];
double dr2 = dx * dx + dy * dy + dz * dz;
double dr = System.Math.Sqrt (dr2);
xdot [i * 6 + 3] += G * m [j] / dr2 * dx / dr;
xdot [i * 6 + 4] += G * m [j] / dr2 * dy / dr;
xdot [i * 6 + 5] += G * m [j] / dr2 * dz / dr;
xdot [j * 6 + 3] -= G * m [i] / dr2 * dx / dr;
xdot [j * 6 + 4] -= G * m [i] / dr2 * dy / dr;
xdot [j * 6 + 5] -= G * m [i] / dr2 * dz / dr;
There are some efficiency improvements we could make in the loop structure here, though they might run the risk of hurting readability for the purposes of this example.
With this, we can add in some control to run this calculation in Model, and connect it to a visualization.
In Model, let’s create a member variable of type NBody called nBody, run setIC, and set up the step routines in TakeStep.
public float M = 1;
public int nBodies = 10;
NBody nBody;
nBody.RK4Step (nBody.x, modelT, dt);
nBody = new NBody ();
nBody.setIC (nBodies, M, G);
We’ve got everything except the visualization. If you wanted to test at this point, you might add a Debug.Log() statement in Model.Update.
To add in a visualization, we could create a prefab gameobject for the visual for each body, Instantiate a bunch of them in Model.Start, and position them to catch up with the threaded model in Model.Update.
Create a sphere in your scene’s Hierarchy panel. Name it Body. Drag it into Project->Assets to make a prefab. Delete it from your scene.
Highlight the prefab body. Remove the collider component in the inspector panel.
In Model, add a public member variable of type GameObject. Name it bodyPRE. Save Model.cs, go back to the scene, and drag your prefab into the now open spot in the inspector window when Model is selected in the scene. Also create an array of gameobjects in the member variables called bodies.
In Model.Start, let’s add a new instance of bodyPRE for every object, and store them in bodies. In Update, set transform.position of each element of bodies.
public float M = 10;
public int nBodies = 1000;
public GameObject bodyPRE;
GameObject [] bodies;
bodies = new GameObject[nBodies];
bodies [i] = Instantiate (bodyPRE);
Vector3 newPos = new Vector3 (
(float)nBody.x [i * 6 + 0],
(float)nBody.x [i * 6 + 2]);
bodies [i].transform.position = newPos;
You may find you want to drop the value of ModelDT, or change the mass, or number of objects. For a large number of objects, it is very likely the spheres being drawn are too big, and you can select the prefab and change its scale in x, y, and z to accommodate this.
Running at DT = 0.01, M=10, and NBodies = 100 (be sure to set in the editor, not in the code, as editor can override public variables!), we get a high number of ejections from the system due to close encounters. Many of these close encounters are in fact numerical artifacts due to a discrete stepping algorithm.
Let’s add a softening factor to our calculation, and link it up to the GUI.
In NBody, add a member variable “soft” to the class, and add “soft*soft” to the value of dr2 in RatesOfChange. In setIC, allow a value to be passed to set soft.
double soft;
public void setIC(int nBodies, double M, double G, double soft) {
double dr2 = dx * dx + dy * dy +
dz * dz + soft*soft;
In Model, add a public variable to the class so that it shows in the editor, and pass it to nBody through setIC.
public float soft = 0.001f;
nBody.setIC (nBodies, M, G, soft);
Run this for different values of soft. Notice that you tend to have fewer ejections using the softening factor. This isn’t always the case–sometimes close encounters, particularly three body encounters, really do result in ejections. The sue of the softening factor, however, should help to limit the number of these that are numerical artifacts.
Try cranking up N to something bigger, like 5,000. Save before running. Notice that the first step may take a substantial amount of time, but that with threading on you can still operate in the Unity editor. Save. Turn off threading. Save. Try to run again (be prepared to have to force quit the editor). Notice that you lose control over the editor while the steps are calculating. Try to press pause in order to regain control of the editor. If you can, quit the model. Turn threading back on. Save. If you can’t, force quit. Reopen the model. Turn threading back on. Save.
Play around with different values of N to see how this threaded version of the model has both better performance through threading at small N and better GUI control when calculating steps at large N.
(As I type this I am force-quitting Unity…)
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Theorem in game theory about whether Bayesian agents can agree to disagree
Aumann's agreement theorem was stated and proved by Robert Aumann in a paper titled "Agreeing to Disagree",[1] which introduced the set theoretic description of common knowledge. The theorem concerns agents who share a common prior and update their probabilistic beliefs by Bayes' rule. It states that if the probabilistic beliefs of such agents, regarding a fixed event, are common knowledge then these probabilities must coincide. Thus, agents cannot agree, that is have common knowledge of a disagreement over the posterior probability of a given event.
4 The non-probabilistic case
The model used in Aumann[1] to prove the theorem consists of a finite set of states
{\displaystyle S}
with a prior probability
{\displaystyle p}
, which is common to all agents. Agent
{\displaystyle a}
's knowledge is given by a partition
{\displaystyle \Pi _{a}}
{\displaystyle S}
. The posterior probability of agent
{\displaystyle a}
{\displaystyle p_{a}}
{\displaystyle p}
{\displaystyle \Pi _{a}}
. Fix an event
{\displaystyle E}
{\displaystyle X}
be the event that for each
{\displaystyle a}
{\displaystyle p_{a}(E)=x_{a}}
. The theorem claims that if the event
{\displaystyle C(X)}
{\displaystyle X}
is common knowledge is not empty then all the numbers
{\displaystyle x_{a}}
are the same. The proof follows directly from the definition of common knowledge. The event
{\displaystyle C(X)}
is a union of elements of
{\displaystyle \Pi _{a}}
{\displaystyle a}
{\displaystyle a}
{\displaystyle p(E|C(x))=x_{a}}
. The claim of the theorem follows since the left hand side is independent of
{\displaystyle a}
. The theorem was proved for two agents but the proof for any number of agents is similar.
Monderer and Samet[2] relaxed the assumption of common knowledge and assumed instead common
{\displaystyle p}
-belief of the posteriors of the agents. They gave an upper bound of the distance between the posteriors
{\displaystyle x_{a}}
. This bound approaches 0 when
{\displaystyle p}
Ziv Hellman[3] relaxed the assumption of a common prior and assumed instead that the agents have priors that are
{\displaystyle \varepsilon }
-close in a well defined metric. He showed that common knowledge of the posteriors in this case implies that they are
{\displaystyle \varepsilon }
-close. When
{\displaystyle \varepsilon }
goes to zero, Aumann's original theorem is recapitulated.
Knowledge which is defined in terms of partitions has the property of negative introspection. That is, agents know that they do not know what they do not know. However, it is possible to show that it is impossible to agree to disagree even when knowledge does not have this property. [4], [5]
Halpern and Kets[6] argued that players can agree to disagree in the presence of ambiguity, even if there is a common prior. However, allowing for ambiguity is more restrictive than assuming heterogeneous priors.
The impossibility of agreeing to disagree, in Aumann's theorem, is a necessary condition for the existence of a common prior. A stronger condition can be formulated in terms of bets. A bet is a set of random variables
{\displaystyle f_{a}}
, one for each agent
{\displaystyle a}
, such the
{\displaystyle \sum _{a}f_{a}=0}
. The bet is favorable to agent
{\displaystyle a}
{\displaystyle s}
if the expected value of
{\displaystyle f_{a}}
{\displaystyle s}
is positive. The impossibility of agreeing on the profitability of a bet is a stronger condition than the impossibility of agreeing to disagree, and moreover, it is a necessary and sufficient condition for the existence of a common prior. [7], [8]
A question arises whether such an agreement can be reached in a reasonable time and, from a mathematical perspective, whether this can be done efficiently. Scott Aaronson has shown that this is indeed the case.[9] Of course, the assumption of common priors is a rather strong one and may not hold in practice. However, Robin Hanson has presented an argument that Bayesians who agree about the processes that gave rise to their priors (e.g., genetic and environmental influences) should, if they adhere to a certain pre-rationality condition, have common priors.[10]
The non-probabilistic case[edit]
^ a b Aumann, Robert J. (1976). "Agreeing to Disagree" (PDF). The Annals of Statistics. 4 (6): 1236–1239. doi:10.1214/aos/1176343654. ISSN 0090-5364. JSTOR 2958591.
^ Monderer, dov; Dov Samet (1989). "Approximating common knowledge with common beliefs". Games and Economic Behavior. 1 (2): 170–190.
^ Hellman, Ziv (2013). "Almost Common Priors". International Journal of Game Theory. 42 (2): 399–410. doi:10.1007/s00182-012-0347-5.
^ Bacharach, Michael (1985). "Some extensions of a claim of Aumann in an axiomatic model of knowledge". Journal of Economic Theory. 37 (1): 167–190. doi:10.1016/0022-0531(85)90035-3.
^ Samet, Dov (1990). "Ignoring ignorance and agreeing to disagree". Journal of Economic Theory. 52 (1): 190–207. doi:10.1016/0022-0531(90)90074-T.
^ Halpern, Joseph; Willemien Kets (2013-10-28). "Ambiguous Language and Consensus" (PDF). Retrieved 2014-01-13.
^ Feinberg, Yossi (2000). "Characterizing Common Priors in the Form of Posteriors". Journal of Economic Theory. 91: 127–179. doi:10.1006/jeth.1999.2592.
^ Samet, Dov (1998). "Common Priors and Separation of Convex Sets". Games and Economic Behavior. 91: 172–174. doi:10.1006/game.1997.0615.
^ Aaronson, Scott (2005). The complexity of agreement (PDF). Proceedings of ACM STOC. pp. 634–643. doi:10.1145/1060590.1060686. ISBN 978-1-58113-960-0. Retrieved 2010-08-09.
^ Hanson, Robin (2006). "Uncommon Priors Require Origin Disputes". Theory and Decision. 61 (4): 319–328. CiteSeerX 10.1.1.63.4669. doi:10.1007/s11238-006-9004-4.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Aumann%27s_agreement_theorem&oldid=1089408585"
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Fit generalized linear mixed-effects model - MATLAB fitglme - MathWorks Switzerland
CovarianceMethod
EBMethod
EBOptions
InitPLIterations
PLIterations
PLTolerance
UseSequentialFitting
Fit generalized linear mixed-effects model
glme = fitglme(tbl,formula)
glme = fitglme(tbl,formula,Name,Value)
glme = fitglme(tbl,formula) returns a generalized linear mixed-effects model, glme. The model is specified by formula and fitted to the predictor variables in the table or dataset array, tbl.
glme = fitglme(tbl,formula,Name,Value) returns a generalized linear mixed-effects model using additional options specified by one or more Name,Value pair arguments. For example, you can specify the distribution of the response, the link function, or the covariance pattern of the random-effects terms.
{\text{defects}}_{ij}\sim \text{Poisson}\left({\mu }_{ij}\right).
\mathrm{log}\left({\mu }_{ij}\right)={\beta }_{0}+{\beta }_{1}{\text{newprocess}}_{ij}+{\beta }_{2}{\text{time}\text{_}\text{dev}}_{ij}+{\beta }_{3}{\text{temp}\text{_}\text{dev}}_{ij}+{\beta }_{4}{\text{supplier}\text{_}\text{C}}_{ij}+{\beta }_{5}{\text{supplier}\text{_}\text{B}}_{ij}+{b}_{i},
{\text{defects}}_{ij}
i
j
{\mu }_{ij}
i
i=1,2,...,20
j
j=1,2,...,5
{\text{newprocess}}_{ij}
{\text{time}\text{_}\text{dev}}_{ij}
{\text{temp}\text{_}\text{dev}}_{ij}
i
j
{\text{newprocess}}_{ij}
i
j
{\text{supplier}\text{_}\text{C}}_{ij}
{\text{supplier}\text{_}\text{B}}_{ij}
i
j
{b}_{i}\sim N\left(0,{\sigma }_{b}^{2}\right)
i
'Distribution','Poisson','Link','log','FitMethod','Laplace', ...
'DummyVarCoding','effects');
t
p
t
Example: 'Distribution','Poisson','Link','log','FitMethod','Laplace','DummyVarCoding','effects' specifies the response variable distribution as Poisson, the link function as log, the fit method as Laplace, and dummy variable coding where the coefficients sum to 0.
1 (default) | scalar value | vector | variable name
Number of trials for binomial distribution, that is the sample size, specified as the comma-separated pair consisting of a scalar value, a vector of the same length as the response, or the name of a variable in the input table. If you specify the name of a variable, then the variable must be of the same length as the response. BinomialSize applies only when the Distribution parameter is 'binomial'.
If you specify 'FitMethod' as 'MPL' or 'REMPL', then the covariance of the fixed effects and the covariance parameters is based on the fitted linear mixed-effects model from the final pseudo likelihood iteration.
CovarianceMethod — Method to compute covariance of estimated parameters
'conditional' (default) | 'JointHessian'
Method to compute covariance of estimated parameters, specified as the comma-separated pair consisting of 'CovarianceMethod' and either 'conditional' or 'JointHessian'. If you specify 'conditional', then fitglme computes a fast approximation to the covariance of fixed effects given the estimated covariance parameters. It does not compute the covariance of covariance parameters. If you specify 'JointHessian', then fitglme computes the joint covariance of fixed effects and covariance parameters via the observed information matrix using the Laplacian loglikelihood.
Example: 'CovarianceMethod','JointHessian'
'FullCholesky' | 'Isotropic' | 'Full' | 'Diagonal' | 'CompSymm' | square symmetric logical matrix | string array | cell array of character vectors or logical matrices
Pattern of the covariance matrix of the random effects, specified as the comma-separated pair consisting of 'CovariancePattern' and 'FullCholesky', 'Isotropic', 'Full', 'Diagonal', 'CompSymm', a square symmetric logical matrix, a string array, or a cell array containing character vectors or logical matrices.
'FullCholesky' Full covariance matrix using the Cholesky parameterization. fitglme estimates all elements of the covariance matrix.
\left(\begin{array}{ccc}{\sigma }_{b}^{2}& 0& 0\\ 0& {\sigma }_{b}^{2}& 0\\ 0& 0& {\sigma }_{b}^{2}\end{array}\right)
where σ21 is the common variance of the random-effects terms.
\left(\begin{array}{ccc}{\sigma }_{b1}^{2}& 0& 0\\ 0& {\sigma }_{b2}^{2}& 0\\ 0& 0& {\sigma }_{b3}^{2}\end{array}\right)
\left(\begin{array}{ccc}{\sigma }_{b1}^{2}& {\sigma }_{b1,b2}& {\sigma }_{b1,b2}\\ {\sigma }_{b1,b2}& {\sigma }_{b1}^{2}& {\sigma }_{b1,b2}\\ {\sigma }_{b1,b2}& {\sigma }_{b1,b2}& {\sigma }_{b1}^{2}\end{array}\right)
For scalar random-effects terms, the default is 'Isotropic'. Otherwise, the default is 'FullCholesky'.
true Estimate a dispersion parameter when computing standard errors
false Use the theoretical value of 1.0 when computing standard errors
'DispersionFlag' only applies if 'FitMethod' is 'MPL' or 'REMPL'.
'Normal' (default) | 'Binomial' | 'Poisson' | 'Gamma' | 'InverseGaussian'
'InverseGaussian' Inverse Gaussian distribution
Example: 'Distribution','Binomial'
'reference' (default) fitglme creates dummy variables with a reference group. This scheme treats the first category as a reference group and creates one less dummy variables than the number of categories. You can check the category order of a categorical variable by using the categories function, and change the order by using the reordercats function.
'effects' fitglme creates dummy variables using effects coding. This scheme uses –1 to represent the last category. This scheme creates one less dummy variables than the number of categories.
'full' fitglme creates full dummy variables. This scheme creates one dummy variable for each category.
EBMethod — Method used to approximate empirical Bayes estimates of random effects
'Auto' (default) | 'LineSearchNewton' | 'TrustRegion2D' | 'fsolve'
Method used to approximate empirical Bayes estimates of random effects, specified as the comma-separated pair consisting of 'EBMethod' and one of the following.
'LineSearchNewton'
'TrustRegion2D'
'Auto' is similar to 'LineSearchNewton' but uses a different convergence criterion and does not display iterative progress. 'Auto' and 'LineSearchNewton' may fail for non-canonical link functions. For non-canonical link functions, 'TrustRegion2D' or 'fsolve' are recommended. You must have Optimization Toolbox™ to use 'fsolve'.
Example: 'EBMethod','LineSearchNewton'
EBOptions — Options for empirical Bayes optimization
Options for empirical Bayes optimization, specified as the comma-separated pair consisting of 'EBOptions' and a structure containing the following.
'TolFun' Relative tolerance on the gradient norm. Default is 1e-6.
'TolX' Absolute tolerance on the step size. Default is 1e-8.
'MaxIter' Maximum number of iterations. Default is 100.
'Display' 'off', 'iter', or 'final'. Default is 'off'.
If EBMethod is 'Auto' and 'FitMethod' is 'Laplace', TolFun is the relative tolerance on the linear predictor of the model, and the 'Display' option does not apply.
If 'EBMethod' is 'fsolve', then 'EBOptions' must be specified as an object created by optimoptions('fsolve').
Indices for rows to exclude from the generalized linear mixed-effects model in the data, specified as the comma-separated pair consisting of 'Exclude' and a vector of integer or logical values.
FitMethod — Method for estimating model parameters
'MPL' (default) | 'REMPL' | 'Laplace' | 'ApproximateLaplace
Method for estimating model parameters, specified as the comma-separated pair consisting of 'FitMethod' and one of the following.
'Laplace' — Maximum likelihood using Laplace approximation
'ApproximateLaplace' — Maximum likelihood using approximate Laplace approximation with fixed effects profiled out
Example: 'FitMethod','REMPL'
InitPLIterations — Initial number of pseudo likelihood iterations
10 (default) | integer value in the range [1,∞)
Initial number of pseudo likelihood iterations used to initialize parameters for ApproximateLaplace and Laplace fit methods, specified as the comma-separated pair consisting of 'InitPLIterations' and an integer value greater than or equal to 1.
'identity' | 'log' | 'logit' | 'probit' | 'comploglog' | 'reciprocal' | scalar value | structure
Link function, specified as the comma-separated pair consisting of 'Link' and one of the following.
g(mu) = mu
This is the default for the normal distribution.
g(mu) = log(mu)
This is the default for the Poisson distribution.
g(mu) = log(mu/(1-mu))
This is the default for the binomial distribution.
'loglog' g(mu) = log(-log(mu))
'probit' g(mu) = norminv(mu)
'comploglog' g(mu) = log(-log(1-mu))
'reciprocal' g(mu) = mu.^(-1)
Scalar value P g(mu) = mu.^P
A structure containing four fields whose values are function handles with the following names:
S.Derivative — Derivative
S.SecondDerivative — Second derivative
S.Inverse — Inverse of link
Specification of S.SecondDerivative can be omitted if FitMethod is MPL or REMPL, or if S is the canonical link for the specified distribution.
The default link function used by fitglme is the canonical link that depends on the distribution of the response.
'Binomial' 'logit'
'Gamma' -1
'InverseGaussian' -2
Example: 'Link','log'
MuStart — Starting value for conditional mean
Starting value for conditional mean, specified as the comma-separated pair consisting of 'MuStart' and a scalar value. Valid values are as follows.
'Normal' (-Inf,Inf)
'Binomial' (0,1)
'Poisson' (0,Inf)
'Gamma' (0,Inf)
'InverseGaussian' (0,Inf)
zeros(n,1) (default) | n-by-1 vector of scalar values
Offset, specified as the comma-separated pair consisting of 'Offset' and an n-by-1 vector of scalar values, where n is the length of the response vector. You can also specify the variable name of an n-by-1 vector of scalar values. 'Offset' is used as an additional predictor that has a coefficient value fixed at 1.0.
'quasinewton' (default) | 'fminsearch' | 'fminunc'
'quasinewton' Uses a trust region based quasi-Newton optimizer. You can change the options of the algorithm using statset('fitglme'). If you do not specify the options, then fitglme uses the default options of statset('fitglme').
'fminsearch' Uses a derivative-free Nelder-Mead method. You can change the options of the algorithm using optimset('fminsearch'). If you do not specify the options, then fitglme uses the default options of optimset('fminsearch').
'fminunc' Uses a line search-based quasi-Newton method. You must have Optimization Toolbox to specify this option. You can change the options of the algorithm using optimoptions('fminunc'). If you do not specify the options, then fitglme uses the default options of optimoptions('fminunc') with 'Algorithm' set to 'quasi-newton'.
Example: 'Optimizer','fminsearch'
structure returned by statset | structure returned by optimset | object returned by optimoptions
Options for the optimization algorithm, specified as the comma-separated pair consisting of 'OptimizerOptions' and a structure returned by statset('fitglme'), a structure created by optimset('fminsearch'), or an object returned by optimoptions('fminunc').
If 'Optimizer' is 'fminsearch', then use optimset('fminsearch') to change the options of the algorithm. If 'Optimizer' is 'fminsearch' and you do not supply 'OptimizerOptions', then the defaults used in fitglme are the default options created by optimset('fminsearch').
If 'Optimizer' is 'fminunc', then use optimoptions('fminunc') to change the options of the optimization algorithm. See optimoptions for the options 'fminunc' uses. If 'Optimizer' is 'fminunc' and you do not supply 'OptimizerOptions', then the defaults used in fitglme are the default options created by optimoptions('fminunc') with 'Algorithm' set to 'quasi-newton'.
If 'Optimizer' is 'quasinewton', then use statset('fitglme') to change the optimization parameters. If 'Optimizer' is 'quasinewton' and you do not change the optimization parameters using statset, then fitglme uses the default options created by statset('fitglme').
The 'quasinewton' optimizer uses the following fields in the structure created by statset('fitglme').
PLIterations — Maximum number of pseudo likelihood iterations
Maximum number of pseudo likelihood (PL) iterations, specified as the comma-separated pair consisting of 'PLIterations' and a positive integer value. PL is used for fitting the model if 'FitMethod' is 'MPL' or 'REMPL'. For other 'FitMethod' values, PL iterations are used to initialize parameters for subsequent optimization.
Example: 'PLIterations',200
PLTolerance — Relative tolerance factor for pseudo likelihood iterations
1e–08 (default) | positive scalar value
Relative tolerance factor for pseudo likelihood iterations, specified as the comma-separated pair consisting of 'PLTolerance' and a positive scalar value.
Example: 'PLTolerance',1e-06
UseSequentialFitting — Initial fitting type
, specified as the comma-separated pair consisting of 'UseSequentialFitting' and either false or true. If 'UseSequentialFitting' is false, all maximum likelihood methods are initialized using one or more pseudo likelihood iterations. If 'UseSequentialFitting' is true, the initial values from pseudo likelihood iterations are refined using 'ApproximateLaplace' for 'Laplace' fitting.
Example: 'UseSequentialFitting',true
Indicator to display the optimization process on screen, specified as the comma-separated pair consisting of 'Verbose' and 0, 1, or 2. If 'Verbose' is specified as 1 or 2, then fitglme displays the progress of the iterative model-fitting process. Specifying 'Verbose' as 2 displays iterative optimization information from the individual pseudo likelihood iterations. Specifying 'Verbose' as 1 omits this display.
vector of nonnegative scalar values
Observation weights, specified as the comma-separated pair consisting of 'Weights' and an n-by-1 vector of nonnegative scalar values, where n is the number of observations. If the response distribution is binomial or Poisson, then 'Weights' must be a vector of positive integers.
In general, a formula for model specification is a character vector or string scalar of the form 'y ~ terms'. For the generalized linear mixed-effects models, this formula is in the form 'y ~ fixed + (random1|grouping1) + ... + (randomR|groupingR)', where fixed and random contain the fixed-effects and the random-effects terms.
Statistics and Machine Learning Toolbox™ notation always includes a constant term unless you explicitly remove the term using -1. Here are some examples for generalized linear mixed-effects model specification.
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Formula and Calculating Days Sales of Inventory (DSI)
What DSI Tells You
DSI vs. Inventory Turnover
Why the DSI Matters
Days Sales of Inventory FAQs
DSI is also known as the average age of inventory, days inventory outstanding (DIO), days in inventory (DII), days sales in inventory, or days inventory and is interpreted in multiple ways. Indicating the liquidity of the inventory, the figure represents how many days a company’s current stock of inventory will last. Generally, a lower DSI is preferred as it indicates a shorter duration to clear off the inventory, though the average DSI varies from one industry to another.
Days sales of inventory (DSI) is the average number of days it takes for a firm to sell off inventory.
DSI is a metric that analysts use to determine the efficiency of sales.
\begin{aligned} &DSI = \frac{\text{Average inventory}}{COGS} \times 365 \text{ days}\\ &\textbf{where:}\\ &DSI=\text{days sales of inventory}\\ &COGS=\text{cost of goods sold}\\ \end{aligned}
DSI=COGSAverage inventory×365 dayswhere:DSI=days sales of inventoryCOGS=cost of goods sold
To manufacture a salable product, a company needs raw material and other resources which form the inventory and come at a cost. Additionally, there is a cost linked to the manufacturing of the salable product using the inventory. Such costs include labor costs and payments towards utilities like electricity, which is represented by the cost of goods sold (COGS) and is defined as the cost of acquiring or manufacturing the products that a company sells during a period. DSI is calculated based on the average value of the inventory and cost of goods sold during a given period or as of a particular date. Mathematically, the number of days in the corresponding period is calculated using 365 for a year and 90 for a quarter. In some cases, 360 days is used instead.
The numerator figure represents the valuation of the inventory. The denominator (Cost of Sales / Number of Days) represents the average per day cost being spent by the company for manufacturing a salable product. The net factor gives the average number of days taken by the company to clear the inventory it possesses.
Two different versions of the DSI formula can be used depending upon the accounting practices. In the first version, the average inventory amount is taken as the figure reported at the end of the accounting period, such as at the end of the fiscal year ending June 30. This version represents DSI value “as of” the mentioned date. In another version, the average value of Start Date Inventory and End Date Inventory is taken, and the resulting figure represents DSI value “during” that particular period. Therefore,
\text{Average Inventory} = \text{Ending Inventory}
Average Inventory=Ending Inventory
\text{Average Inventory} = \frac{(\text{Beginning Inventory} + \text{Ending Inventory})}{2}
Average Inventory=2(Beginning Inventory+Ending Inventory)
COGS value remains the same in both the versions.
Since DSI indicates the duration of time a company’s cash is tied up in its inventory, a smaller value of DSI is preferred. A smaller number indicates that a company is more efficiently and frequently selling off its inventory, which means rapid turnover leading to the potential for higher profits (assuming that sales are being made in profit). On the other hand, a large DSI value indicates that the company may be struggling with obsolete, high-volume inventory and may have invested too much into the same. It is also possible that the company may be retaining high inventory levels in order to achieve high order fulfillment rates, such as in anticipation of bumper sales during an upcoming holiday season.
DSI is a measure of the effectiveness of inventory management by a company. Inventory forms a significant chunk of the operational capital requirements for a business. By calculating the number of days that a company holds onto the inventory before it is able to sell it, this efficiency ratio measures the average length of time that a company’s cash is locked up in the inventory.
However, this number should be looked upon cautiously as it often lacks context. DSI tends to vary greatly among industries depending on various factors like product type and business model. Therefore, it is important to compare the value among the same sector peer companies. Companies in the technology, automobile, and furniture sectors can afford to hold on to their inventories for long, but those in the business of perishable or fast-moving consumer goods (FMCG) cannot. Therefore, sector-specific comparisons should be made for DSI values.
One must also note that a high DSI value may be preferred at times depending on the market dynamics. If a short supply is expected for a particular product in the next quarter, a business may be better off holding on to its inventory and then selling it later for a much higher price, thus leading to improved profits in the long run.
For example, a drought situation in a particular soft water region may mean that authorities will be forced to supply water from another area where water quality is hard. It may lead to a surge in demand for water purifiers after a certain period, which may benefit the companies if they hold onto inventories.
Irrespective of the single-value figure indicated by DSI, the company management should find a mutually beneficial balance between optimal inventory levels and market demand.
A similar ratio related to DSI is inventory turnover, which refers to the number of times a company is able to sell or use its inventory over the course of a particular time period, such as quarterly or annually. Inventory turnover is calculated as the cost of goods sold divided by average inventory. It is linked to DSI via the following relationship:
DSI = \frac{1}{\text{inventory turnover}}\times 365 \text{ days}
DSI=inventory turnover1×365 days
Basically, DSI is an inverse of inventory turnover over a given period. Higher DSI means lower turnover and vice versa.
In general, the higher the inventory turnover ratio, the better it is for the company, as it indicates a greater generation of sales. A smaller inventory and the same amount of sales will also result in high inventory turnover. In some cases, if the demand for a product outweighs the inventory on hand, a company will see a loss in sales despite the high turnover ratio, thus confirming the importance of contextualizing these figures by comparing them against those of industry competitors.
DSI is the first part of the three-part cash conversion cycle (CCC), which represents the overall process of turning raw materials into realizable cash from sales. The other two stages are days sales outstanding (DSO) and days payable outstanding (DPO). While the DSO ratio measures how long it takes a company to receive payment on accounts receivable, the DPO value measures how long it takes a company to pay off its accounts payable. Overall, the CCC value attempts to measure the average duration of time for which each net input dollar (cash) is tied up in the production and sales process before it gets converted into cash received through sales made to customers.
Managing inventory levels is vital for most businesses, and it is especially important for retail companies or those selling physical goods. While the inventory turnover ratio is one of the best indicators of a company’s level of efficiency at turning over its inventory and generating sales from that inventory, the days sales of inventory ratio goes a step further by putting that figure into a daily context and providing a more accurate picture of the company’s inventory management and overall efficiency.
DSI and inventory turnover ratio can help investors to know whether a company can effectively manage its inventory when compared to competitors. A 2014 paper in Management Science, "Does Inventory Productivity Predict Future Stock Returns? A Retailing Industry Perspective," suggests that stocks in companies with high inventory ratios tend to outperform industry averages. A stock that brings in a higher gross margin than predicted can give investors an edge over competitors due to the potential surprise factor. Conversely, a low inventory ratio may suggest overstocking, market or product deficiencies, or otherwise poorly managed inventory–signs that generally do not bode well for a company’s overall productivity and performance.
The leading retail corporation Walmart (WMT) had inventory worth $56.5 billion and cost of goods sold worth $429 billion for the fiscal year 2022. DSI is therefore:
DSI = (56.5/429) x 365= 48.1 days
While inventory value is available on the balance sheet of the company, the COGS value can be sourced from the annual financial statement. Care should be taken to include the sum total of all the categories of inventory which includes finished goods, work in progress, raw materials, and progress payments.
Since Walmart is a retailer, it does not have any raw material, works in progress, and progress payments. Its entire inventory is comprised of finished goods.
What Does a Low Days Sales of Inventory Indicate?
A low DSI suggests that a firm is able to efficiently convert its inventories into sales. This is considered to be beneficial to a company's margins and bottom line, and so a lower DSI is preferred to a higher one. A very low DSI, however, can indicate that a company does not have enough inventory stock to meet demand, which could be viewed as suboptimal.
How Do You Interpret Days Sales of Inventory?
DSI estimates how many days it takes on average to completely sell a company's current inventories.
What Is a Good Days Sale of Inventory Number?
In order to efficiently manage inventories and balance idle stock with being understocked, many experts agree that a good DSI is somewhere between 30 and 60 days. This, of course, will vary by industry, company size, and other factors.
Yasin ,Alan, and George P. Gao. "Does Inventory Productivity Predict Future Stock Returns? A Retailing Industry Perspective." Management Science, Vol. 60, Issue 10, 2014, Pages 2416-2434.
Wall Street Journal. "WMT Financials."
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Generate Untargeted and Targeted Adversarial Examples for Image Classification - MATLAB & Simulink - MathWorks India
Load Network and Image
Untargeted Fast Gradient Sign Method
Targeted Adversarial Examples
This example shows how to use the fast gradient sign method (FGSM) and the basic iterative method (BIM) to generate adversarial examples for a pretrained neural network.
Neural networks can be susceptible to a phenomenon known as adversarial examples [1], where very small changes to an input can cause the input to be misclassified. These changes are often imperceptible to humans.
In this example, you create two types of adversarial examples:
Untargeted — Modify an image so that it is misclassified as any incorrect class.
Targeted — Modify an image so that it is misclassified as a specific class.
Load a network that has been trained on the ImageNet [2] data set and convert it to a dlnetwork.
Extract the class labels.
classes = categories(net.Layers(end).Classes);
Load an image to use to generate an adversarial example. The image is a picture of a golden retriever.
T = "golden retriever";
Resize the image to match the input size of the network.
inputSize = dlnet.Layers(1).InputSize;
img = imresize(img,inputSize(1:2));
title("Ground Truth: " + T)
Prepare the image by converting it to a dlarray.
X = dlarray(single(img),"SSCB");
Prepare the label by one-hot encoding it.
T = onehotencode(T,1,'ClassNames',classes);
T = dlarray(single(T),"CB");
Create an adversarial example using the untargeted FGSM [3]. This method calculates the gradient
{\nabla }_{X}L\left(X,T\right)
of the loss function
L
, with respect to the image
X
you want to find an adversarial example for, and the class label
T
. This gradient describes the direction to "push" the image in to increase the chance it is misclassified. You can then add or subtract a small error from each pixel to increase the likelihood the image is misclassified.
The adversarial example is calculated as follows:
{\mathit{X}}_{\mathrm{adv}}=\mathit{X}+ϵ.\mathrm{sign}\left({\nabla }_{\mathit{X}}\mathit{L}\left(\mathit{X},\mathit{T}\right)\right)
ϵ
controls the size of the push. A larger
ϵ
value increases the chance of generating a misclassified image, but makes the change in the image more visible. This method is untargeted, as the aim is to get the image misclassified, regardless of which class.
Calculate the gradient of the image with respect to the golden retriever class.
gradient = dlfeval(@untargetedGradients,dlnet,X,T);
Set epsilon to 1 and generate the adversarial example.
XAdv = X + epsilon*sign(gradient);
Predict the class of the original image and the adversarial image.
YPred = predict(dlnet,X);
YPred = onehotdecode(squeeze(YPred),classes,1)
YPredAdv = predict(dlnet,XAdv);
YPredAdv = onehotdecode(squeeze(YPredAdv),classes,1)
YPredAdv = categorical
Display the original image, the perturbation added to the image, and the adversarial image. If the epsilon value is large enough, the adversarial image has a different class label from the original image.
showAdversarialImage(X,YPred,XAdv,YPredAdv,epsilon);
The network correctly classifies the unaltered image as a golden retriever. However, because of perturbation, the network misclassifies the adversarial image as a labrador retriever. Once added to the image, the perturbation is imperceptible, demonstrating how adversarial examples can exploit robustness issues within a network.
A simple improvement to FGSM is to perform multiple iterations. This approach is known as the basic iterative method (BIM) [4] or projected gradient descent [5]. For the BIM, the size of the perturbation is controlled by parameter
\alpha
representing the step size in each iteration. This is as the BIM usually takes many, smaller, FGSM steps in the direction of the gradient. After each iteration, clip the perturbation to ensure the magnitude does not exceed
ϵ
. This method can yield adversarial examples with less distortion than FGSM.
When you use untargeted FGSM, the predicted label of the adversarial example can be very similar to the label of the original image. For example, a dog might be misclassified as a different kind of dog. However, you can easily modify these methods to misclassify an image as a specific class. Instead of maximizing the cross-entropy loss, you can minimize the mean squared error between the output of the network and the desired target output.
Generate a targeted adversarial example using the BIM and the great white shark target class.
targetClass = "great white shark";
targetClass = onehotencode(targetClass,1,'ClassNames',classes);
Increase the epsilon value to 5, set the step size alpha to 0.2, and perform 25 iterations. Note that you may have to adjust these settings for other networks.
Keep track of the perturbation and clip any values that exceed epsilon.
delta = zeros(size(X),'like',X);
gradient = dlfeval(@targetedGradients,dlnet,X+delta,targetClass);
delta = delta - alpha*sign(gradient);
delta(delta > epsilon) = epsilon;
delta(delta < -epsilon) = -epsilon;
XAdvTarget = X + delta;
Predict the class of the targeted adversarial example.
YPredAdvTarget = predict(dlnet,XAdvTarget);
YPredAdvTarget = onehotdecode(squeeze(YPredAdvTarget),classes,1)
YPredAdvTarget = categorical
Display the original image, the perturbation added to the image, and the targeted adversarial image.
showAdversarialImage(X,YPred,XAdvTarget,YPredAdvTarget,epsilon);
Because of imperceptible perturbation, the network classifies the adversarial image as a great white shark.
To make the network more robust against adversarial examples, you can use adversarial training. For an example showing how to train a network robust to adversarial examples, see Train Image Classification Network Robust to Adversarial Examples.
Untargeted Input Gradient Function
Calculate the gradient used to create an untargeted adversarial example. This gradient is the gradient of the cross-entropy loss.
function gradient = untargetedGradients(dlnet,X,target)
Y = predict(dlnet,X);
Y = stripdims(squeeze(Y));
loss = crossentropy(Y,target,'DataFormat','CB');
gradient = dlgradient(loss,X);
Targeted Input Gradient Function
Calculate the gradient used to create a targeted adversarial example. This gradient is the gradient of the mean squared error.
function gradient = targetedGradients(dlnet,X,target)
loss = mse(Y,target,'DataFormat','CB');
Show Adversarial Image
Show an image, the corresponding adversarial image, and the difference between the two (perturbation).
function showAdversarialImage(image,label,imageAdv,labelAdv,epsilon)
imgTrue = uint8(extractdata(image));
imshow(imgTrue)
title("Original Image" + newline + "Class: " + string(label))
perturbation = uint8(extractdata(imageAdv-image+127.5));
imshow(perturbation)
title("Perturbation")
advImg = uint8(extractdata(imageAdv));
imshow(advImg)
title("Adversarial Image (Epsilon = " + string(epsilon) + ")" + newline + ...
"Class: " + string(labelAdv))
[1] Goodfellow, Ian J., Jonathon Shlens, and Christian Szegedy. “Explaining and Harnessing Adversarial Examples.” Preprint, submitted March 20, 2015. https://arxiv.org/abs/1412.6572.
[2] ImageNet. http://www.image-net.org.
[3] Szegedy, Christian, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. “Intriguing Properties of Neural Networks.” Preprint, submitted February 19, 2014. https://arxiv.org/abs/1312.6199.
[4] Kurakin, Alexey, Ian Goodfellow, and Samy Bengio. “Adversarial Examples in the Physical World.” Preprint, submitted February 10, 2017. https://arxiv.org/abs/1607.02533.
[5] Madry, Aleksander, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. “Towards Deep Learning Models Resistant to Adversarial Attacks.” Preprint, submitted September 4, 2019. https://arxiv.org/abs/1706.06083.
dlnetwork | onehotdecode | onehotencode | predict | dlfeval | dlgradient
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H∞ norm of dynamic system - MATLAB hinfnorm - MathWorks Switzerland
{H}_{\infty }
\mathit{G}\left(\mathit{s}\right)=\left[\begin{array}{cc}0& \frac{3\mathit{s}}{{\mathit{s}}^{2}+\mathit{s}+10}\\ \frac{\mathit{s}+1}{\mathit{s}+5}& \frac{2}{\mathit{s}+6}\end{array}\right].
{H}_{\infty }
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Signal Features - MATLAB & Simulink - MathWorks Deutschland
Impulsive Metrics
Signal Processing Metrics
Signal features provide general signal-based statistical metrics that can be applied to any kind of signal, including a time-synchronized average (TSA) vibration signal. Changes in these features can indicate changes in the health status of your system. Diagnostic Feature Designer provides a set of feature options .
The statistical features include basic mean, standard deviation, and root mean square (RMS) metrics. In addition, the feature set includes shape factor and the higher order kurtosis and skewness statistics. All these statistics can be expected to change as a deteriorating fault signature intrudes upon the nominal signal.
Shape factor — RMS divided by the mean of the absolute value. Shape factor is dependent on the signal shape while being independent of the signal dimensions.
{x}_{SF}=\frac{{x}_{rms}}{\frac{1}{N}\sum _{i=1}^{N}|{x}_{i}|}
The higher-order statistics provide insight to system behavior through the fourth moment (kurtosis) and third moment (skewness) of the vibration signal.
Kurtosis — Length of the tails of a signal distribution, or equivalently, how outlier prone the signal is. Developing faults can increase the number of outliers, and therefore increase the value of the kurtosis metric. The kurtosis has a value of 3 for a normal distribution. For more information, see kurtosis.
{x}_{kurt}=\frac{\frac{1}{N}\sum _{i=1}^{N}{\left({x}_{i}-\overline{x}\right)}^{4}}{{\left[\frac{1}{N}\sum _{i=1}^{N}{\left({x}_{i}-\overline{x}\right)}^{2}\right]}^{2}}
Skewness — Asymmetry of a signal distribution. Faults can impact distribution symmetry and therefore increase the level of skewness.
{x}_{skew}=\frac{\frac{1}{N}\sum _{i=1}^{N}{\left({x}_{i}-\overline{x}\right)}^{3}}{{\left[\frac{1}{N}\sum _{i=1}^{N}{\left({x}_{i}-\overline{x}\right)}^{2}\right]}^{3/2}}
For more information, see skewness.
Impulsive Metrics are properties related to the peaks of the signal.
Peak value — Maximum absolute value of the signal. Used to compute the other impulse metrics.
{x}_{p}=\underset{i}{\mathrm{max}}|{x}_{i}|
Impulse Factor — Compare the height of a peak to the mean level of the signal.
{x}_{IF}=\frac{{x}_{p}}{\frac{1}{N}\sum _{i=1}^{N}|{x}_{i}|}
Crest Factor — Peak value divided by the RMS. Faults often first manifest themselves in changes in the peakiness of a signal before they manifest in the energy represented by the signal root mean squared. The crest factor can provide an early warning for faults when they first develop. For more information, see peak2rms.
{x}_{crest}=\frac{{x}_{p}}{\sqrt{\frac{1}{N}\sum _{i=1}^{N}{x}_{i}{}^{2}}}
Clearance Factor — Peak value divided by the squared mean value of the square roots of the absolute amplitudes. For rotating machinery, this feature is maximum for healthy bearings and goes on decreasing for defective ball, defective outer race, and defective inner race respectively. The clearance factor has the highest separation ability for defective inner race faults.
{x}_{clear}=\frac{{x}_{p}}{{}^{\left(\frac{1}{N}\sum _{i=1}^{N}{\sqrt{|{x}_{i}|\right)}}^{2}}}
The signal processing metrics consist of distortion measurement functions. System degradation can cause an increase in noise, a change in a harmonic relative to the fundamental, or both.
Signal-to-Noise Ratio (SNR) —Ratio of signal power to noise power
Total Harmonic Distortion (THD) — Ratio of total harmonic component power to fundamental power
Signal to Noise and Distortion Ratio (SINAD) — Ratio of total signal power to total noise-plus-distortion power
For more information on these metrics, see snr, thd, and sinad.
The software stores the results of the computation in new features. The new feature names include the source signal name with the suffix stats.
For information on interpreting feature histograms, see Interpret Feature Histograms in Diagnostic Feature Designer.
kurtosis | skewness | snr | thd | sinad | peak2rms
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SameGame - Wikipedia
(Redirected from Clickomania)
Tile-matching puzzle video game
Find sources: "SameGame" – news · newspapers · books · scholar · JSTOR (November 2010) (Learn how and when to remove this template message)
SameGame for Mac, by Takahiro Sumiya
SameGame (さめがめ) is a tile-matching puzzle originally released under the name Chain Shot! in 1985 by Kuniaki Moribe (Morisuke). It has since been ported to numerous computer platforms, handheld devices, and even TiVo,[1] with new versions as of 2016.
2.1.2 Rules variations
2.2.1 Goal-based scoring
SameGame was originally created as Chain Shot! in 1985 by Kuniaki Moribe. It was distributed for Fujitsu's FM-8 and FM-7 platforms in a Japanese monthly personal computer magazine called Gekkan ASCII. In 1992, the game was ported as SameGame to Unix platforms by Eiji Fukumoto, and to the NEC PC-9801 series by Wataru Yoshioka. In 1993, it was ported to Windows 3.1 by Ikuo Hirohata. This version was translated into English by Hitoshi Ozawa, and is still available from his software archive.[2] In 1994, Takahiro Sumiya ported it to Macintosh. This version has some gameplay differences—three, instead of five, colors—and is probably the most widely distributed of the original series. It was the basis for the Same Gnome and KSame variations created for Linux.
In 2001, Biedl et al. proved that deciding the solvability (whether all blocks can be removed) of 1-column (or 1-row) 2-colour Clickomania can be done in linear time. Deciding the solvability of 2-column, 5-colour Clickomania is NP-Complete. Deciding the solvability of 5-column 3-colour Clickomania is also NP-Complete.[3]
An initial playing field for KSame, part of kdegames
SameGame is played on a rectangular field, typically initially filled with four or five kinds of blocks placed at random. By selecting a group of adjoining blocks of the same color, a player may remove them from the screen. Blocks that are no longer supported will fall down, and a column without any blocks will be trimmed away by other columns always sliding to one side (often the left). The goal of the game is to remove as many blocks from the playing field as possible.
In most versions, there are no time constraints during the game. However, some implementations gradually push the rows upward or drop blocks from above. Sometimes the player can control the number and timing of blocks that drop from above in certain ways. For example, on some implementations for the iOS, this can be done by shaking the device. The game ends if a timer runs out or if no more blocks can be removed. Some versions, including some versions for Windows Mobile, include both portrait and landscape orientations.
In one variation, the game starts with no blocks on the field. Blocks fall down to the playing field, and must be removed before they reach the top. If they reach the top and overflow, the game is over. In some variations, such as Bubble Bang, circles or balls are used instead of blocks—which alters the gameplay, as the balls form different shapes than square blocks.
In three-dimensional variants, the playing field is a cube (containing smaller cubes) instead of a rectangle, and the player has the ability to rotate the cube. "Cubes" for iPhone OS uses this approach.
Some versions allow the player to rotate the playing field 90 degrees clockwise or counter-clockwise, which causes one of two things to happen:
The left and right sides become the bottom and the top, and the blocks fall to the new bottom. The orientation switches between portrait and landscape. NeoSameGame for iPhone OS uses this approach.
The blocks fall to the left or right side, but the player must rotate the field back to portrait orientation (which is fixed). Bubblets Tilt for iPhone OS uses this approach.
In some variations, blocks can be removed when connected to blocks of the same color diagonally, not just horizontally and vertically. Some versions introduce new types of blocks. The different types of blocks interact in various ways with the play field; for example, one type might remove all the blocks in a row. An example of this is the "Revenge mode" in PocketPop Revenge (PocketFun) for iPhone OS.
Rules variations[edit]
The game ends when the playing field is cleared, or if the remaining blocks cannot be removed. At the end of play, the player receives a score.
When the playing field is cleared, instead of ending the game, a new level appears—usually harder, with more block types or lower time limits, or both. The condition for winning may vary between levels. Instead of clearing the whole level, for example, a certain score or a certain number of removed blocks must be reached. When the needed score is reached, in most versions the player is allowed to clear the rest of the level. If the player cannot reach the needed score—or if the timer runs out—the game ends, and the player receives a final score.[citation needed]
In an "endless" variant, the game starts with an empty field. The blocks or balls start falling down; but if they reach the top, new blocks stop falling, so they do not overflow—thus, the game never ends. The player can end the game at any time by waiting for blocks to reach the top, then performing a special action (for example, right-click instead of left-click).
Some versions have player lives.[citation needed] If a player reaches a losing condition one time, the game does not end; instead, a life is lost. If all lives are lost, the game ends.
Swell-Foop, part of GNOME Games, with a score of 1 after move has been made that removed three blocks
Most versions of the game give
{\displaystyle (n-k)^{2}}
points for removing
{\displaystyle n}
tiles at once, where
{\displaystyle k=1}
{\displaystyle 2}
, depending on the implementation. For instance, Insane Game for Texas Instruments calculators uses
{\displaystyle (n-1)^{2}}
; Ikuo Hirohata's implementation uses the formula
{\displaystyle n^{2}-3n+4}
. The Bubble Breaker implementation for Windows Mobile uses the
{\displaystyle n(n-1)}
formula. The 2001 version released by Jeff Reno uses the formula
{\displaystyle n(n-2)}
Some versions also offer a large bonus for removing all blocks from the screen, or leaving no more than a certain number of blocks. Others reduce the final score based on the number of blocks remaining at the end of the game. Some game versions award bonus points for clearing the field quickly, encouraging faster play. The faster the player finishes the level, the bigger the bonus. Still others offer combination, or chain, bonuses for clearing the same color of blocks two or more times in succession.
Another scoring technique awards bonus points for each chain of a certain color that has a certain number of blocks (for example, two red blocks or 11 blue blocks). After receiving the bonus once, sometimes the bonus condition changes. BPop uses this scoring variation.
Some versions have a simple scoring system: each block removed is worth one point, and there is no bonus for removing more than two blocks at a time. This is seen in the Same Pets and Same Hearths variants.
Goal-based scoring[edit]
Some versions award scores based on the attainment of goals. This is typically seen in multi-level versions of the game. There are four primary scoring systems for such games.
In one variation, each level has a target score. The player's score starts at zero, and the player must reach the target score. At the beginning of each level, the player's score is reset to zero; the target score increases with each level.
Other versions have a cumulative target score. In these versions, the player's score carries over from level to level. As a result, if the player substantially exceeds the target score on a given level, they may enter the subsequent level having already met that level's target score, as well. BPop has a cumulative target score.
Some versions maintain the same target score for each level; such variations can be played indefinitely. In such games, the player typically loses due to poor planning or a lapse in concentration. Examples of such games are Same Pets and Same Hearths.
In games without a goal score, like Bonkers for iPhone and SameGameBros for iPhone, the goal is to clear the level completely. The game ends when the player fails to do so.
Example of blocks with a color gradient design
Blocks typically appear as colored squares, circles, or spheres. Some variations use gradient shading to give the illusion of dimensionality. Other tile themes, or skins, include animals, hearts, stars, faces, Lego blocks, and jelly bears. Designs may follow a theme, such as Christmas or monochrome. Most games have only one skin, but others allow choosing from multiple skins.
There is a special visual aspect in some versions; instead of separate blocks, games like iDrops and SameGameManiak feature bordered areas for adjacent blocks of the same color. Some have elaborate tile graphics, featuring pictures or patterns inside the tile, like KSame and Same GNOME.
The SameGame concept can be extended to a "Reveal the picture" game. A picture or photo is behind the blocks; it becomes increasingly visible as blocks are removed, until it is completely revealed. Examples include Same Pets, Same Hearts and the Nissan Cube promotional app for iPhone.
Some games feature animation of one or more game events, such as cleared tiles bursting or exploding, or scoring animations (BPop, Bubblets Tilt).
Some versions display which blocks are selected with a border around them (BPop), jittering of the blocks (BPop), or an increase of the size of the selected blocks (Bubblets Tilt). If the blocks are deselected (usually by dragging away from them, or tapping another block chain or a single block), the highlight is removed.
Versions of SameGame
Chain Shot! Kuniaki Moribe 1985 Fujitsu FM 8/7 · PC-8801 · PC-9800 · N5200 (1988) · Macintosh (1992) The original iteration of the game had a 20×10 playing field and four colors.
Same Game Eiji Fukumoto 1992 Unix The first version titled Same Game; it increased the number of colors to five.
Same Game Wataru Yoshioka (W. Yossi) PC-9801
Same Game Ikuo Hirohata (Japanese)
Hitoshi Ozawa (tr. English) 1993 Windows 3.1 Added an optional large field of 25×15. The large field requires an 800×600 desktop resolution.
Swell Foop Based on Takahiro Sumiya's Macintosh version.
Undake 30: Same Game 1995 SNES Featured Mario franchise-related icons: Mario's head, coins, Super Mushrooms, Fire Flowers, and Yoshi eggs.
Same Game Shimada Kikaku 1997 Game Boy Published by Hudson.[4]
ColorFall Michael LaLena 1998 Java/Browser based Added the concept of levels. Clear levels by removing a fixed number of colors. New colors are added every level. Five different versions are available.
Clickomania! Matthias Schüssler 1998 Windows Board size and number of colors are configurable. Originally the goal was only to clear the playing field, the number of blocks removed in one turn did not affect the score. This is still the default setting.
SameGame Ronald van Dijk 1999 Amiga It has a 15×10 playing field and three colors.
Sega Swirl Scott Hawkins (Sega) 1999 Dreamcast · Adobe Shockwave · Palm OS
MacStones Craig Landrum 1999 Based on Same Gnome.
Cascade 1999 Psion Revo
Mahki 1999 Arcade game, Nintendo DS, Wii, Web browser Included in the Touch Master countertop cabinet arcade game starting with the Touch Master 7000. Re-released with modifications in 2008 as part of TouchMaster 2 for the Nintendo DS, and online as part of "Midway Arcade."
Spore Cubes René Boutin / Spore Productions 2000 Web browser, Windows, ActionScript 3, Palm OS, Pocket PC, iOS, Android (operating system) Inspired by the addictiveness of Clickomania! (see above), this game featured two skill levels which varied the number of colors in the playfield, consisting of 10 x 13 cubes. The original version of the game had randomly selected images behind the cubes, such that when the playfield was cleared, the player could see the entire image.
Maki Christopher G. Stach II December 2000 Java applet/Browser based Based on Mahki. Three difficulty levels, five colors,
{\displaystyle (n-2)^{2}}
scoring, cleared board bonus, online high scoring.
PocketPop PocketFun 2001 Pocket PC Won a number of awards, including Best Game, in Pocket PC Magazine 2001.[5][failed verification]
Bomberman 64 Racjin/Hudson Soft 2001 Nintendo 64 Includes Bomberman-themed SameGame minigame. Four difficulty levels, five colors,
{\displaystyle (n-2)^{2}+n}
scoring.
Jawbreaker 2003 Pocket PC
Bubble Shot FingerFriendlySoft iOS A Bubble Breaker–compatible game where adjacent bubbles visually melt into larger bubbles. Includes additional "Folding" and "Black Hole" modes and static challenges.
Sega Swirl 2 Scott Hawkins (Sega) 2006 Windows The sequel to Sega Swirl, which was only available through GameTap.
bubbles.el Ulf Jasper February 2007 GNU emacs Can display using graphics or text, according to availability
SameGame Steve and Oliver Baker 2008 JavaScript Online version that allows configuration of board size, number of colors and offers a range of alternative tile themes to play with.
Bubble Bang Decane January 2009 Web browser and iOS Three-dimensional game using balls instead of blocks. The iOS version uses Nvidia PhysX for realistic physics. The web browser version requires Unity.
SameGame Alan Alpert July 2009[6] All supported Qt platforms Written as a QML/QtQuick demo.
Pop'Em Drop'Em SAMEGAME Hudson Soft March 23, 2009[7] WiiWare
SameGame Torbjörn Gustafsson February 2009 Android (operating system)
Bubble Drop! Gizmobuddy.com Symbian S60 Includes the ability to selectively remove obstructing bubbles by using "tools", "acid", "fire", or "bomb", and with eight different gameplay modes of three and six colors. Players can submit high scores to a website.
ColorBalls Pistooli March 2010 Haiku OS
Click-o-mania HTML Bugaco January 2011 JavaScript Written in GWT[8]
Cube Crush Gregor Haag June 2011
2016 ActionScript 3
Android (operating system) Written in OpenFL to be cross-platform. Online Highscores. 3 and 4 color mode.[9]
Maki appsburgers September 2011 Android (operating system)
Bubblet Edouard Thiel October 2011 Linux, Mac OS X, Windows Written in C and included in EZ-Draw[10]
Bubblet-js Benoit Favre October 2011 JavaScript Online version, translated from C using EZ-Draw-js[11]
Tapotron Demura Games October 2013 iOS
One More SameGame Dušan Saiko October 2014 QT5 Online score synchronization, multilanguage, installation packages for Android, Windows, Linux[12]
SCRUSH Zafar Iqbal December 2016 Scratch (programming language) Online, Multi-platform, Highscore[13]
samegame1k Gábor Bata February 2017 JavaScript Online version, in 1024 bytes of JavaScript. An entry for the JS1k 2017 code golfing competition[14]
^ Ozawa, Hitoshi. "ISOFT - Home of Japanese software". Retrieved 2010-11-28.
^ Biedl, Therese; Demaine, Erik (2001). "The Complexity of Clickomania". More Games of No Chance. arXiv:cs/0107031. Bibcode:2001cs........7031B.
^ "Same Game for Game Boy - GameFAQs".
^ "pocketfun". pocketfun.co.uk.
^ "Qt Declarative UI SameGame". Nokia. 2009-07-28. Archived from the original on 2014-03-25. Retrieved 2014-03-24.
^ "One WiiWare Game and Two Virtual Console Games Added to Wii Shop Channel". Nintendo. 2009-03-23. Archived from the original on 2009-03-26. Retrieved 2009-03-25.
^ http://gregorhaag.com
^ "EZ-Draw".
^ "EZ-draw-js".
^ https://scratch.mit.edu/projects/136505698/
^ "Samegame1k - SameGame puzzle game in 1024 bytes of JavaScript".
Chain Shot! on Kuniaki Moribe's homepage
Retrieved from "https://en.wikipedia.org/w/index.php?title=SameGame&oldid=1085269044"
|
Here is another way to think about the question: “What is
0!
_nC_r=\frac{n!}{(r!(n-r)!}
How many ways are there to choose all five items from a group of five items? What happens when you substitute into the factorial formula to compute
_5C_5
? Since you know (logically) what the result has to be, use this to explain what
0!
must be equal to.
On the other hand, how many ways are there to choose nothing from a group of five items? And what happens when you try to use the factorial formula to compute
_5C_0
|
Solve Ordinary Differential Equation Using Neural Network - MATLAB & Simulink - MathWorks Deutschland
ODE and Loss Function
Generate Input Data and Define Network
Not all differential equations have a closed-form solution. To find approximate solutions to these types of equations, many traditional numerical algorithms are available. However, you can also solve an ODE by using a neural network. This approach comes with several advantages, including that it provides differentiable approximate solutions in a closed analytic form [1].
Generate training data in the range
x\in \left[0,2\right]
Define a neural network that takes
x
as input and returns the approximate solution to the ODE
\stackrel{˙}{\text{\hspace{0.17em}}\mathit{y}}=-2\mathit{xy}
x
Train the network with a custom loss function.
Compare the network predictions with the analytic solution.
In this example, you solve the ODE
\stackrel{˙}{\text{\hspace{0.17em}}\mathit{y}}=-2\mathit{xy},
\mathit{y}\left(0\right)=1
. This ODE has the analytic solution
\mathit{y}\left(\mathit{x}\right)={\mathit{e}}^{-{\mathit{x}}^{2}}.
Define a custom loss function that penalizes deviations from satisfying the ODE and the initial condition. In this example, the loss function is a weighted sum of the ODE loss and the initial condition loss:
{L}_{\theta }\left(x\right)=‖{\underset{}{\overset{˙}{y}}}_{\theta }+2x{y}_{\theta }{‖}^{2}+k‖{y}_{\theta }\left(0\right)-1{‖}^{2}
\theta \text{\hspace{0.17em}}
is the network parameters,
k
is a constant coefficient,
{\mathit{y}}_{\theta \text{\hspace{0.17em}}}
is the solution predicted by the network, and
\stackrel{˙}{{\mathit{y}}_{\theta \text{\hspace{0.17em}}}}
is the derivative of the predicted solution computed using automatic differentiation. The term
‖\text{\hspace{0.17em}}\stackrel{˙}{{\mathit{y}}_{\theta \text{\hspace{0.17em}}}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2{\mathit{xy}}_{\theta \text{\hspace{0.17em}}}\text{\hspace{0.17em}}{‖}^{2}
is the ODE loss and it quantifies how much the predicted solution deviates from satisfying the ODE definition. The term
‖{\mathit{y}}_{\theta \text{\hspace{0.17em}}}\left(0\right)-1\text{\hspace{0.17em}}{‖}^{2}
is the initial condition loss and it quantifies how much the predicted solution deviates from satisfying the initial condition.
Generate 10,000 training data points in the range
x\in \left[0,2\right]
x = linspace(0,2,10000)';
Define the network for solving the ODE. As the input is a real number
x\in \mathbb{R}
, specify an input size of 1.
featureInputLayer(inputSize,Normalization="none")
dlnet = dlnetwork(layers)
Create the function modelGradients, listed at the end of the example, which takes as inputs a dlnetwork object dlnet, a mini-batch of input data dlX, and the coefficient associated with the initial condition loss icCoeff. This function returns the gradients of the loss with respect to the learnable parameters in dlnet and the corresponding loss.
Train for 15 epochs with a mini-batch size of 100.
Specify the options for SGDM optimization. Specify a learning rate of 0.5, a learning rate drop factor of 0.5, a learning rate drop period of 5, and a momentum of 0.9.
initialLearnRate = 0.5;
Specify the coefficient of the initial condition term in the loss function as 7. This coefficient specifies the relative contribution of the initial condition to the loss. Tweaking this parameter can help training converge faster.
icCoeff = 7;
To use mini-batches of data during training:
Create an arrayDatastore object from the training data.
Create a minibatchqueue object that takes the arrayDatastore object as input, specify a mini-batch size, and format the training data with the dimension labels 'BC' (batch, channel).
ads = arrayDatastore(x,IterationDimension=1);
mbq = minibatchqueue(ads,MiniBatchSize=miniBatchSize,MiniBatchFormat="BC");
By default, the minibatchqueue object converts the data to dlarray objects with underlying type single.
set(gca,YScale="log")
ylabel("Loss (log scale)")
Evaluate the model gradients and loss using the dlfeval and modelGradients functions.
Every learnRateDropPeriod epochs, multiply the learning rate by learnRateDropFactor.
mbq.shuffle
[gradients,loss] = dlfeval(@modelGradients, dlnet, dlX, icCoeff);
% Update network parameters using the SGDM optimizer.
[dlnet,velocity] = sgdmupdate(dlnet,gradients,velocity,learnRate,momentum);
% To plot, convert the loss to double.
D = duration(0,0,toc(start),Format="mm:ss.SS");
% Reduce the learning rate.
if mod(epoch,learnRateDropPeriod)==0
Test the accuracy of the network by comparing its predictions with the analytic solution.
Generate test data in the range
x\in \left[0,4\right]
to see if the network is able to extrapolate outside the training range
x\in \left[0,2\right]
xTest = linspace(0,4,1000)';
To use mini-batches of data during testing:
Create an arrayDatastore object from the testing data.
Create a minibatchqueue object that takes the arrayDatastore object as input, specify a mini-batch size of 100, and format the training data with the dimension labels 'BC' (batch, channel).
adsTest = arrayDatastore(xTest,IterationDimension=1);
mbqTest = minibatchqueue(adsTest,MiniBatchSize=100,MiniBatchFormat="BC");
Loop over the mini-batches and make predictions using the modelPredictions function, listed at the end of the example.
yModel = modelPredictions(dlnet,mbqTest);
Evaluate the analytic solution.
yAnalytic = exp(-xTest.^2);
Compare the analytic solution and the model prediction by plotting them on a logarithmic scale.
plot(xTest,yAnalytic,"-")
plot(xTest,yModel,"--")
legend("Analytic","Model")
ylabel("y (log scale)")
The model approximates the analytic solution accurately in the training range
x\in \left[0,2\right]
and it extrapolates in the range
x\in \left(2,4\right]
with lower accuracy.
Calculate the mean squared relative error in the training range
x\in \left[0,2\right]
yModelTrain = yModel(1:500);
yAnalyticTrain = yAnalytic(1:500);
errorTrain = mean(((yModelTrain-yAnalyticTrain)./yAnalyticTrain).^2)
errorTrain = single
Calculate the mean squared relative error in the extrapolated range
x\in \left(2,4\right]
yModelExtra = yModel(501:1000);
yAnalyticExtra = yAnalytic(501:1000);
errorExtra = mean(((yModelExtra-yAnalyticExtra)./yAnalyticExtra).^2)
errorExtra = single
Notice that the mean squared relative error is higher in the extrapolated range than it is in the training range.
The modelGradients function takes as inputs a dlnetwork object dlnet, a mini-batch of input data dlX, and the coefficient associated with the initial condition loss icCoeff. This function returns the gradients of the loss with respect to the learnable parameters in dlnet and the corresponding loss. The loss is defined as a weighted sum of the ODE loss and the initial condition loss. The evaluation of this loss requires second order derivatives. To enable second order automatic differentiation, use the function dlgradient and set the EnableHigherDerivatives name-value argument to true.
function [gradients,loss] = modelGradients(dlnet, dlX, icCoeff)
y = forward(dlnet,dlX);
% Evaluate the gradient of y with respect to x.
% Since another derivative will be taken, set EnableHigherDerivatives to true.
dy = dlgradient(sum(y,"all"),dlX,EnableHigherDerivatives=true);
% Define ODE loss.
eq = dy + 2*y.*dlX;
% Define initial condition loss.
ic = forward(dlnet,dlarray(0,"CB")) - 1;
% Specify the loss as a weighted sum of the ODE loss and the initial condition loss.
loss = mean(eq.^2,"all") + icCoeff * ic.^2;
gradients = dlgradient(loss, dlnet.Learnables);
The modelPredictions function takes a dlnetwork object dlnet and a minibatchqueue of input data mbq and computes the model predictions y by iterating over all data in the minibatchqueue object.
function Y = modelPredictions(dlnet,mbq)
dlXTest = next(mbq);
% Predict output using trained network.
dlY = predict(dlnet,dlXTest);
YPred = gather(extractdata(dlY));
Y = [Y; YPred'];
Lagaris, I. E., A. Likas, and D. I. Fotiadis. “Artificial Neural Networks for Solving Ordinary and Partial Differential Equations.” IEEE Transactions on Neural Networks 9, no. 5 (September 1998): 987–1000. https://doi.org/10.1109/72.712178.
dlarray | dlfeval | dlgradient
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Home : Support : Online Help : Programming : Audio Processing : Convolution
perform a one-dimensional convolution over audio data
Convolution(audArray, kernel, options)
Array, Vector, or Matrix containing the audio data to perform the convolution on
Array, Vector, or Matrix specifying the convolution kernel/mask
one or more options modifying the convolution operation
The Convolution command performs a one-dimensional convolution over audio data using a convolution kernel (also known as a convolution mask), and returns a new audio object giving the result of this operation.
The audArray parameter specifies the audio data on which to perform the convolution, and must be a dense rectangular Array, Vector, or Matrix with datatype=float[8], and one or two dimensions.
The kernel parameter specifies the convolution kernel in the form of an N-element column Vector or Array, or an Nx1 Matrix or Array of numeric values.
The Convolution operation consists of "sliding" the kernel over each channel of the input data, moving it by one sample each time, computing a new sample value for the output channel.
When the audio data has more than one channel, the kernel can be specified as an NxM Matrix or Array, where M is equal to the number of channels. Each channel of the mask is then used for the corresponding channel of the audio data.
The samples covered by the kernel are multiplied by the corresponding entries (weights) in the kernel, and summed up. Then, the sum is divided by the sum of the weights (or 1 if the weights sum to zero), and the sample corresponding to the center of the kernel is given that value in the output audio. The input audio is not modified.
For an N-element kernel, there will be
\frac{N}{2}-\frac{1}{2}
samples at the beginning and end of the audio data that do not have the required
N-1
neighbors.
The shape=shapeMode option controls which samples of the convolution are to be part of the output:
If shape=valid is specified, those samples with too few neighbors are omitted from the convolution, and thus the output data will be smaller than the input data (by one less than the length of the kernel).
If shape=same is specified, the output data will be the same size as the input data. The treatment of the edge samples is then controlled by the edges=edgeMode option described below.
If shape=full is specified (the default), a full mathematical convolution is performed, in which any position of the kernel that overlaps the data will produce an output sample. Thus, the output data will be larger than the input (by one less than the length of the kernel). When part of the kernel protrudes past the edge of the input data, the values used for samples outside the data are controlled by the exterior=exteriorMode option described below.
The edges=edgeMode option controls what is to be done with the edge samples when shape=same.
If edges=leave is specified, the edge samples are left unmodified (copied from the input into the output data verbatim). Depending on the convolution kernel, there will be an audible difference between these and the processed samples. For example, if the kernel implements a low pass filter, the edge samples will still retain the higher frequency components.
The setting edges=compute (the default) causes the edge samples to be computed, in a manner controlled by the exterior=exteriorMode option described below.
A specific sample value can be specified for the edge samples by giving a numeric value for edgeMode (for example, edgeMode=0.5). In that case, the edge samples get that value (for multi-channel audio data, each channel receives the value).
The exterior=exteriorMode option controls how samples exterior to the data are interpreted when shape=same and edges=compute, or when shape=full.
Specifying exterior=ignore tells Convolution to ignore the exterior samples when part of the kernel falls outside the data. Only the samples within the data are multiplied by the corresponding kernel entries. The result is then reweighted by the ratio of total kernel weight to the sum of the kernel entries falling within the data. This avoids fade-in and fade-out at the extremes of the result.
A specific sample value can be specified for the exterior samples by giving a numeric value for exteriorMode (for example, exterior=0.5). This results in a fade-from/to-DC effect at the edges of the result (preceded/followed by a click if the value is non-zero).
The default if the exterior option is not specified is exterior=0.
The weight=n option overrides the weight by which the weighted sum of samples is multiplied. By default, it is multiplied by the inverse of the sum of the weights of the kernel entries, or by 1 if the kernel entries sum to zero. Specifying weight=n causes the sum to be multiplied by n instead.
Specifying weight=none causes the weighted sum to be used as-is. This corresponds to the mathematical definition of convolution.
A convolution corresponding to the mathematical definition can be obtained by specifying the options shape=full, exterior=0, and weight=none. These are the defaults.
\mathrm{audiofile}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/audio/stereo.wav"\right):
\mathrm{with}\left(\mathrm{AudioTools}\right):
\mathrm{aud}≔\mathrm{Read}\left(\mathrm{audiofile}\right)
\textcolor[rgb]{0,0,1}{\mathrm{aud}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{"Sample Rate"}& \textcolor[rgb]{0,0,1}{22050}\\ \textcolor[rgb]{0,0,1}{"File Format"}& \textcolor[rgb]{0,0,1}{\mathrm{PCM}}\\ \textcolor[rgb]{0,0,1}{"File Bit Depth"}& \textcolor[rgb]{0,0,1}{8}\\ \textcolor[rgb]{0,0,1}{"Channels"}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{"Samples/Channel"}& \textcolor[rgb]{0,0,1}{19962}\\ \textcolor[rgb]{0,0,1}{"Duration"}& \textcolor[rgb]{0,0,1}{0.90531}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\end{array}]
\mathrm{tinny}≔\mathrm{Convolution}\left(\mathrm{aud},〈-1,-1,-1,6,-1,-1,1〉\right)
\textcolor[rgb]{0,0,1}{\mathrm{tinny}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{"Sample Rate"}& \textcolor[rgb]{0,0,1}{22050}\\ \textcolor[rgb]{0,0,1}{"File Format"}& \textcolor[rgb]{0,0,1}{\mathrm{PCM}}\\ \textcolor[rgb]{0,0,1}{"File Bit Depth"}& \textcolor[rgb]{0,0,1}{8}\\ \textcolor[rgb]{0,0,1}{"Channels"}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{"Samples/Channel"}& \textcolor[rgb]{0,0,1}{19968}\\ \textcolor[rgb]{0,0,1}{"Duration"}& \textcolor[rgb]{0,0,1}{0.90558}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\end{array}]
The AudioTools[Convolution] command was updated in Maple 2020.
AudioTools[Modulate]
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Modular_form Knowpia
General definition of modular formsEdit
In general,[1] given a subgroup
{\displaystyle \Gamma \subset {\text{SL}}_{2}(\mathbb {Z} )}
of finite index, called an arithmetic group, a modular form of level
{\displaystyle \Gamma }
{\displaystyle k}
is a holomorphic function
{\displaystyle f:{\mathcal {H}}\to \mathbb {C} }
from the upper half-plane such that the following two conditions are satisfied:
1. (automorphy condition) For any
{\displaystyle \gamma \in \Gamma }
there is the equality
{\displaystyle f(\gamma (z))=(cz+d)^{k}f(z)}
2. (growth condition) For any
{\displaystyle \gamma \in {\text{SL}}_{2}(\mathbb {Z} )}
{\displaystyle (cz+d)^{-k}f(\gamma (z))}
{\displaystyle {\text{im}}(z)\to \infty }
{\displaystyle \gamma ={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}_{2}(\mathbb {Z} )}
3. (cuspidal condition) For any
{\displaystyle \gamma \in {\text{SL}}_{2}(\mathbb {Z} )}
{\displaystyle (cz+d)^{-k}f(\gamma (z))\to 0}
{\displaystyle {\text{im}}(z)\to \infty }
As sections of a line bundleEdit
Modular forms can also be interpreted as sections of a specific line bundles on modular varieties. For
{\displaystyle \Gamma \subset {\text{SL}}_{2}(\mathbb {Z} )}
a modular form of level
{\displaystyle \Gamma }
{\displaystyle k}
can be defined as an element of
{\displaystyle f\in H^{0}(X_{\Gamma },\omega ^{\otimes k})=M_{k}(\Gamma )}
{\displaystyle \omega }
is a canonical line bundle on the modular curve
{\displaystyle X_{\Gamma }=\Gamma \backslash ({\mathcal {H}}\cup \mathbb {P} ^{1}(\mathbb {Q} ))}
The dimensions of these spaces of modular forms can be computed using the Riemann–Roch theorem.[2] The classical modular forms for
{\displaystyle \Gamma ={\text{SL}}_{2}(\mathbb {Z} )}
are sections of a line bundle on the moduli stack of elliptic curves.
Modular forms for SL(2, Z)Edit
Standard definitionEdit
A modular form of weight k for the modular group
{\displaystyle {\text{SL}}(2,\mathbf {Z} )=\left\{\left.{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\right|a,b,c,d\in \mathbf {Z} ,\ ad-bc=1\right\}}
is a complex-valued function f on the upper half-plane H = {z ∈ C, Im(z) > 0}, satisfying the following three conditions:
f is a holomorphic function on H.
For any z ∈ H and any matrix in SL(2, Z) as above, we have:
{\displaystyle f\left({\frac {az+b}{cz+d}}\right)=(cz+d)^{k}f(z)}
f is required to be bounded as z → i∞.
The weight k is typically a positive integer.
For odd k, only the zero function can satisfy the second condition.
The third condition is also phrased by saying that f is "holomorphic at the cusp", a terminology that is explained below. Explicitly, the condition means that there exist some
{\displaystyle M,D>0}
{\displaystyle Im(z)>M\implies |f(z)|<D}
{\displaystyle f}
is bounded above some horizontal line.
The second condition for
{\displaystyle S={\begin{pmatrix}0&-1\\1&0\end{pmatrix}},\qquad T={\begin{pmatrix}1&1\\0&1\end{pmatrix}}}
{\displaystyle f\left(-{\frac {1}{z}}\right)=z^{k}f(z),\qquad f(z+1)=f(z)}
respectively. Since S and T generate the modular group SL(2, Z), the second condition above is equivalent to these two equations.
Since f (z + 1) = f (z), modular forms are periodic functions, with period 1, and thus have a Fourier series.
Definition in terms of lattices or elliptic curvesEdit
If we consider the lattice Λ = Zα + Zz generated by a constant α and a variable z, then F(Λ) is an analytic function of z.
If α is a non-zero complex number and αΛ is the lattice obtained by multiplying each element of Λ by α, then F(αΛ) = α−kF(Λ) where k is a constant (typically a positive integer) called the weight of the form.
The absolute value of F(Λ) remains bounded above as long as the absolute value of the smallest non-zero element in Λ is bounded away from 0.
Eisenstein seriesEdit
{\displaystyle G_{k}(\Lambda )=\sum _{0\neq \lambda \in \Lambda }\lambda ^{-k}.}
Then Gk is a modular form of weight k.
For Λ = Z + Zτ we have
{\displaystyle G_{k}(\Lambda )=G_{k}(\tau )=\sum _{(0,0)\neq (m,n)\in \mathbf {Z} ^{2}}{\frac {1}{(m+n\tau )^{k}}},}
{\displaystyle {\begin{aligned}G_{k}\left(-{\frac {1}{\tau }}\right)&=\tau ^{k}G_{k}(\tau )\\G_{k}(\tau +1)&=G_{k}(\tau )\end{aligned}}}
Theta functions of even unimodular latticesEdit
{\displaystyle \vartheta _{L}(z)=\sum _{\lambda \in L}e^{\pi i\Vert \lambda \Vert ^{2}z}}
{\displaystyle \vartheta _{L_{8}\times L_{8}}(z)=\vartheta _{L_{16}}(z),}
The modular discriminantEdit
The Dedekind eta function is defined as
{\displaystyle \eta (z)=q^{1/24}\prod _{n=1}^{\infty }(1-q^{n}),\qquad q=e^{2\pi iz}.}
Modular functionsEdit
f is meromorphic in the open upper half-plane H.
For every integer matrix
{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
in the modular group Γ,
{\displaystyle f\left({\frac {az+b}{cz+d}}\right)=f(z)}
As pointed out above, the second condition implies that f is periodic, and therefore has a Fourier series. The third condition is that this series is of the form
{\displaystyle f(z)=\sum _{n=-m}^{\infty }a_{n}e^{2i\pi nz}.}
It is often written in terms of
{\displaystyle q=\exp(2\pi iz)}
(the square of the nome), as:
{\displaystyle f(z)=\sum _{n=-m}^{\infty }a_{n}q^{n}.}
This is also referred to as the q-expansion of f. The coefficients
{\displaystyle a_{n}}
are known as the Fourier coefficients of f, and the number m is called the order of the pole of f at i∞. This condition is called "meromorphic at the cusp", meaning that only finitely many negative-n coefficients are non-zero, so the q-expansion is bounded below, guaranteeing that it is meromorphic at q = 0. [3]
Modular forms for more general groupsEdit
The functional equation, i.e., the behavior of f with respect to
{\displaystyle z\mapsto {\frac {az+b}{cz+d}}}
can be relaxed by requiring it only for matrices in smaller groups.
The Riemann surface G\H∗Edit
Let G be a subgroup of SL(2, Z) that is of finite index. Such a group G acts on H in the same way as SL(2, Z). The quotient topological space G\H can be shown to be a Hausdorff space. Typically it is not compact, but can be compactified by adding a finite number of points called cusps. These are points at the boundary of H, i.e. in Q∪{∞},[6] such that there is a parabolic element of G (a matrix with trace ±2) fixing the point. This yields a compact topological space G\H∗. What is more, it can be endowed with the structure of a Riemann surface, which allows one to speak of holo- and meromorphic functions.
{\displaystyle {\begin{aligned}\Gamma _{0}(N)&=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}(2,\mathbf {Z} ):c\equiv 0{\pmod {N}}\right\}\\\Gamma (N)&=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}(2,\mathbf {Z} ):c\equiv b\equiv 0,a\equiv d\equiv 1{\pmod {N}}\right\}.\end{aligned}}}
{\displaystyle \dim _{\mathbf {C} }M_{k}\left({\text{SL}}(2,\mathbf {Z} )\right)={\begin{cases}\left\lfloor k/12\right\rfloor &k\equiv 2{\pmod {12}}\\\left\lfloor k/12\right\rfloor +1&{\text{otherwise}}\end{cases}}}
{\displaystyle \lfloor \cdot \rfloor }
denotes the floor function and
{\displaystyle k}
Line bundlesEdit
Rings of modular formsEdit
For a subgroup Γ of the SL(2, Z), the ring of modular forms is the graded ring generated by the modular forms of Γ. In other words, if Mk(Γ) be the ring of modular forms of weight k, then the ring of modular forms of Γ is the graded ring
{\displaystyle M(\Gamma )=\bigoplus _{k>0}M_{k}(\Gamma )}
Entire formsEdit
New formsEdit
New forms are a subspace of modular forms[10] of a fixed weight
{\displaystyle N}
which cannot be constructed from modular forms of lower weights
{\displaystyle M}
{\displaystyle N}
. The other forms are called old forms. These old forms can be constructed using the following observations: if
{\displaystyle M|N}
{\displaystyle \Gamma _{1}(N)\subseteq \Gamma _{1}(M)}
giving a reverse inclusion of modular forms
{\displaystyle M_{k}(\Gamma _{1}(M))\subseteq M_{k}(\Gamma _{1}(N))}
Cusp formsEdit
Automorphic factors are functions of the form
{\displaystyle \varepsilon (a,b,c,d)(cz+d)^{k}}
which are used to generalise the modularity relation defining modular forms, so that
{\displaystyle f\left({\frac {az+b}{cz+d}}\right)=\varepsilon (a,b,c,d)(cz+d)^{k}f(z).}
{\displaystyle \varepsilon (a,b,c,d)}
is called the nebentypus of the modular form. Functions such as the Dedekind eta function, a modular form of weight 1/2, may be encompassed by the theory by allowing automorphic factors.
^ Lan, Kai-Wen. "Cohomology of Automorphic Bundles" (PDF). Archived (PDF) from the original on 1 August 2020.
^ Milne. "Modular Functions and Modular Forms". p. 51.
^ A meromorphic function can only have a finite number of negative-exponent terms in its Laurent series, its q-expansion. It can only have at most a pole at q = 0, not an essential singularity as exp(1/q) has.
^ Chandrasekharan, K. (1985). Elliptic functions. Springer-Verlag. ISBN 3-540-15295-4. p. 15
^ Kubert, Daniel S.; Lang, Serge (1981), Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 244, Berlin, New York: Springer-Verlag, p. 24, ISBN 978-0-387-90517-4, MR 0648603, Zbl 0492.12002
^ Here, a matrix
{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
sends ∞ to a/c.
^ Gunning, Robert C. (1962), Lectures on modular forms, Annals of Mathematics Studies, vol. 48, Princeton University Press , p. 13
^ Shimura, Goro (1971), Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Tokyo: Iwanami Shoten , Theorem 2.33, Proposition 2.26
^ Milne, James (2010), Modular Functions and Modular Forms (PDF), p. 88 , Theorem 6.1.
^ Mocanu, Andreea. "Atkin-Lehner Theory of
{\displaystyle \Gamma _{1}(N)}
-Modular Forms" (PDF). Archived (PDF) from the original on 31 July 2020.
Apostol, Tom M. (1990), Modular functions and Dirichlet Series in Number Theory, New York: Springer-Verlag, ISBN 0-387-97127-0
Diamond, Fred; Shurman, Jerry Michael (2005), A First Course in Modular Forms, Graduate Texts in Mathematics, vol. 228, New York: Springer-Verlag, ISBN 978-0387232294 Leads up to an overview of the proof of the modularity theorem.
Gelbart, Stephen S. (1975), Automorphic Forms on Adèle Groups, Annals of Mathematics Studies, vol. 83, Princeton, N.J.: Princeton University Press, MR 0379375 . Provides an introduction to modular forms from the point of view of representation theory.
Hecke, Erich (1970), Mathematische Werke, Göttingen: Vandenhoeck & Ruprecht
Rankin, Robert A. (1977), Modular forms and functions, Cambridge: Cambridge University Press, ISBN 0-521-21212-X
Ribet, K.; Stein, W., Lectures on Modular Forms and Hecke Operators (PDF)
Serre, Jean-Pierre (1973), A Course in Arithmetic, Graduate Texts in Mathematics, vol. 7, New York: Springer-Verlag . Chapter VII provides an elementary introduction to the theory of modular forms.
Shimura, Goro (1971), Introduction to the arithmetic theory of automorphic functions, Princeton, N.J.: Princeton University Press . Provides a more advanced treatment.
Skoruppa, N. P.; Zagier, D. (1988), "Jacobi forms and a certain space of modular forms", Inventiones Mathematicae, Springer
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Speech Emotion Recognition - MATLAB & Simulink - MathWorks España
The features used in this example were chosen using sequential feature selection, similar to the method described in Sequential Feature Selection for Audio Features (Audio Toolbox).
Create an audioDatastore (Audio Toolbox) that points to the audio files.
Download and load the pretrained network, the audioFeatureExtractor (Audio Toolbox) object used to train the network, and normalization factors for the features. This network was trained using all speakers in the data set except speaker 03.
The 10-fold cross validation accuracy of a first attempt at training was about 60% because of insufficient training data. A model trained on the insufficient data overfits some folds and underfits others. To improve overall fit, increase the size of the dataset using audioDataAugmenter (Audio Toolbox). 50 augmentations per file was chosen empirically as a good tradeoff between processing time and accuracy improvement. You can decrease the number of augmentations to speed up the example.
Create an audioFeatureExtractor (Audio Toolbox) object. Set Window to a periodic 30 ms Hamming window, OverlapLength to 0, and SampleRate to the sample rate of the database. Set gtcc, gtccDelta, mfccDelta, and spectralCrest to true to extract them. Set SpectralDescriptorInput to melSpectrum so that the spectralCrest is calculated for the mel spectrum.
Extract the training features and reorient the features so that time is along rows to be compatible with sequenceInputLayer.
Define a BiLSTM network using bilstmLayer. Place a dropoutLayer before and after the bilstmLayer to help prevent overfitting.
Define training options using trainingOptions.
\mathit{k}-1
bilstmLayer | trainNetwork | trainingOptions | sequenceInputLayer
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EUDML | An Apéry-like difference equation for Catalan's constant. EuDML | An Apéry-like difference equation for Catalan's constant.
Zudilin, W.. "An Apéry-like difference equation for Catalan's constant.." The Electronic Journal of Combinatorics [electronic only] 10.1 (2003): Research paper R14, 10 p.-Research paper R14, 10 p.. <http://eudml.org/doc/126279>.
@article{Zudilin2003,
author = {Zudilin, W.},
title = {An Apéry-like difference equation for Catalan's constant.},
AU - Zudilin, W.
TI - An Apéry-like difference equation for Catalan's constant.
Symbolic computation (Gosper and Zeilberger algorithms, etc.)
{}_{p}{F}_{q}
Articles by Zudilin
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Unit_hyperbola Knowpia
In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation
{\displaystyle x^{2}-y^{2}=1.}
In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial length
The unit hyperbola is blue, its conjugate is green, and the asymptotes are red.
{\displaystyle r={\sqrt {x^{2}-y^{2}}}.}
Whereas the unit circle surrounds its center, the unit hyperbola requires the conjugate hyperbola
{\displaystyle y^{2}-x^{2}=1}
to complement it in the plane. This pair of hyperbolas share the asymptotes y = x and y = −x. When the conjugate of the unit hyperbola is in use, the alternative radial length is
{\displaystyle r={\sqrt {y^{2}-x^{2}}}.}
The unit hyperbola is a special case of the rectangular hyperbola, with a particular orientation, location, and scale. As such, its eccentricity equals
{\displaystyle {\sqrt {2}}.}
The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction of spacetime as a pseudo-Euclidean space. There the asymptotes of the unit hyperbola form a light cone. Further, the attention to areas of hyperbolic sectors by Gregoire de Saint-Vincent led to the logarithm function and the modern parametrization of the hyperbola by sector areas. When the notions of conjugate hyperbolas and hyperbolic angles are understood, then the classical complex numbers, which are built around the unit circle, can be replaced with numbers built around the unit hyperbola.
Generally asymptotic lines to a curve are said to converge toward the curve. In algebraic geometry and the theory of algebraic curves there is a different approach to asymptotes. The curve is first interpreted in the projective plane using homogeneous coordinates. Then the asymptotes are lines that are tangent to the projective curve at a point at infinity, thus circumventing any need for a distance concept and convergence. In a common framework (x, y, z) are homogeneous coordinates with the line at infinity determined by the equation z = 0. For instance, C. G. Gibson wrote:[2]
For the standard rectangular hyperbola
{\displaystyle f=x^{2}-y^{2}-1}
in ℝ2, the corresponding projective curve is
{\displaystyle F=x^{2}-y^{2}-z^{2},}
which meets z = 0 at the points P = (1 : 1 : 0) and Q = (1 : −1 : 0). Both P and Q are simple on F, with tangents x + y = 0, x − y = 0; thus we recover the familiar 'asymptotes' of elementary geometry.
Minkowski diagramEdit
The Minkowski diagram is drawn in a spacetime plane where the spatial aspect has been restricted to a single dimension. The units of distance and time on such a plane are
units of 30 centimetres length and nanoseconds, or
astronomical units and intervals of 8 minutes and 20 seconds, or
light years and years.
Each of these scales of coordinates results in photon connections of events along diagonal lines of slope plus or minus one. Five elements constitute the diagram Hermann Minkowski used to describe the relativity transformations: the unit hyperbola, its conjugate hyperbola, the axes of the hyperbola, a diameter of the unit hyperbola, and the conjugate diameter. The plane with the axes refers to a resting frame of reference. The diameter of the unit hyperbola represents a frame of reference in motion with rapidity a where tanh a = y/x and (x,y) is the endpoint of the diameter on the unit hyperbola. The conjugate diameter represents the spatial hyperplane of simultaneity corresponding to rapidity a. In this context the unit hyperbola is a calibration hyperbola[3][4] Commonly in relativity study the hyperbola with vertical axis is taken as primary:
The arrow of time goes from the bottom to top of the figure — a convention adopted by Richard Feynman in his famous diagrams. Space is represented by planes perpendicular to the time axis. The here and now is a singularity in the middle.[5]
The vertical time axis convention stems from Minkowski in 1908, and is also illustrated on page 48 of Eddington's The Nature of the Physical World (1928).
The branches of the unit hyperbola evolve as the points
{\displaystyle (\cosh a,\sinh a)}
{\displaystyle (-\cosh a,-\sinh a)}
depending on the hyperbolic angle parameter
{\displaystyle a}
A direct way to parameterizing the unit hyperbola starts with the hyperbola xy = 1 parameterized with the exponential function:
{\displaystyle (e^{t},\ e^{-t}).}
This hyperbola is transformed into the unit hyperbola by a linear mapping having the matrix
{\displaystyle A={\tfrac {1}{2}}{\begin{pmatrix}1&1\\1&-1\end{pmatrix}}\ :}
{\displaystyle (e^{t},\ e^{-t})\ A=({\frac {e^{t}+e^{-t}}{2}},\ {\frac {e^{t}-e^{-t}}{2}})=(\cosh t,\ \sinh t).}
The motion
{\displaystyle \rho =\alpha \cosh(nt+\epsilon )+\beta \sinh(nt+\epsilon )}
has some curious analogies to elliptic harmonic motion. ... The acceleration
{\displaystyle {\ddot {\rho }}=n^{2}\rho \ ;}
thus it is always proportional to the distance from the centre, as in elliptic harmonic motion, but directed away from the centre.[6]
As a particular conic, the hyperbola can be parametrized by the process of addition of points on a conic. The following description was given by Russian analysts:
Fix a point E on the conic. Consider the points at which the straight line drawn through E parallel to AB intersects the conic a second time to be the sum of the points A and B.
For the hyperbola
{\displaystyle x^{2}-y^{2}=1}
with the fixed point E = (1,0) the sum of the points
{\displaystyle (x_{1},\ y_{1})}
{\displaystyle (x_{2},\ y_{2})}
{\displaystyle (x_{1}x_{2}+y_{1}y_{2},\ y_{1}x_{2}+y_{2}x_{1})}
under the parametrization
{\displaystyle x=\cosh \ t}
{\displaystyle y=\sinh \ t}
this addition corresponds to the addition of the parameter t.[7]
Complex plane algebraEdit
Whereas the unit circle is associated with complex numbers, the unit hyperbola is key to the split-complex number plane consisting of z = x + yj, where j 2 = +1. Then jz = y + xj, so the action of j on the plane is to swap the coordinates. In particular, this action swaps the unit hyperbola with its conjugate and swaps pairs of conjugate diameters of the hyperbolas.
In terms of the hyperbolic angle parameter a, the unit hyperbola consists of points
{\displaystyle \pm (\cosh a+j\sinh a)}
, where j = (0,1).
The right branch of the unit hyperbola corresponds to the positive coefficient. In fact, this branch is the image of the exponential map acting on the j-axis. Since
{\displaystyle \exp(aj)\exp(bj)=\exp((a+b)j)}
the branch is a group under multiplication. Unlike the circle group, this unit hyperbola group is not compact. Similar to the ordinary complex plane, a point not on the diagonals has a polar decomposition using the parametrization of the unit hyperbola and the alternative radial length.
^ Eric Weisstein Rectangular hyperbola from Wolfram Mathworld
^ C.G. Gibson (1998) Elementary Geometry of Algebraic Curves, p 159, Cambridge University Press ISBN 0-521-64140-3
^ Anthony French (1968) Special Relativity, page 83, W. W. Norton & Company
^ W.G.V. Rosser (1964) Introduction to the Theory of Relativity, figure 6.4, page 256, London: Butterworths
^ A.P. French (1989) "Learning from the past; Looking to the future", acceptance speech for 1989 Oersted Medal, American Journal of Physics 57(7):587–92
^ William Kingdon Clifford (1878) Elements of Dynamic, pages 89 & 90, London: MacMillan & Co; on-line presentation by Cornell University Historical Mathematical Monographs
^ Viktor Prasolov & Yuri Solovyev (1997) Elliptic Functions and Elliptic Integrals, page one, Translations of Mathematical Monographs volume 170, American Mathematical Society
F. Reese Harvey (1990) Spinors and calibrations, Figure 4.33, page 70, Academic Press, ISBN 0-12-329650-1 .
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Probability, Studymaterial: ICSE Class 11-commerce MATHS, Math - Meritnation
Consider the experiment of throwing a dice. Any of the numbers 1, 2, 3, 4, 5, or 6 can come up on the upper face of the dice. We can easily find the probability of getting a number 5 on the upper face of the dice?
Mathematically, probability of any event E can be defined as follows.
Here, S represents the sample space and n(S) represents the number of outcomes in the sample space.
For this experiment, we have
Sample space (S) = {1, 2, 3, 4, 5, 6}. Thus, S is a finite set.
So, we can say that the possible outcomes of this experiment are 1, 2, 3, 4, 5, and 6.
Number of favourable outcomes of getting the number 5 = 1
Probability (getting 5)
Similarly, we can find the probability of getting other numbers also.
P (getting 1) , P (getting 2) , P (getting 3) , P (getting 4) and
P (getting 6)
Let us add the probability of each separate observation.
This will give us the sum of the probabilities of all possible outcomes.
P (getting 1) + P (getting 2) + P (getting 3) + P (getting 4) + P (getting 5) + P (getting 6) = + + + + + = 1
“Sum of the probabilities of all elementary events is 1”.
Now, let us find the probability of not getting 5 on the upper face.
The outcomes favourable to this event are 1, 2, 3, 4, and 6.
P (not getting 5)
We can also see that P (getting 5) + P (not getting 5)
“Sum of probabilities of occurrence and non occurrence of an event is 1”.
i.e. If E is the event, then P (E) + P (not E) = 1 … (1)
or we can write P(E) = 1 − P (not E)
Here, the events of getting a number 5 and not getting 5 are complements of each other as we cannot find an observation which is common to the two observations.
Thus, event not E is the complement of event E. Complement of event E is denoted by or E'.
Using equation (1), we can write
P (E) + P ( ) = 1
P ( ) = 1 – P (E)
This is a very important property about the probability of complement of an event and it is stated as follows:
If E is an event of finite sample space S, then P ( ) = 1 – P(E) where is the complement of event E.
Now, let us prove this property algebraically.
E ∪ = S and E ∩ =
⇒ n(E ∪ ) = n(S) and n(E ∩ ) = n( )
⇒ n(E ∪ ) = n(S) and n(E ∩ ) = 0 ...(1)
n(E ∪ ) = n(S)
⇒ n(E) + n( ) – n(E ∩ ) = n(S)
⇒ n(E) + n( ) – 0 = n(S) [Using (1)]
⇒ n( ) = n(S) – n(E)
On dividing both sides by n(S), we get
⇒ P( ) = 1 – P(E)
Let us solve some examples based on this concept.
ODDS (Ratio of two complementary probabilities):
Let n be the number of distinct sample points in the sample space S. Suppose, out of these n points, m points are favorable for the occurrence of event A. Hence, the remaining
n-m
points are unfavorable for the occurrence of event A or we can say,
n-m
points are favorable for the occurrence of event A'.
\therefore P\left(A\right)=\frac{m}{n}, P\left(A\text{'}\right)=\frac{n-m}{n}
The ratio of favorable cases to the number of unfavorable cases is known as odds in the favor of event A which is given by
\frac{m}{n-m}
i.e. P(A) : P(A').
The ratio of unfavorable cases to the number of favorable cases is known as odds against the favor of event A which is given by
\frac{n-m}{m}
i.e. P(A') : P(A).
One card is drawn from a well shuffled deck. What is the probability that the card will be
(i) a king?
(ii) not a king?
Let E be the event ‘the card is a king’ and F be the event ‘the card is not a king’.
(i) Since there are 4 kings in a deck.
Number of outcomes favourable to E = 4
Here, the events E and F are complements of each other.
∴ P(E) + P(F) = 1
P(F) = 1 −
If the probability of an event A is 0.12 and B is 0.88 and they belong to the same set of observations, then show that A and B are complementary events.
It is given that P (A) = 0.12 and P (B) = 0.88
Now, P(A) + P(B) = 0.12 + 0.88 = 1
The events A and B are complementary events.
Savita and Babita are playing badminton. The probability of Savita winning the match is 0.52. What is the probability of Babita winning the match?
Let E be the event ‘Savita winning the match’ and F be the event ‘Babita wining the match’.
It is given that P (E) = 0.52
Here, E and F are complementary events because if Babita wins the match, Savita will surely lose the match and vice versa.
P (E) + P (F) = 1
0.52 + P (F) = 1
P (F) = 1 − 0.52 = 0.48
Thus, the probability of Babita winning the match is 0.48.
In a box, there are 2 red, 5 blue, and 7 black marbles. One marble is drawn from the box at random. What is the probability that the marble drawn will be (i) red (ii) blue (iii) black (iv) not blue?
Since the marble is drawn at random, all the marbles are equally likely to be drawn.
Let A be the event ‘the marble is red’, B be the event ‘the marble is blue’ and C be the event ‘the marble is black.
(i) Number of outcomes favourable to event A = 2
(ii) Number of outcomes favourable to event B = 5
∴ P (B)
(iii) Number of outcomes favourable to event C = 7
(iv) We have, P (B)
The event of drawing a marble which is not blu…
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Joaquin is getting a new locker at school and the first thing he must do is decide on a new combination. The three number locker combination can be selected from the numbers
0
21
How many different locker combinations can Joaquin choose if none of the numbers can be repeated?
In this case the common use of the word “combination” conflicts with the mathematical meaning.
With your understanding of permutations, combinations, and factorials, decide if the name “combination lock” is appropriate.
How many mathematical combinations are possible?
_{22}C_3=1540
, but this doesn't make sense for a mechanical lock because it implies dialing the numbers in any order to open the lock.
How many choices would there be if you could repeat a number, but not use the same number twice in a row?
22·21·21 = 9702
Click the link to the right for full version. 10-155 HW eTool (Desmos)
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Recognize and Avoid Round-Off Errors - MATLAB & Simulink - MathWorks Italia
Use Symbolic Computations When Possible
Perform Calculations with Increased Precision
Compare Symbolic and Numeric Results
Plot the Function or Expression
When approximating a value numerically, remember that floating-point results can be sensitive to the precision used. Also, floating-point results are prone to round-off errors. The following approaches can help you recognize and avoid incorrect results.
Performing computations symbolically is recommended because exact symbolic computations are not prone to round-off errors. For example, standard mathematical constants have their own symbolic representations in Symbolic Math Toolbox™:
Avoid unnecessary use of numeric approximations. A floating-point number approximates a constant; it is not the constant itself. Using this approximation, you can get incorrect results. For example, the heaviside special function returns different results for the sine of sym(pi) and the sine of the numeric approximation of pi:
heaviside(sin(sym(pi)))
heaviside(sin(pi))
The Riemann hypothesis states that all nontrivial zeros of the Riemann Zeta function ζ(z) have the same real part ℜ(z) = 1/2. To locate possible zeros of the Zeta function, plot its absolute value |ζ(1/2 + iy)|. The following plot shows the first three nontrivial roots of the Zeta function |ζ(1/2 + iy)|.
fplot(abs(zeta(1/2 + i*y)), [0 30])
Use the numeric solver vpasolve to approximate the first three zeros of this Zeta function:
vpasolve(zeta(1/2 + i*y), y, 15)
Now, consider the same function, but slightly increase the real part,
\zeta \left(\frac{1000000001}{2000000000}+iy\right)
. According to the Riemann hypothesis, this function does not have a zero for any real value y. If you use vpasolve with the 10 significant decimal digits, the solver finds the following (nonexisting) zero of the Zeta function:
old = digits;
vpasolve(zeta(1000000001/2000000000 + i*y), y, 15)
Increasing the number of digits shows that the result is incorrect. The Zeta function
\zeta \left(\frac{1000000001}{2000000000}+iy\right)
does not have a zero for any real value 14 < y < 15:
14.1347251417347 + 0.000000000499989207306345i
For further computations, restore the default number of digits:
Bessel functions with half-integer indices return exact symbolic expressions. Approximating these expressions by floating-point numbers can produce very unstable results. For example, the exact symbolic expression for the following Bessel function is:
B = besselj(53/2, sym(pi))
(351*2^(1/2)*(119409675/pi^4 - 20300/pi^2 - 315241542000/pi^6...
+ 445475704038750/pi^8 - 366812794263762000/pi^10 +...
182947881139051297500/pi^12 - 55720697512636766610000/pi^14...
+ 10174148683695239020903125/pi^16 - 1060253389142977540073062500/pi^18...
+ 57306695683177936040949028125/pi^20 - 1331871030107060331702688875000/pi^22...
+ 8490677816932509614604641578125/pi^24 + 1))/pi^2
Use vpa to approximate this expression with the 10-digit accuracy:
vpa(B, 10)
Now, call the Bessel function with the floating-point parameter. Significant difference between these two approximations indicates that one or both results are incorrect:
besselj(53/2, pi)
Increase the numeric working precision to obtain a more accurate approximation for B:
Plotting the results can help you recognize incorrect approximations. For example, the numeric approximation of the following Bessel function returns:
B = besselj(53/2, sym(pi));
Plot this Bessel function for the values of x around 53/2. The function plot shows that the approximation is incorrect:
fplot(besselj(x, sym(pi)), [26 27])
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Find sources: "Hypotenuse" – news · newspapers · books · scholar · JSTOR (May 2019) (Learn how and when to remove this template message)
2 Calculating the hypotenuse
{\displaystyle c={\sqrt {a^{2}+b^{2}}}.}
{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos 90^{\circ }=a^{2}+b^{2}\therefore c={\sqrt {a^{2}+b^{2}}}.}
{\displaystyle \alpha \,}
{\displaystyle \beta \,}
{\displaystyle c\,}
{\displaystyle b\,}
{\displaystyle {\frac {b}{c}}=\sin(\beta )\,}
{\displaystyle \beta \ =\arcsin \left({\frac {b}{c}}\right)\,}
{\displaystyle \beta \,}
{\displaystyle b\,}
{\displaystyle b\,}
{\displaystyle \alpha \,}
{\displaystyle \beta \,}
{\displaystyle \beta \,}
{\displaystyle \beta \ =\arccos \left({\frac {a}{c}}\right)\,}
{\displaystyle a\,}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Hypotenuse&oldid=1085261400"
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RemovePaletteEntry - Maple Help
Home : Support : Online Help : Programming : Document Tools : RemovePaletteEntry
remove a task from a custom Snippets palette
RemovePaletteEntry(entry, palette=palette_name)
string ; the name of the task to remove from the palette
string ; the name of the Snippets palette from which to remove entry
The RemovePaletteEntry command removes a task from a custom Snippets palette.
If the store=true option was provided when the Snippets palette was created (through the DocumentTools[AddPalette] command), the task will not be included in the palette in subsequent Maple sessions.
The task can be added back to this or another Snippets palette by using the DocumentTools[AddPaletteEntry] command.
To remove a Snippets palette completely, use the DocumentTools[RemovePalette] command.
\mathrm{with}\left(\mathrm{DocumentTools}\right):
\mathrm{AddPaletteEntry}\left("Task_1",\mathrm{palette}="My palette"\right)
\mathrm{AddPaletteEntry}\left("Task_2",\mathrm{palette}="My palette"\right)
\mathrm{RemovePaletteEntry}\left("Task_1",\mathrm{palette}="My palette"\right)
\mathrm{RemovePaletteEntry}\left("Task_2",\mathrm{palette}="My palette"\right)
The DocumentTools[RemovePaletteEntry] command was introduced in Maple 16.
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Stability Conditions of Second Order Integrodifferential Equations with Variable Delay
2014 Stability Conditions of Second Order Integrodifferential Equations with Variable Delay
We investigate integrodifferential functional differential equations
\stackrel{̈}{x}+f\left(t,x,\stackrel{̇}{x}\right)\stackrel{̇}{x}+{\int }_{t-r\left(t\right)}^{t}\mathrm{}a\left(t,s\right)g\left(x\left(s\right)\right)ds=0
with variable delay. By using the fixed point theory, we obtain conditions which ensure that the zero solution of this equation is stable under an exponentially weighted metric. Then we establish necessary and sufficient conditions ensuring that the zero solution is asymptotically stable. We will give an example to apply our results.
Dingheng Pi. "Stability Conditions of Second Order Integrodifferential Equations with Variable Delay." Abstr. Appl. Anal. 2014 (SI66) 1 - 11, 2014. https://doi.org/10.1155/2014/371639
Dingheng Pi "Stability Conditions of Second Order Integrodifferential Equations with Variable Delay," Abstract and Applied Analysis, Abstr. Appl. Anal. 2014(SI66), 1-11, (2014)
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Polynomial with specified roots or characteristic polynomial - MATLAB poly - MathWorks Switzerland
{p}_{1}{x}^{n}+{p}_{2}{x}^{n-1}+...+{p}_{n}x+{p}_{n+1}\text{\hspace{0.17em}}.
\mathrm{det}\left(\lambda I-A\right)={p}_{1}{\lambda }^{n}+\dots +{p}_{n}\lambda +{p}_{n+1}\text{\hspace{0.17em}}.
\left(\lambda -{\lambda }_{1}\right)\left(\lambda -{\lambda }_{2}\right)\dots \left(\lambda -{\lambda }_{n}\right)\text{\hspace{0.17em}}.
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BorderPolynomial - Maple Help
Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : ParametricSystemTools Subpackage : BorderPolynomial
compute the border polynomial of a semi-algebraic system
BorderPolynomial(F, N, P, H, d, R)
The input is a parametric semi-algebraic system whose parameters are the last d variables of R and whose polynomial equations, non-negative polynomial inequalities, (strictly) positive polynomial inequalities, and polynomial inequations are given respectively by F, N, P, and H.
The command BorderPolynomial returns an object of type border_polynomial. It is a list of polynomials of R the product of which is the border polynomial of the input system.
If the output border polynomial only contains the parameters, above each parameter value not canceling the border polynomial, the input parametric system has finitely many solutions. Determining conditions on the parameters for the input system to have a prescribed (finite) number of solutions is achieved by the command RealRootClassification.
If the input system is not sufficiently generic (and in particular if it is not generically zero-dimensional with respect to the d parameters) then the output is set to a special value, as shown in the examples below.
The base field of R is meant to be the field of real numbers. Thus R must be of characteristic zero and must have no parameters (in the sense of the RegularChains library).
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):
R≔\mathrm{PolynomialRing}\left([x,y,s]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
F≔[s-\left(y+1\right)x,s-\left(x+1\right)y]
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{-}\left(\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{-}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}]
\mathrm{bp}≔\mathrm{BorderPolynomial}\left(F,[],[],[],1,R\right)
\textcolor[rgb]{0,0,1}{\mathrm{bp}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{border_polynomial}}
\mathrm{Info}\left(\mathrm{bp},R\right)
[\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{4}}]
The reason why border polynomials must form of type (and cannot just be seen as lists of polynomials) is that under special circumstances, the border polynomial of a parametric semi-algebraic system takes an exceptional value.
The first such case is when the parameters do not appear in the system of polynomials; then there are no border polynomials as in the example below.
R≔\mathrm{PolynomialRing}\left([x,a,b,c]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
\mathrm{bp_nobp}≔\mathrm{BorderPolynomial}\left([{x}^{2}-1],[],[],[],3,R\right)
\textcolor[rgb]{0,0,1}{\mathrm{bp_nobp}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{border_polynomial}}
\mathrm{Info}\left(\mathrm{bp_nobp},R\right)
[\textcolor[rgb]{0,0,1}{1}]
Another special circumstance is that of overdetermined or inconsistent systems, as in the example below
F≔[a{x}^{2}+bx+c,a,b]
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]
\mathrm{bp}≔\mathrm{BorderPolynomial}\left(F,[],[],[],1,R\right)
\textcolor[rgb]{0,0,1}{\mathrm{bp}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{border_polynomial}}
\mathrm{Info}\left(\mathrm{bp},R\right)
[]
A last special circumstance is when the input system has "generically" infinitely many complex solutions, as in the example below (this is because of the d=2).
F≔[a{x}^{2}+bx+c]:
d≔2
\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{2}
\mathrm{bp}≔\mathrm{BorderPolynomial}\left(F,[],[],[],d,R\right)
\textcolor[rgb]{0,0,1}{\mathrm{bp}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{border_polynomial}}
\mathrm{Info}\left(\mathrm{bp},R\right)
[\textcolor[rgb]{0,0,1}{0}]
Yang, L.; Hou, X.; and Xia, B. "A complete algorithm for automated discovering of a class of inequality-type theorems." Science China, F. Vol. 44, (2001): 33-49.
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Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations
2010 Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations
L. K. Kikina, I. P. Stavroulakis
Consider the second-order linear delay differential equation
{x}^{\prime \prime }\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right)=0
t\ge {t}_{0}
p\in C\left(\left[{t}_{0},\infty \right),{ℝ}^{+}\right)
\tau \in C\left(\left[{t}_{0},\infty \right),ℝ\right)
\tau \left(t\right)
\tau \left(t\right)\le t
t\ge {t}_{0}
{\mathrm{lim}}_{t\to \infty }\tau \left(t\right)=\infty
, the (discrete analogue) second-order difference equation
{\Delta }^{2}x\left(n\right)+p\left(n\right)x\left(\tau \left(n\right)\right)=0
\Delta x\left(n\right)=x\left(n+1\right)-x\left(n\right)
{\Delta }^{2}=\Delta \circ \Delta
p:ℕ\to {ℝ}^{+}
\tau :ℕ\to ℕ
\tau \left(n\right)\le n-1
{\mathrm{lim}}_{n\to \infty }\tau \left(n\right)=+\infty
, and the second-order functional equation
x\left(g\left(t\right)\right)=P\left(t\right)x\left(t\right)+Q\left(t\right)x\left({g}^{2}\left(t\right)\right)
t\ge {t}_{0}
P
Q\in C\left(\left[{t}_{0},\infty \right),{ℝ}^{+}\right)
g\in C\left(\left[{t}_{0},\infty \right),ℝ\right)
g\left(t\right)\not\equiv t
t\ge {t}_{0}
{\mathrm{lim}}_{t\to \infty }g\left(t\right)=\infty
{g}^{2}
denotes the 2th iterate of the function
g
{g}^{0}\left(t\right)=t
{g}^{2}\left(t\right)=g\left(g\left(t\right)\right)
t\ge {t}_{0}
. The most interesting oscillation criteria for the second-order linear delay differential equation, the second-order difference equation and the second-order functional equation, especially in the case where
\mathrm{lim}{\mathrm{inf}}_{t\to \infty }{\int }_{\tau \left(t\right)}^{t}\tau \left(s\right)p\left(s\right)ds\le 1/e
\mathrm{lim}{\mathrm{sup}}_{t\to \infty }{\int }_{\tau \left(t\right)}^{t}\tau \left(s\right)p\left(s\right)ds<1
for the second-order linear delay differential equation, and
0<\mathrm{lim}{\mathrm{inf}}_{t\to \infty }\left\{Q\left(t\right)P\left(g\left(t\right)\right)\right\}\le 1/4
\mathrm{lim}{\mathrm{sup}}_{t\to \infty }\left\{Q\left(t\right)P\left(g\left(t\right)\right)\right\}<1
, for the second-order functional equation, are presented.
L. K. Kikina. I. P. Stavroulakis. "Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations." Int. J. Differ. Equ. 2010 (SI2) 1 - 14, 2010. https://doi.org/10.1155/2010/598068
L. K. Kikina, I. P. Stavroulakis "Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations," International Journal of Differential Equations, Int. J. Differ. Equ. 2010(SI2), 1-14, (2010)
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Nonmetals - Course Hero
General Chemistry/Metals, Metalloids, and Nonmetals/Nonmetals
Chemically, nonmetals have high ionization energies so they do not give electrons away easily. Compared to metals, nonmetals have high electronegativity values, which indicates a tendency to strongly pull electrons. For these reasons, when nonmetals react with metals, they generally form ionic compounds. When nonmetals react with other nonmetals, they generally form covalent compounds.
Hydrogen is the lightest element on the periodic table and the only nonmetal element found outside groups 14–17. Hydrogen is the chemical element with the atomic number of 1. It is the smallest, lightest element and the most common atom in the universe. Hydrogen atoms are often part of other compounds. Under standard temperature and pressure, hydrogen atoms react with each other to form hydrogen gas, H2, a diatomic element. Hydrogen gas is highly flammable. Hydrogen is composed of one electron and one proton, and most hydrogen atoms do not contain neutrons. If a hydrogen atom loses its electron, the hydrogen ion (H+) that is formed is a proton.
Hydrogen is commonly produced from methane (CH4) and steam (H2O). At high temperatures, the following reaction takes place:
{\rm{CH}}_4(g)+{2{\rm H}_2\rm O}(g)\rightarrow{\rm{CO}}_2(g)+4{\rm H}_2(g)
Hydrogen is important for synthesis of ammonia (NH3) and in the petroleum industry for processing fossil fuels.
Sulfur is an important element because sulfuric acid is the most commonly used commercial compound in the world, used in artificial fertilizers, detergents, lead-acid automobile batteries, and more. Sulfur is an element from group 16 and has six valence electrons, like oxygen. In compounds, sulfur commonly takes either a –2 or +6 oxidation state. Sulfur commonly takes a +6 oxidation state in compounds with electronegative atoms, such as fluorine. Sulfur has many allotropes. An allotrope is one of the possible physical forms in which an element can exist. The most common allotrope of sulfur is a ring of eight sulfur atoms. In this form, sulfur is a yellow solid at room temperature.
Sulfur was traditionally mined. There are multiple minerals that contain sulfur, with pyrite being a common one. It could even be found in almost pure elemental form in certain locations. Today, most sulfur is obtained from oil and natural gas. Crude oil and natural gas have sulfur atoms bonded to carbons. The carbon-sulfur bonds can be broken, and sulfur can be recovered as hydrogen sulfide (H2S).
<Metalloids>Carbon
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Optimal Inequalities for Power Means
2012 Optimal Inequalities for Power Means
Yong-Min Li, Bo-Yong Long, Yu-Ming Chu, Wei-Ming Gong
We present the best possible power mean bounds for the product
{M}_{p}^{\alpha }\left(a,b\right){M}_{-p}^{1-\alpha }\left(a,b\right)
p>0
\alpha \in \left(0,1\right)
a,b>0
a\ne b
{M}_{p}\left(a,b\right)
p
th power mean of two positive numbers
and
b
Yong-Min Li. Bo-Yong Long. Yu-Ming Chu. Wei-Ming Gong. "Optimal Inequalities for Power Means." J. Appl. Math. 2012 1 - 8, 2012. https://doi.org/10.1155/2012/182905
Yong-Min Li, Bo-Yong Long, Yu-Ming Chu, Wei-Ming Gong "Optimal Inequalities for Power Means," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-8, (2012)
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Home : Support : Online Help : Connectivity : Maple T.A. : MapleTA Package : Overview
Overview of the MapleTA Package
List of MapleTA Subpackages
List of MapleTA Package Commands
The Möbius Assessment software from DigitalEd (formerly Maple T.A.) uses Maple's computation engine to assess student responses. Checking for equivalent results in the domain of mathematics is a hard problem that is best solved using a symbolic language such as Maple. Even simple problems, such as a student answering
y
x
when the expected answer is
x
y
, can be very difficult in a system that does not understand the mathematics behind the student's answer.
While Möbius Assessment has access to all of Maple's mathematical operations, it also has a selection of built-in commands that can be used in algorithms that do not access Maple's engine. This can be used in a broader range of questions. For convenience, these commands are now available in Maple.
For more information on Maple and DigitalEd integration, see Integration with DigitalEd Products: An Overview.
Each command in the MapleTA package can be accessed by using either the long form or the short form of the command name in the calling sequence.
As the underlying implementation of the MapleTA package is a module, it is also possible to use the form MapleTA:-command to access a command from the package. For more information, see Module Members.
The Builtin subpackage is a collection of commands that are natively supported by Maple T.A..
The Overview of the MapleTA Package command was introduced in Maple 18.
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This problem is a checkpoint for finding angles in and areas of regular polygons. It will be referred to as Checkpoint
10
What is the measure of each interior angle of a regular
20
Each angle of a regular polygon measures
157.5º
. How many sides does this polygon have?
Find the area of a regular octagon with sides
5
Ideally, at this point you are comfortable working with these types of problems and can solve them correctly. If you feel that you need more confidence when solving these types of problems, then review the Checkpoint
10
materials and try the practice problems provided. From this point on, you will be expected to do problems like these correctly and with confidence.
If you have an eBook for CCG, login and then click the following link: Checkpoint 10: Finding Angles in and Areas of Regular Polygons
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Transitivity - Maple Help
Home : Support : Online Help : Mathematics : Group Theory : Transitivity
Transitivity( G )
The transitivity of a permutation group
G
is the largest non-negative integer
k
G
k
-transitive. In other words, the transitivity of
G
k
G
k
-transitive, but not
\left(k+1\right)
-transitive. (By convention, the transitivity of an intransitive group is equal to
0
The Transitivity( G ) command returns the transitivity of the permutation group G. The group G must be an instance of a permutation group.
\mathrm{with}\left(\mathrm{GroupTheory}\right):
\mathrm{Transitivity}\left(\mathrm{Alt}\left(4\right)\right)
\textcolor[rgb]{0,0,1}{2}
\mathrm{Transitivity}\left(\mathrm{Symm}\left(4\right)\right)
\textcolor[rgb]{0,0,1}{4}
\mathrm{Transitivity}\left(\mathrm{Alt}\left(5\right)\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{Transitivity}\left(\mathrm{Symm}\left(5\right)\right)
\textcolor[rgb]{0,0,1}{5}
\mathrm{Transitivity}\left(\mathrm{MathieuGroup}\left(23\right)\right)
\textcolor[rgb]{0,0,1}{4}
The GroupTheory[Transitivity] command was introduced in Maple 2016.
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(Redirected from TAP)
TGM+ is a rising garbage mode, similar to Sega's Bloxeed. Players must dig through to survive as they progress through the 999 levels. This mode has speed timings similar to TGM. An internal counter is incremented every time a tetromino is locked down without clearing lines; once this counter reaches
{\displaystyle 13-\left\lfloor {\text{level}}/100\right\rfloor }
, a row of garbage rises from the floor of the playfield, and the counter resets.[d] The garbage follows the fixed pattern shown here, looping every 24 rows.
{\displaystyle {\text{Score}}=(\left\lceil ({\text{Level}}+{\text{Lines}})/4\right\rceil +{\text{Soft}}+(2\times {\text{Sonic}}))\times {\text{Lines}}\times {\text{Combo}}\times {\text{Bravo}}}
{\displaystyle {\text{Score}}=(\left\lceil ({\text{Level}}+{\text{Lines}})/4\right\rceil +{\text{Soft}}+(2\times {\text{Sonic}}))\times {\text{Lines}}\times {\text{Combo}}\times {\text{Bravo}}+\left\lceil ({\text{Level After Clear}})/2\right\rceil +({\text{Speed}}\times 7)}
{\displaystyle \left\lceil \cdot \right\rceil }
is the ceiling function (i.e. the contents are rounded up).
Level After Clear is is the level just after the line clear. This is different from
{\displaystyle ({\text{Level}}+{\text{Lines}})}
for edge cases like reaching 300 in Normal mode, 500 when being torikan-stopped in Death mode, and reaching 999 otherwise.
{\displaystyle {\text{Combo}}={\text{Previous Combo Value}}+(2\times {\text{Lines}})-2}
{\displaystyle {\text{Speed}}={\text{Lock Delay}}-{\text{Active Time}}}
Normal mode multiplies line clear scores by 6 and the player is given a time bonus of
{\displaystyle 1253\times {\text{max}}\left(0,\left\lceil 300-{\text{Seconds}}\right\rceil \right)}
where Seconds is the clear time in seconds.
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EUDML | Improving dense packings of equal disks in a square. EuDML | Improving dense packings of equal disks in a square.
Improving dense packings of equal disks in a square.
Boll, W.David; Donovan, Jerry; Graham, Ronald L.; Lubachevsky, Boris D.
Boll, W.David, et al. "Improving dense packings of equal disks in a square.." The Electronic Journal of Combinatorics [electronic only] 7.1 (2000): Research paper R46, 9 p.-Research paper R46, 9 p.. <http://eudml.org/doc/120990>.
@article{Boll2000,
author = {Boll, W.David, Donovan, Jerry, Graham, Ronald L., Lubachevsky, Boris D.},
keywords = {packings of equal disks; optimal packings},
title = {Improving dense packings of equal disks in a square.},
AU - Boll, W.David
AU - Donovan, Jerry
AU - Graham, Ronald L.
TI - Improving dense packings of equal disks in a square.
KW - packings of equal disks; optimal packings
packings of equal disks, optimal packings
2
Articles by Boll
Articles by Donovan
Articles by Lubachevsky
|
EUDML | Convolutions and products of partially ordered vector-valued positive measures. EuDML | Convolutions and products of partially ordered vector-valued positive measures.
Convolutions and products of partially ordered vector-valued positive measures.
Panaiotis K. Pavlakos
Pavlakos, Panaiotis K.. "Convolutions and products of partially ordered vector-valued positive measures.." Mathematische Annalen 287.2 (1990): 335-342. <http://eudml.org/doc/164689>.
@article{Pavlakos1990,
author = {Pavlakos, Panaiotis K.},
keywords = {partially ordered vector-valued positive measures; partially ordered orderunit-normed space; Fubini-type theorem; -algebras; - algebras; -algebras of type I; Jordan algebras; partially ordered *-involutory -) algebras; semifields; quantum probability},
title = {Convolutions and products of partially ordered vector-valued positive measures.},
AU - Pavlakos, Panaiotis K.
TI - Convolutions and products of partially ordered vector-valued positive measures.
KW - partially ordered vector-valued positive measures; partially ordered orderunit-normed space; Fubini-type theorem; -algebras; - algebras; -algebras of type I; Jordan algebras; partially ordered *-involutory -) algebras; semifields; quantum probability
partially ordered vector-valued positive measures, partially ordered orderunit-normed space, Fubini-type theorem,
{C}^{*}
{W}^{*}
- algebras,
A{W}^{*}
-algebras of type I, Jordan algebras, partially ordered *-involutory
\left({0}^{*}
-) algebras, semifields, quantum probability
Articles by Panaiotis K. Pavlakos
|
Haskell/Recursion - Wikibooks, open books for an open world
Recursion (Solutions)
1 Numeric recursion
1.2 Loops, recursion, and accumulating parameters
1.3 Other recursive functions
2 List-based recursion
3 Don't get TOO excited about recursion...
Recursive functions play a central role in Haskell, and are used throughout computer science and mathematics generally. Recursion is basically a form of repetition, and we can understand it by making distinct what it means for a function to be recursive, as compared to how it behaves.
A recursive function simply means this: a function that has the ability to invoke itself.
And it behaves such that it invokes itself only when a condition is met, as with an if/else/then expression, or a pattern match which contains at least one base case that terminates the recursion, as well as a recursive case which causes the function to call itself, creating a loop.
Without a terminating condition, a recursive function may remain in a loop forever, causing an infinite regress.
Numeric recursionEdit
The factorial functionEdit
Mathematics (specifically combinatorics) has a function called factorial.[1] It takes a single non-negative integer as an argument, finds all the positive integers less than or equal to “n”, and multiplies them all together. For example, the factorial of 6 (denoted as
{\displaystyle 6!}
{\displaystyle 1\times 2\times 3\times 4\times 5\times 6=720}
. We can use a recursive style to define this in Haskell:
Let's look at the factorials of two adjacent numbers:
Example: Factorials of consecutive numbers
Factorial of 6 = 6 × 5 × 4 × 3 × 2 × 1
Factorial of 5 = 5 × 4 × 3 × 2 × 1
Notice how we've lined things up. You can see here that the
{\displaystyle 6!}
includes the
{\displaystyle 5!}
{\displaystyle 6!}
{\displaystyle 6\times 5!}
. Let's continue:
Factorial of 4 = 4 × 3 × 2 × 1
Factorial of 3 = 3 × 2 × 1
Factorial of 2 = 2 × 1
Factorial of 1 = 1
The factorial of any number is just that number multiplied by the factorial of the number one less than it. There's one exception: if we ask for the factorial of 0, we don't want to multiply 0 by the factorial of -1 (factorial is only for positive numbers). In fact, we just say the factorial of 0 is 1 (we define it to be so. Just take our word for it that this is right.[2]). So, 0 is the base case for the recursion: when we get to 0 we can immediately say that the answer is 1, no recursion needed. We can summarize the definition of the factorial function as follows:
The factorial of any other number is that number multiplied by the factorial of the number one less than it.
We can translate this directly into Haskell:
Example: Factorial function
This defines a new function called factorial. The first line says that the factorial of 0 is 1, and the second line says that the factorial of any other number n is equal to n times the factorial of n - 1. Note the parentheses around the n - 1; without them this would have been parsed as (factorial n) - 1; remember that function application (applying a function to a value) takes precedence over anything else when grouping isn't specified otherwise (we say that function application binds more tightly than anything else).
The factorial function above is best defined in a file, but since it is a small function, it is feasible to write it in GHCi as a one-liner. To do this, we need to add a semicolon to separate the lines:
> let factorial 0 = 1; factorial n = n * factorial (n - 1)
Haskell actually uses line separation and other whitespace as a substitute for separation and grouping characters such as semicolons. Haskell programmers generally prefer the clean look of separate lines and appropriate indentation; still, explicit use of semicolons and other markers is always an alternative.
The example above demonstrates the simple relationship between factorial of a number, n, and the factorial of a slightly smaller number, n - 1.
Think of a function call as delegation. The instructions for a recursive function delegate a sub-task. It just so happens that the delegate function uses the same instructions as the delegator; it's only the input data that changes. The only really confusing thing about recursive functions is the fact that each function call uses the same parameter names, so it can be tricky to keep track of the many delegations.
Let's look at what happens when you execute factorial 3:
3 isn't 0, so we calculate the factorial of 2
0 is 0, so we return 1.
To complete the calculation for factorial 1, we multiply the current number, 1, by the factorial of 0, which is 1, obtaining 1 (1 × 1).
To complete the calculation for factorial 2, we multiply the current number, 2, by the factorial of 1, which is 1, obtaining 2 (2 × 1 × 1).
To complete the calculation for factorial 3, we multiply the current number, 3, by the factorial of 2, which is 2, obtaining 6 (3 × 2 × 1 × 1).
(Note that we end up with the one appearing twice, since the base case is 0 rather than 1; but that's okay since multiplying by 1 has no effect. We could have designed factorial to stop at 1 if we had wanted to, but the convention (which is often useful) is to define the factorial of 0.)
When reading or composing recursive functions, you'll rarely need to “unwind” the recursion bit by bit — we leave that to the compiler.
One more note about our recursive definition of factorial: the order of the two declarations (one for factorial 0 and one for factorial n) is important. Haskell decides which function definition to use by starting at the top and picking the first one that matches. If we had the general case (factorial n) before the 'base case' (factorial 0), then the general n would match anything passed into it – including 0. The compiler would then conclude that factorial 0 equals 0 * factorial (-1), and so on to negative infinity (clearly not what we want). So, always list multiple function definitions starting with the most specific and proceeding to the most general.
Type the factorial function into a Haskell source file and load it into GHCi.
Try examples like factorial 5 and factorial 1000.[3]
What about factorial (-1)? Why does this happen?
The double factorial of a number n is the product of all the integers from 1 up to n that have the same parity as n. For example, the double factorial of 8 is 8 × 6 × 4 × 2 = 384, and the double factorial of 7 is 7 × 5 × 3 × 1 = 105. Define a doubleFactorial function in Haskell.
Loops, recursion, and accumulating parametersEdit
Imperative languages use loops in the same sorts of contexts where Haskell programs use recursion. For example, an idiomatic way of writing a factorial function in C, a typical imperative language, would be using a for loop, like this:
Example: The factorial function in an imperative language
Here, the for loop causes res to be multiplied by n repeatedly. After each repetition, 1 is subtracted from n (that is what n-- does). The repetitions stop when n is no longer greater than 1.
A straightforward translation of such a function to Haskell is not possible, since changing the value of the variables res and n (a destructive update) would not be allowed. However, you can always translate a loop into an equivalent recursive form by making each loop variable into an argument of a recursive function. For example, here is a recursive “translation” of the above loop into Haskell:
Example: Using recursion to simulate a loop
go n res
| n > 1 = go (n - 1) (res * n)
go is an auxiliary function which actually performs the factorial calculation. It takes an extra argument, res, which is used as an accumulating parameter to build up the final result.
Depending on the languages you are familiar with, you might have concerns about performance problems caused by recursion. However, compilers for Haskell and other functional programming languages include a number of optimizations for recursion, (not surprising given how often recursion is needed). Also, Haskell is lazy — calculations are only performed once their results are required by other calculations, and that helps to avoid some of the performance problems. We'll discuss such issues and some of the subtleties they involve further in later chapters.
Other recursive functionsEdit
As it turns out, there is nothing particularly special about the factorial function; a great many numeric functions can be defined recursively in a natural way. For example, let's think about multiplication. When you were first learning multiplication (remember that moment? :)), it may have been through a process of 'repeated addition'. That is, 5 × 4 is the same as summing four copies of the number 5. Of course, summing four copies of 5 is the same as summing three copies, and then adding one more – that is, 5 × 4 = 5 × 3 + 5. This leads us to a natural recursive definition of multiplication:
Example: Multiplication defined recursively
mult _ 0 = 0 -- anything times 0 is zero
mult n m = (mult n (m - 1)) + n -- recurse: multiply by one less, and add an extra copy
Stepping back a bit, we can see how numeric recursion fits into the general recursive pattern. The base case for numeric recursion usually consists of one or more specific numbers (often 0 or 1) for which the answer can be immediately given. The recursive case computes the result by calling the function recursively with a smaller argument and using the result in some manner to produce the final answer. The 'smaller argument' used is often one less than the current argument, leading to recursion which 'walks down the number line' (like the examples of factorial and mult above). However, the prototypical pattern is not the only possibility; the smaller argument could be produced in some other way as well.
Expand out the multiplication 5 × 4 similarly to the expansion we used above for factorial 3.
Define a recursive function power such that power x y raises x to the y power.
You are given a function plusOne x = x + 1. Without using any other (+)s, define a recursive function addition such that addition x y adds x and y together.
(Harder) Implement the function log2, which computes the integer log (base 2) of its argument. That is, log2 computes the exponent of the largest power of 2 which is less than or equal to its argument. For example, log2 16 = 4, log2 11 = 3, and log2 1 = 0. (Small hint: read the last phrase of the paragraph immediately preceding these exercises.)
List-based recursionEdit
Haskell has many recursive functions, especially concerning lists.[4] Consider the length function that finds the length of a list:
Example: The recursive definition of length
If you try to load the definition above from a source file, GHCi will complain about an “ambiguous occurrence” when you try to use it, as the Prelude already provides length. In that case, just change the name of the function which you are defining to something else. like length' or myLength.
So, the type signature of length tells us that it takes any type of list and produces an Int. The next line says that the length of an empty list is 0 (this is the base case). The final line is the recursive case: if a list isn't empty, then it can be broken down into a first element (here called x) and the rest of the list (which will just be the empty list if there are no more elements) which will, by convention, be called xs (i.e. plural of x). The length of the list is 1 (accounting for the x) plus the length of xs (as in the tail example in Next steps, xs is set when the argument list matches the (:) pattern).
Consider the concatenation function (++) which joins two lists together:
Example: The recursive (++)
Prelude> "Hello " ++ "world" -- Strings are lists of Chars
This is a little more complicated than length. The type says that (++) takes two lists of the same type and produces another list of the same type. The base case says that concatenating the empty list with a list ys is the same as ys itself. Finally, the recursive case breaks the first list into its head (x) and tail (xs) and says that to concatenate the two lists, concatenate the tail of the first list with the second list, and then tack the head x on the front.
There's a pattern here: with list-based functions, the base case usually involves an empty list, and the recursive case involves passing the tail of the list to our function again, so that the list becomes progressively smaller.
Give recursive definitions for the following list-based functions. In each case, think what the base case would be, then think what the general case would look like, in terms of everything smaller than it. (Note that all of these functions are available in Prelude, so you will want to give them different names when testing your definitions in GHCi.)
replicate :: Int -> a -> [a], which takes a count and an element and returns the list which is that element repeated that many times. E.g. replicate 3 'a' = "aaa". (Hint: think about what replicate of anything with a count of 0 should be; a count of 0 is your 'base case'.)
(!!) :: [a] -> Int -> a, which returns the element at the given 'index'. The first element is at index 0, the second at index 1, and so on. Note that with this function, you're recursing both numerically and down a list[5].
(A bit harder.) zip :: [a] -> [b] -> [(a, b)], which takes two lists and 'zips' them together, so that the first pair in the resulting list is the first two elements of the two lists, and so on. E.g. zip [1,2,3] "abc" = [(1, 'a'), (2, 'b'), (3, 'c')]. If either of the lists is shorter than the other, you can stop once either list runs out. E.g. zip [1,2] "abc" = [(1, 'a'), (2, 'b')].
Define length using an auxiliary function and an accumulating parameter, as in the loop-like alternate version of factorial.
Recursion is used to define nearly all functions to do with lists and numbers. The next time you need a list-based algorithm, start with a case for the empty list and a case for the non-empty list and see if your algorithm is recursive.
Don't get TOO excited about recursion...Edit
Despite its ubiquity in Haskell, one rarely has to write functions that are explicitly recursive. Instead, standard library functions perform recursion for us in various ways. For example, a simpler way to implement the factorial function is:
Example: Implementing factorial with a standard library function
Almost seems like cheating, doesn't it? :) This is the version of factorial that most experienced Haskell programmers would write, rather than the explicitly recursive version we started out with. Of course, the product function uses some list recursion behind the scenes,[6] but writing factorial in this way means you, the programmer, don't have to worry about it.
↑ In mathematics, n! normally means the factorial of a non-negative integer n, but that syntax is impossible in Haskell, so we don't use it here.
↑ Actually, defining the factorial of 0 to be 1 is not just arbitrary; it's because the factorial of 0 represents an empty product.
↑ Interestingly, older scientific calculators can't handle things like factorial of 1000 because they run out of memory with that many digits!
↑ This is no coincidence; without mutable variables, recursion is the only way to implement control structures. This might sound like a limitation until you get used to it.
↑ Incidentally, (!!) provides a reasonable solution for the problem of the fourth exercise in Lists and tuples/Retrieving values.
↑ Actually, it uses a function called foldl to which it “delegates” the recursion.
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Kayla had a
14
-foot rope that she cut into three pieces. Now two of the pieces are the same length, and the third piece is
2
Copy Kayla’s diagram below onto your paper and write an equation that represents the situation. Be sure to remember to define your variable.
2x + 2 = 14
x
= unknown length of the first and second pieces)
Solve your equation and find the length of each of the two equal pieces.
x
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