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Saul_Kripke Knowpia
Modal logicEdit
A Kripke frame or modal frame is a pair
{\displaystyle \langle W,R\rangle }
, where W is a non-empty set, and R is a binary relation on W. Elements of W are called nodes or worlds, and R is known as the accessibility relation. Depending on the properties of the accessibility relation (transitivity, reflexivity, etc.), the corresponding frame is described, by extension, as being transitive, reflexive, etc.
A Kripke model is a triple
{\displaystyle \langle W,R,\Vdash \rangle }
{\displaystyle \langle W,R\rangle }
is a Kripke frame, and
{\displaystyle \Vdash }
is a relation between nodes of W and modal formulas, such that:
{\displaystyle w\Vdash \neg A}
{\displaystyle w\nVdash A}
{\displaystyle w\Vdash A\to B}
{\displaystyle w\nVdash A}
{\displaystyle w\Vdash B}
{\displaystyle w\Vdash \Box A}
{\displaystyle \forall u\,(w\;R\;u}
{\displaystyle u\Vdash A)}
We read
{\displaystyle w\Vdash A}
as "w satisfies A", "A is satisfied in w", or "w forces A". The relation
{\displaystyle \Vdash }
is called the satisfaction relation, evaluation, or forcing relation. The satisfaction relation is uniquely determined by its value on propositional variables.
{\displaystyle \langle W,R,\Vdash \rangle }
{\displaystyle w\Vdash A}
for all w ∈ W,
a frame
{\displaystyle \langle W,R\rangle }
, if it is valid in
{\displaystyle \langle W,R,\Vdash \rangle }
for all possible choices of
{\displaystyle \Vdash }
Consider the schema T :
{\displaystyle \Box A\to A}
. T is valid in any reflexive frame
{\displaystyle \langle W,R\rangle }
{\displaystyle w\Vdash \Box A}
{\displaystyle w\Vdash A}
since w R w. On the other hand, a frame which validates T has to be reflexive: fix w ∈ W, and define satisfaction of a propositional variable p as follows:
{\displaystyle u\Vdash p}
if and only if w R u. Then
{\displaystyle w\Vdash \Box p}
{\displaystyle w\Vdash p}
by T, which means w R w using the definition of
{\displaystyle \Vdash }
. T corresponds to the class of reflexive Kripke frames.
It is often much easier to characterize the corresponding class of L than to prove its completeness, thus correspondence serves as a guide to completeness proofs. Correspondence is also used to show incompleteness of modal logics: suppose L1 ⊆ L2 are normal modal logics that correspond to the same class of frames, but L1 does not prove all theorems of L2. Then L1 is Kripke incomplete. For example, the schema
{\displaystyle \Box (A\equiv \Box A)\to \Box A}
generates an incomplete logic, as it corresponds to the same class of frames as GL (viz. transitive and converse well-founded frames), but does not prove the GL-tautology
{\displaystyle \Box A\to \Box \Box A}
Canonical modelsEdit
The canonical model of L is a Kripke model
{\displaystyle \langle W,R,\Vdash \rangle }
, where W is the set of all L-MCS, and the relations R and
{\displaystyle \Vdash }
{\displaystyle X\;R\;Y}
if and only if for every formula
{\displaystyle A}
{\displaystyle \Box A\in X}
{\displaystyle A\in Y}
{\displaystyle X\Vdash A}
{\displaystyle A\in X}
Kripke semantics has a straightforward generalization to logics with more than one modality. A Kripke frame for a language with
{\displaystyle \{\Box _{i}\mid \,i\in I\}}
as the set of its necessity operators consists of a non-empty set W equipped with binary relations Ri for each i ∈ I. The definition of a satisfaction relation is modified as follows:
{\displaystyle w\Vdash \Box _{i}A}
{\displaystyle \forall u\,(w\;R_{i}\;u\Rightarrow u\Vdash A).}
Carlson modelsEdit
A simplified semantics, discovered by Tim Carlson, is often used for polymodal provability logics. A Carlson model is a structure
{\displaystyle \langle W,R,\{D_{i}\}_{i\in I},\Vdash \rangle }
with a single accessibility relation R, and subsets Di ⊆ W for each modality. Satisfaction is defined as:
{\displaystyle w\Vdash \Box _{i}A}
{\displaystyle \forall u\in D_{i}\,(w\;R\;u\Rightarrow u\Vdash A).}
An intuitionistic Kripke model is a triple
{\displaystyle \langle W,\leq ,\Vdash \rangle }
{\displaystyle \langle W,\leq \rangle }
is a partially ordered Kripke frame, and
{\displaystyle \Vdash }
if p is a propositional variable,
{\displaystyle w\leq u}
{\displaystyle w\Vdash p}
{\displaystyle u\Vdash p}
(persistency condition),
{\displaystyle w\Vdash A\land B}
{\displaystyle w\Vdash A}
{\displaystyle w\Vdash B}
{\displaystyle w\Vdash A\lor B}
{\displaystyle w\Vdash A}
{\displaystyle w\Vdash B}
{\displaystyle w\Vdash A\to B}
{\displaystyle u\geq w}
{\displaystyle u\Vdash A}
{\displaystyle u\Vdash B}
{\displaystyle w\Vdash \bot }
Let L be a first-order language. A Kripke model of L is a triple
{\displaystyle \langle W,\leq ,\{M_{w}\}_{w\in W}\rangle }
{\displaystyle \langle W,\leq \rangle }
is an intuitionistic Kripke frame, Mw is a (classical) L-structure for each node w ∈ W, and the following compatibility conditions hold whenever u ≤ v:
Given an evaluation e of variables by elements of Mw, we define the satisfaction relation
{\displaystyle w\Vdash A[e]}
{\displaystyle w\Vdash P(t_{1},\dots ,t_{n})[e]}
{\displaystyle P(t_{1}[e],\dots ,t_{n}[e])}
holds in Mw,
{\displaystyle w\Vdash (A\land B)[e]}
{\displaystyle w\Vdash A[e]}
{\displaystyle w\Vdash B[e]}
{\displaystyle w\Vdash (A\lor B)[e]}
{\displaystyle w\Vdash A[e]}
{\displaystyle w\Vdash B[e]}
{\displaystyle w\Vdash (A\to B)[e]}
{\displaystyle u\geq w}
{\displaystyle u\Vdash A[e]}
{\displaystyle u\Vdash B[e]}
{\displaystyle w\Vdash \bot [e]}
{\displaystyle w\Vdash (\exists x\,A)[e]}
if and only if there exists an
{\displaystyle a\in M_{w}}
{\displaystyle w\Vdash A[e(x\to a)]}
{\displaystyle w\Vdash (\forall x\,A)[e]}
{\displaystyle u\geq w}
{\displaystyle a\in M_{u}}
{\displaystyle u\Vdash A[e(x\to a)]}
Naming and NecessityEdit
"A Puzzle about Belief"Edit
Saul Kripke CenterEdit
^ Crimmins, Mark (30 October 2013). "Review of Philosophical Troubles: Collected Papers, Volume 1" – via Notre Dame Philosophical Reviews. {{cite journal}}: Cite journal requires |journal= (help)
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Abstract: The present work illustrates a predictive method, based on graph theory, for different types of energy of subatomic particles, atoms and molecules, to be specific, the mass defect of the first thirteen elements of the periodic table, the rotational and vibrational energies of simple molecules (such as , H2, FH and CO) as well as the electronic energy of both atoms and molecules (conjugated alkenes). It is shown that such a diverse group of energies can be expressed as a function of few simple graph-theoretical descriptors, resulting from assigning graphs to every wave function. Since these descriptors are closely related to the topology of the graph, it makes sense to wonder about the meaning of such relation between energy and topology and suggests points of view helping to formulate novel hypotheses about this relation.
Keywords: Graph Theory, Energy, Particles, Atoms, Molecules
{\text{H}}_{2}^{+}
{\text{H}}_{2}^{+}
MD\left(\mu \text{u}\right)=-10.47\times SCBO+54
{N}_{d}=13,{R}^{2}=0.9967,SE=4.79,F=3363
MD\left(\mu \text{u}\right)=-9.60\times \left(Nn+Z\right)+20.104
{N}_{d}=13,{R}^{2}=0.9966,SE=4.86,F=3359
\nu =2B\times \left(N+1\right)
N=0,1,2,3,4,\cdots
\nu =2B\times \left(N+1\right)-4D\times {\left(N+1\right)}^{3}
N=0,1,2,3,4,\cdots
\nu \left({\text{cm}}^{-1}\right)=155.26\times N+185.97
{N}_{d}=7,{R}^{2}=0.9984,SE=14.83,F=3068
N=1,2,3,\cdots
\nu \left({\text{cm}}^{-1}\right)=1264.26\times PCR-1068.38
{N}_{d}=7,{R}^{2}=0.9998,SE=5.16,F=25345
\nu \left({\text{cm}}^{-1}\right)=38.75\times N+48.65
{N}_{d}=14,{R}^{2}=0.9994,SE=1.51,F=77125
N=1,2,3,\cdots
\nu \left({\text{cm}}^{-1}\right)=348.85\times PCR-310.58
{N}_{d}=14,{R}^{2}=0.9989,SE=5.04,F=6931
\nu \left({\text{cm}}^{-1}\right)=7.61\times N+11.45
{N}_{d}=12,{R}^{2}=0.9999,SE=0.0009,F>100000
N=1,2,3,\cdots
\nu \left({\text{cm}}^{-1}\right)=69.57\times PCR-55.30
{N}_{d}=12,{R}^{2}=0.9983,SE=1.21,F=4193
{E}_{v}=\left(N+1/2\right)h{v}_{e},\text{\hspace{0.17em}}\text{\hspace{0.17em}}N=0,1,2,3,\cdots
{E}_{v}=\left(N+1/2\right)h{v}_{e}-{\left(n+1/2\right)}^{2}{h}_{e}{v}_{e}{X}_{e},\text{\hspace{0.17em}}\text{\hspace{0.17em}}N=0,1,2,3,\cdots
{\text{H}}_{2}^{+}
{H}_{2}^{+}
\nu \left({\text{cm}}^{-1}\right)=122.39\times N+161.72
{N}_{d}=12,{R}^{2}=0.9972,SE=24.51,F=3566
N=1,2,3,\cdots
\nu \left({\text{cm}}^{-\text{1}}\right)=1066.1PCR-919.61
{N}_{d}=12,{R}^{2}=0.9999,SE=5.25,F=77862
{\text{H}}_{2}^{+}
N=0,1,2,\cdots
\nu \left({\text{cm}}^{-1}\right)=12416\times N+18229
{N}_{d}=12,{R}^{2}=0.9850,SE=16.62,F=665
N=1,2,3,\cdots
\nu \left({\text{cm}}^{-1}\right)=108125\times PCR-91423
{N}_{d}=12,{R}^{2}=0.9947,SE=9.83,F=1888
{\text{H}}_{2}^{+}
{\text{H}}_{2}^{+}
{\text{H}}_{2}^{+}
{\text{H}}_{2}^{+}
\nu \left({\text{cm}}^{-1}\right)=2040.57\times N+1238.2
{N}_{d}=10,{R}^{2}=0.9998,SE=105.77,F=30709
N=1,2,3,\cdots
\nu \left({\text{cm}}^{-1}\right)=17217\times PCR-16016
{N}_{d}=10,{R}^{2}=0.9995,SE=141.40,F=17179
{\text{H}}_{2}^{+}
EBE-1s\left(\text{eV}\right)=10.60\times {N}^{2}-20.43\times N+23.79
{N}_{d}=12,{R}^{2}=0.9999,SE=4.91,F=43500
N=1,2,3,\cdots
E=-\frac{{Z}^{2}{e}^{2}}{2{a}_{0}{n}^{2}}
HLG\left(\text{eV}\right)=8.03\times Vindex-1.29
{N}_{d}=9,{R}^{2}=0.9979,SE=0.074,F=3392
HLG\left(\text{eV}\right)=-1.73\times Espm02x+10.23
{N}_{d}=9,{R}^{2}=0.9985,SE=0.063,F=4759
Cite this paper: Galvez, J. (2019) A Graph Theoretical Interpretation of Different Types of Energies of Elementary Particles, Atoms and Molecules. Open Journal of Physical Chemistry, 9, 33-50. doi: 10.4236/ojpc.2019.92003.
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\left\{{\mathbf{V}}_{1},{\mathbf{V}}_{2},{\mathbf{V}}_{3}\right\}
{\mathbf{U}}_{1}=\mathbf{A}
{\mathbf{U}}_{2}=\mathbf{B}
{\mathbf{U}}_{3}=\mathbf{C}
Begin by computing the box product
\mathrm{\lambda }=\left[{\mathbf{U}}_{1}{\mathbf{U}}_{2}{\mathbf{U}}_{3}\right]={\mathbf{U}}_{1}·\left({\mathbf{U}}_{2}×{\mathbf{U}}_{3}\right)
|\begin{array}{ccc}3& -2& 4\\ 2& 5& -4\\ 5& 7& 6\end{array}|
Next, obtain the following three cross products.
{\mathbf{U}}_{2}×{\mathbf{U}}_{3}= |\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 2& 5& -4\\ 5& 7& 6\end{array}|
\left[\begin{array}{c}58\\ -32\\ -11\end{array}\right]
{\mathbf{U}}_{3}×{\mathbf{U}}_{1}=|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 5& 7& 6\\ 3& -2& 4\end{array}|
\left[\begin{array}{c}40\\ -2\\ -31\end{array}\right]
{\mathbf{U}}_{1}×{\mathbf{U}}_{2}=|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 3& -2& 4\\ 2& 5& -4\end{array}|
\left[\begin{array}{c}-12\\ 20\\ 19\end{array}\right]
Finally, divide each of the three cross products by
\mathrm{λ}
{\mathbf{V}}_{1}=\frac{1}{194}
\left[\begin{array}{c}58\\ -32\\ -11\end{array}\right]
{\mathbf{V}}_{2}=\frac{1}{194}\left[\begin{array}{c}40\\ -2\\ -31\end{array}\right]
{\mathbf{V}}_{3}=\frac{1}{194}\left[\begin{array}{c}-12\\ 20\\ 19\end{array}\right]
Enter the vectors
{\mathbf{U}}_{1},{\mathbf{U}}_{2},{\mathbf{U}}_{3}
Context Panel: Assign to a Name≻U[1]
〈3,-2,4〉
\stackrel{\text{assign to a name}}{\to }
{\textcolor[rgb]{0,0,1}{U}}_{\textcolor[rgb]{0,0,1}{1}}
〈2,5,-4〉
\stackrel{\text{assign to a name}}{\to }
{\textcolor[rgb]{0,0,1}{U}}_{\textcolor[rgb]{0,0,1}{2}}
〈5,7,6〉
\stackrel{\text{assign to a name}}{\to }
{\textcolor[rgb]{0,0,1}{U}}_{\textcolor[rgb]{0,0,1}{3}}
\mathrm{λ}=\left[{\mathbf{U}}_{1}{\mathbf{U}}_{2}{\mathbf{U}}_{3}\right]
Write a sequence of the names of the three vectors.
Context Panel: Student Multivariate Calculus≻Triple Scalar Product
Context Panel: Assign to a Name≻lambda
{\mathbf{U}}_{1},{\mathbf{U}}_{2},{\mathbf{U}}_{3}
\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{4}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{7}\\ \textcolor[rgb]{0,0,1}{6}\end{array}\right]
\stackrel{\text{scalar triple product}}{\to }
\textcolor[rgb]{0,0,1}{194}
\stackrel{\text{assign to a name}}{\to }
\textcolor[rgb]{0,0,1}{\mathrm{λ}}
Obtain the reciprocal vectors as per the formulas in Table 1.5.1
Common Symbols palette: Cross-product operator
{\mathbf{V}}_{1}=\frac{{\mathbf{U}}_{2}×{\mathbf{U}}_{3}}{\mathrm{λ}}
\stackrel{\text{assign}}{\to }
{\mathbf{V}}_{2}=\frac{{\mathbf{U}}_{3}×{\mathbf{U}}_{1}}{\mathrm{λ}}
\stackrel{\text{assign}}{\to }
{\mathbf{V}}_{3}=\frac{{\mathbf{U}}_{1}×{\mathbf{U}}_{2}}{\mathrm{λ}}
\stackrel{\text{assign}}{\to }
Display the reciprocal vectors
{\mathbf{V}}_{1},{\mathbf{V}}_{2},{\mathbf{V}}_{3}
\left[\begin{array}{c}\frac{\textcolor[rgb]{0,0,1}{29}}{\textcolor[rgb]{0,0,1}{97}}\\ \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{16}}{\textcolor[rgb]{0,0,1}{97}}\\ \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{11}}{\textcolor[rgb]{0,0,1}{194}}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{c}\frac{\textcolor[rgb]{0,0,1}{20}}{\textcolor[rgb]{0,0,1}{97}}\\ \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{97}}\\ \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{31}}{\textcolor[rgb]{0,0,1}{194}}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{c}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{6}}{\textcolor[rgb]{0,0,1}{97}}\\ \frac{\textcolor[rgb]{0,0,1}{10}}{\textcolor[rgb]{0,0,1}{97}}\\ \frac{\textcolor[rgb]{0,0,1}{19}}{\textcolor[rgb]{0,0,1}{194}}\end{array}\right]
\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right):
Define the vectors
{\mathbf{U}}_{1}
{\mathbf{U}}_{2}
{\mathbf{U}}_{3}
{\mathbf{U}}_{1},{\mathbf{U}}_{2},{\mathbf{U}}_{3}≔〈3,-2,4〉,〈2,5,-4〉,〈5,7,6〉:
Apply the BoxProduct command.
\mathrm{λ}≔\mathrm{BoxProduct}\left({\mathbf{U}}_{1},{\mathbf{U}}_{2},{\mathbf{U}}_{3}\right):
Apply the CrossProduct command to obtain the reciprocal vectors as per the formulas in Table 1.5.1
{\mathbf{V}}_{1}≔\mathrm{CrossProduct}\left({\mathbf{U}}_{2},{\mathbf{U}}_{3}\right)/\mathrm{λ}:\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}{\mathbf{V}}_{2}≔\mathrm{CrossProduct}\left({\mathbf{U}}_{3},{\mathbf{U}}_{1}\right)/\mathrm{λ}:\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}{\mathbf{V}}_{3}≔\mathrm{CrossProduct}\left({\mathbf{U}}_{1},{\mathbf{U}}_{2}\right)/\mathrm{λ}:
{\mathbf{V}}_{1},{\mathbf{V}}_{2},{\mathbf{V}}_{3}
\left[\begin{array}{c}\frac{\textcolor[rgb]{0,0,1}{29}}{\textcolor[rgb]{0,0,1}{97}}\\ \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{16}}{\textcolor[rgb]{0,0,1}{97}}\\ \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{11}}{\textcolor[rgb]{0,0,1}{194}}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{c}\frac{\textcolor[rgb]{0,0,1}{20}}{\textcolor[rgb]{0,0,1}{97}}\\ \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{97}}\\ \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{31}}{\textcolor[rgb]{0,0,1}{194}}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{c}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{6}}{\textcolor[rgb]{0,0,1}{97}}\\ \frac{\textcolor[rgb]{0,0,1}{10}}{\textcolor[rgb]{0,0,1}{97}}\\ \frac{\textcolor[rgb]{0,0,1}{19}}{\textcolor[rgb]{0,0,1}{194}}\end{array}\right]
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Local-density_approximation Knowpia
Local-density approximations (LDA) are a class of approximations to the exchange–correlation (XC) energy functional in density functional theory (DFT) that depend solely upon the value of the electronic density at each point in space (and not, for example, derivatives of the density or the Kohn–Sham orbitals). Many approaches can yield local approximations to the XC energy. However, overwhelmingly successful local approximations are those that have been derived from the homogeneous electron gas (HEG) model. In this regard, LDA is generally synonymous with functionals based on the HEG approximation, which are then applied to realistic systems (molecules and solids).
In general, for a spin-unpolarized system, a local-density approximation for the exchange-correlation energy is written as
{\displaystyle E_{\rm {xc}}^{\mathrm {LDA} }[\rho ]=\int \rho (\mathbf {r} )\epsilon _{\rm {xc}}(\rho (\mathbf {r} ))\ \mathrm {d} \mathbf {r} \ ,}
where ρ is the electronic density and εxc is the exchange-correlation energy per particle of a homogeneous electron gas of charge density ρ. The exchange-correlation energy is decomposed into exchange and correlation terms linearly,
{\displaystyle E_{\rm {xc}}=E_{\rm {x}}+E_{\rm {c}}\ ,}
so that separate expressions for Ex and Ec are sought. The exchange term takes on a simple analytic form for the HEG. Only limiting expressions for the correlation density are known exactly, leading to numerous different approximations for εc.
Local-density approximations are important in the construction of more sophisticated approximations to the exchange-correlation energy, such as generalized gradient approximations (GGA) or hybrid functionals, as a desirable property of any approximate exchange-correlation functional is that it reproduce the exact results of the HEG for non-varying densities. As such, LDA's are often an explicit component of such functionals.
Local density approximations, as with GGAs are employed extensively by solid state physicists in ab-initio DFT studies to interpret electronic and magnetic interactions in semiconductor materials including semiconducting oxides and spintronics. The importance of these computational studies stems from the system complexities which bring about high sensitivity to synthesis parameters necessitating first-principles based analysis. The prediction of Fermi level and band structure in doped semiconducting oxides is often carried out using LDA incorporated into simulation packages such as CASTEP and DMol3.[1] However an underestimation in Band gap values often associated with LDA and GGA approximations may lead to false predictions of impurity mediated conductivity and/or carrier mediated magnetism in such systems.[2] Starting in 1998, the application of the Rayleigh theorem for eigenvalues has led to mostly accurate, calculated band gaps of materials, using LDA potentials.[3][4] A misunderstanding of the second theorem of DFT appears to explain most of the underestimation of band gap by LDA and GGA calculations, as explained in the description of density functional theory, in connection with the statements of the two theorems of DFT.
Homogeneous electron gasEdit
Approximation for εxc depending only upon the density can be developed in numerous ways. The most successful approach is based on the homogeneous electron gas. This is constructed by placing N interacting electrons in to a volume, V, with a positive background charge keeping the system neutral. N and V are then taken to infinity in the manner that keeps the density (ρ = N / V) finite. This is a useful approximation as the total energy consists of contributions only from the kinetic energy and exchange-correlation energy, and that the wavefunction is expressible in terms of planewaves. In particular, for a constant density ρ, the exchange energy density is proportional to ρ⅓.
Exchange functionalEdit
The exchange-energy density of a HEG is known analytically. The LDA for exchange employs this expression under the approximation that the exchange-energy in a system where the density is not homogeneous, is obtained by applying the HEG results pointwise, yielding the expression[5][6]
{\displaystyle E_{\rm {x}}^{\mathrm {LDA} }[\rho ]=-{\frac {3}{4}}\left({\frac {3}{\pi }}\right)^{1/3}\int \rho (\mathbf {r} )^{4/3}\ \mathrm {d} \mathbf {r} \ .}
Correlation functionalEdit
Analytic expressions for the correlation energy of the HEG are available in the high- and low-density limits corresponding to infinitely-weak and infinitely-strong correlation. For a HEG with density ρ, the high-density limit of the correlation energy density is[5]
{\displaystyle \epsilon _{\rm {c}}=A\ln(r_{\rm {s}})+B+r_{\rm {s}}(C\ln(r_{\rm {s}})+D)\ ,}
and the low limit
{\displaystyle \epsilon _{\rm {c}}={\frac {1}{2}}\left({\frac {g_{0}}{r_{\rm {s}}}}+{\frac {g_{1}}{r_{\rm {s}}^{3/2}}}+\dots \right)\ ,}
where the Wigner-Seitz parameter
{\displaystyle r_{\rm {s}}}
is dimensionless.[7] It is defined as the radius of a sphere which encompasses exactly one electron, divided by the Bohr radius. The Wigner-Seitz parameter
{\displaystyle r_{\rm {s}}}
is related to the density as
{\displaystyle {\frac {4}{3}}\pi r_{\rm {s}}^{3}={\frac {1}{\rho }}\ .}
An analytical expression for the full range of densities has been proposed based on the many-body perturbation theory. The calculated correlation energies are in agreement with the results from quantum Monte Carlo simulation to within 2 milli-Hartree.
Accurate quantum Monte Carlo simulations for the energy of the HEG have been performed for several intermediate values of the density, in turn providing accurate values of the correlation energy density.[8]
Spin polarizationEdit
The extension of density functionals to spin-polarized systems is straightforward for exchange, where the exact spin-scaling is known, but for correlation further approximations must be employed. A spin polarized system in DFT employs two spin-densities, ρα and ρβ with ρ = ρα + ρβ, and the form of the local-spin-density approximation (LSDA) is
{\displaystyle E_{\rm {xc}}^{\mathrm {LSDA} }[\rho _{\alpha },\rho _{\beta }]=\int \mathrm {d} \mathbf {r} \ \rho (\mathbf {r} )\epsilon _{\rm {xc}}(\rho _{\alpha },\rho _{\beta })\ .}
For the exchange energy, the exact result (not just for local density approximations) is known in terms of the spin-unpolarized functional:[9]
{\displaystyle E_{\rm {x}}[\rho _{\alpha },\rho _{\beta }]={\frac {1}{2}}{\bigg (}E_{\rm {x}}[2\rho _{\alpha }]+E_{\rm {x}}[2\rho _{\beta }]{\bigg )}\ .}
The spin-dependence of the correlation energy density is approached by introducing the relative spin-polarization:
{\displaystyle \zeta (\mathbf {r} )={\frac {\rho _{\alpha }(\mathbf {r} )-\rho _{\beta }(\mathbf {r} )}{\rho _{\alpha }(\mathbf {r} )+\rho _{\beta }(\mathbf {r} )}}\ .}
{\displaystyle \zeta =0\,}
corresponds to the diamagnetic spin-unpolarized situation with equal
{\displaystyle \alpha \,}
{\displaystyle \beta \,}
spin densities whereas
{\displaystyle \zeta =\pm 1}
corresponds to the ferromagnetic situation where one spin density vanishes. The spin correlation energy density for a given values of the total density and relative polarization, εc(ρ,ς), is constructed so to interpolate the extreme values. Several forms have been developed in conjunction with LDA correlation functionals.[10]
Exchange-correlation potentialEdit
The exchange-correlation potential corresponding to the exchange-correlation energy for a local density approximation is given by[5]
{\displaystyle v_{\rm {xc}}^{\mathrm {LDA} }(\mathbf {r} )={\frac {\delta E^{\mathrm {LDA} }}{\delta \rho (\mathbf {r} )}}=\epsilon _{\rm {xc}}(\rho (\mathbf {r} ))+\rho (\mathbf {r} ){\frac {\partial \epsilon _{\rm {xc}}(\rho (\mathbf {r} ))}{\partial \rho (\mathbf {r} )}}\ .}
In finite systems, the LDA potential decays asymptotically with an exponential form. This result is in error; the true exchange-correlation potential decays much slower in a Coulombic manner. The artificially rapid decay manifests itself in the number of Kohn–Sham orbitals the potential can bind (that is, how many orbitals have energy less than zero). The LDA potential can not support a Rydberg series and those states it does bind are too high in energy. This results in the highest occupied molecular orbital (HOMO) energy being too high in energy, so that any predictions for the ionization potential based on Koopmans' theorem are poor. Further, the LDA provides a poor description of electron-rich species such as anions where it is often unable to bind an additional electron, erroneously predicating species to be unstable.[11]
^ Segall, M.D.; Lindan, P.J (2002). "First-principles simulation: ideas, illustrations and the CASTEP code". Journal of Physics: Condensed Matter. 14 (11): 2717. Bibcode:2002JPCM...14.2717S. doi:10.1088/0953-8984/14/11/301.
^ Assadi, M.H.N; et al. (2013). "Theoretical study on copper's energetics and magnetism in TiO2 polymorphs". Journal of Applied Physics. 113 (23): 233913–233913–5. arXiv:1304.1854. Bibcode:2013JAP...113w3913A. doi:10.1063/1.4811539. S2CID 94599250.
^ Zhao, G. L.; Bagayoko, D.; Williams, T. D. (1999-07-15). "Local-density-approximation prediction of electronic properties of GaN, Si, C, and RuO2". Physical Review B. 60 (3): 1563–1572. doi:10.1103/physrevb.60.1563. ISSN 0163-1829.
^ Bagayoko, Diola (December 2014). "Understanding density functional theory (DFT) and completing it in practice". AIP Advances. 4 (12): 127104. doi:10.1063/1.4903408. ISSN 2158-3226.
^ a b c Parr, Robert G; Yang, Weitao (1994). Density-Functional Theory of Atoms and Molecules. Oxford: Oxford University Press. ISBN 978-0-19-509276-9.
^ Dirac, P. A. M. (1930). "Note on exchange phenomena in the Thomas-Fermi atom". Proc. Camb. Phil. Soc. 26 (3): 376–385. Bibcode:1930PCPS...26..376D. doi:10.1017/S0305004100016108.
^ Murray Gell-Mann and Keith A. Brueckner (1957). "Correlation Energy of an Electron Gas at High Density" (PDF). Phys. Rev. 106 (2): 364–368. Bibcode:1957PhRv..106..364G. doi:10.1103/PhysRev.106.364.
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The Normality Criteria of Meromorphic Functions Concerning Shared Fixed-Points
Wei Chen, Qi Yang, Wen-jun Yuan, Hong-gen Tian, "The Normality Criteria of Meromorphic Functions Concerning Shared Fixed-Points", Discrete Dynamics in Nature and Society, vol. 2014, Article ID 654294, 8 pages, 2014. https://doi.org/10.1155/2014/654294
Wei Chen,1 Qi Yang,1 Wen-jun Yuan,2 and Hong-gen Tian1
1School of Mathematics Science, Xinjiang Normal University, Urumqi, Xinjiang 830054, China
2School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, Guangdong 510006, China
We study the normality criteria of meromorphic functions concerning shared fixed-points; we obtain the following: Let be a family of meromorphic functions defined in a domain and a positive integer. For every , all zeros of are of multiplicity at least and all poles of are multiple. If and share in for each pair of functions and , then is normal.
Let be a family of meromorphic functions defined in the domain . If any sequence contains a subsequence that converges spherically locally uniformly in , to a meromorphic function or , we say that is normal in (see [1, 2]).
Let be a meromorphic function in a domain , and . We say is a fixed-point of when . Let and be two meromorphic functions in ; if and have the same zeros (ignoring multiplicity), then we say and share fixed-points.
In , Lu and Gu [3] proved the following results.
Theorem 1. Let be a positive integer and a transcendental. If all zeros of have multiplicity at least , then assumes every finite nonzero value infinitely often.
Theorem 2. Let be a family of meromorphic functions defined in a domain . Suppose that is a positive integer and is a finite complex number. If, for each , all zeros of are of multiplicity at least , and , then is normal in .
In 2011, Hu and Meng [4] extended Theorem 2 as follows.
Theorem 3. Let be a family of meromorphic functions defined in a domain . Let be a positive integer and be a finite complex number. If, for every , and share in , and, for every pair of functions , , all zeros of are of multiplicity at least , then is normal in .
A natural question is the following: what can be said if the finite complex number in Theorem 3 is replaced by the fixed-point ? In this paper, we answer this question by proving the following theorems.
Theorem 4 (main theorem). Let be a family of meromorphic functions defined in a domain . Let be a positive integer. If, for every such that , and share in for every pair of function , , then is normal in .
Theorem 5 (main theorem). Let be a family of meromorphic functions defined in a domain . Let be a positive integer. For every , all zeros of have multiplicity at least and all poles of are multiple. If and share in for every pair of function , , then is normal in .
Theorem 6 (main theorem). Let be a family of meromorphic functions defined in a domain . Let be a positive integer. For every , has only zeros with multiplicity at least and has poles at most . If and share in for every pair of function , , then is normal in .
In order to prove our theorems, we require the following results.
Lemma 7 (see [5, 6]). Let be a family of meromorphic functions in a domain D, and let be a positive integer, such that each function has only zeros of multiplicity at least , and suppose that there exists such that whenever . If is not normal at , then, for each , there exist a sequence of points , , a sequence of positive numbers , and a subsequence of functions such that locally uniformly with respect to the spherical metric in , where is a nonconstant meromorphic function, whose zeros all have multiplicity at least , such that . Moreover, has order at most .
Here as usual, is the spherical derivative.
Lemma 8. Let be a integer and a nonconstant rational meromorphic function such that ; then has at least two distinct zeros.
Proof. Since , set where is a nonzero constant and are positive integers.
For simplicity, we denote . Obviously, .
From (2), we have where is a polynomial and .
Differentiating (3), we get where is a polynomial and .
Next, we assume, to the contrary, that has at most one zero. We distinguish two cases.
Case 1. If has exactly one zero . From (3), we obtain where is a nonconstant. Differentiating (5), we have where is a polynomial and .
From (4) and (6), we obtain then ; this is a contradiction.
Case 2. If has no zero. Then we have from (5), which is a contradiction.
Lemma 9. Let be an integer and a nonconstant rational meromorphic function. If has only zeros with multiplicity at least and all poles of are multiple, then has at least two distinct zeros.
Proof. Assume, to the contrary, that has at most one zero.
Case 1. If is a polynomial, since all zeros of are of multiplicity at least , then we can know that has at least one zero with multiplicity . So has at least one zero and has zeros with multiplicity at least . According to the assumption, we obtain that has only a zero ; then there exists a nonzero constant and an integer such that So we have which, however, has only simple zeros. This is a contraction.
Case 2. If is a rational but not a polynomial. We see where is a nonzero constant and .
For simplicity, we denote From (10), we have where is a polynomial and .
Differentiating (13), we have where , are polynomials and , .
Now we distinguish two subcases.
Case 2.1. Supposing that has exactly one zero , from (13), we obtain Differentiating (16), we have where , are polynomials, , and . From (14) and (17), we have .
Further, we distinguish two subcases.
Case 2.1.1. . From (16), it is easily obtained that . Thus, (13) implies So ; that is, . From (13) and (16), noting that , we have It follows that ; combining this inequality with (11), we obtain which is impossible.
Case 2.1.2. . Next we distinguish two subcases: and .
When , similar to the Case , it follows that ; from (13) and (16), we get a contradiction.
If , by combining (15) and (18), we may give the following inequality: and hence This is a contradiction.
Case 2.2. Suppose that has no zero; then for (16). Similarly with the proof of Case 2.1, we also obtain a contradiction.
Lemma 10. Let be an integer and a nonconstant rational meromorphic function. If has only zeros with multiplicity at least and has poles at most , then has at least two distinct zeros.
Case 1. If is a polynomial, since all zeros of are of multiplicity at least , then we can know that has at least one zero with multiplicity . So we see that has at least one zero and has zeros with multiplicity at least . According to the assumption, we obtain that has only a zero ; then there exists a nonzero constant and an integer such that So we have which, however, has only simple zeros. This is a contraction.
Case 2. If is a rational but not a polynomial, we set where is a nonzero constant and .
Differentiating (29), we have where , are polynomials and and .
Case 2.1. Supposing that has exactly one zero , from (13), we obtain Differentiating (32), we have where , are polynomials, , and . From (30) and (33), we have . From (31) and (34), we see So . Since , we have , so .
Therefore, has no zeros. According to Lemma 8, we have a contradiction.
Lemma 11 (see [7]). Let be a transcendental meromorphic function, and let be a small function such that ; then
Lemma 12. Let be a transcendental meromorphic function; let be a positive integer; let be a polynomial. If all zeros of have multiplicity at least , then has infinitely many zeros.
Proof. We denote then Since is a polynomial, we see . Now we distinguish two cases as follows.(i)If , we have By Lemma 11, we have Thus, (ii)If , then By Lemma 11, we have
From (41) and (44), we can deduce that has infinitely many zeros; thus, has infinitely many zeros.
Lemma 13 (see [4]). Take a positive integer and a nonzero complex number . If is a nonconstant meromorphic function such that has only zeros of multiplicity at least , then has at least two distinct zeros.
Case 1. For , let . If is not normal at , by Lemma 7, there exist a sequence of complex numbers with and a sequence of positive numbers with such that locally uniformly on compact subsets of , where is a nonconstant meromorphic function in .
Here we distinguish two cases.
Case 1.1. Supposing that , is a finite complex number. Then locally uniformly on compact subsets of disjoint from the poles of , where is a nonconstant meromorphic function in , all of whose zeros have multiplicity at least and all poles of which have multiplicity at least . Hence spherically locally uniformly in disjoint from the poles of .
If , since has zeros with multiplicity at least , obviously there is a contradiction. Hence, . Since the multiplicity of all zeros of is at least , so by Lemmas 9 and 12, has at least two distinct zeros.
Suppose that , are two distinct zeros of . We choose a positive number small enough such that and such that has no other zeros in except for and , where
By Hurwitz’s Theorem, for sufficiently large , there exist points , such that
By the assumption in Theorem 5 that and share , it follows that Fix , take , and note , ; we obtain Since the zeros of has no accumulation point, for sufficiently large , we have Therefore, when is large enough, . This contradicts with the facts , , . Thus is normal at 0.
Case 1.2. We may suppose that . We have where . Thus we have Since , , we have On the other hand, for , we have
Thus, we have spherically locally uniformly in disjoint from the poles of .
If , then has no zeros. Of course, also has no poles. Since is a nonconstant meromorphic function of order at most , then there exists constant , , and ; obviously, this is contrary to the case . Hence, .
Since the multiplicity of all zeros of is at least and all poles of which have multiplicity at least , thus, by Lemma 13, has at least two distinct zeros.
By Hurwitz’s Theorem, for sufficiently large there exist points , such that
Similar to the proof of Case 1.1, we get a contradiction. Then, is normal at 0.
From Cases 1.1 and 1.2, we know that is normal at 0; there exists and a subsequence of , and we may still denote it as , such that converges spherically locally uniformly to a meromorphic function or in .
Case i. When is large enough, . Then . Thus, for each , there exists such that if , then for all . Thus, for sufficiently large , , is holomorphic in . Therefore, for all , when , we have By maximum Principle and Montel’s Theorem, is normal at .
Case ii. There exists a subsequence of , and we may still denote it as such that . Since , the multiplicity of all zeros of is at least ; then . Thus, there exists such that is holomorphic in and has a unique zero in . Then converges spherically locally uniformly to a holomorphic function in ; converges spherically locally uniformly to a holomorphic function in . Hence is normal at .
By Cases and , is normal at .
Case 2. For , suppose, to the contrary, that is not normal in . Then there exists at least one point such that is not normal at the point . Then by Lemma 7, there exist a sequence of complex numbers with and a sequence of positive numbers with such that locally uniformly on compact subsets of , where is a nonconstant meromorphic function in , whose zeros all have multiplicity at least . Moreover, has order at most .
From (61) we get also locally uniformly with respect to the spherical metric besides the poles of .
If , then has no zeros. Of course, also has no poles. Since is a nonconstant meromorphic function of order at most , then there exists constant , and . Obviously, this is contrary to the case . Hence .
By Lemmas 12 and 13, we deduce that has at least two distinct zeros. Next we show that it is impossible. Let and be two distinct zeros of . We choose a positive number small enough such that and such that has no other zeros in expect for and , where
Similar to the proof of Case 1, we get a contradiction. This finally completes the proof of Theorem 5.
Similar to the proof of Theorem 5, combining with Lemmas 8 and 10 and Lemma 12, we can prove Theorems 4 and 6 easily; here we omit the proof.
This work was supported by the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University. The third author would like to express his hearty thanks to Chern Institute of Mathematics that provided very comfortable research environments to him while working as visiting scholar. The authors thank the referees for reading the paper very carefully and making a number of valuable suggestions to improve the readability of the paper. Foundation item: Nature Science Foundation of China (11271090); Nature Science Foundation of Guangdong Province (S2012010010121); Graduate Research and Innovation Projects (XJGRI2013131) of Xinjiang Province.
W. H. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964. View at: MathSciNet
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f\left(z\right){f}^{\left(k\right)}\left(z\right)-a
and its normality,” Chinese Quarterly Journal of Mathematics, vol. 24, no. 1, pp. 75–80, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet
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W. L. Zou and Q. D. Zhang, “On value distribution of
\phi \left(z\right)f\left(z\right){f}^{k}\left(z\right)
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Forced Convection Heat Transfer Enhancement by Porous Pin Fins in Rectangular Channels | J. Heat Transfer | ASME Digital Collection
, 3-5-1 Johoku, Hamamatsu 432-8561, Japan
Yang, J., Zeng, M., Wang, Q., and Nakayama, A. (March 5, 2010). "Forced Convection Heat Transfer Enhancement by Porous Pin Fins in Rectangular Channels." ASME. J. Heat Transfer. May 2010; 132(5): 051702. https://doi.org/10.1115/1.4000708
The forced convective heat transfer in three-dimensional porous pin fin channels is numerically studied in this paper. The Forchheimer–Brinkman extended Darcy model and two-equation energy model are adopted to describe the flow and heat transfer in porous media. Air and water are employed as the cold fluids and the effects of Reynolds number (Re), pore density (PPI) and pin fin form are studied in detail. The results show that, with proper selection of physical parameters, significant heat transfer enhancements and pressure drop reductions can be achieved simultaneously with porous pin fins and the overall heat transfer performances in porous pin fin channels are much better than those in traditional solid pin fin channels. The effects of pore density are significant. As PPI increases, the pressure drops and heat fluxes in porous pin fin channels increase while the overall heat transfer efficiencies decrease and the maximal overall heat transfer efficiencies are obtained at
PPI=20
for both air and water cases. Furthermore, the effects of pin fin form are also remarkable. With the same physical parameters, the overall heat transfer efficiencies in the long elliptic porous pin fin channels are the highest while they are the lowest in the short elliptic porous pin fin channels.
porous pin fin channel, forced convection, heat transfer enhancement, CFD simulation, channel flow, flow simulation, flow through porous media, forced convection
Density, Fins, Flow (Dynamics), Fluids, Forced convection, Heat flux, Heat transfer, Porous materials, Pressure drop, Reynolds number, Water, Heat, Flux (Metallurgy), Computational methods, Temperature
Heat Transfer Enhancement by Pin Elements
Strategy for Selection of Elements for Heat Transfer Enhancement
Experimental Research on Convection Heat Transfer in Sintered Porous Plate Channels
Numerical Study of Forced Convection in a Partially Porous Channel With Discrete Heat Sources
Analysis of Forced Convection Enhancement in a Channel Using Porous Blocks
Enhancement of Forced-Convection Cooling of Multiple Heated Blocks in a Channel Using Porous Covers
Thermal Enhancement in Laminar Channel Flow With a Porous Block
Ould-Amer
Forced Convection Cooling Enhancement by Use of Porous Materials
Use of Porous Baffles to Enhance Heat Transfer in a Rectangular Channel
Calculation of Turbulent Flow and Heat Transfer in a Porous-Baffled Channel
, 2005, User Guide, CFX 10.
Forced Convection in a Porous Channel With Discrete Heat Sources
Modified Hazen-Dupuit-Darcy Model for Forced Convection of a Fluid With Temperature-Dependent Viscosity
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Maximum Constrained Approval Bucklin - Electowiki
Maximum Constrained Approval Bucklin (MCAB) is a multiwinner method devised by Kristofer Munsterhjelm, based on Bucklin voting. It uses linear optimization to calculate candidate support by assuming earlier surplus transfers were maximally favorable to the candidate in question, and thus reduces the strategic impact of lowering or raising a winning candidate on a ballot.
MCAB works in multiple rounds, each of which sets an implicit approval cutoff for every ballot. The first round considers first preferences as approved, the second round considers first and second preferences, and so on. For each round, the method evaluates every remaining unelected candidate. The unelected candidate with the greatest support is elected if his support is greater than a Droop quota, and a round may elect more than one candidate.
To determine the support of a particular candidate X, MCAB uses a constraint mechanism when counts implicit approvals. To be Droop proportional, MCAB deweights voters who approve of elected candidates, after they're elected. Unlike BTV and STV, however, MCAB does not directly decide which voters are to be deweighted. Instead it adds a constraint to the next rounds that a Droop quota of voters preferring each elected candidate must be discarded. When counting the support for X in later rounds, it maximizes the possible support for X given those constraints. This is called making the case for X.
Determining which voters to eliminate to maximize the support of a candidate subject to earlier constraints is relatively simple to do by linear programming, but hard to do by hand; MCAB can't be counted entirely by hand.
What ended up as MCAB was initially proposed in 2017[1] and simplified later that year[2]. The method detailed here has been further modified from the EM posts to resist Woodall free riding.
1 Determining the support for X
3.1 Without vote management
3.1.1 First round, r = 1
3.1.2 Second round, r = 2
3.2 With vote management
4.1 Weak invulnerability to Hylland free riding
Determining the support for X[edit | edit source]
{\displaystyle r}
be the rank and round number,
{\displaystyle n}
the number of candidates so far, and
{\displaystyle c_k}
the kth elected candidate. Consider unranked candidates to be ranked equal below every explicitly ranked candidate, i.e. never approved in any round. The linear program for determining the support of candidate X as (n+1)th candidate is:
maximize: sum over all voters v: support[v][n+1]
for i = 1 ... n+1:
(sum over all voters v: support[v][i]) > Droop quota (1)
for all voters v,
for i = 1 ... n+1: (2)
support[v][i] >= 0 if voter v ranks c_i at or
higher than rank r
support[v][i] = 0 otherwise
for all voters v: (3)
(sum over i = 1 ... n+1: support[v][i]) <= v's initial weight
{\displaystyle c_{n+1}}
is provisionally defined as X for the purpose of determining X's support.
The three clauses do the following:
imposes the Droop constraint: that any elected candidate
{\displaystyle c_i}
must have more than a Droop quota's worth of approvals according to the implicit approval cutoff for round r.
defines support:
{\displaystyle c_i}
's support is the number of voters who rank
{\displaystyle c_i}
at or above rank
{\displaystyle r}
defines each voter's budget: no voter can spread more support across the candidates than his ballot's initial weight. The initial weight is 1 per voter for ordinary elections, or some other value in case of a weighted vote.
If the linear program is infeasible, then there's no way for X to obtain support exceeding a Droop quota, and so X is disqualified from being elected in that round. Among the remaining candidates, the candidate with the greatest support is elected.
With the linear programming for determining the support of X defined, the MCAB procedure is this:
Start with round = 1.
Mark every unelected candidate as qualified for the round.
For every unelected undisqualified candidate X:
Solve the linear program to find X's support with r = round.
If every unelected candidate is disqualified by the linear program:
If all ranks have been considered, the method is done.
Otherwise, increment the round number and go to 2.
If not all candidates have zero support, elect the candidate with greatest support and go to 3.
As a shortcut, the procedure can be stopped once every seat has been filled, since every remaining candidate will be disqualified in every round from that point on.
The vote management example from Wikipedia's article on Schulze STV, https://en.wikipedia.org/wiki/Schulze_STV#Scenario
Without vote management[edit | edit source]
The unmanaged ballot set is
The Droop quota for two seats is 30.
First round, r = 1[edit | edit source]
A has 50 votes and is elected. No other candidate passes the Droop quota, and as there are no equal ranks, no A-first surplus can contribute to getting anyone else elected, so the linear program will mark all as disqualified afterwards. The combined surplus of the A-first voters come out to 50-30=20, but we don't know which particular ballots will be eliminated.
Second round, r = 2[edit | edit source]
Making the case for B: The optimum assignment is to allocate 12 of the 20 remaining votes to A>B>C, and then eliminate 8 of the A>C>B ballots. This gives B a support score of 27 (from the B-first ballots) + 12 (from the A>B>C ballots) for a total of 39.
Making the case for C: The optimum assignment is to allocate all 20 remaining votes to A>C>B. This gives C a score of 20 + 13 (from the C>A ballots) = 33.
Both candidates have exceeded the Droop quota, but B has greater support, so B is elected. After this, technically speaking, C gets another shot.
Making the case for C, again: Of the 20 remaining votes from electing A, 3 of these must go to electing B, so that 27 + 3 >= 30. That leaves 30 for C, which is not enough to clear the Droop quota. So C is disqualified.
Now every remaining candidate (namely C) is disqualified and the procedure is over. The winners are A and B.
With vote management[edit | edit source]
The vote-managed ballot set is
25: C>A>B (now includes some dishonest A>C>B voters)
As above, A is elected and then everybody else is disqualified. The surplus is 8 (38 - 30).
Making the case for B: The optimum assignment is to allocate all 8 surplus votes to A>B>C, and then eliminate 4 A>B>C ballots and every A>C>B ballot. Thus B's max support is 27 + 8 = 35.
Making the case for C: The optimum assignment is to allocate all 8 surplus votes to A>C>B and eliminate the remaining A>C>B ballots, as well as every A>B>C ballot. Doing so gives a support of 25 (from the C>A ballots) + 8 = 33.
Both candidates have exceeded the Droop quota, but B has greater support and so is elected.
The case for C again goes as above. Because two Droop quotas have been elected, there are not enough votes left to get anyone else above the Droop quota.
Since A and B were elected in both cases, the vote-management failed.
MCAB passes the following criteria:
Droop proportionality criterion
Invulnerability to Woodall free riding
MCAB fails the following criteria:
Weak invulnerability to Hylland free riding
Weak invulnerability to Hylland free riding[edit | edit source]
Suppose B is elected in round p and then later, the method arrives at round q. A voter who votes B ahead of C will contribute to C's support when the method makes the case for C regardless of whether he voted B ahead or not, unless B would not have been elected in any earlier round without his support. In that respect, Hylland free riding has no impact on MCAB. However, suppose there is another voter who votes B ahead of D. When making the case for D, MCAB needs to allocate a Droop quota of votes towards B since B was elected earlier. The B>C voter makes himself available to cover B's deficit when MCAB makes the case for someone who is not C. Had he not voted for B, he would not be thus available, and perhaps MCAB would have needed to exclude the B>D voter instead, electing C instead of D. So while doing Hylland free riding is more risky than in BTV, it can still pay off, and thus MCAB fails weak invulnerability to Hylland free riding.
Without vote management:
38: A>C>D>B
The first two ranks are the same as in the Schulze STV example, so A and B are elected.
With vote management:
29: C>D>A>B (dishonest A>C>D>B and C>A>D>B voters)
In the first round, A wins. In the second round, the optimal allocation when making the case for B is to remove 22 A>C ballots and 8 A>B ballots. B's score is thus 31. C's score is unchanged, so the free-riding pays off: C wins.
Like BTV, MCAB fails the monotonicity criterion due to a lookahead problem[3]. However, MCAB passes the two criteria above as long as the candidate to elect in a round is chosen (by some method) from the set of candidates with above Droop quota support for that round. Thus, it is possible that a variant that uses a yet unknown lookahead criterion instead of electing the candidate with the greatest support, could pass monotonicity.
↑ Munsterhjelm, K. (2017-01-06). "Bucklin multiwinner method". Election-methods mailing list archives.
↑ Munsterhjelm, K. (2017-09-15). "A simpler vote management-resistant Bucklin LP". Election-methods mailing list archives.
↑ Munsterhjelm, K. (2018-02-18). "Path dependence monotonicity failure in BTV". Election-methods mailing list archives.
Retrieved from "https://electowiki.org/w/index.php?title=Maximum_Constrained_Approval_Bucklin&oldid=15206"
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Haber's rule - Wikipedia
Haber's rule
Toxicology relationship between the concentration of a poisonous gas and duration breathed
In toxicology, Haber's rule or Haber's law is a mathematical statement of the relationship between the concentration of a poisonous gas and how long the gas must be breathed to produce death, or other toxic effect. The rule was formulated by German chemist Fritz Haber in the early 1900s.
Haber's rule states that, for a given poisonous gas,
{\displaystyle tC=k}
{\displaystyle C}
is the concentration of the gas (mass per unit volume),
{\displaystyle t}
is the amount of time necessary to breathe the gas to produce a given toxic effect, and
{\displaystyle k}
is a constant, depending on both the gas and the effect. Thus, the rule states that doubling the concentration will halve the time, for example.
It makes equivalent any two groupings of dose concentration and exposure time that have equivalent mathematical products. For instance, if we assign dose concentration the symbol C, and time the classic t, then for any two dose schema, if C1t1=C2t2, then under Haber's rule the two dose schema are equivalent.
Haber's rule is an approximation, useful with certain inhaled poisons under certain conditions, and Haber himself acknowledged that it was not always applicable. If a substance is efficiently eliminated in the host, for example, then Haber's Law breaks down in the limit of t approaching the order of the half-life of the drug, rewriting the equation as the integral ∫Cdt = constant for arbitrary varying C and elapsed time T. It is very convenient, however, because its relationship between
{\displaystyle C}
{\displaystyle t}
appears as a straight line in a log-log plot.
In 1940, statistician C. I. Bliss published a study of toxicity in insecticides in which he proposed more complex models, for example, expressing the relationship between
{\displaystyle C}
{\displaystyle t}
as two straight line segments in a log-log plot.[1] However, because of its simplicity, Haber's rule continued to be widely used. Recently, some researchers have argued that it is time to move beyond the simple relationship expressed by Haber's rule and to make regular use of more sophisticated models.[2]
^ C. I. Bliss (1940). "The relationship between exposure, time, concentration and toxicity in experiments on insecticides". Annals of the Entomological Society of America. 33: 721–766. doi:10.1093/aesa/33.4.721.
^ F. J. Miller; P. M. Schlosser; D. B. Janszen (August 14, 2000). "Haber's rule: a special case in a family of curves relating concentration and duration of exposure to a fixed level of response for a given endpoint". Toxicology. 149 (1): 22–34. doi:10.1016/S0300-483X(00)00229-8. PMID 10963858.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Haber%27s_rule&oldid=1060662810"
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From probability to geometry (II) - Volume in honor of the 60th birthday of Jean-Michel Bismut - Tome (2009) no. 328
title = {From probability to geometry {(II)} - {Volume} in honor of the 60th birthday of {Jean-Michel} {Bismut}},
TI - From probability to geometry (II) - Volume in honor of the 60th birthday of Jean-Michel Bismut
Dai Xianzhe; Léandre Rémi; Xiaonan Ma; Zhang Weiping (éd.). From probability to geometry (II) - Volume in honor of the 60th birthday of Jean-Michel Bismut. Astérisque, no. 328 (2009), 408 p. http://numdam.org/item/AST_2009__328_/
From probability to geometry (II) - Volume in honor of the 60th birthday of Jean-Michel Bismut - Pages préliminaires
K
Bunke, Ulrich ; Schick, Thomas
Liu, Kefeng ; Xu, Hao
Maillot, Vincent ; Rössler, Damian
The index of projective families of elliptic operators: the decomposable case
Mathai, Varghese ; Melrose, Richard B. ; Singer, Isadore M.
Paradan, Paul-Émile ; Vergne, Michèle
CM
stability and the generalized Futaki invariant II
Paul, Sean Timothy ; Tian, Gang
Calabi-Yau threefolds of Borcea-Voisin, analytic torsion, and Borcherds products
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Modification of Even-A Nuclear Mass Formula
Department of Physics, University of Shanghai for Science and Technology, Shanghai, China.
In this paper we obtain an empirical mass formula of even-A nuclei based on residual proton-neutron interactions. The root-mean-squared deviation (RMSD) from experimental data is at an accuracy of about 150 Kev. While for heavy nuclei, we give another formula that fits the experimental data better (RMSD ≈ 119 Kev). We have successfully described the experimental data of nuclear masses and predicted some unknown masses (like 200Ir not involved in AME2003, the deviation of our predicted masses from the value in AME2012 is only about 82 keV). The predictive power of our formula is more competitive than other mass models.
Residual Proton-Neutron Interactions, Nuclear Masses, Binding Energies
Zhang, J. (2017) Modification of Even-A Nuclear Mass Formula. Journal of Applied Mathematics and Physics, 5, 2302-2310. doi: 10.4236/jamp.2017.511187.
The study of nuclear masses and energy levels has always been one of the most challenging frontiers in the field of nuclear physics. There are two types to describe and understand the nuclear masses, one of which is global relations, and the other is local. Some global nuclear mass models such as Weizäscker model [1] , Duflo-Zuker model [2] , the finite range droplet model [3] , a recent macroscopic-microscopic mass formula [4] [5] [6] etc., successfully produce the measured masses with accuracy at the level of 300 - 600 Kev. However, the global mass models require more physics and more information about nuclear force to get better description of the nuclear masses. On the other hand, the local mass relations, such as the isobaric multiplet mass equation (IMME), the Garvey-Kelson (GK) relations, which use the predicted nuclear masses and the residual proton-neutron interactions to evaluate the mass. It is found that the local mass relations are just approximately satisfied in known masses, so it has a good potential to predict the unknown masses.
In this paper, our purpose is to obtain a residual proton-neutron interactions formula of even-A nuclei from those of neighboring nuclei. In Section II we introduce the residual proton-neutron interactions and obtain our formula based on the proton-neutron interactions between the last proton and the last neutron. Then we introduce two modifications to improve our formula. The RMSD from experimental data is about 150 Kev. And for heavy nuclei, we obtain another formula fits with the experimental data even more precise. With our further refinement of heavy nuclei, the RMSD gets even smaller to about 120 Kev. In Section III we successfully predict some unknown masses. The result shows that the predict power of our formula is competitive with others. In Section IV we discuss and summarize the results of this paper.
2. The Residual Proton-Neutron Interactions
The residual proton-neutron interaction plays an important role in the evolution of collective, deformation and phase transition [7] [8] [9] [10] , so it has attracted many attentions [11] - [17] . The proton-neutron interactions between the last i protons and j neutrons is given by
{V}_{ip-jn}\left(Z,N\right)=B\left(Z,N\right)+B\left(Z-i,N-j\right)-B\left(Z,N-j\right)-B\left(Z-i,N\right).
The famous formula GKL and GKT were derived from the neutron-proton interactions between the last neutron and proton [18] [19] . The relationship between Garvey-Kelson quality is a semi empirical relationship between 6 adjacent nuclear mass. If the interaction between neighboring nuclei changes slowly in the local range, it can be completely counteracted by the addition and subtraction of many adjacent nuclei. Garvey-Kelson mass relationship has two common relationships:
\begin{array}{l}M\left(N,Z+1\right)+M\left(N-1,Z-1\right)+M\left(N+1,Z\right)\\ -M\left(N,Z-1\right)-M\left(N-1,Z\right)-M\left(N+1,Z+1\right)=0,\end{array}
\begin{array}{l}M\left(N,Z-1\right)+M\left(N-1,Z+1\right)+M\left(N+1,Z\right)\\ -M\left(N,Z+1\right)-M\left(N-1,Z\right)-M\left(N+1,Z-1\right)=0,\end{array}
M\left(N,Z\right)
denotes the mass of a nucleus with neutron number N and proton number Z. Equation (2) is called the longitudinal Garvey-Kelson relation (GKL), and Equation (3) the transverse (GKT).
In this section, we use the residual proton neutron interactions between the last proton and the last neutron to form our formula. According to the Equation (1), it is easy to obtain that the residual proton-neutron interactions between the last proton and the last neutron is defined as
\begin{array}{c}{V}_{1p-1n}\left(Z,N\right)=B\left(Z,N\right)+B\left(Z-1,N-1\right)-B\left(Z,N-1\right)-B\left(Z-1,N\right)\\ =M\left(Z,N\right)+M\left(Z-1,N-1\right)-M\left(Z,N-1\right)-M\left(Z-1,N\right)\end{array}
The Garvey-Kelson mass relations require six nuclei, but our formula requires only four. So our formula involves less number of nuclei, its predictions in iterative extrapolations is the more reliable, and its deviations are smaller in the extrapolation process.
In recent years, many papers tried to find formulas to describe and evaluate the nuclear masses, but many of them have a large RMSD. In this work, we focus on the even-A nuclei, through the study on the neighboring nuclei with the database in AME2012 [20] .
For the residual nuclear proton-neutron interactions which
A\ge 42
, we calculate the
\delta {V}_{1p-1n}
as shown in Figure 1. Based on that, we empirically obtained the residual proton-neutron interactions formula of even-A nuclei. The formula is as follows:
\begin{array}{c}\stackrel{¯}{\delta {V}_{1p-1n}}=B\left(Z,N+1\right)+B\left(Z-1,N\right)-B\left(Z,N\right)-B\left(Z-1,N+1\right)\\ \cong \frac{515.6}{{A}^{2}}+\frac{62.78}{A}+0.1079\text{\hspace{0.17em}}\text{keV}\end{array}
\stackrel{¯}{\delta {V}_{1p-1n}}
is the average values of
\delta {V}_{1p-1n}
for nuclei with the same mass number A.
We find that the average binding energy of our predicted mass agrees well with the specific binding energy curve. We successfully describe and predict some even-A nuclear masses by using these equations and some known experimental nuclear masses in AME2012 for calculation of
\delta {V}_{1p-1n}
It can be seen from the Figure 1 that the interaction of proton-neutron is
Figure 1. Circles show that the residual proton-neutron interactions
\delta {V}_{1p-1n}
.The curve is plotted by using the average values of
\delta {V}_{1p-1n}
for nuclei with the same mass number A, expressed as
\stackrel{¯}{\delta {V}_{1p-1n}}
. The smoothed curve are plotted in terms of equation
\stackrel{¯}{\delta {V}_{1p-1n}}\left(A\right)=\frac{515.6}{{A}^{2}}+\frac{62.78}{A}+0.1079\text{\hspace{0.17em}}\text{keV}
for even-A nuclei with
A\ge 42
more stable in the heavy nuclei region than in the light nuclei region.
In order to better describe the quality of the nucleus, we will improve the above formula with some amendments, donated by
\delta {V}_{1p-1n}^{cal}
as the final improvement results [4] [5] [6] . The first is called the Coulomb correction, denoted by
{\Delta }_{C}
{\Delta }_{C}\left(Z,N\right)\approx {a}_{C}\left(-\frac{4}{9}{Z}^{4/3}{A}^{-7/3}-\frac{2}{3}Z{A}^{-4/3}+\frac{4}{9}{Z}^{2}{A}^{-7/3}+\frac{4}{9}{Z}^{1/3}{A}^{-4/3}\right),
the second is called the symmetry energy correction, denoted by
{\Delta }_{sym}
{\Delta }_{sym}\left(Z,N\right)={a}_{sym}\frac{1}{A{\left(2+|IA|\right)}^{3}}+{b}_{sym}{A}^{-1},
I=\left(N-Z\right)/A
{a}_{C}=10.51
{a}_{sym}=20126
{b}_{sym}=-61.25
as parameters [17] [21] .
The revised
\delta {V}_{1p-1n}\left(Z,N\right)
\delta {V}_{1p-1n}^{cal}\left(Z,N\right)=\stackrel{¯}{\delta {V}_{1p-1n}}-{\Delta }_{C}\left(Z,N\right)-{\Delta }_{sym}\left(Z,N\right).
The improvement of these two corrections on our predicted
\delta {V}_{1p-1n}
is about 5 keV. Although the two contributions are small, but with more understanding of the symmetry energy of the nucleus, we believe that these contributes will become more important in the future.
In order to describe the nuclear mass obtained by our theory vividly, we compare the average RMSD of the nuclear mass with the experimental data to represent the difference, and the formula is as follows:
\sigma =\sqrt{\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{\left({M}_{i}^{exp}-{M}_{i}^{cal}\right)}^{2}}.
The RMSD is about 150 Kev. In Figure 2 we show deviations (in units of keV) between our calculated
\delta {V}_{1p-1n}^{cal}
by applying Equations (6) and those experimental data of binding energies compiled in AME2012 [20] . It can be seen that the RMSDs of these
\delta {V}_{1p-1n}
decrease with A. The description is better in the medium mass nucleus and heavy nucleus.
As early as 1960s, the nuclear structure theory predicts the existence of a number of new elements in the long life near the proton number Z = 114 and neutron number N = 184 (i.e. island of super heavy nuclei) and the island of super heavy nuclear plays an important role in the entire nuclear physics field. So for the heavy nuclei, we obtain another formula to describe the mass and it fits more closely with the experimental data. And in order to achieve better result, the different parameters are given between even-even nuclei and odd-odd nuclei, the formula is as follows:
{V}_{1p-1n}\left(A\right)=\frac{a}{{A}^{2}}+\frac{b}{A}+c.
Figure 2. (Color online) Deviations (in units of keV) of our calculated
\delta {V}_{1p-1n}^{cal}
by using Equations (6) with respect to those extracted from experimental binding energies [Equation (4)], for the nuclei with
A\ge 16
When we use the Equation (6) to describe the nuclear masses, the RMSD is about 150 Kev, but if we try the Equation (7) where A > 200, the RMSD is 119 Kev, it shows that our formula of heavy nuclei is more accurate.
Figure 3 displays the difference between the experimental values and calculated values, we compare it with Ref [21] , one can see that our result is better.
3. Mass Predictions
Through above study, we find our formula has a good performance in describing the nuclear masses. In this section, we use our formula and the residual proton-neutron interaction to predict the nuclear mass not obtained in the experiment. Based on the Equation (4), we can obtain
M\left(Z,N\right)=M\left(Z-1,N\right)+M\left(Z,N-1\right)-M\left(Z-1,N-1\right)+\stackrel{¯}{\delta {V}_{1p-1n}}\left(A\right).
The unknown mass
M\left(Z,N\right)
is predicted by using the three nuclei masses around it and the
\delta {V}_{1p-1n}\left(Z,N\right)
we empirical obtained.
Now let’s focus on a few examples of our predictions. Table 1 shows mass excess of some nuclei are not predictive in ame 2003 or ame 2012 databases. These unknown masses are important not only in the context of astrophysics, but also in the nuclear structure. Interestingly, our predicted values show good in comparison with the experimental results. For 182Lu, the deviation of our predicted masses from the value in AME2012 is only ∼63 keV. Three additional
Figure 3. Shows the RMSDs of even-A nuclei. (a) represents the odd-odd nuclei; (b) represents the even-even nuclei. We obtain the even-A nuclear masses from some experimentally known nuclear masses and the residual proton-neutron interactions formula. Comparing calculated values with the AME2012 databases obtain the RMSDs. The triangles are plotted by using the RMSDs of our calculated values. The circles are plotted by using the formula in Ref [21] .
Table 1. Mass excess of some mass nuclei with us and predicted results in the AME2003 database and the AME2012 databsae. (keV).
nuclei are 202Pt, 232Am and 286Ed, the differences between our predicted values and those in AME2012 are approximately 100 keV. It seems our formula shows a great accuracy and can be used predict nuclear masses.
In this paper, we obtain the residual proton-neutron interactions formula to describe and predict the mass of even-A nuclei. In order to improve the accuracy of the
\delta {V}_{1p-1n}
, we use the average value of the
\delta {V}_{1p-1n}
(denoted as
\stackrel{¯}{\delta {V}_{1p-1n}}
modification) and introduce two modifications.
For further understanding of the super heavy nuclei, we use another formula to describe the
\delta {V}_{1p-1n}
, and its results fit the experiment data more accurate, one can see that the RMSD decreases considerably.
Then we investigate the predictive power of these new formulas by numerical experiments. They are competitive with other local mass relations. The deviation of predicted results from experimental values is less compared with other models.
Based on results so far, our method of studying the neighboring nuclei has a good performance. We can predict other unknown masses by using our empirical formula to provide useful reference points for experimental physics.
The author would like to thank G.Y.Gao for reading and commenting of this paper.
[1] Von Weizsäcker, C.F. (1935) Zur Theorie der Kernmassen. Zeitschrift für Physik, 96, 431.
[2] Duflo J. and Zuker, A.P. (1995) Microscopic Mass Formulas. Physical Review C, 52, R23.
https://doi.org/10.1103/PhysRevC.52.R23
[3] Möller P., Myers W. D., Sagawa, H. and Yoshida, S. (2012) New Finite-Range Droplet Mass Model and Equation-of-State Parameters. Physical Review Letters, 108, 052501.
[4] Wang N., Liang Z. Y., Liu, M. and Wu, X. Z. (2010) Mirror Nuclei Constraint in Nuclear Mass Formula. Physical Review C, 82, 044304.
[5] Wang N., Liu, M. and Wu, X. Z. (2010) Modification of Nuclear Mass Formula by Considering Isospin Effects. Physical Review C, 81, 044322.
[6] Mendoza-Temis, J., Hirsch, J. G. and Zuker, A. P. (2010) The Anatomy of the Simplest Duflo–Zuker Mass Formula. Nuclear Physics A, 843, 14.
[7] De Shalit A. and Goldhaber, M. (1953) Mixed Configurations in Nuclei. Physical Review Journals Archive, 92, 1211.
[8] Federman P. and Pittel, S. (1977) Towards a Unified Microscopic Description of Nuclear Deformation. Physics Letters B, 69, 385.
[9] Casten R. F. and Zamfir, N. V. J. (1996) The Evolution of Nuclear Structure: The NpNn Scheme and Related Correlations. Journal of Physics G: Nuclear and Particle Physics, 22, 1521.
[10] Talmi, I. (1962) Effective Interactions and Coupling Schemes in Nuclei. Reviews of Modern Physics, 34, 704.
[11] Brenner, D.S., Wesselborg, C., Casten, R.F., Warner, D.D. and Zhang, J.Y. (1990) Empirical p-n Interactions: Global Trends, Configuration Sensitivity and N=Z Enhancements. Physics Letters B, 243, 1.
[12] Mouze, G., Bidegainberry, S., Rocaboy, A. and Ythier, C. (1993) The Neutron-Proton Interaction Energy of the Valence Nucleons. Il Nuovo Cimento A (1965-1970), 106, 885.
[13] Gao, Z.C., Chen, Y.S. and Meng, J. (2001) Garvey-Kelson Mass Relations and n-p Interaction. Chinese Physics Letters, 18, 1186.
[14] Cakirli R. B., Brenner D. S., Casten, R. F. and Millman, E. A. (2005) Proton-Neutron Interactions and the New Atomic Masses. Physical Review Letters, 94, 092501.
[15] Cakirli R. B. and Casten, R. F. (2006) Direct Empirical Correlation between Proton-Neutron Interaction Strengths and the Growth of Collectivity in Nuclei. Physical Review Letters, 96, 132501.
[16] Breitenfeldt M. et al., (2010) Approaching the N = 82 Shell Closure with Mass Measurements of Ag and Cd Isotopes. Physical Review C, 81, 034313.
[17] Jiao, B.B. (2017) Description and Prediction of Even-A Nuclear Masses Based on Residual Proton-Neutron Interactions.
[18] Garvey, G. T. and Kelson, I. (1966) New Nuclidic Mass Relationship. Physical Review Letters, 16, 197.
[19] Garvey G. T., Gerace W. J., Jaffe R. L., Talmi, I. and Kelson, I. (1969) Set of Nuclear-Mass Relations and a Resultant Mass Table. Reviews of Modern Physics, 41, S1.
https://doi.org/10.1103/RevModPhys.41.S1
[20] Wang, M., Audi, G., Wapstra, A.H., et al. (2012) The Ame2012 Atomic Mass Evaluation. Chinese Physics C, 36, 1603.
[21] Fu, G.J., Lei, Y., Jiang, H., et al. (2011) Description and Evaluation of Nuclear Masses Based on Residual Proton-Neutron Interactions. Physical Review C, 84, 034311.
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Stephen Covey - Wikiquote
Stephen R. Covey (October 24, 1932 – July 16, 2012) was an American author of the bestselling book, The Seven Habits of Highly Effective People, as well as other books.
1.1 The Seven Habits Of Highly Effective People (1989)
1.2 Principle-Centered Leadership (1992)
1.3 First Things First (1994)
1.4 The 8th Habit : From Effectiveness to Greatness (2004)
Live the law of love.
This is a principle-centered approach. It transcends the traditional prescriptions of faster, harder, smarter, and more. Rather than offering you another clock, this approach provides you with a compass — because more important than how fast you're going, is where you're headed.
Trust is the glue that holds everything together. It creates the environment in which all of the other elements — win-win stewardship agreements, self-directing individuals and teams, aligned structures and systems, and accountability — can flourish.
As quoted in What Matters Most : The Power of Living Your Values (2001) by Hyrum W. Smith , p. 111
As quoted in Teaching Sport and Physical Activity : Insights on the Road to Excellence (2003) Paul G. Schempp, p. 79
Foreword to Prisoners of our Thoughts : Viktor Frankl's Principles at Work (2004), by Alex Pattakos, p. x
This statement has also been attributed to James Neil Hollingsworth (AKA: Ambrose Redmoon) in an article entitled "No Peaceful Warriors!" for Gnosis Magazine #21, in 1991.
The Seven Habits Of Highly Effective People (1989)[edit]
The Seven Habits Of Highly Effective People : Restoring the Character Ethic (1989)
...when you get a good night's sleep and wake up ready to produce throughout the day.
Principle-Centered Leadership (1992)[edit]
Ch. 4 : Primary Greatness, p. 58
Ch. 11 : Thirty Methods of Influence
Unless we exercise our power to choose wisely, our actions will be determined by conditions. Our ultimate freedom is the right and power to decide how anybody or anything outside ourselves will affect us.
First Things First (1994)[edit]
First Things First : To Live, to Love, to Learn, to Leave a Legacy (1994)
It's not enough to have values without vision; you want to be good, but you want to be good for something. On the other hand, vision without values can create a Hitler. An empowering mission statement deals with both character and competence; what you want to be and what you want to do in your life.
These quotes were added into an article for First Things First and then transferred here, but many originally among them have already been found to have been paraphrases, rather than exact quotations.
Instead of looking at fragments, try to see the whole picture.
No gardener, no garden!
Often we are so busy with sawing that we forget to sharpen the saw..
Our choices make the legacy to our children.
{\displaystyle \textstyle {\frac {3}{4}}}
of world problems and bewilderment would be lost if we understood our opponents.
There will be real happiness, peace of mind and balance, when living by heart and right-mindedly.
Change happens at the speed of trust. (cf. The SPEED of Trust, 2008)
The 8th Habit : From Effectiveness to Greatness (2004)[edit]
Principles are universal — that is, they transcend culture and geography. They're also timeless, they never change...
Because of the space between stimulus and response, people have the power of choice; therefore, leaders are neither born nor made — meaning environmentally trained and nurtured. They are self-made through chosen responses, and if they choose based on principles and develop increasingly greater discipline, their freedom to choose increases.
Best's enemy is Good.
Russian proverb, used in The Seven Habits Of Highly Effective People
Stephen Covey's official site
ISSSP Profile
"The Shifting Paradigms of Stephen Covey" by Bob Waldrep
IMNO Interview of Stephen R. Covey
A Direct Interview with Steven Covey
In his own words: Stephen Covey
Retrieved from "https://en.wikiquote.org/w/index.php?title=Stephen_Covey&oldid=3032286"
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Help:Musical symbols - Heterodox
2 Note rhythm and tempo
Source: The sonata in B{{music|b}} major has a slow movement in G{{music|#}} minor.
Example: The Presto is marked = 210, but Steblin believes Beethoven meant = 210 instead.
Source: The Presto is marked {{music|quarter}} = 210, but Steblin believes Beethoven meant {{music|quaver}} = 210 instead.
4 (source: {{music|time|4|4}} ) indicates that each measure has four quarter notes.
Example: The ♯ shows up even before the transition to the second subject group.
Source: The {{music|sharp}}{{music|scale|4}} shows up even before the transition to the second subject group.
{\displaystyle \sharp {\hat {4}}}
shows up even before the transition to the second subject group.
Roman numeral analysis, which also includes Latin numerals as in the Nashville Number System (1, 2, ...), spelled out numbers (one, two, ...), and careted or circumflex numerals (scale degree 1, scale degree 2, ...)
Retrieved from "http://wiki.dcldesign.co.uk/index.php?title=Help:Musical_symbols&oldid=9637775"
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Interval-Valued Optimization Problems Involving -Right Upper-Dini-Derivative Functions
Vasile Preda, "Interval-Valued Optimization Problems Involving -Right Upper-Dini-Derivative Functions", The Scientific World Journal, vol. 2014, Article ID 750910, 5 pages, 2014. https://doi.org/10.1155/2014/750910
Vasile Preda 1,2,3
1Faculty of Mathematics and Computer Science, University of Bucharest, 010014 Bucharest, Romania
2Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania
3National Institute of Economic Research, 050711 Bucharest, Romania
We consider an interval-valued multiobjective problem. Some necessary and sufficient optimality conditions for weak efficient solutions are established under new generalized convexities with the tool-right upper-Dini-derivative, which is an extension of directional derivative. Also some duality results are proved for Wolfe and Mond-Weir duals.
Recently, Yuan and Liu [1] considered some new generalized convexity concepts using right upper-Dini-derivative, which is an extension of directional derivative. Thus some optimality and duality results are established for a nondifferentiable multiobjective programming problem. For various approaches relative to generalized convexity, we refer to [2–9].
In many real-life situations data suffer from inexactness. The interval-valued optimization problems are closely related to optimization problems with inexact data. Recently, Wu [10–12] derived optimality conditions and duality results for a multiobjective programming problem with interval-valued objective functions. See also [13] and their references.
In this paper we consider an interval multiobjective optimization problem. Some new optimality conditions and duality results are stated under new generalized convexities with the tool-right upper-Dini-derivative. The paper is organized as follows. In Section 2 some definitions, notations, and some basic arithmetic of interval calculus are given. In Section 3, we state necessary optimality conditions and in Section 4 we present sufficient optimality conditions. The duality results are stated in Sections 5 and 6. The last section gives some conclusions.
Let be the -dimensional Euclidean space and let be its nonnegative orthant. For and we consider the following conventions: Let be an arcwise connected set in Avriel and Zang [14] and Bhatia and Mehra [15] and a real-valued function defined on . Let , and be the arc connecting and in .
Definition 1. The right derivative (or right differential) of with respect to at is defined as
Yuan and Liu [1] give some new generalized convexity with the upper-Dini-derivative concept.
Definition 2. The right upper-Dini-derivative relative to is defined by
Definition 3. is locally arcwise connected at if for any and there exists a positive number , with and a continuous arc s.t. for any . The set is if is at any .
Definition 4 (see [1]). Let be a LAC set and let be a real function defined on . The function is said to be -right upper-Dini-derivative locally arcwise connected with respect to at , if there exist real functions and such that If is -right upper-Dini-derivative locally arcwise connected (with respect to ) at for any , then is called -right upper-Dini-derivative locally arcwise connected (with respect to ) on .
Definition 5 (see [1]). A -dimensional vector-valued function is called -right upper-Dini-derivative arcwise connected (with respect to ) at , if the th component of is -right upper-Dini-derivative arcwise connected (with respect to ) at for , where and. If is -right upper-Dini-derivative arcwise connected (with respect to ) at any , then is called -right upper-Dini-derivative arcwise connected (with respect to ) on .
Definition 6 (see [3]). A -dimensional vector-valued function is called convex-like (with respect to ) on if for all and any , there exists such that .
Definition 7 (see [1]). A -dimensional vector-valued function is called -generalized (strong) pseudoright upper-Dini-derivative arcwise connected (with respect to ) at , if there exists vector-valued function such that , for ; is called -generalized (weak) quasi-right upper-Dini-derivative arcwise connected (with respect to ) at , if there exists vector-valued function such that , for , where .
Lemma 8 (see [16]). Let be a nonempty set and let be a convex-like vector-valued function on . Then either has a solution , or there exists such that the system holds for all , but both are never true at the same time.
Let CBI() be the class of all closed and bounded intervals in . Thus if CBI, we have where and mean lower and upper bounds of . If , then is a real number. Also, let . Then, by definition we have
For a real number , we have
Using [17, 18], we consider some preliminary results about interval arithmetic calculus.
Definition 9. Let and CBI. We say that is less than and write if , .
Definition 10. Let CBI. We say that is less than or equal to and write if and .
Let be a nonempty subset of . A function CBI is called an interval-valued function. In this case, with , , .
We consider the following multiobjective interval-valued optimization problem: with , , , where , for and , . Let be the set of all feasible points of . We put .
Definition 11. Let . We say that is a weak efficient solution of if there exists no such that .
3. Necessary Optimality Conditions
In this section, we establish Fritz John and Karush-Kuhn-Tucker necessary optimality conditions for problem .
Theorem 12 (Fritz John necessary condition). Assume that is an efficient solution for . If, , and are convex-like on with respect to the variable and is upper semicontinuous at for , then there exist , , such that
Proof. We prove firstly that the following system of inequalities has no solution for . We proceed by contradicting. If there exists , a solution of this system, we get that for each there exist and such that for and for each there exists such that Since , for , is semicontinuous at , then is semicontinuous at . Finally we get for any , where . These inequalities contradict that is a weak efficient solution for . Hence the systems (9) and (10) have no solution for . Now we can apply Lemma 8. Since , , and are convex-like on , we obtain that (9) and (10) hold and the proof is complete.
Theorem 13 (Karush-Kuhn-Tucker necessary optimality condition). Let , , and be convex-like on with respect to the variables and , for , be upper semicontinuous at . If there exists such that and is a weak efficient solution for the problem , then there exist , , and satisfying (9), (10), and .
Proof. We suppose . Then, by (9) we obtain Since for , by (10), and there exists such that , by (15), it results , which contradicts (15) and theorem is proved.
In this section we give some Karush-Kuhn-Tucker type sufficient optimality conditions under generalized convexity with upper-Dini-derivative concept.
Theorem 14. Let be a feasible solution of . Assume that there exist and , , such that and are and -right upper-Dini-derivative locally arcwise connected at with respect to , respectively. Also one assumes and hold for all feasible. Then is a weak efficient solution for .
Proof. We suppose to the contrary that is not a weak efficient solution for . Then there exists such that . Now, since , , and , we get
Since and are and -right upper-Dini-derivative locally arcwise connected at , by (18) we obtain
Now, by and (17) we get a contradiction. Thus the theorem is proved.
The next sufficient optimality condition is given in the case of generalized pseudo- and quasi-right upper-Dini-derivative arcwise connected type, where the proof is on the line of the above theorem.
Theorem 15. Let be a feasible solution of . Assume that there exist and , , such that is generalized pseudoright upper-Dini-derivative locally arcwise connected at with respect to and is -generalized quasi-right upper-Dini-derivative locally arcwise connected at with respect to , respectively. Also one assumes that there exist and such that for any . Then is a weak efficient solution for .
5. Wolfe Duality
Relative to we consider the following Wolfe type dual problem: where and such that . Let denote the set of all feasible solutions of and let be the projection of the set on . Now we present weak, strong, and strict converse duality theorems relative to and . The proofs, which will be skipped here, follow the classic lines of multiobjective optimization [4, 7] and interval optimization problems [11].
Theorem 16 (weak duality). Let and be a feasible solution for and , respectively. One supposes that and are and -right upper-Dini-derivative arcwise connected at on with respect to , respectively. Moreover, one assumes and , for all . Then the following cannot hold: .
Theorem 17 (strong duality). Let be a weak efficient solution of , at which the assumptions of Karush-Kuhn-Tucker necessary optimality conditions are satisfied. Then there exist and such that and the objective values of and are equal. Further, if the hypotheses of weak duality theorem hold for all feasible solutions for , then is an efficient solution of .
Theorem 18 (converse duality). Let be a weak efficient solution of . One assumes that and are and -right upper Dini derivative arcwise connected at on with respect to , respectively. If and , for all , then is a weak efficient solution of .
6. Mond-Weir Duality
In this section we consider the following interval multiobjective dual problem, which is Mond-Weir dual type [6]: where , with . Let denote the set of all feasible solutions of and as the projection of the set on .
As in Section 5, we can establish some duality results. Here, we simply state them in the next theorems.
Theorem 19 (weak duality). Let and be a feasible solution for and , respectively. Assume that and are and -right upper-Dini-derivative arcwise connected at on with respect to , respectively. Also one supposes that and , for all . Then the following cannot hold: .
Theorem 20 (strong duality). Let be a weak efficient solution of the interval multiobjective programming problem , at which the assumptions of Karush-Kuhn-Tucker necessary optimality conditions are satisfied. Then there exist , , and , such that is a feasible solution for and the objective values of and are equal. Moreover, if the weak duality result between and holds, then is a weak efficient solution for .
Theorem 21 (converse duality). Let be a weak efficient solution of . Suppose that and are and -right upper-Dini-derivative arcwise connected at on with respect to , respectively. Further, if and , for all , then is a weak efficient solution of .
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Xingping Sheng, "Execute Elementary Row and Column Operations on the Partitioned Matrix to Compute M-P Inverse ", Abstract and Applied Analysis, vol. 2014, Article ID 596049, 6 pages, 2014. https://doi.org/10.1155/2014/596049
Xingping Sheng1
1School of Mathematics and Computational Science, Fuyang Normal College, Fuyang, Anhui, China
We first study the complexity of the algorithm presented in Guo and Huang (2010). After that, a new explicit formula for computational of the Moore-Penrose inverse of a singular or rectangular matrix . This new approach is based on a modified Gauss-Jordan elimination process. The complexity of the new method is analyzed and presented and is found to be less computationally demanding than the one presented in Guo and Huang (2010). In the end, an illustrative example is demonstrated to explain the corresponding improvements of the algorithm.
Throughout this paper we use the following notation. Let and be the dimensional complex space and the set of complex matrices with rank . For a matrix , and are the range and null space of ; and denote the rank and the the conjugate transpose of , while and denote the M-P inverse and Frobenius norm, respectively.
In 1920, Moore [1] defined a new inverse of a matrix by projection matrices. Moore’s definition of the generalized inverse of an matrix is equivalent to the existence of an matrix such that where is the orthogonal projector on . Unaware of Moore’s work, In 1955 Penrose [2] showed that there exists a unique matrix satisfying the four conditions where denotes conjugate transpose. These conditions are equivalent to Moore’s conditions. The unique matrix that satisfied these conditions was known as the Moore-Penrose inverse (abbreviated M-P) and is denoted by . For a subset of the set , the set of matrices satisfied the equations , from among (2) is denoted by . These concepts can be found in Ben-Israel and Greville’s famous book [3] or Campell and Meyer’s book [4]. In their famous books [3, 4], the next statement is valid for a rectangular matrix.
Lemma 1 (see [3, 4]). Let , be the M-P inverse of if and only if is a -inverse of with range and null space .
In the latest fifty years, there have been many famous specialists and scholars, who investigated the generalized inverse and published many articles and books. Its perturbation theories were introduced in [5–7], and the algebraical perturbations were in [8, 9]. Some minors, Cramer rulers, and sums of can be seen in [10–13]. There are a large number of papers [9, 14–18] using various methods, iterative or not, for computing .
One handy method of computing the inverse of a nonsingular matrix is the Gauss-Jordan elimination procedure by executing elementary row operations on the pair to transform it into . Moreover Gauss-Jordan elimination can be used to determine whether a matrix is nonsingular or not. However, one cannot directly use this method on a generalized inverse of a rectangular matrix or a square singular matrix .
Recently, Author [19] proposed a Gauss-Jordan elimination algorithm to compute , which required multiplications and divisions. More recently, Ji [20] improved author’s algorithm [19] and pointed that only multiplications and divisions are required. Following these lines, Stanimirovi and Petkovi in [21] extended the method of [20] to . But these three algorithms need also switching. Guo and Huang [22] executed row and column elementary transformations for computing M-P inverse by applying the rank equalities of matrix . They did not analyze the complexity of their algorithm. In this paper, we first study the total number arithmetic operation of GH-algorithm, then improve it, and present an alternative explicit formula for the M-P inverse of a matrix; the improvements save the total number arithmetic operation. We must point that GH-algorithm and our algorithm do not need to switch blocks of certain matrix in the process of computation.
The paper is organized as follows. The computational complexity of GH-Algorithm 3 for computing M-P inverse is surveyed in the next section. In Section 3, we derive a novel explicit expression for , propose a new Gauss-Jordan elimination procedure for based on the formula, and study the computational complexity of the new approach and Algorithm 7. In Section 4, an illustrative example is presented to explain the corresponding improvements of the algorithm.
2. The Computational Complexity of GH-Algorithm
In [22], Guo and Huang gave a method of elementary transformation for computing M-P inverse by applying the rank equalities of matrix .
Lemma 2. Suppose that , , , . If Then . In particualr when , and .
In [22], the authors also considered an algorithm based on Lemma 2, which was stated as follows.
Algorithm 3. M-P inverse GH-Algorithm is as follows.(1)Compute partitioned matrix .(2)Make the block matrix of become by applying elementary transformations. Meanwhile, the block matrices of and are accordingly transformed. A new partitioned matrix is obtained.(3)Make the block matrices of and be zero matrices by applying matrix , which yields Then
Nevertheless, Guo and Huang [22] did not analyze the complexity of the numerical algorithm. In the following theorem, we will study the total number of arithmetic operations.
Theorem 4. Let ; the total number of multiplications and divisions required in Algorithm 3 to compute M-P inverse is Moreover, is bounded above by .
Proof. It needs multiplications to compute . row pivoting steps and column pivoting steps are needed to transform the partitioned matrix into following the . First row pivoting step involves nonzero columns in . Thus, it needs divisions and multiplications with a total of multiplications and divisions. On the second row pivoting step, there is one less column in the first part of the pair. There are nonzero columns to deal with. These pivoting steps require operations. Following the same idea, the th pivoting steps require operations. So it requires .
For simplicity, assume that . Following the same line, this requires multiplications and divisions on the column pivoting steps.
Then resume elementary row and columns operations on the matrix to transform it into , which requires multiplications, which is the count of .
Therefore, the total number of operations needed for computation of is
Define a function for fixed and . Since , we have which means that is monotonically increasing over when . Therefore takes its maximum value when , which implies
The Gauss-Jordan row and column elimination procedure for the M-P inverse of a matrix by Guo and Huang is based on the partitioned matrix . In this section, we will first propose a modified Gauss-Jordan elimination process to compute and then summarize an algorithm of this method. Finally, the complexity of the algorithm is analyzed.
Theorem 5. Let ; there exist an elementary row operation matrix and an elementary column operation matrix , such that where and are the row and column reduced echelon form of , respectively.
Further, there exists an elementary row operation matrix such that Then
Proof. For , there exist two elementary row and column operation matrices and , such that
It is easy to check that and ; then the matrix is nonsingular, which implies that there exists another elementary row operation matrix such that The above formula also shows that .
Denote that ; it is obvious that and . If we can prove , then and .
In fact, is a full column rank matrix and is an invertible matrix, which implies that .
By deducing, we obtain that
This means that is a 2-inverse of with and . From Lemma 1, we know that .
Remark 6. The representation of in Theorem 5 is consistent with the one in [3], although we use Gauss-Jordan elimination procedure.
According to the representation introduced in Theorem 5, we summarize the following algorithm for computing M-P inverse .
Algorithm 7. M-P inverse-Sheng algorithm is as follows.(1)Input: .(2)Execute elementary row operations on first rows of the partitioned matrix into , where is a reduced row-echelon matrix.(3)Perform elementary column operations on first columns of the partitioned matrix into , where matrix has a reduced column-echelon form.(4)Compute and form the block matrix (5)Execute the elementary row operations on first rows of the partitioned matrix into .(6)Make the block matrices of and be zero matrices by applying elementary row and column transformations, respectively, through matrix , which yields Then .
According to Algorithm 7, the next theorem will analyze the computational complexity of it.
Theorem 8. The total number of multiplications and divisions required for Algorithm 7 to compute M-P inverse of a matrix is
Further, the upper bound of is less than when .
Proof. For a matrix with rank , pivoting steps are needed to make the partitioned matrix into . First pivoting step involves nonzero columns in . Thus, it needs divisions and multiplications with a total number of multiplications and divisions. On the second pivoting step, there is one less column in the first part of the pair. There are nonzero columns to deal with. This pivoting step requires operations. Following the same idea, the pivoting step requires operations. So these pivoting steps require multiplications and divisions to reach the matrix .
Similarly, it needs multiplications and divisions to change the matrix into .
For simplicity, assume that and , which follows from that and are row-echelon and column-echelon reduced matrix, respectively.
multiplications are required to form under the above assumption. Since every row of the partitioned matrix has nonzero elements, each pivoting step needs multiplications and divisions. Thus, it requires multiplications and divisions to obtain the matrix .
Then resume elementary row and columns operations on the matrix to transform it into . The complexity of this process is multiplications, which is the count to compute .
Hence, the total number of complexity of Algorithm 7 is
With fixed and , define a function for . Then we have which implies that is also monotonically increasing over when .
Therefore, when , obtains its maximum value, which yields
Furthermore, we give two remarks: one is explaining the computation speed and the other is how to improve the accuracy of Algorithm 7.
Remark 9. The algorithm in this paper does not need to switch block of certain matrix in the process computation, unlike the existing algorithm in [19–21]. The higher computational complexity is about multiplications and divisions, that is, less than GH-algorithm [22], which requires multiplications and divisions, when they are applied to the case of for .
Remark 10. In order to improve the accuracy of the algorithm, we must select nonzero entries in pivot row and column in each step of the Gauss-Jordan elimination. This improvement is based on the fact that Gauss-Jordan elimination is applied on matrices containing nonnegligible number zero elements.
In this section, we will use a numerical example to demonstrate our results. A handy method is used to compute on a low order matrix.
Example 1. Use Algorithm 7 to compute the M-P inverse of the matrix in [21], where
Solution. Execute elementary row operations on the first four rows of the partitioned matrix ; we have Then perform elementary column operations on the first three columns of matrix , which yields Denote By computing, we have
We execute elementary row operations on the first two rows of the partitioned matrix again; we have
One then resumes elementary row and column operations on , which results in Then we can obtain
This project was supported by NSF China (no. 11226200), Anhui Provincial Natural Science Foundation (no. 10040606Q47), and The University Natural Science Research key Project of Anhui Province (KJ2013A204).
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Ergodicity and Invariance of Flows in Queuing Systems
Ergodicity and Invariance of Flows in Queuing Systems
In this paper, we investigate the flow of customers through queuing systems with randomly varying intensities. The analysis of the Kolmogorov-Chapman system of stationary equations for this model showed that it is not possible to construct a convenient symbolic solution. In this paper an attempt is made to circumvent this requirement by referring to the ergodicity theorems, which gives the conditions for the existence of the limit distribution in the service processes, but do not require knowledge of them.
A Queuing System, An Ergodicity, An Input Flow, A Randomly Varying Intensity
In this paper, we investigate the flow of customers through queuing systems. This work was initiated by the task, which was set in 2000 to the author of the article by the late professor of the Australian National University (Canberra, Australia) Joe Gani. He proposed to calculate a model of a single-server queuing system with a Poisson input flow with a randomly changing intensity and a randomly changing service intensity. The analysis of the Kolmogorov-Chapman system of stationary equations for this model showed that it is not possible to construct a convenient symbolic solution and it is necessary to use numerical methods and to solve approximately a linear system of algebraic equations.
Progress in solving this problem could be obtained only recently, thanks to an appeal to Burke’s theorem on the coincidence of Poisson input and output flow distributions in the
M|M|n|\infty
system. However, the proof of this theorem [1] occurred to be rather cumbersome and inconvenient to extend to other queuing models, including systems in a random environment. Therefore, it was necessary to develop an alternative proof [2] of this theorem, based on the classical work of A.Ya. Khinchin [3] on the representation of Poisson flows.
However, this alternative proof required knowledge of limit distributions in queuing systems, which is not always possible to obtain in a symbolic form. Therefore, in this paper an attempt is made to circumvent this requirement by referring to the ergodicity theorems, which gives the conditions for the existence of the limit distribution in the service processes, but do not require knowledge of them.
2. Equality of Average Intensities of Input and Output Flows in the Queuing System
Consider queuing system A with Poisson input flow of intensity
\lambda
x\left(t\right)
be the number of customers of the input flow in the system A on a half-interval
\left(0,t\right]
x\left(0\right)=0
. Then the following relation is true.
lemma 1. The following almost sure convergence is true:
\frac{x\left(t\right)}{t}\to \lambda ,\text{\hspace{0.17em}}t\to \infty .
\left[t\right]
be an integer part of
t>1
. Then almost surely the following relations are executed:
\frac{x\left(\left[t\right]\right)}{\left[t\right]}\cdot \frac{\left[t\right]}{\left[t\right]+1}\le \frac{x\left(t\right)}{t}\le \frac{x\left(\left[t\right]+1\right)}{\left[t\right]+1}\cdot \frac{\left[t\right]+1}{\left[t\right]},
x\left(\left[t\right]\right)=\underset{i=0}{\overset{\left[t\right]-1}{\sum }}\left(x\left(i+1\right)-x\left(i\right)\right),
where all terms in equality (3) are independent and have Poisson distribution with the parameter
\lambda
. Then from the relations (2), (3) and the strengthened law of large numbers (see, for example, ( [4] , Chapter 8, $4$)) we have the equality (1).
Corollary 1. From Lemma 1 we have the convergence by probability in the relation (1).
y\left(t\right)
the number of customers that came out of the queuing system A on the half-interval
\left(0,t\right]
z\left(t\right)
the number of customers in the system at time t.
Theorem 1. If the random process
z\left(t\right)
is ergodic and stationary, then the convergence by probability is true
\frac{y\left(t\right)}{t}\to \lambda ,\text{\hspace{0.17em}}t\to \infty .
Proof. With probability one, the following equality is performed
x\left(t\right)=y\left(t\right)+z\left(t\right),\text{\hspace{0.17em}}t\ge 0.
In turn, from the ergodicity of the stationary random process
z\left(t\right),\text{\hspace{0.17em}}z\left(0\right)=0
it follows (see, for example, ( [4] , Chapter 10, $4$)) that there is the distribution function
F\left(u\right),\text{\hspace{0.17em}}F\left(0\right)=0,\text{\hspace{0.17em}}F\left(u\right)\to 1,\text{\hspace{0.17em}}u\to \infty
u\ge 0
P\left(z\left(t\right)<u\right)\equiv F\left(u\right)
. We prove now that the following convergence by probability is true
\frac{z\left(t\right)}{t}\to 0,\text{\hspace{0.17em}}t\to \infty .
\epsilon >0
, then the following equality is true:
P\left(\frac{z\left(t\right)}{t}>\epsilon \right)=1-F\left(\epsilon t\right)\to 0,\text{\hspace{0.17em}}t\to \infty ,
and so the convergence by probability in the relation (6) is valid. From the convergence by probability in the relation (6), equality (5) and Corollary 1 we obtain the statement of the theorem 1.
3. Poisson Output Flows in Queuing Systems of General Form
Suppose that the ergodic stationary process
z\left(t\right)
is Markov. Since this process characterizes the number of customers in the queuing system A, then it takes the values
0,1,\cdots
{f}_{k}=P\left(z\left(t\right)=k\right),\text{\hspace{0.17em}}k=0,1,\cdots
and suppose that the intensities of
{\mu }_{k}
customers leaving the system A, in the state
z\left(t\right)=k
Theorem 2. The output flow in queuing system A is Poisson with the intensity
\lambda
Proof. A random sequence of points on the real axis is called a Poisson flow with the intensity
\lambda
if the following conditions ( [3] , page 12, 13), ( [5] , page 20, 35-37) are true: 1) the probability of the existence of the point of flow on the time interval
\left[t,t+h\right)
does not depend on the location of the points of the flow up to the time t (this property is called lack of follow-through and expresses the mutual independence of the flow points into disjoint periods of time); 2) the probability that in the half-interval
\left[t,t+h\right)
the point of flow appears is
\lambda \left(t\right)h+o\left(h\right),\text{\hspace{0.17em}}h\to 0
; 3) the probability of occurrence of two or more flow points in the half-interval
\left[t,t+h\right)
o\left(h\right),\text{\hspace{0.17em}}h\to 0
We use the construction of [2] and prove that the output flow of customers from the queuing system A is Poisson. Indeed, since the random process
z\left(t\right)
is discrete Markov, and the moments of withdrawal of customers from the system A are the moments of jumps down the process
z\left(t\right)
, the output flow of customers from A satisfies the condition a), checking of the conditions b), c) is made by direct calculations. Therefore, the output flow in the system A is Poisson with the intensity
a={\sum }_{k=1}^{\infty }\text{ }{f}_{k}{\mu }_{k}
. Using Theorem 1, we obtain the equality
\lambda =a
Remark 1. In a case the queuing system A is n-server type
M|M|n|\infty
, Theorem 2 is a generalization of the famous theorem of Burke [1] . The peculiarity of this design is the fact, that only ergodic properties of the Markov process
z\left(t\right)
are important, while knowledge of stationary probabilities
{f}_{k},\text{\hspace{0.17em}}k=1,2,\cdots
are not necessary to calculate the intensity of the output Poisson flow.
4. Queuing Systems with Failures
M|M|1|0
queuing system, in which the customers coming on a busy server, is refused. Input flow to this system is a Poisson with the intensity
\lambda
, service times have exponential distribution with the parameter
\mu
. This system is described by a number
z\left(t\right)
of the customers in it at the moment t [5] . The set of states of the Markov process
z\left(t\right)
\left\{0,1\right\}
. Denote transition intensity of the process
z\left(t\right)
from state i to state j by
\alpha \left(i,j\right)
. Then the following equations are true:
\alpha \left(0,1\right)=\lambda ,\text{\hspace{0.17em}}\alpha \left(1,0\right)=\mu ,\text{\hspace{0.17em}}\alpha \left(1,1\right)=\lambda .
Here the transition
0\to 1
corresponds to the arrival of the customer into the empty system. In turn, the transitions between the process states
z\left(t\right)
, leading to the withdrawal of customers from the system have the form
1\to 0,\text{\hspace{0.17em}}1\to 1
1\to 0
corresponds to the withdrawal of the severed customer from the system, and the
1\to 1
corresponds to the withdrawal of the refused customer from the system.
z\left(t\right)
is ergodic for any parameter values
\lambda ,\mu >0
, and its limit distribution has the form
{b}_{0}=\frac{1}{1+\rho },\text{\hspace{0.17em}}{b}_{1}=\frac{\rho }{1+\rho }.
Theorem 2 implies the following statement.
Theorem 3. Stationary flows of the served customers and customers that have been rejected, are Poisson and have intensities
{a}_{1}={b}_{1}\mu ,\text{\hspace{0.17em}}{a}_{2}={b}_{1}\lambda
, respectively. The stationary output flow in the system
M|M|1|0
is Poisson and its intensity
a=\lambda ={a}_{1}+{a}_{2}
Remark 2. The statement of Theorem 3 extends to queuing systems
M|M|n|N
with a limited queue, a finite number of servers, as well as a system with a finite number of flows and a fairly general discipline of their service. In the latter case it is necessary to fulfil the ergodicity condition, which requires that the graph whose nodes are the states of the system, whose edges are transitions between states, satisfies the reachable condition of any node from any other node.
Figure 1. Transition intensities of process
z\left(t\right)
describing the number of customers in the queuing system
M|M|1|0
with failures.
5. Queuing Systems in Random Environment
1) Poisson input flow with randomly varying intensity. Let the time axis
t\ge 0
be split into half-intervals
\left[{T}_{0}=0,{T}_{1}={T}_{0}+{\xi }_{1}\right),\left[{T}_{1},{T}_{2}={T}_{1}+{\xi }_{2}\right),\cdots
where are independent random variables with distribution
with parameter . Introduce discrete Markov chain with set of
states and irreducible transition matrix and stationary probabilities .
Consider the Markov random process , where is a Markov stationary random process that characterizes the intensity of the input flow at the time t, is the number of input flow customers that came to the system on the half-interval . Example of this process is represented in Figure 2. Here is an intensity of a transition from to , and is intensity of transition from to , and is intensity of transition from to , and is intensity of transition from to .
Points of jumps (up) of the process are the moments of customers arrivals to the system. Since the process is Markov, then for a random flow defined by jumps (up) of the component satisfies the condition a) of the Section 3. In turn, due to the stationarity of the Markov process the probability occurrences of process jump in the half-interval equals , where . Thus, it is proved that the input Poisson flow with a randomly varying intensity coincides by the distribution with the Poisson flow having an average intensity .
This circumstance allows us to study the output flows in queuing systems and networks with Poisson input flow having a randomly changing intensity, assuming that the random service intensities in the nodes of the queuing system or network and the random input flow are independent.
2) Queing system in random environment. Consider the system with the service intensity and the Poisson input
Figure 2. Transition intensities of process describing the number of input flow customers that came to the system on the half-interval .
flow with the intensity , that change randomly according to the following rules. Let on each half-interval the input flow to the system be Poisson with the intensity , and the intensity of service is . It is easy to check that under these conditions, the Markov random process is ergodic. Then from Theorem 2 it is easy to establish that the total output flow from a given system is Poisson with the intensity . These constructions allow to consider manifold queuing models with different input flows and distributions of customers service times.
Partially supported by Russian Fund of Basic Researches, project 17-07-00177.
Tsitsiashvili, G.Sh. (2018) Ergodicity and Invariance of Flows in Queuing Systems. Journal of Applied Mathematics and Physics, 6, 1454-1459. https://doi.org/10.4236/jamp.2018.67122
1. Burke, P.J. (1956) The Output of a Queuing System. Operations Research, 4, 699-704. https://doi.org/10.1287/opre.4.6.699
2. Tsitsiashvili, G.Sh. and Osipova M.A. (2018) Generalization and Extension of Burke Theorem. Reliability: Theory and Applications, 13, 59-62.
3. Khinchin, A.Ya. (1963) Works on the Mathematical Queuing Theory. Physmatlit, Moscow. (In Russian)
4. Borovkov, A.A. (1972) Course of Probability Theory. Nauka, Moscow. (In Russian)
5. Ivchenko, G.I., Kashtanov, V.A. and Kovalenko, I.N. (1982) Queuing Theory. Vishaya Shkola, Moscow. (In Russian)
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Lyn asked some of her classmates how many people are normally at home for dinner. Her results, recorded in a histogram, are shown below.
The vertical axis reading for the bar is
1
, so in other words, one classmate has eight or nine people at home for dinner.
Because of this, you cannot know which is the most common number of people home for dinner.
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The pound per square inch or, more accurately, pound-force per square inch (symbol: lbf/in2;[1] abbreviation: psi) is a unit of pressure or of stress based on avoirdupois units. It is the pressure resulting from a force of one pound-force applied to an area of one square inch. In SI units, 1 psi is approximately equal to 6895 Pa.
psi or lbf/in2
1 psi in ... ... is equal to ...
Pounds per square inch absolute (psia) is used to make it clear that the pressure is relative to a vacuum rather than the ambient atmospheric pressure. Since atmospheric pressure at sea level is around 14.7 psi (101 kilopascals), this will be added to any pressure reading made in air at sea level. The converse is pounds per square inch gauge (psig), indicating that the pressure is relative to atmospheric pressure. For example, a bicycle tire pumped up to 65 psig in a local atmospheric pressure at sea level (14.7 psi) will have a pressure of 79.7 psia (14.7 psi + 65 psi).[2][3] When gauge pressure is referenced to something other than ambient atmospheric pressure, then the units would be pounds per square inch differential (psid).
The kilopound per square inch (ksi) is a scaled unit derived from psi, equivalent to a thousand psi (1000 lbf/in2).
ksi are not widely used for gas pressures. They are mostly used in materials science, where the tensile strength of a material is measured as a large number of psi.[4]
The megapound per square inch (Mpsi) is another multiple equal to a million psi. It is used in mechanics for the elastic modulus of materials, especially for metals.[5]
Main article: Orders of magnitude (pressure)
Inch of water: 0.036 psid
Blood pressure – clinically normal human blood pressure (120/80 mmHg): 2.32 psig/1.55 psig
Natural gas residential piped in for consumer appliance; 4–6 psig.
Boost pressure provided by an automotive turbocharger (common): 6–15 psig
NFL football: 12.5–13.5 psig
Atmospheric pressure at sea level (standard): 14.7 psia
Automobile tire overpressure (common): 32 psig
Bicycle tire overpressure (common): 65 psig
Workshop or garage air tools: 90 psig
Air brake (rail) or air brake (road vehicle) reservoir overpressure (common): 90–120 psig
Road racing bicycle tire overpressure: 120 psig
Steam locomotive fire tube boiler (UK, 20th century): 150–280 psig
Union Pacific Big Boy steam locomotive boiler: 300 psig
US Navy steam boiler pressure: 800 psi
Natural gas pipelines: 800–1000 psig
Full SCBA (self-contained breathing apparatus) for IDLH (non-fire) atmospheres: 2216 psig
Nuclear reactor primary loop: 2300 psi
Full SCUBA (self-contained underwater breathing apparatus) tank overpressure (common): 3000 psig
Full SCBA (self-contained breathing apparatus) for interior firefighting operations: 4500 psig
Airbus A380 hydraulic system: 5000 psig
Land Rover Td5 diesel engine fuel injection pressure: 22,500 psi
Ultimate strength of ASTM A36 steel: 58,000 psi
Water jet cutter: 40,000–100,000 psig
The conversions to and from SI are computed from exact definitions but result in a repeating decimal. [6][7]
{\displaystyle P_{\text{Pa}}=P_{\text{PSI}}\times {\frac {(0.45359237~{\text{kg}}\times 9.80665~{\text{m}}/{\text{s}}^{2})/{\text{lbf}}}{(0.0254~{\text{m}}/{\text{in}})^{2}}}}
{\displaystyle P_{\text{PSI}}=P_{\text{Pa}}\times {\frac {(0.0254~{\text{m}}/{\text{in}})^{2}}{(0.45359237~{\text{kg}}\times 9.80665~{\text{m}}/{\text{s}}^{2})/{\text{lbf}}}}}
As the pascal is a very small unit relative to industrial pressures, the kilopascal is commonly used. 1000 kPa ≈ 145 lbf/in2.
Approximate conversions (rounded to some arbitrary number of digits, except when denoted by "≡") are shown in the following table.
^ "Glossary of Industrial Air Cleaning Technology". United Air Specialists, Inc. Archived from the original on August 1, 2011.
^ "Gage v. Sealed v. Absolute pressure" (PDF). Dynisco.
^ "Tensile Strength of Steel and Other Metals". All Metals & Forge Group. Retrieved 2016-07-26. A metal’s yield strength and ultimate tensile strength values are expressed in tons per square inch, pounds per square inch or thousand pounds (KSI) per square inch. For example, a tensile strength of a steel that can withstand 40,000 pounds of force per square inch may be expressed as 40,000 PSI or 40 KSI (with K being the [multiplier] for thousands of pounds). The tensile strength of steel may also be shown in MPa, or megapascal.
^ An example of the use of Mpsi in mechanics for the elastic moduli of several materials
^ BS 350: Part 1: 1974 – Conversion factors and tables. British Standards Institution. 1974. p. 49. ISBN 0 580 08471 X.
^ NIST Special Publication 811 – Guide for the Use of the International System of Units (SI) (PDF). National Institute of Standards and Technology. 2008. p. 66.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Pound_per_square_inch&oldid=1050665111"
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(Redirected from Elliptic Curve Cryptography)
Approach to public-key cryptography
{\displaystyle y^{2}=x^{3}+ax+b,\,}
{\displaystyle \mathrm {Div} ^{0}(E)\to \mathrm {Pic} ^{0}(E)\simeq E,\,}
Cryptographic schemes[edit]
{\displaystyle (\mathbb {Z} _{p})^{\times }}
Domain parameters[edit]
{\displaystyle 2^{m}}
{\displaystyle nG={\mathcal {O}}}
{\displaystyle E(\mathbb {F} _{p})}
{\displaystyle h={\frac {1}{n}}|E(\mathbb {F} _{p})|}
{\displaystyle h\leq 4}
{\displaystyle h=1}
{\displaystyle (p,a,b,G,n,h)}
{\displaystyle (m,f,a,b,G,n,h)}
{\displaystyle \mathbb {F} _{2^{m}}}
{\displaystyle p^{B}-1}
{\displaystyle 2}
{\displaystyle \mathbb {F} _{p}}
{\displaystyle \mathbb {F} _{p^{B}}}
{\displaystyle E(\mathbb {F} _{q})}
{\displaystyle |E(\mathbb {F} _{q})|=q}
{\displaystyle \mathbb {F} _{q}}
{\displaystyle O({\sqrt {n}})}
{\displaystyle \mathbb {F} _{q}}
{\displaystyle q\approx 2^{256}}
Projective coordinates[edit]
{\displaystyle \mathbb {F} _{q}}
{\displaystyle x\in \mathbb {F} _{q}}
{\displaystyle y\in \mathbb {F} _{q}}
{\displaystyle xy=1}
{\displaystyle (X,Y,Z)}
{\displaystyle x={\frac {X}{Z}}}
{\displaystyle y={\frac {Y}{Z}}}
{\displaystyle (X,Y,Z)}
{\displaystyle x={\frac {X}{Z^{2}}}}
{\displaystyle y={\frac {Y}{Z^{3}}}}
{\displaystyle x={\frac {X}{Z}}}
{\displaystyle y={\frac {Y}{Z^{2}}}}
{\displaystyle (X,Y,Z,aZ^{4})}
{\displaystyle (X,Y,Z,Z^{2},Z^{3})}
Fast reduction (NIST curves)[edit]
{\displaystyle p\approx 2^{d}}
{\displaystyle p=2^{521}-1}
{\displaystyle p=2^{256}-2^{32}-2^{9}-2^{8}-2^{7}-2^{6}-2^{4}-1.}
{\displaystyle \mathbb {F} _{p}}
{\displaystyle \mathbb {F} _{p}}
{\displaystyle \mathbb {F} _{2^{m}}}
Elliptic curve cryptography is used by the cryptocurrency Bitcoin.[33] Ethereum version 2.0 makes extensive use of elliptic curve pairs using BLS signatures—as specified in the IETF draft BLS specification—for cryptographically assuring that a specific Eth2 validator has actually verified a particular transaction.[34][35]
Quantum computing attacks[edit]
Invalid curve attack[edit]
Alternative representations[edit]
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Mapped independent suspension - Simulink - MathWorks 日本
\begin{array}{l}{F}_{wzlooku{p}_{a}}=f\left({z}_{{v}_{a,t}}â{z}_{{w}_{a,t}},{\stackrel{Ë}{z}}_{{v}_{a,t}}â{\stackrel{Ë}{z}}_{{w}_{a,t}},{\mathrm{δ}}_{stee{r}_{a,t}}\right)\\ \\ {F}_{w{z}_{a,t}}={F}_{wzlooku{p}_{a}}+{F}_{zasw{y}_{a,t}}\end{array}
\begin{array}{l}{F}_{v{x}_{a,t}}={F}_{w{x}_{a,t}}\\ {F}_{v{y}_{a,t}}={F}_{w{y}_{a,t}}\\ {F}_{v{z}_{a,t}}=â{F}_{w{z}_{a,t}}\\ \\ {M}_{v{x}_{a,t}}={M}_{w{x}_{a,t}}+{F}_{w{y}_{a,t}}\left(R{e}_{w{y}_{a,t}}+{H}_{a,t}\right)\\ {M}_{v{y}_{a,t}}={M}_{w{y}_{a,t}}+{F}_{w{x}_{a,t}}\left(R{e}_{w{x}_{a,t}}+{H}_{a,t}\right)\\ {M}_{v{z}_{a,t}}={M}_{w{z}_{a,t}}\end{array}
\begin{array}{l}{x}_{{w}_{a,t}}={x}_{{v}_{a,t}}\\ {y}_{{w}_{a,t}}={y}_{{v}_{a,t}}\\ {\stackrel{Ë}{x}}_{{w}_{a,t}}={\stackrel{Ë}{x}}_{{v}_{a,t}}\\ {\stackrel{Ë}{y}}_{{w}_{a,t}}={\stackrel{Ë}{y}}_{{v}_{a,t}}\end{array}
Anti-sway bar angular deflection for a given axle and track, Δϴa,t
\begin{array}{l}{\mathrm{θ}}_{0a}={\mathrm{tan}}^{â1}\left(\frac{{z}_{0}}{r}\right)\\ \mathrm{Î}{\mathrm{θ}}_{a,t}={\mathrm{tan}}^{â1}\left(\frac{r\mathrm{tan}{\mathrm{θ}}_{0a}â{z}_{{w}_{a,t}}+{z}_{{v}_{a,t}}}{r}\right)\end{array}
Anti-sway bar twist angle, Ï´a
{\mathrm{θ}}_{a}=â{\mathrm{tan}}^{â1}\left(\frac{r\mathrm{tan}{\mathrm{θ}}_{0a}â{z}_{{w}_{a,1}}+{z}_{{v}_{a,1}}}{r}\right)â{\mathrm{tan}}^{â1}\left(\frac{r\mathrm{tan}{\mathrm{θ}}_{0a}â{z}_{{w}_{a,2}}+{z}_{{v}_{a,2}}}{r}\right)
Anti-sway bar torque, Ï„a
{\mathrm{Ï}}_{a}={k}_{a}{\mathrm{θ}}_{a}
\begin{array}{l}{F}_{zasw{y}_{a,1}}=\left(\frac{{\mathrm{Ï}}_{a}}{r}\right)\mathrm{cos}\left({\mathrm{θ}}_{0a}â{\mathrm{tan}}^{â1}\left(\frac{r\mathrm{tan}{\mathrm{θ}}_{0a}â{z}_{{w}_{a,1}}+{z}_{{v}_{a,1}}}{r}\right)\right)\\ {F}_{zasw{y}_{a,2}}=\left(\frac{{\mathrm{Ï}}_{a}}{r}\right)\mathrm{cos}\left({\mathrm{θ}}_{0a}â{\mathrm{tan}}^{â1}\left(\frac{r\mathrm{tan}{\mathrm{θ}}_{0a}â{z}_{{w}_{a,2}}+{z}_{{v}_{a,2}}}{r}\right)\right)\end{array}
Ï„a
θ0a
Δϴa,t Anti-sway bar angular deflection at axle a, track t
\left[\begin{array}{ccc}{\mathrm{ξ}}_{a,t}& {\mathrm{η}}_{a,t}& {\mathrm{ζ}}_{a,t}\end{array}\right]={G}_{alookup}f\left({z}_{{w}_{a,t}}â{z}_{{v}_{a,t}},{\mathrm{δ}}_{stee{r}_{a,t}}\right)
{\mathrm{δ}}_{whlstee{r}_{a,t}}={\mathrm{δ}}_{stee{r}_{a,t}}+{G}_{alookup}f\left({z}_{{w}_{a,t}}â{z}_{{v}_{a,t}},{\mathrm{δ}}_{stee{r}_{a,t}}\right)
{P}_{sus{p}_{a,t}}={F}_{wzlooku{p}_{a}}\left({\stackrel{Ë}{z}}_{{v}_{a,t}}â{\stackrel{Ë}{z}}_{{w}_{a,t}},{\stackrel{Ë}{z}}_{{v}_{a,t}}â{\stackrel{Ë}{z}}_{{w}_{a,t}},{\mathrm{δ}}_{stee{r}_{a,t}}\right)
{E}_{sus{p}_{a,t}}={F}_{wzlooku{p}_{a}}\left({\stackrel{Ë}{z}}_{{v}_{a,t}}â{\stackrel{Ë}{z}}_{{w}_{a,t}},{\stackrel{Ë}{z}}_{{v}_{a,t}}â{\stackrel{Ë}{z}}_{{w}_{a,t}},{\mathrm{δ}}_{stee{r}_{a,t}}\right)
{H}_{a,t}=â\left({z}_{{v}_{a,t}}â{z}_{{w}_{a,t}}â\mathrm{median}\left(f_susp_dz_bp\right)\right)
{z}_{wt{r}_{a,t}}=R{e}_{{w}_{a,t}}+{H}_{a,t}
\mathrm{WhlPz}={z}_{w}=\left[\begin{array}{cccc}{z}_{{w}_{1,1}}& {z}_{{w}_{1,2}}& {z}_{{w}_{2,1}}& {z}_{{w}_{2,2}}\end{array}\right]
\mathrm{Whl}\mathrm{Re}=R{e}_{w}=\left[\begin{array}{cccc}R{e}_{{w}_{1,1}}& R{e}_{{w}_{1,2}}& R{e}_{{w}_{2,1}}& R{e}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlVz}={\stackrel{Ë}{z}}_{w}=\left[\begin{array}{cccc}{\stackrel{Ë}{z}}_{{w}_{1,1}}& {\stackrel{Ë}{z}}_{{w}_{1,2}}& {\stackrel{Ë}{z}}_{{w}_{2,1}}& {\stackrel{Ë}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlFx}={F}_{wx}=\left[\begin{array}{cccc}{F}_{w{x}_{1,1}}& {F}_{w{x}_{1,2}}& {F}_{w{x}_{2,1}}& {F}_{w{x}_{2,2}}\end{array}\right]
\mathrm{WhlFy}={F}_{wy}=\left[\begin{array}{cccc}{F}_{w{y}_{1,1}}& {F}_{w{y}_{1,2}}& {F}_{w{y}_{2,1}}& {F}_{w{y}_{2,2}}\end{array}\right]
\mathrm{WhlM}={M}_{w}=\left[\begin{array}{cccc}{M}_{w{x}_{1,1}}& {M}_{w{x}_{1,2}}& {M}_{w{x}_{2,1}}& {M}_{w{x}_{2,2}}\\ {M}_{w{y}_{1,1}}& {M}_{w{y}_{1,2}}& {M}_{w{y}_{2,1}}& {M}_{w{y}_{2,2}}\\ {M}_{w{z}_{1,1}}& {M}_{w{z}_{1,2}}& {M}_{w{z}_{2,1}}& {M}_{w{z}_{2,2}}\end{array}\right]
\mathrm{VehP}=\left[\begin{array}{c}{x}_{v}\\ {y}_{v}\\ {z}_{v}\end{array}\right]=\left[\begin{array}{cccc}{x}_{v}{}_{{}_{1,1}}& {x}_{v}{}_{{}_{1,2}}& {x}_{v}{}_{{}_{2,1}}& {x}_{v}{}_{{}_{2,2}}\\ {y}_{v}{}_{{}_{1,1}}& {y}_{v}{}_{{}_{1,2}}& {y}_{v}{}_{{}_{2,1}}& {y}_{v}{}_{{}_{2,2}}\\ {z}_{v}{}_{{}_{1,1}}& {z}_{v}{}_{{}_{1,2}}& {z}_{v}{}_{{}_{2,1}}& {z}_{v}{}_{{}_{2,2}}\end{array}\right]
\mathrm{VehV}=\left[\begin{array}{c}{\stackrel{Ë}{x}}_{v}\\ {\stackrel{Ë}{y}}_{v}\\ {\stackrel{Ë}{z}}_{v}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{Ë}{x}}_{{v}_{1,1}}& {\stackrel{Ë}{x}}_{{v}_{1,2}}& {\stackrel{Ë}{x}}_{{v}_{2,1}}& {\stackrel{Ë}{x}}_{{v}_{2,2}}\\ {\stackrel{Ë}{y}}_{{v}_{1,1}}& {\stackrel{Ë}{y}}_{{v}_{1,2}}& {\stackrel{Ë}{y}}_{{v}_{2,1}}& {\stackrel{Ë}{y}}_{{v}_{2,2}}\\ {\stackrel{Ë}{z}}_{{v}_{1,1}}& {\stackrel{Ë}{z}}_{{v}_{1,2}}& {\stackrel{Ë}{z}}_{{v}_{2,1}}& {\stackrel{Ë}{z}}_{{v}_{2,2}}\end{array}\right]
\mathrm{StrgAng}={\mathrm{δ}}_{steer}=\left[\begin{array}{cc}{\mathrm{δ}}_{stee{r}_{1,1}}& {\mathrm{δ}}_{stee{r}_{1,2}}\end{array}\right]
\mathrm{WhlAng}\left[1,...\right]=\mathrm{ξ}=\left[{\mathrm{ξ}}_{a,t}\right]
\mathrm{WhlAng}\left[2,...\right]=\mathrm{η}=\left[{\mathrm{η}}_{a,t}\right]
\mathrm{WhlAng}\left[3,...\right]=\mathrm{ζ}=\left[{\mathrm{ζ}}_{a,t}\right]
\mathrm{VehF}={F}_{v}=\left[\begin{array}{cccc}{F}_{v}{}_{{x}_{1,1}}& {F}_{v}{}_{{x}_{1,2}}& {F}_{v}{}_{{x}_{2,1}}& {F}_{v}{}_{{x}_{2,2}}\\ {F}_{v}{}_{{y}_{1,1}}& {F}_{v}{}_{{y}_{1,2}}& {F}_{v}{}_{{y}_{2,1}}& {F}_{v}{}_{{y}_{2,2}}\\ {F}_{v}{}_{{z}_{1,1}}& {F}_{v}{}_{{z}_{1,2}}& {F}_{v}{}_{{z}_{2,1}}& {F}_{v}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{VehM}={M}_{v}=\left[\begin{array}{cccc}{M}_{v{x}_{1,1}}& {M}_{v{x}_{1,2}}& {M}_{v{x}_{2,1}}& {M}_{v{x}_{2,2}}\\ {M}_{v{y}_{1,1}}& {M}_{v{y}_{1,2}}& {M}_{v{y}_{2,1}}& {M}_{v{y}_{2,2}}\\ {M}_{v{z}_{1,1}}& {M}_{v{z}_{1,2}}& {M}_{v{z}_{2,1}}& {M}_{v{z}_{2,2}}\end{array}\right]
\mathrm{WhlF}={F}_{w}=\left[\begin{array}{cccc}{F}_{w}{}_{{x}_{1,1}}& {F}_{w}{}_{{x}_{1,2}}& {F}_{w}{}_{{x}_{2,1}}& {F}_{w}{}_{{x}_{2,2}}\\ {F}_{w}{}_{{y}_{1,1}}& {F}_{w}{}_{{y}_{1,2}}& {F}_{w}{}_{{y}_{2,1}}& {F}_{w}{}_{{y}_{2,2}}\\ {F}_{w}{}_{{z}_{1,1}}& {F}_{w}{}_{{z}_{1,2}}& {F}_{w}{}_{{z}_{2,1}}& {F}_{w}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{WhlP}=\left[\begin{array}{c}{x}_{w}\\ {y}_{w}\\ {z}_{w}\end{array}\right]=\left[\begin{array}{cccc}{x}_{w}{}_{{}_{1,1}}& {x}_{w}{}_{{}_{1,2}}& {x}_{w}{}_{{}_{2,1}}& {x}_{{w}_{2,2}}\\ {y}_{w}{}_{{}_{1,1}}& {y}_{w}{}_{{}_{1,2}}& {y}_{w}{}_{{}_{2,1}}& {y}_{w}{}_{{y}_{2,2}}\\ {z}_{wtr}{}_{{}_{1,1}}& {z}_{wtr}{}_{{}_{1,2}}& {z}_{wtr}{}_{{}_{2,1}}& {z}_{wt{r}_{2,2}}\end{array}\right]
\mathrm{WhlV}=\left[\begin{array}{c}{\stackrel{Ë}{x}}_{w}\\ {\stackrel{Ë}{y}}_{w}\\ {\stackrel{Ë}{z}}_{w}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{Ë}{x}}_{{w}_{1,1}}& {\stackrel{Ë}{x}}_{{w}_{1,2}}& {\stackrel{Ë}{x}}_{{w}_{2,1}}& {\stackrel{Ë}{x}}_{{w}_{2,2}}\\ {\stackrel{Ë}{y}}_{{w}_{{}_{1,1}}}& {\stackrel{Ë}{y}}_{{w}_{1,2}}& {\stackrel{Ë}{y}}_{{w}_{2,1}}& {\stackrel{Ë}{y}}_{{w}_{2,2}}\\ {\stackrel{Ë}{z}}_{{w}_{{}_{1,1}}}& {\stackrel{Ë}{z}}_{{w}_{1,2}}& {\stackrel{Ë}{z}}_{{w}_{2,1}}& {\stackrel{Ë}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlAng}=\left[\begin{array}{c}\mathrm{ξ}\\ \mathrm{η}\\ \mathrm{ζ}\end{array}\right]=\left[\begin{array}{cccc}{\mathrm{ξ}}_{1,1}& {\mathrm{ξ}}_{1,2}& {\mathrm{ξ}}_{2,1}& {\mathrm{ξ}}_{2,2}\\ {\mathrm{η}}_{1,1}& {\mathrm{η}}_{1,2}& {\mathrm{η}}_{2,1}& {\mathrm{η}}_{2,2}\\ {\mathrm{ζ}}_{1,1}& {\mathrm{ζ}}_{1,2}& {\mathrm{ζ}}_{2,1}& {\mathrm{ζ}}_{2,2}\end{array}\right]
\mathrm{VehF}={F}_{v}=\left[\begin{array}{cccc}{F}_{v}{}_{{x}_{1,1}}& {F}_{v}{}_{{x}_{1,2}}& {F}_{v}{}_{{x}_{2,1}}& {F}_{v}{}_{{x}_{2,2}}\\ {F}_{v}{}_{{y}_{1,1}}& {F}_{v}{}_{{y}_{1,2}}& {F}_{v}{}_{{y}_{2,1}}& {F}_{v}{}_{{y}_{2,2}}\\ {F}_{v}{}_{{z}_{1,1}}& {F}_{v}{}_{{z}_{1,2}}& {F}_{v}{}_{{z}_{2,1}}& {F}_{v}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{VehM}={M}_{v}=\left[\begin{array}{cccc}{M}_{v{x}_{1,1}}& {M}_{v{x}_{1,2}}& {M}_{v{x}_{2,1}}& {M}_{v{x}_{2,2}}\\ {M}_{v{y}_{1,1}}& {M}_{v{y}_{1,2}}& {M}_{v{y}_{2,1}}& {M}_{v{y}_{2,2}}\\ {M}_{v{z}_{1,1}}& {M}_{v{z}_{1,2}}& {M}_{v{z}_{2,1}}& {M}_{v{z}_{2,2}}\end{array}\right]
\mathrm{WhlF}={F}_{w}=\left[\begin{array}{cccc}{F}_{w}{}_{{x}_{1,1}}& {F}_{w}{}_{{x}_{1,2}}& {F}_{w}{}_{{x}_{2,1}}& {F}_{w}{}_{{x}_{2,2}}\\ {F}_{w}{}_{{y}_{1,1}}& {F}_{w}{}_{{y}_{1,2}}& {F}_{w}{}_{{y}_{2,1}}& {F}_{w}{}_{{y}_{2,2}}\\ {F}_{w}{}_{{z}_{1,1}}& {F}_{w}{}_{{z}_{1,2}}& {F}_{w}{}_{{z}_{2,1}}& {F}_{w}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{WhlV}=\left[\begin{array}{c}{\stackrel{Ë}{x}}_{w}\\ {\stackrel{Ë}{y}}_{w}\\ {\stackrel{Ë}{z}}_{w}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{Ë}{x}}_{{w}_{1,1}}& {\stackrel{Ë}{x}}_{{w}_{1,2}}& {\stackrel{Ë}{x}}_{{w}_{2,1}}& {\stackrel{Ë}{x}}_{{w}_{2,2}}\\ {\stackrel{Ë}{y}}_{{w}_{{}_{1,1}}}& {\stackrel{Ë}{y}}_{{w}_{1,2}}& {\stackrel{Ë}{y}}_{{w}_{2,1}}& {\stackrel{Ë}{y}}_{{w}_{2,2}}\\ {\stackrel{Ë}{z}}_{{w}_{{}_{1,1}}}& {\stackrel{Ë}{z}}_{{w}_{1,2}}& {\stackrel{Ë}{z}}_{{w}_{2,1}}& {\stackrel{Ë}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlAng}=\left[\begin{array}{c}\mathrm{ξ}\\ \mathrm{η}\\ \mathrm{ζ}\end{array}\right]=\left[\begin{array}{cccc}{\mathrm{ξ}}_{1,1}& {\mathrm{ξ}}_{1,2}& {\mathrm{ξ}}_{2,1}& {\mathrm{ξ}}_{2,2}\\ {\mathrm{η}}_{1,1}& {\mathrm{η}}_{1,2}& {\mathrm{η}}_{2,1}& {\mathrm{η}}_{2,2}\\ {\mathrm{ζ}}_{1,1}& {\mathrm{ζ}}_{1,2}& {\mathrm{ζ}}_{2,1}& {\mathrm{ζ}}_{2,2}\end{array}\right]
\mathrm{StrgAng}={\mathrm{δ}}_{steer}=\left[\begin{array}{cc}{\mathrm{δ}}_{stee{r}_{1,1}}& {\mathrm{δ}}_{stee{r}_{1,2}}\end{array}\right]
Anti-sway arm neutral angle, θ0a, at nominal suspension height, in rad.
Anti-sway bar torsion spring constant, ka, in N·m/rad.
|
Data Mining of Spatio-Temporal Variability of Chlorophyll-a Concentrations in a Portion of the Western Atlantic with Low Performance Hardware
1Center for Natural Sciences and Technology (CCNT), Pará State University (UEPA), Belém, Brazil
3Institute of Technology (ITEC), Federal University of Pará (UFPA), Belém, Brazil
4Socio-Environmental Institute and Water Resources (ISARH), Federal Rural University of Amazonia (UFRA), Belém, Brazil
X=\mathrm{log}\frac{\mathrm{max}\left(Rs\left(443,489,510\right)\right)}{Rs\left(555\right)}
X=\mathrm{log}\frac{\mathrm{max}\left(Rs\left(443,489\right)\right)}{Rs\left(547\right)}
\left[Chlorophyll\right]={10}^{\left({a}_{0}+{a}_{1}X+{a}_{2}{X}^{2}+{a}_{3}{X}^{3}+{a}_{4}{X}^{4}\right)}
{a}_{0},{a}_{1},{a}_{2},{a}_{3},{a}_{4}
Skill=1-\frac{{{\displaystyle \sum }}^{\text{}}{\left|{X}_{Model}-{X}_{Obs}\right|}^{2}}{{{\displaystyle \sum }}^{\text{}}{\left(\left|{X}_{Model}-\overline{{X}_{Obs}}\right|+\left|{X}_{Obs}-\overline{{X}_{Obs}}\right|\right)}^{2}}
{r}^{2}=\frac{{\left({\sum }^{\text{}}{X}_{Obs}\cdot {X}_{Model}-{\sum }^{\text{}}{X}_{Obs}{\sum }^{\text{}}\frac{{X}_{Model}}{n}\right)}^{2}}{\left[{\sum }^{\text{}}{X}_{Obs}^{2}-\frac{{\left({\sum }^{\text{}}{X}_{Obs}\right)}^{2}}{n}\right]\left[{\sum }^{\text{}}{X}_{Model}^{2}-\frac{{\left({\sum }^{\text{}}{X}_{Model}\right)}^{2}}{n}\right]}
{X}_{Obs}
{X}_{Model}
\text{Anomaly}=\frac{C-\overline{{C}_{n}}}{\sigma }
\overline{C}
Di Paolo, I.F., Gouveia, N.A., Neto, L.C.F., Paes, E.T., Vijaykumar, N.L. and Santana, Á.L. (2019) Data Mining of Spatio-Temporal Variability of Chlorophyll-a Concentrations in a Portion of the Western Atlantic with Low Performance Hardware. Journal of Software Engineering and Applications, 12, 149-170. https://doi.org/10.4236/jsea.2019.125010
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11. Spyrakos, E., Vilas, L.G., Palenzuela, J.M.T. and Barton, E.D. (2011) Remote Sensing Chlorophyll a of Optically Complex Waters (Rias Baixas, NW Spain): Application of a Regionally Specific Chlorophyll a Algorithm for MERIS Full Resolution Data during an Upwelling Cycle. Remote Sensing of Environment, 115, 2471-2485. https://doi.org/10.1016/j.rse.2011.05.008
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On the Distribution of -Tuples of -Free Numbers
Ting Zhang, Huaning Liu, "On the Distribution of -Tuples of -Free Numbers", Journal of Numbers, vol. 2014, Article ID 537606, 6 pages, 2014. https://doi.org/10.1155/2014/537606
Ting Zhang1 and Huaning Liu 1
1Department of Mathematics, Northwest University, Xi’an, Shaanxi 710069, China
Academic Editor: Andrej Dujella
For k, being a fixed integer ≥2, a positive integer n is called k-free number if n is not divisible by the kth power of any integer >1. In this paper, we studied the distribution of r-tuples of k-free numbers and derived an asymptotic formula.
A positive integer is called square-free number if it is not divisible by a perfect square except . Let be the characteristic function of the sequence of square-free numbers. That is,
Mirsky [2] studied the frequency of pairs of square-free numbers with a given difference and proved the asymptotic formula:
Heath-Brown [3] investigated the number of consecutive square-free numbers not more than and obtained the following result:
Pillai [4] gave an asymptotic formula for
Tsang [5] proved the following.
Proposition 1. Let be distinct integers with and
For we have where is the number of distinct residue classes moduli represented by the numbers .
For , being a fixed integer ≥2, a positive integer is called -free number if is not divisible by the th power of any integer >1. Let be the characteristic function of the sequence of -free integers. Gegenbauer [6] proved that
Mirsky [7] showed that and in [2] Mirsky improved the error term to
Meng [8] further improved this result as follows:
Moreover, some recent results on pairs of -free numbers are given in [9, 10].
In this paper, we will study the distribution of -tuples of -free numbers by using the Buchstab-Rosser sieve and the methods in [5]. Our main result is the following.
Theorem 2. Let be distinct integers with and
Remark 3. Taking in Theorem 2, we immediately get Proposition 1.
The main tool in our argument is the Buchstab-Rosser sieve. Let be a sequence of positive numbers which lie between and and let be a real number. Define . For any square-free number with , , let , where is the Möbius function and when the following set of inequalities is satisfied. Otherwise, .
Similarly, let , where when the following set of inequalities is satisfied. Otherwise, .
From [5] we know that for any positive integer and for any positive integer whose smallest prime factor does not exceed .
In our case we take . Throughout this paper is an unspecified absolute constant.
To prove the theorem we need the following lemma.
Lemma 4. For any and any positive integers , , there exist absolute constants and such that
Proof. This lemma can be proved by using the methods of the lemma in [5] with a slight modification. For completeness we give a proof.
From the result of Ramanujan [11] and the observation we can obtain (19).
For any , the inequalities and imply where is an absolute constant. Now we define
If , we have . While if , we have .
Without loss of generality, we assume that . Then
This ends the proof of (20).
New we prove Theorem 2. Define , , for . Then
Let ; then we have
Define ; then any that is not -free has a divisor ≤. By (18) we have where
It is easy to show that the congruence has solutions modulo . Therefore the congruence has solutions modulo , by the Chinese remainder theorem. Then we have
By (16) we have where
On the other hand, the inequality implies
Now combining (32)–(36) we get
Furthermore, by (16), we have
From (13) of [5] we know that
And by (19) we get
Now from (30), (37), and (41) we immediately get
Choosing from (42) and (44), we immediately deduce that
Using the similar methods we have where
By (15) and (20) we get
Moreover, by using the argument leading to (41), we can have
This work is supported by the National Natural Science Foundation of China under Grant no. 11201370, the Natural Science Foundation of Shaanxi Province of China under Grant nos. 2013JM1017 and 2011JQ1010, and the Natural Science Foundation of the Education Department of Shaanxi Province of China under Grant no. 2013JK0558.
F. Pappalardi, “A survey on
k
-freeness,” in Number Theory, vol. 1 of Lecture Notes Series, pp. 71–88, Ramanujan Mathematical Society, 2005. View at: Google Scholar | MathSciNet
L. Mirsky, “On the frequency of pairs of square-free numbers with a given difference,” Bulletin of the American Mathematical Society, vol. 55, pp. 936–939, 1949. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. R. Heath-Brown, “The square sieve and consecutive square-free numbers,” Mathematische Annalen, vol. 266, no. 3, pp. 251–259, 1984. View at: Publisher Site | Google Scholar
S. S. Pillai, “On sets of square-free integers,” The Journal of the Indian Mathematical Society, vol. 2, pp. 116–118, 1936. View at: Google Scholar
K. M. Tsang, “The distribution of
r
-tuples of squarefree numbers,” Mathematika, vol. 32, no. 2, pp. 265–275, 1985. View at: Publisher Site | Google Scholar | MathSciNet
L. Gegenbauer, “Asymptotische Gesetze der Zahlentheorie,” Denkschriften der Akademie der Wissenschaften zu Wien, vol. 49, pp. 37–80, 1885. View at: Google Scholar
L. Mirsky, “Note on an asymptotic formula connected with r-free integers,” Quarterly Journal of Mathematics, vol. 18, no. 1, pp. 178–182, 1947. View at: Publisher Site | Google Scholar
Z. Z. Meng, “Some new results on k-free numbers,” Journal of Number Theory, vol. 121, no. 1, pp. 45–66, 2006. View at: Publisher Site | Google Scholar
R. Dietmann and O. Marmon, “The density of twins of k-free numbers,” submitted, http://arxiv.org/abs/1307.2481v2. View at: Google Scholar
T. Reuss, “Pairs of k-free numbers, consecutive square-full numbers,” submitted, http://arxiv.org/abs/1212.3150. View at: Google Scholar
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Copyright © 2014 Ting Zhang and Huaning Liu. 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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Elliot has a modern fish tank that is in the shape of an oblique prism, shown at right.
If the slant of the prism makes a
60°
angle with the flat surface on which the prism is placed, what is the volume of water the tank can hold? Assume that each base is a rectangle.
Multiply the area of the base times the height. You can find the height by using the special properties of a
30°
60°
90°
right triangle.
What is the volume of Elliot’s tank in gallons if one cubic foot of water equals
7.48
gallons? Show your steps and work.
945.7\;\text{in}^{3}\cdot\frac{(1\;\text{foot})^{3}}{(12\;\text{in})^{3}}= .547\;\text{ft}^{3}
.547\;\text{ft}^{3}\cdot\frac{7.48\; \text{gallons}}{(1\;\text{foot})^{3}}=...
≈ 4.1
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Investigation of Turbulent Flow and Heat Transfer in Periodic Wavy Channel of Internally Finned Tube With Blocked Core Tube | J. Heat Transfer | ASME Digital Collection
e-mail: wangqw@mail.xtu.edu.cn
Wang, Q., Lin, M., Zeng, M., and Tian, L. (April 23, 2008). "Investigation of Turbulent Flow and Heat Transfer in Periodic Wavy Channel of Internally Finned Tube With Blocked Core Tube." ASME. J. Heat Transfer. June 2008; 130(6): 061801. https://doi.org/10.1115/1.2891219
Three-dimensional complex turbulent flow and heat transfer of internally longitudinally finned tube with blocked core tube and streamwise wavy fin are numerically investigated. The numerical method is validated by comparing the calculated results with corresponding experimental data. The effects of both wave height and wave distance on heat transfer performance are examined. The range of wave height to hydraulic diameter ratio is from 0.61 to 2.45, and that of wave distance to hydraulic diameter ratio is from 3.06 to 14.69, while that of Reynolds number is from 904 to 4520. The computational results demonstrate that the Nusselt number and friction factor increase with the increase of the wave height, while they decrease with the increase of the wave distance. Furthermore, general correlations are proposed to describe the performance of the wavy configuration for
904⩽Re⩽4520
0.61⩽s∕de⩽2.45
6.12⩽l∕de⩽11.02
, with the mean deviations for heat transfer and friction factor correlations being
−2.8%
−1.9%
friction, heat transfer, pipe flow, turbulence, turbulent flow, periodic wavy channel, heat transfer, internally longitudinal finned tube
Friction, Heat transfer, Reynolds number, Turbulence, Waves, Flow (Dynamics), Numerical analysis
Heat Transfer Optimization in Internally Finned Tubes Under Laminar Flow Conditions
Effect of Viscous Dissipation on the Optimization of the Heat Transfer in Internally Finned Tubes
Heat Transfer for Laminar Flow in Internally Finned Pipes With Different Fin Heights and Uniform Wall Temperature
Experimental Study of Turbulent Flow Inside a Circular Tube With Longitudinal Interrupted Fins in the Streamwise Direction
Heat Transfer Enhancement in Latent Heat Transfer Thermal Energy Storage System by Using the Internally Finned Tube
Experimental Study on the Pressure Drop and Heat Transfer Characteristics of Tubes With Internal Wave-like Longitudinal Fins
Experimental Measurements of the Heat Transfer in an Internally Finned Tube
Correlation Equations for Friction Factors and Convective Coefficients in Tubes Containing Bundles of Internal Longitudinal Fins
An Analysis of Entropy Generation through a Circular Duct With Different Shaped Longitudinal Fins for Laminar Flow
Numerical Study of Turbulent Flow and Heat Transfer in Micro-Fin Tubes-Part1, Model Validation
Numerical Investigation of Turbulent Flow and Heat Transfer in Internally Finned Tubes
Effect of Blocked Core-Tube Diameter on Heat Transfer Performance of Internally Longitudinal Finned Tubes
Heat Transfer in a Conjugate Heat Exchanger With a Wavy Fin Surface
Numerical Analysis of Heat Transfer and Fluid Flow in a Three-Dimensional Wavy-Fin and Tube Heat Exchanger
An Experimental Investigation of Heat Transfer and Friction Losses of Interrupted and Wavy Fins for Fin-and-Tube Heat Exchangers
A Numerical Investigation of Louvered Fin-and-Tube Heat Exchangers Having Circular and Oval Tube Configurations
Doormal
Enhancement of the SIMPLE Method for Predicting Incompressible Fluid Flows
Geometry Effects on Turbulent Flow and Heat Transfer in Internally Finned Tubes
|
Exercise 10 | Chemical Equilibrium
\frac{{\left[{\mathrm{PCl}}_{3}\right]}_{\mathrm{eq}} {\left[{\mathrm{Cl}}_{2}\right]}_{\mathrm{eq}}}{{\left[{\mathrm{PCl}}_{5}\right]}_{\mathrm{eq}}}
[Cl2]0 = [PCl3]0 =
\frac{0.40}{1.5}
= 2.7 x 10-1 mol.L-1
From the stoichiometry of the chemical equation: 1 mole of Cl2 reacts with one mole of PCl3 to form one mole of PCl5:
[Cl2]eq = [Cl2]0 – x with x = change in the number of moles of Cl2 per liter
[PCl3]eq = [PCl3]0 – x = [Cl2]0 – x
[PCl5]eq = x
\frac{{\left[{\mathrm{PCl}}_{3}\right]}_{\mathrm{eq}} {\left[{\mathrm{Cl}}_{2}\right]}_{\mathrm{eq}}}{{\left[{\mathrm{PCl}}_{5}\right]}_{\mathrm{eq}}}
\frac{{\left({\left[{\mathrm{Cl}}_{2}\right]}_{0} - \mathrm{x}\right)}^{2}}{\mathrm{x}}
KC x = [Cl2]02 – 2 [Cl2]0 x + x2
⇒ x2 – (2[Cl2]0 + KC) x + [Cl2]02 = 0
⇒ x = 0.40 or x = 0.18
So [Cl2]eq = [Cl2]0 – x = 0.27 - 0.40 < 0 ⇒ this is impossible
or [Cl2]eq = [Cl2]0 – x = 0.27 - 0.18 = 0.08 mol.L-1
|
From meshes to deformation fields | Scalismo
In this tutorial, we show how the deformation fields that relate two meshes can be computed and visualized.
Modelling Shape Deformations (Video)
We will also load three meshes and visualize them in Scalismo-ui.
val meshFiles = new java.io.File("datasets/testFaces/").listFiles.take(3)
}).unzip // take the tuples apart, to get a sequence of meshes and one of meshViews
Representing meshes as deformations
In the following we show how we can represent a mesh as a reference mesh plus a deformation field. This is possible because the meshes are all in correspondence; I.e. they all have the same number of points and points with the same id in the meshes represent the same point/region in the mesh.
Let's say face_0, is the reference mesh:
val reference = meshes.head // face_0 is our reference
Now any mesh, which is in correspondence with this reference, can be represented as a deformation field. The deformation field is defined on this reference mesh; I.e. the points of the reference mesh are the domain on which the deformation field is defined.
The deformations can be computed by taking the difference between the corresponding point of the mesh and the reference:
val deformations : IndexedSeq[EuclideanVector[_3D]] =
reference.pointSet.pointIds.map {
id => meshes(1).pointSet.point(id) - reference.pointSet.point(id)
From these deformations, we can then create a DiscreteVectorField:
val deformationField: DiscreteField[_3D, TriangleMesh, EuclideanVector[_3D]] = DiscreteField3D(reference, deformations)
As for images, the deformation vector associated with a particular point id in a DiscreteVectorField can be retrieved via its point id:
deformationField(PointId(0))
We can visualize this deformation field in Scalismo-ui using the usual show command:
val deformationFieldView = ui.show(dsGroup, deformationField, "deformations")
We can see that the deformation vectors indeed point from the reference to face_1. To see the effect better we need to remove face2 from the ui, make the reference transparent
meshViews(2).remove()
meshViews(0).opacity = 0.3
Exercise: generate the rest of the deformation fields that represent the rest of the faces in the dataset and display them.
Deformation fields over continuous domains:
The deformation field that we computed above is discrete as it is defined only over the mesh points. Since the real-world objects that we model are continuous, and the discretization of our meshes is rather arbitrary, this is not ideal. In Scalismo we usually prefer to work with continuous domains. Whenever we have an object in Scalismo, which is defined on a discrete domain, we can obtain a continuous representation, by means of interpolation.
To turn our deformation field into a continuous deformation field, we need to define an Interpolator and call the interpolate method:
val continuousDeformationField : Field[_3D, EuclideanVector[_3D]] = deformationField.interpolate(interpolator)
The TriangleMeshInterpolator that we use here finds interpolates by finding for each point in the Euclidean space the closest point on the surface and uses the corresponding deformation as the deformation at the given point. The point on the surface is in turn obtained by barycentric interpolation of the corresponding vertex points. As a result of the interpolation, we obtain a deformation field over the entire real space, which can be evaluated at any 3D Point:
continuousDeformationField(Point3D(-100,-100,-100))
Remark: This approach is general: Any discrete object in Scalismo can be interpolated. If we don't know anything about the structure of the domain, we can use the NearestNeighborInterpolator. For most domain, however, more specialised interpolators are defined. To interpolate an image for example, we can use efficient linear or b-spline interpolation schemes.
The mean deformation and the mean mesh
Given a set of meshes, we are often interested to compute mesh that represents the mean shape. This is equivalent to computing the mean deformation
\overline{u}
, and to apply this deformation to them mean mesh.
To compute the mean deformation, we compute for each point in our mesh the sample mean of the deformations at this point in the deformation fields:
val nMeshes = meshes.length
val meanDeformations = reference.pointSet.pointIds.map( id => {
var meanDeformationForId = EuclideanVector3D(0, 0, 0)
val meanDeformations = meshes.foreach (mesh => { // loop through meshes
val deformationAtId = mesh.pointSet.point(id) - reference.pointSet.point(id)
meanDeformationForId += deformationAtId * (1.0 / nMeshes)
meanDeformationForId
val meanDeformationField = DiscreteField3D(reference, meanDeformations.toIndexedSeq)
We can now apply the deformation to every point of the reference mesh, to obtain the mean mesh. To do this, the easiest way is to first genenerate a transformation from the deformation field, which we can use to map every point of the reference to its mean:
val continuousMeanDeformationField = meanDeformationField.interpolate(TriangleMeshInterpolator3D())
val meanTransformation = Transformation((pt : Point[_3D]) => pt + continuousMeanDeformationField(pt))
To obtain the mean mesh, we simply apply this transformation to the reference mesh:
val meanMesh = reference.transform(meanTransformation)
Finally, we display the result:
ui.show(dsGroup, meanMesh, "mean mesh")
Representing meshes as deformations
Deformation fields over continuous domains:
The mean deformation and the mean mesh
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Trigonometry/Soh-Cah-Toa - Wikibooks, open books for an open world
Trigonometry/Soh-Cah-Toa
1 The Mnemonic
2 Trigonometric definitions
2.2 Exercises: (Draw a diagram!)
The MnemonicEdit
{\displaystyle Soh-Cah-Toa}
{\displaystyle \displaystyle {\begin{array}{ccc}\sin(A)&=&\displaystyle {\frac {\text{opposite side}}{\rm {hypotenuse}}}\\\\\cos(A)&=&\displaystyle {\frac {\text{adjacent side}}{\rm {hypotenuse}}}\\\\\tan(A)&=&\displaystyle {\frac {\text{opposite side}}{\text{adjacent side}}}\end{array}}}
Soh-Cah-Toa is a mnemonic for remembering how to use sines, cosines, and tangents to compute lengths in right angle triangles.
Trigonometric definitionsEdit
This triangle has sides a and b. The angle between them, C, is a right angle. The third side, c, is the hypotenuse. Side a is opposite angle A, and side b is adjacent to angle A.
Applying the definitions of the functions, we have:
{\displaystyle \sin(A)={\frac {a}{c}}}
or opposite side over hypotenuse
{\displaystyle \cos(A)={\frac {b}{c}}}
or adjacent side over hypotenuse
{\displaystyle \tan(A)={\frac {a}{b}}}
or opposite side over adjacent side
This can be memorized using the mnemonic 'SOH-CAH-TOA' (sin = opposite over hypothenuse, cosine = adjacent over hypotenuse, tangent = opposite over adjacent).
Video LinkEdit
These links from the Khan Academy are useful for this topic.
Basic Trigonometry Soh-Cah-Toa (sin and cosine)
Basic Trigonometry II Soh-Cah-Toa (starting with tan)
Exercises: (Draw a diagram!)Edit
Exercise: Find sin cos and tan
A right triangle has side a = 3, b = 4, and c = 5.
{\displaystyle \sin(A),\cos(A),\tan(A)}
Exercise: Find a side
A different right triangle has side
{\displaystyle c=6}
{\displaystyle \sin(A)=0.5}
Calculate side a.
"Cosine and Sine"
"Sine Squared plus Cosine Squared"
Retrieved from "https://en.wikibooks.org/w/index.php?title=Trigonometry/Soh-Cah-Toa&oldid=3839656"
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Shape modelling with Gaussian processes and kernels | Scalismo
In this tutorial we learn how to define our own Gaussian processes using analytically defined kernels. Further, we experiment with different kernels that are useful in shape modelling.
Covariance functions (Video)
Constructing kernels for shape modelling (Article)
Enlarging the flexibility of statistical shape models (Article)
In the following we will always visualize the effect of different Gaussian process models, by applying the deformations to a reference mesh. We therefore start by loading the mesh and visualizing it in a separate group.
val referenceMesh = MeshIO.readMesh(new java.io.File("datasets/lowResPaola.ply")).get
val modelGroup = ui.createGroup("gp-model")
val referenceView = ui.show(modelGroup, referenceMesh, "reference")
Modelling deformations using Gaussian processes:
A Gaussian Process is defined by two components: the mean function and the covariance function.
The mean:
As we are modelling deformation fields, the mean of the Gaussian process will, of course, itself be a deformation field. In terms of shape models, we can think of the mean function as the deformation field that deforms our reference mesh into the mean shape.
If the reference shape that we choose corresponds approximately to an average shape, and we do not have any further knowledge about our shape space, it is entirely reasonable to use a zero mean; I.e. a deformation field which applies to every point a zero deformation.
val zeroMean = Field(EuclideanSpace3D, (pt:Point[_3D]) => EuclideanVector3D(0,0,0))
The covariance function:
The covariance function, which is also referred to as a kernel, defines the properties that characterize likely deformations in our model. Formally, it is a symmetric, positive semi-definite function,
k: X \times X \to R^{ d \times d}
, which defines the covariance between the values at any pair of points
x, x'
of the domain. Since we are modelling deformation fields (I.e. vector-valued functions), the covariance function is matrix-valued.
To define a kernel in Scalismo, we need to implement the following methods of the abstract class MatrixValuedPDKernel, which is defined in Scalismo:
abstract class MatrixValuedPDKernel[D]() {
def outputDim: Int;
def domain: Domain[D];
def k(x: Point[D], y: Point[D]): DenseMatrix[Double];
Note: This class is already defined as part of Scalismo. Don't paste it into your code.
The field outputDim determines the dimensionality of the values we model. In our case, we model 3D vectors, and hence outputDimshould be 3. The domain indicates the set of points on which our kernel is defined. Most often, we set this to the entire Euclidean space RealSpace3D. Finally k denotes the covariance function.
The most often used kernel is the Gaussian kernel. Recall that the scalar-valued Gaussian kernel, is defined by the following formula:
k_g(x,x') = \exp^{-\frac{\left\lVert x-x'\right\rVert^2}{\sigma^2} }.
A corresponding matrix-valued kernel can be obtained by multiplying the value with an identity matrix (which implies, that we treat each space dimension as independent). In Scalismo, this is defined as follows:
case class MatrixValuedGaussianKernel3D(sigma2 : Double) extends MatrixValuedPDKernel[_3D]() {
override def outputDim: Int = 3
override def domain: Domain[_3D] = EuclideanSpace3D;
override def k(x: Point[_3D], y: Point[_3D]): DenseMatrix[Double] = {
DenseMatrix.eye[Double](outputDim) * Math.exp(- (x - y).norm2 / sigma2)
This constructions allows us to define any kernel. For the most commonly used ones, such as the Gaussian kernel, there is, however, an easier way in Scalismo. First, the scalar-valued Gaussian kernel is already implemented in Scalismo:
val scalarValuedGaussianKernel : PDKernel[_3D]= GaussianKernel3D(sigma = 100.0)
Further, the class DiagonalKernelallows us to turn any scalar-valued kernel into a matrix-valued kernel, by specifying for each dimension of the output-space a kernel and assuming them to be independent. To obtain the same kernel as defined above, we can write:
val matrixValuedGaussianKernel = DiagonalKernel3D(scalarValuedGaussianKernel, scalarValuedGaussianKernel, scalarValuedGaussianKernel)
In this case, since we are using the same kernel in every space dimension, we can write this even more succinct as:
DiagonalKernel3D(scalarValuedGaussianKernel, 3)
Building the GP :
Now that we have our mean and covariance functions, we can build a Gaussian process as follows:
val gp = GaussianProcess3D[EuclideanVector[_3D]](zeroMean, matrixValuedGaussianKernel)
We can now sample deformations from our Gaussian process at any desired set of points. Below we choose the points to be those of the reference mesh:
val sampleGroup = ui.createGroup("samples")
val sample = gp.sampleAtPoints(referenceMesh)
ui.show(sampleGroup, sample, "gaussianKernelGP_sample")
The result is an instance from the Gaussian Process evaluated at the points we indicated; in this case on the points of the reference mesh.
We can visualize its effect by interpolating the deformation field, which we then use to deform the reference mesh:
val interpolatedSample = sample.interpolate(TriangleMeshInterpolator3D())
val deformedMesh = referenceMesh.transform((p : Point[_3D]) => p + interpolatedSample(p))
ui.show(sampleGroup, deformedMesh, "deformed mesh")
Low-rank approximation
Whenever we create a sample using the sampleAtPoints method of the Gaussian process, internally a matrix of dimensionality
nd \times nd
n
denotes the number of points and
the dimensionality of the output space, is created. Hence if we want to sample from many points we quickly run out of memory.
We can get around this problem by computing a low-rank approximation of the Gaussian process. To obtain such a representation in Scalismo, we can use the method `approximateGPCholesky of the LowRankGaussianProcess object.
This call computes a finite-rank approximation of the Gaussian Process using a Pivoted Cholesky approximation. The procedure automatically chooses the rank (I.e. the number of basis functions of the Gaussian process), such that the given relative error is achieved. (The error is measures in terms of the variance of the Gaussian process, approximated on the points of the reference Mesh). Using this low rank Gaussian process, we can now directly sample continuous deformation fields:
val defField : Field[_3D, EuclideanVector[_3D]]= lowRankGP.sample()
These in turn, can be used to warp a reference mesh, as discussed above:
referenceMesh.transform((p : Point[_3D]) => p + defField(p))
More conveniently, we can visualize the sampled meshes by building again a Point Distribution Model:
val pdm = PointDistributionModel3D(referenceMesh, lowRankGP)
This model can be visualized directly in ScalismoUI.
val pdmView = ui.show(modelGroup, pdm, "group")
Building more interesting kernels
In the following we show a few more examples of kernels, which are interesting for shape modelling.
Kernels from Statistical Shape Models
As discussed previously, a Statistical Shape Model (SSM) in Scalismo is a discrete Gaussian process. We have seen how to interpolate it to obtain a continuously defined Gaussian Process. As any Gaussian process is completely defined by its mean and covariance function, it follows that this is also true for the GP in our statistical shape model.
This allows us to use this sample covariance kernel in combination with other kernels. For example, we often want to slightly enlarge the flexibility of our statistical models. In the following, we show how this can be achieved.
In a first step, we get the Gaussian process from the model an interpolate it.
val pcaModel = StatisticalModelIO.readStatisticalTriangleMeshModel3D(new java.io.File("datasets/lowresModel.h5")).get
val gpSSM = pcaModel.gp.interpolate(TriangleMeshInterpolator3D())
We can then access its covariance function, which is a kernel:
val covSSM : MatrixValuedPDKernel[_3D] = gpSSM.cov
In the next step, we model the additional variance using a Gaussian kernel and add it to the sample covariance kernel.
val augmentedCov = covSSM + DiagonalKernel(GaussianKernel[_3D](100.0), 3)
Finally, we build the Gaussian process with the new kernel.
val augmentedGP = GaussianProcess(gpSSM.mean, augmentedCov)
From here on, we follow the steps outlined above to obtain the augmented SSM.
val lowRankAugmentedGP = LowRankGaussianProcess.approximateGPCholesky(
augmentedGP,
val augmentedSSM = PointDistributionModel3D(pcaModel.reference, lowRankAugmentedGP)
Changepoint kernel:
Another very useful kernel is the changepoint kernel. A changepoint kernel is a combination of different kernels, where each kernel is active only in a certain region of the space.
Here we show how we can define a kernel, which has different behavior in two different regions.
case class ChangePointKernel(kernel1 : MatrixValuedPDKernel[_3D], kernel2 : MatrixValuedPDKernel[_3D])
extends MatrixValuedPDKernel[_3D]() {
override def domain = EuclideanSpace[_3D]
val outputDim = 3
def s(p: Point[_3D]) = 1.0 / (1.0 + math.exp(-p(0)))
def k(x: Point[_3D], y: Point[_3D]) = {
val sx = s(x)
val sy = s(y)
kernel1(x,y) * sx * sy + kernel2(x,y) * (1-sx) * (1-sy)
Let's visualize its effect with two different Gaussian Kernels
val gk1 = DiagonalKernel3D(GaussianKernel3D(100.0), 3)
val gk2 = DiagonalKernel3D(GaussianKernel3D(10.0), 3)
val changePointKernel = ChangePointKernel(gk1, gk2)
val gpCP = GaussianProcess3D(zeroMean, changePointKernel)
val sampleCP = gpCP.sampleAtPoints(referenceMesh)
ui.show(sampleGroup, sampleCP, "ChangePointKernelGP_sample")
As you can see each kernel is now active only on one half of the face.
Symmetrizing a kernel
Quite often, the shapes that we aim to model exhibit a symmetry. This is particularly valid in the case of faces. Therefore when modelling over such shapes, one would want deformation fields that yield symmetric shapes.
Once we obtained a kernel yielding the type of deformations we desire, it is possible to symmetrize the resulting deformation fields by applying the formula below:
where xm is the symmetric point to x around the YZ plane.
The resulting kernel will preserve the same smoothness properties of the deformation fields while adding the symmetry around the YZ plane.
Let's turn it into code:
case class xMirroredKernel(kernel : PDKernel[_3D]) extends PDKernel[_3D] {
override def domain = kernel.domain
override def k(x: Point[_3D], y: Point[_3D]) = kernel(Point(x(0) * -1.0 ,x(1), x(2)), y)
def symmetrizeKernel(kernel : PDKernel[_3D]) : MatrixValuedPDKernel[_3D] = {
val symmetrizedGaussian = symmetrizeKernel(GaussianKernel[_3D](100))
val gpSym = GaussianProcess3D(zeroMean, symmetrizedGaussian)
val sampleGpSym = gpSym.sampleAtPoints(referenceMesh)
ui.show(sampleGroup, sampleGpSym, "ChangePointKernelGP_sample")
Modelling deformations using Gaussian processes:
Building more interesting kernels
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Empirical Analysis of Gross Domestic Product and Coal Import Based on VAR Model
Empirical Analysis of Gross Domestic Product and Coal Import Based on VAR Model
Received: July 3, 2019 ; Accepted: July 28, 2019; Published: July 31, 2019
The speed of China’s economic development is gradually accelerating, and the demand for energy is also constantly increasing, especially the demand for coal. In order to reveal whether the coal imports have an impact on China’s economic development, this paper constructs the VAR(6) model by selecting the quarterly data of coal imports (CIV) and gross domestic product (GDP) from 2002 to 2017, performing ADF (Augmented Dickey-Fuller) stationarity test and Johansen cointegration test. It shows that there is a long-term stable equilibrium relationship between coal imports and GDP. Then the impulse response function is used to obtain the relationship between coal imports and GDP. It is found that the impact of coal imports on GDP is greater than the impact of GDP on coal imports.
Coal Imports, Gross Domestic Product, VAR Model, Impulse Response Function
China’s economy has developed rapidly, and the total amount of GDP has increased year by year. China has become the largest developing country. At the same time of economic development, the problem of energy consumption has become increasingly prominent, especially the amount of coal used. China is a big country in coal use, especially using coal for power generation. Therefore, most of the coal used in China at this stage is imported, so it is particularly important to explore the relationship between coal imports and GDP. Chang Junfeng [1] used the relevant regression analysis method to predict and discuss GDP and coal consumption (price).
Ma Yuanxin [2] dynamically described the long-term equilibrium relationship between coal consumption and economic growth in Shanxi Province through cointegration analysis and Granger causality test. Wu Yongping [3] also described the relationship between coal consumption and economic growth in the world’s major coal-consuming countries through cointegration analysis and Granger causality test. Chen Weidong [4] mainly adopts quantitative analysis to establish a VAR model of coal price and GDP, and dynamically analyzes the long-term impact of coal price on economic growth. Zhou Aiqian [5] analyzed the long-term impact of China’s coal price on economic growth. Xie Changfeng [6] used the panel data model and VAR model to analyze the energy consumption and economic growth in Jiangsu, Zhejiang and Shanghai. In the past, scholars mainly analyzed the relationship between coal prices and economic growth. This paper establishes a VAR model for coal imports and economic growth(GDP), and determines the long-term stable equilibrium relationship between the two through Johansen cointegration test. The impulse response function is used to further explain the dynamics relationship between coal imports and GDP.
The VAR model, also known as the vector auto-regressive model, is commonly used to predict multivariate time series systems and to describe the dynamic effects of random perturbation terms on variable systems. The general form of the VAR(p) model is as follows:
{y}_{t}={A}_{1}{y}_{t-1}+\cdots +{A}_{p}{y}_{t-p}+{B}_{1}{x}_{t}+\cdots +{B}_{r}{x}_{t-r}+{\epsilon }_{t}
{y}_{t}
is a m-dimensional endogenous variable vector,
{x}_{t}
is a d-dimensional endogenous variable vector.
{A}_{1},\cdots ,{A}_{p}
{B}_{1},\cdots ,{B}_{r}
are the parameter matrices to be estimated, p is the lag period of the endogenous variable, r is the lag period of the exogenous variable.
{\epsilon }_{t}
is a random disturbance term, which can be related to the same period, but not autocorrelation.
The Johansen cointegration test includes a trace test and a maximum eigen value test.
The assumption of the trace test is:
H0: at most r cointergration relations H1: m cointergration relations (full rank)
L{R}_{tr}\left(r|m\right)=-T\underset{i=r+1}{\overset{m}{\sum }}\mathrm{ln}\left(1-{\lambda }_{i}\right)
{\lambda }_{i}
is the eigenvalue of row i of the size row, T is the total number of observation periods.
The assumption of the maximum eigenvalue test is:
H0r: there are r cointegration relations H1r: at least r + 1 cointegration relations
\begin{array}{c}L{R}_{\mathrm{max}}\left(r|r+1\right)=-T\mathrm{ln}\left(1-{\lambda }_{r+1}\right)\\ =L{R}_{tr}\left(r|m\right)-LR\left(r+1|m\right),\text{\hspace{0.17em}}r=0,1,\cdots ,m-1\end{array}
According to the VAM(p) form of the VAR(∞) model, the generalized impulse response function of the VAR model is represented by a matrix as:
{\Psi }_{j}^{\left(q\right)}={\sigma }_{jj}^{-1/2}{\Theta }_{q}{\Sigma }_{j},\text{\hspace{0.17em}}\text{\hspace{0.17em}}q=0
2.4. LR Test Statistic
Likelihood ratio test is divided into two models: unconstrained and constrained. The likelihood ratio statistic refers to the difference between the maximum likelihood of the unconstrained model and the constraint model, that is:
LR=2\left({l}_{u}-{l}_{r}\right)~{\chi }^{2}\left(k\right)
{l}_{u}
and represent the maximum likelihood estimates of the unconstrained model and the constrained model for an observed sample condition. k is a positive integer, indicating the degree of freedom of the chi-square distribution, equal to the number of constrains.
2.5. Final Prediction Error
The final prediction error takes the minimum value of p in the formula as the best order of the VAR model.
\text{FPE}\left(p\right)={\stackrel{^}{\sigma }}_{p}^{2}\frac{\left(n+k\right)}{\left(n-k\right)}
{\stackrel{^}{\sigma }}_{p}^{\text{2}}
is the variance estimate of the residual at the time of the lag p period, n is the sample size, and k is the number of parameters to be estimated.
2.6. Information Guidelines
In order to find a balance between the lag period and the degree of freedom, the order is generally determined according to the criteria for the minimum value of AIC (Akaike info criterion), SC (Schwarz criterion) and HQ (Hannan-Quinn criterion) information. The formula is as follows:
\text{AIC}=-2l/n+2k/n
\text{SC}=-2l/n+k\mathrm{ln}n/n
\text{HQ}=-2l/n+2k\mathrm{ln}\left(\mathrm{ln}n\right)/n
k=m\left(rd+pm\right)
is the number estimated parameters, n is the number of observations.
This paper selects the most representative economic indicator GDP (gross domestic product) to represent the status quo of economic development, and uses the index of coal imports to measures the dynamic relationship with GDP. Therefore, this paper takes the GDP and coal import data from the first quarter of 2002 to the fourth quarter of 2017 as the sample time series, and records them as GDP and CIV. Figure 1 shows the trend of GDP and CIV. The data comes from the China Statistical Yearbook and the Energy Comprehensive Database. In order to avoid the influence of the heteroscedasticity of the time series data on the empirical analysis, the original sequence is logarithmized, and the new sequence is recorded as LnGDP and LnCIV.
This paper uses Eviews 7.2 [7] [8] to perform corresponding data analysis.
It can be seen from Figure 1 that the sequence GDP and CIV have obvious trends and are not stable.
The establishment of VAR model theoretically requires time series data to be stable. In this paper, the ADF (Augmented Dickey-Fuller) unit root test is used to test the stationarity of the original sequence and the logarithmized new sequence. The lag order P is selected by SC criterion. The test results are shown in Table 1. From the results of Table 1, it can be seen that the sequence GDP, CIV and the logarithmized new sequence LnGDP and LnCIV did not pass the stationarity test.
In order to make the sequence stable, the logarithmized sequence is first-order differential, and the differenced sequence is recorded as DLnGDP, DLnCIV, and then the ADF unit root test is performed on the two sequences. The test results are shown in Table 2.
Table 1. ADF test of GDP, CIV, LnGDP and LnCIV sequences.
Note: The three items in the test type represent the constant term, the time trend term, and the lag order in the stationarity test, respectively.
Table 2. ADF test of DLnGDP and DLnCIV sequences.
Figure 1. Time series diagram of GDP and CIV.
It can be seen from the results of Table 2 that the sequence after the first-order difference is stable, and both the DLnGDP and the DLnCIV sequences are first-order single-order sequences.
3.3. Recognition of VAR Model
The determination of the lag order is a very important issue when building a VAR model. In general, it is desirable that the lag order is large enough to effectively and completely reflect the dynamic characteristics of the model. However, the larger the lag order becomes, the more the estimated parameters in the model will be. At the same time, it can reduce the freedom of the model. Therefore, we should consider the problem of lag order and degree of freedom at the same time, and find a state of equilibrium. Commonly used methods are LR (likelihood ratio) test, final prediction error (FPE), AIC information criterion, SC information criterion, HQ information criterion, and the optimal lag order is determined by the above method, as shown in Table 3.
It can be seen from the results in Table 3 that among the LR, FPE, AIC, SC and HQ values of the lag order from 0 to 7 orders, there are four criteria that select the lag 6th order, so the order of the model is determined to be 6, and the VAR is established. The estimated results of the model are as follows:
Table 3. Judgment of lag order of VAR model.
Note: “*” indicates the optimal order of choice.
3.4. Stability Test of VAR Model
After estimating the parameters of the VAR model, it is necessary to perform an adaptive test on the model to ensure that the model meets the expected results. The most commonly used method is the reciprocal test of the root of the AR (auto-regressive) characteristic polynomial. The results are shown in Figure 2 and Table 4.
It can be seen from the results of Figure 2 and Table 4 that the reciprocal of all characteristic polynomial roots is less than 1, and within the unit circle, it indicates that the VAR(6) model is stable. The X and Y axes of Figure 2 represent the coefficients of the eigenvalues, respectively.
Since the logarithmic sequence LnGDP and LnCIV are not stable, the VAR model cannot be directly established. However, in the stationarity test of 3.1, the two sequences are known to be first-order single-sequences, so it can be tested by Johansen cointegration. To determine if there is a long-term stable equilibrium relationship between variables, it is assumed that there is cointegration relationship between variables, indicating that the established VAR(6) model is reasonable. The lag of the cointegration test is 5, and the lag order of the VAR model is 6. The Johansen cointegration test is performed on the sequences LnGDP and LnCIV, and the results are shown in Table 5 and Table 6.
It can be seen from the results of Table 5 and Table 6 that Johansen cointegration rank test and maximum eigenvalue test show that there is a cointegration relationship between variable GDP and CIV, and there is a cointegration vector with long-term stable equilibrium relationship. Therefore, the established VAR(6) model is reasonable.
Figure 2. VAR model adaptability test.
Table 4. AR root of VAR model.
Table 5. Johansen cointegration rank test results.
Table 6. Johansen cointegration maximum eigenvalue test results.
In the VAR model, the impulse response function reflects the impact of the model on the dynamic impact of the entire system. This paper changes the two variables of GDP and coal imports to observe the impact on itself and other variables. When giving a one-unit standard deviation of GDP and coal imports, the impulse response function is shown in Figure 3.
As can be seen from the results of the impulse response function of Figure 3, the impact of coal imports on itself reached a maximum of 0.21 in the first phase, then fell to 0.02 in the second phase, and then began to stabilize, reaching a minimum in the fifth phase. Coal imports will interfere with themselves in the
Figure 3. VAR(6) model impulse response diagram.
short term, and in the long run, they cannot ignore their own influence. The impact of coal imports on GDP reached a minimum in the second period, while the adjacent third period was also relatively small. Then it rises slowly, reaches a maximum of 0.26 in the sixth period, and then remains stable. Therefore, regardless of long-term or short-term, the impact of coal imports on GDP is not significant. The impact of GDP on itself reached its maximum in the first period, and then began to decline rapidly, but in the fifth and ninth phases, it quickly rebounded to a larger value, reaching a minimum in the eighth period. On the whole, GDP has a very strong impact on itself, and the change is very large. The impact of GDP on coal imports was relatively stable in the first seven periods, reaching a maximum in the seventh period, followed by a small decline, reaching a minimum in the ninth period. In the long run, GDP has a positive effect on coal imports. Although there are small fluctuations, the overall situation is positive.
In summary, the mutual influence between GDP and coal imports is relatively positive in the long run.
This paper mainly studies the relationship between coal imports and GDP and draws the following conclusions: In the Johansen cointegration test, there is a cointegration relationship between gross domestic product (GDP) and coal imports, and it has a long-term stable and balanced development trend. Therefore,
the VAR(6) model is established and tested by the stationarity test. It can be seen from the reference impulse response image that the increase in coal imports has a positive effect on GDP. In other words, the total value of GDP can be increased by importing coal. In turn, the increase in GDP has also weakly led to an increase in coal imports. It can also be seen that the impact of GDP on coal imports or the impact of coal imports on GDP has a certain time lag effect. As time goes by, this effect will gradually weaken. Through the empirical analysis between coal import volume and GDP, the following suggestions can be made: While increasing coal imports, it is also necessary to increase the utilization rate of coal, which can accelerate GDP growth. As the impact of imported coal on GDP becomes smaller as time goes by, it is necessary to increase efforts to develop new energy sources such as solar energy, nuclear energy and wind energy, transform industrial structure, improve the efficiency of economic development, and make the level of economic development steadily.
Shen, S.C. and Feng, C. (2019) Empirical Analysis of Gross Domestic Product and Coal Import Based on VAR Model. Advances in Pure Mathematics, 9, 619-628. https://doi.org/10.4236/apm.2019.97031
1. Chang, J.F. (2015) Prediction and Discussion of Coal Consumption Based on GDP Growth. Journal of Gansu Sciences, No. 27, 104-107.
2. Ma, Y.X. (2009) Study on the Impact of Energy Consumption and Composition of Shanxi Province on Economic Growth. Northwest University Press, Xi’an.
3. Wu, Y.H., Wen, G.F. and Song, H.L. (2008) Analysis of the Relationship between the World’s Major Coal Consuming Countries and Their National Economic Growth GDP. China Mining Industry, No. 17, 21-25.
4. Chen, W.D. and Dai, D.X. (2014) Analysis of Dynamic Reaction of China’s Coal Price and GDP Based on Eviews Software. Electronic Design Engineering, No. 22, 18-20+24.
5. Zhou, A.Q. (2009) The Impact of Coal Price on China’s Economic Growth. Nanjing Aerospace University Press, Nanjing.
6. Xie, C.F. (2014) Research on the Relationship between Energy Consumption and Economic Growth in Jiangsu, Zhejiang and Shanghai—An Empirical Analysis Based on Panel Data. Nanjing Aerospace University Press, Nanjing.
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Linear Analysis of Tuning Fork - MATLAB & Simulink - MathWorks Nordic
Tuning Fork Structural Model
This example shows how to linearize the structural model of a tuning fork and calculate the time and frequency response when it is subjected to an impulse on one of the tines. This load results in transverse vibration of the tines and axial vibration of the end handle at same frequency. For more details about the structural model, see Structural Dynamics of Tuning Fork (Partial Differential Equation Toolbox).
This example requires Partial Differential Equation Toolbox™ software.
Using Partial Differential Equation Toolbox, create a structural model of the tuning fork.
Use createpde (Partial Differential Equation Toolbox) to construct a structural model.
model = createpde('structural','transient-solid');
Specify the Young's modulus, Poisson's ratio, and mass density to model linear elastic material behavior. Specify all physical properties in consistent units.
Identify faces for applying boundary constraints and loads by plotting the geometry with face labels.
The first mode of vibration of the tines is about 2926 rad/s. You can determine this value by modal analysis (see Structural Dynamics of Tuning Fork (Partial Differential Equation Toolbox)) or from the Bode plot in the Linear Analysis section of this example.
Calculate the corresponding period of oscillations.
T = 2*pi/2926;
Add boundary conditions to prevent rigid body motion when applying an impulse to the tine. Typically, you hold a tuning fork by hand or mount it on a table. A simplified approximation to this boundary condition is fixing a region near the intersection of tines and the handle (faces 21 and 22).
To model an impulse load on the tine, apply a pressure load for 2% of the fundamental period of oscillation T using structuralBoundaryLoad (Partial Differential Equation Toolbox). By using this very short pressure pulse, you ensure that only the fundamental mode of the tuning fork is excited. Specify the label Pressure to use this load as a linearization input.
Te = 0.02*T;
structuralBoundaryLoad(model,'Face',11,'Pressure',5e6,'EndTime',Te,'Label','Pressure');
Set initial conditions for the model using structuralIC (Partial Differential Equation Toolbox).
structuralIC(model,'Displacement',[0;0;0],'Velocity',[0;0;0]);
For this tuning fork model, you want to obtain a linear model from the pressure load on the tine to the y-displacement of the tine tip (face 12) and x-displacement of the end handle (face 6).
To do so, first specify the inputs and the outputs of the linearized model in terms of the structural model. Here, the input is the pressure specified with structuralBoundaryLoad (Partial Differential Equation Toolbox) and the outputs are the y and x degrees of freedom of faces 12 and 6, respectively.
linearizeInput(model,'Pressure');
linearizeOutput(model,'Face',12,'Component','y');
linearizeOutput(model,'Face',6,'Component','x');
In the linearized model, use sys.OutputGroup to locate the outputs associated with each face.
Select one node from each output group to create a model with one input and two outputs.
sys = sys([1,14],:)
sys.OutputName = {'y tip','x base'};
\mathit{M}\stackrel{¨}{\mathit{q}}+\mathit{K}\text{\hspace{0.17em}}\mathit{q}=\mathit{B}×\mathrm{Pressure}
\mathit{y}=\mathit{F}\text{\hspace{0.17em}}\mathit{q}
Use bode to compute the frequency response of this model.
title('Frequency Response from Pressure to Y Tip and X Base')
The plot clearly shows the tine and the end handle vibrate at the same frequency. The first mode is at approximately 2926 rad/s and second resonance is at a frequency approximately six times higher.
Next use lsim to obtain the impulse response of the linearized tuning fork model for 20 periods of the fundamental mode. To limit error due to linear interpolation of pressure between samples, use a step size of Te/10. The pressure is applied for the time interval [0 Te].
Tf = ncycle*T;
h = Te/10;
t = 0:h:Tf;
u(t<=Te) = 5e6;
Plot the transverse displacement at the tine tip and axial displacement of the end handle.
xlim([0,Tf])
legend('Linearized model')
title('Axial Displacement at End of Handle')
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Drag-to-Solve - Maple Help
Home : Support : Online Help : Create Maple Worksheets : Manipulate Expressions : Clickable Math : Drag-to-Solve
Clickable Math: Drag-to-SolveTM
This document provides information and examples on the Drag-to-Solve feature.
Example: Solving a linear equation using Drag-to-Solve:
The Drag-to-Solve feature enables you to drag individual terms in an output equation from one side of the equal sign to the other. This action invokes the Smart Popup feature which displays your available options.
Using the Drag-to-Solve feature allows you to direct what will happen interactively.
Using the Drag-to-Solve feature:
The general procedure for using Drag-to-Solve is as follows:
Click on your Maple input, then press Enter to generate output for the equation you want to solve.
Select a term in the output equation and drag it to the other side of the equality. A Smart Popup window is displayed, previewing the results of your manipulation.
Click on the Smart Popup window to confirm the manipulation.
Repeat steps 2 and 3 until you have your desired result.
Note: Currently, the functionality for this feature is restricted to arithmetic manipulations.
Input the following expression, then press Enter to produce the required output:
5\cdot x-7=3\cdot x+2
3x
from the right side of the equal sign to the left side. A Smart Popup window is displayed, previewing the results of this manipulation.
Drag and drop the factor of
2
, in front of
x
, to the right side of the equal sign.
The equation manipulator provides another tool for solving equations. See Using the Equation Manipulator.
The context panel provides additional options. The list of operations changes as the selection changes. See Context-Sensitive Operations and Clickable Math.
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Quantum Theory and Atomic Structure | General Chemistry 1
Shell vs. subshell
Electron shell: a group of atomic orbitals with the same principal quantum number n (n = 1 ⇒ first shell)
Electron subshell: a group of atomic orbitals with the same principal quantum number n and azimuthal quantum number l
Number of subshells in the first 2 electron shells:
First shell: n = 1 ⇒ l = 0 ⇒ 1 subshell (s subshell)
Second shell: n = 2 ⇒ l = 0, 1 ⇒ 2 subshells (s subshell and p subshell)
The energy states of atoms with 2 or more electrons depend on the values of both n and l (electrons-nucleus + electron-electron interactions). The order of orbital energies is: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s ... You can easily remember this order by using the mnemonic on the right:
The arrangement of electrons in the atomic orbitals of an atom. An atomic orbital can hold a maximum of 2 electrons. The maximum number of electrons in each subshell is as follows:
How to write the electron configuration:
Electrons reside in the lowest energy orbitals available
Each orbital can accommodate a maximum of 2 electrons
The orbitals are filled in the order of the orbital energy
Electron configuration of oxygen and iron:
Oxygen (Z=8) ⇒ 8 protons + neutral ⇒ 8 electrons: 1s2 2s2 2p4
Iron (Z=26) ⇒ 26 protons + neutral ⇒ 26 electrons: 1s2 2s2 2p6 3s2 3p6 4s2 3d6
Electronic Structure Principles
2 electrons in an atom cannot have the same set of 4 quantum numbers (n, l, ml, ms)
First shell: n = 1 ⇒ only 2 possible combinations: (1, 0, 0, +1/2) and (1, 0, 0, -1/2)
This explains why there is a maximum of 2 electrons in the 1s orbital
Electrons fill subshells of the lowest available energy before filling subshells of higher energy. This principle, coming from Aufbau in Geman which means “building-up”, allows to build up the periodic table by successively adding one proton to the nucleus and one electron to the appropriate atomic orbital
Hund's rule:
Orbitals of equal energy, called degenerate orbitals, must all contain one electron with the same spin before they can contain 2 electrons
Electronic structure of carbon (Z=6):
Paramagnetic substance: a substance with unpaired electrons (weakly attracted by a magnetic field)
Diamagnetic substance: a substance without unpaired electrons (not attracted by a magnetic field)
Excited state: a state with higher energy than the ground state. Electrons are promoted from the ground state to an excited state by electromagnetic radiation of intensity hν. The first excited state corresponds to the promotion of the highest energy electron from the ground state to the next available orbital
First excited state of the lithium atom:
Li (1s2 2s1) + hν → Li* (1s2 2p1)
Core vs. Valence Electrons
Core electrons: electrons of the inner energy levels. They do not participate in chemical bonding and form the atomic core with the nucleus
Valence electrons: electrons of the outermost occupied shell of an atom. They are furthest from the positive charge of the nucleus and therefore tend to react more easily than the core electrons
Number of valence electrons in an oxygen atom:
Electron configuration of oxygen: 1s2 2s2 2p4
Outermost occupied shell: n = 2. There are 2 e- in 2s, 4 e- in 2p ⇒ 6 valence electrons
For the electron configuration, we can use an abbreviated form by replacing the electronic configuration of the core electrons by [previous nearest noble gas]
Iron (Z = 26): 1s2 2s2 2p6 3s2 3p6 4s2 3d6 = [Ar] 4s2 3d6
The uncertainty principle states that it is impossible to measure simultaneously the position x and the momentum p = mv of a particle. The more accurately we know one of these values, the less accurately we know the other
What is the formula of the uncertainty principle?
(Δx)(Δp)
\ge \frac{\mathrm{h}}{4\mathrm{\pi }}
Δx = uncertainty in measuring the position
Δp = uncertainty in measuring the momentum
h = Planck constant = 6.63 x 10-34 kg.m2.s-1
The Schrödinger equation is the central equation of quantum theory consistent with both the wave nature of particles and the Heisenberg uncertainty principle. This equation provides wave functions Ψ of the electron’s position associated with allowed energies
An atomic orbital is the wave function Ψ of an electron in an atom. An atomic orbital has a characteristic energy as well as a characteristic electron density distribution
What is electron density and how is it related to wave function?
The electron density is the relative probability of finding an electron at a particular point in space. For one-electron systems, the electron density of an electron in a certain orbital is described by the square of the wave function Ψ2 associated with that orbital
An atomic orbital is defined by 3 quantum numbers: the principal quantum number (n), the azimuthal quantum number (l), and the magnetic quantum number (ml). The electrons that occupy an atomic orbital are defined by their spin quantum number (ms)
What do the quantum numbers describe?
The principal quantum number (n) describes the size of the orbital. The larger n is, the larger the orbital is
The azimuthal quantum number (l) describes the shape of the atomic orbital
The magnetic quantum number (ml) describes the orientation of the orbital in space
The spin quantum number (ms) describes the spin of an electron in an atomic orbital, either -
\frac{1}{2}
or +
What are the possible values of the quantum numbers?
Principal quantum number: n = 1, 2, 3, ...
Azimuthal quantum number: l = 0, 1, 2, … n - 1
Magnetic quantum number: ml = -l, -l + 1, … , -1, 0, 1, … , l - 1, l
Spin quantum number: ms = -
\frac{1}{2}
Which quantum number defines a subshell?
The azimuthal quantum number (l) describes the subshells and thus the shape of the corresponding atomic orbital:
l = 0 ⇒ s orbital (spherical symmetry)
l = 1 ⇒ p orbitals (cylindrical symmetry around its long axis)
l = 2 ⇒ d orbitals
l = 3 ⇒ f orbitals
What is the difference between shell and subshell?
The electron shell is a group of atomic orbitals with the same principal quantum number n (n = 1 ⇒ first shell) while the electron subshell is a group of atomic orbitals with the same principal quantum number n and the same azimuthal quantum number l
How to write the electron configuration?
When we assign electrons to orbitals, we must follow a set of three rules: the Pauli exclusion principle, the Aufbau principle, and the Hund's rule
The Pauli exclusion principle states that 2 electrons in an atom cannot have the same set of 4 quantum numbers (n, l, ml, ms)
The Aufbau principle states that electrons fill subshells of the lowest available energy before filling subshells of higher energy. This principle, coming from Aufbau in Geman which means “building-up”, allows to build up the periodic table by successively adding one proton to the nucleus and one electron to the appropriate atomic orbital
What is the Hund’s rule?
The Hund’s rule states that orbitals of equal energy, called degenerate orbitals, must all contain one electron with the same spin before they can contain 2 electrons
What is the difference between a paramagnetic and a diamagnetic substance?
A paramagnetic substance is a substance with unpaired electrons and therefore weakly attracted by a magnetic field while a diamagnetic substance is a substance without unpaired electrons (not attracted by a magnetic field)
What is the difference between the ground state and an excited state?
The ground state is the lowest energy state of an atom, while an excited state is a state with higher energy than the ground state. Electrons are promoted from the ground state to an excited state by electromagnetic radiation of intensity hν. The first excited state corresponds to the promotion of the highest energy electron from the ground state to the next available orbital
The core electrons are the electrons of the inner energy levels. They do not participate in chemical bonding and form the atomic core with the nucleus
The valence electrons are the electrons of the outermost occupied shell of an atom. They are furthest from the positive charge of the nucleus and therefore tend to react more easily than the core electrons
Quiz - Quantum theory & Atomic structure
Arrangement of electrons - 1
Ionization energy - 1
PES spectrum
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Narrowband communication – Universal-Wiki
The term narrow band, also narrow band communication, English narrow bandis used with different meanings depending on the context in the field of communications technology and in the field of Internet access.
1 Communications Engineering
In the field of communications engineering, the term narrowband describes a transmission channel whose bandwidth is so small that the frequency response can be assumed to be almost constant. This is equivalent to a group delay that is constant throughout the band. In narrowband transmission, it is in principle possible to do without channel equalization because of the frequency-independent group delay. If a transmission channel does not have a constant frequency response, as is the case in wideband communication, the transmitted useful signals are distorted differently depending on the level of their frequencies. Equalizers that reverse these distortions at the receiver of the wanted signals can be implemented using adaptive filters, for example.
In the case of digital data transmissions, a narrow-band transmission is defined equivalently via the symbol rate. Each transmitted symbol requires a certain symbol duration for transmission
{displaystyle T_{s}}
. If the symbol duration is longer than the transmission time
{displaystyle _{mathrm {max} }}
at the transmission channel, a narrowband communication is present:
{displaystyle T_{s}gg _{mathrm {max} }}
It is essential that the definition of a narrowband transmission is not fixed to concrete numerical values of a certain bandwidth, but is oriented to the respective marginal circumstances, such as the physical parameters of a radio link.
From the field of telephone networks and with the advent of Internet access starting in the 1990s, the term narrowband is used to refer to narrowband networks or narrowband access characterized by a bandwidth of less than or equal to 3.1 kHz or a data transmission rate of 64 kbit/s or less.[1] This rigid definition is based, on the one hand, on the bandwidth of analogue fixed network telephony. For the transmission of analogue voice signals via copper wires in good quality, a frequency band of 300 Hz to 3.4 kHz was already defined in the early days of telephony for transmission-related reasons. A similar twin wire is also used for theUK0 interface in the digital transmission of voice signals in ISDN; the required bandwidth here is 40 kHz (for the 2B1Q code with 80 kBaud) or more, depending on the coding. In addition to the signalling channel (D-channel), two so-called basic channels can be transmitted in it, each with a data transmission rate of 64 kbit/s. ITU-T Recommendation I.113 defines the term narrowband service as having a data transmission rate of a basic channel of less than or equal to 64 kbit/s. In mobile telephony, also a narrowband service, digital voice is transmitted at a much lower data rate – in the case of the Full Rate Codec in GSM, the rate is 13 kbit/s. (The bandwidth required here is 2 kbit/s.) The data rate of a base channel is defined by the ITU-T Recommendation I.113. (The bandwidth required here is 2 – 200 kHz, and it can be used by eight connections simultaneously)
Narrowband services for Internet access include analog telephone systems and telephone modems as well as ISDN and mobile networks such as GSM with GPRS. Internet access systems such as DSL belong to broadband Internet access. They require wider frequency bands with more than 40 kHz – with corresponding restrictions on the part of the network operator with regard to line length.
Karl-Dirk Kammeyer, Martin Bossert: Message transmission . 5. Auflage. Vieweg + Teubner, Wiesbaden 2011, ISBN 978-3-8348-0896-7.
↑ Jochen Seitz, Maik Debes, Michael Heubach, Ralf Tosse: Digitale Sprach- und Datenkommunikation: Netze, Protokolle, Vermittlung . Carl Hanser Verlag, 2006, ISBN 978-3-446-22979-2, p. 183.
Retrieved from“https://de.wikipedia.org/w/index.php?title=Schmalbandkommunikation&oldid=164429656”
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6,8,7\frac{1}{2},9,8,8,8,
9,9,10,6,8\frac{1}{2},9,7,8
Copy the set of axes below and create a histogram for the data. (Refer to the glossary in the eBook for assistance, if needed.)
Determine bins appropriate to your data (how wide each bar will be). For this problem use one hour bins.
Think of each bin as containing all the values between the number on the left and the number on the right. For example, a value of
6.5
would be counted with the bar between
6
7
, but a value of
7.0
7
8
Count the number of values in each bin. Draw a bar for each bin. The height of the bar corresponds to the number of values in the bin.
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IsGroup - Maple Help
Home : Support : Online Help : Mathematics : Algebra : Magma : IsGroup
test whether a finite magma is a group
IsGroup( m )
The IsGroup command returns true if the given magma is a group, and returns false otherwise.
A group is an associative magma with an identity element, with respect to which identity each member has a (two-sided) inverse. Alternatively, a group is an associative loop.
\mathrm{with}\left(\mathrm{Magma}\right):
m≔〈〈〈1|2|3〉,〈2|3|1〉,〈3|1|2〉〉〉
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]
\mathrm{IsGroup}\left(m\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
m≔〈〈〈1|2|3〉,〈2|3|3〉,〈3|1|2〉〉〉
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]
\mathrm{IsGroup}\left(m\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
The Magma[IsGroup] command was introduced in Maple 15.
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General components[edit]
Specific methods[edit]
Epidemiology-based method[edit]
{\displaystyle {\begin{aligned}&{\frac {\Pr({\text{Presentation is caused by condition in individual}})}{\Pr({\text{Presentation has occurred in individual}})}}={\frac {\Pr({\text{Presentation WHOIFPI by condition}})}{\Pr({\text{Presentation WHOIFPI}})}}\end{aligned}}}
{\displaystyle \Pr({\text{Presentation is caused by condition in individual}})={\frac {\Pr({\text{Presentation WHOIFPI by condition}})}{\Pr({\text{Presentation WHOIFPI}})}}}
{\displaystyle {\begin{aligned}\Pr({\text{Presentation WHOIFPI}})&=\Pr({\text{Presentation WHOIFPI by condition 1}})\\&{}+\Pr({\text{Presentation WHOIFPI by condition 2}})\\&{}+\Pr({\text{Presentation WHOIFPI by condition 3}})+{\text{etc.}}\end{aligned}}}
{\displaystyle \Pr({\text{Presentation WHOIFPI by condition}})=\Pr({\text{Condition WHOIFPI}})\cdot r_{{\text{condition}}\rightarrow {\text{presentation}}},}
{\displaystyle \Pr({\text{Condition WHOIFPI}})\approx RR_{\text{condition}}\cdot \Pr({\text{Condition in population}}),}
{\displaystyle \Pr({\text{PH in population}})=0.5{\text{ years}}\cdot {\frac {1}{\text{4000 per year}}}={\frac {1}{8000}}}
{\displaystyle \Pr({\text{PH WHOIFPI}})\approx RR_{PH}\cdot \Pr({\text{PH in population}})=10\cdot {\frac {1}{8000}}={\frac {1}{800}}=0.00125}
{\displaystyle {\begin{aligned}\Pr({\text{Hypercalcemia WHOIFPI by PH}})&=\Pr({\text{PH WHOIFPI}})\cdot r_{{\text{PH}}\rightarrow {\text{hypercalcemia}}}\\&=0.00125\cdot 1=0.00125\end{aligned}}}
{\displaystyle \Pr({\text{cancer in population}})=0.5{\text{ years}}\cdot {\frac {1}{\text{250 per year}}}={\frac {1}{500}}}
{\displaystyle \Pr({\text{cancer WHOIFPI}})\approx RR_{\text{cancer}}\cdot \Pr({\text{cancer in population}})=1\cdot {\frac {1}{500}}={\frac {1}{500}}=0.002.}
{\displaystyle {\begin{aligned}&\Pr({\text{Hypercalcemia WHOIFPI by cancer}})\\=&\Pr({\text{cancer WHOIFPI}})\cdot r_{{\text{cancer}}\rightarrow {\text{hypercalcemia}}}\\=&0.002\cdot 0.1=0.0002.\end{aligned}}}
{\displaystyle {\begin{aligned}\Pr({\text{no disease in population}})&=1-\Pr({\text{PH in population}})-\Pr({\text{cancer in population}})\\&{}\quad -\Pr({\text{other conditions in population}})\\&{}=0.997.\end{aligned}}}
{\displaystyle \Pr({\text{no disease WHOIFPI}})=0.997.\,}
{\displaystyle r_{{\text{no disease}}\rightarrow {\text{hypercalcemia}}}=0.0014}
{\displaystyle {\begin{aligned}&\Pr({\text{Hypercalcemia WHOIFPI by no disease}})\\=&\Pr({\text{no disease WHOIFPI}})\cdot r_{{\text{no disease}}\rightarrow {\text{hypercalcemia}}}\\=&0.997\cdot 0.0014\approx 0.0014\end{aligned}}}
{\displaystyle {\begin{aligned}&\Pr({\text{hypercalcemia WHOIFPI}})\\=&\Pr({\text{hypercalcemia WHOIFPI by PH}})+\Pr({\text{hypercalcemia WHOIFPI by cancer}})\\&{}+\Pr({\text{hypercalcemia WHOIFPI by other conditions}})+\Pr({\text{hypercalcemia WHOIFPI by no disease}})\\=&0.00125+0.0002+0.0005+0.0014=0.00335\end{aligned}}}
{\displaystyle {\begin{aligned}&\Pr({\text{hypercalcemia is caused by PH in individual}})\\=&{\frac {\Pr({\text{hypercalcemia WHOIFPI by PH}})}{\Pr({\text{hypercalcemia WHOIFPI}})}}\\=&{\frac {0.00125}{0.00335}}=0.373=37.3\%\end{aligned}}}
{\displaystyle {\begin{aligned}&\Pr({\text{hypercalcemia is caused by cancer in individual}})\\=&{\frac {\Pr({\text{hypercalcemia WHOIFPI by cancer}})}{\Pr({\text{hypercalcemia WHOIFPI}})}}\\=&{\frac {0.0002}{0.00335}}=0.060=6.0\%,\end{aligned}}}
{\displaystyle {\begin{aligned}&\Pr({\text{hypercalcemia is caused by other conditions in individual}})\\=&{\frac {\Pr({\text{hypercalcemia WHOIFPI by other conditions}})}{\Pr({\text{hypercalcemia WHOIFPI}})}}\\=&{\frac {0.0005}{0.00335}}=0.149=14.9\%,\end{aligned}}}
{\displaystyle {\begin{aligned}&\Pr({\text{hypercalcemia is present despite no disease in individual}})\\=&{\frac {\Pr({\text{hypercalcemia WHOIFPI by no disease}})}{\Pr({\text{hypercalcemia WHOIFPI}})}}\\=&{\frac {0.0014}{0.00335}}=0.418=41.8\%\end{aligned}}}
Likelihood ratio-based method[edit]
{\displaystyle {\text{odds}}={\frac {\text{probability}}{1-{\text{probability}}}}}
{\displaystyle {\text{probability}}={\frac {\text{odds}}{{\text{odds}}+1}}}
{\displaystyle \operatorname {Odds} ({\text{PostBT}}_{PH})=\operatorname {Odds} ({\text{PreBT}}_{PH})\cdot LH(BT)=0.595\cdot 7=4.16,}
{\displaystyle \Pr({\text{PostBT}}_{PH})={\frac {\operatorname {Odds} ({\text{PostBT}}_{PH})}{\operatorname {Odds} ({\text{PostBT}}_{PH})+1}}={\frac {4.16}{4.16+1}}=0.806=80.6\%}
{\displaystyle \Pr({\text{PostBT}}_{rest})=100\%-80.6\%=19.4\%}
{\displaystyle \Pr({\text{PreBT}}_{\text{rest}})=6.0\%+14.9\%+41.8\%=62.7\%}
{\displaystyle {\text{Correcting factor}}={\frac {\Pr({\text{PostBT}}_{\text{rest}})}{\Pr({\text{PreBT}}_{\text{rest}})}}={\frac {19.4}{62.7}}=0.309}
{\displaystyle \Pr({\text{PostBT}}_{\text{cancer}})=\Pr({\text{PreBT}}_{\text{cancer}})\cdot {\text{Correcting factor}}=6.0\%\cdot 0.309=1.9\%}
Coverage of candidate conditions[edit]
Machine differential diagnosis[edit]
Alternative medical meanings[edit]
Usage apart from in medicine[edit]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Differential_diagnosis&oldid=1089608436"
|
Copy the axes below onto your paper. Add an appropriate scale and then place and label a point on the graph for each of the products listed below.
Create ordered pairs for each hot dog before plotting.
Decide what information is your independent variable (
x
) and what information is your dependent variable (
y
Dog-Eat-Dog has a supreme hotdog that weighs
80
grams and has
40
grams of fat.
Hot Doggies has a diet hotdog that weighs
50
grams and has only
9
Dog-alicious has a cheap hotdog that weighs
40
30
|
y
x
y
a \cdot x + b
\sigma^2
y \sim N(a \cdot x + b, \sigma^2 ).
a
b
\sigma^2
\theta
\theta = (a, b, \sigma^2)
\theta
X=(x_1, \ldots, x_n)
Y=(y_1, \ldots, y_n)
p(\theta | Y, X) = \frac{p(Y | \theta, X)p(\theta)}{\int P(Y | \theta, X)p(\theta) \, d\theta}
p(Y | \theta, X)
N(a \cdot x + b,\sigma^2)
X, Y
\prod_{i=1}^n p(y_i | \theta, x_i) = \prod_{i=1}^n N(y_i | a \cdot x_i + b, \sigma^2)
p(\theta)
a \sim N(0, 5) \\ b \sim N(0, 10) \\ \sigma^2 \sim logNormal(0, 0.25)
p(\theta \mid Y, X))
Q(\theta' \mid \theta)
\theta
\theta'
\theta
\theta'
\theta
\theta'
import scalismo.sampling.MHSample
import scalismo.sampling.MHDistributionEvaluator
import scalismo.sampling.MHProposalGenerator
import scalismo.sampling.proposals.GaussianRandomWalkProposal
import scalismo.sampling.proposals.MHProductProposal
import scalismo.sampling.ParameterConversion
import scalismo.sampling.loggers.MHSampleLogger
import scalismo.sampling.proposals.MHMixtureProposal
import scalismo.sampling.proposals.MHIdentityProposal
implicit val rng : scalismo.utils.Random = scalismo.utils.Random(42)
// We need this line to seed breeze's random number generator
implicit val randBasisBreeze : breeze.stats.distributions.RandBasis = rng.breezeRandBasis
val errorDist = breeze.stats.distributions.Gaussian(0, sigma2)(rng.breezeRandBasis)
\theta = (a, b, \sigma^2)
To be able to make use of the proposal generators that Scalismo provides, we will also need to define a conversion object, which tells Scalismo how our parameters can be converted to a tuple and back.
implicit object tuple3ParameterConversion extends ParameterConversion[Tuple3[Double, Double, Double], Parameters] {
def from(p: Parameters): Tuple3[Double, Double, Double] = (p.a, p.b, p.sigma2)
def to(t: Tuple3[Double, Double, Double]): Parameters = Parameters(a = t._1, b = t._2, sigma2 = t._3)
p(\theta)
p(Y | \theta, X)
case class LikelihoodEvaluator(data: Seq[(Double, Double)])
extends MHDistributionEvaluator[Parameters] {
override def logValue(theta: MHSample[Parameters]): Double = {
theta.parameters.a * x + theta.parameters.b,
theta.parameters.sigma2
object PriorEvaluator extends MHDistributionEvaluator[Parameters] {
p(y)
Now that the evaluators are in place, our next task is to set up the proposal distributions. In Scalismo, we can define a proposal distribution by implementing concrete subclasses, of the following class
abstract class MHProposalGenerator[A] {
The type A refers to the type of the parameters that we are using. The propose method takes the current parameters and, based on their values, proposes a new one. The method logTransitionProbability yields the logProbability of transitioning from the state represented by the parameter values from to the state represented by the parameter values in to.
Fortunately, we usually do not have to implement these methods by ourselves. Scalismo already provides some proposal generators, which can be flexibly combined to build up more powerful generators.
The most generic one is the GaussianRandomWalkProposal, which takes the given parameters and perturbs them by adding an increment from a zero-mean Gaussian with given standard deviation. The following codes defines a proposal for each of our parameter vectors.
val genA = GaussianRandomWalkProposal(0.01, "rw-a-0.1").forType[Double]
val genB = GaussianRandomWalkProposal(0.05, "rw-b-0.5").forType[Double]
val genSigma = GaussianRandomWalkProposal(0.01, "rw-sigma-0.01").forType[Double]
As we expect the distribution to have more variability in the value of
b
a
, we choose the values for the step size (standard deviation) accordingly. We also provide a tag when defining a proposal generator. This is helpful for debugging and optimizing the chain. Finally, note also that we explicitly specified a type (here Double) of the specified sample.
We can now combine these individual proposal generators to a proposal generator for our Parameter class as follows:
val parameterGenerator = MHProductProposal(genA, genB, genSigma).forType[Parameters]
It might also be a good idea to sometimes only vary the noise genSigma but not the other parameters. To achieve this, we introduce another proposal, the MHIdentityProposal. As the name suggests, it does not do anything, but simply returns the same parameters it gets passed. While this does not sound very useful by itself, by combining it with other proposals we can achieve our goal:
val identProposal = MHIdentityProposal.forType[Double]
val noiseOnlyGenerator = MHProductProposal(identProposal, identProposal, genSigma).forType[Parameters]
We have now two different generators, which generate new parameters given a current set of parameter values. A good strategy is to sometimes vary all the parameters, and sometimes only the noise. This can be done using a MHMixtureProposal:
val mixtureGenerator = MHMixtureProposal((0.1, noiseOnlyGenerator), (0.9, parameterGenerator))
In this case we use the noiseOnlyGenerator 10% of the times and the parameterGenerator 90% of the times.
val chain = MetropolisHastings(mixtureGenerator, posteriorEvaluator)
To be able to diagnose the chain, in case of problems, we also instantiate a logger, which logs all the accepted and rejected samples.
val logger = MHSampleLogger[Parameters]()
The Markov chain has to start somewhere. We define the starting point by defining an initial sample.
val initialSample = MHSample(Parameters(0.0, 0.0, 1.0), generatedBy="initial")
To drive the sampling generation, we define an interator, which we initialize with the initial sample. We also provide the logger as an argument.
val samples = mhIterator.drop(1000).take(5000).toIndexedSeq
The logger that we defined for the chain stored all the accepted and rejected samples. We can use this to get interesting diagnostics.
For example, we can check how often the individual samples got accepted.
println("Acceptance ratios: " +logger.samples.acceptanceRatios)
When running this code we see that the acceptance ratio of the proposal where we vary all the parameters, is around 0.12. The proposal, which only changes the noise value has, as expected, a much higher acceptance ratio of aroun 0.75.
Sometimes it happens that a chain is efficient in the early stages (the burn-in phase), but many samples get rejected in later stages. To diagnose such situations, we can compute the acceptance ratios also only for the last
n
println("acceptance ratios over the last 100 samples: " +logger.samples.takeLast(100).acceptanceRatios)
Such diagnostics helps us to spot when a proposal is not effective and gives us an indication how to tune our sampler to achieve optimal performance.
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Active Shape Model Fitting | Scalismo
In this tutorial we show how we can perform active shape model fitting in Scalismo.
Fitting models to images (Video)
import scalismo.statisticalmodel.asm._
import scalismo.io.{ActiveShapeModelIO, ImageIO}
Active Shape models in Scalismo
Scalismo provides full support for Active Shape models. This means we can use it to learn active shape models from a set of images and corresponding contour, we can save these models, and we can use them to fit images. In this tutorial we will assume that the model has already been built and will only concentrate on model fitting.
We can load an Active Shape Model as follows:
val asm = ActiveShapeModelIO.readActiveShapeModel(new java.io.File("datasets/femur-asm.h5")).get
An ActiveShapeModel instance in Scalismo is a combination of a statistical shape model and an intensity model. Using the method statisticalModel, we can obtain the shape model part. Let's visualize this model:
val modelView = ui.show(modelGroup, asm.statisticalModel, "shapeModel")
The second part of the model is the intensity model. This model consists of a set of profiles, which are attached to specific vertices of the shape model, indicated by the pointId. For each profile, a probability distribution is defined. This distribution represent the intensity variation that we expect for this profile.
The following code shows how this information can be accessed:
val profiles = asm.profiles
val pointId = profile.pointId
val distribution = profile.distribution
Finding likely model correspondences in an image
The main usage of the profile distribution is to identify the points in the image, which are most likely to correspond to the given profile points in the model. More precisely, let
p_i
denote the i-th profile in the model. We can use the information to evaluate for any set of points
(x_1, \ldots, x_n)
, how likely it is that a point
x_j
corresponds to the profile point
p_i
, based on the image intensity patterns
\rho(x_1), \ldots, \rho(x_n)
we find at these points in an image.
To illustrate this, we first load an image:
val image = ImageIO.read3DScalarImage[Short](new java.io.File("datasets/femur-image.nii")).get.map(_.toFloat)
val imageView = ui.show(targetGroup, image, "image")
The ASM implementation in Scalismo, is not restricted to work with the raw intensities, but the active shape model may first apply some preprocessing, such as smooth, applying a gradient transform, etc. Thus in a first step we obtain this preprocess iamge uing the prepocessor method of the asm object:
val preprocessedImage = asm.preprocessor(image)
We can now extract features at a given point:
val point1 = image.domain.origin + EuclideanVector3D(10.0, 10.0, 10.0)
val profile = asm.profiles.head
val feature1 : DenseVector[Double] = asm.featureExtractor(preprocessedImage, point1, asm.statisticalModel.mean, profile.pointId).get
Here we specified the preprocessed image, a point in the image where whe want the evaluate the feature vector, a mesh instance and a point id for the mesh. The mesh instance and point id are needed, since a feature extractor might choose to extract the feature based on mesh information, such as the normal direction of a line at this point.
We can retrieve the likelihood that each corresponding point corresponds to a given profile point:
val featureVec1 = asm.featureExtractor(preprocessedImage, point1, asm.statisticalModel.mean, profile.pointId).get
val probabilityPoint1 = profile.distribution.logpdf(featureVec1)
Based on this information, we can decide, which point is more likely to correspond to the model point. This idea forms the basis of the original m Active Shape Model Fitting algorithm.
The original Active Shape Model Fitting
Scalismo features an implementation of Active Shape Model fitting algorithm, as proposed by Cootes and Taylor.
To configure the fitting process, we need to set up a search method, which searches for a given model point, corresponding points in the image. From these points, the most likely point is select and used as as the corresponding point for one iteration of the algorithm. Once these "candidate correspondences" have been established, the rest of the algorithm works in exactly the same as the ICP algorithm that we described in the previous tutorials.
One search strategy that is already implemented in Scalismo is to search along the normal direction of a model point. This behavior is provided by the NormalDirectionSearchPointSampler
val searchSampler = NormalDirectionSearchPointSampler(numberOfPoints = 100, searchDistance = 3)
In addition to the search strategy, we can specify some additional configuration parameters to control the fitting process:
val config = FittingConfiguration(featureDistanceThreshold = 3, pointDistanceThreshold = 5, modelCoefficientBounds = 3)
The first parameter determines how far away (as measured by the mahalanobis distance) an intensity feature can be, such that it is still chosen as corresponding. The pointDistanceThreshold does the same for the distance of the points; I.e. in this case points which are more than 5 standard deviations aways are not chosen as corresponding points. The last parameters determines how large coefficients of the model can become in the fitting process. Whenever a model parameter is larger than this threshold, it will be set back to this maximal value. This introduces a regularization into the fitting, which prevents the shape from becoming too unlikely.
The ASM fitting algorithm optimizes both the pose (as defined by a rigid transformation) and the shape. In order to allow it to optimize the rotation, it is important that we choose a rotation center, which is approximately the center of mass of the model:
// make sure we rotate around a reasonable center point
val modelBoundingBox = asm.statisticalModel.reference.boundingBox
val rotationCenter = modelBoundingBox.origin + modelBoundingBox.extent * 0.5
To initialize the fitting process, we also need to set up the initial transformation:
// we start with the identity transform
val translationTransformation = Translation3D(EuclideanVector3D(0, 0, 0))
val rotationTransformation = Rotation3D(0, 0, 0, rotationCenter)
val initialRigidTransformation = TranslationAfterRotation3D(translationTransformation, rotationTransformation)
val initialModelCoefficients = DenseVector.zeros[Double](asm.statisticalModel.rank)
val initialTransformation = ModelTransformations(initialModelCoefficients, initialRigidTransformation)
To start the fitting, we obtain an iterator, which we subsequently use to drive the iteration.
val asmIterator = asm.fitIterator(image, searchSampler, numberOfIterations, config, initialTransformation)
Especially in a debugging phase, we want to visualize the result in every iteration. The following code shows, how we can obtain a new iterator, which updates the pose transformation and model coefficients in the ui in every iteration:
val asmIteratorWithVisualization = asmIterator.map(it => {
case scala.util.Success(iterationResult) => {
modelView.shapeModelTransformationView.poseTransformationView.transformation = iterationResult.transformations.rigidTransform
modelView.shapeModelTransformationView.shapeTransformationView.coefficients = iterationResult.transformations.coefficients
case scala.util.Failure(error) => System.out.println(error.getMessage)
To run the fitting, and get the result, we finally consume the iterator:
val result = asmIteratorWithVisualization.toIndexedSeq.last
val finalMesh = result.get.mesh
Evaluating the likelihood of a model instance under the image
In the previous section we have used the intensity distribution to find the best corresponding image point to a given point in the model. Sometimes we are also interested in finding out how well a model fits an image. To compute this, we can extend the method used above to compute the likelihood for all profile points of an Active Shape Model.
Given the model instance, we will get the position of each profile point in the current instance, evaluate its likelihood and then compute the joint likelihood for all profiles. Assuming independence, the joint probability is just the product of the probability at the individual profile points. In order not to get too extreme values, we use log probabilities here (and consequently the product becomes a sum).
def likelihoodForMesh(asm : ActiveShapeModel, mesh : TriangleMesh[_3D], preprocessedImage: PreprocessedImage) : Double = {
val ids = asm.profiles.ids
val likelihoods = for (id <- ids) yield {
val profile = asm.profiles(id)
val profilePointOnMesh = mesh.pointSet.point(profile.pointId)
val featureAtPoint = asm.featureExtractor(preprocessedImage, profilePointOnMesh, mesh, profile.pointId).get
profile.distribution.logpdf(featureAtPoint)
This method allows us to compute for each mesh, represented by the model, how likely it is to correspond to the given image.
val sampleMesh1 = asm.statisticalModel.sample()
println("Likelihood for mesh 1 = " + likelihoodForMesh(asm, sampleMesh1, preprocessedImage))
This information is all that is need to write probabilistic fitting methods methods using Markov Chain Monte Carlo methods, which will be discussed in a later tutorial.
Active Shape models in Scalismo
The original Active Shape Model Fitting
Evaluating the likelihood of a model instance under the image
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Landau_pole Knowpia
In physics, the Landau pole (or the Moscow zero, or the Landau ghost)[1] is the momentum (or energy) scale at which the coupling constant (interaction strength) of a quantum field theory becomes infinite. Such a possibility was pointed out by the physicist Lev Landau and his colleagues.[2] The fact that couplings depend on the momentum (or length) scale is the central idea behind the renormalization group.
Landau poles appear in theories that are not asymptotically free, such as quantum electrodynamics (QED) or φ4 theory—a scalar field with a quartic interaction—such as may describe the Higgs boson. In these theories, the renormalized coupling constant grows with energy. A Landau pole appears when the coupling becomes infinite at a finite energy scale. In a theory purporting to be complete, this could be considered a mathematical inconsistency. A possible solution is that the renormalized charge could go to zero as the cut-off is removed, meaning that the charge is completely screened by quantum fluctuations (vacuum polarization). This is a case of quantum triviality,[3] which means that quantum corrections completely suppress the interactions in the absence of a cut-off.
Since the Landau pole is normally identified through perturbative one-loop or two-loop calculations, it is possible that the pole is merely a sign that the perturbative approximation breaks down at strong coupling. Perturbation theory may also be invalid if non-adiabatic states exist. Lattice gauge theory provides a means to address questions in quantum field theory beyond the realm of perturbation theory, and thus has been used to attempt to resolve this question.
Numerical computations performed in this framework seem to confirm Landau's conclusion that in QED the renormalized charge completely vanishes for an infinite cutoff.[4][5][6][7]
According to Landau, Abrikosov, and Khalatnikov,[8] the relation of the observable charge gobs to the “bare” charge g0 for renormalizable field theories when Λ ≫ m is given by
{\displaystyle g_{\text{obs}}={\frac {g_{0}}{1+\beta _{2}g_{0}\ln \Lambda /m}}\qquad \qquad \qquad (1)}
where m is the mass of the particle and Λ is the momentum cut-off. If g0 < ∞ and Λ → ∞ then gobs → 0 and the theory looks trivial. In fact, inverting Eq.1, so that g0 (related to the length scale Λ−1) reveals an accurate value of gobs,
{\displaystyle g_{0}={\frac {g_{\text{obs}}}{1-\beta _{2}g_{\text{obs}}\ln \Lambda /m}}.\qquad \qquad \qquad (2)}
As Λ grows, the bare charge g0 = g(Λ) increases, to finally diverge at the renormalization point
{\displaystyle \Lambda _{\text{Landau}}=m\exp \left\{{\frac {1}{\beta _{2}g_{\text{obs}}}}\right\}.\qquad \qquad \qquad (3)}
This singularity is the Landau pole with a negative residue, g(Λ) ≈ −ΛLandau /(β2(Λ − ΛLandau)).
In fact, however, the growth of g0 invalidates Eqs.1,2 in the region g0 ≈ 1, since these were obtained for g0 ≪ 1, so that the nonperturbative existence of the Landau pole becomes questionable.
The actual behavior of the charge g(μ) as a function of the momentum scale μ is determined by the Gell-Mann–Low equation[9]
{\displaystyle {\frac {dg}{d\ln \mu }}=\beta (g)=\beta _{2}g^{2}+\beta _{3}g^{3}+\ldots \qquad \qquad \qquad (4)}
which gives Eqs.1,2 if it is integrated under conditions g(μ) = gobs for μ = m and g(μ) = g0 for μ = Λ, when only the term with β2 is retained in the right hand side. The general behavior of g(μ) depends on the appearance of the function β(g).
According to the classification of Bogoliubov and Shirkov,[10] there are three qualitatively different cases:
(a) if β(g) has a zero at the finite value g∗, then growth of g is saturated, i.e. g(μ) → g∗ for μ → ∞;
(b) if β(g) is non-alternating and behaves as β(g) ∝ gα with α ≤ 1 for large g, then the growth of g(μ) continues to infinity;
(c) if β(g) ∝ gα with α > 1 for large g, then g(μ) is divergent at finite value μ0 and the real Landau pole arises: the theory is internally inconsistent due to indeterminacy of g(μ) for μ > μ0.
Landau and Pomeranchuk[11] tried to justify the possibility (c) in the case of QED and φ4 theory. They have noted that the growth of g0 in Eq.1 drives the observable charge gobs to the constant limit, which does not depend on g0. The same behavior can be obtained from the functional integrals, omitting the quadratic terms in the action. If neglecting the quadratic terms is valid already for g0 ≪ 1, it is all the more valid for g0 of the order or greater than unity: it gives a reason to consider Eq.1 to be valid for arbitrary g0. Validity of these considerations at the quantitative level is excluded by the non-quadratic form of the β-function.[citation needed]
Nevertheless, they can be correct qualitatively. Indeed, the result gobs = const(g0) can be obtained from the functional integrals only for g0 ≫ 1, while its validity for g0 ≪ 1, based on Eq.1, may be related to other reasons; for g0 ≈ 1 this result is probably violated but coincidence of two constant values in the order of magnitude can be expected from the matching condition. The Monte Carlo results [12] seems to confirm the qualitative validity of the Landau–Pomeranchuk arguments, although a different interpretation is also possible.
The case (c) in the Bogoliubov and Shirkov classification corresponds to the quantum triviality in full theory (beyond its perturbation context), as can be seen by a reductio ad absurdum. Indeed, if gobs < ∞, the theory is internally inconsistent. The only way to avoid it, is for μ0 → ∞, which is possible only for gobs → 0. It is a widespread belief[by whom?] that both QED and φ4 theory are trivial in the continuum limit.
Phenomenological aspectsEdit
In a theory intended to represent a physical interaction where the coupling constant is known to be non-zero, Landau poles or triviality may be viewed as a sign of incompleteness in the theory. For example, QED is usually not believed to be a complete theory on its own, because it does not describe other fundamental interactions, and contains a Landau pole. Conventionally QED forms part of the more fundamental electroweak theory. The U(1)Y group of electroweak theory also has a Landau pole which is usually considered[by whom?] to be a signal of a need for an ultimate embedding into a Grand Unified Theory. The grand unified scale would provide a natural cutoff well below the Landau scale, preventing the pole from having observable physical consequences.
The problem of the Landau pole in QED is of pure academic interest, for the following reason. The role of gobs in Eqs. 1, 2 is played by the fine structure constant α ≈ 1/137 and the Landau scale for QED is estimated as 10286 eV, which is far beyond any energy scale relevant to observable physics. For comparison, the maximum energies accessible at the Large Hadron Collider are of order 1013 eV, while the Planck scale, at which quantum gravity becomes important and the relevance of quantum field theory itself may be questioned, is 1028 eV.
The Higgs boson in the Standard Model of particle physics is described by φ4 theory (see Quartic interaction). If the latter has a Landau pole, then this fact is used in setting a "triviality bound" on the Higgs mass. The bound depends on the scale at which new physics is assumed to enter and the maximum value of the quartic coupling permitted (its physical value is unknown). For large couplings, non-perturbative methods are required. Lattice calculations have also been useful in this context.[13]
Connections with statistical physicsEdit
A deeper understanding of the physical meaning and generalization of the renormalization process leading to Landau poles comes from condensed matter physics. Leo P. Kadanoff's paper in 1966 proposed the "block-spin" renormalization group.[14] The blocking idea is a way to define the components of the theory at large distances as aggregates of components at shorter distances. This approach was developed by Kenneth Wilson.[15] He was awarded the Nobel prize for these decisive contributions in 1982.
Assume that we have a theory described by a certain function
{\displaystyle Z}
of the state variables
{\displaystyle \{s_{i}\}}
and a set of coupling constants
{\displaystyle \{J_{k}\}}
. This function can be a partition function, an action, or a Hamiltonian. Consider a certain blocking transformation of the state variables
{\displaystyle \{s_{i}\}\to \{{\tilde {s}}_{i}\}}
{\displaystyle {\tilde {s}}_{i}}
must be lower than the number of
{\displaystyle s_{i}}
. Now let us try to rewrite the
{\displaystyle Z}
function only in terms of the
{\displaystyle {\tilde {s}}_{i}}
. If this is achievable by a certain change in the parameters,
{\displaystyle \{J_{k}\}\to \{{\tilde {J}}_{k}\}}
, then the theory is said to be renormalizable. The most important information in the RG flow are its fixed points. The possible macroscopic states of the system, at a large scale, are given by this set of fixed points. If these fixed points correspond to a free field theory, the theory is said to exhibit quantum triviality, and possesses a Landau pole. Numerous fixed points appear in the study of lattice Higgs theories, but it is not known whether these correspond to free field theories.[3]
Large order perturbative calculationsEdit
Solution of the Landau pole problem requires the calculation of the Gell-Mann–Low function β(g) at arbitrary g and, in particular, its asymptotic behavior for g → ∞. Diagrammatic calculations allow one to obtain only a few expansion coefficients β2, β3, ..., which do not allow one to investigate the β function in the whole. Progress became possible after the development of the Lipatov method for calculating large orders of perturbation theory:[16] One may now try to interpolate the known coefficients β2, β3, ... with their large order behavior, and to then sum the perturbation series.
The first attempts of reconstruction of the β function by this method bear on the triviality of the φ4 theory. Application of more advanced summation methods yielded the exponent α in the asymptotic behavior β(g) ∝ gα, a value close to unity. The hypothesis for the asymptotic behavior of β(g) ∝ g was recently presented analytically for φ4 theory and QED.[17][18][19] Together with positiveness of β(g), obtained by summation of the series, it suggests case (b) of the above Bogoliubov and Shirkov classification, and hence the absence of the Landau pole in these theories, assuming perturbation theory is valid (but see above discussion in the introduction ).
^ Landau ghost – Oxford Index
^ Lev Landau, in Wolfgang Pauli, ed. (1955). Niels Bohr and the Development of Physics. London: Pergamon Press.
^ a b D. J. E. Callaway (1988). "Triviality Pursuit: Can Elementary Scalar Particles Exist?". Physics Reports. 167 (5): 241–320. Bibcode:1988PhR...167..241C. doi:10.1016/0370-1573(88)90008-7.
^ Callaway, D. J. E.; Petronzio, R. (1986). "CAN elementary scalar particles exist?: (II). Scalar electrodynamics". Nuclear Physics B. 277 (1): 50–66. Bibcode:1986NuPhB.277...50C. doi:10.1016/0550-3213(86)90431-1.
^ Göckeler, M.; R. Horsley; V. Linke; P. Rakow; G. Schierholz; H. Stüben (1998). "Is There a Landau Pole Problem in QED?". Physical Review Letters. 80 (19): 4119–4122. arXiv:hep-th/9712244. Bibcode:1998PhRvL..80.4119G. doi:10.1103/PhysRevLett.80.4119. S2CID 119494925.
^ Kim, S.; John B. Kogut; Lombardo Maria Paola (2002-01-31). "Gauged Nambu–Jona-Lasinio studies of the triviality of quantum electrodynamics". Physical Review D. 65 (5): 054015. arXiv:hep-lat/0112009. Bibcode:2002PhRvD..65e4015K. doi:10.1103/PhysRevD.65.054015. S2CID 15420646.
^ Gies, Holger; Jaeckel, Joerg (2004-09-09). "Renormalization Flow of QED". Physical Review Letters. 93 (11): 110405. arXiv:hep-ph/0405183. Bibcode:2004PhRvL..93k0405G. doi:10.1103/PhysRevLett.93.110405. PMID 15447325. S2CID 222197.
^ L. D. Landau, A. A. Abrikosov, and I. M. Khalatnikov, Dokl. Akad. Nauk SSSR 95, 497, 773, 1177 (1954).
^ Gell-Mann, M.; Low, F. E. (1954). "Quantum Electrodynamics at Small Distances" (PDF). Physical Review. 95 (5): 1300–1320. Bibcode:1954PhRv...95.1300G. doi:10.1103/PhysRev.95.1300.
^ N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields, 3rd ed. (Nauka, Moscow, 1976; Wiley, New York, 1980).
^ L.D.Landau, I.Ya.Pomeranchuk, Dokl. Akad. Nauk SSSR 102, 489 (1955); I.Ya.Pomeranchuk, Dokl. Akad. Nauk SSSR 103, 1005 (1955).
^ Callaway, D. J. E.; Petronzio, R. (1984). "Monte Carlo renormalization group study of φ4 field theory". Nuclear Physics B. 240 (4): 577. Bibcode:1984NuPhB.240..577C. doi:10.1016/0550-3213(84)90246-3.
^ For example, Callaway, D.J.E.; Petronzio, R. (1987). "Is the standard model Higgs mass predictable?". Nuclear Physics B. 292: 497–526. Bibcode:1987NuPhB.292..497C. doi:10.1016/0550-3213(87)90657-2. Heller, Urs; Markus Klomfass; Herbert Neuberger; Pavols Vranas (1993-09-20). "Numerical analysis of the Higgs mass triviality bound". Nuclear Physics B. 405 (2–3): 555–573. arXiv:hep-ph/9303215. Bibcode:1993NuPhB.405..555H. doi:10.1016/0550-3213(93)90559-8. S2CID 7146602. , which suggests MH < 710 GeV.
^ L.P. Kadanoff (1966): "Scaling laws for Ising models near
{\displaystyle T_{c}}
", Physics (Long Island City, N.Y.) 2, 263.
^ K.G. Wilson(1975): The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys. 47, 4, 773.
^ L.N.Lipatov, Zh.Eksp.Teor.Fiz. 72, 411 (1977) [Sov.Phys. JETP 45, 216 (1977)].
^ Suslov, I. M. (2008). "Renormalization group functions of the φ4 theory in the strong coupling limit: Analytical results". Journal of Experimental and Theoretical Physics. 107 (3): 413–429. arXiv:1010.4081. Bibcode:2008JETP..107..413S. doi:10.1134/S1063776108090094. S2CID 119205490.
^ Suslov, I. M. (2010). "Asymptotic behavior of the β function in the ϕ4 theory: A scheme without complex parameters". Journal of Experimental and Theoretical Physics. 111 (3): 450–465. arXiv:1010.4317. Bibcode:2010JETP..111..450S. doi:10.1134/S1063776110090153. S2CID 118545858.
^ Suslov, I. M. (2009). "Exact asymptotic form for the β function in quantum electrodynamics". Journal of Experimental and Theoretical Physics. 108 (6): 980–984. arXiv:0804.2650. Bibcode:2009JETP..108..980S. doi:10.1134/S1063776109060089. S2CID 7219671.
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Pumpkin Carving Patterns from Photos with Julia
Matt Giamou
Over the last few years, my family and I have rekindled our collective interest in carving jack-o'-lanterns. As a child, I would carve fairly simple templates with my parents, but in 2019 I found a tutorial for creating photo-realistic templates with photo editing software (I used GIMP). The resulting carving of my wife was cool, but it took a lot of manual editing to get to a usable template from a reference photo.
This year, I decided to see what I could do to automate most of the pattern creation steps using Julia's image processing libraries. Inspired by the grumpy mug of my former foster cat Bruno, I got to work.
This tutorial was made with Literate.jl; you can find the source code here.
We'll only be making use of a few image processing packages.
using Images, ImageFiltering, ImageContrastAdjustment
Here's an outline of the steps I used to create a pattern (we'll look at each in detail shortly):
Manual Background Removal: you can get a fairly good template without removing the background, but I found the contrast was improved by doing this on my iPad first. I leave the implementation of an automatic segmentation algorithm for future work.
Grayscale: since I can only control how much light gets through the pumpkin, there's no need for colour.
Equalization: this step spreads out the dynamic range of the pixels to create greater contrast for a clearer image.
Smoothing: real photographs contain too much fine detail for a novice carver like me to recreate, so some form of smoothing filter is needed.
Thresholding: once again, I'm not a skilled artist, so I will only be creating a template with three shades (white, black, and gray). It's possible to have a more shades, or even continuous gradients, but that's well beyond my carving abilities.
Manual Touch-up: unfortunately, the process isn't perfect, and some manual changes on my iPad were needed to simplify the template.
Let's load up the image and make it grayscale.
rgb_image = load("_assets/img/blog/pumpkinizer/bruno_pumpkin_black_background.jpg")
gray_image = Gray.(rgb_image);
Next, we'll apply histogram equalization. Notice how much this simple step has improved the contrast, which will help us to eventually separate Bruno's feature's into three clear shades.
hist_equal = adjust_histogram(gray_image, Equalization(nbins = 256));
Smoothing - Bilateral Filtering
We use a bilateral filter for its edge-preserving properties.
function bilateral_filter(img, n::Int=5, σr=0.5, σd=0.5)
img_filt = zeros(size(img))
for i = 1:size(img, 1)
for j = 1:size(img, 2)
w_sum = 0.0
img_sum = 0.0
for u = max(1, i-n):min(size(img, 1), i+n)
for v = max(1, j-n):min(size(img, 2), j+n)
w_ijuv = exp(-((i-u)^2 + (j-v)^2)/(2*σd^2) - (img[i, j] - img[u, v])^2/(2*σr^2))
w_sum += w_ijuv
img_sum += w_ijuv*img[u, v]
img_filt[i, j] = img_sum/w_sum
return img_filt
The kernel size and smoothing parameters
\sigma_r
σr and
\sigma_d
σd allow for a great deal of control. I eventually settled on the values below.
σr = 3 # Smoothing parameter based on pixel proximity (reduce this for more detail)
σd = 7 # Smoothing parameter based on pixel intensity similarity (reduce this for more detail as well)
kernel_size = 14 # Size of the window to use (make this larger for less detail)
filtered_image = bilateral_filter(hist_equal, kernel_size, σr, σd);
Bruno's face is now blurrier and therefore more easily segmented into simple blobs, but notice how the lines are still fairly crisp (and therefore easy to carve):
Thresholding Operation
The final step was simply a thresholding procedure so that the pattern only featured three shades. Carving with a greater number of shades is beyond my abilities, but more skilled pumpkin pulp-sculptors can easily modify this procedure to support more detailed shading.
function threshold_image(img, black_threshold, gray_threshold)
thresholded_image = zeros(size(img))
for ind in eachindex(img)
if img[ind] < black_threshold
thresholded_image[ind] = 0.0
elseif img[ind] > gray_threshold
After some experimentation, the following parameters were used:
black_threshold = 0.5 # Increase for more black vs. gray
gray_threshold = 0.75 # Increase this for more gray vs. white
thresholded_image = threshold_image(filtered_image, black_threshold, gray_threshold);
The resulting image is now pretty much ready to be saved (with save("template.png", thresholded_image)) and used as a carving pattern:
Manual Touch-Up
The final step before printing and carving involved some manual simplification on my iPad, the re-addition of some whiskers that were lost in translation, and adding some whitespace to save ink.
I set up my workspace and taped the template on the nicest face of a newly-gutted pumpkin:
After a few hours of detailed carving, the un-illuminated Brun-o'-lantern looked pretty terrible:
But I was pretty pleased with the final product when lit up, even though I made some mistakes:
Here's an image of the full pipeline that helps to visualize the changes each step introduces:
Tips to myself (and others) for improving next year's pumpkin:
I carved the wall WAY too thin! Brun-o'-lantern's mouth fell off and I needed to support it with a pin.
The mouth was also just way too small, I need to make sure shapes have better structural integrity.
I need to make smaller holes in the dotting step - is my hole-poking tool too big, or did I just poke too hard?
Projecting a flat image onto a curved surface is never perfect, but the hole in the middle of his head is mostly my fault.
I need to be choosier with my pumpkin: this one was too small and curved for someone with my skill level.
I hope you found this informative! Feel free to use and modify the source code to make your own pumpkin carving templates next Halloween!
© Matt Giamou. Last modified: January 07, 2022. Website built with Franklin.jl and the Julia programming language.
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14 CFR Appendix F to Part 36 - Flyover Noise Requirements for Propeller-Driven Small Airplane and Propeller-Driven, Commuter Category Airplane Certification Tests Prior to December 22, 1988 | CFR | US Law | LII / Legal Information Institute
Appendix F to Part 36 - Flyover Noise Requirements for Propeller-Driven Small Airplane and Propeller-Driven, Commuter Category Airplane Certification Tests Prior to December 22, 1988
14 CFR Appendix F to Part 36 - Flyover Noise Requirements for Propeller-Driven Small Airplane and Propeller-Driven, Commuter Category Airplane Certification Tests Prior to December 22, 1988
F36.1 Scope.
part b - noise measurement
F36.101 General test conditions.
F36.103 Acoustical measurement system.
F36.105 Sensing, recording, and reproducing equipment.
F36.107 Noise measurement procedures.
F36.109 Data recording, reporting, and approval.
F36.111 Flight procedures.
part c - data correction
F36.201 Correction of data.
F36.203 Validity of results.
part d - noise limits
F36.301 Aircraft noise limits.
Section F36.1 Scope. This appendix prescribes noise level limits and procedures for measuring and correcting noise data for the propeller driven small airplanes specified in §§ 36.1 and 36.501(b).
Sec. F36.101 General test conditions.
(a) The test area must be relatively flat terrain having no excessive sound absorption characteristics such as those caused by thick, matted, or tall grass, by shrubs, or by wooded areas. No obstructions which significantly influence the sound field from the airplane may exist within a conical space above the measurement position, the cone being defined by an axis normal to the ground and by a half-angle 75 degrees from this axis.
(b) The tests must be carried out under the following conditions:
(1) There may be no precipitation.
(2) Relative humidity may not be higher than 90 percent or lower than 30 percent.
(3) Ambient temperature may not be above 86 degrees F. or below 41 degrees F. at 33′ above ground. If the measurement site is within 1 n.m. of an airport thermometer the airport reported temperature may be used.
(4) Reported wind may not be above 10 knots at 33′ above ground. If wind velocities of more than 4 knots are reported, the flight direction must be aligned to within ±15 degrees of wind direction and flights with tail wind and head wind must be made in equal numbers. If the measurement site is within 1 n.m. of an airport anemometer, the airport reported wind may be used.
(5) There may be no temperature inversion or anomalous wind conditions that would significantly alter the noise level of the airplane when the noise is recorded at the required measuring point.
(6) The flight test procedures, measuring equipment, and noise measurement procedures must be approved by the FAA.
(7) Sound pressure level data for noise evaluation purposes must be obtained with acoustical equipment that complies with section F36.103 of this appendix.
Sec. F36.103 Acoustical measurement system. The acoustical measurement system must consist of approved equipment equivalent to the following:
(a) A microphone system with frequency response compatible with measurement and analysis system accuracy as prescribed in section F36.105 of this appendix.
(b) Tripods or similar microphone mountings that minimize interference with the sound being measured.
(c) Recording and reproducing equipment characteristics, frequency response, and dynamic range compatible with the response and accuracy requirements of section F36.105 of this appendix.
(d) Acoustic calibrators using sine wave or broadband noise of known sound pressure level. If broadband noise is used, the signal must be described in terms of its average and maximum root-mean-square (rms) value for nonoverload signal level.
Sec. F36.105 Sensing, recording, and reproducing equipment.
(a) The noise produced by the airplane must be recorded. A magnetic tape recorder is acceptable.
(b) The characteristics of the system must comply with the recommendations in IEC 179 (incorporated by reference, see § 36.6).
(c) The response of the complete system to a sensibly plane progressive sinusoidal wave of constant amplitude must lie within the tolerance limits specified in IEC Publication No. 179, dated 1973, over the frequency range 45 to 11,200 Hz.
(d) If limitations of the dynamic range of the equipment make it necessary, high frequency pre-emphasis must be added to the recording channel with the converse de-emphasis on playback. The pre-emphasis must be applied such that the instantaneous recorded sound pressure level of the noise signal between 800 and 11,200 Hz does not vary more than 20 dB between the maximum and minimum one-third octave bands.
(e) If requested by the Administrator, the recorded noise signal must be read through an “A” filter with dynamic characteristics designated “slow,” as defined in IEC Publication No. 179, dated 1973. The output signal from the filter must be fed to a rectifying circuit with square law rectification, integrated with time constants for charge and discharge of about 1 second or 800 milliseconds.
(f) The equipment must be acoustically calibrated using facilities for acoustic freefield calibration and if analysis of the tape recording is requested by the Administrator, the analysis equipment shall be electronically calibrated by a method approved by the FAA.
(g) A windscreen must be employed with microphone during all measurements of aircraft noise when the wind speed is in excess of 6 knots.
Sec. F36.107 Noise measurement procedures.
(a) The microphones must be oriented in a known direction so that the maximum sound received arrives as nearly as possible in the direction for which the microphones are calibrated. The microphone sensing elements must be approximately 4′ above ground.
(b) Immediately prior to and after each test; a recorded acoustic calibration of the system must be made in the field with an acoustic calibrator for the two purposes of checking system sensitivity and providing an acoustic reference level for the analysis of the sound level data.
(c) The ambient noise, including both acoustical background and electrical noise of the measurement systems, must be recorded and determined in the test area with the system gain set at levels that will be used for aircraft noise measurements. If aircraft sound pressure levels do not exceed the background sound pressure levels by at least 10 dB(A), approved corrections for the contribution of background sound pressure level to the observed sound pressure level must be applied.
Sec. F36.109 Data recording, reporting, and approval.
(a) Data representing physical measurements or corrections to measured data must be recorded in permanent form and appended to the record except that corrections to measurements for normal equipment response deviations need not be reported. All other corrections must be approved. Estimates must be made of the individual errors inherent in each of the operations employed in obtaining the final data.
(b) Measured and corrected sound pressure levels obtained with equipment conforming to the specifications described in section F36.105 of this appendix must be reported.
(c) The type of equipment used for measurement and analysis of all acoustic, airplane performance, and meteorological data must be reported.
(d) The following atmospheric data, measured immediately before, after, or during each test at the observation points prescribed in section F36.101 of this appendix must be reported:
(1) Air temperature and relative humidity.
(2) Maximum, minimum, and average wind velocities.
(e) Comments on local topography, ground cover, and events that might interfere with sound recordings must be reported.
(f) The following airplane information must be reported:
(1) Type, model and serial numbers (if any) of airplanes, engines, and propellers.
(2) Any modifications or nonstandard equipment likely to affect the noise characteristics of the airplane.
(3) Maximum certificated takeoff weights.
(4) Airspeed in knots for each overflight of the measuring point.
(5) Engine performance in terms of revolutions per minute and other relevant parameters for each overflight.
(6) Aircraft height in feet determined by a calibrated altimeter in the aircraft, approved photographic techniques, or approved tracking facilities.
(g) Aircraft speed and position and engine performance parameters must be recorded at an approved sampling rate sufficient to ensure compliance with the test procedures and conditions of this appendix.
Sec. F36.111 Flight procedures.
(a) Tests to demonstrate compliance with the noise level requirements of this appendix must include at least six level flights over the measuring station at a height of 1,000′ ±30′ and ±10 degrees from the zenith when passing overhead.
(b) Each test over flight must be conducted:
(1) At not less than the highest power in the normal operating range provided in an Airplane Flight Manual, or in any combination of approved manual material, approved placard, or approved instrument markings; and
(2) At stabilized speed with propellers synchronized and with the airplane in cruise configuration, except that if the speed at the power setting prescribed in this paragraph would exceed the maximum speed authorized in level flight, accelerated flight is acceptable.
Sec. F36.201 Correction of data.
(a) Noise data obtained when the temperature is outside the range of 68 degrees F. ±9 degrees F., or the relative humidity is below 40 percent, must be corrected to 77 degrees F. and 70 percent relative humidity by a method approved by the FAA.
(b) The performance correction prescribed in paragraph (c) of this section must be used. It must be determined by the method described in this appendix, and must be added algebraically to the measured value. It is limited to 5dB(A).
(c) The performance correction must be computed by using the following formula:
\mathrm{\Delta dB}=60-20\phantom{\rule{0ex}{0ex}}{\mathrm{log}}_{10}\left\{\left(11,430-{D}_{50}\phantom{\rule{0ex}{0ex}}\frac{R/C}{{V}_{y}}+50\right\}
D50 = Takeoff distance to 50 feet at maximum certificated takeoff weight.
R/C = Certificated best rate of climb (fpm).
Vy = Speed for best rate of climb in the same units as rate of climb.
(d) When takeoff distance to 50′ is not listed as approved performance information, the figures of 2000 for single-engine airplanes and 1600′ for multi-engine airplanes must be used.
Sec. F36.203 Validity of results.
(a) The test results must produce an average dB(A) and its 90 percent confidence limits, the noise level being the arithmetic average of the corrected acoustical measurements for all valid test runs over the measuring point.
(b) The samples must be large enough to establish statistically a 90 pecent confidence limit not to exceed ±1.5 dB(A). No test result may be omitted from the averaging process, unless omission is approved by the FAA.
Sec. F36.301 Aircraft noise limits.
(a) Compliance with this section must be shown with noise data measured and corrected as prescribed in Parts B and C of this appendix.
(b) For airplanes for which application for a type certificate is made on or after October 10, 1973, the noise level must not exceed 68 dB(A) up to and including aircraft weights of 1,320 pounds (600 kg.). For weights greater than 1,320 pounds up to and including 3,630 pounds (1.650 kg.) the limit increases at the rate of 1 dB/165 pounds (1 dB/75 kg.) to 82 dB(A) at 3,630 pounds, after which it is constant at 82 dB(A). However, airplanes produced under type certificates covered by this paragraph must also meet paragraph (d) of this section for the original issuance of standard airworthiness certificates or restricted category airworthiness certificates if those airplanes have not had flight time before the date specified in that paragraph.
(c) For airplanes for which application for a type certificate is made on or after January 1, 1975, the noise levels may not exceed the noise limit curve prescribed in paragraph (b) of this section, except that 80 dB(A) may not be exceeded.
(d) For airplanes for which application is made for a standard airworthiness certificate or for a restricted category airworthiness certificate, and that have not had any flight time before January 1, 1980, the requirements of paragraph (c) of this section apply, regardless of date of application, to the original issuance of the certificate for that airplane.
[Doc. No. 13243, 40 FR 1035, Jan. 6, 1975; 40 FR 6347, Feb. 11, 1975, as amended by Amdt. 36-6, 41 FR 56064, Dec. 23, 1976; Amdt. 36-6, 42 FR 4113, Jan. 24, 1977; Amdt. 36-9, 43 FR 8754, Mar. 2, 1978; Amdt. 36-13, 52 FR 1836, Jan. 15, 1987; Amdt. 36-16, 53 FR 47400, Nov. 22, 1988; FAA Doc. No. FAA-2015-3782, Amdt. No. 36-31, 82 FR 46131, Oct. 4, 2017]
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Mapped solid axle suspension - Simulink - MathWorks 日本
\begin{array}{l}\left[\begin{array}{c}{\stackrel{¨}{x}}_{a}\\ {\stackrel{¨}{y}}_{a}\\ {\stackrel{¨}{z}}_{a}\end{array}\right]=\frac{1}{{M}_{a}}\left[\begin{array}{c}{F}_{xa}\\ {F}_{ya}\\ {F}_{za}\end{array}\right]+\left[\begin{array}{c}{\stackrel{Ë}{x}}_{a}\\ {\stackrel{Ë}{y}}_{a}\\ {\stackrel{Ë}{z}}_{a}\end{array}\right]Ã\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\frac{1}{{M}_{a}}\left[\begin{array}{c}0\\ 0\\ {F}_{za}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ {\stackrel{Ë}{z}}_{a}\end{array}\right]Ã\left[\begin{array}{c}p\\ 0\\ 0\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ g\end{array}\right]=\left[\begin{array}{c}0\\ p{\stackrel{Ë}{z}}_{a}\\ \frac{{F}_{za}}{{M}_{a}}+g\end{array}\right]\\ \\ \left[\begin{array}{c}\stackrel{Ë}{p}\\ \stackrel{Ë}{q}\\ \stackrel{Ë}{r}\end{array}\right]=\left[\left[\begin{array}{c}{M}_{x}\\ {M}_{y}\\ {M}_{z}\end{array}\right]â\left[\begin{array}{c}p\\ q\\ r\end{array}\right]Ã\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\right]{\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]}^{â1}\\ =\left[\left[\begin{array}{c}{M}_{x}\\ 0\\ 0\end{array}\right]â\left[\begin{array}{c}p\\ q\\ 0\end{array}\right]Ã\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]\left[\begin{array}{c}p\\ 0\\ 0\end{array}\right]\right]{\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]}^{â1}=\left[\begin{array}{c}\frac{{M}_{x}}{{I}_{xx}}\\ 0\\ 0\end{array}\right]\end{array}
\begin{array}{l}{F}_{w{z}_{a,t}}=f\left({z}_{{v}_{a,t}}â{z}_{{w}_{a,t}},{\stackrel{Ë}{z}}_{{v}_{a,t}}â{\stackrel{Ë}{z}}_{{w}_{a,t}},{\mathrm{δ}}_{stee{r}_{a,t}}\right)\\ {M}_{v{z}_{a,t}}=f\left({z}_{{v}_{a,t}}â{z}_{{w}_{a,t}},{\stackrel{Ë}{z}}_{{v}_{a,t}}â{\stackrel{Ë}{z}}_{{w}_{a,t}},{\mathrm{δ}}_{stee{r}_{a,t}}\right)\end{array}
\begin{array}{l}{F}_{v{x}_{a,t}}={F}_{w{x}_{a,t}}\\ {F}_{v{y}_{a,t}}={F}_{w{y}_{a,t}}\\ {F}_{v{z}_{a,t}}=â{F}_{w{z}_{a,t}}\\ \\ {M}_{v{x}_{a,t}}={M}_{w{x}_{a,t}}+{F}_{w{y}_{a,t}}\left(R{e}_{w{y}_{a,t}}+{H}_{a,t}\right)\\ {M}_{v{y}_{a,t}}={M}_{w{y}_{a,t}}+{F}_{w{x}_{a,t}}\left(R{e}_{w{x}_{a,t}}+{H}_{a,t}\right)\\ {M}_{v{z}_{a,t}}={M}_{w{z}_{a,t}}\end{array}
δsteera,t
zva,t, żva,t
zwa,t, żwa,t
xva,t, ẋva,t
xwa,t, ẋwa,t
yva,t, áºva,t
ywa,t, áºwa,t
\left[\begin{array}{ccc}{\mathrm{ξ}}_{a,t}& {\mathrm{η}}_{a,t}& {\mathrm{ζ}}_{a,t}\end{array}\right]={G}_{alookup}f\left({z}_{{w}_{a,t}}â{z}_{{v}_{a,t}},{\mathrm{δ}}_{stee{r}_{a,t}}\right)
ξa,t
ηa,t
ζa,t
{\mathrm{δ}}_{whlstee{r}_{a,t}}={\mathrm{δ}}_{stee{r}_{a,t}}+{G}_{alookup}f\left({z}_{{w}_{a,t}}â{z}_{{v}_{a,t}},{\mathrm{δ}}_{stee{r}_{a,t}}\right)
δwhlsteera,t
{P}_{sus{p}_{a,t}}={F}_{wzlooku{p}_{a}}\left({\stackrel{Ë}{z}}_{{v}_{a,t}}â{\stackrel{Ë}{z}}_{{w}_{a,t}},{\stackrel{Ë}{z}}_{{v}_{a,t}}â{\stackrel{Ë}{z}}_{{w}_{a,t}},{\mathrm{δ}}_{stee{r}_{a,t}}\right)
{E}_{sus{p}_{a,t}}={F}_{wzlooku{p}_{a}}\left({\stackrel{Ë}{z}}_{{v}_{a,t}}â{\stackrel{Ë}{z}}_{{w}_{a,t}},{\stackrel{Ë}{z}}_{{v}_{a,t}}â{\stackrel{Ë}{z}}_{{w}_{a,t}},{\mathrm{δ}}_{stee{r}_{a,t}}\right)
{H}_{a,t}=â\left({z}_{{v}_{a,t}}â{z}_{{w}_{a,t}}â\mathrm{median}\left(f_susp_dz_bp\right)\right)
{z}_{wt{r}_{a,t}}=R{e}_{{w}_{a,t}}+{H}_{a,t}
\mathrm{WhlPz}={z}_{w}=\left[\begin{array}{cccc}{z}_{{w}_{1,1}}& {z}_{{w}_{1,2}}& {z}_{{w}_{2,1}}& {z}_{{w}_{2,2}}\end{array}\right]
\mathrm{Whl}\mathrm{Re}=R{e}_{w}=\left[\begin{array}{cccc}R{e}_{{w}_{1,1}}& R{e}_{{w}_{1,2}}& R{e}_{{w}_{2,1}}& R{e}_{{w}_{2,2}}\end{array}\right]
Track velocity, żw, along wheel-fixed z-axis, in m. Array dimensions are 1 by the total number of tracks on the vehicle.
\mathrm{WhlVz}={\stackrel{Ë}{z}}_{w}=\left[\begin{array}{cccc}{\stackrel{Ë}{z}}_{{w}_{1,1}}& {\stackrel{Ë}{z}}_{{w}_{1,2}}& {\stackrel{Ë}{z}}_{{w}_{2,1}}& {\stackrel{Ë}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlFx}={F}_{wx}=\left[\begin{array}{cccc}{F}_{w{x}_{1,1}}& {F}_{w{x}_{1,2}}& {F}_{w{x}_{2,1}}& {F}_{w{x}_{2,2}}\end{array}\right]
\mathrm{WhlFy}={F}_{wy}=\left[\begin{array}{cccc}{F}_{w{y}_{1,1}}& {F}_{w{y}_{1,2}}& {F}_{w{y}_{2,1}}& {F}_{w{y}_{2,2}}\end{array}\right]
Longitudinal, lateral, and vertical suspension moments at axle a, track t, applied to the wheel at the axle wheel carrier reference coordinate, in N·m. Input array dimensions are 3 by a*t.
\mathrm{WhlM}={M}_{w}=\left[\begin{array}{cccc}{M}_{w{x}_{1,1}}& {M}_{w{x}_{1,2}}& {M}_{w{x}_{2,1}}& {M}_{w{x}_{2,2}}\\ {M}_{w{y}_{1,1}}& {M}_{w{y}_{1,2}}& {M}_{w{y}_{2,1}}& {M}_{w{y}_{2,2}}\\ {M}_{w{z}_{1,1}}& {M}_{w{z}_{1,2}}& {M}_{w{z}_{2,1}}& {M}_{w{z}_{2,2}}\end{array}\right]
\mathrm{VehP}=\left[\begin{array}{c}{x}_{v}\\ {y}_{v}\\ {z}_{v}\end{array}\right]=\left[\begin{array}{cccc}{x}_{v}{}_{{}_{1,1}}& {x}_{v}{}_{{}_{1,2}}& {x}_{v}{}_{{}_{2,1}}& {x}_{v}{}_{{}_{2,2}}\\ {y}_{v}{}_{{}_{1,1}}& {y}_{v}{}_{{}_{1,2}}& {y}_{v}{}_{{}_{2,1}}& {y}_{v}{}_{{}_{2,2}}\\ {z}_{v}{}_{{}_{1,1}}& {z}_{v}{}_{{}_{1,2}}& {z}_{v}{}_{{}_{2,1}}& {z}_{v}{}_{{}_{2,2}}\end{array}\right]
\mathrm{VehV}=\left[\begin{array}{c}{\stackrel{Ë}{x}}_{v}\\ {\stackrel{Ë}{y}}_{v}\\ {\stackrel{Ë}{z}}_{v}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{Ë}{x}}_{{v}_{1,1}}& {\stackrel{Ë}{x}}_{{v}_{1,2}}& {\stackrel{Ë}{x}}_{{v}_{2,1}}& {\stackrel{Ë}{x}}_{{v}_{2,2}}\\ {\stackrel{Ë}{y}}_{{v}_{1,1}}& {\stackrel{Ë}{y}}_{{v}_{1,2}}& {\stackrel{Ë}{y}}_{{v}_{2,1}}& {\stackrel{Ë}{y}}_{{v}_{2,2}}\\ {\stackrel{Ë}{z}}_{{v}_{1,1}}& {\stackrel{Ë}{z}}_{{v}_{1,2}}& {\stackrel{Ë}{z}}_{{v}_{2,1}}& {\stackrel{Ë}{z}}_{{v}_{2,2}}\end{array}\right]
Optional steering angle for each wheel, δ. Input array dimensions are 1 by the number of steered tracks.
\mathrm{StrgAng}={\mathrm{δ}}_{steer}=\left[\begin{array}{cc}{\mathrm{δ}}_{stee{r}_{1,1}}& {\mathrm{δ}}_{stee{r}_{1,2}}\end{array}\right]
\mathrm{WhlAng}\left[1,...\right]=\mathrm{ξ}=\left[{\mathrm{ξ}}_{a,t}\right]
\mathrm{WhlAng}\left[2,...\right]=\mathrm{η}=\left[{\mathrm{η}}_{a,t}\right]
\mathrm{WhlAng}\left[3,...\right]=\mathrm{ζ}=\left[{\mathrm{ζ}}_{a,t}\right]
\mathrm{VehF}={F}_{v}=\left[\begin{array}{cccc}{F}_{v}{}_{{x}_{1,1}}& {F}_{v}{}_{{x}_{1,2}}& {F}_{v}{}_{{x}_{2,1}}& {F}_{v}{}_{{x}_{2,2}}\\ {F}_{v}{}_{{y}_{1,1}}& {F}_{v}{}_{{y}_{1,2}}& {F}_{v}{}_{{y}_{2,1}}& {F}_{v}{}_{{y}_{2,2}}\\ {F}_{v}{}_{{z}_{1,1}}& {F}_{v}{}_{{z}_{1,2}}& {F}_{v}{}_{{z}_{2,1}}& {F}_{v}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{VehM}={M}_{v}=\left[\begin{array}{cccc}{M}_{v{x}_{1,1}}& {M}_{v{x}_{1,2}}& {M}_{v{x}_{2,1}}& {M}_{v{x}_{2,2}}\\ {M}_{v{y}_{1,1}}& {M}_{v{y}_{1,2}}& {M}_{v{y}_{2,1}}& {M}_{v{y}_{2,2}}\\ {M}_{v{z}_{1,1}}& {M}_{v{z}_{1,2}}& {M}_{v{z}_{2,1}}& {M}_{v{z}_{2,2}}\end{array}\right]
\mathrm{WhlF}={F}_{w}=\left[\begin{array}{cccc}{F}_{w}{}_{{x}_{1,1}}& {F}_{w}{}_{{x}_{1,2}}& {F}_{w}{}_{{x}_{2,1}}& {F}_{w}{}_{{x}_{2,2}}\\ {F}_{w}{}_{{y}_{1,1}}& {F}_{w}{}_{{y}_{1,2}}& {F}_{w}{}_{{y}_{2,1}}& {F}_{w}{}_{{y}_{2,2}}\\ {F}_{w}{}_{{z}_{1,1}}& {F}_{w}{}_{{z}_{1,2}}& {F}_{w}{}_{{z}_{2,1}}& {F}_{w}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{WhlP}=\left[\begin{array}{c}{x}_{w}\\ {y}_{w}\\ {z}_{w}\end{array}\right]=\left[\begin{array}{cccc}{x}_{w}{}_{{}_{1,1}}& {x}_{w}{}_{{}_{1,2}}& {x}_{w}{}_{{}_{2,1}}& {x}_{{w}_{2,2}}\\ {y}_{w}{}_{{}_{1,1}}& {y}_{w}{}_{{}_{1,2}}& {y}_{w}{}_{{}_{2,1}}& {y}_{w}{}_{{y}_{2,2}}\\ {z}_{wtr}{}_{{}_{1,1}}& {z}_{wtr}{}_{{}_{1,2}}& {z}_{wtr}{}_{{}_{2,1}}& {z}_{wt{r}_{2,2}}\end{array}\right]
\mathrm{WhlV}=\left[\begin{array}{c}{\stackrel{Ë}{x}}_{w}\\ {\stackrel{Ë}{y}}_{w}\\ {\stackrel{Ë}{z}}_{w}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{Ë}{x}}_{{w}_{1,1}}& {\stackrel{Ë}{x}}_{{w}_{1,2}}& {\stackrel{Ë}{x}}_{{w}_{2,1}}& {\stackrel{Ë}{x}}_{{w}_{2,2}}\\ {\stackrel{Ë}{y}}_{{w}_{{}_{1,1}}}& {\stackrel{Ë}{y}}_{{w}_{1,2}}& {\stackrel{Ë}{y}}_{{w}_{2,1}}& {\stackrel{Ë}{y}}_{{w}_{2,2}}\\ {\stackrel{Ë}{z}}_{{w}_{{}_{1,1}}}& {\stackrel{Ë}{z}}_{{w}_{1,2}}& {\stackrel{Ë}{z}}_{{w}_{2,1}}& {\stackrel{Ë}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlAng}=\left[\begin{array}{c}\mathrm{ξ}\\ \mathrm{η}\\ \mathrm{ζ}\end{array}\right]=\left[\begin{array}{cccc}{\mathrm{ξ}}_{1,1}& {\mathrm{ξ}}_{1,2}& {\mathrm{ξ}}_{2,1}& {\mathrm{ξ}}_{2,2}\\ {\mathrm{η}}_{1,1}& {\mathrm{η}}_{1,2}& {\mathrm{η}}_{2,1}& {\mathrm{η}}_{2,2}\\ {\mathrm{ζ}}_{1,1}& {\mathrm{ζ}}_{1,2}& {\mathrm{ζ}}_{2,1}& {\mathrm{ζ}}_{2,2}\end{array}\right]
\mathrm{VehF}={F}_{v}=\left[\begin{array}{cccc}{F}_{v}{}_{{x}_{1,1}}& {F}_{v}{}_{{x}_{1,2}}& {F}_{v}{}_{{x}_{2,1}}& {F}_{v}{}_{{x}_{2,2}}\\ {F}_{v}{}_{{y}_{1,1}}& {F}_{v}{}_{{y}_{1,2}}& {F}_{v}{}_{{y}_{2,1}}& {F}_{v}{}_{{y}_{2,2}}\\ {F}_{v}{}_{{z}_{1,1}}& {F}_{v}{}_{{z}_{1,2}}& {F}_{v}{}_{{z}_{2,1}}& {F}_{v}{}_{{z}_{2,2}}\end{array}\right]
Longitudinal, lateral, and vertical suspension moment at axle a, track t, applied to the vehicle at the suspension connection point, in N·m. Array dimensions are 3 by a*t.
\mathrm{VehM}={M}_{v}=\left[\begin{array}{cccc}{M}_{v{x}_{1,1}}& {M}_{v{x}_{1,2}}& {M}_{v{x}_{2,1}}& {M}_{v{x}_{2,2}}\\ {M}_{v{y}_{1,1}}& {M}_{v{y}_{1,2}}& {M}_{v{y}_{2,1}}& {M}_{v{y}_{2,2}}\\ {M}_{v{z}_{1,1}}& {M}_{v{z}_{1,2}}& {M}_{v{z}_{2,1}}& {M}_{v{z}_{2,2}}\end{array}\right]
\mathrm{WhlF}={F}_{w}=\left[\begin{array}{cccc}{F}_{w}{}_{{x}_{1,1}}& {F}_{w}{}_{{x}_{1,2}}& {F}_{w}{}_{{x}_{2,1}}& {F}_{w}{}_{{x}_{2,2}}\\ {F}_{w}{}_{{y}_{1,1}}& {F}_{w}{}_{{y}_{1,2}}& {F}_{w}{}_{{y}_{2,1}}& {F}_{w}{}_{{y}_{2,2}}\\ {F}_{w}{}_{{z}_{1,1}}& {F}_{w}{}_{{z}_{1,2}}& {F}_{w}{}_{{z}_{2,1}}& {F}_{w}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{WhlV}=\left[\begin{array}{c}{\stackrel{Ë}{x}}_{w}\\ {\stackrel{Ë}{y}}_{w}\\ {\stackrel{Ë}{z}}_{w}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{Ë}{x}}_{{w}_{1,1}}& {\stackrel{Ë}{x}}_{{w}_{1,2}}& {\stackrel{Ë}{x}}_{{w}_{2,1}}& {\stackrel{Ë}{x}}_{{w}_{2,2}}\\ {\stackrel{Ë}{y}}_{{w}_{{}_{1,1}}}& {\stackrel{Ë}{y}}_{{w}_{1,2}}& {\stackrel{Ë}{y}}_{{w}_{2,1}}& {\stackrel{Ë}{y}}_{{w}_{2,2}}\\ {\stackrel{Ë}{z}}_{{w}_{{}_{1,1}}}& {\stackrel{Ë}{z}}_{{w}_{1,2}}& {\stackrel{Ë}{z}}_{{w}_{2,1}}& {\stackrel{Ë}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlAng}=\left[\begin{array}{c}\mathrm{ξ}\\ \mathrm{η}\\ \mathrm{ζ}\end{array}\right]=\left[\begin{array}{cccc}{\mathrm{ξ}}_{1,1}& {\mathrm{ξ}}_{1,2}& {\mathrm{ξ}}_{2,1}& {\mathrm{ξ}}_{2,2}\\ {\mathrm{η}}_{1,1}& {\mathrm{η}}_{1,2}& {\mathrm{η}}_{2,1}& {\mathrm{η}}_{2,2}\\ {\mathrm{ζ}}_{1,1}& {\mathrm{ζ}}_{1,2}& {\mathrm{ζ}}_{2,1}& {\mathrm{ζ}}_{2,2}\end{array}\right]
\mathrm{StrgAng}={\mathrm{δ}}_{steer}=\left[\begin{array}{cc}{\mathrm{δ}}_{stee{r}_{1,1}}& {\mathrm{δ}}_{stee{r}_{1,2}}\end{array}\right]
T{c}_{t}=\left[\begin{array}{cccc}{x}_{{w}_{1,1}}& {x}_{{w}_{1,2}}& {x}_{{w}_{2,1}}& {x}_{{w}_{2,2}}\\ {y}_{{w}_{1,1}}& {y}_{{w}_{1,2}}& {y}_{{w}_{2,1}}& {y}_{{w}_{2,2}}\\ {z}_{{w}_{1,1}}& {z}_{{w}_{1,2}}& {z}_{{w}_{2,1}}& {z}_{{w}_{2,2}}\end{array}\right]
S{c}_{t}=\left[\begin{array}{cccc}{x}_{{s}_{1,1}}& {x}_{{s}_{1,2}}& {x}_{{s}_{2,1}}& {x}_{{s}_{2,2}}\\ {y}_{{s}_{1,1}}& {y}_{{s}_{1,2}}& {y}_{{s}_{2,1}}& {y}_{{s}_{2,2}}\\ {z}_{{s}_{1,1}}& {z}_{{s}_{1,2}}& {z}_{{s}_{2,1}}& {z}_{{s}_{2,2}}\end{array}\right]
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Colligative_properties Knowpia
In chemistry, colligative properties are those properties of solutions that depend on the ratio of the number of solute particles to the number of solvent particles in a solution, and not on the nature of the chemical species present.[1] The number ratio can be related to the various units for concentration of a solution such as molarity, molality, normality (chemistry), etc. The assumption that solution properties are independent of nature of solute particles is exact only for ideal solutions, which are solutions that exhibit thermodynamic properties analogous to those of an ideal gas, and is approximate for dilute real solutions. In other words, colligative properties are a set of solution properties that can be reasonably approximated by the assumption that the solution is ideal.
Only properties which result from the dissolution of a nonvolatile solute in a volatile liquid solvent are considered.[2] They are essentially solvent properties which are changed by the presence of the solute. The solute particles displace some solvent molecules in the liquid phase and thereby reduce the concentration of solvent and increase its entropy, so that the colligative properties are independent of the nature of the solute. The word colligative is derived from the Latin colligatus meaning bound together.[3] This indicates that all colligative properties have a common feature, namely that they are related only to the number of solute molecules relative to the number of solvent molecules and not to the nature of the solute.[4]
Colligative properties include:
Relative lowering of vapor pressure (Raoult's law)
For a given solute-solvent mass ratio, all colligative properties are inversely proportional to solute molar mass.
Measurement of colligative properties for a dilute solution of a non-ionized solute such as urea or glucose in water or another solvent can lead to determinations of relative molar masses, both for small molecules and for polymers which cannot be studied by other means. Alternatively, measurements for ionized solutes can lead to an estimation of the percentage of dissociation taking place.
Colligative properties are studied mostly for dilute solutions, whose behavior may be approximated as that of an ideal solution. In fact, all of the properties listed above are colligative only in the dilute limit: at higher concentrations, the freezing point depression, boiling point elevation, vapor pressure elevation or depression, and osmotic pressure are all dependent on the chemical nature of the solvent and the solute.
Relative lowering of vapor pressureEdit
A vapor is a substance in a gaseous state at a temperature lower than its critical point. Vapor Pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its solid or liquid state. The vapor pressure of a solvent is lowered when a non-volatile solute is dissolved in it to form a solution.
For an ideal solution, the equilibrium vapor pressure is given by Raoult's law as
{\displaystyle p=p_{\rm {A}}^{\star }x_{\rm {A}}+p_{\rm {B}}^{\star }x_{\rm {B}}+\cdots ,}
{\displaystyle p_{\rm {i}}^{\star }}
is the vapor pressure of the pure component (i= A, B, ...) and
{\displaystyle x_{\rm {i}}}
is the mole fraction of the component in the solution
For a solution with a solvent (A) and one non-volatile solute (B),
{\displaystyle p_{\rm {B}}^{\star }=0}
{\displaystyle p=p_{\rm {A}}^{\star }x_{\rm {A}}}
The vapor pressure lowering relative to pure solvent is
{\displaystyle \Delta p=p_{\rm {A}}^{\star }-p=p_{\rm {A}}^{\star }(1-x_{\rm {A}})=p_{\rm {A}}^{\star }x_{\rm {B}}}
, which is proportional to the mole fraction of solute.
If the solute dissociates in solution, then the number of moles of solute is increased by the van 't Hoff factor
{\displaystyle i}
, which represents the true number of solute particles for each formula unit. For example, the strong electrolyte MgCl2 dissociates into one Mg2+ ion and two Cl− ions, so that if ionization is complete, i = 3 and
{\displaystyle \Delta p=p_{\rm {A}}^{\star }x_{\rm {B}}}
{\displaystyle x_{\rm {B}}}
is calculated with moles of solute i times initial moles and moles of solvent same as initial moles of solvent before dissociation. The measured colligative properties show that i is somewhat less than 3 due to ion association.
Boiling point and freezing pointEdit
Addition of solute to form a solution stabilizes the solvent in the liquid phase, and lowers the solvent's chemical potential so that solvent molecules have less tendency to move to the gas or solid phases. As a result, liquid solutions slightly above the solvent boiling point at a given pressure become stable, which means that the boiling point increases. Similarly, liquid solutions slightly below the solvent freezing point become stable meaning that the freezing point decreases. Both the boiling point elevation and the freezing point depression are proportional to the lowering of vapor pressure in a dilute solution.
These properties are colligative in systems where the solute is essentially confined to the liquid phase. Boiling point elevation (like vapor pressure lowering) is colligative for non-volatile solutes where the solute presence in the gas phase is negligible. Freezing point depression is colligative for most solutes since very few solutes dissolve appreciably in solid solvents.
Boiling point elevation (ebullioscopy)Edit
The boiling point of a liquid at a given external pressure is the temperature (
{\displaystyle T_{\rm {b}}}
) at which the vapor pressure of the liquid equals the external pressure. The normal boiling point is the boiling point at a pressure equal to 1 atm.
The boiling point of a pure solvent is increased by the addition of a non-volatile solute, and the elevation can be measured by ebullioscopy. It is found that
{\displaystyle \Delta T_{\rm {b}}=T_{\rm {b,{\text{solution}}}}-T_{\rm {b,{\text{pure solvent}}}}=i\cdot K_{b}\cdot m}
Here i is the van 't Hoff factor as above, Kb is the ebullioscopic constant of the solvent (equal to 0.512 °C kg/mol for water), and m is the molality of the solution.
The boiling point is the temperature at which there is equilibrium between liquid and gas phases. At the boiling point, the number of gas molecules condensing to liquid equals the number of liquid molecules evaporating to gas. Adding a solute dilutes the concentration of the liquid molecules and reduces the rate of evaporation. To compensate for this and re-attain equilibrium, the boiling point occurs at a higher temperature.
If the solution is assumed to be an ideal solution, Kb can be evaluated from the thermodynamic condition for liquid-vapor equilibrium. At the boiling point the chemical potential μA of the solvent in the solution phase equals the chemical potential in the pure vapor phase above the solution.
{\displaystyle \mu _{A}(T_{b})=\mu _{A}^{\star }(T_{b})+RT\ln x_{A}\ =\mu _{A}^{\star }(g,1\,\mathrm {atm} ),}
where the asterisks indicate pure phases. This leads to the result
{\displaystyle K_{b}=RMT_{b}^{2}/\Delta H_{\mathrm {vap} }}
, where R is the molar gas constant, M is the solvent molar mass and ΔHvap is the solvent molar enthalpy of vaporization.[6]
Freezing point depression (cryoscopy)Edit
The freezing point (
{\displaystyle T_{\rm {f}}}
) of a pure solvent is lowered by the addition of a solute which is insoluble in the solid solvent, and the measurement of this difference is called cryoscopy. It is found that
{\displaystyle \Delta T_{\rm {f}}=T_{\rm {f,{\text{solution}}}}-T_{\rm {f,{\text{pure solvent}}}}=-i\cdot K_{f}\cdot m}
[5] (which can also be written as
{\displaystyle \Delta T_{\rm {f}}=T_{\rm {f,{\text{pure solvent}}}}-T_{\rm {f,{\text{solution}}}}=i\cdot K_{f}\cdot m}
Here Kf is the cryoscopic constant (equal to 1.86 °C kg/mol for the freezing point of water), i is the van 't Hoff factor, and m the molality (in mol/kg). This predicts the melting of ice by road salt.
In the liquid solution, the solvent is diluted by the addition of a solute, so that fewer molecules are available to freeze. Re-establishment of equilibrium is achieved at a lower temperature at which the rate of freezing becomes equal to the rate of liquefying. At the lower freezing point, the vapor pressure of the liquid is equal to the vapor pressure of the corresponding solid, and the chemical potentials of the two phases are equal as well. The equality of chemical potentials permits the evaluation of the cryoscopic constant as
{\displaystyle K_{f}=RMT_{f}^{2}/\Delta _{\mathrm {fus} }H}
, where ΔfusH is the solvent molar enthalpy of fusion.[6]
The osmotic pressure of a solution is the difference in pressure between the solution and the pure liquid solvent when the two are in equilibrium across a semipermeable membrane, which allows the passage of solvent molecules but not of solute particles. If the two phases are at the same initial pressure, there is a net transfer of solvent across the membrane into the solution known as osmosis. The process stops and equilibrium is attained when the pressure difference equals the osmotic pressure.
The osmotic pressure of a solution is directly proportional to its absolute temperature.[7]
These are analogous to Boyle's law and Charles's law for gases. Similarly, the combined ideal gas law,
{\displaystyle PV=nRT}
, has as an analogue for ideal solutions
{\displaystyle \Pi V=nRTi}
{\displaystyle \Pi }
is osmotic pressure; V is the volume; n is the number of moles of solute; R is the molar gas constant 8.314 J K−1 mol−1; T is absolute temperature; and i is the Van 't Hoff factor.
The osmotic pressure is then proportional to the molar concentration
{\displaystyle c=n/V}
{\displaystyle \Pi ={\frac {nRTi}{V}}=cRTi}
The osmotic pressure is proportional to the concentration of solute particles ci and is therefore a colligative property.
As with the other colligative properties, this equation is a consequence of the equality of solvent chemical potentials of the two phases in equilibrium. In this case the phases are the pure solvent at pressure P and the solution at total pressure (P +
{\displaystyle \Pi }
The word colligative (Latin: co, ligare) was introduced in 1891 by Wilhelm Ostwald. Ostwald classified solute properties in three categories:[9][10]
colligative properties, which depend only on solute concentration and temperature and are independent of the nature of the solute particles
additive properties such as mass, which are the sums of properties of the constituent particles and therefore depend also on the composition (or molecular formula) of the solute, and
constitutional properties, which depend further on the molecular structure of the given solute.
^ McQuarrie, Donald, et al. Colligative properties of Solutions" General Chemistry Mill Valley: Library of Congress, 2011. ISBN 978-1-89138-960-3.
^ KL Kapoor Applications of Thermodynamics Volume 3
^ K.J. Laidler and J.L. Meiser, Physical Chemistry (Benjamin/Cummings 1982), p.196
^ Castellan, Gilbert W. (1983). Physical Chemistry (3rd ed.). Addison-Wesley. p. 281. ISBN 978-0201103861. Retrieved 20 July 2019.
^ a b Tro, Nivaldo J. (2018). Chemistry; Structure and Properties (Textbook.) (2nd ed.). Pearson Education. pp. 563–566. ISBN 978-0-134-52822-9.
^ a b T. Engel and P. Reid, Physical Chemistry (Pearson Benjamin Cummings 2006) p.204-5
^ "Van't Hoff's Laws of Osmotic Pressure - QS Study". qsstudy.com. Retrieved 2022-03-08.
^ Engel and Reid p.207
^ W.B. Jensen, J. Chem. Educ. 75, 679 (1998) Logic, History, and the Chemistry Textbook I. Does Chemistry Have a Logical Structure?
^ H.W. Smith, Circulation 21, 808 (1960) Theory of Solutions: A Knowledge of the Laws of Solutions ...
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Estimate transaction cost - Nomadic Labs knowledge center
To measure the cost of a transaction in fiat currency, we calculate as follows
\frac{\text{gas consumed} \times \text{median gas cost} \times \text{tez price}} {1,000,000} = \text{tx price}
We take as an example the median gas cost¹, and the tez price on October 7, 2021:
median gas cost: 0.1 / gas unit
tez price: 6.7
We use the following values for the gas usage:
Hic et Nunc NFT minting: 94,000 gas units
Hic et Nunc NFT transfer: 123,000 gas units
tez transfer: 1,500 gas units
This cost calculation is indicative only. It’s not absolute nor guaranteed; it’s a cost estimate based on figures which are valid at one particular moment.
[1]: We took the median value in a range of 13 gas cost values.
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s
\mathbf{B}\prime \left(s\right)= -\mathrm{τ} \mathbf{N}
By definition, the torsion is
\mathrm{τ}= -∥\mathbf{B}\prime \left(s\right)∥
. Since B is a unit vector,
\mathbf{B}·\mathbf{B}=1
\left(\mathbf{B}·\mathbf{B}\right)\prime =2 \mathbf{B}·\mathbf{B}\prime =0
\mathbf{B}\prime
is orthogonal to B. But B is orthogonal to the plane containing T and N, so
\mathbf{B}\prime
must lie in that plane.
\mathbf{B}·\mathbf{T}=0
\mathbf{B}\prime ·\mathbf{T}+\mathbf{B}·\mathbf{T}\prime =0
\mathbf{B}\prime ·\mathbf{T}
= -\mathbf{B}·\mathbf{T}\prime
= -\mathbf{B}·\left(\mathrm{κ} \mathbf{N}\right)
=0
\mathbf{T}\prime =\mathrm{κ} \mathbf{N}
\mathrm{κ}
\mathbf{B}\prime
, already in the osculating plane, is now orthogonal to T. Hence, it must be proportional to N, with the constant of proportionality taken as
-\mathrm{τ}
, that is, as
∥\mathbf{B}\prime ∥
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Irreducible components of minuscule affine Deligne–Lusztig varieties
Paul Hamacher and Eva Viehmann
We examine the set of
{J}_{b}\left(F\right)
-orbits in the set of irreducible components of affine Deligne–Lusztig varieties for a hyperspecial subgroup and minuscule coweight
\mu
. Our description implies in particular that its number of elements is bounded by the dimension of a suitable weight space in the Weyl module associated with
\mu
of the dual group.
affine Deligne–Lusztig variety, Rapoport–Zink spaces, affine Grassmannian
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Games of gain-ground/Real tennis - Wikiversity
< Games of gain-ground
Previous chapter: Longue paume
Next chapter : Ballon au poing
{\displaystyle {\mathcal {Real~tennis~(or~Courte~paume)}}}
2.1 Chasses
Real tennis is a racquet sport. It is also known as court tennis in the United States, formerly royal tennis in England and Australia, and courte-paume in France.
An example layout of a tennis court.
A valid serve must occur from the serving court before the second gallery line, and hit the service penthouse before dropping in the receiving court, marked by the service line and fault line.
Corresponding chasse lines extend from the centers of the side galleries on both service and hazard ends, including the first, door, second and last. Gallery posts and the net post are marked with circles. Shaded areas are the winning openings, the dedans, grille, and winning gallery. None of these, nor the posts, would be visible with an actual overhead view as depicted.
There are two basic designs in existence today: jeu quarré, which is an older design, and jeu à dedans. The more common real tennis court is jeu à dedans. It is enclosed by walls on all four sides, three of which have sloping roofs ("penthouses") beneath which are various openings ("galleries"), and a buttress that intrudes into the playing area (tambour) off which shots may be played.
The service is always made from the same end of the court (the "service" end); a good service must touch the side penthouse (above and to the left of the server) on the receiver's ("hazard") side of the court before first touching the floor in a marked area on that side. The resulting back-court volleys and the possibility of hitting shots off the sidewalls and the sloping penthouses give many interesting shot choices not available in lawn tennis.
Chasses[edit | edit source]
When the ball bounces twice on the floor at the service end, a "chasse" (chase) is called where the ball made its second bounce and the server gets the chance, later in the game, to "play off" the chasse from the receiving end; but to win the point being played off, their shot's second bounce must be further from the net (closer to the back wall) than the shot they originally failed to reach. A chasse can also be called at the receiving ("hazard") end, but only on the half of that end nearest the net; this is called a "hazard" chasse.
Those areas of the court in which chasses can be called are marked with lines running across the floor, parallel to the net, generally about 1-yard (0.91 m) apart – it is these lines by which the chasses are measured. Additionally, a player can gain the advantage of serving only through skillful play (viz. "laying" a "chasse", which ensures a change of end). This is in stark contrast to lawn tennis, where players alternately serve and receive entire games. In real tennis, the service can only change during a game, and it is not uncommon to see a player serve for several consecutive games till a chasse be made. Indeed, in theory, an entire match could be played with no change of service, the same player serving every point.
When there are two chasses, then the receiver becomes a server, i.e. he crosses side service so-called Dedans (or only one chasse if one of the players reaches 40 or advantage in a game).
Scoring is by fifteens (15, 30, 40) as in lawn tennis. In real tennis, six games wins a set, without the need for a two-game buffer (although some tournaments play up to nine games per set). A match is typically best of three sets, except for the major open tournaments, in which matches are best of five sets for men, and the best of three sets for women.
Various window-like openings below the penthouse roofs offer the player a chance to win the point instantly when the ball is hit into the opening
These "goals" to be aimed for are =
The largest opening, located behind the server, is called the "dedans" and must often be defended on the volley from hard-hit shots, called "forces", coming from the receiving ("hazard") side of the court.
the grille located behind the receiver,
In other cases, these windows in the gallery create a "chasse".
jeu à dedans court,
view toward service end
view toward hazard end
Wikimedia Commons has media related to Category:Jeu de paume.
Retrieved from "https://en.wikiversity.org/w/index.php?title=Games_of_gain-ground/Real_tennis&oldid=2182953"
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Using Neural Networks to Predict Secondary Structure for Protein Folding
Ali Abdulhafidh Ibrahim1, Ibrahim Sabah Yasseen2
1College of Science, Al-Nahrain University, Baghdad, Iraq.
2College of Information Engineering, Al-Nahrain University, Baghdad, Iraq.
Protein Secondary Structure Prediction (PSSP) is considered as one of the major challenging tasks in bioinformatics, so many solutions have been proposed to solve that problem via trying to achieve more accurate prediction results. The goal of this paper is to develop and implement an intelligent based system to predict secondary structure of a protein from its primary amino acid sequence by using five models of Neural Network (NN). These models are Feed Forward Neural Network (FNN), Learning Vector Quantization (LVQ), Probabilistic Neural Network (PNN), Convolutional Neural Network (CNN), and CNN Fine Tuning for PSSP. To evaluate our approaches two datasets have been used. The first one contains 114 protein samples, and the second one contains 1845 protein samples.
Protein Secondary Structure Prediction (PSSP), Neural Network (NN), α-Helix (H), β-Sheet (E), Coil (C), Feed Forward Neural Network (FNN), Learning Vector Quantization (LVQ), Probabilistic Neural Network (PNN), Convolutional Neural Network (CNN)
Ibrahim, A. and Yasseen, I. (2017) Using Neural Networks to Predict Secondary Structure for Protein Folding. Journal of Computer and Communications, 5, 1-8. doi: 10.4236/jcc.2017.51001.
Bioinformatics involves the technology that uses computers for storage, retrieval, manipulation, and distribution of information related to biological macromolecules such as DNA, RNA, and proteins. The use of computers is absolutely essential in mining genomes for information gathering and knowledge building [1] .
Protein structure prediction methods are categorized under bioinformatics which is a broad field that combines many other fields and disciplines like biology, biochemistry, information technology, statistics, and mathematics [2] .
There are four different structure types of proteins, namely the Primary, Secondary, Tertiary and Quaternary structures. The Primary structure contains a sequence of 20 different types of amino acids. It provides the foundation of all the other types of structures. The Secondary structure refers to the arrangement of connections within the amino acid groups to form three different structured classes (H, E, and C) [3] .
PSSP provides a significant first step toward the tertiary structure prediction, as well as offering information about protein activity, relationship, and function. Protein folding, or the prediction of the tertiary structure from linear sequence, is an unsolved and ubiquitous problem that invites research from many fields of study, including computer science, molecular biology, biochemistry and other. Protein secondary structure is also used in a variety of scientific areas, including proteome and gene annotation. Therefore, PSSP remains as an active area of research, and an integral part of protein analysis [4] .
In this research, the authors have proposed five models of NN that has been used, including FNN, LVQ, PNN, CNN and CNN Fine tuning for PSSP. The main objective of this work is to gain an improvement of prediction accuracy (Q3) so that the implementation results show that the proposed model (CNN Fine Tuning) performs better than the other models and looks promising for problems with characteristics similar to that problem (PSSP) by achieving prediction accuracy with Q3 = 90.31%.
In this section, we will introduce dataset description, measures of prediction accuracy.
The first dataset is obtained from matlab math work [5] and from thesis [3] , it is contain 114 protein samples which are divided into training dataset with 75 protein samples and testing dataset with 44 protein sample. It is contains 28.3% α-helix (H), 21.3% β-sheet (E) and 50.4% coil (C). The second dataset is formed of proteins from four different classes. We used this dataset from bioinformatics information Lab from University of Missouri, United States of America, it is contains 1854 protein. The first dataset are used for (FNN, LVQ and PNN) the second one is used for CNN and CNN fine tuning.
2.2. Measures of Prediction Accuracy
We have used one measuring method to evaluate the prediction accuracy of implemented models of NN. The three state accuracy (
{Q}_{3}
) is defined as the percent of residues that have been predicted correctly:
{Q}_{3}=\frac{{N}_{H}+{N}_{E}+{N}_{C}}{{N}_{T}}
{N}_{H}
{N}_{E}
{N}_{C}
are the number of correctly predicted residues of type H, E and C, respectively and NT is the total number of residues in dataset.
{Q}_{3}
concise as useful measure to compare different prediction methods [6] .
In our work, we have used five different structures of Neural Networks including (Feed Forward NN, Learning vector quantization NN, Probabilistic NN). We used a sliding window of size 17 for each structure of NN that which moves through the protein sequence and the output of the network is attained for the residue in the middle of the window; so as a result, the input layer includes 17 × 20 = 340 neurons and output layer contain 3 neurons for each NN structure. During training, it receives the input vectors along with the expected output vectors. When making predictions, it returns output vectors representing the likelihood of each residue being in (H, E or C). Figure 1 illustrates a general structure of NN classifier (for only FNN,PNN and LVQ) that is receiving several input vectors and returning the predicted output vectors, comparing it with what could be the correct (expected output) classification.
3.1. Feed Forward Neural Network
The first structure of NN used are (Feed Forward NN), by using one input layer and two hidden layer with 10 neuron for each layer and one output layer as shown in Figure 2 that illustrates implemented FNN structure using Matlab Version (R2015a).
In FNN the processing units in each hidden layer are fully connected to units in previous layer but not connected to units in the same layer. Only the outputs of the unit are connected to the units of next layer. Therefore there is no feedback in the system [7] .
Figure 1. Classifier (NN) general structure.
Figure 2. Feed forward neural network architecture.
Figure 3. Architecture of PNN [10] .
3.2. Probabilistic Neural Networks (PNN)
The second structures of NN used is (Probabilistic NN), PNN is defined as an implementation of statistical algorithm called Kernel discriminate analysis in which the operations are organized into multilayered feed forward network with four layers: input layer, pattern layer, summation layer and output layer [8] , as shown in Figure 3.
It is usually much faster to train a PNN network than multilayer Perceptron Network but one of the disadvantages of PNN models compared to other networks is that PNN models has a large number of neurons in hidden nodes (pattern layer) due to the fact that there is one neuron for each training line [9] . Our implemented structure for PNN including one input layer with 340 neurons, and pattern layer with 14,151 neurons (one neuron for each amino acid), and three neurons for both summation and output layers as shown in Figure 4 that illustrate implemented PNN structure using Matlab (R2015a).
3.3. Learning Vector Quantization (LVQ)
The Third implemented structure of NN is LVQ; its structure has two layers is a competitive layer and linear layer [11] , as shown in Figure 5.
LVQ is a method for training competitive layers in a supervised manner. The competitive layer learns to classify input vectors in much the same way as the competitive layers of Self-Organizing Feature Maps. The linear layer transforms the competitive layer's classes into target classifications defined by the user. The classes learned by the competitive layer are referred to as subclasses and the classes of the linear layer as target classes [10] . Our implemented structure for LVQ includes one input layer with 340 neurons, and competitive layer with 10 neurons, and three neurons for both linear and output layers, as shown in Figure 6 that illustrates implemented LVQ structure using Matlab (R2015a).
CNN is a multilayer perceptron designed specifically to recognize two dimen-
Figure 4. Implemented Architecture of PNN.
Figure 5. Architecture of LVQ [11] .
Figure 6. Implemented Architecture of LVQ.
sional shapes with a high degree of invariance to translation, scaling, skewing, and other forms of distortion. Figure 7 shows the architectural layout of convolutional network made up of an input layer, four hidden layers, and an output layer. This network is designed to perform image processing (e.g., recognition of handwritten characters) [12] .
In our work we implement two different structures of CNN; the first implemented structure of CNN has six layers. The first layer is the input layer (input: 21 × 15), the second layer is the filter layer (Filter: 21 × 4 × 30), the third layer is the convolutional layer (Convolution: 30 × 12), the forth layer is the pooling layer (Pooling 30 × 6), and the fifth layer is the classifier layer which is (softmax classification). Finally, the last layer is the output layer. Figure 8 describes CNN classifier general structure that is receiving several input vectors and returning predicted output vectors.
The second implemented structure of CNN we used is a CNN fine tuning approach to tune the parameter of the whole model as step to increase the accuracy and find more accurate prediction for the secondary structure of the protein by replace the softmax activation function that is found in Figure 8 of the CNN first structure used by sigmoid activation function and do the back propagation approach for each epoch to tune the parameter.
Figure 7. CNN for image processing such as handwriting recognition [12] .
Figure 8. CNN classifier general structure [12] .
In this section we will display the result of five implemented models of NN, in addition previously are mentioned Q3 as three state accuracy measures will use QH, QE and QC are the percentage of correctly predicted residues observed in class E, H and C, respectively, as shown in Figure 9.
Figure 9 visualizes comparison between the five implemented models of NN. This figure compares the average of QH, QE, QC and Q3 of these three structures and shows that in Feed Forward NN give higher prediction accuracy Q3 than PNN and LVQ and more balanced prediction of three secondary structures and less difference between prediction accuracy of H, E and C. In PNN structure there is high difference between prediction accuracy of (H, E) and C in other side. This is because of class imbalance problem in protein secondary structure datasets which causes that the classifiers give more importance to majority class (C). In LVQ show that this structure can be train and predict only coil (C) so it has higher prediction accuracy (100%) and completely predicted correctly and other classes (H, E) completely predicted wrongly (0%). Finally it has been proven that CNN Non Fine Tuning and CNN Fine Tuning can obtain higher prediction accuracy than all other implemented structures by achieving prediction accuracy (61.694%) for Non Fine Tuning and (90.31%) for Fine Tuning model.
In this work, we presented five structures of NN, including FNN, PNN, LVQ, CNN and CNN fine tuning for prediction of protein secondary structures. The results show that CNN fine tuning network can achieve better performance and can improve prediction accuracy when compared to other structures by achieving prediction accuracy with (90.31%). CNN and CNN Fine Tuning need a
Figure 9. Final comparison of prediction accuracy for all models.
large amount of data in the dataset to maximize their effectiveness, approximately more than 500 protein samples in the dataset. The results also show that FNN network can achieve better prediction accuracy when compared with only PNN and LVQ. For PNN structure, there is high difference between prediction accuracy of (H, E) and C in other side, and PNN has faster training time and large number of neurons in hidden nodes in comparison with another implemented structures. For LVQ, it has the slowest training time in comparison with other structures and this structure can be trained and predicted only in coil (C), so it has higher prediction accuracy (100%) and completely predicts correctly, and other classes (H, E) completely predict wrongly (0%) due to its limited capability to classify complex problem as our problem(PSSP).
[1] Xiong, J. (2006) Essential Bioinformatics. Cambridge University Press, Cambridge.
[2] Buatan, K. (2007) Protein Secondary Structure Prediction from Amino Acid Sequence Using Artificial Intelligence Technique.
[3] Tsilo, L.C. (2008) Protein Secondary Structure Prediction Using Neural Networks. Doctoral Dissertation, Rhodes University, Grahamstown.
[4] Pollastri, G., Martin, A.J., Mooney, C. and Vullo, A. (2007) Accurate Prediction of Protein Secondary Structure and Solvent Accessibility by Consensus Combiners of Sequence and Structure Information. BMC Bioinformatics, 8, 1.
[5] Mathwork. Last Visited 10 June 2016.
http://www.mathworks.com/help/bioinfo/examples/predicting-protein-secondary-structure-using-a-neural-network.html
[6] Singh, M. (2006) Predicting Protein Secondary and Super Secondary Structure.
[7] Schmidt, W.F., Kraaijveld, M.A. and Duin, R.P. (1992) Feedforward Neural Networks with Random Weights. Conference B: Pattern Recognition Methodology and Systems, Proceedings of 11th IAPR International Conference on Pattern Recognition, 2, 1-4.
[8] Rao, P.N., Devi, T.U., Kaladhar, D., Sridhar, G. and Rao, A.A. (2009) A Probabilistic Neural Network Approach for Protein Superfamily Classification. Journal of Theoretical and Applied Information Technology, 6, 101-105.
[9] Sawant, S.S. (2015) Introduction to Probabilistic Neural Network—Used For Image Classifications. College of Engineering and Research, Pune.
[10] Naoum, R.S. and Al-Sultani, Z.N. (2013) Hibrid System of Learning Vector Quantization and Enhanced Resilient Backpropagation Artificial Neural Network for Intrusion Classification. International Journal of Research and Reviews in Applied Sciences (IJRRAS), 14, 2.
[11] Soleiman, E.M. (2014) Intrusion Detection System Using Supervised Learning Vector Quantization. Maleke Ashtar University of Technology, Tehran.
[12] Haykin, S.S., Haykin, S.S., Haykin, S.S. and Haykin, S.S. (2009) Neural Networks and Learning Machines. Vol. 3, Pearson, Upper Saddle River.
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Containers/Guarantees for resources - OpenVZ Virtuozzo Containers Wiki
This page describes how guarantees for resources can be implemented.
How to guarantee a guarantee
It's not obvious at the first glance, but there are only two ways of how a guarantee can be provided:
reserve desired amount in advance
limit consumers to keep some amount free
The first way has the followong disadvantages:
Reservation is impossible for certain resources
such as CPU time, disk or network bandwidth and similar can not be just reserved as their amount instantly increases;
Reserved amount is essentially a limit, but much more strict
cutting off X megabytes from RAM implies that all the rest groups are limited in their RAM consumption;
Reservation reduces containers density
if one wants to run some identical containers, each requiring 100Mb on 1Gb system, reservations can be done for only 10 containers, and starting the 11th is impossible.
On the other hand, limiting of containers can provide guarantees for them as well.
Providing a guarantee through limiting
The idea of getting a guarantee is simple:
if any group
{\displaystyle g_{i}}
requires a
{\displaystyle G_{i}}
units of resource from
{\displaystyle R}
units available then limiting all the rest groups with
{\displaystyle R-G_{i}}
units provides a desired guarantee
{\displaystyle N}
groups in the system this implies solving a linear equation set to get limits
{\displaystyle L_{i}}
{\displaystyle {\begin{cases}L_{2}+L_{3}+\ldots +L_{N}=R-G_{1}\\L_{1}+L_{3}+\ldots +L_{N}=R-G_{2}\\\ldots \\L_{1}+L_{2}+\ldots +L_{N-1}=R-G_{N}\\\end{cases}}}
In a matrix form this looks like
{\displaystyle AL=G\;}
{\displaystyle A={\begin{bmatrix}0&1&1&\cdots &1&1\\1&0&1&\cdots &1&1\\&&\cdots \\1&1&1&\cdots &1&0\\\end{bmatrix}},L={\begin{bmatrix}L_{1}\\L_{2}\\\vdots \\L_{N}\end{bmatrix}},G={\begin{bmatrix}R-G_{1}\\R-G_{2}\\\vdots \\R-G_{N}\end{bmatrix}}}
and thus the solution is
{\displaystyle L=A^{-1}G\;}
Skipping boring calculations, the reverse matrix
{\displaystyle A^{-1}\;}
{\displaystyle A^{-1}={\frac {1}{N-1}}\left(A-(N-2)E\right)}
This solutions looks huge, but the
{\displaystyle L}
vector is calculated in
{\displaystyle O(N)}
void calculate_limits(int N, int *g, int *l)
sum += R - g[i];
l[i] = (sum - (R - g[i]) - (N - 2) * (R - g[i]))/(N - 1);
This approach has only one disadvantage: O(n) time needed to start a new container.
Retrieved from "https://wiki.openvz.org/index.php?title=Containers/Guarantees_for_resources&oldid=10343"
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Complex plane fractal
High-quality overview image of the Burning Ship fractal
High-quality image of the large ship in the left antenna
{\displaystyle z_{n+1}=(|\operatorname {Re} \left(z_{n}\right)|+i|\operatorname {Im} \left(z_{n}\right)|)^{2}+c,\quad z_{0}=0}
{\displaystyle \mathbb {C} }
which will either escape or remain bounded. The difference between this calculation and that for the Mandelbrot set is that the real and imaginary components are set to their respective absolute values before squaring at each iteration. The mapping is non-analytic because its real and imaginary parts do not obey the Cauchy–Riemann equations.[1]
Burning Ship deep zoom to 2.3·10−50
A zoom-in to the lower left of the Burning Ship fractal, showing a "burning ship" and self-similarity to the complete fractal
A zoom-in to line on the left of the fractal, showing nested repetition (a different colour scheme is used here)
High-quality image of the Burning Ship fractal
The Burning Ship fractal featured in the 1K intro "JenterErForetrukket" by Youth Uprising; a demoscene production
Ghost Ship - The Burning Ship fractal rendered using the Nebulabrot technique
High resolution image of the burning ship fractal
The structure of the Burning Ship fractal
Animation of a continuous zoom-out to show the amount of detail for an implementation with 64 maximum iterations
The below pseudocode implementation hardcodes the complex operations for Z. Consider implementing complex number operations to allow for more dynamic and reusable code. Note that the typical images of the Burning Ship fractal display the ship upright: the actual fractal, and that produced by the below pseudocode, is inverted along the x-axis.
x := scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1))
y := scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1, 1))
zx := x // zx represents the real part of z
zy := y // zy represents the imaginary part of z
max_iteration := 100
while (zx*zx + zy*zy < 4 and iteration < max_iteration) do
xtemp := zx*zx - zy*zy + x
zy := abs(2*zx*zy) + y // abs returns the absolute value
zx := xtemp
if iteration = max_iteration then // Belongs to the set
return insideColor
return iteration × color
^ Michael Michelitsch and Otto E. Rössler (1992). "The "Burning Ship" and Its Quasi-Julia Sets". In: Computers & Graphics Vol. 16, No. 4, pp. 435–438, 1992. Reprinted in Clifford A. Pickover Ed. (1998). Chaos and Fractals: A Computer Graphical Journey — A 10 Year Compilation of Advanced Research. Amsterdam, Netherlands: Elsevier. ISBN 0-444-50002-2
Wikimedia Commons has media related to Burning Ship fractal.
About properties and symmetries of the Burning Ship fractal, featured by Theory.org
Burning Ship Fractal, Description and C source code.
Burning Ship with its Mset of higher powers and Julia Sets
Burningship, Video,
Fractal webpage includes the first representations and the original paper cited above on the Burning Ship fractal.
3D representations of the Burning Ship fractal
FractalTS Mandelbrot, Burning ship and corresponding Julia set generator.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Burning_Ship_fractal&oldid=1072671444"
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Some identities of Genocchi polynomials arising from Genocchi basis | Journal of Inequalities and Applications | Full Text
Some identities of Genocchi polynomials arising from Genocchi basis
Dmitry V Dolgy3 &
In this paper, we give some interesting identities which are derived from the basis of Genocchi. From our methods which are treated in this paper, we can derive some new identities associated with Bernoulli and Euler polynomials.
MSC:11B68, 11S80.
As is well known, the Genocchi polynomials are defined by the generating function to be
\frac{2t}{{e}^{t}+1}{e}^{xt}={e}^{G\left(x\right)t}=\sum _{n=0}^{\mathrm{\infty }}{G}_{n}\left(x\right)\frac{{t}^{n}}{n!}\phantom{\rule{1em}{0ex}}\left(\text{see [1–9]}\right)
{G}^{n}\left(x\right)
{G}_{n}\left(x\right)
x=0
{G}_{n}\left(0\right)={G}_{n}
are called the n th Genocchi numbers. From (1), we note that
{G}_{0}=0,\phantom{\rule{2em}{0ex}}{G}_{n}\left(1\right)+{G}_{n}=2{\delta }_{n,1}\phantom{\rule{1em}{0ex}}\left(\text{see [10–16]}\right),
{\delta }_{n,k}
{G}_{n}\left(x\right)={\left(G+x\right)}^{n}=\sum _{l=0}^{n}\left(\genfrac{}{}{0}{}{n}{l}\right){G}_{l}{x}^{n-l}\phantom{\rule{1em}{0ex}}\left(\text{see [6–8, 17]}\right).
Thus, by (2) and (3), we see that
\frac{d}{dx}{G}_{n}\left(x\right)=n{G}_{n-1}\left(x\right),\phantom{\rule{2em}{0ex}}deg{G}_{n}\left(x\right)=n-1.
The n th Bernoulli polynomials are also defined by the generating function to be
\frac{t}{{e}^{t}-1}{e}^{xt}={e}^{B\left(x\right)t}=\sum _{n=0}^{\mathrm{\infty }}{B}_{n}\left(x\right)\frac{{t}^{n}}{n!}\phantom{\rule{1em}{0ex}}\left(\text{see [14–16]}\right)
{B}^{n}\left(x\right)
{B}_{n}\left(x\right)
x=0
{B}_{n}\left(0\right)={B}_{n}
are called the n th Bernoulli numbers. By (5), we get
{B}_{0}=1,\phantom{\rule{2em}{0ex}}{B}_{n}\left(1\right)-{B}_{n}={\delta }_{1,n}\phantom{\rule{1em}{0ex}}\left(\text{see [8, 9, 17]}\right)
{B}_{n}\left(x\right)=\sum _{l=0}^{n}\left(\genfrac{}{}{0}{}{n}{l}\right){B}_{l}{x}^{n-l}=\sum _{l=0}^{n}\left(\genfrac{}{}{0}{}{n}{l}\right){B}_{n-l}{x}^{l}.
The Euler numbers are defined by
{E}_{0}=1,\phantom{\rule{2em}{0ex}}{\left(E+1\right)}^{n}+{E}_{n}=2{\delta }_{0,n}.
The Euler polynomials are defined by
{E}_{n}\left(x\right)={\left(E+x\right)}^{n}=\sum _{l=0}^{n}\left(\genfrac{}{}{0}{}{n}{l}\right){E}_{n-l}{x}^{l}\phantom{\rule{1em}{0ex}}\left(\text{see [7–13, 17]}\right).
{\mathbb{P}}_{n}=\left\{p\left(x\right)\in \mathbb{Q}\left[x\right]\mid degp\left(x\right)\le n\right\}
\left(n+1\right)
-dimensional vector space over ℚ. Probably,
\left\{1,x,\dots ,{x}^{n}\right\}
is the most natural basis for
{\mathbb{P}}_{n}
\left\{{G}_{1}\left(x\right),{G}_{2}\left(x\right),\dots ,{G}_{n+1}\left(x\right)\right\}
is also a good basis for the space
{\mathbb{P}}_{n}
for our purpose of arithmetical applications of Genocchi polynomials. Let
p\left(x\right)\in {\mathbb{P}}_{n}
p\left(x\right)
p\left(x\right)={\sum }_{1\le k\le n+1}{b}_{k}{G}_{k}\left(x\right)
In this paper, we develop some new methods to obtain some new identities and properties of Genocchi polynomials which are derived from the Genocchi basis.
2 Genocchi basis and some identities of Genocchi polynomials
p\left(x\right)\in {\mathbb{P}}_{n}
p\left(x\right)
can be expressed as a ℚ-linear combination of
{G}_{1}\left(x\right),{G}_{2}\left(x\right),\dots ,{G}_{n+1}\left(x\right)
p\left(x\right)=\sum _{1\le k\le n+1}{b}_{k}{G}_{k}\left(x\right)={b}_{1}{G}_{1}\left(x\right)+{b}_{2}{G}_{2}\left(x\right)+\cdots +{b}_{n+1}{G}_{n+1}\left(x\right).
Now, let us define the operator
\stackrel{˜}{\mathrm{△}}
\stackrel{˜}{\mathrm{△}}p\left(x\right)=p\left(x+1\right)+p\left(x\right).
Then, by (10) and (11), we set
g\left(x\right)=\stackrel{˜}{\mathrm{△}}p\left(x\right)=\sum _{1\le k\le n+1}{b}_{k}\left({G}_{k}\left(x+1\right)+{G}_{k}\left(x\right)\right).
\sum _{n=0}^{\mathrm{\infty }}\left\{{G}_{n}\left(x+1\right)+{G}_{n}\left(x\right)\right\}\frac{{t}^{n}}{n!}=\frac{2t}{{e}^{t}+1}{e}^{\left(x+1\right)t}+\frac{2t}{{e}^{t}+1}{e}^{xt}.
\frac{{G}_{n+1}\left(x+1\right)+{G}_{n+1}\left(x\right)}{n+1}=2{x}^{n}.
g\left(x\right)=\stackrel{˜}{\mathrm{△}}p\left(x\right)=2\sum _{1\le k\le n+1}k{b}_{k}{x}^{k-1}.
r\in \mathbb{N}
g\left(x\right)
{g}^{\left(r\right)}\left(x\right)=\frac{{d}^{r}g\left(x\right)}{d{x}^{r}}=2\sum _{1\le k\le n+1}k\left(k-1\right)\cdots \left(k-1-r+1\right){b}_{k}{x}^{k-1-r}.
{g}^{\left(r\right)}\left(0\right)=\frac{{d}^{r}g\left(x\right)}{d{x}^{r}}{|}_{x=0}=2\left(r+1\right)!{b}_{r+1}.
{b}_{r+1}=\frac{1}{2\left(r+1\right)!}\left\{{p}^{\left(r\right)}\left(1\right)+{p}^{\left(r\right)}\left(0\right)\right\},
{p}^{\left(r\right)}\left(a\right)=\frac{{d}^{r}g\left(x\right)}{d{x}^{r}}{|}_{x=a}
n\in \mathbb{N}
p\left(x\right)\in {\mathbb{P}}_{n}
p\left(x\right)={\sum }_{1\le k\le n+1}{b}_{k}{G}_{k}\left(x\right)
{b}_{k}=\frac{1}{2k!}\left({p}^{\left(k-1\right)}\left(1\right)+{p}^{\left(k-1\right)}\left(0\right)\right).
p\left(x\right)={B}_{n}\left(x\right)
. Then by Theorem 1, we get
{B}_{n}\left(x\right)=\sum _{1\le k\le n+1}{b}_{k}{G}_{k}\left(x\right),
{b}_{k}=\frac{1}{2k!}\left\{{p}^{\left(k-1\right)}\left(1\right)+{p}^{\left(k-1\right)}\left(0\right)\right\}=\frac{1}{2k!}{\left(n\right)}_{k-1}\left\{{B}_{n-k+1}\left(1\right)+{B}_{n-k+1}\right\}.
{b}_{k}=\frac{1}{2\left(n+1\right)}\left(\genfrac{}{}{0}{}{n+1}{k}\right)\left\{{\delta }_{n,k}+2{B}_{n-k+1}\right\}.
\begin{array}{rl}{B}_{n}\left(x\right)& =\frac{1}{n+1}\sum _{1\le k\le n-1}\left(\genfrac{}{}{0}{}{n+1}{k}\right){B}_{n-k+1}{G}_{k}\left(x\right)+\frac{1}{2}\left(1+2{B}_{1}\right){G}_{n}\left(x\right)+\frac{1}{2\left(n+1\right)}2{G}_{n+1}\left(x\right)\\ =\frac{1}{n+1}\sum _{1\le k\le n-1}\left(\genfrac{}{}{0}{}{n+1}{k}\right){B}_{n-k+1}{G}_{k}\left(x\right)+\frac{1}{n+1}{G}_{n+1}\left(x\right).\end{array}
n\in \mathbb{N}
{B}_{n}\left(x\right)=\frac{1}{n+1}\sum _{1\le k\le n-1}\left(\genfrac{}{}{0}{}{n+1}{k}\right){B}_{n-k+1}{G}_{k}\left(x\right)+\frac{1}{n+1}{G}_{n+1}\left(x\right).
p\left(x\right)={E}_{n}\left(x\right)\in {\mathbb{P}}_{n}
{E}_{n}\left(x\right)=\sum _{1\le k\le n+1}{b}_{k}{G}_{k}\left(x\right),
{b}_{k}=\frac{1}{2k!}\left\{{p}^{\left(k-1\right)}\left(1\right)+{p}^{\left(k-1\right)}\left(0\right)\right\}=\frac{1}{2k!}{\left(n\right)}_{k-1}\left\{{E}_{n-k+1}\left(1\right)+{E}_{n-k+1}\right\}.
By (8) and (24), we get
\begin{array}{rl}{b}_{k}& =\frac{1}{2\left(n+1\right)}\left(\genfrac{}{}{0}{}{n+1}{k}\right)\left\{2{\delta }_{n-k+1,0}-{E}_{n-k+1}+{E}_{n-k+1}\right\}\\ =\frac{1}{n+1}\left(\genfrac{}{}{0}{}{n+1}{k}\right){\delta }_{n+1,k}.\end{array}
{E}_{n}\left(x\right)=\frac{1}{n+1}{G}_{n+1}\left(x\right).
p\left(x\right)\in {\mathbb{P}}_{n}
p\left(x\right)=\sum _{0\le k\le n}{B}_{k}\left(x\right){B}_{n-k}\left(x\right).
Continuing this process, we get
\begin{array}{rl}\frac{{d}^{k}p\left(x\right)}{d{x}^{k}}& ={p}^{\left(k\right)}\left(x\right)=\left(n+1\right)n\cdots \left(n+1-k+1\right)\sum _{l=k}^{n}{B}_{l-k}\left(x\right){B}_{n-l}\left(x\right)\\ =\frac{\left(n+1\right)!}{\left(n+1-k\right)!}\sum _{l=k}^{n}{B}_{l-k}\left(x\right){B}_{n-l}\left(x\right).\end{array}
{p}^{\left(k-1\right)}\left(1\right)=\frac{\left(n+1\right)!}{\left(n+2-k\right)!}\sum _{l=k-1}^{n}{B}_{l+1-k}\left(1\right){B}_{n-l}\left(1\right).
\begin{array}{rl}{B}_{l+1-k}\left(1\right){B}_{n-l}\left(1\right)& =\left({\delta }_{l+1-k,1}+{B}_{l+1-k}\right)\left({\delta }_{n-l,1}+{B}_{n-l}\right)\\ =\left\{{\delta }_{k,n-1}+{B}_{n-k}+{B}_{n-k}+{B}_{l+1-k}{B}_{n-l}\right\}.\end{array}
{p}^{\left(k-1\right)}\left(1\right)=\frac{\left(n+1\right)!}{\left(n+2-k\right)!}\left\{{\delta }_{k,n-1}+2{B}_{n-k}+\sum _{k-1\le l\le n}{B}_{l+1-k}{B}_{n-l}\right\}.
p\left(x\right)={\sum }_{0\le k\le n}{B}_{k}\left(x\right){B}_{n-k}\left(x\right)
p\left(x\right)=\sum _{1\le k\le n+1}{b}_{k}\left(x\right){G}_{k}\left(x\right),
\begin{array}{rl}{b}_{k}& =\frac{1}{2k!}\left\{{p}^{\left(k-1\right)}\left(1\right)+{p}^{\left(k-1\right)}\left(0\right)\right\}\\ =\frac{\left(n+1\right)!}{2k!\left(n+2-k\right)!}\left\{{\delta }_{k,n-1}+2{B}_{n-k}+2\sum _{l=k-1}{B}_{l+1-k}{B}_{n-l}\right\}.\end{array}
\begin{array}{rcl}p\left(x\right)& =& \frac{n\left(n+1\right)}{12}{G}_{n-1}\left(x\right)+\sum _{1\le k\le n+1}\frac{1}{k}\left(\genfrac{}{}{0}{}{n+1}{k-1}\right){B}_{n-k}{G}_{k}\left(x\right)\\ +\sum _{1\le k\le n+1}\frac{1}{k}\left(\genfrac{}{}{0}{}{n+1}{k-1}\right)\sum _{l=k-1}^{n}{B}_{l+1-k}{B}_{n-l}{G}_{k}\left(x\right).\end{array}
n\in \mathbb{N}
\begin{array}{rcl}\sum _{k=0}^{n}{B}_{k}\left(x\right){B}_{n-k}\left(x\right)& =& \frac{n\left(n+1\right)}{12}{G}_{n-1}\left(x\right)+\sum _{1\le k\le n+1}\frac{1}{k}\left(\genfrac{}{}{0}{}{n+1}{k-1}\right){B}_{n-k}{G}_{k}\left(x\right)\\ +\sum _{1\le k\le n+1}\left(\sum _{k-1\le l\le n}\frac{1}{k}\left(\genfrac{}{}{0}{}{n+1}{k-1}\right){B}_{l+1-k}{B}_{n-l}\right){G}_{k}\left(x\right).\end{array}
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The authors would like to express their gratitude for the valuable comments and suggestions of referees. This research was supported by Kwangwoon University in 2013.
Department of Mathematics Education, Kyungpook National University, Daegu, 702-701, South Korea
Hanrimwon, Kwangwoon University, Seoul, 139-701, South Korea
Dmitry V Dolgy
Division of General Education, Kwangwoon University, Seoul, 139-701, South Korea
Kim, T., Rim, SH., Dolgy, D.V. et al. Some identities of Genocchi polynomials arising from Genocchi basis. J Inequal Appl 2013, 43 (2013). https://doi.org/10.1186/1029-242X-2013-43
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Numerical evaluation of new Mathematical Functions
The evalhf command, which evaluates an expression to a numerical value using the floating-point hardware of the underlying system, has been extended to handle complex numbers. In addition, a number of common commands like map, zip, and basic Matrix arithmetic are now also handled directly by evalhf.
The new fdiff command has been added for computing numerical derivatives. Alternatively, numerical differentiation can be invoked using evalf and D. This feature is particularly useful for mathematical functions for which no symbolic derivative is known.
fdiff(exp(x),x=1);
\textcolor[rgb]{0,0,1}{2.718281828}
fdiff(BesselJ(v,x),[v],[v=1,x=2]) = evalf(D[1](BesselJ)(1,2));
\textcolor[rgb]{0,0,1}{-0.05618076074}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-0.05618076074}
Support for the numerical evaluation of several new mathematical functions has been added. The functions are: Wrightomega, SphericalY (spherical harmonics), and the five Heun functions HeunG, HeunC, HeunB, HeunD, and HeunT.
For more details on this function, see Enhancements to Symbolic Capabilities in Maple 10.
Special code has been added to RootFinding[Analytic] to detect multiple roots. For more details, see Efficiency Improvements in Maple 10.
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Development of a Batch-Type Biogas Digester Using a Combination of Cow Dung, Swine Dung and Poultry Dropping
2{\text{CH}}_{3}{\text{CH}}_{2}{\text{OH}}_{\left(\text{l}\right)}+{\text{CO}}_{2\left(\text{g}\right)}\to {\text{CH}}_{4\left(\text{g}\right)}+2{\text{CH}}_{3}{\text{COOH}}_{\left(\text{l}\right)}
{\text{CH}}_{3}{\text{COOH}}_{\left(\text{ag}\right)}\to {\text{CH}}_{4}{}_{\left(\text{g}\right)}+{\text{CO}}_{2}{}_{\left(\text{g}\right)}
{\text{CO}}_{2\left(\text{g}\right)}+4{\text{H}}_{4}\to {\text{CH}}_{4\left(\text{g}\right)}+{\text{2H}}_{2}{O}_{\left(\text{l}\right)}
{V}_{T}={V}_{s}+{V}_{g}
\begin{array}{l}\text{Daily gas production}\\ =\text{volatile solids content}\times \text{the specific gas yield}\left(\text{solids}\right)\\ =0.6\text{\hspace{0.17em}}\text{kg}/\text{day}\times 0.50\text{\hspace{0.17em}}{\text{m}}^{\text{3}}/\text{kg}=0.\text{3}\text{\hspace{0.17em}}\text{ltrs}/\text{day}\end{array}
\text{Total volume of gas after 25 days}=0.\text{3}\text{\hspace{0.17em}}\text{ltrs}/\text{day}\times \text{25}\text{\hspace{0.17em}}\text{days}=\text{7}.\text{5}\text{\hspace{0.17em}}\text{litres}
\text{Total volume of Digester},\text{}{V}_{T}={V}_{o}\times 1.25=37.5\times 1.25=46.875\text{\hspace{0.17em}}\text{litres}
V=\text{π}{r}^{2}h
{d}^{3}=\frac{16}{\text{π}{S}_{s}}\sqrt{{\left({M}_{t}{k}_{t}\right)}^{2}+{\left({M}_{b}{k}_{b}\right)}^{2}}
Olanrewaju, O.O. and Olubanjo, O.O. (2019) Development of a Batch-Type Biogas Digester Using a Combination of Cow Dung, Swine Dung and Poultry Dropping. International Journal of Clean Coal and Energy, 8, 15-31. https://doi.org/10.4236/ijcce.2019.82002
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2. Gujba, H., Mulugetta, Y. and Azapagic, A. (2015) The Household Cooking Sector in Nigeria: Environmental and Economic Sustainability Assessment. Resources, 4, 412-433. https://doi.org/10.3390/resources4020412
3. Ofoefule, A.U., Uzodinma, E.O. and Onukwuli, O.D. (2009) Biogas Development in Africa. International Journal of Physical Science, 4, 535-539.
4. Lawal, A.K., Ajuebor, F.N. and Ojosu, J.O. (2001) Characteristic of Piggery Wastes Feeds Stock for Determination of Design Parameters to Biogas Digester Plant. Nigerian Journal of Research and Review in Science, 2, 193-198.
5. Onwuliri, F.C., Onyimba, I.A. and Nwaukwu, I.A. (2013) Generation of Biogas from Cow Dung. Bioremediation and Biodegradation, S18, 002. https://doi.org/10.4172/2155-6199.S18-002
6. Ajiboye, A.V., Lasisi, K.H. and Babatola, J.O. (2018) Evaluation of the Effect of Sodium Hydroxide Solution on Biogas Yield of Anaerobic Digestion of Poultry Waste and the Digestate. International Journal of Energy and Water Resources, 2, 23-31. https://doi.org/10.1007/s42108-018-0003-2
7. Otun, T.F., Ojo, O.M., Ajibade, F.O. and Babatola, J.O. (2015) Evaluation of Biogas Production from the Digestion and Co-Digestion of Animal Waste, Food Waste and Fruit Waste. International Journal of Energy and Environmental Research, 3, 12-24.
8. EIA (2016) United State Energy Information Administration. Country Analysis Brief: Nigeria. https://www.eia.gov/beta/international/analysis_includes/countries_long/Nigeria/nigeria.pdf
9. Babatola, J.O. (2008) Comparative Study of Biogas Yield Pattern in Some Animal and Household Wastes. African Research Review, 2, 54-68. https://doi.org/10.4314/afrrev.v2i4.41081
10. Musa, B. and Raji, H.M. (2016) Quantitative and Qualitative Analysis of Biogas Produces from Three Organic Wastes. International Journal of Renewable Energy Research, 6, 299-305.
11. Nwankwo, C.S., Eze, J.I. and Okoyeuzu, C. (2017) Design and Fabrication of 3.60 m3 Household Plastic Bio Digester Loaded with Kitchen Waste and Cow Dung for Biogas Generation. Scientific Research and Essays, 12, 130-141. https://doi.org/10.5897/SRE2017.6516
12. National Bureau of Statistics (2018) Nigerian Gross Domestic Product Report (Q4 and Full Year 2017).
13. Okewale, A.O., Omoruwuo, F. and Ojaigho, R.O. (2016) Alternative Energy Production for Environmental Sustainability. British Journal of Renewable Energy, 1, 18-22. https://doi.org/10.1051/rees/2016040
14. Chibueze, U., Okorie, N., Oriaku, O., Isu, J. and Peters, E. (2017) The Production of Biogas Using Cow Dung and Food Waste. International Journal of Materials and Chemistry, 7, 21-24.
15. MWPS (2000) Manure Characteristics. Mid-West Plan Service MWPS-18.
16. Dioha, I.J., Ikeme, C.H., Nafi’u, T., Soba, N.I. and Yusuf, M.B.S. (2013) Effect of Carbon to Nitrogen Ratio on Biogas Production. International Research Journal of Natural Sciences, 1, 1-10.
17. Nnabuchi, M.N., Akubuko, F.O., Augustine, C. and Ugwu, G.Z. (2012) Assessment of the Effect of Co-Digestion of Chicken Dropping and Cow Dung on Biogas Generation. International Research Journal of Engineering Science, Technology and Innovation, 1, 238-243.
18. Ogur, E.O. and Irungu, P. (2013) Design of a Biogas Generator. International Journal of Engineering Research and Applications, 3, 630-635.
19. Wu, X., Yao, W., Zhu, J. and Miller, C. (2010) Biogas and CH4 Productivity by Co-Digesting Swine Manure with Three Crop Residues as an External Carbon Source. Bioresource Technology, 101, 4042-4047. https://doi.org/10.1016/j.biortech.2010.01.052
20. McCarty, P.L. and Mosey, F.E. (1991) Modelling of Anaerobic Digestion Processes. Water Science & Technology, 24, 17-33. https://doi.org/10.2166/wst.1991.0216
21. De Baere, L. (2000) Anaerobic Digestion of Solid Waste: State-of-the-Art. Water Science and Technology, 41, 283-290. https://doi.org/10.2166/wst.2000.0082
22. Manyi-Loh, C.E., Mamphwell, S.N., Meyer, E.L., Okoh, A.I., Makaka, G. and Simon, M. (2013) Microbial Anaerobic Digestion (Bio-Digesters) as an Approach to the Decontamination of Animal Wastes in Pollution Control and the Generation of Renewable Energy. International Journal of Environmental Research and Public Health, 10, 4390-4417. https://doi.org/10.3390/ijerph10094390
23. Deublein, D.A. (2008) Biogas from Waste and Renewable Resources: An Introduction. John Wiley and Sons, Hoboken.
24. Mata-Alvarez, J. (2003) Fundamentals of the Anaerobic Digestion Process. In: Mata-Alvarez, J., Ed., Biomethanization of the Organic Fraction of Municipal Solid Wastes, IWA Publishing Company, Amsterdam, 202-209.
25. Meegoda, J.N., Li, B., Patel, K. and Wang, L.B. (2018) A Review of the Processes, Parameters, and Optimization of Anaerobic Digestion. International Journal of Environmental Research and Public Health, 15, 2224. https://doi.org/10.3390/ijerph15102224
26. Roddy, D.J. (2012) Biomass and Biofuels—Introduction. In: Sayigh, A., Ed., Comprehensive Renewable Energy, Elsevier, Hoboken, 1-9. https://doi.org/10.1016/B978-0-08-087872-0.00501-1
27. Christy, P.M., Gopinath, L.R. and Divya, D. (2014) Microbial Dynamics during Anaerobic Digestion of Cow Dung. International Journal of Plant, Animal and Environmental Sciences, 4, 86-94.
28. Azman, S. (2016) Anaerobic Digestion of Cellulose and Hemiceullose in the Presence of Humic Acids. Wageningen University, Wageningen.
29. Kigozi, R., Aboyade, A. and Muzenda, E. (2014) Biogas Production Using the Organic Fraction of Municipal Solid Waste as Feedstock. International Journal of Research in Chemical, Metallurgical and Civil Engineering, 1, 54-64.
30. Ukpai, P.A. and Nnabuchi, M.N. (2012) Comparative Study of Biogas Production from Cow Dung, Cow Pea and Cassava Peeling Using 45 Litres Biogas Digester. Advances in Applied Science Research, 3, 1864-1869.
31. Aladjadjiyan, A., Penkov, D., Verspecht, A., Zahariev, A. and Kakanakov, N. (2016) Biobased Fertilizers—Comparison of Nutrient Content of Digestate/Compost. Journal of Agriculture and Ecology Research International, 8, 1-7. https://doi.org/10.9734/JAERI/2016/25217
32. Hansen, C.L. and Cheong, D.Y. (2013) Agricultural Waste Management in Food Processing. In: Kutz, M., Ed., Handbook of Farm, Dairy, and Food Machinery Engineering, Academic Press, Cambridge, 609-661. https://doi.org/10.1016/B978-0-12-385881-8.00023-9
33. House, D. (2006) Complete Biogas Handbook. 3rd Edition, House Press, Culver City, CA, USA, 288 p.
34. Gallert, C. and Winder, J. (2005) Bacterial Metabolism in Wastewater Treatment System. In: Jördening, H.J. and Winter, J., Eds., Environmental Biotechnology: Concepts and Applications, Wiley VCH, Weinheim, Chapter 1. https://doi.org/10.1002/3527604286.ch1
35. Stams, A.J.M. and Plugge, C.M. (2009) Electron Transfer in Syntrophic Communities of Anaerobic Bacteria and Archaea. Nature Reviews Microbiology, 7, 568-577. https://doi.org/10.1038/nrmicro2166
36. Oyeleke, S.B., Onigbajo, H.O. and Ibrahim, K. (2003) Degradation of Animal Wastes (Cattle Dung) to Produce Methane (Cooking Gas). Proceeding of the Eighth Annual Conference of Animal Science Association of Nigeria, Enugu, 12-15 September, 168-169.
37. Deepanraj, B., Sivasubramanian, V. and Jayaraj, S. (2014) Solid Concentration Influence on Biogas Yield from Food Waste in an Anaerobic Batch Digester. Proceeding on International Conference and Utility Exhibition on Green Energy for Sustainable Development, Pattaya, 19-21 March 2014, 1-4.
38. Gaby, J.C., Zamanzadeh, M. and Horn, S.J. (2017) The Effect of Temperature and Retention Time on Methane Production and Microbial Community Composition in Staged Anaerobic Digesters Fed with Food Waste. Biotechnology for Biofuels, 10, 302. https://doi.org/10.1186/s13068-017-0989-4
39. Verspecht, A. and Buysse, J. (2012) Improved Nutrient and Energy Management through Anaerobic Digestion. https://cordis.europa.eu/docs/results/289/289712/final1-summary-inemad-final.pdf
41. Nkoi, B., Lebele-Alawa, B.T. and Odobeatu, B. (2018) Design and Fabrication of a Modified Portable Biogas Digester for Renewable Cooking-Gas Production. European Journal of Engineering Research and Science, 3, 21-29. https://doi.org/10.24018/ejers.2018.3.3.647
42. Vögeli, Y., Lohri, C.R., Gallardo, A., Diener, S. and Zurbrügg, C. (2014) Anaerobic Digestion of Biowaste in Developing Countries—Practical Information and Case Studies. https://www.ircwash.org/sites/default/files/2014-biowaste-eawag.pdf
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Cardinal number - WikiMili, The Best Wikipedia Reader
{\displaystyle \aleph }
(aleph) followed by a subscript, describe the sizes of infinite sets.
{\displaystyle 0,1,2,3,\ldots ,n,\ldots<span typeof="mw:DisplaySpace"> </span>;\aleph _{0},\aleph _{1},\aleph _{2},\ldots ,\aleph _{\alpha },\ldots .\ }
{\displaystyle \aleph _{0}}
In his 1874 paper "On a Property of the Collection of All Real Algebraic Numbers", Cantor proved that there exist higher-order cardinal numbers, by showing that the set of real numbers has cardinality greater than that of N. His proof used an argument with nested intervals, but in an 1891 paper, he proved the same result using his ingenious and much simpler diagonal argument. The new cardinal number of the set of real numbers is called the cardinality of the continuum and Cantor used the symbol
{\displaystyle {\mathfrak {c}}}
{\displaystyle \aleph _{0}}
{\displaystyle (\aleph _{1},\aleph _{2},\aleph _{3},\ldots ).}
{\displaystyle {\mathfrak {c}}}
{\displaystyle \aleph _{1}}
In informal use, a cardinal number is what is normally referred to as a counting number , provided that 0 is included: 0, 1, 2, .... They may be identified with the natural numbers beginning with 0. The counting numbers are exactly what can be defined formally as the finite cardinal numbers. Infinite cardinals only occur in higher-level mathematics and logic.
{\displaystyle \aleph _{0}}
(aleph null or aleph-0, where aleph is the first letter in the Hebrew alphabet, represented
{\displaystyle \aleph }
{\displaystyle \aleph _{0}}
{\displaystyle \aleph _{1}}
{\displaystyle \aleph _{\alpha },}
{\displaystyle \kappa ^{+}\nleq \kappa .}
{\displaystyle |X|+|Y|=|X\cup Y|.}
{\displaystyle (\kappa \leq \mu )\rightarrow ((\kappa +\nu \leq \mu +\nu ){\mbox{ and }}(\nu +\kappa \leq \nu +\mu )).}
{\displaystyle \kappa +\mu =\max\{\kappa ,\mu \}\,.}
{\displaystyle |X|\cdot |Y|=|X\times Y|}
{\displaystyle \kappa \cdot \mu =\max\{\kappa ,\mu \}.}
{\displaystyle |X|^{|Y|}=\left|X^{Y}\right|,}
{\displaystyle \nu ^{\mu }=\kappa }
{\displaystyle \kappa }
{\displaystyle \mu ^{\lambda }=\kappa }
{\displaystyle \nu ^{\lambda }=\kappa }
{\displaystyle \aleph _{0}}
{\displaystyle 2^{\aleph _{0}}.}
{\displaystyle {\mathfrak {c}}}
{\displaystyle 2^{\aleph _{0}}=\aleph _{1}.}
{\displaystyle \kappa }
{\displaystyle \kappa }
{\displaystyle 2^{\kappa }}
. Both the continuum hypothesis and the generalized continuum hypothesis have been proved independent of the usual axioms of set theory, the Zermelo–Fraenkel axioms together with the axiom of choice (ZFC).
{\displaystyle \kappa }
{\displaystyle 2^{\kappa }}
{\displaystyle \kappa <\operatorname {cf} (2^{\kappa })}
In mathematics, the cardinality of a set is a measure of the "number of elements" of the set. For example, the set contains 3 elements, and therefore has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible.
In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A.
In mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.
Freiling's axiom of symmetry is a set-theoretic axiom proposed by Chris Freiling. It is based on intuition of Stuart Davidson but the mathematics behind it goes back to Wacław Sierpiński.
In set theory, König's theorem states that if the axiom of choice holds, I is a set, and are cardinal numbers for every i in I, and for every i in I, then
In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term transfinite was coined by Georg Cantor in 1895, who wished to avoid some of the implications of the word infinite in connection with these objects, which were, nevertheless, not finite. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as infinite numbers. Nevertheless, the term "transfinite" also remains in use.
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal κ is strongly inaccessible if it is uncountable, it is not a sum of fewer than κ cardinals smaller than κ, and implies .
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal κ, or more generally on any set. For a cardinal κ, it can be described as a subdivision of all of its subsets into large and small sets such that κ itself is large, ∅ and all singletons {α}, α ∈ κ are small, complements of small sets are large and vice versa. The intersection of fewer than κ large sets is again large.
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph.
In mathematics, two sets or classes A and B are equinumerous if there exists a one-to-one correspondence between them, that is, if there exists a function from A to B such that for every element y of B, there is exactly one element x of A with f(x) = y. Equinumerous sets are said to have the same cardinality. The study of cardinality is often called equinumerosity (equalness-of-number). The terms equipollence (equalness-of-strength) and equipotence (equalness-of-power) are sometimes used instead.
In mathematics, limit cardinals are certain cardinal numbers. A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This means that one cannot "reach" λ from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear.
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that is a regular cardinal if and only if every unbounded subset has cardinality . Infinite well-ordered cardinals that are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular.
In set theory, one can define a successor operation on cardinal numbers in a similar way to the successor operation on the ordinal numbers. The cardinal successor coincides with the ordinal successor for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same cardinality. Using the von Neumann cardinal assignment and the axiom of choice (AC), this successor operation is easy to define: for a cardinal number κ we have
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers, conventionally written , where is the second Hebrew letter (beth). The beth numbers are related to the aleph numbers, but unless the generalized continuum hypothesis is true, there are numbers indexed by that are not indexed by .
Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ0 and c. The theory was first suggested by Rudy Rucker in his Infinity and the Mind. The details set out in this entry are due to the American mathematician Randall M. Holmes.
In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the continuum and combinatorics on successors of singular cardinals.
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals aimed to extend enumeration to infinite sets.
This is a glossary of set theory.
↑ Dauben 1990 , pg. 54
↑ Weisstein, Eric W. "Cardinal Number". mathworld.wolfram.com. Retrieved 2020-09-06.
↑ Deiser, Oliver (May 2010). "On the Development of the Notion of a Cardinal Number". History and Philosophy of Logic. 31 (2): 123–143. doi:10.1080/01445340903545904. S2CID 171037224.
↑ Enderton, Herbert. "Elements of Set Theory", Academic Press Inc., 1977. ISBN 0-12-238440-7
↑ Friedrich M. Hartogs (1915), Felix Klein; Walther von Dyck; David Hilbert; Otto Blumenthal (eds.), "Über das Problem der Wohlordnung", Math. Ann., Leipzig: B. G. Teubner, Bd. 76 (4): 438–443, doi:10.1007/bf01458215, ISSN 0025-5831, S2CID 121598654, archived from the original on 2016-04-16, retrieved 2014-02-02
↑ Schindler 2014 , pg. 34
↑ Eduard Čech, Topological Spaces, revised by Zdenek Frolík and Miroslav Katetov, John Wiley & Sons, 1966.
↑ D. A. Vladimirov, Boolean Algebras in Analysis, Mathematics and Its Applications, Kluwer Academic Publishers.
Dauben, Joseph Warren (1990), Georg Cantor: His Mathematics and Philosophy of the Infinite , Princeton: Princeton University Press, ISBN 0691-02447-2
Halmos, Paul, Naive set theory . Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
Schindler, Ralf-Dieter (2014). Set theory : exploring independence and truth. Cham: Springer-Verlag. doi:10.1007/978-3-319-06725-4. ISBN 978-3-319-06725-4.
"Cardinal number", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
{\displaystyle \mathbb {N} }
{\displaystyle \mathbb {Z} }
{\displaystyle \mathbb {Q} }
{\displaystyle \mathbb {A} }
{\displaystyle \mathbb {R} }
{\displaystyle \mathbb {C} }
{\displaystyle \mathbb {H} }
{\displaystyle \mathbb {O} }
{\displaystyle \mathbb {R} }
{\displaystyle \mathbb {C} }
{\displaystyle \mathbb {S} }
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Categories All categories Not categorised course notes courses editing email and forums login and accounts recording rooms
Notes. Where can I find the notes for my course?
When you are logged into My.SUPA, your courses will be listed in a box on the right hand side of the first page of My.SUPA. If you can see a list of course categories instead of your courses, log in.
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Look at the relevant week or view the resources list.
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Keyword(s): lecturegraduate schoolnotes
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The $n$-cohomology of representations with an infinitesimal character
n
author = {Casselman, William and Osborne, M. Scott},
title = {The $n$-cohomology of representations with an infinitesimal character},
AU - Osborne, M. Scott
TI - The $n$-cohomology of representations with an infinitesimal character
Casselman, William; Osborne, M. Scott. The $n$-cohomology of representations with an infinitesimal character. Compositio Mathematica, Tome 31 (1975) no. 2, pp. 219-227. http://www.numdam.org/item/CM_1975__31_2_219_0/
[1] F. Aribaud: Une nouvelle demonstration d'un théorème de R. Bott et B. Kostant. Bull. Math. Soc. France 95 (1967) 205-242. | Numdam | MR 236311 | Zbl 0155.06901
[2] P. Cartier: Remarks on 'Lie algebra cohomology and the generalized Borel-Weil theorem' by B. Kostant. Ann. of Math. 74 (1961) 388-390. | MR 142698 | Zbl 0134.03502
[3] W. Casselman: Some general results on admissible representations of p-adic groups. (to appear)
[4] J. Humphreys: Introduction to Lie algebras and representation theory. New York: Springer, 1972. | MR 323842 | Zbl 0254.17004
[5] B. Kostant: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math. 74 (1961) 329-387. | MR 142696 | Zbl 0134.03501
[6] B. Kostant: Lie group representations on polynomial rings. Amer. J. of Math. 85 (1963) 327-404. | MR 158024 | Zbl 0124.26802
[7] M.S. Osborne: Yale University Ph.D. thesis. (1973).
[8] G. Warner: Harmonic analysis on semi-simple groups I. New York, Springer, 1972. | Zbl 0265.22020
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X
, a discrete random variable,
Population Mean (Expectancy)
If two discrete random variables $X$ and $Y$ are independent,
If success,
X = 1
. If failure,
X = 0
$X = $ number of successes in
independent repetitions of the experiment.
X_i
i
th repetition’s success or not.
X_i
follows a Bernoulli Distribution.
$X = $ number of tries up to and including first success
0 < q < 1, \quad n \xrightarrow{} \infty
E(X) = \frac{p}{(1 - q)^2} = \frac{1}{p}
E(X^2) = p\frac{1 - q^2}{(1 - q)^4} = p\frac{1 + q}{(1 - q)^3} = \frac{1 + q}{(1 - q)^2}
Poisson Distribution (To be continued…)
Poisson Distribution is essentially a binomial distribution with very large $n$ (very small time interval).
Exploring Flask Authentication Methods (Part 1)
I am developing a mobile application lately with Swift on iOS, with a provided and working back-end written in Python, with the Flask framework. The back-end project utilises a set of plugins named flask-security to handle login and user-related work.
This is an awesome plugin that works right out of the box. Almost little to no configuration is needed and it works like a charm on the webpage. It utilises a session-based authentication method that stores the key to sensitive information in browsers as cookies, which expires after a certain amount of time for security purposes.
Although session-based authentication is a viable way to set up a web application, this poses certain issues to a mobile client where we cannot store cookies like browsers and shall not record user password and send them out over and over again in every HTTP request.
Therefore, we will need a token-based authentication method, where a JSON Web Token (JWT) is issued the first time we log in. For all following requests, we package this JWT in the header as a way of verifying legit requests. For our particular plugin flask-security, this is done by sending an application-json request instead of x-www-form-urlencoded to the endpoint login.
But, the problem is not that easy. This is where Cross-Site Request Forgery (CSRF) comes into play. If we allow clients to simply authenticate by sending a JSON request with username and password, any other websites can fake that and make malicious requests. How to deal with this and have token-based authentication with CSRF protection at the same time?
Flask-Security 3.0.0 Features
Jane Street Puzzle Jan 2019 - Alter/Nate
Alter/Nate
Two friends, Alter and Nate, have a conversation:
Alter: Nate, let’s play a game. I’ll pick an integer between 1 and 10 (inclusive), then you’ll pick an integer between 1 and 10 (inclusive), and then I’ll go again, then you’ll go again, and so on and so forth. We’ll keep adding our numbers together to make a running total. And whoever makes the running total be greater than or equal to 100 loses. You go first.
Nate: That’s not fair! Whenever I pick a number X, you’ll just pick 11-X, and then I’ll always get stuck with 99 and I’ll make the total go greater than 100.
Alter: OK fine. New rule then, no one can pick a number that would make the sum of that number and the previous number equal to 11. You still go first. Now can we play?
Nate: Um… sure.
Who wins, and what is their strategy?
The first player, Nate, is guaranteed to win in this case.
Our goal is to let Nate get the sum of all
12k + 3
, which would eventually let him get
12 \times 8 + 3 = 99
, a guaranteed win.
Therefore, our strategy for Nate is to let first pick number
3
, which contributes to the offset. Later, for him to increment by
12
each round, if Alter picks any number
x > 1
, Nate can pick the number
2 \leq 12 - x \leq 10
. Otherwise if Alter picks
1
, Nate then picks
1
. Given the rule that two consecutive number cannot add up to the sum of
11
, Alter can only pick
y \leq 9
in this case. Such that in these three rounds
1 + 1 + y \leq 11
(Alter, Nate, Alter). Nate will then pick
1 \leq 12 - 2 - y \leq 9
12k + 3
If Nate can guarantee he reaches a sum of
12k + 3 \quad \forall k \geq 0
, he wins the game by reaching
99
For more puzzles, checkout
Jane Street Puzzles
Swift Thumbnail on the fly
It is quite painful where you have a decent video API, but doesn’t provide you with a thumbnail. This makes it very hard to demonstrate to users what the video looks like a first glance.
Therefore, we have came up with a technique that buffers the first minute of a video and generate a thumbnail on the fly.
Using AVAsset, we can write an extension that dispatches a thread dedicated to generate a thumbnail (also duration of the video in the case below).
func generateThumbnail(completion: @escaping (UIImage?, Int?) -> Void) {
let imageGenerator = AVAssetImageGenerator(asset: self)
let time = CMTime(seconds: 60.0, preferredTimescale: 600)
let times = [NSValue(time: time)]
let duration = Int(self.duration.seconds)
imageGenerator.generateCGImagesAsynchronously(forTimes: times, completionHandler: { _, image, _, _, _ in
completion(UIImage(cgImage: image), duration)
AVAsset - AVFoundation | Apple Developer Documentation
Generate images from AVAsset with AVAssetImageGenerator
for old material, please visit an alternative site.
below is a test on code snippets.
#pramga GCC optimize ("Ofast")
static int fast_io = [] () {
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This is the oul' latest accepted revision, reviewed on 28 April 2022.
Figure of the oul' Earth (radius and circumference)
Global Nav. Here's a quare one for ye. Sat. Whisht now. Systems (GNSSs)
Global Pos. Here's a quare one. System (GPS)
Quasi-Zenith Sat, would ye believe it? Sys. Bejaysus here's a quare one right here now. (QZSS) (Japan)
Discrete Global Grid and Geocodin'
ISO 6709 Geographic point coord, fair play. 1983
NAVD 88 N. G'wan now and listen to this wan. American Vertical Datum 1988
ETRS89 European Terrestrial Ref. Sys. Right so. 1989
Geo URI Internet link to a bleedin' point 2010
Longitude lines are perpendicular to and latitude lines are parallel to the bleedin' Equator.
The geographic coordinate system (GCS) is a bleedin' spherical or ellipsoidal coordinate system for measurin' and communicatin' positions directly on the oul' Earth as latitude and longitude.[1] It is the oul' simplest, oldest and most widely used of the feckin' various of spatial reference systems that are in use, and forms the basis for most others. Sure this is it. Although latitude and longitude form a coordinate tuple like a bleedin' cartesian coordinate system, the bleedin' geographic coordinate system is not cartesian because the feckin' measurements are angles and are not on a planar surface.[2][self-published source?]
A full GCS specification, such as those listed in the bleedin' EPSG and ISO 19111 standards, also includes an oul' choice of geodetic datum (includin' an Earth ellipsoid), as different datums will yield different latitude and longitude values for the oul' same location.[3]
4 Length of a holy degree
The invention of a holy geographic coordinate system is generally credited to Eratosthenes of Cyrene, who composed his now-lost Geography at the oul' Library of Alexandria in the feckin' 3rd century BC.[4] A century later, Hipparchus of Nicaea improved on this system by determinin' latitude from stellar measurements rather than solar altitude and determinin' longitude by timings of lunar eclipses, rather than dead reckonin'. In the oul' 1st or 2nd century, Marinus of Tyre compiled an extensive gazetteer and mathematically plotted world map usin' coordinates measured east from an oul' prime meridian at the oul' westernmost known land, designated the Fortunate Isles, off the coast of western Africa around the bleedin' Canary or Cape Verde Islands, and measured north or south of the island of Rhodes off Asia Minor, what? Ptolemy credited yer man with the bleedin' full adoption of longitude and latitude, rather than measurin' latitude in terms of the length of the bleedin' midsummer day.[5]
Ptolemy's 2nd-century Geography used the feckin' same prime meridian but measured latitude from the Equator instead. Jesus Mother of Chrisht almighty. After their work was translated into Arabic in the oul' 9th century, Al-Khwārizmī's Book of the oul' Description of the feckin' Earth corrected Marinus' and Ptolemy's errors regardin' the oul' length of the Mediterranean Sea,[note 1] causin' medieval Arabic cartography to use a bleedin' prime meridian around 10° east of Ptolemy's line, be the hokey! Mathematical cartography resumed in Europe followin' Maximus Planudes' recovery of Ptolemy's text a holy little before 1300; the bleedin' text was translated into Latin at Florence by Jacobus Angelus around 1407.
In 1884, the United States hosted the oul' International Meridian Conference, attended by representatives from twenty-five nations. Twenty-two of them agreed to adopt the longitude of the oul' Royal Observatory in Greenwich, England as the oul' zero-reference line. Jaykers! The Dominican Republic voted against the feckin' motion, while France and Brazil abstained.[6] France adopted Greenwich Mean Time in place of local determinations by the feckin' Paris Observatory in 1911.
Diagram of the bleedin' latitude (φ) and longitude (λ) angle measurements in the bleedin' GCS.
The "latitude" (abbreviation: Lat., φ, or phi) of a holy point on Earth's surface is the angle between the bleedin' equatorial plane and the straight line that passes through that point and through (or close to) the feckin' center of the feckin' Earth.[note 2] Lines joinin' points of the bleedin' same latitude trace circles on the bleedin' surface of Earth called parallels, as they are parallel to the Equator and to each other. Bejaysus here's a quare one right here now. The North Pole is 90° N; the South Pole is 90° S, bedad. The 0° parallel of latitude is designated the oul' Equator, the feckin' fundamental plane of all geographic coordinate systems. The Equator divides the feckin' globe into Northern and Southern Hemispheres.
The "longitude" (abbreviation: Long., λ, or lambda) of a bleedin' point on Earth's surface is the feckin' angle east or west of a feckin' reference meridian to another meridian that passes through that point. All meridians are halves of great ellipses (often called great circles), which converge at the oul' North and South Poles. Jaysis. The meridian of the bleedin' British Royal Observatory in Greenwich, in southeast London, England, is the oul' international prime meridian, although some organizations—such as the French Institut national de l'information géographique et forestière—continue to use other meridians for internal purposes. The prime meridian determines the oul' proper Eastern and Western Hemispheres, although maps often divide these hemispheres further west in order to keep the oul' Old World on a holy single side. In fairness now. The antipodal meridian of Greenwich is both 180°W and 180°E. This is not to be conflated with the International Date Line, which diverges from it in several places for political and convenience reasons, includin' between far eastern Russia and the oul' far western Aleutian Islands.
The combination of these two components specifies the bleedin' position of any location on the feckin' surface of Earth, without consideration of altitude or depth, Lord bless us and save us. The visual grid on a map formed by lines of latitude and longitude is known as a holy graticule.[7] The origin/zero point of this system is located in the Gulf of Guinea about 625 km (390 mi) south of Tema, Ghana, an oul' location often facetiously called Null Island.
Further information: Figure of the bleedin' Earth, Reference ellipsoid, Geographic coordinate conversion, and Spatial reference system
In order to be unambiguous about the oul' direction of "vertical" and the "horizontal" surface above which they are measurin', map-makers choose a reference ellipsoid with a holy given origin and orientation that best fits their need for the feckin' area to be mapped. They then choose the bleedin' most appropriate mappin' of the spherical coordinate system onto that ellipsoid, called a holy terrestrial reference system or geodetic datum.
Datums may be global, meanin' that they represent the oul' whole Earth, or they may be local, meanin' that they represent an ellipsoid best-fit to only a portion of the feckin' Earth. Whisht now. Points on the oul' Earth's surface move relative to each other due to continental plate motion, subsidence, and diurnal Earth tidal movement caused by the oul' Moon and the feckin' Sun. Stop the lights! This daily movement can be as much as an oul' meter. Arra' would ye listen to this shite? Continental movement can be up to 10 cm a year, or 10 m in an oul' century. A weather system high-pressure area can cause a feckin' sinkin' of 5 mm, like. Scandinavia is risin' by 1 cm a holy year as a result of the meltin' of the bleedin' ice sheets of the oul' last ice age, but neighborin' Scotland is risin' by only 0.2 cm. Jesus, Mary and Joseph. These changes are insignificant if a holy local datum is used, but are statistically significant if an oul' global datum is used.[8]
Examples of global datums include World Geodetic System (WGS 84, also known as EPSG:4326[9]), the oul' default datum used for the oul' Global Positionin' System,[note 3] and the oul' International Terrestrial Reference System and Frame (ITRF), used for estimatin' continental drift and crustal deformation.[10] The distance to Earth's center can be used both for very deep positions and for positions in space.[8]
Local datums chosen by a bleedin' national cartographical organization include the bleedin' North American Datum, the European ED50, and the feckin' British OSGB36. Given an oul' location, the bleedin' datum provides the latitude
{\displaystyle \phi }
{\displaystyle \lambda }
. In the oul' United Kingdom there are three common latitude, longitude, and height systems in use. WGS 84 differs at Greenwich from the bleedin' one used on published maps OSGB36 by approximately 112 m. The military system ED50, used by NATO, differs from about 120 m to 180 m.[8]
The latitude and longitude on a holy map made against a bleedin' local datum may not be the bleedin' same as one obtained from an oul' GPS receiver. Here's another quare one. Convertin' coordinates from one datum to another requires a datum transformation such as an oul' Helmert transformation, although in certain situations a simple translation may be sufficient.[11]
In popular GIS software, data projected in latitude/longitude is often represented as a Geographic Coordinate System. For example, data in latitude/longitude if the oul' datum is the feckin' North American Datum of 1983 is denoted by 'GCS North American 1983'.
Length of an oul' degree[edit]
Main articles: Length of a bleedin' degree of latitude and Length of a bleedin' degree of longitude
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On the GRS80 or WGS84 spheroid at sea level at the Equator, one latitudinal second measures 30.715 meters, one latitudinal minute is 1843 meters and one latitudinal degree is 110.6 kilometers, bejaysus. The circles of longitude, meridians, meet at the oul' geographical poles, with the feckin' west–east width of a feckin' second naturally decreasin' as latitude increases. Me head is hurtin' with all this raidin'. On the Equator at sea level, one longitudinal second measures 30.92 meters, an oul' longitudinal minute is 1855 meters and a feckin' longitudinal degree is 111.3 kilometers. Sure this is it. At 30° a bleedin' longitudinal second is 26.76 meters, at Greenwich (51°28′38″N) 19.22 meters, and at 60° it is 15.42 meters.
On the oul' WGS84 spheroid, the bleedin' length in meters of a feckin' degree of latitude at latitude φ (that is, the number of meters you would have to travel along a north–south line to move 1 degree in latitude, when at latitude φ), is about
{\displaystyle 111132.92-559.82\,\cos 2\varphi +1.175\,\cos 4\varphi -0.0023\,\cos 6\varphi }
Similarly, the feckin' length in meters of a holy degree of longitude can be calculated as
{\displaystyle 111412.84\,\cos \varphi -93.5\,\cos 3\varphi +0.118\,\cos 5\varphi }
(Those coefficients can be improved, but as they stand the bleedin' distance they give is correct within an oul' centimeter.)
An alternative method to estimate the feckin' length of a longitudinal degree at latitude
{\displaystyle \textstyle {\varphi }\,\!}
is to assume an oul' spherical Earth (to get the oul' width per minute and second, divide by 60 and 3600, respectively):
{\displaystyle {\frac {\pi }{180}}M_{r}\cos \varphi \!}
{\displaystyle \textstyle {M_{r}}\,\!}
is 6,367,449 m. Since the bleedin' Earth is an oblate spheroid, not spherical, that result can be off by several tenths of a bleedin' percent; a better approximation of a longitudinal degree at latitude
{\displaystyle \textstyle {\varphi }\,\!}
{\displaystyle {\frac {\pi }{180}}a\cos \beta \,\!}
{\displaystyle a}
{\displaystyle \textstyle {\tan \beta ={\frac {b}{a}}\tan \varphi }\,\!}
; for the GRS80 and WGS84 spheroids, b/a calculates to be 0.99664719, what? (
{\displaystyle \textstyle {\beta }\,\!}
is known as the oul' reduced (or parametric) latitude), would ye swally that? Aside from roundin', this is the exact distance along a parallel of latitude; gettin' the feckin' distance along the oul' shortest route will be more work, but those two distances are always within 0.6 meter of each other if the feckin' two points are one degree of longitude apart.
Like any series of multiple-digit numbers, latitude-longitude pairs can be challengin' to communicate and remember. Therefore, alternative schemes have been developed for encodin' GCS coordinates into alphanumeric strings or words:
the World Geographic Reference System (GEOREF), developed for global military operations, replaced by the bleedin' current Global Area Reference System (GARS).
Open Location Code or "Plus Codes," developed by Google and released into the oul' public domain.
Geohash, a public domain system based on the oul' Morton Z-order curve.
What3words, a bleedin' proprietary system that encodes GCS coordinates as pseudorandom sets of words by dividin' the bleedin' coordinates into three numbers and lookin' up words in an indexed dictionary.
Geographical distance – Distance measured along the bleedin' surface of the bleedin' earth
Linear referencin'
^ The pair had accurate absolute distances within the Mediterranean but underestimated the oul' circumference of the bleedin' Earth, causin' their degree measurements to overstate its length west from Rhodes or Alexandria, respectively.
^ Alternative versions of latitude and longitude include geocentric coordinates, which measure with respect to Earth's center; geodetic coordinates, which model Earth as an ellipsoid; and geographic coordinates, which measure with respect to a plumb line at the oul' location for which coordinates are given.
^ Taylor, Chuck. Whisht now and listen to this wan. "Locatin' a holy Point On the oul' Earth". Archived from the original on 3 March 2016. Here's a quare one for ye. Retrieved 4 March 2014.
^ "Usin' the EPSG geodetic parameter dataset, Guidance Note 7-1", Lord bless us and save us. EPSG Geodetic Parameter Dataset, would ye believe it? Geomatic Solutions, to be sure. Retrieved 15 December 2021.
^ McPhail, Cameron (2011), Reconstructin' Eratosthenes' Map of the oul' World (PDF), Dunedin: University of Otago, pp. 20–24 .
^ Greenwich 2000 Limited (9 June 2011). Jaysis. "The International Meridian Conference". Wwp.millennium-dome.com, to be sure. Archived from the original on 6 August 2012. Retrieved 31 October 2012.
^ American Society of Civil Engineers (1 January 1994), like. Glossary of the Mappin' Sciences. Bejaysus this is a quare tale altogether. ASCE Publications. Sure this is it. p. 224. ISBN 9780784475706.
^ Bolstad, Paul (2012). GIS Fundamentals (PDF) (5th ed.). Atlas books. Sure this is it. p. 102. Here's a quare one. ISBN 978-0-9717647-3-6.
^ "Makin' maps compatible with GPS". Jesus, Mary and Joseph. Government of Ireland 1999. Sufferin' Jaysus. Archived from the original on 21 July 2011. Retrieved 15 April 2008.
, a feckin' desktop planetarium for Linux/KDE. See The KDE Education Project - KStars
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Some exact constants for the approximation of the quantity in the Wallis’ formula | Journal of Inequalities and Applications | Full Text
Some exact constants for the approximation of the quantity in the Wallis’ formula
Senlin Guo1,
Jian-Guo Xu1 &
Feng Qi2
In this article, a sharp two-sided bounding inequality and some best constants for the approximation of the quantity associated with the Wallis’ formula are presented.
MSC:41A44, 26D20, 33B15.
Throughout the paper, ℤ denotes the set of all integers, ℕ denotes the set of all positive integers,
\begin{array}{r}{\mathbb{N}}_{0}:=\mathbb{N}\cup \left\{0\right\},\\ n!!:=\prod _{i=0}^{\left[\left(n-1\right)/2\right]}\left(n-2i\right),\end{array}
{W}_{n}:=\frac{\left(2n-1\right)!!}{\left(2n\right)!!}.
Here in (1), the floor function
\left[t\right]
denotes the integer which is less than or equal to the number t.
The Euler gamma function is defined and denoted for
Rez>0
\mathrm{\Gamma }\left(z\right):={\int }_{0}^{\mathrm{\infty }}{t}^{z-1}{e}^{-t}\phantom{\rule{0.2em}{0ex}}dt.
One of the elementary properties of the gamma function is that
\mathrm{\Gamma }\left(x+1\right)=x\mathrm{\Gamma }\left(x\right).
\mathrm{\Gamma }\left(n+1\right)=n!,\phantom{\rule{1em}{0ex}}n\in {\mathbb{N}}_{0}.
\mathrm{\Gamma }\left(\frac{1}{2}\right)=\sqrt{\pi }.
For the approximation of n!, a well-known result is the following Stirling’s formula:
n!\sim \sqrt{2\pi n}{n}^{n}{e}^{-n},\phantom{\rule{1em}{0ex}}n\to \mathrm{\infty },
which is an important tool in analytical probability theory, statistical physics and physical chemistry.
{W}_{n}
, defined by (2). This quantity is important in the probability theory - for example, the three events, (a) a return to the origin takes place at time 2n, (b) no return occurs up to and including time 2n, and (c) the path is non-negative between 0 and 2n, have the common probability
{W}_{n}
. Also, the probability that in the time interval from 0 to 2n the particle spends 2k time units on the positive side and
2n-2k
time units on the negative side is
{W}_{k}{W}_{n-k}
. For details of these interesting results, one may see [[1], Chapter III].
{W}_{n}
is closely related to the Wallis’ formula.
The Wallis’ formula
\frac{2}{\pi }=\prod _{n=1}^{\mathrm{\infty }}\frac{\left(2n-1\right)\left(2n+1\right)}{{\left(2n\right)}^{2}}
can be obtained by taking
x=\frac{\pi }{2}
in the infinite product representation of sinx (see [[2], p.10], [[3], p.211])
sinx=x\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{{x}^{2}}{{n}^{2}{\pi }^{2}}\right),\phantom{\rule{1em}{0ex}}x\in \mathbb{R}.
\prod _{n=1}^{\mathrm{\infty }}\frac{\left(2n-1\right)\left(2n+1\right)}{{\left(2n\right)}^{2}}=\underset{n\to \mathrm{\infty }}{lim}\left(2n+1\right){W}_{n}^{2},
another important form of Wallis’ formula is (see [[4], pp.181-184])
\underset{n\to \mathrm{\infty }}{lim}\left(2n+1\right){W}_{n}^{2}=\frac{2}{\pi }.
The following generalization of Wallis’ formula was given in [5].
\frac{\pi }{tsin\left(\pi /t\right)}=\frac{1}{t-1}\prod _{i=1}^{\mathrm{\infty }}\frac{{\left(it\right)}^{2}}{\left(it+t-1\right)\left(it-t+1\right)},\phantom{\rule{1em}{0ex}}t>1.
In fact, by letting
x=\left(1-1/t\right)\pi ,\phantom{\rule{1em}{0ex}}t\ne 0
sin\frac{\pi }{t}=\frac{\pi }{t}\left(t-1\right)\prod _{i=1}^{\mathrm{\infty }}\frac{\left(it+t-1\right)\left(it-t+1\right)}{{\left(it\right)}^{2}},\phantom{\rule{1em}{0ex}}t\ne 0.
\frac{\pi }{tsin\left(\pi /t\right)}=\frac{1}{t-1}\prod _{i=1}^{\mathrm{\infty }}\frac{{\left(it\right)}^{2}}{\left(it+t-1\right)\left(it-t+1\right)}
t\ne 0,\phantom{\rule{2em}{0ex}}t\ne \frac{1}{k},\phantom{\rule{1em}{0ex}}k\in \mathbb{Z}.
(12) is a special case of (14). The proof of (12) in [5] involves integrating powers of a generalized sine function.
There is a close relationship between Stirling’s formula and Wallis’ formula. The determination of the constant
\sqrt{2\pi }
in the usual proof of Stirling’s formula (7) or Stirling’s asymptotic formula
\mathrm{\Gamma }\left(x\right)\sim \sqrt{2\pi }{x}^{x-1/2}{e}^{-x},\phantom{\rule{1em}{0ex}}x\to \mathrm{\infty },
relies on Wallis’ formula (see [[2], pp.18-20], [[3], pp.213-215], [[4], pp.181-184]).
{W}_{n}={\left[\left(2n+1\right){\int }_{0}^{\pi /2}{sin}^{2n+1}x\phantom{\rule{0.2em}{0ex}}dx\right]}^{-1}
={\left[\left(2n+1\right){\int }_{0}^{\pi /2}{cos}^{2n+1}x\phantom{\rule{0.2em}{0ex}}dx\right]}^{-1}
and Wallis’ sine (cosine) formula (see [[6], p.258])
{W}_{n}=\frac{2}{\pi }{\int }_{0}^{\pi /2}{sin}^{2n}x\phantom{\rule{0.2em}{0ex}}dx
=\frac{2}{\pi }{\int }_{0}^{\pi /2}{cos}^{2n}x\phantom{\rule{0.2em}{0ex}}dx.
Some inequalities involving
{W}_{n}
were given in [7–12].
In this article, we give a sharp two-sided bounding inequality and some exact constants for the approximation of
{W}_{n}
, defined by (2). The main result of the paper is as follows.
n\in \mathbb{N}
n\ge 2
\sqrt{\frac{e}{\pi }}{\left(1-\frac{1}{2n}\right)}^{n}\frac{\sqrt{n-1}}{n}<{W}_{n}\le \frac{4}{3}{\left(1-\frac{1}{2n}\right)}^{n}\frac{\sqrt{n-1}}{n}.
\sqrt{e/\pi }
4/3
in (20) are best possible.
{W}_{n}\sim \sqrt{\frac{e}{\pi }}{\left(1-\frac{1}{2n}\right)}^{n}\frac{\sqrt{n-1}}{n},\phantom{\rule{1em}{0ex}}n\to \mathrm{\infty }.
Remark 1 By saying that the constants
\sqrt{e/\pi }
4/3
in (20) are best possible, we mean that the constant
\sqrt{e/\pi }
in (20) cannot be replaced by a number which is greater than
\sqrt{e/\pi }
4/3
in (20) cannot be replaced by a number which is less than
4/3
We need the following lemmas to prove our result.
Lemma 1 ([[13], Theorem 1.1])
f\left(x\right):=\frac{{x}^{x+\frac{1}{2}}}{{e}^{x}\mathrm{\Gamma }\left(x+1\right)}
is strictly logarithmically concave and strictly increasing from
\left(0,\mathrm{\infty }\right)
\left(0,\frac{1}{\sqrt{2\pi }}\right)
h\left(x\right):=\frac{{e}^{x}\sqrt{x-1}\mathrm{\Gamma }\left(x+1\right)}{{x}^{x+1}}
\left(1,\mathrm{\infty }\right)
\left(0,\sqrt{2\pi }\right)
Lemma 3 ([[6], p.258])
n\in \mathbb{N}
\mathrm{\Gamma }\left(n+\frac{1}{2}\right)=\sqrt{\pi }n!{W}_{n},
{W}_{n}
Remark 2 Some functions associated with the functions
f\left(x\right)
h\left(x\right)
, defined by (22) and (23) respectively, were proved to be logarithmically completely monotonic in [14–16]. For more recent work on (logarithmically) completely monotonic functions, please see, for example, [17–43].
\frac{3}{e\sqrt{e\pi }}=f\left(\frac{3}{2}\right)\le f\left(n-\frac{1}{2}\right)=\frac{{\left(n-\frac{1}{2}\right)}^{n}}{{e}^{n-1/2}\mathrm{\Gamma }\left(n+1/2\right)}<\frac{1}{\sqrt{2\pi }},\phantom{\rule{1em}{0ex}}n\ge 2,
\underset{n\to \mathrm{\infty }}{lim}\frac{{\left(n-\frac{1}{2}\right)}^{n}}{{e}^{n-1/2}\mathrm{\Gamma }\left(n+1/2\right)}=\frac{1}{\sqrt{2\pi }}.
The lower and upper bounds in (25) are best possible.
By Lemma 3, (25) and (26) can be rewritten respectively as
\frac{3}{{e}^{2}}\le \frac{{\left(n-\frac{1}{2}\right)}^{n}}{{W}_{n}{e}^{n}n!}<\frac{1}{\sqrt{2e}},\phantom{\rule{1em}{0ex}}n\ge 2,
\underset{n\to \mathrm{\infty }}{lim}\frac{{\left(n-\frac{1}{2}\right)}^{n}}{{W}_{n}{e}^{n}n!}=\frac{1}{\sqrt{2e}}.
3/{e}^{2}
1/\sqrt{2e}
{\left(\frac{e}{2}\right)}^{2}=h\left(2\right)\le h\left(n\right)=\frac{{e}^{n}n!\sqrt{n-1}}{{n}^{n+1}}<\sqrt{2\pi },\phantom{\rule{1em}{0ex}}n\ge 2,
\underset{n\to \mathrm{\infty }}{lim}\frac{{e}^{n}n!\sqrt{n-1}}{{n}^{n+1}}=\sqrt{2\pi }.
{\left(e/2\right)}^{2}
and the upper bound
\sqrt{2\pi }
From (27) and (29), we obtain that for all
n\ge 2
\frac{3}{4}\le \frac{\sqrt{n-1}{\left(n-\frac{1}{2}\right)}^{n}}{{W}_{n}{n}^{n+1}}<\sqrt{\frac{\pi }{e}}.
3/4
\sqrt{\pi /e}
in (31) are best possible. From (31) we get that for all
n\ge 2
\sqrt{\frac{e}{\pi }}{\left(1-\frac{1}{2n}\right)}^{n}\frac{\sqrt{n-1}}{n}<{W}_{n}\le \frac{4}{3}{\left(1-\frac{1}{2n}\right)}^{n}\frac{\sqrt{n-1}}{n}.
\sqrt{e/\pi }
4/3
\underset{n\to \mathrm{\infty }}{lim}\frac{\sqrt{n-1}{\left(n-\frac{1}{2}\right)}^{n}}{{W}_{n}{n}^{n+1}}=\sqrt{\frac{\pi }{e}},
The proof is thus completed. □
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Qi F, Guo B-N, Chen C-P: Some completely monotonic functions involving the gamma and polygamma functions. J. Aust. Math. Soc. 2006, 80: 81–88. 10.1017/S1446788700011393
Qi F, Guo S, Guo B-N: Complete monotonicity of some functions involving polygamma functions. J. Comput. Appl. Math. 2010, 233: 2149–2160. 10.1016/j.cam.2009.09.044
Sevli H, Batir N: Complete monotonicity results for some functions involving the gamma and polygamma functions. Math. Comput. Model. 2011, 53: 1771–1775. 10.1016/j.mcm.2010.12.055
Shemyakova E, Khashin SI, Jeffrey DJ: A conjecture concerning a completely monotonic function. Comput. Math. Appl. 2010, 60: 1360–1363. 10.1016/j.camwa.2010.06.017
Koumandos S, Pedersen HL: Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function. J. Math. Anal. Appl. 2009, 355: 33–40. 10.1016/j.jmaa.2009.01.042
Srivastava HM, Guo S, Qi F: Some properties of a class of functions related to completely monotonic functions. Comput. Math. Appl. 2012, 64: 1649–1654. 10.1016/j.camwa.2012.01.016
The authors thank the editor and the referees for their valuable suggestions to improve the quality of this paper. The present investigation was supported, in part, by the Natural Science Foundation of Henan Province of China under Grant 112300410022.
Department of Mathematics, Zhongyuan University of Technology, Zhengzhou, Henan, 450007, China
Senlin Guo & Jian-Guo Xu
Correspondence to Senlin Guo.
All the authors contributed to the writing of the present article. They also read and approved the final manuscript.
Guo, S., Xu, JG. & Qi, F. Some exact constants for the approximation of the quantity in the Wallis’ formula. J Inequal Appl 2013, 67 (2013). https://doi.org/10.1186/1029-242X-2013-67
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PRO-V - Electowiki
Product Voting (PRO-V) is a Single-Winner Cardinal voting systems developed by Aldo Tragni.
The objectives of this voting system is the balance between simplicity, resistance to strategies, elect utilitarian winner and provide the voter with a good representation of interests (range with 5 ratings).
1.2 Ratings scale
2 Proportional vote philosophy
3.1 FAIR-V
Ballots: voter score each candidates with bonus in [x1,x2,x3,x4,x5] and the absence of evaluation is equivalent to x1. Candidates at the beginning all have 1 point.
Counting: bonuses are applied to each candidate (eg x3 means multiplying the candidate's points by 3), and the one with the highest score in the end wins.
The formula used in the count is the following:
{\displaystyle \begin{equation}
C_{i}=\prod V_{ij}
\end{equation}}
Ci = final score of a candidate Ci.
Vij = values of candidate Ci, obtained from the ballots.
However, this formula can return very large results, difficult to manage.
In computer systems the following formula can be used :
{\displaystyle \begin{equation}
C_{i} =\prod \sqrt[n]{V_{ij}}
\end{equation}}
n = total number of votes.
If you have paper ballots then, before counting, you can eliminate ratings x1 and those ratings that appear at least once on all candidates, even in different votes.
Eg given these 3 votes:
A[x1] B[x2] C[x3] D[x3] E[x5]
delete the ratings x1, and x3 that appears at least once on all candidates, making the votes like this:
B[x2] E[x5]
B[x5] D[x2]
A[x5] C[x2] D[x4]
so there is less multiplication to do.
Ratings scale[edit | edit source]
The minimum value of the range is always x1.
In a context with very different options (such as the electoral context) it's better to use an exponential scale, like this:
[x1, x2, x4, x8, x16]
while in contexts with options not very far from each other (such as satisfaction surveys) it's better to use a linear scale of this type:
[x1, x2, x3, x4, x5]
Adapting the scale to the context allows the voter to represent their interests well, maintaining simplicity in the vote (which always has only 5 ratings) and also more resistance to strategies.
PRO-nV: the PRO-V procedure works with ranges of different sizes and n indicates the amount of ratings used in the range.
PRO-3V: uses 3 ratings.
PRO-V: is the default definition, with 5 ratings.
FAIR-9V: uses 9 ratings.
Proportional vote philosophy[edit | edit source]
Voters and candidates are treated as if they were groups of interests. In the case of the voter, interests are what he wants while in the case of candidates, interests are what they will actually do if they are elected (in the real world they can only be estimated by also evaluating the candidate's honesty, competence, sympathy, etc).
Voters and candidates are represented as points in the interest space, and a voter's appreciation of candidates depends on the distance between those points. The minimum distance between two points is 0, while the maximum distance between two points can be infinite.
In the philosophy of proportional voting, there cannot be a better candidate than the one who is 0 distance from the voter, while there can always be a worse candidate. This allows you to use the voter's position as a fixed point to derive proportions between candidates.
E.g. if a voter has these distances from candidates:
A[10] B[20] C[40] D[80] E[100] F[200] G[1000]
then the proportions between candidates will be:
A/A, A/B, A/C, A/D, A/E, A/F, A/G
A[1] B[0.5] C[0.25] D[1.25] E[0.1] F[0.05] G[0.001]
If the ratings to be used are [x1, x2, x4, x8] then the rating will be:
A[x8] B[x4] C[x2] D[x1] E[x1] F[x1] G[x1]
It's noted that the most liked candidates had good representation, while the worst candidates of D, all finished in 1x. However, it would be sufficient to add x16 as a possible rating to obtain a vote like this: A[x16] B[x8] C[x4] D[x2] E[x2] F[x1] G[x1] in which only G was misrepresented.
In the approval votes, instead, the range of distances (utility) is generally stretched until it becomes [0,5] obtaining as a vote:
A[5] B[5] C[5] D[5] E[5] F[4] G[0]
even increasing the range to [0,9] the vote would become:
without actual improvements.
All this points out that, if the distances (utility) between voter and candidates aren't known, then it's not possible to convert an approval vote into a proportional vote, or vice versa.
If the proportions indicated in a proportional vote are to be kept unchanged, then there are no normalizations directly applicable to the vote.
Eg given this vote A[x8] B[x4] C[x2] D[x1] E[x1] if candidates D and E were removed, the vote could be normalized as follows A[x8] B[x2] C[x1] but the proportions with A would not be respected. If instead, candidate A were removed, the vote could be normalized as follows B[x8] C[x4] D[?] E[?] but it would be impossible to know whether candidates D and E deserve x1 or x2.
Even methods that change the weight/power of the vote, multiplying all ratings by a certain value, don't change the proportions between candidates.
A type of indirect normalization applicable is the cumulative one, that is: each voter is assigned a certain amount of points (eg 100 points) which are distributed among the candidates according to the proportions indicated in the vote. By removing a candidate, the distribution of points changes, but the proportions don't change (DV uses this procedure).
FAIR-V[edit | edit source]
The proportional ratings of the PRO-V make the intermediate ratings more used by the voter, because eg. adding these two votes A[x1] B[x2] (B is worth double A) and A[x4] B[x2] (A is worth double B), candidates A and B are equal, unlike the methods that add up the scores. One single point can make a lot of difference. Taking this characteristic of the PRO-V into consideration, and following the analysis of the resistance to strategies in FAIR-V, it can be seen that PRO-V is also resistant to strategies, but not as strong as FAIR-V which uses a range [0,2].
However, the PRO-V procedure is easier to understand than the FAIR-V one, and also offers a wider range of ratings to the voter.
Retrieved from "https://electowiki.org/w/index.php?title=PRO-V&oldid=12847"
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Sequences And Series, Popular Questions: CBSE Class 11-science ENGLISH, English Grammar - Meritnation
Subah & 1 other asked a question
Himanshu Bharadwaj asked a question
What is the next number of the given series:-
4,10,34,94,214,424...
Jishin & 2 others asked a question
the maximum sum of the series 20,58/3,56/3,... is
nandansah07... asked a question
In the A.P. whose common difference is non-zero, the sum of first 3n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2n terms to the nwxt 2n terms is how much?
Gunjan Saxena asked a question
-3/4,3/16,-3/64
Three numbers are in A.P., and the product of them is same as product of smallest and largest. What is the middle number?
Jacob Gomes & 2 others asked a question
2, 6^1/2 ,4.5 ARE THE FOLLOWING TERMS OF AN AP
Oswashbukler . asked a question
let s be the set of integers which are divisible by 5 and let t be the set of integer which are divisible by 7 find the number of positive integers less than 1000 and not in s u t
if x1,x2,x3 ....xn are in H.P then prove that x1x2+x2x3+x3x4+....+xn-1xn =( n-1)x1xn
Layatmika Sahoo asked a question
Karthik G asked a question
a,b,c,d are in increasing GP. AM between a and b is 6. AM between c and d is 54. Then AM between a and d is:
Hifi asked a question
if the sum 3 by 1 square + 5 by 1 square + 2 square + 7 by 1 square + 2 square + 3 square upto 20 terms is equal to K by 21
Bhavya Shukla & 1 other asked a question
Sumit Goyal asked a question
?, 20, 36, 68, 132, 260
Rhea Thankachan Mathew & 2 others asked a question
In an A.P.,s3=6,s6=3,prove that 2(2n+1)Sn+4=(n+4)S2n+1.
Skyhawks asked a question
Surbhi Shrivastava asked a question
If a, b, c, d are in G.P, prove that are in G.P.
Tanya & 2 others asked a question
find the sum of integers from 1 to 100 that are divisible by 2 or 5?
If p , q, r are in G.P and the equation px^2 + 2qx + r = 0 and dx^2 + 2ex + f = 0 have a common root , then show that d/p , e/q , f/r are in A.P.
Aditi Saini asked a question
prove that:2^1/4 . 4^1/8 . 8^1/16............upto infinity=2
why for finding 4 terms in A.P we use (a-3d), (a-d),( a+d), (a-3d)..... not a-2d , a-d,a+d,a+2d ....then also in this list "a" is missing ....?!
If xyz=(1-x)(1-y)(1-z) where 0 is less than equals to x,y,z is less than equals to 1 then the minimum value of x(1-z)+y(1-x)+z(1-y) is :
2. If xyz = \left(1-x\right) \left(1-y\right) \left(1-z\right) where 0\le x,y,z\le 1, then the minimum value of x\left(1-z\right) +y\left(1-x\right) +z\left(1-y\right) is:\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(a\right) \frac{3}{2} \left(b\right) \frac{1}{4} \left(c\right) \frac{3}{4} \left(d\right) \frac{1}{2}
let x=1+a+a2+.............. y=1+b+b2+.......... thn prove that
1+ab+a2b2+.......=xy/x+y-1
Aditya Raina asked a question
An A.P consists of 12 terms whose sum is 354. The ratio of the sum of the even terms to the sum of the odd terms is 32:27. Find the common difference of the progression.
If the pth,qth terms of a G.P. are q,p respectively ,show that (p+q)th term is (qp/pq)1/p-q
Inayat Singhania asked a question
19. Consider a three digit number x1x2x3 such that x1.x2.x3
\in
N. Then the number of positive integral solutions of x1.x2.x3 = 480 is
{\lambda }^{2}
-100, then the sum of the digits of
\left|\lambda \right|
Snehil Kumar asked a question
If a,b,c are in A.P and X,Y,Z are in G.P then prove( Xb-c)x (Yc-a) x(Za-b)=1
In a G.P tp+q=m and tp-q=n. Then find tp.
Wanted solution of
Q.17. Sonia has 55 blocks. She decides to to stack up all the blocks so that each row has one less block than below. She wants to end up with just 1 block on top. How many should she put in the bottom row?
Sushil Prem asked a question
Find the sum tonterms of the sequence, 6, 66, 666, 6666…
7^2+9^2+11^2+............n terms
prove that AM > GM> HM
If in an A.P., Sn= n2p and Sm=m2p, then Sp=?
Q. Find the sum
i. 1.3+2.5+3.7+....+n(2n+1)
ii. .9+.99+.999+.....to n-term.
Anouska Patnaik asked a question
1+(1+2)+(1+2+3)+(1+2+3+4)... find the sum of the series
Arjun Raj asked a question
Find sum of series 243 + 343 + 432 +........upto n terms.
In a G.P. consisting of positive terms , each term equals the sum of the next two terms. Then the common ratio of this progression equals...
a) 1/2 (root 5) b) root 5 c) 1/2 (root 5 - 1) d) 1/2 (1- root 5 )
Let An be a sequence defined by A1=1 then A1+ 2A2 + 3A3+ ... + (n-1)A n-1 = (n^2)An , n is greater than or equal to 2. The value of A786 ? Answer is 1/1572
FIND FOUR NUMBERS IN AP WHOSE SUM IS 20 &SUM OF WHOSE SQUARES IS 120
Find the sum of the sequence 0.4, 0.44, 0.444……. up tonterms.
explaning each step
Aishwarya Khilari & 1 other asked a question
The difference between any two consecutive interior angles of a polygon is 5°.If the smallest angle is 120°, find the number of the sides of the polygon.
if tn=n/(n+1)! then Σ tn is equal to
(where summation is from n=1 to n=20)
In the arithmetic progression 2,5,8....upto 50 terms,and 3,5,7,9....upto 60 terms ,find how many pairs are identical and find the identical pairs.
the sum of the first three terms of a G.P. is to the sum of first six terms as 125:152.Find the common ratio of the G.P.
three numbers are in A.P. and their sum is 15 , if 1, 4 and 19 are added to these numbers respectively the resulting numbers are in G.P.. find the numbers.
Kushagra Saxena asked a question
if a,b,c are in a.p and x,y,z are in g.p. then prove that x^b-c x y ^c-a x z^a-b = 1 descriptive form???????
Taha Husain asked a question
The sum of an infinite number of terms of GP is 4 and the sum of their cubes is 192. Find the series.
Suhas Gupta asked a question
If (xp + yp)/ (xp-1+ yp-1)be the A.M between x and y, then find the value of p.
The sum of first two terms of an infinite G.P. is 5 and each terms is three times the sum of the succeeding terms. Find the G.P.
Find the sum of the infinite series 5, 10/3, 20/9, 40/27..........
Shubham Yadav asked a question
The sum of the reciprocals of the first 11 terms in the harmonic progression series is 110. Determine the 6 terms of the harmonic progression series.
the sum of three numbers in GP is 56. if we subtract 1, 7, 21 from these numbers in that order, we obtain an AP. find the numbers.
Let c and b be the roots of x2-6x-2=0 (cb). If
an=cn-bn where (n=1)
then a10-2a8/2a9 =
Aditya Singhal & 1 other asked a question
If one A.M., A and two geometric means G1 and G2 inserted between any two positive numbers,show that G12/ G2 + G22/ G1 = 2A.
Soumyajit Shome asked a question
Prove that sin 16 degree + cos 16 degree = 1/root 2 ( 3cos 1 degree+sin 1 degree )
if in a gp the (p+q)th term is a and (p-q)th term is b , prove that the pth term is sqrt (ab).
find missing term 2,17,52,......,206
Sweetie... asked a question
pth term of an A.P. is 'a' and qth term is 'b'. prove that sum of (p+q)th term is p+q/2[a+b+(a-b/p-q)]. plz ans needed soon.....
Find the sum to infinity: 0.3 + 0.18 + 0.108 + ...
Rosella Rapunzel asked a question
Let S be the sum, P be the product and R be the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn
Find the sum to n terms of the sequence, 8, 88, 888, 8888, .......... .
Find the sum of 30th term of the sequence 7, 7.7, 7.77, 7.777..................
sum of an infinite gp is 57 and sum of their cube is 9747 find the gp
Find a, b such that 7.2, a, b, 3 are in A.P.
Ans given: a = 5.8, b = 4.4
Sajal Swapnil asked a question
let x=1+a+a^2........ and y=1+b+b^2..... , where modulus(a)
1+ab+a^2b^2.......=xy/(x+y-1)
please i have a test tommorow.....!!
if the sum of the 3 terms of a G.P is 21 and sum of their squares is 189 find the three no.s in G.P.
how to find the solution.. pls help.....
Prove that a,b,c are in A.P. iff 1/bc,1/ca,1/ab are also in A.P.
1.If the AM ,GM and HM of the first and last terms of the series 25,26,27,.....N-1,N are the terms of the series,find the value of N?
2.The sum of all possible products of the first 'N' natural numbers taken two bu two is:
a)1/24n(n+1)(n-1)(3n+2)
b)n(n+1)(n-1)(2n+3)/24
c)n(n+1)(n-1)(2n+1)/6
If An - (An-1) =1 for every positive integer greater than 1, then A1+ A2 + A3... +A100 =? Answer is 5050.A1
THE THIRD TERM OF A G.P IS 2/3 & THE 6TH TRERM IS 2/81 FIND THE 8TH TERM
The sum of the infinite G.P is 15, Sum of squares of 3 terms of the g.p is 45. Find the series.
find 3 numbers in AP whose sum is 24 and sum oftheir cubes is 1968.
Find the sum of sequence, 7, 77, 777, 7777..... to n terms ???
plz ans fast....
Amit Anand asked a question
5+55+555...n terms. Change this term into G.P. then find the sum of this series?
if pth term of a gp is p and qth term is q.prove that the nth term is (p (to the power n-q) /q (to the power n-p) ) to the power 1/p-q
Lt Col Sanjay Kumar asked a question
If (a1,a2,a3......an) are in A>P> with common difference d. then the sum of the series: sin d [cosec a1.cosec a2 + cosec a2.cosec a3 +........+ cosec a(n-1).cosec an ] is what ??
Sigma 3r(2-2r)/(r+1)(r+2)
(A) 1/2 - {3100/100(101)}
(B) 3/2 - {3101/101(102)}
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División de Energías Alternas, Instituto de Investigaciones Eléctricas, Cuernavaca Morelos, México.
Abstract: In this paper it is proposed that the mass of the bodies has its origin and nature in the reciprocal gravitational interactions between them; and also by some kind of effect over the size of the celestial bodies due to the very big distances in space, as seemed each other at a distance. In a Dynamic Theory of Gravitation [1], it is proved that the fundamental velocity is the escape velocity due to the apparent size of the interacting heavenly bodies, which is the medium used by gravity to transmit its effects like propagating force of Nature [2]. Given that is the greatest speed of the Universe, the celestial bodies interact between them in a reciprocal way [3]. Because of that dynamical process all those bodies have an intrinsic property called mass. Then, the mass of any body is a kind of parameter by means of which a measure of the inertial effects can be obtained. That property is different from weight. It is a consequence of the gravitational interactions between any body and all the rest of the heavenly bodies of the Universe, and also by some deep characteristic of the space that separate them.
Keywords: Escape Velocity, Gravitational and Inertial Mass, Weight
It was Galileo who discovered that bodies fall at a rate independent of their mass. He used as a tools an incline plane to slow the fall, a water clock to measure the time, and a pendulum to avoid rolling friction [4] . Newton could use his second law to conclude that the force exerted by gravity over a body is proportional to its mass [5] . However, he was aware that his conclusions might be approximately true; in such a way that the inertial mass of his second law might not be the same as the gravitational mass appearing in the law of gravitation [5] . The question of the equality of both kinds of masses is linked to the problem of the concept of mass as a deep property of the bodies, those of gravity, and also the very big distances in space.
The weight of a body is the gravitational force exerted on it by a source of the gravity attraction. Being a force, is a vector whose direction is the direction of the gravity force; that is, toward the center of the source [5] . When a body of mass m is allowed to fall freely on the Earth’s surface, its acceleration is the gravity acceleration g, and the force acting on it is its weight W.
F=ma
where F is the applied force, m the mass, and a the acceleration, when is applied to a freely falling body, takes the following form
W=m\text{ }g
where both vectors are directed toward the center of the Earth; so that it can therefore write that
W=mg
W and g are the magnitudes of the weight and acceleration vectors, respectively.
To keep an object from falling, it has to exert on it an upward force equal in magnitude to W so as to make the net force zero [5] . It is well known that g is found experimentally to have the same value for all bodies at the same place [5] . From this it follows that the ratio of the weights of two objects must be equal to the ratio of their masses [5] . Therefore a balance, which is an instrument for comparing downward forces, can be used in practice to compare masses. However, it is important to distinguish carefully between weight and mass, because the weight of a body is a vector quantity, in the meantime its mass is a scalar quantity [5] . Besides, the weight is measured with the body at rest, and the mass of the same body is measured in movement.
The relationship between weight and mass is given by the equation (2). Because g varies from point to point on the Earth, W which is the weight of a body of mass m, is different in different localities. Hence, unlike the mass of a body which is an intrinsic property of the body, the weight depends on its location relative to the center of the Earth. Then, the weight of a body is zero in regions of space where the gravitational field or its effects, are null; although the inertial effects, and hence the mass of the body remains unchanged from those on the Earth [5] .
It takes the same force to accelerate a body in a gravity-free space as it does to accelerate it along a horizontal frictionless surface on the Earth; because its mass is the same in each place; but it takes a greater force to hold the body against to pull of the Earth on the earth’s surface than it does high up in space, because its weight is different in each place.
Often, instead of being given the mass, are given the weight of the body on which the forces are exerted. The acceleration a produced by the force F acting on a body whose weight is W can be obtained combining the Equations (1) and (3). Therefore from
F=ma
W=mg
it is easy to obtain that
m=\frac{W}{g}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}F=\left(\frac{W}{g}\right)a
The quantity W/g plays the role of m in the Equation (1), and is in fact the mass of the body whose weight is W [5] .
If the inertial mass of Newton’s second law were not the same as the gravitational mass in the law of gravitation, it would have to write the Newton’s second law as
F={m}_{i}a
and write the law of gravitation as
F={m}_{g}g
where g is the gravity acceleration which is a field depending on position and other masses [4] . Thus, the acceleration at a given point would be
a=\left(\frac{{m}_{g}}{{m}_{i}}\right)g
and would be different for bodies with different values of the ratio mg/mi . In particular, pendulums of equal length would have periods proportional to (mi/mg)1/2 . Newton in experiments with pendulums of equal length but different composition tested that possibility and found no differences in their periods [4] . That result was later verified more accurately by F.W. Bessel [4] , and also by R.v. Eötvos [6] [7] [8] , who proved by a different method that the ratio does not differ from one material to another by more than one part in 109. A few years later, a group under R.H. Dicke [9] [10] improved on Eötvos method by using the gravitational field of the Sun, and Earth’s centripetal acceleration toward the Sun, rather than the rotation of the Earth, to produce a torque on the balance of the Eötvos experiment [4] . The conclusion was that the aluminum and gold fall toward the Sun with the same acceleration differing each other by at most one part in 1011 [9] [10] . It has also been shown, with very much less precision, that neutrons fall with the same acceleration as ordinary matter; and that the gravitational force on electrons in Cooper is the same as on free electrons [4] .
4. Gravity and Mass
In spite of the experimental results by means of which the equality of the gravitational and inertial mass are proved; the question about the real nature of the mass has not an answer. Many teachers and books head of the question assuming that the problem of the mass, as an intrinsic property of the bodies, has already been solved. However, it is clear that the mass is independent of the material, composition, and atomic structure of the bodies; and also is not the same as the weight. So that, even today that problem is centered in the question about the origin and nature of the mass.
In Newton’s Law of Universal Gravitation it is implicit the idea that the gravity force between two celestial bodies is independent of the presence of other heavenly bodies, or the properties of the space between them [5] . Nevertheless, that idea might not be true, because those bodies are not insolated in the space; in such a way that it is important to take into account the gravitational attraction by all the bodies of the Universe which is exerted over any body by means of the escape velocity due to their apparent sizes [2] . So that, when the escape velocity of any body, due to its apparent size, in some region of the space, meets with the respective escape velocity of the rest of the celestial bodies, they pull the body, but that body pull de other bodies, because those velocities due to their respective apparent sizes, are the carriers of their gravitational fields [2] . Given that the gravity force is always attractive, that dynamic process is the responsible for the link which is established between the body and the other bodies. It is proposed here that as a result of that influence the body has the intrinsic property known as its mass. So that, the mass is something produced by gravity as a deep characteristic of the bodies; in such a way that it is possible to conclude that the gravitational and inertial mass are the same thing; as it was proved experimentally by Galileo, Newton, Bessel, Eötvos, and Dicke [4] .
It is possible to assert that the gravitational interactions between the heavenly bodies of the Universe, interact each other in a reciprocal way, and also due to their apparent sizes, as seemed at a distance are the responsible of the intrinsic property of any body known as its mass. That is to say, the gravitational influence is the origin and nature of that important physical property.
From that point of view, as it was said before, perhaps the mass is a kind of parameter by means of which a measure of the inertial effects proper of any body, can be explained.
Also, and this is very important, the reciprocal gravitational interactions play a special role in the structure of the Solar System, the Galaxies, and the Universe itself. In fact, that dynamical process is responsible of the construction of a kind of three-dimensional and complex network by means of lines of gravity force, like those of the magnetic field, which link the heavenly bodies between them as seemed each other at a distance in the space. This is so because the gravity force is always attractive.
In the reciprocal gravitational interactions [2] , in order to obtain the escape velocity it is used the concept of apparent size [2] . Given that none source of gravitational attraction in Nature has size zero, and that the speed of gravitational interactions use as transport medium that velocity, it can be concluded that in Nature can’t exist instantaneous velocities. As the speed of gravitational interactions, using as transport medium the escape velocity due to the apparent size, increases its value with distance [2] , then it is concluded that necessarily there must be a maximum distance among the heavenly bodies; this fact suggests a finite Universe.
Finally, it is well known that A. Einstein was very impressed with the observed equality of gravitational and inertial mass, and that knowledge it can served him as a signpost towards the Principle of Equivalence [4] .
Cite this paper: Palacios, A. (2017) About the Mass. Open Access Library Journal, 4, 1-5. doi: 10.4236/oalib.1102835.
[1] Fierros Palacios, A. (2015). The Small Deformation Strain Tensor as a Fundamental Metric Tensor. Journal of High Energy Physics, Gravitation and Cosmology, 1, 35-47.
[2] Fierros Palacios, A. (2015). Gravitation. Open Access Library Journal.
[3] Fierros Palacios, A. (2015). The Reciprocity Principle in Gravitational Interactions. Open Access Library Journal.
[4] Weinberg, S. (1972) Gravitation and Cosmology. Principles and Applications of the General Theory of Relativity. John Wiley & Sons. Inc., New York, London, Sydney, Toronto.
[5] Resnick, R. and Holliday, D. (1966) Physics. John Wiley & Sons Inc. Hoboken.
[6] Eotvos. R.V. (1890) Math. Nat. Ber. Hungarian 65.
[7] Eotvos, R.V., Pekar, D. and Felcete, E. (1922) Ann. Phys. 68.
[8] Kenner, J. (1935) Hung. Acad. Sci., Vol. 53 Part II.
[9] Dicke, R.H., DeWitt, C. and DeWitt, B.S. (1964) Relativity Groups, and Topology, Gordon and Breach, New York, 167.
[10] Roll, P.G., Krothov, R. and Dicke, R.H. (1967) Ann Phys, New York, Vol. 26, 442.
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Create strip dipole antenna - MATLAB - MathWorks France
Create and View Dipole Antenna
Impedance of Dipole Antenna
Create strip dipole antenna
The dipole object is a strip dipole antenna on the yz- plane.
w=2d=4r
d = dipole
d = dipole(Name,Value)
d = dipole creates a half-wavelength strip dipole antenna on the Y-Z plane.
d = dipole(Name,Value) creates a dipole antenna, with additional properties specified by one or more name-value pair arguments. Name is the property name and Value is the corresponding value. You can specify several name-value pair arguments in any order as Name1, Value1, ..., NameN, ValueN. Properties you do not specify retains their default values.
Length — Dipole length
Dipole length, specified as a scalar in meters. By default, the length is chosen for an operating frequency of 75 MHz.
Dipole width should be less than 'Length'/5 and greater than 'Length'/1001. [2]
FeedOffset — Signed distance from center of dipole
Signed distance from center of dipole, specified as a scalar in meters. The feed location is on yz- plane.
Example: d.Load = lumpedElement('Impedance',75)
Create and view a dipole with 2 m length and 0.5 m width.
d = dipole('Width',0.05)
Calculate the impedance of a dipole over a frequency range of 50 MHz - 100 MHz.
d = dipole('Width',0.05);
loopCircular | monopole | slot | cylinder2strip
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Zener diode - 2D PCM Schematics - 3D Model
Zener diode (12919 views - Electronics & PCB)
A Zener diode is a particular type of diode that, unlike a normal one, allows current to flow not only from its anode to its cathode, but also in the reverse direction, when the so-called "Zener voltage" is reached. Zener diodes have a highly doped p-n junction. Normal diodes will also break down with a reverse voltage but the voltage and sharpness of the knee are not as well defined as for a Zener diode. Also normal diodes are not designed to operate in the breakdown region, but Zener diodes can reliably operate in this region. The device was named after Clarence Melvin Zener, who discovered the Zener effect. Zener reverse breakdown is due to electron quantum tunnelling caused by a high strength electric field. However, many diodes described as "Zener" diodes rely instead on avalanche breakdown. Both breakdown types are used in Zener diodes with the Zener effect predominating under 5.6 V and avalanche breakdown above. Zener diodes are widely used in electronic equipment of all kinds and are one of the basic building blocks of electronic circuits. They are used to generate low power stabilized supply rails from a higher voltage and to provide reference voltages for circuits, especially stabilized power supplies. They are also used to protect circuits from over-voltage, especially electrostatic discharge (ESD).
3D CAD Models - Zener diode
Licensed under Creative Commons Attribution 3.0 (Teravolt (talk) (Transferred by Nk/Originally uploaded by Teravolt)).
A Zener diode is a particular type of diode that, unlike a normal one, allows current to flow not only from its anode to its cathode, but also in the reverse direction, when the so-called "Zener voltage" is reached. Zener diodes have a highly doped p-n junction. Normal diodes will also break down with a reverse voltage but the voltage and sharpness of the knee are not as well defined as for a Zener diode. Also normal diodes are not designed to operate in the breakdown region, but Zener diodes can reliably operate in this region.
The device was named after Clarence Melvin Zener, who discovered the Zener effect. Zener reverse breakdown is due to electron quantum tunnelling caused by a high strength electric field. However, many diodes described as "Zener" diodes rely instead on avalanche breakdown. Both breakdown types are used in Zener diodes with the Zener effect predominating under 5.6 V and avalanche breakdown above.
Zener diodes are widely used in electronic equipment of all kinds and are one of the basic building blocks of electronic circuits. They are used to generate low power stabilized supply rails from a higher voltage and to provide reference voltages for circuits, especially stabilized power supplies. They are also used to protect circuits from over-voltage, especially electrostatic discharge (ESD).
2.1 Surface Zeners
2.2 Subsurface Zeners
3.2 Voltage shifter
A conventional solid-state diode allows significant current if it is reverse-biased above its reverse breakdown voltage. When the reverse bias breakdown voltage is exceeded, a conventional diode is subject to high current due to avalanche breakdown. Unless this current is limited by circuitry, the diode may be permanently damaged due to overheating. A Zener diode exhibits almost the same properties, except the device is specially designed so as to have a reduced breakdown voltage, the so-called Zener voltage. By contrast with the conventional device, a reverse-biased Zener diode exhibits a controlled breakdown and allows the current to keep the voltage across the Zener diode close to the Zener breakdown voltage. For example, a diode with a Zener breakdown voltage of 3.2 V exhibits a voltage drop of very nearly 3.2 V across a wide range of reverse currents. The Zener diode is therefore ideal for applications such as the generation of a reference voltage (e.g. for an amplifier stage), or as a voltage stabilizer for low-current applications.[1]
Another mechanism that produces a similar effect is the avalanche effect as in the avalanche diode.[1] The two types of diode are in fact constructed the same way and both effects are present in diodes of this type. In silicon diodes up to about 5.6 volts, the Zener effect is the predominant effect and shows a marked negative temperature coefficient. Above 5.6 volts, the avalanche effect becomes predominant and exhibits a positive temperature coefficient.[2]
In a 5.6 V diode, the two effects occur together, and their temperature coefficients nearly cancel each other out, thus the 5.6 V diode is useful in temperature-critical applications. An alternative, which is used for voltage references that need to be highly stable over long periods of time, is to use a Zener diode with a temperature coefficient (TC) of +2 mV/°C (breakdown voltage 6.2–6.3 V) connected in series with a forward-biased silicon diode (or a transistor B-E junction) manufactured on the same chip.[3] The forward-biased diode has a temperature coefficient of −2 mV/°C, causing the TCs to cancel out.
Modern manufacturing techniques have produced devices with voltages lower than 5.6 V with negligible temperature coefficients,[citation needed] but as higher-voltage devices are encountered, the temperature coefficient rises dramatically. A 75 V diode has 10 times the coefficient of a 12 V diode.
Zener and avalanche diodes, regardless of breakdown voltage, are usually marketed under the umbrella term of "Zener diode".
Under 5.6 V, where the Zener effect dominates, the IV curve near breakdown is much more rounded, which calls for more care in targeting its biasing conditions. The IV curve for Zeners above 5.6 V (being dominated by Avalanche), is much sharper at breakdown.
The Zener diode's operation depends on the heavy doping of its p-n junction. The depletion region formed in the diode is very thin (<1 µm) and the electric field is consequently very high (about 500 kV/m) even for a small reverse bias voltage of about 5 V, allowing electrons to tunnel from the valence band of the p-type material to the conduction band of the n-type material.
At the atomic scale, this tunnelling corresponds to the transport of valence band electrons into the empty conduction band states; as a result of the reduced barrier between these bands and high electric fields that are induced due to the relatively high levels of doping on both sides.[2] The breakdown voltage can be controlled quite accurately in the doping process. While tolerances within 0.07% are available, the most widely used tolerances are 5% and 10%. Breakdown voltage for commonly available Zener diodes can vary widely from 1.2 volts to 200 volts.
For diodes that are lightly doped the breakdown is dominated by the avalanche effect rather than the Zener effect. Consequently, the breakdown voltage is higher (over 5.6 V) for these devices.[4]
Surface Zeners
The emitter-base junction of a bipolar NPN transistor behaves as a Zener diode, with breakdown voltage at about 6.8 V for common bipolar processes and about 10 V for lightly doped base regions in BiCMOS processes. Older processes with poor control of doping characteristics had the variation of Zener voltage up to ±1 V, newer processes using ion implantation can achieve no more than ±0.25 V. The NPN transistor structure can be employed as a surface Zener diode, with collector and emitter connected together as its cathode and base region as anode. In this approach the base doping profile usually narrows towards the surface, creating a region with intensified electric field where the avalanche breakdown occurs. The hot carriers produced by acceleration in the intense field sometime shoot into the oxide layer above the junction and become trapped there. The accumulation of trapped charges can then cause 'Zener walkout', a corresponding change of the Zener voltage of the junction. The same effect can be achieved by radiation damage.
The emitter-base Zener diodes can handle only smaller currents as the energy is dissipated in the base depletion region which is very small. Higher amount of dissipated energy (higher current for longer time, or a short very high current spike) causes thermal damage to the junction and/or its contacts. Partial damage of the junction can shift its Zener voltage. Total destruction of the Zener junction by overheating it and causing migration of metallization across the junction ("spiking") can be used intentionally as a 'Zener zap' antifuse.[5]
Subsurface Zeners
A subsurface Zener diode, also called 'buried Zener', is a device similar to the Surface Zener, but with the avalanche region located deeper in the structure, typically several micrometers below the oxide. The hot carriers then lose energy by collisions with the semiconductor lattice before reaching the oxide layer and cannot be trapped there. The Zener walkout phenomenon therefore does not occur here, and the buried Zeners have voltage constant over their entire lifetime. Most buried Zeners have breakdown voltage of 5–7 volts. Several different junction structures are used.[6]
Zener diodes are widely used as voltage references and as shunt regulators to regulate the voltage across small circuits. When connected in parallel with a variable voltage source so that it is reverse biased, a Zener diode conducts when the voltage reaches the diode's reverse breakdown voltage. From that point on, the relatively low impedance of the diode keeps the voltage across the diode at that value.[7]
In this circuit, a typical voltage reference or regulator, an input voltage, UIN, is regulated down to a stable output voltage UOUT. The breakdown voltage of diode D is stable over a wide current range and holds UOUT relatively constant even though the input voltage may fluctuate over a fairly wide range. Because of the low impedance of the diode when operated like this, resistor R is used to limit current through the circuit.
In the case of this simple reference, the current flowing in the diode is determined using Ohm's law and the known voltage drop across the resistor R;
{\displaystyle I_{diode}={\dfrac {U_{IN}-U_{OUT}}{R_{\Omega }}}}
The value of R must satisfy two conditions :
R must be small enough that the current through D keeps D in reverse breakdown. The value of this current is given in the data sheet for D. For example, the common BZX79C5V6[8] device, a 5.6 V 0.5 W Zener diode, has a recommended reverse current of 5 mA. If insufficient current exists through D, then UOUT is unregulated and less than the nominal breakdown voltage (this differs to voltage-regulator tubes where the output voltage will be higher than nominal and could rise as high as UIN). When calculating R, allowance must be made for any current through the external load, not shown in this diagram, connected across UOUT.
R must be large enough that the current through D does not destroy the device. If the current through D is ID, its breakdown voltage VB and its maximum power dissipation PMAX correlate as such:
{\displaystyle I_{D}V_{B}<P_{\mathrm {MAX} }}
A load may be placed across the diode in this reference circuit, and as long as the Zener stays in reverse breakdown, the diode provides a stable voltage source to the load. Zener diodes in this configuration are often used as stable references for more advanced voltage regulator circuits.
Shunt regulators are simple, but the requirements that the ballast resistor be small enough to avoid excessive voltage drop during worst-case operation (low input voltage concurrent with high load current) tends to leave a lot of current flowing in the diode much of the time, making for a fairly wasteful regulator with high quiescent power dissipation, only suitable for smaller loads.
These devices are also encountered, typically in series with a base-emitter junction, in transistor stages where selective choice of a device centered around the avalanche or Zener point can be used to introduce compensating temperature co-efficient balancing of the transistor p–n junction. An example of this kind of use would be a DC error amplifier used in a regulated power supply circuit feedback loop system.
Zener diodes are also used in surge protectors to limit transient voltage spikes.
Another application of the Zener diode is the use of noise caused by its avalanche breakdown in a random number generator.
Examples of a Waveform Clipper
Two Zener diodes facing each other in series will act to clip both halves of an input signal. Waveform clippers can be used not only to reshape a signal, but also to prevent voltage spikes from affecting circuits that are connected to the power supply.[9]
Examples of a Voltage Shifter
A Zener diode can be applied to a circuit with a resistor to act as a voltage shifter. This circuit lowers the output voltage by a quantity that is equal to the Zener diode's breakdown voltage.
Examples of a Voltage Regulator
Wikimedia Commons has media related to Zener diodes.
This article uses material from the Wikipedia article "Zener diode", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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Ekman Layer Scrubbing and Shroud Heat Transfer in Centrifugal Buoyancy-Driven Convection | J. Eng. Gas Turbines Power | ASME Digital Collection
Fluid & Acoustic Engineering Laboratory, Aero-Engine Research Institute, Beihang University
e-mail: feng.gao@buaa.edu.cn
This work was conducted at the University of Surrey.
Gao, F., and Chew, J. W. (March 29, 2021). "Ekman Layer Scrubbing and Shroud Heat Transfer in Centrifugal Buoyancy-Driven Convection." ASME. J. Eng. Gas Turbines Power. July 2021; 143(7): 071010. https://doi.org/10.1115/1.4050366
This paper presents large-eddy and direct numerical simulations of buoyancy-driven convection in sealed and open rapidly rotating cavities for Rayleigh numbers in the range 107–109, and axial throughflow Reynolds numbers 2000 and 5600. Viscous heating due to the Ekman layer scrubbing effect, which has previously been found responsible for the difference in sealed cavity shroud Nusselt number predictions between a compressible N–S solver and an incompressible counterpart using the Boussinesq approximation, is discussed and scaled up to engine conditions. For the open cavity with an axial throughflow, laminar Ekman layer behavior of the mean flow statistics is confirmed up to the highest condition in this paper. The Buoyancy number Bo is found useful to indicate the influence of an axial throughflow. For the conditions studied the mean velocities are subject to Ra, while the velocity fluctuations are affected by Bo. A correlation,
Nu′=0.169(Ra′)0.318
, obtained with both the sealed and open cavity shroud heat transfer solutions, agrees with that for free gravitational convection between horizontal plates within 16% for the range of
Ra′
considered.
Cavities, Disks, Ekman dynamics, Flow (Dynamics), Heat transfer, Convection, Temperature, Buoyancy
.http://hdl.handle.net/1805/4441
Numerical Study of Buoyancy-Driven Flow in a Closed Rotating Annulus
,” GPPS, Beijing, China, Paper No.
.https://www.researchgate.net/publication/336315174_Numerical_Study_of_Buoyancy-Driven_Flow_in_a_Closed_Rotating_Annulus
Heat Transfer in H.P. Compressor Gas Turbine Internal Air Systems: Measurements From the Peripheral Shroud of a Rotating Cavity With Axial Throughflow
,” ICFMHTT, Victoria Falls, Zambia, Paper No.
.https://www.researchgate.net/publication/259659507_Heat_Transfer_in_HP_Compressor_Gas_Turbine_Internal_Air_Systems_Measurements_from_the_Peripheral_Shroud_
Computation of Buoyancy-Induced Flow in a Heated Rotating Cavity With an Axial Throughflow of Cooling Air
Prediction of Flow Features in Rotating Cavities With Axial Throughflow by RANS and LES
Prediction of 3d Unsteady Flow and Heat Transfer in Rotating Cavity by Discontinuous Galerkin Method and Transition Model
Velocity Measurements Inside a Rotating Cylindrical Cavity With a Radial Outflow of Fluid
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Critical_distance Knowpia
Critical distance is, in acoustics, the distance at which the sound pressure level of the direct sound D and the reverberant sound R are equal when dealing with a directional source. As the source is directional, the sound pressure as a function of distance between source and sampling point (listener) varies with their relative position, so that for a particular room and source the set of points where direct and reverberant sound pressure are equal constitutes a surface rather than a distinguished location in the room. In other words, it is the point in space at which the combined amplitude of all the reflected echoes are the same as the amplitude of the sound coming directly from the source (D = R). This distance, called the critical distance
{\displaystyle d_{c}}
, is dependent on the geometry and absorption of the space in which the sound waves propagate, as well as the dimensions and shape of the sound source.
— Glenn White and Gary Louie (2005)[1]
{\displaystyle d_{c}={\frac {1}{4}}{\sqrt {\frac {\gamma A}{\pi }}}\approx 0.057{\sqrt {\frac {\gamma V}{RT_{60}}}},}
{\displaystyle \gamma }
is the degree of directivity of the source (
{\displaystyle \gamma =1}
for an omnidirectional source),
{\displaystyle A}
the equivalent absorption surface,
{\displaystyle V}
the room volume in m3 and
{\displaystyle RT_{60}}
the reverberation time of room in seconds. The latter approximation is using Sabine's reverberation formula
{\displaystyle RT_{60}=V/6A}
^ White, Glenn and Louie, Gary (2005). The Audio Dictionary, 3rd edition, p.89. University of Washington Press. Cited in Hodgson, Jay (2010). Understanding Records, p.36. ISBN 978-1-4411-5607-5.
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Generalized Linear Mixed-Effects Models - MATLAB & Simulink - MathWorks Deutschland
What Are Generalized Linear Mixed-Effects Models?
GLME Model Equations
Prepare Data for Model Fitting
Choose a Distribution Type for the Model
Choose a Link Function for the Model
Display the Model
Work with the Model
Inspect and Test Coefficients and Confidence Intervals
Generate New Response Values and Refit Model
Inspect and Visualize Residuals
Generalized linear mixed-effects (GLME) models describe the relationship between a response variable and independent variables using coefficients that can vary with respect to one or more grouping variables, for data with a response variable distribution other than normal. You can think of GLME models as extensions of generalized linear models (GLM) for data that are collected and summarized in groups. Alternatively, you can think of GLME models as a generalization of linear mixed-effects models (LME) for data where the response variable is not normally distributed.
A mixed-effects model consists of fixed-effects and random-effects terms. Fixed-effects terms are usually the conventional linear regression part of the model. Random-effects terms are associated with individual experimental units drawn at random from a population, and account for variations between groups that might affect the response. The random effects have prior distributions, whereas the fixed effects do not.
The standard form of a generalized linear mixed-effects model is
\begin{array}{c}{y}_{i}|b\end{array}\sim Distr\left({\mu }_{i},\frac{{\sigma }^{2}}{{w}_{i}}\right)
g\left(\mu \right)=X\beta +Zb+\delta \text{\hspace{0.17em}},
y is an n-by-1 response vector, and yi is its ith element.
b is the random-effects vector.
Distr is a specified conditional distribution of y given b.
μ is the conditional mean of y given b, and μi is its ith element.
w is the effective observation weight vector, and wi is the weight for observation i.
For a binomial distribution, the effective observation weight is equal to the prior weight specified using the 'Weights' name-value pair argument in fitglme, multiplied by the binomial size specified using the 'BinomialSize' name-value pair argument.
For all other distributions, the effective observation weight is equal to the prior weight specified using the 'Weights' name-value pair argument in fitglme.
g(μ) is a link function that defines the relationship between the mean response μ and the linear combination of the predictors.
δ is a model offset vector.
The model for the mean response μ is
\mu ={g}^{-1}\left(\eta \right)\text{\hspace{0.17em}},
where g-1 is inverse of the link function g(μ), and
{\stackrel{^}{\eta }}_{ME}
is the linear predictor of the fixed and random effects of the generalized linear mixed-effects model
\eta =X\beta +Zb+\delta \text{\hspace{0.17em}}.
A GLME model is parameterized by β, θ, and σ2.
The assumptions for generalized linear mixed-effects models are:
The random effects vector b has the prior distribution:
b|{\sigma }^{2},\theta \sim N\left(0,{\sigma }^{2}D\left(\theta \right)\right)\text{\hspace{0.17em}},
where σ2 is the dispersion parameter, and D is a symmetric and positive semidefinite matrix parameterized by an unconstrained parameter vector θ.
The observations yi are conditionally independent given b.
To fit a GLME model to your data, use fitglme. Format your input data using the table data type. Each row of the table represents one observation, and each column represents one predictor variable. For more information on creating and using table, see Create Tables and Assign Data to Them.
Input data can include continuous and grouping variables. fitglme assumes that predictors using the following data types are categorical:
Character vector or character array
If the input data table contains any NaN values, then fitglme excludes that entire row of data from the fit. To exclude additional rows of data, you can use the 'Exclude' name-value pair argument of fitglme when fitting the model.
GLME models are used when the response data does not follow a normal distribution. Therefore, when fitting a model using fitglme, you must specify the response distribution type using the 'Distribution' name-value pair argument. Often, the type of response data suggests the appropriate distribution type for the model.
Type of Response Data
Suggested Response Distribution Type
Any positive number 'Gamma' or 'InverseGaussian'
GLME models use a link function, g, to map the relationship between the mean response and the linear combination of the predictors. By default, fitglme uses a predefined, commonly accepted link function based on the specified distribution of the response data, as shown in the following table. However, you can specify a different link function from the list of predefined functions, or define your own, using the 'Link' name-value pair argument of fitglme.
Canonical link for the normal distribution.
Canonical link for the Poisson distribution.
Canonical link for the binomial distribution.
A structure containing four fields whose values are function handles:
If 'FitMethod' is 'MPL' or 'REMPL', or if S represents a canonical link for the specified distribution, you can omit the specification of S.SecondDerivative.
When fitting a model to data, fitglme uses the canonical link function by default.
The link functions 'comploglog', 'loglog', and 'probit' are mainly useful for binomial models.
Model specification for fitglme uses Wilkinson notation, which is a character vector or string scalar of the form 'y ~ terms', where y is the response variable name, and terms is written in the following notation.
X1*X2 X1, X2, X1.*X2 (element-wise multiplication of X1 and X2)
Formulas include a constant (intercept) term by default. To exclude a constant term from the model, include –1 in the formula.
For generalized linear mixed-effects models, the formula specification is of the form 'y ~ fixed + (random1|grouping1) + ... + (randomR|groupingR)', where fixed and random contain the fixed-effects and the random-effects terms, respectively.
Suppose the input data table contains the following:
Predictor variables, X1, X2, ..., XJ, where J is the total number of predictor variables (including continuous and grouping variables).
Grouping variables, g1, g2, ..., gR, where R is the number of grouping variables.
The grouping variables in XJ and gR can be categorical, logical, character arrays, string arrays, or cell arrays of character vectors.
Then, in a formula of the form 'y ~ fixed + (random1|g1) + ... + (randomR|gR)', the term fixed corresponds to a specification of the fixed-effects design matrix X, random1 is a specification of the random-effects design matrix Z1 corresponding to grouping variable g1, and similarly randomR is a specification of the random-effects design matrix ZR corresponding to grouping variable gR. You can express the fixed and random terms using Wilkinson notation as follows.
'y ~ X1 + X2' Fixed effects for the intercept, X1, and X2. This formula is equivalent to 'y ~ 1 + X1 + X2'.
'y ~ -1 + X1 + X2' No intercept, with fixed effects for X1 and X2. The implicit intercept term is suppressed by including -1.
'y ~ 1 + (1 | g1)' A fixed effect for the intercept, plus a random effect for the intercept for each level of the grouping variable g1.
'y ~ X1 + (X1 | g1)' Random intercept and slope, with possible correlation between them. This formula is equivalent to 'y ~ 1 + X1 + (1 + X1|g1)'.
'y ~ X1 + (1 | g1) + (-1 + X1 | g1)' Independent random-effects terms for intercept and slope.
For example, the sample data mfr contains simulated data from a manufacturing company that operates 50 factories across the world. Each factory runs a batch process to create a finished product. The company wants to decrease the number of defects in each batch, so it developed a new manufacturing process. To test the effectiveness of the new process, the company selected 20 of its factories at random to participate in an experiment: Ten factories implemented the new process, while the other ten continued to run the old process. In each of the 20 factories, the company ran five batches (for a total of 100 batches), and recorded data on processing time (time_dev), temperature (temp_dev), number of defects (defects), and a categorical variable indicating the raw materials supplier (supplier) for each batch.
To determine whether the new process (represented by the predictor variable newprocess) significantly reduces the number of defects, fit a GLME model using newprocess, time_dev, temp_dev, and supplier as fixed-effects predictors. Include a random-effects intercept grouped by factory, to account for quality differences that might exist due to factory-specific variations. The response variable defects has a Poisson distribution.
defect{s}_{ij}~Poisson\left({\mu }_{ij}\right)
\begin{array}{l}\mathrm{log}\left({\mu }_{ij}\right)={\beta }_{0}+{\beta }_{1}newproces{s}_{ij}+{\beta }_{2}time_de{v}_{ij}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}\text{ }\text{ }\text{ }+{\beta }_{3}temp_de{v}_{ij}+{\beta }_{4}supplier_{C}_{ij}+{\beta }_{5}supplier_{B}_{ij}+{b}_{i}\text{\hspace{0.17em}},\end{array}
defectsij is the number of defects observed in the batch produced by factory i (where i = 1, 2, ..., 20) during batch j (where j = 1, 2, ..., 5).
μij is the mean number of defects corresponding to factory i during batch j.
supplier_Cij and supplier_Bij are dummy variables that indicate whether company C or B, respectively, supplied the process chemicals for the batch produced by factory i during batch j.
Using Wilkinson notation, specify this model as:
'defects ~ 1 + newprocess + time_dev + temp_dev + supplier + (1|factory)'
To account for the Poisson distribution of the response variable, when fitting the model using fitglme, specify the 'Distribution' name-value pair argument as 'Poisson'. By default, fitglme uses a log link function for response variables with a Poisson distribution.
The output of the fitting function fitglme provides information about generalized linear mixed-effects model.
Using the mfr manufacturing experiment data, fit a model using newprocess, time_dev, temp_dev, and supplier as fixed-effects predictors. Specify the response distribution as Poisson, the link function as log, and the fit method as Laplace.
After you create a GLME model using fitglme, you can use additional functions to work with the model.
To extract estimates of the fixed- and random-effects coefficients, covariance parameters, design matrices, and related statistics:
fixedEffects extracts estimated fixed-effects coefficients and related statistics from a fitted model. Related statistics include the standard error; the t-statistic, degrees of freedom, and p-value for a hypothesis test of whether each parameter is equal to 0; and the confidence intervals.
randomEffects extracts estimated random-effects coefficients and related statistics from a fitted GLME model. Related statistics include the estimated empirical Bayes predictor (EBP) of each random effect, the square root of the conditional mean squared error of prediction (CMSEP) given the covariance parameters and the response; the t-statistic, estimated degrees of freedom, and p-value for a hypothesis test of whether each random effect is equal to 0; and the confidence intervals.
covarianceParameters extracts estimated covariance parameters and related statistics from a fitted GLME model. Related statistics include estimate of the covariance parameter, and the confidence intervals.
designMatrix extracts the fixed- and random-effects design matrices, or a specified subset thereof, from the fitted GLME model.
To conduct customized hypothesis tests for the significance of fixed- and random-effects coefficients, and to compute custom confidence intervals:
anova performs a marginal F-test (hypothesis test) on fixed-effects terms, to determine if all coefficients representing the fixed-effects terms are equal to 0. You can use anova to test the combined significance of the coefficients of categorical predictors.
coefCI computes confidence intervals for fixed- and random-effects parameters from a fitted GLME model. By default, fitglme computes 95% confidence intervals. Use coefCI to compute the boundaries at a different confidence level.
coefTest performs custom hypothesis tests on fixed-effects or random-effects vectors of a fitted generalized linear mixed-effects model. For example, you can specify contrast matrices.
To generate new response values, including fitted, predicted, and random responses, based on the fitted GLME model:
fitted computes fitted response values using the original predictor values, and the estimated coefficient and parameter values from the fitted model.
predict computes the predicted conditional or marginal mean of the response using either the original predictor values or new predictor values, and the estimated coefficient and parameter values from the fitted model.
random generates random responses from a fitted model.
refit creates a new fitted GLME model, based on the original model and a new response vector.
To extract and visualize residuals from the fitted GLME model:
residuals extracts the raw or Pearson residuals from the fitted model. You can also specify whether to compute the conditional or marginal residuals.
plotResiduals creates plots using the raw or Pearson residuals from the fitted model, including:
A histogram of the residuals
A scatterplot of the residuals versus fitted values
A scatterplot of residuals versus lagged residuals
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Spearman's rank correlation coefficient - Simple English Wikipedia, the free encyclopedia
In mathematics and statistics, Spearman's rank correlation coefficient is a measure of correlation, named after its maker, Charles Spearman. It is written in short as the Greek letter rho (
{\displaystyle \rho }
) or sometimes as
{\displaystyle r_{s}}
. It is a number that shows how closely two sets of data are linked. It only can be used for data which can be put in order, such as highest to lowest.
The general formula for
{\displaystyle r_{s}}
{\displaystyle \rho =1-{\cfrac {6\sum d^{2}}{n(n^{2}-1)}}}
For example, if you have data for how expensive different computers are, and data for how fast the computers are, you could see if they are linked, and how closely they are linked, using
{\displaystyle r_{s}}
3 If two numbers are the same
Working it out[change | change source]
Step one[change | change source]
To work out
{\displaystyle r_{s}}
you first have to rank each piece of data. We are going to use the example from the intro of computers and their speed.
So, the computer with the lowest price would be rank 1. The one higher than that would have 2. Then, it goes up until it is all ranked. You have to do this to both sets of data.
PC Price ($)
{\displaystyle Rank_{1}}
Speed (GHz)
{\displaystyle Rank_{2}}
A 200 1 1.80 2
B 275 2 1.60 1
C 300 3 2.20 4
D 350 4 2.10 3
E 600 5 4.00 5
Step two[change | change source]
Next, we have to find the difference between the two ranks. Then, you multiply the difference by itself, which is called squaring. The difference is calle{\displaystyle d}
, and the number you get when you square
{\displaystyle d}
{\displaystyle d^{2}}
{\displaystyle Rank_{1}}
{\displaystyle Rank_{2}}
{\displaystyle d}
{\displaystyle d^{2}}
Step three[change | change source]
Count how much data we have. This data has ranks 1 to 5, so we have 5 pieces of data. This number is called
{\displaystyle n}
Step four[change | change source]
Finally, use everything we have worked out so far in this formula:
{\displaystyle r_{s}=1-{\cfrac {6\sum d^{2}}{n(n^{2}-1)}}}
{\displaystyle \sum d^{2}}
means that we take the total of all the numbers that were in the column
{\displaystyle d^{2}}
{\displaystyle \sum }
means total.
{\displaystyle \sum d^{2}}
{\displaystyle 1+1+1+1}
which is 4. The formula says multiply it by 6, which is 24.
{\displaystyle n(n^{2}-1)}
{\displaystyle 5\times (25-1)}
which is 120.
So, to find out
{\displaystyle r_{s}}
, we simply do
{\displaystyle 1-{\cfrac {24}{120}}=0.8}
Therefore, Spearman's rank correlation coefficient is 0.8 for this set of data.
What the numbers mean[change | change source]
This scatter graph has positive correlation. The
{\displaystyle r_{s}}
value would be near 1 or 0.9. The red line is a line of best fit.
{\displaystyle r_{s}}
always gives an answer between −1 and 1. The numbers between are like a scale, where −1 is a very strong link, 0 is no link, and 1 is also a very strong link. The difference between 1 and −1 is that 1 is a positive correlation, and −1 is a negative correlation. A graph of data with a
{\displaystyle r_{s}}
value of −1 would look like the graph shown except the line and points would be going from top left to bottom right.
For example, for the data that we did above,
{\displaystyle r_{s}}
was 0.8. So this means that there is a positive correlation. Because it is close to 1, it means that the link is strong between the two sets of data. So, we can say that those two sets of data are linked, and go up together. If it was −0.8, we could say it was linked and as one goes up, the other goes down.
If two numbers are the same[change | change source]
Sometimes, when ranking data, there are two or more numbers that are the same. When this happens in
{\displaystyle r_{s}}
, we take the mean or average of the ranks that are the same. These are called tied ranks. To do this, we rank the tied numbers as if they were not tied. Then, we add up all the ranks that they would have, and divide it by how many there are.[1] For example, say we were ranking how well different people did in a spelling test.
Test score Rank Rank (with tied)
{\displaystyle {\tfrac {2+3+4}{3}}=3}
{\displaystyle {\tfrac {2+3+4}{3}}=3}
{\displaystyle {\tfrac {2+3+4}{3}}=3}
{\displaystyle {\tfrac {5+6}{2}}=5.5}
{\displaystyle {\tfrac {5+6}{2}}=5.5}
These numbers are used in exactly the same way as normal ranks.
↑ Spearman's Rank at www.statistics4u.info
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Spearman%27s_rank_correlation_coefficient&oldid=8072060"
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9+2(3)
, do you get the same final answer if you add first as you do if you multiply first?
(2·4)·7
, do you get the same final answer if you start with the first two numbers as you do if you start with the last two numbers?
8−4−2
, do you get the same final answer if you start by subtracting the first two numbers as you do if you start with the last two numbers?
Take your time and make sure you subtract in two different ways!
The final answers are not the same. When you subtract
8−4
2
. When you subtract
4−2
6
. As the expression is written, the answer is
2
. In order for the correct answer to be
6
, the expression would have to be written as
8−(4−2)
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A Comparison of Batch, Column and Heap Leaching Efficiencies for the Recovery of Heavy Metals from Artificially Contaminated Simulated Soil
1Chemistry and Environmental Science Division, School of Science and the Environment, Manchester Metropolitan University, Manchester, UK
3Molecular Science Institute, School of Chemistry, University of the Witwatersrand, Johannesburg, South Africa
This paper shows the effect of three different leaching processes and 4 different leaching agents on the extraction of five metals of interest from an artificially contaminated simulated soil (SS). For the first time, it is shown that these processes and extractants could be compared directly, as the soil was a constant variable. The interest of this study is that the recovery of metals that are of importance in the circular economy, have been demonstrated from an unusual resource, soil. Metal reserves are constantly decreasing worldwide and alternative resources becoming topical. Urban mining of contaminated land and/or waste sites, therefore, becomes an attractive choice for metal extraction/recovery. This study has shown that metal extraction of up to 50% efficiency could be achieved. Furthermore, EDTA proved to be the best overall extractant when used in batch leaching processes. However, different metals showed preferential recoveries with specific processes and extractants. Therefore the results suggest that the design of a contaminant-specific leaching process performed in a sequential manner could not only leach the metals, but also achieve reasonable separation of the metals.
Metals, Leaching, Chelants, Depletion, Extraction, Contaminants, Resource
\%\text{metal leached}=\frac{\text{mass ofmetal in supernatant}}{\text{mass of metal originally loaded on substrate}}\times 100
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Mgbeahuruike, L.U., Barrett, J., Potgieter, H.J., van Dyk, L. and Potgieter-Vermaak, S.S. (2019) A Comparison of Batch, Column and Heap Leaching Efficiencies for the Recovery of Heavy Metals from Artificially Contaminated Simulated Soil. Journal of Environmental Protection, 10, 632-650. https://doi.org/10.4236/jep.2019.105038
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2. Gordon, R.B., Bertram, M. and Graedel, T.E. (2006) Metal Stocks and Sustainability. Proceedings of the National Academy of Sciences of the United States of America, 103, 1209-1214. https://doi.org/10.1073/pnas.0509498103
3. Henckens, M.L.C.M., Driessen, P.P.J. and Worrell, E. (2016) The Sustainable Extraction of Primary Zinc and Molybdenum. An Investigation of Measures to Reduce Primary Zinc and Molybdenum Use to a Sustainable Level.
4. Brierley, C.L. (2008) How Will Biomining Be Applied in Future. Transactions of Nonferrous Metals Society of China, 18, 1302-1316. https://doi.org/10.1016/S1003-6326(09)60002-9
5. Kostov, A., Dimitrijevic, M., Tasic, V. and Milosevic, N. (2008) Influence of Pyrometallurgical Copper Production on the Environment. Journal of Hazardous Materials, 164, 892-889. https://doi.org/10.1016/j.jhazmat.2008.08.099
6. Sapsford, D., Cleall, P. and Harbottle, M. (2016) In Situ Resource Recovery from Waste Repositories: Exploring the Potential for Mobilization. Journal Sustainable Metallurgy, 3, 375-392. https://doi.org/10.1007/s40831-016-0102-4
7. Johansson, N., Krook, J. and Eklund, M. (2013) An Integrated Review of Concepts and Initiatives for Mining the Technosphere: Towards a New Taxonomy. Journal of Clean Production, 55, 35-44. https://doi.org/10.1016/j.jclepro.2012.04.007
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9. Dermont, G., Bergeron, M., Mercier, G. and Richer-Lafleche, M. (2008) Soil Washing for Metal Removal: A Review of Physical/Chemical Technologies and Field Applications. Journal of Hazardous Materials, 152, 1-31. https://doi.org/10.1016/j.jhazmat.2007.10.043
10. Anikwe, M.A.N. and Nwaobodo, K.C.A. (2002) Long Term Effect of Municipal Waste Disposal on Soil Properties and Productivity of Sites of Nigeria. Bioresource Technology, 83, 241-250. https://doi.org/10.1016/S0960-8524(01)00154-7
11. Odewande, A.A. and Abimbola, A.F. (2008) Contamination Indices and Heavy Metal Concentrations in Urban Soil of Ibadan Metropolis, Southwestern Nigeria. Environmental Geochemistry and Health, 30, 243-254. https://doi.org/10.1007/s10653-007-9112-2
12. Hultman, J. and Corvellec, H. (2012) The European Waste Hierarchy: From the Sociomateriality of Waste to a Politics of Consumption. Environmental Planning, 44, 2413-2427. https://doi.org/10.1068/a44668
13. Sadhukhan, J., Ng, K.S. and Martinez-Hernandez, E. (2016) Novel Integrated Mechanical Biological Chemical Treatment (MBCT) Systems for the Production of Levulinic Acid from Fraction of Municipal Solid Waste: A Comprehensive Techno-Economic Analysis. Bioresource Technology, 215, 131-143. https://doi.org/10.1016/j.biortech.2016.04.030
14. Grathwohl, P. and Susset, B. (2009) Comparison of Percolation to Batch and Sequential Leaching Tests: Theory and Data. Waste Management Journal, 29, 2681-2688. https://doi.org/10.1016/j.wasman.2009.05.016
15. Jackson, D.R., Garrett, B.C. and Bishop, T.A. (1984) Comparison of Batch and Column Methods for Assessing Leachability of Hazardous Waste. Environmental Science and Technology, 18, 668-673. https://doi.org/10.1021/es00127a007
16. Garrabrants, A.C. and Kosson, D.S. (2005) Leaching Processes and Evaluation Tests for Inorganic Constituent Release from Cement-Based Matrices. In: Spence, R. and Shi, C., Eds., Stabilization and Solidification of Hazardous, Radioactive and Mixed Waste, CRC Press, Boca Raton, 229-280. https://doi.org/10.1201/9781420032789.ch10
17. Al-Abed, S.R., Jegadeesan, G., Purandare, J. and Allen, D. (2008) Leaching Behaviour of Mineral Processing Waste: Comparison of Batch and Column Investigations. Journal of Hazardous Materials, 153, 1088-1092. https://doi.org/10.1016/j.jhazmat.2007.09.063
18. Ham, R.K., Anderson, M.A., Stanforth, R. and Stegmann, R. (1979) Background Study on the Development of a Standard Leaching Test. United States Environmental Protection Agency, Washington DC, EPA/600/2-79/109 (NTIS PB298280).
19. van der Sloot, H.A., van Zomeren, A., Dijkstra, J.J., Hoede, D., Jacobs, J. and Scharff, H. (2003) Prediction of Long Term Leachate Quality and Chemical Speciation for a Predominantly Inorganic Waste Landfill. 9th International Waste Management and Landfill Symposium, Santa Margherita di Pula, 6-10 October 2003, 36-38.
20. Kendall, D.S. (2003) Toxicity Characteristic Leaching Procedure and Iron Treatment of Brass Foundry Waste. Environment, Science and Technology, 37, 367-371. https://doi.org/10.1021/es020621n
21. Wasay, S.A. (1992) Leaching Study of Toxic Trace Elements from Fly Ash in Batch and Column Experiment. Journal of Environmental Science and Health A, 27, 697-712. https://doi.org/10.1080/10934529209375755
22. Dijkstra, J.J., van der Sloot, H.A., Meeussen, J.C.L. and Comans, R.N.J. (2006) Local Chemical Equilibrium Makes Column Test Protocol TS14405 Suitable for Model Predictions. 6th International Conference on the Environmental and Technical Implications of Construction with Alternative Materials Science and Engineering of Recycling for Environmental Protection, Belgrade, 30 May-2 June, 2006, 345-348.
23. Potgieter-Vermaak, S.S., Mgbeahuruike, L.U., van Dyk, L., Barret, J. and Potgieter, J.H. (2017) Predictive Potential of Individual Soil Components on the Leaching Behaviour of Metal Contaminants from a Simulated Soil.
24. OECD (1984) Earthworm, Acute Toxicity Tests. OECD Guidelines for Testing of chemicals, Organisation for Economic Cooperation and Development, Paris, Test No. 207.
25. ISO (1998) Soil Quality: Effects of Pollutants on Earthworms (Eisenia fetida). Part 2: Determination of Effects on Reproduction.
26. DPR (Department of Petroleum Resources) (2002) Environmental Guidelines and Standards for the Petroleum Industry in Nigeria.
27. Roman, R.J., Benner, B.R. and Becker, G.W. (1974) Diffusion Model for Heap Leaching and Its Application to Scale-Up. Transactions of the Society of Mining Engineers, 256, 247-252.
28. Schlitt, W.J. (2006) History of Forced Aeration in Copper Sulphide Leaching. SME Annual Meeting, 27-29 March 2006, St. Louis, MO, 6-17.
29. Soil Conservation Services USDAH (1975) Soil Taxonomy, a Basic System of Soil Classification for Making and Interpreting Soil Surveys. Soil Survey Staff Report, Government Printing Office, Washington DC.
30. Chen, M., Xu, P., Zeng, G., Yang, C., Huaug, D. and Zhang, J. (2015) Bioremediation of Soil Contaminated with Polycyclic Aromatic Hydrocarbons, Petroleum, Pesticides, Chlorophenols and Heavy Metals by Composting: Applications, Microbes and Future Research Needs. Biotechnology Advancement, 33, 745-755. https://doi.org/10.1016/j.biotechadv.2015.05.003
31. McGrath, S.P., Sun, B., Zhao, F.J. and Lombi, E. (2001) Leaching of Heavy Metals from Contaminated Soils Using EDTA. Environmental Pollution, 113, 111-120. https://doi.org/10.1016/S0269-7491(00)00176-7
32. Kordosky, G.A. (1992) Copper Solvent Extraction: The State of the Art. Journal of Metals, 44, 40-45. https://doi.org/10.1007/BF03223049
33. Kovo, G.A., Folasegun, A.D. and Kayode, O.A. (2015) Mechanism on the Sorption of Heavy Metals from Binary Solution by a Low Cost Montmorillonite and Its Desorption Potential. Alexandria Engineering Journal, 54, 757-767. https://doi.org/10.1016/j.aej.2015.03.025
34. Zaleckas, E., Paulauskas, V. and Sendzikiene, E. (2013) Fractionation of Heavy Metals in Sewage Sludge and Their Removal Using Low Molecular Weight Organic Acids. Journal Environmental Engineering Landscape Management, 21, 189-198. https://doi.org/10.3846/16486897.2012.695734
35. Evangelou, M.W.H., Ebel, M. and Schaeffer, A. (2007) Chelate Assisted Phytoextracton of Heavy Metals from Soils. Effect, Mechanism, Toxicity and Fate of the Chelating. Chemosphere, 68, 989-1003. https://doi.org/10.1016/j.chemosphere.2007.01.062
36. Peters, R.W. (1999) Chelant Extraction of Heavy Metals from Contaminated Soils. Journal of Hazardous Materials, 66, 151-210. https://doi.org/10.1016/S0304-3894(99)00010-2
37. Sánchez-Chacón, A.E. and Lapidus, G.T. (1997) Model for Heap Leaching of Gold Ores by Cyanidation. Hydrometallurgy, 44, 1-20. https://doi.org/10.1016/S0304-386X(96)00052-7
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39. Steele, M.C. and Pichtel, J. (1998) Ex-Situ Remediation of a Met-al-Contaminated Superfund Soil Using Selective Extractants. Journal Environmental Engineering—American Society of Chemical Engineers, 124, 639-645. https://doi.org/10.1061/(ASCE)0733-9372(1998)124:7(639)
40. Podyachev, S.N., Sudakova, S.N., Galiev, A.K., Mustafina, A.R., Syakaev, V.V., Shagidullin, R.R., Bauer, I. and Konovalou, A.I. (2006) Synthesis of Tris and Study of Their Complexation with Some Transition Metals. Russian Chemical Bulletin, International Edition, 55, 2000-2007. https://doi.org/10.1007/s11172-006-0542-2
41. Orama, M., Hyvonen, H. and Saarinen, H.A.R. (2002) Complexation of [S,S] and Mixed Stereoisomers of N’N-ethylenediaminedisuccinic Acid (EDDS with F(III), Cu (II), Zn (II) and M (II) Ions in Aqueous Solution. Journal of the Chemical Society Dalton Transactions, 24, 4644-4648. https://doi.org/10.1039/B207777A
42. Tsang, D.W.C., Yip, T.C.M. and Lo, I.M.C. (2009) Kinetic Interactions of EDDS with Soils. 2. Metal-EDDS Complexes in Uncontaminated and Metal-Contaminated Soils. Environmental Science Technology, 43, 837-842. https://doi.org/10.1021/es8020292
43. Tandy, S., Bossart, K., Mueller, R., Ritschel, J., Hauser, L., Schulin, R. and Nowack, B. (2004) Extraction of Heavy Metals from Soils Using Biodegradable Chelating Agents. Environment Science Technology, 38, 937-944. https://doi.org/10.1021/es0348750
44. Schecher, W.D. and McAvoy, D.C. (2001) MINEQL+: A Chemical Equilibrium Modelling System, Version 4.5 for Windows. Environmental Research Software, Hallowell.
45. Koopmans, G.F., Schenkeveld, W.D.C., Song, J., Luo, Y., Japenga, J. and Temminghoff, E.J.M. (2008) Influence of EDDS on Metal Speciation in Soil Extracts: Measurement and Mechanistic Multicomponent Modelling. Environmental Science Technology, 42, 1123-1130. https://doi.org/10.1021/es071694f
46. Elliot, H.A. and Shastri, N.L. (1999) Extractive Decontamination of Metal-Polluted Soils Using Oxalate. Water, Air and Soil Pollution, 110, 335-346. https://doi.org/10.1023/A:1005067404259
47. Martell, A.E. and Smith, R.M. (2003) NIST Critically Selected Stability Constants of Metal Complexes. Version 7.0, NIST, Gaithersburg.
48. Yip, T.C.M., Yan, D.Y.S., Yui, M.M.T., Tsang, D.C.W. and Lo, I.M.C. (2010) Heavy Metal Extraction from an Artificially Contaminated Sandy Soil under EDDS Deficiency: Significance of Humic Acid and Chelant Mixtures. Chemosphere, 80, 416-421. https://doi.org/10.1016/j.chemosphere.2010.03.033
49. Acevedo, F. (2002) Present and Future of Bioleaching in Developing Countries. Electronic Journal of Biotechnology, 52, 56. https://doi.org/10.2225/vol5-issue2-fulltext-10
50. Nowack, B., Hauser, L., Tandy, S. and Schulin, R. (2005) Column Extraction of Heavy Metals from Soils Using the Biodegradable Chelating Agent EDDS. Environmental Science Technology, 39, 6819-6824. https://doi.org/10.1021/es050143r
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Vector_field Knowpia
A portion of the vector field (sin y, sin x)
Vector fields on subsets of Euclidean spaceEdit
Two representations of the same vector field: v(x, y) = −r. The arrows depict the field at discrete points, however, the field exists everywhere.
{\displaystyle (fV)(p):=f(p)V(p)}
{\displaystyle (V+W)(p):=V(p)+W(p)}
define the module of Ck-vector fields over the ring of Ck-functions where the multiplication of the functions is defined pointwise (therefore, it is commutative with the multiplicative identity being fid(p) := 1).
Coordinate transformation lawEdit
{\displaystyle V_{x}=(V_{1,x},\dots ,V_{n,x})}
and suppose that (y1,...,yn) are n functions of the xi defining a different coordinate system. Then the components of the vector V in the new coordinates are required to satisfy the transformation law
{\displaystyle V_{i,y}=\sum _{j=1}^{n}{\frac {\partial y_{i}}{\partial x_{j}}}V_{j,x}.}
Vector fields on manifoldsEdit
A vector field on a sphere
Given a differentiable manifold
{\displaystyle M}
{\displaystyle M}
is an assignment of a tangent vector to each point in
{\displaystyle M}
.[2] More precisely, a vector field
{\displaystyle F}
{\displaystyle M}
into the tangent bundle
{\displaystyle TM}
{\displaystyle p\circ F}
is the identity mapping where
{\displaystyle p}
denotes the projection from
{\displaystyle TM}
{\displaystyle M}
. In other words, a vector field is a section of the tangent bundle.
An alternative definition: A smooth vector field
{\displaystyle X}
{\displaystyle M}
{\displaystyle X:C^{\infty }(M)\to C^{\infty }(M)}
{\displaystyle X}
is a derivation:
{\displaystyle X(fg)=fX(g)+X(f)g}
{\displaystyle f,g\in C^{\infty }(M)}
If the manifold
{\displaystyle M}
is smooth or analytic—that is, the change of coordinates is smooth (analytic)—then one can make sense of the notion of smooth (analytic) vector fields. The collection of all smooth vector fields on a smooth manifold
{\displaystyle M}
{\displaystyle \Gamma (TM)}
{\displaystyle C^{\infty }(M,TM)}
(especially when thinking of vector fields as sections); the collection of all smooth vector fields is also denoted by
{\textstyle {\mathfrak {X}}(M)}
(a fraktur "X").
The flow field around an airplane is a vector field in R3, here visualized by bubbles that follow the streamlines showing a wingtip vortex.
Vector fields are commonly used to create patterns in computer graphics. Here: abstract composition of curves following a vector field generated with OpenSimplex noise.
A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind; the length (magnitude) of the arrow will be an indication of the wind speed. A "high" on the usual barometric pressure map would then act as a source (arrows pointing away), and a "low" would be a sink (arrows pointing towards), since air tends to move from high pressure areas to low pressure areas.
Velocity field of a moving fluid. In this case, a velocity vector is associated to each point in the fluid.
Streamlines, streaklines and pathlines are 3 types of lines that can be made from (time-dependent) vector fields. They are:
streaklines: the line produced by particles passing through a specific fixed point over various times
pathlines: showing the path that a given particle (of zero mass) would follow.
streamlines (or fieldlines): the path of a particle influenced by the instantaneous field (i.e., the path of a particle if the field is held fixed).
Maxwell's equations allow us to use a given set of initial and boundary conditions to deduce, for every point in Euclidean space, a magnitude and direction for the force experienced by a charged test particle at that point; the resulting vector field is the electromagnetic field.
A gravitational field generated by any massive object is also a vector field. For example, the gravitational field vectors for a spherically symmetric body would all point towards the sphere's center with the magnitude of the vectors reducing as radial distance from the body increases.
Gradient field in euclidean spacesEdit
{\displaystyle V=\nabla f=\left({\frac {\partial f}{\partial x_{1}}},{\frac {\partial f}{\partial x_{2}}},{\frac {\partial f}{\partial x_{3}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right).}
{\displaystyle \oint _{\gamma }V(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =\oint _{\gamma }\nabla f(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =f(\gamma (1))-f(\gamma (0)).}
Central field in euclidean spacesEdit
{\displaystyle V(T(p))=T(V(p))\qquad (T\in \mathrm {O} (n,\mathbb {R} ))}
where O(n, R) is the orthogonal group. We say central fields are invariant under orthogonal transformations around 0.
Operations on vector fieldsEdit
Line integralEdit
{\displaystyle \int _{\gamma }V(\mathbf {x} )\cdot \mathrm {d} \mathbf {x} =\int _{a}^{b}V(\gamma (t))\cdot {\dot {\gamma }}(t)\,\mathrm {d} t.}
{\displaystyle \operatorname {div} \mathbf {F} =\nabla \cdot \mathbf {F} ={\frac {\partial F_{1}}{\partial x}}+{\frac {\partial F_{2}}{\partial y}}+{\frac {\partial F_{3}}{\partial z}},}
Curl in three dimensionsEdit
{\displaystyle \operatorname {curl} \mathbf {F} =\nabla \times \mathbf {F} =\left({\frac {\partial F_{3}}{\partial y}}-{\frac {\partial F_{2}}{\partial z}}\right)\mathbf {e} _{1}-\left({\frac {\partial F_{3}}{\partial x}}-{\frac {\partial F_{1}}{\partial z}}\right)\mathbf {e} _{2}+\left({\frac {\partial F_{2}}{\partial x}}-{\frac {\partial F_{1}}{\partial y}}\right)\mathbf {e} _{3}.}
Index of a vector fieldEdit
Physical intuitionEdit
Magnetic field lines of an iron bar (magnetic dipole)
Flow curvesEdit
{\displaystyle \gamma '(t)=V(\gamma (t))\,.}
{\displaystyle {\begin{aligned}\gamma _{x}(0)&=x\\\gamma '_{x}(t)&=V(\gamma _{x}(t))\qquad \forall t\in (-\varepsilon ,+\varepsilon )\subset \mathbb {R} .\end{aligned}}}
Complete vector fieldsEdit
By definition, a vector field is called complete if every one of its flow curves exist for all time.[5] In particular, compactly supported vector fields on a manifold are complete. If
{\displaystyle X}
is a complete vector field on
{\displaystyle M}
, then the one-parameter group of diffeomorphisms generated by the flow along
{\displaystyle X}
exists for all time. On a compact manifold without boundary, every smooth vector field is complete. An example of an incomplete vector field
{\displaystyle V}
{\displaystyle \mathbb {R} }
{\displaystyle V(x)=x^{2}}
. For, the differential equation
{\textstyle x'(t)=x^{2}}
, with initial condition
{\displaystyle x(0)=x_{0}}
, has as its unique solution
{\textstyle x(t)={\frac {x_{0}}{1-tx_{0}}}}
{\displaystyle x_{0}\neq 0}
{\displaystyle x(t)=0}
{\displaystyle t\in \mathbb {R} }
{\displaystyle x_{0}=0}
). Hence for
{\displaystyle x_{0}\neq 0}
{\displaystyle x(t)}
{\textstyle t={\frac {1}{x_{0}}}}
so cannot be defined for all values of
{\displaystyle t}
f-relatednessEdit
Gradient flow and balanced flow in atmospheric dynamics
^ a b Galbis, Antonio & Maestre, Manuel (2012). Vector Analysis Versus Vector Calculus. Springer. p. 12. ISBN 978-1-4614-2199-3. {{cite book}}: CS1 maint: uses authors parameter (link)
^ Tu, Loring W. (2010). "Vector fields". An Introduction to Manifolds. Springer. p. 149. ISBN 978-1-4419-7399-3.
^ Lerman, Eugene (August 19, 2011). "An Introduction to Differential Geometry" (PDF). Definition 3.23.
^ Dawber, P.G. (1987). Vectors and Vector Operators. CRC Press. p. 29. ISBN 978-0-85274-585-4.
^ Sharpe, R. (1997). Differential geometry. Springer-Verlag. ISBN 0-387-94732-9.
Hubbard, J. H.; Hubbard, B. B. (1999). Vector calculus, linear algebra, and differential forms. A unified approach. Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-657446-7.
Warner, Frank (1983) [1971]. Foundations of differentiable manifolds and Lie groups. New York-Berlin: Springer-Verlag. ISBN 0-387-90894-3.
Boothby, William (1986). An introduction to differentiable manifolds and Riemannian geometry. Pure and Applied Mathematics, volume 120 (second ed.). Orlando, FL: Academic Press. ISBN 0-12-116053-X.
Media related to Vector fields at Wikimedia Commons
Online Vector Field Editor
"Vector field", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Vector field — Mathworld
Vector field — PlanetMath
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Please solve this sums plz d _ How many days would 6 HOW many would Mohanrao took 'O - Maths - Direct Proportion and Inverse Proportion - 13227335 | Meritnation.com
Please solve this sums plz
Please solve this sums plz d _ How many days would 6 ? HOW many would Mohanrao took 'O days to finish a book. reading 40 pages every day. How many must hc read in a day to finish it at a Of 4 she takc 4. The of grain in warehouse people. How many days 6000 people Let's learn. Partnership en starting a business enterprise, money is required for an offce, rav amount is called the capital. Often, two or more people put in 'her words, these people business by investing in the rrtnership, all partrMs joint account in a bank. is shared by proportion to the money A tharva d 2100 and 2800 rupees orotit of 3500 rupees. How should G,veslments.
Q1. If 10 students can weed a field in 6 days,
then 1 student can weed a field in
10×6=60days
If 1 worker takes 60 days to weed a field then 6 workers would take=
\frac{60}{6}
=10 days
If there are 15 workers and a single worker can plough a field in 60 days, then 15 workers will take
\frac{60}{15}= 4 days
If Mohan Rao finishes a book in 10 days by reading 40 pages a day then the book contains
10×40=400 pages
If he wishes to finish a book in 8 days then he needs to read
\frac{400}{8}= 50 pages
every day.
Kindly ask different questions in separate threads.
Hope this information has solved your queries.
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Abstract: Graphic design theory research examines how designers can read about and read into designs in order to stimulate growth and change in their own work. It inspires new lines of thoughts and questioning, and opens up new theoretical directions. This study sought to establish the significance and application of graphic design theories in product packaging technology, as packages have minimal time to achieve the goals for which they were created. It therefore generally examined graphic design theories used in day-to-day activities of graphic designers. Adopting a descriptive approach, a sample size of 450 respondents was taken in the Federal University of Technology Akure, with valid responses of 450. Following one hypothesis testing, the study showed a significant relationship between usage and application of graphic design theories and the creation of attractive graphic package designs. The study revealed that creative graphic designers often apply graphic design theories in creating package designs, and that package designs are more appealing or attractive with the application of graphic design theories. It established that graphic design theories guide effective consumer product package designers, both in growth and in practice.
Keywords: Graphic Design, Theory, Packaging, Packaging Technology, Brand
X=\sum fx/N
{X}_{1}^{2}=\underset{n=1}{\sum }\frac{{\left({f}_{o}-{f}_{e}\right)}^{2}}{{f}_{e}}
Cite this paper: Oladumiye, E. (2018) Graphic Design Theory Research and Application in Packaging Technology. Art and Design Review, 6, 29-42. doi: 10.4236/adr.2018.61003.
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Here are the ages of Jahna and her cousins:
11,6,8,4,3,13,
4
Find the mean, median, and mode of their ages.
For help with mean, median, and mode, refer to the Math Notes box in Lesson 1.3.3.
Be sure to put the ages in numerical order (from least to greatest). It will definitely help you when looking at this data!
In this set of data, the mean is
7
, the median is
6
, and the mode is
13
Create two different histograms to represent this data. One of them should have data in two columns, and one should have data in three columns.
For help with histograms, refer to the Math Notes box in Lesson 1.1.1.
For both histograms, it will be best to label the horizontal axes ''Age'' and the vertical axes ''Frequency.''
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Oliynyk, Kateryna1; Tamagnini, Claudio2
1 Department of Civil and Environmental Engineering, University of Perugia, ItalyUniversity of Perugia Dept. of Civil and Env. Engng. via G. Duranti, 93 – 06125 Perugia Italy
2 Department of Civil and Environmental Engineering University of Perugia Italy
Keywords: Hard soils, Soft rocks, Hyperplasticity, Breakage mechanics, Finite deformations, Multiplicative plasticity, Stress–point algorithm, Consistent linearization
Oliynyk, Kateryna 1; Tamagnini, Claudio 2
author = {Oliynyk, Kateryna and Tamagnini, Claudio},
title = {Finite deformation hyperplasticity theory for crushable, cemented granular materials},
TI - Finite deformation hyperplasticity theory for crushable, cemented granular materials
%T Finite deformation hyperplasticity theory for crushable, cemented granular materials
Oliynyk, Kateryna; Tamagnini, Claudio. Finite deformation hyperplasticity theory for crushable, cemented granular materials. Open Geomechanics, Volume 2 (2020), article no. 4, 33 p. doi : 10.5802/ogeo.8. https://opengeomechanics.centre-mersenne.org/articles/10.5802/ogeo.8/
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Lipid index changes in the blood serum of patients with hyperplastic and early neoplastic lesions in the ovaries | Journal of Ovarian Research | Full Text
Lipid index changes in the blood serum of patients with hyperplastic and early neoplastic lesions in the ovaries
Mikołaj Karmowski1,
Krzysztof A Sobiech2,
Jacek Majda3,
Piotr Rubisz1,
Stanisław Han4 &
Andrzej Karmowski1
The authors used the lipid index (WL) to monitor lipid changes before and after surgery. The surgical operation performed was the simultaneous enucleation of a cystic tumor of the hilum ovarii in its entirety (with diagnosis of a simple cyst or teratoma adultum) in groups of 20 patients.
To compare the lipid index WL in the blood serum of patients undergoing surgery treatment at the following times: before and 7 days after surgery, and 6 and 12 months after surgery.
The research material was the blood serum of women aged about 24 years. The authors divided the patients into 3 groups: two groups of 20 women and a control group. The concentrations of the lipid parameters were measured and the lipid index WL was calculated.
Statistically significant differences were found between the lipid index of serum from patients with diagnosed ovarian neoplasms and the index of serum from healthy subjects; differences were demonstrated in the postoperative period, particularly 6 and 12 months after surgery.
The lipid index WL proved useful in diagnosing ovarian neoplasm (simple cysts and teratoma adultum) and in monitoring the postoperative period.
Since 2002, the lipid index has been successfully used in oncogynecological diagnostics. The index is calculated using the concentrations of HDL and LDL lipoprotein, apolipoprotein A1 and B, and triglycerides (TG) [1],[2].
The usefulness of this marker has been demonstrated in monitoring changes in lipid metabolism in perimenopausal women undergoing gynecological surgery, as well as in the assessment of hormone replacement therapy [3]–[5].
Based on these results, it was decided that the objective of the present study would be to determine the diagnostic role of this indicator in the blood serum of women undergoing surgery due to neoplastic hyperplasia within the ovary and parovarian mesonephritic structures. These procedures were performed due to a diagnosis of teratoma: adultum teratoma [6]–[8] or a simple cyst [9]–[11].
The motivation to undertake this research lies in the fact that the authors have found no comprehensive publication in the available literature describing the lipid index in gynecological oncology.
The study included three groups of women, each of up to 20 people.
Group I contained healthy women with an average age of 24.3 years. The criteria for inclusion in Group I were as follows:
no pathological findings in gynecological examination;
normal values of urine analyses, blood counts, and diagnostic enzymology;
BMI between 19.5 and 26 kg/m2.
The exclusion criterion was a history of liver, kidney, or bile-duct disease.
Patients in Group II were operated on because of neoplastic lesions within the ovarian and parovarian structures (appendages). Histopathological examination showed benign hyperplasia of the simple cyst type was found. The following are the criteria for inclusion into the group:
tumor size between 5 and 22 cm, as determined by means of gynecological examination and confirmed by ultrasound examination;
The patients in Group III were operated on because of neoplastic lesions within the appendages (ovary, paroophoron, epoophoron, or other mesonephritic structures). Postoperative histopathological examination revealed a mature teratoma. The indication for surgery in both Groups II and III was tumor size of 5–20 cm in diameter, determined by means of gynecological examination and confirmed by ultrasound examination.
The range of performed operations included enucleation of the cyst in one piece, preserving oncological asepsis and followed by intraoperative pathology consultation.
Biochemical measurements were performed prior to surgery (A), 7 days after surgery (B), and 6 and 12 months after surgery (C and D, respectively).
The research material was blood taken from the basilic vein. The samples obtained from blood serum were analyzed by assessing the concentration of:
apolipoprotein A and B using immunoturbidimetric method with the Orion Diagnostica reagent on a Technicon RA 1000 (USA) analyzer;
triglycerides (TG) using the enzymatic colorimetric method;
lipoprotein HDL cholesterol by precipitation; and
total cholesterol (TCH) using the enzymatic oxidase method.
The lipoprotein LDL cholesterol level was calculated using the Friedewald formula.
Given the lipid metabolism parameters, the lipid index WL was calculated using the formula [1],[2]:
\mathit{WL}=\frac{\left(\mathit{HDL}+\frac{\mathit{TG}}{6}\right)*\frac{\mathit{ApoA}1}{30}*10}{\left(\mathit{LDL}+\frac{\mathit{TG}}{5}\right)*\frac{\mathit{ApoB}}{20}}
The obtained results are presented as arithmetic means and standard deviations, which were statistically analyzed using Student’s t-test and adjusted with the Bonferroni correction, whose significance level for the studied index was 0.05/5 = 0.01.
The study was performed and financed under grant No. 500/03 from Wrocław Medical University, Poland.
Table 1 presents the data on the lipid index WL in the blood serum from the test groups. The results of the statistical analysis are shown in Table 2. It is clear from the data that there was a statistically significant decrease in this parameter in both Groups II and III, compared to the control Group I (p < 0.001). After surgery, Groups II and III showed the highest statistically significant differences between times A and D (p < 0.001). Other results for these two groups between the different time points were highly significant (p < 0.01 and p < 0.001), except for time points A and B, that is, before and 7 days after surgery.
Table 1 Lipid index WL in blood serum in the study groups
The relationship of the percentage of this parameter in both groups compared with the control group is presented in Figure 1. Before the surgery, both groups had lower lipid indices WL, at approximately 31% for Group II and 60% for Group III. After the procedure, a constant upward trend was observed in this parameter, and the data at time D are about 52% for Group II and about 85% for Group III.
Lipid index (%) in each group of patients compared with the control group.
Previous studies using the lipid index WL have shown its usefulness in monitoring hormone replacement therapy in postmenopausal women.
In the search for useful diagnostic indicators in gynecological oncology, a slight deviation from the norm of lipid parameters was found. However, the application of the lipid index WL, which is calculated from HDL, LDL, TG, ApoA1, and ApoB, showed a decrease in this parameter of up to 30% in the serum of patients, as compared with healthy subjects. Further studies have shown differences that depend on the type of gynecological disease.
This information encouraged the authors to conduct further diagnostic tests and monitoring over the 12 months after the operations. Results that are particularly interesting include those for times C and D, where there was an increase in the lipid index WL, slowly reaching the values characteristic of healthy women of similar age and BMI. After one year, indices of growth of about 25% for teratoma adultum and about 20% for simple cysts was found, which correlates with the clinical evaluation of the patients.
It appears that further clinical trials on this indicator which also take into account data regarding physical activity, diet, and risk factors for diseases of civilization will allow for the development of a healthy control system based on the observation of lipid metabolism in women.
MK, KAS, JM, and PR are responsible for writing the manuscript, the conception, and the final design. MK, KAS, and AK collected and assembled the data and wrote the manuscript. PR and SH performed the statistical analysis. MK, KS and SH designed the study and critically revised the study for intellectual content. All authors read and approved the final manuscript.
Wochyński Z, Sobiech KA, Majda J: The assessment of adaptation to physical exercise in soldiers with the use of the lipid index as the qualifying factor for physical fitness tests. Lek Wojsk 2001, 77: 218–221. [in Polish]
Wochyński Z, Sobiech KA, Majda J: Lipid index in the evaluation of physical capacity. Nowiny Lek 2005, 74: 39–44. [in Polish]
Andrzej K, Sobiech KA, Małgorzata P, Halina M, Mikołaj K, Marta G-B, Dorota J: The activity of Arylsulfatase A in patients with benign ovarian tumors. Internet J. Gynecol. Obstet 2007, 6: 2.
Karmowski A, Sobiech KA, Majda J, Karmowski M, Markuszewski M, Kwietniak G, Kotarski A, Kotarska E, Wronecki K: The value of lipid index in the blood serum of perimenopausal women undergoing gynecological surgery. Gin Pol 2004, 75: 847–851. [in Polish]
Karmowski A, Sobiech KA, Markuszewski M, Majda J, Łątkowski K, Karmowski M, Balcerek M, Kotarska E: Values of the lipid indexes in monitoring of the hormone replacement therapy in the postmenopausal women. Adv Clin Ex Med 2005, 14: 725–729. [in Polish]
Bal J, Gabryś MS, Jałocha L: Selected cell molecular pathways in the pathogenesis of ovarian teratomas. Post Hig Med Dośw 2009, 63: 242–249. [in Polish]
Dahl N, Gustavson KH, Rune C, Gustavsson I, Pettersson U: Benign ovarian teratomas: an analysis of their cellular origin. Cancer Genet Cytogenet 1990, 46: 115–23. 10.1016/0165-4608(90)90017-5
Gilks CB: Subclassification of ovarian surface epithelial tumors based on correlation of histologic and molecular pathologic data. Int J Gynecol Pathol 2004, 23: 200–205. 10.1097/01.pgp.0000130446.84670.93
Acs G: Serous and mucinous borderline (low malignant potential) tumors of the ovary. M J Clin Pathol 2005, 123: S13–57.
Funt SA, Hricak H: Ovarian malignancies. Top Magn Reson Imaging 2003, 14: 329–337. 10.1097/00002142-200308000-00005
Hart WR: Mucinous tumors of the ovary: a review. Int J Gynecol Pathol 2005, 24: 4–25.
First Department of Gynecology and Obstetrics, Wrocław Medical University, Wrocław, Poland
Mikołaj Karmowski, Piotr Rubisz & Andrzej Karmowski
Department of Human Biology, University School of Physical Education in Wrocław, Wrocław, Poland
Krzysztof A Sobiech
Department of Laboratory Diagnostics, Fourth Military Hospital, Wrocław, Poland
Hasco-Lek Pharmaceutical Production Company S.A, Wrocław, Poland
Mikołaj Karmowski
Correspondence to Piotr Rubisz.
Karmowski, M., Sobiech, K.A., Majda, J. et al. Lipid index changes in the blood serum of patients with hyperplastic and early neoplastic lesions in the ovaries. J Ovarian Res 7, 90 (2014). https://doi.org/10.1186/s13048-014-0090-6
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King-King Li
We experimentally investigate the memory recall bias of overconfident (underconfident) individuals after receiving feedback on their overconfidence (underconfidence). Our study differs from the literature by identifying the recall pattern conditional on subjects’ overconfidence/underconfidence. We obtain the following results. First, overconfident (underconfident) subjects exhibit [...] Read more.
We experimentally investigate the memory recall bias of overconfident (underconfident) individuals after receiving feedback on their overconfidence (underconfidence). Our study differs from the literature by identifying the recall pattern conditional on subjects’ overconfidence/underconfidence. We obtain the following results. First, overconfident (underconfident) subjects exhibit overconfident (underconfident) recall despite receiving feedback on their overconfidence (underconfidence). Second, awareness of one’s overconfidence or underconfidence does not eliminate memory recall bias. Third, the primacy effect is stronger than the recency effect. Overall, our results suggest that memory recall bias is mainly due to motivated beliefs of sophisticated decision makers rather than naïve decision-making. Full article
(This article belongs to the Special Issue Economics of Motivated Beliefs)
We investigate the effect of the environment dimensionality and different dispersal strategies on the evolution of cooperation in a finite structured population of mobile individuals. We consider a population consisting of cooperators and free-riders residing on a two-dimensional lattice with periodic boundaries. Individuals [...] Read more.
We investigate the effect of the environment dimensionality and different dispersal strategies on the evolution of cooperation in a finite structured population of mobile individuals. We consider a population consisting of cooperators and free-riders residing on a two-dimensional lattice with periodic boundaries. Individuals explore the environment according to one of the four dispersal strategies and interact with each other via a public goods game. The population evolves according to a birth–death–birth process with the fitness of the individuals deriving from the game-induced payouts. We found that the outcomes of the strategic dispersal strategies in the two-dimensional setting are identical to the outcomes in the one-dimensional setting. The random dispersal strategy, not surprisingly, resulted in the worst outcome for cooperators. Full article
Games 2022, 13(3), 36; https://doi.org/10.3390/g13030036 - 29 Apr 2022
Games 2022, 13(2), 28; https://doi.org/10.3390/g13020028 - 31 Mar 2022
r\ge 2
r<2
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Reflect incoming signal - MATLAB - MathWorks Deutschland
Compute Reflected Signals from Two Non-Fluctuating Radar Targets
System object: phased.RadarTarget
Reflect incoming signal
Y = step(H,X,MEANRCS)
Y = step(H,X,UPDATERCS)
Y = step(H,X,MEANRCS,UPDATERCS)
Y = step(H,X,ANGLE_IN,LAXES)
Y = step(H,X,ANGLE_IN,ANGLE_OUT,LAXES)
Y = step(H,X,ANGLE_IN,LAXES,SMAT)
Y = step(H,X,ANGLE_IN,LAXES,UPDATESMAT)
Y = step(H,X,ANGLE_IN,ANGLE_OUT,LAXES,SMAT,UPDATESMAT)
Y = step(H,X) returns the reflected signal Y due to the incident signal X. The argument X is a complex-valued N-by-1 column vector or N-by-M matrix. The value M is the number of signals. Each signal corresponds to a different target. The value N is the number of samples in each signal. Use this syntax when you set the Model property of H to 'Nonfluctuating'. In this case, the value of the MeanRCS property is used as the Radar cross-section (RCS) value. This syntax applies only when the EnablePolarization property is set to false. If you specify M incident signals, you can specify the radar cross-section as a scalar or as a 1-by-M vector. For a scalar, the same value will be applied to all signals.
Y = step(H,X,MEANRCS) uses MEANRCS as the mean RCS value. This syntax is available when you set the MeanRCSSource property to 'Input port' and set Model to 'Nonfluctuating'. The value of MEANRCS must be a nonnegative scalar or 1-by-M row vector for multiple targets. This syntax applies only when the EnablePolarization property is set to false.
Y = step(H,X,UPDATERCS) uses UPDATERCS as the indicator of whether to update the RCS value. This syntax is available when you set the Model property to 'Swerling1', 'Swerling2', 'Swerling3', or 'Swerling4'. If UPDATERCS is true, a new RCS value is generated. If UPDATERCS is false, the previous RCS value is used. This syntax applies only when the EnablePolarization property is set to false. In this case, the value of the MeanRCS property is used as the radar cross-section (RCS) value.
Y = step(H,X,MEANRCS,UPDATERCS) lets you can combine optional input arguments when their enabling properties are set. In this syntax, MeanRCSSource is set to 'Input port' and Model is set to one of the Swerling models. This syntax applies only when the EnablePolarization property is set to false. For this syntax, changes in MEANRCS will be ignored after the first call to the step method.
Y = step(H,X,ANGLE_IN,LAXES) returns the reflected signal Y from an incident signal X. This syntax applies only when the EnablePolarization property is set to true. The input argument, ANGLE_IN, specifies the direction of the incident signal with respect to the target’s local coordinate system. The input argument, LAXES, specifies the direction of the local coordinate axes with respect to the global coordinate system. This syntax requires that you set the Model property to 'Nonfluctuating' and the Mode property to 'Monostatic'. In this case, the value of the ScatteringMatrix property is used as the scattering matrix value.
X is a 1-by-M row array of MATLAB® struct type, each member of the array representing a different signal. The struct contains three fields, X.X, X.Y, and X.Z. Each field corresponds to the x, y, and z components of the polarized input signal. Polarization components are measured with respect to the global coordinate system. Each field is a column vector representing a sequence of values for each incoming signal. The X.X, X.Y, and Y.Z fields must all have the same dimension. The argument, ANGLE_IN, is a 2-by-M matrix representing the signals’ incoming directions with respect to the target’s local coordinate system. Each column of ANGLE_IN specifies the incident direction of the corresponding signal in the form [AzimuthAngle; ElevationAngle]. Angle units are in degrees. The number of columns in ANGLE_IN must equal the number of signals in the X array. The argument, LAXES, is a 3-by-3 matrix. The columns are unit vectors specifying the local coordinate system's orthonormal x, y, and z axes, respectively, with respect to the global coordinate system. Each column is written in [x;y;z] form.
Y is a row array of struct type having the same size as X. Each struct contains the three reflected polarized fields, Y.X, Y.Y, and Y.Z. Each field corresponds to the x, y, and z component of the signal. Polarization components are measured with respect to the global coordinate system. Each field is a column vector representing one reflected signal.
Y = step(H,X,ANGLE_IN,ANGLE_OUT,LAXES), in addition, specifies the reflection angle, ANGLE_OUT, of the reflected signal when you set the Mode property to 'Bistatic'. This syntax applies only when the EnablePolarization property is set to true. ANGLE_OUT is a 2-row matrix representing the reflected direction of each signal. Each column of ANGLE_OUT specifies the reflected direction of the signal in the form [AzimuthAngle; ElevationAngle]. Angle units are in degrees. The number of columns in ANGLE_OUT must equal the number of members in the X array. The number of columns in ANGLE_OUT must equal the number of elements in the X array.
Y = step(H,X,ANGLE_IN,LAXES,SMAT) specifies SMAT as the scattering matrix. This syntax applies only when the EnablePolarization property is set to true. The input argument SMAT is a 2-by-2 matrix. You must set the ScatteringMatrixSource property 'Input port' to use SMAT.
Y = step(H,X,ANGLE_IN,LAXES,UPDATESMAT) specifies UPDATESMAT to indicate whether to update the scattering matrix when you set the Model property to 'Swerling1', 'Swerling2'', 'Swerling3', or 'Swerling4'. This syntax applies only when the EnablePolarization property is set to true. If UPDATESMAT is set to true, a scattering matrix value is generated. If UPDATESMAT is false, the previous scattering matrix value is used.
Y = step(H,X,ANGLE_IN,ANGLE_OUT,LAXES,SMAT,UPDATESMAT). You can combine optional input arguments when their enabling properties are set. Optional inputs must be listed in the same order as the order of their enabling properties.
Create two sinusoidal signals and compute the value of the reflected signals from targets having radar cross sections of
5{m}^{2}
10{m}^{2}
, respectively. Set the radar cross sections in the step method by choosing Input port for the value of the MeanRCSSource property. Set the radar operating frequency to 600 MHz.
sRadarTarget = phased.RadarTarget('Model','Nonfluctuating',...
'MeanRCSSource','Input port',...
x = [cos(2*pi*250*t)',10*sin(2*pi*250*t)'];
y = step(sRadarTarget,x,[5,10]);
disp(y(1:3,1:2))
-0.0249 224.3546
Y=\sqrt{G}\cdot X,
G=\frac{4\pi \sigma }{{\lambda }^{2}}.
\left[\begin{array}{c}{E}_{H}^{\left(scat\right)}\\ {E}_{V}^{\left(scat\right)}\end{array}\right]=\sqrt{\frac{4\pi }{{\lambda }^{2}}}\left[\begin{array}{cc}{S}_{HH}& {S}_{VH}\\ {S}_{HV}& {S}_{VV}\end{array}\right]\left[\begin{array}{c}{E}_{H}^{\left(inc\right)}\\ {E}_{V}^{\left(inc\right)}\end{array}\right]=\sqrt{\frac{4\pi }{{\lambda }^{2}}}\left[S\right]\left[\begin{array}{c}{E}_{H}^{\left(inc\right)}\\ {E}_{V}^{\left(inc\right)}\end{array}\right]
For further details, see Mott [1] or Richards[2].
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ERRATA—Volume LXII
253 ii 36 Wishart, Sir James: before On the accession insert He was M.P. for Portsmouth 1711-15.
18-17 f.e. for and there he died in 1729. read He died on 31 May 1723 (Boyer, Political State of Great Britain, May 1723, p. 571)
13 f.e. Wishart, Sir John: after John insert Lord Pittarrow
257 i 38 Witchell, Edwin: for Nymphsfield read Nympsfield
258 i 9 f.e. Witham, George: for Burton Constable read Constable Burton
259 i 32 Wither, George: for Anne Serle read Mary Hunt, apparently, of Theddon, Hampshire (cf. The Poetry of George Wither, ed. F. Sidgwick, 1902, i. xvi. sq.)
259 ii 4 f.e. for four editions read at least five editions
260 ii 15-17 for in 1617 for private circulation read in 1615 for private circulation. A copy of the private issue is in the Bodleian Library.
265 ii 16 for Hambledon, Surrey read Hambledon, Hampshire
266 ii 22 f.e. after Fidding insert (Theddon)
277 ii 24 Wodelarke, Robert: for 'Acero read 'Cicero
25-26 for 'Etymologiæ' read 'Etymologiarum'
294 ii 35 Wolfe, Arthur, 1st Viscount Kilwarden: for Six years read Five years
295 i 9 for the chief justice read formerly the chief justice
301 i 11 Wolfe, James: for right bank read left bank
302 ii 10 for commanding read cannonading
305 i 31 Wolfe, Reyner: for History of read History or
317 i 25 Wolley, Sir John: for 1572 read 1571.
27 after Elizabeth. insert According to Browne Willis, he was elected M.P. for East Looe in 1571
324 i 20 f.e. Wolseley, William (1640?-1697): for C. D. read C. D-n.
327 ii 38 Wolsey, Thomas: for 16 Nov. read 15 Nov.
329 i 11-12 omit and though he was never consecrated
346 i 44-45 Womock, Laurence: for with the promise of a prebend in Ely Cathedral read On 22 Sept. in the same year, according to Le Neve, he had been installed in the sixth prebendal stall at Ely.
50-52 omit On 22 Sept at Ely
347 i 44 after Bibl. Brit.; insert Cat. Tanner MSS. (Bodleian);
l.l. Wood, Alexander (1725-1807): for His father was the youngest son of Wood read He was the son of Thomas Wood and grandson of Jasper Wood
ii 40 after Edinburgh insert He married Veronica Chalmers
353 i 33-34 Wood, Sir Charles, 1st Viscount Halifax: for lost his seat at Grimsby in 1831, but read in 1831
354 ii 28 Wood, Sir David E.: after 1877. insert He was an unsuccessful candidate for Durham in the conservative interest in 1847.
359 ii 25-26 Wood, James (1672-1759): omit Another son, Robert . . . . Everett Wood [q. v.].
374 i 2 f.e. Wood, Robert (1717?-1771): after i. 289). insert From 1764 to his death he was groom-porter in the royal household.
380 i 20 f.e. Wood, William (1745-1808): for 1807 read 1809
381 ii 13-12 f.e. Wood, William P., Baron Hatherley: for Robert Monsey Rolfe, first baron Cranworth [q. v.]. read Sir George James Turner [q. v.], who was made lord justice in succession to the newly appointed lord chancellor, Robert Monsey Rolfe, first baron Cranworth [q. v.].
384 ii 23 Woodard, Nathaniel: for 1844 read 1894
{\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}}
Wooddeson, Richard: for 1823 read 1822
395 i 36-37 Woodford, Sir John G.: for Sir William Colville read Sir Charles Colville
ii 14 f.e. after the service insert in Oct. 1841. He had been made C.B. in 1815 and K.C.B. in 1838
404 ii 26 Woodley, George: for Maclure's read McClure's
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The Study on the Cycloids of Moving Loops ()
Department of Applied Mathematics, Volgograd State Technical University, Volgograd, Russia.
Tarabrin, G. (2018) The Study on the Cycloids of Moving Loops. Journal of Applied Mathematics and Physics, 6, 817-830. doi: 10.4236/jamp.2018.64070.
y=p\left(x\right)
y=q\left(x\right)
p\left(x\right)>0,q\left(x\right)<0
0<x<H
p\left(0\right)=p\left(H\right)=q\left(0\right)=q\left(H\right)=0
O\left(0,0\right)
{A}_{1}
x=H,y=L
V={V}_{x}i+{V}_{y}j
y=q\left(x\right)
{V}_{x},\text{\hspace{0.17em}}{V}_{y}
V={V}_{x}i+{V}_{y}j
\begin{array}{l}{V}_{x}=v\mathrm{cos}\left(\alpha \right),\\ {V}_{y}=v\left[1+\mathrm{sin}\left(\alpha \right)\right]\end{array}\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⇒\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left\{\begin{array}{l}{V}_{x}=v{\left[1+{\mathrm{tan}}^{2}\left(\alpha \right)\right]}^{-1/2},\\ {V}_{y}=v\left\{1+{\left[1+{\mathrm{tan}}^{2}\left(\alpha \right)\right]}^{-1/2}\mathrm{tan}\left(\alpha \right)\right\}.\end{array}
y=q\left(x\right)
\mathrm{tan}\alpha ={q}^{\prime }\left(x\right)
{V}_{x}=v{\left(1+{\left[{q}^{\prime }\left(x\right)\right]}^{2}\right)}^{-1/2},\text{ }{V}_{y}=v\left\{1+{q}^{\prime }\left(x\right){\left(1+{\left[{q}^{\prime }\left(x\right)\right]}^{2}\right)}^{-1/2}\right\}
y=c\left(x\right)
y=c\left(x\right)
{c}^{\prime }\left(x\right)=\mathrm{tan}\left(\beta \right)
V={V}_{x}i+{V}_{y}j
{V}_{x}/{V}_{y}=\mathrm{tan}\left(\beta \right)={c}^{\prime }\left(x\right)
{V}_{x},\text{\hspace{0.17em}}{V}_{y}
{c}^{\prime }\left(x\right)={q}^{\prime }\left(x\right)+\sqrt{1+{\left[{q}^{\prime }\left(x\right)\right]}^{2}}
c\left(0\right)=0
y=c\left(x\right)\equiv q\left(x\right)+\underset{0}{\overset{x}{\int }}\sqrt{1+{\left[{q}^{\prime }\left(\xi \right)\right]}^{2}}\text{d}\xi ,\text{ }0\le x\le H
l\left(x\right)=\underset{0}{\overset{x}{\int }}\sqrt{1+{\left[{q}^{\prime }\left(\xi \right)\right]}^{2}}\text{d}\xi
y=q\left(x\right)+l\left(x\right)
p\left(x\right)=-q\left(x\right)
L=\underset{0}{\overset{H}{\int }}\sqrt{1+{\left[{q}^{\prime }\left(\xi \right)\right]}^{2}}\text{d}\xi
y\in \left[0,L\right]
y\in \left[L,2L\right]
y=2L-\left[q\left(x\right)+l\left(x\right)\right]
{\left(x-R\right)}^{2}+{y}^{2}={R}^{2}
H=2R
y=p\left(x\right)=-q\left(x\right)\equiv R\sqrt{1-{\left(x/R-1\right)}^{2}},\text{ }0\le x\le 2R
l\left(x\right)=R\left[\text{π}/2+\mathrm{arcsin}\left(x/R-1\right)\right]
R=1
y\in \left[0,\text{\hspace{0.17em}}\text{π}R\right]\text{ }y=R\left[\text{π}/2+\mathrm{arcsin}\left(x/R-1\right)-\sqrt{1-{\left(x/R-1\right)}^{2}}\right],\text{ }0\le x\le 2R
y\in \left[\text{π}R,\text{\hspace{0.17em}}2\text{π}R\right]\text{ }y=R\left[\text{3π}/2-\mathrm{arcsin}\left(x/R-1\right)+\sqrt{1-{\left(x/R-1\right)}^{2}}\right],\text{ }0\le x\le 2R
x=R\left(1-\mathrm{cos}t\right),\text{ }y=R\left(t-\mathrm{sin}t\right)
y=p\left(x\right)=-q\left(x\right)\equiv 4h\left(x/H\right)\left(1-x/H\right),\text{ }0\le x\le H
l\left(x\right)=\underset{0}{\overset{x}{\int }}\sqrt{1+{\left(4h/H\right)}^{2}{\left(1-2\xi /H\right)}^{2}}\text{d}\xi
a,b
{\left(x-a\right)}^{2}/{a}^{2}+{y}^{2}/{b}^{2}=1
H=2a
H=2,h=1.5
a=1,b=0.5
y=p\left(x\right)=-q\left(x\right)\equiv b\sqrt{1-{\left(x/a-1\right)}^{2}},\text{ }0\le x\le 2a
l\left(x\right)=\underset{0}{\overset{x}{\int }}\sqrt{\left\{1-\left[1-{\left(b/a\right)}^{2}\right]{\left(\xi /a-1\right)}^{2}\right\}{\left[1-{\left(\xi /a-1\right)}^{2}\right]}^{-1}}\text{d}\xi ,\text{ }0\le x\le 2a
p\left(x\right)\ne -q\left(x\right)
y\in \left[0,L\right]
y=\underset{0}{\overset{H}{\int }}\sqrt{1+{\left[{q}^{\prime }\left(\xi \right)\right]}^{2}}\text{d}\xi +p\left(x\right)+\underset{x}{\overset{H}{\int }}\sqrt{1+{\left[{p}^{\prime }\left(\xi \right)\right]}^{2}}\text{d}\xi
q\left(x\right)
c\left(x\right)
{q}^{\prime }\left(x\right)
{q}^{\prime }\left(x\right)
q\left(x\right)=\frac{1}{2}\left[c\left(x\right)-\int \frac{\text{d}x}{{c}^{\prime }\left(x\right)}\right]+C
c\left(x\right)
c\left(x\right)=\left\{\begin{array}{l}\left(\sqrt{{a}^{2}+{b}^{2}}-b\right)x/a\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}0\le x\le a,\\ \left(\sqrt{{a}^{2}+{b}^{2}}+b\right)x/a-2b\text{ }\text{if}\text{\hspace{0.17em}}a<x\le 2a.\end{array}
\left(0,0\right)
y=p\left(x\right)=-q\left(x\right)\equiv \left\{\begin{array}{l}bx/a\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}0\le x\le a,\\ 2b-bx/a\text{ }\text{ }\text{if}\text{\hspace{0.17em}}a<x\le 2a.\end{array}
a=1.5,b=1
y=e\left(x\right)
e\left(x\right)<0
0<x<h
e\left(x\right)>0
h<x<H
e\left(h\right)=e\left(H\right)=0
\underset{x\to 0}{\mathrm{lim}}e\left(x\right)=-\infty
\sigma =\epsilon E
{e}^{″}\left(x\right){\left\{1+{\left[{e}^{\prime }\left(x\right)\right]}^{2}\right\}}^{-3/2}=-Px/EI
z\left(x\right)={e}^{\prime }\left(x\right)
{z}^{\prime }\left(x\right){\left[1+{z}^{2}\left(x\right)\right]}^{-3/2}=-Px/EI
z\left(x\right)=\mathrm{sinh}\left[t\left(x\right)\right]
{e}^{\prime }\left(x\right){\left\{1+{\left[{e}^{\prime }\left(x\right)\right]}^{2}\right\}}^{-1/2}=-P{x}^{2}/2EI+C
{e}^{\prime }\left(x\right)\to +\infty
x\to 0+0
C=1
{e}^{\prime }\left(x\right)\to -\infty
x\to H-0
C=1
P=4EI/{H}^{2}
C=1
P=4EI/{H}^{2}
{e}^{\prime }\left(x\right)
{e}^{\prime }\left(x\right)={\left[2\left(x/H\right)\sqrt{1-{\left(x/H\right)}^{2}}\right]}^{-1}-\left(x/H\right){\left[1-{\left(x/H\right)}^{2}\right]}^{-1/2},\text{ }0<x<H
t=\sqrt{1-{\left(x/H\right)}^{2}}
\int {\left[\left(x/H\right)\sqrt{1-{\left(x/H\right)}^{2}}\right]}^{-1}\text{d}x=-H\mathrm{ln}\left\{\left[1+\sqrt{1-{\left(x/H\right)}^{2}}\right]/\left(x/H\right)\right\}
t=1-{\left(x/H\right)}^{2}
\int \left(x/H\right){\left[1-{\left(x/H\right)}^{2}\right]}^{-1/2}\text{d}x=-H\sqrt{1-{\left(x/H\right)}^{2}}
e\left(H\right)=0
y=e\left(x\right)\equiv H\left\{\sqrt{1-{\left(x/H\right)}^{2}}-\frac{1}{2}\mathrm{ln}\left[\frac{1+\sqrt{1-{\left(x/H\right)}^{2}}}{x/H}\right]\right\},\text{ }0<x<H
{H}_{1}=H
{M}_{1}\left({a}_{1},{b}_{1}\right),\text{\hspace{0.17em}}{M}_{2}\left({a}_{2},{b}_{2}\right)
{M}_{1},\text{\hspace{0.17em}}{M}_{2}
b=H\left\{\sqrt{1-{\left(a/H\right)}^{2}}-\frac{1}{2}\mathrm{ln}\left[\left(1+\sqrt{1-{\left(a/H\right)}^{2}}\right)/\left(a/H\right)\right]\right\}
{H}_{2}=kH
k=const>0
y=e\left(x\right)\equiv kH\left\{\sqrt{1-{\left(x/kH\right)}^{2}}-\frac{1}{2}\mathrm{ln}\left[\frac{1+\sqrt{1-{\left(x/kH\right)}^{2}}}{x/kH}\right]\right\},\text{ }0<x<kH
{N}_{1}\left(k{a}_{1},k{b}_{1}\right),\text{\hspace{0.17em}}{N}_{2}\left(k{a}_{2},k{b}_{2}\right)
{N}_{1},\text{\hspace{0.17em}}{N}_{2}
{N}_{1},\text{\hspace{0.17em}}{N}_{2}
{M}_{1}{M}_{2}=\sqrt{{\left({a}_{2}-{a}_{1}\right)}^{2}+{\left({b}_{2}-{b}_{1}\right)}^{2}}
{N}_{1}{N}_{2}=\sqrt{{\left(k{a}_{2}-k{a}_{1}\right)}^{2}+{\left(k{b}_{2}-k{b}_{1}\right)}^{2}}=k\sqrt{{\left({a}_{2}-{a}_{1}\right)}^{2}+{\left({b}_{2}-{b}_{1}\right)}^{2}}
{N}_{1}{N}_{2}/{M}_{1}{M}_{2}=k={H}_{2}/{H}_{1}
k={H}_{2}/{H}_{1}
P,\text{\hspace{0.17em}}E,\text{\hspace{0.17em}}I
H=2\sqrt{EI/P}
H=1
P=4EI
e\left(h\right)=0
h=\eta H
\sqrt{1-{\eta }^{2}}=\left(1/2\right)\mathrm{ln}\left[\left(1+\sqrt{1-{\eta }^{2}}\right)/\eta \right]
\eta \approx 0.288
l\left(x\right)=\underset{h}{\overset{x}{\int }}\sqrt{1+{\left[{e}^{\prime }\left(x\right)\right]}^{2}}\text{d}x,\text{ }h\le x\le H
l\left(x\right)=H\left\{{\pi }_{e}-\left(1/2\right)\mathrm{ln}\left[\left(1+\sqrt{1-{\left(x/H\right)}^{2}}\right)/\left(x/H\right)\right]\right\},\text{ }h\le x\le H
{\pi }_{e}
l\left(h\right)=0
h=\eta H
l\left(h\right)=0
{\pi }_{e}-\left(1/2\right)\mathrm{ln}\left[\left(1+\sqrt{1-{\eta }^{2}}\right)/\eta \right]=0\text{ }⇒\text{ }{\pi }_{e}-\sqrt{1-{\eta }^{2}}=0\text{ }⇒
{\pi }_{e}^{2}+{\eta }^{2}=1
{\pi }_{e}\approx 0.958
L=l\left(H\right)={\pi }_{e}H\text{ }⇒\text{ }{\pi }_{e}=L/H
{\pi }_{e}
H=1
{\pi }_{e}
{\pi }_{e}
x=\eta H
\gamma =2\mathrm{arctan}\left[{e}^{\prime }\left(\eta H\right)\right]\text{ }⇒\text{ }\gamma \approx 0.628\text{π}\sim {113}^{\circ }
\eta ,\text{\hspace{0.17em}}{\pi }_{e},\text{\hspace{0.17em}}\gamma
s\left(x\right)
x\ge h
s\left(x\right)=2\underset{h}{\overset{x}{\int }}e\left(\xi \right)\text{d}\xi \text{ }⇒
s\left(x\right)=\left\{\eta {\pi }_{e}+\frac{x}{H}\left[\sqrt{1-{\left(\frac{x}{H}\right)}^{2}}-\mathrm{ln}\frac{1+\sqrt{1-{\left(x/H\right)}^{2}}}{x/H}\right]\right\}{H}^{2},\text{ }h\le x\le H
S=\eta {\pi }_{e}{H}^{2}
y=q\left(x\right)\equiv -e\left(x\right)
l\left(x\right)
y=H\left[{\pi }_{e}-\sqrt{1-{\left(x/H\right)}^{2}}\right],\text{ }h\le x\le H
y=H\left[{\pi }_{e}+\sqrt{1-{\left(x/H\right)}^{2}}\right],\text{ }h\le x\le H
{\left(x-0\right)}^{2}+{\left(y-{\pi }_{e}H\right)}^{2}={H}^{2}
x=0,y={\pi }_{e}H
\left(h,0\right)
\left(h,2{\pi }_{e}H\right)
L={\pi }_{e}H
L=\text{π}R
S=\eta {\pi }_{e}{H}^{2}
S=\text{π}{R}^{2}
[1] Tarabrin, G.T. (2012) Loop on Elastic Rod. Stroitelnaya Mechanika i Raschet Sooruzheniy, 3, 34-39. (In Russian)
[2] Tarabrin, G.T. (2012) Circulations Cycloids on Moving Counters. Stroitelnaya Mechanika i Raschet Sooruzheniy, 4, 38-43. (In Russian)
[3] Tarabrin, G. (2014) Non-Chrestomathy Problems of Mathematical Physics. Palmarium Academic Publishing, Saarbrucken. (In Russian)
[4] Piskunov, N. (1975) Differential and Integral Calculus. Vol. 1, Mir Publishers, Moscow.
[5] Rabotnov, J.N. (1988) Mechanics of Deformable Solid. Nauka, Moscow. (In Russian)
[6] Timoshenko, S. (1955) Strength of Materials. D. van Nostrand Company, Inc., Princeton, New Jersey, Toronto, New York, London.
[7] Filippov, A.T. (1990) Multiform Soliton. Nauka, Moscow. (In Russian).
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Universe | Special Issue : Gravitational Radiation in Cosmological Spacetimes
Gravitational Radiation in Cosmological Spacetimes
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Special Issue "Gravitational Radiation in Cosmological Spacetimes"
A special issue of Universe (ISSN 2218-1997). This special issue belongs to the section "Gravitation".
Dr. B.P. Bonga (Béatrice)
Institute for Mathematics, Astrophysics and Particle Physics, Radboud University, 6525 AJ Nijmegen, The Netherlands
Interests: gravitational wave theory; resonance effects in neutron stars and black hole spacetimes; gravitational radiation in cosmological spacetimes; early universe cosmology
Special Issue in Universe: Tidal Effects in General Relativity
Gravitational waves are now observed by LIGO-Virgo a few times a month, and it is difficult to imagine that in the early days of General Relativity, their physical reality was debated. However, many prominent scientists including Einstein himself questioned whether gravitational waves truly exist in nature. Theoretical developments to study gravitational waves generated by compact sources such as binary black holes in asymptotically flat spacetimes and the observation of the orbital decay in the Hulse–Taylor binary in the 1970s put an end to this debate. These theoretical advances also allowed the study of the nonlinear effects inherent in General Relativity by studying radiation at future null infinity. Despite these amazing advancements on the theoretical side, gravitational waves in cosmological spacetimes are still mostly studied in the linearized setting. With the increasing sensitivity of gravitational wave observatories, we will observe radiation emitted by sources increasingly far away, and therefore, cosmological effects will become important. Consequently, a fundamental understanding beyond geometrics optics approximation in the linearized setting is needed.
This Special Issue aims to collect and act as a catalyzer for progress in our understanding of gravitational waves emitted by compact sources in cosmological spacetimes—both with and without a cosmological constant.
FLRW spacetimes
Lensing Magnification Seen by Gravitational Wave Detectors
Universe 2022, 8(1), 19; https://doi.org/10.3390/universe8010019 - 30 Dec 2021
In this paper, we studied the gravitational lensing of gravitational wave events. The probability that an observed gravitational wave source has been (de-)amplified by a given amount is a detector-dependent quantity which depends on different ingredients: the lens distribution, the underlying distribution of [...] Read more.
In this paper, we studied the gravitational lensing of gravitational wave events. The probability that an observed gravitational wave source has been (de-)amplified by a given amount is a detector-dependent quantity which depends on different ingredients: the lens distribution, the underlying distribution of sources and the detector sensitivity. The main objective of the present work was to introduce a semi-analytic approach to study the distribution of the magnification of a given source population observed with a given detector. The advantage of this approach is that each ingredient can be individually varied and tested. We computed the expected magnification as both a function of redshift and of the observedsource luminosity distance, which is the only quantity one can access via observation in the absence of an electromagnetic counterpart. As a case study, we then focus on the LIGO/Virgo network and on strong lensing (
\mu >1
(This article belongs to the Special Issue Gravitational Radiation in Cosmological Spacetimes)
Universe 2021, 7(11), 437; https://doi.org/10.3390/universe7110437 - 15 Nov 2021
Cherenkov radiation may occur whenever the source is moving faster than the waves it generates. In a radiation dominated universe, with equation-of-state
w=1/3
, we have recently shown that the Bardeen scalar-metric perturbations contribute to the linearized Weyl tensor [...] Read more.
w=1/3
, we have recently shown that the Bardeen scalar-metric perturbations contribute to the linearized Weyl tensor in such a manner that its wavefront propagates at acoustic speed
\sqrt{w}=1/\sqrt{3}
. In this work, we explicitly compute the shape of the Bardeen Cherenkov cone and wedge generated respectively by a supersonic point mass (approximating a primordial black hole) and a straight Nambu-Goto wire (approximating a cosmic string) moving perpendicular to its length. When the black hole or cosmic string is moving at ultra-relativistic speeds, we also calculate explicitly the sudden surge of scalar-metric induced tidal forces on a pair of test particles due to the passing Cherenkov shock wave. These forces can stretch or compress, depending on the orientation of the masses relative to the shock front’s normal. Full article
Title: Gravitational radiation with non-negative cosmological constant
Authors: José M M Senovilla
Affiliation: Departamento de Física, Universidad del País Vasco UPV/EHU, Apartado 48080, Bilbao, Spain； EHU Quantum Center, Universitu of the Basque Country UPV/EHU
Abstract: The existence of gravitational radiation arriving at null infinity J+ –i.e. escaping from the physical system– is addressed in the presence of a non-negative cosmological constant. The case with vanishing cosmological constant is well understood and relies on the properties of the News tensor field (or the News function) defined at infinity. The situation is drastically different when Lambda >0 where there is no known notion of ‘News’ with similar good properties. In this paper both situations are considered jointly from a tidal point of view, that is, taking into account the strength (or energy) of the curvature tensors. This leads to a novel characterization of gravitational radiation valid for the general case that has been proven to be equivalent, when Lambda = 0, to the standard one based on News. The implications of this result (on asymptotic symmetries, peeling theorem, balance laws, and others) are analyzed in some detail when Lambda > 0. Several explicit illustrative examples are provided.
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Evaluation of DLC Coatings for High-Temperature Foil Bearing Applications | J. Tribol. | ASME Digital Collection
Said Jahanmir,
Mohawk Innovative Technology Inc.
, 1037 Watervliet Shaker Road, Albany, NY 12205
Jahanmir, S., Heshmat, H., and Heshmat, C. (December 2, 2008). "Evaluation of DLC Coatings for High-Temperature Foil Bearing Applications." ASME. J. Tribol. January 2009; 131(1): 011301. https://doi.org/10.1115/1.2991181
Diamondlike carbon (DLC) coatings, particularly in the hydrogenated form, provide extremely low coefficients of friction in concentrated contacts. The objective of this investigation was to evaluate the performance of DLC coatings for potential application in foil bearings. Since in some applications the bearings experience a wide range of temperatures, tribological tests were performed using a single foil thrust bearing in contact with a rotating flat disk up to
500°C
. The coatings deposited on the disks consisted of a hydrogenated diamondlike carbon film (H-DLC), a nonhydrogenated DLC, and a thin dense chrome deposited by the Electrolyzing™ process. The top foil pads were coated with a tungsten disulfide based solid lubricant (Korolon™ 900). All three disk coatings provided excellent performance at room temperature. However, the H-DLC coating proved to be unacceptable at
300°C
due to lack of hydrodynamic lift, albeit the very low coefficient of friction when the foil pad and the disk were in contact during stop-start cycles. This phenomenon is explained by considering the effect of atmospheric moisture on the tribological behavior of H-DLC and using the quasihydrodynamic theory of powder lubrication.
diamond-like carbon, friction, lubricants, machine bearings, protective coatings, tungsten compounds, foil bearings, solid lubricants, coatings, DLC, diamondlike carbon coating, high temperatures, high speeds, coefficient of friction
Coatings, Disks, Foil bearings, Friction, Lubricants, Temperature, Tungsten, Cycles, Diamond films, Tribology, High temperature, Bearings
Principles of Gas Turbine Bearing Lubrication and Design
On the Frictional Damping Characterization of Compliant Bump Foils
Thermal Features of Compliant Foil Bearings: Theory and Experiments
The Application of Foil Seals to a Gas Turbine Engine
Hryniewiscz
Low-Friction Wear-Resistant Coatings for High-Temperature Foil Bearings
Assessment of Tribological Coatings for Foil Bearing Applications
Composition Optimization of Chromium Carbide Based Solid Lubricant Coatings for Foil Gas Bearings at Temperatures to 650°C
Composition Optimization of Self-Lubricating Chromium Carbide Based Comparative Coatings for Use to 760°C
The Effect of Compositional Tailoring on the Thermal Expansion and Tribological Properties of PS300: A Solid Lubricant Composite Coating
Friction and Wear Characteristics of a Modified Composite Solid Lubricant Plasma Sprayed Coating
Proceedings of the 23rd International Surface Engineering Conference
Evaluation of Advanced Solid Lubricant Coatings for Foil Air Bearings Operating at 25 and 500 C
A System Approach to the Solid Lubrication of Foil Air Bearings for Oil-Free Turbomachinery
High Velocity Oxyfuel Deposition for Low Surface Roughness PS304 Self-Lubricating Composite Coatings
Thrust Washer Tribological Evaluation of PS304 Coatings Against Rene 41
Evaluation of Solid Lubricant Coatings for High Temperature Foil Bearings
Evaluation of Coatings for a Large Hybrid Foil/Magnetic Bearing
,” ASME Paper No. IJTC2006-12328.
Coatings for a Large Foil Bearing
Evaluation of DLC Coatings for Foil Bearing Applications
,” ASME Paper No. IJTC20076-44035.
Friction and Wear of Diamond and Diamond Like Carbon Films
Tribological Properties of DLC Film
Tribology of Mechanical Systems
Dumkum
A Multilayer Approach to High Adhesion Diamond Like Carbon Coating on Titanium
Friction, Lubrication and Wear Technology
Friction Control of Diamond Like Carbon Coatings
Design Criteria for Superlubricity in Carbon Films and Related Microstructures
Tribological Performance of Diamond and Diamond Like Carbon Films at Elevated Temperatures
Annealing Effect on the Structure, Mechanical and Tribological Properties of Hydrogenated Diamond Like Carbon Films
Temperature Dependence of Friction Coefficient in Ultrahigh Vacuum for Hydrogenated Amorphous Carbon Coatings
Proceedings of the 2005 World Tribology Congress
Solid Lubricant Materials for High Temperatures
Environmental Effects on the Friction of Hydrogenated DLC Films
Friction of Diamond Like Carbon Films in Different Atmospheres
The Influence of the Temperature Increase in the Tribological Behavior of DLC Films by RF-PECVD
On a Common Tribological Mechanism Between Interacting Surfaces
The Quasi-Hydrodynamic Mechanism of Powder Lubrication: Part I. Lubricant Flow Visualization
The Quasi-Hydrodynamic Mechanism of Powder Lubrication: Part II. Lubricant Film Pressure Profile
On the Cognitive Approach Toward Classification of Dry Triboparticulates
Proceedings of the Leeds-Lyon Symposium on Dissipative Processes in Tribology
, Lyon, France, September 1993,
The Quasi-Hydrodynamic Mechanism of Powder Lubrication: Part III. On Theory and Rheology of Triboparticulates
The Rheology and Hydrodynamics of Dry Powder Lubrication
Lubrication properties of textured CSS-42L bearing steel filled with Sn-Ag-Cu-Ti 3 C 2 under harsh environmental conditions
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1 )The graph of the linear equation 3x-2y=0 passes through the point a ) (2/3, -2/3) b ) (2/3, 3/2) - Maths - Linear Equations in Two Variables - 6917640 | Meritnation.com
The given equation is 3x - 2y = 0
Put x = 1/3 and y = 1/2 in the LHS of above equation, we get
\mathrm{LHS}\quad =\quad \quad 3\quad \times \quad \frac{1}{3}\quad -\quad 2\quad \times \quad \frac{1}{2}\quad =\quad 1\quad -\quad 1\quad =\quad 0\quad \phantom{\rule{0ex}{0ex}}\mathrm{RHS}\quad =\quad 0\phantom{\rule{0ex}{0ex}}\mathrm{so},\quad \mathrm{LHS}\quad =\quad \mathrm{RHS}\phantom{\rule{0ex}{0ex}}\mathrm{so}\quad \mathrm{graph}\quad \mathrm{of}\quad \mathrm{the}\quad \mathrm{equation}\quad \mathrm{passes}\quad \mathrm{through}\quad \left(\frac{1}{3},\quad \frac{1}{2}\right)
When we substitute the remaining two points , we will see that they will not satisfy the given equation.
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Development, Modeling, and Experimental Investigation of Low Frequency Workpiece Vibration-Assisted Micro-EDM of Tungsten Carbide | J. Manuf. Sci. Eng. | ASME Digital Collection
M. P. Jahan,
T. Saleh,
e-mail: mpemusta@nus.edu.sg
Jahan, M. P., Saleh, T., Rahman, M., and Wong, Y. S. (September 20, 2010). "Development, Modeling, and Experimental Investigation of Low Frequency Workpiece Vibration-Assisted Micro-EDM of Tungsten Carbide." ASME. J. Manuf. Sci. Eng. October 2010; 132(5): 054503. https://doi.org/10.1115/1.4002457
This present study intends to investigate the feasibility of drilling deep microholes in difficult-to-cut tungsten carbide by means of low frequency workpiece vibration-assisted micro–electro-discharge machining (micro-EDM). A vibration device has been designed and developed in which the workpiece is subjected to vibration of up to a frequency of 1 kHz and an amplitude of
2.5 μm
. An analytical approach is presented to explain the mechanism of workpiece vibration-assisted micro-EDM and how workpiece vibration improves the performance of micro-EDM drilling. The reasons for improving the overall flushing conditions are explained in terms of the behavior of debris in a vibrating workpiece, change in gap distance, and dielectric fluid pressure in the gap during vibration-assisted micro-EDM. In addition, the effects of vibration frequency, amplitude, and electrical parameters on the machining performance, as well as surface quality and accuracy of the microholes have been investigated. It has been found that the overall machining performance improves considerably with significant reduction of machining time, increase in MRR, and decrease in EWR. The improved flushing conditions, increased discharge ratio, and reduced percentage of ineffective pulses are found to be the contributing factors for improved performance of the vibration-assisted micro-EDM of tungsten carbide.
modeling, vibration-assisted micro-EDM, low frequency vibration, workpiece vibration, machining performance, quality of microholes, tungsten carbide, drilling, electrical discharge machining, micromachining, tungsten alloys, vibrations
Electrical discharge machining, Vibration, Machining, Oscillating frequencies, Tungsten, Drilling
Kimurta
Study of Vibration-Assisted Micro-EDM—The Effect of Vibration on Machining Time and Stability of Discharge
Ultrasonic and Electric Discharge Machining to Deep and Small Hole on Titanium Alloy
A Study of Ultrasonically Aided Micro-Electrical Discharge Machining by the Application of Workpiece Vibration
Experimental Research on Vibration Assisted EDM of Micro-Structures With Non-Circular Cross-Section
The Effect of Ultrasonic Vibration to EDM
Met. Mach. Overseas
Principal Research on the Machining and Discharge Characteristics of Ultrasonic Vibration EDM
Elect. Mach.
Micro-Hole Machining by Vibration-Assisted EDM
Proceedings of the Annual Meeting of Japanese Society of Electrical Machining Engineers
Effects of Ultrasonic Vibration on the Performances in EDM
Study on a New Kind of Combined Machining Technology of Ultrasonic Machining and Electrical Discharge Machining
A Study of Ultrasonic Aided Wire Electrical Discharge Machining
Straight Hole Micro-EDM With a Cylindrical Tool Using a Variable Capacitance Method Accompanied by Ultrasonic Vibration
Study on Vibration-EDM and Mass Punching of Micro-Holes
Hewidy
El-Taweel
Modelling the Performance of ECM Assisted by Low Frequency Vibrations
Fuzzy Logic Control of Microhole Electrical Discharge Machining
Experimental Studies in Electro-Machining
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Are Vaccinations Alone Enough to Curb the Dynamics of the COVID-19 Pandemic in the European Union?
Econometrics 2022, 10(2), 25; https://doi.org/10.3390/econometrics10020025 (registering DOI) - 26 May 2022
I use the data on the COVID-19 pandemic maintained by Our Word in Data to estimate a nonstationary dynamic panel exhibiting the dynamics of confirmed deaths, infections and vaccinations per million population in the European Union countries in the period of January–July 2021. [...] Read more.
I use the data on the COVID-19 pandemic maintained by Our Word in Data to estimate a nonstationary dynamic panel exhibiting the dynamics of confirmed deaths, infections and vaccinations per million population in the European Union countries in the period of January–July 2021. Having the data aggregated on a weekly basis I demonstrate that a model which allows for heterogeneous short-run dynamics and common long-run marginal effects is superior to that allowing only for either homogeneous or heterogeneous responses. The analysis shows that the long-run marginal death effects with respect to confirmed infections and vaccinations are positive and negative, respectively, as expected. Since the estimate of the former effect compared to the latter one is about 71.67 times greater, only mass vaccinations can prevent the number of deaths from being large in the long-run. The success in achieving this is easier for countries with the estimated large negative individual death effect (Cyprus, Denmark, Ireland, Portugal, Estonia, Lithuania) than for those with the large but positive death effect (Bulgaria, Hungary, Slovakia). The speed of convergence to the long-run equilibrium relationship estimates for individual countries are all negative. For some countries (Bulgaria, Denmark, Estonia, Greece, Hungary, Slovakia) they differ in the magnitude from that averaged for the whole EU, while for others (Croatia, Ireland, Lithuania, Poland, Portugal, Romania, Spain), they do not. Full article
Econometrics 2022, 10(2), 15; https://doi.org/10.3390/econometrics10020015 - 25 Mar 2022
Econometrics 2022, 10(1), 10; https://doi.org/10.3390/econometrics10010010 - 21 Feb 2022
Econometrics 2022, 10(1), 9; https://doi.org/10.3390/econometrics10010009 - 16 Feb 2022
{\mathrm{Ice}}_{t}
{\mathrm{CO}}_{2,t}
{\mathrm{Temp}}_{t}
{\mathrm{Ice}}_{t}
{\mathrm{CO}}_{2,t}
{\mathrm{Temp}}_{t}
{\mathrm{Ice}}_{t}
{\mathrm{Ice}}_{t}
{\mathrm{CO}}_{2,t}
{\mathrm{CO}}_{2,t}
{\mathrm{Temp}}_{t}
{\mathrm{Temp}}_{t}
Econometrics 2022, 10(1), 7; https://doi.org/10.3390/econometrics10010007 - 31 Jan 2022
\lambda
\lambda
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Hyers-Ulam stability of a generalized additive set-valued functional equation | Journal of Inequalities and Applications | Full Text
Hyers-Ulam stability of a generalized additive set-valued functional equation
Young Cho3
In this paper, we define a generalized additive set-valued functional equation, which is related to the following generalized additive functional equation:
f\left({x}_{1}+\cdots +{x}_{l}\right)=\left(l-1\right)f\left(\frac{{x}_{1}+\cdots +{x}_{l-1}}{l-1}\right)+f\left({x}_{l}\right)
for a fixed integer l with
l>1
, and prove the Hyers-Ulam stability of the generalized additive set-valued functional equation.
MSC:39B52, 54C60, 91B44.
The theory of set-valued functions has been much related to the control theory and the mathematical economics. After the pioneering papers written by Aumann [1] and Debreu [2], set-valued functions in Banach spaces have been developed in the last decades. We can refer to the papers by Arrow and Debreu [3], McKenzie [4], the monographs by Hindenbrand [5], Aubin and Frankowska [6], Castaing and Valadier [7], Klein and Thompson [8] and the survey by Hess [9].
The stability problem of functional equations originated from a question of Ulam [10] concerning the stability of group homomorphisms. Hyers [11] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [12] for additive mappings and by Th.M. Rassias [13] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by Găvruta [14] by replacing the unbounded Cauchy difference with a general control function in the spirit of Th.M. Rassias’ approach. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [15–17]).
Let Y be a Banach space. We define the following:
{2}^{Y}
: the set of all subsets of Y;
{C}_{b}\left(Y\right)
: the set of all closed bounded subsets of Y;
{C}_{c}\left(Y\right)
: the set of all closed convex subsets of Y;
{C}_{cb}\left(Y\right)
: the set of all closed convex bounded subsets of Y.
{2}^{Y}
we consider the addition and the scalar multiplication as follows:
C+{C}^{\prime }=\left\{x+{x}^{\prime }:x\in C,{x}^{\prime }\in {C}^{\prime }\right\},\phantom{\rule{2em}{0ex}}\lambda C=\left\{\lambda x:x\in C\right\},
C,{C}^{\prime }\in {2}^{Y}
\lambda \in \mathbb{R}
. Further, if
C,{C}^{\prime }\in {C}_{c}\left(Y\right)
C\oplus {C}^{\prime }=\overline{C+{C}^{\prime }}
\lambda C+\lambda {C}^{\prime }=\lambda \left(C+{C}^{\prime }\right),\phantom{\rule{2em}{0ex}}\left(\lambda +\mu \right)C\subseteq \lambda C+\mu C.
Furthermore, when C is convex, we obtain
\left(\lambda +\mu \right)C=\lambda C+\mu C
\lambda ,\mu \in {\mathbb{R}}^{+}
For a given set
C\in {2}^{Y}
, the distance function
d\left(\cdot ,C\right)
and the support function
s\left(\cdot ,C\right)
are respectively defined by
C,{C}^{\prime }\in {C}_{b}\left(Y\right)
, we define the Hausdorff distance between C and
{C}^{\prime }
h\left(C,{C}^{\prime }\right)=inf\left\{\lambda >0:C\subseteq {C}^{\prime }+\lambda {B}_{Y},{C}^{\prime }\subseteq C+\lambda {B}_{Y}\right\},
{B}_{Y}
is the closed unit ball in Y.
The following proposition reveals some properties of the Hausdorff distance.
Proposition 1.1 For every
C,{C}^{\prime },K,{K}^{\prime }\in {C}_{cb}\left(Y\right)
\lambda >0
h\left(C\oplus {C}^{\prime },K\oplus {K}^{\prime }\right)\le h\left(C,K\right)+h\left({C}^{\prime },{K}^{\prime }\right)
h\left(\lambda C,\lambda K\right)=\lambda h\left(C,K\right)
\left({C}_{cb}\left(Y\right),\oplus ,h\right)
be endowed with the Hausdorff distance h. Since Y is a Banach space,
\left({C}_{cb}\left(Y\right),\oplus ,h\right)
is a complete metric semigroup (see [7]). Debreu [2] proved that
\left({C}_{cb}\left(Y\right),\oplus ,h\right)
is isometrically embedded in a Banach space as follows.
C\left({B}_{{Y}^{\ast }}\right)
be the Banach space of continuous real-valued functions on
{B}_{{Y}^{\ast }}
endowed with the uniform norm
{\parallel \cdot \parallel }_{u}
j:\left({C}_{cb}\left(Y\right),\oplus ,h\right)\to C\left({B}_{{Y}^{\ast }}\right)
j\left(A\right)=s\left(\cdot ,A\right)
, satisfies the following properties:
j\left(A\oplus B\right)=j\left(A\right)+j\left(B\right)
j\left(\lambda A\right)=\lambda j\left(A\right)
h\left(A,B\right)={\parallel j\left(A\right)-j\left(B\right)\parallel }_{u}
j\left({C}_{cb}\left(Y\right)\right)
C\left({B}_{{Y}^{\ast }}\right)
A,B\in {C}_{cb}\left(Y\right)
\lambda \ge 0
f:\mathrm{\Omega }\to \left({C}_{cb}\left(Y\right),h\right)
be a set-valued function from a complete finite measure space
\left(\mathrm{\Omega },\mathrm{\Sigma },\nu \right)
{C}_{cb}\left(Y\right)
. Then f is Debreu integrable if the composition
j\circ f
is Bochner integrable (see [18]). In this case, the Debreu integral of f in Ω is the unique element
\left(D\right){\int }_{\mathrm{\Omega }}f\phantom{\rule{0.2em}{0ex}}d\nu \in {C}_{cb}\left(Y\right)
j\left(\left(D\right){\int }_{\mathrm{\Omega }}f\phantom{\rule{0.2em}{0ex}}d\nu \right)
is the Bochner integral of
j\circ f
. The set of Debreu integrable functions from Ω to
{C}_{cb}\left(Y\right)
D\left(\mathrm{\Omega },{C}_{cb}\left(Y\right)\right)
. Furthermore, on
D\left(\mathrm{\Omega },{C}_{cb}\left(Y\right)\right)
\left(f+g\right)\left(\omega \right)=f\left(\omega \right)\oplus g\left(\omega \right)
f,g\in D\left(\mathrm{\Omega },{C}_{cb}\left(Y\right)\right)
\left(\left(\mathrm{\Omega },{C}_{cb}\left(Y\right)\right),+\right)
is an abelian semigroup.
Set-valued functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [19–27]).
In this paper, we define a generalized additive set-valued functional equation and prove the Hyers-Ulam stability of the generalized additive set-valued functional equation.
Throughout this paper, let X be a real vector space and Y be a Banach space.
2 Stability of a generalized additive set-valued functional equation
f:X\to {C}_{cb}\left(Y\right)
. The generalized additive set-valued functional equation is defined by
f\left({x}_{1}+\cdots +{x}_{l}\right)=\left(l-1\right)f\left(\frac{{x}_{1}+\cdots +{x}_{l-1}}{l-1}\right)\oplus f\left({x}_{l}\right)
{x}_{1},\dots ,{x}_{l}\in X
. Every solution of the generalized additive set-valued functional equation is called a generalized additive set-valued mapping.
Note that there are some examples in [28].
\phi :{X}^{l}\to \left[0,\mathrm{\infty }\right)
\stackrel{˜}{\phi }\left({x}_{1},\dots ,{x}_{l}\right):=\sum _{j=0}^{\mathrm{\infty }}\frac{1}{{l}^{j}}\phi \left({l}^{j}{x}_{1},\dots ,{l}^{j}{x}_{l}\right)<\mathrm{\infty }
{x}_{1},\dots ,{x}_{l}\in X
f:X\to \left({C}_{cb}\left(Y\right),h\right)
h\left(f\left({x}_{1}+\cdots +{x}_{l}\right),\left(l-1\right)f\left(\frac{{x}_{1}+\cdots +{x}_{l-1}}{l-1}\right)\oplus f\left({x}_{l}\right)\right)\le \phi \left({x}_{1},\dots ,{x}_{l}\right)
{x}_{1},\dots ,{x}_{l}\in X
. Then there exists a unique generalized additive set-valued mapping
A:X\to \left({C}_{cb}\left(Y\right),h\right)
h\left(f\left(x\right),A\left(x\right)\right)\le \frac{1}{l}\stackrel{˜}{\phi }\left(x,\dots ,x\right)
x\in X
{x}_{1}=\cdots ={x}_{l}=x
in (3). Since
f\left(x\right)
is convex, we get
h\left(f\left(lx\right),lf\left(x\right)\right)\le \phi \left(x,\dots ,x\right),
and if we replace x by
{l}^{n}x
n\in \mathbb{N}
in (5), then we obtain
h\left(f\left({l}^{n+1}x\right),lf\left({l}^{n}x\right)\right)\le \phi \left({l}^{n}x,\dots ,{l}^{n}x\right)
h\left(\frac{f\left({l}^{n+1}x\right)}{{l}^{n+1}},\frac{f\left({l}^{n}x\right)}{{l}^{n}}\right)\le \frac{1}{{l}^{n+1}}\phi \left({l}^{n}x,\dots ,{l}^{n}x\right).
h\left(\frac{f\left({l}^{n}x\right)}{{l}^{n}},\frac{f\left({l}^{m}x\right)}{{l}^{m}}\right)\le \frac{1}{l}\sum _{j=m}^{n-1}\frac{1}{{l}^{j}}\phi \left({l}^{j}x,\dots ,{l}^{j}x\right)
for all integers n, m with
n\ge m
. It follows from (2) and (6) that
\left\{\frac{f\left({l}^{n}x\right)}{{l}^{n}}\right\}
\left({C}_{cb}\left(Y\right),h\right)
A\left(x\right)={lim}_{n\to \mathrm{\infty }}\frac{f\left({l}^{n}x\right)}{{l}^{n}}
x\in X
. Then we claim that A is a generalized additive set-valued mapping. Note that
h\left(\frac{f\left({l}^{n}\left({x}_{1}+\cdots +{x}_{l}\right)\right)}{{l}^{n}},\left(l-1\right)f\left(\frac{{l}^{n}\left({x}_{1}+\cdots +{x}_{l-1}\right)}{{l}^{n}\left(l-1\right)}\right)\oplus \frac{f\left({l}^{n}{x}_{l}\right)}{{l}^{n}}\right)\le \frac{1}{{l}^{n}}\phi \left({l}^{n}{x}_{1},\dots ,{l}^{n}{x}_{l}\right).
h\left(A\oplus B,C\oplus D\right)\le h\left(A,C\right)+h\left(B,D\right)
n\to \mathrm{\infty }
. So, A is a generalized additive set-valued mapping. Letting
m=0
m\to \mathrm{\infty }
in (6), we get the inequality (4).
T:X\to \left({C}_{cb}\left(Y\right),h\right)
be another generalized additive set-valued mapping satisfying (1) and (4). So,
\begin{array}{rcl}h\left(A\left(x\right),T\left(x\right)\right)& =& \frac{1}{{l}^{n}}h\left(A\left({l}^{n}x\right),T\left({l}^{n}x\right)\right)\\ \le & \frac{1}{{l}^{n}}h\left(A\left({l}^{n}x\right),f\left({l}^{n}x\right)\right)+\frac{1}{{l}^{n}}h\left(T\left({l}^{n}x\right),f\left({l}^{n}x\right)\right)\\ \le & \frac{2}{{l}^{n+1}}\stackrel{˜}{\phi }\left({l}^{n}x,\dots ,{l}^{n}x\right),\end{array}
n\to \mathrm{\infty }
x\in X
A\left(x\right)=T\left(x\right)
x\in X
, which proves the uniqueness of A, as desired. □
1>p>0
\theta \ge 0
be real numbers, and let X be a real normed space. Suppose that
f:X\to \left({C}_{cb}\left(Y\right),h\right)
h\left(f\left({x}_{1}+\cdots +{x}_{l}\right),\left(l-1\right)f\left(\frac{{x}_{1}+\cdots +{x}_{l-1}}{l-1}\right)\oplus f\left({x}_{l}\right)\right)\le \theta \sum _{j=1}^{l}{\parallel {x}_{j}\parallel }^{p}
{x}_{1},\dots ,{x}_{l}\in X
A:X\to Y
h\left(f\left(x\right),A\left(x\right)\right)\le \frac{l\theta }{l-{l}^{p}}{\parallel x\parallel }^{p}
x\in X
\phi \left({x}_{1},\dots ,{x}_{l}\right):=\theta \sum _{j=1}^{l}{\parallel {x}_{j}\parallel }^{p}
{x}_{1},\dots ,{x}_{l}\in X
\phi :{X}^{l}\to \left[0,\mathrm{\infty }\right)
\stackrel{˜}{\phi }\left({x}_{1},\dots ,{x}_{l}\right):=\sum _{j=1}^{\mathrm{\infty }}{l}^{j}\phi \left(\frac{{x}_{1}}{{l}^{j}},\dots ,\frac{{x}_{l}}{{2}^{j}}\right)<\mathrm{\infty }
{x}_{1},\dots ,{x}_{l}\in X
f:X\to \left({C}_{cb}\left(Y\right),h\right)
is a mapping satisfying (3). Then there exists a unique generalized additive set-valued mapping
A:X\to \left({C}_{cb}\left(Y\right),h\right)
h\left(f\left(x\right),A\left(x\right)\right)\le \frac{1}{l}\stackrel{˜}{\phi }\left(x,\dots ,x\right)
x\in X
Proof It follows from (5) that
h\left(f\left(x\right),lf\left(\frac{x}{l}\right)\right)\le \phi \left(\frac{x}{l},\dots ,\frac{x}{l}\right)
x\in X
p>1
\theta \ge 0
f:X\to \left({C}_{cb}\left(Y\right),h\right)
A:X\to Y
h\left(f\left(x\right),A\left(x\right)\right)\le \frac{l\theta }{{l}^{p}-l}{\parallel x\parallel }^{p}
x\in X
\phi \left({x}_{1},\dots ,{x}_{l}\right):=\theta \sum _{j=1}^{l}{\parallel {x}_{j}\parallel }^{p}
{x}_{1},\dots ,{x}_{l}\in X
Aumann RJ: Integrals of set-valued functions. J. Math. Anal. Appl. 1965, 12: 1–12. 10.1016/0022-247X(65)90049-1
Debreu G: Integration of correspondences. II. Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability 1967, 351–372. part I
Arrow KJ, Debreu G: Existence of an equilibrium for a competitive economy. Econometrica 1954, 22: 265–290. 10.2307/1907353
McKenzie LW: On the existence of general equilibrium for a competitive market. Econometrica 1959, 27: 54–71. 10.2307/1907777
Hindenbrand W: Core and Equilibria of a Large Economy. Princeton University Press, Princeton; 1974.
Aubin JP, Frankow H: Set-Valued Analysis. Birkhäuser, Boston; 1990.
Castaing C, Valadier M Lect. Notes in Math. 580. In Convex Analysis and Measurable Multifunctions. Springer, Berlin; 1977.
Klein E, Thompson A: Theory of Correspondence. Wiley, New York; 1984.
Hess C: Set-valued integration and set-valued probability theory: an overview. In Handbook of Measure Theory. North-Holland, Amsterdam; 2002. vols. I, II
Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1940. chapter VI, Science edn.
Cascales T, Rodrigeuz J: Birkhoff integral for multi-valued functions. J. Math. Anal. Appl. 2004, 297: 540–560. 10.1016/j.jmaa.2004.03.026
Cardinali T, Nikodem K, Papalini F: Some results on stability and characterization of K -convexity of set-valued functions. Ann. Pol. Math. 1993, 58: 185–192.
Nikodem K: On quadratic set-valued functions. Publ. Math. (Debr.) 1984, 30: 297–301.
Nikodem K: On Jensen’s functional equation for set-valued functions. Rad. Mat. 1987, 3: 23–33.
Nikodem K: Set-valued solutions of the Pexider functional equation. Funkc. Ekvacioj 1988, 31: 227–231.
Nikodem, K: K-Convex and K-Concave Set-Valued Functions. Zeszyty Naukowe 559, Lodz (1989)
Park C, O’Regan D, Saadati R: Stability of some set-valued functional equations. Appl. Math. Lett. 2011, 24: 1910–1914. 10.1016/j.aml.2011.05.017
Piao YJ:The existence and uniqueness of additive selection for
\left(\alpha ,\beta \right)\text{-}\left(\beta ,\alpha \right)
type subadditive set-valued maps. J. Northeast Normal Univ. 2009, 41: 38–40.
Popa D:Additive selections of
\left(\alpha ,\beta \right)
-subadditive set-valued maps. Glas. Mat. 2001, 36(56):11–16.
Lee, K: Stability of functional equations related to set-valued functions. Preprint
SYJ was supported by the 2012 Research Fund of University of Ulsan and had written this paper during visiting the Research Institute of Mathematics, Seoul National University. CP was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299).
Faculty of Electrical and Electronics Engineering, Ulsan College, West Campus, Ulsan, 680-749, Korea
Correspondence to Young Cho.
Jang, S.Y., Park, C. & Cho, Y. Hyers-Ulam stability of a generalized additive set-valued functional equation. J Inequal Appl 2013, 101 (2013). https://doi.org/10.1186/1029-242X-2013-101
generalized additive set-valued functional equation
closed and convex set
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Monetary inflation - Wikipedia
For increases in the general level of prices, see inflation.
Monetary inflation is a sustained increase in the money supply of a country (or currency area). Depending on many factors, especially public expectations, the fundamental state and development of the economy, and the transmission mechanism, it is likely to result in price inflation, which is usually just called "inflation", which is a rise in the general level of prices of goods and services.[1][2]
There is general agreement among economists that there is a causal relationship between monetary inflation and price inflation. But there is neither a common view about the exact theoretical mechanisms and relationships, nor about how to accurately measure it. This relationship is also constantly changing, within a larger complex economic system. So there is a great deal of debate on the issues involved, such as how to measure the monetary base and price inflation, how to measure the effect of public expectations, how to judge the effect of financial innovations on the transmission mechanisms, and how much factors like the velocity of money affect the relationship. Thus there are different views on what could be the best targets and tools in monetary policy.
However, there is a general consensus on the importance and responsibility of central banks and monetary authorities in setting public expectations of price inflation and in trying to control it.
Keynesian economists believe the central bank can sufficiently assess the detailed economic variables and circumstances in real time to adjust monetary policy in order to stabilize gross domestic product. These economists favor monetary policies that attempt to even out the ups and downs of business cycles and economic shocks in a precise fashion.
Followers of the monetarist school think that Keynesian style monetary policies produce many overshooting, time-lag errors and other unwanted effects, usually making things even worse. They doubt the central bank's capacity to analyse economic problems in real time and its ability to influence the economy with correct timing and the right monetary policy measures. So monetarists advocate a less intrusive and less complex monetary policy, specifically a constant growth rate of the money supply.
Some followers of Austrian School economics see monetary inflation as "inflation" and advocate either the return to free markets in money, called free banking, or a 100% gold standard and the abolition of central banks to control this problem.
Currently, most central banks follow a monetarist or Keynesian approach, or more often a mix of both. There is a trend of central banks towards the use of inflation targeting.[2]
1 Quantity theory
3 Austrian view
Main article: Quantity Theory of Money
The monetarist explanation of inflation operates through the Quantity Theory of Money,
{\displaystyle MV=PT}
where M is the money supply, V is the velocity of circulation, P is the price level and T is total transactions or output. As monetarists assume that V and T are determined, in the long run, by real variables, such as the productive capacity of the economy, there is a direct relationship between the growth of the money supply and inflation.
The mechanisms by which excess money might be translated into inflation are examined below. Individuals can also spend their excess money balances directly on goods and services. This has a direct impact on inflation by raising aggregate demand. Also, the increase in the demand for labour resulting from higher demands for goods and services will cause a rise in money wages and unit labour costs. The more inelastic the aggregate supply in the economy, the greater the impact on inflation.
The increase in demand for goods and services may cause a rise in imports. Although this leakage from the domestic economy reduces the money supply, it also increases the supply of money on the foreign exchange market thus applying downward pressure on the exchange rate. This may cause imported inflation.
Modern Monetary Theory, like all derivatives of the Chartalist school, emphasizes that in nations with monetary sovereignty, a country is always able to repay debts that are denominated in its own currency. However, under modern-day monetary systems, the supply of money is largely determined endogenously. But exogenous factors like government surpluses and deficits play a role and allow government to set inflation targets. Yet, adherents of this school note that monetary inflation and price inflation are distinct, and that when there is idle capacity, monetary inflation can cause a boost in aggregate demand which can, up to a point, offset price inflation.[3]
See also: Austrian School § Inflation
The Austrian School maintains that inflation is any increase of the money supply (i.e. units of currency or means of exchange) that is not matched by an increase in demand for money, or as Ludwig von Mises put it:
In theoretical investigation there is only one meaning that can rationally be attached to the expression Inflation: an increase in the quantity of money (in the broader sense of the term, so as to include fiduciary media as well), that is not offset by a corresponding increase in the need for money (again in the broader sense of the term), so that a fall in the objective exchange-value of money must occur.[4]
Given that all major economies currently have a central bank supporting the private banking system, money can be supplied into these economies by means of bank credit (or debt).[5] Austrian economists believe that credit growth propagates business cycles (see Austrian Business Cycle Theory).
^ Michael F. Bryan, On the Origin and Evolution of the Word "Inflation", clevelandfed.org Archived 2008-08-19 at the Wayback Machine
^ a b Jahan, Sarwat. "Inflation Targeting: Holding the Line". International Monetary Funds, Finance & Development. Retrieved 28 December 2014.
^ Éric Tymoigne and L. Randall Wray, "Modern Money Theory 101: A Reply to Critics," Levy Economics Institute of Bard College, Working Paper No. 778 (November 2013).
^ The Theory of Money and Credit, Mises (1912 [1981], p. 272)
^ The Economics of Legal Tender Laws, Jörg Guido Hülsmann (includes detailed commentary on central banking, inflation and FRB)
Monetary Inflation / Quantity Theory
ECB: M3 and CPI
Charts of commodity prices and monetary aggregates
Money and Commodity prices
Bank of England: The Lag from Monetary Policy Actions to Inflation: Friedman Revisited
Retrieved from "https://en.wikipedia.org/w/index.php?title=Monetary_inflation&oldid=1076019088"
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Geometry for Elementary School/Solids - Wikibooks, open books for an open world
Geometry for Elementary School/Solids
Plane shapes Solids Measurements
In this section, we will talk about solids. A three-dimensional space is a space that goes in all directions forever.
3 Other kinds of solids
Solids[edit | edit source]
Solids are shapes that you can actually touch. They have three dimensions, which means that the have length, width and height. These shapes are what make up our daily life, and are very useful. Points on a solid must not be coplanar or colinear. The edge of solids are called the edge, and the surfaces are called faces. The corners, like angles and plane figures, are called vertices.
A solid with only straight edges is called a polyhedron (pol-ee-HEE-dron). The plural form of polehedron is polyhedra (pol-ee-HEE-drah). Your chocolate bars are polyhedra, The Great Pyramids are polyhedra – a lot of things are. We will go into detail about them later.
When dealing with these solid figures, there are two measurements we will need to know: the total surface area and the volume. The former is the sum of the faces of the solid; the latter is how big the solid is.
Polyhedra[edit | edit source]
We will talk about polyhedra in this section. The first thing we will need to know about polyhedra is how to classify them. First, we can classify them by the number of faces:
A tetrahedron is a polyhedron with four faces. Tetrahedrons, interestingly enough, are always rectangular pyramids.
A pentahedron is a polyhedron with five faces.
A hexahedron is a polyhedron with six faces.
The list goes on with the same prefixes as the polygons.
There are no triahedrons. This is because one cannot arrange any polyhedron with three faces. Cylinders don't count as they are not polyhedra.
Secondly, we can classify them by the way they look:
The highlighted parts are the bases.
Prisms are shapes with a flat top and a flat base. The bases must be on parallel planes. Their number of faces is always the same as two added to the number of sides on the base. The number of edges is three times that of the number of sides on the base, while the number of vertices is two times the number of sides on the base. Prisms can be divided by the shape of the base: triangular, quadrilateral, pentagonal, etc. They can be stacked together but cannot be rolled. Note that cylinders are not counted.
Pyramids are shapes with a pointy top and a flat base. Their number of faces is always the same as one added to the number of sides on the base. The number of edges is twice of the number of sides of the base, while the number of vertices is one added to the number of sides on the base. They can neither be stacked together nor rolled. Pyramids can be divided by the shape of the base, like Prisms. Note that cones are not counted.
There is one interesting thing about convex polyhedra. A convex polyhedron is a polyhedron where any line joining any of the vertices will not stick out. According to the Euler's formula, the number of faces plus the number of vertices in minus the number of edges in such a polyhedron is always the same 2. Here is an example to try out.
Look at the pentagonal pyramid on the right. Assuming it is convex, does it follow the Euler's formula?
{\displaystyle {\begin{aligned}F&=5+1=6\\V&=5+1=6\\E&=5\times 2=10\\\therefore F+V-E&=6+6-10\\&=2\end{aligned}}}
{\displaystyle \therefore {\textrm {The\ Euler's\ formula\ holds\ in\ this\ pentagonal\ pyramid.}}}
The final thing we need to know about polyhedra is their total surface area and volume. You will only need to know about prisms for the volume. To find the total surface area of any polyhedron, add up all the faces. Look out for congruent faces and don't forget to check whether you've found all the faces by using the formulae given above – that is, when you are facing a pyramid or prism. Otherwise you can just count. As for the volume of prisms, simply find the area of the base and multiply it by the height. With cuboids, it's easy. Just multiply the length, width/breadth and height. With a cube it's the easiest: cube an edge, any edge!
Other kinds of solids[edit | edit source]
Of course, there are other kinds of solids apart from the polyhedra. They include the sphere, the cone, the cylinder and the ellipsoid. The sphere is a shape that is like a ball. All of the points on a sphere are the same distance from the centre of the sphere. The cone is a shape with a round, flat base and a pointy top. The cylinder is a shape where the top and the base are both circles, and the bases are parallel. The ellipsoid looks like a squashed sphere.
All of the shapes mentioned above can roll. Only the cylinder can be stacked together. You need not learn much about them in elementary school, apart from their names and appearances.
What is the cross-section formed after cutting up this cake?
A cross-section is the shape formed after cutting a solid. Take cutting a birthday cake as an example. When you cut the cake into half, there is a rectangle formed on each half. This is called a cross-section. Some people simply call them sections. This is also correct, but less common. When a series of identical cross-sections can be formed after cutting a solid in the same direction but a different position, uniform cross-sections are formed. This can only be done when cutting a prism or cylinder with the direction parallel to the bases.
Although it is not important to remember the kinds of cross-sections formed, it can be helpful to remember a few. Let us assume that all the shapes are put 'upright' before cutting. When cutting a prism or cylinder, cutting horizonally will result in a series of uniform cross-sections which are congruent to the base. Cutting vertically will result in a number of different rectangles, while cutting obliquely can result in a parallelogram (when four surfaces are involved in a prism), a triangle (when three surfaces are involved in a prism, such as cutting off a corner), what look like arches (in a cylinder where two surfaces are involved), or an ellipse (in a cylinder where only the lateral face is involved).
When cutting a pyramid or cone, cutting horizontally will result in shapes that are similar to the base. (When shapes are similar, it means that they have the same shape but different size. See the chapter on similarity for details.) When cutting a pyramid vertically, a triangle or a parallelogram can be formed depending on the situation. If you cut a corner or cut right across the middle (i.e. three surfaces are involved), then a triangle will be formed. Otherwise, a parallelogram will be formed. On cones, cutting vertically will result in a triangle in the same situation, but otherwise, will result in what looks like an arch. When you cut a pyramid obliquely, the shapes formed can vary a lot!
Finally, if you cut a sphere, you always get a circle. For other shapes, try imagining them yourself!
Retrieved from "https://en.wikibooks.org/w/index.php?title=Geometry_for_Elementary_School/Solids&oldid=4042738"
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28.1: Testing the Value of a Single Mean - Statistics LibreTexts
https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_Statistical_Thinking_for_the_21st_Century_(Poldrack)%2F28%253A_Comparing_Means%2F28.01%253A_Testing_the_Value_of_a_Single_Mean
The simplest question we might want to ask of a mean is whether it has a specific value. Let’s say that we want to test whether the mean BMI value in adults from the NHANES dataset is above 25, which is the lower cutoff for being overweight according to the US Centers for Disease Control. We take a sample of 200 adults in order to ask this question.
One simple way to test for this difference is using a test called the sign test, which asks whether the proportion of positive differences between the actual value and the hypothesized value is different than what we would expect by chance. To do this, we take the differences between each data point and the hypothesized mean value and compute their sign. In our sample, we see that 66.0 percent of individuals have a BMI greater than 25. We can then use a binomial test to ask whether this proportion of positive differences is greater than 0.5, using the binom.test() function in R:
## number of successes = 132, number of trials = 200, p-value = 4e-06
Here we see that the proportion of individuals with positive signs would be very surprising under the null hypothesis of
p=0.5
We can also ask this question using Student’s t-test, which you have already encountered earlier in the book. We will refer to the mean as
\bar{X}
and the hypothesized population mean as
\mu
. Then, the t test for a single mean is:
t = \frac{\bar{X} - \mu}{SEM}
where SEM (as you may remember from the chapter on sampling) is defined as:
SEM = \frac{\hat{\sigma}}{\sqrt{n}}
We can compute this for the NHANES dataset using the t.test() function in R:
## data: NHANES_adult$BMI
## t = 38, df = 4785, p-value <2e-16
This shows us that the mean BMI in the dataset (28.79) is significantly larger than the cutoff for overweight.
28.1: Testing the Value of a Single Mean is shared under a not declared license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.
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Solve the problem below by defining a variable and then writing an equation. If you find this too challenging, then use the
5
-D Process described in this lesson’s Math Notes box to help you get started. State your solution in a sentence.
Jabari is thinking of three numbers. The greatest number is twice as large as the least number. The middle number is three more than the least number. The sum of the three numbers is
75
Draw/Describe:
Rewrite the word problem as an equation or a system of equations.
Let the least number equal
x
A row with 3 columns labeled, left to right, as follows: Least number, x. Middle number, 3 + x. Greatest number, 2 x.
Fourth column added, labeled, x + (3 + x) + 2 x.
Fifth column added, labeled, 75?
Make a guess for what
x
could be. Was your answer higher or lower than
75
? Adjust your next guess accordingly.
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For other uses, see Hexagon (disambiguation).
1 Regular hexagon
2.1 A2 and G2 groups
4 Related polygons and tilings
4.1 Self-crossing hexagons
5 Hexagonal structures
6 Tesselations by hexagons
7 Hexagon inscribed in a conic section
7.1 Cyclic hexagon
8 Hexagon tangential to a conic section
9 Equilateral triangles on the sides of an arbitrary hexagon
10 Skew hexagon
10.1 Petrie polygons
11 Convex equilateral hexagon
11.1 Polyhedra with hexagons
12 Gallery of natural and artificial hexagons
Regular hexagon[edit]
A regular hexagon has Schläfli symbol {6}[1] and can also be constructed as a truncated equilateral triangle, t{3}, which alternates two types of edges.
A step-by-step animation of the construction of a regular hexagon using compass and straightedge, given by Euclid's Elements, Book IV, Proposition 15: this is possible as 6
{\displaystyle =}
2 × 3, a product of a power of two and distinct Fermat primes.
When the side length AB is given, drawing a circular arc from point A and point B gives the intersection M, the center of the circumscribed circle. Transfer the line segment AB four times on the circumscribed circle and connect the corner points.
The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals
{\displaystyle {\tfrac {2}{\sqrt {3}}}}
times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has six rotational symmetries (rotational symmetry of order six) and six reflection symmetries (six lines of symmetry), making up the dihedral group D6. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, and that the regular hexagon can be partitioned into six equilateral triangles.
Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral.
The maximal diameter (which corresponds to the long diagonal of the hexagon), D, is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the inscribed circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base), d, is twice the minimal radius or inradius, r. The maxima and minima are related by the same factor:
{\displaystyle {\frac {1}{2}}d=r=\cos(30^{\circ })R={\frac {\sqrt {3}}{2}}R={\frac {\sqrt {3}}{2}}t}
{\displaystyle d={\frac {\sqrt {3}}{2}}D.}
The area of a regular hexagon
{\displaystyle {\begin{aligned}A&={\frac {3{\sqrt {3}}}{2}}R^{2}=3Rr=2{\sqrt {3}}r^{2}\\[3pt]&={\frac {3{\sqrt {3}}}{8}}D^{2}={\frac {3}{4}}Dd={\frac {\sqrt {3}}{2}}d^{2}\\[3pt]&\approx 2.598R^{2}\approx 3.464r^{2}\\&\approx 0.6495D^{2}\approx 0.866d^{2}.\end{aligned}}}
For any regular polygon, the area can also be expressed in terms of the apothem a and the perimeter p. For the regular hexagon these are given by a = r, and p
{\displaystyle {}=6R=4r{\sqrt {3}}}
{\displaystyle {\begin{aligned}A&={\frac {ap}{2}}\\&={\frac {r\cdot 4r{\sqrt {3}}}{2}}=2r^{2}{\sqrt {3}}\\&\approx 3.464r^{2}.\end{aligned}}}
The regular hexagon fills the fraction
{\displaystyle {\tfrac {3{\sqrt {3}}}{2\pi }}\approx 0.8270}
If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumcircle between B and C, then PE + PF = PA + PB + PC + PD.
It follows from the ratio of circumradius to inradius that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long diagonal of 1.0000000 will have a distance of 0.8660254 between parallel sides.
For an arbitrary point in the plane of a regular hexagon with circumradius
{\displaystyle R}
, whose distances to the centroid of the regular hexagon and its six vertices are
{\displaystyle L}
{\displaystyle d_{i}}
respectively, we have[2]
{\displaystyle d_{1}^{2}+d_{4}^{2}=d_{2}^{2}+d_{5}^{2}=d_{3}^{2}+d_{6}^{2}=2\left(R^{2}+L^{2}\right),}
{\displaystyle d_{1}^{2}+d_{3}^{2}+d_{5}^{2}=d_{2}^{2}+d_{4}^{2}+d_{6}^{2}=3\left(R^{2}+L^{2}\right),}
{\displaystyle d_{1}^{4}+d_{3}^{4}+d_{5}^{4}=d_{2}^{4}+d_{4}^{4}+d_{6}^{4}=3\left(\left(R^{2}+L^{2}\right)^{2}+2R^{2}L^{2}\right).}
{\displaystyle d_{i}}
are the distances from the vertices of a regular hexagon to any point on its circumscircle, then [2]
{\displaystyle \left(\sum _{i=1}^{6}d_{i}^{2}\right)^{2}=4\sum _{i=1}^{6}d_{i}^{4}.}
Example hexagons by symmetry
The six lines of reflection of a regular hexagon, with Dih6 or r12 symmetry, order 12.
The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the regular form is r12 and no symmetry is labeled a1.
The regular hexagon has D6 symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D6), 2 dihedral: (D3, D2), 4 cyclic: (Z6, Z3, Z2, Z1) and the trivial (e)
These symmetries express nine distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order.[3] r12 is full symmetry, and a1 is no symmetry. p6, an isogonal hexagon constructed by three mirrors can alternate long and short edges, and d6, an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. g2 hexagons, with opposite sides parallel are also called hexagonal parallelogons.
Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation. Other hexagon shapes can tile the plane with different orientations.
A2 and G2 groups[edit]
A2 group roots
G2 group roots
The 6 roots of the simple Lie group A2, represented by a Dynkin diagram , are in a regular hexagonal pattern. The two simple roots have a 120° angle between them.
The 12 roots of the Exceptional Lie group G2, represented by a Dynkin diagram are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them.
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into 1⁄2m(m − 1) parallelograms.[4] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. This decomposition of a regular hexagon is based on a Petrie polygon projection of a cube, with 3 of 6 square faces. Other parallelogons and projective directions of the cube are dissected within rectangular cuboids.
Dissection of hexagons into three rhombs and parallelograms
Regular {6} Hexagonal parallelogons
Cube Rectangular cuboid
Related polygons and tilings[edit]
A regular hexagon has Schläfli symbol {6}. A regular hexagon is a part of the regular hexagonal tiling, {6,3}, with three hexagonal faces around each vertex.
A regular hexagon can also be created as a truncated equilateral triangle, with Schläfli symbol t{3}. Seen with two types (colors) of edges, this form only has D3 symmetry.
A truncated hexagon, t{6}, is a dodecagon, {12}, alternating two types (colors) of edges. An alternated hexagon, h{6}, is an equilateral triangle, {3}. A regular hexagon can be stellated with equilateral triangles on its edges, creating a hexagram. A regular hexagon can be dissected into six equilateral triangles by adding a center point. This pattern repeats within the regular triangular tiling.
A regular hexagon can be extended into a regular dodecagon by adding alternating squares and equilateral triangles around it. This pattern repeats within the rhombitrihexagonal tiling.
Hypertruncated triangles
Star figure 2{3}
A concave hexagon
A self-intersecting hexagon (star polygon)
Central {6} in {12}
A skew hexagon, within cube
Dissected {6}
Self-crossing hexagons[edit]
There are six self-crossing hexagons with the vertex arrangement of the regular hexagon:
Self-intersecting hexagons with regular vertices
Center-flip
Hexagonal structures[edit]
Giant's Causeway closeup
From bees' honeycombs to the Giant's Causeway, hexagonal patterns are prevalent in nature due to their efficiency. In a hexagonal grid each line is as short as it can possibly be if a large area is to be filled with the fewest hexagons. This means that honeycombs require less wax to construct and gain much strength under compression.
Irregular hexagons with parallel opposite edges are called parallelogons and can also tile the plane by translation. In three dimensions, hexagonal prisms with parallel opposite faces are called parallelohedrons and these can tessellate 3-space by translation.
Hexagonal prism tessellations
Parallelogonal
Tesselations by hexagons[edit]
Main article: Hexagonal tiling
In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the Conway criterion will tile the plane.
Hexagon inscribed in a conic section[edit]
Pascal's theorem (also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration.
Cyclic hexagon[edit]
The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point.
If the successive sides of a cyclic hexagon are a, b, c, d, e, f, then the three main diagonals intersect in a single point if and only if ace = bdf.[5]
If, for each side of a cyclic hexagon, the adjacent sides are extended to their intersection, forming a triangle exterior to the given side, then the segments connecting the circumcenters of opposite triangles are concurrent.[6]
If a hexagon has vertices on the circumcircle of an acute triangle at the six points (including three triangle vertices) where the extended altitudes of the triangle meet the circumcircle, then the area of the hexagon is twice the area of the triangle.[7]: p. 179
Hexagon tangential to a conic section[edit]
Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then Brianchon's theorem states that the three main diagonals AD, BE, and CF intersect at a single point.
In a hexagon that is tangential to a circle and that has consecutive sides a, b, c, d, e, and f,[8]
{\displaystyle a+c+e=b+d+f.}
Equilateral triangles on the sides of an arbitrary hexagon[edit]
Equilateral triangles on the sides of an arbitrary hexagon
If an equilateral triangle is constructed externally on each side of any hexagon, then the midpoints of the segments connecting the centroids of opposite triangles form another equilateral triangle.[9]: Thm. 1
Skew hexagon[edit]
A regular skew hexagon seen as edges (black) of a triangular antiprism, symmetry D3d, [2+,6], (2*3), order 12.
A skew hexagon is a skew polygon with six vertices and edges but not existing on the same plane. The interior of such an hexagon is not generally defined. A skew zig-zag hexagon has vertices alternating between two parallel planes.
A regular skew hexagon is vertex-transitive with equal edge lengths. In three dimensions it will be a zig-zag skew hexagon and can be seen in the vertices and side edges of a triangular antiprism with the same D3d, [2+,6] symmetry, order 12.
The cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons.
Skew hexagons on 3-fold axes
The regular skew hexagon is the Petrie polygon for these higher dimensional regular, uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections:
Convex equilateral hexagon[edit]
A principal diagonal of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex equilateral hexagon (one with all sides equal) with common side a, there exists[10]: p.184, #286.3 a principal diagonal d1 such that
{\displaystyle {\frac {d_{1}}{a}}\leq 2}
{\displaystyle {\frac {d_{2}}{a}}>{\sqrt {3}}.}
Polyhedra with hexagons[edit]
There is no Platonic solid made of only regular hexagons, because the hexagons tessellate, not allowing the result to "fold up". The Archimedean solids with some hexagonal faces are the truncated tetrahedron, truncated octahedron, truncated icosahedron (of soccer ball and fullerene fame), truncated cuboctahedron and the truncated icosidodecahedron. These hexagons can be considered truncated triangles, with Coxeter diagrams of the form and .
Hexagons in Archimedean solids
There are other symmetry polyhedra with stretched or flattened hexagons, like these Goldberg polyhedron G(2,0):
Hexagons in Goldberg polyhedra
Chamfered tetrahedron
There are also 9 Johnson solids with regular hexagons:
Johnson solids with hexagons
Prismoids with hexagons
Tilings with regular hexagons
Gallery of natural and artificial hexagons[edit]
Assembled E-ELT mirror segments
Saturn's hexagon, a hexagonal cloud pattern around the north pole of the planet
Benzene, the simplest aromatic compound with hexagonal shape.
Hexagonal order of bubbles in a foam.
Crystal structure of a molecular hexagon composed of hexagonal aromatic rings.
Naturally formed basalt columns from Giant's Causeway in Northern Ireland; large masses must cool slowly to form a polygonal fracture pattern
Metropolitan France has a vaguely hexagonal shape. In French, l'Hexagone refers to the European mainland of France.
Hexagonal Hanksite crystal, one of many hexagonal crystal system minerals
Hexagonal barn
The Hexagon, a hexagonal theatre in Reading, Berkshire
Władysław Gliński's hexagonal chess
Pavilion in the Taiwan Botanical Gardens
24-cell: a four-dimensional figure which, like the hexagon, has orthoplex facets, is self-dual and tessellates Euclidean space
Hexagonal tiling: a regular tiling of hexagons in a plane
Hexagram: six-sided star within a regular hexagon
Unicursal hexagram: single path, six-sided star, within a hexagon
Havannah: abstract board game played on a six-sided hexagonal grid
^ Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595, archived from the original on 2016-01-02, retrieved 2015-11-06 .
^ Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40.
^ Dergiades, Nikolaos (2014). "Dao's theorem on six circumcenters associated with a cyclic hexagon". Forum Geometricorum. 14: 243–246. Archived from the original on 2014-12-05. Retrieved 2014-11-17.
^ Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).
^ Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", [1] Archived 2012-05-11 at the Wayback Machine, Accessed 2012-04-17.
^ Dao Thanh Oai (2015). "Equilateral triangles and Kiepert perspectors in complex numbers". Forum Geometricorum. 15: 105–114. Archived from the original on 2015-07-05. Retrieved 2015-04-12.
^ Inequalities proposed in "Crux Mathematicorum", [2] Archived 2017-08-30 at the Wayback Machine.
Look up hexagon in Wiktionary, the free dictionary.
Hexagons are Bestagons an animated youtube video about hexagons
Retrieved from "https://en.wikipedia.org/w/index.php?title=Hexagon&oldid=1077983848"
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Levi used the box plot below to say, “Half of the class walked more than
\mathit{30}
laps at the walk-a-thon.” Levi also knows that his class has more than
20
Do you agree with him? Explain your reasoning. If you do not agree with him, what statement could he say about those who walked more than
30
laps?
If half of the class walked more than thirty laps, then the median would be at
30
Levi wants to describe the portion of students who walked between
20
30
laps (the box). What statement could he say?
The portion of students who walked between
20
30
laps is the interquartile range.
What percentage of the students does this represent?
How could you alter a single data point and not change the graph? How could you change one data value and only move the median to the right?
You could change one of the data points between
30
and the maximum, but keep in the same range of values.
You could add data greater than the current median.
Can you determine how many students are in Levi’s class? Explain why or why not.
No, you cannot determine this information from the data in the box plot.
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Quiz | Chemical Thermodynamics Gibbs Energy Change and Reaction Quotient
Which of the following statements is always true when a reversible chemical reaction has attained equilibrium?
The Gibbs free energy of the system reaches a minimum
All reactants have been converted to products
The forward reaction dominates over the reverse reaction
The reaction quotient equal to 0
Concepts related to this Question
Gibbs Energy Change and Spontaneity of Reactions Gibbs Energy Change and Reaction Quotient
What is the relationship between the equilibrium constant K and the temperature T?
ln K and T are proportional
ln K and -T are proportional
K and ln T are proportional
ln K and 1/T are proportional
\frac{∆{{\mathrm{H}}^{0}}_{\mathrm{rxn}}}{\mathrm{RT}}
\frac{∆{{\mathrm{S}}^{0}}_{\mathrm{rxn}}}{\mathrm{R}}
⇒ a plot of ln K versus
\frac{1}{\mathrm{T}}
is linear (ln K and
\frac{1}{\mathrm{T}}
are proportional)
Van’t Hoff and Clapeyron-Clausius Equations
When does the entropy usually increase?
A molecule is broken into 2 or more molecules
A solid turns into a liquid
A liquid turns into a gas
Entropy S is a measure of the amount of disorder in a system ⇒ the greater the disorder the higher the entropy.
The disorder of a system is greatest when a molecule is broken into 2 or more molecules or when a solid turns into a liquid or when a liquid turns into a gas
Law of Thermodynamics Entropy Changes for Reactions
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ICAR-RCER, Research Centre on Makhana, Darbhanga, India.
The Euryale ferox (Salisb.) or gorgon or makhana is one of the most important non cereal food crops of commerce from wetland ecosystem in India. Flower is cleistogamous (CLS) and predominantly self-pollinated. The variations in floral characters were observed in 10 types of germplasm viz., Manipur-2, Manipur-4, Manipur-7, Manipur-9, Selection-17, Selection-23, Selection-27, Selection-28, Superior Selection-1 and cv. Swarna Vaidehi. In our present study, the number of flowers varied from 8.33 (Manipur-9) to 16.33 (Superior Selection-1) per plant and flowering period was about 40 days. However, peak pollination was observed between 60 - 70 days after transplanting. The weather of August and September were ideal for pollination and fruit set. The temperature and humidity of this period were 29°C - 31°C and 79% - 81%, respectively. Besides cleistogamy (CL), chasmogamy (CH) is also observed after July flowering in Euryale, when crop gets matured, water level considerably goes down, and flowers are generally opened in air. There were rare chances for cross pollination by insect. In later stage, chasmogamous (CHS) flower increases up to 22.50% in October. Seed formation of the CHS flower was very less and seed number varies from July (11.25/fruit) to September (28.33/fruit). Artificial hybridization can be performed in CHS flower. The complete flower development was noticed within 72 - 96 hrs from floral initiation. Therefore, getting of CHS flower outside water is very less. There were strong correlations between number of embryos (r = 0.762), ovary area (longitudinal) (r = 0.681) with the yield of the Euryale plant.
Jana, B. (2018) Flower Characteristics and Pollination Behavior of Euryale ferox (Salisb.). American Journal of Plant Sciences, 9, 722-731. doi: 10.4236/ajps.2018.94057.
\text{Pollination}\left(\%\right)=\frac{\text{Number of Pollinated Embryos}}{\text{Total Number of Embryos}}\times 100
\text{CLSFlower}\left(\%\right)=\frac{\text{Number of Cleistogamous Flowers}}{\text{Total Numberof Flowers}}\times 100
\text{CHSFlower}\left(\%\right)=\frac{\text{Number of Chasmogamous Flowers}}{\text{Total Number of Flowers}}\times 100
[1] Cronquist, A. (1981) An Integrated System of Classification of Flowering Plants. Columbia University Press, New York, 111.
[2] Processed Products: Makhana Phool (2010).
http://www.miltopexports.com/makhana_phool.htm
[3] Mabberley, D.J. (1987) The Plant-Book. Cambridge University Press, Cambridge.
[4] JIT Report (2014) Mission for Integrated Development of Horticulture in Bihar, 31st May to 4th June, 2014. Ministry of Agriculture, Department of Agriculture & Cooperation Krishi Bhawan, New Delhi, 7.
[5] Zhuang, X. (2011) Euryale ferox. The IUCN Red List of Threatened Species. Version 2014.3.
[6] Fuller, D.Q., Qin, L., Zheng, Y., Zhao, Z., Chen, X., Hosoya, LA., et al. (2009) The Domestication Process and Domestication Rate in Rice: Spikelet Bases from the Lower Yangtze. Science, 323, 1607-1610.
[7] Goren-Inbarand, N., Melamed, Y., Zohar, I., Akhilesh, K. and Pappu, S. (2014) Beneath Still Waters—Multistage Aquatic Exploitation of Euryale ferox (Salisb.) during the Acheulian. Internet Archaeology, 37.
[8] IUCN (2011) IUCN Red List of Threatened Species (Ver. 2011.2).
[9] Wu, Z.Y., Raven, P.H. and Hong, D.Y., Eds. (2003) Flora of China. Science Press and Missouri Botanical Garden Press, Beijing and St. Louis.
[10] USDA, ARS, National Genetic Resources Program (2010) Germplasm Resources Information Network—(GRIN) [Online Database]. Beltsville, Maryland.
[11] Das, S., Der, P., Roychoudhury, U., Maulik, N. and Das, D.K. (2006) The Effect of Euryale ferox (Makhana), an Herb of Aquatic Origin, on Myocardial Ischemic Reperfusion Injury. Molecular and Cellular Biochemistry, 289, 55-63.
[12] Li, M.H., Yang, X.Q., Wan, Z.J., Yang, Y.B., Li, F. and Ding, Z.-T. (2007) Chemical Constituents of the Seeds of Euryale ferox. Chinese Journal of Natural Medicines, 5, 24-26.
[13] Duke, J.A. (2010) Phytochemical and Ethnobotanical Databases.
[14] Löhne, C., Borsch, T. and Wiersema, J.H. (2007) Phylogenetic Analysis of Nymphaeales Using Fast-Evolving and Noncoding Chloroplast Markers. Botanical Journal of the Linnean Society, 154, 141-163.
[15] Borsch, T., Löhne, C. and Wiersema, J. (2008) Phylogeny and Evolutionary Patterns in Nymphaeales: Integrating Genes, Genomes and Morphology. Taxon, 57, 1052-1081.
[16] Dkhar, J., Kumaria, S., Rama Rao, S. and Tandon, P. (2012) Sequence Characteristics and Phylogenetic Implications of the nrDNA Internal Transcribed Spacers (ITS) in the Genus Nymphaea with Focus on Some Indian Representatives. Plant Systematics and Evolution, 298, 93-108.
[17] Kumar, L., Gupta, V.K., Jha, B.K., Singh, I.S., Bhatt, B.P. and Singh, A.K. (2011) Status of Makhana (Euryale ferox Salisb.) Cultivation in India-Tech Bull. No. R-32/PAT-21 ICAR-RCER Patna, 31.
[18] UPOV (2006) Color Names of RHS Colour Chart: International Union for the Production of New Varieties of Plants. UPOV, Geneva.
[19] Kumari, A., Singh, I.S., Lokendra, K., Amit, K., Ramesh, K. and Gupta, V.K. (2014) Morphological Characteristics of Makhana Germplasm of Manipur and Darbhanga Conditions. Journal of AgriSearch, 1, 157-170.
[20] Houtte, L. (1853) Flore des serres et des jardin de l’Europe. Vol. 8, 82.
[21] Okada, Y. (1930) Study of Euryale ferox Salisb. VI. Cleistogamous versus Chasmogamous Flower. The Botanical Magazine, Tokyo, 44, 369-373.
[22] Moseley, M.F. and Williamson, P.S. (1984) The Vasculature of the Flower of Euryale ferox. American Journal of Botany, 71, 40-41.
[23] Kadono, Y. and Schneider, E.L. (1987) The Life History of Euryale ferox Salisb. In Southwestern Japan with Special Reference to Reproductive Ecology. Plant Species Biology, 2, 109-115.
[24] Mandal, R.N., Saha, G.S. and Sarangi, N. (2010) Harvesting and Processing of Makhana (Euryale ferox Salisb.)—A Unique Assemblage of Traditional Knowledge. Indian Journal of Traditional Knowledge, 9, 684-688.
[25] Kumar, L., Choudhary, A.K., Bhatt, B.P. and Singh, K.P. (2015) Genetic Divergence in Makhana (Euryale ferox Salisb.). Indian Journal of Horticulture, 72, 365-369.
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RuneScape:Grand Exchange Market Watch/Adjustments (29 September 2012) - The RuneScape Wiki
RuneScape:Grand Exchange Market Watch/Adjustments (29 September 2012)
Adjustment date: 29 September 2012
Added item(s): Fish mask
Purple partyhat 83,100,000 895,254,988
Yellow partyhat 96,300,000 948,567,763
Green h'ween mask 10,600,000 90,439,249
Fish mask – – 4,555,462 Added item
{\displaystyle {div}_{\text{old}}=15.0000}
{\displaystyle {div}_{\text{new}}={div}_{\text{old}}\times {\frac {\sum \left({\frac {p}{q}}\right)_{\text{new}}}{\sum \left({\frac {p}{q}}\right)_{\text{old}}}}}
{\displaystyle {\begin{aligned}\sum \left({\frac {p}{q}}\right)_{\text{old}}&={\text{sum of ratios prior to change}}\\&={\frac {2,146,908,554}{688,800,000}}+{\frac {2,147,483,647}{340,100,000}}+{\frac {1,124,859,082}{114,800,000}}+\dots +{\frac {261,000,577}{31,100,000}}\\&=181.52107486{\text{ (up to 8 d.p.)}}\end{aligned}}}
{\displaystyle {\begin{aligned}\sum \left({\frac {p}{q}}\right)_{\text{new}}&={\text{sum of ratios prior to change}}-{\text{sum of removed ratios}}+{\text{sum of added ratios}}\\&=\sum \left({\frac {p}{q}}\right)_{\text{old}}-{\text{sum of removed ratios}}+{\text{number of added items}}\\&=181.52107486-0+1\\&=182.52107486{\text{ (up to 8 d.p.)}}\end{aligned}}}
{\displaystyle {\begin{aligned}{div}_{\text{new}}&={div}_{\text{old}}\times {\frac {\sum \left({\frac {p}{q}}\right)_{\text{new}}}{\sum \left({\frac {p}{q}}\right)_{\text{old}}}}\\&=15.0000\times {\frac {182.52107486}{181.52107486}}\\&=15.0826{\text{ (4 d.p.)}}\end{aligned}}}
Retrieved from ‘https://runescape.wiki/w/RuneScape:Grand_Exchange_Market_Watch/Adjustments_(29_September_2012)?oldid=35229560’
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(one hundred [and] first)
101 (one hundred [and] one) is the natural number following 100 and preceding 102.
It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one". As an ordinal number, 101st (one hundred [and] first), rather than 101th, is the correct form.
Look up one hundred and one or one hundred one in Wiktionary, the free dictionary.
the 26th prime number, and the smallest above 100.
a palindromic number in base 10, and so a palindromic prime.
a Chen prime since 103 is also prime, with which it makes a twin prime pair.
a sexy prime since 107 and 113 are also prime, with which it makes a sexy prime triplet.
a unique prime, because the period length of its reciprocal is unique among primes.
an Eisenstein prime with no imaginary part and real part of the form
{\displaystyle 3n-1}
the fifth alternating factorial.[1]
a centered decagonal number.[2]
the only existing prime with alternating 1s and 0s in base 10 and the largest known prime of the form 10n + 1.[3]
Given 101, the Mertens function returns 0.[4] It is the second prime having this property.[5]
For a 3-digit number in base 10, this number has a relatively simple divisibility test. The candidate number is split into groups of four, starting with the rightmost four, and added up to produce a 4-digit number. If this 4-digit number is of the form 1000a + 100b + 10a + b (where a and b are integers from 0 to 9), such as 3232 or 9797, or of the form 100b + b, such as 707 and 808, then the number is divisible by 101.[6]
On the seven-segment display of a calculator, 101 is both a strobogrammatic prime and a dihedral prime.[7]
In physics and chemistry, it is the atomic number of mendelevium, an actinide.
In astronomy it is the Messier designation given to the Pinwheel Galaxy in Ursa Major.
According to Books in Print, more books are now published with a title that begins with '101' than '100'. They usually describe or discuss a list of items, such as 101 Ways to... or 101 Questions and Answers About... . This marketing tool is used to imply that the customer is given a little extra information beyond books that include only 100 items. Some books have taken this marketing scheme even further with titles that begin with '102', '103', or '1001'. The number is used in this context as a slang term when referring to "a 101 document" what is usually referred to as a statistical survey or overview of some topic.
Room 101 is a torture chamber in the novel Nineteen Eighty-Four by George Orwell.
Creative Writing 101 by Raymond Carver, "A writer's values and craft. This was what the man (John Gardner) taught and what he stood for, and this is what I've kept by me in the years since that brief but all important time."
See also: 101 (topic)
In American university course numbering systems, the number 101 is often used for an introductory course at a beginner's level in a department's subject area. This common numbering system was designed to make transfer between colleges easier. In theory, any numbered course in one academic institution should bring a student to the same standard as a similarly numbered course at other institutions.[8] The term was first introduced by the University of Buffalo in 1929.[9]
Main article: 101 (disambiguation)
Charter of the French Language, Bill 101.
Taipei 101, the tallest skyscraper in the world from 2004 to 2010.
101st kilometre, a condition of release from the Gulag in the Soviet Union.
Roi Et Province, a province in Thailand. The name is literally 101 in Thai language.
An HTTP status code indicating that the server is switching protocols as requested by the client to do so.
For a new checking account in the US, the number of the first check.
A term used to define the number of keys on a standard 101 key computer keyboard[10]
101 is the main Police Emergency Number in Belgium.
101 is the Single Non-Emergency Number (SNEN) in some parts of the UK, a telephone number used to call emergency services that are urgent but not emergencies. 101 is now available across all areas of England and Wales.[11][12]
iCar 101 is a roadable aircraft design concept.[13]
The Zastava 101 is a compact car by the former Yugoslav automaker.
Highways numbered 101, the longest and most well-known of which are BR-101 and U.S. Route 101.
101, a Depeche Mode album
Vault 101 is the starting area of Fallout 3
Lebanese Red Cross EMS Center at Spears is coded 101.
101 is the identifying number of several infantry units in various militaries across the world, such as the American and Israeli paratrooper brigades.
101 was the tail number of a Polish Air Force Tu-154, which crashed on 10 April 2010 whilst on its final approach to Smolensk North Airport killing all aboard including president Lech Kaczyński and his wife. See 2010 Polish Air Force Tu-154 crash.
"l'ordre des Piliers du 101" is an important student association in Belgium since 1977.
Korean reality girl group survival show Produce 101
TV show Zoey 101
In Hinduism, 101 is a lucky number.
The house number of Samson en Gert in the eponymous Belgian kids' television show.
^ "Sloane's A005165 : Alternating factorials". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 27 May 2016.
^ "Sloane's A062786 : Centered 10-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 27 May 2016.
^ Prime Curios! 101
^ "Sloane's A028442 : Numbers n such that Mertens' function is zero". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 27 May 2016.
^ "Sloane's A100669 : Zeros of the Mertens function that are also prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 29 May 2016.
^ Renault, Marc (November 2006), "Stupid Divisibility Tricks 101 Ways to Stupefy Your Friends", Math Horizons, 14 (2): 18–21, 42, doi:10.1080/10724117.2006.11974676, JSTOR 25678653, S2CID 125269086
^ "Sloane's A134996 : Dihedral calculator primes: p, p upside down, p in a mirror, p upside-down-and-in-a-mirror are all primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 17 December 2020.
^ Forest, J.J.F. (2002) Higher education in the United States: an encyclopedia p.73. ABC-CLIO. ISBN 1-57607-248-7. Retrieved October 2011
^ Engber, Daniel (6 September 2006). "101 101". Slate. Retrieved 9 May 2017.
^ Kozierok, Charles. "101-Key "Enhanced" Keyboard Layout". The PC Guide. Retrieved 29 May 2019.
^ "Report a crime or antisocial behaviour - GOV.UK". www.direct.gov.uk. Retrieved 4 April 2018.
^ Welcome to 101, Home Office, retrieved 5 April 2009
^ iCar 101 - The ultimate roadable aircraft, retrieved 6 August 2010
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Tensor_density Knowpia
In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another (see tensor field), except that it is additionally multiplied or weighted by a power W of the Jacobian determinant of the coordinate transition function or its absolute value. A distinction is made among (authentic) tensor densities, pseudotensor densities, even tensor densities and odd tensor densities. Sometimes tensor densities with a negative weight W are called tensor capacity.[1][2][3] A tensor density can also be regarded as a section of the tensor product of a tensor bundle with a density bundle.
In physics and related fields, it is often useful to work with the components of an algebraic object rather than the object itself. An example would be decomposing a vector into a sum of basis vectors weighted by some coefficients such as
{\displaystyle {\vec {v}}=c_{1}{\vec {e}}_{1}+c_{2}{\vec {e}}_{2}+c_{3}{\vec {e}}_{3}}
{\displaystyle {\vec {v}}}
is a vector in 3-dimensional Euclidean space,
{\displaystyle c_{i}\in \mathbb {R} ^{n}{\text{ and }}{\vec {e}}_{i}}
are the usual standard basis vectors in Euclidean space. This is usually necessary for computational purposes, and can often be insightful when algebraic objects represent complex abstractions but their components have concrete interpretations. However, with this identification, one has to be careful to track changes of the underlying basis in which the quantity is expanded; it may in the course of a computation become expedient to change the basis while the vector
{\displaystyle {\vec {v}}}
remains fixed in physical space. More generally, if an algebraic object represents a geometric object, but is expressed in terms of a particular basis, then it is necessary to, when the basis is changed, also change the representation. Physicists will often call this representation of a geometric object a tensor if it transforms under a sequence of linear maps given a linear change of basis (although confusingly others call the underlying geometric object which hasn't changed under the coordinate transformation a "tensor", a convention this article strictly avoids). In general there are representations which transform in arbitrary ways depending on how the geometric invariant is reconstructed from the representation. In certain special cases it is convenient to use representations which transform almost like tensors, but with an additional, nonlinear factor in the transformation. A prototypical example is a matrix representing the cross product (area of spanned parallelogram) on
{\displaystyle \mathbb {R} ^{2}.}
The representation is given by in the standard basis by
{\displaystyle {\vec {u}}\times {\vec {v}}={\begin{bmatrix}u_{1}&u_{2}\end{bmatrix}}{\begin{bmatrix}0&1\\-1&0\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}=u_{1}v_{2}-u_{2}v_{1}}
If we now try to express this same expression in a basis other than the standard basis, then the components of the vectors will change, say according to
{\textstyle {\begin{bmatrix}u'_{1}&u'_{2}\end{bmatrix}}^{\textsf {T}}=A{\begin{bmatrix}u_{1}&u_{2}\end{bmatrix}}^{\textsf {T}}}
{\displaystyle A}
is some 2 by 2 matrix of real numbers. Given that the area of the spanned parallelogram is a geometric invariant, it cannot have changed under the change of basis, and so the new representation of this matrix must be:
{\displaystyle \left(A^{-1}\right)^{\textsf {T}}{\begin{bmatrix}0&1\\-1&0\end{bmatrix}}A^{-1}}
which, when expanded is just the original expression but multiplied by the determinant of
{\displaystyle A^{-1},}
{\textstyle {\frac {1}{\det A}}.}
In fact this representation could be thought of as a two index tensor transformation, but instead, it is computationally easier to think of the tensor transformation rule as multiplication by
{\textstyle {\frac {1}{\det A}},}
rather than as 2 matrix multiplications (In fact in higher dimensions, the natural extension of this is
{\displaystyle n,n\times n}
matrix multiplications, which for large
{\displaystyle n}
is completely infeasible). Objects which transform in this way are called tensor densities because they arise naturally when considering problems regarding areas and volumes, and so are frequently used in integration.
In contrast to the meaning used in this article, in general relativity "pseudotensor" sometimes means an object that does not transform like a tensor or relative tensor of any weight.
Tensor and pseudotensor densitiesEdit
For example, a mixed rank-two (authentic) tensor density of weight
{\displaystyle W}
transforms as:[5][6]
{\displaystyle {\mathfrak {T}}_{\beta }^{\alpha }=\left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,,}
((authentic) tensor density of (integer) weight W)
{\displaystyle {\bar {\mathfrak {T}}}}
is the rank-two tensor density in the
{\displaystyle {\bar {x}}}
{\displaystyle {\mathfrak {T}}}
is the transformed tensor density in the
{\displaystyle {x}}
coordinate system; and we use the Jacobian determinant. Because the determinant can be negative, which it is for an orientation-reversing coordinate transformation, this formula is applicable only when
{\displaystyle W}
is an integer. (However, see even and odd tensor densities below.)
We say that a tensor density is a pseudotensor density when there is an additional sign flip under an orientation-reversing coordinate transformation. A mixed rank-two pseudotensor density of weight
{\displaystyle W}
{\displaystyle {\mathfrak {T}}_{\beta }^{\alpha }=\operatorname {sgn} \left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)\left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,,}
(pseudotensor density of (integer) weight W)
Even and odd tensor densitiesEdit
The transformations for even and odd tensor densities have the benefit of being well defined even when
{\displaystyle W}
is not an integer. Thus one can speak of, say, an odd tensor density of weight +2 or an even tensor density of weight −1/2.
{\displaystyle W}
is an even integer the above formula for an (authentic) tensor density can be rewritten as
{\displaystyle {\mathfrak {T}}_{\beta }^{\alpha }=\left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,.}
(even tensor density of weight W)
{\displaystyle W}
is an odd integer the formula for an (authentic) tensor density can be rewritten as
{\displaystyle {\mathfrak {T}}_{\beta }^{\alpha }=\operatorname {sgn} \left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)\left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,.}
(odd tensor density of weight W)
Weights of zero and oneEdit
A linear combination (also known as a weighted sum) of tensor densities of the same type and weight
{\displaystyle W}
is again a tensor density of that type and weight.
A product of two tensor densities of any types and with weights
{\displaystyle W_{1}}
{\displaystyle W_{1}}
is a tensor density of weight
{\displaystyle W_{1}+W_{2}.}
The contraction of indices on a tensor density with weight
{\displaystyle W}
again yields a tensor density of weight
{\displaystyle W.}
Matrix inversion and matrix determinant of tensor densitiesEdit
{\displaystyle {\mathfrak {T}}_{\alpha \beta }}
is a non-singular matrix and a rank-two tensor density of weight
{\displaystyle W}
with covariant indices then its matrix inverse will be a rank-two tensor density of weight −
{\displaystyle W}
with contravariant indices. Similar statements apply when the two indices are contravariant or are mixed covariant and contravariant.
{\displaystyle {\mathfrak {T}}_{\alpha \beta }}
is a rank-two tensor density of weight
{\displaystyle W}
with covariant indices then the matrix determinant
{\displaystyle \det {\mathfrak {T}}_{\alpha \beta }}
will have weight
{\displaystyle NW+2,}
{\displaystyle N}
is the number of space-time dimensions. If
{\displaystyle {\mathfrak {T}}^{\alpha \beta }}
{\displaystyle W}
with contravariant indices then the matrix determinant
{\displaystyle \det {\mathfrak {T}}^{\alpha \beta }}
{\displaystyle NW-2.}
The matrix determinant
{\displaystyle \det {\mathfrak {T}}_{~\beta }^{\alpha }}
{\displaystyle NW.}
Relation of Jacobian determinant and metric tensorEdit
Any non-singular ordinary tensor
{\displaystyle T_{\mu \nu }}
{\displaystyle T_{\mu \nu }={\frac {\partial {\bar {x}}^{\kappa }}{\partial {x}^{\mu }}}{\bar {T}}_{\kappa \lambda }{\frac {\partial {\bar {x}}^{\lambda }}{\partial {x}^{\nu }}}\,,}
where the right-hand side can be viewed as the product of three matrices. Taking the determinant of both sides of the equation (using that the determinant of a matrix product is the product of the determinants), dividing both sides by
{\displaystyle \det \left({\bar {T}}_{\kappa \lambda }\right),}
and taking their square root gives
{\displaystyle \left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ={\sqrt {\frac {\det({T}_{\mu \nu })}{\det \left({\bar {T}}_{\kappa \lambda }\right)}}}\,.}
When the tensor
{\displaystyle T}
is the metric tensor,
{\displaystyle {g}_{\kappa \lambda },}
{\displaystyle {\bar {x}}^{\iota }}
is a locally inertial coordinate system where
{\displaystyle {\bar {g}}_{\kappa \lambda }=\eta _{\kappa \lambda }=}
diag(−1,+1,+1,+1), the Minkowski metric, then
{\displaystyle \det \left({\bar {g}}_{\kappa \lambda }\right)=\det(\eta _{\kappa \lambda })=}
−1 and so
{\displaystyle \left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ={\sqrt {-{g}}}\,,}
{\displaystyle {g}=\det \left({g}_{\mu \nu }\right)}
is the determinant of the metric tensor
{\displaystyle {g}_{\mu \nu }.}
Use of metric tensor to manipulate tensor densitiesEdit
Consequently, an even tensor density,
{\displaystyle {\mathfrak {T}}_{\nu \dots }^{\mu \dots },}
of weight W, can be written in the form
{\displaystyle {\mathfrak {T}}_{\nu \dots }^{\mu \dots }={\sqrt {-g}}\;^{W}T_{\nu \dots }^{\mu \dots }\,,}
{\displaystyle T_{\nu \dots }^{\mu \dots }\,}
is an ordinary tensor. In a locally inertial coordinate system, where
{\displaystyle g_{\kappa \lambda }=\eta _{\kappa \lambda },}
it will be the case that
{\displaystyle {\mathfrak {T}}_{\nu \dots }^{\mu \dots }}
{\displaystyle T_{\nu \dots }^{\mu \dots }\,}
will be represented with the same numbers.
{\displaystyle {\mathfrak {T}}_{\nu \dots ;\alpha }^{\mu \dots }={\sqrt {-g}}\;^{W}T_{\nu \dots ;\alpha }^{\mu \dots }={\sqrt {-g}}\;^{W}\left({\sqrt {-g}}\;^{-W}{\mathfrak {T}}_{\nu \dots }^{\mu \dots }\right)_{;\alpha }\,.}
{\displaystyle -W\,\Gamma _{~\delta \alpha }^{\delta }\,{\mathfrak {T}}_{\nu \dots }^{\mu \dots }}
to the expression that would be appropriate for the covariant derivative of an ordinary tensor.
{\displaystyle \left({\mathfrak {T}}_{\nu \dots }^{\mu \dots }{\mathfrak {S}}_{\tau \dots }^{\sigma \dots }\right)_{;\alpha }=\left({\mathfrak {T}}_{\nu \dots ;\alpha }^{\mu \dots }\right){\mathfrak {S}}_{\tau \dots }^{\sigma \dots }+{\mathfrak {T}}_{\nu \dots }^{\mu \dots }\left({\mathfrak {S}}_{\tau \dots ;\alpha }^{\sigma \dots }\right)\,,}
where, for the metric connection, the covariant derivative of any function of
{\displaystyle g_{\kappa \lambda }}
is always zero,
{\displaystyle {\begin{aligned}g_{\kappa \lambda ;\alpha }&=0\\\left({\sqrt {-g}}\;^{W}\right)_{;\alpha }&=\left({\sqrt {-g}}\;^{W}\right)_{,\alpha }-W\Gamma _{~\delta \alpha }^{\delta }{\sqrt {-g}}\;^{W}={\frac {W}{2}}g^{\kappa \lambda }g_{\kappa \lambda ,\alpha }{\sqrt {-g}}\;^{W}-W\Gamma _{~\delta \alpha }^{\delta }{\sqrt {-g}}\;^{W}=0\,.\end{aligned}}}
{\displaystyle {\sqrt {-g}}}
is a scalar density. By the convention of this article it has a weight of +1.
The density of electric current
{\displaystyle {\mathfrak {J}}^{\mu }}
{\displaystyle {\mathfrak {J}}^{2}}
is the amount of electric charge crossing the 3-volume element
{\displaystyle dx^{3}\,dx^{4}\,dx^{1}}
divided by that element — do not use the metric in this calculation) is a contravariant vector density of weight +1. It is often written as
{\displaystyle {\mathfrak {J}}^{\mu }=J^{\mu }{\sqrt {-g}}}
{\displaystyle {\mathfrak {J}}^{\mu }=\varepsilon ^{\mu \alpha \beta \gamma }{\mathcal {J}}_{\alpha \beta \gamma }/3!,}
{\displaystyle J^{\mu }\,}
and the differential form
{\displaystyle {\mathcal {J}}_{\alpha \beta \gamma }}
are absolute tensors, and where
{\displaystyle \varepsilon ^{\mu \alpha \beta \gamma }}
is the Levi-Civita symbol; see below.
The density of Lorentz force
{\displaystyle {\mathfrak {f}}_{\mu }}
(that is, the linear momentum transferred from the electromagnetic field to matter within a 4-volume element
{\displaystyle dx^{1}\,dx^{2}\,dx^{3}\,dx^{4}}
divided by that element — do not use the metric in this calculation) is a covariant vector density of weight +1.
In N-dimensional space-time, the Levi-Civita symbol may be regarded as either a rank-N covariant (odd) authentic tensor density of weight −1 (εα1⋯αN) or a rank-N contravariant (odd) authentic tensor density of weight +1 (εα1⋯αN). Notice that the Levi-Civita symbol (so regarded) does not obey the usual convention for raising or lowering of indices with the metric tensor. That is, it is true that
{\displaystyle \varepsilon ^{\alpha \beta \gamma \delta }\,g_{\alpha \kappa }\,g_{\beta \lambda }\,g_{\gamma \mu }g_{\delta \nu }\,=\,\varepsilon _{\kappa \lambda \mu \nu }\,g\,,}
but in general relativity, where
{\displaystyle g}
is always negative, this is never equal to
{\displaystyle \varepsilon _{\kappa \lambda \mu \nu }.}
{\displaystyle g=\det \left(g_{\rho \sigma }\right)={\frac {1}{4!}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\alpha \kappa }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }\,,}
Pseudotensor – Type of physical quantity
^ Weinreich, Gabriel (July 6, 1998). Geometrical Vectors. pp. 112, 115. ISBN 978-0226890487.
^ Papastavridis, John G. (Dec 18, 1998). Tensor Calculus and Analytical Dynamics. CRC Press. ISBN 978-0849385148.
^ Ruiz-Tolosa, Castillo, Juan R., Enrique (30 Mar 2006). From Vectors to Tensors. Springer Science & Business Media. ISBN 978-3540228875.
^ E.g. Weinberg 1972 pp 98. The chosen convention involves in the formulae below the Jacobian determinant of the inverse transition x → x, while the opposite convention considers the forward transition x → x resulting in a flip of sign of the weight.
^ M.R. Spiegel; S. Lipcshutz; D. Spellman (2009). Vector Analysis (2nd ed.). New York: Schaum's Outline Series. p. 198. ISBN 978-0-07-161545-7.
^ C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). p. 1417. ISBN 0-07-051400-3.
^ Weinberg 1972 p 100.
Spivak, Michael (1999), A Comprehensive Introduction to Differential Geometry, Vol I (3rd ed.), p. 134 .
Kuptsov, L.P. (2001) [1994], "Tensor Density", Encyclopedia of Mathematics, EMS Press .
Charles Misner; Kip S Thorne & John Archibald Wheeler (1973). Gravitation. W. H. Freeman. p. 501ff. ISBN 0-7167-0344-0. {{cite book}}: CS1 maint: multiple names: authors list (link)
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Home : Support : Online Help : Mathematics : Mathematical Functions : MathematicalFunctions Package : Evalf : Overview
Package and command for the numerical evaluation of mathematical expressions, allowing for the use of different numerical methods in the case of Heun and Appell functions
Evalf is both a command and a package of commands for the numerical evaluation of mathematical expressions and functions, numerical experimentation, and fast development of numerical algorithms, taking advantage of the advanced symbolic capabilities of the Maple computer algebra system. This kind of numerical/symbolic environment is increasingly relevant nowadays, when rather complicated mathematical expressions and advanced special functions, as for instance is the case of the Heun and Appell functions, appear more and more in the modeling of problems in science.
The Evalf environment is also an excellent helper for understanding how numerical algorithms work, placing some of the typical numerical approaches used in the literature at the tip of your fingers, in a flexible and friendly manner.
The Evalf command allows, among other things, for the indication of different numerical methods to evaluate the mathematical functions involved in an algebraic expression. In this version, Evalf implements optional arguments for the 10 Heun and 4 Appell functions. For anything else Evalf works the same as the standard evalf command. The options implemented for numerical evaluation, generally speaking, are divided into four groups:
restrictive options;
numerical methods options;
information options;
As a package, Evalf contains the following commands:
QuadrantNumbers
Brief description of the commands of the Evalf package
Add accepts a procedure with a formula that depends on one integer parameter, say
n
, or two of them, and numerically evaluates the formula adding it from
n=0
until the result converges with the current value of Digits, or the value
n=10000\mathrm{Digits}
Evalb works as evalb but can handle boolean expressions involving the functions as And, Or, Not and interpreting the integer types integer, posint, negint, nonposint, nonnegint, even and odd in an extended sense to include floats, for example:
1.
is considered of type integer and
-4.
is considered of type negint and even.
GenerateRecurrence accepts a mathematical function (currently working only with the four Appell functions) and returns a procedure to compute the
n
th coefficient of a power series expansion of the given function around the origin as a function of the
n-1
previous coefficients, where
n
depends on the function given.
PairwiseSummation accepts a one-dimensional Array or a procedure of one argument, say
U
and performs a pairwise summation of
U\left(j\right)
j
from given m to n, or from the lower to the upper bound of the given Array. Pairwise summation is a technique to add floating-point numbers that significantly reduces the accumulated round-off error if compared to adding the numbers one at a time.
QuadrantNumbers accepts a complex number and returns the quadrant of the complex plane where the number is located, or it accepts a list of numbers, and returns an Array of four lists, corresponding to each quadrant, of random complex numbers that are around the numbers indicated.
Singularities accepts a Heun or Appell function, for instance with numerical values for the function's parameters, and returns the singularities of the linear ODE satisfied by the given function.
Zoom is used to zoom within the last plot computed using Evalf for concatenated Taylor expansions used when performing numerical computations of Heun or Appell functions.
As usual, you can load the Evalf package using the with command, or invoke Evalf commands using the long form, e.g. as in Evalf:-Add.
The MathematicalFunctions[Evalf] command was introduced in Maple 2017.
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Variants of axiomatic set theory that allow sets to be elements of themselves
Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation.
The study of non-well-founded sets was initiated by Dmitry Mirimanoff in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non-well-founded sets; he did not regard well-foundedness as an axiom. Although a number of axiomatic systems of non-well-founded sets were proposed afterwards, they did not find much in the way of applications until Peter Aczel’s hyperset theory in 1988.[1][2][3] The theory of non-well-founded sets has been applied in the logical modelling of non-terminating computational processes in computer science (process algebra and final semantics), linguistics and natural language semantics (situation theory), philosophy (work on the Liar Paradox), and in a different setting, non-standard analysis.[4]
In 1917, Dmitry Mirimanoff introduced[5][6][7][8] the concept of well-foundedness of a set:
A set, x0, is well-founded if it has no infinite descending membership sequence
{\displaystyle \cdots \in x_{2}\in x_{1}\in x_{0}.}
In ZFC, there is no infinite descending ∈-sequence by the axiom of regularity. In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC− (that is, ZFC without the axiom of regularity) that well-foundedness implies regularity. In variants of ZFC without the axiom of regularity, the possibility of non-well-founded sets with set-like ∈-chains arises. For example, a set A such that A ∈ A is non-well-founded.
Although Mirimanoff also introduced a notion of isomorphism between possibly non-well-founded sets, he considered neither an axiom of foundation nor of anti-foundation.[7] In 1926, Paul Finsler introduced the first axiom that allowed non-well-founded sets. After Zermelo adopted Foundation into his own system in 1930 (from previous work of von Neumann 1925–1929) interest in non-well-founded sets waned for decades.[9] An early non-well-founded set theory was Willard Van Orman Quine’s New Foundations, although it is not merely ZF with a replacement for Foundation.
Several proofs of the independence of Foundation from the rest of ZF were published in 1950s particularly by Paul Bernays (1954), following an announcement of the result in an earlier paper of his from 1941, and by Ernst Specker who gave a different proof in his Habilitationsschrift of 1951, proof which was published in 1957. Then in 1957 Rieger's theorem was published, which gave a general method for such proof to be carried out, rekindling some interest in non-well-founded axiomatic systems.[10] The next axiom proposal came in a 1960 congress talk of Dana Scott (never published as a paper), proposing an alternative axiom now called SAFA.[11] Another axiom proposed in the late 1960s was Maurice Boffa's axiom of superuniversality, described by Aczel as the highpoint of research of its decade.[12] Boffa's idea was to make foundation fail as badly as it can (or rather, as extensionality permits): Boffa's axiom implies that every extensional set-like relation is isomorphic to the elementhood predicate on a transitive class.
A more recent approach to non-well-founded set theory, pioneered by M. Forti and F. Honsell in the 1980s, borrows from computer science the concept of a bisimulation. Bisimilar sets are considered indistinguishable and thus equal, which leads to a strengthening of the axiom of extensionality. In this context, axioms contradicting the axiom of regularity are known as anti-foundation axioms, and a set that is not necessarily well-founded is called a hyperset.
Four mutually independent anti-foundation axioms are well-known, sometimes abbreviated by the first letter in the following list:
AFA ("Anti-Foundation Axiom") – due to M. Forti and F. Honsell (this is also known as Aczel's anti-foundation axiom);
SAFA ("Scott’s AFA") – due to Dana Scott,
FAFA ("Finsler’s AFA") – due to Paul Finsler,
BAFA ("Boffa’s AFA") – due to Maurice Boffa.
They essentially correspond to four different notions of equality for non-well-founded sets. The first of these, AFA, is based on accessible pointed graphs (apg) and states that two hypersets are equal if and only if they can be pictured by the same apg. Within this framework, it can be shown that the so-called Quine atom, formally defined by Q={Q}, exists and is unique.
Each of the axioms given above extends the universe of the previous, so that: V ⊆ A ⊆ S ⊆ F ⊆ B. In the Boffa universe, the distinct Quine atoms form a proper class.[13]
It is worth emphasizing that hyperset theory is an extension of classical set theory rather than a replacement: the well-founded sets within a hyperset domain conform to classical set theory.
Aczel’s hypersets were extensively used by Jon Barwise and John Etchemendy in their 1987 book The Liar, on the liar's paradox; The book is also a good introduction to the topic of non-well-founded sets.
Boffa’s superuniversality axiom has found application as a basis for axiomatic nonstandard analysis.[14]
^ Pakkan & Akman (1994), section link. sfnp error: no target: CITEREFPakkanAkman1994 (help)
^ Rathjen (2004).
^ Sangiorgi (2011), pp. 17–19, 26.
^ Ballard & Hrbáček (1992).
^ Levy (2002), p. 68. sfnp error: no target: CITEREFLevy2002 (help)
^ Hallett (1986), p. 186.
^ a b Aczel (1988), p. 105.
^ Mirimanoff (1917).
^ Aczel (1988), p. 107.
^ Aczel (1988), pp. 107–8.
^ Nitta, Okada & Tsouvaras (2003). sfnp error: no target: CITEREFNittaOkadaTsouvaras2003 (help)
^ Kanovei & Reeken (2004), p. 303.
Aczel, Peter (1988), Non-well-founded sets, CSLI Lecture Notes, vol. 14, Stanford, CA: Stanford University, Center for the Study of Language and Information, pp. xx+137, ISBN 0-937073-22-9, MR 0940014.
Ballard, David; Hrbáček, Karel (1992), "Standard foundations for nonstandard analysis", Journal of Symbolic Logic, 57 (2): 741–748, doi:10.2307/2275304, JSTOR 2275304.
Barwise, Jon; Etchemendy, John (1987), The Liar: An Essay on Truth and Circularity, Oxford University Press, ISBN 9780195059441
Barwise, Jon; Moss, Lawrence S. (1996), Vicious circles. On the mathematics of non-wellfounded phenomena, CSLI Lecture Notes, vol. 60, CSLI Publications, ISBN 1-57586-009-0
Boffa., M. (1968), "Les ensembles extraordinaires", Bulletin de la Société Mathématique de Belgique, 20: 3–15, Zbl 0179.01602
Boffa, M. (1972), "Forcing et négation de l'axiome de Fondement", Acad. Roy. Belgique, Mém. Cl. Sci., Coll. 8∘, Série II, 40 (7), Zbl 0286.02068
Devlin, Keith (1993), "§7. Non-Well-Founded Set Theory", The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.), Springer, ISBN 978-0-387-94094-6
Finsler, P. (1926), "Über die Grundlagen der Mengenlehre. I: Die Mengen und ihre Axiome", Math. Z., 25: 683–713, doi:10.1007/BF01283862, JFM 52.0192.01 ; translation in Finsler, Paul; Booth, David (1996). Finsler Set Theory: Platonism and Circularity : Translation of Paul Finsler's Papers on Set Theory with Introductory Comments. Springer. ISBN 978-3-7643-5400-8.
Hallett, Michael (1986), Cantorian set theory and limitation of size, Oxford University Press, ISBN 9780198532835.
Kanovei, Vladimir; Reeken, Michael (2004), Nonstandard Analysis, Axiomatically, Springer, ISBN 978-3-540-22243-9
Levy, Azriel (2012) [2002], Basic set theory, Dover Publications, ISBN 9780486150734.
Mirimanoff, D. (1917), "Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles", L'Enseignement Mathématique, 19: 37–52, JFM 46.0306.01.
Nitta; Okada; Tzouvaras (2003), Classification of non-well-founded sets and an application (PDF)
Pakkan, M. J.; Akman, V. (1994–1995), "Issues in commonsense set theory" (PDF), Artificial Intelligence Review, 8 (4): 279–308, doi:10.1007/BF00849061, hdl:11693/25955
Sangiorgi, Davide (2011), "Origins of bisimulation and coinduction", in Sangiorgi, Davide; Rutten, Jan (eds.), Advanced Topics in Bisimulation and Coinduction, Cambridge University Press, ISBN 978-1-107-00497-9
Scott, Dana (1960), "A different kind of model for set theory", Unpublished paper, talk given at the 1960 Stanford Congress of Logic, Methodology and Philosophy of Science
Moss, Lawrence S. "Non-wellfounded Set Theory". Stanford Encyclopedia of Philosophy.
Metamath page on the axiom of Regularity. Fewer than 1% of that database's theorems are ultimately dependent on this axiom, as can be shown by a command ("show usage") in the Metamath program.
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