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Section 15.30: Koszul regular sequences
Comment #1929 by DB on April 25, 2016 at 13:26
It seems like the definition of regular sequence (Tag 00LF) is incompatible with the weaker notions of regular sequence (Koszul-, H1-, quasi-). The former definition excludes 1 as a regular sequence wheras the latter don't.
For instance the lemma above is false if we take f_1 = 1. Similar errors occur elsewhere. For instance in the discussion (Tag 062D) after the definition of Koszul-regular sequence.
OK, thanks for pointing this out. The correct statement probably should be that the sequence is regular provided that the ideal isn't the whole ring. Yes? I will fix this soonish.
We tried to be careful with this, so please point out any other mistakes you see with this. In fact I think the discussion following the definition of a Koszul regular sequence (in Section 15.30) is correct except maybe that in the very last part we need to assume
f = f_1
is not a unit; is that what you mean?
Yes, I think the statement in section 062D is true if
f = f_1
is not a unit. And yes I think the statement is correct if one also assume that the
f_i
does not generate the whole thing.
Is changing the definitions an option? It seems a bit strange to me to impose the extra condition that
f_1, \ldots f_r
should not generate the whole ring in the case for regular sequences but not imposing the condition for the other regularity notions. In for instance EGA IV-1 Ch 0 15.2.2 a regular sequence is allowed to generate the whole ring.
OK, many thanks for the reply.
About the definition of a regular sequence: The reason for the current choice is that in most of the commutative algebra texts I looked at the notion of a regular sequence was defined in this manner. For example, this is how it is defined in:
Eisenbud's commutative algebra
Kaplansky's Ring Theory
Kunz, Introduction to Commutative Algebra and Algebraic Geometry
The notion of a regular sequence is very iffy. It depends on the order of the sequence even in the local situation when the elements all come from the maximal ideal. So making it a notion which is hard to use, may actually be an advantage.
Afterthought: it is "well known" that over a Noetherian local ring, the order of the elements in a regular sequence does not matter (Lemma 10.68.4). But this statement would be false if we changed the definition!
I don't know which definition is "best", but it makes sense to follow the main textbooks on the subject as you say (perhaps together with a warning that also the other definition is common). My guess is that the choice in EGA is made in order to make regular sequences respect localisations.
But my point was rather that if one makes one choice for one of the regularity conditions, then it might save the readers (and the writers) some headache if one makes the same choice for the others. For instance, in Matsumura's "Commutative ring theory", a quasi-regular seqence is not allowed to generate the whole ring (p. 124), which is consistent with the definition of regular sequence given in op. cit.
OK, your point is a good one and there are also plenty of references which define regular sequences without requiring the nonvanishing of the quotient. In fact you are in good company because Burt Totaro suggested the same thing in 2013. So yeah, maybe we'll need to change the definition... Anybody who reads this and agrees, please leave a comment!
Thanks again for the comment. I just fixed it.
The "r" in the 2nd sentence should be an "n".
@#2775: Thanks, fixed here.
4 comment(s) on Section 15.30: Koszul regular sequences
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Lemma 15.9.5 (0ALH)—The Stacks project
Section 15.9: Lifting
Lemma 15.9.5. Let $A$ be a ring, let $I \subset A$ be an ideal. Let $f \in A[x]$ be a monic polynomial. Let $\overline{f} = \overline{g} \overline{h}$ be a factorization of $f$ in $A/I[x]$ such that $\overline{g}$ and $\overline{h}$ are monic and generate the unit ideal in $A/I[x]$. Then there exists an étale ring map $A \to A'$ which induces an isomorphism $A/I \to A'/IA'$ and a factorization $f = g' h'$ in $A'[x]$ with $g'$, $h'$ monic lifting the given factorization over $A/I$.
Proof. We will deduce this from results on the universal factorization proved earlier; however, we encourage the reader to find their own proof not using this trick. Say $\deg (\overline{g}) = n$ and $\deg (\overline{h}) = m$ so that $\deg (f) = n + m$. Write $f = x^{n + m} + \sum \alpha _ i x^{n + m - i}$ for some $\alpha _1, \ldots , \alpha _{n + m} \in A$. Consider the ring map
\[ R = \mathbf{Z}[a_1, \ldots , a_{n + m}] \longrightarrow S = \mathbf{Z}[b_1, \ldots , b_ n, c_1, \ldots , c_ m] \]
of Algebra, Example 10.143.12. Let $R \to A$ be the ring map which sends $a_ i$ to $\alpha _ i$. Set
\[ B = A \otimes _ R S \]
By construction the image $f_ B$ of $f$ in $B[x]$ factors, say $f_ B = g_ B h_ B$ with $g_ B = x^ n + \sum (1 \otimes b_ i) x^{n - i}$ and similarly for $h_ B$. Write $\overline{g} = x^ n + \sum \overline{\beta }_ i x^{n - i}$ and $\overline{h} = x^ m + \sum \overline{\gamma }_ i x^{m - i}$. The $A$-algebra map
\[ B \longrightarrow A/I, \quad 1 \otimes b_ i \mapsto \overline{\beta }_ i, \quad 1 \otimes c_ i \mapsto \overline{\gamma }_ i \]
maps $g_ B$ and $h_ B$ to $\overline{g}$ and $\overline{h}$ in $A/I[x]$. The displayed map is surjective; denote $J \subset B$ its kernel. From the discussion in Algebra, Example 10.143.12 it is clear that $A \to B$ is etale at all points of $V(J) \subset \mathop{\mathrm{Spec}}(B)$. Choose $g \in B$ as in Lemma 15.9.4 and consider the $A$-algebra $B_ g$. Since $g$ maps to a unit in $B/J = A/I$ we obtain also a map $B_ g/I B_ g \to A/I$ of $A/I$-algebras. Since $A/I \to B_ g/I B_ g$ is étale, also $B_ g/IB_ g \to A/I$ is étale (Algebra, Lemma 10.143.8). Hence there exists an idempotent $e \in B_ g/I B_ g$ such that $A/I = (B_ g/I B_ g)_ e$ (Algebra, Lemma 10.143.9). Choose a lift $h \in B_ g$ of $e$. Then $A \to A' = (B_ g)_ h$ with factorization given by the image of the factorization $f_ B = g_ B h_ B$ in $A'$ is a solution to the problem posed by the lemma. $\square$
Comment #1334 by JuanPablo on March 07, 2015 at 16:49
I do not understand why
A/I\cong A'/IA'
In the proof as currently written it does not seem to work.
g=\Delta
the resultant of the polynomials
x^n+\Sigma b_{i}x^{n-i}, x^m+\Sigma c_{j}x^{m-j}
as in example 10.139.13 (tag 00UA) works to make
A\rightarrow B_g
étale and maps to an invertible in
B\rightarrow A/I
(that is, modulo
J
But looking at the universal property of
B_{\Delta}/IB_{\Delta}
it seems to suggest that in any
A/I
algebra with a second factorization
\bar{g}\bar{h}=\bar{g}'\bar{h}'
(\bar{g},\bar{h}),(\bar{g}',\bar{h}')
generating the unit ideals, monic and of the same degree then
(\bar{g},\bar{h})=(\bar{g}',\bar{h}')
, which is in general false.
Thanks very much for pointing this out. I have fixed it here.
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Solar Disinfection Kinetic Design Parameters for Continuous Flow Reactors | J. Sol. Energy Eng. | ASME Digital Collection
Laurence W. Gill,
Department of Civil, Structural and Environmental Engineering,
, Dublin 2, Ireland
e-mail: gilll@tcd.ie
Orlaith A. McLoughlin
e-mail: mclougho@tcd.ie
Gill, L. W., and McLoughlin, O. A. (November 15, 2005). "Solar Disinfection Kinetic Design Parameters for Continuous Flow Reactors." ASME. J. Sol. Energy Eng. February 2007; 129(1): 111–118. https://doi.org/10.1115/1.2391316
The main UV dose-related kinetic parameters influencing solar disinfection have been investigated for the design of a continuous flow reactor suitable for a village-scale water treatment system. The sensitivities of different pathogenic microorganisms under solar light in batch processes have been compared in order to define their relative disinfection kinetics with E. coli used as a baseline organism. Dose inactivation kinetics have been calculated for small scale disinfection systems operating under different conditions such as reflector type, flow rate, process type, photocatalytic enhancement, and temperature enhancement using E. coli K-12 as a model bacterium. Solar disinfection was shown to be successful in all experiments with a slight improvement in the disinfection kinetic found when a fixed
TiO2
photocatalyst was placed in the reactor. There was also evidence that the photocatalytic mechanism prevented regrowth in the post-irradiation environment. A definite synergistic solar UV∕temperature effect was noticed at a temperature of
45°C
. The disinfection kinetics for E. coli in continuous flow reactors have been investigated with respect to various reflector shapes and flow regimes by carrying out a series of experiments under natural sunlight. Finally, photocatalytic and temperature enhancements to the continuous flow process have been evaluated.
microorganisms, water treatment, solar absorber-convertors, chemical reactors, biological techniques, biological effects of ionising radiation, catalysis, photochemistry
Design, Flow (Dynamics), Microorganisms, Solar energy, Sunlight, Ultraviolet radiation, Temperature, Water
Researches on the Effect of Light Upon Bacteria and Other Organisms
Solar Disinfection: Use of Sunlight to Decontaminate Drinking Water in Developing Countries
Elmore-Meegan
Solar Disinfection of Drinking Water Protects Against Cholera in Children Aged Under 6
Effects of Simulated Solar Disinfection on Infectivity of Salmonella Typhimurium
Al-Touati
Solar and Photocatalytic Disinfection of Protozoan, Fungal and Bacterial Microbes in Drinking Water
Photocatalytic Treatment of Water-Soluble Pesticides by Photo-Fenton and TiO2 Using Solar Energy
Mu’llallem
Karahagopian
Water Disinfection by Solar Radiation-Assessment and Application
,” Technical Study 66e, IRDC.
High Performance, Low Cost Solar Collectors for Disinfection of Contaminated Water
Solar Disinfection of Contaminated Water: A Comparison of Three Small-Scale Reactors
Fernández Ibáñez
Malato Rodríguez
Photocatalytic Disinfection of Water Using Low Cost Compound Parabolic Collectors
An Investigation Into the Laws of Disinfection
J. Hyg. (Lond)
Formulation of a Mathematical Model to Predict Solar Water Disinfection
Studies on Photokilling of Bacteria on TiO2 Thin Film
Komvuschara
A New Kinetic Model for Ultraviolet Disinfection of Greywater
Reichsteiner
Wirojanagud
Ajarmeh
SODIS-An Emerging Water Treatment Process
J. Water Supply: Res. Technol.-AQUA
Enhancement of Solar Inactivation of Escherichia Coli by Titanium Dioxide Photocatalytic Oxidation
Photocatalytic Bactericidal Effect of TiO2 on Enterobacter Cloacae: Comparative Study with Other Gram (-) Bacteria
Batch Process Solar Disinfection is an Efficient Means of Disinfecting Water Contaminated With Shigella Dysenteriae Type I
The Inactivation of Microbes by Sunlight: Solar Disinfection as a Water Treatment Process
Bactericidal Action of Illuminated TiO2 on Pure E. Coli and Natural Bacterial Consortia: Post-Irradiation Events in the Dark and Assessment of the Effective Disinfection Time
S-V.
Development and Evaluation of a Reflective Solar Disinfection Pouch for the Treatment of Drinking Water
Trojan Horses of the Microbial World: Protozoa and the Survival of Bacterial Pathogens in the Environment
Axelsson-Olsson
Protozoan Acanthamoeba Polyphaga as a Potential Reservoir for Camplylobacter Jejuni
Association of Vibrio Cholerae With Fresh Water Amoebae
Méndez-Hermida
Castro-Hermida
Effect of Batch-Process Solar Disinfection on Survival of Cryptosporidium Parvum Oocysts in Drinking Water
Photocatalytic Inactivation of E. Coli: Effect of (Continuous-Intermittent) Light Intensity and of (Suspended-Fixed) TiO2 Concentration
, 1999, Standard Methods for the Examination of Water and Wastewater, 20th ed.
Shigella Plays Dangerous Games
Mode of Photocatalytic Bactericidal of Powdered Semiconductor TiO2 on Mutans Streptococci
Quantitative Analysis of Variations in Initial Bacillus Pumilus Spore Densities in Aqueous TiO2 Suspension and Design of a Photocatalytic Reactor
J. Environ. Sci. Health, Part A: Toxic/Hazard. Subst. Environ. Eng.
The Photocatalytic Removal of Bacterial Pollutants From Drinking Water
Photocatalytic Transformation of Pesticides in Aqueous Titanium Dioxide Suspensions Using Artificial and Solar Light: Intermediates and Degradation Pathways
Solar Photocatalysis: A Clean Process for Water Detoxification
Photocatalytic Inactivation of Coliform Bacteria and Viruses in Secondary Effluent
Bactericidal Activity of TiO2 Photocatalyst in Aqueous Media: Toward a Solar-Assisted Water Disinfection System
A Novel TiO2-Assisted Solar Photocatalytic Batch-Process Disinfection Reactor for the Treatment of Biological and Chemical Contaminants in Domestic Drinking Water in Developing Countries
Evaluation of Photocatalytic Disinfection of Crude Water for Drinking-Water Production
EAWAG, SODIS Technical Notes, available from: http://www.sodis.chhttp://www.sodis.ch
Lethal Effects of Heat on Bacterial Physiology and Structure
Enhancement of Solar Water Pasteurisation with Reflectors
Decontamination of Ventilation Systems Using Photocatalytic Air Cleaning Technology
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IndependenceNumber - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Graph Theory : GraphTheory Package : IndependenceNumber
compute independence number
find independent set
IndependenceNumber(G)
IndependenceNumber(G, opt)
MaximumIndependentSet(G)
MaximumIndependentSet(G, opt)
(optional) equation of the form method = m, where m is exact or greedy
IndependenceNumber returns the independence number of the graph G.
MaximumIndependentSet returns a list of vertices comprising a maximum independent set of G.
An independent set of a graph G is a subset S of the vertices of G, with the condition that if any vertices v1 and v2 are both members of S, the graph G must not contain an edge between them. An independent set of G is a clique of the complement of G.
A maximum independent set of G is an independent set of maximum size for the graph G.
The independence number of G is the cardinality of a maximum independent set of G. This is equal to the clique number of the complement of G.
The problem of finding a maximum independent set for a graph is NP-complete, meaning that no polynomial-time algorithm is presently known. The exhaustive search will take an exponential time on some graphs. For a faster algorithm that usually, but not always, returns a relatively large clique see GreedyIndependentSet. This algorithm can also be selected by using the method = greedy option. The default is method = exact.
\mathrm{with}\left(\mathrm{GraphTheory}\right):
G≔\mathrm{CompleteGraph}\left(3,4\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: an undirected unweighted graph with 7 vertices and 12 edge\left(s\right)}}
\mathrm{DrawGraph}\left(G\right)
\mathrm{IndependenceNumber}\left(G\right)
\textcolor[rgb]{0,0,1}{4}
\mathrm{MaximumIndependentSet}\left(G\right)
[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]
This is equivalent to the maximum clique problem on the complement of
G
C≔\mathrm{GraphComplement}\left(G\right):
\mathrm{DrawGraph}\left(C\right)
\mathrm{CliqueNumber}\left(C\right)
\textcolor[rgb]{0,0,1}{4}
\mathrm{MaximumClique}\left(C\right)
[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]
\mathrm{IndependenceNumber}\left(G,'\mathrm{method}'='\mathrm{greedy}'\right)
\textcolor[rgb]{0,0,1}{4}
\mathrm{MaximumIndependentSet}\left(G,'\mathrm{method}'='\mathrm{greedy}'\right)
[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]
The GraphTheory[IndependenceNumber] and GraphTheory[MaximumIndependentSet] commands were updated in Maple 2019.
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Home : Support : Online Help : Connectivity : Database Package : Connection : Close
close the Connection Module
connection:-Close( )
Close frees the resources associated with connection. This happens automatically when connection is garbage collected; however, you can call Close to release the resources immediately.
Any descendant modules of connection are closed when connection is closed. (A module is a descendant of a parent module if it is returned by one of the parent module's exports or if it is a descendant of one of the parent module's descendants.)
\mathrm{driver}≔\mathrm{Database}[\mathrm{LoadDriver}]\left(\right):
\mathrm{conn}≔\mathrm{driver}:-\mathrm{OpenConnection}\left(\mathrm{url},\mathrm{name},\mathrm{pass}\right):
Create a descendant of conn.
\mathrm{res}≔\mathrm{conn}:-\mathrm{ExecuteQuery}\left("SELECT * FROM animals"\right):
Close conn.
\mathrm{conn}:-\mathrm{Close}\left(\right)
Try to use conn.
\mathrm{conn}:-\mathrm{ExecuteQuery}\left("SELECT * FROM animals"\right):
Try using conn's descendant.
\mathrm{res}:-\mathrm{Next}\left(\right)
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Importance of Higher Order Modes and Refined Theories in Free Vibration Analysis of Composite Plates | J. Appl. Mech. | ASME Digital Collection
S. Brischetto,
S. Brischetto
Department of Aeronautics and Space Engineering,
e-mail: salvatore.brischetto@polito.it
E. Carrera Professor of Aerospace Structures and Aeroelasticity
Brischetto, S., and Carrera, E. (October 5, 2009). "Importance of Higher Order Modes and Refined Theories in Free Vibration Analysis of Composite Plates." ASME. J. Appl. Mech. January 2010; 77(1): 011013. https://doi.org/10.1115/1.3173605
This paper evaluates frequencies of higher-order modes in the free vibration response of simply-supported multilayered orthotropic composite plates. Closed-form solutions in harmonic forms are given for the governing equations related to classical and refined plate theories. Typical cross-ply (0 deg/90 deg) laminated panels (10 and 20 layers) are considered in the numerical investigation (these were suggested by European Aeronautic Defence and Space Company (EADS) in the framework of the “Composites and Adaptive Structures: Simulation, Experimentation and Modeling” (CASSEM) European Union (EU) project. The Carrera unified formulation has been employed to implement the considered theories: the classical lamination theory, the first-order shear deformation theory, the equivalent single layer model with fourth-order of expansion in the thickness direction
z
, and the layerwise model with linear order of expansion in
z
for each layer. Higher-order frequencies and the related harmonic modes are computed by varying the number of wavelengths
(m,n)
in the two-plate directions and the degrees of freedom in the plate theories. It can be concluded above all that—refined plate models lead to higher-order frequencies, which cannot be computed by simplified plate theories—frequencies related to high values of wavelengths, even the fundamental ones, can be wrongly predicted when using classical plate theories, even though thin plate geometries are analyzed.
aerospace engineering, laminates, plates (structures), shear deformation, vibrations
Composite materials, Free vibrations, Plates (structures), Plate theory, Shear deformation, Vibration
Composite Materials and Structures: Science, Technology and Applications—A Compendium of Books, Review Papers and Other Sources of Informations
,” NASA Report No. SP-160.
Introduzione alla Tecnica Statistico-Energetica (S.E.A.) per la Dinamica Strutturale e l’Acustica Interna
Free Vibrations of Multilayered Composite Plates
Research on Thick Plate Vibration: A Literature Survey
Study on the Flexural Vibration of Rectangular Thin Plates With Free Boundary Conditions
Transverse Vibration of Thick Polygonal Plates Using Analytically Integrated Trapezoidal Fourier p-Element
Aimmanee
Vibration of Thick Isotropic Plates With Higher Order Shear and Normal Deformable Plate Theories
Exact Solutions for the Free In-Plane Vibration of Rectangular Plates With Two Opposite Edges Simply Supported
Accurate Analytical Type Solutions for the Free In-Plane Vibration of Clamped and Simply Supported Rectangular Plates
Analytical Solution for Vibration of an Incompressibile Isotropic Linear Elastic Rectangular Plate, and Frequencies Missed in Previous Solutions
Missing Frequencies in Previous Exact Solutions of Free Vibrations of Simply Supported Rectangular Plates
Plane Wave Solutions and Modal Analysis in Higher Order Shear and Normal Deformable Plate Theories
On the Free In-Plane Vibration of Isotropic Rectangular Plates
Three-Dimensional Elasticity Solutions for Free Vibrations of Circular Plates: A Polynomial-Ritz Analysis
Discrete Singular Convolution for the Prediction of High Frequency Vibration of Plates
A Hierarchical Functions Set for Predicting Very High Order Plate Bending Modes With Any Boundary Conditions
R. M. E. J.
Free Vibration Analysis of Rectangular Plates With Variable Thickness
/University of Nebraska-Lincoln.
Simple Models of the Energetics of Transversely Vibrating Plates
A Novel Approach for the Analysis of High-Frequency Vibrations
Estimation of Normal Mode and Other System Parameters of Composite Laminated Plates
Accurate Free Vibration Analysis of Completely Free Symmetric Cross-Ply Rectangular Laminated Plates
Free Vibration Analysis of Completely Free Rectangular Plates by the Superposition-Galerkin Method
A Three-Dimensional Free Vibration Analysis of Cross-Ply Laminated Rectangular Plate With Clamped Edges
Free Vibration Analysis of Laminated Plates Using Higher-Order Shear Deformation Theory
Proceedings in Physics
Refined Structural Dynamics Model for Composite Rotor Blades
Higher-Order Zig-Zag Theory for Laminated Composites With Multiple Delaminations
High-Order Layer Wise Mechanics and Finite Element for the Damped Dynamic of Sandwich Composite Beams
Analytical Solutions for Vibrations of Laminated and Sandwich Plates Using Mixed Theory
Mechanics and Finite Elements for the Damped Dynamic Characteristics of Curvilinear Laminates and Composite Shell Structures
On Theories for the Dynamic Response of Laminated Plates
13th ASME and SAE Structures, Structural Dynamics and Materials Conference
, San Antonio, TX, April 10–12.
Vibration Frequencies of Simply Supported Polygonal Sandwich Plates Via Kirchhoff Solutions
Free Vibration Analysis of Composite Sandwich Plates Based on Reddy’s Higher-Order Theory
Vibration of Completely Free Composite Plates and Cylindrical Shell Panels by a Higher-Order Theory
Vibration of Thick Laminated Plates
Geometrically Nonlinear Vibration Analysis of Thin, Rectangular Plates Using the Hierarchical Finite Element Method—II: 1st Mode of Laminated Plates and Higher Modes of Isotropic and Laminated Plates
The Galerkin Element Method Applied to the Vibration of Rectangular Damped Sandwich Plates
A Study of Transverse Normal Stress Effect on Vibration of Multilayered Plates and Shells
A Reissner’s Mixed Variational Theorem Applied to Vibrational Analysis of Multilayered Shells
A Class of Two Dimensional Theories for Multilayered Plates Analysis
Atti Accad. Sci. Torino, Cl. Sci. Fis., Mat. Nat.
Mechanics of Laminated Composite Plates and Shells, Theory and Analysis
Analysis of Thickness Locking in Classical, Refined and Mixed Multilayered Plate Theories
Stress and Free Vibration Analysis of Multilayered Composite Plates
Stability and Vibration of Isotropic, Orthotropic and Laminated Plates According to a Higher Order Shear Deformation Theory
Discussion: “Zeroth-Order Shear Deformation Theory for Laminated Composite Plates” (Ray, M. C., 2003 ASME J. Appl. Mech., 70, pp. 374–380)
Free Flexural Vibrations of Bonded and Centrally Doubly Stiffened Composite Orthotropic Base Plates or Panels
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This problem is a checkpoint for factoring polynomials. It will be referred to as Checkpoint 10B.
Factor each polynomial expression.
x^2−8x+7
y^2−2y−15
7x^2−63
3x^2+10x+8
Check your answers by referring to the Checkpoint 10B materials located at the back of your book.
Ideally, at this point you are comfortable working with these types of problems and can solve them correctly. If you feel that you need more confidence when solving these types of problems, then review the Checkpoint 10B materials and try the practice problems provided. From this point on, you will be expected to do problems like these correctly and with confidence.
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Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Distributions : Triangular
Triangular(a, b, c)
TriangularDistribution(a, b, c)
The triangular distribution is a continuous probability distribution with probability density function given by:
f\left(t\right)={\begin{array}{cc}0& t<a\\ \frac{2\left(t-a\right)}{\left(b-a\right)\left(c-a\right)}& t\le c\\ \frac{2\left(b-t\right)}{\left(b-a\right)\left(-c+b\right)}& t\le b\\ 0& \mathrm{otherwise}\end{array}
a\le c,c\le b,a<b
Note that the Triangular command is inert and should be used in combination with the RandomVariable command.
\mathrm{with}\left(\mathrm{Statistics}\right):
X≔\mathrm{RandomVariable}\left(\mathrm{Triangular}\left(a,b,c\right)\right):
\mathrm{PDF}\left(X,u\right)
{\begin{array}{cc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{a}\\ \frac{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\right)}{\left(\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\right)}& \textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{c}\\ \frac{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{u}\right)}{\left(\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}\right)}& \textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{b}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}
\mathrm{PDF}\left(X,0.5\right)
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\mathrm{Mean}\left(X\right)
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\mathrm{Variance}\left(X\right)
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Brazilian cruzado novo - Wikipedia
A 5000 cruzado banknote overstamped as 5 cruzados novos A 200 cruzados novos banknote overstamped as 200 cruzeiros
50, 100, 200, 500 cruzados novos
Cruzeiro (3rd version)
The Cruzado Novo was the short-lived currency of Brazil between 15 January 1989 and 15 March 1990. It replaced the cruzado in the rate of 1000 cruzados = 1 cruzado novo. It had the symbol
{\displaystyle \mathrm {NCzS} \!\!\!\Vert }
and the ISO 4217 code BRN. In 1990, the cruzado novo was renamed the (third) cruzeiro. This currency was subdivided in 100 centavos.
The redenomination was the result of Plano Verão, which would become one of several heterodox plans in an attempt to stabilize the currency, and the path of redenomination was used to try to circumvent possible legal challenges due to rights established in the currency at that time, as happened in the Bresser Plan.
The method of monetary redenomination would be used again in 1990, when Fernando Collor de Mello assumed the presidency, and this redenomination to cruzeiro was on par with this currency then in circulation, despite the even darker effects of such an economic plan.
Unlike the Cruzeiro Novo denomination of the late 1960s, it was not a transitional pattern between two currency denominations of the same name, with banknotes and coins with that denomination having been released in 1989 and 1990.
Standard circulating stainless-steel coins were issued in denominations of 1, 5, 10 and 50 centavos.
A design for a standard circulation NCz$1 coin was planned for 1990, nicknamed the "Christ's Cross" (Portuguese: Cruz de Cristo) design. However, in March 1990, before the coin was released to the public, the country's currency changed to the Cruzeiro (3rd iteration), so that design never circulated.[1] There are 41 specimens in the Central Bank of Brazil's internal storage, of which there are forty 1990 issues, and a single 1989 issue, the only known specimen of that year.[2] Additionally, there are 15 known samples with collectors, bringing the total to 56 known issues of the coin.[1]
Coins of the Cruzado Novo
NCz$0.01 The "Boiadeiro" design, portrays a cowboy or cattle herder.
NCz$0.05 The "Pescador" design, portrays a fisherman.
NCz$0.10 The "Garimpeiro" design, portrays an artisanal miner.
NCz$0.50 The "Rendeira" design, portrays a lacemaker or weaver.
Unreleased Unreleased NCz$1 The "Cruz de Cristo" design, portrays a cross inside a map of Brazil.
Commemorative[edit]
To celebrate the 100th anniversary of the Republic in Brazil (1889–1989), two Cruzado Novo coins were minted: a NCz$1 circulating coin, and a NCz$200 non-circulating silver coin.
Commemorative coins of the Cruzado Novo
NCz$1 Circulating stainless steel commemorative coin
(missing photo) (missing photo) NCz$200 Non-circulating silver commemorative coin
The first banknotes were overprints on cruzado notes, in denominations of 1, 5 and 10 cruzados novos. Regular notes followed the same year in denominations of 50, 100 and 200 cruzados novos, with the 500 cruzado novo note following in 1990. These banknotes were overprinted with the new name of the currency in 1990. In 1992, the 50 and 100 cruzado novo banknotes were withdrawn. The higher denominations were withdrawn in 1994.
^ a b Pippi, Emerson (17 February 2020). "1 Cruzado Novo com a Cruz de Cristo: novas informações" [1 Cruzado Novo with Christ's Cross: new information] (in Brazilian Portuguese). Retrieved 22 December 2020.
^ Central Bank of Brazil [@bancocentraldobrasil] (4 August 2021). "Foram encontradas 41 moedas 'cruz de Cristo', sendo que uma delas é uma variante de 1989, o único exemplar existente!" [41 'cruz de Cristo' coins were found, one of which being a 1989 variant, the only specimen in existence!] (in Brazilian Portuguese). Retrieved 4 August 2021 – via Instagram.
Rs$ [-]
₢$ [-]
1942–1967 Cruzeiro novo
NCr$ [-]
Cr$ [BRB]
1970–1986 Cruzado
Cz$ [BRC]
1986–1989 Cruzado novo
NCz$ [BRN]
Cr$ [BRE]
1990–1993 Cruzeiro real
CR$ [BRR]
R$ [BRL]
Ratio: 1000 cruzados = 1 cruzado novo Currency of Brazil
15 January 1989 – 15 March 1990 Succeeded by:
Cruzeiro (third iteration)
Reason: currency renaming
Retrieved from "https://en.wikipedia.org/w/index.php?title=Brazilian_cruzado_novo&oldid=1076851442"
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Influence of Internal Combustion Engine Parameters on Gas Leakage through the Piston Rings Area ()
Waleed Momani1,2*
1Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, Rabigh, KSA.
2Department of Mechanical Engineering, FET, Al-Balqa Applied University, Amman, Jordon.
In this work, the influence of internal combustion engine parameters (cylinder-piston clearance, piston head height, the first segment position, gap of the first piston ring and gap of the second piston ring, piston rings’ axial clearance, intake valve debit coefficient) gas leakage from the combustion chamber through the piston rings’ area was investigated. This influence was studied by making an initial forming operation over gas leakage in the analyzed area.
Piston, Clearance, Combustion Engine, Cylinder-Piston, Piston Rings, Chamber, Gas Leakage
Momani, W. (2017) Influence of Internal Combustion Engine Parameters on Gas Leakage through the Piston Rings Area. Modern Mechanical Engineering, 7, 27-33. doi: 10.4236/mme.2017.71003.
*KSA on leave from Faculty of Engineering Technology, Al-Balqa Applied University, Jordan.
The piston ring assembly in an internal combustion engine works as a labyrinth to the combustion gas chamber and to the inferior crankcase. The spaces between the segments are used for gas expansion and elongating the circuit’s length of gases. These reduced sections produce a higher flowing resistance. The labyrinth effect of the piston rings assembly is shown by the variations of gas pressure over the piston ring area.
Several volumes, supposed as constant (for the analysis procedure), make this labyrinth. The volumes are connected through some conduits, the sections of which might be constituted by: gaps of pistons, ring radial clearances of these, piston ring axial clearance and cylinder-piston clearance.
It must be considered that, during a working cycle, the average flexible pressure (radial internal pressure) of the piston ring is sufficient to maintain the contact between the lateral face of the piston ring and the cylinder surface. This last parameter can be neglected, as long as, behind the segment the gas pressure from the piston ring grooves works [1] .
It is real that the flow area of the analyzed sections is not constant because of the segments’ axial movement in the grooves. Two opposite areas are always connected through the gap of piston ring between them, and then through the space created by the momentary piston ring position into its groove (depending on axial clearance a and radial clearance r).
The segment transition from the superior ring groove flank to the inferior one is not instantaneous; it takes 20 - 60 crankshaft revolutions. So, it is very important to establish the instantaneous piston ring position in its groove depending on the rotation angle of the crankshaft.
In order to complete the analysis, a piston with two compression piston rings and a lubrication piston ring was considered [2] . The spaces between the piston, cylinder, segments and the bottom of the piston rings groove, constitute different areas of the labyrinth, hindering gas leakage from the combustion chamber (CC) to the crankcase (Figure 1). The labyrinth is shown as follows:
- 5 regions from 1 to 5 can be defined;
- Region 1 is defined by the volume placed over the first piston ring between the piston and the cylinder to the upper part of the piston’s head (plan A);
- Region 2 is defined by the volume between the internal diameter of the first segment (Dis) and the outer diameter of the ring groove of the piston;
- Region 5 is defined by the volume between the second compression ring, the stripping ring, the piston and the cylinder (this volume is constantly connected to the crankcase);
Figure 1. The geometric model used.
- Plane A is a fictitious plane separating the combustion chamber from region 1;
- Plane B is situated to the superior flank of the first piston ring groove;
- Plane C and plane F are situated to the inferior flank of the first and the second piston ring groove;
- Plane E is to the superior flank of the second piston ring groove.
The forming operation of gas escape along the piston ring area was made using some assumptions: the gas pressure of region 1 and that of the combustion chamber are equal the gas pressure of region 5 is the same as the gas pressure of the crankcase. It was also supposed that different volumes remain constant during a working cycle.
The gas leakage (from the combustion chamber to the crankcase) over the analyzed area was considered rolling and isotherm. The flowing area caused by the piston-cylinder clearance was supposed to be circular, even if the clearance would not be uniformly distributed over the whole circumference during the piston’s secondary movement. This assumption is justified on one hand by the short time of tilting (about 15 - 20 crankshaft revolution degrees) and on the other hand by the fact that the flowing section is the same for the analyzed cases, meaning: [3] [4]
S=\frac{\pi {D}^{2}}{4}-\frac{\pi {D}_{p}^{2}}{4}=\frac{\pi }{4}\left({D}^{2}-{D}_{p}^{2}\right)
The piston-cylinder clearance reduction decreases the piston’s secondary movement (1), improves the combustion chamber sealing and contributes to the durance of the segment by reduced wearing.
The piston ring secondary motions can be divided into piston ring motion in the transverse direction, piston ring rotation, ring lift, and ring twist. These types of motion result from different loads acting on the ring. Loads of this kind is inertia loads arising from the piston acceleration and deceleration, oil film damping loads, loads owing to the pressure difference across the ring, and friction loads from the sliding contact between the ring and cylinder. The piston secondary movement and the gas leakage section illustrated in Figure 2.
The flowing areas made by the gaps of piston rings were caused by the bond between regions 1 - 3 and 3 - 5 through the gap of the two first piston rings. If we don’t consider the cylinder ovals effect, these areas will remain constant:
{S}_{1,3}=\left(\frac{D-{D}_{p}}{2}\right)\cdot {S}_{f1};
{S}_{3,5}=\left(\frac{D-{D}_{p}}{2}\right)\cdot {S}_{{f}_{2}}
The flowing area caused by the axial clearance of the piston ring can be analyzed from the premise that the axial clearance is uniformly distributed on the bottom part of the piston (Figure 3). So, the flowing sections are:
\begin{array}{l}{S}_{1,2}=\pi \cdot {D}_{ms}\cdot {x}_{1}\\ {S}_{2,3}=\pi \cdot {D}_{ms}\cdot \left({\Delta }_{{a}_{1}}-{x}_{1}\right)\\ {S}_{3,4}=\pi \cdot {D}_{ms}\cdot {x}_{2}\\ {S}_{4,5}=\pi \cdot {D}_{ms}\cdot \left({\Delta }_{{a}_{2}}-{x}_{2}\right)\end{array}
Figure 2. The piston secondary movement and the gas leakage section.
Figure 3. Axial piston rings clearance.
x and x are the momentary axial distances between piston ring 1, piston ring 2 and the superior flanks of their grooves.
The determined flow sections made possible the study of gas leakage through the piston rings area, from the combustion chamber to the crankcase. One of the main calculated parameters was the gas pressure from the combustion chamber (based upon gas state equation) [4] [5] .
The used calculation model allows (based upon the general characteristics of the engine) the evaluation of gas pressure distribution along the piston rings’ area, using the general characteristics of the engine. It also allows the evaluation of gas debit in these areas as well as the estimation of gas quantity that reaches the crankcase through the gaps of the piston rings and the gas quantity that comes back into the combustion chamber.
We will show some of the results obtained with the established calculation model.
A quantity of gases which returns into the combustion chamber is gathered in the volume of the upper part of the first segment (piston movement from TDC to BDC). This quantity of gases constitutes an important pollution source. It is better to have this quantity being reduced as for as possible, because these gases are flue gases and their existence causes an incomplete fresh air filling of the entire cylinder volume; thus worsening the combustion.
Region 1 gas volume reduction can be realized either by reducing the piston- cylinder clearance (without causing a sticking) or by approaching, as much as possible, the first segment to the superior part of the piston ring (this is limited by the increase of thermal load of the piston ring).
The piston-cylinder clearance reduction has two effects; first, the decrease of flue gases returning into the combustion chamber, and second, the decease of quantity of gases passing through the crankcase. In this way, the sealing of the combustion chamber and the durance of the piston rings will increase, too.
The diminution of the piston-cylinder clearance has also a beneficial influence on the thermal load of the piston. If the piston-cylinder clearance increases, the gas leakage toward the crankcase will increase (Figure 4) (cylinder wearing could be estimated from the measurements of gas escape quantity). The leaking amount of the gas is proportional to the area of the clearance.
A nominal clearance increase (to warm) from 100 µm to 200 µm causes a 60% increase in gas leakage toward the crankcase.
The bore increase makes the quantity of gas leakage increase, too.
The first piston ring position has also an influence on gas leakage. A thermal unloading of the upper part of the piston will take place if the height of the piston head is reduced.
In this way, putting the first piston as high as possible, the volume of this area is reduced, too.
If we reduce this height to 50%, this would cause a 50% decrease in the gas quantity that returns into the combustion chamber (Figure 5) and would influence, in a minimal way, the gas leakage toward the crankcase (Figure 6).
But we should consider that a lower height of the piston head could cause the sticking of the first piston ring, so that the engine will break.
Figure 4. The bond between the quantity of gas leakage and the piston-cylinder clearance to warm.
Figure 5. The influence of the first piston ring position on gas leakage.
Figure 6. The quantity of gases returning into the combustion chamber depending on the first piston ring position.
Let us now to the influence of the piston rings’ axial clearance. If the segments axial clearance remains within normal limits (recommended in the art), its influence on gas leakage will be minimal.
That is because the axial movement of the piston ring between the flanks of the piston ring groove lasts only a few crankshaft revolution degrees, so that gases don’t have enough time to pass through the piston rings’ area.
When the axial clearance is out of standard limits, its influence on gas leakage is very important. The increase in the distance between the superior flank of the piston ring groove and the superior flank of its piston ring causes the pumping so-called effect. The piston rings axial clearance in flounce over the engine speed is shown in Figure 7.
Finally, let us discuss the influence of the gap of the piston ring. Analyzing the influence first two piston rings gap on gas leakage toward the crankcase, it was established that the gas leakage from the piston rings’ area increases linearly with the gap increase (simultaneously modified) of the two piston rings (Figure 8, a). The variation of the gap of the second piston ring (Figure 8, b) has a bigger influence on gas leakage toward the crankcase than that of the gap of the first piston ring (Figure 8, c). The variation of the intake valve debit coefficient does not have a significant effect on gas leakage.
Figure 7. The piston ring axil clearance influence over the engine speed.
Figure 8. The influence of the gap of the piston rings on gas leakage.
This study allows for an evaluation of gas leakage along the piston rings’ area to the crankcase. The relevant importance and influence of some geometric engine parameters on this gas leakage were shown.
[1] Taylor, C.M. (1998) Automobile Engine Tribology—Design Considerations for Efficiency and Durability. Wear, 221, 1-8.
[2] Heywood, J. (1988) Internal Combustion Engine Fundamentals. McGraw-Hill Education.
[3] Calculations in the Construction of Internal Combustion Engines, 2001, ISBN 973-8198-17-8.
[4] Willard, W. (2004) Pulkrabek Engineering Fundamentals of the Internal Combustion Engine.
[5] Gilleaspie, T.D. (1992) Fundamentals of Vehicle Dynamics.
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The table below shows the results from repeated spins of a mystery spinner. Sketch a possible spinner with the experimental probability written in each part.
40
15
35
5
5
To draw your spinner, it may help to express each number of spins as a percent of the total spins.
\text{For instance, since A is spun 40 out of 100 times, or }\frac{40}{100} \text{, this can be expressed as } 40\%.
So it would make sense for
40\%
of the spinner to be Section A.
Remember that the experimental probability is the number of desired outcomes over the number of total possible outcomes.
Can you find the rest of the experimental probabilities?
Create the mystery spinner with the eTool below.
Click the link at the right for the full eTool version: MC1 7-60 HW eTool
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Overfitting detector - Algorithm details | CatBoost
IncToDec
If overfitting occurs, CatBoost can stop the training earlier than the training parameters dictate. For example, it can be stopped before the specified number of trees are built. This option is set in the starting parameters.
The following overfitting detection methods are supported:
Before building each new tree, CatBoost checks the resulting loss change on the validation dataset. The overfit detector is triggered if the
Threshold
value set in the starting parameters is greater than
CurrentPValue
CurrentPValue < Threshold
CurrentPValue
is calculated from a set of values for the maximizing metric
score[i]
ExpectedInc
ExpectedInc = max_{i_{1} \leq i_{2} \leq i } 0.99^{i - i_{1}} \cdot (score[i_{2}] - score[i_{1}])
x
x = \frac{ExpectedInc[i]}{max_{j \leq i} { } score[j] - score[i]}
CurrentPValue
CurrentPValue = exp \left(- \frac{0.5}{x}\right)
Before building each new tree, CatBoost checks the number of iterations since the iteration with the optimal loss function value.
The model is considered overfitted if the number of iterations exceeds the value specified in the training parameters.
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Jacobson topology of the primitive ideal space of self-similar k-graph C*-algebras
April 2021 Jacobson topology of the primitive ideal space of self-similar k-graph C*-algebras
We describe the Jacobson topology of the primitive ideal space of self-similar
k
-graph
{C}^{\ast }
-algebras under certain conditions.
Hui Li. "Jacobson topology of the primitive ideal space of self-similar k-graph C*-algebras." Rocky Mountain J. Math. 51 (2) 613 - 620, April 2021. https://doi.org/10.1216/rmj.2021.51.613
Received: 2 July 2020; Revised: 29 August 2020; Accepted: 23 September 2020; Published: April 2021
Keywords: C*-algebra , Jacobson topology , primitive ideal , self-similar k-graph
Hui Li "Jacobson topology of the primitive ideal space of self-similar k-graph C*-algebras," Rocky Mountain Journal of Mathematics, Rocky Mountain J. Math. 51(2), 613-620, (April 2021)
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Exercises: The Sun: A Garden-Variety Star | Astronomy | Course Hero
Have your group make a list of all the ways the Sun personally affects your life on Earth. (Consider the everyday effects as well as the unusual effects due to high solar activity.)
Long before the nature of the Sun was fully understood, astronomer (and planet discoverer) William Herschel (1738–1822) proposed that the hot Sun may have a cool interior and may be inhabited. Have your group discuss this proposal and come up with modern arguments against it.
We discussed how the migration of Europeans to North America was apparently affected by short-term climate change. If Earth were to become significantly hotter, either because of changes in the Sun or because of greenhouse warming, one effect would be an increase in the rate of melting of the polar ice caps. How would this affect modern civilization?
Suppose we experience another Maunder Minimum on Earth, and it is accompanied by a drop in the average temperature like the Little Ice Age in Europe. Have your group discuss how this would affect civilization and international politics. Make a list of the most serious effects that you can think of.
Watching sunspots move across the disk of the Sun is one way to show that our star rotates on its axis. Can your group come up with other ways to show the Sun’s rotation?
Suppose in the future, we are able to forecast space weather as well as we forecast weather on Earth. And suppose we have a few days of warning that a big solar storm is coming that will overload Earth’s magnetosphere with charged particles and send more ultraviolet and X-rays toward our planet. Have your group discuss what steps we might take to protect our civilization?
Have your group members research online to find out what satellites are in space to help astronomers study the Sun. In addition to searching for NASA satellites, you might also check for satellites launched by the European Space Agency and the Japanese Space Agency.
Some scientists and engineers are thinking about building a "solar sail"—something that can use the Sun’s wind or energy to propel a spacecraft away from the Sun. The Planetary Society is a nonprofit organization that is trying to get solar sails launched, for example. Have your group do a report on the current state of solar-sail projects and what people are dreaming about for the future.
Describe the main differences between the composition of Earth and that of the Sun.
Describe how energy makes its way from the nuclear core of the Sun to the atmosphere. Include the name of each layer and how energy moves through the layer.
Make a sketch of the Sun’s atmosphere showing the locations of the photosphere, chromosphere, and corona. What is the approximate temperature of each of these regions?
Why do sunspots look dark?
Which aspects of the Sun’s activity cycle have a period of about 11 years? Which vary during intervals of about 22 years?
Summarize the evidence indicating that over several hundreds of years or more there have been variations in the level of the solar activity.
What it the Zeeman effect and what does it tell us about the Sun?
Explain how the theory of the Sun’s dynamo results in an average 22-year solar activity cycle. Include the location and mechanism for the dynamo.
Compare and contrast the four different types of solar activity above the photosphere.
What are the two sources of particles coming from the Sun that cause space weather? How are they different?
How does activity on the Sun affect human technology on Earth and in the rest of the solar system?
How does activity on the Sun affect natural phenomena on Earth?
Table 1 in The Structure and Composition of the Sun indicates that the density of the Sun is 1.41 g/cm3. Since other materials, such as ice, have similar densities, how do you know that the Sun is not made of ice?
Starting from the core of the Sun and going outward, the temperature decreases. Yet, above the photosphere, the temperature increases. How can this be?
Since the rotation period of the Sun can be determined by observing the apparent motions of sunspots, a correction must be made for the orbital motion of Earth. Explain what the correction is and how it arises. Making some sketches may help answer this question.
Suppose an (extremely hypothetical) elongated sunspot forms that extends from a latitude of 30° to a latitude of 40° along a fixed of longitude on the Sun. How will the appearance of that sunspot change as the Sun rotates? (Figure 5 in The Solar Cycle should help you figure this out.)
The text explains that plages are found near sunspots, but Figure 1 in Solar Activity above the Photosphere shows that they appear even in areas without sunspots. What might be the explanation for this?
Why would a flare be observed in visible light, when they are so much brighter in X-ray and ultraviolet light?
How can the prominences, which are so big and ‘float’ in the corona, stay gravitationally attached to the Sun while flares can escape?
If you were concerned about space weather and wanted to avoid it, where would be the safest place on Earth for you to live?
Suppose you live in northern Canada and an extremely strong flare is reported on the Sun. What precautions might you take? What might be a positive result?
Show that the statement that 92% of the Sun’s atoms are hydrogen is consistent with the statement that 73% of the Sun’s mass is made up of hydrogen, as found in Table 2 in The Structure and Composition of the Sun. (Hint: Make the simplifying assumption, which is nearly correct, that the Sun is made up entirely of hydrogen and helium.)
This chapter gives the average sunspot cycle as 11 years. Verify this using Figure 3.
The escape velocity from any astronomical object can be calculated as
{v}_{\text{escape}}=\sqrt{2GM\text{/}R}
. Using the data in Some Useful Constants for Astronomy, calculate the escape velocity from the photosphere of the Sun. Since coronal mass ejections escape from the corona, would the escape velocity from there be more or less than from the photosphere?
From the information in Figure 4 in Solar Activity above the Photosphere, estimate the speed with which the particles in the CME in parts (c) and (d) are moving away from the Sun.
The Sun - A Garden-Variety Star.docx
AST 1112 • Sinclair Community College
Chapter 15 The Sun and 16.docx
AST 191 • Jefferson Community and Technical College
The Sun Student Notes.docx
PHYS 1060 • Snow College
Copy_of_Copy_of_F3_-_The_Formation_of_the_Sun_and_Solar_System
BUSINESS BUS 115 • Rock Canyon High School
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Solid extruded element with geometry, inertia, and color - MATLAB - MathWorks Benelux
Regular Extrusion: Number of Sides
Regular Extrusion: Outer Radius
Regular Extrusion: Length
General Extrusion: Cross-section
General Extrusion: Length
Solid extruded element with geometry, inertia, and color
The Extruded Solid block adds to the attached frame a solid element with geometry, inertia, and color. The solid element can be a simple rigid body or part of a compound rigid body—a group of rigidly connected solids, often separated in space through rigid transformations. Combine Extruded Solid and other solid blocks with the Rigid Transform blocks to model a compound rigid body.
Extruded Solid Visualization Pane
The Extruded Solid block can generate a convex hull geometry representation from an extruded solid. This geometric data can be used to model spatial contact forces.
L-shape sold
Extrusion Type — Shape parameterization to use
Regular (default) | General
Shape parameterization to use. Select Regular or General.
Regular — Translational sweep of a regular polygon cross section with geometry center coincident with the reference frame origin and extrusion axis coincident with the reference frame z axis.
General — Translational sweep of a general cross section with geometry center coincident with the [0 0] coordinate on the cross-sectional XY plane and extrusion axis coincident with the reference frame z axis.
Regular Extrusion: Number of Sides — Number of sides of the extrusion cross-section
3 (default) | scalar with units of length
Number of sides of the extrusion cross-section. The cross-section is by definition a regular polygon—one whose sides are of equal length. The number specified must be greater than two.
Regular Extrusion: Outer Radius — Radius of the inscribed circle of the extrusion cross-section
Radius of the circle that fully inscribes the extrusion cross-section. The cross-section is by definition a regular polygon—one whose sides are of equal length.
Regular Extrusion: Length — Sweep length of the extrusion
Length by which to sweep the specified extrusion cross-section. The extrusion axis is the z-axis of the solid reference frame. The cross-section is swept by equal amounts in the positive and negative directions.
General Extrusion: Cross-section — Cross-section coordinates specified on the XY plane
[1 1; -1 1; -1 -1; 1 -1] (default) | two-column matrix with units of length
Cross-sectional shape specified as an [x,y] coordinate matrix, with each row corresponding to a point on the cross-sectional profile. The coordinates specified must define a closed loop with no self-intersecting segments.
The coordinates must be arranged such that from one point to the next the solid region always lies to the left. The block extrudes the cross-sectional shape specified along the z axis to obtain the extruded solid.
General Extrusion: Length — Sweep length of the extrusion
Select Entire Geometry to export the true geometry of the Extruded Solid block which can be used for other blocks, such as the Spatial Contact Force block.
To enable this option, set Extrusion Type to Regular and select Entire Geometry under the Export.
Convex Hull — Generate the convex hull representation of the true geometry
Select Convex Hull to generate the convex hull representation of the true geometry. This convex hull can be used for contacts by connecting the Spatial Contact Force block.
To enable this option, set Extrusion Type to General and select Convex Hull under the Export.
\left(\begin{array}{ccc}{I}_{xx}& & \\ & {I}_{yy}& \\ & & {I}_{zz}\end{array}\right),
{I}_{xx}=\underset{m}{\int }\left({y}^{2}+{z}^{2}\right)\text{\hspace{0.17em}}dm
{I}_{yy}=\underset{m}{\int }\left({x}^{2}+{z}^{2}\right)\text{\hspace{0.17em}}dm
{I}_{zz}=\underset{m}{\int }\left({x}^{2}+{y}^{2}\right)\text{\hspace{0.17em}}dm
\left(\begin{array}{ccc}& {I}_{xy}& {I}_{zx}\\ {I}_{xy}& & {I}_{yz}\\ {I}_{zx}& {I}_{yz}& \end{array}\right),
{I}_{yz}=-\underset{m}{\int }yz\text{\hspace{0.17em}}dm
{I}_{zx}=-\underset{m}{\int }zx\text{\hspace{0.17em}}dm
{I}_{xy}=-\underset{m}{\int }xy\text{\hspace{0.17em}}dm
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Section 15.71: Hom complexes
Comment #7131 by Hao Peng on March 23, 2022 at 06:55
Just to point out this seems to be a direct consequence of tag 0A5Y.
Dear Hao Peng, can you explain?
H^0
of tag 0A5Y we can get
Hom(K, Hom^\bullet(L, M))\cong Hom(Tot(K\otimes L), M)
functorially.
We can first get a map
Tot(K\otimes Hom^\bullet(K, L))\to L
functorially for any
K, L
: It is just the adjunction map corresponding to
id: Hom^\bullet(K, L)\to Hom^\bullet(K, L)
Then to get a map from
Tot(Hom^\bullet(L, M)\otimes K)\to Hom^\bullet(Hom^\bullet(K, L), M)
, it suffices to show a map from
Tot(Tot(Hom^\bullet(L, M)\otimes K)\otimes Hom^\bullet(K, L))
M
. By associality of
Tot(-\otimes -)
, the map is given by
Tot(Tot(Hom^\bullet(L, M)\otimes K)\otimes Hom^\bullet(K, L))\cong Tot(Hom^\bullet(L, M)\otimes Tot(K\otimes Hom^\bullet(K, L))\to Tot(Hom^\bullet(L, M)\otimes L)\to M
Very good, thanks. In your construction of the evaluation map
Tot(K \otimes Hom(K, L)) \to L
there is a sign (I think) because your are switching the order of the tensors. Then later you use this 2x so hopefully this will match the signs chosen in the proof above.
Why do I think there is no sign matter? Because
Tot(\cdot\otimes \cdot)
is strctly associative(the two orders of tensor both corresponds to a complex with differentials
d^n=\bigoplus_{p+q+r=n}(d_1^p+(-1)^pd_2^q+(-1)^{p+q}d_3^r)
). I guess the sign matter terminates beyond the proof of tag 0A5Y.
Sorry, I finally realized the sign problem. There is truly a sign. But the same method work good for other propositions in this section.
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Chemical Reactions and Chemical Equations - Course Hero
General Chemistry/Reactions in Chemistry/Chemical Reactions and Chemical Equations
A chemical reaction is a process in which atoms from one or more substances rearrange, resulting in a different substance or substances. Chemical reactions involve breaking chemical bonds and forming new ones. A substance that reacts in a chemical reaction is a reactant. A substance that is formed during a chemical reaction is a product. During a chemical reaction, the products are formed from the reactants. To study chemical reactions, scientists categorize them into various classes.
A chemical reaction involves a rearrangement of valence electrons. A valence electron is an electron in the outermost shell of an atom. The nucleus, which contains protons and neutrons, is not affected. Electrons that are not in the valence shell (outermost shell) are also not affected. Furthermore, if there is no rearrangement of electrons, a chemical reaction does not occur. For example, a change of phase, such as melting or freezing, does not affect the arrangement of electrons. A phase change is a physical change, not a chemical reaction.
Consider the chemical reaction that occurs when a magnesium ribbon is heated in a flame. The magnesium burns brightly, forming a white powder, which is magnesium oxide. Chemical reactions are represented by equations in which an arrow points from the left (the reactant side) to the right (the product side). This representation of a chemical reaction in the form of symbols is a chemical equation.
{\rm{Mg}}(s)+{\rm{O_2}}(g)\rightarrow{\rm{2MgO}}(s)+\rm{heat}
In this reaction magnesium and oxygen are the reactants, and magnesium oxide is the product. The equation shows that the reaction of magnesium and oxygen releases energy in the form of heat. A reaction in which energy is released in the form of heat or light is an exothermic reaction. In this chemical reaction, magnesium metal reacts with oxygen to produce the compound magnesium oxide.
A reaction in which energy is absorbed in the form of heat or light is an endothermic reaction.
When writing chemical reactions, elements are often represented by their symbols. For example, when hydrochloric acid is dripped onto a block of zinc, zinc chloride, a white salt, forms, and hydrogen gas is produced. This reaction can be represented by both words and by a chemical equation:
\begin{gathered}\text{Zinc}\;&+&\;\text{Hydrochloric}\;\text{acid}\;&\rightarrow\;&\text{Zinc}\;\text{chloride}\;&+&\;\text{Hydrogen}\;\text{gas}\;\\{\rm{Zn}}(s)&+&\;{\rm{HCl}}({aq})&\rightarrow\;&{\rm{ZnCl}_2}({aq})&+&{\rm H_2}(g)\; (\text{unbalanced})\end{gathered}
Note that the chemical equation is not balanced, because the number of chlorine (Cl) and hydrogen (H) atoms on the left and right sides of the equation are not equivalent. A balanced equation describes a chemical reaction in which the number of atoms of each element involved in the reaction and the total charge are balanced. The letters in parentheses following each reactant and product indicate their phase. The phases include (s) for solid, (l) for liquid, (g) for gas, and (aq) representing a solution in water. The role of the phase does not affect the outcome of balancing chemical reactions.
The example of the reaction between zinc and hydrochloric acid is a molecular equation. In a molecular equation, an ionic compound, such as zinc chloride (ZnCl2), is shown as a molecular formula and not as ions.
<Vocabulary>Balancing Chemical Reactions
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Section 57.15 (0G0F): A category of Fourier-Mukai kernels—The Stacks project
Section 57.15: A category of Fourier-Mukai kernels (cite)
57.15 A category of Fourier-Mukai kernels
Let $S$ be a scheme. We claim there is a category with
Objects are proper smooth schemes over $S$.
Morphisms from $X$ to $Y$ are isomorphism classes of objects of $D_{perf}(\mathcal{O}_{X \times _ S Y})$.
Composition of the isomorphism class of $K \in D_{perf}(\mathcal{O}_{X \times _ S Y})$ and the isomorphism class of $K'$ in $D_{perf}(\mathcal{O}_{Y \times _ S Z})$ is the isomorphism class of
\[ R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes _{\mathcal{O}_{X \times _ S Y \times _ S Z}}^\mathbf {L} L\text{pr}_{23}^*K') \]
which is in $D_{perf}(\mathcal{O}_{X \times _ S Z})$ by Derived Categories of Schemes, Lemma 36.30.4.
The identity morphism from $X$ to $X$ is the isomorphism class of $\Delta _{X/S, *}\mathcal{O}_ X$ which is in $D_{perf}(\mathcal{O}_{X \times _ S X})$ by More on Morphisms, Lemma 37.58.12 and the fact that $\Delta _{X/S}$ is a perfect morphism by Divisors, Lemma 31.22.11 and More on Morphisms, Lemma 37.58.7.
Let us check that associativity of composition of morphisms holds; we omit verifying that the identity morphisms are indeed identities. To see this suppose we have $X, Y, Z, W$ and $c \in D_{perf}(\mathcal{O}_{X \times _ S Y})$, $c' \in D_{perf}(\mathcal{O}_{Y \times _ S Z})$, and $c'' \in D_{perf}(\mathcal{O}_{Z \times _ S W})$. Then we have
\begin{align*} c'' \circ (c' \circ c) & \cong \text{pr}^{134}_{14, *}( \text{pr}^{134, *}_{13} \text{pr}^{123}_{13, *}(\text{pr}^{123, *}_{12}c \otimes \text{pr}^{123, *}_{23}c') \otimes \text{pr}^{134, *}_{34}c'') \\ & \cong \text{pr}^{134}_{14, *}( \text{pr}^{1234}_{134, *} \text{pr}^{1234, *}_{123}(\text{pr}^{123, *}_{12}c \otimes \text{pr}^{123, *}_{23}c') \otimes \text{pr}^{134, *}_{34}c'') \\ & \cong \text{pr}^{134}_{14, *}( \text{pr}^{1234}_{134, *} (\text{pr}^{1234, *}_{12}c \otimes \text{pr}^{1234, *}_{23}c') \otimes \text{pr}^{134, *}_{34}c'') \\ & \cong \text{pr}^{134}_{14, *} \text{pr}^{1234}_{134, *} ((\text{pr}^{1234, *}_{12}c \otimes \text{pr}^{1234, *}_{23}c') \otimes \text{pr}^{1234, *}_{34}c'') \\ & \cong \text{pr}^{1234}_{14, *}( (\text{pr}^{1234, *}_{12}c \otimes \text{pr}^{1234, *}_{23}c') \otimes \text{pr}^{1234, *}_{34}c”) \end{align*}
Here we use the notation
\[ p^{1234}_{134} : X \times _ S Y \times _ S Z \times _ S W \to X \times _ S Z \times _ S W \quad \text{and}\quad p^{134}_{14} : X \times _ S Z \times _ S W \to X \times _ S W \]
the projections and similarly for other indices. We also write $\text{pr}_*$ instead of $R\text{pr}_*$ and $\text{pr}^*$ instead of $L\text{pr}^*$ and we drop all super and sub scripts on $\otimes $. The first equality is the definition of the composition. The second equality holds because $\text{pr}^{134, *}_{13} \text{pr}^{123}_{13, *} = \text{pr}^{1234}_{134, *} \text{pr}^{1234, *}_{123}$ by base change (Derived Categories of Schemes, Lemma 36.22.5). The third equality holds because pullbacks compose correctly and pass through tensor products, see Cohomology, Lemmas 20.27.2 and 20.27.3. The fourth equality follows from the “projection formula” for $p^{1234}_{134}$, see Derived Categories of Schemes, Lemma 36.22.1. The fifth equality is that proper pushforward is compatible with composition, see Cohomology, Lemma 20.28.2. Since tensor product is associative this concludes the proof of associativity of composition.
Lemma 57.15.1. Let $S' \to S$ be a morphism of schemes. The rule which sends
a smooth proper scheme $X$ over $S$ to $X' = S' \times _ S X$, and
the isomorphism class of an object $K$ of $D_{perf}(\mathcal{O}_{X \times _ S Y})$ to the isomorphism class of $L(X' \times _{S'} Y' \to X \times _ S Y)^*K$ in $D_{perf}(\mathcal{O}_{X' \times _{S'} Y'})$
is a functor from the category defined for $S$ to the category defined for $S'$.
Proof. To see this suppose we have $X, Y, Z$ and $K \in D_{perf}(\mathcal{O}_{X \times _ S Y})$ and $M \in D_{perf}(\mathcal{O}_{Y \times _ S Z})$. Denote $K' \in D_{perf}(\mathcal{O}_{X' \times _{S'} Y'})$ and $M' \in D_{perf}(\mathcal{O}_{Y' \times _{S'} Z'})$ their pullbacks as in the statement of the lemma. The diagram
\[ \xymatrix{ X' \times _{S'} Y' \times _{S'} Z' \ar[r] \ar[d]_{\text{pr}'_{13}} & X \times _ S Y \times _ S Z \ar[d]^{\text{pr}_{13}} \\ X' \times _{S'} Z' \ar[r] & X \times _ S Z } \]
is cartesian and $\text{pr}_{13}$ is proper and smooth. By Derived Categories of Schemes, Lemma 36.30.4 we see that the derived pullback by the lower horizontal arrow of the composition
\[ R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes _{\mathcal{O}_{X \times _ S Y \times _ S Z}}^\mathbf {L} L\text{pr}_{23}^*M) \]
indeed is (canonically) isomorphic to
\[ R\text{pr}'_{13, *}( L(\text{pr}'_{12})^*K' \otimes _{\mathcal{O}_{X' \times _{S'} Y' \times _{S'} Z'}}^\mathbf {L} L(\text{pr}'_{23})^*M') \]
as desired. Some details omitted. $\square$
Roy Magen points out that in part (4) of the definition we need to replace the references to Lemmas 37.59.18 and 37.58.13 to more general references avoiding Noetherian hypotheses. Here are the details. The diagonal
\Delta : X \to X \times_S X
X
smooth over over a scheme
S
is a regular immersion by Lemma 31.22.11. A regular immersion is perfect by Lemma 37.58.7. And of course if
i : Z \to Y
is a perfect closed immersion of schemes, then
Ri_*\mathcal{O}_Z = i_*\mathcal{O}_Z
is a perfect object of
D(\mathcal{O}_Y)
almost by definition of being a perfect morphism. We're going to have to add this as a lemma because it isn't a lemma yet as far as I can tell. I will fix this soon.
This is now fixed here.
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The Favored Classical Variables to Promote to Quantum Operators
Department of Physics, Department of Mathematics, University of Florida, Gainesville, FL, USA.
Abstract: Classical phase-space variables are normally chosen to promote to quantum operators in order to quantize a given classical system. While classical variables can exploit coordinate transformations to address the same problem, only one set of quantum operators to address the same problem can give the correct analysis. Such a choice leads to the need to find the favored classical variables in order to achieve a valid quantization. This article addresses the task of how such favored variables are found that can be used to properly solve a given quantum system. Examples, such as non-renormalizable scalar fields and gravity, have profited by initially changing which classical variables to promote to quantum operators.
Keywords: Canonical, Spin, Affine Quantization, Coherent States
1. A Brief Story of Three Valid Quantization Procedures
Conventional phase-space variables, such as p and q, where
-\infty <p,q<\infty
, with Poisson brackets
\left\{q,p\right\}=1
, are natural candidates to promote to basic quantum operators in the procedures that canonical quantization employs. However, traditional coordinate transformations, such as
p\to \stackrel{¯}{p}
q\to \stackrel{¯}{q}
-\infty <\stackrel{¯}{p},\stackrel{¯}{q}<\infty
and Poisson brackets
\left\{\stackrel{¯}{q},\stackrel{¯}{p}\right\}=1
are also qualified, in principle, as natural candidates to promote to basic quantum operators. This article resolves the problem of deciding which pair of classical variables to promote to basic quantum operators in order to achieve a valid quantization.
Dirac gave us the clue about which classical variables to choose for canonical quantization [1], although he did not prove his proposal. Dirac proposed that the proper choice of variables were those that were “Cartesian coordinates”. This is not obvious because phase space does not have a metric to exhibit Cartesian coordinates. To fulfill Dirac’s rule a metric space that admits Cartesian coordinates must be added to these particular coordinates. This request requires that the metric of that two-dimensional space must be given by an expression such as
\text{d}\sigma {\left(p,q\right)}^{2}={\omega }^{-1}\text{d}{p}^{2}+\omega \text{d}{q}^{2}
\omega
. This space admits a simple shift of coordinates, e.g.
p\to p+a
q\to q+b
, where a and b are constants. Moreover, this is a property that we wish to feature, and the shift of these two variables has brought one to a different point on the flat surface that is identical in its surroundings as it was before the shift occurred. Clearly the identity in surroundings for a two-dimensional flat surface is a property that distinguishes it from almost all other surfaces that are not completely flat. Also the Poisson brackets
\left\{q+b,p+a\right\}=\left\{q,p\right\}=1
are automatically fulfilled after the shift in location. If we denote the promotion to quantum operators, e.g.
p\to P
q\to Q
, it also follows that
p+a\to P+a1\text{l}
q+b\to Q+b1\text{l}
\left[Q+b1\text{l},P+a1\text{l}\right]=\left[Q,P\right]=i\hslash 1\text{l}
. In brief, a flat surface and the choice of Cartesian coordinates, with or without a and b shifts, lead to acceptable classical variables to promote to become the basic quantum operators.
The only problem deals with how that flat surface, which is linked to both the classical and quantum realms, comes about. As a potential link to other forms of quantization (the spin and affine versions), we call our flat, two-dimensional surface a “constant zero curvature”.
The Role of Canonical Coherent States
We choose to name the favored classical variables that are promoted to quantum operators as p and q, and the valid quantization of these variables named as P and Q. We confirm this choice by first introducing canonical coherent states given by
|p,q〉\equiv {\text{e}}^{-iqP/\hslash }{\text{e}}^{iqQ/\hslash }|\omega 〉,
\left(Q+iP/\omega \right)|\omega 〉=0
〈p,q|P|p,q〉=〈\omega |\left(P+p1\text{l}\right)|\omega 〉=p
〈p,q|Q|p,q〉=〈\omega |\left(Q+q1\text{l}\right)|\omega 〉=q
, which is a clear connection between the quantum and classical basic variables. We finalize this connection with a Fubini-Study metric [2] that involves a tiny ray distance (minimized over non-dynamical phases) between two infinitely close canonical coherent states given by
\begin{array}{c}\text{d}\sigma {\left(p,q\right)}^{2}\equiv 2\hslash \left[{‖d|p,q〉‖}^{2}-{|〈p,q|d|p,q〉|}^{2}\right]\\ ={\omega }^{-1}\text{d}{p}^{2}+\omega \text{d}{q}^{2}.\end{array}
Observe that this process has given us Cartesian coordinates. These variables clearly are invariant even if we choose
p+a
q+b
. These favored coordinates are Cartesian coordinates and they are promoted to valid basic quantum operators, as Dirac had predicted.
1.2. Spin Quantization
The surface of an ideal three-dimensional ball is two-dimensional and spherical with a constant radius; we can say that it has a “constant positive curvature”. Again, like a flat space, the properties at any point on the spherical surface are exactly like those at any other point on the surface. This is the space on which the spin variables appear. There are three spin operators,
{S}_{1}
{S}_{2}
{S}_{3}
, which belong to the groups
SO\left(3\right)
SU\left(2\right)
. These operators obey certain rules, such as
\left[{S}_{1},{S}_{2}\right]=i\hslash {S}_{3}
, and natural permutations, as well as
{S}_{1}^{2}+{S}_{2}^{2}+{S}_{3}^{2}={\hslash }^{2}s\left(s+1\right)1{\text{l}}_{2s+1}
2s+1
is the dimension of the vectors, and the spin s values are
\left(1,2,3,\cdots \right)/2
. Some basic vectors are
|s,m〉
{S}_{3}|s,m〉=m|s,m〉
-s\le m\le s
\left({S}_{1}+i{S}_{2}\right)|s,m〉=|s,m+1〉
\left({S}_{1}+i{S}_{2}\right)|s,s〉=0
The Role of Spin Coherent States
The spin coherent states are given by
|\theta ,\phi 〉\equiv {\text{e}}^{-i\phi {S}_{3}/\hslash }{\text{e}}^{-i\theta {S}_{2}/\hslash }|s,s〉,
-\pi <\phi \le \pi
0\le \theta \le \pi
q={\left(s\hslash \right)}^{1/2}\phi
p={\left(s\hslash \right)}^{1/2}cos\left(\theta \right)
\begin{array}{c}\text{d}\sigma {\left(\theta ,\phi \right)}^{2}\equiv 2\hslash \left[{‖d|\theta ,\phi 〉‖}^{2}-{|〈\theta ,\phi |d|\theta ,\phi 〉|}^{2}\right]\\ =\left(s\hslash \right)\left[\text{d}{\theta }^{2}+\mathrm{sin}{\left(\theta \right)}^{2}\text{d}{\phi }^{2}\right],\end{array}
\begin{array}{c}\text{d}\sigma {\left(p,q\right)}^{2}\equiv 2\hslash \left[{‖d|p,q〉‖}^{2}-{|〈p,q|d|p,q〉|}^{2}\right]\\ ={\left(1-{p}^{2}/s\hslash \right)}^{-1}\text{d}{p}^{2}+\left(1-{p}^{2}/s\hslash \right)\text{d}{q}^{2}.\end{array}
Equation (4) makes it clear that we are dealing with a spherical surface with a radius of
{\left(s\hslash \right)}^{1/2}
. Equation (5) makes it clear that if
s\to \infty
, in which case both p and q span the real line, we will have recovered properties of canonical quantization.
So far we have obtained surfaces with a constant zero curvature and a constant positive curvature. Could there be more? Could there be surfaces with a “constant negative curvature”?
1.3. Affine Quantization
One of the simplest problems to quantize is an harmonic oscillator for which
-\infty <p,q<\infty
, but it is not so simple if
0<q<\infty
. To solve the second version requires a new method of quantization called affine quantization, which, as we will discover, involves a constant negative curvature. To introduce this procedure let us focus on the classical term
p\text{d}q
which is part of a classical action functional. Instead of these variable’s range being
-\infty <p,q<\infty
, let us assume q is limited to
0<q<\infty
, and we want to change variables. As a first step, let us consider
p\text{d}q=pq\text{d}q/q=pqd\mathrm{ln}\left(q\right)
. While q must be positive,
ln\left(q\right)
covers the whole real line. Although that
p\to pq
q\to ln\left(q\right)
both cover the whole real line, we instead just choose pq and q as our new variables. A potential quantization of this pair of variables could involve
q\to Q
0<Q<\infty
pq\to \left(PQ+QP\right)/2\equiv D
. Note, if
0<Q<\infty
, then P cannot be self adjoint; however, thanks to Q, D can be self adjoint, which is a very important advantage. The two basic operators for affine quantization then are D and Q, for which
\left[Q,D\right]=i\hslash Q
The Role of Affine Coherent States
The affine coherent states, where both q and Q have been chosen dimensionless for simplicity, are given by
|p;q〉\equiv {\text{e}}^{ipQ/\hslash }\text{ }{\text{e}}^{-i\mathrm{ln}\left(q\right)D/\hslash }|b〉,
\left[\left(Q-1\right)+iD/b\hslash \right]|b〉=0
. For these variables we find that
〈p;q|Q|p;q〉=〈b|qQ|b〉=q
〈p;q|D|p;q〉=〈b|D+pqQ|b〉=pq
\begin{array}{c}\text{d}\sigma {\left(p,q\right)}^{2}\equiv 2\hslash \left[{‖d|p,q〉‖}^{2}-{|〈p,q|d|p,q〉|}^{2}\right]\\ ={\left(b\hslash \right)}^{-1}{q}^{2}\text{d}{p}^{2}+\left(b\hslash \right){q}^{-2}\text{d}{q}^{2}.\end{array}
This expression for the Fubini-Study metric is that of a constant negative curvature, an amount of
-2/\left(b\hslash \right)
, which is also geodetically complete [3]. Surfaces that are constant negative curvature are only visible in our three-dimensional space at the “center point” only, namely at
q=1
, where it appears that in one direction the surface goes down and 90 degrees away the surface goes up. In reality that is what happens at all points for a constant negative curvature, but we cannot see that effect. Let us give a
q\to q+{b}^{\prime }
test, with
{b}^{\prime }>0
; we use
{b}^{\prime }
so as to make sure this is different from the b that labels the fiducial vector. In that case, we now have
{\left(b\hslash \right)}^{-1}{\left(q+{b}^{\prime }\right)}^{2}\text{d}{p}^{2}+\left(b\hslash \right){\left(q+{b}^{\prime }\right)}^{-2}\text{d}{q}^{2}
. This version now implies that
-{b}^{\prime }<q<\infty
. Indeed, we can let
{b}^{\prime }\to \infty
and at the same time, let the factor b in the fiducial vector become large (
b\propto {{b}^{\prime }}^{2}
) in which case we can arrange that the negative curvature
-2/b\hslash \to 0
, in such a way that the Fubini-Study metric effectively passes to
{B}^{-1}\text{d}{p}^{2}+B\text{d}{q}^{2}
, namely, the constant zero curvature case, with a non-dynamical positive B. Basically, we have effectively changed the affine operators Q and D to canonical operators, by
Q\to Q+{b}^{\prime }1\text{l}
D\to D+{b}^{\prime }P
so that, as
{b}^{\prime }\to \infty
\left[Q+{b}^{\prime }1\text{l},D+{b}^{\prime }P\right]/{b}^{\prime }=i\hslash \left(Q+{b}^{\prime }1\text{l}\right)/{b}^{\prime }\to \left[Q,P\right]=i\hslash 1\text{l}
Our quantization of classical variables has now been completed. It was shown that classical variables that represent the coordinates of constant positive, zero, and negative curvatures, complete the natural forms of surface and these three divisions include a different variety of classical systems that can be quantized. Affine quantization, as a special procedure to quantize systems, has not yet become universally well-known and exploited; it deserves more attention. Besides the harmonic oscillator with
0<q<\infty
(see [4] ), there are other problems for which affine quantization can work well. Quite recently, a surprisingly transparent version of affine quantization has been used for non-renormalizable covariant scalar fields and for Einstein’s gravity [5].
Others are encouraged to see what affine quantization can do for their own quantization problems.
1Besides
0<q,Q<\infty
, one may also consider
-\infty <q,Q<0
-\infty <q\ne 0,Q\ne 0<\infty
Cite this paper: Klauder, J. (2020) The Favored Classical Variables to Promote to Quantum Operators. Journal of High Energy Physics, Gravitation and Cosmology, 6, 828-832. doi: 10.4236/jhepgc.2020.64055.
[1] Dirac, P.A.M. (1958) The Principles of Quantum Mechanics. Claredon Press, Oxford, 114.
[2] Fubini-Study Metrics. Wikipedia.
[3] http://en.wikipedia.org/wiki/Poincarè_half-plane_model
[5] Klauder, J.R. Using Affine Quantization to Analyze Non-Renormalizable Scalar Fields and the Quantization of Einstein’s Gravity. arXiv:2006.09156.
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((1-6)-alpha-D-xylo)-(1-4)-beta-D-glucan exo-glucohydrolase Wikipedia
((1-6)-alpha-D-xylo)-(1-4)-beta-D-glucan exo-glucohydrolase
In enzymology, a xyloglucan-specific exo-beta-1,4-glucanase (EC 3.2.1.155) is an enzyme that catalyzes the chemical reaction
xyloglucan + H2O
{\displaystyle \rightleftharpoons }
xyloglucan oligosaccharides (exohydrolysis of 1,4-beta-D-glucosidic linkages in xyloglucan)
Thus, the two substrates of this enzyme are xyloglucan and H2O, and its products are xyloglucan oligosaccharides (exohydrolysis of 1,4-beta-D-glucosidic linkages in xyloglucan).
This enzyme belongs to the family of hydrolases, specifically those glycosidases that hydrolyse O- and S-glycosyl compounds. The systematic name of this enzyme class is [(1->6)-alpha-D-xylo]-(1->4)-beta-D-glucan exo-glucohydrolase. This enzyme is also called Cel74A.
Grishutin SG, Gusakov AV, Markov AV, Ustinov BB, Semenova MV, Sinitsyn AP (November 2004). "Specific xyloglucanases as a new class of polysaccharide-degrading enzymes". Biochimica et Biophysica Acta (BBA) - General Subjects. 1674 (3): 268–81. doi:10.1016/j.bbagen.2004.07.001. PMID 15541296.
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Minh Ôn Vũ Ngoc, Yizi Chen, Nicolas Boutry, Joseph Chazalon, Edwin Carlinet, Jonathan Fabrizio, Clément Mallet, Thierry Géraud
Most contemporary supervised image segmentation methods do not preserve the initial topology of the given input (like the closeness of the contours). One can generally remark that edge points have been inserted or removed when the binary prediction and the ground truth are compared. This can be critical when accurate localization of multiple interconnected objects is required. In this paper, we present a new loss function, called, Boundary-Aware loss (BALoss), based on the Minimum Barrier Distance (MBD) cut algorithm. It is able to locate what we call the it leakage pixels and to encode the boundary information coming from the given ground truth. Thanks to this adapted loss, we are able to significantly refine the quality of the predicted boundaries during the learning procedure. Furthermore, our loss function is differentiable and can be applied to any kind of neural network used in image processing. We apply this loss function on the standard U-Net and DC U-Net on Electron Microscopy datasets. They are well-known to be challenging due to their high noise level and to the close or even connected objects covering the image space. Our segmentation performance, in terms of Variation of Information (VOI) and Adapted Rank Index (ARI), are very promising and lead to
{\displaystyle \approx {}15\%}
{\displaystyle \approx {}5\%}
author = {Minh \^On V\~{u} Ng\d{o}c and Yizi Chen and Nicolas Boutry
and Joseph Chazalon and Edwin Carlinet and Jonathan
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Proposition 15.49.2 (07PM)—The Stacks project
Section 15.49: Formal smoothness and regularity
Proposition 15.49.2. Let $A \to B$ be a local homomorphism of Noetherian complete local rings. Let $k$ be the residue field of $A$ and $\overline{B} = B \otimes _ A k$ the special fibre. The following are equivalent
$A \to B$ is regular,
$A \to B$ is flat and $\overline{B}$ is geometrically regular over $k$,
$A \to B$ is flat and $k \to \overline{B}$ is formally smooth in the $\mathfrak m_{\overline{B}}$-adic topology, and
$A \to B$ is formally smooth in the $\mathfrak m_ B$-adic topology.
Proof. We have seen the equivalence of (2), (3), and (4) in Proposition 15.40.5. It is clear that (1) implies (2). Thus we assume the equivalent conditions (2), (3), and (4) hold and we prove (1).
Let $\mathfrak p$ be a prime of $A$. We will show that $B \otimes _ A \kappa (\mathfrak p)$ is geometrically regular over $\kappa (\mathfrak p)$. By Lemma 15.37.8 we may replace $A$ by $A/\mathfrak p$ and $B$ by $B/\mathfrak pB$. Thus we may assume that $A$ is a domain and that $\mathfrak p = (0)$.
Choose $A_0 \subset A$ as in Algebra, Lemma 10.160.11. We will use all the properties stated in that lemma without further mention. As $A_0 \to A$ induces an isomorphism on residue fields, and as $B/\mathfrak m_ A B$ is geometrically regular over $A/\mathfrak m_ A$ we can find a diagram
\[ \xymatrix{ C \ar[r] & B \\ A_0 \ar[r] \ar[u] & A \ar[u] } \]
with $A_0 \to C$ formally smooth in the $\mathfrak m_ C$-adic topology such that $B = C \otimes _{A_0} A$, see Remark 15.40.7. (Completion in the tensor product is not needed as $A_0 \to A$ is finite, see Algebra, Lemma 10.97.1.) Hence it suffices to show that $C \otimes _{A_0} K_0$ is a geometrically regular algebra over the fraction field $K_0$ of $A_0$.
The upshot of the preceding paragraph is that we may assume that $A = k[[x_1, \ldots , x_ n]]$ where $k$ is a field or $A = \Lambda [[x_1, \ldots , x_ n]]$ where $\Lambda $ is a Cohen ring. In this case $B$ is a regular ring, see Algebra, Lemma 10.112.8. Hence $B \otimes _ A K$ is a regular ring too (where $K$ is the fraction field of $A$) and we win if the characteristic of $K$ is zero.
Thus we are left with the case where $A = k[[x_1, \ldots , x_ n]]$ and $k$ is a field of characteristic $p > 0$. Let $L/K$ be a finite purely inseparable field extension. We will show by induction on $[L : K]$ that $B \otimes _ A L$ is regular. The base case is $L = K$ which we've seen above. Let $K \subset M \subset L$ be a subfield such that $L$ is a degree $p$ extension of $M$ obtained by adjoining a $p$th root of an element $f \in M$. Let $A'$ be a finite $A$-subalgebra of $M$ with fraction field $M$. Clearing denominators, we may and do assume $f \in A'$. Set $A'' = A'[z]/(z^ p -f)$ and note that $A' \subset A''$ is finite and that the fraction field of $A''$ is $L$. By induction we know that $B \otimes _ A M$ ring is regular. We have
\[ B \otimes _ A L = B \otimes _ A M[z]/(z^ p - f) \]
By Lemma 15.48.5 we know there exists a derivation $D : A' \to A'$ such that $D(f) \not= 0$. As $A' \to B \otimes _ A A'$ is formally smooth in the $\mathfrak m$-adic topology by Lemma 15.37.9 we can use Lemma 15.49.1 to extend $D$ to a derivation $D' : B \otimes _ A A' \to B \otimes _ A A'$. Note that $D'(f) = D(f)$ is a unit in $B \otimes _ A M$ as $D(f)$ is not zero in $A' \subset M$. Hence $B \otimes _ A L$ is regular by Lemma 15.48.4 and we win. $\square$
Comment #6294 by Ehsan on June 23, 2021 at 04:37
In the statement it is not written what is
k
. Although I know that it is the residue field of
A
but perhaps it is better to write what is
k
explicity?
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Zhiwei Gao, Dexing Kong, Chuanhou Gao, Michael Chen
Distributed Adaptive Synchronization for Complex Dynamical Networks with Uncertain Nonlinear Neutral-Type Coupling
Shi Miao, Li Junmin
Distributed adaptive synchronization control for complex dynamical networks with nonlinear derivative coupling is proposed. The distributed adaptive strategies are constituted by directed connections among nodes. By means of the parameters separation, the nonlinear functions can be transformed into the linearly form. Then effective distributed adaptive techniques are designed to eliminate the effect of time-varying parameters and made the considered network synchronize a given trajectory in the sense of square error norm. Furthermore, the coupling matrix is not assumed to be symmetric or irreducible. An example shows the applicability and feasibility of the approach.
{H}_{\infty }
Formation Control and Obstacle Avoidance for Hybrid Multi-Agent Systems
Dong Xue, Jing Yao, Jun Wang
A new concept of
{H}_{\infty }
formation is proposed to handle a group of agents navigating in a free and an obstacle-laden environment while maintaining a desired formation and changing formations when required. With respect to the requirements of changing formation subject to internal or external events, a hybrid multiagent system (HMAS) is formulated in this paper. Based on the fact that obstacles impose the negative effect on the formation of HMAS, the
{H}_{\infty }
formation is introduced to reflect the above disturbed situation and quantify the attenuation level of obstacle avoidance via the
{H}_{\infty }
-norm of formation stability. An improved Newtonian potential function and a set of repulsive functions are employed to guarantee the HMAS formation-keeping and collision-avoiding from obstacles in a path planning problem, respectively. Simulation results in this paper show that the proposed formation algorithms can effectively allow the multiagent system to avoid penetration into obstacles while accomplishing prespecified global objective successfully.
Yang Yu, Zengqiang Mi
The structural scheme of mechanical elastic energy storage (MEES) system served by permanent magnet synchronous motor (PMSM) and bidirectional converters is designed. The aim of the research is to model and control the complex electromechanical system. The mechanical device of the complex system is considered as a node in generalized coordinate system, the terse nonlinear dynamic model of electromechanical coupling for the electromechanical system is constructed through Lagrange-Maxwell energy method, and the detailed deduction of the mathematical model is presented in the paper. The theory of direct feedback linearization (DFL) is applied to decouple the nonlinear dynamic model and convert the developed model from nonlinear to linear. The optimal control theory is utilized to accomplish speed tracking control for the linearized system. The simulation results in three different cases show that the proposed nonlinear dynamic model of MEES system is correct; the designed algorithm has a better control performance in contrast with the conventional PI control.
Reza Banaei Khosroushahi, Horacio J. Marquez, Jose Martinez-Quijada, Christopher J. Backhouse
This paper details the infinite dimensional dynamics of a prototype microfluidic thermal process that is used for genetic analysis purposes. Highly effective infinite dimensional dynamics, in addition to collocated sensor and actuator architecture, require the development of a precise control framework to meet the very tight performance requirements of this system, which are not fully attainable through conventional lumped modeling and controller design approaches. The general partial differential equations describing the dynamics of the system are separated into steady-state and transient parts which are derived for a carefully chosen three-dimensional axisymmetric model. These equations are solved analytically, and the results are verified using an experimentally verified precise finite element method (FEM) model. The final combined result is a framework for designing a precise tracking controller applicable to the selected lab-on-a-chip device.
Block Empirical Likelihood for Longitudinal Single-Index Varying-Coefficient Model
Yunquan Song, Ling Jian, Lu Lin
In this paper, we consider a single-index varying-coefficient model with application to longitudinal data. In order to accommodate the within-group correlation, we apply the block empirical likelihood procedure to longitudinal single-index varying-coefficient model, and prove a nonparametric version of Wilks’ theorem which can be used to construct the block empirical likelihood confidence region with asymptotically correct coverage probability for the parametric component. In comparison with normal approximations, the proposed method does not require a consistent estimator for the asymptotic covariance matrix, making it easier to conduct inference for the model's parametric component. Simulations demonstrate how the proposed method works.
Adaptive Synchronization of Complex Dynamical Networks with State Predictor
Yuntao Shi, Bo Liu, Xiao Han
This paper addresses the adaptive synchronization of complex dynamical networks with nonlinear dynamics. Based on the Lyapunov method, it is shown that the network can synchronize to the synchronous state by introducing local adaptive strategy to the coupling strengths. Moreover, it is also proved that the convergence speed of complex dynamical networks can be increased via designing a state predictor. Finally, some numerical simulations are worked out to illustrate the analytical results.
An Irregular Flight Scheduling Model and Algorithm under the Uncertainty Theory
Deyi Mou, Wanlin Zhao
The flight scheduling is a real-time optimization problem. Whenever the schedule is disrupted, it will not only cause inconvenience to passenger, but also bring about a large amount of operational losses to airlines. Especially in case an irregular flight happens, the event is unanticipated frequently. In order to obtain an optimal policy in airline operations, this paper presents a model in which the total delay minutes of passengers are considered as the optimization objective through reassigning fleets in response to the irregular flights and which takes into account available resources and the estimated cost of airlines. Owing to the uncertainty of the problem and insufficient data in the decision-making procedure, the traditional modeling tool (probability theory) is abandoned, the uncertainty theory is applied to address the issues, and an uncertain programming model is developed with the chance constraint. This paper also constructs a solution method to solve the model based on the classical Hungarian algorithm under uncertain conditions. Numerical example illustrates that the model and its algorithm are feasible to deal with the issue of irregular flight recovery.
Dynamics Control of the Complex Systems via Nondifferentiability
Carmen Nejneru, Anca Nicuţă, Boris Constantin, Liliana Rozemarie Manea, Mirela Teodorescu, Maricel Agop
A new topic in the analyses of complex systems dynamics, considering that the movements of complex system entities take place on continuum but nondifferentiable curves, is proposed. In this way, some properties of complex systems (barotropic-type behaviour, self-similarity behaviour, chaoticity through turbulence and stochasticization, etc.) are controlled through nondifferentiability of motion curves. These behaviours can simulate the standard properties of the complex systems (emergence, self-organization, adaptability, etc.).
Modeling and Analysis of Epidemic Diffusion with Population Migration
Ming Liu, Yihong Xiao
An improved Susceptible-Infected-Susceptible (SIS) epidemic diffusion model with population migration between two cities is modeled. Global stability conditions for both the disease-free equilibrium and the endemic equilibrium are analyzed and proved. The main contribution of this paper is reflected in epidemic modeling and analysis which considers unequal migration rates, and only susceptible individuals can migrate between the two cities. Numerical simulation shows when the epidemic diffusion system is stable, number of infected individuals in one city can reach zero, while the number of infected individuals in the other city is still positive. On the other hand, decreasing population migration in only one city seems not as effective as improving the recovery rate for controlling the epidemic diffusion.
A New Type of Distributed Parameter Control Systems: Two-Point Boundary Value Problems for Infinite-Dimensional Dynamical Systems
De-Xing Kong, Fa Wu
This survey note describes a new type of distributed parameter control systems—the two-point boundary value problems for infinite-dimensional dynamical systems, particularly, for hyperbolic systems of partial differential equations of second order, some of the discoveries that have been done about it and some unresolved questions.
Analysis of Average Shortest-Path Length of Scale-Free Network
Guoyong Mao, Ning Zhang
Computing the average shortest-path length of a large scale-free network needs much memory space and computation time. Hence, parallel computing must be applied. In order to solve the load-balancing problem for coarse-grained parallelization, the relationship between the computing time of a single-source shortest-path length of node and the features of node is studied. We present a dynamic programming model using the average outdegree of neighboring nodes of different levels as the variable and the minimum time difference as the target. The coefficients are determined on time measurable networks. A native array and multimap representation of network are presented to reduce the memory consumption of the network such that large networks can still be loaded into the memory of each computing core. The simplified load-balancing model is applied on a network of tens of millions of nodes. Our experiment shows that this model can solve the load-imbalance problem of large scale-free network very well. Also, the characteristic of this model can meet the requirements of networks with ever-increasing complexity and scale.
A Data-Based Approach for Modeling and Analysis of Vehicle Collision by LPV-ARMAX Models
Qiugang Lu, Hamid Reza Karimi, Kjell Gunnar Robbersmyr
Vehicle crash test is considered to be the most direct and common approach to assess the vehicle crashworthiness. However, it suffers from the drawbacks of high experiment cost and huge time consumption. Therefore, the establishment of a mathematical model of vehicle crash which can simplify the analysis process is significantly attractive. In this paper, we present the application of LPV-ARMAX model to simulate the car-to-pole collision with different initial impact velocities. The parameters of the LPV-ARMAX are assumed to have dependence on the initial impact velocities. Instead of establishing a set of LTI models for vehicle crashes with various impact velocities, the LPV-ARMAX model is comparatively simple and applicable to predict the responses of new collision situations different from the ones used for identification. Finally, the comparison between the predicted response and the real test data is conducted, which shows the high fidelity of the LPV-ARMAX model.
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Mount&Blade/Skills — StrategyWiki, the video game walkthrough and strategy guide wiki
Mount&Blade/Skills
Skills are divided into three categories: personal, party, and leadership skills. All skills range from level 0-10 (although for party skills, the effective level can exceed this with bonuses). Skills can be improved by spending skill points. Some skill points are given during character creation, one skill point is earned per level up, and also one extra skill point is rewarded per attribute point spent on Intelligence. Each skill has a base attribute that limits the skill to 1/3 of the attribute's value (rounded down). For example, with a STR of 14 Ironflesh may not exceed level 4. Starting at STR 15, Ironflesh may be increased to 5. These attribute limits can be surpassed using bonuses from books, but cannot exceed level 10.
1.1 Ironflesh
1.3 Power Throw
2 Party Skills
2.4 Path-finding
2.6 Wound Treatment
3.3 Prisoner Management
Personal Skills[edit]
These are skills you or a companion can possess. They don't affect the party.
Ironflesh[edit]
Base Attribute: Strength
Effect: +2 HP per level
At early levels it isn't very important, but as you fight tougher enemies every bit of protection helps. This skill has a very large effect on determining whether an NPC survives autoresolved combat.
Effect: +8% melee damage per level
Power Throw[edit]
Effect: +10% thrown weapon damage per level
More powerful throwing weapons cannot be used until this skill is sufficiently high.
Power Draw[edit]
Effect: +14% bow damage for each point (up to 4 plus Power Draw requirement of the bow). Also increases how long the character can aim for before losing accuracy.
This is required for the more powerful bows, and is overall a valuable skill for making archery less difficult. You will benefit from up to four points in this beyond the power draw requirement of the bow (if any).
E.g. if you have a bow that requires Power Draw 2, you may have up to 6 points in this skill in order to add damage to your arrow. If you add a seventh point to this skill, you will not receive an additional +14% damage bonus to your arrow above the maximum 6 points:
{\displaystyle MAXPowerDrawBonus=BowPowerDrawReq.+4}
Base Attribute: Agility
Effect: Each point increases by 40 the limit at which weapon proficiencies can be raised with weapon proficiency points. Also increases the rate at which proficiencies are improved by use in combat.
This skill allows you to continue using weapon proficiency points after you have passed the normal cap.
Weapon Master level
This cap is only for the purpose of spending additional proficiency points; using weapons in combat will continue to raise your skill regardless of what the proficiency cap is set at, but it will do so faster at higher levels of Weapon Master.
Effect: Reduces damage to shields by 8% per level and improves shield speed and coverage.
Due to a bug, raising this skill also raises the Shield skill of enemies, and is therefore a poor choice for characters who make use of ranged weapons. This bug is fixed in Warband.
Cleanup required: Don't need the analysis here, we need the guide
Effect: Increases running speed.
This isn't very obvious early on, but with a few levels it will start to really take effect. This skill is made less important by having a horse, but it's still a useful backup if you lose your horse or if you want to fight on foot. It is also useful during sieges and tournaments, where a horse might not be available. Note that the combined weight of your armor and equipment will reduce the speed bonus your Athletics skill grants.
The goal of the following analysis is to provide an accurate account of the effects of the athletic skill.
To acquire this data, a contributor performed a series of timing measurements from a specific point, A, to a different specific point, B, in the game world. Different encumbrances and athletic skill points were used. Therefore, running speed is not known in units. Speed is only known relative to another based on Athletic skill and encumbrance differences. This contributor did not test for speed increase of more than base running speed. All calculations are an estimate. Agility attribute during the measurements was 15 or 18.
Running speed increase per Athletic skill level is linear.
Running speed change (+/-) per unit encumbrance is linear, and a decrease in running speed from encumbrance occurs at greater than 0. Base running speed = running speed at 0 encumbrance and 0 Athletic skill.
50 units of encumbrance decreases running speed by about 22%, or .44% speed reduction per 1 unit.
1 Athletic skill point is equivalent to about 4.16 units less encumbrance, or 1.83% increase in running speed.
Encumbrance 50 with 0 Athletic skill points reduces base running speed by 22%.
Encumbrance 50 with 2 Athletic skill points reduces base running speed by 18.3%.
Encumbrance 50 with 4 Athletic skill points reduces base running speed by ??% (estimated 14.64%).
Encumbrance 35 with 4 Athletic skill points reduces base running speed by 8.1%.
Effect: Increases riding speed and maneuver, allows riding of more difficult horses.
Horse Archery[edit]
Effect: Reduces penalties for using ranged weapons on a moving horse by 10% per level.
Reduces the aiming reticule spread while mounted, allowing for much greater accuracy. All ranged weapons: bows, crossbows, throwing weapons and firearms are affected by this skill.
Note that the negative effects of being mounted and using ranged weapons are only applied while you are actually moving, thus this skill has no effect if you are mounted but stationary.
Base Attribute: Intelligence
Effect: Every day at midnight lower-level party members gain experience
Experience points are not given to party members that are fully upgraded. If multiple companions have this skill they will each help train party members.
Experience gains per level are as follows:
EXP to each party member per day
This skill also helps you train peasants against bandits faster.
The total experience points given to a soldier is the amount from the training skill multiplied by the number of levels the player or companions are over the soldier to be trained.
For instance, if you are level 30 and have 1 point in training and are training a level 1 recruit you will give 116 experience points to the recruit.
That's 30-1= 29. Training level of 1 gives 4 points of experience.
Stack this with companions and higher skill levels and you will be able to train even the highest tiered troops relatively quickly.
Party Skills[edit]
0-1 pts (+0)
These are skills that each character individually possesses, but they are applied to the party as a whole. If the party leader (you) has these skills, a bonus to the skill is applied as per the table to the right. The leader's bonus is applied to the party even if a companion has a higher skill level than you. An additional (+1) can be earned from certain Books. If characters are listed as Wounded their Party skills will be disabled until they regain some health.
Increases the amount of loot obtained by 10% per skill level.
Looting also increases the quality of the loot. For example, at level 1 looting you will find mostly rusty, broken and cracked prefixes, negative ones. If your looting is 10, you will find more items with positive prefixes such as balanced, reinforced, or tempered. This skill works in village looting, and in any other battle or siege. It also increases the number of cattle obtained when stealing them from a village.
Effect: Makes onscreen tracks appear and gradually become more informative. Also increases the distance at which tracks can be seen.
This skill makes tracks appear on the world map, gradually becoming more accurate and providing additional information. At level 1 the tracks indicate party movement, but little else. As levels progress the accuracy of party size predictions and hours the tracks remain visible for increases. Tracks will also gain colors, allowing you to easily follow a particular path regardless of what other paths it crosses.
Effect: Every 2 points to this raises your battle advantage by 1.
Before each battle your Tactics score is compared to that of the enemy leader, and this determines how many men each side has at the beginning of the battle.
Additionally, if you choose to send your men in while you stay back or are knocked out during battle this skill will have an effect on the aftermath losses. This skill also reduces how many men you need to leave behind if you retreat.
Path-finding[edit]
Effect: Increases party map speed by 3% per level.
Effect: Increases party sight range by 10% per level.
Wound Treatment[edit]
Effect: Increases party healing speed 20% per level. Also applies to the healing of crippled horses in the inventory.
Effect: Each point adds 4% to the chance that struck down troops will be knocked out rather than killed. This is added to a base chance of 25%.
This skill is quite useful for giving your troops the best chance you can to reach higher levels and become effective fighters. It also lessens the annoyance factor of spending top dollar on some new, high tier troops only to have a significant fraction of them be killed in their first battle.
Effect: Heroes regain 5% per level of hit-points lost during a particular battle. This is added to a base rate of 10%.
This skill takes effect both between battle waves and after combat is over. This only heals up to the level of health before you entered battle, so if you go into battle with low health you can only heal back up to that level and no further.
Effect: Build siege equipment and village improvements quicker (-5% on time and price to build improvements)
This skill also affects the cost of building village improvements.
Base Attribute: Charisma
Effect: Reduces trading penalty by 5%
The Trade skill reduces the cost penalty applied to buying and selling. See Trade for more details. Trade also reduces the time it takes to collect taxes for tax collection quests.
Leader Skills[edit]
These skills affect the party, but only if the person with these skills is the party leader.
Effect: Increases inventory size by 6 squares
Very important when pillaging villages or attacking caravans.
Effect: Helps you make other people accept your point of view. There is a random factor involved in determining how successful you are.
Useful for persuading Lords to pay debts or support peace. Also used for persuading lords to rebel and reduce chances that recruited prisoners will run. Perhaps most importantly, it can be used to keep your companions from leaving your group due to low morale.
Prisoner Management[edit]
Effect: Increases prisoner capacity by 5
Needed if you want to recruit or sell prisoners. Also needed for the Bring Prisoners quest. This skill has no effect for NPC lords, who have no limit to the number of prisoners they can take.
Effect: Increases party capacity by 5, boosts party morale by 12, and reduces troop costs and wages by 5%.
The higher your Leadership, the more men you can lead into battle. You'll probably want to put a point towards Charisma every other level or so, and a point towards Leadership every chance you get. Increasing your Renown will also raise the party size cap.
Retrieved from "https://strategywiki.org/w/index.php?title=Mount%26Blade/Skills&oldid=681435"
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(-)-alpha-pinene,NADH:oxygen oxidoreductase Wikipedia
(-)-alpha-pinene,NADH:oxygen oxidoreductase
Alpha-pinene monooxygenase (EC 1.14.13.155) is an enzyme with systematic name (-)-alpha-pinene,NADH:oxygen oxidoreductase.[1] This enzyme catalyses the following chemical reaction
(-)-alpha-pinene + NADH + H+ + O2
{\displaystyle \rightleftharpoons }
alpha-pinene oxide + NAD+ + H2O
Alpha-pinene monooxygenase takes part in catabolism of alpha-pinene.
^ Colocousi A, Saqib KM, Leak DJ (1996). "Mutants of Pseudomonas fuorescence NCIMB 11671 defective in the catabolism of α-pinene". Appl. Microbiol. Biotechnol. 45: 822–830. doi:10.1007/s002530050769.
Alpha-pinene+monooxygenase at the US National Library of Medicine Medical Subject Headings (MeSH)
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Section 47.8 (0BJA): Deriving torsion—The Stacks project
Section 47.8: Deriving torsion (cite)
47.8 Deriving torsion
Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal (if $I$ is not finitely generated perhaps a different definition should be used). Let $Z = V(I) \subset \mathop{\mathrm{Spec}}(A)$. Recall that the category $I^\infty \text{-torsion}$ of $I$-power torsion modules only depends on the closed subset $Z$ and not on the choice of the finitely generated ideal $I$ such that $Z = V(I)$, see More on Algebra, Lemma 15.88.6. In this section we will consider the functor
\[ H^0_{I} : \text{Mod}_ A \longrightarrow I^\infty \text{-torsion},\quad M \longmapsto M[I^\infty ] = \bigcup M[I^ n] \]
which sends $M$ to the submodule of $I$-power torsion.
Let $A$ be a ring and let $I$ be a finitely generated ideal. Note that $I^\infty \text{-torsion}$ is a Grothendieck abelian category (direct sums exist, filtered colimits are exact, and $\bigoplus A/I^ n$ is a generator by More on Algebra, Lemma 15.88.2). Hence the derived category $D(I^\infty \text{-torsion})$ exists, see Injectives, Remark 19.13.3. Our functor $H^0_ I$ is left exact and has a derived extension which we will denote
\[ R\Gamma _ I : D(A) \longrightarrow D(I^\infty \text{-torsion}). \]
Warning: this functor does not deserve the name local cohomology unless the ring $A$ is Noetherian. The functors $H^0_ I$, $R\Gamma _ I$, and the satellites $H^ p_ I$ only depend on the closed subset $Z \subset \mathop{\mathrm{Spec}}(A)$ and not on the choice of the finitely generated ideal $I$ such that $V(I) = Z$. However, we insist on using the subscript $I$ for the functors above as the notation $R\Gamma _ Z$ is going to be used for a different functor, see (47.9.0.1), which agrees with the functor $R\Gamma _ I$ only (as far as we know) in case $A$ is Noetherian (see Lemma 47.10.1).
Lemma 47.8.1. Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. The functor $R\Gamma _ I$ is right adjoint to the functor $D(I^\infty \text{-torsion}) \to D(A)$.
Proof. This follows from the fact that taking $I$-power torsion submodules is the right adjoint to the inclusion functor $I^\infty \text{-torsion} \to \text{Mod}_ A$. See Derived Categories, Lemma 13.30.3. $\square$
Lemma 47.8.2. Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. For any object $K$ of $D(A)$ we have
\[ R\Gamma _ I(K) = \text{hocolim}\ R\mathop{\mathrm{Hom}}\nolimits _ A(A/I^ n, K) \]
in $D(A)$ and
\[ R^ q\Gamma _ I(K) = \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Ext}}\nolimits _ A^ q(A/I^ n, K) \]
as modules for all $q \in \mathbf{Z}$.
Proof. Let $J^\bullet $ be a K-injective complex representing $K$. Then
\[ R\Gamma _ I(K) = J^\bullet [I^\infty ] = \mathop{\mathrm{colim}}\nolimits J^\bullet [I^ n] = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ A(A/I^ n, J^\bullet ) \]
where the first equality is the definition of $R\Gamma _ I(K)$. By Derived Categories, Lemma 13.33.7 we obtain the first displayed equality in the statement of the lemma. The second displayed equality in the statement of the lemma then follows because $H^ q(\mathop{\mathrm{Hom}}\nolimits _ A(A/I^ n, J^\bullet )) = \mathop{\mathrm{Ext}}\nolimits ^ q_ A(A/I^ n, K)$ and because filtered colimits are exact in the category of abelian groups. $\square$
Lemma 47.8.3. Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Let $K^\bullet $ be a complex of $A$-modules such that $f : K^\bullet \to K^\bullet $ is an isomorphism for some $f \in I$, i.e., $K^\bullet $ is a complex of $A_ f$-modules. Then $R\Gamma _ I(K^\bullet ) = 0$.
Proof. Namely, in this case the cohomology modules of $R\Gamma _ I(K^\bullet )$ are both $f$-power torsion and $f$ acts by automorphisms. Hence the cohomology modules are zero and hence the object is zero. $\square$
Let $A$ be a ring and $I \subset A$ a finitely generated ideal. By More on Algebra, Lemma 15.88.5 the category of $I$-power torsion modules is a Serre subcategory of the category of all $A$-modules, hence there is a functor
see Derived Categories, Section 13.17.
Lemma 47.8.4. Let $A$ be a ring and let $I$ be a finitely generated ideal. Let $M$ and $N$ be $I$-power torsion modules.
$\mathop{\mathrm{Hom}}\nolimits _{D(A)}(M, N) = \mathop{\mathrm{Hom}}\nolimits _{D({I^\infty \text{-torsion}})}(M, N)$,
$\mathop{\mathrm{Ext}}\nolimits ^1_{D(A)}(M, N) = \mathop{\mathrm{Ext}}\nolimits ^1_{D({I^\infty \text{-torsion}})}(M, N)$,
$\mathop{\mathrm{Ext}}\nolimits ^2_{D({I^\infty \text{-torsion}})}(M, N) \to \mathop{\mathrm{Ext}}\nolimits ^2_{D(A)}(M, N)$ is not surjective in general,
(47.8.3.1) is not an equivalence in general.
Proof. Parts (1) and (2) follow immediately from the fact that $I$-power torsion forms a Serre subcategory of $\text{Mod}_ A$. Part (4) follows from part (3).
For part (3) let $A$ be a ring with an element $f \in A$ such that $A[f]$ contains a nonzero element $x$ annihilated by $f$ and $A$ contains elements $x_ n$ with $f^ nx_ n = x$. Such a ring $A$ exists because we can take
\[ A = \mathbf{Z}[f, x, x_ n]/(fx, f^ nx_ n - x) \]
Given $A$ set $I = (f)$. Then the exact sequence
\[ 0 \to A[f] \to A \xrightarrow {f} A \to A/fA \to 0 \]
defines an element in $\mathop{\mathrm{Ext}}\nolimits ^2_ A(A/fA, A[f])$. We claim this element does not come from an element of $\mathop{\mathrm{Ext}}\nolimits ^2_{D(f^\infty \text{-torsion})}(A/fA, A[f])$. Namely, if it did, then there would be an exact sequence
\[ 0 \to A[f] \to M \to N \to A/fA \to 0 \]
where $M$ and $N$ are $f$-power torsion modules defining the same $2$ extension class. Since $A \to A$ is a complex of free modules and since the $2$ extension classes are the same we would be able to find a map
\[ \xymatrix{ 0 \ar[r] & A[f] \ar[r] \ar[d] & A \ar[r] \ar[d]_\varphi & A \ar[r] \ar[d]_\psi & A/fA \ar[r] \ar[d] & 0 \\ 0 \ar[r] & A[f] \ar[r] & M \ar[r] & N \ar[r] & A/fA \ar[r] & 0 } \]
(some details omitted). Then we could replace $M$ by the image of $\varphi $ and $N$ by the image of $\psi $. Then $M$ would be a cyclic module, hence $f^ n M = 0$ for some $n$. Considering $\varphi (x_{n + 1})$ we get a contradiction with the fact that $f^{n + 1}x_ n = x$ is nonzero in $A[f]$. $\square$
Comment #5358 by MAO Zhouhang on July 01, 2020 at 03:49
It might be better to either mention that the definition of
D_{I^\infty\mathrm{-torsion}}
is a special case of the one in that section about Serre subcategories, or incorporate a specialized definition here.
@#5358: I think we are referring to the section discussing
D_B(A)
when we first mention this derived category, so I don't see how anybody could be confused. I will change this if another person leaves a comment here indicating how to improve the exposition.
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Structural Changes in Confined Lysozyme | J. Biomech Eng. | ASME Digital Collection
Eduardo Reátegui,
Eduardo Reátegui
Department of Mechanical Engineering, Biostabilization Laboratory,
e-mail: aaksan@me.umn.edu
Reátegui, E., and Aksan, A. (July 27, 2009). "Structural Changes in Confined Lysozyme." ASME. J Biomech Eng. July 2009; 131(7): 074520. https://doi.org/10.1115/1.3171565
Proteins and enzymes can be encapsulated in nanoporous gels to develop novel technologies for biosensing, biocatalysis, and biosynthesis. When encapsulated, certain macromolecules retain high levels of activity and functionality and are more resistant to denaturation when exposed to extremes of
pH
and temperature. We have utilized intrinsic fluorescence and Fourier transform infrared spectroscopy to determine the structural transitions of encapsulated lysozyme in the range of
−120°C<T<100°C
. At cryogenic temperatures encapsulated lysozyme did not show cold denaturation, instead became more structured. However, at high temperatures, the onset of heat denaturation of confined lysozyme was reduced by
15°C
when compared with lysozyme in solution. Altered dynamics of the solvent and pore size distribution of the nanopores in the matrix appear to be key factors influencing the decrease in the denaturation temperature.
biomedical engineering, cryogenics, enzymes, fluorescence, Fourier transform spectra, gels, infrared spectra, molecular biophysics, nanobiotechnology, nanoporous materials
Enzymes, Fluorescence, Fourier transform infrared spectroscopy, High temperature, Proteins, Temperature, Water, Dynamics (Mechanics), Low temperature, Macromolecules, Spectra (Spectroscopy), Heat, Infrared spectra, Biomedical engineering, Cryogenics, Fourier transforms, Molecular biophysics, Nanobiotechnology, Nanopores, Nanoporous materials
Gulcev
Using Sugar and Amino Acid Additives to Stabilize Enzymes Within Sol-Gel Derived Silica
Encapsulation of Protein Molecules in Transparent Porous Silica Matrices Via an Aqueous Colloidal Sol-Gel Process
Molecular Confinement Influences Protein Structure and Enhances Thermal Protein Stability
Crowding and Hydration Effects on Protein Conformation: A Study With Sol-Gel Encapsulated Proteins
Recent Bio-Applications of Sol-Gel Materials
Effects of the Low Temperature Transitions of Confined Water on the Structure of Isolated and Cytoplasmic Proteins
Role of Interfacial Water in Biological Function
Knubovets
Structure, Thermostability, and Conformational Flexibility of Hen Egg-White Lysozyme Dissolved in Glycerol
Design and Structural Analysis of an Engineered Thermostable Chicken Lysozyme
See EPAPS supplementary material at http://dx.doi.org/10.1115/1.3171565http://dx.doi.org/10.1115/1.3171565 E-JBENDY-131-038907for three additional figures pertaining to this work.
Spectroscopy Methods for Analysis of Protein Secondary Structure
A Study of Protein Secondary Structure by Fourier Transform Infrared/Photoacoustic Spectroscopy and Its Application for Recombinant Proteins
Temperature-Induced Changes in Protein Structures Studied by Fourier Transform Infrared Spectroscopy and Global Analysis
Kidchob
Thermal Stability of Lysozyme Langmuir–Schaefer Films by FTIR Spectroscopy
Effects of Freezing on Membranes and Proteins in LNCaP Prostate Tumor Cells
Effect of Water-Wall Interaction Potential on the Properties of Nanoconfined Water
Evolution of the Absorbed Water Layer Structure on Silicon Oxide at Room Temperature
Low-Temperature Glass Transition in Proteins
X. -Q.
Garcia-Sakai
Observation of a Dynamic Crossover in Water Confined in Double-Wall Carbon Nanotubes
Chakrabotry
Baghci
, www.lsbu.ac.uk/water/index.htmlwww.lsbu.ac.uk/water/index.html
The Anomalous Behavior of the Density of Water in the Range, 30 K<T<373 K
Conductivity in Hydrated Proteins
S.-H
Evidence of the Existence of the Low-Density Liquid Phase in Supercooled, Confined Water
Koynova
Tenchov
Modulation of Lipid Phase Behavior by Kosmotropic and Chaotropic Solutes
Silica Gel Surface: Molecular Dynamics of Surface Silanols
I. -S.
A Detailed Model of Local Structure and Silanol Hydrogen Bonding of Silica Gel Surfaces
Infrared Spectroscopy of Biomolecules
Wily-Liss
Larsericsdotter
Thermodynamic Analysis of Proteins Adsorbed on Silica Particles: Electrostatic Effects
Czeslik
Effects of Temperature on the Conformation of Lysozyme Adsorbed to Silica Particles
Cavitation of Water Confined in Hydrophobic Nanoporous Materials
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Brianchon's theorem - Wikipedia
In geometry, Brianchon's theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. It is named after Charles Julien Brianchon (1783–1864).
2 Connection to Pascal's theorem
3 Degenerations
4 In the affine plane
{\displaystyle P_{1}P_{2}P_{3}P_{4}P_{5}P_{6}}
be a hexagon formed by six tangent lines of a conic section. Then lines
{\displaystyle {\overline {P_{1}P_{4}}},\;{\overline {P_{2}P_{5}}},\;{\overline {P_{3}P_{6}}}}
(extended diagonals each connecting opposite vertices) intersect at a single point
{\displaystyle B}
, the Brianchon point.[1]: p. 218 [2]
Connection to Pascal's theoremEdit
The polar reciprocal and projective dual of this theorem give Pascal's theorem.
DegenerationsEdit
3-tangents degeneration of Brianchon's theorem
As for Pascal's theorem there exist degenerations for Brianchon's theorem, too: Let coincide two neighbored tangents. Their point of intersection becomes a point of the conic. In the diagram three pairs of neighbored tangents coincide. This procedure results in a statement on inellipses of triangles. From a projective point of view the two triangles
{\displaystyle P_{1}P_{3}P_{5}}
{\displaystyle P_{2}P_{4}P_{6}}
lie perspectively with center
{\displaystyle B}
. That means there exists a central collineation, which maps the one onto the other triangle. But only in special cases this collineation is an affine scaling. For example for a Steiner inellipse, where the Brianchon point is the centroid.
In the affine planeEdit
Brianchon's theorem is true in both the affine plane and the real projective plane. However, its statement in the affine plane is in a sense less informative and more complicated than that in the projective plane. Consider, for example, five tangent lines to a parabola. These may be considered sides of a hexagon whose sixth side is the line at infinity, but there is no line at infinity in the affine plane. In two instances, a line from a (non-existent) vertex to the opposite vertex would be a line parallel to one of the five tangent lines. Brianchon's theorem stated only for the affine plane would therefore have to be stated differently in such a situation.
The projective dual of Brianchon's theorem has exceptions in the affine plane but not in the projective plane.
Brianchon's theorem can be proved by the idea of radical axis or reciprocation.
^ Coxeter, H. S. M. (1987). Projective Geometry (2nd ed.). Springer-Verlag. Theorem 9.15, p. 83. ISBN 0-387-96532-7.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Brianchon%27s_theorem&oldid=1000083780"
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Amid a major reshuffle, millions of Americans are seeking new employment, and these high-paying, fast-growing positions might be a good match.
We looked for high-paying jobs that are likely to increase using employment predictions for 2020–2030 and wage data from 2020. Some of the top 20 vocations included managerial, technical, and medical jobs such as registered nurses.
Here are 20 high-paying vocations that are expected to grow in popularity over the next ten years:
1. Quality assurance analysts and software developers
By Fotis Fotopoulos on Unsplash
Software development and testing are two of the most promising IT careers. The majority of large IT firms receive work from clients in other countries. The engineers are then assigned to such projects. Every organization has a 60 percent developer vacancy and a 40% tester vacancy.
Between 2020 and 2030: 500,000 new jobs are expected.
In the United States, the average compensation for a software developer is around
120,000** per year and for a tester, it's **
90,000.
Bachelor's degree is the most common educational prerequisite.
While managing the entire operations of a firm or division, a general manager is supposed to enhance efficiency and generate profitability. Managing workers, controlling the budget, implementing marketing tactics, and many other aspects of the firm are all part of the general manager's responsibilities.
The average compensation for a general manager is around $100,000 per year.
By Rusty Watson on Unsplash
They give out medications, manage records, monitor patients, consult with other healthcare experts, and educate patients and their families about health issues. They oversee orderlies, nursing assistants, and licensed practical nurses in addition to providing direct care to their patients.
The average compensation for this job is around $70,000 per year.
Financial managers analyze data and advise top executives on profit-maximizing strategies. Financial managers are in charge of an organization's financial health. They provide financial reports, direct investment operations, and formulate strategies for their organization's long-term financial objectives.
The average compensation for this job is around $134,000 per year.
Management analysts, sometimes known as management consultants, make recommendations to increase the efficiency of a company. They advise executives on ways to make businesses more lucrative by lowering expenses and increasing revenues.
6. Analysts of market research and marketing experts
By Luke Chesser on Unsplash
Influencer marketing, SEO, social media, email marketing, field and event marketing, market research, branding, paid media, content marketing, copywriting, and so on are all examples of marketing specialists.
By Pamela Buenrostro on Unsplash
A lawyer (also known as an attorney, counsel, or counselor) is a licensed practitioner who provides legal advice and representation to others. A lawyer might be young or elderly, male or female, in today's world. Almost one-third of all attorneys are under the age of 35.
Between 2020 and 2030: 70,000 new jobs are expected.
A doctoral or professional degree is the most common educational prerequisite.
8. Managers of computer and information systems
Computer managers are in charge of supervising a company's computer system. Their job includes planning, coordinating, and arranging a company's computer-related operations. Information technology (IT) goals and objectives are developed and implemented by computer managers.
Accountants and auditors help firms and individuals prepare and evaluate financial records, discover possible areas of opportunity and risk, and propose solutions. They make certain that financial records are correct, that financial and data risks are assessed, and that taxes are paid correctly.
10. Teachers at elementary schools
Teachers in elementary schools are responsible for identifying students' problems and strengths at a young age and developing a customized curriculum to prepare them for middle school. Kindergarten pupils from classes 5 through 12 are obliged to be taught by elementary school teachers.
Between 2020 and 2030: 1,00,000 new jobs are expected.
11. Drivers of heavy and tractor-trailer trucks
By Zetong Li on Unsplash
Some drivers of big and tractor-trailer trucks create their routes. Drivers of heavy and tractor-trailer trucks move items from one area to another. The majority of tractor-trailer drivers are long-haul drivers who drive trucks weighing more than 26,000 pounds overall, including the vehicle, passengers, and cargo.
The postsecondary nondegree award is the most common educational prerequisite.
Construction managers are in charge of planning, coordinating, budgeting, and supervising construction projects from start to finish. A Construction Manager performs the following tasks at a high level: Timelines and milestones are used to plan the complete building process. Subcontractors and employees are hired and managed.
A Cybersecurity Analyst, also known as an Information Security Analyst, is in charge of safeguarding a business's networks and computers. They design, assess, and execute security plans to avoid data breaches and protect a company's digital assets.
The average compensation for this job is around $1,00,000 per year.
By Anton Dmitriev on Unsplash
Electrical components and systems are inspected, tested, repaired, installed, and modified by electricians. Electricians operate as contractors in a variety of settings, including homes, companies, and building sites.
A high school diploma or equivalent is the most common educational prerequisite.
A logistician studies and organizes an organization's supply chain, which is the mechanism that transfers a product from source to customer. They are in charge of a product's whole life cycle, which includes how it is procured, distributed, allocated, and delivered. Logisticians are employed in almost every business.
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Effects of B20 on Emissions and the Performance of a Diesel Particulate Filter in a Light-Duty Diesel Engine | J. Eng. Gas Turbines Power | ASME Digital Collection
Amy M. Peterson,
Po-I Lee,
Po-I Lee
Ming-Chia Lai,
Ming-Cheng Wu,
Craig L. DiMaggio,
Craig L. DiMaggio
Hiyang Tang
Peterson, A. M., Lee, P., Lai, M., Wu, M., DiMaggio, C. L., Ng, S., and Tang, H. (August 16, 2010). "Effects of B20 on Emissions and the Performance of a Diesel Particulate Filter in a Light-Duty Diesel Engine." ASME. J. Eng. Gas Turbines Power. November 2010; 132(11): 112802. https://doi.org/10.1115/1.4001068
This paper compares 20% biodiesel (B20-choice white grease) fuel with baseline ultra low sulfur diesel (ULSD) fuel on the emissions and performance of a diesel oxidation catalyst (DOC) and diesel particulate filter (DPF) coupled to a light duty four-cylinder 2.8-l common-rail DI diesel engine. The present paper focuses on the comparison of the fuel effects on loading and active regeneration of the DPF between B20 and ULSD. B20, in general, produces less soot and has lower regeneration temperature, compared with soot loaded with ULSD.
NO2
concentrations before the DPF were found to be 6% higher with B20, indicating more availability of
NO2
to oxidize the soot. Exhaust speciation of the
NO2
availability indicates that the slight increase in
NOx
from B20 is not the dominant cause for the lower temperature regeneration and faster regeneration rate, but the reactivity of the soot that is in the DPF. Formaldehyde concentrations are found to be higher with B20 during regeneration, due to increased oxygen concentrations in the exhaust stream. Finally, the oil dilution effect due to post injection to actively regenerate the DPF is also investigated using a prototype oil sensor and Fourier transform infrared (FTIR) instrumentation. Utilizing an active regeneration strategy accentuates the possibility of fuel oil dilution of the engine oil. The onboard viscosity oil sensor used was in good agreement with the viscosity bench test and FTIR analysis, and provided oil viscosity measurement over the course of the project. The operation with B20 shows significant fuel dilution and needs to be monitored to prevent engine deterioration.
biofuel, catalysts, diesel engines, Fourier transform spectroscopy, infrared spectroscopy, nitrogen compounds, organic compounds, viscosity measurement
Biodiesel, Diesel, Diesel engines, Emissions, Engines, Filters, Fuels, Particulate matter, Soot, Temperature, Viscosity, Viscosity measurement, Exhaust systems, Sensors, Nitrogen oxides, Fourier transform infrared spectroscopy, Catalysts
2002, A Comprehensive Analysis of Biodiesel Impacts on Exhaust Emissions, EPA.
Effects of Biodiesel Blends on Vehicle Emissions
Impact of Biodiesel on NOx Emissions in a Common Rail Direct Injection Diesel Engine
Effect of Biodiesel (B20) on Performance and Emissions in a Single Cylinder HSDI Diesel Engine
P. -I.
Proceedings of the ASME ICE Spring Technical Conference
. Milwaukee, WI.
Impact of Biodiesel Blending on Diesel Soot and the Regeneration of Particulate Filters
Examination of the Oxidation Behavior of Biodiesel Soot
Biodiesel and Fuel Dilution of Engine Oil
Rapid In Situ Measurement of Fuel Dilution of Oil in a Diesel Engine Using Laser-Induced Fluorescence Spectroscopy
The Spray and Engine Combustion Performances of Ethanol-Biodiesel Fuel Blends
|
Market Interest Rates - Course Hero
Introduction to Finance/Interest Rates/Market Interest Rates
Components of Market Interest Rates
The interest rate is affected by many components of the financial system. It is a variable rate and is the amount due per period as a percentage of the amount owed at the end of the previous period. The market interest rate is simply dependent upon the supply of credit and a demand for credit in the marketplace. The market segmentation theory allows for some distinction between interest rates. Market segmentation theory is the belief that long-term and short-term interest rates are separate and not associated with each other, as both are controlled by differing factors. Inflation, which would affect long-term decision-making, would not be the same when looking at short-term-maturation securities.
Within the basic market interest rate is another component, called inflation. Inflation is the continual increase in the average price levels of goods and services. Various forms of inflation exist that further affect interest rates in different ways. Cost-push inflation is a situation that occurs when increased production costs, such as for raw materials and labor, cause elevated price levels. Because this type of inflation is linked to manufacturing costs, it can change rapidly. Natural disasters or powerful labor unions can also alter the cost of production. The more widely known form of inflation is demand-pull inflation. With demand-pull inflation an increase in consumer demand for a product or service causes increased price levels. As overall demand for products outstrips supply, prices go up. It is more difficult to predict speculative inflation, which occurs when people believe prices will continue to rise and they purchase more now to minimize losses and maximize gains. If politicians place restrictions on the export of raw materials, it will lead to an increase in material prices and create cost-push inflation. If the banks attempt to adjust interest rates in anticipation of import/export regulations, it will cause speculative inflation. These various forms of inflation can affect the inflation premium, which is an increase in the normal rate of return to investors to compensate for anticipated inflation. For a bond security to be attractive to investors, its return must be greater than the inflation-adjusted minimally required return.
With all of the inflation components affecting interest rates, a thorough investor should know the potential inflation responses and the causes of inflation. This knowledge will help the investor make informed decisions. Investors may also consider the liquidity preference theory to ensure their financial interests are addressed. Liquidity preference theory is a belief that investors require a higher premium on medium- and long-term investments versus short-term investments to offset investors' desire for greater liquidity.
Treasury Securities as a Baseline
A U.S. Treasury security is considered risk free, as is a Treasury bill, bond, or note issued by the U.S. Treasury. Treasury securities rates are used as the baseline for the calculations of commercial securities that have risks. The risk-free interest rate is the expected rate of return with no risks or financial losses, often estimated from U.S. bonds and treasuries. The risk-free interest rate is a composite of the real rate of interest and a premium for inflation. The idea may sound definite, but investors have found that accepting this without consideration is ineffective and fiscally counterproductive. Though the risk is small, Treasury bonds do have some risk of default. Greece has defaulted on its bond payments previously. Likewise, bond volatility is possible if bonds are traded too aggressively. In short, the risk-free rate of return is not actually free from risk. Investors calculate the real rate of return as the real risk-free rate differs from the stated risk-free rate. There is a formula for the risk-free rate.
\text{Risk-Free Interest Rate}=\text{Real Rate of Interest}+\text{Inflation Premium}
If the Treasury security is paying the risk-free interest rate, then the risk here is inflation. Removing the inflation premium by subtracting it from both sides results in a different formula.
\text{Real Rate of Interest}=\text{Risk-Free Interest Rate}-\text{Inflation Premium}
The real rate of interest is an adjusted interest rate to remove inflation to indicate the real cost of an investment. Real rate of interest is the return with no calculable risk. As an example, if the Treasury security has a return of 3 percent and the current inflation is 2 percent, the real rate of interest with no associated risks is 1 percent. This is an important factor when making decisions against equity investments. The cost of capital when issuing bonds is a function of the risk-free rate and the market rate. The weighted difference between the two rates will dictate the overall cost of building capital for the company. If the risk-free rate improperly calculates the effects of inflation, the total cost of creating the capital can become greater than the actual capital generated, resulting in a loss. If Cogs Inc. were raising capital with a cost of capital rate of 7 percent, the real rate of interest must beat this rate. If inflation is erroneously calculated, then the real rate will be incorrect. If it drops under 7 percent, any money that Cogs Inc. generated as capital would be a net loss. If the real rate of interest is 5 percent and the inflation premium is 2 percent, for example, the risk-free interest rate is 7 percent.
The market interest rate is the rate of interest that a free market can bear. It is dependent on the types of risk associated with the underlying asset. If inflation is the only risk, then the market rate is called the risk-free interest rate. The risk-free interest rate is the expected rate of return with no risks or financial losses. Inflation is the only risk factor associated with this type of security.
\text{Risk-Free Interest Rate}=\text{Real Rate of Interest}+\text{Inflation Premium}
Treasury bonds are generally used as a rate factor for commercial bonds that have a risk of defaulting. For bonds at risk of default, a default risk premium is added to the risk-free rate. A default risk premium is a fee that a borrower is charged to compensate for the possibility that the borrower might default on the loan.
\text{Market Interest Rate}=\text{Real Rate of Interest}+\text{Inflation Premium}+\text{Default Premium}
If the risk-free interest rate is 7 percent and the default risk premium is 1 percent, then the market interest rate is 8 percent.
Some securities have exceptionally long maturity dates. A lot can change in the course of many years or, in some cases, several decades. The market rates can fluctuate significantly over such a long period. To compensate for this form of risk, a maturity risk premium may be included. Maturity risk premium is the extra premium an investor receives for engaging with investments that have a longer duration until maturity. Similarly, some securities are difficult to trade, making their liquidity low. Compare cashing a check, which is a very liquid asset, to selling a house; it could take a year before the cash from selling a house is available to deposit into a bank. To compensate for this form of risk, a liquidity premium is attached. Liquidity premium is an additional financial return expected by investors to offset assets not easily converted to cash.
In cases where maturity and liquidity may include a risk, the maturity risk premium and the liquidity premium are added to the equation.
\begin{aligned}\text{Market Interest Rate}&=\text{Real Rate of Interest}+\text{Inflation Premium}\\&+\text{Default Premium}+\text{Maturity Risk Premium}+\text{Liquidity Premium}\end{aligned}
With a real rate of interest of 3 percent, an inflation premium of 1 percent, a default premium of 1 percent, a maturity risk premium of 2 percent, and a liquidity premium of 1 percent, the market interest rate is 8 percent.
<Supply and Demand in Investment Markets>U.S. Treasury Securities
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Section 10.44: Separable extensions, continued
Lemma 10.44.1. Let $k$ be a field of characteristic $p > 0$. Let $K/k$ be a field extension. The following are equivalent:
the ring $K \otimes _ k k^{1/p}$ is reduced, and
$K$ is geometrically reduced over $k$.
Proof. The implication (1) $\Rightarrow $ (3) follows from Lemma 10.43.6. The implication (3) $\Rightarrow $ (2) is immediate.
Assume (2). Let $K/L/k$ be a subextension such that $L$ is a finitely generated field extension of $k$. We have to show that we can find a separating transcendence basis of $L$. The assumption implies that $L \otimes _ k k^{1/p}$ is reduced. Let $x_1, \ldots , x_ r$ be a transcendence basis of $L$ over $k$ such that the degree of inseparability of the finite extension $k(x_1, \ldots , x_ r) \subset L$ is minimal. If $L$ is separable over $k(x_1, \ldots , x_ r)$ then we win. Assume this is not the case to get a contradiction. Then there exists an element $\alpha \in L$ which is not separable over $k(x_1, \ldots , x_ r)$. Let $P(T) \in k(x_1, \ldots , x_ r)[T]$ be the minimal polynomial of $\alpha $ over $k(x_1, \ldots , x_ r)$. After replacing $\alpha $ by $f \alpha $ for some nonzero $f \in k[x_1, \ldots , x_ r]$ we may and do assume that $P$ lies in $k[x_1, \ldots , x_ r, T]$. Because $\alpha $ is not separable $P$ is a polynomial in $T^ p$, see Fields, Lemma 9.12.1. Let $dp$ be the degree of $P$ as a polynomial in $T$. Since $P$ is the minimal polynomial of $\alpha $ the monomials
\[ x_1^{e_1} \ldots x_ r^{e_ r} \alpha ^ e \]
for $e < dp$ are linearly independent over $k$ in $L$. We claim that the element $\partial P/\partial x_ i \in k[x_1, \ldots , x_ r, T]$ is not zero for at least one $i$. Namely, if this was not the case, then $P$ is actually a polynomial in $x_1^ p, \ldots , x_ r^ p, T^ p$. In that case we can consider $P^{1/p} \in k^{1/p}[x_1, \ldots , x_ r, T]$. This would map to $P^{1/p}(x_1, \ldots , x_ r, \alpha )$ which is a nilpotent element of $k^{1/p} \otimes _ k L$ and hence zero. On the other hand, $P^{1/p}(x_1, \ldots , x_ r, \alpha )$ is a $k^{1/p}$-linear combination the monomials listed above, hence nonzero in $k^{1/p} \otimes _ k L$. This is a contradiction which proves our claim.
Thus, after renumbering, we may assume that $\partial P/\partial x_1$ is not zero. As $P$ is an irreducible polynomial in $T$ over $k(x_1, \ldots , x_ r)$ it is irreducible as a polynomial in $x_1, \ldots , x_ r, T$, hence by Gauss's lemma it is irreducible as a polynomial in $x_1$ over $k(x_2, \ldots , x_ r, T)$. Since the transcendence degree of $L$ is $r$ we see that $x_2, \ldots , x_ r, \alpha $ are algebraically independent. Hence $P(X, x_2, \ldots , x_ r, \alpha ) \in k(x_2, \ldots , x_ r, \alpha )[X]$ is irreducible. It follows that $x_1$ is separably algebraic over $k(x_2, \ldots , x_ r, \alpha )$. This means that the degree of inseparability of the finite extension $k(x_2, \ldots , x_ r, \alpha ) \subset L$ is less than the degree of inseparability of the finite extension $k(x_1, \ldots , x_ r) \subset L$, which is a contradiction. $\square$
Comment #387 by Filip Chindea on December 08, 2013 at 14:26
Is it really obvious that
G^{1/p}
maps to a nonzero element of
k^{1/p} \otimes L
? As far as I know there is a vanishing criterion for elements in a tensor product of modules, but nothing for fields (algebras). I've been thinking at this for some time, but I may be missing something. If the answer is trivial you can delete this altogether after my apologies, anyway thank you for your time.
Yes, this needs a small argument. Say
G
pd
T
. Then the elements
X_1^{e_1}, \ldots, X_r^{e-r} T^e
e < pd
k[X_1, \ldots, X_r, T]
map to
k
-linearly independent elements of
L
. By construction of the tensor product, the same elements map to
k^{1/p}
k^{1/p} \otimes_k L
G^{1/p}
does not map to zero as the
T
-degree of
G^{1/p}
d < pd
. OK?
Thanks; you can delete this. It was just my ridiculous expectation that the irreducibility of
G
T
should turn up while calculating in the tensor product.
OK, I went ahead and added the extra argument. The change is here.
If there is a misunderstanding about what the argument is supposed to be, then probably the argument deserves to be updated and/or extended.
Is it clear that the polynomial
G(T)=P(T,x_2,\ldots,x_r,\alpha)\in k[x_2,\ldots,x_r,\alpha][T]
is non-zero, and that the fact that
\partial P/\partial x_1\neq 0
\partial G/\partial T\neq 0
(for some reason this kind of thing always really confuses me, so this might be a very dumb question)? I thought perhaps it followed from the linear independence of the monomials in the displayed equation, but I can't seem to make that work.
It took me a while to understand your question, but you do have a valid point. I've tried to address your concern by adding a couple of lines explaing that
P
is irreducible as a polynomial in
x_1
k(x_2, \ldots, x_r, \alpha)
which is a purely transcendental extension of
k
. Here is the commit which also includes fixing the other typos you found.
It seems that there is a curse on this proof as it keeps having to be modified! There is a still a last step in the proof that perhaps should be clarified a bit more, namely, why the inseparable degree is lessened...
I looked at the commit, but I'm still confused. How are we concluding that
x_2,\ldots,x_r,\alpha
is algebraically independent before knowing that
x_1
K=k(x_2,\ldots,x_r,\alpha)
? Once we know that
x_1
K
, then because
L
k(x_1,\ldots,x_r)\subseteq K(x_1)
L
K(x_1)
r=\mathrm{trdeg}_k(L)=\mathrm{trdeg}_k(K(x_1))
, which forces
x_2,\ldots,x_r,\alpha
to be algebraically independent. The argument as it is now uses this algebraic independence to prove that
x_1
K
(by proving that the relevant polynomial relation with coefficients in
K
is non-trivial). Also, in the commit, there is a typo: the second
n
is missing from "independent."
There is a nontrivial algebraic relation between
x_1, \ldots, x_r, \alpha
which involves
x_1
P
P
involves
x_1
is proved before we point out that
x_2, \ldots, x_r, \alpha
are algebraically independent. (When you read the commit you have to remove the red lines.) OK?
OK, I guess what I said wasn't good enough because for example, if
P = x_1 Q
Q
, then it wouldn't work. But we also know that
P
is irreducible in the polynomial ring on
r + 1
variables and then it is enough.
Actually the way I think about the situation of the proof is, as soon as we have found
P
k[x_1, \ldots, x_r, T]
a minimal polynomial for
\alpha
k(x_1, \ldots, x_r)
, then I think of the irreducible hypersurface and I replace
k(x_1, \ldots, x_r, \alpha)
by the function field of
V
. Then after we show that
\partial P(y_1, \ldots, y_{r + 1}) / \partial y_1
is nonzero, I think of that as saying that the projection
V \to \mathbf{A}^r
gotten by forgetting the first variable, is generically \'etale, i.e., that the function field extension
k(V) \supset k(\mathbf{A}^r)
is (finite) separable. So certainly, as soon as you agree that at least one variable occurs in a term of
P
with an exponent not divisible by
p
, then I am completely sure that the proof is correct.
The problem is that we also need to keep the proof readable, understandable, etc. A good way to do this would probably be to have a discussion of the relationship between multivariable polynomials and (nonalgebraic) field extensions and then to refer to that. I encourage you to write your own and submit it! Thanks.
The displayed equation should be
Another thing we need is a statement and proof of Gauss's lemma, maybe somewhere in the chapter on fields?
Yes! I understand now. The whole point is that the derivative in
x_1
is non-zero, so
x_1
shows up, and we can write
P(x_1,\ldots,x_r,T)=x_1^dg(x_2,\ldots,x_r,T)+h(x_1,\ldots,x_r,T)
with
d
x_1
d>0
. Written this way, it's then obvious that
P(X,x_1,\ldots,x_r,\alpha)\neq 0
(as a polynomial in
k(x_2,\ldots,x_r,\alpha)[X]
). I don't know why this kind of thing confuses me so much. Thank you for explaining it!
OK, yes, that is a good way to see it. I really appreciate reporting back here. Thanks!
Comment #5562 by DatPham on October 30, 2020 at 10:13
I don't understand why
x_1
k(x_2,\ldots,x_r,\alpha)
. I know that
x_1
P(X,x_2,\ldots,x_n,\alpha)\in k(x_2,\ldots,x_r,\alpha)[X]
. But how do we know this polynomial is nonzero? If we write
P(x_1,\ldots,x_n,T)=X_1^dg(x_2,\ldots,x_n,T)+h(x_1,x_2,\ldots,x_n,T)
d>0
g(x_2,\ldots,x_n,T)\ne 0
h(x_1,x_2,\ldots,x_n,T)
<d
, then it might happen that
g(x_2,\ldots,x_n,\alpha)=0
@#5562: see #764
Comment #5821 by DatPham on December 07, 2020 at 03:19
@#5745: Dear Professor Johan, I did read comment #764 many times, but I still cannot understand why
g(x_2.\ldots,x_n,\alpha)\ne 0
g(x_2,\ldots,x_n,T)\ne 0
{x_2,\ldots,x_n,\alpha}
may be algebraically dependent over k (of course if
x_1
k(x_2,\ldots,x_n,\alpha)
{x_2,\ldots,x_n,\alpha}
has to be algebraically independent but this seems to be circular ...) .
To help me understand the confusion, please read the proof from the beginning and point out the first sentence in the proof where you do not understand the assertion. Because each time I read your comment, I think you are pointing to something in the second paragraph of the proof where we already know that
P \in k[x_1, \ldots, x_r, T]
is monic in
T
\partial P / \partial x_1
is nonzero and
P
is irreducible in
T
k(x_1, \ldots, x_r)
P
x_1
k(x_2, \ldots, x_r, T)
by Gauss lemma. Hence
x_2, \ldots, x_r, \alpha
are algebraically independent. So the polynomial
P(X, x_2, \ldots, x_r, \alpha)
cannot be zero as a polynomial in
X
because this would mean we get an algebraic relation between
x_2, \ldots, x_r, \alpha
by looking at the coefficients of powers of
X
. (I am just repeating the proof here, so this probably doesn't help.)
@#5823: Dear Professor Johan, thank you for being patient with me. In your comment above, I start getting confused from the sentence ''Hence
x_2,\ldots,x_r,\alpha
are algebraically independent...'' I don't understand how we can deduce this from the fact that
P
x_1
k(x_2,\ldots,x_r,T)
Otherwise the transcendence degree of the field generated by
x_1, \ldots, x_r, \alpha
k
would be less than
r
|
Hilbert cube – Violet-Wall INFO
Not to be confused with Hilbert curve.
. . . Hilbert cube . . .
The Hilbert cube is best defined as the topological product of the intervals [0, 1/n] for n = 1, 2, 3, 4, … That is, it is a cuboid of countably infinitedimension, where the lengths of the edges in each orthogonal direction form the sequence
{displaystyle lbrace 1/nrbrace _{nin mathbb {N} }}
If a point in the Hilbert cube is specified by a sequence
{displaystyle lbrace a_{n}rbrace }
{displaystyle 0leq a_{n}leq 1/n}
, then a homeomorphism to the infinite dimensional unit cube is given by
{displaystyle h(a)_{n}=ncdot a_{n}}
List of moths of Togo
|
What Is Capital Gains Yield (CGY)?
Understanding Capital Gains Yield (CGY)
Examples of Capital Gains Yield
Capital Gains Yield FAQs
Capital gains yield is a simple formula to calculate as the only components needed are as follows:
The original price of the security
The current price of the security
That said, the concept doesn't including any income received from the investment.
A capital gains yield is the rise in the price of an investment such as a stock or bond, calculated as the rise in the security's price divided by the original price of the security.
A CGY evaluation does not include dividends; however, depending on the stock, dividends may include a considerable part of the total return in comparison to capital gains.
The total return on a share of common stock includes CGY and dividend yield.
An investment cannot generate CGY if the share price falls below the original purchase price.
Capital gains yield is calculated the same way for a bond as it is for a stock: the increase in the price of the bond divided by the original price of the bond.
Investors must evaluate the total return yield and CGY of an investment. A CGY evaluation does not include dividends; however, depending on the stock, dividends may include a considerable part of the total return in comparison to capital gains.
CGY equals the total return if the investment generates no cash flow. It is the amount of money a stock price is forecast to appreciate or depreciate, and it is the percentage change in the market price of a security over time. However, if a stock decreases in value, it is a capital loss.
\begin{aligned} &\text{Capital Gains Yield} = \frac { \text{P}_1 - \text{P}_0 }{ \text{P}_0 } \\ &\textbf{where:} \\ &\text{P}_0 = \text{original purchase price of the security} \\ &\text{P}_1 = \text{current market price of the security} \\ \end{aligned}
Capital Gains Yield=P0P1−P0where:P0=original purchase price of the securityP1=current market price of the security
For example, Peter buys a share of company ABC for $200 and then sells the share for $220. The CGY for the share in company ABC equals (220-200) / 200 = 10%.
The CGY formula employs the rate of change formula. CGY can be positive, negative, or a capital loss. However, an investment that has a negative CGY may generate profits for an investor. The higher the share price at a specific period, the greater the capital gains indicating higher stock performance.
In addition, the calculation of CGY is related to the Gordon growth model. For constant growth stocks, the CGY is g, the constant growth rate.
Tesla CGY 2020
On December 31, 2019, Tesla stock closed at a price of $83.67. On December 31, 2020, they closed at $705.67.
Thus, Tesla's CGY in 2020 was a whopping 743% ($705.67 - $83.67 = $622 / $83.67).
Nike CGY 2020
On December 31, 2019, Nike stock closed at a price of $101.31. On December 31, 2020, they closed at $141.47.
Therefore, Nike's CGY in 2020 was 46% ($141.47 - $101.31 = $46.16 / $101.31).
Netflix CGY 2020
On December 31, 2019, Netflix stock closed at a price of $323.57. On December 31, 2020, they closed at $540.73.
Thus, Netflix's CGY in 2020 was 67% ($540.73 - $323.57 = $217.16 / $323.57).
CGY is unpredictable and may occur monthly, quarterly, or annually. This format differs from dividends that are set by the company and paid out to shareholders at a predefined period.
An investment cannot generate CGY if the share price falls below the original purchase price. Some stocks pay high dividends and may produce lower capital gains. This occurs because every dollar paid out as a dividend is a dollar the company cannot reinvest into the company.
Other stocks pay lower dividends but may produce higher capital gains. These are growth stocks because profits flow back into the company for growth instead of the company distributing them to shareholders while other stocks pay poor dividends and produce low or no capital gains.
Many investors calculate a security's CGY because the formula shows how much the price fluctuates. This helps an investor to decide which securities are a good investment.
Capital gains may result in paying capital gains taxes. However, investors can offset the taxes by losses or carry it over into the following year.
Capital gains yield is an important metric that all investors need to know how to calculate. Unless you're able to figure out how much a given investment has appreciated, there's no way to tell if has been successful or not.
That said, the limitations of capital gains yield should always be kept in mind. Specifically, capital gains yield doesn't factor in the income received from dividends or interest, so it should not be used as a blind substitute for the total return calculation.
How Do You Calculate the Capital Gains Yield for a Bond?
Capital gains yield is calculated the same way for a bond as it is for a stock: the increase in the price of the bond divided by the original price of the bond. For instance, if a bond is purchased for $100 (or par) and later rises to $120, the capital gains yield on the bond is 20%.
What Is the Difference Between Capital Gains Yield and Current Yield for a Bond?
Capital gains yield measures a given security's rate of appreciation. On the other hand, the current yield is a measure of income.
For a bond, the current yield is an investor's annual interest income dividend by the current price of the bond.
What Is the Difference Between Capital Gains Yield and Holding Period Return?
Capital gains yield does not include income earned on the investment (interest or dividends). On the other hand, holding period return represents the total return earned on an investment (income plus appreciation) during the time it has been held.
Yahoo Finance. "Tesla Historical Data." Accessed June 8, 2021.
Yahoo Finance. "Nike Historical Data." Accessed June 8, 2021.
Yahoo Finance. "Netflix Historical Data." Accessed June 8, 2021.
Current yield is the annual income (interest or dividends) divided by the current price of the security.
What Does Yield On Cost (YOC) Mean?
Yield on Cost (YOC) is a measure of dividend yield calculated by dividing a stock's current dividend by the price initially paid for that stock.
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How to calculate SIP return?
How to use the online SIP calculator?
What are the benefits of mutual fund SIP investment?
SIP calculator disclaimer
Omni's SIP calculator (systematic investment plan calculator) allows you to estimate the returns on your mutual fund SIP investments. You can also use our SIP calculator to know how much you should invest monthly to achieve your investment goal.
Dreaming about retiring early to follow your passion, or planning about your kid's educational expenses? Worry not! Our mutual fund SIP calculator can help you plan your investments so that you can fulfill your dreams and promises. Read on to know what SIP investment is and how to use this calculator to plan your finances.
A systematic investment plan or SIP is a method of investment offered by mutual fund companies. Investors can use this facility to invest a fixed amount periodically in mutual fund schemes of their choice. Apart from SIP, you can also make a one-time investment in mutual funds by investing a lump sum.
A SIP investment is somewhat similar to recurring deposits (RD) in the sense that in both of them, we deposit a fixed amount regularly over a long time period. However, while RDs offer a fixed return as decided by your bank, SIPs offer a varying return as they invest money in mutual funds, which are linked to the stock market. SIPs have the potential to earn higher returns as compared to recurring deposits.
The formula for calculating SIP return is:
\text{M.A} = P \bigg[ \frac{(1 + r)^n - 1 }{ r (1 + r)} \bigg]
P — Amount you invest monthly, i.e., your monthly SIP amount;
M.A — Maturity amount, i.e., the final amount that you will receive at the end of the investment period;
n — Total number of payments you have made during the whole investment period t; and
r — Expected rate of interest per month.
For example, suppose you want to invest Rs. 2,000 per month for 20 years. The expected rate of return is 12% per annum (each year).
As you are making monthly payments (12 per year), the number of payments made in 20 years is:
n = 12 \times t = 12 \times 20 = 240
Your total investment during these 20 years is:
240 \times 2000 = \text{Rs. } 480,000
We can calculate the monthly return as:
r = 12 / (12 \times 100) = 0.01
And determine the maturity amount as:
\text{M.A} = 2000 [ \frac{(1 + 0.01)^{240} - 1}{ 0.01 \times (1 + 0.01)}]
\text{M.A} = \text{Rs. } 1,998,296
This means that if you stay invested for 20 years, you can multiply your investment by 4.2x:
Maturity amount/Total investment = Rs. 1,998,296 / Rs. 480,000 = 4.2.
The rate of return on mutual funds depends on market conditions. It may vary, and so do your estimated returns. You can check out the websites of various mutual fund houses to know the annualized rate of return of their various schemes.
If you are looking for a risk-free investment solution that can generate steady income, we recommend checking the post office monthly income scheme calculator. You can also invest in Atal Pension Yojana scheme or the NPS scheme to secure a regular flow of income in old age.
Let us see how you can use our systematic investment plan calculator to plan your mutual fund investment.
To calculate SIP return: We will choose the same example as in the previous section and show you how easily and quickly you can calculate your SIP return using our online SIP calculator:
Type your monthly SIP amount, i.e., Rs. 2,000, in the first row.
Enter the investment period, say 20 years, in the second row.
Input the expected rate of return per annum, i.e., 12%, in the third row.
The SIP calculator will display your total investment (Rs. 480,000), maturity amount (Rs. 1,998,296), and the factor by which your investment has multiplied (4.2).
If you know your investment goal: To decide how much amount you should invest per month so that you have enough money to fulfill your dream, follow these instructions:
Enter the amount you need to fulfill your dream, say Rs. 500,000, in the section maturity amount.
Type the number of years after which you want the money, for example, 10 years, in the section investment period.
Input the expected rate of return per annum, say 10%.
You will get your monthly SIP amount, i.e., Rs. 2,421.
Note: This systematic investment plan calculator does not take into account exit load and expense ratio.
Some benefits of SIP are:
Power of compounding — If you start your SIPs early in life and stay invested for a longer period, you benefit from compounding. Over the long term, your small investments grow into a massive nest egg.
Inculcating financial discipline — In SIP investment, you are forced to pay a fixed amount at regular intervals. You can start with sums as small as Rs. 500 and then increase your investment as your income increases. All this helps you to inculcate a sense of financial discipline.
A better handle on market fluctuations — When you invest through SIP, you stagger your investments in mutual funds over an extended period. This means you have a better handle on market fluctuations. As you are investing a fixed amount, you buy more fund units when the market is down. If the market is surging, you buy lesser units. Over a longer period, the purchase cost averages out and turns out to be on the lower side. This is known as rupee cost averaging. This benefit is not available when you invest a lump sum. You can also redeem your purchased units using a systematic withdrawal plan.
Better inflation-adjusted return — Equity mutual funds generally offer a better return than saving schemes like fixed deposits. The current inflation rate for India is about 6-7%. Most banks offer fixed deposits at an interest rate of 5-6%, which does not beat inflation. Then you also have to pay income tax on this interest depending upon your income-tax bracket. Investing in mutual funds via the SIP route offers higher returns as well as tax benefits.
Tax benefits of SIP — If you hold your purchased units for more than one year, the income generated from redeeming them becomes a long-term capital gain. In India, long-term capital gain of up to Rs. 1 lakh per year is tax-free. If your income from redeeming your units is more than that, you will have to pay tax at the rate of 10%. Thus, investing in mutual funds offers tax benefits compared to FDs (fixed deposits) or RDs, where the tax can be as high as 30%. Moreover, if you invest in ELSS mutual funds, you can also claim a tax deduction under Section 80C of the I.T. Act.
You should consider the SIP calculator as a model for financial approximation. All payment figures, balances, and interest figures are estimates based on the data you provided in the specifications that are not exhaustive despite our best effort.
How does a systematic investment plan (SIP) work?
In a systematic investment plan, you purchase fund units by investing a fixed amount at regular intervals in the mutual fund schemes of your choice. When the stock market is performing well, you buy fewer units, and when the market is down, you buy more units. As a result, you do not have to time the market. Over a long period, your investment gets compounded and multiplies.
No, not all SIPs are tax-free. When you invest in equity-linked savings schemes (ELSS), you get a tax deduction of up to Rs. 1,50,000 under section 80c of income tax. The return from SIP investment is considered long-term capital gain if the holding period exceeds 12 months. Long-term capital gains up to Rs. 1 lakh per year are tax-exempt.
The main risk factors in SIP investments are:
Since mutual fund investments are subject to market risks, the value of your SIP investment may go down if the market is not performing well.
Selling your fund units and getting your money back may take some time. Some investments also have a lock-in period. As a result, you may run into liquidity risk if you need your money very quickly.
What is the minimum investment in SIP?
INR 500. You can start your investment via SIP with a minimum amount of Rs 500. The minimum amount may vary between different asset management companies.
NAV stands for Net Asset Value. It is the market value of a single mutual fund unit. NAV is calculated at the end of every market day. So, every time you invest in mutual funds, you purchase units at the current NAV.
What is the long-term capital gain tax rate for 2021?
10%. The long-term capital gains (LTCG) are taxed at a flat rate of 10% if the income from investments (equity shares, mutual funds, government securities, etc.) exceeds Rs 1 lakh.
If a mutual fund investor exits or redeems his fund units within a certain time period from the day of purchase, the asset management company charges a fee or penalty. This fee is called exit load in the mutual funds.
Mutual fund houses charge exit load to discourage investors from exiting the scheme before a certain time period. The exit load fee charged by different mutual fund houses is different, and not all fund houses charge an exit fee.
To calculate exit load, follow the given instruction:
Find out the NAV of the fund units and the exit load percent charged by the AMC.
Decide how many units your want to sell.
Multiply the NAV with the number of units and exit load percent.
You have calculated the exit load.
Monthly SIP amount (P)
Investment period (t)
Investment multiplication factor
Determine the true value of growth stocks using the Benjamin Graham intrinsic value calculator.
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Ekedahl–Oort strata of hyperelliptic curves in characteristic 2
Arsen Elkin, Rachel Pries
KEYWORDS: curve, hyperelliptic, Artin–Schreier, Jacobian, $p$-torsion, $a$-number, group scheme, de Rham cohomology, Ekedahl–Oort strata, 11G20, 14K15, 14L15, 14H40, 14F40, 11G10
X
is a hyperelliptic curve of genus
g
k
p=2
. We prove that the de Rham cohomology of
X
decomposes into pieces indexed by the branch points of the hyperelliptic cover. This allows us to compute the isomorphism class of the
2
-torsion group scheme
{J}_{X}\left[2\right]
of the Jacobian of
X
in terms of the Ekedahl–Oort type. The interesting feature is that
{J}_{X}\left[2\right]
depends only on some discrete invariants of
X
, namely, on the ramification invariants associated with the branch points. We give a complete classification of the group schemes that occur as the
2
-torsion group schemes of Jacobians of hyperelliptic
k
-curves of arbitrary genus, showing that only relatively few of the possible group schemes actually do occur.
Bruno Chiarellotto, Alice Ciccioni, Nicola Mazzari
KEYWORDS: syntomic cohomology, cycles, regulator map, rigid cohomology, de Rham cohomology, 14F43, 14F30, 19F27
\mathsc{V}=Spec\left(R\right)
R
be a complete discrete valuation ring of mixed characteristic
\left(0,p\right)
. For any flat
R
\mathsc{X}
, we prove the compatibility of the de Rham fundamental class of the generic fiber and the rigid fundamental class of the special fiber. We use this result to construct a syntomic regulator map
{reg}_{syn}:{CH}^{i}\left(\mathsc{X}∕\mathsc{V},2i-n\right)\to {H}_{syn}^{n}\left(\mathsc{X},i\right)
\mathsc{X}
is smooth over
R
with values in the syntomic cohomology defined by A. Besser. Motivated by the previous result, we also prove some of the Bloch–Ogus axioms for the syntomic cohomology theory but viewed as an absolute cohomology theory.
Zeros of real irreducible characters of finite groups
Selena Marinelli, Pham Tiep
KEYWORDS: real irreducible character, nonvanishing element, Frobenius–Schur indicator, 20C15, 20C33
We prove that if all real-valued irreducible characters of a finite group
G
with Frobenius–Schur indicator
1
are nonzero at all
2
-elements of
G
G
has a normal Sylow
2
-subgroup. This result generalizes the celebrated Ito–Michler theorem (for the prime
2
and real, absolutely irreducible, representations), as well as several recent results on nonvanishing elements of finite groups.
The biHecke monoid of a finite Coxeter group and its representations
Florent Hivert, Anne Schilling, Nicolas Thiéry
KEYWORDS: Coxeter groups, Hecke algebras, representation theory, blocks of permutation matrices, 20M30, 20F55, 06D75, 16G99, 20C08
For any finite Coxeter group
W
, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on
W
. The construction of the biHecke monoid relies on the usual combinatorial model for the
0
-Hecke algebra
{H}_{0}\left(W\right)
, that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits
|W|
simple and projective modules. In order to construct the simple modules, we introduce for each
w\in W
a combinatorial module
{T}_{w}
whose support is the interval
{\left[1,w\right]}_{R}
in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset.
KEYWORDS: shuffle algebra, consecutive pattern avoidance, free resolution, 05E15, 18G10, 16E05, 05A16, 05A15, 05A05
Dragos Ghioca, Liang-Chung Hsia, Thomas Tucker
KEYWORDS: preperiodic points, heights, 37P05, 37P10
a\left(\lambda \right),b\left(\lambda \right)\in ℂ\left[\lambda \right]
{f}_{\lambda }\left(x\right)\in ℂ\left[x\right]
be a one-parameter family of polynomials indexed by all
\lambda \in ℂ
. We study whether there exist infinitely many
\lambda \in ℂ
a\left(\lambda \right)
b\left(\lambda \right)
are preperiodic for
{f}_{\lambda }
$F$-blowups of normal surface singularities
Nobuo Hara, Tadakazu Sawada, Takehiko Yasuda
KEYWORDS: $F$-blowups, Frobenius maps, rational double points, simple elliptic singularities, \textttMacaulay2, 14B05, 14G17, 14E15
F
-blowups of non-
F
-regular normal surface singularities. Especially the cases of rational double points and simple elliptic singularities are treated in detail.
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Sequential minimal optimization - Wikipedia
Optimization algorithm for training support vector machines
Sequential minimal optimization (SMO) is an algorithm for solving the quadratic programming (QP) problem that arises during the training of support-vector machines (SVM). It was invented by John Platt in 1998 at Microsoft Research.[1] SMO is widely used for training support vector machines and is implemented by the popular LIBSVM tool.[2][3] The publication of the SMO algorithm in 1998 has generated a lot of excitement in the SVM community, as previously available methods for SVM training were much more complex and required expensive third-party QP solvers.[4]
Optimization problem[edit]
Main article: Support vector machine
Consider a binary classification problem with a dataset (x1, y1), ..., (xn, yn), where xi is an input vector and yi ∈ {-1, +1} is a binary label corresponding to it. A soft-margin support vector machine is trained by solving a quadratic programming problem, which is expressed in the dual form as follows:
{\displaystyle \max _{\alpha }\sum _{i=1}^{n}\alpha _{i}-{\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}y_{i}y_{j}K(x_{i},x_{j})\alpha _{i}\alpha _{j},}
{\displaystyle 0\leq \alpha _{i}\leq C,\quad {\mbox{ for }}i=1,2,\ldots ,n,}
{\displaystyle \sum _{i=1}^{n}y_{i}\alpha _{i}=0}
where C is an SVM hyperparameter and K(xi, xj) is the kernel function, both supplied by the user; and the variables
{\displaystyle \alpha _{i}}
SMO is an iterative algorithm for solving the optimization problem described above. SMO breaks this problem into a series of smallest possible sub-problems, which are then solved analytically. Because of the linear equality constraint involving the Lagrange multipliers
{\displaystyle \alpha _{i}}
, the smallest possible problem involves two such multipliers. Then, for any two multipliers
{\displaystyle \alpha _{1}}
{\displaystyle \alpha _{2}}
, the constraints are reduced to:
{\displaystyle 0\leq \alpha _{1},\alpha _{2}\leq C,}
{\displaystyle y_{1}\alpha _{1}+y_{2}\alpha _{2}=k,}
and this reduced problem can be solved analytically: one needs to find a minimum of a one-dimensional quadratic function.
{\displaystyle k}
is the negative of the sum over the rest of terms in the equality constraint, which is fixed in each iteration.
Find a Lagrange multiplier
{\displaystyle \alpha _{1}}
that violates the Karush–Kuhn–Tucker (KKT) conditions for the optimization problem.
Pick a second multiplier
{\displaystyle \alpha _{2}}
and optimize the pair
{\displaystyle (\alpha _{1},\alpha _{2})}
When all the Lagrange multipliers satisfy the KKT conditions (within a user-defined tolerance), the problem has been solved. Although this algorithm is guaranteed to converge, heuristics are used to choose the pair of multipliers so as to accelerate the rate of convergence. This is critical for large data sets since there are
{\displaystyle n(n-1)/2}
possible choices for
{\displaystyle \alpha _{i}}
{\displaystyle \alpha _{j}}
The first approach to splitting large SVM learning problems into a series of smaller optimization tasks was proposed by Bernhard Boser, Isabelle Guyon, Vladimir Vapnik.[5] It is known as the "chunking algorithm". The algorithm starts with a random subset of the data, solves this problem, and iteratively adds examples which violate the optimality conditions. One disadvantage of this algorithm is that it is necessary to solve QP-problems scaling with the number of SVs. On real world sparse data sets, SMO can be more than 1000 times faster than the chunking algorithm.[1]
In 1997, E. Osuna, R. Freund, and F. Girosi proved a theorem which suggests a whole new set of QP algorithms for SVMs.[6] By the virtue of this theorem a large QP problem can be broken down into a series of smaller QP sub-problems. A sequence of QP sub-problems that always add at least one violator of the Karush–Kuhn–Tucker (KKT) conditions is guaranteed to converge. The chunking algorithm obeys the conditions of the theorem, and hence will converge.[1] The SMO algorithm can be considered a special case of the Osuna algorithm, where the size of the optimization is two and both Lagrange multipliers are replaced at every step with new multipliers that are chosen via good heuristics.[1]
The SMO algorithm is closely related to a family of optimization algorithms called Bregman methods or row-action methods. These methods solve convex programming problems with linear constraints. They are iterative methods where each step projects the current primal point onto each constraint.[1]
Kernel perceptron
^ a b c d e Platt, John (1998). "Sequential Minimal Optimization: A Fast Algorithm for Training Support Vector Machines" (PDF). CiteSeerX 10.1.1.43.4376.
^ Chang, Chih-Chung; Lin, Chih-Jen (2011). "LIBSVM: A library for support vector machines". ACM Transactions on Intelligent Systems and Technology. 2 (3). doi:10.1145/1961189.1961199. S2CID 961425.
^ Zanni, Luca (2006). "Parallel Software for Training Large Scale Support Vector Machines on Multiprocessor Systems" (PDF).
^ Rifkin, Ryan (2002). Everything Old is New Again: a Fresh Look at Historical Approaches in Machine Learning (Ph.D. Thesis). Massachusetts Institute of Technology. p. 18. hdl:1721.1/17549.
^ Boser, B. E.; Guyon, I. M.; Vapnik, V. N. (1992). "A training algorithm for optimal margin classifiers". Proceedings of the fifth annual workshop on Computational learning theory - COLT '92. p. 144. CiteSeerX 10.1.1.21.3818. doi:10.1145/130385.130401. ISBN 978-0897914970. S2CID 207165665.
^ Osuna, E.; Freund, R.; Girosi, F. (1997). "An improved training algorithm for support vector machines". Neural Networks for Signal Processing [1997] VII. Proceedings of the 1997 IEEE Workshop. pp. 276–285. CiteSeerX 10.1.1.392.7405. doi:10.1109/NNSP.1997.622408. ISBN 978-0-7803-4256-9. S2CID 5667586.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Sequential_minimal_optimization&oldid=1068240249"
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Diophantus_II.VIII Knowpia
The eighth problem of the second book of Arithmetica by Diophantus (c. 200/214 AD – c. 284/298 AD) is to divide a square into a sum of two squares.
Diophantus II.VIII: Intersection of the line CB and the circle gives a rational point (x0,y0).
The solution given by DiophantusEdit
Diophantus takes the square to be 16 and solves the problem as follows:[1]
To divide a given square into a sum of two squares.
To divide 16 into a sum of two squares.
Let the first summand be
{\displaystyle x^{2}}
, and thus the second
{\displaystyle 16-x^{2}}
. The latter is to be a square. I form the square of the difference of an arbitrary multiple of x diminished by the root [of] 16, that is, diminished by 4. I form, for example, the square of 2x − 4. It is
{\displaystyle 4x^{2}+16-16x}
. I put this expression equal to
{\displaystyle 16-x^{2}}
. I add to both sides
{\displaystyle x^{2}+16x}
and subtract 16. In this way I obtain
{\displaystyle 5x^{2}=16x}
{\displaystyle x=16/5}
Thus one number is 256/25 and the other 144/25. The sum of these numbers is 16 and each summand is a square.
Geometrical interpretationEdit
Geometrically, we may illustrate this method by drawing the circle x2 + y2 = 42 and the line y = 2x - 4. The pair of squares sought are then x02 and y02, where (x0, y0) is the point not on the y-axis where the line and circle intersect. This is shown in the adjacent diagram.
Generalization of Diophantus's solutionEdit
Diophantus II.VIII: Generalized solution in which the sides of triangle OAB form a rational triple if line CB has a rational gradient t.
We may generalize Diophantus's solution to solve the problem for any given square, which we will represent algebraically as a2. Also, since Diophantus refers to an arbitrary multiple of x, we will take the arbitrary multiple to be tx. Then:
{\displaystyle {\begin{aligned}&(tx-a)^{2}=a^{2}-x^{2}\\\Rightarrow \ \ &t^{2}x^{2}-2atx+a^{2}=a^{2}-x^{2}\\\Rightarrow \ \ &x^{2}(t^{2}+1)=2atx\\\Rightarrow \ \ &x={\frac {2at}{t^{2}+1}}{\text{ or }}x=0.\\\end{aligned}}}
Therefore, we find that one of the summands is
{\displaystyle x^{2}=\left({\tfrac {2at}{t^{2}+1}}\right)^{2}}
{\displaystyle (tx-a)^{2}=\left({\tfrac {a(t^{2}-1)}{t^{2}+1}}\right)^{2}}
. The sum of these numbers is
{\displaystyle a^{2}}
and each summand is a square. Geometrically, we have intersected the circle x2 + y2 = a2 with the line y = tx - a, as shown in the adjacent diagram.[2] Writing the lengths, OB, OA, and AB, of the sides of triangle OAB as an ordered tuple, we obtain the triple
{\displaystyle \left[a;{\frac {2at}{t^{2}+1}};{\frac {a(t^{2}-1)}{t^{2}+1}}\right]}
The specific result obtained by Diophantus may be obtained by taking a = 4 and t = 2:
{\displaystyle \left[a;{\frac {2at}{t^{2}+1}};{\frac {a(t^{2}-1)}{t^{2}+1}}\right]=\left[{\frac {20}{5}};{\frac {16}{5}};{\frac {12}{5}}\right]={\frac {4}{5}}\left[5;4;3\right].}
We see that Diophantus' particular solution is in fact a subtly disguised (3, 4, 5) triple. However, as the triple will always be rational as long as a and t are rational, we can obtain an infinity of rational triples by changing the value of t, and hence changing the value of the arbitrary multiple of x.
This algebraic solution needs only one additional step to arrive at the Platonic sequence
{\displaystyle [{\tfrac {t^{2}+1}{2}};t;{\tfrac {t^{2}-1}{2}}]}
and that is to multiply all sides of the above triple by a factor
{\displaystyle \quad {\tfrac {t^{2}+1}{2a}}}
. Notice also that if a = 1, the sides [OB, OA, AB] reduce to
{\displaystyle \left[1;{\frac {2t}{t^{2}+1}};{\frac {t^{2}-1}{t^{2}+1}}\right].}
In modern notation this is just
{\displaystyle (1,\sin \theta ,\cos \theta ),}
for θ shown in the above graph, written in terms of the cotangent t of θ/2. In the particular example given by Diophantus, t has a value of 2, the arbitrary multiplier of x. Upon clearing denominators, this expression will generate Pythagorean triples. Intriguingly, the arbitrary multiplier of x has become the cornerstone of the generator expression(s).
Diophantus II.IX reaches the same solution by an even quicker route which is very similar to the 'generalized solution' above. Once again the problem is to divide 16 into two squares.[3]
Let the first number be N and the second an arbitrary multiple of N diminished by the root (of) 16. For example 2N − 4. Then:
{\displaystyle {\begin{aligned}&N^{2}+(2N-4)^{2}=16\\\Rightarrow \ \ &5N^{2}+16-16N=16\\\Rightarrow \ \ &5N^{2}=16N\\\Rightarrow \ \ &N={\frac {16}{5}}\\\end{aligned}}}
Fermat's famous comment which later became Fermat's Last Theorem appears sandwiched between 'Quaestio VIII' and 'Quaestio IX' on page 61 of a 1670 edition of Arithmetica.
Fermat's Last Theorem and Diophantus II.VIII
^ Arithmetica, Diophantus. Book II, problem 8. As paraphrased on p. 24, Diophantus and Diophantine Equations, Isabella Grigoryevna Bashmakova, updated by Joseph Silverman, tr. from Russian by Abe Shenitzer and Hardy Grant. Washington, DC: The Mathematical Association of America, 1997. ISBN 0-88385-526-7. Orig. pub. Moscow: Nauke, 1972. A typo has been corrected in the quote.
^ Bashmakova, pp. 24–25.
^ This solution is II.IX in the numbering of Diophantos of Alexandria: A Study in the History of Greek Algebra, Sir Thomas Little Heath, Cambridge: University of Cambridge Press, 1885. In the numbering of Diophanti Alexandrini Opera Omnia cum Graecis Commentariis, ed. and translated by Paul Tannery, Leipzig: B. G. Teubner, 1893, it is part of II.VIII.
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AreSkewLines - Maple Help
Home : Support : Online Help : Mathematics : Geometry : 3-D Euclidean : Line Segments : AreSkewLines
test if two lines are skew
AreSkewLines(l1, l2)
The routine returns true if the two given line are skew; false if they are not; and FAIL if it is unable to reach a conclusion.
The command with(geom3d,AreSkewLines) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geom3d}\right):
\mathrm{line}\left(\mathrm{l1},[\mathrm{point}\left(\mathrm{A1},0,0,0\right),\mathrm{point}\left(\mathrm{A2},1,1,0\right)]\right)
\textcolor[rgb]{0,0,1}{\mathrm{l1}}
\mathrm{line}\left(\mathrm{l2},[\mathrm{point}\left(\mathrm{B1},0,0,1\right),\mathrm{point}\left(\mathrm{B2},1,-1,1\right)]\right)
\textcolor[rgb]{0,0,1}{\mathrm{l2}}
\mathrm{AreSkewLines}\left(\mathrm{l1},\mathrm{l2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{AreParallel}\left(\mathrm{l1},\mathrm{l2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
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Transform lowpass IIR filter to different lowpass filter - MATLAB iirlp2lp - MathWorks í•œêµ
iirlp2lp
Extend Passband of Lowpass Filter
Lowpass IIR Filter to Different Lowpass Filter Transformation
Transform lowpass IIR filter to different lowpass filter
[num,den] = iirlp2lp(b,a,wo,wt)
[num,den,allpassNum,allpassDen] = iirlp2lp(b,a,wo,wt)
[num,den] = iirlp2lp(b,a,wo,wt) transforms lowpass IIR filter to different lowpass filter.
The prototype lowpass filter is specified with the numerator and denominator coefficients, b and a respectively. The function returns the numerator and denominator coefficients of the transformed lowpass digital filter.
The function transforms the magnitude response from lowpass to a different lowpass. For more details, see Lowpass IIR Filter to Different Lowpass Filter Transformation.
[num,den,allpassNum,allpassDen] = iirlp2lp(b,a,wo,wt) in addition returns the numerator and the denominator coefficients of the allpass mapping filter.
Transform the passband of a lowpass IIR filter by moving the magnitude response from one frequency in the source filter to a new location in the transformed filter.
Generate a least P-norm optimal IIR lowpass filter using the iirlpnorm function. Specify a numerator order of 10 and a denominator order of 6. The function returns the coefficients both in the vector form and in the second-order sections (SOS) form. The output argument g specifies the overall gain of the filter when expressed in the second-order sections form.
[b,a,~,sos,g] = iirlpnorm(10,6, ...
[0 0.0175 0.02 0.0215 0.025 1], ...
[0 0.0175 0.02 0.0215 0.025 1],[1 1 0 0 0 0], ...
[1 1 1 1 10 10]);
Transform Filter Using iirlp2lp
Transform the passband of the lowpass IIR filter using the iirlp2lp function. Specify the filter as a vector of numerator and denominator coefficients.
To generate a lowpass filter whose passband extends out to 0.2Ï€ rad/sample, select the frequency in the lowpass filter at 0.0175Ï€, the frequency where the passband starts to roll off, and move it to the new location.
Moving the edge of the passband from 0.0175Ï€ to 0.2Ï€ results in a new lowpass filter whose peak response in-band is the same as in the original filter, with the same ripple and the same absolute magnitude. The rolloff is slightly less steep and the stopband profiles are the same for both filters. The new filter stopband is a "stretched" version of the original, as is the passband of the new filter.
wc = 0.0175;
[num,den] = iirlp2lp(b,a,wc,wd);
legend(hvft,"Prototype Filter (TF Form)", ...
"Transformed Lowpass Filter")
Alternatively, you can also specify the input lowpass IIR filter as a matrix of coefficients. Pass the scaled second order section coefficient matrices as inputs. Apply the scaling factor g to the first section of the filter.
sosg = sos;
sosg(1,1:3) = g*sosg(1,1:3);
[num2,den2] = iirlp2lp(sosg(:,1:3),sosg(:,4:6),wc,wd);
hvft = fvtool(sosg,[num2 den2]);
legend(hvft,"Prototype Filter (Matrix Form)",...
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{{b}_{0}+{b}_{1}{z}^{â1}+\cdots +{b}_{n}{z}^{ân}}{{a}_{0}+{a}_{1}{z}^{â1}+\cdots +{a}_{n}{z}^{ân}},
b=\left[\begin{array}{ccccc}{b}_{01}& {b}_{11}& {b}_{21}& ...& {b}_{Q1}\\ {b}_{02}& {b}_{12}& {b}_{22}& ...& {b}_{Q2}\\ ⋮& ⋮& ⋮& â±& ⋮\\ {b}_{0P}& {b}_{1P}& {b}_{2P}& \cdots & {b}_{QP}\end{array}\right]
H\left(z\right)=\underset{k=1}{\overset{P}{â}}{H}_{k}\left(z\right)=\underset{k=1}{\overset{P}{â}}\frac{{b}_{0k}+{b}_{1k}{z}^{â1}+{b}_{2k}{z}^{â2}+\cdots +{b}_{Qk}{z}^{âQ}}{{a}_{0k}+{a}_{1k}{z}^{â1}+{a}_{2k}{z}^{â2}+\cdots +{a}_{Qk}{z}^{âQ}},
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{{b}_{0}+{b}_{1}{z}^{â1}+\cdots +{b}_{n}{z}^{ân}}{{a}_{0}+{a}_{1}{z}^{â1}+\cdots +{a}_{n}{z}^{ân}},
a=\left[\begin{array}{ccccc}{a}_{01}& {a}_{11}& {a}_{21}& \cdots & {a}_{Q1}\\ {a}_{02}& {a}_{12}& {a}_{22}& \cdots & {a}_{Q2}\\ ⋮& ⋮& ⋮& â±& ⋮\\ {a}_{0P}& {a}_{1P}& {a}_{2P}& \cdots & {a}_{QP}\end{array}\right]
H\left(z\right)=\underset{k=1}{\overset{P}{â}}{H}_{k}\left(z\right)=\underset{k=1}{\overset{P}{â}}\frac{{b}_{0k}+{b}_{1k}{z}^{â1}+{b}_{2k}{z}^{â2}+\cdots +{b}_{Qk}{z}^{âQ}}{{a}_{0k}+{a}_{1k}{z}^{â1}+{a}_{2k}{z}^{â2}+\cdots +{a}_{Qk}{z}^{âQ}},
Frequency value to transform from the prototype filter, specified as a real positive scalar. Frequency wo must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.
wt — Desired frequency location in transformed target filter
Desired frequency location in the transformed target filter, specified as a real positive scalar. Frequency wt must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.
num — Numerator coefficients of transformed lowpass filter
Numerator coefficients of the transformed lowpass filter, returned as one of the following:
den — Denominator coefficients of transformed lowpass filter
Denominator coefficients of the transformed lowpass filter, returned as one of the following:
Lowpass IIR filter to different lowpass filter transformation takes a selected frequency from your lowpass filter, wo, and maps the corresponding magnitude response value onto the desired frequency location in the transformed lowpass filter, wt. Note that all frequencies are normalized between zero and one and that the filter order does not change when you transform to the target lowpass filter.
When you select wo and designate wt, the transformation algorithm sets the magnitude response at the wt values of your bandstop filter to be the same as the magnitude response of your lowpass filter at wo. Filter performance between the values in wt is not specified, except that the stopband retains the ripple nature of your original lowpass filter and the magnitude response in the stopband is equal to the peak response of your lowpass filter. To accurately specify the filter magnitude response across the stopband of your bandpass filter, use a frequency value from within the stopband of your lowpass filter as wc. Then your bandstop filter response is the same magnitude and ripple as your lowpass filter stopband magnitude and ripple.
The fact that the transformation retains the shape of the original filter is what makes this function useful. If you have a lowpass filter whose characteristics, such as rolloff or passband ripple, particularly meet your needs, the transformation function lets you create a new filter with the same characteristic performance features.
In some cases transforming your filter may cause numerical problems, resulting in incorrect conversion to the target filter. Use fvtool to verify the response of your converted filter.
[1] Nowrouzian, B., and A.G. Constantinides. “Prototype Reference Transfer Function Parameters in the Discrete-Time Frequency Transformations.†In Proceedings of the 33rd Midwest Symposium on Circuits and Systems, 1078–82. Calgary, Alta., Canada: IEEE, 1991. https://doi.org/10.1109/MWSCAS.1990.140912.
[2] Nowrouzian, B., and L.T. Bruton. “Closed-Form Solutions for Discrete-Time Elliptic Transfer Functions.†In [1992] Proceedings of the 35th Midwest Symposium on Circuits and Systems, 784–87. Washington, DC, USA: IEEE, 1992. https://doi.org/10.1109/MWSCAS.1992.271206.
[3] Constantinides, A.G.“Spectral transformations for digital filters.†Proceedings of the IEEE, vol. 117, no. 8: 1585-1590. August 1970.
iirlp2bp | iirlp2bs | iirlp2hp | firlp2lp | firlp2hp
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You can code on any laptop, but for multitasking and other heavy usages, you'll need a well-balanced laptop capable of handling many activities. So, what is a decent coding laptop? It should also have sufficient specs, such as a good keyboard for typing, SSD storage, enough RAM, and a good CPU from Intel, AMD, or Apple.
Here is a list of the greatest laptops available for purchase:
2. MacBook Pro 13-inch:
4. HP Envy 13 (Model 13-ba1047wm):
The LG Gram 17 laptop is a fantastic choice if you require a big, easy-to-read display for your programming needs. Because the screen is not just huge (17 inches), but also has a high resolution, you'll have enough high-quality real estate to work with.
Despite its enormous screen, the LG Gram 17 is surprisingly compact and light (less than 3 pounds), allowing you to enjoy all of the benefits of a large laptop without losing portability. And if you're always on the go, you won't have to worry about continual charging because the Gram has an extremely long battery life — it keeps a full charge for little under 20 hours. Under the hood, you'll find a 10-generation Intel Core i7 CPU with up to 64GB of RAM.
That's a plus. Now for the terrible news: You might not be smitten by the $1,700 price tag or the fact that its touchpad might be better. Some people complain that it is not particularly user-friendly, however this might just be due to the fact that it is new.
The LG Gram 17 laptop's specifications are as follows:
Intel® CoreTM i7-8565U 1.80GHz (base) / 4.60GHz (Turbo Clock)
Hard drive: 256 GB SSD / 512 GB SATA 3 storage
WQXGA 2560 x 1600 resolution on a 17-inch screen. Type of LCD: IPS
Intel UHD Graphics 620 is used for graphics.
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The 2020 MacBook Pro sports an M1 chip and an 8-core CPU, making it an excellent computer for programmers around $1,100. Apps operate smoothly, code compiles swiftly, and the battery life provides enough energy for all-day work on the go — up to 20 hours on a single charge. To increase your productivity, there's a Touch Bar above the keyboard that offers you rapid access to shortcuts and features, which comes in useful pretty frequently.
When typing on the MacBook Pro, you'll have enough keyboard comfort, but it lacks connectors - it has a display port and two Thunderbolt/USB 4 connections. However, for certain developers, this may not be an issue. The 13-inch display may be a challenge for web developers, game developers, and any programmer trying to debug software. However, its modest size makes it an excellent alternative for developers who are frequently on the road.
The following are the specifications for the MacBook Pro 13-inch laptop:
Apple M1 processor 8-core processor (4 performance / 4 efficiency). GPU with 8 cores. Neural Engine with 16 cores.
RAM: A total of 8 GB of unified memory. Upgradeable to 16GB of storage.
SSD 256 GB hard drive (Configurable up to 2TB)
LED-backlit 13.3-inch screen The native resolution is 2560 x 1600.
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While there is no specific order to our list, the 2020 XPS 15 may be the greatest development laptop of the lot. It looks nice, is well-made, and has enough power under the hood thanks to a 10th-generation Intel Core i5 or i7 CPU.
Despite its huge 15.6-inch display, the XPS 15 is relatively light and small, making it easy to transport. And, although you'll have enough of power for coding, trying out games while they're being developed, and so on, you won't have to worry about a lack of battery life, as the XPS 15 is designed to endure without being continually plugged in – roughly 17 hours when running resource-intensive programmes. Another advantage is that the XPS 15 is environmentally friendly, with recycled packaging and 90 percent of the laptop's parts recyclable.
The XPS 15 might cost anywhere from just over
1,000 to well over
2000, depending on your setup.
The following are the specifications for the Dell XPS 15 laptop:
Operating system/OS: Windows 10 Home, with a free upgrade to
Windows 11 when it becomes available.
Processor: Intel® CoreTM i5-10300H (10th Generation) (8MB Cache, up to 4.5 GHz, 4 cores).
8GB DDR4-2933MHz RAM (2x4G)
M.2 PCIe NVMe Solid State Drive with 256GB capacity
15.6-inch Infinity Edge screen with anti-glare coating. The resolution is 1920 x 1200.
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If you're working on a tight budget, the HP Envy 13 has a starting price of $650. Despite its modest price, the Envy has a quality appearance and feel because to its all-metal design, which both attracts attention and provides durability when dragging it around town.
This small laptop, which is very portable but powerful, is powered by an 8th-generation Intel Core i5 or i7 processor. It's also a joy to use, due to a huge trackpad and a fantastic keyboard that enables smooth typing and screen navigation.
Where does the HP Envy 13 let you down? For graphics-intensive apps and developers examining code, the tiny display might be an issue. This is further complicated by the lack of a 4K option. It also features a built-in graphics chip, which may be a detriment to game creators.
The following are the specifications for the HP Envy 13 laptop:
Windows 10 Home is the operating system. 64
11th Generation Processor Processor Intel® CoreTM i5
RAM: DDR4-2666 MHz RAM (8 GB) (onboard)
256 GB PCIe® NVMeTM M.2 SSD hard drive
The diagonal width of the screen is 13.3 inches. 1920 x 1080 is the resolution.
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DSGE模型的Stata实现简介| 连享会主页
A small DSGE model
Specifying the DSGE to dsge
Impulse–responses
今天，在 Stata Blog 看到了一篇文章，使用一个简单范例介绍了 Stata 中如何估计 DSGE 模型的参数。
Source: Estimating the parameters of DSGE models
In this post, I build a small DSGE model that is similar to models used for monetary policy analysis. I show how to estimate the parameters of this model using the new dsge command in Stata 15. I then shock the model with a contraction in monetary policy and graph the response of model variables to the shock.
[^1]: Clarida, R., J. Galí, and M. Gertler. 1999. The science of monetary policy: A new Keynesian perspective. Journal of Economic Literature 37: 1661–1707. [^2]: Woodford, M. 2003. Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton, NJ: Princeton University Press.
A DSGE model begins with a description of the sectors of the economy to be modeled. The model I describe here is related to the models developed in Clarida, Galí, and Gertler (1999)[^1] and Woodford (2003)[^2]. It is a smaller version of the kinds of models used in central banks and academia for monetary policy analysis. The model has three sectors: households, firms, and a central bank.
Households consume output. Their decision making is summarized by an output demand equation that relates current output demand to expected future output demand and the real interest rate.
Firms set prices and produce output to satisfy demand at the set price. Their decision making is summarized by a pricing equation that relates current inflation (that is, the change in prices) to expected future inflation and current demand. The parameter capturing the degree to which inflation depends on output demand plays a key role in the model.
The central bank sets the nominal interest rate in response to inflation. The central bank increases the interest rate when inflation rises and reduces the interest rate when inflation falls.
The model can be summarized in three equations,
\begin{array}{rl}{x}_{t}& ={E}_{t}\left({x}_{t+1}\right)-\left\{{r}_{t}-{E}_{t}\left({\pi }_{t+1}\right)-{z}_{t}\right\}\\ {\pi }_{t}& =\beta {E}_{t}\left({\pi }_{t+1}\right)+\kappa {x}_{t}\\ {r}_{t}& =\frac{1}{\beta }{\pi }_{t}+{u}_{t}\end{array}
{x}_{t}
denotes the output gap. The output gap measures the difference between output and its long run, natural value. The notation
{E}_{t}\left({x}_{t+1}\right)
specifies the expectation, conditional on information a vailable at time
t,
of the output gap in period
t+1.
The nominal interest rate is
{r}_{t},
and the inflation rate is
{\pi }_{t}
. Equation (1) states that the output gap is related positively to the expected future output gap,
{E}_{t}\left({x}_{t+1}\right),
and negatively to the interest rate gap,
\left\{{r}_{t}-{E}_{t}\left({\pi }_{t+1}\right)-{z}_{t}\right\}.
The second equation is the firm's pricing equation; it relates inflation to expected future inflation and the output gap. The para meter
\kappa
determines the extent to which inflation depends on the output gap. Finally, the third equation summarizes the central bank's beha vior; it relates the interest rate to inflation and to other factors, collectively termed
{u}_{t}
The endogenous variables
{x}_{t},{\pi }_{t},
{r}_{t}
are driven by two exogenous variables,
{z}_{t}
{u}_{t}
In terms of the theory,
{z}_{t}
is the natural rate of interest. If the real interest rate is equal to the natural rate and is expected to remain so in the future, then the output gap is zero. The exogenous variable
{u}_{t}
captures all movements in the interest rate that arise from factors other than movements in inflation. It is sometimes referred to as the surprise component of monetary policy.
The two exogenous variables are modeled as first-order autoregressive processes,
\begin{array}{l}{z}_{t+1}={\rho }_{z}{z}_{t}+{\epsilon }_{t+1}\\ {u}_{t+1}={\rho }_{u}{u}_{t}+{\xi }_{t+1}\end{array}
which follows common practice.
In the jargon, endogenous variables are called control variables, and exogenous variables are called state variables. The values of control variables in a period are determined by the system of equations. Control variables can be observed or unobserved. State variables are fixed at the beginning of a period and are unobserved. The system of equations determines the value of state variables one period in the future.
We wish to use the model to answer policy questions. What is the effect on model variables when the central bank conducts a surprise increase in the interest rate? The answer to this question is to impose an impulse ξ~t~ and trace out the effect of the impulse over time.
Before doing policy analysis, we must assign values to the parameters of the model. We will estimate the parameters of the above model using U.S. data on inflation and interest rates with dsge in Stata.
I fit the model using data on the U.S. interest rate and inflation rate. In a DSGE model, you can have as many observable control variables as you have shocks in the model. Because the model has two shocks, we have two observable control variables. The variables in a linearized DSGE model are stationary and measured in deviation from steady state. In practice, this means the data must be de-meaned prior to estimation. dsge will remove the mean for you.
I use the data in usmacro2, which is drawn from the Federal Reserve Bank of St. Louis database.
To specify a model to Stata, type the equations using substitutable expressions.
. dsge (x = E(F.x) - (r - E(F.p) - z), unobserved) ///
(p = {beta}*E(F.p) + {kappa}*x) ///
(r = 1/{beta}*p + u) ///
(F.z = {rhoz}*z, state) ///
(F.u = {rhou}*u, state)
The rules for equations are similar to those for Stata’s other commands that work with substitutable expressions. Each equation is bound in parentheses. Parameters are enclosed in braces to distinguish them from variables. Expectations of future variables appear within the E() operator. One variable appears on the left-hand side of the equation. Further, each variable in the model appears on the left-hand side of one and only one equation. Variables can be either observed (exist as variables in your dataset) or unobserved. Because the state variables are fixed in the current period, equations for state variables express how the one-step-ahead value of the state variable depends on current state variables and, possibly, current control variables.
Estimating the model parameters gives us an output table:
> (p = {beta}*E(F.p) + {kappa}*x) ///
> (r = 1/{beta}*p + u) ///
> (F.z = {rhoz}*z, state) ///
> (F.u = {rhou}*u, state)
(setting technique to bfgs)
Iteration 3: log likelihood = -869.19312 (backed up)
(switching technique to nr)
Iteration 5: log likelihood = -819.0268 (not concave)
Sample: 1955q1 - 2015q4 Number of obs = 244
/structural |
beta | .514668 .078349 6.57 0.000 .3611067 .6682292
kappa | .1659046 .047407 3.50 0.000 .0729885 .2588207
rhoz | .9545256 .0186424 51.20 0.000 .9179872 .991064
rhou | .7005492 .0452603 15.48 0.000 .6118406 .7892578
sd(e.z)| .6211208 .1015081 .4221685 .820073
sd(e.u)| 2.3182 .3047433 1.720914 2.915486
The crucial parameter is {kappa}, which is estimated to be positive. This parameter is related to the underlying price frictions in the model. Its interpretation is that if we hold expected future inflation constant, a 1 percentage point increase in the output gap leads to a 0.17 percentage point increase in inflation.
The parameter β is estimated to be about 0.5, meaning that the coefficient on inflation in the interest rate equation is about 2. So the central bank increases the interest rate about two for one in response to movements in inflation. This parameter is much discussed in the monetary economics literature, and estimates of it cluster around 1.5. The value found here is comparable with those estimates. Both state variables z~t~ and u~t~ are estimated to be persistent, with autoregressive coefficients of 0.95 and 0.7, respectively.
We can now use the model to answer questions. One question the model can answer is, “What is the effect of an unexpected change in the interest rate on inflation and the output gap?” An unexpected change in the interest rate is modeled as a shock to the ut equation. In the language of the model, this shock represents a contraction in monetary policy.
An impulse is a series of values for the shock ξ in (5): (1,0,0,0,0,…). The shock then feeds into the model’s state variables, leading to an increase in u. From there, the increase in u leads to a change in all the model’s control variables. An impulse–response function traces out the effect of a shock on the model variables, taking into account all the interrelationships among variables present in the model equations.
We type three commands to build and graph an IRF. irf set sets the IRF file that will hold the impulse–responses. irf create creates a set of impulse–responses in the IRF file.
. irf set dsge_irf. irf create model1
With the impulse–responses saved, we can graph them:
. irf graph irf, impulse(u) response(x p r u) byopts(yrescale) yline(0)
The impulse–response graphs the response of model variables to a one-standard-deviation shock. Each panel is the response of one variable to the shock. The horizontal axis measures time since the shock, and the vertical axis measures deviations from long-run value. The bottom-left panel shows the response of the monetary state variable, u~t~.
The remaining three panels show the response of inflation, the interest rate, and the output gap. Inflation is in the top-left panel; it falls on impact of the shock. The interest rate response in the upper-right panel is a weighted sum of the inflation and monetary impulse–responses. The interest rate rises by about one-half of one percentage point. Finally, the output gap falls. Hence, the model predicts that after a monetary tightening, the economy will enter a recession. Over time, the effect of the shock dissipates, and all variables return to their long-run values.
In this post, I developed a small DSGE model and described how to estimate the parameters of the model using dsge. I then showed how to create and interpret an impulse–response function.
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Seismic and Hydroacoustic Observations from Underwater Explosions off the East Coast of FloridaSeismic and Hydroacoustic Observations from Underwater Explosions off the East Coast of Florida | Bulletin of the Seismological Society of America | GeoScienceWorld
Seismic and Hydroacoustic Observations from Underwater Explosions off the East Coast of Florida
Ross Heyburn;
Ross Heyburn
AWE Blacknest, Brimpton Common, Reading RG7 4RS, United Kingdom, ross@blacknest.gov.uk
Stuart E. J. Nippress;
Ross Heyburn, Stuart E. J. Nippress, David Bowers; Seismic and Hydroacoustic Observations from Underwater Explosions off the East Coast of Florida. Bulletin of the Seismological Society of America 2018;; 108 (6): 3612–3624. doi: https://doi.org/10.1785/0120180105
In this study, seismic and hydroacoustic signals from underwater explosions in 2001, 2008, and 2016 near Florida are analyzed. These 10,000 lb chemical explosions were detonated by the United States Navy to validate the ability of new classes of ships to withstand explosions. For many of the explosions, the ground‐truth (GT) epicenters are known. These epicenters are used to improve the accuracy of the locations of explosions with no GT data by performing a relative relocation using a Bayesian hierarchical seismic‐event locator. Seismic and hydroacoustic signals are also used to characterize the underwater explosion sources. Bubble pulse modulations, characteristic of underwater explosions, are identified at seismic stations in the United States, and the observed bubble pulse frequency is consistent with published GT information. The absence of clear modulations in the spectra caused by reverberations in the water column means that the depth of the explosions in the water and hence the trinitrotoluene (TNT) equivalent charge weight of the explosion cannot be resolved from the frequency of the bubble pulse modulations. Published estimates of the local magnitudes
ML
and the known charge weights of these explosions are compared with data from previous underwater explosions. A relationship between charge weight and
ML
from previous well‐calibrated explosions detonated in the Dead Sea is shown to provide reasonable estimates of the charge weight once corrected for the salinity of the seawater near Florida. Hydroacoustic signals from the Florida underwater explosions are also observed as H phases on hydrophone sensors near Ascension Island. The bubble pulse is not observed as clearly at the hydrophone sensors at Ascension Island possibly as a result of signal distortion in the shallow water close to Florida. This has implications for event identification using hydrophone stations and demonstrates the importance of combining seismic and hydroacoustic observations.
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Simple Lie group - Wikipedia @ WordDisk
Overview of the classification
Simple Lie groups of small dimension
Simply laced groups
In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces.
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
This article is about the Killing-Cartan classification. For a smaller list of groups that commonly occur in theoretical physics, see Table of Lie groups. For for groups of dimension at most 3, see Bianchi classification.
Together with the commutative Lie group of the real numbers,
{\displaystyle \mathbb {R} }
, and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "special linear group" SL(n) of n by n matrices with determinant equal to 1 is simple for all n > 1.
The simple Lie groups were first classified by Wilhelm Killing and later perfected by Élie Cartan. This classification is often referred to as Killing-Cartan classification.
This article uses material from the Wikipedia article Simple Lie group, and is written by contributors. Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.
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evaluate - Mathematics - TopperLearning.com | er23bpgg
Asked by Rahulsinha1993 | 7th Sep, 2010, 08:50: PM
\int \frac{\mathrm{xdx}}{1+\mathrm{sinx}}=\quad \int \frac{\mathrm{xdx}}{1+\mathrm{sinx}}.\frac{1-\mathrm{sinx}}{1-\mathrm{sinx}}
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\int \frac{x(1-\mathrm{sinx})\mathrm{dx}}{{\mathrm{cos}}^{2}x}
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\quad \int x({\mathrm{sec}}^{2}x-\mathrm{secxtanx})\mathrm{dx}
\mathrm{integrating}\quad \mathrm{by}\quad \mathrm{parts},
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =x\int ({\mathrm{sec}}^{2}x-\mathrm{secxtanx})\mathrm{dx}-\int 1.\int ({\mathrm{sec}}^{2}x-\mathrm{secxtanx})\mathrm{dx}
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\quad x(\mathrm{tanx}-\mathrm{secx})\quad -\quad \int (\mathrm{tanx}-\mathrm{secx})\mathrm{dx}\quad +c
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\quad x(\mathrm{tanx}-\mathrm{secx})-\mathrm{log}\left|\mathrm{secx}\right|+\mathrm{log}\left|\mathrm{secx}+\mathrm{tanx}\right|\quad +c
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =x(\mathrm{tanx}-\mathrm{secx})+\mathrm{log}\left|\frac{\mathrm{secx}+\mathrm{tanx}}{\mathrm{secx}}\right|+c
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =x(\mathrm{tanx}-\mathrm{secx})+\mathrm{log}\left|1+\mathrm{sinx}\right|+c
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The Explosion at the Center of Our Galaxy
Th. Pongprasart, Bang Saphan, Prachuap Kiri Khan, Thailand.
Abstract: At the center of the Milky Way, our black hole may have suddenly changed from supermassive to intermediate-mass status. In doing so, it would have emitted an enormous burst of electromagnetic radiation. Here, the total energy of that burst is calculated and compared with the Fermi bubble data.
Keywords: Black Hole Explosion, Fermi Bubbles
In the electron-positron model [1], black holes naturally separate into two distinct varieties. Those of intermediate mass are supported against gravity by electron degeneracy pressure, while supermassive black holes are supported by ideal gas and radiation pressure. Their equilibrium states are reproduced in Table 1 and Table 2.
A critical mass of
8\times {10}^{6}{\text{M}}_{\odot }
defines the transition region.
Over the course of time, these black holes gradually lose energy by heating ionized gas in the accretion disk. They also suffer the loss of electrons, positrons and low-level radiation directly from the black hole itself. This raises the possibility that our black hole (
4\times {10}^{6}{\text{M}}_{\odot }
, Table 1) may have undergone a spontaneous
transition from supermassive to intermediate-mass status. The radius of the metastable state would have been
R={R}_{s}=1.2\times {10}^{12}\text{\hspace{0.05em}}\text{cm}
. At a point in time millions of years ago, the radius suddenly increased to
R=2.5{R}_{s}=3\times {10}^{12}\text{\hspace{0.05em}}\text{cm}
, releasing the electromagnetic radiation. The remaining leptons settled into the stable quantum state that exists today.
The pressure in a supermassive black hole is given by
P={P}_{\text{gas}}+{P}_{\text{rad}}=\frac{\rho }{m}kT+\frac{a}{3}{T}^{4}
a={\pi }^{2}{k}^{4}/15{\left(\hslash c\right)}^{3}
. In states near the transition region, the gas pressure is far greater than the radiation pressure. This is due to the high number density of leptons. For purposes of calculation, it is simpler to adopt the uniform density model [2], in which case the pressure satisfies
P={P}_{0}\left(1-\frac{{r}^{2}}{{R}^{2}}\right)
Ignoring the radiation pressure leaves the linear ideal gas relation between pressure and temperature, so that
T={T}_{0}\left(1-\frac{{r}^{2}}{{R}^{2}}\right)
The energy density of radiation is
{u}_{\text{rad}}=a{T}^{4}=a{T}_{0}^{4}{\left(1-\frac{{r}^{2}}{{R}^{2}}\right)}^{4}
which yields the total radiant energy
{U}_{\text{rad}}={\displaystyle {\int }_{0}^{R}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{u}_{\text{rad}}4\pi {r}^{2}\text{d}r=1.6\times {10}^{58}\text{\hspace{0.05em}}\text{erg}
M=4\times {10}^{6}\text{\hspace{0.05em}}{\text{M}}_{\odot }\mathrm{,}R=1.2\times {10}^{12}\text{\hspace{0.05em}}\text{cm}
{T}_{0}=1.3\times {10}^{9}\text{\hspace{0.05em}}\text{K}
The energy calculated here is an order of magnitude greater than the current upper estimates for the Fermi bubbles. Nevertheless, given the experimental uncertainties and given the limitations of the model, it may be said that the work is in substantial agreement with observation.
Cite this paper: Dalton, K. (2020) The Explosion at the Center of Our Galaxy. Journal of High Energy Physics, Gravitation and Cosmology, 6, 440-442. doi: 10.4236/jhepgc.2020.63033.
[1] Dalton, K. (2019) Supermassive Black Holes. JHEPGC, 5, 984-988.
[2] Dalton, K. (2014) The Galactic Black Hole. Hadronic J., 37, 241.
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Addition - Wikipedia @ WordDisk
{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}
{\displaystyle \scriptstyle {\text{sum}}}
{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}
{\displaystyle \scriptstyle {\text{difference}}}
{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}
{\displaystyle \scriptstyle {\text{product}}}
{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\text{ }}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}
{\displaystyle {\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}}
{\displaystyle \scriptstyle {\text{base}}^{\text{exponent}}\,=\,}
{\displaystyle \scriptstyle {\text{power}}}
{\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}
{\displaystyle \scriptstyle {\text{root}}}
{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}
{\displaystyle \scriptstyle {\text{logarithm}}}
This article uses material from the Wikipedia article Addition, and is written by contributors. Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.
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Quadratrix_of_Hippias Knowpia
The quadratrix or trisectrix of Hippias (also quadratrix of Dinostratus) is a curve which is created by a uniform motion. It is one of the oldest examples for a kinematic curve (a curve created through motion). Its discovery is attributed to the Greek sophist Hippias of Elis, who used it around 420 BC in an attempt to solve the angle trisection problem (hence trisectrix). Later around 350 BC Dinostratus used it in an attempt to solve the problem of squaring the circle (hence quadratrix).
Quadratrix (red); snapshot of E and F having completed 60% of their motions
Quadratrix as plane curve (a = 1)
Quadratrix as function (a = 1)
Consider a square ABCD with an inscribed quarter circle centered in A, such that the side of the square is the circle's radius. Let E be a point that travels with a constant angular velocity on the quarter circle arc from D to B. In addition the point F travels with a constant velocity from D to A on the line segment AD, in such a way that E and F start at the same time at D and arrive at the same time in B and A. Now the quadratrix is defined as the locus of the intersection of the parallel to AB through F and the line segment AE.[1][2]
If one places such a square ABCD with side length a in a (Cartesian) coordinate system with the side AB on the x-axis and vertex A in the origin, then the quadratix is described by a planar curve
{\displaystyle \gamma :(0,{\tfrac {\pi }{2}}]\rightarrow \mathbb {R} ^{2}}
{\displaystyle \gamma (t)={\begin{pmatrix}x(t)\\y(t)\end{pmatrix}}={\begin{pmatrix}{\frac {2a}{\pi }}t\cot(t)\\{\frac {2a}{\pi }}t\end{pmatrix}}}
This description can also be used to give an analytical rather than a geometric definition of the quadratrix and to extend it beyond the
{\displaystyle (0,{\tfrac {\pi }{2}}]}
interval. It does however remain undefined at the singularities of
{\displaystyle \cot(t)}
except for the case of
{\displaystyle t=0}
where the singularity is removable due to
{\displaystyle \lim _{t\to 0}t\cot(t)=1}
and hence yields a continuous planar curve on the interval
{\displaystyle (-\pi ,\pi )}
To describe the quadratrix as simple function rather than planar curve, it is advantageous to switch the y-axis and the x-axis, that is to place the side AB on y-axis rather than on the x-axis. Then the quadratrix is given by the following function:[5][6]
{\displaystyle f(x)=x\cdot \cot \left({\frac {\pi }{2a}}\cdot x\right)}
Quadratrix compass
The trisection of an arbitrary angle using only ruler and compasses is impossible. However, if the quadratrix is allowed as an additional tool, it is possible to divide an arbitrary angle into n equal segments and hence a trisection (n = 3) becomes possible. In practical terms the quadratrix can be drawn with the help of a template or a quadratrix compass (see drawing).[1][2]
Since, by the definition of the quadratrix, the traversed angle is proportional to the traversed segment of the associated squares' side dividing that segment on the side into n equal parts yields a partition of the associated angle as well. Dividing the line segment into n equal parts with ruler and compass is possible due to the intercept theorem.
For a given angle BAE (≤ 90°) construct a square ABCD over its leg AB. The other leg of the angle intersects the quadratrix of the square in a point G and the parallel line to the leg AB through G intersects the side AD of the square in F. Now the segment AF corresponds to the angle BAE and due to the definition of the quadratrix any division of the segment AF in n equidistant parts yields a corresponding division of the angle BAE into n parts of equal size. To divide the segment AF into n equidistant parts proceed as follows. Draw a ray a with origin in A and then draw n equidistant segments (of arbitrary length) on it. Connect the endpoint O of the last segment with F and draw lines parallel to OF through all the endpoints of the remaining n − 1 segments on AO, these parallel lines divide the segment AF on AD into n equidistant segments. Now draw parallel lines to AB through the endpoints of those segments on AF, these parallel lines will intersects the trisectrix. Connecting those points of intersection with A yields a partition of angle BAE into n parts of equal size.[5]
Since not all points of the trisectrix can be constructed with circle and compass alone, it is really required as an additional tool next to compass and circle. However it is possible to construct a dense subset of the trisectrix by circle and compass, so while you cannot assure an exact division of an angle into n parts without a given trisectrix, you can construct an arbitrarily close approximation by circle and compass alone.[2][3]
Squaring of the circleEdit
Squaring of a quarter circle with radius 1
Squaring the circle with ruler and compass alone is impossible. However, if one allows the quadratrix of Hippias as an additional construction tool, the squaring of the circle becomes possible due to Dinostratus' theorem. It lets one turn a quarter circle into square of the same area, hence a square with twice the side length has the same area as the full circle.
According to Dinostratus' theorem the quadratrix divides one of the sides of the associated square in a ratio of
{\displaystyle {\tfrac {2}{\pi }}}
.[1] For a given quarter circle with radius r one constructs the associated square ABCD with side length r. The quadratrix intersect the side AB in J with
{\displaystyle \left|{\overline {AJ}}\right|={\tfrac {2}{\pi }}r}
. Now one constructs a line segment JK of length r being perpendicular to AB. Then the line through A and K intersects the extension of the side BC in L and from the intercept theorem follows
{\displaystyle \left|{\overline {BL}}\right|={\tfrac {\pi }{2}}r}
. Extending AB to the right by a new line segment
{\displaystyle \left|{\overline {BO}}\right|={\tfrac {r}{2}}}
yields the rectangle BLNO with sides BL and BO the area of which matches the area of the quarter circle. This rectangle can be transformed into a square of the same area with the help of Euclid's geometric mean theorem. One extends the side ON by a line segment
{\displaystyle \left|{\overline {OQ}}\right|=\left|{\overline {BO}}\right|={\tfrac {r}{2}}}
and draws a half circle to right of NQ, which has NQ as its diameter. The extension of BO meets the half circle in R and due to Thales' theorem the line segment OR is the altitude of the right-angled triangle QNR. Hence the geometric mean theorem can be applied, which means that OR forms the side of a square OUSR with the same area as the rectangle BLNO and hence as the quarter circle.[7]
Note that the point J, where the quadratrix meets the side AB of the associated square, is one of the points of the quadratrix that cannot be constructed with ruler and compass alone and not even with the help of the quadratrix compass based on the original geometric definition (see drawing). This is due to the fact that the two uniformly moving lines coincide and hence there exists no unique intersection point. However relying on the generalized definition of the quadratrix as a function or planar curve allows for J being a point on the quadratrix.[8][9]
The quadratrix is mentioned in the works of Proclus (412–485), Pappus of Alexandria (3rd and 4th centuries) and Iamblichus (c. 240 – c. 325). Proclus names Hippias as the inventor of a curve called quadratrix and describes somewhere else how Hippias has applied the curve on the trisection problem. Pappus only mentions how a curve named quadratrix was used by Dinostratus, Nicomedes and others to square the circle. He neither mentions Hippias nor attributes the invention of the quadratrix to a particular person. Iamblichus just writes in a single line, that a curve called a quadratrix was used by Nicomedes to square the circle.[10][11][12]
Although based on Proclus' name for the curve it is conceivable that Hippias himself used it for squaring the circle or some other curvilinear figure, most historians of mathematics assume that Hippias invented the curve, but used it only for the trisection of angles. Its use for squaring the circle only occurred decades later and was due to mathematicians like Dinostratus and Nicomedes. This interpretation of the historical sources goes back to the German mathematician and historian Moritz Cantor.[11][12]
^ a b c Hischer, Horst (2000), "Klassische Probleme der Antike – Beispiele zur "Historischen Verankerung"" (PDF), in Blankenagel, Jürgen; Spiegel, Wolfgang (eds.), Mathematikdidaktik aus Begeisterung für die Mathematik – Festschrift für Harald Scheid, Stuttgart/Düsseldorf/Leipzig: Klett, pp. 97–118
^ a b c Henn, Hans-Wolfgang (2003), "Die Quadratur des Kreises", Elementare Geometrie und Algebra, Verlag Vieweg+Teubner, pp. 45–48
^ a b Jahnke, Hans Niels (2003), A History of Analysis, American Mathematical Society, pp. 30–31, ISBN 0821826239 ; excerpt, p. 30, at Google Books
^ Weisstein, Eric W., "Quadratrix of Hippias", MathWorld
^ a b Dudley, Underwood (1994), The Trisectors, Cambridge University Press, pp. 6–8, ISBN 0883855143 ; excerpt, p. 6, at Google Books
^ O'Connor, John J.; Robertson, Edmund F., "Quadratrix of Hippias", MacTutor History of Mathematics archive, University of St Andrews, p. cur
^ Holme, Audun (2010), Geometry: Our Cultural Heritage, Springer, pp. 114–116, ISBN 9783642144400
^ Delahaye, Jean-Paul (1999),
{\displaystyle \pi }
– Die Story, Springer, p. 71, ISBN 3764360569
^ O'Connor, John J.; Robertson, Edmund F., "Dinostratus", MacTutor History of Mathematics archive, University of St Andrews, p. bio
^ van der Waerden, Bartel Leendert (1961), Science Awakening, Oxford University Press, p. 146
^ a b Gow, James (2010), A Short History of Greek Mathematics, Cambridge University Press, pp. 162–164, ISBN 9781108009034
^ a b Heath, Thomas Little, A History of Greek Mathematics, Volume 1: From Thales to Euclid, Clarendon Press, pp. 182, 225–230
Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA 2010, ISBN 9780883853481, pp. 146–147 (excerpt, p. 146, at Google Books)
Felix Klein: Famous Problems of Elementary Geometry. Cosimo 2007 (Nachdruck), ISBN 9781602064171, pp. 57–58 (excerpt, p. 57, at Google Books) (complete online copy at archive.org)
Wikimedia Commons has media related to Quadratrix of Hippias.
Michael D. Huberty, Ko Hayashi, Chia Vang: Hippias' Quadratrix
Weisstein, Eric W., "Quadratrix of Hippias", MathWorld
O'Connor, John J.; Robertson, Edmund F., "Quadratrix of Hippias", MacTutor History of Mathematics archive, University of St Andrews, p. cur
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Truncated icosidodecahedron - Wikipedia
Elements F = 62, E = 180, V = 120 (χ = 2)
Faces by sides 30{4}+20{6}+12{10}
Conway notation bD or taD
Schläfli symbols tr{5,3} or
{\displaystyle t{\begin{Bmatrix}5\\3\end{Bmatrix}}}
Wythoff symbol 2 3 5 |
Symmetry group Ih, H3, [5,3], (*532), order 120
Rotation group I, [5,3]+, (532), order 60
Dihedral angle 6-10: 142.62°
4-10: 148.28°
4-6: 159.095°
Properties Semiregular convex zonohedron
In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
It has 62 faces: 30 squares, 20 regular hexagons, and 12 regular decagons. It has the most edges and vertices of all Platonic and Archimedean solids, though the snub dodecahedron has more faces. Of all vertex-transitive polyhedra, it occupies the largest percentage (89.80%) of the volume of a sphere in which it is inscribed, very narrowly beating the snub dodecahedron (89.63%) and Small Rhombicosidodecahedron (89.23%), and less narrowly beating the Truncated Icosahedron (86.74%); it also has by far the greatest volume (206.8 cubic units) when its edge length equals 1. Of all vertex-transitive polyhedra that are not prisms or antiprisms, it has the largest sum of angles (90 + 120 + 144 = 354 degrees) at each vertex; only a prism or antiprism with more than 60 sides would have a larger sum. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a 15-zonohedron.
6 Spherical tilings and Schlegel diagrams
7 Geometric variations
8 Truncated icosidodecahedral graph
The name truncated icosidodecahedron, given originally by Johannes Kepler, is misleading. An actual truncation of an icosidodecahedron has rectangles instead of squares. This nonuniform polyhedron is topologically equivalent to the Archimedean solid.
Alternate interchangeable names are:
Truncated icosidodecahedron (Johannes Kepler)
Rhombitruncated icosidodecahedron (Magnus Wenninger[1])
Great rhombicosidodecahedron (Robert Williams,[2] Peter Cromwell[3])
Omnitruncated dodecahedron or icosahedron (Norman Johnson)
Icosidodecahedron and its truncation
The name great rhombicosidodecahedron refers to the relationship with the (small) rhombicosidodecahedron (compare section Dissection).
There is a nonconvex uniform polyhedron with a similar name, the nonconvex great rhombicosidodecahedron.
The surface area A and the volume V of the truncated icosidodecahedron of edge length a are:[citation needed]
{\displaystyle {\begin{aligned}A&=30\left(1+{\sqrt {3}}+{\sqrt {5+2{\sqrt {5}}}}\right)a^{2}&&\approx 174.292\,0303a^{2}.\\V&=\left(95+50{\sqrt {5}}\right)a^{3}&&\approx 206.803\,399a^{3}.\end{aligned}}}
If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest.
Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2φ − 2, centered at the origin, are all the even permutations of:[4]
(±(2φ − 1), ±2, ±(2 + φ)) and
where φ = 1 + √5/2 is the golden ratio.
The truncated icosidodecahedron is the convex hull of a rhombicosidodecahedron with cuboids above its 30 squares, whose height to base ratio is φ. The rest of its space can be dissected into nonuniform cupolas, namely 12 between inner pentagons and outer decagons and 20 between inner triangles and outer hexagons.
An alternative dissection also has a rhombicosidodecahedral core. It has 12 pentagonal rotundae between inner pentagons and outer decagons. The remaining part is a toroidal polyhedron.
These images show the rhombicosidodecahedron (violet) and the truncated icosidodecahedron (green). If their edge lengths are 1, the distance between corresponding squares is φ.
The toroidal polyhedron remaining after the core and twelve rotundae are cut out
The truncated icosidodecahedron has seven special orthogonal projections, centered on a vertex, on three types of edges, and three types of faces: square, hexagonal and decagonal. The last two correspond to the A2 and H2 Coxeter planes.
Spherical tilings and Schlegel diagrams[edit]
The truncated icosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Schlegel diagrams are similar, with a perspective projection and straight edges.
Decagon-centered
Geometric variations[edit]
Within Icosahedral symmetry there are unlimited geometric variations of the truncated icosidodecahedron with isogonal faces. The truncated dodecahedron, rhombicosidodecahedron, and truncated icosahedron as degenerate limiting cases.
Truncated icosidodecahedral graph[edit]
120 (A5×2)
In the mathematical field of graph theory, a truncated icosidodecahedral graph (or great rhombicosidodecahedral graph) is the graph of vertices and edges of the truncated icosidodecahedron, one of the Archimedean solids. It has 120 vertices and 180 edges, and is a zero-symmetric and cubic Archimedean graph.[5]
Schlegel diagram graphs
Bowtie icosahedron and dodecahedron contain two trapezoidal faces in place of the square.[6]
This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
^ Wenninger, (Model 16, p. 30)
^ Williamson (Section 3-9, p. 94)
^ Cromwell (p. 82)
^ Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons Craig S. Kaplan
Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.
Eric W. Weisstein, GreatRhombicosidodecahedron (Archimedean solid) at MathWorld.
Klitzing, Richard. "3D convex uniform polyhedra x3x5x - grid".
* Weisstein, Eric W. "Great rhombicosidodecahedral graph". MathWorld.
Editable printable net of a truncated icosidodecahedron with interactive 3D view
Retrieved from "https://en.wikipedia.org/w/index.php?title=Truncated_icosidodecahedron&oldid=1086280479"
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Energy Flow through Ecosystems | Boundless Biology | Course Hero
Strategies for Acquiring Energy
Autotrophs (producers) synthesize their own energy, creating organic materials that are utilized as fuel by heterotrophs (consumers).
Distinguish between photoautotrophs and chemoautotrophs and the ways in which they acquire energy
Food webs illustrate how energy flows through ecosystems, including how efficiently organisms acquire and use it.
Autotrophs, producers in food webs, can be photosynthetic or chemosynthetic.
Photoautotrophs use light energy to synthesize their own food, while chemoautotrophs use inorganic molecules.
Chemoautotrophs are usually bacteria that live in ecosystems where sunlight is unavailable.
Heterotrophs cannot synthesize their own energy, but must obtain it from autotrophs or other heterotrophs; they act as consumers in food webs.
photoautotroph: an organism that can synthesize its own food by using light as a source of energy
All living things require energy in one form or another since energy is required by most, complex, metabolic pathways (often in the form of ATP ); life itself is an energy-driven process. Living organisms would not be able to assemble macromolecules (proteins, lipids, nucleic acids, and complex carbohydrates) from their monomeric subunits without a constant energy input.
It is important to understand how organisms acquire energy and how that energy is passed from one organism to another through food webs and their constituent food chains. Food webs illustrate how energy flows directionally through ecosystems, including how efficiently organisms acquire it, use it, and how much remains for use by other organisms of the food web. Energy is acquired by living things in three ways: photosynthesis, chemosynthesis, and the consumption and digestion of other living or previously-living organisms by heterotrophs.
Photosynthetic and chemosynthetic organisms are grouped into a category known as autotrophs: organisms capable of synthesizing their own food (more specifically, capable of using inorganic carbon as a carbon source ). Photosynthetic autotrophs (photoautotrophs) use sunlight as an energy source, whereas chemosynthetic autotrophs (chemoautotrophs) use inorganic molecules as an energy source. Autotrophs act as producers and are critical for all ecosystems. Without these organisms, energy would not be available to other living organisms and life itself would not be possible.
Photoautotrophs, such as plants, algae, and photosynthetic bacteria, serve as the energy source for a majority of the world's ecosystems. These ecosystems are often described by grazing food webs. Photoautotrophs harness the solar energy of the sun by converting it to chemical energy in the form of ATP (and NADP). The energy stored in ATP is used to synthesize complex organic molecules, such as glucose.
Chemoautotrophs are primarily bacteria that are found in rare ecosystems where sunlight is not available, such as in those associated with dark caves or hydrothermal vents at the bottom of the ocean. Many chemoautotrophs in hydrothermal vents use hydrogen sulfide (H2S), which is released from the vents, as a source of chemical energy. This allows chemoautotrophs to synthesize complex organic molecules, such as glucose, for their own energy and in turn supplies energy to the rest of the ecosystem.
Chemoautotrophs: Swimming shrimp, a few squat lobsters, and hundreds of vent mussels are seen at a hydrothermal vent at the bottom of the ocean. As no sunlight penetrates to this depth, the ecosystem is supported by chemoautotrophic bacteria and organic material that sinks from the ocean's surface.
Heterotrophs function as consumers in the food chain; they obtain energy in the form of organic carbon by eating autotrophs or other heterotrophs. They break down complex organic compounds produced by autotrophs into simpler compounds, releasing energy by oxidizing carbon and hydrogen atoms into carbon dioxide and water, respectively. Unlike autotrophs, heterotrophs are unable to synthesize their own food. If they cannot eat other organisms, they will die.
Productivity, measured by gross and net primary productivity, is defined as the amount of energy that is incorporated into a biomass.
Explain the concept of primary production and distinguish between gross primary production and net primary production
A biomass is the total mass of living and previously-living organisms within a trophic level; ecosystems have characteristic amounts of biomass at each trophic level.
Net primary productivity (energy that remains in the primary producers after accounting for respiration and heat loss) is available to the primary consumers at the next trophic level.
gross primary productivity: rate at which photosynthetic primary producers incorporate energy from the sun
net primary productivity: energy that remains in the primary producers after accounting for the organisms' respiration and heat loss
Productivity within an ecosystem can be defined as the percentage of energy entering the ecosystem incorporated into biomass in a particular trophic level. Biomass is the total mass in a unit area (at the time of measurement) of living or previously-living organisms within a trophic level. Ecosystems have characteristic amounts of biomass at each trophic level. For example, in the English Channel ecosystem, the primary producers account for a biomass of 4 g/m2 (grams per meter squared), while the primary consumers exhibit a biomass of 21 g/m2.
The productivity of the primary producers is especially important in any ecosystem because these organisms bring energy to other living organisms by photoautotrophy or chemoautotrophy. Photoautotrophy is the process by which an organism (such as a green plant) synthesizes its own food from inorganic material using light as a source of energy; chemoautotrophy, on the other hand, is the process by which simple organisms (such as bacteria or archaea) derive energy from chemical processes rather than photosynthesis. The rate at which photosynthetic primary producers incorporate energy from the sun is called gross primary productivity. An example of gross primary productivity is the compartment diagram of energy flow within the Silver Springs aquatic ecosystem. In this ecosystem, the total energy accumulated by the primary producers was shown to be 20,810 kcal/m2/yr.
Energy flow in Silver Springs: This conceptual model shows the flow of energy through a spring ecosystem in Silver Springs, Florida. Notice that the energy decreases with each increase in trophic level.
Because all organisms need to use some of this energy for their own functions (such as respiration and resulting metabolic heat loss), scientists often refer to the net primary productivity of an ecosystem. Net primary productivity is the energy that remains in the primary producers after accounting for the organisms' respiration and heat loss. The net productivity is then available to the primary consumers at the next trophic level. In the Silver Spring example, 13,187 of the 20,810 kcal/m2/yr were used for respiration or were lost as heat, leaving 7,632 kcal/m2/yr of energy for use by the primary consumers.
\text{TLTE}=\frac { \text{production}\quad \text{at}\quad \text{present}\quad \text{trophic}\quad \text{level} }{ \text{production}\quad \text{at}\quad \text{previous}\quad \text{trophic}\quad \text{level} } \text{x}100
Food web of Lake Ontario: This food web shows the interactions between organisms across trophic levels in the Lake Ontario ecosystem. Primary producers are outlined in green, primary consumers in orange, secondary consumers in blue, and tertiary (apex) consumers in purple. Arrows point from an organism that is consumed to the organism that consumes it. Notice how some lines point to more than one trophic level. For example, the opossum shrimp eats both primary producers and primary consumers.
\text{NPE}=\frac { \text{net}\quad \text{consumer}\quad \text{productivity} }{ \text{assimilation} } \text{x}100
The inefficiency of energy use by warm-blooded animals has broad implications for the world's food supply. It is widely accepted that the meat industry uses large amounts of crops to feed livestock. Because the NPE is low, much of the energy from animal feed is lost. For example, it costs about $0.01 to produce 1000 dietary calories (kcal) of corn or soybeans, but approximately $0.19 to produce a similar number of calories growing cattle for beef consumption. The same energy content of milk from cattle is also costly, at approximately $0.16 per 1000 kcal. Much of this difference is due to the low NPE of cattle. Thus, there has been a growing movement worldwide to promote the consumption of non-meat and non-dairy foods so that less energy is wasted feeding animals for the meat industry.
When toxic substances are introduced into the environment, organisms at the highest trophic levels suffer the most damage.
Other substances that biomagnify are polychlorinated biphenyls (PCBs), which were used in coolant liquids in the United States until their use was banned in 1979, and heavy metals, such as mercury, lead, and cadmium. These substances were best studied in aquatic ecosystems where fish species at different trophic levels accumulate toxic substances brought through the ecosystem by the primary producers. In a study performed by the National Oceanic and Atmospheric Administration (NOAA) in the Saginaw Bay of Lake Huron, PCB concentrations increased from the ecosystem's primary producers (phytoplankton) through the different trophic levels of fish species. The apex consumer (walleye) had more than four times the amount of PCBs compared to phytoplankton. Also, based on results from other studies, birds that eat these fish may have PCB levels at least one order of magnitude higher than those found in the lake fish.
PCB concentration in Lake Huron: This chart shows the PCB concentrations found at the various trophic levels in the Saginaw Bay ecosystem of Lake Huron. Numbers on the x-axis reflect enrichment with heavy isotopes of nitrogen (15N), which is a marker for increasing trophic levels. Notice that the fish in the higher trophic levels accumulate more PCBs than those in lower trophic levels.
photoautotroph. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/photoautotroph. License: CC BY-SA: Attribution-ShareAlike
OpenStax College, Energy Flow through Ecosystems. October 17, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m44887/latest/Figure_46_02_01.jpg. License: CC BY: Attribution
Boundless. Provided by: Boundless Learning. Located at: http://www.boundless.com//biology/definition/gross-primary-productivity. License: CC BY-SA: Attribution-ShareAlike
Boundless. Provided by: Boundless Learning. Located at: http://www.boundless.com//biology/definition/net-primary-productivity. License: CC BY-SA: Attribution-ShareAlike
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ecological pyramid. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/ecological_pyramid. License: CC BY-SA: Attribution-ShareAlike
OpenStax College, Energy Flow through Ecosystems. October 17, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m44887/latest/Figure_46_02_02.png. License: CC BY: Attribution
dichlorodiphenyltrichloroethane. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/dichlorodiphenyltrichloroethane. License: CC BY-SA: Attribution-ShareAlike
biomagnification. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/biomagnification. License: CC BY-SA: Attribution-ShareAlike
Boundless. Provided by: Boundless Learning. Located at: http://www.boundless.com//biology/definition/apex-consumer. License: CC BY-SA: Attribution-ShareAlike
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Home : Support : Online Help : Connectivity : MTM Package : ceil
ceil(M)
For an integer M, ceil(M) is the smallest integer greater than or equal to M.
The ceil(M) function on a matrix, M, computes the element-wise ceil of M. The result, R is formed as R[i,j] = ceil(M[i,j]).
\mathrm{with}\left(\mathrm{MTM}\right):
M≔\mathrm{Matrix}\left(2,3,'\mathrm{fill}'=-3.6\right):
\mathrm{ceil}\left(M\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{-3}& \textcolor[rgb]{0,0,1}{-3}& \textcolor[rgb]{0,0,1}{-3}\\ \textcolor[rgb]{0,0,1}{-3}& \textcolor[rgb]{0,0,1}{-3}& \textcolor[rgb]{0,0,1}{-3}\end{array}]
MTM[round]
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Which Formulation Allows Using a Constant Shear Modulus for Small-Strain Buckling of Soft-Core Sandwich Structures? | J. Appl. Mech. | ASME Digital Collection
Which Formulation Allows Using a Constant Shear Modulus for Small-Strain Buckling of Soft-Core Sandwich Structures?
McCormick School Professor and W. P. Murphy Professor of Civil Engineering and Materials Science
, 2145 Sheridan Road, CEE, Evanston, IL 60208
e-mail: a-beghini@northwestern.edu
Zdeněk P. Bažant McCormick School Professor and W. P. Murphy Professor of Civil Engineering and Materials Science
Alessandro Beghini Graduate Research Assistant
Bažant, Z. P., and Beghini, A. (December 30, 2004). "Which Formulation Allows Using a Constant Shear Modulus for Small-Strain Buckling of Soft-Core Sandwich Structures?." ASME. J. Appl. Mech. September 2005; 72(5): 785–787. https://doi.org/10.1115/1.1979516
Although the stability theories energetically associated with different finite strain measures are mutually equivalent if the tangential moduli are properly transformed as a function of stress, only one theory can allow the use of a constant shear modulus
G
if the strains are small and the material deforms in the linear elastic range. Recently it was shown that, in the case of heterogeneous orthotropic structures very soft in shear, the choice of theory to use is related to the problem of proper homogenization and depends on the type of structure. An example is the difference between Engesser’s and Haringx’s formulas for critical load of columns with shear, which were shown to be energetically associated with Green’s and Almansi’s Lagrangian finite strain tensors. In a previous brief paper of the authors in a conference special issue, it was concluded on the basis of energy arguments that, for constant
G
, Engesser’s formula is correct for sandwich columns and Haringx’s formula for elastomeric bearings, but no supporting experimental results were presented. To present them, is the main purpose of this technical brief.
sandwich structures, buckling, shear modulus, elasticity, deformation
Bearings, Buckling, Sandwich structures, Shear (Mechanics), Shear modulus, Stress, Tensors
Shear Buckling of Sandwich, Fiber-Composite and Lattice Columns, Bearings and Helical Springs: Paradox Resolved
A Correlation Study of Incremental Deformations and Stability of Continuous Bodies
, New York; and republication with updates, Dover, New York, 2003.
,” Theoretical and Applied Mechanics Report No. 04-03/C441s,
, Evanston, IL;
0020-7683, (in press).
Easy-to-Compute Tensors with Symmetric Inverse Approximating Hencky Finite Strain and its Rate
Sandwich Construction: The Bending and Buckling of Sandwich Beams, Plates and Shells
The Handbook of Sandwich Construction
End Compression of Sandwich Column
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Sum of performance supercomputers Nov 2022 | Metaculus
Sum of performance supercomputers Nov 2022
In the seven decades since the invention of the point-contact transistor at Bell Labs, relentless progress in the development of semiconductor devices — Moore’s law — has been achieved despite regular warnings from industry observers about impending limits.
The TOP500 project collects and ranks system performance metrics of the most powerful non-distributed computer systems in the world. The project was started in 1993 and publishes an updated list of the supercomputers twice a year. The first of these updates always coincides with the International Supercomputing Conference in June, and the second is presented at the ACM/IEEE Supercomputing Conference in November.
The TOP500 ranks high-performance computing (HPC) by recording how fast a computer system solves a dense n by n system of linear equations in double precision (64 bits) arithmetic on distributed-memory computers (TOP500, 2019). This is an implementation of the High Performance Computing Linpack Benchmark.
What will the the sum of the level of performance (in exaFLOPS) of the all 500 supercomputers in the TOP500 be according to their November 2022 list?
This question resolves as the sum of performance (at Rmax) in exaFLOPS (1 exaFLOP =
10^{18}
FLOPS) of all supercomputers listed on the November 2022 TOP500 list.
This question resolves ambiguously if TOP500 stops reporting performance in terms of Rmax measured in TFlop/s on the Linpack benchmark.
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Find sources: "Graphical timeline from Big Bang to Heat Death" – news · newspapers · books · scholar · JSTOR (August 2021)
This is the timeline of the Universe from Big Bang to Heat Death scenario. The different eras of the universe are shown. The heat death will occur in around 1.7×10106 years, if protons decay.
Usually, the logarithmic scale is used for such timelines but it compresses the most interesting Stelliferous Era too much as this example shows. Therefore, a double-logarithmic scale s (s*100 in the graphics) is used instead. The minimum of it is only 1, not 0 as needed, and the negative outputs for inputs smaller than 10 are useless. Therefore, the time from 0.1 to 10 years is collapsed to a single point 0, but that does not matter in this case because nothing special happens in the history of the universe during that time.
{\displaystyle s={\begin{cases}\log _{10}\log _{10}year&{\mbox{if }}year>10{\mbox{ , corresponding to }}year=10^{10^{s}}\\0&{\mbox{if }}0.1\leq year\leq 10\\-\log _{10}(-\log _{10}year)&{\mbox{if }}year<0.1{\mbox{ , corresponding to }}year=10^{-10^{-s}}\end{cases}}}
Comparison of log10 and log10log10 scales
log10 year
combination of log10log10 year and
-log10(-log10 year)
100 0 undefined but here forced to 0
10−1 -1 0
10−2 -2 -0.30
10−10 -10 -1
10−100 -100 -2
The seconds in the timescale have been converted to years by
{\displaystyle second/31557600}
using the Julian year.
Big Rip – Cosmological model
Big Freeze – Future scenario if the expansion of the universe will continue forever or not
List of other end scenarios than Heat Death – Theories about the end of the universe
Timeline from Big Bang to the near cosmological future – Visual timeline of the universe
Tiny Graphical timeline from Big Bang to Heat Death – Future scenario if the expansion of the universe will continue forever or not - Timeline uses the log scale for comparison with the double-logarithmic scale in this article.
Fred C. Adams; Greg Laughlin (19 June 2000). The Five Ages of the Universe: Inside the Physics of Eternity. Simon and Schuster. ISBN 978-0-684-86576-8.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Graphical_timeline_from_Big_Bang_to_Heat_Death&oldid=1078142867"
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Will Derek Chauvin be acquitted of murder charges? | Metaculus
Will Derek Chauvin be acquitted of all murder charges?
Derek Chauvin is,
an American former police officer charged with the killing of George Floyd in Minneapolis, Minnesota, on May 25, 2020. During an arrest made by Chauvin and three other officers, he knelt on George Floyd's neck for almost eight minutes while Floyd was handcuffed and lying face down on a street. The death set off a series of protests around the world.
Chauvin was fired by the Minneapolis Police Department the day after the incident. He was initially charged with third-degree murder and second-degree manslaughter; a charge of second-degree murder was later added.
Some have suggested that he will be acquitted of his murder charges. From a Medium post,
There are six crucial pieces of information — six facts — that have been largely omitted from discussion on the Chauvin’s conduct. Taken together, they likely exonerate the officer of a murder charge. [...]
This question resolves positively if Derek Chauvin is acquitted of ALL murder
^†
charges OR all murder charges against him are dropped. Otherwise, it resolves negatively. If he dies before resolution, the question resolves ambiguously.
^†
Only convictions for offences actually called "murder" trigger negative resolution ; conviction for other offences such as manslaughter does not.
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Lemma 6.33.2 (00AL)—The Stacks project
Section 6.33: Glueing sheaves
Lemma 6.33.2. Let $X$ be a topological space. Let $X = \bigcup _{i\in I} U_ i$ be an open covering. Given any glueing data $(\mathcal{F}_ i, \varphi _{ij})$ for sheaves of sets with respect to the covering $X = \bigcup U_ i$ there exists a sheaf of sets $\mathcal{F}$ on $X$ together with isomorphisms
\[ \varphi _ i : \mathcal{F}|_{U_ i} \to \mathcal{F}_ i \]
such that the diagrams
\[ \xymatrix{ \mathcal{F}|_{U_ i \cap U_ j} \ar[r]_{\varphi _ i} \ar[d]_{\text{id}} & \mathcal{F}_ i|_{U_ i \cap U_ j} \ar[d]^{\varphi _{ij}} \\ \mathcal{F}|_{U_ i \cap U_ j} \ar[r]^{\varphi _ j} & \mathcal{F}_ j|_{U_ i \cap U_ j} } \]
Proof. First proof. In this proof we give a formula for the set of sections of $\mathcal{F}$ over an open $W \subset X$. Namely, we define
\[ \mathcal{F}(W) = \{ (s_ i)_{i \in I} \mid s_ i \in \mathcal{F}_ i(W \cap U_ i), \varphi _{ij}(s_ i|_{W \cap U_ i \cap U_ j}) = s_ j|_{W \cap U_ i \cap U_ j} \} . \]
Restriction mappings for $W' \subset W$ are defined by the restricting each of the $s_ i$ to $W' \cap U_ i$. The sheaf condition for $\mathcal{F}$ follows immediately from the sheaf condition for each of the $\mathcal{F}_ i$.
We still have to prove that $\mathcal{F}|_{U_ i}$ maps isomorphically to $\mathcal{F}_ i$. Let $W \subset U_ i$. In this case the condition in the definition of $\mathcal{F}(W)$ implies that $s_ j = \varphi _{ij}(s_ i|_{W \cap U_ j})$. And the commutativity of the diagrams in the definition of a glueing data assures that we may start with any section $s \in \mathcal{F}_ i(W)$ and obtain a compatible collection by setting $s_ i = s$ and $s_ j = \varphi _{ij}(s_ i|_{W \cap U_ j})$.
Second proof (sketch). Let $\mathcal{B}$ be the set of opens $U \subset X$ such that $U \subset U_ i$ for some $i \in I$. Then $\mathcal{B}$ is a base for the topology on $X$. For $U \in \mathcal{B}$ we pick $i \in I$ with $U \subset U_ i$ and we set $\mathcal{F}(U) = \mathcal{F}_ i(U)$. Using the isomorphisms $\varphi _{ij}$ we see that this prescription is “independent of the choice of $i$”. Using the restriction mappings of $\mathcal{F}_ i$ we find that $\mathcal{F}$ is a sheaf on $\mathcal{B}$. Finally, use Lemma 6.30.6 to extend $\mathcal{F}$ to a unique sheaf $\mathcal{F}$ on $X$. $\square$
Comment #3423 by Samir Canning on July 13, 2018 at 13:07
Maybe a slightly shorter proof can be given as follows: define a base for the topology on
X
consisting of all the open sets of each
U_i
and then define
\mathcal{F}
on the base in the obvious way. Then just use tag 009N to get a (unique) sheaf on
X
Thanks Samir! I added this as a second proof. See here.
Comment #5494 by Théo de Oliveira Santos on September 07, 2020 at 00:59
Trivial typo: "for somje
i\in I
3 comment(s) on Section 6.33: Glueing sheaves
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Equivalence Between Short-Time Biphasic and Incompressible Elastic Material Responses | J. Biomech Eng. | ASME Digital Collection
Departments of Mechanical Engineering and Biomedical Engineering,
Department of Bioengineering and Scientific Computing and Imaging Institute,
Ateshian, G. A., Ellis, B. J., and Weiss, J. A. (November 8, 2006). "Equivalence Between Short-Time Biphasic and Incompressible Elastic Material Responses." ASME. J Biomech Eng. June 2007; 129(3): 405–412. https://doi.org/10.1115/1.2720918
Porous-permeable tissues have often been modeled using porous media theories such as the biphasic theory. This study examines the equivalence of the short-time biphasic and incompressible elastic responses for arbitrary deformations and constitutive relations from first principles. This equivalence is illustrated in problems of unconfined compression of a disk, and of articular contact under finite deformation, using two different constitutive relations for the solid matrix of cartilage, one of which accounts for the large disparity observed between the tensile and compressive moduli in this tissue. Demonstrating this equivalence under general conditions provides a rationale for using available finite element codes for incompressible elastic materials as a practical substitute for biphasic analyses, so long as only the short-time biphasic response is sought. In practice, an incompressible elastic analysis is representative of a biphasic analysis over the short-term response
δt⪡Δ2∕∥C4∥∥K∥
Δ
is a characteristic dimension,
C4
is the elasticity tensor, and
K
is the hydraulic permeability tensor of the solid matrix. Certain notes of caution are provided with regard to implementation issues, particularly when finite element formulations of incompressible elasticity employ an uncoupled strain energy function consisting of additive deviatoric and volumetric components.
biomechanics, biological tissues, elasticity, deformation, biorheology, elastic moduli, flow through porous media, permeability, tensors, finite element analysis
Biological tissues, Compression, Constitutive equations, Deformation, Elasticity, Finite element analysis, Permeability, Stress, Tensors, Tension, Porous materials, Cartilage, Fluid pressure, Disks, Elastic analysis
Biphasic Indentation of Articular Cartilage. I. Theoretical Analysis
A Mathematical Analysis for Indentation Tests of Articular Cartilage
Linear Elastic and Poroelastic Models of Cartilage Can Produce Comparable Stress Results: A Comment on Tanck Et Al. (J Biomech 32:153–161, 1999)
Theoretical Stress Analysis of Organ Culture Osteogenesis
Variational and Projection Methods for the Volume Constraint in Finite Deformation Elastoplasticity
Biorheology and Fluid Flux in Swelling Tissues, Ii. Analysis of Unconfined Compressive Response of Transversely Isotropic Cartilage Disc
Application of the U-P Finite Element Method to the Study of Articular Cartilage
Nike3D: A Nonlinear, Implicit, Three-Dimensional Finite Element Code for Solid and Structural Mechanics
,” LLNL Technical Report No. UCRL-MA 105268.
Variationally Derived 3-Field Finite Element Formulations for Quasistatic Poroelastic Analysis of Hydrated Biological Tissues
Penalty Finite Element Analysis for Non-Linear Mechanics of Biphasic Hydrated Soft Tissue Under Large Deformation
Dynamic Analysis of a Fully Saturated Porous Medium Accounting for Geometrical and Material Non-Linearities
A Nonlinear Finite Element Formulation for Axisymmetric Torsion of Biphasic Materials
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Fluid-structure interaction in turbulent flow past cylinder/plate configuration I (First swiveling mode)
The objective of the present contribution is to provide a challenging and well-defined benchmark for fluid-structure interaction (FSI) in turbulent flow to close a gap in the literature. The following list of requirements are taken into account during the definition and setup phase.
First, the test case should be geometrically simple which is realized by a classical cylinder flow configuration extended by a flexible plate structure attached to the backside of the cylinder (see Fig.1).
Second, clearly defined operating and boundary conditions are a must and put into practice by a constant inflow velocity and channel walls. The latter are also evaluated against a periodic setup relying on a subset of the computational domain.
Third, the model to describe the material behavior under load (denoted material model in the following) should be widely used. Although a rubber plate is chosen as the flexible structure, it is demonstrated by additional structural tests that a classical St. Venant-Kirchhoff material model is sufficient to describe the material behavior appropriately.
Fourth, the flow should be in the turbulent regime. Choosing water as the working fluid and a medium-size water channel, the resulting Reynolds number of Re = 30,470 guarantees a sub-critical cylinder flow with transition taking place in the separated shear layers.
Fifth, the benchmark results should be underpinned by a detailed validation process.
For this purpose two dynamic structural tests were carried out experimentally and numerically in order to evaluate an appropriate model to describe the material behavior and to check and evaluate the material parameters of the rubber (Young's modulus, damping). This preliminary work has shown that the St. Venant-Kirchhoff material law is sufficient to describe the deflection of the flexible structure.
After these structural tests, complementary numerical and experimental investigations with flow around the cylinder-plate configuration were performed. Based on optical contactless measuring techniques (particle-image velocimetry and laser distance sensor) the phase-averaged flow field and the structural deformations were determined. These data were compared with corresponding numerical predictions relying on large-eddy simulations and a recently developed semi-implicit predictor-corrector FSI coupling scheme. Both results were found to be in close agreement showing a quasi-periodic oscillating flexible structure (see animation of Fig. 1) in the first swiveling FSI mode with a corresponding Strouhal number of about
{\displaystyle {\text{St}}_{\text{FSI}}=0.11}
Fig. 1: Flow around the flexible structure of the FSI-PfS-1a Benchmark (Click on the figure to see the animation.).
Contributed by: G. De Nayer, A. Kalmbach, M. Breuer — Helmut-Schmidt Universität Hamburg (with support by S. Sicklinger and R. Wüchner from Technische Universität München)
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\dots \mathrm{f1}{!}^{i}\mathrm{f2}{!}^{j}\mathrm{f3}{!}^{k}\dots
i,j,k
\mathrm{f1}
\mathrm{f2}
\mathrm{f3}
\sqrt{\mathrm{\pi }}=\mathrm{\Gamma }\left(\frac{1}{2}\right)
0<i
j,k<0
\mathrm{f1}-\mathrm{f2}-\mathrm{f3}=n
\frac{\left(\genfrac{}{}{0}{}{\mathrm{f1}}{\mathrm{f2}}\right)c\mathrm{f2}!\mathrm{f3}!}{\mathrm{f1}!}
c
is a correction factor depending on
\mathrm{f3}
i,0<j
k<0
\mathrm{f3}-\mathrm{f1}-\mathrm{f2}=n
\frac{c\mathrm{f3}!}{\mathrm{f1}!\mathrm{f2}!\left(\genfrac{}{}{0}{}{\mathrm{f2}}{\mathrm{f1}}\right)}
\dots \mathrm{f1}{!}^{i}\mathrm{f2}{!}^{j}\dots
i,j
\frac{\mathrm{f1}}{\mathrm{f2}}
r
1<|r|
\frac{\mathrm{f2}!\left(\genfrac{}{}{0}{}{\mathrm{f1}}{\mathrm{f2}}\right)\left(\mathrm{f1}-\mathrm{f2}\right)!}{\mathrm{f1}!}
|r|<1
\frac{\mathrm{f2}!}{\mathrm{f1}!\left(\genfrac{}{}{0}{}{\mathrm{f2}}{\mathrm{f1}}\right)\left(\mathrm{f2}-\mathrm{f1}\right)!}
a≔\frac{n!}{k!\left(n-k\right)!}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{!}}{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{!}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\right)\textcolor[rgb]{0,0,1}{!}}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{n}}{\textcolor[rgb]{0,0,1}{k}}\right)
a≔\frac{n\left({n}^{2}+m-k+2\right)\left({n}^{2}+m\right)!}{k!\left({n}^{2}+m-k+2\right)!}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\right)\textcolor[rgb]{0,0,1}{!}}{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{!}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{!}}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{}\left(\genfrac{}{}{0}{}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{k}}\right)}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}
a≔\frac{{m!}^{3}}{\left(3m\right)!}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{{\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{!}}^{\textcolor[rgb]{0,0,1}{3}}}{\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}\right)\textcolor[rgb]{0,0,1}{!}}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{m}}\right)\textcolor[rgb]{0,0,1}{}\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{m}}\right)}
a≔\frac{\mathrm{\Gamma }\left(m+\frac{3}{2}\right)}{\mathrm{sqrt}\left(\mathrm{\pi }\right)\mathrm{\Gamma }\left(m\right)}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{m}\right)}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}}\right)
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Exact Critical Loads for a Pinned Half-Sine Arch Under End Couples | J. Appl. Mech. | ASME Digital Collection
Jen-San Chen, Professor,
Jen-San Chen, Professor
Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10617
e-mail: jschen@ntu.edu.tw
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, January 28, 2004; final revision, May 19, 2004. Associate Editor: R. C. Benson.
J. Appl. Mech. Jan 2005, 72(1): 147-148 (2 pages)
Chen , J., and Lin , J. (February 1, 2005). "Exact Critical Loads for a Pinned Half-Sine Arch Under End Couples ." ASME. J. Appl. Mech. January 2005; 72(1): 147–148. https://doi.org/10.1115/1.1827244
In this note we show that for a pinned half-sine arch under end couples snap-through buckling will occur unsymmetrically if the initial height of the shallow arch is greater than
6.5466r,
where r is the radius of gyration of the cross section. The closed-form expression for the critical couple can be obtained analytically.
shapes (structures), buckling, mechanical stability, nonlinear dynamical systems
Arches, Buckling, Equilibrium (Physics), Stress, Shapes
Fung, Y. C., and Kaplan, A., 1952, “Buckling of Low Arches or Curved Beams of Small Curvature,” NACA Technical Note 2840.
Simitses, G. J., 1986, Elastic Stability of Structures, R. E. Krieger, Malabar, FL, Chap. 7.
Simitses, G. J., 1990, Dynamic Stability of Suddenly Loaded Structures, Springer, New York, Chap. 7.
Erratum: “On Some Issues in Shakedown Analysis” [ASME J. Appl. Mech., 2001, 68 , pp. 799–808]
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{\displaystyle \displaystyle f(x)=g(2\sin x)}
{\displaystyle g'({\sqrt {2}})={\sqrt {2}}}
{\displaystyle \displaystyle f'(\pi /4)}
The question is looking for the derivative of
{\displaystyle f(x)}
. Since this function is definted as a composition of functions, we need to use the chain rule.
Taking the derivative gives
{\displaystyle \displaystyle f'(x)=g'(2\sin(x))(2\cos(x))}
{\displaystyle x=\pi /4}
{\displaystyle {\begin{aligned}f'(\pi /4)&=g'(2\sin(\pi /4))(2\cos(\pi /4))\\&=g'(2\cdot {\tfrac {1}{\sqrt {2}}})\cdot 2\cdot {\tfrac {1}{\sqrt {2}}}\\&=g'({\sqrt {2}})\cdot {\sqrt {2}}\\&={\sqrt {2}}\cdot {\sqrt {2}}\\&=2\end{aligned}}}
MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Chain rule, Pages using DynamicPageList parser function, Pages using DynamicPageList parser tag
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subsop - Maple Help
Home : Support : Online Help : Programming : Operations : Substitution : subsop
substitute for specified operands in an expression
An example involving integrals
subsop(eq1, eq2, ..., eqn, expr)
eq[i]
\mathrm{speci}=\mathrm{expri}
\mathrm{speci}
is an integer or list of integers, and each
\mathrm{expri}
The subsop function is used to replace specified operands of an expression with new values. It will do the simultaneous substitutions specified by the eqi equation arguments in the last argument expr. The result is obtained by replacing op(spec1, expr) by
\mathrm{expr1}
, op(spec2, expr) by
\mathrm{expr2}
, ..., and op(specn, expr) by
\mathrm{exprn}
in expr.
\mathrm{speci}
can be either an integer, or a list of integers. If a list of integers is specified, the integers refer to sub-operands of expr at increasing nesting levels. See op for further details.
The integer(s) comprising each
\mathrm{speci}
must lie in the range
-\mathrm{nops}\left(\mathrm{expr}\right)..\mathrm{nops}\left(\mathrm{expr}\right)
\mathrm{speci}
of 0 is allowed only for function, indexed expression, and series exprs.
If an integer n in a
\mathrm{speci}
is negative, it is considered equivalent to
\mathrm{nops}\left(\mathrm{expr}\right)+n+1
If no eqi are specified, subsop returns its argument with no substitutions.
See also the applyop command which can be used to apply a function to specified operands of an expression.
The action of substitution is not followed by evaluation. In cases where full evaluation is desired, the eval command should be used.
The subsop command is thread-safe as of Maple 15.
p≔{x}^{7}+8{x}^{6}+{x}^{2}-9
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{9}
\mathrm{op}\left(2,p\right)
\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}
\mathrm{subsop}\left(2=y,p\right)
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{9}
\mathrm{subsop}\left(2=-\mathrm{op}\left(2,p\right),p\right)
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{9}
\mathrm{subsop}\left(1=0,p\right)
\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{9}
\mathrm{subsop}\left(1=1,xyz\right)
\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}
\mathrm{subsop}\left(1=\mathrm{NULL},2=z,3=y,[x,y,z]\right)
[\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}]
\mathrm{subsop}\left(0=g,f[a,b,c]\right)
{\textcolor[rgb]{0,0,1}{g}}_{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}}
p≔f\left(x,g\left(x,y,z\right),x\right)
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{subsop}\left([2,3]=w,p\right)
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{w}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{subsop}\left([2,0]=h,[2,3]=w,3=a,p\right)
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{h}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{w}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\right)
\mathrm{subsop}\left(p\right)
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)
You can use subsop and applyop to perform a change of variables in an integral step-by-step.
Int(sin(sqrt(x)),x=0..t);
{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}_{\textcolor[rgb]{0,0,1}{0}}^{\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\sqrt{\textcolor[rgb]{0,0,1}{x}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{x}
Apply the change of variable u = x^(1/2)
subsop( [1,1]=u, (13) );
{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}_{\textcolor[rgb]{0,0,1}{0}}^{\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{x}
We have dx = 2*u*du
subsop( 1=2*u*op(1,(14)), [2,1]=u, (14) );
{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}_{\textcolor[rgb]{0,0,1}{0}}^{\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{u}
applyop( sqrt, [2,2,2], (15) );
{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}_{\textcolor[rgb]{0,0,1}{0}}^{\sqrt{\textcolor[rgb]{0,0,1}{t}}}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{u}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{u}
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Trithionate hydrolase - WikiMili, The Best Wikipedia Reader
In enzymology, a trithionate hydrolase (EC 3.12.1.1) is an enzyme that catalyzes the chemical reaction [1] [2]
trithionate + H2O
{\displaystyle \rightleftharpoons }
thiosulfate + sulfate + 2 H+
Thus, the two substrates of this enzyme are trithionate and H2O, whereas its 3 products are thiosulfate, sulfate, and H+.
This enzyme belongs to the family of hydrolases, specifically those acting on sulfur-sulfur bonds. The systematic name of this enzyme class is trithionate thiosulfohydrolase. This enzyme participates in sulfur metabolism.
Sulfate-reducing microorganisms (SRM) or sulfate-reducing prokaryotes (SRP) are a group composed of sulfate-reducing bacteria (SRB) and sulfate-reducing archaea (SRA), both of which can perform anaerobic respiration utilizing sulfate (SO42–) as terminal electron acceptor, reducing it to hydrogen sulfide (H2S). Therefore, these sulfidogenic microorganisms "breathe" sulfate rather than molecular oxygen (O2), which is the terminal electron acceptor reduced to water (H2O) in aerobic respiration.
Thiosulfate is an oxyanion of sulfur.
Microbial metabolism is the means by which a microbe obtains the energy and nutrients it needs to live and reproduce. Microbes use many different types of metabolic strategies and species can often be differentiated from each other based on metabolic characteristics. The specific metabolic properties of a microbe are the major factors in determining that microbe's ecological niche, and often allow for that microbe to be useful in industrial processes or responsible for biogeochemical cycles.
In enzymology, a sulfite dehydrogenase (EC 1.8.2.1) is an enzyme that catalyzes the chemical reaction
Thiosulfate dehydrogenase is an enzyme that catalyzes the chemical reaction:
In enzymology, a thiosulfate-thiol sulfurtransferase is an enzyme that catalyzes the chemical reaction
In enzymology, an adenylylsulfatase is an enzyme that catalyzes the chemical reaction
In enzymology, a sulfate-transporting ATPase (EC 3.6.3.25) is an enzyme that catalyzes the chemical reaction
In enzymology, a 3'(2'),5'-bisphosphate nucleotidase (EC 3.1.3.7) is an enzyme that catalyzes the chemical reaction
In enzymology, a D-lactate-2-sulfatase (EC 3.1.6.17) is an enzyme that catalyzes the chemical reaction
Thermithiobacillus tepidarius is a member of the Acidithiobacillia isolated from the thermal groundwaters of the Roman Baths at Bath, Somerset, United Kingdom. It was previously placed in the genus Thiobacillus. The organism is a moderate thermophile, 43–45 °C (109–113 °F), and an obligate aerobic chemolithotrophic autotroph. Despite having an optimum pH of 6.0–7.5, growth can continue to an acid medium of pH 4.8. Growth can only occur on reduced inorganic sulfur compounds and elementary sulfur, but unlike some species in other genus of the same family, Acidithiobacillus, Thermithiobacillus spp. are unable to oxidise ferrous iron or iron-containing minerals.
Sulfur is metabolized by all organisms, from bacteria and archaea to plants and animals. Sulfur is reduced or oxidized by organisms in a variety of forms. The element is present in proteins, sulfate esters of polysaccharides, steroids, phenols, and sulfur-containing coenzymes.
Acidithiobacillus caldus formerly belonged to the genus Thiobacillus prior to 2000, when it was reclassified along with a number of other bacterial species into one of three new genera that better categorize sulfur-oxidizing acidophiles. As a member of the Gammaproteobacteria class of Proteobacteria, A. caldus may be identified as a Gram-negative bacterium that is frequently found in pairs. Considered to be one of the most common microbes involved in biomining, it is capable of oxidizing reduced inorganic sulfur compounds (RISCs) that form during the breakdown of sulfide minerals. The meaning of the prefix acidi- in the name Acidithiobacillus comes from the Latin word acidus, signifying that members of this genus love a sour, acidic environment. Thio is derived from the Greek word thios and describes the use of sulfur as an energy source, and bacillus describes the shape of these microorganisms, which are small rods. The species name, caldus, is derived from the Latin word for warm or hot, denoting this species' love of a warm environment.
Acidithiobacillus thiooxidans, formerly known as Thiobacillus thiooxidans until its reclassification into the newly designated genus Acidithiobacillus of the gamma subclass of Proteobacteria, is a Gram-negative, rod-shaped bacterium that uses sulfur as its primary energy source. It is mesophilic, with a temperature optimum of 28 °C. This bacterium is commonly found in soil, sewer pipes, and cave biofilms called snottites. A. thiooxidans is used in the mining technique known as bioleaching, where metals are extracted from their ores through the action of microbes.
The genus Annwoodia was named in 2017 to circumscribe an organism previously described as a member of the genus Thiobacillus, Thiobacillus aquaesulis - the type and only species is Annwoodia aquaesulis, which was isolated from the geothermal waters of the Roman Baths in the city of Bath in the United Kingdom by Ann P. Wood and Donovan P. Kelly of the University of Warwick - the genus was subsequently named to honour Wood's contribution to microbiology. The genus falls within the family Thiobacillaceae along with Thiobacillus and Sulfuritortus, both of which comprise autotrophic organisms dependent on thiosulfate, other sulfur oxyanions and sulfide as electron donors for chemolithoheterotrophic growth. Whilst Annwoodia spp. and Sulfuritortus spp. are thermophilic, Thiobacillus spp. are mesophilic.
Microbial oxidation of sulfur is the oxidation of sulfur by microorganisms to produce energy. The oxidation of inorganic compounds is the strategy primarily used by chemolithotrophic microorganisms to obtain energy in order to build their structural components, survive, grow and reproduce. Some inorganic forms of reduced sulfur, mainly sulfide (H2S/HS−) and elemental sulfur (S0), can be oxidized by chemolithotrophic sulfur-oxidizing prokaryotes, usually coupled to the reduction of oxygen (O2) or nitrate (NO3−).
↑ Lu WP, Kelly DP (1988). "Cellular location and partial purification of the 'thiosulphate-oxidizing enzyme' and 'trithionate hydrolase' from Thiobacillus tepidarius". J. Gen. Microbiol. 134: 877–885. doi: 10.1099/00221287-134-4-877 .
↑ Trudinger PA (1964). "The metabolism of trithionate by Thiobacillus X". Aust. J. Biol. Sci. 17: 459–468.
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Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Distributions : Error
error distribution (exponential power distribution)
Error(a, b, c)
ErrorDistribution(a, b, c)
The error distribution is a continuous probability distribution with probability density function given by:
f\left(t\right)=\frac{{ⅇ}^{-\frac{{\left(\frac{|t-a|}{b}\right)}^{\frac{2}{c}}}{2}}}{b{2}^{\frac{c}{2}+1}\mathrm{\Gamma }\left(\frac{c}{2}+1\right)}
a::\mathrm{real},0<b,0<c
The Error distribution is also known as the exponential power distribution or the general error distribution.
Note that the Error command is inert and should be used in combination with the RandomVariable command.
The Quantile function applied to an error distribution uses a sequence of iterations in order to converge on the desired output point. The maximum number of iterations to perform is equal to 100 by default, but this value can be changed by setting the environment variable _EnvStatisticsIterations to the desired number of iterations.
\mathrm{with}\left(\mathrm{Statistics}\right):
X≔\mathrm{RandomVariable}\left(\mathrm{Error}\left(a,b,c\right)\right):
\mathrm{PDF}\left(X,u\right)
\frac{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\frac{{\left(\frac{|\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}|}{\textcolor[rgb]{0,0,1}{b}}\right)}^{\frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{c}}}}{\textcolor[rgb]{0,0,1}{2}}}}{\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{2}}^{\frac{\textcolor[rgb]{0,0,1}{c}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{c}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}
\mathrm{PDF}\left(X,0.5\right)
\frac{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.5000000000}\textcolor[rgb]{0,0,1}{}{\left(\frac{|\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.5}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}|}{\textcolor[rgb]{0,0,1}{b}}\right)}^{\frac{\textcolor[rgb]{0,0,1}{2.}}{\textcolor[rgb]{0,0,1}{c}}}}}{\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{2.}}^{\textcolor[rgb]{0,0,1}{0.5000000000}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1.}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0.5000000000}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1.}\right)}
\mathrm{Mean}\left(X\right)
\textcolor[rgb]{0,0,1}{a}
\mathrm{Variance}\left(X\right)
\frac{{\textcolor[rgb]{0,0,1}{2}}^{\textcolor[rgb]{0,0,1}{c}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{c}}{\textcolor[rgb]{0,0,1}{2}}\right)}
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The Interstellar Medium | Astronomy | Course Hero
Explain how much interstellar matter there is in the Milky Way, and what its typical density is
Describe how the interstellar medium is divided into gaseous and solid components
Astronomers refer to all the material between stars as interstellar matter; the entire collection of interstellar matter is called the interstellar medium (ISM). Some interstellar material is concentrated into giant clouds, each of which is known as a nebula (plural "nebulae," Latin for "clouds"). The best-known nebulae are the ones that we can see glowing or reflecting visible light; there are many pictures of these in this chapter.
Figure 1. Various Types of Interstellar Matter: The reddish nebulae in this spectacular photograph glow with light emitted by hydrogen atoms. The darkest areas are clouds of dust that block the light from stars behind them. The upper part of the picture is filled with the bluish glow of light reflected from hot stars embedded in the outskirts of a huge, cool cloud of dust and gas. The cool supergiant star Antares can be seen as a big, reddish patch in the lower-left part of the picture. The star is shedding some of its outer atmosphere and is surrounded by a cloud of its own making that reflects the red light of the star. The red nebula in the middle right partially surrounds the star Sigma Scorpii. (To the right of Antares, you can see M4, a much more distant cluster of extremely old stars.) (credit: modification of work by ESO/Digitized Sky Survey 2)
Interstellar clouds do not last for the lifetime of the universe. Instead, they are like clouds on Earth, constantly shifting, merging with each other, growing, or dispersing. Some become dense and massive enough to collapse under their own gravity, forming new stars. When stars die, they, in turn, eject some of their material into interstellar space. This material can then form new clouds and begin the cycle over again.
About 99% of the material between the stars is in the form of a gas—that is, it consists of individual atoms or molecules. The most abundant elements in this gas are hydrogen and helium (which we saw are also the most abundant elements in the stars), but the gas also includes other elements. Some of the gas is in the form of molecules—combinations of atoms. The remaining 1% of the interstellar material is solid—frozen particles consisting of many atoms and molecules that are called interstellar grains or interstellar dust (Figure 1). A typical dust grain consists of a core of rocklike material (silicates) or graphite surrounded by a mantle of ices; water, methane, and ammonia are probably the most abundant ices.
If all the interstellar gas within the Galaxy were spread out smoothly, there would be only about one atom of gas per cm3 in interstellar space. (In contrast, the air in the room where you are reading this book has roughly 1019 atoms per cm3.) The dust grains are even scarcer. A km3 of space would contain only a few hundred to a few thousand tiny grains, each typically less than one ten-thousandth of a millimeter in diameter. These numbers are just averages, however, because the gas and dust are distributed in a patchy and irregular way, much as water vapor in Earth’s atmosphere is often concentrated into clouds.
In some interstellar clouds, the density of gas and dust may exceed the average by as much as a thousand times or more, but even this density is more nearly a vacuum than any we can make on Earth. To show what we mean, let’s imagine a vertical tube of air reaching from the ground to the top of Earth’s atmosphere with a cross-section of 1 square meter. Now let us extend the same-size tube from the top of the atmosphere all the way to the edge of the observable universe—over 10 billion light-years away. Long though it is, the second tube would still contain fewer atoms than the one in our planet’s atmosphere.
While the density of interstellar matter is very low, the volume of space in which such matter is found is huge, and so its total mass is substantial. To see why, we must bear in mind that stars occupy only a tiny fraction of the volume of the Milky Way Galaxy. For example, it takes light only about four seconds to travel a distance equal to the diameter of the Sun, but more than four years to travel from the Sun to the nearest star. Even though the spaces among the stars are sparsely populated, there’s a lot of space out there!
Astronomers estimate that the total mass of gas and dust in the Milky Way Galaxy is equal to about 15% of the mass contained in stars. This means that the mass of the interstellar matter in our Galaxy amounts to about 10 billion times the mass of the Sun. There is plenty of raw material in the Galaxy to make generations of new stars and planets (and perhaps even astronomy students).
Estimating interstellar mass
You can make a rough estimate of how much interstellar mass our Galaxy contains and also how many new stars could be made from this interstellar matter. All you need to know is how big the Galaxy is and the average density using this formula:
\text{total mass}=\text{volume}\times \text{density of atoms}\times \text{mass per atom}
You have to remember to use consistent units—such as meters and kilograms. We will assume that our Galaxy is shaped like a cylinder; the volume of a cylinder equals the area of its base times its height
V={\pi} {R}^{2}h
where R is the radius of the cylinder and h is its height.
Suppose that the average density of hydrogen gas in our Galaxy is one atom per cm3. Each hydrogen atom has a mass of 1.7 × 10−27 kg. If the Galaxy is a cylinder with a diameter of 100,000 light-years and a height of 300 light-years, what is the mass of this gas? How many solar-mass stars (2.0 × 1030 kg) could be produced from this mass of gas if it were all turned into stars?
Recall that 1 light-year = 9.5 × 1012 km = 9.5 × 1017 cm, so the volume of the Galaxy is
V={\pi} {R}^{2}h={\pi} {\left(100,000\times 9.5\times {10}^{17}cm\right)}^{2}\left(300\times 9.5\times {10}^{17}\text{}cm{}\right)=8.0\times {10}^{66}{\text{}cm{}}^{3}
The total mass is therefore
M=V\times \text{density of atoms}\times \text{mass per atom}
8.0\times {10}^{66}{\text{cm}}^{3}\times \left({1\text{atom/cm}}^{3}\right)\times 1.7\times {10}^{-27}\text{kg}=1.4\times {10}^{40}\text{kg}
This is sufficient to make
N=\frac{M}{\left(2.0\times {10}^{30}\text{kg}\right)}=6.9\times {10}^{9}
stars equal in mass to the Sun. That’s roughly 7 billion stars.
You can use the same method to estimate the mass of interstellar gas around the Sun. The distance from the Sun to the nearest other star, Proxima Centauri, is 4.2 light-years. We will see in Interstellar Matter around the Sun that the gas in the immediate vicinity of the Sun is less dense than average, about 0.1 atoms per cm3. What is the total mass of interstellar hydrogen in a sphere centered on the Sun and extending out to Proxima Centauri? How does this compare to the mass of the Sun? It is helpful to remember that the volume of a sphere is related to its radius:
V=\left(4/3\right){\pi} {R}^{3}
The volume of a sphere stretching from the Sun to Proxima Centauri is:
V=\left(4/3\right){\pi} {R}^{3}=\left(4\text{}/{}3\right){\pi} {\left(4.2\times 9.5\times {10}^{17}\text{}cm{}\right)}^{3}=2.7\times {10}^{56}{\text{}cm{}}^{3}
Therefore, the mass of hydrogen in this sphere is:
M=V\times \left(0.1{\text{atom/cm}}^{3}\right)\times 1.7\times {10}^{-27}\text{kg}=4.5\times {10}^{\text{28}}\text{kg}
This is only (4.5 × 1028 kg)/(2.0 × 1030 kg) = 2.2% the mass of the Sun.
Naming the Nebulae
As you look at the captions for some of the spectacular photographs in this chapter and The Birth of Stars and the Discovery of Planets outside the Solar System, you will notice the variety of names given to the nebulae. A few, which in small telescopes look like something recognizable, are sometimes named after the creatures or objects they resemble. Examples include the Crab, Tarantula, and Keyhole Nebulae. But most have only numbers that are entries in a catalog of astronomical objects.
Perhaps the best-known catalog of nebulae (as well as star clusters and galaxies) was compiled by the French astronomer Charles Messier (1730–1817). Messier’s passion was discovering comets, and his devotion to this cause earned him the nickname "The Comet Ferret" from King Louis XV. When comets are first seen coming toward the Sun, they look like little fuzzy patches of light; in small telescopes, they are easy to confuse with nebulae or with groupings of many stars so far away that their light is all blended together. Time and again, Messier’s heart leapt as he thought he had discovered one of his treasured comets, only to find that he had "merely" observed a nebula or cluster.
In frustration, Messier set out to catalog the position and appearance of over 100 objects that could be mistaken for comets. For him, this list was merely a tool in the far more important work of comet hunting. He would be very surprised if he returned today to discover that no one recalls his comets anymore, but his catalog of "fuzzy things that are not comets" is still widely used. When Figure 1 refers to M4, it denotes the fourth entry in Messier’s list.
A far more extensive listing was compiled under the title of the New General Catalog (NGC) of Nebulae and Star Clusters in 1888 by John Dreyer, working at the observatory in Armagh, Ireland. He based his compilation on the work of William Herschel and his son John, plus many other observers who followed them. With the addition of two further listings (called the Index Catalogs), Dreyer’s compilation eventually included 13,000 objects. Astronomers today still use his NGC numbers when referring to most nebulae and star groups.
About 15% of the visible matter in the Galaxy is in the form of gas and dust, serving as the raw material for new stars. About 99% of this interstellar matter is in the form of gas—individual atoms or molecules. The most abundant elements in the interstellar gas are hydrogen and helium. About 1% of the interstellar matter is in the form of solid interstellar dust grains.
interstellar dust: tiny solid grains in interstellar space thought to consist of a core of rocklike material (silicates) or graphite surrounded by a mantle of ices; water, methane, and ammonia are probably the most abundant ices
interstellar medium (ISM): (or interstellar matter) the gas and dust between the stars in a galaxy
nebula: a cloud of interstellar gas or dust; the term is most often used for clouds that are seen to glow with visible light or infrared
AST320_01_Intro
PHYS 1403 • Navarro College
Physic (6).pdf
PHYSICS RNT1 • Western Governors University
18-the-interstellar-medium.pdf
ASTR 209 • Central Luzon State University
SCIENCE 1303 • Northeast Texas Community College
O9FJEO9W- 123449R3-E • Karachi Institute of Technology and Entrepreneurship
Worksheet for Ast Interstel Med.docx
Chapter15-Interstellar-Medium-and-Star-Formation.pdf
In stellar.docx
Lecture 15-14022018.pdf
CHEM F111 • Birla Institute of Technology & Science
CHE 11 • Sto. Niño High School
MECH&AE C163A • University of California, Los Angeles
BSBMGT 506 • Western Sydney University
ASTR MISC • Georgia State University
lect 9 Interstellar Medium
PHYS 1401 • University of Central Arkansas
clustering-interstellar-medium.pdf
SCIENCE 101238 • Arab Unity School
Homework on Interstellar Medium
ASTRONOMY 103 • University of Washington
astr_103_chapter_12.pptx
EE 451 • NUCES - Lahore
ASTRONOMY G9001 • Harvard University
Astronomy Crib Sheet (Final Exam)
ASTR 1960 • Rensselaer Polytechnic Institute
Astronomy Crib Sheet (Exam 2)
Unit 4 (part 1) Interstellar Medium & Star Formation (1)(1).pptx
RES MISC • St. Clair College
Stars, Galaxies, and the Universe â€" Interstellar Dust
10_ism.pdf
ASTR 1303 • Austin Community College District
Unit_Four_Study_Guide_-_Astronomy
ASTR 122 • Johnson County Community College
Astronomy FInal Exam.docx
MA 11 • University Of Arizona
aAstrophy.txt
CS 491 • New Jersey Institute Of Technology
Exoplanets, Star Formation and the Interstellar Medium and Stellar astrophysics.docx
COMPUTER 3707 • Algoma University
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Performance of Two-Equation Turbulence Models for Flat Plate Flows With Leading Edge Bubbles | J. Fluids Eng. | ASME Digital Collection
S. Collie,
, 367 Panama Street, Stanford, CA 94305-2220
M. Gerritsen,
Collie, S., Gerritsen, M., and Jackson, P. (January 24, 2008). "Performance of Two-Equation Turbulence Models for Flat Plate Flows With Leading Edge Bubbles." ASME. J. Fluids Eng. February 2008; 130(2): 021201. https://doi.org/10.1115/1.2829596
This paper investigates the performance of the popular
k-ω
and SST turbulence models for the two-dimensional flow past the flat plate at shallow angles of incidence. Particular interest is paid to the leading edge bubble that forms as the flow separates from the sharp leading edge. This type of leading edge bubble is most commonly found in flows past thin airfoils, such as turbine blades, membrane wings, and yacht sails. Validation is carried out through a comparison to wind tunnel results compiled by Crompton (2001, “The Thin Aerofoil Leading Edge Bubble,” Ph.D. thesis, University of Bristol). This flow problem presents a new and demanding test case for turbulence models. The models were found to capture the leading edge bubble well with the Shear-Stress Transport (SST) model predicting the reattachment length within 7% of the experimental values. Downstream of reattachment both models predicted a slower boundary layer recovery than the experimental results. Overall, despite their simplicity, these two-equation models do a surprisingly good job for this demanding test case.
bubbles, turbulence, turbulence, leading edge bubble, computational fluid dynamics
Boundary layers, Bubbles, Flow (Dynamics), Turbulence, Flat plates, Computational fluid dynamics, Shear (Mechanics), Kinetic energy
The Thin Aerofoil Leading Edge Bubble
,” Ph.D. thesis, University of Bristol.
,” NASA, Technical Report No. 110446.
The Challenging Turbulent Flows Past Downwind Yacht Sails and Practical Application of CFD to Them
Second High Performance Yacht Design Conference
, Auckland.
An Investigation at Low Speed of the Flow Over a Simulated Flat Plate at Small Angles of Attack Using Pitot Static and Hot-Wire Probes
,” NACA, Technical Report No. TN-3876.
Time-Dependent Behavior of a Reattaching Shear Layer
Improved Two-Equation k-ω Turbulence Models For Aerodynamic Flows
,” NASA, Technical Report No. TM-103975.
The Formation of Regions of Separated Flow on Wing Surfaces
, London, Technical Report No. R&M-3122.
Incompressible Flow Past a Flat Plate Aerofoil With Leading Edge Separation Bubble
M. E. M. P.
The Unsteady Structure of Two-Dimensional Separated-and-Reattaching Flows
CFX-International
2003, CFX-5 Solver and Solver Manager Version 5.6, CFX-International.
Robustness of Coupled Algebraic Multigrid for the Navier-Stokes Equations
ICEM-CFD-Engineering
ICEM HEXA 4.2
,” http://www.icemcfd.com/http://www.icemcfd.com/
Experimental Studies of Zero Pressure-Gradient Turbulent Boundary Layer Flow
,” Ph.D. thesis, Kungl Tekniska Högskolan.
, London, Technical Report No. ARC CP-1073.
Low-Reynolds-Number Modeling of Flows Over a Backward Facing Step
A Tensorial Approach to Computational Continuum Mechanics Using Object Orientated Techniques
Nieckele
Numerical Simulations of the Long Recirculation Bubbles Formed in Thin Flat Plates at Shallow Incidence
Proceedings of the EPTT 2006, Escola de Primavera de Transicao e Turbulencia, ABCM
Brazil Society of Mechanical Sciences and Engineering—ABCM
Solution of the Implicit Discretized Fluid Flow Equations by Operator-Splitting
Large Eddy Simulation of Turbulent Channel Flow by One-Equation Modeling
Large Eddy Simulations of the Thin Plate Separation Bubble at Shallow Incidence
,” Ph.D. thesis, Department of Mechanical Engineering, Pontifícia Universidade Católica do Rio de Janeiro, RJ, Brasil, in Portuguese.
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Spectral entropy of signal - MATLAB pentropy
Plot Spectral Entropy of Signal
Plot Spectral Entropy of Speech Signal
Use Spectral Entropy to Detect Sine Wave in White Noise
se = pentropy(xt)
se = pentropy(x,sampx)
se = pentropy(p,fp,tp)
se = pentropy(___,Name=Value)
[se,t] = pentropy(___)
pentropy(___)
se = pentropy(xt) returns the Spectral Entropy of single-variable, single-column timetable xt as the timetable se. pentropy computes the spectrogram of xt using the default options of pspectrum.
se = pentropy(x,sampx) returns the spectral entropy of vector x, sampled at rate or time interval sampx, as a vector.
se = pentropy(p,fp,tp) returns the spectral entropy using the power spectrogram p, along with spectrogram frequency and time vectors fp and tp.
Use this syntax when you want to customize the options for pspectrum, rather than accept the default pspectrum options that pentropy applies.
se = pentropy(___,Name=Value) specifies additional properties using name-value arguments. Options include instantaneous or whole-signal entropy, scaling by white noise entropy, frequency limits, and time limits. You can use Name=Value with any of the input arguments in previous syntaxes.
[se,t] = pentropy(___) returns the spectral entropy se along with the time vector or timetable t. If se is a timetable, then t is equal to the row times of timetable se. This syntax does not apply if Instantaneous is set to false.
pentropy(___) with no output arguments plots the spectral entropy against time. If Instantaneous is set to false, the function outputs the scalar value of the spectral entropy.
Plot the spectral entropy of a signal expressed as a timetable and as a time series.
Generate a random series with normal distribution (white noise).
xn = randn(1000,1);
Create time vector t and convert to duration vector tdur. Combine tdur and xn in a timetable.
t = 0.1:ts:100;
tdur = seconds(t);
xt = timetable(tdur',xn);
Plot the spectral entropy of the timetable xt.
pentropy(xt)
title('Spectral Entropy of White Noise Signal Timetable')
Plot the spectral entropy of the signal, using time-point vector t and the form which returns se and associated time te. Match the x-axis units and grid to the pentropy-generated plots for comparison.
[se,te] = pentropy(xn,t');
te_min = te/60;
plot(te_min,se)
title('Spectral Entropy of White Noise Signal Vector')
xlabel('Time (mins)')
The second input argument for pentropy can represent either frequency or time. The software interprets according to the data type of the argument. Plot the spectral entropy of the signal, using sample rate scalar fs instead of time vector t.
pentropy(xn,fs)
title('Spectral Entropy of White Noise Signal Vector using Sample Rate')
This plot matches the previous plots.
Plot the spectral entropy of a speech signal and compare it to the original signal. Visualize the spectral entropy on a color map by first creating a power spectrogram, and then taking the spectral entropy of frequency bins within the bandwidth of speech.
Load the data x, which contains a two-channel recording of the word "Hello" embedded by low-level white noise. x consists of two columns representing the two channels. Use only the first channel.
Define the sample rate and the time vector. Augment the first channel of x with white noise to achieve a signal-to-noise ratio of about 5 to 1.
load Hello x
t = 1/fs*(0:length(x)-1);
x1 = x(:,1) + 0.01*randn(length(x),1);
Find the spectral entropy. Visualize the data for the original signal and for the spectral entropy.
[se,te] = pentropy(x1,fs);
ylabel("Speech Signal")
plot(te,se)
ylabel("Spectral Entropy")
The spectral entropy drops when "Hello" is spoken. This is because the signal spectrum has changed from almost a constant (white noise) to the distribution of a human voice. The human-voice distribution contains more information and has lower spectral entropy.
Compute the power spectrogram p of the original signal, returning frequency vector fp and time vector tp as well. For this case, specifying a frequency resolution of 20 Hz provides acceptable clarity in the result.
[p,fp,tp] = pspectrum(x1,fs,"spectrogram",FrequencyResolution=20);
The frequency vector of the power spectrogram goes to 22,050 Hz, but the range of interest with respect to speech is limited to the telephony bandwidth of 300–3400 Hz. Divide the data into five frequency bins by defining start and end points, and compute the spectral entropy for each bin.
flow = [300 628 1064 1634 2394];
fup = [627 1060 1633 2393 3400];
se2 = zeros(length(flow),size(p,2));
for i = 1:length(flow)
se2(i,:) = pentropy(p,fp,tp,FrequencyLimits=[flow(i) fup(i)]);
Visualize the data in a color map that shows ascending frequency bins, and compare with the original signal.
imagesc(tp,[],flip(se2)) % Flip se2 so its plot corresponds to the
% ascending frequency bins.
h = colorbar(gca,"NorthOutside");
ylabel(h,"Spectral Entropy")
yticks(1:5)
set(gca,YTickLabel=num2str((5:-1:1).')) % Label the ticks for the ascending bins.
ylabel("Frequency Bin")
Create a signal that combines white noise with a segment that consists of a sine wave. Use spectral entropy to detect the existence and position of the sine wave.
Generate and plot the signal, which contains three segments. The middle segment contains the sine wave along with white noise. The other two segments are pure white noise.
sin_wave = 2*sin(2*pi*20*t')+randn(length(t),1);
x = [randn(1000,1);sin_wave;randn(1000,1)];
t3 = 0:1/fs:30;
title("Sine Wave in White Noise")
Plot the spectral entropy.
pentropy(x,fs)
title("Spectral Entropy of Sine Wave in White Noise")
The plot clearly differentiates the segment with the sine wave from the white-noise segments. This is because the sine wave contains information. Pure white noise has the highest spectral entropy.
The default for pentropy is to return or plot the instantaneous spectral entropy for each time point, as the previous plot displays. You can also distill the spectral entropy information into a single number that represents the entire signal by setting Instantaneous to false. Use the form that returns the spectral entropy value if you want to directly use the result in other calculations. Otherwise, pentropy returns the spectral entropy in ans.
se = pentropy(x,fs,Instantaneous=false)
A single number characterizes the spectral entropy, and therefore the information content, of the signal. You can use this number to efficiently compare this signal with other signals.
Signal timetable from which pentropy returns the spectral entropy se, specified as a timetable that contains a single variable with a single column. xt must contain increasing, finite row times. If the xt timetable has missing or duplicate time points, you can fix it using the tips in Clean Timetable with Missing, Duplicate, or Nonuniform Times. xt can be nonuniformly sampled, with the pspectrum constraint that the median time interval and the mean time interval must obey:
\frac{1}{100}<\frac{\text{Median time interval}}{\text{Mean time interval}}<100.
For an example, see Plot Spectral Entropy of Signal.
Time-series signal from which pentropy returns the spectral entropy se, specified as a vector.
Positive numeric scalar — Sample rate in hertz
\frac{1}{100}<\frac{\text{Median time interval}}{\text{Mean time interval}}<100.
p — Power spectrogram or spectrum of signal
real nonnegative matrix
Power spectrogram or spectrum of a signal, specified as a matrix (spectrogram) or a column vector (spectrum). If you specify p, then pentropy uses p rather than generate its own spectrogram or power spectrogram. fp and tp, which provide the frequency and time information, must accompany p. Each element of p at the i'th row and the j'th column represents the signal power at the frequency bin centered at fp(i) and the time instance tp(j).
For an example, see Plot Spectral Entropy of Speech Signal.
fp — Frequencies for spectrogram p
Frequencies for spectrogram or power spectrogram p when p is supplied explicitly to pentropy, specified as a vector in hertz. The length of fp must be equal to the number of rows in s.
tp — Time information for spectrogram p
vector | duration array | datetime array
Time information for power spectrogram or spectrum p when p is supplied explicitly to pentropy, specified as one of the following:
Vector of time points, whose data type can be numeric, duration, or datetime. The length of vector tp must be equal to the number of columns in p.
duration scalar that represents the time interval in p. The scalar form of tp can be used only when p is a power spectrogram matrix.
For the special case where p is a column vector (power spectrum), tp can be a numeric, duration, or datetime scalar representing the time point of the spectrum.
For the special case where p is a column vector (power spectrum), tp can be a single/double/duration/datetime scalar representing the time point of the spectrum.
Example: "Instantaneous",false,"FrequencyLimits",[25 50] computes the scalar spectral entropy representing the portion of the signal ranging from 25 Hz to 50 Hz.
Instantaneous — Instantaneous time series option
Instantaneous time series option, specified as a logical.
If Instantaneous is true, then pentropy returns the instantaneous spectral entropy as a time-series vector.
If Instantaneous is false, then pentropy returns the spectral entropy value of the whole signal or spectrum as a scalar.
For an example, see Use Spectral Entropy to Detect Sine Wave in White Noise.
Scaled — Scale by white noise option
Scale by white noise option, specified as a logical. Scaling by white noise — or log2n, where n is the number of frequency points — is equivalent to normalizing in Spectral Entropy. It allows you to perform a direct comparison on signals of different length.
If Scaled is true, then pentropy returns the spectral entropy scaled by the spectral entropy of the corresponding white noise.
If Scaled is false, then pentropy does not scale the spectral entropy.
[0 sampfreq/2] (default) | [f1 f2]
Frequency limits to use, specified as a two-element vector containing lower and upper bounds f1 and f2 in hertz. The default is [0 sampfreq/2], where sampfreq is the sample rate in hertz that pentropy derives from sampx.
This specification allows you to exclude a band of data at either end of the spectral range.
full timespan (default) | [t1 t2]
Time limits, specified as a two-element vector containing lower and upper bounds t1 and t2 in the same units as the sample time provided in sampx, and of the data types:
Numeric or duration when sampx is numeric or duration
Numeric, duration, or datetime when sampx is datetime
This specification allows you to extract a time segment of data from the full timespan.
se — Spectral entropy
timetable | double vector
Spectral Entropy, returned as a timetable if the input signal is timetable xt, and as a double vector if the input signal is time series x.
t — time values corresponding to se
Time values associated with se, returned in the same form as the time in se. This argument does not apply if Instantaneous is set to false.
The spectral entropy (SE) of a signal is a measure of its spectral power distribution. The concept is based on the Shannon entropy, or information entropy, in information theory. The SE treats the signal's normalized power distribution in the frequency domain as a probability distribution, and calculates the Shannon entropy of it. The Shannon entropy in this context is the spectral entropy of the signal. This property can be useful for feature extraction in fault detection and diagnosis [2], [1]. SE is also widely used as a feature in speech recognition [3] and biomedical signal processing [4].
The equations for spectral entropy arise from the equations for the power spectrum and probability distribution for a signal. For a signal x(n), the power spectrum is S(m) = |X(m)|2, where X(m) is the discrete Fourier transform of x(n). The probability distribution P(m) is then:
P\left(m\right)=\frac{S\left(m\right)}{{\sum }_{i}S\left(i\right)}\text{.}
The spectral entropy H follows as:
H=-\sum _{m=1}^{N}P\left(m\right){\mathrm{log}}_{2}P\left(m\right)\text{.}
{H}_{n}=-\frac{\sum _{m=1}^{N}P\left(m\right){\mathrm{log}}_{2}P\left(m\right)}{{\mathrm{log}}_{2}N}\text{,}
where N is the total frequency points. The denominator, log2N represents the maximal spectral entropy of white noise, uniformly distributed in the frequency domain.
If a time-frequency power spectrogram S(t,f) is known, then the probability distribution becomes:
P\left(m\right)=\frac{{\sum }_{t}S\left(t,m\right)}{{\sum }_{f}{\sum }_{t}S\left(t,f\right)}.
Spectral entropy is still:
H=-\sum _{m=1}^{N}P\left(m\right){\mathrm{log}}_{2}P\left(m\right)\text{.}
To compute the instantaneous spectral entropy given a time-frequency power spectrogram S(t,f), the probability distribution at time t is:
P\left(t,m\right)=\frac{S\left(t,m\right)}{{\sum }_{f}S\left(t,f\right)}.
Then the spectral entropy at time t is:
H\left(t\right)=-\sum _{m=1}^{N}P\left(t,m\right){\mathrm{log}}_{2}P\left(t,m\right).
[1] Pan, Y. N., J. Chen, and X. L. Li. "Spectral Entropy: A Complementary Index for Rolling Element Bearing Performance Degradation Assessment." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. Vol. 223, Issue 5, 2009, pp. 1223–1231.
[2] Sharma, V., and A. Parey. "A Review of Gear Fault Diagnosis Using Various Condition Indicators." Procedia Engineering. Vol. 144, 2016, pp. 253–263.
[3] Shen, J., J. Hung, and L. Lee. "Robust Entropy-Based Endpoint Detection for Speech Recognition in Noisy Environments." ICSLP. Vol. 98, November 1998.
[4] Vakkuri, A., A. Yli‐Hankala, P. Talja, S. Mustola, H. Tolvanen‐Laakso, T. Sampson, and H. Viertiö‐Oja. "Time‐Frequency Balanced Spectral Entropy as a Measure of Anesthetic Drug Effect in Central Nervous System during Sevoflurane, Propofol, and Thiopental Anesthesia." Acta Anaesthesiologica Scandinavica. Vol. 48, Number 2, 2004, pp. 145–153.
kurtogram | pkurtosis | pspectrum
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Per-unit discrete-time single-phase induction machine field-oriented control - Simulink - MathWorks España
T*=\left({K}_{p_\omega }+{K}_{i_\omega }\frac{{T}_{s}z}{z-1}\right)\left({\omega }_{ref}-{\omega }_{r}\right)
\psi *=\frac{2\pi {f}_{n}{\psi }_{n}}{p\mathrm{min}\left(|{\omega }_{r}|,\frac{2\pi {f}_{n}}{p}\right)}
{i}_{q}*=\frac{\left({L}_{ms}+{L}_{lar}\right)T*}{pa{L}_{ms}\psi *}
{i}_{d}*=\frac{\psi *}{{a}^{2}{L}_{ms}}
\frac{d\theta }{dt}=p{\omega }_{r}+\frac{{a}^{3}{L}_{ms}{R}_{ar}}{{\psi }^{*}\left({L}_{ms}+{L}_{lar}\right)}
\left(\begin{array}{c}{i}_{a}\\ {i}_{b}\end{array}\right)=\left(\begin{array}{cc}{a}^{2}\mathrm{sin}\left(\theta \right)& {a}^{2}\mathrm{cos}\left(\theta \right)\\ \mathrm{cos}\left(\theta \right)& -\mathrm{sin}\left(\theta \right)\end{array}\right)\left(\begin{array}{c}{i}_{d}\\ {i}_{q}\end{array}\right)
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networks(deprecated)/rankpoly - Maple Help
Home : Support : Online Help : networks(deprecated)/rankpoly
(Whitney) rank polynomial of an undirected graph
rankpoly(G, x, y)
rank variable in rank poly
corank variable in rank poly
Important: The networks package has been deprecated.Use the superseding command GraphTheory[RankPolynomial] instead.
n\left(G\right)
= number of vertices,
m\left(G\right)
= number of edges, and
c\left(G\right)
= number of components, one defines rank(G) = n(G) - c(G) and corank(G) = m(G) - rank(G).
The rank polynomial is a sum over all subgraphs H of G of x^(rank(G) - rank(H)) y^corank(H).
{x}^{i}{y}^{j}
in the rank polynomial is thus the number of spanning subgraphs of G having i more components than G and having a cycle space of dimension j.
This routine is normally loaded via the command with(networks) but may also be referenced using the full name networks[rankpoly](...).
\mathrm{with}\left(\mathrm{networks}\right):
G≔\mathrm{complete}\left(4\right):
\mathrm{rankpoly}\left(G,x,y\right)
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{15}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{15}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{16}
\mathrm{rankpoly}\left(G,1,1\right)
\textcolor[rgb]{0,0,1}{64}
\mathrm{rankpoly}\left(G,1,0\right)
\textcolor[rgb]{0,0,1}{38}
\mathrm{rankpoly}\left(G,0,1\right)
\textcolor[rgb]{0,0,1}{38}
\mathrm{rankpoly}\left(G,0,0\right)
\textcolor[rgb]{0,0,1}{16}
GraphTheory[RankPolynomial]
networks(deprecated)[flowpoly]
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FAQ - Olympus Pro
Olympus Pro Partners
Olympus Pro Users
Olympus Pro Article
OlympusDAO Discord
Why do I want to use Olympus Pro?
It allows you to buy your favorite DeFi tokens at a discounted rate. Besides, you are helping the protocol to accumulate their own liquidity in the process. This keeps your goal and the protocol's aligned.
What is the relationship between Olympus Pro and OlympusDAO?
Olympus Pro is a service offered by OlympusDAO. OlympusDAO provides infrastructure, expertise, and exposure to other protocols in setting up the bond mechanism in exchange for a fee.
Do I receive OHM from Olympus Pro bonds?
You do not receive OHM from Olympus Pro bonds. Instead, you get the native governance token the protocol offers. For example, protocol X would reward bonds in X instead of OHM.
Why is the Bond ROI negative?
At times, you'll see that the bond discount turns negative, meaning you'd pay a premium from the market in order to bond. You should not bond during this time, as you can buy the same tokens but at a cheaper price from the market. As time goes on, the discount will slowly increase, until it reaches a positive discount again. The negative discount can be caused by different factors:
High demands for bonds: Bonds offer users the ability to purchase tokens at a market discount. However, this price is dependent on the demands of bonds. When there is a high demand, the bond price goes up, and vice versa. The demand may be so high that it may cause the bond price to inflate above the market price.
Sharp decreases in price: At times, when a token experiences a sharp decrease in price, it takes time for the bond price to decrease to match the new token price. This causes a temporary negative discount, until the bond price matches the market value again.
How is the bond price/discount determined?
The bond discount is determined by the following formula:
bondDiscount = (marketPrice - bondPrice)\ /\ marketPrice
The bond price is determined by the debt ratio of the system and a scaling variable called the Bond Control Variable (BCV). This allows us to control the rate at which the bond price increases. Note that the bond price is independent of the market price, as no data from price oracles are used when determining the bond price.
bondPrice = debtRatio * BCV
The debt ratio is the amount of reward tokens owed to the bonders by the protocol, divided by the total supply of the reward tokens. A higher debt ratio implies a huge demand for bonds, resulting in a higher bond price, and vice versa.
debtRatio = tokenOwed\ /\ tokenSupply
What is the catch? Buying tokens at a discount sounds too good to be true.
Tokens purchased through a bond are not released to the bonder all at once. Instead, they vest linearly over time. This prevents the bonder from selling their tokens all at once for a quick profit.
What parameters can be adjusted?
BCV directly affects the bond price - the higher the BCV, the higher the bond price. As a higher bond price makes bonds less attractive, the protocol can adjust this value to tune the bond capacity.
Maximum Bond Size
This controls the maximum amount of reward tokens a user can purchase through a bond. It is set as a percentage of the total supply and typically ranges between 0.03-0.05% of the total supply.
Bond Vesting Term
A bond vests linearly to the bonder over a length of time, called the bond vesting term. This means the bonder can claim a portion of the reward tokens each day, with all rewards being claimable at the end of the term.
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where - Maple Help
Home : Support : Online Help : Programming : Debugging : where
displays the stack of procedure activations
where(numLevels)
(optional) number of activation levels to display
The where function displays the stack of procedure activations. Starting at the top level, it shows the statement being executed, and the actual parameters passed to the called procedure.
If numLevels is specified, only that many bottommost activations are displayed.
The where function is generally called from within a procedure for debugging purposes. The intent is to be able to determine from where a procedure is being called.
The debugger has a where command which is analogous to the Maple where function, except that it can be invoked interactively by the user from the debugger command line.
\mathrm{where}\left(\right)
Currently at TopLevel.
f := proc(x) g(args) end proc:
g := proc(x) h(args) end proc:
h := proc(x) j(args) end proc:
j := proc(x) where(args) end proc:
f\left(\right)
TopLevel: f()
f: g(_passed)
g: h(_passed)
h: j(_passed)
Currently in j.
f\left(3\right)
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Intersecting_chords_theorem Knowpia
The intersecting chords theorem or just the chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid's Elements.
{\displaystyle |AS|\cdot |SC|=|BS|\cdot |SD|}
{\displaystyle {\begin{aligned}&|AS|\cdot |SC|=|BS|\cdot |SD|\\=&(r+d)\cdot (r-d)=r^{2}-d^{2}\end{aligned}}}
{\displaystyle \triangle ASD\sim \triangle BSC}
More precisely, for two chords AC and BD intersecting in a point S the following equation holds:
{\displaystyle |AS|\cdot |SC|=|BS|\cdot |SD|}
The converse is true as well, that is if for two line segments AC and BD intersecting in S the equation above holds true, then their four endpoints A, B, C and D lie on a common circle. Or in other words if the diagonals of a quadrilateral ABCD intersect in S and fulfill the equation above then it is a cyclic quadrilateral.
The value of the two products in the chord theorem depends only on the distance of the intersection point S from the circle's center and is called the absolute value of the power of S, more precisely it can be stated that:
{\displaystyle |AS|\cdot |SC|=|BS|\cdot |SD|=r^{2}-d^{2}}
where r is the radius of the circle, and d is the distance between the center of the circle and the intersection point S. This property follows directly from applying the chord theorem to a third chord going through S and the circle's center M (see drawing).
The theorem can be proven using similar triangles (via the inscribed-angle theorem). Consider the angles of the triangles ASD and BSC:
{\displaystyle {\begin{aligned}\angle ADS&=\angle BCS\,({\text{inscribed angles over AB}})\\\angle DAS&=\angle CBS\,({\text{inscribed angles over CD}})\\\angle ASD&=\angle BSC\,({\text{opposing angles}})\end{aligned}}}
This means the triangles ASD and BSC are similar and therefore
{\displaystyle {\frac {AS}{SD}}={\frac {BS}{SC}}\Leftrightarrow |AS|\cdot |SC|=|BS|\cdot |SD|}
Next to the tangent-secant theorem and the intersecting secants theorem the intersecting chord theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle - the power of point theorem.
Paul Glaister: Intersecting Chords Theorem: 30 Years on. Mathematics in School, Vol. 36, No. 1 (Jan., 2007), p. 22 (JSTOR)
Bruce Shawyer: Explorations in Geometry. World scientific, 2010, ISBN 9789813100947, p. 14
Hans Schupp: Elementargeometrie. Schöningh, Paderborn 1977, ISBN 3-506-99189-2, p. 149 (German).
Schülerduden - Mathematik I. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, ISBN 978-3-411-04208-1, pp. 415-417 (German)
Intersecting Chords Theorem at cut-the-knot.org
Intersecting Chords Theorem at proofwiki.org
Weisstein, Eric W. "Chord". MathWorld.
Two interactive illustrations: [1] and [2]
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Clutch schedule for a 10-speed transmission - MATLAB
{g}_{x}=\frac{{N}_{{R}_{x}}}{{N}_{{S}_{x}}},
\frac{{g}_{2}}{1+{g}_{2}}
\frac{{g}_{2}\left(1+{g}_{4}\right)}{\left({g}_{2}\left(1+{g}_{4}\right)\right)+{g}_{4}}
\frac{{g}_{2}\left(1+{g}_{3}\right)\left(1+{g}_{4}\right)}{\left({g}_{2}\left(1+{g}_{3}\right)\left(1+g4\right)\right)+{g}_{4}}
7 0 0 1 1 1 1
\frac{\left(1+{g}_{4}\right)\left(1+{g}_{1}+{g}_{2}\left(1+{g}_{3}\right)\right)}{\left(1+{g}_{4}\right)\left(1+{g}_{2}\left(1+{g}_{3}\right)\right)+{g}_{1}}
\frac{\left(1+{g}_{1}+{g}_{2}\right)\left(1+{g}_{4}\right)}{\left(1+{g}_{2}\right)\left(1+{g}_{4}\right)+g1}
\frac{1+{g}_{1}+{g}_{2}}{1+{g}_{2}}
\frac{\left(1+{g}_{1}\right)\left(1+{g}_{4}\right)}{1+{g}_{1}+{g}_{4}}
\frac{{g}_{2}\left(1+{g}_{4}\right)}{1+{g}_{2}}
1+{g}_{4}
\frac{-{g}_{2}{\text{g}}_{\text{3}}\left(\text{1}+{\text{g}}_{\text{4}}\right)}{\left(\text{1}+{\text{g}}_{\text{2}}\right)}
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Effective Placement of a Cantilever Resonator on Flexible Primary Structure for Vibration Control Applications—Part 1: Mathematical Modeling and Analysis | Journal of Vibration and Acoustics | ASME Digital Collection
Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received February 15, 2017; final manuscript received February 9, 2018; published online April 17, 2018. Assoc. Editor: Mohammed Daqaq.
A companion article has been published: Effective Placement of a Cantilever Resonator on Flexible Primary Structure for Vibration Control Applications—Part 2: Model Updating and Experimental Validation
Lundstrom, T., and Jalili, N. (April 17, 2018). "Effective Placement of a Cantilever Resonator on Flexible Primary Structure for Vibration Control Applications—Part 1: Mathematical Modeling and Analysis." ASME. J. Vib. Acoust. October 2018; 140(5): 051003. https://doi.org/10.1115/1.4039531
In this Part 1 of a two-part series, the theoretical modeling and optimization are presented. More specifically, the effect of attachment location on the dynamics of a flexible beam system is studied using a theoretical model. Typically, passive/active resonators for vibration suppression of flexible systems are uniaxial and can only affect structure response in the direction of the applied force. The application of piezoelectric bender actuators as active resonators may prove to be advantageous over typical, uniaxial actuators as they can dynamically apply both a localized moment and translational force to the base structure attachment point. Assuming unit impulse force disturbance, potential actuator/sensor performance for the secondary beam can be quantified by looking at fractional root-mean-square (RMS) strain energy in the actuator relative to the total system, and normalized RMS strain energy in the actuator over a frequency band of interest with respect to both disturbance force and actuator beam mount locations. Similarly, by energizing the actuator beam piezoelectric surface with a unit impulse, one can observe RMS base beam tip velocity as a function of actuator beam position. Through such analyses, one can balance both sensor/actuator performance and make conclusions about optimally mounting the actuator beam sensor/actuator. Accounting for both sensing and actuation requirements, the actuator beam should be mounted in the following nondimensionalized region:
0.4≤e≤0.5
Actuators, Impulse (Physics)
, “Device for Damping Vibrations of Bodies,” U.S. Patent No.
Passive Noise Reduction Using the Modally Enhanced Dynamic Absorber
Topics in Modal Analysis II
Modified Independent Modal Space Control of MDOF Systems
An Observer-Based Piezoelectric Control of Flexible Cartesian Robot Arms: Theory and Experiment
Adaptive Robust Sliding-Mode Control of a Flexible Beam Using PZT Sensor and Actuator
, Taipei, Taiwan, Sept. 2–4, pp.
Tunable Active Vibration Absorber: The Delayed Resonator
Optimization of a Right-Angle Piezoelectric Cantilever Using Auxiliary Beams With Different Stiffness Levels for Vibration Energy Harvesting
Modeling of Piezoelectric Energy Harvesting From an L-Shaped Beam-Mass Structure With an Application to UAVs
Toward Ultrasmall Mass Detection Using Adaptive Self-Sensing Piezoelectrically Driven Microcantilevers
, St. Petersburg, Russia, July 5–8, pp.
Piezoelectric-Based Vibration Control: From Macro to Micro/Nano Scale Systems
, Union College Press, Orlando, FL, Nov. 8–10, pp.
, Prentice Hall, Upper Saddle River,
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Home : Support : Online Help : Connectivity : Database Package : Statement : SetOptions
set options of a Statement module
statement:-SetOptions( opts )
(optional) equations of the form option=integer where option is one of timeout, maxrows, or maxfieldsize
SetOptions sets various options that affect the behavior of statement.
All the options that can be set with SetOptions can also be set when statement is created by a call to CreateStatement.
SetOptions accepts the following options.
The number of seconds statement waits for a query to execute. If the limit is exceeded, an error is raised.
\mathrm{driver}≔\mathrm{Database}[\mathrm{LoadDriver}]\left(\right):
\mathrm{conn}≔\mathrm{driver}:-\mathrm{OpenConnection}\left(\mathrm{url},\mathrm{name},\mathrm{pass}\right):
\mathrm{stat}≔\mathrm{conn}:-\mathrm{CreateStatement}\left(\right):
\mathrm{stat}:-\mathrm{GetOptions}\left(\right)
[\textcolor[rgb]{0,0,1}{\mathrm{timeout}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{maxrows}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{maxfieldsize}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{8192}]
\mathrm{stat}:-\mathrm{SetOptions}\left('\mathrm{timeout}'=10\right);
\mathrm{stat}:-\mathrm{GetOptions}\left(\right)
[\textcolor[rgb]{0,0,1}{\mathrm{timeout}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{maxrows}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{maxfieldsize}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{8192}]
\mathrm{stat}:-\mathrm{SetOptions}\left('\mathrm{maxrows}'=20\right);
\mathrm{stat}:-\mathrm{GetOptions}\left('\mathrm{timeout}','\mathrm{maxrows}'\right)
[\textcolor[rgb]{0,0,1}{\mathrm{timeout}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{maxrows}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{20}]
\mathrm{stat}:-\mathrm{SetOptions}\left('\mathrm{maxfieldsize}'=1000\right);
\mathrm{stat}:-\mathrm{GetOptions}\left('\mathrm{timeout}','\mathrm{maxrows}','\mathrm{maxfieldsize}'\right)
[\textcolor[rgb]{0,0,1}{\mathrm{timeout}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{maxrows}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{20}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{maxfieldsize}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1000}]
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Home : Support : Online Help : Mathematics : Algebra : Magma : IsomorphicCopy
IsomorphicCopy( src, p )
The IsomorphicCopy( 'src', 'p' ) command uses the permutation p to produce an isomorphic copy of the source magma src, in such a way that the permutation p is then an isomorphism from src to the isomorphic copy. The isomorphic copy is returned.
The operation of this command is effected by calling TransportStructure.
If you want to generate a number of isomorphic copies of a magma (or several magmas), you can use the TransportStructure command instead, which allows you to re-use an Array which you pass to the command.
If the permutation p is not supplied, then a randomly generated permutation is used.
\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}]
p≔[2,1,3]
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]
\mathrm{m2}≔\mathrm{IsomorphicCopy}\left(m,p\right)
\textcolor[rgb]{0,0,1}{\mathrm{m2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \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}\end{array}]
\mathrm{AreIsomorphic}\left(m,\mathrm{m2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
The Magma[IsomorphicCopy] command was introduced in Maple 16.
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Apollonian_circles Knowpia
Apollonian circles are two families of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. They were discovered by Apollonius of Perga, a renowned Greek geometer.
Some Apollonian circles. Every blue circle intersects every red circle at a right angle. Every red circle passes through the two points, C and D, and every blue circle separates the two points.
The Apollonian circles are defined in two different ways by a line segment denoted CD.
Each circle in the first family (the blue circles in the figure) is associated with a positive real number r, and is defined as the locus of points X such that the ratio of distances from X to C and to D equals r,
{\displaystyle \left\{X\mid {\frac {d(X,C)}{d(X,D)}}=r\right\}.}
For values of r close to zero, the corresponding circle is close to C, while for values of r close to ∞, the corresponding circle is close to D; for the intermediate value r = 1, the circle degenerates to a line, the perpendicular bisector of CD. The equation defining these circles as a locus can be generalized to define the Fermat–Apollonius circles of larger sets of weighted points.
Each circle in the second family (the red circles in the figure) is associated with an angle θ, and is defined as the locus of points X such that the inscribed angle CXD equals θ,
{\displaystyle \left\{X\mid \;C{\hat {X}}D\;=\theta \right\}.}
Scanning θ from 0 to π generates the set of all circles passing through the two points C and D.
The two points where all the red circles cross are the limiting points of pairs of circles in the blue family.
Bipolar coordinatesEdit
A given blue circle and a given red circle intersect in two points. In order to obtain bipolar coordinates, a method is required to specify which point is the right one. An isoptic arc is the locus of points X that sees points C and D under a given oriented angle of vectors i.e.
{\displaystyle \operatorname {isopt} (\theta )=\{X\mid \angle ({\overrightarrow {XC}},{\overrightarrow {XD}})=\theta +2k\pi \}.}
Such an arc is contained into a red circle and is bounded by points C and D. The remaining part of the corresponding red circle is
{\displaystyle \operatorname {isopt} (\theta +\pi )}
. When we really want the whole red circle, a description using oriented angles of straight lines has to be used
{\displaystyle {\rm {full\;red\;circle}}=\{X\mid \angle ({\overrightarrow {XC}},{\overrightarrow {XD}})=\theta +k\pi \}}
Pencils of circlesEdit
Both of the families of Apollonian circles are pencils of circles. Each is determined by any two of its members, called generators of the pencil. Specifically, one is an elliptic pencil (red family of circles in the figure) that is defined by two generators that pass through each other in exactly two points (C and D). The other is a hyperbolic pencil (blue family of circles in the figure) that is defined by two generators that do not intersect each other at any point.[1]
Radical axis and central lineEdit
Any two of these circles within a pencil have the same radical axis, and all circles in the pencil have collinear centers. Any three or more circles from the same family are called coaxial circles or coaxal circles.[2]
The elliptic pencil of circles passing through the two points C and D (the set of red circles, in the figure) has the line CD as its radical axis. The centers of the circles in this pencil lie on the perpendicular bisector of CD. The hyperbolic pencil defined by points C and D (the blue circles) has its radical axis on the perpendicular bisector of line CD, and all its circle centers on line CD.
Inversive geometry, orthogonal intersection, and coordinate systemsEdit
Circle inversion transforms the plane in a way that maps circles into circles, and pencils of circles into pencils of circles. The type of the pencil is preserved: the inversion of an elliptic pencil is another elliptic pencil, the inversion of a hyperbolic pencil is another hyperbolic pencil, and the inversion of a parabolic pencil is another parabolic pencil.
It is relatively easy to show using inversion that, in the Apollonian circles, every blue circle intersects every red circle orthogonally, i.e., at a right angle. Inversion of the blue Apollonian circles with respect to a circle centered on point C results in a pencil of concentric circles centered at the image of point D. The same inversion transforms the red circles into a set of straight lines that all contain the image of D. Thus, this inversion transforms the bipolar coordinate system defined by the Apollonian circles into a polar coordinate system. Obviously, the transformed pencils meet at right angles. Since inversion is a conformal transformation, it preserves the angles between the curves it transforms, so the original Apollonian circles also meet at right angles.
Alternatively,[3] the orthogonal property of the two pencils follows from the defining property of the radical axis, that from any point X on the radical axis of a pencil P the lengths of the tangents from X to each circle in P are all equal. It follows from this that the circle centered at X with length equal to these tangents crosses all circles of P perpendicularly. The same construction can be applied for each X on the radical axis of P, forming another pencil of circles perpendicular to P.
More generally, for every pencil of circles there exists a unique pencil consisting of the circles that are perpendicular to the first pencil. If one pencil is elliptic, its perpendicular pencil is hyperbolic, and vice versa; in this case the two pencils form a set of Apollonian circles. The pencil of circles perpendicular to a parabolic pencil is also parabolic; it consists of the circles that have the same common tangent point but with a perpendicular tangent line at that point.[4]
Apollonian trajectories have been shown to be followed in their motion by vortex cores or other defined states in some physical systems involving interferential or coupled fields, such photonic or coupled polariton waves.[5] The trajectories arise from the homeomorphic mapping between the Rabi rotation of the full wave function on the Bloch sphere and Apollonian circles in the real space where the observation is made.
^ Schwerdtfeger (1979, pp. 8–10).
^ MathWorld uses “coaxal,” while Akopyan & Zaslavsky (2007) prefer “coaxial.”
^ Akopyan & Zaslavsky (2007), p. 59.
^ Schwerdtfeger (1979, pp. 30–31, Theorem A).
^ Dominici; et al. (2021). "Full-Bloch beams and ultrafast Rabi-rotating vortices". Physical Review Research. 3: 013007. doi:10.1103/PhysRevResearch.3.013007.
Akopyan, A. V.; Zaslavsky, A. A. (2007), Geometry of Conics, Mathematical World, vol. 26, American Mathematical Society, pp. 57–62, ISBN 978-0-8218-4323-9 .
Pfeifer, Richard E.; Van Hook, Cathleen (1993), "Circles, Vectors, and Linear Algebra", Mathematics Magazine, 66 (2): 75–86, doi:10.2307/2691113, JSTOR 2691113 .
Schwerdtfeger, Hans (1979), Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non-Euclidean Geometry, Dover, pp. 8–10 .
Samuel, Pierre (1988), Projective Geometry, Springer, pp. 40–43 .
Ogilvy, C. Stanley (1990), Excursions in Geometry, Dover, ISBN 0-486-26530-7 .
Weisstein, Eric W. "Coaxal Circles". MathWorld.
David B. Surowski: Advanced High-School Mathematics. p. 31
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Generalized optimal subpattern assignment (GOSPA) metric - MATLAB - MathWorks 日本
X=\left[{x}_{1},{x}_{2},â¦,{x}_{m}\right]
Y=\left[{y}_{1},{y}_{2},â¦,{y}_{n}\right]
SGOSPA={\left(GOSP{A}^{p}+S{C}^{p}\right)}^{1/p}
Assuming m ≤ n, GOSPA is:
GOSPA={\left[\underset{i=1}{\overset{m}{â}}{d}_{c}^{p}\left({x}_{i},{y}_{\mathrm{Ï}\left(i\right)}\right)+\frac{{c}^{p}}{\mathrm{α}}\left(nâm\right)\right]}^{1/p}
where dc is the cutoff-based distance and yπ(i) represents the track assigned to truth xi. The cutoff-based distance dc is defined as:
{d}_{c}\left(x,y\right)=\mathrm{min}\left\{{d}_{b}\left(x,y\right),c\right\}
where c is the cutoff distance threshold, and db(x,y) is the base distance between track x and truth y calculated by the distance function. The cutoff based distance dc is the smaller value of db and c. α is the alpha parameter.
SC=SPÃ{n}_{s}^{1/p}
When α = 2, the GOSPA metric can reduce to three components:
GOSPA={\left[lo{c}^{p}+mis{s}^{p}+fals{e}^{p}\right]}^{1/p}
loc={\left[\underset{i=1}{\overset{h}{â}}{d}_{b}^{p}\left({x}_{i},{y}_{\mathrm{Ï}\left(i\right)}\right)\right]}^{1/p}
miss=\frac{c}{{2}^{1/p}}{\left({n}_{miss}\right)}^{1/p}
false=\frac{c}{{2}^{1/p}}{\left({n}_{false}\right)}^{1/p}
[1] Rahmathullash, A. S., A. F. GarcÃa-Fernández, and L. Svensson. "Generalized Optimal Sub-Pattern Assignment Metric." 20th International Conference on Information Fusion (Fusion), pp. 1–8, 2017.
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Section 13.7 (0A8C): Adjoints for exact functors—The Stacks project
Section 13.7: Adjoints for exact functors (cite)
13.7 Adjoints for exact functors
Results on adjoint functors between triangulated categories.
Lemma 13.7.1. Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor between triangulated categories. If $F$ admits a right adjoint $G: \mathcal{D'} \to \mathcal{D}$, then $G$ is also an exact functor.
Proof. Let $X$ be an object of $\mathcal{D}$ and $A$ an object of $\mathcal{D}'$. Since $F$ is an exact functor we see that
\begin{align*} \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(X, G(A[1]) & = \mathop{\mathrm{Mor}}\nolimits _{\mathcal{D}'}(F(X), A[1]) \\ & = \mathop{\mathrm{Mor}}\nolimits _{\mathcal{D}'}(F(X)[-1], A) \\ & = \mathop{\mathrm{Mor}}\nolimits _{\mathcal{D}'}(F(X[-1]), A) \\ & = \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(X[-1], G(A)) \\ & = \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(X, G(A)[1]) \end{align*}
By Yoneda's lemma (Categories, Lemma 4.3.5) we obtain a canonical isomorphism $G(A)[1] = G(A[1])$. Let $A \to B \to C \to A[1]$ be a distinguished triangle in $\mathcal{D}'$. Choose a distinguished triangle
\[ G(A) \to G(B) \to X \to G(A)[1] \]
in $\mathcal{D}$. Then $F(G(A)) \to F(G(B)) \to F(X) \to F(G(A))[1]$ is a distinguished triangle in $\mathcal{D}'$. By TR3 we can choose a morphism of distinguished triangles
\[ \xymatrix{ F(G(A)) \ar[r] \ar[d] & F(G(B)) \ar[r] \ar[d] & F(X) \ar[r] \ar[d] & F(G(A))[1] \ar[d] \\ A \ar[r] & B \ar[r] & C \ar[r] & A[1] } \]
Since $G$ is the adjoint the new morphism determines a morphism $X \to G(C)$ such that the diagram
\[ \xymatrix{ G(A) \ar[r] \ar[d] & G(B) \ar[r] \ar[d] & X \ar[r] \ar[d] & G(A)[1] \ar[d] \\ G(A) \ar[r] & G(B) \ar[r] & G(C) \ar[r] & G(A)[1] } \]
commutes. Applying the homological functor $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}'}(W, -)$ for an object $W$ of $\mathcal{D}'$ we deduce from the $5$ lemma that
\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}'}(W, X) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}'}(W, G(C)) \]
is a bijection and using the Yoneda lemma once more we conclude that $X \to G(C)$ is an isomorphism. Hence we conclude that $G(A) \to G(B) \to G(C) \to G(A)[1]$ is a distinguished triangle which is what we wanted to show. $\square$
Lemma 13.7.2. Let $\mathcal{D}$, $\mathcal{D}'$ be triangulated categories. Let $F : \mathcal{D} \to \mathcal{D}'$ and $G : \mathcal{D}' \to \mathcal{D}$ be functors. Assume that
$F$ and $G$ are exact functors,
$F$ is fully faithful,
$G$ is a right adjoint to $F$, and
the kernel of $G$ is zero.
Then $F$ is an equivalence of categories.
Proof. Since $F$ is fully faithful the adjunction map $\text{id} \to G \circ F$ is an isomorphism (Categories, Lemma 4.24.4). Let $X$ be an object of $\mathcal{D}'$. Choose a distinguished triangle
\[ F(G(X)) \to X \to Y \to F(G(X))[1] \]
in $\mathcal{D}'$. Applying $G$ and using that $G(F(G(X))) = G(X)$ we find a distinguished triangle
\[ G(X) \to G(X) \to G(Y) \to G(X)[1] \]
Hence $G(Y) = 0$. Thus $Y = 0$. Thus $F(G(X)) \to X$ is an isomorphism. $\square$
Comment #2038 by luke on May 12, 2016 at 11:28
In the proof of lemma 13.7.1, functor
\mathop{\rm Hom}\nolimits_{\mathcal{D}'}(W, -)
should be homological instead of cohomological.
Thank you luke, see here.
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Estimator - Simple English Wikipedia, the free encyclopedia
used in mathematical statistics to determine an estimated value
In statistics, an estimator is a rule for calculating an estimate of a given amount based on observed data: thus the rule (the estimator), the amount that is being measured (the estimand) and its result (the estimate) are different.[1]
An estimator of a parameter
{\displaystyle \theta }
{\displaystyle {\hat {\theta }}}
.[2] If the expected value of the estimator is equal to the parameter, then the estimator is called unbiased. Otherwise, it is called biased.[3]
↑ Mosteller, F.; Tukey, J. W. (1987) [1968]. "Data Analysis, including Statistics". The Collected Works of John W. Tukey: Philosophy and Principles of Data Analysis 1965–1986. Vol. 4. CRC Press. pp. 601–720 [p. 633]. ISBN 0-534-05101-4 – via Google Books.
↑ Taylor, Courtney (January 13, 2019). "Unbiased and Biased Estimators". ThoughtCo. Retrieved 2020-09-12.
Fundamentals on Estimation Theory Archived 2020-02-12 at the Wayback Machine
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Estimator&oldid=7739639"
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{}^{136}Xe
{}^{136}Xe
The largest running dark-matter experiment has its first result: No finding yet with the world best sensitivity
Xiangdong Ji, Spokesperson and Project Leader of the PandaX experiment located in the China Jin-Ping underground Laboratory (CJPL), announced the first dark matter search results from the PandaX-II 500 kg liquid xenon detector in the 2016 International Identification of Dark Matter conference at Sheffield, UK in the evening of July 21, 2016 (Beijing Time). He reported that no trace of dark matter was observed with an exposure of 33,000 kg·day of liquid xenon, providing the newest constraints on the existence of dark matter.
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1 November 2006 Estimates for representation numbers of quadratic forms
Valentin Blomer, Andrew Granville
Valentin Blomer,1 Andrew Granville2
2Départment de mathématiques et statistique, Université de Montréal
f
be a primitive positive integral binary quadratic form of discriminant
-D
{r}_{f}\left(n\right)
be the number of representations of
n
f
up to automorphisms of
. In this article, we give estimates and asymptotics for the quantity
\sum _{n\le x}{r}_{f}\left(n\right){}^{\beta }
\beta \ge 0
and uniformly in
D=o\left(x\right)
. As a consequence, we get more-precise estimates for the number of integers which can be written as the sum of two powerful numbers
Valentin Blomer. Andrew Granville. "Estimates for representation numbers of quadratic forms." Duke Math. J. 135 (2) 261 - 302, 1 November 2006. https://doi.org/10.1215/S0012-7094-06-13522-6
Valentin Blomer, Andrew Granville "Estimates for representation numbers of quadratic forms," Duke Mathematical Journal, Duke Math. J. 135(2), 261-302, (1 November 2006)
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IsRightTriangle - Maple Help
Home : Support : Online Help : Mathematics : Geometry : 3-D Euclidean : Plane Functions : IsRightTriangle
test if a given triangle is a right triangle
IsRightTriangle(ABC, cond)
The routine tests if the given triangle ABC is a right triangle. It returns true if ABC is a right triangle; false if it is not; and FAIL if it is unable to reach a conclusion.
If FAIL is returned, and the optional argument is given, the condition that makes ABC a right triangle is assigned to this argument. It will be either of the form
\mathrm{expr}=0
&\mathrm{or}\left(\mathrm{expr_1}=0,\mathrm{expr_2}=0,...,\mathrm{expr_n}\right)
where expr, expri_i are Maple expressions.
The command with(geom3d,IsRightTriangle) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geom3d}\right):
\mathrm{triangle}\left(\mathrm{ABC},[\mathrm{point}\left(A,0,0,0\right),\mathrm{point}\left(B,2,0,0\right),\mathrm{point}\left(C,0,2,0\right)]\right)
\textcolor[rgb]{0,0,1}{\mathrm{ABC}}
\mathrm{IsRightTriangle}\left(\mathrm{ABC}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Consistency criterion - Wikipedia
A voting system is consistent if, whenever the electorate is divided (arbitrarily) into several parts and elections in those parts garner the same result, then an election of the entire electorate also garners that result. Smith[1] calls this property separability and Woodall[2] calls it convexity.
It has been proven a ranked voting system is "consistent if and only if it is a scoring function"[3], i.e. a positional voting system. Borda count is an example of this.
The failure of the consistency criterion can be seen as an example of Simpson's paradox.
As shown below under Kemeny-Young, passing or failing the consistency criterion can depend on whether the election selects a single winner or a full ranking of the candidates (sometimes referred to as ranking consistency); in fact, the specific examples below rely on finding single winner inconsistency by choosing two different rankings with the same overall winner, which means they do not apply to ranking consistency.
1.1.1 First group of voters
1.1.2 Second group of voters
1.1.3 All voters
1.3 Kemeny-Young method
1.3.5 Ranking consistency
1.3.5.1 Informal proof
1.6 Ranked pairs
1.7 Schulze method
Main article: Copeland's method
This example shows that Copeland's method violates the consistency criterion. Assume five candidates A, B, C, D and E with 27 voters with the following preferences:
A > D > B > E > C 3
A > D > E > C > B 2
B > A > C > D > E 3
C > D > B > E > A 3
E > C > B > A > D 3
A > D > C > E > B 3
A > D > E > B > C 1
B > D > C > E > A 3
C > A > B > D > E 3
E > B > C > A > D 3
Now, the set of all voters is divided into two groups at the bold line. The voters over the line are the first group of voters; the others are the second group of voters.
First group of votersEdit
In the following the Copeland winner for the first group of voters is determined.
D [X] 11
E [X] 8
Pairwise election results (won-tied-lost): 3-0-1 2-0-2 2-0-2 2-0-2 1-0-3
Result: With the votes of the first group of voters, A can defeat three of the four opponents, whereas no other candidate wins against more than two opponents. Thus, A is elected Copeland winner by the first group of voters.
Second group of votersEdit
Now, the Copeland winner for the second group of voters is determined.
Result: Taking only the votes of the second group in account, again, A can defeat three of the four opponents, whereas no other candidate wins against more than two opponents. Thus, A is elected Copeland winner by the second group of voters.
All votersEdit
Finally, the Copeland winner of the complete set of voters is determined.
Y A [X] 15
[Y] 12 [X] 15
B [X] 12
C [X] 12
E [X] 15
Result: C is the Condorcet winner, thus Copeland chooses C as winner.
A is the Copeland winner within the first group of voters and also within the second group of voters. However, both groups combined elect C as the Copeland winner. Thus, Copeland fails the consistency criterion.
This example shows that Instant-runoff voting violates the consistency criterion. Assume three candidates A, B and C and 23 voters with the following preferences:
A > B > C 4
C > B > A 4
C > A > B 3
In the following the instant-runoff winner for the first group of voters is determined.
B has only 2 votes and is eliminated first. Its votes are transferred to A. Now, A has 6 votes and wins against C with 4 votes.
Votes in round
Result: A wins against C, after B has been eliminated.
Now, the instant-runoff winner for the second group of voters is determined.
C has the fewest votes, a count of 3, and is eliminated. A benefits from that, gathering all the votes from C. Now, with 7 votes A wins against B with 6 votes.
Result: A wins against B, after C has been eliminated.
Finally, the instant runoff winner of the complete set of voters is determined.
C has the fewest first preferences and so is eliminated first, its votes are split: 4 are transferred to B and 3 to A. Thus, B wins with 12 votes against 11 votes of A.
Result: B wins against A, after C is eliminated.
A is the instant-runoff winner within the first group of voters and also within the second group of voters. However, both groups combined elect B as the instant-runoff winner. Thus, instant-runoff voting fails the consistency criterion.
Kemeny-Young methodEdit
Main article: Kemeny-Young method
This example shows that the Kemeny–Young method violates the consistency criterion. Assume three candidates A, B and C and 38 voters with the following preferences:
1st A > B > C 7
B > C > A 6
2nd A > C > B 8
In the following the Kemeny-Young winner for the first group of voters is determined.
The Kemeny–Young method arranges the pairwise comparison counts in the following tally table:
Pairs of choices
Voters who prefer
Y over X
B C 13 0 3
The ranking scores of all possible rankings are:
A > B > C 10 7 13 30
A > C > B 7 10 3 20
B > A > C 6 13 7 26
B > C > A 13 6 9 28
C > A > B 9 3 10 22
C > B > A 3 9 6 18
Result: The ranking A > B > C has the highest ranking score. Thus, A wins ahead of B and C.
Now, the Kemeny-Young winner for the second group of voters is determined.
A > C > B 8
A C 15 0 7
A > B > C 8 15 7 30
A > C > B 15 8 15 38
B > A > C 14 7 15 36
B > C > A 7 14 7 28
C > A > B 7 15 8 30
C > B > A 15 7 14 36
Result: The ranking A > C > B has the highest ranking score. Hence, A wins ahead of C and B.
Finally, the Kemeny-Young winner of the complete set of voters is determined.
A B 18 0 20
A C 22 0 16
B C 20 0 18
A > B > C 18 22 20 60
A > C > B 22 18 18 58
B > A > C 20 20 22 62
B > C > A 20 20 16 56
C > A > B 16 18 18 52
C > B > A 18 16 20 54
Result: The ranking B > A > C has the highest ranking score. So, B wins ahead of A and C.
A is the Kemeny-Young winner within the first group of voters and also within the second group of voters. However, both groups combined elect B as the Kemeny-Young winner. Thus, the Kemeny–Young method fails the consistency criterion.
Ranking consistencyEdit
The Kemeny-Young method satisfies ranking consistency; that is, if the electorate is divided arbitrarily into two parts and separate elections in each part result in the same ranking being selected, an election of the entire electorate also selects that ranking.
Informal proofEdit
The Kemeny-Young score of a ranking
{\displaystyle {\mathcal {R}}}
is computed by summing up the number of pairwise comparisons on each ballot that match the ranking
{\displaystyle {\mathcal {R}}}
. Thus, the Kemeny-Young score
{\displaystyle s_{V}({\mathcal {R}})}
for an electorate
{\displaystyle V}
can be computed by separating the electorate into disjoint subsets
{\displaystyle V=V_{1}\cup V_{2}}
{\displaystyle V_{1}\cap V_{2}=\emptyset }
), computing the Kemeny-Young scores for these subsets and adding it up:
{\displaystyle {\text{(I)}}\quad s_{V}({\mathcal {R}})=s_{V_{1}}({\mathcal {R}})+s_{V_{2}}({\mathcal {R}})}
Now, consider an election with electorate
{\displaystyle V}
. The premise of the consistency criterion is to divide the electorate arbitrarily into two parts
{\displaystyle V=V_{1}\cup V_{2}}
, and in each part the same ranking
{\displaystyle {\mathcal {R}}}
is selected. This means, that the Kemeny-Young score for the ranking
{\displaystyle {\mathcal {R}}}
in each electorate is bigger than for every other ranking
{\displaystyle {\mathcal {R}}'}
{\displaystyle {\begin{aligned}{\text{(II)}}\quad \forall {\mathcal {R}}':{}&s_{V_{1}}({\mathcal {R}})>s_{V_{1}}({\mathcal {R}}')\\{\text{(III)}}\quad \forall {\mathcal {R}}':{}&s_{V_{2}}({\mathcal {R}})>s_{V_{2}}({\mathcal {R}}')\end{aligned}}}
Now, it has to be shown, that the Kemeny-Young score of the ranking
{\displaystyle {\mathcal {R}}}
in the entire electorate is bigger than the Kemeny-Young score of every other ranking
{\displaystyle {\mathcal {R}}'}
{\displaystyle s_{V}({\mathcal {R}})\ {\stackrel {(I)}{=}}\ s_{V_{1}}({\mathcal {R}})+s_{V_{2}}({\mathcal {R}})\ {\stackrel {(II)}{>}}\ s_{V_{1}}({\mathcal {R}}')+s_{V_{2}}({\mathcal {R}})\ {\stackrel {(III)}{>}}\ s_{V_{1}}({\mathcal {R}}')+s_{V_{2}}({\mathcal {R}}')\ {\stackrel {(I)}{=}}\ s_{V}({\mathcal {R}}')\quad q.e.d.}
Thus, the Kemeny-Young method is consistent with respect to complete rankings.
This example shows that majority judgment violates the consistency criterion. Assume two candidates A and B and 10 voters with the following ratings:
Excellent Fair 3
Poor Fair 2
In the following the majority judgment winner for the first group of voters is determined.
Result: With the votes of the first group of voters, A has the median rating of "Excellent" and B has the median rating of "Fair". Thus, A is elected majority judgment winner by the first group of voters.
Now, the majority judgment winner for the second group of voters is determined.
Result: Taking only the votes of the second group in account, A has the median rating of "Fair" and B the median rating of "Poor". Thus, A is elected majority judgment winner by the second group of voters.
Finally, the majority judgment winner of the complete set of voters is determined.
The median ratings for A and B are both "Fair". Since there is a tie, "Fair" ratings are removed from both, until their medians become different. After removing 20% "Fair" ratings from the votes of each, the sorted ratings are now:
Result: Now, the median rating of A is "Poor" and the median rating of B is "Fair". Thus, B is elected majority judgment winner.
A is the majority judgment winner within the first group of voters and also within the second group of voters. However, both groups combined elect B as the Majority Judgment winner. Thus, Majority Judgment fails the consistency criterion.
This example shows that the minimax method violates the consistency criterion. Assume four candidates A, B, C and D with 43 voters with the following preferences:
A > B > C > D 1
A > D > B > C 6
B > C > D > A 5
C > D > B > A 6
A > B > D > C 8
A > D > C > B 2
C > B > D > A 9
D > C > B > A 6
In the following the minimax winner for the first group of voters is determined.
Pairwise election results (won-tied-lost) 0-0-3 2-0-1 2-0-1 2-0-1
Worst pairwise Defeat (winning votes) 11 12 12 12
Defeat (margins) 4 6 6 6
Opposition 11 12 12 12
Result: The candidates B, C and D form a cycle with clear defeats. A benefits from that since it loses relatively closely against all three and therefore A's biggest defeat is the closest of all candidates. Thus, A is elected minimax winner by the first group of voters.
Now, the minimax winner for the second group of voters is determined.
Result: Taking only the votes of the second group in account, again, B, C and D form a cycle with clear defeats and A benefits from that because of its relatively close losses against all three and therefore A's biggest defeat is the closest of all candidates. Thus, A is elected minimax winner by the second group of voters.
Finally, the minimax winner of the complete set of voters is determined.
Result: Again, B, C and D form a cycle. But now, their mutual defeats are very close. Therefore, the defeats A suffers from all three are relatively clear. With a small advantage over B and D, C is elected minimax winner.
A is the minimax winner within the first group of voters and also within the second group of voters. However, both groups combined elect C as the Minimax winner. Thus, Minimax fails the consistency criterion.
Ranked pairsEdit
This example shows that the Ranked pairs method violates the consistency criterion. Assume three candidates A, B and C with 39 voters with the following preferences:
In the following the Ranked pairs winner for the first group of voters is determined.
Pairwise election results (won-tied-lost): 1-0-1 1-0-1 1-0-1
The sorted list of victories would be:
B (13) vs C (3) B 13
A (10) vs B (6) A 10
A (7) vs C (9) C 9
Result: B > C and A > B are locked in first (and C > A can't be locked in after that), so the full ranking is A > B > C. Thus, A is elected Ranked pairs winner by the first group of voters.
Now, the Ranked pairs winner for the second group of voters is determined.
A (17) vs C (6) A 17
B (8) vs C (15) C 15
A (9) vs B (14) B 14
Result: Taking only the votes of the second group in account, A > C and C > B are locked in first (and B > A can't be locked in after that), so the full ranking is A > C > B. Thus, A is elected Ranked pairs winner by the second group of voters.
Finally, the Ranked pairs winner of the complete set of voters is determined.
A (25) vs C (15) A 24
B (21) vs C (18) B 21
A (19) vs B (20) B 20
Result: Now, all three pairs (A > C, B > C and B > A) can be locked in without a cycle. The full ranking is B > A > C. Thus, Ranked pairs chooses B as winner, which is the Condorcet winner, due to the lack of a cycle.
A is the Ranked pairs winner within the first group of voters and also within the second group of voters. However, both groups combined elect B as the Ranked pairs winner. Thus, the Ranked pairs method fails the consistency criterion.
Schulze methodEdit
This example shows that the Schulze method violates the consistency criterion. Again, assume three candidates A, B and C with 39 voters with the following preferences:
In the following the Schulze winner for the first group of voters is determined.
The pairwise preferences would be tabulated as follows:
Matrix of pairwise preferences
d[X, Y]
Now, the strongest paths have to be identified, e.g. the path A > B > C is stronger than the direct path A > C (which is nullified, since it is a loss for A).
Strengths of the strongest paths
Result: A > B, A > C and B > C prevail, so the full ranking is A > B > C. Thus, A is elected Schulze winner by the first group of voters.
Now, the Schulze winner for the second group of voters is determined.
Now, the strongest paths have to be identified, e.g. the path A > C > B is stronger than the direct path A > B.
Result: A > B, A > C and C > B prevail, so the full ranking is A > C > B. Thus, A is elected Schulze winner by the second group of voters.
Finally, the Schulze winner of the complete set of voters is determined.
Now, the strongest paths have to be identified:
Result: A > C, B > A and B > C prevail, so the full ranking is B > A > C. Thus, Schulze chooses B as winner. In fact, B is also Condorcet winner.
A is the Schulze winner within the first group of voters and also within the second group of voters. However, both groups combined elect B as the Schulze winner. Thus, the Schulze method fails the consistency criterion.
^ John H Smith, "Aggregation of preferences with variable electorate", Econometrica, Vol. 41 (1973), pp. 1027–1041.
^ D. R. Woodall, "Properties of preferential election rules", Voting matters, Issue 3 (December 1994), pp. 8–15.
^ H. P. Young, "Social Choice Scoring Functions", SIAM Journal on Applied Mathematics Vol. 28, No. 4 (1975), pp. 824–838.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Consistency_criterion&oldid=1042335413"
|
Euclid's_theorem Knowpia
Euclid's proofEdit
If q is not prime, then some prime factor p divides q. If this factor p were in our list, then it would divide P (since P is the product of every number in the list); but p also divides P + 1 = q, as just stated. If p divides P and also q, then p must also divide the difference[3] of the two numbers, which is (P + 1) − P or just 1. Since no prime number divides 1, p cannot be in the list. This means that at least one more prime number exists beyond those in the list.
Euler's proofEdit
{\displaystyle \prod _{p\in P_{k}}{\frac {1}{1-{\frac {1}{p}}}}=\sum _{n\in N_{k}}{\frac {1}{n}},}
{\displaystyle P_{k}}
{\displaystyle N_{k}}
{\displaystyle P_{k}.}
{\displaystyle {\begin{aligned}\prod _{p\in P_{k}}{\frac {1}{1-{\frac {1}{p}}}}&=\prod _{p\in P_{k}}\sum _{i\geq 0}{\frac {1}{p^{i}}}\\&=\left(\sum _{i\geq 0}{\frac {1}{2^{i}}}\right)\cdot \left(\sum _{i\geq 0}{\frac {1}{3^{i}}}\right)\cdot \left(\sum _{i\geq 0}{\frac {1}{5^{i}}}\right)\cdot \left(\sum _{i\geq 0}{\frac {1}{7^{i}}}\right)\cdots \\&=\sum _{\ell ,m,n,p,\ldots \geq 0}{\frac {1}{2^{\ell }3^{m}5^{n}7^{p}\cdots }}\\&=\sum _{n\in N_{k}}{\frac {1}{n}}.\end{aligned}}}
{\displaystyle \infty }
{\displaystyle \log \infty }
{\displaystyle x}
{\displaystyle \log x}
{\displaystyle \prod _{n\geq 2}{\frac {1}{1-{\frac {1}{n^{2}}}}}}
{\displaystyle \sum _{p\in P}{\frac {1}{p}}}
{\displaystyle =\log \log \infty }
{\displaystyle x}
{\displaystyle \log \log x}
Erdős's proofEdit
{\displaystyle \left(p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{k}^{e_{k}}\right)s^{2},}
{\displaystyle N\leq 2^{k}{\sqrt {N}}.}
Furstenberg's proofEdit
{\displaystyle S(a,b)=\{an+b\mid n\in \mathbb {Z} \}=a\mathbb {Z} +b.}
{\displaystyle \mathbb {Z} \setminus \{-1,+1\}=\bigcup _{p\mathrm {\,prime} }S(p,0)}
Recent proofsEdit
Proof using the inclusion-exclusion principleEdit
{\displaystyle {\begin{aligned}1+\sum _{i}\left\lfloor {\frac {x}{p_{i}}}\right\rfloor -\sum _{i<j}\left\lfloor {\frac {x}{p_{i}p_{j}}}\right\rfloor &+\sum _{i<j<k}\left\lfloor {\frac {x}{p_{i}p_{j}p_{k}}}\right\rfloor -\cdots \\&\cdots \pm (-1)^{N+1}\left\lfloor {\frac {x}{p_{1}\cdots p_{N}}}\right\rfloor .\qquad (1)\end{aligned}}}
{\displaystyle \sum _{i}{\frac {1}{p_{i}}}-\sum _{i<j}{\frac {1}{p_{i}p_{j}}}+\sum _{i<j<k}{\frac {1}{p_{i}p_{j}p_{k}}}-\cdots \pm (-1)^{N+1}{\frac {1}{p_{1}\cdots p_{N}}}.\qquad (2)}
{\displaystyle 1-\prod _{i=1}^{N}\left(1-{\frac {1}{p_{i}}}\right).\qquad (3)}
{\displaystyle \lfloor x\rfloor }
Proof using de Polignac's formulaEdit
{\displaystyle k!=\prod _{p{\text{ prime}}}p^{f(p,k)}}
{\displaystyle f(p,k)=\left\lfloor {\frac {k}{p}}\right\rfloor +\left\lfloor {\frac {k}{p^{2}}}\right\rfloor +\cdots .}
{\displaystyle f(p,k)<{\frac {k}{p}}+{\frac {k}{p^{2}}}+\cdots ={\frac {k}{p-1}}\leq k.}
{\displaystyle \lim _{k\to \infty }{\frac {\left(\prod _{p}p\right)^{k}}{k!}}=0,}
Proof using the incompressibility methodEdit
{\displaystyle n={p_{1}}^{e_{1}}{p_{2}}^{e_{2}}\cdots {p_{k}}^{e_{k}},}
{\displaystyle p_{i}\geq 2}
{\displaystyle e_{i}\leq \lg n}
{\displaystyle \lg }
{\displaystyle O({\text{prime list size}}+k\lg \lg n)=O(\lg \lg n)}
{\displaystyle N=O(\lg n)}
{\displaystyle \lg \lg n=o(\lg n)}
Stronger resultsEdit
Dirichlet's theorem on arithmetic progressionsEdit
Prime number theoremEdit
{\displaystyle \lim _{x\rightarrow \infty }{\frac {\pi (x)}{x/\ln(x)}}=1.}
{\displaystyle \pi (x)\sim {\frac {x}{\log x}}.}
{\displaystyle \lim _{x\rightarrow \infty }{\frac {x}{\log x}}=\infty .}
Bertrand–Chebyshev theoremEdit
{\displaystyle n>1}
{\displaystyle n<p<2n.}
{\displaystyle \pi (x)}
{\displaystyle \pi (x)}
{\displaystyle x\,}
{\displaystyle \pi (x)-\pi ({\tfrac {x}{2}})\geq 1,}
{\displaystyle x\geq 2.}
{\displaystyle a\mid b}
{\displaystyle a\mid c}
{\displaystyle a\mid (b-c)}
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Experimental Measurement of Minority Carriers Effective Lifetime in Silicon Solar Cell Using Open Circuit Voltage Decay under Magnetic Field in Transient Mode
SGRE Vol.11 No.11 , November 2020
Alain Diasso1, Raguilignaba Sam1,2, Bernard Zouma3, François Zougmoré1
1 Department of physic, Laboratory of Materials and Environment, Units of Sciences and Technology, University of Joseph Ki-Zerbo, Ouagadougou, Burkina Faso.
2 Units of Sciences and Technology, Department of Physic, University of Nazi Boni, Bobo Dioulasso, Burkina Faso.
3 Department of Physic, Units of Sciences and Technology, Laboratory of Thermal and Renewable Energy, University of Ouagadougou, Ouagadougou, Burkina Faso.
Abstract: This manuscript presents a simple method for excess minority carriers’ lifetime measurement within the base region of p-n junction polycrystalline solar cell in transient mode. This work is an experimental transient 3-Dimensionnal study. The magnitude of the magnetic field B is varied from 0 mT to 0.045 mT. Indeed, the solar cell is illuminated by a stroboscopic flash with air mass 1.5 and under magnetic field in transient state. The experimental details are assumed in a figure. The procedure is outlined by the Open Circuit Voltage Decay analysis. Effective minority carrier life-time is calculated by fitting the linear zone of the transient voltage decay curve because linear decay is an ideal decay. The kaleidagraph software permits access to the slope of the curve which is inversely proportional to the lifetime. The external magnetic effects on minority carriers’ effective lifetime is then presented and analyzed. The analysis shows that the charge carrier’s effective lifetime decrease with the magnetic field increase.
Keywords: Carrier Lifetime, Fitting, Magnetic Field, Open Circuit Voltage Decay
The lifetime of minority carriers is an important parameter and its determination is essential to improve solar cells high efficiency. The dark and illuminated characteristics are affected by the charge carriers’ lifetime. This electronic parameter restricts the open-circuit voltage, short circuit courant and the net output power. Therefore, with the increasing interest in photovoltaic energy conversion, fresh attempts have been made in the recent past for accurate lifetime measurement. Suitability for the earlier methods for p-n junctions, e.g. open-circuit voltage decay [1], the reverse recovery [2] [3], photoconductivity decay [4], spectral response [4] has been studied everyone with his advantages.
Among all methods, we are chosen the open-circuit voltage decay method. This method is practice for our experimental conditions. However, two methods are used to induce the initial voltage: the conventional method [1] [5] in which a current is passed through the cell from an external source such as a battery. This is the forward current-voltage decay (F.C.V.D) method. The second is to illuminate the cell using a semi-conductor by a laser [6] or a stroboscope [1] [7]; we call it the photovoltage decay method (P.V.D). In both cases, the voltage is abruptly interrupted and the voltage decay is measured on an oscilloscope. The open-circuit voltage decay technic [1] [8] is a simple non-destructive method for measuring minority carriers’ lifetime in solar cells. In this method, a voltage is induced in the solar cell and is then allowed to decay under open-circuit conditions.
In this study, firstly we present the experimental set-up. Secondly, the experimental conditions will be described and assumptions made will be presented. Thirdly, the method for charge carriers’ effective lifetime measurement will be explained and the results obtained will be presented and analyzed.
The experimental set-up is composed of: a mono-facial silicon solar cell manufactured by MOTCH INDUSTRY; a pulsed light source MINISTROB PHIWE, a digital oscilloscope TECKTRONIX model TDS 10013, a computer INTEL 586, a power supply 0 - 12V DC/6V-12V AC, a teslameter and Helmholtz Coil.
The experimental set-up is presented in Figure 1 and operating mode in Figure 2.
Figure 2. Operating mode.
The experimental system principle operating is the same described by Sam et al. and R. Sam et al. [9] [10]. In additional, we consider the stroboscope flash is near of AM1.5 spectrum and the solar in operating conditions where there are not others fields.
The Kaleidagraph software which is a thoughtfully designed graphing and data analysis application for research scientists permit us to convert complex data obtained during experimentation into transient curves.
The transient response is obtained by disturbance of steady state. While exciting the solar cell, we carry it towards a state characterized by a balance between the phenomena of recombination and generation of pairs electron-hole [10]. This stationary state defined a point1. After a duration Te (stroboscope pulse duration), the excitation is abruptly switched off; the balance between recombination and generation is so broken. A transient response appears and the solar cell towards a new stationary state: its fundamental state. This other state later defines a new point that we call 2. The transient response corresponds to the relaxation state of the sample between two operating points 1 and 2.
Considering the identical optoelectronic properties of all the lot of grains, then the study of phenomena of generation, diffusion and recombination of the charge carriers in the solar cell during the transient state can be described a only one grain. Figure 3 is a representation of a grain under magnetic field.
The calculation of transient voltage decay expression by using boundaries conditions and quasi-neutral base theory gives [11]:
Figure 3. A sample model of grain under magnetic field.
– For exponential zone
V\left(t\right)={V}_{T}Fc\left({k}_{1},{l}^{\ast }{}_{1},{\mu }_{1}\right)r\mathrm{exp}\left(-{\beta }^{\ast }\left(t-Te\right)\right)
This is a time dependent exponential decay function
– For linear zone
V\left(t\right)={V}_{T}\left(-{\beta }^{\ast }\left(t-Te\right)+r\mathrm{ln}\left(1+Fv\left({k}_{1},{l}^{\ast }{}_{1},{\mu }_{1}\right)\right)\right)
This is a linear function of time with a negative slope
-{V}_{T}{\beta }^{\ast }
{\frac{\partial \delta \left(x,y,z,t\right)}{\partial z}|}_{z=0}=\frac{Sf}{{D}^{\ast }}\delta \left(x,y,z=0,t\right)
{\frac{\partial \delta \left(x,y,z,t\right)}{\partial z}|}_{z=H}=\frac{Sb}{{D}^{\ast }}\delta \left(x,y,z=H,t\right)
{\frac{\partial \delta \left(x,y,z,t\right)}{\partial x}|}_{x=\pm a}=\pm \frac{Sgx}{{D}^{\ast }}\delta \left(x=\pm a,y,z,t\right)
{\frac{\partial \delta \left(x,y,z,t\right)}{\partial y}|}_{y=\pm b}=\pm \frac{Sgy}{{D}^{\ast }}\delta \left(x,y=\pm b,z,t\right)
Equations (3)-(6) are boundaries conditions.
Sf
Sb
Sg
are the recombination velocity of minority charge carriers respectively at surfaces z = 0, z = H and x = ±a (or y = ±b). a, b and H are the grain sizes as indicated in Figure 2.
{D}^{\ast }
{L}^{\ast }
are also respectively the electrons diffusion coefficient and length diffusion of charge carriers
3. Effective Lifetime of Minority Charge Carriers’ Measurement
Our approach is based on the linear approximation of transient voltage decay because in low injection it permits to avoid impedances effects [12]. The following curve present the different zones of transient voltage curve.
After identification of linear zone of the curve, we fit this zone and we get a linear regression line indicated in Figure 4.
In Figure 5, the linear regression line equation is found with a good correlation coefficient R. The slope of this linear regression line is related by the effective lifetime of minority charge carriers.
|m|=\frac{{V}_{T}}{{\tau }_{eff}}
Figure 4. Voltage decay with different zones in transient state.
Figure 5. A linear regression line with it equation.
where m is the slope of the transient voltage curve, VT is the thermic voltage and
{\tau }_{eff}
is the charge carrier effective lifetime.
After registration of transient voltage data on digital scope, we use kaleidagraph software to make simulations. Then, we get the curves of transient voltage decay for various magnetic field values under AM1.5 spectrum (Figures 6-13).
Figure 6. Transient voltage decay B = 0 mT.
Figure 7. Transient voltage decay B = 0.01 mT.
Figure 8. Transient voltage decay B = 0.015 mT.
Figure 10. Transient voltage decay B = 0.030 mT.
Figure 12. Transient voltage decay B = 0.04 mT.
Table 1. Minority charge carriers effective lifetime.
RE is correlation coefficient which show the quality of the fit.
By observations of the curves, we note two types of decay: linear decay and exponential decay. Also, we remark that for magnetic field values
B\ge 0.03\text{mT}
, the decay is very fast and the linear decay is major than exponential decay.
After the fitting of all the analyzed curves, we note that the exponential zone is minor than the linear zone. We got ideal types of open circuit transient voltage decay according to Dariwhal and Mahan conditions [7] [13]. Also, this zone of the curves doesn’t present capacitance effects. The results of all transient voltage decay curves linear fitting are noted on Table 1.
The analysis of the results show that the charge carriers’ effective lifetime decreases with the magnetic field. These results are in agreement with theoretical results obtained by Alain et al. [13]. However, we remark that for magnetic values
B\prec 0.02\text{mT}
, the variations are very slow. The decay becomes sudden when magnetic field values exceed
0.02\text{mT}
. Indeed, the charge carriers’ effective lifetime is reduced of 67% for magnetic field values variations
0.02\text{mT}\prec B\prec 0.03\text{mT}
. This decay explain the fast recombination of charge carriers when the magnetic field increase. The extension of space charge region width with the magnetic field increase [14] traduces an important disappearance of charge carriers. Also, the decrease of diffusion length with magnetic field explains these variations of minority charge carriers’ effective lifetime.
In this manuscript, we have developed an experimental technic of measurement of minority charge carriers’ effective lifetime under various magnetic fields with air mass AM1.5. Our approach is based on the method of Open Circuit Transient Voltage Decay. From the experimental set-up described and presented, we got transient voltage decay data and used them to plot the curves. We have done a linear fit on the analyzed curves in their linear zone because this zone is major than exponential zone for all transient voltage curves and did not present impedance effects. These results are in agreement with those obtained and published by other authors in the literature. We validate our results because the theoretical study is in good agreement with the experimental study.
Cite this paper: Diasso, A. , Sam, R. , Zouma, B. and Zougmoré, F. (2020) Experimental Measurement of Minority Carriers Effective Lifetime in Silicon Solar Cell Using Open Circuit Voltage Decay under Magnetic Field in Transient Mode. Smart Grid and Renewable Energy, 11, 181-190. doi: 10.4236/sgre.2020.1111011.
[1] Lederhander, S.R and Giacoletto, L.J. (1955) Measurement of Minority Carrier Lifetime and Surface Effects in Junction Devices. Proceedings of the IRE, 43, 477-483,
[2] Kingston, R.H. (1954) Switching Time in Junction Diodes and Junction Transistors. Proceedings of the IRE, 42, 829-834.
[3] Dean, R.H. and Nusese, C.J. (1971) A Refined Step-Recovery Technique for Measuring Minority Carrier Lifetimes and Related Parameters in Asymmetric p-n Junction Diodes. IEEE Transactions on Electron Devices, 18, 151-158.
[4] Reynolds, J.H. and Meulenberg Jr., A. (1974) Measurement of Diffusion Length in Solar Cells. Journal of Applied Physics, 45, 2582.
[5] Gossick , B.R. (1955) On the Transient Behavior of Semiconductor Rectifiers. Journal of Applied Physics, 26, 1356.
[6] Agarwal, S.K., Muralidharan, R. and Jain, S.C. (1981) Determination of the Minority Carrier Lifetime in the Base of a Back-Surface Field Solar Cell by Forward Current Induced Voltage Decay and Photo Volatage Decay Method. International Workshop on Physics of Semiconductor Devices, Delhi, 23-28 November 1981.
[7] Mahan, J.E., Eksted, T.W., Franck, R.I. and Kaplow, R. (1979) Measurement of Minority Carrier Lifetime in Solar Cells from Photo-Induced Open-Circuit Voltage Decay. IEEE Transactions on Electron Devices, 26, 733-739.
[8] Gossick, B.R. (1953) Post-Injection Barrier Electromotive Force of p-n Junctions. Physical Review Journals Archive, 91, 1012.
[9] Sam, R., Zouma, B., Zougmoré, F., Koalaga, Z., Zoungrana M. and Zerbo. I. (2012) 3D Determination of the Minority Carrier Lifetime and the p-n Junction Recombination Velocity of a Polycrystalline Silicon Solar Cell. IOP Conference Series: Materials Science and Engineering, Ouagadougou, 17-22 October 2011, 1-8.
[10] Sam, R., Kaboré, K. and Zougmoré, F. (2016) A Three-Dimensional Transient Study of a Polycritsalline Silicon Solar Cell under Constant Magnetic Field. International Journal of Engineering Research, 5, 93-97.
[11] Diasso, A., Sam R. and Zougmoré, F. (2020) External Magnetic Field and Air Mass Effects on Carrier’s Effective Lifetime of a Bifacial Solar Cell under Transient State. Research Journal of Applied Sciences, Engineering and Technology, 17, 140-146.
[12] Muralidharan, R. and Jain, S.C. (1982) Determination of the Minority Carrier Lifetime in the Base of a Back-Surface Field Solar Cell by Forward Current-Induced Voltage Decay and Photovoltage Decay Methods. Solar Cells, 6, 157-176.
[13] Dhariwhal, S.R. and Vasu, N.K. (1981) A Generalised Approach to Lifetime Measurement in pn Junction Solar Cells. Solid-State Electronics, 24, 915-927.
[14] Diasso, A., Sam, R., Yacouba, N.T. and Zougmoré, F. (2020) Effects of External Magnetic Field and Air Mass on Space Charge Width Extension of a Bifacial Solar Cell Front Side Illumination. International Journal of Energy and Power Engineering, 9, 29-34.
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Will the NYT end up publishing any articles mentioning SSC or SA in the next year? | Metaculus
Will the NYT end up publishing any articles mentioning SSC or SA in the next year?
Created by Aotho on {{qctrl.question.created_time | dateStr}}.
Published by Aotho on {{qctrl.question.publish_time | dateStr}}.
Will the New York Times end up publishing any articles mentioning Slate Star Codex or Scott Alexander between 2020-07-01 and 2021-07-01?
We already have a Metaculus prediction about if/when such an article is published it will include his full name.
However, that question very much depends on this one, whether they will go ahead with any article in the first place. Since if they are only 1% likely to go ahead then it might be moot whether the name would be included.
It doesn't have to be the currently anticipated article by the currently anticipated NYT author in the currently anticipated topic. Any author's NYT-published article in any topic that mentions either him or his blog is eligible to resolve this question positively.
This question resolves positively if any time between 2020-07-01 00:01 UTC and 2021-07-01 00:01 UTC any article is published on nytimes.com that mentions either "Slate Star Codex", "SlateStarCodex", "slatestarcodex.com", or "Scott Alexander"
^†
Otherwise it resolves negatively at 2021-07-01 00:01 UTC.
^†
And it is clear they are referring to the author of SSC, not any other Scott Alexander.
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Normal (geometry) - Wikipedia @ WordDisk
{\displaystyle P}
{\displaystyle P}
{\displaystyle P.}
This article uses material from the Wikipedia article Normal (geometry), and is written by contributors. Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.
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Section 15.94: Derived completion for Noetherian rings
Lemma 15.94.1. In Situation 15.91.15. If $A$ is Noetherian, then the pro-objects $\{ K_ n^\bullet \} $ and $\{ A/(f_1^ n, \ldots , f_ r^ n)\} $ of $D(A)$ are isomorphic1.
Proof. We have an inverse system of distinguished triangles
\[ \tau _{\leq -1}K_ n^\bullet \to K_ n^\bullet \to A/(f_1^ m, \ldots , f_ r^ m) \to (\tau _{\leq -1}K_ n^\bullet )[1] \]
See Derived Categories, Remark 13.12.4. By Derived Categories, Lemma 13.41.4 it suffices to show that the inverse system $\tau _{\leq -1}K_ n^\bullet $ is pro-zero. Recall that $K_ n^\bullet $ has nonzero terms only in degrees $i$ with $-r \leq i \leq 0$. Thus by Derived Categories, Lemma 13.41.3 it suffices to show that $H^ p(K_ n^\bullet )$ is pro-zero for $p \leq -1$. In other words, for every $n \in \mathbf{N}$ we have to show there exists an $m \geq n$ such that $H^ p(K_ m^\bullet ) \to H^ p(K_ n^\bullet )$ is zero. Since $A$ is Noetherian, we see that
\[ H^ p(K_ n^\bullet ) = \frac{\mathop{\mathrm{Ker}}(K_ n^ p \to K_ n^{p + 1})}{\mathop{\mathrm{Im}}(K_ n^{p - 1} \to K_ n^ p)} \]
is a finite $A$-module. Moreover, the map $K_ m^ p \to K_ n^ p$ is given by a diagonal matrix whose entries are in the ideal $(f_1^{m - n}, \ldots , f_ r^{m - n})$ as $p < 0$. Note that $H^ p(K_ n^\bullet )$ is annihilated by $J = (f_1^ n, \ldots , f_ r^ n)$, see Lemma 15.28.6. Now $(f_1^{m - n}, \ldots , f_ r^{m - n}) \subset J^ t$ for $m - n \geq tn$. Thus by Algebra, Lemma 10.51.2 (Artin-Rees) applied to the ideal $J$ and the module $M = K_ n^ p$ with submodule $N = \mathop{\mathrm{Ker}}(K_ n^ p \to K_ n^{p + 1})$ for $m$ large enough the image of $K_ m^ p \to K_ n^ p$ intersected with $\mathop{\mathrm{Ker}}(K_ n^ p \to K_ n^{p + 1})$ is contained in $J \mathop{\mathrm{Ker}}(K_ n^ p \to K_ n^{p + 1})$. For such $m$ we get the zero map. $\square$
[1] In particular, for every $n$ there exists an $m \geq n$ such that $K_ m^\bullet \to K_ n^\bullet $ factors through the map $K_ m^\bullet \to A/(f_1^ m, \ldots , f_ r^ m)$.
I think what we want and have is that
J^t\supset (f_1^{m-n},\ldots,f_r^{m-n})
m-n\geq tn
, because we want to apply Artin-Rees for
I=J
M=K_n^p
N=\operatorname{Ker}(K_n^p\to K_n^{p+1})
(with the notation of 10.51.2).
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Trophic_level Knowpia
The trophic level of an organism is the position it occupies in a food web. A food chain is a succession of organisms that eat other organisms and may, in turn, be eaten themselves. The trophic level of an organism is the number of steps it is from the start of the chain. A food web starts at trophic level 1 with primary producers such as plants, can move to herbivores at level 2, carnivores at level 3 or higher, and typically finish with apex predators at level 4 or 5. The path along the chain can form either a one-way flow or a food "web". Ecological communities with higher biodiversity form more complex trophic paths.
The word trophic derives from the Greek τροφή (trophē) referring to food or nourishment.[1]
The concept of trophic level was developed by Raymond Lindeman (1942), based on the terminology of August Thienemann (1926): "producers", "consumers" and "reducers" (modified to "decomposers" by Lindeman).[2][3]
Consumers (heterotrophs) are species that cannot manufacture their own food and need to consume other organisms. Animals that eat primary producers (like plants) are called herbivores. Animals that eat other animals are called carnivores, and animals that eat both plants and other animals are called omnivores.
Apex predators by definition have no predators and are at the top of their food web.
In real-world ecosystems, there is more than one food chain for most organisms, since most organisms eat more than one kind of food or are eaten by more than one type of predator. A diagram that sets out the intricate network of intersecting and overlapping food chains for an ecosystem is called its food web.[6] Decomposers are often left off food webs, but if included, they mark the end of a food chain.[6] Thus food chains start with primary producers and end with decay and decomposers. Since decomposers recycle nutrients, leaving them so they can be reused by primary producers, they are sometimes regarded as occupying their own trophic level.[7][8]
The trophic level of a species may vary if it has a choice of diet. Virtually all plants and phytoplankton are purely phototrophic and are at exactly level 1.0. Many worms are at around 2.1; insects 2.2; jellyfish 3.0; birds 3.6.[9] A 2013 study estimates the average trophic level of human beings at 2.21, similar to pigs or anchovies.[10] This is only an average, and plainly both modern and ancient human eating habits are complex and vary greatly. For example, a traditional Eskimo living on a diet consisting primarily of seals would have a trophic level of nearly 5.[11]
Biomass transfer efficiencyEdit
An energy pyramid illustrates how much energy is needed as it flows upward to support the next trophic level. Only about 10% of the energy transferred between each trophic level is converted to biomass.
In general, each trophic level relates to the one below it by absorbing some of the energy it consumes, and in this way can be regarded as resting on, or supported by, the next lower trophic level. Food chains can be diagrammed to illustrate the amount of energy that moves from one feeding level to the next in a food chain. This is called an energy pyramid. The energy transferred between levels can also be thought of as approximating to a transfer in biomass, so energy pyramids can also be viewed as biomass pyramids, picturing the amount of biomass that results at higher levels from biomass consumed at lower levels. However, when primary producers grow rapidly and are consumed rapidly, the biomass at any one moment may be low; for example, phytoplankton (producer) biomass can be low compared to the zooplankton (consumer) biomass in the same area of ocean.[12]
Both the number of trophic levels and the complexity of relationships between them evolve as life diversifies through time, the exception being intermittent mass extinction events.[13]
Fractional trophic levelsEdit
Killer whales (orca) are apex predators but they are divided into separate populations that hunt specific prey, such as tuna, small sharks, and seals.
Food webs largely define ecosystems, and the trophic levels define the position of organisms within the webs. But these trophic levels are not always simple integers, because organisms often feed at more than one trophic level.[14][15] For example, some carnivores also eat plants, and some plants are carnivores. A large carnivore may eat both smaller carnivores and herbivores; the bobcat eats rabbits, but the mountain lion eats both bobcats and rabbits. Animals can also eat each other; the bullfrog eats crayfish and crayfish eat young bullfrogs. The feeding habits of a juvenile animal, and, as a consequence, its trophic level, can change as it grows up.
{\displaystyle TL_{i}=1+\sum _{j}(TL_{j}\cdot DC_{ij})\!}
{\displaystyle TL_{j}}
is the fractional trophic level of the prey j, and
{\displaystyle DC_{ij}}
represents the fraction of j in the diet of i. That is, the consumer trophic level is one plus the weighted average of how much different trophic levels contribute to its food.
In the case of marine ecosystems, the trophic level of most fish and other marine consumers takes a value between 2.0 and 5.0. The upper value, 5.0, is unusual, even for large fish,[16] though it occurs in apex predators of marine mammals, such as polar bears and orcas.[17]
In addition to observational studies of animal behavior, and quantification of animal stomach contents, trophic level can be quantified through stable isotope analysis of animal tissues such as muscle, skin, hair, bone collagen. This is because there is a consistent increase in the nitrogen isotopic composition at each trophic level caused by fractionations that occur with the synthesis of biomolecules; the magnitude of this increase in nitrogen isotopic composition is approximately 3–4‰.[18][19]
{\displaystyle TL_{y}={\frac {\sum _{i}(TL_{i}\cdot Y_{iy})}{\sum _{i}Y_{iy}}}}
{\displaystyle Y_{iy}}
is the catch of the species or group i in year y, and
{\displaystyle TL_{i}}
is the trophic level for species i as defined above.[8]
Humans have a mean trophic level of about 2.21, about the same as a pig or an anchovy.[24][25]
FiB indexEdit
Since biomass transfer efficiencies are only about 10%, it follows that the rate of biological production is much greater at lower trophic levels than it is at higher levels. Fisheries catch, at least, to begin with, will tend to increase as the trophic level declines. At this point the fisheries will target species lower in the food web.[23] In 2000, this led Pauly and others to construct a "Fisheries in Balance" index, usually called the FiB index.[26] The FiB index is defined, for any year y, by[8]
{\displaystyle FiB_{y}=\log {\frac {Y_{y}/(TE)^{TL_{y}}}{Y_{0}/(TE)^{TL_{0}}}}}
{\displaystyle Y_{y}}
is the catch at year y,
{\displaystyle TL_{y}}
is the mean trophic level of the catch at year y,
{\displaystyle Y_{0}}
is the catch,
{\displaystyle TL_{0}}
the mean trophic level of the catch at the start of the series being analyzed, and
{\displaystyle TE}
is the transfer efficiency of biomass or energy between trophic levels.
The FiB index is stable (zero) over periods of time when changes in trophic levels are matched by appropriate changes in the catch in the opposite direction. The index increases if catches increase for any reason, e.g. higher fish biomass, or geographic expansion.[8] Such decreases explain the "backward-bending" plots of trophic level versus catch originally observed by Pauly and others in 1998.[23]
Tritrophic and other interactionsEdit
One aspect of trophic levels is called tritrophic interaction. Ecologists often restrict their research to two trophic levels as a way of simplifying the analysis; however, this can be misleading if tritrophic interactions (such as plant–herbivore–predator) are not easily understood by simply adding pairwise interactions (plant-herbivore plus herbivore–predator, for example). Significant interactions can occur between the first trophic level (plant) and the third trophic level (a predator) in determining herbivore population growth, for example. Simple genetic changes may yield morphological variants in plants that then differ in their resistance to herbivores because of the effects of the plant architecture on enemies of the herbivore.[27] Plants can also develop defenses against herbivores such as chemical defenses.[28]
^ "Definition of Trophic". www.merriam-webster.com. Retrieved 16 April 2017.
^ Lindeman, R. L. (1942). The trophic-dynamic aspect of ecology. Ecology 23: 399–418. link.
^ Heinemann, A. 1926. Der Nahrungskreislauf im Wasser. Verh. deutsch. Zool. Ges., 31: 29-79, link. [Also at: Zool. Anz. Suppl., 2: 29-79.]
^ Science of Earth Systems. Cengage Learning. 2002. p. 537. ISBN 978-0-7668-3391-3.
^ van Dover, Cindy (2000). The Ecology of Deep-sea Hydrothermal Vents. Princeton University Press. p. 399. ISBN 978-0-691-04929-8.
^ a b Lisowski M, Miaoulis I, Cyr M, Jones LC, Padilla MJ, Wellnitz TR (2004) Prentice Hall Science Explorer: Environmental Science, Pearson Prentice Hall. ISBN 978-0-13-115090-4
^ a b c d e Pauly, D.; Palomares, M. L. (2005). "Fishing down marine food webs: it is far more pervasive than we thought" (PDF). Bulletin of Marine Science. 76 (2): 197–211. Archived from the original (PDF) on 14 May 2013.
^ Biodiversity and Morphology: Table 3.5 in Fish on line, Version 3, August 2014. FishBase
^ Yirka, Bob (3 December 2013). "Eating up the world's food web and the human trophic level". Proceedings of the National Academy of Sciences. 110 (51): 20617–20620. Bibcode:2013PNAS..11020617B. doi:10.1073/pnas.1305827110. PMC 3870703. PMID 24297882.
{{cite web |author=Bob Yirka |date=2013-12-03 |title=Researchers calculate human trophic level for first time |website=Phys.org |url=http://phys.org/news/2013-12-human-trophic.html
^ Campbell, Bernard Grant (1 January 1995). Human Ecology: The Story of Our Place in Nature from Prehistory to the Present. p. 12. ISBN 9780202366609.
^ Behrenfeld, Michael J. (2014). "Climate-mediated dance of the plankton". Nature Climate Change. 4 (10): 880–887. Bibcode:2014NatCC...4..880B. doi:10.1038/nclimate2349.
^ a b Pauly, D.; Trites, A.; Capuli, E.; Christensen, V. (1998). "Diet composition and trophic levels of marine mammals". ICES J. Mar. Sci. 55 (3): 467–481. doi:10.1006/jmsc.1997.0280.
^ Gorlova, E. N.; Krylovich, O. A.; Tiunov, A. V.; Khasanov, B. F.; Vasyukov, D. D.; Savinetsk y, A. B. (March 2015). "Stable-Isotope Analysis as a Method of Taxonomical Identification of Archaeozoological Material". Archaeology, Ethnology and Anthropology of Eurasia. 43 (1): 110–121. doi:10.1016/j.aeae.2015.07.013.
^ Millennium Ecosystem Assessment (2005) Ecosystems and Human Well-being: Synthesis Island Press. pp. 32–33.
^ Sethi, S. A.; Branch, T. A.; Watson, R. (2010). "Global fishery development patterns are driven by profit but not trophic level". Proceedings of the National Academy of Sciences. 107 (27): 12163–12167. Bibcode:2010PNAS..10712163S. doi:10.1073/pnas.1003236107. PMC 2901455. PMID 20566867.
^ Branch, T. A.; Watson, Reg; Fulton, Elizabeth A.; Jennings, Simon; McGilliard, Carey R.; Pablico, Grace T.; Ricard, Daniel; Tracey, Sean R. (2010). "Trophic fingerprint of marine fisheries" (PDF). Nature. 468 (7322): 431–435. Bibcode:2010Natur.468..431B. doi:10.1038/nature09528. PMID 21085178. S2CID 4403636. Archived from the original (PDF) on 9 February 2014.
^ a b c Pauly, D; Christensen v, V.; Dalsgaard, J.; Froese, R.; Torres Jr, F. C. Jr (1998). "Fishing down marine food webs". Science. 279 (5352): 860–863. Bibcode:1998Sci...279..860P. doi:10.1126/science.279.5352.860. PMID 9452385. S2CID 272149.
^ "Researchers calculate human trophic level for first time" Phys.org . 3 December 2013.
^ Bonhommeau, S., Dubroca, L., Le Pape, O., Barde, J., Kaplan, D.M., Chassot, E. and Nieblas, A.E. (2013) "Eating up the world's food web and the human trophic level". Proceedings of the National Academy of Sciences, 110(51): 20617–20620. doi:10.1073/pnas.1305827110.
^ Pauly, D.; Christensen, V; Walters, C. (2000). "Ecopath, Ecosim and Ecospace as tools for evaluating ecosystem impact of fisheries". ICES J. Mar. Sci. 57 (3): 697–706. doi:10.1006/jmsc.2000.0726.
^ Kareiva, Peter; Sahakian, Robert (1990). "Letters to Nature:Tritrophic effects of a simple architectural mutation in pea plants". Nature. 35 (6274): 433–434. doi:10.1038/345433a0. S2CID 40207145.
^ Price, P. W. Price; Bouton, C. E.; Gross, P.; McPheron, B. A.; Thompson, J. N.; Weis, A. E. (1980). "Interactions Among Three Trophic Levels: Influence of Plants on Interactions Between Insect Herbivores and Natural Enemies". Annual Review of Ecology and Systematics. 11 (1): 41–65. doi:10.1146/annurev.es.11.110180.000353. S2CID 53137184.
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The ICP22 protein selectively modifies the transcription of different kinetic classes of pseudorabies virus genes | BMC Molecular Biology | Full Text
From: The ICP22 protein selectively modifies the transcription of different kinetic classes of pseudorabies virus genes
The average R r values (
{\overline{\mathbit{R}}}_{\mathbit{r}}\mathbit{=}{\overline{\mathbit{R}}}_{\phantom{\rule{0.25em}{0ex}}\mathbit{\text{us}}\mathbit{1}\mathbf{-}\mathbit{\text{KO}}}/{\overline{\mathbit{R}}}_{\phantom{\rule{0.25em}{0ex}}\mathbit{\text{wt}}}
). The thick solid line depicts the
{\overline{R}}_{r}
values of the total genes, and the thinner lines the
{\overline{R}}_{r}
values of the different kinetic classes of viral genes.
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Before they let you into the dashboard and secure work, you must take a practice exam.
The number of tests you can see is determined by a star ranking system. To earn the most exams, you should aim to keep a 5-star rating. You may make a lot of money here if you are quick to accept projects.
Applying to be a contributor is an excellent method to supplement your income. Companies seeking contributors with your experience will determine the number of possibilities you receive. For every 5-minute test, you may earn
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Serpclix is a platform that pays between
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0.10 per click if you just Google websites and click through to their webpage. It's not overflowing with labor, but given how simple it is, it's worth keeping open.
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OnePlus 10 Pro has been launched with the most powerful Snapdragon processor
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Brazilian cruzado - Wikipedia
"Cruzado" redirects here. For other uses, see Cruzado (disambiguation).
A 10,000 cruzado banknote featuring Carlos Chagas
10, 50, 100, 500, 1000, 5000 and 10,000 cruzados
1, 5 and 10 cruzados
Cruzeiro (2nd version)
The cruzado was the currency of Brazil from 1986 to 1989. It replaced the second cruzeiro (at first called the "cruzeiro novo") in 1986, at a rate of 1 cruzado = 1000 cruzeiros (novos) and was replaced in 1989 by the cruzado novo at a rate of 1000 cruzados = 1 cruzado novo.
This currency was subdivided in 100 centavos and it had the symbol
{\displaystyle \mathrm {CzS} \!\!\!\Vert }
and the ISO 4217 code BRC.
Stainless-steel coins were introduced in 1986 in denominations of 1, 5, 10, 20 and 50 centavos, and 1 and 5 cruzados, with 10 cruzados following in 1987. Coin production ceased in 1988.
Coins of the Cruzado
Cz$0.01
Cz$1
Cz$10
Three designs of commemorative 100 cruzado coins, celebrating the 100th anniversary of the abolition of slavery in the country (the Lei Áurea), were produced in 1988. Although very rare in circulation, the numbers' design was carried over into both Cruzado Novo and the third Cruzeiro.
Commemorative coins of the Cruzado
(missing photo) (missing photo) Cz$100 Design portraying a man, with the word "Axé" and the words "Centenário da Abolição" ("100th anniversary of the abolition"). The years 1888 and 1988 are also inscribed.
(missing photo) (missing photo) Cz$100 Design portraying a woman, with the word "Axé" and the words "Centenário da Abolição". The years 1888 and 1988 are also inscribed.
Cz$100 Design portraying a child, with the word "Axé" and the words "Centenário da Abolição". The years 1888 and 1988 are also inscribed.
Main article: Banknotes of the Brazilian cruzado
The first banknotes were overprints on cruzeiro notes, in denominations of 10, 50 and 100 cruzados. Regular notes followed in denominations of 10, 50, 100 and 500 cruzados, followed by 1000 cruzados in 1987, 5000 and 10,000 cruzados in 1988.
Ratio: 1 cruzado = 1000 cruzeiros Currency of Brazil
28 February 1986 – 15 January 1989 Succeeded by:
Ratio: 1 cruzado novo = 1000 cruzados
Retrieved from "https://en.wikipedia.org/w/index.php?title=Brazilian_cruzado&oldid=1079282641"
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Orientation from magnetometer and accelerometer readings - MATLAB ecompass - MathWorks Switzerland
Determine Declination of Boston
Return Rotation Matrix
Determine Gravity Vector
Track Spinning Platform
Orientation from magnetometer and accelerometer readings
orientation = ecompass(accelerometerReading,magnetometerReading)
orientation = ecompass(accelerometerReading,magnetometerReading,orientationFormat)
orientation = ecompass(accelerometerReading,magnetometerReading,orientationFormat,'ReferenceFrame',RF)
orientation = ecompass(accelerometerReading,magnetometerReading) returns a quaternion that can rotate quantities from a parent (NED) frame to a child (sensor) frame.
orientation = ecompass(accelerometerReading,magnetometerReading,orientationFormat) specifies the orientation format as quaternion or rotation matrix.
orientation = ecompass(accelerometerReading,magnetometerReading,orientationFormat,'ReferenceFrame',RF) also allows you to specify the reference frame RF of the orientation output. Specify RF as 'NED' (North-East-Down) or 'ENU' (East-North-Up). The default value is 'NED'.
Use the known magnetic field strength and proper acceleration of a device pointed true north in Boston to determine the magnetic declination of Boston.
Define the known acceleration and magnetic field strength in Boston.
magneticFieldStrength = [19.535 -5.109 47.930];
properAcceleration = [0 0 9.8];
Pass the magnetic field strength and acceleration to the ecompass function. The ecompass function returns a quaternion rotation operator. Convert the quaternion to Euler angles in degrees.
q = ecompass(properAcceleration,magneticFieldStrength);
e = eulerd(q,'ZYX','frame');
The angle, e, represents the angle between true north and magnetic north in Boston. By convention, magnetic declination is negative when magnetic north is west of true north. Negate the angle to determine the magnetic declination.
magneticDeclinationOfBoston = -e(1)
magneticDeclinationOfBoston = -14.6563
The ecompass function fuses magnetometer and accelerometer data to return a quaternion that, when used within a quaternion rotation operator, can rotate quantities from a parent (NED) frame to a child frame. The ecompass function can also return rotation matrices that perform equivalent rotations as the quaternion operator.
Define a rotation that can take a parent frame pointing to magnetic north to a child frame pointing to geographic north. Define the rotation as both a quaternion and a rotation matrix. Then, convert the quaternion and rotation matrix to Euler angles in degrees for comparison.
Define the magnetic field strength in microteslas in Boston, MA, when pointed true north.
m = [19.535 -5.109 47.930];
a = [0 0 9.8];
Determine the quaternion and rotation matrix that is capable of rotating a frame from magnetic north to true north. Display the results for comparison.
q = ecompass(a,m);
quaterionEulerAngles = eulerd(q,'ZYX','frame')
quaterionEulerAngles = 1×3
r = ecompass(a,m,'rotmat');
theta = -asin(r(1,3));
psi = atan2(r(2,3)/cos(theta),r(3,3)/cos(theta));
rho = atan2(r(1,2)/cos(theta),r(1,1)/cos(theta));
rotmatEulerAngles = rad2deg([rho,theta,psi])
rotmatEulerAngles = 1×3
Use ecompass to determine the gravity vector based on data from a rotating IMU.
Load the inertial measurement unit (IMU) data.
Determine the orientation of the sensor body relative to the local NED frame over time.
orientation = ecompass(sensorData.Acceleration,sensorData.MagneticField);
To estimate the gravity vector, first rotate the accelerometer readings from the sensor body frame to the NED frame using the orientation quaternion vector.
gravityVectors = rotatepoint(orientation,sensorData.Acceleration);
Determine the gravity vector as an average of the recovered gravity vectors over time.
gravityVectorEstimate = mean(gravityVectors,1)
gravityVectorEstimate = 1×3
0.0000 -0.0000 10.2102
Fuse modeled accelerometer and gyroscope data to track a spinning platform using both idealized and realistic data.
Generate Ground-Truth Trajectory
Describe the ground-truth orientation of the platform over time. Use the kinematicTrajectory System object™ to create a trajectory for a platform that has no translation and spins about its z-axis.
accelerationBody = zeros(numSamples,3);
angularVelocityBody = zeros(numSamples,3);
zAxisAngularVelocity = [linspace(0,4*pi,4*fs),4*pi*ones(1,4*fs),linspace(4*pi,0,4*fs)]';
angularVelocityBody(:,3) = zAxisAngularVelocity;
trajectory = kinematicTrajectory('SampleRate',fs);
[~,orientationNED,~,accelerationNED,angularVelocityNED] = trajectory(accelerationBody,angularVelocityBody);
Model Receiving IMU Data
Use an imuSensor System object to mimic data received from an IMU that contains an ideal magnetometer and an ideal accelerometer.
[accelerometerData,magnetometerData] = IMU(accelerationNED, ...
angularVelocityNED, ...
orientationNED);
Fuse IMU Data to Estimate Orientation
Pass the accelerometer data and magnetometer data to the ecompass function to estimate orientation over time. Convert the orientation to Euler angles in degrees and plot the result.
orientation = ecompass(accelerometerData,magnetometerData);
timeVector = (0:numSamples-1).'/fs;
plot(timeVector,orientationEuler)
title('Orientation from Ideal IMU')
Repeat Experiment with Realistic IMU Sensor Model
Modify parameters of the IMU System object to approximate realistic IMU sensor data. Reset the IMU and then call it with the same ground-truth acceleration, angular velocity, and orientation. Use ecompass to fuse the IMU data and plot the results.
'MeasurementRange',20, ...
'Resolution',0.0006, ...
'ConstantBias',0.5, ...
'NoiseDensity',0.004, ...
'BiasInstability',0.5);
'MeasurementRange',200, ...
'Resolution',0.01);
reset(IMU)
[accelerometerData,magnetometerData] = IMU(accelerationNED,angularVelocityNED,orientationNED);
title('Orientation from Realistic IMU')
accelerometerReading — Accelerometer readings in sensor body coordinate system (m/s2)
Accelerometer readings in sensor body coordinate system in m/s2, specified as an N-by-3 matrix. The columns of the matrix correspond to the x-, y-, and z-axes of the sensor body. The rows in the matrix, N, correspond to individual samples. The accelerometer readings are normalized before use in the function.
magnetometerReading — Magnetometer readings in sensor body coordinate system (µT)
Magnetometer readings in sensor body coordinate system in µT, specified as an N-by-3 matrix. The columns of the matrix correspond to the x-, y-, and z-axes of the sensor body. The rows in the matrix, N, correspond to individual samples. The magnetometer readings are normalized before use in the function.
orientationFormat — Format used to describe orientation
Format used to describe orientation, specified as 'quaternion' or 'rotmat'.
N-by-1 vector of quaternions (default) | 3-by-3-by-N array
Orientation that can rotate quantities from a global coordinate system to a body coordinate system, returned as a vector of quaternions or an array. The size and type of the orientation depends on the format used to describe orientation:
'quaternion' –– N-by-1 vector of quaternions with the same underlying data type as the input
'rotmat' –– 3-by-3-by-N array the same data type as the input
The ecompass function returns a quaternion or rotation matrix that can rotate quantities from a parent (NED for example) frame to a child (sensor) frame. For both output orientation formats, the rotation operator is determined by computing the rotation matrix.
The rotation matrix is first calculated with an intermediary:
R=\left[\begin{array}{ccc}& & \\ \left(a×m\right)×a& a×m& a\\ & & \end{array}\right]
and then normalized column-wise. a and m are the accelerometerReading input and the magnetometerReading input, respectively.
To understand the rotation matrix calculation, consider an arbitrary point on the Earth and its corresponding local NED frame. Assume a sensor body frame, [x,y,z], with the same origin.
Recall that orientation of a sensor body is defined as the rotation operator (rotation matrix or quaternion) required to rotate a quantity from a parent (NED) frame to a child (sensor body) frame:
\left[\begin{array}{ccc}& & \\ & R& \\ & & \end{array}\right]\left[\begin{array}{c}\\ {p}_{\text{parent}}\\ \end{array}\right]=\left[\begin{array}{c}\\ {p}_{\text{child}}\\ \end{array}\right]
R is a 3-by-3 rotation matrix, which can be interpreted as the orientation of the child frame.
pparent is a 3-by-1 vector in the parent frame.
pchild is a 3-by-1 vector in the child frame.
For a stable sensor body, an accelerometer returns the acceleration due to gravity. If the sensor body is perfectly aligned with the NED coordinate system, all acceleration due to gravity is along the z-axis, and the accelerometer reads [0 0 1]. Consider the rotation matrix required to rotate a quantity from the NED coordinate system to a quantity indicated by the accelerometer.
\left[\begin{array}{ccc}{r}_{11}& {r}_{21}& {r}_{31}\\ {r}_{12}& {r}_{22}& {r}_{32}\\ {r}_{13}& {r}_{23}& {r}_{33}\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]=\left[\begin{array}{c}{a}_{1}\\ {a}_{2}\\ {a}_{3}\end{array}\right]
The third column of the rotation matrix corresponds to the accelerometer reading:
\left[\begin{array}{c}{r}_{31}\\ {r}_{32}\\ {r}_{33}\end{array}\right]=\left[\begin{array}{c}{a}_{1}\\ {a}_{2}\\ {a}_{3}\end{array}\right]
A magnetometer reading points toward magnetic north and is in the N-D plane. Again, consider a sensor body frame aligned with the NED coordinate system.
By definition, the E-axis is perpendicular to the N-D plane, therefore N ⨯ D = E, within some amplitude scaling. If the sensor body frame is aligned with NED, both the acceleration vector from the accelerometer and the magnetic field vector from the magnetometer lie in the N-D plane. Therefore m ⨯ a = y, again with some amplitude scaling.
Consider the rotation matrix required to rotate NED to the child frame, [x y z].
\left[\begin{array}{ccc}{r}_{11}& {r}_{21}& {r}_{31}\\ {r}_{12}& {r}_{22}& {r}_{32}\\ {r}_{13}& {r}_{23}& {r}_{33}\end{array}\right]\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]=\left[\begin{array}{c}{a}_{1}\\ {a}_{2}\\ {a}_{3}\end{array}\right]×\left[\begin{array}{c}{m}_{1}\\ {m}_{2}\\ {m}_{3}\end{array}\right]
The second column of the rotation matrix corresponds to the cross product of the accelerometer reading and the magnetometer reading:
\left[\begin{array}{c}{r}_{21}\\ {r}_{22}\\ {r}_{23}\end{array}\right]=\left[\begin{array}{c}{a}_{1}\\ {a}_{2}\\ {a}_{3}\end{array}\right]×\left[\begin{array}{c}{m}_{1}\\ {m}_{2}\\ {m}_{3}\end{array}\right]
By definition of a rotation matrix, column 1 is the cross product of columns 2 and 3:
\begin{array}{l}\left[\begin{array}{c}{r}_{11}\\ {r}_{12}\\ {r}_{13}\end{array}\right]=\left[\begin{array}{c}{r}_{21}\\ {r}_{22}\\ {r}_{23}\end{array}\right]×\left[\begin{array}{c}{r}_{31}\\ {r}_{32}\\ {r}_{33}\end{array}\right]\\ \text{ }\text{ }\text{ }=\left(a×m\right)×a\end{array}
Finally, the rotation matrix is normalized column-wise:
{R}_{ij}=\frac{{R}_{ij}}{\sqrt{\sum _{i=1}^{3}{R}_{ij}^{2}}}\text{ },\text{\hspace{0.17em}}\forall j
The ecompass algorithm uses magnetic north, not true north, for the NED coordinate system.
quaternion | ahrsfilter | imufilter
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Constructing and Working with stform Splines - MATLAB & Simulink - MathWorks Italia
Constructing and Working with stform Splines
Introduction to the stform
Construction and Properties of the stform
Working with the stform
A multivariate function form quite different from the tensor-product construct is the scattered translates form, or stform for short. As the name suggests, it uses arbitrary or scattered translates ψ(· –cj) of one fixed function ψ, in addition to some polynomial terms. Explicitly, such a form describes a function
f\left(x\right)=\sum _{j=1}^{n-k}\psi \left(x-{c}_{j}\right){a}_{j}+p\left(x\right)
in terms of the basis function ψ, a sequence (cj) of sites called centers and a corresponding sequence (aj) of n coefficients, with the final k coefficients, an-k+1,...,an, involved in the polynomial part, p.
When the basis function is radially symmetric, meaning that ψ(x) depends only on the Euclidean length |x| of its argument, x, then ψ is called a radial basis function, and, correspondingly, f is then often called an RBF.
At present, the toolbox works with just one kind of stform, namely a bivariate thin-plate spline and its first partial derivatives. For the thin-plate spline, the basis function is ψ(x) = φ(|x|2), with φ(t) = tlogt, i.e., a radial basis function. Its polynomial part is a linear polynomial, i.e., p(x)=x(1)an – 2+x(2)an – 1+an. The first partial derivative with respect to its first argument uses, correspondingly, the basis function ψ(x)=φ(|x|2), with φ(t) = (D1t)·(logt+1) and D1t = D1t(x) = 2x(1), and p(x) = an.
A function in stform can be put together from its center sequence centers and its coefficient sequence coefs by the command
f = stmak(centers, coefs, type);
where type can be specified as one of 'tp00', 'tp10', 'tp01', to indicate, respectively, a thin-plate spline, a first partial of a thin-plate spline with respect to the first argument, and a first partial of a thin-plate spline with respect to the second argument. There is one other choice, 'tp'; it denotes a thin-plate spline without any polynomial part and is likely to be used only during the construction of a thin-plate spline, as in tpaps.
A function f in stform depends linearly on its coefficients, meaning that
f\left(x\right)=\sum _{j=1}^{n}{\psi }_{j}\left(x\right){a}_{j}
with ψj either a translate of the basis function Ψ or else some polynomial. Suppose you wanted to determine these coefficients aj so that the function f matches prescribed values at prescribed sites xi. Then you would need the collocation matrix (ψj(xi)). You can obtain this matrix by the command stcol(centers,x,type). In fact, because the stform has aj as the jth column, coefs(:,j), of its coefficient array, it is worth noting that stcol can also supply the transpose of the collocation matrix. Thus, the command
values = coefs*stcol(centers,x,type,'tr');
would provide the values at the entries of x of the st function specified by centers and type.
The stform is attractive because, in contrast to piecewise polynomial forms, its complexity is the same in any number of variables. It is quite simple, yet, because of the complete freedom in the choice of centers, very flexible and adaptable.
On the negative side, the most attractive choices for a radial basis function share with the thin-plate spline that the evaluation at any site involves all coefficients. For example, plotting a scalar-valued thin-plate spline via fnplt involves evaluation at a 51-by-51 grid of sites, a nontrivial task when there are 1000 coefficients or more. The situation is worse when you want to determine these 1000 coefficients so as to obtain the stform of a function that matches function values at 1000 data sites, as this calls for solving a full linear system of order 1000, a task requiring O(10^9) flops if done by a direct method. Just the construction of the collocation matrix for this linear system (by stcol) takes O(10^6) flops.
The command tpaps, which constructs thin-plate spline interpolants and approximants, uses iterative methods when there are more than 728 data points, but convergence of such iteration may be slow.
After you have constructed an approximating or interpolating thin-plate spline st with the aid of tpaps (or directly via stmak), you can use the following commands:
fnbrk to obtain its parts or change its basic interval,
fnval to evaluate it
fnplt to plot it
fnder to construct its two first partial derivatives, but no higher order derivatives as they become infinite at the centers.
This is just one indication that the stform is quite different in nature from the other forms in this toolbox, hence other fn... commands by and large don't work with stforms. For example, it makes no sense to use fnjmp, and fnmin or fnzeros only work for univariate functions. It also makes no sense to use fnint on a function in stform because such functions cannot be integrated in closed form.
The command Ast = fncmb(st,A) can be used on st, provided A is something that can be applied to the values of the function described by st. For example, A might be 'sin', in which case Ast is the stform of the function whose coefficients are the sine of the coefficients of st. In effect, Ast describes the function obtained by composing A with st. But, because of the singularities in the higher-order derivatives of a thin-plate spline, there seems little point to make fndir or fntlr applicable to such a st.
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Formal logical reconstructions of theoretical propositions attempting to draw a line between science and metaphysics
Ramsey sentences are formal logical reconstructions of theoretical propositions attempting to draw a line between science and metaphysics. A Ramsey sentence aims at rendering propositions containing non-observable theoretical terms (terms employed by a theoretical language) clear by substituting them with observational terms (terms employed by an observation language, also called empirical language).
Ramsey sentences were introduced by the logical empiricist philosopher Rudolf Carnap. However, they should not be confused with Carnap sentences, which are neutral on whether there exists anything to which the term applies. [1]
Distinction between scientific (real) questions and metaphysical (pseudo-)questions
For Carnap, questions such as: “Are electrons real?” and: “Can you prove electrons are real?” were not legitimate questions implying great philosophical/metaphysical import. They were meaningless "pseudo-questions without cognitive content,” asked from outside a language framework of science. Inside this framework, entities such as electrons or sound waves, and relations such as mass and force not only exist and have meaning, but are "useful" to the scientists who work with them. To accommodate such internal questions in a way that would justify their theoretical content empirically – and to do so while maintaining a distinction between analytic and synthetic propositions – Carnap set out to develop a systematized way to consolidate theory and empirical observation in a meaningful language formula.
Distinction between observable and non-observable
Carnap began by differentiating observable things from non-observable things. Immediately, a problem arises: neither the German nor the English language naturally distinguish predicate terms on the basis of an observational categorization. As Carnap admitted, "The line separating observable from non-observable is highly arbitrary." For example, the predicate "hot" can be perceived by touching a hand to a lighted coal. But "hot" might take place at such a microlevel (e.g., the theoretical "heat" generated by the production of proteins in a eukaryotic cell) that it is virtually non-observable (at present). Physicist-philosopher Moritz Schlick characterized the difference linguistically, as the difference between the German verbs "kennen" (knowing as being acquainted with a thing – perception) and "erkennen" (knowing as understanding a thing – even if non-observable). This linguistic distinction may explain Carnap's decision to divide the vocabulary into two artificial categories: a vocabulary of non-observable ("theoretical") terms (hereafter "VT"): i.e., terms we know of but are not acquainted with (erkennen), and a vocabulary of observable terms ("VO"), those terms we are acquainted with (kennen) and will accept arbitrarily. Accordingly, the terms thus distinguished were incorporated into comparable sentence structures: T-terms into theoretical sentences (T-sentences); O-terms into observational sentences (O-sentences).
The next step for Carnap was to connect these separate concepts by what he calls "correspondence rules" (C-rules), which are "mixed" sentences containing both T- and O-terms. Such a theory can be formulated as: T + C = df: the conjunction of T-postulates + the conjunction of C-rules – i.e.,
{\displaystyle [(T_{1}\land T_{2}\land \cdots \land T_{n})+(C_{1}\land C_{2}\land \cdots \land C_{m})]}
. This can be further expanded to include class terms such as for the class of all molecules, relations such as "betweenness," and predicates: e.g., TC ( t1, t2, . . ., tn, o1, o2, . . ., om). Though this enabled Carnap to establish what it means for a theory to be "empirical," this sentence neither defines the T-terms explicitly nor draws any distinction between its analytic and its synthetic content, therefore it was not yet sufficient for Carnap's purposes.
In the theories of Frank P. Ramsey, Carnap found the method he needed to take the next step, which was to substitute variables for each T-term, then to quantify existentially all T-terms in both T-sentences and C-rules. The resulting "Ramsey sentence" effectively eliminated the T-terms as such, while still providing an account of the theory's empirical content. The evolution of the formula proceeds thus:
Step 1 (empirical theory, assumed true): TC ( t1 . . . tn, o1 . . . om)
Step 2 (substitution of variables for T-terms): TC (x1 . . . xn, o1 . . . om)
Step 3 (
{\displaystyle \exists }
-quantification of the variables):
{\displaystyle \exists x_{1}\ldots \exists x_{n}TC(x_{1}\ldots x_{n},o_{1}\ldots o_{m})}
Step 3 is the complete Ramsey sentence, expressed "RTC," and to be read: "There are some (unspecified) relations such that TC (x1 . . . xn, o1 . . . om) is satisfied when the variables are assigned these relations. (This is equivalent to an interpretation as an appropriate model: there are relations r1 . . . rn such that TC (x1 . . . xn, o1 . . . om) is satisfied when xi is assigned the value ri, and
{\displaystyle 1\leq i\leq m}
In this form, the Ramsey sentence captures the factual content of the theory. Though Ramsey believed this formulation was adequate to the needs of science, Carnap disagreed, with regard to a comprehensive reconstruction. In order to delineate a distinction between analytic and synthetic content, Carnap thought the reconstructed sentence would have to satisfy three desired requirements:
The factual (FT) component must be observationally equivalent to the original theory (TC).
The analytic (AT) component must be observationally uninformative.
The combination of FT and AT must be logically equivalent to the original theory – that is,
{\displaystyle F_{T}+A_{T}\Leftrightarrow TC}
Requirement 1 is satisfied by RTC in that the existential quantification of the T-terms does not change the logical truth (L-truth) of either statement, and the reconstruction FT has the same O-sentences as the theory itself, hence RTC is observationally equivalent to TC : (i.e., for every O-sentence: O,
{\displaystyle [TC\models O\Leftrightarrow ^{R}TC\models O]}
). As stated, however, requirements 2 and 3 remain unsatisfied. That is, taken individually, AT does contain observational information (such-and-such a theoretical entity is observed to do such-and-such, or hold such-and-such a relation); and AT does not necessarily follow from FT.
Carnap's solution is to make the two statements conditional. If there are some relations such that [TC (x1 . . . xn, o1 . . . om)] is satisfied when the variables are assigned some relations, then the relations assigned to those variables by the original theory will satisfy [TC (t1 . . . tn, o1 . . . om)] – or: RTC → TC. This important move satisfies both remaining requirements and effectively creates a distinction between the total formula's analytic and synthetic components. Specifically, for requirement 2: The conditional sentence does not make any information claim about the O-sentences in TC, it states only that "if" the variables in are satisfied by the relations, "then" the O-sentences will be true. This means that every O-sentence in TC that is logically implied by the sentence RTC → TC is L-true (i.e., every O-sentence in AT is true or not-true: the metal expands or it does not; the chemical turns blue or it does not, etc.). Thus TC can be taken as the non-informative (i.e., non-factual) component of the statement, or AT. Requirement 3 is satisfied by inference: given AT, infer FT → AT. This makes AT + FT nothing more than a reformulation of the original theory, hence AT Ù FT ó TC.
Carnap took as a fundamental requirement a respect for the analytic–synthetic distinction. This is met by using two distinct processes in the formulation: drawing an empirical connection between the statement's factual content and the original theory (observational equivalence), and by requiring the analytic content to be observationally non-informative.
Carnap's reconstruction as it is given here is not intended to be a literal method for formulating scientific propositions. To capture what Pierre Duhem would call the entire "holistic" universe relating to any specified theory would require long and complicated renderings of RTC → TC. Instead, it is to be taken as demonstrating logically that there is a way that science could formulate empirical, observational explications of theoretical concepts – and in that context the Ramsey and Carnap construct can be said to provide a formal justificatory distinction between scientific observation and metaphysical inquiry.
Among critics of the Ramsey formalism are John Winnie, who extended the requirements to include an "observationally non-creative" restriction on Carnap's AT – and both W. V. O. Quine and Carl Hempel attacked Carnap's initial assumptions by emphasizing the ambiguity that persists between observable and non-observable terms.
Ramsey-style epistemic structural realism
^ David Lewis (1970). "How to Define Theoretical Terms". The Journal of Philosophy. 67 (13): 427–446.
Carnap, R. (1950) "Empiricism, Semantics, and Ontology," in Paul Moser and Arnold Nat, Human Knowledge Oxford University Press. (2003).
Carnap, R. (1966) An Introduction to the Philosophy of Science (esp. Parts III, and V), ed. Martin Gardner. Dover Publications, New York. 1995.
Carnap, R. (2000) [originally: 29 December 1959] "Theoretical Concepts in Science," with introduction by Stathis Psillos. Studies in History and Philosophy of Science 31(1).
Demopoulos, W. "Carnap on the Reconstruction of Scientific Theories," The Cambridge Companion to Carnap, eds. R. Creath and M. Friedman.
Moser, P. K. and vander Nat, A. (2003) Human Knowledge Oxford Univ. Press.
Schlick, Moritz (1918) General Theory of Knowledge (Allegemeine Erkenntnislehre). Trans. Albert Blumberg. Open Court Publishing, Chicago/La Salle, IL. (2002).
Hallvard Lillehammer, D. H. Mellor (2005), Ramsey's legacy, Oxford University Press, p. 109.
Stathis Psillos, "Carnap, the Ramsey-Sentence and Realistic Empiricism", 2000.
"Epistemic Structural Realism and Ramsey Sentences"
"Theoretical Terms in Science"
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What is e on a calculator? – e to the x
How to put e in a calculator? Calculate e to the x
e calculator – examples
Are you solving an equation with Euler's number? Our e calculator is here to help! Our tool allows you to compute e to the power of any number you desire.
Keep on reading if you're still wondering what exactly Euler's number is, what does e mean on a calculator, and how to calculate e to the x 📐🧑🏫
e is one of the most important constants in mathematics. We cannot write e as a fraction, and it has an infinite number of decimal places – just like its famous cousin, pi (π).
e has plenty of names in mathematics. We may know it as Euler's number or the natural number. Its value is equal to 2.7182818284590452353602... and counting! (This is where rounding and approximation become essential.) 🧮
Now that we know what e and its approximate value is, we can start thinking about its possible applications.
We use e in the natural exponential function (eˣ = e power x).
In the eˣ function, the slope of the tangent line to any point on the graph is equal to its y-coordinate at that point.
(1 + 1/n)ⁿ is the sequence that we use to estimate the value of e. The sequence gets closer to e the larger n is - but even if n = infinity, the sequence value is not equal to Euler's number.
We use this equation in compound interest calculations.
e is equal to the result of the following factorial equation:
\scriptsize \qquad \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + ...
e is also a part of the most beautiful equation in mathematics:
e^{iπ} + 1 = 0
Since we already know what's the Euler's number, how about some other numbers we use in physics?
Biot number;
Knudsen number;
Avogadro's number;
Reynolds number; and
f-number 😀
Since we're forced to use an approximation of e, we can simply input the value of e into any calculator.
How does it work in practice? How to calculate e to the power x?
If your calculator doesn't allow symbols, simply enter 2.718281828 (or any rounded form of this number) into your choice value box 👍
In this section, we'll answer the very big question: "How to calculate e to the power eˣ?" using both our calculator and the traditional formula.
The e calculator – it's so simple it doesn't need further explanation. Enter the value of x into the text box and enjoy your results displayed alongside the step-by-step solution 👣
The traditional calculation requires you to choose how many decimal places of the Euler's number you will use.
We decided to use 9 decimal places.
Let's follow an example:
We know that the area up to any x-value is also equal to eˣ:
We'd love to calculate the area up to the e¹⁰ function.
e¹⁰ = 2.718281828¹⁰
2.718281828¹⁰ = 2.718281828 * 2.718281828 * 2.718281828 * ...
2.718281828¹⁰ = 22026.47
And this is how to calculate e to the power of 10.
As you can see, calculating e to the power of x might be pretty troublesome and time-consuming - our tool is a simple solution for that unnecessary problem 🤗
"Exp" is short for "exponential" and is used in the notation exp(x) as another way to write eˣ.
How do you calculate e to the power x without calculator?
You can use the following Taylor series approximation: eˣ = 1 + x + x²/2! + x³/3! + .... Continue calculating and adding terms to get a better approximation.
What is e to the negative infinity?
Zero. Let's say we have e⁻ᴺ, where N is a large number tending toward infinity. Now, given that e⁻ᴺ = 1/eᴺ, as N gets larger, e⁻ᴺ will get smaller, ending up at zero if N = ∞.
What is the derivative of e to the x?
The derivative of eˣ is itself, eˣ. Here is a step-by-step proof:
The equation y = eˣ can be rewritten as ln y = x.
Differentiate both sides of this equation and use the chain rule:
1/y • dy/dx = 1
Since y = eˣ, therefore dy/dx = eˣ.
Result (eˣ)
Learn a different and interesting way to write your arithmetic operations with our Polish notation converter.
Polish Notation Converter
This Pythagorean triples calculator can check if three given numbers form a Pythagorean triple and also generate Pythagorean triples via Euclid's formula!
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Transfer function to coupled allpass - MATLAB tf2ca - MathWorks India
Transfer function to coupled allpass
[d1,d2] = tf2ca(b,a)
[d1,d2,beta] = tf2ca(b,a)
[d1,d2] = tf2ca(b,a) where b is a real, symmetric vector of numerator coefficients and a is a real vector of denominator coefficients, corresponding to a stable digital filter, returns real vectors d1 and d2 containing the denominator coefficients of the allpass filters H1(z) and H2(z) such that
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\left(\frac{1}{2}\right)\left[H1\left(z\right)+H2\left(z\right)\right]
representing a coupled allpass decomposition.
[d1,d2] = tf2ca(b,a) where b is a real, antisymmetric vector of numerator coefficients and a is a real vector of denominator coefficients, corresponding to a stable digital filter, returns real vectors d1 and d2 containing the denominator coefficients of the allpass filters H1(z) and H2(z) such that
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\left(\frac{1}{2}\right)\left[H1\left(z\right)-H2\left(z\right)\right]
In some cases, the decomposition is not possible with real H1(z) and H2(z). In those cases a generalized coupled allpass decomposition may be possible, as described in the following syntax.
[d1,d2,beta] = tf2ca(b,a) returns complex vectors d1 and d2 containing the denominator coefficients of the allpass filters H1(z) and H2(z), and a complex scalar beta, satisfying |beta| = 1, such that
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\left(\frac{1}{2}\right)\left[\overline{\beta }\cdot H1\left(z\right)+\beta \cdot H2\left(z\right)\right]
representing the generalized allpass decomposition.
In the above equations, H1(z) and H2(z) are real or complex allpass IIR filters given by
H1\left(z\right)=\frac{fliplr\left(\overline{\left(D1\left(z\right)\right)}\right)}{D1\left(z\right)},H2\left(1\right)\left(z\right)=\frac{fliplr\left(\overline{\left(D2\left(1\right)\left(z\right)}\right)\right)}{D2\left(1\right)\left(z\right)}
where D1(z) and D2(z) are polynomials whose coefficients are given by d1 and d2.
A coupled allpass decomposition is not always possible. Nevertheless, Butterworth, Chebyshev, and Elliptic IIR filters, among others, can be factored in this manner. For details, refer to Signal Processing Toolbox™ User's Guide.
[d1,d2]=tf2ca(b,a); % TF2CA returns denominators of the allpass.
num = 0.5*conv(fliplr(d1),d2)+0.5*conv(fliplr(d2),d1);
den = conv(d1,d2); % Reconstruct numerator and denonimator.
ca2tf | cl2tf | iirpowcomp | latc2tf | tf2latc
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How Measurements Are Made - Course Hero
General Chemistry/Measurement/How Measurements Are Made
Units of measurement must be based on objective, physical, unchanging standards.
In every scientific discipline, scientists must have a way to make quantitative observations that can be communicated to other scientists. These quantitative observations are normally called measurements. Familiar measurements include distance, volume, mass, time, and temperature.
In order for the measurements to be understood by anyone and everyone, there must be a system that defines the size of standard units of measurement. The size of a given standard unit is arbitrary, but it works as long as everyone agrees on what it is. The standard unit of length, for example, is called a meter (m), and it is defined as the length of the path traveled by light in a vacuum during a time interval of 1/299,792,458 of a second. There is no particular reason requiring a meter to be this length; it could just as easily be defined as the distance light travels in 1/300,000,000 of a second. The key is that the scientific community must agree on what length will be called 1 meter and that there be an objective, unchanging reference for that distance. In this way, scientific measurements can be shared and communicated, understood by all.
For example, the length of a meter was previously defined as the distance between two lines marked on a bar of platinum-iridium alloy kept at the International Bureau of Weights and Measures (BIPM, Bureau International des Poids et Mesures) in France. The rod was kept in a carefully controlled atmosphere, supported on cylinders placed exactly 571 mm from each other. These precautions were taken to preserve the bar and keep it from being deformed in any way, because any change to the length of the rod would have then redefined the length of a meter. Changing the reference standard to be based on the speed of light in a vacuum is a more secure way to ensure that the definition of a meter does not change.
The International System of Units (SI), abbreviated from French Système Internationale d'Unités, is used by all branches of science. This system provides a standardized set of units that can be used by scientists to ensure precise and objective definitions.
A set of units in which some units are defined by their relationship to other units in the system and not by a physical standard is called a system of measurement. There are many systems of measurement in use today. One familiar system is the British Imperial System, which contains units such as the inch (distance), the pound (weight), and the gallon (volume). The British system evolved from units people used as far back as the Middle Ages, which were based on references handy to them. The foot, for example, is so named because it was supposed to be equal to the length of a human foot.
By the 19th century, it was realized that the system needed to be standardized, and Britain passed the Weights and Measures Act of 1824, which created precise and objective definitions. The definitions were updated in 1963, so today an imperial gallon, for example, equals the space occupied by 10 pounds of distilled water at a density of 0.998859 g/mL.
Although its units are now precisely defined, the British Imperial System is not a good choice for scientific measurements. The units are related to one another in inconsistent ways. For example, 1 mile = 1760 yards, 1 yard = 3 feet, and 1 foot = 12 inches. These relationships are difficult to remember and tricky to use in calculations. A better system would have consistent relationships and names that follow simple rules. Fortunately, this system exists. The International System of Units (SI), which in French is Système Internationale d'Unités, is the system of units used by the global scientific community, based on seven fundamental units. A more general form of the International System of Units is the metric system. It improves on the Imperial British System because the units within a particular type of measurement, such as length or mass, are all based on powers of 10 and are therefore more easily related to each other.
Mole mol Quantity of matter
Electric current is a measure of a flow of electric charge. Luminous intensity is the quantity of visible light emitted per unit time per unit solid angle. A mole is the amount of a substance that contains as many particles as 12 grams of pure carbon-12.
Each of the seven fundamental units of measurements in the International System of Units is an SI base unit. All other SI units can be derived from these seven base units and are therefore defined by their relationship to the base units. All the SI base units are defined by objective physical phenomena. Most of these units are expressed as a mathematical expression of the base units. For example, volume is simply measured in cubic meters (m3), but some derived units are specially named. The SI unit of force, for example, is the newton (N). The newton is defined to be the unit of force necessary to accelerate 1 kilogram at a rate of 1 meter per second per second, or
1\;\rm{N}=1\;\rm{kg}\cdot\rm{m/s}^2
. The SI unit of energy is the joule (J), which is equal to the energy transferred by 1 N of force acting on an object for the distance of 1 m. The expression relating these units is therefore
1\;\rm J=1\;\rm N\cdot\rm m=1\;\rm{kg}\cdot\rm m^2/\rm s^2
Examples of Derived SI Units with Names
{\rm{kg}} \cdot {\rm{m/s}}^2
{\rm{N/m}}{^2} = {\rm{kg/}}\left( {{\rm{ms}}^2} \right)
{\rm{N}} \cdot {\rm{m}} = {\rm{kg}} \cdot {\rm{m}}{^2}{\rm{/}}{{\rm{s}}^2}
{\rm{J/s}} = {\rm{kg}} \cdot {\rm{m}}{^2}{\rm{/s}}^3
{\rm{A}} \cdot {\rm{s}}
{\rm{W/A}} = {\rm{kg}} \cdot {\rm{m}}{^2}{\rm{/}}\left( {{{\rm{s}}^3}{\rm{A}}} \right)
Electric potential difference
SI base units can be combined to describe commonly measured properties in science. Examples of equivalencies between units show how units may be derived in different ways.
Finally, prefixes are used to define the magnitude of a given unit. The meter is a length that is convenient to use when describing the width of a room or the height of a person. Instead of creating an entirely new name for a unit of length that is appropriate for describing the distance between two distant cities, the prefix kilo-, meaning 1,000, is added to the base unit meter. The symbol for kilo- is k, so 1 kilometer (km) is equal to 1,000 m. Currently, there are 20 SI prefixes ranging from yocto (y, 10–24) to yotta (Y, 1024).
Larger than the Base Unit
Hecta h 102
Smaller than the Base Unit
Deci d 10–1
Micro µ 10–6
Yocto v 10–24
Prefixes are used in the metric system to increase or decrease units by powers of ten. The prefixes are never used in combination.
Mass, Weight, and Volume
Matter can be described by its mass, weight, and volume. Mass is the amount of matter in an object. Weight is the force of gravity acting on an object. Volume is the three-dimensional space occupied by a gas, a liquid, or a solid.
Mass (m) is a measure of the amount of matter in an object. The kilogram (kg) is the base SI unit of mass. Historically the kilogram has been defined as equal to the mass of the international prototype of the kilogram (IPK). The IPK is a polished cylinder made out of a platinum-iridium alloy, stored under three bell jars at the International Bureau of Weights and Measures in Sèvres, France. It must be stored carefully to protect it from contamination or particles that might settle on it and increase its mass and from physical damage that might chip or mar the prototype and possibly decrease its mass. The kilogram is the only base unit that has a prefix. The masses of materials used in chemistry labs are typically small enough to be measured in grams (g), not kilograms.
The International Prototype of the Kilogram
The international prototype of the kilogram (IPK) is stored under three bell jars in an atmosphere-controlled environment. The IPK is 39 mm in height and 39 mm in diameter.
Mass is not the same thing as weight. Weight (w) is a measure of the force of gravity acting on an object. Because weight is a force, the SI unit of weight is the newton (N). The weight of an object is equal to the mass (m) of the object times the acceleration due to gravity (g):
w=mg
The gravitational acceleration on the moon is 1/6 that of Earth, so 1 kg would weigh 1/6 as much on the moon as it does on Earth:
w_{\rm{moon}}=mg_{\rm{moon}}=m\left(\frac{1}{6}\,g_{\rm{Earth}}\right)=\frac{1}{6}\,{w_{\rm{Earth}}}
Volume (V) is the amount of space occupied by a given mass. The SI unit for volume is a cubic meter (m3). In chemistry a more common unit of volume is the liter (L), which is equal to 0.001 m3. Although the liter is not officially an SI unit (it is part of the metric system), it is defined by an exact mathematical relationship to a base SI unit and is acceptable to use within SI. The standard SI prefixes can all be applied to the liter, so
1\;\rm{mL}=0.001\;\rm{L}
1\;\rm{kL}=1,000\;\rm{L}
, and so forth. A liter is a greater volume than is usually needed for a typical laboratory experiment. Chemists work more often with volumes in the range of 1–1,000 mL. A useful conversion is
1\;\rm{ mL}=1\;\rm{ cm}^3
. The amount of matter contained in a given volume is density (
\rho
). Density is an intensive physical property of matter, which means it does not depend on the amount of matter present. In contrast, mass is an extensive physical property of matter, which means it depends on the amount of matter present. Density is therefore a useful way to characterize a substance. It is calculated by dividing the mass of a substance by its volume and is usually reported in g/mL or g/cm3. For example, the International Prototype of the Kilogram (IPK) is a cylinder with a height (h) of 3.9 cm. It also has a diameter of 3.9 cm, so its radius (r) is 1.95 cm. The density of the prototype can be calculated by dividing its mass by its volume (V):
\rho_{\rm{prototype}}=\frac{m}{V}=\frac{1\;{\rm{kg}}}{\pi{r^2h}}=\frac{1,\!000\;{\rm{g}}}{\pi(1.95\;{\rm{cm}})^2(3.9\;{\rm{cm}})}\approx21\;{\rm{g}}/{\rm{cm}^3}=21\;{\rm{g}}/{\rm{mL}}
Common temperature scales are the Fahrenheit scale, the Celsius scale, and the Kelvin scale. Unlike the Fahrenheit and Celsius scales, the Kelvin scale has no negative temperatures, and no degree symbol is used when temperatures are reported in kelvins.
Two familiar temperature scales in everyday use are the Fahrenheit and Celsius scales. The Fahrenheit temperature scale is based on a freezing point of water of 32°F and a boiling point of water of 212°F at sea level. The interval between the freezing point of water and the boiling point of water is divided into 180 parts, and the size of each part is equal to one Fahrenheit degree. In developing the scale, the Polish-Dutch physicist Daniel Gabriel Fahrenheit (1686–1736) originally designated the freezing point of water to be 30°F and normal human body temperature to be 90°F, but based on more accurate measurements, these values eventually were adjusted.
The Celsius temperature scale is based on a freezing point of water of 0°C and a boiling point of water of 100°C at sea level. The interval is divided into 100 parts so that each part is equal to one Celsius degree. Because there are fewer divisions between the freezing and boiling points of water on the Celsius scale than on the Fahrenheit scale, a Celsius degree is larger than a Fahrenheit degree. Although the Fahrenheit scale is commonly used in the United States, the Celsius scale is appropriate for scientific work. Conversions between the two scales are made according to the following equations. (Note: 5/9 is a reduced fraction of 100/180, which accounts for the different size of a Fahrenheit degree and a Celsius degree.)
\begin{aligned}{}\degree\rm C&=\frac59(\degree\rm F-32)\\{}\degree\rm F&=\left(\frac95\times\degree\!\rm C\right)+32\end{aligned}
As an example of using these conversions, to determine whether the predicted high temperature of 8°C in Montreal, Canada, is warmer or colder than the predicted high of 45°F in Bangor, Maine, convert 8°C to Fahrenheit.
\degree\rm F=\left(\frac95\times8\right)+32=46.4\,\degree\rm F
Alternatively, convert 45°F to Celsius.
\degree\rm C=\frac59(45-32)=7.2\,\degree\rm C
Both methods show that Montreal will be slightly warmer than Bangor. The SI unit of temperature is the kelvin. The Kelvin temperature scale is an absolute temperature scale based on the Celsius scale but shifted so that the lowest possible temperature is 0 K (–273.15°C). The temperature 0 K, called absolute zero, is the minimum possible temperature theoretically achievable, at which there is no particle motion. There are no negative Kelvin temperatures. Notice that no degree symbol is used when temperatures are reported in kelvins. Because a kelvin is the same size as a Celsius degree, converting between them is a simple matter of addition or subtraction.
\rm K=\degree\rm C+273.15
The Fahrenheit, Celsius, and Kelvin Temperature Scales
Fahrenheit, Celsius, and Kelvin temperature scales can all be defined according to the freezing point and boiling point of water. The size of a degree on the Celsius and Kelvin scales is the same. The size of a degree on the Fahrenheit scale is smaller because more degrees are included in the range of temperatures.
<Vocabulary>Accuracy, Precision, and Uncertainty
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Volume -1 Issue -1 | Duke Mathematical Journal
Home > Journals > Duke Math. J. > Advance Publication
Advance Publication | 2022
The advance publication content published here for the Duke Mathematical Journal is in its final form; it has been reviewed, corrected, edited, typeset, and assigned a permanent digital object identifier (DOI). The article's pagination will be updated when the article is assigned to a volume and issue.
Advance publication content can be cited using the date of online publication and the DOI.
Matan Harel, Frank Mousset, Wojciech Samotij
Duke Math. J. Advance Publication, 1-104, (2022) DOI: 10.1215/00127094-2021-0067
KEYWORDS: Nonlinear large deviations, Concentration inequalities, upper tails, Random graphs, arithmetic progressions, 60F10, 05C80
Suppose that X is a bounded-degree polynomial with nonnegative coefficients on the p-biased discrete hypercube. Our main result gives sharp estimates on the logarithmic upper tail probability of X whenever an associated extremal problem satisfies a certain entropic stability property. We apply this result to solve two long-standing open problems in probabilistic combinatorics: the upper tail problem for the number of arithmetic progressions of a fixed length in the p-random subset of the integers and the upper tail problem for the number of cliques of a fixed size in the random graph
{G}_{n,p}
. We also make significant progress on the upper tail problem for the number of copies of a fixed regular graph H in
{G}_{n,p}
. To accommodate readers who are interested in learning the basic method, we include a short, self-contained solution to the upper tail problem for the number of triangles in
{G}_{n,p}
p=p\left(n\right)
{n}^{-1}logn\ll p\ll 1
Duke Math. J. Advance Publication, 1-36, (2022) DOI: 10.1215/00127094-2022-0009
KEYWORDS: projective manifolds, complete intersections, curvature, 53C55, 32Q05, 32Q45
Using the Donaldson–Auroux theory, we construct complete intersections in complex projective manifolds, which are negatively curved in various ways. In particular, we prove the existence of compact simply connected Kähler manifolds with negative holomorphic bisectional curvature. We also construct hyperbolic hypersurfaces, and we obtain bounds for their Kobayashi hyperbolic metric.
Marian Aprodu, Gavril Farkas, Ştefan Papadima, Claudiu Raicu, Jerzy Weyman
KEYWORDS: Koszul module, resonance variety, Metabelian group, Torelli group, 57M07, 14H10
We provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace
K\subseteq {\wedge }^{2}V
as well as a sharp upper bound for its Hilbert function. This purely algebraic statement has interesting applications to the study of a number of invariants associated to finitely generated groups, such as the Alexander invariants, the Chen ranks, and the degree of growth and virtual nilpotency class. For instance, we explicitly bound the aforementioned invariants in terms of the first Betti number for the maximal metabelian quotients of (1) the Torelli group associated to the moduli space of curves, (2) nilpotent fundamental groups of compact Kähler manifolds, and (3) the Torelli group of a free group.
KEYWORDS: rational points of bounded height, non-Archimedean geometry, Wilkie’s conjecture, Pfaffian functions, Noetherian functions, 11U09, 14G05, 03C98, 11D88, 11G50
We consider the problem of counting polynomial curves on analytic or definable subsets over the field
\mathbb{C}\left(\left(t\right)\right)
as a function of the degree r. A result of this type could be expected by analogy with the classical Pila–Wilkie counting theorem in the Archimedean situation.
Some non-Archimedean analogues of this type have been developed in the work of Cluckers, Comte, and Loeser for the field
{\mathbb{Q}}_{p}
, but the situation in
\mathbb{C}\left(\left(t\right)\right)
appears to be significantly different. We prove that the set of polynomial curves of a fixed degree r on the transcendental part of a subanalytic set over
\mathbb{C}\left(\left(t\right)\right)
is automatically finite, but we give examples that show their number may grow arbitrarily quickly even for analytic sets. Thus no analogue of the Pila–Wilkie theorem can be expected to hold for general analytic sets. On the other hand, we show that if one restricts to varieties defined by Pfaffian or Noetherian functions, then the number grows at most polynomially in r, thus showing that the analogue of the Wilkie conjecture does hold in this context.
Convexity estimates for hypersurfaces moving by concave curvature functions
KEYWORDS: fully nonlinear flow, Hypersurface, pinching, convexity estimate, 53C44
We study fully nonlinear geometric flows that deform strictly k-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the k-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve k-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.
Danny Calegari, Fernando C. Marques, André Neves
KEYWORDS: minimal surfaces, hyperbolic manifolds, 53A10
We introduce an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature
\le -1
, by the hyperbolic metric.
KEYWORDS: Kardar–Parisi–Zhang equation, large deviations, Airy point process, Random operators, Stochastic Airy operator, 60F10, 60H25
h\left(t,x\right)
t\to \mathrm{\infty }
h\left(2t,0\right)+\frac{t}{12}
{t}^{2}
{\mathrm{\Phi }}_{-}\left(z\right)
Hodge theory of Kloosterman connections
Javier Fresán, Claude Sabbah, Jeng-Daw Yu
KEYWORDS: L-functions, Kloosterman sums, Galois representations, potential automorphy, ℓ-adic sheaves, exponential motives, irregular Hodge filtration, Fourier transform, connections with irregular singularities, D-modules, mixed Hodge modules, 11G40, 11F80, 11L05, 14F40, 32S35, 32S40
We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their L-functions extend meromorphically to the complex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn out to be “classical,” we compute their Hodge numbers by means of the irregular Hodge filtration on their realizations as exponential mixed Hodge structures. We show that all Hodge numbers are either zero or one, which implies potential automorphy thanks to recent results of Patrikis and Taylor.
A restriction estimate in
{\mathbb{R}}^{3}
using brooms
KEYWORDS: Fourier restriction estimate, 42B10, 42B20
If f is a function supported on the truncated paraboloid in
{\mathbb{R}}^{3}
and E is the corresponding extension operator, then we prove that for all
p>3+3∕13
‖Ef{‖}_{{L}^{p}\left({\mathbb{R}}^{3}\right)}\le C‖f{‖}_{{L}^{\mathrm{\infty }}}
. The proof combines Wolff’s two ends argument with polynomial partitioning techniques. We also observe some geometric structures in the wave packets.
An intersection formula for CM cycles on Lubin–Tate spaces
KEYWORDS: intersection number, CM cycle, Lubin–Tate space, RZ spaces, level structure, infinite level, 14G10, 14G35, 14G40
We give an explicit formula for the arithmetic intersection number of complex multiplication (CM) cycles on Lubin–Tate spaces for all levels. We prove our formula by formulating the intersection number on the infinite level. Our CM cycles are constructed by choosing two separable quadratic extensions
{K}_{1},{K}_{2}
over a non-Archimedean local field F. Our formula works for all cases:
{K}_{1}
{K}_{2}
can be either the same or different, ramified or unramified over F. This formula translates the linear arithmetic fundamental lemma (linear AFL) into a comparison of integrals. As an example, we prove the linear AFL for
{\mathrm{GL}}_{2}\left(F\right)
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