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Table 3 Comparison of psychological measures between left-behind wives and non-left-behind wives (
\overline{\mathbit{\chi }}\mathbf{±}\mathbf{\text{SD}}
Left-behind wives (n = 1893)
Non-left-behind wives (n = 969)
|Cohen d|
CES-D 10.90 ± 10.37 7.93 ± 10.15** 0.29
PSS 17.23 ± 4.22 14.89 ± 6.70** 0.42
PSSS 59.84 ± 11.92 62.78 ± 12.03** 0.25
AC 20.62 ± 6.70 23.22 ± 7.96** 0.35
PC 10.61 ± 5.84 9.53 ± 5.93** 0.18
**Significant difference between left-behind wives and non-left-behind wives, P < 0.01.
CES-D: center for epidemiological studies depression scale; PSS: perceived stress scale; PSSS: perceived social support scale; AC: active coping; PC: passive coping.
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Home : Support : Online Help : Connectivity : Web Features : XMLTools : ParseFile
read an XML document from a file
string; name of file to read
equation(s) of the form option=value where option is one of validate, externaldtd, entities, prolog, or whitespace; specify parsing options
The ParseFile(fileName) command reads XML data from the file fileName.
The file is closed after it is read.
The opts argument can contain one or more equations that set parsing options.
The validate option determines whether a validating or a non-validating parser should be used. Possible values are true and false. By default validate is set to false.
The externaldtd option determines whether the parser will process any external document type definition (DTD) referenced in the file. Note this action typically requires network resources whose availability will therefore affect the parsing time. Possible values are true and false. By default externaldtd is set to true.
The entities option determines whether the parser should resolve entity references. Possible values are name and value. By default entities is set to name.
The prolog option determines whether prolog should be included in the document. Possible values are true and false. By default prolog is set to false.
The whitespace option indicates whether ignorable whitespace should be included in the document. Possible values are true and false. This option is effective only if validate is set to true. By default whitespace is set to true.
\mathrm{with}\left(\mathrm{XMLTools}\right):
\mathrm{file}≔\mathrm{FileTools}:-\mathrm{JoinPath}\left(["help","XMLTools","SimpleDocument3.xml"],\mathrm{base}=\mathrm{datadir}\right)
\mathrm{ParseFile}\left(\mathrm{file}\right)
The XMLTools[ParseFile] command was updated in Maple 18.
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NCERT Solutions Class 12 Physics Chapter 15 Communication System
NCERT Solutions for Class 12 Physics Chapter 15 Communication System
NCERT Solutions for Class 12 Physics Chapter 15 – Free PDF Download
*According to the latest term-wise CBSE Syllabus 2021-22, this chapter has been removed.
The NCERT Solutions for Class 12 Physics Chapter 15 Communication Systems provided here are written by subject experts after extensive research on each and every topic to produce apt and authentic information for the students. If you go through the previous year question papers you will get a clear idea of questions directly asked from the book, in the examination. Hence, referring to these NCERT Solutions for Class 12 Physics during your preparations is a wise decision.
This NCERT Solutions Class 12 Physics presents different varieties of questions like MCQs, fill in the blanks, match the following, true or false, short answer questions along with numerical problems, important formulas, exercises and assignments. The PDF of the NCERT Solutions for Class 12 Physics Chapter 15 helps you to understand the concepts clearly and helps you to retain knowledge for a longer period of time.
Download NCERT Solutions Class 12 Physics Chapter 15 PDF:-Download Here
Q.1: Which of the following frequencies will be suitable for beyond-the horizon communication using sky waves?
(2) 10 MHz
(3) 1 GHz
(4) 1000 GHz
The signal waves need to travel a large distance for beyond – the – horizon communication.
Due to the antenna size, the 10 kHz signals cannot be radiated efficiently.
The 1 GHz – 1000 GHz (high energy) signal waves penetrate the ionosphere.
The 10 MHz frequencies get reflected easily from the ionosphere. Therefore, for beyond – the – horizon communication signal waves of 10 MHz frequencies are suitable.
Q.2: Frequencies in the UHF range normally propagate by means of :
(1) Ground Waves
(2) Sky Waves
(3) Surface Waves
(4) Space Waves
Due to its high frequency, an ultra-high frequency (UHF) wave cannot travel along the trajectory of the ground also it cannot get reflected by the ionosphere. The ultrahigh-frequency signals are propagated through line – of – sight communication, which is actually space wave propagation.
Q.3: Digital signals
(i) Do not provide a continuous set of values
(ii) Represent value as discrete steps
(iii) Can utilize binary system
(iv) Can utilize decimal as well as binary systems
State which statement(s) are true?
(a) (1), (2) and (3) For transferring message signals the digital signals use the binary (0 and 1) system. Such a system cannot utilise the decimal system. Discontinuous values are represented in digital signals.
Q.4: Is it necessary for a transmitting antenna to be at the same height as that of the receiving antenna for line-of-sight communication? A TV transmitting antenna is 81 m tall. How much service area can it cover if the receiving antenna is at the ground level?
Soln: In line – of – sight communication, between the transmitter and the receiver there is no physical obstruction. So, there is no need for the transmitting and receiving antenna to be at the same height.
Height of the antenna, h = 81 m
Radius of earth, R = 6.4 x 106m
d = √2Rh, for range
The service area of the antenna is given by the relation :
A = πd2 = π(2Rh)
= 3.14 x 2 x 6.4 x 106 x 81
= 3255.55 x 106 m2 = 3255.55 = 3256 km2
Q.5: A carrier wave of peak voltage 12 V is used to transmit a message signal. What should be the peak voltage of the modulating signal in order to have a modulation index of 75%?
Amplitude of carrier wave, Ac = 12 V
Modulation index, m = 75% = 0.75
Amplitude of the modulating wave = Am
Modulation index is given by the relation :
\frac{A_{m}}{A_{c}}
Therefore, Am = m.Ac
= 0.75 x 12 V= 9 V
Q.6: A modulating signal is a square wave, as shown in the figure.
The carrier wave is given by c(t)=2sin(8πt)volts.
(1) Sketch the amplitude modulated waveform
(2) What is the modulation index?
The amplitude of the modulating signal, Am = 1v can be easily observed from the given modulating signal.
Carrier wave is given by, c(t) = 2 sin(8nt)
Amplitude of the carrier wave, Ac = 2v
Time period, Tm = 1s
The angular frequency of the modulating signal is given by,
\omega _{m} = \frac{2\pi }{T_{m}}
= 2π rad s-1 …(1)
The angular frequency of carrier signal,
\omega _{c} = 8\pi
rad s-1 …(2)
from eqns.(1) and (2),
\omega _{c} = 4\omega _{m}
The modulating signal having the amplitude modulated waveform is shown in the figure:
(2) Modulation index, m =
\frac{A_{m}}{A_{c}}
\frac{1}{2}
Q.7: For an amplitude modulated wave, the maximum amplitude is found to be 10V while the minimum amplitude is found to be 2V. Determine the modulation index, µ. What would be the value of µ if the minimum amplitude is zero volts?
Maximum Amplitude, Amax = 10 V
Minimum Amplitude, Amin = 2 V
For a wave, modulation index µ, is given by :
µ =
\frac{A_{max} – A_{min}}{A_{max} + A_{min}}
\frac{10 – 2}{10 + 2}
\frac{8}{12}
If Amin = 0,
\frac{A_{max}}{ A_{max}}
= 10/10 = 1
Q.8: Due to economic reasons, only the upper sideband of an AM wave is transmitted, but at the receiving station, there is a facility for generating the carrier. Show that if a device is available which can multiply two signals, then it is possible to recover the modulating signal at the receiver station.
Soln: Let,
\omega _{c}
be the carrier wave frequency
\omega _{s}
be the signal wave frequency
Signal received, V = V1 cos (
\omega _{c}
\omega _{s}
)t
Instantaneous voltage of the carrier wave, Vm = Vc cos
\omega _{c}
V.Vin = V1cos(
\omega _{c}
\omega _{s}
)t. (Vc cos
\omega _{c}
= V1Vc [cos(
\omega _{c}
\omega _{s}
)t . cos
\omega _{c}
\frac{V_{1} V_{c}}{ 2} \left [ cos{( \omega _{c} + \omega _{s}) t + \omega _{c}t} + cos{( \omega _{c} + \omega _{s})t – \omega _{c}t} \right ]
\frac{V_{1} V_{c}}{ 2} \left [ cos{( 2\omega _{c} + \omega _{s}) t + cos\omega _{s}t} \right ]
The low pass filter allows only the high frequency signals to pass through it. The low frequency signal
\omega _{s}
is obstructed by it.
Thus, at the receiving station, we can record the modulating signal,
\frac{V_{1}V_{C}}{2}cos\omega _{s}t
which is the signal frequency.
The transmitter, transmission channel, and receiver are three basic units of a communication system.
Low frequencies cannot be transmitted to long distances. Therefore, they are superimposed on a high-frequency carrier signal by a process known as modulation.
Two important forms of the communication system are Analog and Digital.
Amplitude modulated waves can be produced by application of the message signal and the carrier wave to a non-linear device, followed by a bandpass filter.
Class 12 Physics NCERT Solutions for Chapter 15 Communication Systems
The NCERT Solutions for Class 12 Physics will help you understand that communication systems are simply a collection of systems used for transmission, connection, communication and interconnection. These systems are categorically arranged into three different types on the basis of uses such as the Media, Technology and Application area. Sensors, Transducers, Emitters and Amplifiers are all examples of modern technology that are used as components in the majority of modern devices.
Major examples of general communication systems:
Concepts involved in NCERT Class 12 Chapter 15 Communication System are:
Size of the antenna or aerial
Effective power radiated by an antenna Ex 15.7.3 – Mixing up of signals from different transmitters
To obtain a firm grip over these key concepts of this chapter and subject, students can access the NCERT Solutions that are developed by educational experts in the field. These solutions you to analyze your strengths and weaknesses thus providing a clear strategy to improve your academic performance.
In order to score good marks, it is very important for the students to also solve and get well versed with the NCERT exemplary questions and problems.
NCERT Exemplar for Class 12 Physics Chapter 15
BYJU’S is revolutionising the education sector of the country through its interactive and effective learning methodology that uses videos, animations and info-graphics to teach the students. Our interactive model of teaching helps students learn more effectively than traditional teaching methods.
Stay tuned with BYJU’S to access the NCERT Solutions, download exemplar problems for other classes in PDF and CBSE sample papers.
What are the types of questions asked from Chapter 15 of NCERT Solutions for Class 12 Physics?
The types of questions asked from Chapter 15 of NCERT Solutions for Class 12 Physics are –
1. Very short answers
Students can answer the very short answer type of questions in a single sentence.
Can I download the PDF of NCERT Solutions for Class 12 Physics Chapter 15 for free?
Yes, you can download the PDF of NCERT Solutions for Class 12 Physics Chapter 15 for free from BYJU’S. The solutions are designed based on the latest CBSE syllabus and guidelines. The chapter wise and exercise wise PDF links are provided to help students boost their exam preparation. All the answers are strictly based on the textbook prescribed by the CBSE board. The solutions PDF help students to improve their logical reasoning and analytical thinking skills, which are extremely important for the exam.
What is the concept of amplitude modulation discussed in the Chapter 15 of NCERT Solutions for Class 12 Physics?
Amplitude modulation is a process by which the wave signal is transmitted by modulating the amplitude of the signal. It is often called AM and is commonly used in transmitting a piece of information through a radio carrier wave. Amplitude modulation is mostly used in the form of electronic communication. This concept is explained with various examples in order to provide in-depth knowledge among students. The solutions of Chapter 15 are designed by the faculty at BYJU’S keeping in mind the understanding abilities of students. To get a clear idea about the concepts covered in this chapter, students are advised to download the solutions PDF available in BYJU’S.
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Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g.
{\displaystyle {\sqrt {2}}}
is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the completion of a given space, as explained below.
{\displaystyle x_{1},x_{2},x_{3},\ldots }
{\displaystyle (X,d)}
is called Cauchy if for every positive real number
{\displaystyle r>0}
there is a positive integer
{\displaystyle N}
such that for all positive integers
{\displaystyle m,n>N,}
{\displaystyle d\left(x_{m},x_{n}\right)<r.}
The expansion constant of a metric space is the infimum of all constants
{\textstyle \mu }
such that whenever the family
{\textstyle \left\{{\overline {B}}(x_{\alpha },\,r_{\alpha })\right\}}
intersects pairwise, the intersection
{\textstyle \bigcap _{\alpha }{\overline {B}}(x_{\alpha },\mu r_{\alpha })}
{\displaystyle (X,d)}
is complete if any of the following equivalent conditions are satisfied:
Every Cauchy sequence of points in
{\displaystyle X}
has a limit that is also in
{\displaystyle X.}
Every Cauchy sequence in
{\displaystyle X}
{\displaystyle X}
(that is, to some point of
{\displaystyle X}
The expansion constant of
{\displaystyle (X,d)}
is ≤ 2.[1]
Every decreasing sequence of non-empty closed subsets of
{\displaystyle X,}
with diameters tending to 0, has a non-empty intersection: if
{\displaystyle F_{n}}
is closed and non-empty,
{\displaystyle F_{n+1}\subseteq F_{n}}
{\displaystyle n,}
{\displaystyle \operatorname {diam} \left(F_{n}\right)\to 0,}
then there is a point
{\displaystyle x\in X}
common to all sets
{\displaystyle F_{n}.}
The space Q of rational numbers, with the standard metric given by the absolute value of the difference, is not complete. Consider for instance the sequence defined by
{\displaystyle x_{1}=1}
{\displaystyle x_{n+1}={\frac {x_{n}}{2}}+{\frac {1}{x_{n}}}.}
This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit
{\displaystyle x,}
then by solving
{\displaystyle x={\frac {x}{2}}+{\frac {1}{x}}}
necessarily
{\displaystyle x^{2}=2,}
yet no rational number has this property. However, considered as a sequence of real numbers, it does converge to the irrational number
{\displaystyle {\sqrt {2}}}
The open interval (0,1), again with the absolute value metric, is not complete either. The sequence defined by {
{\displaystyle x_{n}={\tfrac {1}{n}}}
} is Cauchy, but does not have a limit in the given space. However the closed interval [0,1] is complete; for example the given sequence does have a limit in this interval and the limit is zero.
The space Qp of p-adic numbers is complete for any prime number
{\displaystyle p.}
This space completes Q with the p-adic metric in the same way that R completes Q with the usual metric.
{\displaystyle S}
is an arbitrary set, then the set SN of all sequences in
{\displaystyle S}
becomes a complete metric space if we define the distance between the sequences
{\displaystyle \left(x_{n}\right)}
{\displaystyle \left(y_{n}\right)}
{\displaystyle {\tfrac {1}{N}}}
{\displaystyle N}
{\displaystyle x_{N}}
{\displaystyle y_{N}}
{\displaystyle 0}
if there is no such index. This space is homeomorphic to the product of a countable number of copies of the discrete space
{\displaystyle S.}
Some theoremsEdit
Every compact metric space is complete, though complete spaces need not be compact. In fact, a metric space is compact if and only if it is complete and totally bounded. This is a generalization of the Heine–Borel theorem, which states that any closed and bounded subspace
{\displaystyle S}
of Rn is compact and therefore complete.[2]
{\displaystyle (X,d)}
be a complete metric space. If
{\displaystyle A\subseteq X}
is a closed set, then
{\displaystyle A}
is also complete.[3] Let
{\displaystyle (X,d)}
be a metric space. If
{\displaystyle A\subseteq X}
is a complete subspace, then
{\displaystyle A}
is also closed.[4]
{\displaystyle X}
is a set and
{\displaystyle M}
is a complete metric space, then the set
{\displaystyle B(X,M)}
of all bounded functions f from X to
{\displaystyle M}
is a complete metric space. Here we define the distance in
{\displaystyle B(X,M)}
in terms of the distance in
{\displaystyle M}
with the supremum norm
{\displaystyle d(f,g)\equiv \sup\{d[f(x),g(x)]:x\in X\}}
{\displaystyle X}
{\displaystyle M}
{\displaystyle C_{b}(X,M)}
consisting of all continuous bounded functions
{\displaystyle f:X\to M}
{\displaystyle B(X,M)}
and hence also complete.
Theorem[5] (C. Ursescu) — Let
{\displaystyle X}
be a complete metric space and let
{\displaystyle S_{1},S_{2},\ldots }
{\displaystyle X.}
{\displaystyle S_{i}}
{\displaystyle X}
{\textstyle \operatorname {cl} \left(\bigcup _{i\in \mathbb {N} }\operatorname {int} S_{i}\right)=\operatorname {cl} \operatorname {int} \left(\bigcup _{i\in \mathbb {N} }S_{i}\right).}
{\displaystyle S_{i}}
{\displaystyle X}
{\textstyle \operatorname {int} \left(\bigcap _{i\in \mathbb {N} }\operatorname {cl} S_{i}\right)=\operatorname {int} \operatorname {cl} \left(\bigcap _{i\in \mathbb {N} }S_{i}\right).}
For any metric space M, it is possible to construct a complete metric space M′ (which is also denoted as
{\displaystyle {\overline {M}}}
), which contains M as a dense subspace. It has the following universal property: if N is any complete metric space and f is any uniformly continuous function from M to N, then there exists a unique uniformly continuous function f′ from M′ to N that extends f. The space M' is determined up to isometry by this property (among all complete metric spaces isometrically containing M), and is called the completion of M.
The completion of M can be constructed as a set of equivalence classes of Cauchy sequences in M. For any two Cauchy sequences
{\displaystyle x_{\bullet }=\left(x_{n}\right)}
{\displaystyle y_{\bullet }=\left(y_{n}\right)}
in M, we may define their distance as
{\displaystyle d\left(x_{\bullet },y_{\bullet }\right)=\lim _{n}d\left(x_{n},y_{n}\right)}
{\displaystyle p,}
{\displaystyle p}
-adic numbers arise by completing the rational numbers with respect to a different metric.
Topologically complete spacesEdit
Alternatives and generalizationsEdit
Since Cauchy sequences can also be defined in general topological groups, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure. This is most often seen in the context of topological vector spaces, but requires only the existence of a continuous "subtraction" operation. In this setting, the distance between two points
{\displaystyle x}
{\displaystyle y}
is gauged not by a real number
{\displaystyle \varepsilon }
via the metric
{\displaystyle d}
in the comparison
{\displaystyle d(x,y)<\varepsilon ,}
but by an open neighbourhood
{\displaystyle N}
{\displaystyle 0}
via subtraction in the comparison
{\displaystyle x-y\in N.}
It is also possible to replace Cauchy sequences in the definition of completeness by Cauchy nets or Cauchy filters. If every Cauchy net (or equivalently every Cauchy filter) has a limit in
{\displaystyle X,}
{\displaystyle X}
is called complete. One can furthermore construct a completion for an arbitrary uniform space similar to the completion of metric spaces. The most general situation in which Cauchy nets apply is Cauchy spaces; these too have a notion of completeness and completion just like uniform spaces.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Complete_metric_space&oldid=1089262814"
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torch.nn.functional.affine_grid — PyTorch 1.11.0 documentation
torch.nn.functional.affine_grid¶
torch.nn.functional.affine_grid(theta, size, align_corners=None)[source]¶
theta (Tensor) – input batch of affine matrices with shape (
N \times 2 \times 3
) for 2D or (
N \times 3 \times 4
) for 3D
size (torch.Size) – the target output image size. (
N \times C \times H \times W
for 2D or
N \times C \times D \times H \times W
for 3D) Example: torch.Size((32, 3, 24, 24))
align_corners (bool, optional) – if True, consider -1 and 1 to refer to the centers of the corner pixels rather than the image corners. Refer to grid_sample() for a more complete description. A grid generated by affine_grid() should be passed to grid_sample() with the same setting for this option. Default: False
output Tensor of size (
N \times H \times W \times 2
When align_corners = True, 2D affine transforms on 1D data and 3D affine transforms on 2D data (that is, when one of the spatial dimensions has unit size) are ill-defined, and not an intended use case. This is not a problem when align_corners = False. Up to version 1.2.0, all grid points along a unit dimension were considered arbitrarily to be at -1. From version 1.3.0, under align_corners = True all grid points along a unit dimension are considered to be at 0 (the center of the input image).
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Context mixing - Wikipedia
Context mixing is a type of data compression algorithm in which the next-symbol predictions of two or more statistical models are combined to yield a prediction that is often more accurate than any of the individual predictions. For example, one simple method (not necessarily the best) is to average the probabilities assigned by each model. The random forest is another method: it outputs the prediction that is the mode of the predictions output by individual models. Combining models is an active area of research in machine learning.[citation needed]
1 Application to Data Compression
1.1 Linear Mixing
1.2 Logistic Mixing
1.3 List of Context Mixing Compressors
Application to Data Compression[edit]
Suppose that we are given two conditional probabilities,
{\displaystyle P(X|A)}
{\displaystyle P(X|B)}
, and we wish to estimate
{\displaystyle P(X|A,B)}
, the probability of event X given both conditions
{\displaystyle A}
{\displaystyle B}
. There is insufficient information for probability theory to give a result. In fact, it is possible to construct scenarios in which the result could be anything at all. But intuitively, we would expect the result to be some kind of average of the two.
The problem is important for data compression. In this application,
{\displaystyle A}
{\displaystyle B}
are contexts,
{\displaystyle X}
is the event that the next bit or symbol of the data to be compressed has a particular value, and
{\displaystyle P(X|A)}
{\displaystyle P(X|B)}
are the probability estimates by two independent models. The compression ratio depends on how closely the estimated probability approaches the true but unknown probability of event
{\displaystyle X}
. It is often the case that contexts
{\displaystyle A}
{\displaystyle B}
have occurred often enough to accurately estimate
{\displaystyle P(X|A)}
{\displaystyle P(X|B)}
by counting occurrences of
{\displaystyle X}
in each context, but the two contexts either have not occurred together frequently, or there are insufficient computing resources (time and memory) to collect statistics for the combined case.
For example, suppose that we are compressing a text file. We wish to predict whether the next character will be a linefeed, given that the previous character was a period (context
{\displaystyle A}
) and that the last linefeed occurred 72 characters ago (context
{\displaystyle B}
). Suppose that a linefeed previously occurred after 1 of the last 5 periods (
{\displaystyle P(X|A=0.2}
) and in 5 out of the last 10 lines at column 72 (
{\displaystyle P(X|B)=0.5}
). How should these predictions be combined?
Two general approaches have been used, linear and logistic mixing. Linear mixing uses a weighted average of the predictions weighted by evidence. In this example,
{\displaystyle P(X|B)}
gets more weight than
{\displaystyle P(X|A)}
{\displaystyle P(X|B)}
is based on a greater number of tests. Older versions of PAQ uses this approach.[1] Newer versions use logistic (or neural network) mixing by first transforming the predictions into the logistic domain, log(p/(1-p)) before averaging.[2] This effectively gives greater weight to predictions near 0 or 1, in this case
{\displaystyle P(X|A)}
. In both cases, additional weights may be given to each of the input models and adapted to favor the models that have given the most accurate predictions in the past. All but the oldest versions of PAQ use adaptive weighting.
Most context mixing compressors predict one bit of input at a time. The output probability is simply the probability that the next bit will be a 1.
Linear Mixing[edit]
We are given a set of predictions Pi(1) = n1i/ni, where ni = n0i + n1i, and n0i and n1i are the counts of 0 and 1 bits respectively for the i'th model. The probabilities are computed by weighted addition of the 0 and 1 counts:
S0 = Σi wi n0i
P(0) = S0 / S
The weights wi are initially equal and always sum to 1. Under the initial conditions, each model is weighted in proportion to evidence. The weights are then adjusted to favor the more accurate models. Suppose we are given that the actual bit being predicted is y (0 or 1). Then the weight adjustment is:
ni = n0i + n1i
error = y – P(1)
wi ← wi + [(S n1i - S1 ni) / (S0 S1)] error
Compression can be improved by bounding ni so that the model weighting is better balanced. In PAQ6, whenever one of the bit counts is incremented, the part of the other count that exceeds 2 is halved. For example, after the sequence 000000001, the counts would go from (n0, n1) = (8, 0) to (5, 1).
Logistic Mixing[edit]
Let Pi(1) be the prediction by the i'th model that the next bit will be a 1. Then the final prediction P(1) is calculated:
xi = stretch(Pi(1))
P(1) = squash(Σi wi xi)
where P(1) is the probability that the next bit will be a 1, Pi(1) is the probability estimated by the i'th model, and
stretch(x) = ln(x / (1 - x))
squash(x) = 1 / (1 + e−x) (inverse of stretch).
After each prediction, the model is updated by adjusting the weights to minimize coding cost.
wi ← wi + η xi (y - P(1))
where η is the learning rate (typically 0.002 to 0.01), y is the predicted bit, and (y - P(1)) is the prediction error.
List of Context Mixing Compressors[edit]
All versions below use logistic mixing unless otherwise indicated.
All PAQ versions (Matt Mahoney, Serge Osnach, Alexander Ratushnyak, Przemysław Skibiński, Jan Ondrus, and others) [1]. PAQAR and versions prior to PAQ7 used linear mixing. Later versions used logistic mixing.
All LPAQ versions (Matt Mahoney, Alexander Ratushnyak) [2].
ZPAQ (Matt Mahoney) [3].
WinRK 3.0.3 (Malcolm Taylor) in maximum compression PWCM mode [4]. Version 3.0.2 was based on linear mixing.
NanoZip (Sami Runsas) in maximum compression mode (option -cc) [5].
xwrt 3.2 (Przemysław Skibiński) in maximum compression mode (options -i10 through -i14) [6] as a back end to a dictionary encoder.
cmm1 through cmm4, M1, and M1X2 (Christopher Mattern) use a small number of contexts for high speed. M1 and M1X2 use a genetic algorithm to select two bit masked contexts in a separate optimization pass.
ccm (Christian Martelock).
bit (Osman Turan) [7].
pimple, pimple2, tc, and px (Ilia Muraviev) [8].
enc (Serge Osnach) tries several methods based on PPM and (linear) context mixing and chooses the best one. [9]
fpaq2 (Nania Francesco Antonio) using fixed weight averaging for high speed.
cmix (Byron Knoll) mixes many models, and is currently ranked first in the Large Text Compression benchmark,[3] as well as the Silesia corpus [4] and has surpassed the winning entry of the Hutter Prize although it is not eligible due to using too much memory.
^ Mahoney, M. (2005), "Adaptive Weighing of Context Models for Lossless Data Compression", Florida Tech. Technical Report CS-2005-16
^ Mahoney, M. "PAQ8 Data Compression Program".
^ Matt Mahoney (2015-09-25). "Large Text Compression Benchmark". Retrieved 2015-11-04.
^ Matt Mahoney (2015-09-23). "Silesia Open Source Compression Benchmark". Retrieved 2015-11-04.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Context_mixing&oldid=991051541"
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From W. E. Darwin 1 August 1862
Southampton & Hampshire Bank, Southampton
I have got your Lythrum letter, and will send you the 3 kinds tonight. it certainly will be most awfully complicated work with 18 possible crosses.1 there is as much Lythrum as you like to be got here. when I first found a bed, or rather beds along a stream, I gathered haphazards parts of 27 different plants, and examined them when I got home.2 it is very odd the symmetry the the division had
out of the 27 plants
11 of them were long pistilled = Lp
9 — short — = Sp
7 — middle — = Mp.
You see the short are exactly a
\frac{1}{3}
the Lp—2 above the
\frac{1}{3}
, the Mp—2 below the
\frac{1}{3}
If you liked I could gather 90 or 120 or 150 or 300 plants and class them. there is one odd thing if true, that would I should make it less complicated; all the Lp that I have looked at yet are less ripe or advanced than the Sp or the Mp.
This was plain in the 27 plants, (unless of course I have made some hideous mistake)
when the pollen is ripe and the anthers are opening, the filament is crimson and the pollen green— in these 27 plants the filaments were crimson and the pollen green in all the long stamens (only of course of the quite open flowers) both of the Sp and of the Mp.— while in all the 11 heads of the Lp there was not a single red filament or green pollen to be seen—and I looked tolerably carefully through them all.
I have looked at the two pollen of the Lp, and drawn and measured them by Camera Lucida and there is a decided difference.3
there is also difference I think between the pollens of the different kinds, but you will see all that.
I had gathered a lot more yesterday to have another look, but with the family luck my mare tumbled crossing her legs and cut her knee very badly, and in the scrummage all the plants tumbled out of my case
1.1 I have … home. 1.5] crossed pencil
2.2 ‘/27’ added under column of numbers, pencil
3.1 If you … my case 8.3] crossed pencil
CD’s letter has not been found; however, in the letter to Daniel Oliver, 29 [July 1862], CD described the crosses he planned to make with the three forms of Lythrum salicaria. See also letter to W. E. Darwin, 9 July [1862].
William’s observations on the twenty-seven Lythrum plants, dated 26 July 1862, are detailed in his botanical notebook (DAR 117: 36–7). See also William’s earlier observations on Lythrum, dated 13 July and 20 July, in DAR 117: 1, 12–13.
In the letter from W. E. Darwin, 5 August 1862, William enclosed camera lucida drawings, dated 1 August 1862, of the two sets of pollen grains from the long and short stamens of the long-styled form of Lythrum salicaria.
WED has been collecting Lythrum plants. Numerical proportions of the three forms.
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Lucy keeps track of how long it takes her to do the newspaper’s crossword puzzle each day. Her recent times, in minutes, were:
8\ \ 22\ \ 19\ \ 12\ \ 18\ \ 19\ \ 10\ \ 35\ \ 12\ \ 19\ \ 16\ \ 21
What is the median of her data?
The median is the number directly in the center or the average of the two numbers in the middle, when the numbers are arranged in an increasing or decreasing order.
The mean is the sum of all the numbers divided by the number of numbers added together.
17\frac{7}{12}
|
EUDML | New polynomials for which has a solution with almost all real zeros. EuDML | New polynomials for which has a solution with almost all real zeros.
New polynomials
P
{f}^{\text{'}\text{'}}+P\left(z\right)f=0
has a solution with almost all real zeros.
Shin, K.C.
Shin, K.C.. "New polynomials for which has a solution with almost all real zeros.." Annales Academiae Scientiarum Fennicae. Mathematica 27.2 (2002): 491-498. <http://eudml.org/doc/123002>.
author = {Shin, K.C.},
keywords = {entire function; finite order; Schrödinger eigenvalue problem; nonreal potential; existence},
title = {New polynomials for which has a solution with almost all real zeros.},
AU - Shin, K.C.
TI - New polynomials for which has a solution with almost all real zeros.
KW - entire function; finite order; Schrödinger eigenvalue problem; nonreal potential; existence
entire function, finite order, Schrödinger eigenvalue problem, nonreal potential, existence
Oscillation, growth of solutions
Articles by Shin
|
EUDML | On Jordan algebras of Baire type. EuDML | On Jordan algebras of Baire type.
On Jordan algebras of Baire type.
Arzikulov, F.N.
Arzikulov, F.N.. "On Jordan algebras of Baire type.." Vladikavkazskiĭ Matematicheskiĭ Zhurnal 4.3 (2002): 16-21. <http://eudml.org/doc/224587>.
@article{Arzikulov2002,
author = {Arzikulov, F.N.},
keywords = {Jordan algebra; Baire involutive algebra; measurable operator; -algebra; -algebra; -algebra; Peirce decomposition; -algebra; -algebra; -algebra},
title = {On Jordan algebras of Baire type.},
AU - Arzikulov, F.N.
TI - On Jordan algebras of Baire type.
KW - Jordan algebra; Baire involutive algebra; measurable operator; -algebra; -algebra; -algebra; Peirce decomposition; -algebra; -algebra; -algebra
Jordan algebra, Baire involutive algebra, measurable operator,
A{W}^{*}
AJW
OJ
-algebra, Peirce decomposition,
A{W}^{*}
AJW
OJ
Nonassociative topological algebras with an involution
{C}^{*}
{W}^{*}
Nonassociative selfadjoint operator algebras
Articles by Arzikulov
|
EUDML | Boundedness and surjectivity in normed spaces. EuDML | Boundedness and surjectivity in normed spaces.
Boundedness and surjectivity in normed spaces.
Nygaard, Olav
Nygaard, Olav. "Boundedness and surjectivity in normed spaces.." International Journal of Mathematics and Mathematical Sciences 32.3 (2002): 149-165. <http://eudml.org/doc/50610>.
author = {Nygaard, Olav},
keywords = {-boundedness property; -thick set; -boundedness property; -thick set},
title = {Boundedness and surjectivity in normed spaces.},
AU - Nygaard, Olav
TI - Boundedness and surjectivity in normed spaces.
KW - -boundedness property; -thick set; -boundedness property; -thick set
{w}^{*}
-boundedness property,
{w}^{*}
-thick set,
{w}^{*}
{w}^{*}
-thick set
Articles by Nygaard
|
Existence of Subharmonic Solutions for a Class of Second-Order p -Laplacian Systems with Impulsive Effects
2012 Existence of Subharmonic Solutions for a Class of Second-Order
p
-Laplacian Systems with Impulsive Effects
Wen-Zhen Gong, Qiongfen Zhang, X. H. Tang
By using minimax methods in critical point theory, a new existence theorem of infinitely many periodic solutions is obtained for a class of second-order
p
-Laplacian systems with impulsive effects. Our result generalizes many known works in the literature.
Wen-Zhen Gong. Qiongfen Zhang. X. H. Tang. "Existence of Subharmonic Solutions for a Class of Second-Order
p
-Laplacian Systems with Impulsive Effects." J. Appl. Math. 2012 1 - 18, 2012. https://doi.org/10.1155/2012/434938
Wen-Zhen Gong, Qiongfen Zhang, X. H. Tang "Existence of Subharmonic Solutions for a Class of Second-Order
p
-Laplacian Systems with Impulsive Effects," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-18, (2012)
|
On the theory of KM2O-Langevin equations for stationary flows (1): characterization theorem
October, 1999 On the theory of
K{M}_{2}O
-Langevin equations for stationary flows (1): characterization theorem
Yasunori OKABE
In this paper, we introduce a notion of stationarity for the pair of flows in a metric vector space and characterize it in such a way that there exist two relations, called a dissipation-dissipation theorem and a fluctuation-dissipation theorem, among the
K{M}_{2}O
-Langevin matrix associated with the pair of flows.
Yasunori OKABE. "On the theory of
K{M}_{2}O
-Langevin equations for stationary flows (1): characterization theorem." J. Math. Soc. Japan 51 (4) 817 - 841, October, 1999. https://doi.org/10.2969/jmsj/05140817
Keywords: $\mathrm{K}\mathrm{M}_{2}\mathrm{O}$-Langevin matrix , dissipation-dissipation theorem , fluctuation-dissipation theorem , Pair of flows , stationarity
Yasunori OKABE "On the theory of
K{M}_{2}O
-Langevin equations for stationary flows (1): characterization theorem," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 51(4), 817-841, (October, 1999)
|
Intro to Circom - Electron Labs
Circom: Usage/purpose of language syntax and programming practices
This section assumes that you have completed the Circom tutorial for the multiplier circuit given here. Please make sure you have completed the section that explains the power of tau ceremony and proof generation and verification.
Let us now discuss how to make proper use of Circom.
Let's start with something simple/trivial. Say we want to prove the polynomial x^2 + x + 2 = y for a given (x,y). In circom, we write this as:-
signal input y;
x*x + x + 2 === y;
//=== is the constraint operator
//both x and y are provided and we simply apply the polynomial equation
//This is similar to python's assert(x*x + x + 2 = y)
Here, we must first pre-calculate x and y and then provide them as inputs to the circuit. However, Circom allows us to calculate the value of y within the circuit itself. This is done using output signals.
signal output y; //notice y is now an output signal
x*x + x + 2 ==> y;
//notice the use of ==> instead of === here. The ==> operator first calculates
//the value of y and then applies the polynomial constraint
//note that the following statements are the same
y <== x*x + x + 2;
Technically, we only need the === operator. The <== operator is basically Circom's attempt at making the developer's life easier. Also, note that signals are all the same in the underlying cryptography. The separation into input or output signals is virtual and just to make things developer-friendly.
Okay, now let us say we want to prove the polynomial x^3 + x^2 + 1 = y
signal output y;
x*x*x + x*x + 1 ==> y;
This will give an error Non-Quadratic Constraints are not allowed. The reason for this error is that all constraints must be of the form a*b + c i.e. only one multiplication is allowed per constraint. You can have an arbitrary number of additions. No other operators (- or /) are allowed. The reason for this comes from the underlying cryptography (something called "cryptographic pairings").
To solve this issue, we must reduce our polynomial into the form a*b+c. We can do this within the circuit itself by use of intermediary signals, as below:-
signal temp1;
temp1 <== x*x;
temp2 <== temp1*x;
temp2 + temp1 + 1 ==> y;
//this way, all the expressions are reduced to quadratic/linear
This changes the polynomial we are proving. The set of polynomials we are proving in the above circuit is:-
t1 = x*x \newline t2 = t1*x \newline t1 + t2 + 1 = y
How use the modulo operator (%) in Circom?
The underlying cryptography restricts us to just * and + operators. But we obviously want to use all operators. So how can we prove the equation y = x%10?
y <== x%10; //this will give an error of Non-Quadratic Constraints
Alternatively, we could write this as:-
signal input x:
signal input q;
x === q*10 + y;
//here q is the quotient and y is the remainder when x is divided by 10
This solves our problem. However, this way, we have to provide everything as an input signal, which means we have to write a ton of python scripts outside the circuit. Circom allows us to get around this issue by use of the <-- operator.
signal q;
y <-- x%10;
q <-- y\10; //where '\' is the integer division operator
x === q*10 + y; //this works!
What just happened here? The <-- is basically Circom's assignment operator. Please note that the <-- does not set any constraints, hence we are not limited to * or + when using <--. The proper way to use this operator is to perform computations using this operator, and then in the end apply the === operator to set the polynomial constraint.
The <-- operator is basically Circom's attempt at making the developer's life easier.
If you are maths savvy, you might have noted that the constraint x === q*10 + y is not mathematically complete. We must also apply the condition that y<10. We will now discuss how to set this condition through the use of the "less than" circuit.
template LessThan(n) {
assert(n <= 252); //this is required since the altbn prime is upto 254 bits
signal input in[2]; //in[0] & in[1] are integers of upto n bits
component n2b = Num2Bits(n+1);
//Num2Bits takes an integer input and returns it's bitfied array
//the argument n+1 is the max size of the integer input
n2b.in <== in[0]+ (1<<n) - in[1];
out <== 1-n2b.out[n];
You might need pen and paper and need to do a bit of maths to understand how this circuit works. But the basic idea is that you can use the constraints model to describe general computations. One needs to realize that the logic required to describe something in circom is written in a very different way than a typical programming language.
As next steps, it might be worthwhile to check out the Num2Bits template mentioned above, given here. There is a lot of neat stuff in circomlib, circuits that help you achieve general computations. We recommend checking out multiplexers and comparators.
You can also check out this base2^51 multiplier. It takes two numbers, represented as arrays of base2^51 numbers, multiplies them together, and returns an array of base2^51 numbers.
Circom does NOT allow you to apply if/else conditionals on signals. Nor are you allowed to use signals as the termination condition for loops. You can use only compilation time constants (variables in circom) for these, such as n in the lessthan template. This is because otherwise, it would make the constraints dependent on the signals, effectively meaning that the polynomials being proven are dependent on the variables themselves. This is not possible with the current construction of the underlying cryptography. So far in our work, this does not seem to matter.
One more thing, when working with Circom, sometimes it's very useful to be able to see what's inside your witness file. Use the command snarkjs wej witness.wtns witness.json. This command runs on your witness.wtns file and gives you JSON.
One last thought => in theory, the entire Circom language could be replaced by a signals.json file and another constraints.json file that specifies constraints. If you can make constraints.json compatible with Circom's R1CS file, then one could use snarkJS too with this.
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Time-Domain Specifications - MATLAB & Simulink - MathWorks América Latina
Step Command Following
Step Disturbance Rejection
Transient Response Matching
This example gives a tour of available time-domain requirements for control system tuning with systune or looptune.
The TuningGoal.StepTracking requirement specifies how the tuned closed-loop system should respond to a step input. You can specify the desired response either in terms of first- or second-order characteristics, or as an explicit reference model. This requirement is satisfied when the relative gap between the actual and desired responses is small enough in the least-squares sense. For example,
R1 = TuningGoal.StepTracking('r','y',0.5);
stipulates that the closed-loop response from r to y should behave like a first-order system with time constant 0.5, while
R2 = TuningGoal.StepTracking('r','y',zpk(2,[-1 -2],-1));
specifies a second-order, non-minimum-phase behavior. Use viewGoal to visualize the desired response.
This requirement can be used to tune both SISO and MIMO step responses. In the MIMO case, the requirement ensures that each output tracks the corresponding input with minimum cross-couplings.
The TuningGoal.StepRejection requirement specifies how the tuned closed-loop system should respond to a step disturbance. You can specify worst-case values for the response amplitude, settling time, and damping of oscillations. For example,
R1 = TuningGoal.StepRejection('d','y',0.3,2,0.5);
limits the amplitude of
y\left(t\right)
to 0.3, the settling time to 2 time units, and the damping ratio to a minimum of 0.5. Use viewGoal to see the corresponding time response.
You can also use a "reference model" to specify the desired response. Note that the actual and specified responses may differ substantially when better disturbance rejection is possible. Use the TuningGoal.Transient requirement when a close match is desired. For best results, adjust the gain of the reference model so that the actual and specified responses have similar peak amplitudes (see TuningGoal.StepRejection documentation for details).
The TuningGoal.Transient requirement specifies the transient response for a specific input signal. This is a generalization of the TuningGoal.StepTracking requirement. For example,
R1 = TuningGoal.Transient('r','y',tf(1,[1 1 1]),'impulse');
requires that the tuned response from
r
y
look like the impulse response of the reference model
1/\left({s}^{2}+s+1\right)
The input signal can be an impulse, a step, a ramp, or a more general signal modeled as the impulse response of some input shaping filter. For example, a sine wave with frequency
{\omega }_{0}
can be modeled as the impulse response of
{\omega }_{0}^{2}/\left({s}^{2}+{\omega }_{0}^{2}\right)
F = tf(w0^2,[1 0 w0^2]); % input shaping filter
R2 = TuningGoal.Transient('r','y',tf(1,[1 1 1]),F);
Use the TuningGoal.LQG requirement to create a linear-quadratic-Gaussian objective for tuning the control system parameters. This objective is applicable to any control structure, not just the classical observer structure of LQG control. For example, consider the simple PID loop of Figure 2 where
d
n
are unit-variance disturbance and noise inputs, and
{S}_{d}
{S}_{n}
are lowpass and highpass filters that model the disturbance and noise spectral contents.
Figure 2: Regulation loop.
To regulate
y
around zero, you can use the following LQG criterion:
J=li{m}_{T\to \infty }E\left(\frac{1}{T}{\int }_{0}^{T}\left({y}^{2}\left(t\right)+0.05{u}^{2}\right)dt\right)
The first term in the integral penalizes the deviation of
y\left(t\right)
from zero, and the second term penalizes the control effort. Using systune, you can tune the PID controller to minimize the cost
J
. To do this, use the LQG requirement
Qyu = diag([1 0.05]); % weighting of y^2 and u^2
R4 = TuningGoal.LQG({'d','n'},{'y','u'},1,Qyu);
TuningGoal.StepTracking | TuningGoal.StepRejection | TuningGoal.Transient | TuningGoal.LQG
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Acids and Bases - Vocabulary - Course Hero
General Chemistry/Acids and Bases/Vocabulary
able to act as both a proton donor and a proton acceptor
solution being analyzed (titrated) to determine its concentration
compound that can donate a proton to another compound in solution
compound that can accept a proton from another compound in solution
type of graduated tube with a stopcock at the end that allows fine control of the release of the titrant
Common Laboratory Tools
molecule or ion formed when a Brønsted-Lowry base has accepted a proton
molecule or ion remaining after a Brønsted-Lowry acid donates its proton to another molecule or ion
pH at which all of the acid or base molecules in an acidic or basic solution have been neutralized
analysis using the mass of a precipitate formed by a reaction to determine the starting concentration
chemical that undergoes a color change near or at the equivalence point in a titration
ratio of concentration of products to reactants at equilibrium. It can be used to calculate the strength of an acid or a base
having equal concentrations of H+ and OH– ions in a solution
measure of the concentration of H+ (or H3O+) ions in solution;
\rm{pH}=-\log\left[{\rm{H_3O^+}}\right]
measure of the concentration of OH– ions in solution;
\rm{pOH}=-\log\left[{\rm{OH^-}}\right]
acid or base that completely dissociates and has a pH close to one end of the spectrum. Strong acids have a large Ka, and strong bases have a large Kb.
solution of known concentration that is used to neutralize a solution of unknown concentration (the analyte) in order to determine its concentration
quantitative method that relies on measuring the volume of a solution of a known concentration necessary to neutralize a given volume of acid or base
weak acid or base
acid or base in which only a fraction of the molecules dissociates, resulting in a low concentration of H+ or OH– ions and a pH closer to the middle of the range
<Overview>Brønsted-Lowry Acids and Bases
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Every Stieltjes moment problem has a solution in Gel'fand-Shilov spaces
October, 2003 Every Stieltjes moment problem has a solution in Gel'fand-Shilov spaces
Jaeyoung CHUNG, Soon-Yeong CHUNG, Dohan KIM
We prove that every Stieltjes problem has a solution in Gel'fand-Shilov spaces
{\mathcal{S}}^{\beta }
\beta >1
. In other words, for an arbitrary sequence
\left\{{\mu }_{n}\right\}
\varphi
in the Gel'fand-Shilov space
{\mathcal{S}}^{\beta }
with support in the positive real line whose moment
{\int }_{0}^{\infty }{x}^{n}\varphi \left(x\right)dx={\mu }_{n}
for every nonnegative integer
n
This improves the result of A. J. Duran in 1989 very much who showed that every Stieltjes moment problem has a solution in the Schwartz space
\mathcal{S}
, since the Gel'fand-Shilov space is much a smaller subspace of the Schwartz space. Duran's result already improved the result of R. P. Boas in 1939 who showed that every Stieltjes moment problem has a solution in the class of functions of bounded variation. Our result is optimal in a sense that if
\beta \le 1
we cannot find a solution of the Stieltjes problem for a given sequence.
Jaeyoung CHUNG. Soon-Yeong CHUNG. Dohan KIM. "Every Stieltjes moment problem has a solution in Gel'fand-Shilov spaces." J. Math. Soc. Japan 55 (4) 909 - 913, October, 2003. https://doi.org/10.2969/jmsj/1191418755
Keywords: existence , Gel'fand-Shilov space , Stieltjes moment problem
Jaeyoung CHUNG, Soon-Yeong CHUNG, Dohan KIM "Every Stieltjes moment problem has a solution in Gel'fand-Shilov spaces," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 55(4), 909-913, (October, 2003)
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A compliance notice for a pleasure craft of 6 metres (19.7 feet) or less in length gives maximum recommended safe limits for that boat. These limits are:
Maximum number of people who can be on board
Maximum weight (gross load capacity) the pleasure craft is designed to carry, including people, motor, equipment, etc.
Maximum outboard motor weight and horsepower (for an outboard-powered pleasure craft)
This information applies only in good weather. The number of people who can be carried safely depends on the type of pleasure craft, where people and equipment are carried, and weather and water conditions. Operators must not exceed the limits listed on the compliance notice to avoid overloading their pleasure craft.
If your pleasure craft is 6 metres (19.7 feet) or less in length, is powered by an outboard motor, and does not have a compliance notice, use the following formula to calculate the maximum number of people the pleasure craft can carry safely in good weather.
\mathrm{Number of people}=\frac{\mathrm{recommended maximum gross load \left(inkg\right)}-\mathrm{total weight of outboard engine and equipment \left(inkg\right)}}{\mathrm{÷ 75}}
For example, for an outboard-powered boat that is 6 metres (19.7 feet) or less in length with a recommended maximum gross load of 578 kg, an engine weighing 228 kg, and equipment weighing 50 kg, the number of people is 578 kg minus 278 kg divided by 75, which equals four 75-kg/165-lb people (or a total person weight of 4 × 75, or 300 kg/660 lbs).
|
Table 2 Comparison of SF-36 scores between left-behind wives and non-left-behind wives (
\overline{\mathbit{\chi }}\mathbf{±}\mathbf{\text{SD}}
|Cohen’s d|
PF 86.35 ± 13.94 91.82 ± 11.70** 0.43
RP 59.57 ± 39.48 79.39 ± 35.39** 0.53
BP 70.32 ± 14.94 74.97 ± 15.51** 0.31
GH 59.15 ± 18.98 66.35 ± 18.11** 0.39
VT 63.79 ± 16.22 70.47 ± 15.45** 0.42
SF 76.87 ± 17.48 83.99 ± 15.66** 0.43
RE 55.54 ± 39.38 81.70 ± 30.23** 0.75
MH 62.74 ± 14.46 67.80 ± 14.53** 0.35
PCS 68.85 ± 16.60 78.13 ± 16.18** 0.57
MCS 64.73 ± 15.66 75.99 ± 13.04** 0.78
PF: physical functioning; RP: role limitations caused by physical problems; BP: bodily pain; GH: general health perceptions; VT: vitality; SF: social functioning; RE: role limitations caused by emotional problems; MH: mental health; PCS: physical component summary; MCS: mental component summary.
|Cohen’s d|: absolute value of Cohen’s d.
|
torch.nn.functional.conv1d — PyTorch 1.11.0 documentation
torch.nn.functional.conv1d
torch.nn.functional.conv1d¶
torch.nn.functional.conv1d(input, weight, bias=None, stride=1, padding=0, dilation=1, groups=1) → Tensor¶
input – input tensor of shape
(\text{minibatch} , \text{in\_channels} , iW)
weight – filters of shape
(\text{out\_channels} , \frac{\text{in\_channels}}{\text{groups}} , kW)
bias – optional bias of shape
(\text{out\_channels})
. Default: None
stride – the stride of the convolving kernel. Can be a single number or a one-element tuple (sW,) . Default: 1
implicit paddings on both sides of the input. Can be a string {‘valid’, ‘same’}, single number or a one-element tuple (padW,) . Default: 0 padding='valid' is the same as no padding. padding='same' pads the input so the output has the same shape as the input. However, this mode doesn’t support any stride values other than 1.
For padding='same', if the weight is even-length and dilation is odd in any dimension, a full pad() operation may be needed internally. Lowering performance.
dilation – the spacing between kernel elements. Can be a single number or a one-element tuple (dW,) . Default: 1
\text{in\_channels}
>>> inputs = torch.randn(33, 16, 30)
>>> filters = torch.randn(20, 16, 5)
|
The financial metrics return on equity (ROE), and the return on capital employed (ROCE) are valuable tools for gauging a company's operational efficiency and the resulting potential for future growth in value. They are often used together to produce a complete evaluation of financial performance.
ROE is the percentage expression of a company's net income, as it is returned as value to shareholders. This formula allows investors and analysts an alternative measure of the company's profitability and calculates the efficiency with which a company generates profit, using the funds that shareholders have invested.
ROE is determined using the following equation:
ROE = \text{Net Income} \div \text{Shareholders' Equity}
ROE=Net Income÷Shareholders’ Equity
Regarding this equation, net income is comprised of what is earned throughout a year, minus all costs and expenses. It includes payouts made to preferred stockholders but not dividends paid to common stockholders (and the shareholders' overall equity value excludes preferred stock shares). In general, a higher ROE ratio means that the company is using its investors' money more efficiently to enhance corporate performance and allow it to grow and expand to generate increasing profits.
One recognized weakness of ROE as a performance measure lies in the fact that a disproportionate level of company debt results in a smaller base amount of equity, thus producing a higher ROE value off even a very modest amount of net income. So, it is best to view ROE value in relation to other financial efficiency measures.
ROE evaluation is often combined with an assessment of the ROCE ratio. ROCE is calculated with the following formula:
\begin{aligned} &ROCE = \frac{EBIT}{\text{capital employed}} \\ &\textbf{where:}\\ &EBIT=\text{earnings before interest and taxes}\\ \end{aligned}
ROCE=capital employedEBITwhere:EBIT=earnings before interest and taxes
ROE considers profits generated on shareholders' equity, but ROCE is the primary measure of how efficiently a company utilizes all available capital to generate additional profits. It can be more closely analyzed with ROE by substituting net income for EBIT in the calculation for ROCE.
ROCE works especially well when comparing the performance of companies in capital-intensive sectors, such as utilities and telecoms, because unlike other fundamentals, ROCE considers debt and other liabilities as well. This provides a better indication of financial performance for companies with significant debt.
To get a superior depiction of ROCE, adjustments may be needed. A company may occasionally hold cash on hand that isn't used in the business. As such, it may need to be subtracted from the Capital Employed figure to get a more accurate measure of ROCE.
The long-term ROCE is also an important indicator of performance. In general, investors tend to favor companies with stable and rising ROCE numbers over companies where ROCE is volatile year over year.
Learn What Capital Employed Is
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What is the quantum full adder
Just like in classical electronics, where you can make different types of binary adders like half adders, full adders, ripple carry adders etcetera, you can make adders in quantum circuits as well. In this example we will show how a quantum full adder is created and how this adder acts on superposition states.
The full adder adds to input bits A and B plus a carry_input bit and produces the sum and carry_output bits as output. In classical control electronics the full adder has therefore three inputs and four outputs.
Since quantum circuits are reversible, they have an equal amount of input and output qubits, therefore we define a 4-qubit function, where the input qubits are A,B, CarryIn and (zero) and the output qubits are A,B, Sum and CarryOut, see the figure below.
One possible implementation of a 2-bit full adder, using CNOT gates and Toffoli gates is the following:
For completeness we show the truth table for the full adder:
Inputs: q0 = A ; q1 = B ; q2= CarryIn
Outputs: q0 = A ; q1 = B ; q2= SumOut ; q3 = CarryOut
. . Ci B A . Co S B A
T1 0 0 0 0 - 0 0 0 0
The code below shows what happens when we use the quantum full adder to add three qubit states. You can copy and paste this code in your own editor and see what happens when you change the input states A, B and CarryIn to either
\left\lvert 0 \right\rangle
\left\lvert 1 \right\rangle
using the X-gate in the initialization function. In this example we set A = 1, B = 0 and CarryIn = 1, equal to T6 in the truth table.
Examination of the code and results
When we execute the code, setting the number of shots to 1, we get the histogram as shown below the code, which is equal to the expected output (compare to T6 in the truth table). Note: one shot is enough to determine the probability distribution
# qubit definitions
# q[0] --> A
# q[1] --> B
# q[2] --> CarryIn_SumOut
# q[3] --> CarryOut
# initialize inputs to some values
#initialize inputs A=1, B=0 and carry_in=1
{x q[0] | x q[2]}
toffoli q[0],q[1],q[3]
cnot q[0],q[1]
init add
We can also use superposition states and entangled states, such as in the following code where we set A and B to a Bell state and CarryIn to a superposition state. This is similar to setting the input to states T1, T2, T7 and T8 at the same time.
Comparing again the truth table to the histogram of probability amplitudes, we indeed see that all 4 output states that we would expect are generated.
In this example we showed how to create a simple quantum full adder. In the literature you can find other examples to add qubit states, subtract qubit states and execute more complex operations on qubit states.
|
Buildings and Curvature | EMS Press
It was the aim of the meeting to bring together international experts from the theory of buildings, differential geometry and geometric group theory. Buildings are combinatorial structures (simplicial complexes) which can be seen as simultaneously generalizing projective spaces and trees. Already from these examples it is clear that there will be interesting groups acting on buildings. Conversely, groups can be studied using their actions on given buildings. Groups coming up in this context are in particular groups having a
BN
-pair. Examples of such groups include the classical groups, simple Lie groups and algebraic groups (also over local fields), Kac-Moody groups and loop groups. This already indicates that these groups play an important role in many different areas of mathematics such as algebra, geometry, number theory, physics and analysis. Kac-Moody groups correspond to so-called twin buildings, a particularly active area in the theory of buildings. Geometric group theory is concerned with the investigation of group actions on metric spaces using the interplay of group theoretic properties and metric properties like curvature in the sense of Alexandrov, or CAT
(0)
-spaces. The geometric realization of a building is a metric space with interesting curvature properties on which the above mentioned groups as well as their subgroups like uniform lattices or arithmetic groups act in a natural way by isometries. In this respect there are a number of canonical connections between the theory of buildings and geometric group theory. One of the current problems concerns the characterization of buildings as metric spaces. In differential geometry these aspects also play an important role, e.g. in connection with Hadamard manifolds, (simply connected Riemannian manifolds of nonpositive curvature). A special role is played by the Riemannian symmetric spaces and their quotients of finite volume which one wants to characterize geometrically. By considering the fundamental groups, one obtains discrete group actions also studied in geometric group theory. Buildings come up in differential geometry as the compactifications of Riemannian symmetric spaces yielding examples of topological buildings. Asymptotic cones (and ultrapowers) of symmetric spaces present non-discrete affine buildings and create new and interesting relations to model theory. These constructions are important in new proofs of differential geometric rigidity theorems, like Mostow Rigidity and the Margulis Conjecture. This shows that there are close connections between the areas, and this meeting was the first in a number of years in Oberwolfach having these connections as its topic. Geometric group theory has recently introduced interesting aspects into the theory of buildings, in particular the hyperbolic buildings. Conversely, new developments in the theory of buildings, e.g. the twin buildings have interesting group theoretic applications, for example in the theory of
S
-arithmetic groups or in the theory of Kac-Moody groups. All these aspects played an important part in this meeting and the interaction between the participants from different areas was very lively.
Ernst Heintze, Linus Kramer, Bernhard Mühlherr, Bertrand Rémy, Buildings and Curvature. Oberwolfach Rep. 1 (2004), no. 2, pp. 1233–1284
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EUDML | Rationally connected varieties over local fields. EuDML | Rationally connected varieties over local fields.
Rationally connected varieties over local fields.
Kollár, János. "Rationally connected varieties over local fields.." Annals of Mathematics. Second Series 150.1 (1999): 357-367. <http://eudml.org/doc/121428>.
author = {Kollár, János},
keywords = {rationally connected varieties; local fields; unirational variety; chain of rational curves},
title = {Rationally connected varieties over local fields.},
TI - Rationally connected varieties over local fields.
KW - rationally connected varieties; local fields; unirational variety; chain of rational curves
Alena Pirutka,
R
-équivalence sur les familles de variétés rationnelles et méthode de la descente
Viatcheslav Kharlamov, Variétés de Fano réelles
rationally connected varieties, local fields, unirational variety, chain of rational curves
Rational and unirational varieties
Articles by Kollár
|
EUDML | A generalization of Moore-Penrose biorthogonal systems. EuDML | A generalization of Moore-Penrose biorthogonal systems.
A generalization of Moore-Penrose biorthogonal systems.
Matsuura, Masaya
Matsuura, Masaya. "A generalization of Moore-Penrose biorthogonal systems.." ELA. The Electronic Journal of Linear Algebra [electronic only] 10 (2003): 146-154. <http://eudml.org/doc/123705>.
@article{Matsuura2003,
author = {Matsuura, Masaya},
keywords = {linear transformation; Moore-Penrose inverses; Gram matrices; biorthogonal systems; generalized inverse; reflexive -inverses; reflexive -inverses},
title = {A generalization of Moore-Penrose biorthogonal systems.},
AU - Matsuura, Masaya
TI - A generalization of Moore-Penrose biorthogonal systems.
KW - linear transformation; Moore-Penrose inverses; Gram matrices; biorthogonal systems; generalized inverse; reflexive -inverses; reflexive -inverses
linear transformation, Moore-Penrose inverses, Gram matrices, biorthogonal systems, generalized inverse, reflexive
g
-inverses, reflexive
g
-inverses
Articles by Matsuura
|
Rs Aggarwal 2017 for Class 8 Math Chapter 15 - Quadrilaterals
Rs Aggarwal 2017 Solutions for Class 8 Math Chapter 15 Quadrilaterals are provided here with simple step-by-step explanations. These solutions for Quadrilaterals are extremely popular among Class 8 students for Math Quadrilaterals Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rs Aggarwal 2017 Book of Class 8 Math Chapter 15 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rs Aggarwal 2017 Solutions. All Rs Aggarwal 2017 Solutions for class Class 8 Math are prepared by experts and are 100% accurate.
(i) A quadrilateral has ......... sides.
(ii) A quadrilateral has ......... angles.
(iii) A quadrilateral has ......... vertices, no three of which are .........
(iv) A quadrilateral has ......... diagonals.
(v) A diagonal of a quadrilateral is a line segment that joins two ......... vertices of the quadrilateral.
(vi) The sum of the angles of a quadrilateral is.........
(iii) 4, co-linear
(i) How many pairs of adjacent sides are there? Name them.
(ii) How many pairs of opposite sides are there? Name them.
(iii) How many pairs of adjacent angles are there? Name them.
(iv) How many pairs of opposite angles are there? Name them.
(v) How many diagonals are there? Name them.
(i) There are four pairs of adjacent sides, namely (AB,BC), (BC,CD), (CD,DA) and (DA,AB).
(ii) There are two pairs of opposite sides, namely (AB,DC) and (AD,BC).
(iii) There are four pairs of adjacent angles, namely
\left(\angle A,\angle B\right), \left(\angle B, \angle C\right), \left(\angle C,\angle D\right) and \left(\angle D,\angle A\right)
(iv) There are two pairs of opposite angles, namely
\left(\angle A,\angle C\right) and \left(\angle B,\angle D\right)
(v) There are two diagonals, namely AC and BD.
Now, we know that the sum of the angles of a triangle is 180°.
△ABC:\phantom{\rule{0ex}{0ex}}\angle 2+\angle 4+\angle B={180}^{o} ... \left(1\right)
△ADC:\phantom{\rule{0ex}{0ex}} \angle 1+\angle 3+\angle D={180}^{o} ... \left(2\right)
\left(\angle 1+\angle 2+\angle 3+\angle 4\right)+\angle B+\angle D={360}^{o}\phantom{\rule{0ex}{0ex}}
\angle A+\angle B+\angle C+\angle D={360}^{o}\phantom{\rule{0ex}{0ex}}
Hence, the sum of all the angles of a quadrilateral is 360°.
Sum of all the four angles of a quadrilateral is 360°.
\mathrm{Let} \mathrm{the} \mathrm{unknown} \mathrm{angle} \mathrm{be} x°.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}76+54+108+x=360\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}238+x=360\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}x=122
The fourth angle measures 122°.
The angles of a quadrilateral are in the ratio 3 : 5 : 7 : 9. Find the measure of each of these angles.
\mathrm{Let} \mathrm{the} \mathrm{measures} \mathrm{of} \mathrm{the} \mathrm{angles} \mathrm{of} \mathrm{the} \mathrm{given} \mathrm{quadrilateral} \mathrm{be} \left(3x\right)°,\left(5x\right)°,\left(7x\right)° \mathrm{and} \left(9x\right)°. \phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Sum} \mathrm{of} \mathrm{all} \mathrm{the} \mathrm{angles} \mathrm{of} \mathrm{a} \mathrm{quadrilateral} \mathrm{is} {360}^{\mathrm{o}}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\therefore 3x+5x+7x+9x=360\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}24x=360\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}x=15\phantom{\rule{0ex}{0ex}}
\mathrm{Angles} \mathrm{measure}: \phantom{\rule{0ex}{0ex}}\left(3×15\right)°={45}^{°}\phantom{\rule{0ex}{0ex}}\left(5×15\right)°={75}^{°}\phantom{\rule{0ex}{0ex}}\left(7×15\right)°={105}^{°}\phantom{\rule{0ex}{0ex}}\left(9×15\right)°={135}^{°}
A quadrilateral has three acute angles, each measuring 75°. Find the measure of the fourth angle.
Sum of the four angles of a quadrilateral is 360°.
If the unknown angle is
x
°, then:
75+75+75+x=360\phantom{\rule{0ex}{0ex}}x=360-225=135\phantom{\rule{0ex}{0ex}}
Let the three angles measure
x°
Sum of all the angles of a quadrilateral is 360°.
\therefore x+x+x+120=360\phantom{\rule{0ex}{0ex}}3x+120=360\phantom{\rule{0ex}{0ex}}3x=240\phantom{\rule{0ex}{0ex}}x=\frac{240}{3}=80
Let the two unknown angles measure
x°
\therefore 85+75+x+x=360\phantom{\rule{0ex}{0ex}}160+2x=360\phantom{\rule{0ex}{0ex}}2x=360-160=200\phantom{\rule{0ex}{0ex}}x=100
Each of the equal angle measures 100°.
In the adjacent figure, the bisectors of ∠A and ∠B meet in a point P. If ∠C = 100° and ∠D = 60°, find the measure of ∠APB.
\therefore \angle A+\angle B+{60}^{o}+{100}^{o}=360°\phantom{\rule{0ex}{0ex}}\angle A+\angle B=360-100-60=200°\phantom{\rule{0ex}{0ex}}or\phantom{\rule{0ex}{0ex}}\frac{1}{2}\left(\angle A+\angle B\right)=100° ... \left(1\right)\phantom{\rule{0ex}{0ex}}\mathrm{Sum} \mathrm{of} \mathrm{the} \mathrm{angles} \mathrm{of} \mathrm{a} \mathrm{triangle} \mathrm{is} 180°.\phantom{\rule{0ex}{0ex}}\mathrm{In} △APB:\phantom{\rule{0ex}{0ex}} \frac{1}{2}\left(\angle A+\angle B\right)+\angle P=180° \left(\mathrm{because} AP \mathrm{and} PB \mathrm{are} \mathrm{bisectors} \mathrm{of} \angle A \mathrm{and} \angle B\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}U\mathrm{sing} \mathrm{equation} \left(1\right):\phantom{\rule{0ex}{0ex}}100°+\angle P=180°\phantom{\rule{0ex}{0ex}}⇒\angle P=80°\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}
\angle APB=80°
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EUDML | What is the role of twistors in supergeometry. EuDML | What is the role of twistors in supergeometry.
What is the role of twistors in supergeometry.
Cortés, Vicente. "What is the role of twistors in supergeometry.." General Mathematics 5 (1997): 127-133. <http://eudml.org/doc/232072>.
@article{Cortés1997,
author = {Cortés, Vicente},
keywords = {spinor; twistor; supergeometry; -structure; infinitesimal automorphism; -structure},
title = {What is the role of twistors in supergeometry.},
AU - Cortés, Vicente
TI - What is the role of twistors in supergeometry.
KW - spinor; twistor; supergeometry; -structure; infinitesimal automorphism; -structure
spinor, twistor, supergeometry,
G
-structure, infinitesimal automorphism,
G
Spin and Spin
{}^{c}
Twistor methods
Supermanifolds and graded manifolds
Articles by Cortés
|
Computation | Free Full-Text | Hybrid Feedback Control for Exponential Stability and Robust H∞ Control of a Class of Uncertain Neural Network with Mixed Interval and Distributed Time-Varying Delays
Numerical Investigation of a Radially Cooled Turbine Guide Vane Using Air and Steam as a Cooling Medium
LMI-Based Results on Robust Exponential Passivity of Uncertain Neutral-Type Neural Networks with Mixed Interval Time-Varying Delays via the Reciprocally Convex Combination Technique
Chantawat, C.
Botmart, T.
Supama, R.
Weera, W.
Noinang, S.
Hybrid Feedback Control for Exponential Stability and Robust H∞ Control of a Class of Uncertain Neural Network with Mixed Interval and Distributed Time-Varying Delays
Charuwat Chantawat
Rattaporn Supama
Wajaree Weera
Sakda Noinang
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
Department of Mathematics, Faculty of Science, University of Phayao, Phayao 56000, Thailand
Department of Mathematics Statistics and Computer, Faculty of Science, Ubon Ratchathani University, Ubon Ratchathani 34190, Thailand
Academic Editors: Yongwimon Lenbury, Ravi P. Agarwal, Philip Broadbridge and Dongwoo Sheen
(This article belongs to the Special Issue Proceedings of the International Conference in Mathematics and Applications 2020_Mahidol University)
This paper is concerned the problem of robust
{H}_{\infty }
control for uncertain neural networks with mixed time-varying delays comprising different interval and distributed time-varying delays via hybrid feedback control. The interval and distributed time-varying delays are not necessary to be differentiable. The main purpose of this research is to estimate robust exponential stability of uncertain neural network with
{H}_{\infty }
performance attenuation level
\gamma
. The key features of the approach include the introduction of a new Lyapunov–Krasovskii functional (LKF) with triple integral terms, the employment of a tighter bounding technique, some slack matrices and newly introduced convex combination condition in the calculation, improved delay-dependent sufficient conditions for the robust
{H}_{\infty }
control with exponential stability of the system are obtained in terms of linear matrix inequalities (LMIs). The results of this paper complement the previously known ones. Finally, a numerical example is presented to show the effectiveness of the proposed methods. View Full-Text
Keywords: neural networks; H∞ control; hybrid feedback control; mixed time-varying delay neural networks; H∞ control; hybrid feedback control; mixed time-varying delay
Chantawat, C.; Botmart, T.; Supama, R.; Weera, W.; Noinang, S. Hybrid Feedback Control for Exponential Stability and Robust H∞ Control of a Class of Uncertain Neural Network with Mixed Interval and Distributed Time-Varying Delays. Computation 2021, 9, 62. https://doi.org/10.3390/computation9060062
Chantawat C, Botmart T, Supama R, Weera W, Noinang S. Hybrid Feedback Control for Exponential Stability and Robust H∞ Control of a Class of Uncertain Neural Network with Mixed Interval and Distributed Time-Varying Delays. Computation. 2021; 9(6):62. https://doi.org/10.3390/computation9060062
Chantawat, Charuwat, Thongchai Botmart, Rattaporn Supama, Wajaree Weera, and Sakda Noinang. 2021. "Hybrid Feedback Control for Exponential Stability and Robust H∞ Control of a Class of Uncertain Neural Network with Mixed Interval and Distributed Time-Varying Delays" Computation 9, no. 6: 62. https://doi.org/10.3390/computation9060062
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Create biquad or double-biquad antenna - MATLAB - MathWorks América Latina
Create and View Biquad Antenna
Impedance of Biquad Antenna
Create biquad or double-biquad antenna
The biquad antenna is center fed and symmetric about its origin. The default length is chosen for an operating frequency of 2.8 GHz.
The width of the strip is related to the diameter an equivalent cylinder:
w=2d=4r
For a given cylinder radius, use the cylinder2strip utility function to calculate the equivalent width. The default strip dipole is center-fed. The feed point coincides with the origin. The origin is located on the yz- plane.
bq = biquad
bq = biquad(Name,Value)
bq = biquad creates a biquad antenna.
bq = biquad(Name,Value) creates a biquad antenna with additional properties specified by one or more name-value pair arguments. Name is the property name and Value is the corresponding value. You can specify several name-value pair arguments in any order as Name1, Value1, ..., NameN, ValueN. Properties not specified retain their default values.
NumLoops — Number of loops
Number of loops for the biquad, specified as a scalar integer. Setting this property to 4 supports a double biquad antenna.
Example: 'NumLoops',4
Length of two arms, specified as a scalar in meters. The default length is chosen for an operating frequency of 2.8 GHz.
Example: 'ArmLength',0.0206
Width — Biquad arm width
Biquad arm width, specified as a scalar in meters.
Example: 'Width',0.006
ArmElevation — Angle formed by biquad arms to xy- plane
Angle formed by biquad arms to the xy- plane, specified a scalar in meters.
Example: 'ArmElevation', 50
Example: bq.Load = lumpedElement('Impedance',75)
Create a biquad antenna with arm angles at 50 degrees and view it.
bq = biquad('ArmElevation',50);
show(bq)
Calculate the impedance of a biquad antenna over a frequency span 2.5GHz-3GHz.
impedance(bq,linspace(2.5e9,3e9,51));
Create and view a double biquad antenna using default property values.
ant = biquad('NumLoops',4)
biquad with properties:
NumLoops: 4
ArmLength: 0.0305
dipole | dipoleFolded | loopCircular
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Trochoid Knowpia
In geometry, a trochoid (from the Greek word for wheel, "trochos") is a roulette formed by a circle rolling along a line. It is the curve traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line.[1] If the point is on the circle, the trochoid is called common (also known as a cycloid); if the point is inside the circle, the trochoid is curtate; and if the point is outside the circle, the trochoid is prolate. The word "trochoid" was coined by Gilles de Roberval.[citation needed]
A cycloid (a common trochoid) generated by a rolling circle
Basic descriptionEdit
A prolate trochoid with b/a = 5/4
A curtate trochoid with b/a = 4/5
As a circle of radius a rolls without slipping along a line L, the center C moves parallel to L, and every other point P in the rotating plane rigidly attached to the circle traces the curve called the trochoid. Let CP = b. Parametric equations of the trochoid for which L is the x-axis are
{\displaystyle x=a\theta -b\sin \theta \,}
{\displaystyle y=a-b\cos \theta \,}
where θ is the variable angle through which the circle rolls.
Curtate, common, prolateEdit
If P lies inside the circle (b < a), on its circumference (b = a), or outside (b > a), the trochoid is described as being curtate ("contracted"), common, or prolate ("extended"), respectively.[2] A curtate trochoid is traced by a pedal when a normally geared bicycle is pedaled along a straight line.[3] A prolate trochoid is traced by the tip of a paddle when a boat is driven with constant velocity by paddle wheels; this curve contains loops. A common trochoid, also called a cycloid, has cusps at the points where P touches the L.
A more general approach would define a trochoid as the locus of a point
{\displaystyle (x,y)}
orbiting at a constant rate around an axis located at
{\displaystyle (x',y')}
{\displaystyle x=x'+r_{1}\cos(\omega _{1}t+\phi _{1}),\ y=y'+r_{1}\sin(\omega _{1}t+\phi _{1}),\ r_{1}>0,}
which axis is being translated in the x-y-plane at a constant rate in either a straight line,
{\displaystyle {\begin{array}{lcl}x'=x_{0}+v_{2x}t,\ y'=y_{0}+v_{2y}t\\\therefore x=x_{0}+r_{1}\cos(\omega _{1}t+\phi _{1})+v_{2x}t,\ y=y_{0}+r_{1}\sin(\omega _{1}t+\phi _{1})+v_{2y}t,\\\end{array}}}
or a circular path (another orbit) around
{\displaystyle (x_{0},y_{0})}
(the hypotrochoid/epitrochoid case),
{\displaystyle {\begin{array}{lcl}x'=x_{0}+r_{2}\cos(\omega _{2}t+\phi _{2}),\ y'=y_{0}+r_{2}\sin(\omega _{2}t+\phi _{2}),\ r_{2}\geq 0\\\therefore x=x_{0}+r_{1}\cos(\omega _{1}t+\phi _{1})+r_{2}\cos(\omega _{2}t+\phi _{2}),\ y=y_{0}+r_{1}\sin(\omega _{1}t+\phi _{1})+r_{2}\sin(\omega _{2}t+\phi _{2}),\\\end{array}}}
The ratio of the rates of motion and whether the moving axis translates in a straight or circular path determines the shape of the trochoid. In the case of a straight path, one full rotation coincides with one period of a periodic (repeating) locus. In the case of a circular path for the moving axis, the locus is periodic only if the ratio of these angular motions,
{\displaystyle \omega _{1}/\omega _{2}}
, is a rational number, say
{\displaystyle p/q}
{\displaystyle p}
{\displaystyle q}
are coprime, in which case, one period consists of
{\displaystyle p}
orbits around the moving axis and
{\displaystyle q}
orbits of the moving axis around the point
{\displaystyle (x_{0},y_{0})}
. The special cases of the epicycloid and hypocycloid, generated by tracing the locus of a point on the perimeter of a circle of radius
{\displaystyle r_{1}}
while it is rolled on the perimeter of a stationary circle of radius
{\displaystyle R}
, have the following properties:
{\displaystyle {\begin{array}{lcl}{\text{epicycloid: }}&\omega _{1}/\omega _{2}&=p/q=r_{2}/r_{1}=R/r_{1}+1,\ |p-q|{\text{ cusps}}\\{\text{hypocycloid: }}&\omega _{1}/\omega _{2}&=p/q=-r_{2}/r_{1}=-(R/r_{1}-1),\ |p-q|=|p|+|q|{\text{ cusps}}\end{array}}}
{\displaystyle r_{2}}
is the radius of the orbit of the moving axis. The number of cusps given above also hold true for any epitrochoid and hypotrochoid, with "cusps" replaced by either "radial maxima" or "radial minima".
^ Weisstein, Eric W. "Trochoid". MathWorld.
^ "Trochoid". Xah Math. Retrieved October 4, 2014.
^ The Bicycle Pulling Puzzle. YouTube. Archived from the original on 2021-12-11.
Online experiments with the Trochoid using JSXGraph
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Power transmission system with chain and two sprockets - MATLAB - MathWorks América Latina
Sprocket A pitch radius
Sprocket B pitch radius
Chain slack length
Chain damping
Viscous friction coefficient of sprocket A
Viscous friction coefficient of sprocket B
Chain maximum tension
Power transmission system with chain and two sprockets
The Chain Drive block represents a power transmission system with a chain and two sprockets. The chain meshes with the sprockets, transmitting rotary motion between the two. Power transmission can occur in reverse, that is, from driven to driver sprocket, due to external loads. This condition is known as back-driving.
The drive chain is compliant. It can stretch under tension or slacken if loose. The compliance model consists of a linear spring-damper set in a parallel arrangement. The spring resists tensile strain in the chain. The damper resists tensile motion between chain elements.
The spring and damper forces act directly on the sprockets that the chain connects. The spring force is present when one chain branch is taut. The damper force is present continuously. To represent and report a failure condition, the simulation stops and generates an error if the net tensile force in the chain exceeds the specified maximum tension value.
The block accounts for viscous friction at the sprocket joint bearings. During motion, viscous friction causes power transmission losses, reducing chain-drive efficiency. These losses compound due to chain damping. To eliminate power transmission losses in the chain drive, in the Dynamic settings, set the parameters for viscous friction and chain damping to zero.
The tensile strain rate in the chain is the difference between the sprocket tangential velocities, which are each the product of the angular velocity and pitch radii. Mathematically,
\stackrel{˙}{x}={\omega }_{A}{R}_{A}-{\omega }_{B}{R}_{B},
x is the tensile strain.
ωA, ωB are the sprocket angular velocities.
RA, RB are the sprocket pitch radii.
The figure shows the relevant variables.
The chain tensile force is the net sum of the spring and damper forces. The spring force is the product of the tensile strain and the spring stiffness constant. This force is zero when the tensile strain is smaller than the chain slack. The damper force is the product of the tensile strain rate and the damping coefficient. Mathematically,
F=\left\{\begin{array}{cc}-\left(x-\frac{S}{2}\right)k-\stackrel{˙}{x}b,& x>\frac{S}{2}\\ -\stackrel{˙}{x}b,& \frac{S}{2}\ge x\ge -\frac{S}{2}\\ -\left(x+\frac{S}{2}\right)k-\stackrel{˙}{x}b,& x<-\frac{S}{2}\end{array},
S is the chain slack.
k is the spring stiffness constant.
b is the damper coefficient.
The chain exerts a torque on each sprocket equal to the product of the tensile force and the sprocket pitch radius. The two torques act in opposite directions according to these equations:
{\tau }_{A}=-F·{R}_{A}
{\tau }_{B}=-F·{R}_{B}
τA is the torque that the chain applies on sprocket A.
τB is the torque that the chain applies on sprocket B.
In terms of velocity and consideration of friction, these equations apply:
{\omega }_{A}·{R}_{A}={\omega }_{B}·{R}_{B}
\left({\tau }_{A}-{\mu }_{A}·{\omega }_{A}\right){R}_{B}=-\left({\tau }_{B}-{\mu }_{B}·{\omega }_{B}\right){R}_{B}
ωA is the rotational velocity for sprocket A.
ωB is the rotational velocity for sprocket B.
μA is the coefficient of viscous friction for sprocket A.
μB is the coefficient of viscous friction for sprocket B.
The sprocket tooth ratio equals the sprocket pitch radius ratio.
Chain inertia is negligible.
A — Sprocket A
Conserving rotational port associated with sprocket A.
B — Sprocket B
Conserving rotational port associated with sprocket B.
Chain model — Chain compliance and backlash model
Ideal - no chain compliance or backlash (default) | Model chain compliance and backlash
Compliance and backlash model for the block:
Ideal - no chain compliance or backlash — Do not model chain stiffness, damping, or backlash.
Model chain compliance and backlash — Model chain stiffness, damping, and backlash.
If this parameter is set Model chain compliance and backlash, related parameters and variables are enabled.
Sprocket A pitch radius — Sprocket A pitch radius
80 mm (default) | scalar
Radius of the pitch circle for sprocket A. The pitch circle is an imaginary circle passing through the contact point between a chain roller and a sprocket cog at full engagement.
Sprocket B pitch radius — Sprocket B pitch radius
Radius of the pitch circle for sprocket B. The pitch circle is an imaginary circle passing through the contact point between a chain roller and a sprocket cog at full engagement.
Chain slack length — Maximum slack length
Maximum distance the loose branch of the drive chain can move before it is taut. This distance equals the length difference between actual and fully taut drive chains.
If one sprocket is held in place while the top chain branch is taut, then the slack length is the tangential distance that the second sprocket must rotate before the lower chain branch becomes taut.
This parameter is enabled when Chain model is set to Model chain compliance and backlash.
Chain stiffness — Linear spring constant
1e5 N/m (default) | scalar
Linear spring constant in the chain compliance model. This constant describes the chain resistance to strain. The spring element accounts for elastic energy storage in the chain due to deformation.
Chain damping — Linear damping coefficient
5 N/(m/s) (default) | scalar
Linear damping coefficient in the chain compliance model. This coefficient describes the resistance to tensile motion between adjacent elements in the chain. The damper element accounts for power losses in the chain due to deformation.
Viscous friction coefficient of sprocket A — Sprocket A friction
0.001 N*m/(rad/s) (default)
Friction coefficient due to the rolling action of the joint bearing for sprocket A in the presence of a viscous lubricant.
Viscous friction coefficient of sprocket B — Sprocket B friction
Friction coefficient due to the rolling action of the joint bearing for sprocket B in the presence of a viscous lubricant.
These settings are enabled when Chain model is set to Model chain compliance and backlash.
Select whether to constrain the maximum tensile force in the drive chain.
No maximum tension — Chain tension can be arbitrarily large during simulation.
Specify maximum tension — Chain tension must remain lower than a maximum value. If the tension exceeds this value, the simulation generates an error and stops.
When this parameter is set to Specify maximum tension, related parameters are enabled.
Chain maximum tension — Upper tension limit
1e6 N (default)
Maximum allowed value of the tensile force acting in the chain.
This parameter is enabled when both of these conditions are met:
In the Geometry settings, the Chain model parameter is set to Model chain compliance and backlash.
Im the Maximum tension settings, the Maximum tension parameter is set to Specify maximum tension.
Simple Gear | Variable Ratio Transmission
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Possible_world Knowpia
A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their metaphysical status has been a subject of controversy in philosophy, with modal realists such as David Lewis arguing that they are literally existing alternate realities, and others such as Robert Stalnaker arguing that they are not.
Possible worlds are one of the foundational concepts in modal and intensional logics. Formulas in these logics are used to represent statements about what might be true, what should be true, what one believes to be true and so forth. To give these statements a formal interpretation, logicians use structures containing possible worlds. For instance, in the relational semantics for classical propositional modal logic, the formula
{\displaystyle \Diamond P}
(read as "possibly P") is actually true if and only if
{\displaystyle P}
is true at some world which is accessible from the actual world.
Possible worlds play a central role in the work of both linguists and philosophers working in formal semantics. Contemporary formal semantics is couched in formal systems rooted in Montague grammar, which is itself built on Richard Montague's intensional logic.[1] Contemporary research in semantics typically uses possible worlds as formal tools without committing to a particular theory of their metaphysical status. The term possible world is retained even by those who attach no metaphysical significance to them.
Possible worlds are often regarded with suspicion, which is why their proponents have struggled to find arguments in their favor.[2] An often-cited argument is called the argument from ways. It defines possible worlds as "ways how things could have been" and relies for its premises and inferences on assumptions from natural language,[3][4][5] for example:
The central step of this argument happens at (2) where the plausible (1) is interpreted in a way that involves quantification over "ways". Many philosophers, following Willard Van Orman Quine,[6] hold that quantification entails ontological commitments, in this case, a commitment to the existence of possible worlds. Quine himself restricted his method to scientific theories, but others have applied it also to natural language, for example, Amie L. Thomasson in her easy approach to ontology.[7] The strength of the argument from ways depends on these assumptions and may be challenged by casting doubt on the quantifier-method of ontology or on the reliability of natural language as a guide to ontology.
Philosophical issues and applicationsEdit
The ontological status of possible worlds has provoked intense debate. David Lewis famously advocated for a position known as modal realism, which holds that possible worlds are real, concrete places which exist in the exact same sense that the actual world exists. On Lewis's account, the actual world is special only in that we live there. This doctrine is called the indexicality of actuality since it can be understood as claiming that the term "actual" is an indexical, like "now" and "here". Lewis gave a variety of arguments for this position. He argued that just as the reality of atoms is demonstrated by their explanatory power in physics, so too are possible worlds justified by their explanatory power in philosophy. He also argued that possible worlds must be real because they are simply "ways things could have been" and nobody doubts that such things exist. Finally, he argued that they could not be reduced to more "ontologically respectable" entities such as maximally consistent sets of propositions without rendering theories of modality circular. (He referred to these theories as "ersatz modal realism" which try to get the benefits of possible worlds semantics "on the cheap".)[8][9]
Modal realism is controversial. W.V. Quine rejected it as "metaphysically extravagant".[10] Stalnaker responded to Lewis's arguments by pointing out that a way things could have been is not itself a world, but rather a property that such a world can have. Since properties can exist without them applying to any existing objects, there's no reason to conclude that other worlds like ours exist. Another of Stalnaker's arguments attacks Lewis's indexicality theory of actuality. Stalnaker argues that even if the English word "actual" is an indexical, that doesn't mean that other worlds exist. For comparison, one can use the indexical "I" without believing that other people actually exist.[11] Some philosophers instead endorse the view of possible worlds as maximally consistent sets of propositions or descriptions, while others such as Saul Kripke treat them as purely formal (i.e. mathematical) devices.[12]
Explicating necessity and possibilityEdit
At least since Aristotle, philosophers have been greatly concerned with the logical statuses of propositions, e.g. necessity, contingency, and impossibility. In the twentieth century, possible worlds have been used to explicate these notions. In modal logic, a proposition is understood in terms of the worlds in which it is true and worlds in which it is false. Thus, equivalences like the following have been proposed:
False propositions are those that are false in the actual world (for example: "Ronald Reagan became president in 1969").
Necessarily true propositions (often simply called necessary propositions) are those that are true in all possible worlds (for example: "2 + 2 = 4"; "all bachelors are unmarried").[13]
Possible worlds play a central role in many other debates in philosophy. These include debates about the Zombie Argument, and physicalism and supervenience in the philosophy of mind. Many debates in the philosophy of religion have been reawakened by the use of possible worlds.
The idea of possible worlds is most commonly attributed to Gottfried Leibniz, who spoke of possible worlds as ideas in the mind of God and used the notion to argue that our actually created world must be "the best of all possible worlds". Arthur Schopenhauer argued that on the contrary our world must be the worst of all possible worlds, because if it were only a little worse it could not continue to exist.[14] Scholars have found implicit earlier traces of the idea of possible worlds in the works of René Descartes,[15] a major influence on Leibniz, Al-Ghazali (The Incoherence of the Philosophers), Averroes (The Incoherence of the Incoherence),[16] Fakhr al-Din al-Razi (Matalib al-'Aliya)[17] and John Duns Scotus.[16]
The modern philosophical use of the notion was pioneered by David Lewis and Saul Kripke.
^ "Formal Semantics: Origins, Issues, Early Impact". Baltic International Yearbook of Cognition, Logic and Communication. This Proceeding of the Symposium for Cognition, Logic and Communication. Vol. 6. 2011.
^ Lewis, David (1973). Counterfactuals. John Wiley & Sons.
^ Lewis, David (1986). On the plurality of worlds. Wiley-Blackwell.
^ Stalnaker, Robert (1976). "Possible worlds". Noûs. 10 (1). doi:10.2307/2214477. JSTOR 2214477.
^ Kripke, Saul (1972). Naming and necessity. Harvard University Press.
^ See "A Priori and A Posteriori" (author: Jason S. Baehr), at Internet Encyclopedia of Philosophy: "A necessary proposition is one the truth value of which remains constant across all possible worlds. Thus a necessarily true proposition is one that is true in every possible world, and a necessarily false proposition is one that is false in every possible world. By contrast, the truth value of contingent propositions is not fixed across all possible worlds: for any contingent proposition, there is at least one possible world in which it is true and at least one possible world in which it is false." Accessed 7 July 2012.
^ Arthur Schopenhauer, "Die Welt als Wille und Vorstellung," supplement to the 4th book "Von der Nichtigkeit und dem Leiden des Lebens" p. 2222, see also R.B. Haldane and J. Kemp's translation "On the Vanity and Suffering of Life" pp 395-6
^ "Nor could we doubt that, if God had created many worlds, they would not be as true in all of them as in this one. Thus those who could examine sufficiently the consequences of these truths and of our rules, could be able to discover effects by their causes, and, to explain myself in the language of the schools, they could have a priori demonstrations of everything that could be produced in this new world." -The World, Chapter VII
^ a b Taneli Kukkonen (2000), "Possible Worlds in the Tahâfut al-Falâsifa: Al-Ghazâlî on Creation and Contingency", Journal of the History of Philosophy, 38 (4): 479–502, doi:10.1353/hph.2005.0033, S2CID 170995877
^ Adi Setia (2004), "Fakhr Al-Din Al-Razi on Physics and the Nature of the Physical World: A Preliminary Survey", Islam & Science, 2, retrieved 2010-03-02
Brian Skyrms, "Possible Worlds, Physics and Metaphysics" (1976. Philosophical Studies 30)
"Possible Worlds" entry in the Stanford Encyclopedia of Philosophy
"Possible worlds: what they are good for and what they are" — Alexander Pruss.
"Possible Objects" entry in the Stanford Encyclopedia of Philosophy
"Impossible Worlds" entry in the Stanford Encyclopedia of Philosophy
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Basic Topological and Geometric Properties of Orlicz Spaces over an Arbitrary Set of Atoms | EMS Press
Basic Topological and Geometric Properties of Orlicz Spaces over an Arbitrary Set of Atoms
Orlicz spaces \lfg\ over an arbitrary set
\Gamma
, being a natural generalizations of Orlicz sequence spaces are studied. The following problems in these spaces are considered: relationships between the Luxemburg norm and the modular, Fatou property, relationships between the Luxemburg norm and the Orlicz norm, equality of the Orlicz norm and the Amemiya norm, order continuous elements, a formula for the norm in the quotient space
\lfg/\hfg
in terms of the modular
I_\f
for both the Luxemburg and the Orlicz norm, the problem when the equality of the space \lfg\ and its subspace \hfg\ holds, isometric representation of the dual spaces
(\hfg)^*
(\lfg)^*
(\hfgo)^*
(\lfgo)^*
, representation of support functionals, criteria for smooth points and extreme points of
S(\lfg)
and problem of the existence of such points. It is worthy noticing that the problem of the existence of smooth points on
S(\lfg)
depends essentially on the assumption if
\G
is countable or not.
Henryk Hudzik, Lucjan Szymaszkiewicz, Basic Topological and Geometric Properties of Orlicz Spaces over an Arbitrary Set of Atoms. Z. Anal. Anwend. 27 (2008), no. 4, pp. 425–449
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Sommerfeld radiation condition - SEG Wiki
We consider a wave propagation problem in a medium
{\displaystyle D}
bounded by a surface
{\displaystyle \partial D}
. By the Green's function method, we can represent the field at a distinguished location
{\displaystyle {\boldsymbol {x}}_{0}}
using the Green's function method integral equation of the field
{\displaystyle u(\mathbf {x_{0}} )=\int _{D}f(\mathbf {x} )g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )\;dV+\int _{\partial D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} ){\frac {\partial u(\mathbf {x} )}{\partial n}}-u(\mathbf {x} ){\frac {\partial g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )}{\partial n}}\right\}\;dS.}
We can pose boundary value problems by specifying values of either the field
{\displaystyle u(\mathbf {x} )}
(called a Dirichlet condition) or derivatives of the field
{\displaystyle {\frac {\partial u(\mathbf {x} )}{\partial n}}}
, (called a Neumann condition.
Unbounded medium
For this discussion, we consider the boundary condition of an unbounded medium. In this case the distance between the support of the source function
{\displaystyle f}
{\displaystyle \partial D}
is taken to tend to infinity.
In this case the boundary condition is a radiation condition. The basic notion of the radiation condition is that the field is outward propagating from the region of interest described by the the volume integral, and that the wavefield falls off sufficiently rapidly that the surface integral contribution tends to zero.
A mistake that is often made it to imagine that this means that the terms in the integrand cancel. This is not the case. This an asymptotic analysis.
The Helmholtz equation and the Sommerfeld radiation condition
For the problem, in three dimensions, where the Helmholtz equation is the governing equation of the problem, the Sommerfeld radiation condition takes the form that any field
{\displaystyle U({\boldsymbol {x}},\omega )}
that is a slolution to the Helmholtz equation has the following asymptotic behavior
{\displaystyle U({\boldsymbol {x}},\omega )=O(1/r)\qquad }
{\displaystyle \qquad r\rightarrow \infty }
{\displaystyle {\frac {\partial U({\boldsymbol {x}},\omega )}{\partial r}}-{\frac {i\omega }{V(x)}}U({\boldsymbol {x}},\omega )=o(1/r)\qquad }
{\displaystyle \qquad r\rightarrow \infty }
{\displaystyle r}
is the radial distance from the source, and any scatterers of interest.
Retrieved from "https://wiki.seg.org/index.php?title=Sommerfeld_radiation_condition&oldid=161851"
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MEAs for Polymer Electrolyte Fuel Cell (PEFC) Working at Medium Temperature | J. Electrochem. En. Conv. Stor | ASME Digital Collection
I. Gatto,
, via Salita S. Lucia sopra Contesse, 98126 Messina, Italy
e-mail: gatto@itae.cnr.it
A. Saccà,
A. Saccà
A. Carbone,
R. Pedicini,
R. Pedicini
J. Fuel Cell Sci. Technol. Aug 2006, 3(3): 361-365 (5 pages)
Gatto, I., Saccà, A., Carbone, A., Pedicini, R., and Passalacqua, E. (February 8, 2006). "MEAs for Polymer Electrolyte Fuel Cell (PEFC) Working at Medium Temperature." ASME. J. Fuel Cell Sci. Technol. August 2006; 3(3): 361–365. https://doi.org/10.1115/1.2217959
Recently, the CNR-ITAE activity has been addressed to the components development (electrodes and membranes) able to work in medium temperature PEFCs
(80-130°C)
. One of the main problems to work at these temperatures is the proton conductivity loss due to a not full hydration of the membrane. For this reason a study on the modification of perfluorosulphonic membranes (like Nafion) was carried out by introducing different percentages of inorganic oxides (like
SiO2
ZrO2
) in the polymer matrix. These compounds have the function to improve the properties of the materials at high temperature due to their characteristics of softly proton conductor and/or hygroscopicity. The membranes were prepared by the Doctor-Blade casting technique that permits a good check of the thickness and a good reproducibility. A commercial
ZrO2
was used to prepare the membranes varying the inorganic amount between 3 and
20wt%
. The most promising results were obtained at
120°C
with a Nafion-recast membrane loaded with a
10wt%
ZrO2
; a power density value of about
330mW∕cm2
0.6V
was reached. On the other side, an optimization of the electrode structure was carried out, by introducing the inorganic oxide in the catalyst layer in order to improve the performance in the range of considered temperature. By using a spray technique, thin film electrodes with a Pt loading of
0.5mg∕cm2
in the catalyst layer, low PTFE content in the diffusion layer and a 30% Pt/Vulcan (E-Tek, Inc.) as an electro catalyst were prepared. Different amounts of
ZrO2
were introduced in the catalytic layer of the electrodes to increase the working temperature and help the water management of the fuel cell. These electrodes assembled to the modified membrane have shown a better performance at higher cell temperature than standard MEA with a power density of about
330mWcm−2
130°C
proton exchange membrane fuel cells, spray coating techniques, electrochemical electrodes, zirconium compounds, catalysts, membranes, casting, platinum, optimisation, nafion, zirconium oxide, composite membranes, electrodes, PEFCs
Electrodes, Membranes, Temperature
Inorganic/Organic Composite Membranes
Handbook of Fuel Cells, Fundamentals Technology and Applications
Composite Membranes for Medium-Temperature PEM Fuel Cells
Composite Nafion Membranes based on PWA-Zirconia for PEFCs operatine at Medium Temperature
D’Epifanio
Licoccia
Nafion-TiO2 Hybrid Membranes for Medium Temperature Polymer Electrolyte Fuel Cells (PEFCs)
Self-Humidifying Polymer Electrolyte Membranes for Fuel Cells
Analyses of Self-Humidification and Suppression of Gas Crossover in Pt-Dispersed Polymer Electrolyte Membranes for Fuel Cells
Membrane-Supported Nonvolatile Acidic Electrolytes Allow Higher Temperature Operation of Proton-Exchange Membrane Fuel Cells
Parameters of PEM Fuel-Cells Based on New Membranes Fabricated From Nafion, Silicotungstic Acid and Thiophene
Effect of Various Heteropolyacids (HPAs) on the Characteristics of Nafion-HPAS Membranes and Their H2∕O2 Polymer Electrolyte Fuel Cell Parameters
Hybrid Nafion -Silica Membranes Doped with Heteropolyacids for Application in Direct Methanol Fuel Cells
Improvement in the Diffusion Characteristics of low Pt-Loaded electrodes for PEFCs
The Property and Performance Differences Between Catalyst Coated Membrane and Catalyst Coated Diffusion Media
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The workshop covered developments in the field in the last four years. Roughly speaking arithmetic geometry consider algebraic schemes over rings of integers of numberfields. However an important tool is to first extend the base to a p-adic completion. Although both global and local problems matter this time there was a heavy emphasis on p-adic topics. One of them is the deformation-theory of Galois-representations, leading to a proof of Serre's conjecture. here one starts with a global Galois-representation modulo p, then lifts modulo
p^2
, etc. For the lifts one requires certain local conditions (like being unramified outside a given set of places), and the most important and difficult such conditions arise at primes dividing
p
. Here the most important tool is J.M.Fontaine's theory which relates Galois-representations to filtered Frobenius-crystals. Another spectacular progess is the proof (by Ngo) of the fundamental lemma in the theory of automorphic representations. It postulates identities of p-adicorbital integrals and is reduced to a geometric statement about perverse sheaves on Hitchin-fibrations in positive characteristic. Concerning p-adic cohomology theories we are getting closer to a p-adic theory of D-modules, and of overconvergent crystals, over singular schemes. Also the long awaited etale coverings of p-adic period domains have finally been constructed, after it has been understood that they have "holes" which are visible in the Berkovich-space but not in the conventional rigid space. That they exist is suggested by Fontaine's theory. These period domains classify p-divisible groups. Some of them (Drinfeld, Lubin-Tate) can be covered by explicit affinoid domains, thus giving some type of reduction theory for p-divisible groups. There are attempts to extend this to finite flat group-schemes. Concerning K-theory the classical Borel-regulator from K-theory to Deligne-cohomology has been extended to syntomic cohomology, as well as the computation of its values on Eisenstein-symbols. For the l-adic etale theory general finiteness theorems can now be shown for quasi-excellent schemes. A further topic was the theory of p-adic Banach-representations of p-adic Lie-groups. On a more global level we had talks about (p-adic!) constructions of rational points on elliptic curves, the association of K-classes to abelian varieties, and the theory of tame fundamental groups. Finally the theory of small points has been extended to function fields (over numberfields it leads to equidistribution) using tropical geometry.
Gerd Faltings, Johan de Jong, Richard Pink, Arithmetic Algebraic Geometry. Oberwolfach Rep. 5 (2008), no. 3, pp. 1979–2026
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Epitrochoid Knowpia
An epitrochoid (/ɛpɪˈtrɒkɔɪd/ or /ɛpɪˈtroʊkɔɪd/) is a roulette traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is at a distance d from the center of the exterior circle.
The epitrochoid with R = 3, r = 1 and d = 1/2
The parametric equations for an epitrochoid are
{\displaystyle x(\theta )=(R+r)\cos \theta -d\cos \left({R+r \over r}\theta \right),\,}
{\displaystyle y(\theta )=(R+r)\sin \theta -d\sin \left({R+r \over r}\theta \right).\,}
{\displaystyle \theta }
is geometrically the polar angle of the center of the exterior circle. (However,
{\displaystyle \theta }
is not the polar angle of the point
{\displaystyle (x(\theta ),y(\theta ))}
on the epitrochoid.)
Special cases include the limaçon with R = r and the epicycloid with d = r.
The classic Spirograph toy traces out epitrochoid and hypotrochoid curves.
The orbit of the moon, when centered around the sun, approximates an epitrochoid.
The combustion chamber of the Wankel engine is an epitrochoid.
Epitrochoid generator
Weisstein, Eric W. "Epitrochoid". MathWorld.
O'Connor, John J.; Robertson, Edmund F., "Epitrochoid", MacTutor History of Mathematics archive, University of St Andrews
Plot Epitrochoid -- GeoFun
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Protonation - zxc.wiki
Example: acid-base reaction of acetic acid and water. Red arrows: deprotonation of acetic acid; green arrows: protonation of the acetate with formation of acetic acid.
In chemistry, protonation refers to the addition of protons ( hydrogen nuclei / cations ) to a chemical compound as part of an acid-base reaction . One or more positive charges are added to the target molecule, depending on the number of protons transferred . The compound that has taken up the protons is called the protonated compound . The opposite process, the splitting off of protons from a compound, is called deprotonation .
{\ displaystyle \ mathrm {HA \ + \ B \ \ rightleftharpoons \ A ^ {-} \ + \ HB ^ {+}}}
Protonation of compound B by the acid HA , which is deprotonated in the process.
The prerequisite for the process of protonation is the presence of an acid and a base as defined by Brønsted and Lowry . The acid strength - represented by the pK S value - and the base strength ( pK B ) determine whether the balance on the side of the protonated or unprotonated compound is.
The protonation of a compound can be influenced by steric factors.
A positive charge is transferred with the proton, as in the following example, which shows the protonation of ammonia (NH 3 ) by hydrogen chloride (HCl):
{\ displaystyle \ mathrm {HCl \ + \ NH_ {3} \ \ rightleftharpoons \ Cl ^ {-} \ + \ NH_ {4} ^ {+}}}
Hydrogen chloride gives off a proton to the ammonia molecule. This is a negatively charged are chloride - anion and a positively charged ammonium - cation formed.
Protonation is a reaction step that has been widely observed and used. It is often used to activate a chemical compound for subsequent reactions . But they are also used to ionize compounds , for example in the context of a mass spectrometric analysis.
↑ Albert Gossauer: Structure and reactivity of biomolecules an introduction to organic chemistry . John Wiley & Sons, 2006, ISBN 3-906390-29-2 , pp. 572,578 ( limited preview in Google Book Search).
↑ Hartmut Follmann, Walter Grahn: Chemistry for biologists internship and theory . Springer-Verlag, 2013, ISBN 978-3-322-80146-3 , pp. 43 ( limited preview in Google Book search).
↑ James Huheey, Ellen Keiter, Richard Keiter: Inorganic chemistry principles of structure and reactivity . Walter de Gruyter GmbH & Co KG, 2014, ISBN 978-3-11-030795-5 , p. 374 ( limited preview in Google Book search).
↑ Michael Quednau: Applications from Urban Mining to NanoGeoScience . Walter de Gruyter GmbH & Co KG, 2017, ISBN 978-3-11-042287-0 , p. 167 ( limited preview in Google Book search).
↑ Jürgen H. Gross: Mass Spectrometry A Textbook . Springer-Verlag, 2012, ISBN 978-3-8274-2981-0 , pp. 387 ( limited preview in Google Book search).
This page is based on the copyrighted Wikipedia article "Protonierung" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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The y -genus of the moduli space of PGL n -Higgs bundles on a curve (for degree coprime to n )
November 2013 The
y
-genus of the moduli space of
{PGL}_{n}
-Higgs bundles on a curve (for degree coprime to
n
Oscar García-Prada, Jochen Heinloth
Building on our previous joint work with A. Schmitt, we explain a recursive algorithm to determine the cohomology of moduli spaces of Higgs bundles on any given curve (in the coprime situation). As an application of the method, we compute the
y
-genus of the space of
{PGL}_{n}
-Higgs bundles for any rank
n
, confirming a conjecture of T. Hausel.
Oscar García-Prada. Jochen Heinloth. "The
y
{PGL}_{n}
n
)." Duke Math. J. 162 (14) 2731 - 2749, November 2013. https://doi.org/10.1215/00127094-2381369
Received: 2 August 2012; Revised: 1 March 2013; Published: November 2013
Oscar García-Prada, Jochen Heinloth "The
y
{PGL}_{n}
n
)," Duke Mathematical Journal, Duke Math. J. 162(14), 2731-2749, (November 2013)
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Symbolic Powers and Free Resolutions of Generalized Star Configurations of Hypersurfaces
2021 Symbolic Powers and Free Resolutions of Generalized Star Configurations of Hypersurfaces
Kuei-Nuan Lin, Yi-Huang Shen
As a generalization of the ideals of star configurations of hypersurfaces, we consider the a-fold product ideal
{\mathit{I}}_{\mathit{a}}\left({\mathit{f}}_{1}^{{\mathit{m}}_{1}}\cdots {\mathit{f}}_{\mathit{s}}^{{\mathit{m}}_{\mathit{s}}}\right)
{\mathit{f}}_{1},\dots ,{\mathit{f}}_{\mathit{s}}
is a sequence of n-generic forms and
1\le \mathit{a}\le {\mathit{m}}_{1}+\cdots +{\mathit{m}}_{\mathit{s}}
. Firstly, we show that this ideal has complete intersection quotients when these forms are of the same degree and essentially linear. Then, we study its symbolic powers while focusing on the uniform case with
{\mathit{m}}_{1}=\cdots ={\mathit{m}}_{\mathit{s}}
. For large a, we describe its resurgence and symbolic defect. And for general a, we also investigate the corresponding invariants for meeting-at-the-minimal-components version of symbolic powers.
Kuei-Nuan Lin. Yi-Huang Shen. "Symbolic Powers and Free Resolutions of Generalized Star Configurations of Hypersurfaces." Michigan Math. J. Advance Publication 1 - 34, 2021. https://doi.org/10.1307/mmj/20205890
Received: 13 March 2020; Revised: 10 August 2020; Published: 2021
Primary: 13A15 , 13A50 , 13D02 , 14N20 , 52C35
Kuei-Nuan Lin, Yi-Huang Shen "Symbolic Powers and Free Resolutions of Generalized Star Configurations of Hypersurfaces," Michigan Mathematical Journal, Michigan Math. J. Advance Publication, 1-34, (2021)
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IsProcessingInstruction - Maple Help
Home : Support : Online Help : Connectivity : Web Features : XMLTools : IsProcessingInstruction
determine if an expression is an XML processing instruction data structure
IsProcessingInstruction(expr)
The IsProcessingInstruction(expr) command tests whether a Maple expression expr is an XML processing instruction data structure. If expr is an XML processing instruction data structure, the value true is returned. Otherwise, false is returned.
Such expressions can be encountered in XML documents that are read from external sources, or can be generated programmatically by using the XMLTools[XMLProcessingInstruction] constructor.
Every valid XML document begins with the processing instruction
<?\mathrm{xml version}=1.0?>
\mathrm{with}\left(\mathrm{XMLTools}\right):
\mathrm{IsProcessingInstruction}\left(\mathrm{XMLElement}\left("a",["b"="c"],"d","e"\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{IsProcessingInstruction}\left(\mathrm{XMLProcessingInstruction}\left("xml","version=1.0"\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Convert Euler-Rodrigues vector to direction cosine matrix - Simulink - MathWorks España
Rodrigues to Direction Cosine Matrix
Convert Euler-Rodrigues vector to direction cosine matrix
The Rodrigues to Direction Cosine Matrix block determines the 3-by-3 direction cosine matrix from a three-element Euler-Rodrigues vector. The rotation used in this block is a passive transformation between two coordinate systems. For more information on Euler-Rodrigues vectors, see Algorithms.
Euler-Rodrigues vector from which to determine the direction cosine matrix.
Direction cosine matrix determined from the Euler-Rodrigues vector.
\stackrel{⇀}{b}
\stackrel{\to }{b}=\left[\begin{array}{ccc}{b}_{x}& {b}_{y}& {b}_{z}\end{array}\right]
\begin{array}{l}{b}_{x}=\mathrm{tan}\left(\frac{1}{2}\theta \right){s}_{x},\\ {b}_{y}=\mathrm{tan}\left(\frac{1}{2}\theta \right){s}_{y},\\ {b}_{z}=\mathrm{tan}\left(\frac{1}{2}\theta \right){s}_{z}\end{array}
\stackrel{⇀}{s}
Direction Cosine Matrix to Rodrigues | Rodrigues to Quaternions | Rodrigues to Rotation Angles | Quaternions to Rodrigues | Rotation Angles to Rodrigues
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EUDML | Some results on the geometry of full flag manifolds and harmonic maps. EuDML | Some results on the geometry of full flag manifolds and harmonic maps.
Some results on the geometry of full flag manifolds and harmonic maps.
Paredes, Marlio
Paredes, Marlio. "Some results on the geometry of full flag manifolds and harmonic maps.." Revista Colombiana de Matemáticas 34.2 (2000): 57-89. <http://eudml.org/doc/123097>.
@article{Paredes2000,
author = {Paredes, Marlio},
keywords = {flag manifolds; -symplectic metrics; harmonic maps; Hermitian geometry; tournaments; -symplectic metrics},
title = {Some results on the geometry of full flag manifolds and harmonic maps.},
AU - Paredes, Marlio
TI - Some results on the geometry of full flag manifolds and harmonic maps.
KW - flag manifolds; -symplectic metrics; harmonic maps; Hermitian geometry; tournaments; -symplectic metrics
flag manifolds,
\left(1,2\right)
-symplectic metrics, harmonic maps, Hermitian geometry, tournaments,
\left(1,2\right)
-symplectic metrics
Articles by Paredes
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Determining pH and pOH - Course Hero
General Chemistry/Acids and Bases/Determining pH and pOH
The pH of a substance is a measure of the concentration of H+ (or H3O+) ions in solution. The relationship is logarithmic (base 10): the pH value itself is the value of the exponent, x, when the concentration is expressed as
1\times{10}^{-x}{\;\rm{M}}
. In other words, in a substance that has a pH of 5, the concentration of H+ ions is
1\times{10}^{-5}\;{\rm{M}}
. The pOH of a substance is a measure of the concentration of OH– ions in solution. The total of the exponents must equal 14, so in a substance with a pH of 5, the pOH must be 9. When the two values are compared,
1\times{10}^{-5}\;{\rm{M}}
1\times{10}^{-9}\;{\rm{ M}}
, the concentration of H+ is greater than that of OH– by four orders of magnitude. When [H+] is greater than [OH–], the solution is acidic. When the opposite is true, the solution is basic.
The following equation relates pH to the concentration of H+:
\rm{pH}=-\log{[\rm {H}^{+}]}
For example, calculating [H3O+] of 0.20 M acetic acid (CH3COOH) at equilibrium gives a concentration of
1.9\times10^{-3}
. Substitute x for [H+] and [CH3COO–] in the equation for the equilibrium constant, and substitute 0.20 for [CH3COOH]:
K_{\rm {a}}=\frac{\lbrack\rm {H}^{+}\rbrack\lbrack\rm {A}^{-}\rbrack}{\lbrack\rm{HA}\rbrack}=\frac{x^2}{0.20}
Substitute the known value of Ka,
1.77\times10^{-5}
, and rearrange the equation to solve for x:
\begin{aligned}x^2&=(1.77\times10^{-5})(0.20)\\x^2&=3.54\times10^{-6}\\x&=\sqrt{3.54\times10^{-6}}\\&=1.9\times10^{-3}\end{aligned}
The value of x equals the concentration of the hydronium ion.
x=1.9\times10^{-3}=\lbrack{\rm {H}^{+}}\rbrack
The negative logarithm is the pH of the solution.
{\rm{pH}}=-\log(1.9\times10^{-3})=2.7
Thus the pH can be calculated knowing only Ka and the concentration of an acid.
Because pH and pOH must add up to 14, the pOH of the solution can also be determined:
\begin{aligned}\rm{pOH}&=14-\rm{pH}\\&=14-2.7\\&=11.3\end{aligned}
The values of pH and pOH can also be used to calculate [H3O+] or [OH–].
Using pH and pOH to Calculate [H3O+] and [OH–]
What is [H3O+] of a solution with a pOH of 8.3?
First, determine the pH by subtracting pOH from 14.
\begin{aligned}\rm{pH}&=14-\rm{pOH}\\&=14-8.3\\&=5.7\end{aligned}
Substitute this value into the pH equation.
\begin{aligned}\rm{pH}&=-\log[{\rm{H}_{3}\rm{O}^{+}}]\\5.7&=-\log[{\rm{H}_{3}\rm{O}^{+}}]\end{aligned}
Solve for [H3O+].
\begin{aligned}\left[{\rm {H}_{3} O^{+}}\right]&=10^{-5.7}\\&=2.0\times10^{-6}\end{aligned}
Familiar substances have a range of pH values.
In some situations it is desirable to limit the pH change that occurs when an acid or a base is added to a solution. Acidity changes in bodily fluids, for example, must remain within a certain range to avoid health issues. A buffer is a solution that contains significant quantities of an acid and its conjugate base and resists a change in pH when acid or base is added. The pH of a buffer system depends on Ka for the weak acid as well as the initial concentrations of the acid [HA] and the base [A–] that are mixed. The Henderson-Hasselbalch equation can be used to approximate the pH of a buffer solution:
{\rm{pH}}\approx\rm{p}K_{\rm{a}}+\log_{10}\frac{\rm{[A}^{-}]}{[\rm{HA}]}
The term pKa in the equation is defined in terms of the acid ionization constant:
{\rm{pK}_{a}}=-\log{K}_{\rm{a}}
Using the Henderson-Hasselbalch Equation to Calculate the pH of a Buffer
Consider a buffer solution of 0.035 M NH3 and 0.050 M NH4+. NH4+ has a Ka of
5.6\times10^{-10}
. What is the pH of the buffer?
Use the acid ionization constant to calculate pKa.
\begin{aligned}{\rm{p}K_{\rm{a}}}&=-\log{K}_{\rm{a}}\\&=-\log({5.6\times10^{-10}})\\&=9.25\end{aligned}
Substitute pKa and the molarities in the Henderson-Hasselbalch equation to calculate the pH.
\begin{aligned}\rm{pH}&\approx\rm{p}K_{\rm{a}}+\log_{10}\frac{\rm{[A^{-}]}}{[\rm{HA}]}\\&\approx\rm{pK_{a}}+\log_{10}\frac{\rm{[NH}_{3}]}{[{\rm{NH}_{4}}^{+}]}\\&\approx9.25+\log_{10}\frac{[0.035]}{[0.050]}\\&\approx9.1\end{aligned}
The pH of the buffer is 9.1.
The Henderson-Hasselbalch equation can be rearranged to give the ratio of the concentrations of the base [A−] and the acid [HA].
10^{\rm{pH-pK}_{\rm{a}}}=\frac{\rm{[A^{-}]}}{[\rm{HA}]}
This equation can be used to calculate the amount of acid and conjugate base needed to make a buffer of a particular pH.
<Strengths of Acids and Bases>Suggested Reading
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EUDML | Zariski dense subgroups of semisimple algebraic groups with isomorphic -adic closures. EuDML | Zariski dense subgroups of semisimple algebraic groups with isomorphic -adic closures.
Zariski dense subgroups of semisimple algebraic groups with isomorphic
p
-adic closures.
Nguyêñ Quôć Thǎńg. "Zariski dense subgroups of semisimple algebraic groups with isomorphic -adic closures.." Journal of Lie Theory 13.1 (2003): 13-20. <http://eudml.org/doc/122760>.
keywords = {semisimple groups; Zariski dense subgroups; -adic closures; -adic closures},
title = {Zariski dense subgroups of semisimple algebraic groups with isomorphic -adic closures.},
TI - Zariski dense subgroups of semisimple algebraic groups with isomorphic -adic closures.
KW - semisimple groups; Zariski dense subgroups; -adic closures; -adic closures
semisimple groups, Zariski dense subgroups,
p
-adic closures,
p
-adic closures
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EUDML | Ordering Cacti With $n$ Vertices and $k$ Cycles by Their Laplacian Spectral Radii EuDML | Ordering Cacti With $n$ Vertices and $k$ Cycles by Their Laplacian Spectral Radii
Ordering Cacti With
n
k
Cycles by Their Laplacian Spectral Radii
Shu-Guang Guo; Yan-Feng Wang
Shu-Guang Guo, and Yan-Feng Wang. "Ordering Cacti With $n$ Vertices and $k$ Cycles by Their Laplacian Spectral Radii." Publications de l'Institut Mathématique 92(106).112 (2012): 117-125. <http://eudml.org/doc/256256>.
author = {Shu-Guang Guo, Yan-Feng Wang},
title = {Ordering Cacti With $n$ Vertices and $k$ Cycles by Their Laplacian Spectral Radii},
AU - Shu-Guang Guo
AU - Yan-Feng Wang
TI - Ordering Cacti With $n$ Vertices and $k$ Cycles by Their Laplacian Spectral Radii
Articles by Shu-Guang Guo
Articles by Yan-Feng Wang
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Shifted convolution sums for GL(3)×GL(2)
1 October 2013 Shifted convolution sums for
\mathrm{GL}\left(3\right)×\mathrm{GL}\left(2\right)
For the shifted convolution sum
{D}_{h}\left(X\right)={\sum }_{m=1}^{\infty }{\lambda }_{1}\left(1,m\right){\lambda }_{2}\left(m+h\right)V\left(\frac{m}{X}\right),
{\lambda }_{1}\left(1,m\right)
are the Fourier coefficients of an
SL\left(3,\mathbb{Z}\right)
Maass form
{\pi }_{1}
{\lambda }_{2}\left(m\right)
are those of an
SL\left(2,\mathbb{Z}\right)
Maass or holomorphic form
{\pi }_{2}
1\le |h|\ll {X}^{1+\epsilon }
, we establish the bound
{D}_{h}\left(X\right){\ll }_{{\pi }_{1},{\pi }_{2},\epsilon }{X}^{1-1/20+\epsilon }.
The bound is uniform with respect to the shift
h
Ritabrata Munshi. "Shifted convolution sums for
\mathrm{GL}\left(3\right)×\mathrm{GL}\left(2\right)
." Duke Math. J. 162 (13) 2345 - 2362, 1 October 2013. https://doi.org/10.1215/00127094-2371416
Ritabrata Munshi "Shifted convolution sums for
\mathrm{GL}\left(3\right)×\mathrm{GL}\left(2\right)
," Duke Mathematical Journal, Duke Math. J. 162(13), 2345-2362, (1 October 2013)
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Estrone sulfotransferase - Wikipedia
Estrogen sulfotransferase; EST
Estrone sulfotransferase (EST) (EC 2.8.2.4), also known as estrogen sulfotransferase, is an enzyme that catalyzes the transformation of an unconjugated estrogen like estrone into a sulfated estrogen like estrone sulfate. It is a steroid sulfotransferase and belongs to the family of transferases, to be specific, the sulfotransferases, which transfer sulfur-containing groups. This enzyme participates in androgen and estrogen metabolism and sulfur metabolism.
Steroid sulfatase is an enzyme that catalyzes the reverse reaction, the transfer of a sulfate to an unconjugated estrogen.
In enzymology, an EST is an enzyme that catalyzes the following chemical reaction:
3'-phosphoadenylyl sulfate + estrone
{\displaystyle \rightleftharpoons }
adenosine 3',5'-bisphosphate + estrone 3-sulfate
Thus, the two substrates of this enzyme are 3'-phosphoadenylyl sulfate and estrone, whereas its two products are adenosine 3',5'-bisphosphate and estrone 3-sulfate.
The enzyme also catalyzes the same reaction for estradiol, with estradiol sulfate as the product.
Two enzymes have been identified that together are thought to represent estrone sulfotransferase (EST):[1][2]
SULT1A1 (catalyzes the reactions estradiol to estradiol sulfate and, to a lesser extent than SULT1E1, estrone to estrone sulfate)
SULT1E1 (catalyzes the reactions estrone to estrone sulfate and estradiol to estradiol sulfate)
Distribution of STS and EST activities for interconversion of estradiol and estrone in adult human tissues.[3]
As of late 2007, 5 structures have been solved for this class of enzymes, with PDB accession codes 1AQU, 1AQY, 1BO6, 1G3M, and 1HY3.
The systematic name of this enzyme class is 3'-phosphoadenylyl-sulfate:estrone 3-sulfotransferase. Other names in common use include 3'-phosphoadenylyl sulfate-estrone 3-sulfotransferase, estrogen sulfotransferase, estrogen sulphotransferase, oestrogen sulphotransferase, and 3'-phosphoadenylylsulfate:oestrone sulfotransferase.
^ "EC 2.8.2.4 – estrone sulfotransferase and Organism(s) Homo sapiens". BRENDA. Technische Universität Braunschweig. January 2018. Retrieved 10 August 2018. Substrate: 3'-phosphoadenylyl sulfate + estrone
Product: adenosine 3',5'-bisphosphate + estrone 3-sulfate
Commentary (substrate): high activity by SULT1E1, low activity by phenol sulfotransferase SULT1A1, EC 2.8.2.1
^ Miki Y, Nakata T, Suzuki T, Darnel AD, Moriya T, Kaneko C, Hidaka K, Shiotsu Y, Kusaka H, Sasano H (December 2002). "Systemic distribution of steroid sulfatase and estrogen sulfotransferase in human adult and fetal tissues". J. Clin. Endocrinol. Metab. 87 (12): 5760–8. doi:10.1210/jc.2002-020670. PMID 12466383.
Adams JB, Poulos A (1967). "Enzymic synthesis of steroid sulphates. 3. Isolation and properties of estrogen sulphotransferase of bovine adrenal glands". Biochim. Biophys. Acta. 146 (2): 493–508. doi:10.1016/0005-2744(67)90233-1. PMID 4965224.
Rozhin J, Zemlicka J, Brooks SC (1967). "Studies on bovine adrenal estrogen sulfotransferase. Inhibition and possible involvement of adenine-estrogen stacking". J. Biol. Chem. 252 (20): 7214–7220. PMID 903358.
Adams JB, Ellyard RK, Low J (1974). "Enzymic synthesis of steroid sulphates. IX. Physical and chemical properties of purified oestrogen sulphotransferase from bovine adrenal glands, the nature of its isoenzymic forms and a proposed model to explain its wave-like kinetics". Biochim. Biophys. Acta. 370 (1): 160–88. doi:10.1016/0005-2744(74)90042-4. PMID 4473218.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Estrone_sulfotransferase&oldid=993715385"
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The torsion index ofE8 and other groups
15 August 2005 The torsion index of
{E}_{8}
and other groups
We compute Grothendieck's torsion index of a compact Lie group for all simply connected groups and all groups of adjoint type. In particular, the torsion index of the group
{E}_{8}
{2}^{6}·{3}^{2}·5
Burt Totaro. "The torsion index of
{E}_{8}
and other groups." Duke Math. J. 129 (2) 219 - 248, 15 August 2005. https://doi.org/10.1215/S0012-7094-05-12922-2
Burt Totaro "The torsion index of
{E}_{8}
and other groups," Duke Mathematical Journal, Duke Math. J. 129(2), 219-248, (15 August 2005)
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Porohyperviscoelastic Model Simultaneously Predicts Parenchymal Fluid Pressure and Reaction Force in Perfused Liver | J. Biomech Eng. | ASME Digital Collection
Emma C. Moran,
Department of Biomedical Engineering, Wake Forest University School of Medicine, Medical Center Blvd., Winston-Salem, NC 27157;
Virginia-Tech Wake Forest University School of Biomedical Engineering and Sciences, Wake Forest University School of Medicine
, Medical Center Blvd., Winston-Salem,
e-mail: emoran@wakehealth.edu
Smitha Raghunathan,
Douglas W. Evans,
e-mail: devans@wakehealth.edu
Nicholas A. Vavalle,
Nicholas A. Vavalle
e-mail: nvavalle@wakehealth.edu
e-mail: jsparks@wakehealth.edu
Tanya LeRoith,
Department of Biomedical Sciences and Pathology, Virginia-Maryland Regional College of Veterinary Medicine
, Duckpond Drive, Phase II, Virginia Tech (0442), Blacksburg,
e-mail: tleroith@vt.edu
Department of Orthopaedics, Wake Forest University School of Medicine, Medical Center Blvd., Winston-Salem, NC 27157
e-mail: tsmith@wakehealth.edu
A correction has been published: Errata: “Porohyperviscoelastic Model Simultaneously Predicts Parenchymal Fluid Pressure and Reaction Force in Perfused Liver,” [Journal of Biomechanical Engineering, 134(9), 091002]
Moran, E. C., Raghunathan, S., Evans, D. W., Vavalle, N. A., Sparks, J. L., LeRoith, T., and Smith, T. L. (August 27, 2012). "Porohyperviscoelastic Model Simultaneously Predicts Parenchymal Fluid Pressure and Reaction Force in Perfused Liver." ASME. J Biomech Eng. September 2012; 134(9): 091002. https://doi.org/10.1115/1.4007175
Porohyperviscoelastic (PHVE) modeling gives a simplified continuum approximation of pore fluid behavior within the parenchyma of liver tissue. This modeling approach is particularly applicable to tissue engineering of artificial livers, where the inherent complexity of the engineered scaffolds prevents the use of computational fluid dynamics. The objectives of this study were to simultaneously predict the experimental parenchymal fluid pressure (PFP) and compression response in a PHVE liver model. The model PFP matched the experimental measurements (318 Pa) to within 1.5%. Linear regression of both phases of compression, ramp, and hold, demonstrated a strong correlation between the model and the experimental reaction force (
p<0.5
). The ability of this PVE model to accurately predict both fluid and solid behavior is important due to the highly vascularized nature of liver tissue and the mechanosensitivity of liver cells to solid matrix and fluid flow properties.
Biological tissues, Compression, Equilibrium (Physics), Fluid pressure, Fluids, Liver, Stress, Modeling, Gates (Closures), Relaxation (Physics), Finite element model, Testing
,” May 19, 2001, Atlanta, GA.
Organ Procurement and Transplantation Network: National Data, http://optn.transplant.hrsa.gov/http://optn.transplant.hrsa.gov/
The Use of Whole Organ Decellularization for the Generation of a Vascularized Liver Organoid
Soto-Gutierrez
Izamis
Nat. Med. (N.Y.)
Method for the Decellularization of Intact Rat Liver
Large-Scale Whole-Organ Decellularization of Swine Liver for Transplantation
. Available at http://www.multiwebcast.com/aasld/2011/thelivermeeting/13211/http://www.multiwebcast.com/aasld/2011/thelivermeeting/13211/
A Whole-Organ Regenerative Medicine Approach for Liver Replacement
3-D Computational Modeling of Media Flow Through Scaffolds in a Perfusion Bioreactor
Prediction of the Micro-Fluid Dynamic Environment Imposed to Three-Dimensional Engineered Cell Systems in Bioreactors
Appl.Mech. Rev.
Biphasic Creep and Stress-Relaxation of Articular-Cartilage in Compression—Theory and Experiments
ASME J. Biomech.Eng.
Biphasic Poroviscoelastic Simulation of the Unconfined Compression of Articular Cartilage—Part II: Effect of Variable Strain Rates
Dynamic Finite Element Modeling of Poroviscoelastic Soft Tissue
Perialveolar Interstitial Resistance and Compliance in Isolated Rat Lung
. Available at http://jap.physiology.org/content/70/6/2750.longhttp://jap.physiology.org/content/70/6/2750.long
Perfusion Studies of Steady Flow in Poroelastic Myocardium Tissue
Unconfined Compression of White Matter
Effects of Brain Ventricular Shape on Periventricular Biomechanics: A Finite-Element Analysis
Poroviscoelastic Modeling of Liver Biomechanical Response in Unconfined Compression
Leungchavaphongse
A Biphasic Model for Sinusoidal Liver Perfusion Remodeling After Outflow Obstruction
Coupled Modeling of Blood Perfusion in Intravascular, Interstitial Spaces in Tumor Microvasculature
ABAQUS Theory Manual (Version 6.9)
Some Forms of the Strain-Energy Function for Rubber
Effects of Friction on the Unconfined Compressive Response of Articular-Cartilage—A Finite-Element Analysis
Evolutionsstrategie und numerische optimierung
,” Ph.D. thesis, TU Berlin, Berlin, Germany.
Isight 5.0 Component Guide
Dassault Systemes, Vélizy-Villacoublay
Nlpql: A FORTRAN Subroutine for Solving Constrained Nonlinear Programming Problems
Comput.Methods Biomech. Biomed. Eng.
A Prediction of Cell Differentiation and Proliferation Within a Collagen-Glycosaminoglycan Scaffold Subjected to Mechanical Strain and Perfusive Fluid Flow
Brain Mechanics for Neurosurgery: Modeling Issues
Characterizing the Nonlinear Mechanical Response of Liver to Surgical Manipulation
,” PhD thesis, Harvard University, Cambridge, MA.
The Anisotropic Hydraulic Permeability of Human Lumbar Anulus Fibrosus—Influence of Age, Degeneration, Direction, and Water Content
Combined Compression and Elongation Experiments and Non-Linear Modelling of Liver Tissue for Surgical Simulation
Development of In Vivo Constitutive Models for liver: Application to Surgical Simulation
Bulk Modulus and Volume Variation Measurement of the Liver and the Kidneys In Vivo Using Abdominal Kinetics During Free Breathing
Limitation of Finite Element Analysis of Poroelastic Behavior of Biological Tissues Undergoing Rapid Loading
Large Deformation Shear Properties of Liver Tissue
. Available at http://iospress.metapress.com/content/g5r6grjyqc38aprf/http://iospress.metapress.com/content/g5r6grjyqc38aprf/
A Special Theory of Biphasic Mixtures and Experimental Results for Human Annulus Fibrosus Tested in Confined Compression
|
Hc Verma I for Class 12 Science Physics Chapter 21 - Speed Of Light
Hc Verma I Solutions for Class 12 Science Physics Chapter 21 Speed Of Light are provided here with simple step-by-step explanations. These solutions for Speed Of Light are extremely popular among Class 12 Science students for Physics Speed Of Light Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Hc Verma I Book of Class 12 Science Physics Chapter 21 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Hc Verma I Solutions. All Hc Verma I Solutions for class Class 12 Science Physics are prepared by experts and are 100% accurate.
No, it is not advisable to define the length 1 m as the distance travelled by sound in 1/332 s because the speed of sound is affected by various factors such as temperature, humidity and nature of medium. So, it cannot be said that the distance travelled by sound in 1/332 s will be 1 m, less than 1 m or more than 1 m.
We have speed of light = 299792458 m/s
To have a accuracy of 10% the light has to travel 1/10th of a second between the observers so,
Distance travelled by the light in 0.1 s = 0.1×299792458= 29979 km.
The difficulty in separation of that distance will be the curvature of earth. As the earth’s surface is curved, light from one of the experimenters won’t reach the other.
If the wheel is placed away from the focal plane the light returning light rays will fall in an extended area of the wheel, this will let image to appear even when the light ray is blocked by one of the teeth of the wheel.
Distance between the mirrors (D) = 12.0 km = 12 × 103 m
Number of teeth in the wheel (n) = 180
Now we apply Fizeau's apparatus
\mathrm{We} \mathrm{know}, c=\frac{2Dn\omega }{\mathrm{\pi }}\phantom{\rule{0ex}{0ex}}⇒\omega =\frac{c\pi }{2Dn}\mathrm{red}/\mathrm{s}
=\frac{\pi c}{2\mathrm{D}n}×\frac{180}{\pi }\mathrm{deg}/\mathrm{s}
\omega =\frac{3×{10}^{8}}{24×{10}^{3}}\phantom{\rule{0ex}{0ex}}=1.25×{10}^{4} \mathrm{deg}/\mathrm{sec}
Hence, the required angular speed of the wheel for which the image is not seen is
1.25×{10}^{4} \mathrm{deg}/\mathrm{sec}
Distance travelled by light between two reflections from the rotating mirror (D) = 4.8 km = 4.8 × 103 m
Number of faces of the mirror, N = 8
Angular speed of the mirror,
\omega
In Michelson experiment, the speed of light (c) is given by
c=\frac{\omega DN}{2\mathrm{\pi }}
N = number of faces in the polygon mirror
\therefore \omega =\frac{2\mathrm{\pi }c}{DN} rad/\mathrm{s}
=\frac{c}{DN} \mathrm{rev}/\mathrm{s}
=\frac{3×{10}^{8}}{\left(4.8×{10}^{3}×8\right)}
= 7.8 × 103 rev/s
Hence, the required angular speed is 7.8 × 103 rev/s.
Distance between the rotating and the fixed mirror (R) = 16 m
Distance between the lens and the rotating mirror (b) = 6 m
Distance between the source and the lens (a) = 2 m
Mirror is rotated at a speed of 356 revolutions per second
⇒ ω = 356 rev/s= 356 × 2 π rad/sec
Shift in the image (s) = 0.7 m = 0.7 × 103 m
In Foucault experiment, speed of light is given by
c=4{\mathrm{R}}^{2}\frac{wa}{s\left(\mathrm{R}+b\right)}
=\frac{4×\left(16{\right)}^{2}×356×2\mathrm{\pi }×2}{\left(0.7\right)×{10}^{-3}\left(16+6\right)}
Therefore, the required speed of light is 2.975 × 108 m/s.
There is no difficulty if the distance travelled by light is decreased. In this method, light has to travel a large distance of 8.6 km. So, the intensity of the light decreases considerably and the final image becomes dim. If somehow this distance is decreased, the final image is dark due to the increased light intensity.
The advantage of using a polygonal mirror with larger number of faces in the Michelson method is it gives the value that is very near to the accurate value and minimises the error.
If the gas is gradually pumped out, a vacuum will be created inside the closed cylindrical tube, and experimentally, light travels at the fastest speed in vacuum as compared to any other medium.
(a) in vacuum but not in air
Different wavelengths travel at different speeds through different media. In vacuum, the speeds of both the red light and yellow light are same but are different in air due to some optical density of air. Both wavelengths act in a different way in the air.
(b) the image will be shifted a little later than the object
Light rays emitting from a source have to cover some optical distance to form an image of the source on the other side of the lens. So, when a light source is shifted by some distance on the principal axis, then the light rays emitting from the new position of the source take some time to form a shifted image of the object on the other side of the lens. However, this delay is very small because the speed of light has a very larger value.
The speed of light is a fundamental constant, and with respect to any inertial frame, it is independent of the motion of the light source.
(c) Foucault method
Foucault gave the first laboratory method to find the velocity of light. He obtained a value of
2.98×{10}^{8}\mathrm{m}/\mathrm{s}
from his measurements.
Foucault method can be used to measure the speed of light in water. One of the advantage of this method is that one can put some transparent medium (or water) between two mirrors to measure the speed of light in that medium (or water). Foucault observed that the velocity of light in water is less than that in the air.
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StripAttributes - Maple Help
Home : Support : Online Help : Connectivity : Web Features : XMLTools : StripAttributes
remove all attributes from an XML element
remove all comments from an XML element
StripAttributes(xmlTree)
StripComments(xmlTree)
The StripAttributes(xmlTree) command removes all attributes from the XML element xmlTree and returns the resulting XML tree.
The StripComments(xmlTree) command removes all comment structures from the XML element xmlTree and returns the resulting XML tree.
If the input XML tree xmlTree does not have any comments, then the tree is simply returned.
Note: When using these functions, attributes and/or comments are removed at all levels (or subelements) of the XML data structure xmlTree, not only the top-level element that it represents. The resulting XML data structure is completely free of attributes and/or _XML_COMMENT calls.
\mathrm{with}\left(\mathrm{XMLTools}\right):
\mathrm{xmlTree1}≔\mathrm{XMLElement}\left("a",["colour"="red"],["some text",\mathrm{XMLElement}\left("b",["colour"="blue"],"more text"\right)]\right)
\textcolor[rgb]{0,0,1}{\mathrm{xmlTree1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_XML_Element}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_XML_ElementType}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"a"}\right)\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{\mathrm{_XML_Attribute}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_XML_AttrName}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"colour"}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_XML_AttrValue}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"red"}\right)\right)]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{\mathrm{_XML_Text}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"some text"}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_XML_Element}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_XML_ElementType}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"b"}\right)\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{\mathrm{_XML_Attribute}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_XML_AttrName}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"colour"}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_XML_AttrValue}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"blue"}\right)\right)]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{\mathrm{_XML_Text}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"more text"}\right)]\right)]\right)
\mathrm{StripAttributes}\left(\mathrm{xmlTree1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{_XML_Element}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_XML_ElementType}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"a"}\right)\textcolor[rgb]{0,0,1}{,}[]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{\mathrm{_XML_Text}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"some text"}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_XML_Element}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_XML_ElementType}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"b"}\right)\textcolor[rgb]{0,0,1}{,}[]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{\mathrm{_XML_Text}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"more text"}\right)]\right)]\right)
\mathrm{xmlTree2}≔\mathrm{XMLElement}\left("a",["colour"="red"],["some text",\mathrm{XMLElement}\left("b",[],\mathrm{XMLComment}\left("a comment"\right),"more text"\right)]\right)
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\mathrm{StripComments}\left(\mathrm{xmlTree2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{_XML_Element}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_XML_ElementType}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"a"}\right)\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{\mathrm{_XML_Attribute}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_XML_AttrName}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"colour"}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_XML_AttrValue}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"red"}\right)\right)]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{\mathrm{_XML_Text}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"some text"}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{_XML_Element}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_XML_ElementType}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"b"}\right)\textcolor[rgb]{0,0,1}{,}[]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{\mathrm{_XML_Text}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"more text"}\right)]\right)]\right)
XMLTools[RemoveAttributes]
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Nonseasonal Differencing - MATLAB & Simulink - MathWorks Australia
This example shows how to take a nonseasonal difference of a time series. The time series is quarterly U.S. GDP measured from 1947 to 2005.
Load the GDP data set included with the toolbox.
title('U.S. GDP')
The time series has a clear upward trend.
Take a first difference of the series to remove the trend,
\Delta {y}_{t}=\left(1-L\right){y}_{t}={y}_{t}-{y}_{t-1}.
First create a differencing lag operator polynomial object, and then use it to filter the observed series.
D1 = LagOp({1,-1},'Lags',[0,1]);
dY = filter(D1,Y);
plot(2:N,dY)
title('First Differenced GDP Series')
The series still has some remaining upward trend after taking first differences.
Take a second difference of the series,
{\Delta }^{2}{y}_{t}=\left(1-L{\right)}^{2}{y}_{t}={y}_{t}-2{y}_{t-1}+{y}_{t-2}.
D2 = D1*D1;
ddY = filter(D2,Y);
plot(3:N,ddY)
title('Second Differenced GDP Series')
The second-differenced series appears more stationary.
|
Estimate Random Parameter of State-Space Model - MATLAB & Simulink - MathWorks Deutschland
The Zig-Zag Estimation Method
Estimate Random Coefficient Using Zig-Zag Method
Determine Quality of Simulation
This example shows how to estimate a random, autoregressive coefficient of a state in a state-space model. That is, this example takes a Bayesian view of state-space model parameter estimation by using the "zig-zag" estimation method.
Suppose that two states (
{x}_{1,t}
{x}_{2,t}
) represent the net exports of two countries at the end of the year.
{x}_{1,t}
is a unit root process with a disturbance variance of
{\sigma }_{1}^{2}
{x}_{2,t}
is an AR(1) process with an unknown, random coefficient and a disturbance variance of
{\sigma }_{2}^{2}
An observation (
{y}_{t}
) is the exact sum of the two net exports. That is, the net exports of the individual states are unknown.
Symbolically, the true state-space model is
\begin{array}{l}{x}_{1,t}={x}_{1,t-1}+{\sigma }_{1}{u}_{1,t}\\ {x}_{2,t}=\varphi {x}_{2,t-1}+{\sigma }_{2}{u}_{2,t}\\ {y}_{t}={x}_{1,t}+{x}_{2,t}\end{array}
Simulate 100 years of net exports from:
A unit root process with a mean zero, Gaussian noise series that has variance
0.{1}^{2}
An AR(1) process with an autoregressive coefficient of 0.6 and a mean zero, Gaussian noise series that has variance
0.{2}^{2}
{x}_{1,0}={x}_{2,0}=0
Create an observation series by summing the two net exports per year.
rng(100); % For reproducibility
u1 = randn(T,1)*sigma1;
x1 = cumsum(u1);
Mdl2 = arima('AR',phi,'Variance',sigma2^2,'Constant',0);
x2 = simulate(Mdl2,T,'Y0',0);
plot([x1 x2 y])
legend('x_1','x_2','y','Location','Best');
ylabel('Net exports');
\varphi
as if it is unknown and random, and use the zig-zag method to recover its distribution. To implement the zig-zag method:
1. Choose an initial value for
\varphi
in the interval (-1,1), and denote it
{\varphi }_{z}
2. Create the true state-space model, that is, an ssm model object that represents the data-generating process.
3. Use the simulation smoother (simsmooth) to draw a random path from the distribution of the second smoothed states. Symbolically,
{x}_{2,z,t}\sim P\left({x}_{2,t}|{y}_{t},{x}_{1,t},\varphi ={\varphi }_{z}\right)
4. Create another state-space model that has this form
\begin{array}{l}{\varphi }_{z,t}={\varphi }_{z,t-1}\\ {x}_{2,z,t}={x}_{2,z,t-1}{\varphi }_{z,t}+0.8{u}_{2,t}\end{array}
{\varphi }_{z,t}
is a static state and
{x}_{2,z,t}
is an "observed" series with time varying coefficient
{C}_{t}={x}_{2,z,t-1}
5. Use the simulation smoother to draw a random path from the distribution of the smoothed
{\varphi }_{z,t}
series. Symbolically,
{\varphi }_{z,t}\sim P\left(\varphi |{x}_{2,z,t}\right)
{x}_{2,z,t}
encompasses the structure of the true state-space model and the observations.
{\varphi }_{z,t}
is static, so you can just reserve one value (
{\varphi }_{z}
6. Repeat steps 2 - 5 many times and store
{\varphi }_{z}
each iteration.
7. Perform diagnostic checks on the simulation series. That is, construct:
Trace plots to determine the burn in period and whether the Markov chain is mixing well.
Autocorrelation plots to determine how many draws need removing to obtain a well-mixed Markov chain.
8. The remaining series represents draws from the posterior distribution of
\varphi
. You can compute descriptive statistics, or plot a histogram to determine the qualities of the distribution.
Specify initial values, preallocate, and create the true state-space model.
phi0 = -0.3; % Initial value of phi
Z = 1000; % Number of times to iterate the zig-zag method
phiz = [phi0; nan(Z,1)]; % Preallocate
A = [1 0; 0 NaN];
B = [sigma1; sigma2];
Mdl = ssm(A,B,C,'StateType',[2; 0]);
Mdl is an ssm model object. The NaN acts as a placeholder for
\varphi
Iterate steps 2 - 5 of the zig-zag method.
for j = 2:(Z + 1);
% Draw a random path from smoothed x_2 series.
xz = simsmooth(Mdl,y,'Params',phiz(j-1));
% The second column of xz is a draw from the posterior distribution of x_2.
% Create the intermediate state-space model.
Bz = 0;
Cz = num2cell(xz((1:(end - 1)),2));
Dz = sigma2;
Mdlz = ssm(Az,Bz,Cz,Dz,'StateType',2);
% Draw a random path from the smoothed phiz series.
phizvec = simsmooth(Mdlz,xz(2:end,2));
phiz(j) = phizvec(1);
% phiz(j) is a draw from the posterior distribution of phi
phiz is a Markov chain. Before analyzing the posterior distribution of
\varphi
, you should assess whether to impose a burn-in period, or the severity of the autocorrelation in the chain.
Draw a trace plot for the first 100, 500, and all of the random draws.
vec = [100 500 Z];
plot(phiz(1:vec(j)));
title('Trace Plot for \phi');
xlabel('Simulation number');
According to the first plot, transient effects die down after about 20 draws. Therefore, a short burn-in period should suffice. The plot of the entire simulation shows that the series settles around a center.
Plot the autocorrelation function of the series after removing the first 20 draws.
burnOut = 21:Z;
autocorr(phiz(burnOut));
The autocorrelation function dies out rather quickly. It doesn't seem like autocorrelation in the chain is an issue.
Determine qualities of the posterior distribution of
\varphi
by computing simulation statistics and by plotting a histogram of the reduced set of random draws.
xbar = mean(phiz(burnOut))
xbar = 0.5104
xstd = std(phiz(burnOut))
xstd = 0.0988
ci = norminv([0.025,0.975],xbar,xstd); % 95% confidence interval
histogram(phiz(burnOut),'Normalization','pdf');
simX = linspace(h.XLim(1),h.XLim(2),100);
simPDF = normpdf(simX,xbar,xstd);
plot(simX,simPDF,'k','LineWidth',2);
h1 = plot([xbar xbar],h.YLim,'r','LineWidth',2);
h2 = plot([0.6 0.6],h.YLim,'g','LineWidth',2);
h3 = plot([ci(1) ci(1)],h.YLim,'--r',...
[ci(2) ci(2)],h.YLim,'--r','LineWidth',2);
legend([h1 h2 h3(1)],{'Simulation Mean','True Mean','95% CI'});
h.XTick = sort([h.XTick xbar]);
h.XTickLabel{h.XTick == xbar} = xbar;
The posterior distribution of
\varphi
is roughly normal with mean and standard deviation approximately 0.51 and 0.1, respectively. The true mean of
\varphi
is 0.6, and it is less than one standard deviation to the right of the simulation mean.
Compute the maximum likelihood estimate of
\varphi
. That is, treat
\varphi
as a fixed, but unknown parameter, and then estimate Mdl using the Kalman filter and maximum likelihood.
[~,estParams] = estimate(Mdl,y,phi0)
Akaike info criterion: 22.2868
Bayesian info criterion: 25.2974
| Coeff Std Err t Stat Prob
c(1) | 0.53590 0.19183 2.79360 0.00521
| Final State Std Dev t Stat Prob
x(1) | -0.85059 0.00000 -6.45811e+08 0
x(2) | 0.00454 0 Inf 0
estParams = 0.5359
\varphi
is 0.54. Both estimates are within one standard deviation or standard error from the true value of
\varphi
|
WaltherGraph - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Graph Theory : GraphTheory Package : SpecialGraphs : WaltherGraph
construct Walther graph
WaltherGraph()
The WaltherGraph command creates the Walther graph.
The Walther graph is a graph with 25 vertices and 31 edges.
\mathrm{with}\left(\mathrm{GraphTheory}\right):
\mathrm{with}\left(\mathrm{SpecialGraphs}\right):
\mathrm{WG}≔\mathrm{WaltherGraph}\left(\right)
\textcolor[rgb]{0,0,1}{\mathrm{WG}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: an undirected unweighted graph with 25 vertices and 31 edge\left(s\right)}}
\mathrm{ChromaticIndex}\left(\mathrm{WG}\right)
\textcolor[rgb]{0,0,1}{3}
"Walther graph", Wikipedia. http://en.wikipedia.org/wiki/Walther_graph
The GraphTheory[SpecialGraphs][WaltherGraph] command was introduced in Maple 2021.
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(\text{minibatch} , \text{in\_channels} , iT , iH , iW)
(\text{out\_channels} , \frac{\text{in\_channels}}{\text{groups}} , kT , kH , kW)
bias – optional bias tensor of shape
(\text{out\_channels})
stride – the stride of the convolving kernel. Can be a single number or a tuple (sT, sH, sW) . Default: 1
implicit paddings on both sides of the input. Can be a string {‘valid’, ‘same’}, single number or a tuple (padT, padH, padW) . Default: 0 padding='valid' is the same as no padding. padding='same' pads the input so the output has the same shape as the input. However, this mode doesn’t support any stride values other than 1.
\text{in\_channels}
>>> filters = torch.randn(33, 16, 3, 3, 3)
>>> inputs = torch.randn(20, 16, 50, 10, 20)
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Carnot's theorem (thermodynamics) — Wikipedia Republished // WIKI 2
The classical Carnot heat engine
Closed system
Ideal gas
Real gas
State of matter
Phase (matter)
Control volume
Free expansion
Heat engines
Heat pumps
Thermal efficiency
System properties
Note: Conjugate variables in italics
Property diagrams
Intensive and extensive properties
Process functions
Functions of state
Chemical potential / Particle number
Vapor quality
Reduced properties
Material properties
Property databases
Specific heat capacity
{\displaystyle c=}
{\displaystyle T}
{\displaystyle \partial S}
{\displaystyle N}
{\displaystyle \partial T}
{\displaystyle \beta =-}
{\displaystyle 1}
{\displaystyle \partial V}
{\displaystyle V}
{\displaystyle \partial p}
{\displaystyle \alpha =}
{\displaystyle 1}
{\displaystyle \partial V}
{\displaystyle V}
{\displaystyle \partial T}
Carnot's theorem
Clausius theorem
Fundamental relation
Maxwell relations
Onsager reciprocal relations
Bridgman's equations
Table of thermodynamic equations
Free energy
Internal energy
{\displaystyle U(S,V)}
{\displaystyle H(S,p)=U+pV}
Helmholtz free energy
{\displaystyle A(T,V)=U-TS}
Gibbs free energy
{\displaystyle G(T,p)=H-TS}
"Perpetual motion" machines
Entropy and time
Entropy and life
Brownian ratchet
Maxwell's demon
Loschmidt's paradox
Caloric theory
Vis viva ("living force")
Mechanical equivalent of heat
Motive power
Key publications
An Experimental Enquiry
Concerning ... Heat
On the Equilibrium of
Heterogeneous Substances
Reflections on the
Motive Power of Fire
Maxwell's thermodynamic surface
Entropy as energy dispersal
de Donder
von Helmholtz
von Mayer
van der Waals
Order and disorder
Carnot's theorem, developed in 1824 by Nicolas Léonard Sadi Carnot, also called Carnot's rule, is a principle that specifies limits on the maximum efficiency any heat engine can obtain. The efficiency of a Carnot engine depends solely on the temperatures of the hot and cold reservoirs.
Carnot's theorem states that all heat engines operating between two heat reservoirs are less efficient than a Carnot heat engine operating between the same reservoirs. Every Carnot heat engine operating between a pair of heat reservoirs is equally efficient, regardless of the working substance employed or the operation details.
The maximum efficiency is the ratio of the temperature difference between the reservoirs and the temperature of the hot reservoir, expressed in the equation
{\displaystyle \eta _{\text{max}}={\frac {T_{\mathrm {H} }-T_{\mathrm {C} }}{T_{\mathrm {H} }}}}
, where TC and TH are the absolute temperatures of the cold and hot reservoirs, respectively, and the efficiency
{\displaystyle \eta }
is the ratio of the work done by the engine to the heat drawn out of the hot reservoir.
Carnot's theorem is a consequence of the second law of thermodynamics. Historically, it was based on contemporary caloric theory, and preceded the establishment of the second law.[1]
Carnot Engine Efficiency and Carnot's Theorem
Carnot Cycle - An Ideal Heat Engine
1 Proof
1.1 Reversible engines
1.2 Irreversible engines
2 Definition of thermodynamic temperature
3 Applicability to fuel cells and batteries
An impossible situation: A heat engine cannot drive a less efficient (reversible) heat engine without violating the second law of thermodynamics. Quantities in this figure are the absolute values of energy transfers (heat and work).
The proof of the Carnot theorem is a proof by contradiction, or reductio ad absurdum (a method to prove a statement by assuming its falsity and logically deriving a false or contradictory statement from this assumption), as illustrated by the right figure showing two heat engines operating between two thermal reservoirs at different temperature. A heat engine
{\displaystyle M}
with greater efficiency
{\displaystyle \eta _{_{M}}}
is driving a heat engine
{\displaystyle L}
with less efficiency
{\displaystyle \eta _{_{L}}}
, causing the latter to act as a heat pump. However, if
{\displaystyle \eta _{_{M}}>\eta _{_{L}}}
, then the net heat flow would be backwards, i.e., into the hot thermal reservoir:
{\displaystyle Q_{\text{h}}^{\text{out}}=Q<{\frac {\eta _{_{M}}}{\eta _{_{L}}}}Q=Q_{\text{h}}^{\text{in}},}
{\displaystyle Q}
represents heat,
{\displaystyle {\text{in}}}
for input,
{\displaystyle {\text{out}}}
for output, and
{\displaystyle h}
for the hot or high temperature thermal reservoir. This means that heat into the hot reservoir from the engine pair is greater than heat into the engine pair from the hot reservoir (i.e., the hot reservoir continuously gets energy), and it is generally agreed that such a heat transfer is impossible by the second law of thermodynamics.
We begin by verifying the values of work
{\displaystyle W}
and heat
{\displaystyle Q}
depicted in the right figure. First, we must point out an important caveat: the engine
{\displaystyle L}
{\displaystyle \eta _{_{L}}}
is driven as a heat pump by the engine
{\displaystyle M}
with more efficiency
{\displaystyle \eta _{_{M}}}
, and therefore must be a reversible engine.[citation needed] If the engine
{\displaystyle L}
is not reversible, then the device could be built, but the expressions for work and heat flow shown in the figure would not be valid.
For each engine, the absolute value of the energy entering the engine,
{\displaystyle E^{\text{in}}}
, must equal the absolute value of the energy leaving from the engine,
{\displaystyle E^{\text{out}}}
(Otherwise, energy is continuously accumulated in each engine or the conservation of energy is violated.):
{\displaystyle E_{\text{M}}^{in}=Q=(1-\eta _{M})Q+\eta _{M}Q=E_{\text{M}}^{out},}
{\displaystyle E_{\text{L}}^{in}=\eta _{M}Q+\eta _{M}Q\left({\frac {1}{\eta _{L}}}-1\right)={\frac {\eta _{M}}{\eta _{L}}}Q=E_{\text{L}}^{out}.}
These expressions are consistent with the definition of efficiency as
{\displaystyle \eta =W/Q_{h}^{out}}
for the both engines (The second equation above is derived by this efficiency definition,
{\displaystyle W_{L}=-\eta _{M}Q}
, and the conservation of energy on the engine
{\displaystyle L}
.):
{\displaystyle \eta _{M}={\frac {W_{M}}{Q_{h}^{out,M}}}={\frac {\eta _{M}Q}{Q}}=\eta _{M},}
{\displaystyle \eta _{L}={\frac {W_{L}}{Q_{h}^{out,L}}}={\frac {-\eta _{M}Q}{-{\frac {\eta _{M}}{\eta _{L}}}Q}}=\eta _{L}.}
In these expressions, the sign convention is used (+ sign for heat
{\displaystyle Q}
entering an engine, + sign for work
{\displaystyle W}
done by an engine to its surroundings). It may seem odd that a hypothetical heat pump with low efficiency is being used to violate the second law of thermodynamics, but the figure of merit for refrigerator units is not efficiency,
{\displaystyle W/Q_{h}^{out}}
, but the coefficient of performance (COP),[2] which is
{\displaystyle Q_{c}^{out}/W}
. A reversible heat engine with low thermodynamic efficiency
{\displaystyle W/Q_{h}^{out}}
delivers more heat to the hot reservoir for a given amount of work when it is being driven as a heat pump.
Having established that the heat values shown in the right figure are correct, Carnot's theorem may be proven for irreversible and the reversible heat engines.[3]
To see that every reversible engine operating between reservoirs at temperatures
{\displaystyle T_{1}}
{\displaystyle T_{2}}
must have the same efficiency, assume that two reversible heat engines have different values of efficiency
{\displaystyle \eta }
, and let the more efficient engine
{\displaystyle M}
drive the less efficient engine
{\displaystyle L}
as a heat pump. As the figure shows, this will cause heat to flow from the cold to the hot reservoir without any external work or energy (from other than the reservoirs and the engines), which violates the second law of thermodynamics. Therefore, both (reversible) heat engines have the same efficiency, and we conclude that:
All reversible engines that operate between the same two thermal (heat) reservoirs have the same efficiency.
This is an important result because it helps establish the Clausius theorem, which implies that the change in entropy
{\displaystyle S}
is unique for all reversible processes:[4]
{\displaystyle \Delta S=\int _{a}^{b}{\frac {dQ_{\text{rev}}}{T}}}
as the entropy change is the same over all reversible process paths from a state
{\displaystyle a}
to a state
{\displaystyle b}
in a V-T (Volume-Temperature) space. If this integral were not path independent, then entropy would not be a state variable.[5]
Irreversible engines
If one of the engines is irreversible, then it must be the engine
{\displaystyle M}
, placed so that it reversely drives the less efficient but reversible engine
{\displaystyle L}
. But if this irreversible engine is more efficient than the reversible engine, (i.e., if
{\displaystyle \eta _{M}>\eta _{L}}
), then the second law of thermodynamics is violated. Since a Carnot heat engine operating in a Carnot cycle is a reversible engine, we have the first part of Carnot's theorem:
No irreversible engine is more efficient than a Carnot engine operating between the same two thermal reservoirs.
Definition of thermodynamic temperature
Main article: Definition of thermodynamic temperature
The efficiency of the engine is the work divided by the heat introduced to the system or
{\displaystyle \eta ={\frac {w_{\text{cy}}}{q_{H}}}={\frac {q_{H}-q_{C}}{q_{H}}}=1-{\frac {q_{C}}{q_{H}}}}
where wcy is the work done per cycle. Thus, the efficiency depends only on qC / qH.[6]
Because all reversible engines operating between the same heat reservoirs are equally efficient, all reversible heat engines operating between temperatures T1 and T2 must have the same efficiency, meaning the efficiency is a function only of the two temperatures:
{\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})}
In addition, a reversible heat engine operating between temperatures T1 and T3 must have the same efficiency as one consisting of two cycles, one between T1 and another (intermediate) temperature T2, and the second between T2 and T3. This can only be the case if
{\displaystyle f(T_{1},T_{3})={\frac {q_{3}}{q_{1}}}={\frac {q_{2}q_{3}}{q_{1}q_{2}}}=f(T_{1},T_{2})f(T_{2},T_{3}).}
Specializing to the case that
{\displaystyle T_{1}}
{\displaystyle f(T_{2},T_{3})={\frac {f(T_{1},T_{3})}{f(T_{1},T_{2})}}={\frac {273.16\cdot f(T_{1},T_{3})}{273.16\cdot f(T_{1},T_{2})}}.}
{\displaystyle T=273.16\cdot f(T_{1},T)\,}
then the function viewed as a function of thermodynamic temperature, is
{\displaystyle f(T_{2},T_{3})={\frac {T_{3}}{T_{2}}},}
and the reference temperature T1 has the value 273.16. (Of course any reference temperature and any positive numerical value could be used—the choice here corresponds to the Kelvin scale.)
{\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})={\frac {T_{C}}{T_{H}}}}
{\displaystyle \eta =1-{\frac {q_{C}}{q_{H}}}=1-{\frac {T_{C}}{T_{H}}}}
Applicability to fuel cells and batteries
Since fuel cells and batteries can generate useful power when all components of the system are at the same temperature (
{\displaystyle T=T_{H}=T_{C}}
), they are clearly not limited by Carnot's theorem, which states that no power can be generated when
{\displaystyle T_{H}=T_{C}}
. This is because Carnot's theorem applies to engines converting thermal energy to work, whereas fuel cells and batteries instead convert chemical energy to work.[7] Nevertheless, the second law of thermodynamics still provides restrictions on fuel cell and battery energy conversion.[8]
A Carnot battery is a type of energy storage system that stores electricity in heat storage and converts the stored heat back to electricity through thermodynamic cycles.[9]
Chambadal–Novikov efficiency
Heating and cooling efficiency bounds
^ John Murrell (2009). "A Very Brief History of Thermodynamics". Retrieved May 2, 2014. Archive copy at the Internet Archive PDF (142 Archived November 22, 2009, at the Wayback Machine KB)
^ Tipler, Paul; Mosca, G. (2008). "19.2, 19.7". Physics for Scientists and Engineers (6th ed.). Freeman. ISBN 9781429201322.
^ "Lecture 10: Carnot theorem" (PDF). Feb 7, 2005. Retrieved October 5, 2010.
^ Ohanian, Hans (1994). Principles of Physics. W.W. Norton and Co. p. 438. ISBN 039395773X.
^ http://faculty.wwu.edu/vawter/PhysicsNet/Topics/ThermLaw2/ThermalProcesses.html Archived 2013-12-28 at the Wayback Machine, and http://www.itp.phys.ethz.ch/education/hs10/stat/slides/Laws_TD.pdf Archived 2013-12-13 at the Wayback Machine. Both retrieved 13 December 2013.
^ The sign of qC > 0 for the waste heat lost by the system violates the sign convention of heat.
^ "Fuel Cell versus Carnot Efficiency". Retrieved Feb 20, 2011.
^ Jacob, Kallarackel T; Jain, Saurabh (July 2005). Fuel cell efficiency redefined : Carnot limit reassessed. Q1 - Ninth International Symposium on Solid Oxide Fuel Cells (SOFC IX). USA. Archived from the original on 2016-03-04. Retrieved 2013-04-23.
^ Dumont, Olivier; Frate, Guido Francesco; Pillai, Aditya; Lecompte, Steven; De paepe, Michel; Lemort, Vincent (2020). "Carnot battery technology: A state-of-the-art review". Journal of Energy Storage. 32: 101756. doi:10.1016/j.est.2020.101756. ISSN 2352-152X.
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Rd Sharma 2018 for Class 9 Math Chapter 10 - Lines And Angles
Rd Sharma 2018 Solutions for Class 9 Math Chapter 10 Lines And Angles are provided here with simple step-by-step explanations. These solutions for Lines And Angles are extremely popular among Class 9 students for Math Lines And Angles Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma 2018 Book of Class 9 Math Chapter 10 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma 2018 Solutions. All Rd Sharma 2018 Solutions for class Class 9 Math are prepared by experts and are 100% accurate.
{60}^{0}+4x+{40}^{0}={180}^{0}\phantom{\rule{0ex}{0ex}} 4x+{100}^{0}={180}^{0}\phantom{\rule{0ex}{0ex}} 4x={180}^{0}-{100}^{0}\phantom{\rule{0ex}{0ex}} 4x={80}^{0}\phantom{\rule{0ex}{0ex}} x=\frac{{80}^{0}}{4}\phantom{\rule{0ex}{0ex}} x=\overline{){20}^{0}}\phantom{\rule{0ex}{0ex}}
In the given figure, ACB is a line such that ∠DCA = 5x and ∠DCB = 4x. Find the values of ∠DCA and ∠DCB.
\therefore \angle DCA=5x=5×20°=100°\phantom{\rule{0ex}{0ex}}\angle DCB=4x=4×20°=80°
Hence, the values of ∠DCA and ∠DCB are 100∘ and 80∘ respectively.
\frac{1}{2}
\angle QOS+\angle POS=2×90
25°
\mathrm{So}, z + 25°=180°\phantom{\rule{0ex}{0ex}}⇒z=180-25\phantom{\rule{0ex}{0ex}} ⇒z=155°\phantom{\rule{0ex}{0ex}}
155°
\mathrm{So}, y+ 155°=180°\phantom{\rule{0ex}{0ex}}⇒y=180-155\phantom{\rule{0ex}{0ex}} ⇒y=25°\phantom{\rule{0ex}{0ex}}
x=155°, y=25° \mathrm{and} z=155°
\angle 1=36×3=108° \mathrm{and} \angle 2=36×2=72°
\angle 2=120° \left(\mathrm{alternate} \mathrm{interior} \mathrm{angles} \mathrm{are} \mathrm{equal}\right)
\angle 1=\angle 3 \left(\mathrm{corresponding} \mathrm{angles}\right)
\angle 3 \mathrm{and} 120° \mathrm{form} \mathrm{a} \mathrm{linear} \mathrm{pair}.
\angle 3+120°=180°\phantom{\rule{0ex}{0ex}}⇒\angle 3=180-120\phantom{\rule{0ex}{0ex}}⇒\angle 3=60°
\angle 1=\angle 3=60°,\angle 2=120°
{\left(\frac{2}{3}\right)}^{rd}
\perp
\perp
\angle ACD=\angle ACE+\angle ECD\phantom{\rule{0ex}{0ex}}\angle ACD=22°+35°\phantom{\rule{0ex}{0ex}}\angle ACD=57°
\angle AOD,\angle AOC,\angle COB \mathrm{and} \angle BOD
\angle AOD+\angle AOC+\angle COB+\angle BOD=360°
\angle AOC+\angle BOD=85°
\angle COD+85°=180°\phantom{\rule{0ex}{0ex}}⇒\angle COD=180°-85°\phantom{\rule{0ex}{0ex}}⇒\angle COD=95°
90°
\angle 1 \mathrm{and} \angle 2
90-\frac{x}{2}
\angle 1=\angle AOB=110 \left(\mathrm{vertically} \mathrm{opposite} \mathrm{angles}\right)
30°+x+110°=180°\phantom{\rule{0ex}{0ex}}⇒x=180-110-30=40
40°
\frac{y}{x} = 5
\frac{z}{x} = 4
\angle BDC
\angle ABD+\angle BDC=180°
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On framed cobordism classes of classical Lie groups
October, 2003 On framed cobordism classes of classical Lie groups
It is known that any compact connected Lie group with its left invariant framing is framed null-cobordant in the
p
-component for any prime
p\ne 2,3
. In this paper we will prove that the 3-components of
\mathrm{SO}\left(2n+1\right)
Sp\left(n\right)
n\ge 3
n\ne 5,7,11
. Combining this with the previously known results on
\mathrm{SO}\left(2n\right)
SU\left(n\right)
consequently we see that any classical group has at most only the 2-component with some exceptions.
Haruo MINAMI. "On framed cobordism classes of classical Lie groups." J. Math. Soc. Japan 55 (4) 1033 - 1052, October, 2003. https://doi.org/10.2969/jmsj/1191418762
Secondary: 19L20 , 57R15
Keywords: Adams conjecture , classical Lie group , framed manifold , J-morphism , left invariant framing
Haruo MINAMI "On framed cobordism classes of classical Lie groups," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 55(4), 1033-1052, (October, 2003)
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Runs created — Wikipedia Republished // WIKI 2
Runs created (RC) is a baseball statistic invented by Bill James to estimate the number of runs a hitter contributes to their team.
BNTV Saber Shorts: wRC+
TEX@BAL: Rangers score 30 runs against the Orioles
2.1 Basic runs created
2.2 "Stolen base" version of runs created
2.3 "Technical" version of runs created
2.4 2002 version of runs created
2.5 Other expressions of runs created
4 Related statistics
James explains in his book, The Bill James Historical Baseball Abstract, why he believes runs created is an essential thing to measure:
With regard to an offensive player, the first key question is how many runs have resulted from what he has done with the bat and on the basepaths. Willie McCovey hit .270 in his career, with 353 doubles, 46 triples, 521 home runs and 1,345 walks -- but his job was not to hit doubles, nor to hit singles, nor to hit triples, nor to draw walks or even hit home runs, but rather to put runs on the scoreboard. How many runs resulted from all of these things?[1]
{\displaystyle RC={\frac {A\;\times \;B}{C}}}
Basic runs created
{\displaystyle RC={\frac {(H+BB)\times TB}{AB+BB}}}
where H is hits, BB is base on balls, TB is total bases and AB is at-bats.
{\displaystyle RC=OBP\times SLG\times AB}
{\displaystyle RC=OBP\times TB}
where OBP is on-base percentage, SLG is slugging average, AB is at-bats and TB is total bases, however it is worth noting that OBP includes the hit-by-pitch while the previous RC formula does not.
"Stolen base" version of runs created
{\displaystyle RC={\frac {(H+BB-CS)\times (TB+(.55\times SB))}{AB+BB}}}
where H is hits, BB is base on balls, CS is caught stealing, TB is total bases, SB is stolen bases, and AB is at bats.
"Technical" version of runs created
{\displaystyle RC={\frac {(H+BB-CS+HBP-GIDP)\times (TB+(.26\times (BB-IBB+HBP))+(.52\times (SH+SF+SB)))}{AB+BB+HBP+SH+SF}}}
where H is hits, BB is base on balls, CS is caught stealing, HBP is hit by pitch, GIDP is grounded into double play, TB is total bases, IBB is intentional base on balls, SH is sacrifice hit, SF is sacrifice fly, SB is stolen base, and AB is at bats.
2002 version of runs created
Earlier versions of runs created overestimated the number of runs created by players with extremely high A and B factors (on-base and slugging), such as Babe Ruth, Ted Williams and Barry Bonds. This is because these formulas placed a player in an offensive context of players equal to himself; it is as if the player is assumed to be on base for himself when he hits home runs. Of course, this is impossible, and in reality, a great player is interacting with offensive players whose contributions are inferior to his. The 2002 version corrects this by placing the player in the context of his real-life team. This 2002 version also takes into account performance in "clutch" situations.
{\displaystyle H+BB-CS+HBP-GIDP}
{\displaystyle (1.125\times {\mathit {1B}})+(1.69\times {\mathit {2B}})+(3.02\times {\mathit {3B}})+(3.73\times HR)+.29\times (BB-IBB+HBP)+.492\times (SH+SF+SB)-(.04\times K)}
{\displaystyle AB+BB+HBP+SH+SF}
{\displaystyle RC=\left({\frac {(2.4C+A)\;(3C+B)}{9C}}\right)-.9C}
{\displaystyle H_{RISP}-(AB_{RISP}\times BA)+HR_{ROB}-{\frac {AB_{ROB}\times HR}{AB}}}
where RISP is runners in scoring position, BA is batting average, HR is home run, and ROB is runners on base. The subscripts indicate the required condition for the formula. For example,
{\displaystyle H_{RISP}}
Other expressions of runs created
{\displaystyle {\frac {RC}{27}}}
Win Shares is James' attempt to summarize, in one stat, a player's contributions on both offense and defense.
Career leaders in Runs Created
Single-season leaders in Runs Created
Runs Created leaders among active players
Year-by-year leaders in Runs Created
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Total_ring_of_fractions Knowpia
In abstract algebra, the total quotient ring,[1] or total ring of fractions,[2] is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings R that may have zero divisors. The construction embeds R in a larger ring, giving every non-zero-divisor of R an inverse in the larger ring. If the homomorphism from R to the new ring is to be injective, no further elements can be given an inverse.
{\displaystyle R}
{\displaystyle S}
be the set of elements which are not zero divisors in
{\displaystyle R}
{\displaystyle S}
is a multiplicatively closed set. Hence we may localize the ring
{\displaystyle R}
at the set
{\displaystyle S}
to obtain the total quotient ring
{\displaystyle S^{-1}R=Q(R)}
{\displaystyle R}
is a domain, then
{\displaystyle S=R-\{0\}}
and the total quotient ring is the same as the field of fractions. This justifies the notation
{\displaystyle Q(R)}
, which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain.
{\displaystyle S}
in the construction contains no zero divisors, the natural map
{\displaystyle R\to Q(R)}
is injective, so the total quotient ring is an extension of
{\displaystyle R}
For a product ring A × B, the total quotient ring Q(A × B) is the product of total quotient rings Q(A) × Q(B). In particular, if A and B are integral domains, it is the product of quotient fields.
For the ring of holomorphic functions on an open set D of complex numbers, the total quotient ring is the ring of meromorphic functions on D, even if D is not connected.
In an Artinian ring, all elements are units or zero divisors. Hence the set of non-zero divisors is the group of units of the ring,
{\displaystyle R^{\times }}
{\displaystyle Q(R)=(R^{\times })^{-1}R}
. But since all these elements already have inverses,
{\displaystyle Q(R)=R}
In a commutative von Neumann regular ring R, the same thing happens. Suppose a in R is not a zero divisor. Then in a von Neumann regular ring a = axa for some x in R, giving the equation a(xa − 1) = 0. Since a is not a zero divisor, xa = 1, showing a is a unit. Here again,
{\displaystyle Q(R)=R}
The total ring of fractions of a reduced ringEdit
Proposition — Let A be a reduced ring that has only finitely minimal prime ideals,
{\displaystyle {\mathfrak {p}}_{1},\dots ,{\mathfrak {p}}_{r}}
{\displaystyle Q(A)\simeq \prod _{i=1}^{r}Q(A/{\mathfrak {p}}_{i}).}
{\displaystyle \operatorname {Spec} (Q(A))}
is the Artinian scheme consisting (as a finite set) of the generic points of the irreducible components of
{\displaystyle \operatorname {Spec} (A)}
Proof: Every element of Q(A) is either a unit or a zerodivisor. Thus, any proper ideal I of Q(A) is contained in the set of zerodivisors of Q(A); that set equals the union of the minimal prime ideals
{\displaystyle {\mathfrak {p}}_{i}Q(A)}
since Q(A) is reduced. By prime avoidance, I must be contained in some
{\displaystyle {\mathfrak {p}}_{i}Q(A)}
. Hence, the ideals
{\displaystyle {\mathfrak {p}}_{i}Q(A)}
are maximal ideals of Q(A). Also, their intersection is zero. Thus, by the Chinese remainder theorem applied to Q(A),
{\displaystyle Q(A)\simeq \prod _{i}Q(A)/{\mathfrak {p}}_{i}Q(A)}
Let S be the multiplicatively closed set of non-zerodivisors of A. By exactness of localization,
{\displaystyle Q(A)/{\mathfrak {p}}_{i}Q(A)=A[S^{-1}]/{\mathfrak {p}}_{i}A[S^{-1}]=(A/{\mathfrak {p}}_{i})[S^{-1}]}
which is already a field and so must be
{\displaystyle Q(A/{\mathfrak {p}}_{i})}
{\displaystyle \square }
{\displaystyle R}
{\displaystyle S}
is any multiplicatively closed set in
{\displaystyle R}
, the localization
{\displaystyle S^{-1}R}
can still be constructed, but the ring homomorphism from
{\displaystyle R}
{\displaystyle S^{-1}R}
might fail to be injective. For example, if
{\displaystyle 0\in S}
{\displaystyle S^{-1}R}
is the trivial ring.
^ Matsumura 1980, p. 12.
Matsumura, Hideyuki (1980), Commutative algebra
Matsumura, Hideyuki (1989), Commutative ring theory
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Explainer:The Eyesight Improvement Equation - EndMyopia Wiki
This page in a nutshell: Vision improvement is more simple than you think.
Our finest and well-endowed data analysts have looked at all the facts and figures. They figured out this remarkably simple equation:
{\displaystyle good\ close\ up\ habits+(correct\ normalized+active\ focus+distance\ vision)=long\ term\ improvement}
Make sure the four variables to the left are good, and you will improve in 99% of cases.
1 Good close-up habits
2 Correct normalized
3 Active Focus and Distance Vision
3.2 Distance Vision
4.1 Ragging on the Bates method
Good close-up habits
Differentials are essential for successful vision improvement, if you have myopia above -2, and can't see screens clearly without glasses. You cannot wear your full strength prescription glasses for close work, as this is the primary stimulus that elongated your eyeball in the first place. Full strength glasses, when combined with screens are the primary cause of lens-induced myopia.
Differentials are glasses that correct for near vision and are used only for close-up (screens, reading books). They are typically used when someone has more than -2 spherical diopters.[1]
If you're between 0 and -2 D, you can choose not to wear glasses for close-up work on screens, but a weak minus might be required if you work at a longer distance. It is also possible to read books at -2.5 without glasses.
Correct normalized
Normalized should undercorrect you by 0.25 diopters (the minimum difference available for standard corrective lenses) from the correction you would need to see at Emmetropia. Don't reduce more than 0.25, as reducing faster than necessary provides no benefit and can stagnate progress. Vision improvement takes time.
If the original correction was just right, a reduction of 0.25 would take your blur horizon to 4m.
Active Focus and Distance Vision
Active Focus: this one simple trick improves your eyesight, optometrists hate it!
— We should run adverts like this
Active Focus allows you challenge blur and clear your vision, increasing your blur horizon. This is the 'learning to ride a bicycle' part. Once you know how to do Active Focus, you don't usually forget how to do it. It can be very annoying for newcomers, as there isn't really an easy and established way of finding it that works for everybody.
Community:Writings may be of use in helping you figure out Active Focus for yourself.
Distance vision, without a doubt is your primary habit to improve vision. The combination of distance vision with Active Focus is the reason anyone gets back to 20/20 in a relatively efficient and effective manner. There are ways to improve eyesight without distance vision, such as Active Focusing onto screens [citation needed]. However, this is vastly inferior to distance vision habits.
Incorporating distance vision can be done to varying degrees in your life based on many factors. If you're already in a profession or obligation that requires a lot of distance vision, you should be good to go. If not, make a habit of getting distance vision into your life.
Incorporating as little as thirty minutes a day will be good for starting out initially. 2-3 hours a day will be ideal for vision improvement, after which any distance vision beyond that in a day will suffer from diminishing returns. This is important, so you might want to look over your priorities if you are struggling to find the time for distance vision
Average annual improvement (after the initial reduction), about 1 diopter (0.75 to 1.25). Comes out to 0.25 diopters every three months.
Reality of course is that the biology isn’t linear, and our environment changes (winters for example, possibly slower rate of improvement, and often a nice jump after spring).
People tend to overcomplicate vision improvement. EndMyopia is fiendish in that it's relatively simple to understand what to do and start improving.
You will likely experience much faster gains in the beginning due to ciliary spasm you will lose as a result of fixing your close-up habits and wearing differentials. This is good because it shows you a real improvement in your eyesight, and can get you excited for the long-term improvements coming up. You can improve by as much a whole diopter in the first 90 days, after which you should expect your improvements to slow down.[3]
Ragging on the Bates method
For the record, EndMyopia is far more simple and effective than the Bates Method and the millions of eye exercises that are involved there, none of which actually tackle the root cause of the problem. How does placing your palm on your eyes induce myopic defocus, the primary cause for decreasing the axial length of the eye as shown in independent clinical studies?[4] It doesn't, that's how.
The biggest learning curve is Active Focus, after which other stuff is easily addressed through habits and good understanding of the method.
The community is full of people who tried the Bates method and failed
↑ The EndMyopia Blog, https://endmyopia.org/low-myopia-when-can-you-stop-wearing-glasses/
↑ https://endmyopia.org/how-fast-can-i-improve-my-vision/
↑ The EndMyopia Blog, https://endmyopia.org/0-25-to-0-50-improvement-in-90-days-not-enough/
↑ Read SA, Collins MJ, Sander BP (2010). "Human optical axial length and defocus". Invest Ophthalmol Vis Sci. 51 (12): 6262–9. doi:10.1167/iovs.10-5457. PMID 20592235. CS1 maint: multiple names: authors list (link)
Retrieved from "https://wiki.endmyopia.org/index.php?title=Explainer:The_Eyesight_Improvement_Equation&oldid=16323"
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Step response requirement for control system tuning - MATLAB - MathWorks América Latina
\text{Req}\text{.ReferenceModel}=\frac{1/\text{tau}}{s+1/\text{tau}}.
\text{Req}\text{.ReferenceModel}=\frac{{\left(1/\text{tau}\right)}^{2}}{{s}^{2}+2\left(\text{zeta}/\text{tau}\right)s+{\left(1/\text{tau}\right)}^{2}}.
\text{Req}\text{.ReferenceModel}=\frac{1/\text{tau}}{s+1/\text{tau}}.
\text{Req}\text{.ReferenceModel}=\frac{{\left(1/\text{tau}\right)}^{2}}{{s}^{2}+2\left(\text{zeta}/\text{tau}\right)s+{\left(1/\text{tau}\right)}^{2}}.
\text{gap}=\frac{{‖y\left(t\right)-{y}_{ref}\left(t\right)‖}_{2}}{{‖1-{y}_{ref}\left(t\right)‖}_{2}}.
{‖\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}‖}_{2}
f\left(x\right)=\frac{{‖\frac{1}{s}\left(T\left(s,x\right)-{T}_{ref}\left(s\right)\right)‖}_{2}}{\text{RelGap}{‖\frac{1}{s}\left({T}_{ref}\left(s\right)-I\right)‖}_{2}}.
{‖\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}‖}_{2}
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Formulas - Angle
List of formulas used in the Angle protocol and at app.angle.money
Perpetuals (HA)
The margin displayed on your position is computed net of opening fees, i.e.
\texttt{margin} = \texttt{initial margin - position size} \times \texttt{opening fees}
In Angle, leverage is computed as:
\texttt{leverage} = \frac{\texttt{(margin + position Size)}}{\texttt{margin}}
Margin: 10,000 DAI
Position size: 100,000 DAI
\texttt{leverage} = \frac{1 + 10}{1} = 11
The Cash Out Amount represents the amount you should receive in your wallet after closing the perpetual:
\texttt{cash out amount} = \texttt{margin} \pm \texttt{gross PnL - closing fee}
\texttt{grossPnL} = \texttt{position size}\times(1-\frac{\texttt{initialPrice}}{\texttt{currentPrice}})
The PnL displayed on the app represents the gain or loss you would make if closing the position. It is computed net of fees.
\texttt{PnL} = \texttt{cash out amount - initial margin}
\texttt{PnL} = \texttt{gross PnL - closing fee}
Maintenance Margins
DAI: 0.625%
USDC: 0.625%
FEI: 0.625%
FRAX: 0.625%
Margin Ratio Formula In Angle, the margin ratio is computed as:
\texttt{margin ratio} = \frac{\texttt{cash out amount}}{\texttt{position size}}
Est. APR on open positions
In perpetuals/open position page of the app
ANGLE rewards for position holders depends on the Position Size, as what matters for the protocol is how much collateral is hedged. However, the APR is computed from the initial margin, as this is what users bring to the protocol.
Rewards distribution: 5 ANGLE / week / 1,000 DAI in position
ANGLE price: 0.80 DAI
APR = \frac{5\times{52} \times{100} \times{0.80}} {10,000}
APR = 2.08 = 208\%
Depositing Liquidity (SLP)
In the Yield page of the app Both the Slippage and SlippageFee can vary depending on the collateral/stablecoin pair.
SLP can face a slippage when withdrawing funds depending on the collateral ratio of the pool they are withdrawing from. This is put in place to incentivize them to stay in the protocol while it gets re-collateralized.
\texttt{amnt received} = \texttt{amnt withdrawn} \times{(1 - \texttt{slippage})}
The current slippage for SLP can be consulted in the analytics by selecting a pool and looking at Slippage in the Fee Info > SLP section.
Slippage on Fees for SLP
When the protocol gets close to be under-collateralized, it progressively keeps a bigger portion of the fees usually going to SLPs to grow the suplus and be able to pay back stable holders. Note that this doesn't impact the initial deposits of SLPs nor the fees earned up until the start of the slippageFee.
\texttt{fees received} = \texttt{fees to SLPs} \times{(1-\texttt{slippage fee})}
The current slippageFee for SLP can be consulted in the analytics by selecting a pool and looking at SlippageFee in the Fee Info > SLP section.
If you want to know the current protocol parameters in place, you can have a look at the analytics at analytics.angle.money or directly in the SDK.
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Maths of ED25519 - Electron Labs
Welcome to Electron Labs
ZK for Beginners
Intro to Circom
Circom ED25519
Maths of ED25519
Circom Implementation
Batch of Signatures
How to Integrate in your project?
ED25519 signature uses a curve and involves performing operations on this curve. Let's dive deeper.
All operations in this signature scheme are defined under modulo p where
p = 2^{255} - 19
The curve equation is defined as follows:-
ax^2 + y^2 = 1 + dx^2y^2
where a = -1 and d = - 121665/121666
Base Point is defined as (Bx, By) where By = 4/5. Here the / operator represents the inverse modulo operation wrt p. Hence By = 4*mod_inverse(5,p) => By = 46316835694926478169428394003475163141307993866256225615783033603165251855960
Substitute By in the curve equation to calculate Bx = 15112221349535400772501151409588531511454012693041857206046113283949847762202
Defining Point Addition
Say you have two points P and Q on the curve. If we draw a straight line through P and Q, it will intersect the curve at another point R as shown in the diagram. When this happens, we say that R is the sum of points P and Q. This is how pt addition is defined. Now let’s calculate R given P and Q.
Point Addition on the Curve
Given that P + Q = R . R is calculated as follows:-
(x_1, y_1) + (x_2, y_2) = \left( \frac{x_1y_2 + x_2y_1}{1 + dx_1x_2y_1y_2}, \frac{y_1y_2 - ax_1x_2}{1-dx_1x_2y_1y_2} \right)
This formula is derived by finding a point R that lies both on the curve and on the straight line through P and Q. To see detailed derivation, see https://martin.kleppmann.com/papers/curve25519.pdf (although it's for a different curve, but the concept is the same)
The above equation can be re-arranged to a polynomial form (R1CS). Points that satisfy this polynomial will always satisfy the property P + Q = R
Defining Multiplication of a pt on the Curve
Given a point P on the curve, we define another point on the curve Q as the “scalar multiplication” of P and k such that Q =k*P where k is a constant under mod p. Scalar multiplication is defined as the repeated addition of pts:-
k=2, Q = P+P \newline k=4, Q = 2P + 2P \newline k=8, Q = 4P + 4P \newline ... \newline k = 256, Q = 128P + 128P
Hence, we can calculate Q in log(k) steps. Since max(k) < 2^255-19, we need to perform a max of log(2^255-19) ~ 255 steps to calculate a scalar multiple.
Verifying an ED25519 signature
Let’s see some definitions:-
The ED25519 key-pair consists of:
Private Key (integer under mod p) : privKey
Public key (curve point): pubKey = privKey * B where B is the base pt as defined above
The ED22519 signature verification algorithm takes as input a text message msg + the signer's ED25519 public key pubKey + the ED25519 signature {R, s} and produces as output a boolean value (valid or invalid signature). Here s is a scalar, and R is a pt on the curve. ED25519 verification works as follows (with minor simplifications):
EdDSA_signature_verify(msg, pubKey, signature { R, s } ) --> valid / invalid
Calculate h = SHA512(R + pubKey + msg) mod q
Calculate P1 = s * B
Calculate P2 = R + h * pubKey
Here q is the curve order. q = 2^252 + 27742317777372353535851937790883648493
In step one, you must be wondering how can we use pubKey as an input to SHA512 as pubKey is a curve point (not a scalar). pubkey in this step is represented as a "compressed" curve pt i.e only the y-coordinate. In step, 3, pubkey is used as a curve point.
https://martin.kleppmann.com/papers/curve25519.pdf
EdDSA and Ed25519
https://datatracker.ietf.org/doc/html/rfc8032
Circom ED25519 - Previous
Next - Circom ED25519
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Decimate signal using polyphase FIR halfband filter - Simulink - MathWorks 日本
h\left(n\right)=\frac{1}{2\mathrm{Ï}}{â«}_{â\mathrm{Ï}/2}^{\mathrm{Ï}/2}{e}^{j\mathrm{Ï}n}d\mathrm{Ï}=\frac{\mathrm{sin}\left(\frac{\mathrm{Ï}}{2}n\right)}{\mathrm{Ï}n}.
The ideal filter is not realizable because the impulse response is noncausal and not absolutely summable. However, the impulse response of the ideal lowpass filter possesses some important properties that are required of a realizable approximation. Specifically, the ideal lowpass halfband filter’s impulse response is:
equal to 1/2 at n=0. You can see this by using L’Hopital’s rule on the continuous-valued equivalent of the discrete-time impulse response.
g\left(n\right)=\frac{1}{2\mathrm{Ï}}{â«}_{â\mathrm{Ï}}^{â\mathrm{Ï}/2}{e}^{j\mathrm{Ï}n}d\mathrm{Ï}+\frac{1}{2\mathrm{Ï}}{â«}_{\mathrm{Ï}/2}^{\mathrm{Ï}}{e}^{j\mathrm{Ï}n}d\mathrm{Ï}.
g\left(n\right)=\frac{\mathrm{sin}\left(\mathrm{Ï}n\right)}{\mathrm{Ï}n}â\frac{\mathrm{sin}\left(\frac{\mathrm{Ï}}{2}n\right)}{\mathrm{Ï}n}.
The ideal highpass halfband filter’s impulse is:
{\mathrm{â}}^{\infty }
w\left(n\right)=\frac{{I}_{0}\left(\mathrm{β}\sqrt{1â{\left(\frac{nâN/2}{N/2}\right)}^{2}}\right)}{{I}_{0}\left(\mathrm{β}\right)},\text{â}\text{â}0â¤nâ¤N,
To obtain a Kaiser window that represents an FIR filter with stopband attenuation of α dB, use this β.
\mathrm{β}=\left\{\begin{array}{ll}0.1102\left(\mathrm{α}â8.7\right),\hfill & \mathrm{α}>50\hfill \\ 0.5842{\left(\mathrm{α}â21\right)}^{0.4}+0.07886\left(\mathrm{α}â21\right),\hfill & 50â¥\mathrm{α}â¥21\hfill \\ 0,\hfill & \mathrm{α}<21\hfill \end{array}
n=\frac{\mathrm{α}â7.95}{2.285\left(\mathrm{Î}\mathrm{Ï}\right)}
where Δω is the transition width.
Splitting a filter’s impulse response, h(n), into two polyphase components results in an even polyphase component with z-transform
{H}_{0}\left(z\right)=\underset{n}{â}h\left(2n\right){z}^{ân}.
{H}_{1}\left(z\right)=\underset{n}{â}h\left(2n+1\right){z}^{ân}.
H\left(z\right)={H}_{0}\left({z}^{2}\right)+{z}^{â1}{H}_{1}\left({z}^{2}\right).
|
\mathrm{restart};
\mathrm{with}\left(\mathrm{Finance}\right):
If you denote by
{S}_{u}
{S}_{d}
the multiplicative constants for up and down movements in the tree, and by
{P}_{u}
{P}_{d}
the probabilities of the upward and the downward movements, then the stock price
{S}_{p} = {S}_{i, j}
i
-th time step and the
j
-th node is
{S}_{i,j}={S}_{0}{S}_{u}{}^{j}{S}_{d}{}^{i-j}
i ∈\left\{0, 1, ..., n\right\}, j ∈ \left\{0, 1, ..., i\right\}
{P}_{i,j→k}={\begin{array}{cc}{P}_{u}& k=j+1\\ {P}_{d}& k=j\\ 0& \mathrm{otherwis}\end{array}.
{S}_{p}\cdot {S}_{u}\cdot {P}_{u}+{S}_{p}\cdot {S}_{d}\cdot {P}_{d}={S}_{f},
{S}_{f}=\mathrm{exp}\left(r\cdot \mathrm{Δt}\right)\cdot {S}_{p}
is the known forward price of the stock. In a general constant volatility recombining binomial tree
{S}_{u}
{S}_{d}
{S}_{u}={ⅇ}^{q\mathrm{Δt}+\mathrm{σ} \sqrt{\mathrm{Δt}}}
{S}_{d}={ⅇ}^{q\mathrm{Δt}-\mathrm{σ} \sqrt{\mathrm{Δt}}}
for some reasonable value of
q
{S}_{u}:={ⅇ}^{q\mathrm{Δt}+\mathrm{σ} \sqrt{\mathrm{Δt}}}
{\textcolor[rgb]{0,0,1}{S}}_{\textcolor[rgb]{0,0,1}{u}}\textcolor[rgb]{0,0,1}{:=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}
{S}_{d}:={ⅇ}^{q\mathrm{Δt}-\mathrm{σ} \sqrt{\mathrm{Δt}}}
{\textcolor[rgb]{0,0,1}{S}}_{\textcolor[rgb]{0,0,1}{d}}\textcolor[rgb]{0,0,1}{:=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}
{S}_{f}:={ⅇ}^{r\mathrm{Δt}}{S}_{p}
{\textcolor[rgb]{0,0,1}{S}}_{\textcolor[rgb]{0,0,1}{f}}\textcolor[rgb]{0,0,1}{:=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{S}}_{\textcolor[rgb]{0,0,1}{p}}
\mathrm{solve}\left(\left\{{S}_{p}{S}_{u}{P}_{u}+{S}_{p}{S}_{d}{P}_{d}={S}_{f},{P}_{u}+{P}_{d}=1\right\},\left\{{P}_{u},{P}_{d}\right\}\right);
\left\{{\textcolor[rgb]{0,0,1}{P}}_{\textcolor[rgb]{0,0,1}{d}}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}}{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{P}}_{\textcolor[rgb]{0,0,1}{u}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}}\right\}
\mathrm{assign}\left(\right);
{P}_{u};
\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}}
{P}_{d};
\frac{\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}}{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{Δt}}}}}
r:=0.03;
\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{0.03}
\mathrm{σ}≔0.2;
\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{0.2}
N:=20;
\textcolor[rgb]{0,0,1}{N}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{20}
T:=3.0;
\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{3.0}
\mathrm{Δt}:=\frac{T}{N}
\textcolor[rgb]{0,0,1}{\mathrm{Δt}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{0.1500000000}
{S}_{u} = \mathrm{exp}\left(\mathrm{σ}\cdot \sqrt{\frac{T}{n}}\right)
{S}_{d} = \mathrm{exp}\left(- \mathrm{σ}\cdot \sqrt{\frac{T}{n}}\right),
{P}_{u} = \frac{a\cdot \mathrm{exp}\left(r\frac{T}{n}\right) - d}{u - d} = \frac{\mathrm{exp}\left(r\frac{T}{n}\right) - \mathrm{exp}\left(-\mathrm{σ} \sqrt{\frac{T}{n}}\right)}{\mathrm{exp}\left(\mathrm{σ} \sqrt{\frac{T}{n}}\right) - \mathrm{exp}\left(-\mathrm{σ} \sqrt{\frac{T}{n}}\right) }.
This corresponds to the case when
q=0.
q≔0;
\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{0}
\mathrm{CRR}:=\mathrm{BinomialTree}\left(T,N,100,{S}_{u},{P}_{u},{S}_{d},{P}_{d}\right):
\mathrm{TreePlot}\left(\mathrm{CRR},\mathrm{axes}=\mathrm{BOXED},\mathrm{thickness}=3,\mathrm{gridlines}=\mathrm{true}\right);
\mathrm{TreePlot}\left(\mathrm{CRR},\mathrm{axes}=\mathrm{BOXED},\mathrm{thickness}=3,\mathrm{gridlines}=\mathrm{true},\mathrm{scale}=\mathrm{logarithmic}\right);
\mathrm{GetProbabilities}\left(\mathrm{CRR},2,2\right)
\left[\textcolor[rgb]{0,0,1}{0.5097284963}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.4902715037}\right]
{S}_{u} = \mathrm{exp}\left(\frac{\left(r - \frac{{\mathrm{σ}}^{2}}{2}\right)T}{n} + \mathrm{σ}\sqrt{\frac{T}{n}}\right)
\mathit{ }{S}_{d} = \mathrm{exp}\left(\frac{\left(r - \frac{{\mathrm{σ}}^{2}}{2}\right)T}{n} - \mathrm{σ}\sqrt{\frac{T}{n}}\right).
They constructed a binomial model where the first two moments of the discrete and continuous time-return processes match. As a consequence, a probability measure equal to one half results. This corresponds to the case when
q=r-\frac{{\mathrm{σ}}^{2}}{2}
q≔r-\frac{{\mathrm{σ}}^{2}}{2};
\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{0.01000000000}
{S}_{u},{S}_{d};
\textcolor[rgb]{0,0,1}{1.082160674}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.9268535887}
{P}_{u},{P}_{d};
\textcolor[rgb]{0,0,1}{0.5000193723}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.4999806277}
\mathrm{JR}≔\mathrm{BinomialTree}\left(T,N,100,{S}_{u},{P}_{u},{S}_{d},{P}_{d}\right):
\mathrm{TreePlot}\left(\mathrm{JR},\mathrm{axes}=\mathrm{BOXED},\mathrm{thickness}=3,\mathrm{gridlines}=\mathrm{true}\right);
\mathrm{TreePlot}\left(\mathrm{JR},\mathrm{axes}=\mathrm{BOXED},\mathrm{thickness}=3,\mathrm{gridlines}=\mathrm{true},\mathrm{scale}=\mathrm{logarithmic}\right);
\mathrm{restart};
\mathrm{with}\left(\mathrm{Finance}\right):
X:=\mathrm{ItoProcess}\left(0.1,\mathrm{sin}\left(t\right),0.1,x,t\right);
\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{_X}}
\mathrm{Drift}\left(X\left(t\right)\right);
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)
\mathrm{Diffusion}\left(X\left(t\right)\right);
\textcolor[rgb]{0,0,1}{0.1}
T:=\mathrm{ShortRateTree}\left(X,5.0,10\right):
\mathrm{TreePlot}\left(T,\mathrm{thickness}=2,\mathrm{color}=\mathrm{blue},\mathrm{gridlines}=\mathrm{true}\right);
\mathrm{restart};
\mathrm{with}\left(\mathrm{Finance}\right):
r:=0.11;
\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{0.11}
d:=0.04;
\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{0.04}
\mathrm{σ}:=\mathrm{ImpliedVolatilitySurface}\left(0.11-\frac{\left(K-100\right)\cdot 0.001}{10},t,K\right):
T:=\mathrm{ImpliedBinomialTree}\left(100,r,d,\mathrm{σ},3,7\right):
\mathrm{TreePlot}\left(T,\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right);
\mathrm{TreePlot}\left(T,\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true},\mathrm{color}=\mathrm{red}..\mathrm{blue},\mathrm{scale}=\mathrm{logarithmic}\right);
\mathrm{GetProbabilities}\left(T,1,1\right);
\left[\textcolor[rgb]{0,0,1}{0.5000000000}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.5000000000}\right]
\mathrm{GetProbabilities}\left(T,2,1\right);
\left[\textcolor[rgb]{0,0,1}{0.3817424623}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.6182575377}\right]
\mathrm{GetProbabilities}\left(T,2,2\right);
\left[\textcolor[rgb]{0,0,1}{0.6409710334}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.3590289666}\right]
\mathrm{T2}:=\mathrm{BlackScholesBinomialTree}\left(100,r,d,\mathrm{σ}\left(0,100\right),3,7\right):
\mathrm{P1}:=\mathrm{TreePlot}\left(T,\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true},\mathrm{color}=\mathrm{blue}\right):
\mathrm{P2}:=\mathrm{TreePlot}\left(\mathrm{T2},\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true},\mathrm{color}=\mathrm{red}\right):
{\mathrm{plots}}_{\mathrm{display}}\left(\mathrm{P1},\mathrm{P2}\right);
P:=\left(S,T\right)→\mathrm{BlackScholesPrice}\left(100.,S,T,\mathrm{σ}\left(T,S\right),r, d, \mathrm{put}\right);
\textcolor[rgb]{0,0,1}{P}\textcolor[rgb]{0,0,1}{:=}\left(\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{T}\right)\textcolor[rgb]{0,0,1}{→}\textcolor[rgb]{0,0,1}{\mathrm{Finance}}\textcolor[rgb]{0,0,1}{:-}\textcolor[rgb]{0,0,1}{\mathrm{BlackScholesPrice}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{100.}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{S}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{put}}\right)
C:=\left(S,T\right)→\mathrm{BlackScholesPrice}\left(100.,S,T,\mathrm{σ}\left(T,S\right),r, d, \mathrm{call}\right);
\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{:=}\left(\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{T}\right)\textcolor[rgb]{0,0,1}{→}\textcolor[rgb]{0,0,1}{\mathrm{Finance}}\textcolor[rgb]{0,0,1}{:-}\textcolor[rgb]{0,0,1}{\mathrm{BlackScholesPrice}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{100.}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{σ}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{S}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{call}}\right)
P\left(100,1.0\right);
\textcolor[rgb]{0,0,1}{1.62044795}
C\left(100,1.0\right);
\textcolor[rgb]{0,0,1}{8.11597835}
T:=\mathrm{ImpliedBinomialTree}\left(100,r,d,\mathrm{σ},1,200\right):
E:=\mathrm{EuropeanOption}\left(t→\mathrm{max}\left(t-100,0\right),1.0\right):
\mathrm{LatticePrice}\left(E,T,r\right);
\textcolor[rgb]{0,0,1}{8.146123180}
\mathrm{evalf}\left(C\left(100,1.0\right)\right);
\textcolor[rgb]{0,0,1}{8.11597835}
\mathrm{BlackScholesPrice}\left(100.,100,1.0,r,\mathrm{σ}\left(0.0,100\right),d, \mathrm{call}\right);
\textcolor[rgb]{0,0,1}{8.11597835}
E:=\mathrm{EuropeanOption}\left(t→\mathrm{max}\left(t-130,0\right),1.0\right):
\mathrm{LatticePrice}\left(E,T,r\right);
\textcolor[rgb]{0,0,1}{0.1622684190}
\mathrm{evalf}\left(C\left(130,1.0\right)\right);
\textcolor[rgb]{0,0,1}{0.16229691}
\mathrm{BlackScholesPrice}\left(100.,130,1.0,\mathrm{σ}\left(0.0,100\right),r, d, \mathrm{call}\right);
\textcolor[rgb]{0,0,1}{0.18855320}
q:=t→\mathrm{piecewise}\left(t<90,0,t<110,t-90,t<130,130-t,0\right);
\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{→}\textcolor[rgb]{0,0,1}{\mathrm{piecewise}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{90}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{110}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{90}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{130}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{130}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\right)
\mathrm{plot}\left(q,80..150,\mathrm{gridlines},\mathrm{thickness}=3\right);
E:=\mathrm{EuropeanOption}\left(q,1.0\right):
\mathrm{LatticePrice}\left(E,T,r\right);
\textcolor[rgb]{0,0,1}{9.644001241}
A:=\mathrm{AmericanOption}\left(q,0,1.0\right):
\mathrm{LatticePrice}\left(A,T,r\right);
\textcolor[rgb]{0,0,1}{14.07012078}
|
EUDML | Multiple positive solutions for a class of quasilinear elliptic boundary-value problems. EuDML | Multiple positive solutions for a class of quasilinear elliptic boundary-value problems.
Multiple positive solutions for a class of quasilinear elliptic boundary-value problems.
Perera, Kanishka. "Multiple positive solutions for a class of quasilinear elliptic boundary-value problems.." Electronic Journal of Differential Equations (EJDE) [electronic only] 2003 (2003): Paper No. 07, 5 p., electronic only-Paper No. 07, 5 p., electronic only. <http://eudml.org/doc/122830>.
keywords = {-linear -Laplacian problems; positive solutions; nonexistence; multiplicity; variational methods; -linear -Laplacian problems},
title = {Multiple positive solutions for a class of quasilinear elliptic boundary-value problems.},
TI - Multiple positive solutions for a class of quasilinear elliptic boundary-value problems.
KW - -linear -Laplacian problems; positive solutions; nonexistence; multiplicity; variational methods; -linear -Laplacian problems
p
p
-Laplacian problems, positive solutions, nonexistence, multiplicity, variational methods,
p
p
-Laplacian problems
|
EUDML | Change of base for relational variable sets. EuDML | Change of base for relational variable sets.
Change of base for relational variable sets.
Niefield, Susan. "Change of base for relational variable sets.." Theory and Applications of Categories [electronic only] 12 (2004): 248-261. <http://eudml.org/doc/124612>.
keywords = {relational variable set; specification structure; dynamic set; relational presheaf; change of base; exponentiable},
title = {Change of base for relational variable sets.},
TI - Change of base for relational variable sets.
KW - relational variable set; specification structure; dynamic set; relational presheaf; change of base; exponentiable
relational variable set, specification structure, dynamic set, relational presheaf, change of base, exponentiable
Category of relations, additive relations
2
Articles by Niefield
|
Recovering an algebraic curve using its projections from different points. Applications to static and dynamic computational vision | EMS Press
Recovering an algebraic curve using its projections from different points. Applications to static and dynamic computational vision
Jeremy Yirmeyahu Kaminski
We study some geometric configurations related to the projection of an irreducible algebraic curve embedded in
\C \PP^3
onto embedded projective planes. These configurations are motivated by applications to static and dynamic computational vision. More precisely, we study how an irreducible closed algebraic curve
X
\C \PP^3
, which degree is
d
g
, can be recovered using its projections from points onto embedded projective planes. The different embeddings are unknown. The only input is the defining equation of each projected curve. We show how both the embeddings and the curve in
\C \PP^3
can be recovered modulo some action of the group of projective transformations of
\C \PP^3
. In particular in the case of two projections, we show how in a generic situation, a characteristic matrix of the pair of embeddings can be recovered. In the process we address dimensional issues and as a result establish the minimal number of irreducible algebraic curves required to compute this characteristic matrix up to a finite-fold ambiguity, as a function of their degrees and genus. Then we use this matrix to recover the class of the couple of maps and as a consequence to recover the curve. For a generic situation, two projections define a curve with two irreducible components. One component has degree
d(d-1)
and the other has degree
d
, being the original curve. Then we consider another problem.
N
projections, with known projections operators and
N >> 1
, are considered as an input and we want to recover the curve. The recovery can be done by linear computations in the dual space and in the Grassmannian of lines in
\C \PP^3
. Those computations are respectively based on the dual variety and on the variety of intersecting lines. In both cases a simple lower bound for the number of necessary projections is given as a function of the degree and the genus. A closely related question is also considered. Each point of a finite closed subset of an irreducible algebraic curve, is projected onto a plane from a point. For each point the center of projection is different. The projections operators are known. We show when and how the recovery of the algebraic curve is possible, in function of the degree of the curve, and of the degree of the curve of minimal degree generated by the centers of projection. Eventually we show how these questions were motivated by applications to static and dynamic computational vision. A second part of this work is devoted to several applications to this field. The results in this paper solve a long standing problem in computer vision that could not have been solved without algebraic-geometric methods.
Jeremy Yirmeyahu Kaminski, Michael Fryers, Mina Teicher, Recovering an algebraic curve using its projections from different points. Applications to static and dynamic computational vision. J. Eur. Math. Soc. 7 (2005), no. 2, pp. 1–28
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The singly periodic genus-one helicoid | EMS Press
The singly periodic genus-one helicoid
We prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in our paper in Bulletin of the AMS, 29(1):77-84, 1993. Its ends in the quotient are asymptotic to one full turn of the helicoid, and, like the helicoid, it contains a vertical line. Modulo vertical translations, it has two parallel horizontal lines crossing the vertical axis. The nontrivial symmetries of the surface, modulo vertical translations, consist of:
180^\circ
-rotation about the vertical line;
180^\circ
rotation about the horizontal lines (the same symmetry); and their composition.
D. Hoffman, H. Karcher, F. Wei, The singly periodic genus-one helicoid. Comment. Math. Helv. 74 (1999), no. 2, pp. 248–279
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torch.optim¶
torch.optim is a package implementing various optimization algorithms. Most commonly used methods are already supported, and the interface is general enough, so that more sophisticated ones can be also easily integrated in the future.
How to use an optimizer¶
To use torch.optim you have to construct an optimizer object, that will hold the current state and will update the parameters based on the computed gradients.
Constructing it¶
To construct an Optimizer you have to give it an iterable containing the parameters (all should be Variable s) to optimize. Then, you can specify optimizer-specific options such as the learning rate, weight decay, etc.
If you need to move a model to GPU via .cuda(), please do so before constructing optimizers for it. Parameters of a model after .cuda() will be different objects with those before the call.
In general, you should make sure that optimized parameters live in consistent locations when optimizers are constructed and used.
optimizer = optim.Adam([var1, var2], lr=0.0001)
Per-parameter options¶
You can still pass options as keyword arguments. They will be used as defaults, in the groups that didn’t override them. This is useful when you only want to vary a single option, while keeping all others consistent between parameter groups.
For example, this is very useful when one wants to specify per-layer learning rates:
This means that model.base’s parameters will use the default learning rate of 1e-2, model.classifier’s parameters will use a learning rate of 1e-3, and a momentum of 0.9 will be used for all parameters.
Taking an optimization step¶
All optimizers implement a step() method, that updates the parameters. It can be used in two ways:
optimizer.step()¶
This is a simplified version supported by most optimizers. The function can be called once the gradients are computed using e.g. backward().
optimizer.step(closure)¶
Some optimization algorithms such as Conjugate Gradient and LBFGS need to reevaluate the function multiple times, so you have to pass in a closure that allows them to recompute your model. The closure should clear the gradients, compute the loss, and return it.
class torch.optim.Optimizer(params, defaults)[source]¶
Optimizer.load_state_dict
Optimizer.state_dict
Implements Adadelta algorithm.
Implements Adagrad algorithm.
SparseAdam
Implements lazy version of Adam algorithm suitable for sparse tensors.
Implements Adamax algorithm (a variant of Adam based on infinity norm).
Implements Averaged Stochastic Gradient Descent.
Implements L-BFGS algorithm, heavily inspired by minFunc.
Implements NAdam algorithm.
Implements RAdam algorithm.
Implements the resilient backpropagation algorithm.
Implements stochastic gradient descent (optionally with momentum).
How to adjust learning rate¶
torch.optim.lr_scheduler provides several methods to adjust the learning rate based on the number of epochs. torch.optim.lr_scheduler.ReduceLROnPlateau allows dynamic learning rate reducing based on some validation measurements.
Learning rate scheduling should be applied after optimizer’s update; e.g., you should write your code this way:
scheduler = ExponentialLR(optimizer, gamma=0.9)
Most learning rate schedulers can be called back-to-back (also referred to as chaining schedulers). The result is that each scheduler is applied one after the other on the learning rate obtained by the one preceding it.
scheduler2 = MultiStepLR(optimizer, milestones=[30,80], gamma=0.1)
In many places in the documentation, we will use the following template to refer to schedulers algorithms.
>>> scheduler = ...
Prior to PyTorch 1.1.0, the learning rate scheduler was expected to be called before the optimizer’s update; 1.1.0 changed this behavior in a BC-breaking way. If you use the learning rate scheduler (calling scheduler.step()) before the optimizer’s update (calling optimizer.step()), this will skip the first value of the learning rate schedule. If you are unable to reproduce results after upgrading to PyTorch 1.1.0, please check if you are calling scheduler.step() at the wrong time.
Sets the learning rate of each parameter group to the initial lr times a given function.
Multiply the learning rate of each parameter group by the factor given in the specified function.
Decays the learning rate of each parameter group by gamma every step_size epochs.
Decays the learning rate of each parameter group by gamma once the number of epoch reaches one of the milestones.
Decays the learning rate of each parameter group by a small constant factor until the number of epoch reaches a pre-defined milestone: total_iters.
Decays the learning rate of each parameter group by linearly changing small multiplicative factor until the number of epoch reaches a pre-defined milestone: total_iters.
lr_scheduler.CosineAnnealingLR
\eta_{max}
T_{cur}
Chains list of learning rate schedulers.
Receives the list of schedulers that is expected to be called sequentially during optimization process and milestone points that provides exact intervals to reflect which scheduler is supposed to be called at a given epoch.
Sets the learning rate of each parameter group according to cyclical learning rate policy (CLR).
Sets the learning rate of each parameter group according to the 1cycle learning rate policy.
\eta_{max}
is set to the initial lr,
T_{cur}
is the number of epochs since the last restart and
T_{i}
is the number of epochs between two warm restarts in SGDR:
Stochastic Weight Averaging¶
torch.optim.swa_utils implements Stochastic Weight Averaging (SWA). In particular, torch.optim.swa_utils.AveragedModel class implements SWA models, torch.optim.swa_utils.SWALR implements the SWA learning rate scheduler and torch.optim.swa_utils.update_bn() is a utility function used to update SWA batch normalization statistics at the end of training.
SWA has been proposed in Averaging Weights Leads to Wider Optima and Better Generalization.
Constructing averaged models¶
AveragedModel class serves to compute the weights of the SWA model. You can create an averaged model by running:
>>> swa_model = AveragedModel(model)
Here the model model can be an arbitrary torch.nn.Module object. swa_model will keep track of the running averages of the parameters of the model. To update these averages, you can use the update_parameters() function:
>>> swa_model.update_parameters(model)
SWA learning rate schedules¶
Typically, in SWA the learning rate is set to a high constant value. SWALR is a learning rate scheduler that anneals the learning rate to a fixed value, and then keeps it constant. For example, the following code creates a scheduler that linearly anneals the learning rate from its initial value to 0.05 in 5 epochs within each parameter group:
>>> swa_scheduler = torch.optim.swa_utils.SWALR(optimizer, \
>>> anneal_strategy="linear", anneal_epochs=5, swa_lr=0.05)
You can also use cosine annealing to a fixed value instead of linear annealing by setting anneal_strategy="cos".
Taking care of batch normalization¶
update_bn() is a utility function that allows to compute the batchnorm statistics for the SWA model on a given dataloader loader at the end of training:
>>> torch.optim.swa_utils.update_bn(loader, swa_model)
update_bn() applies the swa_model to every element in the dataloader and computes the activation statistics for each batch normalization layer in the model.
update_bn() assumes that each batch in the dataloader loader is either a tensors or a list of tensors where the first element is the tensor that the network swa_model should be applied to. If your dataloader has a different structure, you can update the batch normalization statistics of the swa_model by doing a forward pass with the swa_model on each element of the dataset.
Custom averaging strategies¶
By default, torch.optim.swa_utils.AveragedModel computes a running equal average of the parameters that you provide, but you can also use custom averaging functions with the avg_fn parameter. In the following example ema_model computes an exponential moving average.
>>> ema_avg = lambda averaged_model_parameter, model_parameter, num_averaged:\
>>> 0.1 * averaged_model_parameter + 0.9 * model_parameter
>>> ema_model = torch.optim.swa_utils.AveragedModel(model, avg_fn=ema_avg)
In the example below, swa_model is the SWA model that accumulates the averages of the weights. We train the model for a total of 300 epochs and we switch to the SWA learning rate schedule and start to collect SWA averages of the parameters at epoch 160:
>>> loader, optimizer, model, loss_fn = ...
>>> swa_model = torch.optim.swa_utils.AveragedModel(model)
>>> scheduler = torch.optim.lr_scheduler.CosineAnnealingLR(optimizer, T_max=300)
>>> swa_scheduler = SWALR(optimizer, swa_lr=0.05)
>>> for input, target in loader:
>>> optimizer.zero_grad()
>>> loss_fn(model(input), target).backward()
>>> optimizer.step()
>>> if epoch > swa_start:
>>> swa_model.update_parameters(model)
>>> swa_scheduler.step()
>>> scheduler.step()
>>> # Update bn statistics for the swa_model at the end
>>> # Use swa_model to make predictions on test data
How to use an optimizer
Per-parameter options
Taking an optimization step
How to adjust learning rate
Constructing averaged models
SWA learning rate schedules
Taking care of batch normalization
Custom averaging strategies
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Reciprocity (photography) — Wikipedia Republished // WIKI 2
In photography, reciprocity is the inverse relationship between the intensity and duration of light that determines the reaction of light-sensitive material. Within a normal exposure range for film stock, for example, the reciprocity law states that the film response will be determined by the total exposure, defined as intensity × time. Therefore, the same response (for example, the optical density of the developed film) can result from reducing duration and increasing light intensity, and vice versa.
The reciprocal relationship is assumed in most sensitometry, for example when measuring a Hurter and Driffield curve (optical density versus logarithm of total exposure) for a photographic emulsion. Total exposure of the film or sensor, the product of focal-plane illuminance times exposure time, is measured in lux seconds.
5 Tips to Shoot NIGHT Photos on FILM
Long Exposures on Portra 400 During Blue Hour
Mistakes I made when learning to shoot film
Shooting Expired Film in Brooklyn
Portra 400 at Night - Trying my hand at long exposures on film!
2 In chemical photography
3 Reciprocity failure
4 Schwarzschild law
4.1 Simple model for t > 1 second
5 The Kron–Halm catenary equation
6 Quantum reciprocity-failure model
7 Astrophotography
8 Holography
The idea of reciprocity, once known as Bunsen–Roscoe reciprocity, originated from the work of Robert Bunsen and Henry Roscoe in 1862.[1][2][3]
Deviations from the reciprocity law were reported by Captain William de Wiveleslie Abney in 1893,[4] and extensively studied by Karl Schwarzschild in 1899.[5][6][7] Schwarzschild's model was found wanting by Abney and by Englisch,[8] and better models have been proposed in subsequent decades of the early twentieth century. In 1913, Kron formulated an equation to describe the effect in terms of curves of constant density,[9][10] which J. Halm adopted and modified,[11] leading to the "Kron–Halm catenary equation"[12] or "Kron–Halm–Webb formula"[13] to describe departures from reciprocity.
In chemical photography
In photography, reciprocity refers to the relationship whereby the total light energy – proportional to the total exposure, the product of the light intensity and exposure time, controlled by aperture and shutter speed, respectively – determines the effect of the light on the film. That is, an increase of brightness by a certain factor is exactly compensated by a decrease of exposure time by the same factor, and vice versa. In other words, there is under normal circumstances a reciprocal proportion between aperture area and shutter speed for a given photographic result, with a wider aperture requiring a faster shutter speed for the same effect. For example, an EV of 10 may be achieved with an aperture (f-number) of f/2.8 and a shutter speed of 1/125 s. The same exposure is achieved by doubling the aperture area to f/2 and halving the exposure time to 1/250 s, or by halving the aperture area to f/4 and doubling the exposure time to 1/60 s; in each case the response of the film is expected to be the same.
For most photographic materials, reciprocity is valid with good accuracy over a range of values of exposure duration, but becomes increasingly inaccurate as this range is departed from: this is reciprocity failure (reciprocity law failure, or the Schwarzschild effect).[14] As the light level decreases out of the reciprocity range, the increase in duration, and hence of total exposure, required to produce an equivalent response becomes higher than the formula states; for instance, at half of the light required for a normal exposure, the duration must be more than doubled for the same result. Multipliers used to correct for this effect are called reciprocity factors (see model below).
At very low light levels, film is less responsive. Light can be considered to be a stream of discrete photons, and a light-sensitive emulsion is composed of discrete light-sensitive grains, usually silver halide crystals. Each grain must absorb a certain number of photons in order for the light-driven reaction to occur and the latent image to form. In particular, if the surface of the silver halide crystal has a cluster of approximately four or more reduced silver atoms, resulting from absorption of a sufficient number of photons (usually a few dozen photons are required), it is rendered developable. At low light levels, i.e. few photons per unit time, photons impinge upon each grain relatively infrequently; if the four photons required arrive over a long enough interval, the partial change due to the first one or two is not stable enough to survive before enough photons arrive to make a permanent latent image center.
This breakdown in the usual tradeoff between aperture and shutter speed is known as reciprocity failure. Each different film type has a different response at low light levels. Some films are very susceptible to reciprocity failure, and others much less so. Some films that are very light sensitive at normal illumination levels and normal exposure times lose much of their sensitivity at low light levels, becoming effectively "slow" films for long exposures. Conversely some films that are "slow" under normal exposure duration retain their light sensitivity better at low light levels.
For example, for a given film, if a light meter indicates a required EV of 5 and the photographer sets the aperture to f/11, then ordinarily a 4-second exposure would be required; a reciprocity correction factor of 1.5 would require the exposure to be extended to 6 seconds for the same result. Reciprocity failure generally becomes significant at exposures of longer than about 1 sec for film, and above 30 sec for paper.
Reciprocity also breaks down at extremely high levels of illumination with very short exposures. This is a concern for scientific and technical photography, but rarely to general photographers, as exposures significantly shorter than a millisecond are only required for subjects such as explosions and in particle physics, or when taking high-speed motion pictures with very high shutter speeds (1/10,000 sec or faster).
Schwarzschild law
In response to astronomical observations of low intensity reciprocity failure, Karl Schwarzschild wrote (circa 1900):
"In determinations of stellar brightness by the photographic method I have recently been able to confirm once more the existence of such deviations, and to follow them up in a quantitative way, and to express them in the following rule, which should replace the law of reciprocity: Sources of light of different intensity I cause the same degree of blackening under different exposures t if the products
{\displaystyle I\times t^{0.86}}
are equal."[5]
Unfortunately, Schwarzschild's empirically determined 0.86 coefficient turned out to be of limited usefulness.[15] A modern formulation of Schwarzschild's law is given as
{\displaystyle E=It^{p}\ }
where E is a measure of the "effect of the exposure" that leads to changes in the opacity of the photosensitive material (in the same degree that an equal value of exposure H = It does in the reciprocity region), I is illuminance, t is exposure duration and p is the Schwarzschild coefficient.[16][17]
However, a constant value for p remains elusive, and has not replaced the need for more realistic models or empirical sensitometric data in critical applications.[18] When reciprocity holds, Schwarzschild's law uses p = 1.0.
Since the Schwarzschild's law formula gives unreasonable values for times in the region where reciprocity holds, a modified formula has been found that fits better across a wider range of exposure times. The modification is in terms of a factor the multiplies the ISO film speed:[19]
Relative film speed
{\displaystyle =(t+1)^{(p-1)}\ }
where the t + 1 term implies a breakpoint near 1 second separating the region where reciprocity holds from the region where it fails.
Simple model for t > 1 second
Some models of microscope use automatic electronic models for reciprocity failure compensation, generally of a form for correct time, Tc, expressible as a power law of metered time, Tm, that is, Tc=(Tm)p, for times in seconds. Typical values of p are 1.25 to 1.45, but some are low as 1.1 and high as 1.8.[20]
The Kron–Halm catenary equation
Kron's equation as modified by Halm states that the response of the film is a function of
{\displaystyle It/\psi \ }
, with the factor defined by a catenary (hyperbolic cosine) equation accounting for reciprocity failure at both very high and very low intensities:
{\displaystyle \psi ={\frac {1}{2}}[(I/I_{0})^{a}+(I/I_{0})^{-a}]}
where I0 is the photographic material's optimum intensity level and a is a constant that characterizes the material's reciprocity failure.[21]
Quantum reciprocity-failure model
Modern models of reciprocity failure incorporate an exponential function, as opposed to power law, dependence on time or intensity at long exposure times or low intensities, based on the distribution of interquantic times (times between photon absorptions in a grain) and the temperature-dependent lifetimes of the intermediate states of the partially exposed grains.[22][23][24]
Baines and Bomback[25] explain the "low intensity inefficiency" this way:
Electrons are released at a very low rate. They are trapped and neutralised and must remain as isolated silver atoms for much longer than in normal latent image formation. It has already been observed that such extreme sub-latent image is unstable, and it is postulated that ineffiency is caused by many isolated atoms of silver losing their acquired electrons during the period of instability.
Reciprocity failure is an important effect in the field of film-based astrophotography. Deep-sky objects such as galaxies and nebulae are often so faint that they are not visible to the un-aided eye. To make matters worse, many objects' spectra do not line up with the film emulsion's sensitivity curves. Many of these targets are small and require long focal lengths, which can push the focal ratio far above f/5. Combined, these parameters make these targets extremely difficult to capture with film; exposures from 30 minutes to well over an hour are typical. As a typical example, capturing an image of the Andromeda Galaxy at f/4 will take about 30 minutes; to get the same density at f/8 would require an exposure of about 200 minutes.
When a telescope is tracking an object, every minute is difficult; therefore, reciprocity failure is one of the biggest motivations for astronomers to switch to digital imaging. Electronic image sensors have their own limitation at long exposure time and low illuminance levels, not usually referred to as reciprocity failure, namely noise from dark current, but this effect can be controlled by cooling the sensor.
A similar problem exists in holography. The total energy required when exposing holographic film using a continuous wave laser (i.e. for several seconds) is significantly less than the total energy required when exposing holographic film using a pulsed laser (i.e. around 20–40 nanoseconds) due to a reciprocity failure. It can also be caused by very long or very short exposures with a continuous wave laser. To try to offset the reduced brightness of the film due to reciprocity failure, a method called latensification can be used. This is usually done directly after the holographic exposure and using an incoherent light source (such as a 25–40 W light bulb). Exposing the holographic film to the light for a few seconds can increase the brightness of the hologram by an order of magnitude.
^ Holger Pettersson; Gustav Konrad von Schulthess; David J. Allison & Hans-Jørgen Smith (1998). The Encyclopaedia of Medical Imaging. Taylor & Francis. p. 59. ISBN 978-1-901865-13-4.
^ Geoffrey G. Atteridge (2000). "Sensitometry". In Ralph E. Jacobson; Sidney F. Ray; Geoffrey G. Atteridge; Norman R. Axford (eds.). The Manual of Photography: Photographic and Digital Imaging (9th ed.). Oxford: Focal Press. p. 238. ISBN 978-0-240-51574-8.
^ R.W. Bunsen; H.E. Roscoe (1862). "Photochemical Researches – Part V. On the Measurement of the Chemical Action of Direct and Diffuse Sunlight" (PDF). Proceedings of the Royal Society. 12: 306–312. Bibcode:1862RSPS...12..306B. doi:10.1098/rspl.1862.0069.
^ W. de W. Abney (1893). "On a failure of the law in photography that when the products of the intensity of the light acting and of the time of exposure are equal, equal amounts of chemical action will be produced". Proceedings of the Royal Society. 54 (326–330): 143–147. Bibcode:1893RSPS...54..143A. doi:10.1098/rspl.1893.0060.
^ a b K. Schwarzschild "On The Deviations From The Law of Reciprocity For Bromide Of Silver Gelatine" The Astrophysical Journal vol.11 (1900) p.89 [1]
^ S. E. Sheppard & C. E. Kenneth Mees (1907). Investigations on the Theory of the Photographic Process. Longmans, Green and Co. p. 214. ISBN 978-0-240-50694-4.
^ Ralph W. Lambrecht & Chris Woodhouse (2003). Way Beyond Monochrome. Newpro UK Ltd. p. 113. ISBN 978-0-86343-354-2.
^ Samuel Edward Sheppard & Charles Edward Kenneth Mees (1907). Investigations on the theory of the photographic process. Longmans, Green and Co. pp. 214–215.
^ Erich Kron (1913). "Über das Schwärzungsgesetz Photographischer Platten". Publikationen des Astrophysikalischen Observatoriums zu Potsdam. 22 (67). Bibcode:1913POPot..67.....K.
^ Loyd A. Jones (July 1927). "Photographic Spectrophotometry in the Ultra-Violet Region". Bulletin of the National Research Council: 109–123.
^ J. Halm (Jan 1915). "On the Determination of Fundamental Photographic Magnitudes". Monthly Notices of the Royal Astronomical Society. 75 (3): 150–177. Bibcode:1915MNRAS..75..150H. doi:10.1093/mnras/75.3.150.
^ J. H. Webb (1935). "The Effect of Temperature upon Reciprocity Law Failure in Photographic Exposure". Journal of the Optical Society of America. 25 (1): 4–20. doi:10.1364/JOSA.25.000004.
^ Ernst Katz (1941). Contribution to the Understanding of Latent Image Formation in Photography. Drukkerij F. Schotanus & Jens. p. 11.
^ Rudolph Seck & Dennis H. Laney (1983). Leica Darkroom Practice. MBI Publishing Company. p. 183. ISBN 978-0-906447-24-6.
^ Jonathan W. Martin; Joannie W. Chin; Tinh Nguyen (2003). "Reciprocity law experiments in polymeric photodegradation: a critical review" (PDF). Progress in Organic Coatings. 47 (3–4): 294. CiteSeerX 10.1.1.332.6705. doi:10.1016/j.porgcoat.2003.08.002.
^ Walter Clark (2007). Photography by Infrared – Its Principles and Applications. Read Books. p. 62. ISBN 978-1-4067-4486-6.
^ Graham Saxby (2002). The Science of Imaging. CRC Press. p. 141. ISBN 978-0-7503-0734-5.
^ J.W. Martin et al. "Reciprocity law experiments in polymeric photodegradation: a critical review", Progress in Organic Coatings 47 (2003) pp.306 [2]
^ Michael A. Covington (1999). Astrophotography for the amateur. Cambridge University Press. p. 181. ISBN 978-0-521-62740-5.
^ Fred Rost & Ron Oldfield (2000). Photography with a Microscope. Cambridge University Press. p. 204. ISBN 978-0-521-77096-5.
^ W. M. H. Greaves (1936). "Time Effects in Spectrophotometry". Monthly Notices of the Royal Astronomical Society. 96 (9): 825–832. Bibcode:1936MNRAS..96..825G. doi:10.1093/mnras/96.9.825.
^ W. J. Anderson (1987). "Probabilistic Models of the Photographic Process". In Ian B. MacNeill (ed.). Applied Probability, Stochastic Processes, and Sampling Theory: Advances in the Statistical Sciences. Springer. pp. 9–40. ISBN 978-90-277-2393-2.
^ Collins, Ronald Bernard (1956–1957). "(Page 65 of)". The Journal of Photographic Science. 4–5: 65.
^ J. H. Webb (1950). "Low Intensity Reciprocity-Law Failure in Photographic Exposure: Energy Depth of Electron Traps in Latent-Image Formation; Number of Quanta Required to Form the Stable Sublatent Image". Journal of the Optical Society of America. 40 (1): 3–13. doi:10.1364/JOSA.40.000003.
^ Harry Baines & Edward S. Bomback (1967). The Science of Photography (2nd ed.). Fountain Press. p. 202.
Reciprocity what? – a brief explanation in layman's terms.
Reciprocity charts for slides and black & white
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The theory of minimal surfaces in $M \times \mathbb{R}$ | EMS Press
The theory of minimal surfaces in
M \times \mathbb{R}
In this paper, we develop the theory of properly embedded minimal surfaces in
M \times \mathbb{R}
M
is a closed orientable Riemannian surface. We construct many examples of different topology and geometry. We establish several global results. The first of these theorems states that examples of bounded curvature have linear area growth, and so, are quasiperiodic. We then apply this theorem to study and classify the stable examples. We prove the topological result that every example has a finite number of ends. We apply the recent theory of Colding and Minicozzi to prove that examples of finite topology have bounded curvature. Also we prove the topological unicity of the embedding of some of these surfaces.
Harold Rosenberg, William H. Meeks, The theory of minimal surfaces in
M \times \mathbb{R}
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Singular_point_of_a_curve Knowpia
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.
Algebraic curves in the planeEdit
Algebraic curves in the plane may be defined as the set of points (x, y) satisfying an equation of the form f (x, y) = 0, where f is a polynomial function f : R2 → R. If f is expanded as
{\displaystyle f=a_{0}+b_{0}x+b_{1}y+c_{0}x^{2}+2c_{1}xy+c_{2}y^{2}+\cdots }
If the origin (0, 0) is on the curve then a0 = 0. If b1 ≠ 0 then the implicit function theorem guarantees there is a smooth function h so that the curve has the form y = h(x) near the origin. Similarly, if b0 ≠ 0 then there is a smooth function k so that the curve has the form x = k(y) near the origin. In either case, there is a smooth map from R to the plane which defines the curve in the neighborhood of the origin. Note that at the origin
{\displaystyle b_{0}={\frac {\partial f}{\partial x}},\;b_{1}={\frac {\partial f}{\partial y}},}
so the curve is non-singular or regular at the origin if at least one of the partial derivatives of f is non-zero. The singular points are those points on the curve where both partial derivatives vanish,
{\displaystyle f(x,y)={\frac {\partial f}{\partial x}}={\frac {\partial f}{\partial y}}=0.}
Regular pointsEdit
Assume the curve passes through the origin and write y = mx. Then f can be written
{\displaystyle f=\left(b_{0}+mb_{1}\right)x+\left(c_{0}+2mc_{1}+c_{2}m^{2}\right)x^{2}+\cdots .}
If b0 + mb1 is not 0 then f = 0 has a solution of multiplicity 1 at x = 0 and the origin is a point of single contact with line y = mx. If b0 + mb1 = 0 then f = 0 has a solution of multiplicity 2 or higher and the line y = mx, or b0x + b1y = 0, is tangent to the curve. In this case, if c0 + 2mc1 + c2m2 is not 0 then the curve has a point of double contact with y = mx. If the coefficient of x2, c0 + 2mc1 + c2m2, is 0 but the coefficient of x3 is not then the origin is a point of inflection of the curve. If the coefficients of x2 and x3 are both 0 then the origin is called point of undulation of the curve. This analysis can be applied to any point on the curve by translating the coordinate axes so that the origin is at the given point.[1]
Double pointsEdit
Three limaçons illustrating the types of double point. When converted to Cartesian coordinates as
{\displaystyle \left(x^{2}+y^{2}-x\right)^{2}=\left(1.5\right)^{2}\left(x^{2}+y^{2}\right),}
the left curve acquires an acnode at the origin, which is an isolated point in the plane. The central curve, the cardioid, has a cusp at the origin. The right curve has a crunode at the origin and the curve crosses itself to form a loop.
If b0 and b1 are both 0 in the above expansion, but at least one of c0, c1, c2 is not 0 then the origin is called a double point of the curve. Again putting y = mx, f can be written
{\displaystyle f=\left(c_{0}+2mc_{1}+c_{2}m^{2}\right)x^{2}+\left(d_{0}+3md_{1}+3m^{2}d_{2}+d_{3}m^{3}\right)x^{3}+\cdots .}
Double points can be classified according to the solutions of c0 + 2mc1 + m2c2 = 0.
CrunodesEdit
If c0 + 2mc1 + m2c2 = 0 has two real solutions for m, that is if c0c2 − c12 < 0, then the origin is called a crunode. The curve in this case crosses itself at the origin and has two distinct tangents corresponding to the two solutions of c0 + 2mc1 + m2c2 = 0. The function f has a saddle point at the origin in this case.
AcnodesEdit
If c0 + 2mc1 + m2c2 = 0 has no real solutions for m, that is if c0c2 − c12 > 0, then the origin is called an acnode. In the real plane the origin is an isolated point on the curve; however when considered as a complex curve the origin is not isolated and has two imaginary tangents corresponding to the two complex solutions of c0 + 2mc1 + m2c2 = 0. The function f has a local extremum at the origin in this case.
CuspsEdit
If c0 + 2mc1 + m2c2 = 0 has a single solution of multiplicity 2 for m, that is if c0c2 − c12 = 0, then the origin is called a cusp. The curve in this case changes direction at the origin creating a sharp point. The curve has a single tangent at the origin which may be considered as two coincident tangents.
Further classificationEdit
The term node is used to indicate either a crunode or an acnode, in other words a double point which is not a cusp. The number of nodes and the number of cusps on a curve are two of the invariants used in the Plücker formulas.
If one of the solutions of c0 + 2mc1 + m2c2 = 0 is also a solution of d0 + 3md1 + 3m2d2 + m3d3 = 0 then the corresponding branch of the curve has a point of inflection at the origin. In this case the origin is called a flecnode. If both tangents have this property, so c0 + 2mc1 + m2c2 is a factor of d0 + 3md1 + 3m2d2 + m3d3, then the origin is called a biflecnode.[2]
Multiple pointsEdit
A curve with a triple point at the origin:
{\displaystyle x(t)=\sin 2t+\cos t,\quad }
{\displaystyle y(t)=\sin t+\cos 2t}
In general, if all the terms of degree less than k are 0, and at least one term of degree k is not 0 in f, then curve is said to have a multiple point of order k or a k-ple point. The curve will have, in general, k tangents at the origin though some of these tangents may be imaginary.[3]
Parametric curvesEdit
A parameterized curve in R2 is defined as the image of a function g : R → R2, g(t) = (g1(t),g2(t)). The singular points are those points where
{\displaystyle {\frac {dg_{1}}{dt}}={\frac {dg_{2}}{dt}}=0.}
A cusp in the semicubical parabola
{\displaystyle y^{2}=x^{3}}
Many curves can be defined in either fashion, but the two definitions may not agree. For example, the cusp can be defined on an algebraic curve, x3 − y2 = 0, or on a parametrised curve, g(t) = (t2, t3). Both definitions give a singular point at the origin. However, a node such as that of y2 − x3 − x2 = 0 at the origin is a singularity of the curve considered as an algebraic curve, but if we parameterize it as g(t) = (t2 − 1, t(t2 − 1)), then g′(t) never vanishes, and hence the node is not a singularity of the parameterized curve as defined above.
Care needs to be taken when choosing a parameterization. For instance the straight line y = 0 can be parameterised by g(t) = (t3, 0) which has a singularity at the origin. When parametrised by g(t) = (t, 0) it is nonsingular. Hence, it is technically more correct to discuss singular points of a smooth mapping rather than a singular point of a curve.
The above definitions can be extended to cover implicit curves which are defined as the zero set f −1(0) of a smooth function, and it is not necessary just to consider algebraic varieties. The definitions can be extended to cover curves in higher dimensions.
A theorem of Hassler Whitney[4][5] states
Theorem — Any closed set in Rn occurs as the solution set of f−1(0) for some smooth function f : Rn → R.
Any parameterized curve can also be defined as an implicit curve, and the classification of singular points of curves can be studied as a classification of singular point of an algebraic variety.
Types of singular pointsEdit
Some of the possible singularities are:
An isolated point: x2 + y2 = 0, an acnode
Two lines crossing: x2 − y2 = 0, a crunode
A cusp: x3 − y2 = 0, also called a spinode
A tacnode: x4−y2 = 0
A rhamphoid cusp: x5 − y2 = 0.
^ Th. Bröcker, Differentiable Germs and Catastrophes, London Mathematical Society. Lecture Notes 17. Cambridge, (1975)
Hilton, Harold (1920). "Chapter II: Singular Points". Plane Algebraic Curves. Oxford.
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Linear Inequalities, Popular Questions: CBSE Class 11-humanities MATH, Math - Meritnation
Aarya Talgaonkar asked a question
Questiom 3 please
Q3. Figure shows a square grid of order 3, which of the following is correct formula for total number of squares in a similar grid of order n.
\frac{\mathrm{n}\left(\mathrm{n}+ 1\right)}{2}
\frac{\mathrm{n}\left(\mathrm{n}+ 1\right)\left(2\mathrm{n} +1\right)}{6}
\frac{{\mathrm{n}}^{2}{\left(\mathrm{n}+ 1\right)}^{2}}{4}
d)
\frac{\mathrm{n}\left(\mathrm{n} + 1\right)\left(\mathrm{n} +2\right)}{6}
Arathi took 3 examinations in a year. The
marks obtained by her in the second and third
examinations are more than 5 and 10 respectively
than in the first examination. If her average mark
is at least 80 find the minimum mark that she
should get in the first examination?
an object dropped from cliff falls with a constant acceleration of 10m/s2 find its speed after 2s after it was dropprd
Draw the graph of y=-5 y=5 on the same graph .Are the lines parallel Find the point of integration of two lines Draw the graph of y=-5 y=5 on the same graph .Are the lines parallel Find the point of integration of two lines
wwerock asked a question
show graphically that the solution set of the following system of inequalities is empty
x-2y greater than = 0, 2x-y less than = -2 ,x greater than = 0, y greater than = 0
if 22Pr+1:20pr+2 = 11:52 find r
The number of real values of λ for which the system of linear equations 2x+4y−λz=0 4x+λy+2z=0 λx+2y+2z=0 has infinitely many solutions, is :
a scooter covers 20km/h at a uniform speed of 30km/h what should be its speed for the next 40 km if the average speed is 60km/h
Dear experts, please solve the following inequality
➩ mod(2 ÷ x - 4) is greater than 1, x is not equal to 4.
if log0.3(x-1) < log0.09(x-1), then x lies in the interval....
If x∈w, find the solution set of (3x/5 )- (2x-1/3) >1
Perumalla Sai Utthej asked a question
Draw the graph of the following system of inequalities and mark the solution.
2x + y – 3 ≥ 0
x - 2y ≤ 0
if n-1cr:ncr:n+1cr=6:9:13 find n and r
an object moves along a straight line with an acceleration of 2m/s2 if its initial speed is 10m/s what will be its speed 5s later
Solve the inequality :-
x2 + 1 is greater than or equal to 0.
15x less than 73, where (i)x belongs to N and (ii) Z.
Rupam Saxena & 1 other asked a question
how to solve modulus (2x - 1/x - 1)>2?
Lomas Rishi asked a question
Where is the origin test ??
plezz tell me ..
I think it is the value is (0,0) ..
plezz help me .......
A plumber can be paid under 2 schemes as given below:-
I : $600 and $ 50 per hour.
II: $ 170 per hour .
If the job takes n hours, what values of n does Scheme I give the plumber better wages?
Meera Nair asked a question
solve the inequality and represent it in number line 5-4x/3x2-x-4<4
While drilling a hole in the earth, it was found that the temperature (TC) at x km below the surface of the earth was given by
T = 30 + 25(x - 3), when 3 ≤ x ≤ 15.
Between which depths will the temperature be between 200C and 300C?
PLEASE SHOW ME THE FOLLOWING GRAPHS WITH THE SHADED PORTION :
1) x=0 [ x is greater than or equal to zero ]
2) y=0 [ y is greater than or equal to zero ]
3) x=0 , y=0 [x is greater than or equal to zero , y is greater than or equal to zero]
Solve this please:-
|x-1|+|x-2|+|x-3|>=6
how to factorize:
if the products of the roots of the equation x2-3kx+2e2logk-1=0 is 7, then the roots are real for k=..........
Rakshana Bala asked a question
➩ mod (x) - 1 ÷ mod (x) - 2 is greater than or equal to 0.
{Please explain the whole sum in detail}
Sir, these types of sums are not in our ncert booklet. But I found the same in RD sharma. So, tell me that whether it is right to concentrate in this type of sums or not.
The longest side of a triangle is 3 times the shortest side and the third side
is 2 cm shorter than the longest side. If the perimeter of the triangle is
atleast 61 cm find the minimum length of the shortest side.
Ques no 8 part 2
Pratibha Choudhary asked a question
A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5 cm longer than the second?
[Hint: If x is the length of the shortest board, then x, (x + 3) and 2x are the lengths of the second and third piece, respectively. Thus, x = (x + 3) + 2x ≤ 91 and 2x ≥ (x + 3) + 5]
The graph of the inequalities x=0,y=0,2x+y+6=0 is
1. First quadrant
2.That half of xy plane which lies above the line 2x+y+6=0
If logax,logbx logcx are in A.P. where x not equal to 1, show that c2= (ac)logab
Are the roots real or are they real and distinct?
Ht Ghosh asked a question
Ques no 10 ppl
in the first four paper each of 100 marks, rishi got 95,72,73,83 marks. if he wants an average of greater than or equal to75 marks and less than 80 marks , find the range of marks he should score in the fifth paper.
Please explain step by step , option 4
Phoenix asked a question
The cost and revenue functions of a product are given by C(x) = 2x + 400 and R(x) = 6x + 20 respectively. Where x is the number of items produced by the manufacturer. How many items the manufacturer must sell to realize some profit.
James Bond asked a question
Trupts Rock asked a question
Arundhathy Pradeep asked a question
Solve the inequality (x-2)(x-3) >0
Attitude Girl asked a question
IQ of a person is given by the formula
IQ = (MA/CA)*100,
where MA is the mental age & CA is the chronological age. If 80≤IQ≤140 for a group of 12 yr children, find the range of their mental age.
in how many ways can the letters of the word FRACTION be arranged so that no two vowels are together.
| 4x-5 | <= 1/3
Adeeb Tahir asked a question
Rashmi Varma asked a question
exactly do we use open and closed brackets...??
Matsya asked a question
solution set of the inequality y<0 is :
a) half of XOY plane which lies below x-axis
b) half of XOY-plane which lies above Xaxis
c) half of XOY - plane whichlies below x axis including the points on the x axis
d) none of theese.
correct answer is a)
but i am confused that why we can't take c) ??????????
Harman Shah Singh asked a question
Solve the system of inequalities :-
➩ (1)2x + 5 is less than or equal to 0
(2) x - 3 is less than or equal to zero.
Sir, Please explain the last step to me.
The distance between two towns is 458km.Two trains start from these two stations at the same time and travel towards each other at 45 km per hour and 60km per hour .In how many hours will theybe 38 km apart from each other
Dear experts, I have posted 3 to 4 question of maths. But no any expert is likely to answer them. This delay causes delay in my studies. Please answer that questions so that I could be provided with meaningful help.
Experts, tell me your time of answering question.
Pratik Bharadia asked a question
A point moves such that the sum of its distances from the point (ae,0) and (-ae,0) is 2a. Show that the locus of this point is x2/a2 + y2/a2(1-e2) = 1
A runner covers a distance of 1 mile in 20 minutes. If he runs at the same speed the next day, how many miles will he cover in 3/2 hrs
what is an unbounded set ?????? show that the solution set of the following system of linear inequalities is an unbounded region ; 2x + y is greater than or equal to 8, x + 2y is greater then or equal to 10, x, y is greater then or equal to 0
Sudeshna Das asked a question
2x/ 2x2+5x+2 > 1/x+1
(|x+1| - x)/x < 1
A company manufactures cassettes and its cost equation for a week is ( C = 300 + 1.5 x ), while its revenue equation is (R= 2x ) , where x is the number of cassettes sold. How many cassettes must be sold by the company to get some profit?
A factory contains a series of water tanks, all of the same size .If pump 1 can fill 12 of these tanks in a 12-hour shift and pump 2 can fill 11 tanks in the same time ,then how many tanks can the two pumps fill,working together in 1hour?
Suraj Gupta asked a question
How do we define domain for this log equation? Also, Please explain what conditions are to be taken when base is a variable in log inequalities?
125. logx–1 3 = 2.
Amiable asked a question
Solve inequality : 3x2 - 7x+6 <0 , no links
Tanush & 1 other asked a question
IF 1/8! + 1/9! = x/10! . Then find the value of x .
Please solve this que
Atharv Yeolekar & 1 other asked a question
Solve the equation for x and y: (x - iy) (2 + 3i) = x+2i / 1-i
QUESTION: Solve mod(x-1)+mod(x-2)>equal to 4. In the solution in the image, why have we taken equal to sign also along with < sign in the expression 1<equal to x<2? Is it necessary to take equal to sign also with less than sign(1<equal to x<2) whenever we solve such mod questions? OR IS IT becuase we have equal to sign also in the question . PLS explain this.......
Solve 7x+1<=4x+5 and represent the solution graphically on the number line.
Please answer this fasttt
Solve the following inequations in R.
1. 1/x-1≤2
2. 5x+8/4-x<2
3. x/x-5>1/2
4. 0<-x/2<3
5. |3x-4/2|≤5/12
6. |x-2|/x-2>0
7. |x+2|-x/x<2
8. |2x-1/x-1|>2
9. |x-1|+|x-2|+|x-3|≥6
10. The marks scored by rohit in two tests were 65 and 70. Find the minimum marks he should score in third test to have an average of at least 65 marks.
11. The longest side of a triangle is three times the shortest side and the third side is 2 cm shorter than the longest side if the perimeter of the triangle at least 61cm, find the minimum length of the shortest-side.
12. How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?
13. A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If there are 640 litres of the 8% solution, how many litres of 2% solution will have to be added?
14. The water acidity in a pool is considered normal when he average pH reading of three daily measurements is between 7.2 and 7.8. If the first two pH reading are 7.48 and 7.85, find the range of pH value for the third reading that will result in the acidity level being normal.
15. Show that the solution set of the following linear inequations is empty set:
15.1) x-2y ≥0, 2x-y ≤-2,x ≥0, y ≥0
15.2) x+2y≤3, 3x+4y ≥12, y ≥1, x ≥0, y ≥0
Answer this question right now experts faast
Bipro asked a question
Mohammad Rushan asked a question
Question 29...plss i am a paid member.
Ans meee exp
In the first 4 papers each of hundred marks Ravi got 90, 75, 73, 85 marks.If he wants an average of greater than or equal to 75 marks and less than80 marks, find the range of marks she should score in the fifth paper.
how many different words with or without meaning can be formed using all the letters of the word COMBINE so that
vowels always remain together
no two vowels are together
vowels may occupy odd places
Nishit Shah & 1 other asked a question
Mahima Tyagi asked a question
how do we come to know about sign change bcz in some ques for eg the answer is coming x > - 1 nd the sign is not changing
bt in some sign is chaningingfor eg
= -x > 6
so how do we know about sign change
please do reply....i am having my final exams...so reply fast !!
How can we understand which region we have to shade? answer meas soon as possible.
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On the $\Gamma$-cohomology of rings of numerical polynomials | EMS Press
\Gamma
-cohomology of rings of numerical polynomials
We investigate
\Gamma
-cohomology of some commutative cooperation algebras
E_*E
associated with certain periodic cohomology theories. For KU and
E(1)
, the Adams summand at a prime
p
, and for KO we show that
\Gamma
-cohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique
E_\infty
structures. As a consequence we obtain an
E_\infty
structure for the connective Adams summand. For the Johnson--Wilson spectrum
E(n)
n\geq1
we establish the existence of a unique
E_\infty
structure for its
I_n
Andrew Baker, Birgit Richter, On the
\Gamma
-cohomology of rings of numerical polynomials. Comment. Math. Helv. 80 (2005), no. 4, pp. 691–723
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Amino acids are organic compounds that contain amino[a] (−NH+3) and carboxylate (−CO−2) functional groups, along with a side chain (R group) specific to each amino acid.[1] The elements present in every amino acid are carbon (C), hydrogen (H), oxygen (O), and nitrogen (N) (CHON); in addition sulfur (S) is present in the side chains of cysteine and methionine, and selenium (Se) in the less common amino acid selenocysteine. More than 500 naturally occurring amino acids are known to constitute monomer units of peptides, including proteins, as of 2020[update][2] although only 22 appear in the genetic code, 20 of which have their own designated codons and 2 of which have special coding mechanisms: Selenocysteine which is present in all eukaryotes and pyrrolysine which is present in some prokaryotes.[3][4]
The first few amino acids were discovered in the early 1800s.[7][8] In 1806, French chemists Louis-Nicolas Vauquelin and Pierre Jean Robiquet isolated a compound from asparagus that was subsequently named asparagine, the first amino acid to be discovered.[9][10] Cystine was discovered in 1810,[11] although its monomer, cysteine, remained undiscovered until 1884.[12][10][b] Glycine and leucine were discovered in 1820.[13] The last of the 20 common amino acids to be discovered was threonine in 1935 by William Cumming Rose, who also determined the essential amino acids and established the minimum daily requirements of all amino acids for optimal growth.[14][15]
{\displaystyle \mathrm {p} K_{\mathrm {a} }}
{\displaystyle \mathrm {p} K_{\mathrm {a} }}
{\displaystyle \mathrm {p} K_{\mathrm {a} }}
{\displaystyle \mathrm {p} K_{\mathrm {a} }}
{\displaystyle \mathrm {p} K_{\mathrm {a} }}
{\displaystyle \mathrm {p} K_{\mathrm {a} }=8.3}
{\displaystyle \mathrm {p} K_{\mathrm {a} }}
{\displaystyle \mathrm {p} K_{\mathrm {a} }}
{\displaystyle \mathrm {p} K_{\mathrm {a} }=9.6}
{\displaystyle \mathrm {p} K_{\mathrm {a} }}
{\displaystyle \mathrm {p} K_{\mathrm {a} }=4.1}
{\displaystyle \mathrm {p} K_{\mathrm {a} }=4.5}
{\displaystyle \mathrm {p} K_{\mathrm {a} }=6.3}
{\displaystyle \mathrm {p} K_{\mathrm {a} }=10.4}
{\displaystyle \mathrm {p} K_{\mathrm {a} }>12}
Some nonstandard amino acids are used as defenses against herbivores in plants.[99] For example, canavanine is an analogue of arginine that is found in many legumes,[100] and in particularly large amounts in Canavalia gladiata (sword bean).[101] This amino acid protects the plants from predators such as insects and can cause illness in people if some types of legumes are eaten without processing.[102] The non-protein amino acid mimosine is found in other species of legume, in particular Leucaena leucocephala.[103] This compound is an analogue of tyrosine and can poison animals that graze on these plants.
^ Richard Cammack, ed. (2009). "Newsletter 2009". Biochemical Nomenclature Committee of IUPAC and NC-IUBMB. Pyrrolysine. Archived from the original on 12 September 2017. Retrieved 16 April 2012.
^ Rother, Michael; Krzycki, Joseph A. (1 January 2010). "Selenocysteine, Pyrrolysine, and the Unique Energy Metabolism of Methanogenic Archaea". Archaea. 2010: 1–14. doi:10.1155/2010/453642. ISSN 1472-3646. PMC 2933860. PMID 20847933.
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tensor(deprecated)/conj - Maple Help
Home : Support : Online Help : tensor(deprecated)/conj
complex conjugation of expressions involving complex unknowns
conj(expression, [ [a1, a1_bar], [a2, a2_bar], ... ])
algebraic expression to conjugate
[[a1, a1_bar], [a2, a2_bar], ...]
(optional) list of pairs of conjugates (names of unknowns and their conjugates)
The function conj(expr, [[a1,a1_bar], [a2,a2_bar], ... ]) computes the complex conjugate of the algebraic expression expr by making the following substitutions:
-I is substituted for I (this is the default if only one argument is specified).
For each pair of names,
[\mathrm{ai},\mathrm{ai_bar}]
i=1..n
, ai is substituted for ai_bar and ai_bar is substituted for ai.
The effect of these substitutions is to produce the complex conjugate of an expression which is assumed to contain only real-valued unknowns except for those which are listed in the second argument. The unknowns listed in the second argument are complex-valued and are replaced by their complex conjugate (unknown).
\mathrm{with}\left(\mathrm{tensor}\right):
Suppose that the unknowns a and b are real-valued. Compute the conjugate of a+I*b:
\mathrm{conj}\left(a+Ib\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{I}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}
Notice that since all of the unknowns in the expression `a+I*b' are real, you did not need to specify a second argument in the call to conj (alternatively, you could have passed the empty list: []).
Now suppose that b is complex-valued with complex conjugate b_bar. The conjugate of a+I*b is a-I*b_bar:
\mathrm{conj}\left(a+Ib,[[b,\mathrm{b_bar}]]\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{I}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{b_bar}}
Now suppose that both a and b are complex-valued. Compute the complex conjugate of a+I*b:
\mathrm{conj}\left(a+Ib,[[a,\mathrm{a_bar}],[b,\mathrm{b_bar}]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{a_bar}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{I}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{b_bar}}
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Pantetheine kinase - Wikipedia
In enzymology, a pantetheine kinase (EC 2.7.1.34) is an enzyme that catalyzes the chemical reaction
ATP + pantetheine
{\displaystyle \rightleftharpoons }
ADP + pantetheine 4'-phosphate
Thus, the two substrates of this enzyme are ATP and pantetheine, whereas its two products are ADP and pantetheine 4'-phosphate.
This enzyme belongs to the family of transferases, specifically those transferring phosphorus-containing groups (phosphotransferases) with an alcohol group as acceptor. The systematic name of this enzyme class is ATP:pantetheine 4'-phosphotransferase. This enzyme is also called pantetheine kinase (phosphorylating). This enzyme participates in pantothenate and coa biosynthesis.
Novelli GD (September 1953). "Enzymatic synthesis and structure of CoA". Federation Proceedings. 12 (3): 675–81. PMID 13107738.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Pantetheine_kinase&oldid=1018016124"
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What Is Polynomial Trending?
Polynomial trending describes a pattern in data that is curved or breaks from a straight linear trend. It often occurs in a large set of data that contains many fluctuations. As more data becomes available, the trends often become less linear, and a polynomial trend takes its place. Graphs with curved trend lines are generally used to show a polynomial trend.
Data that is polynomial in nature is described generally by:
\begin{aligned} &y = a + x^n \\ &\textbf{where:}\\ &a = \text{the intercept}\\ &x = \text{the explanatory variable}\\ &n = \text{the nature of the polynomial (e.g. squared, cubed, etc.)}\\ \end{aligned}
y=a+xnwhere:a=the interceptx=the explanatory variablen=the nature of the polynomial (e.g. squared, cubed, etc.)
Understanding Polynomial Trending
Big data and statistical analytics are becoming more commonplace and easy to use; many statistical packages now regularly include polynomial trend lines as part of their analysis. When graphing variables, analysts these days generally use one of six common trend lines or regressions to describe their data. These graphs include:
Each of these parameters has different benefits based on the properties of the underlying data. In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences. They are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science.
They are also used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry.
Real-World Example of Polynomial Trending Data
For example, polynomial trending would be apparent on the graph that shows the relationship between the profit of a new product and the number of years the product has been available. The trend would likely rise near the beginning of the graph, peak in the middle and then trend downward near the end. If the company revamps the product late in its life cycle, we'd expect to see this trend repeat itself.
This type of chart, which would have several waves on the graph, would be deemed to be a polynomial trend. An example of such polynomial trending can be seen in the example chart below:
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The conjugacy problem for Dehn twist automorphisms of free groups | EMS Press
The conjugacy problem for Dehn twist automorphisms of free groups
A Dehn twist automorphism of a group G is an automorphism which can be given (as specified below) in terms of a graph-of-groups decomposition of G with infinite cyclic edge groups. The classic example is that of an automorphism of the fundamental group of a surface which is induced by a Dehn twist homeomorphism of the surface. For
G = F_n
, a non-abelian free group of finite rank n, a normal form for Dehn twist is developed, and it is shown that this can be used to solve the conjugacy problem for Dehn twist automorphisms of
F_n
M. Lustig, M. M. Cohen, The conjugacy problem for Dehn twist automorphisms of free groups. Comment. Math. Helv. 74 (1999), no. 2, pp. 179–200
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Nm Shah 2018 for Class 11 Commerce Economics Chapter 1 - Organisation Of Data
Nm Shah 2018 Solutions for Class 11 Commerce Economics Chapter 1 Organisation Of Data are provided here with simple step-by-step explanations. These solutions for Organisation Of Data are extremely popular among Class 11 Commerce students for Economics Organisation Of Data Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Nm Shah 2018 Book of Class 11 Commerce Economics Chapter 1 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Nm Shah 2018 Solutions. All Nm Shah 2018 Solutions for class Class 11 Commerce Economics are prepared by experts and are 100% accurate.
Prepare a statistical table from the following data taking the class width as 7 by inclusive method:
Class Interval Tally Bars Frequency
Prepare a discrete series from the following data:
Value Tally Bar Frequency
Arrange the following marks in frequency table taking the lowest class interval as (10-20).
Marks Tally Bar Frequency
Change the following into continuous series and convert the series into 'less than' and 'more than' cumulative series :
Marks (mid-values) 5 15 25 35 45 55
No.of students 8 12 15 9 4 2
Mid Value Class Interval Number of Students
Cumulative Series
Marks Number of Students Marks Number of Students
48 + 2 = 50 More than 0
Marks obtained by 24 students in English and Statistics in a class are given below . Prepare two-way frequency distribution table.
S.No 1 2 3 4 5 6 7 8 9 10 11 12
Marks in English 22 23 23 23 23 24 23 25 22 23 24 24
Marks in Statistics 16 16 18 16 16 17 16 19 16 18 18 17
S.No 13 14 15 16 17 18 19 20 21 22 23 24
Marks in Statistics 22 23 24 25 26 27 28 Total
15 − − − − − − 1
16 − − − 9
18 − − − − 4
19 − − − - − 4
In a survey ,it was found that 64 families bought milk ( in litres) in the following quantities in a particular month.
19 16 22 9 22 12 39 19 14 23 6 24 16 18 7
37 30 13 8 15 22 21 32 21 31 17 16 23 12 9
Convert the above data in a frequency distribution making classes of 5-9, 10-14 and so on.
(kg) Tally Bar Frequency
The marks obtained by 20 students in Statistics and Economics are given below . Prepare a bivariate frequency distribution.
Marks in Statistics 10 11 10 11 11 14 12 12 13 10 13 12 11 12 10 14 14 12 13 10
Marks in Economics 20 21 22 21 23 23 22 21 24 25 24 23 22 23 22 22 24 20 24 23
Marks in Economics
Marks in Statistics 20 21 22 23 24 25 Total
10 − − 5
13 − − − − − 3
Prepare 'less than' and 'more than' cumulative frequency distributions of the following date:
Wages (₹) : 140-150 150-160 160-170 170-180 180-190 190-200
No. of workers : 5 10 20 9 6 2
Less than Cumulative Frequency Distribution
More than Cumulative Frequency Distribution
Find out the frequency distribution and 'more than' cumulative frequency table:
Price (₹) below : 10 20 30 40 50 60
Quantity(kg) : 17 22 29 37 50 60
The frequency distribution for the given data is as follows.
Price Class Interval Quantity
Less than 60 0 − 10
If class mid-points in a frequency distribution of a group of persons are: 125, 132, 139, 146, 153, 160, 167, 174, 181 pounds, find:
(a) size of the class intervals, and
(b) the class boundaries.
From the mid-values the class size is calculated using the following formula.
Class Size = Mid value of one class - Mid value of the preceding class
Class Size = 132 - 125 = 7
Now, the upper and lower limit for each of the class intervals is calculated as follows.
\mathrm{Lower} \mathrm{limit} = \mathrm{Mid} \mathrm{value} - \frac{\mathrm{Class} \mathrm{Size}}{2}\phantom{\rule{0ex}{0ex}}\mathrm{or}, \mathrm{Lower} \mathrm{limit} = \mathrm{Mid} \mathrm{value} - \frac{7}{2}\phantom{\rule{0ex}{0ex}}\mathrm{Upper} \mathrm{limit} = \mathrm{Mid} \mathrm{value} + \frac{\mathrm{Class} \mathrm{Size}}{2}\phantom{\rule{0ex}{0ex}}\mathrm{or}, \mathrm{Upper} \mathrm{limit} = \mathrm{Mid} \mathrm{value} + \frac{7}{2}
Mid Value Class Interval
181 121.5 − 128.5
Prepare a continuous series from the following data:
Mid values : 125 135 145 155 165
Frequency : 10 9 12 8 6
Class Size = 135 - 125 = 10
\mathrm{Lower} \mathrm{limit} = \mathrm{Mid} \mathrm{value} - \frac{\mathrm{Class} \mathrm{Size}}{2}\phantom{\rule{0ex}{0ex}}\mathrm{or}, \mathrm{Lower} \mathrm{limit} = \mathrm{Mid} \mathrm{value} - \frac{10}{2}\phantom{\rule{0ex}{0ex}}\mathrm{Upper} \mathrm{limit} = \mathrm{Mid} \mathrm{value} + \frac{\mathrm{Class} \mathrm{Size}}{2}\phantom{\rule{0ex}{0ex}}\mathrm{or}, \mathrm{Upper} \mathrm{limit} = \mathrm{Mid} \mathrm{value} + \frac{10}{2}
Mid Value Class Interval f
|
Translational spring with variable spring stiffness - MATLAB - MathWorks América Latina
Variable Translational Spring
Minimum spring rate
Translational spring with variable spring stiffness
The Variable Translational Spring block represents a translational spring with variable stiffness. A physical signal input port provides the variable spring stiffness. The magnitude of the spring force is equal to the product of the physical signal input and the relative linear displacement between the two translational conserving ports. A minimum spring rate prevents nonphysical values.
The translational spring force satisfies the following expression:
F=\left\{\begin{array}{cc}K\cdot x,& K\ge {K}_{\mathrm{min}}\\ {K}_{\mathrm{min}}\cdot x,& K<{K}_{\mathrm{min}}\end{array}
F is the force transmitted through the spring between the two translational conserving ports.
K is the spring rate coefficient.
Kmin is the minimum allowed spring rate.
x is the relative displacement between the two translational conserving ports according to
x={x}_{init}+{x}_{R}-{x}_{C}
xinit is the initial spring deformation.
xR is the absolute displacement of the translational conserving port R.
xC is the absolute displacement of the translational conserving port C.
The block applies equal and opposite spring forces on the two translational conserving ports. The sign of the spring force acting on port R is equal to the sign of the relative linear displacement between the two ports. A positive linear velocity corresponds to a positive spring force acting on port R, and a negative spring force of equal magnitude acting on port C.
The block limits the value of the variable spring stiffness to remain above zero.
K — Variable spring coefficient, N/m
Physical signal input port associated with variable spring stiffness.
Mechanical translational conserving port associated with the negative spring end. The positive direction is from port R to port C.
Mechanical translational conserving port associated with the positive spring end. The positive direction is from port R to port C.
Minimum spring rate — Spring rate limit value
1 N/m (default) | positive scalar
Minimum value allowed for the spring rate. The physical signal input saturates below the specified value.
Nonlinear Rotational Spring | Nonlinear Translational Spring | Variable Rotational Spring
|
Kirk is helping his grandparents set up their new portable music players. His grandmother, Maude, has
1
jazz album,
2
country western albums, and
5
heavy-metal albums. Kirk’s grandfather, Claude, has
3
classical albums,
2
rap albums, and
7
heavy-metal albums.
If Kirk’s grandparents’ portable music players are on random shuffle mode, who has the greater chance of listening to a heavy-metal album? Explain how you know.
For each grandparent, find the portion of their albums that are heavy-metal. Which portion is larger?
For Claude, the portion of heavy-metal album is
\frac{7}{3+2+7}=\frac{7}{12}
If you compare the two portions as fractions, make sure they have the same denominator. If you have difficulty, try converting both fractions to percentages to compare.
|
Post 2: Generating fake data | R-bloggers
Post 2: Generating fake data
In order to check that an estimation algorithm is working properly, it is useful to see if the algorithm can recover the true parameter values in one or more simulated “test” data sets. This post explains how to build such a test data set for a 2PL model, gives two methods to check that the data are being generated properly, and examines the robustness of the checks by intentionally introducing bugs.
Generating 2PL data
We defined the 2PL IRT model in the previous post in terms of the distribution of
U_{pi}
, the observed response of person
p
i
U_{pi} stackrel{indep}{sim} text{Bernoulli}(pi_{pi})
end{equation*}begin{equation}
ln{frac{pi_{pi}}{1+pi_{pi}}} = a_i ( theta_p – b_i) label{eq:pipi}
where the three parameters
a_i
b_i
theta_p
are interpreted as item discrimination, item difficulty, and person ability parameters respectively.
We can generate fake data by choosing parameter values and following the model definition. In this example we set the item discrimination parameters to be uniformly distributed between 0.5 and 1.5; the item difficulty parameters to be evenly spaced between -3 and 3; the variance of the person ability parameters to be
1.25^2
; and the person ability parameters themselves to be normally distributed around zero. We implement this in R as follows:
## Set the random-number generator seed,
## to make the results reproducible
## set the number of items I and persons P
I.items <- 30
P.persons <- 2000
## set the fixed item and population parameters
a.disc <- 1 + runif(I.items,-0.5,0.5)
b.diff <- seq(-3,3,length=I.items)
## generate the person ability parameters
mean.theta <- 0
sig2.theta <- (1.25)^2
theta.abl <- rnorm(P.persons, mean=mean.theta,
sd=sqrt(sig2.theta))
There are several ways to calculate Equation ref{eq:pipi}. In R, operations on whole vectors are much faster than operations on individual elements of vectors. Therefore the code will be much faster if we avoid looping over the
p
i
values to calculate Equation ref{eq:pipi} directly and instead translate Equation ref{eq:pipi} into a series of operations on matrices.
One way to do this is given below, where we decompose the linear part of Equation ref{eq:pipi} into two terms: an outer product of
boldsymbol theta
boldsymbol a
and a matrix where every row is the same. We then calculate the logit with the native R function plogis, so that P.prob is the matrix corresponding to
pi_{pi}
, and then flip our biased coins with the inverse CDF method to generate
boldsymbol U
## the I x P matrix of response probabilities
term.1 <- outer(theta.abl, a.disc)
term.2 <- matrix( rep(a.disc*b.diff, P.persons),
nrow=P.persons, byrow=TRUE)
P.prob <- plogis(term.1-term.2) # 1/(1 + exp(term.2 - term.1))
## generate the 0/1 responses U as a matrix of Bernoulli draws
U <- ifelse(runif(I.items*P.persons) < P.prob,1,0)
Checking the fake data
Check Method 1: Matching theoretical and empirical moments
One way to check the simulated data is to check that the moments (means, variances, covariances, etc.) of the simulated data agree with the same moments of the simulating model. This approach is especially useful when MCMC is meant to be the first estimation algorithm. Although the 2PL IRT model is itself a well-known model, we illustrate this approach below with 1 dimensional and 2 dimensional moments.
1 dimensional moments
We can use the model to calculate the probability of item
i
being correctly answered by integrating over the person ability parameters: begin{equation} p_i = int_{-infty}^infty frac{exp{a_i(theta - b_i)}} {1 + exp{a_i(theta - b_i)}} cdot f_{text{Normal}}(theta|0,sigma^2_theta) ,dtheta quad . label{eq:1Dtheo} end{equation} A simple estimator of this probability is the average of the observed response across all of the persons: begin{equation} widehat{p}_{i} = frac{1}{P}sum_{p=1}^P U_{pi} quad . label{eq:1Demp} end{equation}
We can use R to calculate the theoretical values from equation ref{eq:1Dtheo} and visually compare them with the empirical estimates from ref{eq:1Demp}.
Equation ref{eq:1Dtheo} can be calculated in R by using the integrate function:
## 1 Dimensional moments
## Initialize a vector to hold the results
theo.1D <- rep(NA, I.items)
## Loop over each item
for( ii in 1:I.items){
## For a given item ii, evaluate Equation 1
## N.B. th.dummy is the integration variable
## a.disc[ii] is the discrimination parameter for item ii
## b.diff[ii] is the difficulty parameter for item ii
theo.1D[ii] <- integrate(
function(th.dummy) {return(
1/(1+exp(-a.disc[ii]*(th.dummy-b.diff[ii])))
* dnorm(th.dummy,mean.theta, sqrt(sig2.theta))
)},-Inf, Inf )$value
Equation ref{eq:1Demp} can be calculated in R by using apply to calculate the mean of every column of the matrix:
## The dimension of U is 2000 (persons) by 30 (items)
#[1] 2000 30
## To calculate the item averages, we 'apply' the 'mean'
## function on the columns of items. We select the columns
## with 'MARGIN=2' (the first margin is the rows, the second
## is the columns).
emp.1D <- apply(U, MARGIN=2, mean)
We can check that the empirical moments from Equation ref{eq:1Demp} match the theoretical moments from Equation ref{eq:1Dtheo} by making a scatter plot:
## Draw a scatter plot with the theoretical values
## on the x-axis and the empirical values on the y-axis.
plot( theo.1D, emp.1D, asp=1,
main='1D Moments',
xlab='Theoretical', ylab='Empirical')
## Draw a 45 degree line through the origin
Since the points lie mostly on the 45 degree line, it looks like the data were generated correctly.
We can use the model to calculate the joint probability of item
i
being correctly answered and item
j
being correctly answered by integrating over the person ability parameters: begin{equation} p_{ij} = int_{-infty}^infty frac{exp{a_i(theta - b_i)}} {1 + exp{a_i(theta - b_i)}} cdot frac{exp{a_j(theta - b_j)}} {1 + exp{a_j(theta - b_j)}} cdot f_{text{Normal}}(theta|0,sigma^2_theta) ,dtheta quad . label{eq:2Dtheo} end{equation} A simple estimator of this probability is the average of the product of observed responses across all of the persons: begin{equation} widehat{p}_{ij} = frac{1}{P}sum_{p=1}^P U_{pi}U_{pj} quad . label{eq:2Demp} end{equation}
Calculation of Equation ref{eq:2Dtheo} and Equation ref{eq:2Demp} use the same ideas as the 1D moments, but the details are more complex. We implement the calculation as follows:
## Initialize vectors to hold the results
## N.B. choose(n,p) calculates the binomial coefficient
## "n choose p"
theo.2D <- rep(NA, choose(I.items, 2) )
emp.2D <- theo.2D
## Generate all possible combinations of items
## N.B. combn(x,y) generates all combinations of x
## taken y at a time.
cmbn.matrix <- combn(I.items, 2)
## where each column of the matrix is a unique combination
dim(cmbn.matrix)
## We iterate over all unique combinations
for( which.cmbn in 1:choose(I.items,2) ) {
## We define the ii and jj for this combination
ii <- cmbn.matrix[1,which.cmbn]
jj <- cmbn.matrix[2,which.cmbn]
## We calculate Equation 3
theo.2D[which.cmbn] <- integrate(
* 1/(1+exp(-a.disc[jj]*(th.dummy-b.diff[jj])))
emp.2D[which.cmbn] <- mean(U[,ii]*U[,jj])
Which again looks pretty good!
Conclusions from Method 1
Based on the 1D and 2D moment scatter plots, it appears that the code to generate the fake data is working well.
Check Method 2: Recovering parameters with an existing estimation method
In the case of a familiar model such as the 2PL IRT model for which many other estimation algorithms have been written, we can also check to see that parameters are recovered by a known algorithm.
In R, the ltm package can be used to recover the item discrimination
a_i
and item difficulty
b_i
parameters. Example code is as follows:
require(ltm)
## ltm library. To do so, un-comment the install.packages
## install.packages('ltm')
## Fit a 2PL IRT model with ltm() and store the resulting
## object in ml.check
ml.check <- ltm( U ~ z1, IRT.param=TRUE )
## help(ltm)
## for details on its syntax.
## Extract out the discrimination and difficulty parameters
ml.a.disc <- coef(ml.check)[,'Dscrmn']
ml.b.diff <- coef(ml.check)[,'Dffclt']
We can check that ltm estimates of
a_i
b_i
match their true values with two scatter plots:
## Generate the unequated scatter plots side-by-side
plot( a.disc, ml.a.disc, asp=1,
xlab="True values", ylab="ltm estimates",
xlim=c(.5,1.6), ylim=c(.5,1.6),
main="ltm Item Discrimination")
plot( b.diff, ml.b.diff, asp=1,
main="ltm Item Difficulty")
The fit between the ltm estimates and the true values is bad.
In this case, the poor fit of the ltm estimates with the true values is because of a latent-space indeterminacy in estimating the 2PL IRT model. Cook and Eignor (1992) give a method to equate two sets of estimates from a 2PL model so that they can be compared. We implement the equating in R as follows:
## The Cook and Eignor (1992) method to equate
## *this* discrimination parameter with *that*
## discrimination parameter.
equate.2pl.a.disc <- function( this.a, this.b,
that.a, that.b ) {
## N.B. that.a is not used, but is included
## for ease of use.
return( this.a * sd( this.b ) / sd( that.b ) )
## *this* difficulty parameter with *that*
## difficulty parameter.
equate.2pl.b.diff <- function( this.a, this.b,
## N.B. this.a and that.a are not used, but are
## included for ease of use.
return( (this.b-mean(this.b))*sd(that.b)/sd(this.b)
+ mean(that.b) )
And then equate this specific example as follows:
equated.a.disc <- equate.2pl.a.disc( ml.a.disc, ml.b.diff,
a.disc, b.diff )
equated.b.diff <- equate.2pl.b.diff( ml.a.disc, ml.b.diff,
A scatter plot of the equated parameters shows that both the code to generate the fake data and the code to equate the parameter estimates works well:
## Generate the equated scatter plots side-by-side
plot( a.disc, equated.a.disc, asp=1,
xlab="True values", ylab="Equated ltm estimates",
main="Equated ltm Item Discrimination")
plot( b.diff, equated.b.diff,
main="Equated ltm Item Difficulty")
Conclusion from checking with an existing method
Based on the equated plots, it appears that the code to generate the fake data is working well.
Robustness of Method 1 and Method 2
In this section, we introduce a bug in the fake data generation code to see how well the two methods detect the bug.
We introduce a sign error as our candidate bug. Instead of subtracting the two terms in the logit, we add them:
## re-set the seed to keep results reproducible
## term.1 and term.2 should be subtracted, not added:
P.prob.buggy <- plogis(term.1 + term.2)
U.buggy <- ifelse(runif(I.items*P.persons) < P.prob.buggy,1,0)
Now we calculate our empirical "buggy" moments:
## Calculate the "buggy" moments
emp.1D.buggy <- apply(U.buggy, MARGIN=2, mean)
emp.2D.buggy <- rep(NA, choose(I.items, 2) )
emp.2D.buggy[which.cmbn] <- mean(U.buggy[,ii]*U.buggy[,jj])
And our "buggy" ltm estimates, which we also equate:
## Calculate the "buggy" ltm estimates
ml.check.buggy <- ltm( U.buggy ~ z1, IRT.param=TRUE )
ml.a.disc.buggy <- coef(ml.check.buggy)[,'Dscrmn']
ml.b.diff.buggy <- coef(ml.check.buggy)[,'Dffclt']
## Equate the "buggy" ltm estimates
equated.a.disc.buggy <- equate.2pl.a.disc(
ml.a.disc.buggy, ml.b.diff.buggy,
a.disc , b.diff )
equated.b.diff.buggy <- equate.2pl.b.diff(
Now we can visualize the effect of the bug on our moment matching and parameter recovery approaches:
## Graph the buggy moments
plot( theo.1D, emp.1D.buggy, asp=1,
main='1D Moments -- Intentional Bug',
xlab='Theoretical', ylab='Empirical (Buggy)')
## Graph the buggy equated ltm estimates
plot( a.disc, equated.a.disc.buggy, asp=1,
main="Equated ltm Item Discrimination -- Intentional Bug",
xlab="True values", ylab="Equated ltm estimates (Buggy)")
plot( b.diff, equated.b.diff.buggy, asp=1,
main="Equated ltm Item Difficulty -- Intentional Bug",
The moment matching method detects this bug very well. The 1D moments have the wrong slope and the 2D moments end up with an interesting shape that is most decidedly not a 45 degree line. This sends a strong signal that something is wrong in the code.
The parameter recovery method does detect the bug, but it sends a weaker signal since the item discrimination parameters seem fine. This is because the form of the scale indeterminacy for the item discrimination parameters is agnostic to this particular bug. Thankfully, that is not the case for the item difficulty parameters.
Based off of the results of both Method 1 and Method 2, we believe that our fake data is being generated correctly.
Note that the details of Method 1 and Method 2 are specific to the 2PL IRT model. In other models, one must derive the analogues of Equations ref{eq:1Dtheo} through ref{eq:2Demp} in order to compare moments, or use other available software and equating methods in order to compare recovered parameters.
|
EUDML | Iterations of concave maps, the Perron-Frobenius theory, and applications to circle packings. EuDML | Iterations of concave maps, the Perron-Frobenius theory, and applications to circle packings.
Iterations of concave maps, the Perron-Frobenius theory, and applications to circle packings.
Peretz, Ronen. "Iterations of concave maps, the Perron-Frobenius theory, and applications to circle packings.." ELA. The Electronic Journal of Linear Algebra [electronic only] 9 (2002): 197-254. <http://eudml.org/doc/122916>.
@article{Peretz2002,
author = {Peretz, Ronen},
keywords = {circle packings; pseudocircle packings; Andreev's theorem; Perron-Frobenius theory; fixed-point theorems},
title = {Iterations of concave maps, the Perron-Frobenius theory, and applications to circle packings.},
TI - Iterations of concave maps, the Perron-Frobenius theory, and applications to circle packings.
KW - circle packings; pseudocircle packings; Andreev's theorem; Perron-Frobenius theory; fixed-point theorems
circle packings, pseudocircle packings, Andreev's theorem, Perron-Frobenius theory, fixed-point theorems
Computational aspects of field theory and polynomials
Inequalities involving eigenvalues and eigenvectors
Norms of matrices, numerical range, applications of functional analysis to matrix theory
2
Rigidity and flexibility of structures
Circle packings and discrete conformal geometry
Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
Articles by Peretz
|
How to measure your treadmill incline and adjust it| NoblePro
This guide provides step-by-step instructions on how to measure and adjust the incline of a treadmill. This is particularly useful when calibrating, and adjusting your treadmill for use with apps that use incline data like Zwift, or our NoblePro Go app.
Incline terminology
There are 3 different ways in which treadmill incline can be defined:
1. Levels (which is usually a number)
3. Degrees (°)
Levels are the number of stops between the minimum incline and the maximum incline of a treadmill.
The level values do not necessarily translate to a percentage (%), or degrees (°) of incline, and you will need to contact the manufacturer for the details. Some manufacturers use levels to enable granular control over the incline without the need to implement a digital system on the display screen.
NoblePro treadmills have 21 levels of incline, from 0 to 20. Each level translates to 0.5% incline.
The percentage incline is defined as, for every 100 metres forward, you climb a number of metres up.
So, for instance, a 2% incline would mean: for every 100 meters you run, you climb 2 meters vertically.
This is the easiest way to think about real-world incline.
NoblePro treadmills have a range of 10% – from 2% to 12%.
We have created ZeroShoes for those who would like to amend this range to 0% – 10%.
And, finally… the most complicated one! The degree of the incline is defined as the angle (in degrees) of a slope in relation to the direction of gravity. So, for instance, at 0° a treadmill’s running board would be level with the ground, and at 2.9° it will have a 5.0% incline.
The good news is that the use of degrees is uncommon, however, it is good to understand the difference. Importantly, that degrees and percentage are not the same thing, but one can be used to calculate the other.
Degrees vs Percentage
There is a direct relationship between degrees and percentage incline which is defined as follows:
To convert from degrees incline to percentage incline use:
\text{percentage} = \mathrm{tan}\left(x°\right) × 100
To convert from percentage incline to degrees incline use:
degrees = {\mathrm{tan}}^{–1}\left(x%/100\right)
To get an accurate incline measurement you will need an inclinometer, also known as a “digital level”.
We have created an inclinometer that you can install on your smartphone. It ships with the NoblePro Go app, and we’ve written a guide on it here.
How to measure the incline
All measurements will be in relation to the floor as this provides a consistent measuring point.
We will be assuming that the floor is level on which the treadmill is placed.
The inclinometer needs to be placed on a flat area on the treadmill and parallel (inline) with the foot rails – see this guide for a photo reference.
Step 1 – Calibrate your incline
To get an accurate measurement it is important to calibrate your incline before starting. This feature is available on most good treadmills with step-by-step guides available for NoblePro treadmills here.
Step 2 – Measure the minimum incline
1. Turn your treadmill on.
2. Keep clear of the running belt and start the treadmill.
3. Put the treadmill on the lowest incline setting, typically “Level 0”.
4. Measure the incline and make note of the value.
Note: It is common for treadmills to have a minimum incline of between 0-2% (0-1.2°) incline.
Step 3 – Measure the maximum incline
3. Put the treadmill on the highest incline setting, typically “Level 20”.
Note: The maximum incline might not be available as a button on the screen so use the incline “+” until it does not change anymore.
Calculate the true incline
To make the calculations a bit easier we have created a spreadsheet for your reference that calculates your actual incline degrees and percentage based on the measurements in step 2 and 3.
(It’s all about the simple things!)
Incline calculator spreadsheet
To dynamically adjust the incline we have added an incline adjustment feature to our NoblePro Go app which will automatically adjust all the incline settings for you! Any related statistics like distance, speed, energy expense, and elevation gained are also tracked in the app. This provides you with a greater deal of flexibility and more accurate running statistics.
How to adjust your 0% incline
To more closely simulate the “energy cost” or “intensity” of outdoor running, treadmills tend to have a 1-2% offset in incline. This means that the minimum level can be between 1-2% when the treadmill is showing 0% (or on level 0).
If you prefer not to have this adjustment from the onset, but the treadmill cannot physically reduce its incline further, you can raise the rear of the treadmill to level it to a true 0%.
We have developed our NoblePro Zero Shoes to do exactly this specifically for our treadmills. By placing the Zero Shoes under the rear wheels it levels the treadmill’s incline to a true 0%.
Here is one of our treadmills standing on ZeroShoes – raising the rear, and giving the treadmill a true 0% incline
PrevPreviousThe bigger treadmill roller size myth, and how roller size affects your treadmill
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Equality (mathematics) - Wikipedia
(Redirected from Equal (math))
Find sources: "Equality" mathematics – news · newspapers · books · scholar · JSTOR (December 2015) (Learn how and when to remove this template message)
In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality between A and B is written A = B, and pronounced A equals B.[1] The symbol "=" is called an "equals sign". Two objects that are not equal are said to be distinct.
{\displaystyle x=y}
means that x and y denote the same object.[2]
{\displaystyle (x+1)^{2}=x^{2}+2x+1}
means that if x is any number, then the two expressions have the same value. This may also be interpreted as saying that the two sides of the equals sign represent the same function.
{\displaystyle \{x\mid P(x)\}=\{x\mid Q(x)\}}
{\displaystyle P(x)\Leftrightarrow Q(x).}
This assertion, which uses set-builder notation, means that if the elements satisfying the property
{\displaystyle P(x)}
are the same as the elements satisfying
{\displaystyle Q(x),}
then the two uses of the set-builder notation define the same set. This property is often expressed as "two sets that have the same elements are equal." It is one of the usual axioms of set theory, called axiom of extensionality.[3]
7 Relation with equivalence, congruence, and isomorphism
9 Equality in set theory
9.1 Set equality based on first-order logic with equality
9.2 Set equality based on first-order logic without equality
These last three properties make equality an equivalence relation. They were originally included among the Peano axioms for natural numbers. Although the symmetric and transitive properties are often seen as fundamental, they can be deduced from substitution and reflexive properties.
Equality as predicateEdit
When A and B may be viewed as functions of some variables, then A = B means that A and B define the same function. Such an equality of functions is sometimes called an identity. An example is
{\displaystyle \left(x+1\right)\left(x+1\right)=x^{2}+2x+1.}
Sometimes, but not always, an identity is written with a triple bar:
{\displaystyle \left(x+1\right)\left(x+1\right)\equiv x^{2}+2x+1.}
An equation is a problem of finding values of some variables, called unknowns, for which the specified equality is true. The term "equation" may also refer to an equality relation that is satisfied only for the values of the variables that one is interested in. For example,
{\displaystyle x^{2}+y^{2}=1}
is the equation of the unit circle.
Approximate equalityEdit
The binary relation "is approximately equal" (denoted by the symbol
{\displaystyle \approx }
) between real numbers or other things, even if more precisely defined, is not transitive (since many small differences can add up to something big). However, equality almost everywhere is transitive.
Relation with equivalence, congruence, and isomorphismEdit
Main articles: Equivalence relation, Isomorphism, Congruence relation, and Congruence (geometry)
In some contexts, equality is sharply distinguished from equivalence or isomorphism.[5] For example, one may distinguish fractions from rational numbers, the latter being equivalence classes of fractions: the fractions
{\displaystyle 1/2}
{\displaystyle 2/4}
are distinct as fractions (as different strings of symbols) but they "represent" the same rational number (the same point on a number line). This distinction gives rise to the notion of a quotient set.
{\displaystyle \{{\text{A}},{\text{B}},{\text{C}}\}}
{\displaystyle \{1,2,3\}}
{\displaystyle {\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3.}
{\displaystyle {\text{A}}\mapsto 3,{\text{B}}\mapsto 2,{\text{C}}\mapsto 1,}
In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties and structure being considered. The word congruence (and the associated symbol
{\displaystyle \cong }
) is frequently used for this kind of equality, and is defined as the quotient set of the isomorphism classes between the objects. In geometry for instance, two geometric shapes are said to be equal or congruent when one may be moved to coincide with the other, and the equality/congruence relation is the isomorphism classes of isometries between shapes. Similarly to isomorphisms of sets, the difference between isomorphisms and equality/congruence between such mathematical objects with properties and structure was one motivation for the development of category theory, as well as for homotopy type theory and univalent foundations.
Logical definitionsEdit
Equality in set theoryEdit
Set equality based on first-order logic with equalityEdit
Set equality based on first-order logic without equalityEdit
In first-order logic without equality, two sets are defined to be equal if they contain the same elements. Then the axiom of extensionality states that two equal sets are contained in the same sets.[8]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Equality_(mathematics)&oldid=1088231902"
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EUDML | Two Families of Mixed Finite Elements for Second Order Elliptic Problems. EuDML | Two Families of Mixed Finite Elements for Second Order Elliptic Problems.
Two Families of Mixed Finite Elements for Second Order Elliptic Problems.
F. Brezzi; J., Jr. Douglas; L.D. Marini
Brezzi, F., Douglas, J., Jr., and Marini, L.D.. "Two Families of Mixed Finite Elements for Second Order Elliptic Problems.." Numerische Mathematik 47 (1985): 217-236. <http://eudml.org/doc/133032>.
author = {Brezzi, F., Douglas, J., Jr., Marini, L.D.},
keywords = {mixed finite elements; asymptotic errors; Raviart-Thomas-Nedelec spaces; computational efficiency},
title = {Two Families of Mixed Finite Elements for Second Order Elliptic Problems.},
AU - Douglas, J., Jr.
AU - Marini, L.D.
TI - Two Families of Mixed Finite Elements for Second Order Elliptic Problems.
KW - mixed finite elements; asymptotic errors; Raviart-Thomas-Nedelec spaces; computational efficiency
Adrian J. Lew, Matteo Negri, Optimal convergence of a discontinuous-Galerkin-based immersed boundary method
M. R. Swager, Y. C. Zhou, Genetic Exponentially Fitted Method for Solving Multi-dimensional Drift-diffusion Equations
Lourenco Beirão Da Veiga, A mimetic discretization method for linear elasticity
Zhangxin Chen, Analysis of mixed methods using conforming and nonconforming finite element methods
Juan Enrique Santos, Ernesto Jorge Oreña, Elastic wave propagation in fluid-saturated porous media. Part II. The Galerkin procedures
P. Peisker, D. Braess, Uniform convergence of mixed interpolated elements for Reissner-Mindlin plates
Gabriel N. Gatica, Analysis of a new augmented mixed finite element method for linear elasticity allowing
{\mathrm{ℝ𝕋}}_{0}
{ℙ}_{1}
{ℙ}_{0}
Zhangxin Chen, Expanded mixed finite element methods for quasilinear second order elliptic problems, II
Jason S. Howell, Noel J. Walkington, Dual-mixed finite element methods for the Navier-Stokes equations
mixed finite elements, asymptotic errors, Raviart-Thomas-Nedelec spaces, computational efficiency
Articles by L.D. Marini
|
Pharmacodynamics - WikiProjectMed
Topics of pharmacodynamics
Pharmacodynamics (PD) is the study of the biochemical and physiologic effects of drugs (especially pharmaceutical drugs). The effects can include those manifested within animals (including humans), microorganisms, or combinations of organisms (for example, infection).
Pharmacodynamics and pharmacokinetics are the main branches of pharmacology, being itself a topic of biology interested in the study of the interactions between both endogenous and exogenous chemical substances with living organisms.
In particular, pharmacodynamics is the study of how a drug affects an organism, whereas pharmacokinetics is the study of how the organism affects the drug. Both together influence dosing, benefit, and adverse effects. Pharmacodynamics is sometimes abbreviated as PD and pharmacokinetics as PK, especially in combined reference (for example, when speaking of PK/PD models).
Pharmacodynamics places particular emphasis on dose–response relationships, that is, the relationships between drug concentration and effect.[1] One dominant example is drug-receptor interactions as modeled by
{\displaystyle {\ce {L + R <=> LR}}}
where L, R, and LR represent ligand (drug), receptor, and ligand-receptor complex concentrations, respectively. This equation represents a simplified model of reaction dynamics that can be studied mathematically through tools such as free energy maps.
Pharmacodynamics: Study of pharmacological actions on living systems, including the reactions with and binding to cell constituents, and the biochemical and physiological consequences of these actions.[2]
1.1 Desired activity
1.3 Therapeutic window
2 Receptor binding and effect
3 Multicellular pharmacodynamics
The majority of drugs either
induce(mimic) or inhibit(prevent) normal physiological/biochemical processes and pathological processes in animals or
inhibit vital processes of endo- or ectoparasites and microbial organisms.
There are 7 main drug actions:[3]
stimulating action through direct receptor agonism and downstream effects
depressing action through direct receptor agonism and downstream effects (ex.: inverse agonist)
blocking/antagonizing action (as with silent antagonists), the drug binds the receptor but does not activate it
stabilizing action, the drug seems to act neither as a stimulant or as a depressant (ex.: some drugs possess receptor activity that allows them to stabilize general receptor activation, like buprenorphine in opioid dependent individuals or aripiprazole in schizophrenia, all depending on the dose and the recipient)
exchanging/replacing substances or accumulating them to form a reserve (ex.: glycogen storage)
direct beneficial chemical reaction as in free radical scavenging
direct harmful chemical reaction which might result in damage or destruction of the cells, through induced toxic or lethal damage (cytotoxicity or irritation)
Some molecular mechanisms of pharmacological agents
The desired activity of a drug is mainly due to successful targeting of one of the following:
Cellular membrane disruption
Chemical reaction with downstream effects
Interaction with enzyme proteins
Interaction with structural proteins
Interaction with carrier proteins
Interaction with ion channels
Ligand binding to receptors:
General anesthetics were once thought to work by disordering the neural membranes, thereby altering the Na+ influx. Antacids and chelating agents combine chemically in the body. Enzyme-substrate binding is a way to alter the production or metabolism of key endogenous chemicals, for example aspirin irreversibly inhibits the enzyme prostaglandin synthetase (cyclooxygenase) thereby preventing inflammatory response. Colchicine, a drug for gout, interferes with the function of the structural protein tubulin, while Digitalis, a drug still used in heart failure, inhibits the activity of the carrier molecule, Na-K-ATPase pump. The widest class of drugs act as ligands that bind to receptors that determine cellular effects. Upon drug binding, receptors can elicit their normal action (agonist), blocked action (antagonist), or even action opposite to normal (inverse agonist).
In principle, a pharmacologist would aim for a target plasma concentration of the drug for a desired level of response. In reality, there are many factors affecting this goal. Pharmacokinetic factors determine peak concentrations, and concentrations cannot be maintained with absolute consistency because of metabolic breakdown and excretory clearance. Genetic factors may exist which would alter metabolism or drug action itself, and a patient's immediate status may also affect indicated dosage.
Undesirable effects of a drug include:
Increased probability of cell mutation (carcinogenic activity)
A multitude of simultaneous assorted actions which may be deleterious
Interaction (additive, multiplicative, or metabolic)
Induced physiological damage, or abnormal chronic conditions
Main article: Therapeutic window
The therapeutic window is the amount of a medication between the amount that gives an effect (effective dose) and the amount that gives more adverse effects than desired effects. For instance, medication with a small pharmaceutical window must be administered with care and control, e.g. by frequently measuring blood concentration of the drug, since it easily loses effects or gives adverse effects.
The duration of action of a drug is the length of time that particular drug is effective.[4] Duration of action is a function of several parameters including plasma half-life, the time to equilibrate between plasma and target compartments, and the off rate of the drug from its biological target.[5]
Receptor binding and effect
The binding of ligands (drug) to receptors is governed by the law of mass action which relates the large-scale status to the rate of numerous molecular processes. The rates of formation and un-formation can be used to determine the equilibrium concentration of bound receptors. The equilibrium dissociation constant is defined by:
{\displaystyle {\ce {L + R <=> LR}}}
{\displaystyle K_{d}={\frac {[L][R]}{[LR]}}}
where L=ligand, R=receptor, square brackets [] denote concentration. The fraction of bound receptors is
{\displaystyle {p}_{LR}={\frac {[LR]}{[R]+[LR]}}={\frac {1}{1+{\frac {K_{d}}{[L]}}}}}
{\displaystyle {p}_{LR}}
is the fraction of receptor bound by the ligand.
This expression is one way to consider the effect of a drug, in which the response is related to the fraction of bound receptors (see: Hill equation). The fraction of bound receptors is known as occupancy. The relationship between occupancy and pharmacological response is usually non-linear. This explains the so-called receptor reserve phenomenon i.e. the concentration producing 50% occupancy is typically higher than the concentration producing 50% of maximum response. More precisely, receptor reserve refers to a phenomenon whereby stimulation of only a fraction of the whole receptor population apparently elicits the maximal effect achievable in a particular tissue.
The simplest interpretation of receptor reserve is that it is a model that states there are excess receptors on the cell surface than what is necessary for full effect. Taking a more sophisticated approach, receptor reserve is an integrative measure of the response-inducing capacity of an agonist (in some receptor models it is termed intrinsic efficacy or intrinsic activity) and of the signal amplification capacity of the corresponding receptor (and its downstream signaling pathways). Thus, the existence (and magnitude) of receptor reserve depends on the agonist (efficacy), tissue (signal amplification ability) and measured effect (pathways activated to cause signal amplification). As receptor reserve is very sensitive to agonist's intrinsic efficacy, it is usually defined only for full (high-efficacy) agonists.[6][7][8]
Often the response is determined as a function of log[L] to consider many orders of magnitude of concentration. However, there is no biological or physical theory that relates effects to the log of concentration. It is just convenient for graphing purposes. It is useful to note that 50% of the receptors are bound when [L]=Kd .
The graph shown represents the conc-response for two hypothetical receptor agonists, plotted in a semi-log fashion. The curve toward the left represents a higher potency (potency arrow does not indicate direction of increase) since lower concentrations are needed for a given response. The effect increases as a function of concentration.
The concept of pharmacodynamics has been expanded to include Multicellular Pharmacodynamics (MCPD). MCPD is the study of the static and dynamic properties and relationships between a set of drugs and a dynamic and diverse multicellular four-dimensional organization. It is the study of the workings of a drug on a minimal multicellular system (mMCS), both in vivo and in silico. Networked Multicellular Pharmacodynamics (Net-MCPD) further extends the concept of MCPD to model regulatory genomic networks together with signal transduction pathways, as part of a complex of interacting components in the cell.[9]
Main article: Toxicodynamics
Pharmacokinetics and pharmacodynamics are termed toxicokinetics and toxicodynamics in the field of ecotoxicology. Here, the focus is on toxic effects on a wide range of organisms. The corresponding models are called toxicokinetic-toxicodynamic models.[10]
^ Duffus, J. (1 January 1993). "Glossary for chemists of terms used in toxicology (IUPAC Recommendations 1993)". Pure and Applied Chemistry. 65 (9): 2003–2122. doi:10.1351/pac199365092003.
^ "Introduction to Pharmacology". PsychDB. 25 March 2018.
^ Carruthers SG (February 1980). "Duration of drug action". Am. Fam. Physician. 21 (2): 119–26. PMID 7352385.
^ Vauquelin G, Charlton SJ (October 2010). "Long-lasting target binding and rebinding as mechanisms to prolong in vivo drug action". Br. J. Pharmacol. 161 (3): 488–508. doi:10.1111/j.1476-5381.2010.00936.x. PMC 2990149. PMID 20880390.
^ Ruffolo RR Jr (December 1982). "Review important concepts of receptor theory". Journal of Autonomic Pharmacology. 2 (4): 277–295. doi:10.1111/j.1474-8673.1982.tb00520.x. PMID 7161296.
^ Dhalla AK, Shryock JC, Shreeniwas R, Belardinelli L (2003). "Pharmacology and therapeutic applications of A1 adenosine receptor ligands". Curr. Top. Med. Chem. 3 (4): 369–385. doi:10.2174/1568026033392246. PMID 12570756.
^ Gesztelyi R, Kiss Z, Wachal Z, Juhasz B, Bombicz M, Csepanyi E, Pak K, Zsuga J, Papp C, Galajda Z, Branzaniuc K, Porszasz R, Szentmiklosi AJ, Tosaki A (2013). "The surmountable effect of FSCPX, an irreversible A(1) adenosine receptor antagonist, on the negative inotropic action of A(1) adenosine receptor full agonists in isolated guinea pig left atria". Arch. Pharm. Res. 36 (3): 293–305. doi:10.1007/s12272-013-0056-z. PMID 23456693. S2CID 13439779.
^ Zhao, Shan; Iyengar, Ravi (2012). "Systems Pharmacology: Network Analysis to Identify Multiscale Mechanisms of Drug Action". Annual Review of Pharmacology and Toxicology. 52: 505–521. doi:10.1146/annurev-pharmtox-010611-134520. ISSN 0362-1642. PMC 3619403. PMID 22235860.
^ Li Q, Hickman M (2011). "Toxicokinetic and toxicodynamic (TK/TD) evaluation to determine and predict the neurotoxicity of artemisinins". Toxicology. 279 (1–3): 1–9. doi:10.1016/j.tox.2010.09.005. PMID 20863871.
Wikimedia Commons has media related to Pharmacodynamics.
Vijay. (2003) Predictive software for drug design and development. Pharmaceutical Development and Regulation 1 ((3)), 159–168.
Werner, E., In silico multicellular systems biology and minimal genomes, DDT vol 8, no 24, pp 1121–1127, Dec 2003. (Introduces the concepts MCPD and Net-MCPD)
Dr. David W. A. Bourne, OU College of Pharmacy Pharmacokinetic and Pharmacodynamic Resources.
Retrieved from "https://mdwiki.org/wiki/Pharmacodynamics"
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Lance forgot his calculator and needed to find the exact value of
1.1^{4}
. He decided to use the binomial expansion by rewriting the expression as
\left(1 + 0.1\right)^{4}
. Use the binomial formula to expand
\left(1 + 0.1\right)^{4}
. Use your result to find the exact value of
1.1^{4}
. (Recall that
0.1^{2} = 0.01, 0.1^{3} = 0.001
= \left(1 + 0.1\right)^{4} = 1^{4} + 4\left(1\right)^{3}\left(0.1\right) + 6\left(1^{2}\right)\left(0.1^{2}\right) + \left(4\right)\left(1\right)\left(0.1^{3}\right) + 0.1^{4}
= 1 + \left(4\right)\left(0.1\right) + \left(6\right)\left(0.01\right) + \left(4\right)\left(0.001\right) + 0.0001
= 1 + 0.4 + 0.06 + 0.004 + 0.0001
= 1.4641
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Riemannian submersions of open manifolds which are flat at infinity | EMS Press
Riemannian submersions of open manifolds which are flat at infinity
We prove that a base
B^{n-k}
of a Riemannian submersion
\pi : {M^n} \to B^{n-k}
is flat, if
M^n
is flat at infinity and
B^{n-k}
is compact. As a corollary we obtain a topological gap-phenomenon for open manifolds of nonnegative sectional curvature (Eschenburg-Schroeder-Strake conjecture).
V. Marenich, Riemannian submersions of open manifolds which are flat at infinity. Comment. Math. Helv. 74 (1999), no. 3, pp. 419–441
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Home : Support : Online Help : Connectivity : Web Features : XMLTools : Print
print an XML document with elements indented
print an XML document with elements indented to a disk file
print an XML document with elements indented to a string
PrintToFile(fileName, xmlTree)
PrintToString(xmlTree)
Maple XML tree or a string; XML document
The Print(xmlTree) command formats an XML document so that its elements are indented as it is displayed on the default display device.
The PrintToFile(fileName, xmlTree) command is similar to the Print command, except that the formatted XML document is written to the file fileName.
The PrintToString(xmlTree) command formats the XML document and returns a Maple string containing the result. This allows you to pass the formatted XML to other procedures that can further process the document.
A string containing valid XML data can be passed in place of a parsed tree data structure, in which case the string is first parsed, and then printed as though the resulting tree were passed directly.
\mathrm{xmls}≔"<a><!-- An XML comment --><b c=\text{'}d\text{'}>Some Text <here></b></a>":
\mathrm{with}\left(\mathrm{XMLTools}\right):
\mathrm{Print}\left(\mathrm{ParseString}\left(\mathrm{xmls}\right)\right)
<b c = 'd'>Some Text <here></b>
XMLTools[ContentModel]
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Benutzer:Dirk Hünniger/wb2pdf – Wikibooks, Sammlung freier Lehr-, Sach- und Fachbücher
< Benutzer:Dirk Hünniger(Weitergeleitet von Benutzer:Dirk Huenniger/wb2pdf)
mediawiki2latex converts MediaWiki markup to LaTeX and, via LaTeX, to PDF. It can be used to export pages from any project running MediaWiki, such as Wikipedia. It is also possible to generate epub and odt output files.
Web Version[Bearbeiten]
You may test mediawiki2latex under the following url
There is a time limit of four hours
{\displaystyle \approx }
2000 pages per request on the server.
There is no limit on the locally installed versions described below.
Installation Instructions[Bearbeiten]
User Manual[Bearbeiten]
Command Line Version[Bearbeiten]
A command line version is currently available as part of the Stretch debian distribution, as well as the current ubuntu distribution.
LaTeX intermediate Code[Bearbeiten]
On Linux you can use the -c command line option with an absolute pathname.
Talk[Bearbeiten]
File:Wb2pdfTalk.ogv
Slides[Bearbeiten]
Poster[Bearbeiten]
File:Wb2pdfPoster.png
In Action[Bearbeiten]
To see it in action look here: Datei:Wb2latexCompilingWikibook2PDF.ogg
Developers[Bearbeiten]
The follwing Link Benutzer:Dirk Huenniger/wb2pdf/details explains some of the inner workings of the software.
Quality and Statistics[Bearbeiten]
A test run in October 2014 processing 4369 featured articles of the English Wikipedia did produce a PDF file in each case. In particular these were all featured articles we were able to find at the beginning of the test.
In May 2020 we looked at the usage of the web server an saw that the 50 requests looked at resulted in the following output:
The failures are believed to be caused by trying to process large books from the wikipedia book namespace exceeding the time limit of four hours.
In December 2018 we also did a test run on 100 featured articles on the English wikipedia. In two cases a PDF was not created. We just ran these two cases once again and got a PDF in each case. The total size of all PDFs was 2.2 GB on disk. 5 GB were downloaded in order to make them. The process took 6 hours and 15 minutes. The computer used was a i5-8250U notebook with 8 GByte of memory running only one instance of mediawiki2latex at a time. The internet downstream speed was 11.6 MBit/s.
The currently largest book we created with mediawiki2latex is 8991 pages.
Abgerufen von „https://de.wikibooks.org/w/index.php?title=Benutzer:Dirk_Hünniger/wb2pdf&oldid=973576“
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Tina’s rectangular living-room floor measures
15
18
If it helps, draw a diagram. Find the area of the rectangle that represents the floor. A rectangle labeled, length, 18 feet, and width, 15 feet.
The carpet Tina likes is sold by the square yard. How many square yards will she need? (
1 \text{ yard}=3\text{ feet}
There are two ways to solve part (b). You can take your answer from part (a) and divide it by the number of square feet
in a square yard, or you can convert the dimensions of the room into yards and then multiply.
If you use the first method, remember to convert your units properly. A square yard is
3
3
feet, or
9
square feet. The same rectangle labeled, 18 feet = 6 yards, and width labeled, 15 feet = 5 yards.
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Option price by Bates model using FFT and FRFT - MATLAB optByBatesFFT - MathWorks Benelux
\mathrm{max}\left(St-K,0\right)
\mathrm{max}\left(K-St,0\right)
\begin{array}{l}d{S}_{t}=\left(r-q-{\lambda }_{p}{\mu }_{J}\right){S}_{t}dt+\sqrt{{v}_{t}}{S}_{t}d{W}_{t}+J{S}_{t}d{P}_{t}\\ d{v}_{t}=\kappa \left(\theta -{v}_{t}\right)dt+{\sigma }_{v}\sqrt{{v}_{t}}d{W}_{t}\\ \text{E}\left[d{W}_{t}d{W}_{t}^{v}\right]=pdt\\ \text{prob(}d{P}_{t}=1\right)={\lambda }_{p}dt\end{array}
\mathrm{ln}\left(1+{\mu }_{J}\right)-\frac{{\delta }^{2}}{2}
\frac{1}{\left(1+J\right)\delta \sqrt{2\pi }}\mathrm{exp}\left\{{\frac{-\left[\mathrm{ln}\left(1+J\right)-\left(\mathrm{ln}\left(1+{\mu }_{J}\right)-\frac{{\delta }^{2}}{2}\right]}{2{\delta }^{2}}}^{2}\right\}
{W}_{t}^{v}
{\lambda }_{p}
{\lambda }_{p}
{f}_{Bate{s}_{j}\left(\varphi \right)}
\begin{array}{l}{f}_{Bates\left(\varphi \right)}=\mathrm{exp}\left({C}_{j}+{D}_{j}{v}_{0}+i\varphi \mathrm{ln}{S}_{t}\right)\mathrm{exp}{\left({\lambda }_{p}\tau \left(1+{\mu }_{J}\right)}^{{m}_{j}+\frac{1}{2}}\left[{\left(1+{\mu }_{j}\right)}^{i\varphi }{e}^{{\delta }^{2}\left({m}_{j}i\varphi +\frac{{\left(i\varphi \right)}^{2}}{2}\right)}-1\right]-{\lambda }_{p}\tau {\mu }_{J}i\varphi \right)\\ {m}_{j}=\left\{\begin{array}{l}{m}_{1}=\frac{1}{2}\\ {m}_{2}=-\frac{1}{2}\end{array}\right\}\\ {C}_{j}=\left(r-q\right)i\varphi \tau +\frac{\kappa \theta }{{\sigma }_{v}{}^{2}}\left[\left({b}_{j}-p{\sigma }_{v}i\varphi +{d}_{j}\right)\tau -2\mathrm{ln}\left(\frac{1-{g}_{j}{e}^{{d}_{j}\tau }}{1-{g}_{j}}\right)\right]\\ Dj=\frac{{b}_{j}-p{\sigma }_{v}i\varphi +{d}_{j}}{{\sigma }_{v}^{2}}\left(\frac{1-{e}^{{d}_{j}\tau }}{1-{g}_{j}{e}^{{d}_{j}\tau }}\right)\\ {g}_{j}=\frac{{b}_{j}-p{\sigma }_{v}i\varphi +{d}_{j}}{{b}_{j}-p{\sigma }_{v}i\varphi -{d}_{j}}\\ {d}_{j}=\sqrt{{\left({b}_{j}-p{\sigma }_{v}i\varphi \right)}^{2}-{\sigma }_{v}^{2}\left(2{u}_{j}i\varphi -{\varphi }^{2}\right)}\\ \text{where for }j=1,2:\\ {u}_{1}=\frac{1}{2},{u}_{2}=-\frac{1}{2},{b}_{1}=\kappa +{\lambda }_{VolRisk}-p{\sigma }_{v},{b}_{2}=\kappa +{\lambda }_{VolRisk}\end{array}
\begin{array}{l}{C}_{j}=\left(r-q\right)i\varphi \tau +\frac{\kappa \theta }{{\sigma }_{v}{}^{2}}\left[\left({b}_{j}-p{\sigma }_{v}i\varphi -{d}_{j}\right)\tau -2\mathrm{ln}\left(\frac{1-{\epsilon }_{j}{e}^{-{d}_{j}\tau }}{1-{\epsilon }_{j}}\right)\right]\\ Dj=\frac{{b}_{j}-p{\sigma }_{v}i\varphi -{d}_{j}}{{\sigma }_{v}^{2}}\left(\frac{1-{e}^{-{d}_{j}\tau }}{1-{\epsilon }_{j}{e}^{-{d}_{j}\tau }}\right)\\ {\epsilon }_{j}=\frac{{b}_{j}-p{\sigma }_{v}i\varphi -{d}_{j}}{{b}_{j}-p{\sigma }_{v}i\varphi +{d}_{j}}\end{array}
\begin{array}{l}Call\left(k\right)=\frac{{e}^{-\alpha k}}{\pi }{\int }_{0}^{\infty }\mathrm{Re}\left[{e}^{-iuk}\psi \left(u\right)\right]du\\ \psi \left(u\right)=\frac{{e}^{-r\tau }{f}_{2}\left(\varphi =\left(u-\left(\alpha +1\right)i\right)\right)}{{\alpha }^{2}+\alpha -{u}^{2}+iu\left(2\alpha +1\right)}\\ Put\left(K\right)=Call\left(K\right)+K{e}^{-r\tau }-{S}_{t}{e}^{-q\tau }\end{array}
\mathrm{ln}\left({S}_{t}\right)-\frac{N}{2}\Delta k
\mathrm{ln}\left({S}_{t}\right)+\left(\frac{N}{2}-1\right)\Delta k
{S}_{t}\mathrm{exp}\left(-\frac{N}{2}\Delta k\right)
{S}_{t}\mathrm{exp}\left[\left(\frac{N}{2}-1\right)\Delta k\right]
Call\left({k}_{n}\right)=\Delta u\frac{{e}^{-\alpha {k}_{n}}}{\pi }\sum _{j=1}^{N}\mathrm{Re}\left[{e}^{-i\Delta k\Delta u\left(j-1\right)\left(n-1\right){e}^{i{u}_{j}}\left[\frac{N\Delta k}{2}-\mathrm{ln}\left({S}_{t}\right)\right]}\psi \left({u}_{j}\right)\right]{w}_{j}
\Delta k\Delta u=\left(\frac{2\pi }{N}\right)
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Opening & Managing Positions - Angle
How to open, modify or close perpetual positions on Angle app
Angle lets you open long collateral/stablecoin leverage positions. This means that if the protocol accepts USDC as collateral, and can issue agEUR, users can open USDC/agEUR long positions, betting that the USD will increase in value against the EUR.
You can read more about this mechanism here.
Perpetuals page
To open a long position on Angle, click on the + Open Positionbutton and choose a collateral/stablecoin pair.
Then, select the amount of collateral you want to send to the protocol as margin for your position. Positions in Angle work similarly than isolated-margin exchanges, where margin is separated between positions.
Now, you can choose your position size or leverage. This is the amount of underlying tokens you will be exposed to. NB: leverage in Angle is computed as
\frac{\texttt{margin + position size}}{\texttt{margin}}
The collateral/stablecoin exchange rate and transaction fees are displayed. Note that the net initial margin of your position will be your initial margin input minus fees.
Clicking the Open position button will prompt you to confirm the transaction. The margin will be sent to the protocol and the leveraged position will be opened.
While it is opened, your position will automatically accrue ANGLE rewards.
If this is the first time opening a position on this collateral/stablecoin pair, you will need to approve your tokens with a transaction or a signature first.
Be careful when opening a position, updating and closing is locked for an hour. More info here.
If you have open positions, you might want to add or remove margin to some of them. Doing so will update your leverage and change your liquidation price. There is no fee for updating the margin of a position.
On the HA positions page, click on the Modify button.
Enter an amount of collateral to add or remove from your position, or change its leverage. You can remove collateral up to reaching the max leverage allowed for the pair of the position. This max leverage can vary from x10 to x100 depending on the pair.
You will find current and updated info about your position.
Then, just confirm the transaction and it will send/withdraw the amount of collateral that was specified.
After the transaction is confirmed, you can go back to the Positions page to see your updated position.
To close a position, click on the Close button of the positions you would like to close.
You can then check your position's opening and current prices, as well as the amount of collateral and potential ANGLE rewards you are going to receive upon closing.
Click on the Close button to confirm the transaction and receive your cash out amount (margin ± PnL) and potential ANGLE rewards.
When opening or closing a position, there is an expert mode to protect for important slippage in price or fees.
Closing Perpetual
Claiming ANGLE rewards
If you have an open position on Angle, it will automatically accrue ANGLE token rewards. You can see how much you can claim by clicking on the Details button at the bottom of the position, and claim your tokens by clicking on the Claim button as shown in the screenshot below.
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The ‘spam comments’ puzzle: tidy simulation of stochastic processes in R | R-bloggers
I love 538’s Riddler column, and the April 10 puzzle is another interesting one. I’ll quote:
This is a great opportunity for tidy simulation in R, and also for reviewing some of the concepts of stochastic processes (this is known as a Yule process). As we’ll see, it’s even thematically relevant to current headlines, since it involves exponential growth.
Solving a puzzle generally involves a few false starts. So I recorded this screencast showing how I originally approached the problem. It shows not only how to approach the simulation, but how to use those results to come up with an exact answer.
The Riddler puzzle describes a Poisson process, which is one of the most important stochastic processes. A Poisson process models the intuitive concept of “an event is equally likely to happen at any moment.” It’s named because the number of events occurring in a time interval of length
x
is distributed according to
\mbox{Pois}(\lambda x)
, for some rate parameter
\lambda
(for this puzzle, the rate is described as one per day,
\lambda=1
How can we simulate a Poisson process? This is an important connection between distributions. The waiting time for the next event in a Poisson process has an exponential distribution, which can be simulated with rexp().
# The rate parameter, 1, is the expected events per day
waiting <- rexp(10, 1)
For example, in this case we waited 0.14 days for the first comment, then 2.8 after that for the second one, and so on. On average, we’ll be waiting one day for each new comment, but it could be a lot longer or shorter.
You can take the cumulative sum of these waiting periods to come up with the event times (new comments) in the Poisson process.
qplot(cumsum(waiting), 0)
Simulating a Yule process
Before the first comment happened, the rate of new comments/replies was 1 per day. But as soon as the first comment happened, the rate increased: the comment could spawn its own replies, so the rate went up to 2 per day. Once there were two comments, the rate goes up to 3 per day, and so on.
This is a particular case of a stochastic process known as a Yule process (which is a special case of a birth process. We could prove a lot of mathematical properties of that process, but let’s focus on simulating it.
The waiting time for the first commentwould be
\mbox{Exponential}(1)
, but the waiting time for the second is
\mbox{Exponential}(2)
\mbox{Exponential}(3)
, and so on. We can use the vectorized rexp() function to simulate those. The waiting times will, on average, get shorter and shorter as there are more comments that can spawn replies.
waiting_times <- rexp(20, 1:20)
# Cumulative time
cumsum(waiting_times)
## [15] 2.9713356 3.0186731 3.1340060 3.2631936 3.2967087 3.3024576
# Number before the third day
sum(cumsum(waiting_times) < 3)
In this case, the first 15 events happened before the third day. Notice that in this simulation, we’re not keeping track of which comment received a reply: we’re treating all the comments as interchangeable. This lets our simulation run a lot faster since we just have to generate the waiting times.
All combined, we could perform this simulation in one line:
sum(cumsum(rexp(20, 1:20)) < 3)
So in one line with replicate(), here’s one million simulations. We simulate 300 waiting periods from each, and see how many happen before the first day.
sim <- replicate(1e6, sum(cumsum(rexp(300, 1:300)) < 3))
mean(sim)
It looks like it’s about 19.1.
Turning this into an exact solution
Why 19.1? Could we get an exact answer that is intuitively satisfying?
One trick to get a foothold is to vary one of our inputs: rather than looking at 3 days, let’s look at the expected comments after time
t
. That’s easier if we expand this into a tidy simulation, using one of my favorite functions crossing().
sim_waiting <- crossing(trial = 1:25000,
observation = 1:300) %>%
mutate(waiting = rexp(n(), observation)) %>%
mutate(cumulative = cumsum(waiting)) %>%
sim_waiting
## trial observation waiting cumulative
## <int> <int> <dbl> <dbl>
## 1 1 1 0.294 0.294
## 3 1 3 0.0790 1.01
## 4 1 4 0.185 1.19
## 10 1 10 0.0488 2.66
We can confirm that the average number of comments in the first three days is about 19.
sim_waiting %>%
summarize(num_comments = sum(cumulative <= 3)) %>%
summarize(average = mean(num_comments))
## average
## 1 18.9
But we can also use crossing() (again) to look at the expected number of cumulative comments as we vary
t
average_over_time <- sim_waiting %>%
crossing(time = seq(0, 3, .25)) %>%
group_by(time, trial) %>%
summarize(num_comments = sum(cumulative < time)) %>%
(Notice how often “solve the problem for one value” can be turned into “solve the problem for many values” with one use of crossing(): one of my favorite tricks).
How does the average number of comments increase over time?
ggplot(average_over_time, aes(time, average)) +
At a glance, this looks like an exponential curve. With a little experimentation, and noticing that the curve starts at
(0, 0)
, we can find that the expected number of comments at time
t
e^t-1
. This fits with our simulation:
e^3 - 1
is 19.0855.
geom_line(aes(y = exp(time) - 1), color = "red") +
labs(y = "Average # of comments",
title = "How many comments over time?",
subtitle = "Points show simulation, red line shows exp(time) - 1.")
Intuitively, it makes sense that on average the growth is exponential. If we’d described the process as “bacteria in a dish, each of which could divide at any moment”, we’d expect exponential growth. The “minus one” is because the original post is generating comments just like all the others do, but doesn’t itself count as a comment.1
Distribution of comments at a given time
It’s worth noting we’re still only describing an average path. There could easily be more, or fewer, spam comments by the third day. Our tidy simulation gives us a way to plot many such paths.
filter(trial <= 50, cumulative <= 3) %>%
ggplot(aes(cumulative, observation)) +
geom_line(aes(group = trial), alpha = .25) +
geom_line(aes(y = exp(cumulative) - 1), color = "red", size = 1) +
y = "# of comments",
title = "50 possible paths of comments over time",
subtitle = "Red line shows e^t - 1")
The red line shows the overall average, reaching about 19.1 at 3 days. However, we can see that it can sometimes be much smaller or much larger (even even more than 100).
What is the probability distribution of comments after three days- the probability there is one comment, or two, or three? Let’s take a look at the distribution.
# We'll use the million simulated values from earlier
num_comments <- tibble(num_comments = sim)
num_comments %>%
ggplot(aes(num_comments)) +
Interestingly, at a glance this looks a lot like an exponential curve. Since it’s a discrete distribution (with values 0, 1, 2…), this suggests it’s a geometric distribution: the expected number of “tails” flipped before we see the first “heads”.
We can confirm that by comparing it to the probability mass function,
(1-p)^np
. If it is a geometric distribution, then because we know the expected value is
e^3-1
we know the rate parameter
p
(the probability of a success on each heads) is
\frac{1}{e^3}=e^{-3}
p <- exp(-3)
filter(num_comments <= 150) %>%
geom_histogram(aes(y = ..density..), binwidth = 1) +
geom_line(aes(y = (1 - p) ^ num_comments * p), color = "red")
This isn’t a mathematical proof, but it’s very compelling. So what we’ve learned overall is:
X(t)\sim \mbox{Geometric}(e^{-t})
E[X(t)]= e^{-t}-1
These are true because the rate of comments is one per day. If the rate of new comments were
\lambda
, you’d replace
t
above with
\lambda t
I don’t have an immediate intuition for why the distribution is geometric. Though it’s interesting that the parameter
p=e^{-t}
for the geometric distribution (the probability of a “success” on the coin flip that would stop the process) is equal to the probability that there are no events in time
t
for a Poisson process.
Conclusion: Yule process
I wasn’t familiar with it when I first tried out the riddle, but this is known as a Yule process. For confirmation of some of the results above you can check out this paper or the Wikipedia entry, among others.
What I love about simulation is how it builds an intuition for these processes from the ground up. These simulated datasets and visualizations are a better “handle” for me for grasp the concepts than mathematical equations would be. After I’ve gotten a feel for the distributions, I can check my answer by looking through the mathematical literature.
If you don’t like the
-1
, you could have counted the post as a comment, started everything out at
X(0)=1
, and then you would find that
E[X(t)]=e^t
. This is the more traditional definition of a Yule process. ↩
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Evolution Equations in Thermoelasticity | Appl. Mech. Rev. | ASME Digital Collection
Song Jiang,, Author,
Inst of Appl Phys and Comput Math, Beijing, Peoples Rep of China
E Racke,, Author,
E Racke,, Author
Univ of Konstanz, Konstanz, Germany
MV Shitikova,, Reviewer
Dept of Struct Mech, Voronezh State Univ of Architec and Civil Eng, ul Kirova 3-75, Voronezh, 394018, Russia
Appl. Mech. Rev. Jan 2002, 55(1): B17-B18
Jiang, , S., Author, Racke, , E., Author, and Shitikova, , M., Reviewer (January 1, 2002). "Evolution Equations in Thermoelasticity." ASME. Appl. Mech. Rev. January 2002; 55(1): B17–B18. https://doi.org/10.1115/1.1445336
thermoelasticity, boundary-value problems, reviews
Boundary-value problems, Thermoelasticity
1R48. Evolution Equations in Thermoelasticity. Monographs and Surveys in Pure and Applied Mathematics, Vol 112. - Song Jiang (Inst of Appl Phys and Comput Math, Beijing, Peoples Rep of China) and E Racke (Univ of Konstanz, Konstanz, Germany). Chapman and Hall/CRC, Boca Raton FL. 2000. 308 pp. ISBN 1-58488-215-8. $84.95.
Reviewed by MV Shitikova (Dept of Struct Mech, Voronezh State Univ of Architec and Civil Eng, ul Kirova 3-75, Voronezh, 394018, Russia).
The authors’ aim is to present a state of the art in the treatment of initial value problems and of initial boundary value problems both in linear and nonlinear thermoelasticity. From the very beginning, the authors restrict themselves by considering the conventional thermoelasticity theory formulated on the principles of the classical theory of heat conduction, and consequently, the heat transport equation of the theory is of parabolic type. The second grave limitation is connected with the boundary conditions used throughout the monograph, namely: the ideal heat exchange between a thermoelastic body and its surrounding medium and heat insulation on the body’s boundary are considered. The more general condition—condition of heat exchange between thermoelastic bodies or between a body and surrounding medium, from which the conditions of constant temperature or heat insulation follow as particular limiting cases—is not investigated at all. But the condition of heat exchange is the most interesting and important for engineering applications, especially in contact problems. That is why this reviewer cannot agree with the authors that “the intended audience includes not only graduate students of both mathematics and physics, but also the foremost expert looking for a survey.” This book may be useful only for students wanting to be familiar with the basics of mathematical aspects of the convectional thermoelasticity theory.
The book includes nine chapters followed by two appendices, lists of main and supplemental references, notation, and an index. The first chapter gives a short summary of the derivation of the equations describing the nonlinear behavior of a thermoelastic body within the framework of the conventional thermoelasticity theory. Using Taylor expansions, the corresponding linearized equations are also written. The well-posedness of the linear initial boundary value problem in the case of the ideal thermal contact between a rigidly clamped body and surrounding medium is discussed in Chapter 2.
The asymptotic behavior as time tends to infinity of such a thermoelastic system with zero exterior forces and heat supply is investigated in linearized one-dimensional formulations in Chapter 3; two- or three- dimensional formulations are investigated in Chapter 4. A local existence theorem for the initial boundary value problem of hyperbolic-parabolic type and for the Cauchy problem is proved in Chapter 5. One-dimensional and three-dimensional thermoelastic nonlinear equations are considered in Chapters 6 and 7, respectively.
Chapter 8 analyzes the evolution of temperature and displacement in an elastic body that may come into contact with a rigid foundation. The system consists of the linearized equations together with the ideal thermal contact between the body and rigid foundation and Signorini’s nonlinear conditions for mechanical contact. In the final chapter, the following problems are briefly described: the linear boundary value problem in the presence of external forces and heat supply, resulting in an additional damping; the far field asymptotic behavior of the solution, as well as a numerical scheme for the numerical solution of the initial boundary value problem.
Thus, the majority of the book, Chapters 2–8, is devoted to the proof of the obvious results about the exponential damping of energy and displacements as time goes to infinity by the use of different mathematical methods. In this reviewer’s opinion, some results dealing with the asymptotic behavior of the desired values at large times can be obtained by the Laplace transformation methods with the corresponding limiting theorems.
Much to this reviewer’s surprise, the authors, when discussing the behavior of the surfaces of strong discontinuity, did not even mention the works by VI Danilovskaya, RB Hetnarski, J Ignaszak, W Nowacki, and many others who have shown that singularities propagate in the stressed-strained thermoelastic medium of the hyperbolic-parabolic type with damping which has exponential character and is defined by the coupling of the strain and temperature fields. All the researchers mentioned above in one way or another investigated the asymptotic behavior of the solutions obtained as
t→∞
x→∞.
As for the thermoelastic contact problems, then the authors completely ignore the results by Barber and his coauthors. That is why this reviewer does not share the opinion of the authors that their book “presents a state-of-the-art treatment of initial boundary value problems in thermoelasticity and includes the most extensive bibliographies on the subject published to date.” Quite to the contrary, the lists of main and additional references are very limited and do not cover a huge amount of monographs and original papers dealing with solving the boundary value problems even in the framework of the conventional thermoelasticity theory, not to mention the extended thermoelasticity theories predicting a finite speed of the propagation of thermal signals.
Applications of the mathematical treatment described in the book are of limited usefulness, and this book cannot attract the attention of engineers and researchers involved with a practical implementation of thermoelasticity. This reviewer thinks that Evolution Equations in Thermoelasticity. Monographs and Surveys in Pure and Applied Mathematics, Vol 1 can be useful only for students who want to have some basic mathematical knowledge in classical thermoelasticity, but it cannot be recommended for purchase by libraries for mechanical or civil engineering departments, or by individuals with an interest in the practical utility of thermoelasticity.
Indentation Problems of Two-Dimensional Anisotropic Thermoelasticity With Perturbed Boundaries
Analytical and Numerical Predictions of Thermoelastic Properties of Carbon Single-Walled Nanotubes
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