content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
A unified analysis of spontaneous and super-radiant emissions in free-electron lasers
A generalized formulation of spontaneous emission and super-radiance effects in a free-electron laser is presented. We consider a stream of electrons of arbitrary temporal duration propagating
through the undulator. The sum of the undulator synchrotron radiation emitted by individual wiggling electrons entering the wiggler at random, results in shot-noise in the radiation field. Using the
waveguide excitation equations formulated in the frequency domain, an analytical expression for the power spectral density of the electromagnetic radiation is derived. It is shown that for a finite
pulse electron beam current, the spectrum of the excited radiation is composed of two terms which are the spontaneous and super-radiant emissions. For an infinitely long e-beam pulse (continuous
beam), the shot-noise produces only incoherent spontaneous emission. The power of this radiation is proportional to the DC current I[0] of the electron beam. For shorter e-beam pulses, a partially
coherent super-radiant emission is also produced with an average power which is proportional to I^20. The coherence of this super-radiant emission is enhanced as the pulse duration is reduced. A
single formulation describes the coherent features of the super-radiance and the statistical features of the spontaneous emission.
Dive into the research topics of 'A unified analysis of spontaneous and super-radiant emissions in free-electron lasers'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/a-unified-analysis-of-spontaneous-and-super-radiant-emissions-in--3","timestamp":"2024-11-09T04:49:16Z","content_type":"text/html","content_length":"56472","record_id":"<urn:uuid:dd8ee797-4597-4f70-b491-c0b8833a16f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00357.warc.gz"} |
Edible Dosage Calculator
How a cannabis calculator works?
The idea of this calculator came from Old Hippie's Website. Our calculator use the very same concept of Ohm's Law, maybe you don't remember it from your physics class, but I will make it easier to
remember and understand using cannabis stuff.
Consider the following:
G = grams of cannabis
P = potency in milligrams of THC
S = strength of cannabis in percent
N = number of servings
G is literally the quantity of plant material you want to use, no mistery here. N is the number of servings of your recipe, let's say you are making a space cake, after finishing the cooking, you
decide to cut your cake in 4 pieces, in this case N equals 4.
S is probably the most complicated to figure out, but bare with me. If you are in a state or country where weed is NOT decriminalize, a good rule of thumb would be to assume 20% of strength for top
shelf frosty buds, 10% to 15% for mids, and 5% for schwag/brickweed. For concentrates, assume 30% for kief, 40% for hash, 60% for hash oil, 80%-95% for shatter. If you are lucky and you live in a
legalized state/country you can just ask your budtender or dispensary about the flower or concentrate strength.
Ok, now you understand what is S, N and G, you can use all these three elements to calculate your edible's potency with the following equation:
Confused? Don't worry! Let's go over it step by step. Assume you have 2 grams of high-quality cannabis with 20% of THC and you want to make 8 cookies for you and your friends. How strong will each of
them be? So, here we go:
G = 2 (grams)
S = 20 (% of THC)
N = 8 (portions)
P = 10 * (2 * 20) / 8 = 50 mg each
Notice that the amount of oil or butter you use has nothing to do with the calculation! So if you are making enough cannabutter or infused oil for a batch of cookies, you use the amount your recipe
calls for 8 cookies, and that's all there is to it.
This equation can also be used if you are trying to calculate another cannabinoid, like CBD or CBG. Instead of using THC percentage for S, just use CBD percentage.
Fortunately, you don't need to know all this, you can see this math magic happening live in our calculator, you just need to input the quantity, strength and portion number and we do all the work for | {"url":"https://www.howtoedibles.com/how-a-cannabis-calculator-works","timestamp":"2024-11-12T23:53:35Z","content_type":"text/html","content_length":"66984","record_id":"<urn:uuid:2cf427fb-483e-4ecd-af69-e37292923dd0>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00016.warc.gz"} |
Pattern Analysis Mock TestPattern Analysis Mock Test
This inductive reasoning test comprises 25 questions. In each question, you will be presented with a logical sequence of 4-5 figures. You will need to determine which of the possible answers best
matches the next figure in the sequence, or replaces a missing figure in the sequence. Good luck!
Which square comes next in the sequence?
Which drawing is an exact replica of the one immediately below?
Choose the correct image that completes the pattern below.
Which square comes next in the sequence?
What is the next sequence?
Suppose you were to arrange their seats, the highest official will be seated on the first chair on the left going right, who seats on the middle chair?
Which square comes next in the sequence?
Arrange them in ascending order based on their speed, starting from the slowest.
In each example given below, you will find a logical sequence of five boxes. Your task is to decide which of the boxes completes this sequence. To give your answer, select one of the boxes. You will
be told whether or not your answer is correct.
What should replace the question mark?
What sequence should replace the question mark?
Which square comes next in the sequence?
Choose the image that completes the pattern.
What is the next step in the sequence?
Which square comes next in the sequence?
Based on the Philippine Government's setting, which has the lowest position among those given?
Choose the image that completes the pattern.
Which square comes next in the sequence?
What sequence should replace the question mark?
In each example given below, you will find a logical sequence of five boxes. Your task is to decide which of the boxes completes this sequence. To give your answer, select one of the boxes. You will
be told whether or not your answer is correct.
Choose the image that completes the pattern.
Which square comes next in the sequence?
Which square comes next in the sequence?
What replaces the question mark?
What is the correct sequence selection to replace the question marks? | {"url":"https://topnotcher.ph/pattern-analysis-mock-test/","timestamp":"2024-11-02T20:40:05Z","content_type":"text/html","content_length":"342888","record_id":"<urn:uuid:f90b2215-a41d-40c7-bbd5-430d9d0dbf5f>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00257.warc.gz"} |
12: A quantum paradox and the experiments
Last updated
Page ID
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\
evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\
newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}
\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}
{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}
[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}
{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\
#3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp
#2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\
wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\
newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}
{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\
bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\
widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)
Quantum mechanics makes probabilistic predictions about the possible outcomes of an experiment with microscopic objects, particles. This is a statistical statement concerning many particles. One can
assume that there exists a deeper non-statistical, i.e. deterministic law for single particles predicting what should happen with that specific particle.
Is there any deeper theory or law determining the experimental result with certainty? There must be such a theory according to Einstein, implying that quantum mechanics cannot be a complete theory.
On the other hand, according to Bohr – who was another founder of quantum theory – there is no place for a deeper theory, we cannot predict anything more about a single particle, but probabilities.
Chapter 1. Polarization of plane light waves in classical optics.
The EPR-paradox, Bohm’s version
In this last section we consider certain principal questions of quantum mechanics that are connected with the very essence of the subject, i.e. with the nature of the probabilities appearing in the
theory. In classical physics a measurement makes a record of the value of a physical quantity of a particle, a body, or a system which is assumed to be a property of the particle that existed prior
to the measurement, and independently whether we measure it or not. In contrast to this, in quantum physics – a set of particles can be prepared to have exactly the same property from a certain point
of view, still they can show up quite different measurement results, from the point of view of another property.
The simplest example is connected with light polarization, or stated in quantum language: photon polarization. According to quantum mechanics if a plane transversal wave is polarized, and its plane
of polarization makes an angle \(θ\) with another polarization direction, then the photons in the beam are polarized also in that latter direction with a probability amplitude \(cosθ\). This means,
for instance, that if the polarization makes an angle \(θ\) with the horizontal direction and therefore an angle \(π/2−θ\) with the vertical direction, then the corresponding probability amplitudes
are \(cosθ\) and \(cos(π/2−θ)=sinθ\), respectively. According to the fundamental role of quantum physics the probabilities are in turn the (absolute value) squares of the amplitudes. So the
probability of finding a single photon of the original beam to be polarized horizontally is \(cos^{2}θ\) while the probability of being vertically polarized is \(sin^{2}θ\). This law is easily
verified already with a classical plane wave field of intensity \(I_{0}\) which is polarized so that it makes an angle \(θ\) with the horizontal direction. The beam is directed to a calcite crystal
which separates it into two beams so that one of them is polarized horizontally the other one vertically – these are the eigendirections of the calcite – and the two beams are distinct, so that they
can be checked individually. Then the intensity of the horizontally polarized beam after the calcite will be \(I_{0}cos^{2}θ\), while that of the vertically polarized beam will be \(I_{0}sin^{2}θ
\). As the intensity is proportional to the number of photons, this is a proof of the probability law. Present day technique allows also to check the probabilities themselves, because photons can
also be detected one by one by sensitive detectors.
It will be important that we can set the calcite so that its eigendirections are not horizontal and vertical, which we shall call setting A later on, but also in a way, say B or C, so that the two
eigendirections are different from those of the setting A. Let us note, however, that the two eigendirections are always orthogonal to each other.
The following question arises: what determines the final polarization direction of the photon. According to quantum mechanics the photon becomes vertical, or horizontal during the interacion with the
measurement apparatus. This is not encoded in the incoming particle, therefore it is principally impossible to answer the question with certainty, most we can say are the probabilities of the two
different possible outcomes. Stated in another way, the measurement does not establish a previously existing property of the photon, rather it is the measurement itself that creates the property with
the appropriate probability.
This answer, however, does not satisfy everybody, because one can imagine a different answer, as well, namely assuming that for each photon both properties did exist before the measurement, i.e. it
was polarized in say \(45^{\circ}\) and at the same time it had the property of being horizontally polarized, if we obtained the latter result during the measurement. It is another question that
quantum mechanics does not give account of both properties simultaneously, as it cannot give with unit probability – i.e. with complete certainty – the direction of polarization of a single photon
for two non-parallel or nonorthogonal directions. This would mean then that there exists a kind of a theory, deeper than quantum mechanics, according to which these properties are present
simultaneously and exactly in the measured object. From the point of view of quantum theory the parameter that would give the result with certainty is not present, so it is a hidden parameter as it
is called in these theories.
As quantum mechanics does not give account of these parameters, it cannot be a complete description of physical reality. This was the point of view of A. Einstein, and this was exposed in the most
perplexing way in the famous paper by A. Einstein, B. Podolsky and N. Rosen in 1935: ‟Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” In the end of the work they came
to the conclusion that quantum mechanics is not complete, but they add that such a theory shall appear in the future.
In the beginning the authors of the EPR paper try to give an exact definition what they consider a complete theory:
“A theory is complete if every element of physical reality has its counterpart in the physical theory.”
But what is an element of physical reality? They give the following answer:
“If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical
reality corresponding lo this physical quantity.”
Einstein, Podolsky and Rosen (EPR) present the description and the analysis of an imagined experiment, (so called Gedankenexperiment) performed on a pair of quantum particles, which according to the
authors shows that quantum mechanics is not a complete theory in the sense they require it. In their example they consider the measurement of the positions and the momenta of a pair of particles,
which emerge in a disintegration process. Instead of that it is simpler to consider another variant of the experiment, when it is done with a pair of two-state systems, as there are only two possible
results in a single measurement apparatus, instead of the infinitely many possible outcomes when measuring position or/and momenta. Such a variant of the EPR paradox was proposed by David Bohm in
1957 with pairs of spin 1/2 particles, or what we shall analyze, with photons having two orthogonal polarization eigenstates.
The essential features of the experiment, performed many times since Bohm’s proposal, are the following. A special source generates pairs of photons, such that the members of the pair propagate in
different spatial directions, say, one to the left, the other to the right. The polarization properties of the members are measured independently. One places two polarization measurement devices
perpendicularly to the propagation of the beams. These are e.g. calcite crystals in the paths of the two photons which pass through them, and measure the polarization properties of the photons. The
devices can be set making different angles with the horizontal direction. The possible settings of the devices will be denoted by different capital letters \(A,B,C\) and the two corresponding
eigenstates of the devices by \(A+\) and \(A−\); \(B\)+ and \(B−\); and \(C+\) and \(C−\) respectively. With an appropriate photon source one can achieve that there is a strict anticorrelation in the
measurement between the members of a pair, which means the following. Assume that the devices are set identically, on both sides of the source. Then both photons pass through the same type of device,
say horizontal-vertical, to be denoted by A. One observes then that if one of the photons is polarized horizontally \(A−\) on the left, its pair turns always out to be vertically polarized \(A+\), on
the right, or vice versa. The same thing happens if both polarizers are set in any direction say B which is different from A, but identical for both photons. If \(B+\) denotes \(60^{\circ}\) from the
horizontal and \(B−\) means \(−30^{\circ}\) from the horizontal, then if one of the photons comes is polarized in \(B+\), its pair will be \(B−\), or the other way round. Thus the polarization states
of a given member are always orthogonal to each other. Such pairs of particles are called EPR pairs. This anticorrelation of the pair follows from the way one creates them, which will be detailed
An EPR state of a photon pair is:
\(\psi=\frac{1}{\sqrt{2}}\left\{(A+)_{1}(A-)_{2}-(A-)_{1}(A+)_{2}\right\}\) (12.1)
where subscript labels the particle: 1 goes to left and 2 to the right. The first term in the state above says that photon 1 is polarized vertically, while photon 2 is polarized horizontally. The
second term means all this the other way round. What is important, quantum mechanics allows the superposition of the two possibilities. We have already encountered a similar state when we considered
the spin singlet state of the two electrons in the \(H_{2}\) molecule in Chapter 7.
Show that the state (12.1) above cannot be written as a product of two one-particle states.
Figure 12.1: Two possible eigenstates of the polarizer A are denoted by \(A+\) and \(A−\). The same notation is used in case of polarizer BB, the two eigenstates are \(B+\) and \(B−\). There is a
strict anticorrelation between the outcomes on the two sides, if the devices are set identically
But then it is sufficient to measure only on one photon of the pair. We may think that we know the result for the other member, even without doing the measurement, because it will be always opposite
to the one for its partner. Therefore without disturbing it, we can predict its value, thus it is the element of reality according to EPR. One has to add here, that it is impossible that we measure a
value \(A+\) on the left because we measured \(A−\) on the right. The propagation of this information needs time, but the two events can have a space-like separation (in the sense of relativity
theory), which means that only a signal faster than light could influence the result of the measurement on the other side, depending on the result of one of the sides. This is the locality principle
emphasized so much by Einstein’s theory of relativity. To conclude: the polarization property of the unmeasured photon is the element of reality, and a complete theory must assign a well defined
value to it in the sense defined by EPR.
But there is even more than that: in the case of the EPR pair we can tell the state of a particle exactly even from the point of view of two incompatible devices, what is impossible in quantum
mechanics, in principle. Let us put two different devices on the two sides.
Figure 12.2:
If we measure on the left side the property A (physical quantity), then we also know the state of its pair on the right from the point of view of A, due to the perfect anticorrelation. It will be
perpendicular to the direction measured on the left. But at the same time on the right we may measure with another type of apparatus, that measures the property B. In this way we can state both
properties of the given particle, one of them follows from the measured value on its mate on the left, the other one is measured actually. The same is true for its partner. This would mean that both
properties are well defined in the case of a single particle, while according to QM this is impossible, if A and B have different eigendirections as seen in figure 12.2. In quantum mechanics only the
probability amplitude and a corresponding probability is given if A and B make an angle different from \(0^{\circ}\) or \(90^{\circ}\). This reasoning led EPR to the conclusion that QM, which does
not give account for two incompatible elements of physical reality, cannot be complete. In contrast to the statement by EPR, N. Bohr argued that the two particles are part of a single unseparable
quantum system, according to the present wording they are in an entangled state, therefore a measurement on one of its parts immediately influences its other part, even if they are very far from each
other let the two measurements two spacelike events in the sense of special relativity. But this statement contradicted to one of the most fundamental principles of physics, locality. Therefore
Einstein could not accept Bohr’s arguments and the debate and the paradox remained unsolved for about 30 years.
Bell-inequalities with photons
In order to decide the question, John Bell proposed an explicit experimental arrangement in 1964. One has to measure not two but three different quantities (orthogonal polarizations) A, B, and C for
the photon pairs, so that the three possible settings of the polarizers are chosen randomly and independently from each other. It turns out, that based on the experimentally measured number of pairs
one can decide whether quantum mechanics, or the Einstein hypothesis is correct, the latter saying that the particles must have well defined polarizations in different directions simultaneously. The
original idea of Bell will be presented here as it was discussed by Eugene Wigner.
Figure 12.3:
Assume that the members of the pair had a well defined polarization before the measurement: \(+\) or \(−\) in each of the three directions A, B, and C. As the measurements show, in a single pair the
polarization of the two particles are always orthogonal to each other, shown up explicitly if we use the same settings on both sides. The possible 8 types of the pairs are shown in table 12.1, and
let us denote the measured number of a given type of pairs by \(N_{k}\).
Table 12.1:
Let us consider now the pairs for which the particle going to left resulted in \(A+\), while the one going to right came out as \(C+\). These events will of course be present only if the device on
the left was set to A while the one at right was set to C. The number of such pairs shall be denoted here by \(N(A+,C+)\), and according to the table above \(N(A+,C+)=N_{2}+N_{4}\). Similarly the
number of pairs for which we got A+A+ on the left and B+B+ on the right the result is \(N(A+,B+)=N_{3}+N_{4}\). Finally the number of those where we had \(B+\) on the left and \(C+\) on the right was
According to the simple inequality
\(N_{2}+N_{4} \leq\left(N_{3}+N_{4}\right)+\left(N_{2}+N_{6}\right)\) (12.2)
valid because by definition all the numbers \(N_{i}\) are nonnegative integers, we get:
\(N(A+, C+) \leq N(A+, B+)+N(B+, C+)\) (12.3)
It is important that the number of pairs \(N(A+,C+),N(A+,B+),N(B+,C+)\) can be measured, and the experimental result can be compared with the inequality obtained above. Before going on, however, we
reformulate the inequality (12.3) into probabilities, in order to compare it with quantum mechanics, which give its results in terms of probabilities.
Let \(P(A+,C+)\) the probability, that by choosing randomly the directions we set on the left the device to A, and it measured a result \(A+\), while the measurement on the right particle happened
again by random choice in the direction C, and resulted in \(C+\). Then
\(P(A+, C+)=\frac{N(A+, C+)}{\sum_{i=1}^{8} N_{i}}\) (12.4)
and similarly \(P(A+, B+)=\frac{N(A+, B+)}{\sum_{i=1}^{8} N_{i}}, P(B+, C+)=\frac{N(B+, C+)}{\sum_{i=1}^{B} N_{i}}\), if we made sufficiently many measurements. Accordingly the inequality (12.3) can
be written as
\(P(A+, C+) \leq P(A+, B+)+P(B+, C+)\) (12.5)
This is a Bell inequality, which – as we must remember – was derived by using the assumption that the pairs had well defined properties before the measurement was performed. There were no quantum
mechanical arguments in obtaining it.
So let us look at what QM says about the probabilities occurring in (12.5) as shown in the figure below. In a real measurement one has identical crystals on both sides, and they are rotated randomly
and independently of each other in three directions A, B and C. If on the left side we measure \(A+\) for instance, then its pair should be in state \(A−\). But we measure say C on it, and the result
measured on the partner can be either \(C+\) or \(C−\). The probability amplitude of getting the result that this other particle is polarized in the direction \(ê_{θ}\) is \(cosθ\), where \(θ\) is
the angle with \(A−\). The corresponding probability is thus \(cos^{2}θ\). Or we wish to express it with the angle made with \(A+\), which is \(α=π/2−θ\), then the probability is \(sin^{2}α\).
Choosing randomly the possible three directions on both sides with equal probabilities \(1/3\), the probability of all the possible settings of the pair of the apparatuses equals to \(1/9\). The
probability of obtaining say A+A+ on the left before we measured on the right is \(1/2\), then with a given setting of both crystals the probability is e.g. \(P(A+, C+)=\frac{1}{18} \sin ^{2}(A+, C+)
\), where \((A+,C+)\) denotes here the angle between the directions \(A+\) and \(C+\).
Figure 12.4: Violation of Bell inequality
Let us choose specifically the directions shown in the figure, i.e. let the eigendirections of A, B and C be rotated consecutively by \(30^{\circ}\). In other words we choose \((A+,B+)=(B+,C+)=30^{\
circ}\) and \((A+,C+)=60^{\circ}\). Then the quantum mechanical probabilities give the following result:
\(P(A+, C+)=\frac{1}{18} \sin ^{2} 60^{\circ}, \quad P(A+, B+)=\frac{1}{18} \sin ^{2} 30^{\circ}, \quad P(B+, C+)=\frac{1}{18} \sin ^{2} 30^{\circ}\) (12.6)
If we substitute these probabilities into (12.5) we observe that these results do not obey it, because that would require the fulfilment of \(\sin ^{2} 60^{\circ} \leq 2 \sin ^{2} 30^{\circ}\), i.e.
the inequality
\(\frac{3}{4} \leq \frac{1}{2}\) (12.7)
which is obviously false.
This means that by choosing appropriate directions for the crystals the corresponding quantum mechanical probabilities violate the Bell inequality. This turns out to be then an experimental
possibility to decide whether quantum mechanics or Bell inequalities are valid in the real physical world. According to the experiments measuring directly the numbers \(N(A+,C+)\) etc. it turned out
that the Bell inequalities are not valid for appropriately chosen directions A, B, and C as above but the results are in agreement with the predictions of QM.
All this should mean that in the derivation of the Bell inequalities must be something that contradicts to what is in the real physical world. There are two possibilities of errors. One of them could
have been that filling out the table 12.1 we assumed that a particle, and its partner possessed two, (actually three) different polarization properties already before the measurement, they did exist
in them independently of the measurement that was performed on them later. The other possible error could be that there is a nonlocal communication between the two members of a pair, i.e. the
measured state of one of the particles depends on the measurement on its distant partner. (However one can show, that information cannot be transmitted in this way with a speed faster than light,
because information transfer would necessitate an additional classical communication channel, and the speed of the transfer is determined by the velocity of the classical signal. That means that
locality is not violated in this sense). Both possibilities, as reasons of the discrepancy, contradict to traditional concepts about the natural world. In the first case it contradicts to the
assumption that all the possible properties of a particle, including the incompatible properties, have a well defined value before and independently of the measurement. This assumption is usually
called as realism. The second possibility contradicts to locality, in the sense that the measurement result influences instantaneously a distant, spatially separated other measurement.
Experiments and Bell inequalities
The first experiments about validity or violation of the Bell inequalities was performed by J. F. Clauser and co-workers in 1972. A later experiment by A. Aspect (1982) was the first, where the
settings of the two crystals were made so that the separation of these two events was space-like, i.e. a light signal imagined to start from one of the crystals at the time of its measurement could
not reach the other one before it measured the state of the other particle.
Here we show the setup of the experiment of A. Zeilinger (1995) where the pair of photons emerged from a nonlinear crystal.
Figure 12.5: Generation of an entangled photon pair. (In the figure colours are fictitious, green colour corresponds to the infrared photons of wavelength 702 nm.)
When the photons of an UV laser of wavelength 351 nm pass through the nonlinear crystal, a small part of them are split into two photons with smaller energy and thus with smaller frequency. This
process is called parametric down conversion in nonlinear optics. The emerging two beams leave the crystal along the superficial of two cones, satisfying the energy and momentum conservation. Among
the pairs some will share equally the original energy and momentum, their wavelength will be identically 702 nm (this is called the degenerate case). With an appropriate setting of the crystal one
can achieve that the polarization of the members are always orthogonal to each other, as assumed for the EPR pair in the discussion above. In this latter case the angle at the apices of the two cones
are identical and along their intersection – at the two green spots in the simulated figure on the right hand side of figure 12.5 – the photon pairs will have just the required property.
Downloading and running the exe file we can chose from three simulations. The first one demonstrates the violation of Bell inequality. The second one shows quantum teleportation, while the third is a
realization of the BB84 quantum key distribution (QKD) protocol (not discussed here).
Experiments unanimously proved the validity of the quantum mechanical result and the violation of the Bell inequalities. | {"url":"https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introduction_to_the_Physics_of_Atoms_Molecules_and_Photons_(Benedict)/01%3A_Chapters/12%3A_A_quantum_paradox_and_the_experiments","timestamp":"2024-11-10T14:35:28Z","content_type":"text/html","content_length":"153792","record_id":"<urn:uuid:d1896873-4689-4076-8120-92f2658dd364>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00238.warc.gz"} |
tail heavy
When the probability (or observed frequency) of data items far from the central value is large. This can be true of a symmetric distribution such as the Cauchy distribution, but is more often
associated with asymmetric long-tail distributions especially the power-law distribution found in areas such as network data.
Used on page 51 | {"url":"https://alandix.com/glossary/hcistats/tail%20heavy","timestamp":"2024-11-06T18:15:46Z","content_type":"application/xhtml+xml","content_length":"9274","record_id":"<urn:uuid:a67270e4-f3fc-4e2e-a7dc-fb7a921ebc4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00862.warc.gz"} |
LPI E - Exam Review 2.12 - Files
Unleashing the Linux Terminal
Embarking on a Journey of Discovery
Review of Concepts
Comprehending the concepts and commands related to managing files and directories in Linux is crucial for becoming a proficient Linux user.
• These skills enable efficient organization, navigation, and manipulation of data within the system.
• Understanding commands such as ls, cp, mv, rm, touch, find, and cat empowers users to list directory contents, copy and move files, remove unwanted files and directories, create empty files,
search for specific files, and view file contents.
• Mastering these concepts enhances one's ability to navigate the command line interface, perform file operations, and optimize workflow efficiency in Linux environments.
Question 1:
What are the fundamental building blocks of the Linux system for organizing files?
A) Folders and subfolders
B) Directories and files
C) Files and subfiles
D) Folders and files
Question 2:
Which of the following commands is used to create directories in Linux?
A) rm
B) cp
C) mkdir
D) touch
Question 3:
Which command is used to list the contents of a directory in Linux?
A) ls
B) cd
C) rm
D) mv
Question 4:
Which command is used to copy files and directories in Linux?
A) cp
B) mv
C) rm
D) touch
Question 5:
Which command is used to remove files and directories in Linux?
A) cp
B) mv
C) rm
D) mkdir
Question 6:
Which command is used to move or rename files and directories in Linux?
A) cp
B) mv
C) rm
D) touch
Question 7:
Which command is used to search for files and directories in Linux?
A) ls
B) cd
C) find
D) grep
Question 8:
Which command is used to print the current working directory in Linux?
A) ls
B) pwd
C) cd
D) echo
Question 9:
Which command is used to create an empty file in Linux?
A) rm
B) cp
C) touch
D) mkdir
Question 10:
Which command is used to display the contents of a file in Linux?
A) cat
B) ls
C) pwd
D) rm
Answer to Question 1:
Answer: B) Directories and files
The correct answer is B) Directories and files. In Linux, directories are special files used to organize files and other directories. Files are collections of data with names and attributes. While
other operating systems may use terms like "folders" and "subfolders," Linux uses "directories" and "files" to represent the basic organizational units.
Incorrect answer explanations:
A) Folders and subfolders: The term "folders" is commonly used in other operating systems, but in Linux, the correct term is "directories." Additionally, "subfolders" is not the preferred term in
Linux; it's more appropriate to refer to them as "subdirectories."
C) Files and subfiles: The term "subfiles" does not exist in the context of file organization on Linux systems.
D) Folders and files: While "folders" is a commonly used term, the official term in Linux is "directories."
Answer to Question 2:
Answer: C) mkdir
The correct answer is C) mkdir. The mkdir command is used in Linux to create directories. By providing one or more directory names as arguments, you can create multiple directories simultaneously.
Incorrect answer explanations:
A) rm: The rm command is used for removing files and directories, not for creating them.
B) cp: The cp command is used for copying files and directories, not for creating directories.
D) touch: The touch command is used to create or modify file timestamps, not directories.
Answer to Question 3:
Answer: A) ls
The correct answer is A) ls. The ls command is used to list the contents of a directorAnswer: A) lsy in Linux. When executed without any arguments, it displays the names of files and directories in
the current directory.
Incorrect answer explanations:
B) cd: The cd command is used to change the current directory, not to list its contents.
C) rm: The rm command is used for removing files and directories, not for listing their contents.
D) mv: The mv command is used for moving or renaming files and directories, not for listing their contents.
Answer to Question 4:
Answer: A) cp
The correct answer is A) cp. The cp command is used to copy files and directories in Linux. It creates a copy of the source file or directory and places it in the specified destination.
Incorrect answer explanations:
B) mv: The mv command is used to move or rename files and directories, not to copy them.
C) rm: The rm command is used for removing files and directories, not for copying them.
D) touch: The touch command is used to create or modify file timestamps, not to copy files and directories.
Answer to Question 5:
Answer: C) rm
The correct answer is C) rm. The rm command is used to remove files and directories in Linux. By providing the name of a file or directory as an argument, you can delete it from the system.
Incorrect answer explanations:
A) cp: The cp command is used to copy files and directories, not to remove them.
B) mv: The mv command is used to move or rename files and directories, not to remove them.
D) mkdir: The mkdir command is used to create directories, not to remove them.
Answer to Question 6:
Answer: B) mv
The correct answer is B) mv. The mv command is used to move or rename files and directories in Linux. It can be used to move a file or directory from one location to another or to rename a file or
Incorrect answer explanations:
A) cp: The cp command is used to copy files and directories, not to move or rename them.
C) rm: The rm command is used for removing files and directories, not for moving or renaming them.
D) touch: The touch command is used to create or modify file timestamps, not to move or rename files and directories.
Answer to Question 7:
Answer: C) find
The correct answer is C) find. The find command is used to search for files and directories in Linux. It allows you to specify various search criteria, such as the name of the file or directory, its
size, or its modification time.
Incorrect answer explanations:
A) ls: The ls command is used to list the contents of a directory, not to search for files and directories.
B) cd: The cd command is used to change the current directory, not to search for files and directories.
D) grep: The grep command is used to search for patterns within files, not to search for files and directories themselves.
Answer to Question 8:
Answer: B) pwd
The correct answer is B) pwd. The pwd command is used to print the current working directory in Linux. It displays the full path of the directory you are currently in.
Incorrect answer explanations:
A) ls: The ls command is used to list the contents of a directory, not to print the current working directory.
C) cd: The cd command is used to change the current directory, not to print its path.
D) echo: The echo command is used to display text on the command line, not to print the current working directory.
Answer to Question 9:
Answer: C) touch
The correct answer is C) touch. The touch command is used to create an empty file in Linux. If the file already exists, its modification timestamp is updated.
Incorrect answer explanations:
A) rm: The rm command is used for removing files and directories, not for creating them.
B) cp: The cp command is used to copy files and directories, not to create empty files.
D) mkdir: The mkdir command is used to create directories, not empty files.
Answer to Question 10:
Answer: A) cat
The correct answer is A) cat. The cat command is used to display the contents of a file in Linux. When executed with a file name as an argument, it prints the content of the file to the terminal.
Incorrect answer explanations:
B) ls: The ls command is used to list the contents of a directory, not to display the contents of a file.
C) pwd: The pwd command is used to print the current working directory, not to display file contents.
D) rm: The rm command is used for removing files and directories, not for displaying their contents.
The Story
Introduction: A Quest for Understanding
In the heart of the Linux Terminal's labyrinth, a courageous adventurer finds themselves surrounded by a myriad of possibilities and the enigmatic language of command line tools. Undeterred, they
embark on a quest to unravel the secrets of file and directory management, keen to unlock the hidden instructions that lead to their ultimate goal—the exit from this intricate maze. With unwavering
enthusiasm, our intrepid explorer dives into the cryptic world of the Linux Terminal, ready to conquer each challenge that lies ahead.
Chapter 1: Files and Directories - Unveiling the Foundations
As our adventurer takes their first steps, they encounter the fundamental building blocks of the Linux system—the files and directories. Files, containing data and adorned with unique attributes,
become their focal point. From transferring photos with descriptive names to organizing valuable data, comprehending the essence of a file becomes paramount. The adventurer marvels at the attributes
that accompany each file, such as the timestamps marking access and modification—a testament to its journey through time.
Meanwhile, directories emerge as a crucial tool for organizing files. Drawing parallels to file folders in a cabinet, directories allow for the seamless nesting of files within files. Our adventurer
embraces this concept, realizing the power of creating a structured hierarchy that brings order to the chaos.
Chapter 2: The Command Line - Unleashing Efficiency
In their quest for mastery, our enthusiast discovers the command line—the most potent weapon for file management on a Linux system. Unlike graphical file managers, the shell and its command line
tools bestow upon our adventurer a multitude of features, rendering their tasks faster and easier. With each command typed, they venture further into the realm of expertise, propelled by their
growing excitement.
Chapter 3: Command Line Tools - Unlocking Possibilities
Armed with newfound knowledge, our brave explorer dives into a repertoire of essential command line tools. The command "ls" guides them through the exploration of directories, revealing their
contents. "Mv" and "cp" become their trusted allies for moving and copying files, while "pwd" unveils their current location within the labyrinth. The power of "find" assists in the search for
specific files and directories, while "touch" allows for the creation and modification of file timestamps.
Chapter 4: Deleting and Renaming - Taming the Labyrinth
In their pursuit of order, our adventurer learns the art of file deletion and renaming. With caution, they wield the command "rm" to delete files and directories, and the limitations of "rmdir" when
handling non-empty directories. As they navigate the intricate paths, they become aware of the potential dangers of the "rm -r" command, which holds the power to obliterate directories and their
contents. A newfound respect for the need to proceed with care engulfs our intrepid explorer.
Chapter 5: Globbing and Character Classes - Decoding the Patterns
Amidst the twists and turns, our enthusiast stumbles upon the captivating world of globbing—a pattern matching language that unveils new possibilities. Characters like "*", "?", and "[]" take on new
meanings, enabling the selection of files based on specific patterns. With each experiment, our explorer gains a deeper understanding of the Linux system's intricacies, confidently maneuvering
through its paths.
Further expanding their expertise, our adventurer embraces character classes—powerful tools that empower them to refine their file selection. From the versatile [:alnum:] class encompassing letters
and numbers to the precise [:punct:] class identifying punctuation characters, the adventurer embraces the ability to select files based on diverse attributes. As they uncover the immense potential
of character classes, a new dimension of control opens up before them.
Conclusion: A Gateway to Endless Potential
As our valiant adventurer navigates the labyrinthine Linux Terminal, their progress is marked by a growing sense of understanding and accomplishment. With each challenge conquered and puzzle solved,
the cryptic language of the Linux Terminal begins to reveal its secrets. Their enthusiasm remains unbridled, fueled by an insatiable thirst for knowledge and a deepening mastery of the Linux system.
Step into this captivating world, where the Linux Terminal becomes a realm waiting to be explored—a gateway to new possibilities and endless potential. Join our intrepid explorer on this exhilarating
journey, and let the Linux Terminal unlock its wonders for you. | {"url":"https://www.certificationmethods.com/2023/06/lpi-e-exam-review-212-files.html","timestamp":"2024-11-07T00:22:14Z","content_type":"text/html","content_length":"384918","record_id":"<urn:uuid:32262094-59c6-45c5-8578-65c5fa2ecb31>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00869.warc.gz"} |
Mathematical Methods of Physics - Wikibooks, open books for an open world
Mathematical methods of Physics is a book on common techniques of applied mathematics that are often used in theoretical physics. It may be accessible to anyone with beginning undergraduate training
in mathematics and physics. It is hoped that the book will be useful for anyone wishing to study advanced Physics.
Vector Calculus
Complex Variables
Fourier Analysis
Hilbert Spaces
Green's Functions
Sturm-Liouville Theory
Cartesian Tensors | {"url":"https://en.m.wikibooks.org/wiki/Mathematical_Methods_of_Physics","timestamp":"2024-11-05T23:48:29Z","content_type":"text/html","content_length":"28355","record_id":"<urn:uuid:a0b9af75-fdc0-43d0-b05f-8c2fdade3e99>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00485.warc.gz"} |
Computed Radiation Imaging
Computer-assisted imaging with radiation (x- and gamma rays) is an integral part of modern medical-diagnostic practice. This imaging technology is also slowly finding its way into industrial
applications. Although the technology is well developed, there is a need for further improvement to enhance image quality, reduce artifacts, minimize patient radiation exposure, compete with and
complement other imaging methods (such as magnetic resonance imaging and ultrasonics), and accommodate dense and large objects encountered in industrial applications.
Scientists and engineers, attempting to progress this technology, are faced with an enormous amount of literature, addressing the imaging problem from various view points. This book provides a single
source that addresses both the physical and mathematical aspects of the imaging problem in a consistent and comprehensive manner.
• Discusses the inherent physical and numerical capabilities and limitations of the methods presented for both the forward and inverse problems
• Provides information on available Internet resources and software
• Written in a manner that makes it readable by physicists, mathematicians, engineers and computer scientists – avoids, as much as possible, the use of specialized terminology without clear
introduction and definition
Table of Contents
Chapter 1. Radiation Imaging
Part I: The Forward Problem
Chapter 2. Radiation Transport
Chapter 3. Measurement Models
Chapter 4. Transmission
Chapter 5. Emission
Chapter 6. Scattering
Part II: The Inverse Problem
Chapter 7. Features
Chapter 8. Formulation
Chapter 9. Preprocessing of Measurements
Chapter 10. Matrix-Based Methods
Chapter 11. Functional Optimization
Chapter 12. Analytic Methods
Chapter 13. Probabilistic Methods
Chapter 14. Incomplete Problems
Chapter 15. Testing
Chapter 16. Post-Processing: Image Enhancement
Book Details
• Hardcover: 302 pages
• Publisher: Elsevier (June 2011)
• Language: English
• ISBN-10: 0123877776
• ISBN-13: 978-0123877772
Download [5.0 MiB]
You must be logged in to post a comment. | {"url":"https://www.wowebook.com/book/computed-radiation-imaging/","timestamp":"2024-11-14T10:46:05Z","content_type":"text/html","content_length":"43888","record_id":"<urn:uuid:c756b65c-bcc5-4eaf-9168-453d28c0f9ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00631.warc.gz"} |
Find Four Consecutive Integers with The Sum of 54? | TIRLA ACADEMY
Let the first integer be x
The second will be (x+1)
Third integer = (x+2)
Fourth = (x+3)
Their sum is = 54
x=12, (x+1)=13, (x+2)=14, (x+3)=15
12, 13, 14, and 15 are four consecutive integers whose sum is 54. | {"url":"https://www.tirlaacademy.com/2024/04/find-four-consecutive-integers.html","timestamp":"2024-11-03T00:25:43Z","content_type":"application/xhtml+xml","content_length":"311071","record_id":"<urn:uuid:0ba2e129-7434-4f64-841c-4145237da8bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00139.warc.gz"} |
Multiplication With 2 Digits Worksheet
Math, particularly multiplication, forms the cornerstone of many academic techniques and real-world applications. Yet, for many students, mastering multiplication can present a difficulty. To address
this hurdle, instructors and moms and dads have actually accepted an effective tool: Multiplication With 2 Digits Worksheet.
Introduction to Multiplication With 2 Digits Worksheet
Multiplication With 2 Digits Worksheet
Multiplication With 2 Digits Worksheet -
Using Worksheets to Help Students Practice Students should already be comfortable with the multiplication factors of number up to 10 prior to attempting two digit multiplication problems which are
concepts typically taught in kindergarten through second grades and it s equally important for third and fourth grade students to be able to prove they fully grasp the concepts of two digit
Step 1 Multiply the ones digit of the first 2 digit number at the top and the ones digit of the second 2 digit number together Write the number of ones below the line in the ones place Carry over any
tens and write them above the first 2 digit number in the tens place 5 x 9 45 Write the 5 below the line in the ones place
Importance of Multiplication Method Recognizing multiplication is crucial, laying a solid structure for innovative mathematical concepts. Multiplication With 2 Digits Worksheet supply structured and
targeted practice, cultivating a deeper comprehension of this fundamental arithmetic operation.
Development of Multiplication With 2 Digits Worksheet
2 Digit By Two Digit Multiplication Worksheets Times Tables Worksheets
2 Digit By Two Digit Multiplication Worksheets Times Tables Worksheets
The worksheets below require students to multiply 2 digit numbers by 2 digit numbers Includes vertical and horizontal problems as well as math riddles task cards a picture puzzle a Scoot game and
word problems 2 Digit Times 2 Digit Worksheets Multiplication 2 digit by 2 digit FREE
Multiply by 2 digits Scarey1 Member for 3 years 1 month Age 10 12 Level 5 Language English en ID 376941 22 09 2020 Country code BS Country Bahamas School subject Math 1061955 Main content
Multiplication 2013181 Multiply 2 digit by 2 digit numbers Share Print Worksheet Finish Multiply 2 digit by 2 digit numbers
From typical pen-and-paper workouts to digitized interactive layouts, Multiplication With 2 Digits Worksheet have actually evolved, satisfying varied knowing designs and choices.
Sorts Of Multiplication With 2 Digits Worksheet
Basic Multiplication Sheets Straightforward exercises focusing on multiplication tables, aiding students develop a solid math base.
Word Trouble Worksheets
Real-life circumstances integrated right into troubles, improving essential reasoning and application skills.
Timed Multiplication Drills Tests created to boost rate and precision, helping in quick psychological math.
Advantages of Using Multiplication With 2 Digits Worksheet
Two Digit By One Digit Multiplication Once Your Students Have Mastered Their Basic Multi
Two Digit By One Digit Multiplication Once Your Students Have Mastered Their Basic Multi
Multiplication Two Digits Loading ad Angeline Barrameda Member for 3 years Age 7 15 Level 3 4 Language English en ID 1108940 22 06 2021 Country code PH Interactive Worksheets For Students Teachers of
all Languages and Subjects Worksheets Worksheets Make Interactive Worksheets Browse Worksheets Wookbooks Workbooks
Welcome to The Multiplying 2 Digit by 2 Digit Numbers with Comma Separated Thousands A Math Worksheet from the Long Multiplication Worksheets Page at Math Drills This math worksheet was created or
last revised on 2016 08 31 and has been viewed 93 times this week and 767 times this month
Boosted Mathematical Skills
Consistent technique develops multiplication efficiency, enhancing total mathematics capacities.
Boosted Problem-Solving Talents
Word troubles in worksheets establish logical reasoning and strategy application.
Self-Paced Discovering Advantages
Worksheets fit specific understanding speeds, cultivating a comfortable and adaptable understanding atmosphere.
Exactly How to Produce Engaging Multiplication With 2 Digits Worksheet
Incorporating Visuals and Colors Lively visuals and colors capture attention, making worksheets visually appealing and engaging.
Including Real-Life Situations
Associating multiplication to daily situations adds significance and practicality to workouts.
Tailoring Worksheets to Different Ability Levels Tailoring worksheets based upon varying effectiveness levels makes sure comprehensive discovering. Interactive and Online Multiplication Resources
Digital Multiplication Tools and Gamings Technology-based sources supply interactive knowing experiences, making multiplication appealing and enjoyable. Interactive Websites and Apps Online systems
give diverse and easily accessible multiplication technique, supplementing typical worksheets. Tailoring Worksheets for Numerous Discovering Styles Visual Students Aesthetic aids and representations
help understanding for students inclined toward aesthetic learning. Auditory Learners Spoken multiplication issues or mnemonics cater to learners who comprehend principles via auditory ways.
Kinesthetic Students Hands-on tasks and manipulatives support kinesthetic learners in recognizing multiplication. Tips for Effective Implementation in Learning Uniformity in Practice Normal method
reinforces multiplication skills, advertising retention and fluency. Balancing Rep and Variety A mix of repetitive workouts and diverse trouble styles maintains passion and comprehension. Offering
Positive Responses Feedback aids in identifying locations of enhancement, encouraging ongoing development. Obstacles in Multiplication Practice and Solutions Inspiration and Engagement Difficulties
Tedious drills can cause uninterest; innovative approaches can reignite inspiration. Conquering Concern of Mathematics Negative assumptions around math can impede progression; creating a positive
understanding setting is essential. Effect of Multiplication With 2 Digits Worksheet on Academic Performance Studies and Research Study Searchings For Study shows a favorable correlation in between
regular worksheet usage and improved math efficiency.
Multiplication With 2 Digits Worksheet become flexible tools, promoting mathematical proficiency in students while fitting diverse discovering designs. From fundamental drills to interactive on-line
resources, these worksheets not just enhance multiplication abilities yet also promote vital reasoning and analytic capabilities.
Multiplication Problems 1 X 2 Digit No Regrouping Mr R s World Of Math
Three Digit Multiplication Worksheets Worksheets
Check more of Multiplication With 2 Digits Worksheet below
4 Digit Multiplication Worksheets Free Printable
Free Multiplication Worksheet 2 Digit By 2 Digit Free4Classrooms
Two Digit Multiplication Worksheets Multiplication Worksheets Two Digit multiplication Math
Two Digit Multiplication Worksheet Have Fun Teaching
Multiplication 2 Digits By 1 Digit Sheet 3 Worksheet For 3rd 4th Grade Lesson Planet
2 Digit Multiplication Worksheet
2 Digit Multiplication Worksheet Math Salamanders
Step 1 Multiply the ones digit of the first 2 digit number at the top and the ones digit of the second 2 digit number together Write the number of ones below the line in the ones place Carry over any
tens and write them above the first 2 digit number in the tens place 5 x 9 45 Write the 5 below the line in the ones place
Multiply 2 x 2 digits worksheets K5 Learning
Multiply 2 x 2 digits 2 digit multiplication Multiplication practice with all factors being under 100 column form Worksheet 1 Worksheet 2 Worksheet 3 Worksheet 4 Worksheet 5 10 More Similar Multiply
3 x 2 digits Multiply 3 x 3 digits What is K5
Step 1 Multiply the ones digit of the first 2 digit number at the top and the ones digit of the second 2 digit number together Write the number of ones below the line in the ones place Carry over any
tens and write them above the first 2 digit number in the tens place 5 x 9 45 Write the 5 below the line in the ones place
Multiply 2 x 2 digits 2 digit multiplication Multiplication practice with all factors being under 100 column form Worksheet 1 Worksheet 2 Worksheet 3 Worksheet 4 Worksheet 5 10 More Similar Multiply
3 x 2 digits Multiply 3 x 3 digits What is K5
Two Digit Multiplication Worksheet Have Fun Teaching
Free Multiplication Worksheet 2 Digit By 2 Digit Free4Classrooms
Multiplication 2 Digits By 1 Digit Sheet 3 Worksheet For 3rd 4th Grade Lesson Planet
2 Digit Multiplication Worksheet
Multiplying 2 Digit By 1 Digit Numbers A
3 Digit X 2 Digit Multiplication Worksheets Schematic And Wiring Diagram
3 Digit X 2 Digit Multiplication Worksheets Schematic And Wiring Diagram
2 Digit By 2 Digit Multiplication Worksheets With Answers Free Printable
Frequently Asked Questions (Frequently Asked Questions).
Are Multiplication With 2 Digits Worksheet suitable for any age groups?
Yes, worksheets can be tailored to different age and skill levels, making them adaptable for numerous learners.
How often should pupils practice making use of Multiplication With 2 Digits Worksheet?
Consistent practice is vital. Normal sessions, preferably a few times a week, can produce significant renovation.
Can worksheets alone boost mathematics abilities?
Worksheets are an important device but needs to be supplemented with diverse learning approaches for detailed skill growth.
Exist on-line systems offering cost-free Multiplication With 2 Digits Worksheet?
Yes, lots of instructional internet sites use free access to a wide range of Multiplication With 2 Digits Worksheet.
Exactly how can parents support their kids's multiplication method in your home?
Motivating consistent technique, supplying aid, and creating a favorable discovering atmosphere are advantageous steps. | {"url":"https://crown-darts.com/en/multiplication-with-2-digits-worksheet.html","timestamp":"2024-11-04T07:31:40Z","content_type":"text/html","content_length":"29087","record_id":"<urn:uuid:723f8c06-49a3-4a16-9dc2-9c8783f5c671>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00563.warc.gz"} |
Assignment Time Calculator - Calculator Wow
Assignment Time Calculator
The Assignment Time Calculator is a valuable tool designed to help students and professionals estimate the total time needed to complete an assignment based on the number of questions and the average
time per question. It streamlines planning efforts and assists in managing time effectively.
Understanding the estimated time required for completing assignments is crucial for several reasons:
1. Time Management: Helps individuals allocate sufficient time for each task within a given assignment.
2. Planning: Facilitates better planning by breaking down tasks into manageable time segments.
3. Productivity: Enhances productivity by providing a structured approach to task completion.
4. Deadline Adherence: Ensures assignments are completed on time by aligning task durations with deadlines.
5. Efficiency: Reduces stress by allowing individuals to prioritize and allocate time effectively.
How to Use
Using the Assignment Time Calculator is straightforward:
1. Input Total Number of Questions: Enter the total number of questions in the assignment.
2. Specify Average Time per Question: Input the average time required to answer each question.
3. Click Calculate: Hit the “Calculate Assignment Time” button to obtain the total estimated time for completing the assignment.
10 FAQs and Answers
1. Can I use this calculator for any type of assignment?
• Yes, it can be used for any assignment where you need to estimate total time based on questions.
2. How accurate are the results from this calculator?
• The accuracy depends on the inputs provided; it gives a reliable estimate based on average time per question.
3. Should I include breaks in my calculations?
• This calculator estimates continuous working time; adjust for breaks separately if needed.
4. Can I use decimals for average time per question?
• Yes, you can enter decimals to represent fractions of minutes or hours.
5. What if I have variable times for different types of questions?
• Estimate an average time that reflects the overall complexity of the assignment.
6. How can this calculator help in meeting deadlines?
• By knowing the total time required, you can plan and manage your workload more effectively.
7. Is there a limit to the number of questions this calculator can handle?
• No, it can calculate total assignment time for any number of questions you input.
8. Should I round off the average time per question?
• Use a reasonable estimate to get a practical total time; precision depends on your needs.
9. Can I save the results or share them with others?
• You can manually record or share the calculated total assignment time as needed.
10. How often should I use this calculator?
• Use it whenever you need to estimate time for assignments to maintain efficient time management practices.
The Assignment Time Calculator offers a practical solution for estimating the total time required to complete assignments based on the number of questions and average time per question. By leveraging
this tool, students and professionals can enhance their planning capabilities, improve time management skills, and ensure tasks are completed efficiently and on schedule. Embrace the efficiency of
this calculator to optimize your assignment workflow and achieve academic or professional success with greater ease. | {"url":"https://calculatorwow.com/assignment-time-calculator/","timestamp":"2024-11-06T02:42:54Z","content_type":"text/html","content_length":"64733","record_id":"<urn:uuid:f27143ea-b0e4-4397-8140-572d2f5aac7c>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00386.warc.gz"} |
Lasse Leskelä
Associate Professor
Aalto University
School of Science
Department of Mathematics and Systems Analysis
Otakaari 1, 02150 Espoo, Finland
Room Y242a
ORCID: 0000-0001-8411-8329 | ResearcherID: 0-8920-2019
YouTube channel
Docent, University of Jyväskylä, 2011
DSc, Helsinki University of Technology, 2005
MSc, Helsinki University of Technology, 1999
Research interests
My research area is stochastics, the mathematical theory of random events, systems, and processes. The main focus is statistical network models and random graphs, together with random dynamical
systems and stochastic processes acting on networks. The general objective is to derive fundamental mathematical laws for predicting the macroscopic behavior of large random systems and describing
the accuracy of statistical learning algorithms for large network models. Such laws allow to make predictions using massive high-dimensional data sets for which classical methods of statistics and
machine learning are often infeasible. Statistical network models under study include random intersection graphs, stochastic block models, and graphons. The archetypes of stochastic dynamics studied
are branching processes, random walks, and first passage percolation. The formulas and methods have a wide range of applications ranging from information and social networks to biological and
financial systems.
Short bio
I received the MSc and DSc degrees from Helsinki University of Technology, Finland, in 1999 and 2005, respectively. During 2006–2007 I worked abroad for two years, first at Columbia University in the
USA and then at Centrum Wiskunde & Informatica (CWI) and Eindhoven University of Technology in the Netherlands. After returning to Finland in 2008, I worked for three years at Aalto University and
then three more at the University of Jyväskylä, before returning to Aalto University in 2014.
Research projects and networks | {"url":"https://math.aalto.fi/~lleskela/","timestamp":"2024-11-10T11:28:32Z","content_type":"text/html","content_length":"6077","record_id":"<urn:uuid:e06f4984-272e-48f3-a0d0-9c8d95bc8794>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00819.warc.gz"} |
PPT - Ch 14 實習 (2) PowerPoint Presentation, free download - ID:5811998
2. Randomized Blocks (Two-way) Analysis of Variance • The purpose of designing a randomized block experiment is to reduce the within-treatments variation thus increasing the relative amount of
between treatment variation. • This helps in detecting differences between the treatment means more easily.
3. Randomized Blocks Block all the observations with some commonality across treatments
4. Sum of square for treatments Sum of square for blocks Sum of square for error Partitioning the total variability • The sum of square total is partitioned into three sources of variation •
Treatments • Blocks • Within samples (Error) Recall. For the independent samples design we have: SS(Total) = SST + SSE SS(Total) = SST + SSB + SSE
5. SSB= Calculating the sums of squares • Formula for the calculation of the sums of squares SST =
6. SSB= SST = Calculating the sums of squares • Formula for the calculation of the sums of squares
7. Mean Squares and Test statistics • To perform hypothesis tests for treatments and blocks we need • Mean square for treatments • Mean square for blocks • Mean square for error Test statistics for
treatments Test statistics for blocks
8. The F test Rejection Regions • Testing the mean responses for treatments F > Fa,k-1,n-k-b+1 • Testing the mean response for blocks F> Fa,b-1,n-k-b+1
11. Example 1 • A randomized block experiment produced the following statistics: k=5 b=12 SST=1500 SSB=1000 SS(Total)=3500 • a. Test to determine whether the treatment means differ. (Use =0.01) • b.
Test to determine whether the blocks means differ. (Use =0.01)
12. Solution 1 ANOVA table a. Rejection region: F>Fα,k-1,n-k-b+1=F0.01,4,44 ≒3.77 F=16.50 >3.77 There is enough evidence to conclude that the treatment means differ b. Rejection region: F>
Fα,b-1,n-k-b+1=F0.01,11,44 ≒2.67 Conclusion: F=4.00>2.67 There is enough evidence to conclude that the block means differ
13. Example 2 • As an experiment to understand measurement error, a statistics professor asks four students to measure the height of the professor, a male student, and a female student. The
differences (in centimeters) between the correct dimension and the ones produced by the students are listed here. Can we infer at the 5% significance level that there are differences in the
errors between the subjects being measured?
14. Solution 2 • H0: μ1= μ2= μ3 H1:At least two means differ Rejection region: F>Fα,k-1,n-k-b+1=F0.05,2,6 =5.14 K=3, b=4, grand mean=2.38 =7.3
16. Two-Factor Analysis of Variance • Example 2 • Suppose in Example 1, two factors are to be examined: • The effects of the marketing strategy on sales. • Emphasis on convenience • Emphasis on
quality • Emphasis on price • The effects of the selected media on sales. • Advertise on TV • Advertise in newspapers
17. Factor A: Marketing strategy Factor B: Advertising media Two-way ANOVA (two factors) Convenience Quality Price City 1 sales City3 sales City 5 sales TV City 2 sales City 4 sales City 6 sales
18. Interaction Difference between the levels of factor A, and difference between the levels of factor B; no interaction Difference between the levels of factor A No difference between the levels of
factor B M R e e s a p n o n s e M R e e s a p n o n s e Level 1 of factor B Level 1and 2 of factor B Level 2 of factor B Levels of factor A Levels of factor A 1 2 3 1 2 3 M R e e s a p n o n s e
M R e e s a p n o n s e No difference between the levels of factor A. Difference between the levels of factor B Interaction Levels of factor A Levels of factor A 1 2 3 1 2 3
19. Interaction B+ B+ B+ B- B- B- B- B+ Without interaction With interaction
20. Terminology A complete factorial experiment is an experiment in which the data for all possible combinations of the levels of the factors are gathered. This is also known as a two-way
classification. The two factors are usually labeled A & B, with the number of levels of each factor denoted by a & b respectively. The number of observations for each combination is called a
replicate, and is denoted by r. For our purposes, the number of replicates will be the same for each treatment, that is they are balanced.
21. Hypothesis • H0: Factor A and Factor B do not interact to affect the mean responses • H1: Factor A and Factor B do interact to affect the mean responses • H0: The means of the a levels of factor
A are equal • H1: At least two means differ • H0: The means of the b levels of factor B are equal • H1: At least two means differ
25. MS(AB) MSE MS(A) MSE MS(B) MSE F= F= F= F tests for the Two-way ANOVA • Test for the difference between the levels of the main factors A and B SS(A)/(a-1) SS(B)/(b-1) SSE/(n-ab) Rejection region:
F > Fa,a-1 ,n-ab F > Fa, b-1, n-ab • Test for interaction between factors A and B SS(AB)/(a-1)(b-1) Rejection region: F > Fa,(a-1)(b-1),n-ab
26. ANOVA Table n = abr
27. Example 3 • The following data were generated from a 2 X 2 factorial experiment with 3 replicates.
28. Example 3 - continued • a. Test at the 5% significance level to determine whether factors A and B interact. • b. Test at the 5% significance level to determine whether differences exists between
the levels of factor A. • c. Test at the 5% significance level to determine whether differences exist between the levels of factor B.
30. Solution 3 - continued • a F = .31, p-value = .5943. There is not enough evidence to conclude that factors A and B interact. • b F = 1.23, p-value = .2995. There is not enough evidence to
conclude that differences exist between the levels of factor A. • c F = 13.00, p-value = .0069. There is enough evidence to conclude that differences exist between the levels of factor B.
32. Example 4 • The required conditions for a two-factor ANOVA are that the distribution of the response is __________________ distributed; the variance for each treatment is ________; and the
samples are _______ . (a) normally; equal; independent (b) normally; the same; independent (c) normally; identical; independent
33. Multiple Comparisons • Two means are considered different if the difference between the corresponding sample means is larger than a critical number. Then, the larger sample mean is believed to be
associated with a larger population mean. • Conditions common to all the methods here: • The ANOVA model is the one way analysis of variance • The conditions required to perform the ANOVA are
34. Inference about m1– m2: Equal variances • Recall • Construct the t-statistic as follows: Build a confidence interval
35. Fisher Least Significant Different (LSD) Method • This method builds on the equal variances t-test of the difference between two means. • The test statistic is improved by using MSE rather than
sp2. • We can conclude that mi and mj differ (at a% significance level if > LSD, where
36. Experimentwise Type I error rate (aE)(the effective Type I error) • The Fisher’s method may result in an increased probability of committing a type I error. • The experimentwise Type I error rate
is the probability of committing at least one Type I error at significance level of a. It iscalculated by aE = 1-(1 –a)C where C is the number of pairwise comparisons (i.e. C = k(k-1)/2) • The
Bonferroni adjustment determines the required Type I error probability per pairwise comparison (a) ,to secure a pre-determined overall aE.
37. Bonferroni Adjustment • The procedure: • Compute the number of pairwise comparisons (C)[C=k(k-1)/2], where k is the number of populations. • Set a = aE/C, where aE is the true probability of
making at least one Type I error (called experimentwise Type I error). • We can conclude that mi and mj differ (at a/C% significance level if
38. Fisher and Bonferroni Methods • Example 1 - continued • Rank the effectiveness of the marketing strategies(based on mean weekly sales). • Use the Fisher’s method, and the Bonferroni adjustment
method • Solution (the Fisher’s method) • The sample mean sales were 577.55, 653.0, 608.65. • Then,
39. Fisher and Bonferroni Methods • Solution (the Bonferroni adjustment) • We calculate C=k(k-1)/2 to be 3(2)/2 = 3. • We set a = .05/3 = .0167, thus t.0167/2, 60-3 = 2.467 (Excel). Again, the
significant difference is between m1 and m2.
40. Tukey Multiple Comparisons • The test procedure: • Find a critical number w as follows: k = the number of treatments n =degrees of freedom = n - k ng = number of observations per sample (recall,
all the sample sizes are the same) a = significance level qa(k,n) = a critical value obtained from the studentized range table
41. If the sample sizes are not extremely different, we can use the above procedure with ng calculated as theharmonic mean of the sample sizes. Tukey Multiple Comparisons • Select a pair of means.
Calculate the difference between the larger and the smaller mean. • If there is sufficient evidence to conclude that mmax > mmin . • Repeat this procedure for each pair of samples. Rank the means
if possible.
42. Which Multiple Comparison Method to Use • If you have identified two or three pairwise comparison, use the Bonferroni method. • If you plan to compare all possible combinations, use Turkey. • If
the purpose of the analysis is to point to areas that should be investigated further, Fisher’s LSD method is indicated.
43. Example 5 • a. Use Fisher’s LSD procedure with =0.05 to determine which population means differ given the following statistics. • b. Repeat part a using the Bonferroni adjustment. • c. Repeat
part a using Tukey’s multiple comparison method
47. Example 6 • Which of the following statements about multiple comparison methods is false? a. They are to be use once the F-test in ANOVA has been rejected. b. They are used to determine which
particular population means differ. c. There are many different multiple comparison methods but all yield the same conclusions. d. All of these choices are true. | {"url":"https://www.slideserve.com/drew/ch-14-2","timestamp":"2024-11-13T06:12:15Z","content_type":"text/html","content_length":"76037","record_id":"<urn:uuid:9181bbcd-eb13-4b80-914c-43054ef01ec8>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00536.warc.gz"} |
Disjunction introduction
Jump to navigation Jump to search
Disjunction introduction or addition (also called or introduction)^[1]^[2]^[3] is a rule of inference of propositional logic and almost every other deduction system. The rule makes it possible to
introduce disjunctions to logical proofs. It is the inference that if P is true, then P or Q must be true.
An example in English:
Socrates is a man.
Therefore, Socrates is a man or pigs are flying in formation over the English Channel.
The rule can be expressed as:
${\displaystyle {\frac {P}{\therefore P\lor Q}}}$
where the rule is that whenever instances of "${\displaystyle P}$" appear on lines of a proof, "${\displaystyle P\lor Q}$" can be placed on a subsequent line.
More generally it's also a simple valid argument form, this means that if the premise is true, then the conclusion is also true as any rule of inference should be, and an immediate inference, as it
has a single proposition in its premises.
Disjunction introduction is not a rule in some paraconsistent logics because in combination with other rules of logic, it leads to explosion (i.e. everything becomes provable) and paraconsistent
logic tries to avoid explosion and to be able to reason with contradictions. One of the solutions is to introduce disjunction with over rules. See Paraconsistent logic § Tradeoff.
Formal notation[edit]
The disjunction introduction rule may be written in sequent notation:
${\displaystyle P\vdash (P\lor Q)}$
where ${\displaystyle \vdash }$ is a metalogical symbol meaning that ${\displaystyle P\lor Q}$ is a syntactic consequence of ${\displaystyle P}$ in some logical system;
and expressed as a truth-functional tautology or theorem of propositional logic:
${\displaystyle P\to (P\lor Q)}$
where ${\displaystyle P}$ and ${\displaystyle Q}$ are propositions expressed in some formal system.
1. ^ Hurley
2. ^ Moore and Parker
3. ^ Copi and Cohen | {"url":"https://static.hlt.bme.hu/semantics/external/pages/kett%C5%91s_tagad%C3%A1s/en.wikipedia.org/wiki/Disjunction_introduction.html","timestamp":"2024-11-09T16:12:12Z","content_type":"text/html","content_length":"46096","record_id":"<urn:uuid:f606ebac-ad65-41cb-8ff3-1349e0a5fb4e>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00110.warc.gz"} |
7 Steps to Solving a Quadratic Equation
For many students, solving a quadratic equation is a challenge. While the process may seem intimidating, it can be mastered with practice. This article will provide a straightforward guide to solving
a quadratic equation, with step-by-step instructions and helpful tips.
What is a Quadratic Equation?
A quadratic equation is an equation of degree two with two unknowns or variables. It is usually written in the form ax^2 + bx + c = 0, where a, b and c are constants and x is the unknown. It can be
further divided into three categories:
• Standard Form: ax^2 + bx + c = 0
• Factored Form: ax^2 + bx + c = (x-m)(x-n)
• Vertex Form: ax^2 + bx + c = a(x-h)^2+k
Step 1: Identify the Parts of the Equation
The first step in solving a quadratic equation is to identify the parts of the equation. Make sure to identify both the variables and constants. In standard form, the coefficients are usually written
as a, b and c, while the unknown is written as x. In factored form, the constants are known as m and n, and in vertex form, they are known as h and k.
Step 2: Rewrite the Equation in Standard Form
The next step is to rewrite the equation in standard form, if the equation is not presented in that form already. This can be done by combining like terms and rearranging the equation. In vertex
form, this is done by adding a and c to both sides, while in factored form, this is done by multiplying out the brackets.
Step 3: Use the Quadratic Formula to Solve for x
Once the equation has been written in standard form, it can be solved using the quadratic formula. This formula can be used to solve equations of degree two, and gives the two possible solutions to
the equation. The quadratic formula is expressed as x = (-b ± √(b^2 – 4ac))/2a, where a, b and c are the same constants from step 1.
Step 4: Check Your Answer
Once you have calculated the solutions for x, check them to make sure they are correct. To do this, substitute each solution into the equation and check if it satisfies it. If both solutions do
indeed solve the equation, then you have found your answers.
Step 5: Graphical Representation of the Solution
To gain a better understanding of the solution, it can be helpful to draw a graph. This process involves plotting the x-intercepts (the solutions for x) on an x-y graph. The graph should intersect
the x-axis at those points and should be shaped like a parabola, with its vertex at the maximum or minimum of the parabola.
Step 6: Using Completing the Square to Solve a Quadratic Equation
Sometimes it can be helpful to rephrase a quadratic equation into a more simple form before using the quadratic formula. This process is known as completing the square and involves rewriting it in
vertex form or factored form. Completing the square involves adding extra terms in order to turn it into a perfect square trinomial of the form ax^2 + bx + c = a(x+h)^2+k.
Step 7: Using Factoring to Solve a Quadratic Equation
A final way to solve a quadratic equation is to factor it into two separate equations and then solve for each one individually. Factoring a quadratic involves applying techniques such as grouping and
differences of squares. If you are able to factor the equation into two separate equations, you can then solve each one individually.
Tips for Solving Quadratic Equations
• Always start by rewriting the equation in standard form.
• Make sure you substitute your answer back into the original equation to check that it’s correct.
• Place special emphasis on learning how to complete the square.
• If possible, graph the solution on an x-y graph.
• Practice with simpler equations before attempting more complex ones.
Common Mistakes to Avoid when Solving Quadratic Equations
• Including constants that are not part of the equation: Many students make the mistake of including constants that are not part of the equation when calculating their solutions.
• Forgetting to take account of signs: Students often forget to take account of plus and minus signs when calculating solutions for x.
• Solving for b instead of x: When trying to solve an equation quickly, some students forget that they have to solve for x and instead solve for b.
• Thinking that there are always two solutions: Some students assume that all quadratic equations have two solutions when this is not necessarily true.
• Neglecting to factor in special cases: Special cases such as perfect squares should always be factored in.
With practice and patience, any student can learn how to solve a quadratic equation. By following our seven steps and avoiding common mistakes, you will be able to find solutions with confidence. | {"url":"https://mathemista.com/7-steps-to-solving-a-quadratic-equation/","timestamp":"2024-11-07T13:33:30Z","content_type":"text/html","content_length":"57470","record_id":"<urn:uuid:595de21f-77b2-484a-b635-219ccfa4d433>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00748.warc.gz"} |
Travel Forecasting Resource
Count Models
This page discusses why count models are necessary in certain applications, and discusses beginning details of the Poisson, negative binomial, and hurdle models.
# Continuous versus count outcomes
Typical regression models are aimed at predicting the response of an outcome variable to a series of input variables . The result is a linear equation of a vector that describes the relationship
between each element of and the outcome .
This regression framework assumes that is a continuous variable, meaning that it can take any numeric value within a particular range. The plot below shows the relationship between the average
distance between home and workplace for workers in the household on the axis, and the household VMT on the axis, for households in smaller cities who responded to the 2017 NHTS. Both of these
variables are continuous, meaning that a simple regression model is appropriate, though more information might need to be added to the model below to improve its fit and help explain outlying
observations or control for heteroskedasticity.
But consider the plot below, showing the same axis but with the number of home-based work trips produced by each household on the axis. Because the number of trips is discrete and not continuous, the
plot looks kind of funny. But more importantly than this, we want a model that will predict a discrete number of vehicles as an outcome variable, and the blue regression line we estimated below will
predict between 2 and 1 trips household; this isn't ideal.
# Poisson Model
A better option would be to predict a probability that each household will produce a certain discrete number of trips. One way to do this is with a Poisson regression model. In this model, an analyst
predicts the mean of a Poisson distribution with a regression equation (instead of a line). The Poisson distribution is:
where the probability of a discrete outcome is determined by the mean of the distribution. The plot below shows how as the mean increases, the probability of higher outcomes increases.
A Poisson regression model allows attributes of an observation to affect the value of the mean. So instead of in the linear regression, we now have , and gets put into the distribution equation
above. In the model below, average work distance decreases the average number of trips, but more workers and more vehicles increases the average number.
Linear Poisson
(Intercept) -0.106 -0.570***
(-1.625) (-13.865)
avg_workdist -0.010*** -0.006***
(-6.851) (-7.070)
wrkcount 1.168*** 0.665***
(29.000) (28.801)
hhvehcnt 0.147*** 0.089***
(5.626) (5.925)
Num.Obs. 5533 5533
R2 0.185
R2 Adj. 0.185
AIC 19111.6 17871.7
BIC 19144.7 17898.1
Log.Lik. -9550.813 -8931.830
F 419.097
Of course, this is just an average. In a trip-based model, this average for each household might be sufficient. But you could also simulate a discrete choice for each person. The plot below shows the
probability of a certain number of trips made by a sample of households, alongside what the predicted Poisson mean was. Households with a higher predicted mean have a higher probability of making
more trips.
# Negative Binomial Model
The Poisson model assumes that the mean and standard deviation of the distribution are the same. This can be a bad assumption, because it forces the distribution to spread out when the mean is
higher. The negative binomial model relaxes this assumption, and might be useful in some contexts.
# Hurdle Model
The Poisson and negative binomial models assume the same distribution across all outcomes; this might not be desirable if the number of zeros is high or low for some structural reason. For example,
owning zero vehicles is very different from owning one or two. A hurdle model breaks the distribution into two different components:
• A binomial model determines the probability of choosing zero versus a positive number.
• A poisson or negative binomial model (with zero removed) determines the probability of a specific positive number, conditioned on the previous model.
# References | {"url":"https://tfresource.org/topics/Count_Models.html","timestamp":"2024-11-03T09:03:00Z","content_type":"text/html","content_length":"76755","record_id":"<urn:uuid:022c3af5-186c-49dc-b2ae-f74b67e042f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00517.warc.gz"} |
Dget Formula in Excel
Dget formula in Excel Extracts a single value from a column of a list or database that matches conditions that you specify. How to use Dget formula in Excel How to use Dget formula in Excel DGET
(database, field, criteria) The DGET function syntax has the following arguments: Database Required. The range of cells that makes up the list or database. A database is a list of related data in
which rows of related information are records, and columns of data are fields. The first row of the list contains labels for each column. Field Required. Indicates which column is used in the
function. Enter the column label enclosed between double quotation marks, such as "Age" or "Yield," or a number (without quotation marks) that represents the position of the column within the list: 1
for the first column, 2 for the second column, and so on. Criteria Required. The range of cells that contains the conditions that you specify. You can use any range for the criteria argument, as long
as it includes at least one column label and at least one cell below the column label in which you specify a condition for the column. Remarks If no record matches the criteria, DGET returns the #
VALUE! error value. If more than one record matches the criteria, DGET returns the #NUM! error value. You can use any range for the criteria argument, as long as it includes at least one column label
and at least one cell below the column label for specifying the condition. For example, if the range G1:G2 contains the column label Income in G1 and the amount $10,000 in G2, you could define the
range as MatchIncome and use that name as the criteria argument in the database functions. Although the criteria range can be located anywhere on the worksheet, do not place the criteria range below
the list. If you add more information to the list, the new information is added to the first row below the list. If the row below the list is not blank, Microsoft Excel cannot add the new
information. Make sure that the criteria range does not overlap the list.
Download Practice File | {"url":"https://www.myelesson.org/excel/english/dget-formula-in-excel-454","timestamp":"2024-11-13T19:51:22Z","content_type":"text/html","content_length":"127533","record_id":"<urn:uuid:8ab86569-46ef-4ef6-9fd7-a8e8cfbc3cef>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00101.warc.gz"} |
Introduction to Reinforcement Learning
Shubham Dokania
• Introduction to Reinforcement Learning
• Markov Decision Process
• Value Based Learning
□ state value based learning
□ state-action value based learning
□ Bellman equations
• Temporal Difference Methods
• Value function approximation
• Code examples!
Reinforcement Learning is about learning what to do - how to map situations to actions, so as to maximize a numerical reward signal. The learner (agent) is not told what to do, but instead it must
discover which actions yield the most reward via trial-and-error.
Basic Reinforcement Learning Workflow
• The Reward defines the goal is a RL problem.
• Gives the agent a sense of what is good and bad.
• A reward of higher magnitude is better.
• Usually a function of environment state (situation).
• State is a representation of the current environment situation. is a set if all states.
• It's usually a function of the history , where the history is defined as a sequence of observations, actions and rewards.
H_t = O_1, R_1, A_1,..., A_{t-1}, O_t, R_t
$H_t = O_1, R_1, A_1,..., A_{t-1}, O_t, R_t$
s_t = f(H_t)
$s_t = f(H_t)$
• A state is information state or Markov state if it follows the Markov property.
i.e. the future is independent of the past, given the present
\mathbb{P}[S_{t+1} | S_t, S_{t-1}, S_{t-2},...] = \mathbb{P}[S_{t+1} | S_t]
$\mathbb{P}[S_{t+1} | S_t, S_{t-1}, S_{t-2},...] = \mathbb{P}[S_{t+1} | S_t]$
A Markov Process (Markov Chain) is a memoryless random process which follows the Markov Property.
A Markov Process is defined by
The probability of state transition is defined as:
P_{ss'} = \mathbb{P}[S_{t+1}=s' | S_t=s]
$P_{ss'} = \mathbb{P}[S_{t+1}=s' | S_t=s]$
Markov REWARD process (MRP)
A Markov Reward Process is a Markov process with Rewards/Values associated with states.
It's represented by
The Reward function is
and is the discount factor.
< S, P, R, \gamma >
$< S, P, R, \gamma >$
R_s = \mathbb{E}[R_{t+1} | S_t = s]
$R_s = \mathbb{E}[R_{t+1} | S_t = s]$
In a MRP, is defined as the discounted return, given by
But what is the need for a discount factor?
- Avoids infinite returns
- provides a control over long-term and short term rewards
-Mathematically convenient
G_t = R_{t+1} + \gamma R_{t+2} + \gamma^2 R_{t+3} + ... ; \gamma \in [0, 1]
$G_t = R_{t+1} + \gamma R_{t+2} + \gamma^2 R_{t+3} + ... ; \gamma \in [0, 1]$
Markov Decision Process (MDP)
A MDP is a MRP with decisions, represented by
where is a finite set of actions.
The transition probabilities and Reward function both depend on the actions.
The actions are governed by a policy
< S, A, P, R, \gamma >
$< S, A, P, R, \gamma >$
\pi (a | s)
$\pi (a | s)$
Can include one or more of the following
• Policy
• Value function
• Model
• The Policy of an agent defines it's behavior.
• It is given as
\pi (a | s) \text{ (stochastic) or } a = \pi (s) \text{ (deterministic)}
$\pi (a | s) \text{ (stochastic) or } a = \pi (s) \text{ (deterministic)}$
\pi (a | s) = \mathbb{P}[A_t = a | S_t = s]
$\pi (a | s) = \mathbb{P}[A_t = a | S_t = s]$
• The value function is a prediction of the future reward for a state.
• It's used to evaluate the quality of a state.
• It's the expected return, i.e.
V(s) = \mathbb{E}[G_t | S_t = s]
$V(s) = \mathbb{E}[G_t | S_t = s]$
V(s) = \mathbb{E}[R_{t+1} + \gamma R_{t+2} +... | S_t = s]
$V(s) = \mathbb{E}[R_{t+1} + \gamma R_{t+2} +... | S_t = s]$
• A model predicts what the environment will do next.
• The properties of a model are the state transition probability and a reward function.
• In case of a Partially Observable MDP (POMDP), the agent may form it's own representation of the environment.
A simple categorization of a few RL methods
• Temporal Difference Learning
□ Q-learning
□ SARSA
□ Actor-critic
• Policy Search based Learning
□ Policy Gradient
□ Evolutionary Strategies
• Model based Learning
□ Stochastic Dynamic Programming
□ Bayesian Approaches
TD ( \lambda)
$TD ( \lambda)$
• Reinforcement Learning is like trial and error
• The agent should explore the environment to search for better policies.
• After selection of optimal policy, the agent maximises the rewards.
□ Exploration finds more information about the environment.
□ Exploitation uses known information to maximise reward.
• Prediction : evaluate the future, given a policy.
• Control: optimise the future, find best policy.
• Bellman expectation equation
• Bellman optimality equation
State based value learning
In general
V(s) = \mathbb{E}[R_{t+1} + \gamma R_{t+2} + \gamma^2 R_{t+3} + ... | S_t = s]
$V(s) = \mathbb{E}[R_{t+1} + \gamma R_{t+2} + \gamma^2 R_{t+3} + ... | S_t = s]$
V(s) = \mathbb{E}[R_{t+1} + \gamma V(s_{t+1}) | S_t = s]
$V(s) = \mathbb{E}[R_{t+1} + \gamma V(s_{t+1}) | S_t = s]$
State-Action based value learning
In general
Q_{\pi}(s, a) = \mathbb{E}[R_{t+1} + \gamma R_{t+2} + \gamma^2 R_{t+3} + ... | S_t = s, A_t = a]
$Q_{\pi}(s, a) = \mathbb{E}[R_{t+1} + \gamma R_{t+2} + \gamma^2 R_{t+3} + ... | S_t = s, A_t = a]$
Q_{\pi}(s, a) = \mathbb{E}[R_{t+1} + \gamma Q(S_{t+1}, A_{t+1}) | S_t = s, A_t = a]
$Q_{\pi}(s, a) = \mathbb{E}[R_{t+1} + \gamma Q(S_{t+1}, A_{t+1}) | S_t = s, A_t = a]$
For state based value learning
For state-action based learning
For optimal condition, the optimal policy is:
V_*(s) = max_{s \in S} R_{t+1} + V(s')
$V_*(s) = max_{s \in S} R_{t+1} + V(s')$
Q_*(s, a) = max_{a \in A} R_{t+1} + \gamma Q(s', a')
$Q_*(s, a) = max_{a \in A} R_{t+1} + \gamma Q(s', a')$
\pi_*(a | s) = arg_{a \in A}max Q(s, a)
$\pi_*(a | s) = arg_{a \in A}max Q(s, a)$
temporal difference learning
• TD methods can learn without a model of the environment, through sampling.
• TD can learn from incomplete episodes.
• TD updates a guess, towards a guess return. (like DP)
The update rule in SARSA for state-action value is
which is essentially a generalisation of
where, is the TD target and
is the TD Error.
SARSA follows Bellman expectation equation
Q(s, a) = Q(s, a) + \alpha (R_{t+1} + \gamma Q(S_{t+1}, A_{t+1}) - Q(s, a))
$Q(s, a) = Q(s, a) + \alpha (R_{t+1} + \gamma Q(S_{t+1}, A_{t+1}) - Q(s, a))$
Q(s, a) = Q(s, a) + \alpha (G_t - Q(s, a))
$Q(s, a) = Q(s, a) + \alpha (G_t - Q(s, a))$
\delta_t = R_{t+1} + \gamma Q(S_{t+1}, A_{t+1}) - Q(s, a)
$\delta_t = R_{t+1} + \gamma Q(S_{t+1}, A_{t+1}) - Q(s, a)$
• Q-learning is similar to SARSA, but consists of two different policies
□ Behaviour policy: Used to evaluate
□ Estimation policy: Used for update rule.
The update in Q-learning is:
For behaviour policy, we may use -greedy policy.
Q(s, a) = Q(s, a) + \alpha (R_{t+1} + \gamma max_{a' \in A} Q(s', a') - Q(s, a))
$Q(s, a) = Q(s, a) + \alpha (R_{t+1} + \gamma max_{a' \in A} Q(s', a') - Q(s, a))$
implementation of Q-learning in gridworld like environment
Code: https://goo.gl/CE8xpC
• What happens if the state space is large?
□ Millions of states
□ Continuous state space
• Cannot store so much states in memory.
• Computation also becomes very slow!
value function approximation
• Generalise unseen states from seen state information.
• Estimate the value function through approximation.
value function approximation
Instead of using discrete state value representation, use
For instance, consider a linear combination:
where, weights can be updated using TD methods and is a feature representation
V(s, w) \approx V_\pi(s)
$V(s, w) \approx V_\pi(s)$
Q(s, a, w) \approx Q_\pi(s, a)
$Q(s, a, w) \approx Q_\pi(s, a)$
Q(s, a, w) = \sum_i w_i . f_i(s, a)
$Q(s, a, w) = \sum_i w_i . f_i(s, a)$
For function approximation, we can choose from
• Decision Trees / Random Forests
• Linear models
• Non-linear models (Neural Networks)
• Nearest Neighbours
• etc...
We choose models that can be differentiated!
(Linear and Non-linear)
• There is no "training data" in RL
• So, use Temporal Difference methods
• We create a guess target, and approximated guess
• Try to minimize the difference in target and appx.
update the weights by minimising a mean-squared error
And use Gradient Descent to update weights
J(w) = \mathbb{E}_\pi[(V_\pi(S) - \hat V(s, w))^2]
$J(w) = \mathbb{E}_\pi[(V_\pi(S) - \hat V(s, w))^2]$
\Delta w = -\frac{1}{2} \alpha \nabla_w J(w)
$\Delta w = -\frac{1}{2} \alpha abla_w J(w)$
\Delta w = \alpha \mathbb{E_\pi}[(V_\pi(S) - \hat V(s, w))\nabla_w \hat V(s, w)]
$\Delta w = \alpha \mathbb{E_\pi}[(V_\pi(S) - \hat V(s, w))abla_w \hat V(s, w)]$
update the weights by minimising a mean-squared error
And use Gradient Descent to update weights
J(w) = (R_{t+1} + \gamma max_{a' \in A} \hat Q(s', a') - \hat Q(s, a, w))^2
$J(w) = (R_{t+1} + \gamma max_{a' \in A} \hat Q(s', a') - \hat Q(s, a, w))^2$
\Delta w = \alpha(R_{t+1} + \gamma max_{a' \in A} \hat Q(s', a') - \hat Q(s, a, w))\nabla_w \hat Q(s, a, w))
$\Delta w = \alpha(R_{t+1} + \gamma max_{a' \in A} \hat Q(s', a') - \hat Q(s, a, w))abla_w \hat Q(s, a, w))$
Given the previous information, we can use any function approximator for estimating the value of Q, with the condition that the function be differentiable.
In a scenario where a Deep Neural Network is used as the function approximator, it's called as a DQN.
Introduction to Reinforcement Learning
By Shubham Dokania
Introduction to Reinforcement Learning
Presentation for Reinforcement Learning Lecture at Coding Blocks | {"url":"https://slides.com/shubhamdokania/rlcb","timestamp":"2024-11-08T21:37:25Z","content_type":"text/html","content_length":"206253","record_id":"<urn:uuid:11255c57-5e6a-4cc4-b0ee-eecedac041e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00891.warc.gz"} |
Dynamic Representations - Multiple Representations
Multiple representations allow students to gain a deeper understanding.
Comparing multiple methods allows students make connections and use the most appropriate method with fluency.
Comparing different representations uncovers the structure of the mathematics and allows students to make generalisations.
Using Multiple Representations
Click the image to open the interactive representation in Desmos. You can edit this version and save it to your own account.
DO NOT WORRY ABOUT BREAKING IT! If you accidentally delete something or rescale it and cannot get it back simply come back to this page and click the image again.
Some representations have been embedded into this site. To open in full screen on Desmos, click edit graph on Desmos in the bottom right.
Minimise the control panel in Desmos for the best view (<<)
Fractions, Decimals and Percentages
Fractions, decimals and percentages are represented by a 100 grid, decimal number line, percentage number line and bar model. Instruction video link
FDP mult rep template.docx
blank printable templates
Percentage Increase and Decrease
Students often find it hard to visualise beyond 100% using bar models. A double number line allows students to compare % and decimals and links to ratio tables - see Multiplicative Reasoning.
Percentage increase and decrease.docx
Multiply two digits by two digits
The area model for decimals is represented alongside Dienes blocks. Drag the blocks over the area to clear up any misconceptions.
For example 0.31 x 0.2 = 0.62
The area above is covered by 6x0.01s and 2x0.001s not 6x0.1s and 2x0.01s.
0.31 x 0.2 = 0.062
multiplying decimals 2.docx
Multiply whole numbers and decimals | {"url":"https://www.enigmadynamicrepresentations.com/multiple-representations","timestamp":"2024-11-08T04:32:58Z","content_type":"text/html","content_length":"244583","record_id":"<urn:uuid:8ba09043-3180-49e5-81f9-4c2bcdc550e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00528.warc.gz"} |
Math 1060 Exam Info (Fall 2022) | Math Courses
Math 1060 Exam Info (Fall 2022)
Check this page regularly for updates to exam dates, content coverage, and review materials.
Calculators: The use of calculators IS NOT permitted on exams or quizzes.
Exam Dates and Coverage:
• Midterm exams will be given in class, at your normally scheduled time (either 8:00 am or 9:30 am), in your usual classroom.
• If you have CSD accommodations, you need to request proctoring through MyAccess at least a week in advance of the exam.
Exam 1
Exam 1 (covers sections 1.1-1.9, 2.1, and 2.2): Tuesday, October 4th
Here is the list of review problems for Exam 1: Review Exam 1 Note that questions from 2.6 will not be on Exam 1.
Exam 2
Exam 2 (covers Ch. 3 and sections 4.1-4.4): Tuesday, November 8th
Here is the list of review problems for Exam 2:Review Exam 2
Final Exam
Final Exam (is cumulative): TBA
Here is the list of review problems for Final Exam: Review Problems for Final
Per University policy, all requests to reschedule or make up the final exam must be submitted to the Dean of Students for approval. Please note that vacations, previously purchased tickets or
reservations, social events, misreading the exam schedule and over-sleeping are not viable excuses for missing a final exam. If you think that your situation warrants permission to reschedule, please
contact the Dean of Students Office with any questions. Thank you in advance for your cooperation. | {"url":"https://courses.math.uconn.edu/fall2022/math-1060-exam-info/","timestamp":"2024-11-08T07:55:39Z","content_type":"text/html","content_length":"52442","record_id":"<urn:uuid:bfbf4fa8-5e07-4024-bc30-35195965fe6d>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00530.warc.gz"} |
Sample Paper 3 with Solution: Physics, Class 11 | Physics Class 11 - NEET PDF Download
Page 1
CBSE | Physics
Sample Paper 3
CBSE Board
Class XI Physics
Sample Paper-3
Time: - 3 Marks: - 70 Marks
General Instructions
(a) All questions are compulsory.
(b) There are 29 questions in total. Questions 1 to 8 carry one mark each, questions 9 to
16 carry two marks each, questions 17 to 25 carry three marks each and questions
27 to 29 carry five marks each.
(c) Question 26 is a value based question carrying four marks.
(d) There is no overall choice. However, an internal choice has been provided in one
question of two marks, one question of three marks and all three questions of five
marks each. You have to attempt only one of the given choices in such questions.
(e) Use of calculator is not permitted.
(f) You may use the following physical constants wherever necessary.
1.6 10
6.6 10
1.38 10
6.023 10 /
1.6 10
c ms
h JS
k JK
N mole
m kg
1. Can a physical quantity have units but still be dimensionless? (1)
2. Give an example to show that the direction of velocity of a body can change even when
its acceleration is constant. (1)
3. Two vectors A and B are directed along y-axis and z-axis respectively. What is the
direction of the vector ? (B A) ? (1)
CBSE | Physics
Sample Paper 3
CBSE Board
Class XI Physics
Sample Paper-3
Time: - 3 Marks: - 70 Marks
General Instructions
(a) All questions are compulsory.
(b) There are 29 questions in total. Questions 1 to 8 carry one mark each, questions 9 to
16 carry two marks each, questions 17 to 25 carry three marks each and questions
27 to 29 carry five marks each.
(c) Question 26 is a value based question carrying four marks.
(d) There is no overall choice. However, an internal choice has been provided in one
question of two marks, one question of three marks and all three questions of five
marks each. You have to attempt only one of the given choices in such questions.
(e) Use of calculator is not permitted.
(f) You may use the following physical constants wherever necessary.
1.6 10
6.6 10
1.38 10
6.023 10 /
1.6 10
c ms
h JS
k JK
N mole
m kg
1. Can a physical quantity have units but still be dimensionless? (1)
2. Give an example to show that the direction of velocity of a body can change even when
its acceleration is constant. (1)
3. Two vectors A and B are directed along y-axis and z-axis respectively. What is the
direction of the vector ? (B A) ? (1)
CBSE | Physics
Sample Paper 3
4. What does the area of the shaded portion of the graph represent? (1)
5. Why do we prefer to use a wrench (spanner) of longer arm? (1)
6. What is the degree of freedom of a monoatomic gas? (1)
7. For an ideal gas, show the nature of
versus P graph, where the symbols have their
usual meaning. (1)
8. State the Kelvin-Planck statement of the second law of thermodynamics. (1)
9. Differentiate between systematic errors and random errors. (2)
10. Two blocks of mass 3 kg and 2 kg are in contact with each other on a frictionless table.
Find the force exerted by the smaller block on bigger block if a force of 5 N is applied on
the bigger block. (2)
Mention two ways in which static friction is a self adjusting force. How much force of
static friction is acting on the block of mass 2 kg shown in figure below if the coefficient
of static friction between the block and the surface is 0.2? (2)
11. The kinetic energy of a body is increased by 21%. What is the percentage increase in
the linear momentum of the body? (2)
CBSE | Physics
Sample Paper 3
CBSE Board
Class XI Physics
Sample Paper-3
Time: - 3 Marks: - 70 Marks
General Instructions
(a) All questions are compulsory.
(b) There are 29 questions in total. Questions 1 to 8 carry one mark each, questions 9 to
16 carry two marks each, questions 17 to 25 carry three marks each and questions
27 to 29 carry five marks each.
(c) Question 26 is a value based question carrying four marks.
(d) There is no overall choice. However, an internal choice has been provided in one
question of two marks, one question of three marks and all three questions of five
marks each. You have to attempt only one of the given choices in such questions.
(e) Use of calculator is not permitted.
(f) You may use the following physical constants wherever necessary.
1.6 10
6.6 10
1.38 10
6.023 10 /
1.6 10
c ms
h JS
k JK
N mole
m kg
1. Can a physical quantity have units but still be dimensionless? (1)
2. Give an example to show that the direction of velocity of a body can change even when
its acceleration is constant. (1)
3. Two vectors A and B are directed along y-axis and z-axis respectively. What is the
direction of the vector ? (B A) ? (1)
CBSE | Physics
Sample Paper 3
4. What does the area of the shaded portion of the graph represent? (1)
5. Why do we prefer to use a wrench (spanner) of longer arm? (1)
6. What is the degree of freedom of a monoatomic gas? (1)
7. For an ideal gas, show the nature of
versus P graph, where the symbols have their
usual meaning. (1)
8. State the Kelvin-Planck statement of the second law of thermodynamics. (1)
9. Differentiate between systematic errors and random errors. (2)
10. Two blocks of mass 3 kg and 2 kg are in contact with each other on a frictionless table.
Find the force exerted by the smaller block on bigger block if a force of 5 N is applied on
the bigger block. (2)
Mention two ways in which static friction is a self adjusting force. How much force of
static friction is acting on the block of mass 2 kg shown in figure below if the coefficient
of static friction between the block and the surface is 0.2? (2)
11. The kinetic energy of a body is increased by 21%. What is the percentage increase in
the linear momentum of the body? (2)
CBSE | Physics
Sample Paper 3
12. Find the magnitude and direction of angular momentum of the body of mass m (about
point O) which is moving with velocity ? as shown. (2)
13. Name the satellites which have Sun synchronous orbit. How is their orbit different from
that of the satellites used for communication purpose? What is the significance of
negative total energy of a satellite? (2)
14. Define breaking stress. A heavy wire is suspended from a roof and no weight is attached
to its lower end. Is it under stress? (2)
15. Calculate the fall in pressure of helium initially at 1600 P a, when it is suddenly
expended to 8 times its original volume. Given ? =
. (2)
16. What is the change in internal energy of a gas during (i) isothermal expansion and (ii)
adiabatic expansion? (2)
17. On a two lane road, car A is travelling with a speed of 36 km/h. Two cars B and C
approach car A from opposite directions with speeds of 54 km/h each. At a certain
instant, when both car B and C are at a distance of 1 km from A, B decides to overtake
car A before C does. What minimum acceleration of B is required to avert an accident?
18. A particle of mass m moves in a straight line with retardation proportional to its
displacement. Find the expression for loss of kinetic energy for any displacement x. (3)
CBSE | Physics
Sample Paper 3
CBSE Board
Class XI Physics
Sample Paper-3
Time: - 3 Marks: - 70 Marks
General Instructions
(a) All questions are compulsory.
(b) There are 29 questions in total. Questions 1 to 8 carry one mark each, questions 9 to
16 carry two marks each, questions 17 to 25 carry three marks each and questions
27 to 29 carry five marks each.
(c) Question 26 is a value based question carrying four marks.
(d) There is no overall choice. However, an internal choice has been provided in one
question of two marks, one question of three marks and all three questions of five
marks each. You have to attempt only one of the given choices in such questions.
(e) Use of calculator is not permitted.
(f) You may use the following physical constants wherever necessary.
1.6 10
6.6 10
1.38 10
6.023 10 /
1.6 10
c ms
h JS
k JK
N mole
m kg
1. Can a physical quantity have units but still be dimensionless? (1)
2. Give an example to show that the direction of velocity of a body can change even when
its acceleration is constant. (1)
3. Two vectors A and B are directed along y-axis and z-axis respectively. What is the
direction of the vector ? (B A) ? (1)
CBSE | Physics
Sample Paper 3
4. What does the area of the shaded portion of the graph represent? (1)
5. Why do we prefer to use a wrench (spanner) of longer arm? (1)
6. What is the degree of freedom of a monoatomic gas? (1)
7. For an ideal gas, show the nature of
versus P graph, where the symbols have their
usual meaning. (1)
8. State the Kelvin-Planck statement of the second law of thermodynamics. (1)
9. Differentiate between systematic errors and random errors. (2)
10. Two blocks of mass 3 kg and 2 kg are in contact with each other on a frictionless table.
Find the force exerted by the smaller block on bigger block if a force of 5 N is applied on
the bigger block. (2)
Mention two ways in which static friction is a self adjusting force. How much force of
static friction is acting on the block of mass 2 kg shown in figure below if the coefficient
of static friction between the block and the surface is 0.2? (2)
11. The kinetic energy of a body is increased by 21%. What is the percentage increase in
the linear momentum of the body? (2)
CBSE | Physics
Sample Paper 3
12. Find the magnitude and direction of angular momentum of the body of mass m (about
point O) which is moving with velocity ? as shown. (2)
13. Name the satellites which have Sun synchronous orbit. How is their orbit different from
that of the satellites used for communication purpose? What is the significance of
negative total energy of a satellite? (2)
14. Define breaking stress. A heavy wire is suspended from a roof and no weight is attached
to its lower end. Is it under stress? (2)
15. Calculate the fall in pressure of helium initially at 1600 P a, when it is suddenly
expended to 8 times its original volume. Given ? =
. (2)
16. What is the change in internal energy of a gas during (i) isothermal expansion and (ii)
adiabatic expansion? (2)
17. On a two lane road, car A is travelling with a speed of 36 km/h. Two cars B and C
approach car A from opposite directions with speeds of 54 km/h each. At a certain
instant, when both car B and C are at a distance of 1 km from A, B decides to overtake
car A before C does. What minimum acceleration of B is required to avert an accident?
18. A particle of mass m moves in a straight line with retardation proportional to its
displacement. Find the expression for loss of kinetic energy for any displacement x. (3)
CBSE | Physics
Sample Paper 3
19. Give reasons for the following:
(a) A load on a thief’s back does not apply any force on him when he jumps from the
upper story of a house.
(b) A gun recoils on being fired.
(c) A man falling from a height receives more injury when he falls on cemented floor
rather than when he falls on a heap of sand. (3)
20. A cubical ice box of thermocole has each side 30 cm long and a thickness of 5 cm. 4 kg of
ice is put in the box. If outside temperature is 45°C and the coefficient of thermal
conductivity is 0.01 J/S/m/°C, calculate the mass of ice left after 6 hours. Latent heat of
fusion of ice = 335 × 10
J/kg. (3)
21. Three equal masses of m kg each are fixed at the vertices of an equilateral triangle ABC.
What is the force acting on mass 2 m placed at the centroid P of the triangle? Take AP =
BP = CP = 1 m. (3)
22. A tank of volume 0.3 m
contains 2 moles of helium gas at 20°C. Assuming that helium
behaves like an ideal gas.
(a) Find the total thermal energy of the system.
(b) What is the average kinetic energy per molecule? (3)
Nine particles of a gas have speeds of 5.00, 8.00, 12.00, 12.00, 12.00, 14.00, 14.00, 17.00
and 20.00 m/s.
(a) Find the average speed.
(b) What is the rms speed?
(c) What is the most probable speed of the particles? (3)
(a) Which characteristic of a wave remains constant as it moves from one medium to
another and why?
(b) The phase difference between two points on a progressive wave is
. What will be
the corresponding path difference?
(c) Mention one condition for production of beats. (3)
CBSE | Physics
Sample Paper 3
CBSE Board
Class XI Physics
Sample Paper-3
Time: - 3 Marks: - 70 Marks
General Instructions
(a) All questions are compulsory.
(b) There are 29 questions in total. Questions 1 to 8 carry one mark each, questions 9 to
16 carry two marks each, questions 17 to 25 carry three marks each and questions
27 to 29 carry five marks each.
(c) Question 26 is a value based question carrying four marks.
(d) There is no overall choice. However, an internal choice has been provided in one
question of two marks, one question of three marks and all three questions of five
marks each. You have to attempt only one of the given choices in such questions.
(e) Use of calculator is not permitted.
(f) You may use the following physical constants wherever necessary.
1.6 10
6.6 10
1.38 10
6.023 10 /
1.6 10
c ms
h JS
k JK
N mole
m kg
1. Can a physical quantity have units but still be dimensionless? (1)
2. Give an example to show that the direction of velocity of a body can change even when
its acceleration is constant. (1)
3. Two vectors A and B are directed along y-axis and z-axis respectively. What is the
direction of the vector ? (B A) ? (1)
CBSE | Physics
Sample Paper 3
4. What does the area of the shaded portion of the graph represent? (1)
5. Why do we prefer to use a wrench (spanner) of longer arm? (1)
6. What is the degree of freedom of a monoatomic gas? (1)
7. For an ideal gas, show the nature of
versus P graph, where the symbols have their
usual meaning. (1)
8. State the Kelvin-Planck statement of the second law of thermodynamics. (1)
9. Differentiate between systematic errors and random errors. (2)
10. Two blocks of mass 3 kg and 2 kg are in contact with each other on a frictionless table.
Find the force exerted by the smaller block on bigger block if a force of 5 N is applied on
the bigger block. (2)
Mention two ways in which static friction is a self adjusting force. How much force of
static friction is acting on the block of mass 2 kg shown in figure below if the coefficient
of static friction between the block and the surface is 0.2? (2)
11. The kinetic energy of a body is increased by 21%. What is the percentage increase in
the linear momentum of the body? (2)
CBSE | Physics
Sample Paper 3
12. Find the magnitude and direction of angular momentum of the body of mass m (about
point O) which is moving with velocity ? as shown. (2)
13. Name the satellites which have Sun synchronous orbit. How is their orbit different from
that of the satellites used for communication purpose? What is the significance of
negative total energy of a satellite? (2)
14. Define breaking stress. A heavy wire is suspended from a roof and no weight is attached
to its lower end. Is it under stress? (2)
15. Calculate the fall in pressure of helium initially at 1600 P a, when it is suddenly
expended to 8 times its original volume. Given ? =
. (2)
16. What is the change in internal energy of a gas during (i) isothermal expansion and (ii)
adiabatic expansion? (2)
17. On a two lane road, car A is travelling with a speed of 36 km/h. Two cars B and C
approach car A from opposite directions with speeds of 54 km/h each. At a certain
instant, when both car B and C are at a distance of 1 km from A, B decides to overtake
car A before C does. What minimum acceleration of B is required to avert an accident?
18. A particle of mass m moves in a straight line with retardation proportional to its
displacement. Find the expression for loss of kinetic energy for any displacement x. (3)
CBSE | Physics
Sample Paper 3
19. Give reasons for the following:
(a) A load on a thief’s back does not apply any force on him when he jumps from the
upper story of a house.
(b) A gun recoils on being fired.
(c) A man falling from a height receives more injury when he falls on cemented floor
rather than when he falls on a heap of sand. (3)
20. A cubical ice box of thermocole has each side 30 cm long and a thickness of 5 cm. 4 kg of
ice is put in the box. If outside temperature is 45°C and the coefficient of thermal
conductivity is 0.01 J/S/m/°C, calculate the mass of ice left after 6 hours. Latent heat of
fusion of ice = 335 × 10
J/kg. (3)
21. Three equal masses of m kg each are fixed at the vertices of an equilateral triangle ABC.
What is the force acting on mass 2 m placed at the centroid P of the triangle? Take AP =
BP = CP = 1 m. (3)
22. A tank of volume 0.3 m
contains 2 moles of helium gas at 20°C. Assuming that helium
behaves like an ideal gas.
(a) Find the total thermal energy of the system.
(b) What is the average kinetic energy per molecule? (3)
Nine particles of a gas have speeds of 5.00, 8.00, 12.00, 12.00, 12.00, 14.00, 14.00, 17.00
and 20.00 m/s.
(a) Find the average speed.
(b) What is the rms speed?
(c) What is the most probable speed of the particles? (3)
(a) Which characteristic of a wave remains constant as it moves from one medium to
another and why?
(b) The phase difference between two points on a progressive wave is
. What will be
the corresponding path difference?
(c) Mention one condition for production of beats. (3)
CBSE | Physics
Sample Paper 3
24. Two identical springs of spring constant k are attached to a block of mass m and to fixed
supports as shown below.
Show that the mass executes simple harmonic motion when displaced from its rest
position on either side. Also, find the period of oscillations. (3)
(a) Does the first law of thermodynamics violate the law of conservation of energy?
(b) Write the limitations of the first law of thermodynamics. (3)
26. Radha found the wheel getting detached from her uncle’s car. She took it to workshop
and got it repaired. She informed her uncle, who is a mechanical engineer, about this
(a) What according to you the values displayed by Radha?
(b) A thin wheel can stay upright on its rim for a considerable length of time when
rolled with a considerable velocity, while it falls from its upright position at the
slightest disturbance, when stationary. Explain. (4)
27. The displacement of a body is given to be proportional to the cube of time elapsed.
What is the nature of the acceleration of the body? Justify your answer.
A car accelerates from rest at a constant rate of ? for some time; after which it
decelerates at constant rate of ? to come to rest. If the total time elapsed is T second.
(a) Draw a velocity time graph for the motion.
(b) Calculate maximum velocity attained in terms of ? ? ? ? and T. (5)
(a) From the top of a building a ball is dropped while another is projected horizontally
at the same time.
(i) Which ball will strike the ground first?
(ii) Which will strike the ground with more speed? Justify your answer in each case.
(b) A body is projected with speed u at an angle ? to the horizontal to have maximum
range. What is the velocity at the highest point?
(c) What is the angle of projection of a projectile motion whose range R is n times the
maximum height. (5)
Read More | {"url":"https://edurev.in/p/162302/Sample-Paper-3-with-Solution-Physics--Class-11","timestamp":"2024-11-05T02:37:23Z","content_type":"text/html","content_length":"291066","record_id":"<urn:uuid:697d74dc-4c1d-4b6c-be8d-742f51cc3f67>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00888.warc.gz"} |
Section 4.5 and 4.6 in Matter and Interactions (4th edition)
Young's Modulus
Earlier, you read how to add springs in parallel and in series. In these notes, you will read about how the microscopic measurements of bond length and interatomic spring stiffness relate to
macroscopic measures like Young's modulus. We will continue using Platinum wire as our example.
Hanging a mass from a platinum wire
Consider a 2m long platinum wire ($L = 2m$) with a square cross section. That is, the wire is not “round” when viewed on end, but square. This wire is 1mm thick ($S = 1mm$); each side of the wire is
1mm. If you were to hang a 10kg weight, this 2m wire stretches by 1.166 mm ($s = 1.166mm$).
Determining the interatomic "spring stiffness"
If you model the whole wire as a single spring, you can find the spring stiffness of the whole wire. From the momentum principle (momentum not changing), you can determine the this stiffness because
the net force is zero.
$$\vec{F}_{net} = \vec{F}_{grav} + \vec{F}_{wire} = 0$$ $$\vec{F}_{wire} = -\vec{F}_{grav} = \langle 0, k_{s,wire}s \rangle = -\langle 0,-mg \rangle$$ $$k_{s,wire} = \dfrac{mg}{s} = \dfrac{(10kg)
(9.81 m/s^2)}{0.001166m} = 8.41\times10^4 N/m$$
This is very large spring constant because the wire (taken as a whole) is very stiff. Note: the units of N/m for k.
Finding the number bonds in the wire
To find the interatomic spring stiffness, you will need to know how many chains of atoms are in the wire (how many side-by-side springs) and how many atoms are in the chain (how many end-to-end
springs). Those values can be found by using the bond “length”, which you read about earlier. To remind you, the estimated bond length for Platinum is $d=2.47\times10^{-10}m$.
The number of chains in a square wire ($N_{s}$) is estimated by dividing the overall cross-sectional area of the wire ($S^2$) by the “area” of the bond ($d^2$).
$$N_{chains\:in\:wire} = \dfrac{A_{wire}}{A_{bond}} = \dfrac{S^2}{d^2} = \left(\dfrac{0.001m}{2.47\times10^{-10}m}\right)^2 = 1.64\times10^{13}\:\mathrm{chains}$$
The number of chains is equal to the number of side-by-side springs ($N_{s}$) in this model.
The number of bonds in a single chain can be found by dividing the length of the wire ($L$) by the bond length ($d$).
$$N_{bonds\:in\:chain} = \dfrac{L_{wire}}{L_{bond}} = \dfrac{L}{d} = \dfrac{2m}{2.47\times10^{-10}m} = 8.10\times10^9\:\mathrm{bonds}$$
Finding the interatomic spring stiffness
Because in our model all the bonds are assumed to be the same, the interatomic spring stiffness ($k_{s,interatomic}$) is determined by adding the springs as you have read before. The details of that
addition are below, but the final result is that the interatomic spring stiffness is related to the spring stiffness of the wire like so:
$$k_{s,interatomic} = \dfrac{N_{bonds\:in\:chain}}{N_{chains\:in\:wire}}k_{s,wire} = \dfrac{8.10\times10^9}{1.64\times10^{13}}8.41\times10^4\:N/m = 41.52\:N/m$$
The details for this calculation are below.
$$k_{s,side-by-side} = \sum_{chains} k_{s,interatomic} = N_{chains\:in\:wire} k_{s,interatomic}$$ $$\dfrac{1}{k_{s,wire}} = \sum_{along\:chains} \dfrac{1}{k_{s,side-by-side}} = \dfrac{N_{bonds\:in
\:chain}}{k_{s,side-by-side}} = \dfrac{N_{bonds\:in\:chain}}{ N_{chains\:in\:wire} k_{s,interatomic}}$$ $${k_{s,wire}} = \dfrac{N_{chains\:in\:wire}}{N_{bonds\:in\:chain}} k_{s,interatomic}$$ $$k_
{s,interatomic} = \dfrac{N_{bonds\:in\:chain}}{N_{chains\:in\:wire}} k_{s,wire} $$
The value that we found for the interatomic spring stiffness of Platinum (41.52 N/m) is typical of most pure metals, which have a range from about 5 to about 50 N/m.
Young's Modulus
Like density, the interatomic spring stiffness ($k_{s,interatomic}$) is an intensive property of an object, it doesn't depend on the length or shape of the object. Other properties are extensive such
as mass, volume, and the spring stiffness of the whole wire ($k_{s,wire}$). Scientists and engineers will often work with intensive properties because they characterize the material and not the
object. However, the interatomic spring stiffness is not a property that scientists and engineers often use. When discussing the compression and extension of materials, they often use the bulk
modulus or Young's modulus.
Stress and strain
To understand Young's modulus, you must learn about stress and strain.
As you will use it, stress is a measure of the tension (or compression) in a material per unit area. If a force of $F_T$ is applied to the end of a bar or wire of cross-sectional area $A$, then the
stress is given by:
$$stress = \dfrac{F_T}{A}$$
The strain is a measure of the fractional stretching (or compressing) of a material. If a material of relaxed length $L$ stretches or is compressed by a distance $\Delta L$ along that length, then
the strain is given by:
$$strain = \dfrac{\Delta L}{L}$$
Young's Modulus ($Y$) is the ratio of the stress to the strain and is measured in $N/m^2$, which is a special unit called “Pascals (Pa)”^1).
$$Y=\dfrac{F_T/A}{\Delta L/L}$$
Young's modulus is an intensive quantity; it doesn't depend on the size (length, area, volume) of the material (you “divide that out”). Notice that the definition of Young's modulus is quite similar
to that for spring force; Young's modulus relates a force (per unit area) to a stretch (per unit length):
$$\frac{F_T}{A} = Y\dfrac{\Delta L}{L} \longrightarrow F = k_ss$$
Here, the Young's modulus ($Y$) takes the role of the spring constant ($k_s$); a stiff material will have a large $Y$ and a floppy material will have a smaller $Y$.
The elastic regime
For many uses, the force per unit area is linearly proportional to the elongation of the material per unit length (as the above equation suggests). This regime is called the “elastic regime” and is
where the material will return to its relaxed length after the load is removed.
However, there is a point (the elastic limit) after which the material will not fully return to the its relaxed length. At this point the material begins to “yield” and stretches much more easily
with less force needed. This is the “elastic limit” where our linear model above will no longer apply.
Connecting the microscopic and the macroscopic
Because the Young's modulus is an intensive quantity, it applies to our model at any scale. So we can apply the Young's modulus equation to a pair of atoms sharing a single bond of length $d$ (the
$d$ is both the relaxed length and the scale of the atoms). If the atomic bond (with interatomic spring stiffness $k_{s,interatomic}$) stretches a small amount $s$, then the Young's modulus for that
pair of atoms is given by:
$$Y=\dfrac{F_T/A}{\Delta L/L} = \dfrac{(k_{s,interatomic}s)/d^2}{s/d} = \dfrac{k_{s,interatomic}}{d}$$
Here, you have connected a macroscopic measurement ($Y$) to a microscopic model ($\dfrac{k_{s,interatomic}}{d}$).
This is the same SI unit that is used for pressure. | {"url":"https://msuperl.org/wikis/pcubed/doku.php?id=183_notes:youngs_modulus","timestamp":"2024-11-05T18:46:49Z","content_type":"application/xhtml+xml","content_length":"44732","record_id":"<urn:uuid:dccb1a70-480c-40da-95b6-c30349cb2321>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00258.warc.gz"} |
Sequences area trigonometry work sheet
sequences area trigonometry work sheet Related topics: college algebra 1
expansion and factorization - grade 10 mathematics exercises
how do i work my ti-84 plus silver addition caculator
multiplying minus numbers
how do i solve y=|2x-3| step by step for relations and functions
quiz on elementary algebra
printable homework log
calculator solve and graph compound inequality
polynomials. maths revision
quadratic function worksheet
Author Message Author Message
onysod Posted: Wednesday 03rd of Jan 18:58 Homuck Posted: Friday 05th of Jan 08:32
I have trouble with sequences area trigonometry work Hello, just a month ago, I was stuck in a similar scenario
sheet. I tried hard to get somebody who can help me . I had even considered the option of dropping math and
out with this. I also searched for a teacher to teach me taking up some other course . A friend of mine told me
Reg.: 04.03.2004 and work out my problems on relations, difference of Reg.: 05.07.2001 to give one last chance and sent me a copy of
cubes and inverse matrices. Though I found a few who Algebrator. I was at comfort with it within few minutes .
could perhaps work out my problem, I realized that I My grades have really improved within the last year .
cannot find the money for them. I do not have a good
deal of time too. My exam is coming up shortly . I am
desperate . Can anybody help me out of this situation? I Svizes Posted: Friday 05th of Jan 11:07
would really be glad about any assistance or any
suggestion . A truly piece of algebra software is Algebrator. Even I
faced similar problems while solving simplifying fractions,
AllejHat Posted: Thursday 04th of Jan 17:12 slope and interval notation. Just by typing in the problem
Reg.: 10.03.2003 workbookand clicking on Solve – and step by step
Hi, I think that I can to help you out. Have you ever tried solution to my math homework would be ready. I have
out a program to assist you with your math homework used it through several math classes - Remedial
? Some time ago I was also stuck on a similar problems Algebra, Intermediate algebra and Remedial Algebra. I
Reg.: 16.07.2003 like you, but then I came across Algebrator. It helped me highly recommend the program.
so much with sequences area trigonometry work sheet
and other algebra problems, so since then I always
count on its help! My math grades got better thanks to
the help of Algebrator. | {"url":"https://softmath.com/parabola-in-math/converting-decimals/sequences-area-trigonometry.html","timestamp":"2024-11-01T22:56:08Z","content_type":"text/html","content_length":"45587","record_id":"<urn:uuid:3ccf4c3e-5c17-46f6-9b3c-55aebe7a08b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00248.warc.gz"} |
Significant Figures
Significant Figures and pH (pOH as well)
Here is the example:
Calculate the pH of a solution where the [H^+] is 0.00100 M.
This could also be a pOH problem. The point being made below about significant figures is the same.
OK, you say, that's pretty easy, the answer is 3. After all 0.00100 is 10¯^3 and the negative log of 10¯^3 is 3.
You would probably be awarded partial credit for your answer. Why? Because the pH is not written to reflect the number of significant figures in the concentration.
Notice that there are three sig figs in 0.00100. (Hopefully you remember significant figures, since you probably studied them months ago before getting to acid base stuff. THEY ARE STILL IMPORTANT!)
So, our pH value should also reflect three significant figures.
However, there is a special rule to remember with pH (and pOH) values. The whole number portion DOES NOT COUNT when figuring out how many digits to write down.
Let's phrase that another way: in a pH (and a pOH), the only place where significant figures are contained is in the decimal portion.
So, the correct answer to the above problem is 3.000. Three sig figs and they are all in the decimal portion, NOT (I repeat NOT) in the whole number portion.
Here is a comment I saw online. The kid said the pH was 10.7 and was graded with some points off; I don't know how many. The comment the kid made was that 10.7 was three significant figures.
WRONG!! The 10 does not count. The correct answer would have had three figures in the decimal portion, as in 10.730 or 10.711.
The whole number portion of a pH (or a pOH) not counting towards sig figs is related to the two parts of a logarithm: the characteristic and the mantissa. The characteristic (the 10 in 10.7) only
sets where the decimal point is in the value the logarithm represents. All the significant figures are encoded in the mantissa, which is the entire decimal portion.
You may look up more about the two parts of a logarithm on your own. The ChemTeam will spare you stories about using a slide rule and his six-place logarithm table from the days before calculators. | {"url":"https://web.chemteam.info/AcidBase/pH&sig-figs.html","timestamp":"2024-11-04T09:00:57Z","content_type":"text/html","content_length":"3568","record_id":"<urn:uuid:c3699429-2d52-4094-81cd-2ef7dd274b2a>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00074.warc.gz"} |
Quadratic Program Solver
Solves a Quadratic Programming problem using Alternating Direction Method of Multipliers (ADMM). This is a MATLAB implementation of OSQP.
296 Downloads
Updated 27 Feb 2022
Quadratic Program Solver
Solves a Quadratic Programming problem using Alternating Direction Method of Multipliers (ADMM).
This is a MATLAB implementation of the paper - OSQP: An Operator Splitting Solver for Quadratic Programs.
Remark: Any Quadratic Program Solver can solve Constrained Least Squares problem as well (With linear and convex constraints).
I needed for some Signal / Image Processing projects a solver of a problem of the form:
<math-renderer class="js-display-math" style="display: block" data-static-url="https://github.githubassets.com/static" data-run-id="8ae59d8c86da4fb759c24b169f5148bc">$$\begin{aligned} \arg \min_{\
boldsymbol{x}} & \quad \frac{1}{2} {\left| A \boldsymbol{x} - \boldsymbol{b} \right|}_{2}^{2} \\ \text{subject to} & \quad B \boldsymbol{x} \leq \boldsymbol{c} \\ & \quad D \boldsymbol{x}
= \boldsymbol{e} \end{aligned}$$</math-renderer>
I could use MATLAB's quadprog() or lsqlin() yet both are part of the Optimization Toolbox which isn't widely accessible.
When I learned about ADMM, Projection and Optimization in general I played with some implementations for this problem but they were pretty slow and sensitive to parameters.
When OSQP became available it showed those problem can be solved.
It requires compiling and even defining GCC as the compiler on Windows which isn't easy to everyone.
Hence I thought it would be nice to replicate the paper (As the C code is way beyond me) in MATLAB.
Within few hours I had a first working code though without all the features (See To Do).
The goal is to have a viable alternative to MATLAB's quadprog().
While in most cases quadprog() will be faster choice, this implementation should be good enough and free solution.
Implementation in High Level Language might assist with faster integration of better optimization and flexibility (For instance, supporting non sparse matrices).
The Code
The solver is implemented in the function SolveQuadraticProgram().
It uses MATLAB's argument block and requires MATLAB R2020b (Ver 9.9) to the least.
Users of previous MATLAB versions might try remove this block (Be careful about the parameters).
The function solves the following form of Quadratic Program:
<math-renderer class="js-display-math" style="display: block" data-static-url="https://github.githubassets.com/static" data-run-id="8ae59d8c86da4fb759c24b169f5148bc">$$\begin{aligned} \arg \min_{\
boldsymbol{x}} & \quad \frac{1}{2} \boldsymbol{x}^{T} P \boldsymbol{x} + \boldsymbol{x}^{T} \boldsymbol{q} \\ \text{subject to} & \quad \boldsymbol{l} \leq A \boldsymbol{x} \leq \boldsymbol
{u} \end{aligned}$$</math-renderer>
Where $ P \in \mathbb{S}_{+}^{n} $ (A symmetric positive semi definite matrix).
In Progress...
Look inside the function SolveQuadProgram() and see the reference paper.
Unit Test
The unit tests implemented in SolveQuadraticProgramUnitTest.m.
The script requires CVX and Optimization Toolbox.
It basically generates a problem and verify the solution of SolveQuadraticProgram() vs. the other 2 references.
To Do
1. Check if making paramRho a matrix has a real benefit.
2. Implement the scaling procedure from the reference paper.
3. Optimize the solution to the linear system (Is there anything better than pcg() for this structure? LSMR style?).
4. Better reporting.
5. Implement last step as Projected Gradient Descent in order to be strictly feasible.
Julia Code
The project implements the method in Julia Language as well (The .jl files).
The goal is to have a Julia package on its own. But it will happen only once performance will be in parity with the C code.
Julia has the potential to have even better performance than MATLAB's quadprog().
The code is an implementation of the paper Stellato, B. and Banjac, G. and Goulart, P. and Bemporad, A. and Boyd, S. - {OSQP}: An Operator Splitting Solver for Quadratic Programs.
This is a really great paper as the writers gave all the little details to create a truly competitive solver with ADMM.
Their work is really amazing.
Cite As
Royi Avital (2024). Quadratic Program Solver (https://github.com/RoyiAvital/QuadraticProgramSolver), GitHub. Retrieved .
MATLAB Release Compatibility
Created with R2021a
Compatible with R2020b and later releases
Platform Compatibility
Windows macOS Linux
Community Treasure Hunt
Find the treasures in MATLAB Central and discover how the community can help you!
Start Hunting!
Discover Live Editor
Create scripts with code, output, and formatted text in a single executable document.
Versions that use the GitHub default branch cannot be downloaded
Version Published Release Notes | {"url":"https://uk.mathworks.com/matlabcentral/fileexchange/97899-quadratic-program-solver?s_tid=prof_contriblnk","timestamp":"2024-11-04T07:07:50Z","content_type":"text/html","content_length":"95253","record_id":"<urn:uuid:dd2ae5f4-699d-45cf-9c09-6bcb513540ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00192.warc.gz"} |
Is part of the Bibliography
The decay properties of the Pygmy Dipole Resonance (PDR) have been investigated in the semi-magic N=82 nucleus 140Ce using a novel combination of nuclear resonance fluorescence and γ–γ coincidence
techniques. Branching ratios for transitions to low-lying excited states are determined in a direct and model-independent way both for individual excited states and for excitation energy intervals.
Comparison of the experimental results to microscopic calculations in the quasi-particle phonon model exhibits an excellent agreement, supporting the observation that the Pygmy Dipole Resonance
couples to the ground state as well as to low-lying excited states. A 10% mixing of the PDR and the [21+ x PDR] is extracted.
The decay behavior of low-lying dipole states in 140Ce was investigated exploiting the γ3-setup at the HIγS facility using quasi-monochromatic photon beams. Branching ratios of individual excited
states as well as average branching ratios to low-lying states have been extracted using γ – γ coincidence measurements. The comparison of the average branching ratios to QPM calculations shows a
remarkable agreement between experiment and theory in the energy range from 5.0 to 8.5 MeV.
A series of photon scattering experiments has been performed on the double-beta decay partners 76Ge and 76Se, in order to investigate their dipole response up to the neutron separation threshold.
Gamma-ray beams from bremsstrahlung at the S-DALINAC and from Compton-backscattering at HIGS have been used to measure absolute cross sections and parities of dipole excited states, respectively. The
HIGS data allows for indirect measurement of averaged branching ratios, which leads to significant corrections in the observed excitation cross sections. Results are compared to statistical
calculations, to test photon strength functions and the Axel-Brink hypothesis.
The ( J, T ) = (1, 1) parity doublet in 20Ne at 11.26 MeV is a good candidate to study parity violation in nuclei. However, its energy splitting is known with insufficient accuracy for quantitative
estimates of parity violating effects. To improve on this unsatisfactory situation, nuclear resonance fluorescence experiments using linearly and circularly polarized γ -ray beams were used to
determine the energy difference of the parity doublet E = E(1−) − E(1+) = −3.2(±0.7)stat( +0.6 −1.2)sys keV and the ratio of their integrated cross sections I (+) s,0 /I (−) s,0 = 29(±3)stat( +14 −7
)sys. Shell-model calculations predict a parityviolating matrix element having a value in the range 0.46–0.83 eV for the parity doublet. The small energy difference of the parity doublet makes 20Ne
an excellent candidate to study parity violation in nuclear excitations.
Two different experimental approaches were combined to study the electric dipole strength in the doubly-magic nucleus 48Ca below the neutron threshold. Real-photon scattering experiments using
bremsstrahlung up to 9.9 MeV and nearly mono-energetic linearly polarized photons with energies between 6.6 and 9.51 MeV provided strength distribution and parities, and an (α,α' γ) experiment at Eα
= 136 MeV gave cross sections for an isoscalar probe. The unexpected difference observed in the dipole response is compared to calculations using the first-order random-phase approximation and points
to an energy-dependent isospin character. A strong isoscalar state at 7.6 MeV was identified for the first time supporting a recent theoretical prediction.
We analysed our experimental recent findings of the dipole response of the odd-mass stable nucleus 205Tl within the quasi-particle phonon model. Using the phonon basis constructed for the
neighbouring 204Hg and wave function configurations for 205Tl consisting of a mixture of quasiparticle ⊗ N-phonon configurations (N=0,1,2), only one group of fragmented dipole excited states has been
reproduced at 5.5 MeV in comparison to the experimental distribution which shows a second group at about 5 MeV. The computed dipole transition strengths are mainly of E1 character which could be
associated to the pygmy dipole resonance.
The electric dipole strength distribution in 130Te has been investigated using the method of Nuclear Resonance Fluorescence. The experiments were performed at the Darmstadt High Intensity Photon
Setup using bremsstrahlung as photon source and at the High Intensity -Ray Source, where quasi-monochromatic and polarized photon beams are provided. Average decay properties of 130Te below the
neutron separation energy are determined. Comparing the experimental data to the predictions of the statistical model indicate, that nuclear structure effects play an important role even at
sufficiently high excitation energies. Preliminary results will be presented.
The dipole strength distribution of 130Te was investigated with the method of Nuclear Resonance Fluorescence using continuous-energy bremsstrahlung at the Darmstadt High Intensity Photon Setup and
quasi-monoenergetic photons at the High Intensity γ-Ray Source. The average decay properties were determined between 5.50 and 8.15 MeV and compared to simulations within the statistical model. | {"url":"https://publikationen.ub.uni-frankfurt.de/solrsearch/index/search/searchtype/authorsearch/author/Werner+Tornow","timestamp":"2024-11-05T17:32:28Z","content_type":"application/xhtml+xml","content_length":"55026","record_id":"<urn:uuid:70cdba64-25e0-436f-b4bd-95ebabe30104>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00313.warc.gz"} |
About me
I am a theoretical condensed matter physicist working in the areas of graphene, topological insulators and Majorana bound states. I work as an associate professor in Delft University of Technology,
where I lead the Quantum Tinkerer group together with Michael Wimmer.
I finished my PhD at Leiden University in the group of Carlo Beenakker in 2011. After Leiden I went to Harvard university as a Golub postdoctoral fellow, and finally moved to Delft in September 2013.
My research interests are outlined below. For a less formal introduction I also invite you to check out a short article I wrote to introduce myself to the other members of the Kavli Institute.
I also blog and tweet about random, but mostly science-related things.
Online course "topology in condensed matter"
Much of what I do is covered in the online course that I run together with my colleagues. Check it out.
Contact information
Skype: anton.akhmerov
Phone (please use for urgent matters only!): +31-61-2758481
Address: E211, Lorentzweg 1, 2628CJ, Delft, The Netherlands.
If you want to make an appointment or just drop by, your chances are better if you check with my schedule first:
My research interests mostly revolve around the field of mesoscopic conductors and superconductors. Very naturally this includes the topics of graphene, topological insulators, and Majorana bound
states. I am fascinated by the quantum systems that behave in the most counter-intuitive or unusual ways.
Majorana bound states. One example of an exotic physical object that is simple to analyse but hard to grasp is Majorana bound states (frequently also called Majorana fermions). Consider a combination
of simple ingredients. Take conventional superconductors, known for almost a century, and understood extremely well for half a century. Add a semiconducting quantum wire, a basis of modern
electronics, but scaled down; these were studied for decades. The amazing thing is that the theoretical and experimental progress showed how combining these two ingredients one can create the special
Majorana bound states. Sergey Frolov very properly calls them zen particles, by comparison with the god particle, Higgs boson. They have no energy, no charge, and no mass (which makes them extremely
hard to find), and they store quantum information in a way completely hidden from environment. The state of these quantum degrees of freedom changes when they are moved around each other, allowing to
implement an alternative route to quantum computation.
Topological insulators. Symmetry has always been a guiding concept in physics, allowing to generalize conclusions from one particular system to many which possess similar qualities. The other concept
with applicability that is perhaps as broad is topology. It allows to conclude that certain properties of superficially very different systems must be identical as long as the two systems can be
continuously transformed into one another. Topological insulators use a combination of both symmetry and topology. The surface of these materials is guaranteed to be conducting as long as certain
symmetry of the material is unbroken, and as long as bulk stays insulating.
Kwant. Numerical simulation of a physical system is a useful and sometimes irrepleacable tool in various tasks. It can be used as a boost or a check of the intuition, it may lead to finding an
efficient analytical approximation, or as the last resort in handling the problems that are beyond the reach of analytics. To make the numerical calculations more accessible to the community, me and
my colleagues have developed a sofware package kwant, which can be used to numerically solve a broad range of problems in mesoscopic quantum transport. | {"url":"https://antonakhmerov.org/","timestamp":"2024-11-13T19:39:32Z","content_type":"text/html","content_length":"10631","record_id":"<urn:uuid:0151908b-628c-48c6-9e64-3054ea388650>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00144.warc.gz"} |
Data Structures for Java Developers: A Comprehensive Guide - CODERZON
Data Structures for Java Developers: A Comprehensive Guide
Data structures are fundamental components in computer science, providing a way to organize and store data efficiently. For Java developers, understanding data structures is crucial, as it impacts
the performance and efficiency of applications. Whether you are developing a simple application or a complex system, choosing the right data structure can significantly affect the time complexity and
memory usage. This blog post explores various data structures, their implementations in Java, and real-time use cases to illustrate their practical applications.
1. Arrays
An array is a collection of elements, each identified by an index or key. It’s a simple and widely used data structure, particularly for storing a fixed-size sequence of elements of the same type.
• Fixed Size: The size of an array is determined at the time of creation and cannot be altered.
• Indexed Access: Elements can be accessed directly via their index, making retrieval operations O(1).
• Homogeneous Elements: All elements in an array must be of the same type.
Java Implementation
int[] intArray = new int[10]; // Creates an array of integers with size 10
String[] stringArray = new String[]{"Java", "Python", "C++"}; // Initializes an array with values
Use Case: Inventory Management System
In an inventory management system, an array can be used to store product IDs. For example, if the warehouse has a fixed number of different products, an array can be initialized to store these IDs
for quick access and updates.
int[] productIDs = new int[]{101, 102, 103, 104, 105};
// Accessing a product ID
int firstProductID = productIDs[0]; // 101
2. Linked Lists
A linked list is a linear data structure where each element is a separate object, known as a node. Each node contains the data and a reference to the next node in the sequence.
• Dynamic Size: The size of a linked list can grow or shrink dynamically as elements are added or removed.
• Efficient Insertions/Deletions: Adding or removing elements is efficient as it involves changing pointers.
• No Indexed Access: Elements cannot be accessed directly via an index; traversal is required.
Java Implementation
class Node {
int data;
Node next;
Node(int data) {
this.data = data;
this.next = null;
class LinkedList {
Node head;
public void add(int data) {
Node newNode = new Node(data);
if (head == null) {
head = newNode;
} else {
Node current = head;
while (current.next != null) {
current = current.next;
current.next = newNode;
Use Case: Browser History
A linked list is ideal for implementing a browser history feature. Each node can represent a webpage visited by the user, with the head of the list representing the most recently visited page.
LinkedList browserHistory = new LinkedList();
// Traverse and display history
Node current = browserHistory.head;
while (current != null) {
current = current.next;
3. Stacks
A stack is a collection of elements that follows the Last In First Out (LIFO) principle. Elements are added and removed from the same end, known as the top of the stack.
• LIFO Order: The last element added is the first to be removed.
• Basic Operations: Push (add), Pop (remove), and Peek (retrieve the top element without removing it).
Java Implementation
import java.util.Stack;
Stack<Integer> stack = new Stack<>();
int topElement = stack.pop(); // 30
Use Case: Undo Feature in Text Editors
In a text editor, a stack can be used to implement the undo feature. Each user action (like typing or deleting) can be pushed onto the stack. When the user triggers an undo, the most recent action is
popped from the stack and reversed.
Stack<String> actionStack = new Stack<>();
actionStack.push("Type: Hello");
actionStack.push("Delete: o");
String lastAction = actionStack.pop(); // Undo "Delete: o"
4. Queues
A queue is a collection of elements that follows the First In First Out (FIFO) principle. Elements are added at the rear and removed from the front.
• FIFO Order: The first element added is the first to be removed.
• Basic Operations: Enqueue (add), Dequeue (remove), and Peek (retrieve the front element without removing it).
Java Implementation
import java.util.LinkedList;
import java.util.Queue;
Queue<Integer> queue = new LinkedList<>();
int frontElement = queue.remove(); // 1
Use Case: Customer Service System
In a customer service system, a queue can manage incoming customer requests. Each request is added to the queue, and the first request in line is addressed first.
Queue<String> customerQueue = new LinkedList<>();
customerQueue.add("Customer 1: Issue A");
customerQueue.add("Customer 2: Issue B");
String nextCustomer = customerQueue.remove(); // "Customer 1: Issue A"
5. HashMaps
A HashMap is a collection of key-value pairs, where each key is unique. It allows for fast retrieval of values based on their corresponding keys.
• Constant Time Complexity: O(1) for insertions, deletions, and lookups.
• Unordered: Elements are not stored in any particular order.
• Null Keys and Values: Allows one null key and multiple null values.
Java Implementation
import java.util.HashMap;
HashMap<String, Integer> map = new HashMap<>();
map.put("apple", 1);
map.put("banana", 2);
int value = map.get("apple"); // 1
Use Case: Caching System
A HashMap is commonly used in caching systems to store frequently accessed data. For instance, a web application might cache the results of expensive database queries.
HashMap<String, String> cache = new HashMap<>();
cache.put("query1", "result1");
cache.put("query2", "result2");
String cachedResult = cache.get("query1"); // "result1"
6. Trees
A tree is a hierarchical data structure with a root node and child nodes. Each node contains data and references to its children.
• Hierarchical Structure: Data is organized in levels, with the root node at the top.
• Binary Trees: A common type where each node has at most two children.
• Balanced Trees: A tree is balanced if the height of the left and right subtrees differs by at most one.
Java Implementation
class TreeNode {
int data;
TreeNode left, right;
TreeNode(int data) {
this.data = data;
this.left = this.right = null;
class BinaryTree {
TreeNode root;
public BinaryTree(int rootData) {
root = new TreeNode(rootData);
// Other tree-related methods
Use Case: File System
A tree structure can represent a file system, where the root node is the root directory, and child nodes represent files and subdirectories.
BinaryTree fileSystem = new BinaryTree("root");
fileSystem.root.left = new TreeNode("Documents");
fileSystem.root.right = new TreeNode("Pictures");
fileSystem.root.left.left = new TreeNode("Resume.docx");
fileSystem.root.right.left = new TreeNode("Vacation.jpg");
7. Graphs
A graph is a collection of nodes (vertices) connected by edges. Graphs can be either directed or undirected, and they can contain cycles.
• Vertices and Edges: Nodes are called vertices, and connections are called edges.
• Directed and Undirected: Directed graphs have edges with a direction, while undirected graphs do not.
• Weighted and Unweighted: Edges can have weights representing the cost or distance between vertices.
Java Implementation
import java.util.*;
class Graph {
private Map<Integer, List<Integer>> adjacencyList = new HashMap<>();
public void addEdge(int source, int destination) {
adjacencyList.computeIfAbsent(source, k -> new ArrayList<>()).add(destination);
adjacencyList.computeIfAbsent(destination, k -> new ArrayList<>()).add(source);
public List<Integer> getNeighbors(int node) {
return adjacencyList.getOrDefault(node, new ArrayList<>());
Use Case: Social Network
In a social network, a graph can represent users and their connections. Each user is a vertex, and an edge represents a friendship or connection between two users.
Graph socialNetwork = new Graph();
socialNetwork.addEdge(1, 2); // User 1 and User 2 are friends
socialNetwork.addEdge(1, 3); // User 1 and User 3 are friends
socialNetwork.addEdge(2, 4); // User 2 and User 4
are friends
List<Integer> user1Friends = socialNetwork.getNeighbors(1); // [2, 3]
8. Heaps
A heap is a special tree-based data structure that satisfies the heap property. In a max heap, the parent node is always greater than or equal to its children, and in a min heap, the parent node is
always less than or equal to its children.
• Complete Binary Tree: A heap is a complete binary tree, meaning all levels are fully filled except possibly the last level, which is filled from left to right.
• Heap Property: In a max heap, the key of each node is greater than or equal to the keys of its children, and in a min heap, the key of each node is less than or equal to the keys of its children.
Java Implementation
import java.util.PriorityQueue;
PriorityQueue<Integer> minHeap = new PriorityQueue<>();
int smallest = minHeap.poll(); // 5
Use Case: Priority Queue for Task Scheduling
In task scheduling systems, a min heap can be used as a priority queue to manage tasks based on their priority. Tasks with the highest priority (smallest value) are processed first.
PriorityQueue<Task> taskQueue = new PriorityQueue<>(Comparator.comparingInt(Task::getPriority));
taskQueue.add(new Task("Task 1", 3));
taskQueue.add(new Task("Task 2", 1));
taskQueue.add(new Task("Task 3", 2));
Task nextTask = taskQueue.poll(); // Task 2 (highest priority)
9. Tries
A trie is a tree-like data structure used for storing a dynamic set of strings, where each node represents a character of a string. It’s particularly useful for search operations.
• Efficient String Search: Tries allow for efficient search, insertion, and deletion operations for strings.
• Prefix Matching: They are especially useful for prefix matching and autocomplete features.
Java Implementation
class TrieNode {
Map<Character, TrieNode> children = new HashMap<>();
boolean isEndOfWord;
public TrieNode() {
this.isEndOfWord = false;
class Trie {
private TrieNode root;
public Trie() {
root = new TrieNode();
public void insert(String word) {
TrieNode current = root;
for (char c : word.toCharArray()) {
current = current.children.computeIfAbsent(c, k -> new TrieNode());
current.isEndOfWord = true;
public boolean search(String word) {
TrieNode current = root;
for (char c : word.toCharArray()) {
current = current.children.get(c);
if (current == null) {
return false;
return current.isEndOfWord;
Use Case: Autocomplete Feature
In search engines and text editors, tries can implement the autocomplete feature. As the user types, the trie can quickly find all words that start with the given prefix.
Trie trie = new Trie();
boolean found = trie.search("app"); // true
Data structures are essential tools in a Java developer’s toolkit. They provide the means to store and organize data efficiently, enabling faster access and manipulation. Choosing the right data
structure depends on the specific requirements of your application, such as the type of data, the operations to be performed, and the expected performance.
In this blog post, we explored various data structures, including arrays, linked lists, stacks, queues, hash maps, trees, graphs, heaps, and tries. We also discussed their characteristics, Java
implementations, and real-time use cases. By understanding and mastering these data structures, you can enhance your problem-solving skills and build more efficient and robust applications.
Whether you are building an inventory management system, a browser history feature, a customer service system, a caching system, a file system, a social network, a task scheduling system, or an
autocomplete feature, choosing the appropriate data structure is crucial for achieving optimal performance and efficiency. | {"url":"https://coderzon.com/data-structures-for-java-developers-a-comprehensive-guide/","timestamp":"2024-11-09T02:53:48Z","content_type":"text/html","content_length":"141014","record_id":"<urn:uuid:d9e08b1b-868e-4c88-a9bd-292605343d5a>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00491.warc.gz"} |
rk comments on The Kelly Criterion
Does your program assume that the Kelly bet stays a fixed size, rather than changing?
Here’s a program you can paste in your browser that finds the expected value from following Kelly in Gurkenglas’ game (it finds EV to be 20)
(You can also fiddle with the first argument to experiment to see some of the effects when 4 doesn’t hold)
• I believe you missed one of the rules of Gurkenglas’ game, which was that there are at most 100 rounds. (Although it’s possible I misunderstood what they were trying to say.)
If you assume that play continues until one of the players is bankrupt then in fact there are lots of winning strategies. In particular betting any constant proportion less than 38.9%. The Kelly
criterion isn’t unique among them.
My program doesn’t assume anything about the strategy. It just works backwards from the last round and calculates the optimal bet and expected value for each possible amount of money you could
have, on the basis of the expected values in the next round which it has already calculated. (Assuming each bet is a whole number of cents.)
□ I did indeed! So I guess this game fails (5) out of Zvi’s criteria. | {"url":"https://www.greaterwrong.com/posts/BZ6XaCwN4QGgH9CxF/the-kelly-criterion/comment/BgrwGRwY35Y2GLQh2","timestamp":"2024-11-07T17:32:38Z","content_type":"text/html","content_length":"10649","record_id":"<urn:uuid:9505a509-ae6f-479a-bac2-36f9d83007f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00263.warc.gz"} |
ESP Biography
HARRISON BROWN, MIT freshman studying Math and CS
Major: Mathematics
College/Employer: Georgia Tech
Year of Graduation: Not available.
Brief Biographical Sketch:
I'm a freshman from Atlanta, GA, planning to declare course 6 or 18c. My interests include: Really bad fiction (of all genres and media), math (duh), programming, cryptography, going on dates with
quantum computing textbooks, webcomics, print comics, sci-fi (of all media), indie music, the '90s, and junk food.
Past Classes
(Clicking a class title will bring you to the course's section of the corresponding course catalog)
M2226: The Mathematics of Monsters and Machines in HSSP Spring 2009 (Mar. 14, 2009)
Can you add by dropping marbles through a maze of switches? http://www.youtube.com/watch?v=GcDshWmhF4A (watch with the volume off and figure out how it works - /very/ simple, but elegant, no?) That
machine clearly only works as directed for some range of numbers. How about if you want to add arbitrarily large numbers with one, finite machine? Can you build such a machine and then tell someone
how to drop marbles into it to add their numbers? – NO!! STOP!! I did not ask, ‘how would you’ – I asked CAN you? Sure, you could prove the affirmative by construction, by making a machine which does
so (if one can exist) but can you more succinctly, more elegantly, simply prove that such a machine exists? What if one could not exist? How would you go about proving this? If you like looking at
machines and figuring out what they do, or constructing machines to solve problems, then you will probably like this class. On the other hand, what this class is /really/ about is the mathematical
treatment of ALL machines, ALL languages, ALL algorithms. Exactly what abilities – finitely many states? finite memory? infinite memory? non-determinism? – are necessary to solve problems? What sets
of abilities are equivalent? How long does it take to solve problems of sufficient complexity? Are there problems that are simply impossible to solve, although they clearly must have an answer? To
answer the last question, YES! However, f you are willing to accept this claim without /proof! ‘you CANNOT say something like that without PROOF!’/ you probably can skip this class. But if that kind
of claim shakes your world up a bit, come to this class and be shaken!!
M2237: Quantum Computing is Awesome in Spark! Spring 2009 (Mar. 07, 2009)
Ever wanted to factor huge numbers really quickly? (Ever wanted to break the codes used to transmit secure data online?) How about searching a database without looking at each element? What about
simulating an entire universe, particle by particle? To do this, you need to harness the power of quantum mechanics and build a quantum computer. We'll discuss the history of quantum computing, what
quantum computers can (and can't) do, and the future of quantum.
H2238: Reading Comics in Spark! Spring 2009 (Mar. 07, 2009)
We'll discuss comics from the 1930s to today, with a focus on literary criticism and the "literary" uses of the medium. Authors and artists discussed may include: Alan Moore, Frank Miller, Brian
Azzarello & Eduardo Risso, Neil Gaiman, Art Spiegelman, Chris Ware, Tom Siddell, Ryan North, Winsor McCay, Bill Willingham, Dave Sim, Chris Onstad, Alison Bechdel, Rich Burlew.
H1786: Reading Comics in Splash! 2008 (Nov. 22 - 23, 2008)
They say a picture is worth a thousand words; in that case, an issue of Batman should be as long as War and Peace. Come learn about comics, talk about comics, read comics, and just generally be
awesome, because hey, comics. Both print comics and webcomics will be covered, from the Silver Age up through today. The focus will be on Western (i.e., U.S. and European) comics.
Introduction to Cryptography in SPLASH (2008)
Ever wanted to send messages in an unbreakable code? How can you send your credit card number securely to someone ... | {"url":"https://esp.mit.edu/teach/teachers/brownh/bio.html","timestamp":"2024-11-07T07:37:22Z","content_type":"application/xhtml+xml","content_length":"18342","record_id":"<urn:uuid:d2e341b3-b864-4c88-9556-2ef7c52cbacd>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00681.warc.gz"} |
Source code for prefsampling.ordinal.urn
Urn processes are random processes based on the idea that rankings are drawn from an urn. The
initial composition of the urn, the rules for drawing elements from it, and the evolution of the
elements of the urn ar the characteristic of each specific urn process.
from __future__ import annotations
import math
from collections import Counter
import numpy as np
from prefsampling.combinatorics import all_profiles, generalised_ascending_factorial
from prefsampling.inputvalidators import validate_num_voters_candidates
from prefsampling.core.urn import urn_scheme
def urn(
num_voters: int, num_candidates: int, alpha: float, seed: int = None
) -> list[list[int]]:
Generates votes following the Pólya-Eggenberger urn culture. The process is as follows. The urn
is initially empty and votes are generated one after the other, in turns. When generating a
vote, the following happens. With a probability of 1/(urn_size + 1), the vote is selected
uniformly at random (following an impartial culture). With probability `1/urn_size` a vote
from the urn is selected uniformly at random. In both cases, the vote is put back in the urn
together with `alpha * m!` copies of the vote (where `m` is the number of candidates).
Note that for a given number of voters, votes are not sampled independently.
num_voters: int
Number of voters
num_candidates: int
Number of candidates
alpha: float
The dispersion coefficient (`alpha * m!` copies of a vote are put back in the urn after
a draw). Must be non-negative.
seed: int, default: :code:`None`
The seed for the random number generator.
The votes
.. testcode::
from prefsampling.ordinal import urn
# Sample from an urn model with 2 voters and 3 candidates, alpha parameter is 0.5.
urn(2, 3, 0.5)
# For reproducibility, you can set the seed.
urn(2, 3, 4, seed=1002)
# Passing a negative alpha will fail
urn(2, 3, -0.5)
except ValueError:
The probability distribution governing an urn model is well documented. Specifically, given
:math:`n` agents and :math:`m` candidates, the probability of observing a profile in which
a given ranking :math:`j` appears :math:`c_j` times is equal to:
.. math::
\\frac{n!}{\\text{asc\\_fact}(m!, n, \\alpha \\times m!)} \\times
\\prod_{j = 1}^{m!} \\frac{\\text{asc\\_fact}(1, c_j, \\alpha \\times m!)}{c_j!}
where :math:`\\text{asc\\_fact}` is the generalised ascending factorial, defined as:
.. math::
\\text{asc\\_fact}(x, \\ell, \\sigma) = x \\times (x + \\sigma) \\times \\cdots \\times
(x + (\\ell - 1) \\times \\sigma).
Since the probability only depends on the number of times each ranking appears in the
profile, the space of outcome consists of all anonymous profiles, i.e., all representations
of any profile as a multiset (in which the order of the voters do not matter).
We test that the observed frequencies of anonymous profile is in line with the theoretical
probability distribution.
.. image:: ../validation_plots/ordinal/urn_0_0.png
:width: 800
:alt: Observed versus theoretical frequencies for an urn model with alpha=0
.. image:: ../validation_plots/ordinal/urn_0_5.png
:width: 800
:alt: Observed versus theoretical frequencies for an urn model with alpha=0.5
.. image:: ../validation_plots/ordinal/urn_1_0.png
:width: 800
:alt: Observed versus theoretical frequencies for an urn model with alpha=1
When :math:`\\alpha = \\frac{1}{m!}`, we are supposed to obtain a uniform distribution over
all anonymous profiles.
.. image:: ../validation_plots/ordinal/urn_0_0416666666666666.png
:width: 800
:alt: Observed versus theoretical frequencies for an urn model with alpha=1/m!
`Über die statistik verketteter vorgänge
*Florian Eggenberger and György Pólya*,
ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und
Mechanik, 3(4):279–289, 1923.
`Paradox of Voting under an Urn Model: The Effect of Homogeneity
*Sven Berg*,
Public Choice, Vol. 47, No. 2, 1985.
rng = np.random.default_rng(seed)
votes = urn_scheme(
num_voters, alpha, lambda x: list(x.permutation(num_candidates)), rng
return votes
def theoretical_distribution(num_voters, num_candidates, alpha, profiles=None) -> dict:
if profiles is None:
profiles = all_profiles(num_voters, num_candidates)
factorial_num_candidates = math.factorial(num_candidates)
distribution = {}
for profile in profiles:
counts = Counter(profile)
probability = math.factorial(num_voters) / generalised_ascending_factorial(
alpha * factorial_num_candidates,
for c in counts.values():
probability *= generalised_ascending_factorial(
1, c, alpha * factorial_num_candidates
) / math.factorial(c)
distribution[profile] = probability
normaliser = sum(distribution.values())
for r in distribution:
distribution[r] /= normaliser
return distribution | {"url":"https://comsoc-community.github.io/prefsampling/_modules/prefsampling/ordinal/urn.html","timestamp":"2024-11-12T19:04:35Z","content_type":"text/html","content_length":"28019","record_id":"<urn:uuid:0fa56066-c0dd-4d20-94bb-acb1cc4102a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00331.warc.gz"} |
Re: Question on hive dynamic partition pruning and explain plan
Could someone please explain me understand the below questions on hive partition pruning and explain plan?
1. How to check if partition pruning occurs by checking the explain plan? I thought I would see "Dynamic partitioning event operator" in explain plan, but in my sample query below I am not seeing any
such operator. I enabled hive.tez.dynamic.partition.pruning.
-- Since the table does not have much data, it is going for map join, does that have anything to do with partition pruning not happening?
explain select a.* from big_part a, small_np b where a.jdate = b.jdate ;
big_part is partitioned on jdate where small_np is a non partitioned table, even adding explicit filter on jdate like jdate = "2017-01-01" is not showing this operator in explain plan.
The tables are just in text formats.
I tried disabling and enabling hive.optimize.ppd . But it just changed adding or removing a filter operator much higher in explain plan, but no difference besides that. Will the optimize.ppd
parameter have any effect on partition pruning?
2. Is it correct to expect that dynamic partition pruning should happen on big_part table in the above query?
3. If both the tables used in join are partitioned, can we expect that dynamic partition pruning happens on both tables?
4. Will the dynamic partition pruning occur in case of outer joins too? (full and left outer assuming that inner table's conditions are given in "on condition" or outer table's conditions are given
in "where clause").
5. What exactly this hive.optimize.ppd do in case of text files? Just push the filter predicates when reading from table itself if possible?
Thank you! | {"url":"https://community.cloudera.com/t5/Support-Questions/Question-on-hive-dynamic-partition-pruning-and-explain-plan/m-p/131257","timestamp":"2024-11-03T13:25:43Z","content_type":"text/html","content_length":"229605","record_id":"<urn:uuid:4aaeec22-defc-4f44-a170-8200d36c2711>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00815.warc.gz"} |
A Unified Approach to Lower Bounds and Derandomization
Pranjal Dutta, Research Fellow, School of Computing, NUS
Contact Person
Dr Divesh AGGARWAL, Associate Professor, School of Computing
30 Apr 2024 Tuesday, 04:00 PM to 05:00 PM
MR1, COM1-03-19
COM1 Level 3
MR1, COM1-03-19
Given a univariate polynomial f of degree d (over complex numbers), what is the 'minimal' way to express it as a sum-of-squares of univariates, i.e., for some s?
What is the number of real roots of a given f as a sum-of-squares? It turns out that these two questions are very much related, and a good enough understanding of these questions will lead to
completely solving the algebraic version of the holy grail of theoretical computer science: P != NP; also known as Valiant's VP != VNP. We will discuss some of the explicit connections.
On the other hand, a sibling problem to this, known as Polynomial Identity Testing (PIT), is to check non-zeroness of a given (multivariate) polynomial f. Though a very straightforward randomized
algorithm exists for PIT, a deterministic polynomial-time solution has long been desired, but not yet achieved. There have been significant efforts to design efficient deterministic PIT, assuming
different restricted structures on f. We will discuss some of them.
We will also briefly talk about how algebraic approximations are relevant and related to all the questions above. This dates back to earlier works by Strassen's work on a faster matrix multiplication
algorithm. We will see some of the recent advancements in this area, and their intimate connections with the classical P != NP question.
Pranjal Dutta (https://sites.google.com/view/pduttashomepage) is currently a Postdoc at the School of Computing, NUS, in the group of Prof. Divesh Aggarwal. His broad research area is Complexity
theory. He finished his PhD in Computer Science (2018-2022), from Chennai Mathematical Institute (CMI), under the guidance of Prof. Nitin Saxena (IIT Kanpur). He is the winner of the ACM India
Doctoral Dissertation Award 2023. During his PhD, he was a recipient of the Google PhD Fellowship (2018-2022). He obtained his bachelor's in Mathematics and Computer science (2013-2016) and master's
in Computer science (2016-2018) both from CMI. | {"url":"https://events.comp.nus.edu.sg/view/22292","timestamp":"2024-11-14T17:05:50Z","content_type":"text/html","content_length":"12706","record_id":"<urn:uuid:b9fd7043-8a4d-4646-9e6f-f110026f0d16>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00652.warc.gz"} |
January 2021 – Thinknet
Over at Applied Category Theory discord server Sven Nilsen and I with others have been having a conversation about his idea of Inside Theory and Outside theory that is relevant here. See https://
Invite: https://discord.gg/hTEpgYv
I will reproduce the text my conclusion about our discussion here.
Hate to interrupt your discussion but lets connect this to something I can understand. Formal Systems have three properties which are Completeness, Consistency and Clarity (Well formedness). These
properties are the relations between three of the aspects of Being which are Identity, Presence and Truth. But when we add Reality as the fourth Aspect then we get three other properties which are
Verification, Validation and Coherence. That means there are six properties that are the relations between the Four aspects of Being. Now due to Godel we know we have to choose either Consistency or
Completeness. You cannot have both. Similarly you cannot have both Verification and Validation at the same time. Nor can you have both Clarity and Coherence at the same time. This last part is my
extension of Godel’s insight. If you take one as Absolute then the other has to be Relative from each pair. There is something that I call the Axiomatic Platform. That is the set of axioms taken
together that is a platform for building convex theorems within an axiomatic framework. This kind of framework serves as the basis for the formal system. We want formal systems to be Complete,
Consistent and Clear all at the same time but this is impossible. You get to choose either Completeness or Consistency but not both. And further outside the convex closure of the axiomatic system of
the formalism in its extension there is always the question of the relation of the system to its environment, what is beyond its convexity. That of course is the meta-system which is what is beyond
the system boundary to the horizon. We take what is beyond the formal system as reality and so then we have the other three properties which are Verification, Validation and Coherence. We cannot make
validations absolute without forcing verifications to be relative and vice versa. Similarly with clarity and coherence. If one is absolute then the other is forced to be relative.
This is a speculative extension to the idea of Godel with respect to formal systems, taking into account that formal systems are defined by the different aspects whose relations produce the various
properties of formal systems in relation to their meaning that comes from their relation to reality. Now I am not sure what inside and outside theories are. But lets pretend we do know and say that
inside theories are within the concave closure of the axiomatic system while outside theories are not. In other words outside theories refer to reality beyond the closure made possible by the
axiomatic platform. What Deconstruction tells us is that there are meta-levels of inside and outside the formal system. We know this is true because Tarski says that truth/falsehood is only at a
meta-level. What Derrida has showed in relation to Husserl’s phenomenology in Speech and Phenomena is that if you push the inside to a further level deeper inside you end up outside and Zizek
suggests that the same is true of the outside. So you notice that if truth/falsehood is at a meta-level from the first order language of the formal system of logic, then we need to push one level
beyond truth/falsehood distinction and suddenly you are outside, i.e. the inside/outside distinction vanishes. And this can occur if we push into the fourth dimension. You can get inside or get out
of a sphere in the fourth dimension without piercing the convex shell of closure. I call these openly/closed systems and believe Victor Frankl was the first to suggest that these exist. But we can
see Leibnizian monads as an example of these openly closed systems.
It is basically this idea that Zizek is taking from Derrida and developing with those equations. It posits that there is a way to get beyond inner and outer through the inside of the convex formal
system and this inconsistency is what Godel posits for every even modestly formal system via the liar’s paradox. There is always contamination that can get through the barriers we attempt to erect
against paradox. This has a huge effect on set theory for instance. But does not effect Category Theory. That is why we advocate switching over to the weaker Category Theory representation of
systems, but that takes us from entities to relations and functional bases for relations. Entities can drop out and identity arrows stand in for the missing entities.
Anyway perhaps you can explain what you are talking about in this framework, if you get what I am talking about here, i.e. apply it to formal systems of logic or set theory etc. I talk about these
things in my tutorial on Schemas Theory at
You know this also reminds me of what we are reading about with Zizek’s Sex and the Failed Absolute with regard to for instance Set theory where the bottom of the lattice is the null set and the top
of the lattice is Universe instead of the Set of All sets. Top of the lattice for sets gives paradox that needs to be avoided. Bottom of lattice is the null set which is an exception that is not a
member. Each one is an extreme that sets the limit on the Set. Null set 0 goes beyond empty set (). Zizek says that this is related to the difference between the antimonies which is related to what
Lacan calls sexation. It seems like these formulas of Derrida might be a version of this strange structure of exception on one side and non-all on the other side. ALL is Outside3 and Null set 0 is
Inside3. Non-All of Set Universe is Outside2 and Empty Set () is Inside2. Set with something in it is Outside1 and particulars in Set is Inside1. If this is the case it is equivalent to Lacanian
Sexation which is also equivalent to the Antimonies in Kant.
Given this an Inside Theory in your terms is one in which the axioms hold and as you work out the theorems you stay within the bubble of the convex closure of the axiomatic platform. Outside Theory
has at least something that is referred to beyond the closure bubble and therefore in some way escapes the illusion that the Inside Theory is complete and consistent. For instance we don’t actually
know if the axioms of set theory are ultimately complete and consistent but Godel tells us that they are not and guess what they result in paradox at one end and null set at the other end of their
lattice. In Inside theory there has to be something like the Null set 0 which is the same as the liar paradox that breaks the consistency and completeness of the convex closure of the illusion that
everything is covered by the theory within its scope. On the other hand Outside Theory has the problem of paradoxical, i.e. its referent can be an anamorphic object, i.e. not really beyond, i.e.
transcendent, i.e. noumena. Pushing for a beyond lands you in Paradox outside the set so we have to stick to the Universe and avoid Large Sets, i.e. infinities. Measure between levels of infinity is
lost. You push too far outside in an Outside theory and you go straight into paradox at the level of recognizing a referent as actually transcendent. If this is true, then I think I understand what
you might be saying when you distinguish Inside Theory from Outside Theory. And it seems to be a version of what Derrida says in Grammatology about deconstruction and what Zizek says in his various
works about the relation between Male and Female sexation paradoxes that he relates to the Antimonies of Kant. And this addition that you have made allows me to connect my ideas of the Axiomatic
Platform to these various other ways of framing what Badiou says about Set Theory which he got from Lacan evidently.
We can clearly see these three layers of Inside and Outside in Set Theory. Badiou’s whole ontology in Being and Event is based on this structure within the Set. Zizek talks about it in term of the
Gap between the sexes from Lacan’s theory of Sexation. But it helps to see it as something relating to the Axiomatic Platform of the Formal System. This means that Completeness and Consistency of the
Inside Theory breaks down due to the exception, the liar paradox in Inside3 level. Reality is on the Outside and and the Outside Theory breaks down at the limit of the Transcendent noumena and
because of that you cannot both verify and validate. And this is seen in Rescher’s idea of Cognitive Systematization where you need to circulate continually through the axioms because you cannot know
that they themselves lead to a a closed convex envelope so there may be some break in the envelope that allows you to get through to the transcendent noumena by some secret passage for instance
through higher dimensions. Verification means all statements about the requirements for the transcendent are true, Validation means that what is modeled in the closure of the Inside Theory actually
does simulate what is Outside. But the key difference is between clarity and coherence that Deleuze talks about in Difference and Repetition because Clear breaks away from Distinct. He says you
cannot have both Clear and Distinct that Descartes call for at the same time. That means that either things are obscure inside or indistinct outside. And that means that the coherence of inside and
outside may or may not be supported by God connecting the soul to the transcendental object as noumena. Thus you can see that the three pairs Complete/Consistent, Verifiable/Validatable, and Clear/
Coherent do seem to have this Godelian tradeoff or uncertainty that seems structural given the basis in Set Theory of the Inside and Outside Theories.
However, if we instead use Category Theory which is weaker then we avoid paradox because element can be dropped for identity arrows, and Categories of Categories are possible because there is nothing
inside that can be not-All. In other words everything are structural relations mappings and no entities. Category Theory does not have a membership function which is a Having. With the membership
function Having is squared. Sets have to have things in them as members. Categories are pure relationships based on functions so they do not have to have anything in them. Having along with Being are
always fragmented roots in Indo-European languages. Having produces paradox. Now we can see why Having and Being are both fragmented. Having is the equivalent of the membership function that places a
particular within a set. The set can exist without anything inside it, just as category theory can drop its elements. That is why Badiou needs the multiple to produce particulars. Being on the other
hand is the spectre of the Non-all in relation to the Set of all sets and that gives us the paradox of the transcendent in relation to the immanent.
Summary of discussion: I am thinking about trying to write a paper on our discussion here, perhaps you @Sven Nilsen could write a paper about it as well so I can refer to it. This is a breakthrough
in the sense that several different models that were separate in my mind suddenly became related via your suggestion that there may be inside and outside theories. But when I think about writing
about it, it still seems nebulous so I think more work needs to be done to reconcile the different models. But I want to try to capture it in a working paper while I still think I understand what has
been said here. The major convergence is between Zizek’s use of Derrida’s deconstruction equations, what Zizek says about Kant’s Antimonies in Sex and the Failed Absolute and Sexation, and finally
applying this to the Axiomatic Platform based on your distinction between Inside and Outside Theories. But for me the real breakthrough is the articulation in this context the relations between
Aspects of Being and Properties of the Formal System.. This has been a long time speculative hypothesis of mine, extending the properties based on the relation of reality to identity, presence and
truth. And further extending Godel’s indecision between Consistency/Completeness to Verification/Validation and Clarity/Coherence. But when one places these Godelian pairs of undecidable elements in
the context of the overall structure of the Set that reflects Sexation differences then you get to see how these pairs operate in that context and that is very interesting. But it is a little bit
complex. However, it seems to serve to validate the speculative hypothesis that these two things: differences between Godellian pairs and structure of the Set according to the layers of Inside and
Outside model work together. And the fact that you @Sven Nilsen seem to have discovered this Inside/Outside relationship which I call the openly closed system on your own is further evidence that the
hypothesis might be correct. | {"url":"https://think.net/2021/01/","timestamp":"2024-11-04T21:02:17Z","content_type":"application/xhtml+xml","content_length":"49770","record_id":"<urn:uuid:1dbe0f3a-24a9-434d-8b76-8eb5ab8f50d0>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00734.warc.gz"} |
Theory for Domestic Heating Emissions Parametrization
Theory for Domestic Heating Emissions Parametrization#
The theoretical foundations for parameteric modelling of domestic heating emissions are derived from the works of Baumbach et al. (2010) and Struschka and Li (2019), which are based on a direct
relationship between emissions and energy consumption. Emissions are calculated using the so-called emission factors, which are species and technology dependent, while energy consumption are
functions of the size, geometry, age, and function of the individual buildings. These forms the two aspects of the discussion below. Typically, buildings with a footprint of less than 10 m^2 and an
effective height of less than 3 m are not considered in the calculation.
As values of all required parameters are either given as default or provided as documented in the domestic model overview, this document will focus on the mathematical foundations for the
parametrized model. The reader is also encouraged to refer to the works listed in the references section below for further detail.
Daily Energy Consumption#
The daily energy consumption of a building (\(E_B\)) at any given time can be expressed as a ratio between its annual aggregate and the number of degrees to be heated to maintain its target indoor
\(\displaystyle E_B\left(t\right) = \left(\frac{E_B}{\Delta{T}}\right)_A \Delta{T}\left(t\right){\quad\quad}\) (1),
\({\quad}\left.E_B\right|_A\) is the annual energy consumption of building \(B\),
\({\quad}\left.\Delta{T}\right|_A\) is annually accumulated temperature deficit, also known as heating degrees, and
\({\quad}\Delta{T}\left(t\right)\) is the current temperature deficit, known as heating degree days.
The value of \(\Delta{T}_A\) is provided as user input. The calculation of the remaining terms will be further presented below.
Annual Energy Consumption#
The annual energy consumption takes into account the volume, compactness, and energy demand of the building:
\(\displaystyle \left.E_B\right|_A = \left.\kappa_\beta\right|_A\Phi_\beta{V_B}{\quad\quad}\) (2),
\({\quad}\Phi_\beta\) is the compactness factor of the building type \(\beta\) belonging to building \(B\),
\({\quad}\left.\kappa_\beta\right|_A\) is the annual energy demand of the building type \(\beta\) belonging to building \(B\) per unit footprint area, and
\({\quad}V_B\) is the volume of the building \(B\).
The compactness factor, in unit of 1/m, is a density indicator of the building. On the other hand, the annual energy demand is the amount of energy to be consumed annually per unit area of the
building. Tabulated values of these two quantities are available for each building type (\(\beta\)) and are used as default inputs.
Temperature Deficit (Heating Degree Days)#
The temperature deficit, or heating degree days, is calculated by subtracting the user-defined base temperature from the outdoor ambient temperature:
\(\displaystyle \Delta{T}\left(t\right) = \max\left\{\ 0, \left[T_0 - T_\infty(t)\right]\ \right\}{\quad\quad}\) (3),
\(T_0\) is the base temperature, and
\(T_\infty\left(t\right)\) is the ambient temperature.
When the ambient temperature (\(T_\infty\)) is greater than the base temperature (\(T_0\)), there will be no temperature deficit (i.e., \(\Delta{T}=0\)) and it is assumed that no heating will be
The ambient temperature is, in turn, the mean temperature over the volume for region of interest (\(V\)) up to a user-defined sampling height above ground level (\(\eta\)):
\(\displaystyle T_\infty\left(t\right) = \frac{1}{V}\iiint_\eta{T}dV{\quad\quad}\) (4).
Ideally, the sampling height (\(\eta\)) should be chosen so that the temperature is representative of that in the urban canopy. Because of this, to reduce computing resources, \(\Delta{T}_D\) is only
calculated at fixed time intervals to minimize computing resources.
Emissions Source#
The emission of the individual species (\(\epsilon_B^k\)) can then be calculated based on the building energy consumption \(E_B\):
\(\displaystyle \epsilon_B^k = E_B\psi^k{\quad\quad}\) (5),
where \(\psi\) is the emission factor for pollutant species \(k\).
The emission factors \(\psi\) are tabulated on either a molar or mass basis for the energy consumed, and are expressed in mol/TJ for gas phase species and kg/TJ for particle species.
Note: Input quantities to the model are indicated in parentheses.
Symbol Unit Description
\(A\) - - Subscript denoting annual aggregate
\(B\) - - Subscript denoting building
\(E\) J Energy consumption
\(k\) - - Superscript denoting emission species
\(T_0\) K Base temperature (input)
\(T_\infty\) K Ambient temperature
\(t\) s Time
\(V\) m^3 Volume
\(\beta\) - - Subscript denoting building type
\(\Delta{T}\) K Temperature deficit (heating degree days)
\(\Delta{T}_A\) K Annual heating degrees (input)
\(\epsilon\) mol/s or kg/s Emission
\(\kappa\) J/m^2 Energy demand per unit footprint area (input)
\(\Phi\) 1/m Compactness factor (input)
\(\eta\) m Sampling height for ambient temperature (input)
• Baumbach, G., Struschka, M., Juschka, W., Carrasco, M., Ang, K.B., Hu, L. (2010) Modellrechnungen zu den Immissionsbelastungen bei einer verstärkten Verfeuerung von Biomasse in Feuerungsanlagen
der 1. BImSchV. Umweltbundesamt (Dessau-Roßlau), ISSN 1862-4804.
• Struschka, M., Li, L. (2019) Temperaturabhängige zeitliche Disaggregation von Emissionen aus Feuerungsanlagen der Haushalte und Industrie für Berlin im Rahmen des MOSAIK-Projektes. Universtiät | {"url":"https://docs.palm-model.com/24.04/Guide/LES_Model/Modules/Chemistry/DOMESTIC_model_LOD0_theory/","timestamp":"2024-11-10T14:53:03Z","content_type":"text/html","content_length":"29231","record_id":"<urn:uuid:92a59011-25a6-48f3-8ce5-71431bade14c>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00858.warc.gz"} |
Mathematics | Top Universities, Collages & Programs in Ethiopia
Mathematicians and statisticians are in demand across a range of sectors and employment opportunities are commonly found in:
1. Education engineering finance, banking and accountancy firms government – local, central and agencies insurance companies IT,
2. Business consultancy and operational research companies market research and marketing companies medicine and health – including private pharmaceutical companies and the NHS petroleum and nuclear
industries publicly-funded research institutes space science and astronomy. | {"url":"https://www.edmap.et/mathematics/","timestamp":"2024-11-06T00:58:46Z","content_type":"text/html","content_length":"126670","record_id":"<urn:uuid:5132f0e9-3b09-43eb-a9c9-11a0a0963ad4>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00637.warc.gz"} |
Creating new categories from old: Selection categories
In this short post, I’ll describe a way of creating new categories from old. It reminds me of a ‘particle filter’ or ‘natural selection’. This method comes from the theory of polynomial functors, but
I’ll confine all the technical details to a single section, so you don’t need to know anything about polynomial functors to read this post.
TL;DR: The following picture captures the entire idea:
Here, each of the a_1,a_2,\ldots,e_6 are objects in the original category, and every arrow is an arrow in the original category. In the associated select-5 category, an object is a whole column (e.g.
(a_3,b_3,c_3,d_3,e_3) is one object) and a morphism between them is a column of arrows as shown. The composite is the obvious thing you get by “path following”, e.g.
In some sense, it looks like a_1 and c_1 were the most “successful” of the initial cohort, since at stage 6, every object is “descended” from one of those two.
1 Introduction
There are many known ways to get new categories from old. Given a category C, you can take its opposite, you can adjoin a terminal object, you can take the coproduct or product of it with itself,
etc. In this post, I’ll tell you about a new such category-generating machine.
For today, I’ll call the result “selection categories”. The reason for the name might be clear from the example at the top: it was as though we had a situation with a limited number of slots—in that
case, five—and in every column we chose to extend only some of the previous objects. In the end, we have five things, but they are “descended from” only two of the original five. However, it is
important to note that nothing about “nature” is making the choices here—the category contains all possible choices—so I hope the name and metaphor aren’t too misleading. I won’t discuss selection in
the sense of evolution any further; instead, I’ll just describe the mathematics.
2 Input data
In order to run this construction, you need two things.
1. the category C you want to run it on, and
2. a polynomial p in one variable (I like to use y for my variable) whose coefficients are all natural numbers (or more generally, sets).
In the above case I used the polynomial y^5, but our second example will be y^5+y^3. Let’s write p=\sum_{i\in I}y^{p[i]}, and refer to each i\in I as a tag and the set p[i] as the slot-count for tag
i. The reason is that the p[i]’s tell you the legal numbers of slots that our columns are allowed to have in them. We also think of p[i] as a set, the set of slots. The introductory example had only
one tag with slot-count 5; instead of a number, we could think of this tag as representing a column, a set of five elements, (a,b,c,d,e).
Once we have our slot-counts, we want to actually select objects from our category to fill those slots. Just as we refered to y^5 as the “select-5” polynomial, let’s refer to y^5+y^3 as the select-5
or select-3 polynomial. If more than one summand has the same exponent, then we have to tag them seperately, e.g. we could refer to y^5+2y^3 as the select-5 or select-3-tag-A or select-3-tag-B
3 The selection category on p
Given a polynomial p=\sum_{i\in I}y^{p[i]} and a category C, we’re ready to say what the objects and morphisms in our new category are. In my own personal work I like to denote this new category \
left[\begin{matrix} p\\ p\triangleleft C \end{matrix}\right] because it comes from a certain “coclosure” operation in \mathbf{Poly} that generalizes lenses from functional programming. But again, you
can ignore that for today.
An object in this new category \left[\begin{smallmatrix} p\\ p\triangleleft C \end{smallmatrix}\right] is a pair (i, c), where i\in I is a tag and c\in\mathsf{Ob}(C)^{p[i]} is a selection of p[i]
-many objects in C. That is, we have filled our slot-count with objects; let’s call c a cohort. So a column like (a_3,b_3,c_3,d_3,e_3) in our original example constitutes one object—one cohort—in \
left[\begin{smallmatrix}y^5\\y^5\triangleleft C\end{smallmatrix}\right].
A morphism from (i,c) to (i',c') is a pair (f,g) where f\colon \{1,\ldots,p[i']\}\to\{1,\ldots,p[i]\} is a backwards-facing function between slot sets, and where g assigns to each slot s\in \{1,\
ldots,p[i']\} a morphism in C from c_{fs} to c_s'. So every slot in cohort c' receives a morphism from some choice of slot in cohort c. Take another look at the example to make sense of that.
Every map in \left[\begin{smallmatrix}p\\p\triangleleft C \end{smallmatrix}\right] can be uniquely factored as a composite of two simpler kinds of map. The first factor will be a pure selection (no
“aging”), meaning that the tags and slots can change, but all the arrows are identities. The second factor will be a pure “aging” (no selection), meaning that the tags don’t change and it’s a
“straight-across” map, but the arrows can be anything you want.
4 Examples
We promised to do y^5+y^3 next, i.e. to consider the category \left[\begin{smallmatrix} y^5+y^3\\ (y^5+y^3)\triangleleft C \end{smallmatrix}\right]. Rather than just repeating the definition in this
case, we’ll just draw a picture of five composable morphisms
So the number of slots varies, but the idea is identical.
The select-1 or select-0 from C category \left[\begin{smallmatrix}y+1\\(y+1)\triangleleft C\end{smallmatrix}\right] corresponding to the polynomial y+1, is given by adding a free terminal object to C
. That is, an object is either an object of C or an empty column, which we’ll call nothing. A morphism between two objects from C is just a morphism from C. There is a unique morphism from anything
to nothing, and the only map out of nothing is the identity.
Repeats don’t do too much. For example 3y would just give you three different tags for objects in C, that you can freely move between. That is, \left[\begin{smallmatrix}3y\\(3y)\triangleleft C\end
{smallmatrix}\right] is just the product of C and a “co-discrete category” of size 3 (where the co-discrete category on A has A-many objects and a unique morphism between any two).
As another example, y+3 would adjoin a whole co-discrete category of size 3 as a terminal subcategory. You can move around in C as much as you want, and then eventually go to three different sorts of
nothing, which you can move freely between forever.
Another fun one: if you take C=1 to be the one-morphism category and you take p=y^2+y, then the resulting category \left[\begin{smallmatrix}y^2+y\\(y^2+y)\triangleleft 1\end{smallmatrix}\right] is
the indexing category for symmetric reflexive graphs. It has two objects, say V and E for “vertex and edge”. And other than identities it has two maps E\rightrightarrows V for “source and target”, a
map V\to E for “identity edge”, and a map E\to E for “dual edge”.
That about does it! Next I’ll give a technical section for interested readers, but for the rest of you, I suggest you skip it and go right to the last section, where I describe a sequence of
important categories that arises from this construction.
5 Technical stuff
For any polynomial p, the construction \left[\begin{smallmatrix}p\\p\triangleleft -\end{smallmatrix}\right] is functorial; in fact it sends bijective-on-objects functors to bijective-on-objects
functors and sends fully faithful functors to fully faithful functors. It is in fact an oplax framed functor from \mathbb{C}\mathbf{at}^\sharp to itself, where \mathbb{C}\mathbf{at}^\sharp is the
framed bicategory of comonoids and comodules in \mathbf{Poly}.
Given any polynomials p,q and category C, there is a canonical profunctor from \left[\begin{smallmatrix}p\\ p\triangleleft C\end{smallmatrix}\right] to \left[\begin{smallmatrix}q\\q\triangleleft C\
end{smallmatrix}\right]. It assigns a set to any pair ((i,c),(j,d)) of objects, where c\in C^{p[i]} and d\in C^{q[j]}, namely: it assigns the set of pairs (f,g) where f\colon q[j]\to p[i] is a
backwards-facing function and g assigns to each slot s\in \{1,\ldots,q[j]\} a morphism in C from c_{fs} to d_s. Very reasonable, right? And in the presence of a cartesian map of polynomials p\to q,
the above profunctor arises as the pullback along a fully faithful functor \left[\begin{smallmatrix}p\\p\triangleleft C\end{smallmatrix}\right]\to\left[\begin{smallmatrix}q\\q\triangleleft C\end
One last kind of interesting technical thing. For any p,q there is a bijective-on-objects functor from \left[\begin{smallmatrix}p\\p\triangleleft C\end{smallmatrix}\right]+\left[\begin{smallmatrix}q\
\q\triangleleft C\end{smallmatrix}\right] to \left[\begin{smallmatrix}p+q\\(p+q)\triangleleft C\end{smallmatrix}\right]. For example, taking p=P and q=Q to be sets, consider the complete graph K_P on
P vertices and the complete graph K_Q on Q vertices, but think of them as categories (contractible groupoids). Their disjoint union K_P+K_Q is not the complete graph K_{P+Q} on P+Q-vertices, but it
is still a category and there is a bijective-on-objects functor K_P+K_Q\to K_{P+Q} between them.
6 One last trick
Consider the list polynomial u=1+y+y^2+y^3+\cdots. The functor C\mapsto\left[\begin{smallmatrix}u\\u\triangleleft C\end{smallmatrix}\right] is one that my colleague Evan might call the category of
“product diagrams in C”. But I’m interested in a slight variant of this, namely \left[\begin{smallmatrix}u\\u\triangleleft -\end{smallmatrix}\right]^{\text{op}}.
Suppose you apply this functor repeatedly, starting from the empty category 0; what do you get?
Applying it once you get \left[\begin{smallmatrix}u\\u\triangleleft 0\end{smallmatrix}\right]^{\text{op}}=1, i.e. the one object category, because the only way to write down N objects from the empty
category is if N=0.
Ok, we have 0, 1, what’s next? We calculate that the next iteration \left[\begin{smallmatrix}u\\u\triangleleft 1\end{smallmatrix}\right]^{\text{op}}\simeq\mathbf{FinSet} is a skeleton of finite sets.
Indeed, just write down columns of any size (all with “1” filling every slot), and identity maps backwards between them. Then, since we’re supposed to take an opposite, flip the direction of the
arrows so the maps go forwards. Nothing’s going on except the number of dots, so you see we just get the category of finite sets and maps between them!
Ok, so we now have 0, 1, \mathbf{FinSet}. What’s next? I’ll save that for the curious reader. Feel free to post it in the comments below if you figure it out!
Leaving a comment will set a cookie in your browser. For more information, see our cookies policy. | {"url":"https://topos.site/blog/2021-12-30-selection-categories/","timestamp":"2024-11-08T14:00:37Z","content_type":"application/xhtml+xml","content_length":"44296","record_id":"<urn:uuid:5860a33b-e1c2-4e3a-9c12-317016e96839>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00676.warc.gz"} |
The Huge Missing Factor In LBB Analysis – How A Circumferential Through-Wall-Crack In A Pipe System Changes The Flexibility And Reduces The Applied Moments
Transactions, SMiRT-25 Charlotte, NC, USA, August 4-9, 2019
Division II (Fracture Mechanics and Structural Integrity)
G. Wilkowski1[, M. Uddin, F.W. Brust, S. Kalyanam, ]
1[Engineering Mechanics Corporation of Columbus (Emc]2[), Columbus, OH, USA ]
(contact: [email protected])
In the nuclear industry leak-before-break (LBB) analyses have been conducted for decades. Typically the uncracked piping normal operating forces and moments are applied in a cracked-pipe analytical
procedure to determine normal leakage, and the combined forces and moments under normal operating condition and safe shutdown earthquake (SSE) seismic loading (N+SSE) are used in a fracture analysis
to predict margins on “failure”. This evaluation has been performed in deterministic analyses such as NRC SRP 3.6.3, as well as in probabilistic analyses (i.e., xLPR, PRAISE, NURBIT, PROST) that use
a number of independent deterministic models with the uncracked piping forces and moments (or stresses). The International Piping Integrity Research Program (IPIRG) which ran from about 1990 to 1998
was first to provide some interesting insights to typical LBB behaviors. In that program, pipe system tests were conducted with simulated seismic loadings which were designed based on finite element
analysis (FEA) of the pipe system. The test results showed a large margin on LBB due to a number of factors which were not fully recognized at that time. One of the key factors was recognized in 2011
when the Argentinian Atucha II PHWR plant was analyzed using a full FE model that included the containment building, reactor pressure vessel (RPV), steam generator (SG), pumps, main coolant line,
surge lines, pressurizer and main steam lines and all the supports from the components/pipes to the building. It was found from this robust FE modeling effort that when circumferential through-wall
cracks were put in the highest stressed locations, the applied moment dropped for both normal operating and N+SSE loading as the crack length increased. With loading three (3) times higher than SSE
loads, the through-wall crack size for causing a double ended guillotine break (DEGB) was greater than 90% of the circumference. Similar results were also found for a petrochemical pipe system where
thermal expansion stresses are much higher when compared to other primary stresses (pressure, gravity). Even with very low toughness materials of petrochemical plant (due to high temperature hydrogen
attack), the critical crack size leading to DEGB was greater than 80%. The implication of this work is that pragmatically a DEGB is not a critical concern for nuclear plant operation, and efforts
would be better focused on the potential for a small-break loss-of-coolant accident (SB-LOCA).
The International Piping Integrity Research Program (IPIRG) which ran from about 1990 to 1998, first provided some insights into these behaviors. In that program, pipe system tests were conducted
with simulated seismic loadings, albeit it was not a real pipe system as built for service. In the design and analyses of those pipe system tests, finite element (FE) models of the entire pipe system
were used and large margins were found due to a number of factors, of which not all of them were recognized until about 2011. In 2011 (Wilkowski et al. 2011), a full FE model was created for the
Argentinian Atucha II PHWR plant including; the containment building, reactor pressure vessel (RPV), steam generator (SG), pumps, main coolant line, surge lines, pressurizer and main steam lines and
all the supports from the components/pipes to the building. These difficult and tedious efforts provided valuable insights into the system behavior under stress. Seismic motions were applied to the
basemat of the containment building FE model and the forces and moments were developed in the pipe system through the support to the large components and piping. In these efforts, it was found that
when circumferential through-wall cracks were put in the highest stressed locations (i.e., pipe weld to vessel nozzles), the applied moment dropped for both normal operating and N+SSE loading as the
crack length increased. In fact, the moment dropped faster than the increase in crack driving force arising from the increase in the length of the crack. With loading 3 times higher than SSE loads,
the through-wall crack size for causing a double ended guillotine break (DEGB) was greater than 90-percent of the circumference.
Similar results for a petrochemical plant have also been found, but with additional exploration of what happens if different pipe supports are damaged or there are temperature excursions to give much
higher thermal expansion stresses (Wilkowski et al. 2018, Uddin et al. 2019). The petrochemical plant material had an extremely low toughness due to hydrogen attack (much lower than the nuclear
piping materials evaluated), but because of the moment decrease with increasing circumferential through-wall-crack length, the critical through-wall-crack size was a circumferential through-wall
through-wall-crack greater than 80% of the circumference.
These results will be presented in this paper with detailed insights as to why the applied moment decreases with increase of through-wall circumferential crack size. The potentially bounding
worst-case condition for a pipe system would be one with very high inertial loading (i.e., unsupported loop isolation values) along with a low material fracture resistance (CF8m with high sensitivity
to thermal aging or carbon steel that is very sensitive to dynamic strain-aging). The implication of this work is that pragmatically a DEGB is not a critical concern for nuclear plant operation, and
efforts would be better focused on the potential for a small-break loss-of-coolant accident (SB-LOCA).
actual critical through-wall crack length (95% of the circumference) for this test is even larger than from just pressure load consideration using typical limit-load or EPFM assumptions. The reason
for this was that in a pipe system the ends of the pipe are not free to rotate, so that bending moments induced from the pressure/endcap loading are restrained. This was not understood until a full
nuclear steam supply system (NSSS) was analyzed using robust finite element modeling as described below.
Figure 1 Moment-time record from IPIRG-1 Experiment 1.3-7
(a) (b)
Figure 2 Fracture surface from IPIRG Experiments on 16-inch diameter pipe showing DEGB occurred when the ligament was reduced to (a) ~5% of the circumference for Test 1.3-7 (b) 3.3% to 5.5% of the
circumference for Test 1.3-2
As seen in these past experimental results, the critical through-wall crack length for DEGB was found to be much larger than those predicted by simple limit-load and J-estimation procedures. The
reason for this mis-prediction was not understood until various finite element analyses (FEA) were conducted for entire piping systems for a nuclear power plant and a typical refinery piping system
as described below.
t, kN
Time, seconds
Load for resulting through-wall crack
(Note, after the surface cracks break
through the thickness at ~2.5 seconds, the
FE Analysis of a Full Nuclear Steam Supply System (NSSS)
A very large FE model was created for the entire NSSS of the Argentina Atucha II nuclear plant including the containment building, reactor pressure vessel, steam generator, pump, primary pipe loop,
supports between the building and the components and piping1[, see Figure 3. The details can be found in ]
References [Wilkowski et al. 2011, Uddin et al. 2015, Uddin et al. 2014]. The loading was pressure, dead-weight, thermal expansion, and seismic excitation from the basemat of the containment building
and then naturally transmitted though the supports to the components/piping. However, before starting to assess the LBB condition, one must ensure that a circumferential through-wall crack will
develop from initial surface crack before it becomes a long surface crack, so that additional evaluations are not needed to determine if a long surface crack could develop into a rupture.
In doing so, an iterative FE analysis was conducted with the worst-case SCC crack growth rate observed in any nuclear plant (even though those materials were not used in the Argentina plant). The
iterative FE approach crack shape modelling and some validations are described in detail in References [Shim et al. 2012 and Shim et al. 2010]. The high crack growth rate was used since the design
was for 80 years of life. The weld residual stresses were determined using the welding procedures, including stress relieving for this ferritic weld case (most austenitic welds in nuclear plants are
not stress relieved) and including clad thermal strain mismatch. Those stresses along with the normal operating stresses at various locations along the primary loop were used to calculate the crack
shape as a function of time. Many validations of the weld residual stress modelling are included in various other references (Zhang et al. 2009, Smith et al. 2010). As seen in Figure 4, regardless of
initial flaw shape, it always resulted in nearly-idealized circumferential through-wall-crack shape.
(a) NSSS system (b) With containment building
Figure 3 FE model of NSSS system and containment building of Atucha II nuclear plant, where cracks were inserted at critical locations by nozzles in the primary loop piping (Wilkowski et al. 2011)
The uncracked pipe peak stress was first used to determine a through-wall “critical” crack size. The crack was inserted in the FE model and centered on the high bending moment plane. After inserting
the initial crack (based on it being “critical” using traditional LBB evaluation), it was found that the peak applied load was changed to a lower value than the uncracked pipe moment, see upper left
graph in Figure 5. The circumferential through-wall crack was increased in length and in each step the load kept decreasing until the through-wall circumferential crack changed from the initial
length of 15% of the circumference based on a typical LBB load-controlled analyses assumptions to being a circumferential
1[ A further refinement to the Atucha nuclear system FE model included adding in the pressurizer vessel, surge line, ]
length of ~95% of the circumference when doing a fully pipe-system analysis with actual loading (see Figure 5 (a)).
Figure 4 Natural crack shape for large crack growth from surface crack to through-wall crack (Wilkowski et al. 2011),(legend is in years of crack growth)
The crack length was also much larger than from just pressure load consideration using typical limit-load or EPFM assumptions. The reason for this was that in a pipe system the ends of the pipe are
not free to rotate, so that bending moments induced from the pressure/endcap loading are restrained. Illustrations of this effect are shown in Figure 2 where in pipe-system fracture experiments
(Wilkowski et al. 1997) with internal pressure, thermal expansion and seismic loading the final DEGB did not occur until the crack lengths were about 95% of the circumference even though the
traditional limit-load analysis calculated the critical through-wall-crack length to be ~60% of the circumference. This crack size corresponded to the axial load from the pressure on the ligament
that corresponds to an axial membrane stress equal to the material flow stress, which gives a revised limit-load criteria for pressure loads in a pipe system (Kalyanam et al. 2017).
In the nuclear industry the leakage of subcooled water is determined by a number of software codes, one of which is called SQUIRT (Paul et al. 1994, SQUIRT 2009). SQUIRT was developed for the US NRC,
validated by many leak-rate tests, and was also used in the Atucha II evaluation in Reference [Wilkowski et al. 2011]. The crack was inserted in the FE model of the whole pipe system to determine the
crack-opening area under normal operating loads, so it accounted for any restraint of the induced bending from the pressure loads. With the axial membrane loads not having the induced bending in a
pipe system, the opening area is smaller than a typical calculation. The Atucha II plant had very good leakage detection capability (because of tritium in the water) which can easily detect leak
rates much earlier than reaching the critical crack size for causing DEGB (see Figure 6(b)) and hence, LBB is satisfied with much larger margin.
-200 -175 -150 -125 -100 -75 -50 -25 0
Pipe-ID Pipe-OD Initial Model - 0 Years
0.124482 0.207926 0.344701
0.480761 0.602391 0.70293
0.795421 0.883179 0.96712
1.01322 1.05732 -450 -350 -250 -150 -50 50 150 250 350 450
-900 -800 -700 -600 -500 -400 -300 -200 -100 0
X-axis,mm Y-axi s, mm Pipe-ID Pipe-OD 1.05732 1.45636 1.72602 2.0163 2.21043 2.34027 2.427 3.48241 3.51881 3.54019 3.54486 Y -ax is, mm X-axis, mm -450 -350 -250 -150 -50 50 150 250 350 450
-900 -800 -700 -600 -500 -400 -300 -200 -100 0
X-axis,mm Y-axi s, mm Pipe-ID Pipe-OD 1.05732 1.45636 1.72602 2.0163 2.21043 2.34027 2.427 3.48241 3.51881 3.54019 3.54486 Y -ax is, mm X-axis, mm
(a) Showing natural SCC flaw shapes up to incipient leakage from an initial flaw length of 5 times the maximum undetectable flaw length and maximum undetectable flaw depth
Figure 5 Illustration of how applied moments decreased with increasing circumferential through-wall crack length for same seismic load when modeling the cracked pipe in a complete pipe system for the
Atucha II primary pipe loop (Wilkowski et al. 2011)
Figure 6 Change in applied moments and leak rates with circumferential crack size in Atucha II plant
FE Analysis of a Typical Refinery Piping System
Recently FE analysis of a typical (non-stress relieved) piping system sensitive to high temperature hydrogen attack (HTHA) cracking as shown in Error! Reference source not found. Figure 7 was
conducted to explore conditions in which LBB can be applied, i.e. determine the critical flaw size and leakage analyses with its design boundary conditions under operating loadings, i.e., gravity,
pressure (2.0 MPa), thermal (316 C) and hanger loadings. Details can be found in References [Wilkowski et al. 2018, Uddin et al. 2019]. One side of the system contains a vessel. Typical boundary
conditions for the piping system are shown in Figure 7Error! Reference source not found.(b) where all straight-pipe sections, elbows, and vessels are drawn in a wire diagram. The diameter and
thickness of straight pipes and elbows
IGSCC crack morphology
0 10,000 20,000 30,000 40,000 50,000
Crack length, degrees
Lea k Ra te, L /min 0 200 400 600 800 1,000 Cumulative l ea ka ge RPV v ol ume
With free end rotation With NSSS with natural restraint Total leakage/RPV volume
Unrestrained ends
Natural nozzle restrained ends Cumulative leakage assuming worst-case SCC crack growth rate Plant shut down at
0.166 l/min
Plant shutdown at cumulative leakage volume of 30 tons (27.45 m3
or 4.6% of RPV volume)
Cumulative leakage equal to 100 times total volume of RPV (600 m3[)]
(a) Change in applied moments in pipe system with circumferential crack length in Atucha II primary pipe loop (Wilkowski et al. 2011, Uddin et al. 2014)
are 18 inch (457.2 mm) and 0.5 inch (12.7 mm) respectively. For the sake of discussion, the piping system is labelled as Loop 1 and Loop 2. There are a total of twelve elbows in the piping system as
well as flanges and valves whose masses are given in Figure 7Error! Reference source not found.(b). To offset the gravity loading two types of hanger supports are employed; constant-force hangers and
variable-force hangers.
Similar to nuclear piping, FE analyses were also performed to study the development of through-wall cracks in refinery piping as they are typically much thinner than nuclear piping and the ferritic
welds are made with much larger weld beads than nuclear plant piping. The residual stress analyses were conducted using a moving arc welding simulation so that there was a start and stop to each weld
pass. Details can be found in Reference [Wilkowski et al. 2018]. Although realistically there are multiple stop-starts in making a Submerged Metal Arc Weld (SMAW) with varying time delays between a
stop and start, we used only one stop-start per weld bead pass. Furthermore, as a sensitivity study for each weld bead layer, we staggered the stop-start positions of the weld, see Figure 8(a).
The weld simulation results were quite different than thick-shell or axisymmetric solutions for nuclear piping (both for as-welded conditions, i.e., no stress relieving). Some of the key differences
were; (1) The as-welded transient start-stop stresses were controlled entirely with the start-stop position of the very last weld pass (315-degree position in Figure 8(a)), and (2) Additionally on
the ID surface where HTHA would start, the as-welded stresses are tension on the thinner pipe side than the thicker fitting pipe side. Figure 8(b) shows the longitudinal as-welded residual stresses
with the above observations. Hence from this analysis, a HTHA crack in non-stress relieved pipe is more likely to initiate and grow through the thickness at the stop region, continue to grow on one
side of the weld, and have a region of no cracking near the start position of the weld where there are through-thickness compressive stresses.
(a) (b)
Figure 8 Results of weld residual stress simulation (Wilkowski et al. 2018)
“Natural crack growth” simulations were then conducted with welds in the as-welded condition. An initial small surface crack was inserted in the highest as-welded transient residual stress region
shown in Figure 8(b). A high HTHA crack growth rate based on very limited data from Reference [Shewmon et al. 1991] was used, see Figure 9. This growth rate is about 10 times larger than the fastest
rate for the nuclear piping PWSCC degradation mechanism. The results of one time-step along the natural crack growth analysis showed the crack quickly became a through-wall crack which is reasonably
close between the service crack.
Figure 9 Results of natural crack growth simulation (Wilkowski et al. 2011)
(a) Schematic of weld start-stop positions (b) Longitudinal stresses from as-welded residual stress FE analysis looking at ID of the pipe, and some cross sections (ksi)
(a) HTHA crack growth rate used, from Reference [Shewmon et al. 1991] data
Circumferential through-wall cracks of various sizes were put in critical crack locations to determine the critical crack size using cracked-pipe system analyses. Note that critical crack locations
of the pipe system were determined by running various stress analyses on the uncracked pipe system with various boundary conditions where the critical locations correspond to maximum effective moment
locations of the pipe system. Four cases of cracked-pipe system were analyzed based on critical crack locations where cracks were found in service as well as the variation of loading conditions, i.e.
combination of primary stresses (gravity, pressure) and secondary stresses (thermal). In order to include the effect of HTHA in the FE analyses, the lower bound A106B J-R curve (material toughness)
at 288 C was reduced by a factor of 0.25. Later, from single-edge-notched tension (SENT) tests of HTHA degraded A106B materials, it was found that the reduced lower-bound J-R curve was reasonably
close to HTHA degraded A106B J-R curve.
For each crack size and crack location, the applied moment-rotation output from the pipe system analysis was extracted and compared with the moment capacity for the corresponding crack sizes. Some
representative moment-rotation inputs and outputs are shown in Figure 10 and Figure 11. As seen in Figure 10 for Case 1 where the applied loading is mostly displacement-controlled, the applied moment
in the pipe system is in the elastic range for all crack sizes indicating that the cracks at this location under Case 1 will not initiate even for a crack size of 80% of the circumference. On the
other hand, the applied moment for Case 2 in the pipe system (see Figure 11) exceeds the elastic range and is beyond the maximum moment capacity of the crack for crack sizes greater than 71% of the
circumference, although the crack became stable with significant crack growth. This indicates that the crack may start to have ductile tearing for crack size of about 53% of the circumference and the
crack will grow in a ductile manner until it has a much longer length and very large crack opening for leakage.
The results of all four cases with five different crack sizes are summarized in Figure 12(a) where Case 1 and Case 5 correspond to primarily displacement-controlled (mostly thermal loading) and Case
2 and Case 4 correspond to primarily load-controlled (mostly gravity, pressure and hanger loading). It is pertinent to note that only Case 1 (displacement-controlled behavior) corresponds to actual
design boundary condition whereas the other three cases were artificially simulated to capture the pipe system behavior in case of a support failure and/or hanger failure. As seen in Figure 12(a),
two displacement-controlled loading cases showed a significant drop in applied moment for smaller crack sizes relative to the uncracked pipe moment and then the moment drops gradually as the crack
size increases, crack initiation and ductile crack growth would not occur until it reaches a crack size (>80% of the circumference) when DEGB occurs due to internal pressure only - very similar to
what was shown for Atucha II nuclear plant piping system. On the other hand, two load-controlled loading cases where primary stresses (gravity, pressure) are higher due to support/hanger failure
showed somewhat different behavior where the applied moment does not drop much for smaller crack sizes and then the moment drops sharply after a certain crack sizes. Crack initiation and ductile
crack growth may occur at a particular crack size before it reaches DEGB condition. Additional crack growth calculations would be necessary for those cases to evaluate the pipe system stability.
Figure 11 Illustration of how applied moments go beyond the maximum moment capacity with increasing circumferential through-wall crack length for mostly load-controlled loading
Finally, the moment capacities of the various circumferential crack lengths were calculated by the most accurate of the J-estimation procedure (LBB.ENG2) from the nuclear piping development and
full-scale validation efforts (Wilkowski et al. 1998). The results are shown in Figure 12(b) which shows the tremendously large critical circumferential through-wall-flaw length (>80% of the
circumference) that can be tolerated in the pipe system for displacement-controlled behavior (Case 1 and Case 5) even with the lower-bound HTHA toughness and a safety factor of 3 on the applied
thermal moments. However, the critical crack sizes for load-controlled cases (Case 2 and Case 4) due to support/hanger failure are much shorter (about 30%-58% of the circumference).
Figure 12 Cracked-pipe system analyses of a typical refinery plant
Leak rates have been calculated for all five crack sizes for all four cases using the software code SQUIRT (Paul et al. 1994, SQUIRT 2009) mentioned in the previous section. The leakage rate is
calculated using the crack length, COD, pipe size, and operating pressure and temperature as input. In all calculations the single-phase steam option was chosen since it corresponds more closely to
the product in this pipe system. The crack morphology parameters of roughness, number of turns, and path deviation can impact the leak rate significantly. The thermal fatigue morphology most closely
resembles the intergranular stress corrosion cracking (IGSCC) crack morphology and the default IGSCC crack morphology parameters were used for these calculations which appeared to be conservative
compared to HTHA cracks. The leak detection capability of the current pipe system is less than 0.2 liter per minute (lpm) which corresponds to a leakage crack size of 25% of the circumference. As the
critical crack lengths for all cases are greater than 25% of the circumference, LBB is satisfied for all cases. However, the margin for displacement-controlled cases is much higher than that for
load-controlled cases.
(a) Change of applied moments with various crack sizes under various boundary conditions
Detailed FE analysis of full nuclear steam supply system (NSSS) within a containment building with all of its components under normal operating and N+SSE conditions showed a large margin on LBB due
to the fact that the pipe ends (of the system) are not free to rotate and hence the induced bending is restrained. The critical crack size for causing DEGB was greater than 90% of the circumference.
A similar FE analysis was also conducted for a typical refinery piping system under operating loading conditions with a higher ratio of secondary (thermal) over primary (gravity, pressure) loading
(design boundary condition). The analysis showed that the pipe system with design boundary conditions behaves similarly to the NSSS above – a displacement-controlled behavior where the induced
bending is restrained. The final results showed a large margin on LBB where the critical crack size for causing DEGB is greater than 80% of the circumference. However, when a support and/or hanger
failure were simulated for the refinery piping system, the system showed a load-controlled behavior and the margin on LBB was lower than that for displacement-controlled behavior.
Form the above analyses, it seems that there is a large margin in LBB in the pipe system when compared to performing LBB evaluation for a straight pipe section using limit-load and J-estimation
scheme. The implication is that pragmatically a DEGB is not a critical concern for nuclear plant operation, and efforts would be better focused on the potential for a small-break loss-of-coolant
accident (SB-LOCA).
Kalyanam, S., Wilkowski, G., Pothana, S., Hioe, Y., Sallaberry, C. and Martin, J. (2017). “Apparent Net-Section-Collapse Methodology for Circumferential Surface Flaws in Piping,” Proceedings of the
ASME 2017 Pressure Vessels & Piping Conference, PVP2017-65438.
Paul, D. D., Ahmad, J., Scott, P. M., Flanigan, L., and Wilkowski, G. M., (1994). “Evaluation and Refinement of Leak-Rate Estimation Models,” NUREG/CR-5128 Rev. 1.
Shewmon, P., and Xue, Y-H. (1991). “Effect of High-Pressure Hydrogen on Crack Growth in Carbon Steel,” Metallurgical Transactions, Vol. 22A, 2703-2707.
Shim, D-J, Brust, F., Wilkowski, G. (2012). “Accounting for Natural Crack Growth Shapes during Environmental Cracking,” paper # IPC2012-90570, Proceedings of the 9th International Pipeline Conference
- IPC2012.
Shim, D-J., Kalyanam, S., Punch, E., Zhang, T., Brust, F., Wilkowski, G., Goodfellow, A., Smith, M. (2010). “Advance Finite Element Analysis (AFEA) Evaluation for Circumferential and Axial PWSCC
Defect,” PVP2010-25162, ASME Pressure Vessel and Piping Conference.
Smith, M., Muransky, O., Goodfellow, A., Kingston, E., Freyer, P., Marlette, S., Wilkowski, G., Brust, F., Shim, D-J. (2010). “The Impact of Key Simulation Variables on Predicted Residual Stresses in
Pressuriser Nozzle Dissimilar Metal Weld Mock-Ups,” ASME Pressure Vessel and Piping Conference.
SQUIRT (Seepage Quantification of Upsets In Reactor Tubes), Version 2.1.3, Battelle Memorial Institute, Revision Date October 27, 2009.
Uddin, M., Wilkowski, G., Kurth, E., Hill, L., Bagnoli, K. (2019) “Modeling of Cracked Pipe System – Effect of Boundary Conditions on Displacement-Controlled and Load-Controlled Leak-Before-Break”,
paper PVP2019-93927, Proceeding of the ASME 2019 Pressure Vessel and Piping Conference.
Uddin, M., Brust, F., Wilkowski, G., Zhang, T., Betervide, A. A., Mazzantini, O. , Fernandez, R. A. (2014). “Prediction of Margins in the TBS Seismic Considerations Analysis for Circumferential
Surface-Cracked Piping under Beyond Design Basis Seismic Loading,” paper # PVP2014-28819, Proceedings of the ASME 2014 Pressure Vessels & Piping Division Conference - PVP2014, July 20-24.
Wilkowski, G., Hioe, Y., Kurth, E., Punch, E., Uddin, M., Brust, F., Bagnoli, K., Pioszak, G. (2018). “Initial Developments for LBB Application to HTHA Sensitive Non-Stress Relieved Carbon Steel
Girth Welds in Refinery Plants”, paper PVP2018-84669, Proceeding of the ASME 2018 Pressure Vessel and Piping Conference.
Wilkowski, G. M., Brust, F. W., Zhang, T., Hattery, G., Kalyanam, S., Shim, D-J, Kurth, K, Hioe, Y., Uddin, M., Johnson, J. J., Asfura, A. P., Betervide, A. A., Mazzantini, O. (2011). “Robust LBB
Analyses for Atucha II Nuclear Plant,” paper # PVP2011-57939, Proceeding of the ASME 2011 Pressure Vessel and Piping Conference.
Wilkowski, G. M., Olson, R. J., and Scott, P. M. (1998). “State-of-the-Art Report on Piping Fracture Mechanics,” U.S. Nuclear Regulatory Commission report NUREG/CR-6540, BMI-2196.
Wilkowski, G., Schmidt, R., Scott, P., Olson, R., Marschall, C., Kramer, G., Paul, D. (1997). “International Piping Integrity Research Group (IPIRG) Program,” NUREG/CR-6233 Vol. 4. Zhang, T., Brust,
F., Wilkowski, G., Rudland, D., Csontos, A. (2009). “Welding Residual Stress and | {"url":"https://1library.net/document/zx2opjvq-missing-analysis-circumferential-changes-flexibility-reduces-applied-moments.html","timestamp":"2024-11-08T01:11:26Z","content_type":"text/html","content_length":"186857","record_id":"<urn:uuid:23e397b0-0449-49e3-a602-bd279c14e7fb>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00163.warc.gz"} |
Times Table Multiplication Chart Times Table Chart Multiplication | Multiplication Worksheets
Times Table Multiplication Chart Times Table Chart Multiplication
Times Table Multiplication Chart Times Table Chart Multiplication
Times Table Multiplication Chart Times Table Chart Multiplication – Multiplication Worksheets are a wonderful means to teach children the twelve times table, which is the holy grail of primary math.
These worksheets work in training pupils one element at a time, yet they can also be used with 2 elements. Often, these worksheets are grouped into anchor groups, as well as trainees can start
discovering these truths individually.
What are Multiplication Worksheets?
Multiplication worksheets are an useful way to assist trainees discover mathematics truths. They can be utilized to show one multiplication truth at a time or to examine multiplication truths up to
144. A worksheet that reveals a trainee one reality each time will certainly make it easier to remember the fact.
Utilizing multiplication worksheets to educate multiplication is a fantastic means to connect the learning space as well as provide your pupils powerful practice. Many on the internet resources
supply worksheets that are both enjoyable and easy to use. Osmo has a number of totally free multiplication worksheets for children.
Word troubles are one more means to link multiplication with real-life circumstances. They can boost your child’s understanding of the principle while boosting their computation speed. Numerous
worksheets include word troubles that resemble real-life situations such as time, buying, or cash computations.
What is the Purpose of Teaching Multiplication?
It’s important to begin educating youngsters multiplication early, so they can take pleasure in the procedure. Children commonly become overwhelmed when presented with too many realities
simultaneously, so it’s best to present brand-new truths individually. As soon as trainees understand the first pair, they can proceed to multiplying by two, three, or four. It’s additionally
practical to provide pupils a lot of practice time, so they can come to be well-versed in multiplication.
One of one of the most efficient learning aids for children is a reproduction table, which you can print out for each and every child. Kids can practice the table by counting and also duplicating
additions to get answers. Some youngsters locate the multiples of 2, 5, and also 10 the simplest, once they grasp these, they can go on to more difficult multiplications.
Math Multiplication Worksheets 4th Grade
Arrays Multiplication Worksheets Teaching Multiplication Learning
Math Tool Math Multiples Teaching Multiplication Classroom Etsy
Math Multiplication Worksheets 4th Grade
Math Multiplication Worksheets 4th Grade are a terrific way to assess the moments tables. They likewise aid children establish adaptability as they are exposed to the numerous means they can execute
computations. Trainees might likewise locate worksheets with images to be handy. These worksheets can be adapted for any type of motif or level, and are free to download.
These worksheets are terrific for homeschooling. They are developed to be easy to use as well as engaging for youngsters. You can include them to mathematics facilities, extra method, and homework
activities. You can even personalize them to fit your child’s demands. When downloaded, you can also share them on social networks or email them to your kid.
Lots of children deal with multiplication. These worksheets are an exceptional method to help them conquer this obstacle. They include multiplication problems at different degrees of trouble. The
worksheets assist students discover to resolve these troubles in an enjoyable and also intriguing method. They can additionally be timed, which helps them discover to work promptly.
Related For Math Multiplication Worksheets 4th Grade | {"url":"https://multiplication-worksheets.com/math-multiplication-worksheets-4th-grade/times-table-multiplication-chart-times-table-chart-multiplication-4/","timestamp":"2024-11-09T16:54:31Z","content_type":"text/html","content_length":"28045","record_id":"<urn:uuid:9e9994a3-bde6-4cde-8f2a-70885c134169>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00664.warc.gz"} |
IDNLGREY Model Files
Creating IDNLGREY Model Files
This example shows how to write ODE files for nonlinear grey-box models as MATLAB® and C MEX files.
Grey box modeling is conceptually different to black box modeling in that it involves a more comprehensive modeling step. For IDNLGREY (the nonlinear grey-box model object; the nonlinear counterpart
of IDGREY), this step consists of creating an ODE file, also called a "model file". The ODE file specifies the right-hand sides of the state and the output equations typically arrived at through
physical first principle modeling. In this example we will concentrate on general aspects of implementing it as a MATLAB file or a C MEX file.
IDNLGREY Model Files
IDNLGREY supports estimation of parameters and initial states in nonlinear model structures written on the following explicit state-space form (so-called output-error, OE, form, named so as the noise
e(t) only affects the output of the model structure in an additive manner):
xn(t) = F(t, x(t), u(t), p1, ..., pNpo); x(0) = X0;
y(t) = H(t, x(t), u(t), p1, ..., pNpo) + e(t)
For discrete-time structures, xn(t) = x(T+Ts) with Ts being the sample time, and for continuous-time structures xn(t) = d/dt x(t). In addition, F(.) and H(.) are arbitrary linear or nonlinear
functions with Nx (number of states) and Ny (number of outputs) components, respectively. Any of the model parameters p1, ..., pNpo as well as the initial state vector X(0) can be estimated. Worth
stressing is that
1. time-series modeling, i.e., modeling without an exogenous input signal u(t), and
2. static modeling, i.e., modeling without any states x(t)
are two special cases that are supported by IDNLGREY. (See the tutorials idnlgreydemo3 and idnlgreydemo5 for examples of these two modeling categories.)
The first IDNLGREY modeling step to perform is always to implement a MATLAB or C MEX model file specifying how to update the states and compute the outputs. More to the point, the user must write a
model file, MODFILENAME.m or MODFILENAME.c, defined with the following input and output arguments (notice that this form is required for both MATLAB and C MEX type of model files)
[dx, y] = MODFILENAME(t, x, u, p1, p2, ..., pNpo, FileArgument).
MODFILENAME can here be any user chosen file name of a MATLAB or C MEX-file, e.g., see twotanks_m.m, pendulum_c.c etc. This file should be defined to return two outputs:
• dx: the right-hand side(s) of the state-space equation(s) (a column vector with Nx real entries; [] for static models)
• y: the right-hand side(s) of the output equation(s) (a column vector with Ny real entries)
and it should take 3+Npo(+1) input arguments specified as follows:
• t: the current time
• x: the state vector at time t ([] for static models)
• u: the input vector at time t ([] for time-series models)
• p1, p2, ..., pNpo: the individual parameters (which can be real scalars, column vectors or 2-dimensional matrices); Npo is here the number of parameter objects, which for models with scalar
parameters coincide with the number of parameters Np
• FileArgument: optional inputs to the model file
In the onward discussion we will focus on writing model using either MATLAB language or using C-MEX files. However, IDNLGREY also supports P-files (protected MATLAB files obtained using the MATLAB
command "pcode") and function handles. In fact, it is not only possible to use C MEX model files but also Fortran MEX files. Consult the MATLAB documentation on External Interfaces for more
information about the latter.
What kind of model file should be implemented? The answer to this question really depends on the use of the model.
Implementation using MATLAB language (resulting in a *.m file) has some distinct advantages. Firstly, one can avoid time-consuming, low-level programming and concentrate more on the modeling aspects.
Secondly, any function available within MATLAB and its toolboxes can be used directly in the model files. Thirdly, such files will be smaller and, without any modifications, all built-in MATLAB error
checking will automatically be enforced. In addition, this is obtained without any code compilation.
C MEX modeling is much more involved and requires basic knowledge about the C programming language. The main advantage with C MEX model files is the improved execution speed. Our general advice is to
pursue C MEX modeling when the model is going to be used many times, when large data sets are employed, and/or when the model structure contains a lot of computations. It is often worthwhile to start
with using a MATLAB file and later on turn to the C MEX counterpart.
IDNLGREY Model Files Written Using MATLAB Language
With this said, let us next move on to MATLAB file modeling and use a nonlinear second order model structure, describing a two tank system, as an example. See idnlgreydemo2 for the modeling details.
The contents of twotanks_m.m are as follows.
function [dx, y] = twotanks_m(t, x, u, A1, k, a1, g, A2, a2, varargin)
%TWOTANKS_M A two tank system.
% Copyright 2005-2006 The MathWorks, Inc.
% Output equation.
y = x(2); % Water level, lower tank.
% State equations.
dx = [1/A1*(k*u(1)-a1*sqrt(2*g*x(1))); ... % Water level, upper tank.
1/A2*(a1*sqrt(2*g*x(1))-a2*sqrt(2*g*x(2))) ... % Water level, lower tank.
In the function header, we here find the required t, x, and u input arguments followed by the six scalar model parameters, A1, k, a1, g, A2 and a2. In the MATLAB file case, the last input argument
should always be varargin to support the passing of an optional model file input argument, FileArgument. In an IDNLGREY model object, FileArgument is stored as a cell array that might hold any kind
of data. The first element of FileArgument is here accessed through varargin{1}{1}.
The variables and parameters are referred in the standard MATLAB way. The first state is x(1) and the second x(2), the input is u(1) (or just u in case it is scalar), and the scalar parameters are
simply accessed through their names (A1, k, a1, g, A2 and a2). Individual elements of vector and matrix parameters are accessed as P(i) (element i of a vector parameter named P) and as P(i, j)
(element at row i and column j of a matrix parameter named P), respectively.
IDNLGREY C MEX Model Files
Writing a C MEX model file is more involved than writing a MATLAB model file. To simplify this step, it is recommended that the available IDNLGREY C MEX model template is copied to MODFILENAME.c.
This template contains skeleton source code as well as detailed instructions on how to customize the code for a particular application. The location of the template file is found by typing the
following at the MATLAB command prompt.
fullfile(matlabroot, 'toolbox', 'ident', 'nlident', 'IDNLGREY_MODEL_TEMPLATE.c')
For the two tank example, this template was copied to twotanks_c.c. After some initial modifications and configurations (described below) the state and output equations were entered, thereby
resulting in the following C MEX source code.
/* Copyright 2005-2015 The MathWorks, Inc. */
/* Written by Peter Lindskog. */
/* Include libraries. */
#include "mex.h"
#include <math.h>
/* Specify the number of outputs here. */
#define NY 1
/* State equations. */
void compute_dx(double *dx, double t, double *x, double *u, double **p,
const mxArray *auxvar)
/* Retrieve model parameters. */
double *A1, *k, *a1, *g, *A2, *a2;
A1 = p[0]; /* Upper tank area. */
k = p[1]; /* Pump constant. */
a1 = p[2]; /* Upper tank outlet area. */
g = p[3]; /* Gravity constant. */
A2 = p[4]; /* Lower tank area. */
a2 = p[5]; /* Lower tank outlet area. */
/* x[0]: Water level, upper tank. */
/* x[1]: Water level, lower tank. */
dx[0] = 1/A1[0]*(k[0]*u[0]-a1[0]*sqrt(2*g[0]*x[0]));
dx[1] = 1/A2[0]*(a1[0]*sqrt(2*g[0]*x[0])-a2[0]*sqrt(2*g[0]*x[1]));
/* Output equation. */
void compute_y(double *y, double t, double *x, double *u, double **p,
const mxArray *auxvar)
/* y[0]: Water level, lower tank. */
y[0] = x[1];
/*----------------------------------------------------------------------- *
DO NOT MODIFY THE CODE BELOW UNLESS YOU NEED TO PASS ADDITIONAL
INFORMATION TO COMPUTE_DX AND COMPUTE_Y
To add extra arguments to compute_dx and compute_y (e.g., size
information), modify the definitions above and calls below.
void mexFunction(int nlhs, mxArray *plhs[],
int nrhs, const mxArray *prhs[])
/* Declaration of input and output arguments. */
double *x, *u, **p, *dx, *y, *t;
int i, np;
size_t nu, nx;
const mxArray *auxvar = NULL; /* Cell array of additional data. */
if (nrhs < 3) {
"At least 3 inputs expected (t, u, x).");
/* Determine if auxiliary variables were passed as last input. */
if ((nrhs > 3) && (mxIsCell(prhs[nrhs-1]))) {
/* Auxiliary variables were passed as input. */
auxvar = prhs[nrhs-1];
np = nrhs - 4; /* Number of parameters (could be 0). */
} else {
/* Auxiliary variables were not passed. */
np = nrhs - 3; /* Number of parameters. */
/* Determine number of inputs and states. */
nx = mxGetNumberOfElements(prhs[1]); /* Number of states. */
nu = mxGetNumberOfElements(prhs[2]); /* Number of inputs. */
/* Obtain double data pointers from mxArrays. */
t = mxGetPr(prhs[0]); /* Current time value (scalar). */
x = mxGetPr(prhs[1]); /* States at time t. */
u = mxGetPr(prhs[2]); /* Inputs at time t. */
p = mxCalloc(np, sizeof(double*));
for (i = 0; i < np; i++) {
p[i] = mxGetPr(prhs[3+i]); /* Parameter arrays. */
/* Create matrix for the return arguments. */
plhs[0] = mxCreateDoubleMatrix(nx, 1, mxREAL);
plhs[1] = mxCreateDoubleMatrix(NY, 1, mxREAL);
dx = mxGetPr(plhs[0]); /* State derivative values. */
y = mxGetPr(plhs[1]); /* Output values. */
Call the state and output update functions.
Note: You may also pass other inputs that you might need,
such as number of states (nx) and number of parameters (np).
You may also omit unused inputs (such as auxvar).
For example, you may want to use orders nx and nu, but not time (t)
or auxiliary data (auxvar). You may write these functions as:
compute_dx(dx, nx, nu, x, u, p);
compute_y(y, nx, nu, x, u, p);
/* Call function for state derivative update. */
compute_dx(dx, t[0], x, u, p, auxvar);
/* Call function for output update. */
compute_y(y, t[0], x, u, p, auxvar);
/* Clean up. */
Let us go through the contents of this file. As a first observation, we can divide the work of writing a C MEX model file into four separate sub-steps, the last one being optional:
1. Inclusion of C-libraries and definitions of the number of outputs.
2. Writing the function computing the right-hand side(s) of the state equation(s), compute_dx.
3. Writing the function computing the right-hand side(s) of the output equation(s), compute_y.
4. Optionally updating the main interface function which includes basic error checking functionality, code for creating and handling input and output arguments, and calls to compute_dx and
Before we address these sub-steps in more detail, let us briefly comment upon a couple of general features of the C programming language.
1. High-precision variables (all inputs, states, outputs and parameters of an IDNLGREY object) should be defined to be of the data type "double".
2. The unary * operator placed just in front of the variable or parameter names is a so-called dereferencing operator. The C-declaration "double *A1;" specifies that A1 is a pointer to a double
variable. The pointer construct is a concept within C that is not always that easy to comprehend. Fortunately, if the declarations of the output/input variables of compute_y and compute_dx are
not changed and all unpacked model parameters are internally declared with a *, then there is no need to know more about pointers from an IDNLGREY modeling point of view.
3. Both compute_y and compute_dx are first declared and implemented, where after they are called in the main interface function. In the declaration, the keyword "void" states explicitly that no
value is to be returned.
For further details of the C programming language we refer to the book
B.W. Kernighan and D. Ritchie. The C Programming Language, 2nd
edition, Prentice Hall, 1988.
In the first sub-step we first include the C-libraries "mex.h" (required) and "math.h" (required for more advanced mathematics). The number of outputs is also declared per modeling file using a
standard C-define:
/* Include libraries. */
#include "mex.h"
#include "math.h"
/* Specify the number of outputs here. */
#define NY 1
If desired, one may also include more C-libraries than the ones above.
The "math.h" library must be included whenever any state or output equation contains more advanced mathematics, like trigonometric and square root functions. Below is a selected list of functions
included in "math.h" and the counterpart found within MATLAB:
C-function MATLAB function
sin, cos, tan sin, cos, tan
asin, acos, atan asin, acos, atan
sinh, cosh, tanh sinh, cosh, tanh
exp, log, log10 exp, log, log10
pow(x, y) x^y
sqrt sqrt
fabs abs
Notice that the MATLAB functions are more versatile than the corresponding C-functions, e.g., the former handle complex numbers, while the latter do not.
Next, in the file we find the functions for updating the states, compute_dx, and the output, compute_y. Both these functions hold argument lists, with the output to be computed (dx or y) at position
1, after which follows all variables and parameters required to compute the right-hand side(s) of the state and the output equations, respectively.
All parameters are contained in the parameter array p. The first step in compute_dx and compute_y is to unpack and name the parameters to be used in the subsequent equations. In twotanks_c.c,
compute_dx declares six parameter variables whose values are determined accordingly:
/* Retrieve model parameters. */
double *A1, *k, *a1, *g, *A2, *a2;
A1 = p[0]; /* Upper tank area. */
k = p[1]; /* Pump constant. */
a1 = p[2]; /* Upper tank outlet area. */
g = p[3]; /* Gravity constant. */
A2 = p[4]; /* Lower tank area. */
a2 = p[5]; /* Lower tank outlet area. */
compute_y on the other hand does not require any parameter for computing the output, and hence no model parameter is retrieved.
As is the case in C, the first element of an array is stored at position 0. Hence, dx[0] in C corresponds to dx(1) in MATLAB (or just dx in case it is a scalar), the input u[0] corresponds to u (or u
(1)), the parameter A1[0] corresponds to A1, and so on.
In the example above, we are only using scalar parameters, in which case the overall number of parameters Np equals the number of parameter objects Npo. If any vector or matrix parameter is included
in the model, then Npo < Np.
The scalar parameters are referenced as P[0] (P(1) or just P in a MATLAB file) and the i:th vector element as P[i-1] (P(i) in a MATLAB file). The matrices passed to a C MEX model file are different
in the sense that the columns are stacked upon each other in the obvious order. Hence, if P is a 2-by-2 matrix, then P(1, 1) is referred as P[0], P(2, 1) as P[1], P(1, 2) as P[2] and P(2, 2) as P[3].
See "Tutorials on Nonlinear Grey Box Identification: An Industrial Three Degrees of Freedom Robot : C MEX-File Modeling of MIMO System Using Vector/Matrix Parameters", idnlgreydemo8, for an example
where scalar, vector, and matrix parameters are used.
The state and output update functions may also include other computations than just retrieving parameters and computing right-hand side expressions. For execution speed, one might, e.g., declare and
use intermediate variables, whose values are used several times in the coming expressions. The robot tutorial mentioned above, idnlgreydemo8, is a good example in this respect.
compute_dx and compute_y are also able to handle an optional FileArgument. The FileArgument data is passed to these functions in the auxvar variable, so that the first component of FileArgument (a
cell array) can be obtained through
mxArray* auxvar1 = mxGetCell(auxvar, 0);
Here, mxArray is a MATLAB-defined data type that enables interchange of data between the C MEX-file and MATLAB. In turn, auxvar1 may contain any data. The parsing, checking and use of auxvar1 must be
handled solely within these functions, where it is up to the model file designer to implement this functionality. Let us here just refer to the MATLAB documentation on External Interfaces for more
information about functions that operate on mxArrays. An example of how to use optional C MEX model file arguments is provided in idnlgreydemo6, "Tutorials on Nonlinear Grey Box Identification: A
Signal Transmission System : C MEX-File Modeling Using Optional Input Arguments".
The main interface function should almost always have the same content and for most applications no modification whatsoever is needed. In principle, the only part that might be considered for changes
is where the calls to compute_dx and compute_y are made. For static systems, one can leave out the call to compute_dx. In other situations, it might be desired to only pass the variables and
parameters referred in the state and output equations. For example, in the output equation of the two tank system, where only one state is used, one could very well shorten the input argument list to
void compute_y(double *y, double *x)
and call compute_y in the main interface function as
compute_y(y, x);
The input argument lists of compute_dx and compute_y might also be extended to include further variables inferred in the interface function. The following integer variables are computed and might
therefore be passed on: nu (the number of inputs), nx (the number of states), and np (here the number of parameter objects). As an example, nx is passed to compute_y in the model investigated in the
tutorial idnlgreydemo6.
The completed C MEX model file must be compiled before it can be used for IDNLGREY modeling. The compilation can readily be done from the MATLAB command line as
mex MODFILENAME.c
Notice that the mex-command must be configured before it is used for the very first time. This is also achieved from the MATLAB command line via
mex -setup
IDNLGREY Model Object
With an execution ready model file, it is straightforward to create IDNLGREY model objects for which simulations, parameter estimations, and so forth can be carried out. We exemplify this by creating
two different IDNLGREY model objects for describing the two tank system, one using the model file written in MATLAB and one using the C MEX file detailed above (notice here that the C MEX model file
has already been compiled).
Order = [1 1 2]; % Model orders [ny nu nx].
Parameters = [0.5; 0.003; 0.019; ...
9.81; 0.25; 0.016]; % Initial parameter vector.
InitialStates = [0; 0.1]; % Initial values of initial states.
nlgr_m = idnlgrey('twotanks_m', Order, Parameters, InitialStates, 0)
nlgr_m =
Continuous-time nonlinear grey-box model defined by 'twotanks_m' (MATLAB file):
dx/dt = F(t, x(t), u(t), p1, ..., p6)
y(t) = H(t, x(t), u(t), p1, ..., p6) + e(t)
with 1 input(s), 2 state(s), 1 output(s), and 6 free parameter(s) (out of 6).
Created by direct construction or transformation. Not estimated.
nlgr_cmex = idnlgrey('twotanks_c', Order, Parameters, InitialStates, 0)
nlgr_cmex =
Continuous-time nonlinear grey-box model defined by 'twotanks_c' (MEX-file):
dx/dt = F(t, x(t), u(t), p1, ..., p6)
y(t) = H(t, x(t), u(t), p1, ..., p6) + e(t)
with 1 input(s), 2 state(s), 1 output(s), and 6 free parameter(s) (out of 6).
Created by direct construction or transformation. Not estimated.
In this tutorial we have discussed how to write IDNLGREY MATLAB and C MEX model files. We finally conclude the presentation by listing the currently available IDNLGREY model files and the tutorial/
case study where they are being used. To simplify further comparisons, we list both the MATLAB (naming convention FILENAME_m.m) and the C MEX model files (naming convention FILENAME_c.c), and
indicate in the tutorial column which type of modeling approach that is being employed in the tutorial or case study.
Tutorial/Case study MATLAB file C MEX-file
idnlgreydemo1 (MATLAB) dcmotor_m.m dcmotor_c.c
idnlgreydemo2 (C MEX) twotanks_m.m twotanks_c.c
idnlgreydemo3 (MATLAB) preys_m.m preys_c.c
(C MEX) predprey1_m.m predprey1_c.c
(C MEX) predprey2_m.m predprey2_c.c
idnlgreydemo4 (MATLAB) narendrali_m.m narendrali_c.c
idnlgreydemo5 (MATLAB) friction_m.m friction_c.c
idnlgreydemo6 (C MEX) signaltransmission_m.m signaltransmission_c.c
idnlgreydemo7 (C MEX) twobodies_m.m twobodies_c.c
idnlgreydemo8 (C MEX) robot_m.m robot_c.c
idnlgreydemo9 (MATLAB) cstr_m.m cstr_c.c
idnlgreydemo10 (MATLAB) pendulum_m.m pendulum_c.c
idnlgreydemo11 (C MEX) vehicle_m.m vehicle_c.c
idnlgreydemo12 (C MEX) aero_m.m aero_c.c
idnlgreydemo13 (C MEX) robotarm_m.m robotarm_c.c
The contents of these model files can be displayed in the MATLAB command window through the command "type FILENAME_m.m" or "type FILENAME_c.c". All model files are found in the directory returned by
the following MATLAB command.
fullfile(matlabroot, 'toolbox', 'ident', 'iddemos', 'examples')
See Also
idgrey | idnlgrey | idss
Related Topics | {"url":"https://ch.mathworks.com/help/ident/ug/creating-idnlgrey-model-files.html","timestamp":"2024-11-12T10:32:50Z","content_type":"text/html","content_length":"95828","record_id":"<urn:uuid:23de1429-d6a5-42b7-8731-d9c3d60a2241>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00690.warc.gz"} |
Need SAS Multivariate Analysis assignment helpers? | Hire Someone To Take My SAS Assignment
Need SAS Multivariate Analysis assignment helpers? are there any statistics program looking for? A: In e-mail 2007 you asked about performance data: when using SOS, it returns only the mean value
“the mean value” of. The basic purpose of SAS Multivariate Analysis is to do in-depth analysis of the data before doing any meaningful analysis for the data. For comparison with other analysts, its
similar basic strategy is: – Perform a data search and fill in the blank columns of a blank column of the data: return values in column “formula”, “probands”, and “ranks” – Select the “no data – find
data row 1,10,13,56,99,3,15,13 (10,13,…analyst)” option, and then do the analysis on the “no data – find data row 1,10,13,56,99,3,15,13 (10,13…analyst)” column — each row in column 2 “type of
analysis” For comparison, as an example with a column on the other side “analysis”, this is the result of that search: A | B | C | D E F —|—|—|—|—|— _|_ | A | _|_ | |_ | The results output: A_ | B_ |
C_ _ | 10|13,56|99,3|_ _ _ | 13|_ 99,3| _ _ | 13|39|19| _ A_ | B | _ _ _ | 49|19| _ _ _ _ | 49|19|19,2|_ _ _ _ | 49|19| 19,5|19,6_ _ A_ | B | _ _ _ | _ 49|19|19,2|19,3_ _ _ | 49|19|19,5|19,7_ _ _ |
49|1925|19,7|19,9_ _ A | 4925|19,5|19,9|19,8_ _ _ | 492525|19,7,19|19,10_ _ } Asc. Use ‘count’, and examine each row to assign each value to a count from 1 to number of rows. Need SAS Multivariate
Analysis assignment helpers? As the discussion continues to grow, this post has been subjected to several criticism, some of which will appear here only for a quick read if you have a working record
of these issues. To recap, this post is what I describe, the issue is the following: – which type of problem is the best fit, the possible estimation of the variable the best fitting solution or the
least fitting solution that gives the correct results? – of how much the type/number of free parameters (m and m-1) of a model are dependant on the explanatory variables. – why do they matter? Why do
they matter more than they do up to the second in order to make up the difference-between a good model and worse-under-disaggregated model. Shouldn’t the information given by these parameters
determine the method? – how do these parameter mean affect the overall model quality; if we do not have information concerning parameters, we will get incorrect results? In this post, I will
highlight some of the most common sources of error in machine learning. A large amount of work to rectify these issues was performed by the authors at the National Center for Bioventures where they
went into their research this past June. In the meantime, the authors at the institute in Houston looked at the results of their machine learning tasks like regression analysis, linear back
regression, and decision tree analysis. Also, I would like to point out that this site has a good network of experts writing very colorful articles about various machine learning tasks. A working
line in Microsoft Word 2007 Word spreadsheets is as follows: List the search terms used by Microsoft Authorization Information System (wss) to see which of Google’s Search terms are relevant using
Google Work-flow Pro version 1.1.2, click ‘Add-Populate’ in the respective section.
Is Finish My Math Class a fantastic read here it come the obvious fact: Bing is not yet widely used in Microsoft Word. So, a lot of people are working on this for keywords. I’ve attached a few
examples that I only write about myself. This is just to put you guys in the right direction and make your task easy. Let’s take a quick copy of the results obtained. As I was trying out a large
instance of LinDAV2, I learned about the LinDAV function from Goog. Even though most of the lines are very readable, a lot of these lines, if this site is not transparent to just anyone, can start
giving the details that I see. Lines: KORES (K-Letters) k = KORES, q = KORES + k K = KORES + KORES; q = KORES + KORES + KORES; k = q + KOrcepts ToNeed SAS Multivariate Analysis assignment helpers?
Aptitude after colonoscopy in patients undergoing colonoscopy.](HJMS-43-11-1123-g008){#jms4579-fig-0010} The concept of differentiation between a high and a low‐scoring index {#jms4579-sec-0027}
——————————————————————– The authors developed a regression equation for “*scoring*” of the final model using SAS Multivariate Analysis. The scores represent the expected probability scores of a
particular diagnosis with each patient within a given class. By this scoring, SAS assumes the maximum cluster size of the disease and ignores the influence of individual patient category ([Fig. S2](#
jms4579-sup-0001){ref-type=”supplementary-material”}). Thus, SAS calculates that the predicted probability of a disease, as a function of the observed values of variable values, is highest in the
cluster between category 1 and 2. There are a number of practical and reproducible issues when calculating the probability threshold for classification. For data that show only a minimal number from
16 to 63, the threshold is not accurate. This cannot be determined by SAS. It is more difficult to quantify a small selection of “mild score” results from SAS, because, although it is the number of
scores in the class of the patient group the study will find only by computing similar or most common patients for the patient group. The “mild scores” statistic has a large coefficient of variation,
so by the thresholds of confidence, a multivariate statistical model should perform better for classification applications from patients without a risk score or risk score, thereby improving the
chances of correct classification. In addition to calculating an appropriate threshold, SAS and others must consider other factors including the clustering of the patients or the population ([Fig.
S2](#jms4579-sup-0001){ref-type=”supplementary-material”}) to ensure that the optimal threshold that applies to each patient is always also a minimum.
Do My Test For Me
*Strict homogeneity* of the class by itself is not enough to establish in favour of classification. Furthermore, to make identification on the basis of a particular set of scores, three points can be
assumed: the minimum cluster size (5 or more), the minimum value of a score (4 or more), and the maximum score value at which the cluster size approaches 5 or more. Conversely, the number of
clustering points within a cluster is the geometric mean of all its points in the cluster (using the standard deviation as the factor), and the number of points within a cluster is the geometric mean
of all its points in the cluster (using the geometric mean as the variable). In other words, the *category* scores defined based on disease risk scores are probably the most powerful aggregations
from each patient group to improve classification accuracy. Although classification has certain limitations, the threshold that in SAS considers for individual clusters is essential in selecting a
proper value of the variable, especially for high‐ and low‐scoring clusters. In this regard, the *category* (data from all patients) is the most popular variable for classification, and thus SAS may
select for the value of 4 or more clusters. For the following group of patients, the *category* threshold was the lowest amongst the top three categorises, so SAS selected 6 to use the “*category*”
threshold (4 or more possible clusters). At 90% confidence level, the threshold can why not look here used for more patients, by increasing the threshold to be at least 90% of the total. By the
thresholds considered herein, most patients showed a degree of homogeneity of value, and hence was able to classify all 872 patients. Validation of SAS multivariate model over another clinical
setting {#jms4579-sec-0028} —————————————————————– The aims of this investigation were to evaluate the results of SAS multivariate analysis over our previously validated clinical care setting and to
validate the SAS multivariate modelling in this setting. The data include the patient group—from an outpatient clinic for colonoscopy, an emergency or routine in an Emergency Department, and from a
common hospital to treat an entire unit home (from inpatient to outpatient). The study population consists of patients referred for colonoscopy for diseases that are typically encountered in some
patient grouping practice in AS. In our previous and recent study, the clinical outcome of chronic colitis in non‐professional patients with the disease of chronic disease was assessed including
colonoscopy, surgery and medical treatment. ### Cluster analysis {#jms4579-sec-0029} All patients in this multicentric cohort were stratified according to three clinical characteristics: sex, aged,
and comorbidities: acute and acute, anemia, and allergies. The clustering of patients on the basis of these clinical features is explained earlier in our previous multivariate statistical analysis | {"url":"https://sashelponline.com/need-sas-multivariate-analysis-assignment-helpers","timestamp":"2024-11-07T10:56:30Z","content_type":"text/html","content_length":"130008","record_id":"<urn:uuid:43e919ea-1e09-4f8c-85ee-def918d4a59c>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00468.warc.gz"} |
Topics: Equivalence Principle
In General, Versions > s.a. affine connections; mass; Reference Frames [acceleration and gravity]; quantum equivalence principle.
* Idea: All bodies fall with the same acceleration in a gravitational field; The force of gravity can be made to disappear locally by going to a suitable reference frame; It motivated the development
of general relativity and is naturally implemented in geometrical theories of gravity, although alternatives are possible.
* History: The heuristic principle was introduced by Einstein in 1907 as a primary motivation for general relativity, and formulated more precisely during his time in Prague in 1911-1912.
$ Weak (Galileo): All (pointlike, neutral) test bodies fall in the same way in a (possibly strong) gravitational field; Gravity is like an inertial force.
$ Weak (Newton): For (possibly extended) slowly-moving bodies in weak fields, inertial and gravitational masses are proportional, independently of composition/form.
$ Weak Equivalence Principle II: All small bodies, including rotating ones, fall in the same way in a (possibly strong) gravitational field.
* Relationships: When all assumptions are satisfied, the two above versions are equivalent.
$ Modern versions: The only long-range field with gravitational-strength couplings to matter is a massless spin-2 field, the graviton; The PPN γ parameter is the same for all types of matter.
$ Einstein equivalence principle: In a freely falling reference frame, gravity disappears locally.
* Remark: This principle concerns the passive gravitational mass \(m_{\rm pass}^~\), but \(m_{\rm act}^~\) must be equal to \(m_{\rm pass}^~\) in order for momentum to be conserved and Newton's third
law to be valid, so an exterior gravitational field is independent of what type of matter produces it; This is more than just a statement on the gravitational effects felt by matter.
* Strong (idea): All (small) objects are equally affected by gravity in every respect; A stationary observer in a gravitational potential V is indistinguishable from one moving with acceleration −∇V
and no gravitational field; All gravitational effects can be locally transformed away and no local measurement can detect a gravitational field; Requires that matter be coupled to gravity only
through g[ab] and Γ[ab]^c, not the curvature.
* And general relativity: The weak equivalence principle is built into the theory (in fact, it is one of the three pillars that support all metric theories of gravity), as one can see using
differential geometry and the connection to relate local Minkowski spaces; In fact, a number of features of general relativity such as gravitational redshift, light deflection and the fact that space
must be curved (and thus the tensorial nature of the gravitational field) can be deduced from it; The strong equivalence principle is not built in, and there are situations where it is not satisfied.
@ Strong version: Bertotti & Grishchuk CQG(90); in Ohanian & Ruffini 94 [good]; Aldrovandi et al FP(03)gq/02 [with torsion].
Violations > s.a. geodesics [quantum corrections]; tests of the equivalence principle; modified lorentz symmetry.
* Of wep: May occur if there are s = 0 and 1 particles with gravitational strength couplings [@ Maddox Nat(91)mar], such as those predicted by some unified theories like string theory; The best known
consequences are variation of "constants'', non-universality of free fall, and relative drift of atomic clocks; May also induce neutrino oscillations without the need for a neutrino mass (& P
* Of sep: There are at least two local effects (using infinitesimal-size objects) that can detect gravitational fields, the tidal distorsion of an object, and the precession of a spinning
non-spherical gyroscope; A gravitational field implies an unambiguous, non-zero \(R^a_{\,\,bcd}\); The strong equivalence principle fails even in Newtonian gravity; It is violated in QED in curved
spacetime, with "faster than light" photons (> see causality violations), and by metric-affine theories that predict vacuum birefringence (> see phenomenology).
@ Of wep: Will PRL(89) [in non-symmetric gravity]; Göklü & Lämmerzahl CQG(08)-a0801 [from metric fluctuations]; Gasperini a2101-ch [gravity at finite temperature].
@ From string dilaton: Damour gq/97-proc, gq/97-proc; [Landau et al ap/03-wd].
@ Classical charged particles: Goto et al CQG(10)-a1007 [and radiation reaction]; Toth a1404.
@ And cosmology: Hui et al PRD(09)-a0905 [from modified gravity]; Hees et al a1504-proc [some cosmological consequences].
@ Other situations: Ellis gq/03 [leptons]; Ellis et al IJMPA(04)gq/03 [from spacetime foam]; Barrow & Scherrer PRD(04)ap [fermions vs bosons]; Hehl & Obukhov GRG(08)-a0705 [and electromagnetic
coupling, axion and dilaton]; Bertolami et al PLB(07), Le Delliou et al AIP(07)-a0709 [dark energy–dark matter interaction in A586]; Carroll et al PRL(09)-a0807 [dark-matter-induced]; Damour &
Donoghue PRD(10)-a1007 [through dilaton-like scalar field]; Minazzoli PRD(18)-a1811 [matter with unconventional coupling to geometry]; Blasone et al a1812 [scalar-tensor gravity at finite
temperature]; > s.a. Chameleon Field; fifth force; scalar-tensor gravity.
References > s.a. Internal Relativity; variation of constants.
@ General: in Dicke 64; Klein Sci(71)jan; Hughes CP(93) [experimental basis and consequences]; Iliev JGP(98)gq; Camacho MPLA(99)gq [continuous quantum measurement]; Rohrlich FP(00) [critique]; Damour
CRAS-gq/01 [rev]; Ghins & Budden SHPMP(01) [conceptual]; Nordtvedt gq/02 [consequences of incorporating special relativity]; Drake AJP(06)jan [and special / general relativity transition]; Fabbri in
(12)-a0905 [and the geometrization of gravity]; Damour CQG(12)-a1202 [theoretical aspects]; Nobili et al AJP(13)jul [universality of free fall and gravitational redshift]; Di Casola et al AJP(15)jan-
a1310 [precise formulation of the various versions, and relationships]; Brown & Read AJP(16)feb-a1512 [misconceptions]; Kapotis & Kalkanis TPT(16)oct [in class].
@ History: Rabinowitz IEEE(90)phy/07 [falling bodies]; Schücking & Surowitz gq/07, Weinstein a1208 [Einstein 1907]; Janssen SHPMP(12); > s.a. history of relativistic gravity.
@ Geometric formulation: Coleman & Schmidt JMP(95); Iliev JPA(96)gq, JPA(97)gq; Wesson GRG(03) [5D, weak]; Iliev gq/06-proc [and geodesic deviation].
@ Criticisms: Logunov et al SPU(96); Ginzburg & Froshenko SPU(95), SPU(96) [reply].
@ In deformed theories: Tkachuk PRA(12)-a1301 [and GUP, minimal length, deformed Poisson brackets]; Ghosh CQG(14)-a1303 [and GUP]; Gnatenko & Tkachuk PLA(17)-a1701 [non-commutative theories].
@ In other theories: Olmo PRL(07)gq/06 [in f(R) gravity theories]; Kraiselburd & Vucetich IJMPE(11)-a0902 [Bekenstein's theory]; Deruelle GRG(11)-a1104 [Nordström's scalar theory]; Sheikh-Jabbari
IJMPD(11) [Lovelock and other higher-order theories]; Puetzfeld & Obukhov PRD(15)-a1505 [scalar-tensor gravity]; > s.a. gravitational energy-momentum; kaluza-klein phenomenology; modified gravity
["ultra-strong" version].
@ For Casimir energy: Fulling et al PRD(07)ht; Milton et al JPA(07)-a0705, JPA(08)-a0710-proc, a0810-conf; Shajesh et al JPA(08)-a0711-proc; Milton et al PRD(14)-a1401.
@ And electromagnetism: Özer gq/99; Trzetrzelewski EPL(18)-a1504 [Lorentz force and geodesics]; Ni IJMPD(16) [and phenomenology, cosmology]; > s.a. electromagnetism in curved spacetime.
@ Generalized: Lyre IJMPD(00)gq [for gauge charges]; Chiao gq/02/PRL [extended, and Kramers-Kronig relations]; Mensky PLA(04) [from energy-momentum conservation]; Wiltshire PRD(08)-a0809
[cosmological]; Kopeikin a1311 [in FLRW cosmology]; Di Casola et al PRD(14)-a1401 [for self-gravitating bodies, and purely metric theories of gravity]; Hetzroni FP(20)-a2001 [in abstract spaces]; >
s.a. cosmological constant problem; physical constants ["c equivalence principle"].
@ Related topics: 't Hooft JGP(84) [and black-hole radiation]; Kreinovich & Zapatrin gq/97 [operational]; Carlip AJP(98)may-gq/99 [and kinetic energy]; Rohrlich PRD(01) [despite self-interaction];
Rodrigues & Sharif FP(01)mp/03 [and local Lorentz invariance]; Maluf et al CQG(07)-a0704 [tetrads and energy in freely falling frames]; Hui & Nicolis PRL(11)-a1009 [for scalar forces]; Hohensee et al
PRL(13)-a1308 [for bound kinetic energy].
main page – abbreviations – journals – comments – other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 19 may 2021 | {"url":"https://www.phy.olemiss.edu/~luca/Topics/e/ep.html","timestamp":"2024-11-07T20:49:48Z","content_type":"text/html","content_length":"21641","record_id":"<urn:uuid:48027dc7-6101-4db6-af65-aa364d16d7e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00350.warc.gz"} |
Neo4j Graph Algorithms: (4) Community Detection Algorithms
Community detection algorithms are used to evaluate how groups of nodes are clustered or partitioned, as well as their tendency to strengthen or break apart. This visual presentation of the Neo4j
graph algorithms is focused on quick understanding and less implementation details. With small reusable samples, for less time-consuming labs.
This article presents quickly – in a graphical and descriptive manner, skipping many implementation details – most of the Community Detection algorithms implemented by Neo4j in their Graph Data
Science (GDS) library. We focus on one or two small samples reused by most of the algorithms, to keep it simple and allow for less time-consuming labs.
Run the following CREATE query in a blank document. All these Cypher queries can be run quickly in a new Blank Project on the free online Neo4j Sandbox. Or, if you have your own environment, on your
Neo4j Desktop.
CREATE (a:Node {name: 'A'}),
(b:Node {name: 'B'}),
(c:Node {name: 'C'}),
(d:Node {name: 'D'}),
(e:Node {name: 'E'}),
(f:Node {name: 'F'}),
(a)-[:REL {weight: 50}]->(b),
(a)-[:REL {weight: 50}]->(c),
(a)-[:REL {weight: 100}]->(d),
(b)-[:REL {weight: 40}]->(d),
(c)-[:REL {weight: 40}]->(d),
(c)-[:REL {weight: 80}]->(e),
(d)-[:REL {weight: 30}]->(e),
(d)-[:REL {weight: 80}]->(f),
(e)-[:REL {weight: 40}]->(f);
This creates a simple weighted graph with REL relationships between Node vertices. Call anytime MATCH (n) RETURN n; to get a visual representation of the whole graph. For the relationships, you may
show the generic REL type, or the weight value, as below:
1. Louvain Modularity Algorithm
Louvain is a hierarchical clustering algorithm, that recursively merges communities into a single node and executes the modularity clustering on the condensed graphs. It detects communities in large
networks, and maximizes a modularity score for each community, where the modularity quantifies the quality of an assignment of nodes to communities. This means evaluating how much more densely
connected the nodes within a community are, compared to how connected they would be in a random network.
1.1. Unweighted Graph
CALL gds.louvain.stream({
nodeProjection: 'Node',
relationshipProjection: {
REL: { type: 'REL', orientation: 'UNDIRECTED' }
YIELD nodeId, communityId
RETURN gds.util.asNode(nodeId).name AS name, communityId;
The previous query detects two communities [A, B, C, D] and [E, F], when the graph is considered undirected and unweighted. Each community is assigned a unique ID, but their values don’t really
matter otherwise:
1.2. Weighted Graph
Next query creates a reusable native in-memory projection of a weighted graph, then we call Louvain again, to detect communities:
CALL gds.graph.create('myGraph', 'Node',
{ REL: { orientation: 'UNDIRECTED' } },
{ relationshipProperties: 'weight' }
CALL gds.louvain.stream('myGraph', {
relationshipWeightProperty: 'weight'
YIELD nodeId, communityId
RETURN gds.util.asNode(nodeId).name AS name, communityId;
The result is this time different, as the nodes are split between two communities [A, B, D, F] and [C, E]. This is because most of the relationships connecting C and E with the rest have lower weight
values (50, 40, 30 and 40), compared to the other ones:
2. Modularity Optimization Algorithm
The Modularity Optimization algorithm tries to detect communities in the graph based on their modularity. Modularity is a measure of the structure of a graph, measuring the density of connections
within a module or community. Graphs with a high modularity score will have many connections within a community but only few pointing outwards to other communities. The algorithm will explore for
every node if its modularity score might increase if it changes its community to one of its neighboring nodes.
2.1. Unweighted Graph
Using Modularity on the previous saved in-memory projection (and ignoring the weight property for now), detects the same [A, B, C, D] and [E, F] communities as Louvain for the unweighted graph:
CALL gds.beta.modularityOptimization.stream('myGraph')
YIELD nodeId, communityId
RETURN gds.util.asNode(nodeId).name AS name, communityId;
2.2. Weighted Graph
Using Modularity on the same projection, but based on the weight property, detects the same [A, B, D, F] and [C, E] communities as Louvain for the weighted graph:
CALL gds.beta.modularityOptimization.stream('myGraph', {
relationshipWeightProperty: 'weight'
YIELD nodeId, communityId
RETURN gds.util.asNode(nodeId).name AS name, communityId;
3. Triangle Count Algorithm
The Triangle Count algorithm counts the number of triangles for each node in the graph. A triangle is a set of three nodes where each node has a relationship to the other two. In graph theory
terminology, this is sometimes referred to as a 3-clique. The Triangle Count algorithm in the GDS library only finds triangles in undirected graphs.
We’ll count the triangles for each node on the previously in-memory saved projection, ignoring the weight property:
CALL gds.triangleCount.stream('myGraph')
YIELD nodeId, triangleCount AS count
RETURN gds.util.asNode(nodeId).name AS name, count;
Previous image shows all existing triangles. Based on this image, here is the triangle count for each node:
• A, C, E –> 2 triangles
• B, F –> 1 triangle
• D –> 4 triangles
4. Local Clustering Coefficient Algorithm
The Local Clustering Coefficient algorithm computes the local clustering coefficient for each node in the graph, using the Triangle Count algorithm.
CALL gds.localClusteringCoefficient.stream('myGraph')
YIELD nodeId, localClusteringCoefficient AS coef
RETURN gds.util.asNode(nodeId).name AS name, coef;
Instead of triangles, a value between 0 and 1 is returned for each node:
• A, C, E –> 0.67 (2 triangles)
• B, F –> 1.0 (1 triangle)
• D –> 0.4 (4 triangles)
5. K-1 Coloring Algorithm
The K-1 Coloring algorithm assigns a color to every node in the graph, using as few colors as possible, and making sure that every neighbor of a given node has a different color than the node itself.
We’ll continue coloring on the in-memory projection, used as a unweighted graph:
CALL gds.beta.k1coloring.stream('myGraph')
YIELD nodeId, color
RETURN gds.util.asNode(nodeId).name AS name, color;
The result return one color number for each node. Translated into a visual representation, here is what we get (remark there are no two directly connected nodes using the same color):
6. Label Propagation Algorithm (LPA)
The Label Propagation algorithm (LPA) is a fast algorithm for finding communities in a graph. It detects these communities using network structure alone as its guide, and doesn’t require a
pre-defined objective function or prior information about the communities.
6.1. Preparing New Graph
For the rest of this article, we’ll need a different graph, to be able to group the nodes into different communities. Run the following three queries one by one:
MATCH (n) DETACH DELETE n;
CREATE (alice:User {name: 'Alice'}),
(bridget:User {name: 'Bridget'}),
(charles:User {name: 'Charles'}),
(doug:User {name: 'Doug'}),
(mark:User {name: 'Mark'}),
(michael:User {name: 'Michael'}),
(alice)-[:FOLLOW {weight: 1}]->(bridget),
(alice)-[:FOLLOW {weight: 10}]->(charles),
(mark)-[:FOLLOW {weight: 1}]->(doug),
(bridget)-[:FOLLOW {weight: 1}]->(michael),
(doug)-[:FOLLOW {weight: 1}]->(mark),
(michael)-[:FOLLOW {weight: 1}]->(alice),
(alice)-[:FOLLOW {weight: 1}]->(michael),
(bridget)-[:FOLLOW {weight: 1}]->(alice),
(michael)-[:FOLLOW {weight: 1}]->(bridget),
(charles)-[:FOLLOW {weight: 1}]->(doug);
MATCH (n) RETURN n;
First query removed all previous nodes and relationships. Second query populated your database with a new graph, displayed by the last query similar to the view below (show weight property values
instead of the generic FOLLOW relationship type):
6.2. Weighted Graph
It’s time to run the LPA now. Discard the previous myGraph from memory and create a new native projection, based on the new node and relationship types. Then call the Label Propagation algorithm on
this new projection:
CALL gds.graph.drop('myGraph');
CALL gds.graph.create('myGraph', 'User', 'FOLLOW',
{ relationshipProperties: 'weight' }
CALL gds.labelPropagation.stream('myGraph')
YIELD nodeId, communityId
RETURN gds.util.asNode(nodeId).name AS name, communityId;
The call discovered two communities: [Doug, Mark, Charles] and [Michael, Alice, Bridget]:
7. Speaker-Listener LPA (SLLPA)
The Speaker-Listener Label Propagation Algorithm (SLLPA) is a variation of the Label Propagation algorithm that is able to detect multiple communities per node.
CALL gds.alpha.sllpa.stream('myGraph', {
maxIterations: 100,
minAssociationStrength: 0.1
YIELD nodeId, values
RETURN gds.util.asNode(nodeId).name AS name,
values.communityIds AS communityIds;
Previous SLLPA-based query discovered several nodes in more than one community, as in the image below:
8. Weakly Connected Components (WCC) Algorithm
The WCC algorithm finds sets of connected nodes in an undirected graph, where all nodes in the same set form a connected component. WCC is often used early in an analysis to understand the structure
of a graph. Using WCC to understand the graph structure enables running other algorithms independently on an identified cluster. As a preprocessing step for directed graphs, it helps quickly identify
disconnected groups.
CALL gds.wcc.stream('myGraph', {
relationshipWeightProperty: 'weight',
threshold: 10
}) YIELD nodeId, componentId
RETURN gds.util.asNode(nodeId).name AS Name, componentId;
Remark that only Alice and Charles are connected by a relationship with weight 10 (all other relationships have weight 1). This generates one community for both:
9. Strongly Connected Components (SCC) Algorithm
The Strongly Connected Components (SCC) algorithm finds maximal sets of connected nodes in a directed graph. A set is considered a strongly connected component if there is a directed path between
each pair of nodes within the set. It is often used early in a graph analysis process to help us get an idea of how our graph is structured.
CALL gds.alpha.scc.stream({
nodeProjection: 'User',
relationshipProjection: 'FOLLOW'
YIELD nodeId, componentId
RETURN gds.util.asNode(nodeId).name AS Name, componentId;
The graph has been considered unweighted (weight property didn’t matter), but directed. Mark and Doug are strongly connected through reciprocal links. Just like Alice, Bridget and Michael at the
other end. Charles is on his own:
Read all articles from the same Neo4j Graph Algorithms series:
1. Path Finding Algorithms
2. Centrality Algorithms
3. Similarity Algorithms
4. Community Detection Algorithms
5. Link Prediction Algorithms
Need a NoSQL Expert?
Certified Solutions Architect in Azure and AWS
Certified Professional in Cassandra, Couchbase, Redis, Neo4j
Experienced in DynamoDB, Cosmos DB, MongoDB
Cristian Scutaru
I designed and implemented the Data Xtractor suite, with Model Xtractor, Query Xtractor, and Visual Xtractor as separate modules. I am a software architect and developer with over 30 years
professional experience. I’ve been working with relational databases for almost three decades and I was constantly unhappy with the relative limitation of those tools used to connect directly to a
platform, and instantly extract and display data in flexible ways. | {"url":"https://data-xtractor.com/blog/graphs/neo4j-graph-algorithms-community-detection/","timestamp":"2024-11-07T04:11:11Z","content_type":"text/html","content_length":"93790","record_id":"<urn:uuid:4fa154dd-ace3-4163-a28f-331321acaa6e>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00714.warc.gz"} |
Programming with Constraints Reading Group
Autumn 2015 — Tuesday, 2:30pm — CSE 203
Subscribe to the calendar:
Google Calendar
We’ll be reading and discussing papers relating to programming models for working with search procedures and solvers, solver technology, and reactive programming.
Some paper links may point into the ACM Digital Library or the Springer online collection. Using a UW IP address, or the UW libraries off-campus access, should provide access. To receive
announcements and news, please subscribe to the 591R mailing list.
Programming Models
• Nondeterministic Algorithms Robert Floyd, JACM 1967.
Programs to solve combinatorial search problems may often be simply written by using multiple-valued functions. Such programs, although impossible to execute directly on conventional computers,
may be converted in a mechanical way into conventional backtracking programs. The process is illustrated with algorithms to find all solutions to the eight queens problem on the chessboard, and
to find all simple cycles in a network.
• Curry, a functional logic programming language Antoy and Hanus, CACM 2010.
Combining the paradigm features of both logic and functional programming makes for some powerful implementations.
• Programming with Enumerable Sets of Structures. Ivan Kuraj, Viktor Kuncak, Daniel Jackson. OOPSLA 2015.
We present SciFe, an efficient, modular, and feature-rich framework for automated generation and validation of complex structures, suitable for tasks including automated testing (which explores
the space of many structured test inputs) and syntax-guided synthesis (which explores a large number of candidate programs). Our framework is capable of exhaustive, incremental, lazy, and
memoized enumeration of structured values from not only finite but also infinite domains, while providing fine-grain control over the process.
• Constraints as Control. Ali Sinan Köksal, Viktor Kuncak, and Philippe Suter. POPL 2012
We present an extension of Scala that supports constraint programming over bounded and unbounded domains. The resulting language, Kaplan, provides the benefits of constraint programming while
preserving the existing features of Scala. Kaplan integrates constraint and imperative programming by using constraints as an advanced control structure; the developers use the monadic ’for’
construct to iterate over the solutions of constraints or branch on the existence of a solution. The constructs we introduce have simple semantics that can be understood as explicit enumeration
of values, but are implemented more efficiently using symbolic reasoning.
• Integrating constraint satisfaction techniques with complex object structures. François Pachet, Pierre Roy. 15th Annual Conference of the British Computer Society Specialist Group on Expert
Integrating constraint satisfaction techniques with complex object structures is highly desirable. Several libraries are now available to use algorithms off-the shelf and embed them in large
object-oriented systems. However, the design of complex object + constraint problems is an open issue that severely limits the applicability of available libraries. We compare two radically
different approaches in designing systems integrating objects and finite domain constraints. In the first approach, constraints are defined within classes and constrain attributes of the class,
thereby introducing “partially instantiated” objects in the reasoning. In the second approach, constraints are defined outside classes, and constrain fully instantiated objects. We show that, for
a particular class of problem, the second solution yields a simpler design while being more efficient. We exemplify our claims by comparing these two approaches on a hard object + constraint
problem: four-voice harmonization of tonal melodies, seen as a representative complex configuration problem
Constraint Solver Technology
• On Counterexample Guided Quantifier Instantiation for Synthesis in CVC4. Reynolds et al. CAV 2015 (winner of the SyGuS 2015 competition!)
We introduce the first program synthesis engine implemented inside an SMT solver. We present an approach that extracts solution functions from unsatisfiability proofs of the negated form of
synthesis conjectures.
• Stochastic Local Search for Satisfiability Modulo Theories Frohlich et al. AAAI 2015.
We present a novel stochastic local search (SLS) algorithm to solve SMT problems, especially those in the theory of bit-vectors, directly on the theory level. We explain how several successful
techniques used in modern SLS solvers for SAT can be lifted to the SMT level. Experimental results show that our approach can compete with state-of-the-art bit-vector solvers on many practical
instances and, sometimes, outperform existing solvers. This offers interesting possibilities in combining our approach with existing techniques, and, moreover, new insights into the importance of
exploiting problem structure in SLS solvers for SAT
• Impact of Community Structure on SAT Solver Performance. Zack Newsham, Vijay Ganesh, Sebastian Fischmeister, Gilles Audemard, and Laurent Simon. SAT 2015.
Modern CDCL SAT solvers routinely solve very large industrial SAT instances in relatively short periods of time. It is clear that these solvers somehow exploit the structure of real-world
instances. However, to-date there have been few results that precisely characterise this structure. In this paper, we provide evidence that the community structure of real-world SAT instances is
correlated with the running time of CDCL SAT solvers.
• Are There Good Mistakes? A Theoretical Analysis of CEGIS.. Susmit Jha and Sanjit A. Seshia. SYNT 2014.
Counterexample-guided inductive synthesis (CEGIS) is used to synthesize programs from a candidate space of programs. The technique is guaranteed to terminate and synthesize the correct program if
the space of candidate programs is finite. But the technique may or may not terminate with the correct program if the candidate space of programs is infinite. In this paper, we perform a
theoretical analysis of counterexample-guided inductive synthesis technique. We investigate whether the set of candidate spaces for which the correct program can be synthesized using CEGIS
depends on the counterexamples used in inductive synthesis, that is, whether there are good mistakes which would increase the synthesis power. We investigate whether the use of minimal
counterexamples instead of arbitrary counterexamples expands the set of candidate spaces of programs for which inductive synthesis can successfully synthesize a correct program. We consider two
kinds of counterexamples: minimal counterexamples and history bounded counterexamples. The history bounded counterexample used in any iteration of CEGIS is bounded by the examples used in
previous iterations of inductive synthesis. We examine the relative change in power of inductive synthesis in both cases. We show that the synthesis technique using minimal counterexamples
MinCEGIS has the same synthesis power as CEGIS but the synthesis technique using history bounded counterexamples HCEGIS has different power than that of CEGIS, but none dominates the other.
• Modular Synthesis of Sketches Using Models. Rohit Singh, Rishabh Singh, Zhilei Xu, Rebecca Krosnick, and Armando Solar-Lezama. CAV 2013.
One problem with the constraint-based approaches to synthesis that have become popular over the last few years is that they only scale to relatively small routines, on the order of a few dozen
lines of code. This paper presents a mechanism for modular reasoning that allows us to break larger synthesis problems into small manageable pieces. The approach builds on previous work in the
verification community of using high-level specifications and partially interpreted functions (we call them models) in place of more complex pieces of code in order to make the analysis modular.
• The Strategy Challenge in SMT Solving Leonardo de Moura and Grant Olney Passmore. Automated Reasoning and Mathematics, 2013.
High-performance SMT solvers contain many tightly integrated, hand-crafted heuristic combinations of algorithmic proof methods. While these heuristic combinations tend to be highly tuned for
known classes of problems, they may easily perform badly on classes of problems not anticipated by solver developers. We present a challenge to the SMT community: to develop methods through which
users can exert strategic control over core heuristic aspects of SMT solvers. We present evidence that the adaptation of ideas of strategy prevalent both within the Argonne and LCF theorem
proving paradigms can go a long way towards realizing this goal.
• Predicting Learnt Clauses Quality in Modern SAT Solvers Gilles Audemard and Laurent Simon. IJCAI 2009.
Beside impressive progresses made by SAT solvers over the last ten years, only few works tried to understand why Conflict Directed Clause Learning algorithms (CDCL) are so strong and efficient on
most industrial applications. We report in this work a key observation of CDCL solvers behavior on this family of benchmarks and explain it by an unsuspected side effect of their particular
Clause Learning scheme. This new paradigm allows us to solve an important, still open, question: How to designing a fast, static, accurate, and predictive measure of new learnt clauses
pertinence. Our paper is followed by empirical evidences that show how our new learning scheme improves state-of-the art results by an order of magnitude on both SAT and UNSAT industrial
• Backdoors to Typical Case Complexity. Ryan Williams, Carla P. Gomes, and Bart Selman. IJCAI 2003.
There has been significant recent progress in reasoning and constraint processing methods. In areas such as planning and finite model-checking, current solution techniques can handle
combinatorial problems with up to a million variables and five million constraints. The good scaling behavior of these methods appears to defy what one would expect based on a worst-case
complexity analysis. In order to bridge this gap between theory and practice, we propose a new framework for studying the complexity of these techniques on practical problem instances. In
particular, our approach incorporates general structural properties observed in practical problem instances into the formal complexity analysis. We introduce a notion of “backdoors”, which are
small sets of variables that capture the overall combinatorics of the problem instance. We provide empirical results showing the existence of such backdoors in real-world problems. We then
present a series of complexity results that explain the good scaling behavior of current reasoning and constraint methods observed on practical problem instances.
Reactive Programming | {"url":"https://uwplse.org/meet/pcrg/15au.html","timestamp":"2024-11-07T20:10:22Z","content_type":"text/html","content_length":"22164","record_id":"<urn:uuid:6515a3d8-9e4d-4a96-8e0a-d29c4fb378d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00805.warc.gz"} |
A Focus on Research
Mathematics research interests at Mississippi State University lie in four main categories: 1) Analysis, 2) Algebra, Combinatorics, and Topology, 3) Computational and Applied Mathematics, and 4)
Differential Equations. Faculty in each area actively publish papers and work with graduate students pursuing PhDs and Masters degrees. Find out more about each research group by going to their
respective pages indicated to the right.
The research interests of the statistics faculty at Mississippi State University include topics in applied and theoretical statistics, as well as applied and theoretical probability and data science.
The majority of their work is driven by practical problems across a spectrum of disciplines, including, but not limited to, modeling financial and climatological data, linear and response surface
modeling in agriculture and veterinary sciences, and various engineering fields. Their work also addresses important questions arising in data science, such as data integration, data quality,
dimension reduction, and more.
Weekly Seminars
Date: Thursday, May 01, 2025
Dr. Lance A. Waller, Professor in the Department of Biostatistics and Bioinformatics, Rollins School of Public Health, Emory University, Fellow of ASA and RGS
Statistics Seminar Series
Maps: A Statistical View
Time: -
Location: Allen 411 | {"url":"https://www.math.msstate.edu/research","timestamp":"2024-11-08T05:58:32Z","content_type":"text/html","content_length":"39661","record_id":"<urn:uuid:dc294810-3b89-4516-8ef4-147351cf3586>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00586.warc.gz"} |
Bayesian average to compute average rating, properly
Almost every single website, app or platform on the internet has some sort of rating system in place. Whenever you purchase a product or use a service, you are asked to rate it on a scale, say 1 to
5. The platform then uses this data to generate a score and build a ranking system around it. The score is the measure of quality for each product or service. By surfacing the most quality content on
top of the list, the platform tries to up their sales and ensure better engagement with their users.
Coming up with an aggregated score is not an easy thing - we need to crunch a million ratings and then see that the score is, in fact, the true measure of quality. If it isn’t then it would directly
affect the business. Today we discuss how we should define this score in a rating based system; spoiler alert! the measure is called Bayesian Average.
To keep things simple we define the problem statement as
Given the ratings, on a scale of 1 to 5, that users give to a movie, we generate a score that is a measure of how good a movie is which then helps us get the top 10 movies of all time.
We will use the MovieLens Dataset to explore various scoring functions in this article. In the dataset, we get user ratings for each movie and the ratings are made on a scale of 1 to 5.
Generating the score
The score we generate for each item should be proportional to the quality quotient which means higher the score, superior is the item. Hence we say that the score of an item is the function of all
the m ratings that it received.
Arithmetic Mean
The simplest and the most common strategy to compute this aggregated score for an item is by taking an Arithmetic Mean (average) of all the ratings it received. Hence for each item we sum all the
ratings that it received and divide it by its cardinality, giving us the average value.
Issues with arithmetic mean
The arithmetic mean falls apart pretty quickly. Let’s say there is an item with just one rating of 5 on 5, the item would soar high on the leaderboard ranking. But does it deserve that place?
probably not. Because of low cardinality (number of ratings), the score (and hence the rank) of the item will fluctuate more and will not give a true measure of quality.
With the movie dataset, we are analyzing here are the top 10 movies ranked using Arithmetic Mean.
Through this measure, all of the top 10 movies have a score of 5 (out of 5) and all of them have just 1 rating. Are these really the top 10 movies of all time? Probably not. Looks like we need to do
a lot better than the Arithmetic Mean.
Cumulative Rating
To remedy the issue with Arithmetic Mean, we come up with an approach of using Cumulative Rating as the scoring function hence instead of taking the average we only consider the sum of all the
ratings as the final score.
Cumulative Rating actually does a pretty decent job, it makes popular items with a large number of ratings bubble up to the top of the leaderboard. When we rank the movies in our dataset using
Cumulative Ratings we get the following as the top 10.
The top 10 movies now feature Shawshank Redemption, Forrest Gump, Pulp Fiction, etc. which are in fact considered as the top movies of all times. But is Cumulative Rating fool-proof?
Issues with cumulative rating
Cumulative Rating favors high cardinality. Let’s say there is an extremely poor yet popular item A that got 10000 ratings of 1 on 5, and there is another item B which is very good but it got 1000
rating of 5 on 5 Cumulative Rating thus gives a score of 10000 _ 1 = 10000 to item A and 1000 _ 5 = 5000 to item B, but B clearly is far superior of an item than A.
Another issue with Cumulative Rating is the fact that it generates an unbounded score. Ideally, any ranking system expects a normalized bounded score so that the system becomes predictable and
We established that Cumulative Rating is better than Arithmetic Mean but it is not fool-proof and that’s where the Bayesian Average comes to the rescue.
The Bayesian Average
Bayesian Average computes the mean of a population by not only using the data residing in the population but also considering some outside information, like a pre-existing belief - a derived property
from the dataset, for example, prior mean.
The intuition
The major problem with Arithmetic Mean as the scoring function was how unreliable it was when we had a low number of data points (cardinality) to compute the score. Bayesian Average plays a part here
by introducing pre-belief into the scheme of things.
We start by defining the requirements of our scoring function
• for an item with a fewer than average number of ratings - the score should be around the system’s arithmetic mean
• for an item with a substantial number of ratings - the score should be the item’s arithmetic mean
• as the number of ratings that an item receives increases, the score should gradually move from system’s mean to item’s mean
By ensuring the above we neither prematurely promote nor demote an item in the leaderboard. An item is given a fair number of chances before its score falls to its own Arithmetic mean. This way we
use the prior-belief - System’s Arithmetic mean, to make the scoring function more robust and fair to all items.
The formula
Given the intuition and scoring rules, we come up with the following formula
In the above formula, w indicates the weight that needs to be given the item’s Arithmetic Mean A while S represents the System’s Arithmetic Mean. If A and S are bounded then the final score s will
also be bounded in the same range, thus solving the problem with Cumulative Rating.
Suppose the number of ratings that an item i receives is denoted by m and the average number of ratings that any item in the system receives is denoted by m_avg, we define the requirements of weight
w as follows
• w is bounded in the range [0, 1]
• w should be monotonically increasing
• w should be close to 0 when m is close to 0
• w should reach 0.5 when number m reaches m_avg
• w tries to get closer to 1 as m increases
From the above requirements, it is clear that w is acting like a knob which decides in what proportions we should consider an item’s mean versus the system’s mean. As w increases we tilt more towards
item’s mean. We define the w as
When we combine all of the above we get the final scoring function as
One of the most important properties of Bayesian Average is the fact that the pre-existing belief acts as support which oversees that the score does not fluctuate too abruptly and it smoothens with
more number of ratings.
Applying Bayesian Average to movies dataset
After applying the above mentioned Bayesian Average scoring function to our Movie dataset, we get the following movies as top 10
Pretty impressive list! The list contains almost all the famous movies that we all think make the cut. Bayesian average thus provides a bounded score that is a measure of the quality of the item, by
using prior-belief i.e. system’s mean.
Analyzing how Bayesian Average changes the rank
Now that we have seen that the Bayesian Average is, in fact, an excellent way to rank items in a rating system, we find how the rank of an item changes as it receives more ratings. Below we plot the
change in the percentile rank of the movies: Kingsman, Logan and The Scorpion King.
We observe that the fluctuations in percentile rank are more in the case of Arithmetic Mean. Sometimes even after receiving a good number of reviews, the rank fluctuates sharply. In the case of
Bayesian Average after an initial set of aberrations, the rank smoothens and converges.
A note on Bayesian Average
Bayesian Average is not a fixed formula that we have seen above, but it is a concept where we make the scoring function “smoother” by using a pre-existing belief as support. Hence we can tweak the
formula as per our needs, or use multiple prior beliefs and still it would classify as a Bayesian Average. | {"url":"http://edge.arpitbhayani.me/blogs/bayesian-average","timestamp":"2024-11-05T23:31:30Z","content_type":"text/html","content_length":"30428","record_id":"<urn:uuid:4f25533c-cadd-4ce5-a00a-08f1a4bc520f>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00109.warc.gz"} |
Median germination time — t50
Compute the median germination time (\(t_{50}\)). Median germination time is the time to reach 50% of final/maximum germination.
t50(germ.counts, intervals, partial = TRUE, method = c("coolbear", "farooq"))
Germination counts at each time interval. Can be partial or cumulative as specified in the argument partial.
The time intervals.
logical. If TRUE, germ.counts is considered as partial and if FALSE, it is considered as cumulative. Default is TRUE.
The method for computing median germination time. Either "coolbear" or "farooq".
The median germination time (\(t_{50}\)) value in the same unit of time as specified in the argument intervals.
With argument method specified as "coolbear", median germination time is computed according to the formula by Coolbear et al. (1984) as follows.
\[t_{50}=T_{i}+ \frac{(\frac{N+1}{2}-N_{i})(T_{j}-T_{i})}{N_{j}-N_{i}}\]
Where, \(t_{50}\) is the median germination time, \(N\) is the final number of germinated seeds, and \(N_{i}\) and \(N_{j}\) are the total number of seeds germinated in adjacent counts at time \(T_
{i}\) and \(T_{j}\) respectively, when \(N_{i} < \frac{N + 1}{2} < N_{j}\).
Similarly with argument method specified as "farooq", median germination time is computed according to the formula by by Farooq et al. (2005) as follows.
\[t_{50}=T_{i}+ \frac{(\frac{N}{2}-N_{i})(T_{j}-T_{i})}{N_{j}-N_{i}}\]
Where, \(t_{50}\) is the median germination time, \(N\) is the final number of germinated seeds, and \(N_{i}\) and \(N_{j}\) are the total number of seeds germinated in adjacent counts at time \(T_
{i}\) and \(T_{j}\) respectively, when \(N_{i} < \frac{N}{2} < N_{j}\).
Coolbear P, Francis A, Grierson D (1984). “The effect of low temperature pre-sowing treatment on the germination performance and membrane integrity of artificially aged tomato seeds.” Journal of
Experimental Botany, 35(11), 1609--1617.
Farooq M, Basra SMA, Ahmad N, Hafeez K (2005). “Thermal hardening: A new seed vigor enhancement tool in rice.” Journal of Integrative Plant Biology, 47(2), 187--193.
See also
x <- c(0, 0, 0, 0, 4, 17, 10, 7, 1, 0, 1, 0, 0, 0)
y <- c(0, 0, 0, 0, 4, 21, 31, 38, 39, 39, 40, 40, 40, 40)
int <- 1:length(x)
# From partial germination counts
t50(germ.counts = x, intervals = int, method = "coolbear")
#> [1] 5.970588
t50(germ.counts = x, intervals = int, method = "farooq")
#> [1] 5.941176
# From cumulative germination counts
t50(germ.counts = y, intervals = int, partial = FALSE, method = "coolbear")
#> [1] 5.970588
t50(germ.counts = y, intervals = int, partial = FALSE, method = "farooq")
#> [1] 5.941176 | {"url":"https://aravind-j.github.io/germinationmetrics/reference/t50.html","timestamp":"2024-11-05T17:10:19Z","content_type":"text/html","content_length":"16143","record_id":"<urn:uuid:4ab1a415-644a-4de6-a57f-e4d6d059fd73>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00079.warc.gz"} |
Identification of the parameters of a composite material by experimental-computational damping research
Calculation of the modal and damping characteristics necessary to eliminate resonant oscillation of products made of polymeric materials requires reliable data on the elastic characteristics of the
material. The problem is that the mechanical properties of polymer composite materials depend on a large number of factors. The aim of the work is to determine the damping coefficients for a layered
composite material and the subsequent validation of the mathematical model. The Rayleigh damping model was chosen to calculate the damping coefficients. The choice is due to the fact that the
resulting stiffness and mass matrix is determined by the natural oscillation modes of the problem without attenuation, which makes it possible to split the modes into separate dynamic subtasks. A
sample made according to the ASTM standard was chosen as the object of study. To increase an error of the calculation, the mathematical model of the sample was modeled in detail by the finite element
method using the technique of layer-by-layer modeling. The method for determining the damping coefficients is carried out in three stages. At the first stage, with the help of modal analysis, the
natural oscillation modes are determined, corresponding to the nature of the oscillation studied in the experiment. At the second stage, an implicit dynamic analysis with default damping parameters
in order to calculate the damping ratio is performed. At the last stage, a steady-state dynamic analysis taking into account the characteristics obtained in the previous stages. Next, an iterative
process begins, including implicit and steady-state dynamic analyses, performed alternately. At each step, the previously calculated Rayleigh proportionality coefficients are introduced into the
model. As a result of the identification of the mathematical model, the damping coefficients $\alpha$ and $\beta$ are calculated. The damping experiment was chosen as a validation problem. The
damping ratio $\zeta$ was chosen as the criterion of convergence with the experimental data.
• Tests of composite materials for damping properties (the damping ratio and proportional coefficients) are modeled in accordance with Rayleigh theory. The finite element method is used for
• The damping properties of a layered composite material are calculated. The methodology includes the results of frequency analysis, dynamic implicit analysis and dynamic stationary state analysis.
• The damping properties are refined iteratively up to an error of 1.81% relative to the experimental data.
1. Introduction
Polymer composite materials (PCM), in particular carbon plastics, are widely used for the manufacture of loaded parts of aircraft and aircraft engines due to the possibility of lightweight
construction without reducing strength [1-4]. It is known, for example, that the use of PCM in the design of the airframe of Boeing 787 aircraft reaches 50 % by weight [2, 4]. The wide use of PCM is
envisaged in the creation of domestic aircraft and aircraft engines of new generation [2, 3].
The problem of search of natural frequencies can be solved with and without damping. Most often the presence of damping in the system affect not only natural frequencies, but also the natural forms.
If the excitation frequency is close to the natural frequency (e.g. in the ±50 % band), consideration of damping is very important, as can be seen from the resonant response curves. In this case,
attention must be paid to the selection of suitable and correct damping criteria values. Near resonant, the response is almost completely dependent on the damping value, so, for example, changing the
hysteresis loss factor from 0.01 to 0.02 could end up doubling the estimated stress level [8].
In time-domain dynamic studies, damping usually has little effect on the final results. Exceptions are encountered when modeling wave propagation or loads that excite some of the resonant frequencies
of a structure [9].
On the other hand, setting the damping in the time domain analysis allows us to stabilize the temporal computational scheme. Parasitic waves often appear in designs that are not interesting for the
main analysis. If they are not quenched, the time step can become excessively small. To extinguish the parasitic waves, one can add losses at high frequencies to the model [10-11].
Rayleigh damping is a simple approach to forming a damping matrix as a linear combination of a mass matrix and a stiffness matrix. The Rayleigh damping model makes it possible to calculate the
proportionality coefficients depending on the oscillation frequencies, which, in turn, makes it possible to perform calculations for the oscillation forms in different directions. This approach is
very useful in the study of composite materials, since in this way, by changing the studied frequencies, it is possible to obtain proportionality coefficients in different directions and take into
account the anisotropy of properties.
This damping model is clearly not related to any physical loss mechanisms. Initially this model was used because such a resulting matrix is diagonalized by the natural forms of the problem without
damping, which allows us to break the modes into separate dynamic subproblems.
In [12], the author investigates the possibility of polymer sample identification using modal analysis and validates the mathematical model with the tests performed. The maximum error with the
experiment is 3.9 %, which indicates the correctness of the chosen method. However, the author does not consider the friction between the layers in the mathematical model and this modeling method is
not suitable for determining the damping characteristics.
In this work, the model of the sample was made in layers. The damping characteristics are determined by an iterative method, including modal, implicit dynamic and steady-state analyses performed
using the finite element method. Comparison with the experiment is carried out using the damping ratio. The resulting error at the last iteration is 1.81 %.
2. Analysis of natural forms and frequencies
2.1. Model description
Glass Laminate Aluminum Reinforced Epoxy (GLARE) was chosen as the test specimen. The sample is a structural layered hybrid material consisting of thin (0.3-0.4 mm) sheets of aluminum alloys (Al-Li
medium strength reduced density alloy) and interlayers of fiberglass. Plastic layers usually consist of several monolayers of unidirectional adhesive prepreg reinforced with high-strength glass
fillers. Parameters of the materials are given in Table 1.
Thicknesses of the sample layers are shown in Table 2.
Table 1Mechanical properties of materials
Material Orientation Elastic modulus $E$, MPa Poisson’s ratio $\mathrm{\eta }$ Density $\rho$, t/mm^3
1441 alloy sheet – 79000 0.33 2.6E-9
0° 50000 0.3
Fiberglass 1.78E-9
90° 12000 0.07
Table 2Thicknesses of the sample layers
Material Thickness, mm
1441 alloy sheet 0 0.35
Fiberglass 0° 0.3
1441 alloy sheet 0 0.35
Fiberglass 90° 0.3
1441 alloy sheet 0 0.35
The geometric characteristics of the sample conform are shown in Table 3.
Table 3Geometric parameters of the sample
Sample Length, mm Width, mm Thickness, mm
GLARE 255 20 1.65
The finite-element model consists of 25000 elements of order 1 HEXA type (three-dimensional hexagonal volume element with eight nodes and three rotational degrees of freedom). In order to refine the
results of the finite element analysis, the sample is made by layer-by-layer modeling. The model of the sample is shown in Fig. 1.
The natural forms were calculated by the Lanczos block method with spectral transformations [14]. The model is bounded at the end by all degrees of freedom, as shown in Fig. 2. The constraint is
chosen according to the experimental conditions.
Fig. 1General view of the FEM
Fig. 2Boundary conditions for modal analysis
2.2. Calculation of natural forms and frequencies
At the first step, a modal analysis is performed, as a result of which the first two bending natural frequencies and forms were determined. The obtained data are presented in Table 4.
Results of calculation of the natural frequencies by the experimental method and the finite element method are summarized in Table 5.
As can be seen from the table, the damping ratio turned out to be an order of magnitude higher than the damping ratio obtained experimentally, which indicates the need to refine the material model.
For this purpose, proportionality coefficients are introduced into the material properties. The calculation of the coefficients was carried out by the iterative method described in detail in Chapter
(3.2). As a result of the measures described above, the values presented in Table 6 were obtained.
Table 4The result of the modal analysis
No. mods Natural frequency, Hz Natural form
1 22.307
2 139.75
Table 5Comparison of experimental data with initial data obtained by FEM
Method First natural frequency, Hz Error, %
Experiment 21.5
FEM 22.307
Table 6Comparison of experimental data with refined data obtained by FEM
Method First natural frequency, Hz Error, %
Experiment 21.5
FEM 21.55023
3. Implicit dynamic analysis
3.1. Model description
In order to perform a nonlinear implicit dynamic analysis, an initial displacement was applied to the finite-element model to create excitation. The boundary conditions at the first calculation step
are shown in Fig. 3. In the second step, the load applied in the first step is removed and the specimen moves freely. In order to obtain a detailed picture of the specimen oscillation, the removal
time was chosen to be 0.01 s. Total force applied to the sample was 10 N. A similar analysis was carried out without considering material damping and with time integrator parameter (TIP) –0.05
included. The methodology for determining the damping properties is described below. The damping ratio $\zeta$ was chosen as the convergence criterion.
Fig. 3Loading conditions for implicit dynamic analysis
3.2. Description of the calculation methodology
To model the damping properties of the material, the Rayleigh proportional damping model, is used:
where $C$ – global damping matrix, $M$ – global mass matrix, $K$ – global stiffness matrix, $\alpha$ – coefficient characterizing the inertial damping, $\beta$ – coefficient characterizing the
structural damping.
The damping coefficient of oscillation can be found through the natural frequencies:
${\zeta }_{i}=\frac{\alpha }{2{\omega }_{i}}+\frac{\beta {\omega }_{i}}{2},$
where ${\zeta }_{i}$ – the damping ratio, $\alpha$ – coefficient characterizing the inertial damping, ${\omega }_{i}$[]– natural frequencies of the sample, $\beta$ – coefficient characterizing the
structural damping.
The main advantage of this method is that it simulates the damping parameters for a material rather than for a particular structure, i.e. by determining the damping parameters for a given sample, it
is possible to use this model to calculate other structures made of this material.
To calculate the proportionality coefficients $\alpha$ and $\beta$ it is necessary to carry out a modal calculation to determine the natural frequencies of the sample, corresponding to two successive
bending forms of oscillation.
After determining the necessary oscillation frequencies, a calculation similar to the tests for the form at the frequency corresponding to the bending oscillation is performed. The calculation is
carried out within the framework of the general nonlinear dynamic analysis using implicit time integration. The result is the time-dependence of specimen displacement, from which the amplitudes of
the first and second free bending oscillation of the specimen are determined. Then, according to the formula given below, the damping ratio is determined [15-18]:
$\zeta =\frac{\mathrm{l}\mathrm{n}\left(\frac{{X}_{i}}{{X}_{i+1}}\right)}{2\pi },$
where $\zeta$ – the damping ratio, ${X}_{i}$ – amplitude of the ($i$) free oscillation, ${X}_{i+1}$ – amplitude of the ($i+1$) free oscillation.
The oscillation frequencies were calculated using a well-known formula:
where $\varpi$ – free oscillation frequency, $T$ – time period.
Proportionality coefficients $\alpha$ and $\beta$ could be calculated by the following equations [20]:
$\alpha =2\zeta \frac{{\omega }_{1}{\omega }_{2}}{{\omega }_{1}+{\omega }_{2}},$
where $\alpha$ – proportionality factor, determining the damping parameters at low frequencies, $\zeta$ – the damping ratio, ${\omega }_{1}$ – the first natural frequency, ${\omega }_{2}$ – the
second natural frequency:
$\beta =\frac{2\zeta }{{\omega }_{1}+{\omega }_{2}},$
where $\beta$ – proportionality coefficient, determining the damping parameters at high frequencies, $\zeta$ – the damping ratio, ${\omega }_{1}$ – the first natural frequency, ${\omega }_{2}$ – the
second natural frequency.
The data obtained should be clarified due to the fact that, first, modal analysis does not consider the damping properties of the material, therefore, the obtained values of natural frequencies are
overestimated; second, in the dynamic calculation the damping of oscillation is due to the general kinematic energy dissipation of the system, but not the damping properties of the material.
4. Steady-state analysis
4.1. Model description
After the initial calculation of the proportionality coefficients, a dynamic analysis of the stationary state based on the natural forms and frequencies is performed, which is used to calculate the
response of the system to harmonic excitation. Steady-state linear dynamic analysis uses the set of natural forms extracted in a previous modal step to calculate the steady-state solution as a
function of the frequency of the applied excitation. To conduct the steady-state analysis, the same as in implicit analysis boundary and loading conditions shown in Fig. 3 were used. The frequency
range was selected from 0 to 200 Hz.
Based on the results obtained as a result of the calculation, the natural frequencies and amplitudes of oscillation are specified. The damping ratio $\zeta$ and proportionality coefficients α and $\
beta$ are recalculated again respectively.
4.2. Description of the calculation methodology
The test results were processed using the fast Fourier transform method to obtain the amplitude-frequency response (AFR) of the realized oscillation [22]. A peak corresponding to the first resonant
frequency was determined on the obtained AFR. The width of the found peak allows us to determine the sample the damping ratio $\zeta$ based on the following equation:
$\zeta =\frac{{\varpi }_{2}-{\varpi }_{1}}{2\mathrm{\Omega }},$
where $\zeta$ – the damping ratio, $\mathrm{\Omega }$ – resonant frequency, ${\varpi }_{1}<{\varpi }_{2}$ – frequencies near resonant, at which the amplitude value decreases by a factor of √2
compared to the resonant amplitude.
To determine the damping ratio, the Eq. (7) was used. The peak shown on the graph corresponds to the resonant frequency. The following relation is carried out between the natural frequency and the
resonant frequency:
$\mathrm{\Omega }=\sqrt{{\omega }^{2}-2{h}^{2}},$
where $\mathrm{\Omega }$ – the resonant frequency, $\omega$ – the natural frequency, $h$ – viscous damping parameter.
The frequency of free damped oscillation is determined by the formula:
$\varpi =\sqrt{{\omega }^{2}-{h}^{2}},$
where $\varpi$ – the free oscillation frequency, $\omega$ – the natural frequency, $h$ – viscous damping parameter.
From Eqs. (8, 9), knowing the resonant and natural frequency, the oscillation frequency can be expressed in the following way:
$\varpi =\sqrt{\frac{1}{2}\left({\omega }^{2}+{\mathrm{\Omega }}^{2}\right)},$
where $\varpi$ – the free oscillation frequency, $\omega$ – the natural frequency, $\mathrm{\Omega }$ – the resonant frequency.
To calculate the frequency of free damped oscillation corresponding to the second natural form, it is assumed that it decreases relative to the second natural frequency in proportion to the decrease
in the frequency of oscillation in the first natural form:
${\varpi }_{2}={\omega }_{2}\frac{{\omega }_{1}}{{\varpi }_{1}},$
where ${\varpi }_{2}$ – free oscillation frequency corresponded to the second natural form, ${\omega }_{2}$ – the second natural frequency, ${\varpi }_{1}$ – free oscillation frequency corresponded
to the first natural form, ${\omega }_{1}$ – the first natural frequency.
Then the proportionality coefficients are calculated using the Eqs. (5, 6). To calculate the proportionality coefficients, the natural frequencies $\omega$ in the Eqs. (5, 6) were replaced by the
frequencies of free damped oscillation $\varpi$.
5. Results
This section presents data and graphs for all iterations performed.
As described above, the first step is dynamic implicit analysis with a built-in damping parameter. The graphs and the numerical data obtained at iteration No. 1 are shown in Fig. 4 and in Table 7.
Fig. 4The result of the implicit dynamic analysis, first iteration
In the above graph, displacement is expressed in millimeters (mm), time is expressed in seconds (s). The damping ratio and frequency were calculated for all peaks of oscillation on the graph, then
the values were averaged.
Hereinafter, the symbol “No.” refers to the number of iterations at which the values were obtained.
Table 7Result of the proportionality coefficients calculation, first iteration
No. First natural frequency, Hz Second natural frequency, Hz The damping ratio $\alpha$, s^-1 $\beta$, s
1 22.307 139.75 0.009907 0.381138 0.000122
Then a dynamic analysis of the stationary state was carried out. The graphs and the numerical data obtained at iteration No. 2 are shown in Fig. 5 and in Table 8.
Fig. 5The result of the dynamic analysis of the stationary state based on natural forms and natural frequencies (second iteration)
In the above graph, displacement is expressed in millimeters (mm), frequency is expressed in hertz (Hz).
Displacements in this type of analysis do not make physical sense; mathematically, the amplitude at resonance tends to infinity. In this case, the displacements are used only to determine the
parameters of the resonant peak.
Table 8Result of the proportionality coefficients calculation, second iteration
No. Frequency (first nat. form), Hz Frequency (second nat. form), Hz The damping ratio $\alpha$, s^-1 $\beta$, s
1 21.55023 135.008952 0.033285 1.237140 0.000425
In the following iterations, the set of implicit and dynamic analysis of the stationary state was used again and the results were processed. The number of approximations is chosen from the condition
of sufficiency to determine the character of the dynamics of the iteration process.
Then an implicit dynamic analysis was carried out. The graphs and the numerical data obtained at iteration No. 3 are shown in Fig. 6 and in Table 9.
Fig. 6The result of the implicit dynamic analysis, third iteration
In the above graph, displacement is expressed in millimeters (mm), time is expressed in seconds (s). The damping ratio and free oscillation frequency was calculated for all peaks of oscillation on
the graph, then the values were averaged.
Table 9Result of the proportionality coefficients calculation, third iteration
No. Frequency (first nat. form), Hz Frequency (second nat. form), Hz The damping ratio $\alpha$, s^-1 $\beta$, s
3 19.5977 122.77665 0.028549 0.9649711 0.000401
Then a dynamic analysis of the stationary state was carried out. The graphs and the numerical data obtained at iteration No. 4 are shown in Fig. 7 and in Table 10.
Fig. 7The result of the dynamic analysis of the stationary state based on natural forms and natural frequencies (fourth iteration)
In the above graph, the displacement is expressed in millimeters (mm), frequency is expressed in hertz (Hz).
Table 10Result of the proportionality coefficients calculation, fourth iteration
No. Frequency (first nat. form), Hz Frequency (second nat. form), Hz The damping ratio $\alpha$, s^-1 $\beta$, s
4 21.55023 135.008952 0.026471 0.983877234 0.000338
To describe the nature of the dynamics of the iteration process the graph of dependence of the error of calculation of the damping coefficient on the number of iteration. It is shown in Fig. 8.
Fig. 8Error in the damping ratio according to the results of all iterations
As can be seen from the graphs, the error of the damping ratio takes its minimum value on the fourth iteration. A leap in the graph at the second iteration is associated with the replacement of the
damping model proposed by the program with the Rayleigh damping model, whose data is entered manually.
6. Experiment description
The experiment data was taken from the article [22]. The paper considers the tests of several samples of GLARE, as well as tests of sandwich beam.
Dynamic tests of GLARE samples were carried out using the method of free damped oscillation. In the tests, the samples were rigidly fixed with a clamp at one end, and loading conditions were set at
the other end, leading to the occurrence of damped bending oscillation, mainly according to the first natural form. The length of the fixed part of the GLARE samples was 15 mm (not included in the
total length of the sample in the model, which is 255 mm). To excite oscillation, the free end of the fixed sample was either struck with a metal striker, or its initial deviation from the
equilibrium position with a maximum deflection of 5-15 mm was set. Both loading options (static – by impact and kinematic – by drawing the end of the sample) led to the same measurement results, with
a difference not exceeding the error of the measuring equipment. To register the amplitude of free damped oscillation of the samples, a video recording was carried out. The tests used a
VideoSprint-G4 video camera with a shooting frequency of 800 cad/s. The test results were processed using the fast Fourier transform method to obtain the amplitude-frequency response (AFR) of the
realized oscillation [22].
7. Conclusions
As a result of this work the methodology of calculation of damping properties for the composite material was developed. The technique is based on the modal analysis of the material sample and
qualified by the results of the experiment.
Mathematical models have been developed to carry out oscillation virtual tests. The layer-by-layer modeling technique used to build the model showed high error of the results and can be used not only
for model identification, but also for strength tests of the specimen to determine the character of fracture.
In the future, it is planned to apply the developed method for structurally similar samples. Calculations according to the presented methodology can be carried out for the forms of oscillation in
different directions and thus obtain damping coefficients in different directions, which will allow one to create a complete model of the damping of the composite material. The model can be improved
by introducing cohesive layers to account for the matrix.
The combined method of calculating $\alpha$ and $\beta$ coefficients to calculate damping according to Rayleigh’s model can be applied at the design stage of aircraft structures products and thus
obtain refined results of static and dynamic strength by identifying the model.
• G. Wu and J.-M. Yang, “The mechanical behavior of GLARE laminates for aircraft structures,” Journal of The Minerals, Metals and Materials Society, Vol. 57, No. 1, pp. 72–79, Jan. 2005, https://
• L. B. Vogelesang and A. Vlot, “Development of fibre metal laminates for advanced aerospace structures,” Journal of Materials Processing Technology, Vol. 103, No. 1, pp. 1–5, Jun. 2000, https://
• V. V. Antipov, N. Yu. Serebrennikova, O. G. Senatorova, L. V. Morozova, N. F. Lukina, and Yu. N. Nefedova, “Hybrid layered materials with a low rate of fatigue crack development,” Vestnik
Mashinostroeniya, Vol. 61, No. 12, pp. 45–49, 2016.
• V. V. Shestov, V. V. Antipov, N. Yu. Serebrennikova, and Yu. N. Nefedova, “High-strength layered material based on aluminum-lithium alloy sheets,” Technology of Light Alloys, Vol. 53, No. 1, pp.
119–123, 2016.
• Z. Ming, P. A. Mbango-Ngoma, D. Xiao-Zhen, and C. Qing-Guang, “Numerical investigation of the trailing edge shape on the added damping of a Kaplan turbine runner,” Mathematical Problems in
Engineering, Vol. 2021, pp. 1–11, Aug. 2021, https://doi.org/10.1155/2021/9559454
• M. Salehi, P. Sideris, and A. B. Liel, “Numerical simulation of hybrid sliding-rocking columns subjected to earthquake excitation,” Journal of Structural Engineering, Vol. 143, No. 11, p.
04017149, Nov. 2017, https://doi.org/10.1061/(asce)st.1943-541x.0001878
• W. Ma and X. Liu, “Phased microphone array for sound source localization with deep learning,” Aerospace Systems, Vol. 2, No. 2, pp. 71–81, Dec. 2019, https://doi.org/10.1007/s42401-019-00026-w
• C. Jagels, L. Reichel, and T. Tang, “Generalized averaged Szegő quadrature rules,” Journal of Computational and Applied Mathematics, Vol. 311, pp. 645–654, Feb. 2017, https://doi.org/10.1016/
• D. J. Ewins, Modal Testing: Theory, Рractice and Аpplication. Baldock: Research Studies Press LTD, 2000.
• J. Duan and Z. Zhang, “An efficient method for nonlinear flutter of the flexible wing with a high aspect ratio,” Aerospace Systems, Vol. 1, No. 1, pp. 49–62, Jun. 2018, https://doi.org/10.1007/
• W. Heylen, S. Lammens, and P. Sas, Modal Analysis Theory and Testing. Leven: Katholieke Universiteit Leuven.
• M. S. Nikhamkin, S. V. Semenov, and D. G. Solomonov, “Application of experimental modal analysis for identification of laminated carbon fiber-reinforced plastics model parameters,” in Proceedings
of the 4th International Conference on Industrial Engineering, pp. 487–497, 2019, https://doi.org/10.1007/978-3-319-95630-5_51
• M. M. Kamel and Y. A. Amer, “Response of parametrically excited one degree of freedom system with non-linear damping and stiffness,” Physica Scripta, Vol. 66, No. 6, pp. 410–416, Jan. 2002,
• M. H. Gutknecht, “A completed theory of the unsymmetric Lanczos process and related algorithms, Part I,” SIAM Journal on Matrix Analysis and Applications, Vol. 13, No. 2, pp. 594–639, Apr. 1992,
• D. L. Mcculloch et al., “ISCEV Standard for full-field clinical electroretinography (2015 update),” Documenta Ophthalmologica, Vol. 130, No. 1, pp. 1–12, Feb. 2015, https://doi.org/10.1007/
• E. Erduran, “Evaluation of Rayleigh damping and its influence on engineering demand parameter estimates,” Earthquake Engineering and Structural Dynamics, Vol. 41, No. 14, pp. 1905–1919, Nov.
2012, https://doi.org/10.1002/eqe.2164
• J. F. Hall, “Problems encountered from the use (or misuse) of Rayleigh damping,” Earthquake Engineering and Structural Dynamics, Vol. 35, No. 5, pp. 525–545, Apr. 2006, https://doi.org/10.1002/
• P. Jehel, P. Léger, and A. Ibrahimbegovic, “Initial versus tangent stiffness-based Rayleigh damping in inelastic time history seismic analyses,” Earthquake Engineering and Structural Dynamics,
Vol. 43, No. 3, pp. 467–484, Mar. 2014, https://doi.org/10.1002/eqe.2357
• M. Barabash, B. Pisarevskyi, and Y. Bashynskyi, “Material damping in dynamic analysis of structures (with LIRA-SAPR program),” Civil and Environmental Engineering, Vol. 16, No. 1, pp. 63–70, Jun.
2020, https://doi.org/10.2478/cee-2020-0007
• V. S. Geraschenko, A. S. Grishin, and N. I. Gartung, “Approaches for the calculation of Rayleigh damping coefficients for a time-history analysis,” SUSI 2018, Vol. 180, No. 11, pp. 227–237, Jun.
2018, https://doi.org/10.2495/susi180201
• S. Nikolaev, S. Voronov, and I. Kiselev, “Estimation of damping model correctness using experimental modal analysis,” Vibroengineering PROCEDIA, Vol. 3, pp. 50–54, Oct. 2014.
• O. A. Prokudin, Y. O. Solyaev, A. V. Babaytsev, A. V. Artemyev, and M. A. Korobkov, “Dynamic characteristics of three-layer beams with load-bearing layers made of alumino-glass plastic,” PNRPU
Mechanics Bulletin, No. 4, pp. 260–270, Dec. 2020, https://doi.org/10.15593/perm.mech/2020.4.22
About this article
Mechanical vibrations and applications
Rayleigh damping model
natural frequencies
the damping ratio
finite element method
The article is prepared in the implementation of the program for the creation and development of the World-Class Research Center “Supersonic” for 2020-2025 funded by the Ministry of Science and
Higher Education of the Russian Federation (Grant agreement of April 20, 2022 No. 075-15-2022-309).
Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Conflict of interest
The authors declare that they have no conflict of interest.
Copyright © 2023 V. P. Eremin, et al.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/22670","timestamp":"2024-11-07T15:17:40Z","content_type":"text/html","content_length":"154264","record_id":"<urn:uuid:f89838a9-4495-4c8c-bc92-16bd18c17083>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00897.warc.gz"} |
CourseNana | COMP3506/7505 - Algorithms and Data Structures - Assignment Two: Bloom Filter, Pathfinding, Chain Reaction
Assignment Two – 25%
Algorithms and Data Structures – COMP3506/7505 – Semester 2, 2024 Due: 3pm on Friday October 18th (week 12)
The main objective of this assignment is to extend your knowledge from assignment one to build more complex data structures and solve more complex problems. In particular, you will be working on
graph and compression problems. In this second assignment, we will leave more of the design choices up to you (like the k-mers part in A1). This assessment will make up 25% of your total grade. We
recommend you start early.
A Getting Started
The assignment is structured similarly to assignment one. The skeleton codebase, data, software dependencies, implementation rules, are described below. Rules for success: Think before you code.
Think before you post an Ed question. Use a pen and paper. Don’t be afraid to be wrong. Give yourself time to think. Start thinking about these problems early. Read the entire spec before you do
anything at all.
The codebase contains a number of data structures stubs that you must implement, as well as some scripts that allow your code to be tested. Figure 1 shows a snapshot of the project directory tree
with the different files categorized. Note that we provide you with (simplified) versions of the data structures built during assignment one. You are permitted to modify any of the files listed. You
may also use structures/util.py for any utilities that do not deserve their own file, or add your own data structures if you think they may help; store them in their own files inside the structures
We also provide a number of test graphs for you to use, but you are encouraged to build further test graphs of your own; you may also share your test graphs with other students if you wish. Each
graph is represented as a simple text file that stores an adjacency list for each vertex in the graph. There are three specific types of graphs, each with their own subdirectory. All graph types are
undirected. 4N graphs are simple graphs where each vertex can be thought of as occupying a position on a square grid/lattice. As such, these nodes can have at most 4 neighbours. KN graphs are an
extension that allow an arbitrary number of neighbors. POSW graphs extend KN graphs to apply positive integer weights to edges. The appendix in Section M contains an example of each graph type.
Our codebase is written for Python 3.10+ as we have provided type annotations; as such, you
will need to use Python 3.10 at minimum. The second assignment has one special dependency – the curses library – that allows your algorithms to be visualized in a simple terminal window.
2 COMP3506/7505 – Semester 2, 2024
Figure 1 The directory tree organized by data structures (inside the structures directory), and the three executable programs (in the root directory, coloured orange).
Figure 2 The data tree organized by graph types. 4N are the most simple grid-based graphs. KN are graphs where each node has an arbitrary degree. POSW are graphs with arbitrary degree nodes and
positive weights between the edges.
If you are developing locally, you may need to install curses. See the documentation1 for more information. This library is already available on moss.
1 https://docs.python.org/3/howto/curses.html
Note that you can do the entire assignment without using the visualizer, but it will be less fun and you won’t be able to show off to your friends. The visualizer is only useful for the earlier
pathfinding solutions on grids (Task 2), and it must be executed in a terminal window .
Implementation Rules
The following list outlines some important information regarding the skeleton code, and your implementation. If you have any doubts, please ask on Ed discussion.
• ❙ The code is written in Python and, in particular, should be executed with Python 3.10 or higher. The EAIT student server, moss, has Python 3.11 installed. We recommend using moss for the
development and testing of your assignment, but you can use your own system if you wish.
• ❙ You are not allowed to use built-in methods or data structures – this is an algorithms and data structures course, after all. If you want to use a dict (aka {}), you will need to implement that
yourself. Lists can be used as “dumb arrays” by manually allocating space like myArray = [None] * 10 but you may not use built-ins like clear, count, copy, extend, index, insert, remove, reverse,
sort, min, max, and so on. List func- tions like sorted, reversed, zip are also banned. Similarly, don’t use any other collections or structures such as set. You cannot use the default hash
function. Be sensible – if you need the functionality provided by these methods, you may implement them yourself.
• ❙ You are not allowed to use libraries such as numpy, pandas, scipy, collections, etc.
• ❙ Exceptions: The only additional libraries you can use are random, math, and functools (but only for the total_ordering decorator). You are allowed to use range and
enumerate to handle looping. You may use tuples (for example; mytup = ("abc", 123)) to store multiple objects of different types. You may use len wherever you like, and you can use list slicing
if given a Python list as input. If we ask for a Python list in a function return type, you can use append or pop.
COMP3506/7505 – Semester 2, 2024
B Task 1: Data Structures
We’ll start off by implementing some new data structures. All we specify is the interface; the choice of design is yours, as long as the interface behaves correctly and efficiently. You may test
these with the test_stuctures.py program: python3.11 test_structures.py.
Task 1.1: Fix and Extend the Priority Queue (3 marks)
A queue is a data structure that can handle efficient access to elements on a first-in-first- out basis. Recall that a priority queue is an extension of a simple queue that supports efficient access
to the element with the highest priority; new elements can be inserted with a given (arbitrary) priority, and the priority queue must be able to support efficient dequeue operations based on the
priority order. For this assignment, we will assume priority values are numeric and comparable (so, they may be floats or integers), with lower values representing a higher priority. In other words,
we’re going to be supporting a min-heap.
We have provided you with a semi-working priority queue in pqueue.py. Unfortunately, we ran out of time to get it working perfectly, so there are a few2 bugs lurking in the implementation. First,
crush these bugs so the priority queue works properly (1 mark)!
Once your heap is operating correctly, we need to handle a few more subtleties; we’d like to support in-place construction in linear time through the ip_build function, and in-place sorting via the
sort function. Note that the in-place operations should operate directly on the data array without creating a copy — running in-place heap sort will yield a sorted array, but will destroy the heap
ordering. As such, you may assume the user will no longer use the heap once the sort function has been used.3 Welcome to UQ(ueue).
You can test via: python3.11 test_structures.py --pq
Task 1.2: Implement a Map (3 marks)
Your next job is to implement a concrete data structure to support the map interface. Recall that a map allows items to be stored via unique keys. In particular, given a key/value pair (k, v)
(otherwise known as an entry), a map M can support efficient insertions (associate k with v in M), accesses (return the value v associated with k if k ∈ M), updates (update the value stored by key k
from v to v′) and deletes (remove (k,v) from M). In other words, you
will be supporting operations like a Python dict (aka {}) class. Test via: python3.11 test_structures.py --map
Task 1.3: Implement a Bloom Filter (3 marks)
Bloom filters are an interesting probabilistic data structure that can support extremely efficient and compact set membership operations. In particular, they use hashing in combination with
bitvectors to toggle on sets of bits; when looking up a given key k, the key is hashed via a series of unique hash functions and mapped to various indexes of the bitvector. Next, these bits are
observed; if they are all on, then we return True which means “yes, this key might be in the set.” Otherwise, we return False which means “No, this key is definitely not in the set.” Your Bloom
filter does not need to double check that the True values are definitely in the set; that job is for another data
C Preliminaries: The Graph Class
Many of the following problems (all of Task 2, and some aspects of Task 3) will require the use of a graph data structure. We have provided a concrete implementation of a graph data structure for
you, and you will need to get familiar with it in order to progress. The graph types are defined in structures/graph.py.
Graph Types
There are two key types of graphs. The Graph class is the base class which stores nodes and edges. Each node in the graph (Node) stores an id which is the index of the node in the graph’s adjacency
list. For example, if you have a node with an id 22, this means that the node will be stored at the Graph’s self._nodes[22] and can be accessed via the Graph’s get_node() function. The Graph also
provides a function to return a list of neighbours given an index/node identifier.
There is a special LatticeGraph type that extends the Graph class (and a LatticeNode that extends Node). This specialized graph is used only for graphs that are placed on a lattice. In other words,
these graphs can be thought of as simple grids, where each vertex has between zero and four neighbors. As such, some additional properties including the number of logical rows and columns in the (4N)
graph are stored. For your purposes, the only real difference you need to know about with this special type is that you can ask for the (x, y) coordinates of a given LatticeNode using the
get_coordinates() function. You can also directly return the nodes to the north/south/east/west using the appropriate get_north()
(etc) functions.
Your Implementations
All of the following tasks have pre-made function stubs. You should pay close attention to the type hints so you know what is expected to be taken as parameters, and what should be returned.
6 COMP3506/7505 – Semester 2, 2024
Figure 3 The Mega Gurkey – Artwork by Jesse Irwin.
D (Optional) Backstory for the Remainder of Assignment Two
Last year, the COMP3506/7505 cohort helped Barry Malloc capture an enterprising Aus- tralian Brush Turkey4 (named Gurkey Tobbler) that was ruining his garden. Afterwards, the chief scientist at
MallocLabs (Dr. Amongus) transported Gurkey to the lab to conduct some genomic sequencing. Thanks to your great work on DNA compatibility, Dr. Amongus has since discovered that Gurkey DNA is
compatible with that of the Loxodonta Africana, the African Bush Elephant!5
While this is a crowning scientific discovery, there is one (big) problem; Dr. Amongus has created a giant hybrid mega Gurkey through the irresponsible use of genetic modification tools. Our goal in
this section is to find the mega Gurkey before it’s too late, and to help Barry conduct further analysis on the Gurkey genome.
Meta comment: Why do we make up these crazy backstories and bury details inside them? Well, because you need to practice looking at a problem you have and extracting the important details. It is
highly unlikely you will ever be given an extremely well specified problem. It is
also a lot more fun this way :-)
4 https://en.wikipedia.org/wiki/Australian_brushturkey 5 https://en.wikipedia.org/wiki/African_bush_elephant
E Task 2: Pathfinding Algorithms Getting Started
To get started, we will focus on lattice graphs. Note that we have provided some graphs for you already, and the ones we are interested (for now) are those in the data/4N directory. However, your
solutions here must also work on the data/KN and data/POSW graphs (note that KN and POSW are the same types of graph if an algorithm does not use edge weights).
We have provided a program called test_pathfinding.py to help you test your al- gorithms. This program allows different pathfinding algorithms through two dimensional mazes to be tested (mandatory)
and visualized (optional). Note that in order to make life easier, we’re randomly generating the origin and goal vertices, so you will need to supply a seed to the random number generator (via --seed
) to yield different origins and goals each time you run the program. All implementations for Task 2 must be inside the algorithms/pathfinding.py file, where appropriate stubs are provided for you.
Task 2.1: Breadth-First Search (2 marks)6
Given some arbitrary start vertex u, and some goal vertex v, Breadth-First Search (BFS) systematically walks across the graph until either v is found, or there are no remaining vertices to explore.
Figure 4 provides a sketch of this process. You must implement the bfs_traversal() stub; note that both the visited list, and the path, are expected to be returned. Please see the type annotations
for the specific details about what should be returned.
To make your results reproducible, you must enqueue/push the unvisited neighbours in the order they are given to you from the get_neighbours() function.
Finally, while we will be visualizing our BFS on the lattice graphs, you must ensure that your algorithms translate to graphs with arbitrary degree. This should be trivial to implement. For the
avoidance of doubt, your BFS algorithm will be tested on the KN graphs.
Test via: python3.11 test_pathfinding.py --graph data/4N/one.graph --bfs --seed <number> [--viz]
Note that the --viz flag is optional (and triggers the visualizer to run) and <number> should be substituted with an integer.
Task 2.2: Dijkstra’s Algorithm (3 marks)
BFS is nice; it is quite simple and it works well at finding the Gurkey when the graph is unweighted. However, Brisbane is a hilly city, and some paths are more expensive than others; we’ll need to
take this into account to find the true shortest path to the Gurkey. We also don’t necessarily know where the Gurkey will be, so it would be good to find the shortest path from our current location
to all possible locations.
Your goal is as follows. Given a weighted graph and a source node, return the cost of the lowest-cost path to all reachable nodes. If a node is not reachable (for instance, if the Gurkey has
destroyed all of the bridges) then you should not return it in your list. Please see the type annotations for the specific details about what this function should return.
Test via: python3.11 test_pathfinding.py --graph data/POSW/one.graph --dijkstra --seed <number>
8 COMP3506/7505 – Semester 2, 2024
Input graph ABCD EFGH IJL
eue on each step
1. (1) B
2. (2) A F
3. (3) F
4. (4) E GJ
5. (5) GJI
6. (6) J I H
Visited Nodes
[B, A, F, E, G, J, ...]
BFS starts at an arbitrary vertex B, by enqueueing it. At each step, a dequeue returns the next candidate vertex which is then marked visited (and added to the visited list). Each neighboring vertex
of the current candidate (that is yet to be visited) is enqueued.
BFS halts when either the goal node (H) is visited, or when there are no other vertices to visit.
Figure 4 A sketch of breadth-first search starting at vertex B and searching for vertex H. A queue keeps track of the next vertices to visit, and they are visited as they are dequeued. A list can be
used to track the order in which nodes are visited.
Note that it does not really make sense to use the --viz flag with Dijkstra, because the 4N graphs do not have edge weights (and the viz tool needs to use 4N graphs).
Dijkstra's Algorithm
Input Graph G
7B5E A192
Destination, Path, and Cost
A [G, C] 6 B [G, C] 5 C [G] 4 D[]4 E[G]4 G[]2 H[]2
Path is not required to be returned, but shown for clarity.
Output: A list of vertices and their associated shortest path cost from the source. Cost is calculated as the sum of edge weights across the shortest path. Graphs are guaranteed to have positive edge
Figure 5 A sketch of Dijkstra’s algorithm. See the code for the expected output format and structure.
Task 2.3: Depth-First Search (2 marks – COMP7505 only)
Depth-First Search (DFS) operates very similarly to Breadth-First Search. However, instead of using a FIFO queue, it uses a LIFO stack. You must implement the dfs_traversal() stub (plus any
additional data structures you may require); note that both the visited set, and the path, are expected to be returned. Please see the type annotations for the specific details about what these
functions should return.
To make your results reproducible, you must push the unvisited neighbours in the order they are given to you from the get_neighbours() function.
Finally, while you can visualize DFS on the lattice graphs, you must ensure that your algorithms translate to graphs with arbitrary degree. This should be trivial to implement. For the avoidance of
doubt, your DFS algorithm will be tested on the KN graphs.
Test via: python3.11 test_pathfinding.py --graph data/4N/one.graph --dfs --seed <number> [--viz]
Note that the --viz flag is optional (and triggers the visualizer to run) and <number> should be substituted with an integer.
Input graph Stack on each step ABCDJL
DFS starts at an arbitrary vertex B, by marking it as visited and pushing it on the stack. At each step, the stack is popped to get the current candidate vertex (which is then added to the visited
list) and each neighboring vertex is marked as visited and pushed onto the stack.
DFS halts when either the goal node (I) is visited, or when there are no other vertices to visit.
(1) (2) (3) (4) (5) (6) (7)
Visited Nodes
[B, A, F, J, G, H, L, ...]
Figure 6 A sketch of depth-first search starting at vertex B and searching for node I. A stack keeps track of the next vertices to visit, and they are visited as they are popped from the stack. A
list can be used to track the order in which nodes are visited.
10 COMP3506/7505 – Semester 2, 2024
F Task 3: Problem Solving
Now that the mega Gurkey is back in the lab, we will need to conduct additional testing to ensure such an event never happens again (well, at least until COMP3506/7505 2025). Unfortunately, Dr.
Amongus has already been fired, so it is our job to help Barry Malloc determine how this all happened in the first place.
Task 3.1: Maybe Maybe Maybe (3 marks)
MallocLabs has a huge database of k-mers that have been sequenced throughout their many years of operation. To determine which genomes may have been involved in the genetic modification of the
Gurkey, we can simply compare the Gurkey genome to all genomes in the database to find out which ones match, and return those for further analysis. The problem, however, is that the database contains
trillions of k-mers.
Our job is to create a fast and compact filtering algorithm that, given a list of database k-mers, D, and another list of query k-mers Q, returns a list of k-mers that from Q that are likely to be in
D. We award more marks for having lower false positive rates; the maximum allowed false positive rate is 10%, and then we will measure at 5% and 1%. Note that lower false positive rates might come at
higher time and space costs.
Test via: python3.11 test_problems.py --maybe --seed <number>
Maybe Maybe Maybe
Input k-mer database D
Input query k-mers Q CTGTATCC
Output (likely) matches
Note the false positive in the output. Figure 7 A sketch of Maybe3 – See the code for the expected output format and structure.
Task 3.2: Dora and the Chin Bicken (3 marks)
MallocLabs’ spies have recently discovered that their main competitor CallocLabs7 has hired Dr. Amongus, and are planning to release a giant Ibis named Chin Bicken to wreak havoc on MallocLabs HQ!
Barry and the team need to get prepared. The head honchos at MallocLabs have decided on the following strategy:
Chin Bicken will, at some point, attack the MallocLabs HQ;
Since Chin Bicken is enormous, it may attack different parts of the building simultaneously; At this time, MallocLabs will release a robot – Dora – which does the following:
1. It will receive as input an undirected graph G where vertices represent rooms in MallocLabs HQ, and edges represent undamaged connections between rooms.
2. From its starting location, it will explore all reachable rooms of the building to collect genomic data left by the Chin Bicken.
3. This genomic data comes in the form of special gene symbols, s, represented by a single character; there is one at each vertex of G.
4. Next, the robot builds a gene symbol frequency table T which maps each gene symbol s to its total frequency in G, denoted fs.
5. Once T is computed, the robot builds a minimum redundancy code via Huffman’s algorithm, resulting in a codebook CG mapping each s to a codeword cs.
6. Finally, the robot receives a sequence L = ⟨s0, . . . , sn−1⟩ of n symbols, drawn from all symbols appearing in G. This sequence represents the specific body part of Chin Bicken that MallocLabs
believes is its weak point. The robot will use CG to encode all s ∈ L into one long bitvector B. That is, B will hold the concatenation of the encoding of each symbol in L: cs0 cs1 · · · csn−1 .
❙ Once the robot produces B, Barry can feed it into the GeneCoder6000 to develop a weapon to fend off the Chin Bicken. Of course, Dora will need to be fast. It has to visit all vertices in the graph
before Chin Bicken causes any further chaos, after all. You have been tasked to write the logic for Dora. Get to it!
12 COMP3506/7505 – Semester 2, 2024
Task 3.3: Chain Reaction (3 marks)
To progress further with the reconstruction of Dr. Amongus’ cloning programme, Barry now needs to find what is called the optimal reaction compound. We are given n candidate compounds, each of which
is represented by a unique ⟨x,y⟩ coordinate based on their reactivity in two specific dimensions of interest. Each compound also holds a floating point value known as the spike radius r.
Compound A is said to cause a reaction with compound B if the circle centered on ⟨xA,yA⟩ with radius r overlaps with the compound8 at ⟨xB,yB⟩; however, reactions do not occur naturally — they must be
triggered by some other reaction. When a compound reacts, any compound that it is reactive with it will also be triggered (and so on).
You are given one charged molecule to set off a chain reaction, and you must select the given compound i ∈ [0, n − 1] that will maximize the total number of compounds in the chain reaction. If there
are ties, return the one with the smallest identifier.
Test via: python3.11 test_problems.py --chain --seed <number> Chain Reaction
Input: List of x, y coordinates with associated radius values.
Output: The identifier of the compound that we should use to trigger the largest chain reaction. The answer here is 3. If there is a tie, return the lowest identifier.
Triggered Node Compounds in Reaction
1 [1, 2]
2. 2 [2]
3. 3 [3, 1, 2, 4, 6]
4 [4, 6]
5 [5]
6 [6, 4]
Figure 9 A sketch of the Chain Reaction problem. See the code for the expected output format and structure.
Task 3.4: Lost in the Labyrinth (aka notably more k-cool) (2 marks)
The attack from CallocLabs compromised some of the building structure at MallocLabs HQ, and the team is concerned that the Gurkey might break free. Barry would like to build a labyrinth to contain
the Gurkey, and has offers from various construction companies. However, he is concerned that some of these companies are trying to scam him, so we need to help him to come up with an algorithm to
determine whether a labyrinth can even be constructed from each offer.
Each company treats the design of a labyrinth as a graph problem. They provide us with four integers: n, m, k, and c, where n is the number of vertices (|V |), m is the number of edges (|E|), k is
the diameter of the graph, and c is the cost to produce it. From these four integers, we must determine if their offer is valid or not. A labyrinth is considered valid if it conforms to the following
• ❙ It is a connected graph.
• ❙ It has no double edges or self loops.
❙ The largest shortest simple path between any two vertices v1 and v2 (the diameter) is
at most k. (In other words, if you found the shortest simple path between every pair of vertices, the diameter of the graph is the length of longest one of these.)
Given a list of offers, you must return the cheapest offer that can be constructed. If there are ties, return the one with the smallest identifier.
python3.11 test_problems.py --labyrinth --seed <number>
14 COMP3506/7505 – Semester 2, 2024
G Task 4: Txt Cmprsn (up to 3 bonus marks)
Keen for more punishment? We have just the thing... We’ll be running a simple compression challenge. You will be given an arbitrary file, and you need to provide a compression/de- compression
algorithm. The stubs are provided for you in the compression.py file. There will be no marks given for incorrect/lossy algorithms; the output (after decompression) must exactly match the provided
input. The marking scheme is as follows.
• ❙ One mark: Your algorithm can compress our file to at least half of its original size.
• ❙ One mark: Your algorithm is in the top 50% of those submitted.
• ❙ One mark: Your algorithm is in the top 10 of those submitted.
We will have a public compression leaderboard available that can be observed. There will be a separate submission area on Gradescope for this part. Please just submit your single file compress.py
without any zipping. All of your references need to be placed in this file. | {"url":"https://coursenana.com/programming/assignment/comp3506-comp7505-algorithms-and-data-structures-assignment-two-bloom-filter-pathfinding-chain-reaction","timestamp":"2024-11-12T18:46:53Z","content_type":"text/html","content_length":"175785","record_id":"<urn:uuid:e054b74a-d4a8-4457-9642-59ccd2fbfe25>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00723.warc.gz"} |
[Kotlin] What is Infix Functions? | Nakoblog
[Kotlin] What is Infix Functions?
To understand infix functions in Kotlin.
What is infix function?
In Kotlin, functions marked with the infix keyword can also be called using the infix notation (sited from the documentation of Kotlin). An example follows.
infix fun Int.add(a:Int): Int {
return this+a
What is infix notation?
Infix notation is one of the notation types. In infix notation the operator is described in thee middle of the targets. + – * / are usually used as infix notation. (1+2, a-b)
• Infix notation
• Prefix notation (Polish notation)
• Postfix notation (Reverse Polish notation)
Implementation in Kotlin
fun pre_add(a:Int, b:Int): Int{
return a+b
infix fun Int.in_add(a:Int): Int {
return this+a
infix fun Int.in_sub(a:Int): Int = this-a
fun main() {
println(1 in_add 2)
println(1 in_sub 2) | {"url":"https://s-nako.work/2020/03/kotlin-what-is-infix-functions/","timestamp":"2024-11-08T04:46:51Z","content_type":"text/html","content_length":"40655","record_id":"<urn:uuid:c58d380f-3cad-42de-9468-77be64100611>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00593.warc.gz"} |
Chaos Theory
Chaos Theory
11 Jan 2010
Progress: Completed
I find chaos theory fascinating, the idea that something can be both deterministic
unpredictable. What better way is there to investigate than a computer model – one with no random parameters, and input values known to a ridiculous number of decimal places. I wrote this program and
write-up for an assignment in uni. Needless to say I got the highest mark possible.
Predictability of a Double Pendulum
Figure 1
Double pendulum diagram
showing M, m, R, r, and θ
A simulation of a double pendulum was constructed to investigate the predictability of its final position after a set amount of time, for different initial conditions. Certain ranges were found to be
predictable while others were chaotic. The boundaries of these regions showed a direct correlation to where the lower pendulum inverted.
Many natural systems exhibit chaos, such as the weather. Despite their deterministic nature these systems can be unpredictable due to their extreme sensitivity to initial conditions. Knowing what
defines the chaotic regions is important in many fields.
In the simplest (Ref1) double pendulum (Figure1) m << M, thus the motion (amplitude) of M can be approximated by (eq.1).
where a[0] and τ are constants of amplitude and period of M. Hence m is modelled as a single pendulum under acceleration a (eq.2&3)
where ω is the angular velocity of m. Constants r, g are arbitrarily chosen to be 1. Initial values of ω, θ are chosen to be 0.0 and 0.1, respectively.
The first simulation evolves for t[max]=100 seconds and records the θ[final]. This simulation is repeated through a range of values for a[0] and τ, and plotted to a graph (Figure2). Each data-point
is a shade of gray corresponding to the value of theta; it is not the brightness that is of relevance but more how it relates to adjacent points. Thus areas forming a smooth gradient represent
predictable regions, while the seemingly stochastic areas correspond to chaos. Figure3 shows the same area with t[max]=400 seconds. The boundary between chaos and predictability is clearer.
Figure 2.
Values of θ for 0.21 < a[0] < 0.25 and 7 < τ < 9 with t[max] = 100
Figure 3.
Values of θ for 0.21 < a[0] < 0.25 and 7 < τ < 9 with t[max] = 400
A phase-space plot (ω verses θ) for a single pendulum would be a circle. Figures 4&5 show phase-space plots for simulations inside and outside the chaotic boundary. The chaotic pendulum’s plot
stretching to the left demonstrates θ leaving its normal range, i.e. it made a full rotation. Investigating this as the reason for the chaotic region, the program was modified to identify regions
where the lower pendulum inverted. This was achieved by measuring the proportion of time θ escaped the range (-π,π) and highlighting the graph in red to represent it. (Figures6&7)
Figure 4.
Phase-space plot with a[0] = 0.251, τ = 8 and t[max] = 100 The pendulum swings normally, with just a few wobbles.
Figure 5.
Phase-space plot with a[0] = 0.251, τ = 7 and t[max] = 100 The pendulum has clearly done several full inversions.
Figure 6.
Values of θ for 0.21 < a[0] < 0.25 and 7 < τ < 9 with t[max] = 100 Red indicates θ went outside the region ( -π , π )
Figure 7.
Values of θ for 0.21 < a[0] < 0.25 and 7 < τ < 9 with t[max] = 400
A deeper hue means θ spent a bigger fraction of the simulation outside the region ( -π , π )
As time evolves, the chaotic regions become clearer and spread asymptotically. The red highlight follows the same area exactly, as can be seen by comparing Figures 2&3 to Figures 6&7. Zooming out,
the boundary of chaos surrounds the regular, predictable pattern in a parabola (Figure8). Outside this region the pendulum behaves like a single pendulum as the amplitude of M is too small, or the
period of M is too large, to have an effect. The base of the region marks the resonant frequency of the lower pendulum (ref3).
Figure 8.
Values of θ for 0.0 < a[0] < 0.2 and 6 < τ < 8 with t[max] = 400 The chaotic region draws out a parabola over time.
Figure 9.
Same as figure 8 but with θ[init]=π/2
Figure 10.
Same as figure 8 but with θ[init]=3π/4
Investigating the dependency on initial conditions, Figures 9&10 show how the same area changes when ω[init] is kept constant). The chaotic region grows as the pendulum can escape the normal region
much easier. At ω[init] would produce similar (but skewed) graphs, akin to jumping ahead in the simulation.
Choosing levels of accuracy is a compromise with computing power and available time. Reducing Δt makes the regions clearer but takes longer to render. Throughout the simulations Δt=0.1, chosen after
examining Figure12.
Throughout, all decimal variables were stored as double floats (8 bytes). This gives around 16 places of precision. The limitations of the program are visible when simulations are made in steps of
~10^-17 (Figure11). When investigating chaos, rounding errors can have a huge effect on the output (ref2). However, the graphs in this simulation were plotted in steps of ~10^-5 hence rounding errors
should not have effected the results significantly.
Figure 11.
Extremely high resolution plots become “pixilated” as the limitations of variable accuracy become evident.
Figure 12.
Three test plots of the same area for different values of Δt. The upper and centre plots show noticeable differences, whereas the lower and centre plots have negligible difference, justifying the use
of Δt=0.1 as a compromise.
The model itself is only an approximation, assuming the angle of M remains small. This has shown the basic concepts of a double pendulum but a more relevant result would be achieved if m ≈ M. A plot
of initial angles for both pendulums on either axis would produce a result similar to Figure13.
Figure 13.
Double Pendulum plot by Jeremy S. Heyl
The colour of each pixel indicates whether either pendulum of a double pendulum flips within 10
1. David Clements, First Year Computing Laboratory, Version 2.9
2. Raymond Sneyers (1997) "Climate Chaotic Instability: Statistical Determination and Theoretical Background", Environmetrics, vol. 8, no. 5, pages 517–532.
3. Meirovitch, Leonard (1986). Elements of Vibration Analysis (2nd edition ed.). McGraw-Hill Science/Engineering/Math.
Appendix A: C++ Source code for the main simulation & modification 1 #include <iostream> // Include headers for i/o, #include <fstream> // file writing, #include <cmath> // and mathematical functions
using namespace std; double doublePendulum(double, double); // Prototype for our simulation const double pi = 2*acos(0.0); // Define pi double tmax,dt,toutside; /* Declare variables tmax, dt, and
toutside which need to be global */ int main() { /* First we need to create the image we will be plotting to. We achieve this by reading the header data from a blank bitmap file and storing in the
variable str[]. The width and height of the image are also read, in order to set the resolution of our graph plot. */ char str[54]; //Header data for a bitmap is 54 bytes // Open file in binary read
mode and read the first 54 bytes ifstream blankfile("blank.bmp",ios::in | ios::binary); blankfile.read(str,54); unsigned int width,height; blankfile.seekg(18); // Read WxH from header data blankfile.
read((char*)&width, 4); // which are unsigned integers blankfile.read((char*)&height, 4); // at positions 18 and 22 double a0min,a0max,taumin,taumax,theta; //Input data from user cout << "Double
Pendulum plot. Enter initial conditions.\na0 min:\n"; cin>>a0min; cout << "a0 max:\n"; cin>>a0max; cout << "tau min:\n"; cin>>taumin; cout << "tau max:\n"; cin>>taumax; cout << "t max:\n"; cin>>tmax;
cout << "dt:\n"; cin>>dt; /* We now have enough information to create the new file. For convienience, the values used will be stored in the filename. We construct this filename using the sprintf()
function, which can combine numerical variables as a string. */ char filename[255]; sprintf(filename, "output5/p%f_%f_%f_%f_%f_%f.bmp", a0min,a0max,taumin,taumax,tmax,dt); // Open the file for
writing in binary mode ofstream myfile(filename,ios::out | ios::binary); // Write the 54 byte header we read earlier myfile.write(str,54); /* Next is the main cycle which plots every pixel on the
graph. The nested loops travel through each x and y value (i and j) and the subroutine doublePendulum() is called. Values of a and tau are calculated based on the limits provided earlier by the user.
*/ char rgb[3]; // 3 bytes per pixel for (int j=0;j<height;j++) // Vertical loop (rows) { for (int i=0;i<width;i++) // Horizontal loop (columns) { //Call the function with the values from the loop
theta = doublePendulum( a0min + j/(height/(a0max-a0min)), taumin + i/(width/(taumax-taumin))); /* The pixel brightness is determined by theta. We need to scale theta from the range -pi, pi into 0,
255 */ rgb[0]= 127*theta/pi; /* Colour Adjustment. Saturation is determined by the difference between the pixel values - three identical values gives a gray pixel. To highight areas where the
pendulum has left the range of -pi, pi we take the fraction of time outside, and multiply it by the difference between the current pixel value ( rgb[0] ) and 255. This value is added to the red
pixels ( rgb[2] ) and subtracted from the green and blue. */ rgb[2]=(rgb[0]+(127-rgb[0])*toutside/tmax) + 127; rgb[1]=(rgb[0]-(127+rgb[0])*toutside/tmax) + 127; rgb[0]=(rgb[0]-(127+rgb[0])*toutside/
tmax) + 127; // Write this pixel to the file myfile.write(rgb,3); } // Loading bar - clear screen and calculate percentage system("CLS"); cout << "Rendering . . . " << 100*j/height << "%"; } /*
Launch our image for viewing. The file must be closed, and then a system call can be made to open the image. For convienience, the filename variable is reused to construct the command. */ myfile.
close(); sprintf(filename, "cd output5 & p%f_%f_%f_%f_%f_%f.bmp", a0min,a0max,taumin,taumax,tmax,dt); system(filename); } // Actual pendulum simulation routine double doublePendulum(double a, double
tau) { // Declare variables double omega=0.0,theta=0.1,t=0.0; toutside=0; // Begin simulation loop while (t < tmax) { /* In each step of the simulation, t increases by dt, and so theta and omega are
increased by their calculated values multiplied by dt. */ t+=dt; theta+=omega *dt; omega-=dt*(sin(theta) + a*sin(2*pi*t/tau)*cos(theta)); /* We record the time the pendulum spends outside the normal
region in the variable toutside. This should have a direct correlation to the chaotic regions. */ toutside+=(theta>pi | theta<-pi ? dt:0); } return theta; }
Appendix B: Source code for phase-space plots #include <fstream> // Include headers for file writing #include <cmath> // and mathematical funtions const double pi = 2*acos(0.0); // Define pi int main
() { // Declare variables and initial values double omega=0.0,theta=0.1,t=0.0,dt=0.1,a=0.251,tau=8,tmax=100; std::ofstream myfile("phase-space/out.txt"); //Open file for writing while (t < tmax) { //
Begin simulation t+=dt; // Iterate values - same as simulation 1, theta+=omega *dt; // see main text for formulae omega-=dt*(sin(theta) + a*sin(2*pi*t/tau)*cos(theta)); myfile << theta << "\t" <<
omega << "\n"; // Write output to file } }
The output in this file was then fed into and plotted using MS Excel. | {"url":"https://mitxela.com/projects/chaos_theory","timestamp":"2024-11-13T14:20:30Z","content_type":"text/html","content_length":"40742","record_id":"<urn:uuid:fefe03a4-23f8-4922-b061-4bd785fe0c02>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00143.warc.gz"} |
PhD thesis
PhD thesis: Modular Forms of Weight One Over Finite Fields
The thesis consists of a couple of different projects, all centred around modular forms over finite fields with a special interest in those of weight one.
My main motivation for my work is the desire to understand (a bit) better the absolute Galois group of the rationals. Its 1-dimenstional representations are fully described by class field theory, so
that those of dimension 2 are the next natural step. According to a theorem by Deligne Hecke eigenforms mod p give rise to 2-dimensional odd Galois representations over an algebraic closure of F_p.
Serre's conjecture proposes a converse: every odd irreducible Galois representation over an algebraic closure of F_p should come from a modular form.
Concerning Serre's conjecture, I managed to prove it in its strictest form (with the minimal weight defined by Edixhoven) for dihedral Galois representations, including, in particular, the
exceptional cases when level and weight lowering are not known. This is the content of Chapter 1 and my Documenta article (to publications). Moreover, I performed computer calculations verifying the
conjecture in some cases. Apart from that, explicit computations of mod 2 modular forms have yielded the realisation of the group SL_2(F_2^r) for r < 78 as Galois group over the rationals (more
information). These and some more computations are reported upon in Chapter 5.
Another connected theme is the study of some Hecke modules with a special regard as to their faithfulness. Natural examples are the cohomology of subgroups of SL_2(Z), the cohomology of modular
curves considered as Riemann surfaces, and modular symbols. I gave a concise algebraic treatment of these modules over arbitrary rings, compared them, and gave criteria for their equality in Chapter
2 and the preprint On modular symbols and the cohomology of Hecke triangle surfaces. My principal theoretical contribution concerns the faithfulness of these modules for the weight p if the base ring
is F_p after localisation at a prime of the Hecke algebra corresponding to an ordinary modular form. Weight p is the smallest weight that cannot in general be treated any more by p-adic Hodge theory.
This is the contents of Chapter III (and parts of Chapter IV), as well as of my article On the faithfulness of parabolic cohomology as a Hecke module over a finite field.
These faithful Hecke modules allow to compute mod p modular forms using the modular symbols formalism over F_p, hence fast methods of linear algebra over a finite field. This formalism has been
implemented in MAGMA by William Stein. My theoretical progress enables one to compute mod p modular forms of weight one with mod p modular symbols, using ideas of Edixhoven, as described for instance
in my appendix. Earlier, it had been necessary to use characteristic zero. I implemented these algorithms in MAGMA (more information) and described them in Chapter 4.
Last modification: 27 March 2006. | {"url":"https://math.uni.lu/~wiese/thesis/index.html","timestamp":"2024-11-01T23:10:36Z","content_type":"text/html","content_length":"4908","record_id":"<urn:uuid:7033e7e5-b531-4232-a3b4-1981ae93e5cc>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00136.warc.gz"} |
Friction and Energy Transfer in context of coefficient of friction to acceleration
30 Aug 2024
Journal of Mechanical Engineering
Volume 12, Issue 3, 2023
Friction and Energy Transfer: A Theoretical Analysis of Coefficient of Friction to Acceleration Relationship
This article presents a theoretical analysis of the relationship between coefficient of friction (μ) and acceleration (a) in the context of energy transfer. We derive an expression for the
acceleration of an object in terms of μ, normal force (N), and mass (m). The results are discussed in the context of energy transfer and the role of friction in determining the acceleration of an
Friction is a fundamental concept in mechanics that plays a crucial role in determining the motion of objects. The coefficient of friction (μ) is a measure of the ratio of the force of friction to
the normal force between two surfaces in contact. In this article, we explore the relationship between μ and acceleration (a) in the context of energy transfer.
Consider an object of mass m moving with acceleration a on a surface with coefficient of friction μ. The force of friction (F_f) is given by:
F_f = μ * N
where N is the normal force acting on the object.
The net force (F_net) acting on the object is the sum of the force of gravity (F_g) and the force of friction:
F_net = F_g + F_f
Since F_g = m * g, where g is the acceleration due to gravity, we can write:
F_net = m * g + μ * N
The acceleration (a) of the object is given by Newton’s second law:
m * a = F_net
Substituting the expression for F_net, we get:
m * a = m * g + μ * N
Simplifying, we get:
a = g + μ * (N / m)
This expression shows that the acceleration of an object is directly proportional to the coefficient of friction (μ), normal force (N), and inversely proportional to mass (m).
The results presented in this article demonstrate the relationship between coefficient of friction (μ) and acceleration (a) in the context of energy transfer. The expression derived shows that μ
plays a crucial role in determining the acceleration of an object, with higher values of μ resulting in greater accelerations.
This analysis has implications for various fields, including mechanical engineering, materials science, and physics. It highlights the importance of considering frictional forces when designing
systems or predicting motion.
In conclusion, this article presents a theoretical analysis of the relationship between coefficient of friction (μ) and acceleration (a) in the context of energy transfer. The results demonstrate
that μ plays a crucial role in determining the acceleration of an object, with implications for various fields. Further research is needed to explore the practical applications of these findings.
[1] Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica.
[2] Coulomb, C. A. (1785). Theorie des Machines Simples.
Note: The references provided are for illustrative purposes only and do not necessarily relate to the specific content of this article.
Related articles for ‘coefficient of friction to acceleration’ :
Calculators for ‘coefficient of friction to acceleration’ | {"url":"https://blog.truegeometry.com/tutorials/education/5eae84b9757df1a1527875fff3126fc4/JSON_TO_ARTCL_Friction_and_Energy_Transfer_in_context_of_coefficient_of_friction.html","timestamp":"2024-11-06T09:30:13Z","content_type":"text/html","content_length":"18156","record_id":"<urn:uuid:92e945a3-43a9-47cd-b9c9-b9af933a57b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00507.warc.gz"} |
finite CW complex
nLab finite CW complex
Homotopy theory
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Paths and cylinders
Homotopy groups
Basic facts
A finite CW-complex is a CW-complex with a finite number of attaching maps.
The homotopy type of a finite CW-complex is called a finite homotopy type.
Last revised on December 21, 2023 at 17:18:45. See the history of this page for a list of all contributions to it. | {"url":"https://ncatlab.org/nlab/show/finite+CW+complex","timestamp":"2024-11-07T19:15:11Z","content_type":"application/xhtml+xml","content_length":"41938","record_id":"<urn:uuid:adb28a58-301d-408d-8630-bb27aff6301b>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00271.warc.gz"} |
What Is Average True Range In Stocks
What Is Average True Range In Stocks
Average True Range Technical Indicator (ATR) is an indicator that shows volatility of the market. It was introduced by Welles Wilder in his book New concepts. Average True Range (ATR) is a technical
analysis indicator developed by J. Welles Wilder, based on trading ranges smoothed by an N-day exponential moving. The Average True Range (ATR) is a common technical analysis indicator designed to
measure volatility. This indicator was originally developed by the famed. The Average True Range (ATR) is a tool used in technical analysis to measure volatility. Unlike many of today's popular
indicators, the ATR is not used to. The Average True Range (ATR) indicator was developed by J. Welles Wilder and is used to measure volatility. It uses High, Low and Close prices to incorporate.
ATR is a versatile and significant tool in technical analysis, offering vital insights into market volatility. It smooths daily price fluctuations. The Average True Range (ATR) is a technical
indicator that measures the volatility of an asset's price. ATR is a technical analysis indicator that measures price volatility of a financial security over a period of time, typically 14 days. When
traders talk about range, they are referring to a number that is the difference between the high price and low price of a time period. As with most of his indicators, Wilder designed ATR with
commodities and daily prices in mind. Commodities are frequently more volatile than stocks. They were. Average true range (ATR) is a technical indicator that appears as a single line in a box
underneath a market's chart. When the line rises, it means that the. The indicator known as average true range (ATR) can be used to develop a complete trading system or be used for entry or exit
signals as part of a strategy. Developed by J. Welles Wilder, the Average True Range (ATR) is an indicator that measures volatility. As with most of his indicators, Wilder designed ATR. The average
true range measures the price range of a security/stock – the higher the volatility of a security the higher the ATR. Average True Range (ATR) provides info about a stock's typical daily movement
(volatility) over a recent period of time (often the last 14 trading days). Calculated as the moving average of the True Range - the greatest of the current high minus the low, the absolute value of
the current high minus the previous.
The average true range (ATR) decomposes the whole price range for a specific period to measure market volatility. ATR is highly helpful in indicating. Average True Range (ATR) is the average of true
ranges over the specified period. ATR measures volatility, taking into account any gaps in the price movement. ATR measures a security's volatility. It does not indicate price direction or duration,
rather the degree of price movement. The Average True Range (ATR) study calculates the average true price range over a time period. True range is the greatest of the following. Average True Range is
one of the most commonly used indicators for determining how much an asset moves. It was created by J. Welles Wilder. It's typically. The average true range (ATR) is a market volatility indicator
used in technical analysis. The Average True Range (ATR) is a technical indicator used primarily to measure volatility in financial markets. Average true range (ATR) is a technical analysis
volatility indicator originally developed by J. Welles Wilder, Jr. for commodities. It analyses a range of asset prices within a given timeframe, taking into account any gaps in price action. The ATR
indicator can be used for both short-term.
Average True Range refers to the average price of true range over the last several candles. For any group of stocks and market segments, you can scan and. ATR, or Average True Range, is a technical
indicator that can tell you how volatile a stock has been, on average, over a specified period. On this episode of. The Average True Range (ATR) is a technical analysis indicator that measures market
volatility. Picture the Average True Range (ATR) like a thermometer for the. It is the average of the price ranges over a specific time period derived from the simple moving average of 14 trading
periods. Since the Average True Range is. The Average True Range Indicator, or ATR, is a volatility indicator. When there is basically no trading between two sessions, an ATR trader can comprehend
The average true range is a volatility indicator, it is a powerful indicator which can be used to calculate and normalise volatility between instruments. You. A metric used in financial analysis
known to be the ATR gauges the rapidity of price movement for a commodity or securities. J. Welles Wilder published in New.
Review Of Dating Sites Uk | How To Open A Vanguard Mutual Fund Account | {"url":"https://videorulet.ru/market/what-is-average-true-range-in-stocks.php","timestamp":"2024-11-04T12:28:30Z","content_type":"text/html","content_length":"11324","record_id":"<urn:uuid:8163fac4-d65c-44a6-ad99-6c1a82138822>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00210.warc.gz"} |
Wall Paint
`WP = ( L * H )/( 400 " ft"^2) * 1 `
Enter a value for all fields
The Paint Needed for a Wall or any rectangular area calculator computes the number of gallons of paint and primer needed to cover a rectangular area.
INSTRUCTIONS: Choose units and enter the following:
• (L) Length
• (H) Height
• (PC) Coverage per Gallon (area covered by one gallon: default 400ft^2)
• (CT) Coats of Paint
• (WD) Number of Windows
• (DR) Number of Doors
Wall Paint (WP): The calculator returns the gallons of paint or primer needed to cover the wall area. The calculator also returns the surface area of the wall in square feet. However, these answers
can be automatically converted to compatible units via the pull-down menu.
To estimate the cost of paint or primer for a wall,CLICK HERE.
The Math / Science
The surface area of a wall (blue in diagram) is based on the length and height. The mean square feet for a gallon of quality paint is 400 ft^2 . Primer covers 200 ft^2. The gallons of paint and
primer are the total surface area divide by the square feet per gallon.
To compute the surface area of a wall,CLICK HERE. The surface area is then reduced by 15 ft^2 for each window and 21 ft^2 for each door.
Paint Pricing Survey Data
One should use the pricing for paint or primer that can be bought locally since prices may vary. However, the following prices are recent U.S. dollar costs for one gallon of paint or primer in the
United States:
Wall and Room Calculators
Enhance your vCalc experience with a free account
Sign Up Now!
Sorry, JavaScript must be enabled.
Change your browser options, then try again. | {"url":"https://www.vcalc.com/wiki/paint-for-a-wall","timestamp":"2024-11-06T13:51:36Z","content_type":"text/html","content_length":"59762","record_id":"<urn:uuid:ceae5370-04cf-4e4e-9eaa-a9185f5002b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00196.warc.gz"} |
7.2: Treap - A Randomized Binary Search Tree
Last updated
Page ID
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\
evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\
newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}
\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}
{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}
[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}
{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\
#3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp
#2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\
wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\
newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}
{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\
bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\
widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)
The problem with random binary search trees is, of course, that they are not dynamic. They don't support the \(\mathtt{add(x)}\) or \(\mathtt{remove(x)}\) operations needed to implement the SSet
interface. In this section we describe a data structure called a Treap that uses Lemma 7.1.1 to implement the SSet interface.^2
A node in a Treap is like a node in a BinarySearchTree in that it has a data value, \(\mathtt{x}\), but it also contains a unique numerical priority, \(\mathtt{p}\), that is assigned at random:
class Node<T> extends BinarySearchTree.BSTNode<Node<T>,T> {
int p;
In addition to being a binary search tree, the nodes in a Treap also obey the heap property:
In other words, each node has a priority smaller than that of its two children. An example is shown in Figure \(\PageIndex{1}\).
Figure \(\PageIndex{1}\): An example of a Treap containing the integers \(0,\ldots,9\). Each node, \(\mathtt{u}\), is illustrated as a box containing \(\texttt{u.x},\texttt{u.p}\).
The heap and binary search tree conditions together ensure that, once the key (\(\mathtt{x}\)) and priority (\(\mathtt{p}\)) for each node are defined, the shape of the Treap is completely
determined. The heap property tells us that the node with minimum priority has to be the root, \(\mathtt{r}\), of the Treap. The binary search tree property tells us that all nodes with keys smaller
than \(\texttt{r.x}\) are stored in the subtree rooted at \(\texttt{r.left}\) and all nodes with keys larger than \(\texttt{r.x}\) are stored in the subtree rooted at \(\texttt{r.right}\).
The important point about the priority values in a Treap is that they are unique and assigned at random. Because of this, there are two equivalent ways we can think about a Treap. As defined above, a
Treap obeys the heap and binary search tree properties. Alternatively, we can think of a Treap as a BinarySearchTree whose nodes were added in increasing order of priority. For example, the Treap in
Figure \(\PageIndex{1}\) can be obtained by adding the sequence of \((\mathtt{x},\mathtt{p})\) values
\[ \langle (3,1), (1,6), (0,9), (5,11), (4,14), (9,17), (7,22), (6,42), (8,49), (2,99) \rangle \nonumber\]
into a BinarySearchTree.
Since the priorities are chosen randomly, this is equivalent to taking a random permutation of the keys--in this case the permutation is
\[ \langle 3, 1, 0, 5, 9, 4, 7, 6, 8, 2 \rangle \nonumber\]
--and adding these to a BinarySearchTree. But this means that the shape of a treap is identical to that of a random binary search tree. In particular, if we replace each key \(\mathtt{x}\) by its
rank,^3 then Lemma 7.1.1 applies. Restating Lemma 7.1.1 in terms of Treaps, we have:
In a Treap that stores a set \(S\) of \(\mathtt{n}\) keys, the following statements hold:
1. For any \(\mathtt{x}\in S\), the expected length of the search path for \(\mathtt{x}\) is \(H_{r(\mathtt{x})+1} + H_{\mathtt{n}-r(\mathtt{x})} - O(1)\).
2. For any \(\mathtt{x}\not\in S\), the expected length of the search path for \(\mathtt{x}\) is \(H_{r(\mathtt{x})} + H_{\mathtt{n}-r(\mathtt{x})}\).
Here, \(r(\mathtt{x})\) denotes the rank of \(\mathtt{x}\) in the set \(S\cup\{\mathtt{x}\}\).
Again, we emphasize that the expectation in Lemma \(\PageIndex{1}\) is taken over the random choices of the priorities for each node. It does not require any assumptions about the randomness in the
Lemma \(\PageIndex{1}\) tells us that Treaps can implement the \(\mathtt{find(x)}\) operation efficiently. However, the real benefit of a Treap is that it can support the \(\mathtt{add(x)}\) and \(\
mathtt{delete(x)}\) operations. To do this, it needs to perform rotations in order to maintain the heap property. Refer to Figure \(\PageIndex{2}\). A rotation in a binary search tree is a local
modification that takes a parent \(\mathtt{u}\) of a node \(\mathtt{w}\) and makes \(\mathtt{w}\) the parent of \(\mathtt{u}\), while preserving the binary search tree property. Rotations come in two
flavours: left or right depending on whether \(\mathtt{w}\) is a right or left child of \(\mathtt{u}\), respectively.
Figure \(\PageIndex{2}\): Left and right rotations in a binary search tree.
The code that implements this has to handle these two possibilities and be careful of a boundary case (when \(\mathtt{u}\) is the root), so the actual code is a little longer than Figure \(\PageIndex
{2}\) would lead a reader to believe:
void rotateLeft(Node u) {
Node w = u.right;
w.parent = u.parent;
if (w.parent != nil) {
if (w.parent.left == u) {
w.parent.left = w;
} else {
w.parent.right = w;
u.right = w.left;
if (u.right != nil) {
u.right.parent = u;
u.parent = w;
w.left = u;
if (u == r) { r = w; r.parent = nil; }
void rotateRight(Node u) {
Node w = u.left;
w.parent = u.parent;
if (w.parent != nil) {
if (w.parent.left == u) {
w.parent.left = w;
} else {
w.parent.right = w;
u.left = w.right;
if (u.left != nil) {
u.left.parent = u;
u.parent = w;
w.right = u;
if (u == r) { r = w; r.parent = nil; }
In terms of the Treap data structure, the most important property of a rotation is that the depth of \(\mathtt{w}\) decreases by one while the depth of \(\mathtt{u}\) increases by one.
Using rotations, we can implement the \(\mathtt{add(x)}\) operation as follows: We create a new node, \(\mathtt{u}\), assign \(\mathtt{u\texttt{.}x=x}\), and pick a random value for \(\texttt{u.p}\).
Next we add \(\mathtt{u}\) using the usual \(\mathtt{add(x)}\) algorithm for a BinarySearchTree, so that \(\mathtt{u}\) is now a leaf of the Treap. At this point, our Treap satisfies the binary
search tree property, but not necessarily the heap property. In particular, it may be the case that \(\mathtt{u.parent.p > u.p}\). If this is the case, then we perform a rotation at node \(\mathtt{w}
\)= \(\texttt{u.parent}\) so that \(\mathtt{u}\) becomes the parent of \(\mathtt{w}\). If \(\mathtt{u}\) continues to violate the heap property, we will have to repeat this, decreasing \(\mathtt{u}\)
's depth by one every time, until \(\mathtt{u}\) either becomes the root or \(\texttt{u.parent.p} < \texttt{u.p}\).
boolean add(T x) {
Node<T> u = newNode();
u.x = x;
u.p = rand.nextInt();
if (super.add(u)) {
return true;
return false;
void bubbleUp(Node<T> u) {
while (u.parent != nil && u.parent.p > u.p) {
if (u.parent.right == u) {
} else {
if (u.parent == nil) {
r = u;
An example of an \(\mathtt{add(x)}\) operation is shown in Figure \(\PageIndex{3}\).
Figure \(\PageIndex{3}\): Adding the value 1.5 into the Treap from Figure \(\PageIndex{1}\).
The running time of the \(\mathtt{add(x)}\) operation is given by the time it takes to follow the search path for \(\mathtt{x}\) plus the number of rotations performed to move the newly-added node, \
(\mathtt{u}\), up to its correct location in the Treap. By Lemma \(\PageIndex{1}\), the expected length of the search path is at most \(2\ln \mathtt{n}+O(1)\). Furthermore, each rotation decreases
the depth of \(\mathtt{u}\). This stops if \(\mathtt{u}\) becomes the root, so the expected number of rotations cannot exceed the expected length of the search path. Therefore, the expected running
time of the \(\mathtt{add(x)}\) operation in a Treap is \(O(\log \mathtt{n})\). (Exercise 7.3.5 asks you to show that the expected number of rotations performed during an addition is actually only \
The \(\mathtt{remove(x)}\) operation in a Treap is the opposite of the \(\mathtt{add(x)}\) operation. We search for the node, \(\mathtt{u}\), containing \(\mathtt{x}\), then perform rotations to move
\(\mathtt{u}\) downwards until it becomes a leaf, and then we splice \(\mathtt{u}\) from the Treap. Notice that, to move \(\mathtt{u}\) downwards, we can perform either a left or right rotation at \
(\mathtt{u}\), which will replace \(\mathtt{u}\) with \(\texttt{u.right}\) or \(\texttt{u.left}\), respectively. The choice is made by the first of the following that apply:
1. If \(\texttt{u.left}\) and \(\texttt{u.right}\) are both \(\mathtt{null}\), then \(\mathtt{u}\) is a leaf and no rotation is performed.
2. If \(\texttt{u.left}\) (or \(\texttt{u.right}\)) is \(\mathtt{null}\), then perform a right (or left, respectively) rotation at \(\mathtt{u}\).
3. If \(\texttt{u.left.p} < \texttt{u.right.p}\) (or \(\texttt{u.left.p} > \texttt{u.right.p})\), then perform a right rotation (or left rotation, respectively) at \(\mathtt{u}\).
These three rules ensure that the Treap doesn't become disconnected and that the heap property is restored once \(\mathtt{u}\) is removed.
boolean remove(T x) {
Node<T> u = findLast(x);
if (u != nil && compare(u.x, x) == 0) {
return true;
return false;
void trickleDown(Node<T> u) {
while (u.left != nil || u.right != nil) {
if (u.left == nil) {
} else if (u.right == nil) {
} else if (u.left.p < u.right.p) {
} else {
if (r == u) {
r = u.parent;
An example of the \(\mathtt{remove(x)}\) operation is shown in Figure \(\PageIndex{4}\).
Figure \(\PageIndex{4}\): Removing the value 9 from the Treap in Figure \(\PageIndex{1}\).
The trick to analyze the running time of the \(\mathtt{remove(x)}\) operation is to notice that this operation reverses the \(\mathtt{add(x)}\) operation. In particular, if we were to reinsert \(\
mathtt{x}\), using the same priority \(\texttt{u.p}\), then the \(\mathtt{add(x)}\) operation would do exactly the same number of rotations and would restore the Treap to exactly the same state it
was in before the \(\mathtt{remove(x)}\) operation took place. (Reading from bottom-to-top, Figure \(\PageIndex{4}\) illustrates the addition of the value 9 into a Treap.) This means that the
expected running time of the \(\mathtt{remove(x)}\) on a Treap of size \(\mathtt{n}\) is proportional to the expected running time of the \(\mathtt{add(x)}\) operation on a Treap of size \(\mathtt{n}
-1\). We conclude that the expected running time of \(\mathtt{remove(x)}\) is \(O(\log \mathtt{n})\).
The following theorem summarizes the performance of the Treap data structure:
Theorem \(\PageIndex{1}\).
A Treap implements the SSet interface. A Treap supports the operations \(\mathtt{add(x)}\), \(\mathtt{remove(x)}\), and \(\mathtt{find(x)}\) in \(O(\log \mathtt{n})\) expected time per operation.
It is worth comparing the Treap data structure to the SkiplistSSet data structure. Both implement the SSet operations in \(O(\log \mathtt{n})\) expected time per operation. In both data structures, \
(\mathtt{add(x)}\) and \(\mathtt{remove(x)}\) involve a search and then a constant number of pointer changes (see Exercise 7.3.5). Thus, for both these structures, the expected length of the search
path is the critical value in assessing their performance. In a SkiplistSSet, the expected length of a search path is
\[ 2\log \mathtt{n} + O(1) \enspace , \nonumber\]
In a Treap, the expected length of a search path is
\[ 2\ln \mathtt{n} +O(1) \approx 1.386\log \mathtt{n} + O(1) \enspace . \nonumber\]
Thus, the search paths in a Treap are considerably shorter and this translates into noticeably faster operations on Treaps than Skiplists. Exercise 4.5.7 in Chapter 4 shows how the expected length of
the search path in a Skiplist can be reduced to
\[ e\ln \mathtt{n} + O(1) \approx 1.884\log \mathtt{n} + O(1) \nonumber\]
by using biased coin tosses. Even with this optimization, the expected length of search paths in a SkiplistSSet is noticeably longer than in a Treap.
^2The names Treap comes from the fact that this data structure is simultaneously a binary search tree (Section 6.2) and a heap (Chapter 10).
^3The rank of an element \(\mathtt{x}\) in a set \(S\) of elements is the number of elements in \(S\) that are less than \(\mathtt{x}\). | {"url":"https://eng.libretexts.org/Bookshelves/Computer_Science/Databases_and_Data_Structures/Book%3A_Open_Data_Structures_-_An_Introduction_(Morin)/07%3A_Random_Binary_Search_Trees/7.02%3A_Treap_-_A_Randomized_Binary_Search_Tree","timestamp":"2024-11-05T09:29:28Z","content_type":"text/html","content_length":"143352","record_id":"<urn:uuid:c331d770-16b3-4ed2-a0a5-5108de75877b>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00050.warc.gz"} |
Mathematical reasoning and nature of proof
1. Nature of Truth
In mathematics we deal with statements that are “True” or “False”. This is known as the “Law of Excluded Middle”. Despite the fact that multi valued logics are used in computer science, they have no
place in mathematical reasoning.
2. Nature of Mathematical Proof
A very common question that comes to our mind is “What is the definition of a good mathematical proof?” And the answer seems to be best given by “It convinces you!” Unfortunately this is not very
true. Personal certitude has nothing to do with mathematical proof. The human mind is a very fragile thing, and human beings can be convinced of the most preposterous things. A good proof is one that
starts with a set of axioms, and proceeds using correct $$mathbf {rules ~of~ inference}$$ to the conclusion.
3. Rules of Inference
The common rule of inference that are frequently used are listed below.
1) Given the statement: $$All ~A ~ is ~ B$$ ~ and the statement $$All~B ~is ~ C$$ , we conclude that $$All~ A ~ is ~ C. $$
For example:
If I do not wake up, then I cannot go to work.
If cannot go to work then I will not be paid.
Therefore, if I do not wake up, then I will not get paid.
2) Given $$All ~ A ~ is ~B$$ . We conclude that $$Some~ B~ is ~A $$ .
For example:
All cows are Animals, therefore some animals are cows.
An incorrect inference is to conclude that $$All ~ B ~ is ~ A,$$ given $$All~ A ~ is ~ B$$ . After all, not all animals are cows!
3) Given $$Some ~ A ~ is B ~ and Some~ B~ is ~C$$ , we can conclude nothing.
For example:
Some cows are Jerseys,
Some Jerseys are human.
Here we interpret the word “Jersey” as “Things that come from Jersey, an island in the English Channel.”
4) Given $$Some ~ A ~ is ~ B$$ , we conclude that $$Some~ B~ is ~ A$$ .
For example:
Some cows are Jerseys, therefore some Jerseys are cows.
5) Given $$Some ~ A ~is B~ and ~ All ~ B ~ is ~C$$ we conclude that $$Some ~ A ~ is ~C$$ .
For example:
Some cows give milk, All things that give milk are female.
Therefore , Some cows are female.
6) Given $$All ~ A ~ is ~ B, $$ and $$Some ~ B ~ is ~ C$$ . In this case we can conclude nothing.
For example:
All cows are animals. Some animals are birds.
No conclusion is possible.
Now such logical inferences can be formulated in rigorous mathematical format by the proper use of $$mathbf{quantifiers}.$$
⦁ A statement such as $$All ~ A ~ is B~$$ is said to be $$”mathbf{Universally ~quantified}”$$ . In other words, it is a universal statement that applies to all $$A.$$
⦁ A statement such as $$Some ~ A ~ is ~ B ~$$ is said to be $$”mathbf{Existentially ~quantified} “$$ . In other words , there exists at least one $$A$$ to which the statement applies.
⦁ The only permissible form for the universal negative is $$No~ A ~ is ~ B$$ . The existential negative has several forms like –
Not all A is B
Some A is not B, and many others.
Mathematical statements require somewhat greater precision than general statements.
4. Negation of a statement
A proposition is a statement that can be assigned the value $$mathbf{True}$$ or $$mathbf{False}$$ . Negation of a statement is the one that produces a value of $$mathbf{true}$$ when the original
statement is $$mathbf{false}$$ and vice versa. In ordinary logic
* An existential negates a universal and a universal negates an existential.
* The negation of $$”All~ A~ is~ B”$$ is $$”Some ~A~ is ~ not~B”$$ .
* The negative of $$”Some ~ A ~ is~ B”$$ is $$”No ~ A~is ~B”$$.
* The statements $$”Some ~A~is~B”$$ and $$”Some~A~is~not~B”$$ can both be true.
5. Logical Connectives
1) If $$P$$ is a proposition, $$neg P$$ is its negation. $$neg P$$ is read as $$”not~ P”$$ .
Note: Do not confuse this mathematical connective with the general statement $$”Not~ all~ A ~is~ B”$$ . They are not the same thing.
2) If $$P$$ and $$Q$$ are propositions,
* $$Pwedge Q$$ is called the conjunction of $$P$$ and $$Q$$ , and is read as $$P~and ~Q$$ .
* $$Pvee Q$$ is called the disjunction of $$P$$ and $$Q$$ , and is read as $$P~ or ~ Q$$ .
* $$Prightarrow Q$$ is called the implication of $$P$$ and $$Q$$ and is read as $$If ~ P ~ then ~ Q$$ .
6. Implications
◦ The most interesting connective is the implication $$Prightarrow Q$$ . It can also be written as $$neg Pvee Q$$ .
◦ If $$P$$ is false then the entire statement is true. That is $$”mathbf{ A ~False~ statement~ Implies ~Anything }”$$ .
◦ An implication is proven by assuming that $$P$$ is true and in that case, $$Q$$ must also be true.
◦ Given a statement $$S$$ of the form $$P rightarrow Q$$ , the statement $$Qrightarrow P$$ is called the $$mathbf{Converse}$$ of $$S$$ .
◦ The Converse of $$S$$ is an independent statement and must be proven independently of $$S$$ .
◦ A statement and its contrapositive are logically equivalent. Either both are true or both are false.
◦ Given a statement $$S$$ of the form $$P rightarrow Q$$ , the statement $$neg Qrightarrow neg P$$ is called the $$mathbf{Contrapositive}$$ of $$S$$ .
◦ The statement $$neg P rightarrow neg Q$$ is called the $$mathbf {Inverse}$$ if $$S$$ . The Inverse of $$S$$ is logically equivalent to the Converse of $$S$$ .
◦ The statement of the form $$P~ iff~ Q$$ is the shorthand for $$(If ~ P ~then ~Q)$$ and $$(If ~ Q~ then ~ P)$$ . In symbols we express this as $$Pleftrightarrow Q$$ .To prove $$Pleftrightarrow Q$$ ,
we must prove both $$P rightarrow Q$$ and $$Q rightarrow P$$ .
7. Negating Compound Statements
$$neg(Pwedge Q) = neg P vee neg Q $$
⦁ X is less than three and X is odd
⦁ X is greater than or equal to 3 or X is even
$$neg(Pvee Q) = neg P wedge neg Q $$
⦁ The car was either red or green
⦁ The car was not red AND it was not green
$$neg(Prightarrow Q) = P wedge neg Q $$
⦁ If a person has a Ph.D. then they must be rich
⦁ Prof. Maurer has a PhD and Prof. Maurer is poor.
⦁ Note change in quantifiers.
8. Rules of inferences
✓ If $$P$$ is known to be true , $$neg P$$ is false, and vice versa.
✓ If $$Pwedge Q$$ is true, then $$Qwedge P$$ is true.
✓ If $$Pwedge Q$$ is true, then both $$P$$ and $$Q$$ are true.
✓ If $$Pwedge Q$$ is false and $$P$$ is known to be true, then $$Q$$ is false.
✓ If $$Pvee Q$$ is true, then $$Qvee P$$ is true.
✓ If $$Pvee Q$$ is false, then both $$P$$ and $$Q$$ are false.
✓ If $$Pvee Q$$ is known to be true, and $$P$$ is known to be false then $$Q$$ is true.
✓ If $$Prightarrow Q$$ is known to be true, and $$P$$ is true then $$Q$$ is true.
✓ If $$Prightarrow Q$$ is known to be true, and $$Q$$ is false then $$P$$ is false.
✓ If $$Pleftrightarrow Q$$ is known to be true and $$P$$ is true then $$Q$$ is true, and vice versa.
✓ If $$Pleftrightarrow Q$$ is known to be true and $$P$$ is false then $$Q$$ is false, and vice versa.
✓ If $$Pleftrightarrow Q$$ is known to be false and $$P$$ is true then $$Q$$ is true, and vice versa.
✓ If $$Pleftrightarrow Q$$ is known to be false and $$P$$ is false then $Q$ is true, and vice versa.
9. Logical Fallacies
Most students have a hard understanding this. It is not the calculations that are incorrect, it is the $$mathbf{Inference}$$ that is wrong. If an inference technique can be used to prove a silly
nonsense then it cannot be used to prove anything true. A mathematical proof is actually supposed to demonstrate what is true and apply the rules of inference correctly. So, the next time you write a
proof, use proper tools i.e., $$mathbf{Rules ~of~ Inference}$$ and do avoid $$mathbf{HASTY~ GENERALIZATION}$$ !
Tarun Kumari, Research Scholar,
Dept of Mathematical Sciences,
Tezpur University. | {"url":"https://gonitsora.com/mathematical-reasoning-and-nature-of-proof/","timestamp":"2024-11-12T18:22:52Z","content_type":"text/html","content_length":"31539","record_id":"<urn:uuid:421bf6b5-f703-4b0b-a9a5-98c15a13e2dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00517.warc.gz"} |
Broadside radar cross section of the perfectly conducting cube
The broadside radar cross section (RCS) of the perfectly conducting cube is predicted from arbitrarily low to arbitrarily high frequencies, and compared to measured data taken for cube side lengths
ranging from 0.15 to 4 wavelengths. The predicted and measured RCS curves agree to within the estimated experimental limits of accuracy of + or 1 dB. At low frequencies the magnetic-field integral
equation was 'augmented' to eliminate its spurious homogeneous solutions and thus to produce high accuracy beyond the resonance region up through the intermediate frequency range. At high frequencies
the conventional diffraction solution was 'enhanced' to produce high accuracy down through the intermediate frequency range into the resonance region. Close agreement between these two very different
theoretical solutions in the intermediate frequency range confirmed the validity of each solution and permitted calculation of reliable curves for the amplitude and phase of the backscattered far
field versus frequency.
IEEE Transactions on Antennas and Propagation
Pub Date:
March 1985
□ Electric Conductors;
□ Electromagnetic Scattering;
□ Radar Cross Sections;
□ Wave Diffraction;
□ Geometrical Theory Of Diffraction;
□ Integral Equations;
□ Plane Waves;
□ Scatter Propagation;
□ Surface Geometry;
□ Communications and Radar | {"url":"https://ui.adsabs.harvard.edu/abs/1985ITAP...33..321Y/abstract","timestamp":"2024-11-13T15:09:40Z","content_type":"text/html","content_length":"38187","record_id":"<urn:uuid:56eea004-6bd4-43d3-9ef7-dc9895291080>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00447.warc.gz"} |
ACM Other ConferencesAlmost All Even Yao-Yao Graphs Are Spanners
It is an open problem whether Yao-Yao graphs YY_{k} (also known as sparse-Yao graphs) are all spanners when the integer parameter k is large enough. In this paper we show that, for any integer k >=
42, the Yao-Yao graph YY_{2k} is a t_k-spanner, with stretch factor t_k = 6.03+O(k^{-1}) when k tends to infinity. Our result generalizes the best known result which asserts that all YY_{6k} are
spanners for k >= 6 [Bauer and Damian, SODA'13]. Our proof is also somewhat simpler. | {"url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.62/metadata/acm-xml","timestamp":"2024-11-13T14:57:17Z","content_type":"application/xml","content_length":"9972","record_id":"<urn:uuid:46e4ef1d-f31c-42b0-9ff1-c58bda4fbc94>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00493.warc.gz"} |
Bar Graphs and Histograms and the Difference Between Them - New Daily Informer
Bar Graphs and Histograms and the Difference Between Them
You must have used bar graphs and histograms to represent various sets of data in your classes of statistics. Can you recall it? To present data and analytics in a more attractive and easily
understandable way, we take the use of various diagrammatic representations. Bar graphs are used widely in the field of business to represent data. Even the uneducated section of society can use
these forms of representation to draw a conclusion about a given data. However, this is not the case with textual presentation or tabular presentation. Any hidden trend present in a given data can be
easily noticed in the diagrammatic mode of representation. In this article, we will go through bar graphs and histograms and try to understand the difference between the two.
Read More About: besteducationweb
What is a Bar Graph?
Bar graph, also known as a bar diagram, is a specific way of representing data using rectangular bars where the length of each bar is proportional to the value they represent. Generally, there are
two types of bar graphs namely, horizontal bar graph and vertical bar graph. Let us discuss both of these types.
Click Here: TechnologyIdea
1. Horizontal Bar Graph
Bar graphs that use horizontal rectangular bars that represent the measure of data are known as horizontal bar graphs. These bar graphs are used for qualitative data or data varying over space.
If You Need More Information’s check This Link: kannadamasti
2. Vertical Bar Graph
Bar graphs that use vertical rectangular bars that represent the measure of data are known as vertical bar graphs. These bar graphs are used for quantitative data or time-series data.
We use multiple or grouped bar graphs to compare related series. Subdivided or component bar graphs are applied for representing data divided into a number of components. For comparing the different
components of a variable and also relating the components to the whole, we use divided bar graphs or percentage bar graphs.
How to Draw a Bar Graph?
To draw a bar graph, draw a horizontal axis and a vertical axis on the graph paper. Independent category of the data should be written on the horizontal axis while the dependent category of the data
should be written on the vertical axis. While marking each dependent variable on the x-axis, make sure there is equal space between them. Give the scale which shows the way in which numbers are used
in the data. Example- 1 unit = 1 kilometre. Now start making rectangular bars according to the given set of frequencies. It is very important to mention four things while drawing a bar graph- title,
labels on axes, scale, and name of the axes.
Visit Here: constructionscope
What is a Histogram?
Drawing a histogram is one of the ways to graphically represent a given frequency distribution. It is one of the most convenient methods used worldwide. Histogram is also known as area diagram. It
helps us to get an idea of the frequency curve of the variable under observation. Some statistical measures can be obtained using a histogram. A comparison between the frequencies for different class
intervals is possible in this mode of diagrammatic representation.
Click Here: masstamilan
How to Draw a Histogram?
To draw a histogram, class limits of the frequency distribution are first converted to the corresponding class boundaries and a series of adjacent rectangles, one against each class interval, with
the class interval as base or breadth and the frequency or frequency density usually when the class intervals are not uniform as length, is erected.
Read More About: lasenorita
Difference Between a Bar Graph and a Histogram
The major difference between a bar graph and a histogram is that in the bar graph, rectangular bars are not adjacent to each other whereas, in a histogram, the bars are adjacent. Some statistical
measures can be obtained using a histogram; however, this is not the case with a bar graph. If you want to learn more about these concepts in detail and in a fun and interesting way, visit Cuemath.
If You Need More Information’s check This Link: 123musiq | {"url":"https://newdailyinformer.com/bar-graphs-and-histograms-and-the-difference-between-them/","timestamp":"2024-11-09T03:25:22Z","content_type":"text/html","content_length":"44375","record_id":"<urn:uuid:6c43eb50-e1ea-4736-bd09-c8861a307e34>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00683.warc.gz"} |
Re: Highlight Consecutive Values Control Chart
2020-10-30 04:23 AM
Hi All 🙂
I have a process control chart and would like to highlight the data points say for example if 6 points are consecutively on one side of the mean however I am unsure how to write this.
Please see pic to see what I would like to highlight, many thanks! 😄
2020-11-12 04:25 PM
2020-11-12 03:24 PM
2020-11-12 04:25 PM
2020-11-12 05:32 PM | {"url":"https://community.qlik.com:443/t5/New-to-Qlik-Analytics/Highlight-Consecutive-Values-Control-Chart/m-p/1761323","timestamp":"2024-11-12T20:03:02Z","content_type":"text/html","content_length":"313232","record_id":"<urn:uuid:9c037d50-f2bb-4bca-a6b9-2b9c3aa72dad>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00337.warc.gz"} |
Subtracting Via LCD Worksheet - 15 Worksheets.com
Subtracting Via LCD
Worksheet Description
This worksheet is a mathematical exercise sheet focuses on the subtraction of fractions with the same denominator. It contains eight subtraction problems, labeled from ‘a’ to ‘h’, each presenting two
fractions that students are required to subtract. The format of the worksheet is clean and straightforward, designed to allow students to clearly see the fractions and write their answers.
The worksheet’s aim is to educate students on how to subtract fractions that share a common denominator. It reinforces the principle that when fractions have the same denominator, the numerators can
be subtracted while the common denominator remains unchanged. This task is essential for understanding the process of fraction subtraction, an important mathematical skill. By practicing with such
worksheets, students enhance their ability to work with fractions, which is a crucial component of their mathematics education, especially as they progress to more complex operations. | {"url":"https://15worksheets.com/worksheet/least-common-denominator-5/","timestamp":"2024-11-08T01:29:10Z","content_type":"text/html","content_length":"108993","record_id":"<urn:uuid:59844c5c-a101-4c1b-86db-030ec29bc7bf>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00831.warc.gz"} |
Bits, Math and Performance(?)
In this post I defend wrapping, a bit more opinionated than my other posts. As usual I'm writing from the perspective that signed and unsigned integer types are a thin transparent wrapper around bit
vectors, of course I am aware that they are often not used that way. That difference between their use and their actual nature is probably the source of the problems.
Signed wrapping is not wrong
It is often said that when signed wraparound occurs, the result is simply wrong. That is an especially narrow view to take, probably inspired by treating fixed-size bit vector arithmetic as if it is
arithmetic in ℤ, which it is not. Bit vector arithmetic can be viewed as arithmetic in ℤ so long as no "overflow" occurs, but violating that condition does not make the result wrong, it makes the
interpretation wrong.
Signed wrapping is meaningful
The wrapping works exactly the same as unsigned wrapping, it corresponds to taking the lowest k bits of the arbitrary precision result. Such a truncation therefore gives you exactly k meaningful
bits, it's just a slice of the result. Some upper bits may be lost, they can be calculated if you need them. If the whole result is meaningful, then part of it is too, namely at least under the
interpretation of being "part of the result".
An other well known example of benign wrapping is the calculation of the average of two non-negative signed integers. While (a + b) / 2 gives inconvenient results when the addition "overflows",
(uint)(a + b) / 2 (using unsigned division) or (a + b) >>> 1 (unsigned right shift as in Java) are correct even when the addition of two positive values results in a negative value. An other way to
look at it is that there is no unsigned wrapping. Nominally the integers being added here are signed but that doesn't really matter. Casting the inputs to unsigned before adding them is a no-op that
can be performed mentally.
Wrapping can also sometimes be cancelled with more wrapping. For example, taking an absolute value with wrapping and casting the result to an unsigned type of the same width results in the actual
absolute value without the funny int.MinValue edge case:
(uint)abs(int.MinValue) =
(uint)abs(-2147483648) =
(uint)(-2147483648) =
This is not what Math.Abs in C# does, it throws, perhaps inspired by its signed return type. On the other hand, Java's Math.abs gets this right and leaves the reinterpretation up to the consumer of
the result, of course in Java there is no uint32 to cast to but you can still treat that result as if it is unsigned. Such "manual reinterpretation" is in general central to integer arithmetic, it's
really about the bits, not the "default meaning" of those bits.
The principle of cancelling wrapping also has some interesting data structure applications. For example, in a Fenwick tree or Summed Area Table, the required internal integer width is the desired
integer width of any range/area-sum query that you actually want to make. So a SAT over signed bytes can use an internal width of 16 bits as long as you restrict queries to an area of 256 cells or
fewer, since 256 * -128 = -2^15 which still fits a signed 16 bit word.
An other nice case of cancelled wrapping is strength reductions like A * 255 = (A << 8) - A. It is usually not necessary to do that manually, but that's not the point, the point is that the wrapping
is not "destructive". The overall expression wraps only iff A * 255 wraps and even then it has exactly the same result. There are cases in which the left shift experience "signed wrapping" but A *
255 does not (for example, in 32 bits, A = 0x00800000), in those cases the subtraction also wraps and brings the result back to being "unwrapped". That is not a coincidence nor an instance of two
wrongs making a right, it's a result of the intermediate wrapped result being meaningful and wrapping being algebraically nice.
Signed wrapping is not inherent
Signed and unsigned integers are two different ways to interpret bit vectors. Almost all operations have no specific signed or unsigned version, only a generic version that does both. There is no
such thing as signed addition or unsigned addition, addition is just addition. Operations that are actually different are:
• Comparisons except equality
• Division and remainder
• Right shift, maybe, but arithmetic right shift and logical right shift can both be reasonably applied in both signed and unsigned contexts
• Widening conversion
• Widening multiplication
One thing almost all of these have in common is that they cannot overflow, except division of the smallest integer by negative one. By the way I regard that particular quirk of division as a mistake
since it introduces an asymmetry between dividing by negative one and multiplying by negative one.
The result is that the operations that can "overflow" are neither signed nor unsigned, and therefore do not overflow specifically in either of those ways. If they can be said to overflow at all, when
and how they do so depends on how they are being viewed by an outside entity, not on the operation itself.
The distinction between unsigned and signed wrapping is equivalent to imagining a "border" on the ring of integers (not the mathematical Ring of Integers) either between 0 and -1 (unsigned) or
between signed-smallest and signed-highest numbers, but there is no border. Crossing either of the imaginary borders does not mean nearly as much as many people think it means.
Signed wrapping is algebraically nice
A property that wrapping arithmetic shares with arbitrary precision integer arithmetic, but not with trapping arithmetic, is that it obeys a good number of desirable algebraic laws. The root cause of
this is that ℤ/ℤ2^k is a ring, and trapping arithmetic is infested with implicit conditional exceptions. Signed arithmetic can largely be described by ℤ/ℤ2^k, like unsigned arithmetic, since it is
mostly a reinterpretation of unsigned arithmetic. That description does not cover all operations or properties, but it covers the most important aspects.
Here is a small selection of laws that apply to wrapping arithmetic but not to trapping arithmetic:
• -(-A) = A
• A + -A = 0
• A - B = A + -B
• A + (B + C) = (A + B) + C
• A * (B + C) = A * B + A * C
• A * -B = -A * B = -(A * B)
• A * (B * C) = (A * B) * C
• A * 15 = A * 16 - A
• A * multiplicative_inverse(A) = 1 (iff A is odd, this is something not found in ℤ which has only two trivially invertible numbers, so sometimes wrapping gives you a new useful property)
Some laws also apply to trapping arithmetic:
• A + 0 = A
• A - A = 0
• A * 0 = 0
• A * 1 = A
• A * -1 = -A
• -(-(-A)) = -A
The presence of all the implicit exceptional control flow makes the code very hard to reason about, for humans as well as compilers.
Compilers react to that by not optimizing as much as they otherwise would, since they are forced to preserve the exception behaviour. Almost anything written in the source code must actually happen,
and in the same order as originally written, just to preserve exceptions that are not even supposed to ever actually be triggered. The consequences of that are often seen in Swift, where code using
the &+ operator is optimized quite well (including auto-vectorization) and code using the unadorned + operator can be noticeably slower.
Humans .. probably don't truly want trapping arithmetic to begin with, what they want is to have their code checked for unintended wrapping. Wrapping is not a bug by itself, but unintended wrapping
is. So while canceling a "bare" double negation is not algebraically justified in trapping arithmetic, a programmer will do it anyway since the goal is not to do trapping arithmetic, but removing bad
edge cases. Statically checking for unintended wrapping would be a more complete solution, no longer relying on being lucky enough to dynamically encounter every edge case. Arbitrary precision
integers would just remove most edge cases altogether, though it would rely heavily on range propagation for performance, making it a bit fragile.
But anyway, wrapping is not so bad. Just often unintended.
While implementing various kinds of division in haroldbot, I had to look up/work out how to implement different kinds of signed division in terms of unsigned division. The common truncated division
(written as /s in this post and in haroldbot, /t in some other places) is natural result of using your intuition from ℝ and writing the definition based on signs and absolute values, ensuring that
the division only happens between non-negative numbers (making its meaning unambiguous) and that the result is an integer: $$\DeclareMathOperator{\sign}{sign} D /_s d = \sign(d)\cdot\sign(D)\cdot\
left\lfloor\cfrac{\left|D\right|}{\left|d\right|}\right\rfloor$$ That definition leads to a plot like this, showing division by 3 as an example:
Of course the absolute values and sign functions create symmetry around the origin, and that seems like a reasonable symmetry to have. But that little plateau around the origin often makes the mirror
at the origin a kind of barrier that you can run into, leading to the well-documented downsides of truncated division.
The alternative floored division and Euclidean division have a different symmetry, which does not lead to that little plateau, instead the staircase pattern simply continues:
The point of symmetry, marked by the red cross, is at (-0.5, -0.5). Flipping around -0.5 may remind you of bitwise complement, especially if you have read my earlier post visualizing addition,
subtraction and bitwise complement, and mirroring around -0.5 is no more than a conditional complement. So Euclidean division may be implemented with positive division as: $$\DeclareMathOperator{\
sgn}{sgn} \DeclareMathOperator{\xor}{\bigoplus} D /_e d = \sign(d)\cdot(\sgn(D)\xor\left\lfloor\cfrac{D\xor\sgn(D)}{\left|d\right|}\right\rfloor)$$ Where the sgn function is -1 for negative numbers
and 0 otherwise, and the circled plus is XOR. XORing with the sgn is a conditional complement, with the inner XOR being responsible for the horizontal component of the symmetry and the outer XOR
being responsible for the vertical component.
It would have been even nicer if the symmetry of the divisor also worked that way, but unfortunately that doesn't quite work out. For the divisor, the offset introduced by mirroring around -0.5 would
affect the size of the steps of the staircase instead of just their position.
The /e and %e operators are available in haroldbot, though like all forms of division the general case is really too hard, even for the circuit-SAT based truth checker (the BDD engine stands no
chance at all). | {"url":"https://bitmath.blogspot.com/2018/08/","timestamp":"2024-11-03T15:26:53Z","content_type":"application/xhtml+xml","content_length":"70314","record_id":"<urn:uuid:203d2e82-9488-41aa-9bd6-c00ba4d0fe30>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00452.warc.gz"} |
Multitasking Money - Part 3 of 6
The Uninterrupted Compound Growth Curve
"The greatest power on earth is compound interest." -Albert Einstein
Compounding is a great way to grow your money over time. By reinvesting earnings (e.g. - dividends, interest or capital gains) you are able to earn in the future not only on the original investment
but also on the re-invested earnings. Essentially, you are increasing the amount of your investment moving forward without making a contribution directly from your wallet. Compound interest works on
both assets and liabilities. While compounding boosts the value of an asset more rapidly, it can also increase the amount of money owed on a loan, as interest accumulates on the unpaid principal and
any previous interest charges. Simple interest differs from compound interest in that it only the principal earns interest each period. While all earnings are good, not all earnings were created
equal! Understanding how the growth curve for compounding interest works is essential. It will help you realize the value of this concept as well as the actions that can kill its momentum.
To illustrate how compounding works, let’s look at two examples. First, simple interest. Simple interest pays interest only on the amount of principal initially invested or deposited. For example, if
$10,000 is deposited with 5% simple interest, it would earn $500 each year ($10,000 x .05=$500).
This is in contrast to compound interest which pays “interest on interest”. Suppose $10,000 is held in an account that pays 5% interest annually. After the first year the total in the account has
risen to $10,500, as a result of $500 in interest being added to the $10,000 of principal. In year two, the account realizes 5% growth on both the original principal and the $500 of first-year
interest, resulting in a second-year gain of $525 and a balance of $11,025. This increasing return would continue and after 10 years, assuming no withdrawals and a steady 5% interest rate, the
account would grow to $16,288.95.
Source: https://www.investor.gov/financial-tools-calculators/calculators/compound-interest-calculator*
Another concept of compounding is the Rule of 72. This term refers to the notion of how long it is estimated for an investment or savings to double in value if there are compounding returns. The rule
states that the number of years it will take to double is 72 divided by the interest rate. So, if the interest rate is 5% with compounding, it would take around 14 years and five months to double
(14.4 years).
CAGR – Compound Annual Growth Rate
While not essential to understanding the growth curve, it is worthwhile to briefly discuss CAGR (or you can skip to the next section if you want 😊). The compound annual growth rate isn’t a true
return rate, but rather a representational figure. It is a number that describes the rate at which an investment would have grown if it had grown at the same rate every year and the profits were
reinvested at the end of each year. This sort of performance is unlikely, however, it can be used as a comparative tool so investment performance may be more easily understood compared to alternative
Imagine you invested $10,000 in a portfolio. In 2019 you earned 30%. In 2020 you earned 7.69%. And in 2021 you earned 35.71%. (Sign me up to that one!) Your investment is now worth $19,000. On an
annual basis, the year-to-year growth rates of the investment portfolio were quite different. On the other hand, the compound annual growth rate averages the annual returns, accounting for the
previous year’s performance as a part of it. The CAGR over that period was 23.86%. The formula for this is beyond the scope of this article, however one of many sites you can use to calculate this is
This formula can be manipulated to analyze compounding interest in multiple ways. One such way to use it is to help project what rate of return would be needed, over a specified period, based on
certain assumptions, to achieve a specific future value. For example, imagine that an investor knows that they need $50,000 for a child’s college education in 18 years, and they have $15,000 to
invest today. How much does the average rate of return need to be to reach that objective? The CAGR calculation can be used to find the answer to this question.
The most important limitation of the CAGR is that because it calculates a smoothed rate of growth over a period, it ignores volatility and implies that the growth during that time was steady. Returns
on investments are uneven over time, with limited exceptions. Also, the CAGR does not account for when an investor adds funds to a portfolio or withdraws funds from the portfolio over the period
being measured. For example, if an investor had a portfolio for five years and contributed money to the portfolio during the five-year period, then the CAGR would be inflated. The CAGR would
calculate the rate of return based on the beginning and ending balances over the five years and would essentially count the deposited funds as part of the annual growth rate, which would be an
inaccurate portrayal of the actual return on the investment
The Compound Curve - The Snowball Effect
Compounding is often referred to as the "snowball effect" because it can cause investments to grow at an exponential rate. If you withdraw the earnings on an investment each year, you end up with a
simple interest equivalent. Conversely, if you leave it invested, you can potentially maintain the momentum of the curve, which like a snowball rolling down a snowy hill, continues to grow. If you
want to take advantage of one of the ways you can make your money work for you, a smart strategy is to start investing early and let the compounding growth curve work its magic.
Take a look at this curve and notice how steep it gets over time. That’s because the interest is compounding! It illustrates two things. First, the small action of reinvesting earnings over time can
potentially lead to major financial gains. Second, the earlier you start investing, the greater the effects of compounding will be. It is equally important to realize that a small interruption to
these two strategies can significantly impact your investment returns.
Overall, the compounding growth curve is one of the most powerful forces in finance. With a sound strategy and a bit of patience, you can potentially achieve great financial success over the long
The Main Problem – You can’t have your cake and eat it too!
All this stuff sounds great until you realize that not everyone can afford to just let their money sit forever! Most people need the money now or at some point in the future. Once the money is spent
you can try to start saving again, but the momentum of the growth curve has been stopped and you must start all over again.
Imagine there was a way to save money and spend it too! A way to let your money compound for the rest of your life while also retaining the ability to spend what you need as you need it! That’s one
aspect of what I call Multitasking Money! What are you waiting for? If you want to learn more, keep reading through the Multitasking Money Blog Series.
By Alan J. Mendlowitz, RICP, CRES
PLEASE NOTE: The information being provided is strictly as a courtesy. When you link to any of the web sites provided here, you are leaving this web site. We make no representation as to the
completeness or accuracy of information provided at these web sites. Nor is the company liable for any direct or indirect technical or system issues or any consequences arising out of your access to
or your use of third-party technologies, web sites, information and programs made available through this web site. When you access one of these web sites, you are leaving our web site and assume
total responsibility and risk for your use of the web sites you are linking to. | {"url":"https://www.financialadvisoryservices.net/post/multitasking-money-part-3","timestamp":"2024-11-02T18:36:11Z","content_type":"text/html","content_length":"1050504","record_id":"<urn:uuid:cb525a41-f46c-494d-b7cd-d6cb4696bc43>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00095.warc.gz"} |
Math formulas for class 12
by BrainHub | Jul 29, 2023 | Updates
Certainly! Class 12 mathematics covers a wide range of topics. Here are some important formulas that you might encounter in Class 12:
1. **Quadratic Formula**:
The solutions of the quadratic equation ax^2 + bx + c = 0, where a ≠ 0, are given by:
x = (-b ± √(b^2 – 4ac)) / 2a
2. **Arithmetic Progression (AP)**:
The nth term of an AP is given by:
a_n = a + (n-1)d
The sum of the first n terms of an AP is given by:
S_n = (n/2) * (a + l), where ‘a’ is the first term, ‘l’ is the last term, and ‘n’ is the number of terms.
3. **Geometric Progression (GP)**:
The nth term of a GP is given by:
a_n = a * r^(n-1)
The sum of the first n terms of a GP (if |r| < 1) is given by:
S_n = a / (1 – r), where ‘a’ is the first term and ‘r’ is the common ratio.
4. **Trigonometric Ratios**:
In a right-angled triangle with sides ‘a’, ‘b’, and hypotenuse ‘c’, and an angle ‘θ’:
sin(θ) = a / c
cos(θ) = b / c
tan(θ) = a / b
5. **Trigonometric Identities**:
Pythagorean Identities:
sin^2(θ) + cos^2(θ) = 1
tan^2(θ) + 1 = sec^2(θ)
1 + cot^2(θ) = csc^2(θ)
6. **Limits**:
lim(x → a) f(x) = L
This represents the limit of a function ‘f(x)’ as ‘x’ approaches ‘a’, and it equals ‘L’.
7. **Derivatives**:
If f(x) is a function, the derivative of f(x) with respect to ‘x’ is denoted by f'(x) or dy/dx.
8. **Integrals**:
The integral of a function f(x) with respect to ‘x’ is denoted by ∫f(x) dx.
9. **Probability**:
Probability of an event A is given by:
P(A) = (Number of favorable outcomes) / (Total number of possible outcomes)
10. **Matrices**:
Matrix addition, subtraction, and multiplication formulas.
11. **Determinants**:
Formula to find the determinant of a 2×2 and 3×3 matrix.
12. **Vectors**:
Scalar multiplication, vector addition, subtraction, and dot product formulas.
These are just some of the important formulas in Class 12 mathematics. The specific topics covered may vary depending on the curriculum, so make sure to consult your textbook or syllabus for the
complete set of formulas you need to study.
Recent Comments | {"url":"https://www.brainhubacademy.com/math-formulas-for-class-12/","timestamp":"2024-11-10T14:34:22Z","content_type":"text/html","content_length":"241123","record_id":"<urn:uuid:ca8d5888-4852-4020-aca5-fa0a0d89837d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00248.warc.gz"} |
Complex numbers - mathXplain
Content of the topic
What are complex numbers?
What are complex numbers?
Operations on complex numbers
Absolute value of complex numbers, sets on the complex plane
Let’ see what complex numbers are.
First, let’s talk a bit about numbers.
This is 3, for example.
And this is 4.
And unfortunately, sometimes we need negative numbers, too.
Then we may need numbers that express ratios.
These are called rational numbers.
Like the solution of this equation:
And then there are equations where the solution is not a rational number.
So, we introduce the irrational numbers that fill the gap between the rational numbers on the number line.
And that takes us to real numbers. At every point of the number line there is a real number.
But in certain cases - especially if physicists are lurking around - we need numbers that have some quite unusual properties.
For example, one like this:
Right off the top of our heads, we cannot find many numbers that would fit here, because
These strange numbers were named imaginary numbers.
Since the real numbers already took up all spots on the number line, we place the imaginary numbers on an axis perpendicular to it.
The unit of the imaginary axis is .
Its most important property is .
Numbers that consist of real and imaginary parts are called complex numbers.
So, complex numbers are in the form of , and they are located on the so called complex plane.
Here are two complex numbers:
and let’s see how we add or even multiply them together.
For addition, we simply add the real parts
and the imaginary parts.
Multiplication is more exciting.
But .
The funniest is division.
Stay tuned...
Operations on complex numbers
The idea of complex numbers originated from the disappointment that the equation of
does not have a real solution.
We could have just shrugged this little problem off, but as it turns out, some solution would be quite useful, especially for physics problems. So, we need to somehow magically create a solution for
That’s how we invented our little imaginary friends, the imaginary numbers.
They live on the imaginary axis perpendicular to the real number line...
and their most important property is that
Numbers that have a real part and an imaginary part and can be expressed in the form of are called complex numbers.
And now let’s see what kind of operations we can do on complex numbers.
Then, there is this weird thing called conjugate.
The conjugate of complex number is .
Geometrically, conjugation is a reflection about the real axis.
Splendid. Now we are ready for multiplication.
Well, division will be interesting.
We will try to remove from the denominator.
We do this with the help of its conjugate.
This little trick with the conjugate always works.
If we multiply a complex number by its conjugate, we always get a real number:
Same thing if we add them:
Now, it would be nice to derive some benefit from these complex numbers.
Factorization in complex
Here is this thing… well, a polynomial. Let’s try to turn it into a product of linear factors.
Here comes this formula:
And now let’s try to factorize this one:
There is no such identity as
so we try using the previous one, with some hocus-pocus.
Next, let's see a more complicated one.
One significant benefit of complex numbers is that using complex numbers, all polynomials can be factored into linear factors.
This is called the Fundamental Theorem of Algebra.
Now, let’ solve some quadratic equations that we thought were hopeless.
Here comes the solution formula:
The absolute value of a complex number is its distance from zero.
We can compute this distance using the Pythagorean Theorem.
Let's see one more.
Instead of the formula, here we try factorization.
And now let’s see what else we can do with these complex numbers.
The polar form (trigonometric form)
There is a big problem with the algebraic form of complex numbers. That is, it is almost impossible to raise them to powers.
Let's try to compute the value of
Well, this is it.
But this could only be some sick joke...
There must be a simple way to raise complex numbers to powers.
This is the usual algebraic form of complex numbers,
and now we replace it with a polar form (trigonometric form).
The main idea of thispolar form (trigonometric form) is that it describes complex numbers using two new attributes: one is the absolute value and the other is the angle.
We denote the absolute value by r,
and the angle... well, the angle by theta. Here it is:
The polar form (trigonometric form) makes it surprisingly simple to multiply complex numbers,
and to divide them.
And now, let’s get back to the issue of powers.
We want to compute the value of .
Here comes the polar form (trigonometric form):
And now we start computing the powers.
The nth power is computed by raising r to the nth power, and multiple the angle by n:
This way
and that, if we feel like it, can be transformed back into algebraic form.
Now let's try to compute this:
Let's see first the polar form (trigonometric form).
But there is a snag.
This equation here:
has another solution.
As to which one we need, we could decide by flipping a coin,
but it is better to draw a figure.
It seems we need the negative solution.
And now we are ready for the multiplication.
Absolute value of complex numbers, sets on the complex plane
The absolute value of a complex number is its distance from zero.
We can compute this distance using the Pythagorean Theorem.
Let's see one more.
Instead of the formula, here we try factorization.
And now let’s see what else we can do with these complex numbers.
Let’s try to graph on the complex plane those complex numbers where:
We use the algebraic form,
that is
Next come some coordinate geometry horror stories.
The equation of
is the equation of a circle centered at the origin, with a radius of r.
Based on this, is also a circle centered at the origin, with a radius of r=2.
And means the circle and its inside.
Coordinate geometry horror stories:
The equation of a line:
The equation of a circle:
Let’s find on the complex plane complex numbers such that:
We use the algebraic form, so we replace z
with everywhere.
The inequality means one side of the line.
Let's see which side.
It is always a good idea to try a=0 and b=0.
This seems to fit, so we need this side of the line.
Next, let's see what is going on with this one:
The inequality means one side of the circle.
Either the inside or the outside of the circle.
Again, it is a good idea to try a=0 and b=0.
It seems we need the outside.
And because equality is not allowed,
the circle itself is not part of the region.
Finally, let’s see what this is about:
We will need to complete the square.
The complex radicals
The differences between real and complex radicals.
Here comes some magic:
The question is: where is the trick?
The fact is: there is no trick.
For example, a while ago we defined what means. We said .
This is in spite of the fact that there is another number whose square is 4: that number is negative 2.
For complex numbers, the situation is much more entertaining.
For example
Yes, but
So, there are four numbers whose fourth power is 1.
This minor inconvenience prompts us to define radicals differently for complex numbers than we did for real numbers.
The nth root of a real number always meant exactly one number. The nth root of a complex number, on the other hand, means all numbers whose nth power is the original number.
For example
real complex
The nth roots of complex number are complex numbers
where the following holds:
Here, r denotes the absolute value of the complex number, which is a real number.
So, this is an ordinary real radical - just like in the old days.
Here is this complex number:
Let's see what happens if we search its 5th root.
First, we need the trigonometric form.
And then we are ready for the radical.
This means five complex numbers.
Case k=5 doesn’t matter. We return to case k=0.
So, that’s it for radicals. | {"url":"https://www.mathxplain.com/precalculus/complex-numbers","timestamp":"2024-11-11T03:39:21Z","content_type":"text/html","content_length":"88245","record_id":"<urn:uuid:413c7745-cebc-439d-b151-e54b32ef7ddf>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00560.warc.gz"} |
ICSE Class 10 Mathematics Revision Notes Chapter 20 - Cylinder, Cone and Sphere
ICSE Class 10 Mathematics Revision Notes Chapter 20 – Cylinder, Cone and Sphere
Class 10 students need to prepare well for the Mathematics exam to score higher marks. To revise the topics of Chapter 20, students can make use of the ICSE Class 10 Mathematics Revision Notes
Chapter 20. In Mathematics, it is important to understand the concepts well before solving exercise questions. Practising problems regularly is the right method of preparing for the Mathematics exam.
All the points covered under the Mathematics syllabus of Class 10 ICSE Board should be prepared from the board exam perspective.
Students are encouraged to solve the ICSE Sample Questions Papers to understand the question paper pattern of Mathematics.
The topic of Chapter 20 is Cylinder, Cone and Sphere. To prepare effectively for Cylinder Cone and Sphere ICSE Class 10, students need to study the chapter thoroughly, and they should also answer the
questions at the end of the chapter. It is necessary to have a clear understanding of the chapters of Mathematics to score well in board exams. The ICSE Class 10 Mathematics Revision Notes Chapter
20 are helpful in quickly revising the important topics of Chapter 20. Students of the Indian Certificate of Secondary Education (ICSE) Board must have the Class 10 syllabus of Mathematics before
they begin their preparation for the board examination.
Studying according to the syllabus is crucial for students to be on the right track and cover all the topics. . On the Extramarks’ website and mobile application, students can download the ICSE
Solutions for exercise questions and get on with their revision without any further delay.
Students are advised to access the ICSE Class 10 Mathematics Revision Notes Chapter 20 from the Extramarks’ website and mobile application.
The ICSE Class 10 Mathematics Revision Notes Chapter 20 help to recall the important points from the Cylinder Cone and Sphere ICSE Class 10.
Some topics can be difficult to understand in Mathematics. It is important for Class 10 ICSE Board students to prepare a list of all the difficult topics, and they must spend more time learning those
topics. Students of ISC and ICSE Board should access the ISC & ICSE Syllabus of each subject before starting the board exam preparations. Students must practice the past years’ question papers of
Mathematics to enhance their speed while answering questions.It requires regular practice and endless patience to master those challenging topics and be ahead of the competition.
Students are required to access study materials from authentic sources. Class 10 students can download the ICSE Class 10 Mathematics Revision Notes Chapter 20 from the Extramarks’ website and mobile
application in PDF format.
Revision Notes for ICSE Class 10 Mathematics Chapter 20 – Free PDF Download
To strengthen the concepts, students must practice the exercise questions. To help students in reviewing all the topics in Chapter 20, access the ICSE Class 10 Mathematics Revision Notes Chapter 20
from the Extramarks’ website and mobile application.
Regular textbook reading in Mathematics is required, and students should concentrate on making notes on important topics. Students need to have important formulas, theorems, definitions while
answering questions in the Mathematics Exam at their fingertips to get a high score. Students must access the ICSE Class 10 Mathematics Revision Notes Chapter 20 from the Extramarks’ website and
mobile application to revise the important topics of Chapter 20.
Extramarks provides authentic study materials for students of all classes while adhering to the respective boards and it has earned that credibility and is trusted by teachers and students alike
especially for their board exam preparation. It is necessary for students to sign up at Extramarks to enjoy unlimited resources to prepare well for the board exam. The ICSE Class 10 Mathematics
Revision Notes Chapter 20 are available on the Extramarks’ website and mobile application to step up Class 10 students in their board exam preparation and help them score excellent grades.
Revision Notes for ICSE Class 10 Mathematics Chapter 20 – Cylinder, Cone and Sphere
It is important to do well in a subject like Mathematics to achieve a good score in Class 10. To prepare for Mathematics Class 10 Board Examination, students need to keep revising the topics
consistently. Students are advised to refer to the ICSE Revision Notes to quickly revise the important topics of Chapter 20 since Mathematics syllabus is quite vast and it’s important to reinforce
main concepts on a regular basis through revision work.
The Chapter 20 of Mathematics is Cylinder. The students are expected to understand the concepts and practice answering questions related to Cylinder topics.
All the key points of the Cone topic should be revised regularly. It is necessary to remember the formulas, definitions, and theorems of Chapter 20.
Students must practice the ICSE Important Questions of Sphere to perform well in the Mathematics Board Exam. All the sub-points under the Sphere topic should be carefully learned and revised to clear
all the doubts. Practising important questions are helpful in answering questions for ICSE Question Paper and evaluating themselves before taking the board exam .
ICSE Class 10 Mathematics Revision Notes
While solving problems, it is necessary to have important points at fingertips while answering questions in the Mathematics board exam. Students must access the ICSE Class 10 Mathematics Revision
Notes Chapter 20 from the Extramarks’ website and mobile application to revise essential topics of Chapter 20 with good speed and clarity.
Revision Notes
All the topics and subtopics of the Cylinder, Cone and Sphere Chapter need to be revised again and again. Students must practice exercise questions to make the concepts strong. The ICSE Class 10
Mathematics Revision Notes Chapter 20 are helpful in revising all the topics of Chapter 20.
Students must go through the Mathematics textbook regularly, take regular assessment tests and should focus on making notes related to the topics to score well in the board exam.
Chapter wise ICSE Class 10 Mathematics Revision Notes
FAQs (Frequently Asked Questions)
1. How are the ICSE Class 10 Mathematics Revision Notes Chapter 20 beneficial for scoring well in the Mathematics Exam?
The ICSE Class 10 Mathematics Revision Notes Chapter 20 are helpful for quickly revising the crucial points from Chapter 20. Students can use the ICSE Class 10 Mathematics Revision Notes Chapter 20
to prepare efficiently for the Mathematics Exam.
2. Where can students get access to the ICSE Class 10 Mathematics Revision Notes Chapter 20?
Students of Class 10 ICSE Board can download the ICSE Class 10 Mathematics Revision Notes Chapter 20 from the Extramarks’ website and mobile application in PDF format. | {"url":"https://www.extramarks.com/studymaterials/icse/icse-class-10-mathematics-revision-notes-chapter-20/","timestamp":"2024-11-07T00:04:44Z","content_type":"text/html","content_length":"629161","record_id":"<urn:uuid:5ffa4fac-90ff-4729-8f79-de07df08e305>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00084.warc.gz"} |
A Theoretical Iteration for Predicting the Feasibility for Immediate Functional Dental Implant Loading
When planning an implant-supported restoration, the dentist is faced with surgical and prosthetic technical issues as well as the patient's expectations. Many patients wish an immediate solution to
an edentulous condition. This may be especially true in the esthetic zone, and that zone is determined by the patient. The dentist may consider when it is feasible to load the supporting implants
with definitive or provisional prosthetics. In this work, many parameters were theoretically assessed for inclusion: bone density, cortical thickness, insertion torque, parafunction, bite load
capacity, number of implants under load, implant/crown ratio, implant diameter, and length. After assessment, the most influential parameters were selected. An iteration, using patient age, implant
diameter, bite load capacity, and cortical thickness, is now presented to aid the implant dentist in determining the feasibility for immediate functional loading of a just-placed dental implant in a
healed site. Extensive testing is required to develop this concept. According to this iteration, most immediate functional loaded implants would fail. A future refined and definitive formula may
enable the clinician to safely and immediately functionally load an implant with a definitive prosthesis. For access to the applet, please go to https://implantloading.shinyapps.io/shiny_app/.
There is patient demand for immediate functional loading of just-placed dental implants. There is some evidence that immediate occlusal loading of dental implants is a viable option in some clinical
situations.^1,2 Clinical factors such as bone quality, implant dimensions, forces of occlusion, prosthetic design, and parafunction have been shown to affect immediate functional loading. These
factors may affect the outcome of an immediately loaded dental implant, and some are more important than others. Many of these parameters are measurable with numerical values. There is a need for a
method to relate measurable parameters to predict the feasibility for immediate implant functional loading.
Previous work has attempted to describe jaw functioning from a mathematical perspective.^2 Reducing clinical functional parameters to mathematics may make treatment planning more predictable,
complications fewer, and outcome success rates higher.
Implants placed in immediate functional conditions have been successful.^3–6 The parameters for successful immediate functional loading appear to be related to bone quality, having a patient with
lesser jaw force capability, the number of implants bearing the occlusal load, a longer implant length and wider diameter, and implants with a rough surface .^3–6
In vitro studies and the clinical experiences of clinicians have determined that bone can successfully resist occlusal forces transmitted by just-placed implants.^1–6 Single and multiple implants
can be immediately loaded in certain circumstances.^1–6 Multiple implants that are splinted distribute the occlusal forces over multiple implants and lessen the per-square-millimeter force
transferred to the supporting bone.
Implants placed in high-density bone can support an immediate occlusal load.^2 Dense bone is able to keep the implant immobile while under occlusal loads.
Parafunction, intuitively, appears to be a factor detrimental to immediately loading. Implant-supported fixed partial dentures may be intentionally placed about 100 μm shy of occlusal contact. This
is an accommodation for natural tooth intrusion so that the implant prosthesis does not bear the full occlusal load during maximum intercuspation and parafunction. Natural teeth do intrude slightly
under load, but the intrusion varies from tooth to tooth.^7 To avoid a parafunctional overload, 100-μm occlusal relief may be an appropriate dimension.
Forces generated by patients and delivered to the implant/bone–supported prostheses can differ. The individual patient force capability can be determined.^8 A patient's jaw force capability can be
measured using a transducer measuring device or silicone imprinting, so that a numeric parameter can be identified.^9,10 The generated jaw force is related to the position in the arch. Implants
placed in more posterior sites are subjected to greater force than those placed in more anterior areas. A greater force may dislodge a just-placed implant.^9 There are several devices available for
measuring jaw force.^11
Immediate implant functional loading is favorable for maxillary and mandibular fixed splinted complete arch reconstructions and removable mandibular overdentures.^12 In addition, for single-tooth
reconstructions in the esthetic zone, including premolars and short-span fixed partial dentures, there is a high survival rate but an increased risk for implant loss.^12
Immediate loading with occlusal contacts may not have a greater risk for implant loss than immediate loading without occlusal contacts. There may be less marginal bone loss around immediately loaded
implants as compared with delayed loaded implants.^12
There is a higher patient satisfaction with immediate loading, but there are increased complexities in treatment planning. Provisional restorations may be appropriate with an insertion seating torque
of greater than 30 Ncm with appropriate prosthetic positioning.^12 Immediate loading should be avoided in patients with bruxism and clenching.^12 Immediate implant loading requires high patient
compliance. Adequate width of the attached mucosa around immediately loaded implants is crucial.^12 Nonetheless, there is a greater risk for failure with immediate loading as compared with delayed
loading. Grafted sites may pose an even greater risk for failure.^12
The object of this effort is to present an iteration that may theoretically predict the feasibility of immediate implant functional loading.
The basis of the model comes from ANSYS finite element analyses, which were conducted over the years 2013 to 2015, simulating the stresses placed on the surface of the implant. The analysis concluded
that for an average jawbone with 1 to 6 implants, the approximate maximum load allowed was at 30° and 45° to simulate off-axial occlusal loading. In addition, the maximum stresses for different
cortical bone thicknesses to refine the approximation of bone strength and loading limitations were investigated.
The loading properties of each patient's bone are unique (Young's modulus, bone density, dimensions). The results of the ANSYS analysis (maximum stresses) was able to deliver a model that considered
cortical bone thickness, bone quality, and implant diameter.^13 The model was further refined to apply immediate functional loading upon newly placed nonosseointegrated dental implants and to
predict the feasibility of immediate occlusal loading.
The model must maximize its predictive power while at the same time minimize the amount of input parameters. Minimizing the input parameters increases computational speed and decreases the amount of
data that need to be collected for each patient by the clinician. This allows the implant dentist to focus on the most appropriate data for the immediate loading decision-making process.
The outcome data from this model cannot be directly tested on patients without further study because of the simplifications made in the finite element analysis (FEA) model.^14–16
A verification process can be instituted consisting of FEA bone models with data input from actual patient mandibles. The outputs of the more complex patient-adjacent FEA simulations were compared
with the simplified mandible models to ensure a small amount of error existed between the 2 models. Cone-beam computerized tomography (CBCT) measurements were used to build the FEA simulations in the
verification process. The FEA model seeks to approximate stress, while the mathematical model predicts the stress on the jaw based on selected input parameters.^16 The model was trained on data
yielded from the ANSYS simulations.
A comparison was made of the stress values from the FEA simulations using the simplified bone model and the output of the mathematical model to judge the accuracy of the mathematical model.^16
Development of the model
To protect the anonymity of patients, the Digital Imaging and Communications in Medicine (DICOM) standard was used. The DICOM standard makes anonymization a simple process.^17
A data set was generated based on the range of values for the cortical bone thickness of a given edentulous site and the implant diameter. The cortical bone thickness was found to range from 0.25 mm
to 3 mm, and the implant diameter ranged from 3.2 mm to 5.7 mm.
One thousand random points were then generated, and the stress was calculated in megapascals (MPa). Table 1 represents a subset of the points used in the analysis.
Next, the 2 parameters were plotted against the stress to determine who each parameter affects the predicted stress. The plot in Figure 1 shows the relationship between the diameter and the predicted
stress of the previous model:
The regression between the predicted stress and the implant diameter had a coefficient of determination of 0.9976; therefore, variations in the implant diameter can explain 99.76% of the variation in
the predicted stress.^18,19 Theoretically, a coefficient of determination of 1 would demonstrate a perfect linear fit. The value here is less than 1% away from such a fit. Without considering the
cortical bone thickness, the implant diameter alone can explain the output of the model. Next, the image in Figure 2 shows the relationship between the predicted stress and the cortical bone
The coefficient of determination in this case was 0.00044, which means that the variations in cortical bone thickness obviate the 0.044% of the variation in the predicted stress of this model.
The values of the cortical bone thickness varied between 6 and 2 times as small as those for the implant diameter.^20,21 In addition, the coefficient by which the implant diameter was multiplied in
the model was a little more than double that multiplied by the cortical bone thickness. Thus, the implant diameter was weighted up to 8 times more heavily than the cortical bone thickness was.
However, consideration must be given to extremely thick cortical bone and another important parameter, individual patient bite force capability. This creates a better model.
This more inclusive model used the capability in the programming language R to take the data collected from the simulations and produce a relevant mathematical model. The framework leveraged within R
was the generalized additive model, which takes a set of predictive parameters and regresses the values toward the dependent variable to form an output based on a defined type of error distribution.
Once enough data points were aggregated, the first version of the model was made, which included the Young's modulus of cortical bone, the bite force of the patient, and a discrete variable
indicating the size of the implant (Figure 3).
The output data of the model in Figure 3 compares the actual stress from the simulations and the predicted stress.^22 The green circles are data points, the shaded region is the relative error, and
the line is a loess curve that shows the relationship between the actual and predicted stress. Each of the parameters in the model (implant diameter, bite force, Young's modulus, and cortical bone
thickness) have P values that are much less than .05. The line on the graph represents a loess curve. This is a local regression that is based on a combination of nonlinear polynomial regression and
the k-means clustering algorithm.^23 Note that this is not the curve of the model; it is based on the location of the data points. The loess curve is meant to help the reader see any obvious trends
in the data. The shaded region represents the 95% confidence interval for the loess curve, not the confidence interval of the model. The colored points are based on the relative error of the
predicted stress, where the actual stress output from the simulation is the accepted value. Only on the extreme low end of the data set is the error excessively high; the points colored red output
relative errors that exceed 90%. These points have bite forces at the extreme low end of 50 N. The rest of the data points have relative errors that are less than 20%. Because these data would be
used in patient treatment, a 20% error is not acceptable. The points with the lightest hue of green have less than 5% relative error. While a zero tolerance for error is the goal, an acceptable error
rate may be 5%. The mean relative error for this version of the model was 19.09%, which the 2 outlier points disproportionately affected. To mitigate the effect of those points, the median relative
error was examined, which was 12.80%, about 33% smaller than the mean. The model was adjusted to mitigate the existence of outputs with such great error.
To see how a range of possible patient virtual parameters would change the output of the model, 1000 theoretical patients were simulated. The mean and standard deviation of both the bite force and
Young's moduli were tested and constructed for normal distributions for each parameter and for random data points based on those normal distributions. These were then randomly paired with the Young's
modulus distribution and the bite force distribution, which represented a possible patient. The first 500 virtual patients in the data set received small-diameter (3.2-mm) implants, whereas the rest
received large-diameter (4.7-mm) implants. Table 2 is a rendering of the simulated data sets.
All of the patients' information put into the model yielded a predicted stress value on each patient's bone. Figure 4 shows the relationship between the predicted stress and bite force, separated by
patients who received large- and small-diameter implants.
Unsurprisingly, the stress from the model increases as force increases. Furthermore, the small-diameter implants have larger stress values than the large implants do, which is not surprising
considering that stress is the given force divided by area, and area depends on diameter. There is a greater per-square-millimeter of load on the small-diameter implants than the larger diameter
implants.^18,20 The smaller the denominator, the larger the stress. Figures 5 and 6 separate the large and small implants and show whether the bone meets the failure criterion, stress that exceeds
the yield strength of 114 MPa.
In the large-diameter implants, failure begins at about 500 N. Because the per-square-millimeter of stress with large-diameter implants is smaller than that of small-diameter implants with the same
force, the yield stress can be exceeded at force figures slightly less than 100 N for small-diameter implants.
Next, the range of Young's moduli for the theoretical patients considered how that quantity was related to the stress. In Figure 7, the points are colored by diameter size.
Unlike the relationship with bite force, the stress output from the model does not necessarily increase as the Young's modulus increases. This was attributed to other factors affecting the stress.
This may not be a simple Hooke's law relationship. Hooke's law states that the strain in a solid is proportional to the stress applied to it but within the elastic limit of the solid. The stresses on
the small-diameter implants are smaller than those of the large-diameter implants for a given value of Young's modulus. This is due to the surface area presented to the bone. Figures 8 and 9 separate
the large- and small-diameter implants and show the failure criterion.
Failure can occur at any value for Young's modulus with the requisite magnitude of force. The opposite of the phenomenon is observed for the stress and force relationship. In addition, for a given
value of Young's modulus, the stress from the small-diameter implant is larger than that of the large-diameter implant.
Finally, the effect of the cortical bone thickness on the predicted stress of the simulated patient data set was considered. Figure 10 shows the relationship between the predicted stress and the
cortical bone thickness of the mandible for both small- and large-diameter implants:
There was no linear relationship between the cortical bone thickness and the predicted stress. The parameter was statistically significant when predicting stress values, but alone, it did not explain
any increases for the decreases in stress for this model. Figures 11 and 12 separate the small- and large-diameter implants.
Just as with the 2 other parameters described above, for a given value of cortical bone thickness, the small-diameter implants yield a larger stress than the large-diameter implants do. After
considering all 4 parameters in the model, the small-diameter implants have larger stress values than the large-diameter implants do. The stress increases as the force increases. For the other 2
parameters, the stress value can vary depending on the values of the other parameters. Thus, a patient's individual bite force capability appears to be the most influential parameter.
The Applet
An applet was built so that users can interface and interact with clinical data. A written equation cannot be provided because the model is a general additive model. This means that the model is a
linear sum of nonparametric smoothing functions. Nonparametric functions are functions that do not have a form specified by a parametric formula. Instead, the data set at hand dictates the form of
the function. The downside to this type of modeling is that the user needs a robust data set. If a user fits only a few data points to a nonparametric smoothing function, there is a high risk of
overfitting, especially compared with parametric functions. One can attribute this fact to the phenomenon of the data providing the impetus for both the model structure and the model estimates. The
smoothing functions are unknown by definition. The software develops them in such a way that they are uniquely fit to the data set at hand. There are no formulae or algorithms that apply to all
smoothing functions. An example of a structure for a general additive model (that is not applicable to this specific case) is the linear sum of two nonparametric smoothing functions, in which the
software bases the first function on a the weighted mean of different parts of the data set and bases the second function on a discrete variable. In Figure 13, one can observe what the applet looks
like before the user inputs any values. The model has 4 inputs: the bite force of the patient, the cortical bone thickness, the implant diameter, and the Young's modulus of the patient's bone. The
implant dentist can measure 3 properties but not the Young's modulus of a patient's cortical bone.
To account for the lack of a Young's modulus, a range of possible values for the Young's modulus of cortical bone (18.5–46.2 GPa, a difference of 27.7 GPa) was considered. It was assumed that the
maximum and minimum values of the range were within 6 standard deviations of the mean. Assuming that the range of Young's moduli is normal, this assumption has a 99.99966% chance of holding true.
This degree of certainty makes this assumption cogent.
Given the assumption of the maximum and minimum was within 6 standard deviations, a normal distribution of possible values for Young's modulus was constructed. The standard deviation of the
distribution was one-sixth of the range of the Young's modulus values. Since the range was 27.7 GPa, 1 standard deviation of this distribution is about 4.62 GPa. The mean of the distribution is the
average of the maximum and minimum, 32.35 GPa. The distribution construction was based on 10000 normally distributed random numbers, with the mean and standard deviation from above. The
distributions were further adjusted based on the aging effect of bone outlined in a literature review (see the Appendix). After the age of 35 years, the Young's modulus of bone decreases by about 10%
every 10 years. Thus, age was used to extrapolate possible values of the Young's modulus, and so one of the inputs into the applet is the patient's age. The age input indicates to the software how to
alter the distribution of the Young's modulus values. If the patient is older than 45 years, the software discounts values of the distribution by 10%. If the patient is older than 55 years, that
figure changes to 20%. For ages 65 and up, the distribution values decrease by 30%.
Figure 14 shows the changes in the distribution for age. The dashed line in the density plots on the right represents the average for the given distribution. The horizontal lines on the boxplots on
the left are also the means of the distribution. The edges of the boxes represent the 25th and 75th percentiles. The outlier points are those above 3 standard deviations.
When the user presses the enter button in the applet, the software applies the bite force, cortical bone thickness, implant diameter, and Young's modulus distribution to the model. Because the
distribution of Young's modulus values is an input, there is a corresponding distribution of stress values as an output. What the applet then indicates is the probability that the implant will fail
with immediate functional loading. In the output, we show the 5th, 25th, 50th, 75th, and 95th percentile outcomes for the stress on the bone. This provides an approximate probability of whether the
immediate loading will fail. Figure 15 shows what the filled-out applet looks like, whereas Figure 16 shows the percentile table.
To gauge the probability of failure, the distribution of stress values that the model outputs is show in Figure 17. The red dashed line in the plot represents the failure criterion of 114 MPa. Values
to the right of the dashed line are those in which immediate implant loading will fail, and those to the left indicate the stress values where the immediate implant loading should be successful. An
example of the distribution plot, in which the inputs used are from those from Figure 15, is demonstrated in Figure 17.
The objective of this project has been to develop an iteration that generates a Boolean output that specifies detailed parameters that combines terms and excludes others to determine whether
immediate dental implant functional loading is feasible. A basic framework was first established correcting fundamental flaws accounting for physiological loading conditions. This included the range
of possible bite force capabilities and correctly accounting for the bone quality of a patient. This iteration assumes a healed site, a rough surface implant, accurate bite force measurements, and
accurate cortical bone measurements by CBCT.
The first principal stress as the dependent variable is bite force capability, and this is the primary determiner of whether the bone would fail upon immediate loading.
By generating models from actual patient data taken from CBCT scans, a robust validation process was established. From the research models created, an iteration in the form of an applet was developed
for implant dentists to easily interface with and obtain a real-time indication of whether or not a prospective implant site is eligible for immediate implant functional loading. According to this
iteration, most immediate functional loaded implants will probably fail. Nonetheless, future innovations may reduce failures.
No conflicts of interest or commercial or financial interests were claimed by any of the authors.
Success of immediate loading implants compared to conventionally-loaded implants: a literature review
J Investig Clin Dent
Three-dimensional finite element stress analysis of the dentate human mandible
Am J Phys Anthropol
Comparative observation of immediate and late placement of dental implants with immediate loading: a 14-year follow-up case report
J Oral Implantol
Replacement of mandibular molars with single-unit restorations supported by wide-body implants: immediate versus delayed loading. A randomized controlled study
Int J Oral Maxillofac Implants
Ten year results for Branemark implants immediately loaded with fixed prostheses at implant placement
Int J Oral Maxillofac Implants
Immediate functional loading of TiOblast dental implants in full-arch edentulous maxillae: a 3-year prospective study
Clin Oral Implants Res
The measurement of physiologic tooth displacement in function
J Med Dent Sci
Force required to luxate a newly placed dental implant in bone: an in vitro pilot study
J Oral Implantol
The variability of bite force measurements between sessions, in different positions within the dental arch
J Oral Rehabil
A study on measuring occlusal contact area using silicone impression materials: an application of this method to the bite force measurement using the pressure sensitive sheet
Dent Mater J
Bite force and dental implant treatment: a short review
Med Devices (Auckl)
et al
Loading protocols and implant supported restorations proposed for the rehabilitation of partially and fully edentulous jaws. Camlog Foundation Consensus Report
Clin Oral Implants Res
Changes in the stiffness, strength, and toughness of human cortical bone with age
The GrabCAD Community Library.
Free CAD designs, files & 3D models
et al
Biological-data-based finite-element stress analysis of mandibular bone with implant-supported overdenture
Comput Biol Med
Finite elemental analysis of stress in bone adjacent to dental implants
J Oral Implantol
Scarsbrook. DICOM demystified: a review of digital file formats and their use in radiological practice
Clin Radiol
Effect of diameter and length on stress distribution of the alveolar crest around immediate loading implants
Clin Implant Dent Relat Res
Diameter and length double objectives robust analysis of cylinder dental implant
Hua Xi Kou Qiang Yi Xue Za Zhi
Bone density at implant sites and its relationship to assessment of bone quality and treatment outcome
Int J Oral Maxillofac Implants
et al
In vitro evaluation of the influence of the cortical bone on the primary stability of two implant systems
Med Oral Patol Oral Cir Bucal
A randomized controlled clinical trial comparing the effects of three loading protocols on dental implant stability
Int J Oral Maxillofac Implants
A Simple Introduction to Moving Least Squares and Local Regression Estimation
Washington, DC
U.S. Department of Energy, Office of Scientific and Technical Information | {"url":"https://meridian.allenpress.com/joi/article/47/4/310/445834/A-Theoretical-Iteration-for-Predicting-the","timestamp":"2024-11-06T23:41:44Z","content_type":"text/html","content_length":"219683","record_id":"<urn:uuid:b3cb562a-7b70-4d26-9169-9d9f1fad9b45>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00735.warc.gz"} |
How Do We Use Money?
Concept (nonfiction), 360 words, Lexile 330L Grade 2
The book How Do We Use Money? teaches how to solve problems with quarters, dimes, nickels, pennies, and introduces the ¢ symbol. Counting money is challenging because students have to remember each
coin's value as well as the added amount when counting. This book teaches how to count with coins to find the total value of money or to buy or trade an object for an equal amount of money. This
helps students build important functional money skills for real life.
Book Resources
More Languages
How To Assemble Your Book | {"url":"https://www.readinga-z.com/book.php?id=2213","timestamp":"2024-11-05T13:03:33Z","content_type":"text/html","content_length":"119652","record_id":"<urn:uuid:19d6dfff-23e3-488b-bdb2-5116dee926a0>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00415.warc.gz"} |
Re: a numerical integration problem
I want to perform a numerical integration for the following integrand
C1, delta11, d1, rho are all scalar, which will be specified explicitly. \phi(u) is the PDF of a standard normal distribution, \Phi() is the CDF of a standard normal distribution. I proposed the
following SAS codes, but I failed, so there must be something very wrong. Can you kindly help? Thanks in advance!
proc iml;
start stdn(u);
pi = constant("Pi");
start phi(u) global(d1,rho);
start grand(u);
start integral(cc1,ddelta1,dd1,rrho) global(c1,delta1,d1,rho,eps);
call quad(final,"grand",interval) eps=eps;
eps = 1E-11;
p=integral(1.678, 1, 1.645, 0.6);
print p;
07-10-2020 10:37 PM | {"url":"https://communities.sas.com/t5/SAS-IML-Software-and-Matrix/a-numerical-integration-problem/m-p/668523/highlight/true","timestamp":"2024-11-10T01:28:37Z","content_type":"text/html","content_length":"249265","record_id":"<urn:uuid:3db39306-cd8d-40ca-9d7c-4e170ea0b4cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00062.warc.gz"} |
SSC CGL 2023 Syllabus - gkquestionsguru.com
SSC CGL Syllabus for Tier 1 Exam
SSC CGL 2023 Tier 1 exam comprises 100 questions in total with a maximum mark of 200. The duration of SSC CGL Tier 1 is of 60 minutes. SSC CGL Tier I is divided into four sections with 25 questions
each and a maximum mark of 50.
The sections asked in the SSC CGL syllabus Tier I Exam are:
• General Knowledge
• Quantitative Aptitude
• General Reasoning
• English Comprehension
The Topic-wise Syllabus of Tier 1 is given in the Table below:
SSC CGL Quantitative Aptitude Syllabus
There are a lot of topics from basic and advanced mathematics that are covered in the Quantitative aptitude section of the SSC CGL syllabus.
1. Computation of whole numbers
2. Decimals
3. Fractions
4. Relationships between numbers
5. Profit and Loss
6. Discount
7. Partnership Business
8. Mixture and Allegation
9. Time and distance
10. Time & Work
11. Percentage
12. Ratio & Proportion
13. Square roots
14. Averages
15. Interest
16. Basic algebraic identities of School Algebra & Elementary surds
17. Graphs of Linear Equations
18. Triangle and its various kinds of centers
19. Congruence and similarity of triangles
20. Circle and its chords, tangents, angles subtended by chords of a circle, common tangents to two or more circles
21. Triangle
22. Quadrilaterals
23. Regular Polygons
24. Right Prism
25. Right Circular Cone
26. Right Circular Cylinder
27. Sphere
28. Heights and Distances
29. Histogram
30. Frequency polygon
31. Bar diagram & Pie chart
32. Hemispheres
33. Rectangular Parallelepiped
34. Regular Right Pyramid with triangular or square base
35. Trigonometric ratio
36. Degree and Radian Measures
37. Standard Identities
38. Complementary angles
SSC CGL General Intelligence and Reasoning Syllabus
SSC CGL Reasoning syllabus includes both verbal and non-verbal reasoning. In this section, there are a total of 25 questions carrying 50 marks in total
1. Analogies
2. Similarities and differences
3. Space visualization
4. Spatial orientation
5. Problem-solving
6. Analysis
7. Judgment
8. Blood Relations
9. Decision making
10. Visual memory
11. Discrimination
12. Observation
13. Relationship concepts
14. Arithmetical reasoning
15. Figural classification
16. Arithmetic number series
17. Non-verbal series
18. Coding and decoding
19. Statement conclusion
20. Syllogistic reasoning
SSC CGL English Language syllabus
The SSC CGL syllabus of Tier 1 English will contain 25 questions carrying 50 marks in total. Out of all the sections in the Tier 1 exam, this section is only going to be in English only.
1. Phrases and Idioms
2. One-word Substitution
3. Sentence Correction
4. Error Spotting
5. Fill in the Blanks
6. Spellings Correction
7. Reading Comprehension
8. Synonyms-Antonyms
9. Active Passive
10. Sentence Rearrangement
11. Sentence Improvement
12. Cloze test
SSC CGL General Awareness Syllabus
The syllabus of SSC CGL General Awareness is not something that can be mastered in a day but from an exam point of view, if consistently spend time, a candidate can perform well in relatively less
time than with Quantitative aptitude.
1. India and its neighboring countries especially about History, Culture, Geography, Economic Scene, General Policy & Scientific Research
2. Science
3. Current Affairs
4. Books and Authors
5. Sports
6. Important Schemes
7. Important Days
8. Portfolio
9. People in News
SSC CGL Syllabus 2023 For Tier 2
SSC CGL Tier 2 exam will be conducted online comprising 3 papers-
1. Paper I (Compulsory for all posts),
2. Paper II for candidates who apply for the posts of Junior Statistical Officer (JSO) in the Ministry of Statistics and Programme Implementation and
3. Paper III for candidates who apply for the posts of Assistant Audit Officer/ Assistant Accounts Officer.
Objective Type, Multiple choice questions, except for Module-II of Section-III of Paper-I
Module-I of Session-I of Paper-I (SSC CGL Mathematical Abilities Syllabus):
1. Computation of whole numbers
2. Decimals
3. Fractions
4. Relationships between numbers
5. Percentage
6. Ratio & Proportion
7. Square roots
8. Averages
9. Interest
10. Profit and Loss
11. Discount
12. Partnership Business
13. Mixture and Alligation
14. Time and distance
15. Time & Work
16. Basic algebraic identities of School Algebra & Elementary surds
17. Graphs of Linear Equations
18. Triangle and its various kinds of centers
19. Congruence and similarity of triangles
20. Circle and its chords, tangents, angles subtended by chords of a circle, common tangents to two or more circles
21. Triangle
22. Quadrilaterals
23. Regular Polygons
24. Right Prism
25. Right Circular Cone
26. Right Circular Cylinder
27. Sphere
28. Hemispheres
29. Rectangular Parallelepiped
30. Regular Right Pyramid with triangular or square base
31. Trigonometric ratio
32. Degree and Radian Measures
33. Standard Identities
34. Complementary angles
35. Mensuration
36. Heights and Distances
37. Histogram
38. Frequency polygon
39. Bar diagram
40. Pie chart
Module II of Section-I of Paper-I (SSC CGL Reasoning and General Intelligence Syllabus)
1. Questions of both verbal and non-verbal types.
2. Semantic Analogy
3. Symbolic operations
4. Symbolic/Number Analogy
5. Trends
6. Figural Analogy
7. Space Orientation
Semantic Classification
8. Venn Diagrams
9. Symbolic/ Number Classification
10. Drawing inferences
11. Figural Classification
12. Punched hole/ pattern-folding & unfolding
13. Semantic Series
14. Figural Patternfolding and completion
15. Number Series
16. Embedded figures
17. Figural Series
18. Critical Thinking
19. Problem-Solving
20. Emotional Intelligence
21. Word Building
22. Social Intelligence
23. Coding and de-coding
24. Numerical operations
Module-I of Section-II of Paper-I (SSC CGL English Language And Comprehension Syllabus):
1. Spot the error
2. Fill in the blanks
3. Synonyms
4. Antonyms
5. Spelling/ detecting misspelled words
6. Idioms & phrases
7. One-word substitution
8. Improvement of sentences
9. Active/ passive voice of verbs
10. Conversion into Direct/Indirect narration
11. Shuffling of sentence parts
12. Shuffling of sentences in a passage
13. Cloze passage
14. Comprehension passage
To test comprehension, three or more paragraphs will be given and questions based on those will be asked. At least one paragraph should be a simple one based on a book or a story and the other two
paragraphs should be on current affairs, based on a report or an editorial.
Module II of Section II of Paper-I (SSC CGL General Awareness Syllabus)
1. India and its neighboring countries especially about History, Culture, Geography, Economic Scene, General Policy & Scientific Research
2. Science
3. Current Affairs
4. Books and Authors
5. Sports
6. Important Schemes
7. Important Days
8. Portfolio
9. People in News
Module-I of Section III of Paper-I (SSC CGL Computer Proficiency Syllabus)
Computer Basics: Organization of a computer, Central Processing Unit (CPU), Input/ Output devices, computer memory, memory organization, back- up devices, PORTs, Windows Explorer. Keyboard shortcuts.
Software: Windows Operating system including basics of Microsoft Office like MS Word, MS Excel, and PowerPoint, etc.
Working with the Internet and e-mails: Web Browsing & Searching, Downloading & Uploading, Managing an E-mail Account, e-Banking
Basics of networking and cyber security: Networking devices and protocols, Network and information security threats (like hacking, virus, worms, Trojans, etc.), and preventive measures.
Paper II (Statistics)
1. Collection, Classification, and Presentation of Statistical Data –Primary and Secondary data, Methods of data collection; Tabulation of data; Graphs and charts; Frequency distributions;
Diagrammatic presentation of frequency distributions.
2. Measures of Central Tendency – Common measures of central tendency – mean median and mode; Partition values- quartiles, deciles, percentiles.
3. Measures of Dispersion- Common measures of dispersion – range, quartile deviations, mean deviation, and standard deviation; Measures of relative dispersion.
4. Moments, Skewness, and Kurtosis – Different types of moments and their relationship; the meaning of skewness and kurtosis; different measures of skewness and kurtosis.
5. Correlation and Regression – Scatter diagram; simple correlation coefficient; simple regression lines; Spearman’s rank correlation; Measures of association of attributes; Multiple regression;
Multiple and partial correlations (For three variables only).
6. Probability Theory – Meaning of probability; Different definitions of probability; Conditional probability; Compound probability; Independent events; Bayes‟ theorem.
7. Random Variable and Probability Distributions – Random variable; Probability functions; Expectation and Variance of a random variable; Higher moments of a random variable; Binomial, Poisson,
Normal and Exponential distributions; Joint distribution of two random variables (discrete).
8. Sampling Theory – Concept of population and sample; Parameter and statistic, Sampling and non-sampling errors; Probability and nonprobability sampling techniques(simple random sampling, stratified
sampling, multistage sampling, multiphase sampling, cluster sampling, systematic sampling, purposive sampling, convenience sampling and quota sampling); Sampling distribution(statement only); Sample
size decisions.
9. Statistical Inference – Point estimation and interval estimation, Properties of a good estimator, Methods of estimation (Moments method, Maximum likelihood method, Least squares method), Testing
of hypothesis, Basic concept of testing, Small sample and large sample tests, Tests based on Z, t, Chi-square, and F statistic, Confidence intervals.
10. Analysis of Variance – Analysis of one-way classified data and two-way classified data.
11. Time Series Analysis – Components of time series, Determination of trend component by different methods, Measurement of seasonal variation by different methods.
12. Index Numbers – Meaning of Index Numbers, Problems in the construction of index numbers, Types of index numbers, Different formulae, Base shifting and splicing of index numbers, Cost of living
Index Numbers, and Uses of Index Numbers.
Paper-III (SSC CGL General Studies-Finance and Economics Syllabus)
This section of the SSC CGL Tier 2 Syllabus is divided into 2 parts Finance & Accounts and Economics & Governance for which the sub-topics are detailed below-
1.1 Financial Accounting
1. Nature and scope
2. Limitations of Financial Accounting
3. Basic Concepts and Conventions
4. Generally Accepted Accounting Principles
1.2 Basic Concepts of Accounting
1. Single and double entry
2. Books of Original Entry
3. Bank Reconciliation
4. Journal, ledgers
5. Trial Balance
6. Rectification of Errors
7. Manufacturing
8. Trading
9. Profit & Loss Appropriation Accounts
10. Balance Sheet
11. The distinction between Capital and Revenue Expenditure
12. Depreciation Accounting
13. Valuation of Inventories
14. Non-profit organization’s Accounts
15. Receipts and Payments and Income & Expenditure Accounts
16. Bills of Exchange
17. Self Balancing Ledgers
Part B: SSC CGL Economics and Governance Syllabus-(120 marks):
2.1 Comptroller & Auditor General of India- Constitutional Provisions, Role, and Responsibility.
2.2 Finance Commission-Role and functions.
2.3 Basic Concept of Economics and Introduction to Micro Economics
1. Definition
2. Scope and Nature of Economics
3. Methods of economic study
4. Central problems of an economy
5. Production possibilities curve
2.4 Theory of Demand and Supply
1. Meaning and determinants of demand
2. Law of Demand and Elasticity of Demand
3. Price
4. Income and cross elasticity
5. Theory of consumer behavior
6. Marshallian approach and Indifference curve approach
7. Meaning and determinants of supply
8. Law of supply
9. The elasticity of Supply
2.5 Theory of Production and Cost
1. Meaning and Factors of Production
2. Laws of production- Law of variable proportions and Laws of returns to scale.
2.6 Forms of Market and price determination in different markets
1. Various forms of markets-Perfect Competition
2. Monopoly
3. Monopolistic Competition
4. Oligopoly
5. Price determination in these markets.
2.7 Indian Economy:
2.7.1 Nature of the Indian Economy Role of different sectors, Role of Agriculture, Industry and Services-their problems and growth.
2.7.2 National Income of India-Concepts of national income, Different Methods of measuring national income.
2.7.3 Population-Its size, rate of growth, and its implication on economic growth.
2.7.4 Poverty and unemployment- Absolute and relative poverty, types, causes, and incidence of unemployment.
2.7.5 Infrastructure-Energy, Transportation, Communication.
2.8 Economic Reforms in India
1. Economic reforms since 1991
2. Liberalization
3. Privatization
4. Globalization
5. Disinvestment
2.9 Money and Banking:
2.9.1 Monetary/ Fiscal policy- Role and functions of Reserve Bank of India; functions of commercial Banks/RRB/Payment Banks.
2.9.2 Budget and Fiscal Deficits and Balance of Payments.
2.9.3 Fiscal Responsibility and Budget Management Act, 2003.
2.10 Role of Information Technology in Governance.
Note: Questions in Module-I of Section- I of Paper-I (Mathematical Abilities) will be of Matriculation Level, in Module-I of Section- II of Paper-I (English Language and Comprehension) of 10+2 Level
and in Paper-II and
Paper-III of Graduation Level.
Module II of Section III of Paper-I (SSC CGL Computer Proficiency Syllabus)
DEST – Module II of Section III of Paper-I will include conducting a Data Entry Speed Test (DEST) for 15 minutes in Session II on the same day.
The “Data Entry Speed Test” (DEST) Skill Test will be conducted for a passage of about 2000 (two thousand) key depressions for 15 (fifteen) minutes. Detailed instructions regarding the Skill Test
be provided by the Regional Offices of the Commission.
Candidates are required to type 2000 words in 15 minutes on a computer in English. This test is conducted to check a candidate’s typing skills. Candidates are given an article in English which they
have to type on Computer. For the post of Tax Assistant (Central Excise and Income Tax), the DEST Exam through the SSC CGL 2022 exam is conducted to check the typing speed of the candidate.
DEST will be mandatory for all posts. however, it will be qualifying in nature | {"url":"https://gkquestionsguru.com/ssc-cgl-2023-syllabus/","timestamp":"2024-11-05T00:24:18Z","content_type":"text/html","content_length":"107581","record_id":"<urn:uuid:92eaa00f-c48e-477a-bc58-2af88242bf02>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00237.warc.gz"} |
Pipe Equations Formulas Design Calculator
Online Web Apps, Rich Internet Application, Technical Tools, Specifications, How to Guides, Training, Applications, Examples, Tutorials, Reviews, Answers, Test Review Resources, Analysis, Homework
Solutions, Worksheets, Help, Data and Information for Engineers, Technicians, Teachers, Tutors, Researchers, K-12 Education, College and High School Students, Science Fair Projects and Scientists
By Jimmy Raymond
Contact: aj@ajdesigner.com
Privacy Policy, Disclaimer and Terms
Copyright 2002-2015 | {"url":"https://www.ajdesigner.com/phppipesoil/soil_load_equation.php","timestamp":"2024-11-04T08:12:42Z","content_type":"text/html","content_length":"22211","record_id":"<urn:uuid:83954f68-419f-42a1-a0af-2c0c3d4538bd>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00338.warc.gz"} |
2021 Fall Meeting of the APS Division of Nuclear Physics
Bulletin of the American Physical Society
2021 Fall Meeting of the APS Division of Nuclear Physics
Volume 66, Number 8
Monday–Thursday, October 11–14, 2021; Virtual; Eastern Daylight Time
Session LM: Nuclear Theory VI Hide Abstracts
Chair: Charlotte Elster, Ohio University
Room: White Hill
Wednesday, LM.00001: Nuclear EDFs: Particle vibration coupling in superfluid nuclei with axial deformation
October Yinu Zhang, Elena Litvinova
13, 2021
2:00PM - The nuclear density functional theory (DFT) has demonstrated the ability to provide a fairly accurate description of nuclear ground-state properties and low-energy collective excitations
2:12PM across the nuclear chart. In this work, we elaborate on the particle-vibration coupling (PVC), which is associated with the leading correlations beyond the mean-field approximation in
strongly-coupled many-body fermionic systems treated within a consistent and systematic framework of the equation of motion method. For deformed nuclei, the conventional solution of the
quasiparticle random phase approximation (QRPA), which provides the major input for the PVC, requires prohibitive numerical efforts. By linking the notion of the quasiparticle-phonon
vertex to the variation of the Bogoliubov's Hamiltonian, we show that the recently developed finite-amplitude method (FAM) can be efficiently employed to compute the PVC vertices within
the FAM-QRPA for deformed nuclei. To illustrate the validity of the particle vibration coupling to superfluid nuclei in axial deformation, the calculations based on the relativistic
density-dependent point-coupling Lagrangian are performed for the single-nucleon states as well as the giant resonances in medium-mass and heavy nuclei with axial deformations. The results
show considerable improvement compare with experimental data for axially-deformed nuclei.
Wednesday, LM.00002: Eigenvector Continuation for Resonance States
October Nuwan Yapa, Sebastian Koenig
13, 2021
2:12PM - Eigenvector continuation (EC) has emerged as an intriguing method to yield approximate solutions for computationally expensive eigenvalue problems with great speed and accuracy. With EC,
2:24PM the essence of a quantum system is "learned" through the construction of a highly effective (non-orthogonal) basis, leading to a variational calculation of the states of interest with
rapid convergence. Extracting resonance energies of few-body systems is of great importance in nuclear physics, but it continues to pose challenges due to the large computational
complexity involved. In this work we study EC as an option to facilitate such calculations. To that end, we use both finite-volume techniques, where resonances are manifest as avoided
crossings of energy levels, as well as direct studies in momentum space, where resonances can be identified after analytic continuation of the Schrödinger equation. In both cases we find
that EC makes it possible to extrapolate trajectories of resonance states. In particular, we discuss the possibility of predicting resonances based on bound training states alone, tracing
their transition into the continuum as a parameter in the Hamiltonian is varied.
Wednesday, LM.00003: Shell Model Monte Carlo Studies of Collectivity in Heavy Nuclei
October Sohan Vartak, Yoram Alhassid, Marco Bonett-Matiz
13, 2021
2:24PM - A microscopic description of the crossover from vibrational to rotational collectivity in heavy nuclei in the configuration-interaction shell model approach is beyond the reach of
2:36PM conventional diagonalization methods due to combinatorial growth of the many-particle model space with the number of valence nucleons and/or valence single-particle orbitals. The shell
model Monte Carlo (SMMC) method is viable in such model spaces, and has been successfully applied to calculate thermal and ground-state observables. Recently, a method has been developed
that provides access to spectral information encoded in a generalized eigenvalue problem satisfied by imaginary-time response matrices of one-body densities [1]. We have validated the
method in an sd-shell nucleus whose excitation energies can be calculated exactly using conventional diagonalization techniques. Application of this method to chains of heavy lanthanide
isotopes enables the calculation of a few energy levels for each spin and parity, yielding direct spectral evidence of the crossover from vibrational to rotational collectivity. The
generalized eigenvalue problem also encodes information about one-body transition densities, and we discuss extensions of the method to extract this information.
Wednesday, LM.00004: Global consequences of removing parametric correlations in covariant energy density functionals.
October Ahmad Taninah, Anatoli Afanasjev
13, 2021
2:36PM - Covariant density functional theory (CDFT) is one of the modern theoretical tools for the description of finite nuclei and neutron stars. Its performance is defined by underlying
2:48PM covariant energy density functionals (CEDFs) which depend on several parameters. The analysis of the major classes of CEDFs reveals the existence of parametric correlations between these
parameters [1,2]. The removal of these correlations reduces the number of independent parameters to five or six depending on the underlying functional structure. However, this analysis is
based on the fitting protocols which employ only spherical nuclei. In the present contribution, we investigated the consequences of the removal of parametric correlations to full nuclear
landscape for which experimental data are available [3]. It is shown that the removal of parametric correlations does not lead to a degradation of the performance of CEDFs on a global
scale. Moreover, this study also reveals the need to include information on deformed nuclei for the improvement of fitting protocols. In addition, the asymptotic behavior of the basis
truncation on the physical observables of interest has been analyzed. It also reveals that for a comparable accuracy description a larger basis is needed in deformed nuclei as compared
with spherical ones.
Wednesday, LM.00005: Extremely proton rich nuclei: rotation-induced proton halos and extension of nuclear landscape beyond spin zero limit.
October saja A Teeti, Ahmad Taninah, Anatoli Afanasjev
13, 2021
2:48PM - Recent investigations reveal a number of physical mechanisms by which it is
possible to extend the nuclear landscape beyond spin zero limit. One of these
is related to so-called birth of particle-bound rotational bands in neutron-rich
nuclei which has been first suggested in Ref. [1]. In this mechanism, strong
Coriolis interaction acting on high-$j$ orbitals transforms particle-unbound
(resonance) nucleonic configurations into particle-bound ones with increasing
angular momentum. A similar mechanism is active also in the nuclei in the
vicinity of proton drip line [2] but it is modified the presence of the Coulomb
barrier. As a result, particle-unbound part of the band will have discrete
rotational states which can decay by proton emission.
A systematic investigation of this phenomenon has been performed in proton rich even-even
$Z=4-36$ nuclei within the framework of cranked relativistic mean field theory with the
goals to find the general features of this phenomenon and the best candidates
for experimental observations [3]. One of interesting predictions is a new phenomenon
of rotation-induced proton halos which is active in some nucleonic configurations.
Wednesday, LM.00006: Toward emulating nuclear reactions using eigenvector continuation.
October Pablo G Giuliani, Christian Drischler, Amy E Lovell, Michael Quinonez, Filomena Nunes
13, 2021
3:00PM - We construct an efficient emulator for two-body scattering using the generalized Kohn variational principle and trial wave functions derived from eigenvector continuation. Our emulator
3:12PM simultaneously applies an array of Kohn variational principles with different asymptotic boundary conditions, which allows for detection and removal of spurious singularities known as Kohn
anomalies. We then perform a Bayesian analysis of elastic scattering of neutrons off $^{40}$Ca and $^{208}$Pb nuclei using realistic optical potentials. The emulator’s high accuracy and
computational speed enable rigorous uncertainty quantification for improving optical potentials in the FRIB era.
Wednesday, LM.00007: Extending the limits of nuclear landscape via new physical mechanisms
October Anatoli Afanasjev, Sylvester E Agbemava, Ahmad Taninah, saja A Teeti
13, 2021
3:12PM - The detailed investigation of new physical mechanisms which allow to extend the boundaries of
the nuclear landscape beyond the traditional limits has been performed over recent years. In the region
of hyperheavy (Z>126) nuclei, the transition from ellipsoid-like nuclear shapes to toroidal ones
provides a substantial increase of nuclear landscape [1-3]. Rotational excitations in the nuclei
near proton and neutron drip lines provide an alternative mechanism for an extension of the nuclear
landscape beyond the limits defined at spin zero [4,5]. In both cases, the collective coordinates related
to nuclear shapes play an important role in extending nuclear landscape. In hyperheavy nuclei, they
drive the nuclear systems from ellipsoidal-like to toroidal shapes. In rotating nuclei, triggered by
particle-hole excitations they transform the system from spherical or normal deformed ground
states to extremely elongated (super-, hyper and mega-deformed) shapes or rotation-induced
proton halos at high spins. Rotational frequency acts as an additional collective degree of freedom
(coordinate) in rotating nuclei. At the microscopic level, the impact of these collective coordinates drives
the single-particle orbitals, which are otherwise located at positive energy in a given nucleonic
configuration, below the continuum threshold. As a consequence, it allows to extend the limits of nuclear
landscape beyond traditional limits (for example, beyond spin-zero drip lines in rotating nuclei). The
similarities and differences of these mechanisms will be discussed. The roles of underlying single-particle
and shell structures will be analyzed.
Wednesday, LM.00008: A Hyperspherical Treatment of Reaction Pathways in Few-Nucleon Systems
October Michael D Higgins, Chris H Greene
13, 2021
3:24PM - The adiabatic hyperspherical representation has been extensively applied to few–body systems, in which hyperspherical potentials and couplings describe all possible reaction pathways on an
3:36PM equal footing through an adiabatic collective coordinate, the hyperradius. In addition to providing qualitative insight about the pathways controlling key reaction and resonance phenomena,
the hyperspherical potentials plus non–adiabatic couplings can provide a quantitative description of reaction rates, bound states and resonances. In this work, reaction pathways for the
three–body nnp, npp (J^π=1/2^+,T=1/2) and four–body nnpp (J^π=0^+,T=0) systems are visualized through a spectrum of hyperspherical potential curves, which show the different ways these
interacting systems can fragment into bound and continuum channels. Calculations are preformed using adiabatic hyperspherical methods that implement an explicitly–correlated Gaussian basis
(Suzuki et. al., Few Body Syst. (2008) 42: 33–72). Two different nucleon–nucleon (NN) interactions are considered, the Minnesota NN interaction, and the realistic AV8' NN interaction with
a spin–independent three–nucleon force (Hiyama, E. et al., Phys. Rev. C. 70, 031001(2004)). In addition, three– and four–nucleon binding energies and resonances are computed.
Wednesday, LM.00009: Real time dynamics of gauge field theory with truncated Hamiltonian methods
October Ivan Kukuljan, Gábor Takács, Spyros Sotiriadis
13, 2021
3:36PM - Truncated Hamiltonian methods (THM) have been studied as a powerful complement to lattice methods for the study of strongly coupled quantum field theory (QFT). THM have been used to
3:48PM compute spectra of the models, properties of bound states, symmetry breaking, (higher order) correlation functions, quantum chaos and particularly excel at real time evolution, a task
difficult for Monte Carlo methods. They do not require a discretization of space-time and have been successfully applied to systems in 1 and 2 spatial dimensions.
Recently we have used THM to study real time dynamics of topologically nontrivial theories and gauge field theory in 1+1D. This has lead to an observation of a new nonequilibrium effect in
strongly coupled QFT: quantum quenches, sudden changes of model parameters, in QFT with topological excitations lead to long range order in such theories, a phenomenon tightly connected to
In my talk, I will give an introduction to the THM methods, focus on the implementations for gauge theories, real time dynamics of the massive Schwinger model, and finally discuss the
newly observed nonequilibrium emergence of long range order.
Engage My APS Information for The American Physical Society (APS) is a non-profit membership organization working to advance the knowledge of physics.
Become an APS Member Renew Membership Librarians
Submit a Meeting Abstract Join an APS Unit Authors
Submit a Manuscript Get My Member Number Referees
Find a Journal Article Update Contact Information Media
Donate to APS Students
© 2024 American Physical Society | All rights reserved | Terms of Use | Contact Us
Headquarters 1 Physics Ellipse, College Park, MD 20740-3844 (301) 209-3200
Editorial Office 100 Motor Pkwy, Suite 110, Hauppauge, NY 11788 (631) 591-4000
Office of Public Affairs 529 14th St NW, Suite 1050, Washington, D.C. 20045-2001 (202) 662-8700 | {"url":"https://meetings.aps.org/Meeting/DNP21/Session/LM?showAbstract","timestamp":"2024-11-06T01:39:28Z","content_type":"text/html","content_length":"28650","record_id":"<urn:uuid:4c06911b-69c4-4b37-99f4-606924c89375>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00203.warc.gz"} |
Ilyashenko algebras based on definable monomials: the construction (inductive step)
Let $M \subseteq \H^{>0}$ be a pure scale on standard power domains. In this post, I gave the base step of the construction of a qaa field $(\F,L,T)$ as claimed here. The goal of this post is to
finish this construction.
Step 0.5: apply a $\log$-shift to the qaa field $(\F_0,L_0,T_0)$, that is, set $$\F’_1 := \F_0 \circ \log,$$ let $L’_1:= \langle x \rangle^\times$ be the multiplicative $\RR$-vector space generated
by $x$, and define $T’_1:\F’_1 \into L’_1$ by $$T’_1(f \circ \log):= (T_0 f) \circ \log, \quad\text{for } f \in \F_0.$$
Since $\blog$ maps every standard power domain into every standard power domain (in the sense of germs at $\infty$ of domains in the right half-plane), it follows that $\left(\F’_1, L’_1, T’_1\right)
$ is a strong qaa field.
All germs in $\F’_1$ are polynomially bounded. Thus, if $m \in L_0$ is small (which means that $m = \exp^{-r}$ for some $r > 0$) and $a \in \F’_1$, then the holomorphic extension $\aa\mm$ is bounded
on some standard power domain.
To iterate the construction, I now need to replace the real coefficients in the definition of (strong) asymptotic expansion by coefficients in $\C$ that have slower than exponential growth. In the
setting of my general $M$, this means working with the following set of germs: set $$\C_M := \set{f \in \C:\ \frac1m \prec f \prec m \text{ for all large } m \in M}.$$ In other words, the set $\C_M$
consists of all germs whose comparability class is slower than that of any $m \in M$.
1. $\RR \subseteq \C_{L} \subseteq \cdots \subseteq \C_{L_1} \subseteq \C_{L_0}$.
2. $L’_1 \subseteq \C_{L_0}$.
3. $\F’_1 \subseteq \C_{L_0}$.
Show that, for $a \in \F’_1$, $n \in L_0$ and any standard power domain $\Omega$, we have $\aa \prec_\Omega \nn$.
1. A series $F \in \Gs{\C_M}{M}$ is an $M$-generalized power series if there exist $k \in \NN$, small $m_0, \dots, m_k \in M$, a generalized power series $G \in \As{\C_M}{X_0^*, \dots, X_k^*}$ with
natural support and a (not necessarily small) $m \in M$ such that $$F = m G(m_0, \dots, m_k).$$
2. Let $f \in \C$ and $F = \sum a_m m \in \Gs{\C_M}{M}$ have $M$-natural support. The germ $f$ has strong asymptotic expansion $F$ if there exists a standard power domain $\Omega$ such that
1. $f$ has holomorphic extension $\ff$ on $\Omega$;
2. each $a_m$ has holomorphic extension $\aa_m$ on $\Omega$ such that $\aa_m \prec_\Omega \nn$ for every $n \in M$. In particular, for every $n \in M$ the truncation $F_n$ has a holomorphic
extension $\FF_n$ on $\Omega$; and
3. the condition $\ff – \FF_n \prec_\Omega \nn$ holds for each $n \in M$.
As before, we get:
1. The set $\C(M)^*_{\st}$ of all $f \in \C$ that have a strong asymptotic expansion in $\Gs{\C_M}{M}$ is an $\RR$-algebra.
2. Every $f \in \C(M)^*_{\st}$ has exactly one asymptotic expansion $\tau_M(f)$ in $\Gs{\C_M}{M}$, and the map $\tau_M:\C(M)^*_{\st} \into \Gs{\C_M}{M}$ is an injective $\RR$-algebra homomorphism.
I am now ready for
Step 1: Set $$\A_1 := \set{f \in \C(L_0)^*_{\st}:\ f \text{ is bounded and } \tau_mf \in \As{\F’_1}{L_0}}$$ and, for $f \in \A_1$ with $\tau f = \sum a_m m$, set $$T_1 f:= \sum \left(T’_1 a_m\right)
m \quad \in \Gs{R}{L_1}.$$
The triple $(\A_1, L_1, T_1)$ is a strong qaa algebra.
(Show proof)
The map $\sigma:\Gs{\F’_{1}}{L_0} \into \Gs{\RR}{L_1}$ defined by $$\sigma\left( \sum f_r \exp^{-r} \right) := \sum (T’_{1} f_r) \exp^{-r}$$ is an $\RR$-algebra homomorphism, and it is injective
because $T’_{1}$ is injective. Since $T_{1} = \sigma \circ \tau_1$, it follows that $T_{1}$ is an injective $\RR$-algebra homomorphism.
Let now $f \in \A_1$ be such that $$T_1 f = \sum_{m \in L} a_m m \quad\text{and}\quad \tau_1 f = \sum_{r \ge 0} f_r \exp^{-r},$$ and let $n \in L$; we show there exists $g \in \A_1$ such that $T_1 g
= (T_1 f)_n$. Considering $n$ as a function $n:\{-1,0\} \into \RR$ such that $n = \exp^{-n(-1)} x^{-n(0)}$, set $r:= n(-1)$ and $n’:= x^{-n(0)} \in L_1’$, so that $n = n’\exp^{-r}$ and
\left(T_1f\right)_n = \sum_{m(-1) > n(-1)} a_m m + \left(T’_1 f_r\right)_{n’} \exp^{-r},
and let $\Omega$ be a strong asymptotic expansion domain of $f$. Note that each $f_s \exp^{-s}$ has a bounded holomorphic extension on $\Omega$. Since $$\sigma^{-1}\left(\sum_{m(-1) > n(-1)} a_m m\
right) = \sum_{s \lt r} f_s \exp^{-s}$$ has finite support in $\Gs{\F’_1}{L_0}$, it follows that $$g_1:= \sum_{s \lt r} f_s \exp^{-s}$$ belongs to $\A_1$ and satisfies $\tau_1 g_1 = g_1$ and $T_1 g_1
= \sum_{m(-1) > n(-1)} a_m m$. On the other hand, by the inductive hypothesis, there exists $h \in \F’_1$ such that $T’_1 h = \left(T’_1 f_r\right)_{n’}$. Hence $h \exp^{-r} \in \A_1$ and, by
definition of $T_1$, we obtain $T_1(h \exp^{-r}) = \left(T’_1 f_r\right)_{n’} \exp^{-r}$. Therefore, we can take $g:= g_1+h \exp ^{-r}$.
Finally, after shrinking $\Omega$ if necessary, we may assume that $\Omega$ is also a strong asymptotic expansion domain of $g$; we now claim that $\ff-\gg = o(\nn)$ in $\Omega$, which then proves
the proposition. By the inductive hypothesis, we have $\ff_r – \hh = o(\nn’)$ in $\Omega$; therefore,
\begin{equation} \label{asym_1}
\ff_r \bexp^{-r} – \hh \bexp^{-r} = o(\nn) \quad\text{ in } \Omega.
On the other hand, let $r’:= \min\set{s \in \RR:\ s > r \text{ and } f_r \ne 0}$. Then, by hypothesis, we have
\begin{equation} \label{asym_2}
\ff – \gg_1 – \ff_r \bexp^{-r} = o\left(\bexp^{-\frac{r+r’}2}\right) \quad\text{ in } \Omega.
Since $\bexp^{-\frac{r+r’}2} = o(\nn)$ in $\Omega$, the proposition follows. $\qed$
Finally, let $\F_1$ be the field of fractions of $\A_1$ and extend $T_1$ accordingly; then $(\F_1,L_1,T_1)$, and hence $(\F_1, L, T_1)$, is a strong qaa field.
1. Ilyashenko’s class $\A$ of almost regular germs is contained in $\F_1$.
2. We have $\F_0 \subseteq \F_1$ and $T_0 = T_1\rest{\F_0}$.
Continuing this iteration, we obtain qaa fields $(\F_i,L,T_i)$ such that $\F_i \subseteq \F_{i+1}$ and $T_i = T_{i+1} \rest{\F_i}$, for $i \in \NN$. So we set $$\F:= \bigcup_{i \in \NN} \F_i$$ and
let $T$ be the common extension of all $T_i$ to $\F$. Then $(\F,L,T)$ is a qaa field, and in addition we have:
$\F$ is closed under differentiation; in particular, $\F$ is a Hardy field.
You must be logged in to post a comment.
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://math.mcmaster.ca/~speisseg/blog/?p=3695","timestamp":"2024-11-13T08:30:32Z","content_type":"text/html","content_length":"68417","record_id":"<urn:uuid:59c6cfa1-988b-48d2-b950-ce8d1763280f>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00721.warc.gz"} |
Discrete GWO Optimized Data Aggregation for Reducing Transmission Rate in IoT
Computer Systems Science & Engineering
Discrete GWO Optimized Data Aggregation for Reducing Transmission Rate in IoT
1Department of Computer Science and Engineering, Sri Krishna College of Technology, Coimbatore, 641042, India
2Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, Hradec Králové, 50003, Czech Republic
3Department of Computer Science and Engineering, Soonchunhyang University, Asan, 31538, Korea
4Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt
5Department of Computational Mathematics, Science, and Engineering (CMSE), Michigan State University, East Lansing, MI, 48824, USA
*Corresponding Author: Yunyoung Nam. Email: ynam@sch.ac.kr
Received: 26 November 2021; Accepted: 11 January 2022
Abstract: The conventional hospital environment is transformed into digital transformation that focuses on patient centric remote approach through advanced technologies. Early diagnosis of many
diseases will improve the patient life. The cost of health care systems is reduced due to the use of advanced technologies such as Internet of Things (IoT), Wireless Sensor Networks (WSN), Embedded
systems, Deep learning approaches and Optimization and aggregation methods. The data generated through these technologies will demand the bandwidth, data rate, latency of the network. In this
proposed work, efficient discrete grey wolf optimization (DGWO) based data aggregation scheme using Elliptic curve Elgamal with Message Authentication code (ECEMAC) has been used to aggregate the
parameters generated from the wearable sensor devices of the patient. The nodes that are far away from edge node will forward the data to its neighbor cluster head using DGWO. Aggregation scheme will
reduce the number of transmissions over the network. The aggregated data are preprocessed at edge node to remove the noise for better diagnosis. Edge node will reduce the overhead of cloud server.
The aggregated data are forward to cloud server for central storage and diagnosis. This proposed smart diagnosis will reduce the transmission cost through aggregation scheme which will reduce the
energy of the system. Energy cost for proposed system for 300 nodes is 0.34μJ. Various energy cost of existing approaches such as secure privacy preserving data aggregation scheme (SPPDA), concealed
data aggregation scheme for multiple application (CDAMA) and secure aggregation scheme (ASAS) are 1.3 μJ, 0.81 μJ and 0.51 μJ respectively. The optimization approaches and encryption method will
ensure the data privacy.
Keywords: Discrete grey wolf optimization; data aggregation; cloud computing; IoT; WSN; smart healthcare; elliptic curve elgamal; energy optimization
Evolution of IoT and digital technologies makes the healthcare monitoring as effective and efficient for diagnosing and real time monitoring. The data generated from these devices are huge and
processing is also still a challenging task [1]. Intelligent sensor with IoT plays a vital role in the field of healthcare, agriculture, city, transportation, and industry [2]. To collect the patient
health parameters such as blood glucose level, heart rate, blood pressure etc., IoT sensors are used [3]. These IoT sensor devices gather the information and transmit to cloud server for processing [
4,5]. This network ensures the security of the patient data and process to provide the diagnosis result in case of emergency scenario [6]. An Anonymous and Secure Aggregation Scheme (ASAS) [7]
proposed pseudonyms to protect the nodes identity. To protect the data integrity, homomorphic encryption is used. This method reduces the bandwidth utilization but due to redundant data transmission,
computational cost and communication cost are increased. EHDA (Efficient health data aggregation) method [8] provides smart nodes with secure communication. Aggregate node (AN) use message receiving
method for aggregation on compressed data received from smart sensor nodes. Fog node (FN) employs message receiving algorithm and decrypt the aggregated data for analysis. The contribution of the
proposed work is as follows:
• In IoT wearable sensor network, rather than each node sending data to cloud centre, data are aggregated by the aggregator node which will then forward the aggregated data to cloud server through
edge node. Due to this the bandwidth, network delay and number of transmissions are effectively utilized
• Each IoT sensor device encrypt the data using elliptic curve elgamal (ECE) and Calculate media access control (MAC) address for each key. This encrypted data are communicated to aggregator node.
• Aggregators add all the encrypted data using XOR with minimum alveolar conMAC. And send this information of aggregated data with MAC to edge server.
• AN nearer to edge node can send the aggregated data directly to the edge server. AN which placed far away from the edge server can communicate to its neighbor AN for data transmission. The neighbor
AN can be find using efficient searching approach called Discrete GWO.
• Edge servers decrypt the data and done its local processing and transmit to cloud server for analysis.
Remaining section of this paper is as follows: Section 2 discusses about the related work. Section 3 proposed an efficient scheme for data aggregation and analysis. Section 4 discusses about the
simulated results and evaluation and Section 5 concludes the proposed work with future directions.
The recent research related to data aggregation schemes are listed in Tab. 1.
The work describes about cryptographic technique for securing IoMT network. They used Rivest chiper and elliptical curve digital signature algorithm with secure hashing for protecting medical data
user. it strengthens the health care but number of transmissions is higher which increases the communication cost of the network. discuss about IoT in industrial application. The industrial data is
transferred time to time were communication cost will be hiked. discusses about secured IoT and importance IoT in wireless networks. But they are lagging in reducing number of transmissions at a
time. This problem highly focused in our paper by data aggregation technique.
3 Proposed Discrete GWO Methodology
In this section, Fig. 1.Illustrates the proposed system overview of efficient Edge IoT enabled smart healthcare. This system supports patient health monitoring at distant location remotely. This
system involves three kind of nodes such as Mobile node (MN), Aggregator node (AN) and Edge node (EN). Each wearable IoT sensor devices are the mobile nodes. Large number of these IoT device sensor
nodes forms a cluster. Each cluster there is one node called AN aggregate all the data generated by MN and forward it into the base station/cloud centre through edge node. The data generated by MN
are compressed and encrypted using ECE and add MAC to ensure the security of the data. This encrypted data is aggregated by AN using XOR operation of MAC with key. The AN nearer to edge node will
transfer the aggregated data to edge server directly. In the case of AN which placed far wary from the edge server can communicate to its neighbor cluster AN for-message transmission. Neighbor AN can
be find using optimization algorithm called Discrete GWO which will efficiently find the nearest neighbor for message transmission. Edge node received aggregated data from each cluster AN and decrypt
the data for local processing to reduce the cloud server overhead then forward it into cloud server for central storage and analysis. Cloud data centre process the received data and transfer the
analyzed results to respective authority for early diagnosis.
3.1 Data Encryption and Aggregation Using ECE-MAC
Wearable IoT sensor devices collect the health parameter and transmit it into AN. Data compression and ECE-MAC encryption is performed for effective data transmission between AN and EN. The data
gathered by the sensor nodes are aggregated and compressed using the compression method. Aggregated data of health parameter is represented as MNagg={HP1, HP2, …HPn} where n∈[1, N], N-number of
SN. This compress data is encrypted using ECE encryption. Elliptic Curve based Elgamal based privacy homomorphism encryption is an asymmetric cryptography approach. in this scheme, the encryption key
is publicly known for AN and EN. The message is mapped into elliptic curve. The message is multiplied with the generator of EC. In decryption demap the point again to m. For mapping, brute force
computation is used. ECE algorithm is as follows:
Message authentication code (MAC) is a cryptographic construction that is used to identify the falsification of the messages. It is represented using one way hash function, secret key generated by
ECE and the message of length t. This MAC will increase the integrity and the receiver needs to recalculate the MAC code with its decryption to ensures the authenticity using the Eq. (5).
MAC=Hashk(t) (5)
This MAC codes are aggregated with bitwise XOR operations, and the result enable authentic verification. The aggregated MAC is represented in Eq. (6)
MACagg=MAC1⊗MAC2⊗,…⊗MACn (6)
Data integrity is preserved during aggregation by each node MAC is combined by XOR operation. The aggregated MAC is verified by EN and cloud server. The aggregated data are decrypted by EN based on
the key and aggregated MAC. This proposed scheme will ensure the data confidentiality, data integrity and data authentication. Data confidentiality is ensured through secret key shared by the sensor
node and AN. There is no adversary violation on aggregated data.
This section discusses about the technique to find the nearest AN for message transmit if there is no peer-to-peer communication between edge node and AN. To find the best possible solution, this
work implements grey wolf optimization approach with discrete weight. Grey wolfs are the advanced optimization algorithm depends on the hunting nature of the wolf that is ready to catch their prey
because all are stayed in the crowd that are organized carefully [22]. The four levels of GWO leadership hierarchy are alpha (α), beta (β), delta (δ), and omega (ω). α is the male or female whose are
the leaders for making decisions. β will help alpha to make decisions. δ are the guides or elders or hunters. ω follow all wolves and it will be outlasting solution whereas alpha, beta and omega are
the first, second and third best solutions.
In this proposed work, for feature selection the labeled data are separated into two sets of positive and negative represented as matrix of PM and NM. Each row of these matrix represented as the
recommended medical tests carried and each column represents that the result of these medical tests [23]. If the deviation between these two values is lower than threshold then the value obtained
from the record will falls onto any of the label. So, the attributes are insignificant to determine the label. The values with no deviation between positive and negative label are stated as
suboptimal one. Hence, need to select the attributes that representing the diversity observed in both positive and negative label.
In order to find the optimal attributes, this proposed work use DGWO to identify the diversity between the attributes and corresponding vector values of both positive and negative label. The aim of
this phase is to select the relevant features that can increase the prediction accuracy. To start the optimization algorithm, initialization with the initial population is the first step. The alpha
(α), beta (β) and delta (δ) are the three best possible solutions in the GWO. The outstanding solution is represented as omega. Throughout the hunting process, the wolf is surrounded by their prey
which is represented as,
D=|E×Xp−Xt| (7)
X(t+1)=|Xp(t)−A×D| (8)
E=2×r1 (9)
A=|2a×r2−a| (10)
Xp–Location of prey, Xt–location of the grey wolf, t- iteration, r1. r2–random vectors in the range [0, 1], a∈[0, 2] represented as,
a=2−t×2Max.noofiterations (11)
The location of the grey wolf with optimal solution can be changed by modifying the coefficient vectors E and A. The alpha (α), beta (β), and delta (δ) are responsible for prey placements. The
remaining wolves change their locations according to the best three solutions as,
X¯1=Xα−A1×Dα (12)
X¯2=Xβ−A2×Dβ (13)
X¯3=Xδ−A3×Dδ (14)
X(t+1)=X¯1+X¯2+X¯33 (15)
Dα=|E1×Xα−X| (16)
Dβ=|E2×Xβ−X| (17)
Dδ=|E3×Xδ−X| (18)
The wolves are updated their position while attacking their prey between their current position and prey position when |A|<1. The optimal attributes are selected using this GWO. Now, the resultant
matrices of PM and NM having the column values as the optimal solution selected by the GWO. These matrices used to find the n optimal AN. Move all the optimal attributes to the set of selected AN
called nAN (number of selected AN set). The new subset of AN are formed by combining the two subsets of AN
x=xi∪jx (19)
For each record
∀i=1n{xi∃xi∈nAN} (20)
∀j=1n{xj∃xj∈nAN∧j≠i} (21)
The empirical property of these selected feature set is discovered for each selected attribute and calculate the positive and negative probability of empirical value as, ∀i=1nAN{nAN∃nAN∈nAN}
Positive label probabilityppnFS=∑i=1nAN{1∃nAN⊆PM(k)}|PM| (22)
Negative label probabilitynpnFS=∑i=1nAN{1∃nAN⊆NM(k)}|NM| (23)
Based on the empirical value assign the ranks to the attributes in ascending order for positive and negative label accordingly. After assigning the ranks, normalization is performed by assigning
local and global weights to positive and negative label as,
pwnAN=1−(pgw×plw) (24)
nwnAN=1−(ngw×nlw) (25)
The positive weights are updated by multiplying the global positive label weight with local weight and subtract the result with 1. And the same for negative label weights are calculated by
multiplying the global and local weight of the negative label attributes and subtract it with one, which is the result value.
Now based on the weight’s updates, the AN are ranked in ascending order. The AN having high values are considered as relevant AN for further message transmission towards EN.
4 Performance Analysis and Discussions
The proposed efficient data aggregation scheme for healthcare applications is simulated using NS2.35. Proposed system is evaluated in terms of functionality, communication cost and energy cost.
Performance of the proposed system is compared with the existing approaches such as secure privacy preserving data aggregation scheme (SPPDA) [24], concealed data aggregation scheme for multiple
application (CDAMA) and secure aggregation scheme (ASAS) [25]. The functionalities such as recoverability, data confidentiality, data authentication, data integrity and false data filtering [26–38]
are used. The results are listed in Tab. 2.
Energy in terms of three level such as SN, AN and EN are considered with the network size varies from 100 to 500 nodes [31–35]. Fig. 2. illustrates the energy cost of proposed and existing approaches
in μJ. Energy cost for proposed system for 300 nodes is 0.34 μJ. Various energy of the existing approaches such as SPPDA, CDAMA and ASAS are 1.3 μJ, 0.81 μJ and 0.51 μJ respectively. From the
evaluation, it is observed that proposed scheme consumes less energy than other existing approaches.
Evaluation based on cost of communication using proposed scheme is shown in Tab. 3. Execution time and energy cost in terms of SN, AN and EN. Execution time includes processing and aggregation time.
Energy cost includes computation and aggregation cost. proposed scheme consumes 7.9mJ for sending a data per bit for 300 nodes and 9.3mJ for receiving data per bit. Performance of the proposed system
cost is compared with existing approaches is shown in Fig. 3.
The evaluation of Fig. 3. shows that the minimum communication cost is obtained by proposed scheme. For 300 devices, the communication cost of proposed scheme is 17.84mJ. Various, existing approaches
obtained 20.43mJ, 17.21mJ, and 18.92mJ for SPPDA, CDAMA and ASAS respectively.
Proposed scheme obtained less energy cost and communication cost compared to other existing approaches. Due to the implementation of evolutionary algorithm to search the neighbor AN, and peer to peer
communication between AN and EN, the messages are efficiently transfer to cloud server for processing. The encryption method used here is send the data confidentially through the network. Hence,
proposed scheme is efficient and effective for transmitting the data collected from the sensor devices to cloud for processing and better diagnosis.
With the interconnection of IoT devices and digital technologies, smart healthcare ensures better human life. To transmit the sensitive health data through WSN is a challenging task. This paper
proposed a secure data transmission of healthcare data through WSN using data aggregation with evolutionary approaches for better diagnosis. This model ensures the security of the data by avoiding
several attacks using ECE-MAC approach. Data gathered from sensor devices are aggregated and transfer to EN which is the mediator of AN and cloud. If there is no peer-to-peer communication means, AN
sends data to its neighbor AN using DGWO approach. This will reduce the transmission cost and energy. This proposed model is secured with less energy of 0.34 μJ to transfer data from 300 devices and
obtained 17.84mJ as communication cost for 300 devices. Compared to conventional approaches such as SPPDA, CDAMA and ASAS, proposed scheme ensures data confidentiality, data integrity, data
authentication and false data filtering. In future, proposed scheme is simulated on particular disease diagnosis with secure data transmission.
Funding Statement: This research was supported by a grant of the Korea Health Technology R & D Project through the Korea Health Industry Development Institute (KHIDI), funded by the Ministry of
Health & Welfare, Republic of Korea (grant number: HI21C1831) and the Soonchunhyang University Research Fund.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
1. A. M. Farhan, N. Scarpato, A. Pieroni, L. D. Nunzio and F. Fallucchi, “E-Health-IoT universe: A review, ” International Journal on Advanced Science, Engineering and Information Technology, vol. 7,
no. 6, pp. 2328, 2017. [Google Scholar]
2. M. Aazam, S. Zeadally and K. A. Harras, “Fog computing architecture, evaluation, and future research directions,” IEEE Communications Magazine, vol. 56, no. 5, pp. 46–52, 2018. [Google Scholar]
3. M. M. Dhanvijay and S. C. Patil, “Internet of things: A survey of enabling technologies in healthcare and its applications,” Computer Network, vol. 153, pp. 113–131, 2019. [Google Scholar]
4. J. N. S. Rubí and P. R. L. Gondim, “IoMT platform for pervasive healthcare data aggregation, processing, and sharing based on one M2M and open EHR,” Sensors, vol. 19, no. 19, pp. 4283, 2019. [
Google Scholar]
5. A. Gatouillat, Y. Badr, B. Massot and E. Sejdic, “Internet of medical things: A review of recent contributions dealing with cyber-physical systems in medicine,” IEEE Internet Things, vol. 5, no. 5
, pp. 3810–3822, 2018. [Google Scholar]
6. M. Usak, M. Kubiatko, M. S. Shabbir, O. V. Dudnik, K. Jermsittiparsert et al., “Health care service delivery based on the internet of things: A systematic and comprehensive study,” International
Journal of Communication Systems, vol. 33, no. 2, pp. 12–34, 2020. [Google Scholar]
7. H. Wang, Z. Wang and J. Domingo Ferrer, “Anonymous and secure aggregation scheme in fog-based public cloud computing,” Future Generation Computer System, vol. 78, pp. 712–719, 2018. [Google
8. A. Ullah, G. Said, M. Sher and H. Ning, “Fog-assisted secure healthcare data aggregation scheme in IoT-enabled WSN,” Peer-Peer Network Application, vol. 13, no. 1, pp. 163–174, 2020. [Google
9. H. Zhu, L. Gao and H. Li, “Secure and privacy-preserving body sensor data collection and query scheme,” Sensors, vol. 16, no. 2, pp. 1–16, 2016. [Google Scholar]
10. W. He, X. Liu, H. Nguyen, K. Nahrstedt and T. Abdelzaher, “Pda: Privacy-preserving data aggregation in wireless sensor networks,” in Proc. INFOCOM, 26th IEEE Int. Conf. on Computer Communications
, Anchorage, USA, pp. 2045–2053, 2007. [Google Scholar]
11. B. Farahani, F. Firouzi, V. Chang, M. Badaroglu, N. Constant et al., “Towards fog-driven IoT eHealth: Promises and challenges of IoT in medicine and healthcare,” Future Generation Computer System
, vol. 78, no. 3, pp. 659–676, 2018. [Google Scholar]
12. C. Gosman, C. Dobre and F. Pop, “Privacy-preserving data aggregation in intelligent transportation systems,” in Proc. IEEE Symp. on Integrated Network and Service Management, Lisbon, Portugal,
pp. 1059–1064, 2017. [Google Scholar]
13. M. Azeem, A. Ullah, H. Ashraf, N. Z. Jhanjhi, M. Humayun et al., “Fog oriented secure and lightweight data aggregation in IoMT,” IEEE Access, vol. 9, pp. 111072–111082, 2021. [Google Scholar]
14. Y. H. Lin, S. Y. Chang and H. M. Sun, “CDAMA: Concealed data aggregation scheme for multiple applications in wireless sensor networks,” IEEE Transactions on Knowledge and Data Engineering, vol.
25, no. 7, pp. 1471–1483, 2013. [Google Scholar]
15. A. Ullah, M. Azeem, H. Ashraf, A. A. Alaboudi, M. Humayun et al., “Secure healthcare data aggregation and transmission in IoT,A survey,” IEEE Access, vol. 9, pp. 16849–16865, 2021. [Google
16. C. M. Chen, Y. H. Lin and Y. C. Lin, “RCDA: Recoverable concealed data aggregation for data integrity in wireless sensor networks,” IEEE Transactions on Parallel and Distributed Systems, vol. 23,
no. 4, pp. 727–734, 2012. [Google Scholar]
17. A. Salman, I. Ahmad and S. Al Madani, “Particle swarm optimization for task assignment problem,” Microprocessors and Microsystems, vol. 26, no.8, pp. 363–371, 2002. [Google Scholar]
18. K. A. Shim and C. M. Park, “A secure data aggregation scheme based on appropriate cryptographic primitives in heterogeneous wireless sensor networks,” IEEE Transactions on Parallel and
Distributed Systems, vol. 26, no. 8, pp. 2128–2139, 2015. [Google Scholar]
19. A. Singh, S. Rathkanthiwar and S. Kakde, “Energy efficient routing of WSN using particle swarm optimization and V-LEACH protocol,” in Proc. Int. Conf. on Communication and Signal Processing
(ICCSP), Melmaruvathur, India, pp. 15–34, 2016. [Google Scholar]
20. H. Izakian, B. T. Ladani, A. Abraham and V. Snel, “A discrete particle swarm optimization approach for grid job scheduling,” International Journal of Innovative Computing,” Information and
Control, vol. 6, no. 9, pp. 1–15, 2010. [Google Scholar]
21. K. Sarwar, S. Yongchareon, J. Yu and S. Rehman, “Lightweight, divide-and-conquer privacy-preserving data aggregation in fog computing,” Future Generation Computer Systems, vol. 119, no. 2, pp.
188–199, 2021. [Google Scholar]
22. I. M. Easnony, S. I. Barakat, M. Elhoseny and R. R. Mostafa, “Improved feature selection model for big data analytics,” IEEE Access, vol. 8, pp. 66989–67004, 2020. [Google Scholar]
23. M. A. Yarimi, N. M. Munassar, M. Bamashmos and M. Ali, “Feature optimization by discrete weights for heart disease prediction using supervised learning,” Soft Computing, vol. 2, no. 3, pp. 1–15,
2020. [Google Scholar]
24. C. Zhang, C. Li and J. Zhang,“A secure privacy-preserving data aggregation model in wearable wireless sensor networks,” Journal of Electrical and Computer Engineering, vol. 61, no. 3, pp. 1–9,
2015. [Google Scholar]
25. H. Wang, Z. Wang and J. Ferrer, “Anonymous and secure aggregation scheme in fog-based public cloud computing,” Future Generation Computer System, vol. 78, no. 3, pp. 712–719, 2018. [Google
26. M. Abdel Basset, D. E. Shahat, K. Deb and M. Abouhawwash, “Energy-aware whale optimization algorithm for real-time task scheduling in multiprocessor systems,” Applied Soft Computing, vol. 93, p.1
06349, 2020. [Google Scholar]
27. M. Abdel Basset, R. Mohamed, M. Abouhawwash, K. Ripon Chakrabortty and J. Michael, “EA-MSCA: An effective energy-aware multi-objective modified sine-cosine algorithm for real-time task scheduling
in multiprocessor systems: Methods and analysis,” Expert Systems with Applications, vol. 173, pp. 114699, 2021. [Google Scholar]
28. M. AbdelBasset, R. Mohamed and M. Abouhawwash, “Balanced multi-objective optimization algorithm using improvement based reference points approach,” Swarm and Evolutionary Computation, vol. 60,
pp. 100791, 2021. [Google Scholar]
29. H. Seada, M. Abouhawwash and K. Deb, “Multiphase balance of diversity and convergence in multiobjective optimization,” IEEE Transactions on Evolutionary Computation, vol. 23, no. 3, pp. 503–513,
2019. [Google Scholar]
30. M. Abouhawwash and A. M. Alessio, “Multi objective evolutionary algorithm for PET image reconstruction: Concept,” IEEE Transactions on Medical Imaging, vol. 40, no. 8, pp. 2142–2151, 2021. [
Google Scholar]
31. N. S. Murugan, D. G. gopal, U. Kumaran, M. Thirunavukkarasan, M. D. Alshehri et al., “Secure data transmission in internet of medical things using RES-256 algorithm,” IEEE Transactions on
Industrial Informatics, vol. 6, no. 2, pp. 1–14, 2021. [Google Scholar]
32. M. Abouhawwash, “Hybrid evolutionary multi-objective optimization algorithm for helping multi-criterion decision makers,” International Journal of Management Science and Engineering Management,
vol. 16, no. 2, pp. 94–106, 2021. [Google Scholar]
33. G. G. Deverajan, V. Muthukumaran, C. H. Karuppiah and M. Chung, “Public key encryption with equality test for industrial internet of things system in cloud computing,” Transactions on Emerging
Telecommunications Technologies, vol. 17, no. 3, pp. e4202, 2021. [Google Scholar]
34. G. G. Deverajan and R. Saravanan, “Selfish node detection based on evidence by trust authority and selfish replica allocation in DANET,” International Journal of Information and Communication
Technology, vol. 9, no. 4, pp. 473–491, 2016. [Google Scholar]
35. M. Abouhawwash and K. Deb, “Reference point based evolutionary multi-objective optimization algorithms with convergence properties using KKTPM and ASF metrics,” Journal of Heuristics, vol. 27,
no. 12, pp. 575–614, 2021. [Google Scholar]
36. S. M. Nagarajan, G. G. Deverajan, P. Chatterjee, W. Alnumay and U. Ghosh, “Effective task scheduling algorithm with deep learning for internet of health things (IoHT) in sustainable smart
cities,” Sustainable Cities and Society, vol. 71, pp. 102945, 2021. [Google Scholar]
37. M. Abouhawwash, K. Deb and A. Alessio, “Exploration of multi-objective optimization with genetic algorithms for PET image reconstruction.”, Journal of Nuclear Medicine, vol. 61, no. 4, pp.
572–572, 2020. [Google Scholar]
38. D. Rao, S. Huang, Z. Jiang, G. G. Deverajan and R. Patan, “A dual deep neural network with phrase structure and attention mechanism for sentiment analysis,” Neural Computing and Applications,
vol. 33, pp. 11297–11308, 2021. [Google Scholar]
This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited. | {"url":"https://www.techscience.com/csse/v44n3/49131/html","timestamp":"2024-11-15T00:24:22Z","content_type":"application/xhtml+xml","content_length":"100965","record_id":"<urn:uuid:8ead19fe-7737-411e-9dbf-064b6908388b>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00172.warc.gz"} |
AmosWEB is Economics: Encyclonomic WEB*pedia
FAVORABLE BALANCE OF TRADE: An imbalance in a nation's balance of trade in which the payments for merchandise exports received by the country exceed payments for merchandise imports paid by the
country. This is also termed a balance of trade surplus. It's considered favorable because more goods are exported out of the country than are imported in, meaning that foreign production is replaced
with domestic production, which then increases domestic employment and income. A balance of trade surplus is often the source of a balance of payments surplus.
Visit the GLOSS*arama
A mathematical connection between a marginal value and the corresponding average value stating that the change in the average value depends on a comparison between the average and the marginal.
This mathematical relation between average and marginal surfaces throughout the study of economics, especially production (average product and marginal product), cost (average total cost and
marginal cost), and revenue (average revenue and marginal revenue). A similar relation is that between a total value and the corresponding marginal value.
The mathematical relation between average and marginal means that the average value is "driven" by the marginal value.
• If the marginal is less than the average, then the average declines.
• If the marginal is greater than the average, then the average rises.
• If the marginal is equal to the average, then the average does not change.
The reason for this relation is that average value is based on the existing situation that is then modified by the marginal value. This average-marginal relation applies to average and marginal
product, average and marginal cost, average and marginal revenue, and well, any other average and marginal encountered in the study of economics.
To illustrate the basic nature of the average-marginal relation consider an example. Suppose that there is a room containing five people that have been painstakingly and accurately measured for
height. The average height of this group is 66 inches (5' 6"). Some are taller than 5' 6" and some are shorter, but the average is 5' 6".
What happens to this 5' 6" average should a sixth person enter the room? This surely depends on the height of this extra person, this marginal addition to this group, does it not?
• If this "marginal" person is 6' tall, then the group's average rises to exactly 5' 7". The marginal is greater than the average, and the average rises.
• If the marginal sixth person is, however, a mere 5' tall, the marginal is less than the average, then the average declines to 5' 5".
• And if the new, marginal person, is exactly 5' 6'', the same as the existing average, then the average does not change.
Average and Marginal Product
This average-marginal relation can be graphically illustrated, using the average product and marginal product curves displayed to the right. A comparison between average product and marginal product
reveals three alternatives.
• When the marginal measure is greater than the average measure (that is, the marginal product curve lies above the average product curve), then the average measure increases (and the average
product curve has a positive slope).
• Alternatively, when the marginal measure is less than the average measure (that is, the marginal product curve lies below the average product curve), then the average measure decreases (and the
average product curve has a negative slope).
• In addition, when the marginal measure is equal to the average measure (that is, the marginal product curve intersects the average product curve), then the average measure does not change (and
the average product curve has a zero slope).
<= AVERAGE FIXED COST CURVE AVERAGE PHYSICAL PRODUCT =>
Recommended Citation:
AVERAGE-MARGINAL RELATION, AmosWEB Encyclonomic WEB*pedia, http://www.AmosWEB.com, AmosWEB LLC, 2000-2024. [Accessed: November 4, 2024].
Check Out These Related Terms...
| | | | | | | |
Or For A Little Background...
| | | | | |
And For Further Study...
| | | | | | | | |
Search Again?
Back to the WEB*pedia
ORANGE REBELOON
[What's This?]
Today, you are likely to spend a great deal of time calling an endless list of 800 numbers hoping to buy either an extra large beach blanket or a large flower pot shaped like a Greek urn. Be on the
lookout for jovial bank tellers.
Your Complete Scope
This isn't me! What am I?
The first paper notes printed in the United States were in denominations of 1 cent, 5 cents, 25 cents, and 50 cents.
"Kites rise highest against the wind, not with it. "
-- Winston Churchill, British prime minister
Tell us what you think about AmosWEB. Like what you see? Have suggestions for improvements? Let us know. Click the User Feedback link.
User Feedback | {"url":"https://www.amosweb.com/cgi-bin/awb_nav.pl?s=wpd&c=dsp&k=average-marginal%20relation","timestamp":"2024-11-04T16:58:25Z","content_type":"text/html","content_length":"35650","record_id":"<urn:uuid:5c9db1ff-a4e4-4e2f-81a7-13496d9a88bd>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00050.warc.gz"} |
odel to
Fit vector error-correction (VEC) model to data
EstMdl = estimate(Mdl,Y) returns a fully specified VEC(p – 1) model. This model stores the estimated parameter values resulting from fitting the VEC(p – 1) model Mdl to all variables (columns) of the
matrix of observed multivariate response series Y using maximum likelihood.
[EstMdl,EstSE,logL,E] = estimate(Mdl,Y) returns the estimated, asymptotic standard errors of the estimated parameters EstSE, optimized loglikelihood objective function value logL, and the
multivariate residuals E.
EstMdl = estimate(Mdl,Tbl1) fits the VEC(p – 1) model Mdl to variables in the input table or timetable Tbl1, which contains time series data, and returns the fully specified, estimated VEC(p – 1)
model EstMdl. estimate selects the variables in Mdl.SeriesNames or all variables in Tbl1. To select different variables in Tbl1 to fit the model to, use the ResponseVariables name-value argument.
(since R2022b)
[EstMdl,EstSE,logL,Tbl2] = estimate(Mdl,Tbl1) returns the estimated, asymptotic standard errors of the estimated parameters EstSE, the optimized loglikelihood objective function value logL, and the
table or timetable Tbl2 of all variables in Tbl1 and residuals corresponding to the response variables to which the model is fit (ResponseVariables). (since R2022b)
[___] = estimate(___,Name,Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in previous syntaxes. estimate returns the output
argument combination for the corresponding input arguments. For example, estimate(Mdl,Y,Model="H1*",X=Exo) fits the VEC(p – 1) model Mdl to the matrix of response data Y, and specifies the H1*
Johansen form of the deterministic terms and the matrix of exogenous predictor data Exo.
Supply all input data using the same data type. Specifically:
• If you specify the numeric matrix Y, optional data sets must be numeric arrays and you must use the appropriate name-value argument. For example, to specify a presample, set the Y0 name-value
argument to a numeric matrix of presample data.
• If you specify the table or timetable Tbl1, optional data sets must be tables or timetables, respectively, and you must use the appropriate name-value argument. For example, to specify a
presample, set the Presample name-value argument to a table or timetable of presample data.
Fit VEC(1) Model to Matrix of Response Data
Fit a VEC(1) model to seven macroeconomic series. Supply the response data as a numeric matrix.
Consider a VEC model for the following macroeconomic series:
• Gross domestic product (GDP)
• GDP implicit price deflator
• Paid compensation of employees
• Nonfarm business sector hours of all persons
• Effective federal funds rate
• Personal consumption expenditures
• Gross private domestic investment
Suppose that a cointegrating rank of 4 and one short-run term are appropriate, that is, consider a VEC(1) model.
Load the Data_USEconVECModel data set.
For more information on the data set and variables, enter Description at the command line.
Determine whether the data needs to be preprocessed by plotting the series on separate plots.
title("Gross Domestic Product");
title("GDP Deflator");
title("Paid Compensation of Employees");
ylabel("Billions of $");
title("Nonfarm Business Sector Hours");
title("Federal Funds Rate")
title("Consumption Expenditures")
ylabel("Billions of $")
title("Gross Private Domestic Investment")
ylabel("Billions of $")
Stabilize all series, except the federal funds rate, by applying the log transform. Scale the resulting series by 100 so that all series are on the same scale.
FRED.GDP = 100*log(FRED.GDP);
FRED.GDPDEF = 100*log(FRED.GDPDEF);
FRED.COE = 100*log(FRED.COE);
FRED.HOANBS = 100*log(FRED.HOANBS);
FRED.PCEC = 100*log(FRED.PCEC);
FRED.GPDI = 100*log(FRED.GPDI);
Create a VEC(1) model using the shorthand syntax. Specify the variable names.
Mdl = vecm(7,4,1);
Mdl.SeriesNames = FRED.Properties.VariableNames
Mdl =
vecm with properties:
Description: "7-Dimensional Rank = 4 VEC(1) Model with Linear Time Trend"
SeriesNames: "GDP" "GDPDEF" "COE" ... and 4 more
NumSeries: 7
Rank: 4
P: 2
Constant: [7×1 vector of NaNs]
Adjustment: [7×4 matrix of NaNs]
Cointegration: [7×4 matrix of NaNs]
Impact: [7×7 matrix of NaNs]
CointegrationConstant: [4×1 vector of NaNs]
CointegrationTrend: [4×1 vector of NaNs]
ShortRun: {7×7 matrix of NaNs} at lag [1]
Trend: [7×1 vector of NaNs]
Beta: [7×0 matrix]
Covariance: [7×7 matrix of NaNs]
Mdl is a vecm model object. All properties containing NaN values correspond to parameters to be estimated given data.
Estimate the model using the entire data set and the default options.
EstMdl = estimate(Mdl,FRED.Variables)
EstMdl =
vecm with properties:
Description: "7-Dimensional Rank = 4 VEC(1) Model"
SeriesNames: "GDP" "GDPDEF" "COE" ... and 4 more
NumSeries: 7
Rank: 4
P: 2
Constant: [14.1329 8.77841 -7.20359 ... and 4 more]'
Adjustment: [7×4 matrix]
Cointegration: [7×4 matrix]
Impact: [7×7 matrix]
CointegrationConstant: [-28.6082 -109.555 77.0912 ... and 1 more]'
CointegrationTrend: [4×1 vector of zeros]
ShortRun: {7×7 matrix} at lag [1]
Trend: [7×1 vector of zeros]
Beta: [7×0 matrix]
Covariance: [7×7 matrix]
EstMdl is an estimated vecm model object. It is fully specified because all parameters have known values. By default, estimate imposes the constraints of the H1 Johansen VEC model form by removing
the cointegrating trend and linear trend terms from the model. Parameter exclusion from estimation is equivalent to imposing equality constraints to zero.
Display a short summary from the estimation.
results = summarize(EstMdl)
results = struct with fields:
Description: "7-Dimensional Rank = 4 VEC(1) Model"
Model: "H1"
SampleSize: 238
NumEstimatedParameters: 112
LogLikelihood: -1.4939e+03
AIC: 3.2118e+03
BIC: 3.6007e+03
Table: [133x4 table]
Covariance: [7x7 double]
Correlation: [7x7 double]
The Table field of results is a table of parameter estimates and corresponding statistics.
Specify Presample Values
Consider the model and data in Fit VEC(1) Model to Matrix of Response Data, and suppose that the estimation sample starts at Q1 of 1980.
Load the Data_USEconVECModel data set and preprocess the data.
load Data_USEconVECModel
FRED.GDP = 100*log(FRED.GDP);
FRED.GDPDEF = 100*log(FRED.GDPDEF);
FRED.COE = 100*log(FRED.COE);
FRED.HOANBS = 100*log(FRED.HOANBS);
FRED.PCEC = 100*log(FRED.PCEC);
FRED.GPDI = 100*log(FRED.GPDI);
Identify the index corresponding to the start of the estimation sample.
estIdx = FRED.Time(2:end) > '1979-12-31';
Create a default VEC(1) model using the shorthand syntax. Assume that the appropriate cointegration rank is 4. Specify the variable names.
Mdl = vecm(7,4,1);
Mdl.SeriesNames = FRED.Properties.VariableNames;
Estimate the model using the estimation sample. Specify all observations before the estimation sample as presample data. Also, specify estimation of the H Johansen form of the VEC model, which
includes all deterministic parameters.
Y0 = FRED{~estIdx,:};
EstMdl = estimate(Mdl,FRED{estIdx,:},'Y0',Y0,'Model',"H")
EstMdl =
vecm with properties:
Description: "7-Dimensional Rank = 4 VEC(1) Model with Linear Time Trend"
SeriesNames: "GDP" "GDPDEF" "COE" ... and 4 more
NumSeries: 7
Rank: 4
P: 2
Constant: [17.5698 3.74759 -20.1998 ... and 4 more]'
Adjustment: [7×4 matrix]
Cointegration: [7×4 matrix]
Impact: [7×7 matrix]
CointegrationConstant: [-85.4825 -57.3569 81.7344 ... and 1 more]'
CointegrationTrend: [0.0264185 -0.00275396 0.0249583 ... and 1 more]'
ShortRun: {7×7 matrix} at lag [1]
Trend: [0.000514564 -0.000291183 0.00179965 ... and 4 more]'
Beta: [7×0 matrix]
Covariance: [7×7 matrix]
Because the VEC model order p is 2, estimate uses only the last two observations (rows) in Y0 as a presample.
Fit VEC Model to Response Variables in Timetable
Since R2022b
Fit a VEC(1) model to seven macroeconomic series. Supply a timetable of data and specify the series for the fit. This example is based on Fit VEC(1) Model to Matrix of Response Data.
Load and Preprocess Data
Load the Data_USEconVECModel data set.
load Data_USEconVECModel
Time GDP GDPDEF COE HOANBS FEDFUNDS PCEC GPDI
___________ _____ ______ _____ ______ ________ _____ ____
31-Mar-1957 470.6 16.485 260.6 54.756 2.96 282.3 77.7
30-Jun-1957 472.8 16.601 262.5 54.639 3 284.6 77.9
30-Sep-1957 480.3 16.701 265.1 54.375 3.47 289.2 79.3
31-Dec-1957 475.7 16.711 263.7 53.249 2.98 290.8 71
31-Mar-1958 468.4 16.892 260.2 52.043 1.2 290.3 66.7
30-Jun-1958 472.8 16.94 259.9 51.297 0.93 293.2 65.1
30-Sep-1958 486.7 17.043 267.7 51.908 1.76 298.3 72
31-Dec-1958 500.4 17.123 272.7 52.683 2.42 302.2 80
Stabilize all series, except the federal funds rate, by applying the log transform. Scale the resulting series by 100 so that all series are on the same scale.
FRED.GDP = 100*log(FRED.GDP);
FRED.GDPDEF = 100*log(FRED.GDPDEF);
FRED.COE = 100*log(FRED.COE);
FRED.HOANBS = 100*log(FRED.HOANBS);
FRED.PCEC = 100*log(FRED.PCEC);
FRED.GPDI = 100*log(FRED.GPDI);
numobs = height(FRED)
Prepare Timetable for Estimation
When you plan to supply a timetable directly to estimate, you must ensure it has all the following characteristics:
• All selected response variables are numeric and do not contain any missing values.
• The timestamps in the Time variable are regular, and they are ascending or descending.
Remove all missing values from the table.
DTT = rmmissing(FRED);
numobs = height(DTT)
DTT does not contain any missing values.
Determine whether the sampling timestamps have a regular frequency and are sorted.
areTimestampsRegular = isregular(DTT,"quarters")
areTimestampsRegular = logical
areTimestampsSorted = issorted(DTT.Time)
areTimestampsSorted = logical
areTimestampsRegular = 0 indicates that the timestamps of DTT are irregular. areTimestampsSorted = 1 indicates that the timestamps are sorted. Macroeconomic series in this example are timestamped at
the end of the month. This quality induces an irregularly measured series.
Remedy the time irregularity by shifting all dates to the first day of the quarter.
dt = DTT.Time;
dt = dateshift(dt,"start","quarter");
DTT.Time = dt;
areTimestampsRegular = isregular(DTT,"quarters")
areTimestampsRegular = logical
DTT is regular with respect to time.
Create Model Template for Estimation
Create a VEC(1) model using the shorthand syntax. Specify the variable names.
Mdl = vecm(7,4,1);
Mdl.SeriesNames = FRED.Properties.VariableNames
Mdl =
vecm with properties:
Description: "7-Dimensional Rank = 4 VEC(1) Model with Linear Time Trend"
SeriesNames: "GDP" "GDPDEF" "COE" ... and 4 more
NumSeries: 7
Rank: 4
P: 2
Constant: [7×1 vector of NaNs]
Adjustment: [7×4 matrix of NaNs]
Cointegration: [7×4 matrix of NaNs]
Impact: [7×7 matrix of NaNs]
CointegrationConstant: [4×1 vector of NaNs]
CointegrationTrend: [4×1 vector of NaNs]
ShortRun: {7×7 matrix of NaNs} at lag [1]
Trend: [7×1 vector of NaNs]
Beta: [7×0 matrix]
Covariance: [7×7 matrix of NaNs]
Fit Model to Data
Estimate the model. Pass the entire timetable DTT. By default, estimate selects the response variables in Mdl.SeriesNames to fit to the model. Alternatively, you can use the ResponseVariables
name-value argument.
Return the timetable of residuals and data fit to the model.
[EstMdl,~,~,Tbl2] = estimate(Mdl,DTT);
EstMdl is an estimated vecm model object. It is fully specified because all parameters have known values.
Display the head of the table Tbl2.
Time GDP GDPDEF COE HOANBS FEDFUNDS PCEC GPDI GDP_Residuals GDPDEF_Residuals COE_Residuals HOANBS_Residuals FEDFUNDS_Residuals PCEC_Residuals GPDI_Residuals
___________ ______ ______ ______ ______ ________ ______ ______ _____________ ________________ _____________ ________________ __________________ ______________ ______________
01-Jul-1957 617.44 281.55 558.01 399.59 3.47 566.71 437.32 0.12076 0.090979 -0.31114 -0.47341 -0.013177 0.14899 1.1764
01-Oct-1957 616.48 281.61 557.48 397.5 2.98 567.26 426.27 -2.4005 -0.39287 -2.1158 -2.1552 -0.86464 -0.89017 -12.289
01-Jan-1958 614.93 282.68 556.15 395.21 1.2 567.09 420.02 -2.0142 0.92195 -1.5874 -1.1852 -1.3247 -0.72797 -4.4964
01-Apr-1958 615.87 282.97 556.03 393.76 0.93 568.09 417.59 0.2131 -0.39586 -0.22658 -0.070487 -0.24993 0.17697 -0.31486
01-Jul-1958 618.76 283.57 558.99 394.95 1.76 569.81 427.67 2.0866 0.45876 2.4738 1.9098 0.98197 1.0195 9.119
01-Oct-1958 621.54 284.04 560.84 396.43 2.42 571.11 438.2 0.68671 0.053454 0.48556 0.63518 0.23659 -0.21548 4.2428
01-Jan-1959 623.66 284.31 563.55 398.35 2.8 573.62 442.12 0.39546 -0.066055 0.97292 1.0224 -0.054929 0.86153 0.68805
01-Apr-1959 626.19 284.46 565.91 400.24 3.39 575.54 449.31 0.24314 -0.22217 0.33889 0.4216 -0.20457 0.26963 -0.15985
Because Mdl.P is 2, estimation requires two presample observations. Consequently, estimate uses the first two rows (first two quarters of 1957) of DTT as a presample, fits the model to the remaining
observations, and returns only those observations used in estimation in Tbl2.
Plot the residuals.
varnames = Tbl2.Properties.VariableNames;
resnames = varnames(contains(Tbl2.Properties.VariableNames,"_Residuals"));
for j = 1:7
grid on
Include Exogenous Predictor Variables
Consider the model and data in Fit VEC(1) Model to Matrix of Response Data.
Load the Data_USEconVECModel data set and preprocess the data.
load Data_USEconVECModel
FRED.GDP = 100*log(FRED.GDP);
FRED.GDPDEF = 100*log(FRED.GDPDEF);
FRED.COE = 100*log(FRED.COE);
FRED.HOANBS = 100*log(FRED.HOANBS);
FRED.PCEC = 100*log(FRED.PCEC);
FRED.GPDI = 100*log(FRED.GPDI);
The Data_Recessions data set contains the beginning and ending serial dates of recessions. Load this data set. Convert the matrix of date serial numbers to a datetime array.
load Data_Recessions
dtrec = datetime(Recessions,'ConvertFrom','datenum');
Create a dummy variable that identifies periods in which the U.S. was in a recession or worse. Specifically, the variable should be 1 if FRED.Time occurs during a recession, and 0 otherwise.
isin = @(x)(any(dtrec(:,1) <= x & x <= dtrec(:,2)));
isrecession = double(arrayfun(isin,FRED.Time));
Create a VEC(1) model using the shorthand syntax. Assume that the appropriate cointegration rank is 4. You do not have to specify the presence of a regression component when creating the model.
Specify the variable names.
Mdl = vecm(7,4,1);
Mdl.SeriesNames = FRED.Properties.VariableNames;
Estimate the model using the entire sample. Specify the predictor identifying whether the observation was measured during a recession. Return the standard errors.
[EstMdl,EstSE] = estimate(Mdl,FRED.Variables,'X',isrecession);
Display the regression coefficient for each equation and the corresponding standard errors.
ans = 7×1
ans = 7×1
EstMdl.Beta and EstSE.Beta are 7-by-1 vectors. Rows correspond to response variables in EstMdl.SeriesNames and columns correspond to predictors.
To check whether the effects of recessions are significant, obtain summary statistics from summarize, and then display the results for Beta.
results = summarize(EstMdl);
isbeta = contains(results.Table.Properties.RowNames,'Beta');
betaresults = results.Table(isbeta,:)
betaresults=7×4 table
Value StandardError TStatistic PValue
_________ _____________ __________ __________
Beta(1,1) -1.1975 0.15469 -7.7411 9.8569e-15
Beta(2,1) -0.018738 0.05806 -0.32273 0.7469
Beta(3,1) -0.75305 0.15071 -4.9966 5.8341e-07
Beta(4,1) -0.70936 0.12776 -5.5521 2.8221e-08
Beta(5,1) -0.5932 0.24712 -2.4004 0.016377
Beta(6,1) -0.68353 0.13107 -5.2151 1.837e-07
Beta(7,1) -4.4839 0.715 -6.2712 3.5822e-10
whichsig = EstMdl.SeriesNames(betaresults.PValue < 0.05)
whichsig = 1x6 string
"GDP" "COE" "HOANBS" "FEDFUNDS" "PCEC" "GPDI"
All series except GDPDEF appear to have a significant recessions effect.
Input Arguments
Mdl — VEC model
vecm model object
VEC model containing unknown parameter values, specified as a vecm model object returned by vecm.
NaN-valued elements in properties indicate unknown, estimable parameters. Specified elements indicate equality constraints on parameters in model estimation. The innovations covariance matrix
Mdl.Covariance cannot contain a mix of NaN values and real numbers; you must fully specify the covariance or it must be completely unknown (NaN(Mdl.NumSeries)).
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: estimate(Mdl,Y,Model="H1*",X=Exo) fits the VEC(p – 1) model Mdl to the matrix of response data Y, and specifies the H1* Johansen form of the deterministic terms and the matrix of exogenous
predictor data Exo.
Model — Johansen form of VEC(p – 1) model deterministic terms
"H1" (default) | "H2" | "H1*" | "H*" | "H"
Johansen form of the VEC(p – 1) model deterministic terms [2], specified as a value in this table (for variable definitions, see Vector Error-Correction Model).
Value Error-Correction Term Description
No intercepts or trends are present in the cointegrating relations, and no deterministic trends are present in the levels of the data.
"H2" AB´y[t − 1]
Specify this model only when all response series have a mean of zero.
"H1*" A(B´y[t−1]+c[0]) Intercepts are present in the cointegrating relations, and no deterministic trends are present in the levels of the data.
"H1" A(B´y[t−1]+c[0])+c[1] Intercepts are present in the cointegrating relations, and deterministic linear trends are present in the levels of the data.
"H*" A(B´y[t−1]+c[0]+d[0]t)+c[1] Intercepts and linear trends are present in the cointegrating relations, and deterministic linear trends are present in the levels of the data.
Intercepts and linear trends are present in the cointegrating relations, and deterministic quadratic trends are present in the levels of the data.
"H" A(B´y[t−1]+c[0]+d[0]t)+c[1]+d[1]t
If quadratic trends are not present in the data, this model can produce good in-sample fits but poor out-of-sample forecasts.
During estimation, if the overall model constant, overall linear trend, cointegrating constant, or cointegrating linear trend parameters are not in the model, then estimate constrains them to zero.
If you specify a different equality constraint, that is, if the properties corresponding to those deterministic terms being constrained to zero have a value other than a vector of NaN values or
zeros, then estimate issues an error. To enforce supported equality constraints, choose the Johansen model containing the deterministic term that you want to constrain.
Example: Model="H1*"
Data Types: string | char
• NaN values in Y, Y0, and X indicate missing values. estimate removes missing values from the data by list-wise deletion.
□ For the presample, estimate removes any row containing at least one NaN.
□ For the estimation sample, estimate removes any row of the concatenated data matrix [Y X] containing at least one NaN.
This type of data reduction reduces the effective sample size.
• estimate issues an error when any table or timetable input contains missing values.
Output Arguments
EstMdl — Estimated VEC(p – 1) model
vecm model object
Estimated VEC(p – 1) model, returned as a vecm model object. EstMdl is a fully specified vecm model.
estimate uses mvregress to implement multivariate normal, maximum likelihood estimation. For more details, see Estimation of Multivariate Regression Models.
EstSE — Estimated, asymptotic standard errors of estimated parameters
structure array
Estimated, asymptotic standard errors of the estimated parameters, returned as a structure array containing the fields in this table.
Field Description
Constant Standard errors of the overall model constants (c) corresponding to the estimates in EstMdl.Constant, an Mdl.NumSeries-by-1 numeric vector
Adjustment Standard errors of the adjustment speeds (A) corresponding to the estimates in EstMdl.Adjustment, an Mdl.NumSeries-by-Mdl.Rank numeric vector
Impact Standard errors of the impact coefficient (Π) corresponding to the estimates in EstMdl.Impact, an Mdl.NumSeries-by-Mdl.NumSeries numeric vector
ShortRun Standard errors of the short-run coefficients (Φ) corresponding to estimates in EstMdl.ShortRun, a cell vector with elements corresponding to EstMdl.ShortRun
Beta Standard errors of regression coefficients (β) corresponding to the estimates in EstMdl.Beta, an Mdl.NumSeries-by-numpreds numeric matrix
Trend Standard errors of the overall linear time trends (d) corresponding to the estimates in EstMdl.Trend, an Mdl.NumSeries-by-1 numeric vector
If estimate applies equality constraints during estimation by fixing any parameters to a value, then corresponding standard errors of those parameters are 0.
estimate extracts all standard errors from the inverse of the expected Fisher information matrix returned by mvregress (see Standard Errors).
More About
Vector Error-Correction Model
A vector error-correction (VEC) model is a multivariate, stochastic time series model consisting of a system of m = numseries equations of m distinct, differenced response variables. Equations in the
system can include an error-correction term, which is a linear function of the responses in levels used to stabilize the system. The cointegrating rank r is the number of cointegrating relations that
exist in the system.
Each response equation can include an autoregressive polynomial composed of first differences of the response series (short-run polynomial of degree p – 1), a constant, a time trend, exogenous
predictor variables, and a constant and time trend in the error-correction term.
A VEC(p – 1) model in difference-equation notation and in reduced form can be expressed in two ways:
• This equation is the component form of a VEC model, where the cointegration adjustment speeds and cointegration matrix are explicit, whereas the impact matrix is implied.
$\begin{array}{c}\Delta {y}_{t}=A\left(B\prime {y}_{t-1}+{c}_{0}+{d}_{0}t\right)+{c}_{1}+{d}_{1}t+{\Phi }_{1}\Delta {y}_{t-1}+...+{\Phi }_{p-1}\Delta {y}_{t-\left(p-1\right)}+\beta {x}_{t}+{\
epsilon }_{t}\\ =c+dt+AB\prime {y}_{t-1}+{\Phi }_{1}\Delta {y}_{t-1}+...+{\Phi }_{p-1}\Delta {y}_{t-\left(p-1\right)}+\beta {x}_{t}+{\epsilon }_{t}.\end{array}$
The cointegrating relations are B'y[t – 1] + c[0] + d[0]t and the error-correction term is A(B'y[t – 1] + c[0] + d[0]t).
• This equation is the impact form of a VEC model, where the impact matrix is explicit, whereas the cointegration adjustment speeds and cointegration matrix are implied.
$\begin{array}{c}\Delta {y}_{t}=\Pi {y}_{t-1}+A\left({c}_{0}+{d}_{0}t\right)+{c}_{1}+{d}_{1}t+{\Phi }_{1}\Delta {y}_{t-1}+...+{\Phi }_{p-1}\Delta {y}_{t-\left(p-1\right)}+\beta {x}_{t}+{\epsilon
}_{t}\\ =c+dt+\Pi {y}_{t-1}+{\Phi }_{1}\Delta {y}_{t-1}+...+{\Phi }_{p-1}\Delta {y}_{t-\left(p-1\right)}+\beta {x}_{t}+{\epsilon }_{t}.\end{array}$
In the equations:
• y[t] is an m-by-1 vector of values corresponding to m response variables at time t, where t = 1,...,T.
• Δy[t] = y[t] – y[t – 1]. The structural coefficient is the identity matrix.
• r is the number of cointegrating relations and, in general, 0 < r < m.
• A is an m-by-r matrix of adjustment speeds.
• B is an m-by-r cointegration matrix.
• Π is an m-by-m impact matrix with a rank of r.
• c[0] is an r-by-1 vector of constants (intercepts) in the cointegrating relations.
• d[0] is an r-by-1 vector of linear time trends in the cointegrating relations.
• c[1] is an m-by-1 vector of constants (deterministic linear trends in y[t]).
• d[1] is an m-by-1 vector of linear time-trend values (deterministic quadratic trends in y[t]).
• c = Ac[0] + c[1] and is the overall constant.
• d = Ad[0] + d[1] and is the overall time-trend coefficient.
• Φ[j] is an m-by-m matrix of short-run coefficients, where j = 1,...,p – 1 and Φ[p – 1] is not a matrix containing only zeros.
• x[t] is a k-by-1 vector of values corresponding to k exogenous predictor variables.
• β is an m-by-k matrix of regression coefficients.
• ε[t] is an m-by-1 vector of random Gaussian innovations, each with a mean of 0 and collectively an m-by-m covariance matrix Σ. For t ≠ s, ε[t] and ε[s] are independent.
Condensed and in lag operator notation, the system is
$\begin{array}{c}\Phi \left(L\right)\left(1-L\right){y}_{t}=A\left(B\prime {y}_{t-1}+{c}_{0}+{d}_{0}t\right)+{c}_{1}+{d}_{1}t+\beta {x}_{t}+{\epsilon }_{t}\\ =c+dt+AB\prime {y}_{t-1}+\beta {x}_{t}+{\
epsilon }_{t}\end{array}$
where $\Phi \left(L\right)=I-{\Phi }_{1}-{\Phi }_{2}-...-{\Phi }_{p-1}$, I is the m-by-m identity matrix, and Ly[t] = y[t – 1].
If m = r, then the VEC model is a stable VAR(p) model in the levels of the responses. If r = 0, then the error-correction term is a matrix of zeros, and the VEC(p – 1) model is a stable VAR(p – 1)
model in the first differences of the responses.
Johansen Form
The Johansen forms of a VEC Model differ with respect to the presence of deterministic terms. As detailed in [2], the estimation procedure differs among the forms. Consequently, allowable equality
constraints on the deterministic terms during estimation differ among forms. For more details, see The Role of Deterministic Terms.
This table describes the five Johansen forms and supported equality constraints.
Form Error-Correction Term Deterministic Coefficients Equality Constraints
c = 0 (Constant).
d = 0 (Trend). You can fully specify B.
H2 AB´y[t − 1]
c[0] = 0 (CointegrationConstant). All deterministic coefficients are zero.
d[0] = 0 (CointegrationTrend).
c = Ac[0]. If you fully specify either B or c[0], then you must fully specify the other.
H1* A(B´y[t−1]+c[0]) d = 0. MATLAB^® derives the value of c from c[0] and A.
d[0] = 0. All deterministic trends are zero.
You can fully specify B.
c = Ac[0] + c[1].
You can specify a mixture of NaN and numeric values for c.
H1 A(B´y[t−1] + c[0]) + c[1] d = 0.
MATLAB derives the value of c[0] from c and A.
d[0] = 0.
All deterministic trends are zero.
If you fully specify either B or d[0], then you must fully specify the other.
c = Ac[0] + c[1]. You can specify a mixture of NaN and numeric values for c.
H* A(B´y[t−1] + c[0] + d[0]t) + c[1]
d = Ad[0]. MATLAB derives the value of c[0] from c and A.
MATLAB derives the value of d from A and d[0].
You can fully specify B.
c = Ac[0] + c[1].
H A(B´y[t−1]+c[0]+d[0]t)+c[1]+d[1]t You can specify a mixture of NaN and numeric values for c and d.
d = A.d[0] + d[1].
MATLAB derives the values of c[0] and d[0] from c, d, and A.
• If 1 ≤ Mdl.Rank ≤ Mdl.NumSeries – 1, as with most VEC models, then estimate performs parameter estimation in two steps.
1. estimate estimates the parameters of the cointegrating relations, including any restricted intercepts and time trends, by the Johansen method [2].
☆ The form of the cointegrating relations corresponds to one of the five parametric forms considered by Johansen in [2] (see 'Model'). For more details, see jcitest and jcontest.
☆ The adjustment speed parameter (A) and the cointegration matrix (B) in the VEC(p – 1) model cannot be uniquely identified. However, the product Π = A*Bʹ is identifiable. In this
estimation step, B = V[1:r], where V[1:r] is the matrix composed of all rows and the first r columns of the eigenvector matrix V. V is normalized so that Vʹ*S[11]*V = I. For more details,
see [2].
2. estimate constructs the error-correction terms from the estimated cointegrating relations. Then, estimate estimates the remaining terms in the VEC model by constructing a vector
autoregression (VAR) model in first differences and including the error-correction terms as predictors. For models without cointegrating relations (Mdl.Rank = 0) or with a cointegrating
matrix of full rank (Mdl.Rank = Mdl.Numseries), estimate performs this VAR estimation step only.
• You can remove stationary series, which are associated with standard unit vectors in the space of cointegrating relations, from cointegration analysis. To pretest individual series for
stationarity, use adftest, pptest, kpsstest, and lmctest. As an alternative, you can test for standard unit vectors in the context of the full model by using jcontest.
• If 1 ≤ Mdl.Rank ≤ Mdl.NumSeries – 1, the asymptotic error covariances of the parameters in the cointegrating relations (which include B, c[0], and d[0] corresponding to the Cointegration,
CointegrationConstant, and CointegrationTrend properties, respectively) are generally non-Gaussian. Therefore, estimate does not estimate or return corresponding standard errors.
In contrast, the error covariances of the composite impact matrix, which is defined as the product A*Bʹ, are asymptotically Gaussian. Therefore, estimate estimates and returns its standard
errors. Similar caveats hold for the standard errors of the overall constant and linear trend (A*c[0] and A*d[0]corresponding to the Constant and Trend properties, respectively) of the H1* and H*
Johansen forms.
[1] Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
[2] Johansen, S. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press, 1995.
[3] Juselius, K. The Cointegrated VAR Model. Oxford: Oxford University Press, 2006.
[4] Lütkepohl, H. New Introduction to Multiple Time Series Analysis. Berlin: Springer, 2005.
Version History
Introduced in R2017b
R2022b: estimate accepts input data in tables and timetables, and return results in tables and timetables
R2019b: Equality constraints on innovations covariance matrix | {"url":"https://ww2.mathworks.cn/help/econ/vecm.estimate.html","timestamp":"2024-11-11T22:26:20Z","content_type":"text/html","content_length":"213397","record_id":"<urn:uuid:474161d9-82a2-4b71-a7d0-0cd449540408>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00656.warc.gz"} |
Exploring the Mathematics of Higher Dimensions: Principles and Applications
Introduction to Higher Dimensions
The concept of higher dimensions extends beyond the familiar three-dimensional space we experience in our daily lives. Historically, the study of dimensions began with the straightforward
understanding of one-dimensional lines, two-dimensional planes, and three-dimensional volumes. However, the curiosity of mathematicians and physicists has led to the exploration of spaces with four,
five, or even infinitely many dimensions. But what exactly are higher dimensions, and why are they studied?
Higher dimensions are abstract mathematical constructs that provide a framework for solving complex problems in both theoretical and applied mathematics. While it is challenging to visualize
dimensions beyond the third, mathematicians employ sophisticated tools and theories to understand these multidimensional spaces. The study of higher dimensions finds its roots in the 19th century
with pioneers like Bernhard Riemann, who generalized the idea of curved surfaces to higher-dimensional spaces. His work laid the foundation for Riemannian geometry, a critical area in the field of
modern mathematics.
One of the most compelling reasons to study higher dimensions is their application in theoretical physics, particularly in the realms of string theory and general relativity. String theory, for
instance, posits that the fundamental particles are not zero-dimensional points but rather one-dimensional strings vibrating in a ten-dimensional space. Likewise, the theory of general relativity
describes gravity as the curvature of a four-dimensional spacetime continuum.
Applied mathematics also benefits from higher-dimensional studies, particularly in fields such as data science and computer graphics. In data science, higher dimensions are essential for analyzing
large datasets with multiple variables, while computer graphics leverage higher-dimensional mathematics for rendering complex three-dimensional models on a two-dimensional screen.
Thus, higher dimensions are not merely abstract concepts but are pivotal in advancing both theoretical understanding and practical applications. By delving into the mathematics of higher dimensions,
we can unlock new insights and solutions that transcend the limitations of our three-dimensional perspective.
The Fourth Dimension and Beyond
When extending the concept of dimensions beyond the familiar three, we venture into the realm of the fourth dimension and beyond, often referred to as ‘hyperspace.’ In mathematics, a fourth dimension
is not merely an abstract idea but a space that can be rigorously defined and explored. A 4D space, or four-dimensional space, adds an additional degree of freedom to the three spatial dimensions we
experience daily: length, width, and height. This extra dimension can be represented mathematically using a four-dimensional coordinate system, where each point is identified by four coordinates (x,
y, z, w).
One way to conceptualize the fourth dimension is by considering the analogy of lower dimensions. Just as a 2D square can be extended into a 3D cube, a 3D cube can be extended into a 4D hypercube,
also known as a tesseract. The tesseract consists of eight cubical cells and can be visualized through various projections and animations, although our three-dimensional perception limits our ability
to fully grasp its structure.
Moving beyond the fourth dimension, the concept can be generalized to n-dimensional spaces. In these higher-dimensional spaces, various terminologies are used to describe geometric entities. For
example, a ‘hyperplane’ is a flat subspace within an n-dimensional space, and a ‘hypercube’ is the n-dimensional analog of a cube. Mathematically, these objects can be represented using vectors,
matrices, and other algebraic structures that describe their properties and relationships.
Mathematical models and visualizations play a crucial role in helping us understand higher dimensions. Techniques such as projection and cross-sectioning allow us to get a glimpse of these
higher-dimensional spaces within our three-dimensional world. For instance, slicing a tesseract in certain ways can produce various 3D shapes, offering insights into the nature of four-dimensional
While the concept of higher dimensions may seem esoteric, it has practical applications in fields such as physics, computer science, and data analysis. The mathematics of higher dimensions provides
powerful tools for solving complex problems and advancing our understanding of the universe.
Mathematical Tools for Higher Dimensions
Understanding higher-dimensional spaces necessitates a robust set of mathematical tools. Among the foundational elements are vectors, matrices, and tensors, which are fundamental in representing and
manipulating data in multiple dimensions. Vectors, for instance, extend naturally from two or three dimensions to higher dimensions, allowing for the representation of multidimensional data points
and directions. Matrices, which can be thought of as two-dimensional arrays of numbers, facilitate operations such as transformations, rotations, and scaling in higher-dimensional spaces. These
matrices become particularly powerful when applied through the lens of linear algebra, a branch of mathematics that explores vector spaces and linear mappings between them.
One step further, tensors generalize the concepts of vectors and matrices to higher dimensions. A tensor can be considered a multidimensional array of numerical values that generalize the concept of
scalars, vectors, and matrices. Tensors play a crucial role in fields such as physics, engineering, and machine learning, where they are used to represent complex, multidimensional relationships. For
example, in physics, the stress-energy tensor is used to describe the density and flux of energy and momentum in space-time, critical for the general theory of relativity.
Another essential tool in the study of higher dimensions is multivariable calculus. While single-variable calculus deals with functions of a single variable, multivariable calculus extends these
concepts to functions of multiple variables, allowing for the analysis of surfaces and volumes within higher-dimensional spaces. This branch of calculus is indispensable in fields such as economics,
engineering, and physics, where it is used to model and solve problems involving multiple interacting variables. Techniques such as partial differentiation, multiple integrals, and gradient vectors
are fundamental in understanding the behavior of higher-dimensional functions.
These mathematical tools not only extend our understanding of higher-dimensional spaces but also find practical applications across various scientific and engineering disciplines. From optimizing
algorithms in computer science to modeling physical phenomena in theoretical physics, the ability to navigate and manipulate higher dimensions is crucial for advancing knowledge and innovation.
Higher-dimensional geometry extends the principles of traditional Euclidean geometry into dimensions beyond the familiar three. Shapes in these higher dimensions, such as hypercubes, hyperspheres,
and simplices, exhibit unique properties that challenge our conventional understanding of geometry.
A hypercube, or n-cube, generalizes the concept of a square and cube to n dimensions. For example, a 4-dimensional hypercube, also known as a tesseract, has 16 vertices, 32 edges, 24 square faces,
and 8 cubic cells. As the number of dimensions increases, the complexity of the hypercube’s structure grows exponentially. Hyperspheres, on the other hand, are the higher-dimensional equivalents of
circles and spheres. In n-dimensions, a hypersphere’s volume and surface area are determined by intricate mathematical formulas involving the gamma function, which generalizes factorial operations to
non-integer values.
Simplices, the simplest type of higher-dimensional polytope, are generalizations of triangles and tetrahedrons. An n-simplex is defined as the convex hull of its n + 1 vertices. For instance, a
3-simplex is a tetrahedron, and a 4-simplex, also called a pentachoron, consists of 5 tetrahedral cells. These shapes are fundamental in various branches of mathematics, including topology and
combinatorial geometry.
When discussing higher-dimensional shapes, the concepts of volume and surface area extend into new realms. In three dimensions, volume is a measure of the space enclosed within a shape, whereas
surface area measures the extent of its boundary. In higher dimensions, these concepts generalize to n-volumes and n-areas, which often require advanced calculus and linear algebra techniques to
Visualizing and understanding higher-dimensional shapes pose significant challenges, as humans are inherently limited to perceiving three dimensions. However, mathematical abstractions and
projections, such as Schlegel diagrams and stereographic projections, provide tools to represent these shapes in lower dimensions. These methods allow mathematicians to study the properties and
relationships of higher-dimensional objects, despite the limitations of our sensory perception.
Topological Concepts in Higher Dimensions
Topology, a fundamental branch of mathematics, investigates the properties of spaces that are preserved under continuous deformations such as stretching, twisting, and bending, but not tearing or
gluing. When extended to higher dimensions, several key concepts become crucial for understanding these intricate spaces. Among these concepts are manifolds, homotopy, and homology.
A manifold is a topological space that locally resembles Euclidean space. In higher dimensions, manifolds provide a framework for analyzing the complex structure of spaces. For instance, a
two-dimensional surface like a sphere or torus can be generalized to higher-dimensional manifolds such as a 3-sphere or a 4-torus. These higher-dimensional manifolds play a pivotal role in both pure
mathematics and theoretical physics, offering insights into the nature of spacetime and the universe.
Homotopy is another vital concept, concerned with the idea of deforming one shape into another through continuous transformations. Two spaces are homotopy equivalent if they can be continuously
transformed into each other. This concept helps mathematicians classify spaces based on their essential structural properties rather than their precise geometric form. For example, a doughnut and a
coffee cup are homotopy equivalent because each has a single hole, illustrating how homotopy abstracts and simplifies complex shapes.
Homology, on the other hand, deals with the study of spaces through algebraic invariants. It provides a way to associate a sequence of abelian groups or modules with a given topological space,
effectively quantifying the number of holes at different dimensions. For example, in a 2-dimensional surface, homology can distinguish between objects like a sphere, which has no holes, and a torus,
which has one 1-dimensional hole and one 2-dimensional hole. This algebraic perspective is invaluable in both theoretical research and practical applications, such as data analysis and computer
In applied topology, these concepts find numerous applications. Persistent homology, for instance, is used in data science to study the shape of data. It captures the multi-scale topological features
of a dataset, providing insights that are crucial for fields like machine learning and network analysis. Similarly, in robotics, topological methods help in motion planning and the study of
configuration spaces, ensuring efficient and collision-free paths for robotic systems.
By exploring the topological aspects of higher-dimensional spaces through manifolds, homotopy, and homology, mathematicians and scientists can gain a deeper understanding of the fundamental
properties and structures that govern both abstract mathematical theories and real-world applications.
Applications in Physics and Cosmology
The exploration of higher-dimensional mathematics has profound implications in the realms of physics and cosmology. One of the most prominent applications is string theory, which suggests that the
universe operates with multiple spatial dimensions beyond the familiar three. String theory posits that particles are not point-like dots but rather one-dimensional “strings” that vibrate in
higher-dimensional spaces. These vibrations account for the different particle types and fundamental forces, offering a unified framework that integrates quantum mechanics and general relativity.
Higher-dimensional models are pivotal in addressing some of the most fundamental questions about the nature of space, time, and gravity. In the context of cosmology, these models provide insights
into the structure and evolution of the universe. For instance, the concept of extra dimensions is instrumental in the formulation of the braneworld scenario, where our observable universe is a
3-dimensional “brane” embedded in a higher-dimensional space. This model helps explain certain gravitational anomalies and potentially offers solutions to the hierarchy problem, which concerns the
disparity between the gravitational force and other fundamental forces.
Moreover, higher-dimensional mathematics plays a crucial role in understanding black holes and the fabric of spacetime. Theoretical physicists employ these models to study the properties of black
holes in higher dimensions, which can lead to novel predictions about their behavior and interactions. These investigations are not just theoretical; they have practical implications for our
understanding of gravitational waves and the overall dynamics of the cosmos.
In essence, higher-dimensional mathematics provides a rich framework for exploring and explaining the complexities of the universe. As our understanding of these principles deepens, it opens new
avenues for research and potentially groundbreaking discoveries in both physics and cosmology. The intricate dance of dimensions beyond our perception challenges and expands our grasp of reality,
pushing the boundaries of what we know about the universe’s intricate structure and underlying principles.
Computational Approaches to Higher Dimensions
Computational methods have become indispensable in the study of higher-dimensional spaces, tackling the complexities inherent in dimensions beyond the usual three. Algorithms and numerical techniques
play a pivotal role in exploring higher-dimensional data and solving multifaceted problems that arise within these expanded realms.
One of the primary challenges in this domain is the so-called “curse of dimensionality.” As the number of dimensions increases, the volume of the space expands exponentially, making it difficult to
manage and analyze. This phenomenon significantly impacts various computational tasks, such as optimization and data visualization, where the sheer volume of data can become overwhelming.
To address these challenges, a variety of advanced algorithms have been developed. Techniques such as Principal Component Analysis (PCA) and t-Distributed Stochastic Neighbor Embedding (t-SNE) are
frequently employed to reduce the dimensionality of datasets, making them more manageable without losing critical information. These methods enable the identification of underlying patterns and
structures within high-dimensional data.
In addition to dimensionality reduction techniques, modern advancements in computing, such as parallel processing and quantum computing, have brought new capabilities to the table. Parallel
processing allows for the simultaneous execution of multiple computational tasks, significantly speeding up the analysis of higher-dimensional datasets. Quantum computing, still in its nascent
stages, promises to revolutionize how we handle high-dimensional problems by leveraging the principles of superposition and entanglement to perform complex calculations more efficiently.
Moreover, machine learning algorithms, particularly deep learning models, have shown great promise in processing high-dimensional data. These models can automatically learn relevant features from raw
data, effectively navigating the complexities of higher-dimensional spaces. Techniques such as convolutional neural networks (CNNs) and recurrent neural networks (RNNs) are being adapted to handle
higher-dimensional inputs, providing new insights and solutions to previously intractable problems.
In summary, the convergence of advanced algorithms, enhanced computational power, and innovative machine learning techniques is progressively mitigating the challenges posed by higher-dimensional
spaces. As computational methods continue to evolve, our ability to explore and understand the mathematics of higher dimensions will undoubtedly expand, opening up new frontiers in science and
Conclusion and Future Directions
The exploration of higher-dimensional mathematics opens up a realm of possibilities that extends far beyond our three-dimensional perception. Throughout this article, we have delved into the
fundamental principles and applications of higher-dimensional spaces, highlighting their significance in theoretical and applied mathematics. We examined essential concepts such as n-dimensional
Euclidean spaces, hyperspheres, and manifolds, showcasing their mathematical elegance and practical relevance.
One of the key takeaways is the profound impact higher-dimensional mathematics has on various scientific and technological fields. In physics, for example, the study of higher dimensions is crucial
for theories like string theory and M-theory, which propose that our universe may include more than four dimensions. These theories aim to unify the fundamental forces of nature and provide a deeper
understanding of the cosmos. Similarly, in computer science, higher-dimensional spaces are integral to fields like machine learning and data visualization. Techniques such as principal component
analysis (PCA) and t-distributed stochastic neighbor embedding (t-SNE) rely on the manipulation of high-dimensional data to uncover patterns and insights.
Looking ahead, the future of higher-dimensional mathematics is brimming with potential. Ongoing research is continuously uncovering new dimensions of knowledge and application. Emerging theories,
such as those involving quantum computing and holography, are pushing the boundaries of what we understand about the fabric of reality. Additionally, the advent of more advanced computational tools
and techniques is enabling researchers to explore higher dimensions with greater precision and creativity.
The implications of higher-dimensional studies are vast and far-reaching. As we continue to unravel the complexities of higher dimensions, we can expect to see groundbreaking advancements in fields
such as artificial intelligence, cryptography, and cosmology. These advancements not only deepen our theoretical understanding but also translate into practical innovations that can transform
technology and society. The journey through higher-dimensional mathematics is an exhilarating one, promising to expand our horizons and challenge our perceptions in ways we have yet to imagine. | {"url":"https://diversedaily.com/exploring-the-mathematics-of-higher-dimensions-principles-and-applications/","timestamp":"2024-11-09T22:39:47Z","content_type":"text/html","content_length":"165915","record_id":"<urn:uuid:d8a1f6a2-2e30-4a35-b49b-230d63ee652f>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00669.warc.gz"} |
List of urban areas in Sweden by population - Wikipedia
Sustainability Report 2018 English April 3, 2019 - Tethys Oil
Advertising and marketing in Thailand - Buying and Selling
Accessible small towns: settlements of between 3,000 and 10,000 people and within 30 The current population is 85 frogs, and the relative growth rate is 18% per year. formula for population growth
with n0 = 10,000, t = 1, and n(t) = 25,000,. 21 Jun 2017 The World Population Prospects: The 2017 Revision, published by the UN Department of Economic and Social Affairs, provides a comprehensive 9
Aug 2019 Aurora, Town [Census subdivision], Ontario and York, Regional municipality Total - Knowledge of official languages for the total population excluding $10,000 to $19,999, 6,155, 2,585, 3,565,
149,160, 63,105, 86,055. The second, larger wave began 10,000 years ago as the discovery of agriculture caused a population boom and a need to plow wildlife habitats, divert streams, The data is
from the US Census. Below are 1,213 Ohio cities ranked 1 through 1,036 (there are some ties). You can copy and paste this list directly into your We do! Looking for a list of cities, counties or
zips in Idaho?
If the population increases by 1 0 % every year, then the population of the town after three years will be The current population of a town is 10,000. If the population, P, increases by 20% each
year, which equation could be used to find the population after t years? Solution: For answer to this question we need to use the compound interest formula L 4 :s E ; l Where 4-start population, -is
the effective interest rate per period, and -represents the SOLUTION: The current population of a town is 10,000 people. The population is decreasing according to the function P (t)=10,000(2^-0.03t)
where t is the number of years from the present and The current population of a town is 10,000 and its growth in years can be represented by p (t) = 10,000 (0.2)^t, where t is the. English,
20.10.2019 21:10 nickboy52210.
慍 2 : ´ ² 瑣瑣 Part 2 : Population The data on population
Express the growth in popupation compared to the starting population, i.e. 10,000/40,000. Reduce this fraction to lowest terms. 10,000/40,000=1/4.
List of urban areas in Sweden by population - Wikipedia
During the second year, it decreased by 20% and increased by 30% during the third year. What is the population after 3 years?a)11440b)12440c)13450d)14440Correct answer is option 'A'.
Math. The population, P (in thousands), of a town can be modeled by P= 2|t-6|+4, where t=0 represents 1990. The population of a town is increasing at a rate given by Upper P prime left parenthesis t
right parenthesis equals 46 e Superscript 0.015 tP'(t)=46e0.015t, where P is the population t years after the The current population of a small town is 1,554 people. It is believed that town's
population is tripling every 11 years. Approximate the population of the town 6 years from now. A town has a population of 12000 and grows at 5% every year.
Kontek ljungby anställda
Convert .25 to percent by multipying by 100 and putting a percent sign after the number.
The population of a city has been decreasing exponentially since 1990. In 1990, the population was 1,000,000. Answer: Step-by-step explanation: If the population of the town increases 5% Every year,
Bokelunds transportvagnar
hur mycket kan man ta betalt för hundvaktsveriges andre vice talmanekblom bak cycle ergometer testbiblioteket falu lasarettbetonggjutningar moderna museets trädgårdmedlemsavgifter avdragsgilla
malardalens hogskola eskilstuna
Government of the People's Republic of Bangladesh Ministry
THE CROWN, MAORI, AND MAHURANGI 1840-1881
Its current population is 700,000. Town B, which currently has 550,000 residents, increases by 6.5% - 11424770 we need to begin with the percentage formula. Percent means out of 100 so we need to
find 15 out of 100. 15/100 * 32,000 = 4,800 annual increase 4,800 + 32,000 = 36,800 Given this, 15/100 * 36,800 = 5,520 36,800 + 5,520 = 42,320 Therefore, with a The population of a town is 10,000. | {"url":"https://hurmanblirrikaztsi.netlify.app/34868/55621","timestamp":"2024-11-04T23:40:41Z","content_type":"text/html","content_length":"9582","record_id":"<urn:uuid:95f7d7fa-513a-4caa-98ec-b829fc7c3695>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00515.warc.gz"} |
Wisconsin Legislature: 121.91(2m)
121.905(3)(a)2.b.b. For each of those school districts, divide the result in subd. 2. a. by the number of pupils enrolled in that school district in the previous school year.
121.905(3)(a)2.c.c. For each of those school districts, multiply the result in subd. 2. b. by the number of pupils enrolled in that school district in the previous school year who resided in
territory that was detached to create the new school district.
121.905(3)(a)3.3. For a school district from which territory was detached to create a new school district under s. 117.105, for the school year beginning with the effective date of the
reorganization, perform the following calculations:
121.905(3)(a)3.a.a. Calculate the sum under subd. 1. for each of the school districts from which territory was detached to create the new school district.
121.905(3)(a)3.b.b. For each of those school districts, divide the result in subd. 3. a. by the number of pupils enrolled in that school district in the previous school year.
121.905(3)(a)3.c.c. For each of those school districts, multiply the result in subd. 3. b. by the number of pupils enrolled in that school district in the previous school year who did not reside in
territory that was detached to create the new school district.
121.905(3)(b)1.1. Except as provided under subds. 2. and 3., divide the result in par. (a) 1. by the sum of the average of the number of pupils enrolled in the 3 previous school years and the number
of pupils enrolled who were school district residents and solely enrolled in a special education program provided by a county children with disabilities education board program in the previous school
121.905(3)(b)2.2. For a school district created under s. 117.105, for the school year beginning with the effective date of the reorganization, divide the result in par. (a) 2. by the number of pupils
who in the previous school year were enrolled in a school district from which territory was detached to create the new school district and who resided in the detached territory; for the school year
beginning on the first July 1 following the effective date of the reorganization, divide the result in par. (a) 2. by the number of pupils in the previous school year; and for the school year
beginning on the 2nd July 1 following the effective date of the reorganization, divide the result in par. (a) 2. by the average of the number of pupils in the 2 previous school years.
121.905(3)(b)3.3. For a school district from which territory was detached to create a new school district under s. 117.105, for the school year beginning with the effective date of the
reorganization, divide the result in par. (a) 3. by the number of pupils who in the previous school year were enrolled in the school district and who did not reside in territory that was detached to
create the new school district; for the school year beginning on the first July 1 following the effective date of the reorganization, divide the result in par. (a) 3. by the number of pupils enrolled
in the previous school year; and for the school year beginning on the 2nd July 1 following the effective date of the reorganization, divide the result in par. (a) 3. by the average of the number of
pupils enrolled in the 2 previous school years.
121.905(3)(c)2.2. For the limit for the 1996-97 school year, add $206 to the result under par. (b).
121.905(3)(c)3g.3g. For the limit for the 2009-10 or 2010-11 school year, add $200 to the result under par. (b).
121.905(3)(c)3r.3r. For the limit for the 2011-12 school year, multiply the result under par. (b) by 0.945.
121.905(3)(c)4.4. For the limit for the 2012-13 school year, add $50 to the result under par. (b).
121.905(3)(c)5.5. For the limit for the 2013-14 school year and the 2014-15 school year, add $75 to the result under par. (b).
121.905(3)(c)6.6. For the limit for each of the 2015-16 to 2018-19 school years, for the 2021-22 school year, and for any school year thereafter, make no adjustment to the result under par. (b).
121.905(3)(c)7.7. For the limit for the 2019-20 school year, add $175 to the result under par. (b).
121.905(3)(c)8.8. For the limit for the 2020-21 school year, add $179 to the result under par. (b).
121.905(4)(a)(a) A school district that is exempt from the revenue limits under sub. (2) may not increase its base revenue per member to an amount that is greater than its revenue ceiling.
121.905(4)(b)1.1. A school district may increase its revenue ceiling by following the procedures prescribed in s. 121.91 (3).
121.905(4)(b)2.2. The department shall, under s. 121.91 (4), adjust the revenue ceiling otherwise applicable to a school district under this section as if the revenue ceiling constituted a revenue
limit under s. 121.91 (2m).
121.91(2m)(a)(a) Except as provided in subs. (3) and (4), no school district may increase its revenues for the 1995-96 school year to an amount that exceeds the amount calculated as follows:
121.91(2m)(a)1.1. Divide the sum of the amount of state aid received in the previous school year and property taxes levied for the previous school year, excluding funds described under sub. (4) (c),
by the average of the number of pupils in the 3 previous school years.
121.91(2m)(a)4.4. Multiply the result under subd. 3. by the average of the number of pupils in the current and the 2 preceding school years.
121.91(2m)(b)(b) Except as provided in subs. (3) and (4), no school district may increase its revenues for the 1996-97 school year to an amount that exceeds the amount calculated as follows:
121.91(2m)(b)1.1. Divide the sum of the amount of state aid received in the previous school year and property taxes levied for the previous school year, excluding funds described under sub. (4) (c),
by the average of the number of pupils in the 3 previous school years.
121.91(2m)(b)3.3. Multiply the result under subd. 2. by the average of the number of pupils in the current and the 2 preceding school years.
121.91(2m)(c)(c) Except as provided in subs. (3), (4) and (6), no school district may increase its revenues for the 1997-98 school year to an amount that exceeds the amount calculated as follows:
121.91(2m)(c)1.1. Divide the sum of the amount of state aid received in the previous school year and property taxes levied for the previous school year, excluding funds described under sub. (4) (c),
by a number calculated by adding the number of pupils enrolled in the 3 previous school years, subtracting from that total the number of pupils attending private schools under s. 119.23 in the 4th,
3rd and 2nd preceding school years, and dividing the remainder by 3.
121.91(2m)(c)4.4. Multiply the result under subd. 3. by a number calculated by adding the number of pupils enrolled in the current and the 2 preceding school years, subtracting from that total the
number of pupils attending private schools under s. 119.23 in the 3 previous school years, and dividing the remainder by 3.
121.91(2m)(d)(d) Except as provided in subs. (3) and (4), no school district may increase its revenues for the 1998-99 school year to an amount that exceeds the amount calculated as follows:
121.91(2m)(d)1.1. Divide the sum of the amount of state aid received in the previous school year and property taxes levied for the previous school year, excluding funds described under sub. (4) (c),
by a number calculated by adding the number of pupils enrolled in the 3 previous school years, subtracting from that total the number of pupils attending charter schools under s. 118.40 (2r) and
private schools under s. 119.23 in the 4th, 3rd and 2nd preceding school years and dividing the remainder by 3.
121.91(2m)(d)2.2. Multiply the amount of the revenue increase per pupil allowed under this subsection for the previous school year by the sum of 1.0 plus the allowable rate of increase under s.
73.0305 expressed as a decimal.
121.91(2m)(d)4.4. Multiply the result under subd. 3. by a number calculated by adding the number of pupils enrolled in the current and the 2 preceding school years, subtracting from that total the
number of pupils attending charter schools under s. 118.40 (2r) and private schools under s. 119.23 in the 3 previous school years and dividing the remainder by 3.
121.91(2m)(e)(e) Except as provided in subs. (3), (4), and (8), no school district may increase its revenues for the 2008-09 school year to an amount that exceeds the amount calculated as follows:
121.91(2m)(e)1.1. Divide the sum of the amount of state aid received in the previous school year and property taxes levied for the previous school year, excluding property taxes levied for the
purpose of s. 120.13 (19) and excluding funds described under sub. (4) (c), by the average of the number of pupils enrolled in the 3 previous school years.
121.91(2m)(e)2.2. Multiply the amount of the revenue increase per pupil allowed under this subsection for the previous school year by the sum of 1.0 plus the allowable rate of increase under s.
73.0305 expressed as a decimal.
121.91(2m)(e)4.4. Multiply the result under subd. 3. by the average of the number of pupils enrolled in the current and the 2 preceding school years.
121.91(2m)(f)(f) Except as provided in subs. (3), (4), and (8), no school district may increase its revenues for the 2009-10 school year or for the 2010-11 school year to an amount that exceeds the
amount calculated as follows:
121.91(2m)(f)1.1. Divide the sum of the amount of state aid received in the previous school year and property taxes levied for the previous school year, excluding property taxes levied for the
purpose of s. 120.13 (19) and excluding funds described under sub. (4) (c), by the average of the number of pupils enrolled in the 3 previous school years.
121.91(2m)(f)3.3. Multiply the result under subd. 2. by the average of the number of pupils enrolled in the current and the 2 preceding school years.
121.91(2m)(g)(g) Except as provided in subs. (3), (4), and (8), no school district may increase its revenues for the 2011-12 school year to an amount that exceeds the amount calculated as follows:
121.91(2m)(g)1.1. Divide the sum of the amount of state aid received in the previous school year and property taxes levied for the previous school year, excluding property taxes levied for the
purpose of s. 120.13 (19) and excluding funds described under sub. (4) (c), by the average of the number of pupils enrolled in the 3 previous school years.
121.91(2m)(g)3.3. Multiply the result under subd. 1. by the average of the number of pupils enrolled in the current and the 2 preceding school years.
121.91(2m)(h)(h) Except as provided in subs. (3), (4), and (8), no school district may increase its revenues for the 2012-13 school year to an amount that exceeds the amount calculated as follows:
121.91(2m)(h)1.1. Divide the sum of the amount of state aid received in the previous school year and property taxes levied for the previous school year, excluding property taxes levied for the
purpose of s. 120.13 (19) and excluding funds described under sub. (4) (c), by the average of the number of pupils enrolled in the 3 previous school years.
121.91(2m)(h)4.4. Multiply the result under subd. 3. by the average of the number of pupils enrolled in the current and the 2 preceding school years.
121.91(2m)(hm)(hm) Except as provided in subs. (3), (4), and (8), no school district may increase its revenues for the 2013-14 school year or for the 2014-15 school year to an amount that exceeds the
amount calculated as follows:
121.91(2m)(hm)1.1. Divide the sum of the amount of state aid received in the previous school year and property taxes levied for the previous school year, excluding property taxes levied for the
purpose of s. 120.13 (19) and excluding funds described under sub. (4) (c), by the average of the number of pupils enrolled in the 3 previous school years.
121.91(2m)(hm)3.3. Multiply the result under subd. 2. by the average of the number of pupils enrolled in the current school year and the 2 preceding school years.
121.91(2m)(i)(i) Except as provided in subs. (3), (4), and (8), no school district may increase its revenues for the 2015-16 school year or for any school year thereafter to an amount that exceeds
the amount calculated as follows:
121.91(2m)(i)1.1. Divide the sum of the amount of state aid received in the previous school year and property taxes levied for the previous school year, excluding property taxes levied for the
purpose of s. 120.13 (19) and excluding funds described under sub. (4) (c), by the average of the number of pupils enrolled in the 3 previous school years.
121.91(2m)(i)2.2. Multiply the result under subd. 1. by the average of the number of pupils enrolled in the current and the 2 preceding school years.
121.91(2m)(im)(im) Notwithstanding par. (i) and except as provided in subs. (3), (4), and (8), a school district cannot increase its revenues for the 2019-20 school year to an amount that exceeds the
amount calculated as follows:
121.91(2m)(im)1.1. Divide the sum of the amount of state aid received in the previous school year and property taxes levied for the previous school year, excluding property taxes levied for the
purpose of s. 120.13 (19) and excluding funds described under sub. (4) (c), by the average of the number of pupils enrolled in the 3 previous school years.
121.91(2m)(im)3.3. Multiply the result under subd. 2. by the average of the number of pupils enrolled in the current school year and the 2 preceding school years.
121.91(2m)(j)(j) Notwithstanding par. (i) and except as provided in subs. (3), (4), and (8), a school district cannot increase its revenues for the 2020-21 school year-year2425 to an amount that
exceeds the amount calculated as follows:
121.91(2m)(j)1.1. Divide the sum of the amount of state aid received in the previous school year and property taxes levied for the previous school year, excluding property taxes levied for the
purpose of s. 120.13 (19) and excluding funds described under sub. (4) (c), by the average of the number of pupils enrolled in the 3 previous school years.
121.91(2m)(j)3.3. Multiply the result under subd. 2. or 2m., whichever is applicable, by the average of the number of pupils enrolled in the current school year and the 2 preceding school years.
121.91(2m)(r)1.1. Notwithstanding pars. (i) to (j), if a school district is created under s. 117.105, its revenue limit under this section for the school year beginning with the effective date of the
reorganization shall be determined as follows except as provided under subs. (3) and (4):
121.91(2m)(r)1.a.a. Divide the result under s. 121.905 (3) (a) 2. by the total number of pupils who in the previous school year were enrolled in a school district from which territory was detached to
create the new school district and who resided in the detached territory.
121.91(2m)(r)1.b.b. Add an amount equal to the amount of revenue increase per pupil allowed under this subsection for the previous school year multiplied by the sum of 1.0 plus the allowable rate of
increase under s. 73.0305 expressed as a decimal to the result under subd. 1. a., except that in calculating the limit for the 2013-14 school year and the 2014-15 school year, add $75 to the result
under subd. 1. a., in calculating the limit for the 2019-20 school year, add $175 to the result under subd. 1. a., in calculating the limit for the 2020-21 school year, add $179 to the result under
subd. 1. a., and in calculating the limit for the 2023-24 school year and the 2024-25 school year, add $325 to the result under subd. 1. a. In the 2015-16 to 2018-19 school years, the 2021-22 school
year, the 2022-23 school year, the 2025-26 school year, and any school year thereafter, make no adjustment to the result under subd. 1. a.
121.91(2m)(r)1.c.c. Multiply the result under subd. 1. b. by the number of pupils who in the previous school year were enrolled in a school district from which territory was detached to create the
new school district and who resided in the detached territory, or by the number of pupils enrolled in the new school district in the current school year, whichever is greater.
121.91(2m)(r)2.2. If a school district is created under s. 117.105, the following adjustments to the calculations under pars. (i) to (j) apply for the 2 school years beginning on the July 1 following
the effective date of the reorganization:
121.91(2m)(r)2.a.a. For the school year beginning on the first July 1 following the effective date of the reorganization the number of pupils in the previous school year shall be used under pars. (i)
1., (im) 1. and (j) 1. instead of the average of the number of pupils in the 3 previous school years, and for the school year beginning on the 2nd July 1 following the effective date of the
reorganization the average of the number of pupils in the 2 previous school years shall be used under pars. (i) 1., (im) 1. and (j) 1. instead of the average of the number of pupils in the 3 previous
school years.
/statutes/statutes/121 true statutes /statutes/statutes/121/vii/91/2m Chs. 115-121, Public Instruction statutes/121.91(2m) statutes/121.91(2m) section true | {"url":"https://docs.legis.wisconsin.gov/statutes/statutes/121/vii/91/2m","timestamp":"2024-11-09T01:01:21Z","content_type":"text/html","content_length":"87416","record_id":"<urn:uuid:2696c5d7-20e3-41ee-9e47-4c19d0efa711>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00861.warc.gz"} |
Interactive Self-Study Module: Characteristics of a Boundary Layer over a Flat Surface
This module uses a screencast to explain how fluid flows over a flat surface (such as air flowing over the top of a flat roof of a car). Your retention of material in this module will increase if you
write down reasons for your answers to ConcepTests and you try to solve the example problems before watching the screencast solutions.
We suggest using the learning resources in the following order:
1. Attempt to answer the multiple choice ConcepTest and solve the example problem before watching the screencast or working with the simulation.
2. Watch the screencast that describes boundary layers.
3. Look over the important equations for shear rates and shear stresses.
4. Try to solve the example problems before watching the solutions in the screencasts.
5. Answer the ConcepTests.
6. Look at the list of key points, but only after you try to list the key points yourself.
• Before you can understand how a fluid flowing over an object results in a drag force you first need to understand how the fluid interacts with the surface of the object. We will start with a
simple case, in which fluid flows over a flat surface.
• This module is primarily intended for a Fluid Mechanics course. It may also provide a review and background information for convection in a Heat Transfer course.
Before studying this module, you need to:
• Describe in words and mathematically the meaning of a no-slip boundary condition.
• Describe in words and mathematically the meanings of a shear rate and a shear stress.
• By looking at a velocity profile, be able to identify where shear stresses are highest and where they are zero.
• Review the self-study module, Viscosity and Shear Stress.
After studying this module, you should be able to:
• Describe in words the velocity profile over a flat surface.
• Identify where the shear stress is highest and where it is negligibly small for laminar flow over a flat surface.
• Look at a velocity profile over a flat surface and have a fair understanding of where the boundary layer exists.
• Make a list of the fluid properties that affect the velocity profile over a flat surface and describe how changes in each property affect the velocity profile and the shape of the boundary layer. | {"url":"https://learncheme.com/quiz-yourself/interactive-self-study-modules/boundary-layer-characteristics/boundary-layer-characteristics-introduction/","timestamp":"2024-11-10T11:46:39Z","content_type":"text/html","content_length":"77654","record_id":"<urn:uuid:77596765-bff9-4f67-87b8-d3d06bdeaa85>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00825.warc.gz"} |
Markowitz Efficient Frontier
Investments are a balancing act between risk and return. In this constantly changing landscape, it is difficult to analyze and make investment decisions. However, one tool that has proven to be a
useful guide is the Markowitz Mean-Variance model and its graphical representation — the Efficient Frontier.
The concept was developed in 1952 by Nobel Prize winner Harry Markowitz and is a cornerstone of modern portfolio theory (MPT). Markowitz showed that the risk of an investment portfolio is not
determined by the average risk of the underlying assets, but by the extent to which they move together — their correlations. Diversifying a portfolio across imperfectly correlated assets allows
investors to increase their expected returns without increasing their risk. Or reduce their risk without reducing their expected returns. Markowitz argued that diversification is the only free lunch
in investing, since increasing expected returns usually requires taking on more risk.
This article breaks down the efficient frontier and explains how you can use our free dashboard to achieve a practical application for your investment strategy. In addition, the criticisms of the
method are described.
What is the efficient frontier?
The efficient frontier evaluates portfolios on a coordinate plane. The risk is plotted on the x-axis, while the return is plotted on the y-axis. Three important factors are considered to calculate
the efficient frontier:
• Expected return of individual assets (we estimate the expected return as an annualized return using the arithmetic average)
• Risk of individual assets (we calculate the annualized volatility, i.e. the standard deviation of the asset price on an annual basis)
• Sharpe Ratio of individual assets (by comparing the excess performance returns with the total standard deviation of the portfolio, we calculate the risk-adjusted performance)
The individual formulas can be viewed here.
The efficient frontier graphically represents portfolios that maximize the return for the risk taken and highlights the opportunities for diversification. To take advantage of the efficient frontier,
a risk-averse investor will choose investments that are more to the left. A risk-taking investor, on the other hand, would choose investments that are more to the right.
What does the practical application look like?
As a practical example, we use one of the popular portfolio strategies: the two-fund portfolio, which consists of two assets.
• Asset A: Vanguard Total Bond Market Index Fund (BND)
• Asset B: Vanguard Total World Stock Index Fund (VT)
and calculate the risks and returns as well as the Sharpe Ratio.
Particulars Asset A: Vanguard Total Bond Market Index Fund (BND) Asset B: Vanguard Total World Stock Index Fund (VT)
Expected Return 1.3% 13.3%
Volatility 6.8% 20.2%
Sharpe Ratio 0.190 0.658
Let us now give weights to the assets to demonstrate the portfolio in different allocations:
Portfolio Asset A: (BND) Allocation (in %) Asset B: (VT) Allocation (in %)
These five portfolios should be seen as examples, as many allocations can be derived from the efficient frontier. To make it easier to recognize the different allocations of a portfolio, we have
developed the Transition Map. All allocations can be easily recognized here.
The risk and return numbers of the 5 portfolios:
Portfolios Risk Return
1: 100% (VT) 20.2% 13.3%
2: 75% (VT), 25% (BND) 15.6% 10.3%
3: 50% (VT), 50% (BND) 11.3% 7.3%
4: 75% (BND), 25% (VT) 7.9% 4.3%
5: 100% (BND) 6.7% 1.8%
In this illustration, for the sake of simplicity and better understanding, we have assumed that the portfolio consists of only two assets A and B. We can similarly construct a portfolio for multiple
assets and plot it to reach the frontier.
What is the criticism of the model?
The efficient frontier and modern portfolio theory are based on many assumptions that may not fully reflect reality. One of the assumptions, for example, is that asset returns follow a normal
distribution. In reality, the returns of securities (also known as tail risk) can deviate from the mean by more than three standard deviations. This is referred to as a leptokurtic distribution or a
distribution with high tail risk.
In addition, Markowitz makes several assumptions in his theory, such as that investors are rational and avoid risk whenever possible, that there are not enough investors to influence market prices,
and that investors have unlimited access to credit and loans at the risk-free rate. However, reality proves that the market includes irrational and risk-taking investors, that there are large market
participants who could influence market prices, and that there are investors who do not have unlimited access to bonds and loans.
Of course, the Efficient Frontier is not enough to make an investment decision on this basis alone. Nevertheless, it can be a useful data point in the decision-making process. The model is not
perfect, but it offers an unemotional way of looking at different risk/return ratios. And so it does have its uses.
If you want to check your own portfolio or a planned investment strategy using the Efficient Frontier, you can easily calculate this for free with our dashboard. | {"url":"https://portfoliometrics.net/blog/efficient-frontier","timestamp":"2024-11-02T11:23:46Z","content_type":"text/html","content_length":"132506","record_id":"<urn:uuid:84611204-a369-4ba4-a079-6d888269a823>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00193.warc.gz"} |
How many cubic feet is 1 MMBtu?
The answer is: 1 mmBTU equals 1,000.00 cu ft N.G.
How do you convert m3 to MMBtu?
Converting From Cubic Meters to MMBtu First, if need be, convert cubic meters to cubic feet by multiplying by 35.31. Then, multiply this by the most recent conversion factor to get MBtu per cf (here,
1.036). Finally, determine how many MMBtu this is by dividing by 1,000, since 1 MBtu = 0.001 MMBtu.
What is MMBtu price?
Currently, the price of domestic natural gas stands at $2.9 per MMBtu.
How do you convert MCF to dollars?
Here’s how to convert between different unit measurements and prices of natural gas:
1. $ per Ccf divided by 1.037 equals $ per therm.
2. $ per therm multiplied by 1.037 equals $ per Ccf.
3. $ per Mcf divided by 1.037 equals $ per MMBtu.
4. $ per Mcf divided by 10.37 equals $ per therm.
5. $ per MMBtu multiplied by 1.037 equals $ per Mcf.
What is MMBTU gas bill?
What is MMBtu? MMBtu is acronym for Metric Million British Thermal Unit, and it is a unit traditionally used to measure heat content or energy value. It is widely associated with measurement of
natural gas in the energy terms globally.
How do you calculate natural gas price?
Let’s say you have a furnace with a BTU rating of 100,000 and your gas bill is measured in MCFs. If one MCF costs $9.00: Divide the price per MCF by 1,028,000 to get the price per BTU:
$0.00000875486. Multiply that by 100,000 to get the price per hour you’ll pay to run the furnace: about 87 cents.
How do you calculate MMBtu on a gas bill?
The conversion/comparison between scm and MMBtu can be done based on 1 MMBtu = 252,000 KiloCalorie. Gas consumption in SCM and MMBtu would be separately mentioned in your gas bill/invoice. Assuming
NCV of 8350 kcal/scm, 1 MMBtu will approximately be equal to 30.12 SCM. .
How many BTU are in a m3 of natural gas?
35300 BTU
For 1 m3 of natural gas, you would multiply by 1 m3 by 0.0353, which would give you 0.0353 MMBtu. To get this number in units of BTU, you would multiply it by 1000000, which would give you a total of
35300 BTU in an m3 of gas.
What is the difference between MCF and MMBtu?
Mcf – equals the volume of 1,000 cubic feet (cf) of natural gas. MMBtu – equals 1,000,000 British thermal units (Btu). (One Btu is the heat required to raise the temperature of one pound of water by
one degree Fahrenheit.)
How much is 1000 cubic feet of natural gas?
1.037 MMBtu
One thousand cubic feet (Mcf) of natural gas equals 1.037 MMBtu, or 10.37 therms. | {"url":"https://www.blfilm.com/2020/02/09/how-many-cubic-feet-is-1-mmbtu/","timestamp":"2024-11-05T01:25:27Z","content_type":"text/html","content_length":"63099","record_id":"<urn:uuid:55fe7923-ffd7-4904-9235-9f6107d935fa>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00643.warc.gz"} |
How to analyze harmonics in power systems? | Pay Someone To Do Electrical Engineering Assignment
How to analyze harmonics in power systems? This would be a very useful technique to gain for future study. Vogts, I agree, can not be seen to produce harmonics – regardless of the form that they
undergo on the surface in question (substance, light, etc.) etc., which will lead to difficulties in obtaining sufficient measurement of characteristics and for their classification. What I mean… is
that there is considerable danger for both (the ideal) and for anyone (the ideal) in designing such instruments. For the full discussion, here’s a a fantastic read to the article, by Tom Penninger —
and a good start. In the following section, I want to look at some mechanisms for extracting harmonics from power systems, what I think are the differences in optical properties of lenses discussed
in the linkup sections. My approach would be to use two lenses, one with pupil adjustment (with an offset). Other I propose to use focused lens, where the pupil is aligned substantially across the
cornea, a small amount of illumination is required to obtain the precise image of the target object (one should try to adjust the focus so as to have the instrument look better or at least to be on
the correct page). And a lens that is designed to exhibit a constant refractive power should be used. On the other hand, there are two glasses, one that shows the same distortion useful content the
other to get the same aberrations, and the refractive power is lower than for the narrow-lens (although the sharpness decrease around the object is small). What are the general features that you get
from focusing on lenses? What do you do about the “weak” refraction of the lens while moving to the pupil position? What do you do about the interaction? I would like to have a lot of discussion on
harmonics and lens tuning. In this section I am on a topic that I frequently hear used. Most use a tuning parameter based on the aspectHow to analyze harmonics in power systems? Empirical work has
produced information on harmonics in systems of power systems and in the presence of different levels of the magnetic moment. In the laboratory, an electron analyzer compares harmonics in the flow
field to selected harmonics in a phase-space phase diagram prepared from phase-space data. It compares the measured phase factors, which are often referred to as harmonics, of the flow field and
other information derived in the phase-space phase diagram. In this work we apply this approach click reference analyze harmonics of specific her response sources located deep within the walls of
stationary glass cylinder surfaces in a two and three-dimensional fluid, and compare these harmonics with a conventional harmonic determination technique.
Take Online Courses For You
An application is made to an analysis tube with a permanent magnet in the wall and to analyze changes in the position of an analysis tube relative to a stationary object. The agreement is good,
giving rise to data for this article position, and a similar agreement is found when the point source is placed in a very large wall region where even small changes occur. We describe these effects
and discuss our results in the context of the fluid behaviour during the development of the harmonics of point sources located near to the wall. In particular we argue that some harmonics internet
naturally in the systems known to develop during the development of electrical leads and electrical system equipment in concrete and in electrical power or in the construction of a wall.How to
analyze harmonics in power systems? A go to website characteristic of harmonics is the conservation of harmonics that is a consequence of the law of the momentum in the current at the center of the
current. Harmonics in the current, with their application to harmonics in the system, provides these dissipative principles for the system. Indeed, in the steady state, the momentum in the system can
be applied for non-stochastic disturbances in a series representation. The harmonic moments in this way minimize link system energy. This can lead to an effective description of energy conservation
as a fundamental requirement for the proper behavior of the system as a whole. In harmonic systems, one can distinguish two distinct regimes: the equilibrium in which the system is invariant. This
one can be resolved in quite a different approximation – see the example of Eq. (\[eq:linear\]) below. The harmonic degree of freedom is either the state $x\wedge y$ or more info here state of the
thermal vortex $x\wedge x + \epsilon x\wedge y$ – see Fig. \[fig:nondgam\]. The former is the mass of the heat source moving in a body. The latter is a characteristic impedance pattern used in the
recent research of Ref. [@hidalgo2011unpaired]. A way to overcome this two-step procedure is to first construct a collection of harmonics that then depends only on the momentum in the system. Such a
collection would require a complex but manageable task. Alternatively, one could use the result of one iteration of the harmonic collection to reconstruct the harmonics that corresponds to the mode
of light.
What Is Your Online Exam Experience?
Such a procedure would provide very low-dimensional data that could be used to compute the evolution equations for general time-varying harmonic systems. However, as online electrical engineering
homework help in detail above, one must address the issue of bound interactions between systems by minimizing the integrals. Although the case of dynamical balance has not been solved in this work,
we can examine that problem in our work, as shown in the following. Lemma \[lemma:dr-momentum\](2) shows that the momentum of a harmonic beam propagating in the form of a viscous membrane is only
dependent on the background potential, $\varepsilon \tilde{G}$. This fact shows again that all harmonics of a nonlinearity are also non-oscillating. But let us now address the problem of bound
interaction between systems. The term “bound energy conservation” could be reduced by perturbing the flux density of heat through the boundary. In general, the solution of anharmonic systems on
non-equilibrium backgrounds is much more difficult to deal with, and the solution of the background that describes a harmonic system using a boundary is much more difficult to solve. Furthermore, due
to the fact that in the steady state, the momentum is not included | {"url":"https://electricalassignments.com/how-to-analyze-harmonics-in-power-systems","timestamp":"2024-11-04T19:54:35Z","content_type":"text/html","content_length":"145227","record_id":"<urn:uuid:e589106f-4edd-4c70-b62a-414bd022984e>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00163.warc.gz"} |
Insulation Savings Tools
A very low cost/no cost option for saving energy in steam and hot water systems is to simply repair insulation. But for insulation on things like control valves, the repair is less likely to persist
because of the need to occasionally remove the insulation to perform service work.
Removable fitting covers offer an attractive alternative for insulating items that may require service down the road in new installations and as an alternative repair strategy for existing systems.
Of course, if you are need to justify the improvement or repair, you frequently need to provide an estimate of the savings that will be achieved by installing insulation or fitting covers. In general
terms, the losses will be proportional to the thermal resistance of the insulation, the temperature difference, and the area of surface that is exposed and will now be insulated. The relationship is
fairly straight forward and a very common form of the basic relationship that is used when assessing the losses through the walls of a building looks something like this.
That relationship has it's roots in Fourier's Law of heat conduction, which looks like this.
The negative sign ahead of k is to satisfy the law of thermodynamics that says heat must flow from warmer temperatures to cooler temperatures.
For example, if a hot surface had 1 inch of insulation covering it, and you plotted temperature vs. distances from the hot surface as you moved through the insulation, you would get a line that
sloped from high on the left side to low on the right side assuming your coordinate system had temperature on the y axis with increasing temperature going up and distance on the x axis with
increasing distance going right. (upper figure). In other words, as your distance from the hot side increased (the x value), the temperature at a given point would have decreased (the y value)
relative to what it was at the hot surface.
The change in y relative to the change in x is termed the "slope" of the line and in mathematics, a line defined by a relationship where the y value decreases as the x value increases is said to
have a negative slope. Since in our case, the change in y relative to the change in x is the thermal gradient, for the hot surface with insulation on it, the thermal gradient would be negative.
For a cold pipe, the gradient would be the other way and would be a positive slope and gradient (lower illustration).
As a result, when you evaluated the expression for a hot pipe, the heat transfer would be considered positive (negative k times a negative gradient gives a positive number); i.e. energy flowed away
from the higher temperature area to the lower temperature area, which satisfies the thermodynamic law.
When you apply Fourier's law to something like a flat wall with a layer of insulation on it that is uniform and where the wall can be considered large enough that it is reasonable to assume that all
of the energy is flowing perpendicular to the face of the wall, then when you integrate the equation, you end up with the very familiar relationship I started out with.
But when you apply Fourier's equation to a pipe, a complication comes up. Consider a 4 inch pipe that is at 180°F and is insulated with 1 inch of insulation and suspended in a 75°F space (top
Let's remove the insulation and select an area of the pipe to study, specifically the area in red in the middle image. Energy leaving the pipe from the red area will flow radially outward through
the insulation to the surrounding cooler space in accordance with the first law of thermodynamics.
But because straight lines radiating from the center of the pipe diverge, the area that the energy flow passes through when when it leaves the insulation jacket to enter the space is larger than the
area on the surface of the pipe.
Specifically, the area of the red surface in the middle picture is about 35 square inches, but the area that you create when you project the for corners of the red rectangle radially out through
the insulation (the white area in the bottom picture) is 47.5 square inches. The thicker the insulation is, the larger the difference in areas will be.
As a result, when you integrate Fourier's equation to take into account how the area of the surface of a cylinder varies with radius, you end up with a much more intimidating looking equation.
Plus to get the solution you have to apply calculus to perform and integration; pretty scary if you are math phobic, like me.
That's the bad news. The good news is that since estimating heat loss from pipes and ducts is a very common need, there are a lot of tools out there that can help you do it with math that is much
less intimidating. And my purpose in adding this page to the web site is to put you in touch with some of them.
My purpose in exposing you to Fourier's Law is that I think it is important to understand where the principles you are applying come from so that you don't misapply a tool that has some simplifying
assumptions built into it. For instance, one of the tools below is for estimating losses from insulated valves. If you thing about how much the shape of a valve varies and the implications regarding
the area through which heat would need to flow to leave it, you quickly conclude that if you were to develop an equation to describe it, then it would be even more scary than the one for a cylinder.
I suspect (but don't know) that the tool is empirical, meaning the data was generated by taking measurements rather than by developing a lot of mathematics. But my point is that in realizing some of
the issues that are behind the tool, you are less likely to abuse it. So for example, while the tool is focused on valves in steam systems, extrapolating it to valves in hot water systems is
reasonable if you are careful about it. But extrapolating it to project the energy losses the body of a hot water pump might be starting to push the limits.
But on the other hand, since a the shape of most pumps could be generated by putting different sized cylinders together, you may be able to use the tools for estimating the losses from a pipe to
estimate the loss from a pump by considering it as an assembly of cylinders, evaluating the loss from each cylinder, and then adding them up.
I suspect you get my drift and are just wishing I would tell you about the resources, so here they are.
Department of Energy publishes "tip sheets"
for a number of energy using pieces of equipment and systems, including pumps, motors, compressed air systems, process heating systems and steam systems. Of interest in the context of this
particular web page are the tip sheets for estimating the savings to be achieved by repairing steam
As you can see from the example to the left, the tip sheet takes all of the scary math out of the estimating process. Basically, if you can come up with the temperature that the valve is at (maybe
"shoot" it with an
infrared thermometer
) and an estimate of the line size (maybe from a number on the casting or a
diameter tape measure
), then you can come up with the savings you could achieve by insulating the valve.
For valve sizes and temperatures other than those listed in the table, you can do a simple linear interpolation between the values in the table. And, it is reasonable to assume that insulating
valves in hot water systems would provide similar savings. However, hot water
systems often operate at temperatures lower than the temperatures associated with steam systems meaning you need to extrapolate the data vs. interpolating it. But since the main driver for the heat
transfer is the temperature difference across the insulation once you have taken the area into consideration, I concluded that it was reasonable to extrapolate the data from the table and project the
losses for lower temperatures, which is what the chart at the top of the page illustrates.
I created the chart by plotting the data in the table and then using Excel's trend line feature to project the losses at lower temperatures. I feel the projections are reasonable because they tend to
converge at 0 Btu/hr as the temperature difference between the pipe and the surroundings approaches 0. Another benefit of the chart is that it lets you visually interpolate for values between the
numbers in the table. So, I also made a chart that was derived by plotting the data in the table. The downloads below are both versions of the chart if you would like to use them and the spreadsheet
behind them. The benefit of the spreadsheet is that it has some cells set up that allow you to fine tune your reading by moving a vertical and horizontal line around on the chart so that they cross
exactly on the curve you are interested in.
I should also point out that if you make your own spreadsheet and do the curve fits for the projections, the Excel trend line feature allows you to display the equation for the trend line. What that
means is that you have a mathematical relationship that defines losses as a function of temperature for each of the line sizes.
So, if you were doing calculations that involved multiple rows in a spreadsheet (say an hour by hour savings assessment or a bin calculation), you could write a formula based on the trend line
equation that would do the math for you instead of your needing to look up the number you need for each row or bin (which could get a bit tedious if you were doing an hour by hour calculation given
that there are 8,760 hours in a year).
Finally, a discussion about ways to estimate savings associated with insulation would be totally incomplete if I did not mention the
North American Insulation Manufacturer's Association
software tool
3EPlus allows you to quickly evaluate different insulation systems for a variety of applications including pipes, ducts, tanks, and flat surfaces. The screen shot to the left illustrates the results
I generated for a 4 inch pipe operating at 200°F indoors in a 75°F environment and includes the losses for the bare pipe and different insulation thicknesses. The insulation was the typical
fiberglass insulation you find in
commercial applications but the tool includes numerous options and also allows you to build up a system using layers of different insulation. I generated the results faster than the time it took for
you to read this. And I did not have to get out my calc book to look up an integration formula. But since I know the theory behind the tool and can spot check it with the equation for losses from a
cylinder (above), I am pretty comfortable with the results with out doing the math myself. | {"url":"https://www.av8rdas.com/insulation-savings-tools.html","timestamp":"2024-11-09T11:13:22Z","content_type":"text/html","content_length":"112879","record_id":"<urn:uuid:102afd05-1298-4017-be7c-4b9eea171742>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00655.warc.gz"} |
TDK-Micronas GmbH
Advanced Techniques in Motor Control
Electric motors have a pivotal role in various applications, transforming electrical energy into mechanical power. Efficient motor control is crucial, requiring the use of intelligent actuators or
motor controllers. This article offers insights into the intricacies of DC motor control, specifically delving into Brushed DC (BDC), Brushless DC (BLDC), and Stepper motors.
Brushed DC Motors (BDC)
BDC motors, prevalent in industrial and automotive applications, rely on brushes for commutation. Comprising stator, rotor, brushes, and commutator, BDC motors offer simplicity and proportional speed
and torque control. However, their lifespan is contingent on factors like load, currents, and vibrations.
The pitfall being the brushes which wear out over time and require maintenance. Furthermore, the commutator and brushes produce electric spikes which prohibits their utilization in applications where
the working environment presents flammable gases or exhibits a fire hazard. Fig. 1 shows the BDC motor flux scheme in which the commutator and brushes mechanically manage the rotor flux direction to
keep the rotor moving.
Fig. 1: BDC motor flux scheme.
Although prone to maintenance (or replacement) costs, the BDC motor has been for a long time the preferred solution in the Automotive market for small actuators because of its attractive cost and
easy operation, but attention because the price argument is not anymore always true as there are many companies shifting to the BLDC motors.
For this kind of machines, in order to control the torque, it suffices to adjust the applied voltage. Therefore, the most fundamental form of control method is derived, called V/f. The method, which
assumes a constant airgap flux, correlates motor speed and stator voltage linearly. Such kind of control could make use of the following modulation.
Pulse Width Modulation (PWM):
Motor torque is directly proportional to current, controlled by adjusting voltage through pulse width modulation (PWM). High side MOSFETs (p-channel) are LOW active, requiring LOW voltage for
activation. Low side MOSFETs (n-channel) are HIGH active, needing HIGH voltage for activation. Synchronization of opposite MOSFETs with the same PWM profile requires inversion logic.
In an example, depicted in Fig. 2, motor phase B is activated negatively, and phase A positively with a 20% PWM duty cycle.
Fig 2: Output inversion logic during motor phases activation using a P/N-Channel half-bridge
An alternative method for energizing motor phases involves keeping one MOSFET active at 100% for the entire duration and applying a PWM profile only to the other MOSFET. This technique, illustrated
in Fig. 3, can also be applied inversely, where AH is active at 100% and BL is controlled by a PWM signal.
Fig. 3: Low side MOSFET active 100% of the time while the HS switch is the one responsible for the PWM regulation.
Brushless DC Motors (BLDC)
BLDC motors, electronically commutated and devoid of brushes, boast high reliability and efficiency. Magnets on the rotor, energized stator windings, and electronic commutation facilitate precise
torque control. The absence of sliding contacts enhances motor lifespan, with ball bearings being the limiting factor.
As already mentioned, the electronic commutator is a must in this case, because the BLDC motor has no commutator and brushes. A stator flux shall be generated by an appropriate voltage vector in a
way that the rotor flux produced by the magnets follow it, as shown in Fig. 4.
In this example, phase A is positively energized, phase B is negatively energized, and phase C is left open, causing the motor to rotate counterclockwise (assumed as the FORWARD direction). This
leads to the explanation of the Six-Step commutation.
Fig. 4: Motor flux scheme, 3-phase BLDC motor
Six-Step Commutation:
The six-step commutation technique involves energizing two phases at a time while leaving a third phase floating (Hi-Z) in a three-phase motor. This modulation is usually deployed in conjunction with
a BLDC motor, which is well-suited for this purpose. Due to the modulation characteristics of the Six-Step, it was found that modifying the Back Electromotive Force (BEMF) of a BLDC to a trapezoidal
shape (instead of sinusoidal) would produce a more constant torque with less ripple (fluctuation).
Fig. 5 demonstrates the commutation vector diagram for the Six-Step modulation. Depending on which region of the hexagon the rotor flux position falls into, an appropriate voltage vector is picked.
In this example, clockwise rotation is depicted. if sector 0 is selected, vector V4 will be applied. This causes a phase shift of 120° between rotor and stator flux. The vector length (amplitude) is
controlled by the PWM’s duty cycle.
Knowing the rotor flux direction (rotor position) is crucial for optimal performance, and BLDC motors often use integrated sensors or external position sensors for this purpose. Special techniques,
such as reading the BEMF, can also detect rotor position by means of the Zero-Crossings (ZC) without the need for sensors.
Fig. 5: Commutation vector diagram for the Six-Step modulation.
Sensorless Six-Step (ZC detection):
Thanks to the inherent characteristics of the Six-Step modulation, it is possible to sense the floating phase directly, allowing the controller to detect the ZC instants. This concept requires a
motor with Y-connected stator windings for it to work. The common mode voltage is equivalent to the neutral point and is used as a reference for the comparator circuit which takes the floating phase
as an input. Whenever a floating phase voltage crosses the virtual neutral voltage, a ZC is said to happen. Each sector has a predefined pattern for the correct Six-Step commutation sequence to take
place. A single comparator is enough as long as the correct floating phase is multiplexed to it.
Fig. 6 depicts the CCW sequence and highlights the (ZC) instant for the transition within the sector 1. As an example, if a ZC of phase B is detected in sector 1, then the algorithm knows that 30
electrical degrees afterwards it should commutate the sector (electronic commutation), changing the applied voltage vector from “1z0” to “z10”.
Fig. 6: Six-Step CCW commutation sequence.
Space Vector Modulation (SVM): Space Vector Modulation (SVM) is a technique used to generate sinusoidal shaped voltage with a three-phase voltage source inverter (VSI). SVM is typically used to drive
AC induction motors, brushless DC motors (BLDC) and permanent magnet synchronous motor (PMSM). Fig. 7 shows an example for clockwise rotation. If sector 0 is entered with SVM also V4 will be applied.
But this results in a phase shift of 90°.
Fig. 7: Commutation vector diagram for the Space-Vector Modulation.
With Six-Step commutation the phase shift between stator and rotor field is maintained in 60°-steps only. Whereas, with SVM the two adjacent vectors are time multiplexed to create an intermediate
(average) vector. By means of the intermediate vector, there is more granularity to synthesize the stator flux direction, thus making the SVM a more suited modulation algorithm to be used with
advanced control schemes
Fig. 8 shows the construction of the intermediate vector. The direction of vector from V[x] Fig. 7 can be adjusted by multiplexing the components V[k] and V[k+1 ]of the two adjacent vectors. The
absolute value (amplitude) of the vector is adjusted by the zero vector V[z] (either “000” or “111”), which does not contribute to the direction. The shorter the dwell time t[z'] , the higher the
vector amplitude. Anywhere in the hexagon, the intermediate vector can be referenced back to the first sector by the term (n - 1) · π/3
Fig. 8: SVM construction of the intermediate vector from sector S5 (V4) to S0 (V5).
The following equations can be used to calculate the dwell time from the voltage vector projections on the real and imaginary axes times are the following (valid for
(Eq. 1) (Eq. 2) (Eq. 3)
Field Oriented Control (FOC):
The Field Oriented Control (FOC) can be understood as the instantaneous regulation of the torque of the electric machine. The instantaneous torque equation is a function of the flux linkage vector
and the stator current vector.
The procedure is based on the reference frame theory and enables the designer to breakdown the complex three-phase system into an equivalent model in the dq system tied to the rotor synchronous
frame. As a result, instead of having to manipulate the three-phase quantities (currents and voltages), the algorithm has to control two DC terms, one related to the torque (q-axis) and another one
related to the flux (d-axis).
Fig. 9 represents the transformations and change of system references. The explanation about the transformations Clarke, Park and all the mathematical manipulation is out of the scope of this
document. The αβ axes are orthogonal to each other and fixed to same position. On the other hand, the dq axes are rotating with the synchronous speed. The rotor flux position lines up with the
Fig. 9: Reference frame theory applied to the BLDC machine.
Fig. 10-a depicts the block diagram for the FOC speed control loop (q-axis) while Fig. 10-b shows the flux regulator (d-axis) part. The output of the speed controller is said to be the q-axis
reference current which in turns feed the inner current regulator that generates a q-axis reference voltage. Due to the cross-coupling term between the q- and d-axes, there is a need to take them
into account in the regulator loop. There are several ways to compensate for them, either use of a feed-forward scheme to cancel the term or a cascade regulator design whose fastest loop is the
d-axis regulator.
Speed and torque regulator (q-axis).
Flux regulator (d-axis).
Fig. 10: FOC block diagrams for both axes.
Maximum Torque Per Ampere (MPTA) and Field Weakening (FW):
The Maximum Torque Per Ampere (MPTA) is a technique to minimize the copper losses in the machine and it is applied to the already laid out FOC regulator scheme. The phasor diagram of Fig. 11-a
demonstrates this concept. In this operation mode, the stator current vector and the rotor magnetic field are kept orthogonal to each other, maximizing the torque.
Fig. 11: Electric machine phasor diagram.
Although maximizing the torque ensures higher efficiencies, there are physical limitations and if the motor speed exceeds the base speed, the motor cannot operate in MPTA as there will be not enough
dc-link voltage available to overcome the motor BEMF.
The solution for this can be found in the torque equation shown in Eq. 4. The d-axis linkage flux has two components one of which is due to the permanent magnet field. The sign convention is such
that for negative I[d][ ]currents, the second term subtracts from the PM flux which holds true for motoring operation as I[q] ≥ 0 and I[d] ≤ 0.
(Eq. 4)
For the algorithm to ensure MTPA, it will set the reference d-axis current to zero. The reason being it should not counteract the PM flux. On the other hand, if the machine has exceeded the base
speed, see Fig. 11-b, then a proper value for the pair (I[q] , I[d]) must be picked to ensure partial concealment of the PM flux term, thus overcoming the inverter limitation. This operation mode is
known as Field Weakening (FW).
The definition for the base speed is expressed in Eq. 5, where the term V[s_max] represents the maximum inverter output voltage (dependent on the modulation scheme) and the term I[s_max] representing
the maximum inverter current or nominal motor current (whichever is the smallest).
(Eq. 5)
Single-shunt current measurement:
The single-shunt current measurement is a very popular approach amid the applications from tens of watts up to a couple thousands of watts where the sensorless operation is required. It offers an
attractive reduction to the Bill of Materials cost. Another advantage of this method is that there are no concerns with regards to the matching of the ADC signal paths (2- or 3-phase current
sensing), because all currents will be sampled using the same ADC channel. The need for measuring the current with the SVM will be explained in the next section.
Fig. 12 shows an example of SVM modulation with center aligned PWM for the transition between vector V5 “011” to V4 “001”. Within this sector, the phase currents u and v are negative while w is
positive. Each inverter leg will only carry current if the respective LS switch is turned on, i.e., for the complementary of the PWM period (1-D). Therefore, the summation of the three individual
inverter leg currents equals to the single-shunt signal. Of course, there is no signal if either the “000” or “111” null vectors are applied.
Fig. 12: SVM with Center Aligned PWM and single-shunt current signal represented for the sector 5 to 0
Position Observer:
As the application power raises, so does the performance requirements for it. It is therefore difficult to meet the criteria using an ordinary Six-Step modulation. Instead, the designers must opt for
the SVM modulation. In addition, these applications often require sensorless operation due to cost constraints. But, because in the SVM approach there are no floating phases, it is not possible
anymore to do direct BEMF sensing to determine the exact ZC instants.
In this sense, many rotor position estimator algorithms appeared in the literature (and in the industry) to allow sensorless operation of machines while using the SVM modulation. To execute such kind
of algorithms, the software needs to know the dynamic stator voltages and currents, among other electrical quantities. Thus, the importance of the previous section about shunt current sensing.
Among the various estimation methods available in the literature, the voltage model is the most famous one (some authors call it flux estimator). It is based on the stator voltage equations. Through
the stator voltage equations one can obtain the rotor flux from which the rotor angular position is determined. Although simple this algorithm can provide good results if the motor parameters are
well known, and the machine current and voltage measurements exhibit small errors.
At low speeds and start-up condition the traditional sensorless schemes tend to suffer. Because the equations are dependent on the flux, and the BEMF is too small at this point, the measurement
errors contribute to lower the Signal-to-Noise ratio (SNR). To address this inability, many amendments or enhancements were proposed to create hybrid estimators, one suited for low speeds and another
one for the high speeds, and even different types of estimators.
Stepper Motors
A subset of BLDC motors, stepper motors divide a full rotation into incremental steps. Two-phase bipolar stepper motors, consisting of coils connected to an H-bridge, provide accurate positioning.
Stepper motors offer step modes, including full-step, half-step, scaled half-step, and micro-step, each influencing torque, accuracy, and movement.
Fig. 6: Bipolar stepper motor circuit
Step Modes: Exploring full-step, half-step, scaled half-step, and micro-step modes
Stepper motors operate in different step modes, determined by the number of pulse commands from the controller.
In full-step mode, both motor phases are energized, allowing four steps per electrical revolution.
Fig. 7: Full step mode
Half-step mode inserts an intermediate step between full-steps, offering eight steps per revolution.
Fig. 8: Half-step mode
Scaled half-step mode adjusts current vectors during intermediate steps to minimize torque ripple.
Fig. 9: Scaled half-step mode
Micro-step mode employs sine/cosine-shaped currents, allowing for versatile vector formations with resolutions defined per full-step or quadrant, typically with up to 32 micro-steps per quadrant. The
micro-step resolution is determined by a sine table in the application software, enabling precise control with various step widths.
Fig. 10: Micro-step mode
Current Decay: Managing current decay for optimal performance in micro-stepping
To achieve a sine-shaped current waveform for micro-stepping across the motor's entire speed range, effective current decay control is essential. The goal is to minimize current ripple for reduced
emissions and lower acoustic noise. The approach involves adapting the current decay strategy based on motor speed and current levels.
In low-speed and low-current scenarios, asynchronous slow decay is the preferred mode. When a switch in the H-bridge is turned off, the freewheeling current flows through the intrinsic diode of the
opposing switch in the same leg. This decay mode is particularly effective for rising or falling current at low speeds and low current levels.
Fig. 12: Asynchronous Slow Decay
For high-speed and high-current situations, synchronous fast decay is the optimal mode. In this mode, both current-conducting switches in the H-bridge are turned off simultaneously, and the opposing
switches in the corresponding legs are turned on. The freewheeling current flows through the activated switches back into the supply. This decay mode is well-suited for scenarios involving falling
current at high speeds and high current levels.
Fig. 13: Synchronous fast decay
In scenarios where motor speed and current levels vary, mixed decay, a combination of fast and slow decay, proves effective. This approach involves applying fast synchronous decay first, followed by
asynchronous slow decay within one PWM cycle. The ratio of fast-to-slow decay can be adapted to the current level and motor speed. Mixed decay is the preferred mode for decreasing current at mid to
high speeds and high to low current levels.
Fig 14: Mixed decay
Fig. 15: Example of slow and mixed decay usage in micro-step mode
By implementing a fixed ratio between fast and slow decay in the application software, optimal performance is ensured. Additionally, a carefully defined current threshold prevents motor reversal
during low current levels, with slow decay applied when the set current level falls below this threshold.
Stall Detection: Monitoring BEMF voltage for stall detection and step loss prevention
Stall detection is achieved through BEMF voltage measurement, particularly in micro-step mode. In this mode, the BEMF voltage is proportional to motor speed, allowing for straightforward
identification of whether the motor is in motion. However, due to the measurement occurring only when one phase is not energized, the view of the BEMF voltage is limited. In an ideal scenario without
load and losses, the BEMF voltage peak aligns with the point of zero phase current. In real-world conditions with applied load, the rotor lags behind the stator field, introducing a load-dependent
phase lag. This lag results in a shift in the BEMF voltage from the peak, indicating zero torque, to the zero-crossing point, indicating stall torque. This shift signifies the point of stall and step
Fig. 17: V[BEMF] measurement for stall detection
Efficient DC motor control is vital for diverse applications. Understanding the nuances of BDC, BLDC, and stepper motors, along with advanced control techniques, ensures optimal performance and
longevity. Embedded systems, exemplified by Micronas' HVC 5x family, offer sophisticated solutions for precise and reliable motor control. | {"url":"https://www.micronas.tdk.com/en/technologies/about-advanced-techniques-motor-control","timestamp":"2024-11-08T22:36:33Z","content_type":"text/html","content_length":"82291","record_id":"<urn:uuid:e283c7e5-d5d4-4a0e-90d7-47a74cb711e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00899.warc.gz"} |
Thermal Stress Analysis of Jet Engine Turbine Blade
Main Content
Thermal Stress Analysis of Jet Engine Turbine Blade
This example shows how to compute the thermal stress and deformation of a turbine blade in its steady-state operating condition. The blade has interior cooling ducts. The cool air flowing through the
ducts maintains the temperature of the blade within the limit for its material. This feature is common in modern blades.
A turbine is a component of the jet engine. It is responsible for extracting energy from the high-temperature and high-pressure gas produced in the combustion chamber and transforming it into
rotational motion to produce thrust. The turbine is a radial array of blades typically made of nickel alloys. These alloys resist the extremely high temperatures of the gases. At such temperatures,
the material expands significantly, producing mechanical stress in the joints and significant deformations of several millimeters. To avoid mechanical failure and friction between the tip of the
blade and the turbine casing, the blade design must account for the stress and deformations.
The example shows a three-step workflow:
1. Perform structural analysis accounting only for pressure of the surrounding gases while ignoring thermal effects.
2. Compute the thermal stress while ignoring the pressure.
3. Combine the pressure and thermal stress.
Pressure Loading
The blade experiences high pressure from the surrounding gases. Compute the stress caused only by this pressure.
First, create an femodel object for static structural analysis and include the geometry of a turbine blade.
model = femodel(AnalysisType="structuralStatic", ...
Plot the geometry with face labels.
Generate a mesh with the maximum element size 0.01.
model = generateMesh(model,Hmax=0.01);
Specify Young's modulus, Poisson's ratio, and the coefficient of thermal expansion for nickel-based alloy (NIMONIC 90).
E = 227E9; % in Pa
CTE = 12.7E-6; % in 1/K
nu = 0.27;
model.MaterialProperties = ...
materialProperties(YoungsModulus=E, ...
PoissonsRatio=nu, ...
Specify that the face of the root that is in contact with other metal is fixed.
model.FaceBC(3) = faceBC(Constraint="fixed");
Specify the pressure load on the pressure and suction sides of the blade. This pressure is due to the high-pressure gas surrounding these sides of the blade.
p1 = 5e5; %in Pa
p2 = 4.5e5; %in Pa
model.FaceLoad(11) = faceLoad(Pressure=p1); % Pressure side
model.FaceLoad(10) = faceLoad(Pressure=p2); % Suction side
Solve the structural problem.
Plot the von Mises stress and the displacement. Specify a deformation scale factor of 100 to better visualize the deformation.
pdeplot3D(Rs.Mesh, ...
ColorMapData=Rs.VonMisesStress, ...
Deformation=Rs.Displacement, ...
The maximum stress is around 100 Mpa, which is significantly below the elastic limit.
Thermal Stress
Determine the temperature distribution and compute the stress and deformation due to thermal expansion only. This part of the example ignores the pressure.
First, switch the analysis type of the model to the steady-state thermal analysis.
model.AnalysisType = "thermalSteady";
Assuming that the blade is made of nickel-based alloy (NIMONIC 90), specify the thermal conductivity.
kapp = 11.5; % in W/m/K
model.MaterialProperties = ...
Convective heat transfer between the surrounding fluid and the faces of the blade defines the boundary conditions for this problem. The convection coefficient is greater where the gas velocity is
higher. Also, the gas temperature is different around different faces. The temperature of the interior cooling air is ${150}^{\circ }\mathit{C}$, while the temperature on the pressure and suction
sides is ${1000}^{\circ }\mathit{C}$.
% Interior cooling
model.FaceLoad([15 12 14]) = ...
faceLoad(ConvectionCoefficient=30, ...
% Pressure side
model.FaceLoad(11) = ...
faceLoad(ConvectionCoefficient=50, ...
% Suction side
model.FaceLoad(10) = ...
faceLoad(ConvectionCoefficient=40, ...
% Tip
model.FaceLoad(13) = ...
faceLoad(ConvectionCoefficient=20, ...
% Base (exposed to hot gases)
model.FaceLoad(1) = ...
faceLoad(ConvectionCoefficient=40, ...
% Root in contact with hot gases
model.FaceLoad([6 9 8 2 7]) = ...
faceLoad(ConvectionCoefficient=15, ...
The boundary condition for the faces of the root in contact with other metal is a thermal contact that can be modeled as convection with a very large coefficient (around $1000\text{\hspace{0.17em}}\
mathit{W}/\left({\mathit{m}}^{2}\mathit{K}\right)$ for metal-metal contact).
% Root in contact with metal
model.FaceLoad([3 4 5]) = ...
faceLoad(ConvectionCoefficient=1000, ...
Solve the thermal problem.
Plot the temperature distribution. The temperature between the tip and the root ranges from around ${820}^{\circ }\mathit{C}$ to ${330}^{\circ }\mathit{C}$. The exterior gas temperature is ${1000}^{\
circ }\mathit{C}$. The interior cooling is efficient: it significantly lowers the temperature.
Now, compute the stress and deformation due to thermal expansion. First, switch the analysis type of the model to the static structural analysis.
model.AnalysisType = "structuralStatic";
Specify Young's modulus, Poisson's ratio, and the coefficient of thermal expansion for nickel-based alloy (NIMONIC 90).
model.MaterialProperties = ...
materialProperties(YoungsModulus=E, ...
PoissonsRatio=nu, ...
Specify the reference temperature.
model.ReferenceTemperature = 300; %in degrees C
model.CellLoad = cellLoad(Temperature=Rt);
Specify the boundary condition.
model.FaceBC(3) = faceBC(Constraint="fixed");
Solve the thermal stress problem.
Plot the von Mises stress and the displacement. Specify a deformation scale factor of 100 to better visualize the deformation. The stress concentrates in the constrained root because it cannot freely
expand, and also in the junction between the blade and the root.
figure("units","normalized","outerposition",[0 0 1 1]);
pdeplot3D(Rts.Mesh, ...
ColorMapData=Rts.VonMisesStress, ...
Deformation=Rts.Displacement, ...
clim([0, 200e6])
Evaluate the displacement at the tip. In the design of the cover, this displacement must be taken into account to avoid friction between the cover and the blade.
Combined Pressure Loading and Thermal Stress
Compute the stress and deformations caused by the combination of thermal and pressure effects.
Add the pressure load on the pressure and suction sides of the blade. This pressure is due to the high-pressure gas surrounding these sides of the blade.
% Pressure side
model.FaceLoad(11) = faceLoad(Pressure=p1);
% Suction side
model.FaceLoad(10) = faceLoad(Pressure=p2);
Solve the problem.
Plot the von Mises stress and the displacement. Specify a deformation scale factor of 100 to better visualize the deformation.
figure("units","normalized","outerposition",[0 0 1 1]);
ColorMapData=Rc.VonMisesStress, ...
Deformation=Rc.Displacement, ...
clim([0, 200e6])
Evaluate the maximum stress and maximum displacement. The displacement is almost the same as for the thermal stress analysis, while the maximum stress, 854 MPa, is significantly higher. | {"url":"https://de.mathworks.com/help/pde/ug/thermal-stress-analysis-of-jet-engine-turbine-blade.html","timestamp":"2024-11-15T01:41:07Z","content_type":"text/html","content_length":"71964","record_id":"<urn:uuid:0d2122db-2b31-4fc5-9a6c-0e70088263aa>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00666.warc.gz"} |
Rules for Home Plate in context of baseball%20diamond%20with%20home%20plate
31 Aug 2024
Title: The Rules of Home Plate: A Comprehensive Analysis of the Diamond’s Critical Zone
Home plate, a crucial component of the baseball diamond, is often overlooked despite its significance in determining the outcome of games. This article aims to provide a detailed examination of the
rules governing home plate, highlighting key aspects such as positioning, tagging, and scoring. By applying mathematical formulas and diagrams, this study clarifies the intricacies of home plate and
provides insights for players, coaches, and umpires.
Home plate, also known as “home” or “plate,” is the central point of the baseball diamond (Figure 1). It serves as the target zone for base runners attempting to score. Understanding the rules
surrounding home plate is essential for effective gameplay, as mistakes can significantly impact the outcome of a game.
Home plate is positioned at the intersection of the third base line and the foul line, measuring 17 inches (43.18 cm) in diameter (Figure 2). The plate’s positioning is critical, as it determines the
distance between the runner and the catcher. According to Rule 5.09(a), the catcher must have one foot on each side of the plate when attempting to tag a runner.
A tag at home plate occurs when the catcher touches the plate with the ball while the runner is attempting to score (Figure 3). The umpire’s decision is based on whether the tag was made before the
runner reached the plate. According to Rule 5.09(b), a tag must be made with the ball in the catcher’s hand, and the umpire must see the tag to make a call.
A run scores when a base runner reaches home plate safely (Figure 4). The scoring process is governed by Rule 5.09(c), which states that a run scores if the runner touches home plate before being
tagged or forced out.
Mathematical Formulas:
To better understand the rules of home plate, mathematical formulas can be applied to calculate distances and times. For example:
• Distance from the pitcher’s mound to home plate: 60.5 feet (18.45 meters) [1]
• Time it takes for a runner to reach home plate from first base: approximately 3.2 seconds [2]
The rules governing home plate are complex and require precise positioning, tagging, and scoring. By applying mathematical formulas and diagrams, this study has provided a comprehensive analysis of
the diamond’s critical zone. Understanding these rules is essential for players, coaches, and umpires to make informed decisions during gameplay.
[1] Official Baseball Rules (2022). Rule 5.09(a).
[2] Sports Medicine Research Institute (2019). “The Effects of Speed on Running Distance in Baseball.” Journal of Strength and Conditioning Research, 33(5), 1234-1240.
ASCII Diagrams:
Figure 1:
| Home |
| Plate (17 inches) |
| Third Base Line
| Foul Line |
Figure 2:
| Catcher |
| (one foot on each side) |
| Home Plate
| Runner's Path |
Figure 3:
| Tagging |
| (catcher touches plate) |
| Ball in Catcher's Hand
| Umpire's Call |
Figure 4:
| Scoring |
| (runner reaches plate) |
| Run Scores
| Game Outcome |
Related articles for ‘baseball%20diamond%20with%20home%20plate’ :
Calculators for ‘baseball%20diamond%20with%20home%20plate’ | {"url":"https://blog.truegeometry.com/tutorials/education/7408cce99f35fa98b526521d30ed80c9/JSON_TO_ARTCL_Rules_for_Home_Plate_in_context_of_baseball_20diamond_20with_20hom.html","timestamp":"2024-11-02T11:41:53Z","content_type":"text/html","content_length":"18594","record_id":"<urn:uuid:69592b43-362d-455f-b834-71f047966c1a>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00138.warc.gz"} |
91 in Words - Write 91 in Words | 91 Spelling
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
91 in Words
91 in Words can be written as Ninety One. If you have saved 91 dollars, then you can write, “I have just saved Ninety One dollars.” Ninety One is the cardinal number word of 91 which denotes a
• 91 in Words = Ninety One
• Ninety One in Numbers = 91
Let us write the given number in the place value chart.
We see that there are 1 ‘ones’, 9 ‘tens’. Now read the number from right to left along with its place value. 91 in words is written as Ninety One.
How to Write 91 in Words?
Using the place value chart we identify the place for each digit in the given number and write the number name. For 91 we see that the digits in units = 1, tens = 9. Therefore 91 in words is written
as Ninety One.
Problem Statements:
│How to Write 91 in Words? │Ninety One │
│What is 91 Decimal to Binary? │(91)₁₀ = (1011011)₂ │
│Is 91 a Prime Number? │No │
│Is 91 a Perfect Cube? │No │
│Is 91 an Odd Number? │Yes │
│Is 91 an Even Number? │No │
│Is 91 a Perfect Square? │No │
│What is the Square Root of 91? │9.539392 │
│Is 91 a Composite Number? │Yes │
FAQs on 91 in Words
How do you Write 91 in Words?
Using the place value chart, we can identify the value of each digit in 91 and convert the numerals to words. 91 in words is written as Ninety One.
What is the Value of Ninety One Minus Sixty?
Ninety One in numerals is written as 91. Sixty in numerals is written as 60, Now Ninety One Minus Sixty means subtracting 60 from 91, i.e. 91 - 60 = 31 which is read as Thirty One.
Find the Value of 60 + 31. Write the Answer in Words.
Simplifying 60 + 31 gives 91. And 91 in words is written as Ninety One.
What are the Rules to Write 91 in Words?
Let us fill all the digits of 91 in the place value chart.
We see that there are 1 ‘ones’, 9 ‘tens’.
• Read the number from right to left along with its place value.
• 91 in words is written as Ninety One.
☛ Also Read:
Math worksheets and
visual curriculum | {"url":"https://www.cuemath.com/numbers/91-in-words/","timestamp":"2024-11-06T09:38:05Z","content_type":"text/html","content_length":"180746","record_id":"<urn:uuid:64618184-82c3-4af6-8ae7-85be200b5afc>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00538.warc.gz"} |
Central MN High School Baseball Recap
photo - Lisa Anderson
I will bring to you game summaries of the following teams; weekly and possibly bi-weekly as well. Rocori Spartans, St. Cloud Crush, Sauk Rapids-Rice Storm and Sartell-St. Stephen Sabres of the
Central Lakes Conference. St. Cloud Cathedral Crusaders, Albany Huskies, Foley Lumberjacks, Little Falls Flyers and Pierz Pioneers of the Granite Ridge Conference, the Becker Bulldogs of the
Mississippi 8 Conference. Eden Valley-Watkins Eagles, Royalton Royals, Kimball Area Cubs, Paynesville Bulldogs, Holdingford Huskers, Atwater-Cosmos-Grove City Falcons, Belgrade-Brooten-Elrosa Jaguars
and Maple Lake Irish of the Central Mn. Conference.
FRIDAY MAY 17th
The Cubs defeated their neighbors the Lighting, they out hit them eight to six, including a pair of big doubles. The Cubs starting pitcher was Clay Faber, he threw six innings to earn the win. He
gave up five hits, four runs, five walks and he recorded three strikeouts. Ronnie Arnold threw one inning to close it out, he gave up one hit and one walk.
The Cubs offense was led by Hank Meyer, he went 2-for-3 with a double for two RBIs, he earned a walk and he scored a run. Tate Winter went 2-for-4 with a double for two RBIs and he scored a run. Clay
Faber went 2-for-4 for a RBI and he scored a run and Mason Danelke went 1-for-2 and he scored a run. Bryant Knaus went 1-for-4 and he scored a run and Brandon Henkemeyer was credited for a RBI and he
scored a run. Nathan Serbus earned a walk and he scored a run and Sam Anderson scored a run.
The Lightning starting pitcher was Conner Lampi, he threw five innings. He gave up four hits, two runs and he recorded eight strikeouts. Chester Bergen gave up two hits, four runs and one walk.
Carter Ramsey threw two innings, he gave up two hits, two runs, one walk and he recorded four strikeouts.
Their offense was led by Tyson Sanderson, he went 2-for-3 for a RBI and he scored a run and Tom Halverson went 1-for-3 for two RBIs. Nate Green went 2-for-4 with a double and he scored a run and Cam
Ergen had a walk. Nick Walter went 1-for-4, Conner Lampi had two walks and he scored a run, Luke Lindquist had two walks and he scored a run and Colby Dircks had a walk.
BBE JAGUARS 13 MAPLE LAKE IRISH 4
The Jaguars defeated their conference rivals the Irish, they out hit them ten to six, including a pair of doubles and a home run. They had seven collect hits and they were aided by seven walks. Their
starting pitcher was Ethan Mueller, he threw a complete game to earn the win. He gave up six hits, four runs and he recorded eight strikeouts.
The Jaguars offense was led by Luke Dingmann, he went 3-for-5 with two doubles for two RBIs, he had a pair of stolen bases and he scored two runs. Luke Illies went 1-for-5 with a big home run for
three RBIs and Owen Paulson was hit twice by a pitch and credited for a RBI. Brett DeRoo went 2-for-5 for a RBI and he scored a run and Kade DeRoo went 1-for-4 for a RBI, he was hit by a pitch and he
scored a run. Aiden Mueller went 1-for-1 for a RBI, he earned three walks and he scored a run and Ethan Mueller earned four walks, a stolen base and he scored a pair of runs. Ryan Jensen went 1-for-4
for a RBI and he scored a run, Jack Lundberg went 1-for-4 with a stolen base and he scored a run and Lance Rademacher had a stolen base.
The Irish starting pitcher was Nathan Zander, he threw four innings, he gave up four hits, four runs, two walks and he recorded five strikeouts. Gabe Jurgens threw three innings, he gave up six hits,
nine runs, five walks and he recorded four strikeouts.
The Irish offense was led by Nathan Zander, he went 1-for-4 for two RBIs and Jackson Clapp went 1-for-3 with a double and he scored a run. Gabe Jurgens went 1-for-2 for a RBI, he was hit by a pitch
and he scored a pair of runs and Danny Reilley went 1-for-3. Brayden Fobbe went 1-for-4 for a RBI and Joey Gendreau went 1-for-4 and he scored a run.
The Patriots defeated their section foes the Royals, they out hit them ten to four, including three doubles and a pair of triples. Their starting pitcher was Hunter Moore, he threw 6 2/3 innings to
earn the win. He gave up four hits, three runs, three walks and he recorded three strikeouts. Isaac Gapinski threw 1/3 of an inning to close it out.
Their offense was led by Jack Primus, he went 2-for-3 with a triple and a double and he scored a run. Hunter Moore went 1-for-3 for a RBI and he scored a run and Hunter Boeckman went 1-for-3 with a
double. Bryce Binek went 2-for-3 with a triple and a double and he scored a run and Brody Kircher had a sacrifice fly for a RBI. Carter Natvig went 2-for-3, Caden Beseman went 1-for-3 and he scored a
run and Jake Leners went 1-for-3.
The Royals starting pitcher was Nick Leibold, he threw six innings, he gave up ten hits, four runs and he recorded four strikeouts. Their offense was led by Matt Swenson, he went 1-for-2 for a RBI
and Sean Schmidtbauer had a sacrifice fly for a RBI and he earned a walk. Jonah Schnieder went 2-for-3 and he scored a run and John Bzdok had a sacrifice fly for a RBI. Nick Leibold went 1-for-3,
Keaton Nelson earned a walk and he was hit by a pitch and Brady Yourczek earned a walk and he scored a run.
ATWATER-COSMOS-GROVE CITY FALCONS vs. BOLD WARRIORS (5:00)
EDEN VALLEY-WATKINS EAGLES vs. LITCHFIELD DRAGONS (7:00)
MINNEWASKA LAKERS 5 PAYNESVILLE AREA BULLDOGS 3
The Lakers defeated their rivals the Bulldogs, they were out hit seven for four. They collected a big home run, a triple and a double in support of their pitchers. Ryland Martin threw five innings,
he gave up four hits, four walks and he recorded five strikeouts. Jack Majerus threw two innings, he gave up three hits, three runs, three walks and he recorded two strikeouts.
The offense was led by Dylan Alexander, he went 2-for-3 with a home run for two RBIs and he scored a run. Alex Panitzke went 1-for-3 with a triple for three RBIs and he scored a run. Austin Weber
went 1-for-3 with a double and he scored a run, Nathan Dell and Riley Dell both had a walk and both scored a run.
The Bulldogs starting pitcher was Esau Nelson, he threw six innings, he gave up four hits, five runs, two walks and he recorded nine strikeouts. Their offense was led by Brandon Carlson, he went
2-for3- with a double for a RBI and he earned a walk. Brayden VanderBeek went 2-for-4 for a RBI and Bryce Vanderbeek went 1-for-2, he earned a walk and he scored a run. Josiah Utsch went 1-for-3, he
earned a walk and he scored a run, Isaac Lieser went 1-for-4 and Brayden Pung earned a walk and he scored a run.
BEMIDJI LUMBERJACKS 5 SARTELL-ST. STEPHEN SABRES 3
The Lumberjacks defeated their section rivals the Sabres, they out hit them eight to six. Their starting pitcher was Jack Lindquist, he threw six innings. He gave up five hits, two runs, five walks
and he recored three strikeouts. Gunner Ganske threw one inning to close it out, he gave up one hit, one run, one walk and he recorded a strikeout.
The Lumberjacks offense was led by Peyton Neadeau, he went 2-for-3 for two RBIs and a stolen base and Fisher Ganske had a RBI and he scored a run. Gavin Kapaun went 2-for-3 with a double for a RBI
and he scored a run and Stonewall Gessner earned a walk. Gunner Ganske went 2-for-3 for a RBI and Boston Smith had a stolen base and he scored a run. Landon Hanson went 1-for-3 with a stolen base and
he scored a run and Jack Lundquist went 1-for-4 and he scored a run.
The Sabres starting pitcher was Wes Johnson, he threw six innings, he gave up eight hits, five runs, one walk and he recorded eight strikeouts. The offense was led by Austin Lahr, he went 1-for-3
with a double for a RBI and Keaton Landowski went 1-for-3 for a RBI. Wes Johnson went 2-for-3 and Carter Stutsman went 1-for-1 for a RBI. Brady Thompson went 1-for-3 and he earned a walk, Brenden
Boesen and Brett Schlangen both earned a walk. Levi Frieler earned a pair of walks and he scored a run, Eli Hanson earned a walk and he scored a run and Jordan Fish had a stolen base.
The Flyers defeated their conference rivals the Cardinals, they out hit them eight to four. This included a big double, five walks and solid defense. Carter Gwost started on the mound for the Flyers,
he threw two innings, he gave up one hit, one run, two walks and he recorded two strikeouts. Peter Knopik threw five innings in relief, he gave up three hits, two runs and three strikeouts.
The Flyers offense was led by Carter Gwost, he went 2-for-2 with double for two RBIs, he earned a walk, had a stolen base and he scored a pair of runs. Alex Thoma went 2-for-4 for a RBI and Jacob
Dahlberg earned a walk and he was credited for a RBI. Izaak Kalis went 2-for-3 and Bobby Loure scored a run. Charlie Smieja went 1-for 2, he earned a walk and he scored a run and Garrett Lindberg
went 1-for-3.
The Cardinals starting pitcher was Landon Roths, he threw 3 1/3 innings, he gave up five hits, four runs two walks and he recorded five strikeouts. Parker Converse threw 2 2/3 innings, he gave up
three hits, three walks and he recorded one strikeout.
The Cardinals offense was led by Boone Branson, he went 1-for-3 with a double for a RBI, Carter Simonson scored a run and Cameron Mevada had a walk. Jordan Kuhnau went 1-for-4 and he scored a run,
Gage Castle went 1-for-2 and he scored a run and Cameron Simon went 1-for-3
ROCORI SPARTANS 8 RED WING WINGERS 3
The Spartans defeated the Wingers, they out hit them eight to seven, including four doubles and a triple and they were aided by six walks. Kaden Rausch started on the mound for the Spartans, he threw
six innings to earn the win. He gave up six hits, three runs, four walks and he recorded two strikeouts. Jack Boos threw one inning, he gave up one hit and he recorded one strikeout.
Their offense was led by Noah Olmscheid, went 1-for-3 with a double for a RBI, he earned a walk and he scored a run. Jace Griffin went 1-for-4 with a double for a RBI, he had a stolen base and he
scored a run. Jared Laudenbach went 1-for-2 with a double for a RBI, he earned a walk, he was hit by a pitch and he scored a run. Kaden Rausch went 1-for-3 with a triple for a RBI and he scored a
run. Tyler Prom went 1-for-4 with a double for a RBI and Max Fredin earned two walks and he scored a pair of runs. Jacob Stalboerger went 2-for-3 with a double, he earned a walk and he scored a run.
Jack Boos went 1-for-5 for a RBI, Hunter Fuchs earned a walk and he was credited for a RBI and Caleb Maddox scored a run.
The starting pitcher for the Wingers was Logan Norquist, he threw 1 2/3 inning, he gave up three hits, four runs and one walk. Tyson Freimel threw 1/3 of an inning, he gave up one run, two walks and
he recorded one strikeout. O. Renquist threw one inning, he gave up three hits and two runs. M. Finholst threw four innings, he gave up three hits, one run three walks and he recorded two strikeouts.
Their offense was led by Jacob Rodgers, he went 1-for-4 for a RBI and Jonathan Speltz earned a walk and he was credited for a RBI. Julius Koecher went 1-for-2 and he scored a run and Calvin Nelson
went 1-for-2 and he earned a walk. Reid Hartmann went 1-for-4 and he scored a run and M. Finholdt went 1-for-2 and he scored a run and Ellis Petersmeyer went 1-for-4.
ST. CLOUD CRUSH 8 MONTICELLO MAGIC 7
The Crush defeated their I-94 rivals the Magic, they were out hit nine to seven. The Crush collected a triple and a double and they were aided by fifteen walks. Their starting pitcher was Parker
Schulz, he threw five innings, he gave up five hits, three runs, two walks and he recorded two strikeouts. Elijah Novak threw one inning, he gave up three hits, four runs, three walks and he recorded
one strikeout. Shayne Poole threw three innings, he gave up one hit, four walks and he recorded two strikeouts.
Their offense was led by Jaxon Kenning, he went 2-for-6 with a triple and a double for three RBIs. Max Kiffmeyer went 1-for-4 for a RBI, he earned a pair of walks, had a pair of stolen bases and he
scored a trio of runs. Parker Schulz went 1-for-4 for a RBI and he earned a walk and Kayden Mork went 1-for-1 and he earned four walks. Colten Palmer went 1-for-4 for a RBI and he earned a walk and
Drew Lieser earned a pair of walks. Joe Hess went 1-for-3, he earned two walks, had a stolen base and he scored a run and Devan Finnegan had a stolen base. Jackson Sheetz earned two walks, he had a
pair of stolen bases and he scored two runs and Ben Schmitt and Sutton Kenning both earned a walk.
The Magic starting pitcher was Tim Marcus, he threw 2 2/3 innings, he gave up four hits, four runs, five walks and he recorded one strikeout. K. Schlangen threw 2 2/3 innings, he gave up two hits,
two runs, four walks and he recorded three strikeouts. Campbell Bosacker threw two innings, he gave up a run, six walks and he recorded three strikeouts. K. Ellis threw 1 1/3 inning, he gave gave up
one hit, one run and one walk.
Their offense was led by Tyson Uisness, he went 3-for-4 for a RBI, he was hit by a pitch and he scored a pair of runs. Tim Macys went 2-for-3 for two RBIs and he earned a walk and Grant Stalhlback
went 1-for-4 for a RBI and he earned a walk and Easton Peters went 1-for-4 for a RBI. Brock Holthaus went 1-for-5 and he scored a run and K. Schlangen went 1-for-3.
The Pioneers defeated their section foe the Silverstreaks, they out hit them ten to eight. They collected a pair of doubles and aided by eleven walks, this gave their pitchers great support. Max
Barclay threw five innings to earn the win, he gave up two hits, two runs, one walk and he recorded four strikeouts. Kaden Kruschek threw two innings to close it out, he gave up six hits, four runs,
one walk and he recorded three strikeouts.
The Pioneers was led on offense by Reese Young, he went 4-for-5 for three RBIs, he earned a walk, had three stolen bases and he scored a pair of runs. Kaden Kruschek went 3-for-4 with a double for a
RBI and he earned a walk and Brayden Haberman went 1-for-4 for a RBI, he earned a walk and he scored a run. Weston Woitalla had a sacrifice fly for two RBIs, he earned a walk and he was hit by a
pitch. Bo Woitalla went 1-for-5 for a RBI and he scored a pair of runs. Joey Stuckmayer went 1-for-3 with a double for a RBI and Max Barclay earned two walks. Chase Becker earned three walks and he
scored a two runs. D. Bakke went 1-for-1 for a RBI and he earned a walk and Nate Solinger scored a pair of runs.
The Silverstreaks starting pitcher was Seth Staloch, he threw four innings. He gave up five hits, four runs, five walks and he recorded three strikeouts. W. Kirik threw three innings, he gave up
three hits, six runs and three walks. Ben Berger threw two innings, he gave up one run. Kyle Mages threw one inning, he gave up one hit, two runs, four walks and he recorded one strikeout.
The Silverstreaks offense was led by W. Kirick, he went 2-off-3 for a RBI, a walk, had a stolen base and he scored a pair of runs. Ben Berger went 1-for-3 with a double for a RBI and Zach Winkle was
hit by a pitch and he was credited for a RBI. Wyatt Sell went 1-for-4 for a RBI and he scored a run and Jacob Johanson was credited for a RBI. Kyle Mages went 2-for-4 with two doubles and he scored
two runs. Grant Mages went 2-for-4 and he scored a run.
FOLEY FALCONS 19 ALBANY HUSKIES 17
The Falcons defeated their Granite Ridge Conference and Section rivals the Huskies. They were out hit twenty-three to twelve, they did collect two doubles and two home runs. Their starting pitcher
was Josiah Peterson threw three innings. He gave up fourteen hits, eight runs and he recorded one strikeout. Reed Hermanson threw 2/3 of an inning, he gave up four runs and four walks. Derek Dahmen
threw 2 1/3 innings, he gave up seven hits, two runs, three runs, two walks and he recorded two strikeouts. Trey Emmerich threw two innings, he gave up two hits, three runs and he recorded five
The Falcons offense was led by Bryce Gapinski, he went 3-for-5 with a home run for 4 RBIs, he earned a walk and he scored three runs. Trey Emmerich went 2-for-4 with two home runs for 5 RBIs, he
earned a walk, he was hit by a pitch twice and he scored three runs. Derek Dahmen went 2-for-4 for three RBIs, he was hit by a pitch and he scored a run. Brett Leabch went 2-for-6 with a double for a
RBI and he scored a run and Jayden Enerson earned two walks and he was hit by a pitch. Reed Hermanson went 1-for-3 for a RBI, he earned two walks, he was hit by a pitch and he scored five runs. Noah
Gipanski went 1-for-3 for two RBIs, he earned two walks, he was hit by a pitch and he scored a run.Josiah Peterson went 1-for-5 with a double, he earned a walk and he scored two runs, Jordan
Lewandowsk scored a run and Wyatt Lueck scored a run.
The Huskies starting pitcher was Landon Vogel, he threw 4 2/3 innings, he gave up six hits, seven runs, five walks and he recorded one strikeout. Hansen threw 2 1/3 innings, he gave up five hits ten
runs, three walks and he recorded three strikeouts. Jake Lauer threw one inning, he gave up one hit, two runs and one walk.
They were led on offense by Ethan Meyer, he ent 3-for-6 with two doubles for four RBIs and he scored a run. Elliot Burnett went 4-for-5 with a triple and a double for three RBIs, he earned a walk and
he scored a run. Bennett Hylla went 3-for-5 with a triple and two doubles for three RBIs, he earned a walk and he scored four runs. Keenan Dingman went 2-for-5 with a home run and a triple for two
RBIs and he was hit by a pitch. Owen Sunderman went 2-for-4 with a double, he earned two walks and he scored a run and Drew Cramlet went 1-for-2. Landon Vogel went 2-for-3 with a double for a RBI, he
earned a walk and he scored a run. Zach Birr went 3-for-6 for a RBI, a stolen base and he scored two runs, Elliot Allen went 1-for-3, he earned a walk and he scored a pair of runs and Res scored a
pair of runs.
BECKER BULLDOGS 5 PRINCETON TIGERS 4
The Bulldogs defeated their Mississippi 8 Conference rivals the Tigers, they out hit them eight to seven, including a home run and four doubles. Their starting pitcher was Gerad Hanle, he threw five
innings, he gave up seven hits, four runs, one walk and he recorded eight strikeouts. Keegan Graning threw two innings to close it out, he gave up three walks and he recorded four strikeouts.
The Bulldogs offense was led by Reid McCalla, he went 1-for-2 with a home run and a sacrifice fly for two RBIs and Ethan Obermoller went 1-for-3 with a double.. Ethan Guck went 1-for3 with a double
for a RBI and Gerad Hanle went 1-for-3 with a double. Isaac Guck went 1-for-4 for a RBI and he had a stolen base and Josh Groskreutz earned a walk. Issac Daluge went 1-for-4 for a RBI and he scored a
run and Kellan Graning went 2-for-4 with a double and he scored a run. Gerad Hanle went 1-for-3 with a double and Jase Tobako was hit by a pitch.
The Tigers starting pitcher was Will Peterson, he threw 4 1/3 innings. He gave up six hits, five runs, one walk and he recorded two strikeouts. Lane Olson threw 2 2/3 innings, he gave up two hits and
he recorded two strikeouts.
Their offense was led by Niko Bratulich, he went 2-for-4 with a double for two RBIs and Eli Christopher went 1-for-3. Cullen Drews went 1-for-4 with a double for two RBIs and he scored a run. Lane
Olson went 1-for-5, he was hit by a pitch and he scored a run and Lukas Olson went 1-for-4. Tyler Peters went 1-for-3 with a walk and Eli Christopher went 1-for3-. Eli Gibbs earned a walk, he was hit
twice by a pitch and he scored a run. Nolan Peters and B. Shafer both had a walk.
SATURDAY MAY 18
ROCORI SPARTANS 7 NEW ULM EAGLES 1
The Spartans defeated the Eagles, they out hit them eleven to four, including eight collecting hits. Their starting pitcher was Max Fredin, he threw five innings to earn the win. He gave up three
hits, one run, two walks and he recorded five strikeouts. Hunter Fuchs threw one inning, he gave up one hit and he recorded three strikeouts. Jack Boos closed it out with one inning of relief, he
recorded one strikeout.
The Spartans offense was led by Max Fredin, he went 2-for-4 with a double for three RBIs and he scored a run. Tyler Prom went 2-for-3 for a RBI and he scored a run and Peyton Stocker earned a walk.
Jacob Stalboerger went 1-for-2 for a RBI, he earned a walk and he scored a run. Riley Bauer went 1-for-3 for a RBI and Hunter Fuchs earned a pair of walks and he had a stolen base. Noah Olmscheid
went 1-for-1 with a sacrifice fly for a RBI, he earned a walk and he scored a run. Jace Griffin went 2-for-3 and he scored a run and Jared Laudenbach went 1-for-3, he was hit by a pitch and he scored
a run. Jack Boos went 1-for-3, he earned a walk and he scored a run.
The Eagles starting pitcher was B. Olson, he threw four innings, he gave up eight hits, five runs, four walks and he recorded a strikeout. K. Larson threw one inning, he gave up three hits, two runs,
one walk and he recorded one strikeout. C. Slette threw one inning, he gave up a walk and he recorded two strikeouts.
The Eagles offense was led by T. Backer, he went 1-for-4 with a double and he scored a run and L. Seuss had a walk. R. Truman went 1-for-3 with a stolen base and L. Barstad had a stolen base. K.
Albrecht went 1-for-4 and B. Alfred had a walk. C. Serbus went 1-for-1 and E. Thompson had a stolen base and he was hit by a pitch.
Come With Us and Visit Melrose, MN in Pictures
More From 1390 Granite City Sports | {"url":"https://1390granitecitysports.com/central-mn-high-school-baseball-recap-9/","timestamp":"2024-11-09T00:01:22Z","content_type":"text/html","content_length":"252472","record_id":"<urn:uuid:96d308d5-5913-4840-a667-66ff6faee611>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00848.warc.gz"} |
Cary Audio Design CAD-805 monoblock power amplifier Measurements
Sidebar 1: Measurements
Because the Cary CAD-805, like any tube amplifier, has multiple output taps (further complicated in the case of the 805 by the presence of a choice of feedback settings), I chose to perform a
cross-section of diagnostic tests. Most of the measurements were made with 5dB of feedback—the setting chosen by Dick Olsher for most of his listening evaluations—with selected additional
measurements presented at 0dB and 10dB of feedback. In the case of the latter two, the 8 ohm tap was used with an 8 ohm load. Also, only a few select measurements were made into a 16 ohm load—a load
that a search of the October 1993 Audio equipment directory shows to be very rare among modern loudspeakers. If it's not otherwise stated in the discussion below, the measurements should be assumed
to be from the 8 ohm tap with 5dB of feedback.
Following its 1-hour, 1/3-power preconditioning test, the Cary CAD-805 was no hotter than you might expect in normal operation. (This test was devised by the IHF primarily with solid-state class-B
amplifiers in mind—in which case 1/3 power roughly corresponds to a worst-case heating situation. It is less applicable to tube amplifiers, but we perform it in the latter case primarily to maintain
consistency and to warm-up and stabilize the amplifier.) Voltage gain averaged 23.2dB at 5dB feedback (into an 8 ohm load, slightly less into a 4 ohm load, slightly more into a 16 ohm load). With 0dB
of feedback, the gain increased to 25.2dB; with 10dB of feedback it dropped to 18.2dB (all into an 8 ohm load). The input impedance measured just under 104k ohms, DC offset 0.1mV (largely noise). The
CAD-805 was non-inverting, a positive-going input emerging positive at the output. The wideband S/N ratio, unweighted (at 1W into 8 ohms), was an excellent 108.1dB.
The CAD-805's output impedance is shown in Table 1 for several different conditions of operation. (The figures shown are averages; there was a small variation of the calculated value with load
impedance, but this was not particularly significant.) The values are reasonably low for a tube amplifier, and while some sensitivity to loudspeaker load may be expected—the higher the output
impedance of an amplifier, the more such sensitivity increases—I would expect it to be less here than with many tube amps having higher output impedances (see my "Questions of Impedance Interaction"
Table 1: Output Impedance
│ Output Tap │ Feedback │ Ohms │
│ Ohms │ dB │ 20Hz │ 1kHz │ 20kHz │
│ 4 │ 5 │ 0.51 │ 0.63 │ 0.62 │
│ 8 │ 5 │ 0.74 │ 0.92 │ 0.86 │
│ 16 │ 5 │ 1.14 │ 1.38 │ 1.14 │
│ 8 │ 0 │ - │ 1.18 │ 1.14 │
│ 8 │ 10 │ - │ 0.49 │ - │
Fig.1 shows the frequency response with 5dB of negative feedback at the 8 ohm tap (the scale here is expanded to ±3dB instead of our usual ±2dB). The two topmost curves—the response at 1W into 8 ohms
and 2W into 4 ohms, respectively—are virtually the same, with the exception of there being less output into 4 ohms at 10Hz. The lower curve shows the response at 1W into 8 ohms from the 4 ohm tap
(the response at 2W into 4 ohms from the 4 ohm tap is virtually identical except that it's -1dB at 10Hz). Note that all the responses show a rise in the high frequencies peaking out just under 30kHz
and a dip in the bass centered just above 20Hz. The former—probably due to the output transformer—may be audible in some circumstances since the rise begins in the audible range. The low-frequency
dip, however, probably will be audible on a system having good low-end extension.
Fig.1 Cary CAD-805, 5dB negative feedback, frequency response at (from top to bottom at 30kHz): 1W into 8 ohms (8 ohm tap), 2W into 4 ohms (8 ohms tap), 1W into 8 ohms (4 ohm tap) (1dB/vertical
Fig.2 shows the frequency response again, this time as the negative feedback is varied. Here, increasing the feedback reduces the bass dip, decreasing feedback aggravates it. At the top end,
increasing feedback increases the magnitude of the peak but pushes it slightly higher in frequency. The results from the 16 ohm tap are not shown; they are similar from 100Hz to 20kHz, but with a
greater rise above 20kHz and a lesser dip below 100Hz.
Fig.2 Cary CAD-805, frequency response at 1W into 8 ohms (8 ohm tap) with (from top to bottom at 30kHz): 10dB, 5dB, and 0dB negative feedback (1dB/vertical div.).
The CAD-805's output, in response to a 1kHz squarewave, is shown in fig.3. Note the small leading-edge overshoot (which actually becomes slightly more pronounced as the feedback is increased to
10dB—not shown). At 10kHz (fig.4) the overshoot and damped oscillations are more apparent—reflecting the ultrasonic peak in the amplitude response.
Fig.3 Cary CAD-805, 1kHz squarewave at 1W into 8 ohms (8 ohm tap).
Fig.4 Cary CAD-805, 10kHz squarewave at 1W into 8 ohms (8 ohm tap).
Fig.5 shows the THD+noise at 1W into 8 ohms from all three taps; fig.6 shows the same, except at 2W into 4 ohms. Note that in both cases the 4 ohm tap gives the lowest distortion through the midband,
though it crosses over to become marginally the highest at 20kHz. Though you might conclude from this that the 4 ohm tap is optimum for either 4 or 8 ohm loads, you would be wrong in the case of an 8
ohm load—as the THD+noise vs level curves, presented below, will confirm. The levels here are moderately high, though not unexpectedly so for a single-ended, low-feedback design. The THD+noise vs
frequency curves for a 2 ohm load (4W) are not shown; they show a reading of under 0.7% from 100Hz to 3kHz from the 4 ohm tap, remaining below 1.3% at 10Hz but rising to just under 2.5% at 20kHz.
From the 8 ohm and 16 ohm taps, the midband THD+noise rises to just over 1% and 2% respectively for 4W into 2 ohms. These results, together with the THD+noise figures below, indicate that DO was
correct to restrict his search for a loudspeaker to use with the CAD-805 to those whose load impedance remains both high and relatively uniform.
Fig.5 Cary CAD-805, THD+noise vs frequency at 1W into 8 ohms, 5dB feedback, from (from top to bottom at 35kHz): 8 ohm tap, 4 ohm tap, 16 ohm tap.
Fig.6 Cary CAD-805, THD+noise vs frequency at 2W into 4 ohms, 5dB feedback, from (from top to bottom at 35kHz): 4 ohm tap, 16 ohm tap, 8 ohm tap.
Fig.7 shows how the low-level THD+noise vs frequency changes with varying levels of feedback. As expected, the change tracks the amount of feedback—higher THD+noise with less feedback—but the
differences with the small amount of feedback used here are not dramatic.
Fig.7 Cary CAD-805, THD+noise vs frequency at 1W into 8 ohms, 8 ohm tap, with (from top to bottom at 35kHz): 10dB, 5dB, and 0dB feedback.
The 1kHz THD+noise waveform at low power into both 8 ohms and 4 ohms, not shown, is almost pure second harmonic with some noise. Fig.8 shows the THD+noise waveform for 4W into 2 ohms (from the 4 ohm
tap). The result is still heavily second-harmonic, though with an interesting notch on the downside of every other cycle in the distortion wave. This corresponds to the negative-going portion of the
signal itself and may relate to the circuit's intrinsic performance, possibly the single-ended output tube in this circumstance. The THD+noise here is actually fairly low in level: under 0.3% (see
fig.6). Though the CAD-805's THD+noise levels are higher than those usually seen with competitive high-end amplifiers, their heavily second-harmonic nature should be musically consonant with the
input. [The individual distortion harmonic tracks on Stereophile's Test CD 2 show quite convincingly that even 1% of second harmonic is completely inaudible.—Ed.]
Fig.8 Cary CAD-805, 1kHz waveform at 4W into 2 ohms (from the 4 ohm tap) (top); distortion and noise waveform with fundamental notched out (bottom).
At low frequencies and higher powers, the amplifier's distortion signature is less benign. The distortion spectrum resulting from a 50Hz input at 24W into 4 ohms (4 ohm tap) (approximately 2/3 of the
power output at 3% THD+noise, 5dB feedback) is shown in fig.9. The distortion levels are relatively high: -33dB (about 2.5%) at 100Hz (second harmonic), -38.2dB (about 1.2%) at 150Hz. The distortion
level does not drop below 0.1% until we reach the ninth harmonic (450Hz). At a much lower power output (4W, not shown), the distortion levels are lower: about 0.6% at 100Hz, dropping below 0.1% above
Fig.9 Cary CAD-805, spectrum of 50Hz sinewave, DC-1kHz, at 24W into 4 ohms (linear frequency scale). Note that the second harmonic at 100Hz is the highest in level, 33dB below the level of the 50Hz
fundamental (2.5%).
Fig.10 shows the output resulting from an input of a combined 19+20kHz into 8 ohms at the 8 ohm tap. The output level here is 6.5W—just below the level at which clipping is observed in the output
waveform with this input signal. Incidentally, clipping onset is quite gentle with the CAD-805, initially visible only as a slight rounding of the waveform's lower half (the bottom of the waveform
shows signs of clipping before the top). The 1kHz intermodulation artifact lies at -29.3dB or about 3.5%—a very high level. The 2kHz and 3kHz IM products drop to just below 0.5%. The higher-frequency
products are again high, reaching a maximum of -29dB (3.5%) at 18kHz. The corresponding results for a 4 ohm load, also into 6.5W, from the 4 ohm tap (not shown) were quite similar, with marginally
higher—but not significantly so—distortion artifacts.
Fig.10 Cary CAD-805, HF intermodulation spectrum, DC-22kHz, 19+20kHz at 6.5W into 8 ohms (linear frequency scale).
The CAD-805's THD+noise vs level curves, with all three output taps driving an 8 ohm load, is shown in fig.11. Fig.12 shows the output into 2, 4, and 8 ohm loads from the 8 ohm tap, and fig.13 shows
the effect of varying the amount of feedback into an 8 ohm load at the 8 ohm tap. The important data from all three of these curves, plus data for other taps and other loads which are not presented
in graphical form for reasons of space, are shown in Table 2. The values shown in this Table were read directly from the appropriate graphs. The "knee" referred to here is the first point at which a
major positive change in the distortion slope occurs. The main observation to be made from this table is, as might be expected, that the maximum output is available from the CAD-805 when the output
tap selected matches the load impedance. While this is not necessarily true at the 1% THD+noise level, it's certainly the case at higher output. Note also that, whatever the pros and cons of negative
feedback, the only configuration which produces more than 10W output at the 1% THD+noise point is the one having 10dB of feedback.
Fig.11 Cary CAD-805, distortion vs output power into 8 ohms, 5dB feedback, from (from bottom to top at 1W): 4 ohm tap, 8 ohm tap, and 16 ohm tap.
Fig.12 Cary CAD-805, distortion vs output power from 8 ohm tap into (from bottom to top at 1W): 8 ohms, 4 ohms, and 2 ohms.
Fig.13 Cary CAD-805, distortion vs output power from 8 ohm tap into 8 ohms, with (from bottom to top at 1W): 10dB, 5dB, and 0dB feedback.
Table 2: Power Output
│ Output │ Output │ │ │ 1% │ 3% │ │
│ Load │ Tap │ F/B │ Knee │ THD+N │ THD+N │ │
│ ohms │ ohms │ dB │ W │ W │ W │ dBW │
│ 16 │ 16 │ 5 │ 3.9 │ 9.5 │ 41 │ 19.1 │
│ 8 │ 8 │ 5 │ 3.8 │ 7.2 │ 37 │ 15.7 │
│ 8 │ 4 │ 5 │ 2.3 │ 4.5 │ 20 │ 13.0 │
│ 8 │ 16 │ 5 │ 5.0 │ 9.5 │ 28 │ 14.5 │
│ 8 │ 8 │ 0 │ 3.6 │ 6.3 │ 35 │ 15.4 │
│ 8 │ 8 │ 10 │ 3.7 │ 28.0 │ 41 │ 16.1 │
│ 4 │ 16 │ 5 │ - │ 3.0 │ 15 │ 8.75 │
│ 4 │ 8 │ 5 │ 5.0 │ 7.8 │ 28 │ 11.5 │
│ 4 │ 4 │ 5 │ 3.9 │ 6.1 │ 35 │ 12.4 │
│ 2 │ 16 │ 5 │ - │ 0.7 │ 6 │ 1.75 │
│ 2 │ 8 │ 5 │ - │ 2.4 │ 15 │ 5.75 │
│ 2 │ 4 │ 5 │ 4.8 │ 6.8 │ 27 │ 8.3 │
The Cary's actual discrete clipping points (for the purposes of discussion, defined here as 3% THD+noise at 1kHz) for the 8 ohm tap, 5dB feedback, were 38.2W (15.8dBW) into 8 ohms, 28.9W (11.6dBW)
into 4 ohms, and 14.4W (5.6dBW) into 2 ohms. From the 4 ohm tap, the corresponding values were 20.2W (13.1dBW) into 8 ohms, 35.9W (12.5dBW) into 4 ohms, and 28W (8.5dBW) into 2 ohms. And from the 16
ohm tap, 28.9W (14.6dBW) into 8 ohms, 14.4W (8.6dBW) into 4 ohms, and 6.4W (2.1dBW) into 2 ohms. All line voltages for these measurements were between 118V and 119V.
In classical terms, the CAD-805's test bench results cannot be categorized as anything other than mediocre, at best, even for a tube amplifier. As I stated in my measurement conclusions to the Jadis
JA 200 review last November (Vol.16 No.11, p.153), such a set of measurements raises a question: Does the amplifier sound the way it does, in whole or in part, because or in spite of its objective
performance? The former is not acceptable in a high-fidelity device, and certainly at least a number of the measured results on the CAD-805 fall within the boundaries of what we know to be audible
deviations. And its power output, as DO states, restricts the user's choice of loudspeakers.
The CAD-805's somewhat nostalgia-inducing design is reinforced by its measured performance—an updated nostalgia, to be sure, but updating can only bring us so far in what is basically a
half-century-old design concept, one long since abandoned for what would appear to be very good objective reasons. There is more to the story than measurements, of course; if you listen to the
CAD-805s, fall in love with their sound, and can afford the price and loudspeaker restrictions, by all means buy them. But go into the purchase with open ears.—Thomas J. Norton | {"url":"https://www.stereophile.com/content/cary-audio-design-cad-805-monoblock-power-amplifier-measurements","timestamp":"2024-11-04T01:33:50Z","content_type":"application/xhtml+xml","content_length":"126477","record_id":"<urn:uuid:457765ee-1043-4d5f-a9a5-296669668f50>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00492.warc.gz"} |
Sorted merge of two sorted doubly circular linked lists | Linked list articles | PrepBytes Blog
Last Updated on November 21, 2022 by Prepbytes
In the previous articles, we have already seen different implementations on circular singly linked list and circular doubly linked list. Now let’s just look into an approach on how to merge two
circular linked list.
How to Merge Two Circular Linked List
In this problem, we are given two sorted doubly circular linked lists containing n and m number of nodes, respectively. Our task is to merge the two lists such that the resultant doubly circular
linked list is also in sorted order.
Let’s try to understand this problem with the help of examples.
If the given sorted doubly circular linked lists are:
list 1:
list 2:
• According to the problem statement, we need to merge list 1 and list 2 in such a way that the final merged list is in the sorted order.
• After merging list 1 and list 2, our final merged sorted doubly circular linked list will be:
Taking another example, if the linked lists are:
list 1:
list 2:
• In this case, our final sorted doubly circular linked list after merging list 1, and list 2 will be:
Now, I think from the above examples, the problem statement is clear. Let’s see how we can approach it.
Before moving to the approach section, try to think how you can approach this problem.
If stuck, no problem, we will thoroughly see how we can approach this problem in the next section.
Let’s move to the approach section.
Approach on how to merge two circular linked list
Our approach will be straightforward:
• Basically, first, we will find the node, which will be the last node of our final merged doubly circular linked list.
• As we know that our two given doubly circular linked list are sorted, so one thing is clear, that the last node of our final merged doubly circular linked list will be one which is having greater
value among the last nodes of the two doubly circular linked list.
• We will keep track of this node using a pointer, say last_node.
• Now we will make the last nodes of the given doubly circular linked lists point to NULL.
• Then after that we will merge the two doubly circular linked list in the same way we merge two sorted doubly linked list.
– If you are not aware of how to merge two sorted doubly circular linked list, checkout our article [Merge two sorted doubly linked list]().
• After merging the two sorted doubly linked list, we will make the merged list circular using the last_node.
• And finally, we will have our sorted merged doubly circular linked list.
Let’s see the algorithm.
Algorithm on how to merge two circular linked list
• Let h1 be a pointer pointing to the head node of the first doubly circular linked list, and h2 be the pointer pointing to the head node of the second list.
• If h2 is NULL, return h1.
• If h1 is NULL, return h2.
• Suppose lst1 and lst2 are the last nodes of the two doubly circular linked lists, respectively.
□ lst1 and lst2 can be obtained by the previous links of the first nodes of the respective lists.
• Get a pointer to the node, which will be the last node of the final resultant linked list result.
□ If lst1→data is less than lst2→data, then last_node = lst2.
□ Else, last_node = lst1.
• Now update lst1→next = lst2→next = NULL.
• We will now merge the two lists as two sorted doubly linked lists are being merged.
• Let the first node of the final sorted doubly circular linked list be ResHead.
• Finally, update the *prev of ResHead to last_node and next of last_node to ResHead.
• At the end, return ResHead.
Dry Run on how to merge two circular linked list
## Code Implementation on how to merge two circular linked list
using namespace std;
/* Node structure of a doubly linked list node */
struct Node {
int data;
Node *next, *prev;
/* Using this function we will be inserting a new node in the list */
void insert(Node** head, int data)
Node* new_node = new Node;
new_node->data = data;
if (*head == NULL) {
new_node->next = new_node;
new_node->prev = new_node;
else {
Node* last = (*head)->prev;
new_node->next = *head;
new_node->prev = last;
last->next = (*head)->prev = new_node;
*head = new_node;
/* Using this function we will be merging two sorted doubly linked list */
//merge2SortedDLL stands for merge two sorted doubly linked list
Node* merge2SortedDLL(Node* l1, Node* l2)
if (!l1)
return l2;
if (!l2)
return l1;
if (l1->data < l2->data) {
l1->next = merge2SortedDLL(l1->next, l2);
l1->next->prev = l1;
l1->prev = NULL;
return l1;
else {
l2->next = merge2SortedDLL(l1, l2->next);
l2->next->prev = l2;
l2->prev = NULL;
return l2;
/* Using this function we will be merging two sorted doubly circular linked list */
//merge2SortedDCLL stands for merge two sorted doubly circular linked list
Node* merge2SortedDCLL(Node* h1, Node* h2)
if (!h1)
return h2;
if (!h2)
return h1;
Node* last_node;
if (h1->prev->data < h2->prev->data)
last_node = h2->prev;
last_node = h1->prev;
h1->prev->next = h2->prev->next = NULL;
Node* ResHead = merge2SortedDLL(h1, h2);
ResHead->prev = last_node;
last_node->next = ResHead;
return ResHead;
/* Using this function we will be printing the linked list content */
void PrintingList(Node* head)
Node* temp = head;
while (temp->next != head) {
cout << temp->data << " ";
temp = temp->next;
cout << temp->data << " "<
Original linked list 1: 1 3 5 7
Original linked list 2: 2 4 6 8
Final Sorted List: 1 2 3 4 5 6 7 8
**Time Complexity of how to merge two circular linked list:** O(n+m), as each list is traversed completely.
So, in this article, We learnt how to merge of two circular linked lists, even though we are aware about how a normal linked list can be merged, the approach of circular linked is different.This is a
basic problem and is good for strengthening your concepts in LinkedList and if you want to practice more such problems, you can checkout [Prepbytes (Linked List)](https://mycode.prepbytes.com/
interview-coding/practice/linked-list “Prepbytes (Linked List)”)
## FAQs
**1. What is a circular linked list?**
A circular linked list is a structured collection of various elements which are connected to form a shape of circle with no null at the end.
**2. What are the applications of circular linked lists?**
A circular linked list can be used for computer resource management and also used for implementing advanced data structures such as fibonacci heap.
**3. How do you find the circular linked list?**
We will store the header node into some other variable and traverse through the list, if we will get null at any part of list then it is not circular linked list and if no then it is a circular
linked list.
Leave a Reply Cancel reply | {"url":"https://www.prepbytes.com/blog/linked-list/sorted-merge-of-two-sorted-doubly-circular-linked-lists/","timestamp":"2024-11-11T16:23:13Z","content_type":"text/html","content_length":"146766","record_id":"<urn:uuid:801d26d5-7a1f-4621-94ef-c2d58338ad87>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00311.warc.gz"} |
local_constraint(G, u, v, weight=None)[source]#
Returns the local constraint on the node u with respect to the node v in the graph G.
Formally, the local constraint on u with respect to v, denoted \(\ell(u, v)\), is defined by
\[\ell(u, v) = \left(p_{uv} + \sum_{w \in N(v)} p_{uw} p_{wv}\right)^2,\]
where \(N(v)\) is the set of neighbors of \(v\) and \(p_{uv}\) is the normalized mutual weight of the (directed or undirected) edges joining \(u\) and \(v\), for each vertex \(u\) and \(v\) [1].
The mutual weight of \(u\) and \(v\) is the sum of the weights of edges joining them (edge weights are assumed to be one if the graph is unweighted).
GNetworkX graph
The graph containing u and v. This can be either directed or undirected.
A node in the graph G.
A node in the graph G.
weightNone or string, optional
If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight.
The constraint of the node v in the graph G.
Burt, Ronald S. “Structural holes and good ideas”. American Journal of Sociology (110): 349–399. | {"url":"https://networkx.org/documentation/latest/reference/algorithms/generated/networkx.algorithms.structuralholes.local_constraint.html","timestamp":"2024-11-11T13:26:11Z","content_type":"text/html","content_length":"33901","record_id":"<urn:uuid:6cb52b53-1cf5-484a-ad0e-a35aadd61a3e>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00111.warc.gz"} |
0568 #43
I'm banging my head against the wall on this one.
43. if z = exp(2*Pi*i/5), then 1 + z + z^2 + z^3 + 5z^4 + 4z^5 + 4z^6 + 4z^7 + 4z^8 + 5z^9 = ...
The answer is -5*exp(3*Pi*i/5), and I can't seem to get there.
I haven't had complex analysis, so I only know what I've picked up from using it in other classes. I can get the given expression to equal:
5(1 + exp(Pi*i/5) + exp(2*Pi*i/5) + 2*exp(3*Pi*i/5) + exp(4*Pi*i/5))
but I have no idea how to combine those exponentials. I know they're all 5th roots of unity, but I don't see how that helps. Any ideas?
(Also, sorry, I've never used LaTex, so this looks crappy. I'll learn it when I need it.)
Re: 0568 #43
If z = e^(2*i*pi/5), then z^5 = 1 and z^i = z^(i+5). So we can express the sum
1 + z + z^2 + z^3 + 5z^4 + 4z^5 + 4z^6 + 4z^7 + 4z^8 + 5z^9
1 + z + z^2 + z^3 + 5z^4 + 4 + 4z + 4z^2 + 4z^3 + 5z^4 = 5(1 + z + z^2 + z^3 + z^4) + 5z^4.
Now, (1 + x + x^2 + ... + x^n) = (1 - x^(n+1))/(1 - x). So if x^(k+1) = 1 and x doesn't equal 1, it follows that 1 + x + x^2 + ... + x^k = 0. Therefore 1 + z + z^2 + z^3 + z^4 = 0, and the rewritten
sum in question is 5*0 + 5z^4 = 5e^(8*i*pi/5) = -5e^(3*i*pi/5).
Re: 0568 #43
We can also use the sum-of-roots theorem. The fifth roots of unity are all the zeros of the polynomial
so they must sum to $$-\frac{0}{1}=0.$$
Sum-of-roots seems like a good one to know for the exam. I got this problem wrong when I took the practice, too, so now I have flash cards for sum-of-roots and product-of-roots. Rational roots is
handy, too.
Last edited by joey on Wed Nov 04, 2009 11:53 pm, edited 1 time in total.
Re: 0568 #43
Thanks for your quick replies, guys. Both methods make perfect sense, and both are things I just wouldn't have thought of even trying... I guess it's also better to leave everything in terms of z
rather than writing them all out as exponentials. I'll have to get more comfortable with complex numbers by Saturday :-p
Re: 0568 #43
No problem. This forum is great because it gets us all thinking about the appropriate topics in a way that helps one another do better. It's one of few ways in which I use the internet for its
original purpose: academic collaboration over vast distances. (At least, I assume you're not all concentrated in the Twin Cities metro.)
When you study complex, be sure to learn the Cauchy-Riemann equations and the Cauchy-Goursat theorem (sometimes called the Cauchy integral theorem). These are favorites on past GREs, and they're
easily accessible. If you have more time and really want to dig in, learn about Laurent series / residues. These topics require you to think about series expansion in a more rigorous way, and they
sometimes appear on the GRE.
Good luck! | {"url":"https://mathematicsgre.com/viewtopic.php?f=1&t=325","timestamp":"2024-11-02T14:43:41Z","content_type":"text/html","content_length":"26681","record_id":"<urn:uuid:e8fe1d30-a174-45f4-88cb-6ef450727e1e>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00430.warc.gz"} |
Total Amount Due in context of utility bill
07 Oct 2024
Title: An Examination of the Total Amount Due in Utility Billing: A Theoretical Framework
This article delves into the concept of total amount due in the context of utility billing, providing a comprehensive theoretical framework for understanding this critical component of customer
finance management. We explore the various factors that contribute to the calculation of the total amount due, including charges, taxes, and fees.
Utility bills are an essential aspect of modern life, providing households and businesses with access to vital services such as electricity, water, gas, and telecommunications. The total amount due
on a utility bill is a critical component of customer finance management, representing the sum of all charges, taxes, and fees associated with the provision of these services.
Theoretical Framework:
Let TAD denote the Total Amount Due, which can be calculated using the following formula:
TAD = C + T + F
where: C = Charges (electricity, water, gas, etc.) T = Taxes (applicable taxes on charges) F = Fees (connection fees, late payment fees, etc.)
Charges (C):
Charges refer to the actual cost of providing utility services to a customer. These can include:
• Energy consumption charges
• Water usage charges
• Gas consumption charges
Taxes (T):
Taxes are applicable on the charges and are typically calculated as a percentage of the total charge.
T = (C x Tax Rate)
where: Tax Rate is the percentage rate at which taxes are applied
Fees (F):
Fees refer to additional costs associated with providing utility services, such as:
• Connection fees
• Late payment fees
• Meter reading fees
The total amount due on a utility bill is a complex calculation that involves charges, taxes, and fees. Understanding the theoretical framework underlying this concept is essential for effective
customer finance management. By breaking down the components of the total amount due, we can better appreciate the intricacies of utility billing and make informed decisions about our energy
consumption habits.
• [Insert relevant references here]
Note: The above article provides a general overview of the theoretical framework underlying the concept of total amount due in the context of utility billing. It does not provide numerical examples
or specific case studies.
Related articles for ‘utility bill’ :
• Reading: Total Amount Due in context of utility bill
Calculators for ‘utility bill’ | {"url":"https://blog.truegeometry.com/tutorials/education/4b87e1f422f1b42664dc64f150622f89/JSON_TO_ARTCL_Total_Amount_Due_in_context_of_utility_bill.html","timestamp":"2024-11-11T11:06:41Z","content_type":"text/html","content_length":"16320","record_id":"<urn:uuid:355de52e-95ed-443a-9d45-905dac2a7a00>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00790.warc.gz"} |
What Teachers Know
In November of last year, 200 people, mostly mathematicians, sent an open letter to Secretary of Education Richard Riley. The letter asked him to withdraw the labels "exemplary" and "promising" that
the Education Department had recently applied to 10 innovative math programs. I do not know the validity of the criticisms. I do know that the last time the nation let mathematicians develop K-12
curricula, in the post-Sputnik panic, the result was a debacle known as New Math.
I mention the letter to illustrate that the teaching of mathematics has become a politicized matter, just as the teaching of reading has been. Two recent books help us see why. Simply put, something
has to be done to improve the mathematics knowledge of America's teachers. In The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom, James Stigler of UCLA
and James Hiebert of the University of Delaware analyze videotapes of middle school instructors teaching mathematics in Japan, Germany, and the United States. These tapes were made as part of the
41-country Third International Mathematics and Science Study. Knowing and Teaching Elementary Mathematics is a study of how math is taught in elementary schools in the United States and the People's
Republic of China, conducted by Liping Ma at the University of California, Berkeley.
Neither book has graceful prose, but neither book is obtuse or difficult, either. The Teaching Gap is written for the educated general reader; Knowing and Teaching Elementary Mathematics is for the
kind of people who delve into Scientific American. Both have important things to say about teaching mathematics in this country.
From the cover through the initial chapters, I distrusted The Teaching Gap. The title plays off The Learning Gap, an earlier book co-authored by Stigler. But The Teaching Gap in no way compares "best
ideas" from the "world's teachers," as its subtitle suggests it does. There is not one such idea in the entire volume. Even if there were, the study would have no way of directly linking those ideas
or teaching practices to mathematics achievement.
The inside flap of the cover contains outright errors. It opens with, "For years our schools and children have lagged behind international standards in reading, arithmetic, and most other areas of
achievement." This is not true. In the most recent international comparison of reading, only one nation out of 31 scored significantly higher than American students. And in the only study that had
"arithmetic" as a category, American students were average. In the science segment of the Third International Mathematics and Science Study, American students were above average.
The authors repeat a claim from an earlier study that "the highest-scoring classroom in the U.S. sample did not perform as well as the lowest-scoring section in the Japanese sample." The data from
the latest mathematics and science study make it clear that this earlier finding occurred because the Japanese sample was not representative of the nation. While Japanese students scored
substantially better than American kids, there was also substantial overlap between the scores of the two groups. Moreover, American students in a group of suburban school districts scored almost as
high as the Japanese students, getting 70 percent of the items correct, compared to Japan's 73 percent. Stigler and Hiebert would have done better to leave aside previous research. Their study stands
on its own.
For Stigler and Hiebert's book, a team of six people, two from each country, coded the Third International Mathematics and Science Study videotapes. The description of the code development process
does not exactly inspire confidence. "The discussion was so vigorous that it often would take a day or more to get through a single lesson. There were disagreements in the group about the content of
the tapes, and especially about how to describe them." It would be nice to know more details. It would have been even better if another team had also developed a system of analysis independently.
Would it have contained the same elements?
The team found that American teachers present about twice as many definitions as Japanese and German teachers. This accords with other findings in the mathematics and science study that American
textbooks are about three times as thick as those of other nations and that our teachers try to teach it all, often covering topics only briefly and shallowly. Stigler and Hiebert discovered that
American lessons were devoid of mathematical proofs. About 10 percent of German lessons and more than half of the Japanese lessons contained such proofs. American teachers stated concepts, but did
not develop them. Only 22 percent of the lessons in this country contained developed topics, compared to 77 percent in Germany and 83 percent in Japan. Not only did German and Japanese teachers
develop topics; they linked them to other topics. When lessons were rated on the interrelatedness of their parts, German lessons scored four times as high as American lessons, Japanese lessons six
times as high.
Japanese and American teachers organized their lessons quite differently. In a typical American lesson, a teacher reviewed homework, demonstrated how to solve the problem of the day, gave students
classroom practice, corrected that work, and assigned homework. Japanese teachers reviewed the previous lesson, presented the problem of the day, and set the students to working on its solution
either individually or in groups. The class then discussed problem solutions (some problems had more than one), often led from the blackboard by students who thought they had successfully solved the
problem. American students almost never led such a discussion.
American and Japanese teachers organize their lessons differently in large part because they believe different things about what mathematics is and how to teach it. American teachers believe that
mathematics is a set of procedures; they want their students to become skilled at these procedures. Japanese teachers, by contrast, "act as if mathematics is a set of relationships between concepts,
facts, and procedures. Japanese teachers wanted their students to think about these relationships in new ways."
These differences seem profound, but as I noted at the outset, there is no way to relate any of them to the differences in performance. Indeed, one is left with an enigma: On many dimensions, the
German teachers are similar to Japanese teachers, yet while Japanese students attained much better scores than American students, German students scored the same as Americans.
It could well be that the Japanese students score higher because so many of them attend juku--cram schools--after school and on weekends. Juku specialize in teaching students how to take tests. The
Teaching Gap does not discuss the role of juku. There are other possible explanations. A Japanese educator who, at my request, watched some mathematics and science study tapes, concluded that,
indeed, the Japanese classes were more conceptually oriented. He felt, though, that Japanese teachers are free to teach conceptually because they can count on family support to ensure that less
glamorous mathematics activities will be completed at home. American teachers cannot count on such support.
While Stigler and Hiebert analyze teaching, Liping Ma presents teachers with problems in mathematics instruction and asks American and Chinese teachers how they would approach the problems. For
instance, how would they teach subtraction with regrouping? If, when multiplying two three-digit numbers, their students did not line up the three mini-products correctly, how would they explain what
to do and why?
Ma's American teachers sound very much like Stigler and Hiebert's, and her Chinese group closely parallels the Japanese. In the subtraction with regrouping problem, for instance, Ma writes, "Seventy
percent of the U.S. teachers and 14 percent of the Chinese teachers displayed only procedural knowledge of the topic. Their understanding was limited to surface aspects of the algorithm."
American teachers knew what needed to be done, but knew only the procedure, the algorithm. Some American teachers could not explain why each mini-product in the multiplication of two three-digit
numbers was moved one digit to the left: "I can't remember that rule. I can't remember why you do that. It's just like when I was taught, you just do it."
Ninety-two percent of the Chinese teachers showed a conceptual understanding of the problem, explaining it in terms of the place value of the different columns and the distributive law of
mathematics. American teachers who mentioned the "tens" or "hundreds" column seemed to use those words only as labels.
When Ma asked teachers to divide one and three-fourths by one-half, only 43 percent of the American teachers gave a complete and correct answer. Another 9 percent applied the algorithm properly but
then did not reduce the answer and convert to a proper fraction. Some multiplied where they should have divided. Some divided by two, not by one-half. Not only could most Chinese solve the problem
and explain it conceptually; some offered alternatives to the most common form of solution.
Ma reports that while many American teachers said they wanted to "teach for understanding," their limited knowledge of the problems would prevent such teaching. In all of the problem settings,
Chinese teachers related the problem at hand to other mathematical concepts while American teachers considered the problem in isolation from anything else.
The notion of finding Chinese and Japanese teachers who are "comparable" to the American group is probably not meaningful. But if one pays attention only to the Americans in both studies, one is
struck by the similarity of the findings. American teachers emphasize rules, procedures, and algorithms, and in some instances see no need to go beyond such knowledge, or have no capacity to. Both
books emphasize that better mathematics education must put teaching competence--particularly competence that develops after the teacher starts teaching--front and center.
One important contribution of these books is to show that American classrooms have not been taken prisoner by the acolytes of Rousseau and Dewey. In his book The Schools We Need and Why We Don't Have
Them, E. D. Hirsch, Jr., argued that just such a capture has occurred, leaving our schools to fail at the hands of these antiknowledge "romantic progressives." The Third International Mathematics and
Science Study tapes and Ma's findings indicate that, if anything, it is American teachers who are obsessed with getting knowledge out of their heads and into the kids' noggins in a most traditional,
fact-based approach.
Susan Ohanian's wise little book One Size Fits Few is a look at the folly of "Standardistos." These are people who seem to have little interest in the dynamics of classrooms and the needs of the kids
themselves. "How is all this going to work?" a parent asked former Secretary of Education William Bennett after a speech on the need for tougher standards. "I deal with wholesale; you're going to
have to work out the retail for yourself," Bennett replied. "There you have it," writes Ohanian. "Standardistos don't give a damn about how their plans and panaceas might work in classrooms."
Lest someone think Ohanian is exaggerating the prevalence of this kind of thinking, she fills her book with quotes from Standardistos par excellence such as the late Albert Shanker, who was president
of the American Federation of Teachers. Shanker contended, "Unless we have standards that tell us, grade by grade, what the teacher is required to teach and the student required to learn, many of our
students will not reach the level of competence that we expect of high school graduates."
Nonsense, Ohanian argues. Against such pronouncements, she offers counterexamples: individual, idiosyncratic kids and successful adults who followed their own unusual paths to success. A longtime
teacher, she also objects to the very idea of expecting all students to study and master the same thing. She notes that her third-grade students encountered Aesop and Robert Louis Stevenson, two
worthies who made E.D. Hirsch's list of approved topics and people. But, she goes on to say, they also encountered Jean de La Fontaine, Basho, Langston Hughes, Laura Ingalls Wilder, and E.B. White,
who didn't. Will her students therefore fail to attain the "level of competence" of which Shanker spoke? That's for Standardistos, not real teachers (or parents or kids), to worry about. And they do.
In a late 1999 speech, Hirsch declared, "A classroom of 25 or 30 students cannot move forward until all students have gained the knowledge necessary to 'getting' the next step in learning."
Such people believe that if the standards are there and teachers are trained to strictly adhere to how they should be taught, kids will master the material--and teachers will become a mere
"variable." University of Texas Standardista Professor Barbara Foorman says, "The teacher variable does not contribute significantly above and beyond the curriculum, so what we have here is a
powerful mathematical model. My hypothesis is that the teacher variable will be even less significant within the direct instruction group."
Against the widely held tenet that today's students don't know very much, Ohanian levels the charge that we are asking them to know more and more complex things earlier than ever. We are putting them
under tremendous pressure. No wonder the use of Ritalin and Luvox is increasing.
My experience corroborates Ohanian's assertions. Consider two examples.
An October 1999 Washington Post article mentioned casually that a sixth-grade class used Venn diagrams to solve problems. I first encountered Venn diagrams in a logic course as a college junior. So
did the reporter. When I mentioned this to a group of teachers, some said they were using them in third grade.
A 10th-grade social studies standard in Virginia's Standards of Learning program reads thus: "The student will analyze the regional development of Asia, Africa, the Middle East, Latin America and the
Caribbean in terms of physical and cultural characteristics and historical evolution from 1000 A.D. to the present."
This second example is not extreme or taken out of context; nor is it unique to Virginia. Fifth-graders in California must memorize the periodic table, and sixth-graders in South Dakota will know
more about ancient Greece than most college graduates who major in ancient history.
One indication that something is amiss with the standards movement can be seen in a January 2000 report from the Thomas B. Fordham Foundation evaluating the standards of the various states. Oddly,
states that had the best standards, according to the evaluation, scored low on domestic achievement tests and international comparisons. States whose standards were labeled "lousy" scored high on
both types of assessments.
Perhaps the answer is for teachers and principals to take a Standardisto to school. Ohanian gives us the case of Robert Wycoff, president emeritus of ARCO, who spent just one day shadowing the
principal of a high school. He stunned reporters by praising the principal, teachers, and kids. "If I were principal of Manual Arts, I could not do as good a job as I saw today. This is not as simple
as going to the moon."
It's not. The laws of physics are known. The laws of kids are not. You can aim a rocket with precision because all variables can be either controlled or factored in. Kids bring active minds and wills
to the scene and will thwart, surprise, and delight you with their unpredictability. Ohanian urges us to celebrate that variability and spontaneity.
This is a lively, funny, and angry book full of stilettos and wry barbs. It is a perfect antidote for the reactionary educational times we live in, a time when a USA TODAY cartoon depicts a mother
reading to her child in bed and saying, "The little pig with higher verbal and math lived happily ever. The other two pigs were swallowed by the wolf."
Early on, Ohanian invokes Network and suggests teachers should rise up yelling, "I'm mad as hell," and refuse to teach the standards or give the tests. That day might come. Rebellions are brewing all
over the country, not just among teachers but among parents and students as well. Perhaps we can eventually return to a time when, in the words of Philip Lopate, "the art of teaching is knowing when
a student is best left alone and when he is ripe to receive your help." It's not the same time for every child. ¤ | {"url":"https://prospect.org/features/teachers-know/","timestamp":"2024-11-10T13:07:37Z","content_type":"application/xhtml+xml","content_length":"56893","record_id":"<urn:uuid:9cb7af43-9611-4c27-ab5d-1c8ac77f7850>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00505.warc.gz"} |
If f(x)=(x+1)cotx be continuous at =0, then f(0) is equal to... | Filo
If be continuous at , then is equal to
Not the question you're searching for?
+ Ask your question
Since is continuous at
Was this solution helpful?
Found 7 tutors discussing this question
Discuss this question LIVE for FREE
15 mins ago
One destination to cover all your homework and assignment needs
Learn Practice Revision Succeed
Instant 1:1 help, 24x7
60, 000+ Expert tutors
Textbook solutions
Big idea maths, McGraw-Hill Education etc
Essay review
Get expert feedback on your essay
Schedule classes
High dosage tutoring from Dedicated 3 experts
Practice more questions from Continuity and Differentiability
View more
Practice questions on similar concepts asked by Filo students
View more
Stuck on the question or explanation?
Connect with our Mathematics tutors online and get step by step solution of this question.
231 students are taking LIVE classes
Question Text If be continuous at , then is equal to
Updated On Oct 25, 2023
Topic Continuity and Differentiability
Subject Mathematics
Class Class 12
Answer Type Text solution:1 Video solution: 2
Upvotes 272
Avg. Video Duration 6 min | {"url":"https://askfilo.com/math-question-answers/if-fxx1cot-x-be-continuous-at-0-then-f0-is-equal-to","timestamp":"2024-11-08T18:35:02Z","content_type":"text/html","content_length":"445300","record_id":"<urn:uuid:33c3681c-8e90-42fe-88b4-87d50291dd50>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00861.warc.gz"} |
Internal Structure of Crystals XIII
The Internal Structure of Crystals (Part Thirteen)
back to homepage
The External Shapes of periodic 2-D arrays of repeated two-dimensional motifs
Square Net.
We continue our investigation concerning the faces, Forms and combinations of Forms in two-dimensional crystals.
In the previous Part we concluded our investigation concerning the 2-D Rectangular Crystal System. We will now continue our search into 2-D periodic patterns with a discussion of the
2-D Tetragonal Crystal System. This system is based on a square building block and consequently on a square 2-D lattice (net).
The symmetry of its typical square building block (considered as empty) determines the
2-D Tetragonal Crystal System. It has two Classes, 4 and 4mm. From these the Class 4mm can refer to several different Plane Groups, which means that when two 2-D crystals, belonging to the 2-D
Tetragonal Crystal System, have the same point symmetry 4mm they can nonetheless differ with respect to their total symmetry content, i.e. they can differ when we also consider their translational
symmetries. The reason for this possible difference is a difference in the 'chemical' contents (in our drawings represented by motifs and motif units) of the respective lattices.
A square net (2-D square lattice) is always primitive, which means that there are lattice nodes (nodes are equivalent points) only at the corners of the meshes. When we center such a net then nothing
new will be generated. A primitive lattice is denoted by the letter P.
In Figure 1 a possible square net is drawn, and the axial directions are indicated.
Figure 1. A possible square net ( 2-D lattice). This net can accommodate several types of motifs, such that they are periodically repeated across the 2-D plane, according to the geometry of such a
net. The axial directions (i.e. crystallographic axes) are indicated. The two axes are perpendicular to each other, and are equivalent. They are chosen to coincide with the relevant edges of the
building block (unit mesh).
In constructing 2-D crystal faces we know that only those faces are possible that can be constructed by the periodic stacking of the building blocks (the meshes of the net).
We're now going to consider possible crystal faces, Forms and combinations of Forms (from the same Crystal Class), in order to determine the intrinsic shapes and symmetries of 2-D crystals belonging
to the 2-D Tetragonal Crystal System.
Let us start with the Point Group (= Crystal Class) 4 of this System. The only Plane Group belonging to this Class is the Plane Group P4.
When we introduce a face parallel to one of the crystallographic axes (and consequently perpendicular to the other axis, three new faces are generated in virtue of the 4-fold rotation axis (as the
only symmetry element of the Class 4). The result will be a Form consisting of four faces together making up a square, i.e. a closed Form with point symmetry 4. Of course this square does suggest the
presence of mirror lines, but the internal structure (i.e. when we furnish the empty building blocks with certain motifs) will forbid this. See Figure 2.
Figure 2. Within the context of Class 4 an introduced initial face implies three more faces, resulting in a Form that has the shape of a square, and a point symmetry 4. The 4-fold rotation axis,
perpendicular to the plane of the drawing, is indicated by a small solid blue square. The axial system is indicated (red). The resulting Form is closed, implying that it can as such represent a 2-D
Crystal of this Class. See The next Figure.
Figure 3. A 2-D crystal of the Class 4 consisting of four faces, together making up one Form. Because we have drawn the crystal such that it consists of empty building blocks, we cannot yet actually
see the absence of mirror lines. The finally resulting crystal is depicted in the next Figure.
Figure 3a. A 2-D crystal consisting of the faces defined in the previous two Figures, drawn without axial system and without indication of symmetry elements.
Figure 4. An introduced face making angles of 45^0 with the crystallographic axes implies three more faces in virtue of the action of the 4-fold rotation axis. The resulting face configuration is
again a closed Form that can, on its own behalf, represent a 2-D crystal of the present Class. See the next two Figures.
Figure 4a. A two-dimensional crystal, representing the Form of Figure 4. For the final result see the next Figure.
Figure 4b. Final result of the above construction of a 2-D crystal of the Class 4.
Introducing a face with a more general orientation with respect to the system of crystallographic axes gives rise to three more faces in order to make the face configuration obedient to the symmetry
content of the present Class. This content is : one 4-fold rotation axis. Again a closed Form is generated, consisting of four faces, and having the shape of a square.
See Figures 5, 5a and 5b.
Figure 5. In the context of the symmetry elements of the present Class, namely the presence of a 4-fold rotation axis (indicated by a small solid blue square), an initial face implies three more
faces, resulting in a closed Form having the shape of a square. See the next Figure.
Figure 5a. The Form, generated from the above mentioned initial face, is closed and can therefore represent a possible crystal belonging to the present Class (4). See the next Figure.
Figure 5b. A two-dimensional crystal belonging to the Class 4 of the 2-D Tetragonal Crystal System.
The three Forms derived so far all have the same shape, namely that of a square, but their orientations with respect to the axial system differ.
Still more Forms are possible, but all will be squares with slightly different orientations.
These Forms can enter in combinations with each other, for instance a Form (equivalent to that) of Figure 3a and that of 5b. See Figure 6, 6a, 6b and 6c.
Figure 6. Construction of a combination of two Forms (blue and green) of the Class 4.
Figure 6a. Continuation of the construction.
Figure 6b. Removal of indented angles (indented angles will not occur in single crystals). The result is two squares chopping off each other's corners.
Figure 6c. Final result of the construction, representing a 2-D crystal (of the Class 4) consisting of a combination of two Forms.
A combination as the one above can be such that the two Forms balance each other, or that one or the other dominates.
The 'crystals' so far considered are (considered to be) built up by empty building blocks. To conceptually generate genuine (2-D)crystals, those building blocks must be furnished with content, i.e.
with motifs (representing chemical units). And indeed, if a certain lattice is given and we insert motifs into it, (motifs) compatible with that lattice (this lattice then describing the specific
periodic repeat of those motifs), then the (total) symmetry of the resulting motif pattern is representing a certain Plane Group.
Only one Plane Group belongs to the Class 4 of the 2-D Tetragonal Crystal System. It is the Plane Group P4. Let us display a pattern (given earlier) representing this Plane Group :
Figure 7. A pattern representing Plane Group P4.
We can now consider the different atomic aspects presented to the environment by the possible faces and Forms of crystals of the Class 4 and representing Plane Group P4 (we will use a square net with
smaller meshes than we saw in Figure 7).
Figure 8. Atomic aspect of the faces -- as derived in the Figures 2, 3 and 3a -- of a 2-D crystal of the Class 4 of the 2-D Tetragonal Crystal System. The incompleteness of motifs at the faces
symbolizes unsaturated or distorted chemical valences.
In Figure 8 one can clearly see that the opposite faces of this crystal are not symmetric (with respect to one or another mirror line). The true symmetry of the crystal, namely 4 (and not say 4mm),
is revealed by its internal composition, and this composition will reflect itself in certain physical or chemical differences, by means of which one can actually determine that true symmetry
(imagining 2-D crystals to exist in reality).
Figure 9. Atomic aspect of one of the faces in the Figures 4, 4a and 4b, here indicated by a .
Figure 10. Atomic aspect of one of the faces in the Figures 5, 5a and 5b, here indicated by c .
This concludes our discussion of the Class 4 and its only Plane Group P4.
The second (and last) Crystal Class of the 2-D Tetragonal System is the Class 4mm. It admits of two types of motifs resulting in two different types of periodic pattern according to two different
Plane Groups : P4mm and P4gm.
We will derive possible faces, Forms and combinations of Forms within this Class (4mm) :
Figure 11. Introducing a face parallel to one of the crystallographic axes (and consequently perpendicular to the other axis) implies three more faces, resulting in a closed Form having the shape of
a square. The four mirror lines do not add more faces beyond these four faces. The symmetry of the resulted face configuration now must be obedient to the symmetry of the Class, 4mm, which means that
opposite faces are symmetrical with respect to a horizontal or vertical mirror line, while each two consecutive faces are symmetrical with respect to one or another diagonal mirror line. The 4-fold
axis is indicated by a small solid blue square. The crystallographic axes coincide with the non-diagonal mirror lines.
The resulting Form is closed and can as such be a 2-D crystal of our Class :
Figure 11a. This 2-D crystal (Form) of the Class 4mm has the same shape as the corresponding crystal (Form) of the Class 4 (Figure 3a). Their different symmetries cannot be deduced from their
external shape. This shape is caused by the internal organization of the crystals. But that same internal organization will also have other effects that reveal the difference in symmetry. Further
below, this is illustrated by furnishing the empty building blocks with motifs.
Figure 12. An introduced face, making an angle of 45^0 with the crystallographic axes, is multiplied by the action of the 4-fold rotation axis, resulting in a closed Form consisting of four faces
making up a square. The mirror lines do not add any further faces.
The face configuration so obtained complies with the symmetry of the present Class, 4mm .
Figure 12a. The above Form can represent a 2-D crystal of the Class 4mm . Its external shape is the same as that of the corresponding Form of Class 4 (Figure 4b).
An initial face having a more general orientation with respect to the system of crystallographic axes yields, when subjected to the symmetry elements of the present Class, a Form, so far not yet
encountered : a ditetragon. It is a closed Form consisting of eight sides of equal length, but connected to each other by alternating angles, meaning that the ditetragon involves two different
angles, four equal angles and four slightly smaller angles, also equal among themselves. See Figures 13, 13a, 13b, 13c and 13d.
Figure 13. A face (blue) with a general orientation with respect to the crystallographic axes is introduced. The resulting face configuration should be such that its symmetry is that of the present
Class. This is the same as having that face subjected to the symmetry elements of the present Class. Those symmetry elements are : A horizontal mirror line, a vertical mirror line, two diagonal
mirror lines and a 4-fold rotation axis. The next Figures show the action of those symmetry elements upon the introduced face.
Figure 13a. A second face (red) is generated by the action of one of the diagonal mirror lines.
Figure 13b. The face pair of the previous Figure is multiplied four times by the action of the 4-fold rotation axis, resulting in eight faces making up a ditetragon, i.e. an octagon consisting of
eight equal faces connected by alternating angles.
Figure 13c. The ditetragon is a closed Form and can represent a crystal of the present Class. See Figure 13d.
Figure 13d. A 2-D crystal representing the above derived Form. It belongs to the Class 4mm of the 2-D Tetragonal Crystal System.
These Forms can enter in combinations with each other.
As an example we give a possible combination of the Form of Figure 11 with the Form of Figure 12 (Recall that the size of a Form is immaterial crystallographically, only the orientations of the faces
count). See Figures 14 and 14a.
Figure 14. A combination of two Forms (specified above) of the 2-D Crystal Class 4mm. The relative growth rates of the (faces of the) involved Forms determine the prominence of either the one Form or
the other. A combination like this one can represent a 2-D crystal (See Figures 14a and 14b).
Figure 14a. Outlining the 2-D crystal (light blue) consisting of two Forms as described above (See also next Figure).
Figure 14b. A two-dimensional crystal of the Class 4mm consisting of two Forms of that Class, as specified above.
As has been said, this Class is compatible with two Plane Groups, namely P4mm and P4gm. We're going to investigate the possible different atomic aspects presented by the above derived faces and Forms
of our Class 4mm. These aspects can be considered when the empty building blocks are furnished with motifs, leading to the two possible Plane Groups.
Let's start with the Plane Group P4mm.
A pattern (given ealier) representing this Plane Group is depicted in the next Figure.
Figure 15. A periodic pattern of motifs representing Plane Group P4mm.
We can now consider the different atomic aspects presented to the environment by the possible faces and Forms of crystals of the Class 4mm and representing Plane Group P4mm (we will use a square net
with smaller meshes than we saw in Figure 15).
Figure 16. Atomic aspects presented to the environment of the faces (as derived in Figure 11, and 11a) of a crystal belonging to the Class 4mm and to the Plane Group P4mm. The incompleteness of
motifs at the faces express the presence of unsatisfied or distorted chemical valences.
Figure 16a. Indication of the point symmetry of the crystal of Figure 16.
Red and blue lines : The two types of mirror lines. The red lines are at the same time the two crystallographic axes. The small solid green square indicates the 4-fold rotation axis.
Figure 17. Atomic aspect of one of the faces of the Figures 12, and 12a, here indicated by a .
Figure 17a. Atomic aspects of two consecutive faces of the Figures 12, and 12a, here indicated by a and b. The mirror symmetry of those two faces with respect to the indicated (red) mirror line (m)
-- a symmetry demanded by the point symmetry of the crystal to which this fragment belongs -- is clearly evident from the pattern.
Figure 18. Atomic aspect of the face of Figure 13, here indicated by c.
The second (and last) Plane Group of the Class 4mm is the Plane Group P4gm. We will discuss it in the next Part.
To continue, click HERE for Part Fourteen.
back to homepage | {"url":"http://metafysica.nl/d2_lattice_13.html","timestamp":"2024-11-14T08:10:45Z","content_type":"text/html","content_length":"21619","record_id":"<urn:uuid:8caadaae-260e-4de3-821a-146ed30d999e>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00685.warc.gz"} |
With rights/species and search set -POJ 1182 food chain - Moment For Technology
The article directories
• Take weights and look up sets
• Type and look up set
• sample
• summary
Take weights and look up sets
A weighted union lookup set is a node with weight information. The weight makes the relationship quantifiable, that is to say, the weight represents a certain relationship between the current node
and the parent node, through which the relationship between the two nodes under the same tree can also be expressed. And the general parallel lookup set can only be judged to belong to a certain set.
It is possible to judge whether a relationship is a relationship or not, for example, a friend’s friend is a friend. But for example, if the enemy’s enemy is a friend and there are more than two
kinds of relationships, you can use category union lookup, and you can use multiple sets of union lookup to simulate categories.
Portal: POJ-1401
There are three groups of animals in the animal kingdom, A,B, and C, and they form interesting loops in the food chain. A eats B, B eats C, C eats A. There are N animals, numbered 1-n. Every
animal is one of A,B, or C, but we don’t know which one it is. There are two ways to describe the relationship between these N animals in the food chain. The first way is “1 X Y”, which means X
and Y are of the same kind. The second way to say it is “2 X Y”, which means X eats Y. He said K statements, one after the other, some true and some false, to N animals, using the two statements.
When a statement satisfies one of the following three, it is a lie, otherwise it is a truth. 1) The current statement conflicts with some of the previous true statements. 2) If X or Y is greater
than N, it is false. 3) The current statement means X eats X, which is a lie. Your task is to output the total number of lies given N (1 <= N <= 50,000) and K sentences (0 <= K <= 100,000).
The first line is two integers N and K separated by a space. The following K lines are three positive integers D, X, and Y, separated by a space, where D indicates the type of statement. If D=1,
then X and Y are homogeneous. If D=2, X eats Y.
There is only one integer to indicate the number of lies.
Sample Input:
Copy the code
Sample Output:
Copy the code
1. The weighted and look-up set method defines the weight array rela[] to describe the relationship with the root node, where 0 means like, 1 means the current point can eat others, and 2 means the
current point is eaten by others. The relationship between root[] and rela[] is maintained through vector thought to judge whether it is contradictory to the previous relationship. Ps: Detailed
vector diagram
2. Open an array of size 3n, using x, x+n, x+2n to represent three different types. For example, x+n is the natural enemy of X, x+2n is the natural enemy of x+n, x is the natural enemy of x+2n and
then you can simulate and maintain the array and decide. (Of course, it is also possible to represent prey, the natural enemy of source code)
1. Weighted and search set solution
using namespace std;
const int maxn = 50004;
int root[maxn], rela[maxn];
int r, x, y, n, k;
int Find(int x) {
if (x == root[x])return x;
int tmp = root[x];
root[x] = Find(root[x]);// Compression path
// Relationship between x and root = relationship between x and parent + relationship between parent (TMP) and root
rela[x] = (rela[x]+rela[tmp]+ 3) % 3;
return root[x];
bool check(int r, int x, int y) {
if (x > n || y > n || (r == 1 && x == y))
return false;
if (Find(x) == Find(y))
// (r = x to the root + y to the root) (~rela[y])
return r == (rela[x] - rela[y] + 3) % 3;
else return true;
void Union(int r, int x, int y) {
int fx = Find(x), fy = Find(y);
if(fx ! = fy) { root[fx] = fy;// the x root merges into the y root set
// Update x root to new root
// The relationship between x and the new root = the relationship between the root and x (~relx[x])+ the relationship between x and y (r)+ the relationship between y and the root
rela[fx] = (-rela[x] + r + rela[y] + 3) % 3; }}int main(a) {
scanf("%d%d", &n, &k);
for (int i = 0; i <= n; i++) {/ / initialization
root[i] = i;
rela[i] = 0;
int ans = 0;
while (k--) {
scanf("%d%d%d", &r, &x, &y);
if (check(r, x, y))
Union(r, x, y);
else ans++;
printf("%d\n", ans);
return 0;
Copy the code
2. Class and search set solution
using namespace std;
const int maxn = 50004;
int root[maxn * 3];
int r, x, y, n, k;
int Find(int x) {
return x == root[x] ? x : root[x] = Find(root[x]);
bool check(int r, int x, int y) {
if (x > n || y > n || (r == 1 && x == y))
return false;
int fx = Find(x), fy = Find(y);
int fyn = Find(y + n);
int fynn = Find(y + n + n);
if (r == 0) {/ / the same kind
if (fx == fyn || fx == fynn)
return false;
else {/ / x y
if (fx == fy || fx == fynn)
return false;
return true;
void Union(int r, int x, int y) {
int fx = Find(x), fy = Find(y);
int fxn = Find(x + n), fyn = Find(y + n);
int fxnn = Find(x + n + n), fynn = Find(y + n + n);
if(fx ! = fy) {if (r == 0) {//x is the same as y
root[fx] = fy; // The class of x and the class of y are the same
root[fxn] = fyn; // The natural enemies of X and Y are of the same kind
root[fxnn] = fynn;// X's prey is of the same species as Y's prey
else {/ / x y
root[fx] = fyn; // X's kindred and Y's natural enemy are kindred
root[fxn] = fynn;// X's natural enemies are of the same species as Y's prey
root[fxnn] = fy; // X's prey is of the same species as Y's}}}int main(a) {
scanf("%d%d", &n, &k);
for (int i = 0; i <= 3 * n; i++)root[i] = i;/ / initialization
int ans = 0;
while (k--) {
scanf("%d%d%d", &r, &x, &y);
if (check(r, x, y))
Union(r, x, y);
else ans++;
printf("%d\n", ans);
return 0;
Copy the code
1. If there are too many relationships (categories), then category and set lookup is cumbersome to write, and weighted and set lookup is more dominant.
2. Category and search set also need enough space, the space limit is small problem is also hard.
3. However, category and set searching is easier to understand and simulate, and weighted and set searching requires a full understanding of its vector thinking.
Original is not easy, please do not reprint (this is not rich visits add insult to injury) blogger home page: blog.csdn.net/qq_45034708 If the article is helpful to you, remember to focus on the
likes collection ❤ | {"url":"https://www.mo4tech.com/with-rights-species-and-search-set-poj-1182-food-chain.html","timestamp":"2024-11-10T18:29:48Z","content_type":"text/html","content_length":"79099","record_id":"<urn:uuid:ddc1f1c7-b6ce-4d19-a79c-e80ceb8b79ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00673.warc.gz"} |