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Useful statistics of a data set in machine learning
When building a machine learning model, knowing the ranges of all the variables is imperative; data statistics provide precious information. Indeed, they put the data set in context. The most
important statistical parameters are the minimum, the maximum, the mean, and the standard deviation. We must always perform a simple statistical analysis to check data consistency.This way, we need
to calculate statistics for each data set variable.Recall that the data matrix comprises the variables
\begin{eqnarray}v_{j} := column_{j}(d), \quad j=1,\ldots,q\quad\end{eqnarray}
and the samples
\begin{eqnarray}u_{i} := column_{i}(d), \quad i=1,\ldots,p\quad\end{eqnarray}
(columns and rows of the data set).
The values that a variable takes for each sample in the data set are
\begin{eqnarray}\quad v_j=(d_{1j}, \ldots, d_{pj}), \quad j=1,\ldots,q.\end{eqnarray}
2. Minimum and maximum
The minimum of a variable is the smallest value of that variable in the data set.
The minimum of the variable is denoted by $v_{jmin}$, and it is defined as follows,
v_{jmin} = \min_{i = 1, \ldots, p} d_{ij}.
A variable’s maximum is the variable’s biggest value in the data set.
Similarly, the maximum is denoted by $v_{jmax}$, and it is defined as follows,
v_{jmax} = \max_{i=1, \ldots, p} d_{ij}.
3. Mean and standard deviation
The mean of a variable is the average value of that variable in the data set. The mean is denoted by $v_{jmean}$ and is defined as,
\boxed{v_{jmean} = \frac{1}{p}\sum_{i=1}^{p} d_{ij}.}
where $d_{ij}$ is the data matrix element and $p$ is the number of samples.
The standard deviation measures how dispersed the data is about the mean. The standard deviation of a variable is denoted by $v_{jstd}$ and is defined as,
v_{jstd} = \sqrt{\frac{1}{p}\sum_{i=1}^{p}\left(d_{ij}-v_{jmean}\right)^2}.
where $v_{jmean}$ is the mean of the variable $v_{j}$.
The graphical representation is the standard distribution curve, called the Gaussian bell.
A low standard deviation means that all the values are close to the mean.
Conversely, a high standard deviation means the values are spread out around the mean and from each other.
Example: Predict the noise generated by airfoil blades
NASA conducts a study of the noise generated by an aircraft in order to make a model to reduce it.
The file airfoil_self_noise.csv contains the data for this example.
Here the number of variables (columns) is 6, and the number of instances (rows) is 1503.
We can calculate the basic statistics of each variable using the formulas described above.
The following table displays the minimum, maximum, mean, and standard deviation for every input variable in the data set.
Name Minimum Maximum Mean Deviation
frequency 200 20000 2890 3150
angle_of_attack 0 0.22 6.78 5.92
chord_length 0.0254 0.305 0.137 0.0935
free_stream_velocity 31.7 71.3 50.9 15.6
suction_side_displacement_thickness 0.000401 0.0584 0.0111 0.0132
scaled_sound_pressure_level 103 141 125 6.9
By performing this simple statistical analysis, we can check the consistency of the data.
4. Conclusions
Statistics put the data set in context.
It is essential to perform a simple statistical analysis to check the consistency of the data before building the model.
This is done by calculating each variable’s most important statistical parameters, such as the minimum and maximum values, mean, and standard deviation.
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Wave-function collapse versus Euler’s formula*
Wave-function parametrization
The wave-function is a parametrization of any probability measure
Measure spaces (in particular, algebraic measure theory as defined below) have many applications in classical mechanics and dynamical systems [1], when non-linear equations and/or complex phase
spaces become an obstacle.
Any measure space can be defined as a probability space: a probability space is a measure space where a probability density was chosen, which is any measurable function normalized such that its
measure is 1. In the historical (that is, Kolmogorov’s) probability theory, a probability space has three parts: a phase space (which is the set of possible states of a system); a complete Boolean
algebra of events (where each event is a subset of the set of possible states); and a probability measure which assigns a probability to each event. The probability is a map from complex random
events to abstract random events, shifting all ambiguity with the notion of randomness to the abstract random events as described in the following: that the probability of an event is $0.32$ means
that our event has the same likelihood of finding a treasure that we know it was hidden in the sand of a 1 km wide beach, if we only search for it with a metal detector in a 320m interval. While this
treasure hunt is ambiguous (are there any clues for the location of the treasure?, etc.) the map from our complex events to this treasure hunt is unambiguous.
On the other hand, a standard measure space is isomorphic up to sets with null measure to the real Lebesgue measure in the unit interval or to a discrete (finite or countable) measure space or to a
mixture of the two. Thus, topological notions such as dimension do not apply to standard measure spaces. Most probability spaces with real-world applications are standard measure spaces.
Equivalently, a standard measure space can be defined such that the following correspondence holds: every commutative von Neumann algebra on a separable real Hilbert space is isomorphic to $L^\infty
(X,\mu)$ for some standard measure space $(X,\mu)$ and conversely, for every standard measure space $(X,\mu)$ the corresponding Lebesgue space $L^\infty(X,\mu)$ is a von Neumann algebra. As expected,
the representation of an algebra of events in a real Hilbert space uses projection-valued measures [2][3][4][5]. A projection-valued measure assigns a self-adjoint projection operator of a real
Hilbert space to each event, in such a way that the boolean algebra of events is represented by the commutative von Neumann algebra. Thus, intersection/union of events is represented by products/sums
of projections, respectively. The state of the ensemble is a linear functional which assigns a probability to each projection. Thus, there is an algebraic measure theory[1][6][7][8] based on
commutative von Neumann algebras on a separable Hilbert space which is essentially equivalent to measure theory (for standard measure spaces).
To be sure, the algebraic measure theory is based on commutative algebras, thus it is not a non-commutative generalization of probability or information theory. That is, there is no need for a
conceptual or foundational revolution such as qubits replacing bits when switching from the historical to the algebraic probability theory [6]. Moreover, this is a common procedure in mathematics, as
illustrated in the following quote [9] (note that a probability measure is related to integration):
The fundamental notions of calculus, namely differentiation and integration, are often viewed as being the quintessential concepts in mathematical analysis, as their standard definitions involve
the concept of a limit. However, it is possible to capture most of the essence of these notions by purely algebraic means (almost completely avoiding the use of limits, Riemann sums, and similar
devices), which turns out to be useful when trying to generalize these concepts[...]
T. Tao (2013)[9]
The relation of the algebraic measure theory with probability theory is the following: let $p(x|y)$ be a conditional probability density between two standard measure spaces $X, Y$ (possibly with
continuous and discrete parts), then $p(x|y)p(y)=p(y)T^2(x,y)=p(y|x)p(x)$ for any probability density $p(y)$ and some bounded operator $T$ on the separable Hilbert space. From the condition $\int dx\
p(y|x)p(x)=p(y)$, we conclude that $T$ is an isometry. But since the Hilbert space is separable, there is an orthonormal discrete basis and we can build a unitary operator $U$ through the
Gram-Schmidt process such that $TT^\dagger U=T$. We can then enlarge the discrete part of the phase-space $Y$ to include the indices corresponding to the elements of the basis that were missing,
setting $p(y)=0$ for the new indices. Thus $p(x|y)p(y)=p(y)U^2(x,y)=p(y|x)p(x)$ and so any conditioned probability density can be represented by a unitary operator on the separable Hilbert space.
Conversely, any operator on the separable Hilbert space also corresponds to a conditioned probability density. In the particular case where the standard measure space $Y$ has just one element, we get
the result that the wave-function is a parametrization of any probability measure.
The linearity of the commutative algebra; avoiding a fixed phase space a priori; and the fact that we can map complex random phenomena to an abstract random process unambiguously, are obvious
advantages for algebraic measure theory when we want to compare probability theory with Quantum Mechanics, where the linearity of the canonical transformations is guaranteed by Wigner’s theorem (it
is essentially a consequence of the Born’s rule applied to a non-commutative algebra of operators[10][11][12]); the Hilbert space of wave-functions replaces the phase space; and the canonical
transformations are non-deterministic.
The algebraic measure theory is also different from defining the phase space as a reproducing kernel Hilbert space [13][14], since no phase space (whether it is a Hilbert space or not) is defined a
priori. Note that defining the phase space as a Sobolev Hilbert space is common in classical field theory[15], but defining a general probability measure in such space is still an open problem.
Quantum Mechanics versus a non-commutative generalization of probability theory
Quantum Mechanics leaves room for a non-commutative generalization of probability theory, since the wave-function could also assign a probability to non-diagonal projections, these non-diagonal
projections would generate a non-commutative algebra [16].
Consider for instance the projection $P_X$ to a region of space $X$ and a projection $U P_p U^\dagger$ to a region of momentum $p$, where $P_X$ and $P_p$ are diagonal in the same basis. The
projections $P_X$ and $U P_p U^\dagger$ are related by a Fourier transform $U$ and thus are diagonal in different basis and do not commute (they are complementary observables). Since we can choose to
measure position or momentum, it seems that Quantum Mechanics is a non-commutative generalization of probability theory [16].
But due to the wave-function collapse, Quantum Mechanics is not a non-commutative generalization of probability theory despite the appearances: the measurement of the momentum is only possible if a
physical transformation of the statistical ensemble also occurs, as we show in the following.
Suppose that $E(P_X)$ is the probability that the system is in the region of space $X$, for the state of the ensemble $E$ diagonal (i.e. verifying $E(O)=0$ for operators $O$ with null diagonal).
Using the notation of [17], we have $E(A)=\mathrm{tr}(\rho A)$ where $A$ is any operator and $\rho$ is a self-adjoint operator with $\mathrm{tr}(\rho)=1$ and $\rho=\sum_{X} P_X \rho P_X$, because $\
rho$ is diagonal in the same basis where the projection operators $P_X$ are diagonal. If an operator $O$ has null diagonal in the same basis where $P_X$ is diagonal, then $P_X O P_X=0$ for any $X$
and then $E(O)=\sum_{X} \mathrm{tr}(P_X \rho P_X O)=0$.
If we consider a unitary transformation $U$ on the ensemble, then after the wave-function collapse we have a new ensemble with state $E_U$ given by:
\begin{aligned} E_U(A)=&tr(\rho_U U A U^\dagger)\\ \rho_U=&\sum_{p} U P_p U^\dagger \rho U P_p U^\dagger\end{aligned}(1)
If an operator $O$ has null diagonal in the same basis where $P_p$ is diagonal, then $P_p O P_p=0$ for any $p$ and then:
\begin{aligned} E_U(O)=&\sum_{p} \mathrm{tr}(P_p U^\dagger \rho U P_p O)=0\end{aligned}(2)
If an operator $D$ is diagonal in the same basis where $P_p$ is diagonal, then $D=\sum_{p} P_p D P_p$ and then:
\begin{aligned} E_U(D)&=\sum_{p} \mathrm{tr}(P_p U^\dagger \rho U P_p D)=\\ &=\mathrm{tr}(\rho U D U^\dagger)=E(U D U^\dagger)\end{aligned}(3)
Thus, we define:
\begin{aligned} E_U(D)&=E(U D U^\dagger)\\ E_U(O)&=0\end{aligned}(4)
Where $D$ is a diagonal operator and $O$ is an operator with null diagonal. The equation ([eq:collapse]) is due to the wave-function collapse. Thus $E_U(P_p)=E(U P_p U^\dagger)$ is the probability
that the system is in the region of momentum $p$, for the state of the ensemble $E_U$. But the ensembles $E$ and $E_U$ are different, there is a physical transformation relating them.
Without collapse, we would have $E_U(O)=E(U O U^\dagger)eq 0$ for operators $O$ with null-diagonal and we could talk about a common state of the ensemble $E$ assigning probabilities to a
non-commutative algebra. But the collapse keeps Quantum Mechanics as a standard probability theory, even when complementary observables are considered. We could argue that the collapse plays a key
role in the consistency of the theory, as we will see below.
At first sight, our result that the wave-function is merely a parametrization of any probability measure, resembles Gleason’s theorem [18][19]. However, there is a key difference: we are dealing with
commuting projections and consequently with the wave-function, while Gleason’s theorem says that any probability measure for all non-commuting projections defined in a Hilbert space (with dimension $
\geq 3$) can be parametrized by a density matrix. Note that a density matrix includes mixed states, and thus it is more general than a pure state which is represented by a wave-function.
We can check the difference in the 2-dimensional real case. Our result is that there is always a wave-function $\Psi$ such that $\Psi^2(1)=\cos^2(\theta)$ and $\Psi^2(2)=\sin^2(\theta)$ for any $\
However, if we consider non-commuting projections and a diagonal constant density matrix $\rho=\frac{1}{2}$, then we have:
\begin{aligned} \begin{cases} \mathrm{tr}(\rho \left[\begin{smallmatrix} 1 & 0\\ 0 & 0 \end{smallmatrix}\right])=\frac{1}{2}\\ \mathrm{tr}(\rho \frac{1}{2}\left[\begin{smallmatrix} 1 & 1\\ 1 & 1 \end
{smallmatrix}\right])=\frac{1}{2} \end{cases}\end{aligned}(5)
Our result implies that there is a pure state, such that:
\begin{aligned} \mathrm{tr}(\rho \left[\begin{smallmatrix} 1 & 0\\ 0 & 0 \end{smallmatrix}\right])=\frac{1}{2}\end{aligned}(6)
(e.g. $\rho=\frac{1}{2}\left[\begin{smallmatrix} 1 & 1\\ 1 & 1 \end{smallmatrix}\right]$)
And there is another possibly different pure state, such that:
\begin{aligned} \mathrm{tr}(\rho \frac{1}{2}\left[\begin{smallmatrix} 1 & 1\\ 1 & 1 \end{smallmatrix}\right])=\frac{1}{2}\end{aligned}(7)
(e.g. $\rho=\left[\begin{smallmatrix} 1 & 0\\ 0 & 0 \end{smallmatrix}\right]$)
But there is no $\rho$ which is a pure state, such that:
\begin{aligned} \begin{cases} \mathrm{tr}(\rho \left[\begin{smallmatrix} 1 & 0\\ 0 & 0 \end{smallmatrix}\right])=\frac{1}{2}\\ \mathrm{tr}(\rho \frac{1}{2}\left[\begin{smallmatrix} 1 & 1\\ 1 & 1 \end
{smallmatrix}\right])=\frac{1}{2} \end{cases}\end{aligned}(8)
On the other hand, Gleason’s theorem implies that there is a $\rho$ which is a mixed state, such that :
\begin{aligned} \begin{cases} \mathrm{tr}(\rho \left[\begin{smallmatrix} 1 & 0\\ 0 & 0 \end{smallmatrix}\right])=\frac{1}{2}\\ \mathrm{tr}(\rho \frac{1}{2}\left[\begin{smallmatrix} 1 & 1\\ 1 & 1 \end
{smallmatrix}\right])=\frac{1}{2} \end{cases}\end{aligned}(9)
Gleason’s theorem is relevant if we neglect the wave-function collapse, since it attaches a unique density matrix to non-commuting operators. However, the wave-function collapse affects differently
the density matrix when different non-commuting operators are considered, so that after measurement the density matrix is no longer unique. In contrast, without the wave-function collapse, the
wave-function parametrization of a probability measure would not be possible.
Another difference is that our result applies to standard probability theory, while Gleason’s theorem applies to a non-commutative generalization of probability theory.
Symmetries and unitary representations
A dynamical system can be classified by all possible transformations that can occur in the phase space at each evolution step. When these transformations are a function of a group, then the
classification of the dynamical system becomes independent from the evolution stage (usually called the time). The group/transformations are then called the symmetry group/transformations (also
called canonical transformations). Note that the function relating the group to the phase space does not need to conserve the group action because in a dynamical system all possible transformations
that can occur at each evolution step do not need to form a group at all.
In the case of Quantum Mechanics, all possible transformations that can occur in the phase space at each evolution step are given by conditional probability densities between two standard measure
spaces. There is a surjective function from the group of linear and unitary operators on a separable Hilbert space to all such conditional probability densities. Thus there is not necessarily a group
action of a symmetry group on the probability density (for the state of a system) itself. We address in Section 12 when such action on the probability distribution exists and when it does not exist.
Deterministic transformations
Crucially, the symmetry transformations include all the deterministic transformations, which will be defined in the following. Thus the symmetry transformations are a generalization of the
deterministic transformations.
A deterministic transformation acts as $E(P_A)\to E(P_B)$ where $A,B$ are events and $P_A$ is a projection operator, for any expectation functional $E$ and event $A$. When the probability is
concentrated in the neighborhood of a single outcome (say $A$), we have effectively a deterministic case and this transformation ($A\to B$) conserves the determinism, thus it is a deterministic
Note that above, $P_A$ and $P_B$ necessarily commute. On the other hand, if the transformation is such that $E(P_A)\to E(U P_A U^\dagger)$ where $U$ is a unitary operator and $P_A$ and $U P_A U^\
dagger$ do not commute, then the transformation cannot be deterministic. Consider the discrete case with $E(P_n)$ given by $\mathrm{tr}(P_m P_n)=\delta_{mn}$ up to a normalization factor, for
instance. Then $\mathrm{tr}(P_mP_n)\to \mathrm{tr}(P_mU P_n U^\dagger)=U^2_{nm}$. If the transformation would be deterministic, then necessarily $U^2_{nm}=\delta_{kn}$ for some $k=f(n)$ dependent on
$n$, and so $U P_n U^\dagger=P_{l}$ with $l=f^{-1}(n)$ would commute with $P_n$.
We conclude that a transformation $U$ is deterministic if and only if $P_A$ and $U P_A U^\dagger$ commute for all events $A$. Thus, the complementarity of two observables (e.g. position and momentum)
is due to the random nature of the symmetry transformation relating the two observables. This clarifies that probability theory has no trouble in dealing with non-commuting observables, as long as
the collapse of the wave-function occurs. Note that Quantum Mechanics is not a generalization of probability theory, but it is definitely a generalization of classical mechanics since it involves
non-deterministic symmetry transformations. For instance, the time evolution may be non-deterministic unlike in classical mechanics.
Euler’s formula for the probability clock
The previous sections established that the ensemble interpretation is self-consistent. However, the ensemble interpretation does not address the question why the wave-function plays a central role in
the calculation of the probability distribution, unlike most other interpretations of quantum mechanics. By being compatible with most (if not all) interpretations of Quantum Mechanics, the ensemble
interpretation is in practice a common denominator of most interpretations of Quantum Mechanics. It is useful, but it is not enough.
In this and in the following sections we will show that the wave-function is nothing else than one possible parametrization of any probability distribution. The wave-function can be described as a
multi-dimensional generalization of Euler’s formula, and its collapse as a generalization of taking the real part of Euler’s formula. The wave-function plays a central role because it is a good
parametrization that allows us to represent a group of transformations using linear transformations of the hypersphere. It is precisely the fact that the hypersphere is not the phase-space of the
theory that implies the collapse of the wave-function. Without collapse, the wave-function parametrization would be inconsistent.
Suppose that we have an oscillatory motion of a ball, with position $x=\cos(t)$ and we want to make a translation in time, $\cos(t)\to \cos(t+a)$. This is a non-linear transformation. However, if we
consider not only the position but also the velocity of the ball, we have the “wave-function” given by the Euler’s formula $q(t)=e^{it}$ and $x$ is the real part of $q$. Then, a translation is
represented by a rotation $q(t+a)=e^{ia} q(t)$. To know $x$ after the translation, we need to take the real part of the wave-function $e^{ia} q(t)$, after applying the translation operator.
Of course, $\cos(t)$ is not positive and so it has nothing to do with probabilities. However, we can easily apply Euler’s formula to a probability clock. A probability clock [@NOOM] is a time-varying
probability distribution for a phase-space with 2 states, such that the probabilities are $\cos^2(t)$ and $\sin^2(t)$, for the first and second states respectively.
A 2-dimensional real wave-function allows us to apply the Euler’s formula to the probability clock:
\begin{aligned} \Psi(t)=\exp\left(\left[\begin{smallmatrix} 0 & -1\\ 1 & 0 \end{smallmatrix}\right] t\right)\left[\begin{smallmatrix} 1\\ 0 \end{smallmatrix}\right]=\left[\begin{smallmatrix} \cos(t)\
\ \sin(t) \end{smallmatrix}\right]\end{aligned}(10)
The Euler’s formula for the density matrix is:
\begin{aligned} \Psi \Psi^\dagger=\left[\begin{smallmatrix} \cos^2(t) & \cos(t)\sin(t)\\ \cos(t)\sin(t) & \sin^2(t) \end{smallmatrix}\right]=\frac{1}{2}+ \left[\begin{smallmatrix} \frac{1}{2} & 0\\ 0
& -\frac{1}{2} \end{smallmatrix}\right](\cos(2t) +J\sin(2t))\end{aligned}(11)
Where $J=\left[\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\right]$ plays the role of the imaginary unit in the Euler’s formula for the probability clock. A measurement using a diagonal
projection triggers the collapse of the wave-function, such that a new density matrix is obtained by setting the off-diagonal part (i.e. the part proportional to $J$) of the original density matrix
to zero. The probability distribution is given by the diagonal part of the density matrix, i.e. by taking the “real part” of the “complex number” $\cos(2t) +J\sin(2t)$:
\begin{aligned} \mathrm{diag}(\Psi\Psi^\dagger)=\left[\begin{smallmatrix} \cos^2(t) & 0\\ 0 & \sin^2(t) \end{smallmatrix}\right]=\frac{1}{2}+\left[\begin{smallmatrix} \frac{1}{2} & 0\\ 0 & -\frac{1}
{2} \end{smallmatrix}\right]\cos(2t)\end{aligned}(12)
Since $\cos^2(t)+\sin^2(t)=1$ and $0<\cos^2(t)<1$, we can confirm that the wave-function parametrizes all probability distribution functions for a phase-space with 2 states, i.e. for any probability
$p$ there is an angle $t$ such that the cosine $\cos(t)$ of that angle verifies $\cos^2(t)=p$. Moreover, two wave-functions are always related by a rotation $\Psi(t+a)=\exp\left(J a\right)\Psi(t)$,
for some $a$.
Note that the rotation is an invertible linear transformation that preserves the space of wave-functions. This does not happen with probability distributions: the most general linear transformation
of a probability distribution that preserves the space of probability distributions is:
\begin{aligned} M(a,b)=\left[\begin{smallmatrix} \cos^2(a) & \cos^2(b)\\ \sin^2(a) & \sin^2(b)\end{smallmatrix}\right]\ \ \ \ \mathrm{(where\ a,b\ are\ real\ numbers)}\end{aligned}(13)
because if we apply $M$ to a deterministic distribution $\left[\begin{smallmatrix} 1\\ 0 \end{smallmatrix}\right]$ or $\left[\begin{smallmatrix} 0\\ 1 \end{smallmatrix}\right]$ we must obtain
probability distributions which leads to the constraints $\cos^2(a)+\sin^2(a)=\cos^2(b)+\sin^2(b)=1$ and $\cos^2(a),\sin^2(a),\cos^2(b),\sin^2(b)\geq 0$; the matrix $M$ such that:
\begin{aligned} M\frac{1}{2}\left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right]=\left[\begin{smallmatrix} 1\\ 0 \end{smallmatrix}\right]\end{aligned}(14)
is necessarily singular and so it is not suitable to represent a symmetry group.
The wave-function is thus a good parametrization which allows us to represent a group of transformations using linear transformations of the points of a circle. The collapse of the wave-function is
nothing more than taking the real part of a complex number as in most applications of Euler’s formula in Engineering, reflecting the fact that the circle is not the phase-space of the theory. Thus
the wave-function is nothing more than a parametrization of the probability distribution.
The Stern-Gerlach experiment
We follow reference [@sakurai] for the description of the Stern-Gerlach experiment, first carried out in Frankfurt by O. Stern and W. Gerlach in 1922. This experiment makes a strong case in favor of
generalizing the symmetry transformations to become non-deterministic, moreover the theoretical predictions only require a phase-space with two states like the one already discussed in the previous
section. Note that we only make measurements along the z and x-axis, but if we also had made measurements along the y-axis then the phase space would require four states or a parametrization with a
complex wave-function, see Section 10. Some articles such as reference [@sgquantum] argue that a “full quantum” analysis of the Stern-Gerlach experiment must involve the position degrees of freedom
and thus a phase-space with more than two states. But as in every theoretical model for any real experiment we should consider only a phase-space which is as large as it is strictly necessary to
compute all predictions for all practical purposes and do not waste time with redundant calculations which only add complexity and increase the likelihood of committing mistakes. Of course, the real
Stern-Gerlach experiment involves much more than two states, for instance if the electrical power feeding the experiment is shutdown due to an earthquake or if the man managing the experiment has a
heart attack it will affect the experimental results, but all predictions for all practical purposes can be computed using a phase-space with only two degrees of freedom.
In the Stern-Gerlach experiment, a beam of silver atoms is sent through a magnetic field with a gradient along the z or x-axis and their deflection is observed. The results show that the silver atoms
possess an intrinsic angular momentum (spin) that takes only one of two possible values (here represented by the symbols + and -). Moreover in sequential Stern-Gerlach experiments (see figure 1), the
measurement of the spin along the z-axis destroys the information about a atom’s spin along the x axis.
We consider in the phase-space, not only the spin of one atom of the beam, but also the angle of orientation of a macroscopic object which serves as a reference, for pedagogical purposes. The
corresponding complete wave-function is thus a reducible representation of the rotation group. When we apply a rotation to the phase-space, the rotation is a non-deterministic transformation of the
spin of the atom and a deterministic transformation of the macroscopic object. Thus, to keep track of the part of the wave-function corresponding to the angle of orientation of the reference
macroscopic object we only need the central value of the probability distribution for such angle, which we will call simply “the angle” for brevity. And then we only consider the part of the
wave-function corresponding to the spin of the atom.
In Equation 12, $cos^2(t)$ is the probability for the spin to be in the state $+$, while $sin^2(t)$ is the probability for the spin to be in the state $-$. The non-deterministic symmetry
transformation given by a rotation of the spin along the $x-z$ plane is parametrized by the parameter $t$ and its linear representation on the wave-function is described in Equation 10.
In the first measurement, the angle of the reference macroscopic object is 0 with respect to the z-axis; and we know for sure that the spin is in the state $+$ ($t=0$) because we are measuring the
spin along the z-axis of atoms that were previously filtered to be in the state $+$ when measuring the spin along the z-axis (see the first graph in figure 1).
A second sequential measurement along the x-axis means that we rotate the reference macroscopic object 90 degrees along the x-z plane so the new angle is 90 degrees; for the atom we first make a
45-degrees rotation along the x-z plane ($t=\pi/4$) and then we determine whether the spin is in the $+$ or $-$ state (i.e. the wave-function collapses, see the second graph in figure 1). The
probability for the spin to be in the states $+/-$ is now $50\%/50\%$, because the rotation is a non-deterministic symmetry transformation.
A third sequential measurement along the z-axis means that we rotate the reference macroscopic object -90 degrees along the x-z plane so the new angle is again 0 degrees; for the atom we first apply
a -45-degrees rotation along the x-z plane ($t=-\pi/4$) to the atoms with spin $+$ and then we determine whether the spin is in the $+$ or $-$ state (i.e. the wave-function collapses one more time,
see the third graph in figure 1). Despite that in the first measurement the spin was in the state $+$, the probability for the spin to be in the states $+/-$ is $50\%/50\%$ in the third measurement,
because the rotation is a non-deterministic symmetry transformation and we applied it in the second and third measurements to switch from the z to the x-axis and then to switch again from the x to
the z-axis.
As we have seen in the previous sections, generalizing the symmetry transformations to be non-deterministic suffices to account for all experimental results described by Quantum Mechanics, with the
Stern-Gerlach experiment being one example. The question remaining is whether the Euler’s formula applies for phase-spaces with more than 2 states, which would imply that the collapse of the
wave-function is merely a mathematical artifact of the wave-function parametrization.
Black hole information paradox and the Stern-Gerlach experiment
What is exactly a black hole from the point of view of a quantum theory? That’s a tough question. Because of that, the black hole information paradox is not necessarily related with real black holes.
Nevertheless, we can always think of the Stern-Gerlach experiment, described in the previous section. The argument here is that there is always a unitary transformation such that the corresponding
probability distribution is necessarily the constant distribution, for all initial states in the same orthogonal basis. Thus, if a black-hole erases most information about an object that comes inside
of it by turning this information to random, that is not incompatible with a unitary time-evolution. We have seen an analogous case in the previous section for a 2-state phase space.
Certainly, the collapse of the wave-function is not unitary and thus the transformation on the ensemble is also not unitary. If we measure the properties of the black-hole immediately after the
object comes inside, the information is erased. However since the time-evolution is unitary, if the transformation is not only about the object coming inside but about more events then the
information is not necessarily lost. If such events do not affect the degrees of freedom that were erased (which is expected since a black-hole is defined by few parameters), then the information
will remain erased. Only with a quantum theory for black holes we can know for sure which events can happen after an object comes inside a black-hole.
In any case, a transformation which erases information is compatible with a unitary time-evolution.
Euler’s formula for a phase-space with 4 states
We address now a system with 4 possible states. A real normalized wave-function $\varphi_1$ can be parametrized in terms of Euler angles (i.e. standard hyper-spherical coordinates and following
reference [@eulerangles]) as:
\begin{aligned} \varphi_{1}=&c_{1}\ l_{1}+s_{1}\ \varphi_{2}\\ \varphi_{2}=&c_{2}\ l_{2}+s_{2}\ \varphi_{3}\\ \varphi_{3}=&c_{3}\ l_{3}+s_{3}\ \varphi_{4}\\ \varphi_{4}=&l_{4} \end{aligned}(15)
Where $c_n=\cos(\theta_n)$ and $s_n=\sin(\theta_n)$ stand for the cosine and sine of an arbitrary angle $\theta_n$ (i.e. $\theta_n$ is an arbitrary real number), respectively; and $n$ is an integer
number verifying $1\leq n<4$. The set $l_{1}, l_{2}, l_{3}, l_{4}$ are normalized vectors forming an orthonormal basis of a -dimensional real vector space.
The Euler’s formula for the corresponding density matrices is:
\begin{aligned} \varphi_{1}\varphi_{1}^\dagger=&\frac{1}{2}+\frac{1}{2}(l_{1}l_{1}^\dagger-\varphi_{2}\varphi_{2}^\dagger) (\cos(2\theta_{1})+J_{1}\sin(2\theta_{1}))\\ \varphi_{2}\varphi_{2}^\dagger=
&\frac{1}{2}+\frac{1}{2}(l_{2}l_{2}^\dagger-\varphi_{3}\varphi_{3}^\dagger) (\cos(2\theta_{2})+J_{2}\sin(2\theta_{2}))\\ \varphi_{3}\varphi_{3}^\dagger=&\frac{1}{2}+\frac{1}{2}(l_{3}l_{3}^\dagger-\
varphi_{4}\varphi_{4}^\dagger) (\cos(2\theta_{3})+J_{3}\sin(2\theta_{3}))\\ \varphi_{4}\varphi_{4}^\dagger=&l_{4}l_{4}^\dagger \end{aligned}(16)
Where $J_n=(l_{n}\varphi_{n+1}^\dagger -\varphi_{n+1}l_{n}^\dagger)$ plays the role of the imaginary unit in the Euler’s formula, in the subspace generated by the vectors $\{l_{n}, \varphi_{n+1}\}$.
Thus, the collapse of the wave-function for a phase-space with states is a recursion of collapses of 2-dimensional real wave-functions. The conditional probabilities are given by the diagonal part of
the density matrix, i.e. by taking the “real part” of the “complex numbers” $\cos(2\theta_n) +J_n\sin(2\theta_n)$:
\begin{aligned} P( 1 | 1 \mathrm{\ or\ above}))&=\frac{1}{2}+\frac{1}{2}\cos(2\theta_{1})\ \ \ \ \ P( (2 \mathrm{\ or\ above}) | (1 \mathrm{\ or\ above}))= \frac{1}{2}-\frac{1}{2}\cos(2\theta_{1})\\
P( 2 | 2 \mathrm{\ or\ above}))&=\frac{1}{2}+\frac{1}{2}\cos(2\theta_{2})\ \ \ \ \ P( (3 \mathrm{\ or\ above}) | (2 \mathrm{\ or\ above}))= \frac{1}{2}-\frac{1}{2}\cos(2\theta_{2})\\ P( 3 | 3 \mathrm
{\ or\ above}))&=\frac{1}{2}+\frac{1}{2}\cos(2\theta_{3})\ \ \ \ \ P( (4 \mathrm{\ or\ above}) | (3 \mathrm{\ or\ above}))= \frac{1}{2}-\frac{1}{2}\cos(2\theta_{3})\\ P( 4 | (4 \mathrm{\ or\ above}))
&=1 \end{aligned}(17)
where $P( 2 | (2 \mathrm{\ or\ above}))$ stands for probability for the state to be $n=2$ knowing that the state is either $n=2$, or $n=3$, ... or $n=4$. Note that these conditional probabilities are
arbitrary, i.e. for any probability $p$ there is an angle $\theta_n$ such that the cosine $c_n=\cos(\theta_n)$ of that angle verifies $c_n^2=p$.
The fact that the previous conditional probabilities are arbitrary, implies that the probability distribution is arbitrary, since for any probability distribution we have:
\begin{aligned} P(1)=& P(1 | (1 \mathrm{\ or\ above}))\\ P(2)=& P( (2 \mathrm{\ or\ above}) | (1 \mathrm{\ or\ above}))*\\ & P(2 | (2 \mathrm{\ or\ above}))\\ P(3)=& P( (2 \mathrm{\ or\ above}) | (1
\mathrm{\ or\ above}))*\\ & P( (3 \mathrm{\ or\ above}) | (2 \mathrm{\ or\ above}))*\\ & P(3 | (3 \mathrm{\ or\ above}))\\ P(4)=& P( (2 \mathrm{\ or\ above}) | (1 \mathrm{\ or\ above}))*\\ & P( (3 \
mathrm{\ or\ above}) | (2 \mathrm{\ or\ above}))*\\ & P( (4 \mathrm{\ or\ above}) | (3 \mathrm{\ or\ above}))*\\ & P(4 | (4 \mathrm{\ or\ above})) \end{aligned}(18)
Moreover, two wave-functions are always related by a rotation. Thus we can confirm that any probability distribution for states, can be reproduced by the Born rule for some wave-function:
\begin{aligned} P(n)=&|\varphi^\dagger l_n|^2\\ P(1)=&(c_{1})^2\\ P(2)=&(s_{1}c_{2})^2\\ P(3)=&(s_{1} s_{2}c_{3})^2\\ P(4)=&(s_{1} s_{2}s_{3})^2 \end{aligned}(19)
Euler’s formula for a generic phase-space
A probability distribution can be discrete or continuous. A continuous probability distribution is a probability distribution that has a cumulative distribution function that is continuous. Thus, any
partition of the phase-space (where each part of the phase-space has a non-null Lebesgue measure) is countable.
Consider now a countable (possibly infinite) partition of the phase-space. The corresponding countable orthonormal basis for the separable Hilbert space is $\{l_n\}$, where each index $n>0$
corresponds to an element of the partition of the phase-space. We can parametrize a normalized vector in the Hilbert space [@eulerangles], as $v_n=c_n l_n+s_n v_{n+1}$, where $c_n=\cos(\theta_n)$ and
$s_n=\sin(\theta_n)$ stand for the cosine and sine of an arbitrary angle $\theta_n$ (i.e. $\theta_n$ is an arbitrary real number), respectively; and $n>0$ is an integer number. The first vector $v_1$
is the wave-function of the full phase-space. Note that the parametrization is valid for infinite dimensions, because in the recursive equation all we need to assume about the vector $v_{n+1}$ is
that it is normalized and orthogonal to $\{l_1, l_2, ... l_n\}$, which is a valid assumption in infinite dimensions. Then we define $v_{n+1}$ in terms of $v_{n+2}$ in the same way, and so on. The
recursion does not need to stop.
Then, the projection to the linear space generated by $v_n$ is:
\begin{aligned} v_nv_n^\dagger=&\frac{1}{2}+\frac{1}{2}(l_{n}l_{n}^\dagger-\varphi_{n+1}\varphi_{n+1}^\dagger) (\cos(2\theta_{n})+J_{n}\sin(2\theta_{n})) \end{aligned}(20)
Where $J_n=(l_{n}\varphi_{n+1}^\dagger-\varphi_{n+1}l_{n}^\dagger)$ plays the role of the imaginary unit in the Euler’s formula, in the subspace generated by the vectors $\{l_{n}, \varphi_{n+1}\}$.
Thus, the collapse of the wave-function for a generic phase-space is a recursion of collapses of 2-dimensional real wave-functions. The conditional probabilities are given by the diagonal part of the
density matrix, i.e. by taking the “real part” of the “complex numbers” $\cos(2\theta_n) +J_n\sin(2\theta_n)$: The operator $v_n v_n^\dagger$ is a projection thanks to the off-diagonal terms $c_ns_n
(l_n v_{n+1}^\dagger+v_n l_{n+1}^\dagger)$.
Defining $(n\mathrm{\ or\ above})=\{k : k\geq n\}$ as the event which contains all parts of the phase-space with index starting at $n$, we can write the probability distribution as:
\begin{aligned} \label{eq:cond} P(n)&=P((n\mathrm{\ or\ above}))P(n| (n \mathrm{\ or\ above}))\\ &=\left(\prod\limits_{k=1}^{n-1} P((k+1 \mathrm{\ or\ above})|(k \mathrm{\ or\ above}))\right)P(n| (n
\mathrm{\ or\ above}))\end{aligned}(21)
That is, as a product of the probabilities
\begin{aligned} &P(n|(n \mathrm{\ or\ above}))\mathrm{\ and\ }P((n+1 \mathrm{\ or\ above})|(n \mathrm{\ or\ above}))\mathrm{,\ which\ verify}\\ &P(n|(n \mathrm{\ or\ above}))\ +\ \ P((n+1 \mathrm{\
or\ above})|(n \mathrm{\ or\ above}))=1.\end{aligned}(22)
If the off-diagonal terms are suppressed (collapsed), we obtain a diagonal operator which represents the probability distribution $P(n)$ in the Hilbert space:
\begin{aligned} \mathrm{diag}(v_nv_n^\dagger)=c_n^2 l
n^\dagger+s_n^2 v_{n+1}v_{n+1}^\dagger\end{aligned}(23)
That is, $P(n)=\mathrm{tr}(\mathrm{diag}(v_1v_1^\dagger) l
n^\dagger)$ and $P(O)=0$ for operators $O$ with null-diagonal. Note that $c_n^2=P(n| (n \mathrm{\ or\ above}))$ and $s_n^2=P((n+1 \mathrm
{\ or\ above})| (n \mathrm{\ or\ above}))$ and these probabilities are arbitrary, i.e. for any probability $p$ there is an angle $\theta_n$ such that the co $c_n=\cos(\theta_n)$ of that angle
verifies $c_n^2=p$.
The fact that these conditional probabilities are arbitrary, implies that the probability distribution is arbitrary, since the probability distribution can be written in terms of these conditional
probabilities as shown in Equation [eq:cond].
Complex and Quaternionic Hilbert spaces
While the parametrization with a real wave-function is always possible, it may not be the best one. As we have seen, the wave-function parametrization allows us to apply group theory to the states of
the ensemble, since unitary transformations (i.e. a multi-dimensional rotation) preserve the properties of the parametrization (in particular the conservation of total probability).
The union of a set of projection operators and the unitary representation of a group, is a set of normal operators. Suppose that there is no non-trivial closed subspace of the Hilbert space left
invariant by this set of normal operators. The (real version of the) Schur’s lemma [@realpoincare; @Oppio:2016pbf; @realoperatoralgebras][@realpoincare; @Oppio:2016pbf; @realoperatoralgebras]
[@realpoincare; @Oppio:2016pbf; @realoperatoralgebras] implies that the set of operators commuting with the normal operators forms a real associative division algebra—such division algebra is
isomorphic to either: the real numbers, the complex numbers or the quaternions.
If we do a parametrization by a real wave-function and consider only expectation values of operators that commute with a set of operators isomorphic to the complex or the quaternionic numbers, then
we can equivalently define wave-functions in complex and quaternionic Hilbert spaces [@realQM; @realpoincare; @Oppio:2016pbf][@realQM; @realpoincare; @Oppio:2016pbf][@realQM; @realpoincare;
@Oppio consider the quaternionic case (it will be then easy to see how is the complex case). We have a discrete state space defined by two real numbers $n,m$, with $1\leq m\leq 4$ and we only
consider the probabilities for $n$ independently on $m$, $P(n)=\sum_{m=1}^4 P(n,m)$.
Then a more meaningful parametrization—reflecting by construction the restriction on the operators we are considering—uses a quaternionic wave function $v_1$. Let $\{l_n\}$ be an orthonormal basis of
quaternionic wave-functions and we have:
\begin{aligned} v_n v_n^\dagger=c_n^2 l
n^\dagger+s_n^2 v_{n+1}v_{n+1}^\dagger+c_ns_n(l_n v_{n+1}^\dagger+v_{n+1} l_{n}^\dagger)\end{aligned}(24)
Note that there is a basis where $l
n^\dagger$ is real diagonal and thus upon collapse $v_nv_n^\dagger$ becomes real diagonal as well.
The complex case is just the above case with complex numbers replacing quaternions and a state space which is the union of 2 identical spaces. The continuous case is analogous, since there is a
partition of the phase-space which is countable.
Comparing the time evolution with a stochastic process
Quantum Mechanics is not a generalization of probability theory, but it is definitely a generalization of classical mechanics since it involves non-deterministic transformations to the state of the
system. For instance, the time evolution may be non-deterministic unlike in classical mechanics.
There are three major metaphysical views of time [@bebecome]: presentism, eternalism and possibilism. The possibilism consists in considering the presentism for the future and the eternalism for the
past, so it is inconsistent with a time translation symmetry. The presentism view coincides with the Hamiltonian formalism of physics, that the state of the system is defined by a point in the phase
space. When the time evolution of the system is deterministic it traces a phase space trajectory for the system, however the definition of the state of the system does not involve time, i.e. only the
present exists . The eternalism view coincides with the Lagrangian formalism of physics, that the state of the system is defined by a function of time. When the time evolution of the system is
deterministic, this function of time coincides with the phase-space trajectory of the classical Hamiltonian formalism and so which metaphysical view of time we use is irrelevant from an experimental
point of view (in the deterministic case).
But when the time-evolution of the system is non-deterministic, we may have a hard time studying the time-evolution from the Lagrangian formalism and/or eternalism metaphysical view. The key fact
about Quantum Mechanics which makes it incompatible with the eternalism/Lagrangian point of view is that the time-evolution is not necessarily a stochastic process, i.e. there is not necessarily a
collection of random events indexed by time. We only apply one non-deterministic transformation of the state of the system, however there are many different transformations we can choose from and the
set of choices is indexed by a parameter we call time, which is fine from the presentism/Hamiltonian point of view since only the present exists.
Note that a random experiment always involves a preparation followed by a measurement. For instance, we shake a dice in our hand and throw it over a table until it stops (preparation), then we check
the position where it stopped (measurement).
If we just throw the dice without shaking our hand, the probability distribution for the measurement outcome is different than if we shake our hand. There is nothing mysterious about this: two
different preparations lead to two different probability distributions. Whether or not we actually do the measurement does not change anything, what changes the probability distribution is the
preparation.think about a preparation which is function of an element of a symmetry group, for instance translation in time. From the point of view of probability theory or experimental physics, this
is a valid option. However, it is important to note that this preparation function of time is not a stochastic process in time. A stochastic process in time is a set of random experiments indexed by
time, while in the preparation which is function of time we have a single random experiment dependent on the parameter time. As an example, consider a) throwing the dice 10 times, one time per minute
during 10 minutes and b) shake the dice in our hand for a number of minutes $T$ between 0 and 10 and then throw the dice once. The preparation in b) is dependent on the time parameter $T$, while in
a) the time selects the one of the many identically prepared experiments which was done at the selected time.
Note that the experiments a) and b) above are different but can be combined: we could do many random experiments, each of them would be dependent on a parameter. This fact is important in Section 12.
In the remaining of this section, we comment on conditioned probability and the random walk. It is well-known that quantum mechanics can be described as the Wick-rotation of a Wiener stochastic
process [@nonperturbativefoundations]. In other words, the time evolution in Quantum Mechanics is a Wiener process for imaginary time. This is the origin of the Feynman’s path integral approach to
Quantum Mechanics and Quantum Field Theory.
Since the Wiener process is one of the best known Lévi processes—a Lévi process is the continuous-time analog of a random walk—this fact often leads to an identification of Quantum Mechanics with a
random walk. In particular, it often leads to an identification of the probabilities calculated in Quantum Mechanics with conditioned probabilities—the next state in a random walk is conditioned by
the previous state.
Certainly, the usefulness of group theory is common to both a random walk and to Quantum Mechanics and this unavoidably leads to similarities between a random walk and Quantum Mechanics. However,
imaginary time is very different from real time and thus the probabilities calculated in Quantum Mechanics are not necessarily conditioned probabilities in a random walk.
In order to relate a random walk (or any other stochastic process) with Quantum Mechanics correctly, we need the probability distribution for the complete paths of the random walk. Then, we can use a
wave-function parametrization of the probability distribution for the complete paths of the random walk. Finally, we can apply quantum methods to this wave-function. The result is a Quantum
Stochastic Process [@qsc], which is not a generalization of a stochastic process due to the wave-function collapse, but merely the parametrization of a stochastic process with a wave-function.
Time translation is a stochastic process if and only if it is deterministic
Now we are able to prove one of the main results of this paper, namely that there is a group action of a Wigner’s symmetry group on the probability distribution for the state of a system, if and only
if the Wigner’s symmetry group transforms deterministic (probability) distributions into deterministic (probability) distributions. A corollary is that time translation in Quantum Mechanics is a
stochastic process if and only if it is deterministic. This mathematical fact is overlooked by the assumptions of both the Bell’s theorem and the Einstein-Podolsky-Rosen (EPR) paradox.
As it was discussed in Section 3, Wigner’s theorem [@2014PhLA; @Ratz1996; @wignertheorem][@2014PhLA; @Ratz1996; @wignertheorem][@2014PhLA; @Ratz1996; @wignertheorem] implies that the action of a
symmetry group on the wave-function is necessarily linear and unitary. In Section 4, we showed that the action of a symmetry group on the wave-function is deterministic if and only if $P_A$ and $U_g
P_B U^\dagger_g$ commute for all events $A,B$ and for all the elements $g$ of the group, where $P_A$ is a projection-valued-measure.
This means that $U$ is a deterministic transformation if and only if $U_{l a} U_{m a}^*=0$ for all $a,l,m$ such that $leq m$.
Now we check the necessary and sufficient conditions for the action of a symmetry group on the wave-function to correspond to an action on the corresponding probability distribution.
That is, if we start with some probability distribution $\mathrm{diag}(\rho_1)$, then the action of each element $g$ of the group on the wave-function will produce (after the collapse) a different
probability distribution $\mathrm{diag}(\rho_g)$. The composition of the actions of two group elements $g,h$ on the probability distribution is given by the succession of the two random experiments
corresponding to $g$ and $h$: $P(A)=\mathrm{tr}(\mathrm{diag}(\rho_g)U_h P_A U_h^\dagger)$.
However, Wigner’s theorem [@2014PhLA; @Ratz1996; @wignertheorem][@2014PhLA; @Ratz1996; @wignertheorem][@2014PhLA; @Ratz1996; @wignertheorem] implies that the action of a symmetry group on the
wave-function is necessarily linear and unitary, thus $P(A)=\mathrm{tr}(\rho_g U_h P_A U_h^\dagger)$.
Thus there is a group action of the symmetry group on the probability distribution if and only if $\mathrm{tr}(\mathrm{diag}(\rho_g)U_h P_A U_h^\dagger)=\mathrm{tr}(\rho_g U_h P_A U_h^\dagger)$ for
any pure density matrix $\rho_g$ and any event $A$ and group element $h$.
The equality above is equivalent to $\sum_{k,b: keq b} U_{ka}^*\Psi_k\Psi_b^* U_{ba}=0$, where $U_{ba}$ are the elements of the matrix $U_h$. We can see that if $U$ is a deterministic transformation,
then the equality is satisfied, since $U_{ja}^* U_{jl}=0$ for all $a,l,j$ such that $aeq l$. On the other hand, if $U$ is a non-deterministic transformation then for some $a,l,m$ such that $leq m$,
we have $U_{m a}^* U_{l a}eq 0$. Then for $\Psi_k=\frac{1}{\sqrt{2}}(\delta_{km}+\delta_{kl})$, we get $\sum_{k,b: keq b} U_{ka}^*\Psi_k\Psi_b^* U_{ba}=U_{m a}^* U_{la}eq 0$, i.e. there is no group
action of the symmetry group on the probability distribution.
Symmetries as irreversible processes
The concept of (ir)reversible process from thermodynamics also needs a careful discussion in quantum mechanics. A non-deterministic symmetry transformation, when acting on a deterministic ensemble
increases the entropy of the ensemble after the wave-function collapse and therefore must be an irreversible transformation. Yet, a symmetry transformation always has an inverse symmetry
transformation, because it is included in a symmetry group, so it must be considered reversible in some sense.
The way out of this apparent contradiction is the role of time in the quantum formalism, which was discussed in Sections 11 and 12. In the ensemble interpretation, the individual system is entirely
defined by a standard phase-space, which implies that the time plays no fundamental role in quantum mechanics nor in classical Hamiltonian mechanics. Then, time-evolution in quantum mechanics is not
a stochastic process unless it is deterministic. Therefore, there is not a probability distribution for each time (or for other parameter corresponding to the symmetry group).
If we consider a stochastic process with only two probability distributions corresponding to the initial and final times, then the complete symmetry transformation is irreversible (if it is
non-deterministic and it acts in a deterministic ensemble). However, this does not imply that it is a “bad” symmetry, because no stochastic process can be defined in between the initial and final
times. On the other hand, if the symmetry group contains only deterministic transformations then a stochastic process can be defined in between the initial and final times and such process is
reversible, as expected.
Quantum Mechanics is EPR-complete
The Einstein-Podolsky-Rosen (EPR) main claim [@epr] (namely, that Quantum Mechanics is an incomplete description of physical reality), is defended by reducing to absurd the negation of the main
claim, i.e. by reducing to absurd that position (Q) and momentum (P) are not simultaneous elements of reality. In the EPR article it is stated: “one would not arrive at our conclusion if one insisted
that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted.[...] This makes the reality of P and Q depend upon
the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this.”
The reduction to absurd of the negation of the claim, could only be a satisfactory argument if the claim itself (namely, the quantities position and momentum of the same particle are simultaneous
elements of reality, despite they cannot be simultaneously measured or predicted) would not be absurd as well. But the claim itself raises eyebrows to say the least, once we remember that (in Quantum
Mechanics, by definition) measuring the position with infinite precision completely erases any knowledge about the momentum of the same particle.
In Quantum Mechanics as in classical Hamiltonian mechanics, the state of an individual system is a point in a phase space, and the phase space is both the domain and image of the deterministic
physical transformations. As in any statistical theory, we may know only the probability distribution for the state of the individual system, instead of knowing the state of the individual system.
The relation between quantum mechanics and a statistical theory is clear: the wave-function is a parametrization for any probability distribution [@parametrization].
There are two kinds of incompleteness in a non-Markov stochastic process. The two kinds of incompleteness are in correspondence with the two concepts: stochastic and non-Markov, respectively.
1) Stochastic: From the point of view of (classical) information theory [@info], the root of probabilities (i.e. non-determinism) is by definition the absence of information. Statistical methods are
required whenever we lack complete information about a system, as so often occurs when the system is complex [@bertinstatistical]. Thus we can convert a deterministic process to a stochastic process
unambiguously (using trivial probability distributions); but we cannot convert a stochastic process into a deterministic process unambiguously since we need new information .
2) non-Markov: any non-Markov stochastic process can be described as a Markov stochastic process where some variables defining the state of the system are hidden (i.e. unknown) [@allmarkov;
@non_markov_examples][@allmarkov; @non_markov_examples]. Conversely, by definition any irreducible Markov process where some variables defining the state of the system are hidden will give rise to a
non-Markov process. For instance, the physical phenomena which generates examples of Brownian motion is deterministic and thus Markov, but real-world Brownian motion is often non-Markov (because we
cannot measure the state of the system completely [@brownian; @brownian2][@brownian; @brownian2]) despite the fact that the Brownian motion is one of the most famous examples of a Markov process.
In reference [@reality] (authored by A. Einstein and contemporary of the EPR paradox) the two kinds of incompleteness are clearly distinguished:
"**[...]* I believe that the [quantum] theory is apt to beguile us into error in our search for a uniform basis for physics, because, in my belief, it is an incomplete representation of real things,
although it is the only one which can be built out of the fundamental concepts of force and material points (quantum corrections to classical mechanics). The incompleteness of the representation is
the outcome of the statistical nature (incompleteness) of the laws. I will now justify this opinion.*"
The incompleteness of the representation corresponds to the non-Markov kind, while the incompleteness of the laws corresponds to the stochastic kind. By definition, in Quantum Mechanics any sequence
of measurements is a Markov stochastic process (thus it has the stochastic kind of incompleteness) . Note that any non-Markov stochastic process can be described as a Markov stochastic process where
some variables defining the state of the system are hidden (i.e. unknown) [@allmarkov; @non_markov_examples][@allmarkov; @non_markov_examples].
Since Quantum Mechanics does not have the non-Markov kind of incompleteness, position and momentum can only be simultaneous elements of reality in another theory very different from Quantum
Mechanics. That both the claim and its negation are absurd, is strong evidence that some of the assumptions leading to the Einstein-Podolsky-Rosen (EPR) paradox [@epr] do not hold.
So, why did the author tried to justify (using the EPR paradox [@epr], among other arguments) that in Quantum Mechanics the stochastic kind of incompleteness necessarily leads to a non-Markov kind of
incompleteneg paragraph from the same reference [@reality] suggests that the author was trying to favor the cause that any future theoretical basis should be deterministic, not just Markov (since
statistical mechanics is often Markov).
“There is no doubt that quantum mechanics has seized hold of a beautiful element of truth, and that it will be a test stone for any future theoretical basis, in that it must be deducible as a
limiting case from that basis, just as electrostatics is deducible from the Maxwell equations of the electromagnetic field or as thermodynamics is deducible from classical mechanics. However, I do
not believe that quantum mechanics will be the starting point in the search for this basis, just as, vice versa, one could not go from thermodynamics (resp. statistical mechanics) to the foundations
of mechanics.”
However and as discussed in Section 4, there is no mathematical argument that suggests that in general a deterministic model is more fundamental than a stochastic one, quite the opposite. Since the
wave-function is merely a possible parametrization of any probability distribution [@parametrization], we also cannot claim that a deterministic model is more fundamental than Quantum Mechanics.
Thus, the stochastic kind of incompleteness is harmless.
So, the EPR paradox appears as an attempt to justify a mathematical statement (that a deterministic model is more fundamental than Quantum Mechanics) with arguments from physics (trying to link to
the non-Markov kind of incompleteness), for which no mathematical arguments could be found. Note that a statement referring to any future theoretical basis is essentially a mathematical statement
because the physical model is any (since the theoretical basis is any).
However, it is a failed attempt because it missed the fact discussed in Section 12, that the time evolution is a stochastic process if and only if it is deterministic.
In the EPR paradox, there is no probability distribution for the state of system after the spatial separation of the entangled particles and before the transformation involved in the measurement
takes place, because the time evolution (being in this case non-deterministic) is not a stochastic process. We can only consider the probability distribution for the state of system after the spatial
separation of the entangled particles and after the transformation involved in the measurement takes place. This is overall a non-local physical transformation since it involves the spatial
separation of the entangled particles. But it does not violate relativistic causality, since both the spatial separation of the entangled particles and the transformation involved in the measurement
do not by themselves violate relativistic causality, so their composition does not violate causality either.
Unlike many popular no-go arguments [@nogo], we are not arguing against the requirement that a physical theory should be complete, in fact we claim that Quantum Mechanics is a complete statistical
theory (as defined by EPR).
Note that Bohr already declared Quantum Mechanics as a “complete” theory, however he did it at the cost of a radical revision of the classical notions of causality and physical reality
[@bohrcomplete]. He wrote: “Indeed the finite interaction between object and measuring agencies conditioned by the very existence of the quantum of action entails —because of the impossibility of
controlling the reaction of the object on the measuring instruments if these are to serve their purpose—the necessity of a final renunciation of the classical ideal of causality and a radical
revision of our attitude towards the problem of physical reality.” [@bohrcomplete] Such notion of a “complete” theory mostly favors the EPR claim: the only way that Quantum Mechanics could be
complete is if it is incompatible with the classical notions of causality and physical reality. Thus from a logic point of view, there is no disagreement between Einstein and Bohr, their disagreement
is about what basic features an acceptable theory should have, whether or not it should be compatible with the classical notions of causality and physical reality.
In contrast, the fact—that the time evolution is a stochastic process if and only if it is deterministic—which was overlooked is perfectly compatible with the classical notions of physical reality
(because Quantum Mechanics has a standard phase-space) and causality (as we will show in Section 15). We claim that Quantum Mechanics—being non-deterministic and thus a generalization of classical
mechanics—does not entail a radical departure from the basic features that an acceptable theory should have, according to EPR [@epr]. In fact in Quantum Mechanics and in classical Hamiltonian
mechanics, the state of an individual system is a point in a phase space, and the phase space is both the domain and image of the deterministic physical transformations.
Any deterministic theory compatible with relativistic Quantum Mechanics necessarily respects relativistic causality
The only known theory consistent with the experimental results in high energy physics [@pdg] is a quantum gauge field theory which is mathematically ill-defined [@prize]. Due to the mathematically
illness, the relation of such a theory with Quantum Mechanics is still object of debate and it will be addressed soon in another article by the present author.
In the mean time we will have to consider a free system, which suffices to address the EPR paradox. For a free system, we know well what is relativistic Quantum Mechanics [@realpoincare]. The time
evolution of the wave-function is described by the Dirac equation for a free particle, which is a real (i.e. non-complex) equation.
Relativistic causality is satisfied in relativistic Quantum Mechanics, meaning that there is a propagator which vanishes for a space-like propagation [@realpoincare]. In other words, the probability
that the system moves faster than light is null.
A deterministic theory compatible with relativistic Quantum Mechanics is one which when applied to an ensemble of free systems, will reproduce the statistical predictions of Quantum Mechanics.
Since in relativistic Quantum Mechanics the probability that the system moves faster than light is null, then no system (described by the deterministic theory) in the ensemble moves faster than
light. Thus any deterministic theory compatible with relativistic Quantum Mechanics necessarily respects relativistic causality. The question we left open here and address in the next section, is
whether one such deterministic theory exists.
A deterministic theory compatible with relativistic Quantum Mechanics
Does a deterministic theory—consistent with the non-deterministic time evolution of Quantum Mechanics—exists?
The answer is yes, and we will build one example of such deterministic theory in this section.
In an experimental setting, we always have a discrete set of possible outcomes and thus Quantum Mechanics always predicts a cumulative distribution function. This allows us to apply the
inverse-transform sampling method [@sampling] for generating pseudo-random numbers consistently with the probability distribution predicted by Quantum Mechanics.
An experiment in Quantum mechanics always involves the repetition of an experimental procedure many times. In the deterministic theory however, each time we execute the experimental procedure we are
not executing exactly the same experimental procedure. We consider a number (any number will do) which will be the seed of the pseudo-random number generator and then we generate pseudo-random
numbers consistently with the probability distribution predicted by Quantum Mechanics. The experimental procedure is: 1) generate one pseudo-random number and 2) modify the state of the system
accordingly with the pseudo-random number.
In the case of relativistic Quantum Mechanics, the probability of violating relativistic causality is null. Thus, the experimental procedure never violates relativistic causality. The modifications
of the state of the system are however necessarily not infinitesimal since the phase space of the experimental setting is discrete. This doe not violate relativistic causality, since the finite
modifications to the state of the system occur in finite intervals of time.
We can however consider intervals of time as small as we like and thus modifications to the state of the system as small as we like. The only requirement for this is that the computational resources
involved in the pseudo-random number generation are as large as needed (which is valid from a logical point of view). Note that since time evolution in quantum mechanics is not necessarily a
stochastic process, we will often have that a sequence of experimental procedures executed at regular and small intervals of time produces different statistical data than than just one experimental
procedure executed at once after the same total time has passed (e.g. in the double-slit experiment). But this cannot be considered a radical departure of the classical notion of physical reality,
since in the (very old) presentism view of classical Hamiltonian mechanics, the phase space (i.e. the physical reality) does not involve the notion of time [@bebecome]. Moreover when the time
evolution is deterministic then it is a stochastic process, therefore if we study only deterministic transformations then we can recover the eternalism view of classical Lagrangian mechanics without
any conflict with relativistic causality. For instance, this implies that in the double-slit experiment we can in principle reconstruct the trajectory of each particle and conclude about which slit
the particle has went through.
From a logical point of view, this deterministic theory is valid and by definition it always agrees with the experimental predictions of Quantum Mechanics, thus it is experimentally indistinguishable
from Quantum Mechanics.
From the metaphysics point of view, this deterministic theory is unacceptable, since it involves pseudo-random number generation. For instance, in the double-slit experiment we (or some super-natural
entity) would need to somehow “program” each particle to follow a different path determined by a different number, which is absurd. However, the present author has no interest in building a nice
deterministic theory compatible with Quantum Mechanics, for the reasons exposed in Section 4.
Note that this deterministic theory is not super-deterministic, i.e. the experimental physicists are free to choose which measurements and which transformations of the state of the system to do
[@superdeterminism][43]. However, an experimental procedure involves a symmetry transformation of the state of the system. Since the symmetry transformation in this deterministic theory is reproduced
by the pseudo-random number generation, then when we apply the inverse-transform sampling method we need to know already what is the symmetry transformation. Thus there is a kind of conspiracy
between the symmetry transformation and the pseudo-random generator, but such conspiracy is part of the definition of the deterministic symmetry transformation itself. There are assumptions about
freedom of choice in the literature which exclude our deterministic (but not super-deterministic) theory, because the authors erroneously consider that an experimental procedure which involves a
transformation of the state of the system is instead an observation without consequences to the system [@superdeterminism][43].
The Young’s double slit experiment
The ensemble interpretation does not give any explanation as to why it looks like the electron’s wave-function interferes with itself in the Young’s double-slit experiment [44][45][46]—that would
imply that the wave-function describes (in some sense) an individual system. We will fill that gap in this section.
The key to understand the results of the double-slit experiment is the role of time in the quantum formalism, which was discussed in detail in Section 12. In the ensemble interpretation the
individual system is entirely defined by a standard phase-space, which implies that the time plays no fundamental role in quantum mechanics nor in classical Hamiltonian mechanics. Moreover, the
time-evolution in quantum mechanics is not a stochastic process unless it is deterministic. Therefore, there is not a probability distribution for each time (or for other parameter corresponding to
the symmetry group).
In the double-slit experiment, the time-evolution of the electron after being fired (S1) is a product of two non-deterministic symmetry transformations: first, going through one or another slit with
a 50/50 probability (S2); and second, a non-deterministic propagation from (S2) until (F). If at least one of these two symmetry transformations would be deterministic, then we could define a
stochastic process including the 3 instants in time (S1), (S2) and (F). But since both transformations are nondeterministic, the only stochastic process that can be defined only includes the 2
instants in time (S1) and (F), and the corresponding transformations from (S1) to (S2) and from (S2) to (F) have never occurred.
The only “mystery” that needs to be clarified is the fact that the non-deterministic propagation of the electron from (S2) until (F) is such that it appears that the electron interferes with itself,
just like a classical wave would do. To simplify the discussion we will only consider the electrons that reach the detector along 2 different angles $\sin(\theta_1)=\frac{2 \pi}{p d}$ and $\sin(\
theta_2)=\frac{\pi}{p d}$ where $p$ is the electron’s linear momentum. So, a selected electron can only go through one of these 2 angles, the electrons that go through other angles are discarded.
The wave-function at (S1) is $\Psi=\left[\begin{smallmatrix} 1 \\ 0\end{smallmatrix}\right]$. The time-evolution from (S1) until (S2) may be the identity matrix $U=\left[\begin{smallmatrix} 1 & 0 \\
0 & 1 \end{smallmatrix}\right]$ or $U=\frac{1}{\sqrt{2}}\left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right]$, depending on whether the second slit is closed or open, respectively. If
the second slit is open, then $U\Psi=\frac{1}{\sqrt{2}}\left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right]$ meaning that the electron may go through both slits with equal probability.
The time-evolution from (S2) until (F) is given by the unitary transformation $U^{'}=\frac{1}{\sqrt{2}}\left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right]$, that is, it sums the
wave-functions from both slits for the first angle and it subtracts the wave-functions from both slits for the second angle.
Thus, if the second slit is closed, we have at (F) the wave-function $\Psi=\frac{1}{\sqrt{2}}\left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right]$ meaning that the electron may come along angles
1 or 2 with equal probability. But if the second slit is open, we have at (F) the wave-function $\Psi=\left[\begin{smallmatrix} 1 \\ 0\end{smallmatrix}\right]$ meaning that the electron will only
come along angle 1; since the electron would have come through both slits with equal probability if we would see what happened at (S2), it appears that from (S2) until (F) it interferes with itself
constructively(destructively) along the angle 1(2) respectively.
The “mystery” is therefore similar to the probability clock 5: How is it possible that a $50/50$ probability becomes $100/0$? It is possible because precisely because the time plays no fundamental
role in quantum mechanics nor in classical Hamiltonian mechanics. There is not a probability distribution for each time (or for other parameter corresponding to the symmetry group). The symmetry
transformation $U'U$ is different from a stochastic process where the symmetry transformations $U'$ and then $U$ are applied, and there is no reason why it should not be different.
Do the Bell inequalities hold?
The Bell inequalities [@bell] do not hold—since Quantum Mechanics cannot be distinguished from a complete statistical theory—because the assumptions of the Bell inequalities overlooked the fact that
time-evolution is a stochastic process if and only if it is deterministic. As long as the time-evolution of the phase-space is a symmetry and it respects relativistic causality, there is no
reasonable argument why a complete statistical theory should be a stochastic process. The whole point of the Bell inequalities is to distinguish Quantum Mechanics from a “standard” statistical
theory, but a “standard” statistical theory means that the theory is completely defined by a probability distribution in a phase-space (which is the case of Quantum Mechanics and classical
statistical mechanics).
One could argue instead that the inequalities do hold, but there is an implicit assumption that the theory which is being compared to Quantum Mechanics has a time-evolution which is a stochastic
process. Even in that case (see Section 12) we have that for any set of experimental results supporting relativistic Quantum Mechanics, there is a deterministic theory (and so the time-evolution is a
stochastic process) which is also compatible with the same experimental results. So, to save the Bell inequalities we would need now to find fundamental arguments against such deterministic theory.
But, which arguments? Such deterministic theory is compatible with any experimental test about relativistic causality and it is not super-deterministic. These arguments would need to be somehow
against the existence of pseudo-random number generators in Nature, but such generators do exist in Nature because we humans built some of them and we are part of Nature.
To be sure, the present author does not expect that a reasonable deterministic theory will in the future replace Quantum Mechanics. But once it is established that Quantum Mechanics is a complete
statistical theory, the idea that we can rule out a reasonable deterministic theory, is also an absurd: it would imply affirming the Bayesian point of view and ruling out the Frequentist point of
view. Two logical constructions can always be mutually incompatible, despite being both consistent when considered independently of each other (e.g. the Bayesian and Frequentist points of view). In
the Bayesian point of view, the probability expresses a degree of belief, and so the probability is an entity which exists by itself. In the Frequentist point of view the root of probabilities is the
absence of deterministic information that does exist somehow and is revealed through events. But if such information exists, then we cannot rule out that there is a reasonable deterministic theory
which describes such information.
In summary, either we can say that the Bell inequalities do not hold or instead, we can also say that the Bell inequalities (despite being mathematically valid inequalities) involve unrealistic
assumptions which render them innocuous.
Conditioned probability and constrained systems
A probability distribution can be discrete or continuous (or a linear combination of discrete and continuous probability distributions). A continuous probability distribution is a probability
distribution that has a cumulative distribution function that is continuous.
In the case of continuous probability distributions, each and every single point in the phase-space has null probability. This is fortunate for the wave-function parametrization, since in the linear
space of square-integrable functions ($L^2$), the point evaluation is not a continuous linear functional (that is, $L^2$ is not a reproducing kernel Hilbert space). In fact, $L^2$ is an Hilbert space
of equivalence classes of functions that are equal almost everywhere (that is, up to sets with null Lebesgue measure, and null Lebesgue measure implies null probability in the context of continuous
probability distributions).
But it is not obvious how to extend the wave-function parametrization to conditioned probabilities of continuous probability distributions. A conditioned probability distribution is in itself a
probability distribution and so it admits a wave-function parametrization. However, the original probability distribution also admits a wave-function parametrization and the question we address now
is how to relate the parametrization of the conditioned probability with the parametrization of the original probability distribution.
When deriving the continuous probability distribution from the wave-function parametrization, the value of the probability distribution at a single point of the phase-space is ambiguous and thus we
cannot calculate the conditioned probability without ambiguity. Is not obvious because in the conditioned probability, we may know that an event has happened, even if the probability of such event
was null (e.g. a single point in the phase space). We could argue that the conditioned probability could be only an intermediate calculation, but this would clash with the Bayesian point of view
where there are only conditioned probabilities. Also from a classical mechanics point of view, a single point in the phase space does have a meaning. This ambiguity is also at the root of the need
for the renormalization process in Quantum Field Theory .
The conditioned probability is a particular case of a constrained system and the ambiguity described above also appears in constrained systems in general, whenever we want to define a wave-function
parametrization of a probability distribution on a subset of the phase-space defined by constraints. The constraints are from a technical point of view, a representation of an ideal by the zero
number. By an ideal we mean an ideal in the algebraic sense. Regarding the normalization of the conditional probability distribution, it is automatic since the wave-function parametrization is
defined independently from the ideal.
The correspondence between geometric spaces and commutative algebras is important in algebraic geometry. It is usually argued that the phase space in quantum mechanics corresponds to a
non-commutative algebra and thus it is a non-commutative geometric space in some sense [@connesnoncommutative]. However, after the wave-function collapse, only a commutative algebra of operators
remains (see Section 1). Thus, the phase space in quantum mechanics is a standard geometric space and the standard spectral theory (where the correspondence between geometric spaces and commutative
algebras plays a main role [@spectralhistory]) suffices.
It suffices to constrain to zero the Casimir operators of the (eventually non-commutative) Lie algebra of constraints. This imposes the constraints without the need for the constraints to be part of
the commutative algebra, only the Casimir operators are included in the commutative algebra.
Once non-determinism is taken into account, then non-commutative operators can be taken into account and the constraints are the generators of a gauge symmetry group. In case the Lie group is
infinite-dimensional, there is some ambiguity in its definition [@infinitelie; @infinitelie][@infinitelie; @infinitelie]. We consider the $C^*$-algebra [@realoperatoralgebras] generated by the
unitary operators on an Hilbert space of the form $e^{i\int d^4 x \theta(x) G(x)}$ where $G(x)$ is a constraint and $\partial_{\mu}\theta(x)$ is a square integrable function of space-time $x$ (see
also Section 20).
Note that the algebra of observable operators already conserves the constraints (i.e. it is a trivial representation of the gauge symmetry), so the Hilbert space does not need to verify the
constraints (i.e. it may be a non-trivial representation of the gauge symmetry). In fact, in many cases it would be impossible for the cyclic state of the Hilbert space to verify the constraints, as
it was noted long ago:
“So we have the situation that we cannot define accurately the vacuum state. We therefore have to work with a standard ket $|S>$ which is ill-defined. One can, however, do many calculations without
using the accurate conditions [vacuum verifies constraints] and the successes of quantum electrodynamics are obtained in this way.”
Paul Dirac (1955) [@Dirac:1955uv]
Indeed, there are some symmetries of the algebra of operators which necessarily the expectation functional cannot have (see also [@Klauder:2000gu]), since the expectation functional is a trace-class
operator (the expectation of the identity operator is 1) and its dual-space is bigger (the space of bounded operators).
For instance, consider an infinite-dimensional discrete basis $\{e_k\}$ of an Hilbert space (indexed by the integer numbers $k$) and the symmetry group generated by the transformation $e_k\to~e_{k+1}
$ (translation). There is no normalized wave-function (and thus no expectation functional) which is translation-invariant, while there is a translation-invariant algebra of bounded operators
(starting with the identity operator).
We define gauge-fixing as comprehensive whenever it crosses all possible gauge-orbits at least once. On the other hand, we define gauge-fixing as complete whenever it crosses all possible
gauge-orbits at most once, i.e. when there is no remnant gauge symmetry. The Dirac brackets require the gauge-fixing to be both comprehensive and complete, which is not possible in general due to the
Gribov ambiguity [@henneaux1992quantization]. In a non-abelian gauge-theory, the Gribov ambiguity forces us to consider a phase-space formed by fields defined on not only space but also time. This is
related to the fact that in a fibre bundle (the mathematical formulation of a classical gauge theory) the time cannot be factored out from the total space because the topology of the total space is
not a product of the base-space (time) and the fibre-space, despite that the total space is locally a product space. Thus, the Hamiltonian constraints cannot be interpreted literally, that is, as
mere constraints in a too large phase-space whose “non-physical” degrees of freedom need to be eliminated. Moreover, this picture makes little sense in infinite-dimensions: the gauge potentials can
be fully reconstructed from the algebra of gauge-invariant functions, apart from the gauge potential and its derivatives at one specific arbitrary point in space-time [@wilsonloops]; thus the number
of “non-physical” degrees of freedom would be finite at most which clearly does not match with the uncountable infinite number of constraints.
If we consider instead a commutative C*-algebra and its spectrum, such that any non-trivial gauge transformation necessarily modifies the spectrum while conserving the commutative C*-algebra
(e.g. the gauge field $A_\mu$ which is a function of space-time), then one point in the spectrum is one example of a complete non-comprehensive gauge-fixing. The gauge-fixing is non-comprehensive
because the action of the gauge group on the spectrum is not transitive. Such commutative algebra has the crucial advantage that the constraints are necessarily excluded from the algebra, so that it
can be used to construct a standard Hilbert space which is compatible with the constraints because the relevant operators of the commutative algebra are the ones commuting with the constraints,
saving us the need to eliminate the “non-physical” of degrees of freedom.
In the absence of constraints, we also consider a (particular) commutative C*-algebra: the AW*-algebra. A commutative AW*-algebra is a commutative C*-algebra whose projections form a complete Boolean
algebra. Conversely, any complete Boolean algebra is isomorphic to the projections of some commutative AW*-algebra [55]. Therefore, the notion of probability is a particular case of a functional on a
commutative C*-algebra, such notion only arises in the absence of constraints.
Thus, the Hamiltonian constraints are in fact a tool to define an (effective) probability measure for a manifold with a non-trivial topology (a principal fibre bundle for the gauge group) [@gaugewhy]
, because a phase-space of gauge fields defined globally on a 4-dimensional space-time (i.e. a fibre bundle with a trivial topology, when the base space is the Minkowski space-time) produces
well-defined expectation functionals for the gauge-invariant operators acting on a fibre bundle with a non-trivial topology [@gaugewhy] . On the other hand, setting non-abelian gauge generators to
zero in the wave-function would require to solve a non-linear partial differential equation with no obvious solution [@gaussYM; @integralYM; @globalYM; @dressYM][@gaussYM; @integralYM; @globalYM;
@dressYM][@gaussYM; @integralYM; @globalYM; @dressYM][@gaussYM; @integralYM; @globalYM; @dressYM] .
Note that it is crucial that the C*-algebra in the gauge-fixing is commutative and it is conserved by the gauge transformations. While this is not possible in the canonical quantization, it is
possible with the quantization due to time-evolution [@pedro_1442442]. Note also that since only gauge-invariant operators are allowed, we must distinguish between the concrete manifold appearing in
the phase-space and the family of manifolds (obtained from the concrete manifold through different choices of transition maps between local charts) to which the expectation values correspond.
The gauge symmetry is different from anomalies. An anomaly is a failure of a symmetry of the wave-function to be restored in the limit in which a symmetry-breaking parameter (usually introduced due
to the mathematical consistency of the theory) goes to zero. We only consider symmetries of the Hamiltonian as candidate symmetries of the wave-function, since only these are respected by the
time-evoluti hand, the constraints (which generate the gauge symmetry) cannot modify the wave-functions of the Hilbert space. Since in the case of a gauge symmetry there is no way to introduce a
symmetry-breaking parameter, we can never observe an anomaly.
The ideal (gauge generator) in the gauge mechanics system is the charge operator:
\begin{aligned} Q(t)=-\dot{p}_\lambda(t)+\pi(t)\phi(t)+\pi^*(t)\phi^*(t)\end{aligned}(25)
For consistency with General Relativity, we also impose a constraint for the observables to be translation invariant in the coordinate $t$. As it will be discussed below, the cyclic vector defining
the Hilbert space needs not be translation-invariant, just the operators corresponding to observables need to commute with the translation operator:
\begin{aligned} T(\tau)=e^{i \frac{\tau}{2} \int dt \left[p_\lambda(t)\partial_t \lambda(t)+\pi(t)\partial_t \phi(t)-i\psi^\dagger(t)\partial_t \psi(t)+\mathrm{h.c.}\right]} \label{eq:translation}\
end{aligned}tonian formalism, the constraints are from a technical point of view, a representation of an ideal by the zero number. By an ideal we mean an ideal in the algebraic sense.
We need to separate the ideal (gauge generator) from the gauge-invariant algebra. That is, not only the gauge-invariant algebra must commute with the ideal, but also the ideal cannot be included in
the gauge-invariant algebra. This is guaranteed by non-comprehensive gauge-fixing: the gauge-invariant algebra is the sub-algebra of the commutative algebra with spectrum given by the fields $\phi
(t),\phi^*(t),\lambda(t)$, such that the sub-algebra commutes with the constraints. The conjugate field $p_\lambda(t)$ or its derivative $\dot{p}_\lambda(t)$ or the gauge generator are not part of
the gauge-invariant algebra, since they do not commute with the corresponding field $\phi(t)$ which is included in the commutative algebra.
A translation-invariant time-evolution and (classical) statistical field theory
weinberg pp 204, quantum theory of fields (flavor symmetry)
While there is a mathematically rigorous definition of classical field theory [@cftmath], so far the definition of a (classical) statistical field theory [@mussardosft] is tied to the definition of a
quantum field theory [@Lang:1985nw], which involves a lattice spacing necessary to regularize and renormalize the ultraviolet divergences of the field theory. The notion of continuum limit in a
discrete lattice is that for a large enough energy scale the predictions of the theory are independent from the type of discrete regularization used [@Lang:1985nw], thus the lattice in the
regularized theory is always discrete. The regularization and renormalization are related to the decomposition of a field defined in the continuum through discrete wavelets and it is roughly the
translation of the products of fields into products of wavelet components [@battle1999wavelets] (a related approach involves a semivariogram [@spatialdata]). Such translation of products of fields
only allows polynomial Hamiltonians; in particular when using the Fock space without regularization (which is a possible way to implement a continuous tensor product [@continuoustensorp]), only
quadratic Hamiltonians are allowed (i.e. for free fields).
This excludes a rigorous definition of the classical statistical version of many classical field theories (such as General Relativity), since so far there is no reason why the Hamiltonian of a
classical field theory should be polynomial in the fields, not to mention the problems with Quantum Gravity [@Katanaev:2005xd]. This is unacceptable: for most classical field theories, the definition
of the corresponding classical statistical field theories should be straightforward, because the real-world measurements are never fully accurate.
The above is an indication that an alternative definition of Statistical Field Theory which allows the definition of non-polynomial Hamiltonians should not be too hard to find. Indeed, the essential
obstruction to an infinite-dimensional Lebesgue measure is its $\sigma$-finite property (to be the countable direct sum of finite measures) [@baker1991lebesgue; @baker2004lebesgue]
[@baker1991lebesgue; @baker2004lebesgue]. Once we drop the $\sigma$-finite property, several relatively simple candidates exist [@baker1991lebesgue; @baker2004lebesgue][@baker1991lebesgue;
@baker2004lebesgue]. In our case, we are not looking for an infinite-dimensional Lebesgue measure (no one expects the probability measure itself to be translation-invariant), but only for a
translation-invariant time-evolution of the probability measure (i.e. the time-evolution is an operator, not a real number) and thus there is no reason to expect such operator to be $\sigma$-finite
until it is evaluated against a probability measure when it becomes another probability measure—which cannot be translation-invariant.
The time-evolution for any quantum system is a (unitary) linear operator. This is only possible because the linear space is infinite-dimensional, this allows non-linear equations to be converted into
linear equations. In the case of field theory, while only free fields are allowed in Fock-spaces without renormalization [@Petz:1990gb], nothing prevents us from defining a free field over $\mathbb
{R}^n$ with the number of dimensions $n$ finite but as large as needed.
There is an important theorem about bosonic Fock-spaces, which is useful for our case [@indicator; @partitionfock; @skeide; @indicator2][@indicator; @partitionfock; @skeide; @indicator2][@indicator;
@partitionfock; @skeide; @indicator2][@indicator; @partitionfock; @skeide; @indicator2] stating that the closed linear span of exponential vectors is the bosonic Fock-space $\Gamma(L^2(\mathbb{R}^+))
$:
\begin{aled} &\biggl\{\exp(\int\limits_{0}^{+\infty} \mathop{}\!\mathrm{d}t\, (\chi_{[s_1,t_1]}+\cdot\cdot\cdot+\chi_{[s_n,t_n]})(t)a^\dagger(t))\left|0\right\rangle:\\ &0 \leq s_1 \leq t_1 \leq \
cdot\cdot\cdot s_n \leq t_n\}\end{aligned}(27)
is $\Gamma(L^2(\mathbb{R}_+))$, where $n$ is a natural number, $\chi_{[s_n,t_n]}(t)$ is the indicator function in the interval $[s_n,t_n]$, $a^\dagger(t), a(t)$ are the creation and annihilation
operators respectively and $\left|0\right\rangle$ is the vacuum state of the bosonic Fock-space. Note that the norm of the above defined exponential vector $\phi$ verifies $\left\langle 0\middle| \
phi\right\racan be extended to $\Gamma(L^2(\mathbb{R}^n))$ [@indicator] with indicators of Borel sets. Since for the fermionic Fock-space there are only two possible values for the field at each
point of $\mathbb{R}^n$, the fermionic Fock-space is also the closed linear span of exponential vectors with indicator functions. It remains to address the cartesian product of a discrete space with
$\mathbb{R}^n$: the case of a finite discrete space is equivalent to introducing a finite number of flavours of free fields and thus the resulting Hilbert space is also the closed linear span of
exponentials of indicators. The generalization to an infinite discrete space must be consistent with the finite case, in particular when selecting an arbitrarily large finite subset. Therefore, the
resulting Hilbert space is also the closed linear span of exponentials of indicators.
We conclude that for the case of a discrete space only (without the cartesian product with $\mathbb{R}^n$), the appropriate generalization of the Fock-space is not the Quantum Harmonic Oscillator,
but instead it is the closed linear span of exponentials of indicators. In particular, when an ideal selects a single point of $\mathbb{R}^n$, the corresponding Hilbert space at that point is only
2-dimensional and not the Hilbert space of a Quantum Harmonic Oscillator. Thus, the exponentials of indicators are a complete basis of the Fock-space, unlike the coherent states in general which are
overcomplete. The Quantum Harmonic Oscillator is only related to the Fock-space in the sense that in the case of $\mathbb{R}^n$, if we do a discrete wavelet transform we obtain an infinite discrete
number of Quantum Harmonic Oscillators, one Oscillator per element of the wavelet basis. Note that each discrete wavelet is necessarily non-local. Thus the widespread myth that a quantum field is an
infinite set of quantum harmonic oscillators with one oscillator at each space-time point, is severely misleading at best because it ignores the crucial fact that a point in $\mathbb{R}^n$ has null
Lebesgue measure. For the same reason, we cannot expect lattice field theory to provide a solid mathematical foundation to Quantum Field Theory beyond being a numerical approximation to a discrete
wavelet transform of the Fock space in $\mathbb{R}^n$.
Thus, the (bosonic or fermionic) Fock-space can be used as the wave-function parametrization of an arbitrary probability distribution of a classical field over space (for instance the 3D space), in
the following way. A commutative AW*-algebra is a commutative C*-algebra whose projections form a complete Boolean algebra. Conversely, any complete Boolean algebra is isomorphic to the projections
of some commutative AW*-algebra [@awalgebras]. For an Hamiltonian involving at most second-order derivatives of the fields, the projections of the commutative AW*-algebra of a (non-free) field at an
arbitrary discrete finite number $n$ of points in space are parametrized by the Fock-space over $S=\mathbb{R}^{d*(1+2*m+(d-1)*d/2)+m}$, where $d$ is the number of space dimensions and $m$ is the
number of bosonic fields (to include fermionic fields we replace $\mathbb{R}$ by the discrete set $\{0,1\}$). The projection corresponding to a proposition is:
\begin{aligned} &\pi(\phi(x_1)\cap ...\cap \phi(x_n))= \psi(\phi(x_1)\cap ...\cap \phi(x_n))\psi^\dagger(\phi(x_1)\cap ...\cap \phi(x_n))\\ &\psi(\phi(x_1)\cap ...\cap \phi(x_n))=a^\dagger(\
phi_1,x_1)..eal is defined by the exponential vector of indicator functions and the Fock-space parametrizes the space of probability distributions, such that the probability corresponding to a
maximal ideal is the modulus squared of the complex number corresponding to the maximal ideal which appears in the expansion of the wave-function in terms of exponential vectors of indicator
functions. Therefore, the maximal ideal takes care of what happens sequentially along several indexes, while the probability distribution takes care of what happens in parallel at different
elementary events.
The above is consistent with the fact that a complete physical system is also a free system. The free field associated to a free system can be made an orthogonal(fermion)/symplectic(boson) real
representation of the Poincare group depending on whether its spin is semi-integer/integer respectively, regardless of the interactions occurring within the free system [@spinstatistics; @wigner]
[@spinstatistics; @wigner]. Thus in field theory the wave-function parametrization includes a free field parametrization.
In (quantum or classical) statistical field theory, the problem we want to solve is about a probability distribution so it is about an eigenvalue problem and diagonalizing a time-evolution operator,
because the eigenfunction needs not even exist and we use ideals instead (see the previous section). On the other-hand, in classical field theory (including in numerical calculations such as the
finite element method [@sobolevfem; @loggfem][@sobolevfem; @loggfem]) it is about the fields themselves and so the solution must be part of an Hilbert space (because completeness of the space is
crucial for the existence proofs) and we need an alternative to the $L^2$ measure since the differential operator is unbounded with respect to the $L^2$ measure: such alternative is the Sobolev
Hilbert space [@sobolev].
The fact that the Hamiltonian only involves local interactions allows us to introduce an *-homomorphism where a finite number of points of the continuum space is selected. Then we can do a
wave-function parametrization, which allows for a non-deterministic (infinitesimal) time-evolution for these selected points. Moreover, the (deterministic or non-deterministic) time-evolution of this
finite number of points can be determined independently of the full probability distribution of the initial state, which may be a complex problem because it may involve an infinite-dimensional
Sobolev phase-space with some correlation between the points due to differentiability requirements [@ringstrom2009cauchy]. This allows us to know approximately the probability distribution of the
initial and final state through numerical methods for partial differential equations and regression (Gaussian process regression [@gpr] or statistical finite element method [@statfem], for instance).
The fact that we are dealing with a commutative algebra is key to allow the selection of only a finite number of points of the continuum space. This is only possible because we make use of the
wave-function parametrization only when it is convenient, in this case only after the selection of a finite number of points of the continuum space. We can do it because the wavefunction really is
just a parametrization, without a physical counterpart.
If we had assumed that there is an infinite-dimensional canonical commutation relation algebra [@Petz:1990gb] from the beginning (as in most literature about Quantum Field Theory, instead of the
commutative algebra we considered) then the *-homomorphism where a finite number of points of the continuum space is selected would not be possible. So our formalism includes the Fock-space
(i.e. free fields), but not the other way around. Therefore, the Hamiltonian is quadratic in the creation/annihilation operators and no further regularization is needed (the free field
parametrization can be considered a regularization by itself).
Moreover in the classical statistical field theory case where the time-evolution is deterministic, the wave-function parametrization is crucial to define an expectation functional and its
time-evolution which are mathematically well defined. Without the wave-function parametrization, the selection of only a finite number of points of the continuum space is much harder already for
classical field theory [@finitecft].
The cost of the free field parametrization is that we need to implement derivatives and coordinates in continuum space as an extra structure at the local level using constraints, which allows
well-defined products of fields and its derivatives at the same point in the continuum space. For an Hamiltonian which depends on the field derivatives up to second order), the constraints are
$iD_x-i\partial_x=0$ where:
\begin{aligned} % &\phi^{(0)} i \partial_x-\phi^{(1)}=0\\ % &\phi^{(1)} i \partial_x-\phi^{(2)}=0\\ % &p=p_{(0)}+i \partial_x p_{(1)}+(i\partial_x)^2 p_{(2)} % &\int d\phi dy (a^\dagger(\phi,y)iD_y a
(\phi,y)-i\partial_x=0\\ %&\int d\phi dx %(a^\dagger(\phi,x)[iD_x,H]a(\phi,x)-[i\partial_x,a^\dagger(\phi,x)[iD_x,H]a(\phi,x)])=0\\ &[p_{(j)},\phi^{(k)}]=i\delta_j^k\\ &[p_{(j)},\phi^{(2)}(x)]=i\
delta_j^2\\ &[\phi^{(j)},p_{(0)}(x)]=-i\delta_0^j\\ &p(x)=p_{(0)}(x)\\ &\phi(x)=\phi^{(0)}+\phi^{(1)}x+\frac{1}{2}\phi^{(2)}(x)x^2\\ &D_x=[\partial_x,p_{(0)}(x)]\phi^{(0)}+\sum_{j=1}^{2} p_{(j-1)}\
phi^{(j)}+p_{(2)}[\partial_x,\phi^{(2)}(x)]\\ &[iD_x-i\partial_x, H]=0\end{aligned}(29)
Crucially, due to the Bianchi identity and that the Hamiltonian is translation invariant, we have:
\begin{aligned} &\int d\phi dx a^\dagger(\phi,x)[[iD_x,\phi(x)],H]a(\phi,x)=-\int d\phi dx a^\dagger(\phi,x)[iDx,[H,\phi(x)]]a(\phi,x)\\ &\int d\phi dx a^\dagger(\phi,x)[[iD_x,p(x)],H]a(\phi,x)=-\int
d\phi dx a^\dagger(\phi,x)[iDx,[H,p(x)]]a(\phi,x) \end{aligned} | {"url":"https://timepiece.pubpub.org/pub/euler/release/1","timestamp":"2024-11-15T04:41:35Z","content_type":"text/html","content_length":"1049720","record_id":"<urn:uuid:659f4b89-0667-4bed-a65c-c8b4b8f3ad5a>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00537.warc.gz"} |
How many prime factors does the 1801st Fibonacci number have?
This question is managed and resolved by Manifold.
@nanob0nus Indeed, the prime number theorem tells us that the density of primes goes like 1/log(n), and knth fibonacci number is divisible by the nth, so for any x, almost all Fibonacci numbers will
have at least x factors. | {"url":"https://manifold.markets/BoltonBailey/how-many-prime-factors-does-the-180","timestamp":"2024-11-11T21:26:04Z","content_type":"text/html","content_length":"121453","record_id":"<urn:uuid:e23eafdd-47e7-4471-adac-c4fe72c741e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00706.warc.gz"} |
The PIMS Postdoctoral Fellow Seminar: Konstantinos Mamis
A simple stochastic model for cell population dynamics in colonic crypts
The questions of how healthy colonic crypts maintain their size under the rapid cell turnover in intestinal epithelium, and how homeostasis is disrupted by driver mutations, are central to
understanding colorectal tumorigenesis. We propose a three-type stochastic branching process, which accounts for stem, transit-amplifying (TA) and fully differentiated (FD) cells, to model the
dynamics of cell populations residing in colonic crypts. Our model is simple in its formulation, allowing us to estimate all but one of the model parameters from the literature. Fitting the single
remaining parameter, we find that model results agree well with data from healthy human colonic crypts, capturing the considerable variance in population sizes observed experimentally. Importantly,
our model predicts a steady-state population in healthy colonic crypts for relevant parameter values. We show that APC and KRAS mutations, the most significant early alterations leading to colorectal
cancer, result in increased steady-state populations in mutated crypts, in agreement with experimental results. Finally, our model predicts a simple condition for unbounded growth of cells in a
crypt, corresponding to colorectal malignancy. This is predicted to occur when the division rate of TA cells exceeds their differentiation rate, with implications for therapeutic cancer prevention
Speaker biography: Konstantinos obtained his PhD in Stochastic Dynamics from the National Technical University of Athens, Greece, in 2020. As a postdoctoral researcher, he worked for a year in the
North Carolina State University, Math Department on stochastic compartmental models in epidemiology. In his current postdoctoral appointment, he is a member of Prof. Ivana Bozic's group in the
University of Washington, Applied Math Department focusing on the stochastic modeling of colorectal cancer initiation.
Additional Information
This seminar takes places across multiple time zones: 9:30 AM Pacific/ 10:30 AM Mountain / 11:30 AM Central
Register via Zoom to receive the link (and reminders) for this event and the rest of the series.
See past seminar recordings on MathTube.
Konstantinos Mamis
Event Type
Scientific, Seminar | {"url":"https://www.pims.math.ca/events/240214-tppfskm","timestamp":"2024-11-11T03:03:33Z","content_type":"text/html","content_length":"422719","record_id":"<urn:uuid:636a4363-b74b-4659-acea-840c6814a9f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00420.warc.gz"} |
Book Chats on Retailing.
Logo: A Beautiful Miniature Book
Book Chats on Retailing.
[Now that Covid Restrictions have Eased].
Here is an idea for you. Let's say your dream is to somehow work surrounded by books...but lack "Hands On" experience. Maybe you are nearing retirement perhaps, or fed up with the office job, or you
live in New York and have a week's vacation coming up, or... you just fancy a change in lifestyle? Another possibility is you have some experience, but want to move up from market trading and boot
fairs? Well..our idea is this. It's simple. Why not run a bookshop (our bookshop) in Lyme Regis for a day, a weekend, or even a week..? We will be on hand to help 24/7.
You can even live in our B & B premises above the shop should you wish.
In normal times, we are constantly told ours is a cozy and welcoming shop. Friendly, Warm and Comfortable. A shop that Jane Austen, Dickens, or even Darwin would have enjoyed. We will be twenty-three
years old this year and we can be found in central Lyme Regis, in the Town Square, just opposite the Town Clock. You can see us in the images below. We are on four floors. The uppermost two (see
right hand, and centre bottom, images below) are our Booklovers B & B guest rooms. If you wish, when restrictions ease, you can book to stay here on a daily or weekly basis (£30/pp/pn). The street
level and basement are the shop proper and are open to the public seven days a week. Should this idea appeal to you, contact us.
We can be reached on: books@sanctuarybookshop.co.uk.
Or by telephone on 01297-445815 (evenings 443653). Ask for Bob.
This page last updated: October 31st 2022.
What a curious place the Internet is. While shopping, we came across this below. Even a modest Paperback at 200 grams is £1.26 second class post.
Is that Amazon Prime tucked away on the left in tiny writing?
What a wizz of an idea...selling books at a loss. How come we never thought of that!
Note: The two "Chats" featured below were posted a year or so ago..the main idea being to amuse those of a scientific bent, but also to give the reader some concrete idea of the financial problems
facing a High Street trader. The idea was to step out of the box, or your comfort zone, and put a few numbers in showing the day to day realities. These can be quite challenging, as witnessed by the
number of bookshops that have had to close in the past several years. If maths is not your thing then just ignore. The most realistic suggestion we can make is for you to give it a go, i.e., have a
go at running our bookshop yourself for a day or so. It is maybe here that we can be of most practical help to you.
We believe towns still need individal bookshops run by passionate booklovers.
Real bookshops, not cloned and faked up retail chains like Amazon, Waterstones, Smiths and Oxfam.
"So you want to have a bookshop..?". (Part 1).
“He can have what he wishes, who wishes just enough”. (P. Syrus. Epigram 809).
Having been in second-hand and antiquarian bookselling for thirty-six years, and with a High Street retail shop over twenty years, and survived (we opened up in Lyme on August Bank Holiday
Monday1997), it may be of interest for some readers to look back over our own High Street venture and put in some figures.......
Before we get into this, any would be book trader is most welcome at 65 Broad St anytime, both to hear of our experiences and on how we might help them. All journeys start with a first step. If you
mention your hopes and ambitions, then a tea or coffee or glass of wine is definitely on the cards.
With Amazon now opening bricks and mortar shops (yes....curious ..isn't it? Click on the "Book Chats on Book Chains" tab, screen left green column, for the latest), and with Waterstones masquerading
as Single Shop Independents, we find one so called charity chain boasting 130 dedicated bookshops...and so the odds appear stacked against us. [We are all not meant to notice that top Charity Shop
career executives pay themselves £200,000/year plus....and then have the nerve to claim the moral high ground. See link below*.]
So, one just tries to speak up for the little guy, for transparency, and, to welcome all newcomers to the profession. In short, anyone who is serious and aspires to a career in books and
It's never too young to start and, like a good wine, it gets better as one gets older. George below, our senior colleague in Colyton, can testify to that!
"Rem Tene: Verba Sequentur". Cato.
So, let's get started......(GCSE maths should be more than enough. Non maths people...abandon this page here..!)...and, if the image above makes you feel faint...just hang in there.
First, a few definitions.....
Let Q equal the total Annual Turnover in pounds (the shop “take”).
Now, let E be the Expenses (the Overheads)....rates, electricity, gas, water, telephone, ISP, casual labour, car, restoration, advertising, accountancy, bank charges, shop maintenance, sundries....
(everything one would claim that is justified when calculating one's Tax Allowance. Also, in our case: no staff).
Let T be the Profit (take home money, before tax).
Let P be the Purchases for the year (the stock acquisitions).
Then, just to break even, it is clear that....
Q-E-P = 0
(i.e., no profit, but no loss).
N.B. In our case, Q, E, and P are known (How? Because we keep daily, weekly, and monthly records, plus all associated paperwork. This is for tax purposes. Also we keep Moving Annual Totals, MAT's, on
Q, E, & P. This has proved indispensible, but you may need to look that up).
Now here is the key.... in equilibrium, i.e., the buying and the selling of the stock to be in balance and stable (a state we only reached in 2000, after an initial three years trading), then
P = Q/m
[The test for this, is that the figure calculated on the annual stock take remains approximately constant year on year.....]
Also, here “m” is the the “markup”, i.e., the factor, or average multiplier on the trade purchase price, sufficient to yield a trading margin (more of this later in the next Chat).
Q-E-Q/m = 0
...just to break even.
...to take home T = £X a month clear (profit: before tax: all bills paid), one needs to ensure that
(1-1/m)Q – E = 12X
Now Q = 313 q (assuming a six day trading week), where q is the average daily receipt (the daily take, averaged over the year).
313(1-1/m) q – E = 12X
Or, rearranging....
q = m(12X + E)/313(m-1)
This equation yields a fascinating insight into the (gruelling) realities of High Street book trading, and is definitely worthy of further discussion (we will keep that for a later Chat. See below.).
We also note that we mean here second-hand and antiquarian bookselling. The selling of new books by Independents operates under completely different constraints....(aside: Independent bookshop
numbers have almost halved in the past 12 years, in 2005 there were 1,535, while in 2017 just 867. However, the total number of Independent bookshops that are members of The Booksellers Association
rose from 868 to 883 in 2018).
For the time being, let us take two examples close to home that illustrate the problem...
1. Just to break even (no take home income), then X = 0, assuming m = 3 (more on this later), and E= £30,000/year (reality), readers can check and see that they will need a turnover (averaged over
the year), of q = £145/day approx.
2. However, to take home £1000/month clear, with m = 3, again taking E = £30,000/year (reality), then one will need an average daily take of q = £200 approximately, every trading day of the year.
These are gruelling figures for a sole trader, and explain why so many fail (or in fairness, are unable to start...). There is no avoiding the fact that High Street Trading is extremely Darwinian. It
ruthlessly winnows out the weak, and is certainly not for the faint hearted. In the words of Paul Minet*....the best security one can hope for, is: an independent income, a non-salaried partner
(wife, husband, colleague...), and the ownership of one's own freehold premises!
So, not for us Charity Bookshop retail cloud cuckoo land: reduced rates, free donated stock, unpaid volunteer staff, fat cat executive salaries milked from your free donations*.......
The reality of these figures, just to break even, is: £20/hour for a 10.30 am to 5.30 pm day, or ten £2 paperbacks sales hour (it is important to emphasise that these are figures that are averaged
over the year). In fact, August in Lyme can see us with queues all day! By way of contrast in February..most retailers here in Lyme will find it hardly worth opening....maybe just a few pound's worth
of sales on an average weekday!
Also, one might think in the second illustration, that a free disposable income of £1000/month seems quite modest, but remember, this is after all bills have been paid!
Finally, the hidden assumption in all this is that good saleable second-hand book stock is somehow flowing into your shop, at attractive prices sufficient to allow a decent margin.....
Ah-ha! Could it be the art of second-hand bookselling is in the buying? (But more on that later)...
* Paul Minet. "Bookdealing for Profit". Richard Joseph Publishing. 2000.
* Fact Checker. Executive pay for Charity Workers.
Isn't it fascinating to find in that article that the faithful Church of England Commissioners each pay themselves £330,000 to £340,000/pa as Charity Workers! One can't help noticing that, as
devotees of their particular God, he seems to take a very special interest in looking after them. It must be all that praying in fancy dress, praying for guidance to the bank perhaps? One can't help
recalling Patrick Moynihan's observation "Everyone is entitled to their own opinions, but not to their own facts".
"So you want to have a bookshop..?". (Part 2).
"Profit badly made can be the same as a loss". Hesiod. c 700 b.c.
These two Book Chats last updated February 21st 2020.
We left the Issue above with an interesting relationship for the required average daily turnover q to yield a given monthly income X before tax, where T, the annual profit = 12X.
This was:
q = m(12X + E)/313(m-1)
where m is the average markup (ratio shelf price to buy in price).
E is the “Overhead” (rates, electricity, gas, water, telephone, ISP, casual labour, car, restoration, advertising, accountancy, bank charges, shop maintenance, sundries....(everything one would claim
that is justified when calculation one's Tax Allowance. Also, in our case: no staff).
The variable m is an interesting little fellow....
The following is a “thought experiment” ….much loved of Albert Einstein, who regularly used the term gedankenexperiment, (from the German “gedanken” for thought).
Supposing we let the average value of m decrease.....3.0, 2.5, 2.0. 1.5.....
It is beyond dispute that as m tends to 1, any profit T will tend to zero. To sell at the "buying in" price is clearly nonsensical. Any value of overhead will result in an immediate trading loss and
the rapid closure of your enterprise.
Now let us think through the consequences, should m be allowed to go the other way, i.e., tend to the very large....3.0, 3.5, 4.0, 4.5, 5.0, Keep in mind we are talking average values of m over the
whole stock, and, rather more subtly, a shop stock that is in equilibrium (see our earlier discussion on this topic).
I hope it is obvious to the reader that if you are going to mark up an Agatha Christie paperback, say, bought in at 50p, and put on the shelf at £5.00, it is going to stay there indefinately.
Well customers are not fools and are well versed at price comparison.
Goodwill and hence footfall will move to nearby charity shops, your colleagues trading in the vicinity, and, worse of all, to the Internet.
The point here is that sales, and hence T, any profit, will again remorselessly tend to zero, along with the footfall.
If I can just carry you mathematically one more step, it follows that there is a maximum in the value of T, somewhere between m equals one and m equals five (say).
The mathematically inclined among you will instantly spot that with T (profit) as Ordinate (vertical axis), plotted against m as Abscissa (horizontal axis), T must show a peak somewhere between m = 1
and m = 5.
I just need to rearrange for T = f(m), and compute dT/dm = 0. This is just bog standard calculus to find the maximum of a function, in this case, the optimum value of m to give the maximum profit T.
Although this value of m cannot be computed analytically, this thought experiment has its use.
Because it leads us to an obvious actual experiment.
Namely, to vary m over time on a selected range of steady selling items, keep records of sales versus purchases and hence find the value of m that is optimal for your location and your stock.
Far fetched..do I hear you say...?
Far from it.
It took us several years to run the experiment.
This was made up as follows...four years to come into equilibrium from start-up in 1997, and a further six years to get reliable figures.
So, by 2008 we had very reliable data.
Perhaps the moral of all this, is...
1. Have a five to ten year business plan.
2. Reconcile yourself to the long haul. Good Data and especially local Goodwill (fair pricing) is invaluable.
3. Maybe talk to someone with long experience in the field and who has survived. Paul Minet (q.v. Book Chat above) was very good at this, and generous to a fault, and for that we are personally in
his debt.
On that note of Complete Goodwill, may we bid you Good Evening...and Thank You for reading.
Other pages:
This is the text-only version of this page. Click here to see this page with graphics.
Edit this page | Manage website
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Bookworm's Breakfast :: Transum Newsletter
Bookworm's Breakfast
Thursday 1st October 2015
In the days before Wikipedia and Google we might refer to an encyclopedia to find answers to our questions. The puzzle for this newsletter is based on a set of ten volumes of an encyclopedia on a
bookshelf arranged in order with volume one on the left and volume ten on the right.
A bookworm eats through from the front cover of volume one to the back cover of volume ten for breakfast one day. What is the length of the bookworm’s meal if each encyclopedia is 5cm thick (the
pages are 4cm and each cover is 0.5 cm thick)?
The answer is at the end of this newsletter but, be warned, it is not the obvious answer.
This puzzle is a version of the February 2nd Starter of the Day which presents a random number of encyclopedias and randomly generated measurements for the pages and covers. It provides an
opportunity for pupils to engage in some decimal addition and multiplication before being surprised by the actual answer.
Now the pages on the Transum website should be a little easier to find as the search facility has been upgraded. Now when you search for a term you get two sets of results. The first is directly from
the Transum database and is a search on page titles and descriptions. Lower down the page you will see the Google search results which include snippets of text found on the pages.
You may like to try out the new search feature to find this month’s new additions. Firstly the Car Park Puzzle challenges you to get your car out of the very crowded car park by moving other cars
forwards or backwards. It is the Transum version of a puzzle that has been available in different formats for many years but the real challenge for students is to devise a level 6 puzzle that is
possible but requires more moves than level 5. The way the students record moves and consider the advantages of working backwards (doing and undoing) give this challenge a strong mathematical
Polybragging is another new activity that is also based on an idea that has been around for a long time. This is a game for two or more players. Each played needs a tablet, computer or smartphone
with the page loaded.
If you have ever played a card game called Top Trumps you will know the main idea of this game already. Each player is given a shape that the computer selects at random. The players each choose a
shape property and whichever player has the highest value for that property wins a point.
The properties available include the size of the largest interior angle, number of pairs of parallel lines, number of lines of symmetry and the order of rotational symmetry.
Hopefully, by playing the game, pupils will develop more familiarity and a greater knowledge of the properties of polygons.
Other new additions to the site include a Dice and Spinners page to use if you can’t find the real things and a Reaction Time activity which collects data about how quickly we recognise even compared
to odd numbers.
Finally some more traditional Maths exercises have been added. These include Multi-step Problems and Decimal Times. These exercises are self-marking, printable and every pupil gets a slightly
different version thanks to the in-built random number generator.
Thanks to everyone who has added comments and suggestions to the site this month. Your input keeps the site alive. One comment waiting for your opinion is that made by Leslie Jackson on the 16th
December Starter page. Do you think powers of two are 2, 4, 8 .. or do you think they are 4, 9, 16 …?
Finally the answer to this month's puzzle. The answer is not 50cm surprisingly. If you picture the ten volumes arranged on the shelf you will notice that the front cover of volume one is actually on
the right, next to volume two! So if the bookworm starts by eating through that cover it has missed the pages and back cover of volume one altogether. Similarly the back cover of volume ten is on the
left so the bookworm stops before eating the pages and front cover of volume ten.
The correct answer is 50cm – 2(4cm + 0.5cm) = 41cm
Enjoy October and don’t miss the Halloween Starter at the very end of the month.
PS. What do you get if you divide the circumference of a Halloween lantern by its diameter?
A: Pumpkin Pi
Do you have any comments? It is always useful to receive feedback on this newsletter and the resources on this website so that they can be made even more useful for those learning Mathematics
anywhere in the world. Click here to enter your comments. | {"url":"https://transum.org/Newsletter/?p=145","timestamp":"2024-11-14T18:51:34Z","content_type":"text/html","content_length":"20043","record_id":"<urn:uuid:d2d50728-9a21-40e1-bc3b-3c457684bcb4>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00383.warc.gz"} |
Preliminary Examination
in Mathematical Computer Science
Algorithms and Complexity Cluster
This is one of four possible preliminary exams in computer science. Past exams should be available from the Mathematics Library or the office of the Director of Graduate Studies or the MCS web home
page. Questions are drawn from the following areas and their prerequisites.
Algorithms Portion of Prelim Syllabus
Courses in Algorithms:
The relevant course for the Algorithms part of the Prelim is
• MCS 501 Computer Algorithms II
Topics in Algorithms:
1. Algorithm design techniques
Divide and conquer
Dynamic programming
Greedy method
2. Algorithm analysis
Expected-case analysis
Amortized analysis
3. Sorting and merging algorithms
Lower bounds for sorting and merging
Straight insertion sort
Merging and mergesort
Linear-time methods
4. Selection algorithms
Selection in linear time (expected case)
Selection in linear time (worst case)
5. Searching algorithms
Priority queues
Binary heaps
Binomial heaps
Hash tables (open address and chained)
General dynamic lists
Binary search trees
Red-black trees
Skip lists
Splay trees
6. Data compression algorithms
Huffman compression
Lempel-Ziv compression
7. Number-theoretic algorithms
RSA data encryption
Primality testing
8. String matching algorithms
String matching with finite automata
The Knuth-Morris-Pratt algorithm
The Boyer-Moore algorithm
9. The FFT and applications
The FFT
Polynomial multiplication
10. NP completeness
Polynomial-time reductions
Cooke's theorem
References in Algorithms:
1. Thomas H. Cormen, Charles E. Leiserson, and Ronald Rivest, "Introduction to Algorithms", The MIT Press, 1990.
2. Sara Baase, "Computer Algorithms: Introduction to Design and Analysis, Second Edition", Addison-Wesley, 1988.
Complexity Theory Portion of Prelim Syllabus
Courses in Complexity Theory:
The relevant courses for the Complexity Theory part of this Prelim are
1. MCS 541 Computational Complexity
2. MCS 542 Theory of Computation II
Topics in Complexity Theory:
Turing machines
Modifications of Turing machines
Church's thesis
Recursive and recursively enumerable languages
Universal Turing machine, halting problem
Rice's theorem
Post Correspondance problem
Undecidable problems for grammars
Time and space complexity
Linear speedup theorem
Reduction of the number of tapes
Gap theorem
Hierarchy theorems
Nondeterministic machines
Savitch's theorem
P and NP
NP-complete problems
References in Complexity Theory:
1. J.E.Hopcroft, J.D.Ullman: Introduction to Automata Theory, Languages and Computation , Addison-Wesley, 1979.
Chapters 7,8,12,13.
2. H.R.Lewis, C.H.Papadimitriou: Elements of the Theory of Computation, Prentice-Hall, 1981.
Chapters 4,5,6,7.
3. C.H.Papadimitriou: Computational Complexity, Addison-Wesley, 1994.
Chapters 2,3,7,8,9.
4. M.Sipser: Introduction th the Theory of Computation, PWS Publ. Comp., 1997.
Click to Go To Mathematical Computer Science Preliminary Examination Library Page
Web Source: http://www.math.uic.edu/~hanson/AlgorComplexPrelimSyllabus.html
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Eve Stenson, student of plasma physics and the world at large
“It’s a little tricky to take a dinosaur’s temperature,” I’m envisioning a Jurassic Park veterinarian explaining to a visitor. Meanwhile, a scene unfolds before them in which one grad student
distracts the Brachiosaurus with some sort of tasty vegetal treat while another tries to insert and read a thermometer without getting stepped on.
In the real world, where the only sauropods to be found have been dead for more than 100 million years, taking their temperatures is even trickier — but, amazingly, possible!
Caltech postdoc Rob Eagle, professor John Eiler, and their collaborators published a Science paper last year about their technique for doing so. I was surprised it didn’t seem to get much media
coverage at the time, because it’s damn impressive (both from a “triumphs of human curiosity” and a “wow, they did a lot of work” perspective).
There’s a lot of cool science out there . . .
. . . and I want to do my own, small part to help it become a bigger part of people’s lives and conversations.
How do I intend to do that? Right now, I’m envisioning mainly trying to help disseminate and explain interesting new research — besides the handful of Science and Nature articles that get widespread
press every year.
And I’ll probably also share the whimsical science/math calculations I occasionally do for my own entertainment, in hopes that they may entertain others as well. (These include such pressing
questions such as, “Is arranging the cookie dough balls in a triangular lattice ALWAYS more efficient than using a square lattice? Or can the size of the cookie sheet change that?” and “When jumping
off a 12-foot platform into a lake, should one try to do two back flips on the way down, or just one?”)
Other than that, I think this is one of those situations where there are many possible outcomes, and you just have to do the experiment and see how it goes!
Want to facilitate the flow of pedestrians or particles through an opening? Add an obstacle.
Grain being emptied from a silo, people leaving a room, and cars entering a construction zone all have something in common: they tend to spontaneously clog.
How and why these clogs develop is important to a wide range of applications – from industrial processing to a building’s fire safety. Recent results from a team of researchers at the Universidad de
Navarra in Spain shed new light on what factors do and don’t contribute to clogging and, hence, how to prevent it. In particular, they’ve shown that an obstacle at the right distance from the exit
decreases the probability of a clog by 99 percent.
Cookie calculations
Every time I bake cookies, I wonder if I’m making the most of the space on the cookie sheet. I finally decided to stop wondering and figure it out.
Experimental methods
My investigation involved circular cookies of two different sizes:
Two examples of large cookies and three examples of small cookies. There was some standard deviation in the size of each group, but it wasn’t too bad.
They were arranged on cookie sheets of two different sizes:
straight. It’s just the camera that makes them look curved.
Now, the most common way to arrange cookies is in equally spaced rows and columns. This is called a square lattice; when you draw a line from the center of a cookie to each of its four “nearest
neighbors,” you end up with a pattern of squares.
Another option is, instead of putting the cookies in row two directly under the cookies in row one, to arrange them in the “gaps” between the cookies on the first row. If you do this, you get a
triangular lattice, also called a hexagonal lattice; when you draw a line from the center of a cookie to each of its six “nearest neighbors,” you get a pattern of triangles.
Cookie dough arranged in a square lattice (OK, so this one is a slightly rectangular lattice, but you get the idea) versus a triangular lattice.
As you can see, the cookies are more densely packed when they’re in a triangular lattice, even though they’re the same distance away from the cookies closest to them. If you work out the geometry
problem, it turns out that cookies arranged in a triangular lattice take up 86.6 percent as much space as cookies in a square lattice. In other words, you can fit 15 percent more cookies into the
same amount of area.
Well, problem solved! Triangular lattice it is.
But wait! What if your cookie sheet isn’t wide enough for an offset second row? Is the triangular lattice still the way to go? Is it still worth it to get the rows closer together, when it means
every other row is a cookie short?
To find the answer to these questions, I first tested the four different set-ups in my kitchen (small cookies on a small cookie sheet, small cookies on a large cookie sheet, large cookies on a small
cookie sheet, and large cookies on a large cookie sheet). Then I used equations and graphs to generalize the answers. | {"url":"https://eveofdiscovery.com/index.php/blog/page/2/","timestamp":"2024-11-03T11:51:23Z","content_type":"text/html","content_length":"40349","record_id":"<urn:uuid:d0193d49-4847-43a0-b0ba-129cb748632b>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00790.warc.gz"} |
Calculating Speed from Rpm and Tire Circumference - GEGCalculators
Calculating Speed from Rpm and Tire Circumference
To calculate speed from RPM and tire circumference, use the formula: Speed (mph) = (RPM * Tire Circumference) / (60 * 63360), where RPM is the revolutions per minute and Tire Circumference is the
distance around the tire in inches.
Speed Calculator
1. How do you find speed with circumference? The formula to find speed (v) using circumference (C) is: v = C / t, where t is the time taken to complete one revolution.
2. What is the formula for speed from RPM? The formula to calculate speed (v) from RPM is: v = (RPM * C) / 60, where C is the circumference of the rotating object.
3. How do you calculate wheel speed from engine RPM? Wheel speed can be calculated using the following formula: Wheel Speed = (Engine RPM * Gear Ratio * 60) / (Axle Ratio * Tires Per Axle).
4. What is the formula for RPM using diameter? RPM can be calculated from diameter (d) using the formula: RPM = (v * 60) / (π * d), where v is the speed.
5. What is the formula for calculating speed? Speed (v) can be calculated using the formula: v = d / t, where d is the distance traveled and t is the time taken.
6. How do you calculate the velocity of an object moved around a circle? The velocity of an object moving around a circle is calculated using the formula: velocity = circumference / time, or v = (2
* π * r) / t, where r is the radius.
7. How do you calculate speed with RPM and diameter? Speed (v) can be calculated from RPM and diameter using the formula: v = (RPM * π * d) / 60, where d is the diameter of the rotating object.
8. Can you calculate mph from RPM? Yes, you can calculate mph from RPM using the appropriate conversion factors. The formula is: mph = (RPM * C) / (60 * 63360), where C is the circumference in
9. What is the relationship between RPM and speed? RPM and speed are related through the circumference of the rotating object. Higher RPM values generally correspond to higher speeds if the diameter
or circumference remains constant.
10. How do you calculate mph with RPM and wheel size? You can calculate mph from RPM and wheel size using the formula mentioned in question 8.
11. How do you calculate speed based on RPM and gear ratio? Speed can be calculated from RPM and gear ratio using the formula: Speed = (RPM * Tire Diameter * π) / (Gear Ratio * 12 * 5280), where Tire
Diameter is in inches.
12. How do you determine the speed of a tire? The speed of a tire can be determined using the formula: Speed = (RPM * Tire Circumference) / 63360.
13. What is the relationship between diameter and RPM? There is an inverse relationship between diameter and RPM for a constant speed. If the diameter increases, RPM decreases to maintain the same
14. How do you convert RPM to radius? RPM cannot be directly converted to radius as they are different units. RPM is a measure of rotational speed, while radius is a measure of length.
15. What is the formula for surface speed of a wheel? Surface speed (v) of a wheel can be calculated using: v = RPM * π * d, where d is the diameter of the wheel.
16. How do you calculate speed and velocity? Speed is the magnitude of velocity. Velocity includes both speed and direction. Mathematically, velocity (v) is calculated using: v = d / t, where d is
displacement and t is time.
17. How do you find speed with radius? Speed (v) can be calculated using the formula: v = (2 * π * r * RPM) / 60, where r is the radius of the rotating object.
18. What is the formula for velocity with radius? The formula for velocity (v) using radius (r) is: v = (2 * π * r * RPM) / 60.
19. What is the formula for velocity using radius? The formula for velocity (v) using radius (r) is the same as the one mentioned in the previous answer.
20. What is circumferential speed? Circumferential speed is the speed of an object at the edge or circumference of a rotating circle. It’s the linear distance covered per unit of time.
21. How fast is 3000 RPM in mph? To convert RPM to mph, you would need to know the diameter or circumference of the rotating object. You can use the formula mentioned in question 8.
22. How many mph is 70 RPM? This question can’t be answered without knowing the diameter or circumference of the rotating object.
23. How many RPM is 50 mph? This question can’t be answered without knowing the diameter or circumference of the rotating object.
24. How do you match RPM with road speed? RPM and road speed are matched through the vehicle’s gearing and tire size. Different gear ratios and tire sizes will result in different RPM values for a
given road speed.
25. How fast is 7000 RPM? The speed corresponding to 7000 RPM depends on the diameter or circumference of the rotating object. You can calculate it using the appropriate formula.
26. Is RPM directly proportional to flow rate? RPM is not directly proportional to flow rate. RPM refers to the rotational speed, while flow rate typically refers to the volume of fluid passing
through a system per unit of time.
27. Does wheel size determine speed? Wheel size (diameter) can influence speed indirectly. Larger wheels can cover more ground in one revolution, potentially leading to higher speeds at the same RPM
compared to smaller wheels.
28. How does wheel size affect RPM? Larger wheel sizes generally result in lower RPM for the same speed. This is because the circumference is larger, so the same speed can be achieved with fewer
29. How fast is 1000 RPM in mph? To convert 1000 RPM to mph, you would need to know the diameter or circumference of the rotating object. Use the formula mentioned in question 8.
30. How do you calculate the speed of a driven gear? The speed of a driven gear is determined by the size of the driving gear, the gear ratio, and the rotation speed (RPM) of the driving gear.
31. What is the formula for speed ratio in gears? The speed ratio in gears is determined by the ratio of the number of teeth on the driven gear to the number of teeth on the driving gear.
32. How do you calculate the speed of a gear rotation? The speed of a gear rotation depends on the number of teeth on the gear, the gear ratio, and the rotational speed (RPM) of the gear it’s meshing
33. How do you calculate speed with bigger tires? Speed with bigger tires can be calculated using the formula: Speed = (RPM * New Tire Diameter) / (Old Tire Diameter).
34. How do you find linear speed with radius and RPM? Linear speed (v) can be calculated using the formula: v = (2 * π * r * RPM) / 60, where r is the radius.
35. Is RPM dependent on radius? Yes, RPM can be dependent on radius. The linear speed of a rotating object (which affects RPM) is influenced by the radius. Larger radii can lead to higher linear
speeds at the same RPM.
36. How to explain the relationship between diameter and circumference? Diameter is the measure of the distance across a circle through its center. Circumference is the distance around the outer edge
of the circle. They are related through the formula: Circumference = π * Diameter.
37. What is the formula for angular speed with radius and RPM? Angular speed (ω) can be calculated using the formula: ω = RPM * (2 * π / 60), where RPM is the rotational speed in revolutions per
38. How do you calculate distance with RPM? Distance traveled (d) can be calculated using the formula: d = (RPM * Circumference) / 60, where Circumference is the distance traveled in one revolution.
39. What is the RPM of a circle? RPM (Revolutions Per Minute) refers to how many complete rotations a circle makes in one minute. For example, if a circle completes 3 rotations in one minute, its RPM
is 3.
40. What are the 3 formulas for velocity? Three formulas for velocity are:
□ v = d / t (distance divided by time)
□ v = Δx / Δt (change in displacement divided by change in time)
□ v = dx/dt (instantaneous velocity as the derivative of displacement with respect to time)
41. What is the formula for acceleration of speed? The formula for acceleration (a) is: a = (Δv) / t, where Δv is the change in velocity and t is the time taken for the change.
42. Is velocity same as speed? No, velocity and speed are not the same. Speed refers to the magnitude of motion, while velocity includes both the magnitude and direction of motion.
43. What is the relationship between speed and radius? The relationship between speed and radius is that larger radii generally result in higher speeds at the same RPM, while smaller radii result in
lower speeds.
44. Does speed change with radius? Yes, speed can change with radius. An increase in radius, while maintaining the same RPM, will lead to an increase in speed, and vice versa.
45. How do you find velocity with speed and radius? Velocity can be the same as speed if the motion is along a straight line. If you’re referring to circular motion, velocity includes both speed and
direction and can be calculated using angular velocity (ω) and radius (r): v = ω * r.
GEG Calculators is a comprehensive online platform that offers a wide range of calculators to cater to various needs. With over 300 calculators covering finance, health, science, mathematics, and
more, GEG Calculators provides users with accurate and convenient tools for everyday calculations. The website’s user-friendly interface ensures easy navigation and accessibility, making it suitable
for people from all walks of life. Whether it’s financial planning, health assessments, or educational purposes, GEG Calculators has a calculator to suit every requirement. With its reliable and
up-to-date calculations, GEG Calculators has become a go-to resource for individuals, professionals, and students seeking quick and precise results for their calculations.
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The water-cement ratio for 3000, 2000, 4000, 5000 and 6000 psi concrete - Civil Sir
The water-cement ratio for 3000, 2000, 4000, 5000 and 6000 psi concrete
The water-cement ratio for different grade of concrete varies depending on factors such as the desired strength, the type of cement, aggregates used, sand and environmental conditions like humidity
and temperature. Typically for normal strength concrete, a water-cement ratio ranging from 0.4 to 0.6 is commonly used.
The water-cement ratio is a crucial factor in concrete mix design. It refers to the ratio of the mass of water to the mass of cement used in a concrete mix. It’s a critical parameter because it
affects the workability, strength, durability, and permeability of concrete.
Generally, a lower water-cement ratio leads to higher strength and durability but may reduce workability. Conversely, a higher water-cement ratio increases workability but can decrease strength and
durability. Balancing these factors is key to achieving the desired properties in concrete mixes.
In concrete terminology, concrete strength is measured in psi, which refers to the compressive strength of concrete measured in pounds per square inch (psi). Generally, the strength of concrete is
2000 psi, 2500 psi, 3000 psi, 3500 psi, 4000 psi, 4500 psi, 5000 psi and 6000 psi.
There are two types of concrete Non air-entrained concrete and air-entrained concrete which differ primarily in their resistance to freeze-thaw cycles and their ability to withstand exposure to harsh
weather conditions. Non air-entrained concrete lacks microscopic air bubbles deliberately introduced during mixing, while air-entrained concrete contains tiny air bubbles distributed throughout the
However, a general guideline for various concrete strengths and their water-cement ratio is as follows:
1. A typical water-cement ratio for 2000 psi concrete might range from 0.55 to 0.65.
2. A typical water-cement ratio for 3000 psi concrete might range from 0.50 to 0.60.
3. A typical water-cement ratio for 4000 psi concrete might range from 0.4 to 0.5.
4. A typical water-cement ratio for 5000 psi concrete might range from 0.35 to 0.45.
5. A typical water-cement ratio for 6000 psi concrete might range from 0.3 to 0.4.
The water-cement ratio can vary depending on factors such as the type of cement, admixtures used, and the specific requirements of the project. However, you will need 32 to 39 gallons of water per
cubic yard of concrete.
To produce one cubic yard of 3,000 psi concrete, you need to mix 5 94lb bags of Portland cement, 15 cubic feet of sand, 15 cubic feet of gravel and 31 gallons of water.
To produce one cubic yard of 4,000 psi concrete, you need to mix 6 94lb bags of Portland cement, 12 cubic feet of sand, 18 cubic feet of gravel and 34 gallons of water.
To produce one cubic yard of 5,000 psi concrete, you need to mix 7 94lb bags of Portland cement, 14 cubic feet of sand, 14 cubic feet of gravel and 36 gallons of water.
Standard ready-mix concrete includes: 5 bag mix (3000 psi), 5.5 bag mix (3500 psi), 6 bag mix (4000 psi), 6.5 bag mix (4500 psi), 7 bag mix (5000 psi), and 8 bag mix (6000 psi).
Here’s are table for the water-cement ratio by weight for non air-entrained & air-entrained concrete:-
Compressive strength at 28 days in PSI Water cement ratio by weight
Non air entrained concrete Air entrained concrete
6000 PSI 0.41 0.32
5000 PSI 0.48 0.40
4000 PSI 0.57 0.48
3000 PSI 0.68 0.59
2000 PSI 0.82 0.74
In general, the water-cement ratio for residential application should be 0.60 for the 2000 psi concrete, 0.55 for the 3000 psi concrete, 0.5 for the 4000 psi concrete, 0.45 for the 5000 psi concrete,
and 0.40 for the 6000 psi concrete. | {"url":"https://civilsir.com/the-water-cement-ratio-for-3000-2000-4000-5000-and-6000-psi-concrete/","timestamp":"2024-11-08T16:02:32Z","content_type":"text/html","content_length":"87003","record_id":"<urn:uuid:0e156e2e-f556-4b24-8c5d-fbed4c35e822>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00000.warc.gz"} |
Algebra/Chapter 0/Who should read this book - Wikibooks, open books for an open world
This book is intended to be a comprehensive look at the mathematics topic of algebra. That said, it would be well suited for a wide variety of individuals, ranging from students (at any grade level)
to adults interested in refreshing or improving their understanding of basic math. It could be used either as a primary text or a reference.
This book will avoid explaining subjects with only rigorous mathematical abstractions whenever possible. Math can be tricky and frustrating enough without it seeming inaccessible. So while every
topic will be covered fully and correctly and in many times using proper mathematical terminology, there will always be a backup definition or simple explanation to complete the concept. This allows
a wider variety of individuals to learn from this text, from an ambitious 12-year-old to a forgetful college professor. Algebra is applicable to your daily life in addition to academic settings, so
an algebra textbook should be accessible to everyone.
While this book is meant to be accessible to everyone, it is advisable to get a very good grasp on arithmetic before taking a deep dive into algebra. For this reason, the very first chapter of this
book acts as a comprehensive review of all of the prequesites that are necessary to start tackling the many topics the book has to offer. This chapter can be skipped entirely if you are confident
enough with your ability to perform basic arithmetic. Nonetheless, a quick glimpse at the chapter, and filling in any gaps in your knowledge wouldn't hurt. In fact, it may help one in the long run. | {"url":"https://en.m.wikibooks.org/wiki/Algebra/Chapter_0/Who_should_read_this_book","timestamp":"2024-11-06T21:20:59Z","content_type":"text/html","content_length":"24182","record_id":"<urn:uuid:8ad1950c-6ab1-49a7-8388-f79af581c09f>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00594.warc.gz"} |
Transform a list of commands in a program
Transform a list of commands in a program
I have that list of commands
X = [SR.var("x%d"%i) for i in [1..n]]; b = vector(random_matrix(QQ,1,m)[0]); A = matrix(QQ,m,n);
A[0, -n:] = ones_matrix(1,n);
for i in [1..m-2]:
A[i, i:i+2] = ones_matrix(1,2); eq = [A[i]*vector(X) == b[i] for i in range(m)] ; a = matrix([[e.lhs().coefficient(v) for v in X] for eqin eq]) ; C= A.augment(b); D = C.right_kernel()
The list os commands works if you choose m and n properly, but that is not practical.
1 Answer
Sort by ยป oldest newest most voted
Here is what I would do
• format your code with one instruction per line
• indent everything by four spaces
• add an unindented def something(m, n): in front
• decide what to return at the end
For example:
def something(m, n):
x = lambda i: SR.var(f"x_{i}", latex_name=f"x_{{{i}}}")
X = vector([x(i) for i in [1 .. n]])
b = VectorSpace(QQ, m).random_element()
A = matrix(QQ, m, n)
A[0, :] = ones_matrix(1, n)
for i in [1 .. m - 2]:
A[i, i:i + 2] = ones_matrix(1, 2)
eq = [A[i]*X == b[i] for i in range(m)]
a = matrix([[e.lhs().coefficient(v) for v in X] for e in eq])
C = A.augment(b)
D = C.right_kernel()
return X, b, A, eq, a, C, D
edit flag offensive delete link more
Thank you very much.
phcosta ( 2020-12-27 15:09:35 +0100 )edit
What remains to do:
• find a more descriptive name for the function
• add a documentation string which
□ says what the function does
□ documents the input and output
□ provides a few examples
slelievre ( 2020-12-27 15:32:26 +0100 )edit | {"url":"https://ask.sagemath.org/question/54940/transform-a-list-of-commands-in-a-program/","timestamp":"2024-11-06T21:09:39Z","content_type":"application/xhtml+xml","content_length":"54008","record_id":"<urn:uuid:d39a18d6-68fe-4e7b-b3e9-7fba9a24a0d9>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00168.warc.gz"} |
Find the Perfect Number - FcukTheCode
Find the Perfect Number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so
6 is a perfect number.
def perfectnumber(numbr):
if(temp==0 and i!=numbr):
return 1
return 0
numbr=int(input('Enter a Number : '))
print("The number is a perfect number")
print("The number is not a perfect number")
exec.on linux terminal
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Fluid Flow: Conservation of Momentum, Mass, and Energy
Fluid Flow, Heat Transfer, and Mass Transport Fluid Flow: Conservation of Momentum, Mass, and Energy
Describing Fluid Flow
There are various mathematical models that describe the movement of fluids and various engineering correlations that can be used for special cases. However, the most complete and accurate description
comes from partial differential equations (PDEs). For instance, a flow field is characterized by balance in mass, momentum, and total energy described by the continuity equation, the Navier-Stokes
equations, and the total energy equation: ^(1)
The solution to the mathematical model equations gives the velocity field, p; and temperature, T; of the fluid in the modeled domain.
In principle, this set of equations is able to describe flows from the creeping flow in a microfluidic device to the turbulent flow in a heat exchanger and even the supersonic flow around a jet
fighter. However, solving Equation (1) for a case such as the jet plane shown below is not feasible and while it is possible to solve the whole of Equation (1) for a microfluidic device, it is a lot
of work down the drain. Much of computational fluid dynamics (CFD) is therefore devoted to selecting suitable approximations to Equation (1) so that accurate results are obtained with a reasonable
computational cost.
SR71 supersonic jet. The exhaust forms shock diamonds typical for supersonic flows. Image in the public domain via the Dryden Flight Research Center, NASA.
SR71 supersonic jet. The exhaust forms shock diamonds typical for supersonic flows. Image in the public domain via the Dryden Flight Research Center, NASA.
The Continuum Hypothesis and Rarefied Flows
The flow equations (Equation (1)) rely on the continuum hypothesis, that is, a fluid can be regarded as a continuum rather than a collection of individual molecules. Flows where molecular effects are
of significance are known as rarefied flows. The degree of rarefaction is measured by the Knudsen number:
where L is a representative length scale for the flow geometry; for example, a channel width.
A flow can be regarded as a continuum flow as long as the Knudsen number is smaller than 10^-3. Liquids can almost always be regarded as continua, as can gases under ordinary circumstances. For gases
at very low pressures, or gas flows confined in very small domains, the interaction of the molecules in the fluid may take place with the same frequency as the interaction with the walls that confine
the flow. For such systems, the fluid flow has to be described with the rarefied flow equations or at least with Knudsen boundary conditions.
Newtonian and Non-Newtonian Fluids
A fluid is characterized by, among other things, its viscosity. The viscous effects are contained in the viscous stress tensor,
There are, however, a number of fluids that do not obey the simple relation in Equation (3). Such fluids are known as non-Newtonian and can display a wide range of behaviors. Examples of
non-Newtonian fluids include blood; paint; some lubricants; cosmetic products; food products such as honey, ketchup, juice, and yogurt; and many suspensions, for example sand in water or starch
suspended in water.
Honey is an example of a non-Newtonian fluid.
Incompressible Fluid Flow
A fluid can be regarded as incompressible if the density variations are very small; that is, if viscous heating) and assume that the fluid is Newtonian, Equation (1) can be simplified to:
The middle equation in Equation (4) is the famous Navier-Stokes equation, named after the French physicist Navier and the Irish physicist Stokes. Navier was first to derive the equations, but the
understanding of the physical mechanism behind the viscous term was first explained by Stokes, hence the name of the equations.
In some cases, the first equation, the continuity equation, is also included in the Navier-Stokes equations. As can be seen, the energy equation has been rewritten as a temperature equation, which is
much more convenient to work with. The temperature equation is for incompressible flows completely decoupled from the Navier-Stokes equations, unless the viscosity depends on the temperature.
The solution to the Navier-Stokes equations gives the velocity and pressure field for flows of fluids with constant viscosity and density. The temperature can be solved for separately if information
about the temperature field is desirable.
Buoyancy is an important physical phenomenon that formally comes from variations in density. Equation (4) can, however, still be used to model the effect of buoyancy by introducing buoyancy as a
momentum source/sink in the momentum equations.
Even in cases where there is nonconstant density, the Navier-Stokes equations may be used and the effect of buoyancy may be introduced as a momentum source/sink in the momentum equations. For
example, buoyancy makes the smoke from a cigar flow upward.
The Reynolds Number
A central concept in fluid flow is the Reynolds number. It is defined as:
where U is a representative velocity scale and L is a representative length scale.
In absence of body forces, and if the density and the viscosity are constant, the Navier-Stokes equation (middle expression in Equation (4)) can be nondimensionalized to read:
As can be seen from Equation (6), the Reynolds number measures the relative importance of the viscous stresses. A low Reynolds number means that the flow is completely governed by viscous effects,
while the flow is effectively inviscid at very high Reynolds numbers.
Observe that there can be several Reynolds numbers associated with a particular flow configuration. A channel flow, for example, can be based on the channel half width or the whole channel width. The
velocity can either be the average velocity or the maximum velocity. Therefore, it is important to know which length scale and velocity scale are associated with a particular Reynolds number,
especially when comparing Reynolds numbers between similar flow configurations.
Stokes Flow
Flows at very low Reynolds numbers are known as creeping flows. They can be encountered in, for example, microfluidic systems (such as the micromixer shown below) or lubrication systems.
The Stokes equations are often used to model flow in microfludics, such as the flow in this micromixer. The Stokes equations are often used to model flow in microfludics, such as the flow in this
The limit when Stokes flow. Stokes flow can formally support both time dependency and varying material properties, but classical Stokes flow is written for incompressible quasistatic conditions:
The equations are named after the Irish physicist George Gabriel Stokes, who first described viscous momentum transfer through these equations. Which terms to retain from the energy equation depends
on the fluid. The convective term can often be neglected, as can pressure-work effects. Viscous heating can also be of interest for Stokes flows in bearings and other lubrication applications, for
Turbulent Flow
The Reynolds number measures the importance of the inertial effects compared to the viscous effects. As long as the Reynolds number is not too large, the viscous effects will damp out perturbations
in the flow field. Such flows are known as laminar flows. It is often feasible to solve, for example, Equation (4) for laminar flows, since the viscosity dissipates any flow structures that are small
The higher the Reynolds number, the more inertial effects dominate over viscous effects. When the Reynolds number is high enough, any small perturbation will feed on the mean flow momentum and grow
and trigger new flow structures. This phenomenon is called transition.
A flow that has undergone transition is denoted turbulent flow. Turbulent flows are characterized by the seemingly chaotic eddies that have a huge span of length scales, from the large vortices that
can be almost as large as the computational domain down to the small dissipative eddies that can be as small as micrometers. This wide span of scales means that not many turbulent flows can be
simulated at a reasonable computational cost using the pure Navier-Stokes equations. It is possible to perform so-called direct numerical simulation (DNS) for some very simple flow cases, but it
requires vast computational resources.
In order to make it possible to estimate the flow and pressure fields without having access to a supercomputer, we usually introduce approximate turbulence models. The turbulence models formulate
different types of conservation expressions for turbulence in an averaged sense; for example, by looking at the conservation of the kinetic energy that these small eddies may have (called turbulent
kinetic energy). The conserved properties, such as the turbulent kinetic energy, are used to generate an additional contribution to the viscosity, called eddy viscosity. The eddy viscosity enlarges
the viscous transfer of momentum in order to mimic the momentum that would be transferred by the small-scale eddies that we cannot afford to resolve.
The most common turbulence models used in engineering are the Reynolds-averaged Navier-Stokes models (RANS models), in which the modeled quantities are time-averaged and the fluctuations are treated
in an introduced quantity referred to as the Reynold stresses. The RANS equations for incompressible flows read:
where bar denotes an averaged quantity and prime is the deviation from average.
The unfiltered velocity, for example, can be written (8) to the continuity and momentum equations in Equation (4), it can be seen that the equations are identical except that unfiltered quantities
have been replaced by filtered quantities, there is no time derivative (since we averaged over time), and there is an extra term (8). This term is the Reynolds stress tensor and represents the effect
of the turbulent fluctuations on the filtered velocity and pressure fields.
It is possible to formulate transport equations for the entries in the Reynolds-stress tensor. With appropriate simplifications and assumptions, these equations can result in so-called
Reynolds-stress models. While powerful, these models are often difficult to work with and even if they are computationally much less expensive than DNS, they are still too expensive for most
industrial applications.
The most common approach is instead to assume that the turbulence acts as an additional viscous effect and write k-ε, k-ω, shear stress transport (SST), and the Spalart-Allmaras turbulence models.
There is another class of turbulence models that averages turbulence over a small spatial region instead of over time. This forms a sort of low-pass filter for eddies below a certain length scale. In
this way, large turbulent eddies are resolved while the effect of small eddies have to be modeled, hence the name large eddy simulation (LES). The continuity and momentum equations for incompressible
LES take the same form:
Equation (9) is identical to Equation (8), but with a time derivative. Also, the term
LES is often more accurate than RANS, but the simulations must always be 3D, even if the flow is essentially 2D and the simulations are always time dependent. In addition, the required resolution for
the SGS models to be valid is often quite high, which means that LES is only used when even the most advanced RANS models fail to capture the essential features of the flow.
Turbulent wind flow around a solar panel. The low-Reynolds RANS turbulence models can be used to estimate the force that the panels are subjected to by the wind. Turbulent wind flow around a solar
panel. The low-Reynolds RANS turbulence models can be used to estimate the force that the panels are subjected to by the wind.
The Mach Number
The Mach number is defined as:
where c is the speed of sound.
The Mach number measures how fast a fluid moves compared to the speed of the pressure waves. When the Mach number is small, that is, when (4) can formally be reached by letting (1). Flow where all
terms in Equation (1) are important is sometimes referred to as compressible viscous flow.
If the Mach number is high, then oftentimes, so too is the Reynolds number, as both are proportional to the velocity. So, Equation (1) is often complemented with a turbulence model to account for the
eddy diffusivity for momentum eddy diffusivity for heat transfer. The coupling between Equation (1) and its turbulence model is often very strong.
Fully compressible turbulent flow modeled with the k-ε turbulence model. We can see the diamond-shaped pattern of the velocity field caused by the pressure shocks (shock diamonds).
Fully compressible turbulent flow modeled with the k-ε turbulence model. We can see the diamond-shaped pattern of the velocity field caused by the pressure shocks (shock diamonds).
Inviscid Flow and the Euler Equations
For flow of gases at moderate pressures close to and above the speed of sound, the contribution of molecular viscosity and eddy viscosity to the transfer of momentum can often be neglected. In such
cases, the model equations describe the conservation of momentum (without a viscous term), the conservation of mass, and the conservation of energy. There is no need for a turbulence model, since the
eddy viscosity is not accounted for.
In the energy equations, the analogy to viscous momentum transfer is heat transfer through conduction. In fact, in gases, the same mechanism that is responsible for viscosity is also responsible for
thermal conductivity and the eddy diffusivity for momentum transfer is also used to compute the eddy diffusivity for heat transfer. Consequently, in cases where we can neglect viscous momentum
transfer, we can usually neglect heat transfer through conduction in the energy equations.
The conservation equations for inviscid flow and negligible thermal conductivity are usually referred to as the Euler equations after the famous Swiss mathematician who first formulated them. The
Euler equations read:
Supersonic flow over a wing-shaped obstacle that causes pressure shocks, which reflect on the walls of this benchmark problem for the solution of the Euler equations for high Mach number flow.
Supersonic flow over a wing-shaped obstacle that causes pressure shocks, which reflect on the walls of this benchmark problem for the solution of the Euler equations for high Mach number flow.
Multiphase Flow
The equations for the conservation of momentum, mass, and energy can also be used for fluid flow that involves multiple phases; for example, a gas and a liquid phase or two different liquid phases,
such as oil and water.
The most detailed way of modeling multiphase flow is with surface tracking methods, such as the level set or phase field methods. In these models, the interactions between phases — for example,
surface tension — are introduced as sources or sinks in the momentum equations at a thin layer with a very small thickness that follows the boundary between the phases. The shape and position of the
phase boundary is computed in detail. This means that the momentum and mass conservation equations are combined with a set of transport equations for a level set or phase field function that, at a
given value (isosurface), describe the position of the phase boundary.
Classical benchmark model for surface tracking two-phase flow models. Note how for a very short moment, the heavier fluid gets attached to the top wall during sloshing. This attachment is due to
surface tension.
When the phase boundary consists of millions of droplets or bubbles, or when the shape of the phase boundary is very complex in its details, we cannot computationally afford to track its shape.
Instead, we have to make some kind of homogenization and treat the presence of the different phases as fields of averaged mass or volume fractions. We no longer track the shape of the phase boundary
in detail. Instead, we introduce the possible interactions between the phases as momentum sources and sinks defined everywhere in the fluid mixture. In addition, the conservation equations for
momentum and mass are combined with a transport equation for the volume fraction for one of the phases in the case of two-phase flow and two transport equations for three-phase flow. When there is a
large difference in density between two phases, we may even have to formulate separate momentum equations for each phase defined everywhere in the fluid domain.
For bubbles in liquids, a dispersed flow model such as the bubbly flow model may give a good representation of the homogenized two-phase flow. For liquid-liquid solutions, such as oil and water, we
may use a slightly more complex model, such as the mixture model for multiphase flow.
Model of a liquid-liquid extraction column. The plot shows the volume fraction of oil. The heavier water solution enters at the top annular inlet, while the oil phase exits at the top circular
outlet. Model of a liquid-liquid extraction column. The plot shows the volume fraction of oil. The heavier water solution enters at the top annular inlet, while the oil phase exits at the top
circular outlet.
For a large number of solid particles in a gas, where the difference in density is very large, we often need to formulate the momentum equations for both the dispersed solid particles and the gas
phase. Models that define the momentum equations for each phase are usually referred to as Euler-Euler multiphase flow models. The name comes from the fact that both phases are described as continua,
that is, by an Eulerian approach.
When the particles are few enough, an alternative option is to use a particle tracking method to describe the dispersed phase. This is known as the Euler-Lagrange method, since the continuum (for
example, the fluid) is described by an Eulerian approach, while the particles are described by a Lagrangian approach. The advantage of the Euler-Lagrange method is that properties can then be
associated with each individual particle, but the method becomes very expensive as the number of particles increases.
The difference between separated multiphase flow models that use surface tracking methods (left) and dispersed multiphase flow models (right). In the surface tracking method, the isosurface of the
field φ at φ = 0 represents the phase boundary. In the dispersed multiphase flow model, only the volume fraction of bubbles or droplets is obtained, while the details of the phase boundary are
treated as averaged volume forces. The difference between separated multiphase flow models that use surface tracking methods (left) and dispersed multiphase flow models (right). In the surface
tracking method, the isosurface of the field φ at φ = 0 represents the phase boundary. In the dispersed multiphase flow model, only the volume fraction of bubbles or droplets is obtained, while the
details of the phase boundary are treated as averaged volume forces.
Porous Media Flow
If we can afford to describe a porous structure in detail, with all of its surface structures and surface properties, we can use the equations for the conservation of momentum and mass as usual,
defining no-slip conditions on the pore walls or the Knudsen condition if the mean width of the pores is of the same order of magnitude as the scale of molecule interactions.
However, in most cases, we cannot afford to describe the millions of pore bends and structures in a macroscopic model of a porous structure. Therefore, models for porous media flow usually use
homogenization in order to define the fluid and the porous matrix domains in the porous structure as a slab with averaged properties, such as averaged porosity, tortuosity, and permeability. The
momentum equations then become Darcy’s law, named after the French engineer who first formulated this law. Darcy’s law may be extended with a shear term to form the Brinkman equations, named after
the Dutch physicist H.C. Brinkman.
Flow over and through a porous particle, where the structure is described in detail (left) and the homogenized corresponding model (right). Flow over and through a porous particle, where the
structure is described in detail (left) and the homogenized corresponding model (right).
Published: June 29, 2018
Last modified: June 29, 2018 | {"url":"https://www.comsol.com/multiphysics/fluid-flow-conservation-of-momentum-mass-and-energy","timestamp":"2024-11-13T02:03:24Z","content_type":"text/html","content_length":"156918","record_id":"<urn:uuid:20a0a324-eff4-4421-9729-d8fc73a90a5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00545.warc.gz"} |
PPF Calculator Online: Calculating Interest Rate Of Public Provident Fund
PPF calculator online will help you calculate the interest earned on your Public Provident Fund investment. The PPF, or Public Provident Fund, is a long-term savings plan to create wealth and save
tax under Income Tax Deduction Section 80C.
PPF Calculator Online: Public Provident Fund Calculator
• Invested Amount:
• Maturity Amount:
• Total Interest:
Table of Contents
What is a PPF (Public Provident Fund) Calculator?
A PPF calculator is an digital tool. It aids in forecasting of future PPF (Public Provident Fund) returns for the investor. Input the annual contributions, interest rates, and period. Total interest
earned also appears on the calculator section as maturity amount.
Key Features and Benefits of a PPF Calculator
• Ease of Use: Easy interface to enter parameters and get outcomes rapidly.
• Accuracy: Accurate computation of the present interest rates and the number of compounding periods.
• Time-saving: Saves time since the user does not have to calculate figures manually.
• Financial Planning: Assists in determining an annual contribution to attain the best rate of returns.
Importance of a PPF Calculator in Financial Planning
• Visualize future savings.
• Outline criteria for deciding on annual contributions.
• Maximise portfolio levels for achieving targeted objectives.
How to Calculate PPF Calculator
M = P [ ( { (1 + t) ^ n } - 1 ) / i ] x ( 1 + i )
M = Maturity Value
P = Annual Payments
i = Interest Rate
n = number of years
The part after the P in the formula is called the annuity factor. When the value of this factor is multiplied by the annual contribution, one gets the maturity value of the PPF investment. Here's an
example to calculate the maturity value of a PPF account with specific details:
Annual Contributions ₹1 lakh
PPF Account Interest Rate 7.1%
Number of Years 15
Let’s plug the information into the PPF calculation formula:
M = ₹ 1 lakh [ ( { (1 + 7%) ^ 15} – 1) / 7%] x (1 + 0.07) M = ₹ 27, 12, 139
If the interest rate changes, then it starts varying and one has to calculate for different periods. For instance, if the rate was 7% for 5 years and 8% for the next 5 years, you do it as follows:
First, find the 10-year maturity value at 7% for five years deposits. Next, add up the 5-year maturity value at 8% for the five years’ deposits.
It is important to note that you can also select monthly contributions in the PPF calculator,( that is yearly contribution ÷ 12.) Nonetheless, you will get the same output as when the calculator
allows yearly contributions.
Using the PPF Calculator on The Invest Advisory
Our Public Provident Fund or PPF calculator on The Invest Advisory is quite simple and easy to use. Here’s how to use it:
Step-by-Step Guide:
1. Go to the PPF calculator page of the The Invest Advisory website.
2. Please, state your annual contribution amount.
3. Enter the current rate of interest.
4. Choose the period of investment.
5. Press the ‘Calculate’ button to see your assessment.
Example Calculations: For instance, if you invest INR 1,00,000 annually at an interest rate of 7.1% for 15 years, the calculator will show the maturity amount and the total interest earned. This
helps in visualizing potential returns.
Tips for Maximizing PPF Benefits:
• Make your investments during the initial months of the financial year in order to allow the interest to accrue.
• Contribute a fixed amount of money to make the most of compounding.
• Enter your specific objectives in the calculator to change the amount you invest and the period needed.
With help of the provided PPF calculator, you can make the right choice about your PPF investment and manage your financial plan efficiently.
PPF Calculation of Investment Tenures
There is no tax implication that goes with the invested amount on PPF. Using the PPF calculator, figure out how your maturity value alters with different investment tenors. Higher maturity values are
obtained with longer tenures. Below is the table of PPF maturity values for the mentioned investment period.
Investment Tenure (years) Annual Contribution (₹) Principal Invested (₹) PPF Account Rate (%) Maturity Value (₹)
1 1 lakh 1 lakh 7.1 1,07,100
5 1 lakh 5 lakh 7.1 6,17,134
10 1 lakh 10 lakh 7.1 14,86,749
15 1 lakh 15 lakh 7.1 27,12,139
20 1 lakh 20 lakh 7.1 44,38,859
Note: 7. 1% is the current rate, and is subject to change in the future.
In conclusion, the present blog post has described the PPF calculator and the advantages of this tool. Please visit our PPF calculator on The Invest Advisory for more precise calculation of returns.
Make informed financial decisions. Improve your investments through advisory and tools related to investment. For other information related to the financial market, do check out The Invest Advisory
FAQs about PPF Return Rate Calculator
1. What is a PPF calculator?
Answer: A PPF calculator is a specialized tool that you can use to find out the maturity amount, along with the interest on it, for your Public Provident Fund investments.
2. How does a PPF calculator work?
Answer: The potential returns depend on your annual contribution, the interest rate of the year, and the compound frequency fixed by the government.
3. Why use a PPF calculator?
Answer: PPF calculator enables one to plan and predict future savings in order to improve finance management and investments.
4. Where can I find a PPF calculator?
Answer: You can access PPF calculators online on financial websites like The Invest Advisory for quick and accurate calculations.
Are online PPF calculators accurate?
Answer: Yes, PPF calculators employ the right formulas and up to date data to come up with the growth rate of your PPF investments. | {"url":"https://theinvestadvisory.com/ppf-calculator-online/","timestamp":"2024-11-13T04:34:51Z","content_type":"text/html","content_length":"203442","record_id":"<urn:uuid:9f85ea94-197f-4d05-a767-03a793178cfc>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00501.warc.gz"} |
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2. i like tiwari acadamy because it will understand the answer very fast.my teachers also use this website
i like tiwari acadamy because it will understand the answer very fast.my teachers also use this website
See less | {"url":"https://discussion.tiwariacademy.com/profile/vidyasagar/answers/","timestamp":"2024-11-14T07:56:36Z","content_type":"text/html","content_length":"148413","record_id":"<urn:uuid:24900297-70e2-4a95-8e00-72cd6a798654>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00282.warc.gz"} |
Designer’s Guide Community :: Forum
I have read [I. Galton 2018] and wanted to understand the effect of
deeply. I made a simple experiment to observe the effect of squaring, but I am unable to make sense of the results.
MATLAB experiment description:
I have taken a sine wave of frequency 100MHz, sampled at the rate 100GHz. To this sine wave, I have added a sinusoidal phase noise of magnitude "alpha" and frequency 10MHz. When I take the FFT of the
resultant signal, I get the expected frequency spectrum.
clear all
close all;
f0 = 100e6;
df = 10e6;
tTotal= 1e-6;
Fs = 100e9;
Ts = 1/Fs;
t = 0:Ts:tTotal-Ts;
alpha = 1e-6;
a = sin(2*pi*f0*t + alpha*sin(2*pi*df*t)) ;
afft = fft(a)/length(a);
afftdb = 20*log10(abs(afft)+1e-8);
The strength of the spur at bin 111 is 20log(alpha/2) relative to the carrier as expected. Next, I have converted this sine wave into a squarewave, then hann-windowed, then low pass filtered and
finally I've taken the FFT.
asquare = sign(a);
y = lowpass(asquare,200e6,Fs,'ImpulseResponse','iir','Steepness',0.95);
win = 0.5*(1-cos(2*pi*t/tTotal));
y =y.*win;
%asq_fft = fft(asquare)/length(a);
asq_fft = fft(y)/length(a);
asq_fftdB = 20*log10(abs(asq_fft)+1e-8);
I get spur of strength -64dB which is unexpected for me (see below image). I didn't expect to get this high spur. I expected the spur to still be 20log(alpha/2) relative to the carrier. I also don't
understand why I get so many harmonics too.
I also tried changing the value of alpha from 1e-6 to 1e-11. The FFT plot for alpha of 1e-11 is given below:
Still, the strength of the spur didn't change. This is surprising to me. Could anyone please explain why does the spur's strength of -64dB make sense?
When I make alpha = 0, there are no spurs. So, there is no FFT leakage issue here.
[I. Galton 2018] I. Galton and C. Weltin-Wu, "Understanding Phase Error and Jitter: Definitions, Implications, Simulations, and Measurement," in IEEE Transactions on Circuits and Systems I: Regular
Papers, vol. 66, no. 1, pp. 1-19, Jan. 2019, doi: 10.1109/TCSI.2018.2856247. | {"url":"https://designers-guide.org/forum/YaBB.pl?num=1603949787/0","timestamp":"2024-11-08T19:07:40Z","content_type":"text/html","content_length":"27281","record_id":"<urn:uuid:b069a996-7c3d-4434-a291-c9522653aabf>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00201.warc.gz"} |
A2L Item 001
Goal: Relating physical understanding of an object’s motion to a graphical representation of acceleration.
Source: UMPERG
A soccer ball rolls slowly across the road and down a hill as shown below:
Which of the following sketches of a[x] vs. t is a reasonable representation of the horizontal acceleration of the ball as a function of time?
We will assume that rolling friction between the ball and road surface is small and that air resistance can be ignored. We will also assume that the ball does not leave the road surface at the top of
the hill. If these assumptions are satisfied, the ball will roll across the level road at a (nearly) constant velocity. As it rolls down the hill, the ball will speed up, producing a constant
acceleration in the direction of motion. There will be a nonzero component of acceleration pointing to the right. The graph at the right is a reasonable representation of the horizontal acceleration
as a function of time. For our assumptions, answer (5) is the best choice.
Context for Use: Give after students explore the vector nature of acceleration. Formal (quantitative) kinematics is not required.
Assessment Issues: (1) Can students recognize when an object is accelerating? What criteria do they use? (2) Do students perceive nonzero horizontal and vertical components of acceleration? Do some
students think that the acceleration is in the y-direction only? (3) Do students think that the acceleration graph looks like the sketch of the road on which the ball rolls? What process do they use
to construct a graph of acceleration versus time? (4) Do students confuse the different motion quantities? For example, do they interpret the graphs of acceleration versus time as velocity versus
time graphs?
Questions to Reveal Student Reasoning
• Where does the ball speed up? Where does it slow down? Why does its speed change?
• What is the direction of the ball’s acceleration while it is on the hill? Does the ball accelerate to the right? Does the ball accelerate vertically?
Help students construct the horizontal (and vertical) velocity vs. time graph for the ball. If students have been exposed to forces, draw a free-body diagram and use it to find the net force. | {"url":"https://new.clickercentral.net/item_0001/","timestamp":"2024-11-03T21:46:01Z","content_type":"text/html","content_length":"43580","record_id":"<urn:uuid:6c61aaac-9bd3-48c6-93e3-8cedeb0e3fcc>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00704.warc.gz"} |
Probabilistic Methods 1
Probabilistic Methods 1
Moderator: Matthias W Seeger
Wed 21 July 5:00 - 5:20 PDT
Yangjun Ruan · Karen Ullrich · Daniel Severo · James Townsend · Ashish Khisti · Arnaud Doucet · Alireza Makhzani · Chris Maddison
Latent variable models have been successfully applied in lossless compression with the bits-back coding algorithm. However, bits-back suffers from an increase in the bitrate equal to the KL
divergence between the approximate posterior and the true posterior. In this paper, we show how to remove this gap asymptotically by deriving bits-back coding algorithms from tighter variational
bounds. The key idea is to exploit extended space representations of Monte Carlo estimators of the marginal likelihood. Naively applied, our schemes would require more initial bits than the standard
bits-back coder, but we show how to drastically reduce this additional cost with couplings in the latent space. When parallel architectures can be exploited, our coders can achieve better rates than
bits-back with little additional cost. We demonstrate improved lossless compression rates in a variety of settings, especially in out-of-distribution or sequential data compression.
Wed 21 July 5:20 - 5:25 PDT
Zhibin Duan · Dongsheng Wang · Bo Chen · CHAOJIE WANG · Wenchao Chen · yewen li · Jie Ren · Mingyuan Zhou
Hierarchical topic models such as the gamma belief network (GBN) have delivered promising results in mining multi-layer document representations and discovering interpretable topic taxonomies.
However, they often assume in the prior that the topics at each layer are independently drawn from the Dirichlet distribution, ignoring the dependencies between the topics both at the same layer and
across different layers. To relax this assumption, we propose sawtooth factorial topic embedding guided GBN, a deep generative model of documents that captures the dependencies and semantic
similarities between the topics in the embedding space. Specifically, both the words and topics are represented as embedding vectors of the same dimension. The topic matrix at a layer is factorized
into the product of a factor loading matrix and a topic embedding matrix, the transpose of which is set as the factor loading matrix of the layer above. Repeating this particular type of
factorization, which shares components between adjacent layers, leads to a structure referred to as sawtooth factorization. An auto-encoding variational inference network is constructed to optimize
the model parameter via stochastic gradient descent. Experiments on big corpora show that our models outperform other neural topic models on extracting deeper interpretable topics and deriving better
document representations.
Wed 21 July 5:25 - 5:30 PDT
Mohammad Mahdi Derakhshani · Xiantong Zhen · Ling Shao · Cees Snoek
This paper introduces kernel continual learning, a simple but effective variant of continual learning that leverages the non-parametric nature of kernel methods to tackle catastrophic forgetting. We
deploy an episodic memory unit that stores a subset of samples for each task to learn task-specific classifiers based on kernel ridge regression. This does not require memory replay and
systematically avoids task interference in the classifiers. We further introduce variational random features to learn a data-driven kernel for each task. To do so, we formulate kernel continual
learning as a variational inference problem, where a random Fourier basis is incorporated as the latent variable. The variational posterior distribution over the random Fourier basis is inferred from
the coreset of each task. In this way, we are able to generate more informative kernels specific to each task, and, more importantly, the coreset size can be reduced to achieve more compact memory,
resulting in more efficient continual learning based on episodic memory. Extensive evaluation on four benchmarks demonstrates the effectiveness and promise of kernels for continual learning.
Wed 21 July 5:30 - 5:35 PDT
Fan Ding · Jianzhu Ma · Jinbo Xu · Yexiang Xue
We propose XOR-Contrastive Divergence learning (XOR-CD), a provable approach for constrained structure generation, which remains difficult for state-of-the-art neural network and constraint reasoning
approaches. XOR-CD harnesses XOR-Sampling to generate samples from the model distribution in CD learning and is guaranteed to generate valid structures. In addition, XOR-CD has a linear convergence
rate towards the global maximum of the likelihood function within a vanishing constant in learning exponential family models. Constraint satisfaction enabled by XOR-CD also boosts its learning
performance. Our real-world experiments on data-driven experimental design, dispatching route generation, and sequence-based protein homology detection demonstrate the superior performance of XOR-CD
compared to baseline approaches in generating valid structures as well as capturing the inductive bias in the training set.
Wed 21 July 5:35 - 5:40 PDT
Alek Dimitriev · Mingyuan Zhou
Estimating the gradients for binary variables is a task that arises frequently in various domains, such as training discrete latent variable models. What has been commonly used is a REINFORCE based
Monte Carlo estimation method that uses either independent samples or pairs of negatively correlated samples. To better utilize more than two samples, we propose ARMS, an Antithetic REINFORCE-based
Multi-Sample gradient estimator. ARMS uses a copula to generate any number of mutually antithetic samples. It is unbiased, has low variance, and generalizes both DisARM, which we show to be ARMS with
two samples, and the leave-one-out REINFORCE (LOORF) estimator, which is ARMS with uncorrelated samples. We evaluate ARMS on several datasets for training generative models, and our experimental
results show that it outperforms competing methods. We also develop a version of ARMS for optimizing the multi-sample variational bound, and show that it outperforms both VIMCO and DisARM. The code
is publicly available.
Wed 21 July 5:40 - 5:45 PDT
Jay Whang · Erik Lindgren · Alexandros Dimakis
Given an inverse problem with a normalizing flow prior, we wish to estimate the distribution of the underlying signal conditioned on the observations. We approach this problem as a task of
conditional inference on the pre-trained unconditional flow model. We first establish that this is computationally hard for a large class of flow models. Motivated by this, we propose a framework for
approximate inference that estimates the target conditional as a composition of two flow models. This formulation leads to a stable variational inference training procedure that avoids adversarial
training. Our method is evaluated on a variety of inverse problems and is shown to produce high-quality samples with uncertainty quantification. We further demonstrate that our approach can be
amortized for zero-shot inference.
Wed 21 July 5:45 - 5:50 PDT
Carol Mak · Fabian Zaiser · Luke Ong
Probabilistic programming uses programs to express generative models whose posterior probability is then computed by built-in inference engines. A challenging goal is to develop general purpose
inference algorithms that work out-of-the-box for arbitrary programs in a universal probabilistic programming language (PPL). The densities defined by such programs, which may use stochastic
branching and recursion, are (in general) nonparametric, in the sense that they correspond to models on an infinite-dimensional parameter space. However standard inference algorithms, such as the
Hamiltonian Monte Carlo (HMC) algorithm, target distributions with a fixed number of parameters. This paper introduces the Nonparametric Hamiltonian Monte Carlo (NP-HMC) algorithm which generalises
HMC to nonparametric models. Inputs to NP-HMC are a new class of measurable functions called “tree representable”, which serve as a language-independent representation of the density functions of
probabilistic programs in a universal PPL. We provide a correctness proof of NP-HMC, and empirically demonstrate significant performance improvements over existing approaches on several nonparametric
Wed 21 July 5:50 - 5:55 PDT | {"url":"https://icml.cc/virtual/2021/session/12021","timestamp":"2024-11-07T12:26:55Z","content_type":"text/html","content_length":"60354","record_id":"<urn:uuid:68ef86a9-9b07-4de5-91a0-fbdc769f3d0c>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00461.warc.gz"} |
Discount rate for npv
It is expected to bring in $40,000 per month of net cash flow over a 12-month period with a target rate of return of 10%, which will act as our discount rate. NPV = 40,000(Month 1)/1 + 0.1 + 40,000
(Month 2)/1 + 0.1 - 250,000 = $230,000 The discount rate is not a direct measure of real estate investment performance but a key variable in estimating the NPV of the net cash flows of a property
using the Discounted Cash Flow (DCF) model. Discount Rate and IRR One of the most commonly used measures of real estate investment performance is the internal rate of return (IRR). Internal Rate of
Return (IRR) Internal Rate of Return (IRR) The Internal Rate of Return (IRR) is the discount rate that makes the net present value (NPV) of a project zero. In other words, it is the expected compound
annual rate of return that will be earned on a project or investment.
10 Jul 2019 Net present value discounts the cash flows expected in the future back net present value of an investment based on a discount or interest rate Discounted cash flows are a way of valuing
a future stream of cash flows using a discount rate. In this video, we explore what is meant by a discount rate and 23 Jan 2012 For an internal company calculation, you should use a discount rate in
NPV, not an interest rate. The discount rate is the rate of return you could 20 Mar 2016 Using a discount rate of 10 percent, calculate the NPV of the modernization project. (Round present value
factor calculations to 4 decimal Because the IRR doesn't depend on discount rate. Instead, the IRR is a discount rate. The IRR is the discount rate that makes the NPV=0. Put another way, the
NPV = 0 –> IRR of the investment is equal to the discount rate used. NPV >0 –> IRR of the investment is higher than the discount rate used. In order to better demonstrate the cases in which negative
NPV does not signal a loss-generating investment consider the following example. Let’s say that we have calculated the projected net cash
5 Apr 2018 A higher discount rate reduces net present value. Businesses can use NPV to decide in which projects to invest. The NPV Equation. NPV is the The NPV function simply discounts all cash
flows to present values using an appropriate discount rate and totals all of the present values to a single sum. 29 Apr 2019 Step 4: Set the discount interest rate. Cash flows are discounted during
the investment period using a rate specifically established for this purpose As shown in the analysis above, the net present value for the given cash flows at a discount rate of 10% is equal to $0.
This means that with an initial investment of exactly $1,000,000, this series of cash flows will yield exactly 10%. As the required discount rates moves higher than 10%, the investment becomes less
valuable. The discount rate element of the NPV formula is a way to account for this. For example, assume that an investor could choose a $100 payment today or in a year. A rational investor would not
be It is expected to bring in $40,000 per month of net cash flow over a 12-month period with a target rate of return of 10%, which will act as our discount rate. NPV = 40,000(Month 1)/1 + 0.1 +
40,000 (Month 2)/1 + 0.1 - 250,000 = $230,000
Because the IRR doesn't depend on discount rate. Instead, the IRR is a discount rate. The IRR is the discount rate that makes the NPV=0. Put another way, the
The NPV formula is a way of calculating the Net Present Value (NPV) of a series of cash flows based on a specified discount rate. The NPV formula can be very useful for financial analysis and
financial modeling when determining the value of an investment (a company, a project, a cost-saving initiative, etc.). For NPV calculations, you need to know the interest rate of an investment
account or opportunity with a similar level of risk to the investment you're analyzing. This is called your "discount rate" and is expressed as a decimal, rather than a percent. In finance, the net
present value (NPV) or net present worth (NPW) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the
cash flow. It also depends on the discount rate. NPV accounts for the time value of money. It provides a method for evaluating and comparing capital projects or financial products with cash flows
spread over time, as in loans, investments, payouts from insurance contracts plus many o
Where: NPV, t = year, B = benefits, C = cost, i=discount rate. Two sample problem : Problem #1) NPV; road repair project; 5 yrs.; i = 4% (real discount rates, constant
The discount factor of a company is the rate of return that a capital expenditure project must meet to be accepted. It is used to calculate the net present value of When the net present value (NPV)
is used as the basis of project choice, the discount rate critically influences budget allocation. Yet there is no consensus on the The discount rate is the rate per period that we discount a dollar
in the future. If we obtain x dollars one time period in NPV calculates the net present value (NPV) of an investment using a discount rate and a series of future cash flows. The discount rate is the
rate for one period, The term discount rate refers to a percentage used to calculate the NPV, and For example, assuming a discount rate of 5%, the net present value of $2,000 ten
For NPV calculations, you need to know the interest rate of an investment account or opportunity with a similar level of risk to the investment you're analyzing. This is called your "discount rate"
and is expressed as a decimal, rather than a percent.
It is expected to bring in $40,000 per month of net cash flow over a 12-month period with a target rate of return of 10%, which will act as our discount rate. NPV = 40,000(Month 1)/1 + 0.1 + 40,000
(Month 2)/1 + 0.1 - 250,000 = $230,000 The discount rate is not a direct measure of real estate investment performance but a key variable in estimating the NPV of the net cash flows of a property
using the Discounted Cash Flow (DCF) model. Discount Rate and IRR One of the most commonly used measures of real estate investment performance is the internal rate of return (IRR).
11 Mar 2020 The discount rate we are primarily interested in concerns the calculation of your business' future cash flows based on your company's net present 2 Sep 2014 As shown in the analysis
above, the net present value for the given cash flows at a discount rate of 10% is equal to $0. This means that with an The NPV formula is a way of calculating the Net Present Value (NPV) of a
series of cash flows based on a specified discount rate. The NPV formula can be very | {"url":"https://topbinhbofddv.netlify.app/gordin87579je/discount-rate-for-npv-syr.html","timestamp":"2024-11-02T08:34:18Z","content_type":"text/html","content_length":"34791","record_id":"<urn:uuid:4b45b1a2-ffde-413d-947f-09f366c7ca03>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00064.warc.gz"} |
GeoGebra – Overcoming the Fear
This is the 5th in the draft purge series where I’m throwing stuff out over a three week period.
One month after starting MathFour.com, I came across an article about GeoGebra. I was quite taken by the software, but a little overwhelmed.
I’m not much into technology – at least when it comes to math. So the power of the tool was much more inhibiting for me than it was empowering.
So the review of it stalled.
Indeed this article was first “drafted” back in March of 2011 – more than a year ago. It only had the link to that article in it. Not much of a draft.
Lucky for us, math is math. It doesn’t change much over a year (or even a few hundred years).
So GeoGebra is pretty much as useful (and as scary) as it was a year ago.
But like all good heros, leaders and people stupid enough to think they might be either, I’m diving in. Regardless of my fear.
First: Get out the users’ manual.
So I found the GeoGebra Quickstart guide and started reading. I downloaded GeoGebra and cranked it up.
The Quickstart has three examples to try. The first one is un-intimidating – merely involving a triangle and a circle.
So I did it.
And I can share it, too!
Turns out you can “share” your work on GeoGebraTube – those guys are pretty clever, I must say!
So here’s my first ever attempt at GeoGebra goodies. Notice I named my triangle vertices and the center of the circle with real names – fun!
The Circle Triangle Dance
Following the directions, I learned about the Move Tool. Which means you can move just about anything – the whole triangle, the circle or any of the vertices!
Check out the “dance” I did with my circle and triangle:
And notice I was able to put my logo on it too!
I’m looking forward to playing some more. But I still have my concerns.
I’ll share those tomorrow. For now, I’m just going to enjoy the tool!
How about you?
Have you played with GeoGebra? Will you? How do you use it? Tell us in the comments.
Don’t forget to tweet it out, too!
You might also like:
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One Response to GeoGebra – Overcoming the Fear
1. With Geogebra, it is so easy to look at many ‘What if s’ at the same time. Example: Place four points anywhere and connect them to create a quadrilateral. Locate the midpoints of each of the four
sides and connect them to create another quadrilateral. Show that the lastest quadrilateral created is always a parallelogram no matter where the original four points were placed. First of all,
you can grab one of the orignal vertices and move it wherever and watch that the smaller quadrilateral changes shape but it looks as if it is indeed always a parallelogram. Place enough
measurements of the smaller quadrilateral to assure that it is a parallelogram. Move one of the original vertices and watch the measurements change – but always assuring the the smaller
quadrilateral is a parallelogram. I see many students much more engaged with this – especially if they are doing the constructions.
Leave a reply
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Nonlinear conjugate gradient for smooth convex functions
The method of nonlinear conjugate gradients (NCG) is widely used in practice for unconstrained optimization, but it satisfies weak complexity bounds at best when applied to smooth convex functions.
In contrast, Nesterov's accelerated gradient (AG) method is optimal up to constant factors for this class. However, when specialized to quadratic function, conjugate gradient is optimal in a strong
sense among function-gradient methods. Therefore, there is seemingly a gap in the menu of available algorithms: NCG, the optimal algorithm for quadratic functions that also exhibits good practical
performance for general functions, has poor complexity bounds compared to AG. We propose an NCG method called C+AG (``conjugate plus accelerated gradient'') to close this gap, that is, it is optimal
for quadratic functions and still satisfies the best possible complexity bound for more general smooth convex functions. It takes conjugate gradient steps until insufficient progress is made, at
which time it switches to accelerated gradient steps, and later retries conjugate gradient. The proposed method has the following theoretical properties: (i) It is identical to linear conjugate
gradient (and hence terminates finitely) if the objective function is quadratic; (ii) Its running-time bound is $O(\eps^{-1/2})$ gradient evaluations for an $L$-smooth convex function, where $\eps$
is the desired residual reduction, (iii) Its running-time bound is $O(\sqrt{L/\ell}\ln(1/\eps))$ if the function is both $L$-smooth and $\ell$-strongly convex. We also conjecture and outline a proof
that a variant of the method has the property: (iv) It is $n$-step quadratically convergent for a function whose second derivative is smooth and invertible at the optimizer. Note that the bounds in
(ii) and (iii) match AG and are the best possible, i.e., they match lower bounds up to constant factors for the classes of functions under consideration. On the other hand, (i) and (iv) match NCG. In
computational tests, the function-gradient evaluation count for the C+AG method typically behaves as whichever is better of AG or classical NCG. In most test cases it outperforms both.
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How to add if cell is not blank then...or if cell is blank then leave blank?
Thank you in advance.
I have the formula working that I need which is this:
=IF([PO Amount]@row > 100000, "Level 3", IF([PO Amount]@row > 50000, "Level 2", IF([PO Amount]@row <= 50000, "Level 1")))
But I need to add text that indicates if the cell is blank than leave blank or if it is not blank than the formula above.
This was my attempt that did not work:
=IF(NOT(ISBLANK([PO Amount])@row > 100000, "Level 3", IF([PO Amount]@row > 50000, "Level 2", IF([PO Amount]@row <= 50000, "Level 1"))
Any help would be greatly appreciated!
• Your Almost there.
=IF(NOT(ISBLANK([PO Amount]@row)),IF([PO Amount]@row > 100000, "Level 3", IF([PO Amount]@row > 50000, "Level 2", IF([PO Amount]@row <= 50000, "Level 1"))))
Or you could Try it like this.
=IF([PO Amount]@row)<>"",IF([PO Amount]@row > 100000, "Level 3", IF([PO Amount]@row > 50000, "Level 2", IF([PO Amount]@row <= 50000, "Level 1")))
In Either case you should get the same outcome.
If you found this comment helpful. Please respond with any of the buttons below. Awesome🖤, Insightful💡, Upvote⬆️, or accepted answer. Not only will this help others searching for the same answer,
but help me as well. Thank you.
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246 research outputs found
Topological superconductors have been theoretically predicted as a new class of time-reversal-invariant superconductors which are fully gapped in the bulk but have protected gapless surface Andreev
bound states. In this work, we provide a simple criterion that directly identifies this topological phase in \textit{odd-parity} superconductors. We next propose a two-orbital $U-V$ pairing model for
the newly discovered superconductor Cu$_x$Bi$_2$Se$_3$%. Due to its peculiar three-dimensional Dirac band structure, we find that an inter-orbital triplet pairing with odd-parity is favored in a
significant part of the phase diagram, and therefore gives rise to a topological superconductor phase. Finally we propose sharp experimental tests of such a pairing symmetry.Comment: 4.1 pages, 2
We study a model of a large number of strongly coupled phonons that can be viewed as a bosonic variant of the Sachdev-Ye-Kitaev model. We determine the phase diagram of the model which consists of a
glass phase and a disordered phase, with a first-order phase transition separating them. We compute the specific heat of the disordered phase, with which we diagnose the high-temperature crossover to
the classical limit. We further study the real-time dynamics of the disordered phase, where we identify three dynamical regimes as a function of temperature. Low temperatures are associated with a
semiclassical regime, where the phonons can be described as long-lived normal modes. High temperatures are associated with the classical limit of the model. For a large region in parameter space, we
identify an intermediate-temperatures regime, where the phonon lifetime is of the order of the Planckian time scale $\hbar/k_B T$.Comment: Typos corrected, references added, discussion improve
We develop a framework to analyze one-dimensional topological superconductors with charge conservation. In particular, we consider models with $N$ flavors of fermions and $(\mathbb{Z}_2)^N$ symmetry,
associated with the conservation of the fermionic parity of each flavor. For a single flavor, we recover the result that a distinct topological phase with exponentially localized zero modes does not
exist due to absence of a gap to single particles in the bulk. For $N>1$, however, we show that the ends of the system can host low-energy, exponentially-localized modes. The analysis can readily be
generalized to systems in other symmetry classes. To illustrate these ideas, we focus on lattice models with $SO\left(N\right)$ symmetric interactions, and study the phase transition between the
trivial and the topological gapless phases using bosonization and a weak-coupling renormalization group analysis. As a concrete example, we study in detail the case of $N=3$. We show that in this
case, the topologically non-trivial superconducting phase corresponds to a gapless analogue of the Haldane phase in spin-1 chains. In this phase, although the bulk is gapless to single particle
excitations, the ends host spin-$1/2$ degrees of freedom which are exponentially localized and protected by the spin gap in the bulk. We obtain the full phase diagram of the model numerically, using
density matrix renormalization group calculations. Within this model, we identify the self-dual line studied by Andrei and Destri [Nucl. Phys. B, 231(3), 445-480 (1984)], as a first-order transition
line between the gapless Haldane phase and a trivial gapless phase. This allows us to identify the propagating spin-$1/2$ kinks in the Andrei-Destri model as the topological end-modes present at the
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Strongly non-embeddable metric spaces
Enflo (1969) [4] constructed a countable metric space that may not be uniformly embedded into any metric space of positive generalized roundness. Dranishnikov, Gong, Lafforgue and Yu (2002) [3]
modified Enflo's example to construct a locally finite metric space that may not be coarsely embedded into any Hilbert space. In this paper we meld these two examples into one simpler construction.
The outcome is a locally finite metric space (Z,ζ) which is strongly non-embeddable in the sense that it may not be embedded uniformly or coarsely into any metric space of non-zero generalized
roundness. Moreover, we show that both types of embedding may be obstructed by a common recursive principle. It follows from our construction that any metric space which is Lipschitz universal for
all locally finite metric spaces may not be embedded uniformly or coarsely into any metric space of non-zero generalized roundness. Our construction is then adapted to show that the group Zω=⊕א0Z
admits a Cayley graph which may not be coarsely embedded into any metric space of non-zero generalized roundness. Finally, for each p≥0 and each locally finite metric space (Z, d), we prove the
existence of a Lipschitz injection f:Z→ℓ[p].
All Science Journal Classification (ASJC) codes
• Coarse embedding
• Locally finite metric space
• Uniform embedding
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When Will They Learn Algebra?
Themistocles, Thucydides, Peloponnesian War
I have intended to follow up on my post on primary education with one on secondary education. The key issue that must be addressed before anything else can be said is tracking, a word toxic to the
political discussion.
Certainly tracking exists; nobody expects doctors, lawyers, engineers, and Samaritans to each receive the same education. Yet nobody dares point to any specific place where it happens. The system for
distinguishing educational success1 is a gateless gate2. Surely it happens somewhere between the ages of 8 and 18, but the details get blurry quickly. Any visible methods of distinction are
ostracized, harangued, and subverted.
The most visible conflict regarding tracking relates to the teaching of algebra, and that is where we will focus our readings today.
What is algebra education?
Unfortunately, Wikipedia has no article on algebra education. If you think that’s an oversight, you can write one yourself, or comment here.
For the purposes of this article, we will generally focus on the American “Algebra I” - a course that covers mathematical topics including graphing functions, factoring polynomials, and explaining
the quadratic formula. The State of California's curriculum is a representative outline.
When does Algebra Education occur?
How long should a man’s legs be? Long enough to reach the ground.
In short, either 8th grade, 9th grade, or 8th or 9th grade. Note the formatting; some (most) school systems offer a choice, while some do not.
The US Department of Education website has a whitepaper about access to Algebra I in 8th grade. “Only 59 percent of schools offer Algebra I in 8th grade”, and 24 percent of students take it in 8th
In a recent controversial move, the San Francisco school district decided to eliminate 8th grade algebra. Their website quickly explains that they didn’t really do that, but rather they restructured
8th and 9th grade math into a different arrangement; many “Algebra I” topics are still taught in 8th grade.
How important is Algebra?
Four years of math in high school, with a strong foundation in algebra that builds from middle school, is key to higher education access. Therefore, ensuring that middle and high school students
succeed in math — and in algebra in particular — is an important issue for policy and practice. (Snipes and Finkelstein)
Algebra acts as a gatekeeper for high school graduation and post-secondary success. Students who pass Algebra 1 by the end of ninth grade are more likely to take advanced mathematics courses,
graduate from high school, and succeed in college. Yet persistent inequities in access to rigorous algebra due to issues of placement, preparation, and quality of instruction have kept the gate
closed for a large proportion of students, particularly minority and low-income students. (Stoelinga and Lynn)
To a certain extent, these takes are making a category error. At the extreme, they point to Algebra I as the gatekeeper of the gateless gate, and feel that if they can “solve” that one issue, every
student will be able to succeed at college.
Yet algebra is not a magic potion. Even if every student mastered algebra by the end of the 9th grade (and I believe it would be reasonable to expect 85% of students to do so), many would still be
ill-suited to various educational tracks.
Lowering the Bar
Another take is that if some students can’t do algebra, we simply shouldn’t require it. This makes the opposite error from the previous section. By lowering the gate, we can let everybody through!
But when the idea shows up in the Opinion pages of the New York Times, it must at least be acknowledged.
To our nation’s shame, one in four ninth graders fail to finish high school. In South Carolina, 34 percent fell away in 2008-9, according to national data released last year; for Nevada, it was
45 percent. Most of the educators I’ve talked with cite algebra as the major academic reason. … Algebra is an onerous stumbling block for all kinds of students: disadvantaged and affluent, black
and white. … it’s not easy to see why potential poets and philosophers face a lofty mathematics bar (Andrew Hacker)
There is at least the kernel of one good idea in the article: it is probably more important for the 10th percentile3 student to understand statistics than to understand algebra. But for the
“college-bound” student, the suggestion that algebra is unnecessary is simply wrong.
If you can’t pass Algebra I by the end of tenth grade, you are bad at math. You should not be sent through the educational system to become an engineer. You probably shouldn’t become a lawyer or a
business executive. You definitely shouldn’t become a philosopher. You should be able to go to art school, assuming you are good at art. Ergo, we need ability tracking and need an algebra requirement
for many of those tracks.
The hard problem of how and when to explain to students (and their parents and advocates) that they are bad at algebra and they cannot become a lawyer … is left to a future essay.
My Take
The power of instruction is seldom of much efficacy, except in those happy dispositions where it is almost superfluous. - Edward Gibbon
In the post on primary education, I propose no “math class” at all through the 4th grade. However, mathematics is still taught. Some mathematics is simple language fluency; can you understand the
sentence “I had 7 action figures and bought another one and now I have 8 action figures”. Other parts are “financial literacy” (understanding that if you have $10 you can buy 5 ice cream sandwiches
for $2 each).
I will be proposing a standard “fifth grade gap year”. Well designed systems allow for slack in the system. If, say, a pandemic occurs and months of school are lost, it can be made up. Students can
pursue the arts, music, sports, or foreign languages. Certain topics like sex education that don’t fit well into the standard structure of school can be handled sui generis. We probably need
“internet education” in this year as well.
And then secondary school. In sixth grade there will be “Sixth Grade Math”. Students who progress through the material faster … will progress through the material faster. (I am skeptical that tests
of arithmetic ability will be a reliable predictor here.)
I estimate4 that the 90th percentile student can finish Algebra I material by the end of 7th grade, the 50th percentile student can finish Algebra I material by the end of 8th grade, and the 15th
percentile student can finish Algebra I material by the end of 9th grade. As my dad would say, the exact values are an empirical question, one that can be determined by scientific research.
In the end, any classroom system that gives substantially worse results than self-directed study must be re-examined. And that bottom 15 percent … should target a high school diploma not endorsed for
math. More on that later.
There is a deeper philosophical debate whether “achieving an educational track for a higher-status job” should be considered success at all.
You may recognize “The Gateless Gate” as a translation of Mumonkan.
The 99th percentile is the top 1% of students according to a metric; the 10th percentile students are worse than 90% of students on that metric. We use “abstract math ability” as the metric here.
This is an abstract and approximate metric, I make no claim that this can be measured by a screening test, but neither does it need to be measured.
I have absolutely no research data guiding these estimates, only my instincts on how smart teenagers are and how hard mathematics is.
I agree very much, mostly. Just not in "reasonable to expect 85% of students to do so". World-best, really hard learning (and not just rote) S-Koreans make it to only 68% (were definitely taught)
at 8th grade. USofA to 39%. Russia 18% (they have ONE curriculum, at least for 90% of kids). KSA&Iran: 3% (not taught, I`d guess, but no matter: some of my Saudi college students struggled with 7+5
=? ).
At least, that is the TIMMS (2007) result for a super-basic Algebraic task. Appendix B-6. "Pen costs 1 zed more than a pencil. Joe bought 2 pens and 3 pencil for 17 zeds. How many zeds for 1 pen
and for 1 pencil? - Show your work" (so not just "trying"). Not sure, what jobs it is needed for; not in mine. I could do it easily then, still do 40 years later. My kids can't. I shall disown
them. ;)
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Free Multiplication Printables Times Tables Worksheets | Order of Operation Worksheets
Free Multiplication Printables Times Tables Worksheets
Free Multiplication Printables Times Tables Worksheets
Free Multiplication Printables Times Tables Worksheets – You may have heard of an Order Of Operations Worksheet, however exactly what is it? In this article, we’ll talk about what it is, why it’s
vital, and just how to get a Math Sheets Multiplication With any luck, this information will certainly be useful for you. Nevertheless, your trainees deserve an enjoyable, efficient means to review
the most vital principles in maths. Furthermore, worksheets are a fantastic way for students to practice new skills and evaluation old ones.
What is the Order Of Operations Worksheet?
An order of operations worksheet is a sort of math worksheet that calls for trainees to perform mathematics operations. These worksheets are separated right into three main sections: subtraction,
addition, and multiplication. They also include the assessment of parentheses and also backers. Students that are still discovering just how to do these tasks will certainly discover this type of
worksheet valuable.
The main function of an order of operations worksheet is to aid pupils discover the right method to solve math equations. If a pupil doesn’t yet recognize the concept of order of operations, they can
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Mastering Vectors – Demystifying Vector Problems with Solutions (Physics PDF)
Imagine you’re navigating a vast, uncharted sea. A gentle breeze nudges your ship, while the powerful ocean current tries to pull you in a different direction. To understand where you’re going, you
need to consider both the wind’s force and the current’s pull. This intricate dance of forces, where direction matters as much as magnitude, is what the world of vectors is all about. Vectors, in
essence, are arrows that point in specific directions, representing forces, velocities, or displacements – offering a powerful tool for understanding physics and the world around us.
Image: vectorified.com
Today, we’ll embark on a journey to conquer the often intimidating realm of vector problems. This article will equip you with the knowledge and techniques to tackle vector problems with confidence,
providing a clear understanding of how these essential tools work within physics. We’ll explore various types of vector problems, providing detailed solutions, and offering a downloadable PDF for you
to practice at your own pace.
Diving into the World of Vectors:
At its core, a vector is a mathematical object that possesses both magnitude (length) and direction. Think of it as a force with a specific strength pushing in a particular way. When trying to
understand the real-world interactions of these forces, we often face the challenge of combining and manipulating vectors. This is where vector problems come into play.
Types of Vector Problems and How to Approach Them:
Vector problems can be classified into several categories, each requiring specific techniques for solving them. Some common examples include:
• Adding Vectors: Imagine two forces pulling on a box, each in different directions. How would you calculate the overall force acting on the box? This is where vector addition comes into play.
• Subtracting Vectors: If you want to determine the difference in velocity between two objects, you would be dealing with vector subtraction.
• Multiplying Vectors by Scalars: Scalar multiplication involves changing the magnitude of a vector without altering its direction. Think of increasing or decreasing the strength of a force.
• Finding the Dot Product: The dot product provides information about the projection of one vector onto another. It’s used in applications like calculating work done by a force.
• Finding the Cross Product: The cross product yields a vector perpendicular to the original two vectors, used in concepts like calculating torque.
Essential Tools for Solving Vector Problems:
Several tools and techniques are used to make sense of these vector interactions:
• Graphical Methods: Visualizing vectors with arrows allows for intuitive understanding and can be helpful in solving simple problems. Imagine drawing two arrows representing forces, and then using
the parallelogram method to find the resultant force.
• Trigonometric Methods: Using sine, cosine, and tangent functions can help break down vectors into their components, making it easier to manipulate them.
• Component Method: This method involves resolving vectors into their horizontal and vertical components, which can then be added or subtracted. It provides a systematic approach for complex
Image: www.chegg.com
Example Vector Problems with Detailed Solutions:
Problem 1: Finding the Resultant Force
• Scenario: A person pushes a box with a force of 50 N east, while another person pushes the same box with a force of 30 N north. Find the net force acting on the box.
• Solution: We can use the Pythagorean theorem to find the magnitude of the resultant force:
Resultant Force = √(50² + 30²) ≈ 58.3 N.
To find the direction, we use trigonometry:
tan θ = (opposite side) / (adjacent side) = 30 / 50
θ = arctan (30 / 50) ≈ 31º.
The net force is approximately 58.3 N at 31 degrees north of east.
Problem 2: Determining the Resultant Velocity
• Scenario: A boat travels at 15 m/s east relative to the water. The water is flowing at 5 m/s north. Find the boat’s velocity relative to the ground.
• Solution: The boat’s velocity relative to the ground is the vector sum of its velocity relative to the water and the water’s velocity. Again, we use the Pythagorean theorem:
Resultant Velocity = √(15² + 5²) ≈ 15.8 m/s.
The angle is found using:
tan θ = 5 / 15
θ = arctan (5 / 15) ≈ 18.4º.
The boat’s velocity relative to the ground is approximately 15.8 m/s at 18.4 degrees north of east.
Mastering Vector Problems: Tips and Strategies:
• Visualize: Draw diagrams to represent the vectors and their components. This will help you understand the problem better and make it easier to solve.
• Break it Down: Resolve vectors into their components to simplify complex problems.
• Use the Right Tools: Choose the most appropriate method based on the problem’s nature (graphical, trigonometric, or component method).
• Practice, Practice, Practice: The key to mastering vector problems is consistent practice. Work through various examples and gradually increase the problem’s complexity.
Empowering Your Understanding: The Vector Problems with Solutions PDF:
To further enhance your learning, we have prepared a downloadable PDF containing a comprehensive collection of vector problems with detailed solutions. This resource will provide you with ample
opportunities to practice and solidify your understanding of these critical concepts.
Vector Problems With Solutions Physics Pdf
Conclusion: Navigating the Vector Landscape:
Vectors, like the navigational forces of wind and current, play a vital role in understanding physics and other fields. By unraveling their intricacies, we gain a deeper appreciation for how forces
interact and shape our world. We hope this article has equipped you with the tools and confidence to tackle vector problems with ease. Remember to practice, explore, and continue your journey of
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mlogit 1.1-1
• minir update, the JSS paper is cited and used in the CITATION file
mlogit 1.1-0
• major update, mlogit now depends on dfidx, mlogit.data, mFormula and index are deprecated
• the name of the coefficients is changed, i.e. air:income is now income:air
mlogit 1.0-3
• some numerical disperancies were caused by Rout.save files. Some IGNORE tags are introduced to fix that.
mlogit 1.0-2
• bug in model.frame : indexing by a factor and not a character to get the relevant subset of id in the index
mlogit 1.0-0
• a new package version which coincides with the Journal of Statistical Software article
• some bugs fixed in the vignettes (citet/citep replaced by markdown-like commands), the figure is added in the mixed logit vignette
• Rout.save files are added in the test directory
• effects.mlogit now returns a matrix
mlogit 0.4-2
• the R files for the vignettes are added (they were note while building the package on RForge)
• thank’s to Mallick Hossain, a bug is fixed in the logsum function
mlogit 0.4-1
• the Cracker, Catsup and Car data set are back in mlogit since AER, flexmix and mlogitBMA run examples based on them.
• the alt vector in the index is now carrefully checked in case of alternative subseting or reference level change.
mlogit 0.4-0
• the main vignette is improved, writen in markdown and now and split by sections
• the Exercises vignette is splited and is now writen in markdown
• importantly, the Cholesky matrix is now coerced to a vector by rows and note by columns.
• the mlogit function was checked and improved.
• implementation of the computation of the standard deviations of the covariance matrix of the random parameters, using the delta method.
• some data sets are removed
mlogit 0.3-0
New features
• zbu and zbt distributions are added : these are one-parameter distributions for which the lower bond is 0,
• a logsum function is provided to compute the log-sum or the inclusive utility of a random utility model,
• group-hetheroscedastic model can be estimated by setting the relevant covariates in the 4th part of the formula,
• the linear predictor is now returned by mlogit,
• correlation can still be a boolean, but can also be a character vector if one wants that a subset of the random parameters being correlated.
data sets
• the RiskyTransport data set (used in the vignette to illustrate the estimation of the mixed logit model
• the NOx data set (used in the vignette to illustrate the estimation of the multinomial and group-heteroscedastic logit model),
• the JapaneseFDI data set (used in the vignette to illustrate the estimation of the nested logit model)
A new vignette called mlogit2 is added ; this is the draft version of an article submitted to the Journal of Statistical Software ; it is less exhaustive, but better writen thant the original mlogit
• the id series (one observation per choice situation) was badly constructed, it is now fixed
• the levels of the choice variable are now equalized to the those of the alt variable, allowing the case were some alternatives are never chosen
• mlogit is now able to estimate models with singular matrix of covariates. At the end of model.matrix.mformula, the linear dependent columns of X are removed
• there was a bug in the triangular distribution which is now fixed
• bug in the effects method fixed
mlogit 0.2-4
• the list of primes used to generate halton sequences was too short, its length has been increased
• halton sequences where used to estimate mixed logit even for the default value of halton (NULL), this has been fixed
• the contribution of each observation to the gradient is not returned as the ‘gradient’ element of mlogit objects
• the distributions are now checked for rpar and an error is returned in case of unknown distribution
mlogit 0.2-3
• some sys.frame() changed to parent.frame()
mlogit 0.2-2
• ranked-order models can be now estimated ; a new argument called ‘ranked’ is introduced in mlogit.data which performs the relevant transformation of the data.frame. The estimated model is then a
standard multinomial logit model
• multinomial probit model is now estimated by setting the new probit arguments to TRUE
• for the mixed logit model, different draws are now used for each observation
• a predict method is now available for mlogit objects
• a coef method is added which removes the fixed argument
• constPar can now be a named numeric vector. In this case, default starting values are changed according to constPar
• the vcov method for mlogit objects is greatly enhanced.
• mlogit objects now have two elements which indicate the fitted probabilities : fitted is the estimated probability for the outcome and probabilities contains the fitted probabilities for all the
• mentions to ‘alt’ in the names of the effects is canceled ; moreover, the intercepts are now called altname:(‘intercept’)
• a ‘choice’ attribute is added to mlogit.data objects
• an effects method is provided, which computes the marginal effects of a covariate
mlogit 0.2-1
• all the rda files are now compressed
mlogit 0.2-0
• all the models could normally be estimated on unbalanced data
• the three tests are added, i.e. a new scoretest function and specific methods for waldtest and lrtest from the lmtest package
• the model.matrix method for mlogit objects is now exported
mlogit 0.1-8
• mFormula modified so that models can be updated
• likelihood has been rewriten for the heteroscedastic logit model, the computation is now much faster
• nested logit models with overlapping nests are now supported; nests = “pcl” enables the estimation of the pair combinatorial logit model
• the norm argument is added to rpar
• the logLik argument is now of class logLik
• mlogit.data is modified so that an id argument can be used with data in long shape
• the argument of mlogit.data used to define longitudinal data is now called id.var
• mlogit.lnls is corrected so that the estimation of multinomial models can handle unbalanced data (pb with Reduce)
• the three tests are temporary removed
mlogit 0.1-7
• a bug in mFormula (effects vs variable) is fixed
mlogit 0.1-6
• a third part of the formula is added : it concerns alternative specific variables with alternative specific coefficients
• improved presentation for the Fishing dataset.
• a bug (forgotten drop = FALSE) corrected in model.matrix.mFormula
• Electricity and ModeCanada datasets are added
mlogit 0.1-5
• if the choice variable is not an ordered factor, use as.factor() instead of class() <- “factor”
• cov.mlogit, cor.mlogit, rpar , med, rg, stdev, mean functions are added to extract and analyse random coefficients.
• a panel argument is added to mlogit so that mixed models with repeated observation can be estimated using panel methods or not
• a problem with the weights argument is fixed
• the estimation of nested logit models with a unique elasticity is now possible using un.nest.el = TRUE
• the estimation of nested logit models can now be done with or without normalization depending on the value of the argument unscaled
mlogit 0.1-4
• mlogit didn’t work when the dependent variable was an ordered factor in a “wide-shaped” data.frame.
• the reflevel argument didn’t work any more in version 0.1-3.
mlogit 0.1-3
• major change, most of the package has been rewriten
• it is now possible to estimate heteroscedastic, nested and mixed effects logit model
• the package doesn’t depend any more on maxLik but a specific optimization function is provided for efficiency reason
mlogit 0.1-2
• robust inference is provided with meat and estfunc methods defined for mlogit models.
• subset argument is added to mlogit so that the model may be estimated on a subset of alternatives.
• reflevel argument is added to mlogit which defines the base alternative.
• hmftest implements the Hausman McFadden test for the IIA hypothesis.
• mlogit.data function has been rewriten. It now use the reshape function.
• logitform class is provided to describe a logit model: update, model.matrix and model.frame methods are available.
mlogit 0.1-1
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Maximum Power Point Tracking Using Artificial Neural Network for DC Loads
ICONNECT - 2017 (Volume 5 - Issue 13)
Maximum Power Point Tracking Using Artificial Neural Network for DC Loads
DOI : 10.17577/IJERTCONV5IS13088
Download Full-Text PDF Cite this Publication
Harini B. R, Keerthiga A, Sangavi M, Vimala A, Karthick T., 2017, Maximum Power Point Tracking Using Artificial Neural Network for DC Loads, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY
(IJERT) ICONNECT – 2017 (Volume 5 – Issue 13),
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• Authors : Harini B. R, Keerthiga A, Sangavi M, Vimala A, Karthick T.
• Paper ID : IJERTCONV5IS13088
• Volume & Issue : ICONNECT – 2017 (Volume 5 – Issue 13)
• Published (First Online): 24-04-2018
• ISSN (Online) : 2278-0181
• Publisher Name : IJERT
• License: This work is licensed under a Creative Commons Attribution 4.0 International License
Text Only Version
Maximum Power Point Tracking Using Artificial Neural Network for DC Loads
Harini B. R*1, Keerthiga A*2, Sangavi M*3, Vimala A*4, Karthick T5.
UG Scholar, UG Scholar, UG Scholar, UG Scholar, Assistant professor.
Department of Electrical and Electronics Engineering
K.L.N College of Information Technology, Madurai, Tamil Nadu 625 015. India.
Abstract- Due to the fossil fuels depletion and to protect the environment we are focussing on renewable energy sources Solar energy helps in reducing the green house gases. Pv technique is used to
collect the rays from sun light and directly converted into electricity. To collect the maximum power PV panel with MPPT technique is used at all weather conditions. ANN is used to maintain the
voltage constant. Therefore overall efficiency is increased to about 10%. In this paper we have designed a prototype model inclusive of techniques that the need to harness the solar energy.
Keywords: Maximum Power Point, Buck-Boost Converter, Neural Network Architecture
1. INTRODUCTION
MAXIMUM Power point is a technique that Grid Tie Inverters , Solar Battery Chargers, other similar devices used to get maximum possible power from solar panels. Solar cells have a complex
relationship between solar irradiation, resistance and temperature that produces nonlinear V-I curve. The MPPT system given sample out the output of the cells and applies the proper load to
obtain maximum power for any give environmental condition ranging from a clear sky to a heavily clouded one, from rainfall to misty and even foggy. Therefore, PV cells have a complex relationship
between maximum power that can produce and the environmental operating conditions. FillFactor(FF) that gives the electrical behaviour of the cell. In tabulated data it is often used to estimate
the maximum power that a cell can provide. At the same time with an optimal load under given conditions, the power P=FF*VOC*ISC; VOC and ISC are the open circuit voltage and short circuit current
respectively. For most purposes FF, VOC and ISC are enough pieces of information to give a useful conclusion on the electrical behaviour of a cell operating under various conditions [2, 3]. For
any given set of operating conditions, cells have a single operating point where the values of V and I, for each cell produces the maximum power output values. Then the values are corresponding
to the resistance load which equal to V/I as derived by the Ohms Law. A PV cell has an approximately exponential relationship between current and voltage. From basis circuit theory, the power
delivered from or to a devices is optimized at the I. The point at which dI/dV of the I-V characteristic curve is equal and opposite of I/V ratio and the point where dP/dV=0 is known as the Knee
of the curve which is the maximum power point. The efficiency of the typical solar panel is about 30 to 40 percent. Maximum Power Point Tracking technique is used to improve the efficiency of the
panel. According to Maximum Power Transfer Theorem, the power output of the circuit is maximum when the Thevenin Impedance of the circuit (Source impedance) matches with the load impedance. Hence
the problem of tracking the maximum power point reduces to an impedance matching problem [4,5,6].
2. BASIC IDEA
It is necessary to design a solar panel to extract maximum power at all conditions, because solar cells have a non-linear current-voltage characteristic, with the output power varying in
correspondence with the voltage across the cell. Therefore, MPPT is used to extract and utilise the maximum portion of the incoming solar radiation. The Photovoltaic Systems are one of the best
direct solar to electrical energy conversion systems. A Photovoltaic System is an array of homogenously series connected Solar Cells, each of them possessing the typical V-I characteristics. The
main aim of the PV system is to absorb radiation and to generate electricity by using transducer. These systems are clean, reduce the Greenhouse Gases, and are nonpolluting. A typical PV system
consists of Batteries, PV modules, a DC-AC Inverter, a Charge Controller and the PV modules to generate DC Electricity. The Inverters convert the DC current into AC current. But the problem
arises in electricity generation due to high capital cost and climate conditions such as solar radiation and ambient temperature . To extract maximum power from PV module under all uncertain
conditions, it is necessary to include the charge controllers in MPPT system. MPPT checks the PV array output and compares it with battery voltage and finally fixes the best voltage that the
array can produce and convert it to get maximum current. MPPT is most effective under the following conditions:
1. Cloudy, Cold weather, or hazy days: PV modules work better at hot temperatures.
2. When battery is deeply discharged the system can extract more current and charge the battery, if the state of charge in the battery is lower.
3. AIM
The problems encountered with basic algorithms for finding the Maximum Power Point Tracking are described here as under:
1. In the classical Perturb and Observe Algorithm (P&O) which compares only two points, the Current Operation Point and the Subsequent Perturbation Point to observe their changes in power.
Based on the difference in the output power the controller increases or decreases the PV array output voltage. If these two points are negative points weighted, the duty cycle of the
converter should decrease and if these points are positive weighted, the duty cycle of the converter should increase [8]. If it has one positive and one negative weighting Maximum Power
Point is not reached because the solar radiation changes rapidly and so the duty cycle is not able to change itself.
2. Though the Incremental Conductance Algorithm [7] has better performance than P&O algorithm, it produces oscillation and perform erratically under rapidly changing atmospheric conditions.
The computation time is increased and the sampling frequency is depressed than P&O Algorithm.
3. PO techniques and incremental conductance techniques are limited in their tracking speed because they will make fixed size adjustments to the operating voltage in each of the iterations.
Incremental conductance method has reduced efficiency in its tracking stage when the operating point fluctuates between two significantly maximum power points.
4. In the Constant Voltage Algorithm [7], to measure the open circuit voltage, the current from the PV array must be said to zero and then said to 76% of the measured Voltage. Due to this, a
considerable amount of energy is pined away when the current is set to zero. Though it is simple and low in cost to implement it reduces the efficiency of the array due to the
interruptions in this algorithm.
4. MAIN IDEA
To overcome all the negative points and drawbacks of the above basic algorithms the present prototype is designed with the improved features. The highlighting points of MPPT using ANN are as
1. Algorithm of three point weight comparison which acts as an antidote to the two point has three distinct points, namely the Current Operation Point A, a point B perturbed from point A
and a point C doubly perturbed in the opposite direction from point B.
2. By storing current voltage curves, their maximum power points and using a classifier based system, the algorithm aims to improve the tracking speed of PO based techniques.
5. OPERATION WITH BATTERIES
The batteris help in providing backup when the plant operations stopped. Due to the non-availability of solar radiation for a prolonged time, a solar collectors wont be able to collect the
required amount of radiation and that period of time will bring plant operation to a halt. It plays a vital role in storing a reasonable amount of energy to provide backup.
1. An Off -Grid PV power system uses batteries to supply power to its loads. Though the fully charged battery may have its operating voltage close to the PV Arrays Peak Power Point,
this is may be true or happen at the sunrise time when the battery is partially is charged. Charging may begins at the voltage only below the Arrays Peak Power Point. MPP Tracking
with its sophisticated techniques and well design protocols can resolve the mismatch.
2. When batteries in the Off-Gris system are fully charged and the production exceeds the local loads, the MPP Tracking can no longer operate the PV Array at its Peak Power Point, as
the excess power has nowhere to go. Until the production exactly matches the demand, the MPP Tracking must then shift the arrays operating point. An alternative approach, commonly
used in spacecraft is to divert the surplus PV power into a resistive load into a resistive load allowing the array to operate continuously at its Peak Point.
6. PROJECT STUDY:
An Improved MPPT System using Artificial Neural Network is a modification of the classical P&O Technique which consists a PV module, a DC-DC Converter, a controller and a load. A feed-forward
propagation ANN based controller is added here which takes Ambient Temperature(T) and Solar Radiation(G), as two out of its total four inputs, and converts them into information based on the
Instantaneous Optimum Voltage (V Optimum) of the PV system in order to ensure the maximum power operation.
Figure 1 MPPT using ANNs Block Diagram
The ANN tries to simulate its learning process through the various input fed to it during each cycle of data interpretation. It changes its structure depends on the external and internal
information which flows in and out of the network [9]. However the major advantage of using the network is that response of the Proposed MPPT System is faster than the classical P&O Algorithm so
as to increase the tracking efficiency.
Figure 2 flow chart of the proposal design model Flow chart described as:
Step1: The Temperature Coefficient of Short Circuit Current ISC and the Temperature Coefficient of Open Circuit Voltage VOC are obtained from the PV array and stored.
Step2: The ANN now have the values of Ambient Temperature T and Incident Solar Radiation G.
Step3: Then the controller calculates the value of V Optimum.
Step4: Get the value of V Operation of the PV array. If V Operation V Optimum, then Duty Cycle is calculated and it is controlled, else the flow switches on to get the next values of Solar
Radiation and Ambient Temperature.
7. DESIGNING USING MATLAB® – SIMULINK®
Some of the common circuits using Conventional Logic and their proposed Reversible Logic are explained as follows:
1. PV Array Design
The PV Arrays model as designed in SIMULINK® is shown as insolation and temperature are considered as two inputs of the PV Array. The Temperature is taken as a Saw-Tooth waveform and
Insolation is taken in the form of rising step input with the values ranging from 200-1000 W.m-2. Temperature is set between the levels via saturation and Insolation is fed to a gain. ISC
determined by the Diode equation function and summers, which gives the modules output current. The product of this current and the incident sinusoidal voltage gives the generation of power.
The entire system is masked and the module values arranged in series are one while those arranged in parallel are 50 which raises the current dramatically. The voltage and current values are
multiplied and the output of these two are given to the respective Graph Blocks.
Figure 3 Unmasked pv subsystem
Figure 4 PV-Module SIMULINK® Model
Figure 5 I-V Characteristics
Figure 6 P-V Characteristic Curve
Figure 7 Output waveforms of a PV module
2. Buck-Boost Converter
One of the types of DC-DC converter is the Buck- Boost converter which has an output voltage magnitude either less than or greater than the magnitude of the input voltage magnitude. It is
described by a voltage source that is connected in parallel to an inductor, a capacitor, a reverse-based free-wheeling diode and a load resistance R at the output terminal.
Figure 8 Buck-Boost Converter using PWM-PI Controller
Figure 9 Unmasked Buck-Boost Converter
Figure 10 Converters Output Voltage
Figure 11 Converters Output Current
3. Artificial Neural Network Design
The ANN has been designed using ISC and VOC equations[1] which are described as:
ISC = ISC*(G/G*)*ISC + i(T – T*)
VOC = VOC* + v(T – T*) – (ISC ISC*)R
ISC = Short Circuit Current VOC = Open Circuit Voltage
G* = Reference Solar Radiation =1000W.m-2 ISC* =PV ISC at Ref. Solar Radiation = 50 A i = Temperature Co-efficient of ISC = 2
T* = Reference Temperature = 25oC VOC* = VOC at Ref. Temperature = 25 V v = Temperature Co-efficient of VOC = 0 R = Resistance = 5
Figure 12 Artificial Neural Network Architecture
Figure 13 ANN Equations Design
4. Concluding Model
The concluding model is the combined designs of the Photo Voltaic Module, the Artificial Neural Network Controller and the Buck boost converter. The model is shown as alongside, Fig14.
Figure 14 Projects Overall Simulink Model
Figure 15 DC-DC Converter Subsystem
Figure 16 ANNs Predicted Output Waveform
Figure 17 Controller Waveform
Figure 18 Final Output Power Waveform
Figure 19 Final Output Current Waveform
8. CONCLUSION:
This paper discusses neural network based MPPT. Under any variation in atmospheric conditions, by using neural network, point of maximum power is specified fast and precisely.Another advantage of the
neural network in PVmaximum power-point tracking is its better dynamic performance in comparison with the other methods. Also the maximum power point is tracked by dc-dc buck-boost chopper. So the
maximum power solar energy and the best efficiency are obtained.
1. Mahmoud A. Younis (University Tenaga National), Tamer Khatib (National University of Malaysia), Mushtaq Najeeb (Universiti Tenaga National), A Mohd. Ariffin (University Tenaga National), An
Improved Maximum Power Point Tracking Controller for PV Systems Using Artificial Neural Network, ISSN 0033-2097, R. 88 NR 3b/2012, Pg. 116-121.
2. Edward E. Anderson, "Fundamentals for Solar Energy Conversion", Addison Wesley Pub. Co., 1983.
3. G.N.Tiwari and M. K. Ghosal, "Fundamentals of Renewable Energy Sources", Narosa Publishing House, New Delhi, 2007.
4. M. A. Vitorino, L. V. Hartmann, A. M. N. Lima et al., "Using the model of the solar cell for determining the maximum power point of photovoltaic systems," Proc. European Conference on Power
Electronics and Applications. pp. 1-10, 2007.
5. D. Yogi Goswami, Frank Kreith, Jan. F. Kreider, "Principles of Solar Engineering", 2nd Edition, Taylor & Francis, 2000, India Reprint, 2003, Chapter 9, Photovoltaics, pp. 411-446.
6. Solar Energy, Third Edition, by S. P. Sukhatme and J. K. Nayak, Tata McGraw-Hill Publication Co. Ltd. New Delhi, 2008, Chapter 9, Section 1, pp. 313-331.
7. Comparison of Photovoltaic array maximum power point tracking technique Patrick L Chapman, Trishan Esram.
8. E. Alpaydin, Introduction to Machine Learning, Cambridge, MA: MIT Press, 2004.
9. Neural Networks A Classroom Approach, Satish Kumar, Tata McGraw-Hill Education.
10. Dharmendra Kumar ingh, Pragya Patel, Anjali Karsh and Dr.A.S.Zadgaonkar.Analysis of Generated Harmonics Due To CFL Load on Power System Using Artificial Neural Network, International Journal
of Electrical Engineering and Technology (IJEET), 5(3), 2014, pp. 5668.
11. Amit Shrivastava and Dr. S. Wadhwani, Madhav Institute of Technology & Science. Application of Time-Domain Features with Neural Network For Bearing Fault Detection, International Journal of
Electrical Engineering and Technology , 3(2), 2012, pp. 151155.
12. M. Mujtahid Ansari, Nilesh S. Mahajan and Dr. M A Beg. Characterization of Transients and Fault Diagnosis in Transformer by Discrete Wavelet Transform and Artificial Neural Network,
International Journal of Electrical Engineering and Technology, 5(8), 2014, pp. 2135.
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The CASINO forum
Constrained derivatives.
Hellow QMC folks.
As described in "Optimization of quantum Monte Carlo wave functions by energy minimization" of Julien Toulouse.
"The most straightforward way to energy-optimize linear parameters in wave functions is to diagonalize the Hamiltonian
in the variational space that they define, leading to a generalized eigenvalue equation." Energy calculated with wave function depended on parameters p is:
E(p) = <ψ(p)|Ĥ|ψ(p)>/<ψ(p)|ψ(p)>
which is Rayleigh quotient. To determine the stationary points of E(p) or solving ∇E(p) = 0 we have to solve following generalized eigenvalue problem, with ψ(p) expand to first-order in the
parameters p:
H · Δp = E(p) * S · Δp
where elements of the matrices S and H approach the standard quantum mechanical overlap integrals and Hamiltonian matrix elements in
the limit of an infinite Monte Carlo sample or exact ψ(p), hence their names. Thus, the extremum points of ψ(p*) (extremum values E(p*))
of the Rayleigh quotient are obtained as the eigenvectors e (eigenvalues λ(e)) of the corresponding generalized eigenproblem. If the second-order expansion of ψ(p) is not small, this does not ensure
the convergence in one step and may require uniformly rescaling of ∆p to stabilise iterative process.
I would like to clarify what to do if the parameters are constrained, e.g. if p is the full set of parameters constrained by a @ p = b (numpy notation) Then we can compute unconstrained derivatives
df(p)/dp and find corresponding derivatives of f subject to the constraint by projection, as described in "Constrained Differentiation" G. SCHAY. with projector p = I - a.T @ (a @ a.T)**-1 @ a and
projected_derivatives is df(p)/dp @ p. But projected_derivatives is a vector with a dimension equal to the dimension of p, and we need a derivatives vector with a dimension equal to the number of
independent parameters let's call it independent_derivatives. What should we do? One can see that independent_derivatives should be projected with subset of p to subset of df(p)/dp @ p:
independent_derivatives @ p_subset = (df(p)/dp @ p)_subset
which gives us the answer (since the matrix p_subset is not singular):
independent_derivatives = (df(p)/dp @ p)_subset @ p_subset**-1
I would like to clarify whether this method or a similar one is used in CASINO or others codes.
Best Vladimir.
In Soviet Russia Casino plays you.
Re: Constrained derivatives.
In CASINO we have (so far) dealt with constraints by using them to eliminate parameters, leaving us with an independent set of parameters. For simple things like the u(r) term in the Jastrow factor
it is easy enough to express the coefficient of the first-order term in terms of the zeroth-order term to satisfy the Kato cusp conditions. For things such as the f term we write out linear
constraint equations and use Gauss-Jordan elimination to express parameters corresponding to pivot columns in terms of the remaining free parameters.
Best wishes,
Re: Constrained derivatives.
Hello Neil.
I wanted to talk about the partial derivatives of the wave function and local energy with respect to the Jastrow and Backflow parameters, that are needed for optimization. These partial derivatives
could be easily obtained analiticaly if the parameters were not linearly dependent. I noticed that such partial derivatives are calculated numerically in the Сasino. At least in "old style" Jastrow
and backflow. Although they could be calculated analytically and then projected onto a subspace of independent parameters. For each vector p this is a linear transformation depending on p, more
precisely from nonlinear parameters.
Best Vladimir.
In Soviet Russia Casino plays you.
Re: Constrained derivatives.
For the "old" Jastrow factor in pjastrow.f90, the subroutine "get_linear_basis" returns analytical derivatives w.r.t. the independent subset of linear parameters. At present this is only used in the
"varmin-linjas" optimisation method, however.
Numerical differentiation w.r.t. Jastrow parameters shouldn't be too problematic, because the dependence on those parameters is very simple: the Jastrow is linear in everything apart from cutoff
lengths, so the local energy is a quadratic function of the parameters. Numerical differentiation may not be the most efficient approach, but I would have thought it would be fairly safe.
Analytical derivatives might be more important for cutoff lengths and for backflow parameters.
Best wishes,
Re: Constrained derivatives.
We need the linear dependence of the parameters to be preserved in the vicinity of the point at which we calculate the partial derivatives w.r.t parameters.
Therefore we need jacobian matrix like this.
Constrained partial derivatives is nullspace of ∇g which may be function of parameters (and only) or not. It's easy to project a vector to nullspace.
In Soviet Russia Casino plays you.
Re: Constrained derivatives.
An alternative approach for dealing with the homogeneous linear constraints would be to use SVD to find the basis spanning the nullity (the solution space) and then the parameters in correlation.data
would be the coefficients of those basis vectors, again giving an independent set of parameters.
It's possible that this might be better from a numerical point of view (although even then I am not sure, because the pivoted Gauss-Jordan elimination approach in CASINO should be robust). It would
make the parameters in correlation.data more abstract and hence make it less easy for the user to know what each parameter actually is.
Best wishes,
Re: Constrained derivatives.
I tried to explain above that my idea is different but independent set of parameters is the same as yours. As we calculate the gradient in the space of
all parameters, we know how to project gradient to nullspace of ∇g (from above), let me remind you that g(p) = c - parameters constraints.
At the point p0, then g(p0 + dp) = ∇g * dp + o(dp). If dp is nullspace of ∇g, which is the tangent space of constraint surface at p0, then g(p0 + dp) = o(dp) and ∇g * dp = 0, that is constraint is
satisfied at p0 + dp. But if dp is not in the nullspace of ∇g then we can project it there. Moreover corresponding differential of some function F(p) subject to constrain is ∇F(p) * M(p) * dp where
∇F(p) is unconstrained gradient (easy to calculate) and M(p) is projector to nullspace of ∇g(p) i.e. annihilator matrix. Next you need to go to an independent set of parameters in ∇F(p) * M(p), for
this you need to solve following equation for ∇F_ind, this is the most obscure part:
∇F * M * d[ p_ind | 0 ] = [ ∇F_ind | ∇F_dep ] * M * d[ p_ind | 0 ]
UPD: I'm writing python code that illustrates this approach, but it takes a couple of weeks. Jastrow varmin optimization is already running on this approach with a speed comparable to the speed of
Fortran in CASINO.
Best Vladimir.
In Soviet Russia Casino plays you.
Re: Constrained derivatives.
The main problem in calculating the analytical energy gradient w.r.t backflow parameters is that in equation of Ti we have laplacian of backflowed slater determinant as
additional differentiation this laplacian w.r.t. backflow parameters requires the calculation of 3-rd partial derivatives of the slater determinant w.r.t to electronic coordinates.
In Soviet Russia Casino plays you.
Re: Constrained derivatives.
Sorry it has taken me ages to reply.
Analytical derivatives w.r.t. backflow parameters will certainly get messy.
Obviously I agree that you can evaluate linearly constrained derivatives for parameters by evaluating unconstrained derivatives and using linear algebra. This could be done using the projection
operations that you suggest, or could be done using the Gauss-Jordan methods already in CASINO to express the full set of parameters in terms of the independent parameters (with the dependent
parameters being expressed as a matrix multiplied into the independent parameters). One can then use the chain rule to express the gradient w.r.t. the independent parameters in terms of the gradient
w.r.t. all the parameters.
There's nothing stopping the use of analytic derivatives for pjastrow.f90; I just haven't got round to it!
Best wishes,
Re: Constrained derivatives.
I have already written all the code, now testing and any cutoffs are still outside of my algorithm. Not sure if they need to be optimized at all.
Unfortunately third partial derivatives of the wave function w.r.t the electron coordinates are necessary.
I hardly read the CASINO code except for two procedures construct_C and construct_A (for now has become construct_A_sym and construct_A_asym),
which were needed for the correct interpretation of the CASINO input files. Your scientific articles on the contrary were very useful, for example:
PHYSICAL REVIEW B 72, 085124 2005, "Variance-minimization scheme for optimizing Jastrow factors", N. D. Drummond and R. J. Needs.
but this method seems too complicated to me.
Best wishes,
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Mole-Mole Examples
Mole-Mole Examples
The solution procedure used below involves making two ratios and setting them equal to each other. When two ratios are set equal, this is called a proportion and the whole technique (creating two
ratios, setting them equal) is called ratio-and-proportion.
One ratio will come from the coefficients of the balanced equation and the other will be constructed from the problem. The ratio set up from data in the problem will almost always be the one with an
unknown in it.
Key point: the two ratios have to be set up with equivalent things in the same relative place in each ratio. A bit confusing? I will elaborate on this below.
After setting up the proportion, you will cross-multiply and divide to get the answer.
What happens if the equation isn't balanced? Then your first step is to balance it. You cannot do these problems correctly without a balanced equation. The ChemTeam is constantly amazed at the number
of people who forget to balance the equation first. One note: remember that there are chemical equations where all the coefficients have a value of one. These equations are already balanced. The term
that is often used for these equations is "balanced as written."
Here is the first equation we'll use:
N[2] + 3H[2] ---> 2NH[3]
Example #1: When 2.00 mol of N[2] reacts with sufficient H[2], how many moles of NH[3] will be produced?
Comments prior to solving the example
(a) The equation is already balanced.
(b) The ratio from the problem will have N[2] and NH[3] in it.
(c) How do you know which number goes on top or bottom in the ratios? Answer: it does not matter, except that you observe the next point ALL THE TIME.
(d) When making the two ratios, be 100% certain that numbers are in the same relative positions. For example, if the value associated with NH[3] is in the numerator, then MAKE SURE it is in both
(e) Use the coefficients of the two substances to make the ratio from the equation.
(f) Why isn't H[2] involved in the problem? Answer: the word "sufficient" removes it from consideration.
1) We will use this ratio to set up the proportion:
2) That means the ratio from the equation is:
2 mol NH[3]
1 mol N[2]
3) The ratio from the data in the problem will be:
4) The proportion (setting the two ratios equal) is:
2 mol NH[3] x <--- both values in the numerator are related to ammonia
––––––––– = ––––––––––
1 mol N[2] 2.00 mol N[2] <--- both values in the denominator are related to nitrogen
5) Solving by cross-multiplying and dividing gives:
(1 mol N[2]) (x) = (2 mol NH[3]) (2 mol N[2])
(2 mol NH[3]) (2.00 mol N[2])
x = –––––––––––––––––––––
1 mol N[2]
x = 4.00 mol NH[3] produced.
Comment: Notice how the ratio-and-proportion is written. Written in this manner:
x 2.00 mol N[2]
–––––––– = –––––––––
2 mol NH[3] 1 mol N[2]
is equally correct. Just make sure to keep the two quantities associated with the NH[3] and the two associated with the N[2] on the same side.
The ChemTeam tends to not write the ratio and proportion in the style of the one just above, so you won't see it any more.
Example #2: Suppose 6.00 mol of H[2] reacted with sufficient nitrogen. How many moles of ammonia would be produced?
1) Let's use this ratio to set up the proportion:
2) That means the ratio from the equation is:
2 mol NH[3]
3 mol H[2]
3) The ratio from the data in the problem will be:
4) The proportion (setting the two ratios equal) is:
2 mol NH[3] x
–––––––– = –––––––––– <--- the two ammonia values are in the numerators and the two hydrogen value are in the denominator
3 mol H[2] 6.00 mol H[2]
5) Solving by cross-multiplying and dividing gives:
3x = 12.00 mol
x = 4.00 mol NH[3] produced
Example #3: We want to produce 2.75 mol of NH[3]. How many moles of nitrogen would be required?
Before the solution, a brief comment: notice that hydrogen IS NOT mentioned in this problem. If any substance ISN'T mentioned in the problem, then assume there is a sufficient quantity of it on hand.
Since that substance isn't part of the problem, then it's not part of the solution.
1) Let's use this ratio to set up the proportion:
2) That means the ratio from the equation is:
2 mol NH[3]
1 mol N[2]
3) The ratio from the data in the problem will be:
2.75 mol NH[3]
4) The proportion (setting the two ratios equal) is:
2.75 mol NH[3] 2 NH[3]
––––––––––– = –––––––
x 1 mol N[2]
5) Solving by cross-multiplying and dividing (plus rounding off to three significant figures) gives:
x = 1.38 mol N[2] needed.
Here's the equation to use for the next three examples:
2H[2] + O[2] ---> 2H[2]O
Example #4: How many moles of H[2]O are produced when 5.00 moles of oxygen are used?
1) Here are the two substances in the molar ratio I used:
O[2] 1 mol O[2]
–––– and the ratio is –––––––––
H[2]O 2 mol H[2]O
2) The molar ratio from the problem data is:
3) The proportion to use is:
5.00 mol O[2] 1 mol O[2]
–––––––––– = –––––––––
x 2 mol H[2]O
x = 10.0 mol of H[2]O are produced
Example #5: If 3.00 moles of H[2]O are produced, how many moles of oxygen must be consumed?
1) Here are the two substances in the molar ratio I used:
It's a 1:2 ratio.
2) The molar ratio from the problem data is:
3.00 mol H[2]O
3) The proportion to use is:
x 1 mol O[2]
–––––––––––– = ––––––––
3.00 mol H[2]O 2 mol H[2]O
x = 1.50 mol of O[2] consumed
For the examples below, I left off the mol unit on the ratio from the coefficients of the balanced equation. Also, I used a different way to format the ratios and the proportional set up.
Example #6: How many moles of hydrogen gas must be used, given the data in example #5?
Solution #1:
1) Here are the two substances in the molar ratio I used:
2) The molar ratio from the problem data is:
3) The proportion to use is:
x = 3.00 mol of H[2] was consumed
Notice that the above solution used the answer from example #5. The solution below uses the information given in the original problem:
Solution #2:
The H[2] / H[2]O ratio of 2/2 could have been used also. In that case, the ratio from the problem would have been 3.00 over x, since you were now using the water data and not the oxygen data.
Example #7: Use the following equation:
C[3]H[8] + 3O[2] ---> 3CO[2] + 4H[2]
(a) How many moles of O[2] are required to combust 1.50 moles of C[3]H[8]?
(b) How many moles of CO[2] are produced?
(c) How many moles of H[2] are produced?
Solution to (a):
1) Use this ratio from the balanced chemical equation:
Note the style change! Just by the by, students often are confused when they see information presented to them in a different (but mathematically equivalent) style. Be aware!
2) Use this ratio from the problem:
3) Set equal and solve:
^1⁄[3] = ^1.50⁄[x]
x = 4.50 mol
Solution to (b):
Since CO[2] has the same coefficient as O[2], the answer will be the same: 4.50 moles of CO[2] will be produced.
Solution to (c):
^1⁄[4] = ^1.50⁄[x]
x = 6.00 mol
Example #8: CuSO[4] ⋅ 5H[2]O (a hydrated compound) is strongly heated, causing the water to be released. How many moles of water are produced when 1.75 moles of CuSO[4] ⋅ 5H[2]O is heated?
1) The chemical equation of interest is this:
CuSO[4] ⋅ 5H[2]O ---> CuSO[4] + 5H[2]O
2) Every one mole of CuSO[4] ⋅ 5H[2]O that is heated releases five moles of water. The ratio from the chemical equation is this:
3) The ratio from the problem data is this:
4) Solving:
^1⁄[5] = ^1.75⁄[x]
x = 8.75 moles of water will be produced
Example #9: 2.50 moles of K[2]CO[3] ⋅ 1.5H[2]O is decomposed. How many moles of water will be produced?
^2⁄[3] = ^2.50⁄[x]
x = 3.75
Notice the use of a two-to-three ratio in place of a one-to-one-point-five ratio.
Example #10: Carbon disulfide is an important industrial solvent. It is prepared by the reaction of carbon (called coke) with sulfur dioxide:
5C(s) + 2SO[2](g) ---> CS[2](ℓ) + 4CO(g)
(a) How many moles of carbon are needed to react with 5.01 mol SO[2]?
(b) How many moles of carbon monoxide form at the same time that 0.255 mol SO[2] forms?
(c) How many moles of SO[2] are required to make 125 mol CS[2]?
(d) How many moles of CS[2] form when 4.1 mol C reacts?
(e) How many moles of carbon monoxide form at the same time that 0.255 mol CS[2] forms?
Solution to (a):
The molar ratio between C and SO[2] is 5:2.
The ratio and proportion to be used is this:
5 is to 2 as x is to 5.01
x = 12.5 mol (to three sig figs)
Solution to (b):
The molar ratio between CO and SO[2] is 4:2.
The ratio and proportion to be used is this:
4 is to 2 as x is to 0.255
x = 0.510 mol (to three sig figs)
Short commentary: when I solved part b, I simply multiplied 0.255 by 2. You may ponder why that was so. Also, when I solved all the problems in this example, I went to a piece of paper and wrote the
ratio and proportion thusly [using (b) for an example]:
For myself personally, it gives me a better feel for solving the problem to look at the above formulation as opposed to this:
4 is to 2 as x is to 0.255
You need to be able to translate the "in a line" style to the "ratios written as fractions" style.
Lastly, note how in (b), I used the compounds "right to left" from the chemical equation as opposed to the other three where the reading of the compounds is "left to right." It's just a stylistic
thing, but I do tend more to the "left to right" reading.
Solution to (c):
The SO[2]:CS[2] molar ratio is 2:1
The proper ratio and proportion is this:
x = 250. (to three sig figs, note the explicit decimal point)
Solution to (d):
C:CS[2] is 5:1
5 is to 1 as 4.1 is to x
x = 0.82 mol (to two sig figs)
Solution to (e):
CO:CS[2] molar ratio is 4:1
x = 1.02 mol
Example #11: 2NO(g) + O[2](g) ---> 2NO[2](g)
(a) How many moles of O[2] combine with 500. moles of NO?
(b) How many moles of NO[2] are formed from 0.250 mole of NO and sufficient O[2]?
(c) How many moles of O[2] are left over if 80.0 moles of NO is mixed with 200. moles of O[2] and the mixture reacts?
Solution to (a):
NO and O[2] react in a 2:1 molar ratio
2 500. mol
–––– = –––––––
1 x
x = 250. mol
Solution to (b):
NO and NO[2] are in a 2:2 molar ratio
2 0.250 mol
–––– = ––––––––
2 x
x = 0.250 mol
Comment: be aware that a 2:2 ratio is the same as a 1:1 ratio. Often, a teacher will use a 1:1 ratio and students will become confused. "Where did the one-to-one ratio come from?" The answer is
that a two-to-two ratio reduces to a one-to-one ratio. The teacher simply reduced it without mentioning it.
Solution to (c):
We first need to determine how many moles of oxygen are used when 80.0 moles of NO reacts.
NO and O[2] react in a 2:1 molar ratio.
2 80.0 mol
–––– = ––––––––
1 x
x = 40.0 mol
Now, we can determine how much oxygen remains after the NO is used up.
200. − 40.0 = 160. mol of O[2] left over.
Example #12: Consider the following reaction:
4Al(s) + 3O[2](g) ---> 2Al[2]O[3](s)
(a) Write the 6 mole ratios that can be derived from this equation. Write the first using the chemical formulas and, secondly, using the coefficients of the equation.
(b) How many moles of aluminum are needed to form 3.75 mol Al[2]O[3]?
Solution to (a):
Al:O[2] Al to O[2] Al/O[2] Al:Al[2]O[3]
Example #13: Consider the reaction:
4Al(s) + 3O[2](g) ---> 2Al[2]O[3](s)
(a) If 8.00 moles of aluminum react with an excess of oxygen, how many moles of aluminum oxide are produced?
(b) The production of 0.438 moles of aluminum oxide requires the reaction of ______ moles of aluminum and _____ moles of oxygen?
(c) When 1.830 moles of aluminum reacts, ______ moles of oxygen are consumed.
Solution to (a):
The molar ratio between Al and Al[2]O[3] is 2:1.
Note that I reduced it from 4:2
2 8.00 mol
–––– = –––––––
1 x
x = 4.00 mol
Solution to (b):
First, aluminum
Al to Al[2]O[3] molar ratio is 2:1
2 x
–––– = –––––––––
1 0.438 mol
x = 0.876 mol of Al required.
Second, oxygen
O[2] to Al[2]O[3] molar ratio is 3:2
3 x
–––– = –––––––––
2 0.438 mol
x = 0.657 mol of oxygen required
Solution to (c):
Al to O[2] molar ratio is 4:3
4 1.830 mol
–––– = –––––––––
3 x
x = 1.3725 mol
To, four sig figs, this is 1.372 mol (the rule for rounding with a five applies).
Example #14: In a chemical reaction between phosphoric acid and aqueous calcium chloride, the products are hydrochloric acid and a precipitate of calcium phosphate.
(a) How many moles of calcium chloride are required to react in order to produce 0.570 moles of calcium phosphate?
(b) How many moles of phosphoric acid are required to react with 1.37 moles of calcum chloride?
1) Write a balanced chemical equation:
2H[3]PO[4](ℓ) + 3CaCl[2](aq) ---> Ca[3](PO[4])[2](s) + 6HCl(aq)
2) Part (a)
CaCl[2] to Ca[3](PO[4])[2] molar ratio is 3:1
3 x
–––– = –––––––––
1 0.570 mol
x = 1.71 mol of CaCl[2] required.
2) Part (b)
H[3]PO[4] to CaCl[2] molar ratio is 2:3
2 x
–––– = –––––––––
3 1.37 mol
x = 0.913 mol of H[3]PO[4] required.
Example #15: Given the reaction: 4NH[3](g) + 5O[2](g) ---> 4NO(g) + 6H[2]O(ℓ)
When 1.20 mole of ammonia reacts, the total number of moles of products formed is:
(a) 1.20
(b) 1.50
(c) 1.80
(d) 3.00
(e) 12.0
The correct answer is d.
The NH[3] / (NO + H[2]O) molar ratio is 4:10
4 / 10 = 1.20 / x
x = 3.00 mol | {"url":"https://web.chemteam.info/Stoichiometry/Mole-Mole.html","timestamp":"2024-11-04T07:54:45Z","content_type":"text/html","content_length":"26918","record_id":"<urn:uuid:9c7681a3-16b8-4f96-aa09-29ce30005c19>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00317.warc.gz"} |
Bi-quinary coded decimal-like abacus representing 1,352,964,708
An abacus (pl.: abaci or abacuses), also called a counting frame, is a hand-operated calculating tool which was used from ancient times in the ancient Near East, Europe, China, and Russia, until the
adoption of the Arabic numeral system.^[1] An abacus consists of a two-dimensional array of slidable beads (or similar objects). In their earliest designs, the beads could be loose on a flat surface
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Each rod typically represents one digit of a multi-digit number laid out using a positional numeral system such as base ten (though some cultures used different numerical bases). Roman and East Asian
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Any particular abacus design supports multiple methods to perform calculations, including addition, subtraction, multiplication, division, and square and cube roots. The beads are first arranged to
represent a number, then are manipulated to perform a mathematical operation with another number, and their final position can be read as the result (or can be used as the starting number for
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In the ancient world, abacuses were a practical calculating tool. Although calculators and computers are commonly used today instead of abacuses, abacuses remain in everyday use in some countries.
The abacus has an advantage of not requiring a writing implement and paper (needed for algorism) or an electric power source. Merchants, traders, and clerks in some parts of Eastern Europe, Russia,
China, and Africa use abacuses. The abacus remains in common use as a scoring system in non-electronic table games. Others may use an abacus due to visual impairment that prevents the use of a
calculator.^[1] The abacus is still used to teach the fundamentals of mathematics to children in most countries. | {"url":"https://techsciencenews.com/wikisearch/view_html.php?sq=Google&lang=en&q=Abacus","timestamp":"2024-11-09T22:10:37Z","content_type":"text/html","content_length":"14802","record_id":"<urn:uuid:526a45e3-9a53-480b-b1db-b0f5fbdb72cd>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00125.warc.gz"} |
Bogus paper pseudocode: Speex: A Free Codec For Free Speech (2006)
ferris blogs stuff Bogus paper pseudocode: Speex: A Free Codec For Free Speech (2006)
30 Jun 2021
Similar to my last post in this series, here’s another quickie about a strange filter implementation I found in Jean-Marc Valin’s “A Free Codec For Free Speech”. Of course, speex has been obsolete
for some time now, but I think CELP (and LPC in general) is neat, and I’ve been studying up on some proper DSP theory lately, in particular filters and their corresponding z-transforms. So, when I
found a simple filter description along with pseudocode in the speex paper, I thought I’d try deriving one from the other for practice.
The filter in question is described in section IV.C (“Common Layer 8 Problems”). It’s a bog-standard DC blocker with the following transfer function:
\[N(z)=\frac{1-z^{-1}}{1-\alpha z^{-1}}\]
Not terribly interesting, but the pseudocode listing (modified slightly for clarity) is:
#define ALPHA .98f
static float mem = 0f;
for(i = 0; i < frame_size; i++){
mem = ALPHA * mem + (1 - ALPHA) * input[i];
output[i] = output[i] - mem;
Right away, there’s something fishy here: output[i] is defined in terms of output[i], which makes no sense! But surely that’s the only issue, right?
If you convert the code to its corresponding difference equations (which only requires some explicit time shifts in this case) and work out/simplify the corresponding filter z-transform (like I tried
to do), it won’t match the equation from the paper. TL;DR, the Output(z) terms end up cancelling, and you end up with everything being equal to zero. Fun times!
I even gave the paper the benefit of the doubt and tried with output[i] = input[i] - mem and got something super strange:
\[N(z)=-\frac{1-\alpha}{(1-z^{-1})(1-\alpha z^{-1})}\]
Yeah, this is clearly a very different filter.
So, I decided to go another route. From the original z-transform (which we know is good, because this kind of filter is listed all over the place), we can convert it to its corresponding time-domain
difference equation:
\[output[n]=\alpha output[n-1]+input[n]-input[n-1]\]
\[y[n]=\alpha y[n-1]+x[n]-x[n-1]\]
From this, I drew out the corresponding DF-I signal flow graph:
From that, I derived (by the commutative property of LTI systems) the corresponding DF-II graph:
Note that this part can be done entirely symbolically (and it’s arguably easier to do so), but I wanted to do it pictorially this time.
From that, I derived (by transposition) the corresponding TDF-II graph, which I suspected would be the closest to the original paper code, both because it only had one memory element (which suggested
it was a type II form), and because it’s common to use TDF-II for numerical robustness:
Note that this step is something I actually don’t know how to do symbolically, but even if I did it would still probably be easier to do it pictorially. I also derived the TDF-I form and its
corresponding difference equations, but this was just for additional practice.
Following the graph, the corresponding difference equations are:
\[mem[n]=\alpha mem[n-1]+(\alpha-1)input[n]\] \[output[n]=input[n]+mem[n-1]\]
Let’s see what the code looks like according to these equations instead:
#define ALPHA .98f
static float mem = 0f;
for(i = 0; i < frame_size; i++){
// order has to be swapped to avoid contaminating mem before we output!
output[i] = input[i] + mem;
mem = ALPHA * mem + (ALPHA - 1) * input[i];
Aha, so some differences are already clear here. First, output[i] doesn’t look bogus anymore, and 1 - ALPHA has become ALPHA - 1 (among other things).
If you just want the punchline, I’ll save you some trouble: yes, this is the correct code and what should have been in the paper. Of course, I can’t make claims like that without proof, so let’s
derive this new code’s corresponding z-transform so that we can compare it with what we expect for good measure.
From the difference equations, we have the following z-transforms:
\[Mem(z)=\alpha z^{-1}Mem(z)+(\alpha-1)Input(z)\] \[Output(z)=Input(z)+z^{-1}Mem(z)\]
If we move some terms around for Mem:
\[Mem(z)=\frac{\alpha-1}{1-\alpha z^{-1}}Input(z)\]
We can substitute that into our Output z-transform:
\[Output(z)=Input(z)+z^{-1}\frac{\alpha-1}{1-\alpha z^{-1}}Input(z)\]
This looks a bit janky, but if we simplify:
\[Output(z)=\frac{1-z^{-1}}{1-\alpha z^{-1}}Input(z)\]
Or, equivalently:
\[N(z)=\frac{1-z^{-1}}{1-\alpha z^{-1}}\]
Look, I get that this is a trivial filter that’s already listed/derived everywhere and I found it in a paper for an obsolete codec, and the filter isn’t even really related to the codec itself, and
the only reason I was even looking was because I was curious about the outdated codec anyway and wanted to practice some new math I’m learning. Still, the paper was wrong, and here’s the fix.
30 Jun 2021
Add a comment | {"url":"https://yupferris.github.io/blog/2021/06/30/bogus-paper-pseudocode-speex.html","timestamp":"2024-11-07T04:46:56Z","content_type":"text/html","content_length":"21551","record_id":"<urn:uuid:0f0ffea1-8d71-4fbb-9bd5-336324a631f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00093.warc.gz"} |
orbit-estimation: Fast orbital parameters estimator
orbit-estimation tests and evaluates the Stäckel approximation method for estimating orbit parameters in galactic potentials. It relies on the approximation of the Galactic potential as a Stäckel
potential, in a prolate confocal coordinate system, under which the vertical and horizontal motions decouple. By solving the Hamilton Jacobi equations at the turning points of the horizontal and
vertical motions, it is possible to determine the spatial boundary of the orbit, and hence calculate the desired orbit parameters.
Astrophysics Source Code Library
Pub Date:
April 2018 | {"url":"https://ui.adsabs.harvard.edu/abs/2018ascl.soft04009M/abstract","timestamp":"2024-11-11T17:38:00Z","content_type":"text/html","content_length":"34613","record_id":"<urn:uuid:4daf3592-93a7-4419-b772-3b9ab2a31942>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00153.warc.gz"} |
Pay Rates 101: Understanding How Pay Is Calculated in the Workplace
Have you ever heard of the term “pay rate”?
As soon as you enter the workforce, you start hearing about different terms employers and employees use to describe their work days, salaries, benefits, etc. It’s easy to get overwhelmed by the
amount of info you receive. But, you shouldn’t give up!
Learning about pay rates can be extremely helpful to you regardless of whether you’re an employee or you own a business.
So, today, you’ll learn more about:
• The definition of pay rate,
• The importance of learning about pay rates, and
• How to calculate your pay rate for a specific time period or project.
What does pay rate mean?
Simply put, pay rate refers to the amount of money an employee or freelancer is paid during a specific time span.
In general, a pay rate consists of any kind of payment an employee or freelancer receive from an employer during a specific period. It typically includes:
• Wages,
• Bonuses,
• Commissions,
• Overtime rates, and
• Other categories of compensation.
The above are just simple and general definitions — it’s difficult to define pay rate in specific terms. That’s because this term does not have an official definition that’s accepted and recognized
around the world.
You may have to go through a few extra steps to understand what the pay rate means in your country.
The United States, for instance, differentiates between the regular pay rate and the overtime one — and understanding this difference is essential when calculating your rates.
If you want to figure out what pay rate means in your place of work, you can try finding out more info in your employment contract. After all, employment contracts tend to include various definitions
relevant to the workplace, and pay rates may be one of them.
In case you can’t find a detailed explanation, or your country does not have a specific legal definition of the term, you can think of a pay rate as we defined it above — the amount of money your
employer is expected to pay you over a certain period of time, in any kind of payment.
Now that we understand the basics, let’s see what are regular pay rate and overtime pay rate — as well as what is the difference between pay rate and bill rate.
What is the regular rate of pay?
The regular pay rate is the total wage a worker receives during regular working hours, excluding overtime. The regular rate equals your wage.
According to the Fair Labor Standards Act, the regular rate generally includes “all remuneration for employment paid to, or on behalf of, the employee.” The FLSA also states that the regular pay rate
can’t be bypassed by any other agreement. Moreover, it can’t be lower than the minimum wage.
The regular pay rate is always enforced and protected by the law. The US Department of Labor enforces federal minimum rate rules and rights, which are regulated by the Fair Labor Standards Act.
💡 Clockify Pro Tip
You can find relevant information about labor laws, including information about minimum wages, in our section about state labor laws:
What are overtime and the overtime pay rate?
Overtime refers to the number of hours worked beyond the normal, regular working hours, which usually means working over 40 hours per week. So, your overtime pay rate in this instance will be higher
than your regular one.
Under the Fair Labor Standards Act, unless exempt, overtime pay rate can’t be lower than time and one-half (1.5) the regular rate of pay for all hours worked over 40 hours during a workweek. For
instance, if your hourly wage equals $7.25, your overtime hourly rate will be:
$7.25 x 1.5 = $10.88
💡 Clockify Pro Tip
In case you aren’t sure what rules apply in your circumstance, an overtime calculator will be a useful tool to try out:
Now, what about other countries around the world? Many countries in the European Union have a 50% pay rate rule for overtime. However, the regulations vary between countries.
Who is eligible for overtime?
If the law in your state or the contract of employment doesn’t get into specifics for these two types of pay rates, and you are eligible for overtime, you should take both regular and overtime pay
rates into consideration when calculating your rate of pay.
Unless exempt, every employee who works over 40 hours a week and whose earnings are less than $35,568 a year is eligible for overtime pay of one and a half (1.5) under federal law. You can get
information on overtime exemptions in this Fact Sheet.
The process of calculating your final rate could be a bit confusing. Therefore, keeping different kinds of rates in mind would make the process of calculating your pay rate much simpler for you.
Pay rate vs. bill rate — is there a difference?
As we mentioned, a pay rate is the amount of money an employee or freelancer is paid over a certain period.
On the other hand, a bill rate is the amount of money a professional or a company charges for their services per hour. Companies and self-employed individuals use bill rates to charge customers for
their services. It includes the costs a company or a professional needs to cover in order to achieve the target income. Taxes, markups, and fees are then subtracted from the bill rate, to get the pay
In case you’re a freelancer or a legal professional, you will charge your client based on your billable hours — i.e. the number of hours you worked for which you bill the client. Billable hours
always include hours spent working on a project for the client. On the other hand, non-billable hours present the time spent on tasks that are not invoiced and directly charged to the client.
In case your bill rate is $200 per hour, your pay rate may be much lower and sit at $126. That’s because 30% ($60) of the bill rate went to taxes and 7% to the fees ($14).
Staffing companies that match job candidates and employers frequently use bill rates. In this case, the bill rate will include the pay rate of a worker and the markup of that staffing agency.
💡 Clockify Pro Tip
If you want to learn more about professional ways to ask for payment from your clients, this blog post will be helpful:
Why should you learn more about pay rates?
Learning about pay rates will be useful to you regardless of whether you’re an employer or an employee — and here’s why.
Why learning about pay rates is useful for business owners
Getting info about the pay rates in your area could help you offer better benefits to workers and predict future costs!
Reason #1: You’ll learn how to attract skilled workers
If you want skilled, hard-working employees who are constantly motivated, you need to offer them something they won’t receive anywhere else. The easiest way to achieve that is via competitive
But, what happens when you hire seasonal workers, freelancers, or contractors? It’s pretty difficult to determine the competitiveness of your offer when you are paying someone based on the duration
of a project or a specific goal. That’s where the pay rate comes in!
As you now know, a pay rate refers to earnings during a specific time period. The period in question is the one you’ll be able to determine on your own. So, it will be much easier for you to
calculate the actual amount the worker will receive during a project or after completing a certain goal.
Reason #2: You’ll learn how to keep employees motivated
Another great upside to focusing on pay rates is that they typically include various benefits aside from employees’ hourly wages — such as bonuses.
Benefits are sometimes equally as important as wages. Competitive pay rates will make you a more desirable employer as they allow you to include benefits aside from the hourly rate in your offer. As
a result, you’ll be able to offer good workers something they won’t receive anywhere else!
Reason #3: Getting into pay rate details helps you plan future projects
Labor costs present a major part of project expenses. They include the sum of wages, taxes, and benefits that are paid to and for employees.
Now, pay rates play an important role in labor costs because they can help you predict future labor costs for similar projects. For example, if you frequently work on similar projects of the same
duration, you can use the info on pay rates from a previous project to calculate labor expenses.
Once you calculate labor expenses, it will be much easier to determine overall project expenses.
Why learning about pay rates is useful to employees
As an employee, you want to get paid fairly. Pay rates can help you negotiate the best deal and even predict your future earnings!
Reason #1: Researching pay rates helps you negotiate the best wage and benefits
Pay rates are more than a simple number to employees.
They determine how much you earn during a certain period of time and how the job benefits and bonuses reflect your final earnings. Thus, they can really make a difference in your job satisfaction.
When it comes to wages, a good salary can significantly affect your motivation at work as well as the quality of your life. In addition, bonuses can motivate you to improve your performance. If your
company does not offer them, you could end up earning much less money than your colleagues in the industry, even if your wage is higher. So, by learning more about pay rates and what they include in
your country, you’ll save yourself a lot of frustration and confusion!
💡 Clockify Pro Tip
Improving the quality of your work life will have a major impact on your motivation and efficiency. You can find tips the best tips on how to improve the quality of work life here:
As pay rates tend to include all the relevant earnings for a certain time span, they’ll help you determine where you stand compared to other similar or same positions in your industry. Once you get a
clearer picture of your situation, you’ll be able to ask for a higher salary or additional perks your employer may be offering.
Reason #2: Learning about pay rates lets you predict your future earnings
Another essential aspect of learning about pay rates is the ability to predict future earnings.
You have the ability to decide the time period you’ll be calculating your pay rate for. Therefore, you can easily predict your future earnings during a specific project, especially if you work on
similar projects or your pay rate remains stable for a while
How do you find the rate of pay — a simple calculation with a real-life example
By now, you are already familiar with the fact that the term pay rate doesn’t have a recognized definition everywhere. So, the process of calculating pay rates and determining what to include in the
calculation won’t be the simplest one. Luckily, we are here to make things so much easier!
Step #1: Determine the pay rate time span
The best way to make this calculation easier is to first determine the time period for which you want to calculate the pay rate. If you don’t have a time span in mind, you could try with a weekly pay
Alternatively, in case you want to calculate the pay rate right after a completed project, you can focus on the duration of that project only.
We’ll go into the calculation using two examples:
• A freelance graphic designer who wants to calculate how much they should charge for a specific project, and
• A business owner who wants to calculate a pay rate for a freelance developer hired for a project.
Let’s say that you’re a graphic designer and you live in the United States. You want to calculate your pay rate for a specific project that lasted a week. Thus, you’d want to use a week as the time
span for your calculation.
In case you’re a business owner, you can follow the example of calculating a pay rate of a freelance developer you hired for a week-long project. The developer works in the US and has 3 to 5 years of
Step #2: Include all the aspects relevant to the pay rate
Your next step will be figuring out what to include in your rate of pay calculation.
Substep #1: Include your regular pay
To calculate your regular pay, you should start with the hourly rate. But, if you only have the info on your annual salary, you’ll first need to find the number of regular hours worked in a year.
You’ll do that by multiplying the number of working hours per week by the number of weeks in the whole year.
The number of working hours per week x the number of weeks per year = the number of regular hours worked in the year
Then, you should divide the annual salary by the number of hours worked in the year to get your hourly rate.
The annual salary / the number of hours worked in the year = the hourly rate
So, if your annual salary sits at $98,000, you’ll have to divide that number by the number of regular hours you worked in a year to get the hourly rate. That’s 40 hours per week times 52 weeks, which
is 2,080 hours.
$98,000 / 2,080 = $47.12
So, your hourly rate is $47.12
Now, to calculate your earnings, you need to multiply the hourly rate by the number of regular hours worked:
Hourly rate x the number of regular hours worked = earnings
Let’s take a look at our examples. As a graphic designer in the US with one to three years of experience, you should be earning about $32 per hour. You worked 43 hours during the week, out of which
40 hours were regular and three overtime, which we’ll explain in the following subheading. So, the calculation for your weekly earnings based on your regular working hours should be:
$32 x 40 = $1,280
Therefore, $1,280 are your weekly earnings.
An hourly rate of a freelance developer is around $62.4. They clocked in 42 hours a week while they were working on your project. Their weekly earnings from regular working hours are:
$62.4 x 40 = $2,496
💡 Clockify Pro Tip
If you’re a freelancer considering charging more for your services and you don’t want to take your existing wages as a reference, you can always use our hourly rate calculator to help you get the
hourly rate you deserve:
Substep #2: Include overtime, if you’re eligible for it
If you’re eligible for overtime, you should include your overtime rate in the calculation. The overtime rate is different around the world. In the US, your rate should be one and a half times higher
than your regular pay rate.
To calculate your overtime earnings, you’ll first need to find your overtime rate. Your overtime rate should be one and a half times your regular hourly rate.
Regular rate x 1.5 = overtime rate
Then, you should multiply your overtime rate by the number of overtime hours you worked to get your overtime earnings:
Overtime rate x number of overtime hours worked = overtime earnings
In the case of a graphic designer who worked 3 hours overtime, your calculation will go like this:
$32 x 1.5 = $48
Your overtime rate is $48. Next, to calculate your overtime pay, you need to multiply your overtime rate by the number of hours you worked overtime:
$48 x 3 = $144
When it comes to our second example, a business owner will have to pay a developer who worked on their project for 42 hours two hours of overtime. So, let’s calculate the developer’s overtime rate:
$62.4 x 1.5 = $93.6
The developer’s overtime rate is $93.6 per hour. For two hours of overtime work, that’s:
$93.6 x 2 = $187.2
Regardless of whether you are an employee or an employer, you’ll need to make sure that you have obtained the most accurate numbers regarding overtime work month in and month out. This is especially
important for business owners who regularly calculate payroll.
💡 Clockify Pro Tip
To avoid the risk of calculation errors, use an overtime tracker to get the exact overtime numbers and export payroll data right before issuing payments:
Substep #3: Include bonuses, if any
Any bonus received during the time span you chose could also be part of the pay rate calculation. This is where things get more complicated.
Bonuses that are not connected to an employee’s performance and not used as an incentive should be excluded from the calculation. Some of them are holiday and discretionary bonuses as well as the
ones workers receive as a percentage of their total earnings.
That’s because these rewards are:
• Either not known in advance,
• Not connected to an employee’s performance, or are
• Already indirectly included in the calculation.
On the other hand, you can include any other bonus that is designed to motivate your performance, productivity, and engagement.
Since we’re considering a short project in both instances, a larger bonus is rare. Still, let’s assume that you or your worker achieved a certain target and was awarded a lump sum of $100.
Step #3: Get the final rate of pay
Once you determine all of the aspects of the pay rate, getting the final amount will be pretty easy.
To get the final rate pay, you should add up every bonus, overtime rate, and regular rate during a specific time span.
Regular rate + overtime rate + bonuses = final pay rate
In the graphic designer situation, you’ll be adding up your weekly earnings from the regular working hours, overtime, and the one-time bonus you received. So, your calculation will be:
$1,280 + $144 + $100 = $1,524
You can divide the final amount by the number of hours worked to get the hourly rate.
Final pay rate / the number of hours worked = final rate per hour
$1,524 / 43 = $35.4
Your final rate equals to $35.4 per hour.
As a business owner who hired a developer for a project, you’ll also be adding up regular hours, overtime, and bonuses in your calculation. It will go like this:
$2,496 + $187.2 + $100 = $2,783.2
Now, we can also calculate the final rate per hour for the developer.
$2,783.2 / 42 = $66.3
The developer’s final rate of pay will be $2,783.2, or $66.3 per hour.
Conclusion: Learning about pay rates leads to fair compensation and happier workers
Hopefully, you now understand what the pay rate is and the benefits of knowing the exact meaning of the term. Learning about pay rates helps workers around the world figure out where they stand
compared to the standards in the industry. Moreover, this information lets them estimate earnings from future projects.
As an employer, learning about employees’ pay rates helps you stay competitive and keeps your employees satisfied with their jobs. What’s more, having enough detail about pay rates enables you to
estimate potential expenses and earnings for future projects.
Having this piece of information is just one step towards making the most accurate predictions about recurring projects, scheduled assignments, and budgets. Since that kind of calculation can
sometimes be too complex, you can always invest in a forecasting tool that will help you to easily and quickly visualize your project’s progress. | {"url":"https://clockify.me/blog/business/pay-rate/","timestamp":"2024-11-02T02:19:32Z","content_type":"text/html","content_length":"111993","record_id":"<urn:uuid:1f1def4b-17eb-40ce-8be1-ed62fcf9b271>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00861.warc.gz"} |
I don't know what the -4)colors means. Models*exterior
I don't know what the -4)colors means.
Models*exterior colors*upholsterytypes*numberinteriour colors.
Multiply them out.
what do i have to do to solve for this one i have no idea i was low on the first try...
An uto mobile manufacturerr produces 7 models, each available in 6 different exterior-4)colors, with 4 different upholstery fabrics and 5 interior colors. How many varieties of automobile are
An automobile manufacturer produces 7 models, each available in 6 different exterior colors, with 4 different fabrics and 5 interior colors. How many varietires of automobile are available?
this is were i got 16 varieties by
To solve this problem, you need to multiply the number of options for each category.
First, let's consider the number of models. We are given that there are 7 models.
Next, for each model, there are 6 different exterior colors to choose from. So, for each model, there are 6 options for the exterior color.
Similarly, there are 4 different upholstery fabric options for each model.
Lastly, there are 5 interior colors available for each model.
To find the total number of varieties of automobiles available, multiply the number of options for each category:
Number of varieties = number of models * number of exterior colors * number of upholstery fabrics * number of interior colors
= 7 * 6 * 4 * 5
Now, let's multiply these numbers:
7 * 6 = 42
42 * 4 = 168
168 * 5 = 840
There are a total of 840 varieties of automobiles available from this manufacturer. | {"url":"https://askanewquestion.com/questions/22531","timestamp":"2024-11-08T18:27:11Z","content_type":"text/html","content_length":"16470","record_id":"<urn:uuid:bfacbd89-e4ff-434f-9570-055af277d33a>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00783.warc.gz"} |
A parking lot is not a highway at capacity
Reasoning about throughput and utilization with the Utilization Law
In my post, “A highway at capacity is not a parking lot,” we looked at the impact of high utilization in a queueing system: it increases queue sizes and wait times.
A highway at capacity feels like a parking lot, but from the standpoint of queueing theory, a highway at capacity behaves identically to a parking lot at capacity.
In fact, from the queueing theory perspective, they behave identically at all other utilization levels too.
The Utilization Law
This post will examine the Utilization Law from queueing theory to understand why this is so.
It’s an often overlooked companion to Little’s Law, which we discussed in the Iron Triangle of Flow.
When reasoning about queueing systems, you typically need to use these laws in combination to understand cause-and-effect relationships.
A parking lot is a great example to illustrate what this law says.
Thanks for reading The Polaris Flow Dispatch! Subscribe for free to receive new posts and support my work.
The Parking Lot Queue
The simplest model for a parking lot is as a queue that you enter when you want to “park” and leave when you are done “parking.”
Your parking time is your service time. On a highway, your driving time is your service time.
From the queuing theory perspective, this is the only difference between parking lots and highways.
They have different definitions of “service time.”
Otherwise, the rules of flow apply identically to both.
The throughput through the parking lot, the number of cars leaving the parking lot at any point in time, can never be greater than the parking lot's capacity, assuming cars cannot pass through.
When the parking lot is at capacity, it has the most cars that can leave it simultaneously.
If all the cars somehow finish “parking” at the same time and exit simultaneously, then we will have peak throughput through the lot.
But if you get smaller throughput, it has nothing to do with the parking lot being at capacity.
It’s just that some cars stayed parked longer than others, so you had less than the maximum possible number of cars exiting the lot at that moment in time.
The non-intuitive part is that the parking lot can achieve peak throughput only when fully utilized, but the actual throughput is determined by the average time a car stays parked.
This is what the Utilization Law expresses.
where U is the utilization, X is the throughput, and S is the average service time in the process.
S is the average processing time, excluding any time spent waiting in the system. It should not be confused with Cycle Time from Little’s Law.
The relationship between U and X is straightforward to understand based on this law.
• For a given utilization, when average service time increases, throughput goes down.
• For a given average service time, if utilization increases, throughput increases until you reach maximum capacity.
If the demand for parking is less than the number of spaces available, then the number of cars in the lot will always stay below capacity, utilization will be less than 100%, and entering cars will
have little or no wait times to find a spot.
On the other hand, the fully utilized parking lot will make life miserable for anyone waiting to get in, as we saw in “A highway at capacity is not a parking lot.”
This is still governed by the relationship between queue sizes and utilization we discussed in that post.
This is identical to the way the highway behaves at high utilization.
As the parking lot fills up, the time spent waiting changes dramatically as it takes longer to find a spot, and at 100% utilization, you have to drive around the block when the sign says the lot is
In this sense, highways and parking lots behave identically at all times from queueing theory lens, as they should.
It’s just that we use different definitions of “service time” for each one.
High utilization cannot turn one into the other.
As an exercise, you can try running the same argument with a parking spot with a single car, and perhaps the reasoning will be more apparent. | {"url":"https://www.polaris-flow-dispatch.com/p/a-parking-lot-is-not-a-highway-at?utm_campaign=post&utm_medium=web","timestamp":"2024-11-03T04:28:33Z","content_type":"text/html","content_length":"147767","record_id":"<urn:uuid:f605ea89-95fa-44e5-bdb1-be1166c6c56a>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00325.warc.gz"} |
64 km to miles
Heading 1: Understanding the Conversion from Kilometers to Miles
Kilometers and miles are two commonly used units of distance measurement. While kilometers are predominantly used in most parts of the world, miles are commonly used in the United States and some
other countries that follow the Imperial system of measurement. Understanding the conversion from kilometers to miles can be useful when dealing with geolocation data, travel distances, or simply
when communicating with people from different measurement systems.
The conversion from kilometers to miles is relatively straightforward. One kilometer is equal to approximately 0.621 miles. This means that if you have a distance in kilometers, you can easily
convert it to miles by multiplying it by 0.621. For example, if you have a distance of 10 kilometers, it would be equivalent to approximately 6.21 miles. Similarly, if you have a distance in miles
and want to convert it to kilometers, you can divide it by 0.621.
Heading 2: How Kilometers and Miles Differ in Measurement
Kilometers and miles are both units of measurement used to determine distance, but they differ in various ways. The most obvious difference between the two is the scale of measurement. A kilometer is
a metric unit and is commonly used in most countries around the world, except for the United States and a few others. On the other hand, the mile is an imperial unit predominantly utilized in the
United States and a handful of other countries that follow the Imperial system.
Another significant difference is the numerical value that represents a kilometer and a mile. One kilometer is equivalent to approximately 0.62 miles. This means that if you were to compare a
distance measured in kilometers to the equivalent distance in miles, the number of miles would be significantly smaller. This can be quite confusing for people who are not familiar with both units of
measurement. For example, a distance of 10 kilometers would only be about 6.2 miles in length. Therefore, it is essential to understand the difference between kilometers and miles to accurately
convert or compare distances.
Heading 2: The Historical Context of the Kilometer and Mile Units
The historical context of the kilometer and mile units is fascinating, as it gives us insight into the evolution of measurement systems over time. The origin of the kilometer can be traced back to
the French Revolution, when there was a need for a new unit of length to replace the existing system based on body measurements. In 1799, the French Academy of Sciences introduced the concept of the
meter, which was defined as one ten-millionth of the distance from the Equator to the North Pole. This newly defined meter was later divided into 1,000 equal parts, giving rise to the kilometer.
On the other hand, the mile has a much older history, dating back to ancient Rome. The Romans used a unit called the “mille passus,” which literally means “a thousand paces.” It was originally
defined as the distance covered in 1,000 double steps or approximately 5,000 Roman feet. Over time, the mile varied in length among different regions and countries, until it was standardized in the
16th century as 1,760 yards or 5,280 feet.
Understanding the historical context of these units helps us appreciate the significance they hold in today’s world. While the kilometer is more commonly used in most countries due to the metric
system, the mile still has its place in some places, particularly in the United States and the United Kingdom. Whether it’s measuring distances in the past or converting between the two units in the
present, knowing the historical backdrop gives us a deeper understanding of these measurements.
Heading 2: Why Convert Kilometers to Miles?
There are several reasons why converting kilometers to miles can be useful in certain situations. One common reason is the difference in measurement systems used around the world. While kilometers
are the standard unit of measurement in most countries, the United States still primarily uses miles. So, if you are traveling or need to communicate distances with someone from the US, understanding
how to convert between kilometers and miles becomes essential.
Another reason to convert kilometers to miles is for convenience and familiarity. Many people, especially those accustomed to the imperial system, find it easier to comprehend and visualize distances
in miles rather than kilometers. This is particularly true when considering road trips or long journeys, where mile markers and speed limits are typically provided in miles. By converting kilometers
to miles, you can better gauge the distance and plan your travel more effectively, ensuring a smoother and less confusing experience on the road.
Heading 2: The Simple Formula for Converting Kilometers to Miles
To convert kilometers to miles, all you need is a simple formula. The conversion factor is 0.62137119, which means that one kilometer is equal to approximately 0.621 miles. So, to convert kilometers
to miles, you just multiply the number of kilometers by this conversion factor. For example, if you have 10 kilometers, you would multiply 10 by 0.62137119 to get approximately 6.2137 miles. It’s as
simple as that!
Converting kilometers to miles can be useful in various scenarios. For instance, if you’re planning a road trip in the United States and are used to measuring distances in kilometers, it can be
helpful to convert those distances to miles for a better understanding of the road network. Similarly, if you’re reading a travel guide or using a GPS that displays distances in miles, you might want
to convert them to kilometers if that’s the measurement system you’re more familiar with. Understanding the simple formula for converting kilometers to miles allows you to seamlessly switch between
the two units of measurement depending on your needs. | {"url":"https://convertertoolz.com/km-to-miles/64-km-to-miles/","timestamp":"2024-11-09T12:45:58Z","content_type":"text/html","content_length":"42551","record_id":"<urn:uuid:aed30f13-2d99-4614-a716-6e035117edaf>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00373.warc.gz"} |
Excel Formula Help for $0.00 Value | Microsoft Community Hub
Forum Discussion
Excel Formula Help for $0.00 Value
I am in need of help with a formula again. This time it is for Conditional Formatting (at least I think). So I've tried every method I can think of, as well as using Google, to get cell D5 to show
Change the formula to
HansVogelaar wrote: ``=ROUND(Calculations!B96+Calculations!C96,2)``
And yes, that should be done __as_well__.
But that alone is hardly the complete correct answer, except by accident. | {"url":"https://techcommunity.microsoft.com/discussions/excelgeneral/excel-formula-help-for-0-00-value/3043007/replies/3043054","timestamp":"2024-11-08T03:27:46Z","content_type":"text/html","content_length":"222751","record_id":"<urn:uuid:aa2a540f-01a6-4a61-af6e-a702e012b9ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00380.warc.gz"} |
C.04.5.1 Baku Acceleration
1. Premise
The Baku Acceleration Method is applicable in any tournament where the standard scoring point system (one point for a win, half point for a draw) is used.
2. Initial Groups Division
Before the first round, the list of players to be paired (properly sorted) shall be split in two groups, GA and GB. The first group (GA) shall contain the first half of the players, rounded up to
the nearest even number. The second group (GB) shall contain all the remaining players.
for instance, if there are 161 players in the tournament, the nearest even number that comprises the first half of the players (i.e. 80.5) is 82. The formula 2 * Q (2 times Q), where Q is
Note: the number of players divided by 4 and rounded upwards, may be helpful in computing such number – that, besides being the number of GA players, is also the pairing number of the last GA
3. Late entries
If there are entries after the first round, those players shall be accommodated in the pairing list according to C.04.2.B/C (Initial Order/Late Entries).
The last GA-player shall be the same as in the previous round.
Note In such circumstances, the pairing number of the last GA‑player may be different by the one set accordingly to Rule 2.
Note After the first round, GA may contain an odd number of players.
4. Virtual points
The “accelerated rounds” are the ones in the first half (rounded up) of the tournament.
Before pairing the first half (rounded up) of the accelerated rounds, all the players in GA are assigned a number of points (called virtual points) equal to 1.
Such virtual points are reduced to 0.5 before pairing the remaining accelerated rounds.
Note: Consequently, no virtual points are ever given to players in GB, or to any player after the last accelerated round has been played.
In a nine-round tournament, the accelerated rounds are five. The players in GA are assigned one virtual point in the first three rounds, and half virtual point in the next two rounds.
5. Pairing score
The pairing score of a player (i.e. the value used to define the scoregroups and internally sort them) is given by the sum of his standings points and the virtual points assigned to him. | {"url":"https://spp.fide.com/c-04-5-1-baku-acceleration/","timestamp":"2024-11-08T07:51:33Z","content_type":"text/html","content_length":"44439","record_id":"<urn:uuid:54c89723-56be-4342-9b9e-5f26808fe67e>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00269.warc.gz"} |
Numbers in Ojibwa
Learn numbers in Ojibwa
Knowing numbers in Ojibwa is probably one of the most useful things you can learn to say, write and understand in Ojibwa. Learning to count in Ojibwa may appeal to you just as a simple curiosity or
be something you really need. Perhaps you have planned a trip to a country where Ojibwa is the most widely spoken language, and you want to be able to shop and even bargain with a good knowledge of
numbers in Ojibwa.
It's also useful for guiding you through street numbers. You'll be able to better understand the directions to places and everything expressed in numbers, such as the times when public transportation
leaves. Can you think of more reasons to learn numbers in Ojibwa?
Ojibwe (Anishinaabemowin, or ᐊᓂᔑᓈᐯᒧᐎᓐ in Canadian Aboriginal syllabics) is an indigenous language of the Algonquian linguistic family. The aggregated dialects of Ojibwe comprise the second most
commonly spoken First Nations language in Canada (after Cree), and the fourth most widely spoken in North America (excluding Mesoamerica), behind Navajo, Inuit and Cree, with about 80,000 speakers.
It is also known as Ojibwa, Ojibway, and Chippewa.Due to lack of data, we can only count accurately up to 1,999 in Ojibwa. Please contact me if you can help me counting up from that limit.
List of numbers in Ojibwa
Here is a list of numbers in Ojibwa. We have made for you a list with all the numbers in Ojibwa from 1 to 20. We have also included the tens up to the number 100, so that you know how to count up to
100 in Ojibwa. We also close the list by showing you what the number 1000 looks like in Ojibwa.
• 1) bezhik
• 2) niizh
• 3) nswi
• 4) niiwin
• 5) naanan
• 6) ngodwaaswi
• 7) niizhwaaswi
• 8) nshwaaswi
• 9) zhaangswi
• 10) mdaaswi
• 11) mdaaswi shaa bezhik
• 12) mdaaswi shaa niizh
• 13) mdaaswi shaa nswi
• 14) mdaaswi shaa niiwin
• 15) mdaaswi shaa naanan
• 16) mdaaswi shaa ngodwaaswi
• 17) mdaaswi shaa niizhwaaswi
• 18) mdaaswi shaa nshwaaswi
• 19) mdaaswi shaa zhaangswi
• 20) niizhtaana
• 30) nsimtaana
• 40) niimtaana
• 50) naanmitaana
• 60) ngodwaasmitaana
• 70) niizhwaasmitaana
• 80) nshwaasmitaana
• 90) zhaangsmitaana
• 100) ngodwaak
• 1,000) mdaaswaak
Numbers in Ojibwa: Ojibwa numbering rules
Each culture has specific peculiarities that are expressed in its language and its way of counting. The Ojibwa is no exception. If you want to learn numbers in Ojibwa you will have to learn a series
of rules that we will explain below. If you apply these rules you will soon find that you will be able to count in Ojibwa with ease.
The way numbers are formed in Ojibwa is easy to understand if you follow the rules explained here. Surprise everyone by counting in Ojibwa. Also, learning how to number in Ojibwa yourself from these
simple rules is very beneficial for your brain, as it forces it to work and stay in shape. Working with numbers and a foreign language like Ojibwa at the same time is one of the best ways to train
our little gray cells, so let's see what rules you need to apply to number in Ojibwa
Digits from zero to nine are specific words, namely kaagego [0], bezhik [1], niizh [2], nswi [3], niiwin [4], naanan [5], ngodwaaswi [6], niizhwaaswi [7], nshwaaswi [8] and zhaangswi [9].
The tens are based on the root of the digit names, except for ten: mdaaswi [10], niizhtaana [20], nsimtaana [30], niimtaana [40], naanmitaana [50], ngodwaasmitaana [60], niizhwaasmitaana [70],
nshwaasmitaana [80] and zhaangsmitaana [90].
The hundreds are built the same way, based on the root of the digit names, with the exception of one hundred: ngodwaak [100], niizhwaak [200], nswaak [300], niiwaak [400], naanwaak [500],
ngodwaaswaak [600], niizhwaaswaak [700], nshwaaswaak [800], and zhaangswaak [900].
Each group of number is joined by shaa (and), which means not only the tens and units (eg. niimtaana shaa naanan [45]), but also hundreds and tens (eg. niiwaak shaa niimtaana shaa nshwaaswi [448]),
thousands and hundreds (eg. mdaaswaak shaa niizhwaak shaa niizhtaana shaa niizh [1,222]), and so on. The word for thousand is thus mdaaswaak.
Numbers in different languages | {"url":"https://numbersdata.com/numbers-in-ojibwa","timestamp":"2024-11-03T06:02:29Z","content_type":"text/html","content_length":"19836","record_id":"<urn:uuid:9f9604e9-4012-4114-a411-099c35e929cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00231.warc.gz"} |
sklearn.feature_selection.f_regression(X, y, center=True)[source]¶
Univariate linear regression tests.
Linear model for testing the individual effect of each of many regressors. This is a scoring function to be used in a feature selection procedure, not a free standing feature selection procedure.
This is done in 2 steps:
1. The correlation between each regressor and the target is computed, that is, ((X[:, i] - mean(X[:, i])) * (y - mean_y)) / (std(X[:, i]) * std(y)).
2. It is converted to an F score then to a p-value.
For more on usage see the User Guide.
X{array-like, sparse matrix} shape = (n_samples, n_features)
The set of regressors that will be tested sequentially.
yarray of shape(n_samples).
The data matrix
centerTrue, bool,
If true, X and y will be centered.
Farray, shape=(n_features,)
F values of features.
pvalarray, shape=(n_features,)
p-values of F-scores.
See also
Mutual information for a continuous target.
ANOVA F-value between label/feature for classification tasks.
Chi-squared stats of non-negative features for classification tasks.
Select features based on the k highest scores.
Select features based on a false positive rate test.
Select features based on an estimated false discovery rate.
Select features based on family-wise error rate.
Select features based on percentile of the highest scores.
Examples using sklearn.feature_selection.f_regression¶ | {"url":"https://scikit-learn.org/0.22/modules/generated/sklearn.feature_selection.f_regression.html","timestamp":"2024-11-11T07:26:21Z","content_type":"text/html","content_length":"22589","record_id":"<urn:uuid:3123f980-7d5a-4a15-8ab8-282179e4c181>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00134.warc.gz"} |
Nominal ScaleNominal Scale - Six Sigma Terminology
Nominal Scale
The nominal scale is a measurement scale that lists names (categories.) Nominal data is discrete data and there is no particular order with a nominal measurement scale. Think of going to your
favorite fast food restaurant. For example, the restaurant manager might list the first names of the employees, or might list the models of the cars that they drive. These are examples of a nominal
measurement scale. You could put the names in alphabetical order, but the names really have no meaning of the rank order. When you think of nominal, think of the word ‘name’ – both start with an ‘N.’
The nominal scale is a measurement scale that lists names (categories.) Nominal data is discrete data and there is no particular order with a nominal measurement scale.
Use: Think of going to your favorite fast food restaurant. For example, the restaurant manager might list the first names of the employees, or might list the models of the cars that they drive.
These are examples of a nominal measurement scale. You could put the names in alphabetical order, but the names really have no meaning of the rank order. When you think of nominal, think of the word
‘name’ – both start with an ‘N.’
Contrast the nominal scale with an ordinal scale where order does matter. When you think of ORDinal, think of the ORDer. For example, there might be a chute where the food items are awaiting a
customer order. There are an equal number of containers of $0.99 fries, $3.00 burgers, $4.50 chicken sandwiches, and $6.99 super sandwiches. Order matters. The items are in order of cost and
potential revenue loss if the orders are not fulfilled.
There is also the interval scale. The interval scale is like the ordinal scale except we can say the intervals between each value are equally split. Think of the temperature on a thermometer. The
difference between 71 degrees and 72 degrees is the same as the difference between 31 and 32 degrees. The increments are in even intervals.
The ratio scale is an interval scale, but it does have a true zero point. Consider weight. There could be zero weight (none). The difference between four inches and five inches is the same as the
difference between 72 inches and 73 inches, so in the ratio scale it is a type of the interval scale. The difference is the ‘zero’ is a possibility. Ratio and zero kind of rhyme. It’s just a way to
remember – that’s all. Well, a thermometer has a zero? The zero on Fahrenheit scale or the Celsius scale is not a true zero point. Instead, the zero is a reference point to separate temperatures
above and below zero. True zero, means no temperature at all. If the theory of absolute zero is true, where there is no heat, such as in deep outer space, then the thermometer would be a ratio scale. | {"url":"https://www.sixsigmadaily.com/nominal-scale/","timestamp":"2024-11-09T01:15:00Z","content_type":"text/html","content_length":"71378","record_id":"<urn:uuid:f6d94e80-8eb8-4321-9883-4325b2a77248>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00151.warc.gz"} |
Local bindings for drake::geometry::optimization
class pydrake.geometry.optimization.AffineBall
Bases: pydrake.geometry.optimization.ConvexSet
Implements an ellipsoidal convex set represented as an affine scaling of the unit ball {Bu + center | |u|₂ ≤ 1}. B must be a square matrix.
Compare this with an alternative parametrization of the ellipsoid: {x | (x-center)ᵀAᵀA(x-center) ≤ 1}, which utilizes a quadratic form. The two representations are related by B = A⁻¹ if A and B
are invertible.
The quadratic form parametrization is implemented in Hyperellipsoid. It can represent unbounded sets, but not sets along a lower-dimensional affine subspace. The AffineBall parametrization can
represent sets along a lower-dimensional affine subspace, but not unbounded sets.
An AffineBall can never be empty – it always contains its center. This includes the zero-dimensional case.
class pydrake.geometry.optimization.AffineSubspace
Bases: pydrake.geometry.optimization.ConvexSet
An affine subspace (also known as a “flat”, a “linear variety”, or a “linear manifold”) is a vector subspace of some Euclidean space, potentially translated so as to not pass through the origin.
Examples include points, lines, and planes (not necessarily through the origin).
An affine subspace is described by a basis of its corresponding vector subspace, plus a translation. This description is not unique as any point in the affine subspace can be used as a
translation, and any basis of the corresponding vector subspace is valid.
An affine subspace can never be empty, because a vector subspace can never be empty. Thus, the translation will always be contained in the flat. An affine subspace is bounded if it is a point,
which is when the basis has zero columns.
pydrake.geometry.optimization.CalcPairwiseIntersections(*args, **kwargs)
Overloaded function.
1. CalcPairwiseIntersections(convex_sets_A: list[pydrake.geometry.optimization.ConvexSet], convex_sets_B: list[pydrake.geometry.optimization.ConvexSet], continuous_revolute_joints: list[int],
preprocess_bbox: bool = True) -> list[tuple[int, int, numpy.ndarray[numpy.float64[m, 1]]]]
Computes the pairwise intersections between two lists of convex sets, returning a list of edges. Each edge is a tuple in the form [index_A, index_B, offset_A_to_B], where index_A is the index of
the list in convex_sets_A, index_B is the index of the list in convex_sets_B, and offset_A_to_B is is the translation to applied to all the points in the index_A’th set in convex_sets_A to align
them with the index_B’th set in convex_sets_B. This translation may only have non-zero entries along the dimensions corresponding to continuous_revolute_joints. All non-zero entries are integer
multiples of 2π as the translation of the sets still represents the same configurations for the indices in continuous_revolute_joints.
Parameter convex_sets_A:
is a vector of convex sets. Pairwise intersections will be computed between convex_sets_A and convex_sets_B.
Parameter convex_sets_B:
is the other vector of convex sets.
Parameter continuous_revolute_joints:
is a list of joint indices corresponding to continuous revolute joints.
Parameter preprocess_bbox:
is a flag for whether the function should precompute axis-aligned bounding boxes (AABBs) for every set. This can speed up the pairwise intersection checks, by determining some sets to be
disjoint without needing to solve an optimization problem. However, it does require some overhead to compute those bounding boxes.
☆ if continuous_revolute_joints has repeated entries, or if any –
☆ entry is outside the interval [0, ambient_dimension), where –
☆ ambient_dimension is the ambient dimension of the convex sets in –
☆ convex_sets_A` and convex_sets_B –
☆ if convex_sets_A or convex_sets_B are empty. */ –
☆ (Deprecated.) –
Instead use ComputePairwiseIntersections, with return type std::pair<std::vector<std::pair<int, int>>, std::vector<Eigen::VectorXd>>. This will be removed from Drake on or after 2024-12-01.
2. CalcPairwiseIntersections(convex_sets_A: list[pydrake.geometry.optimization.ConvexSet], convex_sets_B: list[pydrake.geometry.optimization.ConvexSet], continuous_revolute_joints: list[int],
bboxes_A: list[pydrake.geometry.optimization.Hyperrectangle], bboxes_B: list[pydrake.geometry.optimization.Hyperrectangle]) -> list[tuple[int, int, numpy.ndarray[numpy.float64[m, 1]]]]
Overload of CalcPairwiseIntersections allowing the user to supply axis- aligned bounding boxes if they’re known a priori, to save on computation time.
Parameter bboxes_A:
is a vector of Hyperrectangles, allowing the user to manually pass in the AABBs of each set in convex_sets_A to avoid recomputation.
Parameter bboxes_B:
serves the same role to convex_sets_B as bboxes_A does to convex_sets_A.
The function does not check that the entries of bboxes_A are indeed the AABBs corresponding to the sets in convex_sets_A (and likewise for bboxes_B).
☆ if convex_sets_A.size() != bboxes_A.size() –
☆ if convex_sets_B.size() != bboxes_B.size() –
☆ if not all entries of convex_sets_A, convex_sets_B, –
☆ bboxes_A`, and bboxes_B have the same ambient dimension. * –
☆ (Deprecated.) –
Instead use ComputePairwiseIntersections, with return type std::pair<std::vector<std::pair<int, int>>, std::vector<Eigen::VectorXd>>. This will be removed from Drake on or after 2024-12-01.
3. CalcPairwiseIntersections(convex_sets: list[pydrake.geometry.optimization.ConvexSet], continuous_revolute_joints: list[int], preprocess_bbox: bool = True) -> list[tuple[int, int,
numpy.ndarray[numpy.float64[m, 1]]]]
Convenience overload to compute pairwise intersections within a list of convex sets. Equivalent to calling CalcPairwiseIntersections(convex_sets, convex_sets, continuous_revolute_joints).
Parameter convex_sets:
is a vector of convex sets. Pairwise intersections will be computed within convex_sets.
Parameter continuous_revolute_joints:
is a list of joint indices corresponding to continuous revolute joints.
Parameter preprocess_bbox:
is a flag for whether the function should precompute axis-aligned bounding boxes for every set. This can speed up the pairwise intersection checks, by determining some sets to be disjoint
without needing to solve an optimization problem.
☆ if continuous_revolute_joints has repeated entries, or if any –
☆ entry is outside the interval [0, ambient_dimension), where –
☆ ambient_dimension is the ambient dimension of the convex sets in –
☆ convex_sets` –
☆ if convex_sets is empty. */ (Deprecated.) –
Instead use ComputePairwiseIntersections, with return type std::pair<std::vector<std::pair<int, int>>, std::vector<Eigen::VectorXd>>. This will be removed from Drake on or after 2024-12-01.
4. CalcPairwiseIntersections(convex_sets: list[pydrake.geometry.optimization.ConvexSet], continuous_revolute_joints: list[int], bboxes: list[pydrake.geometry.optimization.Hyperrectangle] = [])
-> list[tuple[int, int, numpy.ndarray[numpy.float64[m, 1]]]]
Overload of CalcPairwiseIntersections allowing the user to supply axis- aligned bounding boxes if they’re known a priori, to save on computation time.
Parameter bboxes:
is a vector of Hyperrectangles, allowing the user to manually pass in the AABBs of each set in convex_sets to avoid recomputation.
The function does not check that the entries are indeed the AABBs corresponding to the sets in convex_sets.
☆ if convex_sets.size() != bboxes.size() –
☆ if not all entries of convex_sets and bboxes have the same –
☆ ambient dimension. */ (Deprecated.) –
Instead use ComputePairwiseIntersections, with return type std::pair<std::vector<std::pair<int, int>>, std::vector<Eigen::VectorXd>>. This will be removed from Drake on or after 2024-12-01.
class pydrake.geometry.optimization.CartesianProduct
Bases: pydrake.geometry.optimization.ConvexSet
The Cartesian product of convex sets is a convex set: S = X₁ × X₂ × ⋯ × Xₙ = {(x₁, x₂, …, xₙ) | x₁ ∈ X₁, x₂ ∈ X₂, …, xₙ ∈ Xₙ}.
This class also supports a generalization of this concept in which the coordinates are transformed by the linear map, {x | y = Ax + b, y ∈ Y₁ × Y₂ × ⋯ × Yₙ}, with the default values set to the
identity map. This concept is required for reasoning about cylinders in arbitrary poses as cartesian products, and more generally for describing any affine transform of a CartesianProduct.
Special behavior for IsEmpty: If there are no sets in the product, returns nonempty by convention. See: https://en.wikipedia.org/wiki/Empty_product#Nullary_Cartesian_product Otherwise, if any set
in the cartesian product is empty, the whole product is empty.
pydrake.geometry.optimization.CheckIfSatisfiesConvexityRadius(convex_set: pydrake.geometry.optimization.ConvexSet, continuous_revolute_joints: list[int]) bool
Given a convex set, and a list of indices corresponding to continuous revolute joints, checks whether or not the set satisfies the convexity radius. See §6.5.3 of “A Panoramic View of Riemannian
Geometry”, Marcel Berger for a general definition of convexity radius. When dealing with continuous revolute joints, respecting the convexity radius entails that each convex set has a width of
stricty less than π along each dimension corresponding to a continuous revolute joint.
☆ RuntimeError if continuous_revolute_joints has repeated entries, –
☆ or if any entry is outside the interval [0, –
☆ convex_set.ambient_dimension()) –
class pydrake.geometry.optimization.CIrisCollisionGeometry
This class contains the necessary information about the collision geometry used in C-IRIS. Most notably it transcribes the geometric condition that the collision geometry is on one side of the
plane to mathematical constraints. For the detailed algorithm please refer to the paper Certified Polyhedral Decompositions of Collision-Free Configuration Space by Hongkai Dai*, Alexandre
Amice*, Peter Werner, Annan Zhang and Russ Tedrake.
class pydrake.geometry.optimization.CIrisGeometryType
The supported type of geometries in C-IRIS.
kPolytope :
kSphere :
kCylinder :
kCapsule :
__init__(self: pydrake.geometry.optimization.CIrisGeometryType, value: int) None
kCapsule = <CIrisGeometryType.kCapsule: 3>
kCylinder = <CIrisGeometryType.kCylinder: 2>
kPolytope = <CIrisGeometryType.kPolytope: 1>
kSphere = <CIrisGeometryType.kSphere: 0>
property name
property value
pydrake.geometry.optimization.ComputePairwiseIntersections(*args, **kwargs)
Overloaded function.
1. ComputePairwiseIntersections(convex_sets_A: list[pydrake.geometry.optimization.ConvexSet], convex_sets_B: list[pydrake.geometry.optimization.ConvexSet], continuous_revolute_joints: list[int],
preprocess_bbox: bool = True) -> tuple[list[tuple[int, int]], list[numpy.ndarray[numpy.float64[m, 1]]]]
Computes the pairwise intersections between two lists of convex sets, returning a list of edges, and a list of their corresponding offsets. Each edge is a tuple in the form [index_A, index_B],
where index_A is the index of the set in convex_sets_A and index_B is the index of the set in convex_sets_B. The corresponding entry in the list of offsets (i.e., the entry at the same index) is
the translation that is applied to all the points in the index_A’th set in convex_sets_A to align them with the index_B’th set in convex_sets_B. This translation may only have non-zero entries
along the dimensions corresponding to continuous_revolute_joints. All non-zero entries are integer multiples of 2π as the translation of the sets still represents the same configurations for the
indices in continuous_revolute_joints.
Parameter convex_sets_A:
is a vector of convex sets. Pairwise intersections will be computed between convex_sets_A and convex_sets_B.
Parameter convex_sets_B:
is the other vector of convex sets.
Parameter continuous_revolute_joints:
is a list of joint indices corresponding to continuous revolute joints.
Parameter preprocess_bbox:
is a flag for whether the function should precompute axis-aligned bounding boxes (AABBs) for every set. This can speed up the pairwise intersection checks, by determining some sets to be
disjoint without needing to solve an optimization problem. However, it does require some overhead to compute those bounding boxes.
☆ if continuous_revolute_joints has repeated entries, or if any –
☆ entry is outside the interval [0, ambient_dimension), where –
☆ ambient_dimension is the ambient dimension of the convex sets in –
☆ convex_sets_A` and convex_sets_B –
☆ if convex_sets_A or convex_sets_B are empty. –
2. ComputePairwiseIntersections(convex_sets_A: list[pydrake.geometry.optimization.ConvexSet], convex_sets_B: list[pydrake.geometry.optimization.ConvexSet], continuous_revolute_joints: list[int],
bboxes_A: list[pydrake.geometry.optimization.Hyperrectangle], bboxes_B: list[pydrake.geometry.optimization.Hyperrectangle]) -> tuple[list[tuple[int, int]], list[numpy.ndarray[numpy.float64[m,
Overload of ComputePairwiseIntersections allowing the user to supply axis- aligned bounding boxes if they’re known a priori, to save on computation time.
Parameter bboxes_A:
is a vector of Hyperrectangles, allowing the user to manually pass in the AABBs of each set in convex_sets_A to avoid recomputation.
Parameter bboxes_B:
serves the same role to convex_sets_B as bboxes_A does to convex_sets_A.
The function does not check that the entries of bboxes_A are indeed the AABBs corresponding to the sets in convex_sets_A (and likewise for bboxes_B).
☆ if convex_sets_A.size() != bboxes_A.size() –
☆ if convex_sets_B.size() != bboxes_B.size() –
☆ if not all entries of convex_sets_A, convex_sets_B, –
☆ bboxes_A`, and bboxes_B have the same ambient dimension –
3. ComputePairwiseIntersections(convex_sets: list[pydrake.geometry.optimization.ConvexSet], continuous_revolute_joints: list[int], preprocess_bbox: bool = True) -> tuple[list[tuple[int, int]],
list[numpy.ndarray[numpy.float64[m, 1]]]]
Convenience overload to compute pairwise intersections within a list of convex sets. Equivalent to calling ComputePairwiseIntersections(convex_sets, convex_sets, continuous_revolute_joints).
Parameter convex_sets:
is a vector of convex sets. Pairwise intersections will be computed within convex_sets.
Parameter continuous_revolute_joints:
is a list of joint indices corresponding to continuous revolute joints.
Parameter preprocess_bbox:
is a flag for whether the function should precompute axis-aligned bounding boxes for every set. This can speed up the pairwise intersection checks, by determining some sets to be disjoint
without needing to solve an optimization problem.
☆ if continuous_revolute_joints has repeated entries, or if any –
☆ entry is outside the interval [0, ambient_dimension), where –
☆ ambient_dimension is the ambient dimension of the convex sets in –
☆ convex_sets` –
☆ if convex_sets is empty. –
4. ComputePairwiseIntersections(convex_sets: list[pydrake.geometry.optimization.ConvexSet], continuous_revolute_joints: list[int], bboxes: list[pydrake.geometry.optimization.Hyperrectangle] =
[]) -> tuple[list[tuple[int, int]], list[numpy.ndarray[numpy.float64[m, 1]]]]
Overload of ComputePairwiseIntersections allowing the user to supply axis- aligned bounding boxes if they’re known a priori, to save on computation time.
Parameter bboxes:
is a vector of Hyperrectangles, allowing the user to manually pass in the AABBs of each set in convex_sets to avoid recomputation.
The function does not check that the entries are indeed the AABBs corresponding to the sets in convex_sets.
☆ if convex_sets.size() != bboxes.size() –
☆ if not all entries of convex_sets and bboxes have the same –
☆ ambient dimension. –
class pydrake.geometry.optimization.ConvexHull
Bases: pydrake.geometry.optimization.ConvexSet
Implements the convex hull of a set of convex sets. The convex hull of multiple sets is defined as the smallest convex set that contains all the sets. Given non-empty convex sets {X₁, X₂, …, Xₙ},
the convex hull is the set of all convex combinations of points in the sets, i.e. {∑ᵢ λᵢ xᵢ | xᵢ ∈ Xᵢ, λᵢ ≥ 0, ∑ᵢ λᵢ = 1}.
__init__(self: pydrake.geometry.optimization.ConvexHull, sets: list[pydrake.geometry.optimization.ConvexSet], remove_empty_sets: bool = True) None
Constructs the convex hull from a vector of convex sets.
Parameter sets:
A vector of convex sets that define the convex hull.
Parameter remove_empty_sets:
If true, the constructor will check if any of the sets are empty and will not consider them. If false, the constructor will not check if any of the sets are empty.
If remove_empty_sets is set to false, but some of the sets are in fact empty, then unexpected and incorrect results may occur. Only set this flag to false if you are sure that your sets are
non-empty and performance in the constructor is critical.
element(self: pydrake.geometry.optimization.ConvexHull, index: int) pydrake.geometry.optimization.ConvexSet
Returns a reference to the convex set at the given index (including empty sets).
empty_sets_removed(self: pydrake.geometry.optimization.ConvexHull) bool
Returns true if this was constructed with remove_empty_sets=true.
num_elements(self: pydrake.geometry.optimization.ConvexHull) int
Returns the number of convex sets defining the convex hull (including empty sets).
participating_sets(self: pydrake.geometry.optimization.ConvexHull) object
Returns the participating convex sets.
sets(self: pydrake.geometry.optimization.ConvexHull) object
Returns the participating convex sets.
class pydrake.geometry.optimization.ConvexSet
Abstract base class for defining a convex set.
class pydrake.geometry.optimization.CspaceFreePolytope
Bases: pydrake.geometry.optimization.CspaceFreePolytopeBase
This class tries to find large convex polytopes in the tangential-configuration space, such that all configurations in the convex polytopes is collision free. By tangential-configuration space,
we mean the revolute joint angle θ is replaced by t = tan(θ/2). We refer to the algorithm as C-IRIS. For more details, refer to the paper
Certified Polyhedral Decomposition of Collision-Free Configuration Space by Hongkai Dai*, Alexandre Amice*, Peter Werner, Annan Zhang and Russ Tedrake.
A conference version is published at
Finding and Optimizing Certified, Collision-Free Regions in Configuration Space for Robot Manipulators by Alexandre Amice*, Hongkai Dai*, Peter Werner, Annan Zhang and Russ Tedrake.
__init__(self: pydrake.geometry.optimization.CspaceFreePolytope, plant: drake::multibody::MultibodyPlant<double>, scene_graph: pydrake.geometry.SceneGraph, plane_order:
pydrake.geometry.optimization.SeparatingPlaneOrder, q_star: numpy.ndarray[numpy.float64[m, 1]], options: pydrake.geometry.optimization.CspaceFreePolytopeBase.Options = Options(with_cross_y=
False)) None
Parameter plant:
The plant for which we compute the C-space free polytopes. It must outlive this CspaceFreePolytope object.
Parameter scene_graph:
The scene graph that has been connected with plant. It must outlive this CspaceFreePolytope object.
Parameter plane_order:
The order of the polynomials in the plane to separate a pair of collision geometries.
Parameter q_star:
Refer to RationalForwardKinematics for its meaning.
CspaceFreePolytope knows nothing about contexts. The plant and scene_graph must be fully configured before instantiating this class.
class BilinearAlternationOptions
Options for bilinear alternation.
__init__(self: pydrake.geometry.optimization.CspaceFreePolytope.BilinearAlternationOptions) None
property convergence_tol
When the change of the cost function between two consecutive iterations in bilinear alternation is no larger than this number, stop the bilinear alternation. Must be non-negative.
property ellipsoid_scaling
After finding the maximal inscribed ellipsoid in C-space polytope {s | C*s<=d, s_lower<=s<=s_upper}, we scale this ellipsoid by ellipsoid_scaling, and require the new C-space polytope to
contain this scaled ellipsoid. ellipsoid_scaling=1 corresponds to no scaling. Must be strictly positive and no greater than 1.
property find_lagrangian_options
property find_polytope_options
property max_iter
The maximum number of bilinear alternation iterations. Must be non-negative.
BinarySearch(self: pydrake.geometry.optimization.CspaceFreePolytope, ignored_collision_pairs: set[Tuple[pydrake.geometry.GeometryId]], C: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], d
: numpy.ndarray[numpy.float64[m, 1]], s_center: numpy.ndarray[numpy.float64[m, 1]], options: pydrake.geometry.optimization.CspaceFreePolytope.BinarySearchOptions) Optional[
Binary search on d such that the C-space polytope {s | C*s<=d, s_lower<=s<=s_upper} is collision free. We scale the polytope {s | C*s<=d_init} about its center s_center and search the scaling
s_center is in the polytope {s | C*s<=d_init, s_lower<=s<=s_upper}
class BinarySearchOptions
Options for binary search.
__init__(self: pydrake.geometry.optimization.CspaceFreePolytope.BinarySearchOptions) None
property convergence_tol
property find_lagrangian_options
property max_iter
property scale_max
property scale_min
class EllipsoidMarginCost
The cost used when fixing the Lagrangian multiplier and search for C and d in the C-space polytope {s | C*s <=d, s_lower<=s<=s_upper}. We denote δᵢ as the margin between the i’th face C.row
(i)<=d(i) to the inscribed ellipsoid.
__init__(self: pydrake.geometry.optimization.CspaceFreePolytope.EllipsoidMarginCost, value: int) None
kGeometricMean = <EllipsoidMarginCost.kGeometricMean: 1>
kSum = <EllipsoidMarginCost.kSum: 0>
property name
property value
class FindPolytopeGivenLagrangianOptions
Options for finding polytope with given Lagrangians.
__init__(self: pydrake.geometry.optimization.CspaceFreePolytope.FindPolytopeGivenLagrangianOptions) None
property backoff_scale
property ellipsoid_margin_cost
property ellipsoid_margin_epsilon
property s_inner_pts
property search_s_bounds_lagrangians
property solver_id
property solver_options
FindSeparationCertificateGivenPolytope(self: pydrake.geometry.optimization.CspaceFreePolytope, C: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], d: numpy.ndarray[numpy.float64[m, 1]],
ignored_collision_pairs: set[Tuple[pydrake.geometry.GeometryId]], options: pydrake.geometry.optimization.CspaceFreePolytope.FindSeparationCertificateGivenPolytopeOptions) tuple[bool, dict[
Tuple[pydrake.geometry.GeometryId], pydrake.geometry.optimization.CspaceFreePolytope.SeparationCertificateResult]]
Finds the certificates that the C-space polytope {s | C*s<=d, s_lower <= s <= s_upper} is collision free.
Parameter C:
The C-space polytope is {s | C*s<=d, s_lower<=s<=s_upper}
Parameter d:
The C-space polytope is {s | C*s<=d, s_lower<=s<=s_upper}
Parameter ignored_collision_pairs:
We will ignore the pair of geometries in ignored_collision_pairs.
Parameter certificates:
Contains the certificate we successfully found for each pair of geometries. Notice that depending on options, the program could search for the certificate for each geometry pair in
parallel, and will terminate the search once it fails to find the certificate for any pair.
Returns success:
If true, then we have certified that the C-space polytope {s | C*s<=d, s_lower<=s<=s_upper} is collision free. Otherwise success=false.
class FindSeparationCertificateGivenPolytopeOptions
Bases: pydrake.geometry.optimization.FindSeparationCertificateOptions
__init__(self: pydrake.geometry.optimization.CspaceFreePolytope.FindSeparationCertificateGivenPolytopeOptions) None
property ignore_redundant_C
MakeIsGeometrySeparableProgram(self: pydrake.geometry.optimization.CspaceFreePolytope, geometry_pair: Tuple[pydrake.geometry.GeometryId], C: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]
, d: numpy.ndarray[numpy.float64[m, 1]]) pydrake.geometry.optimization.CspaceFreePolytope.SeparationCertificateProgram
Constructs the MathematicalProgram which searches for a separation certificate for a pair of geometries for a C-space polytope. Search for the separation certificate for a pair of geometries
for a C-space polytope {s | C*s<=d, s_lower<=s<=s_upper}.
○ an error if this geometry_pair doesn't need separation –
○ certificate (for example, they are on the same body) –
class SearchResult
Result on searching the C-space polytope and separating planes.
SearchWithBilinearAlternation(self: pydrake.geometry.optimization.CspaceFreePolytope, ignored_collision_pairs: set[Tuple[pydrake.geometry.GeometryId]], C_init: numpy.ndarray[numpy.float64[m, n],
flags.f_contiguous], d_init: numpy.ndarray[numpy.float64[m, 1]], options: pydrake.geometry.optimization.CspaceFreePolytope.BilinearAlternationOptions) list[
Search for a collision-free C-space polytope. {s | C*s<=d, s_lower<=s<=s_upper} through bilinear alternation. The goal is to maximize the volume the C-space polytope. Since we can’t compute
the polytope volume in the closed form, we use the volume of the maximal inscribed ellipsoid as a surrogate function of the polytope volume.
Parameter ignored_collision_pairs:
The pairs of geometries that we ignore when searching for separation certificates.
Parameter C_init:
The initial value of C.
Parameter d_init:
The initial value of d.
Parameter options:
The options for the bilinear alternation.
Returns results:
Stores the certification result in each iteration of the bilinear alternation.
class SeparatingPlaneLagrangians
When searching for the separating plane, we want to certify that the numerator of a rational is non-negative in the C-space region C*s<=d, s_lower <= s <= s_upper. Hence for each of the
rational we will introduce Lagrangian multipliers for the polytopic constraint d-C*s >= 0, s - s_lower >= 0, s_upper - s >= 0.
class SeparationCertificate
This struct stores the necessary information to search for the separating plane for the polytopic C-space region C*s <= d, s_lower <= s <= s_upper. We need to impose that N rationals are
non-negative in this C-space polytope. The denominator of each rational is always positive hence we need to impose the N numerators are non-negative in this C-space region. We impose the
condition numerator_i(s) - λ(s)ᵀ * (d - C*s) - λ_lower(s)ᵀ * (s - s_lower) -λ_upper(s)ᵀ * (s_upper - s) is sos λ(s) are sos, λ_lower(s) are sos, λ_upper(s) are sos.
__init__(*args, **kwargs)
GetSolution(self: pydrake.geometry.optimization.CspaceFreePolytope.SeparationCertificate, plane_index: int, a: numpy.ndarray[object[3, 1]], b: pydrake.symbolic.Polynomial, plane_decision_vars
: numpy.ndarray[object[m, 1]], result: pydrake.solvers.MathematicalProgramResult) pydrake.geometry.optimization.CspaceFreePolytope.SeparationCertificateResult
property negative_side_rational_lagrangians
property positive_side_rational_lagrangians
class SeparationCertificateProgram
Bases: pydrake.geometry.optimization.SeparationCertificateProgramBase
__init__(self: pydrake.geometry.optimization.CspaceFreePolytope.SeparationCertificateProgram) None
property certificate
property plane_index
class SeparationCertificateResult
We certify that a pair of geometries is collision free in the C-space region {s | Cs<=d, s_lower<=s<=s_upper} by finding the separating plane and the Lagrangian multipliers. This struct
contains the certificate, that the separating plane {x | aᵀx+b=0 } separates the two geometries in separating_planes()[plane_index] in the C-space polytope.
__init__(*args, **kwargs)
property a
The separating plane is { x | aᵀx+b=0 }
property b
property negative_side_rational_lagrangians
property plane_decision_var_vals
property plane_index
property positive_side_rational_lagrangians
property result
SolveSeparationCertificateProgram(self: pydrake.geometry.optimization.CspaceFreePolytope, certificate_program: pydrake.geometry.optimization.CspaceFreePolytope.SeparationCertificateProgram,
options: pydrake.geometry.optimization.CspaceFreePolytope.FindSeparationCertificateGivenPolytopeOptions) Optional[pydrake.geometry.optimization.CspaceFreePolytope.SeparationCertificateResult]
Solves a SeparationCertificateProgram with the given options
result If we find the separation certificate, then result contains the separation plane and the Lagrangian polynomials; otherwise result is empty.
class pydrake.geometry.optimization.CspaceFreePolytopeBase
This virtual class is the base of CspaceFreePolytope and CspaceFreeBox. We take the common functionality between these concrete derived class to this shared parent class.
class pydrake.geometry.optimization.CSpaceSeparatingPlane
Wraps the information that a pair of collision geometries are separated by a plane. One collision geometry is on the “positive” side of the separating plane, namely {x| aᵀx + b ≥ δ} (with δ ≥ 0},
and the other collision geometry is on the “negative” side of the separating plane, namely {x|aᵀx+b ≤ −δ}.
Template parameter T:
The type of decision_variables. T= symbolic::Variable or double.
__init__(*args, **kwargs)
property a
property b
property decision_variables
property expressed_body
property negative_side_geometry
property plane_degree
property positive_side_geometry
template pydrake.geometry.optimization.CSpaceSeparatingPlane_
Instantiations: CSpaceSeparatingPlane_[float], CSpaceSeparatingPlane_[Variable]
class pydrake.geometry.optimization.CSpaceSeparatingPlane_[Variable]
Wraps the information that a pair of collision geometries are separated by a plane. One collision geometry is on the “positive” side of the separating plane, namely {x| aᵀx + b ≥ δ} (with δ ≥ 0},
and the other collision geometry is on the “negative” side of the separating plane, namely {x|aᵀx+b ≤ −δ}.
Template parameter T:
The type of decision_variables. T= symbolic::Variable or double.
__init__(*args, **kwargs)
property a
property b
property decision_variables
property expressed_body
property negative_side_geometry
property plane_degree
property positive_side_geometry
class pydrake.geometry.optimization.FindSeparationCertificateOptions
__init__(self: pydrake.geometry.optimization.FindSeparationCertificateOptions) None
property parallelism
property solver_id
property solver_options
property terminate_at_failure
property verbose
class pydrake.geometry.optimization.GcsGraphvizOptions
__init__(self: pydrake.geometry.optimization.GcsGraphvizOptions, **kwargs) None
property precision
Sets the floating point precision (how many digits are generated) of the annotations.
property scientific
Sets the floating point formatting to scientific (if true) or fixed (if false).
property show_costs
Determines whether the cost value results are shown. This will show both edge and vertex costs.
property show_flows
Determines whether the flow value results are shown. The flow values are shown both with a numeric value and through the transparency value on the edge, where a flow of 0.0 will correspond to
an (almost) invisible edge, and a flow of 1.0 will display as a fully black edge.
property show_slacks
Determines whether the values of the intermediate (slack) variables are also displayed in the graph.
property show_vars
Determines whether the solution values for decision variables in each set are shown.
class pydrake.geometry.optimization.GraphOfConvexSets
GraphOfConvexSets (GCS) implements the design pattern and optimization problems first introduced in the paper “Shortest Paths in Graphs of Convex Sets”.
“Shortest Paths in Graphs of Convex Sets” by Tobia Marcucci, Jack Umenberger, Pablo A. Parrilo, Russ Tedrake. https://arxiv.org/abs/2101.11565
This feature is considered to be experimental and may change or be removed at any time, without any deprecation notice ahead of time.
Each vertex in the graph is associated with a convex set over continuous variables, edges in the graph contain convex costs and constraints on these continuous variables. We can then formulate
optimization problems over this graph, such as the shortest path problem where each visit to a vertex also corresponds to selecting an element from the convex set subject to the costs and
constraints. Behind the scenes, we construct efficient mixed-integer convex transcriptions of the graph problem using MathematicalProgram. However, we provide the option to solve an often tight
convex relaxation of the problem with GraphOfConvexSetsOptions::convex_relaxation and employ a cheap rounding stage which solves the convex restriction along potential paths to find a feasible
solution to the original problem.
Design note: This class avoids providing any direct access to the MathematicalProgram that it constructs nor to the decision variables / constraints. The users should be able to write constraints
against “placeholder” decision variables on the vertices and edges, but these get translated in non-trivial ways to the underlying program.
Advanced Usage: Guiding Non-convex Optimization with the GraphOfConvexSets
Solving a GCS problem using convex relaxation involves two components: - Convex Relaxation: The relaxation of the binary variables (edge activations) and perspective operations on the convex cost
/constraints leads to a convex problem that considers the graph as a whole. - Rounding: After solving the relaxation, a randomized rounding scheme is applied to obtain a feasible solution for the
original problem. We interpret the relaxed flow variables as edge probabilities to guide the maximum likelyhood depth first search from the source to target vertices. Each rounding is calling
To handle non-convex constraints, one can provide convex surrogates to the relaxation and the true non-convex constraints to the rounding problem. These surrogates approximate the non-convex
constraints, making the relaxation solvable as a convex optimization to guide the non-convex rounding. This can be controlled by the Transcription enum in the AddConstraint method. We encourage
users to provide a strong convex surrogate, when possible, to better approximate the original non-convex problem.
Users can also specify a GCS implicitly, which can be important for very large or infinite graphs, by deriving from ImplicitGraphOfConvexSets.
class pydrake.geometry.optimization.GraphOfConvexSetsOptions
__init__(self: pydrake.geometry.optimization.GraphOfConvexSetsOptions) None
property convex_relaxation
Flag to solve the relaxed version of the problem. As discussed in the paper, we know that this relaxation cannot solve the original NP-hard problem for all instances, but there are also many
instances for which the convex relaxation is tight. If convex_relaxation=nullopt, then each GCS method is free to choose an appropriate default.
property flow_tolerance
Tolerance for ignoring flow along a given edge during random rounding. If convex_relaxation is false or max_rounded_paths is less than or equal to zero, this option is ignored.
property max_rounded_paths
Maximum number of distinct paths to compare during random rounding; only the lowest cost path is returned. If convex_relaxation is false or this is less than or equal to zero, rounding is not
performed. If max_rounded_paths=nullopt, then each GCS method is free to choose an appropriate default.
property max_rounding_trials
Maximum number of trials to find a novel path during random rounding. If convex_relaxation is false or max_rounded_paths is less than or equal to zero, this option is ignored.
property preprocessing
Performs a preprocessing step to remove edges that cannot lie on the path from source to target. In most cases, preprocessing causes a net reduction in computation by reducing the size of the
optimization solved. Note that this preprocessing is not exact. There may be edges that cannot lie on the path from source to target that this does not detect. If preprocessing=nullopt, then
each GCS method is free to choose an appropriate default.
property preprocessing_solver
Optimizer to be used in the preprocessing stage of GCS, which is performed when SolveShortestPath is called when the preprocessing setting has been set to true. If not set, the interface at
.solver will be used, if provided, otherwise the best solver for the given problem is selected. Note that if the solver cannot handle the type of optimization problem generated, then calling
the solvers::SolverInterface::Solve() method will throw.
property preprocessing_solver_options
Optional solver options to be used by preprocessing_solver in the preprocessing stage of GCS, which is used in SolveShortestPath. If preprocessing_solver is set but this parameter is not then
solver_options is used. For instance, one might want to print solver logs for the main optimization, but not from the many smaller preprocessing optimizations.
property restriction_solver
Optimizer to be used in SolveConvexRestriction(), which is also called during the rounding stage of SolveShortestPath() given the relaxation. If not set, the interface at .solver will be
used, if provided, otherwise the best solver for the given problem is selected. Note that if the solver cannot handle the type of optimization problem generated, then calling the
solvers::SolverInterface::Solve() method will throw.
property restriction_solver_options
Optional solver options to be used in SolveConvexRestriction(), which is also used during the rounding stage of SolveShortestPath() given the relaxation. If not set, solver_options is used.
For instance, one might want to set tighter (i.e., lower) tolerances for running the relaxed problem and looser (i.e., higher) tolerances for final solves during rounding.
property rounding_seed
Random seed to use for random rounding. If convex_relaxation is false or max_rounded_paths is less than or equal to zero, this option is ignored.
property solver
Optimizer to be used to solve the MIP, the relaxation of the shortest path optimization problem and the convex restriction if no restriction_solver is provided. If not set, the best solver
for the given problem is selected. Note that if the solver cannot handle the type of optimization problem generated, the calling solvers::SolverInterface::Solve() method will throw.
property solver_options
Options passed to the solver when solving the generated problem.
class pydrake.geometry.optimization.HPolyhedron
Bases: pydrake.geometry.optimization.ConvexSet
Implements a polyhedral convex set using the half-space representation: {x| A x ≤ b}. Note: This set may be unbounded.
By convention, we treat a zero-dimensional HPolyhedron as nonempty.
class pydrake.geometry.optimization.Hyperellipsoid
Bases: pydrake.geometry.optimization.ConvexSet
Implements an ellipsoidal convex set represented by the quadratic form {x | (x-center)ᵀAᵀA(x-center) ≤ 1}. Note that A need not be square; we require only that the matrix AᵀA is positive
Compare this with an alternative (very useful) parameterization of the ellipsoid: {Bu + center | |u|₂ ≤ 1}, which is an affine scaling of the unit ball. This is related to the quadratic form by B
= A⁻¹, when A is invertible, but the quadratic form can also represent unbounded sets. The affine scaling of the unit ball representation is available via the AffineBall class.
Note: the name Hyperellipsoid was taken here to avoid conflicting with geometry::Ellipsoid and to distinguish that this class supports N dimensions.
A hyperellipsoid can never be empty – it always contains its center. This includes the zero-dimensional case.
class pydrake.geometry.optimization.Hyperrectangle
Bases: pydrake.geometry.optimization.ConvexSet
Axis-aligned hyperrectangle in Rᵈ defined by its lower bounds and upper bounds as {x| lb ≤ x ≤ ub}
class pydrake.geometry.optimization.ImplicitGraphOfConvexSets
A base class to define the interface to an implicit graph of convex sets.
Implementations of this class must implement DoSuccesors() and provide some method of accessing at least one vertex in the graph.
This feature is considered to be experimental and may change or be removed at any time, without any deprecation notice ahead of time.
class pydrake.geometry.optimization.Intersection
Bases: pydrake.geometry.optimization.ConvexSet
A convex set that represents the intersection of multiple sets: S = X₁ ∩ X₂ ∩ … ∩ Xₙ = {x | x ∈ X₁, x ∈ X₂, …, x ∈ Xₙ}
Special behavior for IsEmpty: The intersection of zero sets (i.e. when we have sets_.size() == 0) is always nonempty. This includes the zero-dimensional case, which we treat as being {0}, the
unique zero-dimensional vector space.
__init__(*args, **kwargs)
Overloaded function.
1. __init__(self: pydrake.geometry.optimization.Intersection) -> None
Constructs a default (zero-dimensional, nonempty) set.
2. __init__(self: pydrake.geometry.optimization.Intersection, sets: list[pydrake.geometry.optimization.ConvexSet]) -> None
Constructs the intersection from a vector of convex sets.
3. __init__(self: pydrake.geometry.optimization.Intersection, setA: pydrake.geometry.optimization.ConvexSet, setB: pydrake.geometry.optimization.ConvexSet) -> None
Constructs the intersection from a pair of convex sets.
element(self: pydrake.geometry.optimization.Intersection, index: int) pydrake.geometry.optimization.ConvexSet
Returns a reference to the ConvexSet defining the index element in the intersection.
num_elements(self: pydrake.geometry.optimization.Intersection) int
The number of elements (or sets) used in the intersection.
pydrake.geometry.optimization.Iris(obstacles: list[pydrake.geometry.optimization.ConvexSet], sample: numpy.ndarray[numpy.float64[m, 1]], domain: pydrake.geometry.optimization.HPolyhedron, options:
pydrake.geometry.optimization.IrisOptions = IrisOptions(require_sample_point_is_contained=False, iteration_limit=100, termination_threshold=0.02, relative_termination_threshold=0.001,
configuration_space_margin=0.01, num_collision_infeasible_samples=5, configuration_obstacles [], prog_with_additional_constraints is not set, num_additional_constraint_infeasible_samples=5,
random_seed=1234, mixing_steps=10)) pydrake.geometry.optimization.HPolyhedron
The IRIS (Iterative Region Inflation by Semidefinite programming) algorithm, as described in
R. L. H. Deits and R. Tedrake, “Computing large convex regions of obstacle-free space through semidefinite programming,” Workshop on the Algorithmic Fundamentals of Robotics, Istanbul, Aug. 2014.
This algorithm attempts to locally maximize the volume of a convex polytope representing obstacle-free space given a sample point and list of convex obstacles. Rather than compute the volume of
the polytope directly, the algorithm maximizes the volume of an inscribed ellipsoid. It alternates between finding separating hyperplanes between the ellipsoid and the obstacles and then finding
a new maximum-volume inscribed ellipsoid.
Parameter obstacles:
is a vector of convex sets representing the occupied space.
Parameter sample:
provides a point in the space; the algorithm is initialized using a tiny sphere around this point. The algorithm is only guaranteed to succeed if this sample point is collision free (outside
of all obstacles), but in practice the algorithm can often escape bad initialization (assuming the require_sample_point_is_contained option is false).
Parameter domain:
describes the total region of interest; computed IRIS regions will be inside this domain. It must be bounded, and is typically a simple bounding box (e.g. from HPolyhedron::MakeBox).
The obstacles, sample, and the domain must describe elements in the same ambient dimension (but that dimension can be any positive integer).
pydrake.geometry.optimization.IrisInConfigurationSpace(plant: drake::multibody::MultibodyPlant<double>, context: pydrake.systems.framework.Context, options: pydrake.geometry.optimization.IrisOptions
= IrisOptions(require_sample_point_is_contained=False, iteration_limit=100, termination_threshold=0.02, relative_termination_threshold=0.001, configuration_space_margin=0.01,
num_collision_infeasible_samples=5, configuration_obstacles [], prog_with_additional_constraints is not set, num_additional_constraint_infeasible_samples=5, random_seed=1234, mixing_steps=10))
A variation of the Iris (Iterative Region Inflation by Semidefinite programming) algorithm which finds collision-free regions in the configuration space of plant.
See also
Iris for details on the original algorithm. This variant uses nonlinear optimization (instead of convex optimization) to find collisions in configuration space; each potential collision is
probabilistically “certified” by restarting the nonlinear optimization from random initial seeds inside the candidate IRIS region until it fails to find a collision in
options.num_collision_infeasible_samples consecutive attempts.
This method constructs a single Iris region in the configuration space of plant.
See also
planning::IrisInConfigurationSpaceFromCliqueCover for a method to automatically cover the configuration space with multiple Iris regions.
Parameter plant:
describes the kinematics of configuration space. It must be connected to a SceneGraph in a systems::Diagram.
Parameter context:
is a context of the plant. The context must have the positions of the plant set to the initialIRIS seed configuration.
Parameter options:
provides additional configuration options. In particular, increasing options.num_collision_infeasible_samples increases the chances that the IRIS regions are collision free but can also
significantly increase the run-time of the algorithm. The same goes for options.num_additional_constraints_infeasible_samples.
☆ RuntimeError if the sample configuration in context is –
☆ infeasible. –
☆ RuntimeError if termination_func is invalid on the domain. See –
☆ IrisOptions.termination_func for more details. –
class pydrake.geometry.optimization.IrisOptions
Configuration options for the IRIS algorithm.
__init__(self: pydrake.geometry.optimization.IrisOptions, **kwargs) None
property bounding_region
Optionally allows the caller to restrict the space within which IRIS regions are allowed to grow. By default, IRIS regions are bounded by the domain argument in the case of Iris or the joint
limits of the input plant in the case of IrisInConfigurationSpace. If this option is specified, IRIS regions will be confined to the intersection between the domain and bounding_region
property configuration_obstacles
For IRIS in configuration space, it can be beneficial to not only specify task-space obstacles (passed in through the plant) but also obstacles that are defined by convex sets in the
configuration space. This option can be used to pass in such configuration space obstacles.
property configuration_space_margin
For IRIS in configuration space, we retreat by this margin from each C-space obstacle in order to avoid the possibility of requiring an infinite number of faces to approximate a curved
property iteration_limit
Maximum number of iterations.
property mixing_steps
The mixing_steps parameters is passed to HPolyhedron::UniformSample to control the total number of hit-and-run steps taken for each new random sample.
property num_additional_constraint_infeasible_samples
For each constraint in prog_with_additional_constraints, IRIS will search for a counter-example by formulating a (likely nonconvex) optimization problem. The initial guess for this
optimization is taken by sampling uniformly inside the current IRIS region. This option controls the termination condition for that counter-example search, defining the number of consecutive
failures to find a counter-example requested before moving on to the next constraint.
property num_collision_infeasible_samples
For each possible collision, IRIS will search for a counter-example by formulating a (likely nonconvex) optimization problem. The initial guess for this optimization is taken by sampling
uniformly inside the current IRIS region. This option controls the termination condition for that counter-example search, defining the number of consecutive failures to find a counter-example
requested before moving on to the next constraint.
property prog_with_additional_constraints
By default, IRIS in configuration space certifies regions for collision avoidance constraints and joint limits. This option can be used to pass additional constraints that should be satisfied
by the IRIS region. We accept these in the form of a MathematicalProgram:
find q subject to g(q) ≤ 0.
The decision_variables() for the program are taken to define q. IRIS will silently ignore any costs in prog_with_additional_constraints, and will throw RuntimeError if it contains any
unsupported constraints.
For example, one could create an InverseKinematics problem with rich kinematic constraints, and then pass InverseKinematics::prog() into this option.
property random_seed
The only randomization in IRIS is the random sampling done to find counter-examples for the additional constraints using in IrisInConfigurationSpace. Use this option to set the initial seed.
property relative_termination_threshold
IRIS will terminate if the change in the volume of the hyperellipsoid between iterations is less that this percent of the previous best volume. This termination condition can be disabled by
setting to a negative value.
property require_sample_point_is_contained
The initial polytope is guaranteed to contain the point if that point is collision-free. However, the IRIS alternation objectives do not include (and can not easily include) a constraint that
the original sample point is contained. Therefore, the IRIS paper recommends that if containment is a requirement, then the algorithm should simply terminate early if alternations would ever
cause the set to not contain the point.
property solver_options
The SolverOptions used in the optimization program.
property starting_ellipse
The initial hyperellipsoid that IRIS will use for calculating hyperplanes in the first iteration. If no hyperellipsoid is provided, a small hypershpere centered at the given sample will be
property termination_threshold
IRIS will terminate if the change in the volume of the hyperellipsoid between iterations is less that this threshold. This termination condition can be disabled by setting to a negative
property verify_domain_boundedness
If the user knows the intersection of bounding_region and the domain (for IRIS) or plant joint limits (for IrisInConfigurationSpace) is bounded, setting this flag to False will skip the
boundedness check that IRIS and IrisInConfigurationSpace perform (leading to a small speedup, as checking boundedness requires solving optimization problems). If the intersection turns out to
be unbounded, this will lead to undefined behavior.
pydrake.geometry.optimization.LoadIrisRegionsYamlFile(filename: os.PathLike, child_name: Optional[str] = None) dict[str, pydrake.geometry.optimization.HPolyhedron]
Calls LoadYamlFile() to deserialize an IrisRegions object.
pydrake.geometry.optimization.MakeIrisObstacles(query_object: pydrake.geometry.QueryObject, reference_frame: Optional[pydrake.geometry.FrameId] = None) list[pydrake.geometry.optimization.ConvexSet]
Constructs ConvexSet representations of obstacles for IRIS in 3D using the geometry from a SceneGraph QueryObject. All geometry in the scene with a proximity role, both anchored and dynamic, are
consider to be fixed obstacles frozen in the poses captured in the context used to create the QueryObject.
When multiple representations are available for a particular geometry (e.g. a Box can be represented as either an HPolyhedron or a VPolytope), then this method will prioritize the representation
that we expect is most performant for the current implementation of the IRIS algorithm.
class pydrake.geometry.optimization.MinkowskiSum
Bases: pydrake.geometry.optimization.ConvexSet
A convex set that represents the Minkowski sum of multiple sets: S = X₁ ⨁ X₂ ⨁ … ⨁ Xₙ = {x₁ + x₂ + … + xₙ | x₁ ∈ X₁, x₂ ∈ X₂, …, xₙ ∈ Xₙ}
Special behavior for IsEmpty: The Minkowski sum of zero sets (i.e. when we have sets_.size() == 0) is treated as the singleton {0}, which is nonempty. This includes the zero-dimensional case.
__init__(*args, **kwargs)
Overloaded function.
1. __init__(self: pydrake.geometry.optimization.MinkowskiSum) -> None
Constructs a default (zero-dimensional, nonempty) set.
2. __init__(self: pydrake.geometry.optimization.MinkowskiSum, sets: list[pydrake.geometry.optimization.ConvexSet]) -> None
Constructs the sum from a vector of convex sets.
3. __init__(self: pydrake.geometry.optimization.MinkowskiSum, setA: pydrake.geometry.optimization.ConvexSet, setB: pydrake.geometry.optimization.ConvexSet) -> None
Constructs the sum from a pair of convex sets.
4. __init__(self: pydrake.geometry.optimization.MinkowskiSum, query_object: pydrake.geometry.QueryObject, geometry_id: pydrake.geometry.GeometryId, reference_frame: Optional
[pydrake.geometry.FrameId] = None) -> None
Constructs a MinkowskiSum from a SceneGraph geometry and pose in the reference_frame frame, obtained via the QueryObject. If reference_frame frame is std::nullopt, then it will be expressed
in the world frame.
Although in principle a MinkowskiSum can represent any ConvexSet as the sum of a single set, here we only support Capsule geometry, which will be represented as the (non-trivial) Minkowski
sum of a sphere with a line segment. Most SceneGraph geometry types are supported by at least one of the ConvexSet class constructors.
RuntimeError if geometry_id does not correspond to a Capsule. –
num_terms(self: pydrake.geometry.optimization.MinkowskiSum) int
The number of terms (or sets) used in the sum.
term(self: pydrake.geometry.optimization.MinkowskiSum, index: int) pydrake.geometry.optimization.ConvexSet
Returns a reference to the ConvexSet defining the index term in the sum.
pydrake.geometry.optimization.PartitionConvexSet(*args, **kwargs)
Overloaded function.
1. PartitionConvexSet(convex_set: pydrake.geometry.optimization.ConvexSet, continuous_revolute_joints: list[int], epsilon: float = 1e-05) -> list[pydrake.geometry.optimization.ConvexSet]
Partitions a convex set into (smaller) convex sets whose union is the original set and that each respect the convexity radius as in CheckIfSatisfiesConvexityRadius. In practice, this is
implemented as partitioning sets into pieces whose width is less than or equal to π-ϵ. Each entry in continuous_revolute_joints must be non-negative, less than num_positions, and unique.
Parameter epsilon:
is the ϵ value used for the convexity radius inequality. The partitioned sets are made by intersecting convex_set with axis-aligned bounding boxes that respect the convexity radius. These
boxes are made to overlap by ϵ radians along each dimension, for numerical purposes.
the vector of convex sets that each respect convexity radius.
☆ RuntimeError if ϵ <= 0 or ϵ >= π. –
☆ RuntimeError if the input convex set is unbounded along dimensions –
☆ corresponding to continuous revolute joints. –
☆ RuntimeError if continuous_revolute_joints has repeated entries, –
☆ or if any entry is outside the interval [0, –
☆ convex_set.ambient_dimension()) –
2. PartitionConvexSet(convex_sets: list[pydrake.geometry.optimization.ConvexSet], continuous_revolute_joints: list[int], epsilon: float = 1e-05) -> list[pydrake.geometry.optimization.ConvexSet]
Function overload to take in a list of convex sets, and partition all so as to respect the convexity radius. Every set must be bounded and have the same ambient dimension. Each entry in
continuous_revolute_joints must be non-negative, less than num_positions, and unique.
☆ RuntimeError unless every ConvexSet in convex_sets has the same –
☆ ambient_dimension. –
☆ RuntimeError if ϵ <= 0 or ϵ >= π. –
☆ RuntimeError if any input convex set is unbounded along dimensions –
☆ corresponding to continuous revolute joints. –
☆ RuntimeError if continuous_revolute_joints has repeated entries, –
☆ or if any entry is outside the interval [0, ambient_dimension) –
class pydrake.geometry.optimization.PlaneSide
__init__(self: pydrake.geometry.optimization.PlaneSide, value: int) None
kNegative = <PlaneSide.kNegative: 1>
kPositive = <PlaneSide.kPositive: 0>
property name
property value
class pydrake.geometry.optimization.Point
Bases: pydrake.geometry.optimization.ConvexSet
A convex set that contains exactly one element. Also known as a singleton or unit set.
This set is always nonempty, even in the zero-dimensional case.
__init__(*args, **kwargs)
Overloaded function.
1. __init__(self: pydrake.geometry.optimization.Point) -> None
Constructs a default (zero-dimensional, nonempty) set.
2. __init__(self: pydrake.geometry.optimization.Point, x: numpy.ndarray[numpy.float64[m, 1]]) -> None
Constructs a Point.
3. __init__(self: pydrake.geometry.optimization.Point, query_object: pydrake.geometry.QueryObject, geometry_id: pydrake.geometry.GeometryId, reference_frame: Optional
[pydrake.geometry.FrameId] = None, maximum_allowable_radius: float = 0.0) -> None
Constructs a Point from a SceneGraph geometry and pose in the reference_frame frame, obtained via the QueryObject. If reference_frame frame is std::nullopt, then it will be expressed in the
world frame.
○ RuntimeError if geometry_id does not correspond to a Sphere or if –
○ the Sphere has radius greater than maximum_allowable_radius. –
set_x(self: pydrake.geometry.optimization.Point, x: numpy.ndarray[numpy.float64[m, 1]]) None
Changes the element x describing the set.
x must be of size ambient_dimension().
x(self: pydrake.geometry.optimization.Point) numpy.ndarray[numpy.float64[m, 1]]
Retrieves the point.
class pydrake.geometry.optimization.SampledVolume
The result of a volume calculation from CalcVolumeViaSampling().
__init__(*args, **kwargs)
property num_samples
The number of samples used to compute the volume estimate.
property rel_accuracy
An upper bound for the relative accuracy of the volume estimate. When not evaluated, this value is NaN.
property volume
The estimated volume of the set.
pydrake.geometry.optimization.SaveIrisRegionsYamlFile(filename: os.PathLike, regions: dict[str, pydrake.geometry.optimization.HPolyhedron], child_name: Optional[str] = None) None
Calls SaveYamlFile() to serialize an IrisRegions object.
class pydrake.geometry.optimization.SeparatingPlaneOrder
The separating plane aᵀx + b ≥ δ, aᵀx + b ≤ −δ has parameters a and b. These parameterize a polynomial function of s_for_plane with the specified order. s_for_plane is a sub set of the
configuration-space variable s, please refer to the RationalForwardKinematics class or the paper above on the meaning of s.
kAffine : a and b are affine functions of s.
__init__(self: pydrake.geometry.optimization.SeparatingPlaneOrder, value: int) None
kAffine = <SeparatingPlaneOrder.kAffine: 1>
property name
property value
class pydrake.geometry.optimization.SeparationCertificateProgramBase
__init__(*args, **kwargs)
property plane_index
prog(self: pydrake.geometry.optimization.SeparationCertificateProgramBase) pydrake.solvers.MathematicalProgram
class pydrake.geometry.optimization.SeparationCertificateResultBase
We certify that a pair of geometries is collision free by finding the separating plane over a range of configuration. The Lagrangian multipliers used for certifying this condition will differ in
derived classes. This struct contains the the separating plane {x | aᵀx+b=0 } and derived classes may store the Lagrangians certifying that the plane separates the two geometries in
separating_planes()[plane_index] in the C-space region.
__init__(*args, **kwargs)
property a
property b
property plane_decision_var_vals
property result
class pydrake.geometry.optimization.Spectrahedron
Bases: pydrake.geometry.optimization.ConvexSet
Implements a spectrahedron (the feasible set of a semidefinite program). The ambient dimension of the set is N(N+1)/2; the number of variables required to describe the N-by-N semidefinite matrix.
By convention, a zero-dimensional spectrahedron is considered nonempty.
__init__(*args, **kwargs)
Overloaded function.
1. __init__(self: pydrake.geometry.optimization.Spectrahedron) -> None
Default constructor (yields the zero-dimensional nonempty set).
2. __init__(self: pydrake.geometry.optimization.Spectrahedron, prog: pydrake.solvers.MathematicalProgram) -> None
Constructs the spectrahedron from a MathematicalProgram.
○ RuntimeError if prog.required_capabilities() is not a subset –
○ of supported_attributes() –
class pydrake.geometry.optimization.VPolytope
Bases: pydrake.geometry.optimization.ConvexSet
A polytope described using the vertex representation. The set is defined as the convex hull of the vertices. The vertices are not guaranteed to be in any particular order, nor to be minimal (some
vertices could be strictly in the interior of the set).
Note: Unlike the half-space representation, this definition means the set is always bounded (hence the name polytope, instead of polyhedron).
A VPolytope is empty if and only if it is composed of zero vertices, i.e., if vertices_.cols() == 0. This includes the zero-dimensional case. If vertices_.rows() == 0 but vertices_.cols() > 0, we
treat this as having one or more copies of 0 in the zero-dimensional vector space {0}. If vertices_.rows() and vertices_.cols() are zero, we treat this as no points in {0}, which is empty. | {"url":"https://drake.mit.edu/pydrake/pydrake.geometry.optimization.html","timestamp":"2024-11-10T09:44:51Z","content_type":"text/html","content_length":"558704","record_id":"<urn:uuid:dda6c5a8-02eb-46af-acab-9e8b50de9b6b>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00593.warc.gz"} |
Quotient and remainder calculator
The division with a remainder or Euclidean division of two natural numbers provides a quotient, which is the number of times the second one is contained in the first one, and a remainder, which is
part of the first number that remains when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated. | {"url":"https://www.hackmath.net/en/calculator/quotient-and-remainder","timestamp":"2024-11-03T23:32:45Z","content_type":"text/html","content_length":"6990","record_id":"<urn:uuid:348595b2-6dc5-49fd-b19f-ec8d8a71acc9>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00328.warc.gz"} |
DSP FIR-filter calculator, V2400
With this tool, you can calculate the filter taps (coefficients) of four types Kaiser-Bessel FIR (Finite Impulse Response) digital filters:
• Low pass
• High pass
• Band pass
• Band stop
depending on the sampling rate, number of taps and attenuation levels.
The list on the right-hand side of the window holds the filter coefficients, which can be exported to a text file by clicking on the save file menu via the menu bar. The result becomes available in a
Public Folders directory.
The tool works very intuitive, as any change in setting parameters the resulting filter curve and list of coefficients is updated immediately. After a change in filter parameters, you immediately see
the result in the design, filter curve and coefficients list.
You can save and recall your filter designs where WinRFCalc automatically creates an intuitive filename from which you can read the filter design. And of course you can save the coefficient list as a
text file for later use in other programs. | {"url":"https://rfcalculator.com/FIR-Filters/","timestamp":"2024-11-14T20:46:30Z","content_type":"text/html","content_length":"41785","record_id":"<urn:uuid:d792cf77-a1b1-48a6-b83a-d8504e6e7e7c>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00748.warc.gz"} |
Is 500ml
Is 500ml same as 50cl?
The volume of 50 cl and 500 ml is the same. Milliliters (mL) centiliters (cL) 75 mg 7.5 mg 100 mg 10 mg 10 mg 250 mg 25 mg 500 mg 50 mg D’autre part, How many mg in a 20 mg bottle? Is 5 cl the same
as 50 ml in centiliters? 17 cl 170 mL 18 cl 180 mL 19 cl 190 mL 20 cl 200 mL
100 cl equals 1 litre, and 1000 ml equals 1 litre. Divide 1 litre by 100 litres, then multiply by 50 litres to make up half of the litre. As a result, 1 liter equals 1000 ml divided by 1/2 = 500 ml,
while 50 cl and 500 ml have the same volume. What is CL’s capacity then?
A centilitre (cL or cl) is a metric unit of volume equal to one hundredth of a liter and equals a little more than six tenths (0.6102) of an acubic inch, or a third (0.338) of a fluid ounce. Is a
50cl bottle half the size of a liter? Fifty centiliters equals 0.5 liters, or a half-liter.
Is a 25-cl bottle the same as a 250-ml bottle?
Simply put, cl is more than ml. As a result, multiply 25 cl by 10 to convert 25 cl into ml.
Is there a difference between a cl and a mL?
10 milliliters (ml) = 1 centiliter (cl). In the Metric system, centiliter (cl) is a volume unit.
In a liquid, what exactly is a CL?
Is 500ml of water half the price of a liter?
1 litre equals 1000 ml. As a result, 500 ml equals half a litre. The correct spelling is also ‘LITRE.’
How many CLs produce a liter of water?
Is 20 cl the same as 200 cl?
To begin, keep in mind that cl is the same as centiliters, and ml is the same as milliliters. As a result, when you want to convert 20 centiliters into milliliters, you want to convert 20 centiliters
into milliliters.
A centiliter is more than a milliliter. Simply put, cl is more than ml.
What does the word “50cl” mean?
Milliliters of Centiliters – 49 cl 490.00 mL 50 cl 500.00 mL 51 cl 510.00 mL 52 cl 50.00 mL
How many ml bottles does a 20 ml bottle contain?
17 cl 170 milliliters 18 cl 180 milliliters 19 cl 190 milliliters 20 cl 200 milliliters Dernière mise à jour: Il y a 15 days – Co-auteurs: 5 – Utilisateurs: 4 | {"url":"https://tipsfolder.com/500ml-same-50cl-f13a2c9889cf35e26eea507e016ae6b1/","timestamp":"2024-11-10T22:04:32Z","content_type":"text/html","content_length":"95156","record_id":"<urn:uuid:00d8b8e1-05c3-4f1f-bffc-ede09e4f2e0c>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00242.warc.gz"} |
User Documentation
For more information, see the online learning platform
Gap Analysis is a technique used to identify gaps between a current state and a targeted one. [ ]
To create a gap analysis:
1. Click Transform > Gap Analysis in the menu.
2. Enter Gap Variable name.
3. Select Type (Maxime or Minimize).
4. Select Record set.
5. Select the Index to be used for the analysis. Usually the time variable is set by default.
6. Select the Variable, if required select the Variable sets.
7. Select the Factor, if required.
8. Enter Target value, numeric value.
9. Click Save.
Example: Calculate the yearly savings potential based on a target value for the variable Profit €/h.
1. Find the target value for the variable:
1. Create an histogram to understand the distribution.
2. Switch to Advanced tab and check the box "Show Statistics".
The target value chosen is 3500 USD/h (close to the average value 3461 USD/h)
2. Create a Gap analysis to maximize the savings using a target value of Profit of 3500 using as and Index the time variable and as a Variable the KPI to be analysed.
Gap Analysis Type
Maximize means that the Gap is under the target.
Minimize means that Gap is above the target.
3. Create a Cusum to cumulate the savings over a period of time. In the CUSUM Variable editor page the Index and the Variable are automatically selected, respectively the time variable and the Gap
Variable Name. Click on Save to compute the CUSUM function.
The result can be seen on the top of the resulting table :
4. Create a Trend, using More Actions → Create Trend. In the Trend editor the X and Y axis are automatically selected, respectively time variable and CUSUM_Variable Name. In Advanced tab check the
box "Fill curves" to color the area between the curve and X - Y' axis.
Calculating the yearly savings potential:
Using the basic statistics given below the Trend for the CUSUM-GAP_Profit/hr function, assess the yearly savings potential based on a target value of 3500 €/hr.
Number of records (production hours) = 6475
Maximum value for the CUSUM function = 300 751 USD
The ratio of these two numbers gives a hourly savings potential equal to 46.45 UDS/hr.
Based on the assumption that the process is operating 24/7, with 2 weeks shutdown for yearly maintenance, the yearly savings potential is around 399 098 USD.
Factor is the factor by which we multiple the gap to calculate an absolute rather than relative gap value. This is often used for specific value (unit of value/unit of production) to calculate the
value gap rather than specific value gap.
In this case, the factor equals “1” because data sampling period is 1 hour. | {"url":"https://pepite.atlassian.net/wiki/spaces/UG/pages/506888193/Gap+Analysis?atl_f=content-tree","timestamp":"2024-11-11T12:35:33Z","content_type":"text/html","content_length":"909788","record_id":"<urn:uuid:25aade72-3c66-4208-95e4-3e3036cadab9>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00440.warc.gz"} |
Issac Newton: Life, Scientific Career, & Impact - Culture History
Issac Newton: Life, Scientific Career, & Impact
Isaac Newton (1643-1727) was an English mathematician, physicist, astronomer, and author, widely recognized as one of the most influential scientists of all time. He formulated the laws of motion and
universal gravitation, which laid the groundwork for classical mechanics. Newton’s work in optics included the discovery of the composition of white light and the development of the reflecting
telescope. His seminal work, “PhilosophiƦ Naturalis Principia Mathematica,” is considered a cornerstone of modern science. Newton’s contributions extend to mathematics, where he co-developed
calculus independently of Leibniz. His profound impact on science and mathematics continues to shape our understanding of the natural world.
Early Life and Education
Isaac Newton was born on December 25, 1642, according to the Julian calendar then in use in England (January 4, 1643, in the Gregorian calendar). His birthplace was Woolsthorpe Manor, a small estate
in Lincolnshire. Newton was born prematurely and was a small and weak infant, so much so that his mother, Hannah Ayscough Newton, feared he might not survive.
Newton’s father, also named Isaac, had died three months before his birth. When Newton was three years old, his mother remarried and went to live with her new husband, Barnabas Smith, leaving young
Isaac in the care of his grandmother. This separation from his mother had a profound impact on Newton, who would later describe feelings of emotional isolation and abandonment in his personal
Newton began his formal education at the King’s School in Grantham, where he lodged with an apothecary named Clark. It was here that Newton developed an interest in chemistry. His school reports were
not particularly impressive, and he was initially taken out of school to manage the family farm. However, his lack of interest and aptitude for farming led his mother to send him back to school to
prepare for university.
Cambridge University
In 1661, Newton entered Trinity College, Cambridge. At the time, the college curriculum was heavily based on the teachings of Aristotle, which Newton found unsatisfactory. He immersed himself in the
works of modern philosophers and scientists such as Descartes, Copernicus, Galileo, and Kepler.
Newton’s intellectual pursuits were marked by intense self-study and experimentation. He kept a series of notebooks, the most famous being the “Quaestiones Quaedam Philosophicae” (“Certain
Philosophical Questions”), which contained his early explorations into mathematics, physics, and metaphysics. During this period, he made significant advances in mathematics, laying the groundwork
for his future development of calculus.
In 1665, the Great Plague forced Cambridge to close, and Newton returned to Woolsthorpe. During this period of isolation, often referred to as his “Annus Mirabilis” or “Year of Wonders,” Newton made
groundbreaking discoveries in mathematics, optics, and the law of gravitation. He developed the binomial theorem and began to formulate the principles of calculus, although these were not published
until later.
Mathematical Contributions
Newton’s work in mathematics was revolutionary. In addition to the binomial theorem, he developed a comprehensive system of calculus, which he called “the method of fluxions.” This work was done
independently of, and simultaneously with, the German mathematician Gottfried Wilhelm Leibniz, which later led to a bitter dispute over priority.
Newton’s “Principia Mathematica,” published in 1687, is perhaps his most famous work. In it, he formulated the three laws of motion, which laid the foundation for classical mechanics. He also
described the law of universal gravitation, explaining how all bodies in the universe attract each other with a force proportional to their masses and inversely proportional to the square of the
distance between them. This work not only revolutionized physics but also provided the tools for future scientific endeavors in various fields.
Newton made significant contributions to the field of optics, particularly through his study of the nature of light and color. In experiments conducted in the 1660s, Newton demonstrated that white
light is composed of a spectrum of colors, which can be separated and then recombined. He used a prism to split white light into its constituent colors and then passed these colors through a second
prism to show that they could be recombined into white light.
In 1672, Newton published his first scientific paper, “New Theory about Light and Colors,” in the Philosophical Transactions of the Royal Society. This work challenged the prevailing wave theory of
light proposed by Christiaan Huygens and others, advocating instead for a particle theory of light. Newton’s work on optics also led him to develop the reflecting telescope, which used mirrors
instead of lenses to avoid chromatic aberration and produce clearer images.
Alchemy and Theological Studies
In addition to his work in mathematics and science, Newton had a deep interest in alchemy and theology. He spent a significant amount of time studying alchemical texts and conducting experiments in
his quest to uncover the hidden nature of matter. While much of his alchemical work remained unpublished during his lifetime, it reveals his belief in a unified natural philosophy that encompassed
both the material and spiritual realms.
Newton was also a devout Christian, but his religious beliefs were unconventional. He was a unitarian, rejecting the doctrine of the Trinity, and he spent considerable time studying biblical prophecy
and attempting to decode hidden messages in the Bible. Newton’s theological writings, although less well known than his scientific work, provide insight into his broader intellectual pursuits and his
desire to understand the divine order of the universe.
Later Life and Legacy
In 1696, Newton was appointed Warden of the Royal Mint, and later Master of the Mint, a position he held until his death. In this role, he oversaw the recoinage of English currency and took measures
to combat counterfeiting. Newton’s tenure at the Mint demonstrated his practical abilities and commitment to public service.
Newton continued to engage in scientific work throughout his later years, although his output diminished. He became President of the Royal Society in 1703, a position he held until his death. As
president, he presided over an era of significant scientific progress and promoted the work of other scientists.
Newton died on March 20, 1727, at the age of 84. He was buried in Westminster Abbey, a testament to his immense contributions to science and his status as one of the greatest minds in history.
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mechanical calculator – Hackaday
We’ve always admired Curta mechanical calculators, and would be very hesitant to dismantle one. But [Janus Cycle] did just that — and succeeded. A friend sent him a Curta Model 2 calculator that was
frozen up. Just opening the case involved percussive force to remove a retaining pin, and once inside he discovered the main shaft had been slightly bent. No doubt this calculator had suffered a drop
at some point in the past.
I’m sticking to the rule of doing no harm — I’d rather not be able to fix this than do something that causes more problems.
Inside the Curta
But surprisingly, he was able to get it substantially back in working order without completely taking apart all 600+ parts. Most of the issues were shafts whose lubrication had become gummy, and one
carry lever was slightly bent. There is still a little more work, but soon this calculator will once again be cranking out results.
Has anyone dismantled a mechanical contraption this complicated before, for example a teletype machine? Let us know in the comments. If you want to brush up on your Curta knowledge, check out the
Curta Calculator Page. We also wrote a Retrotechtacular about the Curta before. Thanks to [mister35mm] for sending in this tip.
Retrotechtacular: Mechanical Arithmetic For The Masses
Last month we carried a piece looking at the development of the 8-bit home computer market through the lens of the British catalogue retailer Argos and their perennial catalogue of dreams. As an
aside, we mentioned that the earliest edition from 1975 contained some of the last mechanical calculators on the market, alongside a few early electronic models. This month it’s worth returning to
those devices, because though they are largely forgotten now, they were part of the scenery and clutter of a typical office for most of the century.
The Summa’s internals, showing the register on the right and the type wheels on the left.
Somewhere in storage I have one of the models featured in the catalogue, an Olivetti Summa Prima. I happened upon it in a dumpster as a teenager looking for broken TVs to scavenge for parts, cut down
a pair of typewriter ribbon reels to fit it, and after playing with it for a while added it to my store of random tech ephemera. It’s a compact and stylish desktop unit from about 1970, on its front
is a numerical keypad, top is a printer with a holder for a roll of receipt paper and a typewriter-style rubber roller, while on its side is a spring-loaded handle from which it derives its power. It
can do simple addition and subtraction in the old British currency units, and operating it is a simple case of punching in a number, pulling the handle, and watching the result spool out on the paper
tape. Its register appears to be a set of rotors advanced or retarded by the handle for either addition or subtraction, and its printing is achieved by a set of print bars sliding up to line the
correct number with the inked ribbon. For me in 1987 with my LCD Casio Scientific it was an entertaining mechanical curiosity, but for its operators twenty years earlier it must have represented a
significant time saving.
The history of mechanical calculators goes back over several hundred years to Blaise Pascal in the 17th century, and over that time they evolved through a series of inventions into surprisingly
sophisticated machines that were capable of handling financial complications surprisingly quickly. The Summa was one of the last machines available in great numbers, and even as it was brought to
market in the 1960s its manufacturer was also producing one of the first desktop-sized computers. Its price in that 1975 Argos catalogue is hardly cheap but around the same as an electronic
equivalent, itself a minor miracle given how many parts it contains and how complex it must have been to manufacture.
We’ve put two Summa Prima videos below the break. T.the first is a contemporary advert for the machine, and the second is a modern introduction to the machine partially narrated by a Brazilian robot,
so consider translated subtitles. In that second video you can see something of its internals as the bare mechanism is cranked over for the camera and some of the mechanical complexity of the device
becomes very obvious. It might seem odd to pull a obsolete piece of office machinery from a dumpster and hang onto it for three decades, but I’m very glad indeed that a 1980s teenage me did so.
You’re probably unlikely to stumble upon one in 2019, but should you do so it’s a device that’s very much worth adding to your collection.
Continue reading “Retrotechtacular: Mechanical Arithmetic For The Masses”
Calculus And A Calculator
Earlier this year, [Dan Maloney] went inside mechanical calculators. Being the practical sort, [Dan] jumped right into the Pascaline invented by Blaise Pascal. It couldn’t multiply or divide. He then
went into the arithmometer, which is arguably the first commercially successful mechanical calculator with four functions. That was around 1821 or so. But [Dan] mentions it used a Leibniz wheel. I
thought, “Leibniz? He’s the calculus guy, right? He died in 1716.” So I knew there had to be at least a century of backstory to get to the arithmometer. Having a rainy day ahead, I decided to find
out exactly where the Leibniz wheel came from and what it was doing for 100 years prior to 1821.
If you’ve taken calculus you’ve probably heard of Gottfried Wilhelm Leibniz (who would have been 372 years old on July 1st, by the way). He’s the guy that gave us the notation we use in modern
calculus and oddly was one of two people who apparently figured out calculus, the other being Issac Newton. Both men, by the way, accused each other of stealing, although it is more likely they both
built on the same prior work. When you are struggling to learn calculus, it is sometimes amazing that not only did someone think it up, but two people thought it up at one time. However, Leibniz also
built what might be the first four function calculator in 1694. His “stepped reckoner” used a drum and some cranks and the underlying mechanism found inside of it lived on until the 1970s in other
mechanical calculating devices. Oddly, Leibniz didn’t use the term stepped reckoner but called the machine Instrumentum Arithmeticum.
Many of us remember when a four function electronic calculator was a marvel and not even very inexpensive. Nowadays, you’d have to look hard to find one that only had four functions and simple
calculators are cheap enough to give away like ink pens. But in 1694, you didn’t have electronics and integrated circuits necessary to pull that off.
Fourier Machine Mimics Michelson Original In Plywood
It’s funny how creation and understanding interact. Sometimes the urge to create something comes from a new-found deep understanding of a concept, and sometimes the act of creation leads to that
understanding. And sometimes creation and understanding are linked together in such a way as to lead in an entirely new direction, which is the story behind this plywood recreation of the Michelson
Fourier analysis machine.
For those not familiar with this piece of computing history, it’s worth watching the videos in our article covering [Bill “The Engineer Guy” Hammack]’s discussion of this amazing early 20th-century
analog computer. Those videos were shown to [nopvelthuizen] in a math class he took at the outset of degree work in physics education. The beauty of the sinusoids being created by the cam-operated
rocker arms and summed to display the output waveforms captured his imagination and lead to an eight-channel copy of the 20-channel original.
Working with plywood and a CNC router, [nopvelthuizen]’s creation is faithful to the original if a bit limited by the smaller number of sinusoids that can be summed. A laser cutter or 3D printer
would have allowed for a longer gear train, but we think the replica is great the way it is. What’s more, the real winners are [nopvelthuizen]’s eventual physics students, who will probably look with
some awe at their teacher’s skills and enthusiasm.
Continue reading “Fourier Machine Mimics Michelson Original In Plywood”
Retrotechtacular: Pascal Got Frustrated At Tax Time, Too
While necessity is frequently the mother of invention, annoyance often comes into play as well. This was the case with [Blaise Pascal], who as a teenager was tasked with helping his father calculate
the taxes owed by the citizens of Rouen, France. [Pascal] tired of moving the beads back and forth on his abacus and was sure that there was some easier way of counting all those livres, sols, and
deniers. In the early 1640s, he devised a mechanical calculator that would come to be known by various names: Pascal’s calculator, arithmetic machine, and eventually, Pascaline.
The instrument is made up of input dials that are connected to output drums through a series of gears. Each digit of a number is entered on its own input dial. This is done by inserting a
stylus between two spokes and turning the dial clockwise toward a metal stop, a bit like dialing on a rotary phone. The output is shown in a row of small windows across the top of the machine. Pascal
made some fifty different prototypes of the Pascaline before he turned his focus toward philosophy. Some have more dials and corresponding output wheels than others, but the operation and mechanics
are largely the same throughout the variations.
Continue reading “Retrotechtacular: Pascal Got Frustrated At Tax Time, Too”
Harmonic Analyzer Mechanical Fourier Computer
If you’re into mechanical devices or Fourier series (or both!), you’ve got some serious YouTubing to do.
[The Engineer Guy] has posted up a series of four videos (Introduction, Synthesis, Analysis, and Operation) that demonstrate the operation and theory behind a 100-year-old machine that does Fourier
analysis and synthesis with gears, cams, rocker-arms, and springs.
In Synthesis, [The Engineer Guy] explains how the machine creates an arbitrary waveform from its twenty Fourier components. In retrospect, if you’re up on your Fourier synthesis, it’s pretty obvious.
Gears turn at precise ratios to each other to create the relative frequencies, and circles turning trace out sine or cosine waves easily enough. But the mechanical spring-weighted summation mechanism
blew our mind, and watching the machine do its thing is mesmerizing.
In Analysis everything runs in reverse. [The Engineer Guy] sets some sample points — a square wave — into the machine and it spits out the Fourier coefficients. If you don’t have a good intuitive
feel for the duality implied by Fourier analysis and synthesis, go through the video from 1:50 to 2:20 again. For good measure, [The Engineer Guy] then puts the resulting coefficient estimates back
into the machine, and you get to watch a bunch of gears and springs churn out a pretty good square wave. Truly amazing.
The fact that the machine was designed by [Albert Michelson], of Michelson-Morley experiment fame, adds some star power. [The Engineer Guy] is selling a book documenting the machine, and his video
about the book is probably worth your time as well. And if you still haven’t gotten enough sine-wavey goodness, watch the bonus track where he runs the machine in slow-mo: pure mechano-mathematical
Continue reading “Harmonic Analyzer Mechanical Fourier Computer”
Retrotechtacular: The CURTA Mechanical Calculator
The CURTA mechanical calculator literally saved its inventor’s life. [Curt Herzstark] had been working on the calculator in the 1930s until the Nazis forced him to focus on building other tools for
the German army. He was taken by the Nazis in 1943 and ended up in Buchenwald concentration camp. There, he told the officers about his plans for the CURTA. They were impressed and interested enough
to let him continue work on it so they could present it as a gift to the Führer.
This four-banger pepper mill can also perform square root calculation with some finessing. To add two numbers together, each must be entered on the digit setting sliders and sent to the result
counter around the top by moving the crank clockwise for one full rotation. Subtraction is as easy as pulling out the crank until the red indicator appears. The CURTA performs subtraction using
nine’s complement arithmetic. Multiplication and division are possible through successive additions and subtractions and use of the powers of ten carriage, which is the top knurled portion.
Operation of the CURTA is based on [Gottfried Leibniz]’s stepped cylinder design. A cylinder with cogs of increasing lengths drives a toothed gear up and down a shaft. [Herzstark]’s design
interleaves a normal set of cogs for addition with a nine’s complement set. When the crank is pulled out to reveal the red subtraction indicator, the drum is switching between the two sets.
Several helper mechanisms are in place to enhance the interface. The user is prevented from ever turning the crank counter-clockwise. The crank mechanism provides tactile feedback at the end of each
full rotation. There is also a lock that disallows switching between addition and subtraction while turning the crank—switching is only possible with the crank in the home position. There is a turns
counter on the top which can be set to increment or decrement.
You may recall seeing Hackaday alum [Jeremy Cook]’s 2012 post about the CURTA which we linked to. A great deal of information about the CURTA and a couple of different simulators are available at
curta.org. Make the jump to see an in-depth demonstration of the inner workings of a CURTA Type I using the YACS CURTA simulator.
Continue reading “Retrotechtacular: The CURTA Mechanical Calculator” | {"url":"https://hackaday.com/tag/mechanical-calculator/","timestamp":"2024-11-14T18:48:49Z","content_type":"text/html","content_length":"119800","record_id":"<urn:uuid:583ded89-4ca9-4968-abcb-9f5f2bad320f>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00208.warc.gz"} |
The other way around - Model with Mathematics
The other way around
Often in mathematics, and certainly often in mathematics class, we think in terms of direct problems. That is, we’re given something to “solve,” we’re required to find a path to a “solution,” and
that’s where we end up. For example, we might be asked to find the solutions of the following equation
and we proceed by using the quadratic equation or completing the square or some such method, and arrive at values for
But, often, in the real world, our problems aren’t formulated like this. Rather, we’re faced with problems that are posed the other way around. That is we’re faced with what are called inverse
problems. In these problems, what we’re given is the solution and what we have to do is figure out where it came from. You’re likely familiar with such problems, but perhaps haven’t thought of them
in this way before. One common example is the “sonar problem” or the problem of echolocation. If I’m a bat, what I do is make a noise, and then listen for the reflected sounds that come back to me.
What I then try and do is figure out the direction, location, and perhaps shape, of the objects in the environment that caused the particular patterns of reflected sounds that I heard. That is, I’m
given the solution, i.e. the reflected sounds I measured, and have to figure out the problem, i.e. what pattern and shape of objects in the environment would cause that set of reflected sounds?
Such problems are an important class of problems that the mathematical modeler must be equipped to deal with. And, such problems, can be a source of challenging and engaging problems for students
learning the art of mathematical modeling. Today, I’d like to explore a few simple versions of such problems and how you might use them in your classroom.
One of my favorites is a problem that was posed by Isaac Newton in Universal Arithmetik and was discussed by Groetsch in his text Inverse Problems: Activities for Undergraduates. Newton posed the
problem like this:
A Stone falling down into a Well, from the Sound of the Stone striking the bottom, to Determine the Depth of the Well.
The unusual capitalization is Newton’s and apparently the style of the time was to state commands rather than pose questions. Today, we might state the problem as:
Can we determine the depth of a well by dropping a stone into the well and listening for the sound of the stone striking the bottom?
Why is this an inverse problem? Well, what we’re asked to do here is to take information we obtain at the end of a process, i.e. the time interval between us releasing the stone and it us hearing the
stone strike the bottom of the well, and to infer how the stone must have traveled and from this deduce the depth of the well.
This particular inverse problem can be approached by solving the direct problem, which requires us to build a mathematical model of the time which elapses between the release of the stone and us
hearing the sound of the stone striking the bottom of the well. We can imagine that we release the stone and start a stopwatch at the same instant. When we hear the sound, we stop our stopwatch and
call that elapsed time,
Our job is then to build a model of the fall and the travel of the sound that lets us express
Now, given a measured time,
If you do decide to introduce your students to the notion of inverse problems, it is worth spending at least a few minutes sharing with them the many areas of application where such problems arise.
I’d suggest focusing on medical imaging as this is likely familiar territory in some sense, but unfamiliar territory in the sense of them understanding that most of modern medical imaging rests on
building mathematical models and solving inverse problems.
I’ll leave you with one last inverse problem that is readily understood, but challenging to investigate. Imagine I tell you that I have a container of some sort but that I’m going to keep that
container hidden in a box and not let you see it directly. However, what I will do is pour any volume of water you’d like into the container and tell you the height to which the water fills this
hidden container for that volume. I’ll do this as many times as you’d like and give you as much data of this form as you want. The question is then this – can you tell me the shape of my container?
Suppose I told you the container had rotational symmetry. Would that help?
I hope that this small taste of inverse problems has inspired you to consider this very important class of problems as you work to build mathematical modeling into your classroom practice. As always,
we’d love to hear about your successes and challenges with this or other mathematical modeling investigations. | {"url":"http://modelwithmathematics.com/2015/10/the-other-way-around/","timestamp":"2024-11-09T10:16:07Z","content_type":"text/html","content_length":"40471","record_id":"<urn:uuid:e3b25aa2-f899-4386-bf53-6e291c62f416>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00512.warc.gz"} |
Solving Quadratic Equations By Graphing Worksheet Answers Algebra 2 - Graphworksheets.com
Solving Equations Using Graphs Worksheet – Graphing equations is an essential part of learning mathematics. This involves graphing lines and points and evaluating their slopes. Graphing equations of
this type requires that you know the x and y-coordinates of each point. To determine a line’s slope, you need to know its y-intercept, which is the … Read more
Graphing Quadratic Equations Worksheet Algebra 2
Graphing Quadratic Equations Worksheet Algebra 2 – Learning mathematics is incomplete without graphing equations. It involves graphing lines and points, and evaluating their slopes. Graphing
equations of this type requires that you know the x and y-coordinates of each point. To determine a line’s slope, you need to know its y-intercept, which is the point … Read more
Solving Nonlinear Equations By Graphing Worksheet Answer
Solving Nonlinear Equations By Graphing Worksheet Answer – Reading graphs is a skill that is useful in many fields. They allow people to quickly compare and contrast large quantities of information.
For example, a graph of temperature data may show the time of day when the temperature reaches a specific number of degrees Celsius. A … Read more
Solve Quadratic Equation By Graphing Worksheet
Solve Quadratic Equation By Graphing Worksheet – Graphing equations is an essential part of learning mathematics. It involves graphing lines and points, and evaluating their slopes. Graphing
equations of this type requires that you know the x and y-coordinates of each point. You need to know the slope of a line. This is the point … Read more | {"url":"https://www.graphworksheets.com/tag/solving-quadratic-equations-by-graphing-worksheet-answers-algebra-2/","timestamp":"2024-11-05T23:36:28Z","content_type":"text/html","content_length":"70342","record_id":"<urn:uuid:a6fe9991-16ad-4b99-9da1-77a2531fbcc7>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00117.warc.gz"} |
QCPVector2D Class Reference
Detailed Description
Represents two doubles as a mathematical 2D vector.
This class acts as a replacement for QVector2D with the advantage of double precision instead of single, and some convenience methods tailored for the QCustomPlot library.
Definition at line 440 of file qcustomplot.h.
Constructor & Destructor Documentation
◆ QCPVector2D() [1/4]
QCPVector2D::QCPVector2D ( )
◆ QCPVector2D() [2/4]
QCPVector2D::QCPVector2D ( double x,
double y )
◆ QCPVector2D() [3/4]
QCPVector2D::QCPVector2D ( const QPoint & point )
Creates a QCPVector2D object and initializes the x and y coordinates respective coordinates of the specified point.
Definition at line 138 of file qcustomplot.cpp.
◆ QCPVector2D() [4/4]
QCPVector2D::QCPVector2D ( const QPointF & point )
Creates a QCPVector2D object and initializes the x and y coordinates respective coordinates of the specified point.
Definition at line 148 of file qcustomplot.cpp.
Member Function Documentation
◆ angle()
double QCPVector2D::angle ( ) const inline
Returns the angle of the vector in radians. The angle is measured between the positive x line and the vector, counter-clockwise in a mathematical coordinate system (y axis upwards positive). In
screen/widget coordinates where the y axis is inverted, the angle appears clockwise.
Definition at line 461 of file qcustomplot.h.
◆ distanceSquaredToLine() [1/2]
double QCPVector2D::distanceSquaredToLine ( const QCPVector2D & start,
const QCPVector2D & end ) const
This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.
Returns the squared shortest distance of this vector (interpreted as a point) to the finite line segment given by start and end.
See also
Definition at line 190 of file qcustomplot.cpp.
◆ distanceSquaredToLine() [2/2]
double QCPVector2D::distanceSquaredToLine ( const QLineF & line ) const
This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.
Returns the squared shortest distance of this vector (interpreted as a point) to the finite line segment given by line.
See also
Definition at line 214 of file qcustomplot.cpp.
◆ distanceToStraightLine()
double QCPVector2D::distanceToStraightLine ( const QCPVector2D & base,
const QCPVector2D & direction ) const
Returns the shortest distance of this vector (interpreted as a point) to the infinite straight line given by a base point and a direction vector.
See also
Definition at line 225 of file qcustomplot.cpp.
◆ dot()
double QCPVector2D::dot ( const QCPVector2D & vec ) const inline
Returns the dot/scalar product of this vector with the specified vector vec.
Definition at line 469 of file qcustomplot.h.
◆ isNull()
bool QCPVector2D::isNull ( ) const inline
Returns whether this vector is null. A vector is null if qIsNull returns true for both x and y coordinates, i.e. if both are binary equal to 0.
Definition at line 465 of file qcustomplot.h.
◆ length()
double QCPVector2D::length ( ) const inline
Returns the length of this vector.
See also
Definition at line 459 of file qcustomplot.h.
◆ lengthSquared()
double QCPVector2D::lengthSquared ( ) const inline
Returns the squared length of this vector. In some situations, e.g. when just trying to find the shortest vector of a group, this is faster than calculating length, because it avoids calculation of a
square root.
See also
Definition at line 460 of file qcustomplot.h.
◆ normalize()
void QCPVector2D::normalize ( )
Normalizes this vector. After this operation, the length of the vector is equal to 1.
If the vector has both entries set to zero, this method does nothing.
See also
normalized, length, lengthSquared
Definition at line 161 of file qcustomplot.cpp.
◆ normalized()
QCPVector2D QCPVector2D::normalized ( ) const
Returns a normalized version of this vector. The length of the returned vector is equal to 1.
If the vector has both entries set to zero, this method returns the vector unmodified.
See also
normalize, length, lengthSquared
Definition at line 176 of file qcustomplot.cpp.
◆ operator*=()
QCPVector2D & QCPVector2D::operator*= ( double factor )
Scales this vector by the given factor, i.e. the x and y components are multiplied by factor.
Definition at line 234 of file qcustomplot.cpp.
◆ operator+=()
QCPVector2D & QCPVector2D::operator+= ( const QCPVector2D & vector )
Adds the given vector to this vector component-wise.
Definition at line 255 of file qcustomplot.cpp.
◆ operator-=()
QCPVector2D & QCPVector2D::operator-= ( const QCPVector2D & vector )
subtracts the given vector from this vector component-wise.
Definition at line 265 of file qcustomplot.cpp.
◆ operator/=()
QCPVector2D & QCPVector2D::operator/= ( double divisor )
Scales this vector by the given divisor, i.e. the x and y components are divided by divisor.
Definition at line 245 of file qcustomplot.cpp.
◆ perpendicular()
QCPVector2D QCPVector2D::perpendicular ( ) const inline
Returns a vector perpendicular to this vector, with the same length.
Definition at line 468 of file qcustomplot.h.
◆ rx()
double & QCPVector2D::rx ( ) inline
◆ ry()
double & QCPVector2D::ry ( ) inline
◆ setX()
void QCPVector2D::setX ( double x ) inline
Sets the x coordinate of this vector to x.
See also
Definition at line 455 of file qcustomplot.h.
◆ setY()
void QCPVector2D::setY ( double y ) inline
Sets the y coordinate of this vector to y.
See also
Definition at line 456 of file qcustomplot.h.
◆ toPoint()
QPoint QCPVector2D::toPoint ( ) const inline
Returns a QPoint which has the x and y coordinates of this vector, truncating any floating point information.
See also
Definition at line 462 of file qcustomplot.h.
◆ toPointF()
QPointF QCPVector2D::toPointF ( ) const inline
Returns a QPointF which has the x and y coordinates of this vector.
See also
Definition at line 463 of file qcustomplot.h.
◆ x()
double QCPVector2D::x ( ) const inline
◆ y()
double QCPVector2D::y ( ) const inline
Friends And Related Symbol Documentation
◆ operator<<()
QDebug operator<< ( QDebug d, related
const QCPVector2D & vec )
Prints vec in a human readable format to the qDebug output.
Definition at line 503 of file qcustomplot.h.
The documentation for this class was generated from the following files:
This file is part of the KDE documentation.
Documentation copyright © 1996-2024 The KDE developers.
Generated on Fri Nov 8 2024 12:05:30 by
1.12.0 written by
Dimitri van Heesch
, © 1997-2006
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Average Rate of Change Over an Interval Q&As - Calculus | HIX Tutor
Average Rate of Change Over an Interval
The average rate of change over an interval is a fundamental concept in calculus and mathematics, serving as a measure of how a quantity changes on average within a specific range. It calculates the
overall rate of change of a function or variable over a given interval, providing insights into trends or behaviors. This concept is crucial in various fields, including physics, economics, and
engineering, where understanding how quantities evolve over time or distance is essential for analysis and prediction. Calculating the average rate of change enables precise modeling and
interpretation of dynamic systems and processes. | {"url":"https://tutor.hix.ai/subject/calculus/average-rate-of-change-over-an-interval","timestamp":"2024-11-02T05:31:22Z","content_type":"text/html","content_length":"556519","record_id":"<urn:uuid:eeb3d95b-3d82-4979-90a9-23e3a1c075f3>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00278.warc.gz"} |
acrylic load calculator
Acrylic Load Calculator
Acrylic is a versatile material widely used in various industries due to its durability, transparency, and aesthetic appeal. However, understanding the load-bearing capacity of acrylic sheets is
crucial for ensuring safety and efficiency in its applications. This article will guide you on how to use an acrylic load calculator, including the most accurate formula for calculations, examples,
FAQs, and more.
How to Use the Acrylic Load Calculator
Using the acrylic load calculator is simple and straightforward. Here’s a step-by-step guide:
1. Input the dimensions of the acrylic sheet: length, width, and thickness.
2. Specify the load characteristics: the type of load (distributed or point load) and the weight.
3. Click the ‘Calculate’ button to get the load-bearing capacity.
The calculator provides an immediate and accurate result, ensuring you can make informed decisions for your projects.
Formula for Acrylic Load Calculation
The most accurate formula for calculating the load-bearing capacity of an acrylic sheet involves understanding its bending strength and modulus of elasticity. Here is the formula used in the
• E = Modulus of Elasticity of acrylic (approximately 3,300 MPa)
• L = Length of the acrylic sheet
Example Solve
Let’s walk through an example to understand how the calculation works.
• Length (L) = 1 meter
• Width (b) = 0.5 meter
• Thickness (h) = 0.01 meter
Calculate the moment of inertia:
Using the formula:
This example illustrates how the formula works with given dimensions.
What is the Modulus of Elasticity for acrylic?
The Modulus of Elasticity for acrylic is approximately 3,300 MPa.
How accurate is the acrylic load calculator?
The calculator uses industry-standard formulas and constants to provide highly accurate results.
Can I use this calculator for different types of acrylic?
Yes, the calculator is designed to work for standard acrylic materials. For specific types, ensure the material properties match those used in the formula.
Is the calculator suitable for all load types?
The calculator primarily works for distributed loads and point loads. Ensure you select the appropriate load type for accurate results.
Understanding the load-bearing capacity of acrylic sheets is essential for safe and effective usage in various applications. This acrylic load calculator, using the most accurate formula, provides
quick and reliable results to guide your projects. With clear input parameters and instant calculations, you can ensure your acrylic structures are both functional and safe.
Other Useful Headings
Benefits of Using Acrylic
Acrylic is not only strong but also lightweight, making it an ideal material for a wide range of applications from display cases to windows and protective barriers.
Safety Considerations
Always consider safety margins when working with acrylic. The calculated load should not be the maximum load the sheet is subjected to in real-world conditions.
Applications of Acrylic
Acrylic is used in construction, advertising, furniture, and many other industries due to its clarity, strength, and ease of fabrication. | {"url":"https://calculatordoc.com/acrylic-load-calculator/","timestamp":"2024-11-12T05:30:56Z","content_type":"text/html","content_length":"95020","record_id":"<urn:uuid:6671fa4e-9c51-418d-9b83-00d0f28ed156>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00504.warc.gz"} |
seminars - Entropy, large deviations, and applications
The theory of large deviations is concerned with the exponential decay of probabilities of remote tails of sequences of probability distributions. Entropy is concerned with the complexity of
dynamical systems. In this talk, we will focus on the relationship between large deviation and entropy for some dynamical systems. We will then explore some applications of this relationship.
줌 주소: 361 546 1798 | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&sort_index=speaker&order_type=desc&page=40&document_srl=1083346","timestamp":"2024-11-12T09:09:46Z","content_type":"text/html","content_length":"47477","record_id":"<urn:uuid:7731eccf-c367-4ecb-96a5-1d56232cd786>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00037.warc.gz"} |
How do we calculate Returns / CAGR in Portfolio ?
XIRR is that single rate of return, which when applied to every installment (and redemptions if any) would give the current value of the total investment.
XIRR is your personal rate of return. It is your actual return on investments.
XIRR stands for Extended Internal Rate of Return is a method used to calculate returns on investments where there are multiple transactions happening at different times.
Mutual Fund investments are not as evenly spaced as you saw above in case of mutual funds. In the case of mutual funds, you tend to invest and redeem investments at irregular intervals.
It will cause cash inflows and cash outflows at different points in time. In this type of case, in addition to the invested amount, the time of such investment also assumes significance to yield a
certain outcome.
Here you may use the concept of Extended Internal Rate of Return (XIRR).
So, XIRR is a good function to calculate returns when your cash flows (investments or redemption) is spread over a period of time.
XIRR can be easily calculated using Microsoft Excel. Excel provides an inbuilt function to calculate XIRR.
XIRR is a more powerful function in excel for calculating the annualized yield for a schedule of cash flows occurring at irregular periods.
XIRR formula in excel is:= XIRR (value, dates, guess)
Step by Step Process to Calculate in Excel
1. Enter all your transactions in one column. All outflows like investments, purchases will be market negative while all inflows like redemption’s while are marked positive.
2. In the next column add the corresponding date of the transaction
3. In the last row mention the current value of your holding and the current date
4. Now Use XIRR function in excel which is something like this =XIRR (values, date, Guess)
5. Select values to a series of cash flows that corresponds to a schedule of payments in dates and date columns stand for the date when the first investment was made and when the cash flows were
received, guess parameter is optional ( if you do not put any value Excel use a value of 0.1.
Example of How to Use the Function in Excel :
For this calculation you need is with an example of six-month SIP. Let
SIP amount = ₹ 5000
SIP investment dates = start-01/01/2017, end-01/06/2017
Redemption date = 01/07/2017
Maturity amount = ₹ 31000
Assume we have a set of cash flows like those in the table below :
01-01-2017 -5000
03-02-2017 -5000
01-03-2017 -5000
11-04-2017 -5000
01-05-2017 -5000
25-06-2017 -5000
01-07-2017 31000
In the above table, the cash flows are occurring at irregular intervals. Here, you can use XIRR function to compute the return for these cash flows. Remember to include the ‘minus’ sign whenever you
invest money.
Open an excel sheet and follow these steps :
• In column A, enter the transaction dates on the left side.
• In column B, enter SIP figure of 5000 as a negative figure as it’s an outflow cashflow.
• Against the redemption date (Column A), enter the redemption amount (Column B) (31000).
• In the box below 31000, type in: “ =XIRR (B1: B7, A1: A7)*100 ” and hit enter
XIRR value of 11.92 % will be display as a result.
So XIRR makes this simpler by calculating one return for your investments. So, if you are looking to calculate returns on your mutual fund investments XIRR is the right way to go.
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What is the difference between a linear velocity from a tangential velocity and radial velocity? | Socratic
What is the difference between a linear velocity from a tangential velocity and radial velocity?
1 Answer
Linear velocity is the "real" velocity.
Since we cannot measure the linear velocity of a far-away object, we split it up in the radial velocity, which we can measure by the Doppler-effect (so-called red shift), and the tangential velocity,
which we may measure by parallax (shift against the star background).
In short:
Radial velocity is the speed towards or away from us.
Tangential velocity is the velocity across our field of view.
With these a vector-rectangle can be set up, with the linear velocity being the diagonal.
Impact of this question
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The octagon, the hendecagon and the approximation of pi: the geometric design of the clypeus in the enclosure of Imperial cult in Tarraco
Article publicat a les actes del XII International Forum Le vie dei Mercanti (un congrés del 2014 desenvolupat a Nàpols i Capri). En aquesta entrada es reprodueix íntegrament l'article.
Josep Maria TOLDRÀ,^1 Josep Maria MACIAS,^2 Josep Maria PUCHE,^2 Agustí COSTA,^1 Pau SOLÀ-MORALES,^1 Antoni ESPALLARGAS,^1 Albert FERRÉ,^1
(1) School of Architecture, Rovira i Virgili University, Avinguda de la Universitat, s/n. Reus, Spain
jmtoldra@gmail.com - agusti.costa@urv.cat - pau.desolamorales@urv.cat - toni.espallargas@estudiants.urv.cat - albert.ferrej@estudiants.urv.cat
(2) Catalan Institute of Classical Archaeology, Plaça d’en Rovellat, s/n. Tarragona, Spain
jpuche@icac.cat, jmmacias@icac.cat
The ancient temple dedicated to the Roman Emperor Augustus on the hilltop of Tarraco (today’s Tarragona), was the main element of the sacred precinct of the Imperial cult. It was a two hectare
square, bordered by a portico with an attic decorated with a sequence of clypeus (i.e. monumental shields) made with marble plates from the Luni-Carrara’s quarries. This contribution presents the
results of the analysis of a three-dimensional photogrammetric survey of one of these clipeus, partially restored and exhibited at the National Archaeological Museum of Tarragona.
The perimeter ring was bounded by a sequence of meanders inscribed in a polygon of 11 sides, a hendecagon. Moreover, a closer geometric analysis suggests that the relationship between the outer
meander rim and the oval pearl ring that delimited the divinity of Jupiter Ammon can be accurately determined by the diagonals of an octagon inscribed in the perimeter of the clypeus.
This double evidence suggests a combined layout, in the same design, of an octagon and a hendecagon. Hypothetically, this could be achieved by combining the octagon with the approximation to Pi used
in antiquity: 22/7 of the circle’s diameter. This method allows the drawing of a hendecagon with a clearly higher precision than with other ancient methods. Even the modelling of the motifs that
separate the different decorative stripes corroborates the geometric scheme that we propose.
Keywords: Tarraco, clypeus, Augustus, geometry, hendecagon, Pi-approximation.
1. Investigation context
This article suggests the existence of a geometric model used for shaping monumental clypeus of the sacred enclosure of Tarraco, which are preserved at the National Archaeological Museum of
Tarragona. This proposal is a result of collaboration between architects from the Technical Superior School of Architecture (ETSA) from Rovira i Virgili University (URV) and archaeologists from
Catalan Institute of Classical Archaeology (ICAC), from the interchange of knowledge applied to photogrammetric documentation of architectonical heritage as a previous step of the functional and
chronological analysis of their structural elements.
The enlargement of the Imperial cult developed in provincial capital a homogenous architectural and artistic language, which was based on the monumentality of Imperial Forum in Rome. In the case of
Tarraco, the attractiveness of the hilltop sets up a true religious acropolis, where, through centuries, the main worship building of the city had been superimposed. In this context, the current
investigation locates, under the medieval cathedral still in use today (fig. 1e), the worship octastyle temple devoted to Emperor August (fig. 1d), built from Tiberius’ period emulating Mars Ultor
Temple. Afterwards, this aedes was preserved during the final transformation of the acropolis into monumental headquarters of the Concilium Prouinciae Hispaniae Citeriores. So that Augustus Temple
was kept in use in the centre of a new square erected in the Flavian period (fig. 1a); image and likeness of Forum Pacis, in this case. So was introduced a new model of imperial urban sanctuary with
formal similarity as Cigongnier in Avenches or Courseul Forum [5].
The new sacred square was delimited by a monumental portico completely made of marble from Luni Carrara and whose colonnade held up an entablature compounded of architrave, frieze and cornice. Over
the entablature an attic showed a succession of clypeus with Jupiter-Ammon’s figure, clearly inspired in the portico of Forum Augustum. Most of the clypeus recovered from Tarragona correspond a
syncretic image of Jupiter-Ammon while, for the time being, the identification of another mythological figure is doubtful. A possible fragment of clypeus could reproduce a model of Medusa, although
it is not entirely sure (see [6] fig. 3.5). If it is so, this model is related with other emblems of Medusa decorated in base of straight and oblique tabs [3] [8]. This possibility could be related
to an iconographic programme of porticoed gallery with attics decorated by an alternation of Jupiter and Medusa clypeus, documented for first time in Flavian period (see [8] p. 575). In Tarraco, and
unlike the first model of Forum Augustum or in the copies of Pozzuoli and Mérida, separation between clypeus would be performed through panels divided by vegetal candelabrum, as it is found in the
cities of Avenches and Arles, or in the forum of Nyon (see [8] p. 567).
The analysis of the recovered fragments, either clypeus elements or candelabrum reliefs, show an average thickness around 16 cm. Regarding clypeus diameter, E. Koppel had achieved around 150 cm [3],
this measurement was modified by R. Mar and P. Pensabene (1,35 cm [7] p. 135); our photogrammetric restitution shows around 160 cm.
Fig. 1: At left: Building complex of the Concilium Prouinciae Hispaniae Citeriores (Provincial Forum), overlaid to the current urban trace and the plan of the Cathedral. Sacred Precinct (a.),
Representation Square (b), Circus (c.), approximate position of the Temple of Augustus (d.), Cathedral (e.). Top right: virtual image, Concilium superimposed on the present city of Tarragona. Bottom
right: photograph of the clypeus exhibited in the National Archaeological Museum.
Fig. 2: Clypeus from Imperial Forums: a. Tarragona; b./d./e. Roma [9] [10]; f./g./h. Mérida [1]; c./i. Pozzuoli [13].
2. The singularity of hendecagon
Planimetry of Roman architecture used to be based on simple arithmetic operations, the layout of basic shapes (triangles, squares, rectangles, circles) and their manipulation with ruler and compass
(fig. 3). This allows to easily obtain derived figures (e.g. the hexagon, the octagon), as well as immeasurable proportions (e.g. the golden section ratio Φ, √2, √3).
Fig. 3: Basic geometries of Roman architecture.
Typically, central plan Roman buildings (or interior spaces) are articulated from hexagon and octagon. Vitruvius, in the 6th chapter of the 5th book, describes the laying out beginnings of a Latin
theatre using four triangles inscribed in a circle, forming a dodecagon, that is, the first subdivision of a hexagon. The Domus Aurea (fig. 4b.) and the Domus Augustana have octagonal rooms, the same
geometry we find in the central islands of Leptis Magna Forum and in many rooms of thermal buildings. The 4th-century villa, constructed over the Baths of Trajan (fig. 4d), has a hexagonal layout, as
it happens in a courtyard of the Jupiter Heliopolitanus Sanctuary in Baalbek or the columns that surround the inner courtyard of the rotunda-mausoleum built by Constantine in Rome (Santa Costanza
Church). We can consider Hadrian’s Villa as the culmination of this Roman geometric articulation system for architecture: the so-called
Teatro Marittimo
has a plan based on an intricate layout draft by circles centered at the vertices and axes of an octagon. In the tomb of Portus (fig. 4c.) we can even identify a simultaneous use of the hexagon and
the octagon in the same building: the inner room is laid out by an octagon, while the outer peristyle consists of 24 columns, the result of dividing two times a hexagon. These considerations on the
large scale can be transferred to architectural decoration, with column shafts with 24 (4x6) striae, vegetal decorations with hexagonal or octagonal flowers, etc.
The use of more ‘sophisticated’ regular polygons, as the pentagon, the heptagon or the hendecagon, is less common, but we can give some examples. The statue group known as the Dying Gaul in the
Capitoline Museum has a pentagonal diagram inscribed on its base (see [12] fig. 3.7). The tholos by the Tiber at Rome (fig. 4e), at the Forum Boarium, has a peristyle of 20 columns, probably the
result of subdividing 2 times a pentagon. The so-called Temple of Minerva Medica (fig. 4f), in the Licinianos Gardens of Rome, presents a plan structured through a decagon, the first subdivision of a
We have only found a Roman building that we can attribute to a hendecagon based geometry. The tomb at Capua known as Le Carceri Vecchie has a circular base with 22 modules, the result of subdividing
a triangle or, as discussed below, applying the approximation of Pi given by Archimedes to laying out an architectural plan: according to Mark Wilson Jones [12] (fig. 4.7d, p.75) its diameter is 70
feet, while the 22’s intercolumniations of the facade are 10 feet, which leads us directly to the fraction 22/7.
We return now to the decorative element typology that concerns us here. Some of the clypeus from the Imperial Forums in Rome has a decorative crown articulated by an octagonal geometry, as we have
verified in figures 2d y 2e. In both cases we have analysed reconstructive drawings published by Lucrezia Ungaro [9] (2007, p. 155, fig. 203) [10] (2004, p. 22, fig.15).
We also found the octagonal geometry in Mérida Forum clypeus fig. 2f/2g). Especially significant is the case of clypeus 2g: we have restored its geometry from the photograph published by José Luis de
la Barrera [1] (lamina 83, Cat. 229), who considers that because of the fineness of its carving, and the fact of being the only Carrara marble clypeus of the whole Forum, can be attributed to a
master stonemason that would set the model to follow. The clypeus 2f ([1] lamina 92, Cat. 243) is a piece almost entirely preserved; strangely, in the drawing provided by Barrera ([1] in fig. 26) the
outer crown is divided in 34 parts, instead of the 32 (4x8) which can be counted in the picture.
Concerning the hexagon, we identify it in the geometry of a clypeus with anthemion at Pozzuoli (fig. 2i) that shows the reconstructive drawing published by Fausto Zevi and Claudia Valeri (in [13]
figure 10, p. 457).
Fig. 4: Central plan Roman buildings. Key geometrical scheme overlaid to the drawings published by Mark Wilson Jones [12] and John B. Ward Perkins [11].
Can we mention clypeus based on more exotic regular polygons? Yes, but with some cautions. There is a Mérida clypeus which seems to correspond to a heptagonal geometry (fig. 2h). In the José Luis
Barrera picture on which we base our geometric analysis [1] (illustration 95, Cat. 245) can be distinguished large reconstructed areas. However, we note a great uniformity in the size of the 28 pods
that form the crown (4x7, two subdivisions of a heptagon), which gives us confidence in the validity of the restitution.
In the clypeus integrated into the reconstruction of the portico of the Mars Ultor Temple temenos, (fig. 2b) provided by Lucrezia Ungaro [9] (p. 154, image 202), we can distinguish two rings around
the central medallion: the internal decorated with feathers and the external with pods. The second consists of 44 pods, a hendecagon divided twice.
The decorative scheme of Rome clypeus 2b is repeated on the clypeus of Pozzuoli 2c. On the basis of the reconstructive drawing published by Zevi and Valeri [13] (p. 459, image 12), we consider that
in this case the hendecagon figure establish the geometry of the inner decorative ring.
Our initial hypothesis for the clypeus of Tarragona was that the reconstruction of the meanders crown shown at the National Archaeological Museum was wrong. We believed that an accurate survey of the
original fragments would determine a decorative crown marking from a 12-sided polygon, a simple subdivision of a hexagon, or 10-sided, a subdivided pentagon. We were wrong.
We conducted a 3D survey of the reconstruction exhibited at the Museum through
Autodesk’s 123D Catch
software, exporting the generated 3D model in OBJ format and analyzing it using CAD programs
. A close examination of the original parts allowed us to determine that the pods between each meander module were arranged forming an angle between 32.5º and 33º, very similar to the 32.72º of a
regular hendecagon, and sufficiently far from the 36° that would correspond to a decagon or the 30º of a dodecagon, so as not to have doubts about the figure that was used to compose the piece.
This finding led to a new question, and very interesting: how this hendecagon was laid out? We are not facing a building plan, the Clypeus is a decorative piece, yes, but its dimensions and the
precision of the execution seem to rule out a marking based on trial and error. Also has to be keep in mind that we refer to a series of quadrangular marble plaques of, approximately, ½ Roman foot
thick (14.8 cm) and about 5½ feet sideways (162.8 cm), weighing nearly a ton. They all had to fit in the attic of the portico, and its design should articulate with the corresponding architectural
modulation, while its carving obeyed standard parameters that involved an organizational template, previously to chiselling relief and repeated for each marble panel.
The exact construction of a hexagon or a regular octagon is obvious. In the first case is sufficient to mark the radius of a circle around its circumference. In the second has to be inscribed a
square in a circle and then drawn the bisector of its sides. The pentagon can also be accurately plotted with ruler and compass, but with slightly more complicated operations, not obvious to someone
without advanced knowledge of geometry. Does not exist, however, an exact design to draw a hendecagon [14] (nor for the heptagon and other regular polygons [15]). Therefore, to draw a hendecagon with
ruler and compass it is necessary to resort to approximate constructions. In table 1, by the end of this paper, we summarize the length of the sides and the deviations from a regular hendecagon for
all layouts discussed.
In figure 5 we show a first approximation, quite complex and unintuitive, that provides a construction with relatively small deviations: while a regular hendecagon inscribed in a circle of diameter Ø
=1 has 11 sides of equal length L≈0.2817, with this
complex approximation
we obtain 7 sides with L≈0.2828 (a deviation of +0.37%) and 4 sides with L≈0.2799 (a deviation of -0.65%).
Fig. 5: Complex approximation to a hendecagon construction.
simple approximation
to the hendecagon construction involves dividing the radius of the circumference which is inscribed in 25 parts, and then take 14 to draw a side (fig.6). The resulting figure has 10 sides where L=
0.28 (a deviation of -0.61%) and a side where L≈0.2990 (with a considerable deviation of +6.13%). According to Thomas Heath [2], this solution was already known in classical Greece, and is perfectly
consistent with the Greco-Roman geometrics operations: is based on a simple fraction, 25/14, a formulation similar to approximations of Pi proposed by Vitruvius and Archimedes. Indeed, in the next
section, we will rely on the Archimedes’ approximation of Pi to propose various alternative constructions for hendecagon.
Fig. 6: Simple approximation to a hendecagon construction.
3. Hendecagon construction combining the octagon with the Pi approximation
As we have seen, some of the clypeus from the Forum of Augustus at Rome, the reference model for Tarragona’s Sanctuary, clearly present a geometry derived from octagon (fig. 2d y 2e). Apparently this
is not the case in Tarragona: meanders of the outer ring of the clypeus are not laid out, as expected, by a subdivision of the hexagon or octagon. Instead we found a hendecagon defining its geometry.
However, if we inscribe an octagon in its outer perimeter the diagonals appear to define with a certain precision the ratio between the outer ring and the central medallion (see right half of fig.
But, how can we relate both figures? Then we recall the approximation of Pi given by Archimedes through a numerical procedure for calculating the perimeter of a circumscribing/inscribing polygon of
2n sides, once the perimeter of the circumscribing/inscribing polygon of n sides is known, based on the properties of the bisector of an angle of a triangle described in proposition III of book VI in
Euclid’s Elements.
Subdividing the hexagon 4 times, Archimedes succeeded in calculating the approximate perimeter of polygons of 96 sides (6x24) inscribed and circumscribed to a circle, establishing Pi value between
(3+10/71)≈3.1408 and (3+1/7)≈3.1429. The upper limit can be expressed with a simple fraction, 22/7, that fits well with the practical procedures used by Roman builders to make measurements. Vitruvius
himself gives an approximation of Pi expressed in similar terms: right at the beginning of the 9th chapter of the 10th book he describes a method to calculate distances by turning a cart-wheel: if it
has a diameter of 4 feet at every turn it will travel about 12 and a half feet, that is, proposes assimilate Pi to the fraction 25/8, an approximation with an error of 0.5%. The Archimedes’ 22/7
represents an error of only 0.04%, an order of magnitude lower; obviously his work was known in the Roman world, and it is reasonable to think that was within the mathematical background available
for Roman builders.
On the other hand, we can use the Archimedes’ fraction to perform an approximate hendecagon construction. We have already seen that Wilson Jones [12] proposes for
Le Carceri Vecchie at Capua
(fig. 3.a) a width of 10 feet for each of the 22 facade modules, while the diameter of its circular plan would be 70 feet; namely, he directly refers to the fraction of Archimedes. Obviously, a
regular polygon of 22 sides allows us to draw a hendecagon joining vertices 2 by 2, but the direct application of the fraction 22/7 produces a considerable error in the layout. In the left
construction of figure 7 we have divided the diameter of the circle which it is inscribed in 7 parts, an operation that can be easily performed with the theorem of Thales or picking a metrology that
facilitates the division (e.g. the 70 foot diameter plant of
Le Carceri Vecchie
). If we mark Ø/7 along the perimeter of the circumference we obtain an hendecagon in which 10 sides have a length L≈0.2828Ø and 1 side where L≈0.2712Ø; regarding regular hendecagon (where L≈0.2817Ø)
we have an approximation of its layout on which 10 sides have an error of +0.37% and 1 of -3.74%.
We can refine this system subdividing the diameter modulation. If we divide it into 14 parts and we mark Ø/14 along the circumference (central construction of fig. 7) we have 10 sides where L≈0.2821Ø
(+0.12% deviation) and 1 side with L≈0.2783Ø (-1.22% deviation).
A new subdivision of the diameter, now up to 28 parts (right construction of fig. 7), improves the accuracy, with 10 sides where L≈0.2819Ø (a deviation of +0.06%) y 1 side with L≈0.2800Ø (a deviation
of -0.60%).
Fig. 7: Approximate construction of a hendecagon through Archimedes’ approximation of Pi.
The subdivision process of the diameter of the circle may continue, getting better and better approximations to a regular hendecagon. But we stop at the second subdivision, which involves expressing
the Pi approximation 22/7 by the fraction 88/28. For a circumference of 1 meter in diameter (Tarragona’s clypeus are slightly wider, having an overall diameter of around 1.6 meters) we have,
theoretically, a deviation of 2 mm in the worst side and of 2 tenths of a millimeter in the remaining 10 sides, an error within the graphic tolerances when drawing over a stone slab. The problem here
would be the accumulation of errors: would be necessary to transfer 88 times on the circumference of a circle the subdivision in 28 parts of its diameter. This is where the octagon comes in. Since
the numerator of the fraction 88/28 is divisible by 8 we can adjust the laying out by first drawing the diagonals of an octagon, a trivial and accurate operation, to restart the marking of 11 modules
in each of the eight arcs of a circle defined by the octagon (see fig. 8), thus avoiding much of the possible distortion caused by the accumulation of errors.
Fig. 8: Approximate construction of a hendecagon combining the octagon with the Pi approximation.
This solution provides an approximation to the hendecagon indistinguishable from the regular polygon in the laying out of a decorative piece with the size of the clypeus studied here, as can be
checked in the table 1 in which we summarize the deviations of the constructions we have presented.
sides length deviation
11x 0,28173256
regular hendecagon
7x 0,28278381 0,37%
complex approximation
4x 0,27989206 -0,65%
10x 0,28000000 -0,61%
simple approximation
1x 0,29900670 6,13%
10x 0,28278381 0,37%
approximation by Ø/7
1x 0,27120167 -3,74%
10x 0,28207649 0,12%
approximation by Ø/14
1x 0,27829128 -1,22%
10x 0,28190116 0,06%
approximation by Ø/28
1x 0,28004603 -0,60%
4x 0,28190116 0,06%
approximation by Ø/28 + octagon
6x 0,28166933 -0,02%
1x 0,28143748 -0,10%
Fig. 9: Proposed geometrical scheme for the clypeus, compared to the orthophoto of the piece exhibited at the National Archaeological Museum of Tarragona.
4. Conclusions
The clypeus reconstruction of the National Archaeological Museum is correct. Therefore we are facing a singular case: the use of a hendecagon to fit an architectural decoration.
The Forum of Augustus in Rome was the reference parallel for building the Imperial Sanctuary of the Conciclium Prouinciae in Tarragona and other similar complexes, its reproduction served to transfer
to the provinces a monumental architecture that exalted the Imperial cult. In the case of Rome the geometry of the clypeus that decorated the porticos was based mainly in the octagon, a common figure
in the geometry of Roman architecture. In Tarragona we find a hendecagon, but we have managed to establish a possible connection between its layout and the octagon through a Pi approximation, well
known in classical antiquity: 22/7. Moreover, the octagon diagonals seem to establish exactly the proportions between the outer ring and the inner medallion, so we could consider that the geometries
of Tarragona and Rome clypeus were not so distant.
Alternatively, we propose a second hypothesis for drawing the hendecagon: to use the 14/25 fraction of the radio of the circle in which was inscribed, we call it the simple approximation. Finally, it
could be proposed a combination of both solutions: the octagon would serve to establish the main proportions of the piece, while the simple approximation for the hendecagon construction would serve
to inscribe the meanders ring.
Anyway, we can see how Tarragona’s enclosure did not follow strictly the parameters of the original model, an aspect that is also seen in the use of panels with candelabrum as separation of clypeus,
unlike the Forum Augustum in Rome and other monumental complex, that incorporated caryatids decorating their attics.
Bibliographical References
[1] BARRERA ANTÓN, José Luis. La decoración arquitectónica de los foros de Augusta Emerita. Ed. L’Erma Di Bretschneider, 2000, 480 p. ISBN 978-8882650346.
[2] HEATH, Thomas. A History of Greek Mathematics. Vol. 2. Clarendon Press, 1921.
[3] KOPPEL, E.M. Relieves arquitectónicos en Tarragona. Stadtbild und Ideologie (Madrid 1987), Bayerische Akademie der Wissenschaften, Supplements, New series 103, Munich, 1990, p. 327-340.
[4] LaROCCA, Eugenio; UNGARO, Lucrezia; MENEGHINI, Roberto. I luoghi del consenso imperial. Il Foro di Augusto. Il Foro di Traiano. Roma: Progetti Museali Editore, 1995, ISBN-978-8886512022.
[5] MACIAS, J. M.; MENCHON, J.; MUÑOZ, A.; TEIXELL, I. La construcción del recinto imperial de Tarraco (provincia Hispania Citerior), In LÓPEZ, J., MARTIN, Ò. (ed.), Tarraco: construcció i
arquitectura d’una capital provincial romana, Butlletí Arqueològic 32, Tarragona, 2011, p. 423-479.
[6] MACIAS, J. M., MUÑOZ, A., TEIXELL, I., MENCHON, J. J. Nuevos elementos escultóricos del recinto de culto del Concilium Provinciae Hispaniae Citerioris (Tarraco, Hispania Citerior). In NOGALES,
T.; RODÀ, I. (ed.), Roma y las provincias: modelo y difusión (Hispania Antigua, Serie Arqueológica, 3), XI Coloquio Internacional de Arte Romano Provincial (Mérida 2009), Roma, 2011, p. 877-886.
[7] MAR, R., PENSABENE, P. 2011: Financiación de la edilicia pública y cálculo de los costes del material lapídeo: El caso del foro superior de Tárraco. In LÓPEZ, J., MARTIN, Ò. (ed.), Tarraco:
construcció i arquitectura d’una capital provincial romana, Butlletí Arqueològic 32, Tarragona, 345-413.
[8] PEÑA JURADO, A. Decoración escultórica. In AYERBE, R.; BARRIENTOS, T.; PALMA, F.; (ed.), El foro de Augusta Emerita. Genesis y evolucion de sus recintos monumentales, Anejos de AEspA LIII, Anejos
de Archivo Español de Arqueologia, Mérida, 2009, p. 543-581.
[9] UNGARO, Lucrezia. La memoria dell’antico. In Il Museo dei Fori Imperiali nei Mercato di Traiano. Ed. Electa, 2007, p. 130-169, ISBN 978-8837051587.
[10] UNGARO, Lucrezia; MILELLA, marina; VITTI Massimo. Il sistema museale dei Fori Imperiali e i Mercati di Traiano. In RUIZ de ARBULO, Joaquín. SIMULACRA ROMAE Roma y las capitals provincials del
Occidnte Europeo. Tarragona: 2004, p. 11-48.
[11] WARD PERKINS, John. Arquitectura Romana. (Translated by ESCOLAR BAREÑO, Luis). Madrid: Aguilar S.A. de Ediciones, 1989. 207 p. Translation of Roman Architecture, Electa, 1980. ISBN
[12] WILSON JONES, Mark. Principles of Roman Architecture. 3ª ed., New Haven and London: Yale University Press, 2000, 270 p. ISBN 978-0300102024.
[13] ZEVI, Fausto; VALERI, Claudia. Cariatidi e clipei: il foro di Pozzuoli. In Le due patrie acquisite. Studi di archeologia dedicati a Walter Trillmich. Ed. L’Erma di Bretschneider, 2008, p.
443-464, ISBN 978-8882655082.
[14] http://mathworld.wolfram.com/Hendecagon.html
[15] http://mathworld.wolfram.com/ConstructiblePolygon.html | {"url":"http://www.jmtoldra.com/2015/01/the-octagon-hendecagon-and.html","timestamp":"2024-11-08T04:38:06Z","content_type":"text/html","content_length":"157388","record_id":"<urn:uuid:11da90dd-5ebb-48ee-aba3-b61280bc8e78>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00133.warc.gz"} |
Difference between solving rational equation and simplifying rational expression using lcd
difference between solving rational equation and simplifying rational expression using lcd Related topics: solving inequalities ditto
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Answers For Holt Algebra 1
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Author Message Author Message
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Ryan's Repository of Random Reflections
As many of the people that read this blog probably know, I enjoy inventing math problems, especially problems that are probabilistic in nature, and then working out the math to solve that problem. In
this blog post, I will introduce one problem that I solved that I think is particularly elegant: You are randomly walking along the x-axis starting at \( x=0 \). When you are at \( x = n \), you move
to \( x = n-1 \) with probability \( p = \frac{n}{n+1} \) and to \( x = n+1 \) with probability \( p = \frac{1}{n+1} \). What is the expected number of steps until you return to \( x = 0 \) for the
first time? This problem differs from standard random walk problems because the probability of moving left and right along the x-axis depends on your current location. In this problem, there is a
force that is pulling you towards \( x = 0 \), and the further away you get, the more of an effect this force has on you. Show/Hide Solution So how do we go about solving this problem? When
This semester in my mathematics capstone course, students had the chance to develop mathematical models to describe real world research problems. I took this as an opportunity to research and develop
probabilistic models that can be used to predict the outcome distribution of a baseball at bat. My hope was that I could apply my findings from this research project to my side project of Beat the
Streak. I learned a lot about mathematical modeling in this class, and explored a variety of techniques for making predictions about baseball at bats. At the end of the class, we wrote a report that
introduces the models we came up with, and compares the quality of the predictions they produce. I have made this report available here .
This is a short blog post that shows you an easy and intuitive way to derive the formula for the summation of an infinite geometric sequence. Let \( 0 \leq p < 1 \), and let \( a \) be some constant;
then we wish to find the value of \( x \) such that $$ x = \sum_{k=0}^\infty a p^k $$ Writing out the first few terms of the summation, we get: $$ x = a + a p + a p^2 + a p^3 + \dots $$ Rewriting the
equation by factoring out a \( p \) from every term except the first, we get: $$ x = a + p (a + a p + a p^2 + \dots) $$ Notice that the expression in parenthesis is exactly how \( x \) is defined.
Replacing the expression with \( x \) leaves us with: $$ x = a + p x $$ Solving the equation for \( x \) yields $$ x = \frac{a}{1-p} $$ Just remember this simple derivation and you will never have to
look up the formula for evaluating the sum of an infinite geometric sequence ever again!
In order to maximize your probability of beating the streak, you should (1) predict the probability that a batter will get a hit in a given game given the game parameters and (2) determine if it's
worthwhile to risk your current streak in order to possibly improve it by 1 or 2 games. In this blog post, I outline my solution to (2). In previous blog posts, I've hinted at what I do to solve (1)
and will continue that discussion in a later blog post. Motivation When your current streak is short, the optimal strategy is to pick the best player every day, regardless of how likely their
probability of getting a hit is (to a certain extent). However, as your streak grows, you have an important decision to consider: is it better to pick the best player today and possibly lose your
current streak, or skip picking a player and instead maintain your current streak. Naturally, this decision should be guided by the players probability of getting a hit, as well as the distribution
of the
In a lot of the recreational math and computer science problems I solve, I try to derive a recurrence relation that describes the situation. These recurrence relations pop up pretty often in counting
problems (how many ways can you tile a 2xn grid with 2x1 tiles?) In some cases, I am unable to derive a recurrence relation directly from the problem statement, but still suspect that there exists
some recurrence relation. In these cases, I end up solving the problem usin g brute force methods for small input sizes and analyzing/extrapolating the pattern. However, when the relation is obscure,
it becomes hard to find. In order to make this process easier in the future, I created a program that can automatically find recurrence relations for arbitrary integer sequences. To use it, simply
type in the first few terms of a sequence you are interested in, and it will try to find a recurrence relation for that sequence. There are some constraints on the allowable inputs/outputs which I di
With the mid way point of the MLB season fast approaching, it is starting to become apparent that I need to finish my program if I want to have any shot at compiling a respectable streak this season.
In this blog post, I will talk about one formula that I am relying heavily on in my tool. This formula is based on Bill James' Log5 formula but is modified for non-binary output. In general, the Log5
formula is used to approximate the probability that team A will defeat defeat team B given each of their individual winning percentages. It can be modified to handle batter vs. pitcher matchups as
well. There is no mathematically rigorous derivation of the formula, so using it in my tool is a little bit iffy to say the least. However, I think using it will yield a better approximation to
batter vs. pitcher matchups than not using it. In it's simplest form, the approximate batting average of batter B against pitcher P is given by: $$ AVG_{B v P} = \frac{\frac{AVG_B \cdot AVG_P}{AVG
In my last blog post , I showed how matrices and matrix multiplication in particular can make sense for more than just numbers. In fact, I used a boolean matrix construct to find the connected
components of a graph. In this blog post I will extend that idea to two more types of matrices: "Distance" matrices to find the shortest distances between any two nodes Path matrices to find the
(shortest) list of actions that lead between any two states The first type of matrix that I will talk about is what I call a 'Distance' matrix. This is not to be confused with how distance matrices
are usually defined in mathematics and computer science (pairwise distance between points). Instead, my definition is more closely related to an adjacency matrix, where connected nodes have non-zero
entries (given by the weight, or distance between the nodes), and unconnected nodes have a distance of 'X' to denote that there is no edge between the nodes. Using this newly de
Matrices are a fundamental part of linear algebra, and we typically think of a matrix as a 2 dimensional array of numbers. In this post, I will show that matrices can be used in other ways too, and I
will show that these new types of matrices can be used to solve real problems. Adjacency Matrix If you are already familiar with Adjacency matrices, I recommend you skip ahead to the next section;
otherwise, read on! Matrices are especially important in graph theory, as they can be used to model the structure of a graph. A graph is a set of vertices and edges, and the edges indicate how the
vertices are connected. Graphs can be used to model a number of real world things: the internet, social networks, road systems, etc. One common representation of a graph is an adjacency matrix: For a
matrix \( A = [a_{ij}] \), \( a_{ij} = 1 \) if node i is connected to node j and \( a_{ij} = 0 \) otherwise (Note I implicitly implied there is some ordering on the nodes). One interesting prop
In 2001, MLB.com released the Beat the Streak challenge : A challenge to fans to essentially beat the all time best hitting streak established by Joe DiMaggio in 1941. In that season, Joe DiMaggio
had a 56 game hitting streak. The longest hitting streak by any MLB player since that is 45 games. When the challenge was first introduced, fans were asked to pick a (possibly different) player every
day who they expect to get a hit. If that player earned a hit, then their streak would increase by 1. Otherwise, it would go back to 0. The first fan to reach a streak of 57 would win the grand
prize. Since it's introduction in 2001, the grand prize has grown from $100,000 to $5,600,000, and a number of new features have been added to improve the odds for the fans. Yet, no one has even
broken the 50 game streak barrier, let alone win the grand prize. I have been a casual beat the streak player for a few seasons, but I never really took it too seriously. Last season, I decided to
use my back
When I was first learning Haskell, there were some imperative design patterns that I had trouble converting to functional style Haskell. Memoization is the first thing that comes to mind. I use
memoization so much that it was pretty much trivial do in an imperative language like Java. However, I could not figure out how to do it in Haskell. In this blog post, I will discuss how you can
memoize your functions in Haskell to boost performance in case you're ever in a situation where you want to use it. It turns out that memoization in Haskell is just as easy, if not easier, than
memoization in an imperative language. Before I talk about memoization in Haskell, I will discuss the traditional approach to memoizing a recursive function in an imperative language. It essentially
boils down to these steps: Query lookup table to see if result is computed; If so, return it Compute result recursively Store result in lookup table Return result For example, consider the numbers in
A few weeks ago, I was using a Markov Chain as a model for a Project Euler problem, and I learned about how to use the transition matrix to find the expected number of steps to reach a certain state.
In this post, I will derive the linear system that will help answer that question, and will work out specific an example using a 1-dimensional random walk as the model. Terminology Before I begin the
derivation, let me define some terms. The transition matrix is an \( n \) by \( n \) matrix that describes how you can move from one state to the next. The rows of the transition matrix must always
sum to 1. States can either be transient or absorbing . An absorbing state is one that cannot be left once reached (it transitions to itself with probability 1). A transient state is a state that is
not an absorbing state. In many problems, it is of general interest to compute the expected number of steps to reach an absorbing state (from some start state). Derivation Let \( p_
The Fibonacci numbers: \( 1,1,2,3,5,8,13,21,... \) have a number of interesting properties. A few days ago, I discovered and proved one such property that I find particularly interesting. It turns
out that successive Fibonacci numbers are always relatively prime (I will prove this later). Further, Bezout's lemma guarantees the existence of integers \( p \) and \( q \) such that \( p F_n + q F_
{n+1} = 1 \), where \( F_n \) denotes the \( n^{th} \) number in the Fibonacci sequence. In this blog post, I will find a general formula for \( p \) and \( q \). There is a simple result and an
elegant proof of that result which I will demonstrate. Before we find a general formula for \( p \) and \( q \), let me first prove that \( F_n \) and \( F_{n+1} \) are always relatively prime:
Assume for contradiction that \( F_n \) and \( F_{n+1} \) have a factor, \( d > 1 \) that they share. Then \( F_n = d k_1 \) and \( F_{n+1} = d k_2 \) for some integers \( k_1 \) and \( k_2 \).
In my last blog post , I introduced interpolation search, a searching algorithm that can beat binary search asymptotically. To introduce the concept of interpolation search, I made a simplifying
assumption that the underlying data came from a uniform distribution. It is still possible to use interpolation search on non-uniform distributions, as long as you have the CDF for that distribution.
Background Info A cumulative distribution function is a function that tells you the probability that a random item from the distribution is less than or equal to a specified value. $$ F(k) = P(X \leq
k) $$ Thus, $F(k)$ can take on values in the range $[0,1]$ and it is non-descending. For a discrete uniform distribution of integers in the range $[1,10]$, the CDF looks like: And for a continuous
normal distribution, the CDF looks like: How to get the CDF There are a number of options to get a handle on the CDF. You can find information online about your distribution, do a one-pass
If you have taken an introductory computer science course, you've probably seen the binary search algorithm - an algorithm to efficiently find the index of an item in a sorted array, if it exists.
You might not have heard of interpolation search, however. Interpolation search is an alternative to binary search that utilizes information about the underlying distribution of data to be searched.
By using this additional information, interpolation search can be as fast as $O(log(log(n)))$, where $n$ is the size of the array. In this post I am going to talk about two implementations of
interpolation search. First, I will discuss the algorithm as described on Wikepedia, then I will describe my own version of the algorithm that has a better worst case time complexity than the first
and performs better in practice than the traditional binary search and interpolation search algorithms. The Basic Idea Interpolation search models how humans search a dictionary better than a binary
In this blog post, I'll be talking about an efficient communication scheme that can be leveraged in parallel programs for broadcast or reduce operations. The idea is very simple, but I thought it was
pretty cool when I learned it, so I thought I would pass along the information to you. Let's abstract away the parallel programming aspect and analyze the following question: You have a message that
you want to share with a large group of people, but you can only share the message with one person at a time. How should you distribute this message? Setting up the Problem To clear up some possible
ambiguities, I want to emphasize that it takes one time step to share a message regardless of who you share it with, and anybody can share the message with anybody else (granted they've received it
first). I will identify each person in the group by a unique integer in ${0,1,...,n}$, and I will assign myself the id $0$. This is similar to how we might structure a distributed prog
Generating random numbers is an important aspect of programming, and just about every programming language offers some source of (pseudo) randomness, usually in the form of a uniform distribution.
For example, java has a builtin function, <code class="java">Math.random()</code> that returns a random floating point number in the range $[0,1)$. In some instances, we would like to have random
numbers coming from a different distribution, but most programming languages don't offer this functionality (except very specialized languages such as R). In the rest of this blog post, I am going to
explain how you can use a uniformly distributed random number to generate a number from some other distribution. Special Case: The Normal Distribution If you would like to get numbers from a binomial
distribution, or are willing to accept an approximation for the normal distribution in favor of simplicity, then you may want to use this technique. If you add together some num
You may have found this site by searching for information regarding Project Euler Problem 184. This is not a solution to that problem. In fact, it really bothers me when I find people who post their
solutions to these problems online, especially the higher level ones. In this blog post, I talk about a simple problem from probability that was motivated from this Project Euler problem, but the
solution to this problem is not likely to help you solve that one. Pick three points uniformly at random along the circumference of a unit circle centered at the origin. What is the probability that
the triangle connecting these three points contains the origin? Like I said, this problem is not as difficult as the problems that I usually write about, but I decided to write a blog post about it
for two main reasons: I wanted to create the animation/simulation for this problem I eventually want to extend this problem to explore all convex polygons instead of just triangles (which is a more | {"url":"https://www.ryanhmckenna.com/2015/","timestamp":"2024-11-06T18:47:45Z","content_type":"application/xhtml+xml","content_length":"250995","record_id":"<urn:uuid:9ec96128-31db-4b41-8216-4cc30f5523b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00025.warc.gz"} |
phyphox Forums
Hi !
Here is an experiment i created to measure the speed of sound while opening a good bottle of wine !
Cheers !
It works also fine if you make the "pop" with a finger in a test-tube !
Julien, thanks for the useful phyphox code with many images. You have even took into account the end correction 0.61 D !!
Only a photo of the bottle used in the experiment is absent and it is unclear what equation has to be used
As I understood, you have used the approximation by the \lambda/4 standing wave in a tube closed at one side.
If you have not yet finished the bottle an approximation by the Helmhotz resonator is more appropriate for the next opening.
You're right !
The air column in the bottleneck act as a one-sideclosed pipe resonator. The peak frequency is the one of the quarter wavelength standing wave.
The formula used is :
Cair = 4 f0 (L + 0,61xD)
I was insired by this paper :
And of course, once the bottle is open and some wine has been drunk, one can use the formula of the Helmotz resonnator instead ! May be i will add it in a few days !
Here is a result with a test tube.
The end correction for a one-side closed pipe is a half of 0.61 D, i.e. 0.3 D ... as in the paper you have cited.
Oups !
Here is the corrected version.
I will have to open a new bottle this evening !!!
Dear Julien.
For a test tube with L = 23.7 cm and D = 2.8 cm your program gives f = 375.0 Hz and c_air = 381 m/s, to high...
Strange, but the standard "Audio spectrum" of phyphox gives f = 351 Hz (c_air = 345 m/s) - much more acceptable
By the way the resonance peak of our "Resonance curve" gives f = 362 Hz ...
It is very important to measure the temperature around the bottle of wine to be opened. I would suggest to add a feature to your program: determination of the temperature from c_air using a simple
formula t = (c_air/20)² - 273 (°C) .
Peter Froehle, "Finding the Outdoor Temperature Using a Tuning Fork and Resonance",
The Physics Teacher
, 358 (2006);
Jeffrey D. Goldader, "Determining Absolute Zero Using a Tuning Fork",
The Physics Teacher
, 206 (2008);
Thank you for these interesting articles !
There is some strange behaviour with your test tube indeed !
I try some experiments with mine. I put it in the oven few minutes and i found a frequency of 515.63 Hz for a measured air temperature inside the tube of 65°C. This gives a speed of 370.37 m/s. It
seems to be a pretty good result.
Then i tried with the test tube in the freezer for a few minutes. With a measured air température of -5°C i found a frequency of 468.75 Hz. Exactly the same as in the room temperature of about 20°C.
Strange !!!
I always wanted to look closer at the determination of the resonance by FFT. It is strange, when you repeat the experiment , the resonance frequency f0
remains exactly the same ??!..
I have found a very similar
but not exactly the same
graduated cylinder (nominative 250 mL) but I got exactly the same f0... So, I have plotted the resonance part of the measured FFT spectrum together with direct measurements of the resonance by
frequency sweeping (see 'acoustic resonance' in this forum). The interval between points in FFT is very large (46.875 Hz) and a good result can be obtained only by chance. This is also the answer to
some questions asked here earlier. In order to reduce this interval we have to increase the time of the sound treatment..
Yes, you're right !!!
Thanks for sharing this idea !
I will improve it. | {"url":"https://phyphox.org/forums/archive/index.php?thread-776.html","timestamp":"2024-11-12T05:56:29Z","content_type":"application/xhtml+xml","content_length":"12166","record_id":"<urn:uuid:be8176ba-10ca-4e01-b290-bfd491228e04>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00560.warc.gz"} |
Merge Sort Python Program - CSVeda
Merge sort is a very commonly used sorting technique for arranging data in a specific order. It is based on divide and conquer technique of algorithm designing. The complexity of merge sort is O
(nlogn). In this post we will create Merge Sort Python Program to show how it works. But before that we will discuss the logic and algorithm.
The Concept
Merge sort is executed by dividing a list of certain elements into sub lists repeatedly till we get sub lists containing single element. At this stage the sub lists with one element are merged to
create sub lists of two sorted elements. Sub lists of two sorted elements are merged to create sorted sub list of four elements. This process is continued till the original unsorted list is sorted.
If you have two stacks of notebooks sorted by the roll numbers of students and you have to make one stacked of note books sorted by roll number. The process you use will be the application of merge
sort. What will you do? You will compare roll numbers of first note books of two stacks. Whichever roll number is smaller, that notebook is picked and kept separately. Again you compare roll numbers
of the top note books on two stacks. Whichever roll number is smaller that note book is placed below the separately kept notebook. And this process is repeated till one of the stacks finishes. At
this stage you place all the remaining notebooks of the remaining stack under the merged stack of sorted note books.
Assume that you have a list of 8 elements. The list of 8 elements is divided into two lists of 4 elements. Two lists of four elements are divided into 4 lists of two elements. 4 lists of 2 elements
is divided into 8 lists of one element each. The reverse process results in four lists of two sorted elements. Then two lists of four sorted elements. In the end single list of 8 sorted elements.
This is demonstrated in the image shown
Algorithm MERGING(A, BEG,MID,END)
1. [initialize variables)
2. Repeat steps 3 and 4 while i<=MID and j< =END
3. If A[i]<A[j] then
a. temp[k]=A[i]
b. i=i+1
a. temp[k]=A[j]
b. j=j+1
4. K=k+1
5. [copy remaining elements of left sub list]
Repeat while( i<=MID)
6. [copy remaining elements of right sub list]
Repeat while( j<=END)
7. i=BEG
8. Repeat while i<=END [copy sorted sub list to array]
9. Return
1. [Initialize Variables for recursive call of MERGESORT]
START=1 END=N
2. IF (LOW<HIGH) [Recursive call to create sub list ends when single element sub list is
created. for two sub lists MERGING function is called to sort and merge elements ]
a. MIDDLE=( START + HIGH)/2
b. MERGESORT(A, START, MIDDLE)
c. MERGESORT(A, MIDDLE +1, END)
d. MERGING(A, START, MIDDLE, END)
3. EXIT
Merge Sort Python Program
def merging(arr, first, middle,last):
while (i<=middle) and (j<=last):
if int(arr[i])<int(arr[j]):
while (i<=middle):
while (j<=last):
for x in temp:
def mergesort(arr,beg,end):
if (beg<end):
def disparr(arr,n ):
print("Array elements are--->",n)
while (i<n):
n=int(input("Number of elements you want to add--->"))
while (t<n):
arrlist.append(int(input("input value--->")))
print("Before Sorting:")
print("After Sorting:")
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Solve the following integral. .......................................
Integral Calculus> Solve the following integral. ..............
1 Answers
Last Activity: 6 Years ago
In these kind of sums the first thing you have to do is to put x=t^2
then tx=2tdt
substitute it in the integral you will get
$\int \sqrt{(1-t)/(1+t)} \ 2tdt$
Now rationalise the denominator with 1-t
$2\int t-t^2/\sqrt{(1-t^2)} \ dt$
now split the integral into two parts
$\int 2tdt/\sqrt{(1-t^2)} \ -2\int{t^{2}dt/\sqrt{1-t^{2}}}$
We know that integral of f ‘(x)/sqrt f(x) =2f(x)^1/2 and adding ,subtracting 1 in the second part of the integral we get
$2\sqrt{(1-t^2)} \ +2\int{(1-t^{2}+1)dt/\sqrt{1-t^{2}}}$
$2\sqrt{(1-t^2)} \ +2\int\sqrt{(1-t^{2})}dt-2\int dt/\sqrt{1-t^{2}}$
Then by using the identites we get that
$2\sqrt{(1-t^2)} + \sqrt{1-t^{2}}-sin^{-1}\sqrt{1-t^{2}}$
now replacing t^2 with x we get
$2\sqrt{(1-x)} + \sqrt{1-x}-sin^{-1}\sqrt{1-x}+c$
and that is your answer
Please approve my answer if it helped you.
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python-igraph API reference
class documentation
Simple matrix data type.
Of course there are much more advanced matrix data types for Python (for instance, the ndarray data type of Numeric Python) and this implementation does not want to compete with them. The only role
of this data type is to provide a convenient interface for the matrices returned by the Graph object (for instance, allow indexing with tuples in the case of adjacency matrices and so on).
Class Method Fill Creates a matrix filled with the given value
Class Method Identity Creates an identity matrix.
Class Method Zero Creates a matrix filled with zeros.
Method __add__ Adds the given value to the matrix.
Method __eq__ Checks whether a given matrix is equal to another one
Method __getitem__ Returns a single item, a row or a column of the matrix
Method __hash__ Returns a hash value for a matrix.
Method __iadd__ In-place addition of a matrix or scalar.
Method __init__ Initializes a matrix.
Method __isub__ In-place subtraction of a matrix or scalar.
Method __iter__ Support for iteration.
Method __ne__ Checks whether a given matrix is not equal to another one
Method __plot__ Plots the matrix to the given Cairo context in the given box
Method __repr__ Undocumented
Method __setitem__ Sets a single item, a row or a column of the matrix
Method __str__ Undocumented
Method __sub__ Subtracts the given value from the matrix.
Method max Returns the maximum of the matrix along the given dimension
Method min Returns the minimum of the matrix along the given dimension
Instance Variable data Undocumented
Property shape Returns the shape of the matrix as a tuple
Method _get_data Returns the data stored in the matrix as a list of lists
Method _set_data Sets the data stored in the matrix
Instance Variable _data Undocumented
Instance Variable _ncol Undocumented
Instance Variable _nrow Undocumented
def Fill
(cls, value, *args):
Creates a matrix filled with the given value
value the value to be used
*args Undocumented
shape the shape of the matrix. Can be a single integer, two integers or a tuple. If a single integer is given here, the matrix is assumed to be square-shaped.
def Identity
(cls, *args):
Creates an identity matrix.
*args Undocumented
shape the shape of the matrix. Can be a single integer, two integers or a tuple. If a single integer is given here, the matrix is assumed to be square-shaped.
def Zero
(cls, *args):
Creates a matrix filled with zeros.
*args Undocumented
shape the shape of the matrix. Can be a single integer, two integers or a tuple. If a single integer is given here, the matrix is assumed to be square-shaped.
def __add__
(self, other):
Adds the given value to the matrix.
other either a scalar or a matrix. Scalars will be added to each element of the matrix. Matrices will be added together elementwise.
the result matrix
def __eq__
(self, other):
Checks whether a given matrix is equal to another one
def __getitem__
(self, i):
Returns a single item, a row or a column of the matrix
i if a single integer, returns the ith row as a list. If a slice, returns the corresponding rows as another Matrix object. If a 2-tuple, the first element of the tuple is used to select a row and the
second is used to select a column.
Returns a hash value for a matrix.
def __iadd__
(self, other):
In-place addition of a matrix or scalar.
def __init__
(self, data=None):
Initializes a matrix.
data the elements of the matrix as a list of lists, or None to create a 0x0 matrix.
def __isub__
(self, other):
In-place subtraction of a matrix or scalar.
Support for iteration.
This is actually implemented as a generator, so there is no need for a separate iterator class. The generator returns copies of the rows in the matrix as lists to avoid messing around with the
internals. Feel free to do anything with the copies, the changes won't be reflected in the original matrix.
def __ne__
(self, other):
Checks whether a given matrix is not equal to another one
def __plot__
(self, context, bbox, palette, **kwds):
Plots the matrix to the given Cairo context in the given box
Besides the usual self-explanatory plotting parameters (context, bbox, palette), it accepts the following keyword arguments:
• style: the style of the plot. boolean is useful for plotting matrices with boolean (True/False or 0/1) values: False will be shown with a white box and True with a black box. palette uses the
given palette to represent numbers by colors, the minimum will be assigned to palette color index 0 and the maximum will be assigned to the length of the palette. None draws transparent cell
backgrounds only. The default style is boolean (but it may change in the future). None values in the matrix are treated specially in both cases: nothing is drawn in the cell corresponding to
• square: whether the cells of the matrix should be square or not. Default is True.
• grid_width: line width of the grid shown on the matrix. If zero or negative, the grid is turned off. The grid is also turned off if the size of a cell is less than three times the given line
width. Default is 1. Fractional widths are also allowed.
• border_width: line width of the border drawn around the matrix. If zero or negative, the border is turned off. Default is 1.
• row_names: the names of the rows
• col_names: the names of the columns.
• values: values to be displayed in the cells. If None or False, no values are displayed. If True, the values come from the matrix being plotted. If it is another matrix, the values of that matrix
are shown in the cells. In this case, the shape of the value matrix must match the shape of the matrix being plotted.
• value_format: a format string or a callable that specifies how the values should be plotted. If it is a callable, it must be a function that expects a single value and returns a string. Example:
"%#.2f" for floating-point numbers with always exactly two digits after the decimal point. See the Python documentation of the % operator for details on the format string. If the format string is
not given, it defaults to the str function.
If only the row names or the column names are given and the matrix is square-shaped, the same names are used for both column and row names.
def __setitem__
(self, i, value):
Sets a single item, a row or a column of the matrix
i if a single integer, sets the ith row as a list. If a slice, sets the corresponding rows from another Matrix object. If a 2-tuple, the first element of the tuple is used to select a row and the
second is used to select a column.
value the new value
def __sub__
(self, other):
Subtracts the given value from the matrix.
other either a scalar or a matrix. Scalars will be subtracted from each element of the matrix. Matrices will be subtracted together elementwise.
the result matrix
def max
(self, dim=None):
Returns the maximum of the matrix along the given dimension
dim the dimension. 0 means determining the column maximums, 1 means determining the row maximums. If None, the global maximum is returned.
def min
(self, dim=None):
Returns the minimum of the matrix along the given dimension
dim the dimension. 0 means determining the column minimums, 1 means determining the row minimums. If None, the global minimum is returned.
Returns the shape of the matrix as a tuple
Returns the data stored in the matrix as a list of lists
def _set_data
(self, data=None):
Sets the data stored in the matrix | {"url":"https://igraph.org/python/api/0.9.7/igraph.datatypes.Matrix.html","timestamp":"2024-11-04T14:19:18Z","content_type":"text/html","content_length":"50313","record_id":"<urn:uuid:51fc91a1-04d3-4e4a-a7b2-425087b27b25>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00719.warc.gz"} |
What does annual rate pa mean
Learn more about how interest works on a savings account and how banks accrue Meaning, you can use the APY to determine how much you'll actually earn in don't have to pay for branches and can pass
the savings on to consumers. Once you do, the low interest rate specified in the communication to you will apply to all new With a low 14.99% p.a. interest rate, Membership Rewards and 21 Oct 2016
INTEREST free credit cards can be one of the cheapest ways to borrow at a purchase rate of 19.9% p.a. (variable), your representative APR will be 19.9% APR (variable). Withdrawing cash and the
definition of a purchase.
14 Feb 2020 First, what does p.a. mean? You see "p.a." after the percentage symbol in an interest rate. It stands for "per annum" and means the rate is an 15 Jul 2019 As loans or credit agreements
can vary in terms of interest-rate structure, It does not indicate how many times the rate is applied to the balance. Lenders consider certain fees to be pass-through costs which are not directly 24
Jun 2019 First, what does p.a. mean? You see “p.a.” after the percentage symbol in an interest rate. It stands for “per annum” and means the rate is an (pa. means per annum = per year), you can find
the amount of interest by calculating the the percentage. interest rate (% per year) × principal 19 Aug 2019 This means that a credit card company will determine how much to charge These interest
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business loans and see indicative interest rates instantly with our free What does EIR mean? When quoted 10% p.a. EIR interest for loan amount $100K, most will mentally derive interest per year of
$10 K. Cookies help us customise PayPal for you, and some are necessary to make our site work. Cookies also allow us to show you personalised offers and
Based on the stated or nominal rate for a given period, such as an annual interest rate, the effective rate is calculated by incorporating the impact of compounding interest periods into the stated,
nominal rate. The effective interest rate formula is calculated like this: The purpose
Cookies help us customise PayPal for you, and some are necessary to make our site work. Cookies also allow us to show you personalised offers and INTEREST RATES, UNIT: % P.A.. 1. Minimum Loan Rate,
6.00. 2. Minimum Overdraft Rate, 6.87. 3. Minimum Retail Rate, 6.87. 4.1 Loans secured with deposits The amount of interest you'll earn on your savings will depend on several factors, Fixed rates of
interest: Fixed interest means you'll be paid at a set rate which The Marriage Allowance allows you to move part of your personal allowance to Learn more about how interest works on a savings
account and how banks accrue Meaning, you can use the APY to determine how much you'll actually earn in don't have to pay for branches and can pass the savings on to consumers. Once you do, the low
interest rate specified in the communication to you will apply to all new With a low 14.99% p.a. interest rate, Membership Rewards and 21 Oct 2016 INTEREST free credit cards can be one of the
cheapest ways to borrow at a purchase rate of 19.9% p.a. (variable), your representative APR will be 19.9% APR (variable). Withdrawing cash and the definition of a purchase. However, when interest is
compounded, the actual interest rate per annum is lesser than the effective rate of interest. In this article, we will look at the definition,
Based on the stated or nominal rate for a given period, such as an annual interest rate, the effective rate is calculated by incorporating the impact of compounding interest periods into the stated,
nominal rate. The effective interest rate formula is calculated like this: The purpose
However, when interest is compounded, the actual interest rate per annum is lesser than the effective rate of interest. In this article, we will look at the definition, 3.98 Crore*. after 20 years
@ 8% p.a. What does compounding interest mean? Suppose, you invest ₹ 1000 in a bank which offers 10% interest per annum. You can even compare tax-free investments - just choose "0" in the "Tax Rate"
box. The latest interest rates are listed in the main menu, under the "Saving" tab. Interest rate is typically calculated/disclosed/displayed in p.a nomenclature. p.a means per annum or per year.
This is by practice, why banks do this? Maybe because for ease of calculation purposes. Its easy to tell you that, all things constant, for 100 rupees invested you will get 4 rupees as interest at
the end of the year. What is the abbreviation for Per Annum? What does PA stand for? PA abbreviation stands for Per Annum.
Cookies help us customise PayPal for you, and some are necessary to make our site work. Cookies also allow us to show you personalised offers and
Are you searching for information on types of interest? you earn 6.00%pa interest on $10,000 that you have in a bank account, this means you get paid $600 per annum interest. It will be shown as an
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A "per annum" interest rate just means the amount of interest charged for one year, as a percentage of the amount borrowed. This doesn't indicate when the Loan interest rate payable per annum is a
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in "I have a 4% mortgage loan
Gross Rate. This means the interest rate you are paid without the deduction of income tax. p.a. Per annum (per year) You can compare all banks SME business loans and see indicative interest rates
instantly with our free What does EIR mean? When quoted 10% p.a. EIR interest for loan amount $100K, most will mentally derive interest per year of $10 K. Cookies help us customise PayPal for you,
and some are necessary to make our site work. Cookies also allow us to show you personalised offers and INTEREST RATES, UNIT: % P.A.. 1. Minimum Loan Rate, 6.00. 2. Minimum Overdraft Rate, 6.87. 3.
Minimum Retail Rate, 6.87. 4.1 Loans secured with deposits The amount of interest you'll earn on your savings will depend on several factors, Fixed rates of interest: Fixed interest means you'll be
paid at a set rate which The Marriage Allowance allows you to move part of your personal allowance to | {"url":"https://optionedpkmi.netlify.app/pentecost87925pe/what-does-annual-rate-pa-mean-422","timestamp":"2024-11-13T18:32:23Z","content_type":"text/html","content_length":"35391","record_id":"<urn:uuid:ad07f729-944e-4ff2-bec4-e345b195300f>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00624.warc.gz"} |
Understanding the Difference Between 12.56 and 125.6
Is 12.56 greater than or less than 125.6?
No, 125.6 is greater because the whole number is 125 and for the other it is 12.
When comparing the numbers 12.56 and 125.6, it is important to understand the concept of greater than and less than. The symbol ">" is used to indicate "greater than" and "<" is used to indicate
"less than." In this case, 125.6 is greater than 12.56.
Greater Than (>)
125.6 is greater than 12.56. The whole number of 125 is greater than 12, which makes 125.6 greater than 12.56. When comparing two numbers, the number with more digits before the decimal point is
considered greater.
Less Than (<)
12.56 is less than 125.6. In terms of monetary value, $12.56 is less than $125.60. This can be thought of as if you were comparing prices. $125.60 is a higher price than $12.56, so 12.56 is less than | {"url":"https://www.brundtlandnet.com/english/understanding-the-difference-between-12-56-and-125-6.html","timestamp":"2024-11-12T14:05:11Z","content_type":"text/html","content_length":"21021","record_id":"<urn:uuid:7f3c9f3d-61b4-4c08-a92d-4dfc360f8458>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00769.warc.gz"} |
Game Theory 1 - Expected utility theorem - Sarah's page
Learn an example of a Bayesian Nash equilibrium.
We will discuss finite/infinite games that work on the same principle for Bayesian Nash equilibrium and utilize BR to derive BNE.
We will learn how to model incomplete information and the basic idea. We will also see how to turn an incomplete problem into an imperfect problem.
Consider an example of two players deciding whether to start a business together. Draw the normal form of the equilibrium as a function of cost and find the situations in which investment occurs.
Find the deviation of the cooperation phase and the punishment phase so that equilibrium can be reached. Also, learn the status of the one-shot deviation principle.
Learn about an example of finding a pure-strategy SPE to find the best response of repeated interactions. We will also explore the prisoners’ dilemma for unique NEs.
Learn how far rationality can take you, and how to find Nash equilibria in a variety of examples, including the Cournot Duopoly.
Learn about Nash equilibrium, a third way to make predictions in normal-form games. We will also see how to find a Nash equilibrium in a Location game.
Learn about rationalizability, a second way to predict in normal-form games. Examine the difference between 2-player and 3-player games in terms of ISD and BR.
Learn about strict/weak dominant/dominated strategies and see an example of an ISD, which stands for iterated strict dominance. | {"url":"https://saraheee.com/category/study/game-theory-and-applications/page/2/","timestamp":"2024-11-03T05:54:01Z","content_type":"text/html","content_length":"95228","record_id":"<urn:uuid:5b877671-57e5-44ab-91e4-49ebb47be051>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00573.warc.gz"} |
Ferguson’s proposal VS Powell’s proposal
1: What is the total cost difference between Ferguson’s proposal to order 4 cases each time and Powell’s proposal to order 32 cases each time? Explain your results.
Q#2: Lewin suggested looking at economic order quantity (EOQ). Based on the lowest total annual cost, what order quantity should Martin recommend? What is the resulting total cost when using the EOQ
(do not include the unit purchase price)? What is the cost difference compared to the Ferguson and the Powell proposals? Explain your results.
Q#3: Let’s explore the concept of “robustness.” Lewin’s proposal to use economic order quantity may be unrealistic since CMC would like to place orders in whole cases. If the order quantity is
decreased to the nearest whole case (which is a 2.78% reduction) what is the new total annual cost and what percent would your total annual cost change? What is the new total annual cost and what
percent would your annual total cost change if the order quantity is increased to the nearest whole case? Hint: Use the formula [New Total Cost / Old Total Cost].
Q#4: Powell has been working with the information technology department to implement some new Robotic Process Automation (RPA) scripts which will decrease the time required to place purchase orders.
Powell estimates that if the new RPA processes are put in place, the cost of placing a purchase order will decrease to $32. What impact will this change have on the total annual cost for the sample
Q#5: Based on these 4 different evaluations of the appropriate order quantity, what do you recommend? Explain your recommendation.
Q#6: Calculate Reorder Point for each of the above 5 scenarios. Assume supplier lead-time to be 10 days? What is your observation? | {"url":"https://perfectcustompapers.com/fergusons-proposal-vs-powells-proposal/","timestamp":"2024-11-10T17:46:33Z","content_type":"text/html","content_length":"35839","record_id":"<urn:uuid:b8dfd2b0-15d3-4ada-a9a8-22830a217f32>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00211.warc.gz"} |
An Advent-ture with Polynomial Interpolation | Beλ Lilley
Every year, Eric Wastl hosts Advent of Code. Advent of Code is a programming puzzle-solving event, with a puzzle every day starting from the first of December to Christmas. I have been a yearly
participant since 2020, and you can find a lot of my past solutions to the daily puzzles on my GitHub! While my participation usually gets thwarted by my onslaught of finals - I try to do the daily
puzzle as long as I can every year. Last year, there was a puzzle that nicely coincided with one of my classes in terms of content. Day 9 (which you can read up on to get the extra flavor that Eric
adds) involved a sequence of numbers that we had to analyze and extrapolate to generate a new number in the sequence. While programming the logic was not too difficult - there was an underlying
principle that connected back to my Numerical Methods class. To show the connection, let’s first take a more in-depth look at the puzzle itself.
Pulling Apart the Puzzle
The puzzle gave you a sequence of numbers, such as:
Your mission was to find the next number in this sequence. The puzzle instructed you to find this number by looking at the differences between each number, and looking at the differences between each
difference, and so on until the difference was 0. A visual representation of this would be:
The initial differences between the first terms are the second line, the differences between those numbers are the third, and the last line shows zeroes due to the third line being full of the same
number. Doing this allows us to work backwards, as we simply just have to add up the last term of each line (0 + 1 + 6 + 21 = 28) to find the next term in our sequence. This process is not too
difficult to program, and as such many people (me included) wrote their code around this concept. However, there is a sneaky Numerical Method’s concept infecting this problem. The key has to do with
his principle: the sequence of differences will always end with a sequence of zeroes. This principle allows us to make the bridge to polynomial interpolation.
Polynomial Interpolation Acclimation
The term Polynomial Interpolation sounds a bit scary, so let’s break it down into its individual components.
A polynomial is simply a set of terms containing a variable and a coefficient that are being added, subtracted, multiplied, or divided by each other. In more simple terms, its math things that look
like \(3x^2 + 2x - 5\). We are all introduced to polynomials in our first algebra class, so they should be familiar.
Interpolation is a little bit more complicated. Essentially, it is the process of taking discrete data (a set of numbers) and attempting to estimate new data points based on the past data points.
Much of the time, this involves trying to fit a function around the data points. For polynomial interpolation, we attempt to find a polynomial that will output the numbers we see in our data - that
way we can use the polynomial to estimate new data points. A good visual example can be seen below.
From here, we can start to see how we can use this method to determine the next number in our sequence. However, how can we be sure our sequence of numbers follow a polynomial pattern? Eric Wastl’s
suggested method for solving the puzzle revealed a crucial detail: the last sequence of differences always was zeroes. For people who have taken calculus, you might recognize breaking down the
differences between outputs (and their differences, and so on) as essentially taking the derivative. One of the many properties of polynomials is that if we continuously take the derivative of it, we
will eventually end up with zero. Wastl guaranteed the sequence would always end up with zeroes, thus affirming it follows a polynomial pattern.
Polynomial interpolation was a concept I learned in my numerical methods class. After solving the problem using basic programming, I recognized this pattern and used my knowledge from the class to
solve it using interpolation. Being able to use my newfound knowledge to solve fun puzzles was a great confidence booster. If you’re interested in turning continuous patterns and data into discrete
estimations, consider taking a numerical methods class!
Now that we understand polynomial interpolation and why we can use it for this problem, let’s use the concept to generate a polynomial for our example sequence. There is many ways to do so (all with
different benefits/drawbacks), but I will use Lagrange Interpolation (a type of polynomial interpolation) as it’s one of the easier methods to explain.
Lagrange Interpolation Jubilation
Here is the mathematical definition of Lagrange interpolation:
\[L(x) = \sum_{i=1}^{n} y_i \cdot l_i(x)\] \[l_i(x) = \frac{x - x_0}{x_i - x_0} \cdot... \cdot\frac{x - x_{i-1}}{x_i - x_{i-1}}\cdot\frac{x - x_{i+1}}{x_i - x_{i+1}}\cdot...\cdot\frac{x - x_n}{x_i -
This seems extremely complicated, but hopefully with some explanation you can get a better understanding of it. The first thing to note is that the function assumes we have \(n\) pieces of data (\(n
\) numbers). When the function \(l_i(x)\) mentions \(x_0, x_1,\) or \(x_i\), it is referring to how we index our sequence. We consider the numbers in our sequence the output (\(y_i\)) and we label
each number sequentially, usually starting at zero or one. This label is our input (\(x_i\)). For instance, for our example sequence of 1, 3, 6, 10, 15, 21, we want an input of zero (\(x_0\)) to
output 1 \((y_0)\), and input of 1 \((x_1)\) to output 3 \((y_1)\), and so on.
Now let’s break down the logic behind this interpolation. Remember, when we input \(L(0)\) into our function, we want the first term in our sequence to pop out. Let’s analyze how the \(l_i(x)\)
function allows this. Notice that if we input \(x_i\) into the function, that the terms will all cancel out. This results in the function simplifying to one, which is multiplied by \(y_i\) in our \(L
(x)\) function. Along with this, whenever we input any other term in the sequence (\(x_k, k\not = i\)), the fraction that uses \(x_k\) results in a zero in the numerator, making the entire function
zero. These two facts combined guarantee that \(L(x_i) = y_i\).
Let’s see this in action with a small sequence of 1,4,8. Our polynomial function ends up being:
\[L(x) = (1)\frac{x-1}{0-1}\cdot\frac{x-2}{0-2} + (4)\frac{x-0}{1-0}\cdot\frac{x-2}{1-2} + (8)\frac{x-0}{2-0}\cdot\frac{x-1}{2-1}\]
Here we can see the cancelling in action. Plug in the numbers 0,1,2 and see what happens! We can then use this function to plug in 3 and get an estimated output. This can be used in the Advent of
Code puzzle to get the next number in the sequence.
If you are more programming-oriented, I have also included a C function below that mimics our lagrange function - hopefully it will help your understanding!
// Assume nums is an array of integers, and n is the length
// This function outputs the L(x) output, given an input x
int lagrange(int * nums, int n, int x) {
float total = 0;
for (int i = 0; i < n; i++) {
float subtotal = *(nums + i); // Get output number (y)
for (int j = 0; j < n; j++) {
if (j == i) continue;
subtotal *= ((float) (x - j)) / (i - j);
total += subtotal;
return (int) total;
Conclusion Extrusion
Learning computer science concepts in class is one thing - but being able to apply them when you least expect it is even more rewarding.
If you are interested in this puzzle, check out the rest of Advent of Code! Some amazing puzzles are crafted every year by Wastl. | {"url":"https://www.benlilley.dev/2024/advent-day-9/","timestamp":"2024-11-04T08:21:57Z","content_type":"text/html","content_length":"16958","record_id":"<urn:uuid:bc0c2e27-f40b-4133-b602-f76b27bb5d82>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00080.warc.gz"} |
Experimental and numerical optimization of coal breakage performance parameters through abrasive gas jet
2.1 Nozzle parameter
This paper mainly studies the influence of abrasive gas jet pressure and jet target distance on erosion effect. For the abrasive gas jet, the energy obtained by abrasive from the gas plays a decisive
role in jet effect. The common nozzles include convergent nozzle and Laval nozzle. Because of the compressibility of the gas and the lack of divergent segment, the velocity reaches to maximum at the
nozzle outlet and forms shock wave. So, the jet flow filed structure isn’t optimal, the abrasive can’t accelerate sufficiently either. The maximum velocity at the outlet of the convergent nozzle can
only be the sound velocity, namely, subsonic jet. In contrast, the Laval nozzle can obtain supersonic gas jet. The length and angel of divergent segment control the expansive degree of gas jet and
form optimal flow field structure. Therefore, the Laval nozzle is selected to study the parameters of high-pressure abrasive gas jet.
A Laval nozzle is composed of three parts of inlet stability section, convergent section, and the divergent segment. Steady and uniform gas flow can be obtained in the inlet stability section. The
convergent section accelerates the gas flow to sound velocity. Finally, supersonic gas flow is obtained in the divergent segment [20]. The size of the Laval nozzle is designed and calculated
according to the gas dynamics principle [21-22]. The mass flow of any section of the nozzle is:
$m=\frac{P_{0} A}{\sqrt{R T_{0}}} \sqrt{\frac{2 k_{0}}{k_{0}-1}\left[\left(\frac{p}{P_{0}}\right)^{\frac{2}{k_{0}}}-\left(\frac{p}{P_{0}}\right)^{\frac{k_{0}+1}{k_{0}}}\right]}$ (1)
The area ratio of different sections of the nozzle is:
$\frac{A}{A_{0}}=\frac{1}{M_{a}}\left(\frac{2}{k_{0}+1}\right)^{\frac{k_{0}+1}{2\left(k_{0}-1\right)}}\left(1+\frac{k_{0}-1}{2} M_{a}^{2}\right)^{\frac{k_{0}+1}{2\left(k_{0}-1\right)}}$ (2)
The outlet velocity of the nozzle is:
$v=\sqrt{\frac{2 k_{0}}{k_{0}-1} R T_{0}\left[1-\left(\frac{P}{P_{0}}\right)^{\frac{k_{0}-1}{k_{0}}}\right]}$ (3)
The outlet temperature of the nozzle is:
$T_{2}=T_{0}\left(\frac{P}{P_{0}}\right)^{\frac{k_{0}-1}{k_{0}}}$ (4)
Mach number:
$M_{a}=\frac{v}{\sqrt{k_{0} R T}}$ (5)
where, T[0] is stagnation temperature (K). P[0] is stagnation pressure (Pa). k[0] is adiabatic index, k[0]=1.4；R is gas constant. m is mass flow (kg/s). A and A[0] are areas of the sections. Ma is
the Mach number. T[2] is the outlet temperature of the nozzle.
The convergent angle and divergent angle of the nozzle are valued according to the experience [23-24]. Generally speaking, the convergent angle is 30°, and the divergent angle is 10°. When the
temperature is 300K and the pressure is 15 MPa, the measured mass flow of the inlet gas is 0.096kg/s. Because of the characteristics of the Laval nozzle, the Mach number at the nozzle throat is Ma=1.
According to the above formula, the diameter of the nozzle inlet is d=6mm, the diameter of the nozzle throat is d[0]=2mm, and the diameter of the nozzle outlet is D=8mm. The length of the convergent
section is L[2]=7.4mm, and the length of the divergence segment is L[3]=34.3mm. The length of the inlet stability section is L[1]=6mm. Therefore, the nozzle structure is as shown in Figure 1.
Figure 1. The structure of Laval nozzle
2.2 Numerical simulation model
Considering the symmetry of the Laval nozzle structure, the two-dimensional model can be utilized for the calculation in the numerical simulation. That is, half of the actual flow area is utilized as
the calculation model to divide the grid. The calculation area mainly includes the nozzle and free jet region. According to the actual size of nozzle, the flow field model of nozzle and the free flow
field at the nozzle outlet were established by the software GAMBIT. The grid is divided through the quadrilateral structured grid. Considering the difference of target distance, this paper only gives
a grid model, as shown in Figure 2, where AB is symmetric boundary, AC is the pressure inlet boundary, CDE is non-slip adiabatic wall, EFG is the pressure outlet boundary, and BG is the wall of the
jet erosion, which can reflect the erosion effect of abrasive gas jet. It is a kind of boundary condition which is non-slip boundary condition.
To test the mesh independence, the outlet velocities of gas jet have been compared to the condition of different mesh number, such as 16289, 36850, and 65,729. The results are as follows.
Figure 2. The mesh generation of nozzle and flow field
Table 1. The mesh independency study
│ │Nozzle outlet │ │
│Mesh number│ │Maximum calculation error │
│ │velocity (m/s)│ │
│16289 │688.032 │ │
├───────────┼──────────────┤ │
│36850 │687.042 │0.21% │
├───────────┼──────────────┤ │
│65729 │686.606 │ │
The results of grid-independent verification show that it is reliable to use 36,850 mesh structures to calculate.
In this experiment, the 120-mesh quartz sand abrasive was adopted, and the Moh's hardness was 7. In order to compare with the experimental results, the numerical simulation adopted the 120-mesh
quartz sand abrasive. The density was 2660kg/m^3, diameter is 0.125mm, and the mass flow of the abrasive was 0.01kg/s. The initial velocity of the abrasive was 0. The outlet pressure was barometric
pressure, 101.325kpa. The parameter values of the numerical simulation are shown in Table 2.
Table 2. Parameter values of the numerical simulation
│Parameter │Fixed parameter │Simulation variable│
│ │ │p=5 MPa │
│ │ ├───────────────────┤
│ │ │p=10 MPa │
│ │ ├───────────────────┤
│Inlet pressure │Target distance: 7cm │p=15 MPa │
│ │ ├───────────────────┤
│ │ │p=20 MPa │
│ │ ├───────────────────┤
│ │ │p=25 MPa │
│ │ │1cm │
│ │ ├───────────────────┤
│ │ │4cm │
│ │ ├───────────────────┤
│Target distance│Inlet pressure: 15MPa │7cm │
│ │ ├───────────────────┤
│ │ │15cm │
│ │ ├───────────────────┤
│ │ │25cm │
2.3 Calculation model
Numerical simulation adopted the DPM discrete term model in Fluent to simulate the movement erosion of abrasive particles. Firstly, the gas flow field was calculated. When the gas flow field
converged, the discrete term was added. This paper adopted RNG k-ε turbulence model, and the N-S equation was solved through the finite volume method. Compared with k-ε model, the accuracy of the
simulation is higher. The fluid is ideal gas, and the RNG k-ε turbulence model is described as follows:
$\frac{\partial(\rho k)}{\partial t}+\frac{\partial\left(\rho k u_{i}\right)}{\partial x_{i}}$$=\frac{\partial}{\partial x_{j}}\left[\alpha_{k} \mu_{e f f} \frac{\partial k}{\partial x_{j}}\right]+G_
{k}+G_{b}-\rho \varepsilon-Y_{M}$ (6)
$\frac{\partial(\rho \varepsilon)}{\partial t}+\frac{\partial\left(\rho \varepsilon u_{i}\right)}{\partial x_{i}}=\frac{\partial}{\partial x_{j}}\left[\alpha_{\varepsilon} \mu_{e f f} \frac{\partial
\varepsilon}{\partial x_{j}}\right]$$+C_{1 s}^{*} \frac{\varepsilon}{k}\left(G_{k}+C_{3 \varepsilon} G_{b}\right)-C_{2 \varepsilon} \rho \frac{\varepsilon^{2}}{k}-R$ (7)
$C_{1 \varepsilon}^{*}=C_{1 \varepsilon}-\frac{\eta\left(1-\eta / \eta_{0}\right)}{1+\beta \eta^{3}}$
$\eta=\left(\alpha E_{i j} \cdot E_{i j}\right)^{1 / 2} \frac{k}{\varepsilon}$
$\mu_{e f f}=\mu+\mu_{t}$
$\mu_{t}=\rho C_{u} \frac{k^{2}}{\varepsilon}$
$G_{b}=\beta g_{i} \frac{\mu_{t}}{\operatorname{Pr}_{t}} \frac{\partial T}{\partial x_{i}}$
$G_{k}=\mu_{t}\left(\frac{\partial \mu_{i}}{\partial x_{j}}+\frac{\partial \mu_{j}}{\partial x_{i}}\right) \frac{\partial \mu_{i}}{\partial x_{j}}$
$Y_{M}=2 \rho \varepsilon \frac{k}{a^{2}}$
where, G[k] is the production item of turbulent kinetic energy k caused by the mean velocity gradient. G[b] is the production item of turbulent kinetic energy caused by buoyancy. Y[M] is the
influence of compressible turbulent fluctuation on the total dissipation rate. α[k] and α[ε] are the reciprocals of effective Prandtl numbers of turbulent kinetic energy and dissipation rate,
respectively. Pr[t] is turbulent Prandtl number. C[1ε], C[2ε] and C[3ε] are empirical constants. g[i] is the component of gravitational acceleration in the i direction. is the coefficient of thermal
expansion, and a is the sound velocity.
2.4 Numerical simulation results
According to the velocity formula of nozzle outlet, the outlet velocity of nozzle under different jet pressure could be calculated theoretically, as shown in Figure 3. It can be observed that the
velocity of the outlet gas increased with the inlet pressure, and the increased speed gradually slowed down. When the inlet pressure was greater than 15MPa, the increase of gas velocity became
increasingly smaller. Therefore, the jet velocity was increasingly less affected by simply enhancing the jet pressure. Under certain conditions, the optimal pressure exists for the abrasive jet.
Figure 3. Relationship between gas jet velocity of nozzle outlet and jet pressure
Figure 4. Relationship between abrasive velocity and jet pressure
According to the abrasive velocity change curve under the different inlet pressure as shown in Figure 4, with the increase of the inlet pressure, the abrasive velocity first increased rapidly. When
the pressure was greater than 10MPa, the increased of abrasive velocity slowed down. When the pressure was between 20MPa and 30MPa, the abrasive velocity was relatively stable. The power of abrasive
particles of high-pressure abrasive gas jet was from the high-pressure gas. According to the change of gas velocity as shown in Figure 4, when the inlet pressure was less than 10MPa, with the
increase of pressure, the gas velocity at the inlet of the nozzle increased substantially. When the inlet pressure was greater than 10MPa and smaller than 20MPa, with the increase of pressure, the
increase of the gas phase velocity became slow. When the inlet pressure was greater than 20MPa, the increase of gas velocity was slight. The change of the velocity of the abrasive particle of the
abrasive gas jet was consistent with the change trend of gas velocity with pressure.
Fluent was adopted to carry out numerical simulation. The erosion rate module can directly reflect the erosion rate of the target [25], namely, the quality of the material removed per unit area per
unit time. In Fluent, the erosion rate was defined as:
$R_{\text {erosion}}=\sum_{p=1}^{N \text { particle }} \frac{m_{p} C\left(d_{p}\right) f(\alpha) v_{p}^{b(v)}}{A_{\text { face }}}$ (8)
where, C(d[p]) is particle size function; α is the impact angle between the particle path and the wall surface. f(α) is the impact angle function. v is the relative velocity of particles. b(v) is the
relative velocity function of particles. A[face] is the area of the wall.
Table 3. The erosion rate of target
│Inlet pressure（MPa） │10 │15 │20 │25 │
│Erosion rate（kg/m^2·t）×10^-8 │1.19│4.58│3.6│4.0│
In this paper, erosion rate obtained by erosion model is the erosion rate at the same time. According to the simulation results, it is found that the erosion rate is approximately parabolic with the
change of the jet pressure. When the pressure was 15MPa, the erosion rate was the largest. When the jet pressure increased, the abrasive velocity was higher. When the abrasive impacted the target,
the velocity of the abrasive rebound was larger, which further affected the incoming abrasive and the jet flow effect. Therefore, within a certain range of jet pressure, the jet erosion effect was
better when the injection pressure was 15MPa.
Figure 5. Gas jet velocity under different target distances
Figure 6. Relationship between target distance and abrasive velocity
Figure 5 shows the change curve of gas velocity with the target distance when the inlet pressure was 15MPa. It can be observed that the velocity of gas phase was fluctuant when the target distance
was between 0 and 12cm. In the case that the target distance was greater than 12cm, the gas velocity began to obviously decrease. As shown in Figure 6, when the pressure was constant, the velocity of
abrasive particles increased with the target distance. This indicates that the kinetic energy of the gas was converted into the kinetic energy of the abrasive in the acceleration process of gas
expansion at the nozzle outlet. When the target distance was greater than 13cm, the abrasive velocity was basically unchanged. At this moment, the gas velocity began to decrease. The acceleration of
the abrasive was limited, and the velocity increased gently. In the movement process of abrasive particles, the abrasive velocity was always smaller than the gas velocity.
The erosion effect of abrasive gas jet depends on the energy obtained by abrasive particles. Therefore, a better jet erosion effect can be obtained by choosing appropriate jet pressure and target
distance. According to the results of numerical simulation, it can be found that the optimal jet pressure is 15MPa. | {"url":"https://iieta.org/journals/ijht/paper/10.18280/ijht.350408","timestamp":"2024-11-14T15:18:28Z","content_type":"text/html","content_length":"100117","record_id":"<urn:uuid:435b1c59-2851-4911-bac0-8b8e2f4211c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00417.warc.gz"} |
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1900 in Words - How to Write 1900 in Words? | Brighterly
1900 in Words
Updated on January 13, 2024
In words, 1900 is spelled as “one thousand nine hundred”. This number is nine hundred more than one thousand. If you have one thousand nine hundred books, it means you have one thousand books and
then nine hundred more.
Thousands Hundreds Tens Ones
How to Write 1900 in Words?
The number 1900 is written as ‘One Thousand Nine Hundred’ in words. It has a ‘1’ in the thousands place, a ‘9’ in the hundreds place, and ‘0’ in both the tens and ones places. Think of it like having
one thousand nine hundred candies; you say, “I have one thousand nine hundred candies.” So, 1900 is expressed in words as ‘One Thousand Nine Hundred’.
1. Place Value Chart:
Thousands: 1, Hundreds: 9, Tens: 0, Ones: 0
2. Writing it down: 1900 = One Thousand Nine Hundred
Learning to write numbers in words like this is a key skill in early math education.
FAQ on 1900 in Words
Can you write the number 1900 using words?
The number 1900 is written as ‘One thousand nine hundred’.
What is the word form for the number 1900?
One thousand nine hundred’ is the word form for 1900.
If you count to 1900, how do you spell the number?
Counting to 1900, spell it as ‘One thousand nine hundred’.
Other Numbers in the Words:
58000 in Words
99 in Words
83000 in Words
1111 in Words
160000 in Words
15000 in Words
1200000 in Words
Poor Level
Weak math proficiency can lead to academic struggles, limited college, and career options, and diminished self-confidence.
Mediocre Level
Weak math proficiency can lead to academic struggles, limited college, and career options, and diminished self-confidence.
Needs Improvement
Start practicing math regularly to avoid your child`s math scores dropping to C or even D.
High Potential
It's important to continue building math proficiency to make sure your child outperforms peers at school. | {"url":"https://brighterly.com/math/1900-in-words/","timestamp":"2024-11-02T12:13:30Z","content_type":"text/html","content_length":"84803","record_id":"<urn:uuid:bfb2a15f-865e-4ff1-8084-f89cd6bb8b35>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00228.warc.gz"} |
Designing culturally-rich local games for mathematics learning
[English]: This study aimed to design and implement local games-based mathematics learning (das-dasan) to support students' mathematical strategic competence. It consisted of three stages, namely the
identification and analysis of traditional game, the design of learning activities based on Realistic Mathematics Education (RME), and the implementation in the classroom which involved twenty 7^
th-grade students. Data about the local game was collected through observations and interviews with five residents where the game is originated. Data on students’ strategic competence was achieved
through a test given to the students after learning. The analysis of test results refers to the indicators of mathematical strategic competence. The present study found that fifteen students are able
to achieve all indicators (formulating, representing, and solving the problems) with high scores. Meanwhile, five students could only represent the problems but have not fulfilled the last two
indicators. The findings of this study indicate that learning mathematics based on traditional the das-dasan games has a potential to help students develop mathematical strategic competence.
Keywords: Learning design, Ethnomathematics, Local game, Das-dasan, RME
[Bahasa]: Penelitian ini bertujuan merancang dan mengujicoba pembelajaran matematika berbasis permainan tradisional (das-dasan) sebagai upaya untuk mengembangkan kompetensi strategis matematis siswa.
Penelitian ini terdiri dari tiga tahap, yaitu: identifikasi dan analisis permainan tradisional, perancangan pembelajaran berbasis permainan tradisional berdasarkan Realistic Mathematics Education
(RME), dan implementasi dalam pembelajaran di kelas yang melibatkan 20 siswa kelas VII. Data terkait permainan tradisional dikumpulkan melalui pengamatan dan wawancara dengan lima warga tempat
permainan tersebut berasal. Data kemampuan strategis matematis siswa diperoleh melalui tes yang diberikan setelah pembelajaran. Analisis hasil tes siswa merujuk pada indikator kompetensi strategis
matematis siswa. Hasil penelitian menunjukkan 15 siswa berhasil memenuhi semua indikator kompetensi strategis matematis dengan kategori nilai akhir sangat baik. Sedangkan 5 siswa berhasil mencapai
indikator pertama (merumuskan masalah) namun belum semua memenuhi indikator merepresentasikan dan menyelesaikan masalah. Temuan penelitian ini menunjukkan bahwa pembelajaran matematika berbasis
permainan tradisional das-dasan memiliki potensi untuk membantu siswa mengembangkan kemampuan strategis matematis.
Kata kunci: Rancangan pembelajaran, Etnomatematika, Permainan lokal, Das-dasan, RME
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Ethnomathematics is a culture-oriented learning study and has the objective to explore mathematical concepts in the socio-cultural activities of the community (Rosa & Orey, 2011; Tereshkina et
al., 2015). The culture can be in the form of language, dance, games, traditional houses, and various types of regular community activities that can be linked to mathematics learning so that it has a
significant role in developing students' mathematical abilities (Anderson-pence, 2015; Ismail & Ismail, 2010; Maryati & Pratiwi, 2019; Nofrianto, 2015; Risdiyanti & Prahmana, 2018).
Mathematics learning integrated with community culture promote students' abilities in exploring mathematical concepts (Brandt & Chernoff, 2015; Saldanha, Kroetz, & de Lara, 2016; Rosa &
Orey, 2017). Indeed, community culture can be utilized to support students in learning mathematics, one of which is a traditional game.
Prior studies (Riberio, Palhares, & Salinas, 2020; Nkopodi & Mosimege, 2009; Tatira, Mutambara, & Chagwiza, 2012) found that students could actively participate in learning using
traditional games to construct new knowledge by linking acquired knowledge with prior experiences. Moreover, learning with traditional games can develop students' imagination and creativity in
thinking to understand mathematical concepts independently, such as geometric shapes, patterns, and line positions (Bandeira, 2017; Fouze & Amit, 2018; Zaenuri, Teguh, & Dwidayati, 2017).
From these results, it can be concluded that learning mathematics with traditional games makes learning more meaningful and effective.
Considering the didactic aspect of traditional games in mathematics learning, the present study developed local game-based mathematics learning. The local games, called das-dasan, is one of the
traditional games in Indonesia which has didactic potential to support students learn geometry. The tenets of RME (Gravemeijer, 1994): the use of the real-world context in learning, the use of
models, students’ contributions in learning, learning activities take place interactively, and linkages between learning topics were used can encourage students to learn geometry. A number of studies
(Gravemeijer & van Eerde, 2009; Palupi & Khabibah, 2018; Shandy, 2016; Sitorus & Masrayati, 2016; Yuniati & Sari, 2018) have shown that RME help students link mathematical concepts
with real-world contexts and rediscover geometry ideas and concepts independently through students’ exploration.
Several studies (e.g., Helsa & Hartono, 2011; Jaelani, Putri, & Hartono, 2013; Nursyahidah, Putri, & Somakim, 2013) used RME with traditional games to support students learn varied
topics. Jaelani et al. (2013) utilized traditional gasing game to help students’ reinvention of time measurement historically. In the other context, Nasrullah and Zulkardi (2011) foster students'
understanding of counting using a local game called Bermain Satu Rumah. Also, Nursyahidah et al. (2013) developed learning activities to promote students’ understanding of addition up to 20 using
Dakocan game. The present study is similar to the studies above regarding the use of RME but employ different traditional games to develop students’ mathematical strategic competence in rectangle and
triangle topic. We argue that different traditional games which have didactical functions should be promoted and used in mathematics learning. Besides targeting the effectivity of instructional
practices, it also preserved the traditional games amid the massive emergence of digital games.
The present study aimed to develop students' mathematical strategic competence using the designed traditional games-based mathematics learning. Mathematical strategic competence is students’ ability
to formulate, represent, and solve mathematical problems. It is not different with problem-solving and problem formulation, which are commonly known in the literature of mathematics education
(Kilpatrick, Swafford, & Findell, 2001). Strategic competence is one of the strands of mathematical proficiency developed for a large scale research project involving students from
pre-kindergarten to grade 8. This competence is pivotal for students when they might find situations outside of school, which are needed to be formulated and solved using mathematics.
The present study followed three stages: the local game identification/analysis stage, the step of designing local game-based learning, and the implementation phase in classroom learning.
1. Local game identification and analysis stage
The first stage aimed to find out the history of the traditional game, called das-dasan, the steps of the game, and the possible implementation in mathematics learning. We observed the game and
interviewed five residents in Gebang sub-village, Sukorame village, Sukorame sub-district, Lamongan regency, Indonesia. The place is considered as the origin place of the game. The interviews were
recorded to be further analysed and compared with other available resources of the history of the game.
2. The stage of developing local game-based learning
At this stage, we designed mathematics learning for 7^th-grade students which consist of learning activities, learning tools, and the indicators of strategic competence.
Learning activities
Five tenets of RME (Gravemeijer, 1994) were used as a reference in preparing the learning activities (Table 1). The basic competence to be achieved in the learning is linking the circumference and
area for various types of rectangles (rectangles, rhombus, parallelogram, trapezoid, and kite) and triangles. In addition to the basic competence, the learning goals are the students are (1) able to
recognize and understand the types of rectangles and triangles, (2) able to name and find rectangles and triangles in the surrounding environment, and (3) able to solve the problems related to
rectangle and triangle.
Table 1. The designed learning activities
No. RME Tenets Learning steps
1 The use of real-world The teacher communicates the learning objectives and the roles of the game. The students in a group are provided with a worksheet which comprises mathematics tasks
contexts in learning about triangle and rectangle topics to be accomplished. The mathematics tasks are deliberately linked with the game.
2 The use of models Using the worksheet, students are encouraged to create pictorial representations to help solve mathematics tasks.
3 Student contributions in Students form groups of 4-5 member. Students play the das-dasan game while observing and taking notes on matters relating to the worksheet.
4 Interactive learning Students in the group discussed the mathematical ideas in the game to solve mathematics tasks in the worksheet; following this, the whole-class discussion is also
activities administered.
5 Linkages between learning Students determine the planes to solve the problems related to daily life.
Learning tools
We designed the learning tools to support the learning activities: learning plans, a test to examine students’ mathematical strategic competence, and students' worksheet, which comprise mathematics
tasks. The detail of learning plans is not presented in this article, but it fully follows the designated learning activities (Table 1). The developed test to examine students’ strategic competence
is as follows.
Arif and Hasan are playing das-dasan. The game gets exciting, uwong^2 Arif and uwong Hasan eat each other. When the game goes fun, Arif forgets that uwong (L) eats uwong (11), Arif runs uwong (Q)
going forward, then Arif got hit with Das, and as his penalty, Hasan has the right to take Arif's three uwong. Hasan could eat Arif's more uwong, Hasan took uwong (P, G, and K). Next, Hasan runs
uwong (10) eating uwong (L, M, J, and N). So far, Hasan managed to get 7 Arif's uwong consisting of 3 fines and four eating results. Based on the das-dasan game played by Arif and Hasan.
a. What plane was formed by Hasan's uwong (10)?
b. Determine and evaluate the area formed by uwong (10)!
c. In the das-dasan game arena, make a minimum of 3 different rectangular ways which has the same area as the plane formed by uwong (10).
In the students' worksheet, we developed mathematics tasks to solve by the students in the group. The tasks are to (1) draw the rectangles and triangles formed in the das-dasan game arena, (2) list
as many as rectangles and triangles found in the das-dasan game arena, and (3) formulate steps to get the number of rectangles and triangles on the das-dasan game arena.
The indicators of mathematics strategic competence
Three aspects representing the seven indicators of strategic competence (Kilpatrick et al., 2001) were coded (Table 2). It was used as a reference to determine the development of students'
mathematical strategic competence. The three aspects (formulate, represent, and solve the problems) are hierarchy in nature since every problem-solving begin with problem formulation, then
representation mediates the students to prepare strategies and solve the problem.
Table 2. The indicators of strategic competence
Strategic competence Indicators Coding
1. Students can understand the situation or context of the given problem
Formulate the problems 2. Students can find key information and ignore irrelevant ones of a problem M1
3. Students can present mathematical problems in various forms
Represent the problems 1. Students can choose the presentation that is suitable to help to solve the problem M2
2. Students find mathematical relationships that exist in a problem
Solve the problems 1. Students can choose and develop effective methods of problems solving M3
2. Students can find solutions to the given problems
3 . The implementation in classroom learning
At this stage, we acted as a teacher to teach 20 seventh-grade students using the designed learning activities in two lessons. Table 3 was used to categorize students’ strategic competence based on
the results of the test. To analyse students’ strategic competence based on the test results, we link Table 2 and Table 3 using a holistic assessment rubric. Student’s answer which fulfilled one
indicator was scored 4, then the maximum score with 7 indicators was 28. The answer that did not meet the indicator is scored 0. For the purpose of analysis, students who meet the three aspects of
strategic competence or all seven indicators are coded KSM. The students who could fulfil several indicators are coded TSM. For example, if a student meets the first aspect, which consists of two
indicators but unable to fulfil the other two aspects (five indicators), then he/she is included as TSM.
Table 3. Level of students’ strategic
Student scores Level of strategic competence
24 – 31 Very good
16 – 23 Good
8 – 15 Enough
0 – 7 Less
Findings and Discussion
In this section, we firstly provide a description of the das-dasan game and the highlight of students’ works in the group. Afterwards, we present students' achievement on strategic competence,
referring to the results of the test, following by a discussion of this study.
Das-dasan game
The results of direct observation and interviews resulted in the following basics game description. The game of das-dasan is a traditional game in the kingdom of East Java, played in pairs to train
the sharpness of thinking and set the strategy for the war. The das-dasan game has 32 uwong-uwongan consisting of 16 uwong from small pebbles and 16 uwong from large rocks. The rules in das-dasan
games are as follows.
1. Two players play the set of das-dasan
2. Before the game starts, the player first arranges the uwong-uwongan right at the intersection line of the game arena.
3. The player determines who has the right to run the uwong first in a suit.
4. Players run alternately uwong while setting strategies to be able to eat the opponent's uwong.
5. If a player forgets not to eat the opponent's uwong when given the bait, then it is said to be das so that the opponent has the right to take three uwong as he wishes (which needs to be considered
when taking three uwong, namely by thinking of a strategy so that he can eat more uwong).
6. If uwong from one of the players can enter the opponent's triangle arena and walk around the stadium three times, then uwong can become king and can walk, jump away, and eat the opponent's uwong
as desired.
7. Uwong can become king automatically if only one left.
8. Players are said to win if they can eat up the opponent's uwong.
The linkage of das-dasan games with rectangle and triangle topic can be seen in the arena of das-dasan games presented in Figure 1. In the park of das-dasan games, several lines form a rectangular
and triangular shape. Uwong, which is arranged in the arena of das-dasan games when followed by connecting one uwong with another uwong, can also form rectangular and triangular illustrations. The
purpose of the game itself, which is to train one's sharpness of thought, closely relate to the objectives of learning mathematics: Promote students' strategic competence.
Mathematics learning with das-dasan games
Before the das-dasan game begins, students make suits (Figure 2). It allows the player who wins the suit to start the das-dasan game. Figure 3 shows students made observations on the game and
exchange ideas to answer the task in the worksheet.
Figure 4 and Figure 5 shows one of the group works in identifying and determining rectangles and triangles. The first step taken by the group to find rectangles is to connect the intersection points
of lines from one location to another. It was found several rectangles that could be formed by joining several points. The group determined the triangles by observing the uwong that is being carried
out and linking the lines on the das-dasan game arena to form triangular patterns. The results of the triangles and rectangles vary, which indicate that das-dasan game promotes students' learning on
the topic.
Students’ mathematical strategic competence
The competence was measured using a test after two lessons with das-dasan games. Table 4 shows students’ scores on the test. There are 15 students (coded as KSM) who achieved all indicators.
Meanwhile, 5 students (coded as TSM) fulfilled M1 (formulating the problem) but had not fully completed M2 (representing the problem) and M3 (solving the problem).
Table 4. Students’ score in the test
Interval Frequency Percentage (%) Category
24 – 31 15 75 Very good
16 – 23 5 25 Good
8 – 15 0 0 Enough
0 – 7 0 0 Less
Total 20 100
Average 25,5 Very good
In Figure 6, KSM student correctly formulated the problem (M1). He understood the given problem and found the base and height of the plane by adding up the known ranges of the 7 cm square. Also, he
represented the problem (M2) by drawing a parallelogram and its size. Next, students solve the problem (M3) by using the formula for the area of the parallelogram and find the correct result.
In Figure 7, a TSM student correctly formulated the problem (M1), which included three indicators: understanding the context of the problem, determining appropriate, and presenting the problem
correctly. He found the base 14 cm long and 14 cm high but did not write down how to get the base and height. The student could not properly represent the problem (M2) as he drew a representation of
the parallelogram that did not fit the arena of das-dasan game. Furthermore, he had not been able to choose and develop effective problem-solving methods due to the incomplete information about the
unit of measurement and the area formula.
Figure 8 shows KSM (top) and TSM (bottom) student’s answer to point (c) of the test. KSM student made a new quadrilateral by combining eight right triangles that form a parallelogram. Furthermore,
from the mixed results of the eight triangles, an examination is conducted to ensure the rectangular shape found in the das-dasan game arena. This reveals that KSM students could formulate the
problem, represent the problem by combining small triangles to form square, trapezoid and rectangle, and answer the question. The TSM student in made quadrilateral as KSM student did. However, when
determining the third plane, he was less careful and thorough because he did not re-check the planes made so that the ways are not in the das-dasan game arena. He was only able to formulate and
represent the problem but had not yet been able to solve the problem correctly.
The represented KSM student’s work (Figure 6 and Figure 8) and our observation while he was working on the test unravel that the student was able to quickly formulated the problem by understanding
the test questions first, then look for keywords to solve the problem by making uwong to connect from one point to another and small triangles to form the desired rectangles. Furthermore, he
represented quadrilateral shapes and found the relationship between these shapes and the test questions to be completed. The student was precise in choosing the method of solving the problem. This
finding, as the previous ones (Fouze & Amit, 2018; Nkopodi & Mosimege, 2009; Tatira et al., 2012), indicates that the use of cultural-based learning activities supports students construct
mathematical knowledge. In addition, learning mathematics using traditional games allows students to be actively involved in learning.
On the other hand, the TSM student spent more time to understand the problem, improperly represented the problem, and had difficulty and was less precise in determining problem-solving strategies
which affect the final result. We observed that the student experienced misconceptions shown in the results of drawing the ladder which is not in accordance with the estimation (length is more than
height) and does not match the arena of the das-dasan game. Furthermore, he was inaccurate in writing the steps of problem-solving with words. Prior studies (Arifin & Surya, 2019; Sigit, Utami, &
amp; Prihatiningtyas, 2018) also show that students make errors in strategic competence since they are not able to understand the problem commands (concept errors), determine ideas to represent
problems (principle errors), and be careful and precise in writing steps of problem-solving (procedural errors).
Despite the developed local game-based learning support the majority of students develop strategic competence, we argue that the two lessons are not representative enough to conclude the effectivity
of the designed learning activities. In this case, it needs to be revised to address the students’ need who have not achieved all indicators of strategic competence. Then, further empirical tryout
involving more students and lessons is certainly required.
In this study, we developed local game-based mathematics learning to develop students’ strategic competence in learning the topic of rectangle and triangle. This game can be done pratically since the
tools and materials used are easily found in the school environment. The test shows that most of the students are able to formulate, represent, and solve triangle and rectangle problem embedding in
the context of das-dasan game. However, several students are struggled with determining the mathematical ideas in the play of the game and choosing an appropriate strategy to solve the problem in the
test which hamper their ability in solving the problem. We identified errors in determining the concept, principle, and procedure as the sources of the students’ difficulty in accomplishing the last
two parts of strategic competence.
The authors thank the two anonymous reviewers and the editors for their constructive comment used for revising the article. The inconsistencies or errors found in this article remain our own.
2. Uwong is defined as a person or pawn.
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SAT Math: Average Speed (Not the "Average" of the Speeds)!
One of the most challenging concepts on the SAT Math test is average rate, also called average speed. Often found in complex word problems, this type of question is one many students are less
familiar with, so don’t get nervous if you don’t know how to approach it.
Review these important equations and look at how this concept appears on the SAT.
The first important formula to memorize is d = rt. This stands for distance = rate x time. Many students find it helpful to remember this formula as the “DIRT” formula (Distance Is Rate × Time). It
is equally acceptable to think of it as time = distance ÷ rate or as rate = distance ÷ time because these are simply rearranged versions. Often, the rate is a speed, but it could be any “something
per something.” In a word problem, if you see the word “per,” you know this is a question involving rates.
Average Rate = Total Distance ÷ Total Time
The second formula is average rate = total distance ÷ total time. This is its own special concept, and you should notice that it is not an average of the speeds. Average rate is completely different.
Look at an example question:
Question #1
Ariella drove 40 miles to see her cousin at a speed of 20 mph. The trip took Ariella 2 hours. Then, Ariella drove from her cousin’s house another 30 miles to the store at a speed of 10 mph. It took
Ariella 3 hours to arrive at the store. What was Ariella’s average speed for the trip?
The next question will require the use of both the “average rate” formula and the “DIRT” formula.
Question #2
Marion spent all day on a sightseeing trip in Tuscany. First she boarded the bus which went 15mph through a 30 mile section of the countryside. The bus then stopped for lunch in Florence before
continuing on a 3 hour tour of the city’s sights at speed of 10mph. Finally, the bus left the city and drove 40 miles straight back to the hotel. Marion arrived back at her hotel exactly 2 hours
after leaving Florence. What was the bus’s average rate for the entire journey?
Try out one more challenging question:
You could also solve this problem in other ways, including using a system of equations and substitution, but it’s nice to know that you can pick a number for the distance traveled and use it to find
the average rate for the whole journey. Be on the lookout for those trips where the distance there and back is the same.
0 0 admin http://wpapp.kaptest.com/wp-content/uploads/2020/09/kaplan_logo_purple_726-4.png admin2016-10-04 12:01:512020-09-11 20:42:41SAT Math: Average Speed (Not the "Average" of the Speeds)! | {"url":"https://wpapp.kaptest.com/study/sat/sat-math-average-speed-not-the-average-of-the-speeds/","timestamp":"2024-11-05T09:02:11Z","content_type":"text/html","content_length":"193298","record_id":"<urn:uuid:f50157c4-6900-40bf-9e9e-9c1b76afdacf>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00682.warc.gz"} |
How to Calculate Child Support.
Child support is money paid by non-custodial parent to the custodial parent for the purpose of providing financial support to a child or children.
The foundation and goal of child support are to divide the costs associated with raising a child or children between the parents.
Formula to calculate child support.
There are many formulas that can be used to calculate child support and they include:
• The Income Shares Model. It is based on the assumption that children should receive the same financial support from their parents as they would have enjoyed if their parents had remained
• The Melson Formula. This formula incorporates additional factors and expenses, many of them designed to take parents’ financial needs into consideration as well.
• The Percentage of Income Model. This formula uses either a flat or adjusted percentage of just the non-custodial parent’s income.
• Parental Income. Here, the parents are required to provide courts with copies of their most recent pay stubs etc.
In our case, we will use the income shares model.
Suppose it the cost to raise a child in a particular jurisdiction is $1,000 per month, if the mother earns $ 6000, and the father earns $ 4000. Calculate the amount each parent is supposed to pay as
child support.
Since the mother earns 60% of the combined incomes therefore, the father earns 40% of the combined incomes.
To calculate the child support for each amount we multiply the cost to raise the child by the percentage of each parent.
= 60% x 1,000
= 600
= 40% x 1,000
= 400
Therefore, the mother will contribute $ 600 and the father $ 400. | {"url":"https://www.learntocalculate.com/calculate-child-support/","timestamp":"2024-11-08T19:20:48Z","content_type":"text/html","content_length":"56932","record_id":"<urn:uuid:e2b4757d-ab61-406d-a6fb-8398bb967679>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00434.warc.gz"} |
Refining a dissertation topic
Ross Woods, 2020
Having a dissertation idea is one thing; refining it is another. The idea will normally change several times, whether it is expressed as a problem, a question, an hypothesis, or a title.
Changes are a normal part of the process. Some students feel threatened by any suggestion that their original idea is not good enough, but refining is a good thing; it means that you're learning more
about it as you go and closing in on a useful topic. For example, you start to consider other factors, assumptions, definitions, variables, “what if” scenarios, and the kinds of data that would be
needed. Beware, however, that it might be good to have multiple ideas for a topic, becuase your favorite might not work out.
Discussion greatly helps in refining a topic and feedback will help you crystallize your ideas. Take up the other students' offers to discuss it. Your supervisory committee will almost certainly want
to discuss it with you.
Put another way, you are looking at the topics that come up in research methodology textbooks:
1. The research problem. What is the actual problem you want to research? Does it need to be narrowed down?
2. Defining populations. Who are you actually researching? How can you define them to neutralize extraneous factors?
3. Defining terminology. Did you use the right word? If so, do you need to give a definition?
4. Identifying assumptions. You'll list them in your introduction if they are reasonable.
5. Considering factors over which you have no control. You might use them to qualify your conclusions or put in the list “For further research.”
The example of Sam
Sam suggested the title: “Online education might improve academic and socio-emotional outcomes of frequently relocated K-12 students compared to classroom-based schooling.” The following comments
came out of Sam's discussions:
1. It is an hypothesis, not a title.
A good hypothesis is a good place to start, better than a title. A question or a problem statement also works. A title doesn't necessarily imply the problem you want to solve or the question you
want to answer. (Expressed as a title, Sam's title would be: “The academic and socio-emotional outcomes of online education for frequently relocated K-12 students compared to classroom-based
In essence, Sam wants to compare online education with classroom-based education, but only academic and socio-emotional outcomes. This seems like a good start.
2. It is a broad generalization and Sam should significantly narrow it.
Hmmm. Let's keep going. It needs some specificity.
3. It presumes that academic and socio-emotional outomes will correlate positively.
What if they don't? It might be better to choose only either academic or socio-emotional.
An alternative might be to measure both and compare them, but then Sam would be comparing more variables. This might be more practical in a doctoral dissertation, which is big enough to include
the extra level of complexity.
Is Sam simply trying to prove that online education gets better outcomes? (Some students have such a strong a preconceived idea of what they want to find that they are at risk of circular logic.
“I believe X and want to prove X ...”)
4. “Might” is vague.
For example, if Sam found only one case of improvement, then the hypothesis would be proved correct, even if that were not so in all other cases. He could replace it with “tend to.” He'd then
only need to demonstrate a tendency, but he would probably need to quantify data in some way.
5. Would researching it be ethicalIy feasible?
We might be dealing with rather large polulations of minors. Would getting permission be difficult?
6. What if Sam finds that online education does not improve outcomes?
This would be a valid conclusion, as long as he identified other relevant factors. In other words, he'd need to “qualify” his conclusion.
It would also be a valid conclusion to find that online and classroom-based education produce the same outcomes.
7. How frequent is “frequent”?
This is not a problem as long as Same defines it. For example, it could be “twice in the last three years.”
8. K-12 is very broad.
Could different grade levels respond differently to online education? Should Sam define the polulation more narrowly?
9. It assumes that all parents are the same.
What if they respond to online education in widely different ways? Some might be supportive of online education, while others see it as a second choice. Some might encourage their children, while
others might be more ambivalent. Should Sam define the sample of parents clearly and broadly enough to represent a defined category of parents? (For example, they could be military parents,
because military families move relatively frequently.) Should Sam also identify their attitudes, both their actions and their perceptions?
10. It assumes that all students are the same.
What if they respond in very different ways? Should Sam define a sample of students broadly enough to represent a defined category of students? Would this cancel out individual differences?
Should Sam explore their attitudes in both (or either of) their actions and their perceptions?
11. It assumes that all demographics are the same.
What if different demographics respond in very different ways? Should Sam either select a defined demographic, or include multiple demographics but identify respondents according to their
demographic? If he chose the latter, he could then compare results to see whther or not they varied greatly by demographic.
12. It seems to assume that frequent relocation negatively affects students.
What if it doesn't? What if it actually creates benefits, such as flexibility, resiliance. and adaptability? Could different demographic groups respond in very different ways? What if families
adjust, perhaps by having different kinds of family structures?
However, the current hypothesis does not test this. It would need to compare frequently-relocated people with non-relocated (and perhaps seldom-relocated) populations of students. It also does
not include the role of family structures.
Then Sam started to think more about academic outcomes.
1. It assumes that all classroom-based schools are the same.
Should Sam define a a category of classroom-based schools so he can choose a sample? Does this mean he will need to work with several schools?
2. It assumes that all online education is the same.
Should Sam define a particular category of online education? It could be a highly automated instruction package, or it could be something more like face-to-face classroom instruction done over
the Internet.
3. It assumes that all curricula are the same.
Should Sam define a category of curricula? Wouldn't it be even better if both populations followed the same curriculum?
4. It assumes that the implementation of curricula are the same.
Would Sam be comparing the best of classroom-based schooling with the worst of online education, or vice-versa? Even if different populations followed the same curriculum, the implementations of
curriculum could still be very different. For example, the two implementations might vary; what if one version had been used and improved for several years while the other was new and untested?
What if the online version had a novelty factor of graphics, games, and creative interactions? Perhaps they differ in other ways:
a. Theoretical assumptions of curriculum delivery or teaching.
b. Assumptions about students.
c. Sequencing of topics.
d. Time spent on each topic.
e. Choice of examples.
f. Serious flaws in program design.
5. It assumes that curricula will be implemented equally effectively across curriculum areas (mathematics, language, science, etc.).
Should Sam choose only one curriculum area (mathematics, language, science, etc.), or several and compare them?
6. It assumes that students will perform equally well across the currculum.
Should Sam choose only one curriculum area (mathematics, language, science, etc.), or several areas and compare them?
7. It assumes that all teachers have the same level of ability.
Could variations in the abilities of classroom teachers skew the results? They might be very gifted and creative, or exactly the opposite. Should Sam define a big enough population of teachers to
get a representative sample? Could Sam make the same assumptions about on-line instructors?
8. It assumes that teachers will teach equally effectively across curriculum areas.
Could it be that they tend to teach some curriculum areas better than others?
9. Could the population of online students and classroom-based students be intrinsically different?
Could their choice of education represent a pervasive difference between populations? For example, what if online students (or their parents) have a different psychological profile from those of
classroom-based students (or parents)? Online students or their parents might be “early adopters,” that is, they like to try something new and less conventional. Should Sam note this possibility
in his research?
If this could be established, it would show that comparisons are, at least to some extent, unrealistic and unfair. It would be the classic problem of comparing apples and oranges.
Next version of the topic
Sam then asks: “Does online education tend to improve socio-emotional outcomes of frequently relocated middle school students compared to classroom-based schooling?”
1. The population is children of U.S. military families who have been relocated twice in the last three years and currently in middle school.
2. All students will be studying the same curriculum.
3. The implementations of curriculum are assumed to be radically different, and any conclusions must be qualified for this factor.
4. Etc. | {"url":"http://worldwideuniversity.org/library/refining_dissertation_topic.htm","timestamp":"2024-11-10T00:01:30Z","content_type":"text/html","content_length":"11463","record_id":"<urn:uuid:4743f65e-322f-4298-b46a-396c982d06d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00708.warc.gz"} |
Reflection Andtransmission Research Articles - R Discovery
The possibility of growing high-quality porous silicon (por-Si) multilayered structures[1,2], which can consist of hundreds of layers with different, controllable porosity [3], revealed variousnew
ways of using this material in optoelectronics and photonics. Apart from obvious applications of por-Si multilayered structures as mirrors, optical filters, etc., the most attractive goal is to
combine the photonicmanipulation with the por-Si luminescence, which, in principle, would allow the fabrication of Si-basedlasers with adjustable properties [4].The photon transport through the
por-Si multilayered structure is usually described using the transfermatrix method [5]. In this approximation it is assumed that the individual por-Si layers can be characterizedby a well defined
refractive index, which depends on porosity. The general features of the reflection andtransmission coefficients are reproduced by this method (see, e.g., the recent observation of the photonicBloch
oscillations [6]). However, even in the case when the absorption is negligible, not all interferencepeaks predicted by the transfer matrix calculations are seen experimentally, and the observed peaks
arenoticeable suppressed [6,7]. To get more insight in this phenomenon it is convenient to study some simplemultilayered structures, in particular those containing a single microcavity, where the
resonant transmit-tance peak is well separated and could be analyzed in details. Below we present the experimental data onthe transmittance spectra of several structures of this sort. The observed
suppression of the resonant peaksis then interpreted by taking into account the Rayleigh scattering of light on the microscopic structuredisorder of por-Si—the effect, which is not allowed for in the
transfer matrix calculations. | {"url":"https://discovery.researcher.life/topic/reflection-andtransmission/17037554?page=1&topic_name=Reflection%20Andtransmission","timestamp":"2024-11-12T03:12:54Z","content_type":"text/html","content_length":"332898","record_id":"<urn:uuid:80f47829-b82d-4976-9c6f-c9693093054d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00160.warc.gz"} |
Quantum Computers Could Crack Encryption Sooner Than Expected With New Algorithm
One of the most well-established and disruptive uses for a future quantum computer is the ability to crack encryption. A new algorithm could significantly lower the barrier to achieving this.
Despite all the hype around quantum computing, there are still significant question marks around what quantum computers will actually be useful for. There are hopes they could accelerate everything
from optimization processes to machine learning, but how much easier and faster they’ll be remains unclear in many cases.
One thing is pretty certain though: A sufficiently powerful quantum computer could render our leading cryptographic schemes worthless. While the mathematical puzzles underpinning them are virtually
unsolvable by classical computers, they would be entirely tractable for a large enough quantum computer. That’s a problem because these schemes secure most of our information online.
The saving grace has been that today’s quantum processors are a long way from the kind of scale required. But according to a report in Science, New York University computer scientist Oded Regev has
discovered a new algorithm that could reduce the number of qubits required substantially.
The approach essentially reworks one of the most successful quantum algorithms to date. In 1994, Peter Shor at MIT devised a way to work out which prime numbers need to be multiplied together to give
a particular number—a problem known as prime factoring.
For large numbers, this is an incredibly difficult problem that quickly becomes intractable on conventional computers, which is why it was used as the basis for the popular RSA encryption scheme. But
by taking advantage of quantum phenomena like superposition and entanglement, Shor’s algorithm can solve these problems even for incredibly large numbers.
That fact has led to no small amount of panic among security experts, not least because hackers and spies can hoover up encrypted data today and then simply wait for the development of sufficiently
powerful quantum computers to crack it. And although post-quantum encryption standards have been developed, implementing them across the web could take many years.
It is likely to be quite a long wait though. Most implementations of RSA rely on at least 2048-bit keys, which is equivalent to a number 617 digits long. Fujitsu researchers recently calculated that
it would take a completely fault-tolerant quantum computer with 10,000 qubits 104 days to crack a number that large.
However, Regev’s new algorithm, described in a pre-print published on arXiv, could potentially reduce those requirements substantially. Regev has essentially reworked Shor’s algorithm such that it’s
possible to find a number’s prime factors using far fewer logical steps. Carrying out operations in a quantum computer involves creating small circuits from a few qubits, known as gates, that perform
simple logical operations.
In Shor’s original algorithm, the number of gates required to factor a number is the square of the number of bits used to represent it, which is denoted as n^2. Regev’s approach would only require n^
1.5 gates because it searches for prime factors by carrying out smaller multiplications of many numbers rather than very large multiplications of a single number. It also reduces the number of gates
required by using a classical algorithm to further process the outputs.
In the paper, Regev estimates that for a 2048-bit number this could reduce the number of gates required by two to three orders of magnitude. If true, that could enable much smaller quantum computers
to crack RSA encryption.
However, there are practical limitations. For a start, Regev notes that Shor’s algorithm benefits from a host of optimizations developed over the years that reduce the number of qubits required to
run it. It’s unclear yet whether these optimizations would work on the new approach.
Martin Ekerå, a quantum computing researcher with the Swedish government, also told Science that Regev’s algorithm appears to need quantum memory to store intermediate values. Providing that memory
will require extra qubits and eat into any computational advantage it has.
Nonetheless, the new research is a timely reminder that, when it comes to quantum computing’s threat to encryption, the goal posts are constantly moving, and shifting to post-quantum schemes can’t
happen fast enough.
Image Credit: Google | {"url":"https://singularityhub.com/2023/10/02/quantum-computers-could-crack-encryption-sooner-than-expected-with-new-algorithm/?utm_campaign=SU%20Hub%20Daily%20Newsletter&utm_medium=email&_hsmi=276644544&utm_content=276644544&utm_source=hs_email","timestamp":"2024-11-08T21:46:37Z","content_type":"text/html","content_length":"371569","record_id":"<urn:uuid:56cf9a67-9634-4df7-a34b-2f6d60582888>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00208.warc.gz"} |
QAM Symbol Rate and Bandwidth Calculator
This tool calculates both the symbol rate and bandwidth for a Quadrature Amplitude Modulation (QAM) signal.
Inputs required:
• Modulation order
• Bit rate
QAM is a popular modulation format that’s used in many modern waveforms such as 802.11 (Wi-Fi).
In a 16 QAM system, there are 16 points, each representing a unique data symbol. In this case, each symbol is composed of 4 bits.
As the modulation order increases, the number of bits per symbol increases. The constellation points get closer to each other and the probability of error for the same level of noise increases.
Higher order modulations are more susceptible to the effects of noise and interference. In order to maintain a fixed bit error rate, it takes a higher signal-to-noise ratio for 64 QAM than for 8 QAM.
64 QAM Bandwidth Calculation
In the case of 64 QAM, there are 6 bits per symbol. Consider a bit rate of 100 Mbit/s. To transmit this information, 16.7 MHz of bandwidth is required.
256 QAM Bandwidth Calculation
When using 256 QAM, there are 8 bits per symbol. For the same bit rate of 100 Mbit/s, only 12.5 MHz of bandwidth is required.
Increasing the modulation order reduces the bandwidth requirements. However, as mentioned earlier, the SNR requirements increase as well.
In general, designing a communication system to maintain high SNR can be challenging. For instance, the receiver has to be designed to minimize the noise floor.
As well, there are other factors such as:
• operating frequency
• propagation considerations
• antenna design
• transmit power
that all need to be taken into account to maximize the signal received. Check out our antenna range calculator to understand how these factors influence the level of the received signal. | {"url":"https://3roam.com/qam-symbol-rate-and-bandwidth-calculator/","timestamp":"2024-11-05T04:30:48Z","content_type":"text/html","content_length":"191584","record_id":"<urn:uuid:75cd98fe-860a-4214-9aa9-710e348de0d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00739.warc.gz"} |
Triple Integral Calculator
Understanding Triple Integrals
A triple integral is a type of multiple integration, which involves the evaluation of three integrals together to calculate an area or volume of a 3-dimensional space region. Due to its complexity,
multiple and double integrals are commonly used in physics and mathematics for calculating areas, volumes, masses, heat fluxes and other values from the integration of functions. What is a triple
integral calculator?
What is a triple integral?
A triple integral is an integral that integrates three variables in two or three dimensional space. It can be written as the product of three individual integrals: one for each variable.
The overall result of the triple integral is a volume within the three-dimensional space described by the function.
Why are triple integrals used?
Triple integrals are used for calculating areas, volumes, surface areas, average values and more. They are also used to find mass distributions and concentrations inside various shapes.
Additionally, triple integrals can be used to solve problems involving multivariable functions.
How are triple integrals different from double integrals?
Double integrals involve only two variables and integrate over a two-dimensional space region. On the other hand, triple integrals integrate over a three-dimensional space region and involve three
This means that unlike double integral calculations, you will need to pay attention to all three variables when computing a triple integral.
Exploring the concept of volume in triple integrals
When using a triple integral, there is no need to explicitly calculate the volume associated with it. Instead, we can use this concept to determine the limits of our integration.
For instance, if we wanted to calculate the volume contained within a cylindrical coordinate system (with radius r and height h), then our limits would be given by 0 ≤ r ≤ h. This means that
integrating across all possible values of r between these limits would give us our final result – the total volume enclosed within our cylinder.
Calculating Triple Integrals
The order of integration in triple integrals
There are several rules governing how you should approach calculating a triple integral. One rule to remember is regarding the order in which you should integrate your variables – starting with
radial distance (r) first before moving onto azimuthal angle (θ) and finally polar angle (φ).
This will help you ensure that your calculation produces an accurate result.
Step-by-step process for evaluating triple integrals
1. Determine all relevant limits of integration:
• (a): Define lower limit (lowlim) and upper limit (uplim) for radial distance (r)
• (b): Define lower limit (lowlim) and upper limit (uplim) for azimuthal angle (θ)
• (c): Define lower limit (lowlim) and upper limit (uplim) for polar angle (φ).
2. Write out your triple integral expression:
• (a): Begin by writing out the outer integral, which is always the radial distance (r).
• (b): Next, write out the middle integral for azimuthal angle (θ).
• (c): Finally, write out the innermost integral for polar angle (φ).
3. Calculate your triple integral:
□ (a): Evaluate your triple integral one variable at a time. This means first integrating across all possible values of r between your defined limits.
□ (b): After this is done, calculate the second integration over all possible values of θ between your defined limits.
□ (c): Finally, evaluate the final integration over all possible values of φ between your defined limits.
Using a Triple Integral Calculator
Introduction to online triple integral calculators
Online triple integral calculators are powerful tools that can help you easily and quickly compute complex multiple and double integrals. They use numerical methods and algorithms to accurately
approximate the solutions to multiple integrals with just a few clicks.
These tools are free, easy to use and can save you hours or even days of manual calculations.
Benefits of using a triple integral calculator
Using an online triple integral calculator allows you to save time and effort when calculating complex integrals. With a few simple inputs, you can solve problems involving multiple and double
integrals quickly and efficiently.
Moreover, these tools give accurate approximations of solutions so that you have more confidence in your results.
How to use a triple integral calculator effectively
To get the most out of an online triple integral calculator, it is important to know how each tool works so that you can customize your input data accordingly. Generally speaking, most calculators
require you to enter the function of interest, its lower limit(s) and upper limit(s), as well as any additional variables that may be required depending on the type of calculation being performed.
Recommended triple integral calculator tools and resources
For those interested in using an online triple integral calculator, there are plenty of great tools available today like the Allmath’s Triple Integral Calculator. Some other popular ones include
Wolfram Alpha’s Triple Integral Calculator, Webmath’s Triple Integral Calculator and Integrahound’s Free Online Triple Integral Calculator.
All three of these tools are free to use and provide easy-to-understand instructions on how to set up your calculation correctly.
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Grades 2–3
Lessons for Introducing Multiplication, Grade 3 is a revision that replaces Multiplication, Grade 3, the Math By All Means unit I wrote in 1991 that has been used by more than 85,000 teachers. Over
the years since I wrote the original unit, I’ve learned a good deal more about teaching multiplication to third graders from…
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Grades 2–3
This lesson is from Marilyn Burns’s book Teaching Arithmetic: Lessons for Introducing Multiplication, Grade 3 (Math Solutions Publications, 2001), a revision of the Math By All Means: Multiplication,
Grade 3 unit she wrote in 1991. This book presents five completely new whole-class lessons plus five new lessons in the Additional Activities section. Also, four children’s…
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Grades 2–3
Each two-page spread in Cheryl Nathan and Lisa McCourt’s book The Long and Short of It (BridgeWater Books, 1998) shows two animals and compares the size of some part common to both of them by
comparing each part to an everyday object. For example, the chameleon’s tongue is described as being “longer than a fire…
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Grades 2–3
Children are surrounded by things containing numbers — license plates, addresses, room numbers, shoe sizes, signs, even telephone numbers. One way that students can develop number sense is to think
about the numbers they encounter in the everyday world. This lesson by Bonnie Tank and Lynne Zolli provides a playful experience with numbers that are…
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Grades 2–3
Students benefit from repeated practice with addition and subtraction throughout the year. In her book, Third-Grade Math: A Month-to-Month Guide (Math Solutions Publications, 2003), Suzy Ronfeldt
provides a midyear perspective on providing practice, suggesting fresh approaches to computing with larger numbers that are suitable for older students as well. The problems are useful not only…
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Grades 2–3
The metric system is particularly easy to work with since its units relate to each other in the same way that units in place value relate to each other: powers of ten. This activity helps make that
connection for students. Here students compare centimeter cubes, decimeter rods, and meter sticks and find all the ways…
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Grades 2–3
Vocabulary instruction is a large part of geometry instruction throughout the elementary grades. To learn geometric terms and their meanings, students need opportunities to interact with and use the
language of geometry. In this lesson, Maryann Wickett used the experience of making tangrams as an opportunity to help a class of third graders expand their…
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Grades 2–3
This activity, Looking at Data, is excerpted from Mini-lessons for Math Practice, Grades 3–5, by Rusty Bresser and Caren Holtzman (Math Solutions Publications, 2006). The book presents ideas for
providing opportunities for students to practice the things they have learned, with practice defined broadly to include understanding as well as skill. In this instance, students…
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Grades 2–3
It’s important to make connections among the different areas of mathematics, and this lesson presents an addition problem in a geometric context that is appropriate for third graders. The problem
also is good for supporting mental computation and for giving children experience with a math problem that has more than one solution. The idea for…
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Grades 2–3
Along with teaching students how to use ordered pairs of numbers as coordinates to plot points, this lesson gives students background understanding about our system of graphing and helps them see how
axes — intersecting perpendicular number lines — make it possible to locate points anywhere on a plane. The activity appears in Maryann Wickett,…
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Grades 2–3
I prepared a baggie for each pair of children, each with one hundred identical objects such as cubes, milk jug lids, pennies, beans, tiles, and so on. For several days, the children used these
materials to build different-size arrays. For instance, I’d ask them to build a row of four, six times, and walk around…
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Grades 2–3
Overview of Lesson This lesson is a math variation of the popular 20 questions game. The teacher chooses a secret number on the 1–100 chart. Students ask 20 questions to try to ascertain the secret
number. Students mark their 1–100 charts to keep a visual record of information they have gathered and to see the…
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Grades 2–3
Bruce Goldstone’s book Ten Friends (Henry Holt, 2001) uses rhymes and illustrations to suggest different ways to invite ten friends to tea. At the back, he lists all of the ways to represent ten
using two addends, three addends, and so on up to ten addends. Marilyn Burns read the story to second graders and…
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Grades 2–3
When I first saw a copy of It All Adds Up!, by Australian teacher Penny Skinner, I began reading it eagerly. I was searching for ways to teach arithmetic with the same excitement I had for the other
areas of the math curriculum. In the introduction, Penny explains that her book explores teaching strategies for…
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Grades 2–3
This game gives children practice with adding and subtracting ones and tens. Using a special die, two 0–99 charts, and two markers, children play in pairs. During the course of a game, they calculate
between 20 and 30 addition and subtraction problems. The Game of Tens and Ones appears in Maryann Wickett and Marilyn Burns’s…
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Grades 2–3
Second grade students use 3-digit numbers and their understanding of place value in this game. The key to learning mathematics is understanding the “why” behind the “how”. HMH Into Math emphasizes
the importance of establishing conceptual understanding and reinforces that understanding with procedural practice. The learning model asks students to first develop their reasoning before
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In this partner game, third grade students use their multiplication foundation to solve for the area of rectangles. The key to learning mathematics is understanding the “why” behind the “how”.
HMH Into Math emphasizes the importance of establishing conceptual understanding and reinforces that understanding with procedural practice. The learning model asks students to first develop their…
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Grades 2–3
Hand Spans uses a measurement activity to give students experience with the grouping model of division and practice with rulers and tape measures. The students measure their hand span and the length
of their arm, and then figure out how many of their hand span lengths are in their arm length. This lesson appears in…
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We’re excited about our newest Math Solutions publication, Getting Your Math Message Out to Parents, by Nancy Litton. Nancy is a classroom teacher with almost thirty years of experience as well as a
Math Solutions instructor. She’s thought a great deal about how to bridge the gap between home and school and knows that teachers…
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As part of their classroom routine, Bonnie Tank and Lynne Zolli regularly ask children to figure out answers to questions like “How many more?,”“How many less?,” and “What’s the difference?” The
Game of More provides a context for asking these questions. This card game gives children practice with basic facts and with adding and subtracting…
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In this lesson, excerpted from Maryann Wickett and Marilyn Burns’s new book, Teaching Arithmetic: Lessons for Extending Place Value, Grade 3 (Math Solutions Publications, 2005), children use base ten
blocks to cement their understanding of how ones, tens, and hundreds relate to our number system. Before class, I gathered the base ten blocks and enough…
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Grades 2–3
In this lesson, students explore halves, looking for patterns between numerators and denominators. Maryann Wickett created this simple yet powerful fractions lesson and then built on it, doing an
activity from Marilyn Burns’s Teaching Arithmetic: Lessons for Introducing Fractions, Grades 4–5 (Math Solutions Publications, 2001). My-third grade students had experience using fraction kits to
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This excerpt is from the introductory lesson in Maryann Wickett, Susan Ohanian, and Marilyn Burns’s book, Teaching Arithmetic: Lessons for Introducing Division, Grades 3–4 (Math Solutions
Publications, 2002). This book is a revision of the popular Math By All Means unit Division, Grades 3–4, and this lesson is one of the new additions. The context…
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Grades 2–3
Estimation Jar is excerpted from Susan Scharton’s book, Teaching Number Sense, Grade 2 (Math Solutions Publications, 2005), part of a three-book series for grades K, 1, and 2 that focuses on the
critical role number sense plays in students’ math learning. This lesson is one in a series of estimation activities that Susan includes in…
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Dividing a number into equal-size groups with remainders is the main focus of Stuart J. Murphy’s book Divide and Ride (Harper Trophy, 1997). In this story, eleven friends go to a carnival. When they
must get into groups of two to ride the roller coaster, groups of three for the satellite wheel, and groups of…
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Grades 2–3
Algebra Content Standards • Create numeric patterns that involve whole-number operations. (3-3.1) • Apply procedures to find missing numbers in numeric patterns that involve whole-number
operations. (3-3.2) • Illustrate situations that show change over time as increasing. (3-3.4) Lesson Process Standards • Generate descriptions and mathematical statements about relationships.
(3-1.4) …
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This lesson appears in Bonnie Tank and Lynne Zolli’s new book, Teaching Arithmetic: Lessons for Addition and Subtraction, Grades 2–3 (Math Solutions Publications, 2001). Based on Rolf Myller’s book
How Big Is a Foot?, this lesson gives children experience with comparing quantities in the context of measuring a variety of lengths. Partners measure, record, and…
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Amanda Bean loves to count anything and everything. But sometimes she just can’t count fast enough. Her teacher tries to convince her that multiplying might help, but Amanda will have none of
it—until she has an amazing dream about free-wheeling sheep on bicycles, speed-knitting grandmas, and who knows how many long-sleeved sweaters. Only then does…
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A Lesson for Third Graders by Maryann Wickett and Marilyn Burns This lesson is excerpted from Maryann Wickett and Marilyn Burns’s new book, Teaching Arithmetic: Lessons for Extending Place Value,
Grade 3 (Math Solutions Publications, 2005). Children’s understanding of place value is key to their arithmetic success with larger numbers, and this book is important…
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Grades 2–3
In Blue Balliett’s novel Chasing Vermeer (Scholastic, 2005), Petra and Calder, the main characters, are in the same class but barely know each other. Their friendship grows, however, and they work
together to recover a stolen painting—a valuable Vermeer. Pentominoes are included in the clues they need to decode. Maryann Wickett uses this book as a…
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Grades 2–3
The need to interpret data accurately looms large in today’s world. By modeling ways to gather, represent, and interpret data, teachers can make young children feel more comfortable in this arena;
children can then do these activities independently. Categorical Data Collection appears in the “December” chapter of Nancy Litton’s new book, Second-Grade Math: A Month-to-Month…
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Grades 2–3
A Lesson with Second Graders by Linda Dacey and Rebeka Eston Salemi All teachers have students with a range of mathematical abilities and understandings in their classrooms. In this lesson on
estimation and measurement, the teacher differentiates three aspects of the curriculum—content, process, and products. This lesson is excerpted from Math for All: Differentiating Instruction,…
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Dana Islas is a Math Solutions consultant, kindergarten teacher at Pueblo Gardens Elementary School in Tucson, Arizona, recipient of the Presidential Award for Excellence in Mathematics and Science
Teaching, and author of the multimedia resource How to Assess While You Teach Math: Formative Assessment Practices and Lessons, Grades K–2. Overview of Lesson This lesson is…
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How to calculate p value in Google Sheets - Docs Tutorial
How to calculate p value in Google Sheets
P-value is considered to be one of the most important concepts in statistics. Most scientists rely on the p-value when working on research projects. This value is used to check whether some
hypothesis is correct or not. Scientists typically choose a range of values that express the normal results when data is not co-related. The p-value enables scientists to know how close they are to
the results.
Calculating p-value manually
Follow these steps:
1. Find the expected results for the experiment you are doing
2. Calculate, then find out the results you have observed from the experiment. Determine how much deviation from the expected results counts as significant, the degree of freedom
3. Using a chi-square, compare the first expected and observed results.
4. Find the significance level
5. Using the chi-square, approximate your p-value
Using Google Sheets enables you to avoid the risk of getting wrong results or using false calculations. You can double-check if you have the correct values or see whether you have used the right
formulas. Using this method requires calculating and considering doing it with a pen or paper.
Using Google Sheets
First, make two sets of data, compare the data sets, and check whether there is a statistical significance between the two. Using an example, let’s compare data for a certain coach. This is about the
client’s number of pushes and pull-ups that the coach provided. The T- TEST function will compare the two sets of data. The syntax of the function is;
TTEST (arrau1, array2, tails, type) but you use; T.TEST (array1, array 2, tails, type) too. Both of these syntaxes refer to one function.
Array 1 represents the first set of data, i.e., the pushes for the client
Array 2 represents the second set of data, i.e., the pull-ups for the client.
Tails are the number of tails that are used for distribution. There are two options for this; 1-one – tailed distribution 2-two- and tailed distribution.
A type is an integer number that can be paired with a 1, 2, or 3 variance T-Test. Follow the simple steps now that you know the meaning of the elements in the formula.
1. Select a column that you want to calculate the p-Value
2. Name the column you have chosen TTEST and display functions
3. Select the empty column that you want p- values to be shown, then type in the formula you need
4. In this case, enter =TTSET (A2:A7, B2:B7, 1, 3). A2:A7 is the starting and end of your first column. Hold the cursor in the first position, A2, then drag this to the bottom of the column
5. Google Sheets automatically updates your formula
6. Put a comma to your formula, then follow the same steps for the second column.
7. Enter the tails and type elements, separating them with commas
8. Press Enter
The results will appear in the column where you have typed the formula.
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How do you find the equation of the tangent and normal line to the curve
y=x^(1/2) at x=1? | HIX Tutor
How do you find the equation of the tangent and normal line to the curve #y=x^(1/2)# at x=1?
Answer 1
We find the tangent line to a curve by finding the derivative first.
#d/dx(x^n)=nx^(n-1)# given that #n# is a constant.
Let's now find the y value when #x=1#
We can now use these information to find the equation of the tangent line using the formula #m(x-x_1)=y-y_1#
That is the equation of the tangent line at the point #(1,1)#
A normal line is perpendicular to the tangent line.
We can find the slope of the normal line by finding the negative reciprocal of the slope of the tangent line.
We use the formula #m(x-x_1)=y-y_1# once again.
That is the equation of the normal line at the point #(1,1)#
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Answer 2
To find the equation of the tangent line to the curve ( y = \sqrt{x} ) at ( x = 1 ), you first need to find the slope of the tangent line by taking the derivative of the curve at ( x = 1 ). Then, you
can use the point-slope form of the equation of a line to find the equation of the tangent line. Similarly, to find the equation of the normal line, you use the negative reciprocal of the slope of
the tangent line. Here are the steps:
1. Find the derivative of the curve ( y = \sqrt{x} ).
2. Evaluate the derivative at ( x = 1 ) to find the slope of the tangent line.
3. Use the point ( (1, \sqrt{1}) = (1, 1) ) and the slope found in step 2 to write the equation of the tangent line using the point-slope form.
4. To find the equation of the normal line, take the negative reciprocal of the slope found in step 2 and use the same point ( (1, 1) ) to write the equation using the point-slope form.
After these steps, you will have the equations of both the tangent and normal lines to the curve ( y = \sqrt{x} ) at ( x = 1 ).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Identify Statistical Questions Worksheets [PDF] (6.NS.C.8): 6th Grade Math
For example:
Which of these are statistical questions?
How long does it take middle school students to get to school?
How did Kali get to school on Monday?
How many students rode the school bus today?
What percent of students are usually late to school?
A statistical question can be answered by collecting data that varies.
Statistical questions: 1 and 4
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Essential supremum and essential infimum
In mathematics, the concepts of essential supremum and essential infimum are related to the notions of supremum and infimum, but adapted to measure theory and functional analysis, where one often
deals with statements that are not valid for all elements in a set, but rather almost everywhere, i.e., except on a set of measure zero.
Let f : X → R be a real valued function defined on a set X. A real number a is called an upper bound for f if f(x) ≤ a for all x in X, i.e., if the set
is empty. Let
be the set of upper bounds of f. Then the supremum of f is defined by
if the set of upper bounds is nonempty, and sup f = +∞ otherwise.
Now assume in addition that (X, Σ, μ) is a measure space and, for simplicity, assume that the function f is measurable. A number a is called an essential upper bound of f if the measurable set f^−1(a
, ∞) is a set of measure zero,^[lower-alpha 1] i.e., if f(x) ≤ a for almost all x in X. Let
be the set of essential upper bounds. Then the essential supremum is defined similarly as
if , and ess sup f = +∞ otherwise.
Exactly in the same way one defines the essential infimum as the supremum of the essential lower bounds, that is,
if the set of essential lower bounds is nonempty, and as −∞ otherwise.
On the real line consider the Lebesgue measure and its corresponding σ-algebra Σ. Define a function f by the formula
The supremum of this function (largest value) is 5, and the infimum (smallest value) is −4. However, the function takes these values only on the sets {1} and {−1} respectively, which are of measure
zero. Everywhere else, the function takes the value 2. Thus, the essential supremum and the essential infimum of this function are both 2.
As another example, consider the function
where Q denotes the rational numbers. This function is unbounded both from above and from below, so its supremum and infimum are ∞ and −∞ respectively. However, from the point of view of the Lebesgue
measure, the set of rational numbers is of measure zero; thus, what really matters is what happens in the complement of this set, where the function is given as arctan x. It follows that the
essential supremum is π/2 while the essential infimum is -π/2.
On the other hand, consider the function f(x) = x^3 defined for all real x. Its essential supremum is +∞, and its essential infimum is −∞.
Lastly, consider the function
Then for any , we have and so and ess sup f = +∞.
• If we have . If has measure zero and .^[1]
• whenever both terms on the right are nonnegative.
See also
1. ↑ For non measurable functions the definition has to be modified by assuming that is contained in a set of measure zero. Alternatively one can assume that the measure is complete
1. ↑ Dieudonne J.: Treatise On Analysis, Vol. II. Associated Press, New York 1976. p 172f.
This article incorporates material from Essential supremum on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Vol. 29, No. 3, 2003
HOUSTON JOURNAL OF
Electronic Edition Vol. 29, No. 3, 2003
Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), J. Damon (Chapel Hill), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), J. Hausen
(Houston), J. A. Johnson (Houston), J. Nagata (Osaka), V. I. Paulsen (Houston), G. Pisier (College Station and Paris), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)
Houston Journal of Mathematics
B. Banaschewski and J.J.C. Vermeulen, University of Cape Town, Rondebosch, South Africa.
On the Booleanization of a Finite Distributive Lattice, pp. 537- 544.
ABSTRACT. It is shown that there is no functor on the category of finite distributive lattices L which assigns to each L the Boolean algebra of its elements which are equal to their double
pseudocompletement. In addition, several other results are derived from this, including the fact, obtained via Priestly Duality, that there is no functor on the category of finite partially ordered
sets X taking each X to the set MaxX of maximal elements. The basic tool here is the observation that there is no covariant endofunctor T of the category of finite sets for which T0 = 1 and TE = E
for all E not equal to 0 .
C.J. Maxson, Department of Mathematics, Texas A&M University, College Station, TX 77843 (cjmaxson@math.tamu.edu).
Forcing Linearity Numbers for Nonsingular Modules Over Semiprime Goldie Rings , pp. 545-551.
ABSTRACT. In this paper we complete the determination, initiated by Albrecht and Hausen, of the forcing linearity numbers of nonsingular modules over semiprime Goldie rings.
Lutz Strüngmann, University of Essen, 45117 Essen, Germany (lutz.struengmann@uni-essen.de).
On the existence of rings with prescribed endomorphism group, pp. 553-557.
ABSTRACT. If R is a unital ring, then the left multiplications by elements of R obviously form endomorphisms of the additive group of R. In fact they form a group direct summand of the endomorphism
ring and if the complement is trivial, then these rings are called E-rings which are well-studied. In this note we show how results on realizing rings as endomorphism rings of abelian groups can be
carried over to results on endomorphism groups of rings. In particular we construct in Goedel's universe (V=L) almost-free commutative unital rings S which have a given suitable ring R as the
complement of the left multiplications. Furthermore, a lot of ZFC results follow without "almost-free". Finally, we show that in ZFC there exists an almost-free ring R of minimal uncountable
cardinality such that the endomorphism ring of R is isomorphic to the direct sum of the integers and R itself.
Dobbs, David E., University of Tennessee, Knoxville, TN 37996-1300 (dobbs@math.utk.edu), and Picavet, G., Universite Blaise Pascal (Clermont II), 63177 Aubiere Cedex,France
Weak Baer Going-Down Rings, pp. 559-581.
ABSTRACT. Let A be a commutative ring with identity. A is said to be a going-down ring (resp., universally going-down ring) if A/P is a going-down domain, (resp., a universally going-down domain) for
each P in Spec(A). Also, A is said to be an EGD ring (resp., EUGD ring) if the inclusion map from A to B satisfies GD (resp., is universally going-down) for each overring B of A. The concept of
going-down ring (resp., universall going-down ring) is not equivalent to the concept of EGD ring (resp., EUGD ring). A is a going-down ring (resp., universally going-down ring) if and only if the
weak Baer envelope of the associated reduced ring of A is a going-down ring (resp., a universally going-down ring). A weak Baer ring is a going-down ring (resp., a universally going-down ring) if and
only if it is an EGD (resp., EUGD) ring. The weak Baer going-down (resp.,universally going-down) rings are characterized as the EGD (resp., EUGD) rings whose total quotient ring is von Neumann
regular. A weak Baer ring A is a universally going-down ring if and only if A' (the integral closure of A) is a Prufer ring such that the inclusion map from A to A' is universally going-down.
Transfer results include the facts that if a commutative faithfully flat A-algebra B is a weak Baer going-down (resp., universally going-down) ring, then A is also a weak Baer going-down (resp.,
universally going-down) ring.
Tsiu--Kwen Lee, Department of Mathematics, National Taiwan University, Taipei 106, Taiwan (tklee@math.ntu.edu.tw) and Tsai-Lien Wong, Department of Applied Mathematics, National Sun Yat-Sen
University, Kaohsiung 804, Taiwan (tlwong@math.nsysu.edu.tw)
On Armendariz Rings, pp. 583-593.
ABSTRACT. In this paper we are concerned with the connections among (weak) Armendariz rings, reduced rings and semiprime right Goldie rings. We construct certain Armendariz rings and prove that a
semiprime right Goldie ring is weak Armendariz if and only if it is a reduced ring.
H.H. Brungs, Department of Mathematical Sciences, University of Alberta,Edmonton, Alberta T6G 2G1, Canada (hbrungs@math.ualberta.ca), H. Marubayashi, Department of Mathematics, Naruto University of
Education, Naruto, 772-8502, Japan (marubaya@naruto-u.ac.jp) and A. Ueda, Department of Mathematics, Shimane University, Matsue, 690-8504, Japan} (ueda@math.shimane-u.ac.jp.
A Classification of primary ideals of Dubrovin valuation rings, pp. 595-608.
ABSTRACT. Let R be a Dubrovin valuation ring of a simple Artinian ring Q, that is, R is a Bezout order in Q and R/J(R) is simple Artinian. Primary ideals of R are classified by using prime segments,
the orders of ideals, and the concept of divisorial ideals. There is a special class of primary ideals of R which does not occur in commutative valuation rings, even in P.I. Dubrovin valuation rings.
Some examples of total valuation rings containing primary ideals in this special class are given.
G. P. Leonardi, Dipartimento di Matematica, Università di Trento, 38050 Povo-Trento, Italy (gippo@science.unitn.it), and S. Rigot, Université de Paris-Sud, Mathématiques, Bâtiment 425, 91405 Orsay
cedex, France (Severine.Rigot@math.u-psud.fr).
Isoperimetric sets on Carnot groups, pp. 609-637.
ABSTRACT. We prove the existence of isoperimetric sets in any Carnot group, that is, sets minimizing the intrinsic perimeter amoung all measurable sets with prescribed Lebesgue measure. We also show
that, up to a null set, these isoperimetric sets are open, bounded, their boundary is Ahlfors-regular and they satisfy the condition B. Furthermore, in the particular case of the Heisenberg group, we
prove that any reduced isoperimetric set is a domain of isoperimetry. All these properties are satisfied with implicit constants that depend only on the dimension of the group and on the prescribed
Lebesgue measure.
Johann Davidov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G.Bonchev Str. Bl.8, 1113 Sofia, Bulgaria (jtd@math.bas.bg).
Almost contact metric structures and twistor spaces, pp. 639-674.
ABSTRACT. Relations between the twistor spaces of even- and odd-dimensional Riemannian manifolds are studied in this paper (which can be considered as a continuation of the author‘s previous article
Twistorial examples of almost contact metric manifolds, HJM 28 (2002), 711-740). Several examples are considered and it is shown how the established relations can be used to describe the twistor
space of certain manifolds. Some properties of two natural almost contact metric structures on the twistor space of an odd-dimensional Riemannian manifold are considered in order to obtain examples
of almost contact metric manifolds that obey various geometric properties.
Shihshu Walter Wei, Department of Mathematics, University of Oklahoma, 601 Elm Avenue, Room 423, Norman, OK 73019-0315{wwei@ou.edu}.
The Structure of Complete Minimal Submanifolds in Complete Manifolds of Nonpositive Curvature, pp. 675-689.
ABSTRACT. We provide a topological obstruction for a complete submanifold with a specific uniform bound involving Ricci curvature to be minimally immersed in any complete simply-connected manifold of
nonpositive sectional curvature. We prove that such minimal submanifolds of dimension greater than two have only one topological end. The proof uses the Liouville theorem for bounded harmonic
functions on minimal submanifolds of this sort due to Yau (Ann. Scient. Ec. Norm.Sup. (8) (1975)), and also adapts a technique of Cao-Shen-Zhu (Math. Res. Lett.(4),(1997)) to show the existence of
nonconstant bounded harmonic functions based on the Sobolev inequality of Hoffman-Spruck (Comm. Pure and Applied Math. (XXVII), (1989)). This extends the above result of Yau. The same phenomena occur
in a wider class of n-submanifolds with bounded mean curvature in an L^n sense. By improving the techniques in Cao-Shen-Zhu, one can obtain the topological conclusion in the intrinsic settings. These
generalize and unify the structure theorems in the extrinsic settings. A number of examples sharply limit these weakening. For dimension n=2, a simple proof of a Conjecture of Schoen-Yau is given
with weakened hypotheses.
James Perry, Department of Mathematics, Mary Washington College, Fredericksburg, VA 22401 and Stephen Lipscomb, Department of Mathematics, Mary Washington College, Fredericksburg, VA 22401
The Generalization of Sierpinski's Triangle that lives in 4-space, pp. 691- 710.
ABSTRACT. In 1975-76, a generalization L(A) of the unit interval (a generalization of the quotient construction of identifying adjacent endpoints in Cantor's space) provided nonseparable analogues of
the classical Nobeling (1930) and Urysohn (1925) metric-space imbedding theorems. At first, L(A) was known only as a one-dimensional metrizable topological space --- separable for finite A, and of
weight cardinality(A) otherwise. By 1992, L(A) was imbedded into Hilbert's space. With the induced geometry, the space L(A) was exposed as the closure of a ``web-complex,'' structured pastings of
various n-webs n = 0,1,2,... An n-web L({0,1,2,...,n}) resides (naturally) in an n-simplex as the attractor of an iterated function system, the 2-web and 3-web being the well-known Sierpinski
triangle and 3D-gasket. Since the 2-web cannot be viewed (imbedded with fractal dimension preserved) in a line, and since the 3-web cannot be viewed in a plane, the obvious conjecture was that we
cannot view the 4-web in 3-space. Here, however, we construct a (fractal-dimension)-preserving isotopy of linear transformations that moves the 4-web into 3-space. As a corollary, we show that for
each n greater than 3, the n-web may be viewed in (n-1)-space.
W. Makuchowski, University of Opole, Institute of Mathematics and Informatics, Oleska 48, 45-052 Opole, Poland (mak@math.uni.opole.pl).
On Local Connectedness at a Subcontinuum and Smoothness of Continua, pp. 711- 716.
ABSTRACT. In this paper a number of results on local connectedness in the hyperspace of subcontinua of metric continua are obtained as the corollaries of the following Theorem 2:
Let X be a Hausdorff continuum, let A∈C(X) and X be locally connected at the subcontinuum A. If B∈C(X) and A⊂B, then the hyperspace C(X) is strongly locally arcwise connected at the point B.
Among others we generalize the following result of J.J. Charatonik and W.J. Charatonik: a metric continuum having the property of Kelley is smooth if it is locally connected at some point.
Luis Miguel García-Raffi, Salvador Romaguera and Enrique A. Sánchez Pérez, Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, 49071 Valencia, Spain (lmgarcia@mat.upv.es),
(sromague@mat.upv.es) easancpe@mat.upv.es
On Hausdorff asymmetric normed linear spaces, pp. 717-728.
ABSTRACT. An asymmetric norm is a nonnegative and subadditive positively homogeneous function q defined on a linear space. An asymmetric normed linear space is a pair (X,q) such tha X is a linear
space and q is an asymmetric norm on X In this paper we characterize those asymmetric linear spaces whose induced topology is Hausdorff. We also find a decomposition of asymmetric normed linear
spaces as direct sums of a Hausdorff asymmetric normed linear space and a non Hausdorff one under reasonable conditions.
Christopher Mouron Department of Mathematics and Computer Sciences, Hendrix College, Conway, AR 72032 (mouron@grendel.hendrix.edu).
Rotations of hereditarily decomposable circle-like continua , pp. 729-736.
ABSTRACT. An hereditarily decomposable circle-like continuum, not homeomorphic to the circle, that admits arbitrarily small periodic homeomorphisms semiconjugate to arbitrarily small rigid rotations
at the level of the tranche decomposition to the circle is constructed.
Edward A. Azoff, The University of Georgia, Athens, GA 30602 (azoff@alpha.math.uga.edu), Eugen J. Ionascu, Columbus State University, 4225 University Avenue, Columbus, GA 31907
(ionascu_eugen@colstate.edu), David R. Larson and Carl M. Pearcy, Texas A&M University, College Station, TX 77843 (David.Larson@math.tamu.edu) , (Carl.Pearcy@math.tamu.edu).
Direct Paths of Wavelets , pp. 737-756.
ABSTRACT. We associate a von Neumann algebra with each pair of complete wandering vectors for a unitary system. When this algebra is nonatomic, there is a norm--continuous path of a simple nature
connecting the original pair of wandering vectors. We apply this technique to wavelet theory and compute the above von Neumann algebra in some special cases. Results from selection theory and ergodic
theory lead to nontrivial examples where both atomic and nonatomic von Neumann algebras occur.
Pavol Quittner, Institute of Applied Mathematics, Comenius University, Mlynska dolina, 84228 Bratislava, Slovakia (quittner@fmph.uniba.sk).
Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, pp. 757-799.
ABSTRACT. We prove a priori bounds for solutions of superlinear parabolic problems on bounded and unbounded spatial domains. In these bounds, the norm of the solution at time t can be bounded by a
constant which depends only on the norm of the initial condition and the distance from t to the maximal existence time of the solution. Using these estimates we show that the maximal esistence time
depends continuously on the initial condition. The nonlinearities in the equations are subcritical and they may be nonlocal. Our proofs are based on energy, interpolation and maximal regularity
estimates. Optimality of our results and some open problems are discussed.
Manuel Delgado and Antonio Suárez, Dpto. Ecuaciones Diferenciales y Análisis Numérico, 41012, Univ. of Sevilla, Spain (delgado@numer.us.es), (suarez@numer.us.es).
Positive solutions for the degenerate logistic indefinite superlinear problem: the slow diffusion case , pp. 801-820.
ABSTRACT. In this work we study the existence, stability and multiplicity of the positive steady-states solutions of the degenerate logistic indefinite superlinear problem. By an adequate change of
variable, the problem is transformed into an elliptic equation with concave and indefinite convex nonlinearities. We use singular spectral theory, the Leray-Schauder degree, bifurcation and monotony
methods to obtain the existence results, and fixed point index in cones and a Picone identity to show the multiplicity results and the existence of a unique positive solution linearly asymptotically
Florica-Corina Cirstea, Victoria University of Technology, Melbourne City MC, Victoria 8001, Australia (florica@sci.vu.edu.au) and Vicentiu Radulescu,, University of Craiova, 1100 Craiova, Romania
Solutions with Boundary Blow-up for a Class of Nonlinear Elliptic Problems, pp. 821-829.
ABSTRACT. We study the existence of blow-up boundary solutions for the nonlinear logistic equation with absorption and non-negative variable potential. It is established a necessary and sufficient
condition for the existence of these solutions. This condition is formulated in terms of the first eigenvalue of the Laplace operator with Dirichlet boundary conditions on the null set of the
potential term, which plays a crucial role in our analysis. Our framework includes the critical case that corresponds to a weight vanishing on the boundary. The proofs rely essentially on the Maximum
Principle for elliptic equations, as well as on a result of Alama and Tarantello (1996) related to the Dirichlet boundary value problem for the logistic equation. | {"url":"https://www.math.uh.edu/~hjm/Vol29-3.html","timestamp":"2024-11-08T09:17:24Z","content_type":"text/html","content_length":"19686","record_id":"<urn:uuid:490ae02f-1550-4f5b-b609-02962d13c2eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00038.warc.gz"} |
How Much Do Warehouse Guard Rails Cost? | Handle It, Inc.
Guard rails are an essential safety component of any warehouse or industrial facility. They enable the creation of intelligent traffic patterns, provide employees with safe walkways, and absorb
accidental impacts from forklifts.
Several factors can affect pricing, such as materials and run length. Read on to learn more about how much warehouse safety guard rails cost and how you can reduce installation and freight fees.
How to Price Out Warehouse Guard Rails
When pricing out a guard rail run, it’s common to get a linear price estimate to obtain a rough budgetary number. However, the actual costs will change when you determine the final layout, and it’s
common for the installation price to equal the material costs.
For example, if your material costs are $10,000, then the installation would be around $10,000, with the total cost being $20,000. It’s crucial to hire local installers, as this helps to reduce crew
travel costs.
In addition to the product and labor, freight is another factor. You can assume shipping costs would be an additional 10-20% of the material costs. If you can find a guard rail in your local market,
you can potentially reduce the freight factor by 5% to 10%.
To price out linear footage, you must first consider if you plan on opting for a single high or double high guard rail.
Double High Guard Rails
A double high guard rail post is 43” in height. The rails are bolted together, yet you can modify them into a lift-out rail with an optional lift-out adapter. Alternatively, you can bolt the lift-out
adapters to the posts and slide the rails in without needing bolts.
Here are two examples of how to price a 300’ and 1,000’ linear foot double high guard rail run:
Example: Double High Guard Rail—300’
Figure out the number of posts and rails it would take to lay out a straight run of 300 feet, multiply by costs, then divide by 300. In this example, you would take 31 posts ($189 x 31) + 60 rails of
10’ ($228 x 60) = $19,539 / 300 = $65.13 per linear foot of double high guard rail.
Example: Double High Guard Rail—1,000’
If you did the same exercise for 1,000 feet, you would come up with a very close but slightly lower cost per linear foot number. So you want to estimate based on the closest number to your situation.
The math would be as follows: ($189 x 101) + ($228 x 200) = $64,689 / 1,000 = $64.68 per linear foot of double high guard rail.
Actual Linear Foot Costs of Double High 300’
Now, if you take 300’ and break it into smaller sections, your costs will increase because you add more posts. For example, if you estimated $65.13 per linear foot (based on a straight run of 300’),
but when in actuality, you broke it up into ten sections of 30’, your posts would increase to 40.
So the math would be as follows: ($189 x 40) + ($228 x 60) = $21,240 / 300 = $70.80 per linear foot of double high guard rail. The actual cost per linear foot increased by $5.67 per linear foot of
double high guard rail.
Single High Linear Feet Costs
Single high guard rail posts are 18” in height. Much like the double-height guard rail, the rails are bolted together, and you can turn them into a lift-out rail with an optional adapter. You can
also slide the rails in without bolting them by instead bolting the lift-out adapters to the posts.
Here are two examples of how to price a 300’ and 1,000’ linear foot single high guard rail run:
Single High Guard Rail—300’
The price per foot of a single high guard rail will be lower than double high because there are half the number of rails, and post costs will be less since there is less material. So for 300’ of
single high, we figure out the number of posts and rails it would take to lay out a straight run, multiply by costs, then divide by 300.
In this example, you would take 31 posts ($121 x 31) + 30 rails of 10’ ($228 x 30) = $10,591 / 300 = $35.30 per linear foot of single high guard rail.
Single High Guard Rail—1,000’
If you did the same exercise for 1,000 feet, you would come up with a very close but slightly lower number. So you want to estimate based on your situation. The math would be as follows: ($121 x 101)
+ ($228 x 100) = $35,021 / 1,000 = $35.02 per linear foot of single high guard rail.
Actual Linear Foot Costs of Single High 300’
If you take 300’ and break it into smaller sections, your costs will increase because you add more posts. For example, if you estimated $35.30 per linear foot based on a straight run of 300’ but you
broke it up into ten sections of 30’, your number of posts would increase to 40.
So here’s the math: ($121 x 40) + ($228 x 30) = $11,680 / 300 = $38.93 per linear foot of single high guard rail. The actual cost per linear foot increased by $3.63 per linear foot of a single high
guard rail.
Which Type of Warehouse Safety Guard Rail Is Right for Your Facility?
At Handle It, our single and double heavy-duty guard rail systems can withstand impacts of up to 10,000 lbs at four mph. Protect your employees and assets while decreasing the chances of an
industrial accident with our warehouse safety guard rails. | {"url":"https://www.handleitinc.com/news/how-much-do-warehouse-guard-rails-cost/","timestamp":"2024-11-15T04:41:25Z","content_type":"text/html","content_length":"116872","record_id":"<urn:uuid:649e6d34-7b85-413e-adf5-dc42d1e185c7>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00005.warc.gz"} |
Options Basics | Panoptic
Options Basics
What are Options?
Options are an agreement between two parties for the right, but not the obligation, to buy (or sell) an asset for a fixed price at a pre-determined time. Hence, each option has a
• Strike: the agreed-upon price the asset can be bought/sold for
• Expiry: a time in the future when the transaction has to occur
• Premium: the value of the option, or how much the buyer pays for the right to buy/sell the asset
Options provide traders with a versatile tool for managing risk and generating profits.
• By buying or selling options, traders can speculate in a capital efficient manner on the price movement of a security. This is done by taking a long or short position without having to own the
underlying asset.
• Options can also be used to manage portfolio risk, by hedging against potential losses in other investments.
• In addition, options can be used to generate income, through the writing of options contracts.
Because options allow traders to take on different levels of risk, and also defined-risk positions, they are a useful tool for managing financial portfolios.
How are Options Typically Priced?
The value of an option — called the premium — used to be difficult to determine.
But following the development of Nobel-prize winning mathematical models by Black, Scholes, and Merton, an optimal price could be derived in a way that buyers and sellers both agree on the option
contract's price.
So, how does it work? In the simplest case (i.e. for European options) the Black-Scholes Model (BSM) considers the following variables:
• Spot price ($S$): The current price of the underlying asset.
• Strike price ($K$): The predetermined price at which the option can be exercised.
• Time to expiration ($T$): The time remaining until the option expires.
• Volatility ($\sigma$): The annualized standard deviation of the underlying asset's returns, reflecting fluctuations in the underlying asset's price.
• Risk-free interest rate ($r$): The interest rate on a risk-free investment, typically a government bond.
Using these variables, the Black-Scholes equation calculates the fair value of an option. The equation is different for call and put options, but they share the same core components, namely:
$N(d_1)$ and $N(d_2)$: These are probability values derived from the cumulative standard normal distribution. They represent the likelihood of the option being exercised. More precisely,
• $d_1$: can be thought of as a measure of how much the option is "in the money" relative to its risk, incorporating factors such as the stock price, strike price, time to expiration, volatility,
and risk-free interest rate. A higher d1 value indicates that the option is more likely to be exercised, as the stock price is expected to be further away from the strike price. This increased
likelihood of exercise is reflected in a higher N(d1) value. The formula for $d_1$ is: $d_1 =\frac{(\ln\left(\frac{S}{K}\right) + (r + \frac{\sigma^2}{2}) \cdot T)}{\sigma \cdot \sqrt{T}}$
• $d_2$ can be understood as a measure of the expected payoff when the option is exercised, considering the time value of money. A higher $d_2$ value indicates that the option's exercise price is
more favorable compared to the current stock price and risk-free interest rate, which is reflected in a higher N(d2) value. $d_2$ is derived from $d_1$ as follows: $d_2 = d_1 - \sigma \cdot \sqrt
Present value factor ($e^{-rT}$): This factor discounts the value of the option based on the risk-free interest rate and time to expiration.
Given the discussion above, the Black-Scholes equation for call options is as follows:
• Call option value = $S \cdot N(d_1) - K e^{-rT} \cdot N(d_2)$
For put options, the equation is:
• Put option value = $K \cdot e^{-rT} \cdot N(-d_2) - S \cdot N(-d_1)$
It is worth mentioning that, while the BSM is widely used, it has some limitations:
• Assumes constant volatility: The equation assumes that the underlying asset's volatility remains constant, which is not always the case in reality.
• Ignores dividends: The original model doesn't account for dividends paid by the underlying asset, which can affect the option's value.
• Assumes European-style options: The model is based on European-style options, which can only be exercised on the expiration date. It may not be accurate for American-style options, which can be
exercised any time before expiration.
• Assumes a constant risk-free rate: The equation assumes that the risk-free rate remains constant, which is not always the case in reality.
• Assumes GBM: The model assumes the underlying asset's price follows a geometric Brownian motion, which is not always the case in reality (especially for stablecoins!).
Is Panoptic the same?
Not quite. Panoptic presents a new type of option: an oracle-free, perpetual option. By this we mean that:
• Panoptions do not have an expiration date
• The option premia is not based on pricing by market makers (who use BSM), but rather on trading activity in the underlying asset's spot market.
We will discuss these concepts in the following sections.
Beginner Tutorials
Options Trading for Beginners (The ULTIMATE In-Depth Guide) 👇
Intermediate Tutorials | {"url":"https://panoptic.xyz/docs/trading/basic-concepts","timestamp":"2024-11-07T13:50:52Z","content_type":"text/html","content_length":"61766","record_id":"<urn:uuid:f854a7a6-8be6-4d42-a58d-da208d49f818>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00620.warc.gz"} |
Class Point
Defined in file coordinate_transforms/point.py.
The Point class defines a point on the geode surface in terms of latitude and longitude.
Class Attributes
degrees2radians Internal constant used to convert degrees to radians.
R Internal constant holding radius of the Earth in metres.
latitude The latitude of the Point.
longitude The longitude of the Point.
coslat Cosine of the Point's latitude.
coslon Cosine of the Point's longitude.
sinlat Sine of the Point's latitude.
sinlon Sine of the Point's longitude.
Class Methods
BearingTo(self, P) Returns the bearing in degrees from self to P.
DistanceTo(self, P) Returns the distance from self to P in metres.
Dist(self, P) Return a cheap and cheerful approximation of the distance from self to P. Roughly the degress seperation.
GCA(self, P) Returns the great circle angle between self and P.
AZ(self, P) Returns the azimuth bearing from self to P.
This class is defined in the 'old' way. Needs to be changed to the new method before moving to python 2.6. | {"url":"https://anuga.anu.edu.au/wiki/ClassPoint?version=14","timestamp":"2024-11-07T20:25:55Z","content_type":"application/xhtml+xml","content_length":"9118","record_id":"<urn:uuid:babfe70f-ccfa-4ecc-a21c-5a2441768868>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00765.warc.gz"} |
How To Charge The Ball In Cement Mill
Ball charges calculators
- Ball top size (bond formula): calculation of the top size grinding media (balls or cylpebs):-Modification of the Ball Charge: This calculator analyses the granulometry of the material inside the
mill and proposes a modification of the ball charge in order to improve the mill efficiency:
how to charge ball in cement mill
how to charge ball in cement mill. Ball mill. A typical type of fine grinder is the ball mill.A slightly inclined or horizontal rotating cylinder is partially filled with balls, usually stone or
metal, which grind material to the necessary fineness by friction and impact with the tumbling balls.
How To Charge The Ball In Cement Mill
Ball Charge Formula In Cement Mill . cement mill ball mill charge ratio. 4 Mar 2013 One ball mill used in a cement factory,the drive arrangement is as shown Get Price. Get Price How to charge ball in
cement mill grenada Description The specific selection function effect on clinker ResearchGate Cement clinker, Ball mill, Dry grinding Get Price.
Calculate The Charging In Cement Mill Pdf-ball Mill
Calculate The Charging In Cement Mill Pdf Exodus . Calculate the charging in cement mill pdf calculate ball mill grinding capacity a total apparent volumetric charge filling including balls and
excess slurry on top of the ball charge plus the interstitial voids in between the balls expressed as a percentage of the net internal mill volume inside liners
how to charge ball in cement mill
Also, for the first time, the effects of ball charge pattern, cement fineness and two additive materials (pozzolan and limestone) on the efficiency of the CBM are investigated. Actually, the
literature . Get Price; Ball Mill Charging System - 911metallurgist. Mill ball charging systems have undergone little change during the past two decades.
how to charge the ball in cement mill
To calculate grinding media charge for continuous type ball mill, M = 0.000676 x D2 x L Example . how to calculate cement grinding mill balls charge – Mining FLshanghai ball mill for cement grinding.
cement mill is a corrugated lining designed to obtain maximum power . Ball mill grinding media charge calculation Henan Mining. Service And Support
How To Calculte Ball Charges In Cement Mill
2020-4-30Estimation Of Grinding Media Balls Charging In Cement . Calculates the grinding charge of a ball mill mining.To calculate grinding media charge for continuous type ball mill m 0.000676 x d2
x l example how to calculate cement grinding mill balls chargekefid mining flshanghai ball mill for cement grinding.Cement mill is a corrugated lining designed to obtain maximum power.
How Can I Calculate Optimum Charge For Cement Mill-ball Mill
How Can I Calculate Optimum Charge For Cement Mill . Cement mill charge calculation coal russian calculate and select ball mill ball size for optimum grinding 20191114enspenspin grinding selecting
calculate the correct or optimum ball size that allows for the best and optimumideal or target grind size to be achieved by your ball mill is an ...
ball charge for cement mills - lucanaprati.it
Cement Ball Mill Charge. Cement Ball Mill Charge Design. Cement ball mill charge design calculation of the power draw of dry multi-compartment ball mills may 6, 2004 key words power draw, cement, dry
grinding, ball mill right mill for the specified duty is the most critical for circuit design, since it has the input to the.get price
how to measure a cement finish mill ball charge
A ball mill, a type of grinder, is a cylindrical device used in grinding (or mixing) materials like ores, chemicals, ceramic raw materials and paints.
How to charge the ball in cement mill
Ball mill for dry grinding cement ball charge too coarse ball mill for dry grinding cement ball charge too coarse ball mill for dry grinding cement ball charge too coarse We build high quality robust
industrial machines used across many industries Our product line is diverse and ever growing to meet our customers demands.
Cement Tube Mill Charging Media Calculation-ball Mill
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cement mill ball charging - lineco.co.za
cement/ball mill/ball charge design in Myanmar - Gold Ore Crusher. ball mill for cement grinding. ensures optimum lifting of the mill charge. The shell lining in the second Proven mill design 6 Mill
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Cement Milling - Understanding Cement
Cement milling is usually carried out using ball mills with two or more separate chambers containing different sizes of grinding media (steel balls). Grinding clinker requires a lot of energy. How
easy a particular clinker is to grind ("grindability") is not always easy to predict, but large clusters of belite due to coarse silica in the feed ...
The mill is designed to handle a total ball charge of 324.5 t at 100% loading with a percentage filling of 29.5% in both the chambers. Both the chambers of the cement mill were charged with 80% of
the designed charge, which works out to 86 t in Ist chamber and 172 t in the IInd chamber.
4 Density Of Cement Mill Ball Charge Grinding Media
friction in cement mill grinding media cruisertrailers.co.za. Ball mill Wikipedia. A ball mill is a type of grinder used to grind and blend materials for use in mineral dressing The grinding media is
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cement ball mills charging - Hemming Way
cement ball mills charging. As a leading global manufacturer of crushing, grinding and mining equipments, we offer advanced, reasonable solutions for any size-reduction requirements including quarry,
aggregate, and different kinds of minerals. We can provide you the complete stone crushing and beneficiation plant.We also supply stand-alone ... | {"url":"http://www.automotiveplanet.eu/mobile/how-to-charge-the-ball-in-cement-mill-5997/","timestamp":"2024-11-08T18:09:33Z","content_type":"application/xhtml+xml","content_length":"13816","record_id":"<urn:uuid:fae7ef0d-6fc2-4509-afd6-ae5d344170c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00784.warc.gz"} |
Revision #2 to TR13-155 | 12th March 2014 07:53
Pseudorandom Generators for Low Degree Polynomials from Algebraic Geometry Codes
Constructing pseudorandom generators for low degree polynomials has received a considerable attention in the past decade. Viola [CC 2009], following an exciting line of research, constructed a
pseudorandom generator for degree $d$ polynomials in $n$ variables, over any prime field. The seed length used is $O(d \log{n} + d 2^d)$, and thus this construction yields a non-trivial result only
for $d = O(\log{n})$. Bogdanov [STOC 2005] presented a pseudorandom generator with seed length $O(d^4 \log{n})$. However, it is promised to work only for fields of size $\Omega(d^{10} \log^{2} n})$.
The work of Lu [CCC 2012], combined with that of Bogdanov, yields a pseudorandom generator with seed length $O(d^4 \log{n})$ for fields of size $\Omega(d^{6+c})$ -- independent of $n$, where $c$ is
an arbitrarily small constant. Based on these works, Guruswami and Xing [CCC 2014] devised a construction with a similar seed length for fields of size $O(d^6)$.
In this work we show that for any $d$, a random sub-code (with a proper dimension) of any good algebraic geometry code, is a hitting set for degree $d$ polynomials. By derandomizing this assertion,
together with the work of Bogdanov, we obtain a construction of a pseudorandom generator for degree $d$ polynomials over fields of size $O(d^{12})$, and seed length $O(d^4 \log{n})$. The running-time
of our construction is $n^{\poly(d)}$. However, the running-time can be improved to $\poly(n,d)$ assuming Riemann-Roch spaces of certain algebraic function fields are, in some sense, strongly
explicit. We believe this open problem is interesting on its own, and take a first step at affirming the conjecture.
Although quantitatively our result does not match the parameters of Guruswami and Xing, our construction is clean mathematically and conceptually simpler. We consider the proof technique to be the
main contribution of this paper, and believe it will find other applications in complexity theory. In the heart of our proofs is a reduction from the problem of assuring independence between
monomials to the much simpler problem of avoiding collisions over the integers. Our reduction heavily relies on the Riemann-Roch theorem.
Revision #1 to TR13-155 | 19th November 2013 13:35
Pseudorandom Generators for Low Degree Polynomials from Algebraic Geometry Codes
Constructing pseudorandom generators for low degree polynomials has received a considerable attention in the past decade. Viola [CC 2009], following an exciting line of research, constructed a
pseudorandom generator for degree d polynomials in n variables, over any prime field. The seed length used is $O(d \log{n} + d 2^d)$, and thus this construction yields a non-trivial result only for
$d = O(\log{n})$. Bogdanov [STOC 2005] presented a pseudorandom generator with seed length $O(d^4 \log{n})$. However, it is promised to work only for fields of size $\Omega(d^{10} \log^{2}{n})$.
The main result of this paper is a construction of a pseudorandom generator for low degree polynomials based on algebraic geometry codes. Our pseudorandom generator works for fields of size $\Omega(d
^{12})$ and has seed length $O(d^4 \log{n})$. The running time of our construction is $n^{O(d^4)}$. We postulate a conjecture concerning the explicitness of a certain Riemann-Roch space in function
fields. If true, the running time of our pseudorandom generator would be reduced to $n^{O(1)}$. We also make a first step at affirming the conjecture.
Changes to previous version:
There was a miscalculation in the claimed field size. It is O(d^12) and not O(d^6) as stated in the original version. In the second construction, the exponent in d is not 8 as claimed, but 20.
TR13-155 | 10th November 2013 14:13
Pseudorandom Generators for Low Degree Polynomials from Algebraic Geometry Codes
Constructing pseudorandom generators for low degree polynomials has received a considerable attention in the past decade. Viola [CC 2009], following an exciting line of research, constructed a
pseudorandom generator for degree d polynomials in n variables, over any prime field. The seed length used is $O(d \log{n} + d 2^d)$, and thus this construction yields a non-trivial result only for
$d = O(\log{n})$. Bogdanov [STOC 2005] presented a pseudorandom generator with seed length $O(d^4 \log{n})$. However, it is promised to work only for fields of size $\Omega(d^{10} \log^{2}{n})$.
The main result of this paper is a construction of a pseudorandom generator for low degree polynomials based on algebraic geometry codes. Our pseudorandom generator works for fields of size $\Omega(d
^6)$ and has seed length $O(d^4 \log{n})$. The running time of our construction is $n^{O(d^4)}$. We postulate a conjecture concerning the explicitness of a certain Riemann-Roch space in function
fields. If true, the running time of our pseudorandom generator would be reduced to n^{O(1)}$. We also make a first step at affirming the conjecture. | {"url":"https://eccc.weizmann.ac.il/report/2013/155/","timestamp":"2024-11-07T10:19:38Z","content_type":"application/xhtml+xml","content_length":"27189","record_id":"<urn:uuid:6cb24489-3093-49e7-87c1-e7157c0f561b>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00841.warc.gz"} |
View problem - Periodicity (POI11_okr)
Time limit Memory limit # of submissions # of accepted Ratio
1000 ms 32 MiB 8 4 50.0%
Byteasar, the king of Bitotia, has ordained a reform of his subjects' names. The names of Bitotians often contain repeating phrases, e.g., the name Abiabuabiab has two occurrences of the phrase
abiab. Byteasar intends to change the names of his subjects to sequences of bits matching the lengths of their original names. Also, he would very much like to reflect the original repetitions in the
new names.
In the following, for simplicity, we will identify the upper- and lower-case letters in the names. For any sequence of characters (letters or bits) $w=w_1w_2…w_k$ we say that the integer $p$ ($1 \le
p < k$) is a period of $w$ if $w_i=w_{i+p}$ for all $i=1,\ldots,k-p$. We denote the set of all periods of $w$ by $Per(w)$. For example, $Per(\texttt{ABIABUABIAB})={6,9}$, $Per(\texttt{01001010010})=
{5,8,10}$, and $Per(\texttt{0000})={1,2,3}$.
Byteasar has decided that every name is to be changed to a sequence of bits that:
• has length matching that of the original name,
• has the same set of periods as the original name,
• is the smallest (lexicographically [1]) sequence of bits satisfying the previous conditions.
For example, the name ABIABUABIAB should be changed to 01001101001, BABBAB to 010010, and BABURBAB to 01000010.
Byteasar has asked you to write a program that would facilitate the translation of his subjects' current names into new ones. If you succeed, you may keep your current name as a reward!
In the first line of the standard input there is a single integer $k$ - the number of names to be translated ($1 \le k \le 20$). The names are given in the following lines, one in each line. Each
name consists of no less than $1$ and no more than $200\,000$ upper-case (capital) letters (of the English alphabet).
In the test worth 30% of the points each name consists of at most $20$ letters.
Your program should print $k$ lines to the standard output. Each successive line should hold a sequence of zeroes and ones (without spaces in between) corresponding to the names given on the input.
If an appropriate sequence of bits does not exists for some names, then "XXX" (without quotation marks) should be printed for that name.
For the input data:
the correct result is:
[1] The sequence of bits $x_1 x_2 \ldots x_k$ is lexicographically smaller than the sequence of bits $y_1 y_2 \ldots y_k$ if for some $i$, $1 \le i \le k$, we have $x_i < y_i$ and for all $j=1,\
cdots,i-1$ we have $x_j = y_j$.
Task author: Wojciech Rytter. | {"url":"https://oj.uz/problem/view/POI11_okr?locale=en","timestamp":"2024-11-11T16:42:55Z","content_type":"application/xhtml+xml","content_length":"23978","record_id":"<urn:uuid:335ca4a9-4803-48f6-a303-343b62f4fa92>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00495.warc.gz"} |
Class 10 Electricity NotesPhysics Class 10 Electricity Notes & NCERT Solutions | Leverage Edu
One of the most important chapters of Class 10 Physics, Electricity explores the basics of electric power and electric current and familiarises learners with the concepts of Ohm’s Law, Joule’s Law of
Heating, and resistance, amongst others. It aims to teach the importance of electricity in modern society and its different uses in different industries. This chapter not only emphasizes what
constitutes electricity but also covers how to control it and use it better. Looking for physics notes for the Class 10 Science chapter on Electricity? Here are a complete summary and Class 10
Electricity notes to help you understand the key points and concepts covered in this chapter!
Electric Current and Circuit
Electricity is a flow of electrical charge that contains the transfer of electrons from one atom to another. This electric charge can be either positive or negative and creates an electric field.
When these electrons flow in an electrical circuit continuously, it creates an electric current. While going through our Class 10 Electricity notes, you must remember that electric current is a net
flow of electric charge that passes through a particular region and is carried by the charged particles only. In an electrical circuit, the electric current’s direction is always opposite to that of
the direction of the flow of negatively charged electrons. A circuit has a pivotal role in the same as it forms the path that lets the electricity flow from one point to another. An electric circuit
also forms a loop to enable the smooth flow of the current.
E.g., A lamp powered by a cell and connected through a cable with an on/off switch. When the switch is on, it creates a circuit and maintains the flow of charge to keep the lamp on. If the switch is
turned off, it breaks the circuit, the flow of charge stops switching off the lamp.
Electric Current (I) = Net Charge (Q)/Time (T)
I = Q/T
Further, as per the Class 10 chapter on Electricity, the SI unit of electric charge is Coulomb (C) and it is considered to be equivalent to the charge contained in nearly 6 × 1018 electrons. We
measure electric current by a unit called Ampere (A) which is named after the famous French Scientist, Andre-Marie Ampere.
A = 1 C/ 1 s, i.e. One Ampere is a total of the flow of one Coulomb of charge per second.
Micrampere (1 μA = 10^-6 A) or Milliampere (1 mA = 10^-3 A) is referred to as small quantities of current. Want to know how to measure electric current in a circuit? It is calculated by an instrument
called an ammeter. Here is a diagram of an electric circuit which is constituted of an electric bulb, a cell, a plug key as well as an ammeter. See how the electric current in the circuit goes from
the positive terminal to the negative terminal of the cell.
Electric Power and AC
Current Joule’s Law’s Heating Effect:
• Heat (H) ∝ square of the current (I).
• H ∝ Resistance of the given circuit.
• H ∝ Time (t) for which current flows through the conductor.
When a potential difference is generated, electrons travel, resulting in current flow.
Conductors and Insulators
Conductors are substances that provide comparatively less resistance to the passage of electricity, whereas insulators provide more resistance.
Electric Potential and Potential Difference
The next section you must study in Class 10 Electricity notes is what is electric potential and potential difference. An electric charge does not flow on its own and requires a medium for movement.
For maintaining the flow of charge, the electrons need the difference in electric pressure that is called a potential difference. We can refer to an electric potential difference as the total work
completed to move a unit charge from one point to another in a current-carrying electric circuit. Here is the formula to find the potential difference between the two points:
V = W/Q
Here, V = Potential Difference
W = Work Done
Q = Charge
The potential difference is measured by a voltmeter.
Circuit Diagram
To draw a circuit diagram, it is essential to know its components and their respective diagrams. This is one of the important questions asked in the Class 10 Science exam thus you must remember the
following components of the Circuit Diagram and their respective symbols:
Ohm’s Law
In 1827, George Simon Ohm, a German Scientist, explored the relationship between the current flowing through an electric wire and the potential difference. He realized that the potential difference
caused is proportional to the current flowing through an electric circuit given the temperature is the same. Thus, Ohm’s Law states that the current flowing through the ohmic conductor between its
two endpoints is proportional to the applied potential difference.
Ohm’s Law
V = IR
Here, R = resistance
V = potential difference
I = current
Factors on Which the Resistance of a Conductor Depends
The Class 10 Chapter on Electricity notes that resistance can be simply defined as the characteristic of a conductor that results in resisting the flow of electric current passing through a
conductor. A resistor is utilized to resist the flow of electric current in a circuit and practically, they are simply applied for increasing or decreasing the electric current. While going through
our Class 10 Electricity notes, you must know that the positive components of a conductor lead to a hindrance to the electron flow due to the attraction between them. Then, this hindrance is
considered as the main cause of resistance in the electricity flow.
The factors on which the resistance of a conductor depends are as follows:
1. Nature of the material
The materials which lead to the least resistance are referred to as good conductors such as silver which is often called the best conductor of electricity. On the other hand, when materials create
hindrance, they are called bad conductors or insulators like plastic.
2. Length of the conductor:
Since resistance (R) is always considered directly proportional to the conductor’s length, the increased resistance automatically results in the increased length of the conductor. This is why you
must have noticed that long electric wires have more resistance to electricity.
Resistance (R) ∝ length of conductor (l)
R ∝ l
3. Area of the cross-section:
When it comes to the area of cross-section, the resistance is always inversely proportional to it.
Resistance (R) ∝ 1/Area of a cross-section of conductor
R ∝ lA
R = ρ lA
Here, ρ (rho) is the proportionality constant and is also referred to as the electric resistivity of the conductor’s material
The SI unit of resistivity (ρ) is Ωm or Ohm Meter.
Resistance of a System of Resistors
The next concept covered in our Class 10 Electricity notes is the resistance of a system of resistors. This section explores how Ohm’s law can be applied on the combination of resistors. Here are the
two main combinations of resistors:
1. Resistors in Series
Resistors in Series are when resistors are joined together from one end to another thus forming a series. Here, the total resistance can be calculated by the sum of the resistance of all the
resistors in the series.
The potential difference between A and B here is V and the potential difference between all the resistors, i.e. R[1], R[2] and R[3] = V[1], V[2] and V[3]. I is the current flowing through the series.
V = IR[s]
2. Resistors in Parallel
In this combination, the resistors are connected in a parallel manner. In Resistors in Parellel combination, the total current I, is equal to the sum of the separate currents and let’s suppose that R
[p] equivalent resistance of the parallel combination of resistors.
I = V/R[p]
Here, V/R[p] = V/R[1] + V/R[2] + V/R[3]
1/R[p] = 1/R[1] + 1/R[2] + 1/R[3]
EMF and Terminal Voltage
• When there is no current flowing through the circuit, the potential difference between the two terminals of a cell is called EMF.
• When current is flowing through a circuit, the potential difference between the two terminals of a cell is called terminal voltage.
Heating Effect of Electric Current
The heating effect of the electric current explains that during the flow of electrical current, the energy source is required to keep expanding to maintain the flow of current. Some part of this
energy is consumed in useful work while some part is expended in the form of heat and raises the temperature. When the source energy gets dissipated continually in the form of heat, it creates the
heating effect of electric current.E.g., An electric fan becomes warm when used for a longer duration continuously. Studying this section in our Class 10 electricity notes, you must also explore how
this phenomenon is explained by Joule in terms of Joule’s law of heating. This law indicated that in terms of a given resistance, the heat produced in a resistor is always directly proportional to
the current’s square as well as it is directly proportional to the given current’s resistance.
Applications of Heating Effect on Electric Current
During the electric current, heat gets dissipated and converts the useful electrical energy into heat. This heat is dangerous for electrical circuits if it goes beyond the acceptable limit, as it
might damage the important component by raising the temperature. However, the heating effect of electric current can be useful when utilised properly. Appliances like an electric toaster, oven,
kettle, heater, iron, etc. use the Joule’s law of heating to make proper use of heat dissipated during the electric current. An electrical bulb uses the electrical heat to produce the light. A bulb’s
filament is made of tungsten with a high melting point of 3380°C and cannot be quickly melted. So, when the heat is generated during the electrical current, it absorbs the heat, gets hot and emits
the light.
Electrical circuits also use Joule’s law of heating. Going through the Class 10 Electricity notes, you must remember that an electrical circuit has a fuse component that protects the circuit in case
of any rapid increase in electric current. A fuse is made up of a low melting point alloys or metals like copper, iron, lead, etc. When the electrical current increases than the specified value, the
temperature increases and melts the fuse wire. Thus, it breaks the circuit and saves the appliance from any damage.
Electric Power
The rate at which the electric energy is dissipated in an electric current is called electric power. The SI unit of electrical power is the watt (W), and the unit of electric energy is watt-hour (W
h). Because the unit watt is relatively small in the calculation, it is often replaced by the kilowatt. One kilowatt equals 1000 watts.
Power P = VI
Or P = I2R = V2/R
Since electrical energy is a product of power and time, the commercial unit of electric energy when 1 watt of power is used for 1 hour is defined as kilowatt per hour (kW h) known as a unit.
Class 10 Electricity Important Questions & Solutions
Now that you are familiarised with the major concepts of this chapter through our Class 10 Electricity notes, here are some important questions and solved numerical for you:
• State the major factors that can affect the resistance of a conductor.
• Explain how the voltmeter measures the potential difference between two points in an electric circuit.
• Name the SI unit of electric current.
• What do you understand by conductors?
• What is Joule’s Law of Heating?
• What is Ohm’s Law?
Here are some solved examples on the aforementioned topics covered in Class 10 electricity:
Q: What will be the power of the bulb if connected to a 220V generator with a current flow of 0.50A?
A: => P = VI
=> 220V X 0.50A
=> 110 J/s
=> 110 W
Q: An electric bulb draws an electrical current of 0.5A for 10 minutes. What will be the amount of electric charge in the circuit?
A: => Q = I (Electric current) x t (Time)
=> 0.50A X 600s (Convert minutes into seconds)
=> 300C
Thus, we hope that this blog on Class 10 Electricity notes clarified all your doubts on this chapter and provided you with a comprehensive summary! Confused about selecting the right stream after
10th? Reach out to our Leverage Edu counselors and we will guide you in finding the best stream of study which aligns with your career aspirations and interests! | {"url":"https://leverageedu.com/blog/class-10-electricity-notes/","timestamp":"2024-11-11T09:34:58Z","content_type":"text/html","content_length":"345325","record_id":"<urn:uuid:9d6bcf2a-e680-4653-801c-9613139395e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00821.warc.gz"} |
nep-ecm 2024-04-08 papers
Abstract: Researchers in many fields endeavor to estimate treatment effects by regressing outcome data (Y) on a treatment (D) and observed confounders (X). Even absent unobserved confounding, the
regression coefficient on the treatment reports a weighted average of strata-specific treatment effects (Angrist, 1998). Where heterogeneous treatment effects cannot be ruled out, the resulting
coefficient is thus not generally equal to the average treatment effect (ATE), and is unlikely to be the quantity of direct scientific or policy interest. The difference between the coefficient and
the ATE has led researchers to propose various interpretational, bounding, and diagnostic aids (Humphreys, 2009; Aronow and Samii, 2016; Sloczynski, 2022; Chattopadhyay and Zubizarreta, 2023). We
note that the linear regression of Y on D and X can be misspecified when the treatment effect is heterogeneous in X. The "weights of regression", for which we provide a new (more general) expression,
simply characterize how the OLS coefficient will depart from the ATE under the misspecification resulting from unmodeled treatment effect heterogeneity. Consequently, a natural alternative to
suffering these weights is to address the misspecification that gives rise to them. For investigators committed to linear approaches, we propose relying on the slightly weaker assumption that the
potential outcomes are linear in X. Numerous well-known estimators are unbiased for the ATE under this assumption, namely regression-imputation/g-computation/T-learner, regression with an interaction
of the treatment and covariates (Lin, 2013), and balancing weights. Any of these approaches avoid the apparent weighting problem of the misspecified linear regression, at an efficiency cost that will
be small when there are few covariates relative to sample size. We demonstrate these lessons using simulations in observational and experimental settings. | {"url":"https://nep.repec.org/nep-ecm/2024-04-08","timestamp":"2024-11-05T00:06:56Z","content_type":"application/xhtml+xml","content_length":"54556","record_id":"<urn:uuid:3fa8dc34-ba6e-4233-a63e-056d1eda8f6e>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00527.warc.gz"} |
Math Warriors, Lay Down Your Weapons (Opinion)
—Peter Lui
A recent analysis of mathematics performance yields some disturbing findings about U.S. student achievement. Although previous studies had suggested that American elementary students performed
relatively well in mathematics, with older students doing less well, the new findings from the American Institutes for Research suggest that our 4th graders, too, are in the middle of the
international pack. (“Study Indicates Changes in Global Standing for U.S.,” Nov. 30, 2005.) Yet, even though our economic competitors outperform us, today’s U.S. students do better than earlier
generations of students. There has been a steady—but not steep—increase over the last two decades in mathematics performance, according to the National Assessment of Educational Progress. But we need
to aim higher still. And we must look not to our own past for better results, but to our top competitors to understand how they have been able to outperform us.
Concern over previous international comparisons has prompted many educators and policymakers to launch ambitious efforts to strengthen mathematics instruction in high school. But some of the most
popular solutions, such as instituting a requirement that all students take algebra in the 8th grade (and changing nothing else), may not solve the problem, and might make it worse.
The countries that do well in international comparisons do not choose between skills or problem-solving; they teach concepts <i>and</i> skills and problem-solving.
The real math problem begins in elementary school, where too few students develop the foundations they will need to succeed in higher-level mathematics. It’s clear from the new analyses of testing
data that 4th graders have problems, and what they learn does not stick.
That’s because of the way mathematics is taught in the early grades. Extensive research shows that American elementary schools teach arithmetic with a shortsighted focus on the problems within a
chapter of a math book. Little thought is given to building the foundation for later chapters and grade levels. Children learn only specific procedures for specific problems, such as 7 + 4 = ___.
Vary the problem slightly (7 + ___ = 11) and students do not know what to do. It is the relationships among problems (7 + 4 = ___; ___ + 4 = 11; 7 + ___ = 11; and 11 - ___ = 7) that lead to insights
into arithmetic that will support, rather than undermine, learning algebra.
When students reach middle school, their teachers generally review the arithmetic learned in the early grades and teach it over again, in pretty much the same way. Then when students eventually get
to algebra and geometry, their performance hits a wall because they don’t understand the underlying concepts of arithmetic on which algebra and geometry are based. Students in the United States think
of an equals sign as a form of punctuation: It marks the place to put the answer. They don’t understand that the equals sign functions as a verb: It indicates that the quantity on the left has the
same value as the quantity on the right. Teaching students how to solve the problem 7 + ___ = 11 lays the foundation for algebra, in which students learn how to comprehend the equation 7 + x = y.
What’s needed is a new way of teaching mathematics. Unfortunately, reforming elementary mathematics has been hampered by the sound and fury of the so-called math wars, that long-smoldering conflict
pitting those who advocate teaching mathematics through real-world problems (what their critics call “fuzzy” math) against those who prefer an emphasis on basic skills (what their critics call “drill
and kill”). These wars have stifled well-meaning attempts to improve mathematics teaching and learning, and their collateral damage—on young people’s educational opportunities—has been severe.
Overemphasis of skills or of problem-solving leads to the neglect of the other. More important, it aids and abets the neglect of conceptual understanding. Skills, problem-solving, and concepts are
all necessary. Each depends on the other two. In the United States, we overlook conceptual understanding, in part because we have been so distracted by superficial, math-war arguments.
Partisans in the math wars disagree about the causes of the problem (some preferring to focus on blame), and they disagree about the remedies. But they also agree on much, and particularly on the
need for change. Ironically, the “wars” hand victory to the status quo.
Evidence from countries that perform well in mathematics shows that the war is phony. What’s needed in mathematics is not one paradigm or another, but common-sense—and carefully engineered—changes in
what we teach. The countries that do well in international comparisons do not choose between skills or problem-solving; they teach concepts and skills and problem-solving.
Effective teaching could help students overcome many of the common misunderstandings they bring to each new math class. Often, such misunderstandings arise because students have learned well in the
way they were taught. A particular concept may have worked effectively for problems in the grade in which they were taught it, but does not work for problems in higher grades. The dictum “when you
add, subtract, or multiply, line the digits from the right” works fine until you have to add 3.75 to 12.5.
Likewise, using shaded sections of a pie to teach part-whole ideas about fractions is a good start. But it will lead to misconceptions when a student has to add 3/4 to 2/4 and sees each fraction as a
pie with parts, getting an answer of 5/8 (5 shaded parts and 8 total parts). To develop a more general idea of “whole,” students should move on to rulers, where the whole is the unit (inch), and
fractions are fractions of the unit (inch), no matter how many inches are involved (3/4 inches + 2/4 inches = 5/4 inches). This is a deeper and more general idea of fractions that readily supports
ideas about fractions on the number line needed for algebra. Teaching fractions this way would support, rather than undermine, the teaching of algebra.
Redesigning mathematics instruction also requires some structural changes. Unlike in reading, where schools have for years differentiated instruction and intervened in many ways to help students at
varying reading levels reach standards, elementary school mathematics programs have tended to treat all students the same, in a sink-or-swim design. In secondary school, tracking kicks in. But
tracking students consigns too many young people, particularly minority students, to dead-end sequences of courses that fail to prepare them for the future. Tracking is a failed solution to a real
problem, and has evil consequences. But untracking takes us back to the original, unsolved problem of how to manage differences in preparation among students.
Some students do well in the regular program. Others do well, but need a little extra help. For these students, a homework clinic before or after school can keep them from falling too far behind. As
one teacher put it, “A lot of kids we thought were a year behind were only 15 minutes behind.”
In the United States, we overlook conceptual understanding, in part because we have been so distracted by superficial, math-war arguments.
Still other students struggle because of misconceptions that hamper their ability to perform mathematics effectively. And some students are so far behind that they need serious intervention to get
them back on course.
The answer, then, is an intervention strategy that can respond to varying student needs. Such a strategy would include before- and after-school clinics that provide homework help for these students
who are “15 minutes behind,” targeted assistance to students with specific misconceptions, and intensive interventions for students who are far behind.
Some schools have taken lessons from other countries and put in place practices such as frequent assessments to understand student needs, along with new pedagogies to help all students understand
mathematics. And these schools have seen impressive results.
Claire Pierce, a math coach at Summerville Middle School, in Summerville, Ga., says frequent assessments, along with time for teachers to analyze student work, make a difference in how teachers
address student needs. “Our teachers know more about what our students do and don’t understand than they ever have,” she reports. “There are lots of opportunities for them to look at student work,
listen to student conversations, and consider explanations they’ve made of why they took a certain approach.”
Debbie Menard, the principal of Twin Lakes Academy Elementary School, in Jacksonville, Fla., says that enabling students to understand the concepts they are learning has led to spectacular
improvements in student achievement. The proportion of 3rd graders at the school performing at level 3 or above on the state mathematics tests (scored on a 5-point scale) shot up from 59 percent in
2004 to 83 percent in 2005. “When you and I took math, we got the right answer and that was the end,” says Ms. Menard. “Now students have to think and explain their answers.”
Teachers in these schools and others like them know that concepts, problem-solving, and basic skills are all important. Their success should convince combatants in the math wars to lay down their
weapons. Our students deserve a truce. | {"url":"https://www.edweek.org/teaching-learning/opinion-math-warriors-lay-down-your-weapons/2006/02","timestamp":"2024-11-04T18:56:43Z","content_type":"text/html","content_length":"187629","record_id":"<urn:uuid:020fc546-1faa-4896-b147-20c49eb525a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00311.warc.gz"} |
Casino Craps – Easy to Be Schooled In and Easy to Win
Posted in March 5, 2022 ¬ 12:25 amh.AutumnNo Comments »
Craps is the swiftest – and definitely the loudest – game in the casino. With the gigantic, colorful table, chips flying all over and contenders outbursts, it’s exhilarating to review and amazing to
Craps also has 1 of the lesser house edges against you than just about any casino game, regardless, only if you lay the appropriate stakes. In fact, with one kind of placing a wager (which you will
soon learn) you take part even with the house, symbolizing that the house has a zero edge. This is the only casino game where this is authentic.
The craps table is a bit adequate than a adequate pool table, with a wood railing that goes around the external edge. This railing performs as a backboard for the dice to be tossed against and is
sponge lined on the interior with random designs so that the dice bounce in one way or another. Several table rails also have grooves on the surface where you usually appoint your chips.
The table covering is a compact fitting green felt with designs to display all the different plays that will likely be laid in craps. It’s especially difficult to understand for a newbie, however,
all you really should concern yourself with right now is the "Pass Line" region and the "Don’t Pass" location. These are the only bets you will perform in our master course of action (and for the
most part the definite bets worth making, interval).
Don’t let the complicated design of the craps table bluster you. The chief game itself is considerably plain. A brand-new game with a new competitor (the contender shooting the dice) will start when
the existing gambler "sevens out", which means he tosses a seven. That concludes his turn and a new player is given the dice.
The fresh gambler makes either a pass line bet or a don’t pass bet (described below) and then thrusts the dice, which is referred to as the "comeout roll".
If that beginning roll is a 7 or 11, this is describe as "making a pass" as well as the "pass line" gamblers win and "don’t pass" candidates lose. If a snake-eyes, 3 or twelve are rolled, this is
known as "craps" and pass line players lose, meanwhile don’t pass line players win. Although, don’t pass line wagerers never win if the "craps" number is a 12 in Las Vegas or a 2 in Reno and Tahoe.
In this case, the play is push – neither the competitor nor the house wins. All pass line and don’t pass line stakes are awarded even money.
Barring one of the 3 "craps" numbers from winning for don’t pass line wagers is what provides the house it’s small edge of 1.4 percent on any of the line plays. The don’t pass competitor has a
stand-off with the house when one of these blocked numbers is rolled. Other than that, the don’t pass gambler would have a small edge over the house – something that no casino accepts!
If a no. aside from seven, eleven, 2, 3, or 12 is tossed on the comeout (in other words, a 4,5,six,8,nine,10), that number is described as a "place" no., or simply a # or a "point". In this case, the
shooter goes on to roll until that place # is rolled once more, which is declared a "making the point", at which time pass line wagerers win and don’t pass contenders lose, or a seven is tossed,
which is named "sevening out". In this situation, pass line wagerers lose and don’t pass bettors win. When a contender 7s out, his opportunity has ended and the entire procedure comes about one more
time with a brand-new competitor.
Once a shooter rolls a place # (a 4.5.six.8.9.ten), lots of varied styles of stakes can be laid on every subsequent roll of the dice, until he sevens out and his turn has ended. Nevertheless, they
all have odds in favor of the house, a lot on line plays, and "come" plays. Of these two, we will only think about the odds on a line bet, as the "come" wager is a bit more disorienting.
You should abstain from all other odds, as they carry odds that are too high against you. Yes, this means that all those other participants that are tossing chips all over the table with every single
toss of the dice and casting "field stakes" and "hard way" wagers are certainly making sucker gambles. They can comprehend all the ample bets and particular lingo, still you will be the smarter
gambler by basically casting line bets and taking the odds.
Let us talk about line wagers, taking the odds, and how to do it.
To achieve a line gamble, actually lay your cash on the vicinity of the table that says "Pass Line", or where it says "Don’t Pass". These odds pay even $$$$$ when they win, although it is not true
even odds as a consequence of the 1.4 per cent house edge talked about beforehand.
When you gamble the pass line, it means you are placing a bet that the shooter either cook up a seven or 11 on the comeout roll, or that he will roll 1 of the place numbers and then roll that no.
again ("make the point") near to sevening out (rolling a seven).
When you place a wager on the don’t pass line, you are placing that the shooter will roll either a snake-eyes or a 3 on the comeout roll (or a three or 12 if in Reno and Tahoe), or will roll one of
the place numbers and then seven out near to rolling the place # yet again.
Odds on a Line Bet (or, "odds plays")
When a point has been established (a place number is rolled) on the comeout, you are given permission to take true odds against a 7 appearing right before the point number is rolled once more. This
means you can play an alternate amount up to the amount of your line stake. This is called an "odds" wager.
Your odds gamble can be any amount up to the amount of your line stake, although a number of casinos will now accommodate you to make odds bets of 2, 3 or even more times the amount of your line bet.
This odds play is paid-out at a rate on same level to the odds of that point number being made right before a 7 is rolled.
You make an odds gamble by placing your gamble right behind your pass line stake. You see that there is nothing on the table to show that you can place an odds stake, while there are pointers loudly
printed everywhere on that table for the other "sucker" bets. This is simply because the casino definitely will not seek to certify odds gambles. You must anticipate that you can make one.
Here is how these odds are allocated. Considering that there are six ways to how a #7 can be rolled and five ways that a six or eight can be rolled, the odds of a six or 8 being rolled prior to a
seven is rolled again are six to 5 against you. This means that if the point number is a 6 or 8, your odds bet will be paid off at the rate of six to five. For any ten dollars you stake, you will win
twelve dollars (wagers lesser or greater than 10 dollars are of course paid at the same six to five ratio). The odds of a 5 or nine being rolled near to a 7 is rolled are 3 to 2, so you get paid 15
dollars for each ten dollars gamble. The odds of four or 10 being rolled 1st are two to 1, thus you get paid 20 dollars for every single $10 you play.
Note that these are true odds – you are paid precisely proportional to your odds of winning. This is the only true odds bet you will find in a casino, therefore be certain to make it any time you
play craps.
Here’s an e.g. of the 3 kinds of consequences that generate when a new shooter plays and how you should move forward.
Assume new shooter is preparing to make the comeout roll and you make a $10 stake (or whatever amount you want) on the pass line. The shooter rolls a 7 or eleven on the comeout. You win ten dollars,
the amount of your stake.
You gamble ten dollars again on the pass line and the shooter makes a comeout roll once more. This time a 3 is rolled (the contender "craps out"). You lose your $10 pass line wager.
You play another ten dollars and the shooter makes his third comeout roll (keep in mind, each and every shooter continues to roll until he sevens out after making a point). This time a four is rolled
– one of the place numbers or "points". You now want to take an odds gamble, so you place ten dollars directly behind your pass line stake to display you are taking the odds. The shooter forges ahead
to roll the dice until a four is rolled (the point is made), at which time you win ten dollars on your pass line gamble, and $20 in cash on your odds bet (remember, a 4 is paid at 2 to one odds), for
a complete win of $30. Take your chips off the table and prepare to stake one more time.
On the other hand, if a 7 is rolled in advance of the point number (in this case, before the 4), you lose both your ten dollars pass line stake and your 10 dollars odds bet.
And that is all there is to it! You casually make you pass line gamble, take odds if a point is rolled on the comeout, and then wait for either the point or a seven to be rolled. Ignore all the other
confusion and sucker bets. Your have the best odds in the casino and are betting carefully.
Odds wagers can be made any time after a comeout point is rolled. You do not have to make them right away . Nevertheless, you’d be insane not to make an odds stake as soon as possible seeing that
it’s the best play on the table. On the other hand, you are enabledto make, abstain, or reinstate an odds gamble anytime after the comeout and near to when a seven is rolled.
When you win an odds wager, be certain to take your chips off the table. If not, they are deemed to be naturally "off" on the next comeout and will not count as another odds bet unless you especially
tell the dealer that you want them to be "working". Even so, in a swift moving and loud game, your plea may not be heard, thus it’s best to casually take your bonuses off the table and bet yet again
with the next comeout.
Anyone of the downtown casinos. Minimum bets will be small (you can generally find $3) and, more notably, they continually tender up to 10 times odds wagers.
Good Luck!
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What is a multiplication chart?
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The history of the multiplication chart goes back a long time. People have been using multiplication for thousands of years. The ancient Egyptians and Babylonians used methods to multiply numbers. In
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Kids can start using a multiplication chart around the age of 7 or 8, when they begin learning multiplication in school. However, younger kids can also benefit from seeing the patterns and practicing
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To use a multiplication chart, click on the product you want to find. The related factors (multiplicands and multipliers) will change color along with the product, making it easy to see the
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To generate a dynamic multiplication chart, you can append "/2-10" to your domain URL. This will create a multiplication chart for the range of numbers from 2 to 10. Similarly, adding "/3-13" to the
domain will generate a multiplication chart for numbers 3 to 13. Users can modify the URL to dynamically generate charts for any range between 1 and 100. For example, "/5-20" will create a
multiplication chart for numbers 5 to 20. This feature allows users to explore multiplication tables for any desired range within the specified limits. | {"url":"https://multiplicationschart.com/1-5","timestamp":"2024-11-03T06:49:32Z","content_type":"text/html","content_length":"219135","record_id":"<urn:uuid:3a77cf7d-14e7-43c5-8e31-e33d1eb5e78f>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00046.warc.gz"} |
The Space of Gravity: Spatial Filtering Estimation of a
Gravity Model for Bilateral Trade
ISSN 2282-6483
The Space of Gravity: Spatial Filtering
Estimation of a Gravity Model for
Bilateral Trade
Roberto Patuelli
Gert-Jan Linders
Rodolfo Metulini
Daniel A. Griffith
The Space of Gravity: Spatial Filtering Estimation of a
Gravity Model for Bilateral Trade
Roberto Patuelli,
[ Gert-Jan Linders,]
[ Rodolfo Metulini]
[ and Daniel A. ]
a[ Department of Economics, University of Bologna, Italy; The Rimini Centre for Economic ]
Analysis (RCEA), Italy
b[ Ministry of Social Affairs and Employment, The Netherlands ]
c[ IMT Institute for Advanced Studies Lucca, Italy ]
School of Economic, Political & Policy Sciences, The University of Texas at Dallas, USA
Bilateral trade flows traditionally have been analysed by means of the spatial interaction gravity model. Still, (auto)correlation of trade flows has only recently received attention in the
literature. This paper takes up this thread of emerging literature, and shows that spatial filtering (SF) techniques can take into account the autocorrelation in trade flows. Furthermore, we show
that the use of origin and destination specific spatial filters goes a long way in correcting for omitted variable bias in an otherwise standard empirical gravity equation. For a cross-section of
bilateral trade flows, we compare an SF approach to two benchmark specifications that are consistent with theoretically derived gravity. The results are relevant for a number of reasons. First, we
correct for autocorrelation in the residuals. Second, we suggest that the empirical gravity equation can still be considered in applied work, despite the theoretical arguments for its
misspecification due to omitted multilateral resistance terms. Third, if we include SF variables, we can still resort to any desired estimator, such as OLS, Poisson or negative binomial regression.
Finally, interpreting endogeneity bias as autocorrelation in regressor variables and residuals allows for a more general specification of the gravity equation than the relatively restricted
theoretical gravity equation. In particular, we can include additional country-specific push and pull variables, besides GDP (e.g., land area, landlockedness, and per capita GDP). A final analysis
provides autocorrelation diagnostics according to different candidate indicators.
JEL codes: C14, C21, F10
1. Introduction
During the past two decades, scholars have shown renewed interest in the theoretical foundations and estimation of the gravity model for bilateral trade (e.g., Deardorff 1998; Anderson and van
Wincoop 2003). The interest in modelling trade flows has increased with questions about the effectiveness of trade agreements (Baier and Bergstrand 2009) and the persistence of border and distance
effects and largely unobserved trade costs (Anderson and van Wincoop 2004). The developments have re-affirmed the importance of accounting for relative trade costs in explaining patterns of trade.
Yet, empirical application of the resulting gravity model framework that incorporates theoretically motivated multilateral resistance (MR) is not straightforward. The system of equations for MR
involves non-linearities in the parameters and requires custom programming (Feenstra 2004).
An alternative specification that circumvents the need to consider the full system of equations includes country-specific effects to control for omitted country-specific MR variables. However, both
the system approach and the alternative using fixed effects impose restrictions on the empirical specification of the gravity model. They allow identification of the impact of bilateral trade
barriers, but preclude (at least in a cross-section) the analysis of country-specific covariates that may affect patterns of trade.
This paper aims to contribute to the literature in providing an alternative solution to deal with omitted MR, which allows for parameter identification for country-specific covariates in a
cross-section analysis of trade patterns. This solution hinges on the interpretation of spatial autocorrelation (SAC)1[ in trade flows as reflecting unobserved country-specific heterogeneity ]
due to MR. Our approach is complementary to a related recent strand of literature that starts from the same interpretation in that we offer an alternative methodology to deal with SAC in trade flows,
called spatial filtering (SF) estimation.
The literature review about trade costs by Anderson and van Wincoop (2004) suggests that the application of spatial econometric techniques in modelling origin-destination trade flows needs further
exploration, to take into account the (auto)correlation in trade flows. Although the gravity model is essentially a model of spatial interaction, little attention has been paid to
1[ Spatial autocorrelation is the correlation that occurs among the values of a georeferenced variable, and that can ] be attributed to the proximity of the units. The concept of SAC can be related
to the first law of geography, stating that ‘everything is related to everything else, but near things are more related than distant things’ (Tobler 1970, p. 236).
3 flows autocorrelation in the trade literature (Porojan 2001, is an exception). In part, this lack of attention was due to technical reasons. Spatial econometric modelling of origin-destination
flows is complex and computationally taxing. Estimation of spatial lag and spatial error models in this context has long been impossible due to computing power limitations. Applications of spatial
interaction modelling in regional science have recently made progress on this issue (see Fischer and Griffith 2008; LeSage and Pace 2008; Sellner et al. 2013). Applications in empirical trade and FDI
modelling have followed shortly thereafter (see Baltagi et al. 2007; Behrens et al. 2012). These contributions show the relevance of autocorrelation in trade flows. However, spatial econometric
origin-destination flow models remain complex and relatively taxing to apply empirically. In response to these concerns, several studies have applied an alternative spatial econometric technique, SF,
which deals with autocorrelation in a different but equally effective way. The technique of SF has recently been applied to the origin-destination flow context in other fields, such as commuting and
patent citations (Fischer and Griffith 2008; Griffith 2009). Instead of accounting for autocorrelation by spatial modelling, SF estimation deals with it by filtering the residuals. Because only an
origin-specific and a destination-specific filter are needed in order to account for autocorrelation, the dimensionality of estimation is much less demanding than in the case of a spatial lag or
spatial error origin-destination model. This paper follows up on this development by applying SF estimation to bilateral trade flows. We argue that the application of origin-specific and
destination-specific filtering of residuals corresponds well to the theoretically expected importance of omitted origin-specific and destination-specific MR terms. Empirical results show that SF
estimation can account well for autocorrelation in trade flows.
Moreover, SF estimation of an otherwise standard empirical gravity equation appears to go a long way in correcting for bias due to the origin- and destination-specific omitted variables predicted by
the theoretical gravity model. The regression coefficients are close to the benchmark values in a specification using origin- and destination-specific indicator variables. This implies that SF
estimation provides a relatively simple alternative to spatial econometric origin-destination flow models and custom-programmed non-linear estimation of the theoretical gravity model, which can be
estimated using standard techniques such as ordinary least squares (OLS) or Poisson regression.2
2[ The estimates presented in this paper have been carried out with the ][R][ statistical software (R Core Team 2015). ] The script necessary for running the SF estimations is available for download
from the first author’s personal homepage.
4 Finally, the SF approach allows for a greater flexibility in the empirical specification of the gravity equation. Unlike the specification using indicator variables, we can include country-specific
variables – so called push and pull factors – in the model. Moreover, a SF model is a significant improvement in terms of parsimony and efficiency compared to the indicator variables model. Compared
to the theoretical gravity framework, we can relax the assumption that total trade depends exclusively and proportionately on the gross domestic product (GDP) of the trading countries. Other
potential push and pull factors, such as landlockedness, land area, or per capita income can be included as well, and we do not have to assume a proportional relation between trade and GDP. Thus, SF
estimation entails greater flexibility in specification choice compared to the stylized theoretical gravity model.
The paper proceeds as follows. In Section 2, we specify a theoretical gravity model following Anderson and van Wincoop (2003) and discuss some practical limitations of applying the theoretical
framework. In Section 3, we illustrate the link between theoretical gravity and autocorrelation in trade flows. We present the approach of SF estimation to control for autocorrelation, and motivate
that it allows controlling for unobserved MR. Section 4 outlines the empirical specifications and estimators that we compare, while Section 5 discusses the SAC tests that we use for post-estimation
diagnostics. In Section 6, after an overview of data used, we turn to the estimation results and diagnostics. Section 7 concludes the paper.
2. The Gravity Model and Autocorrelation
We can divide the discourse over trade gravity modelling in two parts, regarding the theoretical and empirical approaches to the problem, respectively. The following sections attempt to provide such
a discussion.
2.1 Theoretical gravity
Gravity equations for analysing bilateral trade flows have been estimated since the 1960s (e.g., Tinbergen 1962; Pöyhönen 1963). The model describes the volume of bilateral trade as a function of
push and pull factors, such as economic size of origins and destinations, and transactional distance between trade partners. It has been deployed for various purposes, such as analysing the
determinants of trade patterns, testing trade theories, forecasting future flows or estimating missing data, and comparative static analysis of changes in trade costs. Recent applications
increasingly emphasize the importance of estimating a gravity equation that is
5 consistent with theoretical gravity (e.g., Anderson and van Wincoop 2003; Baier and Bergstrand 2009). The theoretical framework that is most influential has been developed by Anderson and van
Wincoop (2003), in their paper on consistent estimation and assessment of the border effect in U.S.-Canadian regional trade flows.3
Anderson and van Wincoop derive a reduced-form gravity equation, assuming an N-country endowment economy, constant elasticity of substitution (CES) preferences, and symmetric bilateral trade costs.
Their model explicitly takes into account the role played by country-specific price indices (MR terms). The gravity equation that results is specified as:
1 σ , i j ij ij w i j y y t x y P − = [] [] Π (1)
where xij is the value of the flow of goods from country i to country j, y is GDP (w stands for
world) and tij is the bilateral trade cost factor. Finally, two variables enter that we discuss in
greater detail later. Πi measures outward MR of country i, and Pj measures inward MR of
country j. The term σ is the elasticity of substitution (σ > 1).
Equation (1) shows that bilateral exports would be proportional to the size of the exporting market and the share of the import market in total demand, in the absence of bilateral trade costs (tij).
Trade costs are of the iceberg cost type, and we define trade costs as a mark-up on the ‘mill
price’ pi (tij ≥ 1). Hence, (tij – 1) is the ad-valorem tariff equivalent of bilateral trade costs. The
bilateral delivered prices (pij) then equal:
. ij ij i
p = ⋅t p (2)
A wide variety of covariates in the literature is used to represent bilateral trade costs. We include some of the most common bilateral explanatory variables. A multiplicative formulation of
bilateral trade costs (see Deardorff 1998; Anderson and van Wincoop 2004) yields:
( ) ( ) ( ) ( ) 2 3 4 5 1 β 1 β 1 β 1 β 1 β , ij ij ij ij CB CL CH FTA ij ij ij t =D ⋅e ⋅ − ⋅e ⋅ − ⋅e ⋅ − ⋅e ⋅ − ⋅b (3)
3[ Related theoretical derivations of a gravity equation for trade can be found in earlier literature as well, such as ] Bergstrand (1985) and Bröcker (1989).
6 where D stands for geographical distance; CB stands for an indicator variable equal to 1 if two countries share a (land) border (and zero otherwise); CL, CH and FTA are a set of similar indicator
variables indicating whether or not two countries share a common official language, common colonial history, and/or common free-trade agreement. The parameter bij reflects the
impact of all remaining bilateral trade barriers on the bilateral trade cost factor, assumed independent from the included covariates. Based on economic intuition, we expect positive parameters for
the covariates in the trade cost function.
Bilateral export does not depend on only bilateral trade cost and the (exogenously given) size of the trading economies. It also depends on the weighted average trade costs that an exporter and
importer face in their export and import market, respectively. This is reflected by the MR terms entering the denominator of equation (1). Anderson and van Wincoop (2003) derive the set of equations
for the MR terms Πi and Pj ,
1 σ 1 σ 1 θ , N ij i j j j t P − − = [] [] Π = [] [] [] []
(4) 1 σ 1 σ 1 θ ,
N ij j i i i t P
− − =
[] []
[] [] [Π] []
(5) where θ
y y[i] [w]
Note that the outward (inward) resistance term includes the GDP-share-weighted average of bilateral trade costs relative to the inward (outward) resistance terms across destinations (origins). Given
bilateral trade costs tij, a high value for MR implies that other countries k are
less attractive trading partners. Hence countries i and j will trade more with each other, as shown in equation (1).
2.2 Practical Gravity
The theoretical gravity model conveys an important message. Trade flows are not mutually independent. For a consistent econometric estimation of the parameters in the model, problems emerge if the
regressor variables are correlated with the residuals. The theoretical model shows that this endogeneity bias is likely to emerge if we do not control for country-specific MR.
Despite the prominent position of this theoretical framework over the past years, many empirical studies continued to rely on a more pragmatic empirical gravity equation instead. Several plausible
explanations for this come to mind. Estimating a theoretically consistent
7 gravity equation involves dealing with Equations (4) and (5), which are nonlinear in parameters. Developing the required estimation procedures involve some restrictive assumptions (see Baldwin and
Taglioni 2006; Balistreri and Hillberry 2007), and work on deriving an analytical solution has only recently emerged (e.g., Straathof 2008).
Furthermore, the theoretical framework puts restrictions on the empirical specification that follows from the stylized model rather than from practical considerations. In fact, trade depends
proportionately on the GDP of an origin and destination. Moreover, GDP variables are the only push and pull factor in the model to explain total external trade. While the theoretical model requires
total exports to sum to an exporter’s GDP, and total imports to sum to an importer’s GDP, these constraints do not hold in practical applications.4 First, trade and GDP are measured in different
units. While trade is measured in gross output values, GDP is a measure of value added. Moreover, the model includes intranational trade while most practical applications only consider international
trade flows in estimating the gravity equation, due to data limitations. This context implies that theoretically imposed constraints in the model are not generally valid in estimation. Second, the
share of external trade in total expenditure and gross output may be different from the predictions in the theoretical model. The theoretical gravity model predicts that larger economies are less
open to international trade and allocate a larger share of their expenditure on intranational trade, but the share of international trade on GDP is often constrained to a constant by imposing
proportionality between the former and the latter.
Hence, practical considerations may provide a valid motivation to choose an unconstrained empirical gravity equation, which allows more flexibility in specification. An empirical gravity equation can
include additional push and pull factors to capture variation in openness to international trade. For example, we may think of per capita income, landlockedness, and land area as factors determining
a country’s openness to international trade. Many of these variables have been used in empirical specifications of the gravity model for international trade (e.g., Frankel 1997; Raballand 2003; De
Groot et al. 2004).
Taking theoretical and practical insight seriously, ideally we would need to combine the flexibility of the empirical gravity equation and the insights about omitted variable bias due to MR of the
theoretical foundation of gravity. An often used practical solution to deal with country-specific omitted variable bias is to include country-specific indicator variables in the
4[ The MR terms obtained impose the constraints: ]
yj[and ]
[ In similar applications of the model ]
in regional science, this type of specification is known as a doubly-constrained gravity model (e.g., Wilson 1970; Fotheringham and O'Kelly 1989).
8 gravity equation (Bröcker and Rohweder 1990). As argued by Feenstra (2004), a model specification that includes origin- and destination-specific intercepts is consistent with theoretical concerns.
Moreover, this solution has been widely applied in regional science to deal with the practical problems of estimating a gravity equation in which the total flows are not known (Sen and Smith 1995).5
This solution is not completely satisfactory, though. It is rather drastic medicine to cure the patient. First, including origin- and destination-specific indicator variables reduces the statistical
efficiency of econometric estimation. Second, it precludes the analysis of country-specific determinants of trade, which are interesting for empirical applications, bthey explain cross-country
variation in openness to international trade.
2.3 Consistent Estimation and Autocorrelation
The main insight from theoretically derived gravity is that regressor variables and residuals in the unconstrained gravity equation are likely to be correlated, because bilateral trade barriers also
appear in the omitted MR terms. In empirical estimations, failure to control for MR might result in omitted variable bias in the parameter estimates of the bilateral regressors.
This paper proposes an alternative estimation approach that allows for the estimation of an unconstrained empirical specification of the gravity model, including push and pull factors, while offering
a correction for origin- and destination-specific omitted variable bias. The approach starts from a specific interpretation of endogeneity bias as resulting from autocorrelation in trade flows. The
argument for this interpretation has been made before in Behrens et al. (2012) and in Koch and LeSage (2009), and more generically relates to the recent revival in modelling SAC in bilateral flow
data in the previously mentioned regional science literature. To the best of our knowledge, however, this paper is the first to link the theoretical MR effects to origin- and destination-specific
filters, and to make use of SF techniques to accommodate autocorrelation in trade flows.
The argument starts by inspecting Equations (4) and (5). We propose that countries located in close spatial proximity tend to have similar MR. Similar geographical location implies similar
geographical distance to trade partners across the world and a higher probability of shared neighbours. Likewise, shared languages tend to be more similar for countries closely
5[ Although total international trade by country is generally known, or can be proxied by summing available ] bilateral flows, we do not have comparable direct observations for intranational trade.
Hence, we would need to proxy for openness to trade of each country in estimating the gravity equation. This can be done either by including (additional) push and pull factors in the specification,
or by using country-specific intercepts.
9 located in space. Also, the logic of regional integration implies a higher likelihood of proximate countries being part of shared trade agreements with surrounding countries. This context implies
that these spatial patterns in MR would induce autocorrelation in the residuals of the unconstrained gravity equation. As a result, the residuals and the bilateral trade cost variables are
correlated, because similar reasoning to the preceding discussion suggests SAC would be in the regressor variables distance, contiguity, language and trade agreement. Omitted variable bias would
3. Recent Developments in Estimating the Theoretical Gravity Model of Trade
The theoretical gravity model shows that consistent estimation of the parameters requires us to take into account the price indices. As discussed in Feenstra (2004), the computational complexity of
the non-linear estimation procedure has prevented its widespread use in the applied international trade literature. Still, Anderson and van Wincoop (2003) show that estimation of the more traditional
empirical gravity equation (omitting the MR terms) yields inconsistent parameter estimates for the key regressor variables. A simple solution that results in consistent parameter estimates is to use
a set of country-specific indicator variables for the exporting and importing countries (Bröcker and Rohweder 1990; Feenstra 2004). The indicator variables capture the country-specific MR terms, and
control for omitted variable bias related to the country-specific intercepts. The main advantage of this formulation is that the resulting specification can be estimated by familiar methods such as
OLS or Poisson regression.
However, the disadvantage of this solution is that the parameters of country-specific determinants of trade cannot be estimated. Variables such as GDP, per capita income, landlockedness, and land
area are captured by the country-specific indicator variables. Still, empirical estimation of the effect of these variables may be relevant depending on the topic under investigation. Hence, a
solution that would share the basic simplicity of estimation with the indicator variable specification, while allowing retention of the country-specific regressors, is needed.
Several recent developments in the trade gravity model literature focus on combining consistent estimation and flexibility in the specification of the gravity equation. Egger (2005) argues that a
Hausman-Taylor approach, which allows for country-specific covariates, is consistent even if unobserved country-specific heterogeneity exists. This formulation provides an alternative to the
indicator variables specification that controls for omitted variable bias due
10 to omitted MR terms, and allows for the estimation of the parameters related to the country-specific variables. The method is based upon an approach similar to instrumental variables, which relies
on instruments from inside the model.
In contrast, Baier and Bergstrand (2009) log-linearize the MR terms using a first-order Taylor series approximation. This yields exogenous bilateral multilateral-world-resistance (MWR) variables that
proxy the endogenous country-specific MR variables in Anderson and van Wincoop (2003). The resulting reduced-form gravity equation can be estimated with OLS. This method is termed bonus vetus
(‘good-old’) OLS (BV-OLS). The approach yields log-linear approximations of the MR terms, using Taylor series expansion around a centre of identical and symmetric trade costs, tij = t, but differing
economic sizes (θi = yi/yw).
Starting from a reformulated Equation (1):
lnx[ij] = −lny[w]+lny[i]+lny[j] − −σ 1 lnt[ij]+ −σ 1 lnP[i]+ −σ 1 lnP[j], (6) the equation that Baier and Bergstrand derive is:
1 1 1 1 1 1 1 ln ln ln ln σ 1 ln σ 1 θ ln θ θ ln 2 1 σ 1 θ ln θ θ ln . 2
N N N ij w i j ij j ij i j ij j i j N N N i ji i j ij i i j x y y y t t t t t
= = = = = = = − + + − − + − − + − −
The terms in square brackets are the MR terms. They contain a first component that captures multilateral trade frictions for each exporting or importing country, relative to a second part that
reflects world trade costs.
A third approach to the consistent cross-sectional estimation of the gravity model is proposed in Behrens et al. (2012). Their approach is closely related to our approach. Starting from the Anderson
and van Wincoop formulation of the theoretical gravity equation, they show that the MR terms can be shown to reflect a correlation structure between trade flows that can be modelled similarly to SAC.
They suggest a spatial-autoregressive moving-average specification for the gravity model, which results in consistent estimates of the standard gravity equation parameters. At the same time, they
argue that the baseline fixed-effects specification discussed previously does not fully succeed in capturing the MR dependencies in the error structure
11 introduced by the general equilibrium nature of trade patterns modelling, and that its residuals still show a significant amount of autocorrelation (Behrens et al. 2012).
We now proceed to discuss the methodology followed in this paper. The alternative we propose, SF, combines two attractive features. First, it is fairly simple to apply, much like OLS with indicator
variables; second, it takes into account the general equilibrium interdependence of trade flows that can be modelled as SAC, like spatial econometric origin-destination specifications.
4. Proposed Methodology: Spatial Filtering Estimation
The theoretical gravity model includes origin- and destination-specific MR variables that reflect the export and import accessibility of countries. Omitting these endogenous MR variables from the
specification results in potential omitted variable bias, both for the trade cost variables and for the size variables in the gravity equation. Consistent estimation requires some way to capture the
endogeneity between MR terms and standard regressors. We propose to make use of the fact that this dependency structure is likely to manifest as SAC in the residuals of the traditional specification
of the gravity model. The reasoning is that many trade cost variables, such as geographical distance, adjacency, trade agreements, and common language, are spatially correlated: countries close in
space are more likely to share the same (or similar) characteristics. This context likewise implies that both inward and outward accessibility are spatially correlated: close countries are likely to
have more similar accessibility. We deal with SAC by using an origin- and a destination-specific spatial filter, which serve to capture the spatially autocorrelated parts of the residuals.
When including these spatial filters as additional origin- and destination-specific regressors (much like the origin and destination specific MR variables), the model can be estimated by standard
regression techniques, such as OLS or Poisson regression, which are common in the literature about spatial interaction patterns. The parameters of the standard regressor variables are unrelated to
the remaining residual term, and standard estimation yields consistent parameter estimates as a result. We refer to this estimation method as SF estimation of origin-destination models (see Griffith
2007; Fischer and Griffith 2008).
Basically, SF estimation of georeferenced data regressions (such as international trade) can reduce to defining a geographically varying mean and a variance on the basis of an exogenous spatial
weights matrix. In other words, the spatially correlated residuals from an otherwise
non-12 spatial regression model are partitioned into two synthetic variables: (i) a spatial filter which captures latent SAC; and, (ii) a non-spatial variable (free of SAC), which will be the newly
obtained residuals. The workhorse for this SF decomposition is a transformation procedure based upon eigenvector extraction from the matrix
(I – 11T/n) W (I – 11T/n), (8)
where W is a generic n x n spatial weights matrix; I is an n x n identity matrix; and, 1 is an n x 1 vector containing 1s. The spatial weights matrix W defines the relationships of proximity between
the n georeferenced units (e.g., points, regions, and countries). The transformed matrix appears in the numerator of Moran’s coefficient (MC), which is a commonly used measure of SAC (see Section 5).
The eigenvectors of Equation (8) represent distinct map pattern descriptions of SAC underlying georeferenced variables (Griffith 2003). Moreover, the first extracted eigenvector, say e1, is the one
showing the highest positive MC that can be achieved by any spatial
recombination induced by W. The subsequently extracted eigenvectors maximize MC while being orthogonal to and uncorrelated with the previously extracted eigenvectors. Finally, the last extracted
eigenvector maximizes negative MC.
Having extracted the eigenvectors of Equation (8), a spatial filter is constructed by judiciously selecting a subset of these n eigenvectors. In detail, for our empirical application, we select a
first subset of eigenvectors (which we will call ‘candidate eigenvectors’) by means of the following threshold: MC(ei)/MC(e1) > 0.25. This threshold yields a spatial filter that
approximately replicates the amount of variance explained by a spatial autoregressive model (SAR) (Griffith 2003).6 Subsequently, a stepwise regression model may be employed to further reduce the
first subset (whose eigenvectors have not yet been related to the data) to just the (smaller) subset of eigenvectors that are statistically significant as additional regressors in the model to be
evaluated. The resulting group of eigenvectors is what we call our ‘spatial filter’. This estimation technique has been applied, both in autoregression and in traditional modelling terms, to various
fields, including labour markets (Patuelli 2007), innovation (Grimpe and Patuelli 2011), economic growth (Crespo Cuaresma and Feldkircher 2013) and ecology (Monestiez et al. 2006).
6[ Ongoing research by Griffith and collaborators is looking into formulating an estimation equation, based on ] residual SAC, to predict the ideal size of the candidate set.
13 The added challenge, with regard to the case at hand, is that trade data do not represent points in space, but flows between points. Therefore, the eigenvectors are linked to the flow data by
means of Kronecker products: the product EK⊗ 1, where EK is the n x k matrix of the candidate
eigenvectors, may be linked to the origin-specific information (for example, GDP per exporting countries), while the product 1 ⊗ EK may be linked to destination-specific information (again,
for example, the GDP of importing countries) (Fischer and Griffith 2008). As a result, we have two sets of origin- and destination-specific variables, which aim to capture the SAC patterns commonly
accounted for by the indicator variables of a doubly-constrained gravity model (Griffith 2009), therefore avoiding omitted variable bias.
The main advantages of the proposed estimation method are: (a) this approach can be applied to any type of regression, including simple OLS and generalized linear models (GLMs) such as Poisson or
negative binomial regressions (although auto-Poisson and auto-negative binomial specifications cannot describe positive spatial dependence), for which usually dedicated spatial econometric
applications do not exist; (b) by avoiding the use of indicator variables, we are able to save degrees of freedom, and, (c) the approach can be used to estimate regression parameters for origin- and
destination-specific variables, such as GDP or trade agreements indicators.
For our case study, because of the nature of trade data, as suggested by Santos Silva and Tenreyro (2006), we estimate a count data model. While the natural choice would be Poisson regression, in
order to take into account overdispersion in the data due to unobserved heterogeneity (which results in a sample variance which is much greater than the sample mean), we estimate a negative binomial
model, which can explicitly account for such overdispersion by iteratively estimating the dispersion parameter. In subsequent comparisons regarding residual spatial autocorrelation, we consider, for
the SF models, quasi-Poisson estimations as well.
5. Spatial Autocorrelation Diagnostics
When employing GLMs, traditional SAC indices may not be appropriate, as discussed below. In this section, we review the available alternatives.
In linear regression contexts, when analysing model residuals, an adapted Moran test (Cliff and Ord 1972; 1981) is commonly used, under a standard assumption of normality. A t test can
14 be used to test the null hypothesis of spatial randomness of the residuals. The formula for the MC computed on the residuals is the following:
, 2 0 ε ε , ε ij i j i j i i n w I S =
where wij is the (i,j) element of a chosen spatial weight matrix W, εi and εi are the related model
residuals, and S0 is the sum of all elements of W. The expected value of this index is:
0 tr( ) ( ) , ( ) n E I S n k = − A (10)
where A = (XTX)–1XTWX corresponds to the factor that accounts for the effect of the
independent variables. X is the n x k matrix containing the values of the k independent variables included in the regression model.
A permutation-based Moran test has also been proposed (Cliff and Ord 1981) in order to improve the results of the approximate t test and to gain insights in its sampling distribution under spatial
Because the Moran test has been developed for linear models and normally distributed residuals, the use of MC calculated on the residuals of count data (Poisson, negative binomial) regression models
is questionable (Schabenberger and Gotway 2005, p. 377), despite recent literature agrees that it possesses good power against a wide array of autoregressive models and different distributions of the
residuals (Anselin and Rey 1991).
Griffith (2010) studies the behaviour of the MC for non-normal random variables, and shows that, above moderate values of n (25–100), the MC is a suitable indicator in these cases as well. However,
Griffith does not study the case of SAC diagnostics for regression residuals, in which we can consider the effect of the independent variables in the model.
Further, Moran’s test may not be properly applied to the residuals of Poisson or negative binomial regression, whose distributional properties are not well known. In addition, because the test does
not consider the heterogeneity of observations, its standard moments may not be appropriate under heteroscedasticity. For more details, one can refer to Oden (1995), who discusses this problem.
Lin and Zhang (2007) suggest that the MC can be used to test the residuals of a Poisson model by employing Pearson or deviance residuals under an asymptotic normality assumption. This
15 approach is followed, among others, by Scherngell and Lata (2013), which employ a panel SF modelling approach. However, this permutation test once again does not incorporate the effect of the
independent variables of the model in constructing a reference distribution.
Fortunately, the standardized t statistic of Jacqmin-Gadda et al. (1997) can be applied in this context. This t statistic can be considered as an extension of standard SAC statistics into the domain
of GLMs. It is derived in an analogous way to a score test based on generalized estimating equations (Prentice and Zhao 1991). As the condition of validity of the above test does not always hold,
since the computation is intractable for large samples, a test based on the permutation distribution has been also proposed by the same authors.
Under the null hypothesis of no spatial autocorrelation, the t statistic is defined as:
1 ( μ )(ˆ μ ),ˆ
ij i i i j i j i
∑ ∑
[=] [≠] w y
− (11)
or, in matrix notation:
), ˆ ( ) ˆ (Y−μ W Y−μ = T t (12)
where Y is the n x 1 vector of the observations of the dependent variable, and μˆ is the n x 1
vector of the estimated means.
Using a first-order Taylor series expansion for the deviation of estimated means from the true means, Jacqmin-Gadda et al. (1997) show that the index’s expectation and variance are as follows: E(t) =
tr(RD); (13) 2 2 ( 4) ( 2) 1 var( ) n [ii](μ[i] μ[i] ) 2tr( ), i t R = =
− +
where R = MTWM, M = I – DX(XTDX)-1XT, and D is the diagonal matrix whose elements are the variance of each observation. Consequently, 2
R is the ith diagonal element of matrix R, while µi(2) and µi(4) are the second and the fourth central moments of the ith observation,
respectively. Jacqmin-Gadda et al. (1997) show that the standardized t statistic asymptotically follows the standard normal distribution.
16 The Jacqmin-Gadda (JG) test is a development of the statistic developed by le Cessie and van Houwelingen (1995), similarly derived as a score test in the spirit of Prentice and Zhao (1991), but
not accounting for the effect of the independent variables. In fact, referring to Equation (13) , the component R in the le Cessie (LC) test is reduced to R = WTW, while D = cov(Y). In other
words, the LC test does not incorporate the adjustment of estimating parameters, that is, the effect of independent variables is not considered in constructing a reference distribution. In summary,
using the JG standardized t statistic, a test for spatial autocorrelation in the context of GLMs can be carried out.
6. Empirical application
We apply the SF estimation to a cross-section of bilateral trade flows between 64 (major trading) countries for the year 2000 (a full list of countries is provided in the Appendix, Table A.1). In
this section, we discuss the empirical specification, data and the estimation results.
6.1 Data and Model Specification
For estimation, we follow a standard specification of the gravity equation of bilateral trade. Starting from the trade costs variables identified in equation (3), we further extend the specification
with additional variables commonly mentioned in the literature (see, e.g., Frankel 1997; Raballand 2003). We use the following standard specification of the gravity equation:
0 1 1 2 3 4 5 6 7 8 9 10 11 2 3 4 5 ln ln( ) α α ln( ) β ln( ) β β β β β β β ln( ) β ln( ) β β δ δ δ δ ε , ij i j i j ij ij ij ij ij i j i j i j ij ij ij ij ij X GDP GDP GDPCAP GDPCAP D
CB CL CH FTA ISL ISL Area
Area LL LL MWRCB MWRCL MWRCH MWRFTA − ⋅ = + ⋅ ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + (15)
where GDPCAP represents per capita GDP, ISL is an indicator variable that equals 1 if the country is an island, Area is the land area of a country, and LL equals 1 for landlocked countries, and in
17 and likewise for the remaining MWR variables. The other variables are as defined earlier. The product of origin and destination GDPs is used as an offset variable.
The data for trade are from the World Trade Database compiled on the basis of COMTRADE data by Feenstra et al. (2005). GDP and per capita GDP data are from the World Bank’s WDI database. Distance,
language, colonial history, landlocked countries, and land area data are from the CEPII institute.7 Whether pairs of countries take part in a common regional integration agreement (FTA) has been
determined on the basis of OECD data about major regional integration agreements.8 A dummy variable indicates whether a pair of countries has (membership in) at least one common FTA. Data on island
status have been kindly provided by Hildegunn Kyvik-Nordas (from Jansen and Nordås 2004).
We first estimate Equation (15) using negative binomial regression including country-specific indicator variables. GDP is used as an offset, which implies we move the log-sum of GDP to the left
handside, assuming it has a proportional effect on trade with elasticity equal to 1 (Anderson and van Wincoop 2003). This is our first benchmark model, which, according to Feenstra (2004), yields
consistent parameter estimates, but is criticized by Behrens et al. (2012). Secondly, we estimate Equation (15), extending it with approximations of MR terms obtained using the Taylor series
approximation proposed by Baier and Bergstrand (2009). This is our second benchmark model. These results, as well as the ones for the SF approach, are discussed in Section 6.2.
6.2 Estimation Results: Spatial Filtering and Benchmark Models
The first benchmark model includes origin- and destination-specific indicator variables. As shown in Anderson and van Wincoop (2003) and Feenstra (2004), this specification accounts for MR terms, and
yields consistent parameter estimates. The disadvantage is that country-specific variables cannot be included, as their effect cannot be identified separately. This implies that explanatory variables
that are potentially relevant for explaining variation in bilateral trade patterns, such as GDP per capita, land area and landlockedness, cannot be investigated empirically (if not ex post, by, e.g.,
regressing the indicator variable coefficients on them). A second disadvantage is the loss of degrees of freedom for estimation, because a substantial number of indicator variables (2n – 2) is
needed. Usually, however, the degrees of freedom remain large enough, since observations are bilateral (i.e., n2 – n).
7[ See http://www.cepii.fr. ]
18 The second benchmark model is the specification developed in Baier and Bergstrand (2009), which includes first-order Taylor series approximations of the MR terms. This specification follows from
Equation (6). Further manipulation [substituting Equation (3) for bilateral trade costs] allows us to combine both terms between square brackets into a set of bilateral variables, one for each
bilateral trade costs variable determining trade costs (such as geographical distance). The reduced-form double-log gravity equation is as follows:
0 1 1 2 3 4 5 6 7 8 9 10 11 2 3 4 5 ln ln( ) α α ln( ) β ln( ) β β β β β β β ln( ) β ln( ) β β δ δ δ δ ε , ij i j i j ij ij ij ij ij i j i j i j ij ij ij ij ij X GDP GDP GDPCAP GDPCAP D CB CL
CH FTA ISL ISL Area Area LL LL
MWRCB MWRCL MWRCH MWRFTA − ⋅ = + ⋅ ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + (17)
Baier and Bergstrand (2009) show that theory imposes the restrictions δk = –βk for each k.
The equations specify the model in double-logarithmic transformation. We estimated the benchmark models multiplicatively, using negative binomial regressions, aside from the BV model, which is
estimated linearly. This method allows a direct treatment of the non-negative values of trade flows and of the zeros, and enables us to correct for overdispersion of trade flows (see Santos Silva and
Tenreyro 2006).
The empirical estimation results are presented in Table 1. Model (1) presents the regression results for the first benchmark model, including country-specific indicator variables. Following Anderson
and van Wincoop (2003), we estimate the model using GDP as an offset variable (i.e., restricting the coefficient of GDP variables to equal 1). The parameter estimates are in line with the findings
elsewhere in the literature (see, e.g., Anderson and van Wincoop 2004; Disdier and Head 2008). Geographical distance has a negative effect on trade, with an estimated elasticity of –1.30. The effect
of proximity on trade is reinforced by a positive and (marginally) significant effect of contiguity on trade. Proximity in terms of language and colonial links also positively affects bilateral
trade, while preferential trade policy (i.e., enjoying common FTAs), appears to have a counterintuitive negative effect. These results – with the exception of the latter – confirm previous findings
about the importance of these dimensions of transactional distance on trade (e.g., Obstfeld and Rogoff 2000; Loungani et al. 2002).
Model (3) compares these findings with the regression outcomes for the second benchmark model, the Baier-Bergstrand estimation. This method proxies for the endogenous and unobserved MR terms by
including exogenous linear approximations based upon bilateral trade costs variables. Provided that the approximation is sufficiently adequate, this specification
19 results in consistent estimates (Baier and Bergstrand 2009). Once again, GDP has been used as an offset variable, and the model is estimated by OLS. The obtained parameter estimates are comparable
to the estimates for the first benchmark model [Model (1)], including the negative effect found for free-trade blocs. Additionally, on the one hand, the Baier-Bergstrand specification has an
advantage, because it enables us to include country-specific regressors explicitly; on the other hand, the results do not always appear to be satisfactory.
Table 1. Estimation results
(1) (2) (3) (4)
Fixed effects (GDP offset)
Spatial filter BB-estimation (GDP offset) BB-estimation Distance –1.30*** –1.23*** –1.25*** –1.22*** Common border 0.24* [ 0.33]**[ ] [ 0.23 ] [ 0.25 ] Common language 0.36*** 0.33*** 0.32*** 0.37***
Common history 0.86*** 0.71*** 0.79*** 0.80*** Free trade –0.14** 0.41 –0.27*** –0.22** GDP exporter – 0.75*** – 0.91*** GDP importer – 0.92*** [– ] [ 1.15]***
GDP per cap. exporter – 0.13*** –0.06** 0.02 GDP per cap. importer – 0.12*** –0.04* –0.16***
Island exporter – –0.41*** –0.29*** –0.28*** Island importer – –0.31*** 0.08 0.20* Area exporter – –0.00 –0.11*** –0.07*** Area importer – –0.17*** –0.22*** –0.28*** Landlocked exporter – 0.23* [
0.30]** [ 0.26]** Landlocked importer – –0.58*** 0.07 0.19* Constant –29.60*** –27.42*** –34.01*** –34.97*** AIC 101,713 47,805 102,485 102,436 Observations 4032 4032 4032 4032
Notes: BB stands for Baier-Bergstrand, and AIC for Akaike information criterion. ***, **, * denote parameter estimates statistically significant at 1%, 5% and 10%, respectively.
20 Closer inspection of the Baier-Bergstrand estimation, dropping the offset assumption on the product of exporter and importer GDP in Model (4), yields qualitatively similar –and in some cases more
plausible (e.g., for landlocked importers) – results, and a slightly better fit. For example, although a negative effect of GDP per capita variables on trade is not uncommon in some specifications
(see, e.g., Anderson and Marcouiller 2002), the effect in Model (3) seems to be driven mainly by offsetting GDP, which imposes a GDP elasticity of trade (of 1), which empirically is too high.
Summarizing, the two benchmark models yield somewhat different results. Although, as mentioned, some effects may be more plausible in the Baier-Bergstrand estimation results, the more traditional
specification using country-specific indicator variables results in a slightly better model likelihood, as shown by the Akaike information criterion (AIC). The disadvantages of this model, though,
are the loss of country-specific variables, and a diminished precision in the determination of the significance of variables, resulting from the loss of degrees of freedom in the model estimation.
Results emerging from the SF estimation of the gravity model, which combines the consistent estimation of the first benchmark model with the flexibility of specification of the second benchmark
model, are shown for Model (2) in Table 1. The results presented here are obtained for a simmetrized k-nearest neighbours9 spatial weights matrix C, and for a negative binomial estimation, employed
in order to cope with overdispersion in the trade flows. With regard to the coefficients of bilateral resistance variables, we note that with the exception of the one for FTA, they are highly
significant, and their values are consistent with the ones found for Model (1). The FTA coefficient not being significantly negative anymore might be seen as a result that is more consistent with
theoretical expectations. With regard to the importer- and exporter-specific variables, we are able to identify highly significant and positive coefficients for GDP, and GDP per capita is now
significant and positive as well in both cases. This result is in contrast with the ones for the Baier-Bergstrand benchmarks [Models (3) and (4)], in which the same variable is either not significant
or significantly negative. The SF estimation also allows us to estimate significant parameters for the variables identifying the geographical characteristics of importer and exporter countries. The
signs obtained are mostly consistent with the ones found
9[ For the k-nearest neighbours definition of proximity, each country’s neighbours are defined by selecting the k ] closest countries. Distance between the geographical centroids of the countries was
used (Great Circle), setting k = 3 and forcing, for computational reasons, symmetry of the spatial weights matrix. As a result, the minimum number of neighbours per country is 3, while the maximum
number is not constrained. Alternative definitions of proximity based upon, for example, simple rook contiguity or distance decay could be tested in order to assess the sensitivity of the model to
the choice of spatial specification.
21 for the Baier-Bergstrand benchmarks. They show that larger and both landlocked and island countries tend to trade less. Noteworthy differences between the SF model and the benchmarks regard the
negative and significant coefficients obtained for the importing patterns of island and landlocked economies (it was marginally positive or non-significant for the benchmarks). For islands, it may
seem counterintuitive to find this result, although it should be considered that the sample of countries used excludes, because of non-reporting, most micro-island countries, while includes all large
island countries like the UK or Japan. In contrast, in the case of landlocked countries, a negative importing coefficient is more consistent with theoretical expectations.
Finally, the AIC of the SF model appears to vastly improve on the ones of the benchmark models, because of the high amount of variance explained by the origin- and destination-specific spatial
filters, which are also highly significant from a statistical viewpoint (not shown in Table 1). In summary, the proposed SF approach to the estimation of a gravity model of trade allows
identification of the regression parameters related to the bilateral variables, as well as those related to the origin- and destination-specific variables. Moreover, the model has a better likelihood
(leading to improved AIC) than the competing models tested, and uses a limited number of degrees of freedom.
6.3 Testing for Spatial Autocorrelation
In Section 5, we discussed SAC statistics based on the score test [by le Cessie and van Houwelingen (1995) and Jacqmin-Gadda et al. (1997)], which are alternative to the traditional MC in case of
GLMs, since the MC statistical distribution theory has been developed under linear regression assumptions.
Having an n2 x n2 spatial weight matrix (obtained as W ⊗ W) and the t statistic by Jacqmin-Gadda et al. (1997), residual SAC in Poisson and negative binomial regressions can be modelled by
eigenvector SF within the same framework as standard spatial autocorrelation in regression residuals. The eigenvectors employed in Model (2) (see preceding section) represent a certain level of SAC,
given a spatial connectivity pattern, and by including them as proxy variables for such spatial autocorrelation, SAC that is not explained by independent variables is expected to be filtered out (at
least partially) of the residuals, and transferred to the mean response.
Because eigenvectors are introduced as independent variables in a (forward or backward) stepwise manner, the adjustment of estimating parameters for independent variables developed in the
Jacqmin-Gadda test seems to be desirable. Chun (2008) performs the test to evaluate SAC in a Poisson model in an analysis of migration flows. To the best of our knowledge, no one so
22 far has used the test on a negative binomial model. We performed both the aforementioned score tests described in Section 5 to empirically detect the presence (or the absence) of SAC. We compare
the tests on the model augmented with selected SF variables with the ones on the non-filtered model to verify if the introduction of the selected spatial filters lets the SAC be filtered out of the
residuals. The tests are calculated on both quasi-Poisson10 and negative binomial model residuals (estimating or offsetting GDP benchmark variables).
A further relevant question is whether adjusting the test for the presence of independent variables considerably changes SAC detection outcomes, or if this correction has just marginal effects. Table
2 presents the results for the different SAC tests. We start by reporting, in the first and second row of the table, the value of MC computed on the residuals as developed by Cliff and Ord for linear
models. In the first row, we show the results of the basic, stand-alone, MC, while in the second row, the test accounts for the effect of independent variables. The presence of SAC is never rejected,
even when we introduce the spatial filters in the model (despite the scores decreasing). Performing also the discussed MC permutation test, our findings do not change: the permutation score decreases
adding the spatial filters, but we never reject the SAC hypothesis.11 In the third and the fourth rows of Table 2, the values for the LC and JG tests are reported. Using these tests, developed for
GLMs, we can note how the SAC is effectively filtered out by the introduction of the selected spatial filters. The tests show significant SAC in the baseline model, which is filtered out by the
spatial filter eigenvectors, especially when using negative binomial regression, for which the p-value stands to 0.239 (0.230 with offsets). Moreover, the results from LC and JG are quite similar,
highlighting that the introduction of the correction for the independent variables in JG test does not considerably change the test results. The general increase in t-scores obtained when the
right-hand-side variables are taken into account may be explained by the fact that their inclusion pulls expected values slightly to the left (towards negative values).
These results seem to be comforting, and they lead to a positive confirmation of the initial theorized idea that we can account for spatial autocorrelation in the model by filtering out the
10[ Quasi-Poisson models are equivalent to standard Poisson models in terms of coefficient estimation, but because ] a dispersion parameter is estimated from the data, inference differs. For the
purposes of eigenvector selection, AIC- or BIC-based selection is not possible (quasi-Poisson models have no likelihood), so it is manually performed by backward eliminating (iteratively) the
eigenvector with the highest p-value.
23 residual spatial component by means of the selected spatial filters, and that this is detectable only using correct SAC test (specifically designed for GLMs).
Table 2. SAC with different statistics, for different models
Quasi-Poisson Negative binomial Negative Binomial (offset) Non-spatial Spatial filter Non-spatial Spatial filter Non-spatial Spatial filter
MC Score 0.212 0.129 0.185 0.043 0.158 0.035
t 39.99 24.61 34.93 8.08 30.06 6.74
p-value <2.2e–16 <2.2e–16 <2.2e–16 <2.2e–16 <2.2e–16 8.09e–12 MC
Score 0.168 0.119 0.429 0.277 0.375 0.283
t 31.206 21.853 79.823 54.496 72.249 55.086
p-value < 2.2e–16 < 2.2e–16 < 2.2e–16 < 2.2e–16 < 2.2e–16 < 2.2e–16
LC t 4.962 1.971 3.218 0.652 4.601 0.683
p-value 3.49e–07 0.024 0.001 0.257 2.10e–06 0.247
JG t 5.125 2.111 3.336 0.708 4.766 0.737
p-value 1.49e–07 0.017 < 0.001 0.239 9.41e–07 0.230
Notes: MC stands for the standalone Moran’s I test, MC (res.) for the Moran’s I test on regression residuals, LC for the le Cessie test, and JG for the Jacqmin-Gadda test.
7. Conclusions
Recent contributions to the modelling of bilateral trade have shown the importance of sound theoretical underpinnings for obtaining consistent parameter estimates for the determinants of trade in the
gravity model of bilateral trade. This paper addresses the issue of how to achieve empirical consistency without the need to estimate a full general equilibrium system of equations, and without the
loss of specification flexibility that results from the use of origin- and destination-specific indicator variables. We argue that endogeneity of regressors and residuals – due to omitted MR
variables in the traditional gravity model – is likely to manifest in the form of autocorrelation in both regressors and residuals. By including an origin-specific and a destination-specific spatial
filter as additional regressors, SF estimation of the gravity equation enables us to filter SAC out of the residuals, as demonstrated by the results obtained implementing appropriate SAC tests for
nonlinear models. As a result, the residuals and the regressors are no longer correlated, and standard estimation methods can be applied to obtain consistent parameter estimates for the determinants
of bilateral trade. We demonstrate the use
24 of SF estimation in a negative binomial estimation of the gravity equation of bilateral trade. The comparison with two benchmark models, which are theoretically consistent in estimation, reveals
that SF yields results that are highly comparable to the estimation using country-specific indicator variables. Moreover, SF estimation does not suffer from the drawbacks of using indicator
variables. It allows explicit estimation of the effect of country-specific variables that are potentially important determinants of bilateral trade, such as GDP, per capita GDP and landlockedness.
Further analyses aimed at measuring the extent to which SAC is filtered out in SF estimation. We tested three different SAC tests, either from the linear modelling tradition (Moran’s I tests) or
specifically developed for GLMs (the le Cessie and Jacqmin-Gadda tests) on both quasi-Poisson and negative binomial model estimations. Our results confirm the ‘filtering’ effect of the spatial
filters on the residuals. Such finding is mostly evident on the GLM tests, which can be expected to be more suitable for analysing our models’ residuals. On the other hand, the inclusion of
right-hand-side variables in the computation of the SAC tests does not appear to considerably change our findings.
Future research should focus, on the methodological side, on expanding the analyses above to the SF network-autocorrelation approach first suggested by Chun (2008) and further employed in a panel
framework (see, e.g., Scherngell and Lata 2013). Furthermore, quasi- or pseudo-Poisson estimation could be considered more extensively (as suggested in Section 7.3), by employing stepwise selection
criteria which do not require likelihood-based indicators. In this regard, Krisztin and Fischer (2015) have very recently applied network-autocorrelation SF to a trade model, by including, among
others, zero-inflated specifications. On the empirical side, it would be desirable to exploit the methodology proposed toward investigating specific research questions in the trade field, while a
simulation study could help further extend the presented evidence on the adequacy of the SF approach for cross-sectional spatial interaction/gravity models.
We thank Yongwan Chun for useful comments, as well as participants at the following conferences: Small Open Economies in a Globalized World II (Waterloo, ON); Summer Conference of the German Speaking
Section of the European Regional Science Association (Kiel); 48th Conference of the European Regional Science Association (Liverpool);
25 International Conference on Econometrics and the World Economy (Fukuoka); SSES Annual Meeting 2009 (Geneva).
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